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complete domain, the resulting system of equations tends to be full in the sense that
k
u
is
not banded. Even more important is that a Ritz approximation predicts the stiffness of a
system to be higher than the actual stiffness. As a consequence, some static responses tend
to be underestimated and such dynamic responses as natural frequencies would be upper
bounds.
A Functional Form
The Ritz method, as it is often presented, involves the derivatives of a functional, frequently
a weak form, with respect to the free parameters. This is equivalent to the direct use of the
principle of virtual work. Suppose a potential exists for both the internal and external
forces. Then the principle of virtual work can be replaced by the principle of stationary
potential energy. This requires that the potential energy, utilizing kinematically admissible
displacements, be stationary, i.e.,
δ
=
0
.
This can be expressed as
δ
=
∂
∂
=
∂
∂
+
∂
∂
+···+
∂
∂
u
i
δ
u
i
u
1
δ
u
1
u
2
δ
u
2
u
m
δ
u
m
=
0
(7.47)
Since the variations
δ
u
i
are arbitrary, this relation is equivalent to
∂
∂
∂
∂
∂
∂
u
1
=
0
,
u
2
=
0
,
...
u
m
=
0
(7.48)
i.e.,
∂
∂
u
=
0
(7.49)
which are a system of simultaneous equations that can be solved for
u
1
,
...
,
u
m
.
For linear
(
elastic structures,
u
1
,
...
,
u
m
)
will be quadratic in
u
i
,
so that the simultaneous equations
of Eq. (7.48) will be linear algebraic equations for
u
i
,
i.e.,
k
u
u
=
p
u
(7.50)
For a continuum with the kinematic conditions,
=
Du
in
V
and
u
=
u
on
S
u
the potential energy is [Chapter 2, Eq. (2.64)]
1
2
T
E
dV
u
T
p
V
u
T
p
dS
=
−
dV
−
(7.51)
V
V
S
p
and for a beam,
L
L
1
2
(w
)
2
dx
w
]
=
−
p
z
w
−
w
−
EI
dx
[
V
M
(7.52)
on
S
p
0
0
For the beam, use a trial function
w(
x
)
=
N
u
w
=
w
T
N
u
,
the potential energy becomes
Since
N
u
w
w
T
1
N
u
p
z
dx
2
EI
L
0
L
N
T
u
N
u
dx
(
w
)
=
w
−
(7.53)
0
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