1-d numerical analyses of dynamic soil response under surface excitations

One of the main objectives of solving problems in geotechnical engineering is to evaluate the natural soil response under various external loads due to the fact that all civil engineering structures are located on or inside a soil medium. Under steady state or transient conditions, mechanical changes in a saturated soil system as a result of the changes in drainage conditions or the internal structure, are studied by using the "theory of coupled flow and deformation" which essentially considers the deformation of solid skeleton and flow of pore fluid simultaneously. The mathematical formulation behind this theory satisfies the coupled dynamic behavior of porous soils employing the poro-elasticity equations. Although, the actual theory proposes what is called a "fully dynamic (FD) formulation" including all the inertial forces associated with the motion of both solid and fluid parts (i.e. if there is more than one fluid filling in the pores of the porous medium), it is typically preferred to make some simplifiying assumptions in the theory to develop a more practical solution albeit not giving up from the accuracy. Taking the porous soil as a two-phase medium with only a single pore fluid (i.e. as water in the case of soils), these simplified equations are obtained as; i) Partially dynamic (PD) where the inertial terms associated with the pore fluid are neglected, and ii) Quasi-static (QS) where the inertial terms of both solid and the fluid phases are neglected. The dynamic response of the system is then analyzed by the appropriate form of these equations considering the physical structure and loading situation given in the problem. While employing the complex poroelasticity equations helps better understand the problem of evaluating the actual response of soils in geotechnical engineering, it is of more practical value to initiate related studies by taking the soil medium as a single-phase solid material neglecting the porous-structure in the beginning. In this way, pore water pressure is included externally in the system as an 'ad-hoc' way and flow is not taken into account. That said, as it is the easiest way to model the soil behavior, it could be, therefore, pretty much the first approach to obtaining the solution of the dynamic problem, particularly in a one-dimensional (1- D) situation. Along with such a 'simplified' approach, one always chooses to develop analytical solutions to the problem because having a direct solution of a problem allows us to obtain exact results. However, most of the time it is not possible to obtain exact mathematical solutions for complex real-world soil-mechanics problems since soils are nonlinear, heterogeneous and anisotropic materials. Therefore, it is of utmost importance to utilize numerical methods to approximate the actual soil behavior and develop accurate solutions. In this thesis, 1-D numerical solutions to the dynamic response of a soil column under surface excitations are developed using the Finite Element Method (FEM). In the first part of the thesis, the soil is considered as a porous medium, and its linear dynamic behavior is examined through 1-D FE analyses using linear bar-elements. The validity of this numerical study is confirmed by the available analytical solutions. Due to the fact that the actual behavior of the soil is inelastic, our attention is turned to the actual nonlinear behavior. Here, since there can be irreversible deformations taking place in the soil under cyclic loading, the "theory of plasticity" is employed to handle the related calculations. To start off, nonlinear elasto-plastic behavior of a solid material is evaluated by the stress-strain relationship which is then implemented in a MATLAB program in terms of a number of basic constitutive models. These are: the elastic-perfectly plastic model, elastic-linear isotropic hardening plastic model and the elastic-linear combined isotropic and kinematic hardening plastic model, which are considered to be the basis of the nonlinear part of this study. Then a number of single-degree-of-freedom systems (SDOF) are analyzed using such elasto-plastic behavior of the material. The nonlinear finite element analyses in this section involve solution of the stiffness relation using the 'Newton- Raphson Method' and updating its components iteratively at each time-step and calculating and updating the internal force obtained from the elasto-plastic stress-strain relationship as a result of updated degree of freedoms. Then this numerical analysis program, as developed for a SDOF system, is expanded by implementing the equation of motion written in terms of the governing equations of a solid soil element and subsequently of the equations of a porous soil element, into a large-purpose MATLAB code. This is a nonlinear dynamic finite element program developed to analyse a multi degree of freedom soil column under any loading. Our focus was on harmonic response of soil whose stress-strain behavior is governed by elastic-combined hardening plastic model. The FE solution of the nonlinear dynamic analysis of a porous soil column, which is verified for a single element behavior, constitutes the ultimate extent of this thesis study. Interesting results obtained from each chapter are presented at the end of respective sections and summarized in the conclusions. As for future works, it is planned to analyse the dynamic response of the soil with a more realistic nonlinear soil model able to capture the actual cyclic-plastic soil behavior using the gained knowledge on the nonlinear dynamic finite element analysis.