Torsion members - The Behaviour and Design of Steel Structures to EC3 - page 454

When the warping normal stresses σ w vary along the length of the member,
warping shear stresses τ w are induced, in the same way that variations in the
bendingnormalstressesinabeaminducebendingshearstresses(seeSection5.4).
The warping shear stresses can be expressed in terms of the warping shear flow
τ w t =− E d 3 φ
d x 3

s

(α n − α) t d s . (10.42)

0

Information for calculating the warping shear stresses in structural sections is
also given in [5, 6], while a general computer program for their evaluation has
been developed [8].
The warping shear stresses exert a warping torque

E

T w = ρ 0 τ w t d s (10.43)

0

about the shear centre, as shown in Figure 10.12a. It can be shown [14, 15] that
this reduces to
T w =− EI w d 3 φ

d x 3 ,

(10.44)

aftersubstitutingfor τ w andintegratingbyparts.Thewarpingtorque T w givenby
equation 10.44 is related to the bimoment B given by equation 10.39 through
T w = d B

d x .

(10.45)

Sign conventions for T w and B are illustrated in Figure 10.16.
A somewhat simpler explanation of the warping torque T w and bimoment B
can be given for the I-section illustrated in Figure 10.24. This is discussed in
Section10.8.2whereitisshownthatforanequalflangedI-section,thebimoment
B is related to the flange moment M f through
B = d f M f (10.46)

and the warping section constant I w is given by

I z d f
4

I w = (10.47)

in which d f is the distance between the flange centroids.

10.3.2 Elastic analysis

10.3.2.1 Uniform torsion

Some thin-walled open-section members, such as thin rectangular sections and
angle and tee sections, have very small warping section constants. In such cases
it is usually sufficiently accurate to ignore the warping torque T w , and to analyse
the member as if it were in uniform torsion, as discussed in Section 10.2.2.