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Unformatted text preview: The example demonstrates that. for a given reliability goal. the fatigue design factor
that facilitates its attainment is decided by the variabilities of the situation. Furthermore.
the necessary design factor is not a constant independent of the way the concept unfolds.
Rather. it is at‘unetion of a number of seemingly unrelated a priori decisions that are made
in giving deﬁnition to the concept. The involvement of stochastic methodology can be
limited to deﬁning the necessary design factor. in particular. in the example. the design
factor is not a iteration of the design variable 1: rather. 1' follows from the design factor. PROBLEMS Problems 7—l to "LB l are to be solved by detemtirti stic methods. Problems ill—32 to 1—33 are to be
solved by stochastic methods. thlems 7439 to 7—46 are computer problems. Deterministic Problems A Til'41: drill rod was heattreath and ground. The measured hardness was found to be 490 Brinell. Estimate the endurance strength if the rod is used in rotating bending. Estimate 3; for the following materials: (at A15] lﬂ'lfl CD steel. (bl AIS] ltllitl HR steel. (£12024 T3 aluminum. [dt A131 4310 steel heatnested to a tensile strength of 250 kpsi. Estimate the fatigue strength of a mistingbeam specimen made of MSI Hill} hotroiled steel cor
responding to a life of llﬁ kilocycles of stress reversal. Also. estimate the life of the specimen
corresponding to a stress amplitude arse hpsi. The known properties are S... = 66.2 kpsi. in, =
llﬁ lipsi,rn = 0.22.and s; = [1.9!]. Derive Eq. {Tl—lo). For the specimen of Prob. T—3. estimate the strength corresponding to
Sill} cycles. For the interval it)“ 5 N 1: 10" cycles. develop an expression for the fatigue strength (5;. t... for
the polished specimens of 4 [30 used to obtain Fig. 7—10. The ultimate strength is S... = 125 kpsi
and the endurance limit is (ELL. = 49 lip5i Estimate thc endurance strength of a Bloomdimmer rod of Mill lll35 steel having a machined
ﬁnish and heat—treated to a tensile strength of i‘ IllI MPa. No steels are being considered for manufacture of as—forged connecting rods. Unc is AIS! 4340
CrMovNi steel capable of being heatnested to a tensile strength of no kpsi . The other is a plain car
bon steel A13! 1040 with unattainable 5... ol‘ I 13 kpsi. if each rod is to have a size giving an equiva
lent diameter d, omfi’ﬁ in. is there any advantage to using the alloy steel for this fatigue application? A solid round bar. 25 mm in diameter. has a groove 2.54am deep with a 2.5mm radiUs machined
into it. The bar is made of AISI lOlB CD steel and is subjected to a purely reversing torque of
Elli) N m. For the 3H curve of this material. let 1' = 0.9. 388 Mechanical Engineering Design ﬂANALYSIS 79 ﬂANALYSIS 710 ADESIGN ANALYSl‘S E ANALYSIS E ANALYSlS ANALYSES ANALYSIS ennn ANALYSIS E 7” Problem 7—! 1 712 713 714 715
716 717 7—18
719 to) Estimate the number of cycles to failure.
(b) If the bar is also placed in an environment with a temperature of 450C, estimate M?
of cycles to failure. A solid square rod is cantilevered at one end. The rod is 0.8 m long and supports a e t ' versing transverse load at the other end of :l.1 RM. The material is A151 1045 hotr0 
the rod must support this load for 10“ cycles with a factor of safety of 1.5. What dime the square cross section have? Neglect any stress concentrations at the Support end . . . .
that f z 0.9. A rectangular bar is cut from an AISI 018 colddrawn steel flat. The bar is 60 turn wi’ u
thick and has a 12min hole drilled through the center as depicted in Table A—lS—‘l
is concentrically loaded in pushpull fatigue by axial forces Fa. uniformly distri
the width. Using a design factor of not = LS. estimate the largest force Fa that can If,
ignoring column action. II Bearing reacrions R. and R1 are exerted on the shaft shown in the ﬁgure, whi.
1150 revfmin and supports a 10kip bending force. Use a 1095 HR steel. Specify a «. '
using a design factor of ad = L6 fer a life of 3 min. The surfaces are machined. A bar of steel has the minimum properties S, = 276 M'Pa, S, = 413 MPa. and Sm
The bar is subjected to a steady torsional stress of HE MPa and an alternating bee
172 MPa. Find the factor of safety guarding against a static failure. and either the f =. _
guarding against a fatigue failure or the expected life of the part. For the fatigue I = 
(a) Modiﬁed Goodman criterion. I
(b) Gerber criterion. (.9) ASMEelliptic criterion. of 69 MPa. Repeat Prob. 7—1 2 but with a steady torsional stress of 103 MPa. an alternating torsi 69 MPa, and an alternating bending stress of 83 MPa. Repeat Prob. 7—12 but with an alternating torsional stress of 207 MPa. Repeat Prob. ?—12 but with an alternating torsional stress of 103 MPa and a steady . “if.
of 103 MPa. ‘ The cold—drawn A181 1018 steel bar shown in the ﬁgure is subjected to a tensile n.:. between 800 and 3000 Ibf. Estimate the factors of safety in. and n; using (a) a _ _
failure criterion as part of the designer‘s fatigue diagram. and (b) a ASME—elliptic f :91;
criterion as part of the designer's fatigue diagram. Repeat Prob. 7—! 7, with the load ﬂuctuating between —800 and 3000 lbf. Assume no Repeat Prob. 7—17, with the load ﬂuctuating between 800 and —3000 lbf. Assume no" 3722 1723 Fatigue Foilure Resulting irorn Vorioble Loading 389 .‘>n
Kl The ﬁgure shows a formed roundwire cantilever spring subjected to a varying force. The hard
ness tests made on 25 springs gave a minimum hardness of 380 Brinell. it is apparent from the
mounting details that there is no stress concentration. A visual inspection of the springs indicates
that the surface ﬁnish corresponds closely to a hotrolled ﬁnish. What number of applications is
likely to cause failure? Solve using: (a) Modiﬁed Goodman criterion. (in) Gerber criterion. Fm“ = 3n [bf
F .= [5 [bf [11"! The ﬁgure is a drawing of a 3 by 18min latching spring. A preload is obtained during assembly
by shimming under the bolts to obtain an estimated initial deﬂection of 2 mm. The latching oper
ation itself requires an additional deﬂection of exactly 4 mm. The material is ground highcarbon
steel, bent then hardened and tempered to a minimum hardness of 490 Bhn. The radius of the bend
is 3 mm. Estimate the yield strength to be 90 percent of the ultimate strength. to) Find the maximum and minimum latching forces. (b) Is it likely the spring will fail in fatigue? Use the Gerber criterion. Repeat Prob. 21, part b, using the modiﬁed Goodman criterion. The ﬁgure shows the freebody diagram of a connectinglink portion having stress concentration
at three sections. The dimensions are r = 0.25 in. d = 0.75 in. h = 0.50 in. w. = 3.75 in. and
w; = 2.5 in. The forces F ﬂuctuate between a tension of 4 kip and a compression of 16 kip. Ne
glect column action and ﬁnd the least factor of safety if the material is colddrawn AIS! 10l8 steel. 390 Mechanical Engineering Design Problem 7—23 ﬁts N A LY 513 724 The torsional coupling in the figure is composed of a curved beam of square w ‘
welded to an input shaft and output plate. A torque is applied to the shaﬁ and
1 T. The cross section of the beam has dimensions of 5 by 5 mm, and the centroi
describes a curve of the form r = l0 film where r and 6 are in mm and 
(2n 5 6 5 6n). The curved beam has a machined surface with yield and ulti r _ 
of 420 and 770 MP3, respectively. I
(a) Determine the maximum allowable value of T such that the coupling will l1_“ with a factor of safety. it = 3, using the modiﬁed Goodman criterion. (b) Repeat part (a) using the Gerber criterion.
(c) Using Tfound in part (b). determine the factor of safety guarding against Problem 7—24 {Dimensions in mm) ﬂ" N N "' 5 5 725 Repeat Prob. 7—24 ignoring curvature effects on the bending stress. ﬂﬁi N A W 513 726 In the ﬁgure shown. shaftA. made of A131 1010 hot—rolled steel. is welded toa .
subjected to loading by equal and opposite forces F via shaft 3. A theoretical “'
K” of L6 is induced by the 3mrn ﬁllet. The length of shaftA from the ﬁxed so I tion at shaft Bis l m. The load F cycles from 0.5 to2 kN. Problem 726 727 730 731 Foligue Failure Reaching from Variable loading l 391 (a) For shaft A, ﬁnd the factor of safety for inﬁnite life using the modiﬁed Goodman fatigue fail ure criterion.
(b) Repeat part to) using the Gerber fatigue failure criterion. A schematic of a clutchtesting machine is shown. The steel shaft rotates at a constant speed to. An ,I l'.
axial load is applied to the shaft and is cycled from zero to P. The torque T induced by the clutch l l
face onto the shaft is given by i I
T = m l l l 4 l ‘ where D and d are deﬁned in the ﬁgure and f is the coefﬁcient of friction of the clutch face. The I
shaft is machined with 3,. = 800 MP3 and 8,.r = IOOO MPa. The theoretical stress concentration
factors for the ﬁllet are 3.0 and 1.8 for the axial and torsional loading, respectively. l
(a) Assume the load variation P is synchronous with shaft rotation. With f = 0.3. ﬁnd the maxi '
mum allowable load P such that the shaft will survive a minimum of 10‘1 cycles with a factor 1
of safety of 3. Use the modiﬁed Goodman criterion. Determine the corresponding factor of [
safety guarding against yielding. l
(b) Suppose the shaft is nor rotating. but the load P is cycled as shown. With f = 0.3. ﬁnd the
maximum allowable load P so that the shaft will survive a minimum of 10“ cycles with a fac
tor of safety of 3. Use the modiﬁed Goodman criterion. Determine the corresponding factor of
safety guarding against yielding. Friction pad D = 150 mm i For the clutch of Prob. 7—27. the external load P is cycled between 20 W and 80 RN. Assuming
that the shaft is rotating synchronous with the external load cycle. estimate the number of cycles
to failure. Use the modiﬁed Goodman fatigue failure criteria. A ﬂat leaf spring has ﬂuctuating stress of em, = 420 MPa and em“, = 140 MPa applied for
5 (104) cycles. If the load changes to em, = 350 MPa and em = #200 MPa. how many cycles
should the spring survive? The material is A181 1040 CD and has a fully corrected endurance
strength of S“ .= 200 MPa. Assume that f = 0.9. (:1) Use Miner‘s method. (b) Use Manson's method. A machine part will be cycled at :lz48 kpsi for 4 (103) cycles. Then the loading will be changed to
£38 kpsi for 6 (104) cycles. Finally, the load will be changed to 3:32 kpsi. How many cycles of
operation can be expected at this stress level? For the part. Sm = 76 kpsi. f = 0.9. and has a fully
corrected endurance strength of S, = 30 kpsi. {at Use Miner's method. (b) Use Manson‘s method. A rotating~beam specimen with an endurance limit of 50 kpsi and an ultimate strength of 100 kpsi
is cycled 20 percent of the time at 70 kpsi. 50 percent at 55 kpsi. and 30 percent at 40 kpsi. Let
f = 0.9 and estimate the number of cycles to failure. 392 Mechanical Engineering Design Stochastic Problems __
ﬂANALYSIS 732 Solve Prob. 7—1 if the hardness of production pieces is found to be H3 : l I
ADE SIGN 733 The situation is similar to that ofProb. 7—10 wherein the imposed completely re 
E, :2 lSLNtl. 0.20) kN is to be carried by the link with a thickness to be r designer. Use the IOlS colddrawn steel of Prob. 7l0 with Sm : MOLNH
ST, = 370LN(1.0.061). The reliability goal must exceed 0.999. Using the « specify the thickness !. ANALYSI 8 734 A solid round steel bar is machined to a diameter of 1.25 in. A groove % in dee
ii in is cut into the bar. The material has a mean tensile strength of 1 l0 kpsi. A u
bending moment M = I400 lbf in is applied. Estimate the reliability. The size: based on the gross diameter. The bar rotates. 735 Repeat Prob. 7—34. with a completely reversed torsional moment of T = 1400 if ﬁrm A LY S 1 5 736 A l ﬁindiameter hotrolled steel bar has a $411 diameter hole drilled transversely .
bar is nonrotating and is subject to a completely reversed bending moment of in the same plane as the axis of the transverse hole. The material has a mean
58 kpsi. Estimate the reliability. The size factor should be based on the gross size. 7 t for K,. 7'37 Repeat Prob. 136, with the bar subject to a completely reversed torsional moment ui ADEQth 733 The plan view of a link is the same as in Prob. 7—23: however. the forces F 51";
reversed. the reliability goal is 0.998. and the material properties are S.,l = 64”! and S). :2 54LN(I. 0.077) kpsi. Treat F“ as deterministic. and specify the title __ _ Computer Problems 
I ﬂANMVSIS 739 A i by llin steel bar has a 3in drilled hole located in the center. much ‘ Table A—lﬁwl. The bar is subjected to a completely reversed axial load with a L 31: of 1200 lbf. The material has a mean ultimate tensile strength of :u, = 80 kpsi.
to) Estimate the reliability. ' (b) Conduct a computer simulation to conﬁrm your answer to part a. l ADE S lGN 740 From your experience with Prob. 7—39 and Ex. 7—20. you observed that for c w}. axial and bending fatigue. it is possible to
El  Observe the COVs associated with a priori design considerations.
' Note the reliability goal.
I  Find the mean design factor rid which will permit making a geometric design attain the goal using deterministic methods in conjunction with rid. l  Formulate an interactive computer program that will enable the user to ﬁnd rid.  ‘ "
i; properties SW. S... and the load COV must be input by the user. all of the COV _‘
than}. k... kn kg. and K! can be internal. and answers to questions will allow C '
as C ,. and ﬂy, to be calculated. Later you can add improvements. Test your pro : mt; you have already solved. I l 74" When using the Gerber fatigue failure criterion in a stochastic problem. Eqs. (7 ll useful. They are also computationally complicated. It is helpful to have a comptt
procedure that performs these calculations. When writing an executive program.
priate to ﬁnd 5., and (‘5... a simple call to the subroutine does this with a
 f Also. once the subroutine is tested. it is always ready to perform. Write and test mi: Fatigue Foilure Resulting lrom Variable loading 393 Repeat Problem. 7—4] for the ASMEelliptic fatigue failure locus, implementing Eqs. (7—83) and
(784}. Repeat Prob. 7—41 for the SmithDolan fatigue failure locus. implementing Eqs. (7—87) and (7—88}. Write and test computer subroutines or procedures that will implement
(a) Table 7—4. returning a. b. C. and i... (b) Equation (7—19) using Table 7—5. returning kh. (c) Table 7l4, returning or. 19. C. and ii... (0') Equations (7—26) and (7—75). returning lid and C M. Write and test a computer subroutine or procedure that implements Eqs. (7—76) and t 7—77).
returning c}. rig. and Ca. Write and test a computer subroutine or procedure that implements Eq. (7—35) and Tables 7—8 and
7—18. returning f. C“, and It}. Summary of Parts 'I and 2 The ﬁrst recommendation is to reread Chap. 1. With the experience you have gathered
so far. you will gain from doing it. With the meat you have added to the bare bones of
the introductory chapter, it will have a greater meaning. In Sec. 1—3. there are over two
dozen design considerations. We have addressed item 2 in detail. the question of the
strength/stress relationship in a lossof—function for ductile and brittle materials,
for steady and fatigue loading. and for ﬁnite and indeﬁnite life. We have also started
on item 7. reliability. as it applies to stress/strength relationships. In investigating the
stress/strength relations. the reader should now be prepared to  identify the critical location(s). either by inspection. or. if not obvious. by analyzing
the several candidates. and identifying the "worst case.“ 0 Identify the signiﬁcant strength at that location.
 Identify the signiﬁcant stress at that loaation.  Address the question of whether the disparity between stress and strength is sufﬁcient
such that function will be preserved in the face of service loading. This preparation took a long time because an extensive set of ideas and insights had to be identiﬁed in and among your prerequisite studies. and placed in a useful context.
The question of stiffness. distortion. and deflection, item 3. and their inﬂuence on loss of function has also been addressed. The reader should now be prepared to identify 0 The level of distortion that risks loss of function. 0 The location(s) at which loss—offunction due to distortion is possible.
0 The level of distortion present. 0 Whether the difference is sufﬁcient. Some other considerations will be touched on in Part 3. and those just noted will be fur
ther developed for the application at hand. As we proceed into Part 3 our focus becomes
more speciﬁc as we consider particular machine elements and their applications. For now. the reader should feel comfortable with a kit of tools from which an ade—
quacy assessment is devised. Skill I will take on additional substance as applications
unfold. In addition to focus on individual elements. design/synthesis ideas will appear
more often. and skill 2 will take form and grow. ...
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 Fall '09
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