Structural Analysis II Section I.6

Section I.6 Torsion of Thin - Wall Closed Sections

So far, we have considered the torsion of solid and hollow circular sections, thin-walled open sections, and solid noncircular sections. Now we would like to consider the torsion of thin-walled closed sections of various geometry, typical of aircraft wing and fuselage. As before we are interested in calculating the shear stress distribution and overall angle of twist due to applied torque(s). Such analysis can be used to determine, based on a specified material and loading condition, (a) the margin of safety for an existing structure, or (b) the structural sizing such as wall thickness necessary to safely carry the applied loads.

Here are some examples of typical shear stress distributions in thin-walled, single- and multi-cell closed sections under pure torsion.

Derivation

Consider a thin-walled member with a closed cross section subjected to pure torsion.

Examining the equilibrium of a small cutout of the skin reveals that

Angle of Twist

By applying strain energy equation due to shear and Castigliano's Theorem the angle of twist for a thin-walled closed section can be shown to be

Since T = 2qA, we have

If the wall thickness is constant along each segment of the cross section, the integral can be replaced by a simple summation

Here is an example.

Important Observations:

In thin-walled structures failure is governed by buckling instability; therefore, the limiting stress is not necessarily the elastic limit stress.

EXAMPLE PROBLEMS

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