Lab Exercise #5: Tension and Bending with Strain Gages Pre-lab assignment: Yes No Goals: 1. To evaluate tension and bending stress models and Hooke s Law. a. σ = Mc/I and σ = P/A 2. To determine material modulus of elasticity (E) from tension and bending tests. 3. To introduce the use of strain gages. Principles: In an introductory Strength of Materials course, normal stress is one of the fundamental concepts discussed. Normal stress is developed in two ways: from tension due to normal loads or from bending. As seen in Fig. 1a, an object has been subjected to an axial load of magnitude P at both ends. If a cut is made at any transverse plane a-a, there will be a distribution of internal stress across the face as seen in the lower free body diagram of the object. The value of the stress σ is constant (for our purposes) across the cross-sectional area and is a function of the applied axial force. The value of σ can be calculated according to the equation: σ = P/A Figure 1 - An object with an applied axial load and the internal stress. When a beam is subjected to external forces along its length, as seen in Fig. 2a, there are of course bending moments and shear forces produced internally. Bending moment and shear diagrams can be developed to determine the magnitude of the respective values. The moment and shear produce corresponding stresses, which can be calculated with the appropriate techniques. With regards to the bending moment, the type of stress produced is identical in nature to normal stress. However, unlike the stress in uniaxial tension, the stress on the faces of the internal cuts is not uniformly distributed. Consider Fig 2b, which shows a cross-sectional cut of the beam at line a-a. Because of the applied forces on the beam (P and w), there is a bending moment developed along the length of the beam that varies with position. The moment creates stress on Plane a-a that is referred to as bending or flexural stress, but is identical to normal stress because it causes to the longitudinal fibers of the beam to stretch or compress. Figure 2 - Distribution of stress within a loaded beam.
As previously mentioned, the value of the bending stress, σ x is not constant across Plane a-a. It varies along the y-axis in a linear fashion. As seen in Fig. 2c, the value of σ x is zero at the neutral axis of the cross-sectional area. For a rectangular beam, the neutral axis is a horizontal (parallel to z-axis) line passing through the geometric center of the cross-section. From the neutral axis, the value of σ x increases or decreases linearly. When a beam is loaded as in Fig. 2a, the top fibers of the beam are in compression and the bottom fibers of the beam are in tension. The tensile or compressive value of the stress is: σ x = My/I where: M is the bending moment at Plane a-a y is the distance from the neutral axis I is the area moment of inertia The maximum magnitudes of flexural stress will of course lie at the top and bottom surfaces of the beam. In this lab, both scenarios will be examined by applying loading to two different samples. One will be loaded in pure tension and strain will be monitored with strain gages. By use of Hooke s Law, stress can be calculated from the experimental strain values. The calculations for stress can then be compared with the values determined from σ = P/A. In the second scenario, a beam will be loaded in a 4-point loading configuration, inducing bending stress. Using appropriate formulas for longitudinal strain in a beam, stress values can be calculated from the measured strain and compared with expected values of stress from the equation σ x = My/I. Materials: Steel (1018) flat tension specimen (6 ) with two strain gages (see Figure 3) Steel (1018) beam (14 ) with strain gages (see Figure 4) Safety Issues: The Instron 3369 is capable of applying forces up to 11,000 lbs. Keep hands away from the test area during a test or whenever the cross member may move. Pre-lab Several calculations should be performed before the lab exercise is performed. See the Pre-Lab Worksheet and determine the values below: Tension 1. For the tension specimen, calculate the maximum allowable load based on a maximum allowable stress of 16,000 psi. 2. Provide predictions of the strain readings so that you can easily verify your data during the experiment. a. You can do this based on a unit load by allowing the tensile force to be 1 lb. There are two strain gages mounted on the specimen. They are both oriented in the same manner in order to read axial strain and should report the same values. (Two gages are used to increase accuracy by averaging their values.) Calculate the strain that will be reported by each gage when a unit load is applied. Bending 3. Referring to Fig. 4, the value of the bending moment at the midspan of the beam (for the geometry and type of loading) is 2.7P [in lbs]. Using the equation σ x = My/I, predict the value of stress at the top and bottom of the beam assuming P is unit load (1 lb). 4. If the maximum allowable bending stress is ±16,000 psi, calculate the maximum value of P.
5. Using the relationship of E = σ/ε, calculate the expected strain at the top and bottom of the beam using the results from Part 3 (assuming a unit load). This will allow you to easily verify your data during the experiment. Note that all force measurements will be in units of pounds, strain measurements are dimensionless but will be recorded as micro-inches per inch (με), all stresses are in pounds per square inch (psi) and the modulus of elasticity also has units of pounds per square inch (psi). Material properties for steel to use in predictions: E (Young's Modulus) = 30 Mpsi ν (Poisson's Ratio) = 0.29 Allowable bending stress = ± 16,000 psi Allowable shear stress = ± 8,000 psi Procedure: Tension test: For this portion of the testing, the teaching assistant will perform a simple tension test on the Instron universal tester. The testing will be very similar to past experiments. The 1018 sheet metal specimen seen in Fig. 3 will be used. Two strain gages are mounted parallel to the direction of pull and should report the same values. Their results will be averaged for greater accuracy. The test will be run in a slightly different method as compared to previous exercises. Instead of allowing the Instron machine to continuously record force and extension values, the testing will be paused multiple times as the load increases. At each pause, the load and strain values from the gages will be recorded on the attached data sheet. That data will be used for further analysis in the lab report. The teaching assistant will perform the following: 1. After the specimen is mounted and ready for testing, the strain gage values will be balanced ( zeroed ). 2. The applied load will be increased in approximately six increments to the pre-determined maximum load. The load and strain values will be recorded for each increment. 3. The zero value of the gages will be recorded at the end of the test. Bending test: As seen in Fig. 4, the beam will be subjected to what is called a four-point bend test. The test will impose shear and bending moments within the beam. Strain values on the top and bottom of the beam (at midspan) will be measured throughout the testing. As with the tension testing, the test will be performed in increments up to the pre-determined maximum load. The same three steps will be performed as outlined in the tension testing section. Data Analysis and Discussion When testing has been completed: 1. Complete all sections of the Data Sheet, including the column for Expected Strain. Compare the (calculated) Expected Strain with the measured strain. Are they similar? 2. By hand or with Excel, prepare a plot of stress vs. strain for the bending test. Remember to label appropriately. For the value of strain, use the average of the absolute values measured at the top and bottom of the beam. a. Determine E (Young's Modulus) for this specimen from the slope of your plot of stress vs. strain. If creating a graph by hand, create a best fit line by inspection. If using Excel, use the trendline function to create a best fit line. Compare with published values. 3. How well did the experimentally derived value of E compare to published data? 4. What issues may decrease the accuracy of strain gages?
Figure 3 - Details of the tension specimen. Figure 4 - Details of the beam specimen
Pre-Lab Worksheet ENGR 151 Strength of Materials Lab Exercise #5: Tension and Bending with Strain Gages Name: Lab Day - M T W R F Lab Start Time: Tension test 1. Calculate the maximum allowable load for the steel tension specimen if the maximum allowable stress is 16,000 psi. Maximum allowable load lbs 2. Calculate the strain gage value for one pound of applied force (unit loading) Strain gage value με lb of applied force Bending test 3. Given that the bending moment at the top and bottom of the beam at midspan is 2.7P, calculate the value of flexural stress at those locations assuming P is 1 pound. Refer to Fig. 4. σ (top) = psi σ (bottom) = psi 4. Calculate the maximum allowable value of P for the beam according to the arrangement in Fig. 4. Recall that the maximum allowable bending stress is ± 16,000 psi. Maximum allowable load lbs 5. Calculate the expected strain gage readings (top and bottom) for a unit load on the beam. Top με lb of applied force Bottom με lb of applied force
Data Sheet ENGR 151 Strength of Materials Lab Exercise #5: Tension and Bending with Strain Gages Name: Lab Day - M T W R F Lab Start Time: Tension test Zero readings (record after testing): Units Target Load (P) Actual Load Expected Strain 1 Bend Test Zero readings (record after testing): Top gage (#1) Post-Test Calculations Bottom gage (#2) Target Load (P) Actual Load Expected Strain 1 Units 1. Assume E = 30 Mpsi Post-Test Calculations