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6 Answers to Supplementary Problems
Fig. 6.2 Finite difference
discretization of the
fixed-ended rod based on
seven grid nodes: a axial
point load; b load per length
This result is identical to the analytical solution.
2.6 Displacement distribution for a fixed-ended stepped rod structure
The finite difference discretization of the fixed-ended rod is shown in Fig. 6.2 for
seven domain nodes. The single force F 0 is understood as the integral value of a
constant distributed load p 0 , which is acting over a length of , see Fig. 6.2a.
(a) The evaluation of the finite difference approximation according to Eq. (2.56)
at the inner nodes i = 2, . . . , 6 gives:
node 2:
E A I
X
(u 3 − 2u 2 + u 1 ) = 0 ,
(6.15)
node 3:
E A I
X
(u 4 − 2u 3 + u 2 ) = 0 ,
(6.16)
node 4:
E
X
A II − A I
4
(u 5 − u 3 ) +
A I + A II
2
(u 5 − 2u 4 + u 3 )
=
− p 0 = −F 0 ,
(6.17)
node 5:
E A II
X
(u 6 − 2u 5 + u 4 ) = 0 ,
(6.18)
node 6:
E A II
X
(u 7 − 2u 6 + u 5 ) = 0 ,
(6.19)
or in matrix notation under consideration of the boundary conditions: