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The problems of the consolidation of saturated soil under dynamic loading are very complex. At present, numerical methods are widely used in the research. However, some traditional methods, such as the finite element method, involve more degrees of freedom, resulting in low computational efficiency. In this paper, the scaled boundary element method (SBFEM) is used to analyze the displacement and pore pressure response of saturated soil due to consolidation under dynamic load. The partial differential equations of linear problems are transformed into ordinary differential equations and solved along the radial direction. The coefficients in the equations are determined by approximate finite elements on the circumference. As a semianalytical method, the application of scaled boundary element method in soil-structure interaction is extended. Dealing with complex structures and structural nonlinearity, it can simulate two-phase saturated soil-structure dynamic interaction in infinite and finite domain, which has an important engineering practical value. Through the research, some conclusions are obtained. The dimension of the analytical problem can be reduced by one dimension if only the boundary surface is discretized. The SBFEM can automatically satisfy the radiation conditions at infinite distances. The 3D scaled boundary finite element equation for dynamic consolidation of saturated soils is not only accurate in finite element sense but also convenient in mathematical processing.

Dynamic consolidation analysis of foundation is one of the most concerned problems, such as seismic design of large underground caverns, high arch dams, super high-rise buildings, and other large-scale industrial and their response characteristics under mechanical vibration (Chen and He [

For infinite and semi-infinite foundation soils, the main numerical analysis methods are finite element method (FEM), boundary element method, and finite element-boundary element coupling method. The simulation of semi-infinite foundation soil is much more complicated compared to the finite element method mainly because of the large discrete range of foundation soil. In order to obtain enough calculation accuracy, the free degrees of the system increased inevitably. Although the computing power has been greatly improved, the computational units needed for some special cases, such as the accurate analysis of 3D stress wave propagation in anisotropic soils, are still difficult to achieve. Generally, the finite element discrete analysis is used to study foundations around the structure. Free or fixed boundary conditions are used to simulate infinite foundation soil. At the same time, in order to eliminate false reflection on artificial boundary, a transmission boundary model is established, for example, viscous and viscoelastic boundary (Zhang [

The wave attenuation factor is introduced into the shape function to describe the propagation of vibration waves along infinite distances for the infinite element method (Zhao and Valliappan [

Scaled boundary finite element method (SBFEM) is a boundary element method based on finite element method. Compared with the boundary element method, it does not need to solve the fundamental solution, so it can effectively deal with the problem of anisotropic media which is particularly complex and satisfies certain conditions. The scaled boundary finite element method (SBFEM) successfully satisfies Sommer field radiation condition by choosing appropriate similar centers. Therefore, the waves emitted by the source can only dissipate to infinity in the form of de-wave, but not return from infinity in the infinite domain. At present, SBFEM has been applied to the analysis of wave problems in time domain and frequency domain and to the solution of boundary dynamic stiffness matrix of infinite foundation (Wolf and Song [

Taking 3D saturated soil area

Cutaway view of three-dimensional saturated soil: (a) finite region and (b) unbounded media.

Transformation of global Cartesian coordinates to scaled boundary coordinates.

For the 3D infinite saturated soil shown in Figure

In the local coordinate system of the scaled boundary, any point on the bottom ^{e} of

In the scaled coordinates, the relationship between point

Then,

According to Biot’s theory, the governing equations of consolidation of saturated soils include the dynamic equilibrium equation of soil skeleton and the continuous equation of fluid. In the Cartesian coordinate system

The continuity equation of fluid motion in saturated soils in frequency domain is as follows:

For isotropic saturated soils, effective stress, strain, and displacement vector satisfy Hooke's law:

According to equation (

Equation (

In the Cartesian coordinates

From equations (

For the computational element domain ^{e} of

From equations (

By using equations (

Galerkin weighted residual method is adopted in computing unit domain

For the second item in the above equation, by using the partial integral and substituting equations (

By substituting equation (

Then, we substituted

After multiplying

By using equations (

For the continuity equation of fluid in the consolidation equation of saturated soil, the weight function is expressed by the interpolation function which is the same as the pore pressure. Galerkin's method is used in the calculation domain. Similar to the above deduction, the following can be obtained from equation (

By using the partial integral and substituting

The dynamic consolidation equation of saturated soil considering pore water pressure can be obtained by equations (

The displacement and pore pressure vectors are expressed by generalized displacement vectors

The second-order differential equations can be obtained by equations (

The displacement and pore pressure responses of 3D half-space surface subjected to vertical concentrated harmonic loads are investigated. In the calculation, the saturated soil layer is clay layer, and the relevant parameters in [

Using the method presented in this paper, the 3D half-space is divided into two parts. A hemisphere with a radius of 2R is a structure and the rest is a semi-infinite space. According to symmetry, 1/4 half space is taken for analysis, as shown in Figure

SBFEM discretization of 1/4 unbounded half-space.

In order to verify the accuracy of this method, if

Compared present solutions with the reference’s results: (a) real part of vertical displacement and (b) imaginary part of vertical displacement.

Figure

The vertical displacement for the 3D half-space varied with the frequency: (a) real part of vertical displacement and (b) imaginary part of vertical displacement.

Figure

The pore pressure for the interface between the half spec and structure varied with the frequency: (a) real part of vertical displacement and (b) imaginary part of vertical displacement.

In this paper, the scaled boundary element method for single-phase elastic media is extended to the dynamic consolidation of 3D saturated soils with water-soil two-phase coupling. The application of scaled boundary element method in soil-structure interaction is extended. From the theoretical analysis in this paper, it can be seen that

Not the same as single-phase media, the displacement and stress matrices are not only formed in the scaled boundary finite element equation of saturated soil consolidation but also the pore water pressure effect is coupled in the scaled boundary finite element matrix. According to Biot's theory, the governing differential equation of saturated soil consolidation is solved in Cartesian rectangular coordinate system, and the proportional boundary finite element equation in the form of pore pressure and displacement is obtained.

The scaled boundary of dynamic consolidation of saturated soil also has unique advantages. The dimension of the analytical problem can be reduced by one dimension if only the boundary surface is discretized. In addition, similar to the boundary element method, it can automatically satisfy the radiation conditions at infinite distances. However, the method in this paper does not need to solve the fundamental solution and does not involve singular integrals.

As a semi analytical method, the 3D scaled boundary finite element equation for dynamic consolidation of saturated soils is not only accurate in finite element sense but also convenient in mathematical processing. Especially, in dealing with complex structures and structural nonlinearity, it can simulate two-phase saturated soil-structure dynamic interaction in infinite and finite domain, which has important engineering practical value.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was financially supported by the National Science Foundation of China (no. 5206080257), Key Projects of Natural Science Foundation of Jiangxi Province (nos. 20114BAB206012 and 20181BAB216028), and Jiangxi Province Department of Education Science and Technology Research Project (nos. GJJ12629, GJJ171008, and GJJ11253).