Educational material — not for design use. Every number cites its source; see provenance & disclaimers.

What is verified here — and what isn't

This site is built end to end by an AI (Anthropic's Claude) operating documentation and verification systems designed by the project owner. No human reviews the content. What stands between you and a wrong number is the machinery on this page — stated plainly so you can judge it. Everything here is educational material, not for design use.

What the build proves, every time

A page cannot deploy unless every one of these gates passes; a failure names the THING, the relation, and the step.

What the machine cannot prove

Computer algebra proves consequence, not physics. Each THING's derivation begins from steps where physics enters by citation — a constitutive law, a boundary condition, a classical field solution. Those are badged modeling step on every page, and they are the complete list of places where this site asks you to trust a book instead of a proof. Citations are pinned against accessible documents where possible, and the pinning method is recorded below; a citation that says topic-level was not pinned to a section. Corrections land in the errata log — dated, never silent.

Tabulated numbers are cited data, not proof. When a THING reads a value from a published table — the Lewis form factor Y for a gear tooth, from Shigley Table 14-2 — the build proves the machinery (the lookup hits every published point exactly, interpolates linearly between them, and refuses outside the range), but the tabulated numbers themselves come from a book, and that interpolating between rows is legitimate is a cited modeling choice. Y in particular is laid out graphically from the tooth profile and cannot practically be re-derived from first principles, so its cross-check is a second published table, not a computation — while the gear pair's kinematics and force path are re-derived from equilibrium in the test pipeline.

Chaining: what carries across a wire, and what doesn't

A machine is THINGs wired together. On the chain builder and the chaining demo you can bind one THING's output port into another's input and watch a consequence propagate. Each output on those pages carries a provenance disclosure tracing the number back through every relation and every upstream port it depends on, and each page shows an assumptions-in-play panel. Here is exactly what those surfaces do and do not certify.

What the build proves about a chain:

What the machine does not prove: cross-THING modeling consistency. That two THINGs are legitimate to compose is a citation-level modeling judgment, not a proof. When the gearbox's output feeds the shaft, the numbers cross a wire that is dimensionally and kind-legal, but whether the gearbox's ideal, lossless torque and power are really what the physical shaft would see — no bearing drag, no mesh loss, no shock load — rests on each stage's cited assumptions, exactly as it would on a single page. The type system guarantees the units and kinds line up; it cannot certify that the idealizations of two different models agree. That is why every chained number keeps its citations in the provenance disclosure and every stage's assumptions are surfaced in the assumptions panel: the composition is disclosed, not asserted (ADR-0007).

Catalog totals: 37 THINGs · 294 relations (each cited) · 141 derivation identities machine-proven · 87 modeling steps (the audit surface) · 192 parity samples.

Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)

6 relations · 4 identities proven · 2 modeling steps · 2 validity envelopes · 2 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Tup=2FNμ(−ri3+ro3)3(−ri2+ro2)T_{up} = \frac{2 F N \mu \left(- r_{i}^{3} + r_{o}^{3}\right)}{3 \left(- r_{i}^{2} + r_{o}^{2}\right)} — uniform-pressure torque by integration (cited; re-derived in tests)
  • Tuw=FNμ(ri+ro)2T_{uw} = \frac{F N \mu \left(r_{i} + r_{o}\right)}{2} — uniform-wear torque by integration (cited; re-derived in tests)

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §16-5 (frictional-contact axial clutches and brakes): the uniform-pressure and uniform-wear torque derivations, the peak-pressure relation, and the torque-optimal inner radius r_i = r_o/√3; and Table 16-3 (friction materials — coefficient of friction and maximum contact pressure for dry linings). — checked: §16-5 (axial clutches) and Table 16-3 (friction materials) topics confirmed against the published 10th-edition chapter listing, 2026-07-06; the textbook PDF is not web-accessible for a page-exact quote. Both torque formulas, the T_uw ≤ T_up bracket, the peak pressure, and the r_o/√3 optimum are NOT taken on the citation — they are re-derived from first principles by direct integration (∫ μ p(r) r · 2πr dr with p = const and with p·r = const) in pipeline/tests/test_clutch_physics.py. The μ and allowable-pressure knob RANGES are the standard dry-lining values of Table 16-3 (μ ≈ 0.25–0.45, p_max ≈ 1–2 MPa for molded/organic linings); they are cited free knobs, not machine-proven numbers. A friction-lining materials table with full basis/errata provenance is future work belonging to the data/materials/ regime, NOT this page.
  • Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — ch. 18 (clutches and brakes): the same uniform-pressure and uniform-wear axial-clutch analyses and the run-in argument for preferring the uniform-wear result in design. — checked: Topic-level cross-check (axial disk clutch torque; the two pressure models and the mean-radius uniform-wear result are standard and also given in Norton and Mott). Not page-pinned; used to corroborate the treatment and the failure notes, not any single emitted number.

Belt Drive (Flat Belt / Capstan)

4 relations · 2 identities proven · 2 modeling steps · 2 validity envelopes · 5 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Tc=mlv2T_{c} = m_{l} v^{2} — element equilibrium: normal direction
  • T1−Tc=(T2−Tc)eμθwT_{1} - T_{c} = \left(T_{2} - T_{c}\right) e^{\mu \theta_{w}} — element equilibrium: tangential direction, integrated

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §17-2 "Flat- and Round-Belt Drives": the belting equation (F_1 − F_c)/(F_2 − F_c) = e^{fφ} (eq. 17-7), centrifugal tension F_c = (w/g)V² (lettered eq. (e), p. 877), transmitted power H = (F_1 − F_2)V. — checked: Eq. (17-7) and the lettered F_c equation (e) p. 877 pinned against the 10th-edition ch. 17 solutions manual (Oakland University mirror), which cites both by number in problems 17-1/17-2, 2026-06-10; forms cross-checked against 10th-edition course notes (Hashemite University). The capstan ODE is independently solved (sympy dsolve) in pipeline/tests/test_belt_physics.py.
  • 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). — checked: Article title and the T_c = T/3 result confirmed only against third-party course reproductions and search-index snippets of ch. 11, 2026-06-10 — the book itself was not consulted, so no page or article number is pinned. The result is independently re-derived by calculus (sp.diff, sp.solve on the cubic) in pipeline/tests/test_belt_physics.py, which carries the real weight here.

Bolted Joint with Gasket (External Tensile Load)

6 relations · 4 identities proven · 2 modeling steps · 2 validity envelopes · 5 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Fb−Fm=PF_{b} - F_{m} = P — statics: ΣF = 0 on the members
  • Fb−Fikb=Fi−Fmkm\frac{F_{b} - F_{i}}{k_{b}} = \frac{F_{i} - F_{m}}{k_{m}} — compatibility: equal incremental deflection at the interface

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 8 (Screws, Fasteners, and the Design of Nonpermanent Joints): §8-4 the joint constant C = k_b/(k_b + k_m); §8-5 bolt and member forces F_b = F_i + CP, F_m = F_i − (1 − C)P and the separation condition; Table 8-11 metric property classes (proof strength). — checked: Topic-level citation, not page-pinned (the textbook PDF is not web-accessible for a page-exact quote, the same limit as the sibling beam/joint THINGs). The governing results were web-corroborated 2026-07-06 AND re-derived from first principles in pipeline/tests/test_boltedjoint_physics.py (spring load-sharing: bolt and members as parallel springs sharing an equal incremental deflection). The joint constant C = k_b/(k_b + k_m) as the bolt's share of the external load, and the split F_b = F_i + CP / F_m = F_i − (1 − C)P, are confirmed by MechaniCalc's "Bolted Joint Analysis" reference and the University of Portland ME fastener notes (faculty.up.edu/lulay). Proof strength: ISO 898-1 assigns property class 8.8 a proof stress of 600 MPa for 16 mm < d ≤ 72 mm (and 580 MPa for d ≤ 16 mm); Shigley Table 8-11 tabulates Class 8.8 (M16–M36) at S_p = 600 MPa — the default used here. Proof strength is a DIRECT knob (bolt property classes are spec grades, not seeded materials); tensile stress area is a separate knob (M16 ≈ 157 mm², M20 ≈ 245 mm² per ISO 898-1, web-corroborated via eurocodeapplied.com). SIGN NOTE: Shigley writes the member force as F_m = (1 − C)P − F_i (compression negative); this THING takes clamping POSITIVE (F_m = F_i − (1 − C)P), so separation is F_m ≤ 0.

Cantilever Beam (End Load)

5 relations · 4 identities proven · 1 modeling step · 3 validity envelopes · 3 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Mx=P(L−x)M_{x} = P \left(L - x\right) — statics: method of sections

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.5 (flexure formula), §9.3 (cantilever deflections by integration), App. E (section properties). — checked: not section-pinned; cited from general knowledge of the text.

Circular Plate under Uniform Pressure (Clamped vs Simply Supported)

8 relations · 2 identities proven · 5 modeling steps · 2 validity envelopes · 8 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • D=Et312−12ν2D = \frac{E t^{3}}{12 - 12 \nu^{2}} — modeling: plate flexural rigidity and the axisymmetric plate equation D∇⁴w = q
  • δc=a4q64D\delta_{c} = \frac{a^{4} q}{64 D} — clamped edge: w(a)=0 and w′(a)=0 pin the constants
  • σc=3a2q4t2\sigma_{c} = \frac{3 a^{2} q}{4 t^{2}} — clamped: peak radial moment at the edge, σ = 6M/t²
  • δss=a4q(ν+5)64D(ν+1)\delta_{ss} = \frac{a^{4} q \left(\nu + 5\right)}{64 D \left(\nu + 1\right)} — simply supported: w(a)=0, M_r(a)=0 — ν enters through the moment
  • σss=a2q(3ν+9)8t2\sigma_{ss} = \frac{a^{2} q \left(3 \nu + 9\right)}{8 t^{2}} — simply supported: peak moment at the center, σ = 6M/t²

Sources & how they were checked:

  • Timoshenko, S. P., & Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd ed., McGraw-Hill, 1959 — Ch. 3 (Symmetrical Bending of Circular Plates), §15-16: the symmetrical-bending ODE (§15) and the uniformly loaded circular plate (§16), flexural rigidity D = E t³/(12(1-ν²)), the clamped case (max deflection q a⁴/(64 D) at the center, max stress 3 q a²/(4 t²) at the edge) and the simply-supported case (center deflection (5+ν)/(1+ν)·q a⁴/(64 D), center stress 3(3+ν) q a²/(8 t²)). — checked: Section pin corrected §16-17 → §15-16 on 2026-07-06 against published TOC listings of the 2nd ed. (§16 = Uniformly Loaded Circular Plates; §17 = Circular Plate with a Circular Hole at the Center — an annular plate, out of this page's scope; §15 carries the governing ODE). Originally pinned topic-level (textbook PDF not web-accessible that session, as for S02-S05). The results are NOT taken on citation: pipeline/tests/test_plate_physics.py re-derives all four closed forms from the axisymmetric plate ODE D∇⁴w = q — regularity at the center kills the ln r and r² ln r homogeneous terms, then each rim condition (clamped: w(a)=w′(a)=0; simply supported: w(a)=0, M_r(a)=0) is applied independently — and proves the (5+ν)/(1+ν) deflection factor and the (3+ν)/2 stress ratio with ν left SYMBOLIC (the ν-cancellation-proof pattern from compound-cylinder). These are exact analytic facts (uncopyrightable, Feist), matched by the independent ODE solution. Golden (q=100 kPa, a=0.3 m, t=0.01 m, steel E=200 GPa, ν=0.3): σ_c=67.5 MPa, σ_ss=111.375 MPa, δ_ss/δ_c=53/13≈4.077 — hand-checkable arithmetic on the formulas.
  • Young, W. C., Budynas, R. G., & Sadegh, A. M., Roark's Formulas for Stress and Strain, 8th ed., McGraw-Hill, 2012 — Table 11.2 (Flat circular plates of constant thickness, uniform load over the entire plate): case 10a (simply supported edge) and case 10b (fixed/clamped edge), used here as an independent oracle for the four closed forms. — checked: Cross-check only. Roark Table 11.2 cases 10a/10b state the identical maximum deflections and stresses for the uniformly loaded circular plate; the physics test evaluates the closed forms against the independently-integrated ODE solution at sampled geometries and asserts agreement, so the formulas are pinned by re-derivation, not transcription. Topic-level; not page-pinned.

Composite Bar (Core + Sleeve)

10 relations · 3 identities proven · 2 modeling steps · 2 validity envelopes · 7 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • P1+P2=PP_{1} + P_{2} = P — statics: ΣF_axial = 0
  • LP1A1E1=LP2A2E2\frac{L P_{1}}{A_{1} E_{1}} = \frac{L P_{2}}{A_{2} E_{2}} — compatibility: equal elongation (rigid plates)

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — ch. 2 (Axially Loaded Members): statically indeterminate bars, the flexibility/compatibility method, and members sharing load in proportion to axial stiffness A·E. — checked: Topic-level citation, not section-pinned (the textbook PDF is not web-accessible for a page-exact quote, the same limit as the sibling axial/beam THINGs). The governing result — two parallel members under a rigid cap share load in proportion to their axial stiffness A_i E_i, giving P_i = P·A_iE_i /(A_1E_1 + A_2E_2) and equal-strain stresses σ_i ∝ E_i — was web-corroborated 2026-07-06 (engineering statics references for two-material / bolt-and-tube "members in parallel" all give the stiffness-proportional share and δ = PL/ΣAE) AND re-derived from first principles in pipeline/tests/test_composite_physics.py (springs in parallel: k_i = A_iE_i/L, share = k_i/Σk).
  • Hibbeler, R. C., Mechanics of Materials, 10th ed., Pearson, 2017 — ch. 4 (Axial Load): statically indeterminate axially loaded members solved by the compatibility (equal displacement) method. — checked: Topic-level cross-check citation for the equilibrium + compatibility setup. Not section-pinned; the equilibrium equation is elementary statics and the equal-elongation compatibility is re-derived from the rigid-plate constraint in the physics test. Hibbeler ch. 4 is the standard cross-reference for the indeterminate axial-member method used here, consistent with Gere ch. 2.

Compound Cylinder (Shrink Fit)

10 relations · 7 identities proven · 2 modeling steps · 7 validity envelopes · 9 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • uj=rc(νpc+pc(rc2+ro2)−rc2+ro2)Eu_{j} = \frac{r_{c} \left(\nu p_{c} + \frac{p_{c} \left(r_{c}^{2} + r_{o}^{2}\right)}{- r_{c}^{2} + r_{o}^{2}}\right)}{E} — elasticity: Lamé field + plane-stress Hooke at the jacket bore
  • ul=−rc(−νpc+pc(rc2+ri2)rc2−ri2)Eu_{l} = - \frac{r_{c} \left(- \nu p_{c} + \frac{p_{c} \left(r_{c}^{2} + r_{i}^{2}\right)}{r_{c}^{2} - r_{i}^{2}}\right)}{E} — elasticity: the same field with the pressure outside

Sources & how they were checked:

  • 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). — checked: Chapter/article numbers, eq. 181, the superposition sentence, and Art. 41 Prob. 2 pinned against the Internet Archive full text (in.ernet.dli.2015.140195), 2026-06-10 — note the item's DLI metadata mislabels the scan "1930"; its own title and copyright pages read Third Edition, March 1956, so the pins hold for the 3rd edition and earlier-edition numbering was not checked. The book's own worked example (Art. 41, Prob. 2: p = 4,050 psi) is re-computed from the authored contact-pressure relation as a published golden in pipeline/tests/test_shrinkfit_physics.py, and the Lamé fields are independently re-derived there from equilibrium + compatibility.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-16 "Press and Shrink Fits" (radial interference, the general two-material contact-pressure relation and its same-material collapse), §3-14 (stresses in pressurized cylinders), and §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y). — checked: §3-16 title confirmed in the ch. 3 section sequence (3-13…3-19) against published 11th-edition-SI section listings, 2026-06-10 — the 10th edition itself was not re-inspected; numbering taken as stable, anchored by the §3-14 pin the thick-walled-cylinder THING recorded against 10th-edition listings. The general interference relation and the two member stress states at the interface verified against the publisher's companion tutorial (McGraw-Hill, "Machine Design Tutorial 4-17: Press and Shrink Fits", Text Eqs. 4-57…4-60 in 7th-edition numbering, two in the tutorial's "Modified" form), 2026-06-10; the liner-bore assembly stress is the Lamé field of Timoshenko Art. 40, re-derived in pipeline/tests/test_shrinkfit_physics.py, and the same-material collapse is re-proven symbolically there.
  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — prismatic members and section properties (annular area, mass per length). — checked: Topic-level citation, not section-pinned; identical to the annulus citation already carried by the thick-walled-cylinder THING.

Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)

16 relations · 2 identities proven · 3 modeling steps · 3 validity envelopes · 7 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • rn=hlog⁡(rori)r_{n} = \frac{h}{\log{\left(\frac{r_{o}}{r_{i}} \right)}} — modeling: plane sections rotate about the NA; zero net axial force locates it
  • σbi=MciAeri\sigma_{bi} = \frac{M c_{i}}{A e r_{i}} — modeling: moment balance about the centroid fixes the stress constant
  • SF=σyσiSF = \frac{\sigma_{y}}{\sigma_{i}} — definition: safety factor on the governing inner fiber

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-18 (Curved Beams in Bending): the Winkler formula σ = M y/(A e (r_n − y)), the rectangular-section neutral radius r_n = h/ln(r_o/r_i) (Table 3-4), the eccentricity e = r_c − r_n, and the crane-hook combination of direct and bending stress. — checked: Topic + formula pinned against Shigley §3-18 and Table 3-4 (rectangular-section r_n). The Winkler stress is NOT taken on citation: pipeline/tests/test_curved_beam_physics.py re-derives it from first principles — plane-sections strain, the neutral-axis location from ∫σ dA = 0, and the moment balance ∫σ(r_n − r) dA = M — then proves the straight-beam limit (σ_i → Mc/I as r_c/h → ∞) by series expansion, and cross-checks the numeric golden against Roark. The rectangular r_n = h/ln(r_o/r_i) is an exact analytic fact (uncopyrightable, Feist), matched by the independent integral. Golden (r_i=50, r_o=100 mm, b=20 mm, M=1 kN·m): σ_{b,i}=154.5 MPa, σ_o=−97.25 MPa, straight Mc/I=120 MPa, K_i=1.288 — hand-checkable arithmetic on the formulas.
  • Young, W. C., Budynas, R. G., & Sadegh, A. M., Roark's Formulas for Stress and Strain, 8th ed., McGraw-Hill, 2012 — Chapter 9 (Curved Beams): the same Winkler/Bleich curved-beam bending formulas and the rectangular-section neutral-axis radius, used here as an independent oracle. — checked: Cross-check only. Roark Ch. 9 states the identical curved-beam bending stress and the rectangular r_n = h/ln(r_o/r_i); the physics test evaluates the stress two independent ways (the closed form and the raw ∫ definition κ = M/(A e), σ = κ(r_n − r)/r) and asserts they agree, so the formula is pinned by re-derivation, not transcription. Topic-level; not page-pinned.

DC Motor (Permanent Magnet)

5 relations · 4 identities proven · 3 modeling steps · 2 validity envelopes · 2 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Ia=Vs−kmωRaI_{a} = \frac{V_{s} - k_{m} \omega}{R_{a}} — circuit model: KVL with back-EMF
  • T=IakmT = I_{a} k_{m} — torque constant = back-EMF constant (SI)
  • T=Tstall(−ωω0+1)T = T_{stall} \left(- \frac{\omega}{\omega_{0}} + 1\right) — name the datasheet intercepts

Sources & how they were checked:

  • 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. — checked: Section and equation numbers (3.5–3.10) pinned against the 5th-edition indexed full text (Internet Archive), 2026-07-07 — the straight-line statement, the k_t = k_e = k identity, and the half-no-load-speed peak-power statement each read in situ. The line and the peak are independently re-derived from the circuit model (SymPy) in pipeline/tests/test_motor_physics.py; the book does not name a T_stall symbol — the stall intercept kV/R follows from eq. (3.10) at ω = 0, stated honestly here rather than attributed as a labeled formula.
  • Norton, R. L., Design of Machinery, 6th ed., McGraw-Hill, 2020 — §2.19 "Motors and Drivers", permanent-magnet DC motors: the speed–torque curve from stall torque at zero speed to zero torque at no-load speed; the contrast with shunt-, series-, and compound-wound curve shapes; the warning that motors cannot tolerate a full-current zero-speed stall for more than minutes without overheating. — checked: §2.19 content confirmed against a 6th-edition text excerpt (PMDC subsection and the stall-overheating sentence read directly), 2026-07-07. In-book copyright page prints 2018 while the publisher lists the 6th edition as ©2020 — cited by edition, not year alone, for that reason.
  • 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. — checked: Live page fetched and the equations plus the half-speed/half-torque maximum-power statement read directly, 2026-07-07. A legacy static course page — used as the freely accessible corroboration of the peak-power result, with Hughes & Drury §3.4.6 as the textbook anchor.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-12 (power transmission, P = Tω). — checked: Same section pinned previously for torsion-shaft (§3-12 power transmission), 2026-06-10; reused here for the identical relation.

Eccentric Column (Secant Formula)

12 relations · 1 identities proven · 3 modeling steps · 2 validity envelopes · 6 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • ke2=PEIk_{e}^{2} = \frac{P}{E I} — modeling: beam-column with end eccentricity
  • δmid=e(−1+1cos⁡(Lke2))\delta_{mid} = e \left(-1 + \frac{1}{\cos{\left(\frac{L k_{e}}{2} \right)}}\right) — modeling: solve the beam-column BVP (re-derived in pipeline tests)
  • PSFy=PyP SF_{y} = P_{y} — definition: the margin is taken on load (Shigley) — solved numerically, bracketed

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §11.5 (Columns with Eccentric Axial Loads) and §11.6 (The Secant Formula for Columns). — checked: Section numbering pinned against the publisher's WebAssign section list for the Enhanced 9th edition (webassign.net/features/textbooks/ggmechmat9/details.html: "11.5 Columns with Eccentric Axial Loads", "11.6 The Secant Formula for Columns"), 2026-06-10. The midspan deflection and the secant stress formula are independently re-derived from the beam-column ODE with boundary conditions in pipeline/tests/test_eccentric_physics.py.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §4-13 (Columns with Eccentric Loading): the secant equation for the critical load, and the design rule that the factor of safety must be applied to the load because the stress is a nonlinear function of it. — checked: Formulas pinned verbatim against Dr. Ala Hijazi's class notes for the 10th edition (Hashemite University, eis.hu.edu.jo/ACUploads/10526/CH 4.pdf, pp. 21–22), 2026-06-10: δ = e[sec((L/2)√(P/EI)) − 1], σ_c = (P/A)[1 + (ec/k²)sec((L/2k)√(P/EA))], and the secant equation solved for the load — with the note that P "appears twice … needs an iterative process", which is precisely the bracketed root-finding this page performs. Section number §4-13 corroborated topic-level against the publisher chapter-outline deck; not page-pinned against the print TOC.

Euler Column (Buckling)

12 relations · 6 identities proven · 3 modeling steps · 2 validity envelopes · 30 domain guards · 15 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • ke2=PcrEIk_{e}^{2} = \frac{P_{cr}}{E I} — modeling: moment in the perturbed configuration
  • Lke=πL k_{e} = \pi — eigencondition: first nonzero root of sin(kL) = 0
  • σJ=σy−lam2σy24π2E\sigma_{J} = \sigma_{y} - \frac{lam^{2} \sigma_{y}^{2}}{4 \pi^{2} E} — modeling: Johnson parabola, the standard inelastic-buckling fit

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §11.3–11.4 (Euler buckling of ideal pinned columns; effective lengths for the classic end conditions incl. K = 0.699 for fixed–pinned) and §11.7 (critical stress, slenderness limits, elastic vs. inelastic column behavior). — checked: Section numbering pinned against the publisher's WebAssign section list for the Enhanced 9th edition (webassign.net/features/textbooks/ggmechmat9/details.html: 11.3 Columns with Pinned Ends, 11.4 Columns with Other Support Conditions, 11.7 Elastic and Inelastic Column Behavior), 2026-06-10 — this correlates titles to sections; page numbers not checked. The Euler load, effective-length scalings and the σ_y/2 transition are independently re-derived from the buckling ODE in pipeline/tests/test_column_physics.py.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §4-11–§4-12 (Euler long columns with central loading; Johnson's parabolic formula for intermediate-length columns, tangent to the Euler curve at the limiting slenderness (L/k)₁ = √(2π²CE/S_y)). — checked: Formulas pinned verbatim against Dr. Ala Hijazi's class notes for the 10th edition (Hashemite University, eis.hu.edu.jo/ACUploads/10526/CH 4.pdf, pp. 18–19), 2026-06-10: the Johnson parabola σ_cr = S_y − (S_y²/4π²CE)(L/k)², its validity range L/k < (L/k)₁, and the limiting slenderness √(2π²CE/S_y) all match this THING's relations with C = K⁻² folded into the effective slenderness. Section numbers §4-11/§4-12 corroborated topic-level against the publisher chapter-outline deck (column categories in the same order); not page-pinned against the print TOC.

Fixed-Fixed Beam (UDL)

10 relations · 4 identities proven · 4 modeling steps · 3 validity envelopes · 4 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • RA+RB=LwR_{A} + R_{B} = L w — statics: ΣF_y = 0
  • LRB+MA=L2w2+MBL R_{B} + M_{A} = \frac{L^{2} w}{2} + M_{B} — statics: ΣM_A = 0
  • L4w8EI+L2MB2EI=L3RB3EI\frac{L^{4} w}{8 E I} + \frac{L^{2} M_{B}}{2 E I} = \frac{L^{3} R_{B}}{3 E I} — force method: zero deflection at the released end
  • L3w6EI+LMBEI=L2RB2EI\frac{L^{3} w}{6 E I} + \frac{L M_{B}}{E I} = \frac{L^{2} R_{B}}{2 E I} — force method: zero slope at the released end

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — ch. 10 (Statically Indeterminate Beams): the force (flexibility) method, fixed-fixed (clamped) beam reactions by superposition; §5.5 flexure formula; App. E section properties. — checked: Topic-level citation, not section-pinned (the textbook PDF is not web-accessible for a page-exact quote, the same limit as the sibling beam THINGs). The governing results — fixed-end moment M = wL²/12, reactions R = wL/2, midspan moment wL²/24, midspan deflection wL⁴/384EI — were web-corroborated 2026-07-06 against multiple independent statements (standard fixed-end-moment tables and beam formula references give the clamped-beam UDL fixing moment as wL²/12 and the centre deflection as wL⁴/384EI) AND re-derived from first principles in pipeline/tests/test_fixedfixed_beam_physics.py (solving the Euler–Bernoulli BVP EI·v'''' = w with v = v' = 0 at both ends, and independently by the force method).
  • Hibbeler, R. C., Mechanics of Materials, 10th ed., Pearson, 2017 — ch. 12 (Deflection of Beams and Shafts): statically indeterminate beams by the method of superposition; equilibrium of the fixed-fixed beam. — checked: Topic-level cross-check citation for the equilibrium equations (ΣF, ΣM). Not section-pinned; the two equilibrium relations are elementary statics and are re-derived from the free body in the physics test. Hibbeler ch. 12 is the standard cross-reference for the superposition method used here, consistent with Gere ch. 10.

Fixed-Fixed Torsion Shaft (Interior Torque)

10 relations · 4 identities proven · 2 modeling steps · 2 validity envelopes · 6 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • TA+TB=TT_{A} + T_{B} = T — statics: torsional equilibrium
  • TAaGJ=TBbGJ\frac{T_{A} a}{G J} = \frac{T_{B} b}{G J} — compatibility: single-valued rotation at the load point

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §3.8 (statically indeterminate torsional members: a bar fixed at both ends with an intermediate torque, solved by combining equilibrium with the compatibility of the angle of twist); §3.3–3.4 (torsion formula τ = Tr/J, angle of twist TL/GJ). — checked: Topic-level citation, not page-pinned (the textbook PDF is not web-accessible for a page-exact quote, the same limit as the sibling beam/shaft THINGs). The governing result — reaction torques T_A = T·b/L and T_B = T·a/L for an intermediate torque on a doubly built-in shaft — was web-corroborated 2026-07-06 against multiple independent statements (Gere & Goodno §3.8 worked example "bar ACB fixed at both ends"; engineering references give the fixed-end torque split in inverse proportion to the segment lengths) AND re-derived from first principles two independent ways in pipeline/tests/test_fixedfixed_shaft_physics.py (compatibility of twist, and Castigliano's theorem minimising the torsional strain energy over the redundant reaction).
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §5-4 (maximum-shear-stress theory, shear yield S_sy = 0.5 S_y). — checked: Topic-level cross-check citation for the shear-yield criterion (σ_y/2). Not section-pinned; the maximum-shear-stress criterion is standard and is applied identically in the sibling torsion-shaft THING. Consistent with Gere §3.3 for the torsion formula itself.

Flywheel (Solid Rotating Disk)

8 relations · 7 identities proven · 2 modeling steps · 3 validity envelopes · 9 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • σr=ω2ρ(R2−r2)(ν+3)8\sigma_{r} = \frac{\omega^{2} \rho \left(R^{2} - r^{2}\right) \left(\nu + 3\right)}{8} — elasticity: radial equilibrium + compatibility, plane stress
  • σt=ω2ρ(R2(ν+3)−r2(3ν+1))8\sigma_{t} = \frac{\omega^{2} \rho \left(R^{2} \left(\nu + 3\right) - r^{2} \left(3 \nu + 1\right)\right)}{8} — elasticity: hoop component of the same field

Sources & how they were checked:

  • Timoshenko, S. P., & Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, 1970 — ch. 4 (two-dimensional problems in polar coordinates), §32 "Rotating Disks": plane-stress field of a solid uniform disk, σ_r = σ_θ = (3+ν)ρω²R²/8 at the centre, σ_r(R) = 0 at the free rim. — checked: Article number and title pinned against the 3rd-edition full text (Internet Archive scan), 2026-06-10; the field itself is independently re-derived from equilibrium + compatibility in pipeline/tests/test_flywheel_physics.py.
  • Hibbeler, R. C., Engineering Mechanics: Dynamics, 14th ed., Pearson, 2016 — §17.1 (mass moments of inertia; the uniform solid disk I = mR²/2), ch. 18 (planar kinetics, work and energy: E = ½Iω² for rotation about a fixed axis), and ch. 19 (planar kinetics, impulse and momentum: the principle of angular impulse and momentum, I_O·ω₁ + Σ∫M_O dt = I_O·ω₂, §19.1–19.2) — the basis for the constant-torque spin-up time T_d·t = I_z·ω. — checked: §17.1 and the chapter-18/19 titles confirmed against published 14th-edition chapter listings, 2026-07-07 (ch. 19 = "Planar Kinetics of a Rigid Body: Impulse and Momentum"; §19.1 linear and angular momentum, §19.2 principle of impulse and momentum). The disk inertia integral is re-run and the spin-up time t = I_z·ω / T_d is independently re-derived from I·dω/dt = T (SymPy dsolve, from rest) in pipeline/tests/test_flywheel_spinup_physics.py.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 16 (flywheels: energy exchange and sizing) and ch. 1 (design factor against first yield). — checked: Ch. 16 title ("Clutches, Brakes, Couplings and Flywheels") confirmed via 10th-edition course materials, 2026-06-10; the flywheel section number within the chapter is not pinned.
  • Ashby, M. F., Materials Selection in Mechanical Design, 4th ed., Butterworth-Heinemann, 2011 — the "materials for flywheels" case study: maximum kinetic energy per unit mass is governed by the material index σ_y/ρ, independent of the disk's size. — checked: Case-study title and §6.6 numbering confirmed via a published reference index, 2026-06-10; correspondence of that numbering to the 4th edition is not pinned. The index claim itself (geometry cancels; e_max ∝ σ_y/ρ) is machine-proven in the derivation and re-derived in the test pipeline.

Four-Bar Linkage (Position)

2 relations · 4 identities proven · 2 modeling steps · 3 validity envelopes · 2 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • acos⁡(θ2)+bcos⁡(θ3)=ccos⁡(θ4)+da \cos{\left(\theta_{2} \right)} + b \cos{\left(\theta_{3} \right)} = c \cos{\left(\theta_{4} \right)} + d — vector loop, x-component
  • asin⁡(θ2)+bsin⁡(θ3)=csin⁡(θ4)a \sin{\left(\theta_{2} \right)} + b \sin{\left(\theta_{3} \right)} = c \sin{\left(\theta_{4} \right)} — vector loop, y-component

Sources & how they were checked:

  • Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — ch. 2 (Grashof condition), ch. 4 (vector-loop position analysis, the K-constant/Freudenstein closed form, open vs. crossed circuits, transmission angle). — checked: not section-pinned; cited from general knowledge of the text.

Helical Compression Spring

8 relations · 4 identities proven · 1 modeling step · 6 validity envelopes · 12 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Tw=DF2T_{w} = \frac{D F}{2} — statics: cut the wire

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §10-1 (stresses in helical springs: eqs. 10-1 spring index, 10-2/10-3 direct-shear factor), §10-2 (curvature effect: eq. 10-4 Wahl factor K_W, eq. 10-5 Bergsträsser K_B), §10-3 (deflection by Castigliano: eqs. 10-8, 10-9 spring rate), §10-4/Table 10-1 (end conditions; squared-and-ground: N_t = N_a + 2, L_s = d·N_t), §10-5 (stability: eqs. 10-10 to 10-13, Table 10-2 end-condition constant α), and §5-4 (maximum-shear-stress criterion, S_sy = 0.5 S_y). — checked: Equation and table numbers pinned against the publisher's official ch. 10 lecture-slide deck (Istanbul Technical University mirror), whose chapter-outline page also fixes the 10th-edition section pagination (§10-1 p. 518 … §10-5 p. 522), 2026-06-10. The rate formula is independently re-derived from Castigliano's theorem (both energy terms), and the Wahl factor's C → ∞ limit, its ordering above the direct-shear-only factor, and the ≤ ~1.4 % Wahl–Bergsträsser gap are proven, in pipeline/tests/test_spring_physics.py.
  • 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. — checked: Named-attribution source: the factor is universally attributed to Wahl (Shigley eq. 10-4 cites it as "the Wahl factor"); the 1963 monograph is the standard reference. Not independently page-pinned — the formula itself is verified against Shigley eq. 10-4 and exercised numerically in pipeline/tests/test_spring_physics.py.
  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — circular shafts in torsion (the wire's stress state, §3.3–3.5) and prismatic volume/mass. — checked: Topic-level citation — the same Gere sections the torsion-shaft THING cites (not section-pinned there either); the torsion formula this spring reuses is independently re-derived from the stress distribution in pipeline/tests/test_shaft_physics.py.

Impact Loading (Falling Mass, Energy Method)

10 relations · 5 identities proven · 2 modeling steps · 3 validity envelopes · 8 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • W=gmW = g m — weight of the falling mass
  • gm(δi+h)=δi2gm2δstg m \left(\delta_{i} + h\right) = \frac{\delta_{i}^{2} g m}{2 \delta_{st}} — energy balance (cited): potential energy lost = strain energy stored

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §2.8 (impact loading: the falling-collar energy balance, the impact factor, and the suddenly-applied-load n = 2 result), §9.8 (strain energy of bending), §9.10 (deflections produced by impact); the end-loaded cantilever tip deflection WL³/3EI is Ch. 9 material (worked examples / Appendix H). — checked: Beam-section pins corrected 2026-07-06 against the published 9th-edition contents (§9.7 is Nonprismatic Beams, so the original "§9.7-9.8" span was half-wrong; strain energy of bending is §9.8, deflections produced by impact is §9.10). §2.8 (impact loading) re-confirmed against the same listing; the textbook PDF is not web-accessible for a page-exact quote. The impact factor n = 1 + √(1 + 2h/δ_st), the n = 2 suddenly-applied limit, and the σ → √(2mghE/V) strain-energy-density asymptote are all re-derived from the energy balance (not taken on the citation) in pipeline/tests/test_impact_loading_physics.py.
  • Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — impact and energy loads (ch. 7): the same impact-factor result, the strain-energy-density design view (energy absorbed per unit volume, σ²/2E), and the caution that the energy method is unconservative against stress-wave and contact effects. — checked: Topic-level cross-check (impact / energy loading; the impact factor and the energy-per-volume design argument are standard and also given in Shigley §4-17 "Shock and Impact" — corrected from "§4-18" on 2026-07-06 against the published Ch. 4 contents, which end at §4-17 — and Boresi). Not page-pinned; used to corroborate the treatment and the failure notes, not any single emitted number. The member- inertia warn (member not massless) follows Juvinall's stated limitation of the method.
  • 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. — checked: The value 9.80665 m/s² is exact by definition (3rd CGPM, 1901); web-pinned against the NIST SP 330 / SI Brochure conventional value, 2026-07-06. It is displayed as a labeled cited constant, never a knob — the second THING to consume the constants mechanism after shaft-critical-speed.

Planetary (Epicyclic) Gearset

7 relations · 6 identities proven · 0 modeling steps · 0 validity envelopes · 8 domain guards · 12 parity samples

No modeling steps: this THING's derivation is identities all the way down (its physics enters through the relations' cited assumptions instead).

Sources & how they were checked:

  • 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). — checked: not section-pinned; cited from general knowledge of the text.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-13 (planetary gear trains: force, torque, and power flow). — checked: not section-pinned; cited from general knowledge of the text.

Power Screw (Square Thread)

4 relations · 4 identities proven · 1 modeling step · 2 validity envelopes · 6 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • tan⁡(lam)=lπdm\tan{\left(lam \right)} = \frac{l}{\pi d_{m}} — unwrap the helix: thread = inclined plane

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §8-2 "The Mechanics of Power Screws": Fig. 8-6 (unrolled-thread free bodies), eq. (8-1) torque to raise, eq. (8-2) torque to lower, eq. (8-3) self-locking condition πf d_m > l ⇔ f > tan λ, eq. (8-4) efficiency e = Fl/(2πT_R). — checked: Equation and figure numbers pinned against the publisher's official ch. 8 lecture-slide deck (Istanbul Technical University mirror), 2026-06-10, including the intermediate free-body equations (a)–(g) this page's derivation follows. The raise/lower closed forms are independently re-derived from two-equation block equilibrium (N kept explicit) in pipeline/tests/test_powerscrew_physics.py.

Propped Cantilever (UDL)

8 relations · 4 identities proven · 3 modeling steps · 3 validity envelopes · 4 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • RA+RB=LwR_{A} + R_{B} = L w — statics: ΣF_y = 0
  • LRB+MA=L2w2L R_{B} + M_{A} = \frac{L^{2} w}{2} — statics: ΣM_A = 0
  • L4w8EI=L3RB3EI\frac{L^{4} w}{8 E I} = \frac{L^{3} R_{B}}{3 E I} — force method: zero net deflection at the prop

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — ch. 10 (Statically Indeterminate Beams): the force (flexibility) method, propped-cantilever reactions by superposition; §5.5 flexure formula; App. E section properties. — checked: Topic-level citation, not section-pinned (the textbook PDF is not web-accessible for a page-exact quote, the same limit as the sibling beam THINGs). The governing results — prop reaction R_B = 3wL/8, wall reaction R_A = 5wL/8, fixing moment M_A = wL²/8, midspan deflection wL⁴/192EI — were web-corroborated 2026-07-06 against multiple independent statements (testbook.com and prepp.in solved problems give the prop reaction as 3wL/8 = 0.375wL and the fixed-end moment as wL²/8; ScienceDirect Topics "Propped Cantilever") AND re-derived from first principles two independent ways in pipeline/tests/test_propped_physics.py (the force method, and solving the Euler–Bernoulli BVP EI·v'''' = w with dsolve).
  • Hibbeler, R. C., Mechanics of Materials, 10th ed., Pearson, 2017 — ch. 12 (Deflection of Beams and Shafts): statically indeterminate beams by the method of superposition; equilibrium of the propped cantilever. — checked: Topic-level cross-check citation for the equilibrium equations (ΣF, ΣM). Not section-pinned; the two equilibrium relations are elementary statics and are re-derived from the free body in the physics test. Hibbeler ch. 12 is the standard cross-reference for the superposition method used here, consistent with Gere ch. 10.

Rectangular Shaft in Torsion (Saint-Venant)

12 relations · 2 identities proven · 5 modeling steps · 3 validity envelopes · 8 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • τmax=Tab2c1\tau_{max} = \frac{T}{a b^{2} c_{1}} — modeling: membrane analogy (Timoshenko §107); c1(a/b) from the §109 table
  • thetap=TGab3c2thetap = \frac{T}{G a b^{3} c_{2}} — modeling: torsional rigidity G·c2·a·b^3; c2(a/b) from the §109 table
  • θ=Lthetap\theta = L thetap — kinematics: uniform twist along the length
  • ητ=τmaxτround\eta_{\tau} = \frac{\tau_{max}}{\tau_{round}} — definition: efficiency ratios vs the equal-area circle
  • SF=σy2τmaxSF = \frac{\sigma_{y}}{2 \tau_{max}} — maximum-shear-stress criterion

Sources & how they were checked:

  • Timoshenko, S. P., & Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, 1970 — §107 (membrane analogy) and §108-109 (torsion of a bar of rectangular cross section: the coefficient table for maximum stress and torsional rigidity, and the thin-strip limit 1/3). — checked: The primary Timoshenko PDF was not directly reachable this session (archive.org refused the connection), so the coefficients were pinned by a STRONGER route than transcription: every authored row is recomputed from the exact Saint-Venant Fourier series in pipeline/tests/test_rect_torsion_physics.py and agrees within the test's asserted <= 0.001 (worst observed vs the series ~0.0005), and each value also matches Roark's independent closed form (below, within ~0.0005) and the thin-strip limit c1 = c2 = 1/3. These are the exact solution of a classical PDE — uncopyrightable facts confirmed from first principles, not a proprietary fit. Values corroborated against the Roark-attributed table on Wikipedia's "Torsion constant" article, 2026-07-04.
  • 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. — checked: Closed forms fetched from amesweb.info's Roark-cited rectangular-torsion page and implemented independently in test_rect_torsion_physics.py; evaluated per authored row they agree with the Timoshenko table and the exact series within the sources' stated accuracy. Roark's β column cross-read from Wikipedia's "Torsion constant" (which cites Roark 7th ed.), 2026-07-04.
  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §3.3-3.5 (circular shafts in torsion: tau = 16T/(πd^3), theta = TL/(GJ), J = πd^4/32), used here for the equal-area round-shaft comparison. — checked: Standard circular-torsion formulas; re-derived from first principles in the physics test.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y). — checked: Tresca shear-yield criterion; standard, used for the SF readout as on the round torsion-shaft page.

Rolling-Contact Bearing Life (Load–Life + Weibull Reliability)

5 relations · 4 identities proven · 2 modeling steps · 3 validity envelopes · 4 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • L10=1000000(C10P)aL_{10} = 1000000 \left(\frac{C_{10}}{P}\right)^{a} — load–life rating law (cited): L₁₀ = 10⁶(C₁₀/P)^a
  • R=e−(−x0+xRθ−x0)bR = e^{- \left(\frac{- x_{0} + x_{R}}{\theta - x_{0}}\right)^{b}} — three-parameter Weibull life distribution (cited)

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — Ch. 11 (Rolling-Contact Bearings): §11-3 (the load–life relation L₁₀ = (C/P)^a, with a = 3 for ball and 10/3 for roller bearings, C₁₀ rated at 10⁶ revolutions), §11-4 (reliability and life via the three-parameter Weibull distribution), and Table 11-6 (Weibull parameters for two manufacturers' rating lives). — checked: Ch. 11 topics confirmed against the published 10th-edition chapter listing, 2026-07-06. The Weibull parameters x₀ = 0.02, θ = 4.459, b = 1.483 are the Manufacturer-2 row of Table 11-6 (02-series, 10⁶-revolution rating basis — the basis this page uses), transcribed and web-corroborated against two independent reproductions of that table (Bartleby and Chegg solution sets for Shigley 10e Ch. 11); the load–life exponents a = 3 (ball) / 10/3 (roller) are the ISO 281 values also stated in §11-3 and confirmed across multiple bearing-engineering references. These are cited DATA, not machine-proven physics. What IS machine-proven: the Weibull inversion x(R) = x₀ + (θ − x₀)(ln 1/R)^{1/b} is re-derived by solving the Weibull CDF symbolically, and the 10⁶-rev rating basis and the hours conversion are re-derived, in pipeline/tests/test_bearing_physics.py. The textbook PDF is not web-accessible for a page-exact quote.
  • Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — ch. 14 (rolling-element bearings): the same catalog load–life relation and the reliability adjustment for lives other than the 90% rating. — checked: Topic-level cross-check (rolling-contact bearing life; the (C/P)^a load–life law and the reliability adjustment are standard and also given in Norton, Shigley, and the SKF/Timken catalog literature). Not page-pinned; used to corroborate the treatment and the failure notes, not any single emitted number.
  • 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. — checked: NAMED as the origin of the rating framework only — never ingested, reproduced, or quoted. The numeric values used on this page are taken as printed in Shigley (accessible-textbook tier per docs/data-provenance.md); ISO 281 and ABMA are cited to attribute the framework, consistent with the project's fact-extraction legal frame.

Rotating Disk with a Central Bore

9 relations · 4 identities proven · 2 modeling steps · 3 validity envelopes · 11 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • σr=ω2ρ(ν+3)(−R2a2r2+R2+a2−r2)8\sigma_{r} = \frac{\omega^{2} \rho \left(\nu + 3\right) \left(- \frac{R^{2} a^{2}}{r^{2}} + R^{2} + a^{2} - r^{2}\right)}{8} — elasticity: radial equilibrium + compatibility, free bore and free rim
  • σt=ω2ρ(−r2(3ν+1)+(ν+3)(R2a2r2+R2+a2))8\sigma_{t} = \frac{\omega^{2} \rho \left(- r^{2} \left(3 \nu + 1\right) + \left(\nu + 3\right) \left(\frac{R^{2} a^{2}}{r^{2}} + R^{2} + a^{2}\right)\right)}{8} — elasticity: hoop component of the same field

Sources & how they were checked:

  • Timoshenko, S. P., & Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, 1970 — ch. 4 (two-dimensional problems in polar coordinates), §32 "Rotating Disks": the plane-stress annulus with free bore and free rim, peak hoop stress at the bore edge, and the small-hole limit of twice the solid disk's central stress. — checked: Article number and title pinned against the 3rd-edition full text (Internet Archive scan) during the flywheel-disk session, 2026-06-10 — §32 treats the bored disk in the same article as the solid one. The annulus field is independently re-derived from equilibrium + compatibility with free-free boundary conditions in pipeline/tests/test_disk_physics.py.
  • Hibbeler, R. C., Engineering Mechanics: Dynamics, 14th ed., Pearson, 2016 — §17.1 (mass moments of inertia; the annulus I = m(R² + a²)/2) and ch. 18 (planar kinetics, work and energy: E = ½Iω² for rotation about a fixed axis). — checked: §17.1 and chapter-18 titles confirmed against published 14th-edition chapter listings during the flywheel-disk session, 2026-06-10; the annulus inertia integral is re-run in pipeline/tests/test_disk_physics.py.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 16 (flywheels: energy exchange and sizing) and ch. 1 (design factor against first yield). — checked: Ch. 16 title ("Clutches, Brakes, Couplings and Flywheels") confirmed via 10th-edition course materials during the flywheel-disk session, 2026-06-10; the flywheel section number within the chapter is not pinned.
  • 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. — checked: Case-study title and §6.6 numbering confirmed via a published reference index during the flywheel-disk session, 2026-06-10; correspondence of that numbering to the 4th edition is not pinned.

Shaft Critical Speed (Rayleigh + Dunkerley)

9 relations · 4 identities proven · 4 modeling steps · 4 validity envelopes · 5 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • δst=L3W48EI\delta_{st} = \frac{L^{3} W}{48 E I} — reuse the simply-supported center-load deflection
  • ωc2=gδst\omega_{c}^{2} = \frac{g}{\delta_{st}} — Rayleigh single-mass critical speed (cited)
  • ωs2=π4EIAL4ρ\omega_{s}^{2} = \frac{\pi^{4} E I}{A L^{4} \rho} — uniform-shaft first bending mode (cited; re-derived in tests)
  • 1ωcD2=1ωs2+1ωc2\frac{1}{\omega_{cD}^{2}} = \frac{1}{\omega_{s}^{2}} + \frac{1}{\omega_{c}^{2}} — Dunkerley superposition of reciprocal squares (cited)

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §7-6 (critical speeds of rotating shafts: the Rayleigh single-mass estimate ω = √(g/δ), Dunkerley's equation combining reciprocal squares, and the operating-speed margin) and Table A-9 (the simply-supported center-load deflection reused for δ_st). — checked: §7-6 (shaft critical speeds) and Table A-9 (beam deflection cases) topics confirmed against the published 10th-edition chapter/appendix listings, 2026-07-06; the textbook PDF is not web-accessible for a page-exact quote. The Rayleigh formula ω_c = √(g/δ_st), the g-cancellation ω_c = √(48EI/mL³), the first-mode ω_s, and the Dunkerley ≤ Rayleigh bound are all re-derived from first principles (not taken on the citation) in pipeline/tests/test_shaft_critical_speed_physics.py.
  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — App. E section properties (I = πd⁴/64, A = πd²/4 for a solid circular shaft) and the simply-supported beam small-deflection theory behind δ_st. — checked: Topic-level citation, not section-pinned; the section properties and small-deflection limit match the sibling beam and shaft pages. I = πd⁴/64 and A = πd²/4 are re-derived by area integration in the physics test.
  • Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — shaft design and critical (whirling) speed: the same Rayleigh √(g/δ) estimate and the Dunkerley combination, with operating speeds kept away from the resonance band. — checked: Topic-level cross-check (shaft whirling speed; the √(g/δ) form and the Dunkerley rule are standard and also given in Den Hartog, Mechanical Vibrations); not page-pinned. Used to corroborate the Rayleigh/Dunkerley treatment, not any single emitted number.
  • 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. — checked: The value 9.80665 m/s² is exact by definition (3rd CGPM, 1901); web-pinned against the NIST SP 330 / SI Brochure conventional value and the NIST reference on constants, 2026-07-06. It is displayed as a labeled cited constant, never a knob.

Shaft in Torsion (Solid, Circular)

6 relations · 5 identities proven · 1 modeling step · 1 validity envelope · 7 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • γ=rcθL\gamma = \frac{r_{c} \theta}{L} — kinematics: plane sections rotate rigidly

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §3.3–3.5 (circular shafts in torsion: strain kinematics, the torsion formula, angle of twist). — checked: not section-pinned; cited from general knowledge of the text.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-12 (power transmission, P = Tω), §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y). — checked: not section-pinned; cited from general knowledge of the text.

Shaft under Combined Bending + Torsion

7 relations · 2 identities proven · 3 modeling steps · 1 validity envelope · 7 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • σb=32Mπd3\sigma_{b} = \frac{32 M}{\pi d^{3}} — assemble the element: flexure + torsion on one square of surface
  • τmax2=σb24+τt2\tau_{max}^{2} = \frac{\sigma_{b}^{2}}{4} + \tau_{t}^{2} — plane-stress transformation: Mohr's circle radius
  • σvm2=σb2+3τt2\sigma_{vm}^{2} = \sigma_{b}^{2} + 3 \tau_{t}^{2} — distortion energy: von Mises equivalent stress

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 3 (plane-stress transformation and Mohr's circle: the principal stresses σ_A,B = (σ_x+σ_y)/2 ± √(((σ_x−σ_y)/2)² + τ_xy²) and the max-shear radius), §5-4 (maximum-shear-stress theory), §5-5 (distortion-energy theory, the plane-stress von Mises form). — checked: Criterion forms pinned against the 10th-edition ch. 5 solutions manual (Oakland University mirror), 2026-06-10: problems 5-1 … 5-3 apply MSS as n = S_y/(σ₁−σ₃) and DE as σ' = (σ_x² − σ_xσ_y + σ_y² + 3τ_xy²)^½ with the principal-stress radical verbatim. The pages read print no section or equation numbers for these, so the §5-4/§5-5 numbers and the ch. 3 attribution are from the book's structure, not independently pinned. Both criteria are cross-checked against the stress tensor — max shear as the eigenvalue half-spread, and the plane-stress von Mises form proven equivalent to the principal-stress form σ_A² − σ_Aσ_B + σ_B² (the form the manual applies) — and the bracketing ratio [1, 2/√3] proven, in pipeline/tests/test_combined_physics.py.
  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — flexure formula (§5.5) and circular-shaft torsion (§3.3–3.5), the two single-load formulas this THING combines. — checked: Topic-level citation, not section-pinned; §5.5 and §3.3–3.5 match the cantilever-beam and torsion-shaft THINGs' citations exactly. The torsion formula is re-derived from the stress distribution in test_shaft_physics.py, and the round-section flexure form 32M/πd³ is re-derived (I = πd⁴/64 by polar integration, σ = Mc/I) in pipeline/tests/test_combined_physics.py.

Simply Supported Beam (Center Load + UDL)

8 relations · 3 identities proven · 2 modeling steps · 3 validity envelopes · 8 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • MP=Px2M_{P} = \frac{P x}{2} — statics: method of sections, point load
  • Mw=wx(L−x)2M_{w} = \frac{w x \left(L - x\right)}{2} — statics: method of sections, distributed load

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 4 (Deflection and Stiffness): deflections by integration and by superposition, and Table A-9 (beam deflection cases; the simply-supported center-load and uniform-load rows used here). — checked: Pinned against the 10th-edition ch. 4 solutions manual (Oakland University mirror), 2026-06-10: problem 4-9 integrates the simply-supported center-load case from scratch (y_max = Fl³/48EI, symmetry slope condition at midspan — the same derivation our test re-runs), and problems 4-10 … 4-12 cite "Table A-9" by name and case number ("use beams 5 and 6 of Table A-9", "by superposition of beams 6 and 7"), the center-load form recurring as F₂l³/48EI in 4-11. The uniform-load row's 5wl⁴/384EI coefficient was not itself sighted in the manual pages read; both deflection rows are independently re-derived by integrating EI·v″ = M(x) in pipeline/tests/test_ssbeam_physics.py.
  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — flexure formula (§5.5), shear-deflection and small-deflection limits, App. E section properties. — checked: Topic-level citation, not section-pinned; §5.5 and App. E match the cantilever-beam THING's citation, and the shear/small-deflection validity limits are topic-level only. The flexure formula and I = bh³/12 are re-derived from first principles (I by area integration, σ = Mc/I from linear bending strain and moment equilibrium) in pipeline/tests/test_ssbeam_physics.py.

Slider-Crank (Exact Kinematics and Gas Torque)

10 relations · 6 identities proven · 3 modeling steps · 2 validity envelopes · 14 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • l2=q2+r2sin⁡2(θ)l^{2} = q^{2} + r^{2} \sin^{2}{\left(\theta \right)} — geometry: the rod-crank-axis right triangle
  • dispapprox=r(1−cos⁡(θ))+r2(1−cos⁡(2θ))4ldisp_{approx} = r \left(1 - \cos{\left(\theta \right)}\right) + \frac{r^{2} \left(1 - \cos{\left(2 \theta \right)}\right)}{4 l} — binomial two-term approximation (r/l small)
  • lsin⁡(ϕ)=rsin⁡(θ)l \sin{\left(\phi \right)} = r \sin{\left(\theta \right)} — obliquity from the shared transverse offset

Sources & how they were checked:

  • Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — slider-crank position, velocity, and acceleration analysis (the exact closed forms and the r/l two-term series for piston motion) and engine dynamics (the gas-force torque with connecting-rod obliquity, T = F·r·sin(θ+φ)/cosφ). — checked: Topic-level citation, confirmed against the standard treatment of the slider-crank; the textbook PDF is not web-accessible for a page-exact quote this session (same access limit as the S02–S09 mechanics pages). Nothing on this page rests on the citation alone: the velocity and acceleration relations are re-derived by INDEPENDENT differentiation of x(θ) in pipeline/tests/test_slider_crank_physics.py, the two-term approximation's error is bounded across a sweep of r/l there, and the long-rod torque limit T → F·r·sinθ is checked symbolically.
  • 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. — checked: Topic-level cross-check (the slider-crank kinematic and quasi-static force results are standard and identical across Norton, Uicker/Pennock/Shigley, and Wilson & Sadler). Not page-pinned; used to corroborate the treatment, not any single emitted number.

Spur Gear Pair (Lewis Bending)

10 relations · 2 identities proven · 3 modeling steps · 2 validity envelopes · 7 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Kv=V+vbvbK_{v} = \frac{V + v_{b}}{v_{b}} — modeling: Barth velocity factor (empirical, Shigley 14-6b)
  • σbp=KvWtYpbmmod\sigma_{b p} = \frac{K_{v} W_{t}}{Y_{p} b m_{mod}} — modeling: Lewis bending stress; Y_p from Table 14-2
  • SFpσbp=σySF_{p} \sigma_{b p} = \sigma_{y} — definition: bending factor of safety — the pinion governs

Sources & how they were checked:

  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-5/13-7 (involute geometry; interference, eq 13-11), §14-1 (the Lewis bending equation), Table 14-2 (Lewis form factor Y, 20° full-depth, tip load), eq 14-6b (Barth velocity factor, metric). — checked: Topic and formulas pinned 2026-07-04. Velocity-factor equation number corrected 14-4b → 14-6b on 2026-07-06: eq 14-4b is the US-customary (1200+V)/1200 ft/min form; the metric (6.1+V)/6.1 m/s form is eq (14-6b) — corroborated against a published worked-example excerpt of Shigley §14 (mguler.etu.edu.tr Spur_Gear_Example) and the 10th-ed solutions manual's usage. Table 14-2 Y values cross-checked against multiple independent reproductions of the standard 20° full-depth, tip-load Lewis form factors (evolventdesign.com/pages/gear-strength-calculator-lewis confirms the "load near the tip" basis; individual values 12→0.245, 15→0.290, 17→0.303, 20→0.322, 30→0.359, 50→0.409, 400→0.480 confirmed via web search) and are consistent with the tables in Juvinall & Marshek, Fundamentals of Machine Component Design, and other standard references (they all derive from the same standard tooth geometry). The KINEMATICS and FORCE PATH (V, W_t, i) are re-derived from first principles in pipeline/tests/test_gear_physics.py; the Y values are cited data and are NOT machine-proven physics (interpolation between rows is proven; the numbers between rows are asserted by citation — see /verification/).

Stepped Shaft — Shoulder-Fillet Stress Concentration

3 relations · 0 identities proven · 3 modeling steps · 3 validity envelopes · 3 domain guards · 9 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • Kt=ArdbKt = A rd^{b} — modeling: Norton App. C power-law fit (K_t = A(r/d)^b); (A, b) from D/d
  • σmax=Ktσnom\sigma_{max} = Kt \sigma_{nom} — definition: stress-concentration factor amplifies the nominal stress
  • SFσmax=σySF \sigma_{max} = \sigma_{y} — definition: static safety factor — material moves SF, never K_t

Sources & how they were checked:

  • 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. — checked: Read digit-for-digit from Norton App. C Figs C-1/C-2/C-3 (owner-provided scan, 2026-07-04); cross-checked against the independently published Peterson-derived C1..C4 closed-form fits (polynomials in sqrt(h/r) evaluated at 2h/D) for the shoulder fillet in a stepped circular shaft, as tabulated in Pilkey, Formulas for Stress, Strain, and Structural Matrices, and printed as Table 17.1 in Roark's Formulas for Stress and Strain, 7th/8th ed. — the two independently published fits agree within the sources' stated few-percent accuracy over the well-stepped D/d >= 1.5 band (see pipeline/tests/test_stepped_shaft_physics.py). Attribution corrected 2026-07-06: this field originally cited "Roark Table 6-1 case III-2", a designation that exists in no Roark edition (that part/case numbering is Pilkey's); the closed form itself is unchanged. The (A, b) numbers are cited DATA, not machine-proven physics; interpolation between D/d rows is proven in the parity oracle. Shigley Fig. A-15-8/9 is the same chart family and is named as a cross-reference only, never digitized.

Symmetric Two-Bar Truss

13 relations · 8 identities proven · 3 modeling steps · 4 validity envelopes · 16 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • A=πd24A = \frac{\pi d^{2}}{4} — geometry: round-bar section properties
  • eel=FmLAEe_{el} = \frac{F_{m} L}{A E} — Hooke's law: member elongation
  • mtruss=2ALρm_{truss} = 2 A L \rho — mass of the two members

Sources & how they were checked:

  • 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. — checked: Topic-level citation against the published 9th-edition chapter listing, 2026-07-06; the textbook PDF is not web-accessible for a page-exact worked example. The governing results are NOT taken on the citation: the member force F_m = P/(2cos α) is re-derived from joint equilibrium, and the joint deflection δ = P L/(2 A E cos²α) is re-derived two independent ways (unit-load virtual work AND the compatibility-triangle projection) in pipeline/tests/test_truss_physics.py, the two agreeing symbolically. The symmetric two-bar deflection δ_v = e/cos α with e = F_m L/(A E), giving the cos²α form, was web-corroborated 2026-07-06 (Roylance, Mechanics of Materials, MIT — the δ_v = δ/cos θ projection). The δ/L < 0.1 small-displacement warn is the same cited convention used on the beam pages.
  • 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. — checked: Same pinning as the Euler Column page (Dr. Ala Hijazi's 10th-edition class notes, eis.hu.edu.jo/ACUploads/10526/CH 4.pdf, 2026-06-10): the limiting slenderness √(2π²CE/S_y) with C = 1 for pinned–pinned members. The λ_T relation is cross-checked against the Euler Column THING (identical form) in the physics test.
  • 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. — checked: Topic-level cross-check for the compatibility-triangle derivation of the joint deflection (elongation e projected to the joint displacement); not page-pinned (PDF not web-accessible, as on the sibling pages). Used to corroborate the derivation route, which the physics test re-derives from first principles.

Thermal Assembly (Two-Segment Bar Between Rigid Walls)

6 relations · 2 identities proven · 3 modeling steps · 2 validity envelopes · 4 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • eps1=α1dTeps_{1} = \alpha_{1} dT — thermal strain: ε_th = α ΔT
  • F1=F2F_{1} = F_{2} — statics: one internal force path
  • L1α1dT+L2α2dT=F2L2A2E2+F1L1A1E1L_{1} \alpha_{1} dT + L_{2} \alpha_{2} dT = \frac{F_{2} L_{2}}{A_{2} E_{2}} + \frac{F_{1} L_{1}}{A_{1} E_{1}} — compatibility: zero net elongation (rigid walls)

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — ch. 2 (Axially Loaded Members), §2.5 "Thermal Effects, Misfits, and Prestrains": free thermal strain α ΔT, and statically indeterminate bars with temperature changes solved by equilibrium + compatibility. — checked: Topic-level citation, not section-pinned (the textbook PDF is not web-accessible for a page-exact quote, the same limit as the sibling axial THINGs). The governing results — free thermal strain ε = α ΔT, and a fully restrained bar developing σ = E α ΔT (the single-material limit of this two-segment model) — were web-corroborated 2026-07-07 AND re-derived from first principles in pipeline/tests/test_thermal_physics.py, which also reproduces a PUBLISHED worked example (Pytel & Singer, Strength of Materials, 4th ed., Problem 266: a series steel+aluminum bar between unyielding supports heated 100°F, F = 26,691.84 lb, σ_steel = 17,795 psi, σ_al = 13,346 psi) to the pound and psi.
  • 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). — checked: Topic-level cross-check citation for the thermal-stress setup. Not section-pinned; the free expansion + restoring-force superposition is re-derived from first principles in the physics test. Hibbeler ch. 4 is the standard cross-reference for the restrained-thermal-bar method used here, consistent with Gere §2.5. A36 steel α = 6.5 ×10⁻⁶/°F is the value Hibbeler's worked examples use, matching this catalogue's steel-a36 CTE entry.

Thick-Walled Cylinder (Lamé)

7 relations · 5 identities proven · 2 modeling steps · 2 validity envelopes · 11 domain guards · 9 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • σr=A−Br2\sigma_{r} = A - \frac{B}{r^{2}} — elasticity: equilibrium + compatibility, polar coordinates
  • σt=A+Br2\sigma_{t} = A + \frac{B}{r^{2}} — elasticity: hoop component of the same family

Sources & how they were checked:

  • Timoshenko, S. P., & Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, 1970 — ch. 4 (two-dimensional problems in polar coordinates), §28 "Stress Distribution Symmetrical about an Axis": the hollow cylinder under uniform pressure — the Lamé field σ = A ∓ B/r², boundary conditions, bore maximum. — checked: Article number and title pinned against the 3rd-edition full text (Internet Archive scan), 2026-06-10; the Lamé family is independently re-derived from equilibrium + compatibility (and the boundary conditions re-solved) in pipeline/tests/test_cylinder_physics.py.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-14 (stresses in pressurized cylinders) and §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y). — checked: §3-14 "Stresses in Pressurized Cylinders" confirmed against 10th-edition section listings, 2026-06-10; §5-4 matches the citation already used by the torsion-shaft THING.
  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — prismatic members and section properties (annular area, mass per length); thin-walled pressure vessels (the k → 1 limit this THING hands off to). — checked: Topic-level citation, not section-pinned; the k → 1 hand-off limit itself is machine-verified (sp.limit) in the test pipeline.

Thin-Walled Pressure Vessel (Cylinder)

4 relations · 5 identities proven · 0 modeling steps · 2 validity envelopes · 6 domain guards · 6 parity samples

No modeling steps: this THING's derivation is identities all the way down (its physics enters through the relations' cited assumptions instead).

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §8.2–8.3 (thin-walled spherical and cylindrical pressure vessels, including the r/t ≥ 10 thin-shell criterion and the hoop/longitudinal 2:1 ratio). — checked: not section-pinned; cited from general knowledge of the text.

Thin-Walled Tube in Torsion (Bredt)

4 relations · 1 identities proven · 3 modeling steps · 3 validity envelopes · 6 domain guards · 6 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • q=tτq = t \tau — equilibrium of a wall element: shear flow is constant
  • T=2AmqT = 2 A_{m} q — moment resultant: the swept-area identity
  • θ=LST4Am2Gt\theta = \frac{L S T}{4 A_{m}^{2} G t} — Bredt's second formula, by Castigliano

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §3.11 "Thin-Walled Tubes": shear flow, Bredt's torsion formulas τ = T/(2tA_m) and the torsion constant J = 4A_m²t/(∮ds/t), validity limits of the thin-wall approximation. — checked: Section number and title pinned for the 7th/8th editions against the publisher's own edition-comparison table of contents (Cengage, "3.11 Thin-Walled Tubes"), 2026-06-10; a 9th-edition listing shows the topic carried, but the 9e itself was not opened — its section number is inferred from edition continuity, not sighted. Both Bredt formulas are independently re-derived — the swept-area moment integral ∮r⊥ds = 2A evaluated for an ellipse and a rectangle, and the twist by Castigliano on the shear strain energy — in pipeline/tests/test_tube_physics.py.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y). — checked: Same §5-4 already cited by the torsion-shaft and cylinder THINGs; the MSS form n = S_y/(σ₁−σ₃) is confirmed against the 10th-edition ch. 5 solutions manual (Oakland mirror), 2026-06-10 — the §5-4 number and the S_sy = 0.5 S_y statement come from the chapter's known structure, not sighted in the manual pages read.
  • 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. — checked: Classical mathematical result (antiquity; first rigorous proofs 19th century), used here as a hard existence envelope on the (A_m, S) knob pair. The Blåsjö survey is a reader pointer cited from the literature, not consulted or independently pinned this session. Machine-exercised in pipeline/tests/test_tube_physics.py: the stadium-section family realizes any (A, S) with S² ≥ 4πA and its construction goes complex exactly when the inequality fails — the same construction the sim draws.

Torsional Oscillator (Disk on a Shaft)

11 relations · 5 identities proven · 1 modeling step · 3 validity envelopes · 6 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • ωn2=ktJd\omega_{n}^{2} = \frac{k_{t}}{J_{d}} — dynamics: single-DOF simple harmonic motion (cited)

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §3.3–3.5 (circular shafts in torsion: the torsion formula τ = 16T/πd³, the angle of twist θ = TL/GJ, and hence the torsional stiffness k_t = GJ/L). — checked: Chapter/section topic (circular-shaft torsion) confirmed against the published 9th-edition table of contents, 2026-07-06; the torsion formulas themselves are re-derived from the surface-strain kinematics in pipeline/tests/test_torsional_oscillator_physics.py and are shared verbatim with the torsion-shaft page.
  • 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). — checked: §17.1 and the vibrations chapter confirmed against published 14th-edition chapter listings, 2026-07-06; the disk inertia integral I = ½ m R² is re-run symbolically in the physics test.
  • 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). — checked: Classical result attributed to the standard vibration text (the disk-on-shaft torsional pendulum is the archetypal single-DOF example); topic-level, not page-pinned — the textbook is not web-accessible. The result is independently re-derived by the ENERGY method (½ J ω_n² Θ² = ½ k_t Θ²) in the physics test, so nothing rests on the citation alone.
  • 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. — checked: Topic-level (shaft torsional vibration; the one-third-shaft-inertia rule of thumb is standard and also stated in Den Hartog, Mechanical Vibrations); not page-pinned. Used only to cite the lumped-model warn envelope, not any emitted number.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — ch. 7 (shafts and torsional systems), §3-12 (power transmission), and §5-4 (maximum-shear-stress theory, shear yield at S_sy = 0.5 S_y). — checked: Chapter 7 (shafts) and §5-4 (maximum-shear-stress theory) topics confirmed against 10th-edition course materials, 2026-07-06; used for f = ω_n/2π, the period, and the σ_y/2 shear-yield margin — the last of which is re-derived, not taken on faith, in the physics test.

Transverse Shear in Beams (τ = VQ/Ib, Shear Flow, Fastener Spacing)

9 relations · 2 identities proven · 2 modeling steps · 3 validity envelopes · 6 domain guards · 3 parity samples

Where physics enters (everything else on that page follows by machine-proven algebra):

  • τmax=QVIb\tau_{max} = \frac{Q V}{I b} — equilibrium: bending-stress gradient balanced by longitudinal shear
  • Ffastener=qsF_{fastener} = q s — definition: connector force = shear flow × spacing

Sources & how they were checked:

  • Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.8–5.11 (shear stresses in beams of rectangular cross section, the shear formula τ = VQ/Ib, shear flow, and built-up beams / fastener spacing). — checked: Topic-level citation (same source id and §5.x family as simply-supported-beam and cantilever-beam). The shear formula τ = VQ/(Ib), the parabolic distribution, τ_max = 3V/2A, and the shear flow q = VQ/I are ALL re-derived from first principles in pipeline/tests/test_shear_flow_physics.py — the bending-stress-gradient equilibrium integral, the 3/2 peak, and the theorem that the parabola integrates back to exactly V — so nothing here rests on citation alone. The hand golden (V=12 kN, 40×180 mm, s=75 mm → τ_max=2.5 MPa, q=100 N/mm, F=7.5 kN) is arithmetic on those formulas, checkable by hand.
  • Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y). — checked: Tresca shear-yield criterion; standard, the same σ_y/2 basis the torsion-shaft page uses.

Material property values carry their own per-value provenance (source, locator, original units, statistical basis) on the materials page, under the rules in the repository's docs/data-provenance.md.