A worked example: 4m steel beam, 10 kN/m load
A 4-metre steel beam carrying a uniformly distributed load of 10 kN/m, with a moment of inertia of 8000 cm⁴, works out to a maximum bending moment of 20 kN·m, a maximum shear force of 20 kN, and a maximum deflection of 2.0833 mm at midspan.
Bending moment: M = wL² ÷ 8
For a simply-supported beam under a uniform load, the maximum bending moment occurs at midspan: M = wL² ÷ 8. With w = 10 kN/m and L = 4 m, that's 10 × 4² ÷ 8 = 10 × 16 ÷ 8 = 20 kN·m — the load squared with span makes bending moment grow much faster than the load itself as a beam gets longer.
Shear force: V = wL ÷ 2
Maximum shear occurs at each support and equals half the total load on the beam: V = wL ÷ 2 = 10 × 4 ÷ 2 = 20 kN. This is simply half of the beam's total load (10 kN/m × 4 m = 40 kN total), split evenly between the two supports.
Deflection: the formula where section properties matter most
Deflection is δ = 5wL⁴ ÷ (384EI), where E is the material's elastic modulus and I is the beam's moment of inertia. Unlike bending moment and shear, deflection depends on the material stiffness (E) and cross-section shape (I), not just the load and span — the same load and span on a smaller or more flexible section deflects far more. Swapping the 200 GPa steel beam above for an 11 GPa timber beam of similar span and a lighter 4 kN/m load (3m span, moment of inertia 12000 cm⁴) gives a deflection of 3.196 mm, despite a smaller load, because timber's much lower elastic modulus dominates the result.
Why deflection needs its own check, separate from strength
A beam can have plenty of strength margin against bending and shear failure while still deflecting more than a building code allows (commonly expressed as span ÷ 250 or span ÷ 360). Bending moment and shear are strength checks; deflection is a serviceability check — a beam that "holds up" fine can still fail a deflection limit and feel bouncy or crack finishes above it.