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Engineering models you can check.

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Worked example

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Axial response

Inputs

Axial loadF: 10 kN
Cross-sectional areaA: 1000 mm²
Original lengthL: 1000 mm
Elastic modulusE: 200 GPa

Results

Average normal stressσ: 10 MPa
Elastic strainε: 50 µε
Ideal length changeΔL: 0.05 mm

Relation

σ = F / A · ε = σ / E · ΔL = FL / AE

Assumptions

Uniform cross-section · Axial loading · Linear-elastic response

Source

Boston University mechanics notes

Where it stops

Not a stress concentration, not a joint, not a code check.

Pick a field.

Fundamentals

Units, conversions, constants, and quantity integrity.

Unit converterConvert compatible engineering quantities through canonical SI values without losing dimension context.

Mass & densityDerive density and specific volume from mass and volume without relying on a hidden material-property table.

ρ = m / V; v = 1 / ρ

Stress safety marginCompare one applied stress magnitude against one allowable stress and show the resulting factor and margin without asserting material, code, or load-case validity.

Factor = σallow/σapp; margin = factor − 1; utilization = σapp/σallow

Math & geometry

Diagrammed shape, coordinate, and measurement primitives.

Section geometryResolve geometric properties for a small set of explicitly diagrammed cross-sections.

Right-triangle frameResolve hypotenuse, area, and acute angles from two perpendicular legs in a clearly diagrammed triangle.

c = √(a² + b²); A = ab / 2; α = tan⁻¹(b/a)

3D coordinate spanInspect a displacement vector and straight-line span between two three-dimensional points in a coordinate unit.

d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

Cylinder geometryResolve a right circular cylinder’s volume, end area, and lateral surface from its displayed diameter and length.

V = π(D/2)²L; Aend = π(D/2)²; Alateral = πDL

Circular arc & segmentCalculate arc length, chord, sector area, and circular-segment area from radius and central angle.

s = rθ; c = 2r sin(θ/2); Asector = r²θ/2; Asegment = Asector − r²sinθ/2

Pitch circle / PCD layoutCalculate equal-spacing bolt-circle angle, chord, and nominal hole-center coordinates from PCD, hole count, and a start angle.

r = PCD/2; Δθ = 360°/n; chord = 2r sin(π/n); xi = r cos(θ₀+iΔθ); yi = r sin(θ₀+iΔθ)

Regular polygon geometryCalculate regular-polygon perimeter, apothem, circumradius, interior angle, and area from a side count and side length.

P = ns; a = s/[2 tan(π/n)]; R = s/[2 sin(π/n)]; α = (n−2)180°/n; A = aP/2

Statics & mechanics

Bounded member, load, and response models.

Axial responseInspect average stress, elastic strain, and ideal length change for a prismatic member under axial load.

σ = F / A; ε = σ / E; ΔL = FL / AE

Beam deflection & reactionsSet a bounded support and load arrangement, then inspect response quantities with an explicit diagram.

δmax = PL³ / 3EI; Mmax = PL

Elastic stabilityEstimate an ideal elastic critical load while keeping end conditions and model exclusions in view.

Pcr = π²EI / (KL)²

Solid-shaft torsionScreen maximum shear stress, angle of twist, and transmitted power for one uniform solid circular shaft.

J = πD⁴/32; τmax = Tc/J; φ = TL/GJ; P = Tω

Baseline L10 lifeEstimate basic L10 rating life from dynamic rating, one equivalent load, rotational speed, and bearing type.

L10 = (C/P)³; hours = L10·10⁶/(60n)

Torque to preloadEstimate a nominal preload from applied torque, nominal diameter, and a torque coefficient with uncertainty made explicit.

P = T / (K·D); uncertainty band = P(1 ± u)

Combined stressCombine axial and bending normal stresses with one in-plane shear stress, then resolve principal and von Mises screening quantities.

σx = σa + σb; σ₁,₂ = σx/2 ± √[(σx/2)² + τxy²]; σvm = √(σx² + 3τxy²)

Thin-wall vessel stressCalculate ideal hoop and longitudinal membrane stresses for a closed, thin-walled cylindrical pressure vessel from pressure and nominal geometry.

σhoop = pD/(2t); σlong = pD/(4t), closed thin cylinder

Dynamics & motion

Kinematics, energy, momentum, and rotation relationships.

Force & accelerationInspect the net force implied by mass and one-dimensional acceleration under a transparent F = ma relationship.

Fnet = ma

Kinetic energyCalculate translational kinetic energy and momentum from mass and speed, with its introductory model boundary visible.

KE = ½mv²; p = mv

Gravitational potential energyCalculate the change in gravitational potential energy near Earth's surface from mass, height, and gravity.

PE = m g h

Flywheel energy changeCalculate two rotational kinetic-energy states and their difference from inertia and initial/final speeds.

ω = 2πn/60; E = ½Iω²; ΔE = E2 − E1

Screw critical speedCalculate a screw critical-speed relation and operating-speed ratio from root diameter, unsupported length, end-fixity factor, and operating speed.

ncrit = Cs·4.76×10⁶·dr(in)/L(in)²; r = ndeclared/ncrit

Dynamic-braking dutyCalculate peak braking power, duty ratio, derived resistance/current, and source-normal or overhauling average wattage arithmetic without selecting a drive or resistor.

PW = MW·BT; R = Vdc²/PW; DC = tb/tc; DBRW = PW·DC·(1/2 normal, 1 overhauling)

Three-segment RMS torqueCalculate cycle time, absolute peak torque, and RMS torque from three signed torque/time segments without assessing motor thermal capacity.

Trms = √[(T1²t1 + T2²t2 + T3²t3)/(t1 + t2 + t3)]

Motor operating pointCalculate a shaft-power operating point and torque/power reference ratios for a user-classified servo, stepper, or AC motor record without assessing motor capability.

Pshaft = T·ω = T·n·2π/60; rT = T/Tref; rP = Pshaft/Pref

Materials

Source- and condition-aware property context.

Free thermal expansionCalculate ideal free linear length change from a signed coefficient of thermal expansion, initial length, and temperature interval.

ΔL = αL₀ΔT; εth = αΔT

Restrained thermal stressEstimate ideal elastic stress from modulus, linear expansion coefficient, and uniform temperature change under complete restraint.

σth = EαΔT

Fluids

Flow, pressure, and property relationships.

Hydrostatic pressureEstimate the gauge pressure associated with a liquid density and vertical depth under a simple hydrostatic relationship.

pg = ρgh

Flow continuityRelate area, velocity, and volumetric flow for a steady incompressible flow through two displayed cross-sections.

Q = A₁v₁ = A₂v₂

Pipe mean velocityCalculate mean velocity in a circular full-flowing pipe from volumetric flow and inside diameter.

V = 4 Q / (π D²)

Bernoulli equationSolve the pressure change between two flow stations under the ideal steady, incompressible, frictionless Bernoulli relationship.

p₁ + ½ρv₁² + ρgz₁ = p₂ + ½ρv₂² + ρgz₂

Dynamic pressureCalculate Bernoulli dynamic pressure from density and speed as a wind or duct screen.

q = ½ ρ V²

Incompressible orifice flowCalculate incompressible volumetric and mass flow through a sharp-edged orifice-meter geometry using a discharge coefficient.

Q = CdA2√[2(p1−p2)/(ρ(1−β⁴))]; ṁ = ρQ; β = D2/D1

Hydraulic cylinderCalculate ideal extension/retraction area, force, volume, speed, and travel time for one double-acting cylinder from dimensions, pressure, and flow.

Ap = πD²/4; Aa = Ap−πd²/4; F = pA; V = As; v = Q/A; t = s/v

Hydraulic pumpCalculate ideal and actual outlet flow, hydraulic power, shaft-input power, and theoretical input torque from a pump displacement, speed, pressure, and efficiencies.

Qideal = Vd·n; Qdeclared = ηvQideal; Phyd = ΔpQ/600; Pin = Phyd/ηo; Ttheory = ΔpVd/(2π)

Thermal

Heat, energy, and temperature relationships.

Sensible heatCalculate heat transfer from mass, specific heat, and temperature change without applying a hidden property table.

Q = mcΔT

Heat-exchanger ΔTlmResolve the logarithmic mean temperature difference from four terminal temperatures and one selected flow arrangement.

ΔTlm = (ΔT1 − ΔT2) / ln(ΔT1/ΔT2); Q = UAΔTlm, counterflow

Plane-wall conductionCalculate conductive heat rate and thermal resistance across a uniform plane wall from conductivity, geometry, and surface temperatures.

Q̇ = kA(Th − Tc)/L; Rth = L/(kA)

Thermal radiation exchangeCalculate net radiative exchange for a user-defined gray surface facing large isothermal surroundings.

qnet = εσA(Ts⁴ − Tsur⁴); q''net = εσ(Ts⁴ − Tsur⁴)

Convection heat rateCalculate heat rate from a convection coefficient, effective area, and signed temperature difference.

Q̇ = hAΔT

Thermal resistance networkSum hot/cold convection, wall-conduction, and lumped contact terms, then calculate temperature difference at a heat rate.

Rtotal = 1/(hhAh) + L/(kAw) + Rc + 1/(hcAc); ΔT = Q̇Rtotal

Ideal-gas stateCalculate ideal-gas density, specific volume, amount, and mass from absolute pressure, temperature, molar mass, and volume.

pV = nRuT; ρ = pM/(RuT) = p/(RspecT)

Isentropic compressor / turbineCalculate ideal-gas isentropic outlet state and declared-efficiency specific work and power for a compressor or turbine pressure change.

T2s/T1 = (p2/p1)^((γ−1)/γ); wis = cp(T2s−T1); wactual = wis/ηis; P = ṁw

Electrical

Fundamental circuit and power relationships.

Ohm's lawInspect current and resistor power from voltage and resistance in a single ideal DC-resistor relationship.

V = IR; P = VI = V²/R

Three-phase powerApply line-to-line voltage, line current, and power factor to three-phase apparent and real power arithmetic without electrical-system design decisions.

S = √3·VLL·I; P = S·PF

GD&T & quality

Fits, tolerance analysis, inspection, and capability context.

ISO 286 fitsHole and shaft limit deviations, clearance and interference from ISO 286:2010 classes.

cmax = ES − ei; imax = es − EI

Tolerance stackCompare up to six visible independent symmetric contributors using explicit worst-case and root-sum-square stack methods.

TWC = Σ|tᵢ|; TRSS = √(Σtᵢ²), up to six visible contributors

Position deviationCalculate diametrical position deviation from X/Y offsets and compare it with one positional tolerance.

⌀ deviation = 2√(Δx² + Δy²)

MMC bonus & virtual conditionInspect bonus tolerance and a simplified virtual condition for one internal or external feature of size under a MMC position callout.

Bonus = actual hole − MMC hole; VC = MMC hole − ⌀T

Process capability Cp/CpkCalculate Cp, Cpk, and specification-side margins from limits, process mean, and within-process standard deviation.

Cp = (USL−LSL)/(6s); Cpk = min[(USL−x̄)/(3s), (x̄−LSL)/(3s)]

Crossed Gage R&RCalculate crossed-study ANOVA variation components from two operators, two parts, and two trials per operator-part cell.

Crossed ANOVA: SSPart, SSOperator, SSPart×Operator, SSRepeatability · VarRepeat = MSRepeat · VarOperator = max[(MSOperator−MSInteraction)/(a·n),0]

Measurement uncertaintyCombine independent standard uncertainty components and report expanded uncertainty using a coverage factor.

uc = √(uA² + uCal² + uRes² + uEnv²); U = k·uc

Form-control measurement spanReport the measured-extrema span for a flatness, straightness, circularity, or cylindricity screening record.

Observed screening span = user-entered xmax − user-entered xmin

Motion & automation

Declared motion, drive, pneumatic, and fixture relationships.

Trapezoidal motionResolve peak speed, acceleration, and total time for a rest-to-rest symmetric trapezoidal profile with move distance and phase times.

a = s / [ta(ta + tc)]; vmax = a·ta; T = 2ta + tc

Reflected inertiaReflect a load inertia through one ideal reduction ratio and compare it with a motor inertia.

Jref = JL / N²; Jtotal = JM + Jref

Pneumatic cylinder forceCalculate the theoretical extend and retract force from bore, rod, supply pressure, and efficiency factor.

Fextend = P·Abore·η; Fretract = P(Abore − Arod)·η

Clamp-force transferResolve the ideal force transferred through one linkage angle from a known cylinder or actuator force.

Ftransfer = F·sin(θ)·η; Fpivot = F·|cos(θ)|

Lead-screw force & travelResolve ideal raising torque, linear travel rate, and mechanical power from a screw lead, axial force, efficiency, and rotational speed.

T = F·l/(2πη); v = l·n/60; P = Fv

Pneumatic air consumptionEstimate normalized free-air consumption for a double-acting cylinder bore, rod, stroke, operating pressure, and cycle rate.

Vfree = (Aextend + Aretract)s·(Pgauge + 1 bar)

Pneumatic cylinder timeCalculate ideal double-acting cylinder extend/retract volume, speed, and time from an actual volumetric flow at the cylinder.

Ahead = πD²/4; Aannulus = π(D² − d²)/4; v = Qactual/A; t = L/v

Vacuum holding forceCalculate theoretical required holding force for a vertical lift or horizontal transport load case using mass, acceleration, friction, and multiplier.

FTH = m(g + a)M

Manufacturing

Machining and production primitives with bounded process models.

Milling removal rateCalculate theoretical milling material-removal rate from axial depth, radial engagement, and programmed table feed.

MRR = ap·ae·Vf / 1,000

Drilling speed & timeCalculate cutting speed, axial feed rate, and nominal drilling time from drill geometry, spindle speed, feed per revolution, depth, and hole count.

vc = πDcn/1000; vf = frn; Tc = ld·i/(frn)

Turning material removalCalculate theoretical turning material-removal rate from a radial depth of cut, feed per revolution, and cutting speed.

MRR = ap·f·Vc

Milling cutting parametersCalculate spindle speed, table feed, chip load, theoretical removal rate, and specific-force power from milling conditions.

n = 1000vc/(πDc); vf = fz·z·n; MRR = ap·ae·vf/1000; Pc = ap·ae·vf·Kc/(60·10⁶·η)

Sheet-metal bend developmentCalculate bend allowance, bend deduction, outside setback, and a one-bend flat-length estimate from sheet-metal geometry.

BA = (π/180)B(IR + K·MT); OSSB = tan(B/2)(IR + MT); BD = 2OSSB − BA; Flat = L₁ + L₂ − BD

Production metrics & OEECalculate takt, run time, availability, performance, quality, OEE, throughput, capacity, labor content, and utilization from production records.

Run = PPT−ST; A = Run/PPT; P = ICT·N/Run; Q = G/N; OEE = A·P·Q; takt = PPT/D

Fillet weld & heat-inputCalculate equal-leg fillet throat, effective area, direct stress, required leg, and electrical heat input from welding conditions.

t = 0.707a; Aeff = 0.707aLn; τ = F/Aeff; amin = F/(0.707Lτallow n); HI = VI·60·η/(1000v)

Thread pilot & engagementCalculate a basic metric tap drill, engagement length, and a reviewed root-shear-area thread-stripping screening value from dimensions.

Dtap = Dmajor − P·E; Ats = πnLeDsmin[1/(2n)+0.57735(Dsmin−Enmax)]; Fscreen = Atsτallow

Applied systems

Released piping and thermal-system pathways with visible model governance.

Major pipe pressure lossCalculate major friction pressure and head loss from a Darcy friction factor, geometry, density, and mean velocity.

Δp = f(L/D)(ρv²/2); hL = Δp/(ρg)