Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 441

Choose a single collocation point at ξ = 1 / 2 , i.e., x = L / 2. Substitute (8) and (9) into (1),

i v

L 4

and note that w ( x ) = 48 w / . Then, from (1)

1

EI 48

p 0
2 = L 4 w

−

0 (10)

1

and

p 0 L 4
96 EI

w =

(11)

1

From (8), the approximate deflection is

p 0 L 4
96 EI (

2 3 4

w = ξ − ξ + ξ )

3 5 2 (12)

which can be compared to the exact solution, which was derived in the previous chapters.

p 0 L 4
120 EI (

2 3 4 5

w = 4 ξ − 8 ξ + 5 ξ − ξ ) (13)

A comparison of the deflection of (12) and (13) gives

Deflection w

Exact ( p 0 L 4
EI

Approximate ( p 0 L 4
EI

ξ = x/L ) ) Error (%)

/ . ×

10 − 3

. ×

10 − 3

1 4 1 196 1 22 2.0

1 / 22 . 34375 × 10 − 3 2 . 604 × 10 − 3 11.1

3 / 41 . 83105 × 10 − 3 2 . 19 × 10 − 3 19.6

Orthogonal Collocation
The collocation method, which only requires that the residual be evaluated at the col-
location points, would appear to be the simplest of weighted-residual methods. An im-
provement to this procedure is to select judiciously the collocation points. A proper choice
makes the computations more convenient and the results more accurate. One such method,
which is discussed in several textbooks, is the orthogonal collocation technique proposed
by Lanczos 3 (1939). With this method, the collocation points are chosen to be the roots
of orthogonal polynomials such as the Legendre polynomials [Chapter 6, Eq. (6.123)].
Further simplicity is achieved if the constants u i in the trial solutions are replaced by the

values of the trial solution at the collocation points. These values then become the coeffi-
cients to be determined.

3 Cornelius Lanczos was born (Kornel L owry) (1893-1974) and educated in Hungary. After many years of working
in German universities, he moved to Purdue University in 1931. After World War II he joined the engineering
staff at Boeing Airplane Co., then the National Bureau of Standards, and then returned to the academic world
at UCLA. After a brief stint at North American Aviation, in 1954 he took a position in Ireland as a professor of
physics at the Dublin Institute for Advanced Studies. He authored the topics The Variational Principles of Mechanics,
Applied Analysis, Linear Differential Operators, Albert Einstein and the Cosmic World Order, Discourse on Fourier Series,
Numbers Without End, Space Through the Ages, and Einstein Decade: 1905-1915.