Exact closed‐form finite element solution for the bending static analysis of transversely cracked slender elastic beams on Winkler foundation

In this paper, bending problem of a transversely cracked beam resting on elastic foundations is addressed by means of the finite element method. The paper covers the derivation of a new finite element, where the soil is modelled by classical Winkler soil model, and the cracked beam is represented by the simplified computational model, as already widely used for various analyses of such transversely cracked slender beams. The derivation of transverse displacements' interpolation functions for the transversely cracked slender beam and the stiffness matrix, as well as the corresponding load vector for the linearly distributed continuous transverse load are presented. All expressions are derived at in closed symbolic forms, and they allow easy implementation in the existing finite element software. The presented approach is ideal for the effortless modelling of cracked beams in conditions where neither information about the crack's growth nor the stresses at the crack's tip is required. Numerical examples covering several load situations are presented to support the discussed approach. The results obtained are further compared with the values from corresponding coupled differential equations as well as from a large 2D plane finite elements model, where a detailed description of the crack was accomplished. It is evident that the drastic difference in the computational effort is not reflected in any significant difference in the results between the models.