Torsion members - The Behaviour and Design of Steel Structures to EC3 - page 440

t , max

z

Parabolic

b

Linear

y

t

(a) Membrane or
stress function

(b) Cross-section
and shear flow

(c) Shear stress
distribution

Figure 10.8 Uniform torsion of a thin rectangular section.

whence

t / 2

τ xy bz d z = G bt 3
6

d φ
d x

− (10.14)

− t / 2

which is one half of the total torque effect. The other half arises from the shear
stresses τ xz , which have their greatest effects near the ends of the thin rectangle.
Although they act there over only a small length of the order of t (compared with
b for the τ xy stresses), they have a large lever arm of the order of b / 2 (instead of
t / 2), and they make an equal contribution. The total torque can therefore be
expressed in the form of equation 10.8 with

I t ≈ bt 3 / 3. (10.15)

The same result can be obtained by substituting equation 10.12 directly into
equation 10.4. Equation 10.13 indicates that the maximum shear stress occurs at
the centre of the long boundary, and is given by

τ t , max ≈ T t t

I t .

(10.16)

Forstockierrectangularsections,thestressfunction θ variessignificantlyalong
the width b of the section, as indicated in Figure 10.5, and these approximations
may not be sufficiently accurate. In this case, the torsion section constant I t and
the maximum shear stress τ t , max can be obtained from Figure 10.9.

10.2.1.4 Thin-walled open cross-sections

Thestressdistributioninathin-walledopencross-sectionisverysimilartothatin
a narrow rectangular section, as shown in Figure 10.10a. Thus, the shear stresses
areparalleltothewallsofthesection,andvarylinearlyacrossthethickness t ,this
pattern remaining constant around the section except at the ends. Because of this