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

Abstract

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 the mixed 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.

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 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.

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_𝑛. The centre of the footing is subjected to a harmonic moment force, 𝑀𝑒^𝑖𝜔𝑡, with 𝜔 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.

Figure 1 

Description of the model and coordinate system.

2.1. The Dynamic Equations of Elastic Layer under Plane Strain

The wave equations of the elastic layer are𝜕𝜎_𝑥𝐿+𝜕𝑥𝜕𝜏_𝑥𝑧𝐿𝜕𝑧+𝜌_𝐿𝜕²𝑢𝜕𝑡²=0,𝜕𝜏_𝑥𝑧𝐿+𝜕𝑥𝜕𝜎_𝑧𝐿𝜕𝑧+𝜌_𝐿𝜕²𝑤𝜕𝑡²=0, (1a)(1b)

where 𝑢 and 𝑤 are the horizontal and vertical displacements of the soil skeleton, respectively; 𝜎_𝑥𝐿 and 𝜎_𝑧𝐿 are the horizontal and vertical normal stresses, respectively; and 𝜌_𝐿 is single-phase elastic soil density.

The relationship between the stress and the displacement is𝜎_𝑥𝐿=−2𝐺_𝐿1−𝜇_𝐿1−2𝜇_𝐿𝜕𝑢−𝜕𝑥2𝐺_𝐿𝜇_𝐿1−2𝜇_𝐿𝜕𝑤,𝜎𝜕𝑧_𝑧𝐿=−2𝐺_𝐿𝜇_𝐿1−2𝜇_𝐿𝜕𝑢−𝜕𝑥2𝐺_𝐿1−𝜇_𝐿1−2𝜇_𝐿𝜕𝑤,𝜏𝜕𝑧_𝑥𝑧𝐿=−𝐺_𝐿𝜕𝑢+𝜕𝑧𝜕𝑤𝜕𝑥, (2a)(2b)(2c)

where 𝐺_𝐿 and 𝜇_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:𝜕𝜎^′_𝑥+𝜕𝑥𝜕𝜏_𝑥𝑧+𝜕𝑧𝜕𝑝_𝑓𝜕𝜕𝑥+𝜌²𝑢𝜕𝑡²+𝜌_𝑓𝜕²𝑤_𝑥𝜕𝑡²=0,𝜕𝜏_𝑥𝑧+𝜕𝑥𝜕𝜎^′_𝑧+𝜕𝑧𝜕𝑝_𝑓𝜕𝜕𝑧+𝜌²𝑤𝜕𝑡²+𝜌_𝑓𝜕²𝑤_𝑧𝜕𝑡²−=0,𝜕𝑝_𝑓=𝜌𝜕𝑥^𝑓𝑔𝑘_𝑑𝜕𝑤_𝑥𝜕𝑡+𝜌_𝑓𝜕²𝑢𝜕𝑡²+𝜌_𝑓𝑛𝜕²𝑤_𝑥𝜕𝑡²,−𝜕𝑝_𝑓=𝜌𝜕𝑧^𝑓𝑔𝑘_𝑑𝜕𝑤_𝑧𝜕𝑡+𝜌_𝑓𝜕²𝑤𝜕𝑡²+𝜌_𝑓𝑛𝜕²𝑤_𝑧𝜕𝑡²,−𝜕²𝑤_𝑥−𝜕𝜕𝑥𝜕𝑡²𝑤_𝑧=𝜕𝜕𝑧𝜕𝑡²𝑢+𝜕𝜕𝑥𝜕𝑡²𝑤,𝜕𝑧𝜕𝑡 (3a)(3b)(3c)(3d)(3e)

where 𝑢 and 𝑤 are the horizontal and vertical displacements of the soil skeleton, respectively; 𝑤_𝑥 and 𝑤_𝑧 are the horizontal and vertical displacements of water relative to the soil skeleton; 𝜎^′_𝑥 and 𝜎^′_𝑧 are the horizontal and vertical effective normal stresses, respectively; 𝑝_𝑓 is the excess pore pressure; 𝜌 is the mass density of the saturated soil with 𝜌=(1−𝑛)𝜌_𝑠+𝑛𝜌_𝑓; 𝜌_𝑠 and 𝜌_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𝜕𝜎^′_𝑥=−2𝐺1−𝜇1−2𝜇𝜕𝑢−𝜕𝑥2𝐺𝜇1−2𝜇𝜕𝑤,𝜕𝑧𝜕𝜎^′_𝑧=−2𝐺𝜇1−2𝜇𝜕𝑢−𝜕𝑥2𝐺1−𝜇1−2𝜇𝜕𝑤,𝜏𝜕𝑧_𝑥𝑧=−𝐺𝜕𝑢+𝜕𝑧𝜕𝑤𝜕𝑥, (4a)(4b)(4c)

where 𝐺 and 𝜇 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 𝑢=𝑢𝑒^𝑖𝜔𝑡,𝑤=𝑤𝑒^𝑖𝜔𝑡,𝑤_𝑥=𝑤_𝑥𝑒^𝑖𝜔𝑡,𝑤_𝑧=𝑤_𝑧𝑒^𝑖𝜔𝑡,𝜎_𝑥𝐿=𝜎_𝑥𝐿𝑒^𝑖𝜔𝑡,𝜏_𝑥𝑧𝐿=𝜏_𝑥𝑧𝐿𝑒^𝑖𝜔𝑡,𝜎_𝑧𝐿=𝜎_𝑧𝐿𝑒^𝑖𝜔𝑡,𝜎^′_𝑥=𝜎_𝑥𝑒^𝑖𝜔𝑡,𝜏_𝑥𝑧=𝜏_𝑥𝑧𝑒^𝑖𝜔𝑡,𝜎^′_𝑧=𝜎^′_𝑧𝑒^𝑖𝜔𝑡,𝑝_𝑓=𝑝_𝑓𝑒^𝑖𝜔𝑡. (5)

Here, 𝜔 is the circular frequency. The Fourier transform can be written as: 𝜙^∗=ℜ𝜙=^∞_−∞𝜙𝑒^𝑖𝜁𝑥𝑑𝑥. (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:𝑖𝑢^∗(𝜁,𝑧)=−𝐴₁𝜁𝐻_𝐿𝑒^𝑞𝐿𝑧−𝐴₂𝑒^𝐹𝐿𝑧𝜁−𝐵₁𝜁𝐻_𝐿𝑒^{−𝑞𝐿𝑧}−𝐵₂𝑒^{−𝐹𝐿𝑧}𝜁,𝑤^∗(𝜁,𝑧)=𝐴₁𝑞_𝐿𝐻_𝐿𝑒^𝑞𝐿𝑧+𝐴₂𝑒^𝐹𝐿𝑧𝐹_𝐿−𝐵₁𝑞_𝐿𝐻_𝐿𝑒^{−𝑞𝐿𝑧}−𝐵₂𝑒^{−𝐹𝐿𝑧}𝐹_𝐿,𝑒^∗(𝜁,𝑧)=𝐴₁𝑒^𝑞𝐿𝑧+𝐵₁𝑒^{−𝑞𝐿𝑧},𝑖𝜏^∗_𝑥𝑧𝐿(𝜁,𝑧)=2𝐴₁𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿e^𝑞𝐿𝑧−2𝐵₁𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿𝑒^{−𝑞𝐿𝑧}+𝐴₂𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿𝑒^𝐹𝐿𝑧−𝐵₂𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿𝑒^{−𝐹𝐿𝑧},𝜎^∗_𝑧𝐿(𝜁,𝑧)=−2𝐺_𝐿𝐴₁𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿𝑒^𝑞𝐿𝑧+𝐴₂𝑒^𝐹𝐿𝑧+𝐵₁𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿𝑒^{−𝑞𝐿𝑧}+𝐵₂𝑒^{−𝐹𝐿𝑧}, (7a)(7b)(7c)(7d)(7e)

where 𝜕𝑒=𝑢+𝜕𝜕𝑥𝑤,𝐹𝜕𝑧_2𝐿=𝜁²−𝜔²𝐺_𝐿𝜌_𝐿,𝐷_𝐿𝜔=−²1−2𝜇_𝐿𝜌_𝐿2𝐺_𝐿1−𝜇_𝐿,𝑞²_𝐿=𝜁²+𝐷_𝐿,𝐻_𝐿=1𝑞²_𝐿−𝜁²=1𝐷_𝐿. (8)

Similarly, combining (3a)–(3e) and (4a)–(4c) and utilising the Fourier transform, we obtain the solutions of the saturated half-space as follows:𝑖𝑢^∗(𝜁,𝑧)=−𝐴₀𝜁𝐻𝑒^−𝑞𝑧−𝐵₀𝜁𝑁𝑒^−𝜁𝑧−𝐶₀𝑒^−𝐹𝑧𝜁,𝑤^∗(𝜁,𝑧)=−𝐴₀𝑞𝐻𝑒^−𝑞𝑧−𝐵₀𝜁𝑁𝑒^−𝜁𝑧−𝐶₀𝑒^−𝐹𝑧𝐹,𝑝^∗_𝑓(𝜁,𝑧)=𝐴₀𝐸𝑒^−𝑞𝑧+𝐵₀𝑒^−𝜁𝑧,𝑒^∗(𝜁,𝑧)=𝐴₀𝑒^−𝑞𝑧,𝑖𝜏^∗_𝑥𝑧(𝜁,𝑧)=−2𝐴₀𝜁𝑞𝐻𝐺𝑒^−𝑞𝑧−2𝐵₀𝜁²𝑁𝐺𝑒^−𝜁𝑧−𝐶₀𝐹𝜁+𝜁𝐹𝐺𝑒^−𝐹𝑧,𝜎^∗_𝑧(𝜁,𝑧)=−2𝐺𝐴₀𝑞²𝜇𝐻+𝑒1−2𝜇^−𝑞𝑧+𝐵₀𝜁²𝑁𝑒^−𝜁𝑧+𝐶₀𝑒^−𝐹𝑧+𝐴₀𝐸𝑒^−𝑞𝑧+𝐵₀𝑒^−𝜁𝑧, (9a)(9b)(9c)(9d)(9e)(9f)

where 𝑐=𝑛𝑘_𝑑𝜔𝑔𝑛𝑖−𝑘_𝑑𝜔,𝜕𝑒=𝑢+𝜕𝜕𝑥𝑤,𝐹𝜕𝑧₂=𝜁²−𝜔²𝐺𝑐𝜌_𝑓,𝜔+𝜌𝐸=²𝜌_𝑓(𝑐+1),𝜔𝐷𝑐𝐷=²1−2𝜇2𝑐𝜌_𝑓+𝜌_𝑓−𝑐𝜌,𝑞2𝑐𝐺1−𝜇₂=𝜁²1+𝐷,𝐻=𝑞²−𝜁²=1𝐷,𝑁=𝑐+1𝐺𝜁²−𝐹²=𝑐+1𝜔²𝑐𝜌_𝑓.+𝜌 (10)

The transformation as shown in the following is introduced: 𝐴₁=12𝐴₁+𝐵₁,𝐵₁=12𝐴₁−𝐵₁,𝐴₂=12𝐴₂+𝐵₂,𝐵₂=12𝐴₂−𝐵₂. (11)

Equations (7a)–(7e) can be further transformed as below:𝑖𝑢^∗(𝜁,𝑧)=−𝐴₁𝜁𝐻_𝐿ch𝑞_𝐿𝑧−𝐴₂ch𝐹_𝐿𝑧𝜁−𝐵₁𝜁𝐻_𝐿sh𝑞_𝐿𝑧−𝐵₂sh𝐹_𝐿𝑧𝜁,𝑤^∗(𝜁,𝑧)=𝐴₁𝑞_𝐿𝐻_𝐿sh𝑞_𝐿𝑧+𝐴₂sh𝐹_𝐿𝑧𝐹_𝐿+𝐵₁𝑞_𝐿𝐻_𝐿ch𝑞_𝐿𝑧+𝐵₂ch𝐹_𝐿𝑧𝐹_𝐿,𝑒^∗(𝜁,𝑧)=𝐴₁ch𝑞_𝐿𝑧+𝐵₁sh𝑞_𝐿𝑖𝑧,𝜏^∗_𝑥𝑧𝐿(𝜁,𝑧)=2𝐴₁𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿sh𝑞_𝐿𝑧+𝐴₂𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿sh𝐹_𝐿𝑧+2𝐵₁𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿ch𝑞_𝐿𝑧+𝐵₂𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿ch𝐹_𝐿𝜎𝑧,_∗𝑧𝐿(𝜁,𝑧)=−2𝐺_𝐿𝐴₁𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿ch𝑞_𝐿𝑧+𝐴₂ch𝐹_𝐿𝑧+𝐵₁×𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿sh𝑞_𝐿𝑧+𝐵₂sh𝐹_𝐿𝑧. (12a)(12b)(12c)(12d)(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 as𝑤(𝑥,0)=𝑥𝜙−Δ(𝑥)|𝑥|≤𝑏,𝜏_𝑥𝑧𝐿𝑥,−𝐻_𝑛[=0−∞<𝑥<∞,𝑤(𝑥,0)]_𝐿=𝑤()[𝑥,0−∞<𝑥<∞,𝑢(𝑥,0)]_𝐿=𝑢(𝑥,0)−∞<𝑥<∞,𝜏_𝑥𝑧𝐿(𝜁,0)=𝜏_𝑥𝑧(𝜁,0)−∞<𝑥<∞,𝜎_𝑧𝐿(𝜁,0)=𝜎_𝑧(𝜁,0)−∞<𝑥<∞,𝑝_𝑓(𝜁,0)=0−∞<𝑥<∞,𝜎_𝑧𝐿𝑥,−𝐻_𝑛=0|𝑥|>𝑏, (13a)(13b)(13c)(13d)(13e)(13f)(13g)(13h)

where 𝜎_𝑧 and 𝜏_𝑥𝑧 are the normal stress and the shear stress of the soil skeleton, respectively; 𝑝_𝑓 is the pore pressure; 𝑤 is the contact surface displacement between the strip foundation and underlying soil; 𝜙 is the rotation of the centre of the strip foundation; and Δ(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:2𝐴₁𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿sh−𝑞_𝐿𝐻_𝑛+𝐴₂𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿sh−𝐹_𝐿𝐻_𝑛+2𝐵₁𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿ch−𝑞_𝐿𝐻_𝑛+𝐵₂𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿ch−𝐹_𝐿𝐻_𝑛=0,𝐵₁𝑞_𝐿𝐻_𝐿+𝐵₂𝐹_𝐿+𝐴₀𝑞𝐻+𝐵₀𝐶𝜁𝑁+⁰𝐹−=0,𝐴₁𝜁𝐻_𝐿−𝐴₂𝜁+𝐴₀𝜁𝐻+𝐵₀𝐶𝜁𝑁+⁰𝜁2=0,𝐵₁𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿+𝐵₂𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿+2𝐴₀𝜁𝑞𝐻𝐺+2𝐵₀𝜁²𝑁𝐺+𝐶₀𝐹𝜁+𝜁𝐹𝐺=0,−2𝐺_𝐿𝐴₁𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿+𝐴₂+2𝐺𝐴₀𝑞²𝜇𝐻+1−2𝜇+𝐵₀𝜁²𝑁+𝐶₀−𝐴₀𝐸−𝐵₀𝐴=0₀𝐸+𝐵₀=0. (14a)(14b)(14c)(14d)(14e)(14f)

From (14f) we can obtain 𝐵₀=−𝐴₀𝐸. (15)

Utilising the Fourier transform on (12b) and (12e) gives−2𝐺_𝐿𝐴₁𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿ch−𝑞_𝐿𝐻_𝑛+𝐴₂ch−𝐹_𝐿𝐻_𝑛+𝐵₁𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿sh−𝑞_𝐿𝐻_𝑛+𝐵₂sh−𝐹_𝐿𝐻_𝑛=𝜎^∗_𝑧(𝜁,0),𝐴₁𝑞_𝐿𝐻_𝐿sh−𝑞_𝐿𝐻_𝑛+𝐴₂sh−𝐹_𝐿𝐻_𝑛𝐹_𝐿+𝐵₁𝑞_𝐿𝐻_𝐿ch−𝑞_𝐿𝐻_𝑛+𝐵₂ch−𝐹_𝐿𝐻_𝑛𝐹_𝐿=𝑤^∗(𝜁,0). (16a)(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: 𝑇₁₁𝑇₁₂𝑇₁₃𝑇₁₄𝑇₁₅𝑇₁₆𝑇₂₁𝑇₂₂𝑇₂₃𝑇₂₄𝑇₂₅𝑇₂₆𝑇₃₁𝑇₃₂𝑇₃₃𝑇₃₄𝑇₃₅𝑇₃₆𝑇₄₁𝑇₄₂𝑇₄₃𝑇₄₄𝑇₄₅𝑇₄₆𝑇₅₁𝑇₅₂𝑇₅₃𝑇₅₄𝑇₅₅𝑇₅₆𝑇₆₁𝑇₆₂𝑇₆₃𝑇₆₄𝑇₆₅𝑇₆₆⋅𝐴₁𝐵₁𝐴₂𝐵₂𝐴₀𝐶₀=𝜎^∗_𝑧()00000𝜁,0. (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: Γ₁Γ₂Γ₃Γ₄Γ₅Γ₆𝐴₁𝐵₁𝐴₂𝐵₂𝐴₀𝐶₀^𝑇=𝑤^∗(𝜁,0), (18) where Γ₁=𝑞_𝐿𝐻_𝐿sh−𝑞_𝐿𝐻_𝑛,Γ₂=𝑞_𝐿𝐻_𝐿ch−𝑞_𝐿𝐻_𝑛,Γ₃=sh−𝐹_𝐿𝐻_𝑛𝐹_𝐿,Γ₄=ch−𝐹_𝐿𝐻_𝑛𝐹_𝐿,Γ₅=Γ₆=0. (19)

Establishing the following equations by the matrix 𝑇 and the matrix Γ gives 𝑇⋅𝑋=Γ, (20)

where 𝑋 is a 6×1 matrix.

We find from (17) and (18) that the element 𝑋₁ in the first row of the matrix 𝑋 denotes the relationship between the displacement and stress on the strip foundation surface. Thus, when 𝑤^∗(𝜁,0)=𝑓(𝜁)⋅𝜎^∗_𝑧(𝜁,0) (21) we obtain𝑓(𝜁)=𝑋₁, thereby expressing every element in matrix 𝑋 by solving (20). We can obtain𝑓(𝜁) and find that 𝑓(𝜁) and 1/𝜁 are infinitesimal of the same order when 𝜁→∞.

Using the Fourier inverse transform and combining (21), we obtain 𝑤()=1𝑥,02𝜋^∞_−∞𝑤^∗(𝜁,0)𝑒^{−𝑖𝜁𝑥}=1𝑑𝜁𝜋^∞₀𝑓(𝜁)𝜎^∗_𝑧(𝜁,0)sin(𝜁𝑥)𝑑𝜁. (22)

Additionally, 𝜎_𝑧()=1𝑥,02𝜋^∞_−∞𝜎^∗_𝑧(𝜁,0)𝑒^{−𝑖𝜁𝑥}=1𝑑𝜁𝜋^∞₀𝜎^∗_𝑧(𝜁,0)sin(𝜁𝑥)𝑑𝜁. (23)

The dual integral equations of the rocking vibration of an elastic strip foundation on elastic soil with saturated substrata are as follows: ^∞₀𝑓𝜁̃𝜎^∗_𝑧𝜁̃,0sin𝜁̃𝑥𝑑𝜁̃+𝛿𝑏²6𝑥³−𝑥²×^∞₀1+𝜌ℎ𝜔²𝑓𝜁̃/𝐺𝜁̃𝜎^∗_𝑧𝜁̃,0cos𝜁̃𝑑𝜁̃−𝛿𝑏²𝑥^2∞₀1+𝜌ℎ𝜔²𝑓𝜁̃/𝐺𝜁̃²𝜎^∗_𝑧𝜁̃,0sin𝜁̃𝑑𝜁̃−𝛿𝑏²𝑥^∞₀1+𝜌ℎ𝜔²𝑓𝜁̃/𝐺𝜁̃³𝜎^∗_𝑧𝜁̃,0𝑑𝜁̃−𝑏²𝛿^∞₀1+𝜌ℎ𝜔²𝑓𝜁̃/𝐺𝜁̃⁴𝜎^∗_𝑧𝜁̃,0sin𝜁̃𝑥−2𝑑𝜁̃=𝜋𝑥𝜙𝑏²0≤|𝑥|≤1,_∞0𝜎^∗_𝑧𝜁̃,0sin𝜁̃𝑥𝑑𝜁̃=0|𝑥|>1, (24) where 𝜙 is the rotation of the centre of the strip foundation and 𝐷_𝑓 is the flexural stiffness of the foundation.

Simpson’s rule is used to conduct numerical calculation. The dynamic compliance coefficient, 𝐶_𝑀, of the rocking vibration of a strip foundation can be expressed as follows [27]: 𝐶_𝑀=1𝑏𝑎₀. (25)

Defining 𝑓₁=Re[𝐶_𝑀] and 𝑓₂=Im[𝐶_𝑀], we can obtain foundation stiffness 𝐾=𝑓₁/(𝑓²₁+𝑓²₂) and the damping coefficient of the foundation 𝐶=−𝑓₂/(𝑓²₁+𝑓²₂)𝑏₀. Here, 𝑏₀=𝑏𝜔𝜌/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 𝜌_𝑓=0, 𝛿=0, and 𝐻_𝑛=0. The foundation parameters for the degenerated case are 𝑏=2 m, 𝐺=35 MPa, 𝑛=0.35, and 𝜌_𝑠=2650 kg/m³. The variation of the dynamic compliance coefficient 𝐶_𝑀 with dimensionless frequency 𝑏₀ is analysed and is then compared with the numerical results by Luco and Westmann [5] when 𝜇 is 0.25. In Figure 2, “∘” represents the numerical results obtained by Luco and Westmann [5] when 𝜇 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 𝛿 equals zero.

(a)

(b)

Figure 2 

The dynamic compliance coefficient versus the dimensionless frequency. (a) Real part of 𝐶𝑀 versus the dimensionless frequency and (b) imaginary part of 𝐶𝑀 versus the dimensionless frequency.

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_𝐿=35 MPa, 𝜇_𝐿=0.45, and 𝜌_𝐿=1722.5 kg/m³; for a saturated half-space, 𝑛=0.35, 𝐺=35 MPa, 𝑘_𝑑=10⁻⁵ m/s, 𝜌_𝑓=1000 kg/m³, 𝜌_𝑠=2650 kg/m³, and 𝜇=0.25. The dynamic compliance coefficient 𝐶_𝑀 changes over the dimensionless frequency, the state of which is calculated when 𝐻_𝑛=0, 0.2, 1.0, and 2.0 m for 𝛿=10, and the calculation results are shown in Figure 3. Meanwhile, the dynamic compliance coefficient 𝐶_𝑀 changes over the dimensionless frequency, the state of which is calculated when 𝛿=0.0, 0.1, 10, and 1000 for 𝐻_𝑛=0.2, and the calculation results are shown in Figure 4.

(a)

(b)

Figure 3 

The dynamic compliance coefficient 𝐶𝑀 of different single-phase elastic layer thicknesses (𝐻𝑛). (a) Real part of 𝐶𝑀 versus the dimensionless frequency and (b) imaginary part of 𝐶𝑀 versus the dimensionless frequency.

(a)

(b)

Figure 4 

The dynamic compliance coefficient 𝐶𝑀 for different 𝛿. (a) Real part of 𝐶𝑀 versus the dimensionless frequency and (b) imaginary part of 𝐶𝑀 versus the dimensionless frequency.

We can see from Figure 3 that 𝐶_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_𝑀, as 𝑏₀ increases, the real part of 𝐶_𝑀 decreases, but the curve tends to flatten. For the imaginary part of 𝐶_𝑀, as 𝑏₀ increases, the imaginary part of 𝐶_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_𝑀 are essentially coincident when 𝛿=0.0 and 0.1, which can be seen from Figure 4. The curve for 𝛿=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 𝑏₀ when 𝛿≤0.1. It is also demonstrated that the real part of 𝐶_𝑀 is greatly influenced by variations in 𝑏₀ when 𝑏₀<2.8 and that the imaginary part of 𝐶_𝑀 is greatly influenced by variations in 𝑏₀ when 𝑏₀<3.6. However, when 𝑏₀ exceeds the critical values (2.8 for the real part of 𝐶_𝑀 and 3.6 for the imaginary part of 𝐶_𝑀), the curve of 𝐶_𝑀 tends to flatten and 𝐶_𝑀 is only slightly influenced by variations in the dimensionless frequency 𝑏₀. Figure 4 also shows that the dynamic compliance coefficient curve obtained when 𝛿=10 and 1000 is remarkably different from the one obtained when 𝛿=0.0 and 0.1, and the absolute values concerning both the real parts and imaginary parts of the dynamic compliance coefficient when 𝛿=10 and 1000 are larger than the ones when 𝛿=0.0 and 0.1. Moreover, it can be seen from Figure 4 that when 𝛿 is large, the variation of the dynamic compliance coefficient curve with the dimensionless frequency 𝑏₀ tends to be smooth. However, for an average quantity of 𝛿, the variation of the absolute values of the real parts and imaginary parts of 𝐶_𝑀 with 𝑏₀ is significant. Thus, under ordinary circumstances (𝛿=10), we must consider effects exerted by the dimensionless frequency 𝑏₀.

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_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_𝑀 is greatly influenced by variations in the dimensionless frequency 𝑏₀ when 𝑏₀<2.8, and the imaginary part of 𝐶_𝑀 is greatly influenced by variations in the dimensionless frequency 𝑏₀ when 𝑏₀<3.6. (3) When the flexural stiffness of the elastic foundation is comparatively large or when 𝛿≤0.1, the influence of 𝛿 on the rocking vibration can be ignored. (4) When 𝛿>0.1 and as 𝛿 increases, the rocking vibration of the elastic foundation changes significantly, and the absolute values of both the real parts and imaginary parts of 𝐶_𝑀 increase.

Appendix

Expressions for each element of the matrix 𝑇 appearing previously are given by 𝑇₁₁=−2𝐺_𝐿𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿ch−𝑞_𝐿𝐻_𝑛,𝑇₁₂=−2𝐺_𝐿𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿sh−𝑞_𝐿𝐻_𝑛,𝑇₁₃=−2𝐺_𝐿ch−𝐹_𝐿𝐻_𝑛,𝑇₁₄=−2𝐺_𝐿sh−𝐹_𝐿𝐻_𝑛,𝑇₁₅𝑇=0,₁₆𝑇=0,₂₁=2𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿sh−𝑞_𝐿𝐻_𝑛,𝑇₂₂=2𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿ch−𝑞_𝐿𝐻_𝑛,𝑇₂₃=𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿sh−𝐹_𝐿𝐻_𝑛,𝑇₂₄=𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿ch−𝐹_𝐿𝐻_𝑛,𝑇₂₅𝑇=0,₂₆𝑇=0,₃₁𝑇=0,₃₂=𝑞_𝐿𝐻_𝐿,𝑇₃₃𝑇=0,₃₄=1𝐹_𝐿,𝑇₃₅𝑇=𝑞𝐻−𝜁𝑁𝐸,₃₆=1𝐹,𝑇₄₁=−𝜁𝐻_𝐿,𝑇₄₂𝑇=0,₄₃1=−𝜁,𝑇₄₄𝑇=0,₄₅𝑇=𝜁𝐻−𝜁𝑁𝐸,₄₆=1𝜁,𝑇₅₁𝑇=0,₅₂=2𝜁𝑞_𝐿𝐻_𝐿𝐺_𝐿,𝑇₅₃𝑇=0,₅₄=𝐹_𝐿𝜁+𝜁𝐹_𝐿𝐺_𝐿,𝑇₅₅=2𝜁𝑞𝐻𝐺−2𝜁²𝑇𝑁𝐺𝐸,₅₆=𝐹𝜁+𝜁𝐹𝑇𝐺,₆₁=−2𝐺_𝐿𝑞²_𝐿𝐻_𝐿+𝜇_𝐿1−2𝜇_𝐿,𝑇₆₂𝑇=0,₆₃=−2𝐺_𝐿,𝑇₆₄𝑇=0,₆₅=2𝐺𝑞²𝜇𝐻+1−2𝜇−𝜁²,𝑇𝑁𝐸₆₆=2𝐺. (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).