Lab 2 AME 324L

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University Of Arizona *

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324L

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Mechanical Engineering

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Jan 9, 2024

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Beam Bending Aditya Kannan AME 324L 9/18/2023 NOTE TO GRADER: I spoke with the TA in class today and he allowed me to submit the report before midnight instead of 5 PM. Thank you!

Introduction In this experimental investigation, our focus will be on beam stress and bending. We will employ strain gauges strategically positioned along an I-beam and vary their distances from a fixed clamp as well as a dial gage to measure the deflection of the beam in a simple manner. Subsequently, we will subject the beam to diverse loads applied at two different points near its extremity. By collecting data pertaining to strain, load magnitudes, and distances, we will apply Hooke's law and the flexure formula, accompanied by moment diagrams/equations, to determine the strain response of the beam to a load. Discussion of relevant concepts The primary objective of this experiment is to understand the behavior of a beam when subject to bending by loading a cantilevered I-beam and gathering data on its response. In the following paragraphs, we will go over the equations that govern the response of beams to bending moments. The first among them is the flexure formula, as depicted below, where ? represents the bending moment, ? signifies the location for stress assessment, and 𝐼 stands for the second moment of area. 𝜎 = − ?? 𝐼 Now, below is the equation for shear stress due to bending. 𝑉 is the shear stress due to applied loads, ? is the first moment of area and 𝐼 is the second moment of area. 𝜏 = 𝑉? 𝑏𝐼 The distribution of normal stress is linear, where the max stress is at the top and bottom of the beam pointing in opposite directions. The distribution of shear stress is a parabola with the focus located at the central axis of the beam. Deflection is a useful piece of data gathered from this experiment using a dial gauge, and the theoretical deflection can be calculated as below. 𝛿 = ?? 2 6?𝐼 ∗ (3? − ?) Now, here are drawn shear force and bending moment diagrams for cantilever and 3-point bending configuration beams.

Figure 1: Shear and moment diagrams for different configurations of a loaded beam Experimental Procedure A 6061 Aluminum I-Beam, firmly attached to the testing table, is used as the cantilever beam for experimentation. Two strain gauges each are attached at 3 locations on the beam. The measurements were recorded using LabView and a dial gage. After configuring the LabView software and ensuring that the strain gage connections were in good condition, the dial gauge was placed on the beam at a measured location. Then, the position of the beam was adjusted in order to locate the testing machine head above a specified location. The LabView recording was started and the load was applied up to specific values ( 100, 200, 250 ?𝑏𝑠 ) and a photo of the strain and load data from LabView and the dial gage deflection was recorded. This process was performed again with the beam moved to a different position in order to understand how the deflection changed by varying the distance of the load from the cantilever. Results & Calculations Now we calculate the moment of inertia of the I-beam based on measurements taken during the experiment using the following formula: 𝐼 𝑥 = 1 3 𝑏ℎ 3 − 1 12 𝑏 1 ℎ 1 3 − 1 12 𝑏 2 ℎ 2 3 Where:

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• 𝐼? is the moment of inertia about the x-axis. • 𝑏 is the width of the entire I-beam cross-section. • ℎ is the height of the entire I-beam cross-section. • 𝑏 1 and ℎ 1 are the width and height of the smaller rectangular section (the web) of the I-beam. • 𝑏 2 and ℎ 2 are the width and height of each of the larger rectangular sections (the flanges) of the I-beam. Therefore: 𝐼 𝑥 = 1 3 (0.0258)(0.2532) 3 − 1 12 (0.0357)(0.127) 3 − 1 12 (0.0258)(0.063) 3 = ?. ?? ∗ ?? −? ?𝒈 ? ? = ?. ???? ?𝒃 𝒊? ? The area is calculated to be 12.15 𝑖? 2 We take the Young’s Modulus ( ? ) to be 10 6 𝑝𝑠𝑖 . Raw Data from Strain Gauges Table 2: Measured load vs strain at location 1 (0.0131m from cantilever) Load Location 1 (lb) White Red Green Blue Yellow Purple Orange 100 -0.00013 -0.0005 9.93E-06 1.14E-05 0.002114 0.000291 1.22E-05 200 -0.00027 -0.00102 8.29E-06 1.9E-05 0.000459 0.000624 9.6E-06 250 -0.00034 -0.00128 8.53E-06 2.42E-05 0.000595 0.000793 1.82E-05 Load Location 1 (lb) White Red Green Blue Yellow Purple Orange 100 -1.30E-04 -0.0005 8.78E-06 1.04E-05 2.10E-03 2.92E-04 1.06E-05 200 -2.70E-04 -2.70E-03 5.94E-06 1.95E-05 4.57E-04 6.25E-04 1.02E-05 250 -3.30E-04 -0.00127 7.64E-06 2.34E-05 5.96E-04 7.92E-04 1.07E-05 Load Location 1 (lb) White Red Green Blue Yellow Purple Orange 100 -1.32E-04 -0.0005 8.62E-06 9.59E-06 2.07E-03 2.91E-04 8.27E-06 200 -2.70E-04 -0.00102 6.86E-06 1.97E-05 4.60E-04 6.26E-04 1.02E-05 250 -3.35E-04 -0.00128 6.68E-06 2.49E-05 5.96E-04 7.91E-04 1.15E-05 Table 3: Measured load vs strain at location 2 (0.0115m from cantilever) Load Location 2 (lb) White Red Green Blue Yellow Purple Orange 100 4.24E-06 -0.00022 -1.69E-05 -1.74E-06 9.71E-05 -5.80E-05 2.08E-05

200 1.50E-06 -0.00044 -1.65E-05 -3.50E-06 1.83E-04 -2.22E-05 1.67E-05 250 2.12E-06 -0.00055 -1.74E-05 2.06E-06 2.35E-04 -8.33E-07 2.02E-05 Load Location 2 (lb) White Red Green Blue Yellow Purple Orange 100 2.99E-06 -0.00022 -1.69E-05 -1.64E-06 9.56E-05 -5.69E-05 1.96E-05 200 2.02E-06 -0.00044 -1.77E-05 -2.94E-06 1.84E-04 -2.18E-05 1.83E-05 250 1.24E-06 -0.00055 -1.73E-05 1.36E-07 2.34E-04 -9.94E-07 2.03E-05 Load Location 2 (lb) White Red Green Blue Yellow Purple Orange 100 2.41E-06 -0.00022 -1.74E-05 4.61E-08 9.45E-05 -5.92E-05 2.03E-05 200 -2.32E-07 -0.00044 -1.57E-05 -5.49E-07 1.84E-04 -2.13E-05 2.10E-05 250 3.05E-06 -0.00055 -1.65E-05 9.64E-07 2.35E-04 -9.04E-08 1.92E-05 Averaged Trials from Strain Gauges Table 3: Measured load vs average strain at location 1 (0.0131m from cantilever) Load Location 1 (lb) White Red Green Blue Yellow Purple Orange 100 -1.31E-04 -5.00E-04 9.11E-06 1.05E-05 2.09E-03 2.91E-04 1.04E-05 200 -2.70E-04 -1.58E-03 7.03E-06 1.94E-05 4.59E-04 6.25E-04 9.99E-06 250 -3.35E-04 -1.28E-03 7.61E-06 2.42E-05 5.96E-04 7.92E-04 1.35E-05 Table 3: Measured load vs average strain at location 2 (0.0115m from cantilever) Load Location 2 (lb) White Red Green Blue Yellow Purple Orange 100 3.21E-06 -2.18E-04 -1.71E-05 -1.11E-06 9.57E-05 -5.80E-05 2.02E-05 200 1.10E-06 -4.40E-04 -1.66E-05 -2.33E-06 1.84E-04 -2.18E-05 1.87E-05 250 2.14E-06 -5.50E-04 -1.71E-05 1.05E-06 2.35E-04 -6.39E-07 1.99E-05

Data from Dial Gauge Relevant constants used for calculations: 1. ? = 10.276 𝑖? 2. ? 1 = 5.12 𝑖? 3. ? 2 = 5.74 𝑖? 4. ? = 10 6 𝑝𝑠𝑖 5. 𝐴 = 12.15 𝑖? 2 Table 4: Strain gauge data at location 1 (x = 5.12 in) Load (lb) Dial gauge deflection (in) Theoretical deflection (in) 100 0.17 0.410394 200 0.39 0.820787 250 0.54 1.025984 The Youngs Modulus is calculated using the following formula, where ? is the slope of the graph. ? = ? ∗ ? 𝐴 = 409.05 ∗ 10.276 12.15 = 345.95 𝑝𝑠𝑖 Table 5: Strain gauge data at location 2 (x = 5.74 in) Load (lb) Dial gauge deflection (in) Theoretical deflection (in) 100 0.56 0.324028 200 0.96 0.648056 250 1.14 0.81007 y = 409.05x + 33.349 R² = 0.9934 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 Load (lb) Deflection (in) Load vs Deflection

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? = ? ∗ ? 𝐴 = 257.19 ∗ 10.276 12.15 = 217.52 𝑝𝑠𝑖 Error Analysis The hypothesis that the deflection increased as we moved away from the fixed end towards the applied point force was proven by the fact that the deflection increased progressively. Some sources of error include: 1. Zero error from the strain gauges 2. High random error/sensitivity of the strain gauges 3. System not properly insulated from outside disturbances. The effects of the said sources of error were minimized by performing three trials for each measurement. It is important to mention that the huge discrepancy in the theoretical and calculated Youngs Modulus value can be explained by a calculation error. Lab Report Questions Describe the stresses distribution on a cross section of a beam which is subjected to bending moment. Both of normal stress and shear stress. When a beam experiences a bending moment, stress distribution across its cross-section includes normal stress (tension and compression) and shear stress. Normal stress is linear, reaching its maximum on the outermost fibers of the convex and concave sides, while shear stress is parabolic, peaking at the neutral axis with zero values at the top and bottom surfaces. Draw the shear force and bending moment diagrams for cantilever and 3-point bending configuration (Assume you apply load F at the free end of cantilever beam and at the center of a 3-point bending beam. Both of the beams have length L). Answered in previous section. y = 257.19x - 44.705 R² = 0.9994 0 50 100 150 200 250 300 0 0.2 0.4 0.6 0.8 1 1.2 Load (lb) Deflection (in) Load vs Deflection

Plot Deflection Vs Force. From the slope evaluate the Young’s Modulus. Compare the value of Young’s modulus with published value. Answered in previous section. A load of 100 lbf is applied to the center of a rectangular beam made of Aluminum that is ¾ inch wide and 1 ¼ inch tall (Fig.3). What will the stress and deflection be at the center of the beam? Assuming that the beam is fixed on both ends and the force is applied on the center, we begin by defining constants: 𝑏 = 0.75 𝑖? ℎ = 1.25 𝑖? ? = 7.5 𝑖? ? = 10 6 𝑝𝑠𝑖 Now, we find the equation for the bending moment: ? = 1 2 ? ( 1 2 ?) = 1 4 ?? Therefore: 𝜎 = − ?? 𝐼 = − 1 4 ?? ( 1 2 ℎ) 1 12 𝑏ℎ 3 = 960 𝑝𝑠𝑖 Now, we find the deflection with the following formula: 𝛿 = ?? 3 48?𝐼 = 7.2 ∗ 10 −3 ? Give some real life examples where beam bending analysis would be useful. 1. Building construction: Analyzing beams in structural design to ensure the stability and safety of buildings. 2. Bridge engineering: Assessing the behavior of bridge beams under various loads and conditions. 3. Aerospace industry: Analyzing aircraft wings and fuselage beams to ensure they can withstand loads during flight. 4. Automotive design: Evaluating the strength and performance of vehicle chassis and suspension beams. 5. Civil infrastructure: Studying the behavior of beams in dams, tunnels, and pipelines for safety and longevity.