Geometric Nonlinear Analysis of Timoshenko Beams
Résumé
The linear finite element analysis of solids and structures are discussed in the first part of the report. The
finite element formulations for a three dimensional problem is derived and significant issues are addressed.
Three dierent types of elements, the bilinear iso-parametric quadrilaterals, the quadratic triangles and
the linear tetrahedrons are used to solve a linear plate problem and the results are compared.
The analysis of a common geometric nonlinearity encountered in structural mechanics is dealt with in the
second part of the report. A brief introduction to nonlinear analysis is provided, while the geometric nonlinearity
is discussed in detail. Timoshenko beam analysis is considered as the one dimensional version of
Reissner-Mindlin plate theory and the nonlinear strain-displacement relation are treated in an appropriate
way to avoid unrealistic simplifications. The force vector and tangent stiffness matrix are derived and
the formulation is extended to implement the trigonometric basis functions. A major issue in geometric
nonlinear analysis, namely locking, is addressed and reduced order integrations are implemented to avoid
the consequences. The convergence of the model is checked with available analytical solutions. An Euler
method in combination with Newton-Raphson method is used to fully analyze the geometric nonlinearity.
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