DOI QR코드

DOI QR Code

Post-buckling responses of a laminated composite beam

  • Akbas, Seref D. (Department of Civil Engineering, Bursa Technical University)
  • Received : 2017.10.20
  • Accepted : 2018.01.16
  • Published : 2018.03.25

Abstract

This paper presents post-buckling responses of a simply supported laminated composite beam subjected to a non-follower axially compression loads. In the nonlinear kinematic model of the laminated beam, total Lagrangian approach is used in conjunction with the Timoshenko beam theory. In the solution of the nonlinear problem, incremental displacement-based finite element method is used with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The distinctive feature of this study is post-buckling analysis of Timoshenko Laminated beams full geometric non-linearity and by using finite element method. The effects of the fibber orientation angles and the stacking sequence of laminates on the post-buckling deflections, configurations and stresses of the composite laminated beam are illustrated and discussed in the numerical results. Numerical results show that the above-mentioned effects play a very important role on the post-buckling responses of the laminated composite beams.

Keywords

References

  1. 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., Int. J., 25(6), 693-704.
  2. Abualnour, M., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates", Compos. Struct., 184, 688-697. https://doi.org/10.1016/j.compstruct.2017.10.047
  3. Ahouel, M., Houari, M.S.A., Bedia, E.A. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., Int. J., 20(5), 963-981. https://doi.org/10.12989/scs.2016.20.5.963
  4. Akbas, S.D. (2013), "Geometrically nonlinear static analysis of edge cracked Timoshenko beams composed of functionally graded material", Math. Problems Eng.
  5. Akbas, S.D. (2014), "Large post-buckling behavior of Timoshenko beams under axial compression loads", Struct. Eng. Mech., Int. J., 51(6), 955-971. https://doi.org/10.12989/sem.2014.51.6.955
  6. Akbas, S.D. (2015a), "On post-buckling behavior of edge cracked functionally graded beams under axial loads", Int. J. Struct. Stabil. Dyn., 15(4), 1450065. DOI: 10.1142/S0219455414500655
  7. Akbas, S.D. (2015b), "Post-buckling analysis of axially functionally graded three-dimensional beams", Int. J. Appl. Mech., 7(3), 1550047. DOI: 10.1142/S1758825115500477
  8. Akbas, S.D. (2015c), "Large deflection analysis of edge cracked simple supported beams", Struct. Eng. Mech., Int. J., 54(3), 433-451.
  9. Akbas, S.D. (2017a), "Post-buckling responses of functionally graded beams with porosities", Steel Compos. Struct., Int. J., 24(5), 579-589.
  10. Akbas, S.D. (2017b), "Stability of a non-homogenous porous plate by using generalized differantial quadrature method", Int. J. Eng. Appl. Sci., 9(2), 147-155.
  11. Akbas, S.D. (2017c), "Static, Vibration, and Buckling Analysis of Nanobeams", Nanomechanics, InTech.
  12. Akbas, S.D. and Kocaturk, T. (2011), "Post-buckling analysis of a simply supported beam under uniform thermal loading", Sci. Res. Essays, 6(5), 1135-1142.
  13. Akbas, S.D. and Kocaturk, T. (2012), "Post-buckling analysis of Timoshenko beams with temperature-dependent physical properties under uniform thermal loading", Struct. Eng. Mech., Int. J., 44(1), 109-125. https://doi.org/10.12989/sem.2012.44.1.109
  14. Akbas, S.D. and Kocaturk, T. (2013), "Post-buckling analysis of functionally graded three-dimensional beams under the influence of temperature", J. Therm. Stress., 36(12), 1233-1254. https://doi.org/10.1080/01495739.2013.788397
  15. Akgoz, B. and Civalek, O . (2011), "Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations", Steel Compos. Struct., Int. J., 11(5), 403-421. https://doi.org/10.12989/scs.2011.11.5.403
  16. Asadi, H. and Aghdam, M.M. (2014), "Large amplitude vibration and post-buckling analysis of variable cross-section composite beams on nonlinear elastic foundation", Int. J. Mech. Sci., 79, 47-55. https://doi.org/10.1016/j.ijmecsci.2013.11.017
  17. Almitani, K.H. (2017), "Buckling behaviors of symmetric and antisymmetric functionally graded beams", J. Appl. Computat. Mech., 4(2), 115-124.
  18. Baghani, M., Jafari-Talookolaei, R.A. and Salarieh, H. (2011), "Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation", Appl. Math. Model., 35(1), 130-138. https://doi.org/10.1016/j.apm.2010.05.012
  19. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Beg, O.A. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. Part B: Eng., 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  20. Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017a), "A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams", Struct. Eng. Mech., Int. J., 62(6), 695-702.
  21. Bellifa, H., Bakora, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017b), "An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates", Steel Compos. Struct., Int. J., 25(3), 257-270.
  22. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088
  23. Benselama, K., El Meiche, N., Bedia, E.A.A. and Tounsi, A. (2015), "Buckling analysis in hybrid cross-ply composite laminates on elastic foundation using the two variable refined plate theory", Struct. Eng. Mech, Int. J., 55(1), 47-64. https://doi.org/10.12989/sem.2015.55.1.047
  24. Bessaim, A., Houari, M.S., Tounsi, A., Mahmoud, S.R. and Bedia, E.A.A. (2013), "A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets", J. Sandw. Struct. Mater., 15(6), 671-703. https://doi.org/10.1177/1099636213498888
  25. Bessaim, A., Ahmed Houari, M.S., Abdelmoumen Anis, B., Kaci, A., Tounsi, A. and Bedia, A. (2017), "Buckling analysis of embedded nanosize FG beams based on a refined hyperbolic shear deformation theory", J. Appl. Computat. Mech. DOI: 10.22055/JACM.2017.22996.1146
  26. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., Int. J., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  27. Bouderba, B., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2016), "Thermal stability of functionally graded sandwich plates using a simple shear deformation theory", Struct. Eng. Mech., Int. J., 58(3), 397-422.
  28. 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., Int. J., 19(2), 115-126. https://doi.org/10.12989/sss.2017.19.2.115
  29. Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Bedia, E.A.A. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Computat. Methods, 11(6),1350082. https://doi.org/10.1142/S0219876213500825
  30. Bousahla, A.A., Benyoucef, S., Tounsi, A. and Mahmoud, S.R. (2016), "On thermal stability of plates with functionally graded coefficient of thermal expansion", Struct. Eng. Mech., Int. J., 60(2), 313-335. https://doi.org/10.12989/sem.2016.60.2.313
  31. Cardoso, J.B., Benedito, N.M. and Valido, A.J. (2009), "Finite element analysis of thin-walled composite laminated beams with geometrically nonlinear behavior including warping deformation", Thin-Wall. Struct., 47(11), 1363-1372. https://doi.org/10.1016/j.tws.2009.03.002
  32. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S.R. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., Int. J., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  33. Chang, X.P., Zhang, X.D. and Liu, Q.Y. (2011), "Geometrically Nonlinear Analysis of Cross-ply Laminated Composite Beams Subjected to Uniform Temperature Rise", In: Adv. Mater. Res., 335, 527-530.
  34. Chikh, A., Tounsi, A., Hebali, H. and Mahmoud, S.R. (2017), Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT", Smart Struct. Syst., Int. J., 19(3), 289-297. https://doi.org/10.12989/sss.2017.19.3.289
  35. Civalek, O . (2013), "Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches", Compos. Part B: Eng., 50, 171-179.
  36. Cunedioglu, Y. and Beylergil, B. (2014), "Free vibration analysis of laminated composite beam under room and high temperatures", Struct. Eng. Mech., Int. J., 51(1), 111-130.
  37. Draiche, K., Tounsi, A. and Mahmoud, S.R. (2016), A refined theory with stretching effect for the flexure analysis of laminated composite plates", Geomech. Eng., Int. J., 11(5), 671-690. https://doi.org/10.12989/gae.2016.11.5.671
  38. Di Sciuva, M. and Icardi, U. (1995), "Large deflection of adaptive multilayered Timoshenko beams", Compos. Struct., 31(1), 49-60. https://doi.org/10.1016/0263-8223(95)00001-1
  39. Donthireddy, P. and Chandrashekhara, K. (1997), "Nonlinear thermomechanical analysis of laminated composite beams", Adv. Compos. Mater., 6(2), 153-166. https://doi.org/10.1163/156855197X00049
  40. Ebrahimi, F. and Hosseini, S.H.S. (2017), "Surface effects on nonlinear dynamics of NEMS consisting of double-layered viscoelastic nanoplates", Eur. Phys. J. Plus, 132(4), 172. https://doi.org/10.1140/epjp/i2017-11400-6
  41. El-Haina, F., Bakora, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017), "A simple analytical approach for thermal buckling of thick functionally graded sandwich plates", Struct. Eng. Mech., Int. J., 63(5), 585-595.
  42. Emam, S.A. and Nayfeh, A.H. (2009), "Postbuckling and free vibrations of composite beams", Compos. Struct., 88(4), 636-642. https://doi.org/10.1016/j.compstruct.2008.06.006
  43. Felippa, C.A. (2017), "Notes on Nonlinear Finite Element Methods", URL: http://www.colorado.edu/engineering/cas/courses.d/NFEM.d/NFEM.Ch11.d/NFEM.Ch11.pdf
  44. Fouda, N., El-midany, T. and Sadoun, A.M. (2017), "Bending, buckling and vibration of a functionally graded porous beam using finite elements", J. Appl. Computat. Mech., 3(4), 274-282.
  45. Fraternali, F. and Bilotti, G. (1997), "Nonlinear elastic stress analysis in curved composite beams", Comput. Struct., 62(5), 837-859. https://doi.org/10.1016/S0045-7949(96)00301-X
  46. Ganapathi, M., Patel, B.P., Saravanan, J. and Touratier, M. (1998), "Application of spline element for large-amplitude free vibrations of laminated orthotropic straight/curved beams", Compos. Part B: Eng., 29(1), 1-8. https://doi.org/10.1016/S1359-8368(97)00025-5
  47. Ghazavi, A. and Gordaninejad, F. (1989), "Nonlinear bending of thick beams laminated from bimodular composite materials", Compos. Sci. Technol., 36(4), 289-298. https://doi.org/10.1016/0266-3538(89)90043-2
  48. Gunda, J.B. and Rao, G.V. (2013), "Post-buckling analysis of composite beams: A simple intuitive formulation", Sadhana, 38(3), 447-459. https://doi.org/10.1007/s12046-013-0144-2
  49. Gupta, R.K., Gunda, J.B., Janardhan, G.R. and Rao, G.V. (2010), "Post-buckling analysis of composite beams: simple and accurate closed-form expressions", Compos. Struct., 92(8), 1947-1956. https://doi.org/10.1016/j.compstruct.2009.12.010
  50. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  51. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Bedia, E. A.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., 140(2), 374-383.
  52. Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2016), "A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., Int. J., 22(2), 257-276.
  53. Jafari-Talookolaei, R.A., Salarieh, H. and Kargarnovin, M.H. (2011), "Analysis of large amplitude free vibrations of unsymmetrically laminated composite beams on a nonlinear elastic foundation", Acta Mechanica, 219(1), 65-75. https://doi.org/10.1007/s00707-010-0439-x
  54. Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates", Steel Compos. Struct., Int. J., 64(4), 391-402.
  55. Kocaturk, T. and Akbas, S.D. (2011), "Post-buckling analysis of Timoshenko beams with various boundary conditions under non-uniform thermal loading", Struct. Eng. Mech., Int. J., 40(3), 347-371. https://doi.org/10.12989/sem.2011.40.3.347
  56. Kocaturk, T. and Akbas, S.D. (2012), "Post-buckling analysis of Timoshenko beams made of functionally graded material under thermal loading", Struct. Eng. Mech., Int. J., 41(6), 775-789. https://doi.org/10.12989/sem.2012.41.6.775
  57. Kocaturk, T. and Akbas, S.D. (2013), "Thermal post-buckling analysis of functionally graded beams with temperaturedependent physical properties", Steel Compos. Struct., Int. J., 15(5), 481-505.
  58. Kumar, C.N. and Singh, B.N. (2009), "Thermal buckling and postbuckling of laminated composite plates with SMA fibers using layerwise theory", Int. J. Computat. Methods Eng. Sci. Mech., 10(6), 423-429. https://doi.org/10.1080/15502280903108024
  59. Kurtaran, H. (2015), "Geometrically nonlinear transient analysis of thick deep composite curved beams with generalized differential quadrature method", Compos. Struct., 128, 241-250.
  60. Latifi, M., Kharazi, M. and Ovesy, H.R. (2016), "Nonlinear dynamic response of symmetric laminated composite beams under combined in-plane and lateral loadings using full layerwise theory", Thin-Wall. Struct., 104, 62-70.
  61. Li, Z.M. and Qiao, P. (2015a), "Buckling and postbuckling behavior of shear deformable anisotropic laminated beams with initial geometric imperfections subjected to axial compression", Eng. Struct., 85, 277-292. https://doi.org/10.1016/j.engstruct.2014.12.028
  62. Li, Z.M. and Qiao, P. (2015b) "Thermal postbuckling analysis of anisotropic laminated beams with different boundary conditions resting on two-parameter elastic foundations", Eur. J. Mech.- A/Solids, 54, 30-43. https://doi.org/10.1016/j.euromechsol.2015.06.001
  63. Li, Z.M. and Yang, D.Q. (2016), "Thermal postbuckling analysis of anisotropic laminated beams with tubular cross-section based on higher-order theory", Ocean Eng., 115, 93-106. https://doi.org/10.1016/j.oceaneng.2016.02.017
  64. Liu, Y. and Shu, D.W. (2015), "Effects of edge crack on the vibration characteristics of delaminated beams", Struct. Eng. Mech., Int. J., 53(4), 767-780. https://doi.org/10.12989/sem.2015.53.4.767
  65. Loja, M.A.R., Barbosa, J.I., Soares, C.M.M. (2001), "Static and dynamic behaviour of laminated composite beams", Int. J. Struct. Stabil. Dyn., 1(4), 545-560. https://doi.org/10.1142/S0219455401000354
  66. Machado, S.P. (2007), "Geometrically non-linear approximations on stability and free vibration of composite beams", Eng. Struct., 29(12), 3567-3578. https://doi.org/10.1016/j.engstruct.2007.08.009
  67. Mahi, A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Model., 39(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  68. Malekzadeh, P. and Vosoughi, A.R. (2009), "DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges", Commun. Nonlinear Sci. Numer. Simul., 14(3), 906-915. https://doi.org/10.1016/j.cnsns.2007.10.014
  69. Mareishi, S., Rafiee, M., He, X.Q. and Liew, K.M. (2014), "Nonlinear free vibration, postbuckling and nonlinear static deflection of piezoelectric fiber-reinforced laminated composite beams", Compos. Part B: Eng., 59, 123-132. https://doi.org/10.1016/j.compositesb.2013.11.017
  70. Meziane, M.A.A., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions", J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  71. Menasria, A., Bouhadra, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017), "A new and simple HSDT for thermal stability analysis of FG sandwich plates", Steel Compos. Struct., Int. J., 25(2), 157-175.
  72. Mororo, L.A.T., Melo, A.M.C.D. and Parente Junior, E. (2015), "Geometrically nonlinear analysis of thin-walled laminated composite beams", Latin Am. J. Solids Struct., 12(11), 2094-2117. https://doi.org/10.1590/1679-78251782
  73. Oliveira, B.F. and Creus, G.J. (2003), "Nonlinear viscoelastic analysis of thin-walled beams in composite material",Thin-Wall. Struct., 41(10), 957-971. https://doi.org/10.1016/S0263-8231(03)00042-9
  74. Pagani, A. and Carrera, E. (2017), "Large-deflection and postbuckling analyses of laminated composite beams by Carrera Unified Formulation", Compos. Struct., 170, 40-52. https://doi.org/10.1016/j.compstruct.2017.03.008
  75. Pai, P.F. and Nayfeh, A.H. (1992), "A nonlinear composite beam theory", Nonlinear Dyn., 3(4), 273-303. https://doi.org/10.1007/BF00045486
  76. Patel, S.N. (2014), "Nonlinear bending analysis of c composite stiffened plates", Steel Compos. Struct., Int. J., 17(6), 867-890. https://doi.org/10.12989/scs.2014.17.6.867
  77. Patel, B.P., Ganapathi, M. and Touratier, M. (1999), "Nonlinear free flexural vibrations/post-buckling analysis of laminated orthotropic beams/columns on a two parameter elastic foundation", Compos. Struct., 46(2), 189-196. https://doi.org/10.1016/S0263-8223(99)00054-9
  78. Sheinman, I. and Adan, M. (1987), "The effect of shear deformation on post-buckling behavior of laminated beams", J. Appl. Mech., 54(3), 558-562. https://doi.org/10.1115/1.3173069
  79. Shen, H.S., Chen, X. and Huang, X.L. (2016), "Nonlinear bending and thermal postbuckling of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators", Compos. Part B: Eng., 90, 326-335. https://doi.org/10.1016/j.compositesb.2015.12.030
  80. Shen, H.S., Lin, F. and Xiang, Y. (2017), "Nonlinear bending and thermal postbuckling of functionally graded graphenereinforced composite laminated beams resting on elastic foundations", Eng. Struct., 140, 89-97. https://doi.org/10.1016/j.engstruct.2017.02.069
  81. Singh, G., Rao, G.V. and Iyengar, N.G.R. (1992), "Nonlinear bending of thin and thick unsymmetrically laminated composite beams using refined finite element model", Comput. Struct., 42(4), 471-479. https://doi.org/10.1016/0045-7949(92)90114-F
  82. Stoykov, S. and Margenov, S. (2014), "Nonlinear vibrations of 3D laminated composite beams", Math. Problems Eng.
  83. Topal, U. (2017), "Buckling load optimization of laminated composite stepped columns", Struct. Eng. Mech., Int. J., 62(1), 107-111. https://doi.org/10.12989/sem.2017.62.1.107
  84. Valido, A.J. and Cardoso, J.B. (2003), "Geometrically nonlinear composite beam structures: Design sensitivity analysis", Eng. Optimiz., 35(5), 531-551. https://doi.org/10.1080/03052150310001604784
  85. Vinson, J.R. and Sierakowski, R.L. (2002), "The behavior of Structures Composed of Composite Materials", Kluwer Academic Publishers, ISBN 978-140-2009-04-4, Netherlands.
  86. Youzera, H., Meftah, S.A., Challamel, N. and Tounsi, A. (2012), "Nonlinear damping and forced vibration analysis of laminated composite beams", Compos. Part B: Eng., 43(3), 1147-1154. https://doi.org/10.1016/j.compositesb.2012.01.008
  87. 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., Int. J., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693

Cited by

  1. Large deflection analysis of a fiber reinforced composite beam vol.27, pp.5, 2018, https://doi.org/10.12989/scs.2018.27.5.567
  2. Analytical determination of shear correction factor for Timoshenko beam model vol.29, pp.4, 2018, https://doi.org/10.12989/scs.2018.29.4.483
  3. Hygrothermal Post-Buckling Analysis of Laminated Composite Beams vol.11, pp.1, 2018, https://doi.org/10.1142/s1758825119500091
  4. Buckling analysis of sandwich beam rested on elastic foundation and subjected to varying axial in-plane loads vol.34, pp.1, 2018, https://doi.org/10.12989/scs.2020.34.1.075
  5. Buckling and stability analysis of sandwich beams subjected to varying axial loads vol.34, pp.2, 2018, https://doi.org/10.12989/scs.2020.34.2.241
  6. Dynamic analysis of a laminated composite beam under harmonic load vol.9, pp.6, 2020, https://doi.org/10.12989/csm.2020.9.6.563
  7. Monitoring and control of multiple fraction laws with ring based composite structure vol.10, pp.2, 2021, https://doi.org/10.12989/anr.2021.10.2.129
  8. Effect of suction on flow of dusty fluid along exponentially stretching cylinder vol.10, pp.3, 2018, https://doi.org/10.12989/anr.2021.10.3.263
  9. A new procedure for post-buckling analysis of plane trusses using genetic algorithm vol.40, pp.6, 2018, https://doi.org/10.12989/scs.2021.40.6.817