If I have a circular membrane that is fixed by the edges and then I apply a uniform pressure to the membrane (let's say the membrane is the top of a circular chamber that has a rigid wall and flat bottom) by injecting compressed air, what shape will the membrane take? Parabolic? Hyperbolic? Is the shape very dependent if the membrane is parallel to the ground or perpendicular? Will it be different at the edges?
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Without pressure, as a result of gravity it is a catenary in any cross section, https://en.wikipedia.org/wiki/Catenary
It is assumed that the density of the membrane (and therefore its tension) is uniform. The catenary deviates according to the direction of gravity.
The force of pressure is perpendicular to the membrane at all points and would form a perfectly spherical shape if acting alone; depending upon the pressure magnitude and the weight of the membrane this may be an effect that dominates that of gravity.
JMLCarter
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