Finite Element Variational Formulation for Bending Elements with and without Discontinuities

This paper presents a variational formulation of the mechanical behaviour of solids with strong discontinuities adapted to represent strain localization in bending-dominated structural members such as: beams, plates and shells. Its approximation is formulated through finite elements with embedded discontinuities. The paper deals with a formulation which takes into account only the internal strain energy due to bending induced strains (Bernoulli-Euler beams and thin plate theories), and with a more general formulation which also takes into account the internal energy of shear deformation (Timoshenko and thick plate theories). It is shown that the developed energy functional of the solid with discontinuities has as stationary conditions the strong formulation of the problem, i.e., the governing equations of the corresponding boundary value problem. Some advantages of this variational formulation are that the stiffness matrix of the elements: 1) satisfies the inner continuity (moments and shears) at the localization band and the simulated rigid body movements of the finite element parts generated by the discontinuity are consistent, and 2) is symmetric, reducing the involved computational effort and also the number of numerical problems encountered during its application. The development of a local material failure (leading to hinges-like localization zones) is in terms of continuum constitutive models furnished with strain softening capabilities. Representative numerical examples illustrate the performance of the presented formulation.

JUAREZ Gelacio;  AYALA Gustavo;  CASADEI Folco; 

2006-10-19

Associação Portuguesa de Mecânica Teórica, Aplicada e Computacional

JRC33544

https://publications.jrc.ec.europa.eu/repository/handle/JRC33544,