# Bernoulli-Euler Beam Theory

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The overall objective of this lab was to verify Bernoulli-Euler beam theory for the lip-loaded cantilever and use that to calculate the Poisson ratio for the material in this case 6061-T6 aluminum. This was done by having to different objectives for each part of the Lab. Part one was use electrical resistance strain gage, this was to measure the strain a cantilever beam as it undergoes a known deflection. We would then compare the measurements with that of the strains predicted by simple beam theory [1]. The objective for part two was to use two electrical resistance strain gages on the cantilevered beam to measure and calculate the Poisson ratio of the beam. There were several tools need to complete this experiment [1]. First being a strain …show more content…
There were two different parts to this experiment. Part one, utilized a single electrical resistance strain gage in order to measure strains that prove simple Bernoulli-Euler beam theory [1]. Part two, was to use two electrical resistance strain gages which are mounted to the cantilever beam in order to measure the Poisson ratio for the material [1]. We would then use the data collected to calculate the Poisson ratio and compare it to that of 6061-T6 aluminum, along with the uncertainty of the Poisson ratio calculation. We were also able to develop graph, in which related the tip displacement and strain gage output. Before beginning the lab it is important to make sure that we calibrate the strain gages to make sure that were are getting realistic calibrated …show more content…
M is the resultant internal bending moment calculated about the neutral axis of the cross section, y in this case is the thickness of the beam divided by two, and I is the moment of inertia of the cross-sectional area about the neutral axis [4]. The final equation is,
Equation 5 is utilized to calculate the deflection δ of the cantilevered beam at x = L when a load is applied at a point. Since we would not applying a specific load but knowing the displacement we solved this equation for P. There equation came out to be,
After modifying the equations 2, 3, 4, and 6 and forming one equation, we were are then able to derive the equation which represents the Bernoulli-Euler equation to be,
Equation 7, is utilized to relate the strain of a beam to the displacement in which is experience in part one of this lab to the strain that is calculated by the strain gage. We would then use equation 6 to find the uncertainty for the strain output. The equation we used was,
The final equation used was to calculate the Poisson ratio of the material, in this case the beam the strain gage was attached to in part two. The equation was found to be,
Equation 9, will allow people to calculate and relate the Poisson ratio found in the part two of this experiment to the Poisson ratio found for specific

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