# There are accompanying equations provided in the

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There are accompanying equations provided in the experiments procedure section, alsoa more detailed list can be found in the above mentioned “Statics and Strength of Materials” textbook. The equations provided allowed us to calculate the moment of inertia for the beam and then rearrange the beam deflection equation to obtain an equation for the modulus of elasticity. We were also provided with the maximum stress value and used that to determine the maximum allowable load to be placed on the beams for the given location and the different type of beam. Then there are also the alternate methods to measure deflection from a concentrated load. In this experiment we looked at the principal of superposition and Maxwell’s Reciprocity Theorem. Both of these methods involve using two different points on the beam and two weights and then measuring the deflection at the two different locations on the beam and comparing them. Both of these methods are proven and valid methodsin calculating deflection from a concentrated load and in this experiment we were able to use the theory to verify the accuracy of the theoretical values with the measured values of the deflection that we obtained. In our experiment we used the simply supported beam to conduct our verifications of the principal of superposition and Maxwell’s reciprocity theorem and show that with only partial error (most likely attributedto errors in measurement) they hold up.
Theory:Throughout this lab we relied on only a few key equations but they are all fundamental to the process of beam deflection and must be adhered to when undertaking any construction project. Once the measurements of our beams had been taken we had to calculate the Moment of Inertia (I) which takes the product of the base (b) and the height (h) cubed divided by twelve to give us our Moment of Inertia in units of in4:I=bh312The next equation is the deflection due to a concentrated load (δ), sometimes noted as (v) or (y) and is often represented as a negative value because in most cases the deflection moves in the negative direction from the plane of the beam. The deflection is calculated by dividing the product of the concentrated load (P) by the length of the beam(L) cubed by the product of a constant (k) depending on the position of the load on the beam and the type of fixtures on the beam, the Modulus of Elasticity (E) and the previously calculated Moment of Inertia (I):δ=P L3kEIThe Flexure formula is used to determine the bending stress at a specific location on the beam and can be solved in two ways for two slightly different results. To calculate the bending stress at a specific location on the beam it is the product of the internal moment of the beam (M) and the distance from the neutral axis to the specific point (y) divided by the moment of inertia (I):
σ=MyIA similar equation can be used to find the maximum stress at the same specific point byreplacing (y) with the extreme fiber distance (c) to get:σmax=McI

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Term
Fall
Professor
Dombourian
Tags
Cantilever, Deflections of Beams
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