# Benchmark Tests for Solution Procedures for Geometric Non-Linearity

This report is one of a series commissioned by NAFEMS which seek to provide benchmark tests against which finite element systems can be evaluated. The present report considers the solution of large, geometrically non—linear, deflections of truss elements.

## Contents

### 1 First Example: The unrotated Element

• Engineering Strain
• Green Strain
• Logarithmic Strain

### 2 Second Example: Limit Point Behaviour with One Variable

• Description of Model
• Closed Form Solution
• Rotated Engineering Strain
• Green Strain
• Logarithmic Strain
• Numerical Solution Techniques
• Governing Equilibrium Equations
• The Stiffness Matrix
• Taylor Expansions
• Incremental Solution
• Incremental Solution With Residual Correction
• Newton-Raphson (NR) Solution of Non-Linear Equations
• Predictor-Corrector Methods
• The Modified Newton-Raphson Methods (M-NR)
• Modified Newton-Raphson
• Initial Stiffness Method
• AccelarationQuasi-Newton and '~Secant-Newton' Methods
• Line Search
• Arclength Methods
• Automatic Increments
• Detection of Limit Points
• Discussion of Numerical Results

### 3 Third Example: Hardening with One Variable

• Description of Model
• Closed Form Solution
• Discussion of Numerical Solution

### Fourth Example: Bifurcation Problem

• Description of Model
• Closed Form Solution
• Numerical Consequences of Bifurcation Point
• Effect on Stiffness Matrix
• Possible Numerical Responses to a Bifurcation Point
• Inform User
• Confusion Between Bifurcation Point and Limit Points
• Try to Follow Bifurcation Path

### Fifth Example: Limit Point with Two Variables

• Description of Model
• Closed Form Solution
• Discussion of Numerical Results
• Example 5.2 Displacement Control (Large Incs.)
• Example 5.3 Displacement Controll (Small Incs.)
• Arclength Solution

### Sixth Example: Hardening with Two Variables

• Description of Model
• Closed Form Solution
• Discussion of Numerical Results
• Example 6.2 Displacement Control (Large Incs.)
• Example 6.3 Displacement Control (Small Incs.)
• Arclength Solution

### Seventh Example: Snap-Back

• Description of Model
• Closed Form Solution
• Discussion of Numerical Results
• Example 7.2 Displacement Control (Large Incs.)
• Example 7.3 Displacement Control (Small Incs.)
• Arclength Solution

• Appendix A
• Appendix B
• Apendix C

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