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### Circular Membrane Stress and Deflection Calculator and Equations

Circular Membrane Stress and Deflection Design Calculator and Equations

A membrane may be defined as a plate that is so thin that it may ne considered to have no bending rigidity. The only stresses present are in the plane of the surface and are uniform throughout the thickness of the membrane.

Figure 1 shows two views of a circular membrane with the edge clamped under a uniform pressure, p.

The maximum deflection of this membrane is at the center and is given by

Eq. 1

${\delta }_{c}=0.662R\sqrt[3]{\frac{pR}{Et}}$

The deflection of the membrane at a distance, r, from the center is

$\delta ={\delta }_{c}\left[1-0.09{\left(\frac{r}{R}\right)}^{2}-0.1{\left(\frac{r}{R}\right)}^{5}\right]$

The stress at the center of this membrane is

$f=0.423\sqrt[3]{\frac{E{p}^{2}{R}^{2}}{{t}^{2}}}$

while that at the edge is

${f}_{e}=0.328\sqrt[3]{\frac{E{p}^{2}{R}^{2}}{{t}^{2}}}$

Figure 1, Circular Membrane with Clamped Edge

Where

p = pressure
f = calculated stress
E = modulus of elasticity
r = cylindrical coordinate
R = outside radius of circular membrane
t = thickness of membrane
δ = deflection
δc = center deflection of circular membrane
f = stress at center
fe = stress at edge

• Bell Helicopter Structural Design Manual, 1977

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