76
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Fig. 3.24 Finite difference discretization and free-body diagram of the cantilevered Euler–Bernoulli
beam based on seven grid nodes
M Y (X ) = 0
f o r 0 ≤ X ≤
L
3
,
(3.180)
M Y (X ) = F 0
X −
L
3
for
L
3
≤ X ≤
2L
3
,
(3.181)
M Y (X ) = F 0 (L − X )
for
2L
3
≤ X ≤ L .
(3.182)
There are five unknowns to determine, i.e., u 2 , u 3 , u 4 , u 6 , u 7 , which require to state
five equations. Thus, let us state the finite difference approximation of the partial
differential equation in the moment formulation for nodes 2, 3, 4, 5, 6. Doing so,
no fictitious nodes occur at the boundaries and the evaluation of the inner nodes is
sufficient:
node 2:
E I Y
X 2 (u 3 − 2u 2 + u 1 ) = 0 ,
(3.183)
node 3:
E I Y
X 2 (u 4 − 2u 3 + u 2 ) = 0 ,
(3.184)
node 4:
E I Y
X 2 (u 5 − 2u 4 + u 3 ) = −
F 0 L
6
,
(3.185)
node 5:
E I Y
X 2 (u 6 − 2u 5 + u 4 ) = −
F 0 L
3
,
(3.186)
node 6:
E I Y
X 2 (u 7 − 2u 6 + u 5 ) = −
F 0 L
6
,
(3.187)
or in matrix notation under consideration of both support conditions, i.e., u 1 = u 5 =
0: