上海交通大学：《Design and Manufacturing》课程教学资源（参考文献）Shigley's Mechanical Engineering Design ch7 shaft and shaft components.pdf_大学文库

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft T©The McGraw-Hill Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition 348 Mechanical Engineering Design 7-1 Introduction A shaft is a rotating member,usually of circular cross section,used to transmit power or motion.It provides the axis of rotation,or oscillation,of elements such as gears, pulleys,flywheels,cranks,sprockets,and the like and controls the geometry of their motion.An axle is a nonrotating member that carries no torque and is used to sup- port rotating wheels,pulleys,and the like.The automotive axle is not a true axle;the term is a carry-over from the horse-and-buggy era,when the wheels rotated on non- rotating members.A non-rotating axle can readily be designed and analyzed as a static beam,and will not warrant the special attention given in this chapter to the rotating shafts which are subject to fatigue loading. There is really nothing unique about a shaft that requires any special treatment beyond the basic methods already developed in previous chapters.However,because of the ubiquity of the shaft in so many machine design applications,there is some advan- tage in giving the shaft and its design a closer inspection.A complete shaft design has much interdependence on the design of the components.The design of the machine itself will dictate that certain gears,pulleys,bearings,and other elements will have at least been partially analyzed and their size and spacing tentatively determined.Chapter 18 provides a complete case study of a power transmission,focusing on the overall design process. In this chapter,details of the shaft itself will be examined,including the following: ·Material selection ·Geometric layout ·Stress and strength ·Static strength ·Fatigue strength Deflection and rigidity ·Bending deflection Torsional deflection Slope at bearings and shaft-supported elements Shear deflection due to transverse loading of short shafts Vibration due to natural frequency In deciding on an approach to shaft sizing,it is necessary to realize that a stress analy- sis at a specific point on a shaft can be made using only the shaft geometry in the vicin- ity of that point.Thus the geometry of the entire shaft is not needed.In design it is usually possible to locate the critical areas,size these to meet the strength requirements,and then size the rest of the shaft to meet the requirements of the shaft-supported elements. The deflection and slope analyses cannot be made until the geometry of the entire shaft has been defined.Thus deflection is a function of the geometry everywhere, whereas the stress at a section of interest is a function of local geometry.For this rea- son,shaft design allows a consideration of stress first.Then,after tentative values for the shaft dimensions have been established.the determination of the deflections and slopes can be made. 7-2 Shaft Materials Deflection is not affected by strength,but rather by stiffness as represented by the modulus of elasticity,which is essentially constant for all steels.For that reason,rigid- ity cannot be controlled by material decisions,but only by geometric decisions

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components © The McGraw−Hill 351 Companies, 2008 348 Mechanical Engineering Design 7–1 Introduction A shaft is a rotating member, usually of circular cross section, used to transmit power or motion. It provides the axis of rotation, or oscillation, of elements such as gears, pulleys, flywheels, cranks, sprockets, and the like and controls the geometry of their motion. An axle is a nonrotating member that carries no torque and is used to support rotating wheels, pulleys, and the like. The automotive axle is not a true axle; the term is a carry-over from the horse-and-buggy era, when the wheels rotated on nonrotating members. A non-rotating axle can readily be designed and analyzed as a static beam, and will not warrant the special attention given in this chapter to the rotating shafts which are subject to fatigue loading. There is really nothing unique about a shaft that requires any special treatment beyond the basic methods already developed in previous chapters. However, because of the ubiquity of the shaft in so many machine design applications, there is some advantage in giving the shaft and its design a closer inspection. A complete shaft design has much interdependence on the design of the components. The design of the machine itself will dictate that certain gears, pulleys, bearings, and other elements will have at least been partially analyzed and their size and spacing tentatively determined. Chapter 18 provides a complete case study of a power transmission, focusing on the overall design process. In this chapter, details of the shaft itself will be examined, including the following: • Material selection • Geometric layout • Stress and strength • Static strength • Fatigue strength • Deflection and rigidity • Bending deflection • Torsional deflection • Slope at bearings and shaft-supported elements • Shear deflection due to transverse loading of short shafts • Vibration due to natural frequency In deciding on an approach to shaft sizing, it is necessary to realize that a stress analysis at a specific point on a shaft can be made using only the shaft geometry in the vicinity of that point. Thus the geometry of the entire shaft is not needed. In design it is usually possible to locate the critical areas, size these to meet the strength requirements, and then size the rest of the shaft to meet the requirements of the shaft-supported elements. The deflection and slope analyses cannot be made until the geometry of the entire shaft has been defined. Thus deflection is a function of the geometry everywhere, whereas the stress at a section of interest is a function of local geometry. For this reason, shaft design allows a consideration of stress first. Then, after tentative values for the shaft dimensions have been established, the determination of the deflections and slopes can be made. 7–2 Shaft Materials Deflection is not affected by strength, but rather by stiffness as represented by the modulus of elasticity, which is essentially constant for all steels. For that reason, rigidity cannot be controlled by material decisions, but only by geometric decisions

352 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft T©The McGraw-Hill Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition Shafts and Shaft Components 349 Necessary strength to resist loading stresses affects the choice of materials and their treatments.Many shafts are made from low carbon,cold-drawn or hot-rolled steel.such as ANSI 1020-1050 steels. Significant strengthening from heat treatment and high alloy content are often not warranted.Fatigue failure is reduced moderately by increase in strength,and then only to a certain level before adverse effects in endurance limit and notch sensitivity begin to counteract the benefits of higher strength.A good practice is to start with an inex- pensive,low or medium carbon steel for the first time through the design calculations If strength considerations turn out to dominate over deflection,then a higher strength material should be tried,allowing the shaft sizes to be reduced until excess deflection becomes an issue.The cost of the material and its processing must be weighed against the need for smaller shaft diameters.When warranted,typical alloy steels for heat treatment include ANSI1340-50.3140-50,4140.4340.5140.and8650. Shafts usually don't need to be surface hardened unless they serve as the actual joural of a bearing surface.Typical material choices for surface hardening include carburizing grades of ANSI 1020,4320,4820,and 8620. Cold drawn steel is usually used for diameters under about 3 inches.The nom- inal diameter of the bar can be left unmachined in areas that do not require fitting of components.Hot rolled steel should be machined all over.For large shafts requiring much material removal,the residual stresses may tend to cause warping. If concentricity is important,it may be necessary to rough machine,then heat treat to remove residual stresses and increase the strength,then finish machine to the final dimensions. In approaching material selection,the amount to be produced is a salient factor. For low production,turning is the usual primary shaping process.An economic view- point may require removing the least material.High production may permit a volume- conservative shaping method(hot or cold forming.casting),and minimum material in the shaft can become a design goal.Cast iron may be specified if the production quan- tity is high,and the gears are to be integrally cast with the shaft. Properties of the shaft locally depend on its history-cold work,cold forming, rolling of fillet features,heat treatment,including quenching medium,agitation,and tempering regimen. Stainless steel may be appropriate for some environments. 7-3 Shaft Layout The general layout of a shaft to accommodate shaft elements,e.g.gears,bearings,and pulleys,must be specified early in the design process in order to perform a free body force analysis and to obtain shear-moment diagrams.The geometry of a shaft is gen- erally that of a stepped cylinder.The use of shaft shoulders is an excellent means of axially locating the shaft elements and to carry any thrust loads.Figure 7-1 shows an example of a stepped shaft supporting the gear of a worm-gear speed reducer.Each shoulder in the shaft serves a specific purpose,which you should attempt to deter- mine by observation. See Joseph E.Shigley,Charles R.Mischke,and Thomas H.Brown,Jr.(eds-in-chief),Standard Handbook of Machine Design,3rd ed.,McGraw-Hill,New York,2004.For cold-worked property prediction see Chap.29,and for heat-treated property prediction see Chaps.29 and 33

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components 352 © The McGraw−Hill Companies, 2008 Shafts and Shaft Components 349 Necessary strength to resist loading stresses affects the choice of materials and their treatments. Many shafts are made from low carbon, cold-drawn or hot-rolled steel, such as ANSI 1020-1050 steels. Significant strengthening from heat treatment and high alloy content are often not warranted. Fatigue failure is reduced moderately by increase in strength, and then only to a certain level before adverse effects in endurance limit and notch sensitivity begin to counteract the benefits of higher strength. A good practice is to start with an inexpensive, low or medium carbon steel for the first time through the design calculations. If strength considerations turn out to dominate over deflection, then a higher strength material should be tried, allowing the shaft sizes to be reduced until excess deflection becomes an issue. The cost of the material and its processing must be weighed against the need for smaller shaft diameters. When warranted, typical alloy steels for heat treatment include ANSI 1340-50, 3140-50, 4140, 4340, 5140, and 8650. Shafts usually don’t need to be surface hardened unless they serve as the actual journal of a bearing surface. Typical material choices for surface hardening include carburizing grades of ANSI 1020, 4320, 4820, and 8620. Cold drawn steel is usually used for diameters under about 3 inches. The nominal diameter of the bar can be left unmachined in areas that do not require fitting of components. Hot rolled steel should be machined all over. For large shafts requiring much material removal, the residual stresses may tend to cause warping. If concentricity is important, it may be necessary to rough machine, then heat treat to remove residual stresses and increase the strength, then finish machine to the final dimensions. In approaching material selection, the amount to be produced is a salient factor. For low production, turning is the usual primary shaping process. An economic viewpoint may require removing the least material. High production may permit a volumeconservative shaping method (hot or cold forming, casting), and minimum material in the shaft can become a design goal. Cast iron may be specified if the production quantity is high, and the gears are to be integrally cast with the shaft. Properties of the shaft locally depend on its history—cold work, cold forming, rolling of fillet features, heat treatment, including quenching medium, agitation, and tempering regimen.1 Stainless steel may be appropriate for some environments. 7–3 Shaft Layout The general layout of a shaft to accommodate shaft elements, e.g. gears, bearings, and pulleys, must be specified early in the design process in order to perform a free body force analysis and to obtain shear-moment diagrams. The geometry of a shaft is generally that of a stepped cylinder. The use of shaft shoulders is an excellent means of axially locating the shaft elements and to carry any thrust loads. Figure 7–1 shows an example of a stepped shaft supporting the gear of a worm-gear speed reducer. Each shoulder in the shaft serves a specific purpose, which you should attempt to determine by observation. 1 See Joseph E. Shigley, Charles R. Mischke, and Thomas H. Brown, Jr. (eds-in-chief), Standard Handbook of Machine Design, 3rd ed., McGraw-Hill, New York, 2004. For cold-worked property prediction see Chap. 29, and for heat-treated property prediction see Chaps. 29 and 33

354 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft ©The McGraw-Hil Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition Shafts and Shaft Components 351 Axial Layout of Components The axial positioning of components is often dictated by the layout of the housing and other meshing components.In general,it is best to support load-carrying com- ponents between bearings,such as in Fig.7-2a,rather than cantilevered outboard of the bearings,such as in Fig.7-2c.Pulleys and sprockets often need to be mounted outboard for ease of installation of the belt or chain.The length of the cantilever should be kept short to minimize the deflection. Only two bearings should be used in most cases.For extremely long shafts carrying several load-bearing components,it may be necessary to provide more than two bearing supports.In this case,particular care must be given to the alignment of the bearings. Shafts should be kept short to minimize bending moments and deflections.Some axial space between components is desirable to allow for lubricant flow and to pro- vide access space for disassembly of components with a puller.Load bearing com- ponents should be placed near the bearings,again to minimize the bending moment at the locations that will likely have stress concentrations,and to minimize the deflec- tion at the load-carrying components. The components must be accurately located on the shaft to line up with other mating components,and provision must be made to securely hold the components in position.The primary means of locating the components is to position them against a shoulder of the shaft.A shoulder also provides a solid support to minimize deflec- tion and vibration of the component.Sometimes when the magnitudes of the forces are reasonably low,shoulders can be constructed with retaining rings in grooves, sleeves between components,or clamp-on collars.In cases where axial loads are very small,it may be feasible to do without the shoulders entirely,and rely on press fits. pins,or collars with setscrews to maintain an axial location.See Fig.7-2b and 7-2d for examples of some of these means of axial location. Supporting Axial Loads In cases where axial loads are not trivial,it is necessary to provide a means to trans- fer the axial loads into the shaft,then through a bearing to the ground.This will be particularly necessary with helical or bevel gears,or tapered roller bearings,as each of these produces axial force components.Often,the same means of providing axial location,e.g..shoulders,retaining rings,and pins,will be used to also transmit the axial load into the shaft. It is generally best to have only one bearing carry the axial load,to allow greater tolerances on shaft length dimensions,and to prevent binding if the shaft expands due to temperature changes.This is particularly important for long shafts. Figures 7-3 and 7-4 show examples of shafts with only one bearing carrying the axial load against a shoulder,while the other bearing is simply press-fit onto the shaft with no shoulder. Providing for Torque Transmission Most shafts serve to transmit torque from an input gear or pulley,through the shaft,to an output gear or pulley.Of course,the shaft itself must be sized to support the torsional stress and torsional deflection.It is also necessary to provide a means of transmitting the torque between the shaft and the gears.Common torque-transfer elements are: ·Keys ·Splines ·Setscrews

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components 354 © The McGraw−Hill Companies, 2008 Shafts and Shaft Components 351 Axial Layout of Components The axial positioning of components is often dictated by the layout of the housing and other meshing components. In general, it is best to support load-carrying components between bearings, such as in Fig. 7–2a, rather than cantilevered outboard of the bearings, such as in Fig. 7–2c. Pulleys and sprockets often need to be mounted outboard for ease of installation of the belt or chain. The length of the cantilever should be kept short to minimize the deflection. Only two bearings should be used in most cases. For extremely long shafts carrying several load-bearing components, it may be necessary to provide more than two bearing supports. In this case, particular care must be given to the alignment of the bearings. Shafts should be kept short to minimize bending moments and deflections. Some axial space between components is desirable to allow for lubricant flow and to provide access space for disassembly of components with a puller. Load bearing components should be placed near the bearings, again to minimize the bending moment at the locations that will likely have stress concentrations, and to minimize the deflection at the load-carrying components. The components must be accurately located on the shaft to line up with other mating components, and provision must be made to securely hold the components in position. The primary means of locating the components is to position them against a shoulder of the shaft. A shoulder also provides a solid support to minimize deflection and vibration of the component. Sometimes when the magnitudes of the forces are reasonably low, shoulders can be constructed with retaining rings in grooves, sleeves between components, or clamp-on collars. In cases where axial loads are very small, it may be feasible to do without the shoulders entirely, and rely on press fits, pins, or collars with setscrews to maintain an axial location. See Fig. 7–2b and 7–2d for examples of some of these means of axial location. Supporting Axial Loads In cases where axial loads are not trivial, it is necessary to provide a means to transfer the axial loads into the shaft, then through a bearing to the ground. This will be particularly necessary with helical or bevel gears, or tapered roller bearings, as each of these produces axial force components. Often, the same means of providing axial location, e.g., shoulders, retaining rings, and pins, will be used to also transmit the axial load into the shaft. It is generally best to have only one bearing carry the axial load, to allow greater tolerances on shaft length dimensions, and to prevent binding if the shaft expands due to temperature changes. This is particularly important for long shafts. Figures 7–3 and 7–4 show examples of shafts with only one bearing carrying the axial load against a shoulder, while the other bearing is simply press-fit onto the shaft with no shoulder. Providing for Torque Transmission Most shafts serve to transmit torque from an input gear or pulley, through the shaft, to an output gear or pulley. Of course, the shaft itself must be sized to support the torsional stress and torsional deflection. It is also necessary to provide a means of transmitting the torque between the shaft and the gears. Common torque-transfer elements are: • Keys • Splines • Setscrews

356 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft ©The McGraw-Hill Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition Shafts and Shaft Components 353 Splines are essentially stubby gear teeth formed on the outside of the shaft and on the inside of the hub of the load-transmitting component.Splines are generally much more expensive to manufacture than keys,and are usually not necessary for simple torque transmission.They are typically used to transfer high torques.One feature of a spline is that it can be made with a reasonably loose slip fit to allow for large axial motion between the shaft and component while still transmitting torque.This is use- ful for connecting two shafts where relative motion between them is common,such as in connecting a power takeoff(PTO)shaft of a tractor to an implement.SAE and ANSI publish standards for splines.Stress concentration factors are greatest where the spline ends and blends into the shaft,but are generally quite moderate. For cases of low torque transmission,various means of transmitting torque are available.These include pins,setscrews in hubs,tapered fits,and press fits. Press and shrink fits for securing hubs to shafts are used both for torque trans- fer and for preserving axial location.The resulting stress-concentration factor is usu- ally quite small.See Sec.7-8 for guidelines regarding appropriate sizing and toler- ancing to transmit torque with press and shrink fits.A similar method is to use a split hub with screws to clamp the hub to the shaft.This method allows for disassembly and lateral adjustments.Another similar method uses a two-part hub consisting of a split inner member that fits into a tapered hole.The assembly is then tightened to the shaft with screws.which forces the inner part into the wheel and clamps the whole assembly against the shaft. Tapered fits between the shaft and the shaft-mounted device,such as a wheel,are often used on the overhanging end of a shaft.Screw threads at the shaft end then permit the use of a nut to lock the wheel tightly to the shaft.This approach is useful because it can be disassembled,but it does not provide good axial location of the wheel on the shaft. At the early stages of the shaft layout,the important thing is to select an appro- priate means of transmitting torque,and to determine how it affects the overall shaft layout.It is necessary to know where the shaft discontinuities,such as keyways,holes, and splines,will be in order to determine critical locations for analysis. Assembly and Disassembly Consideration should be given to the method of assembling the components onto the shaft,and the shaft assembly into the frame.This generally requires the largest diam- eter in the center of the shaft,with progressively smaller diameters towards the ends to allow components to be slid on from the ends.If a shoulder is needed on both sides of a component,one of them must be created by such means as a retaining ring or by a sleeve between two components.The gearbox itself will need means to physi- cally position the shaft into its bearings,and the bearings into the frame.This is typ- ically accomplished by providing access through the housing to the bearing at one end of the shaft.See Figs.7-5 through 7-8 for examples. Figure 7-5 Arrangement showing bearing inner rings press-fitted to shoft while outer rings float in the 中加00 <00000加9 housing.The axial clearance should be sufficient only to allow for machinery vibrations. Note the labyrinth seal on the right

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components 356 © The McGraw−Hill Companies, 2008 Shafts and Shaft Components 353 Splines are essentially stubby gear teeth formed on the outside of the shaft and on the inside of the hub of the load-transmitting component. Splines are generally much more expensive to manufacture than keys, and are usually not necessary for simple torque transmission. They are typically used to transfer high torques. One feature of a spline is that it can be made with a reasonably loose slip fit to allow for large axial motion between the shaft and component while still transmitting torque. This is useful for connecting two shafts where relative motion between them is common, such as in connecting a power takeoff (PTO) shaft of a tractor to an implement. SAE and ANSI publish standards for splines. Stress concentration factors are greatest where the spline ends and blends into the shaft, but are generally quite moderate. For cases of low torque transmission, various means of transmitting torque are available. These include pins, setscrews in hubs, tapered fits, and press fits. Press and shrink fits for securing hubs to shafts are used both for torque transfer and for preserving axial location. The resulting stress-concentration factor is usually quite small. See Sec. 7–8 for guidelines regarding appropriate sizing and tolerancing to transmit torque with press and shrink fits. A similar method is to use a split hub with screws to clamp the hub to the shaft. This method allows for disassembly and lateral adjustments. Another similar method uses a two-part hub consisting of a split inner member that fits into a tapered hole. The assembly is then tightened to the shaft with screws, which forces the inner part into the wheel and clamps the whole assembly against the shaft. Tapered fits between the shaft and the shaft-mounted device, such as a wheel, are often used on the overhanging end of a shaft. Screw threads at the shaft end then permit the use of a nut to lock the wheel tightly to the shaft. This approach is useful because it can be disassembled, but it does not provide good axial location of the wheel on the shaft. At the early stages of the shaft layout, the important thing is to select an appropriate means of transmitting torque, and to determine how it affects the overall shaft layout. It is necessary to know where the shaft discontinuities, such as keyways, holes, and splines, will be in order to determine critical locations for analysis. Assembly and Disassembly Consideration should be given to the method of assembling the components onto the shaft, and the shaft assembly into the frame. This generally requires the largest diameter in the center of the shaft, with progressively smaller diameters towards the ends to allow components to be slid on from the ends. If a shoulder is needed on both sides of a component, one of them must be created by such means as a retaining ring or by a sleeve between two components. The gearbox itself will need means to physically position the shaft into its bearings, and the bearings into the frame. This is typically accomplished by providing access through the housing to the bearing at one end of the shaft. See Figs. 7–5 through 7–8 for examples. Figure 7–5 Arrangement showing bearing inner rings press-fitted to shaft while outer rings float in the housing. The axial clearance should be sufficient only to allow for machinery vibrations. Note the labyrinth seal on the right

Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft ©The McGra-Hfl 357 Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition 354 Mechanical Engineering Design Figure 7-6 Similar to the arrangement of Fig.7-5 except that the outer bearing rings are preloaded. 咖I0 Figure 7-7 In this arrangement the inner ring of the lefthand bearing is locked to the shaft between a nut and a shaft shoulder.The locknut and washer are AFBMA standard.The snap ring in the outer race is used to positively locate the shaft assembly in the axial direction.Note the floating righthand bearing and the grinding runout grooves in the shaft. Figure 7-8 This arrangement is similar to Fig.7-7 in that the left-hand When components are to be press-fit to the shaft,the shaft should be designed bearing positions the entire so that it is not necessary to press the component down a long length of shaft.This shaft assembly.In this case the inner ring is secured to may require an extra change in diameter,but it will reduce manufacturing and assem- the shaft using a snap ring. bly cost by only requiring the close tolerance for a short length. Note the use of a shield to Consideration should also be given to the necessity of disassembling the compo- prevent dirt generated nents from the shaft.This requires consideration of issues such as accessibility of from within the machine from retaining rings,space for pullers to access bearings.openings in the housing to allow entering the bearing. pressing the shaft or bearings out,etc. 7-4 Shaft Design for Stress Critical Locations It is not necessary to evaluate the stresses in a shaft at every point;a few potentially critical locations will suffice.Critical locations will usually be on the outer surface. at axial locations where the bending moment is large,where the torque is present,and where stress concentrations exist.By direct comparison of various points along the shaft,a few critical locations can be identified upon which to base the design.An assessment of typical stress situations will help

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components © The McGraw−Hill 357 Companies, 2008 354 Mechanical Engineering Design Figure 7–6 Similar to the arrangement of Fig. 7--5 except that the outer bearing rings are preloaded. Figure 7–7 In this arrangement the inner ring of the left-hand bearing is locked to the shaft between a nut and a shaft shoulder. The locknut and washer are AFBMA standard. The snap ring in the outer race is used to positively locate the shaft assembly in the axial direction. Note the floating right-hand bearing and the grinding runout grooves in the shaft. When components are to be press-fit to the shaft, the shaft should be designed so that it is not necessary to press the component down a long length of shaft. This may require an extra change in diameter, but it will reduce manufacturing and assembly cost by only requiring the close tolerance for a short length. Consideration should also be given to the necessity of disassembling the components from the shaft. This requires consideration of issues such as accessibility of retaining rings, space for pullers to access bearings, openings in the housing to allow pressing the shaft or bearings out, etc. 7–4 Shaft Design for Stress Critical Locations It is not necessary to evaluate the stresses in a shaft at every point; a few potentially critical locations will suffice. Critical locations will usually be on the outer surface, at axial locations where the bending moment is large, where the torque is present, and where stress concentrations exist. By direct comparison of various points along the shaft, a few critical locations can be identified upon which to base the design. An assessment of typical stress situations will help. Figure 7–8 This arrangement is similar to Fig. 7--7 in that the left-hand bearing positions the entire shaft assembly. In this case the inner ring is secured to the shaft using a snap ring. Note the use of a shield to prevent dirt generated from within the machine from entering the bearing

358 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 7.Shafts and Shaft T©The McGraw-Hil Mechanical Engineering Elements Components Companies,2008 Design,Eighth Edition Shafts and Shaft Components 355 Most shafts will transmit torque through a portion of the shaft.Typically the torque comes into the shaft at one gear and leaves the shaft at another gear.A free body diagram of the shaft will allow the torque at any section to be determined.The torque is often relatively constant at steady state operation.The shear stress due to the torsion will be greatest on outer surfaces. The bending moments on a shaft can be determined by shear and bending moment diagrams.Since most shaft problems incorporate gears or pulleys that intro- duce forces in two planes,the shear and bending moment diagrams will generally be needed in two planes.Resultant moments are obtained by summing moments as vectors at points of interest along the shaft.The phase angle of the moments is not important since the shaft rotates.A steady bending moment will produce a com- pletely reversed moment on a rotating shaft,as a specific stress element will alter- nate from compression to tension in every revolution of the shaft.The normal stress due to bending moments will be greatest on the outer surfaces.In situations where a bearing is located at the end of the shaft,stresses near the bearing are often not critical since the bending moment is small. Axial stresses on shafts due to the axial components transmitted through heli- cal gears or tapered roller bearings will almost always be negligibly small compared to the bending moment stress.They are often also constant,so they contribute lit- tle to fatigue.Consequently,it is usually acceptable to neglect the axial stresses induced by the gears and bearings when bending is present in a shaft.If an axial load is applied to the shaft in some other way,it is not safe to assume it is negli- gible without checking magnitudes. Shaft Stresses Bending,torsion,and axial stresses may be present in both midrange and alternating components.For analysis,it is simple enough to combine the different types of stresses into alternating and midrange von Mises stresses,as shown in Sec.6-14, p.309.It is sometimes convenient to customize the equations specifically for shaft applications.Axial loads are usually comparatively very small at critical locations where bending and torsion dominate,so they will be left out of the following equa- tions.The fluctuating stresses due to bending and torsion are given by Mac 0a=K I Mmc Om=Kj 1 7-1) Tac ta =Kfs Tmc tm=K: (7-2) where Mm and Ma are the midrange and alternating bending moments,T and Ta are the midrange and alternating torques,and Kf and Kfs are the fatigue stress concen- tration factors for bending and torsion,respectively. Assuming a solid shaft with round cross section,appropriate geometry terms can be introduced for c,I,and J resulting in 32Mo On=Kt πd3 Om =Kt 2Mmm nd3 (7-31 16T ta=Kfsπd 16Tm tm=K:Td形 (7-4

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components 358 © The McGraw−Hill Companies, 2008 Shafts and Shaft Components 355 Most shafts will transmit torque through a portion of the shaft. Typically the torque comes into the shaft at one gear and leaves the shaft at another gear. A free body diagram of the shaft will allow the torque at any section to be determined. The torque is often relatively constant at steady state operation. The shear stress due to the torsion will be greatest on outer surfaces. The bending moments on a shaft can be determined by shear and bending moment diagrams. Since most shaft problems incorporate gears or pulleys that introduce forces in two planes, the shear and bending moment diagrams will generally be needed in two planes. Resultant moments are obtained by summing moments as vectors at points of interest along the shaft. The phase angle of the moments is not important since the shaft rotates. A steady bending moment will produce a completely reversed moment on a rotating shaft, as a specific stress element will alternate from compression to tension in every revolution of the shaft. The normal stress due to bending moments will be greatest on the outer surfaces. In situations where a bearing is located at the end of the shaft, stresses near the bearing are often not critical since the bending moment is small. Axial stresses on shafts due to the axial components transmitted through helical gears or tapered roller bearings will almost always be negligibly small compared to the bending moment stress. They are often also constant, so they contribute little to fatigue. Consequently, it is usually acceptable to neglect the axial stresses induced by the gears and bearings when bending is present in a shaft. If an axial load is applied to the shaft in some other way, it is not safe to assume it is negligible without checking magnitudes. Shaft Stresses Bending, torsion, and axial stresses may be present in both midrange and alternating components. For analysis, it is simple enough to combine the different types of stresses into alternating and midrange von Mises stresses, as shown in Sec. 6–14, p. 309. It is sometimes convenient to customize the equations specifically for shaft applications. Axial loads are usually comparatively very small at critical locations where bending and torsion dominate, so they will be left out of the following equations. The fluctuating stresses due to bending and torsion are given by σa = Kf Mac I σm = Kf Mmc I (7–1) τa = Kf s Tac J τm = Kf s Tmc J (7–2) where Mm and Ma are the midrange and alternating bending moments, Tm and Ta are the midrange and alternating torques, and Kf and Kf s are the fatigue stress concentration factors for bending and torsion, respectively. Assuming a solid shaft with round cross section, appropriate geometry terms can be introduced for c, I, and J resulting in σa = Kf 32Ma πd3 σm = Kf 32Mm πd3 (7–3) τa = Kf s 16Ta πd3 τm = Kf s 16Tm πd3 (7–4)

Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 7. Shafts and Shaft Components © The McGraw−Hill 359 Companies, 2008 356 Mechanical Engineering Design Combining these stresses in accordance with the distortion energy failure theory, the von Mises stresses for rotating round, solid shafts, neglecting axial loads, are given by σ a = (σ2 a + 3τ 2 a ) 1/2 = 32Kf Ma πd3 2 + 3 16Kf sTa πd3 2 1/2 (7–5) σ m = (σ2 m + 3τ 2 m) 1/2 = 32Kf Mm πd3 2 + 3 16Kf sTm πd3 2 1/2 (7–6) Note that the stress concentration factors are sometimes considered optional for the midrange components with ductile materials, because of the capacity of the ductile material to yield locally at the discontinuity. These equivalent alternating and midrange stresses can be evaluated using an appropriate failure curve on the modified Goodman diagram (See Sec. 6–12, p. 295, and Fig. 6–27). For example, the fatigue failure criteria for the modified Goodman line as expressed previously in Eq. (6–46) is 1 n = σ a Se + σ m Sut Substitution of σ a and σ m from Eqs. (7–5) and (7–6) results in 1 n = 16 πd3 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 For design purposes, it is also desirable to solve the equation for the diameter. This results in d = 16n π 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 1/3 Similar expressions can be obtained for any of the common failure criteria by substituting the von Mises stresses from Eqs. (7–5) and (7–6) into any of the failure criteria expressed by Eqs. (6–45) through (6–48), p. 298. The resulting equations for several of the commonly used failure curves are summarized below. The names given to each set of equations identifies the significant failure theory, followed by a fatigue failure locus name. For example, DE-Gerber indicates the stresses are combined using the distortion energy (DE) theory, and the Gerber criteria is used for the fatigue failure. DE-Goodman 1 n = 16 πd3 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 (7–7) d = 16n π 1 Se 4(Kf Ma) 2 + 3(Kf sTa) 21/2 + 1 Sut 4(Kf Mm) 2 + 3(Kf sTm) 21/2 1/3 (7–8)