An analytical solution for stability analysis of unrestrained tapered thin-walled FML profile

Document Type : Research


1 Assistant Professor, Department of civil engineering, University of Kashan, Kashan, Iran.

2 MSc Student in Structural Engineering, Department of civil engineering, University of Kashan, Kashan, Iran.


The main purpose of this study is to compare the lateral buckling behavior of laterally unrestrained Fiber-Metal Laminate (FML) and composite thin-walled beam with varying cross-section under transverse loading. It is supposed that all section walls (the web and both flanges) are composed of two metal layers at the outer sides of the fiber-reinforced polymer laminates. The classical lamination theory and Vlasov’s model for thin-walled cross-section have been adopted to derive the coupled governing differential equations for the lateral deflection and twist angle. Employing an auxiliary function, the two governing equations are reduced to a single fourth-order differential equation in terms of twist angle. To estimate the lateral buckling load, Galerkin’s method is then applied to the resulting torsion equilibrium equation. Eventually, the lateral stability resistance of FML and laminated composite web-tapered I-beam under uniformly distributed load has been compared to each other considering the effects of some significant parameters such as laminate stacking sequence, metal volume fraction, transverse load position, and web tapering ratio. The results show that increasing the metal volume fraction leads to enhance the linear buckling strength of glass-reinforced aluminum laminate I-beam under transverse loading. For the optimum lamination, it is seen that the lateral buckling load increases approximately 25% by raising the metal volume percentage from 0% to 20%.


1. Rajasekaran, S. (1994). Instability of tapered thin-walled beams of generic section. Journal of engineering mechanics, 120(8), 1630-1640. [DOI:10.1061/(ASCE)0733-9399(1994)120:8(1630)]
2. Nam, H. W., Hwang, W., & Han, K. S. (2001). Stacking sequence design of fiber-metal laminate for maximum strength. Journal of Composite Materials, 35(18), 1654-1683. [DOI:10.1106/7NV4-5J5R-XIUJ-PVXT]
3. Lee, J., Kim, S. E., & Hong, K. (2002). Lateral buckling of I-section composite beams. Engineering Structures, 24(7), 955-964. [DOI:10.1016/S0141-0296(02)00016-0]
4. Lee, J., & Kim, S. E. (2002). Free vibration of thin-walled composite beams with I-shaped cross-sections. Composite structures, 55(2), 205-215. [DOI:10.1016/S0263-8223(01)00150-7]
5. Lee, J. (2005). Flexural analysis of thin-walled composite beams using shear-deformable beam theory. Composite Structures, 70(2), 212-222. [DOI:10.1016/j.compstruct.2004.08.023]
6. Vo, T. P., & Lee, J. (2007). Flexural-torsional buckling of thin-walled composite box beams. Thin-walled structures, 45(9), 790-798. [DOI:10.1016/j.tws.2007.06.001]
7. Oh, S. Y., Librescu, L., & Song, O. (2005). Vibration and instability of functionally graded circular cylindrical spinning thin-walled beams. Journal of sound and vibration, 285(4-5), 1071-1091. [DOI:10.1016/j.jsv.2004.09.018]
8. Rajasekaran, S., & Nalinaa, K. (2005). Stability and vibration analysis of non-prismatic thin-walled composite spatial members of generic section. International Journal of Structural Stability and Dynamics, 5(04), 489-520. [DOI:10.1142/S0219455405001714]
9. Magnucka-Blandzi, E. (2009). Critical state of a thin-walled beam under combined load. Applied mathematical modelling, 33(7), 3093-3098. [DOI:10.1016/j.apm.2008.10.014]
10. Vo, T. P., & Lee, J. (2009). Flexural-torsional coupled vibration and buckling of thin-walled open section composite beams using shear-deformable beam theory. International Journal of Mechanical Sciences, 51(9-10), 631-641. [DOI:10.1016/j.ijmecsci.2009.05.001]
11. Moon, C. J., Kim, I. H., Choi, B. H., Kweon, J. H., & Choi, J. H. (2010). Buckling of filament-wound composite cylinders subjected to hydrostatic pressure for underwater vehicle applications. Composite Structures, 92(9), 2241-2251. [DOI:10.1016/j.compstruct.2009.08.005]
12. Ascione, L., Berardi, V. P., Giordano, A., & Spadea, S. (2013). Local buckling behavior of FRP thin-walled beams: a mechanical model. Composite Structures, 98, 111-120. [DOI:10.1016/j.compstruct.2012.10.049]
13. Araújo, A. L., Carvalho, V. S., Soares, C. M., Belinha, J., & Ferreira, A. J. M. (2016). Vibration analysis of laminated soft core sandwich plates with piezoelectric sensors and actuators. Composite Structures, 151, 91-98. [DOI:10.1016/j.compstruct.2016.03.013]
14. Ravishankar, H., Rengarajan, R., Devarajan, K., & Kaimal, B. (2016). Free vibration bahaviour of fiber metal laminates, hybrid composites, and functionally graded beams using finite element analysis. International Journal of Acoustics and Vibration, 21(4), 418-428. [DOI:10.20855/ijav.2016.21.4436]
15. Banat, D., Kolakowski, Z., & Mania, R. J. (2016). Investigations of FML profile buckling and post-buckling behaviour under axial compression. Thin-Walled Structures, 107, 335-344. [DOI:10.1016/j.tws.2016.06.018]
16. Banat, D., & Mania, R. J. (2017). Failure assessment of thin-walled FML profiles during buckling and postbuckling response. Composites Part B: Engineering, 112, 278-289. [DOI:10.1016/j.compositesb.2017.01.001]
17. Banat, D., & Mania, R. J. (2018). Progressive failure analysis of thin-walled Fibre Metal Laminate columns subjected to axial compression. Thin-Walled Structures, 122, 52-63. [DOI:10.1016/j.tws.2017.09.034]
18. Dhaliwal, G. S., & Newaz, G. M. (2017). Compression after impact characteristics of carbon fiber reinforced aluminum laminates. Composite Structures, 160, 1212-1224. [DOI:10.1016/j.compstruct.2016.11.015]
19. Lezgy-Nazargah, M. (2017). A generalized layered global-local beam theory for elasto-plastic analysis of thin-walled members. Thin-Walled Structures, 115, 48-57. [DOI:10.1016/j.tws.2017.02.004]
20. Mohandes, M., & Ghasemi, A. R. (2017). Modified couple stress theory and finite strain assumption for nonlinear free vibration and bending of micro/nanolaminated composite Euler-Bernoulli beam under thermal loading. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231(21), 4044-4056. [DOI:10.1177/0954406216656884]
21. Mohandes, M., Ghasemi, A. R., Irani-Rahagi, M., Torabi, K., & Taheri-Behrooz, F. (2018). Development of beam modal function for free vibration analysis of FML circular cylindrical shells. Journal of Vibration and Control, 24(14), 3026-3035. [DOI:10.1177/1077546317698619]
22. Ahmadi, H., & Rasheed, H. A. (2018). Lateral torsional buckling of anisotropic laminated thin-walled simply supported beams subjected to mid-span concentrated load. Composite Structures, 185, 348-361. [DOI:10.1016/j.compstruct.2017.11.027]
23. Wackerfuß, J., & Kroker, A. M. (2018). An efficient semi-analytical simulation framework to analyse laminated prismatic thin-walled beams. Computers & Structures, 208, 32-50. [DOI:10.1016/j.compstruc.2018.06.010]
24. Asadi, A., Sheikh, A. H., & Thomsen, O. T. (2019). Buckling behaviour of thin-walled laminated composite beams having open and closed sections subjected to axial and end moment loading. Thin-Walled Structures, 141, 85-96. [DOI:10.1016/j.tws.2019.04.005]
25. Ghasemi, A. R., & Mohandes, M. (2019). Comparison between the frequencies of FML and composite cylindrical shells using beam modal function model. Journal of Computational Applied Mechanics, 50(2), 239-245.
26. Soltani, M. (2020). Flexural-torsional stability of sandwich tapered I-beams with a functionally graded porous core. International Journal of Numerical Methods in Civil Engineering, 4(3), 8-20. [DOI:10.52547/nmce.4.3.8]
27. Lezgy-Nazargah, M. (2014). An isogeometric approach for the analysis of composite steel-concrete beams. Thin-Walled Structures, 84, 406-415. [DOI:10.1016/j.tws.2014.07.014]
28. Lezgy-Nazargah, M., & Kafi, L. (2015). Analysis of composite steel-concrete beams using a refined high-order beam theory. Steel and Composite Structures, 18(6), 1353-1368. [DOI:10.12989/scs.2015.18.6.1353]
29. Lezgy-Nazargah, M., Vidal, P., & Polit, O. (2019). A sinus shear deformation model for static analysis of composite steel-concrete beams and twin-girder decks including shear lag and interfacial slip effects. Thin-Walled Structures, 134, 61-70. [DOI:10.1016/j.tws.2018.10.001]
30. Lezgy-Nazargah, M. (2020). A finite element model for static analysis of curved thin-walled beams based on the concept of equivalent layered composite cross section. Mechanics of Advanced Materials and Structures, 1-14. [DOI:10.1080/15376494.2020.1804649]
31. Lezgy-Nazargah, M., Vidal, P., & Polit, O. (2021). A quasi-3D finite element model for the analysis of thin-walled beams under axial-flexural-torsional loads. Thin-Walled Structures, 164, 107811. [DOI:10.1016/j.tws.2021.107811]
32. Einafshar, N., Lezgy-Nazargah, M., & Beheshti-Aval, S. B. (2021). Buckling, post-buckling and geometrically nonlinear analysis of thin-walled beams using a hypothetical layered composite cross-sectional model. Acta Mechanica, 1-18. [DOI:10.1007/s00707-021-02936-3]
33. Vlasov, V. Z. (1959). Thin-walled elastic beams. PST Catalogue, 428.
34. Asgarian, B., Soltani, M., & Mohri, F. (2013). Lateral-torsional buckling of tapered thin-walled beams with arbitrary cross-sections. Thin-walled structures, 62, 96-108. [DOI:10.1016/j.tws.2012.06.007]
35. Soltani, M., Asgarian, B., & Mohri, F. (2019). Improved finite element model for lateral stability analysis of axially functionally graded nonprismatic I-beams. International Journal of Structural Stability and Dynamics, 19(09), 1950108. [DOI:10.1142/S0219455419501086]
36. Soltani, M., & Asgarian, B. (2020). Lateral-Torsional Stability Analysis of a Simply Supported Axially Functionally Graded Beam with a Tapered I-Section. Mechanics of Composite Materials, 1-16. [DOI:10.1007/s11029-020-09859-5]
37. Benyamina, A.B., Meftah, S.A., Mohri, F.: Analytical solutions attempt for lateral torsional buckling of doubly symmetric web-tapered I-beams. Engineering structures. 56, 1207-1219 (2013). [DOI:10.1016/j.engstruct.2013.06.036]
38. Raftoyiannis, I.G. & Adamakos, T. (2010). Critical lateral-torsional buckling moments of steel web-tapered I-beams. The Open Construction and Building Technology Journal, 4(1). [DOI:10.2174/1874836801004010105]
39. Osmani, A., & Meftah, S. A. (2018). Lateral buckling of tapered thin walled bi-symmetric beams under combined axial and bending loads with shear deformations allowed. Engineering Structures, 165, 76-87. [DOI:10.1016/j.engstruct.2018.03.009]
40. Soltani, M., & Sistani, A. (2017). Elastic stability of columns with variable flexural rigidity under arbitrary axial load using the finite difference method. International Journal of Numerical Methods in Civil Engineering, 1(4), 23-31. [DOI:10.29252/nmce.1.4.23]
41. Soltani, M., Asil Gharebaghi, S., & Mohri, F. (2018). Lateral stability analysis of steel tapered thin-walled beams under various boundary conditions. International Journal of Numerical Methods in Civil Engineering, 3(1), 13-25. [DOI:10.29252/nmce.3.1.13]
42. Soltani, M., & Mohammadi, M. (2018). Stability Analysis of Non-Local Euler-Bernoulli Beam with Exponentially Varying Cross-Section Resting on Winkler-Pasternak Foundation. International Journal of Numerical Methods in Civil Engineering, 2(3), 67-77. [DOI:10.29252/nmce.2.3.67]
43. Soltani, M., & Gholamizadeh, A. (2018). Size-dependent buckling analysis of non-prismatic Timoshenko nanobeams made of FGMs rested on Winkler foundation. International Journal of Numerical Methods in Civil Engineering, 3(2), 35-46. [DOI:10.29252/nmce.3.2.35]
44. Soltani, M., & Mohri, F. (2016). Stability and vibration analyses of tapered columns resting on one or two-parameter elastic foundations. International Journal of Numerical Methods in Civil Engineering, 1(2), 57-66. [DOI:10.29252/nmce.1.2.57]
45. Soltani, M. (2017). Vibration characteristics of axially loaded tapered Timoshenko beams made of functionally graded materials by the power series method. International Journal of Numerical Methods in Civil Engineering, 2(1), 1-14. [DOI:10.29252/nmce.2.1.1]
46. Soltani, M. (2020). Finite element modelling for buckling analysis of tapered axially functionally graded Timoshenko beam on elastic foundation. Mechanics of Advanced Composite Structures‎.
47. ANSYS, Version 5.4, Swanson Analysis System, Inc, 2007.