Optimum design of thermally loaded beam-columns for maximum vibration frequency or buckling temperature

With the cross-sectional area function as the design variable, we seek the optimum design of clamped-clamped, thin, linearly elastic beam-columns under thermal load that maximizes the buckling temperature or the fundamental natural frequency of transverse vibrations. The beam-columns have given length, geometrically similar cross-sections of variable size and a given volume of material. A lower and/or upper bound constraint may be prescribed for the cross-sectional area of the beam-columns. To account for possible bimodality of the optimum designs, both the unimodal and bimodal optimality criterion approaches are applied. For a lower bound smaller than a specific value, bimodal optimum designs are obtained for problems of maximum buckling temperature. When maximizing the thermal fundamental natural frequency, optimum designs may be uni- or bimodal depending on the values of the given thermal load and the lower bound on the cross-sectional area. The geometrically unconstrained optimum design for maximum fundamental natural frequency obtained by Olhoff (1976) without consideration of thermal load, is shown to be at the same time an optimum design that maximizes the fundamental natural frequency at any temperature rise, but it is found to be associated with zero buckling load. Moreover, the optimum design maximizing the critical buckling temperature is very similar to that maximizing the mechanical buckling load obtained by Olhoff and Rasmussen (1977). This interesting analogy is explained by studying optimum design for minimum axial compressive force.