Mechanical calculator

Circular Plastic Modulus Calculator

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

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

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 solid 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 4f05a87f961f
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 — ui-ssr is query-result HTML, not a live browser session.
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 — Plastic deformation — circular section
Gere and Goodno, Mechanics of Materials — Plastic deformation — circular section
Sources
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

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Plastic modulusZ = 4 r³/3

Shape factorf = 16/(3π)

iS = π r³/4 is not recomputed here. A rectangle is the rectangular-plastic-modulus page. A hollow circle is its own page.

How to use

Enter the radius

r is positive. It sets the length unit.

Read Z and the shape factor

Z is about a centroidal diameter. f is 16/(3π) for every solid circle.

Example calculations

Common configurations with formula and result.

Radius 2

r = 2

Z = 32/3, S = 2π, f = 16/(3π)

Z = 10.666667, f = 1.697653

Radius 1

r = 1

Z = 4/3, S = π/4, f = 16/(3π)

Z = 1.333333, f = 1.697653

Circular Plastic Modulus calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 4/4 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: For a solid circle, the plastic modulus about a centroidal diameter is Z = 4 r³/3. The elastic modulus comes from the circular-section engine. The shape factor is 16/(3π).
What it calculates: Plastic modulus and shape factor of one solid circle about a centroidal diameter.
Inputs: r
Outputs: Z
S
f
area
Formula: Z = 4 r³/3, S = π r³/4, f = 16/(3π)
Assumptions: Calculation runs locally in the browser; values are not uploaded.
Keep units consistent with the labels on each field.
S = π r³/4 is not recomputed here. A rectangle is the rectangular-plastic-modulus page. A hollow circle is its own page.
Units: r sets the length unit. Z and S are that unit cubed. The shape factor is dimensionless.
Boundary conditions: missing r → MISSING_REQUIRED_INPUT
r ≤ 0 → VALUE_MUST_BE_POSITIVE
Example: r=2 → Z=32/3, S=2π, f=16/(3π)
Validation cases: 3 published on this page · 4/4 tests · Production surface contract 6/6 · View evidence
r=2 → Z=32/3, S=2π, f=16/(3π)
r=1 → Z=4/3, S=π/4, f=16/(3π)
r=0 → VALUE_MUST_BE_POSITIVE
Sources: Hibbeler, Mechanics of Materials — Plastic deformation — circular section
Supports: The plastic modulus of a solid circle about a diameter is 4 r³/3, and the shape factor is 16/(3π).
Gere and Goodno, Mechanics of Materials — Plastic deformation — circular section
Supports: Supports the page formula: Z = 4 r³/3
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the plastic modulus and shape factor of a solid circle.

Supported and not supported

Supported: one solid circle. Z = 4 r³/3 about a centroidal diameter. S comes from the circular-section engine. f = 16/(3π).

Not supported: a rectangle, a hollow circle, and the plastic moment. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.circular_plastic_modulus · tool id: circular-plastic-modulus · pin 1.0.0.

{ "r": 2 }

Share the link with r. The result is not written into the link.

Other calculators in this family: Bending Stress Calculator, Circular Shear Calculator, Hollow Circular Plastic Modulus 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 the shape factor 3/2?

No. A solid circle is 16/(3π), about 1.698. A rectangle is 3/2.

Does this include a hole?

No. A hollow circle is its own page.

Where does S come from?

From the circular-section engine. This page does not keep a second copy of π r³/4.