66
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
node 6: 0 + 0 + u 4 − 4u 5 + 5u 6 − 2u 7 = 0 ,
(3.132)
node 7: 0 + 0 + 0 + 2u 5 − 4u 6 + 2u 7 = −
X
3 F 0
E I Y
,
(3.133)
or in matrix notation:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
7 −4 1 0 0 0
−4 6 −4 1 0 0
1 −4 6 −4 1 0
0 1 −4 6 −4 1
0 0 1 −4 5 −2
0 0 0 2 −4 2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
u 7
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −
X
3F 0
E I Y
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
0
0
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(3.134)
The solution of this linear system of equations gives the unknown nodal values as:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
u 7
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −
F 0 L
3
E I Y
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
144
11
432
23
432
19
216
55
432
73
432
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(3.135)
and the relative error right-hand side of the beam is obtained as:
relative error =
73
432
−
1
3
1
3
× 100 = 49.31% .
(3.136)
From the above calculations, it is easy to derive a general scheme for n nodes (n > 7).
In generalization of Eq. (3.134), the following scheme can be proposed:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
7 −4 1 0 0 0 0 · · · 0
−4 6 −4 1 0 0 0 · · · 0
1 −4 6 −4 1 0 0 · · · 0
0 1 −4 6 −4 1 0 · · · 0
. . . · · ·
· · ·
. . .
0 · · · 0 1 −4 6 −4 1 0
0 · · · 0 0 1 −4 6 −4 1
0 · · · 0 0 0 1 −4 5 −2
0 · · · 0 0 0 0 2 −4 2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
. . .
u n−4
u n−3
u n−2
u n−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −
X
3 F 0
E I Y
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
0
. . .
0
0
0
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(3.137)
where X =
L
n−1
for equidistant spacing.