Low-amplitude dynamic properties for compacted sand-clay mixtures

Transcription

1 Low-amplitud dynamic proprtis for compactd sand-clay mixturs A. Hassanipour, A. Shafi and M.K. Jafari PhD Candidat, Gotchnical Enginring Rsarch Cntr, Intrnational Institut of Earthquak Enginring and Sismology (IIEES), No., Wst Arghavan Av., North Dibaji Str., Thran, Iran. Zip Cod: 97, -mail: Assistant Profssor, Gotchnical Enginring Rsarch Cntr, IIEES, Thran, Iran, -mail: Profssor, Gotchnical Enginring Rsarch Cntr, IIEES, Thran, Iran, -mail: ABSTRACT Shar modulus and damping ratio ar important input paramtrs in dynamic analysis. A sris of rsonant column tsts was carrid out on pur clays and sand-clay mixturs prpard at diffrnt dnsitis to invstigat th ffcts of aggrgat contnt, confining strss, void ratio and clay plasticity on th maximum shar modulus and minimum damping ratio. Tst rsults rvald an incras in th maximum shar modulus of th mixtur with th incras in sand contnt up to %, followd by a dcras byond this valu. It was also found that th maximum shar modulus incrass with confining strss, and dcrass with void ratio. In addition, minimum damping ratio incrass with sand contnt and clay plasticity and dcrass with confining strss. Finally, on th basis of th tst rsults, a mathmatical modl was dvlopd for th maximum shar modulus. Kywords: Maximum shar modulus, Minimum damping ratio, Sand-clay mixtur, Mathmatical modl

2 Introduction Compactd aggrgat-clay mixturs hav bn succssfully usd as th cors of mbankmnt dams. Ths matrials, calld composit clays by Jafari and Shafi [], ar usually broadly gradd and ar composd of clay as th main body with sand, gravl, cobbl or vn bouldrs floating in th clay matrix. Th Miboro and Ohshirakawa dams in Japan [], Taguaza dam in Vnzula [], and Karkhh and Gotvand dams in Iran ar som xampls of dams with cors composd of aggrgatclay mixturs. It is also currnt practic to mploy low prmabl mixturs of high plastic clay with aggrgats as imprvious blankts for wast disposal projcts [-8]. It is gnrally assumd that th coarsr portion of such soils imparts a rlativly high shar strngth, high compactd dnsity and low comprssibility whil th prmability of th soil is govrnd by th proportion and natur of th finr portion. This gnrally rsults in a rlativly srvicabl and troubl fr fill [9]. A rviw of th publishd litratur in monotonic loading rvals that, in gnral, shar strngth ithr incrass with aggrgat contnt or rmains constant until a limiting aggrgat contnt, thn incrass as th aggrgat contnt incrass [-,]. On th othr hand, th comprhnsiv studis of Jafari and Shafi [] and Jafari and Shafi () showd that in th cas of cyclic undraind loading on compactd aggrgat-clay mixturs, th assumption that adding aggrgat to pur clay improvs its mchanical proprtis is qustionabl. It was concludd that whn aggrgat contnt is raisd, cyclic shar strngth would dcras. Furthr numrical invstigations by Shafi [] and Shafi t al. [] rvald that high por prssur build-up in dams with cors composd of aggrgat-clay mixturs may thratn dam stability undr sismic loading. This shows th nd to undrstand diffrnt faturs of aggrgat-clay mixturs bhavior particularly undr dynamic loading. Dtrmination of low-amplitud dynamic proprtis of matrials is th first stp in charactrization of

3 dynamic bhavior. Ths proprtis ar also ssntial paramtrs for ground rspons and soilstructur intraction analyss. At vry low shar strain lvls (lss than - %), shar modulus (G), and damping ratio (D) rmain ssntially constant; shar modulus is at its maximum valu (G max ), and damping ratio at its minimum valu (D min ). Although it appars that D min dos not hav th sam importanc as G max, it is rquird in modling damping ratio whn th gnral damping quation is usd. Svral rsarchrs hav studid rlationships btwn G/Gmax and D. Hardin and Drnvich [] assumd that D is proportional to (-G/Gmax). Othrs associatd D with G/Gmax using a polynomial function [, ]. Non of ths modls, howvr, rflct th complx rlationship btwn plasticity indx (PI) and D. Darndli [] and Stoko t al. [7] modld th hystrtic damping assuming Masing s bhavior [8] and an adjusting function to fit th Masing s damping to th xprimntal data, and thn addd a Dmin trm to obtain th total damping. Th gnral damping quation adoptd for thir study has th following form: D=f(G/ G max ) + D min () whr f(g/ G max ) is a function of normalizd shar modulus. Th most important factors that influnc damping ratio includ shar strain amplitud, man ffctiv confining strss, soil typ and plasticity indx, frquncy of loading, and numbr of loading cycls. Th ffct of plasticity indx chang on damping ratio is complx, howvr th EPRI [7], Stoko t al. [8], and Vuctic t al. [9] found that valus of D min incras with th incras in PI, whil valus of damping ratio dcras at high shar strains with incrasing PI. Earlir studis on damping ratio and shar modulus for composit gravl-clay soils did not show this complx ffct of PI on damping ratio [, ]. As xplaind by Stoko t al. [], on problm with laboratory D masurmnts lis in th idntification of quipmnt-rlatd nrgy loss.

4 On th othr hand, many xprimntal invstigations carrid out on sandy and normally consolidatd clayy soils in arly studis [9-] showd G max was basically rlatd to th man ffctiv principal strss, σ m and void ratio, xprssd by th wll known quation: G max σ ) n = AF( )( () m in which A is an mpirical constant rflcting soil fabric formd through various strss and strain historis, n is mpirically dtrmind xponnt, approximatly qual to. [, ], and F() is void ratio function, which is usually givn by: ( B ) () F( ) = + whr constant B is usually takn as.7 for round-graind sands and.97 for angular-graind sands []. Tabl prsnts a summary of th mpirical formula for normally consolidatd clays. Although composit soils with proprtis btwn cohsiv and granular matrials ar found in natur normously, unlik sands and clays, lss ffort has bn ddicatd toward undrstanding thir dynamic bhavior. This is mainly du to th inhrnt difficultis in charactrizing htrognous mdia. Th invstigation dscribd in this papr ntails a study on th low-amplitud dynamic proprtis of sand-clay mixturs using rsonant column tsts. Low, mdium and high plastic clays wr mixd with diffrnt amounts of sand to xplor th ffct of th soil plasticity and aggrgat contnt on th dynamic proprtis. Th ffct of initial dnsity and confining strss wr also invstigatd by prparing th spcimns at thr diffrnt dnsitis and tsting thm undr thr diffrnt ffctiv confining strsss. Evntually, on th basis of th 8 tst rsults, a mathmatical modl for th maximum shar modulus is prsntd.

5 Tstd matrials and procdur Matrials tstd Low, mdium and high plastic pur clays with nin mixturs of th clays with sand wr usd in this study. Fig. and Tabls (a) and (b) prsnt th grain-siz distribution, and physical proprtis for th clays and sand-clay mixtur. Th sand usd in th study was rtrivd from a rivrbd and composd of subroundd particls with minimum and maximum void ratios of. and.9 rspctivly, and a spcific gravity of.. Fig. shows th grain-siz distribution for th sand. As sn in Fig., all th sand particls ar gratr than. mm, and thy will all rmain on th µm siv. Thus, th plasticity indx of sand-clay mixturs will b that of th clay portion (Tabl b) whn it is dtrmind on th basis of th ASTM D8- []. Spcimn prparation Th spcimn prparation tchniqu was chosn to modl as prcisly as possibl th in situ condition of th cor matrials of mbankmnt dams. All th spcimns, typically 7 mm in diamtr and mm in hight wr prpard, with rlativ compactions (RC) of 8, 88 and 9% and watr contnt of % wt of optimum. Rlativ compaction is dfind as sampl dry dnsity dividd by its maximum dry dnsity obtaind from standard compaction tst mthod [7]. Appropriat amounts of clay and sand for ach layr wr first thoroughly mixd. Each layr was thn mixd with watr at last hours bfor us and sald. Th matrial was pourd in six layrs into a cylindrical mold and compactd. To achiv a gratr uniformity of spcimns, a procdur similar to th undrcompaction tchniqu [8] was usd. For ach layr, th compactiv ffort was incrasd toward th top by incrasing th numbr of blows pr layr. Each layr was thn scord aftr it was compactd for bttr bonding with th nxt layr.

6 Tst procdur Th spcimns wr saturatd with a Skmpton B valu in xcss of 97%. To facilitat th saturation procss, CO was first prcolatd through th spcimns thn d-aird watr was flushd into th spcimns. Lastly, a back prssur of kpa was incrmntally applid to acclrat th saturation rat. Th spcimns wr thn isotropically consolidatd undr ffctiv confining strsss of,, and kpa. Figur shows spcimns' aftr-consolidation void ratios, in trms of sand contnt. Following consolidation, torsional rsonant column tsts wr carrid out undr th spcifications of ASTM D [9], using a fixd-fr typ dvic. Effct of sand contnt on G max and D min Figur prsnts th variation of G max in trms of sand contnt at diffrnt confining strsss ( σ m ) and rlativ compactions (RC). As may b sn, rgardlss of th confining strss and rlativ compaction valu, G max incrass with aggrgat contnt, until a maximum G max is rachd at a sand contnt of %. As th sand contnt continus to incras abov %, G max dcrass. Valljo and Lobo-Gurrro [] also indicatd th sam trnd in unsaturatd mixturs of Ottawa sand with kaolinit clay. Th variation of void ratio against sand contnt (Fig. ) can rasonably justify th bhavior shown in Fig.. As shown in Fig., void ratio dcrass with sand contnt until raching its minimum at %. Byond % sand contnt, whr sandy grains prvails th soil sklton, void ratio gnrally incrass with sand contnt. Prakasha and Chandraskaran [] also showd that a minimum in void ratio is achivd at a sand contnt of 7% in marin sand-clay mixturs. Figur prsnts th variation of D min in trms of sand contnt at diffrnt confining strsss and rlativ compactions. As may b sn, rgardlss of th confining strss and rlativ compaction valu, D min gnrally incrass with sand contnt.

7 Effct of confining prssur on G max and D min Fig. dpicts th variation of G max against confining strss ( σ ). As sn, it appars that th rat of incras in G max with confining strss (that is rflctd by, xponnt n in Eq. ()) is non-linar m and idntical for all th mixturs. Figur prsnts th variation of G P a σ P in trms. ( max / ) /( m / a ) of aggrgat contnt; whr P a is atmosphric prssur and xponnt n in Eq.() is takn as.. As sn, rgardlss of th dnsity and plastic proprtis of th mixtur, G max can b succssfully rlatd to σ by taking n =.. Th ffct of confining strss on D min is shown in Fig. 7. As can b sn, D min m dcrass with confining strss, which is also in accordanc with Zhang t al. [] invstigations on th ffct of confining strss on D min. Effct of clay plasticity and void ratio on G max and D min Th sand was mixd with low, mdium and high plastic clays to invstigat th ffct of clay plasticity on th low-amplitud dynamic proprtis. Th tst rsults prviously prsntd in Figs. and can b usd to xplor th ffct of clay plasticity on G max and D min rspctivly. As shown in Fig., for th mixturs with an idntical rlativ compaction and confining strss, G max dcrass with clay plasticity. Th dcras in G max with clay plasticity can b justifid in th light of void ratio incras with clay plasticity (Fig. ). It is quit vidnt that G max is a function of void ratio (), and dcrass as incrass. Figur 8 prsnts th variation of G max in trms of in diffrnt mixturs usd in this study. Th figur clarly shows that G max is not a function of clay plasticity, but is a function of void ratio. Sinc void ratio incrass with clay plasticity, mixturs containing highr plastic clays show lowr G max. D min also incrass gnrally with clay plasticity (Fig. ). This is in contradict with th Vuctic and Dobry [] curvs that shows matrial damping dcrass with soil plasticity. Howvr, D min masurd in rsonant column apparatus is calld intrnal damping [], and is diffrnt from th matrial damping masurd in cyclic triaxial or simpl shar apparatus. Th intrnal damping occurs 7

8 whn th nrgy is lost in dvloping friction btwn soil particls during strss rvrsals. Figur 9 can also b usd to vrify th ffct of void ratio on D min. As can b sn, D min is almost indpndnt of void ratio. A mathmatical modl for G max It was shown that G max is a function of sand contnt, confining prssur and void ratio. Hnc, to charactriz th low-amplitud dynamic proprtis of aggrgat-clay mixturs, it is ncssary to find an appropriat mathmatical modl in th form of Eq. (). Hrin, a rgrssion analysis basd on th last squar tchniqu is usd to find th valus of constants A and B in Eqs.() and (), assuming n =.. Th analyss ar carrid out by varying B for ach mixtur until achiving a cofficint of dtrmination, R mor than 9%. Th critrion will b satisfid if a valu of.9 is assumd for B. Figur dpicts th variation of. ( max / P a ) /( m / Pa ) G σ in trms of F() for diffrnt mixturs, whr B is qual to.9. As sn, a lin can b succssfully fittd to th data. Th slop of this lin is A / Pa., whr P a is atmosphric prssur, and is qual to kpa. It is intrsting to not that th valu of B obtaind for sand-clay mixturs is vry clos to.97, that is proposd in othr studis for clays (Tabl ). Tabl shows th valus of A and R for ach mixtur whn F() is qual to (.9 ) /( + ). As sn, valu of constant A varis from to which falls wll into th limits proposd in othr invstigations for clays (Tabl ). Figur shows th variation of A against sand contnt. As sn, constant A linarly incrass with aggrgat contnt until a sand contnt of %. As th sand contnt continus to incras abov %, A dcrass. Thus, A can b dscribd as a function of sand contnt (SC) by th following linar quations (Fig.): A = 7 SC + SC. () A = SC + SC 8 () 8

9 Th variation of G max in trms of along with th proposd modl for ach mixtur is shown in Fig.. As sn, th modl succssfully prdicts G max.. Conclusions An xprimntal study was prformd on th compactd pur clays and mixturs of sand-clay to invstigat th ffct of sand contnt, confining strss, dry dnsity and plasticity of th clayy part on th low-amplitud dynamic dformation proprtis using rsonant column tsts. A mathmatical modl was also dvlopd for maximum shar modulus, G max of sand-clay mixturs. Th following conclusions may b drawn basd on this xprimntal study:. G max incrass with aggrgat contnt, until a maximum G max is rachd at a sand contnt of %. As th sand contnt continus to incras abov %, G max dcrass. In addition, G max incrass nonlinarly with confining strss, so that it has a good corrlation with squar root of confining strss. G max is not a function of clay plasticity, but is a function of void ratio. Sinc void ratio incrass with clay plasticity, mixturs containing highr plastic clays show lowr G max ;. D min incrass with sand contnt and soil plasticity, howvr it dcrass with confining strss. In addition, D min is not affctd by th spcimn void ratio;. A mathmatical modl was dvlopd for G max of th mixturs xamind in this study. Th modl is vry similar to Hardin and Drnvich (97) modl for normally consolidatd clays and prdicts G max as (.9 ). σ m A +, whr A is a function of sand contnt, is void ratio and σ m is man ffctiv principal strss. Rfrncs [] Jafari, M.K., Shafi, A.,. Mchanical bhavior of compactd composit clays. Canadian Gotchnical Journal., (): -7. 9

10 [] Asao, I., 9. Th Miboro dam., Procding of 8th Intrnational Congrss on Larg Dams, Edinburgh, UK., Rport, Vol. III: 8-8. [] Shrard, J.L., 98. Building mbankmnt dams in aras of high rainfall. Symposium on Hydrolctric Dvlopmnt in th Amazon Rgion, Sao Paolo, Brazil. -9. [] Abl, W.V., 98. Th influnc of bntonit on th prmability of sandy silts. Nuclar and Chmical Wast Managmnt, :8-88. [] Chapuis, R.P., 99. Sand-bntonit linrs: prdicting prmability from laboratory tsts. Canadian Gotchnical Journal, 7():7-7. [] Lundgrn, T.A., 98. Som bntonit salants in soil mixd blankts. Procdings of th Intrnational Confrnc on Soil Mchanics and Foundation Enginring, Stockholm, Swdn, : 9-. [7] Shafi, A., 8a. Prmability of compactd granul-clay mixturs. Enginring Gology, 97(- ): [8] Pandian, N.S., Nagaraj, T.S., Raju, P.S.R.N., 99. Prmability and comprssibility bhavior of bntonit-sand/soil mixs. Gotchncial Tsting Journal, 8():8-9. [9] Garga, V.K., Madurira, C.J., 98. Compaction charactristics of rivr trrac gravl. Journal of Gotchnical Enginring, (8): [] Muir Wood, D., Kumar G.,. Exprimntal obsrvations of bhaviour of htrognous soils. Mch. Cohsiv-Frictional Matr. : 7-98 [] Patwardhan, A.S., Rao, J.S., Gaidhan, R.B., 97. Intrlocking ffcts and sharing rsistanc of bouldrs and larg siz particls in a matrix of fins on th basis of larg scal dirct shar tsts. Procdings of nd Southast Asian Confrnc on Soil Mchanics. Singapor. -7. [] Shafi, A., Tavakoli, H.R., Jafari, M.K. 8a. Undraind bhavior of compactd sand-clay mixturs undr monotonic loading paths. Journal of Applid Scincs, 8(8), 8-8.

11 [] Valljo, L.E., Zhou, Y., 99. Th mchanical proprtis of simulatd soil-rock mixturs. In Procdings of th th Intrnational Confrnc on Soil Mchanics and Foundation Enginring, Nw Dlhi, India. A.A. Balkma, Rottrdam, th Nthrlands. : 8 [] Jafari, M.K., Shafi, A., 998. Dynamic bhavior of mixd matrials usd for cor of Karkhh dam. Procding of. th Europan Confrnc on Earthquak. Enginring. Paris, Franc, p.79. [] Shafi, A., 8b. Th influnc of clay cor composition on th sismic stability of mbankmnt dams. ICE Gotchnical Journal, (), 8-9. [] Shafi, A., Bahador, M., Bahrami, R., 8b. Application of fuzzy st thory to valuat th ffct of por prssur build-up on th sismic stability of Karkhh mbankmnt dam, Iran. Journal of Earthquak Enginring, (8),9-. [7] Elctric Powr Rsarch Institut (EPRI) (99). Guidlins for dtrmining dsign basis ground motions. Final Rp. No. TR-9, Palo Alto, Calif. [8] Stoko, K. H., II, Hwang, S. K., L, N. J., and Andrus, R. D. (99). Effcts of various paramtrs on th stiffnss and damping of soils at small to mdium strains. Proc., Int. Symp. Prfailur Dformation Charactristics of Gomatrials, Vol., Sapporo, Japan, [9] Vuctic, M., Lanzo, G., and Doroudian, M. (998). Damping at small strains in cyclic simpl shar tst. J. Gotch. Gonviron. Eng., (7), 8-9. [] Sd, H.B., Idriss, I.M., 97. Simplifid procdur for valuating soil liqufaction potntial. Journal of Gotchnical Enginring, 97(9): 9-7. [] Vuctic, M., Dobry, R., 99. Effct of soil plasticity on cyclic rspons. Journal of Gotchnical Enginring, 7():89-7.

12 [] Stoko, K. H., II, Hwang, S. K., Darndli, M. B., and L, N. J. (99). Corrlation study of nonlinar dynamic soil proprtis. Final Rp. to Wstinghous Savannah Rivr Company, Aikn, S.C. [] Hardin, B.O., Drnvich, V.P., 97. Shar modulus and damping in soil: dsign quations and curvs. Journal of th Soil mchanics and Foundation Enginring Division, ASCE, 98(7): 7-9. [] Bordn, R. H., Shao, L., and Gupta, A. (99). Dynamic proprtis of Pidmont rsidual soils. J. Gotch. Eng., (), 8-8. [] Ishibashi, I. and Zhang, X. J. (99). Unifid dynamic shar moduli and damping ratios of sand and clay. Soils Found., (), 8-9. [] Darndli, M. B. (). Dvlopmnt of a nw family of normalizd modulus rduction and matrial damping curvs. PhD dissrtation, Univ. of Txas at Austin, Austin, Tx. [7] Stoko, K. H., II, Darndli, M. B., Gilbrt, R. B., Mnq, F.-Y, and Choi, W.-K. (). Dvlopmnt of a nw family of normalizd modulus rduction and matrial damping curvs. Proc., NSF/PEER Int. Workshop on Uncrtaintis in Nonlinar Soil Proprtis and thir Impact of Modling Dynamic Soil Rspons, Univ. of California at Brkly, Brkly, Calif. [8] Masing, G. (9). Eignspannungn und Vrfstingung bim Mssing. Proc., nd Int. Cong. Of Appl. Mch., -. [9] Drnvich, V.P, Richart, F.E., 97. Dynamic prstraining of dry sands. Journal of Soil Mchanics and Foundations Divisions, Procdings of th Amrican Socity of Civil Enginrs, 9(): -9. [] Hardin, B.O., Richart, F.E., 9. Elastic wav vlocitis in granular soils. Journal of Soil Mchanics and Foundations Division, ASCE, 89(): -.

13 [] Hardin, B.O., Black, W.L., 98. Vibration modulus of normally consolidatd clay: dsign quations and curvs. Journal of th Soil mchanics and Foundation Enginring Division, ASCE, 9(): -9. [] Kokusho, T., 98. Cyclic triaxial tst on dynamic soil proprtis for wid strain rang. Soils and foundations JSSMFE, (): -. [] Sd, H. B., Wong, R. T., Idriss, I. M., and Tokimatsu, K. (98). Moduli and damping factors for dynamic analysis of cohsionlss soils. J. Gotch. Eng., (), -. [] Bobby, O.H., Kalinski, M.E.,. Estimating th shar modulus of gravlly soils. Journal of Gotchnical and Gonvironmntal Enginring, (7): [] Zhou, Y.G., Chn, Y.M.,. Influnc of sismic cyclic loading history on small strain shar modulus of saturatd sands. Soil Dynamics and Earthquak Enginring, (): -. [] ASTM D8. Standard Tst Mthods for Liquid Limit, Plastic Limit, and Plasticity Indx of Soils. Annual Book of ASTM Standards [7] ASTM D 98-a,. Standard Tst Mthods for Laboratory Compaction Charactristics of Soil Using Standard Effort. In Annual Book of ASTM Standards vol..8, sc., ASTM Intrnational, Wst Conshohockn, PA, [8] Ladd, R.S., 978. Prparing tst spcimns using undrcompaction. Gotchnical Tsting Journal, (): 9-9. [9] ASTM D -9.. Standard tst mthods for modulus and damping of soils by th rsonant column mthod. Annual Book of ASTM Standards. [] Valljo, L.E., Lobo-Gurrro, S.,. Th lastic moduli of clays with disprsd ovrsizd particls. Enginring gology, 78():-7.

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15 Tabl. Constants in proposd mpirical quations on maximum shar modulus of normally consolidatd clays (G max and σ m in kpa): G max ( B ) σ m = A +. Rfrnc A B Matrial Hardin and Black (98).97 Clays Hardin and Drnvich (97).97 Clays Marcuson and Wahls (97).97. Kaolinit Bntonit Zn and Umhara (978) ~.97 Rmoldd clay

16 Tabl (a). Physical proprtis of th clays Clay Typ Plasticity Indx, PI (%) Spcific Gravity Low Plastic (CL).7 Mdium Plastic (CM).7 High Plastic (CH).7 Tabl (b). Compaction proprtis of th sampls usd in this study Sand Contnt (%) Plasticity Indx, PI (%) Optimum Moistur Contnt (%) Maximum Dry Dnsity (gr/cm )

17 Tabl. Valus of constant A in Eq. () and R for th mixturs; (G max and B=.9) σ m ar in kpa and Aggrgat Contnt (%) A R

18 Fig.. Grain-siz distribution for th soils usd in th study; CL,CM and CH stands for low, mdium and high plastic clay rspctivly 8 Prcnt Passing Sand CL CM CH... Particl Siz (mm) 8

19 Fig.. Void ratio variations in sand-clay mixturs (a).... σ' m = kpa RC=8% (b).... σ' m = kpa RC=8% (c).... σ' m = kpa RC=8% (d) σ' m = kpa RC=88% 8 () σ' m = kpa RC=88% 8 (f) σ' m = kpa RC=88% 8 (g) σ' m = kpa RC=9% 8 (h) σ' m = kpa RC=9% 8 (i) σ' m = kpa RC=9% 8 9

20 Fig.. Effct of aggrgat contnt on G max (a) (b) (c) σ' m = kpa RC=8% 8 σ' m = kpa RC=88% 8 σ' m = kpa RC=9% 8 (d) () (f) σ' m = kpa RC=8% 8 σ' m = kpa RC=88% 8 σ' m = kpa RC=9% 8 (g) σ' m = kpa RC=8% 8 (h) σ' m = kpa RC=88% 8 (i) σ' m = kpa RC=9% 8

21 Fig.. Effct of aggrgat contnt on D min (a) σ' m = kpa RC=8% (b) σ' m = kpa RC=88% (c) σ' m = kpa RC=9% (d) σ' m = kpa RC=8% () σ' m = kpa RC=88% (f) σ' m = kpa RC=9% (g) σ' m = kpa RC=8% (h) σ' m = kpa RC=88% (i) σ' m = kpa RC=9% 8 8 8

22 Fig.. Effct of confining strss on G max (a) tur RC=8% 8 (b) tur RC=88% 8 (c) tur RC=9% 8 (d) tur RC=8% 8 () tur RC=88% 8 (f) tur RC=9% 8 (g) tur RC=8% 8 (h) tur RC=88% 8 (i) tur RC=9% 8

23 Fig.. Normalization of G max with rspct to confining strss (G max and σ in kpa) m (Gmax/Pa)/(σ'm/Pa). (a) 8 tur RC=8% 8 (Gmax/Pa)/(σ'm/Pa). (b) 8 tur RC=8% 8 (Gmax/Pa)/(σ'm/Pa). (c) 8 tur RC=8% 8 (Gmax/Pa)/(σ'm/Pa). (d) 8 tur RC=88% 8 (Gmax/Pa)/(σ'm/Pa). () 8 tur RC=88% 8 (Gmax/Pa)/(σ'm/Pa). (f) 8 tur RC=88% 8 (Gmax/Pa)/(σ'm/Pa). (g) tur RC=9% 8 (Gmax/Pa)/(σ'm/Pa). (h) tur RC=9% 8 (Gmax/Pa)/(σ'm/Pa). (i) 8 tur RC=9% 8

24 Fig. 7. Effct of confining strss on D min (a) tur RC=8% (b) tur RC=88% (c) tur RC=9% (d) tur RC=8% () tur RC=88% (f) tur RC=9% (g) tur RC=8% (h) tur RC=88% (i) tur RC=9% 8 8 8

25 Fig. 8. Effct of void ratio on G max (a) Pur Clays CH CM CL (b) (a) % Sand-%Clay Mixturs....9 (c) % Sand-%Clay Mixturs....9 (d) (a) 8% Sand-%Clay Mixturs....9

26 Fig. 9. Effct of void ratio on D min (a) 8 Sand Contnt= % CL CM (b) 8 Sand Contnt= % CH Void Ratio Void Ratio (c) 8 Sand Contnt= % (d) 8 Sand Contnt=8 % Void Ratio Void Ratio

27 Fig.. Void ratio function for sand-clay mixturs (a) Sand Contnt= % (b) Sand Contnt= % (Gmax/Pa)/(σ'm/Pa). 8 F()=(.9-) /(+) (Gmax/Pa)/(σ'm/Pa). 8 F()=(.9-) /(+) (c) Sand Contnt= % (d) Sand Contnt=8 % (Gmax/Pa)/(σ'm/Pa). 8 F()=(.9-) /(+) (Gmax/Pa)/(σ'm/Pa). 8 F()=(.9-) /(+) 7

28 Fig.. Constant A as a function of sand contnt A Eq. () Eq. ()....8 Sand Contnt 8

29 Fig.. Accuracy of th modl in prdicting G max (a) (c) (kpa) (kpa) σ' m = (kpa) Pur Clays (kpa) (kpa) σ' m = (kpa) %Sand-%Clay Mixturs Modl Modl (b) (d) (kpa) (kpa) σ' m = (kpa) %Sand-%Clay Mixturs Modl (kpa) (kpa) σ' m = (kpa) Modl 8%Sand-%Clay Mixturs