Torsion members - The Behaviour and Design of Steel Structures to EC3 - page 445

t t t t

Cross-s ec tio n

S lit

Stress
distribution

t

t t

t t

Horizontal
plate

1
4

Membrane

M e m b rane t

t

t t

Membrane
analogy

(a) Slit tube (b) Closed tube

Figure 10.14 Comparison of uniform torsion of open and closed sections.

10.2.2 Elastic analysis

10.2.2.1 Statically determinate members

Somemembersinuniformtorsionarestaticallydeterminate,suchasthecantilever
shown in Figure 10.15. In these cases, the distribution of the total torque T x = T t
can be determined from statics, and the maximum stresses can be evaluated from
equations10.11,10.16,10.18,or10.20,orfromFigures10.7or10.9.Thetwisted
shape of the member can be determined by integrating equation 10.8, using the
sign conventions of Figure 10.16a.Thus, the angle of twist rotation in a region of
constant torque T x will vary linearly in accordance with

φ = φ 0 + T x x

GI t ,

(10.23)

as indicated in Figure 10.15c.

10.2.2.2 Statically indeterminate members

Many torsion members are statically indeterminate, such as the beam shown in
Figure 10.17, and the distribution of torque cannot be determined from statics
alone.Theredundantquantitiescanbedeterminedbysubstitutingthecorrespond-
ing compatibility conditions into the solution of equation 10.8. Once these have
been found, the maximum torque can be evaluated by statics, and the maximum
stress can be determined. The maximum angle of twist can also be derived from
the solution of equation 10.8.