Abstract
This paper establishes the minimum cross-sectional area for an externally convex, hollow, prismatic bar subjected to minimum constraints on the second moment of area and on torsional rigidity. Some properties of the solution established by previous workers are assumed. Prandtl's stress function is expressed as a series in polar coordinates and the cross-sectional shapes are found by the semi-inverse method. The convexity constraint is expressed in a form suitable for application of mathematical programming methods. The optimal shapes were found by quadratic programming and generalized reduced gradient methods. Numerical results are given for typical examples, both for a circular hole and with the hole shape included in the optimization.
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