Elastic Shell-Theory Formulation for Bourdon Tubes

Abstract

The stress-displacement problem is considered for a typical bourdon tube in equilibrium deflection resulting from internal fluid pressure. In contrast with many analyses using approximations and simplifications, here the exact (linear) elastic-shell theory is employed, applied to a tube having elliptical cross section and with central line forming a circular arc. Differential-geometry quantities for the surface are derived as required for shell theory, and the explicit form is obtained for the partial differential equations, referenced to the lines of principal curvature as orthogonal curvilinear coordinates. Equations are presented for the Love-type shell theory and for a Donnell-type theory, both including the interacting bending and membrane effects. The rigid plug at the free end of a bourdon tube creates some complexity in the boundary conditions: These must be considered in-the-large rather than at each boundary point, to obtain the correct number of imposed relations for a determinate problem. The original problem is finally reduced to a specific computational problem for five partial differential equations of order eight, plus certain integral boundary relations, in a rectangular domain.