HomeCalculatorsMechanicalBeams & DeflectionPaired Load at the Load Calculator
Mechanical calculator

Paired Load at the Load Calculator

Deflection under either of two equal loads. δ = P a² (3L − 4a) / (6 E I). 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 · paired loads under either load δ = P a²(3L−4a)/(6 E I); + O3 mpmath tabulated δ.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
paired loads under either load δ = P a²(3L−4a)/(6 E I); + 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 paired loads, and not a station curve elsewhere.
Numerical scope
O2: delta vs a separate-module identity (≤2 ULP). Not midspan under the same paired loads, and not a station curve elsewhere. ≤2 ULP vs O3 applies only to the published tabulated paired-load-at vectors.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Deflection under either of two equal symmetric point loads.
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 · paired loads under either load δ = P a²(3L−4a)/(6 E I); + 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 3fe6f1cc2fa4
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 · 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 — two symmetric concentrated loads
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — symmetric point loads
Sources
Evidence
2 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 under either loadδ = P a² (3L − 4a) / (6 E I)
ia = L/2 matches one midspan load of 2P. The midspan value, between the loads, is a different page.

How to use

1

Enter each load, its distance from the nearer support, the span, the modulus, and the area moment

a must be greater than 0 and no greater than L/2.

2

Read the deflection under a load

The two loads deflect by the same amount.

Example calculations

Common configurations with formula and result.

ϟ

Loads one quarter of the way in

P = 48, a = 1, L = 4, E = 1, I = 1

δ = 48 × 1 × (12 − 4) / 6
δ = 64
ϟ

Both loads at midspan

P = 48, a = 2, L = 4, E = 1, I = 1

δ = (2P) L³ / (48 E I)
δ = 128

Paired Load at the Load calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A simply supported span of length L carries two equal loads P, each at distance a from the nearer support. The deflection under either load is δ = P a² (3L − 4a) / (6 E I).
What it calculates
Deflection under either of two equal symmetric point loads.
Inputs
  • P
  • a
  • L
  • E
  • I
Outputs
  • delta
Formula
δ = P a² (3L − 4a) / (6 E I).
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • a = L/2 matches one midspan load of 2P. The midspan value, between the loads, is a different page.
Units
  • P, a, L, E, and I set the deflection unit.
Boundary conditions
  • missing P, a, L, E, or I → MISSING_REQUIRED_INPUT
  • L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
  • a ≤ 0 or a > L/2 → INVALID_INPUT
Example
P=48 a=1 L=4 E=1 I=1 → δ=64
Validation cases

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

  • P=48 a=1 L=4 E=1 I=1 → δ=64
  • P=48 a=2 L=4 E=1 I=1 → δ=128
  • a=3 L=4 → INVALID_INPUT
Sources
  • Hibbeler, Mechanics of Materials — Deflection of beams — two symmetric concentrated loads
    Supports: The deflection under each load is P a² (3L − 4a) / (6 E I).
  • Gere and Goodno, Mechanics of Materials — Deflection of beams — symmetric point loads
    Supports: Supports the page formula. a = L/2 recovers one midspan load of 2P.
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the deflection under either load when two equal loads sit the same distance from each support.

Supported and not supported

Supported: δ = P a² (3L − 4a) / (6 E I).

Not supported: the midspan value between the loads, and unequal loads. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.paired_load_at_deflection · tool id: paired-load-at · pin 1.0.0.

{ "P": 48, "a": 1, "L": 4, "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.

Is midspan the same number?

Only when the loads meet at midspan. For a = 1 and L = 4, under each load is 64 and midspan is 88.