3.4 Solved Problems
55
The solution of this linear system of equations gives with X =
L
4
the unknown
nodal values as:
u 2 = −
7q 0 L
4
512E I Y
, u 3 = −
43q 0 L
4
1024E I Y
, u 4 = −
39q 0 L
4
512E I Y
, u 5 = −
57q 0 L
4
512E I Y
,
(3.83)
and a much smaller relative error is obtained:
relative error =
57
512
−
1
8
1
8
× 100 = 10.938% .
(3.84)
The graphical comparison between these two approaches and the analytical solution
is given in Fig. 3.13. Depending on the approach, a different behavior, i.e. to overor underestimate the analytical solution is obtained.
3.3 Example: Finite difference approximation of a simply supported beam
loaded by a varying distributed load
Given is a simply supported Euler–Bernoulli beam as shown in Fig. 3.14. The bending stiffness E I Y is constant and the length is equal to L. The simply supported
beam is loaded by a linearly varying distributed load q Z (X ). Derive a finite difference approximation based on five grid points, i.e. an equidistant spacing of X =
L
4
.
Use centered difference schemes where the truncation error is of order X
2 .
Determine
• the displacements at the grid points,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution for
the displacement in the middle of the beam.
3.3 Solution
The function of the distributed load can be expressed as
q Z (X ) = −q 0
1 +
X
L
,
(3.85)
whereas the values at the five grid points are collected in Table 3.4.
The finite difference discretization of the simply supported beam is shown in
Fig. 3.15 where at first only the inner nodes will be considered.
Evaluation of the finite difference approximation of the fourth-order differential
equation according to Eq. (3.9) at the inner nodes i = 2, . . . , 4 gives:
55
The solution of this linear system of equations gives with X =
L
4
the unknown
nodal values as:
u 2 = −
7q 0 L
4
512E I Y
, u 3 = −
43q 0 L
4
1024E I Y
, u 4 = −
39q 0 L
4
512E I Y
, u 5 = −
57q 0 L
4
512E I Y
,
(3.83)
and a much smaller relative error is obtained:
relative error =
57
512
−
1
8
1
8
× 100 = 10.938% .
(3.84)
The graphical comparison between these two approaches and the analytical solution
is given in Fig. 3.13. Depending on the approach, a different behavior, i.e. to overor underestimate the analytical solution is obtained.
3.3 Example: Finite difference approximation of a simply supported beam
loaded by a varying distributed load
Given is a simply supported Euler–Bernoulli beam as shown in Fig. 3.14. The bending stiffness E I Y is constant and the length is equal to L. The simply supported
beam is loaded by a linearly varying distributed load q Z (X ). Derive a finite difference approximation based on five grid points, i.e. an equidistant spacing of X =
L
4
.
Use centered difference schemes where the truncation error is of order X
2 .
Determine
• the displacements at the grid points,
• the analytical solution and
• calculate the relative error between the analytical and finite difference solution for
the displacement in the middle of the beam.
3.3 Solution
The function of the distributed load can be expressed as
q Z (X ) = −q 0
1 +
X
L
,
(3.85)
whereas the values at the five grid points are collected in Table 3.4.
The finite difference discretization of the simply supported beam is shown in
Fig. 3.15 where at first only the inner nodes will be considered.
Evaluation of the finite difference approximation of the fourth-order differential
equation according to Eq. (3.9) at the inner nodes i = 2, . . . , 4 gives: