Abstract
The present paper treats the problem of finding the asymptotic bounds for the globally optimal locations of orthogonal stiffeners minimizing the compliance of a rectangular plate in elastostatic bending. The essence of the paper is the utilization of a method of analysis of orthogonally stiffened rectangular plates first presented by Mazurkiewicz in 1962, and obtained herein in a closed form for several special cases. Asymptotic expansions of the expressions for the deflection field of a stiffened plate are used to derive limit-case globally optimal stiffening layouts for highly flexible and highly rigid stiffeners. A central result obtained in this work is an analytical proof of the fact that an array of flexible enough orthogonal stiffeners of any number, stiffening a simply-supported rectangular plate subjected to any lateral loading, is best to be put in the form of exactly two orthogonal stiffeners, one in each direction.
Similar content being viewed by others
References
Andrianov IV, Lesnichaya VA, Manevich LI (1985) Metod usrednienia v statikie i dinamike rebristikh obolochek (Homogenization methods in statics and dynamics of ribbed shells). Nauka, Moscow
Cheng KT, Olhoff N (1981) An investigation concerning optimal design of solid elastic plates. Int J Solids Struct 17:304–323
Clarkson J (1965) The elastic analysis of flat grillage. Cambridge University Press, Cambridge
Dems K, Mróz Z, Szelag D (1989) Optimal design of rib-stiffeners in disks and plates. Int J Solids Struct 25:973–998
Fletcher HJ, Thorne CJ (1955) Bending of thin rectangular plates. In: Proceedings of the 2nd US national congress on applied mechanics, Ann Arbor, Michigan, 1954
Fuchs MB (1976) Substitute function methods in structural optimization and their application to continuous beams. ScD dissertation, Technion
Fuchs MB, Brull MA (1979) A new strain energy theorem and its use in the optimum design of continuous beams. Comput Struct 10:647–657
Goriupp K (1947) Die dreiseitig gelagerte Rechteckplatte. Arch Appl Mech 16:153–163
Grayhack WT, Mahar TJ (1990) Buckling of rib-stiffened plates: an asymptotic approach. SIAM J Appl Math 50:1126–1133
Grigolyuk EI, Tolkachev VM (1980) Kontaktnyie zadachi teorii plastin i obolochek (Contact problems in the theory of piates and shells). Mashinostroenie, Moscow
Kalamkarov AL (1992) Composite and reinforcement elements of construction. Wiley, New York
Konchkovskii Z (1984) Plity. Staticheskie raschety (Plates. Static calculations). Stroiizdat, Moscow
Lagaros ND, Fragiadakis M, Papadrakakis M (2004) Optimum design of shell structures with stiffening beams. AIAA J 42:175–184
Lam YC, Santhikumar S (2003) Automated rib location and optimization for plate structures. Struct Multidiscip Optim 25:35–45
Mazurkiewicz Z (1962a) Bending and buckling of rectangular plate reinforced transversely by ribs with variable rigidities. Bull Acad Pol Sci Ser Sci Tech 10:231–239
Mazurkiewicz Z (1962b) Buckling of rectangular plate obliquely reinforced by ribs with variable flexural rigidity. Bull Acad Pol Sci Ser Sci Tech 10:329–339
Mróz Z, Rozvany GIN (1975) Optimal design of structures with variable support conditions. J Optim Theory Appl 15:85–101
Nowacki W (1954a) Statecznosc plyt prostokatnych wzmocnionych zebrami. Arch Mech Stosow 6:317–342
Nowacki W (1954b) Zagadnienia statyki i dynamiki plyt wzmocnionych zebrami. Arch Mech Stosow 6:601–638
O’Leary JR, Harari I (1985) Finite element analysis of stiffened plates. Comput Struct 21:973–985
Perchikov N, Fuchs MB (2006) Optimal layouts of stiffeners for plates in bending—topology optimization approach. Paper presented at the 3rd European conference on computational mechanics, solids, structures and coupled problems in engineering, LNEC, Lisbon, 5–8 June 2006
Samsonov AM (1978) The optimal location of a thin rib for an elastic plate. Izv Akad Nauk SSSR, Meh Tverd Tela 1:132–138
Savin GN, Fleishman NP (1964) Plastinki i obolochki s rebrami zhestkosti (Plates and shells with stiffening ribs). Naukova Dumka, Kiev
Schade HA (1940) The orthogonally stiffened plate under uniform lateral load. J Appl Mech 62:143–146
Szczepanik M (2006) Optimization of topology and stiffener locations in 2D structures using evolutionary methods. Paper presented at the 3rd European conference on computational mechanics, solids, structures and coupled problems in engineering, LNEC, Lisbon, 5–8 June 2006
Szilard R (1974) Theory and analysis of plates: classical and numerical methods. Prentice Hall, New York
Timoshenko SP, Woinosky-Krieger S (1959) Theory of plates and shells, 2nd edn. McGraw-Hill, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Perchikov, N. Asymptotic bounds on the globally optimal positions of orthogonal stiffeners for rectangular plates in elastostatic bending. Optim Eng 14, 119–153 (2013). https://doi.org/10.1007/s11081-011-9161-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11081-011-9161-3