HomeCalculatorsPhysicsStatics & StrengthTriangular Load Maximum Calculator
Physics calculator

Triangular Load Maximum Calculator

Largest deflection of a simply supported span with a triangular load. The peak is not at midspan. 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 under a triangular load that is zero at the left support.
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 adf4cc687617
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 · 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 — simply supported beam with a triangular load
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — triangular load
Sources
Evidence
1 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.

Largest deflectionδ = w0 L⁴ √(1−√(8/15)) (2+5√(8/15)) / (450 E I)
Position from the unloaded endx = L √(1−√(8/15))
ix/L is about 0.519. The midspan value 5 w0 L⁴/(768 E I) is a different page.

How to use

1

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

L, E, and I must be positive.

2

Read the largest deflection and its position

x is measured from the left support, where the intensity is zero.

Example calculations

Common configurations with formula and result.

ϟ

Peak of 24 on a span of 2

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

δ = 24 × 16 × √(1−√(8/15)) × (2+5√(8/15)) / 450
δ ≈ 2.504, just above the midspan value 2.5

Triangular Load Maximum calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A simply supported span of length L carries a triangular load that is 0 at the left support and w0 at the right. The largest deflection is δ = w0 L⁴ √(1−√(8/15)) (2+5√(8/15)) / (450 E I), at x = L √(1−√(8/15)) from the left.
What it calculates
Largest deflection of a simply supported span under a triangular load that is zero at the left support.
Inputs
  • w0
  • L
  • E
  • I
Outputs
  • delta
  • x
Formula
δ = w0 L⁴ √(1−√(8/15)) (2+5√(8/15)) / (450 E I).
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • x/L is about 0.519. The midspan value 5 w0 L⁴/(768 E I) is a different page.
Units
  • w0, L, E, and I set the deflection unit.
Boundary conditions
  • missing w0, L, E, or I → MISSING_REQUIRED_INPUT
  • L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
Example
w0=24 L=2 E=1 I=1 → δ just above 2.5
Validation cases

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

  • w0=24 L=2 E=1 I=1 → δ just above 2.5
  • missing I → MISSING_REQUIRED_INPUT
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — simply supported beam with a triangular load
    Supports: The largest deflection is not at midspan. It sits near 0.519L from the zero-intensity end.
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — triangular load
    Supports: Supports the closed form δ = w0 L⁴ √(1−√(8/15)) (2+5√(8/15)) / (450 E I).
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the largest deflection of a simply supported span under a triangular load. Intensity is 0 at the left support and w0 at the right.

Supported and not supported

Supported: the maximum and its position x from the unloaded end.

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

Agent / API notes

Capability id: mechanical.statics.triangular_load_max_deflection · tool id: triangular-load-maximum · pin 1.0.0.

{ "w0": 24, "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.

Where is x measured from?

From the left support, the end where the intensity is zero. x/L = √(1−√(8/15)).