Dynamic elasto-plastic response of symmetrically loaded beams

An approximate model to analyze the dynamic elasto-plastic large deformation response of beams under various symmetrical loading is presented. The present model extends the capabilities of the earlier model (Yankelevsky DZ, Boymel A. Dynamic elasto-plastic response of beams — a new model. Int. J. Impact Engng. 1984;2(4):285–298) and considers the general symmetrical problem with loading along part of the beam. The model beam is composed of two rigid parts interconnected by a gap of zero width, thus yielding a triangular deflection shape. The gap is built of fibers having imaginary length. That imaginary length governs the strains and stresses in the beam, and is determined by requiring equal deflections in the real and in the model beams. The imaginary length is found to be almost constant in the elastic and in the elasto-plastic domains, but depends on the load distribution. Comparisons of predicted maximum deflections with test results show very good correspondence. Apart from the final deflections, which may easily be measured in tests, the model may also calculate the time history of the dynamic reactions, bending moment and membrane force, velocity and acceleration as well as stress and strain distributions in the mid-section of the beam at selected times. The proposed model may be applied to various boundary conditions and may be extended to include strain hardening and rate effects.