Conclusion
From the examples considered above it can be seen that the area of the cross section minimizing the total weight of a simple rod under a given load is in general not unique. Only under particular assumptions on the constraints and the distribution of axial loads is the optimal solution uniquely determined by the principle of minimum weight. Another feature of the optimal solutions is that their properties of regularity (number and location of discontinuities of the solutions and its derivatives) are, in general, not at all obvious from the beginning.
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Communicated by D. D. Joseph
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Velte, W., Villaggio, P. Are the optimum problems in structural design well posed?. Arch. Rational Mech. Anal. 78, 199–211 (1982). https://doi.org/10.1007/BF00280036
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DOI: https://doi.org/10.1007/BF00280036