Mechanical Calculators
Beams, bolts, shafts, and the things that bend.
- Beam DeflectionMaximum deflection for the two textbook cases: cantilever and simply-supported.δ = PL³ / 3EI (cantilever) δ = PL³ / 48EI (simple, center load)
- Strain Gauge OutputBridge output for a strain measurement — quarter, half, or full bridge.Vo ≈ Vex · n · GF · ε / 4 (n = active gauges)
- Bolt Torque from PreloadTightening torque for a target preload — the K-factor shortcut everyone actually uses.T = K · F · d
- Shaft Power ⇄ TorqueP = Tω — convert between power, torque, and rotational speed.P = T · ω ω = 2πN/60
- Gear RatioRatio, output speed, and output torque from tooth counts.i = N₂/N₁ n_out = n_in/i T_out = T_in · i · η
- Coil Spring RateRate of a round-wire helical compression/extension spring, plus spring index and Wahl factor.k = G·d⁴ / (8·D³·n)
- Belt & Pulley SpeedDriven speed, ratio, and belt velocity from pulley diameters.n₂ = n₁ · d₁/d₂ v = π·d₁·n₁/60
- Linear Thermal ExpansionΔL = αLΔT for common structural materials.ΔL = α · L · ΔT
- Axial Stress & Strainσ = F/A and ε = σ/E, with an optional yield-strength safety check.σ = F/A ε = σ/E
- Shaft Torsion (Solid Round)Max shear stress τ = Tr/J for a solid circular shaft, plus optional twist angle.τ = T·r / J J = πd⁴/32 θ = TL/GJ
- Euler Column BucklingCritical buckling load Pcr = π²EI/(KL)² with standard end conditions.Pcr = π²·E·I / (K·L)²
- Bearing L10 LifeBasic rating life in million revolutions and hours — ball or roller exponent.L10 = (C/P)^p p = 3 (ball), 10/3 (roller)
- Flywheel Kinetic EnergyStored rotational energy E = ½Iω² from inertia and speed.E = ½ · I · ω²
- Hydraulic Cylinder ForceExtend and retract force from pressure, bore, and rod diameter.F_ext = P · πD²/4 F_ret = P · π(D² − d²)/4
- Lever Mechanical AdvantageMoment balance about the fulcrum — ideal MA and load force.MA = L_effort / L_load F_load = F_effort · MA
- Kinetic EnergyKE = ½mv² — the energy of anything moving in a straight line.KE = ½ · m · v²
- Gravitational Potential EnergyPE = mgh — energy banked by height.PE = m · g · h
- Projectile Range (No Drag)Range, max height, and flight time on flat ground — vacuum ballistics.R = v²·sin 2θ / g h = v²·sin²θ / 2g t = 2v·sinθ / g
- Friction ForceMaximum static friction μN, with an optional slides-or-holds check.F_max = µ · N
- Mass Moment of Inertia (Shapes)Rotational inertia for the four shapes that cover most machine parts.I_disc = ½mr² I_tube = ½m(r₁²+r₂²) I_sphere = ⅖mr² I_rod = mL²/12
- Rectangular Section ModulusS = bh²/6 and I = bh³/12, with optional bending stress from a moment.S = b·h²/6 I = b·h³/12 σ = M/S
- Thin-Wall Pressure Vessel StressHoop and longitudinal stress σ = Pr/t, with the r/t > 10 validity gate.σ_hoop = P·r/t σ_long = P·r/2t
- Plate Clutch Torque (Uniform Wear)Transmissible torque T = µ·F·n·R_mean for disc clutches and brakes.T = µ · F · n · R_mean R_mean = (r_o + r_i)/2
- Fillet Weld Throat StressShear on the throat, a = 0.707z — the plane where fillet welds actually fail.a = 0.707·z τ = F / (a · l)
- Shaft Critical Speed (Deflection Method)First critical speed from static deflection — the Rayleigh one-liner.f = (1/2π) · √(g/δ) N_c = 60·f
- Natural Frequency (Spring–Mass)f = (1/2π)√(k/m) — the fundamental note of any stiffness and mass.fn = (1/2π) · √(k/m)
- Pendulum PeriodT = 2π√(L/g) — small-swing period of a simple pendulum.T = 2π · √(L/g)
- Speed of Sound in MaterialsLongitudinal bar velocity c = √(E/ρ) for common engineering solids.c = √(E / ρ)
- Reynolds NumberRe = ρVL/µ with one-click air or water properties, or your own fluid.Re = ρ·V·L / µ = V·L / ν
- Pipe Pressure Drop (Darcy-Weisbach)Friction loss in a round pipe — Swamee-Jain friction factor, automatic laminar branch.ΔP = f·(L/D)·½ρV², f = 0.25/[log₁₀(ε/3.7D + 5.74/Re^0.9)]² (laminar: f = 64/Re)
- Pipe Flow VelocityMean velocity in a round pipe from volumetric flow and diameter.V = Q / A, A = πD²/4
- Bernoulli Pressure ChangeDownstream pressure from an upstream state, two velocities, and an elevation change.p₁ + ½ρv₁² + ρgz₁ = p₂ + ½ρv₂² + ρgz₂
- Orifice FlowFlow through a sharp-edged orifice from a pressure differential — Q = C_d·A·√(2ΔP/ρ).Q = C_d · A · √(2·ΔP/ρ)
- Pump Hydraulic & Shaft PowerWater power from flow and head, and the shaft power once efficiency takes its cut.P_hyd = ρ·g·Q·H P_shaft = P_hyd / η
- NPSH AvailableNet positive suction head at the pump inlet — the cavitation margin.NPSHa = (P_surface − P_vapor)/(ρ·g) + z_static − h_friction
- Airflow Units (CFM ⇄ m³/h ⇄ L/s)The three dialects of volumetric airflow, converted exactly.1 CFM = 1.699011 m³/h = 0.471947 L/s (1 ft = 0.3048 m exactly)
- SCFM ⇄ ACFMStandard-to-actual airflow via the ideal-gas temperature and pressure correction.ACFM = SCFM · (P_std / P_act) · (T_act / T_std) (T absolute)
- Drag ForceF = ½ρV²·C_d·A — aerodynamic drag and the power it costs.F_D = ½ · ρ · V² · C_d · A
- Terminal VelocitySteady fall speed where drag balances weight: √(2mg / ρ·C_d·A).V_t = √( 2·m·g / (ρ·C_d·A) )
- Water Hammer (Joukowsky)Pressure surge from a sudden velocity change: ΔP = ρ·a·Δv.ΔP = ρ · a · Δv (valid when closure time < 2L/a)
- Torque ConverterN·m, lbf·ft, lbf·in, kgf·m, and ozf·in — wrench specs in every dialect.1 lbf·ft = 1.355818 N·m 1 kgf·m = 9.80665 N·m
- Energy ConverterJ, kJ, Wh, kWh, BTU, cal, ft·lbf — plus the eV for the physicists.1 BTU(IT) = 1055.056 J 1 kWh = 3.6 MJ 1 cal(th) = 4.184 J
- Power ConverterW, kW, both horsepowers, BTU/h, and tons of refrigeration.1 hp = 745.6999 W 1 hp(metric)/PS = 735.4988 W 1 TR = 3516.853 W
- Length Convertermm to nautical miles, with the PCB designer's mil in between.1 in = 25.4 mm (exact, 1959) 1 nmi = 1852 m
- Log-Mean Temperature DifferenceLMTD for counterflow or parallel-flow heat exchangers from the four terminal temperatures.LMTD = (ΔT₁ − ΔT₂) / ln(ΔT₁/ΔT₂)
- Heat Exchanger DutyQ = ṁ·cp·ΔT for one stream — the first number of every exchanger problem.Q = ṁ · cp · ΔT
- Heat Conduction (Fourier)Steady heat flow through a flat layer — Q = k·A·ΔT/L.Q = k · A · ΔT / L
- Pump Head ⇄ PressureConvert a pressure to metres of head for a fluid, plus static lift.H = P / (ρ·g) + z (ρ = SG × 1000 kg/m³)
- Dew Point & Absolute HumidityDew point from temperature and relative humidity — Magnus formula, plus g/m³ of water.Td = b·γ / (a − γ), γ = ln(RH/100) + a·T/(b+T) (a = 17.625, b = 243.04)
- Basquin S-N Fatigue LifeHigh-cycle fatigue life from the Basquin power law — enter stress amplitude to get cycles, or cycles to get allowable stress.σa = σ'f · (2Nf)ᵇ Nf = ½ · (σa / σ'f)^(1/b)
- Neuber Notch Stress (Local Strain)Estimate elastic-plastic notch-root stress and strain from nominal load using Neuber's rule and a Ramberg-Osgood curve.(Kt·S)² / E = σ·ε , ε = σ/E + (σ/K′)^(1/n′) → solve σ·ε = Kσ·Kε·(S²/E)
- Goodman Mean-Stress CorrectionFold a nonzero mean stress into an equivalent fully-reversed stress and get the Goodman fatigue safety factor, with Gerber for comparison.σₐ/Sₑ + σₘ/Sᵤₜ = 1/n_f (modified Goodman), σₐᵣ = σₐ / (1 − σₘ/Sᵤₜ), Gerber: σₐ/Sₑ + (σₘ/Sᵤₜ)² = 1/n
- Rankine Cycle Efficiency (Ideal)First-cut thermal efficiency, work, and heat for an ideal Rankine steam power cycle from your four state enthalpies.η = wₙₑₜ / qᵢₙ = [(h₁−h₂) − (h₄−h₃)] / (h₁−h₄) , BWR = wₚ / wₜ
- Carnot Efficiency LimitUpper bound on thermal efficiency and refrigeration COP between two reservoir temperatures.η = 1 − T_C / T_H , COP_ref = T_C / (T_H − T_C) , COP_hp = T_H / (T_H − T_C)
- Refrigeration Cycle COPCoefficient of performance for a cooling or heating cycle from duty and compressor power, with the Carnot ceiling for comparison.COP_cool = Q_c / W , COP_heat = (Q_c + W) / W = COP_cool + 1 , COP_Carnot = T_c / (T_h − T_c)
- Heat-Pump COP + Reversible LimitGet heating COP from compressor work plus the Carnot ceiling for the same temperature lift.COP_h = Q̇_H / Ẇ , COP_Carnot = T_H / (T_H − T_C) , η₂ = COP_h / COP_Carnot (T in K)
- Duct Static-Pressure LossFriction loss in a round duct run — Swamee-Jain friction factor on standard air, with in. w.g. per 100 ft for the ASHRAE-chart crowd.ΔP = f·(L/D)·½ρV², V = Q / (πD²/4), f = 0.25/[log₁₀(ε/3.7D + 5.74/Re^0.9)]²
- R-Value / U-Value (Series Wall)Stack up to five layer R-values, get the assembly U-value plus heat flux — SI and US units side by side.R_total = ΣRᵢ U = 1 / R_total q = U · ΔT Q = q · A
- Cooling Tower Approach & RangeGet range, approach, and cooling effectiveness from hot/cold water temps and the ambient wet-bulb.Range = T_hot − T_cold , Approach = T_cold − T_wb , Effectiveness = Range / (T_hot − T_wb) × 100%
- Pump/Fan Affinity LawsScale flow, head/pressure, and power when a centrifugal pump or fan changes speed or impeller diameter.Q₂/Q₁ = r , H₂/H₁ = r² , P₂/P₁ = r³ where r = N₂/N₁ or D₂/D₁
- Damping Ratio → % OvershootTranslate a second-order damping ratio into peak overshoot, peak time, and settling time.%OS = 100 · exp( −π·ζ / √(1−ζ²) ) , tp = π / (ωₙ√(1−ζ²)) , ts ≈ 4 / (ζ·ωₙ)
- Moody Friction Factor (Colebrook)Solve the Colebrook-White equation exactly by iteration — the friction factor behind the Moody chart, without squinting at log-log axes.1/√f = −2 · log₁₀( ε/(3.7·D) + 2.51/(Re·√f) ) (laminar: f = 64/Re)
- Cavitation Number (σ)Dimensionless margin against cavitation for a pump, valve, or hydrofoil operating point.σ = (p − pᵥ) / (½·ρ·V²)
- Compressor Polytropic WorkSpecific work, discharge temperature, and gas power for a polytropic compression — ideal-gas basis, any n you specify.w = (n/(n−1)) · (R·T₁/M) · [(P₂/P₁)^((n−1)/n) − 1] T₂ = T₁ · (P₂/P₁)^((n−1)/n)
- Bolt Shear (Lap Joint)Shear capacity of a bolted lap joint per AISC J3 — LRFD and ASD strengths plus utilization against your applied load.Rn = Fnv · Ab · m · n , Ab = π·d²/4 φ = 0.75 (LRFD) , Ω = 2.00 (ASD)
- Bolt Preload vs YieldTarget preload and installation torque for a fastener at a chosen fraction of proof or yield strength.F_preload = frac · σ_y · A_t , T = K · F · d