HomeCalculatorsMechanicalSection PropertiesSection Modulus Calculator
Mechanical calculator

Section Modulus Calculator

Elastic section modulus of a composite plane area about its centroid. Sx = Ix/cy and Sy = Iy/cx. Ix and Iy come from the area-moment engine. Runs locally. Not bending stress and not beam deflection.

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
Elastic section modulus of a composite plane area about its centroidal x and y axes.
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 e32986c1101c
Legacy regression
6/6 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 · 4/4 CVP boundary · 3/3 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia
  • Gere and Goodno, Mechanics of Materials — Moments of inertia
Sources
Evidence
2 legacy golden · 4 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 3/3 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.

About xSx = Ix / cy
About ySy = Iy / cx
iIx and Iy are shifted to the composite centroid by the area-moment engine. This page does not build a rectangle from b and h, does not return σ = M/S, and does not use the principal axes.

How to use

1

Enter the parts

Each row is centroidal Ix, Iy, A, x, y. A negative area is a hole.

2

Enter the fiber distances

cy is the distance from the centroid to the extreme fiber for Sx. cx is the same distance for Sy. Enter one or both.

3

Read Sx and Sy

A blank distance omits that modulus. Both distances must be positive.

Example calculations

Common configurations with formula and result.

ϟ

Both axes

Ix = 16, Iy = 4, cy = 2, cx = 1

Sx = 16/2, Sy = 4/1
Sx = 8, Sy = 4
ϟ

Sx only

Two unit areas at y = 0 and y = 2, cy = 1

Ix = 4
Sx = 4

Section Modulus calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Sx = Ix/cy and Sy = Iy/cx. Ix and Iy are the centroidal second moments of the composite area. cx and cy are distances from that centroid to the extreme fiber.
What it calculates
Elastic section modulus of a composite plane area about its centroidal x and y axes.
Inputs
  • parts
  • cx?
  • cy?
Outputs
  • Sx
  • Sy
  • Ix
  • Iy
  • x_bar
  • y_bar
  • area
Formula
Sx = Ix/cy, Sy = Iy/cx
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • Ix and Iy are shifted to the composite centroid by the area-moment engine. This page does not build a rectangle from b and h, does not return σ = M/S, and does not use the principal axes.
Units
  • Keep second moment and fiber distance in one consistent unit system. S has dimension length³.
Boundary conditions
  • no parts → MISSING_REQUIRED_INPUT
  • neither cx nor cy → MISSING_REQUIRED_INPUT
  • cx or cy ≤ 0 → VALUE_MUST_BE_POSITIVE
  • Ix < 0 when cy is set, or Iy < 0 when cx is set → INVALID_INPUT
Example
parts=16,4,4,0,0 cx=1 cy=2 → Sx=8, Sy=4
Validation cases

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

  • parts=16,4,4,0,0 cx=1 cy=2 → Sx=8, Sy=4, Ix=16, Iy=4
  • parts=1,0,1,0,0; 1,0,1,0,2 cy=1 → Sx=4, Ix=4
  • parts=16,4,4,0,0 cy=0 → VALUE_MUST_BE_POSITIVE
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia
    Supports: Elastic section modulus S = I/c, with c the distance from the centroid to the extreme fiber
  • Gere and Goodno, Mechanics of Materials — Moments of inertia
    Supports: Supports the page formula: Sx = Ix / cy
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the elastic section modulus of a composite plane area about its centroid.

Supported and not supported

Supported: Sx = Ix/cy and Sy = Iy/cx after the area-moment engine shifts each part to the composite centroid.

Not supported: bending stress, plastic modulus, a rectangle built from b and h, modulus about the principal axes, and beam deflection. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.section_modulus · tool id: section-modulus · pin 1.0.0.

{ "parts": "16,4,4,0,0", "cx": 1, "cy": 2 }

Share the link with parts, cx, and cy. The result is not written into the link.

Other calculators in this family: Area Moment Calculator, Circular Section Calculator, Hollow Circular Section Calculator, Inclined-Axis Moment Calculator, Plane Centroid Calculator, Principal Moment Calculator, Principal Section Modulus Calculator, Rectangular Section Calculator . Explore all Section Properties.

→ Area Moment Calculator Second moment of composite plane areas about the composite centroid. Supply each part's centroidal Ix and Iy. Runs locally. Not a section modulus and not a beam calculator. › → Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid circle from its radius. Runs locally. › → Hollow Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a hollow circle from the outer and inner radii. Runs locally. › → Inclined-Axis Moment Calculator Centroidal area moments about axes rotated by a given angle. Ix, Iy, and Ixy come from the principal-moment engine. Runs locally. Not the principal values, and not a Mohr circle. › → Plane Centroid Calculator Centroid of composite plane areas from each part's area and centroid. Negative area is a hole. Runs locally. Not a moment of inertia and not a beam calculator. › → Principal Moment Calculator Product of inertia and principal moments of a composite plane area. Ix and Iy reuse the area-moment engine. Runs locally. Not a section modulus and not a beam calculator. › → Principal Section Modulus Calculator Elastic section modulus about the principal axes. S1 = I1/c1 and S2 = I2/c2. I1, I2, and θ come from the principal-moment engine. Runs locally. Not Sx and Sy, and not bending stress. › → Rectangular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid rectangle from its width and height. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Does this compute bending stress?

No. It returns the elastic section modulus S = I/c. Bending stress σ = M/S, plastic modulus, and material allowables are out of scope.

Does this build a rectangle from width and height?

A solid rectangle from width and height is the rectangular-section page. This page takes centroidal Ix and Iy you already have. For that rectangle, Sx = b h²/6.

Is this the principal-axis modulus?

No. Sx and Sy use Ix and Iy about axes parallel to x and y. Principal-axis modulus is on the principal-section-modulus page.