ENGR 244 Lab 4

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Dec 6, 2023

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Lab 4: Stress Analysis of Beams Using Strain Gauges Lab Section CI – X Winter 2022 Professor Ahmed Soliman Concordia University Montreal, QC, Canada Tuesday February 15 th , 2022 Table of Contents Nomenclature pg 3 List of Tables pg 3 List of Figures pg 3 Objective pg 4 Introduction pg 4 Procedure pg 4 Results pg 4-7 Discussion and Conclusion pg 8-9 References pg 10 Original Data pg 11

Nomenclature σ = bending stress ε = strain δ ε L = change in longitudinal strain δ ε T = change in transverse strain δP = change in applied load ν = Poisson’s ratio b = base E = elastic modulus G = shear modulus h = height I = moment of inertia M = moment caused by the applied load y = distance from the neutral axis List of Tables Table 1: Strain Indicator reading units are in micro-strain (10 -6 strain) p. 4 Table 2: Bending Stress at each gauge point in MPa p. 6 Table 3: Stress Values for Each Channel Using the Theoretical Elastic Modulus for Steel p. 7 List of Figures Figure 1: Stress vs Strain for gauge channels 1 and 5 (Absolute Values) p. 6

Figure 2: Longitudinal Strain (Channel 5) and Transverse Strain (Channel 6) vs Applied Load (Absolute Values) p. 6 Figure 3: Strain vs Distance from the Horizontal Mid-Plane for the 5060N Load p. 7 Objective: The objective of this lab is to compare the practical values of the stress and strain distribution with their theoretical values. The elastic modulus, Poisson's ratio and the shear modulus will also be calculated using these measurements. Introduction: Bending is a common English word and a term that everyone understands, but what isn’t always known is that bending occurs when the forces applied to the material causes both compressive and tensile stresses over different surfaces [ CITATION Con \l 4105 ]. While bending generally includes many different types of stresses, we will only be looking at bending stress to simplify things a little. This can also be referred to as pure bending, which is bending in the beam when shear force is absent [ CITATION SRa22 \l 4105 ]. Pure bending can be achieved applying two couples to each end of a rod, acting in opposite directions [ CITATION SRa22 \l 4105 ]. Through observation, we can say that bending strain is proportional to the distance of the force applied to the neutral axis of the material [ CITATION Con \l 4105 ]. The neutral axis, also referred to as the neutral surface, is the plane of the object that experiences bending, but does not experience extension or contraction [ CITATION Dep00 \l 4105 ]. Stating the obvious, it can be said that bending takes place in the direction of the force that is placed uniaxially. What isn’t as obvious, but has been studied, is that the material is also either expanded or contracted while being bent depending on whether the stress applied is tensile or compressive [ CITATION Con \l 4105 ]. Using this expansion or contraction, Poisson’s ratio can be figured out which is the constant that relates lateral strain and axial strain [ CITATION Con \l 4105 ]. Since structural design must consider all dimensional changes, it is an important constant to know how different surfaces will react to different forces applied [ CITATION Con \l 4105 ]. Procedure: First, measure the cross section of the bar and record the values of the height and width. A sketch of the bar with the strain gauges should be made and labelled with their number. Make sure no load is applied and then check that the center line is the axis of symmetry for the support beams. Verify that each gauge has a value of 0 (+/- 2) while there is no load applied and adjust it if needed. Now apply a load of 1000N and record the strain measurements of each gauge. Continue doing this, in increments of 1000N, until you reach 5000N. Remove the total load and record the values of the strain once more for verification of calibration. Results: Applied Load, P(N) Channel 1 Channel 2 Channel 3 Channel 4 Channel 5 Channel 6

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1012 -162 -110 -18 83 152 -50 1999 -309 -206 -25 181 306 -93 3003 -476 -309 -30 283 462 -135 4035 -639 -413 -34 391 631 -181 5060 -806 -516 -34 498 794 -225 Unload 4 -2 -5 -2 0 0 Table 1: Strain Indicator reading units are in micro-strain (10 -6 strain) Sample Calculations Formulas used: I = bh 3 /12, σ = My/I, E = σ/ε, ν = |(δ ε T /δP)/(δ ε L /δP)|, σ = Eε and G = E/[2(1+ν)] I = bh 3 /12  I = (0.01897m)(0.03184m) 3 /12 = 51.0276 x10 -9 m 4 FBD Diagram: Shear Diagram: Moment Diagram: σ 1 = My/I  σ 1 = (1012N)(0.0701m)(-0.03184m/2)/(51.0276 x10 -9 m 4 ) = -11.0664 MPa (compression)

σ 2 = My/I  σ 2 = (1012N)(0.0701m)(-0.0101m)/(51.0276 x10 -9 m 4 ) = -7.0208 MPa σ 3 = My/I  σ 3 = (1012N)(0.0701m)(0m)/(51.0276 x10 -9 m 4 ) = 0 σ 4 = My/I  σ 4 = (1012N)(0.0701m)(0.0101m)/(51.0276 x10 -9 m 4 ) = 7.0208 MPa (Tension) σ 5 = My/I  σ 5 = (1012N)(0.0701m)(0.03184m/2)/(51.0276 x10 -9 m 4 ) = 11.0664 MPa σ 6 = My/I  σ 6 = (1012N)(0.0701m)(0.03184m/2)/(51.0276 x10 -9 m 4 ) = 11.0664 MPa Applied Load (N) Stress 1 Stress 2 Stress 3 Stress 4 Stress 5 Stress 6 1012 -22.1328 -14.0415 0 14.0415 22.1328 22.1328 1999 -43.7189 -27.7362 0 27.7362 43.7189 43.7189 3003 -65.6767 -41.6667 0 41.6667 65.6767 65.6767 4035 -88.2469 -55.9858 0 55.9858 88.2469 88.2469 5060 -110.6640 -70.2077 0 70.2077 110.6640 110.6640 Table 2: Bending Stress at each gauge point in MPa 0 100 200 300 400 500 600 700 800 900 0 20 40 60 80 100 120 Stress vs Strain of Channels 1 and 5 Gauge 1 Linear (Gauge 1) Gauge 5 Linear (Gauge 5) Linear (Gauge 5) MicroStrain Stress (MPa) Figure 4: Stress vs Strain for gauge channels 1 and 5 (Absolute Values) E = (σ 2 – σ 1) /(ε 2 - ε 1 )  E = (110.6640MPa – 22.1328MPa)/(806x10 -6 -162 x10 -6 ) = 137.4708GPa

0 1000 2000 3000 4000 5000 6000 0 100 200 300 400 500 600 700 800 900 f(x) = 0.16 x − 5.22 f(x) = 0.04 x + 2.95 Longitudinal and Transverse Strain vs Applied Load Transverse Strain Linear (Transverse Strain) Longitudinal Strain Linear (Longitudinal Strain) Applied Load (N) MicroStrain Figure 5: Longitudinal Strain (Channel 5) and Transverse Strain (Channel 6) vs Applied Load (Absolute Values) δ ε L /δP = (794-152)/(5060-1012) = 0.1586N -1 and δ ε T /δP = (-225 – (-50))/(5060-1012) = -0.04323N -1 ν = |(δ ε T /δP)/(δ ε L /δP)|  ν = |(-0.04323N -1 )/(0.1586N -1 )| = 0.2726 σ = Eε where E = 200GPa  σ = (200 x10 9 Pa)(-162 x10 -6 Pa) = -32.4 MPa (compression) Load (N) Stress 1 (MPa) Stress 2 (MPa) Stress 3 (MPa) Stress 4 (MPa) Stress 5 (MPa) Stress 6 (MPa) 1012 -32.4 -22.0 -3.6 16.6 30.4 -10.0 1999 -61.8 -41.2 -5.0 36.2 61.2 -18.6 3003 -95.2 -61.8 -6.0 56.6 92.4 -27.0 4035 -127.8 -82.6 -6.8 78.2 126.2 -36.2 5060 -161.2 -103.2 -6.8 99.6 158.8 -45.0 Table 3: Stress Values for Each Channel Using the Theoretical Elastic Modulus for Steel G = E/[2(1+ν)]  G = (137.4708GPa)/[2(1+0.2726)] = 54.0118 GPa This value is close to the theoretical G value of 77 GPa.

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-20 -15 -10 -5 0 5 10 15 20 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 f(x) = 50.24 x − 12.8 R² = 1 Channel Strains at 5060N vs Distance from Horizontal Mid-Plane Distance (mm) MicroStrain Figure 6: Strain vs Distance from the Horizontal Mid-Plane for the 5060N Load The experimental value of neutral axis can be determined using the equation of the slope in figure 3. By making y = 0 we find that x is 0.2548mm instead of the theoretical 0mm for pure bending, but the difference is small enough to compare the theoretical and experimental values together. Discussion and conclusion: If we look at figure 3, we can see that the experimental value for the neutral axis is not exactly where it should be in theory, but it is close enough to the theoretical value to say that the neutral axis experiences little to no elongation or contraction. The small discrepancies in values can be caused by many sources, such as: rust, the loads being applied at a slight angle, temperature, non uniform material, uncalibrated strain indicator, non constant load, and damaged material. Some of these sources can account for differences in the experimental vs theoretical stresses. Our experimental E value calculated with the experimental stress was considerably lower than the theoretical value which might be due to the difference in height and width of the beam, and load distance or angle from the center axis. The experimental negative and positive values for channels 1 through 5 were in conformity with the theoretical values, meaning that we had tensile and compressive loads on the correct sides of the horizontal mid-plane as expected. The only exception was for channel 6 which was experimentally calculated as tension (positive), but the theoretical value shows that it is in compression (negative). This is likely because the experimental values were calculated using the distance from the horizontal mid-plane instead of the vertical plane. The transverse strain acts perpendicular to the longitudinal strain and therefore should be calculated using the vertical mid-plane. It is also known that if a surface experiences longitudinal elongation, then the transverse direction will experience contraction [ CITATION Tre22 \l 4105 ]. Since both channels 5 and 6 are on the same

surface, they should have opposite signs and so we know the theoretical value for the stress in channel 6 is correct since it is in compression. In conclusion, we were able to find stress and strain values that, while imperfect, were comparable to the theoretical values. The relationships between the experimental stress and strain values, the elastic modulus, the shear modulus, and Poisson’s ratio were all were as expected even if the values weren’t correct. Despite the many possible sources of error, the experiment was a success to demonstrate these relationships.

References [1] C. U. E. a. C. Science, ENGR 244 Mechanics of Materials Lab Manual, Montreal: Concordia University. [2] S. R. Kumar, "ecourses online," e-Krishi Shiksha, [Online]. Available: http://ecoursesonline.iasri.res.in/mod/page/view.php?id=3662. [Accessed 27 02 2022]. [3] Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526, Mehrdad Negahban and the University of Nebraska, 2000. [Online]. Available: http://emweb.unl.edu/NEGAHBAN/Em325/11-Bending/Bending.htm. [Accessed 27 02 2022]. [4] Trenchlesspedia, "Trenchlesspedia," [Online]. Available: https://www.trenchlesspedia.com/definition/4246/transverse-strain-tensile-force. [Accessed 27 02 2022].

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