fatigue_failure_problems

fatigue_failure_problems - The example demonstrates that....

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7
This is the end of the preview. Sign up to access the rest of the document.

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 7-439 to 7—46 are computer problems. Deterministic Problems A Til-'41: drill rod was heat-treath 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 43-10 steel heat-nested to a tensile strength of 250 kpsi. Estimate the fatigue strength of a misting-beam specimen made of MSI Hill} hot-roiled 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 lip-5i- Estimate thc endurance strength of a Bloom-dimmer 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 Cr-MovNi steel capable of being heat-nested 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 3-H curve of this material. let 1' = 0.9. 388 Mechanical Engineering Design flANALYSIS 7-9 flANALYSIS 7-10 ADESIGN ANALYSl‘S E ANALYSIS E ANALYSlS ANALYSES ANALYSIS ennn ANALYSIS E 7-” Problem 7—! 1 7-12 7-13 7-14 7-15 7-16 7-17 7—18 7-19 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 hot-r0 - 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 cold-drawn steel flat. The bar is 60 turn wi’ u thick and has a 12-min hole drilled through the center as depicted in Table A—lS—‘l is concentrically loaded in push-pull 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 10-kip 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) ASME-elliptic 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" 37-22 17-23 Fatigue Foilure Resulting irorn Vorioble Loading 389 .‘>n Kl The figure shows a formed round-wire 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 hot-rolled 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 18-min 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 high-carbon 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 free-body diagram of a connecting-link 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 cold-drawn AIS! 10l8 steel. 390 Mechanical Engineering Design Problem 7—23 fits N A LY 513 7-24 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 7-25 Repeat Prob. 7—24 ignoring curvature effects on the bending stress. flfii N A W 513 7-26 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 3-mrn 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 7-26 7-27 7-30 7-31 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 clutch-testing 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 7-32 Solve Prob. 7—1 if the hardness of production pieces is found to be H3 : l I ADE SIGN 7-33 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 cold-drawn steel of Prob. 7-l0 with Sm : MOLNH ST, = 370LN(1.0.061). The reliability goal must exceed 0.999. Using the «- specify the thickness !. ANALYSI 8 7-34 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. 7-35 Repeat Prob. 7—34. with a completely reversed torsional moment of T = 1400 if firm A LY S 1 5 7-36 A l fi-in-diameter hot-rolled 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. 1-36, with the bar subject to a completely reversed torsional moment u-i ADEQth 7-33 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 7-39 A i by ll-in steel bar has a 3-in 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 7-40 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 7-4" 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 ASME-elliptic fatigue failure locus, implementing Eqs. (7—83) and (7-84}. Repeat Prob. 7—41 for the Smith-Dolan 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 7-l4, 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 loss-of—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—of-function 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. ...
View Full Document

Page1 / 7

fatigue_failure_problems - The example demonstrates that....

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online