TO DESIGN SHEARING FORCES FOR ELASTIC PLATES

For finding shearing forces by refined theories we use some representations of the work [1], §2, when the elastic plate is isotropic homogeneous with small deformations.

Let the averaged boundary conditions are split with respect to deflection and shearing forces in the bending problems. Then shearing forces satisfy Helmholts' equation with the corresponding boundary conditions. For clearness, consider the case when D=[0,1]x[0,1] and the shearing forces are known on ¶ D. Then the initial system has the following form:

(1)   e D Qa (x₁,x₂) - Qa (x₁,x₂) = Fa (x₁,x₂),

(2)   Qa (x₁,x₂)=f a , (x₁,x₂)Î ¶ D,

where all parameters and Fa are defind in §2, f a are known functions.

Below the iteration scheme (2.7) from [2] is used for solving (1), (2). Thus, we get two sequences of twopoint boundary value problems for ordinary differential equations of the second order with a small parameter.

These problems are solved by the exponentially fitted methods [3] and the generalized finite-difference method [4]. The following results are true:

1. The iteration processes are converged as a geometrical progression for any 0£ r £ 1, where r is an iteration parameter;

2. By these schemes are done numerical experiments which show an effectiveness of the methods. For example, in case, when exact solutions are

,

and e =0.07, m=70, r=0.3, m is a number of iterations, we have

.

[1] T.S. Vashakmadze, Some Problems of Mathematical Theory of Anisotropic Elastic Plates, Tbilisi, 1986 (in Russian).

[2] T.S. Vashakmadze, A Generalized Factorization Method..., Proceeding of sym. on Continuum Mechanics and Related Problems of Analysis, Tbilisi, vol. 1, 1971, 36-45 pp. (in Russian).

[3] E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.