CE 232 : Structural Mechanics

The goal of this course is to give an advanced view of different theories of structural mechanics in the framework of nonlinear solid mechanics. To that purpose, the course is divided in three parts:

1. The first part of the course will concentrate on some specific aspects of finite strain theories of solids. Concepts like nonlinear measures of strain and Piola-Kirchhoff stresses will be introduced to arrive at the fundamental concept of invariance in mechanics. Finite elasticity will be taken as a model theory with this property. (No prior knowledge of nonlinear solid mechanics is required. Only a prior contact with the concepts of infinitesimal strain and stress tensors and equilibrium equations will be assumed.)

2. Next, energy principles will be discussed in detail, starting with the infinitesimal range. Concepts like the principle of virtual work as well as potential and complementary energies in the elastic case will be studied. Extensions to the finite strain elastic theories described in Part 1 will be undertaken next. The nonlinear energy principles thus developed will allow the characterization of the stability in elastic systems. The so-called energy criterion of Lagrange-Dirichlet and its relation with Euler theory will be presented in detail, with a discussion of bifurcation theory due to Koiter characterizing limit and bifurcation points. Extensions to non-conservative systems, including follower loads, will also be discussed in the context of Lyapunov stability.

3. The concepts developed above will be applied to the understanding of classical theories of structural mechanics. Linear classical beam theories (Euler-Bernouilli and Timoshenko beams) will be derived from the 3D equilibrium equations as well as from energy principles via the introduction of the appropriate kinematic assumptions. Large-deflection theories of beam-columns will be similarly developed, leading to the study of buckling of such members. Extensions including torsional and inelastic buckling will be addressed. The same program will be followed for plate theory. First, classical linear plate theories (Kirchhoff and Reissner-Mindlin plates) will be derived from equilibrium and energy considerations. Solution methods will be discussed, including approximate solutions via energy methods. Large-deflection theories (von Karman equations) will be presented, and examples of plate buckling will be solved. Finally, an introduction will be presented of invariant theories of rods and shells, thus connecting with the general geometric framework of nonlinear mechanics. The fundamental concepts behind Cosserat continua will be introduced for the interested student to pursue as a research topic in the future.