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
*B *.

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 table provided.
- 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 people!

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 physically reasonable?

The Geometry Center Calculus Development Team

Copyright © 1996 by The Geometry Center. Last modified: Fri Apr 12 15:44:31 1996