Structural Engineering and Mechanics

  • 제75권6호
  • /
  • Pages.701-711
  • /
  • 2020
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  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Porosity effects on post-buckling behavior of geometrically imperfect metal foam doubly-curved shells with stiffeners

  • 투고 : 2020.02.09
  • 심사 : 2020.04.14
  • 발행 : 2020.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

This papers studies nonlinear stability and post-buckling behaviors of geometrically imperfect metal foam doubly-curved shells with eccentrically stiffeners resting on elastic foundation. Metal foam is considered as porous material with uniform and non-uniform models. The doubly-curved porous shell is subjected to in-plane compressive loads as well as a transverse pressure leading to post-critical stability in nonlinear regime. The nonlinear governing equations are analytically solved with the help of Airy stress function to obtain the post-buckling load-deflection curves of the geometrically imperfect metal foam doubly-curved shell. Obtained results indicate the significance of porosity distribution, geometrical imperfection, foundation factors, stiffeners and geometrical parameters on post-buckling characteristics of porous doubly-curved shells.

키워드

과제정보

The first and second authors would like to thank FPQ (Fidar project Qaem) for providing the fruitful and useful help.

참고문헌

  1. Abualnour, M., Chikh, A., Hebali, H., Kaci, A., Tounsi, A., Bousahla, A. A. and Tounsi, A. (2019), "Thermomechanical analysis of antisymmetric laminated reinforced composite plates using a new four variable trigonometric refined plate theory," Comput. Concrete, 24(6), 489-498. https://doi.org/10.12989/cac.2019.24.6.489.
  2. Abdelaziz, H. H., Meziane, M. A. A., Bousahla, A. A., Tounsi, A., Mahmoud, S. R. and Alwabli, A. S. (2017), "An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions," Steel Compos. Struct., 25(6), 693-704. https://doi.org/10.12989/scs.2017.25.6.693.
  3. Achouri, F., Benyoucef, S., Bourada, F., Bouiadjra, R. B. and Tounsi, A. (2019), "Robust quasi 3D computational model for mechanical response of FG thick sandwich plate", Struct. Eng. Mech., 70(5), 571-589. https://doi.org/10.12989/sem.2019.70.5.571.
  4. Adda Bedia, W.A., Houari, M. S. A., Bessaim, A., Bousahla, A. A., Tounsi, A., Saeed, T. and Alhodaly, M. S. (2019), "A New Hyperbolic Two-Unknown Beam Model for Bending and Buckling Analysis of a Nonlocal Strain Gradient Nanobeams", J. Nano Res., 57, 175-191. https://doi.org/10.4028/www.scientific.net/JNanoR.57.175.
  5. Addou, F. Y., Meradjah, M., Bousahla, A. A., Benachour, A., Bourada, F., Tounsi, A. and Mahmoud, S. R. (2019), "Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT", Comput. Concrete, 24(4), 347-367. https://doi.org/10.12989/cac.2019.24.4.347.
  6. Ahmed, R. A., Fenjan, R. M. and Faleh, N. M. (2019), "Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections", Geomech. Eng., 17(2), 175-180. https://doi.org/10.12989/gae.2019.17.2.175.
  7. Aissani, K., Bouiadjra, M.B., Ahouel, M. and Tounsi, A. (2015), "A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium", Struct. Eng. Mech., 55(4), 743-763. https://doi.org/10.12989/sem.2015.55.4.743.
  8. Alimirzaei, S., Mohammadimehr, M. and Tounsi, A. (2019), "Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions", Struct. Eng. Mech., 71(5), 485-502. http://dx.doi.org/10.12989/sem.2019.71.5.485.
  9. Al-Maliki, A. F., Faleh, N. M. and Alasadi, A. A. (2019), "Finite element formulation and vibration of nonlocal refined metal foam beams with symmetric and non-symmetric porosities", Struct. Monitor. Maintenance, 6(2), 147-159. https://doi.org/10.12989/smm.2019.6.2.147.
  10. Asghar, S., Naeem, M. N., Hussain, M., Taj, M. and Tounsi, A. (2020), "Prediction and assessment of nonlocal natural frequencies of DWCNTs: Vibration analysis", Comput. Concrete, 25(2), 133-144. https://doi.org/10.12989/cac.2020.25.2.133.
  11. Attia, A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2018), "A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations", Struct. Eng. Mech., 65(4), 453-464. https://doi.org/10.12989/sem.2018.65.4.453.
  12. Atmane, H.A., Tounsi, A., Bernard, F .and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369.
  13. Azimi, M., Mirjavadi, S.S., Shafiei, N. and Hamouda, A.M.S. (2017), "Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam", Appl. Phys. A, 123(1), 104. http://dx.doi.org/10.1007/s00339-017-0772-1.
  14. Azimi, M., Mirjavadi, S. S., Shafiei, N., Hamouda, A. M. S. and Davari, E. (2018), "Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution", Mech. Adv. Mater. Struct., 25(6), 467-480. https://doi.org/10.1080/15376494.2017.1285455.
  15. Balubaid, M., Tounsi, A., Dakhel, B. and Mahmoud, S. R. (2019), "Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory", Comput. Concrete, 24(6), 579-586. https://doi.org/10.12989/cac.2019.24.6.579.
  16. Barati, M. R. and Zenkour, A. M. (2018), "Post-buckling analysis of imperfect multi-phase nanocrystalline nanobeams considering nanograins and nanopores surface effects. Compos. Struct., 184, 497-505. https://doi.org/10.1016/j.compstruct.2017.10.019.
  17. Batou, B., Nebab, M., Bennai, R., Atmane, H. A., Tounsi, A. and Bouremana, M. (2019), "Wave dispersion properties in imperfect sigmoid plates using various HSDTs", Steel Compos. Struct., 33(5), 699. https://doi.org/10.12989/scs.2019.33.5.699.
  18. Belbachir, N., Draich, K., Bousahla, A. A., Bourada, M., Tounsi, A. and Mohammadimehr, M. (2019), "Bending analysis of anti-symmetric cross-ply laminated plates under nonlinear thermal and mechanical loadings", Steel Compos. Struct., 33(1), 913-924. https://doi.org/10.12989/scs.2019.33.1.081.
  19. Berghouti, H., Adda Bedia, E. A., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., 7(5), 351-364. https://doi.org/10.12989/anr.2019.7.5.351.
  20. Bellifa, H., Bakora, A., Tounsi, A., Bousahla, A. A. and Mahmoud, S. R. (2017), "An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates", Steel Compos. Struct., 25(3), 257-270. https://doi.org/10.12989/scs.2017.25.3.257.
  21. Berrabah, H. M., Tounsi, A., Semmah, A. and Adda, B. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct. Eng. Mech., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351.
  22. Boukhlif, Z., Bouremana, M., Bourada, F., Bousahla, A. A., Bourada, M., Tounsi, A. and Al-Osta, M. A. (2019), "A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation", Steel Compos. Struct., 31(5), 503-516. https://doi.org/10.12989/scs.2019.31.5.503.
  23. Bourada, F., Bousahla, A. A., Bourada, M., Azzaz, A., Zinata, A. 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.
  24. Boutaleb, S., Benrahou, K. H., Bakora, A., Algarni, A., Bousahla, A. A., Tounsi, A., Tounsi, A. and Mahmoud, S. R. (2019), "Dynamic Analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., 7(3), 189-206. http://dx.doi.org/10.12989/anr.2019.7.3.191.
  25. Boulefrakh, L., Hebali, H., Chikh, A., Bousahla, A. A., Tounsi, A. and Mahmoud, S. R. (2019), "The effect of parameters of visco-Pasternak foundation on the bending and vibration properties of a thick FG plate", Geomech. Eng., 18(2), 161-178. https://doi.org/10.12989/gae.2019.18.2.161.
  26. Chaabane, L. A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F. Z., Tounsi, A., Bousahla, A. A. and Tounsi, A. (2019), "Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation", Struct. Eng. Mech., 71(2), 185-196. https://doi.org/10.12989/sem.2019.71.2.185.
  27. Chen, D., Yang, J. and Kitipornchai, S. (2015), "Elastic buckling and static bending of shear deformable functionally graded porous beam", Compos. Struct., 133, 54-61. https://doi.org/10.1016/j.compstruct.2015.07.052.
  28. Chen, D., Kitipornchai, S. and Yang, J. (2016), "Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core", Thin-Walled Struct., 107, 39-48. https://doi.org/10.1016/j.tws.2016.05.025.
  29. Chikh, A., Bakora, A., Heireche, H., Houari, M. S. A., Tounsi, A. and Bedia, E.A. (2016), "Thermo-mechanical postbuckling of symmetric S-FGM plates resting on Pasternak elastic foundations using hyperbolic shear deformation theory", Struct. Eng. Mech., 57(4), 617-639. https://doi.org/10.12989/sem.2016.57.4.617.
  30. Draiche, K., Bousahla, A. A., Tounsi, A., Alwabli, A. S., Tounsi, A. and Mahmoud, S. R. (2019), "Static analysis of laminated reinforced composite plates using a simple first-order shear deformation theory", Comput. Concrete, 24(4), 369-378. https://doi.org/10.12989/cac.2019.24.4.369.
  31. Draoui, A., Zidour, M., Tounsi, A. and Adim, B. (2019), "Static and dynamic behavior of nanotubes-reinforced sandwich plates using (FSDT)", J. Nano Res., 57, 117-135. https://doi.org/10.4028/www.scientific.net/JNanoR.57.117.
  32. Duc, N. D., Cong, P. H. and Quang, V. D. (2016), "Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory", J. Therm. Stresses, 39(7), 772-794. https://doi.org/10.1080/01495739.2016.1188638.
  33. Duc, N. D. and Quan, T. Q. (2014), "Transient responses of functionally graded double curved shallow shells with temperature-dependent material properties in thermal environment", European J. Mech. A/Solids, 47, 101-123. https://doi.org/10.1016/j.euromechsol.2014.03.002.
  34. Fenjan, R. M., Ahmed, R. A., Alasadi, A. A. and Faleh, N. M. (2019), "Nonlocal strain gradient thermal vibration analysis of double-coupled metal foam plate system with uniform and non-uniform porosities", Coupled Syst. Mech., 8(3), 247-257. https://doi.org/10.12989/csm.2019.8.3.247.
  35. Hussain, M., Naeem, M. N., Tounsi, A. and Taj, M. (2019), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., 7(6), 431-442. https://doi.org/10.12989/anr.2019.7.6.431.
  36. Hellal, H., Bourada, M., Hebali, H., Bourada, F., Tounsi, A., Bousahla, A. A. and Mahmoud, S. R. (2019), "Dynamic and stability analysis of functionally graded material sandwich plates in hygro-thermal environment using a simple higher shear deformation theory", J. Sandwich Struct. Mater. https://doi.org/10.1177/1099636219845841.
  37. Kaddari, M., Kaci, A., Bousahla, A. A., Tounsi, A., Bourada, F., Tounsi, A., Adda Bedia, E.A. and Al-Osta, M.A. (2020), "A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: Bending and Free vibration analysis", Comput. Concrete, 25(1), 37-57. https://doi.org/10.12989/cac.2020.25.1.037.
  38. Khiloun, M., Bousahla, A. A., Kaci, A., Bessaim, A., Tounsi, A. and Mahmoud, S. R. (2019), "Analytical modeling of bending and vibration of thick advanced composite plates using a four-variable quasi 3D HSDT", Eng. Comput., https://doi.org/10.1007/s00366-019-00732-1.
  39. Li, H., Pang, F., Gong, Q. and Teng, Y. (2019), "Free vibration analysis of axisymmetric functionally graded doubly-curved shells with un-uniform thickness distribution based on Ritz method", Compos. Struct., 225, 111145. https://doi.org/10.1016/j.compstruct.2019.111145.
  40. Mahmoudi, A., Benyoucef, S., Tounsi, A., Benachour, A., Adda Bedia, E. A. and Mahmoud, S. R. (2019), "A refined quasi-3D shear deformation theory for thermo-mechanical behavior of functionally graded sandwich plates on elastic foundations", J. Sandwich Struct. Mater., 21(6), 1906-1929. https://doi.org/10.1177/1099636217727577.
  41. Mechab, I., Mechab, B., Benaissa, S., Serier, B and Bouiadjra, B. B. (2016), "Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories", J. Brazilian Soc. Mech. Sci. Eng., 38(8), 2193-2211. https://doi.org/10.1007/s40430-015-0482-6.
  42. Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A. A. and Mahmoud, S. R. (2019), "Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate using energy principle", Steel Compos. Struct., 32(5), 595-610. https://doi.org/10.12989/scs.2019.32.5.595.
  43. Meksi, R., Benyoucef, S., Mahmoudi, A., Tounsi, A., Adda Bedia, E. A. and Mahmoud, S. R. (2019), "An analytical solution for bending, buckling and vibration responses of FGM sandwich plates", J. Sandwich Struct. Mater., 21(2), 727-757. https://doi.org/10.1177%2F1099636217698443. https://doi.org/10.1177/1099636217698443
  44. Mirjavadi, S. S., Rabby, S., Shafiei, N., Afshari, B. M. and Kazemi, M. (2017), "On size-dependent free vibration and thermal buckling of axially functionally graded nanobeams in thermal environment", Appl. Phys. A, 123(5), 315. https://doi.org/10.1007/s00339-017-0918-1
  45. Mirjavadi, S. S., Afshari, B. M., Shafiei, N., Hamouda, A.M.S. and Kazemi, M. (2017), "Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams", Steel Compos. Struct, 25(4), 415-426. https://doi.org/10.12989/scs.2017.25.4.415
  46. Mirjavadi, S. S., Afshari, B. M., Barati, M. R. and Hamouda, A. M. S. (2018), "Strain gradient based dynamic response analysis of heterogeneous cylindrical microshells with porosities under a moving load", Mater. Res. Express, 6(3), 035029. https://doi.org/10.1088/2053-1591/aaf5a2
  47. Mirjavadi, S. S., Afshari, B. M., Khezel, M., Shafiei, N., Rabby, S. and Kordnejad, M. (2018), "Nonlinear vibration and buckling of functionally graded porous nanoscaled beams", J. Brazilian Soc. Mech. Sci. Eng., 40(7), 352. https://doi.org/10.1007/s40430-018-1272-8
  48. Mirjavadi, S. S., Forsat, M., Hamouda, A. M. S. and Barati, M. R. (2019), "Dynamic response of functionally graded graphene nanoplatelet reinforced shells with porosity distributions under transverse dynamic loads", Mater. Res. Exp., 6(7), 075045. https://doi.org/10.1088/2053-1591/ab1552
  49. Mirjavadi, S. S., Forsat, M., Nikookar, M., Barati, M. R. and Hamouda, A. M. S. (2019), "Nonlinear forced vibrations of sandwich smart nanobeams with two-phase piezo-magnetic face sheets", European Physical J. Plus, 134(10), 508. https://doi.org/10.1140/epjp/i2019-12806-8
  50. Mirjavadi, S. S., Afshari, B. M., Barati, M. R. and Hamouda, A. M. S. (2019), "Transient response of porous FG nanoplates subjected to various pulse loads based on nonlocal stress-strain gradient theory", European J. Mech. A/Solids, 74, 210-220. https://doi.org/10.1016/j.euromechsol.2018.11.004
  51. Mirjavadi, S. S., Afshari, B. M., Barati, M. R. and Hamouda, A. M. S. (2019), "Nonlinear free and forced vibrations of graphene nanoplatelet reinforced microbeams with geometrical imperfection", Microsyst. Technol., 25, 3137-3150. https://doi.org/10.1007/s00542-018-4277-4
  52. Mirjavadi, S. S., Forsat, M., Barati, M. R., Abdella, G. M., Hamouda, A. M. S., Afshari, B. M. and Rabby, S. (2019), "Post-buckling analysis of piezo-magnetic nanobeams with geometrical imperfection and different piezoelectric contents", Microsyst. Technol., 25(9), 3477-3488. https://doi.org/10.1007/s00542-018-4241-3
  53. Mirjavadi, S. S., Forsat, M., Barati, M. R., Abdella, G. M., Afshari, B. M., Hamouda, A. M. S. and Rabby, S. (2019), "Dynamic response of metal foam FG porous cylindrical micro-shells due to moving loads with strain gradient size-dependency", European Phys. J. Plus, 134(5), 214. https://doi.org/10.1140/epjp/i2019-12540-3
  54. Nebab, M., Atmane, H. A., Bennai, R. and Tahar, B. (2019), "Effect of nonlinear elastic foundations on dynamic behavior of FG plates using four-unknown plate theory", Earthq. Struct., 17(5), 447-462. https://doi.org/10.12989/eas.2019.17.5.447.
  55. Sahla, M., Saidi, H., Draiche, K., Bousahla, A. A., Bourada, F. and Tounsi, A. (2019), "Free vibration analysis of angle-ply laminated composite and soft core sandwich plates", Steel Compos. Struct., 33(5), 663-679. https://doi.org/10.12989/scs.2019.33.5.663.
  56. Sahu, S. A., Singhal, A. and Chaudhary, S. (2018), "Surface wave propagation in functionally graded piezoelectric material: An analytical solution", J. Intelligent Mater. Syst. Struct., 29(3), 423-437. https://doi.org/10.1177/1045389X17708047.
  57. Semmah, A., Heireche, H., Bousahla, A.A., Tounsi, A. (2019), "Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT", Adv. Nano Res., 7(2), 89-98. https://doi.org/10.12989/ANR.2019.7.2.089
  58. Shafiei, N., Mirjavadi, S. S., Afshari, B. M., Rabby, S. and Hamouda, A. M. S. (2017)", Nonlinear thermal buckling of axially functionally graded micro and nanobeams", Compos. Struct., 168, 428-439. https://doi.org/10.1016/j.compstruct.2017.02.048
  59. Tlidji, Y., Zidour, M., Draiche, K., Safa, A., Bourada, M., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2019), "Vibration analysis of different material distributions of functionally graded microbeam", Struct. Eng. Mech., 69(6), 637-649. https://doi.org/10.12989/SEM.2019.69.6.637
  60. Trinh, M. C., Nguyen, D. D. and Kim, S. E. (2019), "Effects of porosity and thermomechanical loading on free vibration and nonlinear dynamic response of functionally graded sandwich shells with double curvature", Aerosp. Sci. Technol., 87, 119-132. https://doi.org/10.1016/j.ast.2019.02.010.
  61. Tounsi, A., Al-Dulaijan, S.U., Al-Osta, M.A., Chikh, A., Al-Zahrani, M. M., Sharif, A. and Tounsi, A. (2020), "A four variable trigonometric integral plate theory for hygro-thermo-mechanical bending analysis of AFG ceramic-metal plates resting on a two-parameter elastic foundation", Steel Compos. Struct., 34(4), 511-524. https://doi.org/10.12989/SCS.2020.34.4.511
  62. Wattanasakulpong, N and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002.
  63. Yahiaoui, M., Tounsi, A., Fahsi, B., Bouiadjra, R.B. and Benyoucef, S. (2018), "The role of micromechanical models in the mechanical response of elastic foundation FG sandwich thick beams", Struct. Eng. Mech., 68(1), 053. https://doi.org/10.12989/sem.2018.68.1.053.
  64. Zare Jouneghani, F., Dimitri, R., Bacciocchi, M. and Tornabene, F. (2017), "Free vibration analysis of functionally graded porous doubly-curved shells based on the first-order shear deformation theory", Appl. Sci., 7(12), 1252. https://doi.org/10.3390/app7121252.
  65. Zarga, D., Tounsi, A., Bousahla, A. A., Bourada, F. and Mahmoud, S.R. (2019), "Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory", Steel Compos. Struct., 32(3), 389-410. https://doi.org/10.12989/scs.2019.32.3.389.
  66. Zaoui, F. Z., Ouinas, D. and Tounsi, A. (2019), "New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations", Compos. Part B, 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051.
  67. Zhao, J., Xie, F., Wang, A., Shuai, C., Tang, J. and Wang, Q. (2019), "A unified solution for the vibration analysis of functionally graded porous (FGP) shallow shells with general boundary conditions", Compos. Part B Eng., 156, 406-424. https://doi.org/10.1016/j.compositesb.2018.08.115.
  68. Zine, A., Tounsi, A., Draiche, K., Sekkal, M. and Mahmoud, S. R. (2018), "A novel higher-order shear deformation theory for bending and free vibration analysis of isotropic and multilayered plates and shells", Steel Compos. Struct., 26(2), 125-137. https://doi.org/10.12989/scs.2018.26.2.125.

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