MEMS Report for Lab #3. Use of Strain Gages to Determine the Strain in Cantilever Beams

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1 MEMS 1041 Report for Lab #3 Use of Strain Gages to Determine the Strain in Cantilever Beams Date: February 9, 2016 Lab Instructor: Robert Carey Submitted by: Derek Nichols

2 Objective: The objective of this experiment was to measure the strain in a cantilever beam with the use of strain gages. This measured strain was then compared to the theoretical strain found from equations derived from solid mechanics. Theory: When a cantilever beam is subjected to a point force at its free end, the free end is displaced from its equilibrium position. This vertical displacement causes the beam to extend, and the equation for this strain can be derived from knowledge of beams in bending. The experimental setup for the lab can be seen in Figure 1 below. The beam is fixed in place with a point force acting on the end of the beam. Figure 1: The experimental setup of the beam defining variables (Reproduced from ME 1041 Lab #3 handout with permission of the ME Dept., University of Pittsburgh) Summing forces and moments about the clamp end of the beam leads to: F y = 0 = R o,y P R o,y = P M o = 0 = M o PL 2 M o = PL 2 CCW

3 Figure 2: The free body diagram of the beam Figure 3: Cut section of the beam M o = 0 = Px + PL 2 + M(x) M(x) = Px PL EIy = M(x) = Px PL 2 EIy = 1 2 Px2 PL 2 x + C 1 EIy = 1 6 Px3 1 2 PL 2x 2 + C 1 x + C 2

4 Using the boundary conditions that the slope and displacement at the fixed support is zero, the constants of integration can be solved for. [EIy] x=0 = 0 = C 2 [EIy ] x=0 = 0 = C 1 Plugging these back into the deflection equation, the equation can now be solved in terms of deflection, delta. EIy = 1 6 Px3 1 2 PL 2x 2 δ = y(x = L 2 ) = 1 EI (1 6 PL PL 2 3 ) = PL 2 3 Since it is understood that negative deflection is downwards, the negative sign can be removed, and the equation can be used to solve for the elastic modulus. δ = PL 2 3 3EI E = PL 2 3 3δI This cantilever beam is simply a beam in bending. The stress experienced on the top and bottom surfaces of a beam in bending can be expressed as: σ = Eε = Mc I This equation can be used to solve for the strain of the beam by substituting in the expression for the elastic modulus that was derived previously. ε = Mc EI ε = Mc 3 I (PL 2 3δI ) 1 = 3Mcδ PL 2 3 The strain at the surface is seen to be a function of the moment experienced at that portion of the beam. The moment at the strain gages is equal to the force at the end of the beam times the length of beam separating the strain gage and the end of the beam. ε = 3(PL 1) ( t 2 ) δ PL 2 3 3EI

5 ε = 3L 1tδ 2L 2 3 (1) This theoretical strain can then be confirmed in a laboratory with the use of strain gages. Strain gages are long lengths of thin wire that are attached to a beam and have a current running through them. As the beam is subjected to a strain, the wire also strains which changes its resistance. When hooked up to a strain indicator, the strain indicator is able to interpret the changes in resistance and relate it back to the amount of strain that the beam is experiencing. Procedure: A full bridge, half bridge, and quarter bridge were constructed with the strain gages and were used to measure the strain of a beam that they were attached to. The strain gages were assembled as shown in Figures 4 through 6. These strain gages were attached to a beam which was placed in a cantilever beam holder and micrometer mounting display. The strain gages were hooked up to a strain indicator which interpreted the change in resistance from the strain gages and calculated the strain that the beam experienced. The equipment listed in Table 1 was what was needed in order to complete the lab. When the beam was mounted in the beam holder, the micrometer mounting assembly was rotated two full turns which corresponded to 0.05 inches in displacement. For each bridge configuration, the strain was measured for displacements between 0 and 0.5 inches in 0.05 inch increments. Each bridge was tested twice so that the strains for each 0.05 inch displacement could be averaged. Figure 4: Full bridge setup (Reproduced from ME 1041 Lab #3 handout with permission of the ME Dept., University of Pittsburgh)

6 Figure 5: Half bridge (Reproduced from ME 1041 Lab #3 handout with permission of the ME Dept., University of Pittsburgh) Figure 6: Quarter bridge (Reproduced from ME 1041 Lab #3 handout with permission of the ME Dept., University of Pittsburgh) Table 1: List of equipment used in the lab Equipment Model Number Serial Number Cantilever Beam with 4 Strain Gages - - Cantilever Beam with 2 Strain Gages - - Measurements Group P3 Strain Indicator P Cantilever Beam Holder and Micrometer Mounting Assembly Flexor Multimeter HP 34401A US Caliper Fowler -

7 Strain (μ) Summary of Results: The experimental strains were collected from the strain indicator and the theoretical strains were found using Equation 1. Table 2: Data for full bridge Deflection (in) Experimental Average Strain (μ) Theoretical Strain (μ) Percent Difference Between Strains (%) NA Strain vs. Deflection for Full Bridge Measured Strain Theoretical Strain Deflection (in) Figure 7: Experimental and theoretical strains versus deflection for full bridge

8 Strain (μ) Table 3: Data for half bridge Deflection (in) Experimental Average Strain (μ) Theoretical Strain (μ) Percent Difference Between Strains (%) NA Strain vs. Deflection for Half Bridge Experimental Strain Theoretical Strain Deflection (in) Figure 8: Experimental and theoretical strains versus deflection for half bridge

9 Strain (μ) Table 4: Data for quarter bridge Deflection (in) Experimental Average Strain (μ) Theoretical Strain (μ) Percent Difference Between Strains (%) NA Strain vs. Deflection for Quarter Bridge Measured Strain Theoretical Strain Deflection (in) Figure 9: Experimental and theoretical strains versus deflection for quarter bridge Discussion: All of the measured strains closely resemble the theoretical strains as the deflection increases. They all start off with large percent differences for small displacements, but as the deflections increase, the percent differences decrease. These differences are not what I had expected. I would have expected the percent differences to start small and get larger with increased deflection. An explanation for this could be that while deflections are small, the

10 intermolecular bonds are still strong and want to limit strain as much as possible. Of all of the strain gages, the full bridge should be the best since it contains four strain gages, two to measure tension and two to measure compression; however, after observing the data, the full bridge yielded the highest percent differences while the half bridge containing one strain gage for tension and one for compression yielded the lowest percent differences. The theoretical and experimental results were close for all three bridges. As was stated above, the percent differences between the measured and theoretical strains started off large, but eventually became minute. It is important to realize that comparing the theoretical and measured strains leads to a percent difference instead of a percent error because the theoretical value is not the true value. There are many different factors that contribute to the strain experienced by a beam that are not taken into account in Equation 1. Sources of error in the theory or procedure that could explain differences between the theoretical and experimental results are that Equation 1 that is used to solve for the strain in the beam is the result of many assumptions such as that the force from the micrometer acts as a point force at the end of the beam, that the stress and strain relationship is perfectly linear and is related by a constant elastic modulus, and that the cross sectional area of the beam does not change as the beam deflects. Also, the strain gages have resistances that are not exactly what they are listed which could cause some errors in the readings from the strain indicator. Conclusion: This laboratory employed the use of strain gages to measure the strain in a beam and subsequently compared this strain to the theoretical strain calculated with Equation 1, and in doing so, the accuracy of a strain gage was able to be verified. For beams with unusual dimensions or beams under complicated loading conditions, strain gages can be used with confidence to determine the strain for situations where a theoretical strain equation does not exist.

Experiment Five (5) Principal of Stress and Strain

Experiment Five (5) Principal of Stress and Strain Introduction Objective: To determine principal stresses and strains in a beam made of aluminum and loaded as a cantilever, and compare them with theoretical