72
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Table 3.7 Values of the internal bending moment at the grid points (see Fig. 3.22a)
Grid point
Coordinate X
M Y (X )
1
0
F 0 L
2
L
3
2F0 L
3
3
2L
3
F0 L
3
4
L
0
Evaluation of the finite difference approximation of the second-order differential
equation according to Eq. (3.27) at the nodes i = 1, . . . , 3 gives:
node 1:
E2I Y
X 2 (u 2 − 2u 1 + u 0 ) = −F 0 L ,
(3.156)
node 2:
E2I Y
X 2 (u 3 − 2u 2 + u 1 ) = −
2F 0 L
3
,
(3.157)
node 3:
E I Y
X 2 (u 4 − 2u 3 + u 2 ) = −
F 0 L
3
.
(3.158)
The vertical displacement is zero at the left-hand boundary because of the fixed
support and it follows immediately that u 1 = 0 holds. In addition, the rotation is
zero at the fixed support, i.e.
du
dX
1
= 0, and a centered finite difference approach
according to Table 1.1 gives the condition u 0 = u 2 . Introducing these relationships
for the support at the left-hand boundary into the system of equations gives:
node 1: 2u 2 = −
X
2 F 0 L
2E I Y
,
(3.159)
node 2: − 2u 2 + u 3 = −
X
2 F 0 L
3E I Y
,
(3.160)
node 3: u 2 − 2u 3 + u 4 = −
X
2 F 0 L
3E I Y
,
(3.161)
or in matrix notation:
⎡
⎣
2 0 0
−2 1 0
1 −2 1
⎤
⎦
⎡
⎣
u 2
u 3
u 4
⎤
⎦ = −
X
2 F 0 L
E I Y
⎡
⎢
⎣
1
2
1
3
1
3
⎤
⎥
⎦ .
(3.162)
The solution of this linear system of equations gives the unknown nodal values as:
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Table 3.7 Values of the internal bending moment at the grid points (see Fig. 3.22a)
Grid point
Coordinate X
M Y (X )
1
0
F 0 L
2
L
3
2F0 L
3
3
2L
3
F0 L
3
4
L
0
Evaluation of the finite difference approximation of the second-order differential
equation according to Eq. (3.27) at the nodes i = 1, . . . , 3 gives:
node 1:
E2I Y
X 2 (u 2 − 2u 1 + u 0 ) = −F 0 L ,
(3.156)
node 2:
E2I Y
X 2 (u 3 − 2u 2 + u 1 ) = −
2F 0 L
3
,
(3.157)
node 3:
E I Y
X 2 (u 4 − 2u 3 + u 2 ) = −
F 0 L
3
.
(3.158)
The vertical displacement is zero at the left-hand boundary because of the fixed
support and it follows immediately that u 1 = 0 holds. In addition, the rotation is
zero at the fixed support, i.e.
du
dX
1
= 0, and a centered finite difference approach
according to Table 1.1 gives the condition u 0 = u 2 . Introducing these relationships
for the support at the left-hand boundary into the system of equations gives:
node 1: 2u 2 = −
X
2 F 0 L
2E I Y
,
(3.159)
node 2: − 2u 2 + u 3 = −
X
2 F 0 L
3E I Y
,
(3.160)
node 3: u 2 − 2u 3 + u 4 = −
X
2 F 0 L
3E I Y
,
(3.161)
or in matrix notation:
⎡
⎣
2 0 0
−2 1 0
1 −2 1
⎤
⎦
⎡
⎣
u 2
u 3
u 4
⎤
⎦ = −
X
2 F 0 L
E I Y
⎡
⎢
⎣
1
2
1
3
1
3
⎤
⎥
⎦ .
(3.162)
The solution of this linear system of equations gives the unknown nodal values as: