Design sensitivity for arch structures with respect to midsurface shape under static loading. (English) Zbl 0631.49015

We consider in this paper an example of structural optimization in which the structure is a loaded arch and the design variable is the shape of the arch. We concentrate on differentiability of static response with respect to shape changes. After recalling the arch equation with its functional spaces and the optimization problem, we state a differentiability theorem and provide a detailed proof. Numerical use of this result is finally discussed.


49K40 Sensitivity, stability, well-posedness
49K20 Optimality conditions for problems involving partial differential equations
74P99 Optimization problems in solid mechanics
93B35 Sensitivity (robustness)
93C20 Control/observation systems governed by partial differential equations
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[1] Lions, J. L.,Controle Optimal des Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, France, 1980.
[2] Rousselet, B.,Note on Design Differentiability of the Static Response of Elastic Structures, Journal of Structural Mechanics, Vol. 10, pp. 353-358, 1982.
[3] Ciarlet, P. H.,The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, Holland, 1980. · Zbl 0511.65078
[4] Bernadou, M., andDucatel, Y.,Approximation of General Arch Problems by Straight Beam Elements, Numerishe Mathematik, Vol. 40, pp. 1-29, 1982. · Zbl 0508.73069 · doi:10.1007/BF01459071
[5] Budiansky, B., Frauenthal, J. L., andHutchinson, J. W.,On Optimal Arches, Journal of Applied Mechanics, Vol. 36, pp. 880-882, 1969.
[6] Banichuk, N.,Determining the Optimal Form of Curved Elastic Bars, Mekhanika Tverdogo Tela, Vol. 10, pp. 124-133, 1975.
[7] Rozvany, G. I. N.,Optimality Criteria for Grids, Shells, and Arches, Optimization of Distributed Parameter Structures, Edited by E. J. Haug and J. Céa, Sijthoff and Noordhoff, 1980.
[8] Plaut, R., andOlhoff, N.,Optimal Form of Shallow Arches with Respect to Vibration and Stability, Journal of Structural Mechanics, Vol. 11, pp. 81-100, 1983.
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