**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.

- Record the maximum deflection of the beam for every available sort of load distribution.
- Linearly order the loads from those that create the most deflection to those that create the least deflection. Explain geometrically why this ordering makes sense.
- Repeat the experiment with the simply supported beam.
(Recall that now we interpret a concentrated load to be concentrated at
*x=L/2*.) Does the same ordering of loads apply? That is, do the loads that cause maximum (or minimum) deflection in the cantilevered beam also cause maximum (or minimum) deflection in the simple beam? Why or why not?

- Record the maximum deflection of the beam for each available material. Use this data to order the materials from strongest to weakest.
- Look up the modulus of elasticity for aluminum and brass. What is the ratio of their moduli of elasticity? What is the ratio of their maximum deflections? What does this tell us about the way that a beam's deflection depends on the beam's modulus of elasticity?

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.

- Simply by looking at the data on your graph, how does the maximum deflection appear to depend on the width? (For example, does it depend linearly? Quadratically? Exponentially?)
- Repeat the experiment with the same beam and the same
values for the width, but this time use a 1 kg
**uniform**load. Does the maximum deflection curve have a similar or a different shape? How does the deflection for a uniform load compare with the deflection of a concentrated load at*x=L*? Argue that this makes sense on physical grounds.

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.

- Choose values of the length of the beam within the range 0.25 meters and 2.5 meters. Record the maximum deflections of these beams on the graph paper provided. How does the maximum deflection appear to depend on the length?
- The deflection for a cantilevered beam with a concentrated load
at
*x=L*may be found by solving the static beam equation with boundary conditions*w(0)=0, w'(0)=0, w''(L)=0, w'''(L)=mg*. Solve the differential equation (by integrating four times) to find the deflection function*w(x)*and then evaluate the function at*x=L*in order to find the maximum deflection. (Why does the maximum deflection occur here?) You should obtain an expression that depends on*E, I, L*and the concentrated mass,*m*. From this expression, determine how the maximum deflection of the cantilevered beam depends on a power of the length of the beam. **Extra Credit**: a similar computation shows that the maximum deflection of a cantilevered beam under a uniform load depends on the beam's length in the same manner as for a concentrated load. (That is, the dependence is*L^k*for a particular choice of integer*k*. You may wish to verify this!) Is this a generally true statement? Either show that the maximum deflection depends on the length of the cantilevered beam*independent of the type of load*, or else produce a load such that the maximum deflection is not proportional to*L^k*.

The Geometry Center Calculus Development Team

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