4.2 Approximation of the Differential Equations
97
Application of the forward scheme to the left-hand boundary node (i = 1) and the
backward scheme to the right-hand boundary node (i = 1) gives the following two
relationships for the elimination of the rotations at the boundaries:
φ 1 =
4
3
φ 2 −
1
3
φ 3 ,
(4.27)
φ 5 =
4
3
φ 4 −
1
3
φ 3 .
(4.28)
Introducing these relationships in the system of equations given by Eqs. (4.19) till
(4.24) and considering that the load is opposed to the positive Y -direction, the system
of equations can be written in matrix from as given in Eq. (4.30).
The solution of this linear system of equations gives in the middle of the beam
under consideration of ((X =
L
4
) the vertical displacement as:
u 3 = u
L
2
= −
32E I Y + 3 L
2 k s AG
L
2 q 0
4k s AG
64E I Y + L 2 k s AG
.
(4.29)
The analytical solution is given in [9] as u Z
L
2
=
5q 0 L
4
384E I Y
+
q 0 L
2
8k s AG
and the relative
error results for the given numerical values as −97.208%. Thus, the approximation
is not very satisfying and a higher node density is required to improve the results.
⎡
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⎢
⎢
⎢
⎢
⎢
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⎢
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⎢
⎢
⎢
⎢
⎣
0
−
2E I Y
3X 2 − k s AG −
k s AG
2X
2E I Y
3X 2
0
0
−
2k s AG
X 2
−
2k s AG
3X
k s AG
X 2
2k s AG
3X
0
0
k s AG
2X
E I Y
X 2
0
−
2E I Y
X 2 − k s AG −
k s AG
2X
E I Y
X 2
k s AG
X 2
−
k s AG
2X
−
2k s AG
X 2
0
k s AG
X 2
+
k s AG
2X
0
0
k s AG
2X
2E I Y
3X 2
0
−
2E I Y
3X 2 − k s AG
0
0
k s AG
X 2
−
2k s AG
3X
−
2k s AG
X 2
+
2k s AG
3X
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
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⎥
⎥
⎥
⎥
⎥
⎥
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⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
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⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
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⎢
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⎢
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⎢
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⎢
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⎢
⎢
⎢
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⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
φ 2
u 3
φ 3
u 4
φ 4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
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⎢
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⎢
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⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
q 0
0
q 0
0
q 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(4.30)