Acta Mechanica

, Volume 230, Issue 12, pp 4391–4415 | Cite as

Numerical analysis of natural frequency and stress intensity factor in Euler–Bernoulli cracked beam

  • A. AlijaniEmail author
  • M. Kh. Abadi
  • J. Razzaghi
  • A. Jamali
Original Paper


In this paper, the evaluation procedures of the natural frequency and the stress intensity factor in the opening mode are established for the Euler–Bernoulli cracked beam by using (a) a technique in the framework of the finite element method, (b) a group method of data handling (GMDH), and (c) the software ABAQUS software. In the first one, the stiffness and mass matrices of the beam are enriched according to the depth and location of the crack for the determination of the natural frequency. A discrete spring model is used to simulate the crack in the structure based on the energy release rate. The continuity conditions in a cracked element are applied to connect two sub-elements of both sides of the crack. In the second method, the natural frequency and the stress intensity factor are determined using the GMDH algorithm. Design of experiments technique is employed to create an optimum arrangement for the application in the GMDH neural network. A few case studies are examined to investigate the results of the analysis, in addition, to identify the priority and the comparison of the three methods. The procedure of the analysis explains the advantages and limitations of the finite element-based technique, the GMDH method, and ABAQUS.

List of symbols


Crack depth


Constant coefficients of GMDH


Cross-sectional area


Width of the beam

\({\varvec{D}}_{L\left( R \right) }\)

Conversion matrix of left (right) sub-element


Young’s modulus


Flexural stiffness


Height of the beam


Moment of inertia in perfect section


Moment of inertia in cracked section


Stiffness factor of rotational spring


Stress intensity factor


Stiffness matrix of perfect element


Stiffness matrix of left (right) sub-element


Enriched stiffness matrix of left (right) sub-element


Enriched stiffness matrix of rotational spring


Stiffness matrix of cracked element


Length of the element


Length of the beam


Number of output variables of GMDH


Bending moment


Mass matrix of perfect section


Mass matrix of left (right) sub-element


Enriched mass matrix of left (right) sub-element


Enriched total mass matrix


Number of input variables of GMDH


Shape function


Distributed load


Kinetic energy


Kinetic energy of left (right) sub-element


Enriched kinetic energy of (right) sub-element


Total kinetic energy of a cracked element


Displacement vector


Time derivative of displacement vector


Displacement vector of left (right) sub-element


Time derivative of displacement vector of left (right) sub-element


Strain energy


Strain energy in left (right) sub-element


Enriched strain energy in left (right) sub-element


Absorbed potential energy in rotational spring


Total potential energy of a cracked element


Shear force




Time derivative of deflection


Time derivative of the left sub-element deflection


Axial direction


Input variables of GMDH


Crack location in the beam


Crack location in the element


Input vector of GMDH


Output variables of GMDH


Output vector of GMDH


Transverse direction

\(\Delta \phi \)

Change in slope

\(\varepsilon _{x}\)

Axial strain

\(\lambda _{mm}\)

Compliance for bending moment


Poisson’s ratio

\(\rho \)


\(\sigma _{x}\)

Axial stress

\(\phi \)


\({\dot{\phi }}\)

Time derivative of slop

\(\omega \)

Natural frequency



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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Bandar Anzali BranchIslamic Azad UniversityBandar AnzaliIran
  2. 2.Department of Civil EngineeringUniversity of GuilanRashtIran
  3. 3.Faculty of Mechanical EngineeringUniversity of GuilanRashtIran

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