Mechanics Research Communications

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1 Mechncs Reserch Communctons 6 (9) 9 9 Contents lsts vlble t ScenceDrect Mechncs Reserch Communctons journl homepge: Dsplcements nd stresses due to nonunform crculr lodngs n n nhomogeneous cross-nsotropc mterl C.D. Wng, *, C.S. Teng b Deprtment of Cvl nd Dsster Preventon Engneerng, Ntonl Unted Unversty, No., Len-D, Kung-Chng-L, Mo-L, 6, Twn, ROC b Hu-S Townshp, Peng-Hu County, No. -, Hu-S Vllge, 88, Twn, ROC rtcle nfo bstrct Artcle hstory: Receved November 6 Receved n revsed form August 9 Avlble onlne 8 August 9 Keywords: Dsplcement Stress Concl Prbolc Crculr dstrbuton of the vertcl lod Inhomogeneous Cross-nsotropc mterl Ths rtcle presents the solutons for dsplcement nd stress components long the centerlne of nonunform crculr dstrbuton of the vertcl lods n contnuously nhomogeneous cross-nsotropc mterl wth Young s nd sher modul vryng exponentlly wth depth. The nonunform lodng types nclude concl nd prbolc crculr lod. Plnes of cross-nsotropy re ssumed to be prllel to the horontl ground surfce. The proposed solutons cn be obtned by ntegrtng the pont lod solutons n cylndrcl co-ordnte system for n nhomogeneous cross-nsotropc hlf-spce, whch were derved by Wng et l. [Wng, C.D., Teng, C.S., Pn, E., Lo, J.J.,. Dsplcements nd stresses due to vertcl pont lod n n nhomogeneous trnsversely sotropc hlf-spce. Int. J. Rock Mech. Mn. Sc. (), ]. However, the resultng ntegrls of the nonunform crculr soluton for dsplcements nd stresses cnnot be gven n closed form; hence, numercl ntegrtons re requred. Numercl results gree very well wth the nlytcl solutons of dsplcements nd stresses subjected to both present lodng types for homogeneous cross-nsotropc hlf-spce, whch re lso yelded n Appendx A of ths work. In ddton, the proposed solutons re dentcl wth Hrr nd Lovell s [Hrr, M.E., Lovell, C.W. Jr., 96. Vertcl stresses under certn xsymmetrcl lodngs. Hgh. Res. Bord Rec. 9], nd Geddes s [Geddes, J.D., 97. Vertcl stress components produced by xlly symmetrcl subsurfce lodngs. Cn. Geotech. J. (), 8 97] solutons when the medum s sotropy. Two exmples re llustrted to elucdte the effect of nhomogenety, nd the type nd degree of sol nsotropy on the vertcl dsplcement nd vertcl norml stress n the nhomogeneous sotropc/cross-nsotropc sols due to, respectvely, concl nd prbolc crculr dstrbuton of the vertcl lod ctng on the ground surfce. The generted solutons cnnot only smulte the ctul lodng problem but lso provde the relstc strtum n mny felds of engneerng prctce. Crown Copyrght Ó 9 Publshed by Elsever Ltd. All rghts reserved.. Introducton In most prevous theoretcl nlyss of sol behvor, the propertes of sol were ssumed to be homogeneous nd sotropc. However, mny nturl sols, such s flocculted clys, vrved slts or snds, often deposted through geologc process of sedmentton over perod of tme. The effects of deposton, overburden, desccton, etc., cn led geologcl med exhbt both the nhomogeneous nd nsotropc deformblty. The mechncl response of nsotropc mterls wth sptl grdents n composton s clled the nsotropc functonlly grded mterls (FGMs), nd s of consderble nterest n sol/rock mechncs nd foundton engneerng. A very detled survey work on ths topc cn be referred to * Correspondng uthor. Tel.: ; fx: E-ml ddresses: cdwng@nuu.edu.tw, cdwng7@pchome.com.tw (C.D. Wng). 9-6/$ - see front mtter Crown Copyrght Ó 9 Publshed by Elsever Ltd. All rghts reserved. do:.6/j.mechrescom.9.8.

2 9 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) 9 9 Wng et l. (). However, theoretcl understndng of nhomogeneous nd nsotropc phenomen hs not receved much ttenton due to the mthemtcl dffcultes nvolved n these med. Therefore, n elstc subsurfce lodng problem for contnuously nhomogeneous cross-nsotropc hlf-spce wth Young s nd sher modul vryng exponentlly wth depth s relevnt n ths work. In ths rtcle, the lod wth crculr shpe s chosen becuse the solutons produced re of prctcl mportnce n sol/ rock mechncs nd foundton engneerng. Prtculrly, these solutons could hve drect pplcton to problems ssocted wth foundtons under structures such s slos, chmneys, nd tnks contnng lquds (Gerrrd nd Wrdle, 97). Numerous exstng nlytcl/numercl solutons for the nhomogeneous sotropc med owng to generl types of crculr lod (.e., unform, prbolc, etc.) cn be found n Wng et l. (6). Nevertheless, n mny engneerng felds (Hooper, 976; Buer et l., 979; Hemsley, 99; Wng nd Lo,,b), ppled lods re not unformly dstrbuted but more concentrted towrds the center of foundton. Tht mens the lods mght be relstclly smulted s beng dstrbuted s lnerly vryng or s prbol of revoluton. Hence, n ths pper, the dsplcement nd stress components produced by nonunform xsymmetrc lodngs, whch nclude concl nd prbolc crculr dstrbuton of the vertcl lod, ctng n the nteror of n nhomogeneous cross-nsotropc hlf-spce re concerned. For homogeneous cross-nsotropc medum, Gets (98,b) nlytclly nvestgted how sol s cross-nsotropy ffects the surfce dsplcement nd stress dstrbutons when t s cused by xsymmetrc prbolc vertcl surfce lodng. Hnson nd Puj (998,b) estmted the stresses resultng from the combntons of unform, lner, nd qudrtc lodngs, ppled over crculr re on the surfce of cross-nsotropc hlf-spce. As for the nhomogeneous cross-nsotropc mterls due to unform or nonunform xsymmetrc crculr lodngs, the ltertures re very lmted snce the mthemtcl dffcultes encountered n such nlyses. Hooper (97) utled the fnte element method to consder the surfce dsplcement of n nhomogeneous cross-nsotropc hlf-spce owng to prbolc lodng ppled over crculr regon wth O r pc h con con U r, U con con σ, τ r Fg.. A concl crculr dstrbuton of the vertcl lod p c ppled n the nteror of n nhomogeneous cross-nsotropc hlf-spce. O r p c h pr pr U r, U pr pr σ, τ r Fg.. A prbolc crculr dstrbuton of the vertcl lod p c ppled n the nteror of n nhomogeneous cross-nsotropc hlf-spce.

3 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) Young s modulus followng the lner lw (E = E + k). Pn (989, 997) used the vector functons nd the propgtor mtrx method to solve the nduced dsplcement nd stress components of cross-nsotropc nd lyered hlf-spce under generl surfce lods. Wng et l. (6) presented the soluton for dsplcements nd stresses long the centerlne nduced by unform crculr dstrbuton of the vertcl lod n n nhomogeneous cross-nsotropc mterl wth Young s nd sher modul vryng exponentlly wth depth (Ee k, E e k,g e k ). To our knowledge, no nlytcl/numercl soluton for the dsplcements nd stresses n the cross-nsotropc hlf-spce due to concl nd prbolc crculr lod wth Young s nd sher modul vryng exponentlly wth depth hs been proposed. Integrtng the pont lod soluton of Wng et l. (), the dsplcements nd stresses long the xsymmetrc xs (-xs), wth concl nd prbolc crculr lods beng n the nteror of n nhomogeneous cross-nsotropc hlf-spce re generted. Then, the nhomogeneous soluton cn be smplfed to the homogeneous soluton by settng the nhomogenety prmeter to ero. Numercl results revel tht the nduced dsplcement nd stress by the proposed solutons gree very well wth those by the nlytcl solutons. In ddton, the present solutons lso compre wth Hrr nd Lovell s (96), nd Geddes s (97) solutons when the medum s sotropy. Two exmples re llustrted to elucdte the effect of nhomogenety, nd the type nd degree of sol nsotropy on the nduced vertcl dsplcement nd vertcl norml stress long the xsymmetrc xs, cused respectvely by concl nd prbolc crculr dstrbuton of the vertcl lod, on the surfce of the homogeneous/nhomogeneous sotropc/crossnsotropc sols. Tble Dfferences between the homogeneous nd nhomogeneous cross-nsotropc elstc constnts. Homogeneous (Lo nd Wng, 998) Inhomogeneous (Wng et l., ) E E E e k m m m m G G e k Ee k Tble Sol types nd ther elstc propertes (where E = MP, m =. re dopted n the numercl nlyss). Sol type E/E G /E m/m Sol : Isotropy.8 Sol : Cross-nsotropy. Sol : Cross-nsotropy..8 Normled Vertcl Dsplcement U con /p c.... / Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Fg.. Comprson of numercl nd nlytcl results for the normled vertcl dsplcement produced by concl crculr lod long the -xs for Sols.

4 9 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) 9 9. Dsplcement nd stress components n n nhomogeneous cross-nsotropc mterl due to vertcl pont lod In ths work, the solutons for dsplcement nd stress n n nhomogeneous cross-nsotropc mterl due to nonunform crculr dstrbuton of the vertcl lods re drectly ntegrted from Wng et l. s pont lod solutons (). The plnes of nhomogeneous cross-nsotropy re ssumed to be prllel to the horontl ground surfce. The resultng solutons for the rdl nd vertcl dsplcements, nd vertcl norml nd sher stresses n Hnkel domn (n, ) produced by sttc vertcl pont lod, P, ctng t = h (h denotes the bured depth, s seen n Fgs. nd ) n the nteror of n nhomogeneous cross-nsotropc hlf-spce re expressed s follows (Wng et l., 6): U r ¼ P A e u nj h j þ A e ðk u nþj h j D A pc C D e u nðþhþ D A D e u n e ðk u nþh D A D eðk u nþ e unh D A D eðk u nþðþhþ ðþ con Normled Vertcl Dsplcement U /p c.... con Normled Vertcl Dsplcement U /p c.... b / / Sol wth E/E'=, G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Sol wth E/E'=, G'/E'=., ν/ν'= k= k=-. k=-. k=-. con Normled Vertcl Dsplcement U /p c.... c / Sol wth E/E'=., G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Fg.. Effect of nhomogenety prmeter k on the normled vertcl dsplcement produced by concl crculr lod long the -xs for Sol () Sol (b) nd for Sol (c).

5 U ¼ P B e u nj h j þ B e ðk u nþj h j D B pc C D e u nðþhþ D B D e u n e ðk u nþh D B D eðk u nþ e unh D B D eðk u nþðþhþ r ¼ P C e u nj h j þ C e ðk u nþj h D j C pc C D e u nðþhþ D C D e u n e ðk u nþh D C D eðk u nþ e unh D C D eðk u nþðþhþ s r ¼ P D e u nj h j þ D e ðk u nþj h j D þ D pc D e u nðþhþ D þ D D e u n e ðk u nþh D þ D D eðk u nþ e unh D þ D D eðk u nþðþhþ where A A, B B, C C, D D, D, D D, C j (, j = 6), u, nd u cn be referred to Wng et l. (, 6). The dfferences between the homogeneous cross-nsotropc elstc constnts (Lo nd Wng, 998) nd the nhomogeneous ones (Wng et l.,, 6) dopted n ths rtcle re lsted n Tble. It s cler tht, for the present nhomogeneous cross-nsotropc medum, only three engneerng elstc constnts (E, E, nd G ) re ssumed to be exponentlly dependng on the nhomogenety prmeter k (length ); however, two Posson s rtos (m nd m ) remn constnts. In ddton, ccordng to the nhomogenety prmeter k, three dfferent stutons exst s follows: () k >, denotes hrdened surfce, where E, E, nd G decrese wth ncresng depth. () k =, s referred to s the conventonl homogeneous condton (Lo nd Wng, 998). () k <, denotes soft surfce, where E, E, nd G ncrese wth ncresng depth.the dsplcements U r, U, nd stresses r, s r n the physcl domn (r, ) for the nhomogeneous cross-nsotropc hlf-spce cn be obtned by tkng the nverson of Hnkel theorem for U r (Eq. ()), U (Eq. ()), r (Eq. ()), s r (Eq. ()) wth respect to n of order,,, nd, n the followng: U r U r s r ¼ ¼ Z Z n U r J ðnrþ U J ðnrþ n r J ðnrþ s r J ðnrþ C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) dn dn From Eqs. () (), t s noted tht the ntegrnds under the nfnte ntegrls n Eqs. () nd (6) nvolve products of Bessel functon of the frst knd of order n (n =, ), n exponentl functon, nd polynoml, whch cnnot be gven n closed ðþ ðþ ðþ ðþ ð6þ Non-dmensonl Vertcl Norml Stress σ con /p c / Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Fg.. Comprson of numercl nd nlytcl results for the non-dmensonl vertcl norml stress produced by concl crculr lod long the -xs for Sols.

6 96 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) 9 9 form so tht numercl ntegrtons re requred. The detled numercl technques to perform the ntegrton cn be found n Wng et l. (, 6).. Dsplcement nd stress components n n nhomogeneous cross-nsotropc mterl due to nonunform crculr dstrbuton of the vertcl lodngs In ths rtcle, consdertons re gven to the pplcton of forementoned vertcl pont lod solutons to subsurfce crculr res loded n n xlly symmetrcl fshon n n nhomogeneous cross-nsotropc mterl. The subsurfce crculr lodngs nclude two types of nonunform lod re nvestgted s follows... Cse A: A concl crculr dstrbuton of the vertcl lod wth pressure p C Consderng lodng pressure to vry lnerly wth rdl poston from mxmum mgntude p c t the center to ero t the perphery of crculr re of rdus, s depcted n Fg.. The dsplcement nd stress components long the xsymmetrc xs (-xs) re the most commonly quoted n xlly symmetrcl lodng problems (Geddes, 97). The response cn be obtned by double ntegrton of the pont lod solutons between the correct lmts (Brden, 96). Tht Non-dmensonl Vertcl Norml Stress σ con /p c Non-dmensonl Vertcl Norml Stress σ con /p c b / / Sol wth E/E'=, G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Sol wth E/E'=, G'/E'=., ν/ν'= k= k=-. k=-. k=-. Non-dmensonl Vertcl Norml Stress σ con /p c c / Sol wth E/E'=., G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Fg. 6. Effect of nhomogenety prmeter k on the non-dmensonl vertcl norml stress produced by concl crculr lod long the -xs for Sol () Sol (b) nd for Sol (c).

7 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) mens s long s ntegrtng Eqs. () nd (6) for the dsplcement nd stress components wth respect to rdus q from to, nd wth respect to h from to p, the desred dsplcements U con r, U con, nd stresses r con cn be obtned s:, scon r ( ) U con r U con r con s con r Z p ¼ Z p ¼ Z Z U r U r s r q q dqdh qð q Þdqdh ð7þ ð8þ It hs to be stted tht U con r, U con, nd r con, scon r re referred to the -xs thus they re functons of. The nfntesml pont lod t the pont (r = q, h, = h) nvolved n the ntegrton s p ¼ p c q q dqdh. The qunttes Ur, U, r, s r t the -xs (r = ) due to the bove nfntesml lod re tken from Eqs. () nd (6) who re referred n cylndrcl system wth orgn the pont O (q, h, = ). The qunttes t U r, U, r, s r t r =no (r, h, ) (Fg. ) re the sme, due to the xsymmetrc problem (vlues ndependent of the ngle h), wth the vlues occurred from Eqs. () nd (6) t the pont wth rdus r = q n the cylndrcl system wth orgn the pont O (q, h, = ). The vlues of dsplcements nd stresses n Eqs. (7) nd (8) should be clculted numerclly by usng the technques presented n Wng et l. (, 6)... Cse B: A prbolc crculr dstrbuton of the vertcl lod wth pressure p C The second lodng cse nvestgted heren s shown n Fg.. In ths fgure, lod dstrbuted n the form of prbol of revoluton, symmetrclly bout the -xs s studed. The soluton lso cn be obtned by double ntegrton of the pont lod solutons,.e., ntegrtng Eqs. () nd (6) for the dsplcement nd stress components wth respect to q from to, nd wth respect to h from to p, to obtn U pr r U pr r U pr ( ) r pr s pr r Z p ¼ ¼ Z Z p Z q q U r U r q q s r dqdh dqdh, U pr, r pr, nd s pr r In ths rtcle, both proposed solutons provde mthemtcl model for pplctons to the problems n sold mechncs where the mterls re of nhomogenety nd cross-nsotropy. However, the most nterestng results for sol/rock mechncs nd foundton engneerng re the vertcl dsplcements (U con nd U pr ), nd vertcl norml stresses (r con nd r pr ). Numercl results for them re presented n the followng secton. s: ð9þ ðþ Normled Vertcl Dsplcement U pr /p c.... / Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Fg. 7. Comprson of numercl nd nlytcl results for the normled vertcl dsplcement produced by prbolc crculr lod long the -xs for Sols.

8 98 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) 9 9 Normled Vertcl Dsplcement U pr /p c.... Normled Vertcl Dsplcement U pr /p c.... b / Sol wth E/E'=, G'/E'=., ν/ν'= k= k=-. k=-. k=-. Normled Vertcl Dsplcement U pr /p c.... c / / Sol wth E/E'=, G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Sol wth E/E'=., G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Fg. 8. Effect of nhomogenety prmeter k on the normled vertcl dsplcement produced by prbolc crculr lod long the -xs for Sol () Sol (b) nd for Sol (c).. Illustrtve exmples A prmetrc study s conducted to exmne the effect of nhomogenety, nd the type nd degree of mterl nsotropy on the dsplcements nd stresses. The dsplcements nd stresses frequently of mjor nterest re the vertcl components (Geddes, 97); hence, the dstrbutons of vertcl dsplcements (U con nd U pr ) nd vertcl norml stresses (r con nd r pr ) long the centerlne of concl nd prbolc loded crcle ctng on the surfce (h = ) of the nhomogeneous sotropc/crossnsotropc mterls re presented. The effect of the nhomogenety prmeter k, nd the type nd degree of mterl nsotropy, specfed by the rtos E/E, G /E, nd m/m (E nd E re Young s modul n the plne of cross-nsotropy nd n drecton norml to t, respectvely; m nd m re Posson s rtos chrcterng the lterl strn response n the plne of cross-nsotropy to stress ctng prllel or norml to t, respectvely; G s the sher modulus n plnes norml to the plne of cross-nsotropy) on the dsplcements nd stresses s studed. For typcl rnges of cross-nsotropc prmeters, Gets (98) suggested tht the rto E/E for clys rngng from.6 to, nd ws s low s. for snds. However, for the hevly overconsoldted London cly, the rto for E/E s n the rnge of..7, nd for G /E s n.. (Gbson, 97; Lee nd Rowe, 989; Trn nd Lu, 99). The elstc propertes for the three types of sotropc nd cross-nsotropc sols re lsted n Tble. The vlues dopted n Tble for E nd m re MP (Chou nd Bobet, ) nd. (Krkus nd Fowell, ), respectvely. The vrton of proposed solutons for k vres between (homogeneous) to. (k < denotes soft surfce).

9 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) Non-dmensonl Vertcl Norml Stress σ pr /p c / Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Sol wth k= (numercl) Sol wth k= (nlytcl) Fg. 9. Comprson of numercl nd nlytcl results for the non-dmensonl vertcl norml stress produced by prbolc crculr lod long the -xs for Sols. The reson why the stuton wth k > (.e., the hrdened surfce) s not chosen snce t corresponds to n underground wth decresng elstc modul, whch mght not be the usul cse for n erth mterl. Bsed on Eqs. (() (), (7) ()), FORTRAN progrm s wrtten to clculte the dsplcements nd stresses. The computed results of dsplcement nd stress by forementoned numercl technques for concl nd prbolc dstrbuted pressure re respectvely demonstrted n Fgs.. Frstly, the nfluence of nhomogenety, nd the type nd degree of sol nsotropy on the vertcl dsplcement, produced by concl crculr dstrbuton of the vertcl lod s nlyed. Fg. shows the vrtons of the normled vertcl dsplcement (U con =p c ) vs. the non-dmensonl fctor / ( s the rdus of loded crcle) for Sols wth k = (the homogeneous cse). It s observed tht the numercl results re nerly the sme s those obtned from the nlytcl solutons, whch re derved n Appendx A of ths rtcle. Fg. c plot the effect of k (from to -.) on U con =p c for the sotropc Sol, nd cross-nsotropc Sol nd Sol, respectvely. The rnges of U con =p c n Fg. c re wthn.. These fgures revel tht s the degree of nhomogenety of sol ncreses (from k =to.), the mgntude of U con =p c long the -xs lmost decreses for ll sols. In ddton, t s found tht the decrese of G /E (from.8 (Fg. )?. (Fg. b)), nd ncrese of E/E (from (Fg. )?. (Fg. c)), the mgntudes of nduced vertcl dsplcement, U con =p c, re qute dfferent. Tht mens n Fg., the nhomogenety prmeter k, nd the type nd degree of sol nsotropy do hve remrkbly nfluence on the dsplcement, owng to concl crculr dstrbuton of the vertcl lod. Followng, Fg. depcts the nondmensonl vertcl norml stress r con =p c long the -xs for Sols vs. / when k =. The numercl results for the cross-nsotropc Sol nd Sol re n good greement wth the nlytcl solutons, whch re provded n Appendx A. Identclly, the numercl result for the sotropc Sol s well mtchng the nlytcl solutons gven by Hrr nd Lovell (96) nd Geddes (97). Fg. 6 c llustrte the effect of k on r con =p c for Sols, respectvely. The vlues of r con =p c n Fg. 6 c vry from. to.. It s noted tht wth ncresng k from to -., the mgntude of r con =p c nerly decreses n ll sols. Notbly, the non-dmensonl vertcl norml stress, r con =p c, mght be trnsferred to tenson n Sol (Fg. 6) when k =.,.,.; n Sol (Fg. 6b) when k =.; nd n Sol (Fg. 6c) when k =.,.. Secondly, the effect of nhomogenety, nd the type nd degree of sol nsotropy on the vertcl dsplcement cused by prbolc crculr dstrbuton of the vertcl lod s explored. The normled vertcl dsplcement (U pr =p c ) vs. / for Sols wth k = s demonstrted n Fg. 7. Apprently, the numercl results re equvlent to those by the nlytcl solutons (n Appendx A). The normled vertcl dsplcement (U pr =p c ) for Sols wth k from to. s respectvely shown n Fg. 8 c. In these fgures, the vlues of U pr =p c re lso wthn -. (the sme wth those of U con =p c ). As t cn be seen n Fg. 8 tht, s the degree of nhomogenety of sol ncreses (from k =to.), the mgntude of U pr =p c long the -xs stll pproxmtely decreses for ll sols. Fnlly, the non-dmensonl vertcl norml stress (r pr =p c ) vs. / for Sols wth k = s plotted n Fg. 9. Once more, the numercl results of r pr =p c for the sotropc Sol, nd the cross-nsotropc Sol nd Sol re respectvely n excellent greement wth the sotropc solutons of Hrr nd Lovell (96) nd Geddes (97), nd the cross-nsotropc solutons proposed n Appendx A. Fg. c depct the nfluence of k on r pr =p c for Sols, respectvely. The vrtons of r pr =p c n Fg. c re wthn -.6-., whch re lttle dstnct from those of r con =p c n Fg. 6 c. However, the smlr phenomenon cn be found n Sol (Fg. ), Sol (Fg. b), nd Sol (Fg. c) tht wth the ncrese of the degree of nhomogenety (k =to.), the mgntude of r pr =p c decreses. Addtonlly, the nduced tensle stress by ths lodng type would be ppered n ll sols, especlly for the sotropc Sol (Fg. ) wth k =., n extr tensle stress (r pr =p c ¼ :7) s found t / =..

10 9 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) 9 9 Non-dmensonl Vertcl Norml Stress σ pr/p c Non-dmensonl Vertcl Norml Stress σ pr/p c b / / Sol wth E/E'=, G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Sol wth E/E'=, G'/E'=., ν/ν'= k= k=-. k=-. k=-. Non-dmensonl Vertcl Norml Stress σ pr /p c c / Sol wth E/E'=., G'/E'=.8, ν/ν'= k= k=-. k=-. k=-. Fg.. Effect of nhomogenety prmeter k on the non-dmensonl vertcl norml stress produced by prbolc crculr lod long the -xs for Sol () Sol (b) nd for Sol (c). From Fgs., 6, 8,, the effect of nhomogenety on the dsplcements nd stresses owng to concl nd prbolc crculr dstrbuton of the vertcl lod cn be evdently observed. The most nterestng feture perhps s tht for k =,.,. n Sol, for k =. n Sol, nd for k =.,. n Sol, tensle stresses could be nduced by both proposed nonunform lods. These fgures re not only theoretclly but lso prctclly n the felds of engneerng to determne the vertcl dsplcements nd vertcl norml stresses on the lodng xs, resultng from the nonunform crculr dstrbuton of vertcl lods n the nhomogeneous cross-nsotropc mterls.. Conclusons In ths rtcle, solutons re generted for the dsplcement nd stress components long the xsymmetrc xs due to nonunform crculr dstrbuton of the vertcl lodngs n contnuously nhomogeneous cross-nsotropc mterl wth Young s nd sher modul vryng exponentlly wth depth. The nonunform lodng types contn concl nd prbolc crculr lod. The plnes of cross-sotropy re ssumed to be prllel to the boundry surfce. These solutons cn be obtned by ntegrtng the pont lod solutons, whch were developed by Wng et l. (). The numercl technques utled heren nclude the ntegrton over ech of the frst twenty hlf-cycles of Bessel functons, the Guss qudrture

11 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) formul, nd the well-known Smpson s Rule for solvng the present lodng problem. Numercl results gree very well wth the nlytcl soluton of dsplcements nd stresses n homogeneous cross-nsotropc hlf-spce (derved n Appendx A of ths work) when the nhomogenety prmeter k s ero. In ddton, the clculted results re dentcl wth Hrr nd Lovell s (96), nd Geddes s solutons (97) when the mterl s sotropy. A prmetrc study by two llustrtve exmples for London cly hs yelded tht the nhomogenety prmeter k, nd the type nd degree of sol nsotropy do hve profoundly mpct on the vertcl dsplcement nd vertcl norml stress cused by proposed lodng types. The yelded solutons re never mentoned n lterture, nd they could relstclly mtte the ctul strtum of lodng stutons n mny res of engneerng prctce. Furthermore, the present solutons provde new mthemtcl model for the lodng problems n sold mechncs where the mterls re nhomogeneous nd cross-nsotropy. Acknowledgement We cknowledge the funds support by NSC of ROC under Grnt No E-9--MY. Appendx A. The solutons of dsplcements (U p r, Up ) nd stresses (rp, s p r) produced by vertcl pont lod P ctng n the nteror of homogeneous cross-nsotropc hlf-spce ( = h) re quoted from Lo nd Wng (998), nd re re-expressed s: U p r ¼ P p ½gðp dll p dl Þþm ðt p dl T p dlc Þ m ðt p dlb T p dld ÞŠ U p ¼ P p ½m ðgp d þ T m p d T m p db Þ m ðgp d þ T m p dc T m p dd ÞŠ r p ¼ P p ½ðC u m C Þðgp s þ T m p s T m p sb Þ ðc u m C Þðgp s þ T m p sc T m p sd ÞŠ s p r ¼ P p C ½ðu þ m Þðgp s þ T m p s T m p sb Þ ðu þ m Þðgp s þ T m p sc T m p sd ÞŠ where ða:þ ða:þ ða:þ ða:þ C j (, j = 6), m j (j =, ), u nd u of ths cse cn be found n Lo nd Wng (998). g ¼ ðc þc Þ C C ðu Þ,T u ¼ g u þu m u u,t ¼ g u ðu þm Þ,T m ðu u Þðu þm Þ ¼ g u ðu þm Þ,T m ðu u Þðu þm Þ ¼ g u þu m u u. Defnng p d ¼ R, p rr d ¼ ( =,,, b, c, d) n Eqs. (A.) nd (A.), nd p R s ¼, p R s ¼ r ( =,,, b, c, d) n Eqs. (A.) nd R (A.) re respectvely the elementry functons for the dsplcements nd stresses. R ¼ r þ, R ¼ R ( =,,, b, c, d); = u ( h), = u ( h), = u ( + h), b = u + u h, c = u h + u, d = u ( + h). Hence, the solutons for dsplcements nd stresses n homogeneous cross-nsotropc medum due to concl nd prbolc crculr dstrbuton of the vertcl lod cn be drectly ntegrted from forementoned elementry functons (p d p d nd p s p s ). For exmple, the nlytcl solutons of dsplcement nd stress long the xsymmetrc xs (r = ), resultng from concl crculr lod cn be regrouped s the forms of Eqs. ((A.) (A.)) except for the elementry functons (p d p d nd p s p s ) re replced by the dsplcement ntegrl functons c d c d, nd stress ntegrl functons c s c s. The sme s hold for the subjected prbolc one (.e., the dsplcement ntegrl functons re f d f d, nd the stress ones re f s f s ). Therefore, only the dsplcement nd stress ntegrl functons for ech lodng type re presented below. A.. Dsplcement nd stress ntegrl functons for concl crculr dstrbuton of the vertcl lod ppled n the nteror of homogeneous cross-nsotropc hlf-spce c d ¼ p n B n þ n þ n þ n ln A ða:þ þ þ n c d ¼ p þ n n n n B ln A ða:6þ þ þ n n B c s ¼ p þ n ln A ða:7þ þ þ n n B c s ¼ p n þ n ln A ða:8þ þ þ n where n = / ( =,,, b, c, d), nd s the rdus of crculr lodng re.

12 9 C.D. Wng, C.S. Teng / Mechncs Reserch Communctons 6 (9) 9 9 A.. Dsplcement nd stress ntegrl functons for prbolc crculr dstrbuton of the vertcl lod ppled n the nteror of homogeneous cross-nsotropc hlf-spce f d ¼ p þ n þ n þ n ð þ n Þ ln n 6 7 ða:9þ þ þ n f d ¼ p h ð þ n Þ n ð þ n Þ ða:þ f s ¼ pð þ n n þ n Þ ða:þ f s ¼ p þ n þðþn Þ ln n 6 7 ða:þ þ þ n The derved formultons for vertcl stress re dentcl wth the nlytcl solutons gven by Hrr nd Lovell (96), nd Geddes (97) when the medum s sotropy. References Brden, L., 96. Stresses nd dsplcements n cross-nsotropc sol. Géotechnque (), 98. Buer, G.E., Shelds, D.H., Scott, J.D., Nwbuoke, S.O., 979. Norml nd sher mesurements on strp footng. Cn. Geotech. J. 6 (), Chou, W.I., Bobet, A.,. Predctons of ground deformtons n shllow tunnels n cly. Tunnel. Underground Spce Technol. 7 (), 9. Gets, G., 98. Stresses nd dsplcements n cross-nsotropc sols. J. Geotech. Eng. Dv., ASCE 8 (GT),. Gets, G., 98b. Axsymmetrc prbolc lodng of nsotropc hlfspce. J. Geotech. Eng. Dv., ASCE 8 (GT), Geddes, J.D., 97. Vertcl stress components produced by xlly symmetrcl subsurfce lodngs. Cn. Geotech. J. (), Gerrrd, C.M., Wrdle, L.J., 97. Solutons for Pont Lods nd Generled Crculr Lods Appled to Cross-nsotropc Hlf-spce. CSIRO Techncl Pper. Gbson, R.E., 97. The nlytcl method n sol mechncs. Géotechnque (),. Hnson, M.T., Puj, I.W., 998. Elstc subsurfce stress nlyss for crculr foundtons I. J. Eng. Mech., ASCE (), 7 6. Hnson, M.T., Puj, I.W., 998b. Elstc subsurfce stress nlyss for crculr foundtons II. J. Eng. Mech., ASCE (), 7. Hrr, M.E., Lovell Jr., C.W., 96. Vertcl stresses under certn xsymmetrcl lodngs. Hgh. Res. Bord Rec., 9. Hemsley, J.A., 99. Elstosttc deformton of hlf-spce under concl lodng. Int. J. Numer. Anl. Method Geomech. (), Hooper, J.A., 97. Elstc settlement of crculr rft n dhesve contct wth trnsversely sotropc medum. Géotechnque (), Hooper, J.A., 976. Prbolc dhesve lodng of flexble rft foundton. Géotechnque 6 (),. Krkus, M., Fowell, R.J.,. Effects of dfferent tunnel fce dvnce excvton on the settlement by FEM. Tunnel. Underground Spce Technol. 8 (),. Lee, K.M., Rowe, R.K., 989. Deformton cused by surfce lodng nd tunnelng: the role of elstc nsotropy. Géotechnque 9 (),. Lo, J.J., Wng, C.D., 998. Elstc solutons for trnsversely sotropc hlf-spce subjected to pont lod. Int. J. Numer. Anl. Method Geomech. (6), 7. Pn, E., 989. Sttc response of trnsversely sotropc nd lyered hlf-spce to generl surfce lods. Phy. Erth Plnet. Inter. ( ), 6. Pn, E., 997. Sttc Green s functons n multlyered hlf-spces. Appl. Mth. Model. (8), 9. Trn, J.Q., Lu, C.C., 99. Anlyss of subsdence due to pont snk n n nsotropc porous elstc hlf spce. Int. J. Numer. Anl. Method Geomech. (8), 7 9. Wng, C.D., Lo, J.J.,. Elstc solutons of dsplcements for trnsversely sotropc hlf-spce subjected to three-dmensonl bured prbolc rectngulr lods. Int. J. Solds Struct. 9 (8), 8 8. Wng, C.D., Lo, J.J., b. Elstc solutons for stresses n trnsversely sotropc hlf-spce subjected to three-dmensonl prbolc rectngulr lods. Int. J. Numer. Anlyt. Method Geomech. 6 (), Wng, C.D., Teng, C.S., Pn, E., Lo, J.J.,. Dsplcements nd stresses due to vertcl pont lod n n nhomogeneous trnsversely sotropc hlf-spce. Int. J. Rock Mech. Mn. Sc. (), Wng, C.D., Pn, E., Teng, C.S., Hn, F., Lo, J.J., 6. Dsplcements nd stresses due to unform vertcl crculr lod n n nhomogeneous crossnsotropc hlf-spce. Int. J. Geomech., ASCE 6 (),.