Study on one-dimensional consolidation of soil under cyclic loading and with varied compressibility *

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1 Zhuag e al. / J Zhejag v SCI 5 6A(: Joural of Zhejag versy SCIENCE ISSN hp:// E-mal: jus@ju.edu.c Sudy o oe-dmesoal cosoldao of sol uder cyclc loadg ad wh vared compressbly * ZANG Yg-chu ( 庄迎春, XIE Kag-he ( 谢康和 (Deparme of Cvl Egeerg, Zhejag versy, aghou 37, Cha E-mal: dkhxe@ju.edu.c Receved Sep. 3, 3; revso acceped Oc., 3 Absrac: Ths paper preses a sem-aalycal mehod o solve oe dmesoal cosoldao problem by akg cosderao of vared compressbly of sol uder cyclc loadg. I he mehod, sol sraum s dvded eually o layers whle load ad cosoldao me are also dvded o small pars ad me ervals accordgly. The problem of oe-dmesoal cosoldao of sol sraum uder cyclc loadg ca he be deal wh a each me erval as oe-dmesoal lear cosoldao of mul-layered sols uder cosa loadg. The compresso or reboudg of each sol layer ca be judged by he effecve sress of he layer. Whe he effecve sress s larger ha ha he las me erval, he sol layer s compressed, ad whe s smaller, he sol layer rebouds. Thus, approprae compressbly ca be chose ad he cosoldao of he layered sysem ca be aalyed by he avalable aalycal lear cosoldao heory. Based o he sem-aalycal mehod, a compuer program was developed ad he behavor of oe-dmesoal cosoldao of sol wh vared compressbly uder cyclc loadg was vesgaed, ad compared wh he avalable cosoldao heory whch akes o cosderao of vared compressbly of sol uder cyclc loadg. The resuls showed ha by akg he varable compressbly o accou, he rae of cosoldao of sol was greaer ha he oe predced by coveoal cosoldao heory. Key words: Cyclc loadg, Oe-dmesoal cosoldao, Sem-aalycal soluo, Vared compressbly of sol do:.63/jus.5.a4 Docume code: A CLC umber: T47 INTRODCTION * Projec suppored by he Naoal Naural Scece Foudao of Cha (No ad Docoral Subjec Research Fud of Msry of Educao (No , Cha I praccal geoechcal egeerg, sols beeah may srucures, such as ol ad waer aks, hghway embakme, ocea bak, ec., udergo cyclc loadg. Schffma (958 frs obaed a geeral soluo of sol cosoldao cosderg loadgs crease learly wh me. Wlso ad Elgohary (974 preseed a aalycal soluo of oe-dmesoal cosoldao of sauraed sol subjeced o cyclc loadg based o Teragh s lear cosoldao heory. Aloso ad Krek (974 cosdered he seleme of elasc sof sol uder sochasc loadg. Olso (977 aalyed he cosoldao uder me depede loadg. Balgh ad Levadoux (978 developed a smple predco mehod for oe-dmesoal cosoldao of clay layer subjeced o cyclc recagular loadg wh he superposo prcple. More recely, Favare ad Sorao (995, Che e al.(996, ad Gua e al.(3 derved some soluos for dffere ypes of cyclc loadg. I s obvous ha sol sraum wll be compressed ad rebouds whe s subjeced o cyclc loadg. Whe sol sraum s loaded, he effecve sress a ay po wll be creased ad he sol hus compressed, bu s o jus he reverse whe sol sraum s uloaded. I he perod of uloadg, par of he sol he upper layer ear pervous boudary rebouds bu he lower par of he sol ear mpervous boudary sll compressed. Because sol has dffere compressbly whe s compressed or rebouds, s

2 4 Zhuag e al. / J Zhejag v SCI 5 6A(:4-47 ecessary for aalyss o dsgush reasoably whch par of he sol sraum s compressed ad whch rebouds durg cosoldao so ha suable compressbly of sol ca be used. owever, because of he complexy volved, here s so far o cosoldao heory avalable ha akes o cosderao he varaos of compressbly of sol uder cyclc loadg. I hs paper, a sem-aalycal mehod s proposed o solve he problem of oe-dmesoal cosoldao uder cyclc loadg whle akg cosderao of vared compressbly of sol. A releva compuer program s he developed ad used o vesgae he cosoldao behavor of sol wh vared compressbly uder cyclc loadg. The resul of hs mehod s subseuely compared wh he avalable aalycal soluo whou cosderg vared compressbly. MATEMATICAL MODELING The cosoldao problem suded here s show Fg.. A sol sraum wh hckess, vercal permeably coeffce k v, volume compressbly coeffce m v, cosoldao coeffce C v, ad wh a pervous upper boudary ad a mpervous lower boudary, s subjeced o a vercal uform rapeodal cyclc loadg show Fg.a, whch s maxmum load, s cyclc me, α ad β are he coeffces of proporo. Whe α.5, he load becomes ragular oe as show Fg.b; ad whe Pervous Impervous Fg. Model of clay layer o α, chages o suare cyclc loadg, Fg.c. Based o he basc assumpos made Teragh s -D lear cosoldao heory excep for loadg codo ad cosa compressbly (.e he value of m v or C v s cosa durg cosoldao, he dffereal euao goverg he cosoldao process of he clay layer ca be expressed as follows: u C u α β (Nβ.5 β v f( ( whch u s excess porewaer pressure; f(d/d s he rae of loadg. The codos for he soluo of E.( are: ( u u ( > ; ( u ; (3 (mpervous; u (pervous. Accordg o Teragh s effecve sress prcple, he effecve sress σ' ca be relaed o excess porewaer pressure u ad load by σ'+u ( For rapeodal cyclc loadg show Fg.a, he expresso of ca be gve by (a (b (Nβ β (Nβ (c Fg. Loadg ypes (a rapeodal cyclc loadg; (b ragular cyclc loadg; (c recagular cyclc loadg [ ( N β] ( N β ( N β + α α ( N β + α < ( N β + ( α ( [ ( N β ( α ] ( N β + ( α < ( N β + α ( N β + < Nβ (3

3 Zhuag e al. / J Zhejag v SCI 5 6A(: where N,,3,, he umber of cycles. The expressos of oher loadg ypes show Fg.b ad Fg.c ca be obaed from E.(3 by seg he value of α a.5 ad respecvely. SEMI-ANALYTICAL METOD A aalycal soluo o he cosoldao uder cyclc loadg ca be obaed by Laplace rasformao whe he compressbly coeffce s cosa durg he loadg. owever, because sol sraum wll be compressed ad rebouds whe s subjeced o cyclc loadg, ad because sol has dffere compressbly whe s compressed or rebouds, s ecessary for a reasoable aalyss o ake o cosderao he vared compressbly of sol, ha s o dsgush whch par of he sol sraum s compressed ad whch par rebouds durg cosoldao so ha suable compressbly of sol ca be used. Because of he complexy volved, a sem-aalycal mehod smlar o he oe developed by Xe ad Leo (999 ad L e al.(999 for olear cosoldao aalyss s proposed. Frs, sol sraum of hckess s dvded eually o layers of eual hckess / (Fg.3. Le he org of he coordae be a he op of he sol sraum, ad deoe he dsace bewee he org ad he boom of each layer as. Thus,, /,,, 3,,, ad. / / / Pervous Impervous Fg.3 The dvded sol layers sem-aalycal mehod Because he hckess of each layer ca be very small, a ay me, he mea value of each sol parameer of each layer ca sad for he value of he sol parameer of hs layer, ad he error caused by hs approxmao ca be gored. Tha s, he sol parameers of layer ca be respecvely deoed as permeably coeffce k v, volume compressbly coeffce m v, cosoldao coeffce C v (,, 3,,. The akg o cosderao he vared compressbly of sol durg he process of cosoldao, cosoldao me s also dvded o m small ervals ad loadg dvded o m small loads accordgly. Whe he me erval s small eough, boh he coeffce of cosoldao a each small layer ad he load each small me erval ca be ake as cosas. The load k correspodg o me erval k ( k k k, k,,, m, ca be gve by k ( k ( k ( k k ; k,,3,,m (4 For rapeodal cyclc loadg, he expresso of s kow, so k ca he be obaed by subsug E.(3 o E.(4. The dffereal euao goverg he cosoldao of each small layer ca ow be wre as: C u v u (,,,3,, (5 where, u u (, ad C v k v /(m v /γ w, ad γ w s he u wegh of waer, he excess pore waer pressure ad cosoldao coeffce of small layer respecvely ( k k ; k,, 3,, m. C v chages wh me ad deph, ad ca be deduced from mea excess porewaer pressure a me k. The boudary codos for E.(5 are: k : u u : (mpervous; u (pervous (6a : u u + ; u u+ k +,,, 3,, (6b v v( ad he al codo for E.(5 s gve by u u '+ k k (7

4 44 Zhuag e al. / J Zhejag v SCI 5 6A(:4-47 where, u ' s he excess porewaer pressure a me k. Thus, he problem of oe-dmesoal cosoldao of sol wh vared compressbly uder cyclc loadg has bee reduced o he problem of oe-dmesoal lear cosoldao of layered sols. Accordg o Lee e al.(99 ad Xe ad Pa (995, he correspodg soluo of E.(5 ca be gve as βm u C g ( e,,, 3, (8 m m m where, β λ C m m v ; gm ( Am s( µλ m + Bm cos( µλ m b [ u + ( k k ] m( d g Cm b g ( d m [ u + ( k k ] ab Am( C D+ µλ m ab ( Am + Bm + ( Bm Am Bm ( B + A ] (9 ( D+ B+ CA + AmBm( C D+ he defos of parameers λ m, µ, a, b, A m, B m, C, D ca be foud he paper by Xe ad Pa (995. From he above soluo of excess porewaer pressure, he average degree of cosoldao of layer defed erms of seleme ca be gve by: [ ( u]d h Am ( C D + + Bm ( B + A ( m µλ m β d m βm Cme e d d ( The oal average degree of cosoldao of he whole clay sraum defed erms of seleme, s, ca be gve by: s S S m h v [ mv ( ( ud bρ bρ ( Meawhle, he oal average degree of cosoldao defed erms of effecve sress, p, ca be derved as follows: p ( u [ ( ud ] ρ where, b m v ; mv h ρ. ( As ca be see from E.( ad E.(, whch h, ulke ha dcaed by Teragh s -D cosoldao heory, he oal average degree of cosoldao defed erms of seleme ad ha defed erms of effecve sress s o he same,.e., s p, ad wll be he same oly f he value of b,.e., sol compressbly a each layer s eual. owever, as meoed above, sol compressbly of a layer s always chagg uder cyclc loadg. I s herefore ecessary o ake o cosderao he vared compressbly of sol. The chages of compressbly of sol uder cyclc loadg ca be llusraed by Fg.4, where e s he vod rao ad p s he effecve sress. I s obvous ha sol wll repeaedly udergo loadg, uloadg ad reloadg whe s subjeced o cyclc loadg. Accordgly, sol compresses, rebouds ad recompresses aga, ad he compressbly wll be chaged e C c C e A D σ k B C E F log p Fg.4 Compresso curve of sol (e-log p

5 Zhuag e al. / J Zhejag v SCI 5 6A(: freuely. The codo of sol a some elevao ca be judged by comparso of he effecve sress σ k a me k wh σ k+ a me k+. For example, whe σ k+ σ k, meas he sol s loaded (alog sragh le AB, see Fg.4, ad whe rebouds (alog CD, Go o he ex me Yes Keep he parameers varable, C c No Sar Ipu he basc parameers of sraum, loadg ad e-logp Save he resul Dvde sraum ad me Calculae he k Cofrm he boudary codos Cofrm he al codos Calculae he value of C v (,, 3,, Calculae excess pore pressure u (, Effecve sress of k+ > effecve sress of k Calculae degree of cosoldao ( s u (,/ < 8 Ed Yes No Keep he parameers varable, C e Pr he resul Fg.5 The compuao procedure σ k+ mus be smaller ha σ k (.e. σ k+ σ k. The acual compressbly of sol ca hus be chose by hs judgme. sg he soluo show above, a compuer program was developed for hs sem-aalycal mehod wh FORTRAN laguage. Fg.5 s he flow char of hs program. I he compuaos wh he program, sol sraum ca be dvded o 4 small layers, ad he me erval s corolled by o-dmesoal me facor TC v k /, whch ca be.. Whe he sol cosoldao has occurred for a gve me, ad he rao of he sol parameers has become smaller, he me sep ca be made larger o speed up he program calculao. NMERICAL ANALYSIS AND DISCSSIONS Cosoldao behavor There are hree dmesoless varables,.e. C v /, α ad β, ha gover he cosoldao process oe cycle. reflecs he fluece of cosruco me ; α ad β reflec he properes of loadg. Therefore, he oe dmesoal cosoldao behavor of he sol ca be vesgaed by gvg oe of he varables dffere values whle fxg he values of he oher varables, as show by he followg dscussos. Fg.6 shows he fluece of cosruco me o cosoldao, whch, α.3, β.5, ad s varyg. I ca be see ha he loger he cosruco me (.e. he greaer, he slower he rae of cosoldao. I Fg.7,. ad β.5, α s varyg. So he fgure shows he fluece of α,.e. he rae of loadg, o he cosoldao of sol, from whch ca be see ha he fluece s more ad more sgfca as he value of α creases. Fg.8 was ploed o vesgae he fluece of β, whch α.3, ad.. As ca be see from he fgure, wh he decrease of β (.e. he me of oe cycle he cosoldao degree s creased. Addoally, a seady-sae codo s reached afer a suffcely large umber of cycles (e.g. N4 ad N. I ca also be see from hese fgures ha, a gve cycle, he cosoldao degree of each cycle

6 46 Zhuag e al. / J Zhejag v SCI 5 6A(: Fg.6 The fluece of o N7.. Normaled deph (/..4.6 N Excess pore pressure (u/ -.3 ( Fg.7 Ifluece of α o N7 α.4 α.5 α. Fg.9 Excess pore pressure sochroe Comparso wh coveoal heory For he problem of oe-dmesoal cosoldao uder rapeodal cyclc loadg whou cosderg vared compressbly of sol, he aalycal soluos ca be obaed by he mehod of Gua e al.(3 wh Laplace rasformao, Laplace verse rasformao ad he me-delay characerscs. The soluo whe (Nβ (Nβ +α (loadg phase s -.4 N M M T u s ( e 3 BN m αmt (3. whe (Nβ +α (Nβ +(α (cosa loadg phase s β.5.3. M α s M T M T u (e B e N 3 m αmt (4 Fg.8 Ifluece of β o whe (Nβ +(α (Nβ + (uloadg phase s reaches he maxmum value a he ed of he cosa loadg phase ad he mmum value a he begg of he ex cycle (.e. a he ed of he uloadg. Fg.9 preses he excess pore pressure dsrbuo wh deph a he ed of cosa loadg per cycle, whch., α.3, β.5. I shows ha he excess pore pressure decreases wh creasg value of N. M u s 3 m αmt (e + e e α M T ( α M T M T B N whe (Nβ + Nβ (o-loadg phase s M u s 3 m αmt (5

7 Zhuag e al. / J Zhejag v SCI 5 6A(: where, αm T M T ( α M T M T (e e + e B N e (6 αm T αm T M T ( N βm T B N + αm T βm T (e (e e ( e e ( e C v C v m T, T, M π, m,, 3,. The dfferece bewee he coveoal aalycal soluo ad he sem-aalycal soluo developed here ca be see from Fg., where α.3, β.5,.. As dcaed by Fg., he cosoldao behavors he frs cycle of hese wo soluos are almos same, whch ca prove he correcess of he sem-aalycal soluo, ad wh he creasg umber of cycles, he dfferece bewee hese wo soluos becomes more ad more sgfca alhough he rae of cosoldao creases slowly wh he umber of cycles CONCLSION Aalycal soluo Sem-aalycal soluo N5, 6, 7 N 3 4 Fg. Comparso of coveoal aalycal soluo ad sem-aalycal soluo The followg coclusos may be draw from hs sudy:. The soluo of he sem-aalycal soluo developed here for oe-dmesoal cosoldao of sol uder cyclc loadg ad wh vared compressbly shows ha he cosoldao behavor s maly govered by he hree parameers α, β,.. For cyclc rapeodal loadg, he cosoldao rae each cycle maxmes a he ed of he cosa loadg phase ad mmes a he begg of he ex cycle. 3. Vared compressbly of sol ca affec he cosoldao behavor of sol uder cyclc loadg. The rae of sol cosoldao wll be faser ha he coveoal soluo, whe vared compressbly s cosdered. Refereces Aloso, E.E., Krek, R.J., 974. Radomess of seleme rae uder sochasc load. Joural of Geoechcal Egeerg Dvso, ASCE, (EM6:-6. Balgh, M.M., Levadoux, J.N., 978. Cosoldao heory of cyclc loadg. Joural of Geoechcal Egeerg Dvso, ASCE, 4(GT4: Che, J.Z., Zhu, X.R., Xe, K.., Zeg, G.X., 996. Oe Dmesoal Cosoldao of Sof Clay uder Trapeodal Cyclc Loadg. Proc. of he secod Ieraoal Coferece o Sof Sol Egeerg, Najg, p.-6. Favare, M., Sorao, M., 995. A Smplfed Cosoldao Theory Cyclc Loadg Codos. Proc. of I. Symposum o Compresso ad Cosoldao of Clayey Sols, Japa, p Gua, S.., Xe, K.., u, A.F., 3. A aalyss o oe dmesoal cosoldao behavor of sols uder cyclc loadg. Rock & Sol Mechacs, 4(5: ( Chese. Lee, P.K.K., Xe, K.., Cheug, Y.K., 99. A sudy o oe dmesoal cosoldao of layered sysems. Ieraoal Joural for Numercal ad Aalycal Mehods Geomechacs, 6: L, B.., Xe, K.., Yg,.W., Zeg, G.X., 999. Sem-aalycal soluo of oe dmesoal o-lear cosoldao of sof clay uder me-depede loadg. Chese Joural of Geoechcal Egeerg, (3:88-93 ( Chese. Olso, R.E., 977. Cosoldao uder me-depede loadg. Joural of Geoechcal Egeerg Dvso, ASCE, 3(GT:55-6. Schffma, R.L., 958. Cosoldao of Sol uder Tme-depede Loadg ad Varyg Permeably. Proceedgs, ghway Research Board, Washgo, 37: Wlso, N.E., Elgohary, M.M., 974. Cosoldao of sols uder cyclc loadg. Caada Geoechcal Joural, (3:4-43. Xe, K.., Pa, Q.Y., 995. Cosoldao heory of layered sols uder me depede loadg. Chese Joural of Geoechcal Egeerg, 7(5:8-85 ( Chese. Xe, K.., Leo, C.J., 999. A Ivesgao o he Nolear Cosoldao of Layered Sof Sols. Research Repor CE, School of Cvc Egeerg ad Evrome, WS, Nepea, Ausrala.