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Also,
A
is the surface area of the flat element in
th
e
xy
plane,
s
is a c
oo
rdinate along the
perimeter,
S
p
is the total length of the perimeter,
p
V
is the weight, and
p
contains the edge
loads. Introduce the material law
s
=
E
=
ED
u
u
u
T
k
D
u
u
T
p
ds
−
δ
W
=
A
δ
(
−
p
V
)
dA
−
S
p
δ
=
0
(3)
where, for plane stress [Eq. (1.39)],
1
ν
0
Et
E
=
D
ν
10
00
1
−
ν
2
D
=
(4)
1
−
ν
2
so that
.
∂
D
∂
∂ν
D
∂
x
x
x
y
.
+
y
∂
D
(
1
−
ν)
2
D
(
1
−
ν)
2
+
y
∂
∂
y
∂
x
.
.
k
D
u
D
T
ED
u
=
=
(5)
...........
...........
.
∂ν
D
∂
x
∂
D
∂
y
y
y
.
+
x
∂
D
(
1
−
ν)
2
D
(
1
−
ν)
2
+
x
∂
∂
y
∂
x
EXAMPLE 2.6 Differential Equations of Equilibrium for a Beam
In the same manner that the equations of equilibrium of an elastic solid were shown to be
equivalent to the principle of virtual work, the principle can be used to establish governing
differential equilibrium equations for a structural member such as a beam. According to
the principle,
S
p
δ
T
σ
dV
u
T
p
V
dV
u
T
p
dS
−
δ
W
=
V
δ
−
V
δ
−
=
0
(1)
For the engineering beam theory of Chapter 1, Section 1.8, the first integral reduces to
V
δ
T
σ
dV
=
V
(σ
x
δ
x
+
τ
xz
δγ
xz
)
dV
(2)
The strain-displacement relations, which in accordance with the principle of virtual work
must be satisfied by the displacements, are
x
=
∂
u
u
0
+
θ
x
=
z
(3)
∂
γ
xz
=
γ
=
∂
u
z
+
∂w
x
=
θ
+
w
(4)
∂
∂
where from Chapter 1, Eq. (1.98),
u
=
u
0
(
x
)
+
z
θ(
x
)
. Then (2) becomes
V
(σ
x
δ
x
+
τ
xz
δγ
xz
)
dV
V
σ
u
0
dV
δθ
dV
(δθ
+
δw
)
=
δ
+
V
σ
x
z
+
V
τ
dV
x
xz
u
0
+
δθ
+
(δθ
+
δw
)
=
δ
[
N
M
V
]
dx
(5)
x
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