MEMORIAL UNIVERSITY OF NEWFOUNDLAND FACULTY OF ENGINEERING AND APPLIED SCIENCE
Engineering: 4312
Mechanics of Solids I
Lab Test #1 - Cantilever Beam Safety: Please follow all the safety requirements for carrying out experiments. Ask the technician for details. Steel toe boots are a must for the lab.
OBJECTIVES: 1. To measure surface strains in a cantilever beam using electrical strain gages. 2. To measure the deflection of the beam using LPDT. 3. To compare the analytical and experimental values of the stress and deflection in the cantilever beam. 4. To present the results in terms of statistical values. 5. To become acquainted with various items and test equipment for Mechanics of Solids.
TEST SPECIMEN A steel cantilever beam, flat bar cross section, instrumented with 3 strain gages.
EQUIPMENT 1. Structural testing frame. 2. Calibrated weights. 3. Data acquisition unit connected to a personal computer (PC) 4. Linear potentiometric displacement transducer (LPDT) 5. Measuring instruments as required.
PROCEDURE 1. The set-up for this experiment is shown in Figure 1. The strain gauges used are 10 mm long, with a resistance of 120 Ω. 2. Using the appropriate instruments, measure the length, width and depth of the beam. Note that the accuracy needed for the length is less than that for width and depth. Explain why. Also, with the same accuracy used to determine the Length of the beam, measure the distance from the fixed end of the beam to centerline of each gage and the location of the LPDT. Estimate the accuracy of these measurements. Should the distances be measured from the centre of the support width or from the outer face? 3. Check the data acquisition system, the personal computer and driving software. Familiarize yourself with their operation. 4. Start the data acquisition (DA) program and create an electronic file on the computer hard drive for data logging. Using the DA program, set the initial values of strains and deflection to zero at no load. Put the DA program on "monitoring mode". 5. Load the beam (gently) in increments of approximate 0.2 kg, up to 3.0 kg. After placing each load increment, allow the system to stabilize before logging the data. 6. Log the strains and mid-point deflection for each load increment by changing the mode of the DA program from "monitoring" to "log". In the "log" mode, the DA program writes one set of data in the log file every 2 sec. Wait for about 10 seconds and switch back to the monitoring mode. Keep an accurate record of the total applied load at each loading increment for later synchronization with the data in the log file. 7. After reaching the maximum load (3.0 kg) release the load from the beam, in decrements of 0.2 kg until all loads are removed. Again log the strain gauge and LVDT readings for each decrement. 8. Copy the electronic file that contains the results to your own memory stick or diskette. 9. Turn off all equipment, and tidy up.
CALCULTIONS 1. The theoretical values of normal stress can be determined from the flexural formula:
σ=
Mc I
(1)
Where, σ = Normal stress M = Bending moment = P(L-x) c = Distance from N.A. to the extreme fibres (c= h/2 for rectangular cross-section) I = Moment of inertia (I = bh3/12 for solid rectangular cross-section) For this set-up, explain the direction of the above stress and why. 2. Using the measured strains, ε, the experimental values of stress, σ, can be determined from: σ=Eε
(2)
Where, E is the elastic modulus. 3. The theoretical value of deflection (δ) for the cantilever beam is:
δ=
P 2 x ( 3L − x ) 6 EI
(3)
Where, x = Distance measured along the cantilever beam from the fixed end.
ANALYSIS 1. Produce plots of measured strains vs. applied load for all the strain gauges. It is not enough to simply plot the comparisons. Comment on the comparisons. 2. Compare the theoretical and experimental values of stress at the strain gage locations using equations 1 and 2, respectively. When computing the experimental values of stress with equation 2, use the recommended value of E=200,000 MPa. Comment. 3. Compare theatrical and experimental values in a tabular form and separate graphs for each gauge. Produce statistical parameters for the errors (mean, standard deviation, etc.) 4. Obtain Load-Deflection graph (P on the vertical axis) from the experimental data. Comment on the linearity of P-δ relationship (or the lack of it). Approximately measure the slope (the ratio P/δ) of the P-δ line from the graph. Compute the theoretical slope from equation 3 and compare. 5. Using equation 3, determine the experimental value of modulus of elasticity, E, at each load increment. How do these values of E compare with the handbook value of 200,000 MPa for mild steel? If the computed values of E are different from the handbook value, what are the possible reasons?
6. Prepare a report according to the given guidelines. Include all calculations made under “Preliminary Calculations” placed in an Appendix.
Note: Make all calculations (hand written or typed on a computer) legible and professional looking with sufficient margins, titles and good amount of “white space.” The report must be produced the same way as you would a work term report. You may include photographs as well.
REFERENCE R.C. Hibbeler “Mechanics of Materials,” Prentice-Hall
Figure 1 Beam test set-up
Figure 2 Schematic for the data acquisition system
