This course provides a systematic introduction to nonlinear finite element methods in structural mechanics. Building on the linear theory, it develops the variational foundations, nonlinear continuum formulations, and numerical solution strategies required for the analysis of geometrically nonlinear structures and stability phenomena.

1. Variational principles of mechanics

Principle of virtual work for 3D linear elastostatics. Mixed variational formulations: Hellinger–Reissner (two-field) and Hu–Washizu (three-field) principles. Discussion of locking phenomena and mixed formulations through Timoshenko beam models.

2. Introduction to nonlinear finite element analysis

Fundamental concepts of nonlinear structural response, residual force equations, and equilibrium paths. Critical points, bifurcation, and limit points in conservative systems. Introductory nonlinear examples.

3. Formulation of geometrically nonlinear finite elements

Review of nonlinear continuum mechanics. Total Lagrangian formulation of bar, Timoshenko beam and plane stress elements. Consistent linearization of governing equations and derivation of element tangent stiffness matrices.

4. Nonlinear solution methods

Incremental solution strategies including load control and generalized control methods. Newton and Newton-like iterative schemes. Convergence behavior, algorithmic aspects, and implementation issues in nonlinear finite element analysis.

5. Stability analysis of thin structures

Detection and traversal of critical points. Linearized prebuckling analysis and eigenvalue problems. Influence of imperfections on structural stability, numerical solution behavior and postcritical response.

The course combines theoretical foundations with computational implementation and illustrative numerical examples.

• Felippa, C.A., Advanced Finite Element Methods. Lecture Notes, CU Boulder.
• Felippa, C.A., Nonlinear Finite Element Methods. Lecture Notes, CU Boulder.