The membrane analogy is also very useful in analyzing torsion of thin tubes of arbitrary shape with variable thickness that is “small” in comparison to the overall dimensions.

**Thin Walled
Tubes of Arbitrary Shape**

The membrane analogy is
also very useful in analyzing torsion of thin tubes of arbitrary shape with
variable thickness that is “small” in comparison to the overall dimensions.
Because *t* is small, the curvature of the membrane

where *A* is the
area enclosed by the average perimeter. Therefore, the shear stress

and therefore, from a
design standpoint, a circular tube of constant thickness is optimum since it
has the maximum area for a given weight of material to sustain the constant
shear stress.* To determine the angle of twist we can apply, Equation (9.29)

It is important to note
that this analysis, in which we have used the mem-brane analogy can, just as
well, be derived from statics alone. As shown in Figure 9.12, longitudinal
equilibrium of a slice in the *z* direction requires that

Now taking moments
about any arbitrary point *P* in the plane of the cross-section:

where *A* is the
total area enclosed by the cross-section (as in the membrane analogy) and not
the cross-sectional area of the wall of the tube.

Thus the special case
of torsion of a thin tube is *statically determinate*. As we will see
later, this allows us to use this solution as a building block to generate
lower-bound estimates of the plastic behavior of all sorts of cross-sections in
torsion.

Multicell thin tubes,
which occur in box bridges, aircraft, cars, ships, and skyscrapers can be
analyzed with a simple extension of the single-cell approach and the shear flow
concept. An example with small wall thick-nesses is shown in Figure 9.13. From
the membrane analogy

and the twisting moment
equals twice the volume under the membrane or

From Equation (9.40)
assuming *t*_{1}, *t*_{2}, and *t*_{3}
are constant

Again it should be
noted that Equation (9.43) can, as shown in Figure 9.13, be thought of as the
equilibrium condition at a junction in the longitudinal direction where:

and *q* = τ*t*
is a constant in any branch. This is also the continuity equation for fluid
flow in a channel or pipe network stating that the amount of liquid com-ing
into a junction must equal that flowing out. Hence the hydrodynamic analogy and
the term “shear flow” for *q*.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Civil : Principles of Solid Mechanics : Torsion : Thin Walled Tubes of Arbitrary Shape |

**Related Topics **

Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2024 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.