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FIGURE 12.11
Thin-walled cross-section. The centroid and shear center are designated by C and S , respectively.
For circular cross-sections, where no warping occurs, the torsional constant J of Eq. (12.24)
reduces to J
= A (
= A r 2 dA and the tangential shear stress is given by
y 2
+
z 2
)
dA
τ =
M tP r
/
J .
Stresses in Straight Thin-Walled Open Section Beams
Normal Stress
The formula for the normal stress in a thin-walled beam is derived in a manner similar
to that employed for a solid bar except that the out-of-plane deformation due to warping
torsion should be considered. The axial displacement of a point on the middle line of the
wall profile (Fig. 12.11) is given by Eq. (12.4)
dx
where Bernoulli's (Eq. 12.12) and Wagner's (Eq. 12.13) hypotheses are invoked, and the
normal stress is then
z d
dx
y d
v
dx ω
d
u x =
u
E du
T
z d 2
y d 2
d 2
w
dx 2
v
dx 2 ω
φ
dx 2 α
σ x =
dx
(12.46)
where u now is the axial displacement of the origin of a coordinate s which begins at one end
(a free edge) of the middle line of this open cross-section (Fig. 12.11). The stress resultants
of Eq. (12.38) become
N
E =
Au
ω φ
I
M y
E =−
w
v
φ
I zz
I yz
I ω z
(12.47)
M z
E
w +
v
φ
=
I yz
I yy
I ω y
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