Critical Analysis of the Equations of Statics in the Bending Theories of Composite Plates Obtained on the Basis of Varia
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CRITICAL ANALYSIS OF THE EQUATIONS OF STATICS IN THE BENDING THEORIES OF COMPOSITE PLATES OBTAINED ON THE BASIS OF VARIATIONAL PRINCIPLES OF ELASTICITY THEORY 2. PARTICULAR LOW-ORDER THEORIES1
A. P. Yankovskii*
Keywords: composite plates, orthotropic material, statics equations of plates, nonclassical theory of plate bending, dimensionality reduction methods, variation principles, generalized Galerkin method, Reddy– Nemirovskii theory, Margueere–Timoshenko–Naghdi theory In a geometrically linear formulation based on the use of variation principles of elasticity theory, the twodimensional equations of statics and the boundary conditions at the edge of an orthotropic bending plate are obtained within the framework of the classical theory and two low-order nonclassical theories: Reddy– Nemirovskii and Margueere–Timoshenko–Naghdi ones. It is shown that using the hypotheses of the classical theory, the two-dimensional static equations and the corresponding boundary conditions ensure the equilibrium of an arbitrary subarea of the structure and of the entire plate as a rigid whole; within the framework of the Reddy–Nemirovskii theory, this equilibrium is disturbed in moments and forces in the transverse direction; within the framework of the Margueere–Timoshenko–Naghdi theory, the equilibrium is disturbed in the membrane forces and moments. An example of cylindrical bending of rectangular elongate orthotropic and isotropic plates clamped along one longitudinal edge and loaded in the transverse direction on the other longitudinal edge is considered. It is shown that, within the framework of the classical, Reissner, and Ambartsumyan theories, this problem is statically determinate and the shearing force, according to these theories, is determined elementary by the method of sections. According to the Reddy–Nemirovskii theory, this problem is statically indeterminate and has a rather awkward solution. It is demonstrated that, within the framework of the Reddy–Nemirovskii theory, the shearing force is zero on the clamped edge, i.e., the plate as a whole cannot be in equilibrium. In the vicinity of such an edge, there arises a pronounced false boundary effect, which does not take place in reality, because it is located inside the zone of the boundary layer and cannot be calculated by this theory. S. A. Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia * Corresponding author; e-mail: [email protected] 1 Note 1. see [1] Translated from Mekhanika Kompozitnykh Materialov, Vol. 56, No. 4, pp. 649-674, July-August, 2020. Original article submitted August 1, 2019; revision submitted February 17, 2020. 0191-5665/20/5604-0437 © 2020 Springer Science+Business Media, LLC
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Introduction The present work continues the investigation published in [1], where, on the basis of variational principles of elasticity theory in a vectorially linear statement, the equations of statics and the corresponding boundary conditions in force factors were fou
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