HomeCalculatorsMechanicalSection PropertiesPrincipal Section Modulus Calculator
Mechanical 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.

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 principal 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 ee2915a1dd5a
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.

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
  • Gere and Goodno, Mechanics of Materials — Moments of inertia
    Supports: Supports the page formula: S1 = I1 / c1
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: Area Moment Calculator, Circular Section Calculator, Hollow Circular Section Calculator, Inclined-Axis Moment Calculator, Plane Centroid Calculator, Principal Moment Calculator, Rectangular Section Calculator, Section Modulus 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. › → Rectangular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid rectangle from its width and height. Runs locally. › → 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. ›

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.