PSI - Issue 66

H.A.F.A. Santos et al. / Procedia Structural Integrity 66 (2024) 195–204

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H.A.F.A. Santos et al. / Structural Integrity Procedia 00 (2025) 000–000

The discrete spring model of the crack, which neglects its initiation and propagation, is usually adopted in the analysis of the global mechanical performance of cracked beams, where the cracks are regarded as internal rotational springs. A special case is the linear open crack model, in which a non-propagating crack is assumed to be open during the loading process. This model has been extensively applied to study the structural response of cracked beams, as it represents the best trade-off between accuracy and computational effort, in particular when the global response of a structure is of primary interest. Motivated by these considerations, many formulations have been developed based on the discrete spring model, including the rigidity modelling byBiondi et al. [3, 4], in which the singularities in the flexural stiffness, corresponding to concentrated damages, are introduced as negative impulses ( i.e. , Dirac’s delta functions with a negative sign). However, this mathematical representation is not consistent with the definite-positive nature of flexural stiffness. Aimed at overcoming this theoretical flaw, Palmeri et al. [10] developed a physically consistent representation of cracks, in which Dirac’s delta functions with a positive sign are introduced in the bending flexibility of the beam - flexibility modelling approach . A theoretical justification of this approach was subsequently presented by Caddemi et al. [5]. Most of the research works available in the literature on the analysis of cracked beams are focused on uniform beams, i.e. , beams with constant cross-sections. A few execptions are the works by Skrinar et al. [22, 24, 23], in which beams with cross-sections changing along its axial coordinate were considered. On the other hand, thanks to their excellent performance, functionally graded materials (FGMs) have been increas ingly used in the last years in different engineering applications, such as energy electronics, submarines, automotive components, space-station structures, biomedical applications, sensors and thermo-generators, etc.. FGMs are inho mogeneous composites whose material properties vary gradually with respect to spatial coordinates. The material composition can be designed so as to improve the strength, toughness, high-temperature resistance, etc. to meet the desired structural properties and performance. Although several studies can be found in the literature on the numerical modelling of FGM beams with cracks, in particular works that address the free vibration analysis [1, 25, 8, 13, 21], studies on non-uniform FGM beams with cracks are very scarce. Exceptions are the work by Shabani [20], with the focus on the free vibration analysis of cantilevered Timoshenko rectangular beams with a constant height, but an exponential or linear width variation, and, more recently, the work by Nguyen [9], which addresses the free vibration analysis of non-uniform cracked axially FGMbeams. Most of the aforementioned works focus on the derivation of analytical solutions. However, for multi-span beams, complicated loading conditions, complex material constitutive laws, and complex beam geometries, the derivation of analytical solutions may turn out very demanding or even impossible. On the other hand, the rapidly increasing computation power, has led, since the beginning of the seventies, to the development of computational and numerical methods in engineering. Among these methods, the finite element method emerged as the most commonly used tech nique, due to its versatility, capability to handle complex geometries, etc. Among the various formulations of the finite element method, its displacement-based formulation has been the most adopted one. In this formulation, the displace ments are taken as the fundamental unknowns. However, this formulation leads to distributions of bending moments and forces that are discontinuous between elements and that do not satisfy, in general, the equilibrium boundary conditions. As a result, and since for many structural engineering purposes, the stress distribution is very often the paramount information needed, applications of this formulation usually involve an averaging procedure to obtain con tinuous stress distributions for design calculations, which is viewed as a drawback within the framework of structural design. Formulations that avoid the need for these averaging procedures are the so-called equilibrium-based formula tions , which are capable to produce equilibrated solutions. Examples of such formulations applied to one-dimensional beam problems were proposed in, e.g. , [18, 14, 15, 19, 16, 17]. Additionally, when applied to Timoshenko beams, the standard displacement-based formulation may suffer from the so-called shear-locking phenomenon , leading to erroneous solutions. The goal of this work is to propose a simple and effective equilibrium-based finite element formulation for the quasi-static analysis of functionally graded non-uniform Timoshenko beams with multiple and open concentrated cracks. The cracks are modelled using the discrete spring approach in which Dirac’s delta generalized functions are introduced into the bending flexibility of the beams (flexibility modelling approach). The introduced formulation relies on a complementary variational approach and is based on the discretization of the elements’ internal forces