HomeCalculatorsPhysicsStatics & StrengthBeam Moment Maximum Deflection Calculator
Physics calculator

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.

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
Largest deflection of a simply supported span with one concentrated moment at one end, and its location.
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 c4a9e78ae8cf
Legacy regression
4/4 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 · 2/2 CVP boundary · 2/2 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — simply supported beam with a concentrated moment at one end
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — simply supported beam with a concentrated moment at one end
Sources
Evidence
2 legacy golden · 2 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 2/2 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.

Largest deflectionδ = M L² / (9 √3 E I)
Location from the unloaded endx = L / √3
iI comes from a section page. This page does not rebuild a rectangle, a circle, or a hollow circle. Midspan deflection is its own page. A cantilever and a point load are their own pages.

How to use

1

Enter the moment, the span, the modulus, and the area moment

L, E, and I must be positive. M may be negative. Positive M is the positive deflection direction. M, L, E, and I share one unit system, and δ then has the unit of L. x has the unit of L.

2

Read the largest deflection and where it sits

δ = M L²/(9√3 E I). x = L/√3 is measured from the end without the moment, toward the moment. The sign of M is the sign of δ.

Example calculations

Common configurations with formula and result.

ϟ

Span 2

M = 16, L = 2, E = 1, I = 1, in one consistent unit system

δ = 16 × 2² / (9 √3 × 1 × 1)
δ = 64/(9√3), x = 2/√3
ϟ

Same moment at midspan

M = 16, L = 2, E = 1, I = 1 on the beam-moment page

δ_max / δ_mid = 16/(9√3)
δ = 64/(9√3) here, δ = 4 on the midspan page

Beam Moment Maximum Deflection calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A simply supported span of length L carries one concentrated moment M at one end. The largest deflection is δ = M L² / (9 √3 E I), located at x = L/√3 from the end that does not carry the moment. E is the elastic modulus. I is the centroidal area moment, typed in.
What it calculates
Largest deflection of a simply supported span with one concentrated moment at one end, and its location.
Inputs
  • M
  • L
  • E
  • I
Outputs
  • delta
  • x
  • M
  • L
  • E
  • I
Formula
δ = M L² / (9 √3 E I), x = L / √3.
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • I comes from a section page. This page does not rebuild a rectangle, a circle, or a hollow circle. Midspan deflection is its own page. A cantilever and a point load are their own pages.
Units
  • M, L, E, and I set the deflection unit. δ and x have the unit of length when those four are consistent.
Boundary conditions
  • missing M, L, E, or I → MISSING_REQUIRED_INPUT
  • L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
Example
M=16 L=2 E=1 I=1 → δ=64/(9√3), x=2/√3
Validation cases

3 published on this page · 4/4 tests · Production surface contract 6/6 · View evidence

  • M=16 L=2 E=1 I=1 → δ=64/(9√3), x=2/√3
  • M=-16 L=2 E=1 I=1 → δ=-64/(9√3)
  • L=0 → VALUE_MUST_BE_POSITIVE
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — simply supported beam with a concentrated moment at one end
    Supports: The largest deflection is M L² / (9 √3 E I) at x = L/√3 from the end opposite the moment.
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — simply supported beam with a concentrated moment at one end
    Supports: Supports the page formulas: δ = M L² / (9 √3 E I) and x = L / √3
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the largest deflection of a simply supported span with one concentrated moment at one end.

Supported and not supported

Supported: δ = M L²/(9√3 E I) and x = L/√3 from the end without the moment. Positive M is the positive deflection direction. Aliases are moment, couple, span, length, youngs, modulus, Ix, and inertia.

Not supported: the midspan value alone, a cantilever, a point load, a uniform load, and rebuilding I from a section. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.beam_moment_max_deflection · tool id: beam-moment-maximum · pin 1.0.0.

{ "M": 16, "L": 2, "E": 1, "I": 1 }

Share the link with M, L, E, and I. The result is not written into the link.

Other calculators in this family: Angled Pull Calculator, Angled Pull on an Incline Calculator, Area Moment Calculator, Beam Deflection Calculator, Beam Moment Deflection Calculator, Bending Stress Calculator, Cantilever Deflection Calculator, Cantilever Intermediate 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. › → 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. › → Cantilever Intermediate Deflection Calculator Deflection at a concentrated load on a cantilever. δ = P a³ / (3 E I). a is measured from the fixed end. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Where is the deflection reported?

At the largest deflection, x = L/√3 from the end that does not carry the moment. Midspan is a different page.

Does this rebuild the area moment?

No. I is an input. A rectangle, a circle, and a hollow circle stay on their section pages.

Is the midspan value included?

No. Midspan deflection of the same end moment is its own page.