# Elastic Membrane Analogy

Elastic Membrane Analogy

The elastic membrane analogy, which was first published by pioneering aerodynamicist Ludwig Prandtl in 1903, [Prandtl, L.: "Zur torsion von prismatischen stäben", Phys. Zeitschr., 4, pp. 758-770 (1903) ] [Love 1944, article 224, page 322. ] describes the stress distribution on a long bar in torsion. The cross section of the bar is constant along its length, and need not be circular. The differential equation that governs the stress distribution on the 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 the 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.

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 $3T/bt^2$, where T is the torque applied, b is the length of the stretched cross section, and t is the thickness of the cross section.

Notes

References

*cite book | last = Bruhn | first = Elmer Franklin
title = "Analysis and Design of Flight Vehicle Structures
publisher = Jacobs Publishing | date = 1973 | location = Indianapolis
pages = | url = | isbn = 0-9615-2340-9

*cite book | last = Love | first = A. E. H.
authorlink = Augustus Edward Hough Love
title = A Treatise on the Mathematical Theory of Elasticity
publisher = Dover | date = 1944 | location = New York
pages = | url =
isbn = 0-486-60174-9
. Especially Chapter XIV, articles 215 through 224. "This Dover edition, first published in 1944, is an unaltered and unabridged republication of the fourth (1927) edition."

*cite web | last = Lagace | first = Paul A.
title = Unit 10 St. Venant Torsion Theory
work = Structural Mechanics Fall 2002 | publisher = MIT
date = 2001
url = http://ocw.mit.edu/NR/rdonlyres/Aeronautics-and-Astronautics/16-20Structural-MechanicsFall2002/088DEC8D-4ACB-4E3E-9911-F3BB5EB6AF89/0/unit10.pdf
format = pdf | accessdate = 2008-07-03

*cite web | last = Lagace | first = Paul A.
title = Unit 11 Membrane Analogy (for Torsion)
work = Structural Mechanics Fall 2002 | publisher = MIT
date = 2001
url = http://ocw.mit.edu/NR/rdonlyres/Aeronautics-and-Astronautics/16-20Structural-MechanicsFall2002/4BA039AA-752A-47C0-B037-8D379615F602/0/unit11.pdf
format = pdf | accessdate = 2008-07-03

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