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 B .

This is amazing fact that only holds for linear differential equations! Let's see how it is useful:

Question 13

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!

Question 14

Return to the beam simulator one last time and compare the theoretical predictions of the model to actual experimental data for loaded meter sticks.

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?

Return to: Exploring Static Deformations of Beams
Up: Outline

The Geometry Center Calculus Development Team

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