Buckling Analysis of Rectangular Beams Having Ceramic Liners at Its Top and Bottom Surfaces with the help of the Exact Transfer Matrix

Abstract

In this study the elastic buckling behavior of beams with rectangular cross section is studied analytically. It is assumed that both the top and bottom surfaces of the beam are ceramic coated. The aluminum (Al) is chosen as a core material while the aluminum-oxide (Al2O3) is preferred as a liner (face) material. The transfer matrix method based on the Euler-Bernoulli beam theory is employed in the analysis. The exact transfer matrix in terms of equivalent bending stiffness is presented together with the exact buckling equations for hinged-hinged, clamped-hinged, clamped-free, and finally clamped-clamped boundary conditions. After verifying the results for beams without liners, dimensionless buckling loads of the beam with ceramic liners are numerically computed for each boundary condition. The effect of the thickness of the ceramic liner on the buckling loads is also investigated. It is found that a ceramic liner enhances noticeably the buckling loads. As an additional study those effects are also examined for the ratios of elasticity modulus of face material to core material in a wide range.

Keywords

• Exact buckling
• Transfer matrix
• Euler-Bernoulli beam
• Stability
• Sandwich beam
• Critical buckling loads

References

1. Euler, L., Die altitudine colomnarum sub proprio pondere corruentium, Acta Acad Petropol, 1778.
2. Greenhill, A.G., Determination of the greatest height consistent with stability that a vertical pole on mast can be made and of the greatest height to which a tree of given proportions can grow in, Proceedings of the Cambridge Philosophical Society, IV, 1883.
3. Dinnik, A.N., Design of columns of varying cross-section, Transactions of the ASME, 51, 1929.
4. Timoshenko S.P., Gere J.M., Theory of Elastic Stability, McGraw-Hill Companies, 2nd edition, New York; 1961. İnan, M., The Method of Initial Values and the Carry-Over Matrix in Elastomechanics, ODTÜ Publication, No: 20; 1968.
5. Chen W.F., Lui E.M., Structural Stability, Theory and Implementation, Elsevier, New York; 1987.
6. Wang C.M., Wang C.Y., Reddy J.N., Exact Solutions for Buckling of Structural Members, CRC Press, Boca Raton, Fla, USA; 2005.
7. Yoo C.H., Lee S.C., Stability of Structures - Principles and Applications, Butterworth Heinemann, New York; 2011.
8. Smith, C.S., Application of folded plate analysis to bending, buckling and vibration of multilayer orthotropic sandwich beams and panels, Comput Struct, 22(3), 491–497, 1986.

9. Kardomateas, G.A., Three dimensional elasticity solution for the buckling of transversely isotropic rods: the Euler load revisited, ASME J Appl Mech, 62, 346–55, 1995.
10. D’Ottavio, M., Polit, O., Ji, W., Waas, A.M., Benchmark solutions and assessment of variable kinematics models for global and local buckling of sandwich struts, Compos Struct, 156, 125–134,2016.
11. Pi, Y.L., Trahair, N.S., Rajasekaran, S., Energy equation for beam lateral buckling, Journal of Structural Engineering, ASCE, 118(6), 1462-1479, 1992.
12. Wang, C.M., Wang, L., Ang, K.K., Beam-buckling analysis via automated Rayleigh-Ritz method, Journal of Structural Engineering, ASCE, 120(1), 200-211, 1994.
13. Tong, G., Zhang, L., A general theory for the flexural-torsional buckling of thin-walled members I: Energy method, Advances in Structural Engineering, 6(4), 293-298, 2003.
14. Zdravković, N., Gašić, K.M., Savković, K.M., Energy method in efficient estimation of elastic buckling critical load of axially loaded three-segment stepped column, FME Transactions, 41(3), 222-229, 2013.
15. Kundu, B., Ganguli, R., Analysis of weak solution of Euler–Bernoulli beam with axial force, Applied Mathematics and Computation, 298, 247-260, 2017.
16. Barsoum, R.S., Gallagher, R.H., Finite element analysis of torsional and torsional-flexural stability problems, Int J Num Meth Engng, 2, 335-352, 1970.
17. Kabaila, A.P., Fraeijs de Veubeke, B., Stability analysis by finite elements, Air Force Flight Dynamics Laboratory, Tech. Report AFFDL-TR-70-35, 1970. Thomas, J.M., A finite element approach to the structural instability of beam columns, frames and arches, NASA TN D-5782, 1970.
18. He, B., Zhang, H., Stability analysis of slope based on finite element method, I J Engineering and Manufacturing, MECS, 3, 70-74, 2012.
19. Li, J.J., Li, G.Q., Buckling analysis of tapered lattice columns using a generalized finite element, Commun Numer Meth Engng, 20(6), 479–488, 2004.
20. Trahair, N.S., Rasmussen, K.J.R., Finite-element analysis of the flexural buckling of columns with oblique restraints, J Struct Eng, 131(3), 481-487, 2005.
21. Saha, G., Banu, S., Buckling load of a beam-column for different end conditions using multi-segment integration technique, ARPN Journal of Engineering and Applied Sciences, 2(1), 27-32, 2007.
22. Galambos, T.V., Structural Members and Frames. Prentice-Hall International Inc,, New York, N.Y., 1968.
23. López-Aenlle, M., Pelayo, F., Ismael, G., García Prieto, M.A., Fernández-Canteli, A., Buckling of laminated-glass beams using the effective-thickness concept, Composite Structures, 137, 44-55, 2016.
24. He, J.H., Variational iteration method: a kind of nonlinear analytical technique, Int J Nonlin Mech, 34, 699-708, 1999.
25. Coşkun, S.B., Atay, M.T., Determination of critical buckling load for elastic columns of constant and variable cross-sections using variational iteration method, Computers and Mathematics with Applications, 58, 2260–2266, 2009.
26. Coşkun, S.B., Öztürk, B., Advances in Computational Stability Analysis: (Chapter 6) Elastic stability analysis of Euler columns using analytical approximate techniques, 115-132. Edited by Safa Bozkurt Coşkun, InTech ISBN No: 978-953-51-0673-9. DOI: 10.5772/45940, 2012.
27. He, J.H., The homotopy perturbation method for solving boundary problems, Phys Lett A, 350, 87-88, 2006.
28. He, J.H., An elementary introduction to the homotopy perturbation method, Computers and Mathematics with Applications, 57, 410-412, 2009.
29. Coşkun, S.B., Determination of critical buckling load for Euler columns of variable flexural stiffness with a continuous elastic restraint using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 187-193, 2009.
30. Coşkun, S.B., Analysis of tilt-buckling of Euler columns with varying flexural stiffness using homotopy perturbation method, Mathematical Modelling and Analysis, 15(3), 275-286, 2010.
31. Eryılmaz, A., Atay, M.T., Coşkun, S.B., Başbük, M., Buckling of Euler columns with a continuous elastic restraint via homotopy analysis method, Journal of Applied Mathematics, Article ID 341063 (8 pages), 2013.
32. Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, MA, 1994.
33. Yildirim, V., Numerical buckling analysis of cylindrical helical coil springs in a dynamic manner, International Journal of Engineering & Applied Sciences, 1, 20-32, 2009.
34. Yildirim, V., On the linearized disturbance dynamic equations for buckling and free vibration of cylindrical helical coil springs under combined compression and torsion, Meccanica, 47, 1015-1033, 2012.
35. Yildirim, V., Axial static load dependence free vibration analysis of helical springs based on the theory of spatially curved bars, Latin American Journal of Solids and Structures, 13, 2552-2575, 2016.
36. Banerjee, J.R., Exact Bernoulli–Euler static stiffness matrix for a range of tapered beam-columns, International Journal for Numerical Methods in Engineering, 23(9), 1615–1628, 1986.
37. Kacar, İ., Yildirim, V., Free vibration/buckling analyses of noncylindrical initially compressed helical composite springs, Mechanics Based Design of Structures and Machines, 44, 340-353, 2016.
38. Tong, G., Zhang, L., A general theory for the flexural-torsional buckling of thin-walled members II: Fictitious load method, Advances in Structural Engineering, 6(4), 299-308, 2003.
39. Rahai, A.R., Kazemi, S., Buckling analysis of non-prismatic columns based on modified vibration modes, Communications in Nonlinear Science and Numerical Simulation, 13(8), 1721-1735, 2008.
40. Ding, Y., Hou, J., General buckling analysis of sandwich constructions, Comput Struct, 55(3), 485-493, 1995.
41. Chakrabartia, A., Chalaka, H.D., Iqbala, M.A., Sheikhb, A.H., Buckling analysis of laminated sandwich beam with soft core, Latin American Journal of Solids and Structures, 9, 367–381, 2012.
42. Magnucka-Blandzi, E., Magnucki, K., Effective design of sandwich beam with a metal foam core, Thin-Walled Struct, 45, 432-438, 2007.
43. Jasion, P., Magnucki, K., Global buckling of sandwich column with metal foam core, J Sandw Struct Mater, 15(6), 718-732, 2013.
44. Douville, M.A., Grognec, P.L., Exact analysis solutions for the local and global buckling of sandwich beam-columns under various loadings, Int J Solids Struct, 50, 2597-2609, 2013.
45. Galuppi, L.G., Carfagni, R., Buckling of three-layered composite beams with viscoelastic interaction, Composite Structures, 107, 512-521, 2014.
46. Grygorowicz, M., Magnucki, K., Malinowski, M., Elastic buckling of a sandwich beam with variable mechanical properties of the core, Thin-Walled Structures, 2015, 87, 127-132.
47. Sayyad, A.S., Ghugal, Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams, A critical review of literature, Composite Structures, 171, 486-504, 2017.