The Finite Difference Method - Mechanics of Structures: Variational and Computational Methods - page 528

FIGURE P8.9

= / = = ·

m 2 ,

8.9 Find the deflection at x L 2 of the beam of Fig. P8.9. Let P 5 kN, EI 30 000 kN

10 m. Use a variationally based finite difference scheme.
8.10 Show that use of certain types of variational formulations of finite differences can
assure symmetry and positive definiteness of the coefficient matrix of the algebraic
(stiffness) equations.
8.11 Use simple central differences to find the deflections along the beam of Fig. P8.11.

and L =

FIGURE P8.11

Answer:

= (

P 1
P 2
P 3
P 4

5 − 410 w

1

− 46 − 41 w

2

L 3

/ 125 EI )

1 − 46 − 4 w

3

01 − 45 w

4

8.12 A simply supported beam of length L is subjected to a uniformly distributed load of
magnitude p 0 . Find the distribution of deflection along the beam using

(a) conventional finite differences
(b) variationally based finite differences

Answer: Use a subdivision of h = L / 6 . The conventional approach with central

differences gives w = w = 0 and

1 7

− w

5 4100 1

1
1
1
1

1

− − w

46 410

2

p 0 h 4

1 − 46 − 41 w

= ( / EI )

3

01 − 46 − 4 w

4
w 5 001 − 45