Computational analysis of multi-stepped beams and beams with linearly-varying heights implementing closed-form finite element formulation for multi-cracked beam elements Matjaz ˇ Skrinar ⇑ University of Maribor, Faculty of Civil Engineering, Smetanova 17, SI 2000 Maribor, Slovenia article info Article history: Received 5 October 2012 Received in revised form 12 March 2013 Available online 22 April 2013 Keywords: Stepped beams with transverse cracks Transversely cracked beams with linearly varying height Simplified computational model Principle of virtual work Finite element method Stiffness matrix Load vector Transverse displacements abstract The model where the cracks are represented by means of internal hinges endowed with rotational springs has been shown to enable simple and effective representation of transversely-cracked slender Euler–Ber- noulli beams subjected to small deflections. It, namely, provides reliable results when compared to detailed 2D and 3D models even if the basic linear moment–rotation constitutive law is adopted. This paper extends the utilisation of this model as it presents the derivation of a closed-form stiffness matrix and a load vector for slender multi-stepped beams and beams with linearly-varying heights. The principle of virtual work allows for the simple inclusion of an arbitrary number of transverse cracks. The derived at matrix and vector define an ‘exact’ finite element for the utilised simplified computational model. The presented element can be implemented for analysing multi-cracked beams by using just one finite element per structural beam member. The presented expressions for a stepped-beam are not exclusively limited to this kind of height variation, as by proper discretisation an arbitrary variation of a cross-section’s height can be adequately modelled. The accurate displacement functions presented for both types of considered beams complete the der- ivations. All the presented expressions can be easily utilised for achieving computationally-efficient and truthful analyses. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Numerous engineering structures are subjected to degenerative effects during their utilisation. The progressions of cracks can se- verely decrease the stiffness of an element and further lead to the failure of the complete structure. In view of this, it is an impor- tant task of engineers to detect these cracks as soon as possible. However, the efficiency of structural health monitoring depends not only on the data measured but also on the qualities and versa- tilities of computational models regarding mechanical behaviour modelling. Undoubtedly, suitable 2D or 3D meshes of finite ele- ments yield a thorough discretisation of the structure, as well as of the crack and its surroundings. Although this approach is excel- lent when evaluating a structure’s response to a cracked situation (with all the crack’s details known in advance), it becomes quite awkward for inverse problems where the potential crack’s details (presence, location, intensity) are unknown. Consequently, simpli- fied models are more efficient in such situations. The model that has been the subject of numerous research in the past, is the model provided by Okamura et al. (1969). In this model each crack is replaced by a massless rotational linear spring of suitable stiffness and the linear moment-rotation constitutive law is adopted. Each spring connects those neighbouring non- cracked parts of the beam that are modelled as elastic elements. Okamura et al. introduced the earliest definition for rotational linear spring stiffness for a rectangular cross-section. In addition, some other researchers (Dimarogonas and Papadopulus, 1983; Ra- jab and Al-Sabeeh, 1991; Ostachowicz and Krawczuk, 1990; Kra- wczuk and Ostachowicz, 1993; Sundermayer and Weaver, 1993; Hasan, 1995; Skrinar and Pliberšek, 2004) have presented their definitions. This model was successfully implemented for dynamic analy- ses. For a singly-cracked beam Fernández-Sáez and Navarro (2002) presented closed-form expressions for the approximated values of fundamental frequencies, whilst for a beam with multiple cracks several solutions exist for natural frequency calculations: a technique that reduces the order of the determinantal equation (Shifrin and Ruotolo, 1999); a transfer matrix-based method lead- ing to the determinant calculation of a 4 4 matrix (Khiem and Lien, 2001); a fundamental solutions and recurrences formulae- based approach for determining the mode-shapes of non-uniform beams and concentrated masses (Li, 2002). The dynamic response of a cracked cantilever beam subjected to a concentrated moving 0020-7683/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijsolstr.2013.04.005 ⇑ Tel.: +386 2 22 94 358; fax: +386 2 25 24 179. E-mail addresses: [email protected], [email protected] International Journal of Solids and Structures 50 (2013) 2527–2541 Contents lists available at SciVerse ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector
Computational analysis of multi-stepped beams and beams with linearly-varying heights implementing closed-form finite element formulation for multi-cracked beam elements
Jun 29, 2023
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