Information Technology Reference
In-Depth Information
p
=
p
on
S
p
is
u
T
p
T
δ
J
=
S
p
δ
(
p
−
p
)
dS
+
S
u
δ
(
u
−
u
)
dS
=
0
(7.80)
for a continuum and
]
0
+
]
0
=
δ
J
=
[
δw(
V
−
V
)
+
δθ(
M
−
M
)
[
δ
V
(w
−
w)
+
δ
M
(θ
−
θ)
0
(7.81)
on
S
p
on
S
u
for a beam.
Since Eqs. (7.80) and (7.81) are simply global forms of the boundary conditions, as well
as constituting a form of a variational principle, the Trefftz method like Galerkin's method
can be applied to problems for which a variational principle does not necessarily exist.
For Trefftz's method, use the trial solution
u
=
N
p
+
N
u
u
(7.82)
where
N
p
is a vector of particular solutions of the differential equations for the problem. The
term
N
u
u
is a set of linearly independent functions satisfying the homogeneous differential
equations. The parameters
u
are to be chosen to approximate the boundary conditions.
For a beam, choose a trial solution of the form
w(
x
)
=
N
p
+
N
u
w
=
w
0
(
x
)
+
N
u
(
x
)
w
(7.83)
Substitute the trial solution in the variational expression of Eq. (7.81). Note that
δw(
x
)
=
)δ
T
N
u
,
and use
θ
=−
w
,M
w
,
and
V
w
N
u
(
x
w
=
δω
=−
EI
=−
EI
=
δ
)(w
−
w
)
L
0
δ
J
w
T
N
u
(
−
)(w
−
w
)
−
δ
w
T
N
u
EI
(
−
EI
on
S
p
L
0
w
T
N
T
u
w
T
N
T
u
(w
−
w
)
+
−
EI
δ
(w
−
w)
+
EI
δ
=
0
on
S
u
or
w
T
N
u
(w
−
w
)
−
N
u
(w
−
w
)
L
0
+
N
T
(w
−
w
)
L
0
=
N
T
u
δ
(w
−
w)
−
0
(7.84)
u
on
S
p
on
S
u
These relations can be used to determine the unknown parameters
w
.
EXAMPLE 7.11 Beam with Linearly Varying Loading
Apply Trefftz's method to the beam of Fig. 7.1.
As can be observed from Ritz's method solutions for this beam, the particular solution
can be chosen as
p
0
L
4
120
EI
(
4
5
w
0
(
x
)
=
5
ξ
−
ξ
)
(1)
i
v
)
To verify the appropriateness of this
w
(
x
)
,
substitute it into
EI
w
=
p
z
.
For the
N
u
(
x
w
0
term of the trial solution, select the simple polynomial
2
]
w
2
1
N
u
w
=
ξ w
+
ξ
w
=
[
ξξ
(2)
1
2
w
2
so that
p
0
L
4
120
EI
(
2
4
5
w(
x
)
=
ξ w
+
ξ
w
+
5
ξ
−
ξ
)
(3)
1
2
Search WWH ::
Custom Search