HomeCalculatorsPhysicsHollow Circular Plastic Modulus Calculator
Physics calculator

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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Source checked · 4/4 tests · Production surface contract 6/6 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
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 · 4/4 tests · Production surface contract 6/6 · Source checked · v1.0.0
Named expert review
Optional · Not performed
Sources
  • Hibbeler, Mechanics of Materials — Plastic deformation — tubular section
Sources
Evidence
2 golden · 2 boundary · Production surface contract 6/6 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (8 capabilities; 189 remain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

Formulas

Core equations used by this calculator.

Plastic modulusZ = (4/3)(R³−r³)
Shape factorf = Z/S
iS = π(R⁴−r⁴)/(4R) is not recomputed here. A solid circle is the circular-plastic-modulus page. The plastic moment is its own page.

How to use

1

Enter the outer and inner radii

R and r are positive, and r is less than R. They set the length unit.

2

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.

ϟ

Outer 2, inner 1

R = 2, r = 1

Z = 28/3, S = 15π/8
Z = 9.333333, f = 1.584476
ϟ

Outer 4, inner 2

R = 4, r = 2

Z = 224/3, S = 15π
Z = 74.666667, f = 1.584476

Hollow Circular Plastic Modulus calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

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 section
    Supports: The plastic modulus of a hollow circle about a diameter is (4/3)(R³−r³).
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.

Other calculators in this family: Hollow circular section, Circular plastic modulus, Bending stress .

R Hollow circular section › Z Circular plastic modulus › σ Bending stress ›
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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.