Abstract
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.
Highlights
Any degenerative effect that might occur in structures during their utilization can severely decrease structures’ stiffness and potentially lead to their failure
Detailed 2D or 3D meshes of finite elements that certainly offer the finest description of a general structure, as well as of the crack and its surrounding location, appear to be uncomfortable and, simplified models are usually implemented in structural health monitoring techniques
Several papers were devoted to Euler–Bernoulli beam's finite element having an arbitrary number of transverse cracks differing in the mathematical approaches applied to obtain closed-form solutions for stiffness matrix (Biondi and Caddemi (2007); Skrinar (2009 and 2013); Palmeri and Cicirello (2011); Skrinar and Pliberšek (2012))
Summary
Any degenerative effect that might occur in structures during their utilization can severely decrease structures’ stiffness and potentially lead to their failure. Several papers were devoted to Euler–Bernoulli beam's finite element having an arbitrary number of transverse cracks differing in the mathematical approaches applied to obtain closed-form solutions for stiffness matrix (Biondi and Caddemi (2007); Skrinar (2009 and 2013); Palmeri and Cicirello (2011); Skrinar and Pliberšek (2012)). These approaches directly lead to exact displacements at element’s nodes, this changes when the displacements between the nodes are being evaluated. The additional discrete displacement information, in conjunction with novel interpolation functions, offers the direct evaluation of transverse displacements for both sections to the left and right of the crack
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