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Mechanical · Statics Capability ID mechanical.statics.inclined_axis_moment version 1.0.0 UNPUBLISHED

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

Review-policy: engine tested — automated tests, sources, version lock, and evidence published. Supplemental to CVP; not Domain Expert Review. See CVP and Review policy.

Tier: Engine tested — catalog token engine_tested
Calculation version: 1.0.0
Evidence tests: 6/6 calculation tests · golden 3/3 · boundary 3/3 · property 0/0
Engine tested: Yes
Source checked: Yes
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)

Production attestation: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain CURRENT) @ 2026-09-23T10:40:38.140Z
Semantic contract: PASS
Evidence attested at: 2026-09-23T02:27:54.588Z
Last production check: 2026-09-23T02:27:54.588Z
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

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.