## Solving equilibrium equations for stress and deflection functions in a thin elastic shallow shell

##### Abstract

A problem connected with the study of thin elastic shallow shells in the theory of non-linear elasticity is considered. The equilibrium equations are in the form
V4 B1 (x, y) + Eh n4 B2 (x,y)=0
DV 4 B2 (x,y) - n 4 B1 (x,y) = S(x,y)
The entire domain of the shell n bounded in
R3
4 is the biharmonic operator
4 (---)= ( + 2 4 + 4)
( x4 x2 y2 y4)
4 is the pucher 's operator
4 (----)= 2 fd 2 - 2 2 f + 2 + 2 f2
x2 y2 x y x y x2 ( y2
E: Young's modulus of elasticity of the material
h: the uniform thickness of the shell
D= Eh 3 (I- u2)-1 is the flexural rigidity of the material
12
µ=Poisson's ratio
i (x,y) is the stress function
2(x,y) is the deflection function
S(x,y) is the external force on the projection of the shell on xy plane
The entire boundary of the shell in the form dn is assumed clamped so that the deflection and slopes are zero boundaries
The entire domain of the shell l is bounded in R3 having middle surface in Monge's form Z= f (x ,y ). Galerkin's orthogonality conditions are applied to solve the equilibrium equations for stress and deflections. Appropriate forms of orhonormal Fourier's double series are formulated for1 (x, y) and 2(x, y) to satisfy the boundary conditions. Finally the existence and the uniqueness of the solutions are established.