Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 841017, 17 pages
Research Article

Robust Optimal Design of Beams Subject to Uncertain Loads

1Department of Mathematics and Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
2School of Mechanical Engineering, University of KwaZulu-Natal, Durban 4041, South Africa

Received 8 February 2009; Revised 29 June 2009; Accepted 15 October 2009

Academic Editor: Jerzy Warminski

Copyright © 2009 Ismail Kucuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Optimality conditions are derived for the robust optimal design of beams subject to a combination of uncertain and deterministic transverse and boundary loads using a variational min-max approach. The potential energy of the beam is maximized to compute the worst case loading and minimized to determine the optimal cross-sectional shape which results in coupled nonlinear differential equations for the unknown functions except for the case of a variable width beam. The uncertain component of the transverse load acting on the beam is not known a priori resulting in load uncertainty subject only to an norm constraint. Similarly the optimal area function is subject to a volume constraint leading to an isoperimetric variational problem. The min-max approach leads to robust optimal designs which are not susceptible to unexpected load variations as it occurs under operational conditions. The solution methodology is illustrated for the variable width beam by obtaining analytical results for several cases. The efficiency of the optimal designs is computed with respect to a uniform beam under worst case loading taking the maximum deflection as the quantity for comparison. It is observed that the optimal shapes are more than 70% efficient for the examples given in this study.