Project: Bending It Like Beckham

http://numericalmethods.eng.usf.edu/experiments Due April 20, 2011 at 2 PM Project: Bending it like Beckham

Background: A cantilever beam with rectangular cross-section is subjected to an end load. Let’s develop the mathematical model for this problem.

y Weight Strain gage A Strain gage B b x h

Figure: Bending a beam

The deflection of a beam is given by d 2w M (x)  (1) dx2 EI where w = deflection of beam, m M (x) = bending moment at location x , N-m E = Young’s Modulus, Pa I = Second moment of area, m4 The bending moment at location x , is due to an end load in a cantilever beam is given by M (x)  P(L  x) (2) where L = length of beam, m P = force applied at end, N The second moment of area of a rectangular cross section is given by bh3 I  (3) 12 where b = width of beam, m h = thickness of beam, m The normal axial stress, (σ) at location x on the top of the beam is given by h M (x) (4)   2 I and the normal axial strain (ε) at the location x on top of beam is given by

   (5) E

What to do in the lab: 1. Place the unstrained beam apparatus on a stable and flat surface. 2. Set up the strain gauge indicator: a. Take the strain gauge wires and connect them properly (red wire to red terminal, white wire to white terminal, green wire to yellow terminal) to the strain indicator. b. Set BRIDGE pushbutton to up (black) for ¼ bridge. c. Depress AMP ZERO pushbutton. d. Set AMP ZERO control for a reading of +- 0000. e. Depress GAGE FACTOR pushbutton. Set GAGE FACTOR controls to 2.075. (This setting is found in the information given with the strain gauges). f. Lock GAGE FACTOR knob. g. Depress RUN pushbutton. Set BALANCE controls for reading of +- 0000. Lock BALANCE knob. The meter reads in microstrain ( m ) 3. To take measurements for deflection: a. Measure out 20 positions on the beam and take their initial deflection readings b. Add weight to the hanger. c. Wait for the beam to stabilize. d. Take deflection gauge and record various readings along the beam. Record the amount of deflection. (Do not forget to subtract the initial deflection). 4. To take measurements for strain: a. Place the weight on the hanger and read the corresponding strains on the two strain gages. b. Repeat with varying weights. Exercises to do (100 points): First, follow the sample experimental project guidelines. http://numericalmethods.eng.usf.edu/EML3041/homework/i ndex.html

Use MATLAB to solve problems (7 thru 13). Use comments, display commands and fprintf statements, sensible variable names and units to explain your work. Staple all the work in the following sequence. Use SI system of units throughout. Show units at all applicable places.

1. Signed typed affidavit sheet. 2. On engineering graph paper in your own handwriting, write the following a. deflection vs. position data. b. strain vs. weight data for strain gage A c. strain vs. weight data for strain gage B d. any other input data needed for calculations 3. By hand, on a separate sheet of engineering graph paper, derive the expression for deflection as a function of position, x and the applied load. Start from equation (1). 4. By hand, on a separate sheet of engineering graph paper, derive the expressions for the normal strains measured by strain gage A and strain gage B as a function of the applied load. Use equations (2) thru (5) to do this. 5. By hand, on a separate sheet of engineering graph paper, derive the regression formula for Young’s modulus by regressing the deflection vs. location ( x ) data. Note that you are asked to do this as polyfit is not usable. 6. By hand, on a separate sheet of engineering graph paper, derive the regression formula for Young’s modulus by regressing the strain in gage A vs. load data. Rewrite the regression formula if one was regressing the strain in gage B vs. load data. Note that you are asked to do this as polyfit is not usable. Also, note that there is also only one constant of regression in the model. 7. Put all the recorded input data as MATLAB statements in the beginning of the mfile as a separate block. Any changes in the input data should not require one to change any part of the rest of the program. 8. Regress the deflection vs. location ( x ) data from 2(a) to the model you found in #5. Find the best estimate of the Young’s modulus of the material. Illustrate and plot the deflection vs. location data that shows individual data points and the plot of the regression curve. 9. Regress the strain (strain gage A) vs. load data from 2(b) to the model you found in #6. Find the best estimate of the Young’s modulus of the material. Illustrate and plot the force vs. strain data that shows individual data points and the plot of the regression curve. 10. Regress the strain (strain gage B) vs. load data from 2(c) to the model you found in #6. Find the best estimate of the Young’s modulus of the material. Illustrate and plot the force vs. strain data that shows individual data points and the plot of the regression curve. 11. Using the weight used for 2(a) a. solve equation (1) analytically for deflection using MATLAB (Hint: dsolve). b. find the theoretical deflection at the 5th data point of 2(a). Use E=69 GPa for aluminum. c. find the absolute relative percentage difference between the experimental (observed) and the theoretical values. 12. Using the weight from the 5th data point of 2(b) a. find the theoretical strain in strain gage A. Use E=69 GPa for aluminum. b. find the absolute relative percentage difference between the experimental (observed) and the theoretical values. 13. Find the d 2w a. best estimate of at the 5th data point of item 2(a) from the deflection vs. dx2 position data. b. find the absolute relative percentage difference between the experimental d 2w M (observed) value obtained above for and the theoretical value of at the dx2 EI 5th data point of item 2(a). Use E=69 GPa for the theoretical value.