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• Measured and Predicted Loads in Multi-anchor Reinforced Soil Walls in Japan
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• Yoshihisa Miyata・R. J. Bathurst・Takeharu Konami
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• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 1–10, Feb. 2009MEASURED AND PREDICTED LOADS IN MULTI­ANCHORREINFORCED SOIL WALLS IN JAPANYOSHIHISA MIYATAi), RICHARD J. BATHURSTii) and TAKEHARU KONAMIiii)ABSTRACTMore than 3000 multi­anchor walls have been built in Japan over the last decade. The paper brie‰y reviews a total ofeight instrumented wall sections that can be used to estimate anchor loads at the end of construction. Measured loadsare compared to predicted values using equations found in current design guidelines. The comparison shows that thecurrent Japanese and UK design methods to compute anchor loads are reasonably accurate for walls with frictionalbackˆlls provided Ka is calculated using the Rankine equation. For walls with cohesive­frictional backˆlls, current de­sign methods over­predict anchor loads by as much as a factor of two. The eccentricity term in current UK and HongKong design methods is shown to not improve the accuracy of load predictions and it is recommended that this addi­tional complexity be removed from these equations. A new load equation is proposed and constant coe‹cients areback­ˆtted to measured anchor load data. The new method is demonstrated to give quantitatively better predictions ofanchor loads based on the statistics for load bias values computed as the ratio of measured to predicted anchor loads atthe end of construction.Key words: anchor loads, c­q soils, granular soils, multi­anchor wall, reinforced soil walls, retaining walls (IGC:H2/K14)INTRODUCTIONReinforced soil walls in Japan can be broadly classiˆedinto metallic, geosynthetic and multi­anchor categories.Metallic walls are comprised of multiple layers ofhorizontal steel strips. Geosynthetic reinforced soil wallsare constructed using horizontal sheets of relatively ex­tensible polymeric geogrid reinforcement or stiŠer FRPstrap materials. The third category are multi­anchor walls(MAWs) constructed with multiple steel plate anchorsbolted to round bar sections that are attached at the op­posite end to the wall facing. The cumulative number ofall wall structures and number of walls by category inJapan are illustrated in Fig. 1. Multi­anchor walls com­prise the smallest segment of the reinforced soil wall mar­ket in Japan. Nevertheless, in 2002 when the last datawere available, there were more than 3000 structures ofthis type (Ochiai, 2007).Figure 2 shows the details of the key components in theJapanese MAW system. The reinforced concrete panelsare 1.5 m in width, 1 m in height and 180 mm in thick­ness. Pinned connections at the back of the facing panelsare used to attach the anchor rods on 0.75 m centers inthe running length of the wall face. The anchor rods arei)ii)iii)Fig. 1. Cumulative number of reinforced soil walls constructed inJapan (after Ochiai, 2007)smooth circular bars with a 19 mm diameter. Each rod isattached to a plate using a threaded end, washer and nut.The standard steel anchor plates are 300 mm by 300 mm.The internal stability design method for this system isnow well established in Japan and is based on a factor ofsafety approach applied to a plate pullout failuremechanism (PWRC, 2002). The design method requiresAssociate Professor, Department of Civil and Environmental Engineering, National Defense Academy, Yokosuka, Japan (miyamiya—nda.ac.jp).Professor and Research Director, GeoEngineering Centre at Queen's­RMC, Department of Civil Engineering, Royal Military College ofCanada, Ontario, Canada.Okasan Livic Co., Ltd., Tokyo, Japan.The manuscript for this paper was received for review on December 3, 2007; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.12MIYATA ET AL.that the anchor plates be located beyond the potential in­ternal soil failure mechanism described by a single activewedge propagating from the heel of the wall facing(Miyata et al., 2001). This condition was satisˆed for allthe case studies in this paper using equivalent singlesecant friction angles computed from c­q values using theapproach reported by Miyata and Bathurst (2007b).Despite the large number of multi­anchor walls inJapan, comparisons between predicted and measured an­chor loads have not been reported in the open literature.Recently, research reports (written in Japanese) by theJapanese Public Works Research Center (PWRC) havebeen made available to the writers. These reports,together with unpublished data, have allowed us to carryout a preliminary investigation of the accuracy of predict­ed loads using current design methods.In this paper we brie‰y describe a total of seven caseFig. 2.CASE STUDIESThe available multi­anchor wall case studies are sum­marized in Table 1. The wall sections varied in heightfrom 3 to 6 m and were constructed with both frictionaland cohesive­frictional soil backˆlls. In all cases the typeof anchor and spacing in the running length of the wallwas the same (0.75 m). The reinforcement length toheight ratio varied from 0.6 to 2.1. Strain gauges wereattached to the top and bottom of the anchor rods tocompute axial tension along the length of the bars. In allcases the wall deformations at end of construction werewithin serviceability limit criteria in Japan (i.e., 3z ofthe wall height or 300 mm, whichever is less) (PWRC,2002). Details of each wall are described next.Walls MAW­1, ­2 and ­3 (Fig. 3) (PWRC, 1995–inJapanese)Three 6.0­m high walls were constructed at the PublicWorks Research Institute (PWRI) test site in Japan. Thewalls were constructed to investigate the in‰uence ofDetail of multi­anchor wall systemTable 1.studies in which nine instrumented multi­anchor wall sec­tions were constructed and the loads in the anchor rodsmeasured. The measured loads from eight wall sectionsare compared to the values predicted using recommenda­tions in Japanese, UK and Hong Kong guidance docu­ments. Based on the measured load data, an equation isproposed to improve the estimate of maximum loads inmulti­anchor wall systems. The magnitudes of the empiri­cal constants in the new load equation are determined byback­ˆtting to measured load data. The accuracy of cur­rent load equations and the new proposed method isquantiˆed using load bias statistics where load bias is de­ˆned as the ratio of maximum measured load to maxi­mum predicted load in an anchor.Summary of multi­anchor wall case studiesDesignationWall casehistoryProjectdateWallheight,H (m)Soil unitweight,g (kN/m3)Peak frictionangle(b) qtx(degrees)Cohesion, c(kPa)Finescontent(a)(z)MAW­1PWRI Wall19946.016.03606MAW­2PWRI Wall19946.015.430219MAW­3PWRI Wall19946.015.311442MAW­4PWRI Wall19944.015.0MAW­5PWRI Wall19944.015.73828MAW­6aBuildingResearchInstitute(BRI) Wall1999ToyamaField Wall1998MAW­6bMAW­6cMAW­7Reinforcementlength, L (m)L/H4.00.67PWRC(1995)4.01.002.50.633.04.01.1715.033073.55.0(c)6.00.880.7018.03500.2Reference12.82.13Aoyama et al.(2000), Futakiet al. (2000)Kitamuraet al. (2000)(a) particle sizes by massº0.075 mm(b) MAW­1, ­4, ­5, ­6 and ­7 strength data from consolidated­drained triaxial compression tests and are eŠective stress parameters. MAW­2 and ­3strength data from consolidated­undrained triaxial compression tests (cohesion and friction angle are total stress parameters)(c) data at 5­m high wall stage not used due to eŠect of base shakingMULTI­ANCHOR REINFORCED SOIL WALLSFig. 3.Fig. 4.Cross­section for walls MAW­1, ­2 and ­3 (PWRI walls)Cross­section for walls MAW­4 and ­5 (PWRI walls)‰ooding and rapid drawdown on wall response. In thisstudy the anchor loads prior to ‰ooding are used. Thebackˆll soil was a coarse sand (D500.5 mm), ˆne sand(D500.15 mm) and silty sand (D500.1 mm) for wallsMAW­1, ­2 and ­3, respectively. The corresponding ˆnescontents were 6z, 19z and 42z. Strength data fromtriaxial compression tests are summarized in Table 1.Anchor pullout tests were carried out at the end of con­struction and after ‰ooding. Horizontal displacements ofthe wall facing increased during ‰ooding.Walls MAW­4 and ­5 (Fig. 4) (PWRC, 1995–in Japanese)Two 4.0­m high walls were constructed at the PublicWorks Research Institute (PWRI) test site in Japan.These walls were constructed to examine the eŠect offoundation support on wall performance. Other walls inthis series were constructed with geosynthetic reinforce­ment layers and have been described by Bathurst et al.(2008a). The soil type and compaction were the same inboth structures. The base footing shown in the ˆgure wasFig. 5.3Cross­section for walls MAW­6a, ­6b and ­6c (BRI wall)released in a controlled manner to allow horizontal move­ment of the 2.0­m thick depth of foundation soil. In thecurrent investigation the data gathered prior to base foot­ing release (end of construction) are used.Fine sand with a ˆnes content of 8z was used for thebackˆll and foundation soil. Compaction was carried outin 0.25­m lifts. Consolidated­drained triaxial tests werecarried out to estimate strength parameters for the back­ˆll. From these tests a cohesion value of 2 kPa and a peakfriction angle of 38 degrees were deduced.The construction time for each wall in this series isreported as ˆve days. The maximum horizontal displace­ments of the walls occurred at mid­height and wererecorded as 14 and 27 mm for walls MAW­4 and ­5, re­spectively. Hence, both walls were judged to have ex­hibited good performance at the end of construction andto be within serviceability limits.Walls MAW­6a, ­6b and ­6c (Fig. 5) (Aoyama et al.,2000; Futaki et al., 2000)This wall was constructed in three stages on a largeshaking table. The corresponding wall heights were 3, 4and 5 m. At the end of construction of each heightincrement, the structure was subjected to base shaking.The loads at each wall height were recorded prior toshaking. However, data for the 5­m high wall were dis­carded in the current study because the reinforcementloads decreased with depth which is very diŠerent fromall other wall data in this study. The unusual load distri­bution is judged to be result of the staged constructionand shaking increments. The ˆne sand used to constructthis wall has a mean particle size (D50) of 0.2 mm and aˆnes content of 7z.Wall MAW­7 (Fig. 6) (Kitamura et al., 2000)A 16.5­m high wall was constructed as part of a high­way construction project in Toyama prefecture. The wallwas built in 4.5, 6 and 6 m vertical stages with increasing4MIYATA ET AL.Fig. 6. Cross­section for top 6­m high section of wall MAW­7 (Toya­ma Field Wall)anchorage lengths with height of section. The middle 6­mhigh section was constructed one year after the initial 4.5m section. The ˆnal 6­m high section was completed 6months after the completion of the middle section. Thedata for the top 6­m high section at the end of construc­tion are used here. Only loads at the facing panel­anchorconnections were recorded. We assume here that these arethe maximum loads in the anchor rods. The backˆll was aweathered gravelly soil with D5040 mm and a ˆnes con­tent of 0.2z. Out­of­alignment with respect to the verti­cal over the top 6 m section was 170 mm which just satis­ˆed serviceability limits described earlier.CALCULATION OF PREDICTED ANCHOR LOADSA general expression to estimate the maximum load ina multi­anchor wall structure using cohesive­frictionalbackˆll soils and no surcharge loading can be taken fromBS8006 (1995) and expressed as:TmaxKasvSv|2Svc KaKaRØ L|2e» S |2S c Kvvva(1)Here Kacoe‹cient of active earth pressure; svmaxi­mum vertical stress acting at the elevation of the reinfor­cement; Svanchor vertical spacing; csoil cohesion; Rvvertical loadg z L acting at the elevation of the an­chor; zdepth of anchor below the backˆll surface; Llength of anchor, and; eeccentricity. For uniform spac­ing, Eq. (1) predicts that the maximum anchor load willincrease in a generally linear manner with depth.In the UK design guideline (BS8006, 1995), Ka is com­puted according to the Rankine formulation:Ka(1|sin q)/(1{sin q)(2)where qpeak friction angle of the soil. In the Japanesestandard (PWRC, 2002), Ka in Eq. (1) is computed as:Kacos d{cos2 qsin (q{d) sin q1{cos d}2(3)where d2q/3 is the interface friction angle between thesoil and back of the panel facing. Values of Ka are lowerwhen using Eq. (3) compared to Eq. (2).The eccentricity term in Eq. (1) is computed in accor­dance with the Meyerhof method recommended inBS8006 (1995). In Geoguide 6 (2002) the cohesion term isignored which simpliˆes Eq. (1). The use of the Meyerhofapproach to compute vertical stresses will increase loadestimates since eÀ0. In AASHTO (2002) design guide­lines for metallic and geosynthetic reinforced soil walls,eccentricity is not considered. Hence, in the current studythe consequences of setting e0 were also explored. Set­ting ce0 in Eq. (1) is also consistent with the Japanesedesign practice for the calculation of anchor loads in mul­ti­anchor walls (PWRC, 2002).In Geoguide 6 (2002), the Coherent Gravity Method isrecommended to compute loads in anchored systems(i.e., the earth pressure coe‹cient varies linearly from thecoe‹cient of earth pressure­at­rest at the top of the wallto the active earth pressure coe‹cient at a depth of 6 m).This will lead to larger Tmax values than those computedusing Ka. The result is that the Hong Kong method will beeven more conservative for design than the UK approachshown later in the paper. In this paper we limit compari­sons to load equations that use Ka computed using Eqs.(2) and (3).In related work by the ˆrst and second writers on loadpredictions for geosynthetic reinforced soil walls, compu­tations were also carried out using higher peak planestrain friction angles in order to reduce design conser­vatism for loads due to selection of friction angle.However, the diŠerence between peak plane strain fric­tion angles and peak triaxial friction angles for the casestudies in Table 1 is very small and no signiˆcant diŠer­ences in predicted loads resulted. Hence, in this studytriaxial peak strength values are used exclusively. In prac­tice, there are correlations available in the literature toconvert peak friction angles from direct shear box tests totriaxial friction angles if this is required (e.g., Miyata andBathurst, 2007b).It is understood that Eq. (1) and variants that appear indesign guidance documents were originally developed as­suming limit equilibrium and hence applicable to ultimatestate design (i.e., collapse). For ultimate state design us­ing UK and Hong Kong codes, a material factor is ap­plied to the cohesion term in Eq. (1) to increase the mag­nitude of load Tmax. However, there is no data availableto test the accuracy of Eq. (1) against instrumented teststaken to collapse. All of the load measurements in thispaper correspond to operational conditions (end of con­struction) for walls that were judged to have good perfor­mance (i.e., wall deformations and anchor rod axialstrains were within Japanese code serviceability criterialimits). Hence, Eq. (1) is used in this paper with all partialfactors set to unity corresponding to the serviceabilitylimit state condition in BS8006 (1995) and Geoguide 6(2002). This approach represents current practice withrespect to the calculation of anchor loads at end of con­struction for the case study walls investigated. Further­more, it can be argued that the load predictions of mostinterest to design engineers are the loads at end of con­struction which lead to good performance (i.e., satisfyserviceability criteria).MULTI­ANCHOR REINFORCED SOIL WALLSFig. 7. Distribution of Dtmaxratio of maximum anchor load (Tmax) tomaximum anchor load in wall (Tmxmx)MEASURED LOADSAs noted in the earlier description of the case studies,strain gauges were mounted directly on top and bottomof the anchor rods. Hence, axial tensile loads could becomputed even if there was some bending of the anchorrods. The values of normalized maximum tensile load forall anchor rods in the database are plotted against nor­malized depth in Fig. 7. The anchor loads Tmax have beennormalized against Tmxmx which is the maximum anchorload in each wall. Most data can be seen to fall within atrapezoidal envelope. This envelope is diŠerent from agenerally monotonically increasing reinforcement loaddistribution that would result for uniformly spaced rein­forcement layers using Eq. (1). Trapezoidal distributionsfor normalized maximum reinforcement loads in geosyn­thetic and metallic strip reinforced soil walls have beenproposed by Allen et al. (2003, 2004), Miyata andBathurst (2007a, b) and Bathurst et al. (2008a). Thebreak points at z/H0.5 and 0.9 are close to valuesproposed by Allen et al. (2004) for metallic reinforced soilwalls. There is no visually apparent diŠerence in the dis­tribution of data points based on case studies with fric­tional and cohesive­frictional backˆll soils (solid andopen symbols, respectively in Fig. 7).In the ˆeld, multi­anchor walls may be constructedwith surcharge loading. In the current K­stiŠness methoddescribed in the papers by Miyata and Bathurst (2007a, b)and Bathurst et al. (2008a) the normalized distributionfor Dtmax is plotted as a function of (z{S)/(S{H) whereSq/g; q is a uniform distributed surcharge pressure andg is the unit weight of the backˆll soil. However, the ac­curacy of this adjustment for the wall systems in thispaper cannot be veriˆed quantitatively since there is noload data available for multi­anchor walls subjected touniform surcharge loading.COMPARISON OF MEASURED AND PREDICTEDLOADSFigure 8 shows measured versus predicted anchorloads using the current Japanese design method (i.e., set­ting e0 and c0 in Eq. (1) regardless of soil backˆll5Fig. 8. Measured versus predicted maximum anchor loads using cur­rent Japanese design methodtype). Superimposed on the plot is the 1:1 correspon­dence line. The data show that most data points for fric­tional backˆll soil cases (solid symbols) fall close to the1:1 line. However, for c­q soil cases (open symbols) thereis a consistent over­prediction of anchor loads indicatingthat for these backˆll cases the current Japanese methodto compute anchor loads is conservative for design. Simi­lar data plots using diŠerent assumptions in Eq. (1)showed a similar visual spread in data points leading tothe same general qualitative conclusions drawn from Fig.8. For brevity they are not presented here. While there is aclear qualitative appreciation of the accuracy of the cur­rent Japanese method for prediction of anchor loads, aquantitative assessment is required.Table 2 summarizes the statistics for the load biasvalues (ratio of measured to predicted loads). For a per­fect model the mean of bias values is one and the spreadin bias values expressed here as the coe‹cient of variation(COV) is zero. From a practical point of view a mean biasvalue of one or slightly less than one is desirable. An ac­ceptable value for the COV of the bias values is subjec­tive. Columns labeled ``with e0'' and ``with c0''represent calculations with eccentricity and soil cohesionset to zero to simplify computations.Columns 2 and 3 in Table 2(a) show that for frictionalbackˆll wall cases the prediction of maximum anchorloads is reasonably accurate on average since the meanbias values for Tmax are at or close to unity. There is aslight improvement in the bias statistics if eccentricity isignored (compare Column 3 to Column 2). Column 3 cal­culations are equivalent to the current Japanese methodif Ka is calculated using Eq. (2). Column 4 shows that thecurrent Japanese method is slightly non­conservative(i.e., measured anchor load values are 14z higher thanpredicted values on average).The statistics for c­q backˆll soils are presented inTable 2(b). The bias statistics for all cases (i.e., consider­ing eccentricity and setting e0, and considering cohe­sion and setting c0) are much poorer. For the best casewith cÀ0, e0 (Column 3) the mean of the bias values is0.62. Hence, on average, measured maximum anchorageloads are about 60z of the predicted values. The current6MIYATA ET AL.Table 2. Summary of statistics for ratio (bias) of measured to predicted reinforcement loadsa) walls with granular soil backˆll (c0, qÀ0)Statistics for ratio(bias) of measured topredicted loadsEqs. (1) and (2)with e0Eqs. (1) and (3) and e0(PWRC, 2002)Proposed new methodEqs. (4) and (2)a1.212345180.981.001.141.001867656553Eqs. (1) and (3) and e0(PWRC, 2002)Proposed new methodEqs. (4) and (2)a1.02Numberof datapointse Æ0(BS8006, 1995)1MeanCOV (z)Column ªb) walls with cohesive­frictional soil backˆll (cÀ0, qÀ0)Eqs. (1) and (2)Statistics for ratio(bias) of measured topredicted loadscÀ0with c0eÆ0(BS8006, 1995)e0eÆ0withe01234567Mean180.570.620.420.450.501.00COV (z)18797264616246Eqs. (1) and (3) and e0(PWRC, 2002)Proposed new methodEqs. (4) and (2)a1.12Column ªc)Numberof datapointsall walls (cÆ0, qÀ0)Eq. (1)Statistics for ratio(bias) of measured topredicted loadsNumberof datapointscÀ0with c0eÆ0(BS8006, 1995)withe0eÆ0withe01234567Mean360.780.800.700.730.811.00COV (z)36767481787950Column ªJapanese method (Column 6) is even more conservativewith measured loads on average equal to 50z of predict­ed values. However, the spread in bias values is less thanfor the previous case (Column 3). The over­prediction us­ing the current Japanese method is visually apparent forthe c­q data points in Fig. 8. From a practical point ofview there is no beneˆt in considering eccentricity in thecomputation of anchor loads (this is also true for fric­tional soil cases based on the bias statistics in Table 2(a)).Table 2(c) presents bias statistics for all available data.As may be expected, the mean of bias values fall betweenthe two data subsets for walls with frictional and co­hesive­frictional backˆll soils.PROPOSED NEW METHOD TO CALCULATEMAXIMUM ANCHOR LOADSThe following equation is proposed to predict themaximum load in an anchor at the end of construction:Tmaxa1KagHSvDtmaxFc2(4)Here, a is a constant coe‹cient and Dtmaxf (z/H ) is aload distribution factor that is computed from thetrapezoidal envelope in Fig. 7. To simplify calculations,Ka is computed using Eq. (2). Parameter Fc is a cohesionfactor that attenuates the anchor load due to the cohesivecomponent of soil strength and is assumed to vary as:Fc1|lcgH(5)Equations (4) and (5) have been inspired by the structureof similar expressions for geosynthetic reinforced soilwalls proposed by Miyata and Bathurst (2007b) andBathurst et al. (2008a). All other parameters in Eqs. (4)and (5) have been deˆned previously. Inspection of Eq.(4) shows that it is consistent with the current Japanesemethod (PWRC, 2002) for the computation of theaverage anchor load over the entire height of the wall butwith the parameter Ka calculated using Eq. (2), a constantcoe‹cient a and two in‰uence factors, Dtmax and Fc. Thelast two quantities are used to adjust reinforcement loadswith depth and to capture the load attenuation due to thecohesive strength component of the backˆll soil.It should be emphasized that the horizontal earth pres­sure coe‹cient used in Eq. (4) is used here as an indexvalue. We do not imply that the soils in these systems arein an active state. Rather, Ka is used here as a familiargeotechnical quantity that is consistent with the expecta­tion that as soil frictional strength increases, anchor loadswill decrease.Parameters a and l were ˆrst estimated using theMULTI­ANCHOR REINFORCED SOIL WALLSSOLVER utility in Excel with the objective functiontaken as the mean ratio (bias) of Tmax (measured)/Tmax(predicted)1 and using all bias values in the dataset. Inother words, the SOLVER utility was used to search forthe values of coe‹cients a and l that gave a mean biasvalue as close to one as possible. This is a diŠerent ap­proach to the conventional regression analysis methods inwhich the objective function is the sum of the squares ofthe error between predicted and measured data pointsand coe‹cients are computed that minimize this func­tion.Using SOLVER, the best estimates for the coe‹cientsin Eqs. (4) and (5) are a1.12 and l15.2. The value ofa was then adjusted for the two data subsets corre­sponding to frictional and cohesive­frictional backˆll soilcases only to give a mean bias value of unity. The valuesfor frictional backˆll soils only is a1.21 and for co­hesive­frictional soils only is a1.02.A practical consequence of Eq. (5) with l15.2 is thatwalls with c/gHÆ0.06 will not generate any anchorforces. However, the designer must decide if the cohesivesoil strength component is available for the life of thestructure.Predicted versus measured data points are plotted inFig. 9 using Eq. (4) with the values of a and l notedabove. The visual impression is that there is better agree­ment between predicted and measured values since thedata is more closely grouped around the 1:1 correspond­ence line. However, a quantitative assessment of the im­provement using Eq. (4) compared to the currentmethods can be made by examining the statistics for biasvalues as was done earlier.The bias statistics in Table 2(a) show that the proposedapproach to compute reinforcement loads is better thanthe current Japanese (PWRC, 2002) and UK method(BS8006, 1995) for frictional backˆll cases. However,from a practical point of view it can be argued that thecurrent Japanese method would be su‹ciently accurate ifKa was computed using the simpler Rankine formulation(Eq. (2)) as shown in Column 3. For c­q backˆll soil casesthere are quantitative improvements in load predictionaccuracy using the new method. Column 7 in Table 2(b)7shows a mean bias value of unity and a lower spread inthe bias values (e.g., COV46z compared to 62z usingthe current Japanese method). When all data is consi­dered, there is a corresponding quantitative improvementin prediction accuracy as illustrated by the bias statisticsin Column 7 of Table 2(c).As a ˆnal visual check on predicted anchor loads versusmeasured values, anchor load proˆles with depth arepresented in Fig. 10 for all case studies. The plots showthat in some cases there is an under­prediction of loadvalues and in other cases over­prediction. However, inmost cases the agreement between predicted and mea­sured values is visually better using the new calibratedmethod introduced in this paper compared to values com­puted using current UK and Japanese methods.In practice, design engineers may determine the maxi­mum load from all predicted anchor loads in a wall andsize all the anchor rods and plates to a single anchor con­ˆguration that can safely carry this maximum load. Inthis paper the maximum load in the wall is denoted asTmxmx. Figure 11 shows measured versus predicted maxi­mum wall anchor loads in each wall section. Note that adata point may not correspond to the same anchor eleva­tion as can be appreciated from some plots in Fig. 10.Figure 11 shows that there is visually much better agree­ment between predicted and measured maximum wall an­chor loads using the new proposed method than the cur­rent BS8006 (1995) and PWRC (2002) methods.However, while a single anchor design is convenient, fur­ther economies may be possible if the anchor rod and an­chor plate sizes at diŠerent elevations are adjusted (withappropriate factors of safety or partial factors) to carrythe loads computed in accordance with Eq. (4).It is interesting to note that the current method for cal­culation of loads in steel strip walls using the BS8006(1995) Coherent Gravity Method has been shown to beaccurate for frictional soils with qº459in a recent paperby Bathurst et al. (2008b). Similarly good agreement be­tween measured and predicted loads has been reported byBathurst et al. (2009) for metallic reinforced soil walls us­ing the AASHTO (2002) Simpliˆed Method. The reasonfor this good agreement is that the AASHTO method formetallic steel walls was calibrated against measured loaddata from a total of 17 instrumented walls. A similarcalibration exercise for multi­anchor walls has never beencarried out to the best of the writers' knowledge. Themechanical behavior of MAW systems and anchor loadscannot be assumed to be the same as for other metallic re­inforced soil wall systems.CONCLUSIONSFig. 9. Measured versus predicted maximum anchor loads usingproposed new method to compute anchor loadsMeasured loads in multi­anchor walls in Japan havebeen investigated with the objective to quantify the ac­curacy of equations in current design methods that areused to predict loads in these systems and to propose anew load equation if required. The following conclusionscan be made:1. Measurements from a series of eight full­scale wall8MIYATA ET AL.Fig. 10.2.3.4.5.Measured and predicted anchor loadssections showed that the distribution of maximum an­chor loads (Dtmax) was reasonably well captured by atrapezoidal distribution. This distribution is diŠerentfrom the generally monotonically increasing load dis­tribution with depth assumed in current designmethods for anchors placed at constant vertical spac­ing. However, the Dtmax distribution adopted here ap­plies only to non­surcharged walls constructed on aˆrm foundation with good toe support.There was no improvement in load accuracy by con­sidering eccentricity in the calculation of maximumanchor load as recommended in BS8006 (1995) andGeoguide 6 (2002) guidelines. In fact, based on biasstatistics, slightly poorer load predictions occurredwhen eccentricity was considered.For purely frictional backˆll soil cases, the currentJapanese method to predict loads (PWRC, 1995) wasshown to slightly under­predict reinforcement loadson average. However, the method was shown to giveacceptably accurate load predictions based on biasstatistics if the calculation of Ka is based on theRankine formulation.The accuracy of load predictions using current guide­lines (BS8006, 1995; PWRC, 2002) was much poorerwhen multi­anchor walls with cohesive­frictionalbackˆlls were considered.A new design equation with constant coe‹cients back­ˆtted to measured anchor loads has been shown to im­prove maximum anchor load predictions for wallswith c­q backˆll soils on average and to reduce thespread in bias values deˆned as the ratio of measuredto predicted load values.6. For purely frictional backˆll soil cases, there was noimprovement on average using the new proposedmethod when compared to current UK and Japanesemethods using Rankine Ka values. However, there wasa detectable improvement in the accuracy of predic­tions based on the spread in load bias values.The results of anchor load predictions have importantimplications to conventional working stress design sincethe paper has demonstrated the true margin of safety oncomputed anchor loads using current UK and Japanesedesign codes. For frictional soils, current methods (withRankine Ka) are reasonably accurate, while for cohesive­frictional soils there is a hidden factor of safety onaverage of about two using the current Japanese ap­proach.The current study is based on a limited number of in­strumented walls. If the proposed method to compute an­chor loads is used for the design of a project­speciˆc wall,it is important that the structure falls within the heightand geometry parameter range used to calibrate the newload equation. The same is true for the backˆll soilstrength parameters. Furthermore it is assumed that theseMULTI­ANCHOR REINFORCED SOIL WALLSFig. 10.Fig. 11. Comparison of measured and predicted maximum wall an­chor loads (Tmxmx) for all wall sectionswalls were built carefully and with good constructionquality control. For walls falling within the data envelopeused to calibrate the proposed working stress method,wall deformations may be expected to fall within currentserviceability requirements (i.e., not greater than 3z ofthe wall height or 300 mm, whichever is less). Field wallsconstructed to a lower standard of care may result inhigher anchorage loads and greater deformations.Periodic review and recalibration of the proposed equa­tion for maximum anchor load should be carried out as9(continued)more measurements from ˆeld­scale instrumented wallsbecome available.The choice of c and q values to compute reinforcementloads under working stress (end of construction) condi­tions does not imply that the walls are at an ultimate limitstate (collapse condition). Rather these parameters can berecognized as index values to compute parameters Fc andKa. These strength parameters are familiar to geotechni­cal engineers and are readily available. Nevertheless, it isreasonable to assume that as c and q increase, reinforce­ment loads under working stress conditions (end of con­struction) will decrease as predicted using our newproposed method.It is important to note that when making Class A loadpredictions the proposed method will in some cases givevalues that are larger than ``actual'' values and in othercases less. However, on average, predicted and measuredvalues will be the same if the wall falls within the databaseof wall parameters used to calibrate the method. Further­more, the spread in bias values will be less consistent withthe improved accuracy of the general approach. The de­sign engineer can apply factors of safety to load predic­tions (if using a working stress design method) or an ap­propriate load factor(s) (if designing using a limit statesdesign approach). The selection of what factors of safetyor load factors to use in multi­anchor wall design is be­yond the scope of this paper. The focus here has been on10MIYATA ET AL.the improvement of anchor load predictions for walls atthe end of construction, which is the operational condi­tion that is of most interest to design engineers.2)ACKNOWLEDGEMENTSThe work described here was carried out with the fund­ing awarded to the ˆrst author by the Japan Ministry ofDefense. The second author is grateful for the supportprovided by the Public Works Research Center in Japanwhich allowed him the opportunity to work in Japan andcomplete the study described here.NOTATIONcsoil cohesion (Pa)COVcoe‹cient of variationstandard deviation/me­an (dimensionless)eeccentricity (m)Dtmaxload distribution factor (dimensionless)Hheight of wall (m)Kacoe‹cient of active earth pressure(1­sin q/(1{sin q) (dimensionless)Llength of anchor (m)Rvvertical loadg z L acting at the elevation of theanchor (N/m)Svanchor vertical spacing (m)Tmaxmaximum tensile load in anchor (N/m)Tmxmxmaximum anchor load in wall (N/m)zdepth of anchor below the backˆll surface (m)a, lconstant coe‹cients (dimensionless)qpeak friction angle of soil (degrees)Fccohesion factor1­lc/gH (dimensionless)gsoil unit weight (N/m3)svmaximum vertical stress acting at the elevation ofthe reinforcement (Pa)3)4)5)6)7)8)9)10)11)12)13)14)ABBREVIATIONSAASHTOAmerican Association of State Highway andTransportation O‹cials (USA)BRIBuilding Research Institute (Japan)FRPˆber reinforced plasticMAWmulti­anchor wallPWRCPublic Works Research Center (Japan)PWRIPublic Works Research Institute (Japan)REFERENCES1) Allen, T. M., Bathurst, R. J., Holtz, R. D., Walters, D. L. and15)16)17)Lee, W. F. (2003): A new working stress method for prediction ofreinforcement loads in geosynthetic walls, Canadian GeotechnicalJournal, 40(5), 976–994.Allen, T. M., Bathurst, R. J., Holtz, R. D., Lee, W. F. andWalters, D. L. (2004): A new working stress method for predictionof loads in steel reinforced soil walls, ASCE Journal of Geo­technical and Geoenvironmental Engineering, 130(1), 1109–1120.American Association of State Highway and Transportation O‹­cials (2002): Standard Speciˆcations for Highway Bridges, 17thEdition, Washington, DC, USA.Aoyama, K., Kikuchi, N., Konami, T. and Mikami, K. (2000):Full­scaled shaking table test for multi­anchored retaining wall withlarge shear box (Part I), Proc. 35th Japanese Geotechnical SocietyAnnual Meeting, Gifu, Japan, 2213–2214 (in Japanese).Bathurst, R. J., Miyata, Y., Nernheim, A. and Allen, T. M.(2008a): Reˆnement of K­stiŠness method for geosynthetic rein­forced soil walls, Geosynthetics International, 15(4), 269–295.Bathurst, R. J., Nernheim, A. and Allen, T. M. (2008b): Compari­son of measured and predicted loads using the Coherent GravityMethod for steel soil walls, Ground Improvement, 161(3), 113–120.Bathurst, R. J., Nernheim, A. and Allen, T. M. (2009): Predictedloads in steel reinforced soil walls using the AASHTO SimpliˆedMethod, ASCE Journal of Geotechnical and GeoenvironmentalEngineering, 135(2), 177–184.British Standards Institution, BS8006 (1995): Code of Practice forStrengthened/Reinforced Soil and Other Fills, BSI, Milton Keynes,United Kingdom.Futaki, M., Misawa, K. and Tatsumi, T. (2000): Full­scaled shak­ing table test for multi­anchored retaining wall with large shear box(Part II), Proc. 35th Japanese Geotechnical Society Annual Meet­ing, Gifu, Japan, 2215–2216 (in Japanese).Geoguide 6 (2002): Guide to Reinforced Fill Structure and SlopeDesign, Geotechnical Engineering O‹ce, Hong Kong, China.Kitamura, Y., Misawa, K. and Tatsumi, T. (2000): CD Proc. 55thJapan Society of Civil Engineers Annual Meeting, Sendai, Japan,CD­ROM, III­B293, 2p (in Japanese).Miyata, Y. and Bathurst, R. J. (2007a): Evaluation of K­stiŠnessmethod for vertical geosynthetic reinforced granular soil walls inJapan, Soils and Foundations, 47(2), 319–335.Miyata, Y. and Bathurst, R. J. (2007b): Development of K­stiŠnessmethod for geosynthetic reinforced soil walls constructed with c­qsoils, Canadian Geotechnical Journal, 44(12), 1391–1416.Miyata, Y., Fukuda, N., Kojima, K., Konami, T. and Otani, Y.(2001): Design of reinforced soil wall–Overview of design manualsin Japan, Proc. International Symposium on Earth Reinforcement(IS­Kyushu 2001), Fukuoka, Kyushu, Japan, A. A. Balkema, 2,1107–1114.Ochiai, H. (2007): Earth reinforcement technique a role of new geo­technical solutions–memory of IS Kyushu, Proc. InternationalSymposium on Earth Reinforcement (IS­Kyushu 2007), Fukuoka,Kyushu, Japan, A.A. Balkema, 1–23.PWRC (1995): Technical Report on Rational Design Method of Re­inforced Soil Walls, Public Works Research Center, Tsukuba,Ibaraki, Japan, 278p (in Japanese).PWRC (2002): Design Method, Construction Manual and Speciˆ­cations for Multi­Anchored Reinforced Retaining Wall, PublicWorks Research Center, Tsukuba, Ibaraki, Japan, 248p (inJapanese).
• ログイン

• タイトル
• Evaluation of Undrained Shear Strength of Soils from Field Penetration Tests
• 著者
• Yoshimichi Tsukamoto・Kenji Ishihara・Kenji Harada
• 出版
• Soils and Foundations
• ページ
• 11〜23
• 発行
• 2009/02/15
• 文書ID
• 21167
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 11–23, Feb. 2009EVALUATION OF UNDRAINED SHEAR STRENGTH OFSOILS FROM FIELD PENETRATION TESTSYOSHIMICHI TSUKAMOTOi), KENJI ISHIHARAii) and KENJI HARADAiii)ABSTRACTThe evaluation of undrained shear strength of soils is necessary in determining the possibility of occurrence of ‰owdeformation during earthquakes. The present study is aimed at examining the evaluation of undrained shear strengthof silty sands from ˆeld with Swedish weight sounding tests and cone penetration tests. Based on the outcome of theprevious studies on laboratory triaxial tests, the undrained shear strength ratio is deˆned as the undrained shearstrength divided by the initial eŠective major principal stress. The undrained shear strength ratio is then formulatedwith respect to the relative density. The penetration resistances of Swedish weight sounding and cone penetration testsare then formulated with respect to the eŠective overburden stress and relative density, based on laboratory calibrationchamber tests. By combining these formulations, the correlations of the undrained shear strength with Swedishpenetration resistance and cone tip resistance are established. The range of values of penetration resistances indicativeof soil layers susceptible to ‰ow deformation is discussed. The correlations of the undrained shear strength with ˆeldpenetration resistances thus derived are then examined from case history studies. Two case history studies are carriedout with Swedish weight sounding tests at the sites of ‰ow failures induced during the recent earthquakes. A series ofcase history studies are reexamined, which were carried out with Dutch cone penetration tests in the past studies.Key words: sandy soil, sounding, triaxial test, undrained shear strength (IGC: C3/D6)`void redistribution mechanisms', (Kokusho, 2000;Kulasingam et al., 2004; Malvick et al., 2006; andothers).The characterization of soil properties using ˆeldpenetration tests has recently evolved considerably. Theuse of calibration chamber tests allowed the eŠects ofstate parameters such as soil density and overburdenstress as well as grain composition of soils on the ˆeldpenetration resistance to be examined in detail. On theother hand, the eŠects of soil density, conˆning stress andgrain composition of soils on the undrained shearstrength of saturated sands have also been examined ex­tensively by using laboratory triaxial tests. Despite thedi‹culty in estimating the undrained shear strengthmobilized in the ˆeld due to the complex phenomena ofvoid redistribution mechanisms, as mentioned above, itwould be worthwhile to combine such independent stu­dies conducted by laboratory calibration chamber testsand triaxial tests, which would allow the undrained shearstrength to be directly correlated with the ˆeld penetra­tion resistance of soils.The present study is aimed at examining the direct cor­relations of the undrained shear strength of silty sandswith the ˆeld penetration resistances of Swedish weightsounding tests and cone penetration tests.INTRODUCTIONThe procedures for estimating liquefaction resistanceof soils, which are necessary for evaluating the possibilityof occurrence of soil liquefaction during earthquakes,have been examined extensively, based on laboratory cy­clic triaxial tests on frozen intact samples and also on ˆeldpenetration tests including standard penetration tests(SPT), cone penetration tests (CPT) as well as velocitylogging tests. To evaluate the occurrence of ‰ow defor­mation during earthquakes, the procedures for estimat­ing undrained shear strength of soils are necessary andhave also been examined, based on laboratory triaxialtests and also on case history studies involving ˆeldpenetration tests, (Seed, 1987; Seed and Harder, 1990;Ishihara et al., 1990a; Idriss and Boulanger, 2007; andothers). One of the recent ˆndings was that, in thepresence of a soil layer with signiˆcantly lower permeabil­ity overlying the liqueˆable layer, there would be situa­tions where the upward pore water seepage during earth­quakes could lead to localized loosening of the liqueˆablesoil, resulting in the undrained shear strength mobilizedin the ˆeld being much lower than that observed inlaboratory tests, (Idriss and Boulanger, 2007). Such aphenomenon was termed as `water ˆlm generation' andi)ii)iii)Associate Professor, Department of Civil Engineering, Tokyo University of Science, Japan (ytsoil—rs.noda.tus.ac.jp).Professor, ditto.Fudo Tetra Corporation, Japan.The manuscript for this paper was received for review on March 3, 2008; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.1112Fig. 1.TSUKAMOTO ET AL.Typical behaviour in undrained triaxial compression and its characterization, (a) eŠective stress paths and (b) stress­strain relationsUNDRAINED SHEAR STRENGTHIn what follows, the undrained shear strength of siltysands is examined based on the previous studies com­posed of a multiple series of undrained triaxial tests onanisotropically consolidated saturated specimens of cleanˆne sand and various silty sands, (Tsukamoto et al.,2004a; Tsukamoto et al., 2008). The general frameworkand outcome of the previous studies are ˆrst described.The undrained shear strength of silty sands is then formu­lated with respect to the relative density, Dr, in a formsuitable for the purpose of the present study.Flow Condition and Undrained Shear Strength RatioIn the previous studies, the ‰ow conditions were identi­ˆed by categorizing the undrained behaviour in terms ofcontractive and dilative behaviour, as follows. The typi­cal behaviour of anisotropically consolidated sand sub­jected to undrained triaxial compression is schematicallyillustrated in Fig. 1. The eŠective stress paths are shownin Fig. 1(a) in the plots of eŠective mean stress, p?(s?1{2s?3)/3, and deviatoric stress, qs?1|s?3. The stress­strain relations are shown in Fig. 1(b) in the plots of axialstrain, ea, and q. The point A indicates that the specimensare anisotropically consolidated. The specimens wouldthen follow diŠerent eŠective stress paths, dependingupon their density and fabric structures. The dense speci­men shows dilative behaviour, where it follows along thepoints B and C, and the shear stress continues to increase.The points B and C deˆne the state of phase transforma­tion and steady state. This behaviour is categorized as``no ‰ow'', and the undrained shear strength is deˆned asthe shear stress mobilized at the state of phase transfor­mation. On the other hand, the loose specimen showscontractive behaviour, where it follows along the pointsB! and C!, and the shear stress experiences peak andthen reduces. The points B! and C! deˆne the quasi­steady state and steady state. This behaviour is catego­rized as ``‰ow'', and the undrained shear strength is de­ˆned at the quasi­steady state. There is intermediate be­haviour, where it follows along the points B? and C?. Thisbehaviour is categorized as ``‰ow with limited deforma­tion'', and the undrained shear strength is deˆned at thequasi­steady state B?, which is lower than that mobilizedat the steady state C?.The undrained shear strength ratio was then deˆned bynormalizing the undrained shear strength, Susqs cosqs/2, with respect to the initial major eŠective principalstress, s?1c, at consolidation, as follows,1Sus qs cos qs,s?1c 2s?1c(1)where qs and qs are the deviatoric stress and internal fric­tion angle at state of phase transformation.FormulationThe physical properties of the soils examined in thepresent study are summarized in Table 1, (Tsukamoto etal., 2004a, 2008). The grain size distributions of the soilsare also shown in Fig. 2(a). Toyoura sand is clean ˆnesand with no ˆnes, and all the other soils are ˆne sandswith some amount of non­plastic ˆnes. From a numberof tests conducted in the previous studies, the plots ofvalues of Sus/s?1c against the relative density, Dr, werededuced and each plot was identiˆed either as ``‰ow'' oras ``no ‰ow''. The plots for Toyoura sand and Omigawasand are shown in Fig. 3. The primary ˆndings of theprevious studies were as follows.(1) The border determining the conditions of ``‰ow'' and``no ‰ow'' can be given by a unique value of the un­drained shear strength ratio, Sus/s?1c, regardless of thedegree of anistropic consolidation as well as the tri­axial compression/extension modes.(2) This threshold value of the undrained shear strengthratio takes a value ranging from 0.24 to 0.26 forclean sand and ordinary silty sands.In what follows, based on the plots of values of Sus/s?1cagainst Dr for the four diŠerent soils used in the previousstudies, the eŠects of the relative density on the un­drained shear strength are formulated as follows,13UNDRAINED SHEAR STRENGTHTable 1.Physical properties and inferred parameters of soilsLaboratory triaxial testsLaboratory calibration chamber testsCase history studiesToyoura sand Omigawa sand Jamuna river Shirasu Omigawa sand Inage sand Narita sand Tsukidate TannoNo. 1sandNo. 2Speciˆc gravity Gs2.6572.6942.7452.3372.6982.5462.6912.5482.465D50 (mm)0.170.170.190.200.150.150.130.250.19Atterberg limitsNPNPNPNPNPNPNPNPNP08.415.524.010.723.829.136.228.4Maximum void ratio emax0.9731.2821.2021.4421.4951.0211.761.7891.872Minimum void ratio emin0.6070.7960.6020.7660.8840.5090.9950.9470.977emax­emin0.3660.4860.60.6760.6110.5120.7650.8420.8950z25z25z25z25z25z25z3.32.11.731.545.95.42.192.782.11.11.350.98690470890235490620710CswAsw/Aus210225¿313424¿593(215)(615)CcAc/Aus150300¿413338¿473Fines content Fc (z)Dro (z)Aus(Sus/s1c)/(Dr­Dro)Averageunder TCunder TEAswNsw1/(Dr­Dro)Acqc1/(Dr­Dro)222112¿157N.B.) The values of Dr and Dro used for deriving the values of Aus, Asw and Ac are in ratio and not in percentage.SusAus( Dr|Dro)2,s?1cFig. 2. Grain size distributions of soils, (a) traixial tests and (b) cham­ber tests(2)where Dro is the initial oŠset of the relative density, andDr is in ratio and not in percentage. It is found in Fig.3(a) for Toyoura clean sand that there would be no oŠsetof the relative density in this relation, and it can be deter­mined as Dro0. On the other hand, there needs to besome oŠset in this relation for silty sands, as shown inFig. 3(b). The value of Dro25z is adopted for all thesilty sands examined in the present study. This particularvalue was assumed, since the same values need to be as­signed in Eqs. (2), (5) and (8) for the same soils, as de­scribed later. In addition, it is to note here that the adop­tion of this particular value might be restricted to the siltysands with ˆnes content less than 25z. This oŠset of therelative density, Dro, serves as a parameter which deˆnesthe lowest value of the relative density attainable underthe presence of conˆning stress. The value of Dro is ineŠect controlled by the volume compression during con­solidation. Because of its compressibility, the silty sand isfound to show some value of Dro.In the above Eq. (2), in taking account of the eŠects ofeŠective overburden stress, the undrained shear strengthwas normalized with respect to the initial eŠective majorprincipal stress. This would be guaranteed because anynon­linearity between these two values were not detectedwithin the range of s?1c tested. The values of Aus inferredfor the four soils are summarized in Table 1.14TSUKAMOTO ET AL.Fig. 4.Fig. 3. Plots of undrained shear strength ratio Sus/s?1c against relativedensity Dr, (a) Toyoura clean sand and (b) Omigawa silty sandSWEDISH WEIGHT SOUNDING TESTSwedish weight sounding test is frequently used forˆeld inspections on road and railway embankments aswell as for residential house constructions. Since it is rela­tively easy to handle the equipment and to carry out theˆeld tests, this test is also useful for earthquake recon­naissance ˆeld investigations. The details of the test eq­uipment and testing procedure are described byTsukamoto et al. (2004b). The testing procedure consistsof two phases, static and rotational penetration. In theˆrst phase, the screw­shaped point at the tip of the steelrod weighing 5 kg (49 N) is statically penetrated into theground by putting the weights of 2~10 kg (98 N) and 3~25 kg (245 N) to achieve the total weight equal to 100 kg(980 N), while the depth of rod penetration is measured.The value of Wsw is deˆned as the sum of the weights ateach load application. In the second phase, upon holdingthe total weight of 100 kg (980 N), the penetration rod isrotated by using the horizontal bar ˆxed at its top. Thenumber of half a turn necessary to penetrate the rodthrough 25 cm is counted and converted to the value ofNsw (ht/m).In the present study, the penetration resistance, Nsw,observed during Swedish weight sounding tests is exam­ined based on the previous study composed of a multipleseries of calibration pressure chamber tests on clean ˆnesand and various silty sands, (Tsukamoto et al., 2004b).In what follows, the general framework and outcome ofCalibration chamber (after Tsukamoto et al., 2004b)the previous study are ˆrst described. The formulation ofthe penetration resistance, Nsw, against the overburdenstress, s?v, and relative density, Dr, is then re­examinedand rearranged in a manner compatible with the formula­tion of the undrained shear strength as described above.Testing DetailsThe calibration pressure chamber used in the presentstudy is 78.7 cm in diameter and 92.4 cm in depth, asshown in Fig. 4. The important feature of this apparatusis that the vertical stress and horizontal stress can be ap­plied independently to a soil sample in the chamber by in­‰ating the rubber membranes. The dry soil sample waspoured into the chamber by the method of air pluviationand tamping, and various values of the relative density,Dr, were achieved. All the soil samples were normallyconsolidated with the earth pressure coe‹cient Ko rang­ing between 0.2 and 0.3. A series of Swedish weightsounding tests were conducted under various values ofthe relative density, Dr, and vertical and horizontal stress­es, s?v and s?h. The physical properties of the soils used inthe chamber tests are summarized in Table 1. Toyourasand is clean ˆne sand and all the other soils are ˆne sandswith some amount of non­plastic ˆnes. The grain size dis­tributions of these soils are also shown in Fig. 2(b).The outcome of the test results using dry soil samples inthe chamber tests is later used for evaluating the un­drained residual strength of saturated soils. It would beappropriate to assume in the present study that themobilisation of shear strength in dry soil samples wouldin eŠect be equivalent to that in fully saturated soil sam­ples, since there is no eŠect of pore­air and pore­water in­teractions under these two conditions.UNDRAINED SHEAR STRENGTHFormulationThe ˆeld penetration resistance of soils is known to beaŠected mainly by soil density, eŠective overburden stressand grain composition among others, (Meyerhof, 1957;Liao and Whitman, 1986; Skempton, 1986; Ishihara,1993, 1996; Cubrinovski and Ishihara, 1999; Tsukamotoet al., 2004b; and others). In the present study, the eŠectsof soil density and eŠective overburden stress on the Nsw­value are formulated by adopting the same procedure asthat described in the previous study, (Tsukamoto et al.,2004b). The basic procedure of the formulation is brie‰ydescribed below.It was found in the previous study that because thestatic penetration resistance represented by the value ofWsw has some in‰uence on the rotational penetrationresistance, Nsw, especially when the soil density is low andthe overburden stress is small, it is convenient to have anoŠset parameter with respect to the value of Nsw, which isequivalent to the static resistance Wsw100 kg (980 N).The converted overall resistance N?sw was then introducedby deˆning this oŠset as asw as follows,N?swNsw{asw,(asw40).(3)The above Eq. (3) holds true under the phase of rotation­al penetration, where the value of Nsw is larger than 1. Onthe other hand, N?sw would be allowed to take a value lessthan or equal to asw under the phase of static penetrationin a manner that N?swaswWsw (kN). For example, itwould be N?sw20 at Wsw0.5 kN.It then follows that the value of N?sw may be normalizedwith respect to the square root of vertical stress toeliminate the eŠects of overburden stress. The value ofN?sw1 may therefore be deˆned as follows,N?sw1N?sws?oN?sws?v98,s?v(4)where s?o is taken as 98 kPa and s?v is in kPa. The aboveEq. (4) was veriˆed by conˆrming the linear relation be­tween N?sw and s?v, (Tsukamoto et al., 2004b). The linearrelation between N?sw1 and D r2 was then approximately as­sumed in the previous study. However, from the view­point of establishing the relation between Nsw and the un­drained shear strength ratio as formulated in Eq. (2), itwould be preferable to normalize the value of N?sw1 in amanner similar to Eq. (2). The plots of the values of N?sw1against Dr were reproduced for the four soils as shown inFig. 5, and the following relation was also found to workwell,N?sw1Asw( Dr|Dro)2,(5)where Dr is in ratio and not in percentage. It is to notehere that the same values of Dro are assumed in Eqs. (2)and (5), where Dro0 for Toyoura sand and Dro25zfor all the other silty sands. The values of Asw inferred forthe four soils are summarized in Table 1. Herein, the datafor Toyoura clean sand were obtained under the relativedensity ranging from Dr30z to 70z, as shown in Fig.5. The data for the other silty sands were obtained underthe relative density larger than Dr50z. This is due toFig. 5.15Plots of converted values of N?sw1 against relative density Drthe fact that it was almost technically di‹cult to preparedry soil samples with low density in a large chamber.Since the outcome of the laboratory triaxial tests indicat­ed that the borders between ``‰ow'' and ``no ‰ow'' forToyoura clean sand and for the other silty sands are givenat about Dr30z and 60z, respectively, the data of thechamber tests are rather restricted to dense soil samples.It is nonetheless assumed that the expression given in Eq.(5) can be extrapolated reasonably well.It is to note here that in case of Swedish weight sound­ing tests, the chamber size eŠects would be negligiblecompared with cone penetration tests as discussed later,and are not taken into account.CONE PENETRATION TESTThe electric cone penetrometer has been commonlyused for ˆeld investigations. It electrically measures thetip resistance and shaft friction as well as pore pressure byemploying the loadcell and strain­gauged transducers en­cased in the cone tip, and provides a continuous proˆle ofsoil layers. The evaluation of liquefaction potential ofsoils has been examined by establishing the correlationbetween the cone tip resistance and cyclic resistance ofsoils, (Robertson and Campanella, 1985; Seed and DeAlba, 1986; Shibata and Teparaska, 1988; and others).In what follows, based on multiple series of calibrationchamber tests conducted for the present study, the conetip resistance, qc, is formulated against the overburdenstress, s?v, and relative density, Dr, in a manner compati­ble with the formulation of the undrained shear strengthas described above.Testing DetailsThe calibration chamber used for a series of conepenetration tests is the same as that for Swedish weightsounding tests and is shown in Fig. 4. The soil samplesused for the cone penetration tests are Toyoura sand,Omigawa sand No. 2 and Inage sand, as summarized inTable 1. The same procedure as described above for16TSUKAMOTO ET AL.Swedish weight sounding tests is employed in preparationof dry soil samples in the calibration chamber. The soilsamples with three diŠerent values of relative density, Dr,are prepared, and are then subjected to diŠerent combi­nations of vertical stress s?v and horizontal stress s?h,where s?v changes from 30 to 100 kPa, and Ko (s?h/s?v)changes from 0.4 to 1.5.The type of cone penetrometer employed in the presentstudy is commonly used in practice, and has an area of 10cm2 at its tip and an apex angle of 60 degrees. The conepenetrometer is then pushed into the soil sample in thechamber by means of a hydraulic actuator at a constantloading rate of 1 cm/s. The tip resistance measured in thechamber tends to increase with depth and achieves peakat depths of around 40 to 60 cm from the top surface, andsuch peak values of the tip resistance measured in thechamber are denoted as qcc hereafter in the present study.Chamber Size EŠectsIt was commonly accepted in the past studies that thecone tip resistance would be normalized with respect tothe square root of vertical stress, (Jamiolkowski et al.,1988; Tatsuoka et al., 1990). and there would be a linearrelation in the logarithmic plots of relative density, Dr,against qcc/ s?v, as follows,Dr|a{b logØ qs?».ccv(6)Such plots for Toyoura sand, which are obtained fromthe results of the present study, are shown in Fig. 6,where the data are fairly in good agreement with the rela­tion for Toyoura sand obtained by Tatsuoka et al. (1990).In the study by Tatsuoka et al. (1990), the Ko­value waschanged from about 0.3 to 0.5 depending upon its voidratio.It is known that the boundary condition and chambersize exhibit important eŠects on the measured valuesfrom cone penetration tests, (Schnaid and Houlsby,1991; Salgado et al., 1998; Jamiolkowski et al., 2001; andothers). It is rather di‹cult to consider explicitly theeŠects of boundary condition and chamber size. It wassuggested by Jamiolkowski et al. (2001) that given thediameter ratio of RdDc/dc, where Dc is the diameter ofthe chamber and dc is the diameter of the cone tip, thevalue of cone tip resistance, qcc, measured in the chambercan be converted to the value of cone tip resistance, q?c,corresponding to the free ˆeld condition equivalent to RdDc/dc100. The value of Rd imposed in the presentstudy can be determined as RdDc/dc78.7 (cm)/3.57(cm)22.1. It was suggested that the conversion can beimplemented by the expression of q?cc~qcc, where c isthe conversion parameter, which was given as c0.054~D r0.827 for Toyoura sand with the relative density greaterthan Dr34.1z under Rd22.1. The value of c is ineŠect the ratio of ˆeld to chamber penetration resistance,and is greater than 1 at the relative density greater than Dr34.1z due to what is called the chamber size eŠects ex­erted by highly dilatant soil samples such as Toyourasand. The conversion parameter would be dependentupon the grain composition of soils. However, it remainsto be resolved. Based on the study by Salgado et al.(1998), as the diameter ratio Rd changes from 25 to 120,the ratio of chamber to ˆeld penetration resistance wouldvary between 0.5 to 0.9 in case of heavily dilatant samplessuch as silica clean sand, while it would vary between 0.8to 0.98 in case of compressive samples. It would thereforebe reasonable to assume that the ˆeld cone penetrationresistance for compressive silty sands tends to be overesti­mated when the same conversion parameter as Toyouraclean sand was assumed. However, for the sake of sim­plicity, this conversion is applied for all the soil samplesused in the present study.FormulationIt has commonly been accepted to normalize the conetip resistance with respect to the square root of verticalstress, as can be seen in Eq. (6). Herein, in formulatingthe cone tip resistance, the dimensionless parameter, q?c1,is introduced as follows,q?cs?oq?c1q?cq?cs?o,s?vs?os?v 98s?v(7)where s?o is taken as 98 kPa, and q?c and s?v are in kPa. Itthen follows that the values of q?c1 can be plotted againstthe relative density, Dr, as shown in Figs. 7(a), (b) and(c). Also shown in these diagrams are the relations givenas follows,q?c1Ac( Dr|Dro)2,Fig. 6. Logarithmic plots of values of qcc/ s?v against relative densityDr (Toyoura sand)(8)where Dr is in ratio and not in percentage. It is to notehere that the above Eq. (8) is deˆned in the same form asin Eq. (5). It is found that Eq. (8) can represent this rela­tion reasonably well, as shown in Figs. 7(a), (b) and (c),though there seem to be some eŠects of Ko in the case ofOmigawa and Inage silty sands. Here the values of Dro as­sumed for Eq. (8) are the same as those for Eq. (5). Thevalues of Asw inferred for the three soils are also summa­rized in Table 1. Herein, the data for Toyoura clean sand17UNDRAINED SHEAR STRENGTHEVALUATION OF UNDRAINED SHEARSTRENGTHThe undrained shear strength ratio was deˆned as givenin Eq. (1) by normalizing the undrained shear strength,Sus, with respect to the initial eŠective major principalstress, s?1c. However, in usual circumstances except forthe situations where soil densiˆcation is employed for li­quefaction countermeasures, the initial eŠective verticalstress, s?vo, is nearly equivalent to s?1c. Therefore, it wouldbe reasonable to assume that the undrained shearstrength ratio, Sus/s?1c can be replaced by Sus/s?vo. In addi­tion, it was described above that the undrained shearstrength ratio, Sus/s?1c (Sus/s?vo), Swedish penetrationresistance, Nsw, and cone tip resistance, qc, can be formu­lated with respect to the relative density, Dr, in the samemanner as each other, as seen in Eqs. (2), (5) and (8). Bytaking advantage of such compatible formulations andeliminating the eŠects of Dr, the correlations of Sus/s?vowith Nsw and qc can be derived as follows,Sus N?sw1 Nsw{40s?vo CswCsw98,s?voqcSus qc1 ,s?vo Cc Cc 98s?vo(9)(10)where the parameters for vertical stress, s?v, included inEqs. (4) and (7) were all replaced by s?vo. Herein, Csw andCc are the parameters uniquely determined for any givensoil, and CswAsw/Aus and CcAc/Aus. It is to note herethat the cone tip resistance corresponding to the free ˆeldis denoted as qc in Eq. (10). The more robust and versatilecorrelations between the undrained shear strength andˆeld penetration resistance have long been examinedbased on case histories (Ishihara et al., 1990a; andothers). Such direct correlations can be derived as fol­lows,Fig. 7. Plots of corrected values of q?c/ 98s?v against relative densityDr, (a) Toyoura sand, (b) Omigawa sand and (c) Inage sandwere obtained under the relative density ranging from Dr30z to 70z, as shown in Fig. 7(a). The data for theother silty sands were obtained under the relative densitylarger than Dr70z, as shown in Figs. 7(a) and (b). Thisis due to the fact that it was almost technically di‹cult toprepare dry soil samples with low density in a large cham­ber. Since the outcome of the laboratory triaxial tests in­dicated that the borders between ``‰ow'' and ``no ‰ow''for Toyoura clean sand and for the other silty sands aregiven at about Dr30z and 60z, respectively, the dataof the chamber tests are rather restricted to dense soilsamples. It is nonetheless assumed that the expressiongiven in Eq. (8) can be extrapolated reasonably well.SusN?sw Nsw{40,MswMswMswSusqc,Mc98,s?voMcCcCsw,98s?vo(11)(12)where s?vo is in kPa. Msw and Mc are the conversionparameters, where Msw is in per kPa and Mc is dimension­less. These conversion parameters are found to be de­pendent upon the level of vertical stress. The values of Cswand Cc are estimated for the soils used in the presentstudy, as shown in Table 1. Herein, the value of Aus forToyoura sand is assumed as Aus3.3, and those for theother silty sands are assumed to range from 1.5 to 2.1.The values of Csw and Cc are then calculated by dividingthe values of Asw and Ac by the value of Aus thus estimatedfor each soil.The values of Csw and Cc are found to vary among thesoils, as shown in Table 1. Since the state parameters in­‰uencing the soil behaviour, such as soil density andoverburden stress, were incorporated in the formulationand the eŠects of such state parameters were eliminated,18TSUKAMOTO ET AL.Fig. 10. Relation between values of Wsw and Nsw against undrainedshear strength ratio Sus/s?voFig. 8.Plots of values of Csw against void ratio range emax­eminFig. 9.Plots of values of Cc against void ratio range emax­eminthe values of Csw and Cc would only be dependent uponthe grain composition of soils. Some of the recent studieshave shown that the void ratio range, emax­emin, serves as agood parameter in representing the grain composition ofsoils for characterizing the behaviour of silty sands,(Miura et al., 1997; Cubrinovski and Ishihara, 1999,2000a, 2000b). The values of Csw and Cc inferred from thepresent study are therefore plotted against the void ratiorange, as shown in Figs. 8 and 9. The void ratio range forclean sand is generally lower than 0.4, and it becomeslarger as the ˆnes content increases in the soil. Figures 8and 9 suggest that the largest values of Csw and Cc can befound when the soil contains some ˆnes. This would imp­ly according to Eqs. (11) and (12) that the soils containingsome ˆnes exhibit the lowest undrained shear strengthwhen the same values of Nsw or qc are observed at a givendepth in the clean sand deposit and silty sand deposit.As a typical example, the correlations for Toyourasand are illustrated in Figs. 10 and 11 in the plots of theundrained shear strength ratio, Sus/s?vo, against the Swed­ish penetration resistance, Nsw, and the cone tipresistance, qc. Since it was found in the previous studiesthat the threshold value of Sus/s?1c (Sus/s?vo) dividing theconditions of ``‰ow'' and ``no ‰ow'' can be uniquely de­termined ranging typically from 0.24 to 0.26 among siltysands, (Tsukamoto et al., 2004a, 2008), the ranges ofFig. 11. Relation between values of qc against undrained shearstrength ratio Sus/s?vovalues of Nsw and qc indicative of soil layers susceptible to‰ow deformation can also be determined as shown inFigs. 10 and 11.CASE HISTORY STUDIES ON SWEDISH WEIGHTSOUNDING TESTSThe post­earthquake ˆeld reconnaissance investiga­tions were carried out at two locations where the rapidlandslide and the large ‰ow failure of ‰uidized subsurfacesoils occurred recently. In what follows, the relation be­tween the Swedish penetration resistance and undrainedshear strength of soils is examined from these two casehistory studies. Swedish weight sounding tests were con­ducted at the soil layers identical to those failed duringthe earthquakes.Tsukidate LandslideIn the event of Miyagiken­oki Earthquake, which oc­curred on May 26, 2003, the rapid landslide occurred atTsukidate, Miyagi, Japan, on a gentle slope consisting ofreclaimed soil deposits for agricultural purposes (Uzuokaet al., 2005). The location of Tsukidate is shown in Fig.12. The reclaimed soil deposits of 40 metres in width, 80metres in length and 5 metres in depth collapsed and19UNDRAINED SHEAR STRENGTHFig. 12.Location of site of landslide at TsukidateFig. 14.Fig. 15.Grain size distributions of soils at Tsukidate and TannoResults of Swedish weight sounding test at Tsukidate‰owed downstream on a gentle slope with 7 degrees incli­nation, as shown in Figs. 13(a) and (b). The soil used forreclamation was from pyroclastic origin with pumice tuŠ.The physical properties of the soil are summarized inTable 1, and the grain size distribution of the soil isshown in Fig. 14. A series of Swedish weight soundingtests were carried out at this site, and the results of thetest conducted at the original intact soil deposit close tothe top portion of the collapsed area are indicated in Fig.15. It is found that the Swedish penetration resistance atthe possible sliding plane located at around 5.5 metres be­low the ground surface ranges from 0.5 kN to 0.75 kN.The undrained shear strength ratio, Sus/s?vo, of soilsmobilized at this site is estimated to be about 0.12, basedon the following simple expression (Ishihara et al.,1990a),Sus Sus§cos a sin a,s?vo gtHFig. 13. Post­earthquake plan view and cross section of site of land­slide at Tsukidate (a) plan view and (b) cross section(13)where H and a are the depth and angle of subsurface slid­ing surface, and the eŠective vertical stress s?vo is assumedas the overburden stress gtH. The value of Sus/s?vo thus es­timated is also conˆrmed from the results of laboratoryundrained triaxial compression tests conducted on thesoil sample retrieved from this site and ˆeld density meas­urement. The parameters inferred from this case history20TSUKAMOTO ET AL.Table 2.SiteH (m)a (degree)Sus (kPa)5.57120.5 (kN)¿0.75 (kN)0.25 (kN)¿1 (kN)i)Tsukidateii)4.533.5iii)6815iii)69.517.5457TannoMochikoshi No. 1 tailings damMochikoshi No. 2 tailings damHokkaidotailings damiv)`Collapsed' layer`Intact' layerqc (kPa)200¿3002000¿5000(2)(?)6450¿6001.5¿2188450¿600vi)10913.5—71314.5—10128.0150.52.0ChonanShararavi)`Collapsed' layerMay 1vi)`Intact' layer`Collapsed' layerOkulivi)`Intact' layerWsw (kN)/Nsw (ht/m)Reference200¿400v)Metokiiv)FirmaSummary of data from case history studies300¿600500¿1100Ishihara et al. (1990a)Ishihara et al. (1990b)100¿400500¿900N.B.1) i) Miyagiken­oki (May 26, 2003), ii) Tokachi­oki Earthquake (September 26, 2003), iii) Izu­Ohshima­Kinkai Earthquake (January 14, 1978),iv) Tokachi­oki Earthquake (May 16, 1968), v) Chibaken­Toho­oki Earthquake (December 17, 1987), vi) Tajikistan Earthquake (Gissar,January 23, 1989)N.B.2) H: average depth, a: average slope angle, Sus: undrained shear strength, qc: cone tip resistance, Wsw/Nsw: Swedish penetration resistance.N.B.3) ``Metoki'' was the road embankment slump failure, and it was di‹cult to estimate the values of H and a.Fig. 16.Location of site of landslide at Tannostudy are summarized in Table 2.Tanno Flow FailureIn the event of Tokachi­oki Earthquake which oc­curred on September 26, 2003, the ‰uidization and subsi­dence of gently sloped farming ˆelds took place in Tannoarea of Kitami in Hokkaido. The location of the site is in­dicated in Fig. 16. The farming ˆeld of 35 metres wideand 150 metres long subsided due to the eruption of ‰ui­dized subsurface deposits through a couple of ejectionholes located downstream portions of the subsided area.It was found from the interview of the local people thatthe subsided area had been reclaimed with the deposits oflocal volcanic soil for agricultural purposes. The planview and cross section of the site are shown in Fig. 17. Itwas found from the site survey that the bottom surface ofFig. 17. Post­earthquake plan view and cross section of site of land­slide at Tanno (a) plan view and (b) cross sectionthe ‰uidized subsurface deposits forms a round basin in atransverse cross section, and the ‰uidized deposits ‰oweddown swinging leftwards and rightwards along the basinuntil they were erupted at the downstream portions of thesubsided area. The physical properties of the soil are sum­marized in Table 1, and the grain size distribution of thesoil is shown in Fig. 14. A series of Swedish weightsounding tests were carried out at this site, and the resultsof the test conducted at the original intact soil depositclose to the top portion of the collapsed area are indicatedin Fig. 18. It is found that the Swedish penetrationresistance at the ‰uidized layer located at around 3 to 5.5metres below the ground surface ranges from 0.25 kN to1.0 kN. The undrained shear strength ratio, Sus/s?vo, ofUNDRAINED SHEAR STRENGTHFig. 18.Results of Swedish weight sounding test at Tanno21Fig. 19. Plots of values of Sus/s?vo against N?sw1 from case history stu­diessoils mobilized at this site which can be estimated fromEq. (13) is estimated to be as low as 0.05. The value ofSus/s?vo thus estimated is also conˆrmed from the resultsof laboratory undrained triaxial compression tests con­ducted on the soil sample retrieved from this site andˆeld density measurement.Swedish Penetration ResistanceBased on the data derived from the case history studiesas described above, the plots of values of Sus/s?vo againstvalues of N?sw1 are produced as shown in Fig. 19. It is tonote here that the values of N?sw are determined in a man­ner that N?sw is equal to 40 at Wsw1.0 kN. For example,N?sw is equal to 20 at Wsw0.5 kN. The parameter Csw canthen be inferred as the ratio of N?sw1 to Sus/s?vo. It is to notehere that the parameter Csw is considered to be only de­pendent on the grain composition of soils, since theeŠects of overburden stress and soil density are eliminat­ed during the course of its formulation. The values of Cswthus calculated are superimposed on the diagram cor­relating the parameter Csw with the void ratio range, emax­emin, as shown in Fig. 20. The plot for the site of Tsuki­date is located within the range of its correlation derivedfrom the laboratory study conducted in the present study.However, the plot for the site of Tanno is out of rangeprobably because of its extremely low value of undrainedshear strength. It is to note here that this extremely lowvalue was derived based on the average slope angle usingEq. (13). However, as described above, the ‰uidizeddeposits seem to have ‰owed down swinging leftwardsand rightwards along the basin from bottom to top por­tions, indicating that the local failure slope angle mightbe greater than the average angle, due to such basineŠects.CASE HISTORY STUDIES ON CONEPENETRATION TESTSA number of case history studies were reported by Ishi­hara et al. (1990a) on the ‰ow failures of tailings damsduring the past earthquakes in Japan. A series of large­Fig. 20. Plots of values of Csw against void ratio range emax­emin fromcase history studiesscale ‰ow failures of wind­lain collapsible loess depositsin Tajikistan were also reported by Ishihara et al.(1990b), which occurred during the earthquake on Janu­ary 23, 1989. In the present study, the relation betweenthe cone tip resistance and undrained shear strength ofsoils is examined based on the data of undrained shearstrength of soils estimated from each site of ‰ow failureand the results of Dutch cone penetration tests conductedduring the post­earthquake ˆeld reconnaissance investi­gations. The details of the case history studies reportedby Ishihara et al. (1990a, 1990b) are summarized in Table2. The main focus of the past studies was to estimate theundrained shear strength of soils from post­earthquakestability analysis, and to correlate it with the cone tipresistance observed along the depth of `collapsed' debrisof sliding soil mass. Since the present study is focused onthe cone tip resistance of pre­earthquake `intact' soildeposits, the values of cone tip resistance observed at the`intact' portions of the soil deposits, typically located im­mediately below the failure surfaces, are read oŠ from thediagrams reported in these past studies.The values of Sus/s?vo are then plotted against the values22TSUKAMOTO ET AL.Fig. 22. Plots of values of Cc against void ratio range emax­emin fromcase history studiesFig. 21. Plots of values of Sus/s?vo against qc1 from case history studies,(a) `collapsed' layers and (b) `intact' layersof qc1 observed at `collapsed' layers of soil deposits, asshown in Fig. 21(a). The same plots are produced for thevalues of qc1, which are read oŠ from the `intact' layers,as shown in Fig. 21(b). Since the parameter Cc can be in­ferred as the ratio of qc1 to Sus/s?vo, the possible ranges ofCc­values for loess and ordinary silty sand can be calcu­lated from the data shown in Figs. 21(a) and (b). Thevalues of Cc thus calculated are superimposed on the dia­gram correlating the parameter Cc with the void ratiorange, emax­emin, as shown in Fig. 22. Herein, the values ofemax­emin for the silty sands and loess are assumed as 0.5and 0.7. The values of Cc for `collapsed' layers are foundto be extremely lower than those for `intact' layers. Thiswould be reasonable since the `collapsed layers' cor­respond to the loosely deposited debris that collapsed dueto seismic shaking, ‰owed down and ˆnally came to restjust in equilibrium with its stability. The values of Cc for`intact' layers estimated from the case history studies arefound to be within the range of its correlation derivedfrom the laboratory study conducted in the present study.CONCLUSIONSThe undrained shear strength ratio of silty sands wasformulated with respect to the relative density, based onthe outcome of the previous studies on laboratory triaxialtests. The penetration resistances of Swedish weightsounding tests and cone penetration tests were formulat­ed with respect to the eŠective overburden stress and rela­tive density, based on laboratory calibration chambertests. By combining these formulations, the correlationsof the undrained shear strength with Swedish penetrationresistance and cone tip resistance were established. Theexample charts for evaluating the undrained shearstrength from Swedish and cone penetration resistancesfor Toyoura sand were shown, in which the possibleranges of values of penetration resistances indicative ofsoil layers which would be susceptible to ‰ow deforma­tion were also indicated. The correlations of the un­drained shear strength of soils with ˆeld penetrationresistances were then examined based on a number ofcase history studies, and were found to work well.ACKNOWLEDGEMENTSThe laboratory triaxial tests and chamber tests de­scribed in the present study were carried out by a group ofthe past students of the soil mechanics group at TokyoUniversity of Science. The ˆeld investigations at Tsuki­date and Tanno were also carried out with a help of Dr.S. Okada, Prof. S. Yamashita, Dr. T. Hara, Dr. H.Nakazawa, Ms. Y. Tsutsumi and Mr. R. Yamaguchi. Theauthors are grateful to their cooperation.NOTATIONa, b:c:d c:qc (kPa):qcc (kPa):parameters in Eq. (6)conversion parameter for chamber­size eŠectsdiameter of cone tipcone tip resistancecone tip resistance measured in calibrationchamberqc1: converted cone tip resistance: qc/ 98s?v,(unit in kPa)q?c (kPa): cone tip resistance corrected for calibrationchamber size eŠectsUNDRAINED SHEAR STRENGTHAc: parameter correlating converted cone tipresistance and relative density: q?c1/(Dr|Dro)2Asw: parameter correlating converted Nsw­valueand relative density: N?sw1/( Dr|Dro)2Aus: parameter correlating undrained shearstrength ratio and relative density: (Sus/s?1c)/( Dr|Dro)2à(Sus/s?v)/( Dr|Dro)2Cc: parameter correlating converted cone tipresistance and undrained shear strength ratio: q?c1/(Sus/s?1c)Ac/AusCsw: parameter correlating converted Nsw­valueand undrained shear strength ratio: N?sw1/(Sus/s?1c)Asw/AusDc: diameter of chamberDro: constant determining oŠset of relative densityMc: parameter correlating cone tip resistance andundrained shear strength: qc/SusCc 98/s?vo, (unit in kPa)Msw (per kPa): parameter correlating N?sw­value and un­drained shear strength: N?sw/SusCsw/ 98s?vo, (unit in kPa)N?sw: N?swNsw{40 at NswÆ1 and N?swaswWsw(kN) at WswÃ1 kNN?sw1: N?sw 98/s?vo, (unit in kPa)Rd: Dc/dcasw: 40: oŠset parameter against Nsw taking intoaccount the eŠects of static penetrations?o: reference eŠective stress: 98 kPaAbbreviated wordsTC: triaxial compressionTE: triaxial extensionREFERENCES1) Cubrinovski, M. and Ishihara, K. 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(1985): Liquefactionpotential of sands using the CPT, ASCE, III(GT3), 384–403.18) Salgado, R., Mitchell, J. K. and Jamiolkowski, M. (1998): Calibra­tion chamber size eŠects on penetration resistance in sand, Journalof Geotechnical and Geoenvironmental Engineering, ASCE,124(9), 878–888.19) Schnaid, F. and Houlsby, G. T. (1991): An assessment of chambersize eŠects in the calibration of in situ tests in sand, Geotechnique,41(3), 437–445.20) Seed, H. B. (1987): Design problems in soil liquefaction, Journal ofGeotechnical Engineering Division, ASCE, 113(8), 827–845.21) Seed, H. B. and De Alba, P. (1986): Use of SPT and CPT tests forevaluating the liquefaction resistance of sands, Proc. In­Situ Test,ASCE, 281–302.22) Seed, R. B. and Harder, L. F. Jr. (1990): SPT­based analysis of cy­clic pore pressure generation and undrained residual strength, Proc.Seed Memorial Symposium (ed. by Duncan, J. M.), BiTech Pub­lishers, Vancouver, B.C., 351–376.23) Shibata, T. and Teparaska, W. (1988): Evaluation of liquefactionpotential of soils using cone penetration tests, Soils and Founda­tions, 28, 49–60.24) Skempton, A. W. (1986): Standard penetration test procedures andthe eŠects in standards of overburden pressure, relative density,particle size, ageing and overconsolidation, Geotechnique, 36(3),425–447.25) Tatsuoka, F., Zhou, S., Sato, T. and Shibuya, S. (1990): Evalua­tion method of liquefaction potential and its application, TechnicalReport on Seismic Hazards on the Ground in Urban Areas, Minis­try of Education.26) Tsukamoto, Y., Ishihara, K. and Shibayama, T. (2004a): Evalua­tion of undrained ‰ow of saturated sand based on triaxial tests,Proc. 3rd International Conference on Continental Earthquakes,Beijing, China.27) Tsukamoto, Y., Ishihara, K. and Sawada, S. (2004b): Correlationbetween penetration resistance of Swedish weight sounding testsand SPT blow counts in sandy soils, Soils and Foundations, 44(3),13–24.28) Tsukamoto, Y., Ishihara, K. and Kamata, T. (2009): Undrainedshear strength of soils under ‰ow deformation, Geotechnique, inprint.29) Uzuoka, R., Sento, N., Kazama, M. and Unno, T. (2005): Land­slides during the earthquakes on May 26 and July 26, 2003 in Miya­gi, Japan, Soils and Foundations, 45(4), 149–163.
• ログイン

• タイトル
• Effects of Particle Characteristics on the Viscous Properties of Granular Materials in Shear
• 著者
• Tadao Enomoto・Shohei Kawabe・Fumio Tatsuoka・H. di Benedetto・Toshiro Hayashi・A. Duttine
• 出版
• Soils and Foundations
• ページ
• 25〜49
• 発行
• 2009/02/15
• 文書ID
• 21168
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 25–49, Feb. 2009EFFECTS OF PARTICLE CHARACTERISTICS ON THE VISCOUSPROPERTIES OF GRANULAR MATERIALS IN SHEARTADAO ENOMOTOi), SHOHEI KAWABEii), FUMIO TATSUOKAiii),HERVEÁ DI BENEDETTOiv), TOSHIRO HAYASHIv) and ANTOINE DUTTINEvi)ABSTRACTThe viscous properties of a wide variety of unbound granular materials (GMs) were evaluated by drained shear tests.The specimens were reconstituted ones that were loose or dense and air­dried or moist or saturated, of mostly naturalsands and gravels, having diŠerent mean particle diameters, uniformity coe‹cients, ˆnes contents, degrees of particleangularity and particle crushabilities. The tests were mostly triaxial compression (TC) tests and partly plane straincompression tests, both at ˆxed conˆning pressure, and direct shear tests at ˆxed normal pressure. The viscous proper­ties of GMs were evaluated by stepwise changing the loading rate and performing sustained loading (SL) tests duringotherwise monotonic loading (ML) at a constant loading rate. The viscous properties are characterised in terms of therate­sensitivity coe‹cient ( b), the viscosity type parameter (ubr/b ) and the decay parameter (r1). Correlations amongthese parameters and eŠects of particle characteristics on these parameters are analysed. Creep strains are comparedwith residual strains by cyclic loading under otherwise the same TC conditions. As the particles become less angular, asthe grading becomes more uniform and as the particles become less crushable, the viscosity type deviates more fromthe Isotach type (i.e., u1.0) changing toward the P & N type (i.e., uº0) associated with a decrease in b and r1, whilecreep strains by SL decreases and residual strains by many unload/reload cycles increases. It is shown that the loadingrate eŠects observed in the experiments can be simulated well by the three­component model taking into account theeŠects of particle characteristics on the viscous property parameters.Key words: creep deformation, cyclic loading, granular material, particle properties, triaxial compression, viscousproperties (IGC: D6/D7)among many others). The major ˆndings obtained bythese previous studies can be summarised as follows.The viscosity type of geomaterial is not unique butdiŠerent depending on the geomaterial type. Classicalelasto­plastic models cannot describe these diŠerent vis­cosity types of geomaterial. A number of diŠerent elasto­viscoplastic models for geomaterials have been proposed:e.g., the overstress model (Perzyna, 1963). As a moregeneral and ‰exible elasto­viscoplastic model frameworkthat can deˆne and classify these diŠerent viscosity types,the non­linear three­component model (Fig. 1) is em­ployed in the present study. According to this model, thestress (i.e., the measured eŠective stress), s, is obtainedby adding the viscous component, s v, to the inviscid (orrate­independent) component, s f, at the same eir value.The strain rate, ·e, is obtained by adding the irreversible(or inelastic or visco­plastic) component, ·eir, to the hypo­elastic component, ·ee. A combination of E and P bodiesin series represents the classical elasto­plastic models.INTRODUCTIONUnbound granular materials (unbound GMs) (e.g., un­bound sands and gravels) may exhibit signiˆcant rate­dependent behaviour (including signiˆcant creep defor­mation) in ˆeld full­scale cases (e.g., Tatsuoka et al.,2000, 2001; Jardine et al., 2005, 2006; Oldecop andAlonso, 2007). A number of laboratory stress­strain testsshowed that unbound GMs have signiˆcant viscous prop­erties and their trend is similar to that of saturated claysand bound materials (i.e., sedimentary soft rocks andcement­mixed soils) (e.g., Yamamuro and Lade, 1993;Matsushita et al., 1999; Hayano et al., 2001; DiBenedetto et al., 2002; Tatsuoka et al., 1999, 2002, 2004,2006a, 2008a, b; Tatsuoka, 2007; Komoto et al., 2003;Nawir et al., 2003; Aqil et al., 2005; Kongsukprasert andTatsuoka, 2005; Anh Dan et al., 2006; Kiyota and Tat­suoka, 2006; Enomoto et al., 2007a, b; Sorensen et al.,2007; Duttine et al., 2008a; Kongkitkul et al., 2008;i)ii),iii),v),vi)iv)Research Engineer, Public Works Research Institute, Japan (formerly Graduate Student, Departments of Civil Engineering, Universityof Tokyo and Tokyo University of Science).PhD Student, Professor, formerly Graduate Student and Assistant Professor, Department of Civil Engineering, Tokyo University ofScience, Japan (tatsuoka—rs.noda.tus.ac.jp).Professor, D áepartement G áenie Civil et B âatiment, Ecole Nationale des Travaux Publics de l'Etat, France.The manuscript for this paper was received for review on April 11, 2008; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.2526ENOMOTO ET AL.Table 1. EŠects of bounding, grading, particle shape and strain levelon the viscosity type (Tatsuoka et al., 2008a)BoundIsotach( pre­peak)ªTESRA( post­peak)Fig. 1. Non­linear three­component model (Di Benedetto et al., 2002;Tatsuoka et al., 2002)Fig. 2. Four basic viscosity types of geomaterial in shear (Tatsuoka,2008a): with all these types, the same positive stress jump for a stepincrease in the strain rate by a factor of 10 is assumedFour basic viscosity types illustrated in Fig. 2 werefound by recent laboratory shear tests of a number ofdiŠerent geomaterial types. The Isotach type is the mostclassical one. In continuous monotonic loading (ML), theincrement of s v ( Ds v) that develops at any moment byDe ir or D ·e ir or both does not decay with an increase in e irduring subsequent loading. So, the current s v is a uniquefunction of instantaneous e ir and its rate, ·e ir, and thestrength during ML at a constant ·e increases with ·e. Withthe other three types, Ds v decays with e ir towards diŠer­ent residual values. With the TESRA type, Ds v decayseventually toward zero, so the strength during ML at aconstant ·e is essentially independent of ·e. `TESRA'stands for ``temporary eŠects of strain rate and strainacceleration.'' The Combined type is a combination ofthe Isotach and TESRA types. With the Positive andNegative (P&N ) type, a positive Ds v value decaystowards a negative value, so the strength during ML at aconstant ·e decreases with an increase in ·e. Table 1 sum­marises the eŠects of geomaterial type and strain level onthe viscosity type. The arrows indicate that the viscositytype changes with e ir in a single ML test.Moreover, for the description by the model (Fig. 1) ofthe stress­strain behaviour of unbound geomaterials (in­cluding GMs) in drained triaxial compression (TC), triax­ial extension (TE) and plane strain compression (PSC)tests, the eŠective principal stress ratio, Rs?1/s?3, andthe shear strain, ge1|e3, are the relevant stress andUnboundGradingWel­gradedPoorly gradedAngularIsotach (orCombined)ªTESRATESRAªP&NRoundCombinedªTESRAªP&NP&NªP&Nwith instabilityShapeFig. 3. Deˆnition of b and bresidual (illustrated in the case of strain ratechange by a factor of 10) (Tatsuoka et al., 2008a)strain parameters (s and e). Then, by referring to Fig. 3,the viscous properties of unbound geomaterials can becharacterised ˆrstly as follows:Ø »·girafterDRb¥log10 ir·gbeforeR(1)where b is the rate­sensitivity coe‹cient; and DR is thejump in R that takes place upon a step change in the ir­reversible shear strain rate ( ·gireir1 |eir3 ) from ·girbefore to ·girafterwhen RR. In Fig. 3, ·girafter/ ·girbefore§ ·gafter/ ·gbefore10,therefore, bDR/R. Furthermore, diŠerent viscositytypes exhibit diŠerent values of the residual rate­sensitivi­ty coe‹cient, bresidual, deˆned as:Ø »·girafterDRrbresidual¥log10 irRr·gbefore(2)where DRr is the residual value of DR that has taken placeupon a step change in ·gir from ·girbefore to ·girafter and then hasfully decayed during subsequent ML at a constant ·gir; andRr is the R value (when DRr is deˆned) given along thestress­strain curve obtained if ML had continued at theconstant ·gir kept to the original value (i.e., ·g0 in Fig. 3). InFig. 3, ·girafter/ ·girbefore§ ·gafter/ ·gbefore10, therefore, DRrRr¥bresidual. As both b and bresidual values can be diŠerentamong diŠerent geomaterial types that exhibit the sameviscosity type and they may change with strain in a singletest (Tatsuoka et al., 2008a; Kongkitkul et al., 2008; Dut­tine et al., 2008a), it is relevant to deˆne the current vis­cosity type by the viscosity type parameter, u, deˆned as:VISCOUS PROPERTIES OF GRANULAR MATERIALSubresidual/b(3)where bresidual and b are the instantaneous values measuredat the same gir. Then, diŠerent viscosity types can berepresented by diŠerent u values: i.e., Isotach (u1.0);Combined (0.0ºuº1.0); TESRA (u0.0); and P&N(uº0.0). Tatsuoka (2007) and Tatsuoka et al. (2008a)proposed a mathematical expression incorporating theviscosity type parameter, u that can describe these fourviscosity types and transitions among them. Note that u isa function of gir.Lastly, the b value of drained unbound GMs havingmean particle diameters larger than certain values (i.e.,sands and gravels) is basically independent of particle size(Tatsuoka et al., 2006b). With Toyoura sand in drainedTC tests (Nawir et al., 2003) and drained PSC tests (Kon­gkitkul et al., 2007), the eŠects of wet conditions (air­dried or saturated), conˆning pressure and dry density onthe b value are insigniˆcant, if any. The b value of Toy­oura sand is essentially the same in drained TC, TE andPSC tests, as well as drained DS tests (Duttine et al.,2008a), and the eŠects of over­consolidation are insig­niˆcant (Kiyota and Tatsuoka, 2006). Kawabe et al.(2008) also showed that the eŠects of conˆning pressureand over­consolidation on the viscous property ofdrained saturated Albany sand, which exhibits the P&Nviscosity in drained TC, are negligible.Despite these many ˆndings described above, theeŠects of particle characteristics (i.e., grading, particleFig. 4.27shape, particle crushability and so on) on the viscousproperties of a wide variety of unbound GMs as encoun­tered in the ˆeld are still poorly understood. In particu­lar, most of the GMs used in the previous studies arepoorly graded having relatively stiŠ (i.e., less crushable)particles. Furthermore, similarities or diŠerences betweenthe eŠects of particle characteristics on the residualstrains developed by cyclic loading (CL) with unboundGMs and those on the creep strains are poorly under­stood. In view of the above, by using much wider GMtypes than in the previous studies, a series of drained TCtests were performed to evaluate the eŠects of these parti­cle characteristics on the viscous properties of unboundGMs. The results from these TC tests were analysed refer­ring to those from drained PSC and DS tests performedin the previous and present studies. Moreover, a numberof cyclic triaxial tests were performed on air­dried speci­mens for these research objectives. The ageing eŠect is be­yond the scope of this study. The rate­dependency underundrained conditions is also beyond the scope of thepresent study.TEST METHOD AND MATERIALSTest MaterialsFigure 4 shows the particle pictures of GMs representa­tive of those tested in the present study. Figure 5 showsthe grading curves of these GMs and others referred to inthis paper. The values of mean diameter (D50), uniformitySome representative granular materials tested in the present study (*: one unit length0.5 mm)28ENOMOTO ET AL.Fig. 5.Grading curves of granular materials referred to in this papercoe‹cient (Uc) and ˆnes content (FC) are listed next tothe respective material names in the bottom of Fig. 5 andthese values together with the values of speciˆc gravity(Gs), maximum and minimum void ratios (emax and emin)and the b values of these GMs, as well as test methods,are listed in Table A1 of APPENDIX A1. The hydraulicconductivities of these GMs are not listed, as they have nodirect link to the rate­dependent behaviour discussed inthis paper. This is because it is certain that the drainedtests performed on saturated specimens referred to in thispaper were essentially under fully drained conditions.Among those tested, Silica sands Nos. 3, 4, 5, 6 and 8are poorly graded consisting of relatively angular and stiŠparticles having diŠerent D50 values. Mixed silica sandwas produced by mixing these silica sands to have a largerUc value. Coral sand A is a ˆne poorly graded sand con­sisting of relatively angular and stiŠ particles. CorundumA (granular aluminium oxide), Hime gravel and Albanysilica, Ottawa, Ticino, Monterey and SLB (Silver Leight­on Buzzard) sands are poorly graded consisting of rela­tively round and stiŠ particles. MACH is a well­gradedGM with Uc6.3 produced by mixing Albany sand,Corundum A and Hime gravel. Corundum B is a very ˆneround powder of granular aluminium oxide. Glass beadsA, B and C consist of spherical stiŠ particles, havingdiŠerent D50 values. Ishihama Beach sand is poorly grad­ed consisting of sub­angular and stiŠ particles.Shinanogawa riverbed gravel is rather well­graded con­sisting of round gravel particles and relatively angularsand particles (Fig. 4). Model Chiba gravel H (D502.0mm and Uc2.28) is relatively poorly­graded consistingof relatively angular and stiŠ sandstone particles, pro­duced by removing particles larger than 4 mm from origi­nal Chiba gravel. Air­dried specimens were used in stress­controlled TC tests (s?h40 kPa) to evaluate residualdeformation due to sustained and cyclic loading.In comparison with those described above, coral sandB is poorly graded including relatively crushable shellfractions, while Tanno, Inagi and Omigawa sands are in­land weathered sands having relatively angular and rela­tively crushable particles, as noted by an increase in theˆnes content by compaction tests (Kawaharazono, 2007;Seida et al., 2008). Study on the quantiˆcation of particlecrushability is underway and the results will be reportedin the near future.Only saturated loose specimens were produced withIshihama sand. Only saturated dense specimens were pro­duced with Omigawa, Narita, Tan­no and mixed silicasands. Only dense air­dried specimens were producedwith Shinanogawa riverbed gravel (a large specimen) andVISCOUS PROPERTIES OF GRANULAR MATERIALSFig. 6.Parameters to obtain the degree of angularity by Lees (1964)Table 2.Degrees of angularity of representative GMsMaterialsA*MaterialsA*Chiba gravelIshihama beach sandSilica No. 3 sandSilica No. 4 sandSilica No. 5 sand19671705151216191600Inagi sandOmigawa sandTicino sandOttawa sandMonterey sand772768449434297Tanno sandHostun sandCoral sand BSilica No. 6 sandCoral sand AToyoura sand14691435121410701013896Albany silica sandS.L.B. sandHime gravelGlass beads A, B & CCorundum A20916313900Corundum B, MACH and GB A, B and C (small speci­mens). With the other GMs, relatively loose (initial rela­tive density, Dr20¿50z) and relatively dense (Dr65¿95z) specimens were produced by the air­pluviationmethod. Drained TC tests were performed on either air­dried or saturated specimens or both.To quantify the particle shape of many GMs that arerepresentative of those tested in the present study, theirdegrees of angularity, A*, (Lees, 1964) deˆned as followswere measured:nA*S (1809|ai)¥i1xir(4)where a is the angle of a corner; x is the distance from theedge of a corner to the center of the maximum inscribedcircle; and r is the radius of the maximum inscribed circle(Fig. 6). The values of A* obtained by using a representa­tive particle of the respective GM types are listed in Table2. These A* values are consistent with visual impressionsof the particles (Fig. 4).Among the previous studies listed in Table A1, drainedTC tests were performed on; 1) dense moist specimens oftamped original Chiba gravel, which is well­graded con­sisting of angular crushed sandstone particles from aquarry (Anh Dan et al., 2006); 2) dense moist specimensof crushed concrete aggregate consisting of relativelyround, strong and stiŠ granular particles covered with athin weak and soft mortar layer (Aqil et al., 2005; Tat­suoka et al., 2006a); 3) dense air­dried specimens ofModel Chiba gravel A (D500.8 mm and Uc2.1)(Hirakawa, 2003; Hirakawa et al., 2003), produced byremoving particles larger than 5 mm from original Chibagravel; and 4) air­dried specimens produced by compact­ing air­dried powder of Fujinomori clay and kaolin (Li etal., 2004). A series of drained PSC tests were performed29on saturated specimens of Inagi and Narita sands, whichare well­graded relatively angular inland sands obtainedfrom Pleistocene Era deposits having essentially the samegrading while including slightly weathered thereforecrushable particles (more with Inagi sand) (Kawaharazo­no, 2007; Hirakawa et al., 2008); and air­dried specimensof Jamuna River sand, which is a poorly graded sandfrom Bangladesh, consisting of angular particles withsome fraction of mica (Yasin et al., 2003).Triaxial Apparatuses and Test ProceduresStrain­controlled drained TC tests on small specimens(the present study): The major part of the experimentsperformed in the present study used an automated strain­controlled triaxial apparatus (e.g., Santucci de Magistriset al., 1999; Kiyota and Tatsuoka, 2006). The top andbottom ends of saturated or air­dried specimens (7 cm indiameter and 15¿15.5 cm in height) were well­lubricatedby using a 0.3 mm­thick latex rubber smeared with a 0.05mm­thick silicone grease layer (Tatsuoka et al., 1984).External axial strains that were obtained from axial dis­placements of the loading piston measured with a defor­mation transducer arranged outside the triaxial cell arereported in this paper unless otherwise noted. Elasticproperties were evaluated based on axial strains locallymeasured along the specimen's lateral side with a pair oflocal deformation transducer (LDT; Goto et al., 1991).Although the eŠects of bedding error on the externallymeasured axial strains generally cannot be ignored, theireŠects on the parameters describing the viscous proper­ties are negligible (Enomoto et al., 2007a; Tatsuoka et al,2008a; as also shown below). The volume change of satu­rated specimens was obtained by measuring the waterheight in a burette connected to a specimen by using alow­capacity diŠerential pressure transducer. The volumechange of air­dried and moist specimens in strain­con­trolled TC tests was estimated by substituting respectivemeasured values of R (s?1/s?3) and axial strain incre­ments into the modiˆed Rowe's stress­dilatancy relationcalibrated by the corresponding drained TC tests on satu­rated specimens ( see APPENDIX B of Tatsuoka et al.,2008a).The specimens were axially compressed in an automat­ed way using a high precision gear­type axial loading sys­tem at constant cell pressure controlled by using an E/Ptransducer. Isotropic compression was performed at anaxial strain rate of 0.00625z/min towards an eŠectivemean stress p?(s?v{2s?h)/3400 kPa, where s?v and s?hare the eŠective vertical and horizontal principal stresses.At p?50, 100, 200 and 300 kPa in the course of isotrop­ic compression, eight cycles with an axial strain (doubleamplitude) of the order of 0.002z were applied to evalu­ate the elastic property (Fig. 7). The quasi­elastic verticalYoung's modulus, Ev, was evaluated from the average ofthe slopes of hysteresis loops (Fig. 7). Equation (5a) wasˆtted to a full­log plot of Ev|s?v (q{s?h) relation(Hoque and Tatsuoka, 1998), which was used to evaluateelastic axial strain increments in the drained TC tests:30ENOMOTO ET AL.Fig. 7.Typical stress­strain relations (eight very small cycles)EvEv0Ø 98s?»vdeehnvh| e devm(5a)Ø »Ev0s?v¥n0¥Eh0s?hm2Ø »s?v a ¥n0¥s?hm2(5b)Equation (5b), which is part of the hypo­elastic modelproposed by Tatsuoka and Kohata (1995) and Hoque andTatsuoka (1998), was used to estimate elastic lateralstrain increments, where nvh is the elastic Poisson's ratio;a is the parameter representing the inherent anisotropy ofthe stiŠness, which is assumed to be equal to 1.0 in thepresent study; and v0 is the Poisson's ratio when thematerial exhibits isotropic elastic property, which is as­sumed to be equal to 0.168 (Hoque and Tatsuoka, 1998).After drained sustained loading for thirty minutes at p?400 kPa (isotropic), drained TC was started.Strain­controlled drained TC tests on small specimens(the previous studies): Drained TC tests at s?h40 kPawere performed on specimens (d100 mm and h200mm) of; a) moist crushed concrete aggregate compactedat the optimum water content to diŠerent initial dry den­sities (Aqil et al., 2005; Tatsuoka et al., 2006a); and b)compacted air­dried model Chiba gravel A (Hirakawa etal., 2008). The top and bottom ends of the specimenswere in contact with the rigid ‰at faces of stainless steeltop cap and pedestal. Several drained TC tests were per­formed on air­dried specimens (7.5 cm in diameter and 15cm high) of Fujinomori clay and kaolin produced bycompacting air­dried clay powder inside a split mould inten layers with six blows of about 20 kgf per layer usinga 6 cm­diameter hammer (Li et al., 2004; Deng andTatsuoka, 2007; Tatsuoka et al., 2006a). The specimenswere isotropically consolidated to s?h77 kPa and 80 kPa(Fujinomori) and 100 kPa (kaolin).Strain­controlled drained TC tests on large specimens(the present study): A large specimen (30 cm in diameterand 60 cm in height) of Shinanogawa riverbed gravellysoil was produced by compacting air­dried material insidea split mould in six layers using an electric motor­drivenvibrator to an initial dry density equal to 2.05 g/cm3. Thespecimen was isotropically compressed to s?h40 kPa.Anh Dan et al. (2004) performed drained TC tests onlarge moist specimens of the same dimensions as above oforiginal Chiba gravel isotropically compressed to s?h490 kPa. In all these TC tests, the axial strains were ob­tained both externally from axial displacements of theloading piston using a LVDT and locally by using a pairof LDTs (Tatsuoka et al., 1994).Stress­controlled drained triaxial tests on small speci­mens (the present study): To evaluate both creep strainsby sustained loading (SL) and residual strains by cyclicloading (CL), a series of stress­controlled triaxial testswere performed at s?h40 kPa on dense (Dr90z) air­dried specimens (d75 mm and h150 mm) of selectedGMs consisting of relatively angular stiŠ particles (i.e.,Toyoura sand and model Chiba gravel H) and round stiŠparticles (Hime gravel and Corundum A). The specimenswere produced by air­pluviation. The top and bottomends of the specimens were lubricated by the samemethod as described above. The axial and horizontaldeformations were measured locally by a pair of LDTsand a set of three clip gauges. The specimens were axiallycompressed at a constant stress rate of 12 kPa/min.Strain­controlled drained PSC tests (the previous stu­dies): A series of drained PSC tests were performed onrectangular prismatic specimens (20 cm high, 16 cm longand 8 cm wide in the s?h direction) of; 1) air­dried air­pluviated Jamuna River sand (Yasin et al., 2003); and 2)moist and saturated Inagi and Narita sands moist­tampedat the respective optimum water contents (Kawaharazo­no, 2007; Hirakawa et al., 2008). The top and bottomends of the specimens were lubricated. The specimenswere anisotropically compressed at R3.0 towards s?h400 kPa (Jamuna River sand) and at R2.0 towards s?h50 kPa (Inagi and Narita sands). Axial strains weremeasured both externally and locally (with a pair ofLDTs), while lateral strains were measured locally by us­ing two sets of four proximity transducers, each set meas­uring lateral displacements at each sh surface (Yasin andTatsuoka, 2000).Drained direct shear (DS) tests (the present study): Dis­placement­controlled DS tests were performed at a con­stant normal pressure sv (100 kPa) on cubic air­driedspecimens (12 cm~12 cm~12 cm) of poorly gradedsands consisting of relatively angular stiŠ particles(Hostun and Silica No. 6a sands) and relatively roundstiŠ particles (Albany, Monterey and Ottawa sands). Thegrading of Silica No. 6a sand is slightly diŠerent fromthat of Silica No. 6 sand. The specimens were prepared bymulti­layer volume­controlled tamping. During otherwiseML at a constant shear displacement rate (0.055mm/min), several SL tests were performed for two hoursper stage. More details are described by Duttine et al.(2008a).EVALUATION OF VISCOUS PROPERTIESViscosity Type and b ValueFigures 8(a) and (b) show the relationships among Rs?v/s?h and the total axial (or vertical) and volumetric31VISCOUS PROPERTIES OF GRANULAR MATERIALSirFig. 8. Two CD TC tests (s?h400 kPa) of saturated dense silica No. 6 sand: a) R­ev­evol relations, b) R­gir­evol relations, c) zoom­up of test 25 andd) evaluation of rate­sensitivity coe‹cient, bstrains (ev and evol) and those among R and the irreversibleshear and volumetric strains, gir (eirv|eirh) and eirvol (eirv{2eirh), from two typical drained TC tests of dense silicaNo. 6 sand, a poorly graded consisting of relatively angu­lar and stiŠ particles (Fig. 4 and Table 2). Here, eirv is theirreversible vertical strain (ev|eev), where eev is the elas­tic vertical strain evaluated based on Eq. (5a), and eirh isthe irreversible horizontal strain (eh|eeh), where eeh isthe elastic horizontal strain evaluated based on Eq. (5b).In test 24, ML was continued at a constant ·ev. In test 25,·ev was changed stepwise by a factor of up to 200 and SLfor two hours per stage was performed twice during MLat a constant ·ev. It is seen that the peak strengths at diŠer­ent strain rates are essentially the same. Moreover, thestress increment that takes place upon a step change in ·evtends to fully decay with strain during subsequent ML ata constant ·ev in test 25 (Fig. 8(c)). These trends indicatethat the viscosity type is TESRA (Fig. 2). This point canbe conˆrmed by a successful simulation of test 25 by as­suming the TESRA viscosity (Fig. 8(c)); the simulationand the reference stress­strain relation are explainedlater. It may also be seen that the measured ‰ow charac­teristics is not sensitive to the stress changes associatedwith step changes in ·ev, in particular when the specimen is32ENOMOTO ET AL.dilating (Figs. 8(a) and (b)). A stronger trend of contrac­tion rate seen at smaller strain rates in Figs. 8(a) and (b) isconsidered due to contractive volume changes caused bythe volumetric creep mechanism (Kiyota and Tatsuoka,2006).In the results from test 25, any trend that the rate ofvolume increase becomes smaller at higher strain rates,which would have taken place if the specimen had notbeen fully drained, cannot be seen. This result also indi­cates that the specimen were essentially under fullydrained conditions. The above is also the case with thetest results presented in Fig. 9. Di Benedetto et al. (2002),Tatsuoka et al. (2002) and Nawir et al. (2003) also showedthat the trend of rate­dependent stress­strain behaviourof drained saturated specimens of unbound sands andgravels is essentially the same as the one of air­driedspecimens.Figure 8(d) shows the relationship between DR/R and, where ev is the externally measuredlog s( ·ev)after/( ·ev)beforetaxial strain, and between DR/R and log ( ·girafter/ ·girbefore),where g is the shear strain from locally measured axialstrains and measured volumetric strains, from test 25.Nearly the same linear relation is obtained by using thesetwo strain parameters. This result is consistent with thosereported by Enomoto et al. (2007a) and Tatsuoka et al.(2008a). The slope of the linear relation is deˆned as therate­sensitivity coe‹cient b (Eq. (1)). The values of b ob­tained by using ( ·ev)after/( ·ev)before from this and other similartests are used in further analysis.Figure 9 shows results from a drained TC test, similaras test 25 (Fig. 8), of a loose saturated specimen ofanother poorly graded angular sand, silica No. 4. Thesimulation shown in this ˆgure assumed the TESRA vis­cosity as explained later. Many other similar test resultsexhibiting the TESRA viscosity are reported by DiBenedetto et al. (2002) and Tatsuoka et al. (2002, 2008a).Figures 10 and 11 show drained TC test results of air­dried specimens of loose Monterey sand and dense Otta­wa sand, both poorly graded sands consisting of rela­tively round and stiŠ particles. A trend of P&N viscosity(Fig. 2) is obvious already in the pre­peak regime, andthis trend is very obvious in the post­peak regime. Thespecimens are air­dried and drained in these tests, as wellas those described in Figs. 12 and 13. Similar results ofdense Monterey sand and other poorly graded sands con­sisting of round and stiŠ particles exhibiting the P&N vis­cosity are reported in Tatsuoka et al. (2008a). The simu­lation shown in these ˆgures are explained later.Figure 12 shows results from a CD TC test of dense air­dried MACH, a well­graded GM consisting of relativelyround and stiŠ particles. It appears that the viscosity typeis TESRA initially in the pre­peak regime (Fig. 12(b)) andit changes to P&N when approaching the peak stress state(Fig. 12(c)). Yet, this trend of P&N viscosity is noticeablyweaker than those with the original three poorly gradedround GMs (reported by Tatsuoka et al., 2008a). This ob­servation was conˆrmed by simulation of this test takinginto account these trends, presented in Fig. 12, and alsoFig. 9. TESRA viscosity in a CD TC test (s?h400 kPa) on saturated loose Silica No. 4 sand and its simulation: a) whole strain range, b) zoom­upand c) and d) time histories of creep vertical strain33VISCOUS PROPERTIES OF GRANULAR MATERIALSFig. 11. P&N viscosity observed in a CD TC test on dense air­driedOttawa sand and its numerical simulation: a) whole strain rangeand b) zoom­upFig. 10. P&N viscosity in a CD TC test on air­dried loose Montereysand and its numerical simulation: a) whole strain range and b) andc) zoom­upsby the analysis of the viscosity parameters shown later.This test result indicates that the trend of P&N viscositybecomes weaker with an increase in Uc, as summarised inTable 1.Figure 13 shows results from a CD TC test of air­driedreconstituted Shinanogawa riverbed gravelly soil, a well­graded consisting of relatively large round gravel parti­cles and relatively angular sand particles (Fig. 4). It ap­pears that the viscosity type is TESRA in the pre­peak re­gime and this trend can be conˆrmed by simulation as­suming the TESRA viscosity (Fig. 13(b)). In the post­peak regime (Fig. 13(c)), when referring to a continuous­ly smooth relation inferred for continuous ML at a con­stant strain rate (depicted by a broken curve), it is seenthat the viscosity type is obviously TESRA. It seemstherefore that, despite that this gravelly soil comprisesrather round large particles, due to a better inter­particleinterlocking achieved by including relatively angularsmall particles and a large coe‹cient of uniformity, theviscosity type does not become P&N even in the post­peak regime. This test result is consistent with the trendsthat the viscous type deviates more from the P&N type asthe grading becomes less uniform and the particlesbecome more angular, as summarised in Table 1.Non­Linear Three­Component Model and SimulationsDi Benedetto et al. (2002) and Tatsuoka et al. (2002,2008a) showed that the viscous properties of unboundgeomaterials observed in the drained TC and PSC tests atˆxed conˆning pressure can be modelled by the three­component model (Fig. 1) as follows:RRf{Rvf1v1f3(6)v3where Rs?1/s?3(s {s )/(s {s ) (s?v/s?h under theTC and PSC stress conditions); Rf is the inviscid compo­nent of R (sf1/sf3); and Rv is the viscosity componentobtained as R­Rf. Eq. (6) means that the current stress34ENOMOTO ET AL.Fig. 12. CD TC test on dense air­dried MACH and its simulation: a)whole strain range and b) and c) zoom­upsstate sij (s?1, s?3) is represented by point B and the corre­sponding current inviscid stress state sfij (sf1, sf3) by pointF in Fig. 14(a). In the case of Isotach viscosity, Rv in Eq.(6) is obtained as:Rv(gir, ·gir)Rf(gir)¥gv( ·gir)ir1ir3irvirh(7)where gire |e e |e (as eir); and Rv(gir, ·gir) is thecurrent value of Rv, which is a unique function of instan­taneous values of gir and its rate ( ·gir) under loading condi­tions (i.e., when ·gir is kept always positive); and gv( ·gir) isFig. 13. CD TC test on dense air­dried Shinanogawa riverbed gravel:a) whole strain range, b) zoom­up of the initial part and its simula­tion and c) zoom­up of the post­peak behaviourthe viscosity function (Æ0) that is a highly non­linearfunction of ·gir, where gv(0)0; and gv(/)a (a ˆnitepositive value). The incremental form of Eq. (7) is:dRv(gir, ·gir)dsRf(gir)¥gv( ·gir)t(8)ªvdRv when s30 is represented by vector BA in Fig. 14(b).With the other viscosity types (i.e., Combined, TESRAand P&N ), the current Rv when girgir (i.e., [Rv](g ); Eq.ir35VISCOUS PROPERTIES OF GRANULAR MATERIALSFig. 15. Viscosity type parameters as a function of axial strain (modi­ˆed from Fig. 23(a) of Kongkitkul et al., 2008) ( see Table 3 for thetest numbers)Fig. 14. Structures of stress components: a) sij and b) sij{Dsij (Tat­suoka et al., 2008a)(6)) is no longer unique for given instantaneous gir and ·gir,so it cannot be obtained by Eq. (7), but it becomes load­ing history­dependent as follows (Tatsuoka et al., 2008a):girvirf[[dsR (g )¥g ( ·g )t] ¥gf[R ](g )tgirvir(t)decay.g(gir, gir|t)](9a)ir1gdecay.g(gir, gir|t)u(gir){s1|u(gir)t¥[r1(gir)]g |tir(9b)where gdecay.g(gir, gir|t) is the generalised decay function,which is a function of the current gir and the diŠerence be­tween gir and the irreversible strain, t, at which the incre­ment [dsRf(gir)¥gv( ·gir)~](t) developed. r1(gir) is the decayparameter, which is basically a function of gir and it maydecrease with gir under the condition that 0ºr1(gir)Ã1.The parameter r1 denotes the decay rate of a viscous stressincrement that developed when the irreversible strain wasequal to t associated with an increase in the irreversiblestrain from t to the current value eir (i.e., a decay by anamount of eir|t). When r11.0, no decay takes place(i.e., the Isotach viscous property) and the decay rate in­creases with a decrease in r1. More details are explained inTatsuoka et al. (2008a). In the simulations performed inthe present study, for simplicity on one hand and due to alack of reliable data on the other hand, r1(gir) is assumedto be a positive constant with respective GM types. u(gir)br/b is the viscous type parameter (Eq. (3)). Equation(7) (for Isotach viscosity) is obtained by substituting u1.0 into Eqs. (9a) and (b) and the equation for TESRAviscosity by substituting u0.0 into Eqs. (9a) and (b). Inthat case, Eq. (9b) becomes gdecay.g(gir, gir|t)[r1(gir)]g |t.The equations for Combined and P&N types are obtainedirby substituting, respectively, positive values of u less than1.0 and negative values into Eqs. (9a) and (b). Therefore,Eqs. (9a) and (b), as well as Eq. (8), are valid to all theviscosity types depicted in Fig. 2. Tatsuoka et al. (2008a)proposed the following equation for u(gir) that maydecrease with gir under the condition that u(gir)Ã1 in asingle test:« Ø »$uini{uend uini|uendgir{¥cos p¥ ir22guuc(10)where uini, uend, c and giru are the parameters that controlthe manner of viscosity type transition with strain. Figure15 summarises the u|ev relations used in the simulationsdescribed in this paper. As the residual condition couldnot be reached in the drained TC tests performed in thepresent study, the values of uend in these drained TC testsare the u values at ev15z, which may be diŠerent fromthose at the residual state in the DS tests. In the DS testsat a ˆxed shear displacement rate, the residual state wherethe average stress ratio does not change with shear dis­placement can be easily reached. The u values at the resid­ual state of poorly graded relatively angular GMs that ex­hibit the TESRA viscosity with u0.0 in the pre­peak re­gime (e.g., Toyoura, Hostun, Ticino and Silica No. 6asands) are noticeably negative ( see Figs. 12 and 13 ofDuttine et al., 2008a).The simulations presented in Figs. 8 through 13 weremade based on Eqs. (6) through (10) after havingreplaced gir with eirv. This replacement was made becausethe major part of the drained TC tests referred to in thispaper was performed on air­dried specimens withoutmeasuring horizontal strains, for which the analysis ofrate eŠects on the relationship between Rs?1/s?3 and theaxial (vertical) strain, ev is straightforward. The referencecurve (i.e., the Rf­ev curve, representing the elasto­plasticbehaviour; i.e. the stress­strain behaviour of E and Pcomponents connected in series in Fig. 1) in the respec­tive simulations was obtained by explorating the R­ev re­lations measured by continuous ML tests at diŠerent con­stant strain rates toward the one at zero strain rate. In36ENOMOTO ET AL.Table 3.MaterialsViscosity parameters of granular materials in drained TC tests simulated in this studyTestnumberDecayparameter*, r1Toyoura sand0.854Hostun sand0.864Silica No. 3 sand0.824Silica No. 4 sand0.862Silica No. 5 sand0.8570.15Silica No. 6 sand0.9800.2Silica No. 8 sand1.173Mixed silica sand10.582Viscosity type parameter, uuiniuend0.30.00.00.735Coral sand B0.905Tanno sand1.1220.4Ishihama beach sand0.8480.1Inagi sand1.0730.4Omigawa sand0.8740.3$2.060.10.72310|7|0.1|0.60.535|3|0.3|1.0Albany silica sandHime gravelCorundum AMACHMonterey sandOttawa sandS.L.B. sandTicino sand2{34{0.639105~10|8|4|0.110|8|0.3|1.00.82010|3|0.4|1.680.47910|60.0|0.690.765|0.2|0.8100.587|0.2|0.9110.755120.5591371415—10|66.19.3|0.90.935{—|0.6106eu * (z)0.10.5275irc0.1Coral sand AShinanogawa riverbed gravelNote)Consolidatedvoid ratio, ec|0.810|5|0.60.76210|10|0.7|1.00.51710|9|0.4|0.90.86610|50.62210|4|0.1|0.4|1.04.38.01.05.09.05.04.0*: determined in the formulation in terms of ev in z.{: obtained from the simulation of the test results presented in Fig. 25 (after Kongkitkul et al., 2008).$: compacted dry density (g/cm3).Fig. 13(b), the simulation is made of the R and locallymeasured axial strain relation. Table 3 summarises theviscous property parameters used in the simulationspresented in Figs. 8 through 13 together with those ofother drained TC tests described in this paper. Theseparameters were determined so that the measured rate­dependency of stress­strain behaviour is best ˆtted bymodel simulation. Note that these simulations are notmerely curve­ˆtting of respective test results. That is,once these parameters are determined based on resultsfrom tests performed along a certain loading history, themodel is able to predict the rate­dependent stress­strainbehaviour for any other arbitrary loading histories. Yet,it is convenient if approximated values of theseparameters can be estimated only from particle character­istics without performing sophisticated shear tests aspresented in this paper. This is indeed one of the objec­tives of the present study.It may be seen from Figs. 8 through 13 that varioustrends of rate­dependent stress­strain behaviour, exhibit­VISCOUS PROPERTIES OF GRANULAR MATERIALSFig. 16.37Rate­sensitivity coe‹cients b free from the eŠects of pore water plotted against: a) D50, b) Uc and c) FCing diŠerent viscosity types with and without obvious vis­cosity type transition, are all well simulated. A highervalue of uini and a higher value of r1 with MACH thanthose with the three original poorly graded round GMs(Albany sand, Corundum A and Hime gravel) can be seenfrom Table 3, which indicates that the trend of P&N vis­cosity with MACH is weaker than those with these origi­nal round GMs, as discussed earlier related to Fig. 12.ANALYSIS OF VISCOUS PROPERTYIn this section, the correlations between the viscousproperty parameters: i.e., the rate­sensitivity coe‹cient(b, Eq. (1)); the decay parameter (r1, Eq. (9b)); and theviscosity type parameter (Eq. (3)) at eirv15z (uend), andthe particle characteristics: i.e., D50, Uc, ˆnes content(FC) and the degree of particle angularity (A*, Eq. (4))and the correlations among these viscous propertyparameters are analysed.Rate­sensitivity Coe‹cient bThe b values listed in Table A1 are plotted against D50(Fig. 16(a)), Uc (Fig. 16(b)) and FC (Fig. 16(c)). Thesedata are all free from the eŠects of pore water and itschange during shear. With poorly graded GMs consistingFig. 17. EŠects of specimen density and wet conditions (saturated orair­dried) on the b value when the viscosity type is P&Nof relatively round and stiŠ particles that exhibit P&Nalready in the pre­peak regime, only the data of air­driedspecimens are plotted in Fig. 16, while they are comparedwith those of saturated specimens in Fig. 17. The b valueof saturated specimens of the same GM type is only mar­ginally smaller than that of air­dried ones. This result38ENOMOTO ET AL.also indicates that the saturated specimens were essential­ly under fully drained conditions because of the follow­ing. Most of these b values were measured when the speci­mens exhibited dilatant behaviour. Therefore, if underpartially drained conditions, at higher strain rates, thesaturated specimens are stronger and therefore exhibit alarger increase in the strength upon an increase in thestrain rate than it is under fully drained conditions. Con­sequently, the b values when saturated should havebecome larger than those when air­drained. With someother GM types, the b values of saturated specimens havealso been conˆrmed to be essentially the same as those ofair­dried specimens (plotted in Fig. 16). For example,with poorly graded GMs consisting of relatively angularand stiŠ particles that exhibit TESRA in the pre­peak re­gime (i.e., Toyoura and Hostun sands), the b values ofsaturated and air­dried specimens are essentially the same(Nawir et al., 2003).The following trends may be seen from Fig. 16. Firstly,with both angular and round GMs, the eŠects of densityon the b value are generally insigniˆcant. This trend isconsistent with that of Toyoura sand (Nawir et al., 2003).Secondly, the scatter among all the b values is not smallin Fig. 16(a­1). In Fig. 16(a­2), all the data points of theunbound GMs consisting of relatively angular and stiŠ(i.e., less crushable) particles that exhibit TESRA viscosi­ty in the pre­peak regime are located in a zone between apair of horizontal dotted lines. When limited to thesedata, in this very wide range of D50 of about four ordersof magnitude, the eŠects of D50 on the b value are not sig­niˆcant. This result indicates that the eŠect of particlesize is basically insigniˆcant.Thirdly, in Fig. 16(a­2), in comparison with the dataplotted in the zone between a pair of horizontal dottedlines, the b values of GMs consisting of relatively roundand stiŠ particles that exhibit P&N viscosity already inthe pre­peak regime (i.e., Corundum A, Hime gravel,glass beads and Albany, Monterey, Ticino, SLB andOttawa sands) are generally smaller. On the other hand,the b values of GMs consisting of relatively crushableparticles (i.e., crushed concrete, coral sand B and Inagi,Tanno, Narita and Omigawa sands) that exhibit a weakertrend of TESRA viscosity in the pre­peak regime withslower decay of Dsv (i.e., larger r1 values; Table 3 and asshown below) are generally larger. In this respect, Fig. 18shows the relationships between the b value and thedegree of particle angularity, A*, with an index of Ucamong a number of GMs. The data points located abovea horizontal dotted line are those for GMs having eitherUcÀ4 or relatively crushable particles or both. With theother data, the b values are smaller and the variation isrelatively small. Yet, when A*ºabout 500, the b valuetends to decrease with a decrease in A* and the valueswhen A*0 (i.e., glass beads and corundum A) aresmallest. A noticeable scatter in the b value for the sameA* among these data is due largely to a variation in Uc, asdiscussed below. In summary, the eŠects of A* on the bvalue are noticeable, while the eŠects of Uc and particlecrushability are much larger.Fig. 18.Relationships between b and A*Fourthly, the major reason for a noticeable variationin the b values of GMs consisting of relatively angularand stiŠ particles (plotted between two horizontal brokenlines in Fig. 16(a­2)) is the eŠect of Uc. In fact, the bvalues of these GMs tend to decrease with a decrease in Ucwhen Ucºabout 3, as indicated by a pair of brokencurves in Fig. 16(b). The data points located above thiszone are of GMs consisting of relatively crushable parti­cles, while those located in the lower part in this zone andbelow are of GMs consisting of relatively round and stiŠparticles. In comparison, in Fig. 16(c), the scatter of theb values when FC is equal and close to 0 is fairly largewith both angular and round GMs. Furthermore, thephysical meanings of D50 and FC are somehow overlap­ping. For these reasons, the analysis based on FC is notmade in the following.In summary, the b value is basically independent ofD50, while it tends to decrease with a decrease in A*, Ucand particle crushability. The eŠects of Uc and particlecrushability are much larger than those of A*.Decay Parameter r1In the simulations performed in this study, the value ofthis parameter (Eq. 9(b)) was assumed to be a constantwith a given GM type irrespective of viscosity type, asstated earlier, and evaluated by simulation of the respec­tive measured relationships between R and ev (in z).These ri values are listed in Table 3 and plotted inseparate two diŠerent logarithmic scales against D50 andUc in Figs. 19(a) and (b).In Fig. 19(a), the r1 value tends to decrease with an in­crease in D50 with Silica sands, Nos. 3, 4, 5, 6 and 8. InFig. 19(b­2), the data points in a zone surrounded by abroken curve are those of GMs consisting of relatively an­gular and stiŠ particles. In this zone, the r1 value tends todecrease with a decrease in Uc. Moreover, the data pointsplotted in the upper part of Fig. 19(b­2) without namesare those of GMs consisting of relatively crushable parti­cles. A weak trend that the r1 value increases with an in­crease in particle crushability may be seen. These eŠectsof D50 and Uc, as well as particle crushability, are general­VISCOUS PROPERTIES OF GRANULAR MATERIALSFig. 20.Fig. 21.Fig. 19. Decay parameters r1 (in terms of ev in %) plotted against: a)D50 and b) Ucly much smaller than those of particle shape. That is, thedata points in the lower part of Fig. 19(b­2) withoutnames are those of GMs consisting of relatively roundand stiŠ particles. The eŠects of particle shape on the r1value are obvious as a whole in the data plotted in Fig.19(b­2). This trend can be seen more obviously from Fig.20. That is, for the data with Ucº4.0 (plotted below ahorizontal broken line), the r1 value is rather independentof A* when A*Àabout 750, but the r1 value obviously39Relationships between r1 and A*Relationships between uend and A* of granular materialsdecreases with a decrease in A* when A*º about 500.Furthermore, overall eŠects of Uc are noticeable in Fig.20. These trends are consistent with the following obser­vations described earlier. Firstly, the r1 value re‰ects theviscosity type in that r1 is equal to 1.0 with Isotach type,while it decreases as the viscosity type changes fromIsotach toward P&N. Secondly, more coherent geo­materials (e.g., sedimentary soft rock and cement­mixedsoil as well as highly plastic clay) exhibit the Isotach typeviscosity (with r11.0). With unbound GMs, the viscosi­ty type changes from Isotach toward P&N as the micro­structure becomes less stable associated with; a) adecrease in the coordination number (i.e., the averagenumber of contact points between particles per particle)due to a decrease in Uc and a dry density (Duttine et al.,2008b); and b) a decrease in the inter­particle interlockingdue to a decrease in A* (i.e., with a decrease in the parti­cle angularity).Viscosity Type Parameter uendThe viscosity type parameter when ev15z (uend) wasselected as the index representing the viscosity types (Fig.2). Figure 21 shows the relationship between uend and A*40ENOMOTO ET AL.Fig. 23.Relationships between the logarithm of r1 and uendFig. 22. Relationships between the logarithm of r1 and b obtained byassuming b and r1 are constant in a single testFig. 24.with an index of Uc for GMs with Ucº30.4. As no GMsthat exhibit Combined type at ev15z were found in thepresent study, no data point with uendÀ0 is plotted in Fig.21. The uend value is nearly constant, equal to 0.0 (i.e.,TESRA viscosity) when A*À750. When A*º750, uenddecreases with a decrease in A* at a high rate from 0.0toward negative values, indicating that the trend of P&Nviscosity type becomes stronger. The uend value becomesnearly |1.0 when A* reaches 0 (i.e., corundum A).When A* is around 250, the uend value tends to increasewith an increase in Uc, although this trend is less obviousthan the eŠect of A* (i.e., the eŠect of particle shape). Nospeciˆc eŠects of D50 and particle crushability on the uendvalue were found.Correlation among r1, b and uendFigure 22 shows relationship between the values of r1and b obtained by simulations assuming that these valuesare constant in respective TC tests. The r1 value tends todecrease with a decrease in b, associated with changes inthe viscosity type from Isotach toward P&N. A relativelylarge scatter seen in the data is due to that the whole dataconsist of the following three distinct groups showingdiŠerent trends (Fig. 22(b)): 1) GMs consisting of rela­tively angular and stiŠ particles; 2) GMs consisting of rel­Relationships between b and uendatively angular and crushable particles; and 3) GMs con­sisting of relatively round and stiŠ particles. The ˆrst twogroups exhibit separate well­deˆned correlations, whilethe b value for the same r1 value is larger with more crush­able GMs. No independent eŠects of Uc can be seen onthese correlations. With the third group (i.e., GMs con­sisting of relatively round and stiŠ particles), the r1 valuesare very small with a large variation at a nearly constantb.Figure 23 shows the relationships between r1 and uendwith an index of A*. When uend0.0 (i.e., TESRA vis­cosity), the r1 value tends to decrease with a decrease in D50 and Uc (Figs. 19(a) and (b)). When uendº0.0 (i.e., P&Nviscosity), the r1 value obviously decreases with a decreasein uend associated with a decrease in A* (Fig. 21). Yet, forthe same uend value, the r1 value tends to decrease with adecrease in the A* value, showing that the r1 and uend rela­tion is not highly correlated.Finally, Fig. 24 shows the b and uend relation. This dataset consists of the following two groups having totallydiŠerent trends (as cited earlier). When A*À750, the vis­cosity type is TESRA (with uend0) irrespective of A*value (Fig. 21), while the b value signiˆcantly decreasesVISCOUS PROPERTIES OF GRANULAR MATERIALS41with a decrease in Uc and particle crushability (Fig.16(b)). On the other hand, when A*º750, the viscositytype is P&N and this trend becomes stronger (i.e., the uendvalue becomes smaller) with a decrease in A* (Fig. 21),while the b value, which is generally smaller than thosefor TESRA type, is not strongly aŠected by A* (Fig. 18).In summary, the viscous properties of GMs are aŠectedby, at least, particle shape (i.e., A*), grading (i.e., Uc)and particle crushability, while the eŠects of particle size(i.e., D50) are basically insigniˆcant. Generally, as theparticle becomes less angular, more uniform and stiŠer(i.e., less crushable), the values of b, r1 and u generallydecrease. More speciˆcally, the b value is aŠected by Ucand particle crushability more signiˆcantly than A*,while the values of ri and u are aŠected by A* more sig­niˆcantly than Uc and particle crushability.Fig. 25. EŠects of particle shape on the creep deformation (drainedTC tests: data of Toyoura sand added to those reported by Enomo­to et al., 2007a, b; Tatsuoka, 2007; Kongkitkul et al., 2008)Fig. 26. EŠects of particle shape on the creep deformation (drainedDS tests: data of Ottawa and Monterey sands added to thosereported by Kongkitkul et al., 2008)42Fig. 27.ENOMOTO ET AL.Relationships between creep strain by SL at shear stress level0.85 and ``u at the start of SL'' and uend: a) TC tests and b) DS testsCREEP DEFORMATION AND RESIDUALDEFORMATION BY CYCLIC LOADINGCreep DeformationFigure 25(a) shows results from drained TC tests (s?h400 kPa) at ·ev0.0625z/min (constant) with SL for tenhours at several shear stress levels performed on similarlydense air­dried specimens of three relatively round GMs(corundum A, Albany sand and Hime gravel) and threerelatively angular GMs (Toyoura sand, Silica No. 4 sandand coral sand A), all poorly graded consisting of stiŠparticles. Figure 25(b) compares the creep axial strains bySL for ten hours and the shear stress level (i.e., the ratioof Rs?v/s?h to its maximum value, Rmax) at which therespective SL tests were performed. Despite that a varia­tion in the relative density may aŠect this plot, it can benoted that, with similarly dense specimens of similarlypoorly graded GMs consisting of stiŠ particles, the creepstrain by sustained loading (SL) at the same ratio of theprincipal stress ratio to the peak stress ratio becomessmaller as the particles becomes less angular.Figure 26 presents a data set from drained direct shear(DS) tests at a constant s?v (100 kPa) with SL for twohours at diŠerent shear stress levels during otherwise MLat a constant shear displacement rate (0.055 mm/min).This data shows a similar trend as seen in Fig. 25. Thetest materials are two relatively angular GMs that exhibitthe TESRA viscosity in the pre­peak regime and four rel­atively round GMs that exhibit the P&N viscosity in thepre­peak regime, all poorly graded consisting of stiŠ par­ticles. The dilatancy rate is positive at most of the SLstages (Fig. 26(b)). Figure 26(c) compares the creepdeformation by SL for two hours and the shear stress lev­el (i.e., the ratio of RDSt/s to its maximum value,RDS.peak). Also in these DS tests, the creep deformation atthe same shear stress level of the relatively round GMs isnoticeably smaller than the relatively angular ones. Therange of D50 of the tested materials is small, 0.29–0.68mm, and therefore its eŠects on this trend must be insig­niˆcant.Figure 27(a) shows the relationships between the creepaxial strain by SL for 10 hours at a shear stress level equalto 0.85 obtained from Fig. 25(b) and the two viscositytype parameters obtained from the plots in Fig. 15: i.e.,the u value at ev15z (uend) and the u value at the start ofthe respective SL tests, both obtained by the simulationof the overall R­ev relations from the drained TC tests.Figure 27(b) shows similar relationships between thecreep shear displacement by SL for 2 hours at a shearstress level equal to 0.85 obtained from Fig. 26(c) and thetwo viscosity type parameters: i.e., the value at the resid­ual state (uend) and the u value at the start of the respectiveSL tests, both obtained by the simulation of the overallRDS­s relations from the drained DS tests (Kongkitkul etal., 2008). It may be seen that the creep deformationgenerally decreases with a decrease in the viscosity typeVISCOUS PROPERTIES OF GRANULAR MATERIALSFig. 28. Relationship between creep strain by SL and the degree of an­gularity, A*: a) drained TC test and b) drained DS testparameters in both TC and DS tests. As the viscosity typeparameter tends to decrease as the particle shape becomesless angular (Fig. 21), it implies that, at least with poorlygraded GMs consisting of stiŠ particles, the creep defor­mation becomes smaller as the particles become less an­gular. This trend can be seen from Figs. 28(a) and (b).The straight lines in these ˆgures were obtained by linearˆtting. Furthermore, it may be seen from Figs. 27(a­2)and (b­2) that the creep deformation is accurately simu­lated by the proposed model.Residual Deformation by Cyclic LoadingTo examine whether the eŠects of particle shape on thecreep strain by SL and the residual strain by cyclic load­ing (CL) are either similar or diŠerent, a series of stress­controlled drained TC tests were performed applyingboth SL and CL histories under otherwise the same con­ditions on air­dried dense specimens of four types ofGMs consisting of stiŠ particles. The materials are allpoorly graded GMs, relatively angular (Toyoura sandand model Chiba gravel H) or relatively round (corun­dum A and Hime gravel). Figure 29 shows the loadinghistories applied in a pair of drained TC tests, A and B, ats?h40 kPa on Toyoura sand and the test results. Creepshear strains by SL and residual shear strains by CL ap­plied alternatively at diŠerent shear stress levels but atdiŠerent sequences to a pair of very similarly dense speci­43mens were compared. The sustained deviator stress (q)equal to 60, 90, 120 and 150 kPa (Figs. 29(a) and (b)).Each SL or CL history was applied for a period of 50,000seconds, which was 500 cycles of CL at a frequency of 0.1Hz.Figure 30(a) compares the creep shear strain by SL andthe residual shear strain by CL for a short duration (i.e.,the ˆrst 100 seconds or the ˆrst one cycle) and for a longduration (i.e., the whole 50,000 seconds or the whole 500cycles) from these two TC tests. The four data points ofthe respective relations are those obtained at the fourdeviator stress levels. The residual shear strain by the ˆrstcycle of CL is consistently smaller than the creep shearstrain by SL for the ˆrst 100 seconds at any of these devi­ator stress levels. On the other hand, the residual shearstrain by 500 cycles of CL is much larger than the creepshear strain by SL for the same duration (i.e., 50,000 sec­onds), in particular at higher deviator stress levels. Figure30(b) shows the data for a short duration (i.e., the ˆrst100 seconds or the ˆrst one cycle) presented in Fig. 30(a)and similar data for the other three types of GMs havingdiŠerent particle shapes, while Fig. 30(c) shows the datafor a long duration (i.e., the whole 50,000 seconds or thewhole 500 cycle). With all these materials, the residualshear strain by the ˆrst one unload/reload cycle is muchsmaller than the creep shear strain by SL for 100 seconds(except for two data points at small shear stress level), butthe ratio of the residual shear strain by CL to the creepshear strain for the same loading duration largely in­creases with an increase in the loading duration (i.e., withan increase in the number of loading cycles in the CLtests). This trend becomes stronger as the particle shapebecomes less angular.Kongkitkul et al. (2004) showed that the residual strainthat takes place in polymer reinforcement during cyclictensile loading is due totally to viscous properties. On theother hand, as shown above, the eŠects of particle shapeon the creep strain by SL and the residual strain by CLare opposite. This fact suggests that the micro­structurethat is rather stable under stationary boundary stressesbecomes unstable by CL and this trend becomes strongeras the particles become less angular. It seems thereforethat the residual strain that takes place during CL is notdue totally to viscous properties, but it also includes acomponent by inviscid eŠects of CL, which should be afunction of cyclic stress amplitude and loading cycles butnot a function of loading frequency. It is known that theresidual strain that takes place during a given CL historyis not equal to a linear summation of two components byviscous properties and inviscid cyclic loading eŠects (Tat­suoka, 2007). The features of the developments of inelas­tic strain described in this paper are summarized in Table4. The eŠect of void ratio indicated in this table is afterDuttine et al. (2008b).CONCLUSIONSThe following conclusions can be derived from the testresults and analysis presented above:44ENOMOTO ET AL.Fig. 29. Loading histories in a pair of TC tests on air­dried dense Toyoura sand: a) overall loading histories, b) part of loading history of test A, c)overall deviator stress–shear strain relations and d) and e) zoom­up for test A1. The viscous properties of geomaterial can be charac­terized in terms of: a) the rate­sensitivity coe‹cient(b); b) the decay parameter (r1), and c) the viscositytype parameter (u). These parameters are aŠected byparticle characteristics, generally decreasing with adecrease in the uniformity coe‹cient (Uc), the particlecrushability and the degree of particle angularity (A*).The eŠects of mean diameter (D50) on these parametersare generally insigniˆcant.a) The b value is insensitive to changes in dry densityand wet conditions (i.e., air­dried or saturated, ex­cept for clays). The b value is aŠected by Uc andparticle crushability more signiˆcantly than A*(i.e., the particle shape).b) The r1 values of poorly graded granular materials(GMs) consisting of relatively round and stiŠ parti­cles, which exhibit the P&N viscosity in the pre­peak regime, are signiˆcantly smaller than those ofGMs consisting of relatively angular particles,which exhibit the TESRA viscosity. That is, the r1value is aŠected by A* more signiˆcantly than Ucand particle crushability.c) The u value (i.e., the viscosity type) is aŠected byA* more signiˆcantly than Uc and particlecrushability. When uº0, the r1 value tends todecrease with a decrease in the u value.2. With GMs consisting of stiŠ particles, as the particlesbecome less angular:a) the creep shear strain becomes smaller; andb) the increase in the residual shear strain by CL withVISCOUS PROPERTIES OF GRANULAR MATERIALS45Fig. 30. Comparison between residual shear strains by SL and CL from TC tests: a) air­dried dense Toyoura sand and b) and c) eŠects of the parti­cle shape on the relative largeness between residual strains by SL and CLTable 4. General trends of inelastic properties of geomaterial (modi­ˆed from Tatsuoka, 2007)In‰uencingfactorsViscosity IsotachªCombined ªTESRAªP & Ntype (u) (u1)(0ºuº1.0) (u0)(uº0: could beless than |1.0)Particle shape (incase of stiŠ particles) More angularªMore roundGradingcharacteristicsBetter gradedªMore uniformly gradedParticle size(if saturated)Smaller (clay)ªLarger (sand/gravel )Particle crushabilityMore crushableªLess crushableVoid ratioLower void ratioªHigher void ratioInter­particlebondingStrongerªWeakerªNullStrain levelPre­peakªPost­peak(in particular, at residual state)More stable (better bound, better interlocking &­Summary­larger coordination numbers)ªLess stable (lessInter­particle contactbound, weaker interlocking & smaller coordina­point conditiontion numbers)–Deformation bycyclic loading(inviscid)–Creep deformationSmallerªLargerLargerªSmalleran increase in the number of loading cyclesbecomes more signiˆcant than the increase in thecreep shear strain for the same loading duration.3. Residual strains by CL consist of two componentsdue to viscous properties and inviscid CL eŠects. Theinviscid CL eŠect becomes stronger with an increasein the number of loading cycle and the cyclic loadingstress amplitude.ACKNOWLEDGEMENTSThe corundum used in the present study was kindlyprovided by Prof. Gudehus, G., the University ofKarlsruhe, Germany. The study was ˆnancially support­ed by the Grant­in­Aid for Scientiˆc Research (B),KAKENHI No. 18360231, the Japanese Society for Pro­motion of Science. The help of the colleagues of the Geo­technical Laboratory of Tokyo University of Science inperforming the experiment is deeply appreciated.REFERENCES1) Anh Dan, L. Q., Tatsuoka, F. and Koseki, J. (2006): Viscous shearstress­strain characteristics of dense gravel in triaxial compression,Geotechnical Testing Journal, ASTM, 29(4), 330–340.46ENOMOTO ET AL.2) Aqil, U., Tatsuoka, F., Uchimura, T., Lohani, T. N., Tomita, Y.and Matsushima, K. 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(1993): EŠects of strain rate oninstability of granular soils, Geotechnical Testing Journal, 16(3),304–313.56) Yasin, S. J. M. and Tatsuoka, F. (2000): Stress history­dependentdeformation characteristics of dense sand in plane strain, Soils andFoundations, 40(2), 55–74.APPENDIX A1Table A1.MaterialGrading propertiesShinanogawariverbed gravelD5011.25 mm,Uc73.73Hime gravel(air­pluviated)D501.537 mm,Uc3.554, Gs2.682,emax0.759, emin0.515Drained TC and PSC tests with step changes in the strain rateWetconditionDensity1)Test methodRange of ·evbRef.Air­dried(compactionand TC test)rdc2.06 g/cm3Drained TC ats?h40 kPa(large specimen)0.00625–1.25 z/min0.0174Present studyAir­driedec0.639Drained TC ats?h400 kPa0.00625–1.25z/min0.0169Present studyec0.535SaturatedHostun sand(air­pluviated)D500.321 mm,Uc2.012, Gs2.658,emax1.034, emin0.621SaturatedToyoura sand(air­pluviated)D500.180 mm,Uc1.625, Gs2.648,emax0.978, emin0.592SaturatedSilica No. 3 sand(air­pluviated)D501.508 mm,Uc1.691, Gs2.648,emax1.040, emin0.717SaturatedSilica No. 4 sand(air­pluviated)D501.395 mm,Uc1.664, Gs2.648,emax1.008, emin0.688Saturated0.0169ec0.6380.0142ec0.5410.0137ec0.864ec0.686ec0.854ec0.642ec0.824ec0.731ec0.862ec0.691Drained TC ats?h400 kPa0.00625–1.25z/minDrained TC ats?h400 kPa0.00625–1.25z/minDrained TC ats?h400 kPa0.00625–1.25z/minDrained TC ats?h400 kPa0.00625–1.25z/min0.0218Present study0.01940.0211Present study0.01680.0222Present study0.02200.02040.0205Present study48ENOMOTO ET AL.Table A1.MaterialGrading properties(continued)WetconditionDensity1)Test methodRange of ·evbRef.ec0.857Drained TC ats?h400 kPa0.00625–1.25z/min0.0239Present studyDrained TC ats?h400 kPa0.00625–1.25z/minDrained TC ats?h400 kPa0.00625–1.25z/minSilica sand No. 5(air­pluviated)D500.554 mm,Uc2.243, Gs2.646,emax1.079, emin0.671SaturatedSilica No. 6 sand(air­pluviated)D500.290 mm,Uc2.427, Gs2.642,emax1.174, emin0.671SaturatedSilica No. 8 sand(air­pluviated)D500.099 mm,Uc2.235, Gs2.633,emax1.431, emin0.761SaturatedMixed silica sand(air­pluviated)D500.811 mm,Uc13.084, Gs2.639,emax0.899, emin0.473Saturatedec0.582Drained TC ats?h400 kPa0.00625–1.25z/min0.0304Present studyCoral sand A(air­pluviated)D500.170 mm,Uc2.066, Gs2.645,emax0.868, emin0.484Saturatedec0.735Drained TC ats?h400 kPa0.00625–1.25z/min0.0199Present studyCoral sand B(air­pluviated)D500.372 mm,Uc2.153, Gs2.785,emax1.052, emin0.708SaturatedDrained TC ats?h400 kPa0.00625–1.25z/minTanno sand(air­pluviated)D500.168 mm,Uc30.375, Gs2.465,emax1.872, emin0.977Saturatedec1.122Drained TC ats?h400 kPa0.00625–1.25z/min0.0454Present studyInagi sand(air­pluviated)D500.180 mm,Uc20.600, Gs2.836,emax1.348, emin0.784Saturatedec1.073Drained TC ats?h400 kPa0.00625–1.25z/min0.0416Present studyDrained PSC ats?h50 kPa0.025–0.25z/minec0.676ec0.980ec0.742ec1.173ec0.887ec0.538ec0.905ec0.736ec0.7930.02310.0247Present study0.02520.0308Present study0.02880.02140.0334Present study0.03400.0455Moist(w7–13.5z)rd1.554–1.569 g/cm3Saturatedrd1.527–1.561 g/cm3D500.167 mm,Dmax 2.0 mm, Uc16.6,FC19.1z, Gs2.666Saturatedrd1.397–1.556 g/cm3Drained TC ats?h50 kPa0.025–0.25z/min0.0343rd1.509–1.577 g/cm3Drained PSC ats?h50 kPa0.025–0.25z/min0.0424Ishihama beachsand(air­pluviated)D500.344 mm,Uc2.119, Gs2.728,emax0.944, emin0.593Saturatedec0.848Drained TC ats?h400 kPa0.00625–1.25z/min0.0206Present studyOmigawa sand(air­pluviated)D500.172 mm,Uc4.619, Gs2.751,emax1.329, emin0.729Saturatedec0.874Drained TC ats?h400 kPa0.00625–1.25z/min0.0424Present studyAlbany silica sand(air­pluviated)D500.302 mm,Uc2.221, Gs2.671,emax0.804, emin0.505Air­driedec0.723Drained TC ats?h400 kPa0.00625–1.25z/min0.0181Present studyInagi sand(moist compacted)Narita sand(moist compacted)ec0.550Saturatedec0.735ec0.762S.L.B.(Silver LeightonBuzzard) sand(air­pluviated)D500.681 mm,Uc1.43, Gs2.66,emax0.79, emin0.49Air­driedOttawa sand(air­pluviated)D500.174 mm,Uc1.76, Gs2.665,emax0.864, emin0.515Air­driedMonterey sand(air­pluviated)D500.484 mm,Uc1.4, Gs2.64,emax0.86, emin0.55Air­driedTicino sand(air­pluviated)D500.527 mm,Uc1.52, Gs2.68,emax0.96, emin0.59Air­driedCorundum A(air­pluviated)D501.416 mm,Uc1.623, Gs3.900,emax1.066, emin0.865Air­dried0.0488ec0.559ec0.765ec0.587ec0.866ec0.622ec0.935ec0.822SaturatedKawaharazono(2007);Hirakawaet al. (2008)0.01950.0178Drained TC ats?h50 kPa0.00625–1.25z/minec0.517ec0.7550.04360.0193Present study0.0191Drained TC ats?h50 kPa0.00625–1.25z/minDrained TC ats?h50 kPa0.00625–1.25z/minDrained TC ats?h50 kPa0.00625–1.25z/minDrained TC ats?h400 kPa0.00625–1.25z/min0.0243Present study0.02230.0198Present study0.01950.0174Present study0.01720.01500.0132ec0.9480.0105ec0.8200.0131Present study49VISCOUS PROPERTIES OF GRANULAR MATERIALSTable A1.MaterialGrading properties(continued)WetconditionDensity1)Test methodRange of ·evbRef.Corundum B(air­driedcompacted)D500.00163 mm,Uc28.133, Gs4.137,emax3.166, emin1.863Air­driedec1.714Drained TC ats?h50 kPa0.00625–1.25z/min0.0304Present studyMACH(air­pluviated)D501.089 mm,Uc6.343, Gs2.846,emax0.626, emin0.410Air­driedec0.479Drained TC ats?h400 kPa0.00625–1.25z/min0.0200Present studyGlass beads A(air pluviated)D500.4 mm,Uc1.205, Gs2.497,emax0.726, emin0.577Air­driedec0.542Drained TC ats?h50 kPa0.00625–1.25z/min0.0114Present studyGlass beads B(air pluviated)D500.2 mm,Uc1.188, Gs2.497,emax0.783, emin0.593Air­driedec0.582Drained TC ats?h50 kPa0.00625–1.25z/min0.0146Present studyGlass beads C(air pluviated)D500.1 mm,Uc1.093, Gs2.497,emax0.787, emin0.598Air­driedec0.606Drained TC ats?h50 kPa0.00625–1.25z/min0.0174Present studyJamuna river sand(air­pluviated)D500.16, Uc2.1,FC7z; Gs2.7,emax1.173, emin0.690Air­driede00.775and 0.821Drained PSC ats?h100and 400 kPa0.00125–0.125z/min0.0273Yasin et al.(2003)Original Chibagravel(moist compacted)Crushed sandstone;D507.8 mm,Dmax39.6 mm,Uc11.2, Gs2.71Moist(w5z;Sr77z)Dense, ec0.19(two specimens)Drained TC ats?h490 kPa(large specimens)0.0006–0.06z/min0.0335Anh Dan et al.(2004)Model Chibagravel A (air­driedcompacted)Crushed sandstone;D500.8 mm, Dmax5.0mm, Uc2.1, Gs2.74,emax0.727, emin0.363Air­driedec0.584 and0.556 (rd1.760and 1.770 g/cm3)Drained TC ats?h40 kPaCrushed concreteaggregateD505.5–6.5, Uc19,FC1.2–2.1z,Gs2.65,( rd)max1.78 g/cm3Moist(w16.69,17.34z;wopt16.9z)rd1.72,1.75 g/cm3Drained TCs?h20 kPa0.001–0.1z/min0.0536Aqil et al.(2005);Tatsuoka et al.(2006a)Fujinomori clay3)D500.017 mm,Uc§10,PI33,LL62zAir­dried(waf4.29z;Sr.af8.09z)ec1.093Drained TC ats?h77 kPa0.002–0.2z/min0.0444Li et al. (2004);Tatsuoka et al.(2006a)Air­dried(waf2.56z;Sr.af4.68z)ec1.202Drained TC ats?h80 kPa0.003–0.3z/min0.0353Air­dried(waf0.3z;Sr.af0.5z)ec1.38Drained TC ats?h100 kPa0.000046–0.0012z/min0.029Air­dried(waf0.3z;Sr.af0.5z)ec1.41Drained TC ats?h100 kPa0.000049–0.0013z/min0.030Kaolin3)D500.0013 mm,Uc4.32,PI41.6,LL79.6z1) ecconsolidated void ratio.2) waf and Sr.af; measured after each test.3) b value at oven­dried state was inferred based on the respective b­Sr relations.0.008–0.02440.00008z/minHirakawa(2003);Hirakawaet al. (2003)Deng andTatsuoka(2007);Tatsuoka et al.(2006a)
• ログイン

• タイトル
• Strength and Deformation of Soft Rocks under Cyclic Loading Considering Loading Period Effects
• 著者
• D. C. Peckley・Taro Uchimura
• 出版
• Soils and Foundations
• ページ
• 51〜62
• 発行
• 2009/02/15
• 文書ID
• 21169
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 51–62, Feb. 2009STRENGTH AND DEFORMATION OF SOFT ROCKS UNDER CYCLICLOADING CONSIDERING LOADING PERIOD EFFECTSD. C. PECKLEYi) and TARO UCHIMURAii)ABSTRACTCost­eŠective design is the primary motivation for adopting the performance­based design method. This method,however, requires that deformations be reliably estimated. While soft rocks are known to be competent foundationmaterials for large­scale structures, the deformation characteristics of this material when subjected to large cyclic load­ings still have to be understood. In this study, the strength and deformation characteristics of soft rocks under cyclicloadings were investigated by conducting cyclic triaxial tests on natural soft rock samples. The loading histories thatwere applied to these samples were uniform amplitude cyclic loadings with loading periods between 1 s and 9000 s. Thetests revealed that the longer the loading period, the larger is the residual strain accumulated for a certain number ofloading cycles. This dependency of residual strain accumulation on loading period appears to be an intrinsic materialproperty which is irrespective of loading amplitude and water content. From this ˆnding, it can be inferred that theprevailing practice of soft rock cyclic loading tests at 300 s and 9000 s of cyclic loading periods, which are much longerthan that of earthquakes, can result in overestimated residual strains.Key words: cyclic loading, deformations, loading period, soft rocks, triaxial tests (IGC: F7)lates that the cyclic loading should be applied with fre­quency in a range of 0.05 to 1 Hz for general geomateri­als, including soft rocks. It is desirable to examine the cy­clic deformation properties of geomaterials with a fre­quency similar to that of real earthquakes, which isaround 1 Hz or sometimes higher. For example, Shibuyaet al. (1994) investigated the eŠects of cyclic loadingperiod on the deformation of clayey materials, conclud­ing that the Young's modulus and the damping ratio de­pend on the cyclic period due to the creep properties ofthe materials.However, it is not a simple task to apply cyclic loadswith high frequency while precisely controlling the ampli­tude and the wave form, especially for soft rocks whichrequire high amplitudes of load. Therefore, many cyclicloading tests are conducted with lower frequencies, some­times lower than 0.05 Hz. In a round­robin test organizedby Technical Committee 29 of ISSMGE in 1996, cyclicloading periods of 1 s to 100 s were adopted for soft rocks(Yamashita et al., 2001). Recent cyclic loading tests onnatural soft rocks (Indo, 2001) and artiˆcial soft rocks/cement­treated soils (Salas­Monge, 2002) were conductedwith a controlled strain rate of 0.01z/min, which isroughly equivalent to loading period T ranging from 3000s to 6000 s. They implicitly assume that the behavior un­der such loading is independent of strain rate.The speciˆc objective of this paper is to investigate thedeformation and strength characteristics of soft rocks un­OBJECTIVECost­eŠective design is the primary motivation foradopting the performance­based design method. Thismethod, however, requires that deformations be reliablyestimated. While soft rocks are known to be competentfoundation materials for large­scale structures, the defor­mation characteristics of this material when subjected tolarge cyclic loadings still have to be understood. The dis­crepancies between measured and forward­calculated set­tlements of the piers of the Akashi­Kaikyo Bridge afterthe 1995 Kobe Earthquake (Koseki et al., 2001; Kashimaet al., 2000) underscore this point.Previous studies on soft rocks by Nishi (2002), Hayanoet al. (2001), Tatsuoka et al. (2000), Tatsuoka et al.(2003), and Bhandari and Inoue (2005) already estab­lished the rate dependency of soft rocks under monotonicloading. It can be inferred from their results that higherloading strain rates results in higher strength. The resultsof the tests by Nishi (2002) and Hayano et al. (2001), alsoshow neither intermediate stepwise changes in the strainrate nor the application of creep in the loading historyhave any signiˆcant eŠect on strength. Based on these ob­servations, it appears that the strength of soft rocks un­der monotonic loading is a function of the instantaneousloading strain rate at failure.As for cyclic loading, a standard for testing methodsby Japanese Geotechnical Society (JGS 0542–2000) stipu­i)ii)Post­doctoral Researcher, Department of Civil Engineering, University of Tokyo, Tokyo, Japan (dan_peckley—yahoo.com).Associate Professor, Department of Civil Engineering, University of Tokyo, Tokyo, Japan (uchimura—geot.t.u­tokyo.ac.jp).The manuscript for this paper was received for review on July 11, 2007; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.5152PECKLEY AND UCHIMURAder cyclic loading, with particular focus on residual strainaccumulation and loading period eŠects. Note that theloading period is highly dependent on strain rate, as ex­pressed by the relation: Loading period T2~(Doublestrain amplitude)/(strain rate). In many tests with strain­controlled loading systems (Santucci de Magistris et al.,1999; Tatsuoka et al., 2000), loading (whether monotonicor cyclic) is applied/deˆned by strain or deformationrate. In the said previous studies, the loading rates ap­plied were between 0.01z/min to 0.1zmin. In terms ofloading period, this range is between 300 s and 6000 s.TEST MATERIALS, INSTRUMENTATION ANDLOADING HISTORIESTest Materials (Guadalupe TuŠ soft rock, GTF)The GTF soft rock samples were obtained from an ex­cavation for the basement ‰oors of a multi­story buildingnow under construction in Fort Bonifacio, Taguig City,Metro Manila, Philippines. These were extracted byblock sampling (BS) from a sandstone layer at a depth ofaround 12 meters from the existing ground elevation,within an area of around 10 square meters. The blocksamples were then packaged and shipped to the Geotech­nical Engineering Laboratory of the University of Tokyo,Japan, following the recommendations of ASTM D5079–90 (ASTM, 2000) for critical care to minimizemoisture loss and damage in the microstructure of thesamples. The samples were cut and trimmed to L100 mm~W60 mm~H150 mm nominal sizes using a rotary cut­ter.Figure 1 shows photographs of representative samplesfrom the 3 blocks that were tested in this study. The sam­ples from Block A had an average in­situ density of 1.84g/cm3 and average moisture content of 28z; those fromBlock B had an average density of 1.78 g/cm3 andaverage moisture content of 31z. The samples fromBlock C, which were oven­dried, had an average dry den­sity of 1.40 g/cm3. The mean particle diameter D50 ofrepresentative samples from each block, after thoroughcrushing using a mortar and pestle, were similar: 0.15mm to 0.25 mm. The tests on these samples were con­ducted in unsaturated conditions because these weretaken above the water table, which was 8 m below the lay­er where the blocks were extracted.LDTs and Loading SystemsFigure 2 shows a schematic of the test set­up and pho­tograph of a sample with Local Deformation Trans­ducers or LDTs (Goto et al., 1991). These LDTs wereused to measure axial and lateral deformations to avoidbedding error eŠects (Tatsuoka and Kohata, 1994; Tat­suoka et al., 2003). Two 120 mm LDTs were used tomeasure axial (longitudinal) deformations and were at­tached on the 60 mm­wide faces of the specimen. Six 70mm LDTs were used to measure lateral (horizontal)deformations and were arranged such that each of the100 mm­wide faces of the specimen had three of theseLDTs. On each of these faces one LDT was placed at thespecimen mid­height and the other two, 45 mm aboveand below mid­height. The LDTs were attached to thespecimen following the procedure described by Hayanoet al. (2001).As one focus of this study, is the eŠects of loadingperiod or rate, loading systems that can apply cyclic load­ing at diŠerent loading periods were used in the tests. Oneloading system that has this capability is an automatedgear­clutch loading (GCL) system driven by an AC­servomotor (Santucci de Magistris et al., 1999; Tatsuoka et al.,2000). Having a maximum capacity of 50 kN, this load­ing system is strain­controlled and the typical strain ratesapplied range from 0.001z/min to 0.1z/min. It was ob­served that the cyclic stress applied by this machine hasapproximately a saw­tooth proˆle.At a loading period of around 1 s (1 Hz), however, thefeedback and control mechanism of this loading systemFig. 1. Representative GTF samples: GTF02 from Block A, GTF30 from Block B and GTF52 from Block C ­photos taken after test; whitestains/spots are hardened gypsum paste used to cap samples and smoothen surfaces where LDTs are attachedPERIOD OF CYCLIC LOADINGFig. 2.53(a) Schematic diagram of test set­up and (b) photograph of sample with LDTsbecomes unreliable. So far, the maximum cyclic loadingstrain rate that has been applied without compromisingaccuracy in amplitude was 1z/min, which was equiva­lent to a loading period T of around 30 s to 60 s.Since the loading rates that can be applied using a GCLsystem are signiˆcantly slower than the actual loadingrates during earthquakes, oil­hydraulic loading (OHL)systems were also used. One OHL system that was exten­sively used in this study was the combined OHL and GCLsystem at the Koseki Laboratory of the Institute of Indus­trial Science (IIS), University of Tokyo. Also having amaximum capacity of 50 kN, this loading system is stress­controlled and is known to be reliable at loading periodsT10 s (0.1 Hz). Certain di‹culties, however, were en­countered with the use of this machine: (1) control ofloading was through an external load cell and a separatecomputer­controller system; (2) overshooting or under­shooting occurred in the ˆrst half­cycle of loading whenthe loading period is T1 s; (3) the actual amplitude ofloading was signiˆcantly lower than the input amplitude;and (4) even though the input control signal had a saw­tooth proˆle, the actual loading proˆle was sinusoidal.To minimize the eŠect of item (2), the ˆrst cycle of load­ing was set to start with the unloading process. To over­come item (3), preliminary tests to calibrate the input andactual amplitudes were carried out. More informationabout the loading periods that can be applied using theseloading systems can be found in Peckley (2007).Loading HistoriesThe tests were conducted in unsaturated conditions be­cause the samples were taken above the water table,which was 20 m below the existing ground surface. Priorto loading, each GTF sample was consolidated for atleast 20 hours, at an isotropic pressure of 200 kPa–the es­timated in­situ overburden pressure. Figure 3 shows thetypical loading time history applied for each sample. Asshown in the ˆgure, monotonic loading to a static stresscondition q01500 kPa was ˆrst applied in drained con­dition to simulate a condition under the foundations forsuperstructures. During this loading, jumps in strainrates were introduced in an attempt to measure some rate­dependent parameters as described by Hayano et al.(2001). At the static state q0, creep loading was appliedfor one or three days prior to the cyclic loading, exceptfor the ˆrst two samples that were tested for cyclic load­ing.Then, cyclic loading was applied as shown in Fig. 3 andTables 1, 2, and 3. Except for one specimen designated asGTF02, the specimens were subjected to more than onestage of cyclic loading. Here, qd or qd1, qd2, etc. mean thesingle amplitude of the i­th stage of cyclic loading. Table1 tabulates the cyclic loading amplitude qd, number of cy­cles, and loading period T of each cyclic loading stage ap­plied to the samples from Block A; Table 2, those appliedto the samples from Block B; and Table 3, those appliedto the samples from Block C.In between the loading stages applied to the samplesfrom Blocks A, 30 minutes creep at the static state q0 wasapplied to observe if negative creep behavior occurs aftereach loading stage. However, as there was hardly anycreep strain that was measured, this step was skipped forother samples.The last loading stage was monotonic loading to failureat 1z/min. The ultimate strength obtained by this stageis noted as qmax in this paper. In addition, the ``stress ra­54PECKLEY AND UCHIMURAFig. 3.Table 1.GTF02Staged uniform cyclic loading time­historyCyclic loading applied to GTF samples from Block A: amplitude, no. of cycles and periodGTF03GTF05GTF06GTF07GTF08CyclicNo. ofNo. ofNo. ofNo. ofNo. ofNo. ofloadingAmplitude Cycles— Amplitude Cycles— Amplitude Cycles— Amplitude Cycles— Amplitude Cycles—stage Amplitude Cycles—LoadingqdLoadingqdLoadingqdLoadingqdLoadingqdLoadingqdPeriodPeriodPeriodPeriodPeriodPeriodS11256 kPa30 Cyc—60 s1198 kPa 30 Cyc—6000 s261 kPa 60 Cyc—1s1349 kPa 60 Cyc—1s768 kPa60 Cyc—1s558 kPa60 Cyc—1s1003 kPa60 Cyc—1sS3515 kPa60 Cyc—1s815 kPa60 Cyc—1s764 kPa60 Cyc—1sS4261 kPa 60 Cyc— 1020 kPa 60 Cyc—1s1s515 kPa 60 Cyc—1sS560 Cyc—1144 kPa511 kPa 60 Cyc—1s1s265 kPa 60 Cyc—1sS6769 kPa 60 Cyc—1s927 kPa 60 Cyc—1sS7991 kPa 60 Cyc—1s712 kPa 60 Cyc—1sS21522 kPa30 Cyc—73 s1025 kPa 60 Cyc—1s60 Cyc—1s482 kPa1325 kPa 30 Cyc—66 s1316 kPa30 Cyc—60 s1294 kPa 60 Cyc—1s60 Cyc—1sS81025 kPaS91048 kPa 60 Cyc—1s210 kPa 60 Cyc—1sS1060 Cyc—1s1097 kPa 60 Cyc—1s(The average initial moisture contents was 28z, initial average dry density was 1.84 g/cm3, and eŠective conˆning pressure was 200 kPa.)tio'' SR is deˆned as an index of the intensity of cyclicload amplitude relative to the ultimate strength of eachspecimen:SRqd/(qmax­q0)(1)As an attempt to investigate the actual behavior duringearthquakes, irregular cyclic loading tests were per­formed on three GTF samples, namely GTF32, GTF33and GTF35. The combined OHL and GCL system at theKoseki Laboratory was used in these tests. Figure 4shows the irregular cyclic loading time histories that wereapplied to these samples. Although the waveform ofthese loading time histories can be characterized as moreof an irregular periodic loading and do not appear toresemble actual earthquake loading, the frequency con­tents and duration are very similar to those of actualearthquake records.55PERIOD OF CYCLIC LOADINGTable 2.Cyclic loading applied to GTF samples from Block B: amplitude, no. of cycles and periodGTF30GTF31GTF34GTF36GTF40GTF41CyclicNo. ofNo. ofNo. ofNo. ofNo. ofNo. ofloadingCycles— Amplitude Cycles— Amplitude Cycles— Amplitude Cycles— Amplitude Cycles— Amplitude Cycles—stage Amplitude LoadingqdLoadingqdLoadingqdLoadingqdLoadingqdLoadingqdPeriodPeriodPeriodPeriodPeriodPeriodS1275 kPa30 Cyc—1s30 Cyc—1s526 kPa30 Cyc—1s766 kPa30 Cyc—3729 s499 kPa30 Cyc—2214 s763 kPa30 Cyc—33 sS2535 kPa30 Cyc—1s1264 kPa 30 Cyc—1s772 kPa30 Cyc—1s823 kPa4 Cyc—380 s826 kPa4 Cyc—364 s818 kPa4 Cyc—36 sS3758 kPa30 Cyc—1s1008 kPa30 Cyc— 1213 kPa1s4 Cyc—633 s1213 kPa1 Cyc—622 s1201 kPa1 Cyc—59 sS41095 kPa30 Cyc—1s1204 kPa30 Cyc—1s700 kPa4 Cyc—326 s827 kPa4 Cyc—385 s812 kPa4 Cyc—37 sS51353 kPa30 Cyc—1s1332 kPa30 Cyc—1s820 kPa4 Cyc—395 s704 kPa2 Cyc—325 s752 kPaS6(The average initial moisture contents was 31z, initial average dry density was 1.78 g/cm3, and eŠective conˆning pressure was 200 kPa.)Table 3. Cyclic loading applied to GTF samples from Block C: ampli­tude, no. of cycles and periodGTF50GTF51GTF52CyclicNo. ofNo. ofNo. ofloading Amplitude Cycles— Amplitude Cycles— Amplitude Cycles—stageLoadingqdLoadingqdLoadingqdPeriodPeriodPeriodS1Cyc— 1511 kPa 30 Cyc— 1521 kPa 30 Cyc—1517 kPa 30 7162 s9109 ssS2Cyc— 1071 kPa 4 Cyc—1044 kPa 461691070 kPa 4 Cyc—s48 s42 sS31497 kPa 1 Cyc—71 sS41071 kPa4 Cyc—48 s1497 kPa 1 Cyc—61 s1072 kPa4 Cyc—42 s(The specimens were oven­dry, initial average dry density was 1.40g/cm3, and eŠective conˆning pressure was 200 kPa.)TEST RESULTS AND DISCUSSIONSStrengthTable 4 presents the maximum deviator stresses qmaxthat were measured with the GTF samples. Figures 5, 6and 7 show typical stress­strain curves obtained fromthese samples.Looking at the qmax values obtained from the samplesfrom Block A and Figs. 5 and 6, one may be led to con­clude that cyclic loading indeed has no eŠect on strength,regardless of the amplitude or number of cycles applied,as observed by Indo (2001) and Salas­Monge (2002).However, the qmax values of GTF30 and GTF34, whencompared to those of the other samples from Block B, tellotherwise. Shown in Fig. 7 to have failed under cyclicloading, are GTF30 and GTF34 with qmax values that arearound 2,750 kPa while the other samples have qmaxvalues that are as high as 3,200 kPa. As the quasi­elasticYoung's modulus Ei values for GTF30 and GTF34 aresimilar to those of other samples from Block B, one cansurmise that the lower qmax values obtained from GTF30and GTF34 cannot be attributed to sampling disturbance.One can thereby infer that GTF30 and GTF34 failed atdeviator stress levels below the real ``strength'' of thematerial.Comparing the number of cyclic loading stages and theamplitude of loading for GTF30 and GTF34 with thosefor the other samples (summarized in Table 4, detailsprovided in Tables 1 and 2), one can infer that GTF30and GTF34 were the most damaged samples, becausethey had the most severe loading. As a result, the qmaxvalues of GTF30 and GTF34 were lower than those of theothers.The deˆnition of SRqd/(qmax­q0) is based on thepremise that qmax represents the ultimate strength of thematerial, but other values have to be used for GTF30 andGTF34 in place of qmax. As pointed out in previous studiesby Nishi (2002), Hayano et al. (2001) and Bhandari andInoue (2005), the strength or the qmax that can be meas­ured under monotonic loading depends on strain rate,i.e., the higher the strain rate, the higher the strength. Forconsistency with the procedure of measuring qmax that wasadopted in the monotonic loading tests of GTF samples,a new parameter qs is used to represent the original ulti­mate strength of the sample in place of qmax, which is de­ˆned as an expected value of the ultimate strength ob­tained if the samples were tested only under monotonicloading at 1z/min strain rate without cyclic loading.For consistent arguments on the eŠect of SR later on,the value of strength qs should be deˆned for every sam­ple which was subjected with cyclic loading, not onlyGTF30 and GTF34. For this, two samples were selectedfrom each block, which are deemed least damaged by cy­clic loading, and the average values of qmax and Ei of thesetwo samples are calculated asqqmaxrandqEir. Then, qsfor each sample from the same block are determined byusing the expression:56PECKLEY AND UCHIMURAFig. 4.Irregular cyclic loading applied to (a) GTF32, (b) GTF33 and (c) GTF35qsqqmaxrEi/qEir(2)where Ei is the Young's modulus of each sample at q0.Then, for qs given to each sample, the stress ratio SR,which was initially deˆned as SRqd/(qmax­q0), wouldthen be redeˆned as:SRcorrectedqd/(qs|q0)(3)The samples selected from Blocks A and B are markedwith asterisk * in Table 4 respectively. Table 4 alsopresents the strength qs of each sample and the correctedSR for the maximum amplitude qd applied on the sample.Figure 8 shows a plot of the maximum (corrected) SR ap­plied against the ratio qmax/qs, which represents howmuch the ultimate strength of the sample had deteriorat­ed due to the loading process. From this ˆgure, failure be­low the strength qs is possible under a certain cyclic load­ing history with less than 20 cycles when the stress ratio(corrected) of each cycle is greater than 0.8 (GTF30 andGTF34). Failure is also observed under the strength qs insome cases (GTF5, GTF6, and GTF8) after many numberof cyclic loading even with relatively lower stress ratio.Indo (2001) and Salas­Monge (2002) noted that cyclicloading indeed has no eŠect on strength regardless of theamplitude or number of cycles applied, but this is not al­ways true.Residual StrainsIn Fig. 5, the cyclic stress­strain curves of specimensGTF02 and GTF03 are compared. In both GTF02 andthe ˆst cyclic loading stage of GTF03, 30 cycles were ap­plied at almost the same amplitude, but the loadingperiod for GTF02 was 100 times shorter than that ofGTF03. Due to this diŠerence in loading period, it can bereadily observed that the cyclic stress­strain curves of thetwo are diŠerent, and that the accumulated residualstrain of GTF03 after the ˆrst loading stage is larger thanthat of GTF02.Figure 6, on the other hand, compares the cyclic stress­strain curves of GTF07 and GTF08. Similar to the casesof GTF02 and GTF03, the accumulated strain after 30 cy­cles is larger for GTF08 than for GTF07 (the length of thearrow shows the deformation during the ˆrst loadingstage) due to the diŠerence in loading periods. It can alsobe observed from Fig. 6 that creep deformation at thestatic condition q0 after three days is quite small com­pared to the deformations due to the large cyclic loadingapplied. Thus, the diŠerence in the residual deformationdue to cyclic loading with diŠerent period cannot be ex­plained by the creep deformation.Figure 9 plots the accumulation of residual strain atthe end of every cycle until the 30th cycle of the ˆrst cyclicloading stage (S1) of GTF02, GTF03, GTF05, GTF07and GTF08. As GTF02 and GTF03 does not have creepstage before cyclic loading, the average creep strains ofGTF05, GTF06, GTF07 and GTF08 was subtracted fromthose of GTF02 and GTF03 to cancel the possible creepstrain which may occur during the cyclic loading stage. Itis clear from this ˆgure that the longer the loading period,the larger the accumulated residual strain is.57PERIOD OF CYCLIC LOADINGTable 4.BlockBlockAStrength of GTF samplesSampleqmax(kPa)Loading historyMaximumqd (kPa)No. of cyclicloading stages andLoading periodT for max qdStressratioSRGTF01*3323Entirely monotonic—————15001958GTF023286Uniform cyclic12561 stage & 60 s0.7033660.6715002038GTF033620Uniform cyclic15222 stages & 73 s0.7237460.6815002268GTF053287Uniform cyclic10489 stages & 1 s0.5936360.4915012202GTF063288Uniform cyclic114410 stages & 1 s0.6435920.5515072175GTF07*3686Uniform cyclic13496 stages & 1 s0.6236860.6214992286GTF083365Uniform cyclic13252 stages & 66 s0.7137610.5914992278GTF133087Entirely monotonic with creep———3491——2114Average qsBlockBBlockCCorrected Corrected qo (kPa)SRqmaxqs3611Average EiEi(GPa)2165GTF302753Uniform cyclic13535 stages & 1 s1.0831930.8015052227GTF313088Uniform cyclic12642 stages & 1 s0.8031910.7515092226GTF32*3261Irregular cyclic11711 stage & 1 s0.6932610.6915732262GTF33*3221Irregular cyclic12351 stage & 1 s0.7132210.7114772260GTF342780Uniform cyclic13325 stages & 1 s1.0031390.7914502190GTF353040Irregular cyclic13021 stage & 1 s0.8530390.8515032120GTF363137Uniform cyclic12136 stages & 633 s0.7432110.7114962240GTF403109Uniform cyclic12134 stages & 622 s0.7632000.7215042232GTF413013Uniform cyclic12014stages & 59 s0.7931060.7515002166Average qs3173Average Ei2214GTF504239Uniform cyclic15174 stages & 622 s0.5542390.5515022149GTF513900Uniform cyclic15112 stages & 622 s0.6339000.6315021765GTF526617Uniform cyclic15214 stages & 622 s0.3066170.3015022614Fig. 5.Stress­strain curves of GTF02 and GTF03It can be also noted from Fig. 9 that the residual straindue to the ˆrst cycle of loading also appears to be depend­ent on loading period. In fact, for shorter loadingperiods, the residual strain due to the ˆrst cycle becomesFig. 6.Stress­strain curves of GTF07 and GTF08comparable with the increments in strain due to the suc­ceeding cycles. It only becomes signiˆcantly larger thanthose of succeeding cycles when the loading period islong, as observed by Indo (2001) and Salas­Monge58PECKLEY AND UCHIMURAFig. 7.Stress­strain curves of GTF30 and GTF34Fig. 10.Fig. 8.Residual axial strain vs. timeMaximum deviator stress ratio qmax/qs vs. max. SRFig. 11. Summary of loading period eŠects on residual strain accumu­lation in GTF SamplesFig. 9. Accumulation of residual axial strain in GTF samples at q0stress level(2002).The accumulation of residual strain presented in Fig. 9is also plotted with time in Fig. 10. It reveals that the ac­cumulation of residual strain is largely aŠected by the cy­clic loading period (or number of cycles) even if com­pared for the same duration.The eŠect of loading period on residual strain accumu­lation in the GTF samples under diŠerent stress ratios isgeneralized and summarized in Fig. 11, which is a log­logplot of the accumulated residual strain in loading stage S1vs. the loading period applied in this loading stage. In thisˆgure, the accumulated residual strain ear was normalizedusing this equation to cancel the variation of the stiŠnessof each specimen:59PERIOD OF CYCLIC LOADINGFig. 12. Accumulation of residual strain vs. stress ratio–data pointsfrom ˆrst cyclic loading stage (S1)enormar (Ei/q0)ear.(4)From Fig. 11, it can be observed that the dependency ofresidual strain accumulation on loading period is ir­respective of the corrected stress ratio SR. That is, underany stress ratio SR, the longer the loading period, thelarger is the accumulated residual strain for the samenumber of cycles applied. Although limited, the data ob­tained from cyclic loading tests on oven­dried samplesappear to follow the same trend, suggesting that thiseŠect is irrespective of water content.It can be further observed in Fig. 11 that for diŠerentstress ratio levels, unique straight lines can be drawn torepresent the relationship between the logarithm of ac­cumulated residual strain and the logarithm of loadingperiod. As shown, these lines appear to be parallel, sug­gesting that the loading period dependency of residualstrain accumulation is an intrinsic material property.Given this the linear and parallel relationships betweenthe accumulated residual strain and the loading period inlog­log plot at diŠerent stress ratio levels, the data shownin this ˆgure can be summarized by further normalizationthat involves the residual strain­SR relation derived inFig. 12.Figure 12 plots the accumulated residual strainT1s(SR) at diŠerent stress ratio SR levels, at loadingenorm,arperiod T1 s. Note that the data points shown in thisplot are the same data points in Fig. 11 at T1 s. Withthese data points, the following third­order exponentialfunctionT1senorm,(SR)A1eSR/t {A2eSR/t A3eSR/t {y0ar123(5)was introduced as the residual strain­SR relation at T1s and was used to further normalize all the data points inFig. 11. In this residual strain­SR relation, theparameters A1, A2, A3, t1, t2 and t3 are all constants, thevalues of which are determined by an iterative regressionmethod that is based on the Levenberg­Marquardt (LM)algorithm (Originlab Corporation, 2006).Figure 13 shows the normalized data from Fig. 11 us­ing this residual strain­SR relation at T1 s. The datafrom the tests on the samples from Blocks A, B and C ap­Fig. 13.SR­normalized residual strainspear to converge in a band deˆned by Lave}0.5Lave, whereLave is a sort of an average line that is a function of load­ing period T, drawn to ˆt the test data. For example, atest with a loading period of 10000 seconds, which cor­responds to a strain rates of around 0.01z/min, theresidual strain becomes 3.5 times larger than tests con­ducted at T1 s. Given this ˆgure, however, a uniquecorrection factor related to T can be derived for the de­sign process, even though there is some range of error.The practical implications of the ˆnding that longerloading periods result in larger residual deformations arethe following:a) The loading periods in cyclic loading tests of softrocks should be as close as possible to the actualperiods of the cyclic loads to be simulated in the tests.If, for example, the load to be simulated is a seismicload, then the loading period should be as close as pos­sible to the actual predominant seismic loading periodwhich is around 1 s.b) Cyclic loading tests of soft rocks at 1000 seconds to10000 seconds of loading periods, which correspondsto cases with prevailing loading rates of around 0.01z/min to 0.1z/min if the strain amplitude is around 1z, can result in overestimated residual strains com­pared to the actual deformations due to an earth­quake.Going back to Fig. 12, one can observe that SR valuesbelow 0.40 result in relatively small, almost negligibleresidual strains. However, above this value the residualstrain becomes very large, because of the use of an ex­ponential function to represent the relationship betweenSR and accumulated residual strain. On the other hand,the strength ratio qmax/qs in Fig. 8 remains near to 1.0 fora range of SRº0.85. The practical implication of this ob­servation is that the evaluation of residual deformationwithin a range of 0.40ºSRº0.85 is still meaningful,even though qmax is not aŠected by the cyclic loading.This observation also gives credence to one of the rea­sons that was cited for the discrepancies between themeasured and forward­calculated settlements of theAkashi Kaikyo Bridge piers due to the 1995 Kobe Earth­quake (Koseki et al., 2001; Kashima et al., 2000). If in­60PECKLEY AND UCHIMURAdeed the seismic loads were overestimated in the calcula­tions such that stress ratios were higher than actualvalues, then the settlements would also have been overes­timated considering the apparent exponential relation­ship between stress ratio and residual strain. Evidently,another reason for the discrepancies is the loading perioddependency of residual strains, as discussed above.In the tests on GTF samples, no consistent maximumstrain threshold, which Salas­Monge (2002) described asthe strain at which failure appears to consistently occur,was observed (see Figs. 5, 6 and 7). This could be due tothe nature of triaxial compression (TC) tests where shearbands tend to form at any arbitrary diagonal plane, ascan be observed in Fig. 1. Such arbitrariness makes theobservation of a maximum strain threshold di‹cult, un­like in plane strain compression (PSC) tests where the for­mation of shear bands is constrained to a certain plane.StiŠness and DampingIn the succeeding discussions, equivalent shear modu­lus and damping ratio are deˆned following JGS0542–2000: Method for Triaxial Test to DetermineDeformation Properties of Geomaterials (JGS, 2000).See Fig. 14.Figure 15 plots the degradation of the equivalent shearmoduli of the GTF samples from Block A as the numberFig. 14.Deˆnitions of equivalent modulus and damping ratioFig. 15.Geq vs. cycle number of GTF samplesof cycles increases in loading stage S1. As can be inferredfrom the ˆgure, the shear moduli at shorter loadingperiods are generally higher than those at slow loadingrates. It can be noted though that GTF07 and GTF08have nearly the same moduli despite the diŠerence inloading period. This deviation from the general trendcould be due to the diŠerence in the cyclic loading proˆleapplied to the specimens. It is possible that if GTF07 hadbeen subjected to cyclic loading with a saw­tooth proˆlein the same way as GTF08, the observed modulus ofGTF07 could have been higher. As previously discussedin LDTs and Loading Systems, the loading system thatwas used for GTF07 was a hydraulic loading machinethat applies cyclic loading with a sinusoidal proˆle.Figure 16 plots the shear moduli with the double am­plitude of cyclic strain in all cyclic loading stages, for allthe GTF samples from Block A. It is clear from this plotthat higher cyclic strain amplitude results in lower shearmodulus. However, contrary to what is usually assumedin equivalent linear dynamic analysis, this ˆgure showsthat shear modulus is not a unique function of the cyclicstrain amplitude. This observation can be attributed tothe degradation of shear modulus with loading cycle asnoted in Fig. 15. Given this observation, care should beexercised when trying to generate a shear modulus­cyclicFig. 16.Geq vs. shear strain double amplitude of GTF samplesFig. 17.Damping ratio vs. cycle number of GTF samplesPERIOD OF CYCLIC LOADING61strain accumulated for a certain number of loading cy­cles.3) The residual strain due to the ˆrst cycle of loading ap­pears to be dependent on loading period. In fact, forshorter loading periods, the residual strain due to theˆrst cycle becomes comparable with the increments instrain due to the succeeding cycles. It only becomessigniˆcantly larger than those of succeeding cycleswhen the loading period is long.4) The eŠect of loading period on residual strain ac­cumulation appears to be an intrinsic material proper­ty which is irrespective of water content.5) The accumulated residual strain tends to increase ex­ponentially with stress ratio, SRqd/(qmax­q0).Fig. 18. Damping ratio vs. shear­strain double amplitude of GTFsamplesstrain amplitude curve from cyclic loading test results.This is imperative because, in the analysis, groundresponse (stresses and strains) and possible resonanceeŠects depend much on the calculated predominantperiod of the ground, which in turn depends on shearmodulus.Figure 17 shows the variation of the damping ratioswith the number of cycles applied. From this ˆgure, it canbe readily observed that the damping ratios signiˆcantlydrop and remain almost constant after the ˆrst cycle ofloading. As can also be surmised from the ˆgure, thedamping ratios of GTF03, which was subjected to a longloading period at T6000 s, are signiˆcantly higher thanthose subjected to loading periods T1 s, 60 s, and 66 s.At shorter loading periods, however, such loading rateeŠect appears to become less signiˆcant.Figure 18 shows the variation of the damping ratiowith the double amplitude of cyclic strain in all cyclicloading stages, for all the GTF samples from Block A.Consistent to what is usually assumed in equivalent linearanalysis, it can be observed from the ˆgure that there ap­pears to be a unique relationship between damping ratioand cyclic strain amplitude: the higher the cyclic ampli­tude, the higher the damping ratio.CONCLUSIONSBased on the results of the uniform cyclic loading testsof natural GTF soft rocks presented herein, the followingconclusions can be made:On Strength1) Failure of GTF below the strength qs is possible undera certain cyclic loading history with less than 20 cycleswhen the corrected stress ratio of each cycle is greaterthan 0.8. Failure is also observed under the strength qsin some cases after many number of cyclic loadingeven with relatively lower stress ratio.On Residual Strain Accumulation2) The longer the loading period, the larger is the residualOn StiŠness and Damping6) Longer loading periods can result in lower stiŠnessand higher damping ratio.7) While there appears to be a unique relationship be­tween damping ratio and cyclic strain amplitude of cy­clic loading, there is no unique relationship betweenstiŠness and cyclic strain amplitude, because the stiŠ­ness decreases with the number of loading cycles.ACKNOWLEDGMENTSThis research was made possible through a scholarshipgrant that was awarded to the ˆrst author by the JapanMinistry of Education, Culture, Sport, Science, andTechnology (MEXT). The authors are also grateful forthe assistance of Engr. Wilson A. Sy of Aromin & Sy{Associates, Inc. in obtaining GTF soft rock samples fromFort Bonifacio, Metro Manila. Special appreciation isalso extended to Prof. Junichi Koseki of the KosekiLaboratory of IIS, University of Tokyo, and Prof. Fu­mio Tatsuoka of the Soil Mechanics Laboratory of theTokyo University of Science for allowing the authors touse the sample preparation equipment and hydraulic test­ing apparatus of their laboratories.REFERENCES1) ASTM (2000): D 5079–90 Standard practices for preserving andtransporting rock core samples, Annual Book of ASTM Standards,04.08(Soil and Rock), 971–976.2) Bhandari, A. R. and Inoue, J. (2005): Experimental study of strainrate eŠects on strain localization characteristics of soft rocks, Soilsand Foundations, 45(1), 125–140.3) Committee on Uniaxial and Triaxial Compression Tests of Rocks,JGS (1998): Committee Report, Proc. Symposium on Uniaxial andTriaxial Compression Tests of Rocks and Application of TheirResults, Tokyo, Japanese Geotechnical Society (in Japanese).4) Goto, S., Tatsuoka, F., Shibuya, S., Kim, Y. S. and Sato, T.(1991): A simple gauge for local small strain measurements in thelaboratory, Soils and Foundations, 31(1), 169–180.5) Hayano, K., Matsumoto, M., Tatsuoka, F. and Koseki, J. (2001):Evaluation of time­dependent deformation properties of sedimen­tary soft rock and their constitutive modeling, Soils and Founda­tions, 41(2), 21–38.6) Indo, H. (2001): Cyclic triaxial tests on deformation properties ofsoft rocks, Master of Engineering Thesis, University of Tokyo (inJapanese).62PECKLEY AND UCHIMURA7) Japanese Geotechnical Society (JGS 0542–2000): Method for CyclicTriaxial Test to Determine Deformation Properties of Geomateri­als.8) Kashima, N., Fukunaga, S., Saeki, M. and Koseki, J. (2000): Studyon procedures to estimate earthquake­induced residual settlementof large scale bridge foundations (part 2), Proc. 55th Annual Con­ference of Japan Society of Civil Engineers, I­B460, 920–921 (inJapanese).9) Koseki, J., Moritani, T., Fukunaga, S., Tatsuoka, F. and Saeki, M.(2001): Analysis on seismic performance of foundation for AkashiKaikyo Bridge, Proc. 2nd Int. Symposium Pre­failure DeformationCharacteristics of Geomaterials (eds. by Jamiolowski et al.), Balke­ma, 1405–1412.10) Nishi, T. (2002): Time eŠects on deformation and strength charac­teristics of geomaterials and its modeling, Master's Thesis, Univer­sity of Tokyo (in Japanese).11) Originlab Corporation (2006): Origin 7.5 SR6 On­line Manual.12) Peckley, D. C. (2007): Strength and deformation of soft rocks un­der cyclic loading–An empirical study focusing on residual strainaccumulation and loading period eŠects, PhD Dissertation, Univer­sity of Tokyo.13) Salas Monge, R. (2002): EŠects of large amplitude cyclic loading ondeformation and strength properties of cement treated sand,Master of Engineering Thesis, University of Tokyo.14) Santucci de Magistris, F., Koseki, J., Amaya, M., Hamaya, S.,Sato, T. and Tatsuoka, F. (1999): A triaxial testing system to evalu­ate stress­strain behavior of soils for wide range of strain and strainrate, Geotech. Testing J., 22(1), 44–60.15) Shibuya, S., Mitachi, T., Fukuda, F. and Degoshi, T. (1994): Strainrate eŠects on shear modules and damping of normally consolidat­ed clay, Geotechnical Testing Journal, 18(3), 365–375.16) Sub­committee on the English Version Standards for LaboratoryShear Test, Standardization Division, Japanese Geotechnical Soci­ety (JGS) (2000): JGS 0542–2000: Method for triaxial test to deter­mine deformation properties of geomaterials, Standards ofJapanese Geotechnical Society for Laboratory Shear Test, 62–72.17) Tatsuoka, F. and Kohata, Y. (1994): StiŠness of hard soils and softrocks in engineering applications, Keynote Lecture, Proc. Int. Sym­posium of Pre­failure Deformation of Geomaterials (eds. byShibuya et al.), Balkema, 2, 947–1063.18) Tatsuoka, F., Santucci de Magistris, F., Hayano, K., Momoya, Y.and Koseki, J. (2000): Some new aspects of time eŠects on thestress­strain behavior of stiŠ geomaterials, The Geotechnics ofHard Soils­Soft Rocks (eds. by Evangelista and Picarelli), Balke­ma, 1285–1372.19) Tatsuoka, F., Hayano, K. and Koseki, J. (2003): Strength anddeformation characteristics of sedimentary soft rocks in the TokyoMetropolitan Area, Characterization and Engineering Properties ofNatural Soils (eds. by Tan et al.), Swets and Zeitlinger, 1461–1525.20) Yamashita, S., Kohata, Y., Kawaguchi, T. and Shibuya, S. (2001):International round­robin test organized by TC–29, AdvancedLaboratory Stress­Strain Testing of Geomaterials (eds. by Tatsuo­ka, Shubuya and Kuwano), Swets & Zeitilinger Publishers Lisse,65–86.
• ログイン

• タイトル
• The P2S Effect on the Accumulation of Residual Strains in Soft Rocks due to Irregular Cyclic Loading
• 著者
• D. C. Peckley・Taro Uchimura
• 出版
• Soils and Foundations
• ページ
• 63〜74
• 発行
• 2009/02/15
• 文書ID
• 21170
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 63–74, Feb. 2009THE P2S EFFECT ON THE ACCUMULATION OF RESIDUAL STRAINSIN SOFT ROCKS DUE TO IRREGULAR CYCLIC LOADINGD. C. PECKLEYi) and TARO UCHIMURAii)ABSTRACTData and information on the cyclic loading behaviour of soft rocks, especially behaviour under irregular cyclic load­ing, are very limited. This paper shows that the present procedure of estimating residual strain accumulation due to ir­regular cyclic loading using a fatigue model from uniform amplitude cyclic loading can result in underestimated resid­ual strains. Such underestimation occurs because the present procedure fails to take into account the so­called P2SeŠect on the softening behaviour of soft rocks under cyclic loading. The parameter P2S is deˆned as the sum of themagnitudes of the increments in residual strains due to the previous two loading half­cycles. When P2S is large, a largeresidual strain increment can be expected. This paper also shows that taking the P2S eŠect into account can improvethe simulation of residual strain accumulation due to irregular cyclic loading.Key words: irregular cyclic loading, P2S eŠect, soft rocks, triaxial tests (IGC: F7)almost 70z of the metropolis is directly underlain by thesoft rock formation called the Guadalupe TuŠ Formation(GTF). One boundary of this formation is an active faultthat can generate an earthquake with a magnitude of 7.2(Bautista, 2004). A paleoseismic study on the fault thatwas jointly conducted by the USGS and the PhilippineInstitute of Volcanology and Seismology or PHIVOLCS(Nelson et al., 2000) indicated a recurrence interval of200–400 years for magnitude 6–7 earthquakes over thepast 1500 years.In another paper by the same authors (Peckley andUchimura, 2007), which was submitted to thisjournal, it was shown that under uniform cyclic loading,longer loading period can result in signiˆcantly largerresidual strains. In this paper, the behaviour of soft rocksunder irregular cyclic loading is examined, starting with areview of the present methodology of estimating residualstrain accumulation using a fatigue model from uniformamplitude cyclic loading tests.INTRODUCTIONThe minimal settlements of the piers of the Akashi Kai­kyo Bridge, the longest suspension bridge in the world,after the 1995 Kobe Earthquake have demonstrated thatsoft rocks are competent foundation materials for large­scale structures (Yamagata et al., 1996; Koseki et al.,2001; Kashima et al., 2000). To design more cost­eŠec­tive, large­scale foundations on soft rocks, however, thedeformation characteristics of soft rocks when subjectedto large cyclic loading still have to be understood. Theforward calculations of the settlements of the piers of theAkashi­Kaikyo Bridge carried out by Koseki et al. (2001)and Kashima et al. (2000) underscored this point. In thesecalculations, the actual settlements were overestimated bya factor of at least 2.5 times (Koseki et al., 2001) and byas much as 8 times (Kashima et al., 2000). While Kosekiet al. (2001) attributed these discrepancies to overesti­mated seismic accelerations, among other factors, moreempirical studies are still needed because of the paucity ofdata and information regarding cyclic loading behaviourof soft rocks, especially under irregular cyclic loading(Indo, 2001; Salas Monge, 2002; Peckley and Uchimura,2007).The fact that heavily populated metropolitan areassuch as Tokyo, Japan (Tatsuoka et al., 2003) and MetroManila, Philippines (Peckley and Uchimura, 2006) areunderlain by soft rock formations and are known to beseismically active, is another compelling reason for inves­tigating the strength and deformation characteristics ofsoft rocks under cyclic loading. In Metro Manila's case,i)ii)TRIAXIAL CYCLIC LOADING TESTSThe soft rock samples, which were tested for this studyand are referred herein as GTF samples, were obtainedfrom an excavation into the GTF in Fort Bonifacio,Taguig City, Metro Manila. The excavation was for thebasement ‰oors of a multi­story building, which is nowunder construction. These were extracted by block sam­pling from a sandstone layer at a depth of around 12meters from the existing ground elevation. The blocksamples were then packaged and shipped to the Geo­Post­doctoral Researcher, Department of Civil Engineering, University of Tokyo, Tokyo, Japan (dan_peckley—yahoo.com).Associate Professor, Department of Civil Engineering, University of Tokyo, Tokyo, Japan (uchimura—geot.t.u­tokyo.ac.jp).The manuscript for this paper was received for review on July 11, 2007; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.6364PECKLEY AND UCHIMURAtechnical Engineering Laboratory of the University ofTokyo, Japan, following the recommendations of ASTMD 5079–90 (ASTM, 2000) for critical care to minimizemoisture loss and damage in the microstructure of thesamples. The samples were cut and trimmed to L100 mm~W60 mm~H150 mm nominal sizes using a rotarycutter. The unit weight of the samples was around 1.78g/cm3; moisture content was around 31z; and the meandiameter D50 was from 0.20 mm to 0.25 mm (afterthorough crushing). The maximum deviator stress qmax ofthe samples at 200 kPa conˆning pressure and at 1z/minmonotonic loading rate was around 3,200 kPa1.In the tests, two 120 mm Local Deformation Trans­ducers or LDTs (Goto et al., 1991) were used to measurethe longitudinal (axial) deformations and were attachedon the 60 mm­wide faces of the specimen. The LDTs wereattached to the specimen following the procedure de­scribed by Hayano et al. (2001). To measure longitudinal(axial) load, a 50 kN­capacity load cell installed inside thetriaxial cell was used to minimize errors due to frictionalong the loading shaft. Cyclic loading was applied usingan oil­hydraulic loading system at the Koseki Laboratoryof the Institute of Industrial Science (IIS), University ofTokyo.Prior to loading, each specimen was consolidated at anisotropic pressure of 200 kPa, the estimated in­situ over­burden pressure. The tests were unsaturated because thesamples were taken above the water table, which was 20m below the existing ground surface.with increasing amplitude for every loading stage was em­ployed. The loading period for all loading stages was at T1 s. These tests were conducted on samples designatedas GTF30, GTF31 and GTF34. Here, only results of thetests on GTF30 and GTF31, shown in Figs. 2 and 3, areincluded for brevity.Irregular Cyclic Loading TestsThe samples that were subjected to irregular loadingFig. 2.Cyclic stress­strain curve of GTF30Fig. 3.Cyclic stress­strain curve of GTF34Uniform Amplitude Cyclic Loading TestsFigure 1 illustrates the loading time­history of thestaged cyclic loading tests performed in this study. In theprevious uniform amplitude cyclic loading tests that wereperformed (Peckley and Uchimura, 2006), it was ob­served that the accumulation of residual strain is almostlinear with the number of cycles applied, especially athigher loading amplitudes. Thus, for the tests that wereconducted for this study, staged uniform cyclic loadingFig. 1.1Staged uniform cyclic loading time­historyThe tensile strength was not measured. However, since the ratio be­tween the tensile strength and compressive strength of soft rocks canbe as high as 0.20 (Coviello et al., 2005), cyclic loading behaviour con­sidering the tensile strength of soft rocks may have to be explored infuture studies.Fig. 4.Stress and strain time histories of GTF33IRREGULAR CYCLIC LOADINGhistories were GTF32, GTF33 and GTF35 (Peckley,2007), but also for brevity only the results of GTF33 andGTF35 are presented. Figures 4 and 5 show the stress­time histories that were applied on these samples and thecorresponding strain­time histories that were measured.The cyclic stress­strain curves are shown in Figs. 6 and 7.Fig. 5.Stress and strain time histories of GTF35Fig. 6.65FATIGUE MODELING AND SIMULATIONThe basic assumption employed in fatigue modelling isthat the accumulation of residual deformation due to anirregular cyclic loading history can be extrapolated fromthe evolution of deformations under a series of uniformamplitude cyclic loading tests. The use of fatigue modelsto evaluate the behaviour of geomaterials due to irregularcyclic loading already has a long history and has been de­scribed in detail by a number of prominent researchers,among them, Seed and Idriss (1971), Ishihara and Yasu­da (1975), Seed et al. (1975), Tatsuoka and Silver (1981),Tatsuoka et al. (1986), Allotey and El Naggar (2005).Thus, the reader is referred to the works of these resear­chers for a more detailed treatment on the subject.Note, however, that in this study the model derived isapplicable to loading cases where changes in the directionof the principal axis is not signiˆcant and where the ten­sile strength of the material can be neglected.Fatigue ModelThe results of the uniform amplitude cyclic loadingtests (Figs. 2 and 3) can be summarized as shown in Fig.8, employing the deˆnitions for stress ratio SR, residualstrain2 ear, the amplitude of cyclic stress qd, and the devia­tor stress at static condition q0 shown in Fig. 9. The fol­lowing can be observed from Fig. 8:a) The accumulation of residual strains is practicallylinear with the number of cycles applied (CN).b) The residual strains have an exponential, highlynonlinear relationship with stress ratio SR. Shownalso in Fig. 8 is the fatigue model that was obtainedby nonlinear curve ˆtting. Note that in the modelthe above observations were taken into account.In the development of the fatigue model, a nonlinearleast­squares regression method was employed with theNLSF Advanced Fitting Tool of Origin 7.5 (OriginlabCyclic stress­strain curve of GTF33Fig. 8. Comparison between test data and fatigue model (obtained bynonlinear curve ˆtting)2Fig. 7.Cyclic stress­strain curve of GTF35The residual strain ear is measured from the start of the application ofcyclic loading, whereas Dear is the increment in residual strain afterone cycle, or half cycle. See Simnlations Using the Fatigue Model.66PECKLEY AND UCHIMURAFig. 9. Deˆnition of Stress Ratio (SR), qmaxmax. deviator stress(strength) obtained from monotonic loading testFig. 10. Deˆnitions of one cycle, cycle i, cycle amplitude qdi, and in­crement in residual strain Deari, in irregular cyclic loadingCorporation, 2006). This iterative regression method isbased on the Levenberg­Marquardt (LM) algorithm andcan simultaneously ˆt model parameters across data sets.In this case, the ˆtting parameters were k, l, m and n. Thevalues of these parameters determined after a number ofiterations are shown in Fig. 8. In the ˆgure, it can bereadily seen that with these parameter values, the modeland the test results match with each other well. Thecoe‹cient of determination R2 is 0.99.It should be noted that with the assumption that strainaccumulation is linear with CN, it is implicit that there isone­to­one correspondence between residual strain incre­ment and SR. That is, for every value of SR, there canonly be one corresponding value of strain increment.Simulations Using the Fatigue ModelConsistent with the deˆnitions provided for uniformcyclic loading, the deˆnitions for stress ratio SR and in­crement in residual strain provided in Fig. 10 are adoptedfor irregular cyclic loading. As shown, one cycle is de­ˆned with the static stress q0 as reference, irrespective ofwave shape. The stress ratio SR for a cycle is determinedfrom the maximum qd applied in that cycle3, and residualstrain increment is the change in strain after that cycle.Figures 11 and 12 present the results of the simulationsof residual accumulation in samples GTF33 and GTF35using the fatigue model derived in Fatique Model. Asshown, the accumulation of residual strains is poorlysimulated, especially for GTF35. It can be inferred that asstress ratios become higher, the discrepancies betweensimulated and measured residual strains become larger.As the performance at high stress ratio SR values is aprimary concern of this study, to account for and explainthe discrepancies between measured and simulated resid­ual strains are imperative. As a ˆrst step, the measuredand simulated residual strains were plotted together withthe stress and strain time histories, as shown in Figs. 13and 14. Taking note that the SR for one cycle of loadingis derived from the amplitude of the positive half­cycle,one can observe the following from Figs. 13 and 14:3Note that qd corresponds to the amplitude of the positive half cycle.Fig. 11.Accumulated residual strain with cycle number for GTF33Fig. 12.Accumulated residual strain with cycle number for GTF35(1) In the case of GTF33 (Fig. 13), it can be observedthat while half cycle `b' has a higher amplitudethan half cycle `c', the maximum strains and resid­ual strain increments due to these half cycles arealmost the same. It appears that after half cycle`b', softening had occurred, resulting in a residualIRREGULAR CYCLIC LOADING67Fig. 13.Accumulated residual strain with time for GTF33Fig. 15. Deˆnition of half­cycle j and corresponding increment inresidual strain DearjFig. 14.Accumulated residual strain with time for GTF35Fig. 16. GTF35 stress ratio SR & increment in residual strain­cyclenumber history: CN01 to CN37strain due to half cycle `c' that was larger than ex­pected.(2) Similarly, in the case of GTF35 (Fig‚ 14), it is evi­dent that while half cycles `a' and `c' have almostthe same amplitudes–and thus, SR values–theresidual strain increment due to half cycle `c' is sig­niˆcantly larger than that of `a'.Note that both items completely undermine the validityof the implicit assumption that there is one­to­one cor­respondence between stress ratio SR (or amplitude) andresidual strain increment by virtue of the assumed linearrelationship between cycle number CN and residual strainaccumulation. Evidently, the fatigue model that was der­ived cannot be used to simulate residual strain accumula­tion due to irregular cyclic loading, despite the high valueof coe‹cient of determination R2 that was obtained. Athigher stress ratios, the use of the fatigue model resultedin larger underestimation of residual strains. The appar­ent reason for this is that it failed to simulate the soften­ing behaviour that occurred after the application of alarge load impulse, which were the half cycles indicated as`b' in Figs. 13 and 14. This softening behaviour is furtherexamined in the following sections.THE P2S EFFECTHalf­cycle Stress Ratio and Corresponding Increment inResidual StrainsTo facilitate the characterization of the behaviour ob­served in irregular cyclic loadings described above, stress­and strain­time histories are presented in terms of half­cycle stress ratios and their corresponding increment inresidual strains, as illustrated in Fig. 15. In this ˆgure,one loading cycle is analyzed in terms of its componentpositive half­cycle and negative half­cycle. The incrementin residual strain is then decomposed also into two com­ponents: one corresponds to the positive half­cycle, andthe other corresponds to the negative half­cycle. Notethat with these deˆnitions, a negative half­cycle results ina negative increment in the residual strain.When expressed in half­cycle SR and increment inresidual strain, the stress and strain time histories (fromthe start of cyclic loading to t88522 s) in Fig. 5 becomethe SR­cycle number (CN) and increment in residualstrain­CN histories in Fig. 16.68PECKLEY AND UCHIMURAFig. 17. GTF33 stress ratio SR & increment in residual strain­cyclenumber history: CN03 to CN13Fig. 19. GTF35 stress ratio SR & increment in residual strain­cyclenumber history: CN03 to CN13Fig. 18. GTF33 stress ratio SR & increment in residual strain­cyclenumber history: CN33 to CN43Fig. 20. GTF35 stress ratio SR & increment in residual strain­cyclenumber history: CN50 to CN60Relationships between Residual Strain Increment andother Stress­strain ParametersIn the SR­CN and residual strain increment­CN histo­ries of the samples that were subjected to irregular cyclicloading (GTF32, GTF33 and GTF35), a number of seg­ments or windows were selected to further examine thesoftening behaviour observed in Figs. 12 and 13. Thesewindows include those encircled in Figs. 17 through 20.The basic criteria in selecting these time windows were thefollowing:1) When there is a deviation from the trend that thehigher the stress ratio, the larger the increment inresidual strains, as implied in the fatigue model devel­oped in Fatique Model.2) When two half­cycles with essentially the same ampli­tude have remarkably diŠerent increment in residualstrain.The half­cycles where these criteria occurred were thenidentiˆed and were designated either as `c' or `c*', asshown in the ˆgures. To identify the parameter orparameters with which the increment in residual strain ismost strongly linked, all the residual strain incrementsFig. 21. Increment in residual strain Dearc due to half cycle `c' vs.stress ratio SRcdue to these half­cycles were plotted against the currentand previous SR parameters, and against the previousresidual strain increments.Figure 21 plots the increment in residual strain Dearc orIRREGULAR CYCLIC LOADINGFig. 22. Increment in residual strain due to half­cycle `c' Dearc vs. theratio (half cycle SRc/(half cycle SRb–half cycle SR­b)Fig. 23. Increment in residual strain Dearc due to half cycle `c' vs. thesum of the two preceding stress ratiosFig. 24. Increment in residual strain due to half­cycle `c' Dearc vs. P2S`Dearb`{`Dear­b`Dearc* vs. the corresponding stress ratio SRc or SRc* andFig. 22, the increment in residual strain Dearc or Dearc* vs.the corresponding ratio SRc/(SRj­1­SR­j­2). Figure 23 plotsthe increment in residual strain Dearc or Dearc* vs. the sumof the magnitudes of the two preceding stress ratios andFig. 24, plots Dearc or Dearc* vs. the sum of the magnitudesof the increment in residual strain due to the preceding 2Fig. 25.69SRI modulus vs. P2S of selected half­cycles `c' and `c*'half­cycles (`Dearj­1`{`Dearj2`). Relationships betweenDearc or Dearc* and other parameters were also examined(Peckley, 2007), but these did not yield correlations thatare as strong as that shown in Fig. 24.Again for brevity, the sum of the magnitudes of the in­crement in residual strain due to the preceding 2 half­cycles, is referred to from hereon as the preceding 2 half­cycle strains or, simply, P2S. When the quantity SRc/Dearc or SRcDearc* is plotted with P2S, as shown in Fig. 25,a fundamental behaviour in cyclic loading can be in­ferred. Note that the quantity SRc/Dearc can be character­ized as a modulus, referred to as SRI modulus in thisstudy. When P2S is large, the corresponding SRI modu­lus becomes small and large increment in residual strainDearc can be expected. The behaviour just described is oneof the key ˆndings in this study and is referred to here as``the P2S eŠect''. Evidently, this eŠect has to be takeninto account when estimating residual strains in softrocks due to irregular cyclic loading.The P2S EŠect in Uniform Amplitude Cyclic LoadingFigures 26 and 27 present SR­CN and Dear­CN histo­ries of selected loading stages from the tests conducted onsamples designated as GTF30 and GTF31 (see also Peck­ley and Uchimura, 2007).Figure 28, on the other hand, plots the data from allthe cyclic loading stages applied on these samples, includ­ing GTF34, in terms of SRI and P2S.It can be readily observed from Figs. 26 and 27 that theincrement in residual strain due to positive half­cycles in­creases with loading cycle number. The same can be ob­served with the increments in residual strain due to nega­tive half­cycles: the magnitude of these residual strain in­crements increases with cycle number. From Fig. 28, it isapparent that the P2S eŠect also plays an important rolein softening behaviour under uniform amplitude cyclicloading. In fact, the ˆgure appears to suggest that soften­ing behaviour is a unique function of P2S such that acurve can be derived for positive half­cycles and anothercan be derived from negative half­cycles, as shown in theˆgure.Quite a few remarks can be made from these observa­70PECKLEY AND UCHIMURAFig. 26. GTF30 stress ratio SR & increment in residual strain­cyclenumber history: Stage S5Fig. 27. GTF31 stress ratio SR & increment in residual strain­cyclenumber history: Stage S22) Such softening behaviour deviates from what is im­plied in fatigue model developed in Faitgue Modelthat there is a one­to­one correspondence betweenstress ratio and residual strain increment. Note thatif these implied assumptions were true, there shouldbe no change in residual strain increment no matterhow many loading cycles are applied.3) From the foregoing remarks, it can be inferred thatby deˆning residual strain increment as the incre­ment after a full cycle of loading (not as the incre­ment due to a half­cycle), this softening behaviourwas eŠectively hidden. In fact, the observationmade in the Fatigue Model that residual strain ac­cumulates linearly with increasing cycle number im­plies that no softening occurs with the number ofloading cycles applied, which is contrary to whatcan be observed from Fig. 27.4) In other words, when residual strain increment isdeˆned as the increment due to a full cycle, a lineartrend in residual strain accumulation with cyclenumber can still be observed even when softeningoccurs, because in one full cycle of loading, an in­crease in residual strain increment due to a positivehalf­cycle can be cancelled by an increase in themagnitude of the residual strain increment due tothe negative half­cycle.5) Evidently, softening behaviour can be better ana­lyzed when residual strain increment and stress ra­tios are deˆned in terms of half­cycle loading.From all the foregoing remarks and observations madein this section and previous sections, the reason for thelarge discrepancies between the measured residual strainsand the results of simulations using the fatigue modelderived in Fatigue Model is now apparent. The softeningbehaviour in irregular cyclic loading, in which P2S eŠectplays a prominent role, was not fully taken into accountin the fatigue model. It can also be inferred that the P2SeŠect is an important factor to consider when quantifyingcyclic loading degradation.SIMULATIONS CONSIDERING THE P2S EFFECTFig. 28.SRI modulus vs. P2S from uniform cyclic loading teststions:1) The increase in half­cycle residual strain incrementwith cycle number, as shown in Figs. 26 and 27, canbe characterized as the result of cyclic loadingdegradation.With the unique curves derived in Fig. 28, one for posi­tive half cycles and the other for negative halfcycles–simulations of the irregular cyclic loading tests onGTF32, GTF33 and GTF35 can be carried out. For brevi­ty, such curves shall be referred to here as strain softeningcurves, or simply SS curves, and the particular set of SScurves shown in Fig. 28 is referred to as Model 1 SScurves. Since the ˆrst half cycle of loading cannot besimulated using only these strain softening curves, arelationship between the residual strain increment due tothe ˆrst half cycle and the corresponding stress ratio wasalso derived. This relationship is shown in Fig. 29.Figures 30 and 31 show the results of the simulationsperformed for GTF33 and GTF35 using the SS curves ofModel 1 (Fig. 28) and the stress ratio­residual straincurves in Fig. 29. While it can be observed from Figs.30(b) and 31(b) that the model seems to simulate the P2SIRREGULAR CYCLIC LOADINGFig. 29. Residual strain due to the 1st half cycle (GTF30, GTF31,GTF34)Fig. 30. Simulation of GTF33 test using strain softening Model 1 (seeFig. 28)Fig. 31. Simulation of GTF35 test using strain softening Model 1 (seeFig. 28)Fig. 32. Comparison between irregular cyclic loading data and soften­ing Model 1 for ({) half cycles (see Fig. 28), Model 2 includedFig. 33. Comparison between irregular cyclic loading data and soften­ing Model 1 for (|) half cycles (see Fig. 28)a) The SS curve of Model 1 for positive half cycles ap­pears to be above most of the irregular cyclic loadingtests data points.b) The SS curves of Model 1 are too near each other. Forlarger residual strains to accumulate, the SS curvesshould be placed farther apart4.Considering the above conjectures on the discrepanciesbetween simulation results and test data, the SS curve forpositive half cycles in Model 1 was adjusted heuristically5to a position slightly lower than the perceived average ofthe irregular cyclic loading test data points, as shown inFig. 32. The adjusted SS curve for positive half cyclesand the SS curve for negative half cycles in Model 1 wererenamed Model 2 SS curves. Note that no adjustment wasmade on the negative half­cycle SS curve.Figures 34 and 35 show the simulation results using the4eŠect described in the previous chapter, overall the ac­cumulation of residual strains was poorly simulated.Figures 32 and 33 compare the irregular cyclic loadingtest data from GTF32, GTF33, and GTF35 with the SScurves of Model 1. Following are two reasons that couldaccount for the large discrepancies between the test dataand simulation results:715As discussed in The P2S EŠect in Uniform Amplitude Cyclic Loading,an SS curve represents a kind of a modulus that is dependent on theparameter P2S. In these simulations, the idea to estimate the residualstrain given such an SS curve, the half­cycle loading amplitude and theP2S parameter, is explored. Note that the wider the divergence of be­tween the positive half­cycle SS curve and the negative half­cycle SScurve, the larger is the residual strain obtained for one cycle of load­ing.Adjustments were carried out by trial and error taking into accountthe results of earlier simulations.72PECKLEY AND UCHIMURAFig. 34. Simulation of GTF33 test using strain softening Model 2 (seeFigs. 32 and 33)Fig. 35. Simulation of GTF35 test using strain softening Model 2 (seeFigs. 31 and 32)SS curves of Model 2. The following can be observedfrom these ˆgures:1) Unlike for the simulations using the SS curves ofModel 1, the trend in the simulated accumulation ofresidual strain is consistent with the test data. Thesimulated residual strain generally increases with thenumber of cycles.2) The accumulation of residual strains in GTF33 isreasonably replicated. That of GTF35, however, islargely underestimated.3) The P2S eŠect is simulated in all samples, althoughthe simulation results are not quantitatively accurate.Given the conjectures from the ˆrst set of simulationsand the observations made above–speciˆcally observa­tion (2) which can be attributed to the trend among thetest data points that at higher P2S values, there is a widerdivergence between the positive half cycle data points andnegative half cycle data points, as shown in Fig. 36, theSS curves referred to as Model 3 were introduced4,5,6.6The main diŠerence between Model 2 and Model 3 is that for Model 3,there is wider divergence between the SS curve for positive half­cyclesand that for negative half­cycles is wider at higher values of P2S.Fig. 36. Strain softening Model 3 together with test data and Models 1and 2Fig. 37. Simulation of GTF33 test using strain softening Model 3 (seeFig. 36)Fig. 38. Simulation of GTF35 test using strain softening Model 3 (seeFig. 36)Figures 37 and 38 show the simulation results usingModel 3. The following can be observed from theseˆgures:a) Similar to the case of Model 2, the trend in residualstrain accumulation is consistent with the test data.IRREGULAR CYCLIC LOADINGThe residual strain generally increases with the num­ber of cycles.b) The accumulation of residual strains in GTF33 is alsoreasonably replicated. Compared with the simulationusing Model 2, the accumulation of residual strain inGTF35 can also be said to be better simulated.c) As in the case of Model 2, the P2S eŠect is simulated inall samples, but the simulation is not quantitativelyaccurate. It can be also inferred from these results andfrom Fig. 35 that diŠerent loading histories result indiŠerent SS curves.Considering the signiˆcant scatter of irregular test datapoints in the SRI vs. P2S plot shown in Fig. 35, nonlinearcurve ˆtting–involving the half cycle stress ratio SRj, thecorresponding increment in residual strain Dearj, thequantity P2Sj and the time elapsed for half cycle jreferred to here as dtj was carried out to derive other SSfunctions or curves. The simulation results from these SSfunctions were not necessarily any better than the proce­dure involved with Model 3 (Peckley, 2007).CONCLUSIONSBased on the results of tests and simulations that arepresented in this paper, the following conclusions can bemade:(1) The use of fatigue modelling from uniform cyclicloading test data to simulate irregular cyclic load­ing can result in signiˆcant underestimation ofresidual strains. Residual strains are underesti­mated because fatigue modelling does not take intoaccount the eŠect of the preceding two half­cycleresidual strain increments on the softening behav­iour of soft rocks, referred herein as P2S eŠect.The quantity P2S is deˆned as the sum of the mag­nitudes of the residual strain increments due to thepreceding two half­cycles. When P2S is large, alarge residual strain increment can be expected.(2) From the simulation results presented in this paper,it is clear that taking the P2S eŠect into account canimprove the simulation of residual strain accumu­lation due to irregular cyclic loading. Among thesesimulations, the simulations using SS curves fromSRI vs. P2S plot in which the divergence betweenthe SS curve for positive half­cycles and that ofnegative half­cycles becomes wider as the quantityP2S becomes larger, appear to yield results that areclosest to the measured data. It should be noted,however, that the main drawback of this modellingprocedure is that diŠerent loading histories resultin diŠerent SS curves.Considering the said drawback and all the simulationresults presented in this study, it becomes apparent thatsoft rock testing programs should not be limited only touniform cyclic loading tests. Irregular cyclic loading testsare as, if not more, important as uniform cyclic loadingtests. Evidently, there is still much room for improvingthe modelling procedure that was explored in this paper.Aside from the P2S eŠect, another factor or other factors73related to cyclic loading eŠects apparently still have to beidentiˆed and characterized.ACKNOWLEDGMENTSThis research was made possible through a scholarshipgrant that was awarded to the ˆrst author by the JapanMinistry of Education, Culture, Sport, Science, andTechnology (MEXT). The authors are also grateful forthe assistance of Engr. Wilson A. Sy of Aromin & Sy{Associates, Inc. in obtaining GTF soft rock samples fromFort Bonifacio, Metro Manila. Special appreciation isalso extended to Prof. Junichi Koseki of the KosekiLaboratory of IIS, University of Tokyo and Prof. FumioTatsuoka of the Soil Mechanics Laboratory of the TokyoUniversity of Science for allowing the authors to use thesample preparation equipment and hydraulic testing ap­paratus of their laboratories.REFERENCES1) Allotey, N. and El Naggar, M. H. (2005): Cyclic soil degradation/hardening models: A critique, Proc. 16th ICSMGE, Osaka, Japan,785–790.2) ASTM (2000): D 5079–90 Standard practices for preserving andtransporting rock core samples, Annual Book of ASTM Standards,04.08 (Soil and Rock), 971–976.3) Bautista, M. P. (2004): Overview of the MMEIRS Project, http://www.phivolcs.dost/clariˆcation/Leyo's Letter.pdf.4) Coviello, A., Lagioia, R. and Nova, R. 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(2000): Studyon procedures to estimate earthquake­induced residual settlementof large scale bridge foundations (part 2), Proc. 55th Annual Con­ference of Japan Society of Civil Engineers, Section 1 (inJapanese).10) Koseki, J., Moritani, T., Fukunaga, S., Tatsuoka, F. and Saeki, M.(2001): Analysis on seismic performance of foundation for AkashiKaikyo Bridge, Proc. 2nd Int. Symposium Pre­failure DeformationCharacteristics of Geomaterials (eds. by Jamiolowski et al.), Balke­ma, 1405–1412.11) Nelson, A. R., Personius, S. F., Rimando, R. E., Punongbayan, R.S., Tungol, N., Hannah Mirabueno, H. and Rasdas, A. (2000):Multiple large earthquakes in the past 1500 years on a fault inMetropolitan Manila, The Philippines, Bulletin of SeismologicalSociety of America, 90, 73–85.12) Originlab Corporation (2006): Origin 7.5 SR6 On­line Manual.13) Peckley, D. C. (2007): Strength and deformation of soft rocks un­der cyclic loading–an empirical study focusing on residual strain ac­cumulation and loading period eŠects, PhD Dissertation, Univer­sity of Tokyo.14) Peckley, D. C. and Uchimura, T. (2006): Strength and deformation7415)16)17)18)PECKLEY AND UCHIMURAcharacteristics of soft rocks from the Guadalupe TuŠ Formation inMetro Manila when subjected to large cyclic loading, Proc. 3rd In­ternational Conference on Urban Earthquake Engineering, TokyoInstitute of Technology, Japan, 137–144.Peckley, D. C. and Uchimura, T. (2007): Strength and deformationof soft rocks under cyclic loading considering loading periodeŠects, Soils and Foundations, 49(1), 51–62.Salas Monge, R. (2002): EŠects of large amplitude cyclic loading ondeformation and strength properties of cement treated sand,Master of Engineering Thesis, University of Tokyo.Seed, H. B. and Idriss, I. M. (1971): A simpliˆed procedure forevaluating soil liquefaction potential, Journal of SMFE Div.,ASCE, 97(9), 249–274.Seed, H. B., Idriss, I. M., Makdisi, F. and Banerjee, N. (1975):Representation of irregular stress time histories by equivalentuniform stress series in liquefaction analysis, Report No. EERC7529, Univ. of California, EERC, Berkeley.19) Tatsuoka, F. and Silver, M. (1981): Undrained stress­strain behav­iour of sand under irregular loading, Soils and Foundation, 21(1),51–66.20) Tatsuoka, F., Maeda S., Ochi, K. and Fujii, S. (1986): Predictionof cyclic undrained strength of sand subjected to irregular loadings,Soils and Foundations, 26(2), 73–90.21) Tatsuoka, F., Hayano, K. and Koseki, J. (2003): Strength anddeformation characteristics of sedimentary soft rocks in the TokyoMetropolitan Area, Characterization and Engineering Properties ofNatural Soils (eds. by Tan et al.), Swets and Zeitlinger, 1461–1525.22) Yamagata, M., Yasuda, M., Nitta, A. and Yamamoto, S. (1996):EŠects on the Akashi Kaikyo Bridge, Special Issue of Soils andFoundations on Geotechnical Aspects of the January 17, 1995 Hyo­goken­Nambu Earthquake, 179–187.
• ログイン

• タイトル
• Large-scale Monotonic and Cyclic Tests of Interface between Geotextile and Gravelly Soil
• 著者
• G. Zhang・J.-M. Zhang
• 出版
• Soils and Foundations
• ページ
• 75〜84
• 発行
• 2009/02/15
• 文書ID
• 21171
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 75–84, Feb. 2009LARGE­SCALE MONOTONIC AND CYCLIC TESTS OF INTERFACEBETWEEN GEOTEXTILE AND GRAVELLY SOILGA ZHANGi) and JIAN­MIN ZHANGii)ABSTRACTA series of monotonic and cyclic shear tests, as well as pullout tests, were conducted on gravel­geotextile interfacesusing a large­scale apparatus, with development of a new special pullout test element. The macroscopic response ofstress and displacement, as well as the movement and crushing process of soil particles, were observed and measured.The interface exhibited evident strain­softening and aeolotropic normal displacement, which were signiˆcantly in­‰uenced by normal stress. Shear strength decreased and normal displacement increased with increasing number ofshear cycles. Shear deformation was composed of slippage at the contact surface and deformation of the soil con­strained by the geotextile; and the thickness was estimated at 5–6 times the average soil grain size. There was signiˆcantevolution of physical state due to shear application, including soil particle crushing and soil compression, as well asdamage to the geotextile. The pullout test underestimated shear stiŠness of the interface due to signiˆcant deformationof the geotextile itself. Shear strength increased with increasing normal stress, described by a logarithmic equation, ac­cording to the pullout tests, rather than the linear relationship obtained using direct shear tests. Therefore, an ap­propriate test method should be selected with careful consideration of the site conditions.Key words: geotextile, gravelly soil, interface, monotonic and cyclic behavior, pullout test, shear test (IGC:D6/D7/E12)a soil­geotextile interface (e.g., Fannin and Raju, 1993;Bakeer et al., 1998; Aiban and Ali, 2001; Lo, 2003; Longet al., 2007). This test allows for deformation of the geo­textile, and so is closer to reality than shear tests.Whereas, stress and deformation of the interface aretransferred progressively during the test and induces sig­niˆcant nonuniform deformation of the interface. Theramp test is often used to determine properties of soil­geotextile interfaces under small normal stress conditions(Lima et al., 2002; Ling et al., 2002; Gourc and Ramirez,2004).Strength behavior was emphasized in many monotonicand cyclic tests of a soil­geotextile interface (e.g., Lee andManjunath, 2000); a few equations were thereforeproposed to estimate interface strength (e.g., Geroud etal., 1993; Tan et al., 1995). Lo (2003) analyzed the in­‰uence of three­dimensional constrained dilatancy on thepullout resistance factor. The deformation behavior wasalso examined in tests (e.g., Aiban and Ali, 2001). Manytests were conducted on the in‰uence factors of soil­geotextile interfaces (e.g., Eigenbrod et al., 1990; Saleh,2001). The fabric geometry, geosynthetic structure, andsoil particle size were shown to have important eŠects onshear strength (Swan, 1987; Lopes et al., 2001; Lima etal., 2002).INTRODUCTIONGeotextile is an eŠective and widely­used approach toimprove engineering behavior of various earth structures.The static and dynamic response of soil­geotextile sys­tems, due to various loads, may be signiˆcantly aŠectedby monotonic and cyclic behavior of a soil­geotextile in­terface.The direct shear test has been widely used to investigatethe behavior of many kinds of soil­geotextile interfaces(Swan, 1987; Athanasopoulos, 1996; Saleh, 2001; Stoe­wahse et al., 2002; Abu­Farsakh et al., 2007). This typeof test has been improved signiˆcantly in recent years, forexample, the sample size can be maintained constant, andthe slippage displacement at the interface can be separat­ed using measurement of soil particle movements (e.g.,Zhang et al., 2006). The major disadvantage of such atest is the stress concentration at the ends of the interface.The torsion ring test can maintain the strains nearlyuniform within the specimen (Stark and Peoppel, 1994;Tan et al., 1998). However, di‹culties in preparing speci­mens and measuring deformation preclude the torsionring test from broad application, especially in practicalprojects.The pullout test is another important means to examinei)ii)Ph.D, Associate Professor, State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, P R China (zhangga—tsinghua.edu.cn).Professor, ditto.The manuscript for this paper was received for review on May 29, 2008; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.7576G. ZHANG AND J.­M. ZHANGMost available tests have focused on the strength be­havior of the interface between a geotextile andsand/clay. However, the stress­strain behavior, for ex­ample, volumetric change due to cyclic shear, has notbeen investigated adequately. Moreover, the deformationmechanism has not been observed by microscopy, thoughthere were similar studies on steel­type interface (Guler etal., 1999; Zhang and Zhang, 2006b). In addition, behav­ior of the interface between a geotextile and gravelly soil,which has larger grain size than sand, has not been sys­tematically investigated. Such interfaces are a great con­cern because geotextile has been extended to many gravel­ly soil structures such as rockˆll embankments, break­waters, and speed­railways. The gravelly soil­type inter­face has diŠerent features to a sand­type interface, for ex­ample, such type of interface exhibits complex volumetricchange due to dilatancy dependent on only a part of thetangential displacement and shear history (Zhang andZhang, 2006b). Thus, further study is required on themonotonic and cyclic behavior of the interface between ageotextile and gravelly soil.On the basis of serialized monotonic and cyclic testresults of the interface between a steel and gravelly soil(Zhang and Zhang, 2006b), we conducted a series of lar­ge­scale tests on the interface between a geotextile andgravelly soil, including monotonic/cyclic direct shear andpullout tests, with observations on both the macro­ andmicro­ responses. The objectives of this paper are: (1) togive a brief introduction of the tests including the appara­tus, measurements, and materials; (2) to present typicaltest results in both macro­ and micro­ ways; and (3) tosummarize and discuss the behavior and deformationmechanism of such an interface using test results.Fig. 1.Photograph of the test apparatus, TH­20t CSASSIAPPARATUS AND MEASUREMENTSFig. 2.Schematic view of construction of TH­20t CSASSIThe direct shear tests of the soil­geotextile interfacewere conducted using a large­scale apparatus, ``TsingHu­a­20 tonne Cyclic Shear Apparatus for Soil­Structure In­terface'' (TH­20t CSASSI) (Figs. 1 and 2) (Zhang andZhang, 2006a), which was developed especially to exa­mine the monotonic and cyclic behavior of the interfacebetween a gravelly soil and diŠerent kinds of structuralmaterials such as geotextile, steel, and concrete. Any oneof the three normal boundary conditions, namely con­stant stress, constant displacement, and constant stiŠ­ness, can be directly applied to the interface with a highdegree of accuracy, via a modiˆcation of the soil contain­er (Zhang and Zhang, 2006a). The structural materialwas designed with a larger size than the soil surface to en­sure that the size of the interface was maintained duringthe test. A high load capacity up to 200 kN was providedin both directions, tangential and normal to the interface,with automated control at various rates. A soil container(500 mm in length, 360 mm in width) was used for gravel­ly soil. Stresses and displacements of the interface, inboth tangential and normal directions, were measuredautomatically using sensors with an accuracy of 0.1z.The movement and crushing of soil particles was record­ed by a high­resolution camera through a transparent lu­cite window within the soil container (Fig. 2). The lucitesurface, close to the soil, was specially dealt with so that itis smooth decreasing the side friction between the win­dow and soil particles. The movements of the particleswere measured using a microscopic movement­measuringsystem, based on correlation­based image analysis theo­ry, with sub­pixel accuracy (Zhang et al., 2006).When conducting a shear test of a soil­geotextile inter­face, the geotextile was solidly adhered to the structureplate above the soil container. The soil sample was in­stalled in the soil container and compacted to the re­quired dry density in layers (Fig. 2). The soil­structure in­terface was therefore formed for tests. The constant nor­mal stress boundary condition was directly applied to theinterface and maintained by the normal boundary cellduring the test. The container of soil was driven by thehydraulic pressure system to move horizontally withoutvertical movement so that a tangential displacement wasrealized. At the same time, the structure plate with geo­textile was ˆxed in the horizontal direction but was ableINTERFACE BETWEEN GEOTEXTILE AND SOILFig. 3.77Container for pullout test on the basis of TH­20t CSASSIto move freely in the vertical direction, thereby exhibitingthe change of normal displacement.For the TH­20t CSASSI, a new test container was de­veloped for a large­scale geotextile pullout test (Fig. 3);its inner size is 450 mm in length, 360 mm in width, and250 mm in height. A horizontal cut was made in the mid­dle of the container, 360 mm in width, to place the geo­textile; the thickness of the cut can be freely adjustedfrom 0 to 10 mm to accommodate diŠerent types of geo­textiles. The geotextile was larger than the soil surface toensure a constant contact size during the test. There was athick lucite window in one side of the container (Fig. 3),through which the movements and crushing of soil parti­cles were observed and recorded during testing. In pul­lout testing, the container was ˆxed to rails using special­ly designed baŒes (Fig. 3). The constant normal bound­ary condition was applied on the soil surface via the plun­ger of the normal condition cell. The normal displace­ment of the interface was measured via that of the plun­ger. A special holding device was used to drag the geo­textile in a horizontal direction via the horizontalhydraulic pressure system in the pullout test; the loadingrate was adjusted according to the test requirements. Itshould be noted that there was 50 mm from the cut of thecontainer to the load end of the geotextile; thus a 50 mmof geotextile did not make contact with the soil (Fig. 3).Loads and displacements of pullout tests were measuredby the data acquisition system used in shear tests (Fig. 2).MATERIALSA type of nonwoven geotextile, made of nylon, wasused in the tests, which was taken from the geotextile­re­inforced cushion under a breakwater on soft ground ofthe Tianjin Huanghua Harbor, China. This type of geo­textile is widely used in China and is approximately 1.3mm in thickness. The average opening size of this geo­textile, O50, is about 0.04 mm; and O95 is about 0.15mm. The warp direction was used in the tests, with thetensile strength 100 kN/m and modulus 400 kN/m in thisdirection; the limit extensibility is 35z.A conglomerate gravel with very angular particles wasused in the tests as a typical gravelly soil. A homogeneousFig. 4Grain size distributions of soilgrain size distribution was used (Fig. 4) and the gravel drydensity was 1.75 g/cm3. This type of gravel was previous­ly used in tests on a steel­gravel interface (Zhang andZhang, 2006b). Triaxial compression tests indicated thatthe gravel dilated, 5z at the axial strain of 10z, after alittle compression, if the conˆning pressure was small, 0.1MPa; and it exhibited continuous compression, 2z at theaxial strain of 10z, if the conˆning pressure was large,0.8 MPa (Zhang and Zhang, 2006b). In addition, a typeof quartz sand, with known grain size distribution (Fig.4), was used in a few pullout tests; its relative density wascontrolled at 70z.The interface was 500 mm long and 360 mm wide forshear tests, and 450 mm long and 360 mm wide for pul­lout tests; these sizes were maintained invariable duringtests. All tests were displacement controlled at a loadingrate of 1 mm/min. The test conditions, such as the ap­plied displacement and normal stress, were selected byconsidering the limit conditions of the geotextile reinfor­cement of earth structures.SHEAR TEST RESULTSTypical monotonic and cyclic test results of the gravel­geotextile interface are dissertated to discuss the behaviorbased on macro­ and micro­ measurements. Since both78G. ZHANG AND J.­M. ZHANGFig. 5. Monotonic shear test results of the geotextile­gravel interfaceunder constant normal boundary condition. t, shear stress; s,normal stress; u, tangential displacement; v, normal displacementsample size and normal stress were maintained constantduring the shear test, the normal displacement was onlyinduced by changes in shear stress or tangential displace­ment. Thus, it can be regarded as volumetric change dueto dilatancy of the interface, deˆned as positive if the in­terface contracted and negative if dilated.Monotonic Stress­displacement RelationshipFigure 5 shows the monotonic shear test results underconstant normal stress condition. It can be seen thatshear stress increased with monotonically increasing tan­gential displacement and reached a peak value, afterwhich it slightly decreased. This indicated signiˆcantstrain­softening due to shear application. The peak shearstress and initial slope of the tangential stress­displace­ment relationship curve both increased with increasingnormal stress. In addition, the extent of strain­softeningalso increased with increasing normal stress. For exam­ple, strain­softening was negligible if normal stress was50 kPa, but was signiˆcant when normal stress increasedto 400 kPa (Fig. 5). This strain­softening behaviordiŠered signiˆcantly from a steel­gravelly soil interface,which exhibited insigniˆcant strain­softening due to shearapplication (Zhang and Zhang, 2006b), though the gravelwas the same for both interfaces. Moreover, this alsodiŠered signiˆcantly from that of a steel­sand interface,where strain­softening extent decreased with increasingnormal stress (Uesugi and Kishida, 1986). Strain­soften­ing can be attributed to damage to the geotextile inducedby gravel friction during shearing. In other words, thesurface behavior of the geotextile changed due to shearapplication and increasing normal stress signiˆcantly ex­acerbated this change. This was conˆrmed by observa­tions of breakage of the geotextile after shear tests underFig. 6. Shear strength based on direct shear test. tf, shear strength; s,normal stresshigh normal stress.The normal displacement decreased after a small in­crease if the normal stress was small (e.g., 50 or 100 kPa)but gradually increased if normal stress was large (e.g.,400 kPa). Thus, volumetric change due to dilatancy wassigniˆcantly in‰uenced by the magnitude of normalstress, similarly to a steel­gravelly soil interface (Zhangand Zhang, 2006b). In addition, the normal displacementsigniˆcantly changed when shear stress had becomesteady, demonstrating that tangential displacement andvolumetric change may have diŠerent deformationmechanisms due to shear application.In this paper, peak shear strength is deˆned as the max­imum shear stress, and residue strength is deˆned as thestable shear stress due to monotonic shear application un­der constant normal stress condition. Thus, a plot ofshear strength versus normal stress (Fig. 6) was obtainedaccording to the stress­displacement relationship (Fig. 5).The peak shear strength was nearly proportional to nor­mal stress (Fig. 6(a)), similarly to a steel­gravel interface(Zhang and Zhang, 2006b). Therefore, peak shearstrength, tf, can be formulated using Mohr­Coulombstrength criteria with only one parameter, friction angle,f‚ That is,INTERFACE BETWEEN GEOTEXTILE AND SOIL79Fig. 7. Cyclic stress and displacement histories of shear test results un­der constant normal boundary condition. t, shear stress; s, normalstress; u, tangential displacement; v, normal displacement; N,number of shear cyclestfs¥tan f(1)where s is the normal stress. The residue strength can alsobe formulated using a line approximately; however, thederivation became signiˆcant when the normal stress in­creased to 400 kPa (Fig. 6(b)).Cyclic Stress­displacement RelationshipFigures 7 and 8 show the stress­displacement responsesof the interface due to a two­way tangential displacementapplication. The plot of shear stress versus tangential dis­placement looked nearly closed within a single shear cyclebut diŠered with diŠerent numbers of shear cycles (Fig.8). The strain­softening behavior due to cyclic shear ap­plication was negligible after the ˆrst monotonic shear.The normal displacement accumulated in a whole with in­creasing number of shear cycles, but well­regulated va­ried within a single shear cycle (Fig. 7). When shear direc­Fig. 8. Cyclic stress­displacement relationship of shear test results un­der constant normal boundary condition. t, shear stress; s, normalstress; u, tangential displacement; v, normal displacementtion was reversed, the normal displacement always in­creased. This response exhibited a trend similar to the be­havior of a steel­gravel interface (Zhang and Zhang,2006b).There was an asymmetric response of stress and dis­placement in diŠerent shear directions of the interfacedue to a symmetrical tangential displacement application80G. ZHANG AND J.­M. ZHANGFig. 9. Stress ratio change due to cyclic shear application. t1, peakpositive shear stress within a cycle; s, normal stress; N, number ofshear cycles(Fig. 8). For example, shear stress had a smaller absolutemaximum value in the initial shear direction than in thereverse direction; the normal displacement showed moresigniˆcant asymmetry in diŠerent shear directions. Inparticular, normal displacement tended to increase aftera small compression due to unloading, if tangential dis­placement was applied in the initial shear direction; whilethere was a tendency to decrease in the reverse direction.This inconsistency of the stress­displacement relationshipin diŠerent shear directions has previously been describedas ``aeolotropy of interface'', based on results of a steel­gravel interface (Zhang and Zhang, 2006b).The maximum shear stress within a single shear cycledecreased a little with increasing number of shear cycles ifthe normal stress was maintained (Figs. 7 and 8). Thedecrease extent of stress ratio in the monotonic sheardirection (i.e., shear strength) increased signiˆcantly withincreasing normal stress (Fig. 9). This strength behaviorwas signiˆcantly diŠerent from that of a steel­gravel in­terface, where friction angle was maintained with increas­ing number of shear cycles (Zhang and Zhang, 2006b).Microscopic Observations and MeasurementsThe photographs of the geotextile and adjacent soil atthe initial state and fourth shear cycles indicated that sig­niˆcant crushing of soil particles near the geotextile ap­peared due to the application of cyclic shear (Fig. 10).The particle crushing was also found in the tests of thesteel­gravel interface, especially under cyclic shear condi­tions (Zhang and Zhang, 2006b). In addition, the struc­ture plate with geotextile went down gradually as a wholewith increasing number of shear cycles shown in the pho­tographs (Fig. 10) and measured changes in the normaldisplacement using the transducers (e.g., Fig. 7). Thisdemonstrates that some compression of soil near the geo­textile was induced by shear application. Therefore, therewas signiˆcant physical state evolution of soil near thegeotextile, including particle crushing and compressiondue to shear application, similar to a steel­gravel inter­face (Zhang and Zhang, 2006b). Moreover, rather thanthe steel­gravel interface, the damage to geotextile shouldbe involved to the evolution of the physical state of theinterface. These physical evolutions in turn changed theFig. 10. Photographs of the geotextile and soil nearby during a cyclicshear test under constant normal stress of 200 kPa. The loadingconditions are as Fig. 7stress­displacement relationship response. For example,the particle crushing may cause the decrease of shearstrength of the interface (Zhang and Zhang, 2006b).A series of high resolution digital photographs takenthrough the transparent lucite window, recorded the geo­textile and adjacent soil during testing. Characterized us­ing a line segment with a few tracing points, a soil particlecan be tracked by ˆnding the corresponding position oftracing points on the line segment in a series of photo­graphs (Zhang et al., 2006). The movement of a soil parti­cle was obtained by the translation of the midpoint of thesegment and rotation angle of the soil particle from thatof the segment.Figure 11 shows a typical measurement of soil particlemovements near the geotextile. The horizontal transla­tion was deˆned as positive if in the same direction as thesoil container, while vertical translation was positive ifdownward and the rotation was positive if clockwise (Fig.11(a)). The horizontal translations of soil particles werealways opposite to the movement direction of the soilcontainer (Fig. 11). The horizontal translations relativeto the soil container appeared in the zone close to the geo­textile; they decreased with increasing distance from thegeotextile and were negligible by 40 mm from the geotex­tile. These horizontal translations increased with increas­ing tangential displacement, yet were always less than thetangential displacement of the soil container (Fig. 11).This demonstrates that tangential displacement of the in­terface was induced not only by slippage between the geo­textile and adjacent soil at the contact surface, but also bydeformation of the soil constrained by the geotextile. TheINTERFACE BETWEEN GEOTEXTILE AND SOIL81Fig. 11. Soil particle movements relative to the soil container during a monotonic shear test under constant normal stress of 200 kPa. Data point,movements of an individual soil particle; y, distance from the geotextile; dx, horizontal movement; dy, vertical movement; u, rotationvertical translations and rotations of soil particles ap­peared in a complex way but exhibited quite signiˆcantregularity as a whole (e.g., Fig. 11). For example, themajority of soil particles rotated anticlockwise due to theconstraint of the geotextile. Thus, the deformation of thesoil due to constraint of the geotextile was the main con­tribution to volumetric change due to dilatancy, as hasbeen discovered on a steel­gravel interface (Zhang andZhang, 2006b). In addition, it is possible that the mainreason for ``aeolotropy of interface'' was that the initialmonotonic shear application caused structural ae­olotropy of arrangement and dip direction of soil parti­cles near the geotextile (Fig. 11).The measurements showed that there were signiˆcantmovements of soil particles only in a narrow zone close tothe geotextile, demonstrating the interface thickness.From the measurements (Fig. 11), the interface thicknesswas estimated at 5–6 times the average soil grain size (i.e.,about 40mm), similar to the steel­gravel interface (Zhangand Zhang, 2006b).PULLOUT TEST RESULTSLarge­scale pullout tests were conducted on the geo­textile with the gravel on both sides under diŠerent nor­mal stress conditions. It should be noted that the geo­textile was ruptured during the pullout test when normalFig. 12. Geotextile pullout test results with the gravel on both sides. p,average shear resistance; U, pullout displacement; V, normal dis­placement; s, normal stress82G. ZHANG AND J.­M. ZHANGstress was 200 kPa; thus the results are presented only to150 kPa of normal stress. Figure 12 shows the changeprocess of the ``average shear resistance'' and normal dis­placement with increasing pullout displacement. Therethe ``average shear resistance'' is based on the averagesense to be consistent with the normal stress; it is equal tothe total pullout force divided by the area of the contactzone between the geotextile and gravel. Compared withthe direct shear test results (Fig. 5), there were signiˆcantdiŠerences in the pullout tests, including: (1) Therelationship curve between average shear resistance andpullout displacement was signiˆcantly ‰atter than the oneobtained from the direct shear test. For example, averageshear resistance became stable when pullout displacementreached 40 mm while shear stress became stable onlywhen tangential displacement reached 15 mm at normalstress of 100 kPa. (2) The normal displacement becamesigniˆcant only when pullout displacement increased to acertain value; this pullout displacement was far largerthan the one in the shear test. This can be contributed tolow tensile modulus of the geotextile, which caused sig­niˆcant deformation of the geotextile itself. Therefore,the pullout displacement consisted of evident deforma­tion of the geotextile. It should be noted that the defor­mation of the geotextile included that of the 50 mm­longgeotextile that did not contact the soil. It can be conclud­ed that the relative movement on the contact surface be­tween the soil and geotextile occurred progressively dur­ing the pullout test and was signiˆcantly less than the ap­parent pullout displacement. In other words, the shearstress of the interface transferred progressively from loadend to the free end; a larger pullout displacement wasneeded to reach the peak value.The interpretation of pullout tests underestimated theshear stiŠness of the interface, similarly to a clay­geotex­tile interface (Long et al., 2007). In this sense, therelationship of the average shear resistance versus pulloutdisplacement does not accurately describe the stress­strain relationship of the interface. However, it can pro­vide more realistic simulation of behavior of such an in­terface because the deformation of the geotextile can beinvolved.The rotations of several typical soil particles near thegeotextile were measured using image analysis (Fig. 13).These soil particles exhibited signiˆcant rotations whenpullout displacement exceeded 10 mm, and the soil parti­cles near the load end rotated earlier than those near thefree end. This indicated that a soil particle' rotation re­quired a certain pullout displacement, consistent with thechange rule of normal displacement.When the average shear resistance reached an approxi­mately stable state, it can be derived that the deformationof the geotextile also became stable because the pulloutforce was maintained. We conclude that shear stress atthe interface became uniform at this time, with only rela­tive displacement between the geotextile and soil. Thus,the stable value of average shear resistance can be re­garded as the shear strength of the interface under pullouttest conditions; it may be closer to twice the residueFig. 13. Microscopic measurement results of geotextile pullout testwith the gravel on both sides under constant normal stress of 50kPa. U, pullout displacement; u, rotationFig. 14. Shear strength based on the geotextile pullout test results withthe gravel on both sides. pf, pullout strength; s, normal stressstrength of direct shear tests because of two contact sur­faces in pullout tests. Such the strength is deˆned as ``pullstrength,'' denoted as pf, to distinguish from the strengthfrom direct shear tests. In pullout tests, the pull strengthincreased with increasing normal stress in a nonlinearrelationship (Fig. 14); moreover, it was signiˆcantly morethan twice the residue shear strength of direct shear testsunder small normal stress conditions (e.g., 50 kPa) (Fig.6).This strength behavior from pullout tests can be ex­INTERFACE BETWEEN GEOTEXTILE AND SOIL83Fig. 16. Shear strength based on the geotextile pullout test results withthe gravel on one side. pf, pullout strength; s, normal stressFig. 15. Geotextile pullout test results with the gravel on one side. p,average shear resistance; U, pullout displacement; V, normal dis­placement; s, normal stressplained as follows. The interface exhibited signiˆcantdilatancy under low normal stress conditions (Fig. 5),which led to many small local protuberances on the soil­geotextile interface due to ‰exibility of the geotextile.Thus, the soil­geotextile contact area increased sig­niˆcantly requiring a larger pullout force. In other words,shear strength of the interface increased due to dilatancy.This explanation is supported by the large negative nor­mal displacement (i.e., dilative volumetric change) whennormal stress was small (Fig. 12). Increasing normalstress signiˆcantly decreased the dilatancy of the inter­face, for example, the negative normal displacement wassigniˆcantly small when normal stress was 150 kPa (Fig.12); thus shear strength due to dilatancy decreased corre­spondingly.A logarithmic equation was used to describe the non­linear relationship between the pullout strength, pf, andnormal stress, obtained from pullout tests, as follows:pfp0{Dp logØ ps »a(2)where pa is the standard atmosphere. p0 and Dp are theparameter; they are set to 175 kPa and 171 kPa accordingto the test results, respectively. This function showed agood ˆt to the test results when normal stress was notmore than 150 kPa (Fig. 14).Another group of pullout tests were conducted withgravel on one side of the geotextile and sand on the other.The relationship between the average shear resistance andpullout displacement showed that the dilatancy extentdecreased signiˆcantly when gravel was replaced withsand on one side (Fig. 15). Accordingly, there was a sig­niˆcant decrease in the nonlinear extent of the relation­ship between the pull strength and normal stress (Fig.16). It may be because that the sand particles were fairlysmall and so local protuberances were weakened accord­ingly. The logarithmic equation adequately ˆtted therelationship between shear strength and normal stress ofsuch pullout tests (Fig. 16). This demonstrates that therelationship between the pullout strength and normalstress can be described by using an isomorphic expres­sion.DISCUSSIONThe direct shear and pullout tests are the two mostwidely­used approaches to investigate the behavior ofsoil­geotextile interfaces. However, the test results in thepresent paper show that these two types of test methodsyield signiˆcantly diŠerent responses because pullouttests allows for signiˆcant deformation of geotextile,compared with the relative displacement on the interface.The direct shear test can give an entire stress­displace­ment relationship, but it is di‹cult to simulate shearstrength due to dilatancy induced by raised contact areasduring shearing. The pullout test can consider the defor­mation of the geotextile itself and is closer to practicalsituations. However, the stress and deformation are sig­niˆcantly nonuniform on the interface during pullouttests; they can be regarded as uniform only when a stablepullout force is achieved.Therefore, neither direct shear nor pullout tests givecomprehensive understanding of the behavior of the in­terface between a geotextile and gravelly soil. Therefore,the appropriate test method should be selected with care­ful consideration of the site conditions of the interface.Combining both test methods seems an eŠective ap­proach for investigation of the interfaces.CONCLUSIONSA series of large­scale direct shear and pullout testswere conducted to investigate the monotonic and cyclicbehavior of a gravel­geotextile interface. On the basis of84G. ZHANG AND J.­M. ZHANGthe macro­ and micro­ observation results, the main be­haviors of the interface are discovered or summarized asfollows:(1) The interface exhibited signiˆcant strain­softeningdue to damage to the geotextile during shearing;the softening extent was signiˆcantly in‰uenced bynormal stress. The shear strength from the directshear test was proportional to normal stress anddecreased with increasing number of shear cycles.(2) The normal displacement accumulated as a wholewith increasing number of shear cycles, but variedwithin a single shear cycle in a well­regulated man­ner in cyclic shear tests and was dependent on sheardirection. The change of normal displacement wassigniˆcantly in‰uenced by normal stress.(3) Shear deformation of the interface was composedof slippage on the contact surface and deformationof the soil constrained by the geotextile; the volu­metric change due to dilatancy was induced mainlyby the latter. The thickness of the interface was es­timated at 5–6 times the average soil grain size.There was signiˆcant evolution of physical statedue to shear application, including soil particlecrushing and soil compression as well as damage tothe geotextile.(4) The relative movement at the contact surface be­tween the soil and geotextile occurred progressivelyin the pullout test because the geotextile had sig­niˆcant deformation of its own. Thus, shear stiŠ­ness of the interface was underestimated.(5) Rather than direct shear tests, the pullout testsshowed that shear strength of the interface in­creased with increasing normal stress by a non­linear relationship due to dilatancy, described by alogarithmic relationship.(6) Neither direct shear nor pullout tests can give com­prehensive understanding of the behavior of the in­terface between a geotextile and gravelly soil. Anappropriate test method should be selected withcareful consideration of the site conditions.ACKNOWLEDGEMENTSThe project is supported by National Basic ResearchProgram of China (973 Program) (No. 2007CB714108)and National Natural Science Foundation of China (No.50679034, 50778105).REFERENCES1) Abu­Farsakh, M., Coronel, J. and Tao, M. (2007): EŠect of soilmoisture content and dry density on cohesive soil­geosynthetic in­teractions using large direct shear tests, Journal of Materials inCivil Engineering, 19(7), 540–549.2) Aiban, S. A. and Ali, S. M. (2001): Nonwoven geotextile­sabkhaand­sand interface friction characteristics using pullouttests, Geosynthetics International, 8(3), 193–220.3) Athanasopoulos, G. A. (1996): Results of direct shear tests on geo­textile reinforced cohesive soil, Geotextiles and Geomembranes,14(11), 619–644.4) Bakeer, R. M., Abdel­Rahman, A. H. and Napolitano, P. J.(1998): Geotextile friction mobilization during ˆeld pullout test,Geotextiles and Geomembranes, 16(2), 73–85.5) Eigenbrod, K. D., Burak, J. P. and Locker, J. G. (1990): DiŠeren­tial shear movements at soil­geotextile interfaces, Canadian Geo­technical Journal, 27(4), 520–526.6) Fannin, R. J. and Raju, D. M. (1993): On the pullout resistance ofgeosynthetics, Canadian Geotechnical Journal, 30(3), 409–417.7) Giroud, J. P., Darrasse, J. and Bachus, R. C. (1993): Hyperbolicexpression for soil­geosynthetic or geosynthetic­geosynthetic inter­face shear strength, Geotextiles and Geomembranes, 12(3),275–286.8) Gourc, J. P. and Ramirez, R. R. (2004): Dynamics­based interpre­tation of the interface friction test at the inclined plane, Geosyn­thetics International, 11(6), 439–454.9) Guler, M., Edil, T. B. and Bosscher, P. J. (1999): Measurement ofparticle movement in granular soils using image analysis, Journalof Computing in Civil Engineering, 13(2), 116–122.10) Lee, K. M. and Manjunath, V. R. (2000): Soil­geotextile interfacefriction by direct shear tests, Canadian Geotechnical Journal,37(1), 238–252.11) Lima, J., Palmeira, E. M. and Mello, L. G. R. (2002): Interactionbetween soils and geosynthetic layers in large­scale ramp tests, Geo­synthetics International, 9(2), 149–187.12) Ling, H. I., Burke, C., Mohri, Y. and Matsushima, K. (2002):Shear strength parameters of soil­geosynthetic interfaces under lowconˆning pressure using a tilting table, Geosynthetics International,9(4), 373–380.13) Lo, S. R. (2003): The in‰uence of constrained dilatancy on pulloutresistance of strap reinforcement, Geosynthetics International,10(2), 47–55.14) Long, P. V., Bergado, D. T. and Abuel­Naga, H. M. (2007): Geo­synthetics reinforcement application for tsunami reconstruction:Evaluation of interface parameters with silty sand and weatheredclay, Geotextiles and Geomembranes, 25(4–5), 311–323.15) Lopes, P. C., Lopes, M. L. and Lopes, M. P. 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(1998): Sand­geotextileinterface shear strength by torsional ring shear tests, Geotextilesand Geomembranes, 16(3), 161–174.22) Uesugi, M. and Kishida, H. (1986): Frictional resistance at yield be­tween dry sand and mild steel, Soils and Foundations, 26(4),139–149.23) Zhang, G. and Zhang, J.­M. (2006a): A large­scale apparatus formonotonic and cyclic soil­structure interface test, GeotechnicalTesting Journal, 29(5), 401–408.24) Zhang, G. and Zhang J.­M. (2006b): Monotonic and cyclic tests ofinterface between structure and gravelly soil, Soils and Founda­tions, 46(4), 505–518.25) Zhang, G., Liang, D. and Zhang J.­M. (2006): Image analysismeasurement of soil particle movement during a soil­structure in­terface test, Computers and Geotechnics, 33(4–5), 248–259.
• ログイン

• タイトル
• Role of Fly Ash on Strength and Microstructure Development in Blended Cement Stabilized Silty Clay
• 著者
• S. Horpibulsuk・R. Rachan・Y. Raksachon
• 出版
• Soils and Foundations
• ページ
• 85〜98
• 発行
• 2009/02/15
• 文書ID
• 21172
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 85–98, Feb. 2009ROLE OF FLY ASH ON STRENGTH AND MICROSTRUCTUREDEVELOPMENT IN BLENDED CEMENT STABILIZED SILTY CLAYSUKSUN HORPIBULSUKi), RUNGLAWAN RACHANii) and YUTTANA RAKSACHONiii)ABSTRACTThis paper presents the role of ‰y ash on strength and microstructure development in blended cement stabilized siltyclay. Its strength was examined by unconˆned compression test and its microstructure (fabric and cementation bond)by a scanning electron microscope (SEM), mercury intrusion porosimetry (MIP), and thermal gravity (TG) analysis.The ‰occulation of clay particles due to the cation exchange process is controlled by cement content, regardless of ‰yash content. It increases dry unit weight of the stabilized clay with insigniˆcant change in liquid limit. This results in ir­relevant diŠerence in optimum water content (OWC) for the unstabilized and the stabilized clay since OWC of lowswelling silty clay is mainly controlled by liquid limit. It is found from the microstructural and the strength test resultsthat the reactivity of ‰y ash (pozzolanic reaction) is minimal, which is diŠerent from concrete technology. This is possi­bly due to less amount of Ca(OH)2 to be consumed. The role of ‰y ash in cement stabilization is to disperse the largeclay­cement clusters into smaller clusters. Consequently, the reactive surfaces to be interacted with water increase, andhence the cementitious products (inter­cluster cementation bond). To conclude, the strength development in the blend­ed cement stabilized clay is controlled by cementitious products due to combined eŠect: hydration and dispersion.Cementitious products due to hydration are governed by cement content, while cementitious products due to disper­sion by ‰y ash content and ˆneness. Water content of 1.2OWC and 10z replacement ratio are regarded as the eŠectivemixing condition for the stabilization, exhibiting the highest cementitious products.Key words: blended cement, cementation, dispersion, fabric, ‰y ash, hydration, microstructure, pore size distribu­tion, scanning electron microscope, strength, thermal gravity analysis (IGC: D6/M5)curing time, and compaction energy on the engineeringcharacteristics of cement stabilized soils have been exten­sively researched (Terashi et al., 1979, 1980; Clough etal., 1981; Tatsuoka and Kobayashi, 1983; Kamon andBergado, 1992; Uddin, 1994; Nagaraj et al., 1997; Yinand Lai, 1998, Consoli et al., 2000; Kasama et al., 2000;Kitazume et al., 2000; Miura et al., 2001; Horpibulsukand Miura, 2001; Horpibulsuk et al., 2003, 2004a, b,2005, 2006; and others).Many researchers in concrete technology (Owens,1979; Mitsui, et al., 1994; Ollivier and Massat, 1996;Igarashi, et al., 1996; Yang and Su, 2002; Chindaprasirtet al., 2004; and others) have attempted to use waste poz­zolanic materials from industries to reduce the input ofcement. Pozzolanic material generally consists of silica,alumina, ferric oxide etc. These compounds will form acementitious material when combined with cement in thepresence of water. Fly ash is one of the pozzolanic materi­als extracted from ‰ue gases of a furnace fried with coalof Electric Power Plant. Its generation is far in excess ofutilization. It was possible to utilize ‰y ash in geotechni­INTRODUCTIONSoil in northeast Thailand generally consists of twolayers. The upper layer (varying from 1–3 m thickness) iswind­blown and deposited over several decades. It isclayey sand or silty clay with low to moderate strength(12ºNº20, where N is standard penetration number).This upper soil is a problematic soil, which is sensitive tochange in water content (Horpibulsuk et al., 2008b). Itscollapse behavior due to wetting is illustrated by Kohgo etal. (1997) and Kohgo and Horpibulsuk (1999). The lowerlayer is a residual soil, weathered from claystone, consist­ing of clay, silt and sand (Udomchore, 1991). It possessesvery high strength (generally NÀ30) and very low com­pressibility. One of the extensively used soil improvementtechniques for the upper soil is to compact the in­situ soil(in relatively dry state) mixed with cement slurry. Thistechnique is economical because cement is readily availa­ble at reasonable cost in Thailand. Moreover, adequatestrength can be achieved in a short time. EŠects of somein‰uential factors i.e., water content, cement content,i)ii)iii)Associate Professor and Chair of School of Civil Engineering, and Head of Construction Technology Research Unit, Suranaree University ofTechnology, Nakhon Ratchasima, Thailand (suksun—g.sut.ac.th).Lecturer, Department of Civil Engineering, Mahanakorn University of Technology, Bangkok, Thailand.M.Eng. Graduate, School of Civil Engineering, Suranaree University of Technology, Nakhon Ratchasima, Thailand.The manuscript for this paper was received for review on May 26, 2008; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.8586HORPIBULSUK ET AL.cal and geoenvironmental works (Cokca, 1997).Kawasaki et al. (1981) used ‰y ash mixed with cement inthe construction of a man­made island for the HakuchoBridge in Hokkaido. Laboratory investigation demon­strated that ‰y ash can control the volume change of ex­pensive clay (Kehew, 1995). Fly ash­cement admixturecan reduce swelling, compressibility and increase strengthof soil. The incorporation of ‰y ash as partial replace­ment of cement might reˆne the microstructure (fabricand cementation bond) of the cement stabilized clay,resulting in the improvement of engineering properties(strength, compressibility, permeability, and durability).Its role on the microstructure development has not beenwell established so far. It is however fundamental to beexamined for better understanding and analyzing the en­gineering properties of the blended cement stabilizedclay.Microstructural investigation in clay has started since1960 by Aylmore and Quirk (1960); Olsen (1962). Theyhave revealed that the basic element of the microstructureof the natural clay is not the single platelet but domainsconstituted of various platelets aggregated together.Nagaraj et al. (1990) have analyzed the pore size distribu­tion of saturated clay and concluded that the volume ofÀ0.03 micron pores amounts to nearly 90 to 95z of thetotal pore volume. The º0.01 micron pore is the intra­aggregate pore, which is in the stable clusters formed indiŠerent electrolytic environments during deposition andsediment formation. This pore volume accounts for only3 to 5z of the total pore volume. Since net force is likelyto be attractive, these clusters are stable even in the ab­sence of external loading. When such stable units are inclose proximity in the range of separation distance of 0.01to 0.03 micron, the net force of repulsion (R­A) betweensuch units would create an osmotic suction on adjacent‰uid equal in magnitude to the isotropic mean eŠectivestress. When pore ‰uid is subjected to such suction pres­sure at innumerable points, due to several pairs of inter­acting particle units, a large pore (À0.03 micron) can beformed. The separation distance of larger than 0.01micron is referred to as inter­aggregate pore. The porespace between 0.01 and 0.03 micron is designated as inter­aggregate pore between two interacting aggregate, andthe pore space larger than 0.03 micron is large enclosedpore held within a group of clusters by surface tension.Miura et al. (1999) and Yamadera (1999) have analyzedthe change in the micro­structure of natural and remold­ed Ariake clay during K0­consolidation and shearingbased on the cluster theory of Nagaraj et al. (1990). Porespace decreases and the clay clusters become larger as theconsolidation pressure increases. Lapierre et al. (1990)used the pore size distribution from the mercury intrusionporosimetry test to explain the change of the permeabilityof Louiseville clay.When cement is added to clays, calcium ions increase inthe interlayer, resulting in an attraction between the clayparticles and the formation of aggregations of ‰ocs. Thecation exchange process continues until all charges on theinterlayer and edges are satisˆed. This addition is primaryresponsible for enhancing the soil workability, but doesnot result in an increase in strength (Sherwood, 1993;Bell, 1996). Further additions of cement lead to hydra­tion, resulting in formation of cementitious productssuch as calcium silicate hydrates (CSH), calciumaluminate hydrates (CAH), and calcium aluminium sili­cate hydrates (CASH).Abduljauwad (1995) has observed the change inmicrostructure of stabilized soil using scanning electronmicroscopes (SEMs). Keshawarz and Dutta (1993) havereported that the particles of uncemented soil appear as ablocky arrangement of loosely packed particles while thecemented soil shows abundance of tobermorite crystals.Recently, Horpibulsuk et al. (2009) have analyzed thestrength development of cement stabilized clay withwater content, cement content and curing time based onmicrostructural (fabric and cementation bonding) con­siderations. The microstructure was observed by scan­ning electron microscope (SEM) and mercury intrusionporosimetry (MIP), while the cementation bond strengthwas examined from amount of cementitious products(Ca(OH)2 and CSH) by thermal gravity (TG) analysis.They have compared the microstructure of cement stabi­lized clay to that of unstabilized clay and revealed that thecement stabilization markedly enhances the inter­clustercementation bonding and reduces the pore space. Thepores are possibly classiˆed into three categories: airpores (À10 micron), inter­aggregate pores (10–0.01micron) and intra­aggregate pores (º0.01 micron).When clay is mixed with cement and water, the inter­ag­gregate pore volume increases due to the increase incoarse materials (unhydrated cement grains). With time,cementitious products ˆll up the pores, resulting in the in­crease in intra­aggregate pore volume.The present paper attempts to explain the role of ‰y ashin cement stabilization on reˆning the microstructure,and hence the strength improvement. The parameters in­volved in this investigation are molding water content,curing time, replacement ratio and ˆneness of the ‰y ash.The microstructural analyses have been conducted in thispaper via scanning electron microscope, mercury intru­sion porosimetry, and thermal gravity tests.LABORATORY INVESTIGATIONSoil SampleSoil sample is silty clay collected from the campus ofSuranaree University of Technology, Nakhon Ratchasi­ma, Thailand at 3 meter depth. The soil is composed of2z sand, 45z silt and 53z clay. Its speciˆc gravity is2.74. The liquid and plastic limits are in the order of 74and 27 percent, respectively. Based on the Uniˆed SoilClassiˆcation System (USCS), the clay is classiˆed as highplasticity (CH). During sampling the groundwater haddisappeared. Natural water content was 10 percent. Thefree swell test proposed by Prakash and Sridharan (2004)shows that the clay is classiˆed as low swelling type withfree swell ratio (FSR) of 1.0. The FSR is deˆned as the ra­tio of equilibrium sediment volume of 10 g of oven­driedROLE OF FLY ASHTable 1.Chemical composition of silty clay, PC, OFA, and CFAChemical composition (z)Silty clayPCOFACFASiO2Al2O3Fe2O3CaOMgOSO3Na2OK2 OLOI20.107.5532.8926.150.474.92ND3.173.4420.904.763.4165.411.252.710.240.350.9645.6924.5911.2612.152.871.570.072.661.2344.7223.6911.0312.672.631.280.072.871.42Fig. 1.Grain size distributions of the silty clay, PC, OFA, and CFAFig. 2.SEM photo of the natural silty claysoil passing a 425 mm sieve in distilled water (Vd) to thatin kerosene (Vk). This method was employed since it is asimple methodology to obtain an approximate and fairlysatisfactory prediction of the dominant clay mineralogyof soil (Horpibulsuk et al., 2007). Chemical compositionand grain size distribution curve of the silty clay areshown in Table 1 and Fig. 1, respectively. The natural sil­ty clay shows groups of cluster with various sizes (videFig. 2).Cementing MaterialsType I Portland cement (PC), ‰y ash from Mae Mohpower plant in the north of Thailand, and tap water wereused in this study. Chemical composition of PC, original87and classiˆed ‰y ashes (OFA and CFA) is also given inTable 1. The classiˆed ‰y ash was obtained from theoriginal ‰y ash by passing through sieve No. 325. Totalamount of the major components SiO2, Al2O3, and Fe2O3in OFA and CFA are 81.54z and 79.44z, respectively.They are thus classiˆed as class F ‰y ash in accordancewith ASTM C 618. It is noted that there is no signiˆcantdiŠerence in the chemical composition of OFA and CFA.Two ‰y ashes of OFA with the median particle size (D50)of 0.03 mm (30 micron) and CFA with D50 of 0.009 mm(9 micron) were used to replace the cement. Grain sizedistribution curves of PC, OFA, and CFA are also shownin Fig. 1. These curves were obtained from the laser parti­cle size analysis. It is found that grain size distributioncurves of PC and CFA are similar. D50 of PC is 0.01 mm(10 micron) being almost the same as that of CFA.Speciˆc gravities of PC, OFA and CFA are 3.15, 2.33,and 2.54, respectively. The blaine ˆneness of CFA is 510m2/kg as compared with 300 m2/kg of OFA. SEM photosof PC, OFA and CFA are shown in Fig. 3. From thegrain size distribution curves and the SEM photos, it isfound that the particles of the silty clay are much smallerthan those of the cement and the ‰y ashes. The cement isirregular in shape whereas the ‰y ashes are spherical.MethodologyThe silty clay was passed through a 16­mm sieve to re­move coarser particles. It was air­dried for at least threedays and then the water content was adjusted for com­paction test. At least ˆve compaction points were gener­ated. The compaction was carried out according toASTM D 698 and D 1557 in a standard 100­mm diametermold for standard and modiˆed Proctor energies (592.5and 2693.3 kJ/m3), respectively. Compaction curves atthese two energy levels are shown in Fig. 4. The compac­tion characteristics (optimum water content, OWC, andmaximum dry unit weight, gdmax) are 22.4z and 14.6kN/m3 for standard Proctor energy and 17.2z and 17.4kN/m3 for modiˆed Proctor energy.Having obtained the compaction curves, the air­driedclay was thoroughly mixed with 10z binder (cement and‰y ash) and compacted under modiˆed Proctor energy.Liquid and plastic limits of the stabilized samples werealso determined immediately after thorough mixing. This10z binder was proved as suitable for strength improve­ment of this silty clay (Horpibulsuk et al., 2009). The per­centages of ‰y ash to replace the cement (replacement ra­tio) are 0, 10, 20, 30, and 40z by weight of binder for un­conˆned compression test. After 24 hours of compaction,the blended cement stabilized samples were dismantledfrom the mold, wrapped in vinyl bags and stored in a hu­midity chamber of constant temperature (25}29C). Un­conˆned compression test was run on the samples after 7,28, 60, 90, and 120 days of curing. The rate of verticaldisplacement was ˆxed at 1 mm/min. For each curingtime, and combination of water content and replacementratio, at least ˆve samples were tested under the samecondition to check for consistency of the test. In mostcases, the results under the same testing condition were88HORPIBULSUK ET AL.Fig. 3.SEM photos of PC, OFA, and CFATable 2.Curing time(days)Replacementratio (z)SEM PC, PC{OFA, 21 (1.2OWC)and PC{CFA28 and 600, 10, 20 and30MIP PC, PC{OFA, 21 (1.2OWC)and PC{CFA7, 28 and 600, 10, 20 and30PC{OFA, and 14, 17, 21 andPC{CFA24710PC, PC{OFA, 21 (1.2OWC)and PC{CFA7, 28, 60, 90and 1200, 10, 20 and30TestTGFig. 4. Compaction curves of the silty clay under standard and modi­ˆed Proctor energiesreproducible with low mean standard deviation, SD(SD/ šxº9.2z, where šx is mean strength value).Microstructure development of the blended cementstabilized clay samples with replacement ratios between 0and 30z is investigated by the scanning electron micro­scope (SEM), mercury intrusion porosimetry (MIP), andthermal gravity (TG) analyses as summarized in Table 2.The blended cement stabilized samples were broken fromSummary of the microstructural testing programBinderWater content(z)the center into small fragments. The SEM samples werefrozen at |1959C by immersion in liquid nitrogen for 5minutes and evacuated at a pressure of 0.5 Pa at |409Cfor 5 days (Miura et al., 1999; Yamadera, 1999). All sam­ples were coated with gold before SEM (JOELJSM–6400) analysis.Measurement on pore size distribution of the sampleswas carried out using mercury intrusion porosimeter(MIP) with a pressure range from 0 to 288 MPa, capableof measuring pore size diameter down to 5.7 nm (0.0057micron). The MIP samples were obtained by carefullybreaking the stabilized samples with a chisel. Therepresentative samples of 3–6 mm pieces weighing be­tween 1.0–1.5 g were taken from the middle of the89ROLE OF FLY ASHcemented samples. Hydration of the samples was stoppedby freezing and drying, as prepared in the SEM examina­tion. Mercury porosimetry is expressed by the Washburnequation (Washburn, 1921). A constant contact angle (u)of 1409and a constant surface tension of mercury (g) of480 dynes/cm were used for pore size calculation as sug­gested by Eq. (1)D|(4g cos u)/P(1)where D is the pore diameter (micron) and P is the ap­plied pressure (MPa).Thermal gravity (TG) analysis is one of the widely ac­cepted methods for determination of hydration products,which are crystalline Ca(OH)2, CSH, CAH, and CASH,etrringite (Aft phases), and so on (Midgley, 1979). TheCSH, CAH, and CASH are regarded as cementitiousproducts. Ca(OH)2 content was determined based on theweight loss between 450 and 5809C (El­Jazairi andIllston, 1977, 1980; Wang et al., 2004) and expressed as apercentage by weight of ignited sample. When heating thesamples at temperature between 450 and 5809C, Ca(OH)2is decomposed into calcium oxide (CaO) and water as inEq. (2).Ca(OH)2 ª CaO{H2O(2)Due to the heat, the water is lost, leading to thedecrease in overall weight. The amount of Ca(OH)2 canbe approximated from this lost water by Eq. (2), which is4.11 times the amount of lost water (El­Jazairi andIllston, 1977, 1980). The change of the cementitiousproducts can be expressed by the change of Ca(OH)2since they are the hydration products.Cube specimens was crushed into fragmented samplesand freeze­dried as prepared for MIP. Prior to TG test­ing, the dried samples were ground in a ball mill andsieved through 100 mesh (150 mm). TG analysis of thesamples was performed using thermal analyzer. Approxi­mately 10–20 mg of the sample was taken for the analy­sis. The heat rate was maintained at 109C/min and thesample was heated up to 1,0009C.COMPACTION AND UNCONFINEDCOMPRESSION TEST RESULTSFigure 5 shows the compaction curve of the OFA andthe CFA blended cement stabilized clay for diŠerentreplacement ratios (ratios of cement to ‰y ash, C:F) com­pared with that of the unstabilized clay. It is noted thatthe compaction curve of the stabilized clay is insig­niˆcantly dependent upon ˆneness of ‰y ash and replace­ment ratio. Maximum dry unit weight of the stabilizedclay is higher than that of the unstabilized clay whereastheir optimum water content is practically the same. Thischaracteristic is the same as that of cement stabilizedcoarse­ and ˆne­grained soils reported by Horpibulsuk etal. (2006, 2009). Table 3 shows index properties of the sil­ty clay, the cement stabilized clay and the OFA and theCFA blended cement stabilized clay. It is found that plas­tic limit of the blended cement stabilized clay increases asFig. 5. Compaction curves of the OFA and CFA blended cementstabilized clay and the unstabilized clayTable 3. Index properties of the blended cement stabilized clay at 10%binderType ofstabilizationC:FNo stabilizationAtterberg's limits (z)LLPLPI0:074.127.546.6PC100:071.044.826.2PC{CFA80:2060:400:10071.471.773.137.633.329.433.838.443.7PC{OFA80:2060:4071.471.736.333.335.138.4cement content increases (replacement ratio decreases).The increase in plastic limit indicates the ‰occulation ofclay particles caused by the adsorption of Ca2{ ions fromcation exchange process. Fly ash insigniˆcantly aŠectsplastic limit as shown by the results of C:F0:100 andC:F0:0. In other words, ‰y ash does not play any sig­niˆcant role on the ‰occulation. The ‰occulation resultsin the increase in the dry unit weight with insigniˆcantchange in liquid limit. Consequently, optimum watercontents (OWCs) for the unstabilized and the stabilizedsamples are almost the same, since OWC of low swellingclays is mainly controlled by liquid limit (Nagaraj et al.,2006; Horpibulsuk et al., 2008a).Figure 6 shows the strength versus water contentrelationship of the CFA blended cement stabilized clay atdiŠerent replacement ratios after 60 days of curing com­pared with that of the unstabilized clay. The maximumstrengths of the stabilized clay are at about 1.2OWCwhereas the maximum strength of the unstabilized clay isat OWC (maximum dry unit weight). This is because en­gineering properties of unstabilized clay are mainly de­pendent upon the densiˆcation (packing). Even withdiŠerent curing times, the maximum strengths of thestabilized samples are at about 1.2OWC as shown in Fig.7. This characteristic is the same as that of the cement90HORPIBULSUK ET AL.stabilized clay as illustrated by Horpibulsuk et al. (2009).In‰uence of replacement ratio and ‰y ash ˆneness onthe strength development of the blended cement stabi­lized clay compacted at water content (w) of 1.2OWC (w20.9z) for the ˆve curing times is presented in Fig. 8and Table 4. It is of interest to mention that the ‰y ashˆneness slightly aŠects the strength development as illus­trated by the slight diŠerence in strength between theCFA and the OFA blended cement stabilized clay. For allcuring times, the samples with 20z replacement ratio ex­hibit almost the same strength as those with 0z replace­ment ratio. The 30 and 40z replacement samples exhibitlower strength than 0z replacement samples. The sam­ples with 10z replacement ratio exhibit the higheststrength since early curing time. The sudden strength de­velopment with time is not found for all replacement ra­tios. This ˆnding is diŠerent from concrete technologywhere the role of ‰y ash as a pozzolanic material comesinto play after a long curing time (generally after 60days). In other words, the strength of concrete mixedwith ‰y ash is higher than that without ‰y ash after about60 days of curing.MICROSTRUCTURAL TEST RESULTSFig. 6. Strength versus water content relationship of the CFA blendedcement stabilized silty clay at diŠerent replacement ratios and 60days of curingFig. 7. Strength versus water content relationship of the CFA blendedcement stabilized silty clay at C:F80:20 and diŠerent curing timesTable 4.SEM PhotosFigures 9 through 12 show SEM photos of the OFAand the CFA blended cement stabilized clay compacted atw1.2OWC (w21z) and cured for 28 and 60 days atdiŠerent replacement ratios. The ‰y ash particles areFig. 8. Relationship between strength development and replacementratio of the CFA blended cement stabilized silty clay at diŠerentcuring timesStrength of the blended cement stabilized clay at diŠerent replacement ratios and curing timesCompressive strength (kPa)Curing time(days)C:F100:0C:F90:10C:F80:20C:F70:30C:F60:40ClassiˆedOriginalClassiˆedOriginalClassiˆedOriginalClassiˆedOriginal73460}1823465}1423479}3593262}1883257}3013174}2153082}2582803}2272669}195285867}1735916}4715362}4985817}3435701}3785685}3415263}1784821}2574634}153606872}3107138}1096828}4866918}1406437}1866627}1896537}1045821}3015840}141907586}3217353}1037691}2357353}1977043}2177038}1606753} 806316}1006181}1401208432}1118272}4958566}4398271}3368182}4688771}1497901}4587300}3927200}333ROLE OF FLY ASHFig. 9.SEM photos of the OFA blended cement stabilized clay at diŠerent replacement ratios after 28 days of curingFig. 10.SEM photos of the OFA blended cement stabilized clay at diŠerent replacement ratios after 60 days of curingclearly shown among clay­cement clusters especially for30z replacement ratio (C:F70:30) for both curingtimes and both ˆneness (Figs. 9(a), 10(a), 11(a) and9112(a)). It is noted that the hydration products growingfrom the cement grains connect ‰y ash particles and clay­cement clusters together. For the same curing time, the92HORPIBULSUK ET AL.Fig. 11.SEM photos of the CFA blended cement stabilized clay at diŠerent replacement ratios after 28 days of curingFig. 12.SEM photos of the CFA blended cement stabilized clay at diŠerent replacement ratios after 60 days of curingˆner the ‰y ash, the lesser the pore space. It is moreoverfound that some of the surfaces of ‰y ash particles arecoated with layers of amounts of hydration products.However, they are still smooth with diŠerent curingtimes. This ˆnding is diŠerent from concrete technologywhere the precipitation in the pozzolanic reaction is indi­ROLE OF FLY ASH93Fig. 14. Pore size distribution of the CFA blended cement stabilizedclay at diŠerent replacement ratios and curing timesFrom this observation, it is thus possible to conclude thatthe pozzolanic reaction is minimal for strength develop­ment in the blended cement stabilized clay.Fig. 13. Pore size distribution of the OFA blended cement stabilizedclay at diŠerent replacement ratios and curing timescated by the etching on ‰y ash surface (Fraay et al., 1989;Berry et al., 1994; Xu and Sarker, 1994; Chindapasirt,2005). This is because the input of cement in concrete ishigh enough to produce a relatively high amount ofCa(OH)2 to be consumed for pozzolanic reaction. Itswater to binder ratio (W/B) is generally about 0.2–0.5,providing a strength higher than 30 MPa (30,000 kPa) at28 days of curing, whereas for ground improvement, theW/B is much lower. In this study, it is 2.1 (W/B21z/10z) at 1.2OWC, which is about 4–10 times diŠerence.Pore Size DistributionFigures 13 and 14 show the pore size distribution of theOFA and the CFA blended cement stabilized clay atdiŠerent curing times and replacement ratios. It is evidentthat for a particular curing time, pore size distribution ismainly controlled by ‰y ash ˆneness. The total porevolume is lesser for the CFA blended cement stabiliza­tion. This observation is in agreement with the SEM one.In case of the OFA blended cement stabilization, the totalpore volume increases with replacement ratio. This is be­cause OFA particles are coarser than the clay and PC par­ticles. The increase in replacement ratio thus increasescoarser particles, resulting in the increase in pore volume.The same is not for the CFA blended cement stabiliza­tion. The pore size distribution for all replacement ratiosis almost identical since the grain size distribution and D50of PC and CFA are practically the same. Even with thedistinct diŠerence in pore size distribution, the strengthsof the CFA blended cement stabilized clay are slightlyhigher than those of the OFA blended cement stabilizedclay. Moreover, it is found that for all curing times,94HORPIBULSUK ET AL.although total pore volumes of the OFA blended cementstabilized clay at C:F90:10 are higher than those of thecement stabilized clay (without ‰y ash), the strengths ofthe OFA blended cement stabilized clay are higher. Thisimplies that the strength of blended cement stabilized clayis not directly dependent upon only pore size distribution.However, it might control permeability and durability.With time, the total pore and large pore (À0.1 micron)volumes decrease while the small pore (º0.1 micron)volume increases. This is due to the growth of cementiti­ous products ˆlling up pores. The growth of cementitiousproducts would be depicted in the next section.Last but not least, it is notable that the small pore(º0.1 micron) volumes for both OFA and CFA blendedcement stabilized clay are higher than those of the cementstabilized clay. This implies that a number of large clay­cement clusters possessing large pore space reduce whenboth OFA and CFA are utilized. In other words, the ‰yashes disperse large clay­cement clusters into smallclusters, resulting in the increase in small pore volume.Thermal Gravity AnalysisEŠect of water content on cementitious products isclearly explained by Table 5, which shows Ca(OH)2 ofthe OFA and the CFA blended cement stabilized clay at10z replacement ratio (C:F90:10) after 7 days of cur­ing for diŠerent water contents. For both OFA and CFAblended cement stabilization, the highest Ca(OH)2(highest degree of hydration) is at w1.2OWC, associ­ated with the highest strength. At w17z (OWC), eventhough its dry unit weight is the highest (total porevolume is lowest) (Fig. 5), its Ca(OH)2 is lesser than thatat 1.2OWC, hence lower strength. The water content of1.2OWC is thus regarded as the suitable state where theair pore volume is minimal. For wº1.2OWC, the airpores cause the discontinuity in the cement paste resultingin less hydration. The greater the air pore volume (thelower the water content), the lower the strength. ForwÀ1.2OWC (degree of saturation close to 1.0), the soilwater/cement ratio, w/C governs the strength develop­ment (Miura et al., 2001; Horpibulsuk et al., 2003, 2005,2006). As such, for the same input of cement, the higherthe water content, the lower the hydration products.Table 6 shows Ca(OH)2 of the blended cement stabi­lized clay at w1.2OWC for diŠerent replacement ratiosand curing times. It is found from Tables 5 and 6 thatCa(OH)2 increases with ˆneness for all water contents,replacement ratios, and curing times. For a particularwater content and curing time, Ca(OH)2 for both OFAand CFA blended cement stabilized clay decreases withreplacement ratio only when the replacement ratio is inexcess of a certain value. This ˆnding is diŠerent fromconcrete technology in which Ca(OH)2 decreases sig­niˆcantly with the increase in ˆneness and replacementratio (Berry et al., 1989; Sybertz and Wiens, 1991; Harriset al., 1987; Chindapasirt, 2005, 2006; and others) due topozzolanic reaction. The highest Ca(OH)2 is at 10zreplacement ratio (C:F  90:10) for both OFA and CFAblended cement stabilization and for all curing times. Forreplacement ratios higher than 10z, Ca(OH)2 decreaseswith replacement ratio. Ca(OH)2 at 20z replacement ra­tio is almost the same as that at 0z replacement ratio.This ˆnding is associated with the strength test resultsTable 6. Ca(OH)2 of the blended cement stabilized clay at diŠerentreplacement ratios and curing timesCuringtime(days)7286090Table 5. Ca(OH)2 of the blended cement stabilized clay at diŠerentwater content for C:F90:10 after 7 days of curingWater content (z)14z17z21z24z14z17z21z24z(0.8 OWC)(OWC)(1.2 OWC)(1.4 OWC)(0.8 OWC)(OWC)(1.2 OWC)(1.4 OWC)Fly ashWeight loss (z)Ca(OH)2 (z)CFACFACFACFAOFAOFAOFAOFA1.591.631.701.621.571.611.651.556.526.726.976.676.436.616.776.35120Ca(OH)2 (z)ReplacementratioC:FFly ashTest(CombinedeŠect)100:090:1080:2070:30—CFACFACFA6.676.976.796.396.676.005.344.670.000.971.451.7290:1080:2070:30OFAOFAOFA6.776.666.126.005.344.670.771.321.45100:090:1080:2070:30—CFACFACFA6.796.966.816.576.796.115.434.750.000.851.381.8290:1080:2070:30OFAOFAOFA6.836.766.466.115.434.750.721.331.70100:090:1080:2070:30—CFACFACFA6.827.166.926.686.826.145.464.770.001.021.461.9190:1080:2070:30OFAOFAOFA6.896.816.536.145.464.770.751.351.76100:090:1080:2070:30—CFACFACFA7.077.286.946.677.076.365.664.950.000.911.281.7290:1080:2070:30OFAOFAOFA7.076.836.626.365.664.950.711.171.67100:090:1080:2070:30—CFACFACFA7.087.296.966.707.086.375.664.960.000.921.301.7490:1080:2070:30OFAOFAOFA7.096.856.686.375.664.960.721.191.72InducedHydration (dispersingeŠect)ROLE OF FLY ASHthat the 10z replacement ratio gives the highest strengthand the strengths for 0z and 20z replacement ratios arepractically the same for all curing times. It is thus con­cluded that cementious products mainly control thestrength development. In other words, the strengths ofthe blended cement stabilized clay having diŠerent mixingcondition (binder content, replacement ratios, and curingtime) could be identical as long as cementitious productsare the same.From SEM and MIP observation, it is possible to men­tion that the role of ‰y ash is to disperse large clay­cementclusters formed when interacted with water into smallerclusters. The higher the replacement ratio, the better thedispersion. Consequently, the reactive surfaces increase,resulting in the increase in cementitious products as illus­trated by dispersion induced Ca(OH)2 (vide Table 6). It isthe diŠerence in Ca(OH)2 of the blended cement stabi­lized clay due to the combined eŠect (hydration and dis­persion) and due to hydration. Ca(OH)2 due to combinedeŠect is directly obtained from TG test on the blended ce­ment stabilized sample. Ca(OH)2 due to hydration is alsoobtained from TG test on the cement stabilized samplehaving the same cement content as the blended cementstabilized sample. For simplicity, Ca(OH)2 due to hydra­tion at any cement content can be estimated from knownCa(OH)2 of cement stabilized clay at a speciˆc cementcontent by assuming that the change in the cementitiousproducts is directly proportional to the input of cement(Sinsiri et al., 2006). Thus, Ca(OH)2 due to hydration (H)for any replacement ratio at a particular curing time is ap­proximated in the form.ØF100HT~ 1|»(3)where T is known Ca(OH)2 of the cement stabilized clay(0z replacement ratio) obtained from TG test, and F isthe replacement ratio expressed in percentage. Sinsiri etal. (2006) have shown that Ca(OH)2 of the cement pastewith ‰y ash is always lower than Ca(OH)2 of the cementpaste without ‰y ash, resulted from Ca(OH)2 consump­tion for pozzalanic reaction. The same is not for theblended cement stabilized clay. It is found that Ca(OH)2due to combined eŠect is higher than that due to hydra­tion for all replacement ratios and curing times. The dis­persion induced Ca(OH)2 increases with ‰y ash ˆnenessand the replacement ratio for all curing times.DISCUSSIONCement, ‰y ash, and soil are particulate materials,which are composed of individual units. The particulatematerials can be regarded as either non­interacting or in­teracting materials dependent upon the absence orpresence of physico­chemical interactions with the pore‰uid. In case of the blended cement stabilization, cementand clay are interacting materials with water. Fly ash,silt, and sand are non­interacting materials, primarily dueto their low speciˆc surface and non­electrical nature ofsurfaces.95Fig. 15. Strength development in the CFA blended cement stabilizedclay and its generalizationSince cement and clay are interacting materials, whenclay is mixed with cement and water, clay and cementparticles would group together into large clay­cementclusters. Fly ashes as a non­interacting material can dis­perse the large clay­cement clusters into smaller clusters,resulting in the increase in reactive surfaces, and hencecementitious products.Figure 15 shows the strength development (qD/q28)with time of the CFA blended cement stabilized claywhere qD is the strength after D days of curing and q28 isthe 28­day strength. It is found that the relationship ispractically the same for all replacement ratios and veryclose to that proposed for cement admixed clay by Hor­pibulsuk et al. (2003). This evidence reinforces the con­clusion from the microstructural observation thatstrength development with time for stabilization with andwithout ‰y ash is the same and mainly controlled byhydration with minimal pozzolanic reaction.From the present investigation, it is of interest to dis­cuss herein that strength development in the blended ce­ment stabilized clay is dependent upon the cementitousproducts mainly due to combined eŠect (hydration anddispersion). Cementitious products due to hydration aregoverned by the input of cement while those due to dis­persion mainly by ‰y ash content (replacement ratio).10z replacement ratio shows the highest cementitiousproducts due to hydration but lowest cementitiousproducts due to dispersion. Whereas 40z of the replace­ment ratio shows the lowest cementitious products due tohydration but highest cementitious products due to dis­persion. The 10z replacement ratio exhibits higheststrength and can be regarded as the most eŠective because96HORPIBULSUK ET AL.the cementitious products due to the combined eŠect arethe highest. As such, it is worthwhile to mention that be­sides the application of ‰y ash as a replacement materialas studied in this paper, it can be considered as a dispers­ing material added into cement to increase the cementiti­ous products, and hence strength. The suitable additionalratio is however needed to be further examined.CONCLUSIONSThis paper presents the role of ‰y ash on the strengthand microstructure development in the blended cementstabilized clay. The following conclusions can be ad­vanced from this study.1. The ‰occulation of clay particles due to the cationexchange process is controlled by cement content,regardless of ‰y ash content. It results in the in­crease in dry unit weight with insigniˆcant change inliquid limit. Hence, OWCs of stabilized and un­stabilized silty clay (low swelling clay) are practical­ly the same.2. The surfaces of ‰y ash in the blended cement stabi­lized clay are still smooth for diŠerent curing timesand ˆneness, suggesting that pozzolanic reaction isminimal. Fly ash is considered as a dispersingmaterial in the blended cement stabilized clay. Thisis diŠerent from the application of ‰y ash as a poz­zolanic material in concrete structure in whichCa(OH)2 from hydration is much enough to be con­sumed for pozzolanic reaction.3. From the microstructural investigation, it is con­cluded that the role of ‰y ash as a non­interactingmaterial is to disperse the cement­clay clusters withlarge pore space into smaller clusters with smallerpore space. The dispersing eŠect by ‰y ash increasesthe reactive surfaces, and hence the increase indegree of hydration as clearly illustrated by the in­crease in the induced Ca(OH)2 with replacement ra­tio and ˆneness.4. The increase in cementitious products with time isobserved not only from TG test results, but alsofrom the pore size distribution. With time, the largepore (À0.1 micron) and total pore volumesdecrease while the small pore (º0.1 micron)volumes increase. This shows the growth of thecementitious products ˆlling up the large pores.5. The macro­scale observation on the strength de­velopment with time reinforces the conclusion fromthe microstructural observation that the strengthdevelopment of blended cement stabilized clay ismainly due to hydration with minimal pozzolanicreaction.6. The strength development in the blended cementstabilized clay is dependent upon cementitiousproducts mainly attributed to the combined eŠect(hydration and dispersion). The water content ofabout 1.2OWC and the 10z replacement ratio areregarded as the most eŠective mixing condition forthe stabilization, exhibiting the highest cementitiousproducts.ACKNOWLEDGEMENTThe authors would like to acknowledge the ˆnancialsupport provided by the Commission on Higher Educa­tion (CHE) and the Thailand Research Fund (TRF) underthe contract MRG5080127. Facilities, equipments andˆnancial support provided from Suranaree University ofTechnology are appreciated. The authors are indebted toDr. Theerawat Sinsiri, School of Civil Engineering, Sur­anaree University of Technology for his technical advicein cement and concrete technology.REFERENCES1) Abduljauwad, S. N. (1995): Improvement of plasticity and swellpotential of calcareous expansive clays, Geotechnical EngineeringJournal, Southeast Asian Geotechnical Society, 26(1), 3–16.2) Aylmore, L. A. G. and Quirk, J. P. (1960): Domain or turbostraticstructure of clays, Nature, 187, 1046.3) Bell, F. G. 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• ログイン

• タイトル
• Equations of State in Soil Compression Based on Statistical Mechanics
• 著者
• Masaharu Fukue・C. N. Mulligan
• 出版
• Soils and Foundations
• ページ
• 99〜114
• 発行
• 2009/02/15
• 文書ID
• 21173
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 99–114, Feb. 2009EQUATIONS OF STATE IN SOIL COMPRESSIONBASED ON STATISTICAL MECHANICSMASAHARU FUKUEi) and CATHERINE N. MULLIGANii)ABSTRACTThere have been many theories and interpretations of the compression process for soil. However due to the complex­ity of the compression process for a wide range of soil states, i.e., suspended, liquid, plastic, semisolid or solid stateslike rock, the interaction between the states or the transition from one state to another, there have often been misun­derstandings and di‹culties regarding the application of these theories. In this study, an ideal particle distribution inan ideal ground system is derived to ˆnd the equation of state, a general compression process for a wide variety of soiltypes and states. In other words, it is a general relationship between the void ratio and eŠective overburden pressure interms of interparticle energy and is thus a ``law of soil.'' The principle is based on the law of energy distribution instatistical mechanics. The derived equations can be applied to describe the void ratio proˆle and the compression proc­ess of various soils. Case studies show that this theoretical approach agrees well with the experimental data obtainedfor several soils such as sediments, Mexican City clay, sand and others.Key words: compression, interparticle energy, law of soil, particle distribution, void ratio (IGC: D5)view points (Yoon et al., 2004; Park et al., 2004; Giasi etal., 2003; Tsuchida, 2001; Burland, 1990; Butterˆeld,1979). However, at present, due to its complexity, theremay be some misunderstandings for the compression ofsoils.It is known that the state of ˆne soils changes in termsof water content. The boundaries have been provided bythe liquid, plastic and shrinkage limits. This concept canbe extended from the suspended state (Imai, 1980; Imai,1981; Been and Sill, 1981) to sedimentary rock. Thus, soilparticles with the entire soil system are varied in their na­ture, from suspended, to deposited, consolidated andcemented states, though the boundaries are not oftenclear. For example, the deˆnition of the bottom surfaceunder marine conditions has not been established yet, be­cause it is di‹cult to describe the general aspect of theboundary between suspended and deposited states there.Fukue et al. (1987) found that there is a very loose sedi­ment layer on the top of the sea bottom from sedimenta­tion experiments and ˆeld observations. Since the voidratio of this layer decreases abruptly with depth, itscharacterization is often di‹cult. Fukue et al. (1987) useda concept of average void ratio to characterize this thintop layer and found that the void ratio decreases to halfof the surface void ratio at a depth of about 5 cm. Thus,this layer has diŠerent characteristics from the suspendedsolid systems or sediments, possibly according to thetypes of interactions. Herein, suspended solid systemsconsist of solid particles without contact and interactingINTRODUCTIONThe process of soil consolidation in nature takes placeover a period of many thousands of years. This processcan be expressed in terms of volume, pressure and energystate, i.e., interparticle energy. Di‹culties within soilmechanics exist because of the varied nature of the soilstate in terms of interparticle energy, and because thereare no ``equations of state'' for soil materials. This ismainly because of the compressible and irreversible na­ture of the interactions of soil particles.The ˆrst impression is that the void ratio (e) proˆle inthe soil is determined by the overburden pressure (p).However, this pressure is also controlled by the fabric ofthe soils, which can be related to the void ratio. There­fore, many studies have been carried out to describe andexpress the e­log p relationship for soils (Skempton,1944; Nishida, 1959; Oswald, 1980; Fukue and Okusa,1987). Because this relationship can provide the state forsoils in terms of volume (volume ratio or void ratio) andpressure (usually eŠective overburden pressure), it couldbe used to describe the shear strength characteristics(Schoˆeld and Wroth, 1968; Leroueil and Vaughan,1990; Burland, 1990). Nevertheless, the relationships be­tween volume and pressure obtained have been alwaysempirical, and consequently the interpretations have beenmade based on empirical relationships. This has providedmany theories and interpretations for the compressionprocess and compression index of soils from diŠerenti)ii)Department of Marine Civil Engineering, Tokai University, Shizuoka, Japan (fukue—scc.u­tokai.ac.jp).Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Quebec, Canada.The manuscript for this paper was received for review on May 9, 2008; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.99100FUKUE AND MULLIGANonly by repulsive forces such as dispersed montmoril­lonite clay particles in distilled water, while sediments ex­hibit an ordinary structure similar to soil. Furthermore, itwas found that the void ratio of the top layers of sedi­ments (about 5 cm) can be given by a simple empiricalformula using the average void ratio system of sediments(Fukue and Okusa, 1987; Fukue et al., 1987).In this study, the empirical formula, an average voidratio–depth relationship obtained previously, is shown tobe derived theoretically from statistical mechanics andcan be applied to other states of soils, such as suspendedsoils, ordinary plastic soils, peat, sand, etc. However, itsapplication shows that the compression curve, e­log pcurve, will not have a straight portion (Fukue and Okusa,1987), although it seems to be so, and that the compres­sion curve through the boundaries from one to anotherstate would be irregular (Imai, 1980; Imai, 1981). Thus,the phenomena observed experimentally have often con­fused engineers and researchers. Since soil is a compressi­ble and irreversible material, it is quite di‹cult to treat itthermodynamically. This is because the compressionprocess produces dissipation energy, and unknown varia­bles such as entropy have to be introduced. This may beone of the reasons why the state equation of soil has notbeen developed.In this study, soil systems consisting of elements withdiŠerent void ratios are considered. Considering thateach element is subjected to the corresponding overbur­den pressure, the system can be converted to the compres­sion process, as illustrated in Fig. 1. The advantage inutilizing the soil system instead of a process is to minimizethe number of unknown variables.Characterizing the overburden pressure to void ratio,the relationship (e­log p) between the void ratio and theeŠective overburden pressure becomes unique. This iswhy the void ratio­pressure relationships for various soilssuch as sands and clays are similar. This also indicatesthat there is a ``law of soil'' that exists in a similar way tothe ``law of atmosphere'' established in statisticalmechanics (Reif, 1965).In this study, an ideal particle distribution is used toderive a general compression process. Void ratio(volume), pressure and energy terms are taken into con­Fig. 1.Concept of process and system in a soil columnsideration. Since the equation of state is the most basicvolume–pressure relationship for substances, to describethe state during compression, as is well known for idealgases, the development of a relationship makes it easierfor soils to be treated in a more scientiˆc manner. Casestudies are also used to illustrate the applicability of thisapproach.BACKGROUNDGeotechnical engineering has often been based on thetesting of sampled soils. The basic concept is that thereloading for sampled soil will return it to the in­situ con­dition, because it is assumed that the compression index,Cc is constant for the original and reloaded soil, as illus­trated in Fig. 2. This concept has been widely acceptedand used for the prediction of settlement and the conceptof shear strength, though further modiˆcations have beenmade using reconstituted soils which can provide a stan­dard compression line, such as an intrinsic compressionline (ICL) and a sedimentation compression line (SCL)for shallow marine deposits (Butterˆeld, 1979; Leroueiland Vaughan, 1990; Burland, 1990). The ICL and SCLmay correspond to compression for remoulded soils andnatural compression lines, respectively. This is also animportant subject in this study and will be discussed later.On the other hand, the empirical facts have shown thatthe samples taken from diŠerent depths have diŠerentvalues of Cc, as illustrated in Fig. 3(a) and (b). As aresult, the compression index becomes a function of theinitial void ratio prior to the loading for testing. Manyreferences such as Nishida (1959) and Oswald (1980) haveshown that this relationship is close to linear as knownempirically. If the empirical relationships are correct,there might be an inconsistency between Figs. 2 and 3.Researchers and engineers who realized this incon­sistency have thought that the inconsistency resultedfrom the disturbance of the samples (Mesri et al., 1975;LaRochelle et al., 1980; Leroueil and Vaughan, 1990).Therefore, relatively low values of Cc have been obtainedfor the disturbed samples. It is true that for highlycemented or sensitive soils, the value of Cc will decreaseFig. 2. Illustration of a general e­log p relationship in soil compres­sion101EQUATION OF STATEFig. 3.Hypothetical compression behaviour of sampled soils after stress release and prediction of natural compressionwith disturbance. This can be explained by strain soften­ing during compression, as explained in later sections.However, the dependency of the initial void ratio on Cc isnot due to the disturbance of the sample, but is an inher­ent property of soils. This will be discussed in this paper.It has been shown that there is a good relationship be­tween Cc and the initial void ratio adjusted to the watercontent of the liquid limit (Burland, 1990). In this case, itcan be emphasized that Cc will vary as if the relationshipis a function of soil type. However, if Cc is dependent onthe initial void ratio alone, then the in‰uence of soil typewill vanish. This becomes clear by the existence of auniversal formula representing the e­log p relationshipfor varied states and types of soils. The above­mentionedis very important not only for prediction of soil proper­ties and behaviour including settlement and shearstrength, but also for the fundamentals in soil mechanicsand geotechnical engineering.The authors consider that the present situation mayresult from a lack of theoretical examination based on theempirical data. As mentioned earlier, the application ofthermodynamics is not practical, because of manyunknown factors. Statistical mechanics can be easier toapply for a soil system for many reasons, as shownthroughout this paper.FUNDAMENTAL RELATIONSHIP BETWEENPRESSURE, VOLUME AND POTENTIAL ENERGYThe conventional relationship in geotechnical engineer­ing for void ratio and pressure is often given in the fol­lowing form:gszp1{e(1)Where p is the eŠective overburden pressure, e is the voidratio, gs is the submerged unit weight of the solid and z isthe soil depth under consideration. In Eq. (1), a criticalassumption is that the void ratio is constant over depth.Therefore, this will not be applied to the variable void ra­tio e.Therefore, a generalized relationship between eŠectivepressure and void ratio can be expressed as:p(z)gsz1{ še(z)(2)where p(z) is the overburden pressure as a function of thedepth z, and še(z) is not constant, but variable with theproˆle of the average void ratio of a soil column withdepth z, as shown in Fig. 4. In Fig. 4, the diŠerence be­tween average void ratio and true void ratio with depth zis clear. Thus, the proˆle of the overburden pressure is in­dependent of the true void ratio, but is directly dependenton the average void ratio proˆle. If the proˆle of še is de­termined, then the proˆle of p can be obtained when thee­p or e­log p relationship is derived from the e and šerelationship (Fukue and Okusa, 1987).e(z)dzfše(z)z(3)Equation (2) can be rewritten as p(z) (1{ še(z))mgz/Vs,where Vs is the volume of the solid portion of the soil andcan be assumed to be unity, because the unity in the term(1{ še(z)) resulted from the assumption that the volume of102FUKUE AND MULLIGANFig. 4. Deˆnition of average void ratio when the true void ratio varieswith depththe solid portion is unity. Therefore, the term (1{ še(z))provides the volume of a soil element and mgz is propor­tional to the potential energy of the soils having a volumeof 1{ še(z). Although one may think that mgz is thepotential energy, this is not the case because z does not in­dicate the depth or height of the center of gravity. Thiswill be discussed in a later section.The relationship can be represented by Eq. (2)?p(z)V(z)mgz(2)?Fig. 5.or more simply as,pVE?pFlowchart of the theoretical development in this study(2)!Where E?pmgz. This is proportional to the potentialenergy of the soil element. Equation (2)! suggests that thecharacteristics of the soil volume change can be expressedby pVvariable, and can be compared to the equation ofstate for an ideal gas, i.e., pVconstant, in which themolecules have no interaction. In fact, this comparison isnot essential, because the former deals with a soil systemand the latter describes an ideal gas element. However,even if a soil element was used, the relationship for a soilelement can be more complicated due to the existence ofthe interactions between particles. Thus, the diŠerencebetween the two substances may be due to the interactionbetween particles for soil or gas molecules. If the descrip­tion here is true, Eq. (2) would be the most fundamentaland substantial expression, i.e., the state equation of soil.This is one of the most important aspects in this studyand this will be established through this study. Therefore,one of the objectives of this study is to examine Eq. (2)?or Eq. (2)! to establish the še(z) formula as an equation ofstate for soil.ENERGY DISTRIBUTIONTheoretical development in this study is based onstatistical mechanics which may not be familiar to mostreaders. Therefore, it seems that the derivation of equa­tions is a little complicated, as shown in Fig. 5.The ˆrst problem is to derive the most probable poten­tial energy distribution of particles. This is achieved byusing a constant total potential energy with elevation andthe total number of particles. In this formulation, the in­terparticle energy is included in the potential energy bychanging the mass of the particles. It will appear in theparameter based on the experimental results.The next step is to transform the number of particles tovoid ratio. The concept of average void ratio system isconveniently used in this conversion. The average voidratio system is also theoretically converted into the truevoid ratio system. These void ratios are expressed as afunction of potential energy which is also a function ofdepth if it is needed to use. The eŠective pressure is ex­pressed as a function of the void ratio. Thus, the relation­ship between pressure, volume and potential energy is ob­tained.Probability of Particle DistributionConsider a system with a large number of particles nand that the particles are enclosed in a large number ofimmobile containers as shown in Fig. 6. The ˆrst contain­er at the bottom contains n1 number of particles at anenergy level of e1. The next level has n2 particles at anenergy level of e2 and so on. Since the total number ofparticles is n, the number of possible distributions, V,based on Maxwell­Boltzmann statistics, is given by:Vn!/n1!n2!n3! . . . nr! . .(4)Now, the most probable distribution with a constant totalenergy E can be obtained at the maximum value of V. Ifone particle is moved from the second container to thethird container and if another particle is moved from the103EQUATION OF STATEFig. 6.Particle distribution into piled boxessecond to the bottom container, the new distribution isgiven by:V?n!/(n1{1)!(n2|2)!(n3{1)!n4! . . . nr! . .(5)The ratio of Eq. (5) to Eq. (4) is then:n2(n2|1)(n1{1)(n3{1)Then if n1, n2 and n3 are very large numbers, Eq. (6)becomes:(7)Since there is no change in energy in the system, then theratio V?/V in Eq. (7) must be equal to 1 and then thesteady state of the system can be obtained as:n1/n2n2/n3(8)Including the other particles will give the followingrelationship:n1/n2n2/n3n3/n4n4/n5. . .ni/njnj/nk . . . (9)dnr ln nr{(nr/nr)dnrt{0d ln V0|S s|S s(ln nr{1)dnrtTherefore, the most appropriate proˆle can be given byEq. (10) and is illustrated in Fig. 7:nrn1 exp (|mer)(10)where er (r1, 2, 3 . . . r . . .) is the energy state corre­sponding to nr and m is a physical constant to be discussedlater.Partition FunctionThe Stirling approximation for a large n for Eq. (4) canbe expressed as:(12)If no energy is added, then Eq. (12) is equal to 0 (Tolman,1979). If a number of particles (dnr) are added to a groupof particles with an energy level of er, then the energy lev­el of that group changes from nrer to (nr{dnr)er. If it is as­sumed that the total energy is constant, then :dES erdnr0(13)where E is the total energy.If the total number of particles is constant, then:rn3n1 exp (|me3),(11)If an increment of ln V is taken, and n! and Snr can be as­sumed to be constants, then:S dn 0This leads directly to:n2n1 exp (|me1),ln Vln (n!)|S ln (nr!)ln (n!)|S nr ln nr{S nr(6)(n2)2V?/Vn1n3Probable distribution of particlesln (n!)|S (nr ln nr|nr)n!/n1!n2!n3! . . . nr! .V?/V(n1{1)!(n2|2)!(n3{1)!n4! . . . nr!Fig. 7.(14)To maximize V in Eq. (12), the following equation is re­quired:S ln n dn 0rr(15)Equations (13), (14), and (15) do not include the totalnumber of particles and the total energy E. To relatethese equations then to these parameters, two constantsmust be introduced, the dimensionless l and m which is areciprocal of the energy term. Multiplying Eq. (13) by mgives:S me dn 0rrSimilarly, multiplying Eq. (14) by l leads to:(16)104FUKUE AND MULLIGANS l dn 0(17)rAdding Eqs. (15), (16) and (17) gives:S (ln n {l{me )dn 0rrr(18)following equation is obtained:TdSS erdnr(29)On the otherhand, from Eqs. (27), (12) and (20), the leftside of Eq. (29) is then:TdST(kd ln V)orln nr{l{mer0(19)Rearranging Eq. (19), gives:nrexp (|l) exp (|mer)(20)where the total number of particles isnn1{n2{n3{. . . nr{. . .exp (|l)sexp (|me1){exp (|me2){. . .tnexp (|l)S exp (|mer)(21)Then,exp (|l)n/S exp (|mer)(22)By substituting Eq. (22) into Eq. (20), the following fun­damental equation used in statistical mechanics (Reif,1965) is obtained:n exp (|mer)S exp (|mer)(23)nrn exp (|mer)P(24)PS exp (|mer)(25)nrwhere,which is the partition function for the energy system.Physical Meaning of mThe change in internal energy of a substance resultingfrom a volume change is thermodynamically expressed bydETdS|pdV(26)where E is the internal energy, T is the absolute tempera­ture, S is the entropy, p is the pressure and V is thevolume. S is deˆned by:Sk ln V(27)where k is the Boltzmann constant. From the microscopicpoint of view, the internal energy equals Sernr. There­fore, the change in internal energy is given as:dES erdnr{S nrder(28)A Comparison of Eqs. (13) and (28) indicates that the sec­ond term on the right hand side of Eq. (28) is due to avolume change, because the number of particles remainsconstant. Therefore, the ˆrst term of the right side of Eq.(26) corresponds to the ˆrst term of Eq. (28). Thus theØ»kT« |S ln sexp (|l) exp (|me )tdn $kT« |S s|l|me tdn $(30)kT |S ln nrdnrrrrrFrom Eqs. (17) and (30), thenTdSkTm S erdnr(31)From Eqs. (29) and (31), the following importantrelationship is obtained:m1/kT(32)If this is substituted into Eq. (24), then the result is thewell known energy distribution of Maxwell­Boltzmann:n exp (|er/kT )Pnr(33)where nr is the number of particles at an energy level of er.Particle DistributionFor a given soil system, the partitioning function, P ex­pressed by Eq. (25) is a constant value. Therefore, the dis­tribution of particles given by Eq. (33) is governed by theenergy level, e, where n, k and T are regarded as con­stants.If a particle exists at height h from ground level, asshown in Fig. 7, and assuming that there are no interpar­ticular forces acting between the particles, the potentialenergy is then mgh, where m is the mass of the particleand g is the gravitational acceleration. However, the as­sumption above cannot be valid, since there are particleinteractions. By using the concept of imaginary particles,as described later, this can be corrected. At a potentialenergy level of er, the number of particles in a soil volumewith a unit area and thickness of dh is expressed as:nr(h)dhn exp (|mgh/kT )dhP(34)Using the h­z relationship where h(zc|z) as shown inFig. 8, then Eq. (34) becomes:n exp (|mg(zc|z)/kT )dzPnr(z)dz(35)where zc is the total height or thickness of the soil groundsystem, which can represent the ``size of the system.''105EQUATION OF STATEbitrary void ratio at depth z, a function F?r is deˆned as:DISTRIBUTION OF VOID RATIOAverage Void Ratio SystemFor a system with equi­dimensional particles ofvolume, Vp, for each particle, then the average void ratioin a soil column (Fig. 4) which has a unit area A andthickness of z, is expressed as:Vv Az|VsVsVsše(z)(36)where Vs and Vv are the volumes of the solid and porespaces, respectively. Expressing the number of particlesin the soil column by N, then:VsNVp(37)From Eqs. (36) and (37),Azše(z)|1NVp(38)Average Void Ratios as Boundary ConditionsThe void ratio e0 in which z is very small (dz) is depend­ent on the soil structure. The value is assumed to be an ar­bitrary constant and herein accords with the surface voidratio, as shown in Fig. 8. However, in this study, e0 canbe deˆned for any initial void ratio prior to loading. Thevoid ratio of a soil column whose thickness varies with zis deˆned as the average void ratio of the system še. As faras the void ratio decreases down to the emin at zzc, thevoid ratio of the total column (ec) will be the minimumaverage void ratio, as shown in Fig. 8, and is thus deˆnedas the minimum average void ratio.a) Surface void ratio, eoThe void ratio of a very thin surface soil layer, withthickness of zo(dz) is deˆned as the surface void ratio.Expressing the number of particles as No, the surface voidratio is deˆned as:(39)b) Minimum average void ratio, ecAzcec |1N c Vpše(z)|eceo|ec(41)Substituting Eqs. (38) to (40) into Eq. (41), gives the fol­lowing:No(Ncz|Nzc)(42)F?rN(Nczo|Nozc)To derive Fr in terms of potential energy, the distributionof particles given by Eq. (35) can be used. For example,the number of particles in a column of thickness z can beexpressed in two ways, i.e., by integrating Eq. (35) overdepth and by using the average number of particles at thecenter of gravity of the column as shown in Fig. 9. Theintegration gives:zcwhere zÀ0. If z is taken between 0 and zc for a given soilsystem as shown in Fig. 8, then še will vary from a particu­lar value at the surface to the average void ratio for theentire soil system. The two extreme void ratios at z0and zzc, then will be used as the boundary conditions toobtain a relationship between the average void ratio andthe energy term.Azoeo|1NoVpF?rNCnfexp s(|mgh)/kTtdhzc|zCn(kT/mg)[exp s|mg(zc|z)kTt|exp mgzc/kT )]where C is a constant which equals the reciprocal value ofthe partition function. It is noted that the partition func­tion P expressed by Eq. (25) is obviously constant for agiven soil energy system.Using the average number of particles, the number ofparticles contained in the column with a depth of z isgiven by:N šnzCnz exp s|mg(zc|zG)/kTtwhere Nc is the total number of particles at this void ratio.Relationship between Void Ratio and Potential Energya) Average void ratio systemThe size of the system with an average void ratio isgiven by the range from eo to ec which is deˆned as theminimum average void ratio and equals the average voidratio of a total soil system. For a soil state with an ar­(44)where zG is the depth at the center of gravity of thecolumn with depth of z and šn is the average number ofparticles over the depth z.From Eqs. (43) and (44), z can be expressed then as:zkT/mg[exp s|mg(zc|z)/kTt|exp (mgzc/kT )](45)exp s|mg(zc|zG)/kTtSimilarly, No, Nc, zo and zc are also expressed in twoways. By integrating Eq. (35) over depth, from z0 to zzo, No can be expressed as:NoCn(kT/mg)[exp s|mg(zc|zo)/kTt|exp (mgzo/kT )](46)In terms of the average number of particles, the equationis:NoCnzo exp s|mg(zc|zGo)/kTt(40)(43)(47)Where zGo is the depth at the center of gravity of thecolumn of depth zo. From Eqs. (46) and (47), zo can be ex­pressed as:|exp (|mgzc/kT )t]kT/mg[exp s|mg(zc|zo)/kTtexp s|mg(zc|zGo)/kTt(48)zo For the entire system, Nc, after integration from z0 to zzc, can be expressed, as:106FUKUE AND MULLIGANFig. 8.Interpretation of average void ratio systemNoCn(kT/mg)s1|exp (mgzc/kT )tFig. 9.(49)and with the average number at the center of gravity, zGcas:NoCnzc exp s|mg(zc|zGc)/kTt(50)From Eqs. (49) and (50), zc is expressed by:(kT/mg)s1|exp (mgzc/kT )texp s|mg(zc|zGc)/kTtzc (51)b) Relationship between average void ratio and potentialenergySubstituting Eqs. (43), (45), (46), (48), (49) and (51)into Eq. (42) gives:še(z)|ec exp (|mgzG/kT )|exp (|mgzGc/kT )eo|ec exp (|mgzGo/kT )|exp (|mgzGc/kT )(52)Since zGo is close to zero and zGc is relatively large, thenEq. (52) can be simpliˆed into the following:še(z)|ecexp (|mgzG/kT )eo|ec(53)exp (|mgRz/kT )exp (|bz)(54)where RzzG and bmgR/kT.Since bz is smaller than 2.0, as derived by eemin in Eq.(55), the calculated error between Eqs. (52) and (54) is lessthan 0.011z.DISCUSSIONProˆle of Average Void RatioAn evaluation of Eq. (54) was performed experimen­tally by sedimentation tests by Fukue et al. (1987). Theprocedure is brie‰y described in later section. The averagevoid ratios were measured on the sediments in water. Theresults showed that b(mgR/kT ) was almost constantfor a bentonite clay sediment proˆle in sea water asshown in Fig. 10. Therefore, the value of mgR/kT isalmost constant. The true void ratio proˆle shown in theˆgure was converted theoretically using the theoreticalrelationship between average void ratio and true void ra­Potential energy of a particle at the center of gravitytio (Fukue and Okusa, 1987), which shows that the voidratio decreases rapidly within a depth of 5 cm.It was noted that the sediments consolidated undervery low self­weight and that the particles were usually‰occulated. Therefore, the energy system of the sedi­ments would be diŠerent from that of the lower, moreconsolidated sediments. Nevertheless, the derived equa­tions were feasible for expressing the proˆles of both theupper and lower sediments (Fukue et al., 1987; Fukueand Okusa, 1987).Average Void Ratio ­ Overburden Pressure RelationshipAs mentioned previously, the relationship between theaverage void ratio and the overburden pressure is ex­pressed by Eq. (2), while the average void ratio is given inEqs. (53) and (54). Therefore, Eq. (2) provides a state ofthe soil in terms of potential energy of the particles in it.Equation (53) or Eq. (54) or Eq. (2) is applicable as wasshown in Fig. 10.Compression ProcessThe equations theoretically derived from this studywere given experimentally in Fukue and Okusa (1987).The average void ratio was mathematically converted tothe corresponding true (ordinary) void ratio, using Eq.(3), where the true void ratio is deˆned as the convention­al void ratio of an element at an arbitrary depth. The truevoid ratio proˆle is expressed by:e|emin0.119{0.881(1|bz) exp (|bz)eo|emin(55)where e is the void ratio of an element as a function of zand emin is the minimum void ratio of a given element(Fukue and Okusa, 1987). In Eq. (55), if eemin, bzbecomes two. This indicates that bz will change from zeroto two for any soil system. Equation (55) gives a form ofthe void ratio proˆle with depth. To obtain Eq. (55), thefollowing theoretical relationship (Fukue and Okusa,1987) is used:eminec|0.135(eo|ec)(56)The equations obtained are also applied to describe the107EQUATION OF STATEconsolidation behaviour of soils. In order to apply Eq.(55) to this behaviour, the eo is taken as the initial void ra­tio of the soil sample. The void ratio­pressure relation­ship can be obtained using Eqs. (2) and (55) where z is nolonger considered as depth but as a variable to express in­terparticular energy, as described later. The theoretical e­log p relationships obtained from Eqs. (2) and (55) agreewith the experimental results shown in Fig. 11. The bvalue is constant for each loading, unloading and reload­ing. This indicates from an energy point of view that thesoil types are diŠerent for each type of loading since thesize of the energy system is diŠerent as will be describedlater.The set of equations suggests that the e­log p curve maynot overlap at any point if the initial void ratio, e0 isdiŠerent. The lower the initial void ratio, the lower theslope which is conventionally expressed as compressionindex. The empirical relationships between the initialvoid ratio or initial water content and compression indexare well known. This trend is seen in Fig. 11, though thediŠerence between the slopes which result from the diŠer­ent initial void ratios is small.On the other hand, there is the concept that the soilstate will return to the natural, unsampled condition byreloading the sampled soil. This is usually assumed inmany engineering applications, testing soil samples, con­solidation theory and others. However, it is apparently incontradiction to the above mentioned, i.e., dependencyon the initial condition.Thus, there is the possibility that any sampled soil isdiŠerent from soils under natural conditions. Althoughthe in‰uence of the stress release by sampling has oftenbeen explained from the disturbance of the sample, theremay be inherent properties of sampled and natural soilswhich should be understood as the diŠerent soil systemswith diŠerent initial states.Figure 12 shows the compression behaviour of Toy­oura sand and an oil sand. The data of the oil sand wasobtained from Roberts (1964). The lines indicated in theˆgure are theoretical, by assuming initial and minimumvoid ratios and b values. The frictional resistance of theoil sand is weaker than ordinary sands, because of the oilbetween the particles. For ordinary sands, the b value issmaller than that of the oil sand. In fact, the compressionbehaviour of sands is similar to clayey soils (Coop, 1990),which may imply that there is a universal expression forsoil compression.Cubrinovski and Ishihara (2002) stated that emax­emin canprovide valuable and unique information about thematerial properties of sandy soils and it can be particu­larly eŠective in evaluating the potential of compressibili­ty and contractiveness of cohesionless soils. This prob­ably is dependent on Eq. (57) obtained in this study.Since bz, deˆned as relative energy or also as normalizedenergy ranging from 0 to 2.0 as was described earlier, therelationship between Dr and bz is unique as shown in Fig.13. The simple concept of relative energy is that bz0 atthe initial state or loosest state, and that bz2.0 at the ˆ­nal state for a compression process or the densest state ina soil proˆle system.Though we assumed that emaxe0, they are not alwaysequal. The e0 state can be obtained at any time with un­loading, which is not emax which is the void ratio at theloosest state of sand. Therefore, it is important to knowthe state of sand in terms of Dr in Eq. (57), the pressure,and the initial states (void ratio). Thus, the states of sandcorrelate with an energy state deˆned in this study. Amore detailed examination of the relative density for sandcan be possible using Eq. (57).CONCEPT OF IMAGINARY PARTICLESSoil System and Compression ProcessAs shown in Figs. 10 to 12, the compression behaviourof soils or the change in soil state under compression isexpressed by the equations derived in this study. It is im­portant that in the cases shown in these ˆgures, the com­pression behaviour or change in state is basically de­scribed by the constant b. It is important to realize thatFig. 10 is an application of Eq. (53) for describing a soilproˆle, i.e., the soil system (Fukue et al., 1987). Theˆgure was an example of many experiments obtained by along term sedimentation experiment. The average voidratio was obtained by settling the slurry samples 18 timesinto a one­liter glass cylinder with a certain amount ofwater. A few grams of the slurry sample were poured intothe cylinder and left until the settlement of the depositwas ˆnished. Then the volume of the sediment was meas­ured. The ˆrst average void ratio was calculated using theRelative Density of SandThe state of sand is often expressed by the relative den­sity, which is deˆned using Eq. (55) and by taking emaxe0, we obtain1|Dre|emin0.119{0.881(1|bz) exp (|bz) (57)eo|eminwhere Dr is the relative density of sand, which is a stateparameter indicating how dense a given sand sample is.Thus the relative density of sand can be expressed as afunction of an energy term b(mgR/kT ) and z.Fig. 10.Average void ratio proˆle in a sedimentation test108FUKUE AND MULLIGANFig. 12. Compression curves of a Japanese standard sand, and oilsand (data by Roberts, 1964)Fig. 11. Compression curves of marine soils, including unloading andreloading processes (data from Silva and Jordan, 1984)volume and dry weight of the sample poured. Next, a fewgrams of slurry sample were poured in to the cylinder.After the settlement of the deposits was ˆnished, theaverage void ratio of the ˆrst and second deposits was ob­tained. These procedures were repeated until the asymp­totic average void ratio was obtained, as shown in Fig.10. Thus the thickness of the deposits increases. The exis­tence of an asymptotic void ratio indicates that the com­pression apparently stops at a given void ratio, i.e., ec oremin of this sediment. Thus, emin is deˆned as the void ratiowhere the deformation stops under the present loadingstress. In addition, it is a transition point from this loosestate to the next stage. The b value was determined for thebest ˆt using the asymptotic void ratio and the assumedsurface void ratio. Though the b value has a physicalmeaning; at present it is used as a parameter that connectsthe initial void ratio with emin. It should be noted that thedeposit obtained is the top layer of sediment soil, whichcan be described by a b value of approximately 0.4 cm|1for various types of soil samples and natural sedimentswith diŠerent ranges of the average void ratio (Fukue etal., 1987). Thus, a transition at a very loose state can befound at z0.5 cm, because z2.0/0.4. This is almostindependent of the type of soil.Figures 11 and 12 show an application for expressingthe consolidation behaviour of soil, i.e., the compressionprocess. The initial void ratio, e0 is not di‹cult to esti­mate from the e­log p relationships. The emin is a littledi‹cult to predict, because it is dependent on the geomet­ry of the particles. However, the ˆrst trial is to use 0.5 forthe emin. Figure 12 shows that the emin of oil sand for thebest ˆt is low, because oil can reduce the closest packingvoid ratio. This emin provides the transition point to thenext compression stage under greater loads. Thus, theequations derived in this study can be applied to describeboth the soil system and soil processes. Therefore, it maybe possible to compare artiˆcial and natural compressionbehaviour using the equations.Imaginary ParticlesFor the derivation of Eq. (53), it was assumed that mwas the mass of a particle and that the potential energy isonly dependent on the elevation of the particle. However,this is not the case. It is obvious that the mass m calculat­ed from the experimental b is not the actual value for theparticle, but it includes the eŠects of interactions betweenparticles.Therefore, it must be considered that the potentialenergy deˆned by mgh is not only dependent on the eleva­tion of the particle, but also on the interparticular forcesacting between the particles or interparticle energy. Thisis due to the relationship of the interactions with the voidratio proˆle. This is analogous to the situation of poten­tial energy of a mass suspended on a spring, depending onthe elasticity of the spring.Therefore, in this study the particle which can be im­aginatively represented by the experimental b is deˆned asan ``imaginary particle'' in any soil system. For example,the diŠerent b values for the respective curves, shown inFig. 11 provide the diŠerent sizes of imaginary particlesrelating to the mass, respectively.APPLICATION TO VARIOUS TYPES OF SOILSInteractions between ParticlesThere are many types of interactions between soil par­ticles, which depend upon mineralogy and size distribu­tion of particles, interfacial phenomena between particlesand pore liquids and interactions namely, the contact be­tween particles, as known in soil physics, soil chemistry,soil mechanics and geotechnical engineering. The b valuewill be in‰uenced by these factors and typical examplescan be as follows:a) The cementation between particles will lower the bvalue if the void ratio is the same. The cementation ofEQUATION OF STATEFig. 13.Relationship between relative density Dr and bzsoils has not been fully understood. It has been ex­plained by factors such as salt for quick clay (Rosen­qvist, 1953), organic matter (Pusch and Arnold,1969), amorphous materials (glass in volcanic soils),carbonates (Fukue et al., 1999; Fukue andNakamura, 1996; Imai et al., 2006) and others. If thecementation is relatively strong at higher void ratios,it will be broken down easily during compression andthen, the b value will be increased largely. It is notedthat an increasing b value means weaker interactions.b) The change in the interfacial forces due to the changein properties of pore water will cause the interactionsbetween particles. A good example can be obtainedfor an active clay­ electrolyte system. For example,the mechanical properties of bentonite clay will bedrastically changed by adding salt. Therefore, if theelectrolyte solution is allowed to penetrate into thesoil, then the interactions are weakened (Fukue et al.,1986). Consequently, the b value will increase.c) For the case of a sand­clay mixture, the mechanicalproperties of the soil are dependent on the fabric. Ifthere is no signiˆcant contact between sand fractions,the soil may behave like a clay soil. However, if thecontacts between sand fractions will be formed duringthe consolidation of clay fractions, the soil is morelikely a sand with cohesion (Fukue et al., 1986). Theseprovide a change in the b value. The development offrictional resistance due to the formation of contactsbetween sand fractions will decrease the b value.Since the relationship between undrained shearstrength, su and eŠective stress, p is often expressed assu/ptan q (constant), the e­log p curve becomesparallel to the e­log su curve, when p and su are plotted us­ing the same axis. The q is deˆned as the angle of un­drained shearing resistance. Thus, the b value can also becorrelated to the undrained shear strength. The details ofthis will be shown in another paper under preparation.Change in the Energy System during CompressionThe change in the energy system will aŠect the changein partition function, P and m, resulting from the changein the type of interactions between particles. However, inthe fundamental equation, Eq. (35), the P has been elimi­nated. Therefore, the change in the energy system will in­109‰uence the value of m, as well as the void ratios, such aseo and emin.Actual compression curves for an undisturbed soilsample can be demonstrated in Fig. 14. The data plottedare typical examples of Mexico City clays, obtained byZeevaert (1957). Other data obtained by him and forcemented or sensitive clays obtained by many otherresearchers show similar trends (Mesri et al., 1975;LaRochelle et al., 1980; Burland, 1990). The experimen­tal data show that the compression for undisturbed sam­ple breaks the bonds between particles, consequentlycausing a disturbance during compression. Thisphenomenon is strain softening. Strain softening is,therefore, expressed by an increasing b value. If the sam­ple is initially disturbed, the original compression curvemay have a greater b value in comparison to an un­disturbed sample. This means that the compression in­dex, Cc will decrease with the degree of disturbance, asused to evaluate sample quality. It is important to notethat actual compression curves exist between the twostandard curves for remolded (non­bonded) and un­disturbed samples without strain softening. Both lines aresimilar to the ICL and SCL deˆned by Burland (1990).The increasing Cc is due to the strain softening which maybe dependent on the initial void ratio and cementationdegree.A big question arises here as to whether or not thebreaking of bonds between particles will occur, and howmuch loading rate is required for that to occur in theˆeld. Because natural compression of sediments undernewly deposited particles may not cause the breaking ofbonds, it cannot always be assumed that the laboratorytest can predict ˆeld performance. Only a relatively highloading rate will cause strain softening to occur due tomicro­fractures (breaking of bonds) which may not be al­ways achievable in the ˆeld. If so, then it is doubtful thatit is beneˆcial to use the experimental Cc values forprediction of settlement in the ˆeld.Thus, with constant initial and minimum void ratiosand diŠerent values of b, the compression curves basedon the characteristics of the equations, Eqs. (2), (53) and(55), are all parallel. An example is demonstrated in Fig.15 which shows compression curves with a constant b.,i.e., b1 and b2, indicated by the solid lines, and also com­pression curves with a variable b changing from b1 to b2,indicated by the broken lines, where b1Àb2. The curvewith the variable b changing from b2 to b1 is also feasible.TRANSITION FROM ONE STATE TO ANOTHERThere is no doubt that there may be a transition in thestate from soil to rock. This is because the interaction be­tween particles is quite diŠerent for both materials. Simi­larly, there may be no doubt that the suspended solidssystem, (i.e., the dispersed particles system in water) isdiŠerent from the sediments deposited in water. Thereaders may think that suspended solids are not soils.However, we may think that sediments are also suspend­ed and settling while interacting. In fact, the settling ve­110FUKUE AND MULLIGANlocity of sediments can be measured and there is no con­solidation due to the excess pore water pressure. There­fore, the diŠerence between suspended solids and sedi­ments results from the diŠerent interactions for bothmaterials. In the dispersed system, the repulsions betweenthe particles overcome the attractions. On the otherhand, settled particles may be aggregated with nodominant repulsive forces between the particles.Marine soils often have a higher water content than theliquid limit. In general, the top 1 or 2 m of the sedimentlayers have a water content higher than the liquid limit.These soils have very loose but relatively ˆrm structures.These soils will be easily collapsed even under a laterallyconˆned condition. Therefore, the deformation proper­ties above and below the liquid limit may be diŠerent.Thus, if the deformation properties will change, theremay be a transition during compression. In this study, thepossible transitions can be found using the equations der­ived with the experimental results.Transition at the Liquid LimitIn the process of natural consolidation, there are sometransitions from one form to another. For example, clayand shale can be cited as a transition from soil to rock.Other transitions can exist in suspended systems anddeposited surface sediments. These transitions can be de­scribed as the change in the type of interparticle energyfrom one to another. The simplest example is the transi­tion from uncemented sand to sandstone.In the case of sediments, they usually have a structureof ‰oc­like aggregates. The pressure to the self weightcounteracts the pressure between the ‰ocs. The compres­sion and/or consolidation ˆrst occurs without collapsingthe ‰ocs under a very low overburden pressure. The mac­ro­pores formed by ‰ocs will then collapse under theirown weight during the sedimentation of new ‰ocs. Oncethis has occurred, the interparticle interactions becomedominant. Therefore, there must be a transition occur­ring in this case.The liquid limit of soil is also an example of a transi­tion, which can be deˆned as a transition from liquid toplastic states. To examine this, a consolidation test on acommercially available clay using centrifugal force wasperformed at Kyoto University. The test specimen of 10.2~50~16.8 cm (W~L~H) was consolidated for 17hours in which no deformation was observed, under agravity of 20 g. The soil sample had an initial water con­tent of 65z (liquid limit; 43z and plastic limit; 23z).The arm length of the centrifuge was 2.5 m. Water con­tent was measured on the sliced sample from the consoli­dated specimen. The experimental results are shown inFig. 16 along with the theoretical relationships obtainedin this study. From the experimental results, two sets ofconstant parameters, e0, emin and b are determined. Be­tween the two curves using the two sets of parameters, thetransition is clearly seen near a void ratio (1.17) at theliquid limit of the supplemental (theoretical) curves forsoils having a water content higher and lower than the liq­uid limit. Since the sample used was initially homogene­Fig. 14. The e­log p curves for a strongly cemented, volcanic soil inMexico City (data from Zeevaert, 1957)ous, there was little scattering for this consolidationresult. Thus, the transition obtained at the liquid limit islikely to be real. It is emphasized that without theoreticalcurves, the experimental result cannot be properly under­stood and would only be a collection of empirical data.Transition at a Low Void Ratio and High PressureAn example of the transition of marine sediments isshown in Fig. 17. The data were obtained from the ma­rine deposited soil of North Hokkaido (Aoyagi, 1978).The lines indicated in Fig. 17 express ordinary soils ob­tained using the values, e04, emin0.5 and b1.5~10|5cm|1 for the shallower sediments and e00.5, emin0 andb1.7~10|6 for the deeper sediments which can be clas­siˆed as sedimentary rocks. The theoretical curve was de­termined as the best ˆt using these constant parameters,e0, emin and b. If any of these is not properly determined,the theoretical line will deviate entirely from the ex­perimental points. This apparently shows that a void ra­tio of 0.5 is almost the densest packing state for particu­late materials in nature, and that compression beyondthis void ratio leads to the deformation or creep of parti­cles (Fukue and Okusa, 1987). With the theoreticalrelationships, the transition between them is apparent,which is similar to Fig. 16. The deviation of the datafrom the computed line may be due to the varied natureof sediments with respect to grain size, mineralogy andother constituents, because of the considerable interval ofdepth.Since compression of soil may stop at once at theclosest packing state of constituents, where no more slid­ing between particles occurs, continuous compressionmay provide a signiˆcant creep deformation of particlesthemselves, i.e., formation of the next stage of sediment.Aoyagi (1978) mentioned that in the compression ofmontmorillonite clay, there is a transitional state at aporosity of 30z, which is close to the emin for this stage.Thus, the emin is the void ratio where the deformation willstop immediately during continuous compression load­ing. This will occur when the type of interactions andstructure will change, as was shown in this study.In Fig. 17, the maximum depth of the shallower sedi­111EQUATION OF STATEFig. 15. Compression curves with variable b values in relation tostrain softening and hardeningments is approximately 1330 m. The eŠective pressurecalculated from Eq. (2) at the maximum depth is approxi­mately 9200 kPa. The emin for the shallower sediments isdeˆned as the initial void ratio of the next deeper sedi­ments.The soil states related to the stages can be demonstrat­ed in Fig. 18. The zero order stage shown in Fig. 18(a)can be identiˆed as the dispersed particles in water. Theparticles are suspended due to the relatively strong repul­sion. For example, this stage is found as suspended ben­tonite clay particles dispersed in distilled water. Basically,this stage does not exist for sand. Figure 18(b) illustratesthe most top layer under water, when the ‰occulatedstructure with macro pores. This layer is identiˆed as adrifting layer at the ˆrst stage, with a thickness of about 5cm (Fukue et al., 1987). The macro­pores disappear whenthe average void ratio reaches ec, where the soil state istransformed into the next stage, i.e., the second stage, asshown in Fig. 18(c). A soil in this stage has a water con­tent higher than the liquid limit. It is noted that this typeof soils usually have a thickness, about a few meters inmarine conditions. Since the water content for this typeof soil is higher than the liquid limit, wL, it becomes liq­uid like easily by disturbance. The ordinary soils can bedeˆned as the soil with a water content lower than the liq­uid limit, as shown in Fig. 18(d). This type of soil is most­ly encountered in engineering practice. The followingcompression process, i.e., diagenesis for a long time, willproduce the sedimentary rock. This stage can be illustrat­ed in Fig. 18(f). As was shown through this study, theequations derived can be applied to express the compres­sion behaviour of these types of soils shown in Fig. 18.The b and bz to be determined can be used to characterizethe soil state.COMPRESSION INDEXThe compression index is deˆned as the slope of thelinear portion of e­log p curve and is used for estimatingthe settlement of ground. If we deˆne the slope of e­log prelationship by de/d log p, we obtain the followingFig. 16. Results of the change in void ratio of a commercially availa­ble clay during a centrifugal consolidation testrelationship (Fukue and Okusa, 1987),ded log pC?c2.3(2|E)E(eo|ec)s1{ec{(eo|ec)AtA1{ec{(eo|ec)(1{E)A(58)where, Aexp (|E), Ebz. The following theoreticalrelationship can then be used,emin{0.135eo1.135ec (59)Therefore, the compression index can be deˆned as themaximum of the C?c value. Strictly speaking, the maxi­mum value of Cc' is a little greater than the compressionindex, because the conventional compression index is themaximum slope of the assumed straight portion of e­logp curve, but the maximum value by Eq. (58) is taken asthe tangent to the nonlinear relationship. In fact, it is not­ed that there are no linear portions for the e­log prelationships for soils (Fukue and Okusa, 1987). Since themaximum value of C?c is given at approximately E0.5(Fukue and Okusa, 1987), then Eq. (58) can be expressedby substituting Eq. (59) asCc|0.926(eo|emin)s1{0.653eo{0.347emint1{0.921eo{0.079emin(60)Thus, Cc is expressed in terms of e0, which may not large­ly diŠer from those obtained by many researchers, assummarized by Yoon et al. (2004) and Giasi et al. (2003).The diŠerence between Eq. (60) and empirical relation­ships is their degree of linearity. Although most empiricalrelationships are linear, there is usually no theoretical ba­sis for this and there is always some scattering of the data.When soils are saturated, the initial void ratio, e0 caneasily be converted into water content, w, as,weo/Gs~100z(61)where Gs is the speciˆc gravity of particles. Substitutingthis into Eq. (60), the following is obtained,112FUKUE AND MULLIGANCc 0.926(wGs/100|emin)(1{0.653wGs/100{0.347emin)1{0.921wGs/100{0.079emin(62)|Fig. 17. Void ratio proˆles of deep sediments (Aoyagi, 1978), and the­oretical relationships showing the transition between themFig. 18.Taking soil systems which can be expressed by an eminof 0.5, the relationship between Cc and w can be ob­tained. It is noted that the assumption of the emin providesfor soils existing in the normal soil stage. Herein it is not­ed that the emin is deˆned as the void ratio which providesa Cc value of zero. Therefore, it can be regarded as theclosest packing state of soil particles under a stress range.Figure 19 shows the values of Cc for various soil sam­ples and empirical relationships (after Mesri and Rok­hsar, 1974) and the theoretical relationship using Eq.(62). For the theoretical relationship, the value of Gs wasassumed to be 2.65. It can be seen that the theoreticalrelationship agrees well with the empirical relations andshows the relations for various soils. Figure 19 impliesIllustration of various soil systems and types of interactions with approximate b values. wn: natural water content, wL: liquid limitEQUATION OF STATE113Fig. 19. Theoretical and empirical relationships for compression index as a function of initial water content. The empirical relationships are adapt­ed from Mesri and Roksar (1974)that many kinds of marine clays and sensitive clays havehigher Cc values than the theoretical ones. This is becauseof strain softening during compression, as shown in Fig.14. Canadian muskeg has a lower Cc than the theoreticalone. This may be the result of organic fragments as con­stituents and structural characteristics, such as emin. If theemin is greater than 0.5, the theoretical Cc is lower. Thedeviation may be due to behaviour including strainsoftening and the eŠect of sand content during compres­sion (Fukue and Okusa, 1985). If the sand content is rela­tively high, frictional resistance will be developed duringvolume change. Accordingly, it is likely then that Cc willtend to lower to the corresponding initial void ratio of thesand.There are many empirical relationships between thecompression index and Atterberg limits, as summarizedby Giasi et al. (2003). These include the assumption thatthe natural water content is higher for ˆne soils with ahigher liquid limit. However, there is no theoretical basisfor the compression index to be based on liquid limits. Inthis study, it has been shown that there is basis for therelationship of compression index with the initial andminimum void ratios.CONCLUSIONSDi‹culties in soil mechanics exist since there is no``equations of state for soil.'' This is because soil is ahighly compressible and irreversible material with inter­actions between the constituents. Since a state of soilshould be expressed in terms of void ratio, pressure andenergy terms, the equations proposed will make it easy todeal with soils in a more scientiˆc manner. In this man­ner, some misunderstandings such as regarding compres­sion behaviour of soils can be eliminated. This theoreticalapproach was successfully applied to a wide variety ofsituations and soils.In summary, the following conclusions can be made:a) There is a universal relationship for volume and eŠec­tive pressure, which is derived using statisticalmechanics.b) The general compression curve for soils can be deter­mined by initial and ˆnal (minimum) void ratios and aconstant b value which is related to the potentialenergy of a soil element.c) The general relationships and experimental resultsimply that diŠerent soil systems in nature, such as sus­pended, deposited (‰occulated), liquid­like (watercontent higher than the liquid limit), consolidated (or­dinary soils) and strongly cemented soils (rock), canbe distinguished from each other by their transitionbehaviour. DiŠerent soil systems have been expressedby the corresponding parameters, i.e., e0, emin, and bvalues.d) The standard compression index (without strainsoftening or hardening) is only dependent on the ini­tial and ˆnal (minimum) void ratios. The initialcementation magnitude will provide the b value.e) Strain softening and hardening during compressionwill lead to a change in the b value.f) Thus, consolidation behaviour can be quantitativelyanalyzed in detail using this constant and its change.g) Compression index is a primary function of e0 andemin, whereas strain hardening and softening are sec­114FUKUE AND MULLIGANondary factors. The classiˆcation of the soil accord­ing to the liquid limit may be aŠected by the watercontent (void ratio) at sedimentation, which may thenappear that the Atterberg limit in‰uences the Cc.h) The developed formula can be applied for varioussoils including sand.i) From the formula, it is shown that the relative densityof sand has a physical meaning.j) Through this study, it is concluded that the developedformula constitutes the state equation for soils.ACKNOWLEDGEMENTSThe centrifugal consolidation test was performed byDr. K. Kita, Tokai University, at the Disaster PreventionResearch Institute, Kyoto University. The authors thankDr. K. Kita and Prof. M. Kamon, Kyoto University, fortheir cooperation. Special thanks are given to the lateProf. G. Imai for his encouragement throughout thisstudy.REFERENCES1) Aoyagi, K. (1978): Diagenesis of marine argillaceous sediments,The Memoirs of the Geological Society of Japan, 15, 3–14 (inJapanese).2) Been, K. and Sills, G. C. (1981): Self­weight consolidation of softsoils: an experimental and theoretical study, G áeotechnique, 31(4),519–535.3) Burland, J. B. (1990): On the compressibility and shear strength ofnatural clays, G áeotechnique, 40(3), 329–347.4) Butterˆeld, R. (1979): A natural compression law for soils,G áeotechnique, 29(4), 469–480.5) Coop, M. R. (1990): The mechanics of uncemented carbonatesands, G áeotechnique, 40(4), 607–626.6) Cubrinovski, M. and Ishihara, K. (2002): Maximum and minimumvoid ratio characteristics of sands, Soils and Foundations, 42(6),65–78.7) Fukue, M., Okusa, S. and Nakamura, T. (1986): Consolidation ofsand­clay mixtures, Consolidation of Soils, ASTM, STP 892,627–641.8) Fukue, M. and Okusa, S. (1987): Compression law of soils, Soilsand Foundations, 27, 23–34.9) Fukue, M., Yoshimoto, N. and Okusa, S. (1987): General charac­teristics of upper soil sediments, Marine Geotechnology, 7, 15–36.10) Fukue, M. and Nakamura, T. (1996): EŠects of carbonate oncementation of marine soils, Marine Georesources and Geotechnol­ogy, 14, 37–45.11) Fukue, M., Nakamura, T. and Kato, Y. (1999): Cementation ofsoils due to calcium carbonate, Soils and Foundations, 39(6),55–64.12) Giasi, C. I., Cherubini, C. and Paccapelo, F. (2003): Evaluation ofcompression index of remoulded clays by means of Atterberglimits, Bull. Engineering Geology. Env., 62, 333–340.13) Imai, G. (1980): Settling behaviour of clay suspensions, Soils andFoundations, 20(2), 61–77.14) Imai, G. (1981): Experimental studies on sedimentation mechanismand sediment formation of clay materials, Soils and Foundations,21(1), 7–20.15) Imai, G., Komatsu, Y. and Fukue, M. (2006): Consolidation yieldstress of Osaka­Bay Pleistocene clay with reference to calcium car­bonate content, ASTM, STP 1482, 89–97.16) LaRochelle, P., Sarrailh, J., Tavenas, F., Roy, M. And Leroueil,S. (1980): Causes of sampling disturbance and design of a new sam­pler for sensitive soils, Can. Geotech. J., 18(1), 52–66.17) Leroueil, S. and Vaughan, P. R. (1990): The general and congruenteŠects of structure in natural soils and weak rocks, G áeotechnique,40(3), 467–488.18) Mesri, G. and Rokhsar, A. (1974): Theory of consolidation forclay, J. Geotechnical Engineering Division., ASCE, 100 (GT8),889–904.19) Mesri, G., Rokhsar, A. and Bohor, B. F. (1975): Composition andcompressibility of typical samples of Mexico City clay, G áeotech­nique, 25(3), 507–554.20) Moran, D. E. et al. (1958): Study of deep soil stabilization by verti­cal sand drains, OTS Report, DB151692, Bureau of Yards andDocks, Dept. of the Navy, Washington DC.21) Nishida, Y. (1959): A brief note on compression index of soil, J.Soil Mechanics and Foundation Div., ASCE, 82(SM3), 1–14.22) Oswald, R. H. (1980): Universal compression index equation, J.Geotechnical Engrg. Div., ASCE, 106, 1179–1199.23) Park, J. H. and Koumoto, T. (2004): New compression index equa­tion, Journal of Geotechnical and Geoenvironmental Engineering,ASCE, 130, 223–226.24) Pusch, R. and Arnold, M. (1969): The sensitivity of artiˆciallysedimented organic­free Illitic clay, Engineering Geology, 3(2),135–145.25) Reif, F. (1965): Fundamentals of Statistical and Thermal Physics,McGraw­Hill, New York, 211p.26) Roberts, J. E. (1964): Sand compression as a factor in oil ˆeld sub­sidence. Sc.D. Thesis, Dept. Civil. Engineering, MIT, Boston, MA.27) Rosenqvist, J. Th. (1953): Considerations on the sensitivity of Nor­wegian quick clays, G áeotechnique, III(5), 195–200.28) Schoˆeld, A. N. and Wroth, C. P. (1968): Critical State SoilMechanics, McGraw Hill.29) Silva, A. J. and Jordan, S. A. (1984): Consolidation properties andstress history of some deep sea sediments, Seabed Mechanics, Int.UTAM. (ed. by B. Dennesss), Graham & Trotman, 25–39.30) Skempton, A. W. (1944): Notes on the compressibility of clays,Quarterly Journal of the Geological Society of London, C(Parts aand 2), 119–135.31) Tolman, R. C. (1979): The Principles of Statistical Mechanics,Dover Publications, Inc. New York, 660p.32) Tsuchida, T. (2001): General interpretation on natural void ratio­overburden pressure relationship of marine deposits, Soils andFoundations, 41(1), 127–143 (in Japanese).33) Yoon, G. L., Kim, B. T. and Jean, S. S. (2004): Empirical correla­tions of compression index for marine clay from regression analy­sis, Can. Geotech. J., 41, 1213–1221.34) Zeevaert, L. (1957): Foundation design and behaviour of TowerLatino America in Mexico City, G áeotechnique, 7, 115–133.
• ログイン

• タイトル
• Bearing Capacity of Shallow Foundations in a Low Gravity Environment
• 著者
• Taizo Kobayashi・Hidetoshi Ochiai・Yusuke Suyama・Shigeru Aoki・Noriyuki Yasufuku・Kiyoshi Omine
• 出版
• Soils and Foundations
• ページ
• 115〜134
• 発行
• 2009/02/15
• 文書ID
• 21174
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 115–134, Feb. 2009BEARING CAPACITY OF SHALLOW FOUNDATIONSIN A LOW GRAVITY ENVIRONMENTTAIZO KOBAYASHIi), HIDETOSHI OCHIAIii), YUSUKE SUYAMAiii),SHIGERU AOKIiv), NORIYUKI YASUFUKUv) and KIYOSHI OMINEv)ABSTRACTAs a basic study for future lunar/planetary explorations and the in­situ resource utilization missions, bearing capaci­ty characteristics of shallow footing systems in a low gravity environment were investigated. A series of model loadingtests on a simulated lunar soil (lunar soil simulant) and Toyoura sand were conducted on an aircraft that ‰ew in para­bolic paths to generate partial gravity ˆelds. As a result of the model tests, it became clear that bearing characteristics,including the coe‹cient of subgrade reaction and ultimate bearing capacity of the lunar soil simulant in a low gravityenvironment is hardly in‰uenced by the gravity levels, while Toyoura sand exhibits a high dependence on gravity.From the observation of the failure mechanisms, it was found that the gravity dependence seems to correlate well withsoil compressibility. To rationally explain the dependence of ultimate bearing capacity on gravity, theoretical evalua­tions were attempted in the framework of the upper bound method. The proposed calculation method not only makesit possible to correlate quantitatively the failure mode with dependence on gravity, but also may allow us to predict theultimate bearing capacity in the lunar surface environment.Key words: bearing capacity, foundation, low gravity, lunar surface, regolith, upper bound method (IGC: B8/E3)such as low gravity, high vacuum, extreme temperaturesand radiation. From a geotechnical engineering stand­point, since the self weight of soil signiˆcantly aŠectssoil deformation and collapse, of considerable impor­tance is the fact that gravity on the Moon is approximate­ly one­sixth that of Earth's gravity. Klein and White(1990) conducted simple experiments concerning granular‰ow with rotating drum test apparatuses on NASA'sKC­135 aircraft during variable gravity maneuvers, andpresented the eŠects of gravity on angles of repose. Boleset al. (1997) carried out soil cutting experiments on thesame aircraft, and presented the results of excavatingforce measurements in the low gravity environment.Sture et al. (1998) conducted triaxial compression tests ongranular materials in a microgravity environment duringthe Space Shuttle missions, and examined stress­strainrelationships at low eŠective conˆning stresses.Kobayashi et al. (2008) performed model tests on a wheel­terrain system on an aircraft, and investigated the mobili­ty performance of a lunar/planetary rover in low gravityconditions.Although these studies will be valuable for predictingthe actual soil behavior in an extraterrestrial environmentthey are still limited. It should be emphasized again thatINTRODUCTIONIn January, 2004, President George W. Bush an­nounced a new vision for human and robotic space explo­ration that he named ``A Renewed Spirit of Discovery''.Since the announcement, lunar and planetary explorationprograms have been taking shape in several countries. InJapan, a long­term vision for the space development pro­grams (JAXA 2025), one of which is to explore the moonand utilize in­situ resources, was o‹cially announced bythe Japan Aerospace Exploration Agency in April, 2005.The lunar exploration and the in­situ resource utilizationwill involve various operations, including soil excava­tions, mining and foundation works for extraterrestrialfacilities. Successful operations rely on an understandingof the soil­machine and/or soil­structure interactionproblems. In short, knowledge of the geotechnical prop­erties of the regolith (soil on the moon/planet) is of fun­damental importance in evaluating the feasibility of themissions.Needless to say, conditions and constraints on the lu­nar surface are quite diŠerent from a terrestrial environ­ment. Besides diŠerences in the surface material, the lu­nar surface is subject to harsh environmental conditionsi)ii)iii)iv)v)Assistant Professor, Department of Civil and Structural Engineering, Kyushu University, Japan (t­koba—civil.kyushu­u.ac.jp).Vice President, Kyushu University, Japan.Formerly Graduate Student, Graduate School of Engineering, Kyushu University, Japan.Research Engineer, Shimizu Institute of Technology, Shimizu Corporation, Japan.Associate Professors, Department of Civil and Structural Engineering, Kyushu University, Japan.The manuscript for this paper was received for review on July 7, 2008; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.115116KOBAYASHI ET AL.an understanding of soil behavior in a low gravity en­vironment, especially in terms of problems with soil­machine and/or soil­structure interaction systems, is ofoverriding importance, and has not yet been fully estab­lished.This paper attempts to examine the eŠects of gravity onthe load­settlement characteristics of shallow foundationsystems. A series of model loading tests on a Japanese lu­nar soil simulant and Toyoura standard sand were con­ducted on an aircraft during variable gravity maneuvers,and the bearing capacity characteristics revealed from thediŠerences in soil type and the gravity condition arepresented. Moreover, the upper bound calculationmethod, which takes soil compressibility into account, isnewly proposed to quantitatively evaluate the dependenceof ultimate bearing capacities on gravity.The bearing capacity problems relate to various situa­tions in surface operations, such as a touchdown of alanding module, supporting of a structure and machine,or even the walking of an astronaut. This study is expect­ed to be pioneering work that provides a fundamental un­derstanding of regolith­structure interaction characteris­tics in a low gravity environment.MATERIALSSoils that cover the lunar/planetary surface are called``regolith''. Physical properties of lunar regolith havebeen measured in­situ by robots and astronauts, in thelaboratory on returned samples, and by remote sensingduring the Luna, Surveyor and Apollo missions. The de­tailed results of these measurements and the estimates arecompiled in the Lunar Sourcebook (Carrier et al., 1991).In this book, Carrier et al. (1991) pointed out that the lu­nar regolith is distinctly diŠerent from terrestrial materi­als. For one thing, its mineral composition is limited.Fewer than a hundred minerals have been found on theMoon, compared to several thousand on Earth. Further­more, geotechnical properties of lunar soil tend to fall ina fairly narrow range due to the absence of atmosphereand the lack of water and organic materials in theregolith. The typical values of the physical properties oflunar soil presented in the Lunar Sourcebook have beenrearranged in Table 1.For experimental studies through soil mechanics ap­proaches, regolith simulants, which are made of terres­trial­based soils that mimic real regolith, are often useddue to limited availability of real regolith. In recent years,several lunar soil simulants have been manufactured.MLS­1 (e.g., Weiblen et al., 1990), JSC­1 (e.g., McKay etTable 1.al., 1994; Willman et al., 1995; Klosky et al., 2000) andFJS­1 (e.g., Kanamori et al., 1998) are widely used for ex­perimental studies. The materials used in this study werea Japanese lunar soil simulant (FJS­1), manufactured byShimizu Corporation, and Toyoura sand. The lunar soilsimulant is a sandy material that mimics the lunar sam­ples brought back by the Apollo missions. Targeted prop­erties of the material include chemical composition, par­ticle density, particle size distribution, and shearstrength. The major raw material is basaltic lava rockmined from the Mt. Fuji area, and Ilmenite and Olivinewere added for simulating the chemical composition ofthe actual lunar soil. Toyoura sand is frequently used asthe standard Japanese sandy material in geotechnical stu­dies.Table 2 shows the physical properties of the materialsused in this study. Compared to Toyoura sand, the lunarsoil simulant exhibits a higher density. It seems thatheavy raw materials contained in the lunar stimulant,such as Ilmenite and Olivine, lead to such a high densityof the soil. As to the void ratio, diŠerence in minimumvoid ratio, emin, can be seen when comparing the materi­als, while the maximum void ratio, emax, remains con­stant. The emin of the lunar soil simulant is lower than thatof Toyoura sand, indicating that the lunar soil simulantcan be packed more densely. As described later, thisseems to be because the lunar soil simulant is well­gradedand has a wider range of particle size distribution com­pared to Toyoura sand. Figure 1 shows the ScanningElectron Microscope (SEM) photographs of the materi­als. As can be seen from Fig. 1(a), the grains of the lunarsoil simulant are very jagged and of irregular shapes. Thisis because the simulant is made by crushing the rawmaterials with a hydraulic rock clipper, a jaw crusher anda rotary impact mill. Carrier et al. (1973) presented somephotographs of real lunar soil, and mentioned that manyTable 2.Physical properties of the soilsSoil propertiesLunar soilsimulant(FJS­1)ToyourasandSoil particle density, rs (g/cm3)Maximum bulk density, rmax (g/cm3)Minimum bulk density, rmin (g/cm3)Maximum void ratio, emaxMinimum void ratio, eminEŠective grain size, D10 (mm)Mean grain size, D50 (mm)Coe‹cient of uniformity, UcCoe‹cient of curvature, U?c2.951.492.020.980.460.0140.1011.431.302.651.341.640.980.620.210.261.330.98Typical values of lunar soil properties (Carrier et al. (1991), rearranged by the authors)Depth range(cm)Bulk densityr (g/cm3)Relative densityDr (z)Void ratioeCohesionc? (kN/m2)Friction angleq? (degree)0–150–3030–600–601.50}0.051.58}0.051.74}0.051.66}0.0565}374}392}383}31.07}0.070.96}0.070.78}0.070.87}0.070.44–0.620.74–1.12.4–3.81.3–1.941–4344–4752–5548–51BEARING CAPACITY IN LOW GRAVITY ENVIRONMENTFig. 1.SEM photographs of the soil particlesFig. 2.Grain size distributions(in some cases, most) of the particles are not compact,but exhibit irregular shapes and surface textures. The lu­nar soil is formed primarily by two processes, comminu­tion and aggregation, as a result of meteorite impacts onthe lunar surface (Carrier et al., 1973). On the otherhand, the surface textures of Toyoura sand are compara­tively smooth and the grains are chipped and rounded asshown in Fig. 1(b). The grain size distributions arepresented in Fig. 2. The lunar soil simulant can be classi­ˆed as well­graded, silty sand, while Toyoura sand ispoorly graded sand. The grain size distribution of the lu­nar soil simulant falls within the range of the lunar sam­ples obtained from Apollo 11, 12, 14 and 15 missions as117illustrated by the broken lines.Strength characteristics of the materials were examinedthrough triaxial compression tests. Cylindrical specimensof 100 mm in height and 50 mm in diameter were used.The samples of the lunar soil simulant were prepared in adry state to have bulk densities of 1.77 g/cm3 and 1.95g/cm3, representing relative densities of 60z and 90z,respectively. The samples of Toyoura sand were also pre­pared to have relative densities of 60z and 90z, havingbulk densities of 1.51 g/cm3 and 1.61 g/cm3, respectively.These samples were tested at conˆning stresses rangingfrom 2.0 to 98.1 kN/m2.Stress­strain relationships obtained from the compres­sion tests are presented in Fig. 3. As can be seen fromFig. 3(a), the lunar soil simulant in a dense state exhibitsdistinct peak strengths, followed by signiˆcant strainsoftening behavior. By comparing Fig. 3(a) with Fig.3(b), it appears that residual strengths of the sand pre­pared in a dense state roughly equal those in a mediumstate at each conˆning stress level. These behaviors implythat the lunar soil simulant can demonstrate high strengthdue to the component of work done by its dilatancy in ad­dition to its friction by inter­particle interactions. Inother words, the lunar soil simulant seems to have strongdependence on stress and density. On the other hand, dis­tinct peak strength and/or strain softening behavior arenot seen in most of the cases of Toyoura sand (Figs. 3(c)and (d)). In terms of volumetric strain, the lunar soilsimulant exhibits in‰ection points at around the peakstrengths, and the trend of dilations is then suppressed,while Toyoura sand continues to dilate. The visual obser­vation of soil deformations during the tests also revealedthat distinct shear planes were generated in the lunar soilsimulant, while the samples of Toyoura sand demonstrat­ed barrel­shaped deformations and no shear planes untilthe end. It seems that the strain softening and in‰ectionpoints that appeared in the lunar soil simulant are due tothe generation of shear planes.The peak strengths of the stress­strain curves are plot­ted on a p?­q plane as shown in Fig. 4, in which p? and qare the eŠective mean principal stress and the deviatorstress, respectively. The p?­q relationship for each sampleis plotted on a linear line, indicating that nonlinearity offailure envelopes does not need to be taken into accountfor this range of stress levels. The cohesion and the inter­nal friction angles determined by the failure envelopes arelisted in Table 3. Despite the dry conditions, both materi­als exhibit cohesive strength. In particular, with the lunarsoil simulant, its cohesive strength shows a high depend­ence on density, demonstrating that the more densely thematerial is packed, the greater the cohesive strength willbe. This appears to be because of the apparent cohesioncaused by the particle interlocking of the deformedgrains. Furthermore, the internal friction angles of the lu­nar soil simulant are comparatively greater than those ofToyoura sand, indicating a considerable dependence ondensity again. It appears that the distinctive dilatancycharacteristic of the lunar soil simulant mentioned earlieris yielding such higher internal friction angles.118KOBAYASHI ET AL.Fig. 3.Triaxial compression test resultsTable 3.Bulk densities and strength parameters of the soils usedMaterialsFig. 4.Failure points in p?­q eŠective stress planeRelativeBulkCohesionInternaldensitydensity c? (kN/m2) ˆction angle3q? (degree)Dr (z) r (g/cm )Lunar soil simulant(FJS­1)90601.951.777.541.4450.141.6Toyoura sand90601.611.513.043.0442.738.1To investigate the material compressibility, one­dimen­sional compression tests were conducted using an appara­tus shown in Fig. 5. The size of the specimen is the sameas that of the triaxial compression tests, i.e., cylindricalspecimens of 100 mm in height and 50 mm in diameter.The soil samples were compressed in a rigid mold so thatBEARING CAPACITY IN LOW GRAVITY ENVIRONMENTthe materials could deform only in the axial direction.The samples were prepared to have the relative densitiesof 60z and 90z, which are the same values as those usedin the triaxial tests. Figure 6 shows the results of the one­dimensional compression tests, and the e­log pc relation­ships are presented in Fig. 7 along with the actual lunarFig. 5.Compression test setup119soil data (Carier et al., 1991) obtained from the Apolloand Luna missions. As shown in Fig. 6, the lunar soilsimulant in a dense state is more compressed than Toy­oura sand in a medium state, indicating that the lunar soilsimulant has extremely high compressibility. It is interest­ing to note here that a compression can occur in the lunarsoil simulant even in densely packed conditions. In theprevious paragraph, it was mentioned that great dilation(volume expansion) can be seen in the simulant duringshearing. At the same time, the lunar soil simulant alsoexhibits large contractions during a loading process thatdoes not accompany any shearing. As such, it can be con­cluded that the major feature of the lunar soil simulant isthat the mode of deformation changes signiˆcantly ac­cording to the type of loading.From the soil tests, it was shown that the lunar soilsimulant (FJS­1) resembles the actual lunar soil in parti­cle density, bulk density, particle size distribution and in­ternal friction angle, etc. The lunar soil simulant can,therefore, be expected to demonstrate mechanical charac­teristics similar to those of actual lunar soil. Needless tosay, not all of the bearing characteristics can be predictedby this material. However, comparisons of the two kindsof materials, i.e., FJS­1 and Toyoura sand, which havediŠerent mechanical characteristics, will make it possibleto make clear the mechanism of gravity dependence ofthe bearing capacity.PARABOLIC FLIGHT AND PARTIAL GRAVITYFig. 6.Typical results of the one­dimensional compression testsFig. 7.e­log pc relationshipsThe loading experiments of a model shallow footingwere performed on an aircraft that ‰ew in parabolic pathsto generate partial gravity ˆelds. A photo of the aircraftand of the test conditions inside the cabin are as shown inFigs. 8 and 9, respectively. The ‰ight path and the gravityvariations during the ‰ight maneuver are described inFig. 10. As shown in the ˆgure, the aircraft starts to ac­celerate in a concave path, and then the thrust is suspend­ed when the aircraft attains su‹cient speed, after whichpartial gravity is maintained for approximately 20 to 40seconds in a convex path. The rapid accelerations beforeand after the period of partial gravity generate 1.5 g to 2 gfor approximately 20 to 30 seconds. The gravitationalFig. 8.Aircraft for partial gravity experiment120KOBAYASHI ET AL.Fig. 9.Cabin in the aircraftFig. 11.Fig. 10.Parabolic ‰ight maneuverforce is measured in three components by accelerometers;fore and aft acceleration (Gx), lateral acceleration (Gy),and head to foot acceleration (Gz) on the body­ˆxed coor­dinate system. The ˆgure indicates that Gx and Gy arealmost zero and that only Gz can occur during the para­bolic maneuver regardless of how the body of the aircraftinclines. Therefore, it is unnecessary to be concerned withthe eŠects of the ‰ight pattern.MODEL TEST APPARATUS AND EXPERIMENTALPROCEDUREA series of model loading tests were performed with afooting installation system ( see Fig. 11). The test condi­tions, including model footing size, soil box size andloading rate, were required to be determined based on therestrictions of the payload allotment for the ‰ights. Ac­cordingly, the experiments were designed to be somewhatsmall compared to common laboratory model experi­ments. However, the experimental system used in thisstudy is su‹cient to achieve the research objective be­cause the preliminary experiments conˆrmed that bound­ary and grain size eŠects can be avoided.Model test apparatusThe soil box is constructed of 10 mm­thick acrylicboards, and the upper end of the soil box is covered withan aluminum clamp so as to prevent the boards frombending. The size of the model ground is 400 mm inwidth, 160 mm in height and 50 mm in depth. In order toreduce the eŠects of friction between the soils and theboards, 0.02 mm thick latex membranes were laid be­tween them, and silicon grease was applied between theboards and the membranes.The model grounds were prepared by using a shakingtable equipped with a vibrator. The desired densities ofthe model grounds were achieved by applying vibrationto the table on which the soil box was resting, and ap­plying uniform surcharge until the ground surface settledto the prescribed line on the side of the soil box. Themodel grounds were prepared to have the relative densi­ties of 60z and 90z for each soil, as was the case withthe triaxial compression tests previously discussed. Aslisted in Table 3, the bulk densities of the lunar soilsimulant were 1.77 g/cm3 and 1.95 g/cm3, and those ofthe Toyoura sand were 1.51 g/cm3 and 1.61 g/cm3. Thepreparation was performed in a laboratory on the groundto adjust the initial ground conditions so that the diŠer­ence in soil behavior induced by the gravity condition canbe revealed.Figure 11(a) shows a schematic of the model appara­BEARING CAPACITY IN LOW GRAVITY ENVIRONMENTtus. The model footing is made of aluminum and the sizeof the base is 20 mm in width and 50 mm in length. Apiece of sandpaper was attached to the base so that themodel would have a rough footing. The model footingwas designed to penetrate the model ground surface at aconstant rate of 3.0 mm/s. The model footing (as seen inFig. 11(b)) and the data logging system were mounted ona sturdy rack as illustrated in Fig. 11(c). To keep theground conditions constant, multiple soil boxes werereserved in a separate rack, from which a new box wastaken to replace the used box after each test. Shock ab­sorbers are attached to the separate rack to avoid distur­bances of model ground by shock and vibration of theaircraft. Moreover, heaving of the reserved modelgrounds was prevented by ˆxing spacers onto the groundsurfaces. Consequentially, the settlement and/or distur­bance were not observed during the ‰ights, though wewere apprehensive that the reserved model grounds mightsettle down by high­sustained acceleration of the aircraft.Therefore, the in‰uence of the gravity change and vibra­tion of the aircraft on the initial ground condition can bedisregarded.Fig. 12.121Six variations of gravity were targeted in the tests,namely 0 g, 1/6 g, 1/2 g, 3/4 g, 1 g and 2 g. The gravitylevels of 0 g, 1/6 g, 1/2 g and 3/4 g were achieved via theparabolic ‰ight pattern of the aircraft, while the gravitylevel of 2 g was achieved via the circular ‰ight pattern.The 1 g gravity level was achieved by performing thesame tests in a laboratory on the ground.EXPERIMENTAL RESULTSLoad­Settlement RelationshipsA series of the model tests described above was con­ducted using both the lunar soil simulant and Toyourasand. Examples of load­settlement relationships arepresented in Fig. 12. Settlement, S, is normalized by thefooting width, B, and the load is expressed as load inten­sity, q, representing the average footing contact pressure.Figure. 12 shows that the load­settlement curves exhibit­ed peak load intensities which were followed by a drop, inall cases, except in the case of the lunar soil simulant in amedium state (Fig. 12(a)). Such a trend of the curvesseems to indicate that a general shear failure mode, whichTypical examples of load­settlement relationships122KOBAYASHI ET AL.is usually observed in the case of a shallow footing ondense sand, was taking place beneath the footing.However, in the case of the lunar soil simulant, distinctslip lines were not visually observed in this settlementrange (S/B0–50z). Therefore, it is too premature todetermine what type of deformation took place in themodel grounds based only on these load­settlementcurves. Detailed observation results of the soil deforma­tion are discussed later. In Fig. 12, it is also apparent thatthe shapes of the curves for Toyoura sand (Figs. 12(c)and (d)) varied with the gravity levels, while the shapes ofthe curves for the lunar soil simulant (Figs. 12(a) and (b))were not as much aŠected by gravity. This suggests thatthe degree to which gravity aŠects the bearing characteris­tics depends on the soil used.EŠects of Gravity on Bearing Capacity CharacteristicsIn a load­settlement relationship, a peak load is gener­ally deˆned in terms of ultimate bearing capacity, qu, asthe maximum allowable load for a given structure. A less­er load before reaching the ultimate bearing capacity isdeˆned as allowable bearing capacity, qa. The amount ofsettlement with respect to an arbitrary allowable bearingcapacity is usually estimated based on a parameter calledthe coe‹cient of subgrade reaction, Ks, which is deˆnedby a linear slope of the load­settlement curve. Based onthe assumption that the ground is a semi­inˆnite elasticbody, the coe‹cient of subgrade reaction Ks, can be theo­retically expressed as follows:Fig. 13.Subgrade reaction versus gravitational acceleration levelE0(1|n2)IpBKs (1)in which E0: deformation modulus, n: Poisson's ratio, Ip:shape factor and B: footing breadth. Figure 13 shows theeŠects of gravity on the coe‹cient of subgrade reaction,Ks. It is clear from this ˆgure that gravity hardly in­‰uences Ks for the lunar soil simulant (Fig. 13(a)),whereas Ks for Toyoura sand varies proportionally withthe gravity levels (Fig. 13(b)). As can be seen in the triaxi­al compression test results (Fig. 3), it is generally knownthat deformation modulus, E0 (slope at an early stage ofthe stress­strain curve) of granular materials is aŠected byconˆning stresses. Therefore, Ks also must be aŠected bygravity since Ks is a function of E0 as expressed in Eq. (1).While the test result of Toyoura sand appears to cor­respond with this eŠect, the dependence of Ks on gravityis not seen for the lunar soil simulant, although the stressdependence of E0 can be seen in Figs. 3(a) and (b). Thissuggests that one­dimensional compressions, which ac­company no lateral movement, took place beneath thefootings on the lunar soil simulant. This suggestion canalso be easily inferred from the fact that a spring constantis independent of the gravity levels.Figure 14 shows the eŠects of gravity on ultimate loadintensity, qu. In the case of the lunar soil simulant in amedium state (Fig. 14(a)), the in‰ection points that ap­peared on the load­settlement curves are plotted as qu,since distinct peaks were not observed. This ˆgure showsthat the qu values of Toyoura sand are in proportion toFig. 14.Ultimate load intensity versus gravitational acceleration levelBEARING CAPACITY IN LOW GRAVITY ENVIRONMENT123is likely to collapse in a low gravity environment. A com­parison between Figs. 15(a) and (b) shows that the foot­ings resting on the lunar soil simulant in low gravity con­ditions require larger settlements to collapse than on Toy­oura sand. Thus, from a standpoint of geotechnical de­sign for foundation works, it can be concluded that thebearing capacities on the lunar surface will be safer thanin the terrestrial condition because the bearing capacitiesof the lunar soils will not be reduced even when the loadsof structures become one­sixth in the lunar gravity condi­tion.Fig. 15. Normalized settlement at collapse versus gravitational ac­celeration levelthe gravity levels, regardless of how it is packed, whereasno clear proportionality between the gravity levels andthe qu values can be found in the case of the lunar soilsimulant partly due to the dispersion of the data. As tothe case of Dr60z for the lunar soil simulant, gravityhardly in‰uences the qu. As to Dr90z, even if linearityon the relationships is assumed, the slope (sensitivity ofgravity toward qu) is still low compared to that of Toy­oura sand. As discussed later in more details, in classicalbearing capacity theories, the ultimate bearing capacity isin proportion to gravity. It is interesting that, while Toy­oura sand exhibits such a proportional trend, it does nothold true for the lunar soil simulant. Accordingly, thisimplies that the classical bearing capacity theories cannotbe applied to the lunar surface.Soil collapses due to loading of the footings are accom­panied by some settlements, Sc (where Sc reŠers to a set­telement at collapse). Figure 15 shows the relationshipbetween gravity and the normalized settlements at col­lapse, Sc/B. In Fig. 15(a), it appears that Sc/B for the lu­nar soil simulant is independent of gravity when gravity isless than 1 g, while it becomes somewhat greater whengravity reaches 2 g. In the meantime, Fig. 15(b) showsthat gravity signiˆcantly in‰uences Sc/B for Toyourasand, especially under a partial gravity condition. Alsoconsidering that the ultimate bearing capacity of Toyourasand exhibits high stress dependence, we can then statethat a ground that consists of terrestrial sandy materialsObservation of Soil DeformationsThrough the parabolic ‰ight experiments, it becameclear that the bearing characteristics of the shallow foot­ings in a low gravity environment vary signiˆcantly de­pending on the materials. In this section, the diŠerencesin soil behavior are examined through observation of thesoil. By using the Particle Image Velocimetry (PIV) tech­nique, we have illustrated the displacement vectors of thesoils during the footing loading processes (Fig. 16). Thedisplacement vectors of the soil particles were traced atintervals of 2.4 mm of settlement (S/B12z) between 0and 9.6 mm (S/B0–48z). Although deformationmechanism may vary depending on the gravity condi­tions, this paper focuses on the diŠerences in the defor­mation between materials and, therefore, explores defor­mation ˆelds of the simulant and Toyoura sand of Dr90z under a 1 g condition as the typical deformationmechanism. The uppermost charts in the ˆgure, whichrepresent the vectors just after the installations, showthat a general shear failure mode was seen in Toyourasand (Fig. 16(b)), while the lunar soil simulant formed acompression region below the footing (Fig. 16(a)), fol­lowed by a deformation ˆeld that indicates a generalshear failure. That is, the lunar soil simulant exhibits aphased failure mode where a local failure is ˆrst causeddue to compression and is followed by a general shearfailure. With some varying degrees, general soils exhibitgradual failure development. That is to say, a stage ofelastic deformation is ˆrst presented, which then developsinto a stage of local shear and cracking before reachingthe ˆnal stage of general shear failure. Therefore, it canbe said that diŠerences in failure mechanisms betweenmaterials are due to the speed and the degrees of the tran­sitions from stage to stage. In the advanced stages of thefooting installation, however, downward vectors can beseen and the compression still continues in the lunar soilsimulant, while all soil particles under the footing movessideways in the case of Toyoura sand. Moreover, throughan examination of the process after the peak load in thelunar soil simulant, it was found that the magnitude ofthe vectors moving sideways in the lunar soil simulant isrelatively smaller than that of Toyoura sand, indicatingthat the general shear failure mode seen in the lunar soilsimulant is not as perfect as that in Toyoura sand. Fromthis observation, it can be concluded that the lunar soilsimulant generates primarily a punching or local shearfailure mode even in a dense state, while Toyoura sand124KOBAYASHI ET AL.Fig. 16.Displacement vectors of the soils during the footing installationsexhibits a general shear failure mode, and that this diŠer­ence aŠects the bearing capacity characteristics. As illus­trated in Fig. 6 previously, the lunar soil simulant ex­hibits higher compressibility than Toyoura sand even in adense state. Such high compressibility can be attributedto the fact that the lunar soil simulant is much wider inthe particle size distribution and more compactable thanToyoura sand, and due to the breakage of its deformedsoil particles.Discussions on the Failure MechanismThe general shear failure mode can be characterized bythree zones, namely the active failure zone, the transitionzone and the passive failure zone, as described in Fig.17(a). Soils below the footing are forced downward toform a soil wedge (active failure zone). Soils around thewedge are forced outward in a fan­shape. This zone, inwhich the direction of the major principal stresses shiftfrom a passive to an active failure state, is referred to asthe transitional zone. Furthermore, the passive failurezone spreads upward until the failure surface extendsthrough the ground surface, causing the surface to swell.At that moment, the zone moving upward is doing workagainst gravity, and this appears to be the major factor indetermining the dependence of the ultimate bearingcapacity on gravity. In the case of the local shear failuremode as described in Fig. 17(b), downward displace­ments are predominant and, thus, there is no workagainst gravity. In Fig. 16(a), although the outwardand/or upward soil displacements can be seen in the lunarsoil simulant, the magnitude of such displacements issmall compared to that in Toyoura sand, demonstratingthat the eŠect of gravity was not present as prominently.In other words, gravity tends to have less eŠect on thebearing characteristics on the ground where a punchingor local shear failure mode is generated.Examining from the standpoint of energy balance asexplained here, we can rationally explain the diŠerencesin the degree of dependence on gravity as observed in theexperiments and to infer that it is attributed to the com­BEARING CAPACITY IN LOW GRAVITY ENVIRONMENT125than the conventional peak strength parameters, to thecalculations. However, it seems impossible to explainthat the eŠect of gravity varies depending on the materialsbecause his and other conventional methods always as­sume a general failure mode. Although various resear­chers (e.g., Johnson, 1989; Carrier et al., 1991; Ettouneyand Benaroya, 1992) have attempted to attain coherentresults by multiplying the bearing capacity factors in theTerzaghi's bearing capacity formula by the shape factorsof foundations, such a treatment assumes that the bear­ing capacities are always in proportion to gravity and isstill unable to explain the experimental results obtained inthis study. Alternatively, our attempt will be unique anduseful in that gravity dependence is analytically correlat­ed to soil compressibility.Fig. 17.Schematics of observed failure modepressibility of the materials.THEORETICAL EVALUATION OF GRAVITYDEPENDENCEThe previous section qualitatively explained, from astandpoint of energy balance during soil deformations,that the failure mechanism is related to the dependence ofthe bearing capacity on gravity. In this section, therelationship between gravity dependence and the failuremechanism is quantitatively examined by means of limitanalysis to rationally explain the ultimate load intensitiesobtained in the ‰ight experiments.In this paper, the upper bound method is applied forthe bearing capacity problems of shallow footings on c­qsoils. In the general upper bound method, collapse loadsof the footings are obtained by assuming a failuremechanism that satisˆes a kinematically admissible veloc­ity ˆeld, and then by equating the total rate of the internalenergy dissipation within that failure region to the totalrate of the external work done by the force on the footingand soil weight in motion. As mentioned above, the con­cept of the upper bound method is based on energybalance in an assumed failure mechanism and, thus, isconsidered to be a powerful tool in exploring gravity de­pendence.Parkins (1995) conducted model footings loading testson an American lunar soil simulant in a geotechnical cen­trifuge and theoretically predicted the ultimate bearingcapacities through conventional calculation methods in­cluding the limit equilibrium method, the slip­linemethod, and the upper bound method. In his paper, hepoints out that the conventional theories tend to sig­niˆcantly overestimate the experimental values, and thatadequate predictions can be made by introducing the de­pendence of the strength parameters on stress, ratherUpper Bound Method for Shallow Footings on HighlyCompressible SoilsSuppose that the materials have apparent cohesion, c,and internal friction angle, q?, and the footing bases arecompletely rough to meet the experimental conditions.Therefore, the so­called Plandtl mechanism is to beformed below the footing. Since soil deformations aresymmetrical, we can focus only on the right half of thefailure mechanism as shown in Fig. 18. The Plandtl­typegeneral shear failure mechanism commonly consists ofthree distinct regions, namely, the active failure zone(zone oab), the transitional zone (zone abc) and the pas­sive failure zone (zone ace). The transitional zone abcforms a fan­shaped radial shear region that is partitionedby log­spiral curve bc and is composed of a sequence ofsmall rigid triangles which form an angel of du at point a.On log­spiral curve bc radius vector, r, is expressed as rr0 exp (u tan q?), where r0 is the length of line ab. Angle j,which determines the dimension of active failure zoneoab is not yet established. As for angle h, we can say thathp/4|q?/2, by considering that the edge of the foot­ing (point a) is a stress singular point and that the groundsurface (line ae) corresponds with one of the directions ofthe principal stresses. The general shear failure mechan­ism usually forms the passive failure zone from the tran­sitional zone toward the ground surface. In this paper, byincorporating a straight line, called ``the failure bound­ary surface,'' which makes an angle of b from the groundsurface, and by assuming that no deformation occurs onthe outside of the failure boundary surface, we attempt toexplain the punching or local shear failure mechanism inrelation to soil compressibility. This is based on the no­tion that the failure region can be controlled by changingthe value of b according to the soil compressibility.Major factors in contraction of sandy materials can begenerally classiˆed into two mechanisms: 1) formationchange of the soil particles, and 2) particle breakage.Deformed soil particles such as the lunar soil simulantmay cause signiˆcant particle breakage, resulting in mak­ing a decisive impact on the failure mechanism. Althoughthe clariˆcation of the contraction mechanism is an im­portant problem to determine the value of b, the diŠer­ence between the formation change and the particle126KOBAYASHI ET AL.the overburden pressures ( see APPENDIX B). By equat­ing the total internal energy dissipation rate to the totalexternal work rate ( see APPENDIX C), ultimate bearingcapacity, q0, for the proposed failure mechanism can beexpressed as follows:1rNgBNg(j, b, q?)2q0cNc(j, b, q?){(2)where r: bulk density of the soil, Ng: gravitational ac­celeration level, B: footing breadth. Nc(j, b, q?) and Ng(j, b, q?) represent the bearing capacity factors withrespect to cohesion and self weight of soil, respectively,and are given as a function of j, b and q? as follows:When bºh (when b is comparatively small and failureboundary surface cd is inside passive failure zone ace):f1{f2{f3g1(3)|g2{g3{g4{0.5h12g1 cos j(4)Nc(j, b, q?)`bºhNg(j, b, q?)`bºhFig. 18.Assumed failure mechanismbreakage has not been taken into consideration in thispaper.The assumption that no deformation generates on theoutside of the failure boundary surface yields occurrenceof ``convergence'' of the volume along the line ad.However, it can be said that the solutions from this calcu­lation will provide upper bound values, because the ve­locity ˆelds within the assumed failure zone still satisˆesthe boundary and the compatibility conditions of the ve­locity.In the upper bound method, an equation related to thecollapse load is built by calculating the total internalenergy dissipation rate and the total external work rate inthe assumed failure mechanism, and by equating the sumof these values. Furthermore, the solutions are obtainedby changing the geometrical parameters that determinethe failure region to minimize the collapse load.Energy dissipates along the velocity discontinuity. Ve­locity discontinuities shown in Fig. 18 are as follows: 1)boundary surface ab formed between active failure wedgeoab and transitional zone abc; 2) boundary surface bcformed between the interior side of transitional zone abcand its exterior side; and 3) boundary surface cd (which isseen only if Bºh), formed between the passive failurezone and the exterior side. The calculations of the energydissipations for velocity discontinuities are detailed inAPPENDIX A.The total external work rate is based on two concepts:work done by the footing load and work done by the selfweight of soil wedges. Now, in this paper, based on thesupposition that the overburden pressures correspondingto the depth are in eŠect on failure boundary surface ad,a formulation is achieved by taking into consideration therate of work done by the failure boundary surface againstWhen hÅbÅp|j (when b is comparatively large andfailure boundary surface cd is inside transitional zoneabc):f1{f4g1(5)|g2{g5{0.5h22g1 cos j(6)Nc(j, b, q?)`hÅbÅp|jNg(j, b, q?)`hÅbÅp|jwheresin j cos q?cos (j|q?)(7)exp s2(p|j|h) tan q?t|1tan q?(8)sin (h|b) cos q? exp s2(p|j|h) tan q?tcos (h|b{q?)(9)f1f2f3exp s2(p|j|b) tan q?t|1tan q?(10)cos j cos q?cos (j|q?)(11)sin j cos j cos q?cos (j|q?)(12)f4g1g2BEARING CAPACITY IN LOW GRAVITY ENVIRONMENTsin j{3 cos j tan q?{(sin h{3 cos h tan q?) exp s3(p|j|h) tan q?t21{9 tan q?(13)cos h sin (h|b) cos q?exp s3(p|j|h) tan q?tcos (h|b{q?)(14)sin j{3 cos j tan q?{(sin b{3 cos b tan q?) exp s3(p|j|b) tan q?t21{9 tan q?(15)h1sin b cos2 q?s(2|sin q?) cos (h|b){sin q? cos (h{b)texp s3(p|j|h) tan q?tcos2 (h|b{q?)(16)h2sin bs2{(cos 2b|1) sin q?texp s3(p|j|h) tan q?t(17)g3g4g5The upper bound solution for the ultimate bearingcapacity is a problem of minimization of q0 in Eq. (2).Speciˆcally, the upper bound value of q0 is obtained byspecifying the soil parameters and the angle of b, and bychanging the angle of h to minimize q0. This minimiza­tion procedure can be easily executed by using an optimi­zation tool available in most spreadsheet software pack­ages. In this study, the Solver optimization tool inMicrosoft Excel was used.Soil Governing ParametersA division of both sides of Eq. (2) by cohesion yields anon­dimensionalized ultimate bearing capacity as fol­lows:q0Nc(j, b, q?){GNg(j, b, q?)c(18)where G is a non­dimensional parameter expressed as GrNgB/2c, and this equation shows that a non­dimen­sionalized ultimate bearing capacity can be expressed as alinear combination of bearing capacity factors and G.At this point, we perform the ``dimensional analysis''in this problem. The dimensional analysis allows us toderive the relationship between physical quantities thatgovern the deformation ˆelds. When c, q? and r are chos­en as the physical quantities of soil properties, the ulti­mate bearing capacity can be non­dimensionalized by c orrNgB/2, and can be expressed as the functions of non­dimensionalized parameters as follows:Ø127»rNgBq0hc, q? hc(G, q?)c2cØ»(19)Øq02c1hrNgB/2, q? hrNgB/2, q?rNgB/2rNgBG»(20)Equations (19) and (20) demonstrate that the non­dimen­sionalized ultimate bearing capacity is expressed as afunction of G and q?. This means that, even if eachparameter that constitutes G changes individually, thenon­dimensionalized ultimate bearing capacity is un­changed, as long as both G and q? remain constant. Con­sequently, general results that can apply to anyphenomenal scale can be obtained. As Chen (1975)mentioned, the soil behaves essentially as a cohesiveweightless medium if G is small. If G is large, the soilweight, rather than its cohesion, is the principal source ofbearing strength. Therefore, G can be termed a ``soilgoverning parameter''. The soil governing parameter alsosuggests that reducing the gravity level to 1/6 has thesame eŠect on the bearing capacity as scaling down thefooting breadth to 1/6 or increasing the cohesion by afactor of six. Equations (18) and (19) show that q0/c de­pends only on q? and G and, therefore, the followinganalyses were performed using these parameters as in­dependent variables.Calculation results and DiscussionsThe calculation results of q0/c obtained from theproposed upper bound analysis are as shown in Fig. 19.The solutions were obtained upon changing the angle of bwhich regulates the dimension of failure regions. The so­lution for the case of b09(Fig. 19(a)) agrees with theconventional result that is obtained by assuming a generalshear failure mode as Chen (1975), etc. presented. Theˆgure shows that the ultimate bearing capacity decreasesas b increases, and the sensitivities of G and q? toward q0/c becomes weaker. To examine this further, the sensitiv­ities of G and b toward q0/c for highly frictional materi­als (q?409and 509), such as the lunar soils (simulant),are illustrated in Fig. 20. This ˆgure reveals that the non­dimensionalized ultimate bearing capacity is hardly in­‰uenced by G, regardless of the angle of b, when G is ap­proximately less than 0.1. This means that gravity has lesseŠect when the footing breadth is narrow or the soil ishighly cohesive. In practice, however, careful attention isrequired for a discussion on G since the amount of appar­ent cohesion, particularly in dry sandy soils, is compara­tively smaller than that of cohesive soils and may vary sig­niˆcantly depending on how a regression line of thefailure envelope is drawn, as a result of which the soilgoverning parameter also varies greatly. From a practicalpoint of view, it has to be said that G has many uncertain­ties. Thus, it must be noted that the discussions on G hereprovide no more than theoretical ˆndings. From a theo­retical point of view, the following discussions are at­tempted to clarify the relationships between parameters.In the case of the lunar soil simulant of Dr90z wherer1.95 g/cm3 and c7.54 kN/m2 ( see Table 3), footing128Fig. 19.KOBAYASHI ET AL.Calculation results for the assumed failure mechanismsFig. 21. Soil governing parameter versus non­dimensionalized ulti­mate load intensityFig. 20.EŠect of b and rNgB/2c on ultimate bearing capacitybreadth, B, must be approximately less than 0.08 m inorder to satisfy Gº0.1 in the 1 g condition (N1). Inother words, gravity hardly in‰uences the bearing charac­teristics of the densely packed simulant if B is less than0.08 m. In the case of Dr60z where r1.77 g/cm3 andc1.44 kN/m2 (also see Table 3), B must be less than0.02 m. The lunar soil parameters recommended in theLunar Sourcebook ( see Table 1) (where the depth rangeis the average value between 0 and 60 cm, r1.66 g/cm3and c1.66 kN/m2) also require Bº0.02 m. Accord­ingly, it can be concluded that the bearing capacities ofshallow footings with the widths of several tens of cen­timeters or more are signiˆcantly aŠected by gravityregardless of the failure mode.Now, the experimental relationships shown in Fig. 14are non­dimensionalized in terms of G and qu/c to rear­range the experimental results as shown in Fig. 21. Thenon­dimensionalization of qu is achieved by using the co­hesion shown in Table 3. As previously mentioned in Eq.(18), q0/c can be expressed in a straight line with a slopeof Ng and an intercept of Nc. Therefore, Ng and Nc can berevealed through linear approximations of the ex­perimental data. As shown in Fig. 21, it turns out that thelunar soil simulant of Dr60z yielded Nc21.8 and Ng60.0, and in the case of Dr90z, it yielded Nc38.7and Ng337.8. Toyoura sand yielded Nc6.3 and Ng378.0 when Dr60z, and Nc13.9, and Ng694.9when Dr90z. Since q0/c becomes more susceptible toBEARING CAPACITY IN LOW GRAVITY ENVIRONMENT129Fig. 23. Reduction ratio of ultimate bearing capacities in 1/6 g tothose of 1 gFig. 22.EŠect of b on bearing capacity factorsgravity as the slope of the linear relationship becomesgreater, it can be said that the magnitude of Ng representsthe sensitivity of gravity against the bearing capacity. Themagnitude of Ng for the lunar soil simulant is small com­pared to that of Toyoura sand, suggesting that the lunarsoil simulant is less dependent on gravity than Toyourasand.Figure 22 shows the relationships between the internalfriction angles and the bearing capacity factors obtainedfrom the upper bound calculations. Although the calcula­tion results should show dependence on G, no considera­ble diŠerence in the bearing capacity factors is seen withinthe range of 0.01ÅGÅ10.0. Figures 22 (a) and (b)demonstrate that the angle of b signiˆcantly in‰uencesnot only Ng but also Nc. This suggests that the diminish­ment of the failure region that is associated with the in­crease in b results in a reduction of the total rate of exter­nal work by the self weight of the soil particles, as well asa reduction of the failure region that generates internalenergy dissipation. Moreover, the ˆgure also shows thatthe sensitivities of q? toward both bearing capacity fac­tors become greater as b becomes smaller. In addition tothe theoretical relationships, experimental values of Ngand Nc for the lunar soil simulant are plotted on thecharts with respect to each internal friction angle ( seeTable 3). These ˆgures show that experimental value of, while Ng is found inNc is found in the vicinity of b509the vicinity of b459. This means that the proposed up­per bound calculation when b45 to 509allows us totake into account the simulant's dependence on gravityand to predict the ultimate bearing capacity on the lunarsoil simulant. The proposed upper bound analysis is in­teresting in that the soil compressibility (material condi­tion) and the gravity dependence (environmental condi­tion) can be rationally associated to one another.Although a method of estimating b as part of the calcula­tion procedure has not yet been proposed, it may becomepossible to correlate b with a soil compressibilityparameter since it determines the domain of the failureregion.Through the application of the method of stress char­acteristics, Kobayashi et al. (2005, 2007) demonstratedhow much reduction of the collapse loads for founda­tions and retaining walls could be estimated in the lunargravity environment as opposed to the 1 g condition. Inthis paper, the relationship between the soil governingparameter, G, and the reduction ratio, R, whichrepresents the ratio of the ultimate bearing capacityreduction in the 1/6 g condition against that in the 1 gcondition, is identiˆed using the proposed upper boundmethod. Figure 23 shows the calculation results. Thehorizontal axis of the charts represents the G values in the1 g condition. Figure 23(a) shows the variations of R with130KOBAYASHI ET AL.respect to G and q? in the cases where the optimal valueof b for the simulant is set at 459. According to theresults, reduction ratio, R, is hardly in‰uenced by Gwhen it is less than 0.01, while the larger reductions areseen as G increases to around 0.01 or greater. It can alsobe seen from the ˆgure that the reduction trend becomessteeper as q? increases. Figure 23(b) shows the results ofR with respect to G and b in the case of q?499, which isthe recommended value of the lunar soils for the depthrange of 0 to 60 cm ( see Table 1). This ˆgure illustratesthat the reduction trend becomes more moderate when bis 459or greater. What is noteworthy is that the reductiontrend does not always become steeper as b decreases.That is to say, no variation can be seen in the reductionratio, and the reduction curve can be expressed only as afunction of G once the failure mechanism reaches thegeneral failure mode. Both of these ˆgures clearly indi­cate that R is very much dependent on G. As can be un­derstood from Eq. (18), an increase in G means that theeŠect of the soil weight on bearing capacity becomes rela­tively greater and, thus, the dependence on gravitybecomes more notable as G increases. In other words, itcan be said that the reduction ratio is determined by thebalance of the contribution ratio of the cohesion and thatof the self weight of the soils in Eq. (18). Generally, awider footing breadth is more favorable when attaininggreater bearing capacity. However, as the ˆgure shows,the wider the footing breadth is, the greater G becomes,which then results in a higher reduction of bearing capaci­ty in a low gravity environment. According to the ``scaleeŠect'' as it is generally known, Ng decreases as B in­creases. The fact that the reduction ratios vary dependingon G may be positioned as another ``scale eŠect'' in termsof gravity dependence.CONCLUSIONSThe main conclusions of this study are summarized asfollows:1) From the triaxial compression tests, it became clearthat the lunar soil simulant exhibits a signiˆcant de­pendence on stress and density compared to Toyourasand. In a dense state, the lunar soil simulant exhibitsnot only a large internal friction angle, but also highlyapparent cohesion. As a result of the one­dimensionalcompression tests, it was found that the lunar soilsimulant has higher compressibility than that of Toy­oura sand, indicating that the deformation mechanismof the lunar soil simulant changes signiˆcantly accord­ing to the type of loading. That is to say, the lunar soilsimulant exhibits a great dilation during shearing,while also showing large contractions during isotropiccompressions.2) As a result of the model loading experiments of a shal­low footing on an aircraft that ‰ew in parabolic paths,it became clear that gravity hardly in‰uences thecoe‹cient of subgrade reaction, Ks, of the lunar soilsimulant, while that of Toyoura sand varies propor­tionally with the gravity levels. A similar trend can beseen for ultimate load intensities, qu. That is, qu ofToyoura sand is again in proportion to the gravity lev­els, while no obvious proportionality can be seen inthe lunar soil simulant. Moreover, the amount offooting settlement at collapse in the lunar soilsimulant is independent of the gravity levels in a lowgravity environment, while Toyoura sand shows sig­niˆcant gravity dependence.3) From the observation of the soil deformations usingthe PIV technique, it became clear that the lunar soilsimulant demonstrates a punching or local shearfailure mode even in a dense state, while Toyoura sandexhibits a general shear failure mode. In the case ofthe punching or local shear failure mode, downwardsoil displacements are predominant and the workagainst gravity exhibited by the outward and/or up­ward soil displacements are rather small compared tothose in the general shear failure mode. Therefore, theaforementioned experimental results can be explainedby the failure mechanism because gravity has lesseŠect on the bearing characteristics when the soilgenerates a punching or local shear failure mode, andthe level of dependency on gravity is closely related tothe compressibility of the materials.4) To theoretically investigate the relationships betweengravity dependence and soil compressibility, a modi­ˆed upper bound method was proposed by incorporat­ing the concept of ``failure boundary surface,'' whichforms an angle of b from the ground surface. Theproposed method makes it possible to quantitativelycorrelate the failure mode with gravity dependence.Moreover, it was found that the in‰uence of gravitybecomes less, regardless of the failure mechanism,when the footing breadth is approximately less than0.02 m. The proposed upper bound method also sug­gests that the ultimate bearing capacity is hardlyreduced in the lunar gravity condition (1/6 g condi­tion) when the soil governing parameter, G, is lessthan 0.01, while the reduction becomes notable whenG increases to around 0.01 or greater. The comparisonof the calculation results with the experimental resultsdemonstrates that the ultimate bearing capacity of thelunar soil simulant can be evaluated by tuning b45to 509, indicating that the proposed method with anappropriate value of b would allow us to predict theultimate bearing capacity in the lunar surface environ­ment.ACKNOWLEDGEMENTSThis study was carried out as part of ``GroundResearch Announcement for Space Utilization'' promot­ed by Japan Aerospace Exploration Agency and JapanSpace Forum, and the authors are grateful for their sup­port for this study.REFERENCES1) Boles, W. W., Scott, W. D. and Connolly, J. F. (1997): Excavation131BEARING CAPACITY IN LOW GRAVITY ENVIRONMENT2)3)4)5)6)7)8)9)10)11)12)13)14)15)16)17)forces in reduced gravity environment, Journal of Aerospace En­gineering, American Society of Civil Engineers, 10(2), 99–103.Carrier, W. D. III, Mitchell, J. K. and Mahmood, A. (1973): Thenature of lunar soil, Journal of Soil Mechanics and FoundationsDivision, American Society of Civil Engineers, 99 (SM10),813–832.Carrier, W. D. III, Olhoeft, G. and Mendell, W. W. (1991): Chap­ter 9; Physical properties of lunar surface, Lunar Sourcebook,Cambridge University Press, 475–567.Chen, W.­F. (1975): Chapter 6; Bearing capacity of strip footings,Limit Analysis and Soil Plasticity, Development in GeotechnicalEngineering 7, Elsevier Scientiˆc Publishing Company, 220–222.Ettouney, M. M. and Benaroya, H. (1992): Regolith mechanics,dynamics, and foundations, Journal of Aerospace Engineering,American Society of Civil Engineers, 5(2), 214–229.Johnson, S. W. (1989): Extra­terrestrial Facilities Engineering, En­cyclopedia of Physical Science and Technology, 1989 Yearbook,Academic Press, Inc., 90–123.Kanamori, H., Udagawa, S., Yoshida, T., Matsumoto, S. andTakagi, K. (1998): Properties of lunar soil simulant manufacturedin Japan, Proc. 6th International Conference on Engineering, Con­struction and Operations in Space, American Society of Civil En­gineers, 462–468.Klein, S. P. and White, B. R. (1990): Dynamic shear of granularmaterial under variable gravity conditions, AIAA Journal, Ameri­can Institute of Aeronautics and Astronautics, 28(10), 1701–1702.Klosky, J. L., Sture, S., Ko, H. Y. and Barnes, F. (2000): Geo­technical behavior of JSC­1 lunar soil simulant, Journal of Aero­space Engineering, American Society of Civil Engineers, 13(4),133–138.Kobayashi, T., Ochiai, H., Yasufuku, N. and Omine, K. (2005):Prediction of soil collapse by lunar surface operations in reducedgravity environment, Proc. 15th International Conference of theInternational Society for Terrain­Vehicle Systems, InternationalSociety for Terrain­Vehicle Systems, (on CD­ROM).Kobayashi, T., Ochiai, H. and Yasufuku, N. (2007): Mechanicalproperties and bearing capacity of lunar surface, Proc. 13th AsianRegional Conference on SMGE, 1(Part 1), 149–152.Kobayashi, T., Ochiai, H., Yamakawa, J., Aoki, A., Matsui, K.and Miyahara, A. (2008): Mobility characterization of planetaryrover in reduced gravity environment, Space Technology and Ap­plications International Forum–STAIF 2008, AIP ConferenceProceedings, American Institute of Physics, (969), 769–775.McKay, D. S., Carter, J. L., Boles, W. W., Allen, C. C. andAllton, J. H. (1994): JSC­1: A new lunar soil simulant, Proc. 4thInternational Conference on Engineering, Construction and Opera­tions in Space, American Society of Civil Engineers, 857–866.Perkins, S. W. (1995): Bearing capacity of highly frictional materi­al, Geotechnical Testing Journal, American Society for Testing andMaterials, 18(4), 450–462.Sture, S., Costes, N. C., Batiste, S. N., Lankton, M. R., AlShibli,K. A., Jeremic, B., Swanson, R. A. and Frank, M. (1998): Mechan­ics of granular materials at low eŠective stresses, Journal of Aero­space Engineering, American Society of Civil Engineers, 11(3),67–72.Weiblen, P. W., Murawa, M. J. and Reid, K. J. (1990): Prepara­tion of simulants for lunar surface materials. Engineering, Proc.2nd International Conference on Engineering, Construction andOperations in Space, American Society of Civil Engineers, 98–106.Willman, B. M., Boles, W. W., McKay, D. S. and Allen, C. C.(1995): Properties of lunar soil simulant JSC­1, Journal of Aero­space Engineering, American Society of Civil Engineers, 8(2),77–87.bc and cd. Angular parameters that determine the geo­metry of the failure mechanism, namely, j, h and b, aregiven as shown in Fig. 18(a) and ultimate bearing capaci­ty, q0, and overburden pressures according to depth level,qb( y), are applied on lines oa and ad, respectively, asshown in Fig. 18(b). Value of r0 represents the length ofline ab. When the footing penetrates the soil, the footingand the soil wedge underneath it move downward with avelocity of V0. In transition zone abc, which is composedof a sequence of small rigid triangles that form an angleof u at point a, the i­th small triangle moves with a veloc­ity of V1, i. Furthermore, zone acd move in the form ofrigid bodies with a velocity of V2 in directions that forman angle of q? with the discontinuity lines as shown inFig. 18(b). As described in Chen's book (1975), the rateof energy dissipation is given by multiplying the length ofdiscontinuity line by c times the velocity diŠerence acrossthe line multiplied by cos q?.Along ab: Based on the geometrical relationship, the rela­tive velocity of the ˆrst small triangle in the transitionalzone with respect to soil wedge oab, V0ª1, 1 can be ex­pressed as follows:V0ª1, 1V1, 1 cos jcos (j|q?)(21)Since relative velocity V0ª1, 1, represents an incrementaldisplacement, the rate of energy dissipation along ab isgiven asDabcV1, 1r0 f1(22)wheresin j cos q?cos (j|q?)f1In the case where bºh (when b is comparatively smalland failure boundary surface, cd, is in passive failurezone ace);In transitional zone abc: the radius vector and the veloc­ity of the i­th segment are expressed as rr0 exp (u tanq?) and V1, iV1, 1 exp (u tan q?), respectively. Given thatdu is inˆnitesimal, the relative velocity of V1, i with respectto V1, i{1 can be approximated as follows:duV1, icos q?V1, iª1, i{1(23)Thus, the rate of energy dissipation along the i­th radiusvector yields criV1, idu, and, consequently, the rate ofenergy dissipation in transition zone abc is given as:DabccV1, 1r0fp|j|h0exp (2u tan q?)du1cV1, 1r0f22(24)whereAPPENDIX A: RATE OF INTERNAL ENERGYDISSIPATIONIn Fig. 18, the internal energy dissipations can be seenin transitional zone abc, and along boundary surfaces ab,exp s2(p|j|h) tan q?t|1tan q?f 2Along bc: the arc length of the discontinuity in the i­thsegment is expressed as ridu/cos q?, and the rate of132KOBAYASHI ET AL.energy dissipation along the arc is given as criV1, idu.Therefore, the rate of energy dissipation along bc is givenas:fDbdcV1, 1r0p|j|hDbccV1, 1r00exp (2u tan q?)du1 cV1, 1r0f22(25)It can be said that Eq. (25) coincides with Eq. (24).Along cd: the length of line ac is, r0 exp s(p|j|h) tanq?t, so the length of line cd is given as:sin (h|b)r0 exp s(p|j|h) tan q?tcos (h|b{q?)Lcd(26)Since V2V1, 1 exp s(p|j|h) tan q?t, the rate of energydissipation along cd is expressed as:DcdcV1, 1r0 f3sin (h|b) cos q? exp s2(p|j|h) tan q?tcos (h|b{q?)f0exp (2u tan q?)du(28)whereexp s2(p|j|b) tan q?t|1tan q?f 4Along bd: resetting the integral domain du in Eq. (25) tothe range from 0 to p|j|b yields a rate of energy dissi­pation along bd as follows:Wabc1rNgV1, 1r202|1cV1, 1r0 f42(29)The total rate of external work consists of three com­ponents, namely, work done by the footing load, workdone by the self weight of the soil wedges and work doneby the failure boundary surface against the overburdenpressures.The rate of external work done by the footing load: thevelocity of the footing and soil wedge oab is:V0 cos q?V1, 1cos (j|q?)Wfq0V1, 1r0g1In the case where hÅbÅp|j (when b is comparativelylarge and failure boundary surface cd is in transitionalzone abc);In transitional zone abd: the rate of energy dissipation inthe transitional zone can be derived in the similar manneras that applied in Eq. (24). The integral domain of du inEq. (24) is reset to the range from 0 to p|j|b, and therate of energy dissipation is given as:1 cV1, 1r0f42exp (2u tan q?)du(30)Thus, by multiplying the footing load, the rate of exter­nal work done by the footing load is given as:f3DabdcV1, 1r00APPENDIX B: RATE OF EXTERNAL WORK(27)wherep|j|bfp|j|bfp|j|h0(31)wherecos j cos q?cos (j|q?)g1In soil wedge aob: the rate of external work done by soilwedge aob is given by multiplying its self weight by thevertical component of the velocity as follows:Waob1rNgV1, 1r 20g22(32)wheresin j cos j cos q?cos (j|q?)g2In the case where bºh;In transitional zone abc: the rate of external work doneby the self weight of the i­th small triangle is:Wi 1rNgV1, ir 2i cos (u{j)du2(33)Substituting rir0 exp s(p|j|h) tan q?tand V1, iV1, 1exp su tan q?tin Eq. (33) and integrating it with thewhole area, the rate of external work done by the selfweight in the transitional zone can be expressed as:cos (u{j) exp (3u tan q?)du1rNgV1, 1r 20g32wheresin j{3 cos j tan q?{(sin h{3 cos h tan q?) exp s3(p|j|h) tan q?t21{9 tan q?g3In passive failure zone acd: in the geometrical condition, the self weight of triangle acd is:(34)133BEARING CAPACITY IN LOW GRAVITY ENVIRONMENT1 2 sin (h|b) cos q?r0exp s2(p|j|h) tan q?tcos (h|b{q?)2(35)Also, the vertical component of V3 is:V3|V1, 1 cos h exp s(p|j|h) tan q?t(36)Thus, the rate of external work in passive failure zone acd is given as:Wacd|1rNgV1, 1r20g42(37)wherecos h sin (h|b) cos q?exp s3(p|j|h) tan q?tcos (h|b{q?)g4In the case where hÅbÅp|j;In transitional zone abd: the rate of external work done by the transitional zone can be derived in the similar manner asthat applied in Eq. (34). The integral domain of du in Eq. (34) is reset to the range from 0 to p|j|b, and the rate ofexternal work is:Wabd1rNgV1, 1r 202|fp|j|b0cos (u{j) exp (3u tan q?)du1rNgV1, 1r 20g52(38)wheresin j{3 cos j tan q?{(sin b{3 cos b tan q?) exp s3(p|j|b) tan q?t21{9 tan q?g5The rate of external work done by the failure boundarysurface against the overburden pressures: suppose thatthe stresses in triangle ade is in a state of at­rest, and theoverburden pressures with respect to the depth levels areapplied on failure boundary surface ad. The vertical andhorizontal earth pressures at depth z are:svrNgz(39)shK0svK0rNgz(40)where sv, sh: vertical and horizontal earth pressures, re­spectively, K0: coe‹cient of earth pressure at rest. Sinceline ad forms an angle of b from the ground surface, thenormal and shear stresses along line ad are:1sb rNgz s1{cos 2b{K0(1|cos 2b)t2tb 1rNgz(1|K0) sin 2b2DWad, bºh1rNgzV1, 1 exp s(p|j|h) tan q?t2~[(1{K0) cos (h|b){(1|K0) cos (h{b)](43)Based on the geometry of triangle acd the length of thefailure boundary surface Lad is:Ladcos q?r0 exp s(p|j|h) tan q?tcos (h|b{q?)(44)By integrating Eq. (43) along line ad the rate of externalwork done by the failure boundary surface is given as:Wad(bºh)|(41)1rNgV1, 1r 204sin b cos2 q?[(1{K0) cos (h|b){(1|K0) cos (h{b)]cos2 (h|b{q?)~(42)where sb, tb: normal and shear stresses along line, ad, re­spectively.In the case where bºh;The work is given as a scalar product of a stress vectorand a velocity vector. Assuming that the failure boundarysurface moves with a velocity of V2V1, 1 exp s(p|j|h)tan q?t, the rate of work done by a unit length on line adat any depth, z, is:~exp s3(p|j|h) tan q?t(45)The lunar surface is considered to be under a normallyconsolidated condition. In this paper, the coe‹cient ofthe earth pressure at rest, K0 is considered to followJaky's formula, K01|sin q?. Hence, Eq. (45) is rear­ranged as follows:Wad(bºh, K 1|sin q?)|0where1rNgV1, 1r 20h14(46)134KOBAYASHI ET AL.The length of r0 can be expressed with footing width, B asr0B/2 cos j. Substituting this relation in Eq. (50) andsin b cos2 q?s(2|sin q?) cos (h|b){sin q? cos (h{b)t rearranging the terms with respect to ultimate bearingcapacity, q0, the following formula can be obtained.cos2 (h|b{q?)h1~exp s3(p|j|h) tan q?tq0cNc(j, b, q?)`bºh{In the case where hÅbÅp|j;In the transitional zone, the velocity vector is or­thogonal to the radius vector and, thus, the rate of workdone by the unit length on failure boundary surface ad atany depth z is given as a scalar product of sb and V2, thatis:1rNgBNg(j, b, q?)`bºh2(51)wheref1{f2{f3,g1|g2{g3{g4{0.5h1Ng(j, b, q?)`bºh2g1 cos jNc(j, b, q?)`bºh1In the case where hÅbÅp|j;rNgzV1, 1 exp s(p|j|h) tan q?tThe equivalent expression of the energy balance can as2~[1{K0{(1|K0) cos 2b](47) be follows:DWad, hÅbÅp|jBy integrating Eq. (47) along line ad the rate of externalwork done by the failure boundary surface is given as:1Wad(hÅbÅp|j)| rNgV0r20h24(48)Dab{Dabd{DbdWf{Waob{Wabd{WadThat is,11cV1, 1r0 f4{ cV1, 1r0 f42211q0V1, 1r0g1{ rNgV1, 1r 20g2| rNgV1, 1r 20g52212| rNgV1, 1r 0h2(53)4cV1, 1r0 f1{whereh2sin bs2{(cos 2b|1) sin q?t~exp s3(p|j|h) tan q?tAPPENDIX C: SOLUTIONS OF THE ASSUMEDFAILURE MECHANISMBy rearranging the terms of Eq. (53) with the relationshipof r0B/2 cos j, ultimate bearing capacity, q0, is given asfollows:In the case where bºh;Equating the total internal energy dissipation rate tothe total external work rate gives the following relation:Dab{Dabc{Dbc{DcdWf{Waob{Wabc{Wacd{Wad(49)That is,11cV1, 1r0 f2{ cV1, 1r0 f2{cV1, 1r0 f32211q0V1, 1r0g1{ rNgV1, 1r20g2| rNgV1, 1r20g322112| rNgV1, 1r0g4| rNgV1, 1r20h1(50)24cV1, 1r0 f1{(52)q0cNc(j, b, q?)`hÃbÃp|j{1rNgBNg(j, b, q?)`hÃbÃp|j2wheref1{f4,g1|g2{g5{0.5h2.Ng(j, b, q?)`hÃbÃp|j2g1 cos jNc(j, b, q?)`hÃbÃp|j(54)
• ログイン

• タイトル
• Evaluating Model Uncertainty of an SPT-based Simplified Method for Reliability Analysis for Probability of Liquefaction
• 著者
• C. H. Juang・S. Y. Fang・W. H. Tang・E. H. Khor・G. T.-C. Kung・J. Zhang
• 出版
• Soils and Foundations
• ページ
• 135〜152
• 発行
• 2009/02/15
• 文書ID
• 21175
• 内容
• SOILS AND FOUNDATIONSJapanese Geotechnical SocietyVol. 49, No. 1, 135–152, Feb. 2009EVALUATING MODEL UNCERTAINTY OF AN SPT­BASEDSIMPLIFIED METHOD FOR RELIABILITY ANALYSISFOR PROBABILITY OF LIQUEFACTIONC. HSEIN JUANGi),ii), SUNNY YE FANGiii), WILSON H. TANGiv),ENG HUI KHORv), GORDON TUNG­CHIN KUNGvi) and JIE ZHANGvii)ABSTRACTIn this paper, an innovative procedure is developed for estimating the uncertainty of an empirical geotechnicalmodel. Here, the Youd et al. (2001) method, a deterministic model for liquefaction triggering evaluation, is examinedfor its model uncertainty. The procedure for evaluating this model uncertainty involves two steps: 1) deriving a Bayesi­an mapping function based on a database of case histories, and 2) using the calibrated Bayesian mapping function as areference to back­ˆgure the uncertainty of the model. Details of the developed procedure within the framework of theˆrst­order reliability method (FORM) are presented. Using FORM with the calibrated model uncertainty, theprobability of liquefaction can be readily determined, and thus, the results presented in this paper extend the use of theYoud et al. (2001) method.Key words: case history, liquefaction, model uncertainty, probability, reliability, standard penetration test (IGC:D7/E8)parameters, and in this regard, reliability analysis usingFORM may be performed. A rigorous reliability analysisrequires the knowledge of parameter uncertainty as wellas the knowledge of model uncertainty. Once the modeluncertainty of the Youd et al. (2001) method is deter­mined, the deterministic solution obtained from thispopular method can be readily extended to theprobabilistic solution using the well­established FORManalysis. Thus, the results of this paper can extend the useof the Youd et al. (2001) method from a deterministic so­lution to both deterministic and probabilistic solutions.Furthermore, a new procedure for evaluating model un­certainty is developed in this paper, which is, by itself, asigniˆcant contribution to the theoretical side of thegeneral reliability analysis.Model uncertainty of a geotechnical model, particu­larly for those limit state models that are deˆned empiri­cally, is in general di‹cult to determine. Phoon and Kul­hawy (2005) pointed out that model uncertainty may beestimated if a su‹ciently large and representative data­base is available. In the present study, the database of li­quefaction/no­liquefaction case histories compiled andINTRODUCTIONThis paper deals with two related problems; one ischaracterization and estimation of model uncertainty, theuncertainty of a given geotechnical model, and the otheris reliability analysis of liquefaction potential of soils us­ing First Order Reliability Method (FORM). Here, a newprocedure is developed for estimating model uncertaintywithin the framework of FORM (Ang and Tang, 1984).As an example to demonstrate this new procedure, a sim­pliˆed model based on Standard Penetration Test (SPT)for evaluating liquefaction potential of soils, originatedby Seed and Idriss (1971) and updated by Youd et al.(2001), is examined.The SPT­based simpliˆed method documented inYoud et al. (2001) is generally recognized as the state­of­the­art method for liquefaction evaluation. The Youd etal. (2001) method is a deterministic method, and liquefac­tion potential determined by this method is generally ex­pressed as a factor of safety. In many occasions,however, it is desirable to determine the probability of li­quefaction to account for the uncertainty in the inputi)ii)iii)iv)v)vi)vii)Professor, Department of Civil Engineering, Clemson University, South Carolina, USA (hsein—clemson.edu).Chair Professor, Department of Civil Engineering, National Central University, Taiwan.Project Engineer, Ardaman & Associates, FL, USA (formerly Research Assistant, Clemson University, Clemson, South Carolina, USA).Chair Professor, Department of Civil Engineering, Hong Kong University of Science and Technology, Hong Kong.Technical StaŠ, ANSYS, Inc., Probabilistic Design and Optimization Group, PA, USA.Assistant Research Fellow, Sustainable Environment Research Center, National Cheng Kung University, Taiwan (formerly Postdoctoral Fel­low, Clemson University, Clemson, South Carolina, USA).Research Assistant, Department of Civil Engineering, Hong Kong University of Science and Technology, Hong Kong.The manuscript for this paper was received for review on July 7, 2008; approved on November 27, 2008.Written discussions on this paper should be submitted before September 1, 2009 to the Japanese Geotechnical Society, 4­38­2, Sengoku,Bunkyo­ku, Tokyo 112­0011, Japan. Upon request the closing date may be extended one month.135136JUANG ET AL.re­assessed by Cetin (2000) and Cetin et al. (2004), whichinclude estimated statistics (mean and standard devia­tion) of the input parameters, is used for evaluating themodel uncertainty of the Youd et al. (2001) method.The new procedure for estimating model uncertaintyinvolves two steps. First, a mapping function is derivedby means of Bayes' theorem using a database of casehistories. The mapping function allows for an interpreta­tion of probability of liquefaction (PL) based on a relia­bility index (b) calculated from a reliability analysis usingFORM that considers only parameter uncertainty, theuncertainty associated with each input variable in thelimit state model. The procedure for developing such PL­b mapping function through a calibration with ˆeld ob­servations was previously reported by Juang et al. (1999,2000). In the second step, the probabilities obtained fromthe calibrated PL­b mapping function are used as a refer­ence to ``back­ˆgure'' the uncertainty of the limit statemodel. This procedure for estimating model uncertaintywas reported by Juang et al. (2004, 2006). Whereas theframework established by these previous studies is fun­damentally sound, there is a drawback; the issue of priorprobability in the development of PL­b mapping functionwas not addressed, and possible variation in the calibrat­ed mapping function and model uncertainty and theireŠects on the ˆnal probability of liquefaction were notexamined. In this paper, the previous procedures arecombined and reˆned into the new procedure.In a simpliˆed model for liquefaction potential evalua­tion such as Youd et al. (2001), the seismic loading is ex­pressed as cyclic stress ratio (CSR) and the liquefactionresistance is expressed as cyclic resistance ratio (CRR). Tomeasure the potential for liquefaction, factor of safety(FS), deˆned as the ratio of CRR over CSR, is tradition­ally employed. Alternatively, use of probability of li­quefaction to measure liquefaction potential has alsobeen suggested (for examples, Christian and Swiger,1975; Liao et al., 1988; Youd and Noble, 1997; Toprak etal., 1999; Juang et al., 2002; Cetin et al., 2004). However,in the analysis of a future case using these empirical equa­tions, uncertainties or variations in the input variables, ifexist, cannot be entered into the equations (because theyare not required in these equations), and thus, the ob­tained probability for this future case could be subject toerror if the variations in the input variables are sig­niˆcant. To account for the variations in the input varia­bles, a reliability analysis of soil liquefaction may be con­ducted (for example, Haldar and Tang, 1979). To thisend, a reliability analysis to determine the probability ofliquefaction using FORM is desirable.In order to have a realistic estimate of the probabilityof liquefaction using FORM, it is essential to consider ex­plicitly both the uncertainty in the input variables and theuncertainty in the limit state model. The uncertainty inthe input variables (parameters) is problem­speciˆc, andshould be evaluated by the user applying the proposedmethod. Nevertheless, some guidance for assessing inputparameter uncertainty is provided later in this paper. Onthe other hand, the uncertainty in the limit state model isone main focus of this paper. Once the model uncertaintyis characterized, the analysis using FORM for assessingthe probability of liquefaction can be readily performed,which is another main focus of this paper.Because the analysis using FORM has a strong theoret­ical basis (Ang and Tang, 1984; Baecher and Christian,2003), the limitations of the proposed approach (tech­nique) arise mostly from the assumptions made in thecalibration of the limit state model. Thus, to apply theproposed technique to practical problems, it is essentialto accommodate the following limitations:1) To compute reliability index (Ang and Tang, 1984)using FORM, the limit state model is assumed to belinear,2) All input random variables are assumed to be log­normally distributed,3) No correlation is assumed between model uncer­tainty and input variables, and4) The model uncertainty of the limit state model, theYoud et al. (2001) method, is calibrated using adatabase of liquefaction case histories compiled byCetin et al. (2004), and as such, the proposed tech­nique is most applicable to future cases that aresimilar in nature to the cases in the database.Fortunately, the database consists of cases from manydiŠerent earthquakes in diŠerent parts of the world andwith a variety of soil conditions, and thusly, the proposedtechnique is applicable to a broad range of seismic andsoil conditions. Further discussion of these limitations(assumptions) is presented later as appropriate.SPT­BASED SIMPLIFIED MODEL FORLIQUEFCATION EVALUATIONIn this paper, the SPT­based simpliˆed model by Youdet al. (2001) is examined for its model uncertainty. Thissimpliˆed model has been, and is still, widely used for li­quefaction potential evaluation in the United States andthroughout much of the world. A brief summary of thismodel is presented to set the stage for the discussion ofmodel uncertainty. In this method, the cyclic stress ratio(CSR) that is adjusted to reference conditions of Mw7.5and eŠective stress s?v100 kPa, denoted as CSR7.5, s,may be expressed as (this form is modiˆed slightly fromthe original form by Seed and Idriss, 1971; the modiˆedform has appeared previously in Juang et al., 2002; Idrissand Boulanger, 2004; Juang et al., 2006; see additionaldiscussion presented later):Ø s?s »Ø ag »(r )/MSF/KCSR7.5, s0.65vvmaxds(1)where svthe total overburden stress at the depth of in­terest (kPa), s?vthe eŠective stress at the depth of in­terest (kPa), gthe unit of the acceleration of gravity,amaxthe peak horizontal ground surface acceleration(amax/g is dimensionless), rdthe depth­dependent stressreduction factor (dimensionless), MSFthe magnitudescaling factor (dimensionless), and Ksthe overburdenstress adjustment factor for the calculated CSR (dimen­EVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHODsionless). The parameter rd is a function of depth wherecyclic stress ratio is calculated, the parameter MSF is afunction of moment magnitude Mw, and the parameterKs is a function of the eŠective stress s?v. For amax, the ge­ometric mean is preferred for use in engineering practice,although use of the larger of the two orthogonal peak ac­celerations is conservative and allowable (Youd et al.,2001).For routine practice and no critical projects, the fol­lowing equations may be used to estimate the values of rd(Liao and Whitman, 1986; Youd et al., 2001):for dº9.15 m,rd1.0|0.00765drd1.174|0.0267d for 9.15 mºdÃ20 m(2a)(2b)where dthe depth of interest (m). The variable MSFmay be calculated with the following equation (Youd etal., 2001):MSF(Mw/7.5)|2.56(3)It should be noted that diŠerent formulas for rd and MSFhave been proposed by many investigators (e.g., Youd etal., 2001; Idriss and Boulanger, 2004; Cetin et al., 2004).To be consistent with the Youd et al. (2001) method, useof Eqs. (2) and (3) is required.As noted previously, the variable Ks is a stress adjust­ment factor used to adjust CSR to the eŠective overbur­den stress of s?v100 kPa. This is diŠerent from the over­burden stress correction factor (CN) that is applied to theSPT blow count (N60), which is described later. The ad­justment factor Ks is deˆned as follows (Hynes and Ol­sen, 1999; Youd et al., 2001):Ks(s?v/Pa)( f|1)(4)where f§0.6 to 0.8. For routine practice and no criticalprojects, f0.7 may be assumed, and thus, the exponentin Eq. (4) would be |0.3.Finally, it should be noted that in the formulation ofthe Youd et al. (2001) method, the factors MSF and Kswere applied to the term cyclic resistance ratio (CRR). Inother words, CRR was multiplied by the term Ks and theterm MSF (both as a multiplier) before comparing withCSR. In this paper, however, both Ks and MSF are ap­plied to the original CSR as a divisor, as shown in Eq. (1),and the corrected CSR, in terms of CSR7.5, s, is then com­pared with CRR for assessing liquefaction potential. Thetwo approaches have the same eŠect but Eq. (1) ispreferred (Juang et al., 2002; Idriss and Boulanger 2004;Juang et al., 2006) because it is desirable to lump theeŠect of MSF and Ks with seismic load parameters andoverburden pressures so that the term CRR would only bedependent on corrected SPT blow count.For the convenience of presentation hereinafter, theadjusted cyclic stress ratio CSR7.5, s is simply labeled asCSR whenever no confusion would be caused by suchuse. For liquefaction potential evaluation, CSR is com­pared with cyclic resistance ratio (CRR). In the SPT­based model by Youd et al. (2001), the CRR is calculatedas:CRR13750N1, 60cs11{{|(5)2135[10¥N1, 60cs{45] 20034|N1, 60cswhere N1, 60cs (dimensionless) is the clean­sand equivalenceof the overburden stress­corrected SPT blow count, de­ˆned as (Youd et al., 2001):N1, 60csa{bN1, 60(6)where a and b are coe‹cients to account for the eŠect ofˆnes content (FC) and both are a function of FC; andN1, 60 is the SPT blow count normalized to the referencehammer energy e‹ciency of 60z and eŠective overbur­den stress of 100 kPa:N1, 60CNN60(7)where N60the SPT blow count at 60 percent hammerenergy e‹ciency and corrected for rod length, samplerconˆguration, and borehole diameter (Skempton, 1986;Youd et al., 2001) andCN( Pa/s?v)0.5Ã1.7(8)where Paatmosphere pressure (§100 kPa). The coe­‹cients, a and b, in Eq. (6) are related to ˆnes content(FC) as follows (Youd et al., 2001):a0 for FCÃ5zexp [1.76|(190/FC2)] for 5zºFCº35z5.0 for FCÆ35z(9)b1.0 for FCÃ5z[0.99{(FC1.5/1000)] for 5zºFCº35z1.2 for FCÆ35z(10)In reference to CSR deˆned in Eq. (1), the equation forCRR (Eq. (5)) deˆnes the boundary curve in a two­dimen­sional liquefaction evaluation chart. In the deterministicapproach, factor of safety (FS), deˆned as FSCRR/CSR, is used to measure liquefaction potential. Intheory, liquefaction is said to occur if FSÃ1, and no li­quefaction if FSÀ1. The entire process of determiningCSR, CRR, and FS through the use of Eqs. (1) through(10) is the SPT­based deterministic model adopted in thispaper. This set of equations collectively is referred to asthe modiˆed Youd et al. (2001) model. The modiˆcationto the original Youd et al. (2001) method is very minorand the two methods yield the same factor of safety forany given case. Nevertheless, the two do not have exactlythe same formulation, and to avoid the unnecessary con­fusion, this deterministic model is referred to as the mo­diˆed Youd et al. (2001) model.As noted previously, the Youd et al. (2001) model iswidely used. However, it was created by a large group ofexperts in a liquefaction workshop (Youd et al., 1997)with diverse opinions. The uncertainty of this model wasnever evaluated, and thus, it should be of signiˆcant con­tribution to evaluate the model uncertainty of this model.In the sections that follow, the model uncertainty of themodiˆed Youd et al. (2001) model is evaluated, and thereliability analysis using the calibrated model uncertaintyis presented and discussed.138JUANG ET AL.PROCEDURE FOR EVALUATING MODELUNCERTAINTYIn the context of reliability analysis, the liquefactionboundary curve may be taken as a limit state. Accordingto Juang et al. (2006), the limit state model of liquefac­tion triggering may be expressed as:h(x)c*CRR|CSR0(11)where x is a vector of input variables that consist of soiland seismic parameters that are required in the calcula­tion of CRR and CSR (Eqs. (1) through (10)), andh(x)º0 indicates liquefaction. The random variable c isemployed to describe the model uncertainty of the limitstate model. Use of a single random variable to describethe model uncertainty is appropriate because the onlydata available for calibration is in the form of binary ˆeldobservation of liquefaction or no liquefaction. The ran­dom variable c is referred to herein as the model uncer­tainty factor, or simply model factor. The reader isreferred to Ang and Tang (1984) and Juang et al. (2006)for background and use of a model factor to characterizethe model uncertainty.It should be noted that while the form of Eq. (11) ap­pears to be simple at the ˆrst glance, the formulations ofCSR and CRR are highly nonlinear, and thus, the func­tion h(x) is actually highly nonlinear with respect to thebasic input variables that are required in the calculationsof CSR and CRR. In the subsequent reliability analysisusing FORM, these basic variables and their correlationsare considered directly in the analysis.The ultimate goal of this paper is to present a proce­dure for determining the probability of liquefaction usingthe well­established reliability method. The foundationof such reliability analysis is the knowledge of the modeluncertainty of the adopted limit state model (in thispaper, it is the modiˆed Youd et al. model, representedcollectively by Eqs. (1) through (10)). To begin with, thevariables of the limit state model, those that are requiredfor the calculation of CSR and CRR, are ˆrst discussed.Random Variables in the Modiˆed Youd et al. (2001)ModelCSR expressed in Eq. (1) is a function of amax, Mw, s?v,sv, and rd, since MSF is a function of Mw and Ks is a func­tion of s?v as noted previously. The ˆrst four variables,amax, Mw, s?v, and sv are assumed to be random variablesin the reliability analysis to be presented. Selection ofamax, Mw as a random variable is obvious; they accountfor the uncertainty in the seismic loading. Selection of thevariables s?v and sv as a random variable is to account forthe possible uncertainty in the unit weight of soil and thedepth of ground water table. The variable rd is a derivedparameter that is a function of depth (Eq. (2)). Althoughthe depth to liqueˆable layer ( see Eq. (2)) in a particularcase is not necessary a ``certain'' value, CSR and CRR inthe modiˆed Youd et al. (2001) model are evaluated forthe soil at the same given depth, and thus, the variable rdmay be treated as a non­random variable. Of course,there is signiˆcant uncertainty in the value of rd deter­mined from Eq. (2). Similarly, there is signiˆcant uncer­tainty in the value of MSF and Ks determined from Eqs.(3) and (4), respectively. These are the uncertainties of the``component'' models, as Eqs. (2), (3), and (4) are the in­tegral part of the modiˆed Youd et al. (2001) model. Theuncertainty in the component models (Eqs. (2), (3), and(4)) is eventually re‰ected in the uncertainty of the entiremodel, and thus, in this paper, only the model uncertain­ty of the entire modiˆed Youd model is to be calibrated.Although it may not be ideal to lump the uncertainties ofthe component models into the uncertainty of the entiremodiˆed Youd model, it is a necessity as the only datathat are available for model calibration are the binaryˆeld observations (liquefaction or no liquefaction) thatre‰ect the combined eŠects of CSR and CRR.Through similar reasoning, the variation in the CRRdetermined from Eq. (5) and the associated equations(Eqs. (6) through (10)) may be attributed to two randomvariables, N1, 60 and FC. Again, the uncertainty in thecomponent models (Eqs. (6) through (10)) is not calibrat­ed separately; rather, they are considered integral part ofthe modiˆed Youd et al. (2001) model.Based on the above discussions, a total of six randomvariables, including N1, 60, FC, Mw, amax, s?v, and sv, areidentiˆed in the modiˆed Youd et al. (2001) model. Theuncertainties in these variables, referred to as ``parameteruncertainties,'' are an essential element in a reliabilityanalysis and must be fully addressed. In this paper, thesevariables are assumed to be lognormally distributed ran­dom variables, although other distribution such as nor­mal distribution may also be used. The use of lognormaldistribution is based on two aspects: ˆrst, the measuredgeotechnical parameters are often modeled well with log­normal distribution (JeŠeries et al., 1988); and second,the lognormal distribution prevents negative parametervalues. Previous study (Juang et al., 2000) has shown thatthe diŠerence between the results of using the lognormaldistribution versus the normal distribution in the reliabil­ity analysis is quite modest. Furthermore, even with thispossible diŠerence, the ``induced'' error, if any, can beconsidered as an integral part of the model uncertainty ofthe entire modiˆed Youd model.Based on the above discussion of the random variablesof the modiˆed Youd model, the limit state deˆned in Eq.(11) may be re­written as follows:h(x)c*CRR|CSRh(c, N1, 60, FC, Mw, amax, s?v, sv)0(12)Procedure for Estimating Model FactorThe earlier version of the procedure for estimating orcalibrating model factor in a limit state model such as Eq.(12) has previously described by Juang et al. (2006). Abrief summary is provided in the following. The premiseof this procedure is that the probability of liquefactioncan be inferred from observed ground performanceswithout the knowledge of model uncertainty. In thisregard, Juang et al. (1999) showed that a mapping­func­EVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHODtion that relates the probability of liquefaction ( PL) to re­liability index (b) can be established by applying Bayes'theorem to observed performance data:P( b`L)P(L)P( b`L)P(L){P( b`NL)P(NL)PLP(L`b)(13)where P(L`b)probability of liquefaction for a given b;P( b`L)probability of b, given that liquefaction did oc­cur; P( b`NL)probability of b, given that liquefactiondid not occur; P(L)prior probability of liquefaction;P(NL)prior probability of no­liquefaction.It should be noted that ``sample bias'' (i.e., the bias ina sample or database where the number of liqueˆed casesis greater than the number of non­liqueˆed cases becauseof choice in sampling; for example, see Liao et al., 1988)is not an issue in the derived mapping function. The twoconditional probability functions, P( b`L) and P( b`NL),are derived using liqueˆed data subset and non­liqueˆeddata subset separately. These functions are equally ap­plicable to the population of sites analyzed. Estimationof the prior probabilities, P(L) and P(NL), however, is adiŠerent story. The latter, estimation of the priorprobabilities, is a challenging issue, which is a key ele­ment of this paper and is discussed later.To derive the PL­b mapping function based on Eq.(13), reliability analyses are performed for all cases in thedata set assuming that the model factor is a constant, c1. In other words, reliability analyses are performed con­sidering only the parameter uncertainty as the model un­certainty is assumed to be non­existent. This assumptionis out of necessity because at this point, the knowledge (ormore precisely, statistical characterization) of model fac­tor c is not available. Even though the reliability index bis calculated without the knowledge of the model factor,the probability of liquefaction inferred for a given bbased on the developed PL­b mapping function is consi­dered an adequate approximation of the ``true''probability because the mapping function is calibratedfor this very deˆnition of b using observed ground per­formance data.For convenience of presentation, the reliability indexcalculated with the assumption that the model factor is aconstant, c1, is hereinafter denoted as b1. For this b1,the probability of liquefaction is inferred from the devel­oped Bayesian mapping function, rather than from thenominal concept (i.e., PL1|F( b1) where F is the cu­mulative standard normal distribution function). Theprobability inferred from the Bayesian mapping functionwith a given b1, denoted as PL1, is considered accurate, asreasoned previously, while the probability based on thenominal concept might be subject to error if the ``cor­rect'' model uncertainty is not included in the reliabilityanalysis. If the model factor is known (i.e., with adequatestatistical characterization) and incorporated in the relia­bility analysis, the calculated reliability index, denoted asb3, and the corresponding nominal probability, denotedas PL3, will be accurate theoretically.Under the premise that the probability of liquefactioncan be inferred from observed ground performance, the139probability of liquefaction PL1 inferred from the Bayesianmapping function can then be used as a reference for esti­mation or calibration of the model factor. The idea ofthis calibration is to ˆnd a set of statistical parameters(for example, the mean and standard deviation) of themodel factor c such that the nominal probability PL3 ob­tained from the FORM analysis matches the probabilityPL1 inferred from the Bayesian mapping function that hasbeen calibrated with observed performance data. To im­plement this idea, each of the 201 cases in the data set isanalyzed for PL1 (through a reliability analysis for b1 withan assumption that cconstant1) and PL3 (through areliability analysis for b3 with an assumption that canundetermined random variable). By means of a trial­and­error process with varying statistical parameters, themodel factor in Eq. (12) can be estimated based onminimization of the root­mean­square­error (RMSE) de­ˆned below:NRMSES (Pi1L3|PL1)2N(14)where N is the number of cases in the data set (in thisstudy, N201).In this paper, the above procedure is applied to es­timating the model uncertainty of the modiˆed Youd etal. (2001) model. An improvement on this procedure ismade to better characterize the prior probability in Eq.(13). The eŠect of the variation of the Bayesian mappingfunction on the estimated model factor is also investigat­ed. Once fully calibrated, the model factor can be usedalong with the knowledge of parameter uncertainties inthe reliability analysis of a future case, and the accuratenominal probability of liquefaction can be determinedwith a routine reliability analysis that can be easily im­plemented in a spreadsheet.MODEL FACTOR CALIBRATED BASED ONPERFORMANCE DATADatabase of Liquefaction Case HistoriesThe source of liquefaction/no­liquefaction case histo­ries used in the present study was compiled anddocumented by Cetin (2000), which was later reported inCetin et al. (2004). Cetin (2000) examined a large collec­tion of liquefaction/no­liquefaction case histories frompublished and unpublished records. His ˆnal databaseconsisted of 201 cases (89 non­liqueˆed cases and 112 li­queˆed cases; note that 3 marginal liquefaction caseswere treated as liqueˆed cases herein). These cases werederived from earthquakes from diŠerent parts of theworld. The soils in these case histories ranged from cleangravels and sands to silt mixtures (sandy and clayey silts).The depths at which the cases were reported ranged from1.05 m to 20.5 m. The corrected SPT blow count N1, 60ranged from 2 to 64.3, and the ˆnes content in percentranged from 0 to 92. The vertical eŠective and totalstresses s?v and sv in kPa were in the ranges of 8 to 199,and 15.5 to 384, respectively. The peak horizontal ground140JUANG ET AL.surface acceleration amax ranged from 0.09 g to 0.7 g. Theearthquake's moment magnitude Mw ranged from 5.9 to8.0.Model Uncertainties and Correlations among Input Vari­ablesIn the reliability analysis presented herein, the inputrandom variables are assumed to follow a lognormal dis­tribution, as noted previously. A lognormal distributionrequires knowledge of the mean and standard deviation.For each case history in the database, both the mean andthe standard deviation (or the coe‹cients of variation)are available; they were estimated by Cetin (2000) basedon limited ˆeld data. The reader is referred to Cetin(2000) for additional detail of these case histories andparameter variations.It is noted that the correlations among the six inputrandom variables are also incorporated in the reliabilityanalysis in the present study. To deal with correlated log­normally distributed random variables, the equivalentnormal variables are ˆrst obtained, followed by a trans­formation to the uncorrelated normal space. The readeris referred to the literature (e.g., Der Kiureghian and Liu,1985; Haldar and Mahadevan, 2000) for details regardingthe treatment of correlated non­normal random variablesin the FORM analysis.The correlation coe‹cients may be estimated empiri­cally using statistical methods. Except for the pair of amaxand Mw, the correlation coe‹cient between each pair ofvariables used in the limit state model is estimated basedon an analysis of the actual data in the database. The cor­relation coe‹cient between amax and Mw is taken to be 0.9,which is based on statistical analysis of the simulated datagenerated from the attenuation relationships (Juang etal., 1999). This correlation is suitable for back­analysis ofcase histories where amax is obtained through the attenua­tion relationship established for a given earthquake (Mw).In a forward analysis of a future case subject to uncertainsources, this correlation could be much lower, and thuslower correlation coe‹cient should be used accordingly.The coe‹cients of correlation among the six input varia­bles are shown in Table 1. These values are consideredappropriate for back­analysis of the case histories in thedatabase.Although the details are not shown herein, a series ofsensitivity analyses were carried out to examine the eŠectof varying the coe‹cients of correlation of these pairs(for example, the coe‹cient of correlation between amaxand Mw, ra , M 0.6 instead of 0.9; the coe‹cient of cor­relation between N1, 60 and s?v, rN , s?0.5 instead of 0.3).Based on the results of reliability analyses of 201 cases inthe database, the diŠerence in the calculated reliability in­dex b between rN , s?0.5 and 0.3 is about 1z and theresulting diŠerence in the calculated probabilities, interms of root­mean­square error (RMSE), is 0.002. Simi­larly, based on the results of reliability analyses of 201cases in the database, the diŠerence in the calculated reli­ability index b between ra , M 0.6 and 0.9 is about 6zand the resulting diŠerence in the calculated probabili­maxw1, 601, 60vmaxwvTable 1. Coe‹cients of correlation among the six input variables(after Juang et al., 1999, 2008b)VariableN160FCs?vsvamaxMwN160100.30.300FC010000s?v0.3010.900sv0.300.9100amax000010.9(1)Mw00000.9(1)1Variable(1)This is estimated based on local attenuation relationships calibrated togiven historic earthquakes (Juang et al., 1999). This is suitable for relia­bility analysis of a case history, as in the post­event investigation. Thecorrelation of these two parameters at a locality subjected to uncertainsources, as in the analysis of a future case, could be much lower andeven negligible. In such cases, the joint distribution of amax and Mw maybe developed (Juang et al., 2008a), which can provide a more accurateestimate of the correlation.ties, in terms of RMSE, is 0.011. These diŠerences areconsidered relatively insigniˆcant since they are withinthe ``precision'' of the procedure for the model factorcalibration.It should be noted that the correlation matrix as shownin Table 1 must be symmetric and ``positive deˆnite''(Phoon, 2004). If this condition is not satisˆed, a nega­tive variance might be obtained, which would contradictthe deˆnition of the variance. For the correlation matrixshown in Table 1, the diagonal entries of the matrix ofCholesky factors are all positive; thus, the condition of``positive deˆniteness'' is satisˆed.The model factor, which is a random variable, is oftenassumed to follow lognormal distribution (e.g., Phoonand Kulhawy, 2005; Juang et al., 2006). Although themodel factor may also be assumed to follow normal dis­tribution, use of lognormal distribution is preferred inthis study because all other input variables are also mo­deled with lognormal distribution, which makes it easierto code the FORM procedure. Use of lognormal distribu­tion also avoids, in theory, any possibility of a negativemodel factor. Furthermore, the variation in the calculat­ed reliability index caused by the assumed distribution ofmodel factor is eventually re‰ected in the model factorthat is calibrated with observed performance data. Withthe assumption of lognormal distribution, the characteri­zation of this model factor c is thus reduced to the task ofdetermining the mean mc and standard deviation sc (or itscoe‹cient of variation).For reliability analysis using FORM in this study, nocorrelation is assumed between the model factor c andeach of the input variables in Eq. (12). This assumption isdeemed appropriate because: (1) the only data availablefor model calibration is the binary ˆeld observations of li­quefaction or no­liquefaction and thus, the model factorshould ``operate'' only at this level of detail, and (2) evenif some degree of correlation does exist (e.g., Phoon andKulhawy (2005) and Phoon et al. (2006) reported weak toEVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHOD141moderate correlations in their analyses of observedcapacities of foundations), the eŠect of not incorporatingsuch correlation in the reliability analysis would havebeen ``compensated'' in the calibration process andre‰ected in the calibrated model factor. In other words,the assumption of ``zero correlation'' between model fac­tor c and other six input variables may be considered aspart of the entire model, and the error, if any, as a resultof this assumption will be re‰ected in the model uncer­tainty of the entire model.Reliability Analysis, Bayesian Mapping Function, andInferred Model FactorFor each case in the data set of 201 cases, the FORManalysis based on the adopted limits state model (Eq. (12)and all the associated equations, Eqs. (1) through (10)) isˆrst conducted considering parameter uncertainties butnot model uncertainty, and a reliability index b1 is ob­tained. Repeating this analysis for all 201 cases, and thenapplying Eq. (13), the following PL­b1 mapping functionis obtained under the assumption that prior probabilities,P(L)P(NL):11{a exp [bb1]P LFig. 1. The eŠect of the COV component of the model factor on thenominal probabilities determined through the FORM analysis(15)where a0.67 and b1.89 are the curve­ˆttingcoe‹cients (R20.95 for this least­square regression). Itshould be noted that the b1 (reliability index) computedfor all cases in the database range from |3.95 to 4.90;thus, practically Eq. (15) is applicable to all cases, as theb1 values in this range corresponds to the probabilitiesranging from 0 to 1.It is noted that the prior probabilities, P(L) and P(NL),are di‹cult to determine, and when there is no prior in­formation, the assumption of P(L)P(NL) is not un­reasonable. The scenario of P(L)»P(NL) is examinedlater. Using the developed mapping function, the condi­tional probability of liquefaction for a given b1 can be in­ferred. Thus, the reference probability (PL1) for each ofthe 201 cases is obtained. These reference probabilitiescan be used to back­ˆgure the model factor, as outlinedpreviously.As reported in the previous work (Juang et al., 2006),the eŠect of the variation of COV_c, the coe‹cient ofvariation of model factor c, on the ˆnal probability (PL3)obtained from the FORM analysis is relatively small com­pared to the eŠect of the variation of the mean of modelfactor (mc) on the ˆnal probability. To re­conˆrm thisresult, a series of sensitivity analysis is performed in thisstudy. First, for each of the 201 cases, the FORM analysisis performed assuming each of the four scenarios: (a) mc1.0 and COV_c0.0, (b) mc1.0 and COV_c0.1, (c) mc1.0 and COV_c0.2, and (d) mc1.0 and COV_c0.3.Under each scenario, the reliability analysis is repeatedfor all 201 cases, and the diŠerence between the nominalprobabilities (PL3) obtained from the FORM analysis andthe Bayesian probabilities (PL1) obtained from Eq. (15),in terms of RMSE as deˆned in Eq. (14), is calculated.Figure 1 shows the calculated RMSE values for the fourFig. 2. Optimum mean model factors calibrated with diŠerent as­sumed COVsscenarios. The results re­conˆrm that the eŠect of thevariation of COV_c on the ˆnal probability (PL3) ob­tained from the FORM analysis is relatively insigniˆcant,although the minimum RMSE occurs approximately atCOV_c0.2. Thus, in the subsequent analysis for themean model factor, COV_c may be assumed to be 0without incurring much error.For the adopted limit state model (the modiˆed Youdet al. model), the mean of the model factor is determinedto be mc0.92 under the assumption of COV_c0. Tofurther investigate the eŠect of COV_c, the analysis isrepeated for the scenarios of COV_c0.1 and 0.2 (recall­ing that the optimum COV_c is approximately at 0.2).Under the assumption of COV_c0.1, the optimum mc isdetermined to be 0.93, and under the assumption ofCOV_c0.2, the optimum mc is determined to be 0.94.The results of these analyses are shown in Fig. 2. To seethe diŠerence among the nominal probabilities ( PL3) ob­tained through the FORM analysis incorporating thesethree characterizations of model factor, (a) mc0.92 andCOV_c0.0, (b) mc0.93 and COV_c0.1, (c) mc0.94and COV_c0.2, all 201 cases in the database are ana­142JUANG ET AL.mate weighting factors that should be used to adjust theeŠect of sample bias so that an unbiased ˆnal regressionmodel can be developed. To this end, they employed ex­pert opinions and a sophisticated Bayesian updating tech­nique and conducted sensitivity analyses to produce anestimate of weighting factors. To minimize the regressionmodel variance or maximize the likelihood function toaccount for the eŠect of choice­based sample bias, Cetinet al. (2002) deˆned weighting factors as follows:WLQp/QsWNL(1|Qp)/(1|Qs)(16a)(16b)whereFig. 3. Comparisons of nominal probabilities calculated with threemodel factors calibrated with diŠerent assumed COV values (r1)lyzed, and the results are compared, as shown in Fig. 3.The RMSE between the scenario of mc0.92 and COV_c0.0 and the scenario of mc0.93 and COV_c0.1 basedon 201 cases is 0.006, and the RMSE between the scenarioof mc0.92 and COV_c0.0 and the scenario of mc0.94and COV_c0.2 is 0.018. Little diŠerence in the calculat­ed nominal probabilities (PL3) is seen among the resultsobtained by using diŠerent characterizations of the modelfactor that was calibrated separately with diŠerent as­sumed COV_c values.In summary, for the adopted limit state model, themean of the model factor is determined to be mc0.92 un­der the assumption of COV_c0, or alternatively withthe assumption of COV_c0.2 (which is approximatelyan optimum value), the mean of the model factor isfound to be mc0.94. This characterization of the modelfactor is considered satisfactory in re­producing theBayesian probabilities inferred from the observed perfor­mance data. The error of the assumption of COV_c0appears to have been adequately ``realized'' in thecalibration process that led to the outcome of mc0.92.The eŠect of prior probability is examined next.EŠect of Prior Probability on the Inferred Model FactorFor convenience of subsequent discussions in referenceto Eq. (13), a term called prior probability ratio, r, is de­ˆned: rP(L)/P(NL). The model factor determinedbased on the reference probability inferred from Bayesi­an mapping function may be aŠected by the assumptionof the prior probabilities or the prior probability ratio.An estimate of the r value is thus essential. In this paper,an attempt is made to estimate this ``non­informative''prior based on expert opinions and simulation resultsreported by Cetin et al. (2002).Estimation of Prior Probability RatioCetin et al. (2002) performed a comprehensive study onthe issue of sample bias. In their studies, they tried to esti­WLweighting factor to apply to liqueˆed cases inthe sample,WNLweighting factor to apply to non­liqueˆed casesin the sample,Qpproportion of liquefaction sites in the popula­tion, andQsproportion of liquefaction sites in a sample.It is noted that the proportion of liquefaction sites inthe population (Qp) that contains a given sample is gener­ally unknown. On the other hand, the proportion of li­quefaction sites in the given sample (Qs) can readily be de­termined. Cetin (2000) surveyed experts for opinionsabout weighting factors, and the results indicated the ra­tio of WNL over WL fell in the range of 1 to 3, with themost common range from 1.5 to 2.0. This agrees with theintuition that the eŠect of non­liqueˆed cases should beincreased to compensate the fact that sampling is general­ly biased toward liqueˆed cases in ˆeld investigation. Fur­thermore, based on the axiom of the probability theory,both Qp and Qs must fall in the range of 0 to 1. Mathe­matically, it can be proven from Eq. (16) that WLº1 andWNLÀ1. For the unknown population that contains thedata set employed by Cetin et al. (2002), which includes112 liqueˆed cases and 89 non­liqueˆed cases, the maxi­mum likelihood analysis conducted by Cetin et al. (2002)using the information of the biased sample yielded WL0.8 and WNL1.2, and the ratio of WNL/WL1.5.The results obtained by Cetin et al. (2002) provide a ba­sis for estimation of the prior probability ratio in thepopulation. In this regard, it is noted that the term Qp de­ˆned in Eq. (16) is the probability of liquefaction P(L) inthe population, which is essentially the prior probabilityrequired for the development of Bayesian mapping func­tion using a sample ( see Eq. (13)). Thus, the priorprobability P(L) can be determined with the followingequation that is derived based on Eq. (16):11{(WNL/WL)[(1|Qs)/Qs]P(L)Qp(17)Using the results of WL0.8 and WNL1.2 for a samplewith Qs0.56 (112 liqueˆed cases in a sample of 201cases) obtained by Cetin et al. (2002), the value of Qp isdetermined to be 0.46, and thus, the prior probability ra­tio r is determined to be 0.85.Recall that the expert opinions yielded a range of 1.0 toEVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHOD3.0 for the ratio WNL/WL, while the most probable valueobtained by Cetin et al. (2002) is 1.5. Based on theseresults, an approximate distribution, namely, triangulardistribution, of the variable WNL/WL may be constructedwith a minimum at 1.0, a maximum at 3.0, and a mode at1.5. With the assumption of a triangular distribution ofWNL/WL, instead of a constant (presumably, the mostprobable estimate), the variables Qp and r become ran­dom variables and their distributions can be derived. Nu­merical solutions of the ˆrst moment and the second mo­ment of the prior probability ratio (r) yield the followingresults: the mean mr0.82 and the standard deviation sr0.179.Furthermore, the eŠect of possible variation in the esti­mated most probable value of WNL/WL is examined. Us­ing a mode of 1.3 in the assumed triangular distributionof WNL/WL, instead of 1.5, yields mr0.85 and sr0.183; using a mode of 1.7 yields mr0.78 and sr0.172.Approximately, a 13z change in the estimated mode(most probable value) of the variable WNL/WL results inless than 5z change in both mr and sr. Thus, the estimateof mr0.82 and sr0.18 that was based on results of thecomprehensive study by Cetin et al. (2002) appears to bequite reasonable. It is interesting to note that the assump­tion of a prior probability ratio of r1 used in the initialanalysis presented previously actually came within onestandard deviation of the most probable estimate (mode0.85) or the mean (0.82).With the knowledge of prior probability ratio, mr0.82 and sr 0.18, the model factor for the adopted limitstate model (the modiˆed Youd et al. model) can be re­calibrated. For example, with the prior probability ratioof rmr0.82, the calibration using the database of 201cases yields the optimum mean model factor mc0.96 un­der the assumption of COV_c0. For an additional con­ˆrmation that the assumption of COV_c0 would not in­cur much error, this calibration is re­performed with theassumption of COV_c0.2, and the optimum meanmodel factor becomes mc0.98. Figure 4 compares thenominal probabilities (PL3) calculated for the 201 casesFig. 4. Comparisons of nominal probabilities calculated with twomodel factors calibrated with diŠerent assumed COV values (r0.82)143using the two sets of model factor statistics, (a) mc0.96and COV_c0, and (b) mc0.98 and COV_c0.2.Again, little diŠerence between the two sets of nominalprobabilities is observed from the results shown in Fig. 4,indicating the appropriateness of assuming COV_c0 formodel factor calibration using the observed performancedata. An estimate of the variation of PL3, denoted as sP ,as a result of this assumption may be made based on theRMSE shown in Figs. 3 and 4. This variation is estimatedto be sP §0.02.L3L3EŠect of Prior Probability Ratio on Model FactorBecause the prior probability ratio is shown to be a ran­dom variable with mr0.82 and sr0.18, instead of aconstant, it is essential to investigate the eŠect of the priorprobability ratio on the back­ˆgured model factor. Tothis end, a series of analyses using diŠerent assumed rvalues (ranging from mr|3sr0.28 to mr{3sr1.36) areconducted. With each assumed r value, a Bayesian map­ping function is obtained and a model factor is back­ˆgured using the approach described previously. Figure 5shows the computed relationship between the model fac­tor (the optimum mc at an assumed COV_c0) and theprior probability ratio. Curve­ˆtting of the data shown inFig. 5 using the least square principle yields (R20.999,standard error0.004):mc1.45|0.78rØ r{0.48»(18)Equation (18) quantiˆes the eŠect of the prior probabilityratio on the inferred model factor. The standard error ofthe estimate by Eq. (18) is considered negligible. At themean rmr0.82, Eq. (18) yields mc0.96, which is prac­tically the same as the mean model factor determinedpreviously from a direct calibration analysis. It is interes­ting to note that according to Eq. (18), mc0.95 at themode r0.85, and thus, the diŠerence between the modelfactor calibrated using the mode (r0.85) and the mean(r0.82) is quite negligible. Furthermore, Eq. (18) yieldsmc0.92 at r1, which is equal to the mean model factorobtained in the initial calibration analysis under the as­sumption of r1. Consistent results presented above in­dicate the soundness and robustness of Eq. (18).Fig. 5. Relationship between the model factor (the optimum mc at theassumed COV_c0) and the prior probability ratio144JUANG ET AL.Nominal Probability Based on the Calibrated ModelFactorWith the knowledge of the limit state model, the modeluncertainty ( mc0.96 and COV_c0 at the mean r0.82), the case­speciˆc parameter uncertainty, and thecorrelations among the input variables, the FORM analy­sis can be performed for a given case, and the reliabilityindex and the nominal probability of liquefaction (PL3, orsimply, PL hereinafter) can be determined.It should be emphasized that the probability deter­mined through the FORM analysis is a point estimate,meaning that PL is a single value for a given case.However, because of the variation of the prior probabil­ity ratio r and its eŠect on the calibrated model factor (asre‰ected in Eq. (18)), it would be of interest to investigatepossible variation in the calculated PL accordingly. Tothis end, it is noted that the mean model factor ( mc) deter­mined from Eq. (18) is actually the mean of mc, which isdenoted herein as šmc. The variation in the mean modelfactor (mc) as a result of the variation in the estimated r(which is itself a random variable) can be derived fromEq. (18). This variation, in terms of standard deviation ofthe mean model factor, sm , is derived using the ˆrst orderanalysis (Ang and Tang, 1984) as follows:cs2m c`` «$&f 2 2|0.37 2 2sr sr&r(r{0.48)2(19)where f is the function ( mcf(r)) deˆned in Eq. (18). Sub­stituting rmr0.82 and sr0.18 into Eq. (19), the stan­dard deviation of the mc is obtained: sm 0.04.Thus, for a future case, the most probable probabilityof liquefaction PL can be determined through a FORManalysis that considers the model uncertainty ( mc šmc0.96 and COV_c0), the case­speciˆc parameter uncer­tainties, and the correlations among the input variables.Whereas the PL determined by the FORM analysis for agiven case is a point estimate, the variation in PL is possi­ble due to the variation in the estimated model factorstatistics ( mc and COV_c) and/or the variation in the esti­mated parameter uncertainty statistics (mean mx andcoe‹cient of variation COV_xi of the six input variablesxi, i1, 6). Since the eŠects of the variation in COV_c andCOV_xi are generally negligible, the variation of the cal­culated PL due to the variation in the estimated mc and mxmay be expressed as follows:ciis2P L` ` ` `&PL 2 2&PLsm {&mC&mxciwhere2i1, 6s2mxi(20)sP standard deviation of the calculated PL,sm standard deviation of the mean of variable xi, andmx mean of variable xi.LxiiIt is further noted that in geotechnical engineering prac­tice, the mean and standard deviation of an input varia­ble are almost always treated as point estimates, andthus, sm 0 can be assumed. This follows that Eq. (20)can be reduced into:xis2P |cL` `&PL 2 2s&mC m(21)cwhere sP |c is the standard deviation of the PL causedonly by the variation of mc. Equation (21) can further beapproximated as:L` ` Ø »&PLDPLsm §sm&mCDmcsP L|ccc(22)By taking Dmc2sm , Eq. (22) is reduced to (after Dun­can, 2000):c`Ø DP2 »`P |P2LsP §L|c{L|L(23)whereP{L probability of liquefaction obtained through aFORM analysis that uses a model factor of mcšmc{1sm 1.0 with COV_c0, andP|L probability of liquefaction obtained through aFORM analysis that uses a model factor of mcšmc|1sm 0.92 with COV_c0.ccBy deˆnition of the derivative, the approximation inEq. (22) or Eq. (23) is acceptable as long as Dmc is smallenough. Results of the analysis of limited cases in thedatabase ( see EXAMPLE APPLICATION presented inthe next section) conˆrm that Dmc is indeed small enoughand thus Eq. (23) is valid. It should be noted that approx­imate formulation such as Eq. (23) has previously beenreported by Hassan and WolŠ (1999), Duncan (2000),and Gutierrez et al. (2003) for other geotechnical applica­tions.Finally, recall that the variation in the calculated PL asa result of the assumption of COV_c0 is approximatelyequal to 0.02 (due to adequate Bayesian calibration ofmodel factor at the assumed COV_c). Assuming that thetwo sources of variation, caused by the assumption ofCOV_c0 and by the variation in the estimated meanmodel factor ( mc), are independent from each other, thevariation in the calculated PL can be further combinedinto:sP  0.022{s2P |cLL(24)In summary, Eq. (23) is an approximate solution thatcan be used to estimate the variation in the mean PLcaused by the variation in the model factor of the adoptedlimit state model (the modiˆed Youd et al. model). Toevaluate Eq. (23) for sP |c, only two FORM analyses (us­ing mc0.92 and 1.0 separately) are needed. Finally, thetotal variation in the calculated PL can be expressed as astandard deviation deˆned in Eq. (24) by further con­sidering possible variation due to the assumption ofCOV_c0.LESTIMATION OF PARAMETER UNCERTAIMTYFOR RELIABILITY ANALYSIS USING FORMThe results presented in the previous sections have es­tablished a comprehensive and yet practical framework145EVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHODfor conducting reliability analysis to determine theprobability of liquefaction. The section that follows im­mediately will present an example application using apractical tool (i.e., a spreadsheet that implements the en­tire reliability analysis framework). In the current sec­tion, the objective is to provide some guidance for prac­ticing engineers on the estimation of parameter uncer­tainty that is required for FORM analysis of a speciˆccase.In general, the evaluation of parameter uncertainty fora speciˆc case is the duty of the engineer in charge. Foreach input variable that is required in the limit state equa­tion (Eq. (12)), this process involves the estimation of themean and standard deviation if the variable is assumed tofollow normal or lognormal distribution. The engineerusually can make a pretty good estimate of the mean of avariable even with limited data. This probably has to dowith the well­established statistics theory that the ``sam­ple mean'' is a best estimate of the ``population mean.''Thus, the following discussion focuses on the estimationof standard deviation of each input random variable.Duncan (2000) suggested that the standard deviation ofa random variable may be obtained by one of the follow­ing three methods: 1) direct calculation from data, 2) esti­mate based on published coe‹cient of variation (COV),and 3) estimate based on the ``three­sigma rule.'' The ˆrsttwo methods are straightforward. In the last method, theknowledge of the highest conceivable value (HCV) andthe lowest conceivable value (LCV) of the variable is usedto calculate the standard deviation s as follows (Duncan,2000):HCV|LCV6s(25)It should be noted that the engineer tends to under­esti­mate the range of a given variable (and thus, the standarddeviation), particularly if the estimate was based on verylimited data and judgment was required. Thus, for asmall sample size, a value of less than 6 should be usedfor the denominator in Eq. (25). Whenever in doubt, asensitivity analysis should be conducted to investigate theeŠect of diŠerent levels of uncertainty (in terms of COV)of a particular variable on the results of reliability analy­sis.Typical ranges of COVs of the input variables accord­ing to the published data are listed in Table 2. It shouldbe noted that the COVs of the earthquake parameters,amax and Mw, listed in Table 2 are based on values report­ed in the published databases of case histories whererecorded strong ground motions and/or locally calibrat­ed data were available. The COV of amax based on generalattenuation relationships could easily be as high as 0.50(Haldar and Tang, 1979). According to Youd et al.(2001), for a future site in the U.S., the variable amax maybe estimated using one of the following methods:1) Using empirical correlations of amax with the earth­quake magnitude, the distance from the seismicenergy source, and local site conditions.2) Performing local site response analysis (e.g., usingTable 2. Typical coe‹cients of variation of input variables (Juang etal., 2008b)a)b)RandomVariableTypical rangeof COVa)N1, 600.10–0.40FCs?vsvamaxMw0.05–0.350.05–0.200.05–0.200.10–0.20b)0.05–0.10ReferencesHarr (1987);Gutierrez et al. (2003);Phoon and Kulhawy (1999)Gutierrez et al. (2003)Juang et al. (1999)Juang et al. (1999)Juang et al. (1999, 2008b)Juang et al. (1999, 2008b)The word ``typical'' here implies the range approximately bounded bythe 15th percentile and the 85th percentile, estimated from case histo­ries in the existing databases such as Cetin (2000). Published COVsare also considered in the estimate given here. The actual COV valuescould be higher or lower, depending on the variability of the site andthe quality and quantity of data that are available.The range is based on values reported in the published databases ofcase histories where recorded strong ground motions and locallycalibrated data were available. However, the COV of amax based ongeneral attenuation relationships or ampliˆcation factors could easilybe as high as or over 0.50. The reader is referred to Juang et al.(2008b) for further discussion of this issue.SHAKE or similar software) to account for localsite eŠects.3) Using the USGS National Seismic Hazard webpages and the NEHRP ampliˆcation factors.Further discussion on the ˆrst two methods is beyond thescope of this paper, as this is best handled by the engineerin charge for a speciˆc case. The third method, the am­pliˆcation factor approach, is brie‰y discussed in the fol­lowing. The USGS National Hazard Maps (Frankel etal., 2002) provide rock outcrop peak ground acceleration(PGA) for a speciˆed locality based on latitude/longi­tude. The USGS National Hazard Maps web site (USGS,2002a) provides PGA value at each of the six seismic haz­ard levels, which corresponds to earthquake returnperiods of 4975, 2475, 975, 475, 224, and 108 years, re­spectively. Thus, for a given locality, a PGA can be ob­tained for a speciˆed probability of exceedance in an ex­posure time from this USGS web site.For liquefaction analysis, the rock PGA needs to beconverted to peak ground surface acceleration at the site,amax. Ideally, the conversion should be carried out basedon site response analysis. Various simpliˆed proceduresare also available for an estimate of amax (e.g., Gutierrezet al., 2003; Stewart et al., 2003). As an example, a sim­pliˆed procedure for estimating amax, perhaps in the sim­plest form, is expressed as follows:amaxFa(PGA)(26)where Fa is the ampliˆcation factor, which, in a simplestform, may be expressed as a function of rock PGA andthe NEHRP site class. Figure 6 shows an example of asimpliˆed chart for the ampliˆcation factor. The NEHRPsite classes used in Fig. 6 are based on the mean shearwave velocity of soils in the top 30 m, as listed in Table 3.Other simpliˆed solutions for the ampliˆcation factor in­clude regression equations developed by Stewart et al.146JUANG ET AL.Fig. 6. Ampliˆcation factor as a function of rock PGA and theNEHRP site class (after Gutierrez et al., 2003; Juang et al., 2008b)Table 3. Site classes (categories) in NEHRP provisions (NEHRP,1998)NEHRPCategoryDescription of soil conditions withrespect to shear wave velocityAHard rock with measured mean shear wave velocity in thetop 30 m, švsÀ1,500 m/sRock with 760 m/sº švsÃ1,500 m/sDense soil and soft rock with 360 m/sº švsÃ760 m/sStiŠ soil with 180 m/sº švsÃ360 m/sSoil with švsÃ180 m/s or any proˆle with more than 3 m ofsoft clay (plasticity index PIÀ20, water content wÀ40zand undrained shear strength suº25 kPa)Soils requiring a site­speciˆc study, e.g., liqueˆable soils,highly sensitive clays, collapsible soils, organic soils, etc.BCDEF(2003).The rock outcrop PGA is generally assumed to followlognormal distribution (Kramer and Mayˆeld, 2007). Theampliˆcation factor Fa also follows lognormal distribu­tion (Stewart et al., 2003). Therefore, the variable amaxcan also be characterized with lognormal distribution,and thusly in a simpliˆed solution, the mean and stan­dard deviation of amax can easily be determined based onthe mean and standard deviation of PGA and Fa. Forpractical applications, this simpliˆed solution is ap­propriate.For reliability analysis of a future site in a speciˆed lo­cality in the U.S., the magnitude of Mw can also be der­ived from the USGS web pages through a de­aggregation(USGS, 2002b). The task of seismic hazard de­aggrega­tion involves the determination of earthquakeparameters, principally magnitude and distance, for usein a seismic­resistant design. The seismic hazard curvepresented in the USGS web page is de­aggregated to exa­mine the ``contribution to hazard'' (in terms of fre­quency) as a function of magnitude and distance. Theseplots of ``contribution to hazard'' as a function of mag­nitude and distance are useful for specifying design earth­quakes. On the available de­aggregation plots from theUSGS web site, the height of each bar represents the per­cent contribution of that magnitude and distance pair (orbin) to the speciˆed probabilities of exceedance. The dis­tribution of the heights of these bars (i.e., frequencies) isessentially a joint probability mass function of magnitudeand distance. When this joint mass function is ``integrat­ed'' along the axis of distance, the probability mass ordistribution function of the magnitude is obtained.In summary, the PGA and Mw may be obtained for agiven site at a speciˆed hazard level. The selected PGA isconverted to amax, and the pair of amax and Mw is then usedin the liquefaction evaluation. For reliability analysis, themean value and the standard deviation (and thus, thecoe‹cients of variation) of amax and Mw can be deter­mined from their respective distributions. If such distri­butions are not available, the coe‹cients of variation forthese two seismic parameters may be estimated usingTable 2 as a guide. It should be noted that the ranges ofCOV listed in Table 2 are estimated based on publisheddatabases of case histories where recorded strong groundmotions and locally calibrated data are available.However, the COV of amax based on general attenuationrelationships or ampliˆcation factors for a given site con­sidering all possible ground motions at all hazard levelscould easily be as high as or over 0.50. For situations likethat, it is vital to construct the joint distribution of amaxand Mw, considering all possible ground motions at allhazard levels. Such approach is, however, beyond thescope of this paper; the interested reader is referred toJuang et al. (2008a) for this issue.EXAMPLE APPLICATIONThis example concerns a non­liqueˆed case in the data­base. Field observation of the site, designated as Araha­ma (Cetin et al., 2004), indicated no evidence of liquefac­tion during the 1978 Miyagiken­Oki earthquake (Mw6.7, amax0.1 g). The mean values of seismic and soilparameters at the critical depth (5.0 m) are given as fol­lows: N1, 6014.1, FC0z, s?v44.9 kPa, sv85.0kPa, amax0.1 g, and Mw6.7. The correspondingcoe‹cients of variation of these parameters are assumedto be 0.191, 0.0, 0.185, 0.206, 0.2, and 0.1, respectively.Reliability analysis using FORM with the knowledge ofthe model factor ( mc šmc0.96 and COV_c0) yields areliability index of b31.592 and the nominal probabilityof liquefaction of PLPL30.056. As noted previously,the result of the FORM analysis is a point estimate. Thissolution may be obtained easily with a computer code(Yang, 2003) or a simple spreadsheet (Low and Tang,1997; Phoon, 2004; Juang et al., 2006), as shown in Fig.7. The spreadsheet developed speciˆcally for this FORManalysis of liquefaction potential is available from theˆrst author upon request.To estimate the variation of PL, two additional FORManalyses with diŠerent mc values ( mc šmc|1sm 0.92 andmc šmc{1sm 1.0 separately) are performed, which yields|for this case, P {L 0.047 and P L  0.067. Thus, accord­ing to Eq. (23), the variation of the mean PL caused bythe variation in the estimated mc is determined to be sP |c0.0099. Although the variation of PL appears quiteccL147EVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHODFig. 7.Spreadsheet that implements the entire FORM analysis (after Juang et al., 2008b)Table 4.Summary of the sensitivity analysis of Eq. (23)sP |cLStep SizeVariant of Eq. (23) (for diŠerent step sizes)Dmc2smcDmc4smDmc6smArahama Site1978 Miyagiken­OkiearthquakeKobe No. 35 site1995 Hyogoken­NambuearthquakeSan Juan B­5 Site1977 Argentinaearthquake|sP |c`P {L |P L `/20.00990.02460.0314c||sP |c`P {{L |P L `/40.01010.02480.0317c`/6sP |c`P {{{|P |||LL0.01030.02490.0322LLLsmall in this non­liqueˆed case, it actually represents ap­proximately 18z of variation from the mean of PL0.056.To examine the eŠect of the ``step size'', Dmc, on theapproximation in Eqs. (22) and (23), the same problem isanalyzed with two diŠerent step sizes, (a) Dmc4sm and||(b) Dmc6sm . In the ˆrst case, sP |c`P {{L |P L `/4{{where P L is the probability of liquefaction obtainedthrough a FORM analysis that use mc šmc{2sm 1.04,is the probability of liquefaction obtained withand P ||Lmc šmc|2sm 0.88. In the second case, sP |c`P {{{|L`/6 where P {{{P |||is the probability of liquefactionLLobtained through a FORM analysis that use mc šmc{3sm1.08, and P |||is the probability of liquefaction ob­Ltained with mc šmc|3sm 0.84. For the same case as de­scribed previously, the two alternatives that used greaterstep sizes yield practically the same sP |c§0.01, as shownccLccLccLin Table 4. It is noted that the results of the sensitivityanalysis for two other cases (presented later) are also in­cluded in Table 4. These results verify the validity of Eq.(23) that was previously examined and reported by otherinvestigators (Hassan and WolŠ, 1999; Duncan, 2000;Gutierrez et al., 2003).Finally, the variation in the calculated PL, in terms ofstandard deviation, can be calculated with Eq. (24),which yields sP 0.022. By taking an approximation ofthe mean plus and minus 3 times standard deviation, theprobability of liquefaction for this case (assuming that itis predicted before the event) would fall approximately inthe range of 0.0 to 0.123. This result suggests that li­quefaction is extremely unlikely to occur at this site whensubjected to the 1978 Miyagiken­Oki earthquake (Mw6.7, amax0.1 g), which agrees with ˆeld observation ofno liquefaction.L148JUANG ET AL.logistic regression analysis. The other is the empiricalmodel established by Cetin et al. (2004) based on a moresophisticated regression analysis with Bayesian updating.In the model by Youd and Noble (1997), the probabil­ity of liquefaction is calculated with the following equa­tion:COMPARISON WITH EXISTING METHODSThe probabilities of liquefaction of the case analyzedpreviously along with additional examples are calculatedwith two existing empirical models. One is the empiricalmodel established by Youd and Noble (1997) based onln [PL/(1|PL)]|7.633{2.256Mw|0.258(N1, 60cs){3.095(ln CSR)(27)where CSR is not ``adjusted'' by MSF and Ks. In other words, CSR in Eq. (27) is calculated as (Seed and Idriss, 1971;Seed et al., 1985):CSR0.65Ø s?s »Ø ag »(r )vmaxv(28)dIn the more sophisticated Bayesian regression model by Cetin et al. (2004), the probability of liquefaction is calculat­ed with the following equation:^ «_]N1, 60(1{0.004FC)|13.32 ln (CSR)|29.53 ln (Mw)|3.70 lnPLF |Table 5.vab2.70where CSR is deˆned in Eq. (28), Pa is the atmosphericpressure (§100 kPa) and F is the cumulative standardnormal distribution function.It should be noted that a direct comparison of theFORM solution with the results obtained from the twoempirical models (Eqs. (27) and (29)) is not entirelymeaningful because that in the two regression­basedmodels, only the representative values (or the meanvalues) of the input variables are entered into the respec­tive equations (Eqs. (27) and (29)), whereas with theFORM solution, the variation of the input variables, thecorrelations among the input variables, and the modeluncertainty are all directly incorporated in the reliabilityanalysis. Nevertheless, this comparison is still desirable asØ s?P »{0.05FC{16.85 $`a(29)it may provide some indication on the performance of theFORM solution presented.For the same Arahama case in the 1978 Miyagiken­Okiearthquake (Mw6.7, amax0.1 g) presented previously,the probability of liquefaction is PL0.050 calculatedfrom Eq. (27) (Youd and Noble, 1997), and PL0.005calculated from Eq. (29) (Cetin et al., 2004). Recall thatthe FORM solution yielded a mean PL of 0.056, and apossible range of 0.0 to 0.123. Thus, for this case, theresults obtained using the three methods are consistentwith each other, all suggesting that liquefaction is ex­tremely unlikely to occur at this site when subjected to the1978 Miyagiken­Oki earthquake. This prediction agreeswell with the ˆeld observation of no liquefaction.Basic data from three case histories and the probabilities of liquefaction determined with three diŠerent methodsBasic soil data of the critical layerExample CaseDepth(m)Probability of liquefactionCOV (required only in theFORM analysis)Mean valuesvs?vFCN1, 60 (z) (kPa)(kPa)N1, 60FCs?vsvYoud andNoble (1997)Cetin et al.(2004)This studymean, m (m}3s)Site: Arahama1978 Miyagiken­OkiMw6.7, amax0.1 gCOV of amax0.20(Tohno and Yasuda, 1981;Cetin et al., 2004)5.014.1044.985.00.19100.185 0.2060.0500.0050.056(0. to 0.123)Site: Kobe No. 351995 Hyogoken­NambuMw6.9, amax0.5 gCOV of amax0.15(Cetin et al., 2004)4.519.0848.672.60.1370.250.098 0.1170.5470.9850.829(0.73 to 0.92)Site: San Juan B­51977 ArgentinaMw7.4, amax0.2 gCOV of amax0.075(Idriss et al., 1979;Cetin et al., 2004)2.914.5338.145.60.007 0.333 0.085 0.1070.3770.3280.165(0.05 to 0.28)EVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHODFig. 8. Comparison of three probability­based methods with 20 casehistoriesTo further compare the three methods, another two ex­amples are worked out, including one liqueˆed case andone non­liqueˆed case. The liqueˆed case analyzed is theKobe No. 35 site in the 1995 Hyogoken­Nambu earth­quake (Mw6.9, amax0.5 g), and the non­liqueˆed caseis the San Juan B­5 site in the 1977 Argentina earthquake(Mw7.4, amax0.2 g). The analysis presented previouslyis repeated for the two cases, and the results are shown inTable 5. For the liqueˆed case (Kobe No. 35), PL0.547calculated from Eq. (27) (Youd and Noble, 1997), and PL0.985 calculated from Eq. (29) (Cetin et al., 2004). TheFORM solution in this paper yields a mean of 0.829 and arange of 0.73 to 0.92. The solutions obtained usingFORM (this study) and the Cetin et al. (2004) methodsuggest that the site is very likely to experience liquefac­tion when subjected to the ground shaking the level of the1995 Hyogoken­Nambu earthquake, which agrees withthe ˆeld observation. The Youd and Noble (1997) methodis less accurate than the other two methods for this lique­ˆed case.For the non­liqueˆed case (San Juan B­5), the Cetin etal. (2004) method yields PL0.328, whereas the Youdand Noble (1997) method yields PL0.377. The FORMsolution in this study yields a mean of 0.165 and a range149of 0.05 to 0.28 for the San Juan B­5 case, which comparesfavorably to the solutions by Cetin et al. (2004) and Youdand Noble (1997) for this non­liqueˆed case.Finally, Fig. 8 shows additional comparison of thethree probability­based methods using 20 case histories.These cases include 6 non­liqueˆed cases and 14 liqueˆedcases, taken from published records from the 1976Guatemala earthquake, the 1977 Argentina earthquake,the 1978 Miyagiken­Oki earthquake, the 1971 San Fer­nando earthquake, the 1979 Imperial Valley earthquake,the 1994 Northridge earthquake, and the 1995 Hyo­goken­Nambu earthquake (Cetin et al., 2004). Similarresults as presented previously can be observed. As shownin Fig. 8(a), the probabilities of liquefaction computedwith the Youd and Noble method and the proposedmethod (FORM analysis) in this study are quite consis­tent. However, for liqueˆed cases included in zone A, thePL values obtained in this study are higher than those ob­tained with the Youd and Noble method, which indicatesthat the proposed method (FORM) is more accurate. Fornon­liqueˆed cases included in zone B, the PL values ob­tained in this study are lower than those obtained with theYoud and Noble method, which again indicates that theproposed method is more accurate.As shown in Fig. 8(b), in all but one liqueˆed cases(zone A), the PL values computed by the Cetin et al.(2004) method are all practically equal to 1.0, indicatingthat predictions made with this method for these liqueˆedcases are accurate. For the same liqueˆed cases, the PLvalues computed by the proposed method are also veryhigh (zone A), indicating that the proposed method isalso accurate in this regard. On the other hand, for thenon­liqueˆed cases (included in zone B), the PL valuescomputed by the Cetin et al. (2004) method are sig­niˆcantly higher than those obtained with the proposedmethod, which is less desirable. Overall, the proposedmethod yields the most desirable results among the threemethods examined.Based on the results presented, it appears that the Cetinet al. (2004) method has a tendency to produce a higherestimate of the probability of liquefaction. This tendencyis biased toward the conservative side—it is more likely tocorrectly predict liqueˆed cases. On the other hand, theCetin et al. (2004) method is likely to over­estimate theprobability of liquefaction of non­liqueˆed cases, whichmay not be desirable as the sites that are suitable for de­velopment would be wrongly judged to be unsuitable,and unnecessary ground improvement project could havebeen suggested based on incorrect prediction of theprobability of liquefaction. The solutions by the Youdand Noble (1997) method are quite consistent with theFORM solutions presented in this paper. Overall, theFORM solutions appear to be able to produce reasonableestimates of the probability of liquefaction, either in li­queˆed cases or non­liqueˆed cases.It should be noted that the comparison of the threemethods made herein is only approximate and based onlimited cases. In particular, the parameter uncertaintieswere not included in the Youd and Noble (1997) method150JUANG ET AL.and the Cetin et al. (2004) method, as they were not re­quired in Eqs. (27) and (29), respectively. On the otherhand, the FORM solution considers explicitly both modeland parameter uncertainties.SUMMARYThe SPT­based simpliˆed method recommended inYoud et al. (2001) has been examined for its model uncer­tainty within the framework of the ˆrst order reliabilitymethod. Strictly speaking, the model uncertainty deter­mined and presented in this paper is not exactly the modeluncertainty of this SPT­based model because several as­sumptions and adjustments were made in the modelcalibration process. These included: 1) the Youd et al.(2001) method was modiˆed slightly, as described previ­ously, and the entire limit state model was deˆned by Eqs.(1) through (10); 2) all input random variables for the cal­culation of CSR and CRR were assumed to be lognormal­ly distributed; 3) reliability analysis was conducted usingFORM; 4) correlation between the model uncertainty cand the basic input variables of the limit state model wasassumed to be negligible; and 5) non­informative priorregarding sample disparity in the database of case histo­ries, reported by Cetin et al. (2002), was employed. Anyerror induced from these assumptions/adjustments iseventually re‰ected in the overall model error (uncertain­ty) that is calibrated with ˆeld observations. In otherwords, the uncertainties in the component models, andthose induced by the adjustments/assumptions made, arelumped into the overall model uncertainty. Thus, thecalibrated model bias factor c is for the entire ``package''with all these adjustments/assumptions, and not just themodel uncertainty of the original Youd et al. (2001)method.It should be noted that for each case in the databasethat is used for model calibration, the CSR computedwith the peak horizontal ground surface acceleration andthe CRR computed based on the SPT blow counts have anoticeable margin of error. In other words, the calibratedmodel uncertainty may be aŠected by the uncertainty inthe case history data. However, the deterministic model(Youd et al., 2001) that evaluates the liquefaction poten­tial using CSR and CRR is widely accepted, the databaseof case histories by Cetin et al. (2004) is considered themost updated and accurate by the profession, and the un­certainty in the input parameters in each case in the data­base is included in the calibration within the frameworkof the well­accepted ˆrst order reliability method(FORM), therefore, the proposed FORM analysis frame­work developed through this comprehensive calibrationprocess is considered to be satisfactory.Using the entire calibrated package as a whole, theFORM analysis that considers the variation of the inputvariables, the correlations among the input variables, andthe model uncertainty, as illustrated previously in theEXAMPLE APPLICATION section, can produce areasonable estimate of the probability of liquefaction,either in liqueˆed cases or non­liqueˆed case. The entireprocess can easily be implemented in a spreadsheet forpractical application. Moreover, possible variation of thecomputed probability of liquefaction can easily be deter­mined with only two additional spreadsheet solutions.The spreadsheet is available from the ˆrst author uponrequest.CONCLUSIONS1. A new procedure has been developed and veriˆed withwhich the uncertainty of a geotechnical model can beeŠectively characterized. This procedure involves twosteps, (a) deriving a Bayesian mapping function basedon a database of case histories, and (b) back­ˆguringmodel uncertainty by means of the calibrated Bayesi­an mapping function. Results of an extensive series ofanalyses show that this procedure is eŠective for es­timating model uncertainty of an SPT­based modelusing observed ˆeld liquefaction performances. Thedeveloped approach is considered innovative as theuncertainty of a semi­empirically established modelfor liquefaction evaluation can be quantiˆed so that amore realistic reliability analysis can be performed.2. Regardless of what the prior probability ratio r isused, the eŠect of the variation of COV_c (coe‹cientof variation of the model factor c) on the ˆnal nomi­nal probability PL3 obtained from the FORM analysisis shown to be relatively insigniˆcant. At r1, themean of the model factor that represents the uncer­tainty of the modiˆed Youd et al. (2001) model isback­ˆgured to be mc0.92 under the assumption ofCOV_c0; or alternatively with the assumption ofCOV_c0.2 (which is approximately an optimumvalue), the mean of the model factor is found to be mc0.94. The diŠerence between the PL3 values calculat­ed with these two statistical characterizations of modelfactor, in terms of the root­mean­square­error(RMSE) using 201 cases, is quite small (approximatelyequal to 0.02). The assumption of COV_c0 can thusbe made for back­ˆguring mc without incurring mucherror, as the eŠect of such assumption appears to havebeen ``compensated'' in the calibration of mc.3. The prior probability ratio r is estimated in this paperbased on the ˆndings of the comprehensive study ofweighting factors that were used to correct the eŠect ofchoice­based sampling bias by Cetin et al. (2004).Based on a series of sensitivity analyses using the ˆnd­ings by Cetin et al. (2004), the variable r is character­ized with a mean of mr0.82 and a standard deviationof sr0.18. The assumption of r1 used in the previ­ous study by Juang et al. (2006) and the preliminaryanalysis in this paper is found to be within one stan­dard deviation of the most probable estimate (mode0.85) or the mean ( mr0.82).4. The mean of the model factor, mc, calibrated with ob­served performances is found to be dependent on theprior probability ratio r, as re‰ected in Eq. (18). Be­cause r is a random variable (mr0.82, sr0.18), theuncertainty in r will lead to the uncertainty in theEVALUATING MODEL UNCERTAINTY OF AN SPT­BASED SIMPLIFIED METHODcalibrated mc, regardless of the assumption that thecoe‹cient of variation of the model factor COV_c0.The variation of mc, in terms of standard deviation, asa result of the uncertainty in the estimated r, is foundto be sm 0.04.5. For a future case, the probability of liquefaction PLcan be determined through a FORM analysis that con­siders the model uncertainty ( mc šmc0.96 and COV_c0 inferred at rmr0.82), the case­speciˆcparameter uncertainties, and the correlations amongthe input variables. Whereas the PL determined by theFORM analysis for a given case is a point estimate, thevariation in the calculated PL can be caused by the un­certainty in the estimated model factor. Equations(23) and (24) provide a means for an estimate of thevariation in the calculated PL, in terms of standarddeviation, caused by the uncertainty in the estimatedmodel factor.6. Example application of the FORM analysis with thecalibrated model factor presented in this paper showsthat the procedure is easy to apply, particular with aspreadsheet solution. This is encouraging as the proce­dure also has a sound theoretical basis. This proceduremay be used for evaluating the probability of liquefac­tion in a routine practice. Further validation of the de­veloped procedure using additional ground perfor­mance data, however, is desirable. Additional com­parison with existing probabilistic models using moreground performance data should also be made.7. Using the entire package as a whole, the FORM analy­sis that considers the variation of the input variables,the correlations among the input variables, and themodel uncertainty can produce reasonable estimatesof the probability of liquefaction, either in liqueˆedcases or non­liqueˆed cases. Thus, the results present­ed in this paper have extended the use of the Youd etal. (2001) method from being a deterministic model tobeing capable of providing both deterministic andprobabilistic solutions.cACKNOWLEDGEMENTThe study on which this paper is based was supportedby the National Science Foundation through GrantCMS­0218365 under program director Dr. Richard J.Fragaszy. This ˆnancial support is greatly appreciated.The opinions expressed in this paper do not necessarilyre‰ect the view and policies of the National Science Foun­dation. The third and last authors appreciate the ˆnancialsupport in part by the Research Grant Council of HongKong through Grant No. HKUST 620206. 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