Dynamic interaction analysis of soil and structure under the effect of earthquake load using finite element method with scaled boundary

Master's Thesis, Civil Engineering, Soil and Foundation Mechanics

September 1390

Abstract: Dynamic interaction analysis of soil and structure (due to the mutual effect of the structure and its supporting soil on each other) plays a fundamental role in the design of the structure under seismic load. The soil-structure system can be subjected to dynamic and static loads. In the case of static loading, the unlimited environment of the soil can be limited, so that the boundaries are considered far enough from the structure. But in the case of dynamic loading, this method cannot be used, because the boundaries, instead of allowing the received waves to pass from the wave propagation source to infinity, reflect them towards the structure and the soil, and this reflection affects the behavior and reaction of the soil-structure system. Therefore, to analyze this soil-structure interaction, the soil is divided into a regular finite soil part that can model the nonlinear behavior of the soil and a regular infinite part that continues to infinity. The finite part of the soil can be modeled by the finite element method. In an infinite medium, there is an important rule in wave dynamics: waves moving toward infinity are not returned to the domain. The boundary conditions must be able to model the wave motion in reality.

A high-order boundary condition is developed to model wave propagation in an unbounded domain under seismic loads. This boundary condition is applicable for scalar and vector waves with arbitrary geometry and inhomogeneous materials. The formulation in terms of the continuous gap solution is used in the analysis of the dynamic stiffness matrix for the unbounded domain. The matrix of the coefficients of the continuous gap method is recursively obtained from the equation of the finite element method with a scaled boundary in terms of dynamic stiffness, which can be integrated with the finite element. Standard methods in structure dynamics can be used directly to obtain responses in time and frequency domains. Numerical and analytical examples show the high rate of convergence and efficiency of this high degree boundary condition. In this thesis, for the first time, the dynamic interaction analysis of soil and structure under seismic loads has been developed using continuous crack solution and modal analysis (using reduced basis functions). Definition of the problem

In a soil-structure system located in a semi-infinite environment, the volume of the soil is much greater than the volume of the structure, so in general, the structure interacts with the surrounding soil. This means that the behavior of each has an effect on the other. Therefore, the analysis of the structure alone is not allowed and in many important situations (for example, seismic fluctuations) when the load is applied to the soil around the structure, the effect of the soil should be considered, meaning that the area around the structure should be modeled in any case. Since the seismic interaction of soil and structure is closely related to the security of many engineering projects (such as nuclear power plants) in terms of earthquake resistance, this interaction is considered a very important issue in earthquake engineering. Also, due to the effect of energy reflection from the unbounded environment and non-uniform movements of the ground on the response of the structure, the importance of the dynamic interaction of the soil and the structure in the behavior of giant structures against earthquakes is significant. Limited and unlimited soil by the common boundary of the soil – The structures (Figure 1-1) are separated from each other. An artificial boundary (Figure 1-1) is introduced at a distance far enough from the structure, where the field response is expected to be damped, and appropriate boundary conditions are defined on it. Depending on the accuracy of the analysis, this boundary can be defined further or closer to the structure. The nodes on the structure are denoted by s and the nodes on the common boundary between the soil and the structure are denoted by b. Dynamic loading can be introduced into the system as an external load or seismic load (Figure 1-1).

Soil and structure system can be subjected to dynamic and static loads.. To analyze the response of the structure under static loading, as mentioned, an artificial boundary can be introduced at a distance far enough from the structure, where the domain response is expected to decay. This results in a limited environment for soil that can be modeled as a structure. The entire elemented system, including soil and structure, is modeled by finite element or finite difference methods, and with the development and advancement of finite element software, the finite element problem can be easily analyzed. But in the case of dynamic loading, this method cannot be used, because the artificial boundaries, instead of allowing the received waves to pass from the wave propagation source to infinity, reflect them towards the structure and the soil, and this reflection affects the behavior and reaction of the soil and structure system. If the soil is unlimited, it should play the role of an energy absorber [1]. Therefore, to analyze this soil-structure interaction, the soil is divided into two parts: finite irregular soil, which can model the nonlinear behavior of the soil, and regular infinite soil, which continues to infinity and has linear behavior. Therefore, as you can see in Figure 1-1, the soil can be divided into two parts:

  the area close to the structure, irregular soil with nonlinear behavior called the limited area

  the area far from the structure, regular soil with linear behavior called the unconfined area

the limited part of the soil can be modeled by the finite element method. In an infinite medium, there is an important rule in wave dynamics: waves moving toward infinity are not returned to the domain. The boundary conditions must be able to model the wave motion in reality. The condition of bringing the displacement to zero at infinity is not sufficient] [i][. The boundary condition applied at infinity must have the ability to transfer energy completely from the limited domain to the unlimited domain and can prevent the reflection of the waves that come into contact with the boundary into the environment. This boundary condition is called reflective boundary condition[2]. The most challenging part in the dynamic interaction analysis of soil and structure is to obtain suitable reflective boundary conditions for engineering problems with large dimensions. The unbounded soil domain can be modeled in such a way that the soil dynamics is completely separated from the structure dynamics. To better understand the issue, wave propagation in a one-dimensional infinite domain is given below. In a real model, by applying a dynamic force (impact force R) at the beginning of the rod (point A), the wave motion starts from the beginning of the rod and propagates to infinity and decays (Figure 2-1 (a)). To model this state, the rod is cut and placed at the place where the harness is cut. When the wave starts moving and hits the existing restraint, the wave is returned. As long as the returned wave has not reached the starting point (point A), the behavior of point A is similar to reality, and since the wave reaches that point, the behavior of point A collapses (Figure 1-2 (b)). This is also the behavior in the unlimited soil. The model should be such that the wave is directed towards infinity and does not return to the structure. Therefore, the problem in modeling the dynamic interaction of the soil and the structure is in finite element software, which, for the reasons mentioned, do not have the ability to correctly analyze the dynamic behavior of the infinite domain. We are looking for a way to create a model close to reality for the dynamic analysis of an unlimited domain under seismic load, and we are also looking for a method that will bring us closer to the desired goal as reliably and quickly with high efficiency and accuracy.

1-2-  The concept of dynamic soil-structure interaction

To express the importance of soil-structure interaction, the dynamic response of a structure built on rock has been compared with the same structure built on soil. Here, only the issue is examined qualitatively. Two completely identical structures with a rigid bed (including the base and surrounding walls) are shown in Figure 1-3 (a). The soil layer is placed on top of the rock. Since the distance between the two structures is small, it can be assumed that the wave that reaches the two structures located on soil and rock from the epicenter of the earthquake is the same. For simplicity, a horizontal movement that expands vertically is chosen.