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Dynamic torsional response of an end bearing pile in saturated poroelastic medium
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Computers and Geotechnics 35 (2008) 450–458 www.elsevier.com/locate/compgeo
Dynamic torsional response of an end bearing pile in saturated poroelastic medium Kuihua Wang a, Zhiqing Zhang b
a,*
, Chin Jian Leo b, Kanghe Xie
a
a Department of Civil Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China School of Engineering, University of Western Sydney, Locked Bag 1797 Penrith South DC, Sydney, NSW 1797, Australia
Received 13 October 2006; received in revised form 4 May 2007; accepted 28 June 2007 Available online 22 August 2007
Abstract A comprehensive analytical solution is developed in this paper to investigate the torsional vibration of an end bearing pile embedded in a homogeneous poroelastic medium and subjected to a time-harmonic torsional loading. The poroelastic medium is modeled using Biot’s two-phased linear theory and the pile using one-dimensional elastic theory. By using the separation of variables technique, the torsional response of the soil layer is solved first. Then based on perfect contact between the pile and soil, the dynamic response of the pile is obtained in a closed form. Numerical results for torsional impedance of the soil layer are presented first to portray the influence of wave modes, slenderness ratio, pile–soil modulus ratio and poroelastic properties. A comparison with the plane strain theory is performed. The selected numerical results are obtained to analyze the influence of the major parameters on the torsional impedance at the level of the pile head. Finally, the dynamic torsional impedance of this study is compared with that for floating pile in elastic soil. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Saturated poroelastic medium; Torsional vibration; End bearing pile; Impedance; Dynamic response
1. Introduction The pile–soil interaction is a complicated dynamic contact problem. Pile foundations of machines and structures are often exposed to dynamic loads such as traffic, machinery, wind or impact of switch current. In most cases, piles are liable to undergo some degree of torsion due to the eccentricity of these applied loads and the study of the pile subjected to transient torsional load is important in the context of dynamic foundation design. Over the past many years, several studies have been undertaken to investigate the torsional impedance of the pile. For example, Poulos [1] studied the static torsional response of the pile. Stoll [2] conducted experimental investigations on pile under static torsional loading. Randolph [3] analyzed the torsional
*
Corresponding author. Tel.: +86 571 87952163; fax: +86 571 87952165. E-mail address: [email protected] (Z. Zhang). 0266-352X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.06.013
stiffness of the pile embedded in elastic soil with either homogeneous modulus, or modulus proportional to depth. Guo and Randolph [4] studied the torsional response of piles in non-homogeneous soil media. Novak and Howell [5] analyzed the torsional vibration of an end bearing pile embedded in viscoelastic soil medium. Rajapakse and Shah [6] solved the axisymmetric torsional vibrations of an elastic pile and hemispherical foundation embedded in homogeneous elastic half-space. Using Green’s functions for a time-harmonic buried loading configuration, Rajapakse and Shah [7] also investigated the axial, lateral, rocking, coupled and torsional impedances of an elastic pile embedded in homogeneous soil. With a coupled FEM–BEM approach, Tham et al. [8] studied the torsional vibration of single piles embedded in a layered half-space. Militano and Rajapakse [9] used Laplace transform to derive the frequency domain solutions for an elastic pile subjected to transient torsional and axial loading in multi-layered elastic soil. Furthermore, they obtained the time-domain solutions by means of numerical inverse Laplace transform
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
presented by Stefest [10] to investigate the time-domain characteristics of the response. Most of the previous studies on piles had regarded the material of the soil medium as being a single phase continuum. However, in practice, soil is generally a multi-phase medium which reduces to a two-phase medium in its fluid-saturated state. As early as 1956, Biot [11] presented a propagation theory of elastic waves and its general solutions for fluid-saturated porous medium. The theory has wide applications in engineering, including the dynamic response of piles in saturated medium. Zeng and Rajapakse [12] analyzed the timeharmonic response of an axially loaded elastic pile embedded in a homogeneous poroelastic medium. Chen and Wang [13] studied the torsional vibration of an elastic foundation bonded to the surface of the saturated half-space. Using Hankel integral transform and integral equation techniques, Cai et al. [14] analyzed the dynamic response of a cylindrical elastic pile embedded in a homogeneous poroelastic medium halfspace and subjected to torsional loading, but this solution was only suitable to the floating pile and required a formidable computational effort. Notwithstanding the work above, it appears that no analytical solution corresponding to the torsional vibration characteristics of an end bearing pile embedded in saturated soil has yet been reported in the literature until now. Moreover, in many cases, the pile may rest on a rigid bedrock(usually named as an end bearing pile) and the study of vibration characteristics of an end bearing pile is important in the guidance for engineering application. In light of this, the objective of this paper is to derive a closed form analytical solution for analyzing the dynamic response of an end bearing pile subjected to time-harmonic torsional load in a fluid-saturated soil. Using the solution developed, a parametric study has been undertaken to assess the influence of various soil and pile parameters on the torsional impedance of the pile. The results of the parametric study are also presented in this paper. 2. Governing equations and general solution The problem studied in this paper is the torsional vibration of a pile embedded in saturated soil medium and the geometric model is shown in Fig. 1. The main assumptions adopted in this paper are: (1) The pile is vertical, elastic, end bearing, circular in cross section and has a perfect contact with the surrounding soil; (2) the soil is a linearly elastic, homogeneous and isotropic layer, and the soil medium is a two-phase material consisting of soil grains and fluid. The layer overlies a rigid bedrock; (3) the vibration is harmonic; (4) the free surface of the soil has no normal or shear stresses and there is no displacement occurring at the bottom of the layer; (5) Only the circumferential displacements are considered. 2.1. Dynamic soil reactions for the saturated soil According to the dynamic consolidation theory of Biot for a completely saturated medium, the dynamic equilib-
451
T0 eiωt
r
z
H
saturated soil
Pile
f ( z)
2r0
rigid base
Fig. 1. Model of soil–pile interaction.
rium equation for the saturated soil layer in polar coordinate system can be expressed as: orrh orhz rrh o2 o2 þ þ2 ¼ q 2 uh ðr; z; tÞ þ qf 2 wh ðr; z; tÞ or oz r ot ot
ð1Þ
where rrh and rhz are the non-zero stresses, uh(r, z, t) and vh(r, z, t) are, respectively, the circumferential displacements of the soil skeleton and that of the pore fluid, wh(r, z, t) = / [vh(r, z, t) uh(r, z, t)] is the circumferential displacements of pore fluid relative to the soil skeleton; / is the porosity. qs and qf are the mass densities of soil particle and fluid, respectively; q = (1 /)qs + /qf, the mass density of soil; To solve this problem, Eq. (1) can be expressed in terms of displacements uh(r, z, t) and wh(r, z, t) as: lr2 uh ðr; z; tÞ
uh ðr; z; tÞ o2 o2 l ¼ q u ðr; z; tÞ þ q wh ðr; z; tÞ h f r2 ot2 ot2 ð2Þ
where l is the Lame constant of the solid skeleton; 2 2 2 r2 ¼ oro 2 þ 1r oro þ ozo 2 ¼ 1r oro r oro þ ozo 2 . The pore fluid equilibrium equation can be written as: g owh ðr; z; tÞ o2 q o2 1 oP þ qf 2 uh ðr; z; tÞ þ f 2 wh ðr; z; tÞ ¼ k ot r oh ot / ot ð3Þ where P is excess pore water pressure; g and k are the fluid viscosity and intrinsic permeability of soil, respectively. Due to the symmetry of the pure torsional vibration of medium about the vertical axis, the motion is independent of angle h. For the harmonic motion, uh(r, z, t) = uh(r,z)eixt ixt ixt ixt = uhpeffiffiffiffiffiffi ffi and wh(r, z, t) = wh(r,z)e = wh e , where i ¼ 1, x is circular frequency of excitation and t is time. Then, Eqs. (2) and (3) can be written as:
452
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
uh l ¼ ðqx2 uh þ qf x2 wh Þ r2 g q ðiwÞwh qf x2 uh f x2 wh ¼ 0 k /
lr2 uh
ð4Þ ð5Þ
Substituting Eq. (5) into Eq. (4), the equilibrium equation of circumferential motion uh can be written as: uh x2 qf x r 2 uh 2 ¼ qþ ð6Þ uh ig=ðkqf Þ x=/ r l Variables r and z can be separated by assuming uh ¼ RðrÞZðzÞ
ð7Þ
The substitution for uh from Eq. (7) into Eq. (6) results in: 1 0 1 x2 qf x 00 00 qþ RZ R Z þ R Z þ RZ 2 RZ ¼ l r r ig=ðkqf Þ x=/ ð8Þ Eq. (8) can now be split into two ordinary differential equations: d2 Z þ h2 Z ¼ 0 dz2 r2
ð9Þ
d2 R dR ð1 þ q2 r2 ÞR ¼ 0 þr 2 dr dr
ð10Þ
where constants h and q satisfy the following relationship: x2 qf x 2 2 qþ q ¼h ð11Þ ig=ðkqf Þ x=/ l
The shear stress amplitude srh corresponding to Eq. (16) can be expressed as: 1 X ouh uh Dn qn K 2 ðqn rÞ cosðhn zÞ ð18Þ srh ¼ l ¼ l or r n¼1 where K2(qnr) denotes the modified Bessel functions of the second kind of the second order. It is mathematically convenient at this stage to introduce the dimensionless quantities: rffiffiffi r0 r z q n ¼ Hhn ¼ p ð2n 1Þ; r ¼ ; z ¼ ; r0 ¼ ; a0 ¼ r0 x; h H H H l 2 q gr0 f ¼ f ; b ¼ pffiffiffiffiffiffi qn ¼ Hqn ; q q k ql where r0 and H are the radius and length of the pile, respectively. After introducing the dimensionless variables, Eq. (17) becomes: f a0 a20 q 2 2 qn ¼ hn 2 1 þ ð19Þ r0 ib= qf a0 =/ Assuming that the pile and soil layer have no separation, the circumferential displacement uh at the pile–soil interface and the resisting moment p(z) per unit of length of the pile can be expressed respectively: 1 X uh ðr0 ; zÞ ¼ Dn K 1 ðqnr0 Þ cosðhnzÞ ð20Þ pðzÞ ¼
ð12Þ
RðrÞ ¼ AK 1 ðqrÞ þ BI 1 ðqrÞ
ð13Þ
where I1(qr) and K1(qr) are the modified Bessel functions of the first and second kind of the first order, respectively; A,B, C and D can be obtained from the boundary conditions. Referring to Fig. 1, it can be observed that the boundary conditions of the soil layer can be written as: uh ðr ! 1; zÞ ¼ 0
ð14Þ
shz ðr; 0Þ ¼ 0;
ð15Þ
uh ðr; H Þ ¼ 0
Furthermore, it is noted from expression (14) that I1(qr) ! 1 when r ! 1, so constant B should vanish to zero. The expression (15) requires C = 0 and hn ¼ ð2n1Þp where n = 1, 2H 2, 3 . . .. Then the general solution of Eq. (6) can be written in a series expansion as: 1 X Dn K 1 ðqn rÞ cosðhn zÞ ð16Þ uh ¼ n¼1
q2n
¼
h2n
x2 qf x qþ ig=ðkqf Þ x=/ l
and Dn(n = 1, 2, 3,. . .) are a series of constants.
ð17Þ
r20 srh dh ¼ 2pr0 lr0
1 X
Dn qn K 2 ðqnr0 Þ cosð hnzÞ
n¼1
ð21Þ Following the definition of Novak et al. [15], the complex torsional impedance related to a unit of length of the pile can be expressed as: k w ¼ 2plr0
qn K 2 ðqnr0 Þr0 K 1 ðqnr0 Þ
ð22Þ
The torsional impedance can be further expressed in terms of its real and imaginary parts as: k w ¼ lr0 ðS w1 þ iS w2 Þ
ð23Þ
where all except i is real. The dimensionless parameters Sw1 and Sw2 represent real stiffness and damping parts of the soil reaction, respectively. 2.2. Impedance of a pile For a pile subjected to harmonic torsional loading, the twist angle, U(z,t) = U(z)eixt is governed by the following one-dimensional equation of motion: Gp
where
2p 0
It is not difficult to show that the general solutions of Eqs. (9) and (10) can be expressed as: ZðzÞ ¼ C sinðhzÞ þ D cosðhzÞ
n¼1
Z
o2 f ðzÞ ixt o2 ixt bUðzÞe c þ 4 e ¼ q ½UðzÞeixt p r20 oz2 ot2
ð24Þ
where Gp, qp, and U(z) are the shear modulus, mass density and the twist angle amplitude at depth z of the pile, respectively; f(z) denotes amplitude of the shear stress related to a unit area of the pile shaft applied by the soil.
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
Due to a perfect connection between the pile and the soil, the value of contact traction f(z) should be equal to the shear stress at r = r0 of the soil and can be expressed as: o uh ðr; zÞ ½uh ðr; zÞ f ðzÞ ¼ srh jr¼r0 ¼ l or r r¼r0
¼ l
1 X
Dn qn K 2 ðqn r0 Þ cosðhn zÞ
ð25Þ
parts. The first part is free vibration characteristics of the pile, as shown in the first and second terms in the righthand side of Eq. (27). Because of the reaction from the soil, the second part represents the forced vibration characteristics of the pile, as is shown in the third term in the righthand side of Eq. (27). Substituting U(z) from Eq. (27) into Eq. (26) yields:
n¼1
The substitution for f(z) from Eq. (25) into Eq. (24) results in: 1 d2 UðzÞ qp x2 UðzÞ 4l X þ ¼ Dn qn K 2 ðqn r0 Þ cosðhn zÞ ð26Þ dz2 Gp r20 n¼1 Gp The solution of Eq. (26) is: rffiffiffiffiffiffi rffiffiffiffiffiffi qp qp UðzÞ ¼ a1 cos xz þ a2 sin xz Gp Gp 1 X þ wn cosðhn zÞ
453
wn ¼
4lqn K 2 ðqn r0 ÞDn
q x2 Gp r20 h2n Gp p
ð28Þ
Since a perfect connection between the pile and the soil, the circumferential displacement of the pile must be the same as that of the soil given by Eq. (20). Therefore, the successive condition at the pile–soil interface can be written as: r0 /ðzÞ ¼ uh jr¼r0
ð27Þ
n¼1
where a1 and a2 are the coefficients which remain to be determined later from the boundary conditions. wn (n = 1, 2, 3. . .) are also a series of constants. During the vibration, the pile and the soil interact, so the vibration characteristics should be expressed in two
ð29Þ
Substituting Eqs. (16) and (27) into Eq. (29) leads to: ( ) rffiffiffiffiffiffi rffiffiffiffiffiffi X 1 qp qp xz þ a2 sin xz þ wn cosðhn zÞ r0 a1 cos Gp Gp n¼1 ¼
1 X
Dn K 1 ðqn r0 Þ cosðhn zÞ
ð30Þ
n¼1
13.0
(a) Stiffness
16
12.5
(a) Stiffness n =1 n =3 n =5
15
Sw1
Sw1
14
b b b b
12.0
13
11.5
=50 =100 =1000 =10000
11.0 12
10.5 11
10.0 10 9 0.0
9.5 0.0 0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
a0
3.0
a0 18
20
(b) Damping (b) Damping
15
16
Sw2
Sw2
12
12
9
8
6
4
3
0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
a0 Fig. 2. Torsional stiffness and damping parameters for various modes n.
0 0.0
0.5
1.0
1.5
a0 Fig. 3. Torsional stiffness and damping parameters for various poroelastic material properties b (/ = 0.4, qf ¼ 0:5, H/r0 = 20, mode number n = 1).
454
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
By invoking the orthogonality of eigenfunctions cos(hn z)(n = 1, 2, 3 . . .), multiplying cos(hnz) on both sides of Eq. (30), and then integrating over the interval z = [0,H], the undetermined coefficient Dn is found to be: Dn ¼
Z
1 Ln En
) 1 X UðzÞ ¼ a1 cosðkzÞ þ nn cosðhnzÞ n¼1 ( ) 1 X þ a2 sinðkzÞ fn cosðhnzÞ
ð31Þ
P cosðhn zÞdz:
qffiffiffiffi qp xz P ¼ a1 cos where Ln ¼ 0 cos2 ðhn zÞdz ¼ H2 , Gp
qffiffiffiffi qp 4lq K ðq r Þ þa2 sin xz , En ¼ r10 K 1 ðqn r0 Þ þ G r nðh22 qnp x0 2 Þ . Gp RH
p 0
n
Gp
It is again mathematical convenient at this stage to introduce the following dimensionless quantities: ¼ Gp =l; l
rffiffiffiffiffi rffiffiffiffiffiffi qp p 1 q xz ¼ a0z ¼ kz; r0 l Gp
4 qn K 2 ð qnr0 Þ ; En ¼ r0 En ¼ K 1 ð qnr0 Þ þ 2 r0 ðhn l k2 Þ
where
1 1 nn ¼ mn sinðk hn Þ þ sinðk þ hn Þ k hn k þ hn
1 1 fn ¼ mn fcosðk þ hn Þ 1g þ fcosðk hn Þ 1g k þ hn k hn
mn ¼
ð32Þ
Now, the dimensionless boundary conditions at the base of and top of the pile are: dUðzÞ T 0H Uðz ¼ 1Þ ¼ 0; ¼ ð35Þ dz G I where I p ¼ pr40 =2 denotes polar moment inertia of the pile. T0 eixt is the harmonic excitation force acting at the pile 15
(a) Stiffness H/r0=10 H/r0=20 H/r0=40 H/r0=100 plane strain case
12.0 11.5
13
11.0
12
10.5
11
10.0
10
1.0
1.5
the result of this paper the result of Novak [5]
14
Re (kT)
12.5
0.5
p p
z¼0
13.0
Sw1
4qn K 2 ðqnr0 Þ r0 l ðh2n k2 ÞEn ð34Þ
The amplitude of the twist angle of the pile is then given by:
9.5 0.0
ð33Þ
n¼1
H
0
p ¼ qp =q; q
(
9 0.0
2.0
a0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
a0
12
16
(b) Damping 10 12
Im (kT)
Sw2
8 6
8
4 4
2 0 0.0
0.5 .5
1.0
a0
1.5
2.0
Fig. 4. Torsional stiffness and damping parameters for different slenderf ¼ 0:5, b ¼ 1000, mode number n = 1). ness ratio H/r0 (/ = 0.4, q
0 0.0
0.5
1.0
1.5
a0 Fig. 5a. Comparison of the torsional impedance of this paper with that of p ¼ 1:3; qf = 0; l ¼ 300). Novak [5] for an elastic soil (H/r0 = 40; q
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
head. By substituting Eq. (35) into Eq. (33), the variables, a1, a2 are obtained: a1 ¼
T 0H tan k k Gp I p
ð36aÞ
T 0H k Gp I p
ð36bÞ
a2 ¼
so that the dimensionless torsional impedance at the top end of the pile is given by: kT ¼ ¼
3T 0 3 16lr0 Uðz
¼ 0Þ
3p l kr 0
: 1 1 P P 32 1 þ nn tan k þ fn n¼1
ð37Þ
n¼1
3. Numerical results and discussion In this section, numerical results are presented to highlight the dynamic performance of pile–soil system by using Eqs. (23) and (37). Firstly, the torsional impedance of the soil layer is analyzed. Then the torsional impedance at the top end of the pile is investigated.
3.1. Torsional impedance of soil layer Fig. 2 shows the effect of the mode n on soil reactions. f ¼ 0:5, The parameters used in calculation are: / = 0.4, q b ¼ 1000, H/r0 = 20. The definition of the torsional impedance is based on Novak et al. [15]. The real and imaginary parts of the torsional impedance of the soil represent the real stiffness and damping of the soil, respectively. It can be observed from Fig. 2a that the stiffness depends much on the mode number in the dimensionless frequency range of 0 < a0 < 1.0. When the excitation frequency increases further, the influence of the mode number becomes less important. Fig. 2b shows the variations of damping with the excitation frequency and mode number. The damping represents the dissipation of the energy. When the frequency is in the low range, the damping is very little. However, with the further increase of frequency, the damping becomes proportional to the excitation frequency but the mode number has negligible influence on the response. This behavior is similar to the vertical vibration of the soil reported by Nogami and Novak [16]. Fig. 3 shows the variations of torsional impedance of the saturated soil with the different poroelastic parameter b. It can be seen that the stiffness parameter Sw1 is almost
90
30
80 b =10 b =20 b =100 b=1000
24
μ =10 μ =100 μ = 500 μ =1000 μ = 2000 μ = 5000
70 60
Re (kT)
27
Re (k T)
455
21
50 40 30
18
20 15 12 0.0
10 0.5
1.0
1.5
2.0
2.5
0 0.0
3.0
0.5
1.0
a0
1.5
2.0
2.5
0.5
1.0
a0
1.5
2.0
2.5
a0 20
28 24
16
20
Im (kT)
12
Im (kT)
16 12
8
8
4 4 0 0.0
0.5
1. 1.0
1.5
a0
2.0
0.5
3.0
Fig. 5b. Torsional impedance of elastic pile in poroelastic medium (H/ p ¼ 1:3; q f ¼ 0:5; / = 0.4). r0 = 20; q
0 0.0
(H/r0 = 20; Fig. 6. Variation of torsional impedance of elastic pile with l f ¼ 0:5; / = 0.4; b ¼ 100). p ¼ 1:3; q q
456
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
independent of b in the frequency range a0 = 0 to 1.0. Then with the increase of the frequency, the stiffness parameter Sw1 decreases with increasing b. However, Sw1 shows negligible changes for b > 1000 when compared with b ¼ 1000. It is also noted that b has negligible influence on the damping parameter Sw2. The effect of the slenderness ratio on the response is shown in Fig. 4. In order to compare this solution with f ¼ 0 is set the plane strain solution [15], the parameter q to zero to simulate the special case of the elastic solution in homogenous soil. It can be seen that the slenderness ratio H/r0 has marked influence on the stiffness parameter Sw1 and damping parameter Sw2 in the range of low frequencies. As the frequency increases, the torsional impedance becomes less dependent on H/r0 and is closer to the plane strain case. It can also be seen that the torsional impedance obtained by present solution is in good agreement with the plane strain case, when the H/r0 is greater than about forty. In other words, the plane strain solution is accurate, when H/r0 > 40. It is also found that the plane strain solution underestimates the stiffness in the range of low frequencies when H/r0 < 20. The effect of slenderness
ratio on damping is small for a0< 1.0 and negligible for a0 > 1.0. 3.2. Impedance of end bearing pile In the analysis of structures supported by pile foundations, the complex impedance at the pile head is often used to gauge the vibration characteristics of the pile–soil system. The real part of the complex impedance represents the real stiffness, while the imaginary part of the complex impedance represents the radial damping of the system. The complex impedance is influenced by the frequency of excitation x, pile slenderness ratio H/r0, pile modulus ratio and other material properties. In order to verify the corl rectness of the solution proposed in this paper, the problem is reduced to the torsional vibration of an end bearing pile embedded in elastic medium and compared with the correp ¼ 1:3, ¼ 300, q sponding result of Navak [5]. H/r0 = 40, l qf = 0. Fig. 5a shows that the real and imaginary parts of this paper are close to that of Novak’s. Four groups of poroelastic materials are considered in the following analysis, the dimensionless properties are: p ¼ 1:3, q f ¼ 0:5 and / = 0.4, with ¼ 1000, q H/r0 = 20, l b ¼ 10, 20, 100 and 1000, respectively. The influence of
8.6 22
8.5
a0=0.1 a0=0.3 a0=0.5
8.3
a0=0.1 a0=0.3 a0=0.5
21
20
8.2
Re (kT)
Re (kT)
8.4
8.1
19
8.0 18
7.9 7.8
0
10
20
30
40
50
60
70
80
90
100
17
H/r0
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100 00
H/r0
0.6
1.5
0.5 1.2
Im (k kT)
0.4
Im (kT)
0.9
0.3
0.6
0.2
0.3
0.1 0.0
0
10
20
30
40
50
60
70
80
90
100
H/r0 Fig. 7a. Variation of torsional impedance of elastic pile with slenderness f ¼ 0:5: / = 0.4; b ¼ 100; l qp ¼ 1:3; q ¼ 100). ratio H/r0 (
0.0
0
10
20
30
40
50
H/r0
Fig. 7b. Variation of torsional impedance of elastic pile with slenderness f ¼ 0:5: / = 0.4; b ¼ 100; l qp ¼ 1:3; q ¼ 500). ratio H/r0 (
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458 25 the result of Militano [9], H/r0=10 the result of this paper, H/r0=10 the result of Militano [9], H/r0=40 the result of this paper, H/r0=40
Re (kT)
20
15
10
5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
1.5
2.0
2.5
3.0
a0
30 25 20
Im (kT)
the poroelastic parameter b on the complex impedance with the increasing dimensionless frequency a0 is shown in Fig. 5b. It can be seen from Fig. 5 that the real and imaginary parts of the torsional impedance shows the negligible variations with the frequency in the range a0 = 0–1.5 for all four materials. As the frequency increases, the poroelastic medium with a larger value of b has lower stiffness and higher dissipation of energy. However, Re (kT) and Im (kT) show negligible changes for b > 100 when compared with b ¼ 100. and a0 on the The influence of pile–soil modulus ratio l torsional stiffness is shown in Fig. 6. It can be seen that has a significant influence the pile–soil modulus ratio l on the response. The real parts of torsional impedance with the decrease with increasing frequency for a given l increases. It is decrease becoming more pronounced as l also noted that the real stiffness of the pile becomes higher increases for a given a0. The imaginary parts of the toras l sional impedance increase with increasing frequency for both flexible and rigid piles. For a given a0, the radial . Such behaviors for damping increases with increasing l an end bearing pile are similar to that of the pile embedded in homogenous soil layer [6].
457
15 10
38
5
36
a0=0.1 a0=0.3 a0=0.5
Re (kT)
34
0 0.0
32
0.5
1.0
a0
Fig. 8. Comparison of torsional impedance of end bearing pile with that of floating pile by Militano of [9] for an elastic soil ( qp ¼ 1:3; qf = 0; ¼ 500). l
30 28 26 24 0
10
20
30
40
50
60
70
80
90
100
H/r0 2.25 2.00 1.75
Im (kT)
1.50 1.25 1.00
Figs. 7 shows the variations of the dimensionless torsional impedance with slenderness ratio H/r0 for different ¼ 100, 500 and 1000. It is seen that the real a0 and for l parts of the dimensionless torsional impedance show a steep decrease with the slenderness ratio and the imaginary parts show a steep increase. Both the real and imaginary parts asymptote to constant values. Thus there is a critical and a0, above which the real slenderness ratio for given l and imaginary parts could be deemed to have converged to constant values. It is also noted that the critical slender ratio and the critical ness ratio increases with increasing l slenderness ratio is less than 40 as the pile becomes much stiffer.
0.75
3.3. Comparison with solution for a floating pile in elastic soil
0.50 0.25 0.00 0
10
20
30
40
50
60
70
80
90
100
H/r0 Fig. 7c. Variation of torsional impedance of elastic pile with slenderness f ¼ 0:5: / = 0.4; b ¼ 100; l qp ¼ 1:3; q ¼ 1000). ratio H/r0 (
In this section, the torsional impedance of this paper is compared with that of Militano’s [9]. Because Militano studies the torsional vibration characteristics of floating pile embedded in elastic soil, so the problem of this study is reduced to the torsional vibration of pile embedded in elastic soil in the subsequent analysis. And two cases are
458
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
considered to investigate the difference and sameness ¼ 500, between these two solutions, the parameters are: l p ¼ 1:3, qf = 0, with H/r0 = 10 and 40, respectively. q Fig. 8 shows the comparison of torsional impedance of end bearing pile with that of floating pile by Militano. It can be seen that these two solutions have great difference when H/r0 = 10. However, as the increase of the slenderness ratio, such as H/r0 = 40, these two solutions have good agreement. Therefore, the torsional vibration characteristics of an end bearing pile are close to that of a floating pile for large slenderness ratio. 4. Conclusions By considering the end bearing pile embedded in a saturated soil as a pile–soil dynamic interaction problem, an analytical solution for the torsional impedance of the saturated soil layer and pile has been derived. An extensive parameters study has been conducted to investigate the influence of major parameters. From the numerical results, conclusions can be drawn that both the real and imaginary parts of the torsional impedance of the soil depend much on the mode number and slenderness ratio in the low range of frequency, but have negligible influence in the higher frequency range. The poroelastic parameter b has some influence on the real part (stiffness) of the torsional impedance of the soil, but has negligible effect on the imaginary part (damping). The comparison with the plane strain theory shows that plane strain solution underestimates stiffness of the soil and overestimates the damping of the soil at low excitation frequencies. The selected numerical results for torsional impedance of elastic pile show that modulus ratio and slenderness ratio greatly influence the response. While the poroelastic property b has negligible effect on the response in the low frequency range, it has some influence on the response in the higher frequencies. Furthermore, the influence of the slenderness ratio has a critical value, such that when the slenderness ratio is above the critical value, the torsional
impedance of the pile converges to a constant and the corresponding torsional vibration characteristics is close to that of a floating pile. Acknowledgements This research is supported by the National Natural Science Foundation of China (Grant No. 50279047). References [1] Poulos HG. Torsional response of piles. J Geotech Eng Div, ASCE 1975;101(GT10):1019–35. [2] Stoll UW. Torque shear test on cylindrical friction piles. J Civil Eng Div, ASCE 1972;42(4):63–4. [3] Randolph MF. Piles subjected to torsion. J Geotech Eng Div, ASCE 1981;107(GT8):1095–111. [4] Guo W, Randolph MF. Torsional piles in non-homogeneous media. Comput Geotech 1996;19(4):265–87. [5] Novak M, Howell JF. Torsional vibration of pile foundations. J Geotech Eng Div, ASCE 1977;103(GT4):271–85. [6] Rajapakse RKND, Shah AH, et al. J Earthquake Eng Struct Dyn 1987;15:279–97. [7] Rajapakse RKND, Shah AH. Impedance curves for an elastic pile. Soil Dyn Earthquake Eng 1989;8(3):145–52. [8] Tham LG, Cheung YK, Lei ZX. Torsional dynamic analysis of single piles by time-domain BEM. J Sound Vibr 1994;174(4):505–19. [9] Militano G, Rajapakse RKND. Dynamic response of a pile in a multi-layered soil to transient torsional and axial loading. Geotechnique 1999;49(1):91–109. [10] Stefest H. Numerical inversion of Laplace transforms. Commun Assoc Comput Mach B 1970;13(1):47–9. [11] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. I: Low-frequency range. J Acoust Soc Am 1956;28:168–78. [12] Zeng X, Rajapakse RKND. Dynamic axial load transfer from elastic bar to poroelastic medium. J Eng Mech 1999;125(9):1048–55. [13] Chen L, Wang G. Torsional vibration of elastic foundation on saturated media. Soil Dyn Earthquake Eng 2002;22(3):223–7. [14] Cai Y, Chen G, Xu C, et al. Torsional response of pile embedded in a poroelastic medium. Soil Dyn Earthquake Eng 2006;26(12):1143–8. [15] Novak M, Nogami T, Aboul-Ella F. Dynamic soil reactions for plane strain case. J Eng Mech Div, ASCE 1978;104(EM4):953–9. [16] Nogami T, Novak M. Soil–pile interaction in vertical vibration. J Earthquake Eng Struct Dyn 1976;4:277–93.
Computers and Geotechnics 35 (2008) 450–458 www.elsevier.com/locate/compgeo
Dynamic torsional response of an end bearing pile in saturated poroelastic medium Kuihua Wang a, Zhiqing Zhang b
a,*
, Chin Jian Leo b, Kanghe Xie
a
a Department of Civil Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China School of Engineering, University of Western Sydney, Locked Bag 1797 Penrith South DC, Sydney, NSW 1797, Australia
Received 13 October 2006; received in revised form 4 May 2007; accepted 28 June 2007 Available online 22 August 2007
Abstract A comprehensive analytical solution is developed in this paper to investigate the torsional vibration of an end bearing pile embedded in a homogeneous poroelastic medium and subjected to a time-harmonic torsional loading. The poroelastic medium is modeled using Biot’s two-phased linear theory and the pile using one-dimensional elastic theory. By using the separation of variables technique, the torsional response of the soil layer is solved first. Then based on perfect contact between the pile and soil, the dynamic response of the pile is obtained in a closed form. Numerical results for torsional impedance of the soil layer are presented first to portray the influence of wave modes, slenderness ratio, pile–soil modulus ratio and poroelastic properties. A comparison with the plane strain theory is performed. The selected numerical results are obtained to analyze the influence of the major parameters on the torsional impedance at the level of the pile head. Finally, the dynamic torsional impedance of this study is compared with that for floating pile in elastic soil. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Saturated poroelastic medium; Torsional vibration; End bearing pile; Impedance; Dynamic response
1. Introduction The pile–soil interaction is a complicated dynamic contact problem. Pile foundations of machines and structures are often exposed to dynamic loads such as traffic, machinery, wind or impact of switch current. In most cases, piles are liable to undergo some degree of torsion due to the eccentricity of these applied loads and the study of the pile subjected to transient torsional load is important in the context of dynamic foundation design. Over the past many years, several studies have been undertaken to investigate the torsional impedance of the pile. For example, Poulos [1] studied the static torsional response of the pile. Stoll [2] conducted experimental investigations on pile under static torsional loading. Randolph [3] analyzed the torsional
*
Corresponding author. Tel.: +86 571 87952163; fax: +86 571 87952165. E-mail address: [email protected] (Z. Zhang). 0266-352X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.06.013
stiffness of the pile embedded in elastic soil with either homogeneous modulus, or modulus proportional to depth. Guo and Randolph [4] studied the torsional response of piles in non-homogeneous soil media. Novak and Howell [5] analyzed the torsional vibration of an end bearing pile embedded in viscoelastic soil medium. Rajapakse and Shah [6] solved the axisymmetric torsional vibrations of an elastic pile and hemispherical foundation embedded in homogeneous elastic half-space. Using Green’s functions for a time-harmonic buried loading configuration, Rajapakse and Shah [7] also investigated the axial, lateral, rocking, coupled and torsional impedances of an elastic pile embedded in homogeneous soil. With a coupled FEM–BEM approach, Tham et al. [8] studied the torsional vibration of single piles embedded in a layered half-space. Militano and Rajapakse [9] used Laplace transform to derive the frequency domain solutions for an elastic pile subjected to transient torsional and axial loading in multi-layered elastic soil. Furthermore, they obtained the time-domain solutions by means of numerical inverse Laplace transform
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
presented by Stefest [10] to investigate the time-domain characteristics of the response. Most of the previous studies on piles had regarded the material of the soil medium as being a single phase continuum. However, in practice, soil is generally a multi-phase medium which reduces to a two-phase medium in its fluid-saturated state. As early as 1956, Biot [11] presented a propagation theory of elastic waves and its general solutions for fluid-saturated porous medium. The theory has wide applications in engineering, including the dynamic response of piles in saturated medium. Zeng and Rajapakse [12] analyzed the timeharmonic response of an axially loaded elastic pile embedded in a homogeneous poroelastic medium. Chen and Wang [13] studied the torsional vibration of an elastic foundation bonded to the surface of the saturated half-space. Using Hankel integral transform and integral equation techniques, Cai et al. [14] analyzed the dynamic response of a cylindrical elastic pile embedded in a homogeneous poroelastic medium halfspace and subjected to torsional loading, but this solution was only suitable to the floating pile and required a formidable computational effort. Notwithstanding the work above, it appears that no analytical solution corresponding to the torsional vibration characteristics of an end bearing pile embedded in saturated soil has yet been reported in the literature until now. Moreover, in many cases, the pile may rest on a rigid bedrock(usually named as an end bearing pile) and the study of vibration characteristics of an end bearing pile is important in the guidance for engineering application. In light of this, the objective of this paper is to derive a closed form analytical solution for analyzing the dynamic response of an end bearing pile subjected to time-harmonic torsional load in a fluid-saturated soil. Using the solution developed, a parametric study has been undertaken to assess the influence of various soil and pile parameters on the torsional impedance of the pile. The results of the parametric study are also presented in this paper. 2. Governing equations and general solution The problem studied in this paper is the torsional vibration of a pile embedded in saturated soil medium and the geometric model is shown in Fig. 1. The main assumptions adopted in this paper are: (1) The pile is vertical, elastic, end bearing, circular in cross section and has a perfect contact with the surrounding soil; (2) the soil is a linearly elastic, homogeneous and isotropic layer, and the soil medium is a two-phase material consisting of soil grains and fluid. The layer overlies a rigid bedrock; (3) the vibration is harmonic; (4) the free surface of the soil has no normal or shear stresses and there is no displacement occurring at the bottom of the layer; (5) Only the circumferential displacements are considered. 2.1. Dynamic soil reactions for the saturated soil According to the dynamic consolidation theory of Biot for a completely saturated medium, the dynamic equilib-
451
T0 eiωt
r
z
H
saturated soil
Pile
f ( z)
2r0
rigid base
Fig. 1. Model of soil–pile interaction.
rium equation for the saturated soil layer in polar coordinate system can be expressed as: orrh orhz rrh o2 o2 þ þ2 ¼ q 2 uh ðr; z; tÞ þ qf 2 wh ðr; z; tÞ or oz r ot ot
ð1Þ
where rrh and rhz are the non-zero stresses, uh(r, z, t) and vh(r, z, t) are, respectively, the circumferential displacements of the soil skeleton and that of the pore fluid, wh(r, z, t) = / [vh(r, z, t) uh(r, z, t)] is the circumferential displacements of pore fluid relative to the soil skeleton; / is the porosity. qs and qf are the mass densities of soil particle and fluid, respectively; q = (1 /)qs + /qf, the mass density of soil; To solve this problem, Eq. (1) can be expressed in terms of displacements uh(r, z, t) and wh(r, z, t) as: lr2 uh ðr; z; tÞ
uh ðr; z; tÞ o2 o2 l ¼ q u ðr; z; tÞ þ q wh ðr; z; tÞ h f r2 ot2 ot2 ð2Þ
where l is the Lame constant of the solid skeleton; 2 2 2 r2 ¼ oro 2 þ 1r oro þ ozo 2 ¼ 1r oro r oro þ ozo 2 . The pore fluid equilibrium equation can be written as: g owh ðr; z; tÞ o2 q o2 1 oP þ qf 2 uh ðr; z; tÞ þ f 2 wh ðr; z; tÞ ¼ k ot r oh ot / ot ð3Þ where P is excess pore water pressure; g and k are the fluid viscosity and intrinsic permeability of soil, respectively. Due to the symmetry of the pure torsional vibration of medium about the vertical axis, the motion is independent of angle h. For the harmonic motion, uh(r, z, t) = uh(r,z)eixt ixt ixt ixt = uhpeffiffiffiffiffiffi ffi and wh(r, z, t) = wh(r,z)e = wh e , where i ¼ 1, x is circular frequency of excitation and t is time. Then, Eqs. (2) and (3) can be written as:
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K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
uh l ¼ ðqx2 uh þ qf x2 wh Þ r2 g q ðiwÞwh qf x2 uh f x2 wh ¼ 0 k /
lr2 uh
ð4Þ ð5Þ
Substituting Eq. (5) into Eq. (4), the equilibrium equation of circumferential motion uh can be written as: uh x2 qf x r 2 uh 2 ¼ qþ ð6Þ uh ig=ðkqf Þ x=/ r l Variables r and z can be separated by assuming uh ¼ RðrÞZðzÞ
ð7Þ
The substitution for uh from Eq. (7) into Eq. (6) results in: 1 0 1 x2 qf x 00 00 qþ RZ R Z þ R Z þ RZ 2 RZ ¼ l r r ig=ðkqf Þ x=/ ð8Þ Eq. (8) can now be split into two ordinary differential equations: d2 Z þ h2 Z ¼ 0 dz2 r2
ð9Þ
d2 R dR ð1 þ q2 r2 ÞR ¼ 0 þr 2 dr dr
ð10Þ
where constants h and q satisfy the following relationship: x2 qf x 2 2 qþ q ¼h ð11Þ ig=ðkqf Þ x=/ l
The shear stress amplitude srh corresponding to Eq. (16) can be expressed as: 1 X ouh uh Dn qn K 2 ðqn rÞ cosðhn zÞ ð18Þ srh ¼ l ¼ l or r n¼1 where K2(qnr) denotes the modified Bessel functions of the second kind of the second order. It is mathematically convenient at this stage to introduce the dimensionless quantities: rffiffiffi r0 r z q n ¼ Hhn ¼ p ð2n 1Þ; r ¼ ; z ¼ ; r0 ¼ ; a0 ¼ r0 x; h H H H l 2 q gr0 f ¼ f ; b ¼ pffiffiffiffiffiffi qn ¼ Hqn ; q q k ql where r0 and H are the radius and length of the pile, respectively. After introducing the dimensionless variables, Eq. (17) becomes: f a0 a20 q 2 2 qn ¼ hn 2 1 þ ð19Þ r0 ib= qf a0 =/ Assuming that the pile and soil layer have no separation, the circumferential displacement uh at the pile–soil interface and the resisting moment p(z) per unit of length of the pile can be expressed respectively: 1 X uh ðr0 ; zÞ ¼ Dn K 1 ðqnr0 Þ cosðhnzÞ ð20Þ pðzÞ ¼
ð12Þ
RðrÞ ¼ AK 1 ðqrÞ þ BI 1 ðqrÞ
ð13Þ
where I1(qr) and K1(qr) are the modified Bessel functions of the first and second kind of the first order, respectively; A,B, C and D can be obtained from the boundary conditions. Referring to Fig. 1, it can be observed that the boundary conditions of the soil layer can be written as: uh ðr ! 1; zÞ ¼ 0
ð14Þ
shz ðr; 0Þ ¼ 0;
ð15Þ
uh ðr; H Þ ¼ 0
Furthermore, it is noted from expression (14) that I1(qr) ! 1 when r ! 1, so constant B should vanish to zero. The expression (15) requires C = 0 and hn ¼ ð2n1Þp where n = 1, 2H 2, 3 . . .. Then the general solution of Eq. (6) can be written in a series expansion as: 1 X Dn K 1 ðqn rÞ cosðhn zÞ ð16Þ uh ¼ n¼1
q2n
¼
h2n
x2 qf x qþ ig=ðkqf Þ x=/ l
and Dn(n = 1, 2, 3,. . .) are a series of constants.
ð17Þ
r20 srh dh ¼ 2pr0 lr0
1 X
Dn qn K 2 ðqnr0 Þ cosð hnzÞ
n¼1
ð21Þ Following the definition of Novak et al. [15], the complex torsional impedance related to a unit of length of the pile can be expressed as: k w ¼ 2plr0
qn K 2 ðqnr0 Þr0 K 1 ðqnr0 Þ
ð22Þ
The torsional impedance can be further expressed in terms of its real and imaginary parts as: k w ¼ lr0 ðS w1 þ iS w2 Þ
ð23Þ
where all except i is real. The dimensionless parameters Sw1 and Sw2 represent real stiffness and damping parts of the soil reaction, respectively. 2.2. Impedance of a pile For a pile subjected to harmonic torsional loading, the twist angle, U(z,t) = U(z)eixt is governed by the following one-dimensional equation of motion: Gp
where
2p 0
It is not difficult to show that the general solutions of Eqs. (9) and (10) can be expressed as: ZðzÞ ¼ C sinðhzÞ þ D cosðhzÞ
n¼1
Z
o2 f ðzÞ ixt o2 ixt bUðzÞe c þ 4 e ¼ q ½UðzÞeixt p r20 oz2 ot2
ð24Þ
where Gp, qp, and U(z) are the shear modulus, mass density and the twist angle amplitude at depth z of the pile, respectively; f(z) denotes amplitude of the shear stress related to a unit area of the pile shaft applied by the soil.
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
Due to a perfect connection between the pile and the soil, the value of contact traction f(z) should be equal to the shear stress at r = r0 of the soil and can be expressed as: o uh ðr; zÞ ½uh ðr; zÞ f ðzÞ ¼ srh jr¼r0 ¼ l or r r¼r0
¼ l
1 X
Dn qn K 2 ðqn r0 Þ cosðhn zÞ
ð25Þ
parts. The first part is free vibration characteristics of the pile, as shown in the first and second terms in the righthand side of Eq. (27). Because of the reaction from the soil, the second part represents the forced vibration characteristics of the pile, as is shown in the third term in the righthand side of Eq. (27). Substituting U(z) from Eq. (27) into Eq. (26) yields:
n¼1
The substitution for f(z) from Eq. (25) into Eq. (24) results in: 1 d2 UðzÞ qp x2 UðzÞ 4l X þ ¼ Dn qn K 2 ðqn r0 Þ cosðhn zÞ ð26Þ dz2 Gp r20 n¼1 Gp The solution of Eq. (26) is: rffiffiffiffiffiffi rffiffiffiffiffiffi qp qp UðzÞ ¼ a1 cos xz þ a2 sin xz Gp Gp 1 X þ wn cosðhn zÞ
453
wn ¼
4lqn K 2 ðqn r0 ÞDn
q x2 Gp r20 h2n Gp p
ð28Þ
Since a perfect connection between the pile and the soil, the circumferential displacement of the pile must be the same as that of the soil given by Eq. (20). Therefore, the successive condition at the pile–soil interface can be written as: r0 /ðzÞ ¼ uh jr¼r0
ð27Þ
n¼1
where a1 and a2 are the coefficients which remain to be determined later from the boundary conditions. wn (n = 1, 2, 3. . .) are also a series of constants. During the vibration, the pile and the soil interact, so the vibration characteristics should be expressed in two
ð29Þ
Substituting Eqs. (16) and (27) into Eq. (29) leads to: ( ) rffiffiffiffiffiffi rffiffiffiffiffiffi X 1 qp qp xz þ a2 sin xz þ wn cosðhn zÞ r0 a1 cos Gp Gp n¼1 ¼
1 X
Dn K 1 ðqn r0 Þ cosðhn zÞ
ð30Þ
n¼1
13.0
(a) Stiffness
16
12.5
(a) Stiffness n =1 n =3 n =5
15
Sw1
Sw1
14
b b b b
12.0
13
11.5
=50 =100 =1000 =10000
11.0 12
10.5 11
10.0 10 9 0.0
9.5 0.0 0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
a0
3.0
a0 18
20
(b) Damping (b) Damping
15
16
Sw2
Sw2
12
12
9
8
6
4
3
0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
a0 Fig. 2. Torsional stiffness and damping parameters for various modes n.
0 0.0
0.5
1.0
1.5
a0 Fig. 3. Torsional stiffness and damping parameters for various poroelastic material properties b (/ = 0.4, qf ¼ 0:5, H/r0 = 20, mode number n = 1).
454
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
By invoking the orthogonality of eigenfunctions cos(hn z)(n = 1, 2, 3 . . .), multiplying cos(hnz) on both sides of Eq. (30), and then integrating over the interval z = [0,H], the undetermined coefficient Dn is found to be: Dn ¼
Z
1 Ln En
) 1 X UðzÞ ¼ a1 cosðkzÞ þ nn cosðhnzÞ n¼1 ( ) 1 X þ a2 sinðkzÞ fn cosðhnzÞ
ð31Þ
P cosðhn zÞdz:
qffiffiffiffi qp xz P ¼ a1 cos where Ln ¼ 0 cos2 ðhn zÞdz ¼ H2 , Gp
qffiffiffiffi qp 4lq K ðq r Þ þa2 sin xz , En ¼ r10 K 1 ðqn r0 Þ þ G r nðh22 qnp x0 2 Þ . Gp RH
p 0
n
Gp
It is again mathematical convenient at this stage to introduce the following dimensionless quantities: ¼ Gp =l; l
rffiffiffiffiffi rffiffiffiffiffiffi qp p 1 q xz ¼ a0z ¼ kz; r0 l Gp
4 qn K 2 ð qnr0 Þ ; En ¼ r0 En ¼ K 1 ð qnr0 Þ þ 2 r0 ðhn l k2 Þ
where
1 1 nn ¼ mn sinðk hn Þ þ sinðk þ hn Þ k hn k þ hn
1 1 fn ¼ mn fcosðk þ hn Þ 1g þ fcosðk hn Þ 1g k þ hn k hn
mn ¼
ð32Þ
Now, the dimensionless boundary conditions at the base of and top of the pile are: dUðzÞ T 0H Uðz ¼ 1Þ ¼ 0; ¼ ð35Þ dz G I where I p ¼ pr40 =2 denotes polar moment inertia of the pile. T0 eixt is the harmonic excitation force acting at the pile 15
(a) Stiffness H/r0=10 H/r0=20 H/r0=40 H/r0=100 plane strain case
12.0 11.5
13
11.0
12
10.5
11
10.0
10
1.0
1.5
the result of this paper the result of Novak [5]
14
Re (kT)
12.5
0.5
p p
z¼0
13.0
Sw1
4qn K 2 ðqnr0 Þ r0 l ðh2n k2 ÞEn ð34Þ
The amplitude of the twist angle of the pile is then given by:
9.5 0.0
ð33Þ
n¼1
H
0
p ¼ qp =q; q
(
9 0.0
2.0
a0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
a0
12
16
(b) Damping 10 12
Im (kT)
Sw2
8 6
8
4 4
2 0 0.0
0.5 .5
1.0
a0
1.5
2.0
Fig. 4. Torsional stiffness and damping parameters for different slenderf ¼ 0:5, b ¼ 1000, mode number n = 1). ness ratio H/r0 (/ = 0.4, q
0 0.0
0.5
1.0
1.5
a0 Fig. 5a. Comparison of the torsional impedance of this paper with that of p ¼ 1:3; qf = 0; l ¼ 300). Novak [5] for an elastic soil (H/r0 = 40; q
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458
head. By substituting Eq. (35) into Eq. (33), the variables, a1, a2 are obtained: a1 ¼
T 0H tan k k Gp I p
ð36aÞ
T 0H k Gp I p
ð36bÞ
a2 ¼
so that the dimensionless torsional impedance at the top end of the pile is given by: kT ¼ ¼
3T 0 3 16lr0 Uðz
¼ 0Þ
3p l kr 0
: 1 1 P P 32 1 þ nn tan k þ fn n¼1
ð37Þ
n¼1
3. Numerical results and discussion In this section, numerical results are presented to highlight the dynamic performance of pile–soil system by using Eqs. (23) and (37). Firstly, the torsional impedance of the soil layer is analyzed. Then the torsional impedance at the top end of the pile is investigated.
3.1. Torsional impedance of soil layer Fig. 2 shows the effect of the mode n on soil reactions. f ¼ 0:5, The parameters used in calculation are: / = 0.4, q b ¼ 1000, H/r0 = 20. The definition of the torsional impedance is based on Novak et al. [15]. The real and imaginary parts of the torsional impedance of the soil represent the real stiffness and damping of the soil, respectively. It can be observed from Fig. 2a that the stiffness depends much on the mode number in the dimensionless frequency range of 0 < a0 < 1.0. When the excitation frequency increases further, the influence of the mode number becomes less important. Fig. 2b shows the variations of damping with the excitation frequency and mode number. The damping represents the dissipation of the energy. When the frequency is in the low range, the damping is very little. However, with the further increase of frequency, the damping becomes proportional to the excitation frequency but the mode number has negligible influence on the response. This behavior is similar to the vertical vibration of the soil reported by Nogami and Novak [16]. Fig. 3 shows the variations of torsional impedance of the saturated soil with the different poroelastic parameter b. It can be seen that the stiffness parameter Sw1 is almost
90
30
80 b =10 b =20 b =100 b=1000
24
μ =10 μ =100 μ = 500 μ =1000 μ = 2000 μ = 5000
70 60
Re (kT)
27
Re (k T)
455
21
50 40 30
18
20 15 12 0.0
10 0.5
1.0
1.5
2.0
2.5
0 0.0
3.0
0.5
1.0
a0
1.5
2.0
2.5
0.5
1.0
a0
1.5
2.0
2.5
a0 20
28 24
16
20
Im (kT)
12
Im (kT)
16 12
8
8
4 4 0 0.0
0.5
1. 1.0
1.5
a0
2.0
0.5
3.0
Fig. 5b. Torsional impedance of elastic pile in poroelastic medium (H/ p ¼ 1:3; q f ¼ 0:5; / = 0.4). r0 = 20; q
0 0.0
(H/r0 = 20; Fig. 6. Variation of torsional impedance of elastic pile with l f ¼ 0:5; / = 0.4; b ¼ 100). p ¼ 1:3; q q
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independent of b in the frequency range a0 = 0 to 1.0. Then with the increase of the frequency, the stiffness parameter Sw1 decreases with increasing b. However, Sw1 shows negligible changes for b > 1000 when compared with b ¼ 1000. It is also noted that b has negligible influence on the damping parameter Sw2. The effect of the slenderness ratio on the response is shown in Fig. 4. In order to compare this solution with f ¼ 0 is set the plane strain solution [15], the parameter q to zero to simulate the special case of the elastic solution in homogenous soil. It can be seen that the slenderness ratio H/r0 has marked influence on the stiffness parameter Sw1 and damping parameter Sw2 in the range of low frequencies. As the frequency increases, the torsional impedance becomes less dependent on H/r0 and is closer to the plane strain case. It can also be seen that the torsional impedance obtained by present solution is in good agreement with the plane strain case, when the H/r0 is greater than about forty. In other words, the plane strain solution is accurate, when H/r0 > 40. It is also found that the plane strain solution underestimates the stiffness in the range of low frequencies when H/r0 < 20. The effect of slenderness
ratio on damping is small for a0< 1.0 and negligible for a0 > 1.0. 3.2. Impedance of end bearing pile In the analysis of structures supported by pile foundations, the complex impedance at the pile head is often used to gauge the vibration characteristics of the pile–soil system. The real part of the complex impedance represents the real stiffness, while the imaginary part of the complex impedance represents the radial damping of the system. The complex impedance is influenced by the frequency of excitation x, pile slenderness ratio H/r0, pile modulus ratio and other material properties. In order to verify the corl rectness of the solution proposed in this paper, the problem is reduced to the torsional vibration of an end bearing pile embedded in elastic medium and compared with the correp ¼ 1:3, ¼ 300, q sponding result of Navak [5]. H/r0 = 40, l qf = 0. Fig. 5a shows that the real and imaginary parts of this paper are close to that of Novak’s. Four groups of poroelastic materials are considered in the following analysis, the dimensionless properties are: p ¼ 1:3, q f ¼ 0:5 and / = 0.4, with ¼ 1000, q H/r0 = 20, l b ¼ 10, 20, 100 and 1000, respectively. The influence of
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Fig. 7b. Variation of torsional impedance of elastic pile with slenderness f ¼ 0:5: / = 0.4; b ¼ 100; l qp ¼ 1:3; q ¼ 500). ratio H/r0 (
K. Wang et al. / Computers and Geotechnics 35 (2008) 450–458 25 the result of Militano [9], H/r0=10 the result of this paper, H/r0=10 the result of Militano [9], H/r0=40 the result of this paper, H/r0=40
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the poroelastic parameter b on the complex impedance with the increasing dimensionless frequency a0 is shown in Fig. 5b. It can be seen from Fig. 5 that the real and imaginary parts of the torsional impedance shows the negligible variations with the frequency in the range a0 = 0–1.5 for all four materials. As the frequency increases, the poroelastic medium with a larger value of b has lower stiffness and higher dissipation of energy. However, Re (kT) and Im (kT) show negligible changes for b > 100 when compared with b ¼ 100. and a0 on the The influence of pile–soil modulus ratio l torsional stiffness is shown in Fig. 6. It can be seen that has a significant influence the pile–soil modulus ratio l on the response. The real parts of torsional impedance with the decrease with increasing frequency for a given l increases. It is decrease becoming more pronounced as l also noted that the real stiffness of the pile becomes higher increases for a given a0. The imaginary parts of the toras l sional impedance increase with increasing frequency for both flexible and rigid piles. For a given a0, the radial . Such behaviors for damping increases with increasing l an end bearing pile are similar to that of the pile embedded in homogenous soil layer [6].
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Figs. 7 shows the variations of the dimensionless torsional impedance with slenderness ratio H/r0 for different ¼ 100, 500 and 1000. It is seen that the real a0 and for l parts of the dimensionless torsional impedance show a steep decrease with the slenderness ratio and the imaginary parts show a steep increase. Both the real and imaginary parts asymptote to constant values. Thus there is a critical and a0, above which the real slenderness ratio for given l and imaginary parts could be deemed to have converged to constant values. It is also noted that the critical slender ratio and the critical ness ratio increases with increasing l slenderness ratio is less than 40 as the pile becomes much stiffer.
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In this section, the torsional impedance of this paper is compared with that of Militano’s [9]. Because Militano studies the torsional vibration characteristics of floating pile embedded in elastic soil, so the problem of this study is reduced to the torsional vibration of pile embedded in elastic soil in the subsequent analysis. And two cases are
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considered to investigate the difference and sameness ¼ 500, between these two solutions, the parameters are: l p ¼ 1:3, qf = 0, with H/r0 = 10 and 40, respectively. q Fig. 8 shows the comparison of torsional impedance of end bearing pile with that of floating pile by Militano. It can be seen that these two solutions have great difference when H/r0 = 10. However, as the increase of the slenderness ratio, such as H/r0 = 40, these two solutions have good agreement. Therefore, the torsional vibration characteristics of an end bearing pile are close to that of a floating pile for large slenderness ratio. 4. Conclusions By considering the end bearing pile embedded in a saturated soil as a pile–soil dynamic interaction problem, an analytical solution for the torsional impedance of the saturated soil layer and pile has been derived. An extensive parameters study has been conducted to investigate the influence of major parameters. From the numerical results, conclusions can be drawn that both the real and imaginary parts of the torsional impedance of the soil depend much on the mode number and slenderness ratio in the low range of frequency, but have negligible influence in the higher frequency range. The poroelastic parameter b has some influence on the real part (stiffness) of the torsional impedance of the soil, but has negligible effect on the imaginary part (damping). The comparison with the plane strain theory shows that plane strain solution underestimates stiffness of the soil and overestimates the damping of the soil at low excitation frequencies. The selected numerical results for torsional impedance of elastic pile show that modulus ratio and slenderness ratio greatly influence the response. While the poroelastic property b has negligible effect on the response in the low frequency range, it has some influence on the response in the higher frequencies. Furthermore, the influence of the slenderness ratio has a critical value, such that when the slenderness ratio is above the critical value, the torsional
impedance of the pile converges to a constant and the corresponding torsional vibration characteristics is close to that of a floating pile. Acknowledgements This research is supported by the National Natural Science Foundation of China (Grant No. 50279047). References [1] Poulos HG. Torsional response of piles. J Geotech Eng Div, ASCE 1975;101(GT10):1019–35. [2] Stoll UW. Torque shear test on cylindrical friction piles. J Civil Eng Div, ASCE 1972;42(4):63–4. [3] Randolph MF. Piles subjected to torsion. J Geotech Eng Div, ASCE 1981;107(GT8):1095–111. [4] Guo W, Randolph MF. Torsional piles in non-homogeneous media. Comput Geotech 1996;19(4):265–87. [5] Novak M, Howell JF. Torsional vibration of pile foundations. J Geotech Eng Div, ASCE 1977;103(GT4):271–85. [6] Rajapakse RKND, Shah AH, et al. J Earthquake Eng Struct Dyn 1987;15:279–97. [7] Rajapakse RKND, Shah AH. Impedance curves for an elastic pile. Soil Dyn Earthquake Eng 1989;8(3):145–52. [8] Tham LG, Cheung YK, Lei ZX. Torsional dynamic analysis of single piles by time-domain BEM. J Sound Vibr 1994;174(4):505–19. [9] Militano G, Rajapakse RKND. Dynamic response of a pile in a multi-layered soil to transient torsional and axial loading. Geotechnique 1999;49(1):91–109. [10] Stefest H. Numerical inversion of Laplace transforms. Commun Assoc Comput Mach B 1970;13(1):47–9. [11] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. I: Low-frequency range. J Acoust Soc Am 1956;28:168–78. [12] Zeng X, Rajapakse RKND. Dynamic axial load transfer from elastic bar to poroelastic medium. J Eng Mech 1999;125(9):1048–55. [13] Chen L, Wang G. Torsional vibration of elastic foundation on saturated media. Soil Dyn Earthquake Eng 2002;22(3):223–7. [14] Cai Y, Chen G, Xu C, et al. Torsional response of pile embedded in a poroelastic medium. Soil Dyn Earthquake Eng 2006;26(12):1143–8. [15] Novak M, Nogami T, Aboul-Ella F. Dynamic soil reactions for plane strain case. J Eng Mech Div, ASCE 1978;104(EM4):953–9. [16] Nogami T, Novak M. Soil–pile interaction in vertical vibration. J Earthquake Eng Struct Dyn 1976;4:277–93.