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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 ﬁniteelement
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 shearforce and bendingmoment 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')
34 Repeat Prob. 3—3 using singularity functions exclusively (for reactions as well).
35 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 36 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 37 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 38 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 39 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 310 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 312 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—13 314 3—15 316 Problem 3— l6 317 31 8
31 9
320 Problem 3—20 tood and Stress Analysis 125 A llrnmdiameter 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) cost3 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 threecircle 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, rectangularcross
section beam with a concentrated load F, as depicted in the ﬁgure. ...
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 Spring '08
 NamıkÇıblak
 Shear Stress, Stress Analysis, repeat prob.

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