APPLICATION OF CLOSED FORM SOLUTIONS TO BEAMS PROBLEMS WITH SOFTENING DISCONTINUITIES

Abstract

The development of variational formulations for thin and thick beams with softening discontinuities adapted to represent strain localization is presented. The formulation for thin beams takes into account the internal strain energy due to bending induced strains, in which the discontinuities are considered as softening hinges. For the case of thick beams, the formulation takes into account the internal strain energy due to bending and shear strains, in which the discontinuities are considered as softening hinges/dislocations. These formulations include the embedded discontinuity model, in which the discontinuity and the strain concentration are lumped into a zero-thickness localization zone. The non-linear behavior of the material is described by a discrete constitutive model, in which linear or exponential softening is considered. The boundary value problems and closed form solutions are developed from the variational formulations and the conventional methods of calculus of variations. The application of closed form solutions to practical examples modeled numerically guarantees the accuracy of the developed solutions.