Summary
This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.
Zusammenfassung
Die vorliegende Arbeit enthält einen Ueberblick über die elementaren Theorien der Biegung von geraden und krummen Stäben, von Platten und Schalen unter Verwendung von asymptotischen Approximationen der dreidimensionalen linearisierten Grundgleichungen der Elastizität, Zur Festlegung der Grössenordnung der massgebenden Funktionen wurde die Maximum-Norm verwendet. Die asymptotischen Entwicklungen sind wie üblich in Abhängigkeit des kleinen geometrischen Parameters gegeben, der die dünne Struktur charakterisiert. Die meisten Ideen und Resultate sind gut bekannt. Dennoch in den mehrparametrigen Fällen, so etwa bei kleinen Krümmungen (flache Strukturen) oder bei der expliziten Berücksichtigung des kleinen Belastungparameters, der zur Linearisierung der elastizitätsgleichungen verwendet wurde, führt die Diskussion über die Gültigkeitsgrenzen der verschiedenen Theorien zu einigen neueren und interessanten Aspekten. Ausserdem können die in der vorliegenden Arbeit dargelegten Hauptideen über mehrparametrige Entwicklungen zur Diskussion und zur Herleitung wertvoller Resultate über das Verhalten dünner Tragwerke in komplizierten Fällen wie mässige und starke Anisotropie, dynamische Probleme, Stabilität usw. übertragen werden.
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Sayir, M., Mitropoulos, C. On elementary theories of linear elastic beams, plates and shells (review paper). Journal of Applied Mathematics and Physics (ZAMP) 31, 1–55 (1980). https://doi.org/10.1007/BF01601703
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DOI: https://doi.org/10.1007/BF01601703