The Finite Difference Method - Mechanics of Structures: Variational and Computational Methods - page 525

and

γ
−

2

12 γ

− β 02 γ Symmetric

0 − 2 β − 2 γ

0 0 − β 0 γ

0 0 0 − 2 β − 12 γ

ν − β 0 0002 γ

E ∗

K =

0 ν 0 000 − βγ

β − β α ν − α γ

0 002

− α α α − β α − − β γ

2 0 20 2 4

0 − α 0 α 000 − 1 0 − 2 β 2 γ

0 0 β 0 − βν 00 − 1 0 0 γ

0 0 − α α β − β 00 0 − 20 − βα

0 0 0 − α 0 α 0 0 0 0 − 10 − βγ

with E ∗ =

2

h 2 , 2.

From the dimensions of the plate in Chapter 6, Fig. 6-14 and the mesh layout in Fig. 8.12,
we have h

tAE /( 1 − ν ) α = ( 1 + ν)/ 2 , β = ( 1 − ν)/ 2 , and γ = ( 3 − ν)/

30 GN/m 2 ,

= 1 . Assign the numerical values t = 0 . 2m, E = ν = 0 . 0 .

Solve equation (12) to compute the displacements at the nodal points

u x u y

10 − 4

1 − 2 . 5 × 0

10 − 4 10 − 4

2 − 2 . 02 × − 0 . 44 ×

10 − 4 30 − 0 . 55 ×

10 − 4 10 − 4

4 − 0 . 059 × − 0 . 183 ×

10 − 4 10 − 4

5 − 0 . 0648 × − 0 . 48 ×

10 − 4 60 0 . 062 ×

. ×

10 − 4

− . ×

10 − 4 70 15 0 25

10 − 4 10 − 4

80 . 11 × − 0 . 55 ×

10 − 4 90 − 0 . 74 ×

The displacement pattern is shown in Fig. 8.14. It is similar to that of the finite element
solution in Chapter 6, but the numerical values are lower than those for the finite element
method. This means that the finite difference method appears to “stiffen” the structure. For
a better result, a finer mesh should be employed.

It is, of course, possible to employ global (integral) formulations other than the principle
of virtual work as the basis of finite difference approximations. For example, such weighted
residual approaches as Galerkin's method are quite suitable. If the Hellinger-Reissner func-
tional of Chapter 2 is utilized, both displacement and force variables occur in this mixed
variational functional and, hence, in the derived finite difference equations. See Noor and
Schnobrich (1973), Noor, et al. (1973), and Pian (1971) for finite difference formulations
utilizing mixed methods.