Equations have been derived to fill a gap in previously published standard collections of stress-and-strain formulas: These are equations for deflection of a thin circular membrane that is clamped rigidly around its periphery and subjected to differential pressure. Related equations were described in "Deflection and Stress in Preloaded Square Membrane," (GSC-13367, Vol. 15, No. 9 (September 1991), page 96 and "Deflection and Stress in Preloaded Rectangular Membrane," (GSC-13561), Vol. 18, No. 3 (March 1994), page 100.

As in the cases of the square and rectangular membranes, the derivation of the equations for the circular membrane follows a strain-energy/virtual deflection approach, which is common in stress-and-strain problems of this kind. The displacements of the membrane under load are initially assumed to be of the form

(1)

and

(2)

where *r* is the radial coordinate, *a* is the radius of the clamping edge, *w* is the transverse displacement (that is, the deflection perpendicular to the nominal membrane plane) at radius *r,**w*_{0} is the maximum transverse displacement, *u* is the radial displacement at radius *r,*and *c*_{1} and *c*_{2} are constants.

The radial and transverse strains are given, respectively, by

(3)

and

(4)

The strain energy associated with stretching of the membrane is given by

(5)

where *E* is Young's modulus, *h* is the thickness of the membrane, and ν is Poisson's ratio.

To calculate the deflection of the membrane, one must solve the foregoing equations to find *c*_{1},*c*_{2}, and *w*_{0}. First, one substitutes the right sides of equations (1) through (4) for the corresponding terms in equation (5). Using the resulting form of equation (5), one finds *c*_{1}and *c*_{2} by imposing the requirements that

(6)

and

(7)

Next, one imposes the requirement that the change in work done by the differential pressure acting through a virtual displacement equals the change in strain energy associated with the virtual displacement. If the virtual displacement is chosen to be *δw *∝* δw*_{0}, then this requirement is expressed by the equation

(8)

where *q* is the differential pressure on the membrane. The solution for the maximum displacement is

(9)

where

(10)

* This work was done by Alfonso Hermida of *Goddard Space Flight Center. GSC-13783.