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Bending stress

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Introduction: Often times when performing analysis for static equilibrium, assumptions are made in order to simplify calculations. One common assumption is that the body in question is perfectly rigid; that is it does not have a tendency to deform despite the magnitude of any externally applied force. But if all engineering applications adopted this convention without consideration for deformation during design, the result would be a product that not only operates very inefficiently, but in certain cases even dangerously. Hence, it is necessary to impose limits on the validity of the assumption for rigidity. It is known that any material, given ample loading conditions and force, will deform and ultimately fail. But before this occurs, the material undergoes several stages of deformation, namely: Elastic Deformation, Yielding, Plastic Deformation, Necking, and Failure. For engineering purposes, we wish to limit the deformation of the material to the elastic region, up to the point of yielding. In the elastic region, stress and strain, which are both material dependent characteristics, are linearly related in accordance to Hooke's Law. The constant of proportionality is defined as Young's Modulus of Elasticity and represents the ratio of stress to strain at every point in the elastic region. Hence, for a measured value of strain that can be obtained experimentally using a strain gauge, multiplying by Young's Modulus will give the corresponding stress in the structural element. This becomes convenient because by definition the stress in a sample cannot be measured with ease. This is because stress represents a distributed force that acts non-uniformly across an area. However, using Hooke's Law provides an average value for the stress in a member. With this information, if a design element created from a given material must support a given load, its cross sectional area can be adjusted accordingly. Similarly, if the cross sectional area is given the maximum allowable force in the member is simultaneously known. The above fact is used to investigate and confirm the accuracy of the bending equation, which relates the bending moment at a point, the second moment of area for a sample, the stress in the sample, and the relative position of a measurement with respect to a neutral axis. This is done by applying two loads symmetrically about an inverted T-beam to induce bending. Due to the symmetry of the application, the deformation caused by the bending should generate identical values of strain in the strain gauges that are mounted around the beam. When the resulting strain is plotted against the relative position of the gauge with respect to the neutral axis, the plot should show that the neutral axis undergoes no strain. Furthermore, since the second moment of area, bending moment, and the relative position are all known, the bending equation can be used to obtain an experimentally determined value for the stress in the sample. Comparison to the value given by Young's modulus should in theory yield identical values.
Procedure: 1.
To begin, the loading frame was examined to record any relevant information pertaining to the experiment. These values include the second moment of area (I), young's Modulus for Aluminum (E), and the position from the top of the inverted T-beam to the neutral axis. 2.
The thumbwheel attached to the load cell was then adjusted to apply a preload of 100 N to the beam. The resulting strain values for gauges 1-9 displayed on the computer screen were recorded under the column corresponding to a load of 0 on Table 1. 3.
The control on the front of the load cell was adjusted to zero the force shown on the digital force display. 4.
The thumbwheel was again changed to apply a load of 100 N (now a total of 200 N) to the beam. Strain readings from the computer were recorded under the column for a load of 100 N on Table 1. 5.
Step 4 was repeated as the load increased from 100N to 500N in 100N increments. The strain for each load was recorded in its corresponding column. 6.
To obtain values for Table 2, any strain caused by the preload was subtracted from all values in that row. This procedure corrects for the preload and ensures that the strain measurements for all 9 gauges are 0 for a load of 0N. Furthermore, loads from Table 1 were converted to bending moments about the center of the beam where the gauges are mounted. The values can be obtained by applying the conditions for static equilibrium on the beam to solve for support reactions and then performing bending moment analysis at the midpoint of the beam. 7.
As previously mentioned, because the beam is expected to deform symmetrically and the positions of gauges 2-9 correspond to the same vertical position on either side of the beam, these strain values are averaged and inputted into Table 3. Since gauge 1 was mounted on the top of the beam its position is unique so any strain recorded in gauge 1 is not averaged. 8.
Lastly, measurements were taken to record the positions of the gauges and thickness of the beam for comparison to their given values. This was accomplished using a vernier.

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