Refined differential equations of deflections in axial symmetrical bending problems of spherical shell and their singular perturbation solutions

Abstract

This paper deals with the research of accuracy of differential equations of deflections. The basic idea is as follows. Firstly, considering the boundary effect the meridian midsurface displacement u=0, thus we derive the deflection differential equations; secondly we accurately prove that by use of the deflection differential equations or the original differential equations the same inner forces solutions are obtained; finally, we accurately prove that considering the boundary effect the meridian surface displacement u=0 is an exact solution. In this paper we give the singular perturbation solution of the deflection differential equations. Finally we check the equilibrium condition and prove the inner forces solved by perturbation method and the outer load are fully equilibrated. It shows that perturbation solution is accurate. On the other hand, it shows again that the deflection differential equation is an exact equation.

The features of the new differential equations are as follows:

• 1. The accuracies of the new differential equations and the original differential equations are the same.

• 2. The new, differential equations can satisfy the boundary conditions simply.

• 3. It is advantageous to use perturbation method with the new differential equations.

• 4. We may obtain the deflection expression(w)and slope expression (dw/da) by using the new differential equations.

The new differential equations greatly simplify the calculation of spherical shell. The notation adopted in this paper is the same as that in Ref. [1]