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be constant over the width
b
, the axial equilibrium requirements are
τ
bdx
=
dF
or
σ
1
b
dF
dx
=
1
b
d
x
dx
τ
=
dA
(12.40)
A
0
=
A
0
σ
x
dA
. Substitution of
where
F
σ
x
from Eq. (12.39) into Eq. (12.40) yields
A
0
)
dA
M
y
I
yy
+
M
z
I
yz
I
yy
I
zz
M
y
I
yz
+
M
z
I
zz
I
yy
I
zz
1
b
A
N
I
z
−
I
y
−
τ
=
+
E
(α
T
(12.41)
−
I
yz
−
I
yz
A
0
=
A
0
dA, I
z
=
A
0
z dA, I
y
=
A
0
ydA
.
The conditions of equilibrium relating the bending moments to the shear forces can be
expressed as [Chapter 1, Eq. (1.113)]
()
=
where
d
/
dx, A
0
dM
Ty
dx
M
y
=
=
V
z
,
V
z
+
V
Tz
with
V
Tz
=
(12.42)
dM
Tz
dx
M
z
=−
V
Ty
=−
V
y
,
V
y
−
with
V
Ty
=
Equation (12.41) then can be written as
A
0
)
dA
V
z
I
yy
−
V
y
I
yz
I
yy
I
zz
V
z
I
yz
−
V
y
I
zz
I
yy
I
zz
1
b
A
N
I
z
−
I
y
−
τ
=
+
E
(α
T
(12.43)
−
I
yz
−
I
yz
A
0
or
A
0
)
dA
I
y
I
zz
−
I
z
I
yz
I
z
I
yy
−
I
y
I
yz
1
b
A
N
I
yz
V
y
I
yz
V
z
τ
=
+
+
−
(α
E
T
I
yy
I
zz
−
I
yy
I
zz
−
A
0
These expressions should be regarded as reasonable approximations, since the assump-
tion that
is constant over the width is often not valid. More accurate shear stresses can be
calculated using a finite element implementation of theory of elasticity, as discussed later.
τ
Shear Stress Due to Torsional Moment
For solid cross-sections, the shear stresses due to torsion are those resulting from
M
tP
(Eq. 12.9). These shear stresses are given in Chapter 1, Eq. (1.143). In calculating these
stresses, the warping of the cross-section must be considered. The longitudinal displace-
ment in the
x
direction due to warping is [Eq. (12.5c)]
u
x
=
ω(
y, z
)ψ
(12.44a)
ψ
=−
φ
. Also, if the displacements due to bending are not
From Wagner's hypothesis,
considered [Eq. (12.6)],
u
y
=−
φ
z,
u
z
=
φ
y
(12.44b)
Then
G
∂
φ
∂ω
∂
z
y
+
∂
u
x
∂
u
y
∂
τ
=
G
γ
=
=−
G
y
+
(12.45a)
xy
xy
x
G
∂
φ
∂ω
∂
y
u
x
∂
z
+
∂
u
z
∂
τ
xz
=
G
γ
xz
=
=−
G
z
−
(12.45b)
x
which are in agreement with Chapter 1, Eqs. (1.142) and (1.143).
Although formulas for the warping function
ω
are available [Pilkey, 1994] for a few cross-
ω
sectional shapes,
can be computed for arbitrary shapes as discussed in Section 12.2.2.
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