HomeCalculatorsPhysicsStatics & StrengthCantilever Moment at the Couple Calculator
Physics calculator

Cantilever Moment at the Couple Calculator

Deflection at a concentrated moment on a cantilever. δ = M a² / (2 E I). a is measured from the fixed end. 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
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
Deflection at a concentrated moment on a cantilever, at a stated distance from the fixed end.
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 609f936a602d
Legacy regression
3/3 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 · 1/1 CVP boundary · 1/1 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — cantilever with an intermediate moment
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — cantilever with a couple
Sources
Evidence
2 legacy golden · 1 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 1/1 invalid · Artifact integrity PASS
STALE · Build schema 1.0.0 ready · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld
Semantic contract
PASS
Full verification

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

Formulas

Core equations used by this calculator.

Deflection at the momentδ = M a² / (2 E I)
ia = L recovers the tip-moment page. The free-end deflection when a is shorter than L is a different page.

How to use

1

Enter the moment, its distance from the fixed end, the length, the modulus, and the area moment

L, E, and I must be positive. a must be greater than 0 and no greater than L.

2

Read the deflection at the moment

The sign of M is the sign of δ.

Example calculations

Common configurations with formula and result.

ϟ

Moment at mid-length

M = 4, a = 1, L = 2, E = 1, I = 1

δ = 4 × 1² / 2
δ = 2
ϟ

Moment at the free end

M = 4, a = 2, L = 2, E = 1, I = 1

δ = M L² / (2 E I)
δ = 8

Cantilever Moment at the Couple calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A cantilever of length L is fixed at one end and carries one concentrated moment M at distance a from the fixed end. The deflection at the moment is δ = M a² / (2 E I).
What it calculates
Deflection at a concentrated moment on a cantilever, at a stated distance from the fixed end.
Inputs
  • M
  • a
  • L
  • E
  • I
Outputs
  • delta
Formula
δ = M a² / (2 E I).
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • a = L recovers the tip-moment page. The free-end deflection when a is shorter than L is a different page.
Units
  • M, a, L, E, and I set the deflection unit.
Boundary conditions
  • missing M, a, L, E, or I → MISSING_REQUIRED_INPUT
  • L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
  • a ≤ 0 or a > L → INVALID_INPUT
Example
M=4 a=1 L=2 E=1 I=1 → δ=2
Validation cases

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

  • M=4 a=1 L=2 E=1 I=1 → δ=2
  • M=4 a=2 L=2 E=1 I=1 → δ=8
  • a=3 L=2 → INVALID_INPUT
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — cantilever with an intermediate moment
    Supports: The deflection at the moment is M a² / (2 E I).
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — cantilever with a couple
    Supports: Supports the page formula: δ = M a² / (2 E I) at the couple.
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the deflection at one concentrated moment on a cantilever. a is the distance from the fixed end.

Supported and not supported

Supported: δ = M a² / (2 E I). a = L is the tip-moment page.

Not supported: the free-end deflection when a is shorter than L, and a simply supported span. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.cantilever_moment_at_deflection · tool id: cantilever-moment-at · pin 1.0.0.

{ "M": 4, "a": 1, "L": 2, "E": 1, "I": 1 }

Other calculators in this family: Angled Pull Calculator, Angled Pull on an Incline Calculator, Area Moment Calculator, Beam Deflection Calculator, Beam Moment Deflection Calculator, Beam Moment Maximum Deflection Calculator, Bending Stress Calculator, Cantilever Deflection Calculator . Explore all Statics & Strength.

→ Angled Pull Calculator Pull at an angle above the horizontal that puts a block on a level surface at impending slip. The normal is W − P sinα. Runs locally. › → Angled Pull on an Incline Calculator The pull at an angle above the horizontal that puts a block on an incline at impending slip. α = 0 is horizontal. α = θ is parallel to the plane. Runs locally. › → 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. › → Beam Deflection Calculator Midspan deflection of a simply supported span with one concentrated load. δ = P L³/(48 E I). I is an input. Runs locally. › → Beam Moment Deflection Calculator Midspan deflection of a simply supported span with one end moment. δ = M L²/(16 E I). I is an input. Runs locally. › → Beam Moment Maximum Deflection Calculator Largest deflection of a simply supported span with one end moment. δ = M L²/(9√3 E I) at x = L/√3. I is an input. Runs locally. › → Bending Stress Calculator Elastic bending stress from a moment and a section modulus. S can also be formed from Ix and c. Runs locally. › → Cantilever Deflection Calculator Free-end deflection of a cantilever with one concentrated load at the tip. δ = P L³/(3 E I). I is an input. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this the free-end deflection?

Only when a = L. Otherwise the free end moves farther, and that value is a different page.