Interaction of Pile-Soil-Pile in Battered Pile Groups under Statically Lateral Load

Transcription

1 Interaction of Pile-Soil-Pile in Battered Pile Group under Statically Lateral Load H. Ghaemadeh 1*, M. Alibeikloo 2 1- Aitant Profeor, K. N. Tooi Univerity of Technology 2- M.Sc. Student, K. N. Tooi Univerity of Technology * ghaemadeh@kntu.ac.ir Abtract Battered pile foundation uually are ued for improving the deformation characteritic of pile foundation ubjected to lateral load. The aim of thi reearch i to develope a imple and acceptable method enabling uer to calculate the interaction factor of a battered pile group under tatically lateral load. For thi purpoe, the governing differential equation have been derived by uing one dimenional continuou model. In thi modeling, Winkler method ha been ued in which the reaction of oil againt pile deformation i modeled by the pring along the pile length. The cloed form olution of governing equation have been obtained uing appropriate boundary condition. The reult are compared with other method and the parametric tudy of interaction factor i performed. Keyword: pile-oil-pile interaction, battered pile, lateral load. 1. INTRODUCTION Battered pile uually are ued in pile group under lateral load. Since the pile are ued in group, it i neceary to conider the effect of each pile on another pile in the group. For thi purpoe the pile-oil-pile interaction factor can be ued in form of the ratio of an unloaded pile ettlement to the ettlement of a loaded pile becaue of oil deformation. During the lat few decade, everal reearcher have tudied the pile-oil-pile interaction factor of vertical pile under lateral load. Poulo, 1971 [1] calculated interaction factor by uing continuum model and Mindlin equation. Nogami and Paulon, 1985 [2] and Hariharan and Kumaraamy, 1982 [3] repreented interaction factor for vertical pile under lateral load by uing Winkler model. Other model which ha been ued for pile-oil-pile interaction factor i Hybrid model. Thi model ha been obtained by Focht and Koch, 1973 [4] by combining continuum model of Poulo and nonlinear analyi of P-Y curve. In all method which are conducted for pile-oilpile interaction factor of laterally loaded pile the preence of receiver pile i not conidered and are only ued for vertical pile. Mot reearche that conider the behavior of pile group with battered pile are numerical. Rajahree and Sitharam, 21 [5] have developed a finite element method in which the nonlinear oil behavior i repreented and the pile are battered. Alo a implified method of numerical analyi ha been developed by Kitiyodom and Matumoto, 22 [6] to etimate the deformation and load ditribution of pile group with battered pile. The only reearch that conidered the interaction factor of battered pile ha been reported by Poulo and Davi, 198 [7]. In Poulo reearch, it ha been aumed for implicity that the interaction factor for two battered pile are identical with thoe for vertical pile at ome equivalent pacing. In thi paper a imple and acceptable method i developed for interaction factor of battered pile under lateral load uing theoretical method

2 x The 4 th International Conference on Geotechnical Engineering and Soil Mechanic, 2. PILE MODEL To calculate the pile-oil-pile interaction factor of laterally loaded battered pile a one dimenional continuou model i developed in which the reaction of oil i modeled by Winkler pring (K ) along the pile length (ee Figure 1). The tiffne of thee pring i function of oil and pile parameter for vertical pile [8] and alo batter angle of pile for battered pile. The pile and oil are aumed linear elatic, homogeneou and iotropic. Pile are cylindrical, long, compreible and battered in a ame plane. Q x K Batter Figure 1. Winkler pring for pile oil interaction ÏB > - 45< y < y Ô K = BK. (1 ) 1 Batter Æ B = - = ÌB = y = 9 Ô ÓB < < y < 45 (1) K -.53 ÈEp = d. E, d = 1.67Í Æ.9< d < 1.2 ÎE (2) Where L i pile length, E i elaticity module of the pile, p E i elaticity module of the oil, G i hear module of the oil, r i pile radiu and y i batter angle of the pile (ee Figure 2). Q Q x Q x -y +y Negative batter pile Vertical pile Poitive batter pile Figure 2. Different kind of battered pile under lateral load - 2 -

3 In battered pile a lateral load on a ource pile (loaded pile) caue a deflection on a receiver pile (unloaded pile) which can be reolved in axial and normal direction of the receiver pile. The diplacement of receiver pile becaue of lateral diplacement of ource pile can be written in the form below: Ïu Ì ÔÓw ji Ô Ô ji Ô = { a a }.[ u ] nn na ii (3) Where u i lateral diplacement of receiver pile (pile j) due to lateral load of ource pile ji (pile i), w i axial diplacement of receiver pile (pile j) due to lateral load of ource pile (pile i), ji a i normal normal interaction factor of pile oil pile and nn a i axial normal interaction factor na of pile oil pile. Calculating interaction factor mentioned above ha four conecutive tep Diplacement of pile under lateral load The ource pile i ubjected to a lateral (normal) load at it head. Lateral diplacement i obtained by conidering one element of a pile (ee Figure 3). Equilibrium equation i written along the pile length and by applying boundary condition for the frictional pile and fixed to the cap, lateral diplacement of the pile i obtained: Ql K u = e + Æ (4) ( l l ) l = 4 -l ' Batter 11( ').co( ') in( ') K 4. E. I Batter p p Where Q i lateral (normal to the pile axe) act at the pile head. M V x' d' ' V+dV M+dM u 11 K.u 11 Batter Figure 3. An element of battered pile for modeling battered pile ubjected to lateral load 2.2. Soil diplacement around pile At the location of the unloaded (receiver) pile, if thi pile were not preent, the oil diplacement would be [8]: 1 E 7 2 p 11( ', ) 11( '). f(, q) f(, q).34(1 co q). E È Ê ˆ u = u Æ r = + Í Á Î Ë D -1 (5) - 3 -

4 q S Figure 4. The location of pile in pile group under lateral load 2.3. Lateral diplacement of receiver pile The preence of receiver pile modifie (uually reduce) the normal diplacement of oil caued by normal diplacement of ource pile. Since oil diplacement i in the normal direction of ource pile, it induce axial and normal diplacement in receiver pile. For thi tep, normal diplacement of receiver pile i calculated. With the pring reaction being proportional to the relative diplacement, normal equilibrium of an element of receiver pile i written a (In thi paper it i aumed that ource pile i negative battered pile and receiver pile i poitive battered pile (ee Figure 5)): Q x y 1 y 2 Figure 5. Soil diplacement along normal direction of ource pile 2 ln - p ' u21( ') = f(, q).co( y1+ y 2).( Q '. K in( l ')) (( 21co( ') 21 in( '))] Batter n + e l C lp + D lp (6) 2 In equation (6), C' 21 and D' 21 are calculated by uing boundary condition in which normal force and rotation angle at the head of receiver pile i ero. 3 l C = D = QK. (7) 4 n Batter 3 4 lp l l Where n i for negative battered pile and p i for poitive battered pile. Normal normal interaction factor i defined a: u a nn = u ( = ) ( = ) (8) The reult obtained by equation (8) are compared with thoe obtained by Poulo and Davi [7] for vertical pile. A can be een in Figure 6a and 6b by conidering preence of receiver pile the normal-normal interaction factor i reduced but without conidering preence of receiver pile the reult are very cloe with thoe obtained by Poulo and Davi

5 1 2 t The 4 th International Conference on Geotechnical Engineering and Soil Mechanic,.8.4 with preence of receiver pile with preence of receiver pile.6 without preence of receiver pile without preence of receiver pile Poulo & Davi (198) Poulo & Davi (198) αnn.4 αnn.2.2 θ= (a) (b) Figure 6. Comparion of normal-normal interaction factor between preented analyi and the reult of Poulo and Davi for vertical pile (ψ1=ψ2, L/D=25 Ep/E =1, ν =.5) θ= Axial diplacement of receiver pile With the pring reaction being proportional to the relative diplacement, equilibrium of an element of receiver pile i written in ' direction, ee Figure 7. Axial diplacement of receiver pile i obtained a: l w = f(, q).in( y + y )[ B e + e ( C co l ' + D in l ')] (9) -L' p -l' n 21 KBatter n l C D D B C D 2 E. A. l G p p p p 21 = 1 -G 21, 21 =, 2 21 =- ( 21-21), G= 1 +G L. K K Batter ' (1) A p K t u ( )in(y + y ) 11 A - (A )d Figure 7. An element of receiver pile for conidering axial deflection p p When pile i ubjected to axial load, tiffne of pring for modeling oil reaction i [9]: 2p G K =, rm = 2.5 L(1 -n ) rm ln( ) r Where n i Poion ratio of the oil. Axial normal interaction factor i defined a: (11) w21( = ) a na = u ( = ) 11 (12) - 5 -

6 3. PARAMETRIC STUDY The reult of parametric tudy of interaction factor have been hown in Figure 7 to Figure 11. It ha been hown that by increaing batter angle, normal-normal interaction factor increae until batter angle about 15 (ψ 1 = ψ 2 =3 ) and then by increaing batter angle it decreae. Thee different value for normal-normal interaction factor of battered pile under lateral load are becaue of different behavior of negative and poitive battered pile againt lateral load. It i obviou that by increaing batter angle, axial-normal interaction factor increae too. By increaing relative tiffne (E p /E ) both normal-normal and axial-normal interaction factor decreae, becaue by increaing thi ratio, the changing rate of diplacement of receiver pile i le than ource pile. Since pile are long and flexible different value of L/D have no effect on normal-normal interaction factor but by increaing L/D axial-normal interaction factor decreae..8.6 ψ= ψ=1 ψ=2 ψ=3 αnn Figure 8. Normal normal interaction factor veru for different value of batter angle (L/D=25 E p /E =1, ν =.5).8.6 ψ=1 ψ=2 ψ=3 αna Figure 9. Axial normal interaction factor veru For different value of batter angle (L/D=25 Ep/E=1, ν =.5) - 6 -

7 1.8 Ep/E=1 Ep/E=1 Ep/E=1.6 αnn Figure 1. Normal normal interaction factor veru for different value of Ep/E (L/D=25 ψ1=ψ2=3, ν =.5).1.8 Ep/E=1 Ep/E=1 Ep/E=1.6 αna Figure 11. Axial normal interaction factor veru for different value of Ep/E (L/D=25 ψ1=ψ2=3, ν =.5).8.6 L/D=1 L/D=2 L/D=3 L/D=4 αna Figure 12. Axial normal interaction factor veru for different value of L/D (Ep/E =1 ψ1=ψ2=3, ν =.5) 4. CONCLUSIONS A imple analytical method ha been preented for calculating the diplacement of ingle pile and interaction factor of pile in pile group with battered pile. The bai of the model i a generalied Winkler-type model for pile-oil and pile-oil-pile interaction analyi. The method - 7 -

8 preented herein ha been conidered the preence of receiver pile and interaction of it with oil. So interaction factor obtained by thi method are le than thoe obtained by Poulo and Davi and then thi method ugget more economical deign. The method developed here permit key parameter to be evaluated through cloed-form expreion. A it can be een in parametric tudy part, the mot effective parameter in interaction factor of pile group with battered pile i batter angle. In battered pile lateral diplacement of ource pile caue both axial and lateral diplacement of receiver pile. So chooing an optimum batter angle for pile group need more conideration and the diplacement of receiver pile hould be conidered in both direction (axial and normal). REFERENCES 1. Poulo, H.G., (1971), The diplacement of laterally loaded pile,. J. Soil. Mech.Found. 97 (5), pp. 711~731; 2. Nogami, T. and abd Paulon, S.K., (1985), Tranfer matrix approach for nonlinear pile group repone analyi, Int. J. Numer and Anal. Meth in Geomech. 5, pp. 289~316; 3. Hariharan, M. and Kumaraamy, K., (1982), Analyi of pile group ubjected to lateral load, Proc., 3rd Int. Conf. On Behavior of Offhore Struct 2, Hemiphere Publihing Corp., Wahington, D.C, pp. 383~39; 4. Focht, J. A. and Koch, K. J., (1973), Rational analyi of the lateral performance of offhore pile group, Proc. 5th Offhore Tech. Conf., 2, Offhore Technology Conference, Dalla, Tex, pp. 71~78; 5. Rajahree, S.S. and Sitharam, T.G., (21), Nonlinear finite element modeling of batter pile under lateral load, J. Geotechnical and Geoenviromental Engineering, ASCE. 127, pp. 64~612; 6. Kitiyodom, P. and Matumoto, T. (22), A implified analyi method for piled raft and foundation with batter pile, J. Geotech.Eng. Bangkok, Thailand, 23(1), pp. 47~6; 7. Poulo, H.G. and Davi, E. H., (198), Pile Foundation Analyi and Deign, New York, Willy; 8. Randolph, M.F., (1981), The repone of flexible pile to lateral loading, Geotechnique, 31(2), pp. 247~259; 9. Randolph, M.F. and Worth, C.P., (1978), Analyi of Deformation of Vertically Loaded Pile, J. Geotech. Engrg. ASCE. 14(12), pp. 1465~1488; - 8 -