Homework1 - 3—20 Load and Stress Analysts 121 Summary The...

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Unformatted text preview: 3—20 Load and Stress Analysts 121 Summary The ability to quantify the stress condition at a critical location in a machine element is an important skill of the engineer. Why? Whether the member fails or not is assessed by comparing the (damaging) stress at a critical location with the corre sponding material strength at this location. This chapter has addressed the description of stress. Stresses can be estimated with great precision where the geometry is sufficiently simple that theory easily provides the necessary quantitative relationships. In other cases, approximations are used. There are numerical approximations such as finite element analysis (PEA, see Chap. 19), whose results tend to converge on the true val- ues. There are experimental measurements, strain gauging, for example, allowing infer— ence of stresses from the measured strain conditions. Whatever the method(s), the goal is a robust description of the stress condition at a critical location. The nature of research results and understanding in any field is that the longer we work on it, the more involved things seem to be, and new approaches are sought to help with the complications. As newer schemes are introduced, engineers, hungry for the improvement the new approach promises, begin to use the approach. Optimism usually recedes, as further experience adds concerns. Tasks that promised to extend the capabilities of the nonexpert eventually show that expertise is not optional. 1n stress analysis, the computer can be helpful if the necessary equations are avail- able. Spreadsheet analysis can quickly reduce complicated calculations for parametric studies, easily handling “what if ” questions relating trade—offs (e.g., less of a costly material or more of a cheaper material). It can even give insight into optimization opportunities. When the necessary equations are not available, then methods such as FEA are attractive, but cautions are in order. Even when you have access to a powerful FEA code, you should be near an expert while you are learning. There are nagging questions of convergence at discontinuities. Elastic analysis is much easier than elastic—plastic analysis. The results are no better than the modeling of reality that was used to formu- late the problem. Chapter 19 provides an idea of what finitelelement analysis is and how it can be used in design. The chapter is by no means comprehensive in finite-element theory and the application of finite elements in practice. Both skill sets require much exposure and experience to be adept. PROBLEMS The symbol W is used in the various figure parts to specify the weight of an element. If not given, assume the parts are weightless. For each figure part, sketch a free—body diagram of each element, including the frame. Try to get the forces in the proper directions, but do not compute magnitudes. Using the figure part selected by your instructor, sketch a free—body diagram of each element in the figure. Compute the magnitude and direction of each force using an algebraic or vector method, as specified. Find the reactions at the supports and plot the shear-force and bending-moment diagrams for each of the beams shown in the figure on page 123. Label the diagrams properly. I22 Mechanical Engineering Design Problem 34 (d) y 0.15—m radius (a) Problem 3—2 x (I?) (11') 3-4 Repeat Prob. 3—3 using singularity functions exclusively (for reactions as well). 3-5 Select a beam from Table A—9 and find general expressions for the loading, shear—force, bending- moment, and support reactions. Use the method specified by your instructor. toad ond Stress Analysis 323 e" -“ 2 W 4 kN/m | 200 N 300 N I itU m 0.1m 0.]3m ().l m D 0 AB C— 0 7 x .r A C 200 mm ISO mm lSO mm RI R: (b) (a) \ _~,- 5 kN 1.8 m I 2 m 0 7 x Problem 3—3 A E R1 R: (6) iv 2 kN 4 kN 0.9 m 0.9 m 3-6 A beam carrying a uniform load is simply supported with the supports set back a distance a from the ends as shown in the figure. The bending moment at x can be found from summing moments to zero at section x: 1 1 EM : M+ 5100:; +x)2 v Eu,an :0 or m n M : EUJC — (a +x)‘j where w is the loading intensity in lbf/in. The designer wishes to minimize the necessary weight of the supporting beam by Choosing a setback resulting in the smallest possible maximum bend- ing stress. (a) If the beam is configured with a : 0.06 m, l : 0.25 m, and w = 18 kN/m, find the magni— tude of the severest bending moment in the beam. (12) Since the configuration in part (a) is not optimal, find the optimal setback a that will result in the lightestrweight beam. x -—> u‘tri + x) w. kN/Iit l Problem 3—6 124 Mechanical Engineering Design 3-7 An artist wishes to construct a mobile using pendants, string, and span wire with eyelets as showu in the figure. (a) At what positions w, x, y, and z should the suspension strings be attached to the span wires? (5) ls the mobile stable? If so, justify; if not, suggest a remedy. Problem 3—7 3-8 For each of the plane stress states listed below, draw a Mohr’s circle diagram properly labeled, find the principal normal and shear stresses, and determine the angle from the x axis to 0.. Draw stress elements as in Fig. 3—] 1c and d and label all details. (a) ox 2 12, o}. 2 6,1”. 2 4 cw (B?) OX 2 16,5). 2 9,1,”. 2 5 ccw - (C) ox 210,0), 2 24,1”. 2 6 cow (d) ox 2 9, o). = 19, 1'1}, 2 8 cw 3-9 Repeat Prob. 378 for: (a) a, 2 24, (r). 2 l2, 1:“ 2 7 cow (b) a). 2 6, a". 2 —5. I“. 2 8 cow (C) ex 2 —8,0'y 2 7, r” 2 6 cw (d) 0x = 9, CI} 2 —6, r” 2 3 cw 3-10 Repeat Prob. 3—8 for: (a) 0x = 20, a}. 2 —10,rx_\. 2 8 cw (b) o; 2 30, 0,. = 210, r“ = 10 ccw (c) 01.2 710,0), 2 18,:1, 2 9 CW ((1)012 —12,oy 2 22,5». 212 cw 3-] 1 For each of the stress states listed below, find all three principal normal and shear stresses. Draw a complete Mohr‘s three—circle diagram and label all points of interest. (a) 0A 210,5) 2 —4 (b) Ox 2 10,110. 2 4 cow (C) crx 2 —2, 0'}: 2 28, I“. 2 4 CW ((1)0). 2 l0, 0y 2 —30,1’Xy: 10 cow 3-12 Repeat Prob. 3—11 for: (a) U; = —80, a), 2 —30, r“ =' 20 cw (b) o;— 2 30, 07,. 2 760, ti” = 30 cw (c) aJ 2 40, a; 2 —30, IU 2 20 ccw (d) ox = 50,172 2 —20, I“. 2 30 cw 3—13 3-14 3—15 3-16 Problem 3— l6 3-17 3-1 8 3-1 9 3-20 Problem 3—20 tood and Stress Analysis 125 A llrnm-diameter steel tension rod is 1.8 In long and carries a load of 9 kN. Find the tensile stress, the total deformation, the unit strains, and the change in the rod diameter. Twin diagonal aluminum alloy tension rods 15 mm in diameter are used in a rectangular frame to prevent collapse. The rods can safely support a tensile stress of 135 MPa. If the rods are inir tially 3 111 in length, how much must they be stretched to develop this stress? Electrical strain gauges were applied to a notched specimen to determine the stresses in the notch. The results were ex : 0.0021 and 6,. : 70.00067. Find (71 and cry if the material is carbon steel. An engineer wishes to determine the shearing strength of a certain epoxy cement. The problem is to devise a test specimen such that thejoint is subject to pure shear. Thejoint shown in the fig- ure, in which two bars are offset at an angle 9 so as to keep the loading force F centroidal with the straight shanks, seems to accomplish this purpose. Using the contact area A and designating Sm as the ultimate shearing strength, the engineer obtains F S“, : A cos 9 A The engineer’s supervisor, in reviewing the test results, says the expression should be F l W 5,,“ : E (1+ Etan2 6) cost-3 Resolve the discrepancy. What is your position? The state of stress at a point is o, = *2. a}. 2 6, a; : —4, r” : 3. 7.3-: = 2, and r2} = *5 MPa. Determine the principal stresses, draw a complete Mohr’s three-circle diagram, labeling all points of interest, and report the maximum shear stress for this case. Repeat Prob. 347 with a, : 10, a, : 0. a, z 10, a, = 20,1), : 40$, and a, : 0 MPa. Repeat Prob. 3—17 with ox : 1,07,. : 4, crZ : 4,15,). : 2,131 : —4, and I” : 72 MPa. The Roman method for addressing uncertainty in design was to build a copy of a design that was satisfactory and had proven durable. Although the early Romans did not have the intellectual tools to deal with scaling size up or down, you do. Consider a simply supported, rectangular-cross section beam with a concentrated load F, as depicted in the figure. ...
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This note was uploaded on 02/22/2010 for the course MECHANICAL ME 342 taught by Professor Namıkçıblak during the Spring '08 term at Yeditepe Üniversitesi.

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Homework1 - 3—20 Load and Stress Analysts 121 Summary The...

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