62
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Table 3.5 Values of the internal bending moment at the grid points (see Fig. 3.17)
Grid point
Coordinate X
M Y (X )
1
0
0
2
L
6
−
F0 L
12
3
L
3
−
F0 L
6
4
L
2
−
F0 L
4
5
2L
3
−
F0 L
6
6
5L
6
−
F0 L
12
7
L
0
where X =
L
n−1
for equidistant spacing.
Let us look now on the second approach which is based on the second-order partial
differential equation for the moment distribution, see Table 3.1, 3rd form of the PDE.
Using a centered difference scheme (O((X
2
)) for the second-order derivative (see
Table 1.1), one can state the following approximation:
E I Y
d
2 u
dX 2 = E I Y
u i+1 − 2u i + u i−1
X 2
= −M Y (X ) .
(3.111)
Before further evaluating the last equation, let us look on the function of the moment
distribution. The moment balance between the external load and the internal bending
moment gives finally the following relation (0 ≤ X ≤ L/2):
M Y (X ) = −
F 0
2
X ,
(3.112)
where the values at the seven grid points are summarized in Table 3.5.
Evaluation of the finite difference approximation of the second-order differential
equation according to Eq. (3.111) at the inner nodes i = 2, . . . , 6 gives:
node 2:
E I Y
X 2 (u 3 − 2u 2 + u 1 ) =
F 0 L
12
,
(3.113)
node 3:
E I Y
X 3 (u 4 − 2u 3 + u 2 ) =
F 0 L
6
,
(3.114)
node 4:
E I Y
X 2 (u 5 − 2u 4 + u 3 ) =
F 0 L
4
,
(3.115)