Torsional vibrations of elastic foundation on saturated media

Soil Dynamics and Earthquake Engineering 22 (2002) 223±227

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Torsional vibrations of elastic foundation on saturated media Longzhu Chen, Guocai Wang* School of Civil Engineering and Mechanics, Shanghai Jiaotong University, Shanghai 200030, People's Republic of China Accepted 27 February 2001

Abstract A comprehensive analytical solution is developed to examine the torsional vibration of an elastic foundation on a semi-in®nite saturated elastic medium for the ®rst time. First, the governing equations of saturated media are solved by use of Hankel transform techniques. Then, based on the assumption that the contact between the foundation and the half-space is perfectly bonded, this dynamic mixed boundary-value problem can lead to dual integral equations, which are further reduced to the standard Fredholm integral equations of the second kind and solved by numerical procedures. Numerical examples are given at the end of the paper. The numerical results indicate that the response of the elastic foundation strongly depends on the material and geometrical properties of both the saturated soil-foundation system and the load acting on the foundation. In most of the cases, the dynamic behavior of an elastic foundation on saturated media signi®cantly differs from that of a rigid plate bearing on the elastic half-space. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Saturated media; Elastic foundation; Torsional vibration; Mixed boundary-value condition; Dynamic response

1. Introduction In soil±structure interaction problems, the dynamic compliance of foundations on a half-space plays a key role when considering problems such as the seismic response of buildings subjected to earthquakes, the transmission of vibrations and noise through foundations and vibration of entire buildings, dams, machine foundations and similar large structures. A considerable amount of work has been done on this problem. For example, Luco and Westmann [1] considered the vertical, horizontal and rotatory vibration of a rigid footing bonded to an elastic half-space. Tsai [2] studied the torsional vibrations of a circular disk on an in®nite transversely isotropic medium. Kassir et al. [3] used Hankel transforms and analyzed the vertical vibration of a circular footing on a saturated half-space. Ressiner and Sagoci [4] obtained the characteristics of the torsional vibration by means of spheroidal wave functions. Gladwell [5] considered the forced tangential and rotatory vibration of a rigid circular disc on a semi-in®nite solid. Bougacha et al. [6] analyzed the wave propagation in layered, ¯uid-®lled poroelastic media. Halpern and Christiano [7] derived the compliance functions for vertical and rocking motions of a rigid square plate bearing on the surface of a liquidsaturated poroelastic half-space. Recently, Bo and Hua [8]

solved the transient interaction of a loaded elastic plate resting on a poroelastic half-space. In the work previously referred to, most formulations of dynamic interaction problems regard the material of the half-space as being either elastic or viscoelastic solid and the foundation as a rigid one. However, in most cases, it may be more appropriate to regard the half-space as a two-phase medium and the foundation as an elastic one. It is worth mentioning that the solutions corresponding to the torsional vibration of elastic foundations on saturated elastic halfspace are not reported in the literature. Therefore, the main objective of this paper is to solve this problem and to ®nd out to what degree the responses are signi®cantly different from that of a rigid circular plate bearing on the elastic media. 2. General solutions of governing equations Consider the model shown in Fig. 1. The half-space is a porous, ¯uid-saturated medium and its constitutive behavior follows Biot's two-phased linear theory [9,10]. The governing differential equations for saturated media, when the inertia-coupling term between solid matrix and pore ¯uid is neglected, are as follows [11,12]:

mDW 1 …l 1 m†grad…div…W†† 1 grad Pw ˆ r1 U 1 r2 W …1† * Corresponding author. E-mail address: [email protected] (G. Wang).

_ ˆ r1 U grad Pw 1 b…W_ 2 U†

0267-7261/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0267-726 1(02)00012-X

…2†

224

L. Chen, G. Wang / Soil Dynamics and Earthquake Engineering 22 (2002) 223±227

variables: r ˆ

r ; r0

r w ˆ

z ˆ

rw ; r

r 1 ˆ

r0 b; b ˆ p rm Fig. 1. Description of the model and coordinate system.

n _ P ˆ0 n div U_ 1 …1 2 n†div W_ 2 Ew w

…3†

Here W and U are the displacement vectors of solid and ¯uid, respectively, Pw the pore pressure, Ew the volumetric deformation modulus of ¯uid, n the porosity coef®cient of the medium, l , m the Lame's constants of the solid skeleton, r1 ˆ nrw ; r2 ˆ …1 2 n†rs ; b ˆ nrw g=k; (r w, r s are the mass density of pore water and solid particle, respectively), g the gravitational acceleration, k the Darcy's permeability coef®cient of the medium. Overdots denote the derivatives of ®eld variables with respect to time t. An elastic circular foundation rests on the half-space. The z-axis of the cylindrical coordinate system coincides with the vertical axis of symmetry of the foundation. Torsional motion is induced by the loading T eivt ; where v is the p angular velocity of steady-state vibration and i ˆ 21: Due to the symmetry of the problem, the motion is independent of angle u , and the only non-vanishing component of the displacement vector is the soil tangential displacement wu …r; z†eivt and the ¯uid tangential displacement vu …r; z†eivt : With these notations, Eqs. (1)±(3) can be written as     1 2 …4† m 7 2 2 wu ˆ 2 rw v2 vu 1 rv2 wu r 2

  ibv vu ˆ 2 rw v2 vu 1 r1 v2 wu n

…5†

72 ˆ

22 1 2 22 1 2 1 2 r 2r 2r 2z

r ˆ r1 1 r2 ; vu ˆ n…uu 2 wu † is the tangential displacement of the ¯uid relative to the solid. For brevity, the harmonic time factor eivt is suppressed from now on. The only non-zero bulk stress components of the soil are 2w tz u ˆ m u 2z   2wu wu 2 tr u ˆ m 2r r

…6†

…7†

It is convenient to introduce the dimensionless constants and

wu v ; vu ˆ u ; r0 r0 r r aˆ r v; m 0

w u ˆ

r1 ; r

t zu ˆ

tzu ; m

t ru ˆ

tr u m

where r0 is the radius of the foundation. After introducing the dimensionless variables, Eqs. (4)± (7) can be written as     1 7 2 2 2 w u ˆ 2 r w a2 vu 1 a2 w u …8† r  iba v ˆ r w a2 vu 1 r 1 a2 w u n u

t zu ˆ

2w u 2z 

t ru ˆ

…9† …10†

2w u w 2 u 2r r

 …11†

We denote the Hankel transform of the ®rst order of a function f…r; z† by Z1 f 1 …p; z† ˆ r f…r; z†J1 …pr†dr 0

and its inverse transform by Z1 pf 1 …p; z†J1 …pr†dp f…r; z† ˆ 0

where J1(pr) is the ®rst-order Bessel function of the ®rst kind. By means of Hankel transform of the ®rst order, Eqs. (8)± (10) yield 2p2 w1u 1

where

z ; r0

d2 1 wu ˆ 2r w a2 v1u 2 a2 w1u dz2

 1 iba v ˆ r w a2 v1u 1 r 1 a2 w1u n u

t1zu ˆ

d 1 w dz u

…12†

…13† …14†

By taking the radiation condition into account, i.e. no incoming waves present in the system, the solutions of Eqs. (12) and (13) are found to be w1u ˆ A e2jz

…15†

nr a A e2jz v1u ˆ  1 ib 2 r 1 a

…16†

L. Chen, G. Wang / Soil Dynamics and Earthquake Engineering 22 (2002) 223±227

in which j 2 ˆ p2 2 s 2 ;

s2 ˆ

!

r 1 r 2 a3 r a 2 a2 = 1 2 1  ir ib

trivial manipulation, Eqs. (18) and (19) can be written as ! Z1 2T d sin dah  1 f cos dah ; pA…p†J1 …pr †dp ˆ r pr f a 0



s is the dimensionless complex wave number associated with the rotational wave. A is an arbitrary function of p. Substitute Eq. (15) into Eq. (14) to yield

t1zu

ˆ 2Aj e

2jz

…17†

3. Integral equations of torsional vibrations of elastic foundation The interaction of the soil-foundation system belongs to the class of mixed boundary-value problems. Here we assume that the contact between the foundation and the underlying half-space is perfectly bonded. The dimensionless boundary conditions at the surface (z ˆ 0) of the halfspace can be stated as follows: w u …r ; 0† ˆ ru…0†; 0 # r # 1

…18†

t zu …r; 0† ˆ 0; r . 1

…19†

mf

2

2u 2u ˆ rf 2 2 2z 2t

Considering the harmonic vibration, Eq. (20) can be written as 22 u 1 d2 a2 u ˆ 0 …21† 2z2 p in which d ˆ …rf m†=…rmf †: Here m f and r f are the shear modulus and density of the elastic foundation, respectively. The boundary conditions at z ˆ 2h are described by

Tˆ

…22†

Zr0 0

2pr 2 t dr

…23†

where the shearing stress 2u t ˆ mf r ; 2z

Z1 0

(24a)

pA…p†jJ1 …pr †dp ˆ 0; r . 1

…24b†

in which the dimensionless constants T T ˆ ; mr03

r f ˆ

rf r

The solution of the dual integral equations (24a) and (24b) has been considered by Erdelyi and Sneddon [14], and Noble [15]. Based on the procedure suggested by Robertson [16], we de®ne the following integral representation: Z1 4 pA…p†j ˆ pf q…t†sin…pt†dt …25† p 0 Substituting Eq. (25) into Eqs. (24a), (24b), Eq. (24b) is automatically satis®ed and Eq. (24a) is equivalent to the following integral equation: ! 2 Z1 2T d sin dah 0  q…t† 1 q…t†k …t; t†dt ˆ t cos dah 1 p 0 pr f af in which

…20†

uˆf

0 # r # 1

…26†

where u (0) satis®es the following equation [13]: 2

225

h h ˆ r0

where h is the height of the foundation and f the amplitude of the torsional angle. Combining with Eqs. (15), (17), (21)± (23) and after some

k 0 …t; t† ˆ

Z1  p 0

j

 2 1 sin…pt†sin…pt†dp

Observing that there is no torque at the side of the foundation, it has the following identity: Z1 Z 1  2u  m f r3 dr 1 r2 t zu …r ; 0†dr ˆ 0 …27† 2z zˆ0 0 0 By virtue of Eqs. (21)±(23), Eq. (27) can be written as Z1 16 tq…t†dt   T pr f atgdah 0 …28† ˆ 1 2d f cos dah Substituting Eq. (28) into Eq. (26), we get the Fredholm integral equation of the second kind 2 Z1 t …29† q…t† 1 q…t†k…t; t†dt ˆ p 0 cos dah where the kernel k…t; t† ˆ k 0 …t; t† 2

16dtgdah t r f a

The above question can be reduced to the torsional vibration of a massless rigid disk on an elastic half-space [16] if we put d ˆ 0; h ˆ 0 and r w ˆ 0: So the torsional vibration of a rigid foundation is a special case of that of the elastic one.

226

L. Chen, G. Wang / Soil Dynamics and Earthquake Engineering 22 (2002) 223±227

If we denote the static torsional angle amplitude

f0 ˆ

3 T 3  T ˆ 16 mr03 16

and the static tangential displacement w u0 ˆ rf, the dynamic compliance coef®cient of the elastic foundation can be expressed as CT ˆ

f ˆ f0

1 3pr f atgdah 1 32d

3

Z1 0

…30† tq…t†dt

cos dah

and the tangential displacement of the foundation can be expressed as u…0† 1 32dtgdah Z1 w u …r ; 0† ˆ 1 ˆ tq…t†dt …31† w u0 …r ; 0† pr f a f cos dah 0 4. Numerical examples After obtaining the solutions of the Fredholm integral equation of the second kind from Eq. (29), we can calculate the dynamic response of the foundation by using Eqs. (30) and (31). As an example, the dimensionless parameters of

the media are: n ˆ 0:35; h ˆ 0:5; r w ˆ 0:478; r f ˆ 1:21 and b ˆ 2; 10. In order to explore the in¯uence of the rigidity of the foundation on the impedance functions and the tangential displacement, three values of the dimensionless parameter, d , are considered, namely d ˆ 0; 0:2; 0:3: The value of d indicates the rigidity of the foundation relative to that of the half-space. The stiffer the foundation is, the smaller the value of d . The real and imaginary parts of the dynamic compliance coef®cient and the tangential displacement of the elastic foundation which vary with the dimensionless frequency a are shown in Figs. 2±4. From Fig. 2, we can see that the value of the real part of CT changes with the increase of the value of d , but is smaller than for the rigid case …d ˆ 0†: We can also see that the value of the real part of CT increases with the increase of the vibrational frequency at the beginning and then it begins to decrease after passing a small unconspicuous peak. Fig. 3 shows that the value of the imaginary part of CT changes also with the increase of the value of d , but is bigger than for the rigid case …d ˆ 0†: From Fig. 3, it also shows that the value of the imaginary part of CT decreases ®rst with the increase of the value of a, then begins to increase after passing the peak. The real part of CT designates the stiffness of the foundation and the imaginary one the damping of the foundation. From Figs. 2 and 3, it is

Fig. 2. The real part of CT vs a.

Fig. 3. The imaginary part of CT vs a.

L. Chen, G. Wang / Soil Dynamics and Earthquake Engineering 22 (2002) 223±227

227

Fig. 4. The tangential displacement vs a.

hard to determine whether it is strength or weak to the total response of the vibration of the foundation. But Fig. 4 indicates that the tangential displacement of the foundation increases with the increase of the value of d and of the vibration frequency. From Figs. 2±4, it is also found that the value of b has some in¯uence on the dynamic response of the foundation. The differences shown in Figs. 2±4 can be attributed to the inter-granular energy losses in the solid phase, the viscous resistance to the ¯ow of the pore ¯uid and the deformation of the elastic foundation. All of these in¯uence factors cannot be neglected when analyzing the dynamic response of the foundation subjected to torsional loadings. 5. Concluding remarks The dynamic interaction problems of harmonic torsional vibration of an elastic foundation, which is perfectly bonded to the surface of the saturated half-space, have been presented for the ®rst time. Numerical values of the impedance functions (stiffness and damping coef®cients) and of the tangential displacement have been computed for a wide range of applied frequencies to show the in¯uence of the material parameters of the soil and foundation. From the numerical results, conclusions can be drawn that the response of the elastic foundation strongly depends on the material and geometrical properties of both the saturated soil-foundation system and the load acting on the foundation. Such differences can be attributed to the applied torque, the presence of the pore ¯uid and the foundation's rigidity. These should not be neglected in determining the response of structures to dynamic loadings, particularly the earthquake and dynamic machine loadings. It should be emphasized that the solutions presented in this paper are only applicable to the condition that the foundation is laid upon the surface of the half-space. Other conditions such as the foundation embedded into the medium are very important from the viewpoint of practical application and will be addressed in the future studies.

Acknowledgements This research is supported by the National Natural Science Foundation of China (Grant No. 50079027).

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