The elastic membrane analogy, also known as the soap-film analogy, was first published by pioneering aerodynamicist

Ludwig Prandtl Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of ...

in 1903. It describes the

stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...

distribution on a long bar in

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...

. The cross section of the bar is constant along its length, and need not be circular. The

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...

applied, b is the length of the stretched cross section, and t is the thickness of the cross section. It can be shown that the

for the deflection surface of a homogeneous membrane, subjected to uniform lateral pressure and with uniform surface tension and with the same outline as that of the

cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...

of a bar under

, has the same form as that governing the stress distribution over the cross section of a bar under

. This analogy was originally proposed by

in 1903.Prandtl, L.: "Zur torsion von prismatischen stäben", Phys. Z., 4, pp. 758-770 (1903).

Other applications

Prandtl's stretched-membrane concept was used extensively in the field of electron tube ("vacuum tube") design (1930's to 1960's) to model the trajectory of electrons within a device. The model is constructed by uniformly stretching a thin rubber sheet over a frame, and deforming the sheet upwards with physical models of electrodes, impressed into the sheet from below. The entire assembly is tilted, and steel balls (as electron analogs) rolled down the assembly and the trajectories noted. The curved surface surrounding the "electrodes" represents the complex increase in field strength as the electron-analog approaches the "electrode"; the upward distortion in the sheet is a close analogy to field strength.

References

* *. 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." * {{DEFAULTSORT:Membrane Analogy Mechanics Solid mechanics Structural analysis Analogy