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Typical 1D problems involve bars or beams under
axial loading. In order to test our various algorithms we define
a reference problem composed of a beam fixed at one end and free and the other
with an applied axial load at the free end, as seen in
Figure 3.1.
We use this general problem for all of our analyses in the
upcoming sections.
Figure 3.1:
Reference problem in 1D.

This beam, of length , is fixed at the right end
and has a compressive axial force applied on the left,
corresponding to an axial stress
. The beam is homogeneous
and has a crosssectional area , density
and Young's modulus .
The beam wave speed given by

(3.1) 
is thus .
In 1D the analytical solution to a beam problem can be
obtained by solving the wave equation

(3.2) 
with initial conditions

(3.3) 
and boundary conditions

(3.4) 
where is the displacement and is the wave speed.
The solution to this problem is represented by the xt (displacementtime) diagram
in Figure 3.2.
The displacement, velocity and stress profiles, for a sample point at ,
are shown in Figure 3.3.
Figure 3.2:
xt diagram in 1D.
Figure 3.3:
Analytical solution for displacement , velocity and stress
in the middle of the beam for the 1D wave problem described in
Figure 3.1.

Next: MultiTime Step Nodal Subcycling
Up: 1D ANALYSIS AND RESULTS
Previous: 1D ANALYSIS AND RESULTS
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Mariusz Zaczek
20021013