CantileversTrey Talley C’13

Department of Physics and AstronomySewanee: The University of the South, Sewanee, TN

IntroductionCantilevers can be found all around us, and they include objects such as signs, diving boards, and horizontal flagpoles. The cantilever beam is a key concept in civil engineering, and it is used in the construction of buildings, bridges, and balconies. An equation for the deflection of cantilever beams from certain loads was discovered in 1750 by Euler and Bernoulli, and this equation can be used to deeply explore the physics of the cantilever beam.

Loaded Cantilever Beams

Euler-Bernoulli Beam Theory

Young’s Modulus

Acknowledgements

Figure 1: A loaded cantilever beam with a concentrated load P at the free end.

Bernoulli and Euler put together this theory in 1750. The theory relates and allows us to calculate the load carrying and deflection characteristics of cantilever beams, and the theory is widely used by engineers in the construction of cantilevers. The Eiffel Tower and Ferris Wheel were constructed using this theory; and they proved the validity of this mathematical theory on large scales. Before this theory, there was considerable doubt that a mathematical product could be trusted for practical safety applications.

For infinitesimal strains and small deflections:

For a uniformly distributed load:

Young’s Modulus, E, is the measure of the stiffness of a material. It can be calculated using the Euler-Bernoulli cantilever beam equation if the load and beam deflection are known. It can also be calculated using the Young’s Modulus equation if the stress and strain on the material is known:

Figure 4: This is my cantilever setup with an aluminum ruler and weight added at the free end.

Thanks to Dr. Peterson, Rodger, Jim, and the P-Team C’13.

• I initially tested a simple loaded cantilever using an aluminum ruler.

ω = distributed loady = beam deflectionI = second moment of inertiaE = Young’s Modulus of material

My Experiment (Cont.)• I added intervals of 25 grams of mass to the end of the cantilever, and recorded the deflection in the y as a function of the distance:

A cantilever is simply a horizontal beam that is anchored only at one end, and this beam can carry a load. When a cantilever is loaded , there will be a deflection in the y-axis. A cantilever beam can be load in the following ways: a concentrated load P at the free end, a concentrated load P at any point, a uniformly distributed load, a uniform varying load.

I used the equation for the deflection of a uniformly loaded cantilever beam and entered it into mathematica, which gave me a fit for the straight (unloaded) cantilever. I then used the equation for the deflection of a cantilever beam with a concentrated load at the free end, and mathematica gave me a fit for the cantilever with a 175 gram load at the end.

Strain GaugesA strain gauge is a device used to measure the strain of an object. It measures the strain from the resistance change in the Wheatstone Bridge that is inside the strain gauge. I soldered wires to the strain gauge, and then connected the other ends of the wires to a data plug that connected to the computer, which displayed the strains in μe (micro strains).

Holography

A hologram can be taken of a small cantilever just like any object. I took a double exposure hologram of a small cantilever, adding only 1 gram to the free end for the second exposure. With your eye you could not see a change in the shape of the cantilever, but looking at the double exposure hologram, you could clearly see the deflection of the cantilever after adding the gram to the end.

My Experiment

I then set the coefficients in the cantilever equations for the straight and 175 loaded cantilevers equal to coefficients of the mathematica fit equations, and I solved for the average of the Young’s Modulus, E:

For Straight:

For 175 gram load:

120 Ohm Quarter Bridge