, The Beam,

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Zvxrwbua1997

Engineering

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Please Look at the syllabus on how to write the report for the Experiment,,,

Use the instructions for the experiment and write it in the report but please please please don't copy and paste write your own words or paraphrase,,,

I uploaded images from 1-3, please use them in the report,,

Image 1 is explain how to write the numbers from the screenshot, there are 3 (red) columns the first column from the left is is the Load, the second column is the extension, the final column is the gauge reading if the gauge pointer passes 0 like in the 7th screenshot 1st run from the image (beam screenshot) it becomes 10.

Image 2 shows the measurements for the beam in image 3,,,

Image 4-10 are just photos from the lecture for explaining the Instructions for the lab and showing which formulas to us and how to graph,,,

Image 11-12 is showing the beam in action,,

Thank you :)

**** Don't worry about the number of pages just whatever it takes to finish the report in detail ****

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Experiment III The Beam OBJECTIVES The objectives of this experiment are (a) to determine the stress, deflection and strain of a simply supported beam under load, and (b) to experimentally verify the beam stress and flexure formulas. THEORY Structural members are usually designed to carry tensile, compressive, or transverse loads. A member which carries load transversely to its length is called a beam. In this experiment, a beam will be symmetrically loaded as shown in Fig. III-1(a), where P is the applied load. Note that at any cross section of the beam there will be a shear force V (Fig. III-1(b)) and moment M (Fig. III-1c). Also, in the central part of the beam (between the loads P/2) V is zero and M has its maximum constant value. Notice the sign convention of a positive moment, M, causing a negative (downward) deflection, y. If in this part a small slice EFGH of the beam is imagined to be cut out, as shown, then it is clear that the external applied moment, M, must be balanced by internal forces (stresses) at the sections (faces) EF and GH. For M applied as shown in Fig. III-2(a), these forces would be compressive near the top, EG, and tensile near the bottom, FH. Since the beam material is considered elastic, these forces would deform the beam such that the length EG would tend to become shorter, and FH would tend to become longer. The first fundamental assumption of the beam theory can be stated as follows: “Sections, or cuts, which are plane (flat) before deformation remain plane after deformation.” Thus, under this assumption, the parallel and plane sections EF and GH will deform into plane sections E ′F ′ and G ′H ′ which will intersect at point O, as shown in Fig. III-2(b). Since E ′F ′ and G′H ′ are no longer parallel, they can be thought of as being sections of a circle at some radial distance from O. Convince yourself of this by drawing a square on an eraser and observe its shape when you bend the eraser. Since the forces near E ′G ′ are compressive, and those near F ′H ′ are tensile, there must be some radial distance r where the forces are neither compressive nor tensile, but zero. This axis, N-N, is called the neutral axis. Notice that N-N is not assumed to lie in the center of the beam. Consider an arc of distance +η, from the neutral axis, or distance r + η from O (Fig. III-2(b)). At this radius, the length of arc is l ′ =(r + η) Δθ. As shown in Fig. III-2(a), the length of the arc was l before the deformation. This length is also equal to rΔθ (because at N-N there are no forces to change the length). Thus, the strain at distance +η from the neutral axis can be found by: ε = η l′ - l (r + η )Δθ - rΔθ = = l rΔθ r III-1 (III-1) L bb a c P 2 P 2 ym (a) V P 2 + M - x (b) Pa 2 x (c) Figure III-1. Symmetrically Loaded Beam (a), with Shear Force Diagram (b) and Bending Moment Diagram (c) III-2 Figure III-2. Stresses and Strains of a Beam III-3 In other words, the axial strain is proportional to the distance from the neutral axis. It is remarked that this strain is positive, because positive η was taken on the tensile side of N-N in Fig. III2(b). Had η been taken in the opposite direction, then the strain would have been negative, as appropriate for the compressive side. The second fundamental assumption is that Hooke’s Law applies both in tension and compression with the same modulus of Elasticity. Thus, from Eqs. (I-3) and (III-1), σ = E η (III-2) r If c is the maximum distance from the neutral axis (largest positive or negative value of η), then the maximum stress (compressive or tensile) is given by σm = Ec/r, and Eq. (III-2) can also be written as σ = σm η (III-3) c That is, the stress at a section EF or GH, due to applied moment M, varies linearly from zero at the neutral axis to some maximum value σm (positive or negative) when η = c. To obtain the beam stress formula, it remains to define where the neutral axis is located, and to relate σm to M. To locate the neutral axis, it is observed that the tensile and compressive forces on a section are equal to the stress times a differential element of area, as shown in Fig. III-2(c). For static equilibrium, the sum (or integral) of all these internal forces must be zero. That is, dF = ∫ σ dA = A σ m η dA = 0 ∫ c A where, the integrals are over the whole cross-sectional area. Thus, it is seen that the neutral axis is located such that the first moment of area about it is zero; that is, the neutral axis passes through the centroid of the cross-sectional area. In Fig. III-2(c), a rectangular area was used for illustration; however, any shape of vertically symmetric cross-sectional area is valid for the area integral. In a similar fashion, the moment due to all the forces is the sum (or integral) of the forces times their moment arms about the neutral axis, and this must be equal to the external applied moment. Thus, M = ∫ ηdF = ∫ ησdA = σm c ∫ η dA 2 (III-4) If I is defined as the second moment of area about the neutral axis, commonly called the moment of inertia, I = ∫ η dA 2 (III-5) then Eq. (III-4) can be written as: σm = M Mc = I Z III-4 (III-6) where Z = I/c is the section modulus, which depends only on the cross-sectional geometry of the beam. Equation (III-6) is the beam stress equation which relates the maximum (compressive or tensile) stress to the applied moment. Notice its similarity to Equation (I-1), the stress equation for uniaxial tension. It is understood, of course, that σm is the maximum bending stress at a particular location, x, along the beam. In general, both σm and M are functions of x, and are related by Eq. (III-6). The remaining question about the beam concerns its degree of deformation, or flexure. That is, how is the radius of curvature, r, related to the moment M (or load P)? From calculus, it can be shown that the curvature of a function y(x) is given by d2 y 1 dx 2 = d2 y 3 r )2 (1 + dx 2 Thus, if x is the distance along the beam, y will be the deflection as indicated in Fig. III-1(a). For most beams of practical interest, this deflection will be small, so that the slope dy/dx will be very small compared to 1. Hence, a very good approximation is 1 d2 y = r dx 2 But, since σm = Ec/r = Mc/I, there results the differential equation of the elastic curve: d2 y EI 2 = M(x) dx (III-7) To obtain the elastic curve of the beam, y(x), and the maximum deflection, ym, it is necessary to integrate Eq. (III-7) using the moment function M(x) in Fig. III-1(c). Thus, using M(x) = Px/2 for 0 ≤ x ≤ a and M(x) = Pa/2 for a ≤ x ≤ a + b, it is found that y(x) = y ( x) = P ⎛ x 3 ax(a + b) ⎞ ⎟, 2EI ⎜⎝ 6 2 ⎠ for 0 ≤ x ≤ a P ⎛ a 3 ax(2a + b) ax 2 ⎞ + , 2EI ⎜⎝ 6 2 2 ⎟⎠ for a ≤ x ≤ a + b and that the maximum deflection at x = a + b/2 is - ym = Pa a 2 ab b2 ( + + ) 2EI 3 2 8 (III-8) In particular, for a = b = L/3, - ym = 23 Pa 3 23 PL3 = 48EI 1296EI III-5 (III-9) Although the above stress and flexure formulas are quite simple, it took some of the best minds of the 17th and 18th centuries to derive them correctly. Part of the difficulty in obtaining the correct results at that time was that there were no methods of verifying the results experimentally. Today, with the advent of sensitive displacement dial gauges, the verification is more convenient. Beam Deflection Dial indicator Beam to Be Tested Figure III-3. MTS Insight Tensile Testing Machine PROCEDURE 1. The test will be conducted on a 1018 steel beam (E = 30x106 psi) using the MTS testing machine. The position of the beam in the testing machine is shown in Fig. III-3. 2. Observe that the cross-section of the I-beam is not a symmetric; one flange of the beam is thicker than the other. Measure carefully the cross-sectional dimensions and the location of the loading and support points. 3. Carefully place the beam into the MTS machine if it is not already installed. Align the 12inch black marks on the beam with the roller supports of the lower bending fixture. Make sure the beam is squarely resting and centered on the lower support with the strain gauge facing down. III-6 4. Enter the TestWorks 4 software by double clicking on the icon on the desktop. a. When prompted, make sure the Name field under the User Login says “306A_lab”. b. Click OK to login. c. Under the Open Method dialog, select “exp-3 4 Point Flex Mod X”. d. Now, select the Motor Reset button in the bottom right corner by clicking on it. 5. Zero the “load” readout by right clicking on the “Load cell” icon and selecting “zero channel”. 6. Next we will use the handset to position the upper bending fixture over the beam. a. Enable the handset by pressing the “unlock” button at the top right of the handset b. Slowly lower the crosshead using the down arrow until the fixture is NEARLY touching the beam. c. Make sure not to pinch the strain gauge lead wires between the fixture and the beam. d. While observing the digital load readout on the screen, use the thumb wheel of the handset to lower the fixture onto the beam. Watch for the load reading to increase when the upper loading component of the fixture makes contact with the beam. e. Now, slowly raise the fixture with the thumb wheel until only a very slight pre-load of approximately 0.2 lb is applied. f. When finished, return control to the computer software by locking the handset using the same button as before. Note that the beam has a strain gauge bonded to its surface. See Figure III-4. This strain gauge has a strain gauge factor of 2.14 ± 0.5% and 350 ohm resistance. Connect the strain gauge lead wires to the “#1 Strain” channel of the grey DAQ box. Data acquisition is conducted by the LabVIEW software. 7. Take the magnetic base holding the dial indicator, and position it with the dial indicator in the center of the beam on the bottom side. Make sure the dial indicator is not touching the strain gauge. Also, be VERY CAREFUL not to damage the dial indicator while doing this! When the dial indicator is in position, lock it to the MTS frame by activating the magnetic base. Finally, zero the dial indicator by turning the dial of the indicator. Note the divisions on the dial indicator and the accuracy with which it can be read. 8. Start the LabVIEW software. a. Double click the LabVIEW icon on the desktop, or follow the procedure suggested for Experiment II. b. Select “Open” and double click “exp2&3-Strain Mod 9-15 LV7.1”. c. Press the white arrow near the upper left corner of the screen to start the strain gauge acquisition. d. Press the “Zero Strain” button to zero-out the strain indicator. 9. You are now ready to run the experiment. a. Press the Green Arrow on the TestWorks 4 GUI. b. Give the sample ID for your group. III-7 c. Under the Required Inputs dialog, enter 0.125 for the thickness and 0.500 for the width. DO NOT press OK yet! 10. You will load the beam up to 1000 lb in increments of 100 lb. To do this: a. “OK” the Required Inputs dialog from above and the test will begin. b. When the digital load readout reaches approximately 100 lb, press the “Pause” button to temporarily stop the test. c. Record the actual load on your data sheet from the digital readout. d. Toggle over to the LabVIEW screen and record the strain reading from the software. e. Record the deflection of the beam from the dial indicator. f. Return to the TestWorks 4 screen and press the “Pause” button again to resume the test. g. Continue pausing and taking readings approximately every 100 lb from 0 to 1000 lb. 11. At 1000 lb the test will stop. Collect your last data point before pressing “OK” at the last dialog. 12. Before repeating the experiment, make sure your load readout, strain indicator, and dial indicator are all re-zeroed. 13. Repeat steps 7 through 9. Figure III-4. Beam with Bonded Strain Gauge REPORT REQUIREMENTS 1. Determine the neutral axis, moment of inertia, and section modulus of the beam crosssection. 2. Draw shear force and bending moment diagrams for the beam for the maximum load. 3. Compute the maximum bending stress and deflection of the beam for the maximum load. 4. Plot deflection versus load, and also stress versus load. Each graph should contain a curve based completely on theoretical calculations, and another using the experimental data points. III-8 5. By error calculation, determine if the beam theory is vindicated within the precision of the instruments. REFERENCES [1] Beckwith, T. G., Buck, N. L. and Marangoni, R. D., Mechanical Measurements, AddisonWesley. [2] Popov, E.P. Mechanics of Materials, Prentice-Hall Inc. [3] Thomas, G.B., and Finney R.L., Calculus and Analytical Geometry, Addison-Wesley, III-9 • An ability to communicate effectively with a range of audiences. • An ability to function effectively on a team whose members together provide leadership, create a collaborative and inclusive environment, establish goals, plan tasks, and meet objectives. • An ability to use the techniques, skills and modern engineering tools necessary for engineering practice (k). INSTRUCTIONAL MATERIALS, HANDOUTS AND TEXTBOOKS • Required: Experiment instructions provided by the instructor in PDF format* • Recommended: J. P. Holmam, “Experimental Methods for Engineers”, 8th Edition, McGraw-Hill, 2012 • Recommended: J. Chaudhuri and A. Marathe, “Materials and Mechanics Laboratory Experiments”, Cognella Academic Publishing, 2014 * Students are required to bring a printed copy of the experiment instructions to the class, or any device capable of reading and opening the PDF files (cell phones are not allowed) GRADING STANDARDS, AND CRITERIA Grading Individual Laboratory Reports 60 % Class Participation and Attendance 10% Final Exams 30 % Final Grade Scheme Over/or 90.00% A Between 89.99 % and 80.00% B Between 79.99 % and 70.00% C Between 69.99 % and 60.00% D Less Than 59.99 % F Note: The above ranges for the course letter grades may be adjusted to lower limits based on the overall performance of the class that will be determined at the end of the semester. +/grading will not be used in this course. ATTENDANCE POLICY • Attendance: Attendance is mandatory. Attending the class on time and participating in class lectures and discussions are essential to successful completion of this class. Since much of the learning in this course is experiential, you must be in class to experience it. If you are not in class, you cannot participate; thus, any more than 2 unexcused absences will result in a reduction in the class participation and attendance component of the grade and may endanger your ability to complete the course. • Students who must miss class to represent the university or to participate in a universitysponsored activity must notify the class instructor in writing a minimum of two weeks in advance of the absence. 2|Page Fall 2018 • Whether an absence is excused or unexcused is determined solely by the instructor with the exception of absences due to religious observance and officially approved activities described above. Late Assignments Late submissions of reports will not be collected unless prior arrangements are made with the instructor in a timely manner. INSTRUCTION FOR LAB REPORT WRITING • Submit your report in .PDF formation through Titanium. • Use professional language. • DO NOT use content/figures, from any source (Books or Internet) without proper referencing. • DO NOT copy word for word from the textbook or experiment instructions. Use your own wording. Lab Report Writing Content (Total Point: 100) Section Points Contents ▪ Course Name and Number ▪ Title of Experiment ▪ Lab Sec. ▪ Name ▪ Lab partner’s name ▪ Date ▪ Submitted to An overview of the entire experiment including: ▪ Should be no more than 200 words ▪ Objectives, approach and primary results ▪ Briefly answer: o What was done? o How was it done? o What were your basic results? o How does your result compare to that of theory and/or other sources? Title / Cover page 2 Abstract 10 List of Symbols and Units 2 ▪ ▪ ▪ Variables Name of Variables Units Introduction 10 ▪ ▪ A brief review of the problem Objectives Theory 15 ▪ ▪ Theoretical background Proper figures & complete set of Equations 3|Page Fall 2018 Test Setup and Procedure Results ▪ ▪ ▪ ▪ Complete description of the setup Schematic of setup / image of the setup with proper labelling Specimen description using test standards used Complete step by step procedure ▪ ▪ ▪ Explanations of how reduced results were obtained Results (tabulated form) Plots with titles ▪ ▪ ▪ Interpretation of data Error analysis Comparison of results to establish standard values and discussion ▪ ▪ Conclusions reached based on results & discussion Recommendations if appropriate ▪ Numerically number list of all used references and cite within the report body ▪ Datasheets, Sample Calculations, additional figures, charts etc. ▪ All reports must follow the ‘’ Lab Report Writing Format’’. 15 18 Discussion 15 Conclusions 4 References 2 Appendix 2 Format 5 Lab Report Writing Format • • • • • • • • • Use 1 1⁄2 spacing for texts and equations. For all texts and equations, you must use the font Times New Roman, font size 12. For section titles, use Times New Roman, size 14 with boldface. Use 1-inch left margin, and 3⁄4 margin on all other sides (right, top, and bottom). Use Justification on both left and right margins. All equations must be centred with equations number: (1), (2), (3) with right justification. Try to use present tense in writing your report. For all pages except the cover page, you should have headers/footers such that: o Upper left corner: EGME 306A o Upper right corner: Experiment name o Lower left corner: Your name o Lower right corner: Page #/Total pages For all graphics and tables: o Numbered consecutively and captioned o Caption = 9 pt. boldface sans serif typeface, uppercase, centered below the graphic 4|Page Fall 2018 NO SMOKING I ol ©-0 +0+ 0 0.000 l xio 100 DIV 10 -0.00) -1.5 lo7 199 000068 2 8 1 4 301. 403 503 604 6. n.oooool 0000 35 ..0023 cool05 -0039-000141 .0055 000/75 .0072 0002|| .0086 000247 olol 100028) 01/7000319 0132 000351 0145 0160 703 804 903 997 / K** 0 (2 G 쿠 H (3 9:0+0+ 0 x 10 LOAP DEFLECTION U STRAN 1.5 -.00000! lo7 e 1 301. 403 0000 35 .00 23 1000068 cool05 0039 1.000141 q ON 00015 * 1018 Steel 10/10/2018 Experiment 3 3 trials 30 screenshots total don't put the screenshots in the lab report only use the measurements in the screenshots in the report Measure the lengths of the beam at the end of the experiment x 1,80 0.252 2, 0.494 32297 40,197 9, 1.011 6.0.405 7.1.008
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