Basic Engineering Mechanics

How to Calculate the Reactions at the Supports of a Beam?

This is the first step in analysing a beam member, and it involves calculating the reaction forces at the supports (supports A and B in the below example) due to the loads applied on the beam. These reactions are also very important to calculate the internal forces (Shear force & Bending Moments)  and thus to provide optimum capacities at those sections via the size of the beam (B X D) and the steel reinforcement in the beam. 

When solving a problem like this we want to first remember that the beam is static; meaning it is not moving. From simple physics, this means that the sum of the forces in all the vertical (Y ; for sign convention) direction equals zero (i.e. the total downward forces equal the total upward forces). A second formula to remember is that the sum of the moments about any given point is equal to zero. This is because the beam is static and therefore not rotating.

To determine the reactions at supports, follow these simple steps:

1) The sum of vertical forces equals 0 (ΣF_y = 0) : Sum the forces in the Y (vertical) direction and let the sum equal zero. Consider a simple example of a 4m beam with a pin supoort at A and a roller support at B. The free body diagram is shown below where A_y and B_y are the vertical reactions at the supports:

Include all forces including reactions and normal loads such as point loads. So if we sum the forces in the Y direction for the above example, we get the following equation: 

A_y + B_y = 20kN     _____ {1}

2) The sum of moments about any point equals 0 (ΣM = 0) : All we need to know about moments about any point is that they are they are equal to the force multiplied by the distance from a point (i.e. the force x distance from point). Using the equilibrium equation of the moments about  point A one can deduce;

(4m x A_y) - (20kN x 2m) = 0

→ 4.A_y = 40 kN

→ A_y = 10kN

Using equation {1},B_y =10 kN

We have used the two above equations (sum of moments equals zero and sum of vertical forces equals zero) and calculated that the reaction at support A is 10 kN and the reaction at support B 10kN. This makes sense as the point load is right in the middle of the beam, meaning both supports should have the same vertical forces (i.e. it is symmetric).