Civil & Structural Engineering

Beam Reaction Calculator

Calculate support reactions, maximum shear, and maximum bending moment for common simply supported and cantilever beam cases.

Formula RA = P(L − a)/L; RB = Pa/LReviewed Sep 8, 2026

Beam reactions are the support forces needed to keep a loaded beam in static equilibrium. This calculator handles the common determinate cases used in first-pass beam checks: simply supported beams with an off-center point load or full-span uniform load, and cantilevers with a point load or full-span uniform load. It returns reactions in kilonewtons and also shows the maximum shear and bending moment needed for follow-up stress or deflection checks.

Calculation Bench

Beam

Load

01Span Length

L · Distance between supports, or fixed end to free end for a cantilever.

02Point Load

P · Concentrated force applied to the beam.

03Load Position from A

a · Distance from support A or the fixed end to the point load.

Solution

Enter beam span and loading to calculate support reactions.

RA = P(L − a)/L; RB = Pa/L

Formula Sheet

R_A = \dfrac{P(L-a)}{L},\quad R_B = \dfrac{Pa}{L}

R_A = R_B = \dfrac{wL}{2}

R = P,\quad M = Pa

R = wL,\quad M = \dfrac{wL^2}{2}

RAReaction at Support A
RBReaction at Support B
RFixed Support Reaction
MFixed-End Moment
PPoint Load
wUniform Load
LSpan Length
aLoad Position
VmaxMaximum Shear
MmaxMaximum Bending Moment

Variables & Units

Symbol	Variable	Description	Common Units
RA	Reaction at Support A	Upward reaction at the left support.	N, kN, lbf
RB	Reaction at Support B	Upward reaction at the right support for a simply supported beam.	N, kN, lbf
R	Fixed Support Reaction	Vertical reaction at the fixed support of a cantilever.	N, kN, lbf
M	Fixed-End Moment	Moment that the fixed support must resist.	N·m, kN·m, lbf·ft
P	Point Load	Concentrated force applied to the beam.	N, kN, lbf
w	Uniform Load	Load intensity spread evenly across the full span.	N/m, kN/m, lbf/ft
L	Span Length	Distance between supports, or fixed end to free end for a cantilever.	m, ft, in
a	Load Position	Distance from support A or the fixed end to the point load.	m, ft, in
Vmax	Maximum Shear	Largest internal shear force magnitude for the selected case.	N, kN, lbf
Mmax	Maximum Bending Moment	Largest internal bending moment magnitude for the selected case.	N·m, kN·m, lbf·ft

How to Use This Calculator

01Select the beam support condition: Simply Supported for two supports, or Cantilever for a fixed end and free end.
02Select the load case: Point Load for a concentrated force, or Uniform Load for a load spread across the full span.
03Enter the span length. For a point load, enter the load position measured from support A or the fixed end.
04Include beam self-weight, equipment weight, and permanent loads in the load input when they matter; the calculator does not add them automatically.
05Select Calculate to get support reaction A, reaction B when applicable, maximum shear, and maximum bending moment.
06Use the result with the Bending Stress Calculator or Beam Deflection Calculator if you need the next design check.

How the Formula Works

For simply supported beams, the reactions are found from the two static equilibrium equations: the vertical forces must sum to zero, and the moments about either support must sum to zero. A point load closer to support A creates a larger reaction at A and a smaller reaction at B; a centered point load splits equally.

For a full-span uniform load, the distributed load is first converted to an equivalent resultant force W = wL acting at midspan. On a simply supported beam that gives equal reactions, while on a cantilever the fixed support carries the full resultant and a fixed-end moment.

The maximum shear and bending moment are included because reactions are usually only the first step. For these standard cases, the maximum bending moment is Pab/L for an off-center simply supported point load, wL²/8 for a simply supported uniform load, Pa for a cantilever point load, and wL²/2 for a cantilever uniform load.

Worked Example 01

Simply supported beam with an off-center point load

Known

Span Length (L): 6 m
Point Load (P): 20 kN
Load Position (a): 2 m from support A

Formula

RA = P(L − a)/L; RB = Pa/L

Substitution

RA = 20 × (6 − 2) / 6 = 13.33 kN; RB = 20 × 2 / 6 = 6.67 kN

Result

RA = 13.33 kN, RB = 6.67 kN, Mmax = 26.67 kN·m

Because the load is closer to support A, support A carries about two-thirds of the load while support B carries one-third. The reactions still sum to the 20 kN applied load.

Worked Example 02

Simply supported beam with a uniform load

Known

Span Length (L): 4 m
Uniform Load (w): 5 kN/m

Formula

RA = RB = wL/2

Substitution

Total load = 5 × 4 = 20 kN; RA = RB = 20 / 2 = 10 kN

Result

RA = 10 kN, RB = 10 kN, Mmax = 10 kN·m

A full-span uniform load acts like a 20 kN resultant at midspan, so both supports share the load equally.

Worked Example 03

Cantilever with a point load at the free end

Known

Span Length (L): 3 m
Point Load (P): 10 kN
Load Position (a): 3 m from fixed end

Formula

R = P; M = Pa

Substitution

R = 10 kN; M = 10 × 3 = 30 kN·m

Result

Fixed reaction = 10 kN, fixed moment = 30 kN·m

The fixed support must carry the entire vertical load and resist the load times its lever arm from the wall.

Applications

01Finding support reactions before drawing a shear and moment diagram
02Checking simple joists, lintels, beams, shelves, and cantilever brackets during preliminary sizing
03Getting maximum bending moment for a Bending Stress Calculator input
04Comparing centered and off-center load placement on a simply supported beam

Assumptions

01Beam is statically determinate: simply supported with two vertical reactions, or a single fixed-end cantilever.
02Loads are vertical and downward, and reactions are reported as upward magnitudes.
03Uniform load is applied across the full span, not over a partial length.
04Point load position is measured from the left support for simply supported beams and from the fixed end for cantilevers.
05Beam self-weight should be included as part of the uniform load if it matters for the check.

Where This Model Stops

01Not for continuous beams, propped cantilevers, fixed-fixed beams, overhangs, partial distributed loads, trapezoidal loads, or multiple point loads.
02Does not apply load combinations, load duration factors, live-load reduction, impact factors, or serviceability limits.
03Does not check member strength, deflection limits, connection design, bearing, lateral-torsional buckling, or code compliance.
04Results are preliminary statics values only. Structural design decisions should be verified by a qualified engineer using the governing code and project load combinations.

References

[1]
Beam Loads - Support Force Calculator
Engineering ToolBox
Equilibrium method for support forces: sum of forces and sum of moments.
[2]
Beams Supported at Both Ends - Continuous and Point Loads
Engineering ToolBox
Reference formulas for simply supported uniform-load reactions and moments.
[3]
Simply Supported Beam Reactions
CivilKits
Point-load and uniform-load reaction equations for simply supported beams.
[4]
Cantilever Beam with Uniformly Distributed Load
EasyEngineeringCalc
Cantilever full-span uniform-load reaction and fixed-end moment equations.

Frequently Asked Questions

What is a beam reaction?

A beam reaction is the support force or moment required to balance the applied loads. For a simple beam, reactions are usually found from force equilibrium and moment equilibrium.

Why are RA and RB different for an off-center load?

The support closer to the load carries more of it because the moment balance gives that support a shorter lever arm to the applied load. If the point load is exactly at midspan, the reactions become equal.

Can I include beam self-weight?

Yes. Add beam self-weight to the uniform load intensity if it is significant. For example, a 0.4 kN/m beam plus a 2.0 kN/m service load can be entered as 2.4 kN/m.

Is this enough to design a beam?

No. Reactions are only one part of beam design. You still need strength, shear, deflection, bearing, connection, load-combination, and code checks.

Why are maximum moment and shear shown on a reaction calculator?

Support reactions are often the first equilibrium result, but maximum shear and bending moment are the values usually carried into stress, section modulus, and deflection checks.

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