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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.

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 principal 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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain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.

I1 axisS1 = I1 / c1
I2 axisS2 = I2 / c2
iI1, I2, and θ come from the principal-moment engine. This page does not return Sx or Sy, does not draw a Mohr circle, and does not compute σ = M/S.

How to use

1

Enter the parts

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

2

Enter the fiber distances

c1 is measured along the I1 axis. c2 is measured along the I2 axis. Enter one or both.

3

Read S1 and S2

θ is the I1 axis, counterclockwise from +x. A blank distance omits that modulus.

Example calculations

Common configurations with formula and result.

ϟ

Axes aligned with x and y

Ix = 4, Iy = 1, c1 = 2, c2 = 1

S1 = 4/2, S2 = 1/1
S1 = 2, S2 = 1, θ = 0°
ϟ

S1 only

Two unit areas on the diagonal, c1 = 1

I1 = 4, θ = −45°
S1 = 4

Principal Section Modulus calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
S1 = I1/c1 and S2 = I2/c2. I1 and I2 are the principal moments. c1 and c2 are distances from the centroid to the extreme fiber along those axes.
What it calculates
Elastic section modulus of a composite plane area about its principal axes.
Inputs
  • parts
  • c1?
  • c2?
Outputs
  • S1
  • S2
  • I1
  • I2
  • theta_deg
  • x_bar
  • y_bar
  • area
Formula
S1 = I1/c1, S2 = I2/c2
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • I1, I2, and θ come from the principal-moment engine. This page does not return Sx or Sy, does not draw a Mohr circle, and does not compute σ = M/S.
Units
  • Keep second moment and fiber distance in one consistent unit system. S has dimension length³.
Boundary conditions
  • no parts → MISSING_REQUIRED_INPUT
  • neither c1 nor c2 → MISSING_REQUIRED_INPUT
  • c1 or c2 ≤ 0 → VALUE_MUST_BE_POSITIVE
  • I1 < 0 when c1 is set, or I2 < 0 when c2 is set → INVALID_INPUT
Example
parts=4,1,0,2,0,0 c1=2 c2=1 → S1=2, S2=1, θ=0
Validation cases

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

  • parts=4,1,0,2,0,0 c1=2 c2=1 → S1=2, S2=1, I1=4, I2=1, theta_deg=0
  • parts=0,0,0,1,0,0; 0,0,0,1,2,2 c1=1 → S1=4, I1=4, theta_deg=-45
  • parts=4,1,0,2,0,0 c1=0 → VALUE_MUST_BE_POSITIVE
Sources
  • Hibbeler, Engineering Mechanics: Statics — Moments of inertia
    Supports: Elastic section modulus about a principal axis is I/c, with c measured from the centroid along that axis
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the elastic section modulus about the principal axes of a composite plane area.

Supported and not supported

Supported: S1 = I1/c1 and S2 = I2/c2 after the principal-moment engine finds I1, I2, and θ.

Not supported: Sx and Sy, bending stress, plastic modulus, a Mohr-circle drawing, and beam deflection. /calc/mechanical is not open.

Agent / API notes

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

{ "parts": "4,1,0,2,0,0", "c1": 2, "c2": 1 }

Share the link with parts, c1, and c2. The result is not written into the link.

Other calculators in this family: Principal moments, Section modulus .

I1 Principal moments › S Section modulus ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this Sx and Sy?

No. Sx and Sy use Ix and Iy about axes parallel to x and y. This page uses I1 and I2.

Does this compute bending stress?

No. It returns the elastic section modulus about the principal axes. Bending stress, plastic modulus, and material allowables are out of scope.

Where is the fiber distance measured?

From the composite centroid, along the principal axis. θ tells you which way the I1 axis points.