Engineering software manual for the 66 active product modules shipped under /products/*. This document describes purpose, governing methods, design-code support, UI maturity, and known gaps. It complements the homogenization contract in Homogenization-Roadmap.md.

Audience: engineers evaluating PhyCalcPro for design work, and developers extending modules.

Disclaimer: All modules produce indicative results unless explicitly marked beta with implemented code checks. Nothing here replaces licensed professional review or official code compliance certification.

Equations (authoring): Use LaTeX in $…$ for inline math and `

for display blocks (or$\dots$/$\dots$). The site converts these to KaTeX. Use \frac{\partial^2 w}{\partial x^2}for partial derivatives (slash shorthand is normalized at load time). List items like- Torque: $T=\dots$` render as labeled display equations.

Table of contents

1. Platform architecture
2. Module inventory
3. Module reference — compiled from docs/modules/*.md at build time
4. Maturity & numerical methods
5. Gaps & roadmap

1. Platform architecture

1.1 Navigation and layout

• Single products nav: src/app/products/layout.tsx renders the category sub-bar with module dropdowns (Sidebar). Category layouts are passthrough wrappers — no nested module nav.
• Module chrome: Each calculator page uses CalculatorLayout with a workspace under the products category bar:
  • Inputs column — parameters, mesh controls, calculate/save (CalculatorInputPanel where adopted).
  • Results column — plots, metric cards, engineering checks, export (CalculatorResultsShell / ExportableReport).
  • Optional Design Summary rail (summary prop) — persistent sticky checklist that can update live from inputs (bearings selection, plain bearings, housing).
• Layout contract: All product pages pass explicit inputs / results props to CalculatorLayout. The legacy left / center / right API is removed; npm run validate:layout enforces the contract in CI. Optional summary is allowed.

1.2 Calculation pipeline

Standard module contract (see homogenization roadmap):

page.tsx
  useStandardCalculation(moduleId, onRegionUnits?)   // or useCalculatorModule
  CalculatorLayout(moduleId, inputs, results)
  calculate → solver engine → wrapResult(output)
  *Results → ExportableReport(moduleId=…)

• wrapResult attaches a CalculationSpec with design-code checks via withCalculationSpec.
• Specialized evaluators (full check mapping): beams, columns, gears, combined-loading, welds.
• Generic evaluator (evaluators/generic.ts): all other catalogued modules — maps solver output fields (safety factor, utilization, life margin, fatigue SF, etc.) to catalog checks via MODULE_FIELD_OVERRIDES. Flagship modules with extended mappings: shafts, bearings, compression-springs, extension-springs, torsion-springs, rivets, welds.

1.3 Design codes

Global selector:

Changing design code sets default units via useDesignCodeUnits / moduleProfiles.ts (on code change, not every render). Field unit selectors still expose all units for the dimension unless restrictToProfile is set.

1.4 Units

• Field definitions: src/lib/units/moduleProfiles.ts (expansion modules profiled; legacy gaps remain for trusses, cost-estimator, cam-toolpaths).
• Preferred input widget: CalculatorUnitField + calculatorNumberInputClass.
• Metric display: CalculatorMetricCard / formatEngineeringValue for auto scientific notation when $|v|\ge 1000$ or $|v|<0.01$.
• Temperature: affine conversions among °C, K, and °F (not a simple offset-only scale).

1.5 Export

ExportableReport with moduleId enables:

• Structured PDF reports via src/lib/export/structuredReport.ts — title block, optional named sections (Design Summary, ISO 281 / film / housing factors, arrangement, recommendation), curated inputs, metric results, engineering checks, formula steps, and embedded chart images (collectChartImages from Plotly). Flat resultRows remain supported for older modules. Excel mirrors section groups as extra sheets when present.
• CSV export from solver output.
• Quality checklist from moduleQualityDefaults.
• Engineering checks panel when calculationSpec is present.

Charts use EngineeringPlot with separate yLabel, unitLabel, xLabel, xUnit and data-export-plot for high-resolution capture.

1.6 Testing & verification

• Vitest (npm test) — unit tests and externally sourced benchmarks (Shigley, Roark, AISC, ISO 6336/281, VDI 2230, EN 13906, etc.) under src/lib/**/**/*.test.ts.
• Verification CI — npm run test:verification runs 38 JSON cases in src/data/verification/ against moduleSolverRegistry.ts (64 solvers registered).
• Bootstrap — npx tsx scripts/bootstrap-verification.ts generates JSON from seeds in verificationSeeds.ts.
• Engineer sign-off — validation-master-checklist.md lists validation tasks for all 62 modules; springs also have spring-modules-user-tasks.md.
• FEM regression — analytical comparisons for beam equilibrium, column buckling, and plate shear-locking in src/lib/structural/__tests__/.

CI benchmark modules (38 JSON cases, 34 modules): beams, bearings, bevel-gears, bolts, circular-plates, columns, combined-loading, compression-springs (×2), corrosion, extension-springs, fatigue, frames, gears, hydraulics, impact, internal-gears-rack (×2), keys-splines, pipes, plain-bearings (×2), power-screws (×2), rivets, rotation, shafts, shells, suspension, timing-belts, tolerance, torsion-springs, trusses, unit-converter, v-belts, vessels, vibrations, welds.

1.7 Release tiers

CalculatorLayout shows catalog validationStatus and a computed release tier from benchmark stats (ReleaseTierBadge). Solvers are registered for 61 modules; promote modules toward verified by adding JSON cases and completing the master validation checklist.

1.8 Design workflow layer

Every calculator page receives a shared Auto-design / Validate / Compare toolbar through CalculatorLayout. Tab order is fixed: Auto-design (size from targets) → Validate (forward check) → Compare (ranked alternatives with Apply). User-facing names and button labels live in src/lib/design-workflows/workflowModeLabels.ts; internal IDs remain design, check, and select.

The workflow registry (src/lib/design-workflows/moduleDesignWorkflows.ts) provides:

• required design inputs to define before sizing,
• automatic sizing targets,
• computed reference-design candidate comparisons,
• standard/catalog tables to consult,
• linked downstream modules,
• expert notes and explicit gaps.

The computed candidate engine (src/lib/design-workflows/computedCandidates.ts) supplies numerical candidate rows for every active module family. These rows use best-available first-principles or standard screening equations (for example beam stress/deflection, shaft von Mises stress, Lewis gear bending, ISO 281 bearing life, spring shear stress, pressure hoop stress, pump-down time, thermal conductance, coil field and battery cooling flow).

This is the platform layer needed for MITCalc-style worksheets. As of the full rollout:

• designModeRegistry.ts maps every module ID to a category design solver (catalog sweep, reverse sizing, or optimization screen).
• computedCandidates.ts calls the registry so the advisor shows live ranked candidates from page userInputs.
• Calculate branches on workflow mode: Validate runs the forward solver only; Auto-design applies the best registry candidate then re-runs validation; Compare ranks options without auto-apply (Apply in the advisor loads a row and switches to Validate).
• Shared helpers: sweepCatalogForUtilization, materialCatalogService, scripts/scaffold-design-mode.mjs.
• Full mode behavior: see docs/Design-Workflow-Reference.md.

Count: 61 modules with real design paths · 2 validate-only tools · 1 profiles page (section-from-required-I).

1.9 Persistence & cross-calculator handoff

• Local projects — src/lib/localProjects.ts saves inputs/results per module; /projects dashboard lists and reloads saved work.
• Cloud sync — optional Supabase workspace sync via /api/workspaces/models when authenticated.
• Cross-calc handoff — crossCalcHandoff.ts + CrossCalcHandoffBanner carry gear outputs → shaft sizing → bearing selection on linked pages.

2. Module inventory

Homogenization snapshot

Validation catalog status

3. Module reference

Per-module write-ups live in docs/modules/{moduleId}.md. Each entry covers purpose, physics & theory, governing equations, numerical method, inputs, outputs, design codes & checks, assumptions & limitations, and references. Browse individually at /documentation/modules/{moduleId}.

Structural engineering

Beam Analysis (beams) — beta

Purpose

Analyze one-dimensional prismatic beams under point loads, uniformly distributed loads (UDL), and applied moments. The module computes shear force, bending moment, slope, deflection, and bending stress along the span, then compares results against application-specific allowable stress and deflection limits with optional AISC 360 and EN 1993-1-1 screening checks.

Physics & theory

Euler–Bernoulli beam theory relates curvature to bending moment through $\kappa =M/(EI)$, where $E$ is Young's modulus and $I$ is the second moment of area about the bending axis. For small deflections, the governing differential equation is $EI\frac{d^{4}w}{d{x}^4}=q(x)$, with boundary conditions set by the support type (simply supported, cantilever, or fixed–fixed).

Shear force $V$ and bending moment $M$ are obtained by equilibrium: $V=\frac{dM}{dx}$ and $q=\frac{dV}{dx}$. Bending stress at distance $c$ from the neutral axis follows $\sigma =\frac{Mc}{I}$. The solver uses a finite-element discretization of the beam with Hermite shape functions, enforcing displacement and slope continuity at nodes while applying concentrated and distributed loads through equivalent nodal forces.

Static equilibrium is verified by comparing the sum of vertical reactions to applied vertical load. Mesh density (meshSegments) controls spatial resolution; low segment counts may underpredict peak stress near concentrated loads.

Sign conventions follow standard structural practice: downward applied loads are positive, upward reactions positive, and sagging moments positive for horizontal beams. For cantilevers, maximum moment occurs at the fixed support; for simply supported beams with central point load, peak moment occurs at mid-span $M_{\max }=PL/4$. These closed-form benchmarks are used internally to validate the FEM solver against classical cases.

Application presets (lifting beam, machine frame, crane bridge) adjust load factor, allowable stress ratio, and deflection limit without implementing full code load combinations — the engineer must apply appropriate factors per the governing standard before relying on pass/fail flags.

Governing equations

For a simply supported beam with central point load $P$:

Numerical method

1D beam FEM (beam-fem solver): the span is meshed into meshSegments elements. Stiffness matrices are assembled for the selected support condition, loads are mapped to the global force vector, and the linear system is solved for nodal displacements and rotations. Post-processing differentiates shape functions to obtain shear, moment, slope, deflection, and stress along $x$.

The solver pipeline executes in four stages: (1) validate geometry and material inputs; (2) build element stiffness matrices for the selected support type and assemble the global system; (3) apply equivalent nodal loads for point, UDL, and moment entries; (4) back-substitute for reactions and differentiate shape functions along the mesh. Results export as evenly spaced stations along the span for plotting in CalculatorResultsShell.

Physics checks report staticEquilibriumResidual (difference between total applied vertical load and sum of vertical reactions) and finiteValues flag. Warnings are issued when meshSegments < 20 because peak stress near point loads requires adequate mesh refinement.

Inputs

Outputs

• Diagrams: Shear $V(x)$, moment $M(x)$, slope, deflection $w(x)$, stress $\sigma (x)$ along the span
• Peaks: maxShear, maxMoment, maxDeflection, maxStress with station location
• Reactions: Support reaction forces and moments per constraint type
• Equilibrium: staticEquilibriumResidual from physicsChecks
• Code checks: Bending utilization, shear utilization, LTB utilization, deflection utilization per selected design code
• Meta: solverMeta.meshSegments, solverMeta.warnings, application context notes when preset selected

Design codes & checks

• Indicative: Roark / Euler–Bernoulli beam theory
• US: ASME BTH-1, B30.20 (lifting context); AISC 360 Ch. F/G (stress and deflection screening)
• EU: EN 13001, FKM analytical strength; EN 1993-1-1 §6.2
• ISO: ISO 8686, ISO 12100

Assumptions & limitations

• Linear elastic, prismatic cross-section; no large deflection or plasticity.
• 1D beam model — not a building-code member design check.
• LTB uses simplified unbraced length = span unless overridden.
• Shear check uses rectangular-web estimate from $I$ and $c$.
• Application presets adjust targets but do not implement full standard clauses.

Verification

• CI: beams-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Roark, R. J., Young, W. C., & Budynas, R. G. Formulas for Stress and Strain, 8th ed. McGraw-Hill.
2. Gere, J. M., & Goodno, B. J. Mechanics of Materials, 9th ed. Cengage.
3. AISC. Specification for Structural Steel Buildings (ANSI/AISC 360-22).
4. EN 1993-1-1:2005. Eurocode 3 — Design of steel structures — Part 1-1: General rules and rules for buildings.
5. Cook, R. D., et al. Concepts and Applications of Finite Element Analysis, 4th ed. Wiley — beam element formulation.
6. Hibbeler, R. C. Structural Analysis, 10th ed. Pearson — shear and moment diagrams.

Frame Analysis (frames)

Purpose

Perform two-dimensional elastic frame analysis for rigid-jointed structures composed of prismatic members. The module assembles global stiffness matrices, applies nodal loads and support constraints, and returns member end forces, joint reactions, and stress utilizations for machine and industrial frame screening.

Physics & theory

A plane frame member carries axial force, shear, and bending moment. Each member contributes a 6×6 (or condensed) stiffness matrix in local coordinates relating end forces to end displacements. Coordinate transformation maps local stiffness to global axes before assembly into the structure stiffness matrix $\mathbf{K}$.

Equilibrium requires $\mathbf{Ku}=\mathbf{F}$, where $\mathbf{u}$ collects nodal translations and rotations and $\mathbf{F}$ collects applied loads and fixed-end equivalents. Member stresses are recovered from axial stress $\sigma =N/A$ and bending stress $\sigma =Mc/I$, combined for utilization screening.

The solver assumes small displacements and linear elastic material behavior. P–Δ effects, plastic hinges, and semi-rigid connection stiffness are not modeled unless explicitly added in future releases.

Support conditions are applied at nodes: fixed (restrained translation and rotation), pinned (translation restrained, rotation free), or roller (one translation free). Member end releases and semi-rigid connections are not modeled — all joints are treated as rigid unless a member is flagged as pinned.

Nodal loads and member distributed loads superpose linearly. The solver rejects structures with insufficient restraints or zero-stiffness members.

Governing equations

Numerical method

Direct stiffness method: nodes and members define the mesh. Element stiffness matrices are transformed and assembled; boundary conditions eliminate constrained DOFs. The reduced linear system is solved via Gaussian elimination or equivalent sparse solver. Member end forces are back-calculated from nodal displacements.

Inputs

Outputs

• Nodal displacements and rotations
• member axial force, shear, and end moments
• joint reactions
• member stress utilization
• equilibrium check residuals.

Design codes & checks

• Indicative: Member stress utilization, joint reaction equilibrium
• US/EU/ISO: Application-dependent; presets reference industrial equipment standards

Assumptions & limitations

• 2D plane frame only; no out-of-plane buckling or torsion.
• Prismatic members, linear elastic behavior.
• Rigid joints unless connection flexibility is added externally.
• Does not replace licensed structural design per building codes.

References

1. Hibbeler, R. C. Structural Analysis, 10th ed. Pearson.
2. McCormac, J. C., & Brown, R. H. Structural Analysis, 5th ed. Cengage.
3. EN 1993-1-1:2005. Eurocode 3 — General rules.
4. ISO 12100:2010. Safety of machinery — General principles for design.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Truss Analysis (trusses)

Purpose

Determine axial forces in two-dimensional pin-jointed truss members under nodal loading. The module identifies tension and compression members, flags zero-force links, and reports axial stress utilization against allowable values for preliminary truss sizing.

Physics & theory

Truss members are assumed two-force elements carrying only axial force $N$ along the member axis. At each pin joint, equilibrium $\sum F_x=0$ and $\sum F_y=0$ must hold. Because moments cannot be transferred at pins, the structure stiffness matrix involves only translational DOFs (two per node in 2D).

Axial stress is $\sigma =N/A$. Tension members are limited by yield or net-section rupture in detailed design; compression members require buckling checks handled separately in the Column Buckling module. The solver uses the direct stiffness method with bar elements of stiffness $EA/L$.

Indeterminate trusses are solved by the same matrix approach; degree of indeterminacy must be matched by sufficient supports and member connectivity.

External supports restrain selected nodal translations; internal joints are ideal pins with no moment capacity. Loads apply only at nodes as concentrated forces — member self-weight and distributed loads must be converted to equivalent nodal forces by the user.

The solver validates connectivity, positive member areas, and sufficient boundary restraints before assembling the stiffness matrix.

Governing equations

Numerical method

Bar-element FEM: each member contributes axial stiffness in global coordinates after direction-cosine transformation. The assembled system is solved for nodal displacements; member forces $N=(EA/L)(u_j-u_i)$ are recovered. Zero-area or disconnected members produce singular systems and are rejected at validation.

Inputs

Outputs

• Member axial force (signed tension/compression), axial stress, utilization ratio, reaction forces at supports, deformed shape (optional visualization).

Design codes & checks

• Indicative: Member axial utilization
• US: AISC 360 tension/compression member context (screening only)
• EU: EN 1993-1-1 member rules (screening)

Assumptions & limitations

• Pin joints, members connected at centroidal axes.
• No joint eccentricity, secondary bending, or buckling in this module.
• 2D planar truss; no 3D spatial truss.
• Linear elastic; no cable slack or compression-only release logic unless configured.

References

1. Hibbeler, R. C. Structural Analysis, 10th ed. Pearson.
2. AISC. Steel Construction Manual, 16th ed.
3. EN 1993-1-1:2005. Eurocode 3 — Tension and compression members.
4. ISO 10721:1997. Steel structures — Static analysis and design.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Column Buckling (columns) — beta

Purpose

Evaluate elastic stability of slender compression members using finite-element buckling analysis and compare applied axial load to Euler critical load and code column curves. Supports fixed, pinned, and guided end conditions with optional initial imperfection for practical capacity estimates.

Physics & theory

When a straight column is compressed, lateral deflection grows once the axial load exceeds the critical value. Euler's formula for a pinned-pinned column gives $P_{\mathrm{cr}}={\pi }^{2}EI/L_{\mathrm{eff}}^2$, where $L_{\mathrm{eff}}=KL$ depends on end restraint through the effective length factor $K$.

Real columns fail below $P_{\mathrm{cr}}$ due to residual stresses, initial curvature, and material yield interaction. Design codes replace pure Euler buckling with column curves relating normalized slenderness $\lambda$ to buckling reduction factor $\chi$. The FEM solver assembles a geometric stiffness matrix ${\mathbf{K}}_g$ proportional to axial load and solves the eigenvalue problem $(\mathbf{K}+\lambda {\mathbf{K}}_g)\mathbf{v}=0$ for buckling modes.

End restraint sets the effective length factor $K$: pinned–pinned ($K=1$), fixed–fixed ($K=0.5$), and cantilever ($K=2$) bracket most practical cases. Initial imperfection amplifies lateral deflection below $P_{\mathrm{cr}}$ and reduces the usable capacity in code column curves.

The solver validates positive length, area, and stiffness before eigenvalue extraction and rejects disconnected or unrestrained models.

Governing equations

Numerical method

Linear buckling FEM (femSolver): the column is meshed along its length. Elastic stiffness $\mathbf{K}$ and geometric stiffness ${\mathbf{K}}_g(P)$ are assembled for the selected end conditions. The lowest positive eigenvalue yields critical load and buckling mode shape. Post-processing compares $P/P_{\mathrm{cr}}$ to AISC 360 §E3 and EN 1993-1-1 §6.3 curve checks.

Inputs

Outputs

• Critical load $P_{\mathrm{cr}}$, buckling mode shape, slenderness ratio, utilization per selected code
• Euler safety factor $P_{\mathrm{cr}}/P$.

Design codes & checks

• Indicative: Euler buckling $P_{\mathrm{cr}}/P$
• US: AISC 360-22 Chapter E (flexural buckling)
• EU: EN 1993-1-1 §6.3 buckling curves
• ISO: ISO 10721 compression member context

Assumptions & limitations

• Elastic buckling eigenvalue; inelastic column curves applied post-hoc per code.
• Single-axis flexural buckling; no torsional or flexural-torsional modes unless section data extended.
• Uniform prismatic section along length.
• Validated against Euler closed-form for standard end conditions.

Verification

• CI: columns-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Timoshenko, S. P., & Gere, J. M. Theory of Elastic Stability, 2nd ed. McGraw-Hill.
2. AISC. Specification for Structural Steel Buildings (ANSI/AISC 360-22), Chapter E.
3. EN 1993-1-1:2005. Eurocode 3 — Buckling of members.
4. Gere, J. M., & Goodno, B. J. Mechanics of Materials, 9th ed.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Plate Bending (plates)

Purpose

Analyze bending of thin rectangular plates under uniform pressure or point loads with various edge boundary conditions. Computes maximum deflection, bending moments, and membrane/bending stresses for flat plate components in machinery housings, panels, and structural decks.

Physics & theory

Kirchhoff–Love plate theory extends beam bending to two dimensions. For a thin plate of thickness $t$, flexural rigidity is $D=E{t}^3/[12(1-{\nu }^2)]$. The biharmonic equation $D{\nabla }^{4}w=p$ governs out-of-plane deflection $w(x,y)$ under transverse pressure $p$.

Bending moments relate to curvature: $M_x=-D\left(\frac{{\partial }^{2}w}{\partial x^2}+\nu \frac{{\partial }^{2}w}{\partial y^2}\right)$. Maximum stress at the surface is $\sigma =6M/t^2$ for pure bending. Edge conditions — simply supported (SS), clamped (C), or free — strongly influence peak deflection and stress concentration at corners.

For rectangular plates, Navier or Levy series solutions exist for simply supported edges; the solver uses a finite-element discretization on a rectangular mesh for general boundary mixes.

Each edge can be simply supported (SS), clamped (C), or free, independently on all four sides. Mixed edge conditions change moment distribution and peak deflection significantly compared with fully simply supported plates.

Transverse pressure and point loads superpose linearly in elastic analysis. The solver validates positive plate dimensions and flexural rigidity before meshing.

Governing equations

Numerical method

2D plate FEM on a structured rectangular mesh (femSolver). Mindlin–Reissner or Kirchhoff plate elements assemble stiffness from $D$ and mesh geometry. Transverse loads are applied as consistent nodal forces. The linear system yields nodal deflections; moments and stresses are recovered by differentiation of shape functions.

Inputs

Outputs

• Deflection field $w(x,y)$, maximum deflection, bending moments $M_x
• M_y$, maximum bending stress, utilization vs allowable stress and deflection limits.

Design codes & checks

• Indicative: Plate bending stress and deflection screening
• US: ASME BPVC Section VIII, Div. 1 flat plate context (screening)
• EU: EN 13445 flat ends and plates (screening)

Assumptions & limitations

• Thin plate theory ($t/L<1/20$ typically); thick-plate shear deformation not included.
• Linear elastic, small deflection.
• Flat plate only; no stiffeners or large membrane stretching.
• Fewer centralized validation benchmarks than beam/column modules.

References

1. Timoshenko, S., & Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd ed. McGraw-Hill.
2. Roark, R. J., Young, W. C., & Budynas, R. G. Formulas for Stress and Strain, 8th ed.
3. Ugural, A. C. Stresses in Plates and Shells, 4th ed. CRC Press.
4. ASME BPVC Section VIII, Division 1 (flat plate design rules).
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Combined Loading (combined-loading) — beta

Purpose

Evaluate combined axial, bending, torsion, and shear stresses in a rectangular cross-section and compute von Mises equivalent stress and safety factor. Used for quick screening of machine elements and structural members under multiaxial loading without full 3D FEA.

Physics & theory

Real components rarely experience a single stress mode. Axial force $F$ produces uniform normal stress ${\sigma }_a=F/A$. Bending moment $M$ creates linear normal stress ${\sigma }_b=Mc/I$. Torque $T$ generates torsional shear ${\tau }_t=Tc/J$ for a rectangular section using the thin-wall approximation $J\approx w{h}^3/3$. Direct shear from transverse force adds ${\tau }_v=V/A$.

Normal stresses from axial and bending load superpose: $\sigma ={\sigma }_a+{\sigma }_b$. For ductile materials under combined normal and shear stress, the von Mises (distortion energy) criterion gives equivalent stress ${\sigma }_{\mathrm{vm}}=\sqrt{{\sigma }^2+3{\tau }^2}$. Safety factor is $SF={\sigma }_y/{\sigma }_{\mathrm{vm}}$.

Stress components are evaluated at the section centroid for a prismatic rectangular cross-section. The module assumes elastic behavior and does not model local buckling, stress concentrations, or warping restraint — use dedicated beam or shell analysis when those effects govern.

Inputs must specify positive width, height, and material yield strength; zero-area sections are rejected at validation.

Governing equations

Numerical method

Closed-form evaluation: section properties $A$, $I_{xx}$, and $J$ are computed from rectangular width and height. Individual stress components are calculated algebraically; von Mises stress and safety factor follow directly. Design status flags safe, warning, or critical based on threshold ratios (SF ≥ 2 safe, ≥ 1.25 warning).

Inputs

Outputs

• Section properties $A$, $I_{xx}$, $J$
• stress components
• von Mises stress
• safety factor
• design status.

Design codes & checks

• Indicative: Von Mises combined stress
• US: AISC 360-22 Chapter H (combined forces)
• EU: EN 1993-1-1 Clause 6.2.1 equivalent stress
• ISO: ISO 10828 equivalent stress methods

Assumptions & limitations

• Solid rectangular section; not I-beams, tubes, or arbitrary profiles.
• Elastic linear superposition; no buckling or local instability.
• Torsion uses rectangular approximation; thin-wall or circular sections need dedicated checks.
• Shear stress from transverse force is averaged over area (not parabolic distribution).

Verification

• CI: combined-loading-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed. McGraw-Hill.
2. Gere, J. M., & Goodno, B. J. Mechanics of Materials, 9th ed.
3. AISC. Specification for Structural Steel Buildings (ANSI/AISC 360-22), Chapter H.
4. EN 1993-1-1:2005. Eurocode 3 — Clause 6.2.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Load Case Manager (load-case-manager)

Purpose

Orchestrate multiple structural load cases and compute envelope results (maximum/minimum stress, deflection, and utilization) across cases. Enables design-by-envelope workflows where the governing load combination is identified without re-running individual module calculations manually.

Physics & theory

Structural design requires evaluating several load scenarios — dead, live, wind, seismic, thermal, and operational loads — each factored per the governing code. Rather than solving a single load vector, envelope analysis tracks the extremum of each response quantity across all defined cases: $R_{\mathrm{env}}={\max }_i|R_i|$ or signed envelopes for asymmetric checks.

Linear elastic structures satisfy superposition:

Each stored case preserves the originating module inputs and factored results. Envelope utilizations identify the governing case for each check without re-solving the underlying structural model.

Load factors follow design-code presets (Indicative, US, EU, ISO); users must confirm factor sets match the project load combination requirements.

Governing equations

Numerical method

Orchestration layer over module solvers: each load case invokes the underlying structural engine (beams, frames, etc.) or stores imported results. Envelope logic scans result arrays and metric summaries to extract governing values. No independent FEM — numerical depth is in aggregation and comparison logic.

Inputs

Outputs

• Envelope stress utilization, governing load case ID, per-case utilizations, max/min deflection envelopes, summary table for export.

Design codes & checks

• Indicative: Envelope stress utilization
• US/EU: Load combination factors per AISC/ASCE 7 or EN 1990 (user responsibility)

Assumptions & limitations

• Linear elastic superposition unless cases explicitly marked nonlinear.
• User responsible for correct load factors and combination rules per code.
• Does not perform load pattern optimization or automatic code combination generation.
• Envelope of nonlinear results may be non-conservative.

References

1. ASCE/SEI 7-22. Minimum Design Loads and Associated Criteria for Buildings.
2. EN 1990:2002. Eurocode — Basis of structural design.
3. AISC. Specification for Structural Steel Buildings (ANSI/AISC 360-22), Chapter B.
4. ISO 8686:1989. Cranes — Design principles for loads and load combinations.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Circular Plates (circular-plates)

Purpose

Compute deflection and bending stress in solid circular plates under uniform transverse pressure with simply supported or clamped outer edges. Combines Roark closed-form benchmarks with an axisymmetric finite-difference solver for mesh-controlled accuracy.

Physics & theory

Axisymmetric circular plates under uniform pressure $p$ exhibit radially symmetric deflection $w(r)$. Flexural rigidity is $D=E{t}^3/[12(1-{\nu }^2)]$. For a clamped edge ($w=0$, $dw/dr=0$ at $r=a$), peak deflection at center scales as $w_{\max }\propto p{a}^4/D$; for simply supported edges, boundary moments vanish and deflection is larger.

Roark's tabulated coefficients provide quick screening: clamped plate $w_{\max }=\alpha p{a}^4/D$ with $\alpha \approx 0.0138$; simply supported $\alpha \approx 0.171$. Maximum bending stress at the surface follows $\sigma \approx \beta p{a}^2/t^2$ with $\beta \approx 0.75$ for uniform pressure.

The axisymmetric FDM solver discretizes the biharmonic operator on a radial grid, iterating until convergence between applied pressure and plate curvature.

Outer-edge support is modeled as either clamped ($w=0$, $dw/dr=0$ at $r=a$) or simply supported ($w=0$, $M_r=0$ at $r=a$). Switching between these edge conditions changes center deflection by an order of magnitude and shifts the location of peak bending stress from center to edge.

The solver requires positive radius, thickness, and flexural rigidity; non-axisymmetric loading and annular plates are outside scope.

Governing equations

Numerical method

Dual approach: (1) Roark closed-form coefficients for benchmark comparison; (2) axisymmetric Kirchhoff FDM on a radial line with meshSegments (4–64). Jacobi-style iteration (~800 steps) enforces boundary conditions. FEM deflection error vs Roark is reported as femDeflectionErrorPercent.

Inputs

Outputs

• Maximum deflection and stress, flexural rigidity $D$
• Roark benchmark values
• FEM–Roark error percentage, mesh segment count.

Design codes & checks

• Indicative: Plate deflection and bending stress screening
• US: ASME BPVC UG-34 flat head context (screening)
• EU: EN 13445 flat ends (screening)

Assumptions & limitations

• Solid circular plate; annular plates use simplified extensions only.
• Thin Kirchhoff plate theory; no transverse shear deformation.
• Uniform pressure; no point loads or thermal gradients.
• Linear elastic, small deflection.

Verification

• CI: circular-plates-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Roark, R. J., Young, W. C., & Budynas, R. G. Formulas for Stress and Strain, 8th ed., Case 11.
2. Timoshenko, S., & Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd ed.
3. Ugural, A. C. Stresses in Plates and Shells, 4th ed.
4. ASME BPVC Section VIII, Division 1, UG-34.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Power transmission

V-Belt Drive (v-belts)

Purpose

Size classical V-belt drives by computing belt length, wrap angles, power capacity, speed ratio, and estimated pretension for a two-pulley layout. Screens belt selection against transmitted power with friction-based tight/slack side tension estimates.

Physics & theory

Flat and V-belt drives transmit torque through friction on pulley wrap arcs. The belt speed is $v=\pi Dn/60$ (m/s with $D$ in m, $n$ in rpm). Open belt length for center distance $C$ and pulley diameters $D_1,D_2$ follows the standard layout formula accounting for straight spans and arc lengths.

Euler's belt equation relates tight side tension $F_1$ to slack side $F_2$: $F_1/F_2=e^{\mu \theta }$, where $\mu$ is friction coefficient and $\theta$ is wrap angle in radians on the driver pulley. Power capacity depends on allowable belt tension, speed, and wrap — the solver uses a simplified capacity model scaled by belt class factor and service factor.

Power transmission elements operate under cyclic tension, bending, and contact stresses. Service factors account for driver type (motor vs engine), daily operating hours, and shock loading. Belt slip occurs when required friction capacity exceeds available wrap; chain drives depend on proper lubrication and sprocket tooth count for rated life.

Center distance adjustment affects belt length and wrap angle simultaneously — the solver uses the standard open-drive length formula assuming coplanar shafts and parallel pulley grooves.

Governing equations

Numerical method

Closed-form classical belt equations. Wrap angles computed from geometry via $\theta =\pi -2\arcsin [(D_2-D_1)/(2C)]$. Power capacity estimated from belt factor, belt speed, service factor, and exponential friction term. Pretension estimated as average of tight and slack tensions.

Inputs

Outputs

• Belt length, wrap angles (driver/driven), belt speed, power capacity and utilization, speed ratio, driven speed, pretension estimate.

Design codes & checks

• Indicative: Power capacity utilization, minimum wrap angle
• US: Gates/McGraw belt design handbook methods (screening)
• ISO: ISO 4184 classical V-belt sections (reference)

Assumptions & limitations

• Two-pulley open drive; no idlers or quarter-turn layouts (see Multi-Pulley module).
• Steady-state, no belt creep dynamics or temperature derating beyond service factor.
• Flat friction model; V-belt wedge effect absorbed in beltFactor.
• Does not select specific belt cross-section from catalog tables automatically.

Verification

• CI: v-belts-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 17.
2. Marks' Standard Handbook for Mechanical Engineers, 12th ed., McGraw-Hill.
3. ISO 4184:1992. Classical V-belts and pulleys.
4. Gates Corporation. Drive Design Manual.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Timing Belt Drive (timing-belts)

Purpose

Size synchronous (toothed) belt drives by computing pitch length, number of teeth, belt speed, transmitted power, and shaft loads. Positive engagement eliminates slip, making timing belts suitable for positioning and high-ratio compact drives.

Physics & theory

Timing belts mesh with pulley teeth at a defined pitch $p$. Pitch diameter relates to tooth count: $D=pN/\pi$. Belt length for two pulleys includes tooth engagement arcs plus tangent spans. Unlike friction belts, power capacity is limited by tooth shear, belt tensile strength, and pulley tooth bending — the module applies manufacturer-style screening factors.

Speed ratio $i=N_{\mathrm{driven}}/N_{\mathrm{driver}}$ is exact (no slip). Radial load on shafts combines belt tension from power transmission and centrifugal effects at high speed. Pretension must prevent tooth jump under peak torque while limiting bearing loads.

Power transmission elements operate under cyclic tension, bending, and contact stresses. Service factors account for driver type (motor vs engine), daily operating hours, and shock loading. Belt slip occurs when required friction capacity exceeds available wrap; chain drives depend on proper lubrication and sprocket tooth count for rated life.

Center distance adjustment affects belt length and wrap angle simultaneously — the solver uses the standard open-drive length formula assuming coplanar shafts and parallel pulley grooves.

Governing equations

Numerical method

Closed-form geometry and power screening per timing belt check templates. Tooth count and pitch determine pulley diameters; belt length rounded to whole tooth pitches. Power utilization compared against rated power adjusted by service, width, and speed factors.

Inputs

Outputs

• Pitch length, tooth count, pulley diameters, belt speed, power utilization, estimated belt tension, shaft load components.

Design codes & checks

• Indicative: Power capacity and tension screening
• ISO: ISO 5296 synchronous belt drives (reference pitch systems)

Assumptions & limitations

• Two-pulley layout; no idler pulleys or back-side wrap.
• Screening-level rating — not a substitute for manufacturer software (Gates, Conti).
• Neglects belt stiffness dynamics and resonance at high speed.
• Standard trapezoidal or curvilinear tooth profiles per selected pitch family.

Verification

• CI: timing-belts-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 17.
2. ISO 5296:2012. Synchronous belt drives — Pulleys.
3. Gates Corporation. Poly Chain GT Carbon Design Manual.
4. Budynas, R. G., Nisbett, J. K. Shigley's Mechanical Engineering Design, 11th ed.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Roller Chain Drive (roller-chains)

Purpose

Size roller chain drives by computing sprocket geometry, chain speed, transmitted power, tension, and estimated service life. Supports strand selection and power capacity screening for industrial machinery drives.

Physics & theory

Roller chains transmit power through sprocket tooth engagement. Chain pitch $p$ and number of teeth $N$ define pitch diameter $D=p/\sin (\pi /N)$. Chain speed $v=\pi Dn/60$. Power $P=Fv$ relates to chain tension $F$, which includes centrifugal and chordal action effects at high speeds.

Chain life depends on lubrication, alignment, load spectrum, and pitch selection. ANSI/ISO power rating tables provide allowable power vs speed for each chain number; the module applies service factors and strand count to estimate utilization and life cycles.

Power transmission elements operate under cyclic tension, bending, and contact stresses. Service factors account for driver type (motor vs engine), daily operating hours, and shock loading. Belt slip occurs when required friction capacity exceeds available wrap; chain drives depend on proper lubrication and sprocket tooth count for rated life.

Center distance adjustment affects belt length and wrap angle simultaneously — the solver uses the standard open-drive length formula assuming coplanar shafts and parallel pulley grooves.

Governing equations

Numerical method

Closed-form sprocket and length equations with tabulated power ratings per chain size. Life estimate from load ratio relative to catalog rating, adjusted by lubrication and service factors. Multi-strand capacity scales approximately linearly with strand count.

Inputs

Outputs

• Pitch diameters, chain speed, chain tension, power utilization, estimated chain life, length in pitches.

Design codes & checks

• Indicative: Power capacity utilization, chain life estimate
• US: ANSI/ASME B29.1 roller chain standards
• ISO: ISO 606 short-pitch precision roller chains

Assumptions & limitations

• Steady power transmission; shock loads require additional service factor.
• Horizontal or near-horizontal drives; vertical lifts need tension adjustment.
• Catalog ratings assume adequate lubrication and sprocket tooth count ≥ 17 (recommended).
• Does not analyze silent (inverted tooth) or leaf chains.

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 17.
2. ANSI/ASME B29.1-2011. Precision Power Transmission Roller Chains.
3. ISO 606:2015. Short-pitch transmission precision roller chains.
4. Renold. The Complete Guide to Chain.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Multi-Pulley Layout (multi-pulley)

Purpose

Compute total belt or chain length and wrap angles for drives with three or more pulleys in a single plane. Supports layout verification before detailed power rating in the V-Belt or Roller Chain modules.

Physics & theory

Multi-pulley drives route a single belt or chain around several shafts. Total length equals sum of straight tangent segments between pulley pairs plus arc lengths on each pulley. Wrap angle on each pulley depends on incoming and outgoing tangent directions, which are determined by pulley centers and diameters in the layout plane.

Minimum wrap angle governs friction capacity on friction belts — values below ~120° require idler pulleys or larger diameters. For timing belts and chains, wrap angle affects chordal action and engagement but not slip. The module treats pulleys as circles in 2D with user-specified center coordinates and diameters.

Power transmission elements operate under cyclic tension, bending, and contact stresses. Service factors account for driver type (motor vs engine), daily operating hours, and shock loading. Belt slip occurs when required friction capacity exceeds available wrap; chain drives depend on proper lubrication and sprocket tooth count for rated life.

Center distance adjustment affects belt length and wrap angle simultaneously — the solver uses the standard open-drive length formula assuming coplanar shafts and parallel pulley grooves.

Governing equations

Numerical method

Geometric layout solver: pulley centers and diameters define tangent lines between adjacent pulleys in the routing order. Arc lengths computed from wrap angles derived from vector geometry. Belt length summed; minimum wrap flagged if below threshold.

Inputs

Outputs

• Total belt/chain length, per-pulley wrap angle (degrees), minimum wrap angle, tangent segment lengths.

Design codes & checks

• Indicative: Total belt length, minimum wrap angle screening

Assumptions & limitations

• Coplanar pulleys only; no 3D skew or quarter-turn twist.
• Circular pulleys; no crowned or flanged geometry effects.
• Does not compute power capacity — use with V-Belt or Chain modules.
• Routing order must be specified correctly by user.

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 17.
2. Marks' Standard Handbook for Mechanical Engineers, 12th ed.
3. Gates Corporation. Heavy-Duty V-Belt Drive Design Manual.
4. ISO 4184:1992. Classical V-belts and pulleys.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Machine design

Shaft Design (shafts)

Purpose

Analyze rotating shafts under combined bending, torsion, and axial loads using 1D FEA. Supports stepped/hollow geometry, configurable bearing supports, transverse forces, stress concentrations, fatigue screening (Marin + Goodman), FEA critical speed, and bearing reaction handoff.

Physics & theory

Power-transmitting shafts experience bending from belt/gear forces, torsion from transmitted torque, and occasional axial thrust. Stress at any section combines normal and bending stress with torsional shear; von Mises equivalent stress governs static yield checks for ductile materials.

Rotating shafts subject bending to fully reversed fatigue; torsion is typically steady. Critical (whirling) speed is estimated from FEA mass and bending stiffness (first lateral modes).

Governing equations

Fatigue (Indicative/US):

Numerical method

1D shaft FEM: Hermite beam elements (12 DOF) with axial, torsion, and biaxial bending. Stepped diameter and hollow sections via segment mesh. Pin or fixed supports at user-defined bearing positions. Lumped-mass eigen iteration for critical speed.

Inputs

Outputs

• T(x), M(x), V(x), ${\sigma }_{\mathrm{vm}}(x)$, deflection, slope, critical speed, fatigue SF
• Bearing reactions and slope utilization
• Governing failure mode (static / fatigue / deflection / slope / whirling)

Design codes & checks

• Indicative: von Mises static, deflection, critical speed margin, Goodman fatigue
• US: AGMA 6001 interface loads (context via gear handoff)
• EU: DIN 743 full worksheet — not yet implemented; use Indicative fatigue for screening

Assumptions & limitations

• Linear elastic Timoshenko/Euler shaft model; no 3D fillet FEA
• Kt from Peterson/Shigley approximations, not DIN 743-2 tables
• Critical speed: first two lateral modes; gyroscopic/damping omitted
• DIN 743 influence factors (K₁, K₂, K₃, β, K_V) not yet integrated

Verification

• CI: shafts-indicative-01.json
• Vitest: src/lib/machine/shafts/engine.test.ts
• Engineer sign-off: validation-master-checklist.md (Machine → shafts)

Cross-module handoff

• Publishes alternating/mean stress to fatigue module after calculate
• Receives gear/pulley loads from upstream calculators (manual today)

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 7.
2. Peterson, R. E. Stress Concentration Factors.
3. DIN 743:2012 (EU target standard — partial integration planned).

Gear Design (gears) — beta

Purpose

Design and rate spur and helical gear pairs for bending and contact (pitting) strength. Combines Lewis bending screening with ISO 6336 Method B/C factors including dynamic load $K_V$, zone factor $Z_H$, elasticity factor $Z_E$, and contact ratio factor $Y_{\epsilon }$.

Physics & theory

Gear teeth convert rotation and torque through involute meshing. Transmitted tangential force at the pitch circle is $W_t=2T/d$, where $T$ is torque and $d$ is pitch diameter. Lewis equation estimates bending stress in a tooth treated as a cantilever: $\sigma =W_t{K}_V/(bm{Y}_J)$, with module $m$, face width $b$, and form factor $Y_J$.

Contact (Hertzian) stress between mating teeth limits pitting life. ISO 6336 expresses contact stress ${\sigma }_H$ with factors for load sharing, geometry, lubrication, and material. Scuffing and micropitting are not fully evaluated in indicative mode.

Governing equations

Numerical method

Closed-form ISO 6336 and Lewis screening via solveGearDesign. Input power, speed, module, face width, tooth counts, and material limits feed factor calculations. Results include bending and contact utilization, geometry summary, and pitch-line velocity.

Inputs

Outputs

• Tangential force, pitch-line velocity, bending stress and utilization, contact stress and utilization, geometry (centers, diameters), factor breakdown.

Design codes & checks

• Indicative: Lewis bending and simplified Hertzian contact
• ISO: ISO 6336-1/2/3 Method B/C rating (screening)
• US: AGMA 2101-D04 (reference context)

Assumptions & limitations

• External spur/helical pair; no internal gears or planetary sets (see dedicated modules).
• Indicative scuffing and bending fatigue screening; full AGMA/ISO factor sets not included.
• Uniform load distribution along face width unless $K_{H\beta }$ specified.
• No microgeometry (profile modification) analysis.

Verification

• CI: gears-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. ISO 6336-1:2019. Calculation of load capacity of spur and helical gears — Part 1: Basic principles.
2. ISO 6336-2:2019. Part 2: Calculation of surface durability (pitting).
3. ISO 6336-3:2019. Part 3: Calculation of tooth bending strength.
4. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 13–14.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Bearing Selection (bearings)

Purpose

Rolling-element bearing screening per ISO 281 (basic and modified rating life) and ISO 76 static load check. Multi-manufacturer catalog (SKF, FAG, NSK, Timken, NTN) with application profiles, series/sealing filters, and representative C, C₀, geometry, and limiting speed.

Physics & theory

Basic rating life L₁₀ is the life in revolutions (or hours at speed n) exceeded by 90% of bearings under constant equivalent load P = C:

Modified rating life (ISO 281:2007 / SKF):

where aSKF (ISO 281 aISO) is computed from viscosity ratio κ = ν/ν₁, contamination factor eC (ηc), and fatigue load limit Pu (catalog datasheet Pu when available; otherwise estimated as 0.025C for ball, 0.03C for roller).

Life model ceiling (screening, opt-in):

Shaft FEM handoff can publish bearing slopes (rad) → misalignment (mrad) for the ceiling path.

Static safety (ISO 76):

Paired arrangements (O / X / T): treated as first-class engineering objects with:

Loads are split per bearing; system life is governed by the Weibull combination of station modified lives.

Variable load (ISO 281-1): optional two-step spectrum computes equivalent load and Palmgren-Miner combined life.

Inputs

Outputs

• Equivalent loads P and P₀ (paired per-station when applicable)
• Modified Lnm and basic L₁₀ with first-class factors: a₁, aSKF (≡ aISO), κ, eC (ηc), ν / ν₁, Pu/P
• Arrangement analysis: preload, Ka/Kr/Km, axial displacement δa, thermal preload shift, O/X/T rigidity comparison
• Defect frequencies BPFO / BPFI / BSF / FTF (screening geometry)
• Grease life L₁₀h and/or relubrication interval tf
• Adjusted reference speed n_θ and friction energy / CO₂ screening
• Side-by-side OEM compare under the same duty
• Intelligent recommendation with life / safety / cost and Explain Recommendation (Engineering Advisor narrative)
• Persistent Design Summary right rail (bearing type, loads, Lnm, safety, ISO 281, catalog, status) — updates continuously as inputs change
• PDF / Excel packages with named sections: Design Summary, Inputs, Catalog, ISO 281 factors, Arrangement, Recommendation, Checks, Formulas, Standards, Charts
• Mounted kit BOM (housing SNL/UCP/FY/SAF-class + seal + grease)
• Dynamic utilization P/C, static safety C₀/P₀, speed margin n_lim/n
• Minimum load (skidding), SKF-inspired Mrr/Msl friction torque and power loss (screening)
• Recommended shaft/housing fits and differential ring-ΔT operating clearance
• Cross-OEM interchange candidates (same bore/type/C class)
• Required C and C₀ for target life
• Governing failure mode

Design codes & checks

• ISO 281:2007 — Basic and modified rating life
• ISO 76 — Static load rating screening
• Catalog limiting speed (grease) and reference speed (oil) where listed
• Defect frequencies — kinematic screening (verify Z, Bd against OEM for CM)

Assumptions & limitations

• Constant load and speed unless variable spectrum is enabled
• Pu from catalog with explicit datasheet vs C₀-ratio provenance; user override available
• Representative catalog — not full vendor databases
• Friction is screening Mrr/Msl — not full SKF four-component model
• ISO 16281 and stress-life paths are screening only — not vendor GBLM, AFC, Bearinx, or full ISO 16281:2025
• Housing SKUs are screening-class (SNL/UCP/…) — not full OEM mounted-product databases
• CO₂ from friction power only — not a product LCA
• Duplex paired C catalog multipliers use tandem axial convention (screening)
• Temperature derating on C above 120 °C (screening factor)

Verification

• CI: bearings-indicative-*.json (multiple rolling cases)
• Gold harness: npm run test:bearings-gold / Vitest bearingsGold.test.ts (screening_reference; vendor cases pending paste)
• Vitest: engine.test.ts, industryParity.test.ts, lifeModelCeiling.test.ts, auxMounted.test.ts
• Engineer sign-off: validation-master-checklist.md (Machine → bearings) — pending ±5% vendor gold

Design workflow

• Validate: ISO 281/76 forward check on selected designation.
• Auto-design: Ranks catalog entries by manufacturer, application profile, life utilization, static SF, and speed margin within bore limit. Design mode passes the UI lubricant type, ISO VG, operating temperature, and contamination into ISO 281 screening (not fixed VG68 / normal_clean).
• Handoff: Receives Fr₀/Fr₁, span, slopes from shafts (auto-apply); Fa may still need gear/user input (planar FEM warning).
• System wizard: Size locating + floating pair and apply both designations.

References

1. ISO 281:2007 — Dynamic load ratings and rating life
2. ISO 76 — Static load ratings
3. Shigley, Ch. 11; SKF Rolling Bearings Catalogue

Cam Design (cams)

Purpose

Analyze cam–follower kinematics and kinetics: displacement, velocity, acceleration, pressure angle, and contact stress for a specified cam profile and follower type. Used for motion control mechanism screening in machine design.

Physics & theory

A cam imparts prescribed motion to a follower through shaped surface contact. The displacement curve $s(\theta )$ defines follower position vs cam angle. Velocity and acceleration follow $v=ds/dt=\omega ds/d\theta$ and $a={\omega }^2{d}^{2}s/d{\theta }^2$. Smooth acceleration profiles (cycloidal, modified trapezoidal) reduce impact and wear compared to sharp velocity corners.

Pressure angle $\varphi$ is the angle between follower motion direction and the normal to the cam profile. High pressure angles increase side thrust on the follower and risk binding. Contact stress between cam and follower uses Hertzian line or point contact depending on follower geometry (flat-faced, roller, or mushroom).

Governing equations

Numerical method

Kinematic differentiation of user-defined or standard motion laws (constant velocity, SHM, cycloidal). Pressure angle computed at each cam angle step. Contact force from follower mass, spring force, and inertia $F=ma+ks+F_{\mathrm{ext}}$. Hertzian contact stress screened against material allowable.

Inputs

Outputs

• Displacement, velocity, acceleration plots
• max pressure angle
• contact force
• contact stress
• torque required.

Design codes & checks

• Indicative: Pressure angle limit, cam contact stress screening

Assumptions & limitations

• 2D planar cam; no 3D spatial cams or conjugate surface optimization.
• Rigid cam and follower; no compliance or lubrication film analysis.
• Single-dwell motion programs; multi-segment profiles user-defined.
• Manufacturing eccentricity and wear not modeled.

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 16.
2. Norton, R. L. Design of Machinery, 6th ed. McGraw-Hill.
3. Chen, F. Y. Mechanics and Design of Cam Mechanisms. Pergamon.
4. Hertz, H. On the Contact of Elastic Solids (contact stress foundation).
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Flywheel Design (flywheels)

Purpose

Size flywheels for energy storage and speed regulation by computing required moment of inertia, rim stress, and energy capacity for a specified speed fluctuation or power pulse. Used in presses, engines, and cyclic machinery.

Physics & theory

A flywheel stores kinetic energy $E=\frac{1}{2}I{\omega }^2$. For a rim-dominated disk, $I\approx m{r}^2$ where $m$ is rim mass and $r$ is mean radius. Energy change between max and min speed during a cycle is $\Delta E=\frac{1}{2}I({\omega }_{\max }^2-{\omega }_{\min }^2)$.

Coefficient of speed fluctuation $C_s=({\omega }_{\max }-{\omega }_{\min })/\omega$ links inertia to cyclic energy input/output. Rim stress from centrifugal loading approximates hoop tension $\sigma =\rho r^2{\omega }^2$ for thin rings; solid disk models use radial and tangential stress distributions.

Governing equations

Numerical method

Closed-form energy–inertia relations. Required $I$ computed from specified $\Delta E$ and speed limits. Geometry (rim thickness, width, hub bore) iterated to achieve target inertia while checking rim stress utilization against material allowable.

Inputs

Outputs

• Required moment of inertia, rim mass, stored energy, rim stress, speed fluctuation coefficient, stress utilization.

Design codes & checks

• Indicative: Rim stress utilization, energy storage capacity

Assumptions & limitations

• Axisymmetric rotation; no blade or spoke dynamic stress analysis.
• Thin-rim approximation for hoop stress; hub and spoke contributions simplified.
• No burst containment or safety guard requirements.
• Constant angular deceleration during energy release not enforced.

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 15.
2. Spotts, M. F., & Shoup, T. E. Design of Machine Elements, 8th ed.
3. Marks' Standard Handbook for Mechanical Engineers, 12th ed.
4. Peterson, R. E. Stress Concentration Factors (rotor burst context).
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Bevel Gear Screening (bevel-gears)

Purpose

Screen straight and spiral bevel gear sets for geometry, pitch cone dimensions, and bending/contact strength using adapted spur gear rating methods. Provides preliminary sizing before detailed Gleason/Klingelnberg analysis.

Physics & theory

Bevel gears transmit power between intersecting axes, typically at 90°. Pitch cone geometry relates pinion and gear tooth counts through shaft angle $\Sigma$. Mean cone distance $R_m$ and mean module $m_m$ define the virtual spur gear equivalent used for strength screening.

Tangential force acts at the mean pitch circle on the pitch cone. Bending and contact stresses use ISO 6336–style factors applied to the virtual cylindrical gear dimensions. Face width is limited by cone length and should not exceed $R_m/3$ without detailed analysis.

Governing equations

Numerical method

Virtual spur gear transformation followed by gear rating checks shared with the spur gear module. Cone geometry computed from tooth counts and shaft angle; strength factors applied at mean section.

Inputs

Outputs

• Pitch cone angles, mean diameters, cone distance, tangential force, bending/contact utilization.

Design codes & checks

• Indicative: Lewis/ISO-style bending and contact screening
• ISO: ISO 10300 bevel gear load capacity (reference)
• US: AGMA 2003 bevel gear rating (reference)

Assumptions & limitations

• Straight or zerol bevel screening; spiral angle effects simplified.
• Virtual gear method — not full bevel-specific ISO 10300 factor set.
• Assumes proper mounting and lapping; no deflection under load.
• No scuffing or lapping contact pattern analysis.

Verification

• CI: bevel-gears-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. ISO 10300-1:2014. Calculation of load capacity of bevel gears.
2. AGMA 2003-D19. Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel, Zerol Bevel and Spiral Bevel Gear Teeth.
3. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed.
4. Maitra, G. M. Handbook of Gear Design, 2nd ed. McGraw-Hill.
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Worm Gear Drive (worm-gears)

Purpose

Screen worm and worm-wheel drives for efficiency, sliding velocity, contact stress, and thermal load. Worm drives provide high speed reduction in compact envelopes but generate significant sliding and heat.

Physics & theory

A worm is a helical gear with one or few teeth (threads); the worm wheel mates at 90° shaft angle. Lead angle $\lambda$ and friction angle $\varphi$ determine efficiency: $\eta =\tan \lambda /\tan (\lambda +\varphi )$ for driving worm. Self-locking occurs when $\lambda <\varphi$, preventing back-driving.

Sliding velocity along tooth flanks is high compared to spur gears, limiting efficiency and promoting wear. Contact stress uses Hertzian line contact with equivalent radii from worm and wheel geometry. Heat generation $Q=P(1-\eta )$ must be dissipated to avoid oil breakdown.

Governing equations

Numerical method

Closed-form geometry and efficiency calculations. Friction coefficient from material pair and lubrication. Contact stress screened against allowable; thermal power loss computed for oil bath sizing guidance.

Inputs

Outputs

• Gear ratio, efficiency, self-locking flag, sliding velocity, contact stress utilization, heat loss (kW).

Design codes & checks

• Indicative: Efficiency, contact stress utilization
• DIN: DIN 3996 worm gear load capacity (reference)
• ISO: ISO/TR 14521 worm gear rating (reference)

Assumptions & limitations

• Cylindrical worm with throated wheel; no double-enveloping geometry.
• Steady-state thermal balance not fully solved — heat loss is screening only.
• Wear and pitting life not computed to ISO/TR 14521 full method.
• Manufacturing tolerance effects on contact pattern omitted.

References

1. ISO/TR 14521:2020. Gears — Calculation of load capacity of worm gears.
2. DIN 3996:2016. Tragfähigkeitsberechnung von Zylinderschneckengetrieben.
3. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 13.
4. Dudley, D. W. Handbook of Practical Gear Design, 2nd ed.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Planetary Gear Set (planetary-gears)

Purpose

Size planetary (epicyclic) gear trains by selecting sun, planet, and ring tooth counts for a target ratio while checking assembly, planet spacing, and approximate strength balance. Used for compact high-ratio reducers and automatic transmissions.

Physics & theory

A basic planetary set has sun gear $S$, planet gears $P$, and ring gear $R$ with carrier $C$. Fundamental speed relation: ${\omega }_R{N}_R+{\omega }_S{N}_S={\omega }_C(N_S+N_R)$ for internal ring mesh. Gear ratio depends on which element is held fixed.

Tooth count constraint: $N_R=N_S+2N_P$ for equally spaced planets. At least two planets require $(N_S+N_R)/\mathrm{numPlanets}$ integer. Planet–ring and planet–sun meshes share load; planet bearing load and equal spacing are design constraints.

Governing equations

Numerical method

Integer tooth search for target ratio within bounds. Validates assembly condition and planet spacing. Approximate torque sharing assigns equal planet load; strength screening uses per-planet tangential force vs allowable.

Inputs

Outputs

• Sun, planet, ring tooth counts, actual ratio, ratio error, assembly validity, approximate planet load.

Design codes & checks

• Indicative: Actual ratio vs target, assembly constraint check

Assumptions & limitations

• Single-stage planetary; no compound or multi-stage trains.
• Full ISO 6336 planet load sharing factors not applied.
• Planet carrier stiffness and pin bearing loads simplified.
• Helical planets require additional axial load analysis.

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 13.
2. Müller, H. W. Epicyclic Drive Trains. Wayne State University Press.
3. ISO 6336 series (planet gear load sharing context).
4. AGMA 6123-B06. Design Manual for Enclosed Epicyclic Gear Drives.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Gear Ratio Design (gear-ratio-design)

Purpose

Search integer tooth-count combinations to achieve a target speed ratio within specified tolerance. Optimizes for compactness, balanced wear, or minimum total teeth subject to interference and manufacturing constraints.

Physics & theory

Gear ratio for external spur gears is $i=N_g/N_p={\omega }_p/{\omega }_g$. Only integer tooth counts are manufacturable, so exact ratios are approximated: $i_{\mathrm{actual}}=N_g/N_p\approx i_{\mathrm{target}}$. Error $ϵ=|i_{\mathrm{actual}}-i_{\mathrm{target}}|/i_{\mathrm{target}}$ must fall within tolerance for synchronized drives.

Minimum tooth counts avoid undercut in standard involute profiles (typically $N\ge 17$ for 20° pressure angle without profile shift). Hunting tooth combinations (where common factors of $N_p$ and $N_g$ exceed 1) distribute wear unevenly — coprime tooth counts are preferred for long life.

Governing equations

Numerical method

Exhaustive or bounded integer search over tooth count ranges. Filters candidates by minimum teeth, interference, and ratio error. Ranks solutions by total teeth, center distance, or hunting tooth preference.

Inputs

Outputs

• Ranked tooth-count pairs, actual ratio, ratio error, center distance, hunting tooth flag.

Design codes & checks

• Indicative: Ratio error screening

Assumptions & limitations

• External spur pair; internal or compound trains not searched.
• No profile shift or helical overlap considered.
• Center distance assumes standard involute with zero backlash.
• Does not verify bending/contact capacity — use Gear Design module.

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 13.
2. AGMA 917-B97. Design Manual for Parallel Shaft Fine-Pitch Gearing.
3. ISO 21771:2007. Cylindrical involute gears and gear pairs — Concepts.
4. Buckingham, E. Analytical Mechanics of Gears. Dover.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Plain Bearings (plain-bearings)

Purpose

Screen hydrodynamic journal and thrust pad bearings (ISO 7902 / ISO 12130 / ISO 12131 screening) with Sommerfeld number, minimum film thickness, power loss, and temperature rise. Supports preliminary bearing design before detailed Reynolds equation solution.

Physics & theory

In a journal bearing, rotating shaft (journal) separates from the bushing by a lubricant film when sufficient speed generates hydrodynamic pressure. The Sommerfeld number $S=(\mu n/p)(r/c{)}^2$ characterizes operation, where $\mu$ is viscosity, $n$ is speed, $p$ is unit load, $r$ is radius, and $c$ is radial clearance.

Minimum film thickness $h_{\min }$ occurs near the maximum pressure arc; it must exceed composite surface roughness to avoid boundary contact. Eccentricity ratio $\epsilon$ is interpolated from Raimondi–Boyd $\epsilon (S)$ charts (full journal, $L/D\approx 1$ screening). Power loss is viscous shear in the film. Outlet temperature uses a light 2–3 pass ΔT ↔ viscosity iteration (Walther screening scale on the user viscosity); short-bearing / single-zone limits remain.

Governing equations

Numerical method

Sommerfeld + Raimondi–Boyd $\epsilon (S)$, iterative mean-film temperature viscosity. Inputs: diameter, length, clearance, load, speed, viscosity, ambient temperature. Outputs: $S$, $h_{\min }$, eccentricity, power loss, specific load, outlet T, shaft/housing fit recommendation.

Inputs

Outputs

• Sommerfeld number, eccentricity ratio, minimum film thickness, film parameter / specific load, power loss, outlet temperature
• Live Design Summary rail (S, $h_{\min }$, specific load, outlet T, status)
• Deterministic plain advisor (L/D, clearance, viscosity, pad count rationale + alternatives)
• Sectioned PDF / Excel (Design Summary, film factors, recommendation)
• Status banner with ε, film ratio, load-limit highlights

Design codes & checks

• ISO 7902 — Hydrodynamic plain journal screening
• ISO 12130 / 12131 — Tilting-pad / thrust pad screening
• Specific load and temperature screening limits

Assumptions & limitations

• Full journal, steady-state; oil catalog + Walther ν(T); light ΔT ↔ viscosity iteration (not full flow heat balance).
• Raimondi–Boyd ε(S) interpolated for L/D ∈ {0.25…1.5} — still not full finite-length Reynolds solution.
• No dynamic instability (oil whirl/whip) analysis.
• No MITCalc-IV-class sliding material + oil flow database.

Design workflow

• Validate / Calculate: Forward ISO 7902/12130 screening with mode-aware Calculate label.
• Auto-design: Available via design workflow where wired.

References

1. Hamrock, B. J., Schmid, S. R., & Jacobson, B. O. Fundamentals of Fluid Film Lubrication, 2nd ed. CRC Press.
2. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 12.
3. ISO 7902-1:2020. Calculation of plain bearings — Hydrodynamic plain journal bearings.
4. Bassani, R., & Piccigallo, B. Hydrostatic Lubrication. Elsevier.
5. PhyCalcPro verification benchmarks in src/data/verification/ (plain-bearings-indicative-*.json).

Brakes & Clutches (brakes-clutches)

Purpose

Calculate friction torque capacity, energy dissipated per stop or engagement, and thermal screening for disk and drum brakes and clutches. Supports single-plate and multi-plate configurations.

Physics & theory

Friction devices transmit torque through normal force $F_n$ and coefficient of friction $\mu$: $T=n\mu F_n{r}_{\mathrm{eff}}$, where $n$ is number of friction surfaces and $r_{\mathrm{eff}}$ is effective radius (mean radius for uniform pressure assumption).

Energy per engagement is $E=T\Delta \theta$ for angular displacement $\Delta \theta$, or $E=\frac{1}{2}I{\omega }^2$ for full stop from speed $\omega$. Repeated engagements heat friction surfaces; average power dissipation $P=E\cdot f_{\mathrm{cycle}}$ must not exceed material and coolant limits.

Governing equations

Numerical method

Closed-form friction torque and energy relations. Safety factor applied to required vs available torque. Thermal screening compares energy per cycle to allowable surface temperature rise (simplified lumped model).

Inputs

Outputs

• Friction torque capacity, torque utilization, energy per stop, average dissipated power, thermal warning flags.

Design codes & checks

• Indicative: Friction torque capacity, energy per stop screening

Assumptions & limitations

• Uniform pressure or uniform wear assumption — user selects model.
• Dry or wet friction $\mu$ from tables; no dynamic $\mu$ vs speed/temperature.
• No detailed transient thermal FEA of friction surfaces.
• Vibration, chatter, and fade not modeled.

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 16.
2. SAE J2681. Brake Effectiveness — Vehicle Analysis.
3. Newcomb, T. P., & Spurr, R. T. A Technical History of the Motor Car. (Brake fundamentals)
4. ISO 7649:1988. Brakes — Friction materials — Classification.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Springs

Compression Springs (compression-springs)

Purpose

Design helical compression springs per EN 13906-1 and Shigley methods:

Physics & theory

A helical compression spring wound from wire diameter $d$ on mean coil diameter $D$ with $n$ active coils behaves as a linear spring with rate $k=G{d}^4/(8D^{3}n)$, where $G$ is shear modulus. Wahl factor $K_w=(4C-1)/(4C-4)+0.615/C$ with spring index $C=D/d$ corrects for curvature and direct shear in maximum wire shear stress $\tau =K_w\cdot 8FD/(\pi d^3)$.

EN 13906-1 allowable shear for cold-coiled springs is ${\tau }_{\mathrm{zul}}=0.56R_m$, where $R_m$ follows size-effect fit $R_m=A/d^m$ for standard wire grades. Buckling when free length $L_0$ exceeds $2.63D/\nu$ with end condition coefficient $\nu$.

Optional fatigue screening uses characteristic shear fatigue strength ${\tau }_{k0}$ with life-class reduction and Goodman mean-stress correction when minimum deflection is specified (life classes VL/LH/MH/HH).

Governing equations

Numerical method

Closed-form EN 13906-1 / Shigley equations. Wire ultimate from Shigley Table 10-4 fits or springWireCatalog.ts (EN 10270 / ASTM stock). Active coil mass computed for surge frequency. Fatigue via en13906Fatigue.ts when enabled.

Inputs

Outputs

• Spring rate, solid height, loaded length, solid height clearance, max load, shear stress, static SF
• Surge frequency and margin, buckling limit, spring index, Wahl factor
• Optional fatigue SF and utilization; governing failure mode
• Load–deflection and stress plots; spring outline preview

Design codes & checks

• Indicative: Shear stress utilization, solid height, surge margin, fatigue life (when enabled)
• EU: EN 13906-1 cold-coiled helical compression springs
• US: SAE AMS spring wire specifications (reference)

Design workflow

• Validate: Forward check on entered geometry.
• Auto-design: Sweeps springWireCatalog wire diameters and active coils within max OD for target rate and stress.
• Compare: Ranked wire/coil alternatives with Apply.

Assumptions & limitations

• Circular wire, closed and ground ends (solid height includes 2d end allowance).
• Fatigue uses simplified τk0 + Goodman screening — verify critical designs against EN 13906 nomographs.
• Surge margin requires operating frequency input; default 10× margin target.
• Not for extension or torsion springs (see dedicated modules).

Verification

• CI: compression-springs-indicative-01.json, compression-springs-indicative-fatigue-01.json
• Vitest: src/lib/springs/compression-springs/engine.test.ts, en13906Fatigue.test.ts
• Engineer sign-off: spring-modules-user-tasks.md, validation-master-checklist.md

References

1. EN 13906-1:2013. Cylindrical helical springs — Part 1: Compression springs.
2. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 10.
3. Wahl, A. M. Mechanical Springs, 2nd ed. McGraw-Hill.
4. ASTM A228/A227/A229. Steel Wire for Mechanical Springs.

Extension Springs (extension-springs)

Purpose

Design helical extension (tension) springs including initial tension, hook stress, spring rate, EN 13906 fatigue screening, and wire catalog selection. Used for assemblies requiring pull force with near-zero free length.

Physics & theory

Extension springs are wound with initial coiled tension $F_i$ that must be overcome before coils separate. Total force at extension $x$ is $F=F_i+kx$, with rate $k=G{d}^4/(8D^{3}n)$ identical to compression spring formula.

Maximum shear stress in the body uses Wahl correction on the coil body load. Hook stress concentrations often govern failure; standard hooks (machine, cross-over, extended) use empirical stress factors $K_{\mathrm{hook}}$. Initial tension is user-specified or estimated from the manufacturable limit (Shigley screening).

Governing equations

Numerical method

Closed-form rate and body stress with Wahl factor. Hook factors from wireStrength.ts. Fatigue on body stress range when minimum extension is set. Auto-design sweeps catalog wire sizes and coil counts for target rate, hook SF, and optional fatigue margin.

Inputs

Outputs

• Spring rate, initial tension, max manufacturable $F_i$, force at extension
• Body and hook shear stress and separate safety factors
• Coil bind length, extended length, surge frequency
• Optional fatigue SF; governing failure mode
• Load–extension plot (F = Fi + kx)

Design codes & checks

• Indicative: Body shear utilization, hook stress SF, surge margin, fatigue life (when enabled)
• EU: EN 13906-2 extension springs (reference)

Design workflow

• Validate: Forward check with hook type and Fi validation flag.
• Auto-design: Wire/coil sweep from springWireCatalog for target rate, max force, hook SF.
• Handoff: Fatigue module receives body shear as alternating stress input.

Assumptions & limitations

• Hook stress uses empirical factors — not a substitute for hook FEA on critical applications.
• Initial tension validated against manufacturable estimate; not auto-sized.
• Fatigue simplified per EN 13906-2 screening; full hook fatigue nomograph not embedded.

Verification

• CI: extension-springs-indicative-01.json
• Vitest: src/lib/springs/extension-springs/engine.test.ts
• Engineer sign-off: spring-modules-user-tasks.md

References

1. EN 13906-2:2013. Cylindrical helical springs — Part 2: Extension springs.
2. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 10.
3. Wahl, A. M. Mechanical Springs, 2nd ed.
4. Associated Spring Raymond. Design Handbook.

Torsion Springs (torsion-springs)

Purpose

Design helical torsion springs loaded by bending in the coil wire (typically via legs). Computes spring rate, curvature-corrected coil bending stress, leg stress estimate, EN 13906 fatigue screening, and wire catalog selection.

Physics & theory

Torsion springs store energy through wire bending rather than torsion shear along the coil axis. Spring rate in terms of angle $\theta$ is:

(Shigley Eq. 10-37), for $n_a$ active coils. Bending stress uses curvature factor $K_b$ on the mean-diameter stress:

Legs act as cantilever beams; leg bending stress is estimated separately. Allowable bending stress screening uses ${\sigma }_{\mathrm{zul}}=0.8R_m$.

Governing equations

Numerical method

Closed-form bending-based rate and stress with Shigley curvature factor. Optional EN 13906 bending fatigue when minimum wind angle is specified. Auto-design sweeps wire diameter, coil count, and leg length for target rate and bending SF.

Inputs

Outputs

• Spring rate (N·m/rad), torque at angle, coil bending stress with $K_b$
• Leg force and leg bending stress estimate, static SF
• Optional fatigue SF; spring index, governing failure mode
• Torque–angle and stress–angle plots

Design codes & checks

• Indicative: Coil bending stress utilization, fatigue life (when enabled)
• EU: EN 13906-3 torsion springs (reference)

Design workflow

• Validate: Forward check on entered geometry and angle.
• Auto-design: Wire/coil/leg sweep for target rate (N·m/rad) and bending SF.
• Handoff: Fatigue module receives coil bending stress.

Assumptions & limitations

• Circular wire; rectangular wire requires different section modulus.
• Leg stress uses simplified cantilever model; coil–leg junction not FEA’d.
• Rate formula updated to Shigley Eq. 10-37 (re-baseline saved projects from older builds).
• Fatigue simplified per EN 13906-3 screening.

Verification

• CI: torsion-springs-indicative-01.json
• Vitest: src/lib/springs/torsion-springs/engine.test.ts
• Engineer sign-off: spring-modules-user-tasks.md

References

1. EN 13906-3:2013. Cylindrical helical springs — Part 3: Torsion springs.
2. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 10.
3. Wahl, A. M. Mechanical Springs, 2nd ed.
4. Spring Manufacturers Institute. Handbook of Spring Design.

Fasteners & connections

Bolt Calculator (bolts)

Purpose

Analyze threaded fasteners including power screw efficiency, bolt pattern stiffness, and VDI 2230 single-bolt preloaded joint worksheet. Computes tensile, shear, bearing utilization and preload margin for mechanical joints.

Physics & theory

Bolted joints clamp parts together with initial preload $F_i$ from torque $T=K{F}_{i}d$, where $K$ is nut factor. External tensile load $F_e$ shares between bolt and members by stiffness: bolt load increment $\Delta F_b=F_e{k}_b/(k_b+k_m)$. Separation occurs when preload is lost.

Shear may be carried by friction (when clamped) or bolt shank/threads in bearing. Combined tension and shear uses interaction criteria per AISC J3 or VDI 2230. Power screws convert torque to axial force with efficiency $\eta =\tan \lambda /\tan (\lambda +\varphi )$ for square/Acme threads.

Connections transfer load through bearing, shear, tension, and friction paths depending on joint configuration. Preload in bolted joints reduces joint separation and can allow friction to carry shear; without adequate preload, bolts carry full shear in bearing against hole walls.

FEM-based bolt analysis resolves member and bolt stiffness for load sharing; VDI 2230 provides a systematic worksheet for high-fidelity preloaded joints including embedding loss and tightening scatter.

Governing equations

Numerical method

Dual paths: (1) Power screw and pattern analysis via FEA stiffness (femSolver); (2) VDI 2230 worksheet for high-fidelity single-bolt joints with embedding, thermal, and tightening scatter. Validators enforce thread and geometry consistency.

Inputs

Outputs

• Bolt and member load sharing, tensile/shear/bearing utilization, preload safety margin, torque recommendation
• VDI 2230 assembly preload range.

Design codes & checks

• Indicative: Tensile, shear, bearing utilization
• US: AISC 360-22 Chapter J3
• EU: EN 1993-1-8, VDI 2230 Part 1

Assumptions & limitations

• Linear elastic joint behavior; no gasket creep long-term model unless VDI embedding used.
• VDI 2230 is single-bolt centric; patterns use simplified stiffness superposition.
• Power screw FEA validated against Shigley benchmarks.
• Does not replace licensed pressure vessel or nuclear QA bolt procedures.

References

1. VDI 2230 Part 1:2015. Systematic calculation of highly stressed bolted joints.
2. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 8.
3. AISC. Specification for Structural Steel Buildings (ANSI/AISC 360-22), Chapter J3.
4. EN 1993-1-8:2005. Design of joints.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Weld Group Analysis (welds) — beta

Purpose

Analyze weld groups under direct shear, torsion, and eccentric loading by computing throat shear stress distribution and combined throat stress utilization per AWS D1.1 and EN 1993-1-8 screening methods.

Physics & theory

Fillet welds are sized by effective throat $a_e=0.707a$ for equal-leg fillets. Throat area resists shear; normal stress on throat is often neglected for fillet welds in simplified analysis. For a weld group of total throat area $A_w$, direct shear is $\tau =V/A_w$.

Eccentric load $e$ creates moment resisted by weld group polar moment $J_w$ about the group centroid: combined shear $\tau =\sqrt{{\tau }_V^2+{\tau }_M^2}$. Common patterns (rectangle, circle, line) have tabulated $J_w$ formulas. Allowable throat shear is typically $0.3F_u$ (AWS) or partial factor per EN.

Connections transfer load through bearing, shear, tension, and friction paths depending on joint configuration. Preload in bolted joints reduces joint separation and can allow friction to carry shear; without adequate preload, bolts carry full shear in bearing against hole walls.

FEM-based bolt analysis resolves member and bolt stiffness for load sharing; VDI 2230 provides a systematic worksheet for high-fidelity preloaded joints including embedding loss and tightening scatter.

Governing equations