Information Technology Reference
In-Depth Information
difference method that must satisfy all boundary conditions. The principle of virtual work
formulation assures that the coefficient matrix for the system of algebraic equations is
symmetric and positive definite.
This coefficient matrix for the system of equations can be formed by assembling it in the
same fashion as a global stiffness matrix is assembled. Indeed, the variationally based finite
difference procedure is quite similar to the finite element method and, in some circum-
stances, can even lead to the same equations [Bushnell, 1973]. Sometimes, finite differences
are treated as being a special case of finite element methods. It follows, that one could
establish physical element models that correspond to the finite difference discretization.
Although the finite difference and finite element methods can be considered to be related,
the finite element procedure is clearly the most dominant of the methods in use in structural
mechanics today and has been successfully implemented into powerful, general purpose
computer programs. Several example problems will be used to illustrate the development
and application of variationally based finite differences.
EXAMPLE 8.4 Beam with Linearly Varying Loading
Consider the beam of Fig. 8.6 that was treated in Example 8.2 with conventional finite
differences.
The principle of virtual work, for a beam with no concentrated loads applied on its ends,
takes the form
L
L
w δw dx
δ
W
=
EI
p z δw
dx
=
0
(1)
0
0
Division of the integration into six segments leads to
L i
5
w
i
δw
i
δ
W
=
(
EI
p z δw i )
dx
=
0
0
i
=
0
w and EI be constant over each interval and let L 0 and L 5 be L
Let
/
10 and L 1 through L 4
/
be L
5. Upon integration over the length, the expression for the principle of virtual work
becomes
L
10
L
5
L
10
δ
W
=
δ
W 0 +
W 1 + δ
W 2 + δ
W 3 + δ
W 4 ) +
δ
W 5 =
0
(2)
with
W i = δw
)w
i
δ
(
EI
p i δw i
(3)
i
Insertion of the central difference expression
1
h 2
w
i
=
(w
2
w
+ w
)
(4)
i
1
i
i
+
1
into (3) gives
EI
5 4 [
δ
W i =
[
δw i 1
2
δw i + δw i + 1 ]
w i 1
2
w i + w i + 1 ]
p i δw i
(5)
The q uantity p i c an be assi gn ed the val u e of the distri bu ted load at point i . Thus, p 0 =
p 0 , p 1 =
4 p 0
/
5 , p 2 =
3 p 0
/
5 , p 3 =
2 p 0
/
5 , p 4 =
p 0
/
5 , and p 5 =
0
.
The displacement bound-
w
= w 0 = w
=
.
w 0 =
w 1
= w
ary conditions are
0
The condition
0 leads to
1 . The virtual
0
5
Search WWH ::




Custom Search