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Asymptotic bounds on the globally optimal positions of orthogonal stiffeners for rectangular plates in elastostatic bending. (English) Zbl 1293.74341

Summary: 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 Z. Mazurkiewicz [“Buckling of rectangular plate obliquely reinforcement transversely by ribs with rigidity”, Bull. Pol. Acad. Sci., Tech. Sci. 10, 329–339 (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.

MSC:

74P05 Compliance or weight optimization in solid mechanics
74K20 Plates
74E30 Composite and mixture properties
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[1] 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
[2] Cheng KT, Olhoff N (1981) An investigation concerning optimal design of solid elastic plates. Int J Solids Struct 17:304-323 · Zbl 0457.73079 · doi:10.1016/0020-7683(81)90065-2
[3] Clarkson J (1965) The elastic analysis of flat grillage. Cambridge University Press, Cambridge
[4] Dems K, Mróz Z, Szelag D (1989) Optimal design of rib-stiffeners in disks and plates. Int J Solids Struct 25:973-998 · Zbl 0711.73172 · doi:10.1016/0020-7683(89)90017-6
[5] Fletcher, HJ; Thorne, CJ, Bending of thin rectangular plates, Ann Arbor, Michigan, 1954
[6] Fuchs MB (1976) Substitute function methods in structural optimization and their application to continuous beams. ScD dissertation, Technion · Zbl 0404.73043
[7] 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 · Zbl 0404.73043 · doi:10.1016/0045-7949(79)90008-7
[8] Goriupp K (1947) Die dreiseitig gelagerte Rechteckplatte. Arch Appl Mech 16:153-163 · Zbl 0031.42601
[9] Grayhack WT, Mahar TJ (1990) Buckling of rib-stiffened plates: an asymptotic approach. SIAM J Appl Math 50:1126-1133 · Zbl 0698.73041 · doi:10.1137/0150067
[10] Grigolyuk EI, Tolkachev VM (1980) Kontaktnyie zadachi teorii plastin i obolochek (Contact problems in the theory of piates and shells). Mashinostroenie, Moscow
[11] Kalamkarov AL (1992) Composite and reinforcement elements of construction. Wiley, New York
[12] Konchkovskii Z (1984) Plity. Staticheskie raschety (Plates. Static calculations). Stroiizdat, Moscow
[13] Lagaros ND, Fragiadakis M, Papadrakakis M (2004) Optimum design of shell structures with stiffening beams. AIAA J 42:175-184 · doi:10.2514/1.9041
[14] Lam YC, Santhikumar S (2003) Automated rib location and optimization for plate structures. Struct Multidiscip Optim 25:35-45 · doi:10.1007/s00158-002-0270-7
[15] 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 · Zbl 0137.44002
[16] 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 · Zbl 0137.44001
[17] Mróz Z, Rozvany GIN (1975) Optimal design of structures with variable support conditions. J Optim Theory Appl 15:85-101 · Zbl 0277.73073 · doi:10.1007/BF00933023
[18] Nowacki W (1954a) Statecznosc plyt prostokatnych wzmocnionych zebrami. Arch Mech Stosow 6:317-342 · Zbl 0059.18301
[19] Nowacki W (1954b) Zagadnienia statyki i dynamiki plyt wzmocnionych zebrami. Arch Mech Stosow 6:601-638 · Zbl 0059.18302
[20] O’Leary JR, Harari I (1985) Finite element analysis of stiffened plates. Comput Struct 21:973-985 · Zbl 0587.73115 · doi:10.1016/0045-7949(85)90210-X
[21] 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
[22] Samsonov AM (1978) The optimal location of a thin rib for an elastic plate. Izv Akad Nauk SSSR, Meh Tverd Tela 1:132-138
[23] Savin GN, Fleishman NP (1964) Plastinki i obolochki s rebrami zhestkosti (Plates and shells with stiffening ribs). Naukova Dumka, Kiev
[24] Schade HA (1940) The orthogonally stiffened plate under uniform lateral load. J Appl Mech 62:143-146 · JFM 66.1377.01
[25] 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
[26] Szilard R (1974) Theory and analysis of plates: classical and numerical methods. Prentice Hall, New York · Zbl 0295.73053
[27] Timoshenko SP, Woinosky-Krieger S (1959) Theory of plates and shells, 2nd edn. McGraw-Hill, New York
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