Section I.3
**
Section
I.3 Elastic
Membrane
Analogy
**

Prandtl showed that the Laplace equation describing the torsion of an elastic
member is identical to that used to describe the
deflection of an elastic membrane subjected to a uniform pressure.
The elastic membrane analogy allows the solution of a torsion
problem to be determined in a simpler way than that found by the theory of
elasticity which requires the availability of the
warping function.

**Elastic Membrane and Twisted Bar
Relationships**
Consider a tube which has the same cross-sectional boundary as the bar. For example, if the
bar has a solid square cross section of side dimension b, then the tube will
have a hollow square cross section of side dimension b as well. Next we stretch
an elastic membrane over the tube's cross section and apply internal pressure.
The deflected shape of the membrane helps us visualize the stress pattern in
the bar under torsion.

The analogy can be viewed as follows:

- Lines of equal deflection on the membrane (contour
lines) correspond to shearing stress lines of the twisted
bar.
- The direction of a particular shear stress resultant at a
point is at right angle to the maximum slope of the membrane at
the same point.
- The slope of the deflected membrane at any point, with
respect to the edge support plane is proportional in magnitude
to the shear stress at the corresponding point on the bar's cross
section.
- The applied torsion on the twisted bar is proportional to
twice the volume included between the deflected membrane and
plane through the supporting edges.

**Examples**
Keep in mind that the slope at a point on the deflected membrane, and not the
displacement from the base, is the parameter that is
related to the shear stress in the bar. In
all of the following examples, the slope is zero at the very top of the
membrane, therefore the stress is zero, not the maximum, at the same location on
bar's cross section.

To Index Page of
Pure Torsion
To Section
I.2
To Section I.4