Simple Formulation of the Ultimate Lateral Resistance of Single Pile in Cohesive Soils based on Active Pile Length

Transcription

1 6 th Internatinal Cnference n Earthquake Getechnical Engineering 1-4 Nvember 215 Christchurch, New Zealand Simple Frmulatin f the Ultimate Lateral Resistance f Single Pile in Chesive Sils based n Active Pile Length M. R. Aglipay 1, K. Knagai 2, T. Kiyta 3, H. Kykawa 4 ABSTRACT Flexible piles subjected t a lateral lad defrm prminently near the grund surface and the pile defrmatin decreases with increasing depth. This significant defrmatin is called the active pile length, L a. During a nnlinear event, a sil wedge is pushed up in the passive regin alng this active pile length. T investigate if the active pile length can describe the ultimate side sil reactin, a numerical simulatin f single pile embedded in hmgeneus chesive sil is dne using the 3D OpenSeesPL. The elast-plastic behavir f the sil is mdeled using the Vn Mises multi surface kinematic plasticity mdel while the pile is mdeled using elastic beam-clumn elements. Frm the results f the rigrus slutin, a simplified methd t define the ultimate lateral resistance f single piles in chesive sils using the key parameter, L a, is presented. This methd prvides mre practical apprach in the seismic design and assessment f piles. Intrductin Piles are usually used as deep fundatins fr imprtant structures, r structures supprted by weak sils. These piles are mst susceptible t external lateral lads like seismic mtins. With the mere presence f tw elements, i.e., the sil and the pile, the lateral resistance f piles is generally gverned by the sil-pile interactin. The mvements f gruped piles and their side sils are mutually dependent such that when the piles heads are laterally laded, they are defrmed relative t the defrmatin f the surrunding sil and vice versa. Fr flexible piles cmmnly used in engineering practice, the defrmatin f a vertical beam is bserved t be significantly prminent in the regin near the grund surface and decreases with increasing depth (Knagai 23). This regin f significant defrmatin is defined as the active pile length, L a, where the piles can be described as a cantilever beam assuming fixity at the negligible defrmatin pint. In this study, the pint f negligible defrmatin is defined as the 3% f the maximum pile head defrmatin. This L a, nting the stiffness f the pile relative t the surrunding sil, is a parameter reflective f the sil-pile interactin. A number f cases varying the pile and sil stiffness was numerically simulated t establish the relatinship between the active pile length and the rati f the pile stiffness t the sil stiffness. During nnlinear events like large seismic excitatins, a sil wedge is pushed up alng L a. This sil wedge alng L a is indicative f the ultimate lateral resistance. Thus, it can be inferred that the ultimate lateral pile resistance can be described by a simple parameter 1 PhD Student, Institute f Industrial Science, University f Tky, Tky, Japan, aglipay@iis.u-tky.ac.jp 2 Prfessr, Grad Schl f Urban Innvatin, Ykhama Natinal Univ, Ykhama, Japan, knagai@ynu.ac.jp 3 Prfessr, Institute f Industrial Science, University f Tky, Tky, Japan, kiyta@iis.u-tky.ac.jp 4 Kajima Crpratin, Tky, Japan, kykawa@kajima.cm

2 reflective f the sil pile interactin. Many researches (Elgamal et. al. 29, Lu et. al 26, Wang et. al. 214, etc.) n sil pile interactins have been dne especially in the advent f high cmputing pwers where rigrus slutins can be dne fr any cmplex sil-pile cnfiguratin. Hwever, it is still warranted especially in the engineering practice that simple yet high caliber slutins can be made available. Therefre, a simplified expressin using L a as a key parameter t describe the ultimate lateral resistance f single pile embedded in chesive sil is presented fr mre practical apprach in the seismic design and assessment f piles. Numerical Simulatin The finite element analyses were perfrmed using a freeware called the OpenSeesPL develped by Lu et al. (26) frm University f Califrnia, San Dieg. This sftware is specifically develped t simulate respnse f laterally laded fundatin system in three dimensin. The fllwing are the descriptin f the gemetrical cnfiguratin f the sil-pile cnfiguratin cnsidered in this study. Gemetry f the Mdel In view f symmetry, a half mesh is studied as shwn in Figure 1. A single end bearing pile f length, L p =3m is embedded in the sil bx with a.3m prtrusin abve the grund surface t apply the lateral lad. The dimensins f the sil bx is m in the lngitudinal side (x-axis), 79.38m in the transverse directin (y-axis) and 31m in depth (zaxis). The difference in the clrs f the sil medium indicates the difference in the vertical meshing. The range layer f the sil indicates finer meshing t have a better simulatin near the grund surface. On the ther hand, the blue layer indicates carser meshing cmpared t the blue layer. In ttal, the sil-pile cnfiguratin mdel has 3536 (sil) and 231(pile and links) ndes and 29 (sil) and 23 (pile and links) elements. The fllwing bundary cnditins are implemented: (1) the bttm f the dmain is fixed in all axes (x, y and z axis), (2) the ndes in the side planes and the back are fixed in the x and y axes and free in the z axis and (3) t mdel the full mesh 3D slutin, the ndes in the plane f symmetry are fixed in y-axis and free in x and z axis. Legend sil pile links m 25m 6m Figure 1. Sil-Pile Cnfiguratin Sil Mdeling In OpenSeesPL, the elast-plastic behavir f the undrained clay is captured using Prevst mdel (Prevst 1977). This mdel uses the multi-surface kinematic plasticity cncept (Mrz 1967) incrprating the vn Mises yield surface as seen in Figure 2. In this case, the plasticity is bserved nly in the deviatric stress-strain relatinship. The

3 vlumetric stress-strain is nt affected by the deviatric respnses and exhibits a linear elastic relatinship. Simply put, the cnstitutive mdel used simulates the shear behavir f clay, in mntnic r cyclic, independent f the cnfinement change. The backbne f the shear stress-strain curves f the clay material is mdeled using the hyperblic relatinship (Kndner 1963) defined by tw parameters: (1) initial tangent mdulus, E i and (2) ultimate shear strength, (σ 1 -σ 3 ) ult. This hyperblic relatinship is given by the fllwing equatin: ε σ 1 σ 3 = (1) 1 ε + E ( σ σ ) i 1 3 ult OpenSeesPL uses the user-defined PressureIndependMultiYield fr clay and has a predefined material prperties fr clays as sft, medium and stiff. (a) Vn Mises multi-surface (b) Hysteretic shear respnse Figure 2. Vn Mises multi surface kinematic plasticity mdel (after Lu et. al. 26) Pile Mdeling In this study, pile in elastic case is cnsidered and uses the element type elasticbeamclumn. This is based n the beam-clumn element frmulatin fr elastic case defined by the fllwing parameters: (1) Yung s mdulus (2) pisn rati and (3) mment f inertia. Detailed infrmatin can be fund in the OpenSees User Manual (Mazzni et al. 26). Cases Cnsidered In this study, a number f static pushver tests fr single end bearing pile embedded in a hmgeneus clay were simulated. The static pushver test was cnducted using a displacement cntrl at pile head, where it is cnsidered fixed. A lateral displacement is applied at the pile head at an increment f.1m until it reaches the final lad f 1.m. A ttal f 21 cases are cnsidered in this study. The diameter f a circular pile was varied frm.3 t 1.2m, while the Yung s mdulus f the pile, E p, varied frm 3GPa t 2GPa. Table 1 summarizes the different gemetric and material prperties f the piles used in the simulatin study. These piles are embedded in sft, medium and stiff clay. The material parameters f the crrespnding clay materials are given in Table 2 frm the pre-defined material library f OpenSeesPL.

4 Table 1. Pile parameters cnsidered Diameter, d (m) Yung s mdulus, E p (GPa) Pile stiffness, EI p (N-m 2 ) E E E E E + 9 Diameter, d (m) Yung s mdulus, E p (GPa) Pile stiffness, EI p (N-m 2 ) E E + 8 Clay Undrained Chesin c u (kpa) Table 2. Sil parameters cnsidered Shear mdulus G max (MPa) Bulk mdulus B (MPa) Permeability ceff. (m/s) Unit weight, γ (kn/m 3 ) Sft E Medium E Stiff E Active Pile Length In this study, L a is defined frm the grund surface dwn t the pint where the defrmatin is equal t 3% f the pile head defrmatin. Alng this length, the pile can be described as a cantilever beam, assuming fixity fr the deeper regin f negligible defrmatin. In cmmn engineering practice, Chang s frmula (Chang 1959) given by Equatin 2 is used t define the characteristic length, L = 1/ β. k d β = (2) h 4 4 EI p where in Equatin 2, k h : cefficient f hrizntal subgrade reactin, d: pile diameter r width and EI p : pile stiffness. Hwever, we nte that in k h d is nt an inherent prperty f the sil. Thus, Knagai (23) prpsed a mre ratinal expressin given in Equatin 3. This expressin cnsiders the stiffness f the pile relative t the surrunding sil. L EI p 4 = (3) G where EI p : pile stiffness and G: sil shear mdulus. The L a is clsely investigated n by varying the parameters directly affecting it such as the pile stiffness and sil stiffness. This is assumed t exhibit a prprtinal relatinship with L a, ( La = al ). The prgressive L a fr the single piles f varius EI p embedded in sft, medium and stiff clay are shwn in Figure 3. The y-axis indicates the depth frm the grund surface. In this figure, it can be bserved that fr the same surrunding sil, the higher the EI p, the lnger the L a. Cnversely, fr the same EI p, the nes embedded in sfter clay has lnger L a than the nes embedded in stiff clay. Differences lie in the rate f change f L a with pile head displacements depending n the EI p

5 and crrespnding surrunding sil medium. T prvide a clear cupled actin f the pile stiffness and sil stiffness n the active pile length, L frm Equatin 3 must be defined. The shear mdulus is derived accunting fr the shear degradatin frm the increase in the shear strain,γ, due t prgressive pile head defrmatin. The derived L is pltted against L a fr varius applied pile head displacements, particularly at.1m,.1m,.5m and 1.m, as seen in Figure 4. It can be seen that there is indeed a linear relatinship between L and L a fr varius pile head displacements. Therefre frm Figure 4, L a =α L can be established d=.3m Ep=3GPa d=.5m Ep=3GPa d=.7m Ep=3GPa d=1.m Ep=3GPa -25 d=1.2m Ep=3GPa d=.3m Ep=2GPa d=.5m Ep=2GPa Lateral Pile Head Displacement, u y (m) (a) Piles in sft clay d=.3m Ep=3GPa d=.5m Ep=3GPa -2 d=.7m Ep=3GPa d=1.m Ep=3GPa d=1.2m Ep=3GPa d=.3m Ep=2GPa -25 d=.5m Ep=2GPa Lateral Pile Head Displacement, u y (m) (b) Piles in medium clay d=.3m Ep=3GPa d=.7m Ep=3GPa d=1.2m Ep=3GPa d=.5m Ep=2GPa d=.5m Ep=3GPa d=1.m Ep=3GPa d=.3m Ep=2GPa Lateral Pile Head Displacement, u y (m) (c) Piles in stiff clay Figure 3. Prgressive active pile lengths fr pile f different stiffness embedded in (a) sft clay, (b) medium clay and (c) stiff clay uy=.1m uy=.1m uy=.5m uy=1.m Linear (uy=.1m) Linear (uy=.1m) Linear (uy=.5m) Linear (uy=1.m) R² =.9553 R² =.9495 R² =.9514 R² = L = (EI/G) uy=.1m uy=.15m uy=.1m uy=.2m 14 uy=.25m uy=.3m uy=.35m uy=.4m 12 uy=.5m uy=1.m uy=.75m L =(EI/G max ).25 Figure 4. Relatinship f L t L a Figure 5. Relatinship f initial L t L a Hwever, it is nted that in the field, G max is easily btained. T relate this t L a, Figure 5 is pltted. It can be seen that there is als a linear relatinship with L a and initial L (L a =α L ), where alpha is just the slpe dependent n the applied lateral pile head displacement.

6 Ultimate Lateral Pile Resistance The numerical simulatins shw that upn applicatin f the lateral lad, a sil wedge is prgressively frmed at the passive regin. In Figure 6, the lateral frce at the pile head is given by the black line. The pile resistance based n the L a at.1m (1%<γ<5%) pile head displacement is given by the blue line, where the sil wedge is deemed t be frmed. The side sil reactin is derived by the difference f the lateral frce at the pile head and the pile resistance. It can be bserved frm the side sil reactin curve, a cnstant line appears. This is where the ultimate lateral pile reactin is derived fr all cases. Otani (26) investigated the failure pattern fr a laterally laded pile embedded in medium dense sand using an X-ray. His findings shw that the shape f the failure wedge is a cnic shape cntrary t the cnventinal pyramidal wedge type. This study attempts t relate this cnic shape t the mbilized failure wedge fr laterally laded piles embedded in chesive sil. Since the cnic wedge is mbilized alng the L a, the height f the cne is defined as L a. Lateral Lad (kn) 4 35 Lateral Frce, P Pile Resistance, uy=.1m 3 Side Sil Resistance Lateral Pile Head Defrmatin, u y (m) L a 2r Vlume f Cne: Vl cne = (1/3)πr 2 L a Lateral Surface Area: A lat = (1/2)h(2πr) 2πr h Figure 6. Lad defrmatin curve Figure 7. Cnic Prperties P v L a θ W c Figure 8. Sil Defrmatin Mechanism Based n the energy rate equatins, the gverning equatin is given by Equatin 4. The rate f ttal external wrk dne is seen at the left hand side resulting frm the lateral lad, P and the lifting f sil weight in the wedge, W. The right hand side f Equatin 4 is the ttal internal energy rate dissipatin at the failure surface given by C which is due t the chesin at the failure surface. P v sinθ W v csθ = C v csθ (4) In Equatin 4, v : the virtual velcity which can be later n cancelled and θ: half angle f the dip f the failure surface with respect t the vertical pile and dependent upn r. Hwever, the

7 wedge that is discussed herein is a 3D upside-dwn cne, and the average θθ fr all directins, which is t be larger than that fr 2D case, can be less dependent n r. In light f this gemetric feature f the 3D cne, the dependence f equivalent csθ and sinθ in Equatin 4 fr a 3D failure wedge are tentatively ignred and designated as C and C 1, hereafter, fr the sake f simple discussin fr predminant parameter t describe ultimate side sil reactin. The weight f the sil wedge is given by the prduct f the unit weight f the sil and the vlume f the cnic sil wedge while C is given by the prduct f the undrained chesin and the lateral surface area f the cnic wedge. Substituting the cnic prperties t Equatin 4, the ultimate frce PP frm the side-sil wedge is thus given by P = C γ Vcne + C1 cu A (5) lat where, V cne = (1/3)πr 2 L a, r: radius f the cne, c u : undrained chesin and γ:unit weight. A lat be apprximated by: A a 2π (6) lat rl a Exactly speaking, the cefficient αα depending n γγ and lies within the range frm 1 t 2 1+ (2r / La ), and thus shuld be larger than 1. Hwever the actual circumference f the cne s base is nt cmpletely a circle, and can be smaller than 2πr. Thus fr the sake f simplicity, Equatin 5 is rewritten here as: 1 2 P = C γ πr La + C1 cu 2πrL (7) a 3 Frm the stress rati distributin in the passive regin as seen in Figure 9, the area f plastic frmatin given by the red clr fr piles f different diameters embedded in the same type f clay is almst the same. But the area f plastic frmatin fr the same type f pile increases with increasing strength f clay. Thus, it is assumed that r is prprtinal t the sil prperties: undrained chesin and unit weight as expressed by Equatin 8 c r = λ u (8) γ where λ is the prprtinal cnstant. Substituting Equatin 8 in Equatin 7, it is nted that L c 2 a u P (9) γ Based n Equatin 9, the lateral frce is a functin f the active pile length, undrained chesin and unit weight. Thus, these parameters are pltted against the ultimate lateral pile resistance as seen in Figure 1.

8 sft clay medium clay stiff clay d =.5m d = 1.m d = 1.2m d =.5m d = 1.m d = 1.2m d =.5m d = 1.m d = 1.2m Figure 9. Stress rati distributin fr piles with diameter, d=.5, 1. and 1.2m embedded in sft, medium and stiff clay in XY plane (tp view) 35 y = x R² = Pultimate sft clay medium clay stiff clay cu2la/γ Figure 1. Relatinship f P ultimate with L a,c u and γ It can be seen that there is a high crrelatin between these parameters and the ultimate lateral pile resistance. There is a unique line fr all the cases cnsidered in this study varying the pile stiffness and surrunding sil. The ultimate lateral pile resistance can be empirically described by the fllwing expressin Pultimate = 3.68π c u2 La γ (1) Cnclusins Fr flexible piles, the active pile length is established t be gverned by the stiffness f the pile relative t the surrunding sil stiffness. Particularly, there is a linear relatinship with (EI p /G).25 fr varius lateral pile head displacements. This L a can be easily defined by the initial L dependent n the maximum shear mdulus and applying the slpe f the line dependent n the applied pile head displacement. The failure wedge in the passive regin can be described by a cnic shape. Thugh a mre thughtful derivatin is needed t shw the predminant parameters, the numerical simulatins shw the ptential f the idea f the use f active pile length tgether with ther imprtant sil parameters such as the sil unit weight and the chesin t estimate the ultimate lateral resistance f the side sil. The verificatin f this idea shall be presented in the future papers. This simplified expressin fr the ultimate lateral pile resistance fr flexible single end bearing piles can be useful fr mre practical

9 apprach in the seismic and assessment f piles. This idea can be extended t a mre cmplicated scenari i.e. nn-hmgeneus sil, fr grup piles, etc. Acknwledgments The authr expresses thanks t the Japanese gvernment (Mnbukagakush: MEXT) fr making this study pssible thrugh their financial supprt. References Chang, YL. Discussin n lateral piles laded tests. Feagin Trans, ASCE, 1937; 1959: Elgamal, A, Lu J, Yang Z, Shantz T. Scenari-fcused three-dimensinal cmputatinal mdeling in gemechanics. Prc. 4th Internatinal Yung Getechnical Engineers Cnference, ISSMGE, 29. Knagai K, Yin Y, Murn Y. Single beam analgy fr describing sil-pile grup interactin. Sil Dynamics and Earthquake Engineering 23; 23: Kndner, RL. Hyperblic stress-strain respnse: chesive sils. Jurnal f Sil Mechanics and Fundatins Divisin, 1963; 89: Lu J, Yang Z, Elgamal A. Openseespl 3D Lateral Pile-Grund Interactin ver. 1. User's Manual. Department f Structural Engineering, University f Califrnia, San Dieg, La Jlla, CA, 26. Mazzni S, McKenna F, Fenves GL. Open System fr Earthquake Engineering Simulatin User Manual, Pacific Earthquake Engineering Research Center, University f Califrnia, Berkeley, 26. Mrz, Z. On the descriptin f anistrpic wrk hardening. Jurnal f Mechanics and Physics f Slids 1967; 15: Otani J, Dang Pham K, San J. Investigatin f failure patters in sand due t a laterally laded pile using X- RAY CT. Sils and Fundatins 26; 46(4): Prevst JH. Mathematical mdelling f mntnic and cyclic undrained clay behavir. Int J Numer Anal Methds Gemech, 1977; 1(2): Wang N, Elgamal A, Lu J. Three-Dimensinal Finite Element Mdeling f Pile and Pile Grup System Respnse. Sil Behavir Fundamentals t Innvatins in Getechnical Engineering 214; pp