Intenatonal Jounal of GEOMATE, Apl, 8 Vol. 4, Issue 44, pp. -7 Geotec., Const. Mat. & Env., DOI: https://do.og/.66/8.44.785 ISS: 86-98 (Pnt), 86-99 (Onlne), Japan AAYSIS OF AXIA OADED PIE I MUTIAYERED SOI USIG ODA EXACT FIITE EEMET MODE *Chnnapat Buachat, Chayanon Hansapnyo and Wosak anok-ukulcha 3, Cente of Excellence fo atual Dsaste Management, Depatment of Cvl Engneeng, Faculty of Engneeng, Chang Ma Unvety, Thaland; 3 Aan Insttute of Technology, Thaland *Coespondng Autho, Receved: 9 June 7, Revsed: 9 ov. 7, Accepted: 3 Dec. 7 ABSTRACT: The nodal exact dsplacement based fnte element method fo analyzng axally loaded ple embedded n multlayeed of fnte depth of elastc sol s pesented. The condton of shape functon by whch exact value may be epoduced at the nodal ponts egadng a few numbe of elements s nvestgated. The examned shape functons whch satsfy the homogeneous govenng equatons n each laye of elastc sol ae ntoduced to obtan the so-called exact element stffness matx. Then the stffness matx of poposed shape functon was constucted va total potental enegy pncple. The esults obtaned fom poposed fnte element wee compaed wth analytcal solutons fom lteatue. Axal foce and dsplacement solutons of the ple embedded n multlayeed sol obtaned fom poposed fnte element model show exact ageement wth analytcal solutons and data fom the avalable lteatue. eywods: Axally loaded ple, Dsplacement method, Fnte element, Multlaye sol. ITRODUCTIO In ths study, the dsplacement of axally loaded ple embedded n mult-laye sol s solved va poposed fnte element pocedue. The nodal exact shape functon concept suggested n [] [4] ae used to constuct the stffness matx and equvalent nodal foce ncopoate wth fxed-pont teaton algothm to solve nonlnea algebac equatons. In each teaton step, the coeffcents of dffeental equaton descbe ple settlement behavo wee estmated ealy va the component of stffness matx and nodal dsplacement obtaned fom pevous teaton step. Examples of elastostatc ple embedded n multlayeed sol subjected to qua-statc pont load on topsol level wee analyzed [5]. The esults fom poposed element ae compaed wth analytcal solutons obtaned fom [6] to vefy the accuacy of poposed ple element.. MATHEMATICA FORMUATIO In ths secton, the govenng equaton of axsymmetc poblem wll be deved. Then the fnte element fomulaton and soluton scheme to obtan the nodal dsplacement wll be descbed.. Poblem Defnton The analys condes a ngle ccula cosssecton ple [5], wth adus p and total length p embedded n a total of hozontal sol laye (Fg. ). The ple s subjected to an axal foce Q t at the ple head whch s flush wth the gound suface. The ple tself cosses m layes (m < ). All sol layes ae assumed to extend to nfnty n the adal decton, and the bottom laye ( th laye) also extends to nfnty n downwads decton (halfspace) as shown n Fg.. The sol medum n any laye ( th laye, whee =,, ) s assumed to be elastc and sotopc mateal, wth elastc popetes descbed by sol shea modulus G and Posson s ato. The vetcal depth fom the gound suface to the bottom of any laye s denoted by H. Hence, the thckness of each laye s computed by the dffeence of bottom depth H H wth H =. The ple s assumed to behave as an elastc column wth Young s modulus E p. The Posson s ato of the ple mateal s neglected.. Govenng Dffeental Equatons Snce the cylndcal ple settlement poblem n Fg. s axsymmetc. Hence, we use the system of cylndcal coodnates (-z coodnate) to ndcate any poton n ple and sol bodes. The ogn of cylndcal coodnate concdes wth the cente of ple coss secton at the ple head level. The vetcal (potve n the downwad decton) coodnate z- axs concdes wth ple axs. The non-slp condtons between ple suface and suoundng sol and between sol layes ae assumed. The vetcal dsplacement u z(,z) at any pont n the sol s epesented as the poduct of two functons n and z coodnates as follows:
Intenatonal Jounal of GEOMATE, Apl, 8, Vol. 4, Issue 44, pp. -7 suoundng sol by defnng the stans fom dsplacement functons n Eq. (), and pescbe the vaaton of total potental enegy wth espect to w and equal to zeos [6]. The govenng dffeental equaton due to vaaton wth espect to w fo the ple and sol below the ple tp s as follow: dw E A t k w z (4) whee E = E p and A = A p when m (along ple axs), and E = + G and A = p when (m + ) < (sol below the ple tp). The elastc constant + G s a functon of Posson s ato and shea modulus G of sol: G G v v (5) z Fg. Axally loaded ple and mult-laye sol, modfed fom [5], u z w z () whee w(z) s the vetcal dsplacement of the ple at any pont along ple axs, and () s the sol dsplacement decay functon n the adal decton. To compute the stan and stess n the elastc sol medum, the dsplacement n the adal component s assumed to be small compaed wth vetcal dsplacement n Eq. (),.e. u =. Hence, the nonzeo stan components n elastc sol medum can be expessed as: uz w z zz z z z uz u w z z () and stess n sol medum can be computed followng the Hooke s aw: s s zz zz s Gs z z Gs (3) whee λ s and G s ae the elastc constants of sol. Then, the calculus of vaatons s used to obtan the govenng dffeental equaton n a ple and ote that the coeffcents k, and t epesent the shea and compesve estances of sol mass aganst ple settlement. Both k, and t ae a functon of decay functon and elastc popetes of sol as follow: d k G d p d (6) t ( G) d p (7) The govenng dffeental equaton fo the sol suoundng the ple can be obtaned by takng the vaaton of total potental enegy wth espect to equals to zeo: d d d d p whee (8) s (9) p n m s H s H m G w dz () H dw ns ( G ) dz H dz ()
Intenatonal Jounal of GEOMATE, Apl, 8, Vol. 4, Issue 44, pp. -7 and the soluton of Eq. (8) wth bounday condtons () = at extend to nfnty, and () = at = p s a zeo ode modfed Bessel functon of the second knd: p ( ) ; p () whee a = (E A + t ), the ntegaton constants B and C n Eqs. (6) and (9) can be detemned analytcally fom the pocedue poposed n [5]..3 Fnte Element Fomulaton Substtutng decay functon () nto Eqs. (6) and (7), obtan the explct fomula fo coeffcents k and t n tems of modfed Bessel functon of the second knd, zeo and fst odes [6]: k G (3) t p G (4) whee the coeffcent s the ato between the modfed Bessel functon of the second knd of the fst-ode and zeo ode,.e.: (5) The geneal soluton of Eq. (4) s gven by: cosh nh w z B z C z (6) whee B and C ae ntegaton constant. The chaactestc paamete of ple and sol nteacton s expessed as follow: k E A t (7) ote that the dmenon of paamete s an nveon of length. An axal foce Q (z) at a depth z n the th laye s obtaned defned as: dw Q z E A t (8) dz o explctly n the fom: nh cosh Q z a B z C z (9) Fg. Axally loaded ple element and element degees of feedom, modfed fom [4] In the pevous woks [5, 6]; ntegaton constants B and C fo all layes wee solved by applyng dectly the bounday condtons fo the ple load head, bottom end, and laye ntefaces to the algebac system of equatons. Then the authos mplfed the calculaton by the detemnant of the matx of the algebac system. The equaton at all sol ntefaces can also be constucted automatcally though the assembly of fnte element stffness. The bounday condtons at all sol ntefaces need not be constucted sepaately. Hence, n ths secton, the fomula of stffness matx va total potental enegy and exact ntepolaton functon wll be descbed and wll be used to test wth analytcal poblems fom lteatue. Conde one-dmenonal element n Fg., whch epesents the poton of ple embedded n any one laye of suoundng sol govened by Eq. (4). Assumng that sol suoundng ple element s n an elastc condton fo the whole length. Shea estance of sol s epesented by equvalent sol spng coeffcent k. Ple element n Fg. composes of two nodes at top and bottom, numbeng wth node and, espectvely. The total potental enegy of ths sol-ple element subjected to an equvalent nodal foces P and P s defned as the sum of ntenal potental enegy (stan enegy) and the extenal potental enegy due to extenal load as follow [4]: 3
Intenatonal Jounal of GEOMATE, Apl, 8, Vol. 4, Issue 44, pp. -7 dw dz k w dz E A t Pw P w s () dz whee w(z) s the vetcal ple dsplacement at depth z whee z. The fst vaaton of Eq. () leads to: d w dw E A t dz dz dz k w wdz P w P w () ote that the ogn of vetcal coodnate along the ple axs n potental functon, Eq. (), s now moved to the top node of ple poton, nstead of ple head on gound level. The nodal dsplacement at the top and bottom nodes ae denoted by w and w, espectvely. Suppose that the ple poton at the th laye s condeed, the ple length can be computed fom dffeent of bottom depth between neaby sol laye,.e. = H H -. Applyng the appopate Gauss-Geen theoem to Eq. () and settng δп =, gves the dffeental equaton fo equlbum mla to Eq. (4), and a set of natual bounday condtons as follows dw P E A t () dz z dw P E A t (3) dz z The natual bounday condton at fst node, Eq. () s mla to axal load expeson n Eq. (8)..4 Exact Intepolaton Functon To constuct the system of algebac equatons wth espect to nodal dsplacement, the tal soluton of w(z) n Fg. s ntoduced n the fom: w z z w z w (4) The shape functons n Eq. (4) ae taken fom a homogeneous soluton of Eq. (4),.e. z nh z (5) nh z nh z (6) nh whee the dmenonless paamete β =. In pesent wok, the poposed fnte element pocedue to solve Eq. (4) fo a gven load Q t (Fg. ) wll be explaned n next secton. Then, the esults fom poposed nodal exact element wll be compaed wth an avalable analytcal soluton n [5, 6]..5 Iteaton Scheme Accodng to Eqs. () and (), the shape paamete of decay functon, namely n Eqs. (8) and (9) depends on ple settlement w(z). Hence, the fnte element dscetzaton of Eq. () wth tal dsplacement functon n Eq. (4) leads to the steady steady-state set of non-lnea algebac equatons as follow: ww f (7) In whch stffness matx s a non-lnea functon of nodal dsplacement w. The extenal nodal load f s pesented n tem of specfed vecto. To solve the nodal soluton fom Eq. (7), we employed the fxed pont teaton technque as follows [7]: n ( n) w f (8) whee (n-) = (w (n-) ), n =,, ; s stffness matx evaluated fom nodal soluton w at pevous teaton step. Usually, teaton pocess n Eq. (8) s epeated untl the value of nodal soluton w conveged. In ths wok, we wll use convegence ctea of the paamete nstead. Hence, the convegence ctea fo all cases ae: n n 5 (9) ote that the convegence ctea used n Eq. (9) s mla to ctea used n [5, 6] and also esults n the convegence of nodal soluton w n Eq. (7)..6 Devaton of Element Stffness Matx Refe to the fst vaaton of stan enegy tems n Eq. (), the element stffness matx fo ple and sol can be expessed as follows: 4
Intenatonal Jounal of GEOMATE, Apl, 8, Vol. 4, Issue 44, pp. -7 d d ple a b ab dz dz E A t dz (3) sol ab a b k dz (3) whee the ndces a and b epesent element nodal numbe, anged fom to..6. Element stffness matx Substtutng shape functons fom Eq. (5) and (6) nto element stffness fomulaton n Eq. (3) and (3), the component of element stffness matces can be explctly expessed as a csch coth a csch coth ple ple ple ple and k coth csch sol sol k csch coth sol sol (3) (33) Hence, the total stffness matx of laye, whch s the summaton of stffness n Eqs. (3) and (33), can be defned as: coth csch a csch coth (34) whee paamete a was aleady defned n Eq. (9). The element stffness matces n Eq. (34) ae assembled to fom global stffness n Eq. (7)..6. Stffness of bottom-most laye At the bottom-most laye (laye ), the thckness s assumed to be nfnty, and nodal dsplacement at the bottom most pont s pescbed to zeo. Hence, the element stffness matx n Eq. (34) has to be edefned. Convegence of hypebolc functon when shows that the element stffness matx fo th laye s mla to penalty spng coeffcent [8] attached to th degee of feedom, w, wth the followng fom: k a E A t (35) The convegence of hypebolc functon also shows that the dsplacement n the th laye can be ntepolated va the fom below: cosh nh w z w z z (36) The ntepolaton of w n Eqs. (4) and (36) ae then used to compute axal foce accodng to Eq. (8). ote that coodnate z n Eqs. (4) (36) s local coodnate defned n th -laye..6.3 Calculaton of decay paamete The value of decay paamete n any teaton step of Eq. (8) can be evaluated fom nodal dsplacement soluton w n each step. Substtutng the dsplacement functon, Eq. (4), nto Eqs. () and () obtaned: coth csch w w G ms ww csch coth (37) Gs w csch coth w w E ns ww csch coth (38) Es + w whee E G ; the values m s and n s ae then substtuted n Eq. (9) to compute the value of the paamete. 3. UMERICA EXAMPES In ths secton, two numecal examples ae pesented to llustate the effectveness of nodal exact fnte element poposed n the pevous secton. Results fom poposed element ae vefed ung analytcal solutons avalable n [5, 6]. 3. Ple wth Ideal Rgd End Beang In ths example, we study the behavo of ple n homogeneous sol (one laye) subjected to ple head load as shown n Fg. 3. The ato of ple elastc modulus and sol shea modulus s set to be E p/g s = 3, and Posson s ato v s =.4999. The ple tp s assumed to est on a gd laye and ple damete B = p =. m. ote that ths poblem was aleady solved n [6], and epeat hee to vefy ou poposed fnte element model. Because the value of Posson s ato s close to.5, the modulus λ s s appoach nfnty, accodng to Eq. (5). Hence, ths poblem wll set the value of modulus λ s equals to 5
Intenatonal Jounal of GEOMATE, Apl, 8, Vol. 4, Issue 44, pp. -7 zeo and eplace the value of G s by equvalent sol shea modulus [5]: * s s s G.75G.5v (39) Fgue 4 shows the elaton between the nomalzed ple head stffness vesus nomalzed ple length p/b. The nomalzed ple head stffness s defned as the ato of load at ple head vesus settlement at ple head nomalzed by E pb,.e. = Q t/(w E pb) whee w = settlement at ple head. The plot n Fg. 4 was obtaned wth the fnte element analys poposed n ths wok and fom the analytcal method n the pevous study [5]. The nomalzed ple head stffness n Fg. 4 deceases wth nceang of nomalzed ple length. The esults of fnte element analys pesented n ths wok ae n good ageement wth the pevous study n [6]. 3. Mcople (Italy) Ths example pesents the case of mco-ple, whch was nstalled n a complex sol pofle [6]. The sol pofle and ple length ae shown n Fg. 5. Ple damete and length ae equal to. m and 9 m, espectvely. Modulus of elastcty of ple s appoxmately 7 GPa. In all sol layes, the Posson s ato was assumed to be.3. The values of sol depth, shea modulus, and Posson s ato of Fg. 5 ae lsted n Table. The numecal test was pefomed ung fou poposed nodal exact elements wth fou actve degees of feedom. The value of stffness fo bottom most sol was calculated accodng to Eq. (35). Fg. 5 Sol Pofle of Italy case, modfed fom [5] Fg. 3 Ple est on gd laye, modfed fom [6] Table Input popetes fo the analys of mcople tested n Italy (B =. m, p = 9 m, E p = 7 GPa) omalzed ple head stffness,.8.75.7.65 Ple est on gd base Pesent study Seo and Pezz [5] aye H (m) G (MPa) ν 9..3 9 45..3 3 45..3 4 53..3 Fgue 6 shows the calculated ple head settlement vesus nput ple head load. Fgue 7 shows measued and calculated load-tansfe cuves fo appled load equal to 5, 5, and 5 k. These fgues show that thee s vey good ageement between the poposed fnte element and analytcal soluton n the lteatue [5]..6 5 5 75 omalzed ple length, p/b Fg. 4 omalzed ple head stffness vesus nomalzed ple length of gd-end ple 4. COCUSIO The fnte element model fo ple embedded n multlayeed elastc sol subjected to axal load s poposed. The system of nonlnea algebac 6
Intenatonal Jounal of GEOMATE, Apl, 8, Vol. 4, Issue 44, pp. -7 equatons constucted fom poposed fnte element has been solved va fxed pont teaton technque. The values of the nodal soluton ae ecalculated untl the shape paamete of decay functon conveged. Ple head settlement (mm) Fg. 6 oad-settlement cuve at ple head (Italy case) oad (k) 3.5.5.5 3 4 5 oad (k) 6 5 4 3 oad-settlement cuve (Italy case) Pesent study Seo and Pezz [5] oad tansfe cuve (Italy case) Pesent (5 k) Pesent (5 k) Pesent (5 k) Seo and Pezz (5 k) Seo and Pezz (5 k) Seo and Pezz (5 k) 5 5 Depth (m) Fg. 7 oad-tansfe cuve fo ple head load equal to 5, 5, and 5 k (Italy case) umecal examples fo statc load ple embedded n multlayeed elastc sol wee tested by poposed fnte element compae wth avalable analytcal solutons fom lteatue. Two poblems, composed of ple estng on the gd base, and ple embedded n fou layes sol wth nfnte bottom depth wee solved to obtan ple settlement and load tansfe cuves. The numecal test ndcates that the poposed fnte element method ae vey good ageement wth the avalable analytcal soluton poposed n the lteatue. 5. ACOWEDGEMETS The autho gatefully acknowledges the fnancal suppot eceved fom Thaland Reseach Fund (ew Reseache Gant TRG58866). 6. REFERECES [] Tong P, Exact solutons of cetan poblems by fnte element method, AIAA J., Vol. 7, 969, pp. 78-8. [] anok-ukulcha W, Dayawansa PH, and aasudh P, An exact fnte element model fo deep beams, Int. J. of Stuc, Vol., Jan. 98, pp. -7. [3] Ma H, Exact solutons of axal vbaton poblems of elastc bas, Int. J. ume. Meth. Eng., Vol. 75, 8, pp. 4-5. [4] Buachat C, Hansapnyo C, and Sommanawat W, An exact dsplacement based fnte element model fo axally loaded ple, Int. J. of GEOMATE, Sept., Vol. (5), 6, pp. 474-479. [5] Seo, H and Pezz, M, Analytcal solutons fo a vetcally loaded ple n multlayeed sol, Geomechancs and Geoengneeng: An Intenatonal Jounal, Vol. (), 7, pp. 5 6. [6] Seo, H, Pezz, M, and Salgado, R, Settlement analys of axally loaded ple, Intenatonal Confeence on Case Hstoes n Geotechncal Engneeng, Vol. 7, 8, pp. 8. [7] Chapa SC, Appled umecal Methods wth Matlab fo Engnees and Scentsts. McGaw- Hll, 5. [8] Zenkewcz OC and Taylo R, The Fnte Element Method. Vol. I. Oxfod: Buttewoth- Henemann,. Copyght Int. J. of GEOMATE. All ghts eseved, ncludng the makng of copes unless pemson s obtaned fom the copyght popetos. 7