Torsion members - The Behaviour and Design of Steel Structures to EC3 - page 444

Constant
shear
flow t t C , S C , S

x

x

y

y

z

z

Relative warping displacement

Relative warping displacement
(due to shear)

d

(due to twisting) 0

- d s

d s

d x

G

(a) Warping displacements
due to twisting only

(b) Warping displacements
due to shear only

Figure 10.13 Warping of a closed section.

The membrane analogy can also be used for thin-walled closed sections, by
imaginingthattheinner-sectionboundaryisfixedtoaweightlesshorizontalplate
supported at a distance equivalent to τ t t above the outer-section boundary by
the transverse pressures and the forces in the membrane stretched between the
boundaries, as shown in Figure 10.14b. Thus the slope (τ t t )/ t of the membrane
is substantially constant across the wall thickness, and is equivalent to the shear
stress τ t , while twice the volume A e τ t t under the membrane is equivalent to the
uniform torque T t given by equation 10.20.
The membrane analogy is used in Figure 10.14 to illustrate the dramatic
increasesinthestiffnessesandstrengthsofthin-walledclosedsectionsoverthose
ofopensections.Itcanbeseenthatthetorque(whichisproportionaltothevolume
under the membrane) is much greater for the closed section when the maximum
shear stress is the same. The same conclusion can be reached by considering the
effectiveleverarmsoftheshearstresses,whichareofthesameorderastheoverall
dimensions of the closed section, compared with those of the same order as the
wall thickness of the open section.
The behaviour of multi-cell closed-section members in uniform torsion can be
determined by using the warping displacement continuity condition for each cell,
asindicatedinSection10.7.1.Ageneralmatrixmethodofcarryingoutthisanalysis
by computer has been given in [7], and a computer program has been developed
[8].Alternatively, the membrane analogy can be extended by considering a set of
horizontal plates at different heights, one for each cell of the cross-section.