8.5 The Galerkin Method
483
above integral result of the Galerkin equation need only some modification, the
corresponding constants a 1 , a 2 and the corresponding formula still can be found.
If putting m =
q
D
, u = w, then Eq. (8.5.23) is the basic differential equation
of the small deflection of thin plate bending in elastic mechanics [it is called the
Lagrange(’s) equation] and the four edges clamped boundary conditions of rectangular thin plate, where, q is the load per unit area on the neutral surface, D is the
bending stiffness of the thin plate, w is the deflection of the thin plate. In the case that
q is the uniformly distributed load, the maximum deflection of the thin plate appears
in the symmetric center of the thin plate, if a = b, then the analytical solution is
w max = 0.02016
qa
4
D
However, the maximum deflection of the first and the second approximate solution
of the Galerkin method are respectively
w 1 max =
49
2304
qa
4
D
= 0.02126736
qa
4
D
w 2 max =
20713
1025280
qa
4
D
= 0.02020229
qa
4
D
Comparing with the analytical solution, the relative errors are 5.492863% and
0.209753% respectively.
If b = 2a, then the analytical solution and the second approximate solution of the
Galerkin method are respectively
w max = 0.040640
qa
4
D
, w 2 max = 0.0403113
qa
4
D
Comparing with the analytical solution, the relative error is 0.808792%. It follows
that for such problems, the second approximate solution of the Galerkin method has
the very good approximateness, therefore the expression (8.5.25) within a certain
range almost can be used as the analytical solution formula.
It should be pointed out that the choice method of the coordinate functions is not
unique, for instance, it can choose in this example
u =
∞
m=1
∞
n=1
a mn
1 − cos
mπ(x + a)
a
1 − cos
nπ(y + b)
b
as the coordinate function, it satisfies the boundary conditions. If m, n only choose
the first terms, then there is
u = a 11
1 − cos
π(x + a)
a
1 − cos
π(y + b)
b
483
above integral result of the Galerkin equation need only some modification, the
corresponding constants a 1 , a 2 and the corresponding formula still can be found.
If putting m =
q
D
, u = w, then Eq. (8.5.23) is the basic differential equation
of the small deflection of thin plate bending in elastic mechanics [it is called the
Lagrange(’s) equation] and the four edges clamped boundary conditions of rectangular thin plate, where, q is the load per unit area on the neutral surface, D is the
bending stiffness of the thin plate, w is the deflection of the thin plate. In the case that
q is the uniformly distributed load, the maximum deflection of the thin plate appears
in the symmetric center of the thin plate, if a = b, then the analytical solution is
w max = 0.02016
qa
4
D
However, the maximum deflection of the first and the second approximate solution
of the Galerkin method are respectively
w 1 max =
49
2304
qa
4
D
= 0.02126736
qa
4
D
w 2 max =
20713
1025280
qa
4
D
= 0.02020229
qa
4
D
Comparing with the analytical solution, the relative errors are 5.492863% and
0.209753% respectively.
If b = 2a, then the analytical solution and the second approximate solution of the
Galerkin method are respectively
w max = 0.040640
qa
4
D
, w 2 max = 0.0403113
qa
4
D
Comparing with the analytical solution, the relative error is 0.808792%. It follows
that for such problems, the second approximate solution of the Galerkin method has
the very good approximateness, therefore the expression (8.5.25) within a certain
range almost can be used as the analytical solution formula.
It should be pointed out that the choice method of the coordinate functions is not
unique, for instance, it can choose in this example
u =
∞
m=1
∞
n=1
a mn
1 − cos
mπ(x + a)
a
1 − cos
nπ(y + b)
b
as the coordinate function, it satisfies the boundary conditions. If m, n only choose
the first terms, then there is
u = a 11
1 − cos
π(x + a)
a
1 − cos
π(y + b)
b