# Optimal design of an elastic beam on an elastic basis

Aplikace matematiky (1986)

• Volume: 31, Issue: 2, page 118-140
• ISSN: 0862-7940

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## Abstract

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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.

## How to cite

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Chleboun, Jan. "Optimal design of an elastic beam on an elastic basis." Aplikace matematiky 31.2 (1986): 118-140. <http://eudml.org/doc/15442>.

@article{Chleboun1986,
abstract = {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.},
author = {Chleboun, Jan},
journal = {Aplikace matematiky},
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; optimal design; concentrated forces and moments; continuous load; cost functional; H(sup 2)-norm of the deflection curve; L(sup 2)-norm of the normal stress; primary and dual formulations; elastic beam; elastic foundation; existence; convergence},
language = {eng},
number = {2},
pages = {118-140},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Optimal design of an elastic beam on an elastic basis},
url = {http://eudml.org/doc/15442},
volume = {31},
year = {1986},
}

TY - JOUR
AU - Chleboun, Jan
TI - Optimal design of an elastic beam on an elastic basis
JO - Aplikace matematiky
PY - 1986
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 31
IS - 2
SP - 118
EP - 140
AB - 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.
LA - eng
KW - 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; optimal design; concentrated forces and moments; continuous load; cost functional; H(sup 2)-norm of the deflection curve; L(sup 2)-norm of the normal stress; primary and dual formulations; elastic beam; elastic foundation; existence; convergence
UR - http://eudml.org/doc/15442
ER -

## References

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1. M. S. Bazaraa C. M. Shetty, Nonlinear Programming, Theory and Algorithms, (Russian translation - Mir, Moskva 1982.) (1982) MR0671086
2. D. Begis R. Glowinski, 10.1007/BF01447854, Applied Mathematics & Optimization, 2 (1975), 130-169. (1975) MR0443372DOI10.1007/BF01447854
3. R. Courant D. Hilbert, Methoden der matematischen Physik I, Springer-Verlag 1968, 3. Auflage. (1968) MR0344038
4. S. Fučík J. Milota, Mathematical Analysis II, (Czech - University mimeographed texts.) SPN Praha 1975. (1975)
5. I. Hlaváček, Optimization of the shape of axisymmetric shells, Aplikace matematiky, 28 (1983), 269-294. (1983) MR0710176
6. I. Hlaváček I. Bock J. Lovíšek, 10.1007/BF01442173, Applied Mathematics & Optimization, 1984, 111-143. (1984) MR0743922DOI10.1007/BF01442173
7. J. Chleboun, Optimal Design of an Elastic Beam on an Elastic Basis, Thesis (Czech). MFF UK Praha, 1984. (1984)
8. J. Nečas I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier, Amsterdam, 1981. (1981) MR0600655
9. S. Timoshenko, Strength of Materials, Part II, D. Van Nostrand Company, Inc. New York 1945. (Czech translation, Technicko-vědecké nakladatelství, Praha 1951.) (1945)

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