UNIFIED ENGINEERING Systems Problem 3 3/11/04 ObjectivesAfter completing this systems problem, you should be able to: • Apply Bernoulli Euler beam theory to compute deflections and strains in a beam • Carry out a simple experiment using apparatus such as dial gauges and strain gauges Due Date March 18, 12 PM. The lab will be conducted in team of 2-3 people,arranged according to your schedules, but the pre-lab and post-lab write-ups are individual assignments – each student should hand in his or her own pre-lab and post-lab. Make certain that you have all the relevant information and data before you leave the laboratory. The usual collaboration policy applies. Introduction In lecture we have discussed the derivation of the expressions for the stresses and deformations of simple beams. These expressions are based on a set of modeling assumptions, including the Euler-Bernoulli hypothesis. This lab is designed to provideyou with an opportunity to verify that simple beam theory provides a good prediction of the deformation and stresses in a beam and hence to explore the underlying assumptions. The laboratory consists of three related experiments, each of which probes one of the predictions of beam theory. The laboratory exercise will take place in the Civil and Environmental Undergraduate Structures Teaching Laboratory . ApparatusThe apparatus consists of a mild steel beam, approximately 36" long, with across-section of 0.375" deep by 1.25" wide. The beam is supported on rollers at its ends and is loaded centrally by hanging weights from a chain suspended from a yoke clamped to the beam. The provided weights are approximately 6 lbs. Each, exact weights are marked. Strain gauges are attached on the top and bottom surfaces of the beam. Strain gauge monitors are available to monitor the strain gauges. The strain gauges are monitored individually in a quarter bridge circuit. A dial gauge is availableto monitor the displacement at a fixed point along the beam. Assume that the steel of which the beam is made has a Young's modulus of 200 GPa ±5 GPa Pre-lab exercise Calculate or otherwise obtain an expression for the deflection of a beam simplysupported at its ends and point-loaded at its center as a function of the beamdimensions, material properties, and applied load. Obtain expressions for the strains on the top and bottom surfaces of the beam as a function of the beam dimensions, material properties, applied load and axial position along the beam. Have these expressionsavailable in the lab so that you can quickly compare theory with experiment. Given the dimensions listed above, calculate the applied load that will result in a stress of 150 MPa 1in the beam, assuming that the span is the entire length of the beam. This load should not be exceeded in conducting the lab. ExperimentsThree separate experiments will be conducted using the same apparatus. Vernier calipers and meter rules are available to allow you to measure the beam dimensions and distances between the supports and loading points. Do not forget to take this data for each experiment below. Also take the time to inspect the fixture, loading apparatus, and instrumentation to evaluate any sources of error. The lab work should take no more than 60 minutes, and a well-organized group may take substantially less time. Note: You will need to adjust the apparatus between experiments, make sure that the support points are secure and properly aligned before applying load. Note: It is highly recommended that you perform at least some preliminary calculations while you perform each experiment. This will allow you to identify anygross errors in the way you have conducted the experiment, for instance if the displacement or strain readings are faulty, or if you have incorrectly measured a dimension. See also the directions for balancing and calibrating the strain gaugemonitors at the end of this set of lab instructions. 1. Verification of load-displacement and load-strain relationships.Place the beam on its supports so that the strain gauges are on the upper and lower surfaces. Apply load centrally to the beam and measure the resulting center point displacement using the dial gauge. Also measure the strains on the top and bottom surfaces using the strain gauges. Note, to avoid damaging the strain gauges they will need to be positioned slightly offset from the loading point. This offset should be accounted for in calculating the strains via beamtheory (how do the strains vary along the length of a simply supported beam?). Similarly, if the dial gauge cannot be placed directly at the point of loadapplication, and this offset will need to be accounted for. Make measurements at 5-7 load levels during loading and 3-5 load levels during unloading. Tabulate andplot the displacement and strain data as a function of load. 2. Verification of role of cross-sectional properties. Rotate the beam through 90° so that the strain gauges are on the side faces, and the depth of the beam is now the long cross-sectional dimension. Repeatexperiment 1, recording load, displacement, and strain at a sufficient number ofdata points to produce a curve. Tabulate and plot the displacement and straindata as a function of load. 23. Verification of span-displacement relationship. Reorient the beam so that the strain gauges are on the top and bottom of the beam. (a) Reduce the span to approximately 75% of the original and repeatexperiment 1, tabulating displacement and strain data. (b) Reduce the span to 50% of the original and repeat the measurements. Post-lab write-upThe post-lab write-up should be an individual effort. In particular, the graphing of data and its interpretation should be done separately. The lab write-up should take no more than 2 hours. 1. Using graph paper or a computer graphing application, produce neat, labeled, and titled versions of the data that was taken in the experiments 1) through 3) described above. Estimate the errors associated with taking the data and include appropriate error bars with your data points. 2. From the slope of the graphs compare the experimental data with the predictions of beam theory for how: (a) Deflection varies with applied load for a fixed span(b) Strain varies with applied load for a fixed spanÊÁË Pˆ˜¯d(c) Beam stiffness varies with second moment of area, I ÊÁË Pˆ(d) Beam stiffness varies with span˜¯d 3. Comment
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