Engineering reference · V0.8

Engineering Formula Library

A searchable desk reference for common mechanical-design equations. Each formula states a compact assumption so you can decide whether it matches the problem before using it.

Family	Relationship	Formula	Scope / assumption
Axial	Normal stress	`σ = F / A`	Uniform axial load; nominal stress.
Axial	Normal strain	`ε = ΔL / L`	Small engineering strain.
Axial	Hooke’s law	`σ = Eε`	Linear-elastic uniaxial material response.
Axial	Axial elongation	`δ = FL / (AE)`	Uniform prismatic member.
Bending	Flexure stress	`σ = My / I`	Elastic bending; neutral-axis geometry known.
Bending	Section modulus form	`σmax = M / Z`	Z = I/c.
Bending	Curvature	`κ = M / (EI)`	Euler–Bernoulli beam theory.
Beams	Simply supported, center load	`δmax = PL³ / (48EI)`	Point load at midspan.
Beams	Cantilever, end load	`δmax = PL³ / (3EI)`	Point load at free end.
Beams	Simply supported, UDL	`δmax = 5wL⁴ / (384EI)`	Uniform distributed load over full span.
Beams	Cantilever, UDL	`δmax = wL⁴ / (8EI)`	Uniform distributed load over full span.
Sections	Rectangle area	`A = bh`	Rectangle.
Sections	Rectangle Iₓ	`Iₓ = bh³ / 12`	Centroidal axis parallel to width.
Sections	Solid circle I	`I = πd⁴ / 64`	Any centroidal diameter.
Torsion	Circular-shaft shear	`τ = Tr / J`	Uniform circular shaft.
Torsion	Angle of twist	`θ = TL / (JG)`	Linear-elastic circular shaft.
Torsion	Solid circular polar moment	`J = πd⁴ / 32`	Solid circular section.
Torsion	Hollow circular polar moment	`J = π(dₒ⁴−dᵢ⁴)/32`	Concentric tube.
Springs	Compression spring rate	`k = Gd⁴ / (8D³n)`	Round-wire helical compression spring.
Springs	Spring index	`C = D / d`	Mean coil diameter / wire diameter.
Springs	Wahl factor	`Kᵥ = (4C−1)/(4C−4) + 0.615/C`	Curvature/direct-shear correction.
Thermal	Linear expansion	`ΔL = αLΔT`	Constant coefficient over temperature interval.
Thermal	Thermal strain	`εₜ = αΔT`	Unconstrained free strain.
Power	Rotational power	`P = Tω`	Consistent units.
Power	Speed conversion	`ω = 2πn / 60`	n in rpm, ω in rad/s.
Kinematics	Tangential speed	`v = ωr`	Rigid rotation.
Kinematics	Centripetal acceleration	`a = v²/r = ω²r`	Circular motion.
Pressure vessels	Thin-wall hoop stress	`σh = pD / (2t)`	Thin cylindrical wall approximation.
Pressure vessels	Thin-wall longitudinal stress	`σL = pD / (4t)`	Closed-end thin cylindrical wall.
Fasteners	Nut-factor torque	`T = KFd`	Concept-stage torque–preload relation.
Tolerances	Worst-case bilateral stack	`tWC = Σ\|tᵢ\|`	Simple independent linear dimension chain.
Tolerances	RSS stack	`tRSS = √Σ(tᵢ²)`	Statistical screening with distribution assumptions.
Fits	Thermal diameter	`D₁ = D₀(1 + αΔT)`	Linear isotropic expansion approximation.
Mechanics	Friction	`Ff ≤ μN`	Coulomb dry-friction model.
Mechanics	Work	`W = F·s`	Constant force aligned with displacement.
Mechanics	Kinetic energy	`Ek = ½mv²`	Translational point mass.
Mechanics	Rotational kinetic energy	`Er = ½Jmω²`	Jm is mass moment of inertia.
Dynamics	Newton’s second law	`ΣF = ma`	Translational equation of motion.
Dynamics	Rotational equation	`ΣM = Jmα`	About a fixed axis using mass moment of inertia.
Power	Mechanical rotational power	`P = Tω`	Ideal mechanical shaft power.
Power	Angular speed from rpm	`ω = 2πn / 60`	n in revolutions per minute.
Power	Power in kW	`P(kW) = T·n / 9549.3`	T in N·m; n in rpm.
Gears	Simple external gear ratio	`i = z₂/z₁ = n₁/n₂`	Basic two-gear external mesh.
Gears	Pitch diameter	`d = mz`	Metric module reference geometry.
Gears	Center distance	`a = (d₁+d₂)/2`	Basic unshifted external pair.
Gears	Circular pitch	`p = πm`	Metric spur-gear reference geometry.
Gears	Base diameter	`dᵦ = d cos α`	Involute reference geometry.
Belts	No-slip speed ratio	`n₁D₁ ≈ n₂D₂`	Use working/pitch diameters.
Belts	Belt linear speed	`v = πDn / 60`	D in metres; n in rpm.
Belts	Open-belt length	`L ≈ 2C + π(D₁+D₂)/2 + (D₂−D₁)²/(4C)`	Approximate open belt geometry.
Bearings	Basic L10 life	`L₁₀ = (C/P)ᵖ`	Million revolutions; p depends on bearing type.
Bearings	Basic life in hours	`L₁₀h = 10⁶/(60n) · (C/P)ᵖ`	Constant speed; basic rating-life model.
Buckling	Euler critical load	`Pcr = π²EI / (KL)²`	Ideal slender elastic column.
Buckling	Effective length	`Le = KL`	Idealized end-restraint factor.
Buckling	Radius of gyration	`r = √(I/A)`	Section property.
Buckling	Slenderness ratio	`λ = KL/r`	Column slenderness measure.
Safety	Scalar factor of safety	`n = S/σ`	Strength and stress must refer to the same failure criterion.
Safety	Allowable stress	`σallow = S/nrequired`	Simple allowable-stress form.
Safety	Utilization	`U = σ/σallow`	Scalar screening ratio.

Use formulas as models, not facts detached from assumptions. Boundary conditions, units, load path, material behavior, safety factors, standards and manufacturing variation still have to be checked.