MODELING OF THE STRESS STATE IN THIN ISOTROPIC PLATES
DOI:
https://doi.org/10.31734/architecture2022.23.005Keywords:
mathematical model, thin isotropic plates, automatic differentiationAbstract
A method of calculating the stress state in thin isotropic rectangular arbitrarily loaded plates for various boundary conditions has been developed.
Solution of the problem is reduced to solution of a differential equation of the fourth order in particular derivatives
, (1)
where is Laplace’s differential operator; w – deflection of the plate; q – transverse load applied to the upper surface of the plate and D is bending rigidity of the plate.
Solution of the equation (1) is presented in the form of a sum of its particular solution w⁎ and the general solution wо of corresponding uniform equation
(2)
General solution wо is presented as a sum of products of unknown coefficients and shape functions .
(3)
Coefficients are treated as degrees of freedom of the plate.
Similarly, a particular solution is given as a sum of products of force functions and other unknown coefficients.
Such approach allows to satisfy conditions at the edge and on the surface of the plate.
Boundary conditions are performed in the separate nodes at the plate edge (in each node two boundary conditions are written). In each node on the plate surface only one condition is written.
A program which automatically generates and places nodes at the edge and on the surface of the plate has been developed.
The expression (3) is called a function of the plate deflection state. State functions of other static and kinematic quantities are obtained from formula (3) by Automatic Differentiation.
Distributions of deflection, tangent displacements, moments and shearing forces are obtained on the whole area of the plate.
In this paper, two variants of the plate are considered: symmetrical and nonsymmetrical one. A plate is considered symmetrical if it satisfies the conditions of symmetry of the plate contour and boundary conditions also the conditions of symmetry of the external load and mechanical properties. Although if one of them is not performed the plate is nonsymmetrical.
For a symmetrical plate, the results are presented in the form of plots built in the central and edge cross sections of the plate. The results are compared with numerical ones obtained with the help of Package ABEQUS in the form of space plots.
It is shown that kinematic boundary conditions are performed exactly with analytical and numerical approaches. Instant results are not coincided as statistical boundary ones.
For nonsymmetrical plates, the results are given in the form of contour plots on the whole area of the plate.
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