6.1 Introduction

As explained in Chapter 5，under normal excavation conditions，excavation⁃induced stress and deformation are engendered by unbalanced forces acting on the wall due to the removal of soils within the excavation zone.The magnitude of unbalanced forces is influenced by many factors：the conditions of soil layers，the table and pressures of groundwater，the excavation depth，the excavation width and so on.Theoretically，the finite element method is capable of simulating these factors and therefore the results derived from the method would be more accurate than those derived from simplified methods or the beam on elastic foundation method.The theories on which the finite element method is based，however，are rather complicated and the data to be processed both before and after analysis are enormous.What's more，some of the theories are not fully developed.To apply a program based on the finite element method，analysts are required to be well equipped with comprehensive geotechnical knowledge and experience.All this adds confusion and trouble for analysts.Considering the complexity of the finite element method and that any small neglect is likely to lead to wrong results，the results of the finite element method should be examined by other methods，the simplified methods，for example，to ensure the reasonability of the results.

Besides，some researchers write the governing equation in the form of an explicit finite difference equation and solve it by dynamic relaxation.This method solves the velocity and movement through the movement equation by assigning a damping value close to the critical damping.The strain rate is then obtained from velocity and used to solve the new stress increment.The process continues till the unbalanced forces are in equilibrium or the system reaches a steady state.The main theory on which the finite difference method is based is not the same as that of the finite element method.However，other theories，such as constitutive laws of soil，drained or undrained behaviors，determination of soil parameters，simulations of excavation are identical with the finite element method.

In terms of stress type（see Section 6.6.1），the finite element method can be split into the total stress analysis and the effective stress analysis methods.Concerning the analysis of drained conditions（see Section 6.6.2），the finite element method can also be divided into the undrained analysis method，drained analysis method，and the partially drained analysis method.With the finite element method，the undrained behaviors of clayey soils can be analyzed by total stress analysis，effective stress analysis，and coupled analysis.

Although many of the theoretical descriptions in this chapter are complicated，they are necessary knowledge for excavation analyses using the finite element method，and are also applicable to other problems concerning geotechnical engineering.Readers are advised to run programs according to the contents of this chapter to examine the correctness of analyses.

Although this chapter tries to elucidate the application of the finite element method on deep excavation in details，there remains much to be explored.Readers interested in the method can see the related references.Besides，considering the complexity of the analysis process of the finite element method and the comprehensive knowledge required by analysts，this chapter is recommended for readers above the graduate level.