On the Bending Analysis of Multi-Cracked Slender Beams with Continuous Height Variations

Abstract

IntroductionAs the cracks are degenerative effects that might severely impact the behaviour of engineering structures, their immediate detection is vitally important for safety reasons. However, the efficiency of structural health monitoring is a combination of accurate data measurements and the versatility of mathematical representation of mechanical behaviour. Although suitable 2D or 3D meshes of finite elements yield the best description of the crack and its surroundings, this approach is advantageous only when all the crack's details are known in advance. Consequently, simplified models are more efficient for inverse problems where none of the potential crack's details (presence, location, intensity) is known.The appropriate simplified model that has been the subject of numerous researches in the past, is the model provided by Okamura et al. [1]. This model simulates cracks by massless rotational linear springs. Each spring connects those neighbouring non-cracked parts of the beam that are modelled as elastic elements and the linear moment-rotation constitutive law is adopted. Such a mathematical model allows for all the essential data (transverse displacements and inner forces) to be evaluated with adequate accuracy.Okamura's computational model was effectively implemented in finite element solutions for the computation of transverse displacements. Initially, Gounaris and Dimarogonas [2] presented a numerical procedure for the computation of a beam element with a single transverse crack. Afterwards, closed-form solutions for the static transverse displacements and stiffness matrix of a beam's finite element having an arbitrary number of transverse cracks have been derived at by several authors by implementing different mathematical methods. The Dirac delta function was implemented twice: by Biondi and Caddemi [3] in regard to the rigidity, and by Palmeri and Cicirello [4] in regard to the flexibility. Sequential solutions of coupled differential equations were implemented by Skrinar [5], while Skrinar and Pliberšek [6] implemented the principle of virtual work.Due to its simplicity (the depth and the location are the only crack's parameters required) the "discrete spring" model has been intensively implemented in vibration analysis of cracked

Enhanced transverse displacements analysis of transversely cracked beams with a linear variation of width due to axial tensile forces

“…where k EI represents of the slope the flexural rigidity function i. e., $${{{E{I}_{L}-E{I}_{0}}\over{L}}}$$ , and EI 0 and EI L represent flexural rigidities at the start and the end of the beam, respectively. Although the last equation can be introduced into Equation (4) prior to any further derivations, this leads to logarithmic terms with negative arguments in the derived solutions for simple bending, causing numerical issues [6]. However, these problems are completely avoided by introducing the coefficient k EI , which further yields more abbreviate forms of the derived expressions.…”

On the Application of the Simplified Crack Model in the Bending, Free Vibration and Buckling Analysis of Beams with Linear Variation of Widths

“…The majority of the research has been limited to elements with constant rectangular cross-sections. Skrinar and Imamović [20] presented a numerical study in which they studied bending of beams of various heights' variations along the length. In their investigation, they implemented a multistepped multi-cracked beam finite element (Skrinar [21]) where the genuine continuous variation of height was modelled by an adequate series of steps.…”