Torsion members - The Behaviour and Design of Steel Structures to EC3 - page 460

of the flanges in opposite directions, with a series of flexural hinges (frictionless
or plastic) in each flange. Thus warping torsion plastic collapse can be analysed
by using the methods discussed in Section 5.5.5 to analyse the flexural plastic
collapse of the flanges.
Warpingtorsionplastichingesoftenformatsupportsoratpointsofconcentrated
torque,asindicatedinFigure10.29.Examplesofwarpingtorsionplasticcollapse
mechanisms are shown in Figures 10.20 and 10.21.
At plastic collapse, each flange becomes a mechanism which is statically
determinate, so that it can be analysed by using statics to determine the plas-
tic collapse torques. For the member shown in Figure 10.29, each flange forms
a mechanism with plastic hinges at the concentrated torque and the right-hand
support and a frictionless hinge at the left-hand support where warping is unre-
strained. If the concentrated torque α w T acts at mid-length, then the flanges
collapse when

α w ( T / d f ) L / 4 = M fp + M fp / 2 (10.62)

so that the warping torsion plastic collapse load factor is given by

α w = 6 M fp d f / TL (10.63)

Values of the warping torsion plastic collapse load factor α w for a number of
example-torsion members are given in Figures 10.20 and 10.21.

10.3.3.3 Plastic analysis of non-uniform torsion

Ithasnotbeenpossibletodevelopasimplebutrigorousmodelfortheanalysisof
the plastic collapse of members in non-uniform torsion where both uniform and
warping torsion are important.This is because

(a) different types of stress (shear stresses τ t , τ w , and normal stress σ w ) are
associated with uniform and warping torsion collapse,
(b) the different stresses τ t , τ w , and σ w are distributed differently across the
section, and
(c) the uniform torque T t and the warping torque T w are distributed differently
along the member.

A number of approximate theories have been proposed the simplest of which is
the Merchant approximation [19] according to which the plastic collapse load
factor α x is the sum of the uniform and warping torsion collapse load factors,
so that

α x = α t + α w (10.64)