Chapter 18 Shafts and Axles Dr. A. Aziz Bazoune King Fahd University of Petroleum & Minerals Mechanical Engineering Department

Chapter Outline 18-1 Introduction ……….922 18-2 Geometric Constraints ……….927 18-3 Strength Constraints ……….933 18-4 Strength Constraints – Additional Methods ……….940 18-5 Shaft Materials ……….944 18-6 Hollow Shafts ……….944 18-7 Critical Speeds (Omitted) ……….945 18-8 Shaft Design ……….950

LECTURE 29 18-1 Introduction ……….922 18-2 Geometric Constraints ……….927 18-3 Strength Constraints ……….933

18-1 Introduction In machinery, the general term “shaft” refers to a member, usually of circular cross-section, which supports gears, sprockets, wheels, rotors, etc., and which is subjected to torsion and to transverse or axial loads acting singly or in combination. An “axle” is a non-rotating member that supports wheels, pulleys,… and carries no torque. A “spindle” is a short shaft. Terms such as lineshaft, headshaft, stub shaft, transmission shaft, countershaft, and flexible shaft are names associated with special usage.

Considerations for Shaft Design Deflection and Rigidity (a) Bending deflection (b) Torsional deflection (c) Slope at bearings and shaft supported elements (d) Shear deflection due to transverse loading of shorter shafts Stress and Strength (a) Static Strength (b) Fatigue Strength (c) Reliability

Considerations for Shaft Design The geometry of a shaft is that of a stepped cylinder bending. Gears, bearings, and pulleys must always be accurately positioned Common Torque Transfer Elements Keys Splines Setscrews Pins Press or shrink fits Tapered fits

Common Types of Shaft Keys.

Common Types of Shaft Keys.

Common Types of Shaft Pins.

Common Types of Shaft Pins.

Common Types of Retaining or Snap Rings.

Common Types of Splines.

Rigid Shaft Coupling.

Figure 18-2 Choose a shaft configuration to support and locate the two gears and two bearings. (b) Solution uses an integral pinion, three shaft shoulders, key and keyway, and sleeve. The housing locates the bearings on their outer rings and receives the thrust loads. (c) Choose fanshaft configuration. (d) Solution uses sleeve bearings, a straight-through shaft, locating collars, and setscrews for collars, fan pulley, and fan itself. The fan housing supports the sleeve bearings.

18-3 Strength Constraints The design of a shaft involves the study of Stress and strength analyses: Static and Fatigue Deflection and rigidity Critical Speed

Static or Quasi-Static Loading on Shaft

Static or Quasi-Static Loading on Shaft The stress at an element located on the surface of a solid round shaft of diameter d subjected to bending, axial loading, and twisting is Normal stress Shear stress Non-zero principal stresses

Static or Quasi-Static Loading on Shaft Von Mises stress Maximum Shear Stress Theory

Static or Quasi-Static Loading on Shaft Under many conditions, the axial force F in Eqs. (6-37) and (6-38) is either zero or so small that its effect may be neglected. With F = 0, Eqs. (6-37) and (6-38) become Von Mises stress (6-41) Maximum Shear Stress Theory (6-42)

Static or Quasi-Static Loading on Shaft Substitution of the allowable stresses from Eqs. 6-39 and 6-40 we find (6-43) Von Mises stress (6-44) (6-45) Maximum Shear Stress Theory (6-46)

Fatigue Strength 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. 7–14, p. 361. 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

Fatigue Strength The fluctuating stresses due to bending and torsion are given by where Mm and Ma are the midrange and alternating bending moments, Tm and Ta are the midrange and alternating torques, and Kf and Kfs 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

Assuming a solid shaft with round cross section, appropriate geometry terms can be introduced for c, I, and J resulting in

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