A Study For Dynamic Interaction Between Foundation And The Complex Multi-layered Soil

The evaluation of the dynamic impedance of the foundation is the key issue for the dynamic soil-structure interaction analysis. By mean of substructure method, it can be incorporated into the general finite element program to obtain the dynamic response of supper structure and the subgrade reaction excited by the loads, such as earthquake, explosion and so on. A lot of theoretical research and analysis show that the dynamic characteristics of the complex multi-layered soil have a very important influence on the foundations as well as the supper structure, especially in case of anisotropic soil. Many researchers and engineers have realized this point, and have carried out related study, and a variety of numerical algorithms have been put forward for the solution of the dynamic stiffness of multi-layered soil. From the point of view of present development situation of the dynamic structure-soil interaction, most of the existing algorithms have limitations, such as in the layer number and thickness of layered soil, the anisotropy characteristics of soil, the embedded foundation and so on.Based on the space transformation of the dynamic equations of the multi-layered soil, combined with the precise integration method and the application of the dual equation, this paper presents a hybrid numerical method to solve the dynamic impedance matrices of the surface or embedded foundations resting on the multi-layered soil. The proposed method overcomes the limitations of the existing algorithms, and has following features:(1) It has extensive applicability for arbitrary horizontal multi-layered soil. There is no limitation to the layer number, thickness of layers and the material property.(2) It has a high efficient for the matrices in the algorithm has small dimension.(3) The precise integration method used in the proposed method can ensure the high accuracy. In a sense that, the accuracy of the proposed method only depend on the precision of computer used.(4) The computation is always stable, because the numerical implementation is based on algebraic matrix operation.The main research contents and achievements in this paper are as follows:1. For the isotropic multi-layered soil, the Hankel transform is used to transform the wave motion equations from frequency-spatial domain into frequency wave-number domain, and the wave motion equations is decoupled into in-plane motion and out-of-plane motion. By introducing a dual form vector, they are transformed into first-order ordinary differential equations which can be solved by precise integration method. Finally, the Green’s functions for the general plane wave motion problem and the three-dimensional wave motion problem of the isotropic multi-layered soil are evaluated by the inverse Fourier transform and Hankel transform respectively. The dynamic impedance matrices of the strip foundation and arbitrary-shaped foundation at the surface of the isotropic multi-layered soil are obtained by using the Green’s functions. The accuracy of the proposed method is verified by numerical examples.2. For the anisotropic multi-layered soil, the Fourier transform is used to transform the wave motion equations from frequency-spatial domain into frequency wave-number domain as a second-order ordinary differential equation. By introducing a dual form vector, they are transformed into first-order ordinary differential equations which can be solved by precise integration method. Finally, the Green’s functions for the general plane wave motion problem and the three-dimensional wave motion problem of the anisotropic multi-layered soil are evaluated by the inverse Fourier transform. The dynamic impedance matrices of the strip foundation and arbitrary shaped foundation at the surface of the anisotropic multi-layered soil are obtained by using the Green’s functions. A research is carried out to study the influence of the anisotropy of the multi-layered soil on the dynamic impedance of the foundation. The results show that, the anisotropy of the multi-layered soil has a significant influence on the dynamic foundation-soil interaction.3. The study on embedded foundation has more significance to the engineering practice than the surface foundation. Extending the hybrid algorithm for solving the surface Green’s functions of the multi-layered soil, a hybrid numerical is presented to solve the interior Green’s functions of the multi-layered soil. Then the dynamic impedance functions of embedded strip foundation and arbitrary-shaped embedded foundation are evaluated by combing the flexible volume method. The accuracy of the proposed method is verified by numerical examples.4. Using the proposed hybrid algorithm for solving the dynamic response of the multi-layered soil, the dynamic interaction analysis between adjacent foundations is carried out. Plenty of parametric studies have been carried out to clarify the effects of layer depth, soil damping, spacing between adjacent foundations, and the wave propagation velocity on the dynamic behavior of three-dimensional foundation-soil-foundation interaction. The results show that, the heterogeneous and anisotropy of the multi-layered soil has great influence on the dynamic foundation-soil-foundation interaction.5. The stress distribution of the multi-layered soil beneath the rigid foundation by the applied concentrated load is solved based on the dynamic impedance function of the foundation. The effect of the exciting frequency of applied load and the anisotropy of the soil on the soil reaction under the foundation has been studied. 6. The discrete dynamic impedance function of rigid foundation in the frequency spatial domain is fitted by the Pade series as a continued matrix valued rational function based on the solution in the frequency domain. The mixed-variables which include forces and displacements are introduced, and the problem transforms to derive the numerical solution of a first order ordinary differential equation in time domain which can be solved by the precise integration method. The dynamic response of the arbitrary-shaped foundation resting on the multi-layered soil in time domain is obtained.