HomeCalculatorsMechanicalBeams & DeflectionCantilever Load at a Station Calculator
Mechanical calculator

Cantilever Load at a Station Calculator

Deflection at station x for a cantilever load at a. x = a is the load page. x = L is the free-end page. 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 · cantilever intermediate load at station (two-piece); + O3 mpmath tabulated δ.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
cantilever intermediate load at station (two-piece); + O3 mpmath tabulated δ.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (main, alias, signed, awkward, invalid-domain) · Matrix
Deferred
Not tip-only under the same intermediate load, and not a simply supported span.
Numerical scope
O2: delta vs a separate-module identity (≤2 ULP). Not tip-only under the same intermediate load, and not a simply supported span. ≤2 ULP vs O3 applies only to the published tabulated cantilever-load-at vectors.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Deflection at a stated station for a cantilever load at distance a from the wall.
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 · cantilever intermediate load at station (two-piece); + O3 mpmath tabulated δ.· 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 b7cacbc34f20
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 · 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 · 1/1 CVP boundary · 3/3 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — cantilever with an intermediate load
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — cantilever intermediate load
Sources
Evidence
3 legacy golden · 1 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 3/3 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.

Between the wall and the loadδ = P x² (3a − x) / (6 E I)
Between the load and the tipδ = P a² (3x − a) / (6 E I)
ix = a recovers P a³ / (3 E I). x = L recovers the free-end page.

How to use

1

Enter the load, its position, the station, the length, the modulus, and the area moment

a and x must lie on the cantilever, up to the free end.

2

Read the deflection at x

Past the load the beam is straight.

Example calculations

Common configurations with formula and result.

ϟ

At the load

P = 48, a = 1, x = 1, L = 2, E = 1, I = 1

δ = P x² (3a − x) / (6 E I)
δ = 16
ϟ

At the free end

P = 48, a = 1, x = 2, L = 2, E = 1, I = 1

δ = P a² (3x − a) / (6 E I)
δ = 40

Cantilever Load at a Station calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A cantilever of length L carries one load P at distance a from the wall. For x ≤ a, δ = P x² (3a − x)/(6 E I). For x ≥ a, δ = P a² (3x − a)/(6 E I).
What it calculates
Deflection at a stated station for a cantilever load at distance a from the wall.
Inputs
  • P
  • a
  • x
  • L
  • E
  • I
Outputs
  • delta
Formula
Two-piece elastic curve. x = a is the load page. x = L is the free-end page.
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • x = a recovers P a³ / (3 E I). x = L recovers the free-end page.
Units
  • P, a, x, L, E, and I set the deflection unit.
Boundary conditions
  • missing P, a, x, L, E, or I → MISSING_REQUIRED_INPUT
  • L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
  • a or x outside (0, L] → INVALID_INPUT
Example
P=48 a=1 x=2 L=2 E=1 I=1 → δ=40
Validation cases

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

  • P=48 a=1 x=1 L=2 E=1 I=1 → δ=16
  • P=48 a=1 x=2 L=2 E=1 I=1 → δ=40
  • P=48 a=1 x=0.5 L=2 E=1 I=1 → δ=5
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — cantilever with an intermediate load
    Supports: x = a gives P a³ / (3 E I) and x = L gives P a² (3L − a) / (6 E I).
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — cantilever intermediate load
    Supports: Supports the two-piece curve, including the straight portion past the load.
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the deflection at a stated station when the cantilever load is at distance a from the wall.

Supported and not supported

Supported: the two-piece curve. x = a is the load page. x = L is the free-end page.

Not supported: a simply supported span. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.cantilever_load_at_deflection · tool id: cantilever-load-at · pin 1.0.0.

{ "P": 48, "a": 1, "x": 2, "L": 2, "E": 1, "I": 1 }

Other calculators in this family: Beam Deflection Calculator, Beam Moment Deflection Calculator, Beam Moment Maximum Deflection Calculator, Cantilever Deflection Calculator, Cantilever Intermediate Deflection Calculator, Cantilever Moment at the Couple Calculator, Cantilever Moment Deflection Calculator, Cantilever Outer Uniform Calculator . Explore all Beams & Deflection.

→ 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. › → 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. › → 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. › → Cantilever Moment Deflection Calculator Free-end deflection of a cantilever with one concentrated moment. δ = M L²/(2 E I). I is an input. Runs locally. › → Cantilever Outer Uniform Calculator Tip deflection when a uniform load starts away from the wall. δ = w [3 L⁴ − a³ (4L − a)] / (24 E I). Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

What is halfway to a load at mid-length?

For P = 48, a = 1, x = 0.5, and L = 2 the deflection is 5.