A Model for Nonlinear Total Stress Analysis With Consistent Stiffness and Damping Variation

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1 A Model fo Nonline Totl Stess Anlysis With Consistent Stiffness nd Dmping Vition M.J. Aefi, M. Cubinovski & B.A. Bdley Univesity of Cntebuy, New Zelnd SUMMARY: The ppe discusses modelling of cyclic stess-stin behviou of soil, in pticul simple model tht cn poduce desied stiffness nd hysteetic dmping fo given stin level s obseved in lbotoy testing is fomulted. The unloding-eloding eltionship is developed fo totl stess seismic site esponse nlysis with ppopite dmping t lge stin. The constitutive model employs hypebolic eqution s the bckbone cuve, nd uses modifiction of the extended Msing unloding-eloding eltionship leding to coect mesued modulus eduction nd dmping cuves simultneously. A qusi-sttic cyclic loding of incesing mplitude is used to demonstte the model s pefomnce nd its cpbility to llow impoved modelling of the mgnitude of enegy dissiption bsed on n expeimentl pogm on ntive sndy soils fom Chistchuch, New Zelnd. Keywods: Msing ule, Hypebolic eqution, Mteil dmping 1. INTRODUCTION The dynmic esponse of soil deposits beneth site, commonly efeed to s site esponse, hs significnt influence on the gound motion hzd of engineeed stuctues. The popeties tht typiclly need to be detemined in ode to chcteize pticul soil site include she modulus, G, nd mteil dmping tio, h, mongst othes. She modulus epesents the she stiffness of the soil nd cn essentilly be consideed s the slope of the she stess - she stin eltionship nd is denoted s tngent she modulus, G t. It cn lso be ppoximted s degee of inclintion of loop in the cse of dynmic lodings s illustted in Figue 1 nd in this cse, it is known s secnt she modulus, G s. Dmping tio, h, is mesue of the popotion of dissipted enegy to the mximum etined enegy duing single cycle of she defomtion o simply mesue of bedth of the loop (Fig. 1). Figue 1. Secnt she modulus, G s, nd mteil dmping tio, h, duing cyclic loding

2 The eltionship between secnt she modulus, G s, nd she stin mplitude is commonly chcteised by she modulus eduction cuves (e.g. Fig. 2). Also, the nonlineity in the stess-stin eltionship, which leds to enegy dissiption pe cycle, esults in the mteil dmping tio, h incesing with she stin (Figue 2b). Nomlized She Modulus, G/G mx () g She Stin, % She Stin, % Figue 2. () Nomlized modulus eduction cuve nd, nonline mteil dmping tio cuve Mthemticl models, which e cpble of pedicting soil esponse in futue possible ethqukes, e equied in ode to theoeticlly undestnd locl site effects. Depending to the desied level of ccucy nd simplicity, thee genel bod clsses of soil models hve been poposed, nmely equivlent line models, cyclic totl stess nonline models (Hdin nd Dnevich, 1972, Rmbeg nd Osgood, 1943), nd dvnced constitutive models which incopotes poe pessue genetion (Moz, 1967, Momen nd Ghboussi, 1982, Dflis, 1986, Kbilmny nd Ishih, 199, Gutieez et l., 1993, Cubinovski nd Ishih, 1998). Equivlent line nlysis is the simplest nd most widely employed scheme but hs sevel impotnt limittions. Whees, dvnced constitutive models cn epesent mny detils of dynmic soil behviou, but numeous pmetes which must be detemined though lbotoy nd field tests limit its use fo mny common pcticl poblems (Kme, 1996). Cyclic totl stess nonline models cn ppoximtely simulte the ctul stess-stin pth duing cyclic loding nd theefoe epesent the she stength of the soil fo engineeing puposes. These models genelly hve bckbone cuve nd set of unloding-eloding ules which cn epesent the totl stess behviou of the soil (e.g. Kme, 1996). Genelly, the shpe of the bckbone cuve is detemined by the mximum she modulus, G mx, she stength, τ mx, nd sevel cuve-fitting pmetes. Dendeli (21) nd Phillips nd Hshsh (29) poposed modified hypebolic eqution s bckbone cuve bsed on n elie wok by Hdin nd Dnevich (1972): Dmping, h (%) F bb Gmx = τ = (1) α Gmx 1+ β τ mx whee τ is she stess; is she stin, α nd β e dimensionless fctos. In the oiginl fom poposed by Hdin nd Dnevich (1972) α = β = 1. In this model, the efeence stin is defined s: τ mx = (2) Gmx Using equtions (1) nd (2) the nomlized modulus eduction cuve cn be evluted s:

3 G G mx 1 = 1+ β α (3) A pcticl poblem with the definition of efeence stin,, is tht the she stength is often not vilble. Theefoe, pseudo-efeence stin is poposed to be used fo low to modete stin levels (Stewt et l., 28). The pseudo-efeence stin is defined fom lbotoy modulus eduction cuve s the she stin t which G/G mx =.5 (Figue 2.). This definition is esulted fom hypebolic fits of modulus eduction cuve ccoding to eqution (3). The dvntge of using the pseudo-efeence stin is tht in the bsence of mteilspecific tests, empiicl eltionships exist to pedict it fom othe stte pmetes (Dendeli, 21). α nd β e fitting pmetes genelly tken s 1 nd 1, espectively. If α vlue is dopted s gete thn one, the she stess eches mximum vlue t = (Fig. 1). Bckbone Unloding Reloding (4) (g,t ) < 1 =1 > 1 (1) t, (kp) t, (kp) (2) () (3) g m Figue 3. () Effect of α on hypebolic cuve, hypebolic bckbone cuve nd Msing unloding-eloding bnches Msing ules o modified Msing ules e often used in conjunction with the bckbone cuve to descibe the unloding-eloding behviou of soil. In the modified Msing ule, if stess evesl occus t point defined by (, τ ), the stess-stin cuve is identicl to the shpe of bckbone cuve but enlged by fcto of n = 2 (Fig. 3b), which is given by: τ τ n = F bb n (4) If the unloding o eloding cuve exceeds the mximum pst stin nd intesects the bckbone cuve, it follows the bckbone cuve until the next evesl point. If n unloding o eloding cuve cosses n unloding o eloding cuve fom the pevious cycle, the pth follows tht of the pevious cycle. By pplying Msing ule to hypebolic model, Ishih (1996) showed tht the dmping tio t ech stin level cn be obtined s: ln = + π π h (5)

4 It cn be seen tht dmping tio conveges to 2/π =.637 when she stin mplitude becomes infinitely lge. The mgnitude of dmping pedicted by Msing ule is not suppoted by expeimentl test esults obseved in the couse of this study nd vlues epoted by othes (Hdin nd Dnevich, 1972, Ishih, 1996, Dendeli, 21, Stewt et l., 28). Figue 4. illusttes tht using Msing citei the e of the hysteesis loop is gete thn tht mesued by expeimentl test dt esulting in oveestimtion of dmping tio especilly fo highe she stin level (Fig. 4b). The oveestimtion of hysteetic dmping induced by employing Msing citei cn unconsevtively led to undeestimtion of some of the seismic esponse pmetes especilly in the highe fequency nge (Stewt et l., 28, Silv et l., 2). A solution of the foementioned dmping poblem with Msing citei hs been poposed by Phillips nd Hshsh (29). Bsed on n elie wok by Dendeli (21), Phillips nd Hshsh (29) developed eduction fcto to modify the unloding-eloding equtions fom those of the Msing citeion. Eqution 6 pesents the functionl fom fo the dmping eduction fcto poposed by Phillips nd Hshsh (29): 4 Expeiment Msing ule 3 Chistchuch Snd t, (kp) Dmping, h(%) 2 1 () Figue 4. Compison of expeimentl nd Msing-bsed clculted hysteesis loop (), dmping tio cuve p3 G m F( m) = p1 p2 1 G (6) mx whee G m is secnt modulus coesponding to the mximum she stin level, m, nd p 1, p 2 nd p 3 e nondimensionl coefficients selected to obtin the best possible fit with the tget dmping cuve. This dmping eduction fcto given by eqution (6) is used in the unloding-eloding eltionship given by eqution (7) (Hshsh et l., 21): G mx G ( ) ( ) G ( ) τ = F 2 mx mx m. 2 + τ + m m 1+ β + β + β (7) While functionl fom in equtions (6) nd (7) povide n impoved mtch between the expeimentl nd mthemticl dmping cuves, the tngent modulus t the point of evesl educes with the eduction fcto nd theefoe is not equl to G mx. This is inconsistent with expeimentl evidence fo snd behviou unde cyclic loding (Hdin nd Dnevich, 1972, Hdin, 1978). Given soil model fo symmeticl lodings, Pyke (1979) poposed n ltentive unloding-eloding ule in

5 which the Msing coefficient n cn devite fom two, in ode to extend the Msing model fo use with iegul lodings. A fcto n gete thn two llows simultion of cyclic hdening, while cyclic softening cn be modelled by ssuming vlue of n less thn two (Lo Pesti et l., 26). The objective of this ppe is to illustte tht the sme ide cn be employed similly, to simulte ny tget dmping tio cuve by modifying the Msing citeion. 2. HYSTERETIC DAMPING FORMULATION The bckbone descibing the monotonic stess-stin cuve is modelled using the modified hypebolic eltionship stted in eqution (1) in which β is ssumed to be equl to 1. The cyclic behviou, o unlodingeloding bnches, hs been modelled using modified vesion of Msing citeion. A pmete, φ, is intoduced fo the unlod-elod cuves. Moeove, the pmete, n, is llowed to vy depending on the desied level of hysteetic dmping. To peseve the simplicity of the solution poposed by Msing (1926), s well s chieving bette geement between the expeimentl nd modelled hysteetic dmping, two conditions need to be stisfied. Fist, the unloding-eloding cuves fo symmeticl peiodic nd cyclic lodings should fom closed loop fo ny level of she stin nd moeove, be simil in shpe to tht of the initil loding cuve. Second, the tngent she modulus on ech evesl point should ssume vlue equl to the initil tngent modulus fo the initil loding cuve, G mx. To meet the fist condition, unlod-elod equtions e equied in which the she stess t evesl points be equl but with opposite signs; in othe wods, two points A(, τ ) nd B(-, -τ ) should fll on the unloding nd eloding bnches. This cn be confimed by expnding eqution (1.4) nd using eqution (1) to obtin: τ τ n Gmx = n 1+ n φ (8) It is obvious tht points A nd B both fll on the bove cuve consideing Msing coefficient n, to be equl to two. Howeve, this my not be tue fo n bity vlue of n. Solving eqution (8b) fo φ, nd fo genel n- vlue, nd enteing point B in the eqution yields: φ + ln ln 1 n = (9) Theefoe, ny dopted combintion of φ nd n which stisfies eqution (9) will esult to closed loop hysteesis (Fig. 5). The next step would involve the e of the loop to be in geement with mteil dmping cuve, the fome epesenting mesue of the hysteetic dmping. A best n vlue cn edily be obtined by itetion, mtching the dmping tio fom expeimentl test esults nd the one clculted by unloding-eloding ule. Once n is dopted, cuvtue vible φ, cn be obtined using eqution (9). It cn be shown tht the deivtive of the unlod-elod eqution t the evesl points is equl to the initil tngent modulus nd hence the second condition emins vlid. Fo she stin levels lge thn efeence she stin, coefficient α in the bckbone cuve cn be shown s n ltentive pmete to be consideed long with Msing coefficient n in ode to mtch the expeimentl dmping cuve with the mthemticl model.

6 8 n =.6, f =.66 n = 1.2, f =.83 n = 2.4, f = t, (kp) Figue 5. Vying Msing coefficient, n, nd cuvtue pmete, φ Hence to summise, the poposed constitutive model utilizes eqution (1.1) fo the monotonic behviou nd equtions (1.8) nd (1.9) fo the unloding-eloding equtions in ode to impove the mtch between the expeimentl nd modelled hysteetic dmping. 3. COMPARISON OF THE PROPOSED RELATIONSHIP WITH EXPERIMENT A seies of cyclic dined tixil tests hve been conducted on snd obtined fom Fitzgeld Avenue Site in Chistchuch, New Zelnd. The confining pessue ws kept constnt t vlue of 1 kp fo ll tests exmined below. Compession-extension loding cycles wee imposed by Sevo contolle t constnt fequency unde dined conditions. The foce pplied to the specimen nd the displcement t the top of the specimen ws ecoded in ddition to cell pessue nd volume chnge. The moist tmping method ws used fo specimen peption. A totl of 1 lyes of pedetemined quntities of moist soil wee woked into pescibed thickness. CO 2 ws pecolted up though the specimen using pessue to pomote full specimen stution. De-eted wte ws flushed though the smple nd the smple ws then stuted using 1 kp bck pessue nd left ovenight. The soil smple ws consideed stuted when the Skempton B-vlue ws equl to o moe thn.96. Ech specimen ws consolidted isotopiclly unde confining pessue σ c = 1 kp. The specimens wee then loded with.1 Hz sinusoidl cyclic loding. Stndds of Jpnese geotechnicl society (2) fo lbotoy she tests e employed s guideline to mesue the dynmic popeties of tested mteil. Figue 6 illusttes the nomlized modulus eduction nd mteil dmping cuves e plotted only fo clen snds with vying eltive density. To obtin the G/G mx esults fom expeimentl esults the secnt Young s moduli obtined fom cyclic tixil tests wee conveted to secnt she moduli ssuming Poisson s tio, ν =.1. In ode to simulte the expeimentl modulus eduction cuves shown in Figue (6) with the hypebolic eqution given by eqution (3), lest sque eo method is employed to estimte the equied pmetes in the hypebolic model. Ech cuve pesented in Figue (6) is consideed septely in this cse; theefoe unique set of pmetes (α, ) fo hypebolic eqution is defined fo ech test. Hdin nd Dnevich (1972) illustted tht consideing some ssumptions, eltionship between she stin nd dmping cn be deived. Hence simil hypebolic eqution cn be estblished fo mteil dmping cuve:

7 h h mx = 1+ (1) whee h mx, is the mximum vlue of the dmping tio which is suggested s between 25% nd 33% fo clen snds. A good fit between the eqution (1) nd mteil dmping cuve equies evluting diffeent set of cuvtue coefficient, α nd efeence stin. Theefoe, simil ppoch to the modulus eduction cuve is tken to fit eqution (1) to expeimentl dmping esults. It is to be noted tht eqution (1) implies tht t vey low stin nge the dmping tio is close to zeo, hence smll stin dmping must be septely ccounted fo in the viscous dmping mtix fo time domin solutions; this is shown with the solid line in Fig. 7b. This is in conty to lbotoy findings s illustted by dotted points in Fig. 7b nd lso shown by Stokoe et l. (1999); but since the ovedmping due to employing Msing s ules occus t highe stin nges, it is sufficient fo the pupose of this wok to use the bove eqution. 1 2 () Chistchuch snd, s' = 1 kp G/G mx D~44.2% D~54.4% D~66.1% D~73.3% D~85.2% Dmping tio, h(%) Figue 6. () Nomlized modulus eduction, nd mteil dmping cuves fo Chistchuch snd Figue 7 illusttes compison between the stess-stin nd dmping tio-stin obtined using the foementioned eltionships in compison to expeimentl esults fo pticul specimen, FB () 2 Model Expeiment She stess, t(kp) Test No.8 D ~ 44.2% = 1 g =.1% Dmping tio, h(%) 15 1 Test No.8 D ~ 44.2% =.93 g =.1% Figue 7. Compison of hypebolic stess-stin bck-bone cuve () nd mteil dmping cuve with expeimentl test esults fo test FB-8

8 4. RESULTS A symmeticl cyclic she stin time seies ws employed to evlute the pefomnce of the poposed model. Subsequent stin cycles e obtined s twice the mplitude of the pevious cycle (Fig. 8). Bsed on the imposed she stin time seies in Fig 8, the she stess esponse in Fig 8.b ws computed using the pocedue explined in section Symmeticl cyclic she stin time-histoy 4 Computed she stess She stin, g(%) She stess (kp) () Time (s) Time (s) Figue 8. () Input she-stin time histoy, computed she stess time histoy fo test FB-5 Fig. 9 pesents stess-stin hysteesis loops geneted by employing eithe Msing citei o modified Msing citei explined bove. The solid line epesents the modified Msing behviou developed hee nd the dshed line of the conventionl Msing ule. It is illustted tht the bedth of the loops fo highe stin levels e smlle fo modified eltionships implying lowe hysteetic dmping. This diffeence between the two methods in dmping is explicitly illustted in Fig. 9b, which lso illusttes the expeimentl test esults. Tixil esults fo pticul test (FB-5) e epesented by open cicles. It cn be seen tht modified Msing ule is cpble of cptuing expeimentl dmping tios quite well. It is to be noted tht the tngent she modulus t the evesl points e equl to initil she modulus, G mx. This is not ffected by incese of cyclic she stin mplitude. 6 4 Hysteesis loop fo test FB Modified Msing Msing Expeiment She stess g(%) 2-2 Dmping tio, h(%) () Modified Msing Msing She stin, g(%) She stin, g(%) Figue 9. () Compison of stess-stin hysteesis loops, nd dmping tio cuves using Msing nd modified Msing citeion

9 The Msing coefficient, n, is not constnt nd theefoe vies with incese of she stin level. In Figue 1, n- vlues e plotted ginst she stin fo snds of vible eltive densities. A smoothe plot cn be obtined ssuming smlle she stin incements in the input time histoy intoduced in Figue 8. An n-vlue smlle thn two is equied to cicumvent the ovedmping issue intoduced by employing Msing citeion. It is envisged tht simil pocedue cn be cied out fo diffeent types of soils hving vying conditions in ode to develop eltionship between n-vlue nd she stin mplitude. The computed n-vlues cn then be employed diectly in 1-D site esponse nlyses to bette simulte the expeimentl modulus nd dmping cuves fo wide stin nge. A functionl fom cn be obtined fo n-vlue in tems of she stin, eltive density, fines content etc. cying out simil pocedues. 2 D ~ 54.4% 1.5 D ~ 66.1% n-vlue 1 D ~ 73.3% D ~ 85.2% Figue 1. Msing fcto vesus she stin level fo diffeent clen snd with vible eltive density 5. CONCLUSIONS A new simple eqution ws poposed fo modelling of unloding-eloding bnches of cyclic stess-stin hysteesis loops fo sndy soils. The poposed model uses the hypebolic model s the bckbone to epesent the modulus eduction cuve. A simple cyclic she stin time seies ws employed to compute the she stesses nd evlute the pefomnce of the model. It ws shown tht the eqution is cpble of cptuing ny desied level of enegy dissiption s function of she stin in contst to conventionl models which tend to oveestimte dmping. A pmete φ, cn be obtined fo ny given she stin mplitude in ode to mtch the dmping poduced by the numeicl model with the obseved behvio in the lbotoy. Theefoe, both the modulus eduction nd dmping cuves cn be simulted simultneously. It ws illustted tht Msing coefficient n, should be smlle thn two, in ode to mtch the hysteetic dmping with the mesued stin dependent dmping. AKCNOWLEDGEMENT The fist utho ws suppoted by the Univesity of Cntebuy Doctol Scholship nd the New Zelnd Geotechnicl Society Scholship duing the study t the Univesity of Cntebuy, New Zelnd. Also the suppot by EQC nd ECAN is gtefully cknowledged.

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