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Please Look at the Lab Format 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 (Station A) these are the numbers from the experiment please use them in the report and Station D use the 7 inch reading because it is missing from station A,,,

Don't forget to use the formulas in the Experiment 4 instructions in the report :)

Thank you :)

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

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LABORATORY REPORT CONTENT For almost all of the experiments, the laboratory reports should include the following (total of 20 points): • Title Page (0.5 points) • Table of Content (0.5 points) • Abstract (2 points) • List of Symbols and Units (2 points) • Theory (3 points) • Procedures and Experimental Setup (2 points, with colored pictures) • Sample Calculation and Error Analysis (3 points, Error Analysis may not exist for some experiments) • Results (2 points with Table of Results and/or Figures) • Discussion and Conclusion (4 points) • References (0.5 points, e.g. textbooks, journal papers. Do NOT reference Wikipedia.) • Appendix (0.5 points, raw data sheet and hand calculation) LABORATORY REPORT FORMAT • Use 1 ½ 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 ¾ margin on all other sides (right, top, and bottom). • Use justification on both left and right margins. • All equations must be centered with equations number: (1), (2), (3) with right justification • Try to use present tense in writing your report. • For all pages, 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 • • • • • Title Page: o Please include: course number, course title, name of the experiment, your name, Group name, your lab partner’s names, date the report due date, submission date Abstract: o Abstract should be between ½ pages to ¾ pages. You should clearly state the objective of the experiment in the very first sentence. You must also briefly answer a) What was done? b) How was it done? c) What were your basic results? d) How is your result compare to that of theory and/or other sources? List of Symbols and Units: o You should clearly write variables, name of the variables, and units in three column format. Theory: o With books and other sources, you must provide background information that helps in analyzing your data. You should include theoretical information for all of the equations that you used in analyzing your data. Procedures and Experimental Setup: o Concisely describe procedures and setup in your own words (do not copy from lab 1 • • • • handouts). Number the procedure in chronological order. Please place a couple of colored photos to better illustrate your procedure of the experiment. Sample Calculations and Error Analysis: o “Number” the sample calculation that you are analyzing in chronological order. o This number should correspond to the number in the error analysis. Results: o Make sure you have titles, axis labels with units in all tables and figures. Discussion: o Explain how your results relate to the theory. Similarities and differences between your results and that of others can be used to confirm your conclusions. You must explain in detail some sources of error. If your result disagrees with the published source, try to explain possible sources of error. If it agrees, you must also explain how you obtained the accurate results. Conclusion: o For the concluding paragraph, you must discuss the most important overall result and explain what you have accomplished. Remember, this “Discussion and Conclusion” section weighs more than any other section for a good reason (4 points). 2 EXPERIMENT IV The Column OBJECTIVES The objectives of this experiment are (a) to study the crushing of short structural members, and buckling behavior of long columns under axial compressive loading, and (b) to determine the critical loads of columns. THEORY Columns are structural members loaded in compression. In the axial tension and beam tests, the load carrying ability of the structural members was directly related to the yield or ultimate strength of the material. This is also the case for a short column where the crushing strength of the column depends on the compressive yield and ultimate strength for the material. However, for long columns, another effect must be considered: that of static stability. A tension member of any length is statically stable because the tensile forces tend to align the member along the axis of loading, and thus reduce any misalignment. A compression member, however, may be statically unstable because any misalignment of the compressive forces, or lack of straightness or uniformity of the member, tends to increase with increasing loads. Convince yourself of this effect by holding the two ends of a pencil eraser, and apply a compressive load at the two end points. Consider now the slender rod in Fig. IV-1, free to rotate at the ends, and loaded axially with load P. If a lateral load Q is applied, the elastic rod will deflect by -y, and if P is not too large, it will return to its original shape when Q is removed. In this condition, the rod is stable with respect to P. If P is gradually increased, and Q is successively applied and released, it is expected that some value of P will be reached where the elastic forces in the rod are no longer able to bring the rod back to its original position when Q is removed; instead, it remains in its deflected position under the action of P. In this condition, the rod is neutrally stable, and the value of P is called the critical load, Pcr. Any slight further increase in P would cause the rod to become unstable, and to collapse. Figure IV-1 Illustration of Column Deflection Under Load To derive the equations needed for analyzing the above phenomenon, consider a rod in neutral stability. From Fig. IV-1, it is seen that the moment at any section is M = −Pcry (using the same sign convention as for the beam experiment). But, from beam theory, M = EId²y/dx²; thus, the equation describing the deflected shape of the neutrally stable rod is IV-1 d2 y EI 2 + Pcr y = 0 dx (IV-1) with the boundary conditions that y = 0 at x = 0 and x = L. The general solution of Eq. (IV-1) is y(x) = A sin x λ + B cos x (IV-2) λ where λ2 = EI/Pcr. The boundary condition at x = 0 can be met by taking B = 0, but the one at x = L can be met only for characteristic values of λ for which sin(L/λ) = 0; or for L/λ = nπ, n = 1, 2, 3.... Thus, solving this equation, the smallest value of the critical load is obtained for n = 1: Pcr = π 2 EI L2 (IV-3) Notice that this value depends only on the stiffness property, E, of the material, and not on the strength of the material. This result is known as Euler’s equation, and it is applicable to long slender elastic columns. In Eq. (IV-3), I is the minimum moment of inertia of the cross-section of the rod, which may be replaced by Ak², where k is the corresponding “radius of gyration.” With this substitution, Euler’s equation takes the form σ cr = π 2E (L / k) (IV-4) 2 where σcr = Pcr/A is the critical stress. The plot for Euler’s equation is shown by the solid curve on the right side of the graph in Figure IV-2. σ cr σy Short Intermediate Long Empirical Equation Euler Equation σ pl ( Le / k ) max Le / k Figure IV-2 Effect of the Slenderness Ratio on the Critical Stress of Columns IV-2 The ratio L/k is called the “slenderness ratio”. It provides the measure by which a compressive member is judged long or short. Euler’s equation is based on the assumption that the material of the column remains in the elastic range until buckling occurs. For this reason, the plot of Euler’s equation starts at a value of L/k , identified as (L/k)max, which corresponds to a critical stress equal to the proportional limit of the material, σpl (see Fig. IV-2). (L/k)max is calculated by using Equation IV-4 and setting σcr = σpl: ( L / k ) max = 2π 2 E σ pl The solid horizontal line on the left side of the graph in Figure IV-2 belongs to short compression members that fail by crushing with a critical stress equal to the yield point, σy. Intermediate members that are neither “short” nor “long” may fail by a combination of crushing and buckling. Their behavior is represented by the dashed curve in Figure IV-2. Empirical equations are used to predict the critical stress of an intermediate column. The length, L, in Euler’s equation is not necessarily the geometric length of the column. L is a function of the end conditions imposed on the column. It is the “effective length” of an equivalent column having zero moments at its ends. Figure IV-3 shows the relation of L to the geometric length, l, for a number of column configurations. L is the distance between points having zero moment (where d²y/dx² = 0). Figure IV-3 Effective Length of Columns In practice, most columns are of intermediate length, and various empirical formulas have been developed. For example, for structural steel, the American Institute of Steel Construction recommends the following parabolic type formula for the working stress in “PSI”. IV-3 Pcr = 17,000 - 0.485 L    A k 2 0 < L < 120 k For aluminum alloys, the Aluminum Company of America (Alcoa) recommends the following linear type formula (not including the safety factor) in “PSI”. P cr = B - D  L    A k where the constants, B and D, depend on a particular alloy and are given in the following table. Table IV-1. Column Properties for Selected Aluminum Alloys Alloy 53ST 17ST 27ST 2014-T4 6061-T4 6061-T6 B (PSI) 38,400 43,800 62,500 35,200 15,700 38,300 D (PSI) 287 350 596 251 74 202 (L/k)max 90 83 70 89 128 63 If L/k is greater than (L/k)max, the Euler’s formula is recommended. PROCEDURE In this experiment, the critical load will be determined for eight various lengths of 6061-T6 aluminum alloy tubes (i.e. 12- and 10-inch long columns, 9-, 8-, 6-, and 5-inch intermediate columns, 4- and 3-inch short columns). 1. Carefully measure and record the length, I.D., and O.D. dimensions of the tubes using a digital calipers. Record the precision of the measurement device. 2. Read and understand all safety information in the MTS manual that is electronically available on the lab computers. 3. Ensure “Emergency Stop” is released and not activated. 4. Turn on the PC and the power switch at the bottom panel of the MTS frame. 5. Ensure the column test fixtures are installed on the MTS frame. 6. Double-click the TestWorks 4 icon on the PC display. TestWorks 4 is the software that operates the MTS and collects information from the extensometer. 7. Log in under “306A lab” (no password required). 8. Click on “Open Method” drop down list and highlight “Exp-3 4 Point Flex Mod 9-18A”. Then, click on OK box. 9. Click on the “Calibrate Device” icon and highlight the load cell serial number associated with the load cell that is installed on your MTS machine. Then, click Calibrate. 10. Click "Next" and wait until the process is completed. Then click Finish. IV-4 11. Click OK in the device calibrate box. 12. Use the handset control to raise the cross head, making certain that the top fixture has enough clearance to fit the aluminum alloy tube column between free inserts of MTS fixture. 13. Place the aluminum tube column/member with the fixed-fixed fittings in the testing machine. The machine fixtures and components are shown in Figure IV-4. 14. Use the handset control to lower the cross head to hold the aluminum tube in place without causing any load on the MTS load cells. 15. Zero the cross head indicator on the MTS display. 16. Identify a point midway between the two ends of the column, and measure the distance between the reference bar and the aluminum tube column (Figure IV-5). 17. In this experiment, we measure long columns deflections with the digital calipers and use a dial indicator to measure deflection of intermediate and short columns. 18. The subsequent deflections are measured with respect to the undeflected position of the specimen. While the calipers or the dial indicator showing this undeflected position, press the “zero” button on the calipers to initialize it, or rotate the outer ring of the dial to set it at zero. 19. Increase the load gradually, and slightly prod the column (load Q in Fig. IV-1). When some deflection occurs (stays), rotate the column, if necessary, so the deflection is always in a single direction (release load if needed). For consistent measurements, impose the same direction of deflection on all samples. See Figures IV-5 and 6. Note: Rotate the swing arm that holds the dial indicator in the desired direction of deflection (Figure IV-7). It may be required to rotate the swing arm again in the desired direction of deflection, as you increase the load in the subsequent steps. 20. Increase the load successively, every time 10 lb for long columns, 50 lb for intermediate columns and 75 lb for short columns until the specimen fails. The load must be read continuously as the column will suddenly buckle when the critical load is reached. Determination of the critical load by this method requires careful coordination of the group members. Produce a curve showing the critical load (Figure IV-8). Note: The load will stop increasing once the critical load is reached. There is no need to continue increasing the MTS crosshead movement and measuring the corresponding deflection after the critical load and/or failure of the material is established. 21. Repeat the process with other end conditions, namely pinned-pinned and fixed-pined for all the samples provided to your team. IV-5 Bottom Fixture with Swing Arm Top Fixture Fixed End Inserts Pinned End Inserts Tube End Inserts Pinned and Fixed Inserts 12-inch and 6-inch Reference Bars Figure IV-4 Column Fixtures and Components Reference Bar For long tubes Figure IV-5 Caliper Measurement IV-6 Figure IV-6 Dial Indicator Measurement for Short and Intermediate Tubes Fixed End Insert on Bottom Swing Arm with Reference Bar Figure IV-7 Bottom Fixture on MTS IV-7 Figure IV-8 Determination of Critical Load REPORT REQUIREMENTS Tabulate the critical load and length for each column (include end fittings). Determine the moment of inertia and radius of gyration. Tabulate the critical stress and slenderness ratio. For each end conditions, plot the experimental critical stress versus slenderness ratio points and the ALCOA points on a graph. Also, superimpose the theoretical curve for long columns. 5. Compare the experimental results with theory and discuss the validity of Euler’s equation. Discuss differences between the theory and experiment for short and intermediate columns, and discuss relationship of data to the aluminum material properties (Young’s modulus, yield strength, ultimate strength, and proportional limit). 1. 2. 3. 4. REFERENCES [1] Beer, Ferdinand P. and E. R. Johnson, Jr., Mechanics of Materials, McGraw-Hill Book Company, New York. [2] Gere, J.M. and S.P. Timoshenko, Mechanics of Materials, PWS-Kent. IV-8
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