26
2 Investigation of Rods in the Elastic Range
The evaluation of the cases presented in Table 2.4 reveals that this derivation gives
identical results as Eq. (2.54), i.e. an identical solution as a finite element approach.
2.4 Solved Problems
2.1 Finite difference approximation of a cantilevered rod with a single force
based on five domain nodes
Given is a cantilevered rod of length L with constant tensile stiffness E A as shown
in Fig. 2.9. The rod is loaded by a single force F 0 at its right-hand boundary. Use five
domain nodes of equidistant spacing, i.e. =
L
4
, for the finite difference approximation. Use only finite difference approximations of second-order accuracy for the
nodal evaluations and boundary conditions. Perform the evaluations (a) starting from
the second-order partial differential equation and (b) as an alternative starting from
the first-order partial differential equation. Determine
• the horizontal displacements at the nodes,
• the analytical solution at X = L and
• calculate the relative error between the analytical and finite difference solution at
X = L.
• Derive a general FD scheme for n nodes (n > 5).
2.1 Solution
The finite difference discretization of the simply cantilevered rod is shown in
Fig. 2.10.
(a) Let us start with the second-order partial differential equation as given in
Table 2.1. Application of a centered difference approximations of second-order accuracy (see Table 1.1), one can state the following FD scheme for node i:
E A
2 (−u i−1 + 2u i − u i+1 ) = R i
p X =0
= 0 .
(2.64)
Since we have four unknown nodal displacements (u 2 , . . . , u 5 ), we must state four
equations to solve the problem. Thus, we could write the FD approximation in
Eq. (2.64) for the four nodes i = 1, . . . , 4 or for the four nodes i = 2, . . . , 5. Both
approaches are possible. Let us write the FD scheme in the following for nodes
Fig. 2.9 Cantilevered rod
loaded by a single force
2 Investigation of Rods in the Elastic Range
The evaluation of the cases presented in Table 2.4 reveals that this derivation gives
identical results as Eq. (2.54), i.e. an identical solution as a finite element approach.
2.4 Solved Problems
2.1 Finite difference approximation of a cantilevered rod with a single force
based on five domain nodes
Given is a cantilevered rod of length L with constant tensile stiffness E A as shown
in Fig. 2.9. The rod is loaded by a single force F 0 at its right-hand boundary. Use five
domain nodes of equidistant spacing, i.e. =
L
4
, for the finite difference approximation. Use only finite difference approximations of second-order accuracy for the
nodal evaluations and boundary conditions. Perform the evaluations (a) starting from
the second-order partial differential equation and (b) as an alternative starting from
the first-order partial differential equation. Determine
• the horizontal displacements at the nodes,
• the analytical solution at X = L and
• calculate the relative error between the analytical and finite difference solution at
X = L.
• Derive a general FD scheme for n nodes (n > 5).
2.1 Solution
The finite difference discretization of the simply cantilevered rod is shown in
Fig. 2.10.
(a) Let us start with the second-order partial differential equation as given in
Table 2.1. Application of a centered difference approximations of second-order accuracy (see Table 1.1), one can state the following FD scheme for node i:
E A
2 (−u i−1 + 2u i − u i+1 ) = R i
p X =0
= 0 .
(2.64)
Since we have four unknown nodal displacements (u 2 , . . . , u 5 ), we must state four
equations to solve the problem. Thus, we could write the FD approximation in
Eq. (2.64) for the four nodes i = 1, . . . , 4 or for the four nodes i = 2, . . . , 5. Both
approaches are possible. Let us write the FD scheme in the following for nodes
Fig. 2.9 Cantilevered rod
loaded by a single force