3.4 Solved Problems
57
Table 3.4 Values of the distributed load at the grid points (see Fig. 3.15)
Grid point
Coordinate X
q Z (X )
1
0
−q 0
2
L
4
−
5
4 q 0
3
L
2
−
3
2 q 0
4
3L
4
−
7
4 q 0
5
L
−2q 0
Fig. 3.15 Finite difference discretization of the cantilevered Euler–Bernoulli beam shown in
Fig. 3.14
node 2:
E I Y
X 3 (u 4 − 4u 3 + 6u 2 − 4u 1 + u 0 ) = −
5q 0
4
,
(3.86)
node 3:
E I Y
X 3 (u 5 − 4u 4 + 6u 3 − 4u 2 + u 1 ) = −
3q 0
2
,
(3.87)
node 4:
E I Y
X 3 (u 6 − 4u 5 + 6u 4 − 4u 3 + u 2 ) = −
7q 0
4
.
(3.88)
The vertical displacement is zero at both ends and it can be immediately concluded
that u 1 = u 5 = 0. The fictitious nodes i = 0 and i = 6 outside the domain can be
eliminated based on the boundary condition that the moment must be equal to zero
at the supports, i.e. M Y (X = 0) = M Y (X = L) = 0. Application of the bending
differential equation in the form with the bending moment according to Table 3.1,
i.e. E I Y
d
2 u
dX 2 = −M Y , and the centered finite difference approximation of the secondorder derivative according to Table 1.1, the following two conditions can be derived:
E I Y
d
2 u
dX 2
1
= E I Y
u 2 − 2u 1 + u 0
X 2
= 0 ,
(3.89)
E I Y
d
2 u
dX 2
5
= E I Y
u 6 − 2u 5 + u 4
X 2
= 0 ,
(3.90)