On the derivation of symbolic form of stiffness matrix and load vector of a beam with an arbitrary number of transverse cracks

Skrinar, Matjaž; Pliberšek, Tomaž

doi:10.1016/j.commatsci.2011.07.013

Cited by 14 publications

(7 citation statements)

References 30 publications

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“…In the FE proposed by Skrinar [15,16], cubic splines are used to represent the field of transverse displacements in each uncracked region of the beam, while the additional kinematic and static unknowns arising at each crack have been eliminated with the help of compatibility and equilibrium equations combined with the Hooke's law for the rotational springs simulating the cracks. However, this study has only considered slender Euler-Bernoulli beams in bending and masses lumped at the two nodes of the resulting FE, which may limit its applicability.…”

Section: Introductionmentioning

confidence: 99%

An efficient two-node finite element formulation of multi-damaged beams including shear deformation and rotatory inertia

Donà

Palmeri

Lombardo

et al. 2015

Computers & Structures

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Section: Introductionmentioning

confidence: 99%

An efficient two-node finite element formulation of multi-damaged beams including shear deformation and rotatory inertia

Donà

Palmeri

Lombardo

et al. 2015

Computers & Structures

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“…Afterwards, several papers were devoted to Euler-Bernoulli beam's finite element having an arbitrary number of transverse cracks differing in the principles of mechanics applied to obtain closed-form solutions of the genuine governing differential equation for transverse displacements. The static transverse displacements and various forms of stiffness matrix were thus obtained by implementing the Dirac delta function either in regard to the rigidity (Biondi and Caddemi [16]) or flexibility (Palmeri and Cicirello [17]), sequential solutions of coupled differential equations (Skrinar [18]), as well as the virtual work principle (Skrinar and Pliberšek [19]).…”

Section: Introductionmentioning

confidence: 99%

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

Skrinar

2019

Period. Polytech. Civil Eng.

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Adequate computational models are crucial for a reliable representation of the mechanical behaviour of structural elements and, therefore, numerous investigations are oriented towards the modelling of cracked structures. This paper studies the behaviour of transversely cracked beams of rectangular cross-sections with linearly-varying widths. The governing differential equations of bending are analytically solved for various beams with different boundary conditions. These simplified model’s solutions are further validated by the results from corresponding 3D finite models of the considered structures. Furthermore, strain and kinetic energy, as well as the work done by an external axial compressive force P are evaluated from the computed transverse displacement functions. These values allowed for estimations of the first eigenfrequency as well as the buckling load. These structural’s parameters were additionally evaluated by implementing dedicated polynomial functions for some cases considered.The results from the simplified model have exhibited very good agreement with the results from more detailed 3D FE models for all performed analyses. The simplified model thus yields an adequate, as well as accurate, approach for the modelling of cracked beams with a linear variation of width in engineering situations, where cracks have to be considered during analysis. The results from the simplified model have exhibited very good agreement with the results from more detailed 3D FE models for all performed analyses. The simplified model thus yields an adequate, as well as accurate, approach for the modelling of cracked beams with a linear variation of width in engineering situations, where cracks have to be considered during analysis.

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“…This simplified model was also included in numerous finite element solutions. Several different approaches were applied to obtain closed‐form solutions for the static transverse displacements and stiffness matrix of a beam's finite element having an arbitrary number of transverse cracks (Biondi and Caddemi, Palmeri and Cicirello, Skrinar, and Skrinar and Pliberšek). The solutions for multistepped beams and beams with linearly varying heights also exist (Skrinar).…”

Section: Introductionmentioning

confidence: 99%

Exact closed‐form finite element solution for the bending static analysis of transversely cracked slender elastic beams on Winkler foundation

Skrinar

Imamović

2018

Num Anal Meth Geomechanics

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Summary In this paper, bending problem of a transversely cracked beam resting on elastic foundations is addressed by means of the finite element method. The paper covers the derivation of a new finite element, where the soil is modelled by classical Winkler soil model, and the cracked beam is represented by the simplified computational model, as already widely used for various analyses of such transversely cracked slender beams. The derivation of transverse displacements' interpolation functions for the transversely cracked slender beam and the stiffness matrix, as well as the corresponding load vector for the linearly distributed continuous transverse load are presented. All expressions are derived at in closed symbolic forms, and they allow easy implementation in the existing finite element software. The presented approach is ideal for the effortless modelling of cracked beams in conditions where neither information about the crack's growth nor the stresses at the crack's tip is required. Numerical examples covering several load situations are presented to support the discussed approach. The results obtained are further compared with the values from corresponding coupled differential equations as well as from a large 2D plane finite elements model, where a detailed description of the crack was accomplished. It is evident that the drastic difference in the computational effort is not reflected in any significant difference in the results between the models.

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On the derivation of symbolic form of stiffness matrix and load vector of a beam with an arbitrary number of transverse cracks

Cited by 14 publications

References 30 publications

An efficient two-node finite element formulation of multi-damaged beams including shear deformation and rotatory inertia

An efficient two-node finite element formulation of multi-damaged beams including shear deformation and rotatory inertia

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

Exact closed‐form finite element solution for the bending static analysis of transversely cracked slender elastic beams on Winkler foundation

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