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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
  • 24 Downloads

Abstract

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

a

Crack depth

\(a_{i}\)

Constant coefficients of GMDH

A

Cross-sectional area

b

Width of the beam

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

Conversion matrix of left (right) sub-element

E

Young’s modulus

\(EI_{0}\)

Flexural stiffness

h

Height of the beam

\(I_{0}\)

Moment of inertia in perfect section

\(I_{c}\)

Moment of inertia in cracked section

\(k_{sp}\)

Stiffness factor of rotational spring

\(K_{I}\)

Stress intensity factor

\({\varvec{K}}_{0}\)

Stiffness matrix of perfect element

\({\varvec{K}}_{L(R)}\)

Stiffness matrix of left (right) sub-element

\({\varvec{K}}_{L(R)}^{c}\)

Enriched stiffness matrix of left (right) sub-element

\({\varvec{K}}_{sp}^{c}\)

Enriched stiffness matrix of rotational spring

\({\varvec{K}}_{t}^{c}\)

Stiffness matrix of cracked element

\(l_{e}\)

Length of the element

L

Length of the beam

m

Number of output variables of GMDH

M

Bending moment

\({\varvec{M}}_{0}\)

Mass matrix of perfect section

\({\varvec{M}}_{L(R)}\)

Mass matrix of left (right) sub-element

\({\varvec{M}}_{L(R)}^{c}\)

Enriched mass matrix of left (right) sub-element

\({\varvec{M}}_{t}^{c}\)

Enriched total mass matrix

n

Number of input variables of GMDH

\({\varvec{N}}\)

Shape function

q

Distributed load

T

Kinetic energy

\(T_{L(R)}\)

Kinetic energy of left (right) sub-element

\(T_{L(R)}^{c}\)

Enriched kinetic energy of (right) sub-element

\(T_{t}^{c}\)

Total kinetic energy of a cracked element

\({\varvec{u}}\)

Displacement vector

\({\dot{{\varvec{u}}}}\)

Time derivative of displacement vector

\({\varvec{u}}_{L(R)}\)

Displacement vector of left (right) sub-element

\({\dot{{{\varvec{u}}}}}_{L(R)}\)

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

U

Strain energy

\(U_{L(R)}\)

Strain energy in left (right) sub-element

\(U_{L(R)}^{c}\)

Enriched strain energy in left (right) sub-element

\(U_{sp}\)

Absorbed potential energy in rotational spring

\(U_{t}^{c}\)

Total potential energy of a cracked element

V

Shear force

w

Deflection

\({\dot{w}}\)

Time derivative of deflection

\({\dot{w}}_{L(R)}\)

Time derivative of the left sub-element deflection

x

Axial direction

\(x{}_{i}\)

Input variables of GMDH

\(x_{0}\)

Crack location in the beam

\(x_{c}\)

Crack location in the element

\({\varvec{X}}\)

Input vector of GMDH

\(y_{i}\)

Output variables of GMDH

\({\varvec{Y}}\)

Output vector of GMDH

z

Transverse direction

\(\Delta \phi \)

Change in slope

\(\varepsilon _{x}\)

Axial strain

\(\lambda _{mm}\)

Compliance for bending moment

v

Poisson’s ratio

\(\rho \)

Density

\(\sigma _{x}\)

Axial stress

\(\phi \)

Slope

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

Time derivative of slop

\(\omega \)

Natural frequency

Notes

References

  1. 1.
    Mohamadi, M.: Extended Finite Element Method For Fracture Analysis of Structures. Blackwell Publishing, Oxford (2008)CrossRefGoogle Scholar
  2. 2.
    Irwin, G.R., Kies, J.A.: Critical energy rate analysis of fracture strength. J. Weld. 33, 193–198 (1954)Google Scholar
  3. 3.
    Irwin, G.R.: Analysis of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. 24, 361–364 (1957)Google Scholar
  4. 4.
    Kienzler, R., Herrmann, D.: On material forces in elementary beam theory. J. Appl. Mech. 53, 561–564 (1986)CrossRefGoogle Scholar
  5. 5.
    Ngo, D., Scordelis, A.C.: Finite element analysis of reinforced concrete beams. J. Am. Concr. Inst. 64, 52–163 (1967)Google Scholar
  6. 6.
    Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. J. Geotech. 29, 47–65 (1979)CrossRefGoogle Scholar
  7. 7.
    Owen, D.R.J., Fawkes, A.J.: Engineering Fracture Mechanics: Numerical Methods and Applications. Pine Ridge Press, Swansea (1983)Google Scholar
  8. 8.
    Owen, D.R.J.: Hinton E Finite Element Plasticity, Theory and Practice. Pine Ridge Press, Swansea (1980)Google Scholar
  9. 9.
    Bazant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other Quasi Brittle Materials. CRC Press Publishing, London (1998)Google Scholar
  10. 10.
    Moes, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. J. Numer. Methods Eng. 46, 131–150 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. J. Numer. Methods Eng. 45, 601–620 (1999)zbMATHCrossRefGoogle Scholar
  12. 12.
    Sih, G.C.: Handbook of stress intensity factors for researchers and engineerings. Lehigh University, Bethlehem (1973)Google Scholar
  13. 13.
    Chang, R.: Static finite element stress intensity factors for annular cracks. J. Nondestr. Eval. 2, 119–124 (1981)CrossRefGoogle Scholar
  14. 14.
    Tada, H., Paris, P.C., Irwin, G.R.: The stress analysis of crack handbook. Paris Productions & (Del Research Corp.) (1985)Google Scholar
  15. 15.
    Kienzler, R., Herrmann, G.: An elementary theory of defective beams. J. Acta Mech. 62, 37–46 (1986)zbMATHCrossRefGoogle Scholar
  16. 16.
    Ricci, P., Viola, E.: Stress intensity factors for cracked T-sections and dynamic behavior of T-beams. J. Eng. Fract. Mech. 73, 91–111 (2006)CrossRefGoogle Scholar
  17. 17.
    Kumar, P.: Elements of Fracture Mechanics. McGraw-Hill, McGraw Hill Education (India) Private Limited, Bengaluru (2009)Google Scholar
  18. 18.
    Qian, G.L., Gu, S.N., Jiang, J.S.: The dynamic behavior and crack detection of a beam with a crack. J. Sound Vib. 2, 233–243 (1990)CrossRefGoogle Scholar
  19. 19.
    Leissa, A.W., Qatu, M.S.: Vibrations of Continuous Systems. McGraw-Hill, New York (2011)Google Scholar
  20. 20.
    Rao, S.S.: Mechanical vibrations. Prentice Hall, New York (2011)Google Scholar
  21. 21.
    Logan, D.L.: A First Course in the Finite Element Method. Chris Carson, Toronto (2007)Google Scholar
  22. 22.
    Yokoyama, T., Chen, M.C.: Vibration analysis of edge-cracked beams using a line-spring model. J. Eng. Fract. Mech. 59, 403–409 (1998)CrossRefGoogle Scholar
  23. 23.
    Gudmundsun, P.: The dynamic behavior of slender structures with cross-sectional cracks. J. Mech. Phys. Solids 1, 329–345 (1983)CrossRefGoogle Scholar
  24. 24.
    Pandey, A.K., Biswas, M., Samman, M.M.: Damage detection from change in curvature mode shapes. J. Sound Vib. 145, 321–335 (1994)CrossRefGoogle Scholar
  25. 25.
    Lele, S.P., Maiti, S.K.: Modelling of transverse vibration of short beams for crack detection and measurement of crack extension. J. Sound Vib. 257, 559–583 (2002)CrossRefGoogle Scholar
  26. 26.
    Silva, J.M., Gomes, A.J.: Experimental dynamic analysis of cracked free-free beams. J. Exp. Mech. 30, 20–25 (1990)CrossRefGoogle Scholar
  27. 27.
    Kim, J.T., Stubbs, N.: Crack detection in beam-type structures using frequency data. J. Sound Vib. 259, 145–160 (2003)CrossRefGoogle Scholar
  28. 28.
    Swamidas, A.S.J., Yang, X.F., Seshadri, R.: Identification of cracking in beam structures using Timoshenko and Euler formulations. J. Eng. Mech. 130, 1297–1308 (2004)CrossRefGoogle Scholar
  29. 29.
    Dobeling, S.W., Farrar, C.R., Prime, M.B.: A summary review of vibration-based damage identification methods. J. Shock Vib. Dig. 30, 91–105 (1998)CrossRefGoogle Scholar
  30. 30.
    Cawly, P., Adams, R.D.: The locations of defects in structures from measurements of natural frequencies. J. Strain Anal. 14, 49–57 (1979)CrossRefGoogle Scholar
  31. 31.
    Friswell, M.I., Penny, J.E.T., Wilson, D.A.L.: Using vibration data and statistical measures to locate damage in structures. J. Anal. Exp. Modal Anal. 9, 239–254 (1994)Google Scholar
  32. 32.
    Narkis, Y.: Identification of crack location in vibrating simply supported beams. J. Sound Vib. 172, 549–558 (1994)zbMATHCrossRefGoogle Scholar
  33. 33.
    Pandey, A.K., Biswas, M.: Damage detection in structures using changes in flexibility. J. Sound Vib. 169, 3–17 (1994)zbMATHCrossRefGoogle Scholar
  34. 34.
    Ratcliffe, C.P.: Damage detection using a modified Laplacian operator on mode shape data. J. Sound Vib. 204, 505–517 (1997)CrossRefGoogle Scholar
  35. 35.
    Ostachowicz, W.M., Krawczuk, M.: Analysis of the effect of cracks on the natural frequencies of a cantilever Beam. J. Sound Vib. 150, 191–201 (1991)CrossRefGoogle Scholar
  36. 36.
    Binici, B.: Vibration of beam with multiple open cracks subjected to axial force. J. Sound Vib. 287, 277–295 (2005)CrossRefGoogle Scholar
  37. 37.
    Sergio, H.S., Daniel, J.I.: Continuous model for the transverse vibration of cracked Timoshenko beams. J. Trans. ASME 124, 310–320 (2002)Google Scholar
  38. 38.
    Shafiei, M., Khaji, N.: Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load. J. Acta Mech. 221, 79–97 (2011)zbMATHCrossRefGoogle Scholar
  39. 39.
    Neves, A.C., Simões, F.M.F., Pinto da Costa, A.: Vibrations of cracked beams: discrete mass and stiffness models. J. Comput. Struct. 168, 68–77 (2016)CrossRefGoogle Scholar
  40. 40.
    Ivakhnenko, A.G.: Heuristic self-organization in problems of engineering cybernetics. J. Automat. 6, 207–219 (1970)CrossRefGoogle Scholar
  41. 41.
    Ivakhnenko, A.G.: Polynomial theory of complex systems. J. Trans. Syst. 1, 364–378 (1971)MathSciNetGoogle Scholar
  42. 42.
    Kutuk, M.A., Atmaca, N., Guzelbey, I.H.: Explicit formulation of SIF using neural networks for opening mode of fracture. J. Eng. Struct. 29, 2080–2086 (2007)CrossRefGoogle Scholar
  43. 43.
    Nariman-zadeh, N., Darvizeh, A., Darvizeh, M., Gharababaei, H.: Modelling of explosive cutting process of plates using GMDH-type neural network and singular value decomposition. J. Mater. Process. Technol. 128, 80–87 (2002)zbMATHCrossRefGoogle Scholar
  44. 44.
    Besarati, S.M., Myers, P.D., Covey, D.C., Jamali, A.: Modelling friction factor in pipeline flow using a GMDH-type neural network. J. Cogent Eng. 2, 1–14 (2015)CrossRefGoogle Scholar
  45. 45.
    Montgomery, D.: Design and Analysis of Experiments. Wiley, New York (2013)Google Scholar
  46. 46.
    Box, G., Hunter, J., Hunter, W.: Statistics for Experiments. Wiley, New York (2005)zbMATHGoogle Scholar
  47. 47.
    Ranjit, K.R.: A Primer on the Taguchi Method. Society of Manufacturing Engineers, Michigan (2010)zbMATHGoogle Scholar
  48. 48.
    Eriksson, M., Andersson, P., Burman, A.: Using design of experiments techniques for an efficient finite element study of the influence of changed. International ANSYS Conference, vol. 2, pp. 63–72 (1998)Google Scholar
  49. 49.
    Alijani, A., Mastan Abadi, M., Darvizeh, A., Kh. Abadi, M.: Theoretical approaches for bending analysis of founded Euler–Bernoulli cracked beams. J. Arch. Appl. Mech. 88: 875–895 (2018)CrossRefGoogle Scholar
  50. 50.
    Mottaghian, F., Darvizeh, A., Alijani, A.: A novel finite element model for large deformation analysis of cracked beams using classical and continuum-based approaches. J. Arch. Appl. Mech. (2018).  https://doi.org/10.1007/s00419-018-1460-0 CrossRefGoogle Scholar
  51. 51.
    Zarrinzadeh, H., Kabir, M.Z., Deylami, A.: Extended finite element fracture analysis of a cracked isotropic shell repaired by composite patch. J. Fatigue. Fract. Eng. Mater. Struct. 39, 1352–1365 (2016)CrossRefGoogle Scholar

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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