Maximum beam moment from a centre point load
For a simply supported beam with a central point load, the largest bending moment is at midspan. You can find it by considering only the left half of the beam.
What the formula is saying
Each reaction is P/2. Its lever arm to midspan is L/2. Multiplying them gives (P/2)(L/2) = PL/4.
Read the symbols in plain language
- P
- Point loadN
- L
- Spanm
Sort out the units first
Use P in N and L in m for N·m. 20 kN = 20,000 N. Divide N·m by 1,000 for kN·m.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
- P · Point load
- 20000 N
- L · Span
- 4 m
Find the left support reaction
Symmetry gives half of P at each support.
(20000) ÷ 2 = 10000 NFind the distance to midspan
The midpoint is half the span from the left support.
(4) ÷ 2 = 2 mMultiply force by its lever arm
This is the sagging-moment magnitude at midspan.
(10000) × (2) = 20000 N·m
Does this worked answer make sense?
At ideal simple supports the bending moment is zero. The worked maximum is 20 kN·m at midspan.
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Use these new values. Work it out first, then check your answer.
- P · Point load
- 12000 N
- L · Span
- 6 m
Find: Maximum beam moment from a centre point load
A hint, not the answer
Each reaction is P/2. Its lever arm to midspan is L/2. Multiplying them gives (P/2)(L/2) = PL/4.
Use P in N and L in m for N·m. 20 kN = 20,000 N. Divide N·m by 1,000 for kN·m.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Find the left support reaction
Symmetry gives half of P at each support.
(12000) ÷ 2 = 6000 NFind the distance to midspan
The midpoint is half the span from the left support.
(6) ÷ 2 = 3 mMultiply force by its lever arm
This is the sagging-moment magnitude at midspan.
(6000) × (3) = 18000 N·m
Avoid the common trap
PL/4 is not the cantilever formula. It also does not apply unchanged when the point load moves away from the centre.
When this method applies — and when it does not
An ideal straight beam with the supports and load stated here. Loads are magnitudes; self-weight is omitted unless already included. This is an equilibrium result, not a check of strength, deflection or stability. Enter the nonnegative magnitude of the downward load; sign conventions for internal moments are explained separately.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Lesson updated: · Worked examples checked against the implemented formula; not an independent engineering certification.