In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 172

for the closed section to
τ v t = τ vc t − V z
I y

s

zt d s , (5.29)

0

sinceitcannolongerbesaidthat τ v iszeroat s = 0,thisnotbeingafreeend(the
closed section has no free ends). Mathematically, τ vc t is a constant of integration.
Substituting equation 5.28 or 5.29 into equation 5.27 leads to

τ vo d s

( 1 / t ) d s ,

τ vc t =− (5.30)

which allows the circulating shear flow τ vc t to be determined.
The shear stress distribution in any single-cell closed section can be obtained
by using equations 5.29 and 5.30. The shear centre of the section can then be
determinedbyusingequations5.24and5.25asforopensections.Fortheparticular
case of the rectangular box shown in Figure 5.20,

d f t w
8

τ vc t = V z
I y

+ d f t f b
4

, (5.31)

and the resultant shear flow shown in Figure 5.20b is symmetrical because of the
symmetryofthecross-section,whiletheshearcentrecoincideswiththecentroid.
Aworkedexampleofthecalculationoftheshearstressesinathin-walledclosed
section is given in Section 5.12.11.
The shear stress distributions in multi-cell closed sections can be determined
byextendingthismethod,asindicatedinSection5.10.3.Ageneralmatrixmethod
of analysing the shear flows in thin-walled closed sections has been described
[11,12].Thiscanbeusedforbothopenandclosedsections, includingcomposite
and asymmetric sections, and sections with open and closed parts. It can also be
used to determine the centroid, principal axes, section constants, and the bending
normal stress distribution.

5.4.5 Shear lag

Intheconventionaltheoryofbending,shearstrainsareneglectedsothatitcanbe
assumed that plane sections remain plane after loading. From this assumption
follow the simple linear distributions of the bending strains and stresses dis-
cussed in Section 5.3, and from these the shear stress distributions discussed in
Sections 5.4.1-5.4.4. The term shear lag [13] is related to some of the discrep-
ancies between this approximate theory of the bending of beams and their real
behaviour,andinparticular,referstotheincreasesofthebendingstressesnearthe
flange-to-web junctions, and the corresponding decreases in the flange stresses
away from these junctions.
Theshearlageffectsnearthemid-pointofthesimplysupportedcentrallyloaded
I-section beam shown in Figure 5.21a are illustrated in Figure 5.22. The shear