Geodesic
Optimal design of an elastic beam on an elastic basis
Applications of Mathematics, Tome 31 (1986) no. 2, pp. 118-140
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An elastic simply supported beam of given volume and of constant width and length, fixed on an elastic base, is considered. The design variable is taken to be the thickness of the beam; its derivatives of the first order are bounded both above and below. The load consists of concentrated forces and moments, the weight of the beam and of the so called continuous load. The cost functional is either the $H^2$-norm of the deflection curve or the $L^2$-norm of the normal stress in the extemr fibre of the beam. Existence of solutions of optimization problems in both the primary and dual formulations of the state problem is proved. For both formulations, approximate problems are introduced and convergence of their solutions to those of the continuous problem is established. Theoretical conclusions are corroborated by an illustrative example.
An elastic simply supported beam of given volume and of constant width and length, fixed on an elastic base, is considered. The design variable is taken to be the thickness of the beam; its derivatives of the first order are bounded both above and below. The load consists of concentrated forces and moments, the weight of the beam and of the so called continuous load. The cost functional is either the $H^2$-norm of the deflection curve or the $L^2$-norm of the normal stress in the extemr fibre of the beam. Existence of solutions of optimization problems in both the primary and dual formulations of the state problem is proved. For both formulations, approximate problems are introduced and convergence of their solutions to those of the continuous problem is established. Theoretical conclusions are corroborated by an illustrative example.
DOI : 10.21136/AM.1986.104192
Classification : 73k40, 74B05, 74K10, 74P99
Keywords: optimal design; concentrated forces and moments; continuous load; cost functional; $H^2$-norm of the deflection curve; $L^2$-norm of the normal stress; primary and dual formulations; elastic beam; elastic foundation; existence; convergence 
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Chleboun, Jan. Optimal design of an elastic beam on an elastic basis. Applications of Mathematics, Tome 31 (1986) no. 2, pp. 118-140. doi: 10.21136/AM.1986.104192

[1] M. S. Bazaraa C. M. Shetty: Nonlinear Programming, Theory and Algorithms. (Russian translation - Mir, Moskva 1982.) | MR

[2] D. Begis R. Glowinski: Application de la méthode des éléments finis à l'approximation d'un problème de domaine optimal. Méthodes de résolution des problèmes approchés. Applied Mathematics & Optimization, 2 (1975), 130-169. | DOI | MR

[3] R. Courant D. Hilbert: Methoden der matematischen Physik I. Springer-Verlag 1968, 3. Auflage. | MR

[4] S. Fučík J. Milota: Mathematical Analysis II. (Czech - University mimeographed texts.) SPN Praha 1975.

[5] I. Hlaváček: Optimization of the shape of axisymmetric shells. Aplikace matematiky, 28 (1983), 269-294. | MR

[6] I. Hlaváček I. Bock J. Lovíšek: Optimal control of a variational inequality with applications to structural analysis. Optimal design of a beam with unilateral supports. Applied Mathematics & Optimization, 1984, 111-143. | DOI | MR

[7] J. Chleboun: Optimal Design of an Elastic Beam on an Elastic Basis. Thesis (Czech). MFF UK Praha, 1984.

[8] J. Nečas I. Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam, 1981. | MR

[9] S. Timoshenko: Strength of Materials, Part II. D. Van Nostrand Company, Inc. New York 1945. (Czech translation, Technicko-vědecké nakladatelství, Praha 1951.)

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