Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 458

∂/∂

where it has been assumed that no applied forces occur on the boundaries. From w = 0,

we obtain the set of equations
EI L
0

L

N T
u

N u dx p z N u dx

w − = 0

0

(7.54)

k u w − p u = 0

or

k u w = p u

As is to be expected, we have obtained the same relations found from the principle of
virtual work with

δ W = 0 .

Solution of Differential Equations
As described above, the Ritz method uses an approximation function of the form of Eq. (7.9)
to satisfy

, stationary. The successful application of the
Ritz method depends on the construction of the expression

δ W = 0 or to make a functional,

δ W = 0 or the functional ,

frequently a weak form, for which the derivatives, with respect to u i , are taken to make

it stationary. For the case that the operator L of Eq. (7.1) is linear, self-adjoint and positive
definite and the essential boundary conditions are homogeneous, a functional of the form

u T Lu 2 u T f

( u ) =

L (

− ) dx (7.55)

can be obtained. Although the expression of Eq. (7.55) is written for one-dimensional prob-
lems, it applies to three-dimensional problems if the longitudinal coordinate x is replaced
by the volume V

. It can be shown (Proble m 7.18) that the stationary value of this integral,

δ( u ) = 0 , is equivalent to solving Lu − f = 0 . Thus, to solve the differential equations

Lu − f = 0 , form the functional ( u ) of Eq. (7.55), employ a trial solution of Eq. (7.9), take

derivatives of with respect to u i as in Eq. (7.48), and solve the resulting set of simultaneous

linear algebraic equations for u i .

7.4.2 Galerkin's Method

Galerkin's method is frequently treated from a variational viewpoint. For the mechanics of
solids, Galerkin's method is widely used and is considered to be one of the most viable tech-
niques available. It applies to problems for which the governing equations are expressed
in differential equation (local) form. This includes boundary value and eigenvalue prob-
lems, such as static, stability, vibration, and even geometrically nonlinear problems. Trial
solutions which satisfy all boundary conditions are normally required by the method.
Return to the weighted-residual formulation of Section 7.3.6. For a linear elastic solid,
the residual is given by Eq. (7.11) as

D T ED

R = u + p V (7.56)

Galerkin's expression of Eq. (7.28) becomes

N u ( D T EDN u u + p V ) dV = 0 (7.57a)

V

or

N u D T EDN u dV N u p V u + dV = 0

V V

(7.57b)

− =

k u u p u 0