Calculation of orthotropic plates for creep taking into account shear deformations

A system of differential equations is obtained for calculating the creep of orthotropic plates taking into account the deformations of the transverse shear. The basic hypothesis is a parabolic change in tangential stresses over the thickness of the plate. An example of the calculation is given for a GRP plate hinged on the contour under the action of a uniformly distributed load.


Introduction
The classical theory of plates, constructed on the hypothesis of the straight normal, describes well the stress-strain state of only those thin plates in which the deformations of transverse shear xz  and yz  are significantly smaller than the rotation angles w x   and w y   . There is a large number of papers devoted to creep calculation of plates based on the Kirchhoff-Love theory, including [1][2]. In isotropic plates, the shear angles are extreme near the reference contour, points and lines of application of concentrated and distributed along the lines forces, as well as near the notches [3]. The classical theory gives the greatest errors in stress values in these zones. However, if the thin plate is anisotropic and the transverse shear moduli are significantly smaller than the longitudinal elastic moduli in directions parallel to the median plane, then the shear angles xz  and yz  are comparable with the rotation angles far from the indicated regions, i.e. in the inner region of the plate. Because of this, the classical theory leads to noticeable errors at all points of the plate. In the literature there is a large number of different versions of refined theories that differ not only in the hypotheses and factors used, but also in approaches to their construction.

Methods
According to [3,4], the least errors are given by theories based on the hypothesis of a parabolic change in the stresses τ xz and τ yz over the plate thickness. One of these theories, which also takes into account the normal stresses σ z in addition to the parabolic distribution of the tangential stresses, is presented in the paper of S.A. Ambartsumian [4]. In this paper, we generalize theory of S.A. Ambartsumian for viscoelastic material.
Following [4], the tangential stress distribution is taken as: where φ = φ (x, y) is the desired shift function with the force dimension. The relationship between strains and stresses will be written as: The stresses xz  , yz  and z  are related by the equilibrium equation: Substituting (1) into (4), we obtain: The integration of equation (5) yields: The function C(x, y) is determined from the boundary conditions on the plate surfaces: which leads to the formula: and to the integral equation of equilibrium of the plate element hdxdy with respect to the z axis: Equation (9) allows expressing the stresses σ z explicitly through the load: ; .
From the physical point of view, the deformations γ xz and γ yz represent the sum of elastic deformations and creep strains: Substituting (1) into (12) and equating (12) to (11) then, we obtain: Integration of the equality (13) with respect to z gives: The median plane of the plate is assumed to be non-deformable, i.e. for z = 0 u = 0, from which f 1 (x, y) = 0. Similarly, for displacements v, we can write: Deformations ε x , ε y and γ xy are defined as follows: The bending and twisting moment will be written in the form: https://doi.org/10.1051/matecconf/201819601002 XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering   /2  2  2  2  2  2  *  2  2  2  2  /2   4  1  5 1 ; 0 This solution exactly satisfies the boundary conditions at constraints on the edges of tangential displacements: 0 at ; 0 at .
The functions q(x, y) and q*(x, y) can be expanded in a Fourier series by the formulas:     The coefficients of the expansion are determined by the formulas: https://doi.org/10.1051/matecconf/201819601002 XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering The time variation of the maximum stress σ x is shown in Figure 2. It can be seen from the presented graph that when calculating with the corrected theory, the stresses in the plate are higher in comparison with the Kirchhoff-Love theory.

Discussion
It is known that for isotropic plates the classical theory of plates is applicable if the following relation holds [6]: where a is the smallest plate size in plan.