About this site

Educational use only

Everything here exists to teach. Nothing on this site may be used for real engineering design. Material values are nominal published figures of mixed statistical bases (labeled per value); relations carry idealizing assumptions (labeled per relation); no professional review applies. For real design work, use current controlled sources (MMPDS, ASTM/ASME standards, supplier certifications) under qualified engineering judgment.

Who makes this

This site is built end to end by an AI (Anthropic's Claude) — the THINGs, derivations, prose, simulations, and tests — operating documentation and verification systems designed by the project owner. No human reviews the content. Accuracy rests on programmatic verification: a build that proves every derivation identity, refuses unverifiable math, and discloses exactly where physics enters by citation rather than proof. The verification page lists what the machine proves, per THING, and what it cannot. Corrections land in the errata log — dated, never silent.

How the numbers are made trustworthy (for learning)

• Every relation is dimensionally checked and carries a citation and validity envelope.
• Every derivation line is verified by a computer algebra system at build time — lines where physics enters (free-body statements, boundary conditions, cited field solutions) are explicitly badged as modeling steps, and the test pipeline independently re-derives them from first principles where possible.
• Every widget value is computed by code generated from the same verified math, and checked against high-precision reference values before deploy.
• Every material value records its source, exact locator, original units, and basis (specified minimum / design minimum / typical).

Source attributions

• shigley — Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §4-11–§4-12 (Euler long columns and the Johnson parabola for intermediate columns, tangent to the Euler curve at the limiting slenderness √(2π²CE/S_y)). Cited here for the transition slenderness λ_T and the compression-member hand-off to the Johnson regime, which this page cross-links to Euler Column rather than re-implementing.
• juvinall — Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — shaft design and torsional vibration: when the shaft's own inertia is not small compared with the attached rotor, a lumped one-disk model over-predicts the natural frequency, and roughly one-third of the shaft inertia should be added to the disk.
• iso — ISO 281:2007, Rolling bearings — Dynamic load ratings and rating life; and ABMA Standards 9 and 11 — the standards that define the basic dynamic load rating C and the L₁₀ = (C/P)^p rating-life framework used throughout the bearing industry.
• gere — Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — axially loaded members and trusses (§2.1–2.3: two-force members, statically determinate joints), displacements by the unit-load/energy method (§9.8–9.9), and the small-displacement assumption underlying linear truss analysis. Euler buckling of pinned columns is §11.3, reused from the Euler Column page.
• khurmi — Khurmi, R. S., & Gupta, J. K., Theory of Machines, rev. ed., S. Chand — ch. 11 (Belt, Rope and Chain Drives), art. "Condition for the Transmission of Maximum Power": P is maximum when T_c = T/3, i.e. v* = √(T/3m).
• timoshenko — Timoshenko, S. P., Strength of Materials, Part I: Elementary Theory and Problems, 3rd ed., Van Nostrand, 1955 — statically determinate trusses and joint displacements by the elongation geometry (the compatibility triangle), the classical treatment cross-checked here.
• roark — Young, W. C., & Budynas, R. G., Roark's Formulas for Stress and Strain, 7th ed., McGraw-Hill, 2002, Ch. 10 (Table 10.7 in the 8th ed.) — solid rectangular section in torsion: the closed forms K = a·b^3·[1/3 − 0.21(b/a)(1 − b^4/12a^4)] and tau_max at the midpoint of the longer side.
• hibbeler — Hibbeler, R. C., Mechanics of Materials, 10th ed., Pearson, 2017 — ch. 4 (Axial Load), thermal stress: a fully restrained bar heated by ΔT carries σ = E α ΔT; a compound restrained bar is solved by superposition (free expansion, then the restoring force that returns the length).
• timoshenko-som2 — Timoshenko, S., Strength of Materials, Part II: Advanced Theory and Problems, 3rd ed., Van Nostrand, 1956 (first published 1930) — ch. VI "Deformations Symmetrical About an Axis", Art. 40 (the thick-walled cylinder: Lamé field, internal/external pressure, displacements) and Art. 41 "Stresses Produced by Shrink Fit" (eq. 181: the same-material contact pressure; built-up cylinders under internal pressure behave as one solid wall with shrink stresses superposed; gun barrels; pre-yielding a solid tube as the analogous trick).
• hughes — Hughes, A., & Drury, B., Electric Motors and Drives: Fundamentals, Types and Applications, 5th ed., Newnes/Elsevier, 2019 (ISBN 978-0-08-102615-1) — ch. 3 "D.C. Motors": §3.3 motional e.m.f. and equivalent circuit (T = kI, E = kω, with k_t = k_e = k in SI); §3.4.1 no-load speed ω₀ = V/k under the stated constant-flux assumption; §3.4.3 "Behaviour when loaded", eq. (3.10) — the straight torque–speed line; §3.4.6 "Maximum output power" — peak mechanical power at half the no-load speed.
• norton-dom — Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — ch. 9 (gear trains; epicyclic/Willis analysis).
• mit-motors — MIT 2.007 (Design and Manufacturing I) motor tutorial, "D.C. Motor Torque/Speed Curve", MIT Center for Innovation in Product Development, 1999 — §3.1–3.2: the linear torque–speed equations and the explicit statement that maximum output power occurs at half the stall torque and half the no-load speed.
• hibbeler-dyn — Hibbeler, R. C., Engineering Mechanics: Dynamics, 14th ed., Pearson, 2016 — §17.1 (mass moments of inertia; the uniform solid disk I = ½ m R² about its spin axis) and ch. 22 (undamped free vibration: a single mass on a spring oscillates at ω_n = √(k/m), the translational analogue of the torsional system here).
• ashby — Ashby, M. F., Materials Selection in Mechanical Design, 4th ed., Butterworth-Heinemann, 2011 — the "materials for flywheels" case study: kinetic energy per unit mass governed by the material index σ_y/ρ; the bored rotor keeps the index but loses a factor on the coefficient.
• norton — Norton, R. L., Machine Design: An Integrated Approach, Appendix C "Stress-Concentration Factors", Figs C-1 (axial tension), C-2 (bending), C-3 (torsion), pp. 1028-1029 — fitted equations K_t = A·(r/d)^b for a shoulder fillet in a stepped circular shaft, credited to Peterson's charts.
• wahl — Wahl, A. M., Mechanical Springs, 2nd ed., McGraw-Hill, 1963 — the curvature-correction factor K_W = (4C−1)/(4C−4) + 0.615/C and the torsion-bar model of the helical spring.
• nist — BIPM, The International System of Units (SI Brochure), 9th ed., 2019, and NIST Special Publication 330 (2019): the standard acceleration of gravity gₙ = 9.80665 m/s², a conventional value fixed BY DEFINITION (3rd CGPM, 1901) — not a measurement.
• uicker — Uicker, J. J., Pennock, G. R., & Shigley, J. E., Theory of Machines and Mechanisms, 5th ed., Oxford University Press, 2017 — slider-crank position/velocity/acceleration and static force analysis; used as an independent cross-check on the kinematic closed forms and the gas-torque expression.
• isoperimetric — The classical isoperimetric inequality — every simple closed plane curve satisfies S² ≥ 4πA, with equality only for the circle. See e.g. Blåsjö, V., "The Isoperimetric Problem", American Mathematical Monthly 112 (2005), 526–566.
• timoshenko-vib — Timoshenko, S. P., Young, D. H., & Weaver, W., Vibration Problems in Engineering, 4th ed., Wiley, 1974 — the torsional pendulum / disk-on-shaft system: a rotor of mass moment J on a shaft of torsional stiffness k_t oscillates in simple harmonic motion at ω_n = √(k_t/J).
• mil-hdbk-5j — MIL-HDBK-5J (31 Jan 2003), Table 3.2.3.0(b1), Design Mechanical and Physical Properties of 2024 Aluminum Alloy Sheet and Plate, p. 3-71 (omega, lb/in.3 = 0.100)
• united-aluminum-2024 — United Aluminum, '2024 Aluminum Alloy' datasheet, typical mechanical properties for T3 temper: Tensile Strength 70 ksi, Yield 50 ksi, Elongation 16% (averages, explicitly 'not guaranteed' and 'should not be used for design purposes')
• asm-desk-ed-1998 — ASM Metals Handbook Desk Edition (Davis, J.R., ed., ASM International, 1998), linear thermal-expansion table for aluminum alloys, 2024 row: 12.7 x 10^-6/degF (22.8 x 10^-6/degC), mean over 20-100 degC (68-212 degF). Read from the AmesWeb reproduction of the ASM table.
• jeelix-6061 — Jeelix, '6061-T6 Aluminum Yield Strength' engineering guide: typical design value 276 MPa (40 ksi); consistent with ASM-handbook-derived typical values widely quoted (and consistent with MatWeb)
• nasa-cr-123773 — NASA CR-123773, Materials Data Handbook: Aluminum Alloy 7075 (2nd Ed., April 1972), Sec. 9.1: density 0.101 lb/in3 at 68 F (Sec. 9.11 gives specific gravity 2.80 g/cm3 at 20 C). This is the MIL-HDBK-5 lineage value; MIL-HDBK-5J Table 3.7.6.0(b1) lists the same omega = 0.101 lb/in3.
• gabrian-7075 — Gabrian International, '7075 Aluminum Alloy: Properties' datasheet, Mechanical Properties table, 7075-T6/-T651: Yield Strength 503 MPa | 73000 psi.
• cda-c26000 — Copper Development Association, C26000 Alloy datasheet (alloys.copper.org), Physical Properties: density 0.308 lb/cu in. at 68°F (specific gravity 8.53)
• azom-6341 — AZoM, 'Cartridge Brass UNS C26000', Article ID 6341 (azom.com): Poisson's ratio 0.34
• uofm-civl1101 — University of Memphis CIVL 1101/1112 course notes, 'Properties of Concrete' (Concrete Material Properties slides, p. 6/10): reinforced concrete with normal aggregates is about 150 lb/ft3 (pcf); if 5 pcf is allowed for steel, plain concrete w = 145 pcf
• aci-318-formula — ACI 318 elastic modulus formula for normal-weight concrete, Ec = 57,000*sqrt(f'c) psi (ACI 318-08 Sec. 8.5.1; ACI 318-19 Eq. 19.2.2.1.b), as quoted with worked example by CivilEngPro, 'Modulus of Elasticity of Concrete'; ACI 318 itself is paywalled and was not fetched
• gilson-c469 — Gilson Company blog, 'Modulus of Elasticity & Poisson's Ratio of Concrete: Beyond Compressive Strength' (re ASTM C469 testing): 'typical ranges of Poisson's ratio values for concrete are 0.15 to 0.20 for normal weight concrete'
• astm-a48 — ASTM A48/A48M, Standard Specification for Gray Iron Castings, Class 30: specified minimum tensile strength 30 ksi [207 MPa] (paywalled; number verified via foundry datasheets quoting the spec)
• penticton-a48-c30 — Penticton Foundry, Gray Iron ASTM A48 Class 30 Data Sheet: compressive strength 109 ksi (752 MPa)
• asm-via-amesweb — AmesWeb 'Poisson's Ratio of Cast Iron' (citing Callister 2007 and Davis, ASM Metals Handbook Desk Edition 1998): gray irons ~0.26 (grades G1800/G3000/G4000 all listed 0.26)
• celanese-zytel-101l — Celanese, Zytel 101L NC010 Nylon Resin Product Data Sheet, rev. 2023-04-04, 'Other properties' table, Density, ISO 1183, dry
• ensinger-tecamid-66 — Ensinger TECAMID 66 natural (PA66 extruded stock shapes), US product datasheet, Tensile strength at break, ASTM D638, 73 F
• astm-a240 — ASTM A240/A240M, Type 304 (UNS S30400), specified minimum 0.2% offset yield strength, annealed; spec is paywalled so value verified via Penn Stainless 'Stainless Steel Plate 304/304L ASTM A240' product page quoting the A240 minimums ('minimum yield strength at 0.2% of 30 ksi')
• rolled-alloys-304 — Rolled Alloys, '304/304L Data Sheet' (rev. 08/23), Physical Properties table: 'Density 0.285 lb/in3'
• ak-steel-304 — AK Steel Corporation, '304/304L Stainless Steel Product Data Sheet', doc. 304/304L-S-8-01-07 (7/07), Physical Properties: 'Modulus of Elasticity, ksi (MPa): 28.0 x 10^3 (193 x 10^3) in tension'
• kmac-304 — K-mac Distribution, '304 Stainless Material Technical Data Sheet' (AISI Type 304, ASM-style typical property listing): Poisson's Ratio 0.29
• sandmeyer-304 — Sandmeyer Steel Company, 'Alloy 304/304L (UNS S30400/S30403)' plate datasheet, 'Mean Coefficient of Thermal Expansion' table, 68-212 degF (20-100 degC) row: 9.2 x 10^-6 in/in-degF (16.6 x 10^-6 cm/cm-degC).
• azom-6130 — AZoM, 'AISI 1045 Medium Carbon Steel', Article ID 6130, Physical Properties table: 7.87 g/cc (0.284 lb/in3)
• interlloy-1045 — Interlloy Pty Ltd, '1045 Medium Tensile Carbon Steel Bar' datasheet, 'Typical Mechanical Properties - Normalised Condition' table (grade per AS 1442-1992 1045): Yield Strength 410 MPa, Elongation in 50mm 22%, Impact Izod 54 J, Hardness 187 HB / Rc 10
• asm-ready-ref — AmesWeb 'Thermal Expansion Coefficient of Steel by Grade' table (citing Cverna, F., ASM Ready Reference: Thermal Properties of Metals, ASM International, 2002), AISI 1045 (annealed) row: 11.6 x 10^-6/degC over 0-100 degC (32-212 degF).
• astm-a36 — ASTM A36/A36M, Standard Specification for Carbon Structural Steel — specified minimum yield strength (thickness <= 8 in.). Verified via SSAB ASTM A36 mill datasheet quoting the spec minimum.
• fe-handbook-9-2 — NCEES FE Reference Handbook 9.2, Mechanics of Materials, 'Table 1 - Typical Material Properties', p. 80: Steel density 0.282 lb/in3 (7.8 Mg/m3)
• amesweb-metals — AmesWeb 'Thermal Expansion Coefficient (CTE) of Metals' table (citing Callister, W.D., Materials Science and Engineering: An Introduction, 7th ed., Wiley, 2007; and Oberg et al., Machinery's Handbook, 29th ed., Industrial Press, 2012, pp. 376-377), row 'Steel Alloy A36': 11.7 x 10^-6/degC (6.5 x 10^-6/degF), room-temperature average.
• wood-handbook-fpl-gtr-282 — USDA Forest Products Laboratory, Wood Handbook: Wood as an Engineering Material, FPL-GTR-282, Chapter 5 (Senalik & Farber), Table 5-3b, p. 5-12, Douglas-fir Coast, 12% MC

Values are compiled as individual facts with citation (see docs/data-provenance.md in the repository). No source PDFs are redistributed and no source's table layout is reproduced. MIL-HDBK-5J is cited as a publicly released government handbook; it is not claimed to be public domain.

Errata log

No corrections yet. When a published value is corrected, the change is dated and logged here — never silently edited.

Licenses

Code: MIT. Prose, derivations, figures, and the curated dataset: CC BY 4.0 — reuse with attribution welcome, especially by other educators.