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 (7-4Budynas−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 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 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 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 σa = Kf 32Ma πd3 σm = Kf 32Mm πd3 (7–3) τa = Kf s 16Ta πd3 τm = Kf s 16Tm πd3 (7–4)