Finite element variational formulation for beams with discontinuities

This paper presents a variational formulation of the mechanical behaviour of beams with strong discontinuities, enhanced to simulate the strain localization process. The considered strain localization zones represent the formation of dislocations and hinges in beams. The presented general formulation applies to thick beams, which takes into account the internal strain energy due to bending and shear, and also a simpler formulation which takes into account only bending induced strains which applies to thin beams. It is shown that the developed energy functional for the beams with discontinuities has as stationarity conditions the strong formulation of the associated boundary value problem. As illustration, the energy functionals for Timoshenko and Euler–Bernoulli beams with embedded discontinuities are approximated by finite elements with embedded discontinuities. The development of a local material failure (leading to hinge-like strain localization zones) is in terms of continuum constitutive models furnished with strain softening capabilities. To show the validity of this formulation and its consistency with its continuum counterpart, representative numerical examples illustrating the performance of the proposed formulation are presented.

► General variational formulation for beams with inner discontinuities.
► Hierarchy of particular variational formulations derived from the general formulation.
► Finite element approximations developed for Timoshenko and Bernoulli–Euler beams.
► Symmetric element matrices, leading to guaranteed stable and convergent solutions.
► These elements allow the numerical simulation of collapse progression of frames.