Torsional vibrations of a cylindrical foundation embedded in a saturated poroelastic half-space

Abstract

Considering the interactions between an embedded foundation and saturated soil, the torsional vibrations of a cylindrical foundation embedded in a saturated poroelastic medium are analyzed in this paper. Both a rigid foundation and an elastic foundation are considered. Assuming both the side surface and the bottom surface of the foundation are perfectly bonded to soil, the reaction torques that the side soil and bottom soil acting on the foundation can be gained from basic dynamic equations of the poroelastic medium. According to the dynamic equilibrium equations of a foundation under harmonic torque, the torsional vibrations of an embedded cylindrical foundation are presented. Besides, the angular amplitude of the foundation, the equivalent stiffness and damping coefficients of the soil are expressed explicitly. Selected examples are presented to investigate the influence of relevant parameters on the torsional vibrations.

Keywords

embedded foundation / saturated soil / rigid foundation / elastic foundation / torsional vibration

Introduction

The dynamic response problems of foundations resting on the surface of soil have been investigated by many researches. However, in practice, many foundations are usually embedded in soil, and experiments indicate that embedment has significant effect on the dynamic response of foundations. So it is necessary to analyze the interaction problem between an embedded foundation and soil. Pak [ 1] and Selvadurai [ 2] studied the torsional and rotary vibrations of a rigid disc embedded in elastic half space, respectively. The vertical vibrations of an arbitrarily embedded rigid plate were also considered by Pak and Gobert [ 3]. Rahman [ 4] analyzed the Reissner-Sagoci problem of half space under buried torsional force. All studies mentioned above assumed the height of the foundation is very small and omitted the interactions between the side surface of the foundation and soil. Assuming the height of the foundation is a given value, Novak [ 5] investigated the vertical vibration of partially embedded circular footings, and pointed out that there is a decrease in resonant amplitudes and an increase in resonant frequencies with increasing embedment depth and increasing density of the backfill. Novak and Sachs [ 6] further studied the torsional and coupled vibrations of embedded footings. After conducting field experiments, Sankaran [ 7] studied the vibration problems of an embedded foundation subjected to harmonic torsional loading, and drew the conclusion similar to that of Novak. Apsel and Luco [ 8] derived the impedance functions for a massless foundation embedded in a layered half space by an integral equation technique. Resorting to a combination of boundary element method and finite-element method, Spyrakos [ 9] investigated the dynamic response of a flexible massive strip-foundation embedded in a layered soil.

Since Biot first established the theory of wave propagation in a fluid-filled, poroelastic solid, many researchers used this theory and studied the dynamic response of foundations embedded in saturated media. Philippacopoulos [ 10] and Senjuntichai [ 11] gained the Green’s functions for buried point source and buried line source in poroelastic half space, respectively. By using Hankel integral transforms, Zeng and Rajapakse [ 12] considered the steady-state vertical vibrations of a rigid circular disk embedded at a finite depth below the free surface of a poroelastic medium. However, Zeng did not take into account the height of the foundation. Senjuntichai and Sapsathiarn [ 13] investigated vertical harmonic vibrations of a flexible circular plate embedded in a multilayered poroelastic half space, and did not considered the height of the foundation either. Applying the similar assumptions using by Novak, Hu and Cai [ 14, 15] studied the steady-state vertical and rocking vibrations of a rigid, cylindrical foundation embedded in a poroelastic soil, respectively. In these two papers, the soil underlying foundation base was treated as a homogeneous poroelastic half space and the soil along the side of the foundation was considered to consist of a series of infinitesimally thin layers. Taking into account the actual situation that bedrock often underlies the soil, Hu and Cai [ 16] further studied the vertical vibration of rigid embedded foundations in saturated soil overlying bedrock.

It is worth mentioning that torsional vibrations of a cylindrical foundation embedded in saturated soil have not been reported in the literature. The main objective of present paper is to solve this problem. Both the interaction between the bottom surface of foundation and saturated soil and the interaction between the side surface of foundation and saturated soil are taken into account. Besides, a rigid cylindrical foundation and an elastic cylindrical foundation are also be considered.

General solutions to basic dynamic equations

Considering the model shown in Fig. 1. The z-axis of the cylindrical coordinate system coincides with the vertical symmetry axis of the foundation. The cylindrical foundation is embedded in a saturate poroelastic half-space. The radius of the foundation is a. Torsional motion is induced by load T{e}^{i\omega t} , where \omega is the circular frequency of the harmonic vibration and i is the imaginary unit. Due to the symmetry of the problem, the motion is independent of \theta , and the only non-vanishing components of the displacement vector are u_{\theta }{e}^{i\omega t} and w_{\theta }{e}^{i\omega t} . For brevity, the harmonic time factor e^{i\omega t} is suppressed later.

During analyze the dynamic response of footings partially embedded in a single-phase elastic soil, Novak and Sachs [ 6] adopted the fundamental assumptions proposed by Baranov [ 17]. In present paper, several similar hypotheses are introduced, those are: 1) The saturated half-space are treated as two parts, one is the saturated layer that surrounding the embedded foundation, and the other is the half-space that underlying the foundation. 2) The surrounding saturated layer is an independent saturated layer composed of a series of infinitesimally thin independent elastic layer. 3) In each thin independent saturated layer, the tangential displacement is independent of z. 4) The foundation and the saturated soil are perfectly bonded. 5) The interaction between the bottom surface of foundation and saturated soil is independent of the depth of embedment.

Basing on the former assumptions, the equations of motion for side saturated layer can be written as:

(1) G\left({\nabla }^{\mathrm{2}}-{\displaystyle \frac{\mathrm{1}}{r^{\mathrm{2}}}}\right)u_{\theta }=\rho {\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_{\theta }}{\partial t^{\mathrm{2}}}}+{\rho }_f{\displaystyle \frac{{\partial }^{\mathrm{2}}{w}_{\theta }}{\partial t^{\mathrm{2}}}},

(2) {\rho }_f{\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_{\theta }}{\partial t^{\mathrm{2}}}}+{\displaystyle \frac{{\rho }_f}{n}}{\displaystyle \frac{{\partial }^{\mathrm{2}}{w}_{\theta }}{\partial t^{\mathrm{2}}}}+{\displaystyle \frac{\eta }{k}}{\displaystyle \frac{\partial w_{\theta }}{\partial t}}=\mathrm{0},

Here G is Lame’s constant of the soil skeleton. w_{\theta }=n(v_{\theta }-u_{\theta }) , u_{\theta } and v_{\theta } are the tangential displacements of solid and fluid, respectively. n is the porosity coefficient of the medium. {\nabla }^{\mathrm{2}}={\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial r^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{r}}{\displaystyle \frac{\partial }{\partial r}} . {\rho }_s , {\rho }_f are the mass densities of the soil skeleton and the pore water, respectively. \rho =(\mathrm{1}-n){\rho }_s+n{\rho }_f . k is the permeability coefficients of saturated soil, and \eta is the viscosity coefficient.

The constitutive relations of the soil skeleton can be expressed as:

(3a) {\tau }_{z\theta }=G{\displaystyle \frac{\partial u_{\theta }}{\partial z}},

(3b) {\tau }_{r\theta }=G\left({\displaystyle \frac{\partial u_{\theta }}{\partial r}}-{\displaystyle \frac{u_{\theta }}{r}}\right),

where {\tau }_{z\theta } and {\tau }_{r\theta } are the shear stress components.

It is convenient to introduce the dimensionless frequency as a_0=a\omega \sqrt{\rho /G} , and other dimensionless variables as:

\overline{r}={\displaystyle \frac{r}{a}},\overline{z}={\displaystyle \frac{z}{a}},\overline{h}={\displaystyle \frac{h}{a}},{\overline{u}}_{\theta }={\displaystyle \frac{u_{\theta }}{a}},{\overline{w}}_{\theta }={\displaystyle \frac{w_{\theta }}{a}},\overline{\rho }={\displaystyle \frac{{\rho }_f}{\rho }},{\overline{\rho }}_d={\displaystyle \frac{{\rho }_d}{\rho }},\overline{\tau }={\displaystyle \frac{\tau }{G}},\overline{k}=\sqrt{{\displaystyle \frac{\rho }{G}}}{\displaystyle \frac{\eta }{{\rho }_{f}k}}a,\overline{T}={\displaystyle \frac{T}{G{a}^{\mathrm{3}}}},T_s={\displaystyle \frac{T}{G{a}^{\mathrm{3}}},}{T}_b={\displaystyle \frac{T}{G{a}^{\mathrm{3}}}},{\overline{G}}_d={\displaystyle \frac{G_d}{G}}

.

After introducing the dimensionless variables, Eqs. (1)-(3) can be written as:

(4) \left({\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial {\overline{r}}^{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{\overline{r}}}{\displaystyle \frac{\partial }{\partial \overline{r}}}-{\displaystyle \frac{\mathrm{1}}{{\overline{r}}^{\mathrm{2}}}}\right){\overline{u}}_{\theta }=-a_{\mathrm{0}}^{\mathrm{2}}{\overline{u}}_{\theta }-a_{\mathrm{0}}^{\mathrm{2}}\overline{\rho }{\overline{w}}_{\theta },

(5) i\overline{k}{\overline{w}}_{\theta }=a_{\mathrm{0}}{\overline{u}}_{\theta }+{\displaystyle \frac{a_{\mathrm{0}}}{n}}{\overline{w}}_{\theta },

(6a) {\overline{\tau }}_{z\theta }={\displaystyle \frac{\partial {\overline{u}}_{\theta }}{\partial \overline{z}}},

(6b) {\overline{\tau }}_{r\theta }={\displaystyle \frac{\partial {\overline{u}}_{\theta }}{\partial \overline{r}}}-{\displaystyle \frac{{\overline{u}}_{\theta }}{\overline{r}}}.

From Eqs. (4) and (5), following expression can be gained:,

(7) {\overline{r}}^{\mathrm{2}}{\displaystyle \frac{{\partial }^{\mathrm{2}}{\overline{u}}_{\theta }}{\partial {\overline{r}}^{\mathrm{2}}}}+\overline{r}{\displaystyle \frac{\partial {\overline{u}}_{\theta }}{\partial \overline{r}}}+\left(S^{\mathrm{2}}{\overline{r}}^{\mathrm{2}}-\mathrm{1}\right){\overline{u}}_{\theta }=\mathrm{0},

where S=\sqrt{a_{\mathrm{0}}^{\mathrm{2}}+a_{\mathrm{0}}^{\mathrm{3}}\overline{\rho }/\left(i\overline{k}-a_{\mathrm{0}}/n\right)} .

The solution of Eq. (7) can be expressed as:

(8) {\overline{u}}_{\theta }=A_{\mathrm{1}}{H}_1^{(1)}(S\overline{r})+B_{\mathrm{1}}{H}_1^{(2)}(S\overline{r}),

where H_1^{(1)}\left(S\overline{r}\right) and H_1^{(2)}\left(S\overline{r}\right) are the Hankel functions of the first order, first and second kind, respectively; A_{\mathrm{1}} and B_{\mathrm{1}} are the constants to be determined.

As there is only outward transmitted wave in the saturated layer, from the asymptotic characteristic of Hankel function, it is easily found that parameter A_{\mathrm{1}} equals to zero. So Eq. (8) can be expressed as:

(9) {\overline{u}}_{\theta }=B_{\mathrm{1}}{H}_1^{(2)}\left(S\overline{r}\right).

After denoting the torsional angle of the foundation as \phi , the tangent displacement of the foundation can be formulated as:

(10) {\overline{u}}_{\theta }=\overline{r}\phi (\overline{z}).

For the reason of the embedded foundation are bonded to the side saturated layer, there is no relative displacement at the interface. Comparison of Eqs. (9) and (10) yields, with \overline{r}=\mathrm{1} , the parameter B_{\mathrm{1}} is formulated as:

(11) B_{\mathrm{1}}=\phi (\overline{z})/H_1^{(2)}(S).

Substituting Eqs. (9) and (11) into Eq. (6b), the shear stress {\overline{\tau }}_{r\theta } can be drawn as:

(12) {\overline{\tau }}_{r\theta }\left(\overline{r},\overline{z}\right)={\displaystyle \frac{\phi (\overline{z})}{H_{\mathrm{1}}^{(\mathrm{2})}(S)}}\left[S{H}_0^{(2)}\left(S\overline{r}\right)-{\displaystyle \frac{\mathrm{2}}{\overline{r}}}{H}_1^{(2)}\left(S\overline{r}\right)\right].

Reaction torques

Rigid cylindrical foundation

When the foundation is rigid, the tangential displacement of foundation is independence of z, so the shear stress {\overline{\tau }}_{r\theta } at the interface of foundation and side saturated layer can be expressed as:

(13) {\overline{\tau }}_{r\theta }\left(\overline{r}=\mathrm{1},\overline{z}\right)={\displaystyle \frac{\phi }{H_{\mathrm{1}}^{(\mathrm{2})}(S)}}[S{H}_0^{(2)}(S)-\mathrm{2}{H}_1^{(2)}(S)],

where \phi is the amplitude of angular displacement of the rigid foundation.

Integration of moment r{\tau }_{r\theta } along the circumference of the cylinder yields the dimensionless reaction torque that the side saturated soil acting on the foundation:

(14) {\overline{T}}_s=-{\displaystyle {\int }_{\mathrm{0}}^{\overline{h}}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}\mathrm{\pi }}{\overline{\tau }}_{r\theta }}}\mathrm{d}\phi \mathrm{d}\overline{z}=\mathrm{2}\mathrm{\pi }\overline{h}\phi \left[\mathrm{2}-S{\displaystyle \frac{H_{\mathrm{0}}^{(\mathrm{2})}(S)}{H_{\mathrm{1}}^{(\mathrm{2})}(S)}}\right],

where T_s is the reaction torque that the side saturated layer acting on the embedded foundation.

Equation (14) can be expressed in complex number form as follows:

(15) {\overline{T}}_s=\mathrm{2}\mathrm{\pi }\overline{h}\phi \left(\mathrm{2}-S{\displaystyle \frac{J_{\mathrm{0}}{J}_{\mathrm{1}}+Y_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\right)+i\mathrm{2}\mathrm{\pi }\overline{h}\phi \left(S{\displaystyle \frac{J_{\mathrm{1}}{Y}_{\mathrm{0}}-J_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\right).

Here, J_{\mathrm{0}},J_{\mathrm{1}} are Bessel functions of the first kind of order zero and one respectively, and Y_{\mathrm{0}},Y_{\mathrm{1}} are Bessel functions of the second kind of order zero and one.

The torsional vibrations of a rigid disk resting on saturated half space has been investigated in Ref. [ 18], and the relation between the torsional angle and the acting torque can be expressed as follows:

(16) \overline{T}={\displaystyle \frac{\mathrm{16}}{\mathrm{3}}}{\displaystyle \frac{\phi }{C_T}}={\displaystyle \frac{\phi f_{\mathrm{1}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}-i{\displaystyle \frac{\phi f_{\mathrm{2}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}},

where T is the torque that applied on the disk, C_T is the dynamic compliance coefficients of the disk, and f_{\mathrm{1}}=\mathrm{3}\mathrm{Re}(C_T)/\mathrm{16} , f_{\mathrm{2}}=\mathrm{3}\mathrm{Im}(C_T)/\mathrm{16} .

Basing on the assumption that the interaction between the bottom surface of foundation and saturated soil is independent of the embedment depth, the reaction torque that the bottom saturated half space acting on the embedded foundation can be expressed as:

(17) {\overline{T}}_b={\displaystyle \frac{\phi f_{\mathrm{1}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}-i{\displaystyle \frac{\phi f_{\mathrm{2}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}{\overline{T}}_b.

Elastic cylindrical foundation

To establish the basic dynamic equation of an embedded elastic cylindrical foundation, an infinitesimally thin body whose height equals to \mathrm{\Delta }z is selected. As \mathrm{\Delta }z is very small, the torsional angle of the infinitesimal body can be assumed independent of z, so the reaction torque that the side saturated soil acting on this body can be expressed as:

(18) {\overline{T}}_{si}(\overline{z})=-{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{\Delta }\overline{z}}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}\mathrm{\pi }}{\overline{\tau }}_{r\theta }}}\mathrm{d}\phi \mathrm{d}\overline{z}=\mathrm{2}\mathrm{\pi }\mathrm{\Delta }\overline{z}\phi (\overline{z})\left(\mathrm{2}-{\displaystyle \frac{S{H}_0^{(2)}(S)}{H_1^{(2)}(S)}}\right).

From vibration mechanics [ 19], the magnitude of a twisting moment at a cross-section of uniform circular shaft when it is subjected to a torque can be expressed as:

(19) \overline{T}(\overline{z})={\displaystyle {\int }_{\mathrm{0}}^1\overline{\tau }\cdot \mathrm{2}\mathrm{\pi }}{\overline{r}}^{\mathrm{2}}\mathrm{d}\overline{r}={\displaystyle \frac{\mathrm{\pi }{G}_d}{\mathrm{2}G}}{\displaystyle \frac{\mathrm{d}\phi }{\mathrm{d}\overline{z}}},

where G_d is the shear modulus of the elastic foundation.

After denoting the twist moments at the upper and lower surface of the infinitesimal body as T_{\mathrm{1}} and T_{\mathrm{2}} , the dynamic equation of the infinitesimal body can be formulated as:

(20) T_{\mathrm{1}}-T_{\mathrm{2}}-T_{si}=\left[{\rho }_d\mathrm{\Delta }z{\displaystyle {\int }_{\mathrm{0}}^a\mathrm{2}\mathrm{\pi }{r}^{\mathrm{3}}\mathrm{d}r}\right]{\displaystyle \frac{{\partial }^{\mathrm{2}}\phi }{\partial t^{\mathrm{2}}}},

where {\rho }_d is the mass density of the elastic foundation.

Substituting T_{\mathrm{1}} , T_{\mathrm{2}} , T_{si} into Eq. (20) and introducing the dimensionless variables, following expression can be gotten:

(21) {\displaystyle \frac{\mathrm{\pi }{G}_d}{\mathrm{2}G}}{\displaystyle \frac{\left[{\left({\displaystyle \frac{\mathrm{d}\phi }{\mathrm{d}\overline{z}}}\right)}_{\mathrm{1}}-{\left({\displaystyle \frac{\mathrm{d}\phi }{\mathrm{d}\overline{z}}}\right)}_{\mathrm{2}}\right]}{\mathrm{\Delta }\overline{z}}}-\mathrm{2}\mathrm{\pi }\phi (\overline{z})\left(\mathrm{2}-{\displaystyle \frac{S{H}_0^{(2)}(S)}{H_1^{(2)}(S)}}\right)={\displaystyle \frac{{\rho }_d\mathrm{\pi }{a}^2}{\mathrm{2}G}}{\displaystyle \frac{{\partial }^{\mathrm{2}}\phi }{\partial t^{\mathrm{2}}}}.

Considering the exciting torque is a harmonic load, and, in the limit of \mathrm{\Delta }\overline{z}\to \mathrm{0} , Eq. (21) becomes:

(22) {\displaystyle \frac{{\partial }^{\mathrm{2}}\phi }{\partial {\overline{z}}^{\mathrm{2}}}}+X^{\mathrm{2}}\phi =\mathrm{0},

where X=\sqrt{{\varsigma }^{\mathrm{2}}{a}_{\mathrm{0}}^{\mathrm{2}}+\mathrm{4}{\displaystyle \frac{G}{G_d}}\left[\left({\displaystyle \frac{S(J_{\mathrm{0}}{J}_{\mathrm{1}}+Y_{\mathrm{0}}{Y}_{\mathrm{1}})}{J_{\mathrm{1}}^{\mathrm{2}}+Y_{\mathrm{1}}^{\mathrm{2}}}}-\mathrm{2}\right)-{\displaystyle \frac{i(J_{\mathrm{1}}{Y}_{\mathrm{0}}+J_{\mathrm{0}}{Y}_{\mathrm{1}})}{J_{\mathrm{1}}^{\mathrm{2}}+Y_{\mathrm{1}}^{\mathrm{2}}}}\right]} , \varsigma =\sqrt{G/\rho }/\sqrt{G_d/{\rho }_d} , Here parameter \varsigma is the ratio of the shear wave velocities between the saturated soil and elastic foundation.

The solution of the ordinary differential Eq. (22) can be expressed as:

(23) \phi (\overline{z})=A_{\mathrm{2}}\cos \left(X\overline{z}\right)+B_{\mathrm{2}}\sin \left(X\overline{z}\right),

where A_{\mathrm{2}} and B_{\mathrm{2}} are integral constants to be determined.

The dimensionless torque acting in the foundation at a certain position can be expressed as:

(24) \overline{T}(\overline{z})=\mathrm{2}\mathrm{\pi }{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}{\overline{r}}^{\mathrm{2}}}{\overline{G}}_d\overline{r}{\displaystyle \frac{\partial \phi }{\partial \overline{z}}}\mathrm{d}\overline{r}={\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}{\displaystyle \frac{\partial \phi }{\partial \overline{z}}}.

The torque and torsional angle in the bottom surface of the foundation can be denoted as T_{\mathrm{0}} and {\phi }_{\mathrm{0}} .

Combining Eqs. (23) and (24), parameters A₂ and B₂ can be expressed as:

(25) A_{\mathrm{2}}=\phi ,_{\mathrm{0}}

(26) B_{\mathrm{2}}={\overline{T}}_{\mathrm{0}}/\left({\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}X\right).

Substituting Eqs. (25) and (26) into Eq. (23), the torsional angle of the embedded elastic foundation can be expressed as:

(27) \phi (\overline{z})={\phi }_{\mathrm{0}}\left[\cos \left(X\overline{z}\right)+{\displaystyle \frac{{\overline{T}}_{\mathrm{0}}}{{\phi }_{\mathrm{0}}}}{\displaystyle \frac{\sin \left(X\overline{z}\right)}{{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}X}}\right].

Substituting Eq. (27) into Eq. (10), the shear stress {\overline{\tau }}_{r\theta } at the interface of embedded elastic foundation and side saturated soil is given as:

(28) {\overline{\tau }}_{r\theta }\left(\overline{r}=\mathrm{1},\overline{z}\right)={\phi }_{\mathrm{0}}\left[\cos \left(X\overline{z}\right)+{\displaystyle \frac{{\overline{T}}_{\mathrm{0}}}{{\phi }_{\mathrm{0}}}}{\displaystyle \frac{\sin \left(X\overline{z}\right)}{{\displaystyle \frac{\pi }{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}X}}\right]\left({\displaystyle \frac{S{H}_{\mathrm{0}}^{(\mathrm{2})}(S)}{H_{\mathrm{1}}^{(\mathrm{2})}(S)}}-\mathrm{2}\right).

Integration of moment r{\tau }_{r\theta } along the circumference of the cylinder yields the dimensionless reaction torque that the side saturated layer acting on the elastic foundation:

(29) {\overline{T}}_s=-{\displaystyle {\int }_{\mathrm{0}}^h{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}\pi }{\overline{\tau }}_{r\theta }\mathrm{d}\phi }d\overline{z}}=\mathrm{2}\mathrm{\pi }\overline{h}{\phi }_{\mathrm{0}}\left(\mathrm{2}-{\displaystyle \frac{S{H}_{\mathrm{0}}^{(\mathrm{2})}(S)}{H_{\mathrm{1}}^{(\mathrm{2})}(S)}}\right)\left[{\displaystyle \frac{\sin \left(X\overline{h}\right)}{X\overline{h}}}-{\displaystyle \frac{{\overline{T}}_{\mathrm{0}}}{{\phi }_{\mathrm{0}}}}{\displaystyle \frac{\cos \left(X\overline{h}\right)-\mathrm{1}}{{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}{X}^{\mathrm{2}}\overline{h}}}\right]=\mathrm{2}\mathrm{\pi }\overline{h}{\phi }_{\mathrm{0}}\left(\mathrm{2}-S{\displaystyle \frac{J_{\mathrm{0}}{J}_{\mathrm{1}}+Y_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}+S{\displaystyle \frac{J_{\mathrm{1}}{Y}_{\mathrm{0}}-J_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\mathit{i}\right)\left[{\displaystyle \frac{\sin \left(X\overline{h}\right)}{X\overline{h}}}-{\displaystyle \frac{{\overline{T}}_{\mathrm{0}}}{{\phi }_{\mathrm{0}}}}{\displaystyle \frac{\cos \left(X\overline{h}\right)-\mathrm{1}}{{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}{X}^{\mathrm{2}}\overline{h}}}\right].

The reaction torque that the saturated half space acting on the bottom of embedded elastic foundation is the same as Eq. (17).

Torsional vibration of an embedded foundation

The equation of torsional vibrations of an embedded cylindrical foundation about the vertical axis of symmetry is:

(30) I_T\ddot{\phi }=T(t)-T_s(t)-T_b(t),

where I_T is the mass inertia moment of the foundation, T(t) is the excitation torque, T_s(t) and T_b(t) are the reaction torques that the saturated soil acting on the foundation.

Introducing the dimensionless variables, Eq. (30) can be expressed as:

(31) {\displaystyle \frac{\rho a^{\mathrm{2}}{\overline{I}}_T}{G}}\ddot{\phi }+D_T\dot{\phi }+K_T\phi =\overline{T}(t),

where {\overline{I}}_T={\displaystyle \frac{I_T}{\rho a^{\mathrm{5}}}} , K_T and D_T are the equivalent stiffness coefficient and equivalent damping coefficient of the soil, respectively. For an embedded rigid foundation,

(32) K_T=\mathrm{2}\mathrm{\pi }\overline{h}\left(\mathrm{2}-S{\displaystyle \frac{J_{\mathrm{0}}{J}_{\mathrm{1}}+Y_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\right)+{\displaystyle \frac{f_{\mathrm{1}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}\mathrm{and}{D}_T={\displaystyle \frac{\mathrm{2}\mathrm{\pi }\overline{h}\left(S{\displaystyle \frac{J_{\mathrm{1}}{Y}_{\mathrm{0}}-J_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\right)-{\displaystyle \frac{f_{\mathrm{2}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}}{\omega }},

and for an embedded elastic foundation,

K_T=\mathrm{2}\mathrm{\pi }\overline{h}\left\{\left(\mathrm{2}-S{\displaystyle \frac{J_{\mathrm{0}}{J}_{\mathrm{1}}+Y_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\right)\left[{\displaystyle \frac{\sin \left(X\overline{h}\right)}{X\overline{h}}}-{\displaystyle \frac{f_{\mathrm{1}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}{\displaystyle \frac{\cos \left(X\overline{h}\right)-\mathrm{1}}{{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}{X}^{\mathrm{2}}\overline{h}}}\right]-S{\displaystyle \frac{J_{\mathrm{1}}{Y}_{\mathrm{0}}-J_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}{\displaystyle \frac{f_{\mathrm{2}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}{\displaystyle \frac{\cos \left(X\overline{h}\right)-\mathrm{1}}{{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}{X}^{\mathrm{2}}\overline{h}}}\right\}+{\displaystyle \frac{f_{\mathrm{1}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}},

(33) D_T=\mathrm{2}\mathrm{\pi }\overline{h}\left\{S{\displaystyle \frac{J_{\mathrm{1}}{Y}_{\mathrm{0}}-J_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\left[{\displaystyle \frac{\sin \left(X\overline{h}\right)}{X\overline{h}}}-{\displaystyle \frac{f_{\mathrm{1}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}{\displaystyle \frac{\cos \left(X\overline{h}\right)-\mathrm{1}}{{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}{X}^{\mathrm{2}}\overline{h}}}\right]+\left(\mathrm{2}-S{\displaystyle \frac{J_{\mathrm{0}}{J}_{\mathrm{1}}+Y_{\mathrm{0}}{Y}_{\mathrm{1}}}{J_{\mathrm{1}}{J}_{\mathrm{1}}+Y_{\mathrm{1}}{Y}_{\mathrm{1}}}}\right){\displaystyle \frac{f_{\mathrm{2}}}{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}{\displaystyle \frac{\cos \left(X\overline{h}\right)-\mathrm{1}}{{\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}{\displaystyle \frac{{\overline{\rho }}_d}{{\varsigma }^{\mathrm{2}}}}{X}^{\mathrm{2}}\overline{h}}}\right\}/\omega -{\displaystyle \frac{f_{\mathrm{2}}/\omega }{f_{\mathrm{1}}^{\mathrm{2}}+f_{\mathrm{2}}^{\mathrm{2}}}}.

So the torsional angular amplitude of the embedded foundation can be written as:

(34) A_T={\displaystyle \frac{\overline{T}(t)}{\sqrt{{\left(K_T-a_{\mathrm{0}}^{\mathrm{2}}{\overline{I}}_T\right)}^{\mathrm{2}}+{(D_T\omega )}^{\mathrm{2}}}}},

and the phase shift can be expressed as:

(35) \theta =\arctan \left({\displaystyle \frac{D_T\omega }{K_T-a_{\mathrm{0}}^{\mathrm{2}}{\overline{I}}_T}}\right).

For an embedded elastic foundation whose shear modulus is a infinitely large quantity, parameter X equals to zero, then Eq. (33) is the same as Eq. (32). This indicates that a rigid foundation is a special case of an elastic foundation.

To evaluate the effect of different embedment on the equivalent stiffness and damping of the soil, the equivalent stiffness and damping coefficients with \overline{h}=\mathrm{0} is selected as a scale, the curves of equivalent stiffness ratio K_T(h)/K_T(\mathrm{0}) and equivalent damping ratio D_T(h)/D_T(\mathrm{0}) can be drawn respectively.

The above problem can be reduced to the torsional vibrations of a cylindrical foundation embedded in single-phase half space if setting {\rho }_f=\mathrm{0} .

Numerical examples and discussions

To verify the correctness of the solution proposed in this paper, the problem is reduced to the torsional vibrations of a rigid cylindrical foundation embedded in a single-phase medium, and compared with the corresponding results of Novak [ 6], which was gained through approximate formulas. The parameters of single-phase elastic medium are from Ref. [ 6]. Three different embedment depths are selected and the results are shown in Fig. 2. From this figure, it can be seen that the curves show good agreement.

Parameters of the saturated soil are shown in Table 1.

Basing on the equivalent stiffness coefficient and equivalent damping coefficient of the soil when a cylindrical foundation resting on saturated soil, the curves of K_T(h)/K_T(\mathrm{0}) and D_T(h)/D_T(\mathrm{0}) are shown as Figs. 3-5. To explore the effect of embedment depth on the torsional vibration of an embedded cylindrical foundation, five different \overline{h} are selected. Figure 6 shows the torsional angular amplitude of the embedded foundation, Fig. 7 reveals the difference of the torsional angular amplitude of the foundation embedded in single-phase medium and saturated half space, and Fig. 8 gives the lag phase angle of the embedded foundation.

Figure 3 gives the curves of the equivalent stiffness ratio for various dimensionless frequencies and embedment depths, respectively. Figure 3(a) shows, for a given frequency, the equivalent stiffness ratio of the soil increases linearly with the embedment depth. Moreover, the larger the dimensionless frequency is, the larger is the slope of the line. Figure 3(b) shows that the equivalent stiffness ratio fluctuates slightly with the dimensionless frequency. And the larger the embedment depth is, the more obvious is the fluctuation.

Figure 4 exhibits the curves of the equivalent damping ratio for various dimensionless frequencies and embedment depths, respectively. Figure 4(a) reveals, for a given frequency, the equivalent damping ratio of the soil increases linearly with the embedment depth. Moreover, the smaller the dimensionless frequency is, the smaller is slope of the line. Figure 4(b) indicates that the equivalent damping ratio decreases along with the dimensionless frequency. When the dimensionless frequency is smaller than 0.5, the decrease speed is very fast. Furthermore, Fig. 4(b) shows that the larger the embedment depth is, the larger is the equivalent damping ratio.

Figures 5 gives the curves of K_T(h)/K_T(\mathrm{0}) and D_T(h)/D_T(\mathrm{0}) for various \varsigma . From Fig. 5(a), it can be seen that, for a given embedment depth, the curve of equivalent stiffness ratio shows obvious fluctuation along with the dimensionless frequency, especially the value of \varsigma is larger. The equivalent damping ratio decreases along with the dimensionless frequency, and \varsigma has almost no effect on the equivalent damping ratio, which is indicated in Fig. 5(b).

Figure 6 shows that the resonant amplitude of an embedded foundation decreases drastically with the increase of the embedment depth. At the same time, the resonant frequency increases with the embedment depth. Furthermore, when the embedment depth is larger than the diameter of the foundation, the resonant of foundation is not obvious.

Figure 7 is the torsional angular amplitudes of a rigid foundation embedded in a single-phase medium and saturated half space for various embedment depths. From this figure, it can be found that, at the same embedment depth, the resonant amplitude of a single-phase medium is larger than that of a saturated half space, and there is no obvious difference between the resonant frequencies.

Figure 8 gives the lag phase angle curves of an embedded elastic foundation for various {\overline{I}}_T , \overline{h} and \varsigma . It can be found from Fig. 8 that the lag phase angle increases along with the dimensionless frequency. Figure 8(a) reveals that, for a given embedment depth, the larger the mass inertia moment of the foundation is, the larger is the lag phase angle. Figure 8(b) shows that, for a given \varsigma , when the dimensionless frequency is larger than 1, the larger the embedment depth is, the larger is the lag phase angle. Figure 8(c) indicates that parameter \varsigma has no effect on the lag phase angle.

Conclusions

The dynamic response of a cylindrical foundation embedded in saturated soil subjected to a harmonic torsional loading has been presented in this paper. Both a rigid foundation and an elastic foundation are considered. During the analytic process, the reaction torques that the saturated soil acting on the side surface and bottom surface of the foundation are also taken into account. Numerical comparison result indicates that present paper’s result coincides with that of Novak. Selected numerical examples are presented for an embedded elastic foundation to examine the influence of relevant parameters on the results. Based on the results of parametric studies, the following conclusions are made:

1) For a given dimensionless frequency, the equivalent stiffness ratio and the equivalent damping ratio of the saturated soil increase linearly with the embedment depth. Moreover, the larger the dimensionless frequency is, the faster is the equivalent stiffness ratio increase speed, and the slower is the equivalent damping ratio increase speed.

2) For an embedded elastic foundation, when the embedment depth is a constant, the equivalent stiffness ratio curve shows obvious fluctuation along with the dimensionless frequency, especially the shear wave speeds’ ratio \varsigma is larger. However, the equivalent damping ratio decreases along with the dimensionless frequency, and parameter \varsigma has no effect on the equivalent damping ratio.

3) The resonant amplitude of an embedded cylindrical foundation decreases drastically with the embedment depth. At the same time, the resonant frequency increases with the embedment depth.

4) The lag phase angle of an embedded foundation increases along with the dimensionless frequency. The mass inertia moment of foundation and the embedment depth have significant effect on the lag phase angle, while the shear wave speeds’ ratio \varsigma has almost no effect on the lag phase.

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