Analysis of Deep to Moderately Deep Beams Using Timoshenko Theory and 2D Plane-Stress Elasticity
- 1 School of Mechanical Engineering, National Technical University of Athens, Athens, Greece
Abstract
This study examines rectangular cantilever beams, clamped at one end and subjected to a transverse tip load at the other, with aspect ratios L / h = 1 ÷ 5 , a range in which shear deformation and stress field nonuniformities have a pronounced influence on structural behavior. A plane-stress finite element formulation is employed to capture the full two-dimensional elastic response, and the results are systematically compared with closed-form solutions derived from Timoshenko beam theory. The comparison highlights the limitations of classical beam assumptions within this aspect ratio range. Based on numerical evidence, approximate expressions for the maximum deflection under end loading are proposed for selected values of Poisson’s ratio, offering improved accuracy for moderately deep and deep beams. In addition, a MATLAB® code is provided for estimating the maximum deflection for arbitrary values of Poisson’s ratio.
- Elishakoff, I. (2019) Who Developed the So-Called Timoshenko Beam Theory? Mathematics and Mechanics of Solids , 25, 97-116. https://doi.org/10.1177/1081286519856931
- Timoshenko, S.P. (1921) LXVI. On the Correction for Shear of the Differential Equatio n for Transverse Vibrations of Prismatic Bars . The London , Edinburgh , and Du blin Philosophical Magazine and Journal of Science , 41, 744-746. https://doi.org/10.1080/14786442108636264
- Timoshenko, S.P. (1922) X. On the Transverse Vibrations of Bars of Uniform Cross-section . The London , Edinburgh , and Dublin Philosophical Magazine and Journal of Science , 43, 125-131. https://doi.org/10.1080/14786442208633855
- Xia, G. (2022) Generalized Foundation Timoshenko Beam and Its Calculating Methods. Archive of Applied Mechanics , 92, 1015-1036. https://doi.org/10.1007/s00419-021-02090-1
- Öchsner, A. (2021) Classical Beam Theories of Structural Mechanics. Springer. https://doi.org/10.1007/978-3-030-76035-9
- Conway, H.D., Chow, L. and Morgan, G.W. (1951) Analysis of Deep Beams. Journal of Applied Mechanics , 18, 163-172. https://doi.org/10.1115/1.4010271
- Chow, L., Conway, H.D. and Winter, G. (1953) Stresses in Deep Beams. Transactions of the American Society of Civil Engineers , 118, 686-702. https://doi.org/10.1061/taceat.0006784
- Theocaris, P.S. (1959) The Stress Distribution in a Semi-Infinite Strip Subjected to a Concentrated Load. Journal of Applied Mechanics , 26, 401-406. https://doi.org/10.1115/1.4012052
- Theocaris, P.S. (1964) The Method of Isostatics Applied to Rectangular Bars with Uniform Loading. International Journal of Engineering Science , 2, 1-19. https://doi.org/10.1016/0020-7225(64)90007-2
- Benthem, J.P. (1963) A Laplace Transform Method for the Solution of Semi-Infinite and Finite Strip Problems in Stress Analysis. The Quarterly Journal of Mechanics and Applied Mathematics , 16, 413-429. https://doi.org/10.1093/qjmam/16.4.413
- Horvay, G. and Born, J.S. (1957) Some Mixed Boundary-Value Problems of the Semi-Infinite Strip. Journal of Applied Mechanics , 24, 261-268. https://doi.org/10.1115/1.4011507
- Johnson, M.W. and Little, R.W. (1965) The Semi-Infinite Elastic Strip. Quarterly of Applied Mathematics , 22, 335-344. https://doi.org/10.1090/qam/187479
- Cowper, G.R. (1966) The Shear Coefficient in Timoshenko’s Beam Theory. Journal of Applied Mechanics , 33, 335-340. https://doi.org/10.1115/1.3625046