Torsion members - The Behaviour and Design of Steel Structures to EC3 - page 455

10.3.2.2 Warping torsion

Verythin-walledopen-sectionmembershaveverysmalltorsion-sectionconstants
I t , while some of these, such as I- and channel sections, have significant warping
sectionconstants I w .Insuchcases,itisusuallysufficientlyaccuratetoanalysethe
member as if the applied torque were resisted entirely by the warping torque, so
that T x = T w .Thus, the twisted shape of the member can be obtained by solving

− EI w d 3 φ

d x 3 = T x ,

(10.48)

as

x x x

T x d x d x d x + A 1 x 2
2

− EI w φ = + A 2 x + A 3 , (10.49)

0 0 0

where A 1 , A 2 , and A 3 are constants of integration whose values depend on the
boundary conditions.
For statically determinate members, there are three boundary conditions from
which the three constants of integration can be determined. As an example of
thethreecommonlyassumedboundaryconditions,thestaticallydeterminatecan-
tilevershowninFigures10.1cand10.25isanalysedinSection10.8.3.Thetwisted
shape of the cantilever is shown in Figure 10.25c, and it can be seen that the
maximumangleoftwistrotationoccursattheloadedendandisequalto TL 3 / 3 EI w .
For statically indeterminate members such as the beam shown in Figure 10.26,
there is an additional boundary condition for each redundant quantity involved.
These redundants can be determined by substituting the additional boundary

Warping normal stresses w

Bimoment d f M f

y

x M f Warping shear stresses w

Flange shears V f

z

d f

v f

V f

M f

y

d f

C , S

V f

z (a) Rotation of

cross-section

(b) Bimoment and warping stresses

Figure 10.24 Bimoment and stresses in an I-section member.