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3.4 Torsion of a narrow rectangular strip 63

The analogy therefore provides an extremely useful method of analysing torsion bars
possessing irregular cross-sections for which stress function forms are not known.
Hetenyi2 describes experimental techniques for this approach. In addition to the strictly
experimental use of the analogy it is also helpful in the visual appreciation of a parti-
cular torsion problem. The contour lines often indicate a form for the stress function,
enabling a. solution to be obtained by the method of Section 3.1. Stress concentrations
are made apparent by the closeness of contour lines where the slope of the membrane is
large. These are in evidence at sharp internal corners, cut-outs, discontinuities etc.

on of a narrow rectangular strip

In Chapter 9 we shall investigate the torsion of thin-walled open section beams; the
development of the theory being based on the analysis of a narrow rectangular
strip subjected to torque. We now conveniently apply the membrane analogy to the
torsion of such a strip shown in Fig. 3.9. The corresponding membrane surface has
the same cross-sectional shape at all points along its length except for small regions
near its ends where it flattens out. If we ignore these regions and assume that the
shape of the membrane is independent of y then Eq. (3.1 1) simplifies to

!!.!!! - -2G- d0
dx2 - dz
Integrating twice
d0
4 = -G-.x~ + BX+ c
dz
Substituting the boundary conditions = 0 at x = ft/2 we have

(3.26)
I_

t’

__ X
I I

Fig. 3.9 Torsion of a narrow rectangular strip.