Improved finite element of a transversely cracked straight beam with an additional degree of freedom

Abstract

A new finite element with an additional degree of freedom at the crack location is derived for static bending analysis of a transversely cracked uniform slender beam. In the simplified computational model, which is based on Euler-Bernoulli's theory of small displacements, the crack is represented by a linear rotational spring connecting two elastic parts. The derivation of the transverse displacements, the coefficients of the stiffness matrix as well as the load vector for uniformly distributed load along the whole beam element was based on the utilization of polynomial interpolation functions of the fourth degree and all derived expressions were obtained in the closed form. The novelty of the new model, by comparison to the previously presented simplified finite element models, is that the transverse displacements functions obtained by utilization of the newly presented interpolation functions for the case of uniform continuous transverse load along whole beam element, as well as the functions of the bending moments and transverse forces, are accurate. The values obtained by the simplified model also exhibited good agreement in additional comparison with the results from more demanding and more detailed 2D models.