Abstract
The refined theory of elastic thin and thick plates is constructed by the asymptotic method for reducing three-dimensional (3D) equations of linear elasticity to two-dimensional ones without the use of any assumptions. The resulting refined theory is much more complicated than the known classical Kirchhoff theory: the required values of the refined theory vary in thickness of the plate by more complex laws; the system of partial differential equations of the refined theory has a higher order than the system of equations of the classical theory. A comparison of the obtained theory with the popular refined theory of Timoshenko and E. Reissner, taking into account the transverse shear deformation is made. It is shown that the inclusion only of the transverse shear deformation is insufficient. In addition to the transverse shear deformation, many additional terms having the same order as the transverse shear deformation must be taken into account.
Highlights
Elastic plates are widely used in various areas of modern engineering
The first rough estimate of the error in the classical theory of plates was obtained in [1], where it was shown that the error of Kirchhoff’s plate theory is the value of the order of h/R (R is the characteristic size of the plate, h is its half - thickness)
We write out the refined relations of elasticity of Tymoshenko [3] for bending problem taking into account the transverse shear deformation
Summary
Elastic plates are widely used in various areas of modern engineering. For example, in civil construction plates are used as floors, walls, slab foundations, etc. Well known Kirchhoff’s plate theory is constructed by the use of this method. Where s is the variability of the stress-strain state or characteristic length of the strain pattern Another method of constructing the theory of plates is the asymptotic method, which makes it possible to use the smallest thickness of the plate most fully. The implementation of the asymptotic method consists of the following steps: we perform a scale extension with respect to independent variables in 3D equations of the theory of elasticity in such a way that differentiation with respect to them does not lead to a significant increase or decrease in the unknown functions; we write all the unknown quantities in a dimensionless form and find the asymptotic orders of the unknown quantities relative to each other. Performing in the 3D equations of elasticity theory, asymptotic stretching of the scale along the coordinate lines and taking into account the asymptotic representation of desired quantities we reduce 3D equations to equations in the theory of plates
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