We have learned a great deal about how the bending of a beam depends on
the beam's load, material properties, cross section, and manner of
support. Engineers use the static beam equation and the ideas that we
have explored as a basis for understanding the static deformations of
more complicated structures. As you have seen, integration plays a
key role in an engineer's ability to analyze these structures.
What we have not yet addressed, however, is an important mathematical
feature of the static beam equation that helps engineers to
approximate real loads by examining combinations of the idealized
loads that we have been studying. The important observation is that
the static beam equation is a linear differential equation .
Of the many important characteristics of linear differential
equations, the one we will be concerned about is called linearity
of solutions . For us, this means that if we know how a beam
bends under the load distribution q(x) and we also know how
the beam bends under the load distribution, p(x) , then we
also know how the beam bends under the loads q(x) + p(x), q(x) + 2
p(x), -q(x)+0.3 p(x), and, in fact, any load that may be
expressed as A q(x) + B p(x) for some numbers A and
This is amazing fact that only holds for linear
differential equations! Let's see how it is useful:
- Pick any material and dimensions for your beam.
You may make your beam any (reasonable) dimensions, but long beams
will work best for this experiment because they will deflect more.
Look up the density of the material
that you chose, and use this to compute the mass of your beam.
- Let's figure out how much the beam will deflect due to its own weight.
Again go to the beam simulator.
Because our beam has a uniform cross section, choose the
"Uniform" load distribution and type in the mass of the beam as
the "Load Mass." As best you can, estimate the deflection of your beam at
at least four points along the beam's length. Record these
locations and deflection in the
- Now pick any load distribution except for
the uniform distribution, type in
any load mass, and simulate the bending of the beam under your
chosen load. Estimate the deflection of your beam at
the same points along the beam that you chose above. Record these
deflection in your chart.
- Without changing your load or load mass, click the button
which will ask the beam simulator to add in the deflection of the
beam due to its own weight. Now simulate the bending of the beam and, again,
estimate and record the deflections of the beam at the chosen
points along the beam.
- What is the relationship between the deflections for the combined load
and the deflections for each load considered independently?
- For a more general theoretical analysis of this problem,
let u be the solution of the static beam equation for
the load q(x) (this means that u'''' = q/(EI) )
and let v be the solution of the static beam equation for
the load p(x) (thus v'''' = p/(EI) ).
What is the deflection function that solves the
static beam equation for the load q(x) + p(x) ?
What about for the load 3 q(x) - p(x) ? How do you know?
For every mathematical model of a physical process, one must ask the
questions "How valid is this model?", "What are the limitations of
this model?" and "Can we use this model to predict actual
(experimentally collected) data?" This is especially important if you
are going to use the theory to build a skyscraper holding thousands of
Return to the beam simulator one last time and
compare the theoretical predictions of the model to
actual experimental data for loaded meter
Describe your observations and state (and defend) whether you think
that the static beam equation is a valid model with predictive
properties. Does the theory predict the deformation of beams better
for small deflections or for large deflections? What are sources of
error between the theory presented in this module and the reality of
actual beams? Can you find limitations to the theory? In other
words, can you find situations that the theory predicts that are not
Return to: Exploring Static Deformations of Beams
The Geometry Center Calculus Development Team
Copyright © 1996 by The Geometry Center.
Last modified: Fri Apr 12 15:44:31 1996