14
2 Investigation of Rods in the Elastic Range
Fig. 2.3 Axially loaded rod:
a strain and b stress
distribution
Fig. 2.4 Rod loaded with a
distributed load p X (X ) and a
displacement boundary
condition u 0 or an end load
F 0
The problem is described according to Table 2.2 in the domain X ∈ [0, L] by the
following partial differential equation
E A
d
2 u X
dX 2 = −p X (X )
(2.4)
and the boundary conditions
u X (X = 0) = 0 and u X (X = L) = u 0
(2.5)
or
u X (X = 0) = 0 and
du X
dX
X = L
=
F 0
E A
.
(2.6)
In a more formal way, Eq. (2.4) can be written as
E AL{u X } = −p X (X ) ,
(2.7)
where the differential operator is given by L =
d
2
dX 2 . Taking from Table 1.1 the centered difference scheme of second-order accuracy, a finite difference approximation
of the partial differential equation (2.4) can be written for node i as
1 :
1 To simplify the notation, the index X is omitted in the following derivations.