Verified calculation capability

Hollow circular plastic modulus Calculation Capability

Plastic modulus and shape factor of a hollow circle. The elastic modulus comes from the hollow-circular-section engine.

Mechanical · Statics Capability ID mechanical.statics.hollow_circular_plastic_modulus version 1.0.0 UNPUBLISHED

Model

Plastic modulus

Plastic modulus: Z = (4/3)(R³−r³)
Shape factor: f = Z/S

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`R`	Outer radius	—	input / output
`r`	Inner radius, less than R	—	input / output
`area`	—	—	output
`S`	—	—	output
`Z`	—	—	output
`f`	—	—	output
`model`	—	—	output
`solver`	—	—	output

Assumptions / Validity

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

Assumptions

Hollow circle about a centroidal diameter
Z = (4/3)(R³−r³)
Shape factor Z/S
Not a solid circle, a rectangle, or a plastic moment

Example

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

R=2 r=1 → Z=28/3, S=15π/8

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: 4/4 calculation tests · golden 2/2 · boundary 2/2 · 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Production attestation: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (8 capabilities; 189 remain CURRENT) @ 2026-09-23T21:48:15.717Z
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/hollow-circular-plastic-modulus' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"R":2,"r":1},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "R": 2,
    "r": 1
  },
  "calculation_version": "1.0.0"
}

POST /api/v1/calc/hollow-circular-plastic-modulus. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: hollow-circular-plastic-modulus
Engine: hollow-circular-plastic-modulus
REST: POST /api/v1/calc/hollow-circular-plastic-modulus
MCP: execute_calculation with capability_id: "mechanical.statics.hollow_circular_plastic_modulus"
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, Mechanics of Materials — Plastic deformation — tubular section

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.hollow_circular_plastic_modulus"},"_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.hollow_circular_plastic_modulus","inputs":{"R":2,"r":1},"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.