Prandtl Elastic-Membrane (Soap-Film) Analogy:

The equation that described the small displacement of a plane elastic membrane subject to lateral pressures is identical in mathematical form to the differential equation which find the stress in a twisted bar. Therefore, the displacement function of the membrane is mathematically equivalent to the stress function, provided that the boundary shape of the membrane is identical to the boundary shape of the cross-section of the torsion member. We let the physical and mathematical procedures that lead to a complete analogy proposed by Prandtl, between the membrane problem and torsion problem.

Consider an opening in the (x, y) plane that has the same shape as the cross-section of the torsion bar to be investigated. Cover the opening with a homogeneous elastic membrane, such as a soap film, and apply pressure

Figure

At the boundary the deflection of the membrane is zero. Mention the tensile force per unit length of the boundary line of the membrane by S and let p be the lateral pressure on membrane whereas θ is the angle of twist per unit length of the bar subjected to torque T. For differential equations for the deflection of membrane and equation for determination of stress in torsion bar to be identical :

                                                             p / S = 2G θ