Abstract
This paper deals with the development of numerical methods for determining the strongest cantilever beam having a constant volume of the beam material. An analytical method for calculating the properties of the cross section shaped as a solid regular polygon is formulated. For carrying out structural analyses of the cantilever beam, the stress resultants are computed and the differential equation of the elastic curve is derived, in which the effect of shear deformation as well as bending moment is included. The optimal geometries of the strongest beam are determined from the results of the structural analysis. Extreme stress and deflection are chosen as the decision criteria for determining the strongest beam. By using the geometries of the strongest beam obtained in this study, a design example of the minimum weight beam, which can sustain the subjected load with the minimum volume of the beam material, is presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carnahan, B., Luther, H. A., and Wilkes, J. O. (1969). Applied numerical methods. John Wiley & Sons, NY, USA.
Cox, S. J. and Overton, M. I. (1992). “On the optimal design of columns against buckling.” SIAM Journal on Mathematical Analysis, Vol. 23, pp. 287–325.
Gere, J. M. and Timoshenko, S. P. (1997). Mechanics of materials. PWS Publishing Company, USA.
Haftka, R. T., Grdal, Z., and Kamat, M. P. (1990). Elements of structural optimization. Klwer Academic Publisher, Dordrecht, Netherlands.
Keller, J. B. (1960). “The shape of the strongest column.” Archive for Rational Mechanics and Analysis, Vol. 5, No. 1, pp. 275–285.
Keller, J. B. and Niordson, F. I. (1966). “The tallest column.” Journal of Mathematics and Mechanics, Vol. 16, pp. 433–446.
Lee, B. K., Carr, A. J., Lee, T. E., and Kim, I. J. (2006). “Buckling loads of columns with constant volume.” Journal of Sound and Vibrations. Vol. 294, Nos. 1 and 2, pp. 381–387.
Lee, B. K., Lee, T. E., and Shin, S. C. (2009a). “Strongest beams having constant volume supported by clamped-clamped and clampedhinged ends.” Journal of the Korean Society of Civil Engineers, Vol. 29, No. 3A, pp. 251–258 (in Korean).
Lee, B. K. and Oh, S. J. (2000). “Elastica and buckling loads of simple tapered columns with constant volume.” International Journal of Solids and Structures, Vol. 37,Issue 18, pp. 2507–2518.
Lee, B. K., Oh, S. J., and Lee, T. E. (2009b). “Strongest static arches with constant volume.” Journal of the Korean Society of Civil Engineers, Vol. 29, No. 5A, pp. 477–486 (in Korean).
Lee, B. K., Oh, S. J., Lee, T. E., and Park, J. S. (2011). “Minimum weight beams with shear stain energy.” KSCE Journal of Civil Engineering, Vol. 16, No. 1, pp. 145–154.
Taylor, J. E. (1967). “The strongest column — Energy approach.” Journal of Applied Mechanics, ASME, Vol. 34, No. 2, pp. 486–487.
Wilson, J. F., Holloway, D. M., and Biggers, S. B. (1971). “Stability experiments on the strongest columns and circular arches.” Experimental Mechanics, Vol. 11, No. 4, pp. 303–308.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lee, B.K., Lee, T.E. & Jung, Y.S. Numerical methods for determining strongest cantilever beam with constant volume. KSCE J Civ Eng 16, 169–178 (2012). https://doi.org/10.1007/s12205-012-1383-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-012-1383-1