Mechanical calculator

Bending Stress Calculator

Q: Does this build a rectangle from b and h?

No. Width and height stay on the rectangular-section page. Bring the section modulus here.

Q: What if I have Ix and the distance to the fiber?

Omit S and pass Ix with c. S is Ix/c from the section-modulus engine, then σ = M/S.

Q: Is this the plastic modulus?

No. This is elastic flexure. The plastic modulus of a rectangle is its own page.

Elastic bending stress from a moment and a section modulus. S can also be formed from Ix and c. 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 · σ = M/S on one centroidal axis; S given or S = Ix/c; + O3 mpmath tabulated σ.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: σ = M/S on one centroidal axis; S given or S = Ix/c; + O3 mpmath tabulated σ.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (S, Ix, alias, awkward, invalid-domain) · Matrix
Deferred: Not plastic modulus, not a rectangle from b·h, and not beam deflection.
Numerical scope: O2: σ = M/S vs a separate-module identity (≤2 ULP). Not plastic modulus, not a rectangle from b·h, and not beam deflection. ≤2 ULP vs O3 applies only to the published tabulated bending-stress vectors. Not plastic modulus.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Elastic bending stress on one centroidal axis from M and S.
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 · σ = M/S on one centroidal axis; S given or S = Ix/c; + O3 mpmath tabulated σ.· 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 2dc6e2ca9a7c
Legacy regression: 6/6 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status: STALE · attestation snapshot core 45bf7b58ef79 ≠ live cvp_core_sha256 2dc6e2ca9a7c (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 — Bending — flexure formula
Gere and Goodno, Mechanics of Materials — Bending — flexure formula
Sources
Evidence: 3 legacy golden · 3 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

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

Formulas

Core equations used by this calculator.

Flexureσ = M/S

Section modulus from IxS = Ix/c

iEnter S directly, or omit S and pass Ix with c. That second path calls the section-modulus engine. A rectangle built from b and h stays on the rectangular-section page.

How to use

Enter the moment and the modulus

M is the bending moment. S is the elastic section modulus at the fiber of interest. S must be positive.

Read the stress

σ = M/S. The sign of M is the sign of the stress at that fiber.

Example calculations

Common configurations with formula and result.

Moment 12, modulus 4

M = 12, S = 4, in one consistent unit system. σ then has the unit of M/S.

σ = 12/4

σ = 3

From Ix and c

M = 24, Ix = 36, c = 3, with S omitted. The page form can instead take S = Ix/c = 12 directly.

S = 36/3, σ = 24/12

σ = 2

Bending Stress 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: On one centroidal axis, the elastic bending stress is σ = M/S. S is the elastic section modulus at the fiber. A positive moment keeps the stress sign of that fiber.
What it calculates: Elastic bending stress on one centroidal axis from M and S.
Inputs: M
S
Outputs: sigma
M
S
Formula: σ = M/S. If S is omitted, S = Ix/c from the section-modulus engine.
Assumptions: Calculation runs locally in the browser; values are not uploaded.
Keep units consistent with the labels on each field.
Enter S directly, or omit S and pass Ix with c. That second path calls the section-modulus engine. A rectangle built from b and h stays on the rectangular-section page.
Units: M and S set the stress unit. σ has the unit of M/S.
Boundary conditions: missing M → MISSING_REQUIRED_INPUT
missing S and missing Ix → MISSING_REQUIRED_INPUT
S ≤ 0 or c ≤ 0 → VALUE_MUST_BE_POSITIVE
Ix < 0 → INVALID_INPUT
Example: M=12 S=4 → σ=3
Validation cases: 4 published on this page · 6/6 tests · Production surface contract 6/6 · View evidence
M=12 S=4 → σ=3
M=24 Ix=36 c=3 → S=12, σ=2
M=-12 S=4 → σ=-3
S=0 → VALUE_MUST_BE_POSITIVE
Sources: Hibbeler, Mechanics of Materials — Bending — flexure formula
Supports: The elastic bending stress at a fiber is the moment divided by the elastic section modulus at that fiber.
Gere and Goodno, Mechanics of Materials — Bending — flexure formula
Supports: Supports the page formula: σ = M/S
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the elastic bending stress on one centroidal axis.

Supported and not supported

Supported: σ = M/S. S can be typed in, or formed as Ix/c by the section-modulus engine. The sign of M is the sign of the stress at that fiber.

Not supported: a rectangle built from b and h, transverse shear, beam deflection, and the plastic modulus. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.bending_stress · tool id: bending-stress · pin 1.0.0.

{ "M": 12, "S": 4 }

Share the link with M and S. The result is not written into the link.

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

Frequently asked questions

Key distinctions behind the calculation.

Does this build a rectangle from b and h?