Support Reaction Calculator: Beam Reaction Forces

How to Find Beam Reaction Forces

Draw the free body diagram. Replace each support with its reactions: one force for a roller, two for a pin, and two forces plus a moment for a fixed support.
Replace distributed loads with a single resultant: the load times its length, acting at the middle of a uniform load.
Take moments about one support (ΣM = 0) to solve for the reaction at the other.
Sum vertical forces (ΣF_y = 0) to get the remaining reaction, and ΣF_x = 0 for any horizontal reaction.

Worked Examples

Point load off center

A 10 m simply supported beam (pin at A, roller at B) carries a 20 kN load 3 m from A.

Moments about A: R_B × 10 − 20 × 3 = 0, so R_B = 6 kN.
Vertical forces: R_A + 6 − 20 = 0, so R_A = 14 kN.
Check: the support nearer the load takes more of it, and 14 + 6 = 20 kN.

In general, for a load P at distance a from A on a span L: R_B = Pa/L and R_A = P(L − a)/L.

Example 1

Simply supported beam with a uniform load

A 6 m beam, pinned at the left end and on a roller at the right, carries 10 kN/m over its full length.

R_A = R_B: 30 kN
M_max: 45 kN·m

›Solution

Reactions: Total load = 10 × 6 = 60 kN, shared equally: R_A = R_B = 30 kN.
Shear: Shear starts at +30 kN, falls in a straight line, crosses zero at midspan (3 m) and ends at −30 kN.
Moment: Maximum moment is at the zero-shear point: M_max = wL²/8 = 10 × 6² / 8 = 45 kN·m at midspan. The diagram is a parabola.

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Example 2

Cantilever with a uniform load

A 3 m cantilever, fixed at the left end and free at the right, carries 5 kN/m over its full length.

R: 15 kN
M_fixed: 22.5 kN·m

›Solution

Reactions: Vertical reaction R = 5 × 3 = 15 kN; fixed-end moment M = wL²/2 = 5 × 3² / 2 = 22.5 kN·m.
Shear: Shear is 15 kN at the fixed end and falls linearly to zero at the free end.
Moment: Moment magnitude is largest at the support, 22.5 kN·m (hogging), and drops to zero at the free tip along a parabola.

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Reactions for Common Beams

Beam and load	Reactions
Simply supported, uniform load w	R_A = R_B = wL/2
Simply supported, point load P at distance a from A	R_A = P(L − a)/L, R_B = Pa/L
Cantilever, uniform load w	R = wL, M = wL²/2 at the fixed end
Cantilever, point load P at free end	R = P, M = PL at the fixed end

For trusses, the same equilibrium equations give the support reactions before you solve the members; the truss calculator does both.

Support Reaction FAQ

How do you calculate support reactions?

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Draw the free body diagram, then apply the three equilibrium equations: sum of horizontal forces = 0, sum of vertical forces = 0, and sum of moments about any point = 0. Taking moments about one support removes that support’s unknowns.

How many reactions does each support have?

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A roller has one (perpendicular to the surface), a pin has two (horizontal and vertical), and a fixed support has three (horizontal, vertical and a moment).

What if there are more unknown reactions than equations?

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The beam is statically indeterminate and equilibrium alone is not enough. This calculator still solves it, because the stiffness method also uses the beam’s deformation.

Why is a reaction negative?

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A negative result means the reaction acts opposite to the direction you assumed, for example a support pulling down on an overhanging beam.

Support Reaction Calculator