Arc-Length Method with Numerical Examples

The arc-length method is a numerical technique for solving nonlinear structural mechanics problems, particularly suited for handling strong nonlinear behaviors such as buckling and post-buckling responses. It introduces an arc-length parameter to control load increments, overcoming convergence difficulties near limit points that commonly occur in traditional load-controlled methods.

Core Algorithm Concept The arc-length method treats both the load factor and displacement increments as unknown variables, with constraint equations (e.g., fixed "arc-length" iteration path) enabling automatic adjustment of increment steps. This approach allows the solution to pass through limit points and capture the complete load-displacement path of structures.

Typical Implementation Scenarios Column buckling analysis: Demonstrates how the arc-length method traces post-buckling equilibrium paths. Arch structure instability: Illustrates limit point snapping phenomena in load-displacement curves. Material nonlinear problems: Large deformation analysis using hyperelastic models like rubber materials.

Code implementation for arc-length method examples typically involves key stages: initial stiffness calculation, incremental step adjustment strategies (e.g., spherical arc-length method), iterative correction procedures, and convergence criteria. Through practical examples, one can visually compare differences between the arc-length method and Newton-Raphson iterations, highlighting its advantages in complex nonlinear problems.