Transverse Displacements of Transversely Cracked Beams with a Linear Variation of Width Due to Axial Tensile Forces

  • Matjaž SkrinarEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 124)


This paper considers the analysis of transverse displacements of slender beams with a linear variation of width and a single-sided transverse crack, subjected to axial tensile forces. When analysing such structures, the application of detailed 2D or 3D finite element meshes is undoubtedly the best solutions possible. However, in inverse identification of potential cracks such comprehensive models are actually not the most suitable solutions. Therefore, the presented studies implement a simplified model where the crack is represented by means of an internal hinge endowed with a rotational spring. In the first part of the research presented, solutions from the simplified model’s governing differential equations are obtained. The purpose of this study is to demonstrate the model’s ability to adequately describe the considered phenomenon and to derive an appropriate rotational spring definition. The second part of the research discussed is devoted to modelling of the phenomenon by the simple one-dimensional beam finite element. Afterwards, the implementation and the quality of the results are being presented through a comparative case study that complements the derivations.


Beam finite element Stiffness matrix and load vector Linear variation of width Transverse cracks Transverse displacements Axial tensile forces 



The author acknowledges the partial financial support from the Slovenian Research Agency (research core funding No. P2-0129 (A) “Development, modelling and optimization of structures and processes in civil engineering and traffic”).


  1. 1.
    Okamura, H., Liu, H.W., Chu, C.-S., Liebowitz, H.: A cracked column under compression. Eng. Fract. Mech. 1(3), 547–564 (1969)CrossRefGoogle Scholar
  2. 2.
    Reddy, J.N.: An Introduction to the Finite Element Method, 2nd edn. McGraw-Hill (1994)Google Scholar
  3. 3.
    Skrinar, M.: Computational analysis of multi-stepped beams and beams with linearly-varying heights implementing closed-form finite element formulation for multi-cracked beam elements. Int. J. Solids Struct. 50(14–15), 2527–2541 (2013)CrossRefGoogle Scholar
  4. 4.
    Skrinar, M.: On the application of the simplified crack model in the bending, free vibration and buckling analysis of beams with linear variation of widths. Periodica Polytech. Civil Eng. (2019).
  5. 5.
    Smith, I.M., Griffiths, D.V.: Programming the finite element method. Wiley, Chichester (1997)zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of Civil Engineering, Transportation Engineering and ArchitectureUniversity of MariborMariborSlovenia

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