HomeCalculatorsMechanicalStrength of MaterialsPlastic Moment Calculator
Mechanical calculator

Plastic Moment Calculator

Fully plastic moment from a yield stress and a plastic modulus. One centroidal axis. 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 · Mp = σ_y Z on one centroidal axis; + O3 mpmath tabulated Mp.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Mp = σ_y Z on one centroidal axis; + O3 mpmath tabulated Mp.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (unit, alias, Zx, awkward, invalid-domain) · Matrix
Deferred
Not elastic S, not transverse shear, and not beam deflection. Does not rebuild the section.
Numerical scope
O2: Mp = σ_y Z vs a separate-module identity (≤2 ULP). Not elastic S, not transverse shear, and not beam deflection. ≤2 ULP vs O3 applies only to the published tabulated plastic-moment vectors. Not elastic S.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Fully plastic moment on one centroidal axis from a yield stress and a plastic modulus.
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 · Mp = σ_y Z on one centroidal axis; + O3 mpmath tabulated Mp.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-27.o2-o3
Verification revision
2026-09-27.o2-o3 · 1/1 property · digest e75cf6bcabb6
Legacy regression
5/5 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status
STALE · attestation snapshot core a82456dbd32e ≠ live cvp_core_sha256 e75cf6bcabb6 (2026-09-27.o2-o3)
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 3/3 CVP boundary · 5/5 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Mechanics of Materials — Plastic deformation — plastic moment
  • Gere and Goodno, Mechanics of Materials — Plastic deformation — plastic moment
Sources
Evidence
2 legacy golden · 3 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

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

Formulas

Core equations used by this calculator.

Plastic momentMp = σ_y Z
iBring Z from a plastic-modulus page. This page does not rebuild a rectangle, a circle, or a hollow circle.

How to use

1

Enter the yield stress and the plastic modulus

σ_y must be positive. Z must be positive. Both use the same force and length units.

2

Read the moment

Mp = σ_y Z. The moment has the unit of stress times length cubed.

Example calculations

Common configurations with formula and result.

ϟ

Rectangle Zx with σy = 250

σ_y = 250, Z = 18

Mp = 250 × 18
Mp = 4500
ϟ

Solid circle of radius 2

σ_y = 250, Z = 32/3

Mp = 250 × 32/3
Mp = 8000/3

Plastic Moment calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
On one centroidal axis, the fully plastic moment is Mp = σ_y Z. Z is the plastic modulus. σ_y is the yield stress in the same force and length units.
What it calculates
Fully plastic moment on one centroidal axis from a yield stress and a plastic modulus.
Inputs
  • sigma_y
  • Z
Outputs
  • Mp
  • sigma_y
  • Z
Formula
Mp = σ_y Z.
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • Bring Z from a plastic-modulus page. This page does not rebuild a rectangle, a circle, or a hollow circle.
Units
  • σ_y and Z set the moment unit. Mp has the unit of σ_y × Z.
Boundary conditions
  • missing σ_y or Z → MISSING_REQUIRED_INPUT
  • σ_y ≤ 0 or Z ≤ 0 → VALUE_MUST_BE_POSITIVE
Example
sigma_y=250 Z=18 → Mp=4500
Validation cases

3 published on this page · 5/5 tests · Production surface contract 6/6 · View evidence

  • sigma_y=250 Z=18 → Mp=4500
  • sigma_y=250 Z=32/3 → Mp=8000/3
  • sigma_y=0 Z=18 → VALUE_MUST_BE_POSITIVE
Sources
  • Hibbeler, Mechanics of Materials — Plastic deformation — plastic moment
    Supports: The fully plastic moment on a centroidal axis is the yield stress times the plastic modulus of that axis.
  • Gere and Goodno, Mechanics of Materials — Plastic deformation — plastic moment
    Supports: Supports the page formula: Mp = σ_y Z
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the fully plastic moment on one centroidal axis.

Supported and not supported

Supported: Mp = σ_y Z. Z is typed in. Aliases are Fy, yield_stress, yield, Zx, and plastic_modulus.

Not supported: rebuilding a rectangle, a circle, or a hollow circle, and transverse shear. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.plastic_moment · tool id: plastic-moment · pin 1.0.0.

{ "sigma_y": 250, "Z": 18 }

Share the link with sigma_y and Z. The result is not written into the link.

Other calculators in this family: Bending Stress Calculator, Circular Plastic Modulus Calculator, Circular Shear Calculator, Hollow Circular Plastic Modulus Calculator, Hollow Circular Shear Calculator, Rectangular Plastic Modulus Calculator, Rectangular Shear Calculator, Transverse Shear Calculator . Explore all Strength of Materials.

→ Bending Stress Calculator Elastic bending stress from a moment and a section modulus. S can also be formed from Ix and c. Runs locally. › → Circular Plastic Modulus Calculator Plastic modulus and shape factor of a solid circle. The elastic modulus comes from the circular-section engine. Runs locally. › → Circular Shear Calculator Neutral-axis shear stress of one solid circle. The area moment comes from the circular-section engine. Runs locally. › → 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. › → Hollow Circular Shear Calculator Neutral-axis shear stress of one hollow circle. The area moment comes from the hollow-circular-section engine. Runs locally. › → Rectangular Plastic Modulus Calculator Plastic moduli and shape factor of a solid rectangle. Elastic Sx and Sy come from the rectangular-section engine. Runs locally. › → Rectangular Shear Calculator Neutral-axis shear stress of one solid rectangle. The area moment comes from the rectangular-section engine. Runs locally. › → Transverse Shear Calculator Shear stress at one line from the shear force, the first moment, the area moment, and the width. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Does this rebuild the section?

No. Width, height, and radii stay on the plastic-modulus pages. Bring Z here.

Which axis is this?

One centroidal axis. For a rectangle, pass Zx or Zy from the rectangular plastic-modulus page.

Is this transverse shear?

No. Transverse shear is its own page.