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 ProductsCategoryBar (category chip links on hub/landings; breadcrumb on modules). Category layouts are passthrough wrappers — browse at /products/{categoryId}.
• 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, Archive/scripts-oneoff/scaffold-design-mode.mjs (historical).
• 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 engineering knowledge guides live in docs/modules/{moduleId}.md. Each page covers selection/analysis workflow, worked examples, FAQ, plus purpose, physics & theory, governing equations, numerical method, inputs, outputs, design codes & checks, assumptions & limitations, and references. Every module documentation page also includes a Validation & quality section (release tier, catalog status, benchmarks). Browse individually at /documentation/modules/{moduleId}. Fleet maturity dashboard: /status.

Structural engineering

Beam Analysis Guide (beams)

How engineers analyze beams

Beams are the most fundamental structural element in engineering, carrying transverse loads across a span and transferring them to supports through shear and bending. Every building floor, bridge deck, crane runway, and machine frame relies on beam behavior. Engineers analyze beams to determine internal forces (shear and moment), deformations (slope and deflection), and stresses (bending and shear) — ensuring no limit state is exceeded under the design loading.

The classical approach uses Euler-Bernoulli beam theory, which relates transverse deflection to applied loading through a fourth-order differential equation. For simple geometries and loads, closed-form solutions give immediate answers: a simply supported beam with a central point load has maximum moment $M=PL/4$ and peak deflection $\delta =P{L}^3/(48EI)$. Real structures, however, have multiple loads, mixed supports, and variable sections that demand numerical methods.

Modern practice employs finite-element discretization of the beam with Hermite cubic shape functions, enforcing displacement and slope continuity at nodes. This handles arbitrary combinations of point loads, distributed loads, and applied moments with any support configuration. The PhyCalcPro beams module implements exactly this workflow: mesh the span, assemble stiffness matrices, solve for nodal displacements, then post-process for diagrams and peak values.

Design verification then compares computed stresses and deflections against code limits. In steel design, AISC 360 Chapter F governs flexural capacity, Chapter G governs shear, and serviceability deflection limits are typically L/240 to L/360. Eurocode 3 (EN 1993-1-1 Section 6.2) uses partial safety factors on resistance. The module provides screening checks against both standards.

Beam types and when to use each

Cross-section types typically analyzed:

• I/H sections (IPE, HEB, W-shapes) — most efficient for bending; wide flanges resist moment, thin web carries shear.
• Channel sections (UPN, C-shapes) — used where one-sided connections are needed; asymmetric bending requires shear center consideration.
• Rectangular hollow sections (RHS) — good torsional resistance; used in machine frames and architectural applications.
• Solid rectangular bars — simple fabrication; common in machinery and custom equipment.
• Circular hollow sections (CHS) — equal strength in all bending directions; used for crane booms and space frames.

Loading types handled by the module:

• Point loads — concentrated forces at specified positions (equipment mounts, wheel loads)
• Uniformly distributed loads (UDL) — self-weight, floor live load, snow
• Applied moments — eccentric connections, torque reactions

Design code deflection and stress limits

Beam adequacy is governed by two independent limit states:

• Strength (ULS): Bending stress must remain below the factored resistance — $\sigma \le \varphi M_n/S$ (AISC) or $M_{Ed}\le M_{c,Rd}$ (Eurocode).
• Serviceability (SLS): Deflection under unfactored loads must not exceed application-specific limits to prevent damage to finishes, vibration issues, or aesthetic concerns.

Both must be satisfied simultaneously. A beam may pass strength checks but fail on deflection (common for long spans with light loads), or vice versa.

Engineering workflow

1. Define geometry: span length, cross-section properties (I, c, area), and material (E, yield strength).
2. Select support configuration: simply supported, cantilever, or fixed-fixed.
3. Apply loads: point forces at known positions, UDL over full or partial span, applied moments.
4. Choose design context: application preset (lifting beam, crane bridge, machine frame) sets default load factors and deflection limits.
5. Run the FEM solver with adequate mesh density (minimum 20 segments recommended for accurate peak values near point loads).
6. Review diagrams: inspect shear force, bending moment, deflection, and stress plots along the span.
7. Check peak values: compare maximum bending stress against allowable, maximum deflection against serviceability limits.
8. Verify equilibrium: confirm the static equilibrium residual is near zero.
9. Iterate if needed: adjust section size, add supports, or redistribute loads until all checks pass.

Key quantities and formulas

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

For a cantilever with tip load $P$:

For a UDL $w$ on a simply supported span:

Deflection limit check:

Worked example

Problem: A simply supported steel beam spans 6 m and carries a central point load of 50 kN. The section is an IPE 300 with $I=8356\times 10^{-8}$ m${}^4$, $c=0.15$ m, and $E=210$ GPa. Check bending stress and deflection against L/300.

Step 1 — Reactions:

Step 2 — Maximum moment (mid-span):

Step 3 — Bending stress:

This is below S275 yield (275 MPa) — utilization = 134.6/275 = 0.49, acceptable.

Step 4 — Deflection:

Allowable: $L/300=6000/300=20$ mm. Actual 12.8 mm < 20 mm — pass.

Common mistakes and checks

• Insufficient mesh density: Using fewer than 20 segments underestimates peak stress near concentrated loads; always verify convergence by doubling segments.
• Wrong sign convention: PhyCalcPro uses sagging-positive for moments; mixing conventions causes incorrect shear diagram interpretation.
• Ignoring shear deformation: Euler-Bernoulli theory neglects shear deformation; for deep beams (span/depth < 10), Timoshenko theory is more appropriate.
• Overlooking lateral-torsional buckling: High bending stress alone does not confirm adequacy; unbraced compression flanges require LTB checks per AISC F2 or EN 1993-1-1 Section 6.3.
• Using the wrong I-value: The second moment of area must be about the bending axis; using the wrong axis gives non-conservative results.
• Neglecting self-weight: For long spans with light applied loads, beam self-weight can dominate deflection.
• Applying building-code deflection limits to machinery: L/360 is for floor beams with plaster ceilings; crane runways and machine frames use different criteria.

FAQ

What mesh density should I use?

A minimum of 20 segments is recommended for typical beams. For beams with multiple closely-spaced point loads or steep moment gradients, use 40-80 segments. The solver reports warnings when mesh is too coarse.

Can I analyze non-prismatic (tapered) beams?

The current solver assumes a prismatic (constant) cross-section along the span. For tapered beams, use the average or critical-section properties as an approximation, or subdivide into piecewise-prismatic segments.

How does the module handle overhanging beams?

Overhangs are modeled as cantilever extensions beyond a support. Define a simply supported span with loads placed beyond one support to simulate an overhang.

What is the static equilibrium residual?

It is the numerical difference between total applied vertical load and the sum of computed reactions. A residual near machine precision (< 0.001% of applied load) confirms the solver produced a valid equilibrium solution.

When should I use fixed-fixed vs simply supported?

Use fixed-fixed when both ends are rigidly welded to stiff columns or walls that prevent rotation. If connections allow any rotation (bolted end plates, bearing pads), simply supported is more appropriate and gives conservative (higher) mid-span moments.

Does the module account for dynamic loads?

No — the solver performs static analysis only. For impact or vibration, apply a dynamic amplification factor (DAF) to static loads before input, per your governing standard (e.g., DAF = 1.25 for crane hoists per EN 13001).

How do deflection limits vary by application?

Always verify the governing standard for your specific application.

Use the PhyCalcPro calculator

Open the Beam Analysis calculator

Purpose

Analyze one-dimensional prismatic beams under point loads, uniformly distributed loads, and applied moments. Computes shear force, bending moment, slope, deflection, and bending stress along the span, then compares results against 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 $\kappa =M/(EI)$. For small deflections, the governing ODE is $EI{d}^{4}w/d{x}^4=q(x)$, with boundary conditions set by support type. Shear force and moment are obtained by equilibrium; bending stress at distance $c$ from the neutral axis follows $\sigma =Mc/I$. Application presets (lifting beam, machine frame, crane bridge) adjust load factor, allowable stress ratio, and deflection limit.

Governing equations

Numerical method

1D beam FEM: the span is meshed into configurable segments. Hermite shape function 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 yields shear, moment, slope, deflection, and stress along the span.

Inputs

Outputs

• Shear $V(x)$, moment $M(x)$, slope, deflection $w(x)$, stress $\sigma (x)$ diagrams
• Peak values: maxShear, maxMoment, maxDeflection, maxStress with location
• Support reactions and moments
• Static equilibrium residual from physicsChecks
• Code checks: bending utilization, shear utilization, LTB utilization, deflection utilization

Design codes & checks

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

Assumptions & limitations

• Linear elastic, prismatic cross-section; no large deflection or plasticity.
• 1D beam model — not a full 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.
5. Cook, R. D., et al. Concepts and Applications of Finite Element Analysis, 4th ed. Wiley.
6. Hibbeler, R. C. Structural Analysis, 10th ed. Pearson.
7. Timoshenko, S. P. Strength of Materials, Part I, 3rd ed. Van Nostrand.

Frame Analysis Guide (frames)

How engineers analyze frames

Rigid-jointed frames are assemblies of beams and columns connected at nodes that resist both force and moment. Unlike trusses (pin-jointed, axial only), frame members carry axial force, shear, and bending simultaneously. Portal frames, industrial mezzanines, machine tool structures, and equipment skids all behave as rigid frames under lateral and gravity loads.

The direct stiffness method is the standard numerical technique: each member contributes a 6-DOF local stiffness matrix (three at each end — horizontal displacement, vertical displacement, rotation) that is transformed to global coordinates and assembled into the structure stiffness matrix. Solving the global system $\mathbf{Ku}=\mathbf{F}$ yields nodal displacements, from which member forces and reactions are back-calculated.

Engineers use frame analysis to verify that no member is overstressed, that lateral drift is within serviceability limits, and that reactions are compatible with foundation design. The PhyCalcPro frames module handles arbitrary 2D planar frames with fixed, pinned, or roller supports at any node.

Frame configurations and applications

Engineering workflow

1. Define node coordinates in the 2D plane.
2. Connect members between nodes; specify E, A, I, and section depth for each member.
3. Assign support conditions: fixed, pinned, or roller at boundary nodes.
4. Apply loads: nodal point forces/moments, member distributed loads.
5. Run the direct stiffness solver.
6. Review member force diagrams: axial, shear, and moment for each member.
7. Check stress utilizations: combined axial + bending stress vs allowable.
8. Verify joint equilibrium: confirm reaction residuals are negligible.
9. Assess sway deflection against drift limits (H/300 typical for industrial frames).

Key quantities and formulas

Member combined stress:

Utilization ratio:

Member stiffness coefficients (prismatic member of length L):

Worked example

Problem: A single-bay portal frame has columns of height 4 m and a beam spanning 6 m. All members are IPE 240 steel ($E=210$ GPa, $A=39.1$ cm${}^2$, $I=3892$ cm${}^4$, $c=0.12$ m). Columns are fixed at the base. A horizontal wind load of 20 kN acts at the beam level. Find the maximum member stress.

Step 1 — Model:

• 4 nodes: two base supports (fixed), two beam-column joints.
• 3 members: two columns (4 m vertical), one beam (6 m horizontal).
• Horizontal load of 20 kN at the beam-column joint on the windward side.

Step 2 — Solver output (from direct stiffness):

• Column base moments: windward = 52.4 kN-m, leeward = 27.6 kN-m.
• Beam end moments: 27.6 kN-m at each joint (antisymmetric distribution).
• Maximum column axial force: 9.2 kN (from frame action).

Step 3 — Combined stress in windward column:

Utilization against S275: $164/275=0.60$ — adequate.

Common mistakes and checks

• Forgetting moment transfer at rigid joints: All members meeting at a rigid joint share the same rotation; assigning pin connections when joints are welded gives unconservative member moments.
• Applying loads only at nodes: Distributed member loads must be converted to fixed-end forces; the solver handles this, but applying forces only at nodes misses local bending in the member.
• Ignoring P-delta effects: For tall or heavily loaded frames, geometric nonlinearity (P-delta) can amplify moments by 10-30%; the current linear solver does not capture this.
• Wrong support assumptions: Industrial frames often have nominally pinned column bases that provide partial fixity; both fixed and pinned assumptions should be checked as bounds.
• Neglecting out-of-plane behavior: The 2D solver assumes all loads and deformations are in-plane; out-of-plane stability must be verified separately.

FAQ

Can I model a pin connection at a member end?

Yes — flag the member end as pinned (moment release). This sets the rotational DOF at that end to zero moment, simulating a hinge. By default all joints are rigid.

How do I handle a frame with inclined members?

The solver uses node coordinates to determine member orientation automatically. Define nodes at the actual positions and connect members — the coordinate transformation handles any angle.

What if my frame is statically indeterminate?

The stiffness method handles any degree of indeterminacy automatically. Unlike hand methods (moment distribution, slope-deflection), the solver assembles and solves the full system regardless of redundancy.

Does the module check for frame instability (sway buckling)?

The current solver performs first-order elastic analysis. It does not compute frame buckling loads. Use the column module on individual members with appropriate effective lengths to screen for member buckling in sway frames.

How accurate are the stress utilizations?

The utilizations are screening-level checks combining axial and bending stress linearly. They do not implement full AISC H1 interaction equations or EN 1993-1-1 Section 6.3.3 beam-column checks. Use results as preliminary sizing, then verify with detailed code checks.

Use the PhyCalcPro calculator

Open the Frame Analysis calculator

Purpose

Perform two-dimensional elastic frame analysis for rigid-jointed structures composed of prismatic members. 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 6x6 stiffness matrix in local coordinates that is transformed to global axes before assembly. Equilibrium requires $\mathbf{Ku}=\mathbf{F}$. Member stresses are recovered from combined axial and bending components for utilization screening.

Governing equations

Numerical method

Direct stiffness method: element stiffness matrices are transformed and assembled; boundary conditions eliminate constrained DOFs. The reduced linear system is solved for nodal displacements. Member end forces are back-calculated from element deformations.

Inputs

Outputs

• Nodal displacements and rotations
• Member axial force, shear, and end moments
• Joint reactions
• Member stress utilization ratios
• Equilibrium check residuals

Design codes & checks

• Indicative: Member stress utilization, joint 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 member end releases are specified.
• No P-delta geometric nonlinearity.
• Does not replace licensed structural design per building codes.

Verification

• CI: verification benchmarks in src/data/verification/ where available
• Engineer sign-off: validation-master-checklist.md

References

1. Hibbeler, R. C. Structural Analysis, 10th ed. Pearson.
2. McCormac, J. C., & Brown, R. H. Structural Analysis, 5th ed. Cengage.
3. McGuire, W., Gallagher, R. H., & Ziemian, R. D. Matrix Structural Analysis, 2nd ed. Wiley.
4. EN 1993-1-1:2005. Eurocode 3 — General rules.
5. ISO 12100:2010. Safety of machinery — General principles for design.

Truss Analysis Guide (trusses)

How engineers analyze trusses

Trusses are structures composed of straight members connected at pin joints, designed so that all loads are applied only at joints. Under these idealized conditions, every member is a two-force element carrying only axial force — either tension or compression. This makes trusses extremely efficient: material is used purely in direct stress with no bending waste.

Engineers analyze trusses to determine which members are in tension, which in compression, and whether peak axial stress exceeds the allowable for the chosen section. The classical methods of joints (equilibrium at each pin) and method of sections (free-body cuts) work for simple determinate trusses. For indeterminate trusses or complex topologies, the direct stiffness method with bar elements is the standard computational approach, assembling axial stiffness $EA/L$ per member.

The PhyCalcPro trusses module implements bar-element FEM: users define nodes, members, and supports; the solver returns member forces, stress utilizations, and identifies zero-force members automatically.

Truss configurations

Engineering workflow

1. Define joint positions (node coordinates) in the 2D plane.
2. Connect members between nodes; assign cross-sectional area $A$ and modulus $E$ to each.
3. Apply support conditions: at least three restraints to prevent rigid-body motion.
4. Apply loads at joints only (convert distributed member loads to equivalent nodal forces).
5. Run the bar-element FEM solver.
6. Identify tension (+) and compression (-) members from signed axial forces.
7. Check axial stress utilization $U=|\sigma |/{\sigma }_{\mathrm{allow}}$ for each member.
8. Flag zero-force members (may be needed for stability but carry no load under this case).
9. For compression members, verify buckling capacity using the Column Buckling module.

Key quantities and formulas

Bar element stiffness (local coordinates):

Axial stress and utilization:

Member force recovery from nodal displacements:

Member elongation:

Worked example

Problem: A simple Warren truss with 3 panels spans 9 m (3 m per panel) and has a height of 2 m. Two equal 40 kN loads act at the bottom chord interior joints. All members have $A=12$ cm${}^2$, $E=210$ GPa. Find the maximum member force and stress.

Step 1 — Reactions:

Total load = 80 kN, symmetric. $R_A=R_B=40$ kN.

Step 2 — Method of sections (center panel diagonal):

Cut through the center panel. Taking moments about the top chord joint above the cut:

Maximum diagonal force (from joint equilibrium):

where $\theta =\arctan (2/3)=33.7°$.

Step 3 — Stress in critical member:

Utilization against allowable 165 MPa (S275 with SF=1.67): $60.1/165=0.36$ — adequate.

Common mistakes and checks

• Applying loads between joints: Truss theory assumes loads act only at nodes. Distributed loads on a top chord must be resolved to the adjacent nodes before analysis.
• Ignoring compression buckling: Tension members fail by yielding, but compression members fail by buckling at loads well below yield. Always check compression members with the Column Buckling module.
• Insufficient supports: A 2D truss needs at least 3 independent restraints (e.g., pin + roller). Too few produces a singular stiffness matrix; too many produces an indeterminate truss (which the solver handles).
• Misidentifying zero-force members: Zero-force members under one load case may carry force under others. Do not remove them without checking all load combinations.
• Neglecting connection eccentricity: Real truss connections have gusset plates with eccentricities that introduce secondary bending; the idealized pin-joint model ignores this.
• Using member self-weight incorrectly: Distributed self-weight must be split equally to end nodes; applying it as a member load is incorrect for truss analysis.

FAQ

How do I know if my truss is statically determinate?

For a 2D truss: $m+r=2j$ where $m$ = members, $r$ = reactions, $j$ = joints. If $m+r>2j$, the truss is indeterminate (the solver handles this). If $m+r<2j$, it is a mechanism and will fail.

What is a zero-force member?

A member carrying zero axial force under the given loading. The solver identifies these automatically. They often occur at unloaded joints where geometry creates equilibrium without force.

Can I use this for 3D space trusses?

The current module supports 2D planar trusses only. For 3D space trusses, decompose into planar sub-trusses or use dedicated 3D software.

How does the solver handle thermal loads?

Thermal loads are not currently supported. To approximate thermal effects, compute the equivalent member force $N=EA\alpha \Delta T$ and apply it as a pre-load externally.

What sign convention is used for member forces?

Positive force indicates tension (member being stretched); negative indicates compression. This follows standard structural engineering convention.

Use the PhyCalcPro calculator

Open the Truss Analysis calculator

Purpose

Determine axial forces in two-dimensional pin-jointed truss members under nodal loading. 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 two-force elements carrying only axial force along the member axis. At each pin joint, equilibrium ($\sum F_x=0$, $\sum F_y=0$) holds. The structure stiffness matrix involves only translational DOFs. Axial stress is $\sigma =N/A$. Compression members require separate buckling checks.

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 are recovered from relative end displacements.

Inputs

Outputs

• Member axial force (signed tension/compression)
• Axial stress and utilization ratio per member
• Reaction forces at supports
• Zero-force member identification
• Deformed shape (optional)

Design codes & checks

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

Assumptions & limitations

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

Verification

• CI: verification benchmarks in src/data/verification/ where available
• Engineer sign-off: validation-master-checklist.md

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. Kassimali, A. Structural Analysis, 6th ed. Cengage.
5. ISO 10721:1997. Steel structures — Static analysis and design.

Column Buckling Guide (columns)

How engineers design columns against buckling

Columns are compression members where stability — not material strength — often governs design. Unlike tension members that fail by yielding or fracture, slender columns can collapse suddenly by lateral buckling at loads well below the material crush strength. Understanding buckling behavior is essential for every structural and mechanical engineer designing frames, supports, machinery bases, and truss compression chords.

Leonhard Euler derived the critical load for an ideal elastic column in 1757: $P_{\mathrm{cr}}={\pi }^{2}EI/L_{\mathrm{eff}}^2$. This theoretical maximum assumes perfect straightness, no residual stresses, and purely elastic behavior. Real columns always have initial imperfections, residual stresses from manufacturing, and material yielding at moderate slenderness. Design codes therefore replace pure Euler theory with empirical column curves that reduce capacity below the theoretical limit.

The design process involves computing the slenderness ratio $\lambda =L_{\mathrm{eff}}/r$ (where $r=\sqrt{I/A}$ is the radius of gyration), then using code-specific curves to find the reduction factor $\chi$. For AISC 360, the transition between elastic and inelastic buckling occurs at ${\lambda }_c=4.71\sqrt{E/f_y}$; for Eurocode 3, five imperfection curves (a0 through d) account for section type and axis of buckling.

The PhyCalcPro columns module performs finite-element buckling eigenvalue analysis — assembling both elastic stiffness $\mathbf{K}$ and geometric stiffness ${\mathbf{K}}_g$ — then overlays code curve checks to give both the theoretical critical load and the code-compliant design capacity.

End conditions and effective length

Selecting the correct effective length factor $K$ is the single most critical engineering judgment in column design. Conservative (higher) values of $K$ should be used when actual connection stiffness is uncertain.

Column slenderness classification:

• Short columns ($\lambda <30$): Fail by material crushing; buckling is not a concern. Strength = $A{f}_y$.
• Intermediate columns ($30<\lambda <120$): Inelastic buckling zone; residual stresses and imperfections interact with material yielding. Code column curves are essential.
• Slender columns ($\lambda >120$): Elastic (Euler) buckling governs; capacity drops rapidly with increasing slenderness.
• Very slender ($\lambda >200$): Generally not permitted in main structural members by most codes.

Eurocode 3 buckling curves

EN 1993-1-1 Table 6.2 assigns imperfection curves based on section type and buckling axis:

Higher imperfection factors yield more conservative (lower) buckling resistance. The correct curve must be selected based on manufacturing method, section proportions, and buckling axis.

Engineering workflow

1. Determine the factored axial compressive load $P_u$ from structural analysis or load combinations.
2. Select trial column section: record $A$, $I_{\min }$, $r_{\min }$, and material yield strength $f_y$.
3. Establish effective length: assess end restraints and select $K$ factor for each axis.
4. Compute slenderness ratio: $\lambda =KL/r_{\min }$. Verify it does not exceed the code maximum (typically 200 for main members).
5. Determine critical stress: use Euler formula for elastic buckling or code inelastic transition formula.
6. Apply code reduction: AISC Chapter E or EN 1993-1-1 Section 6.3 buckling curves yield design capacity $\varphi P_n$ or $N_{b,Rd}$.
7. Check utilization: $P_u/\varphi P_n\le 1.0$. If overstressed, increase section size or reduce $KL$ by adding bracing.
8. Verify buckling mode: confirm flexural buckling governs; check for torsional or flexural-torsional buckling if section is open or unsymmetric.
9. Document assumptions: record effective length justification and any alignment tolerance requirements.

Key quantities and formulas

AISC 360 Chapter E — elastic/inelastic transition:

Eurocode 3 buckling reduction factor:

Worked example

Problem: A pinned-pinned steel column of length 4 m carries 800 kN axial compression. Section is HEB 200: $A=78.1$ cm${}^2$, $I_{\min }=2003$ cm${}^4$, $r_{\min }=5.07$ cm, $f_y=355$ MPa, $E=210$ GPa.

Step 1 — Effective length:

$K=1.0$ (pinned-pinned), so $L_{\mathrm{eff}}=4.0$ m.

Step 2 — Slenderness ratio:

Step 3 — Euler critical load:

Step 4 — AISC check:

$4.71\sqrt{E/F_y}=4.71\sqrt{210000/355}=114.6$. Since $\lambda =78.9<114.6$, inelastic buckling governs:

Step 5 — Utilization:

The column has adequate capacity with 50% utilization.

Common mistakes and checks

• Using the wrong axis: Always check buckling about the weak axis (minimum $r$); buckling occurs about the axis of least resistance unless bracing prevents it.
• Assuming K = 1.0 for all cases: Fixed-free columns (K = 2.0) have four times lower critical load than pinned-pinned; mis-classifying end conditions is the most common source of unconservative design.
• Ignoring slenderness limits: Codes limit $KL/r$ to 200 for main members. Exceeding this means the column is too slender for reliable performance regardless of calculated capacity.
• Forgetting combined loading: Columns in frames always have some bending moment from frame action or eccentric connections; use interaction equations (AISC H1 or EN 1993-1-1 Section 6.3.3) when moment is present.
• Applying Euler formula to stocky columns: Euler theory is unconservative for short columns that yield before buckling; always use code column curves that capture the inelastic transition.
• Neglecting initial imperfection: Real columns have out-of-straightness tolerances (L/1000 typical); this is already embedded in code curves but matters for FEM eigenvalue interpretation.

FAQ

What is the difference between Euler buckling and code column curves?

Euler buckling gives the theoretical elastic critical load for a perfect column. Code column curves (AISC Chapter E, EN 1993-1-1 Section 6.3) reduce this value to account for residual stresses, initial imperfections, and inelastic behavior. Always use code curves for design; use Euler as a theoretical upper bound.

How do I choose the correct imperfection curve in Eurocode 3?

EN 1993-1-1 Table 6.2 assigns curves a0 through d based on section type (hot-rolled H, welded box, etc.) and buckling axis. For example, a hot-rolled HEB section buckling about the weak axis typically uses curve b with imperfection factor $\alpha =0.34$.

Can the module handle combined axial and bending (beam-columns)?

The columns module focuses on pure axial buckling. For combined loading, use the combined-loading module or apply AISC H1/EN 1993-1-1 interaction equations manually with the axial capacity from this module.

What does the buckling mode shape tell me?

The mode shape shows the lateral deformation pattern at the critical load. A half-sine wave indicates first-mode flexural buckling (most common). Higher modes or unusual shapes may indicate the column is partially braced or has non-uniform properties.

When does torsional buckling govern over flexural buckling?

Torsional and flexural-torsional buckling govern for open sections (channels, angles, tees) and doubly-symmetric sections with very thin flanges. The current module addresses flexural buckling; check AISC E4 or EN 1993-1-1 Section 6.3.1.4 for torsional modes.

How sensitive is column capacity to effective length?

Very sensitive — capacity scales as $1/(KL{)}^2$. Reducing effective length by 50% (e.g., adding a mid-height brace) quadruples the Euler critical load. This makes bracing the most cost-effective way to increase column capacity.

What is the difference between AISC LRFD and ASD for columns?

LRFD (Load and Resistance Factor Design) uses $\varphi P_n$ with $\varphi =0.9$ and factored loads. ASD (Allowable Stress Design) uses $P_n/\Omega$ with $\Omega =1.67$ and service loads. Both give similar designs; LRFD is more rational for combined loading but ASD remains common in practice.

Use the PhyCalcPro calculator

Open the Column Buckling calculator

Purpose

Evaluate elastic stability of slender compression members using finite-element buckling analysis. Compares 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 gives $P_{\mathrm{cr}}={\pi }^{2}EI/L_{\mathrm{eff}}^2$. Real columns fail below this due to residual stresses, initial curvature, and material yield interaction. The FEM solver assembles elastic stiffness $\mathbf{K}$ and geometric stiffness ${\mathbf{K}}_g$ proportional to axial load, then solves the eigenvalue problem $(\mathbf{K}+\lambda {\mathbf{K}}_g)\mathbf{v}=0$ for buckling modes.

Governing equations

Numerical method

Linear buckling FEM: the column is meshed along its length. Elastic stiffness and geometric stiffness matrices are assembled for selected end conditions. The lowest positive eigenvalue yields critical load and buckling mode shape. Post-processing compares utilization to AISC 360 Chapter E and EN 1993-1-1 Section 6.3 curves.

Inputs

Outputs

• Critical load $P_{\mathrm{cr}}$ and buckling mode shape
• Slenderness ratio $\lambda$
• Euler safety factor $P_{\mathrm{cr}}/P$
• Code utilization per selected design standard
• Buckling curve classification

Design codes & checks

• Indicative: Euler buckling $P_{\mathrm{cr}}/P$
• US: AISC 360-22 Chapter E (flexural buckling)
• EU: EN 1993-1-1 Section 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.
• Uniform prismatic section along length.
• Linear elastic material model.
• Validated against Euler closed-form for standard end conditions.
• Does not replace full building-code member design with all interaction checks.

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 in compression.
4. Galambos, T. V., & Surovek, A. E. Structural Stability of Steel, 5th ed. Wiley.
5. Gere, J. M., & Goodno, B. J. Mechanics of Materials, 9th ed. Cengage.
6. Salmon, C. G., Johnson, J. E., & Malhas, F. A. Steel Structures: Design and Behavior, 5th ed. Pearson.
7. Ziemian, R. D. Guide to Stability Design Criteria for Metal Structures, 6th ed. Wiley.

Plate Bending Guide (plates)

How engineers analyze flat plates

Flat plates are two-dimensional structural elements that resist transverse loading through bending — appearing as machinery housings, pressure vessel covers, floor panels, electronic enclosures, and structural decks. Unlike beams that bend in one plane, plates develop bending moments in two orthogonal directions simultaneously, creating a biaxial stress state that requires careful analysis.

Kirchhoff-Love plate theory extends Euler-Bernoulli beam bending to two dimensions. The governing biharmonic equation $D{\nabla }^{4}w=p$ relates plate deflection $w(x,y)$ to transverse pressure $p$ through the flexural rigidity $D=E{t}^3/[12(1-{\nu }^2)]$. Edge boundary conditions — simply supported, clamped, or free — profoundly influence both the deflection magnitude and the distribution of bending moments.

For rectangular plates with all edges simply supported, the Navier double Fourier series provides an exact analytical solution. Mixed boundary conditions (e.g., two edges clamped, two free) require numerical methods. The PhyCalcPro plates module uses finite-element plate elements on a structured rectangular mesh to handle arbitrary edge condition combinations.

Edge conditions and their effects

Engineering workflow

1. Define plate geometry: length $a$, width $b$, thickness $t$.
2. Specify material: elastic modulus $E$ and Poisson's ratio $\nu$.
3. Set edge boundary conditions independently for all four edges.
4. Apply transverse loading: uniform pressure or concentrated point loads.
5. Choose mesh density (more segments = more accuracy near stress concentrations).
6. Run the plate FEM solver.
7. Review deflection contour: confirm maximum is within the allowable limit.
8. Check bending stresses: compare peak ${\sigma }_x$ and ${\sigma }_y$ against material allowable.
9. For pressure vessel applications, verify against ASME BPVC or EN 13445 flat plate rules.

Key quantities and formulas

Bending moments per unit width:

Surface bending stress:

Navier solution for SSSS plate under uniform pressure $p$:

where $\alpha$ depends on aspect ratio $a/b$ (e.g., $\alpha \approx 0.00406$ for a square plate).

Worked example

Problem: A square steel plate 500 mm x 500 mm, thickness 10 mm, all edges clamped, subjected to 0.1 MPa uniform pressure. $E=200$ GPa, $\nu =0.3$. Find maximum deflection and stress.

Step 1 — Flexural rigidity:

Step 2 — Maximum deflection (clamped square plate, Roark coefficient $\alpha =0.00126$):

Step 3 — Maximum bending stress (center, Roark coefficient $\beta =0.0513$):

Well below typical steel allowable (150+ MPa). The 10 mm plate is adequate.

Common mistakes and checks

• Violating thin-plate assumptions: Kirchhoff theory requires $t/a<1/20$. For thick plates, transverse shear deformation becomes significant and Mindlin-Reissner theory is needed.
• Confusing plate and membrane behavior: Large deflections (> 0.5t) engage membrane stretching that dramatically stiffens the plate. The linear solver will overestimate deflections in this regime.
• Ignoring Poisson coupling: Unlike beams, plates develop transverse moments due to Poisson's ratio. A plate bent about x also develops $M_y=\nu M_x$.
• Underestimating corner effects: Clamped plates develop stress concentrations at corners; ensure mesh is fine enough near boundaries.
• Applying wrong boundary conditions: A bolted plate edge behaves between simply supported and clamped depending on bolt spacing and flange stiffness.

FAQ

When is thin-plate theory valid?

When plate thickness is less than 1/20 of the shorter span dimension. Below this ratio, transverse shear deformation contributes less than 5% to deflection.

How does aspect ratio affect plate behavior?

As a plate becomes long and narrow ($a/b>3$), it approaches cylindrical bending where the center strip behaves like a beam spanning the shorter direction. One-way plate strip theory then gives adequate results.

Can I analyze plates with stiffeners?

Not directly — the module solves unstiffened flat plates. Model stiffened plates as equivalent orthotropic plates with modified rigidities, or analyze the plate panel between stiffeners separately.

What is the difference between this module and the circular plates module?

This module handles rectangular plates with per-edge boundary conditions. The circular plates module is specialized for axisymmetric round plates with radial symmetry, using different governing equations and solution methods.

How do I handle a plate with a central hole?

The current solver does not support plates with cutouts. For plates with holes, apply stress concentration factors from Roark's tables to the peak stress results, or use full 2D FEA software.

Use the PhyCalcPro calculator

Open the Plate Bending calculator

Purpose

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

Physics & theory

Kirchhoff-Love plate theory extends beam bending to two dimensions. Flexural rigidity $D=E{t}^3/[12(1-{\nu }^2)]$ and the biharmonic equation $D{\nabla }^{4}w=p$ govern out-of-plane deflection. Bending moments relate to curvature; maximum stress at the surface is $\sigma =6M/t^2$. Edge conditions strongly influence peak deflection and stress.

Governing equations

Numerical method

2D plate FEM on a structured rectangular mesh. Kirchhoff or Mindlin-Reissner 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 $w(x,y)$ and 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 rules (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 (deflection < 0.5t).
• Flat plate only; no stiffeners or large membrane stretching.
• Rectangular geometry only; no irregular shapes or cutouts.

Verification

• CI: verification benchmarks in src/data/verification/ where available
• Engineer sign-off: validation-master-checklist.md

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. EN 13445-3:2021. Unfired pressure vessels — Part 3: Design.

Combined Loading Guide (combined-loading)

How engineers analyze combined loading

Real engineering components rarely experience a single type of loading. A shaft carries both torque and bending; a bracket sustains axial tension plus moment from eccentricity; a machine frame member sees compression, shear, and torsion simultaneously. Combined loading analysis determines the equivalent stress state from all contributions and compares it against the material yield criterion.

The von Mises (distortion energy) criterion is the standard approach for ductile materials under multiaxial stress. It states that yielding begins when the distortion energy equals that at uniaxial yield: ${\sigma }_{\mathrm{vm}}=\sqrt{{\sigma }^2+3{\tau }^2}$ for plane stress with one normal and one shear component. This single equivalent stress allows engineers to compare a complex multiaxial state against simple tensile test data.

The PhyCalcPro combined-loading module evaluates all stress components for a rectangular cross-section — axial, bending, torsion, and direct shear — combines them via von Mises, and returns a safety factor and design status. This provides rapid screening without requiring full 3D FEA for common prismatic machine elements.

Loading types and superposition

Normal stresses superpose algebraically: $\sigma ={\sigma }_a+{\sigma }_b$. Shear stresses from torsion and transverse shear superpose at the critical point. The von Mises criterion then combines normal and shear.

Engineering workflow

1. Identify all external loads on the component: forces, moments, and torques.
2. Select the critical cross-section (typically where moment is maximum or section is smallest).
3. Define rectangular section dimensions: width $w$ and height $h$.
4. Compute section properties: $A=wh$, $I=w{h}^3/12$, $J\approx w{h}^3/3$ (rectangular approximation).
5. Calculate individual stress components from each load.
6. Superpose normal stresses (axial + bending) and shear stresses (torsion + direct shear).
7. Apply the von Mises criterion to get equivalent stress.
8. Compute safety factor: $SF={\sigma }_y/{\sigma }_{\mathrm{vm}}$.
9. Assess design status: safe (SF >= 2), warning (1.25-2), or critical (< 1.25).

Key quantities and formulas

Stress components:

Von Mises equivalent stress (plane stress):

Safety factor:

Section properties for rectangular cross-section:

Worked example

Problem: A rectangular steel bar (50 mm wide x 80 mm tall, ${\sigma }_y=350$ MPa) carries simultaneously: axial tension 120 kN, bending moment 8 kN-m, and torque 3 kN-m.

Step 1 — Section properties:

Step 2 — Stress components:

Step 3 — Von Mises stress:

Step 4 — Safety factor:

Design status: warning (1.25 < SF < 2.0). Adequate for static loading but marginal for cyclic applications.

Common mistakes and checks

• Forgetting to add axial and bending normal stresses: These act on the same face and must be algebraically summed before applying von Mises.
• Using Tresca instead of von Mises without noting the difference: Tresca (maximum shear stress) is more conservative by up to 15%. PhyCalcPro uses von Mises.
• Applying to brittle materials: Von Mises is valid for ductile materials. For cast iron, ceramics, or concrete, use Mohr-Coulomb or Rankine criteria instead.
• Ignoring stress concentration factors: The module computes nominal stress. At notches, holes, or fillets, multiply by the stress concentration factor $K_t$ from charts.
• Using the wrong J for non-circular sections: The torsion constant for rectangular sections is approximate; for thin-walled or open sections, torsional behavior differs significantly.
• Neglecting fatigue for cyclic loads: Static safety factor alone does not ensure fatigue life. Use modified Goodman or S-N curve approaches for repeated loading.

FAQ

Why use von Mises instead of maximum principal stress?

Von Mises (distortion energy) correlates better with experimental yield data for ductile metals. Maximum principal stress (Rankine) is appropriate for brittle materials. For steel and aluminum, von Mises is the standard.

Can I use this for circular cross-sections?

The module uses rectangular section formulas for $J$. For circular shafts, the standard $J=\pi d^4/32$ applies — you would need to input equivalent rectangular dimensions or use the result conceptually.

What safety factor should I target?

Depends on application:

Does this handle fatigue (alternating loads)?

No — the module computes static equivalent stress only. For fatigue, separate the mean and alternating stress components and apply a fatigue criterion (modified Goodman, Soderberg, or S-N curves).

When does combined loading analysis replace FEA?

For simple prismatic sections under known loads at a single critical section, this closed-form approach is exact. FEA is needed for complex geometries, stress concentrations, contact, or thermal gradients.

What is the design status threshold logic?

• Safe: SF >= 2.0 — adequate margin for most static applications
• Warning: 1.25 <= SF < 2.0 — acceptable only with well-defined loads
• Critical: SF < 1.25 — redesign required

Use the PhyCalcPro calculator

Open the Combined Loading calculator

Purpose

Evaluate combined axial, bending, torsion, and shear stresses in a rectangular cross-section. Computes von Mises equivalent stress, safety factor, and design status for quick screening of machine elements and structural members under multiaxial loading.

Physics & theory

Normal stresses from axial and bending superpose: $\sigma ={\sigma }_a+{\sigma }_b$. Shear from torsion adds ${\tau }_t=Tc/J$. For ductile materials, the von Mises criterion combines all components: ${\sigma }_{\mathrm{vm}}=\sqrt{{\sigma }^2+3{\tau }^2}$. Safety factor is yield strength divided by equivalent stress.

Governing equations

Numerical method

Closed-form evaluation: section properties are computed from 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 SF thresholds.

Inputs

Outputs

• Section properties $A$, $I_{xx}$, $J$
• Individual stress components (${\sigma }_a$, ${\sigma }_b$, ${\tau }_t$, ${\tau }_v$)
• Von Mises equivalent stress
• Safety factor
• Design status (safe/warning/critical)

Design codes & checks

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

Assumptions & limitations

• Solid rectangular section only; 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 (not parabolic distribution).
• No stress concentration factors applied; user must account for notches externally.

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. Cengage.
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. Dowling, N. E. Mechanical Behavior of Materials, 4th ed. Pearson.

Circular Plates Guide (circular-plates)

How engineers analyze circular plates

Circular plates appear throughout engineering: pressure vessel flat heads, flange blind covers, manhole closures, piston crowns, diaphragm sensors, and optical mirror substrates. Their axisymmetric geometry under uniform pressure produces radially symmetric deflection and stress fields, enabling elegant one-dimensional solutions that are both fast and highly accurate.

The governing equation for an axisymmetric circular plate under uniform pressure reduces from the full biharmonic to an ordinary differential equation in the radial coordinate $r$. Roark's classical tabulated coefficients provide immediate deflection and stress answers for standard boundary conditions — clamped or simply supported outer edges. These closed-form solutions serve as verification benchmarks for numerical methods.

The PhyCalcPro circular-plates module implements a dual approach: Roark closed-form coefficients for instant benchmark values, plus an axisymmetric finite-difference solver on a radial grid for mesh-controlled accuracy and visualization of the deflection profile. The FDM-vs-Roark error percentage is reported to confirm numerical convergence.

Boundary conditions and behavior

Key behavioral differences:

• Clamped plates develop fixed-edge moments that reduce center deflection but create high stress at the boundary.
• Simply supported plates allow free rotation at the edge, resulting in larger center deflection but more uniform stress distribution.
• Switching from clamped to simply supported can increase deflection by a factor of 4-5 for a given pressure.

Engineering workflow

1. Define plate geometry: outer radius $a$ and uniform thickness $t$.
2. Specify material: elastic modulus $E$ and Poisson's ratio $\nu$.
3. Select edge boundary condition: clamped or simply supported.
4. Apply uniform transverse pressure $p$ (positive = towards plate from one side).
5. Choose radial mesh segments (4-64; higher for convergence study).
6. Run the dual solver (Roark + FDM).
7. Compare FDM result against Roark benchmark; confirm error < 5%.
8. Check maximum deflection against allowable (e.g., L/300 of diameter).
9. Check maximum bending stress against material allowable or code limit.

Key quantities and formulas

Axisymmetric plate governing equation:

Flexural rigidity:

Clamped circular plate center deflection (Roark):

Simply supported center deflection:

Maximum bending stress:

where $\beta \approx 0.75$ (clamped edge) or $\beta \approx 1.24$ (SS, center) for $\nu =0.3$.

Worked example

Problem: A clamped circular steel plate of radius 200 mm and thickness 12 mm carries 0.5 MPa uniform pressure. $E=200$ GPa, $\nu =0.3$. Determine maximum deflection and stress.

Step 1 — Flexural rigidity:

Step 2 — Center deflection (clamped):

Step 3 — Maximum bending stress (at clamped edge):

Well below typical steel allowable. The plate design is adequate.

Step 4 — FDM verification: With 16 radial segments, FDM gives $w=0.393$ mm (error = 0.5% vs Roark).

Common mistakes and checks

• Using simply supported coefficients for a welded plate: A plate welded around its circumference is closer to clamped than simply supported. Using SS coefficients overestimates deflection conservatively but underestimates edge stress.
• Exceeding thin-plate limits: If $t/a>1/10$, thick-plate (Mindlin) effects become significant. The module assumes thin Kirchhoff theory.
• Large deflection regime: When $w_{\max }>0.5t$, membrane stretching stiffens the plate. Linear theory overestimates deflection in this regime.
• Ignoring thermal loads: Circular plates in hot environments (exhaust covers, boiler heads) develop thermal stresses not captured by pressure-only analysis.
• Low mesh density: Fewer than 8 radial segments can produce >5% error vs Roark. Always verify convergence.

FAQ

What is the FDM-Roark error percentage?

It measures how closely the finite-difference numerical solution matches Roark's exact closed-form result. Error below 2% confirms the mesh is adequately refined. Higher errors prompt increasing meshSegments.

Can I analyze annular plates (with a central hole)?

The current module solves solid circular plates. Annular plates have different boundary conditions at the inner radius and require modified Roark coefficients. This is a planned enhancement.

How do I model a bolted flange cover?

A bolted cover with a gasket is partially constrained — between clamped and simply supported. Analyze both extremes and use the more conservative result. The bolt circle radius defines the effective plate radius.

When should I use ASME UG-34 instead of this module?

ASME UG-34 provides mandatory rules for pressure vessel flat heads with specific attachment details and stress intensification factors. Use UG-34 for code-stamped vessels; use this module for preliminary sizing and non-code applications.

What Poisson's ratio should I use?

Steel: 0.3; aluminum: 0.33; copper: 0.34; glass: 0.22; concrete: 0.15-0.20. The value significantly affects the simply supported solution but has less impact on the clamped case.

How does plate thickness affect deflection?

Deflection scales as $1/t^3$ (through the rigidity $D$). Doubling thickness reduces deflection by a factor of 8 — making thickness the most powerful design lever for plate stiffness.

Use the PhyCalcPro calculator

Open the Circular Plates calculator

Purpose

Compute deflection and bending stress in solid circular plates under uniform transverse pressure with clamped or simply supported 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 exhibit radially symmetric deflection $w(r)$. Flexural rigidity $D=E{t}^3/[12(1-{\nu }^2)]$ scales resistance. Roark's coefficients give quick screening; the FDM solver provides convergence-verified numerical results. Clamped plates deflect ~5x less than simply supported.

Governing equations

Numerical method

Dual approach: (1) Roark closed-form coefficients for benchmark comparison; (2) axisymmetric Kirchhoff FDM on a radial line with configurable segments (4-64). Jacobi-style iteration enforces boundary conditions. FDM-vs-Roark error percentage is reported for convergence verification.

Inputs

Outputs

• Maximum deflection and bending stress
• Flexural rigidity $D$
• Roark benchmark values
• FDM-Roark error percentage
• Radial deflection profile

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 only; no point loads or thermal gradients.
• Linear elastic, small deflection ($w<0.5t$).

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., Table 11.2.
2. Timoshenko, S., & Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd ed. McGraw-Hill.
3. Ugural, A. C. Stresses in Plates and Shells, 4th ed. CRC Press.
4. ASME BPVC Section VIII, Division 1, UG-34.
5. EN 13445-3:2021. Unfired pressure vessels — Part 3: Design.

Cylindrical Shells Guide (shells)

How engineers design cylindrical shells

Cylindrical shells are the most common pressure-containing geometry in engineering: boilers, chemical reactors, storage tanks, pipelines, heat exchangers, and rocket casings are all cylindrical shells. Under internal pressure, the shell develops biaxial membrane stresses — hoop (circumferential) and axial (longitudinal) — that govern wall thickness selection.

Thin-shell membrane theory provides elegant closed-form results when $t/r<0.1$: hoop stress ${\sigma }_h=pr/t$ and axial stress ${\sigma }_a=pr/(2t)$ for closed-ended cylinders. The hoop stress is always twice the axial stress from pressure alone, making circumferential failure the governing mode. When external axial loads or bending moments are superimposed, von Mises equivalent stress determines the combined safety margin.

The PhyCalcPro shells module evaluates membrane stresses from internal pressure combined with external axial force and bending moment, then computes von Mises equivalent stress and safety factor. This provides rapid pressure vessel screening without full finite-element shell analysis.

Shell loading scenarios

Engineering workflow

1. Define shell geometry: mean radius $r$, wall thickness $t$, and length $L$.
2. Specify end condition: open or closed (closed adds axial membrane stress from pressure).
3. Apply internal pressure $p$.
4. Add external loads if present: axial force $F$ and/or bending moment $M$.
5. Specify material allowable stress or yield strength.
6. Run the membrane stress solver.
7. Review individual stress components: hoop, axial, bending.
8. Check von Mises equivalent stress against allowable.
9. Verify safety factor meets design requirements (typically SF >= 3.5 for ASME vessels).
10. For code compliance, cross-check with ASME BPVC or EN 13445 thickness formulas.

Key quantities and formulas

Hoop (circumferential) stress from internal pressure:

Axial (longitudinal) membrane stress for closed ends:

Additional axial stress from external loads:

Von Mises equivalent stress (biaxial, no shear):

where ${\sigma }_x={\sigma }_a+{\sigma }_{a,\mathrm{ext}}$ is the total axial stress.

ASME minimum thickness:

where $S$ = allowable stress and $E_j$ = joint efficiency.

Worked example

Problem: A closed-ended steel cylinder has mean radius 500 mm, wall thickness 8 mm, and operates at 2 MPa internal pressure. An external axial tension of 200 kN is also applied. Material yield = 250 MPa. Evaluate the safety factor.

Step 1 — Hoop stress:

Step 2 — Axial stress (pressure + external load):

Step 3 — Von Mises equivalent stress:

Step 4 — Safety factor:

Adequate for industrial service (SF > 2.0). For ASME code compliance, compare against code allowable stress (typically $S={\sigma }_u/3.5$ or ${\sigma }_y/1.5$).

Common mistakes and checks

• Confusing inner radius with mean radius: ASME formulas use inner radius; membrane theory uses mean radius $r=r_i+t/2$. Small difference for thin shells but matters for accuracy.
• Forgetting the 2:1 hoop-to-axial ratio: Hoop stress is always twice axial for pressure alone. Longitudinal welds (resisting hoop stress) are the critical joints.
• Ignoring open vs closed ends: Open cylinders have no axial membrane stress from pressure. Specifying wrong end condition changes the stress state significantly.
• Applying to thick shells: When $t/r>0.1$, Lame's thick-cylinder equations are required. Membrane theory underestimates inner-surface stress.
• Overlooking buckling under external pressure: External pressure causes shell buckling (a stability problem), not a membrane stress problem. This module does not handle external pressure collapse.
• Not accounting for corrosion allowance: Design thickness = calculated thickness + corrosion allowance (1-3 mm typical). Always add corrosion allowance before specifying fabrication thickness.

FAQ

When is thin-shell theory valid?

When $t/r<0.1$ (wall thickness less than 10% of mean radius). Most industrial pressure vessels satisfy this. For thick cylinders, use Lame's equations which account for radial stress variation through the wall.

How do I determine allowable stress for ASME vessels?

ASME BPVC Section II, Part D provides allowable stresses by material and temperature. For carbon steel at ambient: $S\approx \min ({\sigma }_u/3.5,{\sigma }_y/1.5)$. Joint efficiency $E_j$ accounts for weld quality (1.0 for full RT, 0.85 for spot RT).

Does this module handle nozzle reinforcement?

No — nozzle openings create stress concentrations and require separate reinforcement calculations per ASME UG-37 or EN 13445 Section 9. Use this module for the basic shell away from discontinuities.

What about thermal stresses in the shell?

Thermal gradients through the wall thickness create bending stresses not captured by membrane theory. For significant temperature differences ($\Delta T>50$ K through the wall), perform a thermal stress analysis separately.

Can I evaluate external pressure (vacuum) vessels?

External pressure causes buckling, not membrane yielding. The module does not perform buckling analysis. Use ASME UG-28 charts or EN 13445 Section 8 for external pressure design.

How does bending moment affect the shell?

Bending moment creates alternating tension and compression around the circumference, added to the membrane axial stress. The peak stress occurs at the extreme fiber: ${\sigma }_b=M/(\pi r^{2}t)$. This is included in the von Mises calculation.

Use the PhyCalcPro calculator

Open the Cylindrical Shells calculator

Purpose

Screen thin cylindrical shells under internal pressure, axial force, and bending moment using membrane theory plus von Mises combined stress. Provides rapid pressure vessel screening for wall thickness adequacy and safety factor evaluation.

Physics & theory

For a thin cylinder of mean radius $r$ and wall thickness $t$, hoop stress from internal pressure is ${\sigma }_h=pr/t$. Closed ends add axial membrane stress ${\sigma }_a=pr/(2t)$. External axial load and bending further modify the axial stress. Von Mises equivalent stress combines all components for yield screening.

Governing equations

Numerical method

Closed-form membrane stress evaluation with simplified beam bending deflection estimate via the shell engine solver. No iterative solution required — direct algebraic computation from inputs.

Inputs

Outputs

• Hoop stress, axial stress, bending stress components
• Von Mises equivalent stress
• Safety factor
• Indicative deflection estimate
• Design status message

Design codes & checks

• Indicative: Membrane + von Mises screening
• US: ASME BPVC Section VIII (pressure vessel context, screening)
• EU: EN 13445 (unfired pressure vessel context, screening)

Assumptions & limitations

• Thin-shell membrane theory ($t/r<0.1$).
• No nozzles, knuckles, or geometric discontinuities.
• No buckling under external pressure.
• No thermal stress or fatigue analysis.
• User must supply appropriate load combinations and safety factors.

Verification

• CI: verification benchmarks in src/data/verification/ where available
• 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., Ch. 13.
2. ASME BPVC Section VIII, Division 1 — pressure vessel shell design.
3. EN 13445-3:2021. Unfired pressure vessels — Part 3: Design.
4. Bednar, H. H. Pressure Vessel Design Handbook, 2nd ed. Van Nostrand Reinhold.
5. Moss, D. R., & Basic, M. Pressure Vessel Design Manual, 4th ed. Elsevier.

Power transmission

V-Belt Drive Design Guide (v-belts)

How engineers select V-belt drives

V-belt drives are the workhorse of low-to-medium power transmission. They are inexpensive, tolerant of misalignment, and provide shock absorption through belt compliance. The design process involves matching the belt cross-section and pulley diameters to the required power while ensuring adequate wrap angle on the smaller pulley and manageable belt tension.

The fundamental physics is Euler's capstan equation: the ratio of tight-side to slack-side tension depends exponentially on the friction coefficient and wrap angle. A V-belt gains friction advantage over a flat belt because the V-groove wedges the belt against both flanks, effectively multiplying the friction coefficient.

Types and configurations

Narrow-section belts (3V, 5V, 8V) provide higher power density for the same center distance. PhyCalcPro uses a generalized belt capacity model scaled by belt class factor.

Engineering workflow

1. Determine power and speed — Motor power, driver rpm, and driven rpm establish the speed ratio and required belt capacity.
2. Apply service factor — Multiply transmitted power by the service factor (1.0–1.6 depending on driver type, driven load, and daily hours) to get the design power.
3. Select belt cross-section — Use manufacturer horsepower charts or ISO 4184 tables to choose a section that can carry the design power at the driver speed.
4. Size pulleys — Driver and driven pitch diameters set the speed ratio $i=D_2/D_1$. Minimum pulley diameter is limited by belt bending fatigue.
5. Compute belt length and center distance — The standard open-drive formula gives the pitch length from pulley diameters and center distance.
6. Check wrap angle — Minimum wrap on the smaller pulley should be at least 120 degrees; below that, capacity derates significantly.
7. Estimate tensions — Euler's equation gives tight-side and slack-side tensions; pretension is the average.

Key quantities and formulas

Belt pitch length (open drive)

Euler's belt equation and transmitted power

where $F_1$ is tight-side tension, $F_2$ is slack-side tension, $\mu$ is the effective friction coefficient, and $\theta$ is the wrap angle in radians.

Speed ratio and wrap angle

Belt speed

Worked example

Problem: A 7.5 kW motor at 1750 rpm drives a centrifugal pump at 875 rpm through a B-section V-belt. Center distance 500 mm.

1. Speed ratio: $i=1750/875=2.0$. Choose $D_1=125$ mm, $D_2=250$ mm.
2. Belt speed: $v=\pi \times 125\times 1750/60000=11.45$ m/s.
3. Belt length: $L=2\times 500+\pi (125+250)/2+(250-125{)}^2/(4\times 500)=1000+589+7.8=1597$ mm. Select standard length 1600 mm (B63).
4. Wrap angle on driver: ${\theta }_1=\pi -2\arcsin (125/1000)=180-14.4=165.6$ degrees — adequate.
5. Service factor: 1.2 (motor to pump, 8–16 h/day). Design power: $7.5\times 1.2=9.0$ kW.
6. From B-section capacity at 1750 rpm: single belt rated at approximately 5.5 kW after wrap correction. Need 2 belts: $2\times 5.5=11.0$ kW capacity. Utilization: 82 % — acceptable.
7. Tight-side tension ($\mu =0.50$, $\theta =2.89$ rad): $F_1/F_2=e^{1.45}=4.26$. Net pull $F_1-F_2=9000/11.45=786$ N. Slack tension $F_2=241$ N, tight $F_1=1027$ N.

Common mistakes and checks

• Insufficient wrap angle — With large speed ratios the small pulley wrap can drop below 120 degrees, severely derating belt capacity. Use an idler or increase center distance.
• Ignoring belt speed limits — Most classical V-belts should not exceed 25–30 m/s; centrifugal tension grows with the square of speed and reduces effective pull.
• Wrong number of belts — Running a single belt above its rated capacity shortens life dramatically. Always size for the design power, not the nominal power.
• Neglecting pulley alignment — Misaligned pulleys cause uneven belt wear and premature failure. Angular misalignment should be less than 0.5 degrees.
• Forgetting service factor — An un-factored design may pass at rated power but fail under start-up, shock, or extended daily operation.

FAQ

How do I determine the service factor?

Service factors are tabulated by driver type (electric motor, IC engine) and driven machine (fan, pump, compressor, crusher). AGMA, Gates, and ISO 4184 all publish tables. Typical range is 1.0 (uniform load, motor) to 1.8 (heavy shock, engine).

What is the minimum recommended wrap angle?

120 degrees on the smaller pulley is the practical minimum. Below this, belt slip becomes likely and capacity is derated by the wrap correction factor $K_{\theta }$. For reliable drives, target at least 150 degrees.

Can I use the calculator for synchronous (timing) belts?

No. Timing belts use tooth engagement rather than friction and have different capacity models. Use the dedicated Timing Belt module for synchronous drives.

How does belt speed affect capacity?

Power capacity rises with belt speed up to about 20–25 m/s, then plateaus and eventually decreases as centrifugal tension consumes a larger fraction of the allowable belt tension. Optimal belt speed is typically 15–25 m/s.

Should I tension V-belts to a specific value?

Belt manufacturers recommend setting pretension so that the belt deflects approximately 1.5 mm per 100 mm of free span under a specified force. This corresponds to the average of tight-side and slack-side tensions at rated load.

Use the PhyCalcPro calculator

Size classical V-belt drives with length, wrap, and capacity screening in the V-Belt Drive Calculator.

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

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. The V-groove increases the effective friction by a factor $1/\sin (\beta /2)$, where $\beta$ is the groove angle (typically 34–40 degrees), making V-belts far more compact than flat belts for a given power.

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. ISO 4184:1992. Classical V-belts and pulleys.
3. Gates Corporation. Drive Design Manual.
4. Marks' Standard Handbook for Mechanical Engineers, 12th ed., McGraw-Hill.
5. Childs, P. R. N. Mechanical Design Engineering Handbook, 2nd ed., Ch. 14.

Timing Belt Drive (timing-belts)

How engineers size synchronous belt drives

Timing belts use toothed engagement between belt and pulley to transmit power without slip. Unlike V-belts, the positive mesh guarantees exact speed ratios, making timing belts the standard choice for positioning systems, packaging machines, and compact high-ratio drives. Sizing a timing belt means selecting a pitch family, choosing pulley tooth counts, computing pitch length, and verifying that the belt's rated power exceeds the service-adjusted demand.

Belt types and pitch families

Trapezoidal profiles (XL, L, H) are traditional; curvilinear profiles (HTD, GT) improve tooth load distribution and reduce ratcheting risk.

Engineering workflow

1. Determine required power, speed, and ratio.
2. Apply a service factor for driver type (motor, engine) and shock.
3. Select pitch family from manufacturer power-rating charts.
4. Choose driver and driven tooth counts to achieve the desired ratio.
5. Calculate pitch length from center distance and pulley diameters.
6. Round pitch length to the nearest whole-tooth increment.
7. Verify wrap angle on the small pulley (minimum 60 deg for 6 teeth in mesh).
8. Check belt speed against manufacturer limits (typically 40–80 m/s).
9. Confirm shaft radial loads for bearing selection.

Key quantities and formulas

Pitch diameter from tooth count:

Belt pitch length for a two-pulley open drive:

Belt linear speed and transmitted power:

Speed ratio (exact, no slip):

Worked example

A 5M HTD belt drives a packaging roller at 3:1 reduction. Driver pulley has 20 teeth, driven has 60 teeth, center distance is 250 mm.

• Pitch diameters: $D_1=5\times 20/\pi =31.83$ mm, $D_2=5\times 60/\pi =95.49$ mm.
• Pitch length: $L_p=2(250)+\pi (31.83+95.49)/2+(95.49-31.83{)}^2/(4\times 250)=704.2$ mm, rounded to 710 mm (142 teeth).
• Belt speed at 1750 rpm: $v=\pi \times 31.83\times 1750/60000=2.92$ m/s — well within limits.

Common mistakes and checks

• Ignoring service factors: a 1.0 kW motor can demand 1.6 kW design power under heavy shock.
• Too few teeth in mesh: wrap angle below 60 deg risks tooth jump under peak torque.
• Not rounding to whole pitches: fractional tooth lengths create misalignment.
• Exceeding belt speed limits: above manufacturer limit, centrifugal tension dominates.
• Neglecting shaft loads: belt tension produces radial bearing loads that must enter bearing selection.

FAQ

What minimum tooth count avoids excessive wear?

Most manufacturers recommend at least 14–16 teeth on the small pulley for HTD profiles; fewer teeth increase tooth stress and chordal speed variation.

Can timing belts replace roller chains?

Yes, for clean environments where lubrication is impractical. Timing belts are quieter and maintenance-free but have lower shock tolerance than chains.

How does belt width affect power capacity?

Rated power scales roughly linearly with belt width. Wider belts share tooth load across more area, directly raising capacity.

Do timing belts need tensioning?

Yes. Pretension prevents tooth skip under peak loads. Automatic tensioners or slotted motor bases maintain proper tension as belts wear.

When should I choose curvilinear over trapezoidal profiles?

Curvilinear (HTD/GT) profiles distribute load more evenly across tooth flanks and resist ratcheting at lower wrap angles — preferred for all new designs above 1 kW.

Use the PhyCalcPro calculator

Open the Timing Belt Drive calculator to enter tooth counts, pitch, center distance, and operating speed. The tool returns pitch length, belt speed, power utilization, belt tension, and shaft load components — ready for bearing and frame design.

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 $i=N_2/N_1$ is exact (no slip). Radial load on shafts combines belt tension from power transmission and centrifugal effects at high speed.

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.

Roller Chain Drive Design Guide (roller-chains)

How engineers select roller chain drives

Roller chain drives offer positive engagement (no slip), high efficiency (97–99 %), and the ability to transmit high torque in compact space. They are preferred over belt drives when slip-free synchronization is required, when the environment is too hot or oily for belts, or when the drive must carry very high loads at moderate speeds.

Design starts from the required power and speed, then selects a chain pitch and sprocket tooth counts that keep chain tension within the catalog rating while providing adequate service life — typically 15,000 hours for industrial duty.

Types and configurations