142
6 Answers to Supplementary Problems
Fig. 6.10 Finite difference discretization of the cantilevered Euler–Bernoulli beam (imposed displacement) based on five grid nodes
node 4:
E I Y
3 (u 6 − 4u 5 + 6u 4 − 4u 3 + u 2 ) = 0 .
(6.169)
Under consideration of the boundary conditions at the left-hand end, i.e., u 1 = 0
and u 0 = u 2 , and the conditions at the right-hand boundary, i.e., u 5 = −u 0 and
u 6 = 2u 5 − u 4 , the following matrix scheme can be stated:
⎡
⎣
7 −4 1
−4 6 −4
1 −4 5
⎤
⎦
⎡
⎣
u 2
u 3
u 4
⎤
⎦ = −u 0
⎡
⎣
0
1
−2
⎤
⎦ .
(6.170)
The solution of this linear system of equations gives the unknown nodal values as:
⎡
⎣
u 2
u 3
u 4
⎤
⎦ = −u 0
⎡
⎢
⎣
1
11
7
22
7
11
⎤
⎥
⎦ = −u 0
⎡
⎢
⎣
0.09090909
0.31818182
0.63636364
⎤
⎥
⎦ .
(6.171)
The analytical solution can be taken from [2] as
u(X ) =
1
2
X
L
3
−
3
2
X
L
2
u 0 ,
(6.172)
and the relative error is obtained, for example, at X =
3L
4
as:
relative error =
7
11
−
81
128
81
128
× 100 = 0.56% .
(6.173)
6 Answers to Supplementary Problems
Fig. 6.10 Finite difference discretization of the cantilevered Euler–Bernoulli beam (imposed displacement) based on five grid nodes
node 4:
E I Y
3 (u 6 − 4u 5 + 6u 4 − 4u 3 + u 2 ) = 0 .
(6.169)
Under consideration of the boundary conditions at the left-hand end, i.e., u 1 = 0
and u 0 = u 2 , and the conditions at the right-hand boundary, i.e., u 5 = −u 0 and
u 6 = 2u 5 − u 4 , the following matrix scheme can be stated:
⎡
⎣
7 −4 1
−4 6 −4
1 −4 5
⎤
⎦
⎡
⎣
u 2
u 3
u 4
⎤
⎦ = −u 0
⎡
⎣
0
1
−2
⎤
⎦ .
(6.170)
The solution of this linear system of equations gives the unknown nodal values as:
⎡
⎣
u 2
u 3
u 4
⎤
⎦ = −u 0
⎡
⎢
⎣
1
11
7
22
7
11
⎤
⎥
⎦ = −u 0
⎡
⎢
⎣
0.09090909
0.31818182
0.63636364
⎤
⎥
⎦ .
(6.171)
The analytical solution can be taken from [2] as
u(X ) =
1
2
X
L
3
−
3
2
X
L
2
u 0 ,
(6.172)
and the relative error is obtained, for example, at X =
3L
4
as:
relative error =
7
11
−
81
128
81
128
× 100 = 0.56% .
(6.173)