3.4 Solved Problems
73
Table 3.8 Values of the internal bending moment at the grid points (see Fig. 3.22b)
Grid point
Coordinate X
M Y (X )
1
0
F 0 L
2
L
4
3F0 L
4
3
L
2
F0 L
2
4
3L
4
F0 L
4
5
L
0
⎡
⎣
u 2
u 3
u 4
⎤
⎦ = −
F 0 L
3
E I Y
⎡
⎢
⎣
1
36
5
54
7
36
⎤
⎥
⎦ ,
(3.163)
and the relative error right-hand side of the beam is obtained as [11]:
relative error =
7
36
−
3
16
3
16
× 100 = 3.70% .
(3.164)
Let us now repeat the derivation for five nodes. The evaluation of Eq. (3.155) at the
five grid points is summarized in Table 3.8.
Evaluation of the finite difference approximation of the second-order differential
equation according to Eq. (3.27) at the nodes i = 1, . . . , 4 gives
8 :
node 1:
E2I Y
X 2 (u 2 − 2u 1 + u 0 ) = −F 0 L ,
(3.165)
node 2:
E2I Y
X 2 (u 3 − 2u 2 + u 1 ) = −
3F 0 L
4
,
(3.166)
node 3:
E
2I Y +I Y
2
X 2 (u 4 − 2u 3 + u 2 ) = −
F 0 L
2
,
(3.167)
node 4:
E I Y
X 2 (u 5 − 2u 4 + u 3 ) = −
F 0 L
4
,
(3.168)
or in matrix notation under consideration of the conditions at the left-hand boundary:
⎡
⎢
⎢
⎣
2 0 0 0
−2 1 0 0
1 −2 1 0
0 1 −2 1
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
u 2
u 3
u 4
u 5
⎤
⎥
⎥
⎦ = −
X
2 F 0 L
E I Y
⎡
⎢
⎢
⎢
⎣
1
2
3
8
1
3
1
4
⎤
⎥
⎥
⎥
⎦
.
(3.169)
8 It should be noted here that the second moment of area is discontinuous at node 3. As a workaround,
we simply take the arithmetic mean at this node, i.e.,
2I Y +I Y
2
.