Minimum weight design of beams against failure under uncertain loading by convex analysis

Under operational conditions, some loads acting on a beam are known (deterministic loads), but there usually exist other loads the magnitude and distribution of which are unpredictable (uncertain loads). If the uncertainty in the loading is not taken into account in the design, the likelihood of failure increases. In the present study beams are designed for minimum weight subject to maximum stress and buckling load criteria and under deterministic and uncertain transverse loads. The uncertain load, which is subject to a constraint on its L ₂ norm, is determined to maximize the normal stress using a convex analysis. The location of the maximum stress is determined under the combination of deterministic and worst-case uncertain loads. The minimum weight design is obtained by determining the minimum cross-sectional area subject to stress and buckling load constraints. Results are given for a number of problem parameters including the axial load, elastic foundation modulus and uncertainty levels.