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where the internal axial force
N
, bending moment
M
, and shear force
V
have been defined
as
N
=
A
σ
x
dA
M
=
A
σ
x
zdA
V
=
A
τ
xz
dA
(6)
Thus, (2) reduces to
V
δ
T
σ
dV
T
s
dx
=
x
δ
(7)
where, as in Chapter 1, Section 1.8,
d
x
00
0
d
x
1
00
d
x
u
0
w
θ
N
V
M
0
x
γ
κ
=
=
s
(8)
=
D
u
u
In a similar fashion, the integral over the body force
p
V
in (1) can be rewritten as
u
T
p
V
dV
u
T
p
dx
V
δ
=
x
δ
(9)
where
p
x
p
z
m
p
=
(10)
Here,
p
x
and
p
z
are force per unit length, and
m
is the moment intensity, i.e., the moment
per unit length.
The final integral of (1) represents the virtual work performed by surface loads. If these
surface forces are applied at the ends
(
)
0
,L
of the beam, they would be concentrated forces,
and the integral would reduce to
=
δ
u
T
s
L
0
u
T
p
dS
S
p
δ
(11)
where
N
V
M
s
=
(12)
are the concentrated forces at the ends. This expression is readily adjusted to account for
in-span concentrated loads.
In summary, the principle of virtual work expression for beams would appear as
−
δ
u
T
s
L
0
x
δ
T
s
dx
u
T
p
dx
−
δ
W
=
−
x
δ
=
[
δ
u
0
,x
N
+
(δθ
+
δw
)
V
+
δθ
,x
M
]
dx
,x
x
−
δ
M
L
0
−
x
(δ
u
0
p
x
+
δw
p
z
+
δθ
m
)
dx
u
0
N
+
δw
V
+
δθ
=
0
(13)
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