Membrane analogy
Encyclopedia
The elastic membrane analogy, also known as the soap-film analogy, was first published by pioneering aerodynamicist Ludwig Prandtl in 1903.
It 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.
, where T is the torque
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 differential equation
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
of a bar under torsion
, has the same form as that governing the stress distribution over the cross section of a bar under torsion
.
This analogy was originally proposed by Ludwig Prandtl in 1903.
It describes the stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
distribution on a long bar in torsion
Torsion (mechanics)
In solid mechanics, torsion is the twisting of an object due to an applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius....
. The cross section of the bar is constant along its length, and need not be circular. The differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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, where T is the torqueTorque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
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 differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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 (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
of a bar under torsion
Torsion (mechanics)
In solid mechanics, torsion is the twisting of an object due to an applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius....
, has the same form as that governing the stress distribution over the cross section of a bar under torsion
Torsion (mechanics)
In solid mechanics, torsion is the twisting of an object due to an applied torque. In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius....
.
This analogy was originally proposed by Ludwig Prandtl in 1903.