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 • For all texts and equations, you must use the font Times New Roman, font size 12. Use 1 ½ spacing for texts and equations. • • 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: Upper left corner: EGME 306A Upper right corner: Experiment name Lower left corner: Your name ○ Lower right corner: Page # /Total pages Title Page: ○ 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: о 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: ○ 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 Results: "Number" the sample calculation that you are analyzing in chronological order. This number should correspond to the number in the error analysis. о Make sure you have titles, axis labels with units in all tables and figures. Discussion: ○ 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/n 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 +7, from the neutral axis, or distance r+ 7 from O (Fig. III-2(b)). At this radius, the length of arc is l'=(r + n) A¤. As shown in Fig. III-2(a), the length of the arc was / before the deformation. This length is also equal to r^0 (because at N-N there are no forces to change the length). Thus, the strain at distance +ŋ from the neutral axis can be found by: η l' ε = = I (r + η)Δθ - Δθ ΔΟ η r III-1 (III-1) V P 22 Pa 2 M a P 2 b + Ут (a) 22 (b) (c) ✗ Figure III-1. Symmetrically Loaded Beam (a), with Shear Force Diagram (b) and Bending Moment Diagram (c) III-2 E (a). H M (compression) (c) M E N -AB- N (b) ¿A dFm odA H' (tension) Figure III-2. Stresses and Strains of a Beam III-3 M 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 7 was taken on the tensile side of N-N in Fig. III- 2(b). Had ŋ 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), η σ= ε r (III-2) If c is the maximum distance from the neutral axis (largest positive or negative value of 7), then the maximum stress (compressive or tensile) is given by σm = Ec/r, and Eq. (III-2) can also be written as η σ= σm C (III-3) 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 ŋ = 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 = SodA - Om с √ndA = 0 A 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 = √ndF = nodA ŋodA = √ n² dA σms C (III-4) If I is defined as the second moment of area about the neutral axis, commonly called the moment of inertia, I = √n² dA In then Eq. (III-4) can be written as: = 6 т (III-5) Mc M (III-6) I Z III-4 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 σ 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 dx² r d² y 3 (1 + dx² 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 r d² y dx² But, since σm = Ec/r = Mc/I, there results the differential equation of the elastic curve: d² y ΕΙ dx² = M(x) (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)= P x³ ax(a+b) for 0≤x≤a 2EI 6 2 3 y(x)= P a ax(2a+b) ax 2EI 6 2 + for a≤x≤a+b 2 2 and that the maximum deflection at x = a + b/2 is Pa a²+ab+b² - Ym 2EI 3 2 In particular, for a = b = L/3, - Ym 48EI 23 Pa³ 23 PL³ 1296EI III-5 (III-8) (III-9)