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6 Answers to Supplementary Problems
6.2 Answers for Problems from Chap. 2
2.3 Finite difference approximation of a cantilevered rod with distributed load
based on five domain nodes
The linear system of equations reads as:
E A
X
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
2 −1 0 0
−1 2 −1 0
0 −1 2 −1
0 0 −1 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
X p 0
X p 0
X p 0
X p 0
X p 0
4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(6.5)
or solved for the nodal unknowns:
⎡
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
13
4
X
2 p 0
E A
11
2
X
2 p 0
E A
27
4
X
2 p 0
E A
7
1
X
2 p 0
E A
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
(6.6)
The analytical solution can be taken from [1] as u 5 =
p 0 L
2
2E A
whereas the finite difference solution gives u 5 =
7
16
p 0 L
2
E A
≈ 0.4375 ×
p 0 L
2
E A
.
2.4 Refined finite difference approximation of a cantilevered rod with distributed load
In generalization of Eq. (6.5), the following scheme can be proposed:
E A
X
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
2 −1 0 0 · · · 0
−1 2 −1 0 · · · 0
0 −1 2 −1 · · · 0
. . . · · ·
· · ·
. . .
0 · · · 0 −1 2 −1
0 · · · 0 0 −2 2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
. . .
u n−1
u n
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
X p 0
X p 0
X p 0
. . .
X p 0
X p 0
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(6.7)
where X =
L
n−1
for equidistant spacing.
pg