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Mixed boundary-value analysis of rocking vibrations of an elastic strip foundation on elastic soil with saturated substrata.

1. Introduction

Dynamic interaction between structural foundations and the underlying soil, both in theory and in practice, is widely studied in the field of geotechnical engineering and has important implications in power machine design and fundamental analysis of foundations under seismic loads. To simplify the boundary-value problem, in the early theoretical studies, the soil under the forced vibration of the foundation was often assumed to be a single-phase linear elastic medium [1-6].

However, soil is generally a two-phase material consisting of a solid skeleton and pores, which are filled with fluid. Such materials are commonly known as poroelastic materials in mechanics literature. After Biot established a theory of propagation of elastic waves in a fluid-saturated porous solid [7,8] in 1956, the research significance on vibration characteristics of foundation on saturated soil became apparent. Lin [9] studied the vertical and rocking vibrations of an elastic circular plate lying on a single-phase viscoelastic medium. Iguchi and Luco [10] studied the dynamic response of a massless flexible circular plate supported on a layered viscoelastic half-space, obtaining the vertical and rocking impedance of the flexible plate and the numerical solution of contact stress beneath the plate. Halpern and Christiano [11, 12] evaluated compliance functions for the harmonic rocking and vertical motions of rigid permeable and impermeable plates bearing on a poroelastic half-space. Kassir and Xu [13] studied the mixed boundary-value problem of the vibration of a rigid strip foundation on a fluid-saturated porous half-space. Jin and Liu [14, 15] analysed the dynamic response of a rigid disk on a saturated half-space subjected to harmonic horizontal and rocking excitation. Li [16] studied the vertical vibration of a rigid strip foundation on saturated soil. Finally, a parametric study by Ma et al. [17] examined the influences of dimensionless frequency, dynamic permeability, and Poisson's ratio on saturated soil under a rocking rigid strip footing.

Most of the results reported previously concern the dynamic interaction between the rigid structural foundation and the underlying saturated half-space. As research advances in this field of mechanics, a more realistically analytical model becomes increasingly necessary. In fact, the soil of the earth's surface, because of differences in structure and sedimentation, usually has an apparent stratification, formed naturally over the course of history. During foundation construction, underlying soil is routinely reinforced via a variety of methods, inevitably leading to some degree of soil stratification. In practice, the underlying soil will have different physical properties (porosity, permeability, etc.), which have a layered distribution in depth. In researching dynamic interactions of soil and a structural foundation, considering the underlying soil as a homogeneous elastic or saturated medium is not sufficiently accurate. Taking into account the presence of groundwater, the soil below the groundwater level should be considered as saturated soil and the soil above the groundwater level may be regarded as an ideal, single-phase elastic layer. As for the structural foundation, assuming it to be an elastic body is more accurate than assuming it to be a rigid body.

Based on the Biot theory of elastic waves in fluid-saturated porous medium, Philippacopoulos [18] studied the vertical vibration of a rigid circular disk resting on a saturated layered half-space. Bougacha et al. [19, 20] analysed the dynamic stiffness coefficients of rigid strip and circular foundations on a saturated layered half-space using spatially semi-discrete finite element technology. Rajapakse and Senjuntichai [21] presented an exact stiffness matrix method to evaluate the dynamic response of a multilayered poroelastic medium due to time-harmonic loads and fluid sources applied in the interior of the layered medium. Yang et al. [22] neglected the fluid inertia force exerted on the soil skeleton as proposed in the works of Zienkiewicz et al. [23] and studied the steady state response of an elastic soil layer and a saturated layered half-space. Chen [24] explored the characteristics of vertical vibration of both rigid and elastic circular plates on elastic soils with saturated substrata, utilising the Hankel transform to solve the wave equations. Furthermore, the torsional and rocking vibration characteristics of a rigid circular plate on elastic soil with saturated substrata were studied by Wang [25] and Fu [26], respectively, and the effects of the thickness of the elastic layer and the vibration frequency on the plate's dynamics were analysed. The previously listed literature reviews do not present a study of the dynamics between a vibrating elastic strip foundation and elastic soil with saturated substrata.

In this paper, a novel study is presented to make up a deficiency. The wave equations concerning both the single-phase elastic layer and the saturated half-space a resolved using a Fourier integral transform technique. Then, the dual integral equations of the rocking vibration of an elastic strip foundation are established according to mixed boundary conditions. The dynamic compliance coefficient's variation curve with the dimensionless frequency is obtained by applying Simpson's rule to conduct numerical calculation, and the effects of the elastic layer's thickness and the elastic characteristic parameters of the foundation on the rocking vibration are analysed.

2. The Dynamic Equations and Their Solutions

Soil and water weight are ignored; the soil is considered to be isotropic and the water incompressible. This paper studies the plane strain problem for an infinite-length strip with a footing of width 2b and an elastic layer thickness of [H.sub.n]. The centre of the footing is subjected to a harmonic moment force, [Me.sup.iwt] with [omega] denoting circular frequency. The horizontal direction is established as the x-axis and the vertical direction as the z-axis, and the origin of the coordinate is placed at the interface of the elastic soil and the saturated soil. The model is shown in Figure 1.

2.1. The Dynamic Equations of Elastic Layer under Plane Strain. The wave equations of the elastic layer are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1b)

where u and w are the horizontal and vertical displacements of the soil skeleton, respectively; [[sigma].sub.xL] and [[sigma].sub.zL] are the horizontal and vertical normal stresses, respectively; and [[rho].sub.L] is single-phase elastic soil density.

The relationship between the stress and the displacement is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2c)

where [G.sub.L] and [[mu].sub.L] are the shear modulus and Poisson's ratio of single-phase elastic soil, respectively.

2.2. The Dynamic Equations of Saturated Half-Space under Plane Strain. The basic dynamic equations of saturated half-space are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3e)

where u and w are the horizontal and vertical displacements of the soil skeleton, respectively; [w.sub.x] and [w.sub.z] are the horizontal and vertical displacements of water relative to the soil skeleton; [sigma]'x and [[sigma]'.sub.z] are the horizontal and vertical effective normal stresses, respectively; [p.sub.f] is the excess pore pressure; [rho] is the mass density of the saturated soil with [rho] = (1 - n) [[rho].sub.s] + n[[rho].sub.f]; [[rho].sub.s] and [[rho].sub.f] are the densities of the soil and water, respectively; and n is the porosity of the saturated soil.

The equations of stress and displacement are

[partial derivative][[sigma]'.sub.x] = -[2G(1 - [mu])/[1 - 2[mu]]] [[partial derivative]u/[partial derivative]x] - [2G[mu]/[1 - 2[mu]]][[partial derivative]w/[partial derivative]z], (4a)

[partial derivative][[sigma]'.sub.z] = -[2G[mu]/[1 - 2[mu]]] [[partial derivative]u/[partial derivative]x] - [2G(1 - [mu])/[1 - 2[mu]] [[partial derivative]w/[partial derivative]z], (4b)

[[tau].sub.xz] = -G([[partial derivative]u/[partial derivative]z] + [partial derivative]w/[partial derivative]x), (4c)

where G and [mu] are the shear modulus and Poisson's ratio of the saturated soil, respectively.

2.3. The Solutions of the Dynamic Equations. For a simple harmonic load, the displacement, stress, and excess pore pressure may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Here, w is the circular frequency. The Fourier transform can be written as:

[[phi].sup.*] = [Real part] = [[integral].sup.[infinity].sub.-[infinity]] [phi][e.sup.i[zeta]x]dx. (6)

Combining (1a)-(1b) and (2a)-(2c) and utilising the Fourier transform, we can obtain the solutions of the single-phase elastic layer as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7e)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Similarly, combining (3a)-(3e) and (4a)-(4c) and utilising the Fourier transform, we obtain the solutions of the saturated half-space as follows:

i[[bar.u].sup.*] ([zeta], z) = -[A.sub.0][zeta][He.sup.-qz] - [B.sub.0][zeta][Ne.sup.-[zeta]z] - [[C.sub.0][e.sup.- Fz]/[zeta]], (9a)

[[bar.w].sup.*] ([zeta], z) = -[A.sub.0]q[He.sup.-qz] - [B.sub.0][zeta][Ne.sup.- [zeta]z] - [[C.sub.0][e.sup.-Fz]/F], (9b)

[[bar.P].sup.*.sub.f] ([zeta], z) = [A.sub.0][Ee.sup.-qz] + [B.sub.0][e.sup.- [zeta]z], (9c)

[[bar.e].sup.*] ([zeta], z) = [A.sub.0][e.sup.-qz], (9d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9e)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9f)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

The transformation as shown in the following is introduced:

[A.sub.1] = [1/2] ([[bar.A].sub.1] + [[bar.B].sub.1]), [B.sub.1] = [1/2] ([[bar.A].sub.1] - [[bar.B].sub.1]),

[A.sub.2] = [1/2] ([[bar.A].sub.2] + [[bar.B].sub.2]), [B.sub.1] = [1/2] ([[bar.A].sub.2] - [[bar.B].sub.2]). (11)

Equations (7a)-(7e) can be further transformed as below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],(12b)

[[bar.e].sup.*]([zeta], z) = [[bar.A].sub.1][chq.sup.L]z + [[bar.B].sub.1][shq.sub.L]z, (12c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12e)

3. The Mixed Boundary-Value Problem of the Rocking Vibration of an Elastic Strip Foundation on Elastic Soil with Saturated Substrata and Boundary Conditions

It is assumed that the contact between the elastic strip foundation and the saturated soil is smooth and that the surface of the saturated soil is pervious. The boundary conditions are expressed as

[bar.w](x, 0) = x[phi] - [bar.[DELTA]](x) [absolute value of x] [less than or equal to] b, (13a)

[[bar.[tau]].sub.xzL] (x - [H.sub.n]) = 0 -[infinity] < x < 00, (13b)

[[[bar.w](x, 0)].sub.L] = [bar.w](x, 0) - [infinity] < x < [infinity], (13c)

[[[bar.u](x, 0)].sub.L] = [bar.u](x, 0) - [infinity] < x < [infinity], (13d)

[[bar.[tau]].sub.xzL] ([zeta], 0) = [[bar.[tau]].sub.xz] ([zeta], 0) - [infinity] < x < [infinity], (13e)

[[bar.[sigma]].sub.zL] ([zeta], 0) = [[bar.[sigma]].sub.z] ([zeta], 0) - [infinity] < x < [infinity], (13f)

[[bar.p].sub.f] ([zeta], 0) = 0 - [infinity] < x < [infinity], (13g)

[[bar.[sigma]].sub.zL] (x, -[H.sub.n]) = 0 [absolute value of x] > b, (13h)

where [[bar.[sigma]].sub.z] and [[bar.[tau]].sub.xz] are the normal stress and the shear stress of the soil skeleton, respectively; [[bar.p].sub.f] is the pore pressure; [bar.w] is the contact surface displacement between the strip foundation and underlying soil; [phi] is the rotation of the centre of the strip foundation; and [bar.[DELTA]] (x) is the deflection of the strip foundation relative to the centre.

Combining (9a)-(9f) and (12a)-(12e) and applying the Fourier transform to (13b)-(13h), we can obtain the following relationships:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14a)

[[bar.B].sub.1][q.sub.L][H.sub.L] + [[[bar.B].sub.2]/[F.sub.L]] + [A.sub.0]qH + [B.sub.0][zeta]N + [[C.sub.0]/F] = 0, (14b)

-[[bar.A].sub.1][zeta][H.sub.L] + [[[bar.A].sub.2]/[zeta]] [A.sub.0][zeta]H + [B.sub.0][zeta]N + [[C.sub.0]/[zeta]] = 0, (14c)

2[[bar.B].sub.1][zeta][q.sub.L][H.sub.L][G.sub.L] + [[bar.B].sub.2] ([[F.sub.L]/[zeta]] + [[zeta]/[F.sub.L]]) [G.sub.L] + 2 [A.sub.0][zeta]qHG (14d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14e)

[A.sub.0]E + [B.sub.0] = 0. (14f)

From (14f) we can obtain

[B.sub.0] = -[A.sub.0]E. (15)

Utilising the Fourier transform on (12b) and (12e) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16b)

Substituting (15) into (14b), (14c), (14d), and (14e), then (16a), (14a), (14b), (14c), (14d), and (14e) may be transformed into the following matrix form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

The expression of each element of matrix T can be seen in the appendix.

The displacement of the strip foundation surface can be expressed in the following matrix form:

[[[GAMMA].sub.1] [[GAMMA].sub.2] [[GAMMA].sub.3] [[GAMMA].sub.4] [[GAMMA].sub.5] [[GAMMA].sub.6]] [[[[bar.A].sub.1] [[bar.B].sub.1] [[bar.A].sub.2] [[bar.B].sub.2] [A.sub.0] [C.sub.0]].sup.T] (18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Establishing the following equations by the matrix T and the matrix [GAMMA] gives

T x X = [GAMMA], (20)

where X is a 6 x 1 matrix.

We find from (17) and (18) that the element [X.sub.1] in the first row of the matrix X denotes the relationship between the displacement and stress on the strip foundation surface. Thus, when

[[bar.w].sup.*] ([zeta], 0) = f([zeta]) - [[bar.[sigma]].sup.*.sub.z] ([zeta], 0) (21)

we obtain f([zeta]) = [X.sub.1], thereby expressing every element in matrix X by solving (20). We can obtain f([zeta]) and find that f([zeta]) and 1/[zeta] are infinitesimal of the same order when [zeta] [right arrow] [infinity].

Using the Fourier inverse transform and combining (21), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Additionally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

The dual integral equations of the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where [phi] is the rotation of the centre of the strip foundation and [D.sub.F] is the flexural stiffness of the foundation.

Simpson's rule is used to conduct numerical calculation. The dynamic compliance coefficient, [C.sub.M], of the rocking vibration of a strip foundation can be expressed as follows [27]:

[C.sub.M] = 1/b[a.sub.0]. (25)

Defining [f.sub.1] = Re[[C.sub.M]] and [f.sub.2] = Im[[C.sub.M]], we can obtain foundation stiffness K = [f.sub.1]/([f.sup.2.sub.1] + [f.sup.2.sub.2]) and the damping coefficient of the foundation C = -[f.sub.2]/([f.sup.2.sub.1] + [f.sup.2.sub.2]) [b.sub.0]. Here, [b.sub.0] = b[omega] [square root of [pi]/G] is the dimensionless frequency.

4. Verifications and Numerical

Example Analysis

The rocking vibration solution of an elastic strip foundation on elastic soil with saturated substrata can be degenerated to the single-phase elastic half-space case by defining [[rho].sub.f] = 0, [delta] = 0, and [H.sub.n] = 0. The foundation parameters for the degenerated case are b = 2 m, G = 35 MPa, n=0.35, and [[rho].sub.s] = 2650 kg/[m.sup.3]. The variation of the dynamic compliance coefficient [C.sub.M] with dimensionless frequency [b.sub.0] is analysed and is then compared with the numerical results by Luco and Westmann [5] when [mu] is 0.25. In Figure 2, "[degrees]" represents the numerical results obtained by Luco and Westmann [5] when [mu] is 0.25. Both of the results derived from Figure 2 are consistent and verify the feasibility and accuracy of the calculating methods described in this paper. Meanwhile, the rigid foundation is considered as a special case of the elastic foundation when [delta] equals zero.

Concerning the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata, the physical and mechanical parameters of a single-phase elastic layer are [G.sub.L] = 35 MPa, [[mu].sub.L] = 0.45, and [[rho].sub.L] = 1722.5 kg/[m.sup.3]; for a saturated half-space, n = 0.35, G = 35 MPa, [k.sub.d] = [10.sup.-5] m/s, [[rho].sub.f] = 1000 kg/[m.sup.3], [[rho].sub.s] = 2650 kg/[m.sup.3], and [mu] = 0.25. The dynamic compliance coefficient CM changes over the dimensionless frequency, the state of which is calculated when [H.sub.n] = 0, 0.2, 1.0, and 2.0 m for 5 = 10, and the calculation results are shown in Figure 3. Meanwhile, the dynamic compliance coefficient [C.sub.M] changes over the dimensionless frequency, the state of which is calculated when [delta] = 0.0,0.1,10, and 1000 for [H.sub.n] = 0.2, and the calculation results are shown in Figure 4.

We can see from Figure 3 that [C.sub.M] decreases with an increase in the elastic layer thickness, which indicates that the presence of elastic soil can reduce the vibration to the foundation. Meanwhile, in the given parameters for the real part of [C.sub.M], as [b.sub.0] increases, the real part of [C.sub.M] decreases, but the curve tends to flatten. For the imaginary part of [C.sub.M], as [b.sub.0] increases, the imaginary part of [C.sub.M] first decreases and then increases, finally becoming smooth.

For the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata, the curves of [C.sub.M] are essentially coincident when [delta] = 0.0 and 0.1, which can be seen from Figure 4. The curve for [delta] = 0.0 is the one of the dynamic compliance coefficients of the rocking vibration of a rigid strip foundation on elastic soil with saturated substrata. Therefore, it can be inferred that the rocking vibration of an elastic and a rigid strip foundation on elastic soil with saturated substrata has the similar dynamic characteristics in variations of [b.sub.0] when [delta] [less than or equal to] 0.1. It is also demonstrated that the real part of [C.sub.M] is greatly influenced by variations in [b.sub.0] when [b.sub.0] < 2.8 and that the imaginary part of [C.sub.M] is greatly influenced by variations in [b.sub.0] when [b.sub.0] < 3.6. However, when [b.sub.0] exceeds the critical values (2.8 for the real part of [C.sub.M] and 3.6 for the imaginary part of [C.sub.M]), the curve of [C.sub.M] tends to flatten and [C.sub.M] is only slightly influenced by variations in the dimensionless frequency [b.sub.0]. Figure 4 also shows that the dynamic compliance coefficient curve obtained when [delta] = 10 and 1000 is remarkably different from the one obtained when [delta] = 0.0 and 0.1, and the absolute values concerning both the real parts and imaginary parts of the dynamic compliance coefficient when [delta] = 10 and 1000 are larger than the ones when [delta] = 0.0 and 0.1. Moreover, it can be seen from Figure 4 that when [delta] is large, the variation of the dynamic compliance coefficient curve with the dimensionless frequency [b.sub.0] tends to be smooth. However, for an average quantity of [delta], the variation of the absolute values of the real parts and imaginary parts of [C.sub.M] with [b.sub.0] is significant. Thus, under ordinary circumstances ([delta] = 10), we must consider effects exerted by the dimensionless frequency [b.sub.0].

5. Conclusions

In this paper, an analytical solution for the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata is developed. The solution is based on dual integral equations, which are formulated from Biot's equations of dynamic poroelasticity by means of the Fourier transform in combination with mixed boundary conditions. Validation of the analytical solution for dry soil is based on the solution presented by Luco and Westmann [5].

Our conclusions of this study are as follows. (1) The dynamic compliance coefficient [C.sub.M] decreases with an increase in the elastic layer thickness, which indicates that the presence of elastic soil can reduce the vibration to the foundation. (2) The real part of dynamic compliance coefficient [C.sub.M] is greatly influenced by variations in the dimensionless frequency [b.sub.0] when [b.sub.0] < 2.8, and the imaginary part of [C.sub.M] is greatly influenced by variations in the dimensionless frequency [b.sub.0] when [b.sub.0] < 3.6. (3) When the flexural stiffness of the elastic foundation is comparatively large or when [delta] < 0.1, the influence of [delta] on the rocking vibration can be ignored. (4) When [delta] > 0.1 and as [delta] increases, the rocking vibration of the elastic foundation changes significantly, and the absolute values of both the real parts and imaginary parts of [C.sub.M] increase.

http://dx.doi.org/10.1155/2013/124782

Appendix

Expressions for each element of the matrix T appearing previously are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.1)

Acknowledgment

The work presented in this paper is partly supported by the National Natural Science Foundation of China (Grant nos. 51179171, 51079127, and 51279180).

References

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Xiaohua Ma, (1,2) Zhenyu Wang, (3) Yuanqiang Cai, (3) and Changjie Xu (3)

(1) Geotechnical Engineering Institute, Zhejiang University, Hangzhou 310058, China

(2) College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China

(3) Zhejiang Geofore Geotechnical Engineering Co., Ltd., Hangzhou 310013, China

Correspondence should be addressed to Zhenyu Wang; wzyu@zju.edu.cn

Received 23 October 2012; Accepted 24 February 2013

Academic Editor: Juan Torregrosa
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Title Annotation:Research Article
Author:Ma, Xiaohua; Wang, Zhenyu; Cai, Yuanqiang; Xu, Changjie
Publication:Journal of Applied Mathematics
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Date:Jan 1, 2013
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