2.2 Constant Material and Geometry Parameters
15
E A
u i+1 − 2u i + u i−1
X 2
= −p i ,
(2.8)
or as
E A
X
(−u i+1 + 2u i − u i−1 ) = R i ,
(2.9)
where the equivalent nodal force R i , resulting from a distributed load p(X ), is
in general given for an inner node i as: R i =
x/2
−x/2 p( ˆ
x)d ˆ
x. In this integral, the
local coordinate x has its origin at the location of node i. In the case of the boundary nodes, the equivalent nodal loads must be calculated as R 1 =
x/2
0
p( ˆ
x)d ˆ
x or
R n =
0
−x/2 p( ˆ
x)d ˆ
x in order to completely distribute the entire load p(X ) to the
nodes.
Let us assume for simplicity that the distributed load shown in Fig. 2.4 is zero
( p(X ) = 0) and introduce n = 5 nodes, i.e. two boundary nodes and three inner
nodes. The evaluation of the finite difference approximation according to Eq. (2.9)
at the inner nodes i = 2, . . . , 4 gives:
node 2:
E A
X
(−u 3 + 2u 2 − u 1 ) = 0 ,
(2.10)
node 3:
E A
X
(−u 4 + 2u 3 − u 2 ) = 0 ,
(2.11)
node 4:
E A
X
(−u 5 + 2u 4 − u 3 ) = 0 .
(2.12)
Considering at both ends displacement boundary conditions, i.e. Eq. (2.5), the system
of equations (2.10) till (2.12) can be written with u 1 = 0 and u 5 = u 0 as
node 2:
E A
X
(0 + 2u 2 − u 3 ) = 0 ,
(2.13)
node 3:
E A
X
(−u 2 + 2u 3 − u 4 ) = 0 ,
(2.14)
node 4:
E A
X
(−u 3 + 2u 4 ) =
E A
X
u 0 ,
(2.15)
or in matrix notation as
E A
X
⎡
⎣
2 −1 0
−1 2 −1
0 −1 2
⎤
⎦
⎡
⎣
u 2
u 3
u 4
⎤
⎦ =
⎡
⎣
0
0
E A
X
u 0
⎤
⎦ .
(2.16)