Modelling of the elastoplastic dynamics of longitudinally reinforced wall beams based on a time-explicit central difference method

A numerical analytical method for modelling the elastoplastic deformation of longitudinally reinforced wall beams with isotropically strengthened composite component materials, which enables a solution of the corresponding elastoplastic problem to be obtained at discrete instants of time using an explicit scheme, is developed by a time step method involving central finite differences. In the case of linear elastic composite component materials in the reinforced beams, the proposed model reduces to Bolotin's well-known structural model of the mechanics of composites. An initial-boundary-value problem of the dynamic deformation of longitudinally reinforced flexible wall beams is formulated in the von Karman approximation with consideration of their weakened resistance to transverse shearing. Equations and relations corresponding to two versions of Timoshenko's theory are obtained from a few positions. An explicit “cross” scheme for numerical integration of the initial-boundary-value problem posed, which is consistent with the step-by-step scheme used to model the elastoplastic deformation of a composite beam material, is constructed. Calculations of the dynamic and quasistatic bending behaviour of reinforced wall beams during the linear elastic and elastoplastic deformation of the composite component materials are performed. It is found that the classical theory is totally unacceptable for performing such calculations (except for beams of very small relative height) and that the first version of Timoshenko's theory gives adequate results only in the case of linear elastic composite component materials. Use of the second version of Timoshenko's theory is recommended as more accurate for calculations of the elastoplastic deformation of reinforced wall beams.