CivilStructural AnalysiseasyFE ExamCoursework

Bending Moment Under a Point Load

A simply supported beam spans $L = 10\ \text{m}$ . A single point load of $P = 40\ \text{kN}$ acts $3\ \text{m}$ from the left support.

Find the bending moment at the location of the load.

Given

P	40 kN	Point load
L	10 m	Span
a	3 m	Distance from left support to load

Hint 1

Find the support reactions first, then cut the beam at the load.

Hint 2

R_A = Pb/L, where b is the distance from the load to the far support.

Hint 3

The standard result is M = Pab/L.

Worked solution — try the problem first

Reactions. Taking moments about the right support:

$R_A = P\frac{b}{L} = 40 \times \frac{7}{10} = 28\ \text{kN}$

The reaction nearer the load is the larger one — note $R_A$ uses $b = 7$ m, the distance to the far support.

Moment at the load. Cut just left of the load and take moments:

$M = R_A \times a = 28 \times 3 = 84\ \text{kN·m}$

$\boxed{M = 84\ \text{kN·m}}$

Check with the standard formula:

$M = \frac{Pab}{L} = \frac{40 \times 3 \times 7}{10} = 84\ \text{kN·m} \ \checkmark$

For a single point load the moment peaks at the load, so this is also the maximum moment in the beam.

Concepts:Bending moment diagramsSupport reactionsPoint loads

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