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Machine Elements · Chapter 6 of 10 · Advanced

Shafts and Shaft Components

A rotating shaft sees steady torque but fully reversed bending every turn, so it is a fatigue problem first. Sizing one brings together everything from the previous five chapters.

Readiness check

This chapter assembles the earlier tools into a shaft design. Tick only what you can do closed-notes.

Find bending moment and torque on a shaft.
Recall the endurance limit and the Goodman idea.
Use stress-concentration factors K_f and K_fs.
Compute a deflection from a load and stiffness.
Rearrange an equation to solve for a diameter.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit fatigue and the Goodman line in Chapter 5.

3 or more weak itemsReview torsion in Mechanics of Materials, Chapter 3.

The core idea

On a rotating shaft the bending stress reverses every revolution while the torque holds steady. A single equation combines these into a diameter that meets a fatigue factor of safety.

M_a reversed, T_m steadyd = {(16n/π)[2K_fM_a/S_e + √3 K_fsT_m/S_ut]}^1/3

Because the shaft spins, a fixed bending load becomes a fully reversed alternating stress, so M_a is the alternating moment and its mean is zero. A constant drive torque is a steady mean, so T_m carries no alternating part. The distortion-energy Goodman shaft equation folds the stress-concentration factors, the endurance limit, and the ultimate strength into one expression for the required diameter at the critical section. Deflection and critical speed are then checked separately.

The skill works when: you assign bending to the alternating term and torque to the mean term, then size with the shaft equation.

The skill breaks down when: reversed bending is treated as steady, or the running speed nears the critical speed.

The concept. A shaft on two bearings carries a gear load (bending) and transmits torque. As it spins, the bending stress at any fibre fully reverses, while the torque stays steady, the signature of a shaft fatigue problem.

The skills, taught in order

Five skills lay out the shaft, classify its loads, size it for fatigue, and check deflection and critical speed.

6.1 Shaft layout and loads

Gears and pulleys put transverse loads and torque onto a shaft supported by two bearings. A bending-moment diagram locates the critical section, usually at a shoulder or keyway where both the moment and a stress concentration are large.

6.2 Reversed bending and steady torque

On a rotating shaft, a stationary bending load produces a fully reversed stress, so M_a is alternating with zero mean. A constant transmitted torque is a steady mean T_m with no alternating part. Keeping this split straight is the key to the shaft equation.

Load	Character on a rotating shaft	Component
Bending moment	fully reversed each revolution	alternating M_a
Torque	steady drive torque	mean T_m
Axial	usually small	often neglected

6.3 The DE-Goodman shaft equation

Combining von Mises stresses with the Goodman line gives the diameter directly: d = {(16n/π)[2K_fM_a/S_e + √3 K_fsT_m/S_ut]}^1/3. The fatigue stress-concentration factors K_f (bending) and K_fs (torsion) act on their respective terms.

6.4 Deflection and critical speed

A shaft must also be stiff: too much deflection misaligns gears and bearings. And every shaft has a critical speed where it resonates in bending; the first critical speed follows ω = √(g/δ) from the static deflection δ. Running speed should stay well clear of it.

6.5 Shaft components

Keys, shoulders, retaining rings, and press fits locate and drive the parts on a shaft. Each adds a stress concentration, so the critical section is usually where a feature and a high moment coincide. Generous fillet radii reduce K_f and buy fatigue life cheaply.

Engineering connection: the gears of Chapter 10 and the bearings of Chapter 9 mount on the shaft sized here, and their loads set its bending moment.

Worked example 1: sizing a shaft for fatigue

A rotating shaft carries a fully reversed bending moment M_a = 150 N·m and a steady torque T_m = 120 N·m at a shoulder. With S_e = 200 MPa, S_ut = 690 MPa, K_f = 1.7, K_fs = 1.5, and a target factor of safety n = 2, find the required diameter by the DE-Goodman equation.

Figure 1. At the shoulder the reversed bending and steady torque combine through the DE-Goodman equation, with the concentration factors raising the effective stress, to set the diameter.

ProblemFind the diameter for the shaft section in Figure 1 to reach n = 2.
Given / findM_a = 150 N·m = 150 000 N·mm, T_m = 120 N·m = 120 000 N·mm, S_e = 200 MPa, S_ut = 690 MPa, K_f = 1.7, K_fs = 1.5, n = 2. Find d.
AssumptionsRotating shaft, so bending is fully reversed and torque is steady; DE-Goodman criterion; critical section at the shoulder.
ModelUse the DE-Goodman diameter equation with the bending term on S_e and the torque term on S_ut.
Equationsd = {(16n/π)[2K_fM_a/S_e + √3 K_fsT_m/S_ut]}^1/3
SolveBending term: 2(1.7)(150 000)/200 = 2550. Torque term: √3(1.5)(120 000)/690 = 452. Sum = 3002. Then d³ = (16·2/π)(3002) = 10.19 × 3002 = 30 580, so d = 31.3 mm → 32 mm.
CheckThe bending term dominates because reversed bending is the fatigue driver and S_e is much smaller than S_ut. Rounding up to 32 mm keeps the factor of safety at or above 2.
ConclusionOne equation turns the fatigue work of Chapter 5 into a diameter. The shoulder fillet should then be generous to justify the assumed K_f.

Result. Required d = 31.3 mm; choose 32 mm.

Worked example 2: the first critical speed

A shaft carrying a rotor deflects 0.50 mm under the rotor's static weight at midspan. Estimate the first critical speed in rad/s and rev/min.

Figure 2. The static deflection under the rotor weight sets the first critical speed: the more the shaft sags, the lower the speed at which it resonates.

ProblemFind the first critical speed for the shaft in Figure 2.
Given / findδ = 0.50 mm = 0.0005 m, g = 9.81 m/s². Find ω_c and N_c.
AssumptionsSingle concentrated mass; shaft mass negligible; the static deflection sets the stiffness.
ModelThe first critical speed is ω_c = √(g/δ); convert to rev/min with N = 60ω/2π.
Equationsω_c = √(g/δ)N_c = 60 ω_c/(2π)
Solveω_c = √(9.81/0.0005) = √19 620 = 140 rad/s. N_c = 60 × 140/(2π) = 1338 rev/min.
CheckA smaller deflection (a stiffer shaft) would raise the critical speed. The operating speed should stay below about 0.75 N_c or above about 1.4 N_c to avoid resonance.
ConclusionStiffness sets the critical speed, so a shaft sized only for stress must still be checked for resonance before the design is final.

Result. ω_c = 140 rad/s, so N_c = 1338 rev/min.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Treating bending as steady	Shaft sized as a static part	"Does the shaft rotate under this bending load?"	A rotating shaft sees fully reversed bending: M_a, not M_m.
Swapping the strength in each term	Bending divided by S_ut	"Which term is alternating, which is mean?"	Alternating bending uses S_e; mean torque uses S_ut.
Skipping critical speed	Shaft resonates at running speed	"Is the operating speed near N_c?"	Check ω = √(g/δ) and keep clear of it.
Sharp shoulder fillet	K_f higher than assumed	"Is the fillet radius generous enough?"	Use a larger radius to match the assumed K_f.

Practice ladder

Level 1 · Direct skill

A solid shaft of 30 mm diameter carries a steady torque of 200 N·m. Find the surface shear stress.

Show answer

τ = 16T/πd³ = 16(200 000)/(π·30³) = 3 200 000/84 823 = 37.7 MPa.

Level 2 · Mixed concept

For the shaft of Worked Example 1, the target factor of safety rises to n = 3. Roughly what diameter is needed?

Show answer

d scales with n^1/3: d = 31.3 × (3/2)^1/3 = 31.3 × 1.145 = 35.8 mm, so choose 36 mm. A 50 percent larger factor of safety needs only about 15 percent more diameter.

Level 3 · Independent problem

A shaft deflects 0.20 mm under its rotor. Find the first critical speed in rev/min.

Show answer

ω_c = √(9.81/0.0002) = √49 050 = 221.5 rad/s. N_c = 60 × 221.5/(2π) = 2115 rev/min. Less sag means a higher critical speed.

Level 4 · Transfer to real engineering

Pick a real shaft (a car axle, a fan shaft, a drill spindle). Identify where it would fail in fatigue and whether its running speed is near a critical speed.

What good work looks like

A critical section at a shoulder or keyway, the recognition that bending reverses each turn, and a sense of whether the operating speed sits safely below the first critical speed.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I put bending in the alternating term and torque in the mean term."

"Give me three shaft load cases; I will identify the critical section."

"What diameter do I need?" Setting up and solving the shaft equation is the skill.

"Is the speed safe?" Computing the critical speed and comparing is the point.

Portfolio task

Size a shaft section for a real drive: find its bending moment and torque, apply the DE-Goodman equation, choose a standard diameter, and check the first critical speed.

Must include: the alternating and mean loads, a DE-Goodman diameter, and a critical-speed check.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Why is bending alternating on a rotating shaft?

Each fibre moves from tension to compression and back every revolution, fully reversing the stress.

2. Which strength goes with each shaft term?

The alternating bending term uses S_e; the mean torque term uses S_ut.

3. Write the first critical speed.

ω_c = √(g/δ), from the static deflection δ.

4. Where is the critical section usually?

At a shoulder or keyway where a high moment and a stress concentration coincide.

5. How does diameter scale with the factor of safety?

As n^1/3, so large safety gains need only modest size increases.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the shaft diameter and critical speed from a blank page.

+3 daysSize a second shaft section.

+7 daysCarry the shaft into fasteners and joints, Chapter 7.

+30 daysMount the gears and bearings of Chapters 9 and 10 on this shaft.

Textbook mapping

Item	Mapping
Primary source	Budynas and Nisbett, Shigley's Mechanical Engineering Design, Chapter 7 (Shafts and Shaft Components)
Cross-reference	Norton, Ch. 9 · Dynamics, Ch. 10 (vibration)
Core topics	6.1 Shaft layout · 6.2 Reversed bending and steady torque · 6.3 DE-Goodman equation · 6.4 Deflection and critical speed · 6.5 Shaft components
Engineering connection	The shaft carries the gears and bearings sized in later chapters.
Read next	Chapter 7: Screws, Fasteners, and Joints.