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Worked Examples
Every tool on this site reproduces at least one published example as its default case, and the tools are checked against those printed answers. The pages below give the problem, the inputs to enter and the answers the source prints, with a link that opens the tool. 315 examples.
Structures (95)
- AISC Connection Checks
- Bolted plate on a gusset: AISC 360-16 bolt shear, bearing and tearout, effective bolt-group strength, tension and block shear (Purdue CE 470 Example 3.1)
- Beam Deflection
- Simply supported beam under a uniform load: δmax = 5wL⁴/(384EI) for a W12x26
- Beam Stress
- Peery Chapter 7 Example 1: Z-section stringer in unsymmetrical bending (Fig. 7.6)
- Beam-Column with Crippling
- NACA Report 1255 Z-stringer with its sheet as a beam-column: exact Timoshenko moments, crippling and Johnson–Euler interaction
- Bearing Capacity and Settlement
- 2 m square footing in a c-φ soil: Terzaghi, Meyerhof, Hansen and Vesić side by side with USACE EM 1110-1-1905's factors, allowable pressure and settlement
- Bolt Preload and Torque
- Bolted cast-iron pressure-vessel cover: stiffnesses, joint constant, preload and the number of bolts for a load factor (Shigley Example 8-4)
- Box Beam Shear Flow
- Bruhn's two-cell hexagonal torque box: cell shear flows, interior web flow and rate of twist (A6.10 Example 1)
- Center Crack
- Centre crack under a general in-plane state: K_I, K_II from the crack-frame far field and G = (K_I² + K_II²)/E
- Inclined centre crack, β = 30°, plane strain: K_I = σcos²β√(πa), K_II = σ sinβ cosβ√(πa), G and Irwin's plastic zone
- Westergaard / Irwin: centre crack under uniaxial tension, K_I = σ√(πa) = 88.62 ksi√in
- Circular Inclusion
- Circular inclusion tool, empty-hole limit: Kirsch general field K_hoop = 3 − S_2/S_1 = 2.737
- Goodier (1933): rigid circular inclusion under uniaxial tension, plane stress ν = 0.3, radial stress 1.51 S at the pole
- Circular and Annular Plates
- Annular steel plate, simply supported outside with a free central hole, under uniform pressure: exact solution against the Wahl–Lobo coefficients
- Column Buckling Tables
- Pinned 12-ft A-36 W8x31 column: Euler load above the yield load, with the Johnson and AISC E3 values at the same slenderness (Hibbeler Example 13.1)
- Contact Stress
- AGMA 2001-D04: spur gear pair at the pitch point against the Grade 1 through-hardened allowable at 300 HB
- Hertz and Johnson: steel sphere on a flat, contact size, pressure and subsurface shear maximum
- Continuous Beam and Moment Distribution
- Three equal 20-ft spans of a W12x26 under 1.0 kip/ft dead and 1.5 kip/ft patterned live load: stiffness solution, Hardy Cross table, three-moment equations and the live-load envelope
- Crack Growth
- Paris-law life of a centre crack in 2024-T3 with the FAA Walker constants
- Crippling Strength
- Extruded 2024-T4 tee LS 236 of NACA TN 3784 Table 9: Gerard's T-type crippling formula against the test
- Curved Panel Buckling
- Curved 2024-T3 fuselage skin panel in compression: NACA TN 3783 curvature parameter, transition-curve coefficient and margin
- Cylindrical Shell Buckling
- Thin 7075-T6 cylinder under axial compression and bending: NASA SP-8007 lower-bound buckling stresses and the interaction margin
- Diagonal Tension Web
- NACA TN 2661 example 1: thin-web beam I-40-4Da in incomplete diagonal tension (Kuhn's engineering theory)
- Drawn Cutout
- Drawn circular cutout: boundary-element K_t = 2.997 against Kirsch's 3
- Drawn elliptic cutout, a/b = 2 across the load: boundary-element K_t = 4.984 against Inglis's 5
- Drawn obround slot 4:1 across the load: boundary-element K_t = 5.09 against Inglis's tip-radius estimate 5.0
- Two drawn equal holes with centres three radii apart under tension across the line: boundary-element 3.266 against the two-hole series 3.264
- Effective Sheet Width
- Effective width of a 0.040 in 2024-T3 sheet at a stringer stress of 35 ksi: von Kármán, Marguerre, Koiter, Winter and the plasticity factor
- Elliptic Hole General Load
- Elliptic hole under equal biaxial tension: 2S·a/b at the ends of the major axis
- Elliptic Hole Uniaxial Stress
- Inglis (1913): elliptic hole under uniaxial tension, K_t = 1 + 2a/b = 5 for a/b = 2
- Elliptic Inclusion
- Hardiman's elliptic inclusion tool in the empty-hole limit: Inglis K_t = 1 + 2a/b = 5
- Fatigue Spectrum Generation
- Flight-by-flight fatigue spectrum of a four-engine transport from the VGH gust and maneuver counts of NASA Memorandum 1-17-59L
- Hole in Plate Bending
- Goodier (1936): round hole in a plate under uniaxial bending, K = (5 + 3ν)/(3 + ν)
- Inertia and Gyroscopic Loads
- Twin-engine light airplane in the 14 CFR 23.371 yaw condition: item inertia loads and the engine gyroscopic couple
- Landing Gear Loads
- Landing-gear loads of a 2,400 lb Part 23 airplane: 23.473 descent velocity, the drop-test energy balance, the Appendix C conditions and the strut trunnion loads
- Large-Deflection Plates
- Levy's simply supported square plate under uniform pressure: large-deflection centre deflection and stresses at p a⁴/(E t⁴) = 132
- Lateral-Torsional Buckling
- AISC Manual Table 3-2: W12×26 beam, F_y = 50 ksi, unbraced length 10 ft
- Local Buckling Coefficients
- Local buckling of a 2024-T3 flange with one edge free: exact coefficient against TN 3781 Table 7
- Lug Strength
- AFFDL-TR-69-42 Section 9.14.1.3: design of a double-shear lug joint for 37,900 lb
- AFFDL-TR-69-42 Section 9.18: fatigue life of a 7075-T6 lug under an axial stress cycle
- AFFDL-TR-69-42 Section 9.6: double-shear lug joint analysed lug by lug (Table 9-1)
- Masonry Wall and Pier
- TM 5-809-3 wall design example 1: 12 in CMU wall under wind and an eccentric axial load, one #6 bar at 24 in
- Membrane Shell Stresses
- Steel cylinder with hemispherical heads under 100 psi: membrane stresses and the junction discontinuity by NASA TR R-103
- Modal Estimates
- Aluminium cantilever with a tip mass: exact frequency against Rayleigh's and Dunkerley's estimates
- Mohr's Circle
- Hibbeler Example 9.3: principal stresses of a plane-stress element by Mohr's circle
- Neuber Hyperbolic Notch
- Neuber deep hyperbolic groove in a round bar, a/ρ = 5, ν = 0.3: tension, bending and torsion
- Neuber deep hyperbolic notch in a flat bar, a/ρ = 5 (Noda, Takase and Monda 1996 Table 1)
- Open Section Shear Flow
- Megson Example 20.3: shear flow in an idealized channel of four booms under 4.8 kN, and its shear centre
- Orthotropic Plate Bending
- Rib-stiffened aluminium plate as an orthotropic plate: Huber's equation by the Lévy series, the smeared rigidities of Timoshenko Art. 86 and the affine reduction to Table 8
- Pin-Loaded Hole
- Bickley (1928): pin-loaded hole in an infinite plate, cosine pressure, side hoop stress 0.81 σ_br
- Plate Bending
- Simply supported 20 × 10 in aluminium plate under 2 psi: Timoshenko's Table 8 coefficients from the Navier and Lévy series
- Plate Buckling
- NACA TN 3781: simply supported plate under combined compression and shear, R_c + R_s² = 1
- Timoshenko and Gere: buckling coefficients of clamped plates, plates in shear and in pure bending
- Timoshenko and Gere: simply supported plate in uniaxial compression, k = 4
- Plate Vibration
- Simply supported 20 × 10 × 0.1 in aluminium plate: the first six natural frequencies by Leissa's exact formula, with Warburton's estimate and the in-plane-force shift
- Pressurization Loads
- Pressurized cabin of a transport fuselage: 14 CFR 25.365 design pressures, skin loads and the frame share by Flügge's shell-ring interaction
- Pressurized Hole
- Lamé pressurized hole superposed on Kirsch: 3σ + p at the edge
- Random Vibration Fatigue
- NAVMAT P-9492 random vibration on a 100 Hz, Q = 10 structure: response rms, Miles' equation, spectral moments and the fatigue damage by Dirlik, narrow band, Steinberg and Wirsching–Light
- Random Vibration Response
- GEVS qualification random vibration on a 100 Hz component: profile rms, Miles' equation against the exact SDOF integral, 3σ levels and exceedance
- Retaining Wall and Footing
- Das's cantilever retaining wall (Example 7.1) and 2 m square footing (Examples 3.1 and 3.2): overturning, sliding, bearing and allowable load
- Rotating Disk With Hole
- Rotating disk with a central hole (Timoshenko and Goodier): hoop stress at the bore and the factor 2 limit
- Round Hole Finite Strip
- Howland (1930): round hole in a finite strip, D/W = 0.5, K_tg = 4.348
- Round hole in a finite strip, D/W = 0.25: K_tg = 3.248, K_tn = 2.436 against Heywood's 2 + (1 − D/W)³
- Round Hole General Load
- Round hole under a general in-plane state: σ_max = 3S_1 − S_2 on the S_2 axis
- Round hole under equal biaxial tension: hoop stress 2S all round the edge
- Round hole under pure shear: ±4τ at ±45° from the load axes
- Round Hole Uniaxial Stress
- Kirsch (1898): round hole in an infinite plate under uniaxial tension, K_t = 3 and −1
- Rounded Square Hole
- Rounded square turned 45° (diamond) under tension along a diagonal: 7.8, against Inglis 1 + 2√(d/ρ)
- Savin's rounded square hole (m = −1/6) under uniaxial tension parallel to a side
- Savin's rounded square hole under equal biaxial tension: K_t = 6.0 at the corners
- Row of Holes
- Infinite row of holes under tension across the row, D/s = 0.5: K_tn 1.621 against the Pilkey fit 1.628
- Infinite row of holes under tension across the row, D/s = 1/3: K_tn 2.031 against the Pilkey fit 2.032
- Infinite row of holes under tension along the row, D/s = 0.5: K_tg 2.139 against the Pilkey fit 2.139
- Sandwich Panel
- Timoshenko and Woinowsky-Krieger Table 8: simply supported square plate under uniform load
- Section Properties
- Timoshenko and Goodier: torsion constants of the square, circular and tubular sections
- Semicircular Edge Notch
- Ling (1947) / Maunsell (1936): semicircular edge notch in a semi-infinite plate under tension parallel to the edge, K_t = 3.065
- Shear Lag in Box Beams
- Kuhn and Chiarito's box beam of NACA Report 739 Figure 50: substitute single-stringer shear-lag analysis by the recurrence formula (Tables I to VII)
- Shear-Resistant Web
- Shear-resistant 2024-T3 web, 10 in deep with single angle stiffeners at 6 in: buckling stress by NACA TN 2661 equation (32) and the stiffener inertia by four published criteria
- Shock Response Spectrum
- Irvine's 50 g, 11 ms half-sine shock: the Q = 10 shock response spectrum by Smallwood's ramp-invariant filter, its closed forms and the MIL-STD-810H test spectrum
- Slope Stability
- EM 1110-2-1902 end-of-construction embankment: Simplified Bishop, Modified Swedish and Spencer on the manual's circle, with the critical-circle search
- Spectrum Fatigue Life
- ASTM E1049-85 §5.4.4: rainflow counting of the nine-reversal history A to I
- MIL-HDBK-5J Figure 3.2.3.1.8(a): best-fit S/N equation of unnotched 2024-T3 sheet
- Sphere and Cone Buckling
- NASA SP-8019 truncated cone: 7075-T6 frustum under axial compression and bending with the SP-8019 correlation factors
- Spherical Cavity
- Spherical cavity under remote uniaxial tension (Southwell and Gough 1926; Goodier 1933): K_t = (27 − 15ν)/(2(7 − 5ν))
- Stick Model Lumping
- Tapered 180 in wing with an engine at station 3: lumped stick model, harmonic-mean segment stiffness, and the Myklestad and Holzer frequency passes
- Stiffened Panel Compression
- NACA Report 1255 numerical example: wrinkling strength of a 2024-T3 Z-stiffened compression panel
- Thermal Bending of Plates
- Aluminium plate 100 °F hotter on its top face: free curvature, clamped-plate stress and the simply supported plate's membrane-shaped deflection (Timoshenko Art. 14 and 49)
- Timber Beam and Column
- 4 × 12 timber beam on a 14 ft span and a 6 × 6 column: NDS adjustment factors, Ylinen stability factors C_L and C_P (Wood Handbook chapter 9)
- Truss and Frame Analysis
- Three-bar truss by the direct stiffness method: displacements, reactions and bar forces (McGuire, Gallagher and Ziemian Examples 3.1 and 3.2)
- Two Holes
- Haddon (1967) / Ling (1948): two equal holes, gap of one radius, tension along the line of centres, K_tg = 2.650
- Haddon (1967): two equal holes, gap of one radius, tension across the line of centres, K_tg = 3.264
- Ling (1948) / Haddon (1967): two equal holes, gap of two radii, tension across the line of centres, K_tg = 3.066
- Two unequal holes (radius ratio 2, gap one small radius) under tension across the line: 4.051 on the small hole
- V-n Diagram and Load Factors
- V-n diagram of a 2,400 lb Part 23 normal-category airplane: 23.337 maneuver factors, 23.335 design speeds and the 23.341 gust load factors
- Weld Group Stress
- Eccentrically loaded fillet-weld group: 50 kN fitting on a channel by the weld-as-line method (Shigley Example 9-1)
- Wing Span Loads
- Light-airplane tapered wing at a 3.8 g maneuver: Schrenk's lift distribution with inertia relief, and the root shear, bending moment and torque
Mechanical Design (37)
- Bearing Life
- SKF 6210 angular-contact ball bearing under combined load: equivalent load, L10 life, reliability and the rating required (Shigley Example 11-3)
- Belt, Clutch and Brake Torque
- V-belt drive by Euler's relation, disc clutch by uniform wear and uniform pressure, long-shoe drum brake and band brake with the stop energy
- Bolted Flange Joint
- NPS 8 class 150 weld-neck flange with a spiral-wound gasket at 150 psi: Appendix 2 bolt loads, hub factors, flange stresses and rigidity
- Compressed Air System Sizing
- 100 hp two-compressor shop at 100 psig: demand, leaks, capacity at altitude, receiver, loop header, condensate and the year's energy (DOE Sourcebook methods)
- Fins and Extended Surfaces
- Aluminium straight rectangular fin 3 × 50 × 100 mm in air: fin parameter, tip temperature, heat rate, efficiency, effectiveness and the Harper–Brown corrected length
- Flow Measurement: Orifices, Nozzles, Weirs and Flumes
- A 4-in Schedule 40 water line with a β = 0.6 orifice plate and flange taps at 100 gpm, and the NBS 90° V-notch weir example (1 ft head, 1 ft notch height, 2.5 ft channel)
- Formula with Units
- Buckingham Pi groups: pendulum period, drag on a body and pump power
- NIST SP 811 factors: 1,000 lbf on 2 in² = 500 psi = 3.44738 MPa
- NIST SP 811 factors: kinetic energy of 2 slug at 10 ft/s = 100 ft·lbf = 135.58 J
- NIST SP 811 factors: stress in a 0.5 in rod under 1,000 lbf, shown in ksi and MPa
- Gear Stress
- AGMA rating of a 17/52-tooth spur mesh at 4 hp: bending and contact stresses, S_F and S_H for pinion and gear (Shigley Example 14-4)
- Interference and Press Fit
- 2 in steel shaft pressed in a 4 in hub: Lamé contact pressure, hub stresses, holding torque, shrink temperature and release speed
- Journal Bearing and Wear
- 2-in full journal bearing at 1,800 rpm and 500 lb: Reynolds-equation solution against the NACA Report 1157 short-bearing forms and Petroff
- Keys, Splines and Couplings
- 2 in shaft with a 1/2 in square key under 3,000 lb-in: key shear and crushing, required length, keyseat geometry and the keyseated-shaft torque
- Limits, Fits and Tolerances
- 50 mm H7/g6 sliding fit and 25 mm H7/p6 interference fit from the ISO 286-1 formulas, checked against the printed tables
- Machining Parameters
- Turning a 2-in 4140 bar at 500 rpm: cutting speed against the TC 9-524 and NAVEDTRA tables, removal rate, unit-power horsepower, Taylor tool life and the minimum-cost and maximum-production speeds
- Multi-DOF Vibration and Critical Speed
- Three equal masses on equal springs (the closed-form uniform chain), a three-disc torsional train on 2 in and 1.5 in steel segments, and a 2 in steel shaft with two discs on a 30 in span
- Noise Exposure
- OSHA Technical Manual noise dose: 95 dBA for 2 hours and 90 dBA for 3 hours, TWA, hearing-conservation action level and an NRR 29 protector
- Open-Channel Flow
- Trapezoidal channel of Chow's Examples 6-2 and 10-1: normal depth, critical depth and the M1 backwater profile behind a dam
- Pipe Flow
- Moody's chart illustration: head loss and pressure drop in 200 ft of 6-in asphalted cast-iron pipe at 6 ft/s (White Example 6.7)
- Pressure Relief Valve and Rupture Disk Sizing
- Air relief valve at 150 psig set, 10 % overpressure, 5,000 lb/h: required area by the nozzle equation, API 526 orifice G, inlet 3 % check, built-up back pressure and reaction force
- Pressure Vessel
- 48 in pressure vessel at 120 psi: membrane stresses, Lamé check and the ASME VIII-1 UG-27 / UG-32 thickness and MAWP
- Pump and Fan Performance
- Textbook-style centrifugal pump against a 40 ft static head: curve fit, operating point, specific speed, NPSH and the 1,450 rpm affinity scaling
- Radiation View Factors
- Two aligned 1 × 2 m plates 1 m apart: Hamilton–Morgan view factor (NACA TN 2836 Table 12) and the grey-body network with re-radiating surroundings
- Room Acoustics and Noise Control
- A 30 x 20 x 10 ft office: Sabine, Eyring and Millington reverberation times from the UFC 3-450-01 absorption table, the treatment area for T60 = 0.6 s, and the NC rating of an office machine at 10 ft
- SDOF Vibration
- Plate supporting a pump: steady-state amplitude of a single-degree-of-freedom system under a harmonic force (Rao Example 3.1)
- Shaft Design
- Stepped shaft on two bearings: Marin endurance limit and DE-Goodman factor (Shigley's method)
- Sheet Metal Bend Allowance and Press Tonnage
- 16-gauge steel channel with a return lip: setback, bend allowance, flat pattern, springback and press-brake tonnage by the FAA handbook method
- Spring Design
- No. 16 music-wire compression spring: torsional yield strength, yield load, rate, solid and free lengths, buckling and pitch (Shigley Example 10-1)
- Spur and Helical Gear Geometry
- 20/40-tooth helical pair, 2.5 mm module, 20°, 15° helix: dimensions, span and pin measurements, contact ratio and sliding
- Steam System and Boiler Efficiency
- 50,000 lb/h natural-gas boiler at 150 psig: heat-loss efficiency from 3 % O2 and a 450 °F stack, blowdown and deaerator balance, trap, insulation and backpressure-turbine savings
- Thermal Stress and Conduction
- Steel bolt in an aluminum sleeve heated 65 °C, a single-pane window, a copper rod fin and a thermocouple bead: thermal stress, conduction network, fin and lumped transient (Hibbeler 4.12, Çengel 3-2, Incropera 3.8 and 5.1)
- Thread Engagement and Strength
- 1/2-13 UNC-2A bolt in a 2B nut: H28 limits of size, thread shear areas, stripping strengths and the required length of engagement
- Tolerance Stack-Up
- Scholz (1995) methods on the tool's five-part shaft-and-housing chain: worst case, RSS, Bender and Cpk
- Unit Calculator Pad
- Twelve-line unit pad: a rod stress, a pipe flow, a car's kinetic energy and the head of one atmosphere, every factor from NIST SP 811
- Units and Sig Figs Tutor
- NIST SP 811 factor-label chains: 60 mi/h to m/s, 14.7 psi to kPa, 1 Btu to J, with the rounding reasoning of Appendix B.7.2
- Water Hammer and Pipe Expansion
- 1 km 12-in steel main closed in 1 s: wave speed, Joukowsky rise and the characteristics solution; a 100 ft steam line heated 300 °F: expansion, anchor force and bends
Aerodynamics (9)
- Aircraft Drag Build-Up and Trim Drag
- Light twin at 10,000 ft and 180 kt: DATCOM component drag build-up, polar, trim drag at 25 % MAC and Korn drag divergence
- Aircraft Performance
- 2,400 lb light airplane from its drag polar: (L/D)max, top speed, best climb, ceilings, Breguet range and endurance, take-off and landing over 50 ft
- Airfoil and Wing Lift
- NACA 2412 wing of aspect ratio 8, taper 0.5 and 3° washout at 5° angle of attack: thin-airfoil section and Prandtl's lifting line at Multhopp stations
- Compressible Flow Tables
- Mach 2 in air: the Report 1135 Table II row, the 10° oblique shock and the Fanno and Rayleigh states
- Fan Noise Estimate
- NACA TN 2968 Table I: rotational noise of a 3-blade, 8 ft propeller at M_t = 0.5 and 1,000 hp, 300 ft away at 105°
- NACA TN 2968 Table I: rotational noise of a 4-blade, 8 ft propeller at sonic tip speed (M_t = 1.0) and 1,000 hp
- Longitudinal Static Stability
- Light airplane: stick-fixed neutral point, static margin, trim and the NACA Report 711 elevator-gradient criterion
- Propeller and Jet Thrust
- NACA Report 640 propeller 5868-9, three blades, 10 ft, set 20° at 0.75R, 1,000 rpm at 100 mph: blade-element momentum theory against the measured curves
- Standard Atmosphere and Airspeeds
- 35,000 ft at 250 knots calibrated airspeed: the 1976 standard atmosphere and the airspeed chain of NACA Report 837
HVAC and Energy (16)
- Control Loop Tuning
- FOPDT process K = 2, τ = 10 min, θ = 2 min: ultimate gain, six tuning rules side by side and the SIMC PI closed loop with its margins
- Duct Sizing
- Four-segment supply duct: Colebrook friction against the Moody chart and the ASHRAE equal-friction diameter
- Energy Savings and Payback
- BPA Regression for M&V guide Section 6: degree-day regression of classroom heat-pump bills
- DOE Motor Systems Tip Sheet #1: energy saved by a NEMA Premium motor
- Envelope UA Compliance
- 2,000 ft² single-storey house in Chicago (climate zone 5): total-UA trade-off against the DOE-reprinted IECC U-factors, a 2x6 R-21 + R-5 wall from its layers, degree-day energy and a Glaser condensation check
- Fire Dynamics Calculations
- NUREG-1805 Example 2-1: hot gas layer temperature and smoke layer height in a naturally ventilated room (MQH, Yamana and Tanaka)
- Generator and UPS Sizing
- Standby generator for a 480 V facility: running load with growth, step loading with a soft-started 100 hp chiller, voltage dip from X′_d, fuel and tank
- Heat Exchanger Sizing and Pressure Drop
- Water-to-water 1-2 shell-and-tube exchanger: energy balance, LMTD with the Bowman–Mueller–Nagle F, Kern shell side, shell sized to the area, both pressure drops
- Industrial Ventilation and Exhaust Hoods
- Two-hood local exhaust system (flanged slot over a bench, canopy over a hot process): hood flows, branch sizing, junction balance by design, fan pressure and the MIL-HDBK-1003/17C xylene dilution case
- Lighting and Circuit Sizing
- 40 by 30 by 10 ft office with twenty 4,000 lm cosine down-lights: zonal-cavity CU by flux transfer against the UFC 30 fc criterion, and a 16 A continuous 120 V branch circuit on 100 ft of 12 AWG copper
- Occupant Load and Egress Capacity
- Two-storey assembly-over-business building: occupant loads, exits and egress widths against the code factors, separation and travel-distance checks, and the evacuation time by the hydraulic model
- Plumbing Pipe Sizing
- Small building with 10 flush-valve closets: Hunter's demand curve, the BMS 79 pressure budget, a 4-in stack with its vent and the building drain
- Psychrometric Calculator
- ASHRAE Fundamentals: moist air at 25 °C dry bulb, 50 % relative humidity, sea level (SI)
- ASHRAE Fundamentals: moist air at 75 °F dry bulb, 50 % relative humidity, sea level
- Sprinkler Hydraulics
- Twelve-head light-hazard tree: calculation sheet from the remote head to the hydrant-test supply curve
- Voltage Drop and Short Circuit
- 480 V, 1,500 kVA radial system: feeder voltage drop, bus fault currents by the ohmic method and the TM 5-811-14 per-unit example
Quality (17)
- Acceptance Sampling
- MIL-STD-105E (ANSI/ASQ Z1.4): lot of 2,000, inspection level II, AQL 1.0 → code K, n = 125, Ac 3 Re 4
- Montgomery Chapter 15: OC curve, AOQL and ATI of the single sampling plan n = 89, c = 2
- Montgomery Section 15.2.4: designing a single plan for AQL 1 %, α 0.05, LTPD 6 %, β 0.10
- Montgomery Section 15.3: double sampling plan n1 = 50, c1 = 1, r1 = 3, n2 = 100, c2 = 3
- Dimensional Capability
- Montgomery Table 6.1: capability of the hard-bake flow width (Cp, Cpk, ppm)
- FMEA
- AIAG-VDA 2019 Action Priority table: RPN and AP of published worked rows
- MIL-STD-1629A Appendix A §50.7: failure mode and item criticality numbers
- GD&T Inspection
- Four-hole pattern at MMC: position deviation, bonus, datum shift and composite feature-relating fit; flatness and roundness by least squares and minimum zone
- Gauge R&R
- AIAG MSA gauge R&R study: three appraisers, ten parts, three trials by the Average and Range method and by ANOVA
- Pareto and Root Cause
- Montgomery Example 7.3: c chart of nonconformities on printed circuit boards
- Montgomery Figure 5.17: Pareto chart of the tank defect data
- NIST/SEMATECH 6.3.3.1: c chart of defects on 25 wafers
- SPC and Capability
- Montgomery Example 6.1: X̄–R chart and Cp of hard-bake flow width
- Montgomery Example 7.1: p chart of nonconforming orange-juice cans
- Montgomery Example 7.3: c chart of nonconformities on printed circuit boards
- NIST e-Handbook 6.3.2.2: I–MR chart of ten batch flow rates
- Variables Sampling (Z1.9)
- MIL-STD-414 Example B-3: double specification limit, variability unknown, standard deviation method (lot of 40, level IV, AQL 1 %)
Reliability (24)
- Accelerated Life
- Meeker and Escobar Example 19.5: Device-A Arrhenius-lognormal fit with a use-temperature cell of no failures
- NIST/SEMATECH 8.4.2.2: Arrhenius-lognormal maximum likelihood fit of a three-temperature test
- Nelson's Class-B insulation data (SAS RELIABILITY procedure): Arrhenius-lognormal fit with quantiles at 130 °C
- ReliaSoft ALTA reference: Arrhenius-Weibull and Eyring-Weibull fits of a 393/408/423 K test
- Fault Tree Analysis
- Pressure tank rupture of the NUREG-0492 Fault Tree Handbook: minimal cut sets, top-event probability and importances
- HALT and HASS Planning
- MIL-HDBK-344A screening strength: 4 thermal cycles at 5 °C/min over 100 °C and 5 minutes at 5 Grms with detection efficiency 0.855
- Hazard and Safety Analysis
- Hazard log by MIL-STD-882E, a LOPA with two protection layers and the LANL SCRAM SIL determination, and a 1oo2 / 1oo1 / 1oo1 safety function verified
- Maintainability and Availability
- Five-subsystem series system of MIL-HDBK-338B Figure 5.8-1: MTTR, maximum repair time, the availability family, a repaired 1-of-2 pair and the MIL-HDBK-470A demonstration plan
- PCB Vibration and Thermal Cycling Fatigue
- 6 × 8 in FR-4 board, simply supported, 0.04 g²/Hz: Miles response, Steinberg 3σ displacement limit, three-band Miner life, lead frame and plated-through hole
- Reliability Growth
- MIL-HDBK-189 Appendix C / 189C Table VI: Crow-AMSAA fit to 27 failures of two prototypes, time terminated at 300 h
- MIL-HDBK-189 Appendix C 20.2.2.4: failure-terminated tank test, 26 failures in miles
- MIL-HDBK-189C Table VII: Crow-AMSAA fit to grouped aircraft failure counts
- ReliaSoft reference: Crow-AMSAA on 22 failure times, failure terminated at 620 h, with Fisher and Crow bounds
- ReliaSoft reference: six systems tested concurrently, equivalent single system and projection to 3,000 h
- Solder Joint Fatigue
- IPC-SM-785 Appendix A: CC1820 chip capacitor solder joints, 25 to 85 °C daily cycles, tested at 0 to 100 °C
- System Reliability
- MIL-HDBK-338B §5.3.8.1: number of spares for a 95 % confidence of no stock-out
- MIL-HDBK-338B §6.3.2–6.3.3: equal and ARINC apportionment of a 0.90 requirement
- MIL-HDBK-338B §6.4.4.2.1.3: reliability of a bridge-type network with repeated elements
- MIL-HDBK-338B §7.5.5.3: cold standby redundancy to reach R(100 h) = 0.98
- Weibull Life Analysis
- Freireich 6-MP leukaemia remission times: Kaplan-Meier survival with Greenwood standard errors
- Kececioglu's rank-adjustment example: median ranks of three failures among two suspensions
- Nelson's diesel generator fan data: Weibull MLE with 58 suspensions and Fisher-matrix bounds
- ReliaWiki MLE example: two-parameter Weibull by maximum likelihood on six failure times
- ReliaWiki RRY and RRX examples: median-rank regression of six failure times
Statistics (33)
- A/B Test
- A/B test of 5.0 % against 5.6 % conversion in 10,000 visitors each: z test, Newcombe interval, posterior probability and sample size
- ANOVA and DOE
- Box, Hunter and Hunter Table 5.2: pilot plant 2³ factorial, temperature, concentration and catalyst
- Montgomery DOE Example 3.1: plasma etching, one-way ANOVA with Tukey and LSD
- Montgomery DOE Example 5.1: battery life, material × temperature two-way ANOVA
- Montgomery DOE Example 6.2: filtration rate, unreplicated 2⁴ factorial with Lenth's method
- Data Summary and Outliers
- Anscombe 1973: the quartet, four data sets with the same summary statistics
- Grubbs 1950: residuals of the semi-diameter of Venus (NIST 1.3.5.17.2)
- NIST e-Handbook 1.3.5.17.1: Grubbs' test on the Tietjen–Moore uranium data
- NIST e-Handbook 1.3.5.17.3: Rosner's generalized ESD test on 54 values
- NIST e-Handbook 1.4.2.8.3: heat flow meter calibration data (ZARR13), summary and tests
- Shapiro and Wilk 1965: W test on the weights of eleven men
- Distribution Fitting
- MIL-HDBK-17-1F Section 8.3.7 Problem 1: Weibull fit and B-basis value of 30 compressive strengths
- MIL-HDBK-17-1F Section 8.3.7 Problem 2: normal B-basis value of 20 strengths
- MIL-HDBK-17-1F Section 8.3.7 Problem 3: lognormal B-basis value of 30 strengths
- MIL-HDBK-17-1F Section 8.3.7 Problem 5: nonparametric B-basis by Hanson–Koopmans for 15 strengths
- Forecasting
- FPP2 §7.1: simple exponential smoothing of Saudi Arabian oil production 1996–2013
- FPP2 §7.2: Holt's linear trend method on Australian air passengers 1990–2016
- FPP2 §7.3: Holt-Winters additive and multiplicative methods on Australian international visitor nights 2005–2015
- Hypothesis Tests
- Cohen's sample sizes for power 0.80 at α = 0.05: t tests, χ² and one-way ANOVA
- Montgomery Example 3.1: one-way ANOVA of plasma etch rate at four RF power levels with Tukey HSD
- Student's sleep data: paired, Welch and pooled t tests of two soporific drugs
- Logistic Regression and Classification
- UCLA graduate admissions: logistic regression of admit on GRE, GPA and institution rank (R's glm output reproduced)
- Pareto and Root Cause
- Agresti Table 3.2: gender by party identification cross-tab (1991 GSS) with X², G² and adjusted residuals
- Regression and Curve Fitting
- Montgomery, Peck and Vining Examples 2.1–2.8: shear strength of rocket propellant against propellant age
- NIST StRD Longley: ill-conditioned multiple regression of employment on six economic predictors
- NIST StRD Misra1a: nonlinear least squares y = b1 (1 − exp(−b2 x)) from both certified starts
- NIST StRD Norris: straight-line calibration of an ozone monitor (36 points, certified values)
- NIST StRD Pontius: quadratic calibration of a load cell (40 points, certified values)
- Sample Size and Power
- Two-sample t test for a standardized difference of 0.5 at α = 0.05 and 80 % power: 64 per group by the exact noncentral t (R's power.t.test root 63.77)
- Survey Analysis
- Agresti Table 2.5: gender × party identification (2000 GSS)
- Agresti Table 3.2: gender × party identification (1991 GSS), χ² and G² tests
- Cochran's sample size and the Krejcie–Morgan table at a 5 % margin
- Fisher's tea-tasting experiment: exact test on a 2 × 2 table
Uncertainty (7)
- Sensitivity Analysis
- Clemen and Reilly Chapter 5: Eagle Airlines one-way sensitivity, tornado and break-even analysis
- Saltelli's Ishigami function: first-order and total Sobol' indices against the analytic values
- Sobol' g-function with a = (0, 1, 4.5, 9, 99, 99, 99, 99): analytic Sobol' indices
- Uncertainty Budget
- Calibration of a 10 kg weight against a class F2 standard: EA-4/02 uncertainty budget, U = 59 mg, and the ILAC-G8 conformity decision against the class M1 limits
- Uncertainty Propagation
- GUM Annex H.1: calibration of a 50 mm end gauge by comparison (uncertainty budget, ν_eff, k₉₉, U₉₉)
- JCGM 101 §9.2.2 (Table 2): additive model with four normal inputs, GUM against Monte Carlo
- JCGM 101 §9.2.3 (Table 3, Annex E): additive model with four rectangular inputs
Operations Research (24)
- Decision Analysis
- Heizer, Render and Munson: Thompson Lumber plant-size decision
- Hillier and Lieberman: the Goferbroke Company drill-or-sell decision with a seismic survey
- Inventory Policy
- Cachon and Terwiesch Chapter 12: O'Neill Hammer 3/2 wetsuit newsvendor
- Chopra and Meindl Chapter 12 Examples 12-1 to 12-8: safety inventory and the periodic-review order-up-to level
- Heizer and Render Chapter 12 Example 10: Nathan Manufacturing production order quantity
- Heizer and Render Chapter 12 Example 11: Wohl's Discount Store all-units quantity discounts
- Heizer and Render Chapter 12 Example 1: ABC classification of Silicon Chips inventory
- Hillier and Lieberman §18.3: economic order quantity for the television speakers
- Hillier and Lieberman §18.3: speaker EOQ with planned shortages
- Nahmias Chapter 5: Harvey's Specialty Shop (Q, r) policy with a stock-out cost
- Job Shop Scheduling and Sequencing
- Five-job, four-machine job shop: eight dispatching rules, Giffler-Thompson active schedules, the disjunctive-graph bound and the exact branch and bound
- Linear Programming
- Hillier and Lieberman: California Manufacturing Co. factory and warehouse decisions (binary IP)
- Hillier and Lieberman: Wyndor Glass Co. product mix with shadow prices and ranging
- Winston: Dakota Furniture desks, tables and chairs with the LINDO sensitivity report
- Winston: Telfa Corporation tables and chairs (pure integer program)
- Multi-Criteria Decision (AHP, TOPSIS)
- Four-concept trade study: AHP consistency, weighted sum, TOPSIS, ELECTRE I, PROMETHEE II and the weight sensitivity
- Network Flow and Routing
- Nine-node network: Dijkstra's label table, Edmonds–Karp augmenting paths and the minimum cut
- Queueing and Capacity
- Hillier and Lieberman: County Hospital emergency room, one doctor (M/M/1) against two (M/M/2)
- Sensitivity Analysis
- Hillier and Lieberman Goferbroke: EMV of drilling against the prior probability of oil
- Shift Scheduling
- Union Airways agent scheduling: five shifts covering ten periods at least daily cost (Hillier and Lieberman Section 3.4)
- Transportation and Assignment
- Hillier and Lieberman Section 9.1: the P&T Company transportation problem
- Taha Example 5.3-1: SunRay Transport by northwest corner, least cost, Vogel and the transportation simplex
- Winston Section 7.5: the Machineco assignment problem by the Hungarian method
- Winston Section 7.6: the Widgetco transshipment problem
Project Management (10)
- Cost Risk Analysis
- NASA Cost Estimating Handbook Appendix G: roll-up of eleven correlated lognormal WBS elements
- Crew Productivity and Labor Estimating
- Five-task concrete foundation estimate: crew-hour cost build-up, production efficiency delay factor, UFC overtime premium, chained schedule, earned hours and the EP 1110-1-8 crane rate
- Delay and Claims Analysis
- Eight-activity project with an owner delay, an overlapping contractor delay and severe weather: impacted as-planned, time-impact windows, collapsed as-built, literal and functional concurrency, and the DCAA manual's Eichleay example
- Earned Value
- DOE EVM Tutorial Module 6: ACME Home Building status and estimates at completion
- Lipke 2003 Table 1: earned schedule of an early-finish project
- Lipke 2003 Table 2: earned schedule of a late-finish project at August
- Project Schedule
- Hillier and Lieberman Chapter 22: Reliable Construction Co. critical path and PERT
- Winston Section 8.4: Widgetco project network, CPM and PERT
- Resource Loading and Leveling
- Twelve-activity building project with a crew of 4 and 2 technicians: resource histograms, Burgess–Killebrew leveling within float and the resource-constrained schedule
- Risk Register and Quantitative Risk
- Eight-risk register on a $10 M, 24-month baseline: DoD RIO Guide scoring, expected monetary value and the Monte Carlo S-curves
Business (22)
- Break-Even and Pricing
- Garrison Chapter 5: Acoustic Concepts cost-volume-profit analysis
- Garrison Chapter 5: Sound Unlimited multiproduct break-even
- Horngren Chapter 3: Emma Frost's GMAT Success package, break-even with taxes
- Horngren Chapter 3: GMAT Success and GRE Guarantee sales-mix break-even
- Mankiw Chapter 5: midpoint method for the price elasticity of ice-cream demand
- Pindyck and Rubinfeld: arc elasticity (§2.4) and the markup pricing rule (Example 10.1)
- Capacity and Throughput
- Five-step production line at 400 units per shift: takt, step capacities and the bottleneck, flow time by the Kingman chain against a discrete-event simulation, WIP by Little's law, OEE with the six losses, a nine-task line balance and the newsvendor capacity cushion
- Cohort Retention and CLV
- Fader and Hardie 2007 Table 1: shifted-beta-geometric fit to the High End segment
- Fader and Hardie 2007 Table 1: shifted-beta-geometric fit to the Regular segment
- Fader and Hardie note 017: sBG fitted to five cohorts, with the DERL of note 018
- Gupta, Lehmann and Stuart 2004 Table 1: customer lifetime value as a margin multiple
- Customer Metrics (NPS)
- Net Promoter Score of an illustrative 400-response file: Rocks' Adjusted Wald interval, segment and period tests, key drivers
- Financial Ratios and Statement Analysis
- Three-period illustrative manufacturer: liquidity, leverage, efficiency and profitability ratios, DuPont, Altman Z, Beneish M and free cash flow
- Job Quote Builder
- Garrison's Yost Precision Machining: job cost sheet of job 2B47
- Horngren, Datar and Rajan Chapter 13: Astel Computers Provalue cost-plus price
- Horngren, Datar and Rajan Chapter 4: Robinson Company job WPP 298 under normal and actual costing
- Loss Reserving
- ChainLadder vignette: RAA triangle by Mack, the LDF exhibit with a 1.05 tail and the ODP bootstrap
- England and Verrall 2001 Table 6.1: chain-ladder reserve and ODP prediction error of a triangle with a negative increment
- Friedland 2010: U.S. Industry Auto reported claims by development, expected claims, Bornhuetter–Ferguson and Cape Cod
- Mack 1993: chain-ladder reserves and standard errors of the Taylor–Ashe triangle
- Lost Profits and Business Interruption
- Six-month interruption of an illustrative retail business: growth-trend but-for revenue with seasonal indices, cost regression, prejudgment interest, ex post and ex ante values, the business-income forms and the coinsurance penalty
- Pricing Waterfall
- Marn and Rosiello's Castle Battery Power-Lite: pocket price waterfall, pocket discount and profit levers
Finance (8)
- Cost Estimating
- GAO-20-195G Appendix VII: unit learning curve at 80 % and cumulative-average curve at 90 %
- NASA Cost Estimating Handbook v4 Table C-5: linear CER for a storage tank
- NASA Cost Estimating Handbook v4 Table N-7: present value, NPV, SIR and ROI of a ten-year program
- NASA Cost Estimating Handbook v4 Tables C-6 and C-7: power model fitted to unit costs
- Depreciation (MACRS)
- Seven-year property of 100,000 under MACRS: Publication 946's half-year percentages, the switch from 200 % declining balance, and the tax shield against the book methods
- Economic Damages
- NCHS United States Life Tables 2020: life expectancy and survival probabilities of a 40-year-old
- Investment Appraisal (NPV, IRR, Payback)
- Five-year series −100,000 then +30,000 at 8 %: NPV, IRR, MIRR, paybacks, EAA, SIR and AIRR with the NIST Handbook 135 factors
- Retirement Planner
- Retirement plan at 40 for a 65–95 horizon: Monte Carlo success probability, Pfau's maximum withdrawal rate and the Bengen / Trinity rolling-start test
Thermal Fluids (3)
- Convection Correlations
- Water at 300 K in a 25 mm tube at 1 m/s: Reynolds and Prandtl numbers, the turbulent correlations side by side and the heat rate to a 50 °C wall
- Fluid Property Lookup
- Superheated steam at 1 MPa and 300 °C: IAPWS-IF97 region 2 against the NIST WebBook (IAPWS-95) row
- Power and Refrigeration Cycles
- Regenerative Brayton cycle at a pressure ratio of 10 with 85 % / 90 % compressor and turbine efficiencies: air with variable specific heats against the cold-air standard
Civil (9)
- Concrete Slab and Footing Design
- FEMA P-751 interior spread footing (11 × 11 ft × 26 in under 387 / 98 kips): one-way and punching shear, flexure, 9 No. 8 bars
- Culvert and Storm Drain Hydraulics
- HDS-5 Design Guideline 1.3 circular culvert (200 ft³/s, 54 in concrete pipe) and the HEC-22 Example 7-3 storm drain energy grade line
- Earthwork Volumes and Mass Haul
- NAVEDTRA 14337 Table 10-2: cumulative yardage of a 2,000-ft road and its mass diagram with a 500-ft free haul
- Horizontal and Vertical Curves and Sight Distance
- Simple curve R 1,000 ft, Δ 30°, PI 12+50: elements, PC / PT and the 100-ft staking table; the 600-ft crest at 50 mph against stopping sight distance
- Lifting and Rigging Loads
- 20,000 lb load on a 4-leg bridle at about 60° with an offset centre of gravity: leg tensions by the DOE two-leg-carry rule, the load-angle factor and the required sling rating
- Pile Capacity and Lateral Load
- HP 12x74 at the FHWA GEC 12 North Abutment: Nordlund shaft and toe resistance in three sand layers, factored resistance, uplift, group, neutral plane, Broms and driving formulas
- Reinforced Concrete Beam and Column
- 12 × 24 in beam with 3 No. 8 bars and a 16 × 16 in tied column with 8 No. 8 bars: flexure, shear, cracked section and the interaction diagram by strain compatibility
- Steel Beam and Column Design
- W12x26 A992 beam-column, 20 ft long and braced at 10 ft, under 20 kips, 60 and 3 kip-ft and 20 kips shear: AISC 360-16 E3, F2, F6, G2 and H1-1a (LRFD)
- Wind, Snow and Seismic Loads
- Two-story 100 × 60 ft building: wind pressures, roof snow with a lower-roof drift, ELF base shear and load combinations