Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 445

For the beam of Fig. 7.1, suppose there is a single domain, and Eq. (8) of Example 7.1 is
used to approximate the displacement. Then Eq. (7.24) becomes
1 EI 48 d

L 4 w

− p 0 ( 1 − ξ) ξ = 0 (3)

1

0

i.e.,

EI 48

p 0
2 = L 4 w 1 −

0 (4)

This is the same result obtained in Example 7.1.

7.3.5 Orthogonality Methods
A variety of techniques can be classified as orthogonality methods. Define a set of linearly
independent functions Ψ j ( x ) ,r = 1 , 2 , ... ,n, in domain V . The integrals

Ψ j RdV = 0 j = 1 , 2 , ... ,n (7.26a)

V

or

ΨR dV = 0

V

which form a system of n equations, are referred to as the orthogonality conditions . For a
p -dimensional problem, let Ψ ( k ) be a p × p diagonal matrix of functions for k = 1 , 2 , ... n .

The orthogonality method requires that

Ψ ( k ) R dV = 0 k = 1 , 2 , ... n (7.26b)

V

with the total number of equations being np . Equations (7.26) are obtained from Eq. (7.14)
by choosing h to be the identity mapping. There are several useful methods, e.g., Galerkin's, 7
least squares, and method of moments, which employ the orthogonality condition of
Eq. (7.26).

7.3.6 Galerkin's Method

If Ψ j in Eq. (7.26a) are chosen to be the m trial functions N ui , then

N ui L u i N ui f dV

N ui RdV = − = 0 i = 1 , 2 , ... ,m (7.27)

V V

For a p -dimensional problem, Galerkin's method requires the mp conditions

N u 1 i R 1 dV

=···= N u 2 i R p dV ,m or V N u R dV

= 0 ,i = 1 , 2 , ... = 0 , i.e.,

N u ( N u ( N u f dV

LN u

u − f ) dV = LN u ) dV u − = 0

(7.28)

V V V

k u u − p u = 0

7 Boris Grigorievich Galerkin (1871-1945) was a Russian engineer, graduate of the Petersburg Technological
Institute. He lectured at several colleges in the St. Petersburg area, including Leningrad University, where he
became dean of the structural engineering department. He contributed to several challenging problems, such as
the curvature of thin plates. His approximate solution of differential equations is utilized today for the solution
of many applied mechanics problems.