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1. Chapter 1 Shafts MWU Collage of Engineering Mechanical Engineering Department By Abebe A.

2. 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 rotating elements such as wheels, pulleys,… and carries no torque. A “spindle” is a short shaft. Terms such as line shaft, head shaft, stub shaft, transmission shaft, countershaft, and flexible shaft are names associated with special usage. 1-1 Introduction

3. Stub shaft- A shaft that is integral with a motor, engine or prime mover and is of a size, shape, and projection as to permit easy connection to other shafts . Line shaft- A shaft connected to a prime mover and used to transmit power to one or several machines Jackshaft- (Sometimes called countershaft ). A short shaft that connects a prime mover with a line shaft or a machine Flexible shaft- A connector which permits transmission of motion between two members whose axes are at an angle with each other 1-1 Introduction

4. Considerations for Shaft Design 1. Stress and Strength (a) Static Strength (b) Fatigue Strength (c) Reliability 2. Deflection and Rigidity (a) Bending deflection (b) Torsional deflection (c) Slope at bearings and shaft supported elements 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 .

5. 1.2 Shaft Materials Deflection is not affected by strength(σ or τ), but rather by stiffness(EI) 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. 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 General Considerations for material selection : It should have high strength. It should have good machinability. It should have low notch sensitivity factor . It should have good heat treatment properties. It should have high wear resistant properties. When a shaft of high strength is required , then an alloy steel such as nickel, nickel-chromium or chrome-vanadium steel is used.

6. Shaft Materials 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 . For 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 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 .

7. 3. 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 Keys Splines Setscrews Pins Press or shrink fits Tapered fits Many of these devices are designed to fail if the torque exceeds acceptable operating limits, protecting more expensive components.

8. The design of a shaft involves the study of Stress and strength analyses: Static and Fatigue Deflection and rigidity Critical Speed Strength Constraints The following stresses are induced in the shafts : 1. Shear stresses due to the transmission of torque ( i.e. due to torsional load). 2. Bending stresses (tensile or compressive) due to the forces acting upon machine elements like gears, pulleys etc. as well as due to the weight of the shaft itself. 3. Stresses due to combined torsional and bending loads.

9. Design of Shafts The shafts may be designed on the basis of 1. Strength, and 2. Rigidity and stiffness . In designing shafts on the basis of strength, the following cases may be considered : ( a ) Shafts subjected to twisting moment or torque only, ( b ) Shafts subjected to bending moment only, ( c ) Shafts subjected to combined twisting and bending moments, and ( d ) Shafts subjected to axial loads in addition to combined torsional and bending loads Shafts Subjected to Twisting Moment Only When the shaft is subjected to a twisting moment (or torque) only, then the diameter of the shaft may be obtained by using the torsion equation. We know that T = Twisting moment (or torque) acting upon the shaft, J = Polar moment of inertia of the shaft about the axis of rotation,  = Torsional shear stress, and r = Distance from neutral axis to the outer most fiber = d / 2; where d is the diameter of the shaft.

10. Shafts Subjected to Bending Moment Only When the shaft is subjected to a bending moment only, then the maximum stress (tensile or compressive ) is given by the bending equation. We know that M = Bending moment, I = Moment of inertia of cross-sectional area of the shaft about the axis of rotation, = Bending stress, and y = Distance from neutral axis to the outer-most fibre .   We know that for a round solid shaft, moment of inertia Substituting these values in equation We also know that for a hollow shaft, moment of inertia, Substituting

11. Shafts Subjected to Combined Twisting Moment and Bending Moment It must be designed on the basis of the two moments simultaneously. Various theories have been suggested to account for the elastic failure of the materials when they are subjected to various types of combined stresses. Maximum shear stress theory or Guest's theory . used for ductile materials ( mild steel). Maximum normal stress theory or Rankine’s theory . used for brittle materials ( cast iron) When the shaft is subjected to combination of loads, the principal stress and principal shear stress are obtained by constructing Mohr’s circle as shown in Fig. below. The normal stress is denoted by while the shear stress, by . We will consider two cases for calculating the value of .   Mohr’s Circle Case I In this case, the shaft is subjected to a combination of axial force, bending moment and torsional moment Case II In this case, the shaft is subjected to a combination of bending and torsional moments without any axial force.

12. The Mohr’s circle is constructed by the following steps: Select the origin O . Plot the following points : Join . The point of intersection of and is E . Construct Mohr’s circle with E as center and as radius .   The principal stress is given by,   The principal shear stress is given by   ( i ) Maximum Principal Stress Theory The maximum principal stress is . Since the shaft is subjected to bending and torsional moments without any axial force   Substituting or (ii) Maximum Shear Stress Theory The principal shear stress is   Shafts Subjected to Combined Twisting Moment and Bending Moment symbollicaly  

13. Shafts Subjected to Combined Twisting Moment and Bending Moment The expression is known as equivalent twisting moment and is denoted be . The equivalent twisting moment may be defined as that twisting moment, which when acting alone, produces the same shear stress ( ) as the actual twisting moment. By limiting the maximum shear stress ( ) equal to the allowable shear stress ( ) for the material   e expression is known as equivalent bending moment and is denoted by . The equivalent bending moment may be defined as that moment which when acting alone produces the same tensile or compressive stress ( ) as the actual bending mom   In case of a hollow shaft, the equations may be written as

14. Shafts Subjected to Fluctuating Loads(fatigue load) In the previous articles we have assumed that the shaft is subjected to constant torque and bending moment . But in actual practice , the shafts are subjected to fluctuating torque and bending moments . In order to design such shafts like line shafts and counter shafts, the combined shock and fatigue factors must be taken into account for the computed twisting moment ( T ) and bending moment ( M ). Thus for a shaft subjected to combined bending and torsion, the equivalent twisting moment and equivalent bending moment respectively = Combined shock and fatigue factor for bending, and = Combined shock and fatigue factor for torsion.  

15. Shafts Subjected to Axial Load in addition to Combined Torsion and Bending Loads When the shaft is subjected to an axial load ( F ) in addition to torsion and bending loads as in propeller shafts of ships and shafts for driving worm gears, then the stress due to axial load must be added to the bending stress ( ). We know that bending equation is   stress due to axial load Resultant stress (tensile or compressive) for solid shaft,

16. In case of a hollow shaft, the resultant stress Shafts Subjected to Axial Load in addition to Combined Torsion and Bending Loads In case of long shafts (slender shafts) subjected to compressive loads, a factor known as column factor (α) must be introduced to take the column effect into account . ∴ Stress due to the compressive load, The value of column factor (α) for compressive loads may be obtained from the following relation : This expression is used when the slenderness ratio ( L / K ) is less than 115. When the slenderness ratio ( L / K ) is more than 115, then the value of column factor may be obtained from the following relation :

17. In general, for a hollow shaft subjected to fluctuating torsional and bending load, along with an axial load, the equations for equivalent twisting moment ( Te ) and equivalent bending moment ( Me ) may be written as Shafts Subjected to Axial Load in addition to Combined Torsion and Bending Loads It may be noted that for a solid shaft, k = 0 and . When the shaft carries no axial load, then F = 0 and when the shaft carries axial tensile load, then α = 1.  

18. Design of Shafts on the basis of Rigidity In some applications, the shafts are designed on the basis of either torsional rigidity or lateral rigidity . A transmission shaft is said to be rigid on the basis of torsional rigidity, if it does not twist too much under the action of an external torque. Similarly , the transmission shaft is said to be rigid on the basis of lateral rigidity, if it does not deflect too much under the action of external forces and bending moment Torsional rigidity The torsional deflection may be obtained by using the torsion equation

19. Lateral rigidity: It is important in case of transmission shafting and shafts running at high speed , where small lateral deflection would cause huge out-of-balance forces. The lateral rigidity is also important for maintaining proper bearing clearances and for correct gear teeth alignment. If the shaft is of uniform cross-section, then the lateral deflection of a shaft may be obtained by using the deflection formulae as in Strength of Materials. But when the shaft is of variable cross-section, then the lateral deflection may be determined from the fundamental equation for the elastic curve of a beam , i.e . Design of Shafts on the basis of Rigidity