Central Point Load on a Simply Supported Beam: Formulas

One load in the middle of one span: the case every lab bench test recreates and every quick check reaches for.

The equation

M_max = PL/4 under the load
V = R = P/2
δ_max = PL³/48EI at midspan

P is the concentrated load, in kN
L is the span between supports, in meters
E is the modulus of elasticity
I is the second moment of area
δ is the vertical deflection

Concentration is expensive: the same total load spread along the span would produce half this moment, which is why point loads on joists and purlins deserve respect.

Worked example

A 6 m span with EI = 16000 kNm² carries 20 kN at midspan. Reactions are 10 kN each, the moment under the load is 20 × 6/4 = 30 kNm, and the deflection is 20 × 6³/(48 × 16000) = 0.0056 m, or 5.6 mm. Spread the same 20 kN evenly along the span instead and the moment halves to 15 kNm while the deflection drops to 3.5 mm: concentration is what hurts.

When to use it

The same elastic limits as the uniform case apply, plus one of its own: the load must actually be concentrated. A bearing plate a few hundred millimetres long still counts; a load spread over a third of the span does not, and sits somewhere between this case and the uniform one. Check local effects at the load point too, since crushing and web checks arrive before flexure on short spans.

Related Calculators and Reference

Beam Deflection Studio: run this case with your own numbers.
Section Properties Calculator: the I behind the deflection.
Point Load at Any Position: the general version when the load leaves midspan.
RC Beam Design Calculator: turn PL/4 into reinforcement.
American Society of Civil Engineers: loads and structural standards.