20
2 Investigation of Rods in the Elastic Range
Fig. 2.7 Rod with linear varying cross-section area discretized by five nodes in a finite difference
approach
v i+
1
2
= k i+
1
2
du
dX
i+
1
2
= k i+
1
2
u i+1 − u i
,
(2.37)
v i−
1
2
= k i−
1
2
du
dX
i−
1
2
= k i−
1
2
u i − u i−1
X
.
(2.38)
Thus, the finite difference approximation of the problem given in Eq. (2.35) can be
written for node i as:
1
X
−k i−
1
2
u i−1 + (k i−
1
2
+ k i+
1
2
)u i − k i+
1
2
u i+1
= 0 .
(2.39)
It should be noted here that the last equation was multiplied by . This is in general
required to obtain from distributed loads the equivalent nodal force R i , cf. Sect. 2.2.
In addition, the last equation was multiplied by ‘-1’ to make the comparison with the
finite element approach easier.
In order to simplify the explanations of the different approaches and possibilities,
let us consider in the following a rod with linear varying cross-sectional area as
shown in Fig. 2.7. This structure should be discretized by five nodes and either a
displacement u 0 or a force F 0 should be prescribed at the right-hand end.
Evaluation of the finite difference scheme given in Eq. (2.39) for the inner nodes
shown in Fig. 2.7 gives:
node 2:
E
X
−
A 1 + A 2
2
u 1 +
A 1 + A 2
2
+
A 2 + A 3
2
u 2 −
A 2 + A 3
2
u 3
= 0 ,
(2.40)
node 3:
E
X
−
A 2 + A 3
2
u 2 +
A 2 + A 3
2
+
A 3 + A 4
2
u 3 −
A 3 + A 4
2
u 4
= 0 ,
(2.41)