A Cyclic Shear-Volume Coupling and Pore Pressure Model for Sand

Transcription

1 Missori University of Science and Technology Scholars' Mine nternational Conferences on Recent Advances in Geotechnical Earthqake Engineering and Soil Dynamics Second nternational Conference on Recent Advances in Geotechnical Earthqake Engineering & Soil Dynamics Mar 11th - Mar 15th A Cyclic Shear-Volme Copling and Pore Pressre Model for Sand Peter M. Byrne University of British Colmbia, Vancover, B.C., Canada Follo this and additional orks at: Part of the Geotechnical Engineering Commons Recommended Citation Byrne, Peter M., "A Cyclic Shear-Volme Copling and Pore Pressre Model for Sand" (1991. nternational Conferences on Recent Advances in Geotechnical Earthqake Engineering and Soil Dynamics This Article - Conference proceedings is broght to yo for free and open access by Scholars' Mine. t has been accepted for inclsion in nternational Conferences on Recent Advances in Geotechnical Earthqake Engineering and Soil Dynamics by an athoried administrator of Scholars' Mine. This ork is protected by U. S. Copyright La. Unathoried se inclding reprodction for redistribtion reqires the permission of the copyright holder. For more information, please contact scholarsmine@mst.ed.

2 \ Proceedings: Second nternational Conference on Recent Advances in Geotechnical Earthqake Engineering and Soil Dynamics, March 115, 1991, St. Lois, Missori, Paper No A Cyclic Shear-Volme Copling and Pore Pressre Model for Sand Peter M. Byrne Professor of Civil Engineering, University of British Colmbia, Vancover, B.C., Canada SYNOPSS: A to parameter incremental shear-volme copling eqation is presented for sand. The eqation is based pon experimental data and gives predictions that are in excellent agreement ith data over a range of relative densities and stress conditions. Empirical expressions for the to parameters based on incorporated in a simple shear pore pressre element model and the predictions of the model are compared ith both satrated ndrained cyclic strain and cyclic load tests. t is fond that, provided a threshold strain is incorporated, the model predictions are in very good agreement ith the laboratory data over a ide range of stress and density conditions. The element model is also calibrated against field experience dring earthqakes, and predicts pore pressre rise and liqefaction behavior in close agreement ith crrent design practice. The model can easily be calibrated to represent any cyclic loading data and is appropriate for incorporation in "loose copled" dynamic analyses procedres sch as those employed by Finn and his colleages. NTRODUCTON Cyclic shear loading can indce significant volmetric compression strains in nsatrated sands hich can reslt in ndesirable grond settlements and possible damage to strctres. n satrated sands, sch loading can indce pore pressre rise and liqefaction hich may reslt in severe damage to strctres. Cyclic loading may arise from a nmber of cases, inclding earthqakes, ice loading, blasting, machine vibration, ind and ave loading. Experimental evidence indicates that volmetric compression strains are indced by cyclic shear strain de to a copling beteen the shear and volmetric response of sand. These volmetric strains are plastic in natre rather than elastic as they are not recovered at the end of a loading cycle. A rigoros effective stress dynamic analysis of soil strctres comprised of sandy material reqires a stress-strain la that incldes shear-volme copling effects for repeated load cycles. Sch a stress-strain la is very complex and ill reqire many parameters to adeqately model the observed laboratory and field behavior nder cyclic loading conditions. A simple effective stress analysis approach as first proposed by Martin et al. ( The basis of the approach is an eqation linking the increment of volmetric strain per cycle of load ith the shear strain occrring dring that particlar cycle. For a drained condition, the increments can be simply added to give the accmlated volmetric strain ith nmber of cycles as carried ot by Finn and Byrne (1976. For an ndrained condition the increment of volmetric strain ill lead to a rise in poreater pressre that can be compted by imposing volme constraints together ith an elastic rebond modls. Pore pres sre compted in this ay can be incorporated in a simple incremental elastic dynamic response analysis in hich the tangent stiffness is modified ith both the level of shear strain and the pore pressre rise. Since the pore pressres can only be compted after each cycle or 1/2 cycle of strain as the analysis proceeds, this procedre is referred to as loose-copled. The first effective stress dynamic analyses by this procedre ere presented by Finn, Byrne and Martin (1976. The procedre has since been extensively developed by Finn et al. (1986. A key factor in the loose-copled effective stress approach is the cyclic shear-v.olme copling eqation. Martin et al. proposed a 4 parameter eqation based on laboratory data on a single sand at a single relative density. Finn and Byrne (1976 sggested an additional eqation for predicting volme changes at other relative densities. A detailed examination of the Martin et al. eqation shos that it is not generally stable. Herein, an alternative to parameter eqation is proposed that gives excellent agreement ith measrements over a range of relative densities. The parameters can easily be derived from cyclic loading tests, or can be estimated from relative density or penetration vales based pon available data. The parameters can be sed in analysis to predict expected plastic volme changes and settlements nder dry or drained conditions and/or pore pressre rise and liqefaction of satrated sands in either an effective or total stress dynamic analysis. CYCLC SHEAR-VOLUME COUPLNG EQUATON Martin et al. (1975 proposed the folloing incremental shear-volme copling eqation for sand nder simple shear loading: 47

3 ( 1 in hich r c 1. c 2 c'. c. the increment of volmetric strain in percent per cycle of shear strain, the accmlated volmetric strain from previos cycles in per cent, the amplitde of shear strain in per cent for the cycle in qestion, and constants for the sand in qestion at the relative density nder consideration. <1: : (/ U...J cr t- ::;;: = 6a. t5<j <1: :: Crystal Silica Sand Dr= 45% The basic data sed by Martin et al. to determine their eqation is shon in Fig. 1 and comprises accmlated volmetric strain verss nmber of cycles from simple shear tests on crystal silica sand condcted at three different levels of shear strain and a relative density of 45%. Fig. 2. CYCLC SHEAR STRAN AMPLTUDE, y% ncremental volmetric strain crves from the data of Fig Crystal Siliec Sand Dr 11 45fo <1:. 6 : Vl 2 :.8 :::;; :::...J CYCLES Fig. 1. Volmetric strains from constant amplitde cyclic simple tests. Test data from Martin et al. (1975. t may be seen from Fig. 1 that the volmetric strain increases ith the level of shear strain applied, and that for the same level of shear strain, the rate of accmlation of volmetric strain redces ith nmber of cycles. Martin et al. plotted this same data in an incremental form as shon in Fig. 2, and it indicates that the volmetric strain per cycle, t.ev depends pon the crrent level of applied strain as ell as the accmlated volmetric strain, i.e., the accmlated volmetric strain is the hardener that controls the plastic volme change in the crrent cycle. Also, in this form it is not necessary that the shear strain be the same for every cycle in a loading seqence. Hoever, from this figre, it is difficlt to express the data in eqation form. <l <1: : (/ <1:.2 : : (/ :::;; LL. :::...J.1...J LL. - : a. :E : Fig. 3. s.-, Or Str o,.'j, -o 2 VOLUMETRC STRAN, Ev% Alternative volmetric strain crves from the data of Fig. 1. strain per cycle, t. v is plotted verss the accmlated volmetric strain, v for the three levels of shear strain. f the axes of Fig. 3 are divided by the shear strain, the three crves of Fig. 3 collapse to the single crve in the dimensionless plot shon in Fig. 4. This crve is ell represented by here cl shon. C 1 EXP ( -C 2 (.Y r. 8 and C 2 (2. 5 for the data An alternative form to plot the data of Fig. 1 is shon in Fig. 3 in hich the volmetric The parameter C 1 volme change. controls the amont of 48

4 -- <] MFS Dolo points from y =.1,.2,.3% D, = 45%!!?..: -- < cr < cr -- C/ cr -- ::: --' NUMBER OF CYCLES Fig. 4. Normalied strains. incremental volmetric For the first cycle of loading v hence c 1 and The data also shos that the accmlated volmetric strain at the end of 15 niform cycles is abot 5 times greater than for the lst cycle, hence ( 3 1 Fig. 5. Relationship beteen volmetric strain ratio and nmber of cycles for dry sands. Test data from Tokimats and Seed (1987. accmlated volmetric strain at any specific nmber of cycles is knon. Hoever, it is sggested that the second constant C 2 be preserved as it gives greater flexibility in matching the data if a more complete history of accmlated volmetric strain is available. Tokimats and Seed (1987 have,.;...fisented accmlated volmetric strains da,. fter 15 cycles for a range of cyclic shear strains and relative densities and these are shon in Fig. 6. The solid lines represent the c = ( V1, 1 5 r This eqation may be preferable to eqation (3 in many instances, becase there is considerable data on ( v 15 as a fnction of relative density. The parameter C 2 controls the shape of the accmlated volme change ith nmber of cycles. The predicted shape is shon in Fig. 5 and is in good agreement ith the Martin et al. data as ell as the Tokimats and Seed (1987 data. The Tokimats and Seed data is for a range of relative densities, hile the Martin et al. data is jst for Dr = 45%. Since the shape is the same for all densities the parameter C 2 is a constant fraction of C 1 for all relative densities and can be prescribed as: ( 4 C 2 =.4/C 1 ( 5 -- <.. -- ::: ;; cr -- C/ cr -- ::: --' CYCLC SHEAR STRAN, y% Model + MFS 15 Cycles The fndamental incremental shear-volme copling eqation therefore involves only one constant, c,, hich depends on the density of the sand and can be simply assessed if the Fig. 6. Relationship beteen volmetric strain and shear strain for dry sands. Test data from Silver and Seed (

5 Tokimats and Seed interpretation of the data, and based on their lines, and sing Eqs. (4 and (5, the folloing vales of C1 and C 2 are compted: Table 1. C 1 and C 2 from Relative Density Dr ( e1,/r c1 c The model predictions sing these vales of c 1 and C 2 are also shon in Fig. 6 and are seen to be in excellent agreement ith the data. The Martin et al. data is also shon on this figre and denoted by MFS, and is seen to lie belo the Tokimats and Seed data. The c 1 vale can be expressed in eqation form as follos: in hich Dr is in %, and ( 6 C 2 =.4/C 1 ( 7 Tokimats and Seed also sho vales of accmlated volmetric strains after 15 cycles as a fnction of normalied standard penetration test vales (N 1 6 Their data is not shon as it jst involves a conversion from Dr to (N 1 l o The conversion beteen relative density and (N 1, sed by Tokimats and Seed can be approximated in the range 3 < Dr < 9 by: Based on their data and sing Eqs. (4 and (6, the vales of C 1 and C 2 are as follos: Table 2. C 1 and C 2 from SPT N Vales (N 1 o ( e1,/r C c ( 8 dimensional series of strain plses at any depth in a sand stratm. ntegration of sch strains ill give the settlement at any point ithin the stratm. For a random pattern of strain cycles it is appropriate to modify the basic eqation to compte the volmetric strains per 1/2 cycle as follos: (llevl 1/2 cycle (11 The volmetric strains can also be sed to compte pore pressre rise and liqefaction and this is discssed in the next section. VOLUMETRC STRAN AND PORE PRESSURE RSE f the pores of the sand are satrated ith ater and if the ater has not sfficient time to drain dring the cycles of loading, the pore pressre ill rise and liqefaction may occr. The pore pressre rise for satrated ndrained conditions can be compted from volme compatibility as follos: in hich ll v ll ll e ( 12 the total incremental change in volmetric strain per 1/2 cycle the elastic incremental change in volmetric strain per 1/2 cycle the plastic incremental volmetric strain per 1/2 cycle no for simple shear conditions here M llo M ( 13 the change in vertical effective stress per 1/2 cycle the constrained rebond stress tangent modls of skeleton effective the sand The volmetric strains referred to in the previos section and in particlar Eq. (11 are not recoverable and are therefore plastic strains, and in the discssion to follo ill be given the sperscript p. The C 1 and C 2 vales can be expressed in eqation form as follos: For satrated ndrained conditions ll v and hence from Eqs. (12 and (13, ' c 1 c (N 1.4/C 1 ( 9 ( 1 The volmetric strain eqation can be incorporated in a dynamic analysis to compte the volmetric strains arising from any one- (14 f there is no change in total stress then llov =, and the change in poreater pressre llv= -llo, hence 5

6 6 = M 6e:e ( 15 Knoing 6e:e from Eq. (11 for any knon half cycle of strain, the pore pressre rise per half cycle can be compted from Eq. (15. The pore pressre generated, g, by any specified pattern of strain cycles can be compted by simply smming the pore pressre increments, i.e., g = 6. The appropriate rebond effective stress constrained tangent modls, M appears to depend only pon the level of effective stress and not the relative density, and can be prescribed as follos: M o K P (_y_ m m a Pa (16 Vales of Km = 16 and m =.5 give modli that are in good agreement ith vales reported by Martin et al. (1975 as ell as reslts of liqefaction tests. CYCLC STRAN CONTROLLED RESPONSE A pore pressre model for predicting simple shear response of a sand element nder prescribed cycles of shear strain can be developed from the eqations presented in the previos sections and these have been incorporated in the compter code SSLQ (Byrne, 199. The prpose here is to check this model against laboratory data. Pore pressre rise from cyclic strain controlled tests are shon in Fig. 7 and indicate that there is a threshold shear strain belo hich plastic volmetric strains and pore pressre rise ill not occr in ndrained tests. These tests involve a very large nmber of cycles. t as fond that the model based on Eq. 11 overpredicted the volmetric strains and pore pressres. A correction to the shear strain to accont for threshold strain is necessary to obtain reasonable agreement ith the data.!;i ::: o...tf Q. ( l.o ,-fr-!r , Model Pred1C1n --- / / n2cycln / Y. Thret.hold t.trotn amplitde / Lbborotory Tst Data Montere'1 - Sand / J Specimen Spectmen :j No. Type o / o M-1 Hollo J M ll Hollo r,. o---.,1 o M-12 Hollo ' o M l Sohd // / Q M-17 Solid / ---- '"?K :;/"" _l After Chno..,""" o o/ Yt 1. etol.,l984 _.,.,. / OL-oo-OOa SHEAR STRAN AMPLTUDE, r% Fig. 7. Excess poreater pressre bild-p verss shear strain amplitde, resonant colmn tests. The threshold strain effect is acconted for by specifying an effective or plastic shear strain, 7*, to be sed in Eq. follos: in hich 1* = 1-1t 1t the threshold strain ( 11 as ( 17 Model predictions ith 1t = and.5% are shon in Fig. 7. 1t = clearly overpredicts the pore pressre response hile 1t =.5% gives a loer bond to the response. 1t =.2% gave a best fit to the data bt is not shon on the figre for clarity. Strain controlled cyclic triaxial tests reported by NRC (1985 and attribted to Dobry are shon in Fig. 8 and indicate that little pore pressre is generated for 1 cycles of strain if the cyclic strain is less than.1%. 1t =.5% gives an pper bond to this data. Based pon the data of these to figres a compromise 1t =.5% as selected for calibration ith the reslts of load controlled tests... -tl', LO :::.6 ::l C/ C/ ll , ,.4.2 Strain- controlled Cyclic Triaxial Tests Model n Ocycles Prediction / Yt =.5% /. / / / /"" ;' ll. OL_------_L j 3 Fig. 8. SHEAR STRAN, l'c% Excess poreater pressre bild-p. Test data from NRC (1985. CYCLC LOAD CONTROLLED RESPONSE n predicting the cyclic load controlled ndrained response of satrated sand sing the proposed model, it is necessary to introdce a shear-stress strain la in order to compte shear strains from the applied shear stresses. Shear Stress-Strain La Nmeros researchers (Seed & driss (197; Hardin & Drnevich (1972; Tokimats & Seed (1987, have proposed shear stress-strain relations for sand. The relations are nonlinear bt are generally expressed sing a strain compatible secant modls that is sally specified in terms of a maximm shear modls, Gmax and a modls redction factor that depends pon the level of shear stress 51

7 or strain. Based on Seed and driss (197, the maximm shear modls, Gmax hich occrs at shear strain vales of less than lo-% can be expressed as: hich o Gmax = 21.7 (K P (!!!.. 5 max a Pa (18 (K,lmax a modls parameter that depends on the density or (N 1 the sand 6 vale of Pa = atmospheric pressre in the nits sed and -b".4..3 t= <t :: 1/ 1/ :: t- 1/ ::::i NUMBER OF CYCLES 1 Laboratory Dolo o Dr= 82% o Dr 68% Dr= 54% o = the mean normal effective stress Seed and driss (197 data on (k 2 lmax indicates that it can be expressed as a fnction of Dr by the folloing eqation: Fig. 9. Cyclic stress ratio verss nmber of cyclic to initial liqefaction. Test data from shaking table tests, De Alba et al. (1976. (k,lmax ( 19 in hich Dr is in %. n terms of (N 1 sggest that, 6, Seed et al. ( ' 3 (K 2 lmax = 2(N 1 6 ( 2 Eqations (18, (19, and/or (2 allo Gmax to be compted hen the effective stresses and (N 1 6 or relative density are knon. Hardin and Drnevich (1972, Seed et al. (1986 and Tokimats and Seed (1987 propose modls redction crves that allo the appropriate strain compatible secant modls to be compted. The Hardin and Drnevich approach is sed here as it gives reslts similar to Seed et al. and is readily expressed in mathematical form as follos: G = Gmax = (- l+rh ( 21 in hich rh = hyperbolic strain. The details involved in compting rh are given in the Appendix. Liqefaction Resistance TN Too The characteristic shape of the liqefaction resistance crves can be better examined in terms of a dimensionless stress ratio tn/t 15 verss the nmber of cycles to liqefaction and this is shon in Fig. 1. Both the laboratory data and the model predictions are for a range of relative densities and normal stresses. t as fond that the characteristical shape as strongly dependent on the threshold strain vale assmed and this is shon in the figre. For a threshold strain vale of ero, as as initially considered, a poor fit as obtained (not shon. The best fit as obtained sing a threshold strain Yt =.1%. An adeqate fit is obtained ith Yt =.5%, and this vale is sed in all other model predictions LEGEND Sd et ol 198 Sd,979-,,4% Seed,1979-D,82% o Garoo 8 McKoy,l984-,5% 6 Shibata et ol.,1972 Toyoro Sond-,1% shihara 8 Wotonabt, 1976 Naigata Sand-, 54,.. Vatd Data - Tailinos Sand TN Cycltc stress ratio to cam hqetoction in N cycles The shear stress-strain eqations ere incorporated in the SSLQ program. This allos the shear strain to be compted for the prescribed shear stress, and accont is taken of the rising pore pressre and its effect on the shear modls NUMBER OF CYCLES TO LQUEFACTON 1 The predicted liqefaction resistance crves are compared ith laboratory measred vales for three different relative densities in Fig. 9. t may be seen that both the characteristic shape of the predicted crves as ell as the actal stress ratio vales are in good agreement ith the measrements over the range of relative densities. Fig. 1. Relationship beteen shear stress level and the nmber of cycles to case initial liqefaction. Liqefaction resistance crves presented by Seed et al. ( 1985 and based on field observations dring past earthqakes are presented in Fig. 11. These crves presently represent the state of the practice and are based on stress ratios compted from the earthqake, 52

8 normalied standard penetration resistance vales (N 1 6 at the sites in qestion, and field evidence of liqefaction. The chart lines are considered to represent the field resistance for M7. 5 earthqakes casing 15 load cycles. The model prediction for initial liqefaction, g/o = 1 in 15 cycles, is shon as the dashed line in Fig. 11. t is generally in.6 LtQefoction ith }L =:2% =: 1"/o :::3 /o.8-6 ::: " f n. 4 a.. 2 a CYCLE RATO, nlnt -bo. - i= < ex:.5.4 (/.3 (/ ex: (/ _.J Fig. 11. rp ooo Field Oa a ' Liqefaction No Liqefaction No LiQe fo ctio n C Relationship beteen cyclic stress ratio, (N 1 6 vale and liqefaction, M=7.5. Test data from Seed et al. (1984. Fig. 12. Rate of pore pressre cyclic simple shear. from Seed et al. (1976 bild-p in Test data The generated pore pressre ratio as a fnction of factor of safety against tiggering liqefaction is shon in Fig. 13. The factor of safety is defined as the ratio of the stress ratio to case liqefaction to the applied stress ratio, and the pore pressres are examined at n = n 1. Also shon on the figres is laboratory data on a range of sand and gravels. The model reslts follo the trend of the measrements and plot near the pper bond for sands. gl , r---r-- B f < Grovel ([vo Mtl'ltl 1988 [::J Sand (Tck''Ch 8 Ycl.,,l!'l,!9631 Model 5 otl-;:o----:-l:::.::::;:j te 2 22 q 21 cr FACTOR: OF SAFE T'r' AGANST LtOUEFACT!ON, FSl close agreement ith the field data except at the higher (N 1 6 vales, here the model predictions lie belo the Seed line. The loest seed line is for a cyclic strain amplitde of 3%. nitial liqefaction occrs at strains of less than. 5%, and for the denser material ith the high (N 1 6 vale, additional cycles old be reqired to indce 3% strains and this may accont for the divergence in the predicted and observed response shon. At the loer densities, large strains occr as soon as the initial liqefaction condition is reached, so that the crves for all strains converge as shon at loer (N 1 6 vales. The characteristic shape of the pore pressre rise crve ith nmber of constant amplitde load cycles is examined in Fig. 12. The nmber of cycles is expressed in dimensionless form as the ratio of the crrent cycle nmber, n, to the nmber of cycles to case initial liqefaction, n 1. t may be seen that the model prediction lies ithin the measred data. Fig. 13. Excess Poreater Pressre and SUMMARY Factor of Safety Against Liqefaction. Test data from Marcson et al. (199. A simple 2 parameter incremental shear-volme copling eqation has been presented for sand. The eqation is based pon laboratory data and gives predictions that are in very good agreement ith laboratory data over a range of relative densities. The model parameters can be obtained from laboratory tests or they may be estimated from existing data if the relative density or (N 1 the sand is knon. 6 vale of The shear-volme copling eqation is incorporated in a cyclic simple shear pore pressre element model hich incldes an elastic rebond modls eqation that allos the excess pore pressres to be compted for any prescribed cyclic shear strain history. A comparison ith laboratory cyclic strain test data indicates that there is a threshold 53

9 shear strain, Tt' hose vale is in the range 2. to 2 % belo hich plastic volmetric strain and pore pressre rise does not occr. The existing data sggests that Tt =.5% is appropriate. The model is extended to cyclic loading tests by the introdction of a shear stress-strain la in hich the shear modls is modified for both the crrent strain and excess poreater pressre. This allos the appropriate shear strain to be compted for the crrent cycle. The model predictions are compared ith laboratory cyclic load tests and field experience dring earthqakes, and fond to be generally in excellent agreement both in terms of trends and in terms of specific vales. The model parameters are easily obtained from specific test data sing the interactive compter code SSLQ. Parameters that ill match existing design crves based on (N 1 6 vales are bilt into SSLQ. These parameters may then be incorporated in dynamic analysis programs sch as DLQ (Byrne and Yan, 199, based on Finn, Byrne and Martin (1976, or they cold be incorporated in a modified version of TARA-3 (Finn et al., ACKNOWLEDGEMENT The athor is gratefl to Mr. Li Yan for his critical revie of the paper and to Mrs. K. Lamb for typing and presentation. The athor also acknoledges the financial spport of NSERC Canada. REFERENCES Byrne, P.M., "SSLQ: A Compter Code for Predicting the Simple Shear Response of a Sand Element to Cyclic Loading", Soil Mechanics Series No. 145, Dept. of Civil Engineering, University of British Colmbia, Agst 199. Byrne, P.M. and Yan, L., "D-LQ: A Compter Code for Predicting the Effective Stress Dimensional Response of Soil Layers to Seismic Loading", Soil Mechanics Series No. 146, Dept. of Civil Engineering, University of British Colmbia, September 199. Chng, R.M., Yokel, F.Y. and Drnevich, V.P., "Evalation of Dynamic Properties of Sands by Resonant Colmn Testing", Geotechnical Testing Jornal, ASTM, Vol. 7, No. 2, Jne , p p De Alba, P., Seed, H.B. and Chan, C.K., "Sand Liqefaction in Large-Scale Simple Shear Tests", Jornal of the Geotechnical Engineering Division, ASCE, Vol. 12, No. GT9, September 1976, pp Finn, W.D. Liam and Byrne, P.M., "Estimating Settlements in Dry Sands Dring Earthqakes", Canadian Geotechnical Jornal, 1976, Vol. 13, No.4. Finn, W.D. Liam, Byrne, P.M. G. R., "Seismic Response and of Sands", Jornal of the Eng. Division, ASCE, No and Martin, Liqefaction Geotechnical GT8, Agst Finn, W.D. Liam, Yogendrakmar, M., Yoshida, N. and Yoshida, H., "TARA-3: A Program to Compte the Response of 2-D Embankments and Soil-Strctre nteraction Systems to Seismic Loadings", Dept. of Civil Engineering, University of British Colmbia, Vancover, B.C., Canada, Hardin, B.O. and Drnevich, V.P., "Shear Modls and Damping Crves in Soils: Design Eqations and Crves", Jornal of the Soil Mech. and Fondations Division, ASCE, 1972, No. SM7. Marcson, W.E., Hynes, M.E. and Franklin, A. G., "Evalation and Use of Residal Strength in Seismic Safety Analysis of Dams", Earthqake Spectra, Vol. 6, No. 3, Agst 199. Martin, G.R., Finn, W.D. Liam and Seed, H.B., "Fndamentals of Liqefaction Under Cyclic Loading", Jornal of the Geotechnical Eng. Division, ASCE, May 1975, Vol. 11, No. GT5. National Research Concil, Committee on Earthqake Engineering, Commission on Engineering and Technical Systems, "Liqefaction of Soils Dring Earthqakes", National Academy Press, Washington, D.C., Seed, H.B. and driss,.m., "Soil Modli and Damping Factors for Dynamic Response Analysis", Rpt. No. UCB/EERC-7/1 University of California, Berkeley: 197. Seed, H.B., Martin, P.P. and Lysmer J., "Pore-Water Pressre Changes Drin Soil Liqefaction", Jornal of the Geotechnical Engineering Division, ASCE, Vol. 12, No. GT4, April 1976, pp Seed, H.B., Wong, R.T., driss,.m. and Tokimats, K., "Modli and Damping Factors for Dynamic Analyses of Cohesionless Soils", Jornal of Geotechnical Eng., November 1986, Vol. 112, No. 11. Seed, H.B., Tokimats, K., Harder, L.F. and Chng, R., "nflence of SPT Procedres in Soil Liqefaction Resistance Evalations", Jornal of Geotechnical Eng., ASCE, 1985, Vol. 111, No. 12. Tokimats, K. and Seed, H.B., "Evalation of Settlements in Sands De to Earthqake Shaking", Jornal of Geotechnical Eng., ASCE, 1987, Vol. 113, No. 8, pp

10 APPENDX Shear Modls The appropriate secant specified by Eqation namely: shear ( 21 modls as in the text, G = max ( lh (Al here lh _J_ [ 1 + a EXP(-b L_ l ref ref (A2 a -.2 log Ncyc (A3 b.16 and ref 1 max1 max (A4 1 max l+k K [(-- osin<jl 2 - ( v. 2 l l / 2 (AS here Ko the at-rest pressre coefficient <P' the effective friction angle of o' v the sand given by. <P' <jl: - ll.<jl' log(o/pa (A6 <P, 32 + (N 1 and 6 /3 (A7 ll.<jl' =.18 (N 1 o (AS the vertical effective stress f a and b are taken to be ero, then G = max ( 1-1 cyc1 1 maxl (A9 The modls redction crves G/Gmax defined by either Eq. (Al or (A9 essentially fall ithin the modls redction band sggested by Seed et al. ( Eqation (Al as sed in all predictions. 55