70
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
u 7
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −
F 0 L
3
E I Y
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
36
17
216
4
27
25
108
35
108
91
216
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(3.152)
and the relative error right-hand side of the beam is obtained as:
relative error =
91
216
−
1
3
1
3
× 100 = 26.39% .
(3.153)
From the above calculations, it is easy to derive a general scheme for n nodes (n > 4).
In generalization of Eq. (3.151), the following scheme can be proposed:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 0 0 0 0 · · · 0
−2 1 0 0 0 · · · 0
1 −2 1 0 0 · · · 0
0 1 −2 1 0 · · · 0
. . . · · ·
· · ·
. . .
0 · · · 0 1 −2 1 0
0 · · · 0 0 1 −2 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
. . .
u n−1
u n
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −F 0 L
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
1 −
1
n−1
1 −
2
n−1
1 −
3
n−1
. . .
1 −
n−3
n−1
1 −
n−2
n−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(3.154)
where X =
L
n−1
for equidistant spacing.
3.5 Finite difference approximation of a stepped cantilevered Euler–Bernoulli
beam with a single force based on four or five domain nodes
Given is a stepped Euler–Bernoulli beam of length L with a bending stiffness of
E(2I Y ) in the range 0 ≤ X ≤ L/2 and a value of E I Y in the range L/2 ≤ X ≤ L as
shown in Fig. 3.21. The beam is loaded by a single force F 0 at its right-hand boundary.
Use four or five domain nodes of equidistant spacing, i.e. X =
L
3
or X =
L
4
, for
the finite difference approximation. Use only finite difference approximations of
second-order accuracy for the nodal evaluations and boundary conditions. Perform
the evaluations (a) starting from Eq. (3.27) and as an alternative considering (b)
Eq. (3.25). Determine the vertical displacements at the nodes and compare your
result with the analytical solution.
3.5 Solution
(a) The finite difference discretization of the cantilevered beam is shown in Fig. 3.22
for four and five domain nodes.
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
u 6
u 7
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −
F 0 L
3
E I Y
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
36
17
216
4
27
25
108
35
108
91
216
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(3.152)
and the relative error right-hand side of the beam is obtained as:
relative error =
91
216
−
1
3
1
3
× 100 = 26.39% .
(3.153)
From the above calculations, it is easy to derive a general scheme for n nodes (n > 4).
In generalization of Eq. (3.151), the following scheme can be proposed:
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 0 0 0 0 · · · 0
−2 1 0 0 0 · · · 0
1 −2 1 0 0 · · · 0
0 1 −2 1 0 · · · 0
. . . · · ·
· · ·
. . .
0 · · · 0 1 −2 1 0
0 · · · 0 0 1 −2 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
u 2
u 3
u 4
u 5
. . .
u n−1
u n
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= −F 0 L
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
1 −
1
n−1
1 −
2
n−1
1 −
3
n−1
. . .
1 −
n−3
n−1
1 −
n−2
n−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(3.154)
where X =
L
n−1
for equidistant spacing.
3.5 Finite difference approximation of a stepped cantilevered Euler–Bernoulli
beam with a single force based on four or five domain nodes
Given is a stepped Euler–Bernoulli beam of length L with a bending stiffness of
E(2I Y ) in the range 0 ≤ X ≤ L/2 and a value of E I Y in the range L/2 ≤ X ≤ L as
shown in Fig. 3.21. The beam is loaded by a single force F 0 at its right-hand boundary.
Use four or five domain nodes of equidistant spacing, i.e. X =
L
3
or X =
L
4
, for
the finite difference approximation. Use only finite difference approximations of
second-order accuracy for the nodal evaluations and boundary conditions. Perform
the evaluations (a) starting from Eq. (3.27) and as an alternative considering (b)
Eq. (3.25). Determine the vertical displacements at the nodes and compare your
result with the analytical solution.
3.5 Solution
(a) The finite difference discretization of the cantilevered beam is shown in Fig. 3.22
for four and five domain nodes.