ANALYSIS OF PILE EQUIVALENT ANCHORAGE LENGTH FOR ELEVATED PILE CAPS UNDER LATERAL LOAD

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna ANALYSIS OF PILE EQUIVALENT ANCHORAGE LENGTH FOR ELEVATED PILE CAPS UNDER LATERAL LOAD M ZHOU Wancheng YUAN and Yue ZHANG Ph.D Canddate, State Key Laboratory for Dsaster Reducton n Cvl Engneerng, Tongj Unversty, Shangha, Chna Professor, State Key Laboratory for Dsaster Reducton n Cvl Engneerng, Tongj Unversty, Shangha, Chna Lecturer, School of Constructon and Cvl Engneerng X an Unversty of Scence and Technology, X an, Chna Emal: zhoum@yahoo.cn, yuan@mal.tongj.edu.cn ABSTRACT : Calculaton model of equvalent anchorage length for elevated ple caps under lateral load was proposed based on the nvarablty of basc perod. Ple-sol nteracton was consdered by elastc subgrade reacton method and usng the power seres method solved the curve dfferental equaton. Varety rules of anchorage length for elevated ple caps wth the changes of sol propertes, dameter-spacng rato of ple, ple-per stffness rato etc. were studed. The emprcal equaton of ple equvalent fxed length for elevated ple caps under lateral load was gven n the paper. The man factors affectng the equvalent anchorage length of ple consst of sol propertes and ple-per stffness rato. So the paper can provde some theoretcal foundaton for smplfyng calculaton and conceptual desgn of brdges. KEYWORDS: Lateral load, Elevated ple caps, Fxng length. INTRODUCTION Recently elevated ple caps have been wdely appled n long-span brdges, wharfs and offshore ol platforms. Elevated ple caps can penetrate the soft stratum to reach the deeper stratum and ts ablty of bearng vertcal loads s well. The researches n ths area are relatvely rch, but the study on workng performance of elevated ple caps under lateral wnd load s not yet perfect. So ts workng performance and response under horzontal load, such as wave force, sesmc forces, shp mpact load, brake force etc, become the concerned problem. On the statc and dynamc analyss for ple foundaton, many researchers adopted wnkler model to smulate sol-structure nteracton and determned equvalent sol sprng (Fg.(b))[ Harry G. Poulos. (999), Rollns et al. (005)]. Although Ples and sol can be fnely smulated by the method, but the dsadvantage s that element number of fnte model s enormous and the method s dffcult to master. Another way s drectly to fx ple n a certan depth under ground or scour lne and gnore the effect of sol around ples(fg.(c)).ths method s smple, clear and easy to be appled n the practcal engneerng. Whether the value of ple anchorage length s reasonable has great nfluence on the superstructure responses. Murat[Murat Dclel et al. (005)]determned the ple equvalent anchorage length for a certan ntegral abutment by push-over. An ntal value should be hypotheszed and then t needs to repeat the process of tral calculaton untl the ple top bendng moment and angle of equvalent system are equal to that of refned model. Y. Chen[Y. Chen(997)] summarzed several methods to determne the ple equvalent anchorage length and dscussed the lmtaton of emprcal formula gven by AASHTO LRFD Specfcaton. But hs research s only to H shaped steel ples. The smple cubc equaton to solve ple equvalent anchorage length s gven n the lterature[nchrp REPORT 89 (00)], but the emprcal formula s only sutable for sngle ple model wth the same dameter ple or column. The paper s based on the theory that the prmtve perod of equvalent system s equal to the prototype and the elastc subgrade reacton method s used to consder ple-sol nteracton. Analytcal method to solve equvalent anchorage length for elevated ple-group caps s proposed. Ths method can precsely calculate the response of superstructure and reduce calculaton workload. Therefore, the method can be appled for concept desgn of brdges.. DEFLECTON DIFFERENTIAL EQUATION OF PILE UNDER LATERAL LOAD

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna Fgure Ple-sol model The ple s supposed as elastc component embedded n sol. When the ple at the depth z produces lateral dsplacement x, sol resstance of ple sde at the depth z s q z.when the hypothess of M-method s adopted, the foundaton coeffcent ncreases lnearly wth the depth. Namely, the deflecton dfferental equaton of the ple can be gven as: d x EI m( z) b zx 0 dz + = (.) where x s the horzontal dsplacement of the ple; EI s rgdty of the ple; b s the wdth of the ple; z s the depth of the ple; mz ( ) s the functon whch reflects the trend of the sol level foundaton coeffcent ncreasng wth the depth. The soluton of Eqn.. can be descrbed as followng power seres: x = az (.) = 0 where a s undetermned coeffcent. Gettng the frst to forth dervatve of above formula we substtute them nto Eqn... Accordng to analytc theory of dfferental equaton, the power seres representaton of the ple dsplacement, rotaton, moment and shear force can be solved[hu Ren-l(987)].. EQUIVALENT ANCHORAGE LENGTH OF THE PILE FOR ELEVATED PILE CAP.. Basc Prncples For the regulaton system of beam brdges wth elevated ple-group caps, the mass of the superstructure can be concentrated on the per top. Then the whole brdge could be smplfed as a sngle per system(such as Fg. (a)). If gnorng the response of ples under the scour lne, the ple can be fxed at a certan depth under the scour lne and the system would be further smplfed as an equvalent system as Fg. (b). The Ple equvalent anchorage length L e can be calculated based on the assumpton that the basc perod of equvalent system s equal to that of sngle per system. To solve the basc perod of the sngle per system, elastc subgrade reacton method s adopted to consder the ple-sol nteracton. Flexblty method s used to solve the structure flexblty matrx, and dsplacement of per crown s solved by power seres method. In terms of the boundary condtons of ple tps have great mpact on the calculated results, the calculaton model would be dvded nto two types: one s the end of ple beng supported on the sol or rock layer (model ), the other s the end of ple beng embedded n the rock (model )... The End of Ple Is Supported on Sol or Rock

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna.. The basc perod of sngle per system Fg. shows the deformaton of the sngle per system wth the end of ples beng supported on the sol or rock. The stffness of column and each sngle ple s assumed as E I, EI respectvely; the length of ples below and above the mudlne s h, h ; the heght of per column s h ;the abutment thckness s h ;horzontal force on per crown s H ;the dsplacement of per crown s Δ ; the stackng mass of per crown s M ;b, a and β are used to represent vertcal and horzontal dsplacement of ple cap, angle of ple round coordnate orgn respectvely. To the sngle partcle system two degrees of freedom ( horzontal and vertcal) should be consdered and flexblty method s used to solve the structure flexblty matrx. Then natural frequency of the system can be calculated. (a) Sngle per system (b) Equvalent system Fgure Calculatng dagram of model Under a unt horzontal force, the lateral dsplacement and angle at the top of every sngle ple s respectvely HH, ; under a unt moment, the lateral dsplacement and angle at the top of every sngle ple s MH respectvely HM,. The calculaton formulas can be wrtten as: l0 (0) (0) (0) = + l + l + HH 0 MH 0 HH EI (.) l0 (0) (0) = + l + MH 0 HM EI (.) l0 (0) = + (.) EI where (0) (0) (0),, and (0) are the flexbltes of a sngle ple at the mudlne; ther physcal meanng s same HH HM MH as explaned n structural mechancs and can be calculated by Eqn..: (0) ( B D BD) + k( BD BD) h HH α EI ( AB AB) + k ( AB AB) h (0) ( AD AD) + k ( AD AD) h MH α EI ( AB AB) + k ( AB AB) h (0) ( B C BC) + k ( BC BC ) h HM α EI ( AB AB) + k ( AB AB) h (0) ( AC AC ) + k ( AC AC ) h (.) α EI ( AB AB) + k ( AB AB) h where A, B, C, D (=,)are dmensonless parameters that can be solved by Eqn..;α s deformaton

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna coeffcent of foundaton; kh s the nfluence factor of sol resstance due to the rotaton of ple surface whch can be calculated by Eqn..5. k /( ) h = c0i0 α EI (.5) where c0 s the foundaton coeffcent of ple basement sol; I, I0 s respectvely the sectonal moment of nerta and bottom surface moment of nerta of the ple under ground or local scour lne; E s the elastc modulus of concrete. When the unt dsplacement occurs at the bottom of the ple cap, the calculaton of nternal force produced on ple top s as follows: If the unt dsplacement occurred along ple axs, axal force produced on ple top s denoted by ρ ; when the unt transverse dsplacement occurred along perpendcular to ple-axs, shear produced on ple top s ρ ; when the unt transverse dsplacement occurred along perpendcular to ple-axs, moment produced on ple top s ρ ; when the unt angle occurred, moment produced on ple top s ρ, all of these can be calculated by Eqn..6~.9. ρ = (.6) ( l + ξh)/( EA) + /( C A ) 0 0 0 where A s the cross sectonal area of ples; E s the elastc modulus of ples; A 0 s the projected area at the bottom of the ple whch dffused wth ϕ / angle from the mudlne to the ple s bottom and ϕ s nternal frcton angle; C0 s vertcal elastc resstance coeffcent of foundaton at the plane of ple bottom; ξ s correcton coeffcent. ρ = /( ) HH MH (.7) ρ = /( ) MH HH MH (.8) ρ = /( ) HH HH MH (.9) Based on the hypothess that the relatve poston among every ple top does not change after the absolutely rgd platform occurrng dsplacement, the rotaton angle of every ple top and platform s same. If the platform subjected to an external force, b, a and β s taken as the vertcal dsplacement, horzontal dsplacement and the rotaton angle about the coordnate orgn respectvely. If ples are ntercepted at the bottom of the ple cap, then the ple cap can be treated as a free body and ts equlbrum equatons of all external force and nternal force can be wrtten as: aγ ba + bγbb + βγbβ N = 0 aγaa + bγab + βγaβ H = 0 (.0) aγ + bγ + βγ M = 0 βa βb ββ where γ ba, γ aa, γ β a s respectvely the whole sum of vertcal force, horzontal force and reverse moment of ple top acted on ple cap when the unt horzontal dsplacement occurred at the ple cap; γ bb γ ab γ βb s respectvely the whole sum of vertcal force, horzontal force and reverse moment of ple top acted on ple cap when the unt vertcal dsplacement occurred at the ple cap; γ bβ γ aβ γ ββ s respectvely the whole sum of vertcal force, horzontal force and reverse moment of ple top acted on ple cap when the unt rotaton angle occurred at the ple cap; N, H and M are external loads. If the vertcal ple group s arranged symmetrcally, let b =, a = 0, β = 0 ; b = 0, a =, β = 0 ; b= 0, a= 0, β = and respectvely substtute nto Eqn..0.The dsplacements of ple cap can be calculated: a = H M (.) β = γ γ γ γ γ (.) = (.) ( γ ββ γaβ )/( γaaγββ γaβ) ( aam a β H)/( aa ββ a β ) b N / γ bb

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna where γ bb = nρ, γ aa = nρ, γ β = γβ = ρ, γββ = nρ + ρ Kx ; n s the total number of ple foundaton; x s the dsplacement from coordnate orgn to every ple axs; K s the root number of th row ple. 0 Let the flexblty matrx s 0 and the self-vbraton frequency of the system can be calculated by Eqn... ( ) (, ) ω = m + m ± ( m + m ) mm (.) where s horzontal dsplacement of partcle under horzontal unt generalzed force; s vertcal dsplacement of partcle under vertcal unt generalzed force; the two varables can be calculated respectvely by Eqn..5~.6; m and m are the horzontal and vertcal generalzed masses. = a+ sn β ( h + h ) + xh (.5) where a, β s respectvely the horzontal dsplacement and angle of ple cap bottom under the unt horzontal force acted on the per top; xh s the dsplacement under unt force when the ple bottom s fxed, whch reflects the flexblty of the ple. = b+ h /( E A) (.6) where b s the vertcal dsplacement of ple cap under the unt vertcal force acted on per top. The meanngs of other parameters are the same as before... The soluton of equvalent anchorage length The equvalent system s llustrated as Fg. (b). Its natural vbraton frequency can be calculated by Eqn.., n whch the flexblty coeffcent can be solved by Eqn..7~.8. NN VV = ds + ds k ds EI + EA (.7) GI where M, N, V s respectvely the moment, axal force and shear of structure when the unt horzontal force acted on the ple top; k s correcton coeffcent that s nduced by the shear stress msdstrbuton on the secton. = h /( E A ) + ( h + Le ) /( nea) (.8) where A, A s respectvely the secton area of ple and column; n s the total number of ple foundaton; Le s equvalent ple length; the meanngs of other parameters are the same as before. Let the basc frequency of the equvalent system equal to that of the orgnal system, the equvalent anchorage length of ple can be solved... The End of Ple Is Embedded n Rock a a n Fg. (a) shows the deformaton of a sngle per system wth the end of ples beng embedded n rock. The meanngs of parameters n Fg. are the same as Fg. (a) sngle per system (b) equvalent system Fgure Calculatng dagram of model

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna Accordng to the same calculaton method and formulas as model, the equvalent anchorage length (model ) of ple-group can be solved. The dfferent between the two models s that the calculaton method of flexblty coeffcents HH, HM, MH and are dfferent. Because the dsplacement and angle of ple toe are zero, flexblty coeffcents should be calculated by Eqn..9: (0) ( BD BD ) HH = α EI ( AB AB ) (0) ( AD AD ) MH = α EI ( AB AB ) (0) ( BC BC ) HM α EI ( AB AB ) (0) ( AC AC ) = (.9) α EI ( AB AB ) where α s deformaton coeffcent of foundaton; A, B, C and D (=,)are dmensonless parameters that can be solved by Eqn... THE SENSITIVITY ANALYSIS OF PARAMETERS Takng a ple foundaton wth elevated ple cap(fg.)as an example, the senstvty analyss of parameters s conducted for the equvalent anchorage length of ples. The geometry of the foundaton s as follows: ple-group s conssted of 6 hole-drllng pourng ples; the outsde dameter of each ple s.0m; the length of the per s.5m; the concentrated mass of per top s 5000 tons; the thckness of the ple cap s.0m; the elastc modulus of concrete(e) s.9e0pa. The meanngs of other parameters (Fg. and Tab..) are as follows: Sp s the ple spacng; D s the dameter of ple; h s the bured depth of ple foundaton; h s the protruded length of ple foundaton; h s the heght of per; I s the moment of nerta of cross sectons of the per; m s the parameter whch reflects the varaton trend of the sol level foundaton coeffcent wth the ncrease of depth. To analyze the varaton trends of the equvalent anchorage length L e and Le of the foundaton n longtudnal and transverse drecton wth m, the dmensonless parameter D/ Sp(the rato of ple dameter and per spacng), dmensonless parameter h/ h(the rato of bured depth of ple foundaton and protruded length of ple foundaton),dmensonless parameter xh / a(the rato of ant-push rgdty of ple group and that of per), three cases are desgned to study the parametrc senstvty of these parameters. The selected values of each varable parameter n these three cases are lsted n Tab... Table. The selected values of each varable parameter parameter m h I D/ Sp h/ h Item ( N / m ) ( m ) ( m ) 5.00E6~ 0.5 Case 6.6.50 0.785.00E7 ~0. 5.00E6~.66~ Case 0.00.50 0.785.00E7 6.6 5.00E6~ 0.5 0.6 Case 6.6.50.00E7 ~0. ~0.870 Fgure The layout of the elevated ple cap (Unt :m)

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna In Fg.5~Fg.7, the dmensonless parameters R R (the raton of the equvalent anchorage length of ple and ple dameter n longtudnal and transverse drecton) are selected as vertcal coordnate, trend curves of R R wth the varaton of other parameters are plotted. Fg.5 represents the model change law of R R wth the m and D/ Spunder Case ; Fg.6 represents the model change law of R R wth the m and h/ h under Case ; Fg.7 represents the model change law of R R wth the xh / a and D/ Spunder Case. In ths example, the number of ples n longtudnal drecton s less than those n transverse drecton and under the same case R s greater than R. But the foundaton sol s harder, the dfference s less. When m s larger than.0e7 N / m, R s almost equal to R. When the number of ples s large, both R and R are nsenstve to the change of D/ Sp(Fg.5, Fg.7). It s clear that the value of m s one of the man factors that nfluence the equvalent anchorage length of ples. The analyss results show that the equvalent anchorage length of ples s ~5 tmes of ple dameter f the foundaton sol s hard plastc cohesve sol, hard cohesve sol, medum sand or gravel sand. Parameter a s the horzontal dsplacement of the ple cap bottom when unt horzontal force acted on the top of per and t reflects the flexblty of the ple group. Parameter x H s the flexblty of per column. Parameter xh / a s the rato of ant-push rgdty of the ple group and that of the per. The nfluence of ple spacng, ple dameter, bured depth of ple foundaton, the value of m, heght of per column and sze of per column can be reflected n parameter xh / a. Fg.7 shows that R and R decease monotonously wth the ncrease of ple-group s stffness proporton n the whole system and the trend curve can be ftted wth a negatve exponental functon. Accordng to regresson analyss, the relaton between the equvalent anchorage length Le of ple and parameter xh / a satsfes power functon: B Le / D= A( a/ xh) (.) where A s regresson coeffcent and B s ndex of correlaton; D s ple dameter. In the example the value range of A s [.866,.05] and the value range of B s [0.5, 0.69]. 7.6 6.0 R =L e /D 7. 6.8 6. 6.0 5.6 5..8. D/Sp =0.5 D/Sp =0.00 D/Sp =0. 0. 0.8..6.0..8. m (.00E+07N/m ) R =L e /D 5.6 5..8. D/Sp =0.5 D/Sp =0.00 D/Sp =0. 0. 0.8..6.0..8. m (.00E+07N/m ) (a)in longtude drecton (b) In transverse drecton Fgure 5 Trend curves of R wth m, D/ Sp R =L e /D 7.5 6.0 m = 0.50E+07 N/m m =.50E+07 N/m m =.00E+07 N/m m =.00E+07 N/m R =L e /D 7 6 5 m = 0.50E+07 N/m m =.50E+07 N/m m =.00E+07 N/m m =.00E+07 N/m R = L e /D 8 7 6 D/Sp = 0.5 D/Sp = 0.00 D/Sp = 0. R = L e /D 6.0 5.5 5.0 D/Sp = 0.5 D/Sp = 0.00 D/Sp = 0..5 5.5.0.6...0.8 5.6 6. h/h.6...0.8 5.6 6. h/h 0. 0.8 0. 0.6 0.0 0. 0.8 0.5 x H /a.0 0.8 0. 0.6 0.0 0. 0.8 0.5 x H /a (a)in longtude drecton (b) In transverse drecton (a)in longtude drecton (b) In transverse drecton Fgure 6 Trend curves of R wth h/ h, m Fgure 7 Trend curves of R wth x / a, D/ Sp H

The th World Conference on Earthquake Engneerng October -7, 008, Bejng, Chna 5. CONCLUSIONS An equvalent anchorage length model of ples s proposed for ple foundatons wth elevated ple caps under lateral loads. The equvalent anchorage length of ples s determned by keepng the fundamental perod of the ple foundatons unchangeable. The sngle partcle calculaton model s appled to determne the equvalent ple anchorage length for ple foundatons wth elevated ple caps n ths paper. Based on the m-method of lnear elastc subgrade reacton method, the analytc calculaton method s gven. The parametrc senstvty analyss results show that the property of sols and ple-per stffness rato are the man nfluencng factors of equvalent ple anchorage length. For ple foundatons wth elevated ple caps, the number of ples n longtude drecton and transverse drecton s always not equal. Under the same case n the drecton of fewer ples, Le s larger than the other drecton. But f the foundaton sol s hard, the dfference can be gnored. When the number of ples s small, L would ncreases wth the ple dameter-ple spacng raton ( D/ Sp). When the number of ple s large, e Le s nsenstve to the change of D/ Sp. An emprcal formula, whch s obtaned by regresson analyss based on the results of parametrc senstvty analyss, s proposed to calculate the equvalent ple anchorage length for elevated ple caps. ACKNOWLEDGEMENTS Ths research s supported by the Natonal Scence Foundaton of Chna under Grant No.50778, Key Projects n the Natonal Scence & Technology Supportng Program n the Eleventh Fve-year Plan Perod under Grant No. 006BAG0B0, and Scence & Technology Program for Transportaton Development n West Chna under Grant No. 006885. REFERENCES Harry G. Poulos. (999).Approxmate computer analyss of ple groups subjected to loads and ground movements. Internatonal Journal for Numercal and Analytcal Methods n Geomechancs :0, 0-0. Rollns, Kyle M., Gerber, Travs M., Lane, J. Dusty, Ashford, Scott A. (005).Lateral resstance of a full-scale ple group n lquefed sand. Journal of Geotechncal and Geoenvronmental :,5-5. Murat Dclel, M.ASCE. (005).Integral abutment-backfll behavor on sand sol-pushover analyss approach. Journal of Brdge Engneerng ASCE 005:MAY/JUNE, 5-6. Y. Chen. (997).Assessment on ple effectve lengths and ther effects on desgn-i assessment. Computers & Structures 6:,65-86. NCHRP REPORT 89.(00).Desgn of Hghway Brdges for Extreme Events, Transportaton Research Board, Washngton, D.C. Hu Ren-l.(987).Analyss and Desgn of Brdge Ple, Chna Ralway Publshng House, Bejng.