Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Vibration of bio-inspired laminated composite beams under varying axial loads

  • Tharwat Osman (Department of Engineering Mathematics, Faculty of Engineering) ;
  • Salwa A. Mohamed (Department of Engineering Mathematics, Faculty of Engineering) ;
  • Mohamed A. Eltaher (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Mashhour A. Alazwari (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Nazira Mohamed (Department of Engineering Mathematics, Faculty of Engineering)
  • Received : 2022.02.15
  • Accepted : 2023.12.11
  • Published : 2024.01.10
⟨ Previous Next ⟩

Abstract

In this article, a mathematical model is developed to predict the dynamic behavior of bio-inspired composite beam with helicoidal orientation scheme under variable axial load using a unified higher order shear deformation beam theory. The geometrical kinematic relations of displacements are portrayed with higher parabolic shear deformation beam theory. Constitutive equation of composite beam is proposed based on plane stress problem. The variable axial load is distributed through the axial direction by constant, linear, and parabolic functions. The equations of motion and associated boundary conditions are derived in detail by Hamilton's principle. Using the differential quadrature method (DQM), the governing equations, which are integro-differential equations are discretized in spatial direction, then they are transformed into linear eigenvalue problems. The proposed model is verified with previous works available in literatures. Parametric analyses are developed to present the influence of axial load type, orthotropic ratio, slenderness ratio, lamination scheme, and boundary conditions on the natural frequencies of composite beam structures. The present enhanced model can be used especially in designing spacecrafts, naval, automotive, helicopter, the wind turbine, musical instruments, and civil structures subjected to the variable axial loads.

Keywords

References

  1. Abdelrahman, A.A. and Eltaher, M.A. (2020), "On bending and buckling responses of perforated nanobeams including surface energy for different beams theories", Eng. Comput., 1-27. https://doi.org/10.1007/s00366-020-01211-8.
  2. Abdelrahman, A.A., Esen, I., Ozarpa, C. and Eltaher, M.A. (2021), "Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory", Appl. Mathem. Modelling, 96, 215-235. https://doi.org/10.1016/j.apm.2021.03.008.
  3. Alazwari, M.A., Daikh, A.A., Houari, M.S.A., Tounsi, A. and Eltaher, M.A. (2021), "On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations", Steel Compos. Struct., 40(3), 389-404. https://doi.org/10.12989/scs.2021.41.6.787.
  4. Almitani, K.H., Mohamed, N., Alazwari, M.A., Mohamed, S.A. and Eltaher, M.A. (2022), "Exact solution of nonlinear behaviors of imperfect bioinspired helicoidal composite beams resting on elastic foundations", Mathematics, 10(6), 887. https://doi.org/10.3390/math10060887.
  5. Baakeel, F., Eltaher, M.A., Basha, M.A., Melibari, A. and Abdelrhman, A.A. (2023), "Static and modal analysis of bioinspired laminated composite shells using numerical simulation", Adv. Aircraft Spacecraft Sci., 10(4), 347. https://doi.org/10.12989/aas.2023.10.4.347
  6. Bizzi, A., Fortaleza, E.L. and Guenka, T.S. (2021), "Dynamics of heavy beams: Closed-form vibrations of gravity-loaded Rayleigh-Timoshenko columns", J. Sound Vib., 116259. https://doi.org/10.1016/j.jsv.2021.116259.
  7. Dabbagh, A., Rastgoo, A. and Ebrahimi, F. (2019), "Finite element vibration analysis of multi-scale hybrid nanocomposite beams via a refined beam theory", Thin-Wall. Struct., 140, 304-317. https://doi.org/10.1016/j.tws.2019.03.031.
  8. Daikh, A.A., Drai, A., Houari, M.S.A. and Eltaher, M.A. (2020), "Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes", Steel Compos. Struct., 36(6), 643-656. https://doi.org/10.12989/scs.2020.36.6.643.
  9. Daikh, A.A., Belarbi, M.O., Ahmed, D., Houari, M.S.A., Avcar, M., Tounsi, A. and Eltaher, M.A. (2023), "Static analysis of functionally graded plate structures resting on variable elastic foundation under various boundary conditions", Acta Mechanica, 234(2), 775-806. https://doi.org/10.1007/s00707-022-03405-1.
  10. Daraei, B., Shojaee, S. and Hamzehei-Javaran, S. (2021), "Analysis of stationary and axially moving beams considering functionally graded material using micropolar theory and Carrera unified formulation", Compos. Struct., 271, 114054. https://doi.org/10.1016/j.compstruct.2021.114054.
  11. Ding, H.X. and She, G.L. (2021), "A higher-order beam model for the snap-buckling analysis of FG pipes conveying fluid", Struct. Eng. Mech., 80(1), 63-72. http://dx.doi.org/10.12989/sem.2021.80.1.063.
  12. Ebrahimi, F. and Barati, M.R. (2016), "A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment", Appl. Phys. A, 122(9), 1-14. https://doi.org/10.1007/s00339-016-0322-2.
  13. Eltaher, M.A., Mohamed, S.C. and Melaibari, A.A. (2020a), "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.
  14. 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.
  15. Eltaher, M.A., Abdelrahman, A.A. and Esen, I. (2021), "Dynamic analysis of nanoscale Timoshenko CNTs based on doublet mechanics under moving load", Europ. Phys. J. Plus, 136(7), 1-21. https://doi.org/10.1140/epjp/s13360-021-01682-8.
  16. Esen, I., Daikh, A.A. and Eltaher, M.A. (2021), "Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubes under moving point load", Europ. Phys. J. Plus, 136(4), 1-22. https://doi.org/10.1140/epjp/s13360-021-01419-7.
  17. Gao, G., Sun, N., Shao, D., Tao, Y. and Wu, W. (2021), "A unified analysis for the free vibration of the sandwich piezoelectric laminated beam with general boundary conditions under the thermal environment", Shock Vib., 2021. https://doi.org/10.1155/2021/1328886.
  18. Gunasekaran, V., Pitchaimani, J. and Chinnapandi, L.B.M. (2020a), "Vibro-acoustics response of an isotropic plate under non-uniform edge loading: An analytical investigation", Aeros. Sci. Technol., 105, 106052. https://doi.org/10.1016/j.ast.2020.106052.
  19. Gunasekaran, V., Pitchaimani, J. and Chinnapandi, L.B.M. (2020b), "Analytical investigation on free vibration frequencies of polymer nano composite plate: Effect of graphene grading and non-uniform edge loading", Mater. Today Commun., 24, 100910. https://doi.org/10.1016/j.mtcomm.2020.100910.
  20. 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.
  21. Hamed, M.A., Abo-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. Comput., 36(4), 1929-1946. https://doi.org/10.1007/s00366-020-01023-w.
  22. Hendi, A.A., Eltaher, M.A., Mohamed, S.A., Attia, M.A. and Abdalla, A.W. (2021), "Nonlinear thermal vibration of pre/postbuckled two-dimensional FGM tapered microbeams based on a higher order shear deformation theory", Steel Compos. Struct., 41(6), 787-802. https://doi.org/10.12989/scs.2021.41.6.787.
  23. Ibrahim, S.M., Alsayed, S.H., Abbas, H., Carrera, E., Al-Salloum, Y.A. and Almusallam, T.H. (2014), "Free vibration of tapered beams and plates based on unified beam theory", J. Vib. Control, 20(16), 2450-2463. https://doi.org/10.1177/1077546312473766.
  24. 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", Int. J. Solids Struct., 42(14), 4220-4238. https://doi.org/10.1016/j.ijsolstr.2004.12.011.
  25. Karamanli, A. (2017), "Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory", Compos. Struct., 174, 70-86. https://doi.org/10.1016/j.compstruct.2017.04.046.
  26. Karamanli, A. (2018), "Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory", Compos. Struct., 189, 127-136. https://doi.org/10.1016/j.compstruct.2018.01.060.
  27. Karamanli, A. and Aydogdu, M. (2019), "On the vibration of size dependent rotating laminated composite and sandwich microbeams via a transverse shear-normal deformation theory", Compos. Struct., 216, 290-300. https://doi.org/10.1016/j.compstruct.2019.02.044.
  28. Katili, I., Syahril, T. and Katili, A.M. (2020), "Static and free vibration analysis of FGM beam based on unified and integrated of Timoshenko's theory", Compos. Struct., 242, 112130. https://doi.org/10.1016/j.compstruct.2020.112130
  29. Khodabakhshi, P. and Reddy, J.N. (2017), "A unified beam theory with strain gradient effect and the von Karman nonlinearity", ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 97(1), 70-91. https://doi.org/10.1002/zamm.201600021.
  30. Kundu, B. and Ganguli, R. (2020), "Closed-form solutions of nonuniform axially loaded beams using Lie symmetry analysis", Acta Mechanica, 231(11), 4421-4444. https://doi.org/10.1007/s00707-020-02773-w.
  31. Leissa, A.W. and Kang, J.H. (2002), "Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses", Int. J. Mech. Sci., 44(9), 1925-1945. https://doi.org/10.1016/S0020-7403(02)00069-3.
  32. Li, X.Y., Wang, X.H., Chen, Y.Y., Tan, Y. and Cao, H.J. (2020), "Bending, buckling and free vibration of an axially loaded Timoshenko beam with transition parameter: Direction of axial force", Int. J. Mech. Sci., 176, 105545. https://doi.org/10.1016/j.ijmecsci.2020.105545.
  33. Liu, H. and Zhang, Q. (2021), "Nonlinear dynamics of two-directional functionally graded microbeam with geometrical imperfection using unified shear deformable beam theory", Appl. Mathem. Modelling, 98, 783-800. https://doi.org/10.1016/j.apm.2021.05.029.
  34. Lu, L., She, G.L. and Guo, X. (2021), "Size-dependent postbuckling analysis of graphene reinforced composite microtubes with geometrical imperfection", Int. J. Mech. Sci., https://doi.org/10.1016/j.ijmecsci.2021.106428.
  35. MARAS, S. and SENSOY, A.T. (2023), "Estimating the effect of certain manufacturing parameters for fiber laminated composites: A validated DQM model integrated with RSM.", Eng. Anal. Bound. Elements, 155, 169-181. https://doi.org/10.1016/j.enganabound.2023.06.007.
  36. Maras, S. and Yaman, M. (2023), "Investigation of dynamic properties of GLARE and CARALL hybrid composites: Numerical and experimental results", Eng. Anal. Bound. Elements, 155, 484-499. https://doi.org/10.1016/j.enganabound.2023.06.026.
  37. Maras, S. and Yaman, M. (2022), "Free vibration analysis of fiber-metal laminated composite plates using differential, generalized and harmonic quadrature methods: Experimental and numerical studies", Eng. Comput., 39(6), 2326-2349. https://doi.org/10.1108/EC-08-2021-0490.
  38. Maras, S., Yaman, M., Sansveren, M.F. and Reyhan, S.K. (2018), "Free vibration analysis of fiber metal laminated straight beam", Open Chemistry, 16(1), 944-948. https://doi.org/10.1515/chem-2018-0101.
  39. Melaibari, A., Khoshaim, A.B., Mohamed, S.A. and Eltaher, M.A. (2020), "Static stability and of symmetric and sigmoid functionally graded beam under variable axial load", Steel Compos. Struct., 35(5), 671-685. https://doi.org/10.12989/scs.2020.35.5.671.
  40. Mohamed, N., Mohamed, S.A., Abdelrhmaan, A.A. and Eltaher, M.A. (2023), "Nonlinear stability of bio-inspired composite beams with higher order shear theory", Steel Compos. Struct., 46(6), 759. https://doi.org/10.12989/scs.2023.46.6.759.
  41. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2022a), "Nonlinear static stability of imperfect bio-inspired helicoidal composite beams", Mathematics, 10(7), 1084. https://doi.org/10.3390/math10071084.
  42. Mohamed, S.A., Mohamed, N. and Eltaher, M.A. (2022b), "Bending, buckling and linear vibration of bio-inspired composite plates", Ocean Eng., 259, 111851. https://doi.org/10.1016/j.oceaneng.2022.111851.
  43. Naguleswaran, S. (1991), "Vibration of a vertical cantilever with and without axial freedom at clamped end", J. Sound Vib., 146(2), 191-198. https://doi.org/10.1016/0022-460X(91)90758-C.
  44. Ng, T.Y., Lam, K.Y., Liew, K.M. and Reddy, J.N. (2001), "Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading", Int. J. Solids Struct., 38(8), 1295-1309. https://doi.org/10.1016/S0020-7683(00)00090-1.
  45. 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.
  46. Patil, H.H., Pitchaimani, J. and Eltaher, M.A. (2023), "Buckling and vibration of beams using Ritz method: Effects of axial grading of GPL and axially varying load", Mech. Adv. Mater. Struct., 1-14. https://doi.org/10.1080/15376494.2023.2185711.
  47. Poloskei, T. and Szekrenyes, A. (2021), "Dynamic stability analysis of delaminated composite beams in frequency domain using a unified beam theory with higher order displacement continuity", Compos. Struct., 272, 114173. https://doi.org/10.1016/j.compstruct.2021.114173.
  48. Priyanka, R., Twinkle, C.M. and Pitchaimani, J. (2021), "Stability and dynamic behavior of porous FGM beam: influence of graded porosity, graphene platelets, and axially varying loads", Eng. Comput., 1-20. https://doi.org/10.1007/s00366-021-01478-5.
  49. Sarkar, K., Ganguli, R., Ghosh, D. and Elishakoff, I. (2016), "Closed-form solutions and uncertainty quantification for gravity-loaded beams", Meccanica, 51(6), 1465-1479. http://dx.doi.org/10.1007/s11012-015-0314-x.
  50. Schafer, B.E. and Holzach, H. (1985), "Experimental research on flexible beam modal control", J. Guidance, Control Dyn., 8(5), 597-604. https://doi.org/10.2514/3.20028.
  51. Schafer, B. (1985), "Free vibrations of a gravity-loaded clampedfree beam", Ingenieur-archiv, 55(1), 66-80. https://doi.org/10.1007/BF00539551
  52. Shao, D., Wang, Q., Tao, Y., Shao, W. and Wu, W. (2021), "A unified thermal vibration and transient analysis for quasi-3D shear deformation composite laminated beams with general boundary conditions", Int. J. Mech. Sci., 198, 106357. https://doi.org/10.1016/j.ijmecsci.2021.106357.
  53. She, G.L., Liu, H.B. and Karami, B. (2021), "Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets", Thin Wall. Struct., 160, 107407. https://doi.org/10.1016/j.tws.2020.107407.
  54. Simsek, M. and Reddy, J.N. (2013), "Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory", Int. J. Eng. Sci., 64, 37-53. https://doi.org/10.1016/j.ijengsci.2012.12.002.
  55. Valle, J., Fernandez, D. and Madrenas, J. (2019), "Closed-form equation for natural frequencies of beams under full range of axial loads modeled with a spring-mass system", Int. J. Mech. Sci., 153, 380-390. https://doi.org/10.1016/j.ijmecsci.2019.02.014.
  56. Wang, G.X., Ding, H. and Chen, L.Q. (2020), "Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams", J. Sound Vib., 474, 115265. https://doi.org/10.1016/j.jsv.2020.115265.
  57. Xi, L.Y., Li, X.F. and Tang, G.J. (2013), "Free vibration of standing and hanging gravity-loaded Rayleigh cantilevers", Int. J. Mech. Sci., 66, 233-238. https://doi.org/10.1016/j.ijmecsci.2012.11.013.
  58. Yokoyama, T. (1990), "Vibrations of a hanging Timoshenko beam under gravity", J. Sound Vib., 141(2), 245-258. https://doi.org/10.1016/0022-460X(90)90838-Q.
  59. Zhang, Y.W., She, G.L. and Eltaher, M.A. (2023), "Nonlinear transient response of graphene platelets reinforced metal foams annular plate considering rotating motion and initial geometric imperfection", Aeros. Sci. Technol., 142, 108693. https://doi.org/10.1016/j.ast.2023.108693.
  60. Zhang, Y.Y., Wang, Y.X., Zhang, X., Shen, H.M. and She, G.L. (2021), "On snap-buckling of FG-CNTR curved nanobeams considering surface effects", Steel Compos. Struct. 38(3), 293- http://dx.doi.org/10.12989/scs.2021.38.3.293.