Stress, Strain, and Hooke's Law
Machine design begins with translating an applied load into a stress that can be compared against a material's strength. Normal stress (σ) is force acting perpendicular to a cross-section divided by that area, σ = F/A, and it can be tensile or compressive. Shear stress (τ) is force acting parallel to a cross-section divided by the area it acts over, τ = F/A for direct shear, and it also arises from torsion and transverse bending loads. Strain is the resulting deformation: normal strain ε is the change in length divided by original length, and shear strain γ is the angular distortion of a small element.
For most engineering metals below yield, stress and strain are linearly related through Hooke's law: σ = E·ε, where E is the modulus of elasticity (Young's modulus), and τ = G·γ, where G is the shear modulus. The two moduli are related through Poisson's ratio ν by G = E / [2(1+ν)], so for steel (E ≈ 200 GPa, ν ≈ 0.3) the shear modulus works out to roughly 77 GPa. Every stress-based design calculation in this article, from shaft torsion to bolt stretch, ultimately traces back to these two linear relationships.
Factor of Safety and Failure Theories for Ductile Materials
A factor of safety (FS or n) is the ratio of a material's failure strength to the stress the part actually experiences: FS = S/σ. For a ductile metal under a simple uniaxial load, the relevant strength is the yield strength Sy, and static failure is defined as the onset of yielding, not fracture. Real machine elements are rarely under simple uniaxial stress, though — a shaft carries bending and torsion simultaneously, a bracket sees combined tension and shear — so a failure theory is needed to reduce a multiaxial stress state to a single equivalent stress that can be compared to Sy.
Two theories dominate ductile-material design. The maximum shear stress theory (Tresca criterion) says yielding begins when the maximum shear stress in the part reaches the shear yield strength, τmax = (σ1 − σ3)/2 = Sy/2, using the maximum and minimum principal stresses. The distortion energy theory (von Mises criterion) says yielding begins when the energy stored in distorting the material's shape reaches a critical value, expressed through the von Mises stress:
σ′ = √[ ((σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²) / 2 ]
For the common case of plane stress with a normal stress σ and a shear stress τ acting together (as in a shaft under bending and torsion), this simplifies to σ′ = √(σ² + 3τ²). Failure is predicted when σ′ = Sy.
| Theory | Failure criterion | Character |
|---|---|---|
| Maximum shear stress (Tresca) | σ1 − σ3 = Sy | Simpler, more conservative (predicts yield at lower stress) |
| Distortion energy (von Mises) | σ′ = Sy | Matches test data more closely, used by FEA software |
Brittle materials (cast iron, ceramics, ordinary concrete) are governed by fracture rather than yielding and use different criteria, most commonly the maximum normal stress theory or the Coulomb-Mohr theory, because brittle materials have little or no ability to redistribute stress plastically before breaking.
Fatigue Failure: The S-N Curve and the Goodman Diagram
Many machine elements fail after a large number of load cycles at stresses well below the static yield strength — this is fatigue. Fatigue behavior is characterized by an S-N curve (also called a Wöhler curve), which plots the alternating stress amplitude a specimen can survive against the number of cycles to failure, N, typically on a log-log or semi-log axis. Many steels exhibit a knee in the curve around one to ten million cycles, below which the curve flattens into an endurance limit, Se — a stress amplitude the material can theoretically sustain indefinitely. For steels with an ultimate strength Sut below about 1400 MPa (200 ksi), a common approximation is Se′ ≈ 0.5·Sut for the polished, unnotched laboratory specimen. Non-ferrous metals like aluminum generally have no true endurance limit and are instead rated for a specific finite life, commonly 5×10⁸ cycles.
The laboratory endurance limit Se′ must be corrected for real-world conditions using a set of empirical factors: Se = ka·kb·kc·kd·ke·Se′, where ka accounts for surface finish (machined and forged surfaces are far worse than polished), kb for part size, kc for the type of loading (axial, bending, or torsional), kd for operating temperature, and ke for reliability. A rough-cast surface or a large-diameter shaft can easily cut the usable endurance limit to a third or less of the polished-specimen value, which is why fatigue-critical shafts are typically ground or polished at fillets and other stress-raiser locations.
Real loads are rarely fully reversed; they usually have both a mean component σm and an alternating component σa superimposed. The modified Goodman diagram plots σa on the vertical axis against σm on the horizontal axis, drawing a straight line from Se on the alternating-stress axis to Sut on the mean-stress axis. Any combination of σa and σm that falls below this line is predicted to survive infinite life; combinations above it are predicted to fail. The design equation, incorporating a factor of safety n, is:
σa/Se + σm/Sut = 1/n
This single relationship is the backbone of fatigue design for shafts, springs, and connecting rods, because it lets a designer evaluate a load that never fully reverses (such as a shaft that always rotates the same direction under a steady bending moment plus a superimposed vibratory component) against both the material's fatigue strength and its ultimate strength in one check.
Shaft Design: Torsion, Combined Loading, and Sizing
Power-transmission shafts are sized primarily from torque and bending moment. Pure torsion produces a shear stress that varies linearly from zero at the center to a maximum at the outer surface: τ = T·c/J, where T is the applied torque, c is the outer radius, and J is the polar moment of inertia, J = π·d⁴/32 for a solid circular shaft of diameter d. Substituting gives the familiar sizing form τmax = 16T/(π·d³).
Most real shafts carry bending as well as torsion — from gear separating forces, belt tension, or an overhung load — producing a bending stress σ = 32M/(π·d³) at the outer fiber, superimposed on the torsional shear stress. Combining these through the distortion energy theory (σ′ = √(σ² + 3τ²)) and solving for the required diameter gives the standard combined-loading shaft equation:
d³ = (16 / π·σallow) · √(4M² + 3T²)
where σallow = Sy/n is the allowable stress for the chosen factor of safety n. Stress concentration factors Kf for keyways, shoulders, and grooves must be applied to M and T before using this equation for any location other than a smooth, unnotched section, and a fatigue version of this equation (using Se in place of Sy and separating mean and alternating moment/torque components through the Goodman relationship) is used whenever the load varies with shaft rotation, which is the normal case for a rotating shaft under a steady bending moment.
Worked example (pure torsion sizing): A shaft made from a steel with Sy = 300 MPa must transmit a steady torque of T = 500 N·m with no significant bending, at a design factor of safety of n = 2.5 against yielding.
Using the distortion energy theory, the shear yield strength is τy = Sy/√3 = 300/1.732 = 173.2 MPa.
The allowable shear stress is τallow = τy/n = 173.2/2.5 = 69.3 MPa.
Solving the torsion equation for diameter: d³ = 16T/(π·τallow) = (16 × 500)/(π × 69.3×10⁶) = 8000/2.176×10⁸ = 3.676×10⁻⁵ m³.
d = (3.676×10⁻⁵)^(1/3) ≈ 0.0333 m ≈ 33.3 mm, which would be rounded up to the next standard stock size, typically 35 mm.
Critical Speed and Shaft Dynamics
Every shaft-and-rotor assembly has a natural frequency of lateral vibration; when the shaft's rotating speed approaches this frequency, small residual imbalance can drive the shaft into large-amplitude whirling — this speed is the first critical speed. For a simplified single-mass model, the first critical speed in rpm can be estimated from the static deflection δst that the same mass would produce if it simply hung under gravity at that location:
ncrit ≈ 30/π · √(g/δst) (with g and δst in consistent units, giving ncrit in rpm)
A shaft with a large static deflection (long, slender, heavily loaded) has a low critical speed, while a short, stiff, well-supported shaft has a high one. Standard design practice keeps the operating speed below roughly 70 to 80 percent of the first critical speed for a shaft that runs below resonance ("rigid" operation), or well above the first critical (and through it quickly during startup) for turbomachinery designed to run supercritically. Multi-mass or continuous-shaft systems require a full Rayleigh-Ritz or finite element modal analysis rather than the single-mass approximation, but the single-mass estimate is the standard first-pass hand check for a shaft carrying one or two concentrated loads such as a gear or a coupling.
Gear Fundamentals: Terminology, the Lewis Equation, and AGMA
A spur gear's geometry is described by its pitch diameter d (the diameter of the imaginary rolling cylinder), its number of teeth N, and either its diametral pitch Pd = N/d (US units, teeth per inch of pitch diameter) or its metric module m = d/N. The standard pressure angle is 20 degrees, defining the angle at which tooth contact force is transmitted. Other key terms include the addendum and dedendum (tooth height above and below the pitch circle), the base circle from which the involute tooth profile is generated, and the contact ratio, which describes how many tooth pairs share the load at any instant.
The classical hand check for tooth bending strength is the Lewis bending stress equation:
σ = Wt·Pd / (F·Y) (US customary units)
where Wt is the transmitted tangential load at the pitch line, F is the face width, and Y is the dimensionless Lewis form factor, which depends on the number of teeth and captures the tooth's shape as a cantilever beam.
| Number of teeth | Lewis form factor Y (20° full-depth) |
|---|---|
| 12 | 0.245 |
| 17 | 0.303 |
| 24 | 0.337 |
| 50 | 0.408 |
| Rack | 0.485 |
Fewer teeth produce a thinner, weaker tooth root, so smaller pinions govern gear-set bending design. The basic Lewis equation ignores dynamic effects, load sharing, and manufacturing quality, so modern practice replaces it with the AGMA (American Gear Manufacturers Association) bending and pitting resistance equations, which apply additional factors for overload (Ko), dynamic effects from tooth-to-tooth velocity (Kv), load distribution across the face width (Km), and a geometry factor (J) that supersedes the simple Lewis Y. The Lewis equation remains valuable as a quick hand check and as the conceptual foundation the AGMA method builds on, but final gear-set sizing for the PE exam and for real hardware should reference the AGMA J-factor method.
Rolling-Element Bearings: Load Ratings and L10 Life
Rolling-element bearings are rated by two numbers: the basic dynamic load rating, C, is the constant radial load a bearing can theoretically sustain for one million revolutions with 90 percent of a population surviving; the basic static load rating, C0, limits permanent deformation under load at very low speed or standstill. Under a combined radial and axial (thrust) load, the two are reduced to a single equivalent dynamic load, P = X·Fr + Y·Fa, where X and Y are radial and thrust factors published in the bearing manufacturer's catalog and depend on the ratio of thrust to radial load.
Bearing fatigue life is predicted with the L10 life equation:
L10 = (C/P)^k × 10⁶ revolutions
where the life exponent k = 3 for ball bearings and k = 10/3 for roller bearings, reflecting their different contact geometry (point contact for balls, line contact for rollers). To convert to hours at a running speed of n rpm: L10h = L10 / (60·n).
Worked example: A deep-groove ball bearing has a catalog dynamic load rating C = 15 kN. It carries an equivalent dynamic load P = 3 kN at a shaft speed of n = 1750 rpm. Find the L10 life.
L10 = (C/P)³ × 10⁶ = (15/3)³ × 10⁶ = 5³ × 10⁶ = 125 × 10⁶ revolutions.
L10h = 125×10⁶ / (60 × 1750) = 125×10⁶ / 105,000 ≈ 1190 hours.
At 2,000 operating hours per year (roughly one shift, five days a week), this bearing's rated life is under a year, which would likely prompt the designer to select a larger bearing, reduce the load, or accept a shorter service interval and plan for scheduled replacement. Bearing selection in practice also weighs speed limits (the DN value — bore diameter in mm times rpm), lubrication method, mounting fit and preload, and misalignment tolerance, all of which affect the achievable life beyond the catalog L10 number.
Bolted Joints: Preload and Joint Stiffness
A properly designed bolted joint is tightened to a preload, Fi, well before any external service load is applied, and this preload — not the bolt's ultimate strength — governs how the joint behaves under load. When an external tensile load P is applied to a preloaded joint, it does not add directly to the bolt tension; instead, it is split between the bolt and the clamped members according to their relative stiffness. Defining the joint stiffness constant C = kb/(kb + km), where kb is the bolt stiffness and km is the stiffness of the clamped members, the bolt load and the clamped-member load become:
Fbolt = Fi + C·P and Fmember = Fi − (1 − C)·P, valid as long as Fmember remains positive (the joint has not separated).
Because the clamped members (steel flanges, gaskets, castings) are typically much stiffer than the relatively slender, longer-gauge-length bolt, C is often only 0.2 to 0.3, meaning the bolt only sees 20 to 30 percent of any additional external load — the preload is doing most of the work of keeping the joint together. This is also why a correctly preloaded bolt in a properly designed joint rarely fails from the static external load; it is far more likely to fail from fatigue if the joint separates or from insufficient preload allowing slip or leakage.
In practice, preload is most often set by torque, using T = K·Fi·d, where d is the nominal bolt diameter and K is an empirical nut factor (about 0.2 for standard steel fasteners with as-received or lightly oiled finish, lower for lubricated or coated threads, higher for dry or galled threads). Recommended preload is commonly specified as roughly 75 percent of the bolt's proof load for connections that may be reused, and up to 90 percent of proof load for permanent, non-reused joints, both chosen to maximize the resistance to loosening and fatigue while leaving margin below the bolt's yield strength.