Lab 4 Strain Measurement Lab Manual

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MEEN 3210 MEASUREMENTS STRAIN MEASUREMENT LABORATORY Robert Benton and Kendrick Aung Introduction Experimental determination of stress is important when designing components with complex geometry and/or loading. To verify a given design, nondestructive tests of a prototype may be performed using strain gages. Once strain is determined experimentally, the stress may be calculated using the stress-strain relationship for the material. Although there are various failure theories for the ductile and brittle materials subjected to static and dynamic loads, each of these theories predicts failure when the stress in a given material exceeds various predetermined values. As such, stress is a major consideration in design of mechanical components. Analytical methods exist for relatively simple and common geometric cases such as shafts, beams, and cylinders. However for cases with stress concentration or other complex geometry, these analytical methods are not accurate. Numerical methods such as finite element analysis (FEA) may be applied in such cases. However, finding the appropriate FEA model (or mesh) for a given situation can be more of an art than an exact science. As such, strain measurements should be used to verify stresses in physical prototypes of the design. Strain Measurement Three common tools are available for experimental determination of stress and strain, namely, photoelasticity, stress coat application, and strain gage application. Photoelasticity Although photoelasticity is most easily applied in cases of two-dimensional components, techniques also exist for more complex geometry. When polarized light is passed through a transparent material, visible bands appear as contour lines within the material to indicate the level of stress in the material. This is because the speed of light in the medium is a function of stress. Determination of stress using photoelasticity requires relatively expensive equipment when compared to the methods described below. In addition, prototypes of the part must be prepared especially for photoelastic measurement. Stress Coat Application The application of a stress coat is a fairly simple concept for which students should have an intuitive sense. A stress coat is used to determine the principal stress directions by observation. A brittle lacquer is applied to the surface of the prototype (note: the stress coat is applied in liquid form and allowed to dry). Because the coating is brittle, the

coating will crack perpendicular to the direction of maximum tensile stress (maximum principal stress). Strain Gage Application Strain gages may be applied to any free surface of a prototype component. In many cases, the component may be tested in service with the strain gages applied. The main limitations include environmental limitations required to avoid corrosion motion limitations due to the requirement that wire leads be connected between the strain gages and measurement instruments (for example, rotating components would present a challenge). In these cases, service conditions should be simulated on the prototype during strain gage experiments. Strain gage measurements are based on small changes in resistance of the gage when strained. As such, a bridge circuit is employed to measure change in resistance. Although the bridge can be cheaply constructed from scratch using precision resistors, specialized bridge-balancing units and strain indicators are most often used in practice to ensure precise measurements. Specialized analog-to-digital- conversion (AD) cards may also be purchased with built-in bridge circuits for dynamic strain gage measurement and data logging via personal computer (PC). Because strain gage resistance and prototype/gage strain are functions of temperature, temperature is an important consideration in strain gage application. If used to determine stress, the thermal growth in the component should not be measured by the strain-gage bridge circuit. Bridges with more than one strain gage leg may be used to aid in temperature compensation. In addition, self-temperature compensation gages are commonly available for application with steel and aluminum alloys (designated 06 and 13, respectively). Resistors in the bridge will also change resistance with temperature. Once the gage system is connected, temperature tests can be performed to determine error due to temperature in the measurement system. During any experiment where environmental factors may affect the results, temperature variation and other environmental conditions should be recorded along with strain measurements for future analysis. Theory The resistance in a wire in Equation (1) R = ρL A Eq. (1) where  is the wire material’s resistivity, L is the wire’s length, and A is the wire’s cross sectional area. When a strain is placed on the wire, its resistance changes due to changes in resistivity, length, and cross-sectional area. As a result, measurement of the wire’s resistance can be used to indicate strain. A strain gage is a specially designed metallic element that is very sensitive to strain in a given direction (see discussion and figures in [1], [2], and [3]). Strain gage application procedures are described on pages 17-23 of [1], which should be read at this time. Once the gage is applied and connection wires have been made, the gage must be placed in a

bridge circuit in order to measure changes in the gage’s resistance (see page 24 of [1]). For this lab, specially designed strain gage instruments described in [1] will be used to implement the bridge circuits. Often, specially oriented sets of strain gage “rosettes” are applied to measure strain in various directions at a single location (see pages 31 and 32 of [1] for examples of various strain gages and strain gage rosettes). Juvinal [3] gives equations necessary to find principal strains from measured strains in chapter 5. Objectives The main objectives of the laboratory are: 1. To familiarize students with industry standard strain measurement techniques 2. To determine Modulus of Elasticity of three materials using a cantilever beam 3. To compare experimental and theoretical values of Modulus of Elasticity 4. To compare experimental and theoretical deflection values Experiment 1: Application of a Strain Gage on a beam Required Equipment and Supplies  Bulletin 309D: Student Manual for Strain Gage Technology [1]  Student Strain Gage Application Kit  Sandpaper  Degreaser  M-Prep Conditioner A  M-Prep Neutralizer 5A  Gauze and Cotton Swabs  Cellophane tape  M-Bond 200 Kit  M-Coat A  Leadwire  Drafting tape  Soldering Station with Soldering Supplies  Student Strain Gage Practice Patterns and Student Strain Gages  Steel or Aluminum Bar Stock Procedure 1. Select a piece of steel or aluminum bar stock. 2. Prepare the surface of the bar stock as described on pages 18 and 19 of [1]. 3. Obtain a student strain gage practice pattern from the instructor and practice the strain gage application procedure described on pages 20 and 21. Special care should be taken to replace the caps on all chemicals to avoid spillage or evaporation. 4. Repeat steps 2 and 3 until the practice pattern in successfully applied to the bar stock surface. 5. Obtain a student strain gage from the instructor and repeat the procedures described on pages 18-21.

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Figure 1 Materials and Soldering Station for Strain Gage Mounting Possible discussions: Why is the cleaning process before mounting the strain gage important? Experiment 2: Strain Measurement and Application Required Equipment and Supplies  Model P-3500 Portable Digital Strain Indicator  Connecting wires  Beam with Strain Gage  Calipers and/or Ruler  Test Weights Procedure 1. Select a beam with strain gage (record the strain gage resistance and gage factor for future reference). 2. Clamp the beam to the table or place the beam in a vice. Note that care should be taken to clamp the beam far enough from the strain gage to minimize the strain induced by the clamping forces. Recall that the vice forces may induce strain due to the effects of Poisson’s ratio. 3. Measure the dimensions of the beam required to calculate the stress at the location of the strain gage. Record data in Table 1. 4. Connect the wires to the P-3500 Portable Digital Strain Indicator as indicated inside the lid of the Indicator unit. 5. Use the procedure inside the lid of the P-3500 unit to set the appropriate gage factor for the strain gage(s).

6. With no weight attached to the beam, use the procedure inside the lid of the P-3500 unit to balance the bridge circuit. To avoid unnecessary wear on the equipment, please unlock the balance control knobs before attempting to adjust the knob positions. 7. Once the bridge is balanced, lock the balance control knob on the P-3500 unit and do not attempt to adjust the knob for the remainder of the lab period. 8. Apply weights supplied by instructor to the beam made of steel and record the strain for each weight. Record data in Table 2. 9. Repeat the measurements in step 8 using the beam made of Plexiglas. Record the measured data in Table 2. Figure 2 An Experimental Setup for Measuring Strain Results and Discussions Result I: Comparison of Stress Using the Published Value of Young’s Modulus 1. Use the stress strain relationship (Hooke’s law) to calculate the experimental stress (  exp ) using the measured strain and Young’s Modulus of the beam material from the Machine Design text book. σ exp = E text ∗ ε exp Eq. (2) where,  exp is the experimental stress, E text is the value of modulus of elasticity from the textbook and  exp is the experimental strain. 2. Calculate the values of the theoretical bending stress (  theo ) for each weight using the bending stress formula from the Machine Design text and compare with the experimental stress values found in step 1.

σ theo = Mc I Eq. (3) Where  theo is the theoretical stress, M is the moment experienced by the strain gage, c is the half of the beam height, and I is the moment of inertia of the beam. 3. Using a table, compare the two values of stresses for each load and compute the error (in absolute value and percentage) as defined in Eq. (4). error = σ theo − σ exp , error ( % )= error σ theo 100 Eq. (4) 4. Plot strain versus load on a scatter plot. Plot both stress values as a function of load on the same scatter plot. 5. Provide some discussions on the results such as magnitude of the error, sources of errors, etc. Possible discussions: Do the values of experimental and theoretical stress agree? What is the maximum error/difference between the two values? Does the plot of strain versus load show linearity and hence support the validity of Hooke’s Law? What are the sources of errors? Result II: Experimental Determination of Young’s Modulus of the Beam Materials For each material, do the following 1. Draw a scatter plot (x-y plot) using experimental values of strain (  ) as y and values of weight (P) as x in Excel. Draw a trend line using linear regression. Using the value of the slope of the trend line, determine the Young’s Modulus of each material based on the equation given below. Eq. (5) is obtained by combining equations (2) and (3). ε = Mc E exp I = PLc E exp I = C 1 P Eq. (5) Where C 1 = Lc/E exp I is the proportionality constant which is the slope of the curve between the experimental strain (  ) and load (P). Since C 1 depends on the properties of the beam (L, c and I) that are measured in the lab and the value of modulus of elasticity (E), modulus of elasticity (E) can be determined experimentally using the value of the slope obtained from the plot and the measured dimensions of the beam.

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2. Compare the experimental value of Young’s Modulus of each material from step 1with the value given for that material in the Machine Design text book. 3. Compute the absolute difference (error) between the experimental and textbook values of Young’s Modulus for each material. Determine the percentage difference and give the results in a table. 4. Discuss the differences between the experimental and textbook values of Young’s Modulus. Provide some discussions on the reasons for errors. Possible discussions: Does experimental Young’s Modulus value agree with the value given in the textbook? What are the sources of errors? Based on the results of the experiment, what can you conclude about the values of modulus of elasticity given in the textbook? Result III: Statistical Determination of Young’s Modulus of the Beam Materials 1. Compute the experimental value of modulus of elasticity for each value of measured strain and weight for each material according to equation (5) rearranged as Eq. (6). E exp = Mc εI Eq. (6) 2. Compute the values of average and standard deviation of Young’s Modulus for each material. 3. Assuming the measured data follows Gaussian probability distribution, compute the 95% confidence interval of the value of Young’s Modulus for each material. 4. Give all the results including means and standard deviations as well as the textbook values in a table. 5. Compare the experimental value of Young’s Modulus of each material with the value given for that material in the Machine Design text book. Is the textbook value within the range of the confidence interval of experimental Young’s Modulus? Discuss the results and provide reasons for errors. Possible discussions: Does Young’s Modulus value obtained by statistics agree with the value given in the textbook? Does Young’s Modulus value obtained by statistics agree with the value obtained in the Results II section? Is the textbook value within the range of experimental Young’s Modulus values? What are the sources of errors? Table 1: Dimensions of the Beams

Dimensions (unit) Beam Material Steel________ Aluminum_______ Brass________ Width Length Height Location of a strain gauge from an open end Table 2: Strain Measurements of Three Beams Steel_______ Aluminum________ Brass_________ Weight (unit) Strain (  ) Weight (unit) Strain (  ) Weight (unit) Strain (  ) 500 gm 100 gm 200 gm 1000 gm 200 gm 400 gm 1500 gm 300 gm 600 gm 2000 gm 400 gm 800 gm 2500 gm 500 gm 1000 gm Table 3: Comparison of Experimental and Theoretical Stress Values Material Stress Result Experimental Theoretical Error Error (%) (unit) (unit) (unit) Aluminum Steel Brass Table 4: Comparison of Experimental and Textbook Values of Modulus of Elasticity for each material Material Modulus of Elasticity Result Experimental Textbook Error Error (%) (unit) (unit) (unit) Aluminum Steel Brass Table 5: Comparison of Statistical Results of Modulus of Elasticity

Material Experimental Results (unit) Textbook Value (unit) Average Range Aluminum Steel Brass Experiment 3: Strain and Deflection Measurements in a Three-point Bending Test Figure 3 An Experimental Setup of a Three-Point Bending Test Figure 4 Shear Force and Bending Moment Diagrams for a Three-Point Bending Test Procedure 1. Select the beam to be used and record the strain gage factor for future reference.

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2. Measure the dimensions of the beam and record data in Table 6. 3. Place the beam onto the support bars and level it. 4. Connect the strain gage wires to the P3500 strain gage indicator and calibrate the indicator as shown on the inside of the lid of the indicator unit. 5. Apply the weights at the designated distance (mid-point of the beam) and record data (strain and deflection) in Table 7. 6. Compare the measured values of defection with the theoretical values obtained using Eq. (7). σ = Mc I δ = PL 3 48 EI Eq. (7) 7. Discuss the differences between theoretical and experimental results. Possible discussions: Does experimental and theoretical values of deflection agree with each other? What are the sources of errors? Table 6: Dimensions of the Beams Dimensions (unit) Beam Material Steel Width Length Height Location of a strain gauge Table 7: Strain and Deflection Measurements for the Beam Load (in lb) Load (in gm) Strain Deflection (unit) 2 0907.18 gm 4 1814.37 gm 6 2721.55 gm 8 3628.74 gm Table 8 : Comparison of Experimental and Theoretical Deflection Values Material Load Deflection (unit) (unit) Experimental Theoretical Steel References

[1] Bulletin 309D: Student Manual for Strain Gage Technology , Vishay Measurements Group, Inc., Education Division, 1992. [2] J. P. Holman, Experimental Methods for Engineers , 7 th ed., McGraw-Hill, 2001. [3] Robert C. Juvinall and Kurt M. Marshek, Fundamentals of Machine Component Design , 3 rd ed., John Wiley & Sons, Inc., New York, 2000.