## A study of some fundamental equations for the deformation of a variable thickness plate

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The approach to the problem of a variable thickness plate used in this paper is different from the usual approach in that this paper starts with general stress-strain relations and a generalized form of the position vector as used by Green and Zerna in "Theoretical Elasticity".

They use

R̅=L[ r̅ (θ₁,θ₂)+ λθ₃a̅₃(θ₁,θ₂)]

where θ₁,θ₂, and θ₃ are curvilinear coordinates with θ₁ and θ₂ being the coordinates of the middle surface and λ=t/L being a constant for a plate of constant thickness t.

This paper takes λ = λ(θ₁,θ₂) as a function of θ₁ and θ₂ so that the variable thickness may be taken into account. General tensor notation is used so as to work independent of coordinate systems.

Making simplifying assumptions only when necessary, the equations of equilibrium and stress-strain relations are derived in terms of tensors connected with the middle surface as was done by Green and Zerna for a constant thickness plate. The additional terms obtained in these equations due to the variation in λ help us to evaluate the effects of the varying thickness.

Expressions for stress are developed and they include the effects of transverse shear deformation and normal stress as well as the variation in thickness. These expressions are very much like those used by Essenburg and Naghdi in a paper presented at the Third U.S. National Congress of Applied Mechanics, June, 1958. However, they assumed the form for the stresses while the present paper arrived at their assumed forms with some additional terms after starting with general stress-strain relations.

Using the notation of Green and Zerna, a set of nine equations involving the nine unknowns, m ^{αβ}, w, n^{αβ}, and v^{α} is derived and under appropriate boundary conditions, this set will yield a solution to the problem which will be better than the classical solution.

Two problems are solved and numerical results are obtained and compared with the classical solutions. One of the problems involves a rectangular plate clamped on one edge with a uniform shear load on the other. The other problem involves a circular ring plate clamped on the outer edge with a uniform shear load on the inner edge. A much better correlation for the deflection of the middle surface is obtained for the rectangular than for the circular ring plate. The deflection at the inner edge of the ring plate obtained by the theory of this paper is over twice that obtained in the classical solution of the same problem.

In the previously mentioned set of nine fundamental equations, we have the stress resultants n^{αβ} and the deflections v^{α}. With appropriate boundary conditions, these equations could lead to a study of in-plane forces and buckling of variable thickness plates, a field in which not much progress has been made. This paper does not include any numerical work in this direction. It is felt, however, that one of the principal contributions of this paper to the literature is that the set of nine fundamental equations includes the stress resultants in n^{αβ} thus enabling us to study the effect of in-plane forces as well as that of transverse shear deformation, normal stress, and surface tractions.