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Experiment B3 Beam Bending Procedure Overview Theoretical Background

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Experiment B3 Beam Bending Procedure Overview Theoretical Background University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2019 Experiment B3 Beam Bending Procedure Deliverables: checked lab notebook, Brief technical memo Overview When a beam is rigidly mounted on one end, and a load is applied to another end, we call it a “cantilever beam”. Such configurations show up in a wide variety of engineering applications. Diving boards, airplane wings, balconies, and cranes are all common examples of cantilever beams. In this lab, you will examine the bending of an aluminum cantilever when a load is applied to the end. Specifically, you will hang weights from the end of the beam and measure the resultant deflection and strain. Theoretical Background When a load is applied to one end, the beam is bent downward, and the top of the beam becomes stretched in tension, while the bottom is compressed. An exaggerated illustration of this effect is shown in Figure 1. Load x y(x) Tension Compression Figure 1 – A schematic of a cantilever beam with a load applied. The vertical displace of the beam y as a function of the horizontal distance from the fixed mount x is given by the equation F y(x) = x3 − 3Lx2 , (1) 6EI ( ) where F is the load force, E is Young’s modulus, L is the total length of the beam, and I is the “second moment of area” or “area moment of inertia. As you can imagine, thicker beam are more difficult to bend. Furthermore, beams that are taller in the y-direction are also more difficult to B3 – Beam Bending 1 Last Revision: 2/5/19 University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2019 bend than flat beams. This geometric effect is accounted for be the second moment of area I, which takes into account the cross-sectional shape of the beam. Equations for I for the different beam geometries used in this lab are shown in Figure 2. U-Channel Centroid: 2nd moment of Area: th 2h2t dt 2 3 3 h + w 2thh + dtw tw d h c = I = 2hw − 2d(h − tw ) 3 w i-Beam b a Centroid: 2nd moment of Area: a H ah3 + b(H 3 − h3 ) c = I = h H 2 12 Square Tube Centroid: 2nd moment of Area: 4 4 S1 S1 − S2 S2 S1 c = I = 2 12 Figure 2 – Different beam geometries and formulas for calculating the centroid and second moment of area. The load will also induce stress and strain on the beam, with the top in tension and bottom in compression, as illustrated in Fig. 1. Somewhere between the top and bottom, there is a portion of the beam that is neither stretched nor compressed. This sliver of material is known as the “neutral surface” of the beam. The maximum strain typically occurs on top and/or bottom of the beam. The strain ε at a distance l from the load F (on the top and/or bottom of the beam) is given be the formula cFl ε = , (2) EI where c is distance in the y-direction to the neutral surface of the beam. Formulas for calculating c using the beam’s cross-sectional geometry are given in Fig. 2. Note that the strain and displacement for a given load depends on the c and I, which are essentially geometric properties of the beam. Thus, the stiffness of the beam is highly dependent on the beam’s cross-sectional profile. B3 – Beam Bending 2 Last Revision: 2/5/19 University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2019 Experimental Method Similar to the Beerless Pong exercise in Lab I, you will measure the strain using four strain gauges in a Wheatstone Bridge configuration (2 on top, 2 on bottom). You will also use a dial indicator to measure the vertical displacement of the beam at a given location. A schematic of the experimental set-up is shown below in Figure 3. Dial Indicator x BEAM l Strain Gauges Hanging Weights F = mg Figure 3 – A schematic of the cantilever beam experiment. Strain gauges in a Wheatstone bridge configuration are used to measure the strain and a dial indicator is used to measure the vertical displacement as a function of load. The load is applied by hanging metal weights from the end of the beam. Part I: Displacement and Strain vs. Load Safety First – This lab involves heavy metal weights. You must wear closed-toe shoes while performing this lab! 1. Sketch the experimental apparatus in your lab notebook. Make a note that the beams are made of 6063 Aluminum. 2. Make sure the dial indicator is a few inches from the end of the beam, near where the load will be applied. 3. Measure the length of the beam L, the distance from the load to the strain gauges l, and the distance from the base to the dial indicator x. Record these values in your lab notebook. 4. Locate the small sample piece of the beam. Use the dial calipers and/or micrometer to measure the cross-sectional dimensions. Sketch the cross section in your lab notebook. Copy down the corresponding formulas from Fig. 2 into your lab notebook. 5. Measure the resistances of the four strain gauges. They should all have the same value (either 120 or 350 Ohms). B3 – Beam Bending 3 Last Revision: 2/5/19 University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2019 6. Connect the strain gauges to the blue Vishay P3 strain gauge amplifier in a “full bridge” configuration. Refer to the diagram inside the lid of the P3. The “T” stands for tension, which are the SGs on top. The “C” stands for compression, which are the strain gauges on the bottom. 7. Check that the P3 is programmed correctly for a full bridge: a. Press the “CHAN” button and ensure the correct channel is set to “Active”. (All other channels should be inactive.) b. Press the “BRIDGE” button and ensure the correct channel is set to “FB 4 active”. c. Press the “K” button and set the gauge factor to 2.13. 8. Press the “BAL” button and balance the bridge. You should see the strain go to zero. 9. Make a table in your lab notebook with a column for mass, a column for strain, and a column for vertical displacement. Be sure to note the units! 10. For the first point in your table, record the strain and initial value on the dial indicator y0 with no load on the end of the beam or m = 0. (You will need to subtract off this initial value from all of subsequent dial indicator measurements to get the displacement.) 11. Hang the empty hanger on the end of the beam and record the strain and value on the dial indicator. Note that the strain is in units of “microstrain” and the displacement is in increments of 0.001 inches. 12. Incrementally increase the mass and measure the strain and displacement. You should record at least 7 data points. 13. Remove the weights from the end of the beam. Check to make sure the strain returned to zero. 14. Disconnect the strain gauge wires from the P3. Part II: Displacement vs. Position You will now measure the displacement y as a function of the horizontal position x for a constant load F = mg. 1. Using the largest load from Part I, measure the displacement when the load is added. 2. Loosen the lever next to the dial indicator and slide it a few centimeters away from the load. (Use both hands to loosen and move the dial indicator.) Tighten the lever and measure the displacement when the load is added. 3. Repeat this until you have measured the displacement at 10 positions or more. 4. Return the dial indicator to its original position when you are finished. B3 – Beam Bending 4 Last Revision: 2/5/19 University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2019 Data Analysis and Deliverables Using LaTeX or MS Word, make the following items and give them concise, intelligent captions. Additionally, write a paragraph separate from the caption describing what you did in lab and how it relates to the plot/table. Any relevant equations should go in this paragraph. For the following deliverables, you will need to look up Young’s modulus E for the aluminum beams on the McMaster-Carr website. Part numbers can be found in Appendix A. On the McMaster website, enter the part number, then click “Product Detail” to see the specific material properties. For this lab, please use SI units. 1. Make a plot of the measured displacement y (units of meters) as a function of load F (units of Newtons). Plot the theoretical curve given by Eq. (1) on top of the measured data. 2. Make a plot of the measured strain ε (unitless) as a function of load F (units of Newtons). Plot the theoretical curve given by Eq. (2) on top of the measured data. 3. Make a plot of the measured displacement y (units of meters) as a function of distance x (units of meters). Plot the theoretical curve given by Eq. (1) on top of the measured data. Be sure to note what the load is your caption. Talking Points – Discuss the following in your paragraphs. • Using the part numbers in Appendix A, look up Young’s modulus E for the beams on the vendor’s website. Be sure to include the vendor website in your references.
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