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11.1 Introduction

Structural nonlinearities may be caused by a number of factors and are broadly classified into two categories: geometric and material. In nonlinear response the total load is no longer proportional to the total displacement. Individual factors causing nonlinearities may be grouped as follows:

1. Large rotations Result in a nonlinear force displacement relationship, and incremental effect is computed as the geometric stiffness matrix.

2. Large displacements Require updating of the original equilibrium equations and involve not only updating geometry at every computational time step but also calculating the initial stress stiffness matrix, representing the effect of realignment of current internal stresses because of displacements.

3. Nonlinear stress strain law Occurs in rubberlike materials as well as in most metals, elastomers, and some composites when properties are unequal in tension and compression.

4. Large strain Occurs in plastics, some metals, rubbers, and elastomers.

The first two items belong to the large deformation category, involving geometrical nonlinearity, whereas the last two items belong to material nonlinearity. Descriptions of numerical algorithms for solution of nonlinear problems are given next with small strain assumption.

11.2 Geometric Nonlinearity

For geometric nonlinearity, a Newton Raphson iterative procedure is used in which the elastic stiffness matrix is supplemented with the geometric stiffness matrix K G, so that both large displacements and rotations, as well as the effect of in-plane stretching, are taken into consideration; strains are assumed to be small in the formulation given herein.

For static analysis, the solution algorithm is as follows.

Let n equal the number of load increments and i

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