Module 2 · Lesson 2.4
Finding the reactions
Using the three equations in the right order, and checking the answer.
Why this matters
Reactions are the gateway to everything else. Get them wrong and every shear force, bending moment and deflection that follows is wrong too — usually without any obvious warning.
By the end of this lesson you should be able to
- Choose a moment point that removes an unknown
- Solve for both reactions on a simply supported beam
- Check the result independently
- Recognise and explain uplift
The method is always the same. Draw the free body. Replace distributed loads with resultants. Take moments about one support to find the other reaction. Then use vertical equilibrium to find the first. Finally, check with an equation you have not already used.
Try it
Beam laboratory
Change the beam, then reveal one diagram at a time. Predict each shape before you reveal it.
Beam type
m
kN (0 to remove)
m from left
kN/m (0 to remove)
Reveal
Worked example
Reactions for a beam with a UDL and a point load
Given
- Simply supported span of 8.0 m, pin at A (left), roller at B (right)
- Uniformly distributed load of 5 kN/m over the whole span
- Point load of 40 kN at 3.0 m from A
Find
The vertical reactions at A and B.
Assumptions
- Self-weight is included in the UDL
- The beam is straight and the supports are at its ends
Practice
A simply supported beam spans 6.0 m. A single point load of 24 kN acts 2.0 m from the left support. What is the reaction at the left support?
Practice
A simply supported beam spans 10.0 m. It carries 20 kN at 3.0 m and 30 kN at 7.0 m from the left support. What is the left-hand reaction?
Practice
A cantilever 3.0 m long carries a UDL of 10 kN/m over its whole length plus a 15 kN point load at the free end. What is the vertical reaction at the fixed support?
Practice
A beam 6.0 m long has a pin at x = 0 and a roller at x = 4.0 m, with a 20 kN load at the free end x = 6.0 m. What is the reaction at the pin? (A negative answer means uplift.)
Summary
- Moments about one support removes that reaction from the equation
- Then vertical equilibrium gives the other
- Always check with an equation you have not used
- A negative reaction means uplift and needs a physical holding-down detail
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint