Hollow Circular Plastic Modulus Calculator
Plastic modulus of a hollow circle from the outer and inner radii. The elastic modulus comes from the hollow-circular-section engine. Runs locally.
Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance
- Input interpretation
- Enter values to calculate.
- Result
- —
- Assurance
- Engineering
- Declared partition coverage
- PASS · 2/2 declared partitions (domain, invalid-domain) · Matrix
- Known limitations
- Physics L1 gap capability
- Core CVP does not include live graph, viewport, or pointer interaction.
- Model
- Plastic modulus and shape factor of one hollow circle about a centroidal diameter.
- Scope
- Calculation runs locally in the browser; values are not uploaded.
- Verification
- Engine tested · Source checked · v1.0.0 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
- Versions
- Calculation 1.0.0 · CVP protocol 1.0.0-proposed
- CVP identity
- 0/0 property · digest 33a80edfbee0
- Legacy regression
- 4/4 tests · Production surface contract 6/6
- Trust layers
- Verification VERIFIED · Production STALE · overall VERIFIED_STALE
- CVP status
- STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
- Reference
- O1 model · O2 expected_values · O2 numerical_behavior
- Interfaces
- PASS · UI (SSR) / REST / MCP
- Supplemental domain review
- Not performed
- Named expert review
- Not performed
- CVP suite
- 1/1 golden · 2/2 CVP boundary · 2/2 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
- Sources
- Hibbeler, Mechanics of Materials
- Gere and Goodno, Mechanics of Materials
- Evidence
- 2 legacy golden · 2 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 2/2 invalid · Artifact integrity PASS
- Semantic contract
- PASS
Full verification
Formulas
Core equations used by this calculator.
How to use
Enter the outer and inner radii
R and r are positive, and r is less than R. They set the length unit.
Read Z and the shape factor
Z is the first moment of the whole area about a centroidal diameter. f is Z divided by the elastic modulus at the outer fiber.
Example calculations
Common configurations with formula and result.
Outer 2, inner 1
R = 2, r = 1
Outer 4, inner 2
R = 4, r = 2
Hollow Circular Plastic Modulus calculator specification
Version 1.0.0 · Engine tested
- Engine tested 4/4 tests · Production surface contract 6/6
- Named expert review Not performed
- Calculation version 1.0.0
- Definition
- For a hollow circle, the plastic modulus about a centroidal diameter is Z = (4/3)(R³−r³). The elastic modulus comes from the hollow-circular-section engine.
- What it calculates
- Plastic modulus and shape factor of one hollow circle about a centroidal diameter.
- Inputs
- R
- r
- Outputs
- Z
- S
- f
- area
- Formula
Z = (4/3)(R³−r³), S = π(R⁴−r⁴)/(4R), f = Z/S- Assumptions
- Calculation runs locally in the browser; values are not uploaded.
- Keep units consistent with the labels on each field.
- S = π(R⁴−r⁴)/(4R) is not recomputed here. A solid circle is the circular-plastic-modulus page. The plastic moment is its own page.
- Units
- R and r set the length unit. Z and S are that unit cubed. The shape factor is dimensionless.
- Boundary conditions
- missing R or r → MISSING_REQUIRED_INPUT
- R ≤ 0 or r ≤ 0 → VALUE_MUST_BE_POSITIVE
- r ≥ R → INVALID_INPUT
- Example
- R=2 r=1 → Z=28/3, S=15π/8
- Validation cases
3 published on this page · 4/4 tests · Production surface contract 6/6 · View evidence
- R=2 r=1 → Z=28/3, S=15π/8, area=3π
- R=4 r=2 → Z=224/3, S=15π, area=12π
- r=R → INVALID_INPUT
- Sources
- Hibbeler, Mechanics of Materials — Plastic deformation — tubular sectionSupports: The plastic modulus of a hollow circle about a diameter is (4/3)(R³−r³).
- Gere and Goodno, Mechanics of Materials — Plastic deformation — tubular sectionSupports: Supports the page formula: Z = (4/3)(R³−r³)
- Hibbeler, Mechanics of Materials — Plastic deformation — tubular section
- Calculation version
- 1.0.0
Background
Interpretation and common distinctions.
Find the plastic modulus and shape factor of a hollow circle.
Supported and not supported
Supported: outer radius R and inner radius r, with r < R. Z = (4/3)(R³−r³). S comes from the hollow-circular-section engine. f = Z/S.
Not supported: a solid circle, a rectangle, and the plastic moment. /calc/mechanical is not open.
Agent / API notes
Capability id: mechanical.statics.hollow_circular_plastic_modulus · tool id: hollow-circular-plastic-modulus · pin 1.0.0.
{ "R": 2, "r": 1 }
Share the link with R and r. The result is not written into the link.
Related tools
Other calculators in this family: Bending Stress Calculator, Circular Plastic Modulus Calculator, Circular Shear Calculator, Hollow Circular Shear Calculator, Plastic Moment Calculator, Rectangular Plastic Modulus Calculator, Rectangular Shear Calculator, Transverse Shear Calculator . Explore all Strength of Materials.
Frequently asked questions
Key distinctions behind the calculation.
Is this a solid circle?
A solid circle is the circular-plastic-modulus page. This page keeps a hole.
Does a thin tube approach 4/π?
The shape factor falls from the solid-circle value toward 4/π as the wall gets thin. This page reports Z/S for the radii you enter.
Does this return the plastic moment?
No. It returns Z and the shape factor. The plastic moment is its own page.