Structural Analysis Fundamentals / Forces, moments and equilibrium
Finding the reactions
Using the three equations in the right order, and checking the answer.
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This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
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Read the lesson register and outstanding review notesWhy 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
For these simply supported beam examples, draw the free body and replace distributed loads with resultants. Take moments about one support to find the other reaction, then use vertical equilibrium. Check the arithmetic by taking moments about the other support; this is a consistency check using the same equilibrium conditions.
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