In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 170

(b) if the section is of a channel type, then the shear centre lies outside the web
and the centroid inside it, and
(c) ifthesectionconsistsofasetofconcurrentrectangularelements(tees,angles,
and cruciforms), then the shear centre lies at the common point.

A general matrix method for analysing the shear stress distributions and for
determiningtheshearcentresofthin-walledopen-sectionbeamshasbeenprepared
[11, 12].
Workedexamplesofthedeterminationoftheshearcentrepositionaregivenin
Sections 5.12.8 and 5.12.10.

5.4.4 Thin-walled closed cross-sections

The shear stress distribution in a thin-walled, closed-section beam is similar to
that in an open-section beam, except that there is an additional constant shear
flow τ vc t aroundthesection.Thisadditionalshearflowisrequiredtopreventany
discontinuity in the longitudinal warping displacements u which arise from the
shear straining of the walls of the closed sections. To show this, consider the slit
rectangularboxwhoseshearflowdistribution τ vo t duetoaverticalshearforce V z is
asshowninFigure5.18a.Becausethebeamisnottwisted,thelongitudinalfibres
remain parallel to the centroidal axis, so that the transverse fibres rotate through
angles τ vo / G equal to the shear strain in the wall, as shown in Figure 5.19.These
rotationsleadtothelongitudinalwarpingdisplacements u showninFigure5.18b,
the relative warping displacement at the slit being E

0 (τ vo / G ) d s .

b

y 0

S

Relative warping
displacement

C

E

E

vo
G

y

E d s

O

d f

S

0

C

O x

V z

y

Slit

z

z

(a) Shear flow vo t (b) Warping displacements due to shear stress

Figure 5.18 Warping of a slit box.