Structural Engineering and Mechanics

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Transfer matrix formulations and single variable shear deformation theory for crack detection in beam-like structures

  • Bozyigit, Baran (Department of Civil Engineering, Dokuz Eylul University) ;
  • Yesilce, Yusuf (Department of Civil Engineering, Dokuz Eylul University) ;
  • Wahab, Magd Abdel (CIRTech Institute, Ho Chi Minh City University of Technology (HUTECH))
  • Received : 2019.03.08
  • Accepted : 2019.09.17
  • Published : 2020.01.25
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Abstract

This study aims to estimate crack location and crack length in damaged beam structures using transfer matrix formulations, which are based on analytical solutions of governing equations of motion. A single variable shear deformation theory (SVSDT) that considers parabolic shear stress distribution along beam cross-section is used, as well as, Timoshenko beam theory (TBT). The cracks are modelled using massless rotational springs that divide beams into segments. In the forward problem, natural frequencies of intact and cracked beam models are calculated for different crack length and location combinations. In the inverse approach, which is the main concern of this paper, the natural frequency values obtained from experimental studies, finite element simulations and analytical solutions are used for crack identification via plots of rotational spring flexibilities against crack location. The estimated crack length and crack location values are tabulated with actual data. Three different beam models that have free-free, fixed-free and simple-simple boundary conditions are considered in the numerical analyses.

Keywords

References

  1. Abdelhakim, K., Mohammed Sid Ahmed, H., Bousahla, A., Tounsi, A. and Hassan, S. (2018), "Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory", Struct. Eng. Mech., 65, 621-631. https://doi.org/10.12989/sem.2018.65.5.621.
  2. Ahouel, M., Houari, M.S.A., Bedia, E.A.A. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., 20(5), 963-981. https://doi.org/10.12989/SCS.2016.20.5.963.
  3. Ait Yahia, S., Hassen, A.A., Mohammed Sid Ahmed, H. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143.
  4. Al Rjoub, Y.S. and Hamad, A.G. (2017), "Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method", KSCE J. Civil Eng., 21(3), 792-806. https://doi.org/10.1007/s12205-016-0149-6.
  5. Attar, M. (2012), "A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions", J. Mech. Sci., 57(1), 19-33. https://doi.org/10.1016/j.ijmecsci.2012.01.010.
  6. Barad, K.H., Sharma, D.S. and Vyas, V. (2013), "Crack Detection in Cantilever Beam by Frequency based Method", Procedia Eng., 51, 770-775. https://doi.org/10.1016/j.proeng.2013.01.110.
  7. Belabed, Z., Bousahla, A., Mohammed Sid Ahmed, H., Tounsi, A. and Hassan, S. (2018), "A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate", Earthq. Struct., 14(2), 103-115. https://doi.org/10.12989/eas.2018.14.2.103.
  8. Belkorissat, I., Mohammed Sid Ahmed, H., Tounsi, A., Adda Bedia, E.A. and Hassan, S. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063.
  9. Bickford, W.B. (1982), "A consistent higher order beam theory", Development Theoretical Appl. Mech., 11, 137-150.
  10. Bouafia, K., Kaci, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., 19(2), 115-https://doi.org/126.10.12989/SSS.2017.19.2.115.
  11. Bourada, F., Bousahla, A., Bourada, M., Azzaz, A., amina, Z. and Tounsi, A. (2019), "Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory", Wind Struct., 28(1), 19-30 https://doi.org/10.12989/was.2019.28.1.019
  12. Bozyigit, B. and Yesilce, Y. (2018), "Investigation of natural frequencies of multi-bay and multi-storey frames using a single variable shear deformation theory", Struct. Eng. Mech., 65(1), 9-17 https://doi.org/10.12989/SEM.2018.65.1.009
  13. Bozyigit, B. and Yesilce, Y. (2018), "Natural frequencies and harmonic responses of multi-story frames using single variable shear deformation theory", Mech. Res. Commun., 92, 28-36. https://doi.org/10.1016/j.mechrescom.2018.06.007.
  14. Dansheng, W., Hongping, Z., Chuanyao, C. and Yong, X. (2007), "An impedance analysis for crack detection in the Timoshenko beam based on the anti-resonance technique", Acta Mechanica Solida Sinica, 20(3), 228-235. https://doi.org/10.1007/s10338-007-0727-8.
  15. El-Sayed, T.A. and Farghaly, S.H. (2017), "A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods", Shock Vib., 2017, 23. https://doi.org/10.1155/2017/8186976.
  16. Faiza Klouche, L.D., Mohamed Sekkal, Abdelouahed Tounsi and S.R. Mahmoud (2017), "An original single variable shear deformation theory for buckling analysis of thick isotropic plates", Struct. Eng. Mech., 63(4), 439-446 https://doi.org/10.12989/sem.2017.63.4.439
  17. Fekrazadeh, S. and Khaji, N. (2017), "An analytical method for crack detection of Timoshenko beams with multiple open cracks using a test mass", Europe J. Environ. Civil Eng., 21(1), 24-41. https://doi.org/10.1080/19648189.2015.1090929.
  18. Ghasemi, H., Park, H.S. and Rabczuk, T. (2017), "A level-set based IGA formulation for topology optimization of flexoelectric materials", Comput. Method Appl. M., 313, 239-258. https://doi.org/10.1016/j.cma.2016.09.029.
  19. Ghasemi, H., Park, H.S. and Rabczuk, T. (2018), "A multi-material level set-based topology optimization of flexoelectric composites", Comput. Method Appl. M., 332, 47-62. https://doi.org/10.1016/j.cma.2017.12.005.
  20. Gillich, G.-R., Furdui, H., Abdel Wahab, M. and Korka, Z.-I. (2019), "A robust damage detection method based on multi-modal analysis in variable temperature conditions", Mech. Syst. Signal Process., 115, 361-379. https://doi.org/10.1016/j.ymssp.2018.05.037.
  21. Hachemi, H., Abdelhakim, K., Mohammed Sid Ahmed, H., Bourada, M., Tounsi, A. and Hassan, S. (2017), "A new simple three-unknown shear deformation theory for bending analysis of FG plates resting on elastic foundations", Steel Compos. Struct., 25(6), 717-726.10.12989/scs.2017.25.6.717.
  22. Han, S.M., Benaroya, H. and Wei, T. (1999), "Dynamics of transversely vibrating beams using four engineering theories", J. Sound Vib., 225(5), 935-988. https://doi.org/10.1006/jsvi.1999.2257.
  23. Heyliger, P.R. and Reddy, J.N. (1988), "A higher order beam finite element for bending and vibration problems", J. Sound Vib., 126(2), 309-326. https://doi.org/10.1016/0022-460X(88)90244-1.
  24. Jena, P.K. and Parhi, D.R. (2015), "A Modified Particle Swarm Optimization Technique for Crack Detection in Cantilever Beams", Arabian J. Sci. Eng., 40(11), 3263-3272. https://doi.org/10.1007/s13369-015-1661-6.
  25. Khaji, N., Shafiei, M. and Jalalpour, M. (2009), "Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions", J. Mech. Sci., 51(9), 667-681. https://doi.org/10.1016/j.ijmecsci.2009.07.004.
  26. Khatir, S. and Abdel Wahab, M. (2019), "Fast simulations for solving fracture mechanics inverse problems using POD-RBF XIGA and Jaya algorithm", Eng. Fracture Mech., 205, 285-300. https://doi.org/10.1016/j.engfracmech.2018.09.032.
  27. Khatir, S., Abdel Wahab, M., Boutchicha, D. and Khatir, T. (2019), "Structural health monitoring using modal strain energy damage indicator coupled with teaching-learning-based optimization algorithm and isogoemetric analysis", J. Sound Vib., 448, 230-246. https://doi.org/10.1016/j.jsv.2019.02.017.
  28. Khatir, S., Dekemele, K., Loccufier, M., Khatir, T. and Abdel Wahab, M. (2018), "Crack identification method in beam-like structures using changes in experimentally measured frequencies and Particle Swarm Optimization", Comptes Rendus Mecanique, 346(2), 110-120. https://doi.org/10.1016/j.crme.2017.11.008.
  29. Khatir, S.B.I., Serra, R., Wahab, M.A. and Tawfiq, K. (2016), "Numerical study for single and multiple damage detection and localizaton in beam-like structures using BAT algorithm", J. Vibroeng., 18(1), 202-213.
  30. Khiem, N.T. and Huyen, N.N. (2017), "A method for crack identification in functionally graded Timoshenko beam", Nondestructive Test. Evaluation, 32(3), 319-341. https://doi.org/10.1080/10589759.2016.1226304.
  31. Khnaijar, A. and Benamar, R. (2017), "A new model for beam crack detection and localization using a discrete model", Eng. Struct., 150, 221-230. https://doi.org/10.1016/j.engstruct.2017.07.034.
  32. Kindowa-Petrova, D. (2014), "Vibration-based methods for detecting a crack in a simply supported beam", J. Theoretical Appl Mech., 44(4), 69-82. https://doi.org/10.2478/jtam-2014-0023
  33. Le, T. C., Phung Van, P., Thai, C. H., Nguyen-Xuan, H., and Abdel Wahab, M. (2018), "Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory." Compos. Struct., 184, 633-649. https://doi.org/10.1016/j.compstruct.2017.10.025
  34. Lee, J.W. and Lee, J.Y. (2016), "Free vibration analysis using the transfer-matrix method on a tapered beam", Comput. Struct., 164, 75-82. https://doi.org/10.1016/j.compstruc.2015.11.007.
  35. Lee, J.W. and Lee, J.Y. (2017), "Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression", Int. J. Mech. Sci., 122, 1-17. https://doi.org/10.1016/j.ijmecsci.2017.01.011.
  36. Lee, J.W. and Lee, J.Y. (2018), "An exact transfer matrix expression for bending vibration analysis of a rotating tapered beam", Appl. Math. Modell., 53, 167-188. https://doi.org/10.1016/j.apm.2017.08.022.
  37. Lee, J.W. and Lee, J.Y. (2018), "A transfer matrix method for in-plane bending vibrations of tapered beams with axial force and multiple edge cracks", Struct. Eng. Mech., 66(1), 125-138. https://doi.org/10.12989/SEM.2018.66.1.125
  38. Lele, S.P. and Maiti, S.K. (2002), "Modellin of transverse vibration of short beams for crack detection and measurement of crack extension", J. Sound Vib., 257(3), 559-583. https://doi.org/10.1006/jsvi.2002.5059.
  39. Levinson, M. (1981), "A new rectangular beam theory", J. Sound Vib., 74(1), 81-87. https://doi.org/10.1016/0022-460X(81)90493-4.
  40. Lin, H.P. and Chang, S.C. (2005), "Free vibration analysis of multi-span beams with intermediate flexible constraints", J. Sound Vib., 281(1), 155-169. https://doi.org/10.1016/j.jsv.2004.01.010.
  41. Liu, J., Shao, Y. and Zhu, W. (2017), "Free vibration analysis of a cantilever beam with a slant edge crack", Proceedings of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sci., 231(5), 823-843. https://doi.org/10.1177/0954406216631006.
  42. Moezi, S.A., Zakeri, E. and Zare, A. (2018), "Structural single and multiple crack detection in cantilever beams using a hybrid Cuckoo-Nelder-Mead optimization method", Mech. Syst. Signal Process., 99, 805-831. https://doi.org/10.1016/j.ymssp.2017.07.013.
  43. Mohammed, S.A., H., Tounsi, A., Bessaim, A. and Hassan, S. (2016), "A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., 22(2), 257-276. https://doi.org/10.12989/scs.2016.22.2.257.
  44. Mohammed Sid Ahmed, H., Youcef, M., Heireche, H., Bousahla, A., Tounsi, A. and Hassan, S. (2018), "A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory", Smart Struct. Syst., 21(4), 397-405. https://doi.org/10.12989/sss.2018.21.4.397.
  45. Mouffoki, A., Houari, M.S.A. and Tounsi, A. (2017), "Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory", Smart Struct. Syst., 20(3), 369-383. https://doi.org/10.12989/SSS.2017.20.3.369.
  46. Mungla, M.J., Sharma, D.S. and Trivedi, R.R. (2016), "Identification of a Crack in Clamped-Clamped Beam using Frequency-based Method and Genetic Algorithm", Procedia Eng., 144, 1426-1434. https://doi.org/10.1016/j.proeng.2016.05.174.
  47. Nahvi, H. and Jabbari, M. (2005), "Crack detection in beams using experimental modal data and finite element model", J. Mech. Sci., 47(10), 1477-1497. https://doi.org/10.1016/j.ijmecsci.2005.06.008.
  48. Nandwana, B.P. and Maiti, S.K. (1997), "Detection of the location and size of a crack in stepped cantilever beams based on measurements of natural frequencies", J. Sound Vib., 203(3), 435-446. https://doi.org/10.1006/jsvi.1996.0856.
  49. Nanthakumar, S.S., Lahmer, T. and Rabczuk, T. (2013), "Detection of flaws in piezoelectric structures using extended FEM", Int. J. Numer. Meth. Eng., 96(6), 373-389. https://doi.org/10.1002/nme.4565.
  50. Nanthakumar, S.S., Lahmer, T., Zhuang, X., Zi, G. and Rabczuk, T. (2016), "Detection of material interfaces using a regularized level set method in piezoelectric structures", Inverse Probl. Sci. En., 24(1), 153-176. https://doi.org/10.1080/17415977.2015.1017485.
  51. Narkis, Y. (1994), "Identification of Crack Location in Vibrating Simply Supported Beams", J. Sound Vib., 172(4), 549-558. https://doi.org/10.1006/jsvi.1994.1195.
  52. Ostachowicz, W.M. and Krawczuk, M. (1991), "Analysis of the effect of cracks on the natural frequencies of a cantilever beam", J. Sound Vib., 150(2), 191-201. https://doi.org/10.1016/0022-460X(91)90615-Q.
  53. Phung Van, P., Thai, C.H., Nguyen-Xuan, H., and Abdel Wahab, M. (2019), "Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis", Compos. Part B Eng., 164, 215-225. https://doi.org/10.1016/j.compositesb.2018.11.036.
  54. Phung Van, P., Le, T. C., Nguyen-Xuan, H. and Abdel Wahab, M. (2018), "Nonlinear transient isogeometric analysis of FG-CNTRC nanoplates in thermal environments", Compos. Struct., 201, 882-892. https://doi.org/10.1016/j.compstruct.2018.06.087
  55. Rajasekaran, S. and Khaniki, H.B. (2018), "Free vibration analysis of bi-directional functionally graded single/multi-cracked beams", J. Mech. Sci., 144, 341-356. https://doi.org/10.1016/j.ijmecsci.2018.06.004.
  56. Reddy, J. (1984), "A simple higher-order theory for laminated composite plates", J. Appl. Mech., 51, 745-752. https://doi.org/10.1115/1.3167719
  57. Rizos, P.F., Aspragathos, N. and Dimarogonas, A.D. (1990), "Identification of crack location and magnitude in a cantilever beam from the vibration modes", J. Sound Vib., 138(3), 381-388. https://doi.org/10.1016/0022-460X(90)90593-O.
  58. Rosales, M.B., Filipich, C.P. and Buezas, F.S. (2009), "Crack detection in beam-like structures", Eng. Struct., 31(10), 2257-2264. https://doi.org/10.1016/j.engstruct.2009.04.007.
  59. Salima Abdelbari, L.H.H.A., Abdelhakim Kaci and Abdelouahed Tounsi (2018), "Single variable shear deformation model for bending analysis of thick beams", Struct. Eng. Mech., 67(3), 291-300. https://doi.org/10.12989/SEM.2018.67.3.291
  60. Shimpi, R.P. (2002), "Refined plate theory and its variants", AIAA J., 40(1), 137-146. https://doi.org/10.2514/2.1622
  61. Shimpi, R.P., Patel, H.G. and Arya, H. (2006), "New First-Order Shear Deformation Plate Theories", J. Appl. Mech., 74(3), 523-533. https://doi.org/10.1115/1.2423036.
  62. Shimpi, R.P., Shetty, R.A. and Guha, A. (2017), "A simple single variable shear deformation theory for a rectangular beam", Proceedings of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sci., 231(24), 4576-4591. https://doi.org/10.1177/0954406216670682.
  63. Vu-Bac, N., Duong, T.X., Lahmer, T., Zhuang, X., Sauer, R.A., Park, H.S. and Rabczuk, T. (2018), "A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures", Comput. Method Appl. M., 331, 427-455. https://doi.org/10.1016/j.cma.2017.09.034.
  64. Wu, J.-S. and Chang, B.-H. (2013), "Free vibration of axial-loaded multi-step Timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method", Europ. J. Mech. A/Solids, 38, 20-37. https://doi.org/10.1016/j.euromechsol.2012.08.003.
  65. Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams including surface stress effects", Smart Struct. Syst., 21(1), 65-74. https://doi.org/10.12989/SSS.2018.21.1.065.
  66. Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/SEM.2015.54.4.693.
  67. Zhou, Y.-L. and Abdel Wahab, M. (2017), "Cosine based and extended transmissibility damage indicators for structural damage detection", Eng. Struct., 141, 175-183. http://dx.doi.org/10.1016/j.engstruct.2017.03.030.
  68. Zhou, Y.-L., Maia, N. and Abdel Wahab, M. (2016), "Damage detection using transmissibility compressed by principal component analysis enhanced with distance measure", J. Vib. Control, 24(10), 2001-2019. https://doi.org/ 10.1177/1077546316674544.
  69. Zhou, Y.-L., Maia, N.M.M., Sampaio, R. and Wahab, M.A. (2016), "Structural damage detection using transmissibility together with hierarchical clustering analysis and similarity measure", Struct. Health Monitor., 16(6), 711-731. https://doi.org/10.1177/1475921716680849
  70. Zidi, M., Mohammed Sid Ahmed, H., Tounsi, A., Bessaim, A. and Hassan, S. (2017), "A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams", Struct. Eng. Mech., 64(2), 145-153. https://doi.org/10.12989/sem.2017.64.2.145.

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