Transversely Cracked Beams with Quadratic Function’s Variation of Height

Abstract

This paper considers the modelling of transversely cracked slender beams by a new one-dimensional finite element where the cross-section’s height variation is given as a general quadratic function. The derivations of a closed-form stiffness matrix and a load vector are based on a simplified computational model where the cracks are represented by means of internal hinges endowed with rotational springs that take into account the cross-section’s residual stiffness. These derived at expressions efficiently upgrade the stiffness matrix as well as the load vector coefficients which were previously presented for the non-cracked beams. Although the derivations were rather straightforward, all the derived terms are written entirely in closed-symbolic forms. These solutions, together with the solutions of second order governing differential equation of bending thus define an ‘exact’ finite element for the implemented simplified computational model. The derivations are complemented by two comparative case studies. They demonstrated that elaborated solutions may be effectively implemented for structural analyses as the presented expressions produced excellent results that were confirmed independently by more thorough 3D models.