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Dynamic response of pile groups embedded in a poroelastic medium
Soil Dynamics and Earthquake Engineering 23 (2003) 235–242 www.elsevier.com/locate/soildyn
Dynamic response of pile groups embedded in a poroelastic medium J.H. Wang*, X.L. Zhou, J.F. Lu School of Civil Engineering and Mechanics, Shanghai JiaoTong University, Shanghai 200030, People’s Republic of China Accepted 30 September 2002
Abstract The dynamic response of pile groups embedded in a homogeneous poroelastic medium and subjected to vertical loading is considered. The piles are represented by compressible beam-column elements and the porous medium uses Biot’s three-dimensional elastodynamic theory. The dynamic impedance of pile groups can be computed directly by using pile – soil– pile dynamic interaction factors. The axial forces and pore pressures along the length of pile groups are computed by superposition method, which greatly reduces the computational time for the direct analysis of pile groups. Parametric studies are conducted for various conditions of pile groups. The superposition method is proposed for the dynamic response analysis of pile groups that is computationally feasible for practical applications. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Pile groups; Poroelastic medium; Pile–soil interaction
1. Introduction The dynamic responses of piles and pile groups embedded in an elastic half-space and subjected to harmonic loads have been the subject of much research in recent years. These were prompted by the design of nuclear plants, machine foundation and other facilities. Wolf and Von Arx [1] used an axisymmetric finite element method and Salinero [2] used a boundary element method obtain the dynamic displacement field of pile groups. Nogami [3] presented a simple approach to analyze the flexural responses of pile groups and the characteristics of pilesoil-pile interaction in flexural vibration. Sen [4] used boundary element formulation for the dynamic analysis of axially and laterally loaded pile groups. Mamoon [5] used a hybrid boundary element formulation and rigorous boundary element formulation to evaluate the impedance and compliance functions of piles and inclined pile groups. Kaynia [6] provided elastic solutions for pile groups in layered half space. Green’s functions for layered media evaluated numerically by the application of integral transform techniques. Haldar [7] used approximate methods to analyze the pile group under vertical, torsional and lateral loading. Gazetas and Makris [8,9] proposed a simple model * Corresponding author. Tel.: þ 86-21-62933082. E-mail address: [email protected] (J.H. Wang).
for calculating the dynamic interaction factor between piles in homogeneous elastic soil. George [10] used a beam-onDynamic-Winkler-Foundation simplified model and a Green’s function based rigorous method to determine the dynamic response of pile groups. Kuai and Shen [11] used simplified method for calculating horizontal dynamic impedances of pile groups in layered media. For a better soil systems you need non-linearity and residual pore water pressure. You are still using an elastic soil and including ‘elastic’ transient pore water pressure. In this paper the analysis extended to a poroelastic solid. The first theory of propagation of wave in fluid-saturated porous medium was developed by Biot [12,13]. Biot’s consolidation equations satisfy not only the requirements for the deformation of the solid matrix but also the Darcy’s law for the fluid flowing in the medium. Zeng [14] extended the classical Muki and Sternberg formulation [15] to analyze steady-state dynamic response of an axially loaded elastic bar partially embedded in a homogeneous poroelastic medium. It is noted that solutions corresponding to pile groups embedded in a poroelastic medium is not reported in the literatures. The objective of this paper is to study the effect of pile groups under vertical loading in a poroelastic medium. The dynamic response of single pile in homogeneous poroelastic medium use Muki and Sternberg Method. The load transfer problem is formulated in terms of a Fredholm integral equation of the second kind.
0267-7261/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0267-7261(02)00224-5
236
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Dynamic interaction factor approach is used to generate a pile group response from a single pile response. Selected numerical results for vertical impedance, axial force and pore pressure profiles are presented to portray the influence of pile stiffness and pile distance. The results presented can be used in practice as a guide in obtaining the dynamic stiffness and damping of foundations with a large number of piles.
The continuity conditions at the plane of loading at z ¼ z and 0 # r , 1 are ui ðr; zþ Þ 2 ui ðr; z2 Þ ¼ 0
i ¼ r; z
pf ðr; zþ Þ 2 pf ðr; z2 Þ ¼ 0 þ
2
ð4cÞ
szr ðr; zþ Þ 2 szr ðr; z2 Þ ¼ 0
ð4dÞ
2
szz ðr; z Þ 2 szz ðr; z Þ ¼ PðrÞ
Consider the axisymmetric response of a poroelastic medium. According to the consolidation theory of Biot for a completely saturated medium, the governing equations can be expressed in terms of displacements and the pore pressure as follows [12,13,16] 1 ›e ›p G 72 2 2 ur þ ðl þ GÞ 2 f ¼ ru€ r ð1aÞ ›r ›r r ›e ›p 2 f ¼ ru€ z ð1bÞ G72 uz þ ðl þ GÞ ›r ›z ›p 1 2 f ¼ 0 w_ r þ rf u€ r ð1cÞ kd ›r
›p 1 2 f ¼ 0 w_ z þ rf u€ z kd ›z
›u szz ¼ le þ 2G z 2 pf ›z ›ur ›u þ z szr ¼ G ›z ›r _z qz ¼ w
ð2aÞ
ð2cÞ
pf ðr; 0Þ ¼ 0
i ¼ r; z
The application of Hankel integral transforms of zero-order or first-order to Eqs. (1a) – (1d) and (2a)– (2c) can solve the governing equations of motion. The general solutions for uz, szz and pf can be expressed as " ð1 a j uz ¼ ðA eaz 2 B1 e2az Þ þ ðA ejz 2 B2 e2jz Þ h 1 rv2 2 0 # þA4 edz þ B4 e2dz jJ0 ðjrÞdj
szz ¼
ð1 0
þ
! 2Ga2 b ðA1 eaz þ B1 e2az Þ þl2 h h ! 2Gj2 2 1 ðA2 ejz þ B2 e2jz Þ 2 2GjðA3 edz rv2 #
"
2B3 e2dz Þ jJ0 ðjrÞdj pf ¼
ð1 b 0
ð6aÞ
h
ð6bÞ
ðA1 eaz þ B1 e2az Þ þ A2 ejz þ B2 e2jz jJ0 ðjrÞdj ð6cÞ
Where j denotes Hankel transform parameter; J0 ðjrÞ is Bessel functions of zero-order; A1 – A4 ; B1 – B4 are arbitrary functions to be determined from above boundary and continuity conditions.
ð2bÞ
in which, qz is the fluid discharge. The motion is assumed to be time harmonic with a factor pffiffiffiffi of eivt ; where v is frequency of the motion and i ¼ 21: For brevity, the term eivt is suppressed from all expressions in the sequel. The boundary conditions for the free and permeable surface at z ¼ 0 and 0 # r , 1 are
szi ðr; 0Þ ¼ 0
ð4eÞ
For a uniformly distributed vertical load of radius a acting in the z direction, it can be shown that 8 > < P0 ; r # a PðrÞ ¼ pr 2 ð5Þ > : 0; r.a
ð1dÞ
where ur and uz are the displacements of the solid matrix in the r and z directions, respectively; wr and wz are the fluid displacements relative to the solid matrix in the r and z directions, respectively; pf is the pore pressure; r rs and rf are mass densities of the bulk material, the soil and the pore fluid, respectively; r ¼ ð1 2 nÞrs þ nrf ; n is the porous ratio; l and G are Lame constants; k0d is dynamic permeability; 72 is the Laplace operator; e is the dilatation; overdots denote the derivatives of field variables with respect to time t. The constitutive relations of a homogenous poroelastic material can be expressed as
ð4bÞ
qz ðr; z Þ 2 qz ðr; z Þ ¼ 0 þ
2. Governing equations
ð4aÞ
ð3aÞ ð3bÞ
3. Single pile We now consider a cylindrical elastic pile of radius a and length L ða=L p 1Þ partially embedded in a poroelastic medium as shown in Fig. 1. Following Muki’s method, the system is decomposed into an extended poroelastic halfspace and fictitious pile as shown in Fig. 2. In what follows we treat the extended embedding medium as a threedimensional poroelastic continuum. In contrast, we regard the fictitious pile as a one-dimension elastic continuum as far as its constitutive law and equilibrium conditions are concerned [14].
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 1. Geometry of the pile and embedding medium: (a) fictitious pile; (b) extended poroelastic medium.
The governing equation for time harmonic axial motion in the fictitious pile is d2 up ðzÞ v2 rpp qðzÞ þ up ðzÞ þ ¼0 2 p AEpp Ep dz
ð7Þ
where up ðzÞ is the axial displacement of the fictitious pile; rpp is density of the fictitious pile; Epp is Young’s modulus of the fictitious pile; qðzÞ is the shear force per unit length in the z direction on the shaft of the pile Pp; A is area of the fictitious pile. Ep p ¼ Ep 2 Es
ð8aÞ
rpp ¼ rp 2 ð1 2 nÞrs
ð8bÞ
where EP and ES are Young’s modulus of the pile and soil, respectively; rp and rs are mass density of the pile and soil, respectively. The force– displacement relationship for the fictitious pile is dup ðzÞ Np ðzÞ ¼ Epp A dz
0,z,L
where Np ðz; tÞ ¼ Np ðzÞeivt is the axial pile force.
ð9Þ
Fig. 3. Vertical impedance of single pile in a poroelastic medium.
With the aid of displacement and strain influence functions due to a uniform body-force distribution over the closed disk, The component of displacement and strain in the fictitious pile are given by up ðzÞ ¼ ½N0 þ Np ð0Þ^uð0; zÞ 2 Np ðLÞ^uðL; zÞ 2
ðL 0
qðjÞ^uðj; zÞdj
ð10Þ
^ zÞ 2 Np ðLÞ1ðL; ^ zÞ 1ðzÞ ¼ ½N0 þ Np ð0Þ1ð0; 2
ðL 0
^ j; zÞdj qðjÞ1ð
ð11Þ
1^ is found to possess merely finite jump discontinuities at z ¼ j for each fixed z in the open interval ð0; LÞ: 1ðjþ ; zÞ 2 1ðj2 ; zÞ ¼ 2
ð1 2 2mÞð1 þ mÞ EAð1 2 mÞ
ð12Þ
Muki and Zeng adopted the condition that the axial strain of the fictitious pile is equal to the average of the extensional strain of the cross section of the extended half space. Accordingly, fictitious pile is governed by the stress –strain relation 1ðzÞ ¼
Fig. 2. The model of single pile in a poroelastic medium.
237
Np ðzÞ Epp A
ð13Þ
Fig. 4. Normalized impedance for vertically loaded 3 £ 3 pile group (L=d ¼ 15; EP =ES ¼ 1000; rS =rP ¼ 0:7).
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J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 7. Identification of data in Figs. 8–19.
Fig. 5. Impedances of 3 £ 3 pile groups embedded in a poroelastic medium ðEP =ES ¼ 1000Þ:
Manipulation of Eqs. (10) – (13) together with the displacement compatibility condition described above results in the following Fredholm’s integral equation of the second kind N ðzÞ ^ jþ 2 zÞ 2 1ð ^ j2 2 zÞ þ p Np ðzÞ½1ð EPp A ðL r p p Av 2 ð L ^ j; zÞ ›1ð ^ j; zÞdj þ N p ðjÞ dj 2 1ð ›j E Pp A 0 0 ðj ðL ^ j; zÞdj Np ðhÞdh þ rpp Av2 1ð 0 "0 1 ðL ›u^ ðj; 0Þ 1 rpp Av2 N p ðjÞ dj 2 c1 0 c 1 Ep p A ›j # ðL ðj u^ ðj; 0Þdj Np ðhÞdh 0
The axial force and pore pressure can be determined by NðzÞ ¼ 2 N0 s^ð0; zÞ þ
ðL 0 ðL 0
s^ðj; zÞdj
0 ðL 0
Np ðjÞ
0
ðj 0
rpp Av2 ›s^ðj; zÞ dj 2 ›j Epp A
Np ðhÞdh 2 rpp Av2 up ð0Þ
s^ðj; zÞdj
pf ðzÞ ¼N0 p^ ð0; zÞ 2 ðL
ðL
ðL
ð17aÞ
N p ðjÞ
0
p^ ðj; zÞdj
ðj 0
rpp Av2 ›p^ ðj; zÞ dj þ ›j Epp A
Np ðhÞdhrpp Av2 up ð0Þ
p^ ðj; zÞdj
ð17bÞ
0
^ zÞ þ rpp Av2 ¼ N0 1ð0;
ðL 0
^ j; zÞdj 1ð
1 N u^ ð0; 0Þ c1 0
ð14Þ
4. Vertical response of pile group
where c1 ¼ 1 2 rpp Av2
ðL 0
4.1. Impedance of pile group u^ ðj; 0Þdj
ð15Þ
Once the fictitious pile force Np ðzÞ is known, the top of displacement of the pile can be determined by " ðL rpp Av2 1 ›u^ ðj; 0Þ N0 u^ ð0; 0Þ 2 Np ðjÞ dj þ up ð0Þ ¼ c1 ›j Ep p A 0 # ðL ðj u^ ðj; 0Þdj Np ðhÞdh ð16Þ 0
With the superposition method the dynamic impedance of group of N identical piles individual connected through a rigid cap is calculated by superimposing the interactions factors between individual pile pairs. Dynamic group effects can be treated approximately using complex interaction factors. The interaction factor a between two piles is defined as displacement of reference pile due to load on adjacent
0
Fig. 6. Compliances of 3 £ 3 pile groups embedded in a poroelastic medium ðEP =ES ¼ 1000Þ:
Fig. 8. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 9. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
pile, normalized by displacement of reference pile under its own load. Based on Dobry and Gazetas’s method [17], for vertical oscillation, the interaction factor is written as
1 s 2ð1=2Þ b vs ivs aðs; 0Þ < pffiffi exp 2 s 2 CS CS 2 d
n X j¼1
uij ¼
1 X aij Fj KSV
ðj ¼ 1; 2; …nÞ
n X
Fj
impedance of pile group F K G V ¼ uG
ð21Þ
4.2. Axial fore and pore pressure
ð19Þ
where K SV is the normalized dynamic impedance of single pile; Fj is the load acting on the j pile heads; aij ðj ¼ 1; 2; …nÞ is the dynamic interaction factor between j pile and i pile, aij ¼ 1: With a rigid cap and these loads are not equal, the total loads of the pile group are F¼
Fig. 11. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
ð18Þ
where bs is hysteretic soil damping; CS is the shear wave velocity of soil; s is distance of two piles; d is single pile diameter. To generate group stiffness for the case of a rigid cap, the displacements of all heads must be equal to unity uG ¼ u i ¼
239
ð20Þ
j¼1
In order to analyze the axial fore and pore pressure of pile groups vary with along pile length, we consider the ith pile of N piles. In adopting the method of the compatibility condition for dynamic load transfer problems for poroelastic medium, Fredholm’s integral equation of the second kind of the ith pile can be expressed as N p ðzÞ Nip ðzÞ½1^ii ðjþ 2 zÞ 2 1^ii ðj2 2 zÞ þ i EPip Ai " n ðLj X rpjp Aj v2 ›1^ji ðj; zÞ þ Njp ðjÞ dj 2 ›j EPjp Aj 0 j¼1 ð Lj ðj ð Lj 1^ji ðj; zÞdj Njp ðhÞdh þ rppj Aj v2 1^ji ðj; zÞdj 0 0 "0 ›u^ ji ðj; 0Þ 1 ð Lj 1 rpjp Aj v2 Njp ðjÞ dj 2 c1 0 c1 Eppj Aj ›j ## ð Lj ðj u^ ji ðj; 0Þdj Njp ðhÞdh 0
Solving Eqs. (19) and (20) can obtain the displacement of pile group uG and load Fi of single pile. The dynamic
0
n ðLj X 1 1^ji ðj; zÞdj· Nj0 u^ ji ð0; 0Þ Nj0 1^ji ð0; zÞ þ rpjp Aj v2 ¼ c1 0 j¼1 ð22Þ The ith pile’s axial fore and pore pressure can be expressed as N ð Lj X ›s^ji ðj;zÞ Ni ðzÞ ¼ 2Nj0 s^ji ð0;zÞþ Njp ðjÞ dj ›j 0 j¼1
ðj rpjp Aj v2 ðLj s^ji ðj;zÞdj Njp ðhÞdh 2 rpjp Aj v2 upj ð0Þ Epjp Aj 0 0 ð Lj s^ji ðj;zÞdj ð23Þ
2
Fig. 10. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
0
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J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 12. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
pfi ðzÞ ¼
N ð Lj X ›p^ ji ðj;zÞ Nj0 p^ ji ð0;zÞ2 Njp ðjÞ dj ›j 0 j¼1
ðj rpjp Aj v2 ðLj p^ ji ðj;zÞdj Njp ðhÞdh Epjp Aj 0 0 ð Lj 2 þrpjp Aj v upj ð0Þ p^ ji ðj;zÞdj þ
ð24Þ
0
5. Numerical results and discussions
Fig. 14. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
rf =rS ¼ 0:8; n ¼ 0:4; L=a ¼ 20; m ¼ 0:4: There is a reasonably good agreement between the present result and those of Zeng. The vertical impedances of 3 £ 3 pile groups obtained by the degeneration of a poroelastic medium to an ideal elastic medium are portrayed in Fig. 4 and compared with those obtained by Mamoon [5]. The impedances have been normalized with respect to the respective single-pile static stiffness multiplied by the number of piles in the group. The accuracy of the present formulation and its numerical results are confirmed by the above comparisons.
The normalized axial impedance K V ; normalized frequency a0; normalized axial force N z and normalized average pore pressure p f are, respectively, represented by K V ¼
P0 ; Gaup ðzÞ
a0 ¼
vd ; CS
NðzÞ N z ¼ ; P0
ð25Þ
Apf ðzÞ p f ¼ P0 For the vertical vibration of single pile embedded in a poroelastic medium, the results of this study are compared with those of Zeng [14]. Fig. 3 shows the comparison between the real and imaginary parts of the axial impedances for a pile. EP =ES ¼ 1000; rP =rS ¼ 1:2;
Fig. 15. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
Fig. 13. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
Fig. 16. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
241
Fig. 17. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
Fig. 19. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
In the remainder of this paper, the influence of pile stiffness ratio and spacing diameter ratio on the dynamic response of pile group in a poroelastic medium are investigated. The normalized properties are m ¼ 0:4; l ¼ G ¼ G0 ð1 þ 2bs ; iÞ; bs ¼ 0:05; n ¼ 0:4; rf =rS ¼ 0:8; rP =rS ¼ 1:2; L=d ¼ 20: Figs. 5 and 6 show the variation in the real and imaginary parts of the vertical pile group impedances of 3 £ 3 pile groups embedded in a poroelastic medium for s/d ¼ 2, 5,10 with respect to the normalized frequency parameter a0 (Fig. 7). The influence of pile stiffness ratio EP =ES and spacing diameter ratio s=d on the axial forces and pore pressures of 3 £ 3 pile groups in a poroelastic medium are shown in Figs. 8 –19 for low frequency range (v ¼ 1.64). Figs. 8 –10 and Figs. 14 –16 show the profiles of normalized axial forces N z along the pile length with EP =ES ¼ 1000; 100 and s/d ¼ 2, 5, 10. The outer piles carry greater loads than the central piles, with the corner piles carrying the most. The axial forces gradually decrease as the pile spacing diameter ratio increases. The axial forces on central piles of the square pile group are very small. It can also be seen that the axial force profiles have some dependence on pile stiffness ratio. The load transfer along the pile is rapid for smaller values of stiffness ratio, resulting in negligible load transfer at the base. Figs. 11 – 13 and Figs. 17 – 19 show the profile of normalized average pore pressures p f along the pile length with EP =ES ¼ 1000; 100 and s/d ¼ 2, 5, 10. The pore pressures generation is mostly confined to
the upper part and the lower part of the piles. The pore pressures on the outer of pile group are larger than those located central, with pore pressures on corner piles occurring the most. The pore pressures gradually decrease as the pile spacing ratio increases.
6. Conclusion The Muki method is used to evaluate the dynamic impedance, axial force and pore pressure of single pile in a poroelastic medium. The dynamic interaction factor approach successfully predicts the dynamic impedance of pile groups. In order to analyze axial force and pore pressure of pile group along pile length, the superposition method is used to obtain Fredholm’s integral equation of the second kind of the pile group in a poroelastic medium. Using numerical quadrature can solve the integral equation. Comparisons with the existing solutions available for single pile embedded in a poroelastic medium and pile group embedded in elastic medium confirms the accuracy of the proposed formulation. The solution presented may be extended to analyze the dynamic response of pile groups in layered saturated soils.
Acknowledgements The work presented in this paper was supported by the National Science Foundation of China, grant number is 59879012.This support is greatly appreciated. First author would like to thank Professor Radoslaw L. Michalowski for his kindly discussions during author’s visit at Department of Civil and Environmental Engineering, University of Michigan.
References
Fig. 18. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
[1] Wolf JP, Arx GAV. Impedance functions of a group of vertical piles. Proceedings of the ASCE Specially Conference on Earthquake Engineering and Soil Dynamics, Pasadena, CA II; 1978. p. 1024–41.
242
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[2] Salinero IS. Dynamic stiffness of pile groups: approximate solution. Geotechnical Engineering Report GR83-5, Department of Civil Engineering, University of Texas at Austin; 1983. [3] Nogami T. Flexural responses of grouped piles under dynamic loading. Earthquake Engng Struct Dyn 1985;13:321–36. [4] Sen R, Davies TG, Banerjee PK. Dynamic analysis of piles and pile groups embedded in homogeneous soils. Earthquake Engng Struct Dyn 1985;13:53–65. [5] Mamoon SM, Kaynia AM, Banerjee PK. Frequency domain dynamic analysis of piles and pile groups. J Engng Mech 1990;116(10): 2237–57. [6] Kaynia AM, Kausel E. Dynamics of piles and pile groups in layered soil media. Soil Dyn Earthquake Engng 1991;10(8):386 –401. [7] Haldar SS, Bose SK. Dynamic soil stiffness on a pile in a vibrating group of floating piles. Soil Dyn Earthquake Engng 1992;11:125– 32. [8] Gazetas G, Makris N. Dynamic pile–soil–pile interaction, Part I: analysis of axial vibration. Earthquake Engng Struct Dyn 1991;20: 115–32. [9] Makris N, Gazetas G. Dynamic pile–soil –pile interaction, Part II: lateral and seismic response. Earthquake Engng Struct Dyn 1992;21: 145–62.
[10] Gazetas G, Fan K, Kaynia A. Dynamic response of pile groups with different configurations. Soil Dyn Earthquake Engng 1993;12: 239 –57. [11] Kuai XC, Shen PS. Simplified method for calculating horizontal dynamic impedances of pile groups in layered media. J Vibr Engng 1998;11(3):258– 64. [12] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid, I, low frequency rang. J Acoust Soc Am 1956;28(2): 168 –78. [13] Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 1962;33(4):1482–98. [14] Zeng X, Rajapakse RK. Dynamic axial load transfer from elastic bar to poroelastic medium. J Engng Mech 1999;125(9):1048–55. [15] Muki R, Sternberg E. Elastostatic load-transfer to a half-space from a partially embedded axially loaded rod. Int J Solids Struct 1970;6: 69– 90. [16] Yang J, Wu SM. Transient solutions of axisymmetric dynamic problem for inhomogeneous two-phase media. Acta Mech Sinica 1996;28(3):308– 17. [17] Dobry R, Gazetas G. Simple method for dynamic stiffness and damping of floating pile groups. Geotechnique 1988;38(4):557 –74.
Dynamic response of pile groups embedded in a poroelastic medium J.H. Wang*, X.L. Zhou, J.F. Lu School of Civil Engineering and Mechanics, Shanghai JiaoTong University, Shanghai 200030, People’s Republic of China Accepted 30 September 2002
Abstract The dynamic response of pile groups embedded in a homogeneous poroelastic medium and subjected to vertical loading is considered. The piles are represented by compressible beam-column elements and the porous medium uses Biot’s three-dimensional elastodynamic theory. The dynamic impedance of pile groups can be computed directly by using pile – soil– pile dynamic interaction factors. The axial forces and pore pressures along the length of pile groups are computed by superposition method, which greatly reduces the computational time for the direct analysis of pile groups. Parametric studies are conducted for various conditions of pile groups. The superposition method is proposed for the dynamic response analysis of pile groups that is computationally feasible for practical applications. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Pile groups; Poroelastic medium; Pile–soil interaction
1. Introduction The dynamic responses of piles and pile groups embedded in an elastic half-space and subjected to harmonic loads have been the subject of much research in recent years. These were prompted by the design of nuclear plants, machine foundation and other facilities. Wolf and Von Arx [1] used an axisymmetric finite element method and Salinero [2] used a boundary element method obtain the dynamic displacement field of pile groups. Nogami [3] presented a simple approach to analyze the flexural responses of pile groups and the characteristics of pilesoil-pile interaction in flexural vibration. Sen [4] used boundary element formulation for the dynamic analysis of axially and laterally loaded pile groups. Mamoon [5] used a hybrid boundary element formulation and rigorous boundary element formulation to evaluate the impedance and compliance functions of piles and inclined pile groups. Kaynia [6] provided elastic solutions for pile groups in layered half space. Green’s functions for layered media evaluated numerically by the application of integral transform techniques. Haldar [7] used approximate methods to analyze the pile group under vertical, torsional and lateral loading. Gazetas and Makris [8,9] proposed a simple model * Corresponding author. Tel.: þ 86-21-62933082. E-mail address: [email protected] (J.H. Wang).
for calculating the dynamic interaction factor between piles in homogeneous elastic soil. George [10] used a beam-onDynamic-Winkler-Foundation simplified model and a Green’s function based rigorous method to determine the dynamic response of pile groups. Kuai and Shen [11] used simplified method for calculating horizontal dynamic impedances of pile groups in layered media. For a better soil systems you need non-linearity and residual pore water pressure. You are still using an elastic soil and including ‘elastic’ transient pore water pressure. In this paper the analysis extended to a poroelastic solid. The first theory of propagation of wave in fluid-saturated porous medium was developed by Biot [12,13]. Biot’s consolidation equations satisfy not only the requirements for the deformation of the solid matrix but also the Darcy’s law for the fluid flowing in the medium. Zeng [14] extended the classical Muki and Sternberg formulation [15] to analyze steady-state dynamic response of an axially loaded elastic bar partially embedded in a homogeneous poroelastic medium. It is noted that solutions corresponding to pile groups embedded in a poroelastic medium is not reported in the literatures. The objective of this paper is to study the effect of pile groups under vertical loading in a poroelastic medium. The dynamic response of single pile in homogeneous poroelastic medium use Muki and Sternberg Method. The load transfer problem is formulated in terms of a Fredholm integral equation of the second kind.
0267-7261/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0267-7261(02)00224-5
236
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Dynamic interaction factor approach is used to generate a pile group response from a single pile response. Selected numerical results for vertical impedance, axial force and pore pressure profiles are presented to portray the influence of pile stiffness and pile distance. The results presented can be used in practice as a guide in obtaining the dynamic stiffness and damping of foundations with a large number of piles.
The continuity conditions at the plane of loading at z ¼ z and 0 # r , 1 are ui ðr; zþ Þ 2 ui ðr; z2 Þ ¼ 0
i ¼ r; z
pf ðr; zþ Þ 2 pf ðr; z2 Þ ¼ 0 þ
2
ð4cÞ
szr ðr; zþ Þ 2 szr ðr; z2 Þ ¼ 0
ð4dÞ
2
szz ðr; z Þ 2 szz ðr; z Þ ¼ PðrÞ
Consider the axisymmetric response of a poroelastic medium. According to the consolidation theory of Biot for a completely saturated medium, the governing equations can be expressed in terms of displacements and the pore pressure as follows [12,13,16] 1 ›e ›p G 72 2 2 ur þ ðl þ GÞ 2 f ¼ ru€ r ð1aÞ ›r ›r r ›e ›p 2 f ¼ ru€ z ð1bÞ G72 uz þ ðl þ GÞ ›r ›z ›p 1 2 f ¼ 0 w_ r þ rf u€ r ð1cÞ kd ›r
›p 1 2 f ¼ 0 w_ z þ rf u€ z kd ›z
›u szz ¼ le þ 2G z 2 pf ›z ›ur ›u þ z szr ¼ G ›z ›r _z qz ¼ w
ð2aÞ
ð2cÞ
pf ðr; 0Þ ¼ 0
i ¼ r; z
The application of Hankel integral transforms of zero-order or first-order to Eqs. (1a) – (1d) and (2a)– (2c) can solve the governing equations of motion. The general solutions for uz, szz and pf can be expressed as " ð1 a j uz ¼ ðA eaz 2 B1 e2az Þ þ ðA ejz 2 B2 e2jz Þ h 1 rv2 2 0 # þA4 edz þ B4 e2dz jJ0 ðjrÞdj
szz ¼
ð1 0
þ
! 2Ga2 b ðA1 eaz þ B1 e2az Þ þl2 h h ! 2Gj2 2 1 ðA2 ejz þ B2 e2jz Þ 2 2GjðA3 edz rv2 #
"
2B3 e2dz Þ jJ0 ðjrÞdj pf ¼
ð1 b 0
ð6aÞ
h
ð6bÞ
ðA1 eaz þ B1 e2az Þ þ A2 ejz þ B2 e2jz jJ0 ðjrÞdj ð6cÞ
Where j denotes Hankel transform parameter; J0 ðjrÞ is Bessel functions of zero-order; A1 – A4 ; B1 – B4 are arbitrary functions to be determined from above boundary and continuity conditions.
ð2bÞ
in which, qz is the fluid discharge. The motion is assumed to be time harmonic with a factor pffiffiffiffi of eivt ; where v is frequency of the motion and i ¼ 21: For brevity, the term eivt is suppressed from all expressions in the sequel. The boundary conditions for the free and permeable surface at z ¼ 0 and 0 # r , 1 are
szi ðr; 0Þ ¼ 0
ð4eÞ
For a uniformly distributed vertical load of radius a acting in the z direction, it can be shown that 8 > < P0 ; r # a PðrÞ ¼ pr 2 ð5Þ > : 0; r.a
ð1dÞ
where ur and uz are the displacements of the solid matrix in the r and z directions, respectively; wr and wz are the fluid displacements relative to the solid matrix in the r and z directions, respectively; pf is the pore pressure; r rs and rf are mass densities of the bulk material, the soil and the pore fluid, respectively; r ¼ ð1 2 nÞrs þ nrf ; n is the porous ratio; l and G are Lame constants; k0d is dynamic permeability; 72 is the Laplace operator; e is the dilatation; overdots denote the derivatives of field variables with respect to time t. The constitutive relations of a homogenous poroelastic material can be expressed as
ð4bÞ
qz ðr; z Þ 2 qz ðr; z Þ ¼ 0 þ
2. Governing equations
ð4aÞ
ð3aÞ ð3bÞ
3. Single pile We now consider a cylindrical elastic pile of radius a and length L ða=L p 1Þ partially embedded in a poroelastic medium as shown in Fig. 1. Following Muki’s method, the system is decomposed into an extended poroelastic halfspace and fictitious pile as shown in Fig. 2. In what follows we treat the extended embedding medium as a threedimensional poroelastic continuum. In contrast, we regard the fictitious pile as a one-dimension elastic continuum as far as its constitutive law and equilibrium conditions are concerned [14].
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 1. Geometry of the pile and embedding medium: (a) fictitious pile; (b) extended poroelastic medium.
The governing equation for time harmonic axial motion in the fictitious pile is d2 up ðzÞ v2 rpp qðzÞ þ up ðzÞ þ ¼0 2 p AEpp Ep dz
ð7Þ
where up ðzÞ is the axial displacement of the fictitious pile; rpp is density of the fictitious pile; Epp is Young’s modulus of the fictitious pile; qðzÞ is the shear force per unit length in the z direction on the shaft of the pile Pp; A is area of the fictitious pile. Ep p ¼ Ep 2 Es
ð8aÞ
rpp ¼ rp 2 ð1 2 nÞrs
ð8bÞ
where EP and ES are Young’s modulus of the pile and soil, respectively; rp and rs are mass density of the pile and soil, respectively. The force– displacement relationship for the fictitious pile is dup ðzÞ Np ðzÞ ¼ Epp A dz
0,z,L
where Np ðz; tÞ ¼ Np ðzÞeivt is the axial pile force.
ð9Þ
Fig. 3. Vertical impedance of single pile in a poroelastic medium.
With the aid of displacement and strain influence functions due to a uniform body-force distribution over the closed disk, The component of displacement and strain in the fictitious pile are given by up ðzÞ ¼ ½N0 þ Np ð0Þ^uð0; zÞ 2 Np ðLÞ^uðL; zÞ 2
ðL 0
qðjÞ^uðj; zÞdj
ð10Þ
^ zÞ 2 Np ðLÞ1ðL; ^ zÞ 1ðzÞ ¼ ½N0 þ Np ð0Þ1ð0; 2
ðL 0
^ j; zÞdj qðjÞ1ð
ð11Þ
1^ is found to possess merely finite jump discontinuities at z ¼ j for each fixed z in the open interval ð0; LÞ: 1ðjþ ; zÞ 2 1ðj2 ; zÞ ¼ 2
ð1 2 2mÞð1 þ mÞ EAð1 2 mÞ
ð12Þ
Muki and Zeng adopted the condition that the axial strain of the fictitious pile is equal to the average of the extensional strain of the cross section of the extended half space. Accordingly, fictitious pile is governed by the stress –strain relation 1ðzÞ ¼
Fig. 2. The model of single pile in a poroelastic medium.
237
Np ðzÞ Epp A
ð13Þ
Fig. 4. Normalized impedance for vertically loaded 3 £ 3 pile group (L=d ¼ 15; EP =ES ¼ 1000; rS =rP ¼ 0:7).
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J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 7. Identification of data in Figs. 8–19.
Fig. 5. Impedances of 3 £ 3 pile groups embedded in a poroelastic medium ðEP =ES ¼ 1000Þ:
Manipulation of Eqs. (10) – (13) together with the displacement compatibility condition described above results in the following Fredholm’s integral equation of the second kind N ðzÞ ^ jþ 2 zÞ 2 1ð ^ j2 2 zÞ þ p Np ðzÞ½1ð EPp A ðL r p p Av 2 ð L ^ j; zÞ ›1ð ^ j; zÞdj þ N p ðjÞ dj 2 1ð ›j E Pp A 0 0 ðj ðL ^ j; zÞdj Np ðhÞdh þ rpp Av2 1ð 0 "0 1 ðL ›u^ ðj; 0Þ 1 rpp Av2 N p ðjÞ dj 2 c1 0 c 1 Ep p A ›j # ðL ðj u^ ðj; 0Þdj Np ðhÞdh 0
The axial force and pore pressure can be determined by NðzÞ ¼ 2 N0 s^ð0; zÞ þ
ðL 0 ðL 0
s^ðj; zÞdj
0 ðL 0
Np ðjÞ
0
ðj 0
rpp Av2 ›s^ðj; zÞ dj 2 ›j Epp A
Np ðhÞdh 2 rpp Av2 up ð0Þ
s^ðj; zÞdj
pf ðzÞ ¼N0 p^ ð0; zÞ 2 ðL
ðL
ðL
ð17aÞ
N p ðjÞ
0
p^ ðj; zÞdj
ðj 0
rpp Av2 ›p^ ðj; zÞ dj þ ›j Epp A
Np ðhÞdhrpp Av2 up ð0Þ
p^ ðj; zÞdj
ð17bÞ
0
^ zÞ þ rpp Av2 ¼ N0 1ð0;
ðL 0
^ j; zÞdj 1ð
1 N u^ ð0; 0Þ c1 0
ð14Þ
4. Vertical response of pile group
where c1 ¼ 1 2 rpp Av2
ðL 0
4.1. Impedance of pile group u^ ðj; 0Þdj
ð15Þ
Once the fictitious pile force Np ðzÞ is known, the top of displacement of the pile can be determined by " ðL rpp Av2 1 ›u^ ðj; 0Þ N0 u^ ð0; 0Þ 2 Np ðjÞ dj þ up ð0Þ ¼ c1 ›j Ep p A 0 # ðL ðj u^ ðj; 0Þdj Np ðhÞdh ð16Þ 0
With the superposition method the dynamic impedance of group of N identical piles individual connected through a rigid cap is calculated by superimposing the interactions factors between individual pile pairs. Dynamic group effects can be treated approximately using complex interaction factors. The interaction factor a between two piles is defined as displacement of reference pile due to load on adjacent
0
Fig. 6. Compliances of 3 £ 3 pile groups embedded in a poroelastic medium ðEP =ES ¼ 1000Þ:
Fig. 8. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 9. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
pile, normalized by displacement of reference pile under its own load. Based on Dobry and Gazetas’s method [17], for vertical oscillation, the interaction factor is written as
1 s 2ð1=2Þ b vs ivs aðs; 0Þ < pffiffi exp 2 s 2 CS CS 2 d
n X j¼1
uij ¼
1 X aij Fj KSV
ðj ¼ 1; 2; …nÞ
n X
Fj
impedance of pile group F K G V ¼ uG
ð21Þ
4.2. Axial fore and pore pressure
ð19Þ
where K SV is the normalized dynamic impedance of single pile; Fj is the load acting on the j pile heads; aij ðj ¼ 1; 2; …nÞ is the dynamic interaction factor between j pile and i pile, aij ¼ 1: With a rigid cap and these loads are not equal, the total loads of the pile group are F¼
Fig. 11. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
ð18Þ
where bs is hysteretic soil damping; CS is the shear wave velocity of soil; s is distance of two piles; d is single pile diameter. To generate group stiffness for the case of a rigid cap, the displacements of all heads must be equal to unity uG ¼ u i ¼
239
ð20Þ
j¼1
In order to analyze the axial fore and pore pressure of pile groups vary with along pile length, we consider the ith pile of N piles. In adopting the method of the compatibility condition for dynamic load transfer problems for poroelastic medium, Fredholm’s integral equation of the second kind of the ith pile can be expressed as N p ðzÞ Nip ðzÞ½1^ii ðjþ 2 zÞ 2 1^ii ðj2 2 zÞ þ i EPip Ai " n ðLj X rpjp Aj v2 ›1^ji ðj; zÞ þ Njp ðjÞ dj 2 ›j EPjp Aj 0 j¼1 ð Lj ðj ð Lj 1^ji ðj; zÞdj Njp ðhÞdh þ rppj Aj v2 1^ji ðj; zÞdj 0 0 "0 ›u^ ji ðj; 0Þ 1 ð Lj 1 rpjp Aj v2 Njp ðjÞ dj 2 c1 0 c1 Eppj Aj ›j ## ð Lj ðj u^ ji ðj; 0Þdj Njp ðhÞdh 0
Solving Eqs. (19) and (20) can obtain the displacement of pile group uG and load Fi of single pile. The dynamic
0
n ðLj X 1 1^ji ðj; zÞdj· Nj0 u^ ji ð0; 0Þ Nj0 1^ji ð0; zÞ þ rpjp Aj v2 ¼ c1 0 j¼1 ð22Þ The ith pile’s axial fore and pore pressure can be expressed as N ð Lj X ›s^ji ðj;zÞ Ni ðzÞ ¼ 2Nj0 s^ji ð0;zÞþ Njp ðjÞ dj ›j 0 j¼1
ðj rpjp Aj v2 ðLj s^ji ðj;zÞdj Njp ðhÞdh 2 rpjp Aj v2 upj ð0Þ Epjp Aj 0 0 ð Lj s^ji ðj;zÞdj ð23Þ
2
Fig. 10. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
0
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J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
Fig. 12. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
pfi ðzÞ ¼
N ð Lj X ›p^ ji ðj;zÞ Nj0 p^ ji ð0;zÞ2 Njp ðjÞ dj ›j 0 j¼1
ðj rpjp Aj v2 ðLj p^ ji ðj;zÞdj Njp ðhÞdh Epjp Aj 0 0 ð Lj 2 þrpjp Aj v upj ð0Þ p^ ji ðj;zÞdj þ
ð24Þ
0
5. Numerical results and discussions
Fig. 14. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
rf =rS ¼ 0:8; n ¼ 0:4; L=a ¼ 20; m ¼ 0:4: There is a reasonably good agreement between the present result and those of Zeng. The vertical impedances of 3 £ 3 pile groups obtained by the degeneration of a poroelastic medium to an ideal elastic medium are portrayed in Fig. 4 and compared with those obtained by Mamoon [5]. The impedances have been normalized with respect to the respective single-pile static stiffness multiplied by the number of piles in the group. The accuracy of the present formulation and its numerical results are confirmed by the above comparisons.
The normalized axial impedance K V ; normalized frequency a0; normalized axial force N z and normalized average pore pressure p f are, respectively, represented by K V ¼
P0 ; Gaup ðzÞ
a0 ¼
vd ; CS
NðzÞ N z ¼ ; P0
ð25Þ
Apf ðzÞ p f ¼ P0 For the vertical vibration of single pile embedded in a poroelastic medium, the results of this study are compared with those of Zeng [14]. Fig. 3 shows the comparison between the real and imaginary parts of the axial impedances for a pile. EP =ES ¼ 1000; rP =rS ¼ 1:2;
Fig. 15. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
Fig. 13. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 1000Þ:
Fig. 16. Axial forces of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
J.H. Wang et al. / Soil Dynamics and Earthquake Engineering 23 (2003) 235–242
241
Fig. 17. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
Fig. 19. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
In the remainder of this paper, the influence of pile stiffness ratio and spacing diameter ratio on the dynamic response of pile group in a poroelastic medium are investigated. The normalized properties are m ¼ 0:4; l ¼ G ¼ G0 ð1 þ 2bs ; iÞ; bs ¼ 0:05; n ¼ 0:4; rf =rS ¼ 0:8; rP =rS ¼ 1:2; L=d ¼ 20: Figs. 5 and 6 show the variation in the real and imaginary parts of the vertical pile group impedances of 3 £ 3 pile groups embedded in a poroelastic medium for s/d ¼ 2, 5,10 with respect to the normalized frequency parameter a0 (Fig. 7). The influence of pile stiffness ratio EP =ES and spacing diameter ratio s=d on the axial forces and pore pressures of 3 £ 3 pile groups in a poroelastic medium are shown in Figs. 8 –19 for low frequency range (v ¼ 1.64). Figs. 8 –10 and Figs. 14 –16 show the profiles of normalized axial forces N z along the pile length with EP =ES ¼ 1000; 100 and s/d ¼ 2, 5, 10. The outer piles carry greater loads than the central piles, with the corner piles carrying the most. The axial forces gradually decrease as the pile spacing diameter ratio increases. The axial forces on central piles of the square pile group are very small. It can also be seen that the axial force profiles have some dependence on pile stiffness ratio. The load transfer along the pile is rapid for smaller values of stiffness ratio, resulting in negligible load transfer at the base. Figs. 11 – 13 and Figs. 17 – 19 show the profile of normalized average pore pressures p f along the pile length with EP =ES ¼ 1000; 100 and s/d ¼ 2, 5, 10. The pore pressures generation is mostly confined to
the upper part and the lower part of the piles. The pore pressures on the outer of pile group are larger than those located central, with pore pressures on corner piles occurring the most. The pore pressures gradually decrease as the pile spacing ratio increases.
6. Conclusion The Muki method is used to evaluate the dynamic impedance, axial force and pore pressure of single pile in a poroelastic medium. The dynamic interaction factor approach successfully predicts the dynamic impedance of pile groups. In order to analyze axial force and pore pressure of pile group along pile length, the superposition method is used to obtain Fredholm’s integral equation of the second kind of the pile group in a poroelastic medium. Using numerical quadrature can solve the integral equation. Comparisons with the existing solutions available for single pile embedded in a poroelastic medium and pile group embedded in elastic medium confirms the accuracy of the proposed formulation. The solution presented may be extended to analyze the dynamic response of pile groups in layered saturated soils.
Acknowledgements The work presented in this paper was supported by the National Science Foundation of China, grant number is 59879012.This support is greatly appreciated. First author would like to thank Professor Radoslaw L. Michalowski for his kindly discussions during author’s visit at Department of Civil and Environmental Engineering, University of Michigan.
References
Fig. 18. Pore pressures of 3 £ 3 pile groups along depth ðEP =ES ¼ 100Þ:
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