Static Deformations

While the integration of the static beam equation and the application of boundary conditions is not too difficult for constant loads, the algebra become more involved (but still manageable) for loads that vary across the beam's surface. Instead of having you solve dozens of differential equations, in this section we present a set of questions for you to explore, using an interactive "beam simulator" that will solve the beam equation and to graph the deflection function. As before, we will focus on four main ideas: how does the deflection of a beam depend on the load, the beam cross section, the beam material, and the support.

Note: To avoid unnecessary complication, in this section we will ignore the load on the beams coming from their own mass. Consequently, when using the simulator to answer the follow questions, DO NOT add in the deflection coming from the weight of the beam.


Question 9

Using the beam simulator, simulate the bending of a brass cantilevered beam of length 1 meter, width 0.05 meter, and height of 0.01 meter, subject to a load mass of 2 kg.

Question 10

Using the beam simulator, simulate the bending of a cantilevered beam of length 1 meter, width 0.05 meter, and height of 0.01 meter, subject to a uniform load mass of 2 kg.

Question 11

Earlier it was claimed that the amount of deflection of a piece of 2x4 lumber will depend on whether the 2" or 4" edge of the beam is laid flat. This question pursues this assertion.

The amount of material in a beam of length L and with a rectangular cross section depends only on the area of the cross section. If the area of the cross section is held constant, then there is range of beams (from wide-but-short to skinny-but-tall) that contain the same amount of material, and therefore cost approximately the same.

Using the beam simulator, simulate the bending of a cantilevered aluminum beam of length 1 meter and with constant area 0.0004 square meters under a concentrated load of 1 kg. Let the width of the beam vary according to the entries in the column of the chart you were given. For each value of the width, compute the height such that the cross sectional area is 0.0004 square meters, and record the maximum deflection of the cantilevered beam on the graph paper provided.


Question 12

[NOTE to instructors: This question may be skipped without loss of continuity.]

Intuitively, a long cantilevered beam will bend more than a short beam of the same material and cross section. This question asks the question: "can we determine how the maximum deflection of a cantilevered beam depends on the length of the beam?"

Using the beam simulator, simulate the bending of a cantilevered aluminum beam of width 0.05 meters and height 0.02 meters subject to a concentrated load mass of 1.5 kg.


Next: Conclusions
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The Geometry Center Calculus Development Team

Copyright © 1996 by The Geometry Center. Last modified: Fri Apr 12 15:48:21 1996