Lateral stresses caused by uniform rectangular area loads on a cross-anisotropic backfill

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1 Wang, C. D. (). Géotechnque 5, No. 9, 5 [do:./geot ] TECHNICAL NOTE Lateral tree caued by unform rectangular area load on a cro-anotropc backfll C. D. WANG* KEYWORDS: anotropy; degn; earth preure; retanng wall; tre analy; theoretcal analy Manucrpt receved July 5; reved manucrpt accepted July. Dcuon on th paper cloe on May 9, for further detal ee p.. * Department of Cvl and Dater Preventon Engneerng, Natonal Unted Unverty, Tawan, ROC. INTRODUCTION Surcharge load appled to the backfll of a retanng tructure produce addtonal lateral tree that, for workng load level, can be predcted ung the theory of elatcty. Thee urcharge load mght are from wheel, ralway track, hghway pavement or foundaton of adjacent buldng, and could be modelled a pont load, lne load, trp load or area load. In a conventonal elatcty calculaton the backfll materal aumed to be a homogeneou, lnearly elatc and otropc contnuum. Neverthele, numerou tude have recogned that the elatc properte n the horontal and vertcal plane mght be dfferent. Hence the effect of anotropc deformablty on the lateral tree nduced by urcharge loadng hould be taken nto conderaton. Poulo & Dav (9) provded a large number of oluton for cro-anotropc meda, reultng from varou type of urface load. Th clacal text alo gave two extenve appendce (by Gerrard & Harron) contanng oluton for crcular loaded area. Thee oluton are ueful, a the effect of a crcular loaded regon could be practcally the ame a that of a quare one. A detaled tudy of urcharge-nduced lateral tree on rgd retanng tructure wth cro-anotropc backfll wa made by Wang (5), who derved analytcal oluton for varou horontal and vertcal urcharge load. Thee ncluded a horontal/ vertcal pont load, a horontal/vertcal nfnte lne load, a horontal/vertcal unform trp load, and varou lnear and non-lnear varyng trp load. However, n the cae of a backfll wth an rregularly haped loaded area the trp loadng oluton propoed by Wang (5) mght not be very utable. For th reaon, the preent artcle decrbe analytcal oluton for the lateral tree caued by unform horontal and vertcal rectangular area load on a croanotropc backfll. An arbtrary rregularly haped area can be handled by approxmatng t wth a utable number of rectangle, and ung uperpoton. The backfll materal aumed to be homogeneou, lnearly elatc and cro-anotropc. The retanng wall vertcal, wth horontal backfll, and the plane of croanotropy are parallel to the urface of the backfll. Two further mplfyng aumpton are made: (a) the wall doe not move; and (b) the wall perfectly mooth (there no hear tre between the wall and the ol). Under thee condton, the lateral tre nduced on the wall can be calculated by conderng an elatc half-pace carryng two load of equal magntude (e.g. Fang, 99). The magnary load would caue equal and oppote normal dplacement on a plane mdway between t and the real urcharge load, thu enforcng the dered ero-horontal-dplacement boundary condton at the retanng wall. Conequently, the horontal tre on the wall twce that nduced n an elatc half-pace (Fang, 99). However, n realty a large tre concentraton mght be developed around the lower corner of a retanng wall n contact wth the backfll. Addtonally, the theory of elatcty utled n th nvetgaton doe not conder the trength of the ol or the varaton of t tffne under dfferent tre tate. Alo, the aumpton of a perfectly mooth wall retrctve, and lmt the applcablty of the elatcty method to practcal applcaton. Neverthele, a ere of experment conducted at the Iowa Engneerng Experment Staton (Spangler, 9, 9a, 9b; Spangler & Mckle, 95; Spangler & Handy, 9) and by Teragh (95) confrmed the fact that doublng the horontal tre n an elatc half-pace could provde a good approxmaton to meaured value of earth preure on retanng wall (Fang, 99). In th tudy, lateral tre oluton for rectangular area load appled to a cro-anotropc backfll are obtaned by ntegratng the pont load oluton of Wang & Lao (999). Fg. (a) depct an appled horontal/vertcal pont load, P/ Q, actng at a pont wth coordnate x ¼ a, y ¼ c, ¼ relatve to a cro-ecton through the retanng wall, whoe heght H aumed to be large relatve to a and c. Once oluton for the pont load cae have been obtaned, they are ued to generate oluton for the lateral tre nduced by a horontal/vertcal fnte lne load (Fg. (b)), and a horontal/vertcal unform rectangular area load (Fg. (c)). Illutratve example are then ued to clarfy the nfluence of the type and degree of materal anotropy, the loadng dtance from the retanng wall (a and c), the dmenon of the loaded area (l and w), and the loadng drecton (horontal or vertcal). CASE A: LATERAL STRESS CAUSED BY A HORIZONTAL/VERTICAL POINT LOAD Referrng to Fg. (a), the exact oluton for the horontal tre (ó p xx ) due to a horontal pont load P and a vertcal pont load Q on the urface of a cro-anotropc half-pace can be recat from Wang & Lao (999) a 5

2 5 WANG y y y O c a Q P x w O c P l x a Q l w Q O c P x a l H σ p h H σ l h H (a) (b) (c) Fg.. Lateral tre caued by three type of horontal/vertcal urcharge load on a cro-anotropc backfll: (a) pont load cae; (b) fnte lne load cae; (c) unform rectangular area load cae ó p xx ¼ P ( kðm u m u Þ A u ð m m ðu u Þ p u p u A p u ) p þ u p m þ u m þ u þ Q kðm u m u Þ A ðu p u p Þ A ðp p Þ ðp p Þ ð u u m þ u m þ u () For a perfectly mooth rgd retanng wall, the lateral tre (ó p h ) aumed to be twce a large a that computed from the cro-anotropc pont load oluton, ó p h ¼ P ð ( kðm u m u Þ A u m m ðu u Þ p u p u A p u ) p þ u p m þ u m þ u þ Q kðm u m u Þ A ðu p u p Þ A ðp p Þ ðp p Þ ð u u m þ u m þ u () In th equaton, (a) A j (, j ¼ ) are the elatc modul or elatcty contant of the medum (Lao & Wang, 99), and can be expreed n term of the fve ndependent elatc contant for a cro-anotropc medum a E ðe=e9þí9 A ¼, ð þ íþ í ðe=e9þí9 Eí9 A ¼ í ðe=e9þí9 A ¼ E9ð íþ í ðe=e9þí9, E A ¼ G9, A ¼ þ ð íþ where E Young modulu n the horontal drecton, E9 Young modulu n the vertcal drecton, í Poon rato for the effect of horontal tre on complementary horontal tran, í9 Poon rato for the effect of vertcal tre on horontal tran, and G9 the hear modulu n the vertcal plane (Lee & Rowe, p99). (b) u ¼ ffffffffffffffffffffffffffffffff A =A and u, are the root of the charactertc equaton u u þ t ¼ () where ¼ [A A A (A þ A )]=A A and t ¼ A =A. A the tran energy aumed to be potve defnte n the medum, the value of elatc contant () (c) are retrcted. Hence there are three categore of the charactertc root u,, a follow. Cae u, ¼ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff p [ ffffffffffffffffffffffffffffffffffff ( t) ] are two real dtnct root when t.. Cae p u, ¼ ffffffffffffff pffffffffffffff =, = are equal real root when t ¼ (.e. complete otropy). Cae u ¼ qffffffffffffffffffffffffffffffff p þ ffff t qffffffffffffffffffffffffffffffffffffffff p þ ffff t ¼ ª ä, u ¼ ª þ ä are two complex conjugate root when t,. k ¼ A þ A A A (u u ), m j ¼ (A þ A )u j A u j A p ¼ x þ a R, ¼ A A u j (A þ A )u j ( j ¼, ); p ¼ x þ a (x þ a) R R (R þ ) þ (x þ a) (R þ ) R (R þ ), p ¼ R, p ¼ R (R þ ) (x þ a) (R þ ) R (R þ ),

3 qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff R ¼ (x þ a) þ (y þ c) þ, ¼ u ( ¼,, ). In th nvetgaton, p p ( ¼,, ) are defned a the elementary tre functon. The lateral tre caued by a horontal/vertcal fnte lne load (Cae B) and a horontal/vertcal unform rectangular area load (Cae C) can be obtaned drectly by ntegratng the elementary tre functon of the pont load oluton. CASE B: LATERAL STRESS CAUSED BY A HORIZONTAL/VERTICAL FINITE LINE LOAD Now conder the tuaton hown n Fg. (b): a horontal/vertcal fnte lne load parallel to the y-ax, wth an ntenty P l /Q l (force per unt length) and length w, actng from (a, c, ) to(a, c + w, ), on a cro-anotropc backfll. The analytcal oluton for the lateral tre, ó l h, can be derved by ntegratng the elementary tre functon of the pont load oluton (p p ) wth repect to y between the lmt and w, a follow. d d d d ð w 5 ¼ p p p p 5dy (5) The explct oluton for the lateral tre due to a unform rectangular area load can be regrouped n the ame form a equaton (), except that P/Q and the elementary tre functon p p ( ¼,, ) are replaced, repectvely, by P /Q and the tre ntegral functon e e ( ¼,, ), a follow. qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ðn þ n Þ þ n e ¼ ln þ þ n qffffffffffffffffffffffffffffffffffffffffffffffffffffff n þ n þ þ n qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ðn þ n Þ þ ðn þ n Þ þ ln qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff n þ ð n þ n Þ þ " e ¼ n p þ ffffffffffffffffffffffffffffffffffffffffffffffffffff n þ n þ þ ðn þ n þ ðn þ n Þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff5 þ ðn þ n Þ þ n þ þ ðn þ n Þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff þ n þ ð n þ n Þ þ The explct oluton for the lateral tre due to a fnte lne load can be regrouped n the ame form a equaton (), qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff5 except that P/Q and the elementary tre functon p p þ ðn þ n Þ þ ðn þ n Þ þ ( ¼,, ) are replaced, repectvely, by P l /Q l and the tre ntegral functon d d ( ¼,, ), a follow. x þ a c c þ l d ¼ ðx þ aþ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff5 () ðx þ a ðx þ aþ þ ðc þ l Þ þ c þ Þ þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 9 >< c ðx þ aþ þ c þ ðc þ lþ ðx þ aþ þ ðc þ lþ þ >= d ¼ ðxþaþ h qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff h qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff () >: ðx þ aþ þ c ðx þ aþ þ c þ ðx þ aþ þ ðc þ lþ ðx þ aþ þ ðc þ lþ þ >; c c þ l d ¼ ðx þ aþ ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff þ q qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff5 () ðx þ aþ þ c þ ðx þ aþ þ ðc þ lþ þ d ¼ UNIFORM RECTANGULAR AREA LOADS ON A CROSS-ANISOTROPIC BACKFILL 59 c ðx þ aþ þ c c þ l ðx þ aþ þ ðc þ lþ h h 9 >< c ðx þ aþ þ c þ ðc þ lþ ðx þ aþ þ ðc þ lþ þ >= ðx þ aþ h qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff h qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff þ >: ðx þ aþ þ c ðx þ aþ þ c þ ðx þ aþ þ ðc þ lþ ðx þ aþ þ ðc þ lþ þ >; Þ () () (9) CASE C: LATERAL STRESS CAUSED BY A HORIZONTAL/VERTICAL UNIFORM RECTANGULAR AREA LOAD Fgure (c) depct a unformly dtrbuted horontal/ vertcal rectangular area load wth an ntenty P /Q (force per unt area) and wdth l, on a cro-anotropc backfll. The analytcal oluton for the lateral tre, ó u h, can be obtaned by ntegratng the tre ntegral functon of the fnte lne load oluton (d d ) wth repect to x between the lmt and l a follow. e d e e e ð l 5 ¼ d d d 5 dx () e ¼ tan n n pffffffffffffffffffffffffffffffffffffffffffffffffffff n þ n þ tan ðn þ n Þn qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ðn þ n Þ þ n þ tan n ðn þ n Þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff n þ ð n þ n Þ þ þ tan ðn þ n Þðn þ n Þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ðn þ n Þ þ ðn þ n Þ þ ()

4 WANG a / l /P 5 Effect of c / w caued by horontal unform rectangular load for otropc/cro-anotropc backfll c / w for E / E, νν / G / G (Sol ) c / w for E / E νν / G / G (Sol ) c / w for E / E, νν / G / G (Sol ) c / w for E / E, νν / G / G (Sol ) c / w for E / E νν / G / G (Sol ) c / w for E / E, νν / G / G (Sol ) c / w 5 for E / E, νν / G / G (Sol ) c / w 5 for E / E νν / G / G (Sol ) c / w 5 for E / E, νν / G / G (Sol ) (a) a / l /Q 5 Effect of c / w caued by vertcal unform rectangular load for otropc/cro-anotropc backfll c / w for E / E, νν / G / G (Sol ) c / w for E / E νν / G / G (Sol ) c / w for E / E, νν / G / G (Sol ) c / w for E / E, νν / G / G (Sol ) c / w for E / E νν / G / G (Sol ) c / w for E / E, νν / G / G (Sol ) c / w 5 for E / E, νν / G / G (Sol ) c / w 5 for E / E νν / G / G (Sol ) c / w 5 for E / E, νν / G / G (Sol ) Fg.. Effect of c/w on nduced lateral tre caued by: (a) horontal rectangular area load; (b) vertcal rectangular area load

5 UNIFORM RECTANGULAR AREA LOADS ON A CROSS-ANISOTROPIC BACKFILL c / w /P Effect of a / l caued by horontal unform rectangular load for otropc/cro-anotropc backfll a / l for E / E, νν / G / G (Sol ) a / l for E / E νν / G / G (Sol ) a / l for E / E, νν / G / G (Sol ) a / l for E / E, νν / G / G (Sol ) a / l for E / E νν / G / G (Sol ) a / l for E / E, νν / G / G (Sol ) a / l 5 for E / E, νν / G / G (Sol ) a / l 5 for E / E νν / G / G (Sol ) a / l 5 for E / E, νν / G / G (Sol ) 5 (a) c / w /Q Effect of a / l caued by vertcal unform rectangular load for otropc/cro-anotropc backfll a / l for E / E, νν / G / G (Sol ) a / l for E / E νν / G / G (Sol ) a / l for E / E, νν / G / G (Sol ) a / l for E / E, νν / G / G (Sol ) a / l for E / E νν / G / G (Sol ) a / l for E / E, νν / G / G (Sol ) a / l 5 for E / E, νν / G / G (Sol ) a / l 5 for E / E νν / G / G (Sol ) a / l 5 for E / E, νν / G / G (Sol ) 5 (b) Fg.. Effect of a/l on nduced lateral tre caued by: (a) horontal rectangular area load; (b) vertcal rectangular area load

6 WANG e ¼ e þ tan n n tan tan n þ n n þ tan tan tan þ tan n n n n n þ n þ tan n þ n n þ n pffffffffffffffffffffffffffffffffffffffffffffffffffff n þ n þ n þ n qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ðn þ n Þ þ n þ ðn þ n n qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff Þ n þ ð n þ n Þ þ n þ n qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ðn þ n Þ ðn þ n Þ þ ðn þ n Þ þ () that the nfluence of ol anotropy on ó u h =P and ó u h =Q become le apparent a c=w ncreae. The fgure demontrate that the lateral tree are ntenely affected by the type and degree of ol anotropy (E/E9), the value of c=w, and the loadng drecton (horontal or vertcal). Smlarly, Fg (a) and (b) how the effect of the other non-dmenonal rato n =n ¼ (a= )=(l= ) ¼ a=l ¼,, 5 on the nduced lateral tre for horontal and vertcal rectangular area load repectvely. From Fg. (a) t can be een that ó u h =P ncreae wth decreang E/E9 n the cae of a=l ¼ ; however, when a=l ¼ and 5, the order of ó u h =P are revered. Fg. (b) ndcate that ó u h =Q ncreae wth ncreang E/E9 for all a=l. Fgure and clearly reveal the nfluence of the type and degree of materal anotropy (E/E9), the loadng dtance from the retanng wall (a, c), the dmenon of the loaded area (l, w), and the loadng drecton (horontal or vertcal) on the lateral tre. They alo how that when a unform rectangular area load appled to a cro-anotropc backfll, the lateral tre cannot be accurately computed by ung prevou oluton for trp loadng (Wang, 5). where n ¼ a=, n ¼ l=, n ¼ c= and n ¼ w= ( ¼,, ) are non-dmenonal factor (functon of a, c, l, w and ). ILLUSTRATIVE EXAMPLES Th ecton preent a parametrc tudy to confrm the propoed oluton and to demontrate the effect of the type and degree of backfll anotropy, the loadng dtance from the retanng wall, the dmenon of the loaded area, and the loadng drecton, on the lateral tre. Concernng typcal range of cro-anotropc elatcty parameter, Gaeta (9) ummared expermental data regardng deformatonal cro-anotropy of clay and and, and concluded that the rato E/E9 ranged from. to for clay, and wa a low a. for and. Table lt the elatc properte and root type of the three hypothetcal backfll materal ued n th parametrc tudy. Sol and are cro-anotropc, wherea Sol otropc. The value adopted for E and í are 5 MPa and. repectvely. Baed on the derved equaton, a Mathematca program wa wrtten to calculate the nduced lateral tre under a pont load (equaton ()), a fnte lne load (equaton (), () (9)), and a unform rectangular area load (equaton (), () ()). The reult dcued here are the lateral tree caued by horontal and vertcal unform rectangular area load, for Sol. Fgure (a) and (b) depct the effect of the non-dmenonal rato n =n ¼ (c= )=(w= ) ¼ c=w ¼,, 5 on the nduced lateral tre for horontal and vertcal rectangular area load repectvely. The two fgure how that the nduced non-dmenonal lateral tree ó u h =P (caued by a horontal load) and ó u h =Q (caued by a vertcal load) ncreae wth ncreang E/E9 (Sol! Sol ), and decreae wth ncreang c=w (from! 5). It alo oberved Table. Elatc properte and root type for the otropc and cro-anotropc backfll Sol type E/E9 í/í9 G/G9 Root type Sol. Cro-anotropy... Dtnct Sol. Iotropy... Equal Sol. Cro-anotropy.. Complex CONCLUSIONS In th tudy, analytcal oluton have been developed for the horontal tre on a mooth, rgd retanng wall wth cro-anotropc elatc backfll, due to unformly dtrbuted horontal and vertcal load appled over a rectangular area on the urface of the backfll. The plane of cro-anotropy are aumed to be parallel to the horontal urface of the backfll. It ha been hown that the calculated lateral tree are profoundly affected by the type and degree of ol anotropy, the loadng dtance from the retanng wall, the dmenon of the loaded area, and the loadng drecton. In partcular, llutratve example have been gven for two hypothetcal cro-anotropc and (Sol and ), and an otropc and (Sol ). The lateral tre reultng from an rregularly haped area load can be computed by dvdng the loaded area nto many rectangle; nfluence from thee ub-area are then upermpoed. The preent oluton could offer a valuable reference for the degn of relatvely rgd retanng tructure under workng urcharge load. REFERENCES Fang, H. Y. (99). Foundaton engneerng handbook, nd edn. New York: Chapman & Hall. Gaeta, G. (9). Stree and dplacement n cro-anotropc ol. J. Geotech. Engng Dv. ASCE, No. GT, Lee, K. M. & Rowe, R. K. (99). Deformaton caued by urface loadng and tunnelng: the role of elatc anotropy. Géotechnque 9, No., 5. Lao, J. J. & Wang, C. D. (99). Elatc oluton for a tranverely otropc half-pace ubjected to a pont load. Int. J. Numer. Anal. Method Geomech., No., 5. Poulo, H. G. & Dav, E. H. (9). Elatc oluton for ol and rock mechanc. New York: Wley. Spangler, M. G. (9). The dtrbuton of normal preure on a retanng wall due to a concentrated urface load. Proc. t Int. Conf. Sol Mech. Found. Engng, Cambrdge, MA, pp.. Spangler, M. G. (9a). Lateral preure on retanng wall caued by upermpoed load. Proc. th Annual Meetng of the Hghway Reearch Board, Wahngton, DC, 5 5. Spangler, M. G. (9b). Horontal preure on retanng wall due to concentrated urface load. Iowa Engneerng Experment Staton, Bulletn. Spangler, M. G. & Handy, R. L. (9). Sol engneerng, th edn. New York: Harper & Row. Spangler, M. G. & Mckle, J. L. (95). Lateral preure on

7 UNIFORM RECTANGULAR AREA LOADS ON A CROSS-ANISOTROPIC BACKFILL retanng wall due to backfll urface load. Hghway Re. Board Bull.,. Teragh, K. (95). Anchored bulkhead. Tran. ASCE, 9,. Wang, C. D. (5). Lateral tre caued by horontal and vertcal urcharge trp load on a cro-anotropc backfll. Int. J. Numer. Anal. Method Geomech. 9, No.,. Wang, C. D. & Lao, J. J. (999). Elatc oluton for a tranverely otropc half-pace ubjected to bured aymmetrc-load. Int. J. Numer. Anal. Method Geomech., No., 5 9.