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The remaining stiffness coefficients are obtained in a similar fashion. The relationship be-
tween the displacements at
b
and
c
and the applied forces at
b
and
c
becomes
√
2
4
√
2
4
V
xb
P
xb
1
+
0
0
√
2
4
√
2
4
V
zb
P
zb
1
+
0
−
1
EA
L
=
(7)
√
2
4
−
√
2
4
0
0
1
+
V
xc
P
xc
−
√
2
4
√
2
4
0
−
1
1
+
V
zc
P
zc
K
V
=
P
Of course, the same result can be obtained directly from the principle of virtual work, or
the principle of stationary potential energy. For example, consider the use of the principle
of stationary potential energy. The potential energy is given by
4
=
U
i
+
U
e
=
U
i
−
P
j
V
j
(8)
j
=
1
=
.
with
P
xb
, P
zb
, P
xc
, P
zc
P
1
, P
2
, P
3
, P
4
Then use of
=
∂
∂
V
1
+
∂
∂
V
2
+
∂
∂
V
3
+
∂
∂
δ
=
0
V
1
δ
V
2
δ
V
3
δ
V
4
δ
V
4
(9)
along with the strain energy of (4), l
e
ads to the stiffness relations of (7) again.
The relationship of (7) i.e.,
KV
=
P
,
can be solved for the displacements
V
.
3.1.2
The Unit Displacement Method
The
unit displacement
or
dummy displacement method
serves almost the same purpose as
Castigliano's theorem, part I. This
me
thod, which is described in Chapter 3 of the first edition
of this topic, determines the force
P
j
at a given point necessary to maintain equilibrium in
a structure under a known state of stress
σ
.
3.2
The Principle of Complementary Virtual Work Related Theorems
The principle of complementary virtual work and its corollaries are very useful for hand cal-
culations of structural analysis, but they are somewhat difficult to systematize for computer
solutions of large-scale systems. This is in stark contrast to the principle of virtual work
theorems which, because of the need for imposing geometric compatibility, are not well
suited for hand calculations, yet—as will be seen in Chapter 5—are readily systematized
for the solution of large-scale systems.
According to the principle of complementary virtual work, a solid satisfies the kinemat-
ical conditions if the sum of the external complementary virtual work and the internal
complementary virtual work is zero for statically admissible virtual stresses. The principle
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