Elastic Membrane Analogy

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First published by pioneering aerodynamicist Ludwig Prandtl in 1903, the elastic membrane analogy describes the stress distribution on a non-circular bar in torsion. The differential equation that governs the stress distribution on a bar in torsion is of the same form as the equation governing the shape of a membrane under differential pressure. Therefore, in order to discover the stress distribution on a non circular bar, all one has to do is cut the shape of the cross section out of a piece of wood, cover it with a soap film, and apply a differential pressure across it. Then the slope of the soap film at any area of the cross section is directly proportional to the stress in the bar at the same point on its cross section.

[edit] Application to thin-walled, open cross sections

While the membrane analogy allows the stress distribution on any cross section to be determined experimentally, it also allows the stress distribution on thin-walled, open cross sections to be determined by the same theoretical approach that describes the behavior of rectangular sections. Using the membrane analogy, any thin-walled cross section can be "stretched out" into a rectangle without affecting the stress distribution under torsion. The maximum shear stress, therefore, occurs at the edge of the midpoint of the stretched cross section, and is equal to $3 T / b t 2$ , where T is the torque applied, b is the length of the strected cross section, and t is the thickness of the cross section.