Mechanical 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.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance · Simply supported midspan δ = P L³/(48 E I); + O3 mpmath tabulated δ.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: Simply supported midspan δ = P L³/(48 E I); + O3 mpmath tabulated δ.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (midspan, alias, signed, awkward, invalid-domain) · Matrix
Deferred: Not a cantilever, not a distributed load, and not an indeterminate beam.
Numerical scope: O2: delta vs a separate-module identity (≤2 ULP). Not a cantilever, not a distributed load, and not an indeterminate beam. ≤2 ULP vs O3 applies only to the published tabulated beam-deflection vectors.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Midspan deflection of a simply supported span with one concentrated load at midspan.
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 · Simply supported midspan δ = P L³/(48 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 c9388cb0380b
Legacy regression: 4/4 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status: STALE · attestation snapshot core e21b7482734b ≠ live cvp_core_sha256 c9388cb0380b (2026-09-27.o2-o3)
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 · 2/2 CVP boundary · 4/4 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 — simply supported beam with a concentrated load at midspan
Gere and Goodno, Mechanics of Materials — Deflection of beams — simply supported beam with a concentrated load at midspan
Sources
Evidence: 2 legacy golden · 2 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Midspan deflectionδ = P L³ / (48 E I)

iI comes from a section page. This page does not rebuild a rectangle, a circle, or a hollow circle. A cantilever and a distributed load are their own pages.

How to use

Enter the load, the span, the modulus, and the area moment

L, E, and I must be positive. P may be negative. Positive P is the positive deflection direction. P, L, E, and I share one unit system, and δ then has the unit of L.

Read the midspan deflection

δ = P L³/(48 E I). The sign of P is the sign of δ.

Example calculations

Common configurations with formula and result.

Span 2

P = 48, L = 2, E = 1, I = 1, in one consistent unit system

δ = 48 × 2³ / (48 × 1 × 1)

δ = 8

Same deflection at twice the span

P = 96, L = 4, E = 1, I = 16. The load and I change with the span.

δ = 96 × 4³ / (48 × 1 × 16)

δ = 8

Beam Deflection calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 4/4 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: A simply supported span of length L carries one concentrated load P at midspan. The midspan deflection is δ = P L³ / (48 E I). E is the elastic modulus. I is the centroidal area moment, typed in.
What it calculates: Midspan deflection of a simply supported span with one concentrated load at midspan.
Inputs: P
L
E
I
Outputs: delta
P
L
E
I
Formula: δ = P L³ / (48 E I).
Assumptions: Calculation runs locally in the browser; values are not uploaded.
Keep units consistent with the labels on each field.
I comes from a section page. This page does not rebuild a rectangle, a circle, or a hollow circle. A cantilever and a distributed load are their own pages.
Units: P, L, E, and I set the deflection unit. δ has the unit of length when those four are consistent.
Boundary conditions: missing P, L, E, or I → MISSING_REQUIRED_INPUT
L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
Example: P=48 L=2 E=1 I=1 → δ=8
Validation cases: 3 published on this page · 4/4 tests · Production surface contract 6/6 · View evidence
P=48 L=2 E=1 I=1 → δ=8
P=-48 L=2 E=1 I=1 → δ=-8
L=0 → VALUE_MUST_BE_POSITIVE
Sources: Hibbeler, Mechanics of Materials — Deflection of beams — simply supported beam with a concentrated load at midspan
Supports: The midspan deflection is P L³ / (48 E I) when the load sits at the center of a simply supported span.
Gere and Goodno, Mechanics of Materials — Deflection of beams — simply supported beam with a concentrated load at midspan
Supports: Supports the page formula: δ = P L³ / (48 E I)
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the midspan deflection of a simply supported span with one concentrated load.

Supported and not supported

Supported: δ = P L³/(48 E I). Positive P is the positive deflection direction. Aliases are load, force, F, span, length, youngs, modulus, Ix, and inertia.

Not supported: a cantilever, a distributed load, a deflection away from midspan, and rebuilding I from a section. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.beam_deflection · tool id: beam-deflection · pin 1.0.0.

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

Share the link with P, L, E, and I. The result is not written into the link.

Other calculators in this family: 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, Cantilever Outer Uniform Calculator . Explore all Beams & Deflection.

Frequently asked questions

Key distinctions behind the calculation.

Where is the deflection reported?

At midspan, under the concentrated load. A point elsewhere on the span is not this page.

Does this rebuild the area moment?

No. I is an input. A rectangle, a circle, and a hollow circle stay on their section pages.

Is a cantilever included?

No. A cantilever and a distributed load are their own pages.

Is this a separate beam engine for agents?

No. REST/MCP resolve this discovery id to mechanical.beam.deflection with a simply-supported midspan point-load prefill. Sibling load cases share that Capability.