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Methodology · Engineering
Engineering Methodology
An engineering answer carries a unit, a sign convention and a tolerance. Those three are part of the result, not footnotes to it — so this page documents the chain every calculator runs, the stop rule each solver uses, and the published work each model is checked against.
§ 1
The dimensional chain
Every physical model follows the same chain: the declared unit is converted to base SI once, the constitutive law is evaluated there, and the result is converted back for display. Nothing is ever computed in a mixed unit system.
The chain matters because most engineering errors are unit errors, not algebra errors. A bar is 10⁵ Pa; a litre per minute is 1/60,000 m³/s; a viscosity quoted in mPa·s is 10⁻³ Pa·s. Each of those factors is applied once, on the line that needs it, and the reverse conversion happens only in the formatter.
Sign conventions are stated per calculator rather than globally, because they genuinely differ: downward loads are positive on the beam pages, compression is positive on the column and spring pages, and heat flow is positive into the control volume. A page that silently flipped one of those would still return a plausible number.
§ 2
Solver tolerances
A solver that does not publish its stop rule is asking to be trusted. These are the actual caps and tolerances in the code — read from the implementations, not from the model names.
| Solver | Method | Cap | Stop rule |
|---|---|---|---|
| Colebrook friction | Fixed-point substitution, seeded at f = 0.02 | 80 passes | |Δf| < 1e-10 — seven passes on the default pipe run |
| Internal rate of return | Bisection on a log-spaced grid of 5,000 brackets | 200 iterations | |NPV| < 1e-12 or bracket width < 1e-12 |
| Bond yield to maturity | Newton with a bisection fallback when a step leaves the bracket | 80 iterations | residual < 1e-12 × max(1, market price) |
| Kepler's equation | Newton on E − e·sin E = M | until convergence | |ΔE| < 1e-14 |
| Finite elements (CST) | Direct assembly and solve, plane stress or plane strain | nx 4–18, ny 3–12 (≤ 432 CST triangles) | Direct factorisation — no tolerance, but a singular matrix throws |
| Lattice Boltzmann | Collide–stream on a D2Q9 lattice | ≤ 1,000 steps | Relaxation time clamped to τ ∈ [0.501, 2.5] for stability |
| Re-entry thermal | Implicit 1-D conduction with a radiative boundary | steps = ceil(horizon ÷ dt) | dt floored at 0.01 s; coarse steps raise a warning |
| Conjugate heat transfer | Implicit tridiagonal (Thomas) sweep per layer | steps from dt ∈ [0.05, 20] s | Unconditionally stable; accuracy set by the step, not the scheme |
| Load flow | Newton–Raphson on the polar Jacobian, Ybus assembled from line data | 30 iterations | max |ΔP|, |ΔQ| < 1e-9 p.u. — the Jacobian is re-formed every pass |
| Normal depth (Manning) | Bisection on flow against depth; bracket expanded ×1.6 when the guess cannot carry the flow | 90 passes, 40 expansions | Fixed pass count — the midpoint of the final bracket is returned |
| Wet bulb | Bisection on saturated enthalpy between dew point and dry bulb | 60 passes | Fixed pass count — monotone in temperature, so no tolerance is needed |
| Shortcut distillation | Underwood root by bisection, then McCabe–Thiele stage stepping from the top composition | 120 root passes · 200 stages | Stop when x ≤ x_b; a column that never reaches it is returned at the 200-stage cap |
§ 3
The fidelity ladder
Engineering models trade cost for fidelity in a fixed order. The catalog sits on every rung — 37 of 193 pages run an actual solver, the rest are closed forms or table conversions.
| Rung | Cost | What it buys | Pages |
|---|---|---|---|
| Closed form | O(1) | One evaluation of a published relationship — no mesh, no iteration, no tolerance. | 153 |
| Table lookup | O(1) | A discrete code or unit factor applied affinely; exact up to one rounding. | 2 |
| Iterative solve | O(iterations) | The root of a residual: friction factor, eccentric anomaly, flutter boundary. | 12 |
| Matrix / eigen | O(n³) | An assembled stiffness, covariance or plane-wave basis, factorised and solved. | 15 |
| Time march | O(steps × cells) | A field advanced in time: conduction, lattice gas, RANS relaxation. | 7 |
| Monte Carlo | O(paths × steps) | A sampled distribution where the exact integral has no closed form. | 3 |
The ladder is not a quality ranking. A closed-form beam deflection is exact within Euler–Bernoulli theory and is the right answer for a hand calculation; a finite-element model is the right answer when the geometry stops being a beam. What changes is the assumption set, and each page states which one it is using.
§ 4
Citation index
Derived from the reference each calculator publishes. All 193 engineering calculators resolve to a standard or a paper here.
ACI 318Building Code Requirements for Structural Concrete
Anchor BoltBeam ShearColumn Interaction DiagramConcrete Anchor DesignConcrete Beam DesignConcrete ColumnConcrete Torsion CheckFooting DesignMat Foundation (Rigid Method)Pile Cap DesignRebar DevelopmentSeismic DetailingShear WallSlender ColumnTwo-Way Slab
15
IEC60027-3 · 60038 · 60062 · 60228 · 60255-151 · 60364-5 · 60534-2 · 60617 · 60831 · 60909-0 · 62548
Control ValveDecibelFlashing Control ValvePower Factor CorrectionProtective Device CoordinationRC Time ConstantResistor Color CodeShort-CircuitSolar PanelThree-Phase PowerVoltage DividerWire Gauge
12
ISO281 · 4413 · 5167 · 6946 · 80000-5 · 9613-1/-2 · 9906 · 10211 · 12241
Aeroacoustic NoiseBearing LifeDecibelHeat ConductionHydraulic CylinderIdeal Gas LawInsulation ThicknessOrifice FlowPump PowerThermal BridgeUnit Converter
11
NFPANational Electrical Code (NFPA 70) · Electrical Safety in the Workplace (NFPA 70E)
Arc FlashBox FillCable AmpacityConduit FillGrounding ConductorMotor FLAOptional Method Service LoadService LoadVoltage DropWire Gauge
10
ASHRAEHandbook — Fundamentals (psychrometrics, loads, coils, degree-day method)
Building Energy EstimateCooling Coil SelectionCooling LoadDuct Fitting LossesEconomiser SavingHeat LossPsychrometricRoom Load
8
AISC 360Specification for Structural Steel Buildings
Base PlateBeam Deflection & StressBolt TorqueFillet WeldPortal Frame AnalysisSteel Connection DesignSteel Member Code Check
7
ASCE/SEI 7Minimum Design Loads and Associated Criteria
6
Colebrook · Darcy · WeisbachFriction factor and pipe loss
6
IAPWSIF97 steam tables
6
ASTMA228 · B258 · C680 · D2435 · E739
5
Incropera & DeWittFundamentals of Heat and Mass Transfer
5
AISCShapes Database v15.0
4
Gere & Goodno · TimoshenkoMechanics of Materials / Theory of Elastic Stability
4
NISTHandbook 44 · SP 800-186 · TN 2109
4
Przemieniecki · CookTheory of Matrix Structural Analysis
4
AISCDesign Guide 1 — Base Plate and Anchor Rod Design
3
AndersonFundamentals of Aerodynamics
3
CENEN 1993-1-1 Design of steel structures
3
Blasius · Colburn · Dittus–Boelter · Churchill–ChuConvective heat-transfer correlations
2
Cited once each: AASHTO · Abernethy · ACI · AISC · AISI · Ames Research Staff · API · ASHRAE · ASM · ASME · MSS · AWC · Beér · WSGG · BIPM · Bishop · Box, Hunter & Hunter · BRE · BS 7910 · CEN · Chow · CIBSE · Converse & Labarre · Cook et al. · CSA Group · CTI · ASHRAE · Das · Einstein · Euler–Eytelwein · Fenske · Underwood · Gilliland · Grainger & Stevenson · Griffiths · Hasofer & Lind · Ditlevsen · Hetényi · IEEE / ANSI C57 · IES · Janbu · Johnson & Joannopoulos · Krüger et al. · Lockhart & Martinelli · Manning · Gauckler · Metropolis et al. · Meyer & Nachtsheim · Scheffé · Meyerhof · Montgomery · Moran & Shapiro · Nakajima · NEMA · NRL · Ongaro & Ousterhout · Oppenheim & Schafer · PCI · Peukert · Pozar · Rao · Timoshenko · RCSC · Reynolds · Launder & Spalding · Seed & Idriss · Shannon · Standards Australia · Sutton & Graves · Terzaghi · Theodorsen · Dowell · TMS 402 · Tomlinson · Vallado · White · Ziegler & Nichols · Cohen & Coon.
§ 5
The register
All 193 engineering calculators with the relationship each one implements and the standard it is implemented against.
Aeroacoustic Noise Calculatoraeroacoustic-turbulence-noise-engine
W=α ρ_0 (u'^(8))/(c_0^(5))L^(2), α≈38; ε=C_μ^(3/4)(k^(3/2))/(L), k= 32 u'^2; η=((ν^(3))/(ε))^(1/4); f_(peak)=0.2 (U)/(L)
Proudman, "The generation of noise by isotropic turbulence," Proc. R. Soc. A 214 (1952); Lighthill, Proc. R. Soc. A 211 (1952); Pope, Turbulent Flows (2000), ch. 6 (Kolmogorov scales); ISO 9613-2:1996 (atmospheric attenuation).
Aeroelastic Flutter Calculatornonlinear-aeroelastic-flutter-boundary-engine
M q + C q + Kq = F_(aero)(q, q), det(A - λ I) = 0, flutter at ℜ(λ) = 0
Theodorsen (1935) unsteady aerodynamics; Dowell et al., "A Modern Course in Aeroelasticity".
Airfoil Lift & Drag Calculatoraerodynamic-airfoil-lift-drag-matrix
α_(L0) = -(1)/(π)∫_0^(π)(dz)/(dx) (cosθ_0 - 1) dθ_0, C_l = 2π (α - α_(L0))
Thin-airfoil theory: Anderson, Fundamentals of Aerodynamics (6th ed.), Ch. 4; zero-lift angles tabulated in Abbott & von Doenhoff, Theory of Wing Sections.
Anchor Bolt Calculatoranchor-bolt-calculator
N_(cb) = (A_(Nc))/(A_(Nco)) ψ_(ed,N) 24λ√(f'_c) h_(ef)^(1.5)
ACI 318-19 Chapter 17, §17.6.1 (steel), §17.6.2 (breakout) and §17.6.3 (pullout).
Angle and Channel Propertiesangle-channel-properties
Centroidal properties for angle and channel sections.
ANOVA Calculatoranova-calculator
F = (MS_(between))/(MS_(within)), η^2 = (SS_(between))/(SS_(total))
One-way analysis of variance; F distribution with (k−1, N−k) degrees of freedom.
Arc Flash Calculatorarc-flash-calculator
E = 4.184 C_f E_n ((t)/(0.2))((610)/(D))^(x)
IEEE 1584-2002, Guide for Performing Arc-Flash Hazard Calculations; NFPA 70E for the PPE categories.
AS 3600 Concrete Shearas3600-concrete-check
AS 3600 cl. 8.2.4.1 concrete shear with the β1 size effect.
ASCE 7 Wind Load Calculatorasce-wind-load-calculator
q_z = 0.613 K_z K_(zt) K_d V^2, p = q_z G C_p - q_z GC_(pi)
ASCE 7-22 Chapter 26 (velocity pressure) and Chapter 27 (main wind-force-resisting system).
ASME Pipe Thickness Calculatorasme-pipe-thickness-calculator
t = (P D)/(2 (S E W + P Y)) + c
ASME B31.3 Process Piping, §304.1.2 (straight pipe under internal pressure).
Bandwidth & Data Throughput Calculatordata-throughput-bandwidth-core
C = B log_2 (1 + SNR), FSPL(dB) = 32.44 + 20log_(10) f_(MHz) + 20log_(10) d_(km), N_(dBm) = -174 + 10log_(10) B + NF
Shannon-Hartley channel-capacity theorem (Shannon, 1948); free-space path loss per ITU-R P.525; thermal-noise floor −10·log₁₀(kT₀) = −174 dBm/Hz at T₀ = 290 K (Boltzmann constant, Johnson-Nyquist noise).
Base Plate Calculatorbase-plate-calculator
A_1 = (P_u)/(0.65 × 0.85 f'_c √(A_2/A_1))
AISC 360 §J8 for bearing; AISC Design Guide 1 for the plate thickness.
Base Plate Designbase-plate-design
AISC DG1 cantilever model for a concentrically loaded base plate.
Battery Runtime Calculatorbattery-runtime-calculator
t = (C_(20)((C_(20))/(20I))^(k-1)DoD f_T f_a)/(I)
Peukert's equation (1897); depth, temperature and age derating follow IEEE 485 battery-sizing practice.
Beam Deflection & Stress Calculatorbeam-deflection-calculator
EI(d^4w)/(dx^4) = q(x), δ_(max) = (5wL^4)/(384EI) (simply supported, UDL), M = EI(d^2w)/(dx^2)
Timoshenko beam theory; Gere & Goodno, "Mechanics of Materials"; AISC 360 serviceability provisions.
Beam Shear Calculatorbeam-shear-calculator
φ V_n = 0.75(V_c+V_s), V_s=A_v f_y d/s; V_c= cases(0.17λ√(f'_c)+N_u/6A_g)b_wd,&A_v≥ A_(v,min)\(0.66λ_sλρ_w^(1/3)√(f'_c)+N_u/6A_g)b_wd,&A_v<A_(v,min) cases
ACI 318-19 / 318-19(22) Table 22.5.5.1 and §§22.5.5, 22.5.10, 9.6.3 and 9.7.6; λs per Table 22.5.5.1.3.
Bearing Life Calculatorbearing-life-calculator
L_(10) = ((C)/(P))^p, L_(10h) = (L_(10) · 10^6)/(60n)
ISO 281:2007 rolling bearing rating life; a1 reliability factors.
Bolt Torque Calculatorbolt-torque-calculator
T = K F d; A_t = (π)/(4)(d - (0.9743)/(n))^2
RCSC Specification for Structural Joints, Table J3.1 for minimum pretension and §8.2 for turn-of-nut; AISC 360 Table J3.2 for tensile strength.
Box Fill Calculatorbox-fill-calculator
V = (n_u + n_g)A + A + 2dA + cA + fA
NFPA 70 §314.16, Table 314.16(A) for box volumes and Table 314.16(B) for the conductor allowance.
Brayton Cycle Calculatorbrayton-cycle-calculator
η_(ideal) = 1 - (1)/(r_p^((k-1)/k))
Cold-air-standard Brayton analysis; the isentropic relations and the efficiency definitions are standard.
Bridge Load Ratingbridge-load-rating
AASHTO MBE load rating: RF = (φRn − γDC·DC − γDW·DW)/(γLL(LL+IM)).
Building Energy Estimatebuilding-energy-model
E = (UA HDD 24)/(1000 η)(1-g) + (UA CDD 24)/(1000 COP)(1+g)
ASHRAE Fundamentals, degree-day method.
Cable Ampacity Calculatorcable-ampacity-calculator
I_(allow)=min(I_(table)f_(temp)f_(conductors),I_(terminal)); I_(design)=1.25I_(continuous); I_(design)≤ I_(OCPD)≤min(I_(allow),I_(240.4(D)))
NFPA 70 (NEC) Tables 310.16, 310.15(B)(1), 310.15(C)(1), §§110.14(C), 210.19(A)(1), 240.4(D); resistance from Chapter 9, Table 8.
Cable Pulling Tensioncable-pulling-tension
Capstan equation through bends; sidewall pressure at the radius.
Chiller Plant Sizingchiller-plant-sizing
Tons, COP and kW/ton conversions.
Coil Geometry Calculatorcoil-geometry-calculator
A = V / v_(face); f = 0.316 Re^(-0.25); Δ p_w = (f(L)/(d_i) + Σ K)(ρ v^2)/(2)
Darcy–Weisbach with the Blasius friction factor; air-side correlation calibrated to published finned-coil curves.
Cold-Formed Steel Checkcold-formed-steel-check
AISI S100 §E2 with φc = 0.85.
Column Buckling Calculatoreuler-bernoulli-column-buckling-matrix
P_(cr) = (π^(2) E I)/((K L)^(2)), r = (I)/(A), λ = (K L)/(r), σ_(cr) = (P_(cr))/(A)
Euler elastic buckling of a prismatic column; Timoshenko & Gere, Theory of Elastic Stability; effective-length factors K per standard end-condition cases (e.g. AISC Table C-A-7.1).
Column Interaction Diagramcolumn-interaction-diagram
P_n, M_n from a sweep of c, (1)/(P_n) = (1)/(P_(nx)) + (1)/(P_(ny)) - (1)/(P_0)
ACI 318-19 §22.2 (stress block), Table 21.2.2 (φ transition), Bresler reciprocal load method.
Compressible Flow & Shock Wave Calculatorcompressible-flow-isentropic-shock-wave-engine
(T_0)/(T)=1+(γ-1)/(2)M^2, M_2^2=(1+(γ-1)/(2)M_1^2)/(γ M_1^2-(γ-1)/(2)), (p_2)/(p_1)=1+(2γ)/(γ+1)(M_1^2-1)
NACA Report 1135, "Equations, Tables, and Charts for Compressible Flow" (Ames Research Staff, 1953); Anderson, Modern Compressible Flow, 3rd ed., chs. 3-4.
Computational Fluid Dynamics Calculatorcomputational-fluid-dynamics-solver
Re=(ρ V D)/(μ), (1)/(√(f))=-2log_(10) ((ε/D)/(3.7)+(2.51)/(Re√(f))), h_f=f(L)/(D)(V^2)/(2g), Δ p=ρ g h_f
Colebrook, J. Inst. Civ. Eng. 11 (1939); Darcy-Weisbach equation as given in White, Fluid Mechanics, 7th ed., ch. 6; Moody, Trans. ASME 66 (1944).
Concrete Anchor Designanchor-design-concrete
N_(cb) = (A_(Nc))/(A_(Nco))ψ_(ed) N_b, N_b = 10λ_a√(f'_c) h_(ef)^(1.5)
ACI 318-19 §17.6.2 (breakout), §17.6.3 (pullout), §17.7 (shear), φ = 0.70.
Concrete Beam Design Calculatorconcrete-beam-design-calculator
a = (A_s f_y)/(0.85 f'_c b), φ M_n = φ A_s f_y (d - (a)/(2))
ACI 318-19 §22.2 (equivalent rectangular stress block), §9.6.1.2 (minimum steel), §21.2 (tension control).
Concrete Calculatorconcrete-calculator
V = L W T (1+w); c = (w)/(w/c); 27 = V_w + V_c + V_(air) + V_(fa) + V_(ca)
ACI 211.1, Standard Practice for Selecting Proportions for Normal, Heavyweight and Mass Concrete, Tables 6.3.3, 6.3.4 and 6.3.6.
Concrete Column Calculatorconcrete-column-calculator
φ P_(n,max) = 0.65 · 0.80 [0.85 f'_c (A_g - A_(st)) + f_y A_(st)]
ACI 318-19 §22.4.2 (tied columns), §10.6.1.1 (steel limits), §6.2.5 (slenderness).
Concrete Torsion Checkconcrete-torsion-calculator
ACI 318 §22.7.4.1 torsion threshold.
Conduit Fill Calculatorconduit-fill-calculator
fill = (n × A_(cond))/(A_(raceway)) ≤ 0.40
NFPA 70 Chapter 9 Table 1 (allowance), Table 4 (raceway area) and Table 5 (conductor area); the nipple exception is §310.14(A)(2).
Confidence Interval Calculatorconfidence-interval-calculator
x ± z (σ)/(√(n))
Normal approximation to the sampling distribution of the mean.
Consolidation Settlement Calculatorfem-settlement
s = (C_c H)/(1+e_0)log_(10) ((σ'_0+Δσ)/(σ'_0))
Terzaghi one-dimensional consolidation; ASTM D2435 for Cc and cv.
Control Valve Calculatorcontrol-valve-calculator
C_v = Q (G_f)/(Δ P); Δ P_(ch) = F_L^2(P_1 - F_f P_v); Q = 1360 C_v P_1 Y (x)/(G_g T Z)
IEC 60534-2-1 and ISA-75.01.01; recovery factor and terminal ratio from the valve manufacturer's data sheet.
Convection Calculatorconvection-calculator
h = (Nu k)/(L)
Blasius and Colburn for the plate, Dittus-Boelter for the pipe, Churchill-Chu for the vertical plate.
Cooling Coil Selection Calculatorcoil-selection-calculator
ADP = t_w + Δ t_(app); BF = (t_2 - ADP)/(t_1 - ADP); Q = m_(da)(h_1 - h_2)
ASHRAE Handbook — Fundamentals, psychrometrics and cooling-coil apparatus dew point; Carrier psychrometric chart construction.
Cooling Load Calculatorcooling-load-calculator
Q_s = 1.2 Q c_p Δ T, Q_l = 1.2 Q Δ W 2501
ASHRAE psychrometrics; Magnus formula for saturation pressure.
Cooling Tower and Chiller Plantcooling-tower-sizing
E = (Q)/(h_(fg)), B = (E)/(COC - 1)
Latent heat of vaporisation at about 2,450 kJ/kg; blowdown from the cycles of concentration.
Crane Load Calculatorcrane-load-calculator
ASCE 7 §4.9 lateral force; impact factor on the vertical.
CSA S16 Steel Checkcsa-steel-check
CSA S16 cl. 13.3.1 compression and cl. 13.2 tension, φ = 0.90.
Daylighting Calculatordaylighting-calculator
Split-flux daylight factor, BRE guidance.
Decibel Calculatordecibel-calculator
L = 10log_(10)(P)/(P_0), L = 20log_(10)(A)/(A_0), L_(Σ) = 10log_(10) (10^(L_1/10) + 10^(L_2/10)), Δ L = -20log_(10)(r_2)/(r_1)
IEC 60027-3 (logarithmic quantities); ISO 9613-1 for atmospheric attenuation.
Discrete Fourier Transform (DFT) Calculatordiscrete-fourier-transform-signal-matrix
X[k] = Σ_(n=0)^(N-1) x[n] e^(-j 2π k n / N), x[n] = (1)/(N)Σ_(k=0)^(N-1) X[k] e^(+j 2π k n / N), Σ_(n=0)^(N-1) |x[n]|^(2) = (1)/(N)Σ_(k=0)^(N-1) |X[k]|^(2)
Oppenheim & Schafer, Discrete-Time Signal Processing (DFT, Parseval's relation); Cooley–Tukey radix-2 FFT (Math. Comp. 19, 1965); window definitions from Harris, "On the use of windows", Proc. IEEE 66 (1978).
Distillation Calculatordistillation-calculator
N_(min) = (ln[(x_D)/(1-x_D)·(1-x_B)/(x_B)])/(lnα)
Fenske (1932), Underwood (1948), Gilliland (1940); the Eduljee (1975) fit to the Gilliland correlation.
Duct Fitting Lossesduct-fitting-losses
Δp = K·0.5·ρ·V²; K from ASHRAE fitting tables.
Duct Network Calculatorduct-network-calculator
Δ p = Σ ( f (L)/(D) + Σ K ) (ρ v^2)/(2)
Darcy–Weisbach friction plus local loss coefficients; standard duct design practice.
Duct Sizing Calculatorduct-sizing-calculator
D = √(4Q/(π v)), Δ p = f (L)/(D) (ρ v^2)/(2)
Darcy–Weisbach with the Swamee-Jain approximation to Colebrook–White.
Eccentric Bolt Groupbolt-group-eccentric
f_d = (P)/(n), f_t = (M r)/(Σ(x^2 + y^2)), f = (f_(dx)+f_(tx))^2 + (f_(dy)+f_(ty))^2
AISC Manual Part 7, elastic method (conservative against the instantaneous centre method).
Economiser Saving Calculatoreconomiser-calculator
E_(net) = (Q · h)/(COP) - P_(fan) · h
ASHRAE air-side economiser analysis; screening estimate.
Elastoplastic Damage Calculatornonlinear-elastoplastic-damage-core
ε = ε^e + ε^p, σ = E(ε - ε^p), f = | σ| - (σ_(y0) + Hα) ≤ 0, Δγ = (f_(trial))/(E + H), σ = (1 - D) σ
Small-strain J2 (von Mises) plasticity with linear isotropic hardening and scalar (Lemaitre-type) damage; Simo & Hughes, Computational Inelasticity (radial return); Lemaitre, A Course on Damage Mechanics; ZSoil CPDM manual (combined damage D = 1−(1−D_c)(1−sD_t)); GetFEM small-strain plasticity notes.
Elliptic Curve Cryptography Calculatorcryptographic-elliptic-curve-scalar-field-engine
Q = dG, λ = (y_2 - y_1)/(x_2 - x_1), x_3 = λ^(2) - x_1 - x_2, y_3 = λ(x_1 - x_3) - y_1 p, y^(2) ≡ x^(3) + ax + b p
SEC 1: Elliptic Curve Cryptography (Certicom, v2.0) short-Weierstrass group law and double-and-add scalar multiplication; NIST SP 800-186; secp256k1 parameters from SEC 2.
Eurocode 3 Steel Checkeurocode-steel-check
N_(b,Rd) = (χ A f_y)/(γ_(M1)), χ = (1)/( + √( ^2 - λ^2))
EN 1993-1-1 §6.2 and §6.3.1, γM0 = γM1 = 1.0.
Fan Curve Calculatorfan-curve-calculator
Q = √(a/(K + b)), Q ∝ n, Δ p ∝ n^2, P ∝ n^3
Fan affinity laws; backward-curved centrifugal curve approximation.
Fatigue Life Calculatorfatigue-life-calculator
S_e = k_a k_b k_c (0.5 S_u); S_f = a N^b; D = Σ (n_i)/(N_i)
Shigley's Mechanical Engineering Design, §6-9 to §6-12 for the Marin factors and the S-N construction; ASTM E739 for the stress-life method.
Fillet Weld Calculatorfillet-weld-calculator
R_n = 0.60 F_(EXX) (0.707 w) l
AISC 360 §J2.4 for fillet weld strength, Table J2.4 for minimum size and §J2.2b for maximum size.
Fin Efficiency Calculatorfin-efficiency-calculator
η_f = (tanh(mL_c))/(mL_c), m = (2h)/(kt)
The hyperbolic solution for a straight rectangular fin with a convective tip (Incropera & DeWitt).
Finite Element Method (FEM) Calculatorfinite-element-method-mesh-solver
ε = Bu, σ = D ε, K_e = t A B^ TDB, KU = F, σ_(vM) = σ_x^2 - σ_xσ_y + σ_y^2 + 3τ_(xy)^2
Constant-strain triangle (CST) plane-elasticity formulation and global assembly K_e = t A Bᵀ D B, K U = F; Zienkiewicz, Taylor & Zhu, The Finite Element Method; Cook, Malkus, Plesha & Witt, Concepts and Applications of Finite Element Analysis; Logan, A First Course in the Finite Element Method.
Finite Element Stress Calculatorfinite-element-stress-tensor-solver
ε = Bu, σ = D ε, K_e = t A B^( )DB, σ_(vM) = σ_(xx)^2 - σ_(xx)σ_(yy) + σ_(yy)^2 + 3τ_(xy)^2
Constant-strain triangle plane elasticity and the finite-element patch test; Zienkiewicz & Taylor, The Finite Element Method; Cook, Malkus & Plesha, Concepts and Applications of Finite Element Analysis; MIT OCW / solidmechanics.org linear-elasticity notes.
Flange Rating (ASME B16.5)flange-rating-calculator
P_(allow) = f(class,material,T)
ASME B16.5 Table 2, carbon steel and austenitic stainless groups.
Flash Steam Calculatorflash-steam-calculator
x = (h_(f1) - h_(f2))/(h_(fg2))
Energy balance across the trap; saturated properties from IAPWS-IF97.
Flashing Control Valve Calculatorflashing-valve-calculator
C_v = (Q)/(F_L√((P_1 - F_F P_v)/G))
IEC 60534-2-1, industrial-process control valves, sizing equations for incompressible fluid flow.
Floor Vibration Checkvibration-serviceability
AISC Design Guide 11 frequency criterion.
Footing Design Calculatorfooting-design-calculator
A = (1.1P)/(q_a); V_u = P_u - q_u(c+d)^2
ACI 318-19 for one-way shear (§22.5), two-way shear (§22.6), flexure and the shrinkage minimum; allowable bearing is a geotechnical input.
Fractional Factorial Calculatorfractional-factorial-calculator
I = ABCD; e = (2)/(n)Σ_i x_i y_i; R = min |w|
Box, Hunter & Hunter, Statistics for Experimenters, ch. 12 (fractional factorials, resolution and blocking).
Fracture Mechanics Calculatorfracture-mechanics-calculator
K_I = Y σ√(π a); K_r = (K_I)/(K_(mat)); L_r = (σ)/(σ_Y)
BS 7910:2019, Guide to methods for assessing the acceptability of flaws in metallic structures, Option 1.
Frame Analysis Calculatorframe-analysis-calculator
K_e u = F; (K_e - K_g)u = F; det(K_e - λ K_g) = 0
Direct stiffness method for 2D frames; consistent geometric stiffness (Cook et al., Concepts and Applications of Finite Element Analysis, ch. 11).
Frame Load Combinations Calculatorframe-combinations-calculator
q_z = 0.613 K_z K_(zt) K_d V^2; C_s = (S_(DS))/(R/I_e); 1.2D + 1.0E + L
ASCE 7-22 §26.10 (velocity pressure), §12.8.1.1 (base shear) and §2.3.1 (strength combinations).
Frame Model Calculatorframe-model-calculator
node x y [support]; member i j [w=] [mass=] [P=@]; (K_e - K_g)φ = ω^2 Mφ
Direct stiffness method; Guyan reduction of massless rotations; Jacobi eigensolver for the generalised problem.
Gas Radiation Calculatorgas-radiation-calculator
ε_g = Σ_i a_i(T) (1 - e^(-b_i pL))
Three-term weighted-sum-of-gray-gases fit; flux in the Hottel form.
Gear Ratio Calculatorgear-ratio-calculator
i = (N_(driven))/(N_(driver)), i_(total) = i_1 i_2, ω_(out) = (ω_(in))/(i_(total)), T_(out) = T_(in) i_(total) η^(n)
Standard involute spur-gear kinematics; AGMA 2001 for load ratings.
Grounding Conductor Calculatorgrounding-conductor-calculator
GEC = min (Table 250.66, electrode cap )
NFPA 70 Table 250.66, Table 250.122, §250.66(A) to (C).
Grounding Grid Calculatorgrounding-grid-design
E_(touch) = (1000 + 1.5 C_s ρ_s)(0.116)/(√(t_s))
IEEE 80 §8 step and touch criteria; simplified Schwarz grid resistance.
Harmonic Analysisharmonic-analysis
THD and the K-factor transformer rating.
Heat Conduction Calculatorheat-conduction-calculator
Q = (kAΔ T)/(L), R = (L)/(kA), q'' = (Q)/(A), U = (1)/(RA), T_i = T_(hot) - QR_1
Incropera & DeWitt, "Fundamentals of Heat and Mass Transfer", ch. 3; ISO 6946 for building U-values.
Heat Exchanger Calculatorheat-exchanger-calculator
Q = mc_pΔ T, LMTD = (Δ T_1-Δ T_2)/(ln(Δ T_1/Δ T_2)), NTU = (UA)/(C_(min)), ε = (Q)/(Q_(max))
Incropera & DeWitt, "Fundamentals of Heat and Mass Transfer", ch. 11.
Heat Loss Calculatorheat-loss-calculator
Q = U A Δ T + 0.33 ACH V Δ T
CIBSE and ASHRAE steady-state heating load method.
Heat Transfer Calculatorradiative-conjugate-heat-transfer-matrix
ρ c_p(∂ T)/(∂ t) = (∂)/(∂ x) (k(∂ T)/(∂ x)), q^( )_(conv) = h (T_g - T_s), q^( )_(rad) = εσ (T_(env)^(4)-T_s^(4)), J_i = ε_iσ T_i^(4) + (1-ε_i)Σ_j F_(ij)J_j
Stefan-Boltzmann radiative exchange and the gray-enclosure radiosity method (Incropera & DeWitt, Fundamentals of Heat and Mass Transfer; Modest, Radiative Heat Transfer). 1D transient conduction discretized in conservative finite-volume form with implicit Euler; σ = 5.670374419×10⁻⁸ W·m⁻²·K⁻⁴ (CODATA/NIST).
HSS Section Propertieshss-section-properties
Section properties from geometry; closed-section torsion.
Hydraulic Cylinder Calculatorhydraulic-cylinder-force-calculator
A_b = (π d_b^2)/(4), A_a = (π (d_b^2 - d_r^2))/(4), F = p A η, v = (Q)/(A)
Pascal's principle and continuity; ISO 4413 hydraulic fluid power — general rules and safety requirements.
Hypersonic Reentry Heating Calculatorhypersonic-reentry-thermal-protection-solver
q_c = C (ρ_∞)/(R_n) V_∞^(3), ρ c_p (∂ T)/(∂ t) = (∂)/(∂ x) (k (∂ T)/(∂ x)), -k(∂ T)/(∂ x) |_(s) = q_c - εσ (T_s^(4)-T_(ref)^(4))
Sutton & Graves, “A General Stagnation-Point Convective-Heating Equation for Arbitrary Gas Mixtures,” NASA TR R-376 (1971); 1D transient conduction solved by implicit finite-volume marching as in NASA NTRS TPS thermal-response analyses. Earth-air coefficient C ≈ 1.83×10⁻⁴ (SI).
Ideal Gas Law Calculatorideal-gas-law-calculator
PV = nRT, n = (PV)/(RT), ρ = (PM)/(RT), R = 8.314462618 J/(mol·K)
ISO 80000-5; CODATA 2018 value of the molar gas constant.
Insulation Thickness Calculatorinsulation-thickness-calculator
q = (Δ T)/((ln(r_2/r_1))/(2π k) + (1)/(2π r_2 h))
Cylindrical conduction with a surface film coefficient, ASTM C680 and ISO 12241 practice.
Lateral Pile Analysislateral-pile-analysis
Hetényi beam on an elastic foundation; long-pile closed form.
Lattice Boltzmann Flow Calculatorlattice-boltzmann-mesoscopic-flow-core
ν=c_s^2(τ- 12), c_s^2= 13; f_i^(eq)=w_iρ[1+(e_i )/(c_s^2)+((e_i )^2)/(2c_s^4)-(u )/(2c_s^2)]; E(t)=E_0 e^(-2ν k^2 t)
Kruger et al., The Lattice Boltzmann Method: Principles and Practice (Springer, 2017); Bhatnagar, Gross & Krook, Phys. Rev. 94 (1954); Taylor & Green, Proc. R. Soc. A 158 (1937).
Layered Slope Stability Calculatorlayered-slope-stability-calculator
F = (Σ [c b + (W - ub)tanφ]/n_α)/(Σ W tanα)
Janbu simplified method with the f₀ correction (Janbu 1973), on a non-circular surface.
Lighting Calculatorlighting-calculator
RCR = (5h(L+W))/(LW); N = (E A)/( CU LLF)
IES Lighting Handbook, lumen method; the cavity-ratio definition and the CU tables are in IES LM-57 and the IES Fundamentals volume.
Liquefaction Assessmentliquefaction-assessment
Seed–Idriss simplified CSR with the CRR curve for (N1)60cs.
Masonry Wall Calculatormasonry-wall-calculator
(f_a)/(F_a) + (f_b)/(F_b) ≤ 1
TMS 402-16 §8.3.4 for axial compression and §8.3.3 for flexural compression, allowable-stress design.
Mat Foundation (Rigid Method)mat-foundation-fem
q = (P)/(A) ± (M_x)/(S_x) ± (M_y)/(S_y)
ACI 336.2R rigid-mat method; ACI 318 §22.6 punching shear.
Motor FLA Calculatormotor-fla-calculator
I_(cond) = 1.25 I_(FLA); I_(prot) = k I_(FLA)
NFPA 70 Article 430: Table 430.250 and 430.248 for FLA, §430.22, §430.32, §430.52 and §430.83.
Motor Sizing Calculatormotor-sizing-calculator
T_(peak) = T_(load) + Jα, P = (Tω)/(η)
Newton’s second law for rotation; standard servo and induction motor sizing practice.
Motor Starting Voltage Dipmotor-starting-calculator
IEEE 399 (Brown Book) motor-starting voltage dip: dip ≈ kVA_start / (kVA_fault + kVA_start), with NEMA MG 1 locked-rotor current.
Navier-Stokes (RANS) Calculatornavier-stokes-reynolds-averaged-closure-core
u_j ∂ u_i∂ x_j = -(1)/(ρ) ∂ p∂ x_i + ν∇^2 u_i - ∂ u_i'u_j'∂ x_j, u^+ = (1)/(κ)ln y^+ + A^+
Reynolds (1895) averaging; Launder & Spalding (1974) wall-function treatment.
Network Latency Calculatordistributed-consensus-network-latency-solver
q = N/2 + 1, A_(quorum) = Σ_(j=q)^(N) Nj p^( j)(1-p)^( N-j), ρ = (λ)/(μ), W_q = (ρ)/(μ-λ)
Raft majority-commit and timing rules (Ongaro & Ousterhout, “In Search of an Understandable Consensus Algorithm,” 2014); Paxos prepare/accept phases (Lamport, “Paxos Made Simple,” 2001); binomial quorum availability; M/M/1 queue mean wait W_q = ρ/(μ−λ) (Kleinrock, Queueing Systems).
NPSH Calculatornpsh-calculator
NPSH_a = (p_(atm) - p_v)/(ρ g) + z - h_f
Hydraulic Institute standards for rotodynamic pumps; standard cavitation margin practice.
OEE Calculatoroee-calculator
OEE = A × P × Q
Total productive maintenance (Nakajima) definition of OEE.
Ohm's Law Calculatorohms-law-calculator
V = IR, P = VI = I^2R = (V^2)/(R), cost = (P t)/(1000)× tariff
Ohm's law; Joule's law; standard DC circuit analysis.
Open Channel Flow Calculatoropen-channel-flow-calculator
Q = (1)/(n) A R^(2/3) S^(1/2)
Manning's equation; roughness values from Chow, Open-Channel Hydraulics, and FHWA HDS-3.
Optimal Design Calculatoroptimal-design-calculator
max |X'X|; D-eff = (|X'X|^(1/p))/(n); Var( y) = σ^2 x'(X'X)^(-1)x
Coordinate exchange (Meyer & Nachtsheim, 1995); Scheffé mixture models (Scheffé, 1958).
Optional Method Service Load Calculatorservice-load-optional-calculator
L = min(G, 10,000) + 0.4 (G - 10,000) + L_(HVAC)
NFPA 70 §220.82, optional feeder and service load calculations.
Orbital Mechanics Calculatororbital-mechanics-keplerian-state-engine
M = E - esin E, r = (a(1-e^(2)))/(1+ecosν), v = μ((2)/(r)-(1)/(a)), T = 2π (a^(3))/(μ)
Classical two-body orbital mechanics: Kepler’s equation and the vis-viva (energy) integral. Vallado, Fundamentals of Astrodynamics and Applications; Curtis, Orbital Mechanics for Engineering Students.
Orifice Flow Calculatororifice-flow-calculator
Q = C_d A (2Δ P)/(ρ); m = C_d A P_1 (k)/(RT)((2)/(k+1))^(k+1)/(2(k-1))
ISO 5167 for orifice plates; the compressible-flow expression is the isentropic choked-nozzle result from Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow.
Photonic Band Structure Calculatorphotonic-crystal-band-structure-solver
Σ_(G') (k+G)·(k+G') ε^(-1)_(G-G') H_(G') = ((ω)/(c))^(2) H_(G), ω = (ω a)/(2π c)
Plane-wave expansion of the Maxwell eigenproblem in periodic dielectrics; Johnson & Joannopoulos, Opt. Express 8 (2001); Busch et al., C. R. Physique 3 (2002). Empty-lattice (free-photon) band folding used as the analytic validation case.
PID Tuning Calculatorpid-tuning-calculator
K_c = 1.2 (τ)/(Kθ); T_i = 2θ; T_d = 0.5θ
Ziegler & Nichols (1942); Cohen & Coon (1953); Rivera, Morari & Skogestad (1986) for IMC.
Pile Cap Designpile-cap-design
ACI 318 §22.6 punching at d/2 from the column face.
Pile Capacity Calculatorpile-capacity-calculator
Q_(ult) = q_p A_p + f_s π D L
Meyerhof (1976) as summarised in FHWA deep-foundation design guidance.
Pile Group Calculatorpile-group-calculator
E_g = 1 - (θ[(n-1)m + (m-1)n])/(90 mn)
Converse-Labarre for group efficiency; undrained block failure from the group perimeter and base.
Pile Settlement Calculatorpile-settlement-calculator
s = (q B (1 - ν^2) I)/(E_s) + (Q L)/(2 A E_p)
Equivalent raft method (Tomlinson), with the load spreading at 1 in 4 from two-thirds of the pile length.
Pipe Friction Loss Calculatorpipe-friction-calculator
V = (Q)/(A), h_f = f(L)/(D)(V^2)/(2g), Δ P = ρ g h_f
Darcy (1857); Weisbach (1845); Colebrook-White (1939).
Pipe Stress Analysispipe-stress-analysis
Δ = α L Δ T, S = (3 E D Δ)/(L^2), S_A = f(1.25 S_c + 0.25 S_h)
ASME B31.3 §302.3.5 allowable stress range; guided-cantilever screening formula.
Pipe Support Loadspipe-support-loads
Pipe, insulation and contents over the span.
Plasma (MHD) Field Calculatorplasma-magnetohydrodynamic-field-solver
V_A = (B_0)/(√(μ_0ρ)), c_s = (γ p)/(ρ), β = (2μ_0 p)/(B_0^(2)), c_(f,s)^(2) = (1)/(2) [(c_s^(2)+V_A^(2)) ± (c_s^(2)+V_A^(2))^(2)-4c_s^(2)V_A^(2)cos^(2)θ ]
NRL Plasma Formulary (2023), Alfvén and magnetosonic speeds; standard ideal-MHD dispersion relation (e.g., Boyd & Sanderson, The Physics of Plasmas).
Portal Frame Analysisportal-frame-analysis
Direct-stiffness linear static analysis under AISC 360 §C2, 12-degree-of-freedom 3D beam element.
Power Factor Correction Calculatorpower-factor-correction-calculator
Q_c = P(tanφ_1 - tanφ_2), C = (Q_c)/(2π f V^2)
IEEE 18 (shunt power capacitors); IEC 60831.
Power Flow Calculatorpower-flow-calculator
P_i = Σ_j V_i V_j (G_(ij)cosθ_(ij) + B_(ij)sinθ_(ij))
Grainger & Stevenson, Power System Analysis; the Newton-Raphson formulation follows the standard polar Jacobian.
Pressure Vessel Shellpressure-vessel-shell
t = (P R)/(S E - 0.6P)
ASME Boiler and Pressure Vessel Code, Section VIII Division 1, UG-27(c)(1).
Prestressed Concrete Stressesprestressed-concrete-calculator
Elastic fibre stresses with effective prestress.
Process Capability Calculatorprocess-capability-calculator
C_p = (USL - LSL)/(6σ), C_(pk) = (min(USL - μ, μ - LSL))/(3σ)
Standard capability indices; Cpk 1.33 minimum for existing processes, 1.67 for new ones.
Projectile Motion Calculatorprojectile-motion-calculator
x = v_0cosθ t, y = h_0 + v_0sinθ t - (1)/(2)gt^2, t_f = (v_y + √(v_y^2 + 2gh_0))/(g), R = v_0cosθ t_f
Standard vacuum ballistics; Newtonian kinematics.
Protective Device Coordinationprotective-device-coordination
t = (TMS · 0.14)/((I/I_s)^(0.02) - 1), m = (t_(up))/(t_(down))
IEC 60255-151 standard inverse curve; margin convention from common protection practice.
Psychrometric Calculatorpsychrometric-calculator
W = (0.621945 p_w)/(p-p_w); h = 1.006 T + W(2501 + 1.86 T); ln p_(ws) = (C_8)/(T)+C_9+C_(10)T+C_(11)T^2+C_(12)T^3+C_(13)ln T
ASHRAE Handbook — Fundamentals 2017, Chapter 1: Equations 5 and 6 (Hyland–Wexler), 20, 32 and 33.
Pump and System Curvepump-system-curve
H_(pump) = H_0 - aQ^2, H_(system) = H_(static) + kQ^2
Quadratic pump and system curves; affinity laws Q ∝ N, H ∝ N², P ∝ N³.
Pump Power Calculatorpump-power-calculator
P_h = ρ g Q H, P_s = (P_h)/(η_p), P_m = (P_s)/(η_m), n_s = (n√(Q))/(H^(0.75))
Hydraulic Institute HI 14.6 / ISO 9906 pump performance standards.
Quantum Monte Carlo Calculatorquantum-monte-carlo-many-body-solver
E_0 = (d)/(2) [ω + (N-1) ω^(2)+gN ], E_(VMC) = _T | H | _T _T | _T = E_(loc)(R) _(| _T|^(2))
Harmonic-interaction (Moshinsky) model exact solution; variational Monte Carlo and the Metropolis algorithm — Metropolis et al., J. Chem. Phys. 21 (1953); Needs et al., J. Phys.: Condens. Matter 22 (2010).
Quantum Tunneling Calculatorquantum-tunneling-probability-density-engine
T = [ 1 + (V_0^(2) sinh^(2)(κ L))/(4E(V_0-E)) ]^(-1), κ = (2m(V_0-E))/(ℏ^(2)), E < V_0
Griffiths, Introduction to Quantum Mechanics, finite rectangular barrier transmission (exact boundary-matching solution); WKB opaque-barrier limit T ≈ 16·E(V0−E)/V0²·e^(−2κL).
Radiation Calculatorradiation-calculator
q = (σ (T_1^4 - T_2^4))/((1-ε_1)/(ε_1 A_1) + (1)/(A_1 F_(12)) + (1-ε_2)/(ε_2 A_2))
Stefan-Boltzmann law with the gray-diffuse two-surface enclosure; σ from CODATA 2018.
Rankine Cycle Calculatorrankine-cycle-calculator
η = ((h_1 - h_2) - (h_4 - h_3))/(h_1 - h_4)
Saturated steam properties from IAPWS-IF97; the cycle is the simple ideal Rankine with isentropic efficiencies applied to the endpoints.
Rankine Regeneration Calculatorrankine-regeneration-calculator
y = (h_(fw) - h_f)/(h_2 - h_f)
Energy balance on an open feedwater heater, with saturated properties from IAPWS-IF97.
Rankine Reheat Calculatorrankine-reheat-calculator
η = ((h_1 - h_2) + (h_3 - h_4) - w_p)/((h_1 - h_6) + (h_3 - h_2))
Saturated properties from IAPWS-IF97 with a constant-cp superheat approximation.
RC Time Constant Calculatorrc-time-constant-calculator
τ = RC, V_C(t) = V_(in)(1 - e^(-t/τ)), f_c = (1)/(2πτ), E = (1)/(2)CV^2
Standard first-order linear network solution; IEC 60617 for RC filter conventions.
Rebar Development Calculatorrebar-development-calculator
l_d = (f_y ψ_t ψ_e ψ_s ψ_g)/(20 λ√(f'_c)) d_b
ACI 318-19 Table 25.4.2.3 for the simplified equations and §25.5 for lap splices.
Relativistic Orbital Precession Calculatorrelativistic-orbital-precession-matrix
Δ = (6π G M)/(c^(2) a (1-e^(2))) = (6πμ)/(c^(2)p), p = a(1-e^(2))
Einstein’s 1915 weak-field perihelion advance from the Schwarzschild geodesic (first post-Newtonian order). Einstein, “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie” (1915); Weinberg, Gravitation and Cosmology; Misner, Thorne & Wheeler, Gravitation.
Relief Valve Calculatorrelief-valve-calculator
A = (W)/(C K_d P_1) (TZ)/(M)
API 520 Part I, Sizing, Selection and Installation of Pressure-Relieving Devices; orifice letters from API 526.
Resistor Color Code Calculatorresistor-color-code-calculator
R = (10d_1 + d_2)· 10^(m) (4-band), R = (100d_1 + 10d_2 + d_3)· 10^(m) (5-band)
IEC 60062, "Marking codes for resistors and capacitors".
Response Surface Calculatorresponse-surface-calculator
y = b_0 + b_1x_1 + b_2x_2 + b_(12)x_1x_2 + b_(11)x_1^2 + b_(22)x_2^2; x_s = -(1)/(2)B^(-1)b
Box & Wilson (1951), 'On the Experimental Attainment of Optimum Conditions'; Myers, Montgomery & Anderson-Cook, Response Surface Methodology.
Retaining Wall Calculatorretaining-wall-calculator
K_a = (1-sinφ)/(1+sinφ); P_a = (1)/(2)K_aγ H^2; FS_(ot) = (Σ M_r)/(Σ M_o)
Rankine earth pressure theory; stability factors and the bearing equation follow Das, Principles of Foundation Engineering, and the usual 1.5 working-stress targets.
Reynolds Number Calculatorreynolds-number-calculator
Re = (ρ V D)/(μ), f = (64)/(Re) (laminar), (1)/(√(f)) = -2log_(10) ((ε)/(3.7D) + (2.51)/(Re√(f)))
Reynolds (1883); Colebrook (1939); Moody (1944) chart.
Room Load Calculatorhvac-room-load
Q = Σ U A Δ T + SHGC · A_g · G + Q_(int) + 1.2 · V · 1.005 · Δ T
Steady-state cooling load; ASHRAE Fundamentals for the air properties.
Sample Size Calculatorsample-size-calculator
n = ((z_(1-α/2) + z_(1-β))^2 σ^2)/(Δ^2)
Standard power analysis for a two-sided test; normal approximation.
Section Buildersection-builder
I_x = (b h^3 - (b - t_w) h_w^3)/(12), S_x = (I_x)/(h/2), r_x = (I_x)/(A)
Parallel-axis theorem applied to the rectangles that make up the shape.
Section Properties Calculatorsection-properties-calculator
I_(rect) = (bh^3)/(12), I_(circ) = (π d^4)/(64), S = (I)/(c), r = (I)/(A), σ = (M)/(S)
Gere & Goodno, "Mechanics of Materials", appendix tables; EN 1993-1-1 for steel section properties.
Seismic Base Shear Calculatorseismic-base-shear-calculator
C_s = min((S_(DS))/(R/I_e), (S_(D1))/(T (R/I_e))) ≥ C_(s,min), V = C_s W
ASCE 7 equivalent lateral force procedure (§12.8); site coefficients from the hazard maps.
Seismic Detailing Calculatorseismic-detailing-calculator
Σ M_(nc) ≥ 1.2 Σ M_(nb)
ACI 318-19 Chapter 18: §18.6.2, §18.6.4, §18.7.3 and §18.8.4.
Seismic Response Calculatorkinetic-seismic-structural-response-node
M u + C u + Ku = -Mr u_g(t), S_a = ω^2 S_d, S_v = ω S_d
Linear MDOF base-excitation equation of motion with Newmark-β average-acceleration integration (β = 1/4, γ = 1/2) and pseudo-spectral ordinates; Chopra, Dynamics of Structures (base excitation, response spectrum); FEMA/NEHRP Recommended Seismic Provisions (modal response-spectrum analysis).
Series & Parallel Resistance Calculatorseries-parallel-resistance-calculator
R_(series) = Σ R_i, (1)/(R_(parallel)) = Σ (1)/(R_i), P = I^2R
Kirchhoff's circuit laws; standard passive-network analysis.
Service Load Calculatorservice-load-calculator
L = 3,000 + 0.35 (L_(light) - 3,000) + …
NFPA 70 Article 220 Part III: Table 220.12, Table 220.42, §220.53, Table 220.55, §220.54 and §220.60.
Shear Wall Calculatorshear-wall-calculator
V_c = 0.17λ√(f'_c) h d + (N_u d)/(4 l_w), φ V_n = 0.75(V_c + V_s)
ACI 318-19 §11.5.3; minimum horizontal steel §11.6.2.
Short-Circuit Calculatorshort-circuit-calculator
I''_k = (c U_n)/(√(3) Z_k), i_p = κ √(2) I''_k
IEC 60909-0 radial network; adiabatic check from IEC 60364-5-54.
Slender Column Calculatorslender-column-calculator
δ = (C_m)/(1 - P_u/(0.75P_c)) ≥ 1, P_c = (π^2 EI)/((kl_u)^2)
ACI 318-19 §6.6.4 (moment magnification), §6.6.4.4.4 (effective stiffness).
Slope Stability Calculatorslope-stability-calculator
F = (Σ [(c'b + (W - ub)tanφ')/(m_α)])/(Σ W sinα)
Bishop (1955), simplified method; the ru pore-pressure simplification is standard practice.
Snow Drift Calculatorsnow-drift-calculator
h_d = 0.43 l_u^(1/3)(p_g + 10)^(1/4) - 1.5
ASCE 7-22 §7.7.1, evaluated in feet and psf and converted.
Snow Load Calculatorsnow-load-calculator
p_f = 0.7 C_e C_t I_s p_g; p_s = C_s p_f; h_d = 0.43 l_u^(1/3)(p_g+10)^(1/4)-1.5
ASCE/SEI 7-22, Chapter 7: Equations 7.3-1 and 7.4-1, Tables 7.3-1, 7.3-2 and 1.5-1, Figure 7.4-1, §7.3.4 minimum and §7.7.1 drift.
Solar Panel Calculatorsolar-panel-calculator
P_(array) = (E_(daily))/(PSH (1-η_(loss))), N = (P_(array)· 1000)/(P_(panel)), C_(batt) = (E_(daily)· autonomy)/(DoD)
IEC 62548 (PV array design); NREL PVWatts derate conventions.
Soot and Ash Radiation Calculatorparticulate-radiation-calculator
ε = 1 - e^-(k_(gas) + k_(soot) + k_(ash))L
Soot k = 3.6 f_v/λ (Beér); ash k = 1.5 c/(ρ d); gas from the WSGG model.
Space Truss Calculatorspace-truss-calculator
K u = F, N = (EA)/(L)(Δ u c_x + Δ v c_y + Δ w c_z)
Direct stiffness method, three degrees of freedom per node, pin-jointed members.
Spring Rate Calculatorspring-rate-calculator
k = (G d^4)/(8 D^3 N), C = (D)/(d), K_w = (4C-1)/(4C-4) + (0.615)/(C), τ = K_w(8FD)/(π d^3)
Wahl (1963) coil spring correction; Shigley's Mechanical Engineering Design ch. 10; ASTM A228 / A231 material shear moduli.
Steam Pipe Calculatorsteam-pipe-calculator
A = ( m v_g)/(V); d = (4A)/(π)
IAPWS-IF97 saturated steam properties; velocity sizing practice from Spirax Sarco and Crane TP-410.
Steam Trap Calculatorsteam-trap-calculator
m = 0.65 A√(2ρΔ P); x = (h_(f1) - h_(f2))/(h_(fg2))
Liquid orifice discharge for trap capacity; flash from the saturated steam table (IAPWS-IF97).
Steel Connection Designsteel-connection-design
R_n = F_(nv) A_b n, R_(brg) = 2.4 d t F_u, R_(block) = min(0.6F_uA_(nv) + F_uA_(nt), 0.6F_yA_(gv) + F_uA_(nt))
AISC 360-22 §J3.2, §J3.6, §J3.10, §J4.3; φ = 0.75.
Steel Fire Resistance Checkfire-resistance-check
θ_(cr) = 39.19ln ((1)/(0.9674 μ_0^(3.833)) - 1) + 482
EN 1993-1-2 §4.2.4 (critical temperature) and §4.2.5.2 (unprotected heating).
Steel Member Code Checksteel-member-code-check
M_p = F_y Z_x, L_p = 1.76 r_y√(E/F_y), φ M_n = 0.9 M_n, utilisation = (demand)/(capacity)
AISC 360-22, chapters F (flexure), G (shear), E (compression) and L (serviceability); EN 1993-1-1, 6.2 and 6.3.
Steel Section Databasesteel-section-database
AISC Shapes Database v15.0, converted to SI; curated doubly-symmetric W-shapes.
Storage Tank Shellstorage-tank-design
API 650 §5.6 one-foot method for shell thickness.
Stormwater Runoff Calculatorstormwater-runoff-calculator
Q = C i A; Q_(depth) = ((P - 0.2S)^2)/(P + 0.8S)
Rational method (ASCE / FHWA HEC-22); NRCS curve number from TR-55; time of concentration from Kirpich.
Structural Dynamics Calculatormulti-degree-of-freedom-structural-dynamics-solver
M u + C u + Ku = -Mr a_g(t), (K - ω_i^2 M) φ_i = 0
Modal analysis and Newmark-β direct integration of MDOF systems; Chopra, Dynamics of Structures; Bathe, Finite Element Procedures (Newmark average-acceleration method, β = 1/4, γ = 1/2); shear-building closed-form frequencies ω_i = 2√(k/m)·sin((2i−1)π/(2(2n+1))).
Structural Reliability Calculatorstructural-reliability-calculator
β = (μ_R - μ_D - μ_L)/(√(σ_R^2 + σ_D^2 + σ_L^2)); P_f = (-β)
Hasofer & Lind (1974); Ditlevsen & Madsen, Structural Reliability Methods; EN 1990 for the β = 3.5 calibration.
Thermal Bridge Calculatorthermal-bridge-calculator
H = ψ·L·ΔT; ψ from the detail, ISO 10211 or the manufacturer.
Thermal Expansion Calculatorthermal-expansion-calculator
Δ L = α L Δ T, ε = α Δ T, σ = E α Δ T, F = σ A
Standard isotropic thermoelasticity; α and E values from ASM Handbook and MatWeb material tables.
Three-Phase Power Calculatorthree-phase-power-calculator
S = √(3) V_L I_L, P = Scosφ, Q = Ssinφ, V_(ph) = (V_L)/(√(3)) (star), I_(ph) = (I_L)/(√(3)) (delta)
IEC 60038 (standard voltages); balanced-load power relations per IEEE 141.
Timber Beam Calculatortimber-beam-calculator
f_b = (M)/(S) ≤ F_b'; F_b' = F_b C_D C_F C_V C_L C_r; Δ = (5wL^4)/(384EI)
NDS for Wood Construction (AWC), §3.3 and Tables 2.3.2, 4A and 4.3.1; reference values from the NDS Supplement.
Tolerance Stack-Up Calculatortolerance-stackup-calculator
T_(wc) = Σ t_i, T_(rss) = √(Σ t_i^2)
Worst-case (arithmetic) and statistical (root-sum-square) tolerance analysis, standard practice in mechanical design.
Torsional Vibrationtorsional-vibration
Free–free torsional mode of a uniform shaft.
Transformer Sizing Calculatortransformer-sizing-calculator
S = (P)/(cosφ); I = (S)/(√(3) V); I_(sc) = (I_(FL))/(Z%/100)
ANSI C57 and IEEE C57.12 for ratings and impedance; the ASA regulation approximation from IEEE Std 141 (Red Book).
Transient Heat Calculatortransient-heat-calculator
θ = e^(-t/τ) or θ_0 = C_1 e^(-ζ_1^2 Fo)
Lumped capacitance and the one-term approximation of the exact series; coefficients from Incropera & DeWitt Table 4.1.
Truss Analysis Calculatortruss-analysis-calculator
K·u = F, N = (EA)/(L)(Δ ucosθ + Δ vsinθ)
Direct stiffness method for a two-dimensional pin-jointed truss.
Turbulent Boundary Layer Calculatorturbulent-boundary-layer-spectral-closure-engine
Re_τ = (u_τδ)/(ν), y^+ = (y u_τ)/(ν), E_(11)(k_1) = (k_1 E_(11)^(HIT))/(u_τ^2)((y)/(δ))^(n)
Pope (2000), "Turbulent Flows"; the closure coefficients follow a 2025 JFM spectral formulation.
Two-Phase Flow Calculatortwo-phase-flow-calculator
Δ P = φ_l^2 Δ P_l, X = ((1-x)/(x))^(0.9)((ρ_g)/(ρ_l))^(0.5)((μ_l)/(μ_g))^(0.1)
Lockhart-Martinelli separated flow; Zivi void correlation.
Two-Phase Relief Calculatortwo-phase-relief-calculator
G = 0.66 (P_1)/(v_0 ω), ω = 9.1
API 520 Part I Appendix C, the omega method for flashing service.
Two-Way ANOVA Calculatortwo-way-anova-calculator
y_(ijk) = μ + α_i + β_j + (αβ)_(ij) + ε_(ijk)
Two-way analysis of variance with interaction, balanced design.
Two-Way Slab Calculatortwo-way-slab-calculator
M_0 = (w_u l_2 l_n^2)/(8), v_u = (V_u)/(b_o d) ≤ φ v_c
ACI 318-19 §8.10 (direct design method) and §22.6.5.2 (punching).
Unit Converterunit-converter
y = (x · f_(from) + o_(from) - o_(to))/(f_(to))
SI Brochure (BIPM) for metric definitions; NIST Handbook 44 and international yard-and-pound agreement for US customary definitions (1 in = 25.4 mm, 1 lb = 0.45359237 kg); ISO 80000 for BTU and thermochemical calorie.
Vibration Calculatoroscillatory-kinetic-vibration-engine
X = (F_0)/(√((k - mω^2)^2 + (cω)^2)), φ = atan2(cω, k - mω^2), M = (1)/(√((1-r^2)^2 + (2ζ r)^2)), r = (ω)/(ω_n)
Steady-state response of a damped SDOF oscillator to harmonic forcing; Rao, Mechanical Vibrations; MIT OCW 2.003/18.03 forced-vibration notes.
Voltage Divider Calculatorvoltage-divider-calculator
V_(out) = V_(in)(R_2)/(R_1+R_2), V_(loaded) = V_(in)(R_2 R_L)/(R_1 + R_2 R_L), R_(th) = R_1 R_2
Thévenin equivalent of a resistive network; IEC 60038 for nominal low-voltage levels.
Voltage Drop Calculatorvoltage-drop-calculator
V_d = 2IRL (single phase), V_d = √(3)IRL (three phase), % = (V_d)/(V)× 100
NEC Chapter 9 Table 8 (conductor DC resistance); NEC 210.19(A) informational note on 3% / 5% voltage drop.
Waveguide Attenuation Calculatorelectromagnetic-waveguide-attenuation-core
f_(c,10) = (c)/(2a√(ε_r)), β = k_0^(2)-k_c^(2), R_s = (ωμ_0)/(2σ), α_c = (R_s (2b k_c^(2)+a k_0^(2)))/(a b β k_0 η_0), α_(dB/m) = 8.686 α_c
Pozar, Microwave Engineering, 4th ed., TE10 attenuation of a rectangular waveguide (Ch. 3); conductor loss cross-checked against NIST Technical Note 2109; surface resistance R_s = √(ωμ0/2σ).
Weibull Reliability Calculatorweibull-reliability-calculator
R(t) = e^(-(t/η)^β), h(t) = (β)/(η)((t)/(η))^(β-1), MTBF = η (1+(1)/(β))
Abernethy, "The New Weibull Handbook", 5th ed.
Wire Gauge Calculatorwire-gauge-calculator
d = 0.127 · 92^((36-n)/39) mm, A = (π d^2)/(4), R = (ρ L)/(A), V_d = 2IRL
ASTM B258 (AWG diameters); NEC Article 310.16 (ampacity tables); IEC 60228 for metric equivalents.
§ 6
Verification & safety
A solver is only as good as the case it can be checked against. Every model ships with a benchmark it must reproduce before its page is published.
01
Analytic benchmarks
The FE beam must reproduce PL³/3EI at the cantilever tip and 5wL⁴/384EI at midspan; the shock solver must reproduce the NACA 1135 normal-shock tables at M₁ = 2.
02
Patch and conservation tests
A constant-strain patch must return the exact displacement field; energy and mass balances are checked to round-off on every marching scheme.
03
Mesh and step refinement
Where no closed form exists, the answer must converge under refinement — a result that moves with the mesh is reported as unresolved, not as a number.
Safety boundary. These tools are educational. They do not replace design codes, safety factors, peer review or a licensed professional engineer’s stamp, and a result that has not been checked against the governing code is not a design. See the disclaimer.
193 engineering calculators · reviewed 2026-10-23Finance methods: methodology/finance · How they compute: the calculation system