Equation and convention
Define symbols, signs, axes and the quantity being calculated.
Technical transparency
How Formulaxis validates its formulas
Every tool is linked to an equation, assumptions, a benchmark, automated tests and identified sources.
42published tools or engines audited
25distinct sources
1correction from the audit
Protocol
Six checks before publication
Dimensional consistency
Convert to SI and check every result dimension.
Authoritative source
Compare against a course, book, standard or official catalogue.
Independent check
Use a second source or an analytical identity.
Numerical benchmark
Test a reference case, conversions and geometric limits.
Scope of validity
State what is included, omitted or needs a complete design check.
A correct formula is not automatically a code check.
ASME B31.3 also covers materials, components, coefficients, fabrication, assembly, examination, inspection and testing. Simplified piping tools make no B31.3 compliance claim.
Audit dated 16 July 2026
Register of checked formulas
| Calculation | Status | Scope | Conclusion |
|---|---|---|---|
| Solid rectangle propertiesA = bh · Ix = bh³/12 · Iy = hb³/12 · W = I/c · r = √(I/A) | Verified | Fundamental relation | Analytical identities and scaling laws verified.Ideal solid section without fillets or chamfers. |
| Solid circle propertiesA = πd²/4 · Ix = Iy = πd⁴/64 · W = πd³/32 · r = d/4 | Verified | Fundamental relation | Symmetry and diameter power laws verified.Ideal solid circular section. |
| Circular tube propertiesd = D − 2t · A = π(D²−d²)/4 · Ix = Iy = π(D⁴−d⁴)/64 · W = 2I/D | Verified | Fundamental relation | Inner-section subtraction and symmetry verified.Concentric tube with uniform wall thickness. |
| Rectangular tube propertiesA = bh − bihi · Ix = (bh³−bihi³)/12 · Iy = (hb³−hibi³)/12 | Verified | Fundamental relation | Difference-of-rectangles calculation verified.Corner radii omitted in the analytical model. |
| I/H section propertiesA = ΣAi · G = ΣAiGi/A · I = Σ(Ii + Aidi²) · W = I/c | Verified | Fundamental relation | Decomposition and parallel-axis theorem verified.Web fillets and rolled radii omitted. |
| Channel-section propertiesA = ΣAi · x̄ = ΣAixi/A · I = Σ(Ii + Aidi²) · W = I/c | Verified | Fundamental relation | Offset centroid and centroidal axes verified.Idealized profile with rectangular component plates. |
| T-section propertiesA = ΣAi · ȳ = ΣAi yi/A · I = Σ(Ii + Aidi²) | Verified | Fundamental relation | Composite-section and centroid calculation verified.Fillets omitted. |
| Angle-section propertiesA = A1 + A2 − Arecouvrement · G = ΣAiGi/A · Ix, Iy par composition · Ixy = ΣAi(xi−x̄)(yi−ȳ) · I1,2 = (Ix+Iy)/2 ± √[((Ix−Iy)/2)²+Ixy²] · θp = ½ atan2(−2Ixy, Ix−Iy) | Verified | Fundamental relation | Composition, signed product of inertia, tensor invariants and principal-axis diagonalization verified.Root and toe radii omitted. |
| Standard profile libraryA, G, Ix, Iy, Ixy, W et r recalculés depuis la géométrie nominale | Verified | Catalogue recalculation | Analytical identities, symmetry, principal axes and scaling laws verified.Simplified values do not replace contractual catalogue or standard properties. |
| Linear mass and weightV = ALn · m′ = ρA · m = ρV · P = mg | Verified | Fundamental relation | Dimensions, standard gravity and conversions verified.Uniform density and constant section. |
| Bolt tightening torqueT = KFd | Verified | Pre-sizing | The simplified torque-preload relation is verified with an explicitly entered K factor.Friction-sensitive estimate; it does not replace torque/clamp-force testing or a qualified tightening procedure. |
| Bolt proof load and preloadAt = π(d−0.9382P)²/4 · Fproof = SpAt · Fi = rFproof | Verified | Pre-sizing | Metric tensile stress area, proof load and target preload are related without conflating proof stress, yield strength and tensile strength.Static pre-sizing per bolt; relaxation, tightening scatter, fatigue and joint stiffness still require verification. |
| Bolt tensile stressAt = π(d−0.9382P)²/4 · σ = Fb/At · Fproof = SpAt · ηproof = Fb/Fproof · Mproof = Fproof/Fb | Verified | Fundamental relation | Stress and proof-load ratios use the total axial force actually carried by one bolt.Nominal static axial tension; load distribution, joint stiffness, bending, fatigue and stress concentrations are excluded. |
| Bearing equivalent dynamic loadP = XFr + YFa | Verified | Pre-sizing | Radial and axial contributions are combined with explicitly entered X and Y factors.Factors and piecewise rules come from applicable catalogue data; no universal table or full ISO 281 implementation. |
| Bearing L10 basic rating lifeL10 = (C/P)^p · p = 3 or 10/3 · L10h = 10^6 L10/(60n) | Verified | Pre-sizing | Ball/roller exponents and the hour conversion are verified without infinity at zero speed.Basic rating life at 90% statistical reliability, not guaranteed service life or modified rating life. |
| Bearing static safety factorP0 = X0Fr + Y0Fa · s0 = C0/P0 | Verified | Pre-sizing | Both input modes give the neutral C₀/P₀ ratio; zero P₀ is not applicable.X₀/Y₀ differ from X/Y; no universal threshold or automatic safety verdict. |
| Spur gear forcesFt = 2T/d · Fr = Ft tan(α) · Fn = Ft/cos(α) | Verified | Fundamental relation | Tangential, radial and normal components were verified from coherent torque, pitch diameter and pressure angle values.Ideal quasi-static external spur-gear model; dynamics, friction, axial force, profile modifications and strength checks are excluded. |
| Gear pitch diameters and center distanced1 = m z1 · d2 = m z2 · a = (d1+d2)/2 = m(z1+z2)/2 | Verified | Fundamental relation | The relations d = mz and a = (d₁ + d₂)/2 were verified for a standard pair with positive integer tooth counts.Nominal standard geometry without profile shift, backlash, tolerances or operating-center-distance modification. |
| Bending normal stressσmax = |M|/W | Verified | Fundamental relation | Navier elastic-bending relation verified.Simple elastic bending and modulus for the checked fibre. |
| Axial normal stress and elongationσ = N/A · ε = σ/E · ΔL = NL/(EA) · k = EA/L | Verified | Fundamental relation | Centred axial loading and Hooke’s law verified.Prismatic member, small strain and linear-elastic material. |
| Classical beam deflectionFL³/(48EI) · 5qL⁴/(384EI) · FL³/(3EI) · qL⁴/(8EI) | Verified | Fundamental relation | Deflections, slopes, reactions and moments for all four cases verified.Euler–Bernoulli, small deflection and idealized supports. |
| Euler elastic bucklingLe = KL · r = √(Imin/A) · λ = Le/r · Pcr = π²EImin/Le² | Verified | Fundamental relation | Elastic bifurcation load and slenderness verified.Sufficiently slender perfect member; no code compliance claimed. |
| Circular-shaft torsionJ = π(D⁴−d⁴)/32 · τmax = T(D/2)/J · θ = TL/(GJ) · kt = GJ/L | Verified | Fundamental relation | Saint-Venant torsion for solid and hollow shafts verified.Uniform circular section, elastic behaviour and omitted stress concentrations. |
| von Mises equivalent stressσ = σa + σb · σVM = √(σ²+3τ²) · σVM = √{[(σ1−σ2)²+(σ2−σ3)²+(σ3−σ1)²]/2} | Verified | Fundamental relation | Plane-stress and principal-stress forms verified.Criterion mainly for ductile materials under static loading. |
| Torsion shaft pre-sizingDmin = ∛[16T/(πτadm(1−k⁴))] | Verified | Pre-sizing | Analytical inversion of τmax verified.Fatigue, notches, stiffness and commercial sizes require separate checks. |
| Torque, power and rotational speedP = Tω · ω = 2πn/60 | Verified | Fundamental relation | Power relation and rpm-to-rad/s conversion verified.Quantities on the same shaft, steady state and losses omitted. |
| Gear ratioi = Z2/Z1 · n2 = n1/i · T2 = T1iη | Verified | Fundamental relation | Single external-gear-stage kinematics and power balance verified.Single stage, no dynamics or backlash, entered overall efficiency. |
| Gear module and diametral pitch conversionPd = 25.4/m · m = 25.4/Pd | Verified | Fundamental relation | The reciprocal relation m·Pd = 25.4 was verified in both directions with m in millimetres and Pd in teeth per inch.Tooth-size convention conversion only; it does not determine tooth counts, diameters or geometric compatibility. |
| Pulley ratio and belt speedD1n1 = D2n2 · v = πD1n1/60 · T2 = T1iη | Verified | Fundamental relation | Tangential speeds and power balance verified.Pitch diameters, no slip and simplified overall efficiency. |
| Pressure, force and effective areaF = pA | Verified | Fundamental relation | Pressure-force-area relation and conversions verified.Uniform pressure without friction or back pressure. |
| Cylinder extension and retraction forcesAs = πD²/4 · Ar = π(D²−d²)/4 · F = pAη | Verified | Pre-sizing | Full and annular areas and theoretical force verified.Simplified efficiency; back pressure and dynamics omitted. |
| Cylinder extension and retraction speedsv = Q/A | Verified | Pre-sizing | Volumetric continuity applied to both chambers verified.Constant flow, leakage and compressibility omitted. |
| Cylinder stroke timeV = AL · t = V/Q | Verified | Pre-sizing | Chamber volumes and theoretical travel times verified.Constant flow; acceleration, switching and compressibility omitted. |
| Hydraulic powerPh = ΔpQ · Pentrée = Ph/η | Verified | Fundamental relation | Pressure-flow product and efficiency balance verified.Steady incompressible flow with entered overall efficiency. |
| Reynolds numberRe = ρvDₕ/μ · Re = vDₕ/ν · ν = μ/ρ · Dₕ = 4A/Pₘ | Verified | Fundamental relation | Both Reynolds formulations, the viscosity relationship and unit conversions were verified.Quantities evaluated at the same fluid state; appropriate characteristic length, Newtonian fluid and internal-flow regime thresholds used only as guidance. |
| Ohm’s law, power and energyR = U/I · P = UI · E = Pt | Verified | Fundamental relation | Relations and Wh/kWh conversion verified.DC or resistive load; power factor omitted. |
| Linear thermal expansionεth = αΔT · ΔL = αL0ΔT · L = L0+ΔL | Verified | Fundamental relation | Free expansion with constant coefficient verified.A restrained part requires thermal-stress analysis. |
| Flow in a circular pipeA = πD²/4 · Q = vA | Verified | Fundamental relation | Inside area, volumetric flow and conversions verified.Full pipe and known mean velocity; no pressure loss calculated. |
| Major pressure loss in a pipeA = πD²/4 · Q = vA · Re = ρvD/μ · f = 64/Re · 1/√f = −2log10(ε/(3.7D)+2.51/(Re√f)) · Δp = f(L/D)ρv²/2 · hf = Δp/(ρg) | Verified | Fundamental relation | Continuity, Reynolds number, Darcy friction factor, signed Colebrook–White residual and Darcy–Weisbach major loss verified; the transition range remains flagged as indicative.Straight full pipe, steady flow, constant properties and fluid treated as incompressible. For air, the approximation is only acceptable when density variation remains small. The solver only accepts inputs for which the equation has a finite positive solution within the residual tolerance; this is not a normative roughness limit. |
| Thin-pipe membrane stressesDm = Do−t · σθ = pDm/(2t) · σz = pDm/(4t) · σVM = √(σθ²−σθσz+σz²) | Verified | Fundamental relation | Membrane equilibrium and plane-stress state verified.Straight closed thin-wall pipe; not a B31.3 calculation. |
| Membrane wall pre-sizingtp = pDo/(2S E + p) · tn = tp+c | Corrected | Pre-sizing | Formula corrected to exactly invert the stress calculator’s mean-diameter model.Educational pre-sizing; tolerances, code coefficients and external loads remain out of scope. |
| Belt length and center distance between two pulleyse = (D2−D1)/2 · β = asin(e/C) · α1 = π−2β · α2 = π+2β · L = 2√(C²−e²) + D1α1/2 + D2α2/2 | Verified | Fundamental relation | Exact external-tangent geometry, arc lengths and the monotonic inverse solution were verified.Open belt, parallel shafts, pitch diameters and pitch length; no idler, tension, elasticity or power sizing. |
Correction identified
Theoretical pressure-pipe wall thickness
Previous approximation
tp = p·Do/(2·S·E)Mean-diameter-consistent form
tp = p·Do/(2·S·E + p)The new relation exactly inverts σθ = p(Do−t)/(2t). A reciprocal test now confirms that the calculated thickness returns the entered limit when E = 1.
Traceability
Sources used for validation
2.001 Mechanics & Materials I — Lecture Notes
Massachusetts Institute of Technology — MIT OpenCourseWare
Checks fundamental relations for axial loading, stress states, thin-walled vessels, bending, deflection and torsion.
Open source →University Physics Volume 2 — Ohm’s Law
OpenStax — Rice University
Checks R = V/I and documents its validity for ohmic behaviour.
Open source →University Physics Volume 2 — Electrical Energy and Power
OpenStax — Rice University
Checks P = V·I and E = P·t.
Open source →University Physics Volume 2 — Thermal Expansion
OpenStax — Rice University
Checks free linear expansion ΔL = αL₀ΔT and the limitation introduced by restraint.
Open source →University Physics Volume 1 — Fluid Dynamics
OpenStax — Rice University
Checks volumetric flow Q = A·v and incompressible continuity.
Open source →University Physics Volume 1 — Pascal’s Principle and Hydraulics
OpenStax — Rice University
Checks the pressure-force-area relation and force ratios in an ideal hydraulic system.
Open source →B31.3 Process Piping
ASME · 2024
Scope reference showing that code-compliant piping design also covers materials, components, fabrication, assembly, examination, inspection and testing.
The public page is used for code scope, not to reproduce proprietary equations or claim B31.3 compliance.Open source →ISO 898-1:2013 — Mechanical properties of fasteners made of carbon steel and alloy steel — Part 1
ISO · 2013
Scope reference for mechanical properties, proof stress and property classes of bolts, screws and studs.
The public page identifies the standard and its scope; no proprietary tabulated values are reproduced.Open source →ISO 16047:2005 — Fasteners — Torque/clamp force testing
ISO · 2005, amended 2012
Scope reference for torque/clamp-force testing and the sensitivity of results to friction conditions.
The public page is used for test scope without reproducing a proprietary procedure.Open source →Fastener Design Manual — NASA Reference Publication 1228
NASA · 1990
Public reference for the simplified torque-preload relation, friction factors and bolted-joint design principles.
Open source →ISO 281:2007 — Rolling bearings — Dynamic load ratings and rating life
ISO · 2007
Scope reference for basic dynamic load ratings, equivalent dynamic load and L10 rating life.
Published edition at stage 90.92; a DIS revision is under development. No proprietary table or full ISO 281 implementation is reproduced.Open source →ISO/TR 1281-1:2021 — Rolling bearings — Explanatory notes on ISO 281 — Part 1
ISO · 2021, corrected 2024
Explanatory notes for ISO 281 rating-life terminology and scope.
Open source →ISO 76:2006 — Rolling bearings — Static load ratings
ISO · 2006
Scope reference for basic static load rating, equivalent static load and static factor s₀.
The 2006 edition and Amendment 1:2017 are published; a CD revision is under development. No proprietary table is reproduced.Open source →ISO 76:2006/Amd 1:2017 — Rolling bearings — Static load ratings — Amendment 1
ISO · 2017
Published amendment to ISO 76:2006 for basic static load ratings.
Open source →ISO/TR 10657:2021 — Explanatory notes on ISO 76
ISO · 2021
Explanatory notes for static load ratings and the scope of ISO 76.
Open source →New SKF IEC offers easy online calculations
SKF
Public manufacturer reference showing that calculations use data and factors specific to the selected bearing.
Open source →Gear Technical Reference — Gear Forces and Calculation of Gear Dimensions
Kohara Gear Industry — KHK Gears
Checks spur-gear force relations and standard pitch geometry.
Open source →Elements of Metric Gear Technology
SDP/SI — Stock Drive Products / Sterling Instrument
Cross-reference for force components, d = mz, center distance and reciprocal module–diametral-pitch conversion.
Open source →Le Système international d’unités (SI)
Bureau international des poids et mesures · 9e édition
Reference for coherent units, symbols and conversions.
Open source →Sections and Merchant Bars — European product ranges
ArcelorMittal Europe — Long Products
Checks European I, H and U series and nominal dimensions. Formulaxis properties remain identified as recalculated when simplified geometry is used.
Open source →Fluid Mechanics — Frank M. White
McGraw Hill
Cross-reference for Reynolds number, Darcy–Weisbach, the Darcy friction factor, Colebrook–White and the limits of incompressible pipe-flow modelling.
Mechanics of Materials — Gere & Goodno
Cengage
Cross-reference for axial loading, bending, torsion, deflection and elastic buckling.
Roark’s Formulas for Stress and Strain
McGraw Hill · 9th edition
Cross-reference for classical beam cases and section properties.
Shigley’s Mechanical Engineering Design — Budynas & Nisbett
McGraw Hill
Cross-reference for the Von Mises criterion, torsion and shaft pre-sizing.
Theory of Elastic Stability — Timoshenko & Gere
McGraw Hill
Cross-reference for Euler critical load and the limits of the perfect-column model.
Terminology
Preferred engineering terms
pressure
pNormal force per unit area. Lower-case p avoids confusion with power P.
effective area
AArea over which the pressure difference produces the resultant force.
Accepted synonyms: working areanormal stress
σStress component normal to the considered plane.
axial normal stress
σNormal stress produced by a centred axial force in a member.
Accepted synonyms: axial stressshear stress
τStress component tangent to the considered plane.
circumferential stress
σθMembrane stress tangent to the circumference of a pipe or shell.
Accepted synonyms: hoop stresslongitudinal stress
σzMembrane stress parallel to the longitudinal axis of the pipe.
Accepted synonyms: axial membrane stressvon Mises equivalent stress
σVMScalar quantity from the distortion-energy criterion, mainly used for ductile materials.
allowable stress
σadm ou SLimit selected according to material, temperature and the applicable design method.
second moment of area
ISection property with dimension L⁴ used in bending and buckling.
Accepted synonyms: area moment of inertiapolar second moment of area
JGeometric property used in Saint-Venant torsion of circular sections.
Accepted synonyms: polar area momentsection modulus
WRatio I/c used to relate bending moment to extreme-fibre normal stress.
Accepted synonyms: elastic section modulusrotational speed
nRevolutions per unit time, distinct from angular velocity ω in rad/s.
Accepted synonyms: rotational frequencycylinder extension and retraction
Extension uses the full piston area; retraction uses the annular rod-side area.
Accepted synonyms: push and pulltheoretical membrane thickness
tpThickness from a simplified thin-membrane model before tolerances, allowances and code requirements.