Mechanical Engineering Design · Part 3 — Design of Mechanical Elements

Shafts and Shaft Components

Applying the fatigue and deflection tools of earlier chapters to the rotating shaft: material choice and layout, the distortion-energy design equations for combined bending and torsion (Goodman, Gerber, ASME-elliptic, Soderberg), deflection and slope limits, critical-speed estimation by Rayleigh and Dunkerley, keys, pins, retaining rings, and standardized limits and fits.

7–1Introduction

Learning objectives

Distinguish a torque-transmitting rotating shaft from a nonrotating axle and identify likely failure modes.
Construct shaft free-body diagrams in two transverse planes and obtain resultant bending moments and internal torque.
Select static, fatigue, stiffness, and critical-speed equations consistent with the load history.
Size a trial shaft section, refine local concentration factors, and verify yielding, deflection, and dynamics.
Specify component retention and fits using traceable material, tolerance, bearing, and manufacturing data.

Shaft as a load path

A shaft transmits torque while locating gears, pulleys, couplings, or bearings. An axle primarily supports transverse load without transmitting torque. Begin with a free-body diagram of the whole shaft: resolve belt or gear forces into orthogonal planes, solve bearing reactions, then cut sections to obtain \(M_y(x)\), \(M_z(x)\), and \(T(x)\). The resultant bending moment is \(M(x)=\sqrt{M_y^2+M_z^2}\).

Loads create stress; material and processed-part data provide strength. A satisfactory stress safety factor does not establish adequate stiffness, fatigue life, wear resistance, or critical speed.

Physical meaning

A surface point on a rotating shaft passes from tension to compression under a stationary transverse load, so steady spatial bending becomes fully reversed material stress. Steady transmitted torque normally remains a mean shear stress.

Problem-solving strategy

Recognize whether the requested result concerns layout, local strength, fatigue, stiffness, speed, or a fit.
Draw the shaft and attached components, define positive torque and transverse axes, and solve reactions in both planes.
Locate shoulders, keyways, grooves, maximum moments, and torque changes; these cues identify candidate critical sections.
Choose a trial diameter and supplied or traceable material, notch, endurance, and reliability data; calculate before selecting stock.
Iterate concentration and size factors, then audit yielding, fatigue, slope, deflection, critical speed, assembly, and tolerances.

7–2Shaft Materials

Strength, stiffness, and processing

Steel shaft stiffness changes little with grade because \(E\) is nearly grade-independent, whereas yield and fatigue strengths depend strongly on alloy, heat treatment, surface finish, size, and reliability. Increasing strength may reduce a stress-sized diameter but does not materially reduce elastic deflection at fixed geometry.

Use certified mill or supplier data for the actual condition and temperature. Machining sequence, case hardening, residual stress relief, corrosion protection, and journal finish can govern reliability. Large stock may require rough machining, stress relief, and finish machining to control distortion.

Data and assumptions

Educational values of \(S_y\), \(S_{ut}\), and \(S_e\) are supplied inputs, not universal properties. Production design must trace them to a material specification, heat, diameter range, process, surface condition, temperature, and required reliability.

7–3Shaft Layout

Layout before calculation

Place high transverse loads near bearings, keep spans and overhangs short, and provide an explicit axial load path. Usually one bearing locates the shaft axially while the other accommodates thermal growth. Shoulders, spacers, locknuts, rings, keys, splines, and fits must permit assembly without forcing every component across a long interference surface.

A generous fillet reduces local stress but may conflict with a bearing or gear corner; relief grooves and chamfered spacers can resolve the conflict. A key offers orientation and simple manufacture but weakens both shaft and hub. Splines distribute torque and may permit sliding, but require controlled alignment and specialized tooling.

7–4Shaft Design for Stress

Nominal and local fluctuating stress

For a solid circular section, nominal bending and torsional stresses vary with \(d^{-3}\). Use fatigue factors \(K_f\) and \(K_{fs}\) for alternating components and an explicitly justified treatment of mean stresses. They are not automatically equal to theoretical elastic factors \(K_t\) and \(K_{ts}\).

Alternating and mean stresses

\[\sigma_a=K_f\frac{32M_a}{\pi d^3},\quad \sigma_m=K_f\frac{32M_m}{\pi d^3},\quad \tau_a=K_{fs}\frac{16T_a}{\pi d^3},\quad \tau_m=K_{fs}\frac{16T_m}{\pi d^3}\]

Distortion-energy equivalents and Goodman check

\[\sigma_a'=\sqrt{\sigma_a^2+3\tau_a^2},\qquad \sigma_m'=\sqrt{\sigma_m^2+3\tau_m^2},\qquad \frac{1}{n_f}=\frac{\sigma_a'}{S_e}+\frac{\sigma_m'}{S_{ut}}\]

The linear Goodman relation is a common conservative screening model for ductile metals under proportional loading. Gerber is less conservative and data-dependent; Soderberg uses yield strength for the mean intercept. Any fatigue result must be paired with a maximum-load yielding check, for example \(n_y=S_y/\sigma_{\mathrm{vm,max}}\).

EXAMPLE 1Iterating a conveyor shaft diameter

A solid steel conveyor shaft has fully reversed bending \(M_a=450\ \mathrm{N\!\cdot\!m}\), steady torque \(T_m=300\ \mathrm{N\!\cdot\!m}\), and a \(30\ \mathrm{mm}\) trial diameter. Given \(K_f=1.60\), \(K_{fs}=1.35\), \(S_e=220\ \mathrm{MPa}\), \(S_{ut}=650\ \mathrm{MPa}\), and \(S_y=530\ \mathrm{MPa}\), find the Goodman factor, resize for \(n_d=1.50\), select the next whole-millimetre diameter, and recheck.

Recognition strategy

Stationary transverse load on a rotating shaft signals alternating bending, while steady power transmission signals mean torsion. The shared \(d^{-3}\) dependence permits a direct diameter iteration.

Given information

Positive torque follows the right-hand rule; moments are magnitudes at the critical shoulder.
Material strengths and concentration factors are supplied design inputs requiring later drawing and certificate verification.

Model and interpretation

Model a solid, linearly elastic circular section with \(M_m=T_a=0\), ignore residual and axial stress, and combine normal and shear components by distortion energy before applying Goodman.

Step-by-step solution

1. Convert loads consistently. Use \(M_a=450000\ \mathrm{N\!\cdot\!mm}\) and \(T_m=300000\ \mathrm{N\!\cdot\!mm}\), so stress emerges in \(\mathrm{N/mm^2}=\mathrm{MPa}\).

2. Calculate local components. \(\sigma_a=32(1.60)(450000)/[\pi(30)^3]=271.6\ \mathrm{MPa}\), while \(\tau_m=16(1.35)(300000)/[\pi(30)^3]=76.39\ \mathrm{MPa}\).

3. Apply the fatigue locus. Here \(\sigma_a'=271.6\ \mathrm{MPa}\) and \(\sigma_m'=\sqrt{3}(76.39)=132.3\ \mathrm{MPa}\), giving \(n_f=[271.6/220+132.3/650]^{-1}=0.695\).

4. Iterate and select size. Since every computed stress scales as \(d^{-3}\), \(d_{\mathrm{req}}=30(1.50/0.695)^{1/3}=38.77\ \mathrm{mm}\). Select \(d=40\ \mathrm{mm}\), yielding \(n_f=0.695(40/30)^3=1.65\).

Verification

Units: Both moment inputs use \(\mathrm{N\!\cdot\!mm}\); division by \(\mathrm{mm^3}\) produces \(\mathrm{N/mm^2}\), so each stress and strength ratio is dimensionless.

Signs and direction: Reversed bending has zero mean and a positive amplitude; steady positive torque contributes only mean shear. Magnitudes enter distortion energy, so reversing the torque sign would not change this scalar factor.

Plausibility: Increasing diameter from \(30\) to \(40\ \mathrm{mm}\) should multiply safety factor by \((4/3)^3=2.370\); \(0.695(2.370)=1.65\), independently confirming the iteration.

Final result: \(\boxed{d_{\mathrm{req}}=38.77\ \mathrm{mm};\ \text{select }40\ \mathrm{mm},\ n_f\approx1.65}\)

Engineering interpretation

The \(30\ \mathrm{mm}\) trial is fatigue-unsafe. Its maximum-load local yield factor cannot be determined until theoretical static factors \(K_t\) and \(K_{ts}\), or a justified local elastic analysis, are provided. The selected size still requires updated notch and size factors, tolerances, and stiffness verification.

7–5Deflection Considerations

Strength does not guarantee alignment

After the stepped geometry is established, solve elastic deflection in both transverse planes and combine vector components. Bearing slope, gear mesh misalignment, seal motion, and torsional twist may govern even when stress is acceptable. Use \(EI\,y''=M(x)\) with consistent signs and boundary conditions; local fillets usually matter to stress but little to gross stiffness.

Uniform scaling from a slope or deflection result

\[d_{\mathrm{new}}=d_{\mathrm{old}}\left(\frac{n_d\,y_{\mathrm{old}}}{y_{\mathrm{allow}}}\right)^{1/4},\qquad \theta=\sum_i\frac{T_iL_i}{G_iJ_i}\]

The fourth-root correction is valid only when all relevant diameters scale together, supports and loads are unchanged, and linear elasticity applies. Vendor bearing and gear limits, including housing flexibility, must replace educational limits in released work.

EXAMPLE 2Resizing a mixer shaft for bearing slope

A linear beam model of a geometrically similar stepped mixer shaft predicts a bearing slope magnitude of \(0.00140\ \mathrm{rad}\) at a \(35.0\ \mathrm{mm}\) reference journal. A supplied bearing limit is \(0.000800\ \mathrm{rad}\), and the project applies a deflection design factor \(n_d=1.10\). Find the uniform diameter scale, the required journal, and a practical selected size.

Recognition strategy

The model is already solved and the question changes all diameters uniformly, so the \(I\propto d^4\) scaling relation is more auditable than rebuilding the complete beam model.

Given information

Reference journal \(35.0\ \mathrm{mm}\), calculated slope \(0.00140\ \mathrm{rad}\), supplied limit \(0.000800\ \mathrm{rad}\).
Loads, spans, supports, elastic modulus, and all diameter ratios remain fixed during scaling.

Model and interpretation

Treat the shaft as an Euler–Bernoulli stepped beam in the small-deflection range. Slope sign identifies rotation direction, but alignment acceptance uses its absolute magnitude.

Step-by-step solution

1. Form the utilization ratio. Including the project margin, the stiffness demand is \(n_d\theta_{\mathrm{old}}/\theta_{\mathrm{allow}}=1.10(0.00140)/0.000800=1.925\).

2. Convert demand to diameter scale. Because slope is inversely proportional to \(I\) and \(I\propto d^4\), the uniform multiplier is \(r_d=(1.925)^{1/4}=1.1779\).

3. Resize the governing journal. The mathematical minimum is \(d_{\mathrm{req}}=35.0(1.1779)=41.23\ \mathrm{mm}\), before stock, bearing-bore, and shoulder decisions.

4. Select and predict performance. Choose a \(42.0\ \mathrm{mm}\) journal if compatible with the bearing catalog. The predicted slope is \(0.00140(35/42)^4=0.000675\ \mathrm{rad}\).

Verification

Units: Radians and their ratio are dimensionless; the fourth-root multiplier is dimensionless, and multiplying it by the original millimetre diameter correctly returns millimetres.

Signs and direction: A negative beam-analysis slope would indicate opposite section rotation, not extra margin. The absolute slope is compared with the symmetric alignment limit.

Plausibility: The selected \(42\ \mathrm{mm}\) shaft predicts \(0.000675\ \mathrm{rad}\), below the factored allowable \(0.000800/1.10=0.000727\ \mathrm{rad}\), so rounding upward preserves margin.

Final result: \(\boxed{r_d=1.1779,\ d_{\mathrm{req}}=41.23\ \mathrm{mm};\ \text{select }42.0\ \mathrm{mm}}\)

Engineering interpretation

The stiffness-driven journal is larger than a simple stress result may suggest. The complete stepped model must be rerun because bearing, seal, and gear seat choices may prevent exact uniform scaling.

7–6Critical Speeds for Shafts

Static sag as a dynamic clue

Mass eccentricity excites lateral vibration. Rayleigh's energy estimate uses a plausible mode shape and generally approaches the first natural frequency from above; Dunkerley combines single-mass contributions and is generally conservative. Both are screening models, not substitutes for rotordynamic analysis when gyroscopic effects, bearing coefficients, damping, variable speed, or fluid forces matter.

Rayleigh estimate for discrete weights

\[\omega_1=\sqrt{\frac{g\sum_i W_i y_i}{\sum_i W_i y_i^2}},\qquad N_1=\frac{60\omega_1}{2\pi}\]

EXAMPLE 3Screening a fan rotor operating speed

A shaft beam model under gravity gives downward static deflections of \(0.350\ \mathrm{mm}\) at a \(180\ \mathrm{N}\) fan wheel and \(0.280\ \mathrm{mm}\) at a \(120\ \mathrm{N}\) coupling. Using Rayleigh's discrete-weight estimate, calculate the first critical speed and determine whether continuous operation at \(1200\ \mathrm{rpm}\) satisfies a project separation requirement \(N_1/N_{\mathrm{op}}\ge1.25\).

Recognition strategy

Discrete rotating weights and their gravity sag are supplied, which directly cue Rayleigh's first-mode screening relation and a subsequent speed-separation decision.

Given information

\(W_1=180\ \mathrm{N}\), \(y_1=0.000350\ \mathrm{m}\); \(W_2=120\ \mathrm{N}\), \(y_2=0.000280\ \mathrm{m}\); \(g=9.81\ \mathrm{m/s^2}\).
Downward deflection is taken positive with each positive weight; shaft distributed mass and damping are omitted.

Model and interpretation

The static deflected shape approximates the first bending mode of a linear, stationary-support rotor. The result is an undamped natural-speed estimate, not a predicted vibration amplitude.

Step-by-step solution

1. Evaluate the numerator sum. \(\sum W_i y_i=180(0.000350)+120(0.000280)=0.0966\ \mathrm{N\!\cdot\!m}\).

2. Evaluate the denominator sum. \(\sum W_i y_i^2=180(0.000350)^2+120(0.000280)^2=3.1458\times10^{-5}\ \mathrm{N\!\cdot\!m^2}\).

3. Calculate and convert frequency. \(\omega_1=\sqrt{9.81(0.0966)/(3.1458\times10^{-5})}=173.6\ \mathrm{rad/s}\), hence \(N_1=60(173.6)/(2\pi)=1657\ \mathrm{rpm}\).

4. Make the auditable decision. The separation ratio is \(1657/1200=1.381\), which exceeds the supplied minimum \(1.25\); the screening calculation therefore passes.

Verification

Units: The Rayleigh quotient has units \((\mathrm{m/s^2})(\mathrm{N\!\cdot\!m})/(\mathrm{N\!\cdot\!m^2})=\mathrm{s^{-2}}\); its square root is \(\mathrm{rad/s}\).

Signs and direction: Weights and their downward static deflections use the same positive direction, keeping both energy sums positive. Mixing signed modal ordinates without a consistent mode convention would corrupt the estimate.

Plausibility: An equivalent sag \(g/\omega_1^2\) is about \(0.326\ \mathrm{mm}\), lying between the two supplied sags, which supports the computed magnitude.

Final result: \(\boxed{N_1\approx1657\ \mathrm{rpm},\quad N_1/N_{\mathrm{op}}=1.381\ \text{; project screen passes}}\)

Engineering interpretation

The simple screen provides only a 38 percent speed gap. Commissioning run-up, balance quality, bearing and foundation stiffness, shaft mass, damping, and vendor rotordynamic limits still require review before release.

7–7Miscellaneous Shaft Components

Keys, pins, rings, and shoulders

Size a key from transmitted tangential force \(F=2T/d\), then check key shear, key and hub bearing, shaft weakening, and fatigue at the keyway. Pins and setscrews suit limited torque or locating duty; retaining rings require groove capacity and create severe local concentration. Catalog ratings apply only to the specified groove, material, installation, speed, temperature, and load direction.

Prefer standard cutters, radii, key sizes, ring grooves, and bearing seats. The weakest replaceable component may intentionally act as a serviceable fuse, but that decision must not create uncontrolled fragments or secondary damage.

7–8Limits and Fits

Fit is a statistical assembly requirement

Clearance fits permit motion, transition fits may clear or interfere, and interference fits transmit load by contact pressure. Basic size alone is not a tolerance. Specify a recognized fit designation and edition, surface finish, geometry controls, temperature basis, and inspection method. Verify minimum and maximum clearance from the independent hole and shaft limits.

Chapter recap

Resolve applied forces and reactions in two planes before calculating resultant bending and internal torque.
Distinguish nominal from local stress, alternating from mean loading, and fatigue from maximum-load yielding.
Iterate shaft geometry because concentration and endurance modifiers depend on the selected dimensions and processing.
Verify stiffness and critical speed separately from strength, then integrate components, fits, manufacture, and inspection.

Before moving on: You should be able to identify critical sections, size and recheck a shaft, interpret slope and speed margins, and state which material, catalog, standard, and test data remain to be verified.