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A
σ
x
dA
u
=
δ
Axial Force
x
N
A
σ
x
ydA
A
σ
x
zdA
dA
δθ
z
+
δθ
y
+
δψ
+
−
A
σ
x
ω
Moments
M
z
M
y
M
ω
(Bimoment)
A
τ
xy
dA
A
τ
xz
dA
δ(v
−
θ
z
)
+
δ(w
+
θ
y
)
+
Shear Forces
V
y
V
z
xz
y
xy
z
dA
−
∂ω
∂
+
∂ω
∂
Primary Torsional
Moment
δφ
x
+
τ
−
τ
z
y
A
M
tP
Secondary
Torsional Moment
(from warping
shear)
dA
dx
τ
xy
∂ω
∂
y
+
τ
xz
∂ω
δ(φ
x
+
ψ)
+
(12.9)
∂
z
A
M
tS
The definitions of the stress resultants (internal forces and moments) have been identified
in Eq. (12.9) as
N
=
A
σ
x
dA,
M
z
=−
A
σ
x
y dA,
M
y
=
A
σ
x
zdA
M
ω
=
A
σ
x
ω
dA,
V
y
=
A
τ
xy
dA,
V
z
=
A
τ
xz
dA
xz
y
xy
z
dA
−
∂ω
∂
+
∂ω
∂
M
tP
=
τ
−
τ
z
y
A
dA
xy
∂ω
∂
xz
∂ω
∂
=
τ
y
+
τ
M
tS
z
A
Thus, Eq. (12.9) leads to the internal virtual work expression for the bar
x
{
−
δ
W
i
=
N
δ
+
M
z
δκ
+
M
y
δκ
+
M
ω
δκ
ω
+
V
y
δγ
+
V
z
δγ
+
M
tP
δγ
+
M
tS
δγ
}
dx
z
y
y
z
tP
tS
-------- -------- ----------
(12.10)
The stress resultants and corresponding strains are summarized in the following table:
Stress
Resultant
Strain
u
Longitudinal
N
=
κ
z
=
θ
z
,
κ
y
=
θ
y
Bending
M
z
,M
y
γ
tP
=
φ
x
,
γ
tS
=
φ
x
+
ψ
κ
ω
=
ψ
Torsion
M
tP
,M
tS
,M
ω
,
γ
y
=
v
−
θ
z
,
γ
z
=
w
+
θ
y
Direct Shear
V
y
,V
z
Simplifying Kinematic Assumptions
Bending: Bernoulli's Hypothesis
Neglecting the transverse shear strains, i.e.,
γ
=
v
−
θ
=
0
y
z
(12.11)
=
w
+
θ
γ
=
0
z
y
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