3. Nonlinear Numerical Analysis (Elasto-plastic Analysis)

Chapter 1) What Is Nonlinear Numerical Analysis? 

• Introduction to nonlinear numerical (FEM) analysis 
• Understanding the working of nonlinear 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

In conventional soil mechanics, the deformation problem is treated as a linear problem, and the stability-related problem is treated as a rigid plasticity problem.

However, actual soil behavior is elasto-plastic nonlinear behavior, so the assumption of elasticity simplifies soil behavior greatly.

In order to properly deal with the ground stability problem, it must be treated as an Elasto-plastic nonlinear problem, which is possible by numerical analysis using a computer.

The method of solving nonlinear problems is completely different from linear equations.

The nonlinear stress-strain relationship is not a linear relationship as shown in Figure 3.2(a), and thus the load-displacement relationship is also nonlinear as shown in Figure 3.2(b).

Since the stress-strain relationship is in a nonlinear state, it can be expressed as follows. Here, 𝐷𝑒𝑝 is referred to as an Elasto-plastic constitutive matrix.

If we assume nonlinear behaviour as a piecewise linear behaviour by dividing it into small sections, the nonlinear equation can be expressed using the incremental concept, ∆.

Therefore, the form of the overall finite element equation assembled by the element stiffness matrix is as follows.

In linear analysis, 𝐾𝐺 remains constant during the analysis, while in nonlinear analysis, 𝐾𝐺 varies depending on the magnitude of the deformation.

Nonlinear equations are generally solved using incremental analysis techniques that assume approximate linear behaviour in the load increment interval.

Some examples of these solution methods include the Tangent stiffness method (e.g., Newton-Raphson method), Initial stiffness method (e.g., Modified Newton-Raphson method), and so on.

We will not go deeper into the actual solution methods, and instead, we will use software to practice Elasto-plastic nonlinear analysis.