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Then the force
P
necessary to generate vertical displacement
V
is
=
∂
U
i
∂
3
2
EAV
L
P
V
=
(3)
This force-displacement relationship can be “post-processed” to obtain in
fo
rmation about
the member forces, stresses, and deformations. Equation (3) gives
V
=
2
PL
/
3
EA
.
Use of
v
=
V
cos
α
gives the elongation of any bar. Also, the axial force
N
in each bar is found
from
EA
v
EAV
L
N
=
=
cos
α
(4)
L
EXAMPLE 3.2 Stiffness Matrix for a Spring
Suppose a linear elastic spring is placed between points
a
and
b
in a structure. The strain
energy for this spring will be
N
u
U
i
=
(1)
2
where
N
is the force in the spring, and
u
is the deformation experienced by the spring,
i.e.,
u
=
u
b
−
u
a
,
where
u
a
and
u
b
are the extensions at
a
and
b
.
From the definition of an
elastic spring, the force in the spring is given by
N
=
k
u
=
k
(
u
b
−
u
a
)
(2)
where
k
is the spring constant or spring rate. Substituting (2) into (1), the strain energy
becomes
N
u
k
2
(
2
U
i
=
=
u
b
−
u
a
)
(3)
2
From Castigliano's theorem, part I, the tensile forces
N
a
and
N
b
at the ends
a
and
b
are
N
a
=
∂
U
i
u
a
=
ku
a
−
ku
b
∂
(4)
N
b
=
∂
U
i
u
b
=−
ku
a
+
ku
b
∂
These are the expected results, since, from the conditions of equilibrium,
N
N
a
.
In (4), Castigliano's theorem, part I has been applied to a case where the forces,
N
a
and
N
b
,
are not applied loads. These forces should be interpreted as the forces that would have to
be applied to generate the displacements
u
a
and
u
b
.
=
N
b
=−
In matrix notation, (4) can be expressed
as
N
a
N
b
u
a
u
b
k
aa
u
a
u
b
k
−
k
k
ab
=
=
(5)
−
k
k
k
ba
k
bb
p
=
k
v
The constants
k
ij
are called the
stiffnesses
of the structure (spring), while the associated
matrix
k
is referred to as the
stiffness matrix
. Several characteristics of the stiffnesses are of
interest. Note that if
u
a
is zero and
u
b
is unity,
N
a
=
k
ab
and
N
b
=
k
bb
.
We conclude that
k
ij
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