CVP verified calculation capability

Inclined-axis moments Calculation Capability

Centroidal moments about axes rotated by a given angle. Ix, Iy, and Ixy come from the principal-moment engine. Not the principal values, and not a Mohr circle.

Mechanical · Statics Capability ID mechanical.statics.inclined_axis_moment version 1.0.0 CVP VERIFIED CURRENT Engineering

Model

Ix′

Ix′: Ix′ = (Ix + Iy)/2 + (Ix − Iy)/2 · cos 2θ − Ixy · sin 2θ
Iy′: Iy′ = (Ix + Iy)/2 − (Ix − Iy)/2 · cos 2θ + Ixy · sin 2θ
Ixy′: Ixy′ = (Ix − Iy)/2 · sin 2θ + Ixy · cos 2θ

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`parts`	Rows Ix,Iy,Ixy,A,x,y	—	input
`theta_deg`	Rotation of the primed axes, degrees, counterclockwise from +x	—	input / output
`area`	—	—	output
`x`	—	bar	output
`y`	—	bar	output
`Ix`	—	—	output
`Iy`	—	—	output
`Ixy`	—	—	output
`Ix_prime`	—	—	output
`Iy_prime`	—	—	output
`Ixy_prime`	—	—	output
`n`	—	—	output
`model`	—	—	output
`solver`	—	—	output

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

θ is counterclockwise from +x and is an input, not a solved principal angle
Ix′ + Iy′ = Ix + Iy about the composite centroid
Negative area is a hole
Not I1 or I2, not a Mohr-circle drawing, not section modulus, and not beam deflection

Example

One numerical check. The calculator page holds worked steps and FAQ.

parts=4,1,0,2,0,0 theta_deg=0 → Ix′=4, Iy′=1, Ixy′=0

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
Assurance: Engineering
Declared partition coverage: PASS · 2/2 declared partitions (domain, invalid-domain)
Calculation version: 1.0.0
Verification: 6/6 calculation tests · 3/3 legacy golden · 3/3 legacy boundary
Production surface contract: 6/6
Versions: Calculation 1.0.0 · CVP protocol 1.0.0-proposed
CVP identity: 0/0 property · digest 743e1a8a564c
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session.
CVP suite: 1/1 golden · 3/3 CVP boundary · 3/3 invalid · 1/1 cross-interface · 6/6 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.0 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 6/6.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/inclined-axis-moment' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"parts":"4,1,0,2,0,0","theta_deg":0},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "parts": "4,1,0,2,0,0",
    "theta_deg": 0
  },
  "calculation_version": "1.0.0"
}

POST /api/v1/calc/inclined-axis-moment. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: inclined-axis-moment
Engine: inclined-axis-moment
REST: POST /api/v1/calc/inclined-axis-moment
MCP: execute_calculation with capability_id: "mechanical.statics.inclined_axis_moment"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.0)
Versioned output schema: Immutable pin (1.0.0)

References

Human-readable sources plus CVP reference evidence classes.

Sources

Hibbeler, Engineering Mechanics: Statics — Moments of inertia for an area about inclined axes
Gere and Goodno, Mechanics of Materials — Moments of inertia for an area about inclined axes

Reference evidence classes: O1 model · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"mechanical.statics.inclined_axis_moment"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"mechanical.statics.inclined_axis_moment","inputs":{"parts":"4,1,0,2,0,0","theta_deg":0},"calculation_version":"1.0.0"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.0.0) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.