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i.e., the Euler equation for (9) is
5
2
ψ
5
2
ψ
1
−
=−
(12)
1
with boundary conditions
ψ
1
=
0at
y
=±
2
.
The solution to (12) is
sinh
√
10
1
2
y
sinh
√
40
sinh
5
1
2
ψ
(
y
)
=
−
/
(13)
1
and the final approximate solution is given by (3) with
ψ
1
of (13).
7.4.4 Extended Methods
Extended Ritz's Method
Since Ritz's method can be based on the principle of virtual work, i
t i
s required that the
trial functions satisfy the displacement boundary conditions, i.e.,
u
u
on
S
u
. It is possible
to extend this principle and, hence, the method, to relax the conditions to be satisfied by
the trial functions. Then this extended Ritz's method provides a solution that fulfills the
displacement boundary conditions approximately, as well as the conditions of equilibrium
and the static boundary conditions.
In order to change the formulation such that the trial solution need not satisfy the dis-
placement boundar
y
conditions, we will supplement the virtual work expressions with the
global form of
u
=
−
u
=
0on
S
u
.
Thus, we will add the integral
p
T
W
=
(
u
−
u
)
dS
(7.67)
S
u
to the work expression. Then for a continuum, the virtual work of Eq. (7.43) can be extended
to
T
σ
dV
u
T
p
V
u
T
p
dS
p
T
−
δ
W
=
V
δ
−
V
δ
dV
−
S
p
δ
−
δ
(
u
−
u
)
dS
=
0
(7.68a)
S
u
and for a beam, Eq. (7.44) would be adjusted to become
L
L
w
δw
dx
]
0
−
δ
W
=
EI
−
p
z
δw
dx
−
[
V
δw
+
M
δθ
0
0
on
S
p
]
0
=
−
δ
[
V
(w
−
w)
+
M
(θ
−
θ)
0
(7.68b)
on
S
u
We will convert the beam expression into a form expressed in terms of displacements
only. For a unifo
rm b
eam, set
θ
=−
w
,
M
w
, and
V
w
.
=−
EI
=−
EI
Also, set all applied
displacements
(w
,
θ)
and forces
(
V, M
)
on the boundaries equal to zero. This reduces the
variational principle to the form
L
L
w
(
)δw
(
δw
(
)
+
w
(
−
δ
W
=
EI
x
x
)
dx
−
p
z
(
x
)δw(
x
)
dx
+
[
x
)
EI
w(
x
x
)
EI
δw(
x
)
0
0
−
δw
(
w
(
)
−
w
(
δw
(
]
0
=
x
)
EI
x
x
)
EI
x
)
0
(7.69)
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