Beams - Mechanics of Structures: Variational and Computational Methods - page 745

Shear Stress
The expression for the shear stress is derived by substituting Eq. (12.52) into Eq. (12.40) in
the form

s

1
t

d σ x

dx

1
t

d σ x

dx tds

τ = dA =

A 0 0

s

A 0 N
At

I yz I y
t I yy I zz −

I yy I z − I yz I z

t I yy I zz −

I zz I y −

I ω
tI ωω

1
t I yz M y − I yz M z +

) tds (12.54) = + M tS − E (α T

0

where t is the wall thickness, A 0 is the area between the s coordinate origin 0, a free edge,
and s , the point of interest,

s
0 ω

I y = I z = I ω = y dA, z dA,

A 0 ω

dA = tds

A 0 A 0

d M

and M tS = dx is called the warping torque, or the secondary torsional moment

(Eq. 12.9). In Eq. (12.54), the shear stress due to warping is

ω /

I ω
tI ωω

I ω
I ωω

τ ω = M tS or q = τ ω t = M tS (12.55)

where q is called the shear flow .
An alternative form is provided by the substitution (Eq. 12.42)

M y = V z , M z =− V y

The shear stress in a thin-walled open section contains two distinct modes. The first mode
of shear stress, which is due to the bending, non-uniform axial deformation (restrained
warping), and the thermal gradient, is given in Eq. (12.54). The second mode is due to a
pure twisting of the beam during which the cross-sections are free to warp. These stresses,
which can also be expressed as shear flows, are given in Eqs. (12.45). Summation of the
stresses of Eqs. (12.54) and (12.45) gives the total shear stresses on the cross-section.

Shear Center
The shear center S of a cross-section is the y, z location through which the plane of the
resultant of the applied loadings must pass so that no twisting moment is developed.
Hence, by definition, the twisting moment due to the shear force and the shear stress about
an arbitrary point P must vanish, i.e., (Fig. 12.11)
s L

τ tr p ds + V z ( y S − y P ) − V y ( z S − z P ) = 0 (12.56)

0

where is the shear stress due to the shear forces V y and V z acting at the shear center S ,

and s L is the total length of the wall profile.
Now insert

τ

from Eq. (12.54) (with the axial, thermal, and constrained warping effects
neglected) into Eq. (12.56) and integrate. This leads to
y S −

τ

V z − z S − V y =

I yy I − I yz I I zz I − I yz I

ω z ω y ω y ω z

y P + z P + 0

I yy I zz − I yz I yy I zz − I yz