2. Linear Numerical Analysis (Elastic Analysis)

Chapter 1) What Is Linear Numerical Analysis? 

• Introduction to linear numerical (FEM) analysis 
• Understanding the matrix method of analysis   

Chapter 2) Example 

• Modelling, boundary condition setting, applying load using Midas GTS NX software to simulate the uniaxial compression test on a granite rock specimen. 
• Analysis and see the results for the same.  

Chapter 3) Comparison of results.  

• Compare the results with those obtained manually and using the program.  

Summary

Ground materials exhibit elastic behavior only within a limited range of deformation in the early stages. In compacted clay or dense sand, the elastic range generally increases.

Due to the convenience of having fewer required properties, there have been many attempts to interpret the behavior of ground materials using an elastic model throughout the entire range.

Elastic analysis has two attractions related to geotechnical engineering. The first is that it provides mechanical intuition when there are no alternative solutions to a problem, and it also helps to derive new and improved solutions.

Ground behavior generally exhibits nonlinearity, heterogeneity, and anisotropy, but it has often been assumed to be linear elastic in traditional approaches.

However, since most ground deformations involve plastic behavior, the cases where linear elasticity can be applied to actual ground problems are limited.

In the case of linear behavior, the system equation for the total vector relationship between displacement and load is expressed as follows.

Here, [K_G] is the stiffness matrix, {U_G} is the displacement vector, and {R_G} is the load vector.

The subscript G denotes the global system. In linear analysis, the magnitude of [K_G] remains constant during analysis.

The stiffness matrix in linear analysis can be defined by two elastic parameters that is E and ν.

The solution of the finite element equation requires the computation of the inverse of the stiffness matrix [K_G].

Most programs use Gaussian elimination for two-dimensional analysis and iterative methods for three-dimensional problems to compute the inverse.

When an engineer uses a program, the computer performs these operations to solve the finite element equations.

Since it is a complex process, we will skip the solution process for linear numerical analysis equations using Gaussian elimination in this lecture.