CEE370L_Lab_05_Shi, He

pdf

School

University of Hawaii *

*We aren’t endorsed by this school

Course

370

Subject

Mechanical Engineering

Date

Feb 20, 2024

Type

pdf

Pages

Uploaded by SuperDangerPuppy48

CEE 370 Mechanics of Materials Lab BEAM BENDING THEORY He Shi University of Hawaii at Manoa CEE 370L Mechanics of Materials Nov 26, 2023 1

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 EXECUTIVE SUMMARY During an experiment, an aluminum beam was examined to validate the principles of beam bending. The theory suggests that a beam made of flexible & homogeneous material with an equivalent modulus of elasticity in tension and compression will deform equally under the same stress. The moment of inertia (I) was calculated for bending around the horizontal centroid axis using the provided properties of the beam section. The moment about the maximum load at the plane section was determined using a shear force and bending moment diagram. Stresses were computed using the moment of inertia, moment about the plane surface, and distance from the centroid axis. It was proven that the distributed loads on the top and bottom flanges were very close to each other, thus confirming the theory. During the beam experiment, loads were applied which caused sagging. As a result, throughout the beam, there was a positive bending moment. Gauges on the plane section of the beam indicated that a plane section within the beam before bending remains plane after bending throughout a constant moment section of the beam. This proves that the beam bending theory is accurate. The experiment concluded that the material is linearly elastic, as the relationships between stress and strain were directly proportional. When a load was applied, the top of the beam was in compression while the bottom of the beam was in tension. The lowest sagging point of the beam showed the largest strains, and therefore had the largest stresses.

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 1 Introduction 1.1 Background A transverse load is applied to an aluminum beam in order to test the beam bending theory. A number of electrical resistance strain gauges are attached at various points on the beam to measure strain. The purpose of the testing is to confirm the relationship between moment and bending stresses. 1.2 Reason for Experiment The purpose of this lab was to test the deflection limits of beams that are installed in buildings. The experiment provided valuable information to engineers regarding how beams support and resist loads, which enables them to ensure the safety and stability of structures. Finding the maximum values of quantities and their locations along the beam is essential for beam design. 1.3 Theory When a beam is subjected to loads that are transverse to its length, there is no axial load applied. However, when a transverse load is applied, a relationship can be found between the bending moment and the transverse deflection of the beam. The beam is made of a linearly elastic material, which means it follows Hooke's Law. Therefore, a small deflection should cause small angles due to the deflection. The moment-curvature equation ( is determined by the bending moment divided by the modulus of elasticity times the moment of inertia (also known as the flexural rigidity). The equation given for the moment-curvature is:

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 where M is the bending moment and EI is the flexural rigidity. The flexure formula ( is determined by the bending moment multiplied by the distance from the neutral distance divided by the inertia of the cross section. The equation given for the flexure formula is: where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia of the cross section 1.4 Objective The goal of this laboratory experiment was to confirm the correlation between bending moments and bending stresses and assess whether the stress distribution aligns with the principles of beam bending theory.

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 2 Approach 2.1 Test Setup and Instrumentation The experiment was done with a load cell. There were a total of 20 strains on the steel, five on the back, front, top, and bottom. Each of the strains were 6 inches apart from each other. The strains are then hooked up to three different boxes. The load cell co impressed the aluminum beam at 0 kips to 2 kips in 1 increment and back down to 0 kips

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 2.2 Test Specimens Material Length (inches) Length Between Load Points (inches) Tube Steel 96.00 48.00 2.3 Test Procedure • Load applied to specimen: total transverse load - 0, 1, 2, 1, 0 kips • At each load level, record the microstrain reading from each gage

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 The moment of inertia, span length, and length between load points of the tube steel were measured and recorded. Then, the measurements were placed in the frame of the RIEHLE machine. After that, 20 different electrical resistance strains were applied to the front, back, top, and bottom of the object. The data was collected from the steel specimen at 0 kips, 20 times for each strain. Later, a force of 1 kip was loaded and 20 measurements were recorded from each side. Finally, the entire process was repeated three more times for forces of 2 kips, 1 kip, and 0 kips. 3 Results 3.1 Overview Table 3-1.1: Measured Strains Load(kips) Strain(ksi) T1 T2 T3 T4 T5 0 -0.000003 -0.000002 -0.000002 -0.000002 0 1 0.000088 0.00009 0.00009 0.000083 0.000068 2 0.000182 0.000182 0.000184 0.000168 0.000139 1 0.00009 0.000093 0.000092 0.000084 0.000071 0 -0.000005 -0.000002 -0.000003 -0.000003 0.000002 U1 U2 U3 U4 U5 0 0 0 0 0 -0.000004 1 -0.000089 -0.000088 -0.000089 -0.000074 -0.000068 2 -0.000175 -0.000172 -0.00017 -0.000148 -0.000131 1 -0.000094 -0.00009 -0.000083 -0.000074 -0.000068 0 -0.000005 0 0.000011 0.000003 0 F1 F2 F3 F4 F5 0 -0.000005 -0.000003 -0.000003 -0.000003 0

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 1 0.00006 0.000031 0 -0.000032 -0.00006 2 0.000119 0.000059 -0.000003 -0.000064 -0.000121 1 0.000058 0.000028 -0.000005 -0.000036 -0.000065 0 -0.000006 -0.000006 -0.000004 -0.000005 -0.000003 B1 B2 B3 B4 B5 0 -0.000004 -0.000002 0 -0.000005 -0.000001 1 0.000057 0.000029 -0.000003 -0.000036 -0.000062 2 0.000119 0.000061 -0.000001 -0.000065 -0.000119 1 0.000059 0.00003 -0.000003 -0.000038 -0.000065 0 -0.000006 -0.000004 -0.000003 -0.000006 -0.000003 Table 3-1.2: Measured Stress Load(kips) Stress(ksi) T1 T2 T3 T4 T5 0 -0.087 -0.058 -0.058 -0.058 0.000 1 2.552 2.610 2.610 2.407 1.972 2 5.278 5.278 5.336 4.872 4.031 1 2.610 2.697 2.668 2.436 2.059 0 -0.145 -0.058 -0.087 -0.087 0.058 U1 U2 U3 U4 U5 0 0.000 0.000 0.000 0.000 -0.116 1 -2.581 -2.552 -2.581 -2.146 -1.972 2 -5.075 -4.988 -4.930 -4.292 -3.799 1 -2.726 -2.610 -2.407 -2.146 -1.972 0 -0.145 0.000 0.319 0.087 0.000 F1 F2 F3 F4 F5 0 -0.145 -0.087 -0.087 -0.087 0.000

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 1 1.740 0.899 0.000 -0.928 -1.740 2 3.451 1.711 -0.087 -1.856 -3.509 1 1.682 0.812 -0.145 -1.044 -1.885 0 -0.174 -0.174 -0.116 -0.145 -0.087 B1 B2 B3 B4 B5 0 -0.116 -0.058 0.000 -0.145 -0.029 1 1.653 0.841 -0.087 -1.044 -1.798 2 3.451 1.769 -0.029 -1.885 -3.451 1 1.711 0.870 -0.087 -1.102 -1.885 0 -0.174 -0.116 -0.087 -0.174 -0.087 Figure 3-1.1 stresses in the top and bottom gages at mid-span for each load level Figure 3-1.1 Stresses of the Top and Bottom Strain Gages In Figure 3-1.1, the graph depicts the relationship between the load at each level and the stresses on the top and bottom. The blue line in the graph represents the top stress data, while the

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 red line represents the bottom stress data. It's important to note that the graph only pertains to the mid-span portion, specifically T1 and U1. Figure 3-1.2: Average Stresses of the Front and Back Strains Figure 3-1.2 shows the relationship of the distance from the neutral axis to the average stresses of the front and back strains. The graph only shows the data collected from the 2 kips load. The graph also shows the stresses from T1 and U1.

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 Figure 3-1.3: Top and Bottom Stresses from Mid-Span to Support In Figure 3-1.3, we can see how the top and bottom stresses vary in relation to the distance from mid-span to the support. The blue line represents the data collected for the top stress while the red line represents the data collected for the bottom stress. 4 Analysis 4.1 Tube Steel Specimen Analysis Based on the data collected during the experiment, it was found that the force with the highest load (2 kips) resulted in the most strain. The strain gages located at the top of the specimen showed the largest strain, while those at the bottom showed the smallest. Moreover, the front and back strain gages displayed a deviation of the data going from positive to negative.

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 In Figure 3-1.1, there are two linear lines in blue and red colors. The blue line represents the top stresses and moves towards the positive direction, while the red line represents the bottom stresses and moves towards the negative direction. The theoretical stress would show two linear lines originating from the origin and extending towards the positive or negative direction. Upon comparing the measured stress with the theoretical stress, we observe that both the top and bottom measured stresses align with the theoretical stress. However, upon closer inspection of the red line, we can see a gap indicating that the data collected from the bottom strain was larger when reverting to 2 kips to 1 kip than the first time we recorded the bottom data. The graph in Figure 3-1.2 represents the average stress of the front and back, which is shown by a semi-straight line. The theoretical stress should be a straight line through the origin. Comparing these two stresses, the measured stress shows a slight curve at the end, but the data was still able to create a line. In Figure 3-1.3, there are two lines going towards zero. The blue line indicates the top stresses collected from the mid-span to the support, while the red line indicates the bottom stresses. The theoretical stress will show a linear line in the first half and then begin to travel to zero. Comparing the stresses, both the measured stresses align with the theoretical stress. Based on what we learned in class, a plane will be perpendicular to the neutral axis of the theoretical lines if they are correct. Based on the graphs shown, all theoretical lines almost or perfectly line up with the measured lines. The stress distribution agrees with the beam bending theory because the theory states that cross-sections of the beam must remain plane during

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023 bending. The tube steel remained symmetric to the longitudinal axis during the experiment, and there were not any significant errors in the results of the experiment. 5 Conclusions/Recommendations Based on the data collected, the following conclusions can be made: - The top strains had the highest number of strains, while the bottom strains showed the smallest number of strains. - Both strains data were observed when the load was at 2 kips. - The front and back strain gauges showed both positive and negative data. - All the graphs almost or perfectly aligned with the theoretical stress. - As the steel beam remained unchanged after the experiment and the stresses aligned with the theoretical, the stress distribution corroborates the beam bending theorem. - No significant errors were discovered during the experiment. - Plane sections remained plane in constant moment portions of the beam From the experiment, recommendations for the lab would be to use a more accurate loader and to use a beam that has not been in many previous experiments. This could lower the margin of error in the lab.

CEE 370L Mechanics of Materials Laboratory 1 Fall 2023