Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 459

Thus, the unknowns u are obtained by solving

k u u = p u (7.58)

i v

N u

In the case of a uniform beam, R = ( EI w ) − p z and w = w , so that Galerkin's relations

would be

L L

EI N i u ) N u ( N u dx w − p z dx = 0

0 0

(7.59a)

− =

k u w p u 0

or

k u w = p u (7.59b)

Remember that it is necessary that the basis functions N u satisfy all of the boundary condi-
tions. They must also be sufficiently differentiable.
Galerkin's method can also be viewed from the standpoint of a variational technique.
R ecall from Chapter 1 or 2 that the solid continuum relations of Eq. (7.5a), D T E Du +

D T ED

p V = 0 , is an equilibrium expression. Thus, the residual (Eq. 7.56) R = u + p V is

an “out-of-balance” force. A reasonable variational integral would seem to be

u T R dV

V δ

=

0 (7.60)

since this is of the form of work. Thus,

u T D T ED

V δ

( u + p V ) dV = 0 (7.61)

N u

With u = u , Eq. (7.61) becomes

u T N u ( D T ED N u

V δ

+

p V )

=

u dV 0 (7.62a)

or

u T dV dV

δ N u ( D T ED N u ) N u p V

u + = 0

(7.62b)

V V

k u − p u

so that u can be obtained from the familiar linear relationship

k u u = p u (7.63)

These are the same expressions given in Eq. (7.57), which were based on Galerkin's
weighted residual method.
Galerkin's relations of Eq. (7.59) for a uniform beam are readily derived from a variational
integral. Begin with

L
0 (

i v

EI w − p z )δw dx = 0 (7.64)

which, with w = N u w , becomes

w T L
0

N u EI N i u dx N u p z dx

L

δ

w − = 0 (7.65a)

0