Mixed Boundary-Value Analysis of Rocking Vibrations of an Elastic Strip Foundation on Elastic Soil with Saturated Substrata

The dynamic response of an elastic strip foundation lying on elastic soil with saturated substrata is greatly affected by pore pressure induced by a rocking moment. In this paper, we explore the mixed boundary-value problem of the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata via Biot dynamic equations. First, the wave equations concerning both the single-phase elastic layer and the saturated half-space are solved using a Fourier integral transform technique. The dual integral equations of the rocking vibration of an elastic strip foundation are established according to themixed boundary conditions. Finally, the relationship of the dynamic compliance coefficient with the dimensionless frequency is obtained by applying Simpson’s rule to conduct numerical calculation. We also analyse the influences of the elastic layer’s thickness and elastic characteristic parameters of the foundation on the rocking vibration.


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][2][3][4][5][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 semidiscrete 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 are solved 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.

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 2 and an elastic layer thickness of   .The centre of the footing is subjected to a harmonic moment force,   , with  denoting circular frequency.The horizontal direction is established as the x-axis and the vertical direction as the zaxis, 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 where  and  are the horizontal and vertical displacements of the soil skeleton, respectively;   and   are the horizontal and vertical normal stresses, respectively; and   is singlephase elastic soil density.
The relationship between the stress and the displacement is where   and   are the shear modulus and Poisson's ratio of single-phase elastic soil, respectively.

The Dynamic Equations of Saturated Half-Space under
Plane Strain.The basic dynamic equations of saturated halfspace are: where  and  are the horizontal and vertical displacements of the soil skeleton, respectively;   and   are the horizontal and vertical displacements of water relative to the soil skeleton;    and    are the horizontal and vertical effective normal stresses, respectively;   is the excess pore pressure;  is the mass density of the saturated soil with  = (1 − )  +   ;   and   are the densities of the soil and water, respectively; and  is the porosity of the saturated soil.
The equations of stress and displacement are where  and  are the shear modulus and Poisson's ratio of the saturated soil, respectively.

The Solutions of the Dynamic Equations.
For a simple harmonic load, the displacement, stress, and excess pore pressure may be expressed as Here,  is the circular frequency.The Fourier transform can be written as: Combining (1a)-( 1b) and (2a)-(2c) and utilising the Fourier transform, we can obtain the solutions of the singlephase elastic layer as follows: where Similarly, combining (3a)-(3e) and (4a)-(4c) and utilising the Fourier transform, we obtain the solutions of the saturated half-space as follows: * (, ) =  0  − , (9d) where The transformation as shown in the following is introduced: Equations ( 7a)-(7e) can be further transformed as below: (12e)

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 where   and   are the normal stress and the shear stress of the soil skeleton, respectively;   is the pore pressure;  is the contact surface displacement between the strip foundation and underlying soil;  is the rotation of the centre of the strip foundation; and Δ() 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: From (14f) we can obtain Utilising the Fourier transform on (12b) and (12e) gives Substituting ( 15) into (14b), (14c), (14d), and (14e), then (16a), (14a), (14b), (14c), (14d), and (14e) may be transformed into the following matrix form: The expression of each element of matrix  can be seen in the appendix.
The displacement of the strip foundation surface can be expressed in the following matrix form: where Establishing the following equations by the matrix  and the matrix Γ gives where  is a 6 × 1 matrix.We find from ( 17) and ( 18) that the element  1 in the first row of the matrix  denotes the relationship between the displacement and stress on the strip foundation surface.Thus, when  * (, 0) =  () ⋅  *  (, 0) we obtain() =  1 , thereby expressing every element in matrix  by solving (20).We can obtain() and find that () and 1/ are infinitesimal of the same order when  → ∞.
Using the Fourier inverse transform and combining (21), we obtain Additionally, The dual integral equations of the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata are as follows: where  is the rotation of the centre of the strip foundation and   is the flexural stiffness of the foundation.Simpson's rule is used to conduct numerical calculation.The dynamic compliance coefficient,   , of the rocking vibration of a strip foundation can be expressed as follows [27]: Defining  1 = Re[  ] and  2 = Im[  ], we can obtain foundation stiffness  =  1 /( 2 1 +  2 2 ) and the damping coefficient of the foundation  = − 2 /( 2 1 +  2 2 ) 0 .Here,  0 = √/ is the dimensionless frequency.

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   = 0,  = 0, and   = 0.The foundation parameters for the degenerated case are  = 2 m,  = 35 MPa,  = 0.35, and   = 2650 kg/m 3 .The variation of the dynamic compliance coefficient   with dimensionless frequency  0 is analysed and is then compared with the numerical results by Luco and Westmann [5] when  is 0.25.In Figure 2, "∘" represents the numerical results obtained by Luco and Westmann [5] when  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  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   = 35 MPa,   = 0.45, and   = 1722.5kg/m 3 ; for a saturated half-space,  = 0.35,  = 35 MPa,   = 10 −5 m/s,   = 1000 kg/m 3 ,   = 2650 kg/m 3 , and  = 0.25.The dynamic compliance coefficient   changes over the dimensionless frequency, the state of which is calculated when   = 0, 0.2, 1.0, and 2.0 m for  = 10, and the calculation results are shown in Figure 3.Meanwhile, the dynamic compliance coefficient   changes over the dimensionless frequency, the state of which is calculated when  = 0.0, 0.1, 10, and 1000 for   = 0.2, and the calculation results are shown in Figure 4.
We can see from Figure 3 that   decreases with an increase in the elastic layer thickness, which indicates that

Figure 1 :
Figure 1: Description of the model and coordinate system.

Figure 2 :Figure 3 :
Figure 2: The dynamic compliance coefficient versus the dimensionless frequency.(a) Real part of   versus the dimensionless frequency and (b) imaginary part of   versus the dimensionless frequency.