Steel and Composite Structures

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Static stability and of symmetric and sigmoid functionally graded beam under variable axial load

  • Melaibari, Ammar (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Khoshaim, Ahmed B. (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Mohamed, Salwa A. (Department of Engineering Mathematics, Faculty of Engineering, Zagazig University) ;
  • Eltaher, Mohamed A. (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University)
  • Received : 2020.01.20
  • Accepted : 2020.05.23
  • Published : 2020.06.10
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Abstract

This manuscript presents impacts of gradation of material functions and axial load functions on critical buckling loads and mode shapes of functionally graded (FG) thin and thick beams by using higher order shear deformation theory, for the first time. Volume fractions of metal and ceramic materials are assumed to be distributed through a beam thickness by both sigmoid law and symmetric power functions. Ceramic-metal-ceramic (CMC) and metal-ceramic-metal (MCM) symmetric distributions are proposed relative to mid-plane of the beam structure. The axial compressive load is depicted by constant, linear, and parabolic continuous functions through the axial direction. The equilibrium governing equations are derived by using Hamilton's principles. Numerical differential quadrature method (DQM) is developed to discretize the spatial domain and covert the governing variable coefficients differential equations and boundary conditions to system of algebraic equations. Algebraic equations are formed as a generalized matrix eigenvalue problem, that will be solved to get eigenvalues (buckling loads) and eigenvectors (mode shapes). The proposed model is verified with respectable published work. Numerical results depict influences of gradation function, gradation parameter, axial load function, slenderness ratio and boundary conditions on critical buckling loads and mode-shapes of FG beam structure. It is found that gradation types have different effects on the critical buckling. The proposed model can be effective in analysis and design of structure beam element subject to distributed axial compressive load, such as, spacecraft, nuclear structure, and naval structure.

Keywords

References

  1. Abdelrahman, A.A., Eltaher, M.A., Kabeel, A.M., Abdraboh, A. M. and Hendi, A.A. (2019), "Free and forced analysis of perforated beams", Steel Compos. Struct., 31(5), 489-502. https://doi.org/10.12989/scs.2019.31.5.489.
  2. Abdulrazzaq, M.A., Fenjan, R.M., Ahmed, R.A. and Faleh, N.M. (2020), "Thermal buckling of nonlocal clamped exponentially graded plate according to a secant function based refined theory", Steel Compos. Struct., 35(1), 147-157. https://doi.org/10.12989/scs.2020.35.1.147.
  3. Akbas, S.D. (2018), "Geometrically nonlinear analysis of functionally graded porous beams", Wind Struct., 27(1), 59-70. https://doi.org/10.12989/was.2018.27.1.059.
  4. Almitani, K.H., Abdelrahman, A.A. and Eltaher, M.A. (2019), "On forced and free vibrations of cutout squared beams", Steel Compos. Struct., 32(5), 643-655. https://doi.org/10.12989/scs.2019.32.5.643.
  5. Almitani, K.H., Abdelrahman, A.A. and Eltaher, M.A. (2020), "Stability of perforated nanobeams incorporating surface energy effects", Steel Compos. Struct., 35(4), 555-566. https://doi.org/10.12989/scs.2020.35.4.555.
  6. Aminbaghai, M., Murin, J. and Kutis, V. (2012), "Modal analysis of the FGM-beams with continuous transversal symmetric and longitudinal variation of material properties with effect of large axial force", Eng. Struct., 34, 314-329. https://doi.org/10.1016/j.engstruct.2011.09.022.
  7. Aminbaghai, M., Murin, J., Kutis, V., Hrabovsky, J., Kostolani, M. and Mang, H.A. (2019), "Torsional warping elastostatic analysis of FGM beams with longitudinally varying material properties", Eng. Struct., 200, 109694. https://doi.org/10.1016/j.engstruct.2019.109694
  8. Ansari, R., Hassani, R. and Torabi, J. (2019), "Mixed-type formulation of higher-order shear deformation theory for vibration and buckling analysis of FG-GPLRC plates using VDQFEM", Compos. Struct., 111738. https://doi.org/10.1016/j.compstruct.2019.111738.
  9. Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F. (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006.
  10. Arioui, O., Belakhdar, K., Kaci, A. and Tounsi, A. (2018), "Thermal buckling of FGM beams having parabolic thickness variation and temperature dependent materials", Steel Compos. Struct., 27(6), 777-788. https://doi.org/10.12989/scs.2018.27.6.777.
  11. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603.
  12. Benoy, M.B. (1969), "An energy solution to the buckling of rectangular plates under non-uniform in-plane loading", Aeronaut. J., 73(707), 974-977. https://doi.org/10.1017/S0001924000051423.
  13. Bert, C.W. and Devarakonda, K.K. (2003), "Buckling of rectangular plates subjected to nonlinearly distributed in-plane loading", Int. J. Solids Struct., 40(16), 4097-4106. https://doi.org/10.1016/S0020-7683(03)00205-1.
  14. Duan, W.H. and Wang, C.M. (2008), "Exact solution for buckling of columns including self-weight", J. Eng. Mech., 134(1), 116-119. https://doi.org/10.1061/(ASCE)0733-9399(2008)134:1(116)
  15. Eisenberger, M. (1991), "Buckling loads for variable cross-section members with variable axial forces", Int. J. Solids Struct., 27(2), 135-143. https://doi.org/10.1016/0020-7683(91)90224-4.
  16. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090.
  17. Eltaher, M.A., Alshorbagy, A.E. and Mahmoud, F.F. (2013a), "Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams", Compos. Struct., 99, 193-201. https://doi.org/10.1016/j.compstruct.2012.11.039.
  18. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2013b), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88, https://doi.org/10.1016/j.compstruct.2012.09.030.
  19. Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F.A. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Math. Comput., 229, 283-295. https://doi.org/10.1016/j.amc.2013.12.072.
  20. Eltaher, M.A., Khater, M.E., Park, S., Abdel-Rahman, E. and Yavuz, M. (2016), "On the static stability of nonlocal nanobeams using higher-order beam theories", Adv. Nano Res., 4(1), 51-64. https://doi.org/10.12989/anr.2016.4.1.051.
  21. Eltaher, M.A., Mohamed, N., Mohamed, S.A. and Seddek, L.F. (2019), "Periodic and nonperiodic modes of postbuckling and nonlinear vibration of beams attached to nonlinear foundations", Appl. Math. Model., 75, 414-445. https://doi.org/10.1016/j.apm.2019.05.026.
  22. Eltaher, M.A., Mohamed, S.A. and Melaibari, A. (2020). Static stability of a unified composite beams under varying axial loads. Thin-Wall. Struct., 147, 106488. https://doi.org/10.1016/j.tws.2019.106488.
  23. Eltaher, M.A. and Mohamed, S.A. (2020), "Buckling and Stability Analysis of Sandwich Beams subjected to Varying Axial Loads", Steel Compos. Struct., 34(2), 241-260, https://doi.org/10.12989/scs.2020.34.2.241.
  24. Emam, S. and Eltaher, M.A. (2016), "Buckling and postbuckling of composite beams in hygrothermal environments", Compos. Struct., 152, 665-675. https://doi.org/10.1016/j.compstruct.2016.05.029.
  25. Emam, S.A., Eltaher, M.A., Khater, M.E. and Abdalla, W.S. (2018), "Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load", Appl. Sci., 8(11), 2238. https://doi.org/10.3390/app8112238.
  26. Farajpour, A., Shahidi, A.R., Mohammadi, M. and Mahzoon, M. (2012), "Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics", Compos. Struct., 94(5), 1605-1615. https://doi.org/10.1016/j.compstruct.2011.12.032.
  27. Hamed, M.A., Eltaher, M.A., Sadoun, A.M. and Almitani, K.H. (2016), "Free vibration of symmetric and sigmoid functionally graded nanobeams", Appl. Phys. A, 122(9), 829. https://doi.org/10.1007/s00339-016-0324-0.
  28. Hamed, M.A., Sadoun, A.M. and Eltaher, M.A. (2019), "Effects of porosity models on static behavior of size dependent functionally graded beam", Struct. Eng. Mech., 71(1), 89-98. https://doi.org/10.12989/sem.2019.71.1.089.
  29. Hamed M.A., Mohamed, S.A. and Eltaher, M.A. (2020a), "Buckling analysis of sandwich beam rested on elastic foundation and subjected to varying axial in-Plane Loads", Steel Compos. Struct., 34(1), 75-89. https://doi.org/10.12989/scs.2020.34.1.075.
  30. Hamed M.A., Abu-bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2020b), "influence of Axial Load Function and Optimization on Static Stability of Sandwich Functionally Graded Beams with Porous Core", Eng. with Comput., 1-14. https://doi.org/10.1007/s00366-020-01023-w.
  31. Hamed M.A, Mohamed, N.A. and Eltaher, M.A. (2020c), "Stability buckling and bending of nanobeams including cutouts", Eng. with Comput., 1-14. https://doi.org/10.1007/s00366-020-01063-2.
  32. Kahya, V. and Turan, M. (2017), "Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory", Compos. Part B: Eng., 109, 108-115. https://doi.org/10.1016/j.compositesb.2016.10.039.
  33. Kang, J.H. and Leissa, A.W. (2005), "Exact solutions for the buckling of rectangular plates having linearly varying in-plane loading on two opposite simply supported edges", International J. Solids Struct., 42(14), 4220-4238. https://doi.org/10.1016/j.ijsolstr.2004.12.011.
  34. Karamanli, A. and Aydogdu, M. (2019a), "Buckling of laminated composite and sandwich beams due to axially varying in-plane loads", Compos. Struct., 210, 391-408. https://doi.org/10.1016/j.compstruct.2018.11.067.
  35. Karamanli, A. and Aydogdu, M. (2019b), "Free vibration and buckling analysis of laminated composites and sandwich microbeams using a transverse shear-normal deformable beam theory", J. Vib. Control, 1077546319878538. https://doi.org/10.1177/1077546319878538.
  36. Karami, B., Shahsavari, D., Nazemosadat, S.M.R., Li, L. and Ebrahimi, A. (2018), "Thermal buckling of smart porous functionally graded nanobeam rested on Kerr foundation", Steel Compos. Struct., 29(3), 349-362. https://doi.org/10.12989/scs.2018.29.3.349.
  37. Khorshidi, M.A., Shariati, M. and Emam, S.A. (2016), "Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory", Int. J. Mech. Sci., 110, 160-169. https://doi.org/10.1016/j.ijmecsci.2016.03.006 .
  38. Kim, H.J., Yoon, K. and Lee, P.S. (2020), "Continuum mechanics based beam elements for linear and nonlinear analyses of multilayered composite beams with interlayer slips", Compos. Struct., 111740. https://doi.org/10.1016/j.compstruct.2019.111740.
  39. Lou, J., He, L., Wu, H. and Du, J. (2016), "Pre-buckling and buckling analyses of functionally graded microshells under axial and radial loads based on the modified couple stress theory", Compos. Struct., 142, 226-237. https://doi.org/10.1016/j.compstruct.2016.01.083.
  40. Mijuskovic, O., Coric, B. and Scepanovic, B. (2014), "Exact stress functions implementation in stability analysis of plates with different boundary conditions under uniaxial and biaxial compression", Thin-Wall. Struct., 80, 192-206. https://doi.org/10.1016/j.tws.2014.03.006.
  41. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2018), "Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations", Int. J. Nonlinear Mech., 101, 157-173. https://doi.org/10.1016/j.ijnonlinmec.2018.02.014.
  42. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2019), "Energy equivalent model in analysis of postbuckling of imperfect carbon nanotubes resting on nonlinear elastic foundation", Struct. Eng. Mech., 70(6), 737-750. https://doi.org/10.12989/sem.2019.70.6.737.
  43. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2020), "Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model", Eng. with Comput., 1-14. https://doi.org/10.1007/s00366-020-00976-2.
  44. Mohammadimehr, M., Mohammadi-Dehabadi, A.A., Akhavan Alavi, S.M., Alambeigi, K., Bamdad, M., Yazdani, R. and Hanifehlou, S. (2018), "Bending, buckling, and free vibration analyses of carbon nanotube reinforced composite beams and experimental tensile test to obtain the mechanical properties of nanocomposite", Steel Compos. Struct., 29(3), 405-422. https://doi.org/10.12989/scs.2018.29.3.405.
  45. Murin, J., Aminbaghai, M., Hrabovsky, J., Kutis, V., Paulech, J., and Kugler, S. (2016), "Elastostatic and modal and buckling analysis of spatial FGM beam structures", Proceedings of the VII European Concress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)", National Technical University of Athens (NTUA), 2016, Paper-Nr. 4649, 28 S.
  46. Osmani, A. and Meftah, S.A. (2018), "Lateral buckling of tapered thin walled bi-symmetric beams under combined axial and bending loads with shear deformations allowed", Eng. Struct., 165, 76-87. https://doi.org/10.1016/j.engstruct.2018.03.009.
  47. Panda, S.K. and Ramachandra, L.S. (2010), "Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads", Int. J. Mech. Sci., 52(6), 819-828. https://doi.org/10.1016/j.ijmecsci.2010.01.009.
  48. Polit, O., Merzouki, T. and Ganapathi, M. (2018), "Elastic stability of curved nanobeam based on higher-order shear deformation theory and nonlocal analysis by finite element approach", Finite Elem. Anal. Des., 146, 1-15. https://doi.org/10.1016/j.finel.2018.04.002.
  49. Rahmani, O., Hosseini, S.A.H. and Parhizkari, M. (2017), "Buckling of double functionally-graded nanobeam system under axial load based on nonlocal theory: an analytical approach", Microsystem Technologies, 23(7), 2739-2751. DOI 10.1007/s00542-016-3127-5.
  50. Ravindran, A. and Bhaskar, K. (2019), "Three-dimensional analysis of composite FGM rectangular plates with in-plane heterogeneity", Int. J. Mech. Sci., 160, 386-396. https://doi.org/10.1016/j.ijmecsci.2019.07.004.
  51. Reddy, J.N. (1984), "A simple higher-order theory for laminated composite plates", J. Appl. Mech., 51(4), 745-752. https://doi.org/10.1115/1.3167719
  52. Robinson, M.T.A. and Adali, S. (2016), "Variational solution for buckling of nonlocal carbon nanotubes under uniformly and triangularly distributed axial loads", Compos. Struct., 156, 101-107. https://doi.org/10.1016/j.compstruct.2016.01.026.
  53. Sedighi, H.M., Keivani, M. and Abadyan, M. (2015a), Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: corrections due to finite conductivity, surface energy and nonlocal effect", Compos. Part B: Eng., 83, 117-133. https://doi.org/10.1016/j.compositesb.2015.08.029.
  54. Sedighi, H.M., Daneshmand, F. and Abadyan, M. (2015b), "Modified model for instability analysis of symmetric FGM double-sided nano-bridge: corrections due to surface layer, finite conductivity and size effect", Compos. Struct., 132, 545-557. https://doi.org/10.1016/j.compstruct.2015.05.076.
  55. She, G.L., Yuan, F.G. and Ren, Y.R. (2017), "Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory", Appl. Math. Model., 47, 340-357. https://doi.org/10.1016/j.apm.2017.03.014.
  56. Shu, C. (2012). Differential quadrature and its application in engineering. Springer Science & Business Media.
  57. Simsek, M. and Reddy, J.N. (2013), "A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory", Compos. Struct., 101, 47-58. https://doi.org/10.1016/j.compstruct.2013.01.017
  58. Singh, S.J. and Harsha, S.P. (2019), "Buckling analysis of FGM plates under uniform, linear and non-linear in-plane loading", J. Mech. Sci. Technol., 33(4), 1761-1767. DOI 10.1007/s12206-019-0328-8
  59. Wang, J.T.S., Biggerst, S.B. and Dickson, J.N. (1984), "Buckling of composite plates with a free edge in edgewise bending and compression", AIAA J., 22(3), 394-398. https://doi.org/10.2514/3.48460.

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