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.

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?

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.

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?

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.

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.

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.

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 .

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The Geometry Center Calculus Development Team