Simply Supported Beam Reactions

\sum F_{y} = 0 \sum M = 0

A simply supported beam has a pin and a roller (3 unknown scalars in 2D with no axial load often 2). Take moments about one support to find the other reaction, then ΣFy. Check with the opposite moment.

$R_{A}, R_{B}$ (N): Vertical reactions
$L$ (m): Span
$P$ (N): Point load

Step-by-step solved example

Beam length 8 m, pin at A, roller at B. Point load 20 kN at 2 m from A. Find reactions.

Simply supported beam: pin + roller. ΣM about A eliminates RA.

1. ΣM_A
8 RB = 20×2 → RB = 5 kN.
$R_{B} = 5 t e x t k N$
2. ΣFy
RA + 5 = 20 → RA = 15 kN.
$R_{A} = 15 t e x t k N$

Answer: RA = 15 kN, RB = 5 kN

10 practice questions

0/10 correct

1.Midspan point load P on span L: each reaction is

2.Uniform w on simple span: each reaction is

3.A roller on a horizontal beam (2D) cannot resist

4.Cantilever fixed at A, load P at the free end. RA =

5.Taking ΣM about A eliminates

6.P = 12 kN at 3 m from A on a 9 m simple beam. RB =

7.Negative reaction means the support

8.Internal hinge in a beam adds

9.Check: RA + RB must equal

10.A couple M applied at midspan of a simple beam produces reactions