Physics calculator

Overhang Span, Uniform, Center Triangle, Midspan Point, Offset, Paired, and Equal End Moments Calculator

Deflection at a stated station on an overhanging beam: overhang tip P, uniform w, center triangle w0, midspan point Pm, offset Po at ao, paired Pp at ap, and equal end moments M. Superposition of the listed elastic curves on the span. Runs locally.

Instant result

Result

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Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: Deflection at a stated station for this overhang-span superposition.
Scope: Calculation runs locally in the browser; values are not uploaded.
Verification: Engine tested · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: Hibbeler, Mechanics of Materials — Deflection of beams — superposition
Gere and Goodno, Mechanics of Materials — Deflection of beams — superposition
Sources

Formulas

Core equations used by this calculator.

Deflection at xδ = δ_overhang + δ_uniform + δ_center + δ_point + δ_offset + δ_paired + δ_M

iThe check station recovers −12 + 5 + 3.2 + 8 + 5.5 + 11 + 2.

How to use

Enter the loads, the station, the overhang, the span, the modulus, and the area moment

x must lie strictly between the supports. a is the overhang; other length symbols are span stations.

Read the deflection

A downward overhang load lifts the span; span loads and sagging moments keep their usual signs.

Example calculations

Common configurations with formula and result.

Check station

P = 48, Pm = 48, Po = 48, Pp = 48, w = 24, w0 = 24, M = 4, a = 1, ao = 0.5, ap = 0.5, x = 1, L = 2, E = 1, I = 1

δ = δ_overhang + δ_uniform + δ_center + δ_point + δ_offset + δ_paired + δ_M

δ = 22.7

Overhang Span, Uniform, Center Triangle, Midspan Point, Offset, Paired, and Equal End Moments calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 2 published cases
Named expert review Not performed
Calculation version 1.0.0

Review policy

Definition: A beam of span L has an overhang a beyond the right support, with load P at the overhang end, a uniform load w on the span, a center-peaked triangle load of peak w0 on the span, a midspan point load Pm, an offset span point load Po at ao from the left support, a symmetric paired point load Pp at distance ap from each support, and equal sagging end moments M on the span. Deflection at station x on the span is the sum of those elastic curves.
What it calculates: Deflection at a stated station for this overhang-span superposition.
Inputs: P
Pm
Po
Pp
w
w0
M
a
ao
ap
x
L
E
I
Outputs: delta
Formula: δ = δ_overhang + δ_uniform + δ_center + δ_point + δ_offset + δ_paired + δ_M. x = L/2 recovers the listed midspan pages.
Assumptions: Calculation runs locally in the browser; values are not uploaded.
Keep units consistent with the labels on each field.
The check station recovers −12 + 5 + 3.2 + 8 + 5.5 + 11 + 2.
Units: The inputs set the deflection unit.
Boundary conditions: a required input missing → MISSING_REQUIRED_INPUT
L ≤ 0, E ≤ 0, or I ≤ 0 → VALUE_MUST_BE_POSITIVE
x ≤ 0 or x ≥ L → INVALID_INPUT
Example: P = 48, Pm = 48, Po = 48, Pp = 48, w = 24, w0 = 24, M = 4, a = 1, ao = 0.5, ap = 0.5, x = 1, L = 2, E = 1, I = 1 → δ=22.7
Validation cases: 2 published on this page
P = 48, Pm = 48, Po = 48, Pp = 48, w = 24, w0 = 24, M = 4, a = 1, ao = 0.5, ap = 0.5, x = 1, L = 2, E = 1, I = 1 → δ=22.7
x out of range → INVALID_INPUT
Sources: Hibbeler, Mechanics of Materials — Deflection of beams — superposition
Supports: The listed actions superpose on the span.
Gere and Goodno, Mechanics of Materials — Deflection of beams — superposition
Supports: Supports adding the elastic curves.
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the deflection at a stated station for this superposition: overhang tip P, uniform w, center triangle w0, midspan point Pm, offset Po at ao, paired Pp at ap, and equal end moments M.

The overhang length is a. The check midspan recovers −12 + 5 + 3.2 + 8 + 5.5 + 11 + 2.

Agent / API notes

Capability id: mechanical.statics.overhang_span_uniform_center_triangle_point_offset_paired_end_moments_at_deflection · tool id: overhang-span-uniform-center-triangle-point-offset-paired-end-moments-at · pin 1.0.0.

{
  "P": 48,
  "Pm": 48,
  "Po": 48,
  "Pp": 48,
  "w": 24,
  "w0": 24,
  "M": 4,
  "a": 1,
  "ao": 0.5,
  "ap": 0.5,
  "x": 1,
  "L": 2,
  "E": 1,
  "I": 1
}

Other calculators in this family: Overhang span, uniform, center triangle, midspan point, offset, and paired .

Frequently asked questions

Key distinctions behind the calculation.

What does the check midspan mean?

With E = I = 1 and the listed check loads, midspan recovers the sum of the component midspan values. It gives δ = −12 + 5 + 3.2 + 8 + 5.5 + 11 + 2 = 22.7.

Are a, ao, and ap the same?

No. a is the overhang beyond the right support. ao is the offset load distance from the left support. ap is the distance of each paired load from the nearer support.