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is the force at
i
due to a unit displacement at
j,
with the displacement at
i
equal to zero.
Also, observe that
k
ij
=
k
ji
,
i.e., the stiffness matrix is symmetric.
EXAMPLE 3.3 Strain Energy in Terms of the Stiffness Matrix
Su
ppose a linear solid is acted upon by a system of gradually applied forces
P
1
, P
2
,
...
P
k
,
...
P
n
,
with corresponding displacements
V
1
,V
2
,
...
V
k
,
...
V
n
.
The strain energy is equal to the
external work.
n
1
2
U
i
=
P
k
V
k
(1)
k
=
1
In matrix notation,
1
2
V
T
P
1
2
P
T
V
U
i
=
=
(2)
=
...
,V
n
]
T
,
P
=
...
, P
n
]
T
.
where
V
The stiffness coefficients
K
ij
for this
solid are defined in the same fashion as in Example 3.2. Upper case letter
K
is used to
indicate that the stiffness matrix is for the whole s
ys
tem, whereas a lower case
k
denotes
the stiffness matrix for a single element. The force
P
k
at coordinate
k
would be
[
V
1
V
2
,
[
P
1
P
2
,
n
P
k
=
K
kj
V
j
or
P
=
KV
(3)
j
=
1
The strain energy of (2) would then appear as
1
2
V
T
KV
=
U
i
(4)
Transpose both sides of (4) to obtain
1
2
V
T
K
T
V
U
i
=
(5)
Equating (4) and (5) gives
K
T
K
=
(6)
.
The partial derivative of (4), with respect to an arbitrary displacement
V
k
,
gives
=
Thus, the system matrix
K
, is symmetric, i.e.,
K
kj
K
jk
n
∂
U
i
V
k
=
K
kj
V
j
(7)
∂
j
=
1
which, according to (3), is the same as Castigliano's theorem, part I. The partial derivative
of (7), with respect to a displacement
V
h
,
gives
2
U
i
∂
V
k
=
K
kh
(8)
∂
V
h
∂
which is a stiffness coefficient. The stiffness coefficient is equal to the second derivative of
the strain energy with respect to the displacements at
k
and
h
.
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