HomeCalculatorsMechanicalBeams & DeflectionTriangular Load at a Station Calculator
Mechanical calculator

Triangular Load at a Station Calculator

Deflection at station x for a triangular load that is zero at the left support. x = L/2 is the midspan 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 · one-sided triangle at station δ = w0[(7L³/360)x−(L/36)x³+x⁵/(120L)]/(EI); + O3 mpmath tabulated δ.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
one-sided triangle at station δ = w0[(7L³/360)x−(L/36)x³+x⁵/(120L)]/(EI); + O3 mpmath tabulated δ.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (main, alias, signed, awkward, invalid-domain) · Matrix
Deferred
Not midspan under the same triangle, and not a cantilever.
Numerical scope
O2: delta vs a separate-module identity (≤2 ULP). Not midspan under the same triangle, and not a cantilever. ≤2 ULP vs O3 applies only to the published tabulated triangular-load-at vectors.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Deflection at a stated station for a one-sided triangular load on a simply supported span.
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 · one-sided triangle at station δ = w0[(7L³/360)x−(L/36)x³+x⁵/(120L)]/(EI); + 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 e8401e119987
Legacy regression
2/2 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 — triangular load
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — triangular load
Sources
Evidence
1 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.

Deflection at xδ = w0 [(7 L³/360) x − (L/36) x³ + x⁵/(120 L)] / (E I)
ix = L/2 recovers 5 w0 L⁴/(768 E I).

How to use

1

Enter the peak intensity, the station, the span, the modulus, and the area moment

x must lie strictly between the supports, measured from the zero-intensity end.

2

Read the deflection

The sign of w0 is the sign of δ.

Example calculations

Common configurations with formula and result.

ϟ

Midspan

w0 = 24, x = 1, L = 2, E = 1, I = 1

δ = w0 [(7 L³/360) x − (L/36) x³ + x⁵/(120 L)] / (E I)
δ = 2.5

Triangular Load at a Station calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A simply supported span carries a triangular load that is 0 at the left support and w0 at the right. At station x from the left, δ = w0 [(7 L³/360) x − (L/36) x³ + x⁵/(120 L)] / (E I).
What it calculates
Deflection at a stated station for a one-sided triangular load on a simply supported span.
Inputs
  • w0
  • x
  • L
  • E
  • I
Outputs
  • delta
Formula
δ = w0 [(7 L³/360) x − (L/36) x³ + x⁵/(120 L)] / (E I).
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • x = L/2 recovers 5 w0 L⁴/(768 E I).
Units
  • w0, x, L, E, and I set the deflection unit.
Boundary conditions
  • missing w0, x, L, E, or I → MISSING_REQUIRED_INPUT
  • L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
  • x ≤ 0 or x ≥ L → INVALID_INPUT
Example
w0=24 x=1 L=2 E=1 I=1 → δ=2.5
Validation cases

2 published on this page · 2/2 tests · Production surface contract 6/6 · View evidence

  • w0=24 x=1 L=2 E=1 I=1 → δ=2.5
  • x=2 L=2 → INVALID_INPUT
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — triangular load
    Supports: The curve is zero at both supports and 5 w0 L⁴/(768 E I) at midspan.
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — triangular load
    Supports: Supports the closed form measured from the zero-intensity end.
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the deflection at a stated station for a triangular load that is zero at the left support.

Agent / API notes

Capability id: mechanical.statics.triangular_load_at_deflection · tool id: triangular-load-at · pin 1.0.0.

{ "w0": 24, "x": 1, "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 Load at a Station Calculator, Cantilever Moment at the Couple Calculator, Cantilever Moment Deflection 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 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. › → 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. ›

Frequently asked questions

Key distinctions behind the calculation.

Does midspan match the triangular-load page?

Yes. x = L/2 gives δ = 2.5 for w0 = 24 and L = 2.