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L et x
= ξ +
x a , where x a is the x coordinate of the left end of the element. Then
p i z =
2
4 p 0 [ a i
ξ
+
b i
ξ +
c i ] where
L
2
x a
L 2
b i = L
2
x a
L 2
x a
L
a i =−
c i =
x 6
90 L 2 +
x 5
30 L
Lx 3
9
L 2 x 2
6
p 0
EI
Answer:
Exact solution:
w =
+
7.35 Use a finite element procedure to solve the torsion problem of Problem 7.24.
7.36 Use a beam with distributed load p z (
, a fixed left end, and a pinned right end to
demonstrate the equivalence of the Galerkin and Ritz methods.
x
)
Hint:
Begin with the potential energy
L
EI
2
d 2
dx 2 2
dx
w
w |
=
p z w
M
x
=
L
0
Introduce
w =
N u w
, and set
∂/∂w i =
0 for Ritz's method. Use integration by
parts to find the Galerkin formulation.
Answer: For the beam problem, if the trial function satisfies all of the bound-
ary conditions (essential and natural), Ritz's method is equivalent to Galerkin's
method. In general, Galerkin's method can be employed even if the operator of
the governing differential equation does not exhibit particular properties, such
as being positive definite. As a result, it might seem that the Galerkin method is
more generally applicable and should be favored over Ritz's method. On the other
hand, in the case where the operator is linear, symmetric, and positive definite, a
weak form can be formulated easily and Ritz's method is preferable. The require-
ment on the order of the continuity of the trial solution need not be so high for
Ritz's method as for Galerkin's method. Also, the trial solution for Ritz's method
only needs to satisfy the displacement (essential) boundary conditions, while for
Galerkin's method, both the displacement and the static (natural) conditions must
be satisfied.
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