EFFECT OF CONSOLIDATION RATIOS ON MAXIMUM DYNAMIC SHEAR MODULUS OF SANDS

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1 Octobr 12-17, 28, Bijing, China EFFECT OF CONSOLIDATION RATIOS ON MAXIMUM DYNAMIC SHEAR MODULUS OF SANDS Xiaoming YUAN 1 Jing SUN 2 and Rui SUN 3 1 Profssor, Dpt. of otchnical Enginring, Institut of Enginring Mchanics, Harbin. China 2 Profssor assistant, Dpt. of otchnical Enginring, Institut of Enginring Mchanics, Harbin. China 3 Profssor, Dpt. of otchnical Enginring, Institut of Enginring Mchanics, Harbin. China yxmim@163.com, imsr@163.com ABSTRACT : Th shar modulus is th most basic paramtr and can b attaind by th xprimnts in th fild or in th laboratory. Th maximum dynamic shar modulus obtaind in th laboratory is gnrally for th cass of th isotropic consolidation. Th most advancd apparatus for tsting th dynamic shar modulus in small strain rang in th laboratory now is th rsonant column dvic. Howvr, th most xisting rsonant column dvics ar only suitabl for spcimns undr isotropic consolidation. Thrfor, th ffct of anisotropic consolidation on th maximum dynamic shar modulus is still a qustion to b discussd furthr. A formula for calculating th incrmnt of th maximum dynamic shar modulus of anisotropically-consolidatd sands is prsntd in th papr. Th nw rsonant column tsting dvic with th anisotropic consolidation function is mployd to attain th formula for calculating th incrmnt of th maximum dynamic shar modulus. Th rsults hr indicat: (1) Th ffct of th anisotropic consolidation on th maximum dynamic shar modulus is quit rmarkabl and cannot b nglctd; (2) A suitabl form to show this ffct is to us th powr function of th incrmnt of th consolidation ratio, i.. (k c -1) B ; (3) Th variation of th maximum dynamic shar modulus for th cass of k c >1 can b xprssd by th rlativ incrmnt formula, i.. Δ m /,m = 1+.66(k c -1).54 ; (4) Th formula prsntd abov mans that th maximum dynamic shar modulus shows a mor rapid ris in th intrval of k c nar to 1 and a slowr ris in th intrval of k c far away from 1; (5) Th incrasing dgr of th maximum shar modulus du to k c >1 is significantly largr than that dscribd by th Hardin and Black s formula,.g. th incrasing dgr for k c from 1 to 2 in th papr is about 66% whil only 15% by th Hardin and Black s formula; (6) Th consolidation ratio also should b on of th important rasons on th obvious diffrnc btwn in th fild and laboratory dtrmination of th maximum dynamic shar modulus of th soils.. KEYWORDS: Anisotropic consolidation; Maximum dynamic shar modulus; Incrmnt formula; Sands 1. INTRODUCTION In th soil proprty, th shar modulus is th most basic paramtr and can b attaind by th tsts in th fild or in th laboratory (Hardin and Black, 1968, 1969; Zn and Higuchi, 1984; Sd and Wong, 1986; Amini t al, 1988; Yu t al, 1988; H, 1997; Chn t al, 22). Th most advancd apparatus for tsting th dynamic shar modulus in small strain rang in th laboratory now is th rsonant column dvic bcaus of its advantags in th simply mchanical principl, clar strss condition and convnint opration and th small dviation of tsting rsults. Howvr, th most xisting rsonant column dvics ar only suitabl for spcimns undr isotropic consolidation. Bcaus th consolidation ratio, k c, is about from 1.4 to 3 in th actual subsoil, som rsarchrs and nginrs try to mploy Hardin and Black s formula (1968,1969) to dscrib th maximum shar modulus of soils undr th diffrnt Consolidation ratio k c. In trms of th dynamic triaxial tsts, som rsarchrs (H, 1997) hav raisd doubt whthr th formula is corrct in dscribing th soil maximum dynamic shar modulus undr th anisotropic consolidation. Howvr, ths rsults ar only a fw and th dynamic triaxial tsts ar basically suitabl for th modrat and larg dformation of soils. Thrfor, th ffct of consolidation ratios on th maximum dynamic shar modulus is still a qustion to b discussd furthr.

Octobr 12-17, 28, Bijing, China 2 TEST APPARATUS To idntify th maximum dynamic shar modulus of soils undr anisotropic consolidation a nw rsonant column dvic as shown in Fig.

2 Octobr 12-17, 28, Bijing, China 2 TEST APPARATUS To idntify th maximum dynamic shar modulus of soils undr anisotropic consolidation a nw rsonant column dvic as shown in Fig.1 is dvlopd in th Institut of Enginring Mchanics, China Earthquak Administration, in 22. As usual th rsonant column is fixd-fr typ, but thr ar two spcial dsigns for prforming th tsts of dviatoric strss consolidation. On is a spcial transmission mchanism to apply th vrtical static dviatoric forc to th soil spcimns without ccntric forc. Anothr is to st a manipulator insid th prssur vssl to solv th unstabl problm during fitting th spcimn and loading th vrtical dviatoric strss to th soil spcimn. Both dsigns nsur that soil spcimn is just right in th axial lin of th load. Aftr fitting th spcimns and applying th confining prssur and th additional vrtical strss, th manipulator can loos by oprating outsid. Figur 1 Th rsonant column dvic usd in tsts 3 SOIL SAMPLES AND TEST PROCEDURE Two kinds of sands, Th Fujian standard sand and Harbin sand of China, ar mployd in th tsts of th papr. Th physical spcifications of th sands ar listd in Tabls 1 and 2 and th grain compositions of th sands ar shown in Figs.2 and 3, rspctivly. From th grain-siz distribution, both two kinds of sands blong to th mdium compact sands. Prcnt finr by wight Tabl 1 Physical spcification of th Fujian sand Spcific gravity Maximum void ratio Minimum void ratio Uniformity cofficint Curvatur cofficint 2.66g/cm Tabl 2 Physical spcification of th Harbin sand Spcific gravity Maximum void ratio Minimum void ratio Uniformity cofficint Curvatur cofficint 2.62g/cm rain siz (mm) Figur 2 Th grading distribution of th Fujian sand.1 Prcnt finr by wight rain siz (mm) Figur 3 Th grading distribution of th Harbin sand In th tsts, thr rlativ dnsitis of th sampls, D r =72.8%, D r =6% and D r =3%, ar mad for th sands. Thr confining strsss, σ 3 =1kPa, 2kPa and 3kPa, and fiv consolidation ratios, k c =1, 1.2, 1.5, 1.7 and.1

3 Octobr 12-17, 28, Bijing, China 2., ar mployd. In som cass, th tsts for k c =2.4 ar prformd. For ach confining strss, fiv or six idntical sand sampls of 3.91cm 8cm ar usd rspctivly for th tsts of th diffrnt consolidation ratio. Aftr finishing th consolidation, th gradd loads of th torsional momnt ar conductd to th soil sampls. By th fr vibration mthod, th dynamic shar modulus of th sampls is finally obtaind. 4 EXPERIMENTAL RESULTS Th tsts on th rlation of th dynamic shar modulus,, and th dynamic shar strain,, ar conductd for th diffrnt rlativ dnsitis of th sampls. Th hyprbolic quation, =1/(a+b), is usd to obtain th rgrssion curv of th dynamic shar modulus vrsus th shar strain. Th maximum dynamic shar modulus, max, can b obtaind from th rgrssion quation by, which also quals to 1/a and th rciprocal of th ordinat intrcption of 1/~ as wll. Th maximum dynamic shar modulus max undr th diffrnt consolidation ratio can b includd by two parts:,m, rprsnting th maximum dynamic shar modulus for k c =1 and Δ m, rprsnting th incrmnt of th maximum dynamic shar modulus du to k c >1. In th papr, th focus is placd on th rlativ incrmnt of th maximum dynamic shar modulus, Δ m /,m, rfrring to th ffct of consolidation ratios on th maximum dynamic shar modulus. 4.1 Rsults of th Fujian Sand Th typical rsult on th tst data and th rgrssion curvs of 1/~ for th Fujian sand of Dr=.6 undr thr confining strsss is dmonstratd in Fig.4 and it can b sn that th dviation of th tst data is quit small. All of th othr rsults show th sam good bhavior. According to th abov dtrmind maximum dynamic shar modulus, th rlations btwn th maximum dynamic shar modulus and th confining strss ar xhibitd in Fig.5 by dual logarithm plot and, th rlations btwn th maximum dynamic shar modulus and th void ratio ar xhibitd in Fig.6. By rgrssing th data for th cass of k c =1 in Figs.5 and 6 a formula for th maximum dynamic shar modulus of th Fujian sand in k c =1 is formd as 2 ( ). 5, m = 117 σ (MPa) (1) Th form of Eq.(1) is consistnt with th Hardin and Black s formula xcpt th slight diffrnc in th first cofficint, which is 117 hr and 12 in th Hardin and Black s. Furthrmor, it can b sn that all of th lins for th diffrnt k c in Fig.5 ar basically paralll and also, th sam phnomna occurs in Fig.6. Thrfor, th form of Δ m /,m can b takn to show th variation of th maximum dynamic shar modulus undr th diffrnt k c. Th rlation ofδ m /,m ~k c can b takn as th following quation Δ m B = C ( k c 1) (2), m whr C and B ar cofficints to b dtrmind. It should b noticd that cofficint C quals to th rlativ incrmnt of th maximum dynamic shar modulus whn k c =2, and th cofficint B rfrs to th curvatur of Δ m /,m ~k c -1.

4 Octobr 12-17, 28, Bijing, China k =1. c k c = / (MPa) k c = / (MPa) k c = /.6 (1/MPa).4 1/ (MPa).6.4 1/ (MPa) k c =1.5.2 k c =1.7.2 k c =1.5.2 k c = k c =2. 1kPa 2kPa 3kPa 1kPa 2kPa 3kPa 1/ (MPa) k c =2. 1kPa 2kPa 3kPa 1kPa 2kPa 3kPa Figur 4 Th 1/~ for th Fujian sand Figur 5 Rlations btwn max and σ 3 max (MPa) σ 3 =1kPa kc=2. kc= max (MPa) σ 3 =2kPa kc=2. kc=1. max (MPa) σ 3 =3kPa kc=2. kc= Figur.6 Rlations btwn max and To dtrmin th cofficints B and C, th ffcts of th confining strssσ 3 and th void ratio onδ m /,m ar illustratd in Figs.7 and 8, rspctivly. It can b sn that th ffct of th confining strssσ 3 can b nglctd as shown in Fig.7 and th ffct of th void ratio should b considrd as shown in Fig.8. It mans that it is only possibl that B and C ar th function of th void ratio. Thrfor, for ach kind of th rlativ dnsity of th sand, B and C can b takn as th avrag valu of th various confining strsss and basd on this, th

5 Octobr 12-17, 28, Bijing, China rlations of th rlativ incrmnt of th maximum dynamic shar modulus, Δ m /,m, and th incrmnt of consolidation ratio, k c -1, for thr void ratios ar illustratd in Fig.9. It can b sn from Fig.9 that th thr lins ar basically paralll, which mans that cofficint B is indpndnt on th void ratio. Thn, th cofficint B can b takn as.458, th avrag valu of.472,.443 and.46 as shown in Fig.9. From Fig.9, it can also b sn that th variation of th cofficint C mainly rsults from th void ratios. By rgrssing th data btwn th cofficint C and th void ratio as shown in Fig.1, th cofficint C is obtaind as following C = 1.85 (3) Finally, th rlativ incrmnt of th maximum shar modulus du to k c >1 can b xprssd as Δ m = 1.85 ( k c.458 1) (4), m Δ m /,m kc =1.2 kc =1.5 kc =1.7 kc =2. Dr= σ 3 (kpa) Δ m /,m (%) kc= 1.2 kc= 1.5 kc= 1.7 kc= 2. σ 3 =1kPa Δ m /,m (%) kc=2. Dr= σ 3 (kpa) Δ m /,m (%) kc=2. σ 3 =2kPa Δ m /,m (%) kc =1.2 kc =1.5 kc =1.7 kc =2. Dr=.3 Δ m /,m (%) kc =1.2 kc =1.5 kc =1.7 kc =2. σ 3 =3kPa σ 3 (kpa) Figur 7 Effcts of th confining strssσ 3 onδ m /,m Figur 8 Effcts of th void ratio onδ m /,m Δm/,m (%) 1 Dr=.3 Dr=.6 Dr=.728 C C=1.85* kc-1 Figur 9 Rlations btwn Δ m /,m and k c Figur1 Th rgrssion of th cofficint C Th maximum shar modulus of th Fujian sand undr th diffrnt consolidation ratio now can b obtaind by combination of Eqs.(1) and (4). Th comparison of th maximum shar modulus btwn th prsntd formula and th tst data for D r =.6 is listd in Tabl 3 and it can b sn that th rrors ar quit small.

6 Octobr 12-17, 28, Bijing, China Tabl 3 Comparison of max (MPa) btwn th prsntd formula and th tst data σ 3 kpa 1 Tst Formula Error (%) 2 Tst Formula Error (%) 3 Tst Formula Error (%) k c = Furthrmor, it can b sn from Fig.1 that th ffct of void ratios on th cofficint C is not vry notabl. To simplicity, th ffct of void ratios on th incrmnt of th maximum shar modulus can b nglctd and thn, th maximum shar modulus of th Fujian sand undr diffrnt consolidation ratios now can b xprssd by avraging th cofficints of C and B in thr dnsitis and in this cas, C=.593 and B=.458. Fig.11 shows th comparison btwn th calculatd and tstd maximum shar modulus in this cas and it can b sn from it that th powr function of k c -1 is a quit suitabl form for dscribing th variation of th maximum dynamic shar modulus du to k c >1 bcaus all of th rrors btwn th tst data and rgrssion quation is nough small in nginring sns max / k c Figur 11 Comparison btwn th prsntd formula and th tst data for Fujian sands 4.2 Rsults of th Harbin Sand Som typical rsults on th tst data and th rgrssion curvs of 1/~ undr two confining strsss for th Harbin sand with Dr=.6 ar illustratd in Fig.12. According to th abov sam procdur, th rlativ incrmnt of th maximum shar modulus for th Harbin sand of Dr=.6 du to k c >1 can b obtaind as th sam form as th Fujian sands, but th cofficints C=.735 and B=.626. Th comparison btwn th formula and th tst data for Harbin sands is illustratd in Fig.13 and it indicats that th powr function of k c -1 also is a quit suitabl form for dscribing th variation of th maximum dynamic shar modulus for th Harbin sand du to k c > σ3 =2kPa σ3 =3kPa.5.1 Figur 12 Th 1/~for th Harbin sand 4.3 Rcommndd Formula of This Papr kc=1. kc=2. kc=1. kc=2. Δm/m (%) 1 1 σ3=2kpa σ3=3kpa kc-1 Figur 13 Comparison for Harbin sands

7 Octobr 12-17, 28, Bijing, China By avraging abov xprimntal rsults of two kinds of sands, a rcommndd formula for xprssing th rlativ incrmnt of th maximum dynamic shar modulus undr th diffrnt k c is prsntd by 5 DISCUSSIONS 5.1 Comparison with th Hardin and Black s Formula (5) max.54 = ( k c 1) Bcaus th ffct of consolidation ratios on th maximum dynamic shar modulus is not vry clar, som rsarchrs and nginrs lik to us th Hardin and Black s formula to calculat th maximum dynamic shar modulus of soils undr th anisotropic consolidation conditions. Ifσ 2 =σ 3 andσ 1 = k c σ 3, according to th Hardin and Black s formula (1968), Δ m /,m can b writtn as Th comparison of Eq.(6) with Eq.(5) in this papr is illustratd in Fig.14. Δ m 2 + k c.5 = ( ) 1 (6) 3, m Δ m /,m Hardin and Black This papr k c Figur 14 Comparison of Δ m /,m ~k c btwn th Hardin and Black s and this papr s First, it should b noticd that compard with Eq.(5) in th papr, Eq.(6) shows a quit diffrnt form as shown in Fig.15. Th rlation of Δ m /,m ~k c in Eq.(6) is a narly linar incras in th maximum dynamic shar modulus in th intrval of k c =1 to 3. Whil in th papr, Δ m /,m is th function of k c -1 to th powr of B. Th cofficint B in Eq.(5) is.54, much lss than 1, which mans Δ m /,m has a mor rapid ris whn k c is nar to 1 and thn has a slowr ris with th incrasing of k c as shown in Fig.15. Scond, it also can b sn from Fig.15 that th ffct of consolidation ratios on th maximum dynamic shar modulus of sand is significant. For k c =1.5 and k c =2. in Fig.15, Δ m /,m by th formula prsntd in th papr is 45% and 66%, rspctivly, and whil Δ m /,m by Eq.(6) is only about 8% and 15%, rspctivly. Th incrasing dgr of th maximum dynamic shar modulus for k c =2 in th papr is not as small as th valu of 15% dscribd by th Hardin and Black s formula. 5.2 Comparison with th Othr Rsults H (1997) conducts th dynamic triaxial tsts for th undisturbd cohsiv soils and th disturbd sandy soils undr th dviatoric strsss. By mploying th tst data of Tabl 3 in his papr, w can attain th rlativ incrmnt of th comprssion modulus, i.. ΔE m /E,m. From his rsults, ΔE m /E,m for k c =2 is about 4%-1% highr than thos for k c =1. In trms of th formula of E=2(1+ν), Δ m /,m should b quit nar to ΔE m /E,m although th Poisson ratio ν prhaps is slightly diffrnt for th soil spcimns undr th diffrnt consolidation ratio. Thn, it can b dducd that Δ m /,m for th cass of k c =2 in his papr may b about 4%-1%. This is coincidnt in quality with th rsults in this papr. Som rsults (Pitilakis t al, 1992; Jiang, 199) rval that th maximum dynamic shar modulus by th rsonant column tsts is always blow th valus by th vlocity tsts in th fild, and in many cass, th diffrnc btwn in-situ and in th laboratory is 1%-2% (Jiang, 199). Som rsarchrs imagin that th

8 Octobr 12-17, 28, Bijing, China rason is th tim factor in th consolidation or is that th soil to b tstd in th laboratory is disturbd to a crtain xtnt. Howvr, it sms that ths xplanations ar not prfct bcaus th vidnc is not nough. From th abov rsults in th papr, th maximum dynamic modulus will ris to a grat xtnt if th actual anisotropic strsss ar considrd. Thrfor, th diffrnc in th consolidation ratio also should b on of th important factors to caus th significant dviation of th maximum dynamic shar modulus btwn th fild and laboratory tsts. If thr is an opportunity, mor dtaild comparison of rsults btwn th rcommndd formula in th papr and in-situ tsts should b conductd furthr to validat Eq.(5). 6 CONCLUSIONS By th rsonant column tsts th ffct of th consolidation ratios on th maximum dynamic shar modulus for two kinds of sands is invstigatd and th rcommndd formula for calculating th incrmnt of th maximum dynamic shar modulus for th cass of k c >1 is obtaind. Th conclusions of th papr can b summarizd as following: 1. Th ffct of th anisotropic consolidation on th maximum dynamic shar modulus is quit rmarkabl and cannot b nglctd. 2. A suitabl form to show this ffct is to us th powr function of th incrmnt of th consolidation ratio, i.. (k c -1) B. 3. Th variation of th maximum dynamic shar modulus for th cass of k c >1 can b xprssd by th rlativ incrmnt formula, i.. Δ m /,m = 1+.66(k c -1) Th formula prsntd abov mans that th maximum dynamic shar modulus shows a mor rapid ris in th intrval of k c nar to 1 and a slowr ris in th intrval of k c far away from Th incrasing dgr of th maximum shar modulus du to k c >1 is significantly largr than that dscribd by th Hardin and Black s formula,.g. th incrasing dgr for k c from 1 to 2 in th papr is about 66% whil only 15% by th Hardin and Black s formula. 6. Th consolidation ratio also should b on of th important rasons on th obvious diffrnc btwn in th fild and laboratory dtrmination of th maximum dynamic shar modulus of th soils. REFERENCES Amini F, Tawfig KS and Aggour MS (1988), Cohsionlss soil bhavior undr random xcitation conditions, Journal of otchnical Enginring, ASCE, 114(.8): Chn CL, Hu ZQ, Xi DY and Fng ZY (22), Rsarch on dynamic charactristic of sand in foundation of Xiabandi Dam, Proc. 6 th Chins National Confrnc on Soil Dynamics, Nanjing, China, (in Chins) Hardin BO and Black WL (1968), Vibration modulus of normally consolidatd clay, Journal of th Soil Mchanics and Foundations Division, ASCE, 94( 2): Hardin BO and Black WL (1969), Vibration modulus of normally consolidatd clay (closur), Journal of th Soil Mchanics and Foundations Division, ASCE, 95(6): H CR (1997), Dynamic triaxial tst on dynamic modulus and damping ratio, Chins Journal otchnical Enginring, 19(2): (in Chins) Jiang ST and Wang XX (199), Dynamic modulus and damping ratio of undisturbd soils in Zhngzhou ara, Proc. 3 th Chins National Confrnc on Soil Dynamics, Shanghai, China, (in Chins) Pitilakis KD, Anastassiadis A and Raptakis D (1992), Fild and laboratory dtrmination of dynamic proprtis of natural soil dposits, Proc. 1 th World Confrnc on Earthquak Enginring, Balkma, Rottrdam, Sd HB, Wong RT, Idriss IM and Tokimatsu K (1986), Modulus and damping factors for dynamic analysis of cohsionlss soils, Journal of th Soil Mchanics and Foundations Division, ASCE, 112(11): Yu PJ, Liang YX and Qin WQ (1988), Dynamic modulus and damping ratio of disturbd sands, Chins Journal otchnical Enginring, 1(4): (in Chins) Zn K and Higuchi Y (1984), Prdiction of vibratory shar modulus and damping ratio for cohsiv soils, Proc. 8 th World Confrnc on Earthquak Enginring, Vol. 3, San Francisco, 23-3.

15. Stress-Strain behavior of soils

15. Stress-Strain behavior of soils 15. Strss-Strain bhavior of soils Sand bhavior Usually shard undr draind conditions (rlativly high prmability mans xcss por prssurs ar not gnratd). Paramtrs govrning sand bhaviour is: Rlativ dnsity Effctiv