In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 171

x

Element δ s × δ x × t

v

Projection of s, x
axes onto plane of
element
δ s × δ x

v

v

δ s Change in warping

displacement due to
shear strain

s

v

=

δ u δ s

G

δ x

Figure 5.19 Warping displacements due to shear.

Thickness t f

y

d f

V z

Thickness t w

b

z

(b) Total shear flow

(a) Circulating shear flow vc t
(slit box shear flow vo t
shown in Fig. 5.18a)

v t vc t vo t

= +

Figure 5.20 Shear stress distribution in a rectangular box.

However,whentheboxisnotslit,asshowninFigure5.20,thisrelativewarping
displacement must be zero.Thus

τ v
G d s = 0,

(5.27)

in which total shear stress τ v can be considered as the sum

τ v = τ vc + τ vo (5.28)

of the slit box shear stress τ vo (see Figure 5.18a) and the shear stress τ vc due to a
constant shear flow τ vc t circulating around the closed section (see Figure 5.20a).
Alternatively,equation5.16fortheshearflowinanopensectioncanbemodified