Steel and Composite Structures

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A computational shear displacement model for vibrational analysis of functionally graded beams with porosities

  • Atmane, Hassen Ait (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Bernard, Fabrice (Laboratoire de Genie Civil et Genie Mecanique INSA de Rennes) ;
  • Mahmoud, S.R. (Department of Mathematics, Faculty of Science, King Abdulaziz University)
  • Received : 2014.11.18
  • Accepted : 2015.01.18
  • Published : 2015.08.25
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Abstract

This work presents a free vibration analysis of functionally graded metal-ceramic (FG) beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. For this purpose, a simple displacement field based on higher order shear deformation theory is implemented. The proposed theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The most interesting feature of this theory is that it accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors. In addition, it has strong similarities with Euler-Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. By employing the Hamilton's principle, governing equations of motion for coupled axial-shear-flexural response are determined. The validity of the present theory is investigated by comparing some of the present results with those of the first-order and the other higher-order theories reported in the literature. Illustrative examples are given also to show the effects of varying gradients, porosity volume fraction, aspect ratios, and thickness to length ratios on the free vibration of the FG beams.

Keywords

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  41. A novel and simple HSDT for thermal buckling response of functionally graded sandwich plates vol.62, pp.4, 2017, https://doi.org/10.12989/sem.2017.62.4.401
  42. A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams vol.62, pp.6, 2017, https://doi.org/10.12989/sem.2017.62.6.695
  43. Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory vol.19, pp.6, 2015, https://doi.org/10.12989/sss.2017.19.6.601
  44. A new shear deformation plate theory with stretching effect for buckling analysis of functionally graded sandwich plates vol.24, pp.5, 2017, https://doi.org/10.12989/scs.2017.24.5.569
  45. Free vibrations of laminated composite plates using a novel four variable refined plate theory vol.24, pp.5, 2017, https://doi.org/10.12989/scs.2017.24.5.603
  46. An original single variable shear deformation theory for buckling analysis of thick isotropic plates vol.63, pp.4, 2017, https://doi.org/10.12989/sem.2017.63.4.439
  47. A simple analytical approach for thermal buckling of thick functionally graded sandwich plates vol.63, pp.5, 2015, https://doi.org/10.12989/sem.2017.63.5.585
  48. An efficient shear deformation theory for wave propagation in functionally graded material beams with porosities vol.13, pp.3, 2015, https://doi.org/10.12989/eas.2017.13.3.255
  49. A four variable refined nth-order shear deformation theory for mechanical and thermal buckling analysis of functionally graded plates vol.13, pp.3, 2015, https://doi.org/10.12989/gae.2017.13.3.385
  50. Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory vol.20, pp.3, 2015, https://doi.org/10.12989/sss.2017.20.3.369
  51. A new and simple HSDT for thermal stability analysis of FG sandwich plates vol.25, pp.2, 2015, https://doi.org/10.12989/scs.2017.25.2.157
  52. A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams vol.64, pp.2, 2015, https://doi.org/10.12989/sem.2017.64.2.145
  53. Free vibration of functionally graded plates resting on elastic foundations based on quasi-3D hybrid-type higher order shear deformation theory vol.20, pp.4, 2017, https://doi.org/10.12989/sss.2017.20.4.509
  54. An analytical solution for bending and vibration responses of functionally graded beams with porosities vol.25, pp.4, 2015, https://doi.org/10.12989/was.2017.25.4.329
  55. An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates vol.25, pp.3, 2015, https://doi.org/10.12989/scs.2017.25.3.257
  56. A novel and simple higher order shear deformation theory for stability and vibration of functionally graded sandwich plate vol.25, pp.4, 2017, https://doi.org/10.12989/scs.2017.25.4.389
  57. A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates vol.64, pp.4, 2015, https://doi.org/10.12989/sem.2017.64.4.391
  58. Nonlocal-strain gradient forced vibration analysis of metal foam nanoplates with uniform and graded porosities vol.5, pp.4, 2017, https://doi.org/10.12989/anr.2017.5.4.393
  59. A new quasi-3D HSDT for buckling and vibration of FG plate vol.64, pp.6, 2015, https://doi.org/10.12989/sem.2017.64.6.737
  60. An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.693
  61. A new simple three-unknown shear deformation theory for bending analysis of FG plates resting on elastic foundations vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.717
  62. An original HSDT for free vibration analysis of functionally graded plates vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.735
  63. Vibration analysis of thick orthotropic plates using quasi 3D sinusoidal shear deformation theory vol.16, pp.2, 2015, https://doi.org/10.12989/gae.2018.16.2.141
  64. Development of super convergent Euler finite elements for the analysis of sandwich beams with soft core vol.65, pp.6, 2015, https://doi.org/10.12989/sem.2018.65.6.657
  65. A novel four variable refined plate theory for wave propagation in functionally graded material plates vol.27, pp.1, 2018, https://doi.org/10.12989/scs.2018.27.1.109
  66. A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory vol.21, pp.4, 2015, https://doi.org/10.12989/sss.2018.21.4.397
  67. Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates vol.14, pp.6, 2015, https://doi.org/10.12989/gae.2018.14.6.519
  68. Bending of FGM rectangular plates resting on non-uniform elastic foundations in thermal environment using an accurate theory vol.27, pp.3, 2018, https://doi.org/10.12989/scs.2018.27.3.311
  69. Analytical investigation of bending response of FGM plate using a new quasi 3D shear deformation theory: Effect of the micromechanical models vol.66, pp.3, 2018, https://doi.org/10.12989/sem.2018.66.3.317
  70. Free vibration of FGM plates with porosity by a shear deformation theory with four variables vol.66, pp.3, 2015, https://doi.org/10.12989/sem.2018.66.3.353
  71. A novel four-unknown quasi-3D shear deformation theory for functionally graded plates vol.27, pp.5, 2015, https://doi.org/10.12989/scs.2018.27.5.599
  72. A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates vol.28, pp.1, 2015, https://doi.org/10.12989/scs.2018.28.1.099
  73. Geometrically nonlinear analysis of functionally graded porous beams vol.27, pp.1, 2015, https://doi.org/10.12989/was.2018.27.1.059
  74. Single variable shear deformation model for bending analysis of thick beams vol.67, pp.3, 2015, https://doi.org/10.12989/sem.2018.67.3.291
  75. Effect of homogenization models on stress analysis of functionally graded plates vol.67, pp.5, 2018, https://doi.org/10.12989/sem.2018.67.5.527
  76. A novel quasi-3D hyperbolic shear deformation theory for vibration analysis of simply supported functionally graded plates vol.22, pp.3, 2015, https://doi.org/10.12989/sss.2018.22.3.303
  77. Analysis of wave propagation and free vibration of functionally graded porous material beam with a novel four variable refined theory vol.15, pp.4, 2018, https://doi.org/10.12989/eas.2018.15.4.369
  78. A refined quasi-3D hybrid-type higher order shear deformation theory for bending and Free vibration analysis of advanced composites beams vol.27, pp.4, 2015, https://doi.org/10.12989/was.2018.27.4.269
  79. An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models vol.69, pp.2, 2015, https://doi.org/10.12989/sem.2019.69.2.231
  80. Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory vol.28, pp.1, 2015, https://doi.org/10.12989/was.2019.28.1.019
  81. Dynamic and wave propagation investigation of FGM plates with porosities using a four variable plate theory vol.28, pp.1, 2015, https://doi.org/10.12989/was.2019.28.1.049
  82. Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections vol.17, pp.2, 2019, https://doi.org/10.12989/gae.2019.17.2.175
  83. Strain gradient based dynamic response analysis of heterogeneous cylindrical microshells with porosities under a moving load vol.6, pp.3, 2015, https://doi.org/10.1088/2053-1591/aaf5a2
  84. Vibration response and wave propagation in FG plates resting on elastic foundations using HSDT vol.69, pp.5, 2015, https://doi.org/10.12989/sem.2019.69.5.511
  85. Free vibration of imperfect sigmoid and power law functionally graded beams vol.30, pp.6, 2019, https://doi.org/10.12989/scs.2019.30.6.603
  86. Nonlocal strain gradient model for thermal stability of FG nanoplates integrated with piezoelectric layers vol.23, pp.3, 2015, https://doi.org/10.12989/sss.2019.23.3.215
  87. Assessing the Effects of Porosity on the Bending, Buckling, and Vibrations of Functionally Graded Beams Resting on an Elastic Foundation by Using a New Refined Quasi-3D Theory vol.55, pp.2, 2015, https://doi.org/10.1007/s11029-019-09805-0
  88. Dynamic response of metal foam FG porous cylindrical micro-shells due to moving loads with strain gradient size-dependency vol.134, pp.5, 2015, https://doi.org/10.1140/epjp/i2019-12540-3
  89. Nonlocal strain gradient thermal vibration analysis of double-coupled metal foam plate system with uniform and non-uniform porosities vol.8, pp.3, 2015, https://doi.org/10.12989/csm.2019.8.3.247
  90. Finite element formulation and vibration of nonlocal refined metal foam beams with symmetric and non-symmetric porosities vol.6, pp.2, 2015, https://doi.org/10.12989/smm.2019.6.2.147
  91. Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions vol.33, pp.1, 2019, https://doi.org/10.12989/scs.2019.33.1.133
  92. Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT vol.24, pp.4, 2015, https://doi.org/10.12989/cac.2019.24.4.347
  93. Effect of nonlinear elastic foundations on dynamic behavior of FG plates using four-unknown plate theory vol.17, pp.5, 2015, https://doi.org/10.12989/eas.2019.17.5.447
  94. Effect of variable elastic foundations on static behavior of functionally graded plates using sinusoidal shear deformation vol.12, pp.24, 2019, https://doi.org/10.1007/s12517-019-4871-5
  95. Wave dispersion properties in imperfect sigmoid plates using various HSDTs vol.33, pp.5, 2015, https://doi.org/10.12989/scs.2019.33.5.699
  96. Nonlocal strain gradient effects on forced vibrations of porous FG cylindrical nanoshells vol.8, pp.2, 2015, https://doi.org/10.12989/anr.2020.8.2.149
  97. A numerical method for dynamic characteristics of nonlocal porous metal-ceramic plates under periodic dynamic loads vol.7, pp.1, 2020, https://doi.org/10.12989/smm.2020.7.1.027
  98. Thermal buckling of nonlocal clamped exponentially graded plate according to a secant function based refined theory vol.35, pp.1, 2020, https://doi.org/10.12989/scs.2020.35.1.147
  99. Transversely isotropic thin circular plate with multi-dual-phase lag heat transfer vol.35, pp.3, 2015, https://doi.org/10.12989/scs.2020.35.3.343
  100. Nonlocal nonlinear dynamic behavior of composite piezo-magnetic beams using a refined higher-order beam theory vol.35, pp.4, 2015, https://doi.org/10.12989/scs.2020.35.4.545
  101. Post-buckling of higher-order stiffened metal foam curved shells with porosity distributions and geometrical imperfection vol.35, pp.4, 2020, https://doi.org/10.12989/scs.2020.35.4.567
  102. Assessment of transient vibrations of graphene oxide reinforced plates under pulse loads using finite strip method vol.25, pp.6, 2015, https://doi.org/10.12989/cac.2020.25.6.575
  103. Dynamic response of size-dependent porous functionally graded beams under thermal and moving load using a numerical approach vol.7, pp.2, 2015, https://doi.org/10.12989/smm.2020.7.2.069
  104. Numerical investigation on scale-dependent vibrations of porous foam plates under dynamic loads vol.7, pp.2, 2020, https://doi.org/10.12989/smm.2020.7.2.085
  105. Influence of the distribution shape of porosity on the bending of FGM beam using a new higher order shear deformation model vol.26, pp.2, 2020, https://doi.org/10.12989/sss.2020.26.2.253
  106. Porosity effects on post-buckling behavior of geometrically imperfect metal foam doubly-curved shells with stiffeners vol.75, pp.6, 2015, https://doi.org/10.12989/sem.2020.75.6.701
  107. Nonlocal nonlinear stability of higher-order porous beams via Chebyshev-Ritz method vol.76, pp.3, 2015, https://doi.org/10.12989/sem.2020.76.3.413
  108. A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory vol.136, pp.2, 2015, https://doi.org/10.1140/epjp/s13360-021-01238-w
  109. Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory vol.10, pp.3, 2015, https://doi.org/10.12989/anr.2021.10.3.281
  110. Investigation on the dynamic response of porous FGM beams resting on variable foundation using a new higher order shear deformation theory vol.39, pp.1, 2015, https://doi.org/10.12989/scs.2021.39.1.095
  111. Analyzing dynamic response of nonlocal strain gradient porous beams under moving load and thermal environment vol.26, pp.1, 2015, https://doi.org/10.12989/gae.2021.26.1.089
  112. Investigating dynamic response of nonlocal functionally graded porous piezoelectric plates in thermal environment vol.40, pp.2, 2021, https://doi.org/10.12989/scs.2021.40.2.243
  113. Numerical forced vibration analysis of compositionally gradient porous cylindrical microshells under moving load and thermal environment vol.40, pp.6, 2021, https://doi.org/10.12989/scs.2021.40.6.893