2.1 The Basics of a Rod
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Table 2.3 Different boundary conditions and corresponding reactions for a continuum rod (deformation occurs along the X -axis)
. . .
X
u X (X = 0) = 0
. . .
F
R
X
u
L
0
u X (X = L) = u 0
L
F
R
L
F 0
E A
du X (L)
dX
= N X (L) = F 0
L
ux
Fig. 2.2 Internal reactions
for a continuum rod
N X (X ) = E A
du X (X )
dX
= −p 0 X + c 1 .
(2.2)
The internal reactions in a rod become visible if one cuts—at an arbitrary location
X —the member in two parts. As a result, two opposite oriented normal forces N X
can be indicated. Summing up the internal reactions from both parts must result in
zero. Their positive direction is connected with the direction of the outward surface
normal vector and the orientation of the positive X -axis, see Fig. 2.2.
Once the internal normal force N X is known, the normal stress σ X can be calculated:
σ X (X ) =
N X (X )
A
.
(2.3)
Application of Hooke’s law (see Table 2.1) allows us to calculate the normal strain
ε X . Typical distributions of stress and strain in a rod element are shown in Fig. 2.3.
It can be seen that both distributions are constant over the cross section.
2.2 Constant Material and Geometry Parameters
Let us consider in the following a loaded rod of length L as shown in Fig. 2.4. The
tensile stiffness E A, i.e. the material and geometric parameters, are considered to
be constant within this chapter. The left-hand end is fixed and the right-hand side is
either loaded by a prescribed displacement u or a single force F. Along the length
of the rod is a distributed load p X (X ) acting.