1. What is a Shaft ?

A shaft is a rotating machine element used to transmit power from one part of a machine to another. It supports rotating components like gears, pulleys, flywheels, and sprockets.

Key idea:
A shaft must safely withstand torsion (twisting) and bending loads.

2. Functions of Shaft

• Transmit power and motion
• Support rotating machine elements
• Maintain proper alignment
• Withstand torsional and bending stresses

3. Types of Shafts

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(a) Transmission Shafts

• Used to transmit power between source and machine
• Examples: Line shaft, countershaft

(b) Machine Shafts

• Integral part of machines
• Examples: Crankshaft, camshaft

(c) Axles

• Support rotating elements
• Do not transmit torque

4. Materials for Shafts

Common materials used:

• Mild Steel (low cost, good strength)
• Alloy Steel (high strength, fatigue resistance)
• Stainless Steel (corrosion resistance)

👉 Properties required:

• High strength
• Good fatigue resistance
• Toughness
• Machinability

5. Loads Acting on Shaft

A shaft is subjected to:

(a) Torsional Load

• Due to power transmission
• Causes shear stress

(b) Bending Load

• Due to pulleys, gears, weights
• Causes bending stress

(c) Axial Load (sometimes)

• Due to thrust forces

6. Design Considerations

While designing a shaft, consider:

• Strength
• Rigidity (deflection and twist limits)
• Fatigue loading
• Stress concentration (keyways, shoulders)
• Critical speed
• Wear and corrosion

7. Torsion Equation

The basic torsion relation is:

$\frac{T}{J} = \frac{\tau}{R} = \frac{G\theta}{L}$JT​=Rτ​=LGθ​

Where:

• $T$T = Torque
• $J$J = Polar moment of inertia
• $\tau$τ = Shear stress
• $R$R = Radius
• $G$G = Modulus of rigidity
• $\theta$θ = Angle of twist
• $L$L = Length of shaft

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8. Design of Solid Shaft

For a solid circular shaft:$$T = \frac{\pi}{16} \tau d^3$$T=16π​τd3

👉 Diameter of shaft:$$d = \left(\frac{16T}{\pi \tau}\right)^{1/3}$$d=(πτ16T​)1/3

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9. Design of Hollow Shaft

For a hollow shaft:$$T = \frac{\pi}{16} \tau \frac{(D^4 – d^4)}{D}$$T=16π​τD(D4−d4)​

Where:

• $D$D = Outer diameter
• $d$d = Inner diameter

👉 Hollow shafts are lighter and more efficient.

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10. Combined Bending and Torsion

When both loads act:

Equivalent Torque:

$$T_e = \sqrt{T^2 + M^2}$$Te​=T2+M2​

Equivalent Bending Moment:

$$M_e = \frac{1}{2} \left[M + \sqrt{M^2 + T^2}\right]$$Me​=21​[M+M2+T2​]

Where:

• $M$M = Bending moment
• $T$T = Torque

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11. Shaft Design Based on Strength

• Use maximum shear stress theory (Guest theory)
• Ensure:

$$\tau_{max} \leq \tau_{allowable}$$τmax​≤τallowable​

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12. Shaft Design Based on Rigidity

Limit angle of twist:$$\theta = \frac{TL}{GJ}$$θ=GJTL​

👉 Typical limits:

• Transmission shaft: ≤ 1° per 20 diameters

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13. Stress Concentration

Occurs at:

• Keyways
• Shoulders
• Grooves

👉 Reduced by:

• Fillets
• Smooth transitions
• Proper design

14. Shaft Couplings and Keys

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• Keys: Connect shaft to rotating elements
• Couplings: Join two shafts

15. Advantages of Proper Shaft Design

• Safe power transmission
• Longer service life
• Reduced failure
• Efficient operation

16. Common Failures of Shafts

• Fatigue failure
• Torsional shear failure
• Bending failure
• Wear and corrosion

17. Applications

• Automobiles (crankshaft, drive shaft)
• Machine tools
• Power transmission systems
• Industrial machinery

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