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Thus, the displacement of point
a
is
PL
V
=
(6)
EA
(
1
+
2 cos
3
α)
The internal forces
N
in each bar are found from th
e s
tresses (3) in terms of
V
(which is
given by (6) as a function of the known applied load
P
) by using
N
.
An alternative solution to this problem is obtained by using the principle in the form
−
δ
=
σ
A
W
i
is written in terms of the internal forces. For the kine-
matically admissible virtual displacement, the internal virtual work is
W
i
−
δ
W
e
=
0
,
in which
δ
−
δ
W
i
=
N
bar1
δ
V
+
N
bar2
δ
V
cos
α
+
N
bar3
δ
V
cos
α.
From
−
δ
W
i
=
δ
W
e
,
P
=
N
bar1
+
N
bar2
cos
α
+
N
bar3
cos
α
(7)
This, as is to be anticipated from the principle of virtual work, is the equilibrium relationship
that would be obtained by summing forces in the vertical direction. As noted previously in
this section, in order to evaluate the internal forces, one turns to Hooke's law. The internal
forces in terms of the unknown displacement
V
are
EAV
L
EA
V
EA
L
cos
2
N
bar1
=
,
N
bar2
=
N
bar3
=
V
cos
α
=
α
(8)
L
/
cos
α
Substitution of the relations of (8) into (7) gives (6) again.
EXAMPLE 2.5 In-Plane Deformation of a Flat Element
The principle of virtual work as represented by Eq. (2.54) can be specialized for two-
dimensional problems such as the in-plane deformation of flat element (plate) lying in
the
xy
plane. For this case, the virtual work relation [Eq. (2.54)]
S
p
δ
V
δ
T
σ
dV
u
T
p
V
dV
u
T
p
dS
−
δ
W
=
−
V
δ
−
=
0
(1)
becomes
S
p
δ
A
δ
T
s
dA
u
T
p
V
dA
u
T
p
ds
−
δ
W
=
−
A
δ
−
=
0
(2)
or
S
p
δ
u
T
(
u
D
T
s
u
T
p
ds
−
δ
W
=
A
δ
−
p
V
)
dA
−
=
0
where, from Chapter 1,
t
2
t
2
t
2
n
x
=
2
σ
x
dz,
n
y
=
2
σ
y
dz,
n
xy
=
2
τ
xy
dz
t
t
t
−
−
−
u
x
u
y
u
v
,
T
T
u
T
u
D
T
,
δ
=
δ(
)
=
δ
=
=
Du
u
∂
x
0
n
x
n
y
n
xy
,
D
=
0
∂
y
s
=
∂
y
∂
x
p
V
x
p
V
y
p
x
p
y
p
V
=
p
=
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