Document Type : Research Paper

Author

Department of Civil Enginnering, Enugu State, University of Science And Technology - Nigeria, Agbani, Enugu State, Nigeria.

Abstract

This paper presents analytical solutions for the buckling of thick beams. The Bernoulli-Euler beam theory (BEBT) overestimates their critical buckling load. This paper has derived a cubic polynomial shear deformation beam buckling theory (CPSDBBT) from first principles using the Euler-Lagrange differential equation (ELDE). It develops closed-form solutions to differential equations using the finite sine transform method. The formulation considers transverse shear deformation and satisfies the transverse shear stress-free boundary conditions. The governing equation is developed from the energy functional, Õ, by applying the ELDE. The domain equation is obtained as an ordinary differential equation (ODE). The finite sine transformation of the ODE transforms the thick beam, which is considered an algebraic eigenvalue problem. The solution gives the buckling load Nxx at any buckling mode n. The critical buckling load Nxx cr occurs at the first buckling mode and is presented in depth ratios to span (h/l). It is found that and agrees with previous solutions using shear deformable theories. For (a moderately thick beam), the Nxx cr is 2.50% lower than the value predicted using BEBT, confirming the overestimation by BEBT. The Nxx cr agrees with previous solutions, implying the shear deformation has been adequately accounted for, and the BEBT overestimates the Nxx cr. The value of Nxx cr found agrees with previous values in the literature.

Graphical Abstract

Highlights

• Closed form solutions are derived for the buckling of thick beams under in-plane loading.
• The buckling peoblem is derived using cubic polynomial shear deformation theory.
• For thin beams ( ), Nxx cr has a negligible difference from the BEBT results.
•  For  (thick beam), Nxx cr obtained is 13.44% lower than the value predicted by BEBT.

Keywords

Main Subjects

###### References
1. S .P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag., 41 (1921) 742-746. https://doi.org/10.1080/14786442108636264
2. P. Timoshenko, G.M. Gere Theory of elastic stability 2nd Edition, McGraw Hill Book Company, New York,1961.
3. C. Ike Timoshenko beam theory for the flexural analysis of moderately thick beams-variational formulation and closed-form solutions, Ital. J. Eng. Sci., 63 (2019) 34-45. http://dx.doi.org/10.18280/ti-ijes.630105
4. R. Cowper On the accuracy of Timoshenko beam theory, ASCE J. Eng. Mech., 94 (1968) 1447-1453.
5. M. Ghugal, A single variable parabolic shear deformation theory for flexure and flexural vibration of thick isotropic beams, Proc. Third Intl Conf. Struct. Eng., Mech. Comput., Cape Town, South Africa, 2007, 77-78.
6. M. Ghugal R. Sharmam, A hyperbolic shear deformation theory for flexure and vibration of thick isotropic beams, Int. J. Comput Methods, 6 (2009) 585-604. https://doi.org/10.1142/S0219876209002017
7. S. Sayyad, Y.M. Ghugal, Flexure of thick beams using hyperbolic shear deformation theory, Int. J. Mech. 5 (2011) 111-122.
8. C. Ike, Fourier series method for finding displacements and stress fields in hyperbolic shear deformable thick beams subjected to distributed transverse loads, J. Comput. Appl. Mech., 53 (2022) 126-141. https://doi.org/10.22059/jcamech.2022.332719.658
9. M. Ghugal, R.P. Shimpi, A Trigonometric shear deformation theory for flexure and free vibration of thick isotropic beams. Proc. Struct. Eng. Convention (SEC2000), IT Bombay, Mumbai, India, 2000, 255-263,
10. V. Krishna Murty, Towards a consistent beam theory, AIAA J., 22 (1984) 811-816.
11. Levinson, A new rectangular beam theory, J. Sound Vib., 74 (1984) 81-87. https://doi.org/10.1016/0022-460X(81)90493-4
12. M. Ghugal, A new refined bending theory for thick beam including transverse shear and transverse normal strain effects. Departmental Report, Applied Mechanics Department, Government College of Engineering, Avrangabad, India, 2006, 1 – 96.
13. S. Sayyad, Comparison of various refined beam theories for the bending and free vibration analysis of thick beams, Appl. Comput. Mech., 5 (2011) 217-230.
14. M. Ghugal, R.P. Shimpi, A review of refined shear deformation theories for isotropic and anisotropic beams, J. Reinf. Plast. Compos., 20 (2001) 255-272. https://doi.org/10.1177/073168401772678283
15. P. Shimpi, P.J. Gurprasad, K.S. Pakhere, Simple two variable refined theory for shear deformable isotropic rectangular beams, J. Appl. Comput. Mech., 6 (2019) 394-415. https://doi.org/10.22055/jacm.2019.29555.1615
16. S. Sayyad, Y.M. Ghugal, Single variable refined beam theories for the bending, buckling and free vibration of homogeneous beams, Appl. Comput. Mech. 10 (2016) 123-138
17. A. Oguaghamba, C.C. Ike, E.U. Ikwueze, I.O. Ofondu, Finite Fourier sine integral transform method for the elastic buckling analysis of doubly symmetric thin-walled beams with Dirichlet boundary conditions, ARPN J. Eng. Appl. Sci., 14 (2019) 3968-3974.
18. A. Oguaghamba, C.C. Ike, E.U. Ikwueze, I.O. Ofondu, Ritz variational method for solving the elastic buckling problems of thin-walled beams with bisymmetric cross-sections, Math. Model. Eng. Probl., 10 (2023) 129-139. https://doi.org/10.18280/mmep.100114
19. N. Onah, C.U. Nwoji, M.E. Onyia, B.O. Mama, C.C. Ike. Exact solutions for the elastic buckling problem of moderately thick beam, J. Compos. Adv. Mater., 30 (2020) 83-93. https://doi.org/10.18280/rcma.300205
20. O. Mama, O.A. Oguaghamba, C.C. Ike, Finite Fourier sine transform method for finding natural frequencies of flexural vibration of moderately thick beams, Proc. 2020 Sustainable Eng. Ind. Tech. Conf., 2020, A5-1 – A5-5.
21. R. Heyliger, J.N. Reddy, A higher order beam finite element for bending and variation problems, J. Sound Vib. 126 (1988) 309-326. https://doi.org/10.1016/0022-460X(88)90244-1
22. Karama, K.S. Afaq, S. Mistou, Mechanical behaviour of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity, Int. J. Solids Struct., 40 (2003) 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9