Physics calculator

Section Modulus Calculator

Q: 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.

Q: Does this build a rectangle from width and height?

No. Enter the centroidal Ix and Iy you already have. For a rectangle, Īx = b h³/12 and the extreme-fiber distance is h/2, so Sx = b h²/6.

Q: 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.

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 Engine tested · Source checked · 6/6 tests · Production surface contract 6/6 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 6/6 tests · Production surface contract 6/6 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: Hibbeler, Engineering Mechanics: Statics — Moments of inertia
Sources
Evidence: 2 golden · 4 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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain CURRENT) @ 2026-09-23T10:40:38.140Z
Semantic contract: PASS

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

Enter the parts

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

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.

Read Sx and Sy

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

Example calculations

Common configurations with formula and result.

Both axes

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

Sx = 16/2, Sy = 4/1

Sx = 8, Sy = 4

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

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

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
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, Principal section modulus, Principal moments .

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Frequently asked questions

Key distinctions behind the calculation.

Does this compute bending stress?