CEE370L_Test5_Report_Vergara, Kailah Reign

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CEE 370 Mechanics of Materials Lab Beam Bending Theory Kailah Reign Vergara University of Hawaii at Manoa CEE 370L Mechanics of Materials November 14, 2023

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) EXECUTIVE SUMMARY During the laboratory experiment, a large steel beam specimen with simple support was exposed to a transverse load. The steel beam was equipped with multiple electrical resistance strain gauges at different locations to track the strain in the steel. These gauges were employed to assess the specimen's response under various loads. By calculating the theoretical stresses based on the measured microstrains, we could observe the behavior of the plot. The outcomes of the test will be utilized to validate the correlation between bending moment and bending stresses, commonly referred to as the beam bending theory. Page

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) TABLE OF CONTENTS EXECUTIVE SUMMARY………………...……………………………………………………………...i 1 INTRODUCTION ............................................................................................................................. 1 1.1 B ACKGROUND .............................................................................................................................. 1 1.2 R EASON FOR E XPERIMENT ........................................................................................................... 1 1.3 T HEORY ........................................................................................................................................ 1 1.4 O BJECTIVE ................................................................................................................................... 1 2 APPROACH ...................................................................................................................................... 1 2.1 T EST S ETUP AND I NSTRUMENTATION ........................................................................................... 1 2.2 T EST S PECIMENS .......................................................................................................................... 2 2.3 T EST P ROCEDURE ......................................................................................................................... 3 3 RESULTS ........................................................................................................................................... 3 3.1 ............................................................................................................................................................... 3 3.2 ............................................................................................................................................................... 3 4 ANALYSIS ......................................................................................................................................... 3 4.1 ............................................................................................................................................................... 4 4.2 ............................................................................................................................................................... 4 5 CONCLUSIONS/RECOMMENDATIONS ..................................................................................... 4 6 REFERENCES .................................................................................................................................. 5 APPENDIX TABLE OF FIGURES FIGURE 1.3.1: DIAGRAM FOR A TYPICAL STEEL BEAM SUBJECTED TO AN EXTERNAL LOAD FIGURE 2-2: DIAGRAM OF THE BEAM WITH THE DISTANCES AND LAYOUT OF THE VARIOUS GAGES FIGURE 2-3: CROSS SECTION OF THE BEAM Figure 2-4: BEAM BENDING THEORY TEST SETUP Figure 3.1: Stress vs Load Relationship of Top and Bottom Gages Figure 3.2: Average Stress vs Distance Relationship of Front and Back Gages Figure 3.3: Stress vs Distance Relationship from Center of Beam Page

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CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) 1 Introduction 1.1 Background Beam bending arises when external forces act on a beam, inducing deformation. This phenomenon is prevalent in the construction of diverse structures. To assess a beam's ability to bear loads, shear force and bending moment diagrams are employed to pinpoint areas of peak shear force and moment. Steel, widely utilized for beams, exhibits proficiency in both tension and compression. Its versatility in being shaped into various forms with distinct capabilities further enhances its appeal. By scrutinizing stress distribution across the cross-section, the highest stress occurs farthest from the neutral axis. Additionally, when assessing stress variation along the beam span, stress peaks align with maximum bending moments. 1.2 Reason for Experiment The objective of conducting this laboratory experiment was to aid a civil engineering professional in the design phase by identifying the most suitable variation of a specific material for a project. The stress-strain curve generated through the experiment provides essential characteristics that play a vital role in the material selection process. This careful consideration is crucial for ensuring the safety and stability of the structure being worked on. The broader public stands to gain valuable insights from the outcomes of this experiment. Knowledge of how various materials behave and respond to applied forces can instill confidence in the public regarding the safety and structural integrity of their surrounding environment. 1.3 Theory Page 1

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) The fundamental principle of beam bending theory, also known as Bernoulli Euler beam theory, plays a crucial role in mechanical engineering. It elucidates the behavior of a beam when subjected to external loads. As a beam undergoes deformation or bending under loading, we can compute the resulting stresses and deflections. Notably, any cross-section of the beam that is initially perpendicular to the neutral axis will maintain this perpendicular orientation throughout the deformation process. The core tenet of beam bending theory asserts that a beam will deform in a manner that minimizes its potential energy under the given external loads. This principle holds as long as the cross-section remains constant during bending, and the material exhibits consistent mechanical properties in all directions. Figure 1.3.1: Diagram for a typical steel beam subjected to an external load 1.4 Objective The goal of this laboratory experiment was to confirm the correlation between bending moment and bending stresses. Furthermore, the validity of the assumption that plane sections remain flat will be assessed. Page 2

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) 2 Approach 2.1 Test Setup and Instrumentation This test used a steel beam and electrical gages to identify the strain in the beam. Figure 2-2: Diagram of the beam with the distances and layout of the various gages. Figure 2-3: Cross section of the beam. Page 3

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CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) Figure 2-4: Beam Bending Theory Test Setup Test frame: Simple supports (reaction anchors), Hydraulic jack with two points of contact Test Instrument: Electrical resistance strain gages Test materials: Steel tube 2.2 Test Specimens Table 2.1: Materials and their properties Table 2.1: Materials and their properties Material Span length (in) Height (in) Depth (in) E (ksi) I (in^4) Vertical Strain Gage Distance (in) Horizontal Strain Gage Distance (in) Steel Beam 96 6 3 29000 13.4 6 1 2.3 Test Procedure Initially, the sensors were calibrated to ensure accurate data collection. The first load of 0 kips was subsequently applied to the beam, and shortly thereafter, the Page 4

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) microstrain from the top sensors was read aloud. This step was then repeated for loads of 1, 2, 1, and 0 kips. In total, 5 strains were recorded for each of the 5 loads. This entire process was then repeated three more times for each of the strain gage positions (bottom, front, and back). Page 5

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) 3 Results 3.1 Stress vs. Load Load T1 U1 0 -0.087 0 1 2.552 -2.581 2 5.278 -5.075 1 2.61 -2.726 0 -0.145 -0.145 0 0.5 1 1.5 2 2.5 -6 -4 -2 0 2 4 6 0 -2.58 -5.08 -2.73 -0.15 -0.09 2.55 5.28 2.61 -0.15 f(x) = − 2.51 x − 0.1 f(x) = 2.7 x − 0.12 T1 Linear (T1) U1 Linear (U1) Load (in kips) Stress (in ksi) Figure 3.1: Stress vs Load Relationship of Top and Bottom Gages 3.2 Stress vs Distance From Neutral Axis Table 3.2: Experimental Stress Load (kips) Experimental Stress Average T1 T2 T3 T4 T5 0 -0.087 -0.058 -0.058 -0.058 0 -0.0522 1 2.552 2.61 2.61 2.407 1.972 2.4302 2 5.278 5.336 5.336 4.872 4.031 4.9706 1 2.61 2.697 2.668 2.436 2.059 2.494 0 -0.145 -0.058 -0.087 -0.087 0.058 -0.0638 Page 6

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CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) U1 U2 U3 U4 U5 0 0 0 0 0 -0.116 -0.0232 1 -2.581 -2.552 -2.581 -2.146 -1.972 -2.3664 2 -5.075 -4.988 -4.93 -5.046 -3.799 -4.7676 1 -2.726 -2.61 -2.407 -2.146 -1.972 -2.3722 0 -0.145 0 0.319 0.087 0 0.0522 F1 F2 F3 F4 F5 0 -0.145 -0.087 -0.087 -0.087 0 -0.0812 1 1.74 0.986 0 -0.928 -1.74 0.0116 2 3.451 1.711 -0.087 -1.856 -3.509 -0.058 1 1.682 0.812 -0.145 -1.044 -1.885 -0.116 0 -0.174 -0.174 -0.116 -0.145 -0.087 -0.1392 B1 B2 B3 B4 B5 0 -0.116 -0.058 0 -0.145 0.029 -0.058 1 1.653 0.841 -0.087 -1.044 -1.798 -0.087 2 3.451 1.943 -0.029 -1.885 -3.451 0.0058 1 1.711 0.87 -0.087 -1.102 -1.885 -0.0986 0 -0.174 -0.116 -0.087 -0.174 -0.087 -0.1276 Distance Ex. Stress Theo. Stress 3 5.278 7.164 2 3.451 4.776 1 1.74 2.388 0 -0.058 0 -1 -1.8705 -2.388 -2 -3.48 -4.776 -3 -5.075 -7.164 Page 7

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) -4 -3 -2 -1 0 1 2 3 4 -8 -6 -4 -2 0 2 4 6 8 7.164 4.776 2.388 0.000 -2.388 -4.776 -7.164 5.28 3.45 1.74 -0.06 -1.87 -3.48 -5.08 f(x) = 2.39 x − 0 f(x) = 1.73 x − 0 Experimental Stress Linear (Experimental Stress) Theoretical Stress Linear (Theoretical Stress) Y-Distance from Origin Axis (in inches) Stress (in ksi) Figure 3.2: Average Stress vs Distance Relationship of Front and Back Gages 3.3 Stress vs Distance from Center Of Beam X-Dist. Ex T1-T5 Ex U1-U5 Theo T1- T5 Theo U1- U5 0 5.278 -5.075 7.164 -7.164 6 5.336 -4.988 7.164 -7.164 12 5.336 -4.93 7.164 -7.164 18 4.872 -4.292 6.716 -6.716 24 4.031 -3.799 5.373 -5.373 Page 8

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) 0 5 10 15 20 25 30 -10 -7.164 -7.164 -7.164 -6.716 -5.373 7.164 7.164 7.164 6.716 5.373 -5.08 -4.99 -4.93 -4.29 -3.8 5.28 5.34 5.34 4.87 4.03 f(x) = 0.01 x² − 0.08 x − 7.07 f(x) = − 0.01 x² + 0.08 x + 7.07 f(x) = 0 x² − 0.01 x − 5.07 f(x) = − 0 x² + 0.06 x + 5.24 Experimental T1-T5 Logarithmic (Experimental T1-T5) Power (Experimental T1-T5) Power (Experimental T1-T5) Power (Experimental T1-T5) Polynomial (Experimental T1-T5) Experimental U1-U5 Polynomial (Experimental U1-U5) Theoretical T1-T5 Polynomial (Theoretical T1-T5) Theoretical U1-U5 Polynomial (Theoretical U1-U5) X-Distance from midspan (in inches) Stress (in ksi) Figure 3.3: Stress vs Distance Relationship from Center of Beam Page 9

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CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) 4 Analysis Observing trends becomes possible by plotting both theoretical and measured data results obtained from the strain gauges. Linear behavior is evident in the stress vs. load relationship for both the top and bottom gauges at mid-span. At maximum load, the average stress plot across the height of the section, derived from the front and back gauges, displays an increasing linear trend. As height increases, so does the average stress. Analysis of stress vs. distance from the center of the beam indicates a decrease in stress as the distance from the center load increases. Notably, in the constant moment section of a beam, plane sections deviate from their original flatness. The conducted tests validate beam bending theory, accounting for the interplay between bending moment, curvature, and material properties. Within the constant moment segment, where the bending moment remains constant along the beam's length, cross- sectional deformation is non-uniform. This results in an alteration of the angle between the normal to the original plane section and the normal to the deformed plane section, as evident in the measured results. The stress distribution aligns with beam bending theory, assuming linear elasticity and small deformations. The maximum normal stress occurs at the furthest distance from the neutral axis, specifically at the top and bottom of the beam cross-section. In summary, experimental measurements of stress distribution in the beam corroborate theoretical predictions from beam bending theory. Nonetheless, some deviations from these predictions arise due to factors such as non-uniform material properties, loading conditions, and geometric imperfections in the cross- section. Page 10

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) Page 11

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) 5 Conclusions/Recommendations This laboratory experiment focused on applying a load to a steel tube, utilizing electrical strain gauges to validate the connection between bending moment and bending stresses. The assessment also addressed the assumption regarding the persistence of plane sections. - Linear behavior was evident in the stress vs. load relationship for the top and bottom gauges at mid-span. - Under maximum load, the plot of average stress from the front and back across the section height displayed a progressively linear trend. Increasing height correlated with an increase in average stress. - Analysis of stress vs. distance from the center of the beam indicated a reduction in stress as the distance from the center load increased. - In the constant moment segment of a beam, plane sections failed to maintain their planarity. This segment, where the bending moment remains constant along the beam's length, resulted in non-uniform deformation of the cross-section. This led to a change in the angle between the normal to the original plane section and the normal to the deformed plane section, as observed in the measurements. - The stress distribution in the beam aligned with beam bending theory, assuming linear elasticity and small deformations. - Deviations from theoretical predictions might be attributed to factors such as non- uniform material properties, loading conditions, and geometric imperfections in the cross-section. Page 12

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CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) 6 References Gere, James M., 2001. Mechanics of Materials 5 th Edition . Pacific Grove, California: Brooks/Cole. “CEE 370L - Mechanics of Solids Laboratory Stress-Strain Relationships for Various Materials Tension Test.” University of Hawaii at Manoa Dept. of Civil & Environmental Engineering. Page 13

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) Appendix A-1 Equations and Sample Calculations Experimental stress: ? = E*ε_x Theoretical stress: ? =-MY/I Moment of Inertia: I=(bh^3)/12 Page 14

CEE 370L Mechanics of Materials Fall 2020 Laboratory (No.) A-2 Raw Data RAW DATA Load (kips) Strain (macrostrain)(10^-6) T1 T2 T3 T4 T5 0 -3 -2 -2 -2 0 1 88 90 90 83 68 2 182 184 184 168 139 1 90 93 92 84 71 0 -5 -2 -3 -3 2 U1 U2 U3 U4 U5 0 0 0 0 0 -4 1 -89 -88 -89 -74 -68 2 -175 -172 -170 -174 -131 1 -94 -90 -83 -74 -68 0 -5 0 11 3 0 F1 F2 F3 F4 F5 0 -5 -3 -3 -3 0 1 60 34 0 -32 -60 2 119 59 -3 -64 -121 1 58 28 -5 -36 -65 0 -6 -6 -4 -5 -3 B1 B2 B3 B4 B5 0 -4 -2 0 -5 1 1 57 29 -3 -36 -62 2 119 67 -1 -65 -119 1 59 30 -3 -38 -65 0 -6 -4 -3 -6 -3 Page 15

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