This chapter is an introduction to the requirement and the various problems that will be encountered during numerical modelling in geotechnical engineering. A large scale tunneling work in Australia will be used to illustrate the necessity to use numerical methods in some real life engineering problems. After that, the authors will introduce a series of numerical problems that may be encountered during the use of commercial engineering programs. Such problem cases may arise from various sources, and engineers are strongly advised to understand the basic principle of each commercial program and to assess the program output with care before accepting the results of analysis. Finally, some of the more important governing differential equations for geotechnical problems are discussed.

Finite element method is the most popular and important numerical method in geotechnical engineering analysis and design. In this chapter, the basic formulation for finite element method will be introduced, followed by the various procedures in the actual implementation of the method which are illustrated by several sample Fortran programs. The style of Fortran programming, mesh generation, bandwidth/profile minimizer as well as the development of modern thick plate element are used for the illustration of the various techniques required in developing a finite element program.

In this chapter, the ultimate limit state of a system is considered by means of limit equilibrium, plasticity slip line method, limit analysis and DLO methods. The basic plasticity formulation for the slip line method is given, which is applied to some classical geotechnical problems. Following this, the three major geotechnical problems are unified under the extremum principle by the plasticity formulation. There is also discussion on the basic formulation for the DLO method and the limitations of the method or the commercial program.