HomeCalculatorsMechanicalBeams & DeflectionPaired Load Deflection Calculator
Mechanical calculator

Paired Load Deflection Calculator

Midspan deflection of two equal loads, each at distance a from the nearer support. δ = P a (3 L² − 4 a²) / (24 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 · symmetric paired loads midspan δ = P a (3L²−4a²)/(24 E I); + O3 mpmath tabulated δ.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
symmetric paired loads midspan δ = P a (3L²−4a²)/(24 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 a single offset load, not a cantilever, and not a station curve.
Numerical scope
O2: delta vs a separate-module identity (≤2 ULP). Not a single offset load, not a cantilever, and not a station curve. ≤2 ULP vs O3 applies only to the published tabulated paired-load-deflection vectors.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Midspan deflection of a simply supported span with 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 · symmetric paired loads midspan δ = P a (3L²−4a²)/(24 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 ae9a60631e46
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.

Midspan deflectionδ = P a (3 L² − 4 a²) / (24 E I)
ia = L/2 puts both loads at midspan and matches one load of 2P on the beam-deflection page.

How to use

1

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

L, E, and I must be positive. a must be greater than 0 and no greater than L/2.

2

Read the midspan deflection

The sign of P is the sign of δ.

Example calculations

Common configurations with formula and result.

ϟ

Loads one quarter of the way in on a span of 4

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

δ = 48 × 1 × (48 − 4) / 24
δ = 88
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 Deflection 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 midspan deflection is δ = P a (3 L² − 4 a²) / (24 E I).
What it calculates
Midspan deflection of a simply supported span with two equal symmetric point loads.
Inputs
  • P
  • a
  • L
  • E
  • I
Outputs
  • delta
Formula
δ = P a (3 L² − 4 a²) / (24 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 puts both loads at midspan and matches one load of 2P on the beam-deflection 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=88
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=88
  • 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: Midspan δ = P a (3 L² − 4 a²) / (24 E I) when each load is a distance a from the nearer support.
  • 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 midspan deflection of a simply supported span with two equal loads, each the same distance from the nearer support.

Supported and not supported

Supported: δ = P a (3 L² − 4 a²) / (24 E I). a = L/2 matches a single midspan load of 2P.

Not supported: unequal loads, a past midspan, and a cantilever. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.paired_load_deflection · tool id: paired-load-deflection · 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.

What if a is past midspan?

The two positions would cross. Keep a at most L/2, measured from each support toward the center.