348 Mechanical Engineering Design PROBLEMS Problems marked with an asterisk (*) are linked to problems in other chapters, as summarized in Table 1 1 of Sec. 1 16, p. 24. Problems 6 1 to 6 63 are to be solved by deterministic methods. Problems 6 64 to 6 78 are to be solved by stochastic methods. Problems 6 71 to 6 78 are computer problems. Deterministic Problems 6 1 10-mm drill rod was heat-treated and ground. The measured hardness was found to be 300 Brinell. Estimate the endurance strength in MPa if the rod is used in rotating bending. 6 2 Estimate S e in kpsi for the following materials: (a) ISI 1035 CD steel. (b) ISI 1050 HR steel. (c) 24 T4 aluminum. (d) ISI 4130 steel heat-treated to a tensile strength of 235 kpsi. 6 3 steel rotating-beam test specimen has an ultimate strength of 1 kpsi. Estimate the life of the specimen if it is tested at a completely reversed stress amplitude of 70 kpsi. 6 4 steel rotating-beam test specimen has an ultimate strength of 1600 MPa. Estimate the life of the specimen if it is tested at a completely reversed stress amplitude of 900 MPa. 6 5 steel rotating-beam test specimen has an ultimate strength of 230 kpsi. Estimate the fatigue strength corresponding to a life of 150 kcycles of stress reversal. 6 6 Repeat Prob. 6 5 with the specimen having an ultimate strength of 1100 MPa. 6 7 steel rotating-beam test specimen has an ultimate strength of 150 kpsi and a yield strength of 135 kpsi. It is desired to test low-cycle fatigue at approximately 500 cycles. Check if this is possible without yielding by determining the necessary reversed stress amplitude. 6 8 Derive Eq. (6 17). Rearrange the equation to solve for N. 6 9 or the interval 10 3 N 10 6 cycles, develop an expression for the axial fatigue strength (S f ) ax for the polished specimens of 4130 used to obtain ig. 6 10. The ultimate strength is S ut = 125 kpsi and the endurance limit is (S e ) ax = 50 kpsi. 6 10 Estimate the endurance strength of a 1.5-in-diameter rod of ISI 1040 steel having a machined finish and heat-treated to a tensile strength of 110 kpsi. 6 11 Two steels are being considered for manufacture of as-forged connecting rods. One is ISI 4340 Cr-Mo-Ni steel capable of being heat-treated to a tensile strength of 260 kpsi. The other is a plain carbon steel ISI 1040 with an attainable S ut of 113 kpsi. If each rod is to have a size giving an equivalent diameter d e of 0.75 in, is there any advantage to using the alloy steel for this fatigue application? 6 12 1-in-diameter solid round bar has a groove 0.1-in deep with a 0.1-in radius machined into it. The bar is made of ISI 10 CD steel and is subjected to a purely reversing torque of 1800 lbf in. or the S-N curve of this material, let f = 0.9. (a) Estimate the number of cycles to failure. (b) If the bar is also placed in an environment with a temperature of 750, estimate the number of cycles to failure. 6 13 solid square rod is cantilevered at one end. The rod is 0.6 m long and supports a completely reversing transverse load at the other end of ±2 kn. The material is ISI 1080 hot-rolled steel. If the rod must support this load for 10 4 cycles with a factor of safety of 1.5, what dimension should the square cross section have? Neglect any stress concentrations at the support end.
atigue ailure Resulting from Variable Loading 349 6 14 rectangular bar is cut from an ISI 10 cold-drawn steel flat. The bar is 2.5 in wide by 3 8 in thick and has a 0.5-in-dia. hole drilled through the center as depicted in Table 15 1. The bar is concentrically loaded in push-pull fatigue by axial forces a, uniformly distributed across the width. Using a design factor of n d = 2, estimate the largest force a that can be applied ignoring column action. 6 15 solid round bar with diameter of 2 in has a groove cut to a diameter of 1.8 in, with a radius of 0.1 in. The bar is not rotating. The bar is loaded with a repeated bending load that causes the bending moment at the groove to fluctuate between 0 and 25 000 lbf in. The bar is hot-rolled ISI 1095, but the groove has been machined. Determine the factor of safety for fatigue based on infinite life and the factor of safety for yielding. 6 16 The rotating shaft shown in the figure is machined from ISI 10 CD steel. It is subjected to a force of = 6 kn. ind the minimum factor of safety for fatigue based on infinite life. If the life is not infinite, estimate the number of cycles. Be sure to check for yielding. 500 175 Problem 6 16 Dimensions in millimeters 25 D. 35 D. 3 R. 50 D. 25 D. 180 280 6 17 The shaft shown in the figure is machined from ISI 1040 CD steel. The shaft rotates at 1600 rpm and is supported in rolling bearings at and B. The applied forces are 1 = 2500 lbf and 2 = 1000 lbf. Determine the minimum fatigue factor of safety based on achieving infinite life. If infinite life is not predicted, estimate the number of cycles to failure. lso check for yielding. 8 in 8 in 8 in 1 2 1 1 4 in 1 5 8 in 1 7 8 in 1 3 4 in 1 1 4 in Problem 6 17 10 in 10 in 3 in 1 2 in ll fillets 1 in R. 16 1 2 in B 6 18 Solve Prob. 6 17 except with forces 1 = 10 lbf and 2 = 2400 lbf. 6 19 Bearing reactions R 1 and R 2 are exerted on the shaft shown in the figure, which rotates at 950 rev/min and supports an 8-kip bending force. Use a 1095 HR steel. Specify a diameter d using a design factor of n d = 1.6 for a life of 10 hr. The surfaces are machined. = 8 kip 10 in 5 in 5 in d/5 R. Problem 6 19 d R 1 d/10 R. 1.4 d 1 in R2 d
350 Mechanical Engineering Design 6 bar of steel has the minimum properties S e = 40 kpsi, S y = 60 kpsi, and S ut = 80 kpsi. The bar is subjected to a steady torsional stress of 15 kpsi and an alternating bending stress of 25 kpsi. ind the factor of safety guarding against a static failure, and either the factor of safety guarding against a fatigue failure or the expected life of the part. or the fatigue analysis use: (a) Modified Goodman criterion. (b) Gerber criterion. (c) SME-elliptic criterion. 6 21 Repeat Prob. 6 but with a steady torsional stress of kpsi and an alternating bending stress of 10 kpsi. 6 22 Repeat Prob. 6 but with a steady torsional stress of 15 kpsi, an alternating torsional stress of 10 kpsi, and an alternating bending stress of 12 kpsi. 6 23 Repeat Prob. 6 but with an alternating torsional stress of 30 kpsi. 6 24 Repeat Prob. 6 but with an alternating torsional stress of 15 kpsi and a steady bending stress of 15 kpsi. 6 25 The cold-drawn ISI 1040 steel bar shown in the figure is subjected to a completely reversed axial load fluctuating between 28 kn in compression to 28 kn in tension. Estimate the fatigue factor of safety based on achieving infinite life, and the yielding factor of safety. If infinite life is not predicted, estimate the number of cycles to failure. 25 mm 6-mm D. Problem 6 25 10 mm 6 26 Repeat Prob. 6 25 for a load that fluctuates from 12 kn to 28 kn. Use the Modified Goodman, Gerber, and SME-elliptic criteria and compare their predictions. 6 27 Repeat Prob. 6 25 for each of the following loading conditions: (a) 0 kn to 28 kn (b) 12 kn to 28 kn (c) 28 kn to 12 kn 6 28 The figure shows a formed round-wire cantilever spring subjected to a varying force. The hardness tests made on 50 springs gave a minimum hardness of 400 Brinell. It is apparent from the mounting details that there is no stress concentration. 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. (b) Gerber criterion. 12 in max = 40 lbf min = lbf Problem 6 28 3 in D. 8
atigue ailure Resulting from Variable Loading 351 6 29 The figure is a drawing of a 4- by -mm latching spring. preload is obtained during assembly by shimming under the bolts to obtain an estimated initial deflection of 2 mm. The latching operation 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 4 mm. Estimate the yield strength to be 90 percent of the ultimate strength. (a) ind the maximum and minimum latching forces. (b) Is it likely the spring will achieve infinite life? 140 Problem 6 29 Dimensions in millimeters Section 4 6 30 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 = 0.40 in, h = 0.50 in, w 1 = 3.50 in, and w 2 = 3.0 in. The forces fluctuate between a tension of 5 kip and a compression of 16 kip. Neglect column action and find the least factor of safety if the material is cold-drawn ISI 1018 steel. r h Problem 6 30 w 1 w 2 d Section 6 31 Solve Prob. 6 30 except let w 1 = 2.5 in, w 2 = 1.5 in, and the force fluctuates between a tension of 16 kips and a compression of 4 kips. 6 32 or the part in Prob. 6 30, recommend a fillet radius r that will cause the fatigue factor of safety to be the same at the hole and at the fillet. 6 33 The torsional coupling in the figure is composed of a curved beam of square cross section that is welded to an input shaft and output plate. torque is applied to the shaft and cycles from zero to T. The cross section of the beam has dimensions of 3 16 3 16 in, and the centroidal axis of the beam describes a curve of the form r = 0.75 + 0.4375 θ/π, where r and θ are in inches and radians, respectively (0 θ 4π ). The curved beam has a machined surface with yield and ultimate strength values of 60 and 110 kpsi, respectively. (a) Determine the maximum allowable value of T such that the coupling will have an infinite life with a factor of safety, n = 3, using the modified Goodman criterion. (b) Repeat part (a) using the Gerber criterion. (c) Using T found in part (b), determine the factor of safety guarding against yield.
352 Mechanical Engineering Design T 3 16 Problem 6 33 T 3 4 2.5 (Dimensions in inches) 6 34 Repeat Prob. 6 33 ignoring curvature effects on the bending stress. 6 35 part is loaded with a combination of bending, axial, and torsion such that the following stresses are created at a particular location: Bending: xial: Torsion: Completely reversed, with a maximum stress of 60 MPa Constant stress of MPa Repeated load, varying from 0 MPa to 50 MPa ssume the varying stresses are in phase with each other. The part contains a notch such that K f,bending = 1.4, K f,axial = 1.1, and K f,torsion = 2.0. The material properties are S y = 300 MPa and S u = 400 MPa. The completely adjusted endurance limit is found to be S e = 0 MPa. ind the factor of safety for fatigue based on infinite life. If the life is not infinite, estimate the number of cycles. Be sure to check for yielding. 6 36 Repeat the requirements of Prob. 6 35 with the following loading conditions: Bending: xial: Torsion: luctuating stress from 40 MPa to 150 MPa None Mean stress of 90 MPa, with an alternating stress of 10 percent of the mean stress 6 37* to or the problem specified in the table, build upon the results of the original problem to determine 6 46* the minimum factor of safety for fatigue based on infinite life. The shaft rotates at a constant speed, has a constant diameter, and is made from cold-drawn ISI 1018 steel. Problem Number Original Problem, Page Number 6 37* 3 68, 137 6 38* 3 69, 137 6 39* 3 70, 137 6 40* 3 71, 137 6 41* 3 72, 138 6 42* 3 73, 138 6 43* 3 74, 138 6 44* 3 76, 139 6 45* 3 77, 139 6 46* 3 79, 139
atigue ailure Resulting from Variable Loading 353 6 47* to or the problem specified in the table, build upon the results of the original problem to determine 6 50* the minimum factor of safety for fatigue based on infinite life. If the life is not infinite, estimate the number of cycles. The force is applied as a repeated load. The material is ISI 1018 CD steel. The fillet radius at the wall is 0.1 in, with theoretical stress concentrations of 1.5 for bending, 1.2 for axial, and 2.1 for torsion. Problem Number Original Problem, Page Number 6 47* 3 80, 139 6 48* 3 81, 140 6 49* 3 82, 140 6 50* 3 83, 140 6 51* to or the problem specified in the table, build upon the results of the original problem to determine 6 53* the minimum factor of safety for fatigue at point, based on infinite life. If the life is not infinite, estimate the number of cycles. The force is applied as a repeated load. The material is ISI 1018 CD steel. Problem Number Original Problem, Page Number 6 51* 3 84, 140 6 52* 3 85, 141 6 53* 3 86, 141 6 54 Solve Prob. 6 17 except include a steady torque of 2500 lbf in being transmitted through the shaft between the points of application of the forces. 6 55 Solve Prob. 6 18 except include a steady torque of 20 lbf in being transmitted through the shaft between the points of application of the forces. 6 56 In the figure shown, shaft, made of ISI 10 hot-rolled steel, is welded to a fixed support and is subjected to loading by equal and opposite forces via shaft B. theoretical stress-concentration factor K ts of 1.6 is induced by the 1 8-in fillet. The length of shaft from the fixed support to the connection at shaft B is 2 ft. The load cycles from 150 to 500 lbf. (a) or shaft, find the factor of safety for infinite life using the modified Goodman fatigue failure criterion. (b) Repeat part (a) using the Gerber fatigue failure criterion. 7 -in dia 8 1 in Problem 6 56 10 mm 1 in 1 8 in fillet Shaft Shaft B
354 Mechanical Engineering Design 6 57 schematic of a clutch-testing machine is shown. The steel shaft rotates at a constant speed ω. n axial load is applied to the shaft and is cycled from zero to P. The torque T induced by the clutch face onto the shaft is given by T = f P(D + d) 4 where D and d are defined in the figure and f is the coefficient of friction of the clutch face. The shaft is machined with S y = 1 kpsi and S ut = 145 kpsi. The theoretical stress-concentration factors for the fillet are 3.0 and 1.8 for the axial and torsional loading, respectively. ssume the load variation P is synchronous with shaft rotation. With f = 0.3, find the maximum allowable load P such that the shaft will survive a minimum of 10 6 cycles with a factor of safety of 3. Use the modified Goodman criterion. Determine the corresponding factor of safety guarding against yielding. R = 0.1 in d = 1.2 in Problem 6 57 P riction pad D = 6 in 6 58 or the clutch of Prob. 6 57, the external load P is cycled between 4.5 kips and 18 kips. ssuming 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. 6 59 flat leaf spring has fluctuating stress of σ max = 360 MPa and σ min = 160 MPa applied for 8 (10 4 ) cycles. If the load changes to σ max = 3 MPa and σ min = 0 MPa, how many cycles should the spring survive? The material is ISI 10 CD and has a fully corrected endurance strength of S e = 175 MPa. ssume that f = 0.9. (a) Use Miner s method. (b) Use Manson s method. 6 60 rotating-beam specimen with an endurance limit of 50 kpsi and an ultimate strength of 140 kpsi is cycled percent of the time at 95 kpsi, 50 percent at 80 kpsi, and 30 percent at 65 kpsi. Let f = 0.8 and estimate the number of cycles to failure. 6 61 machine part will be cycled at ±350 MPa for 5 (10 3 ) cycles. Then the loading will be changed to ±260 MPa for 5 (10 4 ) cycles. inally, the load will be changed to ±225 MPa. How many cycles of operation can be expected at this stress level? or the part, S ut = 530 MPa, f = 0.9, and has a fully corrected endurance strength of S e = 210 MPa. (a) Use Miner s method. (b) Use Manson s method. 6 62 The material properties of a machine part are S ut = 85 kpsi, f = 0.86, and a fully corrected endurance limit of S e = 45 kpsi. The part is to be cycled at σ a = 35 kpsi and σ m = 30 kpsi
atigue ailure Resulting from Variable Loading 355 for 12 (10 3 ) cycles. Using the Gerber criterion, estimate the new endurance limit after cycling. (a) Use Miner s method. (b) Use Manson s method. 6 63 Repeat Prob. 6 62 using the Goodman criterion. Stochastic Problems 6 64 Solve Prob. 6 1 if the ultimate strength of production pieces is found to be Sut = 1030LN (1, 0.0508) MPa. 6 65 The situation is similar to that of Prob. 6 14 wherein the imposed completely reversed axial load a = 3.8LN(1, 0.) kip is to be carried by the link with a thickness to be specified by you, the designer. Use the 10 cold-drawn steel of Prob. 6 14 with Sut = 68LN (1, 0.28) and Syt = 57LN(1, 0.058) kpsi. The reliability goal must exceed 0.99. Using the correlation method, specify the thickness t. 6 66 solid round steel bar is machined to a diameter of 32 mm. groove 3 mm deep with a radius of 3 mm is cut into the bar. The material has a mean tensile strength of 780 MPa. completely reversed bending moment M = 160 N m is applied. Estimate the reliability. The size factor should be based on the gross diameter. The bar rotates. 6 67 Repeat Prob. 6 66, with a completely reversed torsional moment of T = 160 N m applied. 6 68 1 1 2 -in-diameter hot-rolled steel bar has a 3 16 -in diameter hole drilled transversely through it. The bar is nonrotating and is subject to a completely reversed bending moment of M = 1500 lbf in in the same plane as the axis of the transverse hole. The material has a mean tensile strength of 76 kpsi. Estimate the reliability. The size factor should be based on the gross size. Use Table 16 for K t. 6 69 Repeat Prob. 6 68, with the bar subject to a completely reversed torsional moment of 00 lbf in. 6 70 The plan view of a link is the same as in Prob. 6 30; however, the forces are completely reversed, the reliability goal is 0.998, and the material properties are S ut = 64LN(1, 0.045) kpsi and S y = 54LN(1, 0.077) kpsi. Treat a as deterministic, and specify the thickness h. Computer Problems 6 71 1 4 by 1 1 2 -in steel bar has a 3 4 -in drilled hole located in the center, much as is shown in Table 15 1. The bar is subjected to a completely reversed axial load with a deterministic load of 10 lbf. The material has a mean ultimate tensile strength of S ut = 80 kpsi. (a) Estimate the reliability. (b) Conduct a computer simulation to confirm your answer to part a. 6 72 rom your experience with Prob. 6 71 and Ex. 6 19, you observed that for completely reversed axial and bending fatigue, it is possible to Observe the COVs associated with a priori design considerations. Note the reliability goal. ind the mean design factor n d that will permit making a geometric design decision that will attain the goal using deterministic methods in conjunction with n d. ormulate an interactive computer program that will enable the user to find n d. While the material properties S ut, S y, and the load COV must be input by the user, all of the COVs associated with