Sources and scope for this page: Roark's Formulas for Stress and Strain (Roark's, 8th edition (2011)), covering the stress, bending, torsion and column formulas; and Shigley's Mechanical Engineering Design (Shigley's, 11th edition (2020)), covering the power transmission, gear, spring and bearing formulas (one more source is tabulated below). No independent source check is recorded for this page: the content rests on the sources listed and on the working shown on the page. Full sources, values and limits on this page.
Page last changed 2026-10-19 · based on Roark's, 8th edition (2011)
Engineering Formulas by Subject
| Subject[1] | Quantity[1] | Formula[1] | Variables[1] | Notes[1] |
|---|---|---|---|---|
| Stress # | Stress | σ = F ÷ A | σ stress, F force, A cross-sectional area | Basic definition; the starting point for every strength check |
| Stress # | Strain | ε = ΔL ÷ L₀ | ε strain, ΔL change in length, L₀ original length | Dimensionless; often quoted in microstrain (×10⁻⁶) |
| Stress # | Hooke's law | σ = E · ε | E modulus of elasticity | Valid only below the proportional limit |
| Stress # | Poisson's ratio | ν = −ε_lateral ÷ ε_axial | Typically 0.3 for metals, 0.4 for plastics | Why a bar gets thinner when pulled |
| Bending # | Bending stress | σ = M · c ÷ I | M moment, c distance to extreme fibre, I second moment of area | Also written σ = M ÷ S where S = I/c |
| Bending # | Section modulus | S = I ÷ c | S elastic section modulus | Governs stress; I governs deflection |
| Bending # | Beam deflection, central point load | δ = P · L³ ÷ (48 · E · I) | P load, L span | Simply supported |
| Bending # | Beam deflection, uniform load | δ = 5 · w · L⁴ ÷ (384 · E · I) | w load per unit length | Simply supported |
| Bending # | Cantilever, end point load | δ = P · L³ ÷ (3 · E · I) | P load, L length | Same load and span as a simple beam gives 16× the deflection |
| Torsion # | Torsional shear stress | τ = T · r ÷ J | T torque, r radius, J polar moment of inertia | Maximum at the outer surface |
| Torsion # | Polar moment of inertia, solid shaft | J = π · d⁴ ÷ 32 | d shaft diameter | Twice the planar moment of inertia |
| Torsion # | Angle of twist | θ = T · L ÷ (G · J) | G shear modulus, L length | θ in radians |
| Torsion # | Power transmitted by a shaft | P = 2π · n · T ÷ 60 | n rotational speed in rpm, T torque in N·m | Gives power in watts |
| Columns # | Euler buckling load | Pcr = π² · E · I ÷ (K · L)² | K effective length factor | Valid while the buckling stress is below about half yield |
| Columns # | Slenderness ratio | λ = K · L ÷ r | r radius of gyration = √(I/A) | Governs whether buckling or yielding governs |
| Pressure # | Hoop stress, thin cylinder | σ = p · D ÷ (2 · t) | p internal pressure, D diameter, t wall | Valid when t is under about a tenth of D |
| Pressure # | Longitudinal stress, thin cylinder | σ = p · D ÷ (4 · t) | Half the hoop stress | Why a cylinder splits lengthwise, not around |
| Pressure # | Spherical shell stress | σ = p · D ÷ (4 · t) | Uniform in all directions | Half the hoop stress of a cylinder of the same size |
| Fluids # | Reynolds number | Re = ρ · v · D ÷ µ | ρ density, v velocity, D diameter, µ dynamic viscosity | Laminar below ~2 300, turbulent above ~4 000 |
| Fluids # | Darcy-Weisbach pressure drop | ΔP = f · (L ÷ D) · ρ · v² ÷ 2 | f friction factor | Rises with the square of flow rate in turbulent flow |
| Fluids # | Bernoulli's equation | p + ½ρv² + ρgh = constant | h elevation | Energy conservation along a streamline |
| Fluids # | Flow through a pipe | Q = A · v | Q volumetric flow rate | Continuity: what goes in comes out |
| Heat # | Conduction, plane wall | Q = k · A · ΔT ÷ t | k thermal conductivity, t thickness | Fourier's law |
| Heat # | Convection | Q = h · A · ΔT | h convective heat transfer coefficient | Newton's law of cooling |
| Heat # | Sensible heat | Q = m · cp · ΔT | cp specific heat capacity | The heat to change temperature without a phase change |
| Heat # | Thermal expansion | ΔL = L · α · ΔT | α coefficient of linear thermal expansion | Roughly 12 µm/m·°C for steel |
| Heat # | Fin efficiency parameter | m = √(h · P ÷ (k · A)) | P perimeter, A cross-section | Governs how far heat travels along a fin |
| Machining # | Spindle speed from surface speed | n = SFM × 3.82 ÷ D | D tool diameter in inches | The basic machine setting calculation |
| Machining # | Surface speed from spindle speed | SFM = π · D · n ÷ 12 | D in inches, n in rpm | The inverse relationship |
| Machining # | Milling table feed | F = fz · z · n | fz feed per tooth, z number of teeth | Independent of depth and width of cut |
| Machining # | Material removal rate, milling | MRR = ap · ae · F | ap depth, ae width of cut | Volume removed per minute |
| Machining # | Theoretical surface roughness, turning | Ra ≈ f² ÷ (32 · r) | f feed per rev, r tool nose radius | Why feed matters more than speed for finish |
| Machining # | Machining time | T = L ÷ F | L length of cut, F feed rate | Add tool approach and retract for the actual cycle time |
| Threads # | Metric tensile stress area | As = π/4 · (d − 0.9382P)² | d major diameter, P pitch | ISO 898-1 definition |
| Threads # | Unified tensile stress area | As = 0.7854 · (D − 0.9743/n)² | D major diameter, n threads per inch | ASME definition; note the different coefficient |
| Threads # | Pitch diameter | d₂ = d − 0.64952 · P | P pitch | Unified and metric 60° threads |
| Threads # | Tap drill, approximate | drill = d − P | d major diameter, P pitch | Gives about 75% thread engagement; round to the nearest standard drill |
| Electrical # | Ohm's law | V = I · R | V voltage, I current, R resistance | The fundamental relationship of circuit analysis |
| Electrical # | Power, resistive | P = V · I = I² · R = V² ÷ R | P power in watts | Which form to use depends on which two quantities are known |
| Electrical # | Resistance of a conductor | R = ρ · L ÷ A | ρ resistivity, L length, A area | Why long thin conductors drop more voltage |
| Electrical # | Voltage drop, single phase | Vd = 2 · ρ · L · I ÷ A | The 2 accounts for the return path | Use √3 instead of 2 for three-phase |
| Electrical # | Three-phase power | P = √3 · V · I · PF | PF power factor | V is the line-to-line voltage |
| Gears # | Gear ratio | i = n_driven ÷ n_driver | Teeth, not diameters | Speed divides by i; torque multiplies by it |
| Gears # | Gear pitch diameter | D = N ÷ DP | N number of teeth, DP diametral pitch | Inches. For metric, D = N × module |
| Gears # | Module from diametral pitch | m = 25.4 ÷ DP | m in millimetres | Reciprocal relationship — inch and metric gears do not interchange |
| Springs # | Compression spring rate | k = G · d⁴ ÷ (8 · D³ · N) | G shear modulus, d wire diameter, D mean coil diameter, N active coils | Wire diameter is to the fourth power |
| Springs # | Spring force | F = k · x | x deflection | Hooke's law for a spring |
| Springs # | Spring index | C = D ÷ d | Practical range 4 to 12 | Governs manufacturability and stress concentration |
| Bearings # | Bearing L10 life | L10 = (C ÷ P)^p × 10⁶ | C dynamic rating, P equivalent load, p = 3 ball / 10/3 roller | Life in revolutions that 90% of bearings reach |
| Bearings # | Bearing life in hours | L10h = L10 ÷ (60 · n) | n speed in rpm | The figure used for selection |
| Fits # | Interference from temperature | ΔT = δ ÷ (α · D) | δ diametral interference, α expansion coefficient | For shrink-fit assembly |
| Fits # | Worst-case tolerance stack | T = Σ tᵢ | Sum of individual tolerances | Guaranteed; expensive |
| Fits # | Statistical tolerance stack | T = √(Σ tᵢ²) | Root-sum-square | Valid for centred, in-control processes only |
| Material # | Weight of a section | W = A · L · ρ | A area, L length, ρ density | Use kg/m³ with m, or lb/in³ with in |
| Material # | Density from weight and volume | ρ = m ÷ V | The definition | Steel is 7.85 g/cm³ = 0.2836 lb/in³ |
| Material # | Modulus from stress and strain | E = σ ÷ ε | In the linear region only | Roughly 205 GPa for every steel |
| Material # | Hardness to tensile, steels | UTS ≈ 500 × HB (MPa) | Approximate, ±10% | A rule of thumb, not a specification |
Formulas are stated in consistent SI units unless the note says otherwise. Converting to imperial is not simply a matter of changing the numbers — several of these formulas carry constant coefficients that assume SI (Euler's buckling load, the thin-cylinder stresses, the deflection formulas in terms of E and I) and others carry coefficients that assume a particular unit system built into the constant, such as n = SFM × 3.82 ÷ D, where the 3.82 only works with inches and feet per minute.
Every formula here is an idealisation. Elastic formulas assume small deflections and a constant section; the thin-cylinder stresses assume a wall under a tenth of the diameter; Euler buckling assumes the stress at buckling is below about half the yield strength. Using a formula outside its assumptions is the most common source of a plausible but wrong answer.
The Three Formulas Worth Knowing by Heart
Most engineering estimates come down to three relationships, and knowing them makes it possible to check almost any number someone hands you.
σ = F ÷ A. Stress is force divided by area. It converts a load into a stress in one step, and comparing that stress against a yield strength is the first check on any part.
δ = P·L³ ÷ (48·E·I). Deflection of a simply supported beam under a central point load. The cube on the span and the fact that E·I sits underneath is the whole of beam design: span matters far more than load, and stiffness is a product of material and section.
P = 2π·n·T ÷ 60. Power from torque and speed. It is what connects a motor's rating to the torque available at a given speed, and it explains why a gearbox trades speed for torque while leaving power unchanged.
Everything else on this page is a variation on these three applied to a particular geometry or a particular physical process.
Unit Consistency Is Not Optional
More engineering answers are wrong because of units than because of physics. Three cases account for most of it.
Metres against millimetres. In deflection formulas the length is cubed, so a length entered in millimetres where metres were expected makes the answer wrong by 10⁹ — a factor of a billion. Always check the unit the formula's constant assumes.
Mass against force. The kilogram is a mass and the newton is a force, related by g = 9.80665 m/s². Treating a kilogram-force as a newton understates a load by roughly a factor of ten. The same trap exists in imperial units between the pound and the pound-force, though there the two are numerically equal under standard gravity and the error is easier to miss.
Hard-coded constants. Some constants in these formulas are unit conversions in disguise. The 3.82 in the milling speed formula, the √3 in three-phase power, and the 0.7854 in the unified stress area are all unit-specific. Substitute a different unit system and the constant changes even though the physics does not.
Frequently Asked Questions
- What is the formula for bending stress?
- σ = M·c ÷ I, where M is the bending moment, c the distance from the neutral axis to the extreme fibre and I the second moment of area. It is more often written σ = M ÷ S, where S = I/c is the section modulus — a tabulated property for standard sections.
- What is the beam deflection formula?
- For a simply supported beam with a central point load, δ = P·L³/(48·E·I). For a uniformly distributed load it is 5w·L⁴/(384·E·I), and for a cantilever with an end load it is P·L³/(3·E·I). All three carry the cube of span, which is why span drives section size so strongly.
- How do I calculate spindle speed?
- n = SFM × 3.82 ÷ D, where D is the tool diameter in inches. Going the other way, SFM = π·D·n ÷ 12. The 3.82 is the constant 12/π, so the formula only works with surface speed in feet per minute and diameter in inches.
- What is the difference between stress and strain?
- Stress is force per unit area, in pascals or psi. Strain is the fractional change in length, dimensionless. They are related by Hooke's law, σ = E·ε, but only below the proportional limit — past yield the relationship stops being linear and the formula no longer applies.
- How do I convert torque to power?
- P = 2π·n·T ÷ 60 with n in rpm and T in newton-metres, giving watts. In imperial units, horsepower = torque in lb·ft × rpm ÷ 5,252. The two constants differ because the unit systems do — the underlying relationship is the same.
- Can I use these formulas with imperial units?
- Not without changing the constants. Several formulas carry unit-specific coefficients: the 3.82 in the spindle speed formula, the √3 in three-phase power, and the different coefficients in the metric and unified tensile stress areas. The physics is the same but the constant is not — always check what units a formula's coefficient assumes.
Related
Value Sources
References and calculation methods used for this page are listed below. Numbered markers connect table columns where present. A reference listing does not itself establish that every value has been independently checked.
| # | Source | Type | Revision / method |
|---|---|---|---|
| [1] | Published handbook mechanical properties for engineering materials | reference | compilations as published 2024–2026 |
Data Sources
| Standard | Revision | What it covers on this page |
|---|---|---|
| Roark's Formulas for Stress and Strain | Roark's, 8th edition (2011) | the stress, bending, torsion and column formulas |
| Shigley's Mechanical Engineering Design | Shigley's, 11th edition (2020) | the power transmission, gear, spring and bearing formulas |
| ASME B1.1 and ISO 898-1 | ASME B1.1-2019, ISO 898-1:2013 | the thread stress area and pitch diameter formulas |
Verification scope
No page-specific independent source review is recorded here. The listed references and formulas explain the basis of the compilation; automated consistency checks do not verify the original sources or establish product conformity.
Formulas are idealisations with stated assumptions — small deflections, constant section, stress below the proportional limit for elastic formulas, thin walls for the cylinder stresses. Constants embedded in some formulas are unit-system specific. Where a value will be used for design or acceptance, verify it against the governing standard rather than relying on a reference formula.
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