Max Bending Moment: Uniform Load

wL²/8 is the most-used number in structural engineering. Spread a load evenly along a simply supported beam (its own weight, a floor, snow on a roof, a run of water-filled pipe) and the moment diagram is a parabola peaking at midspan with the value wL²/8. A 5 kN/m load on a 4 m span gives M = 5000 × 16 ÷ 8 = 10 000 N·m. The square on L is the important part: stretch the span by 50% and the moment goes up by 125%.

Enter w as force per unit length: this calculator borrows the N/m unit family, which also offers kN/m, lbf/ft and lbf/in. Two traps. First, converting an area load to a line load requires the tributary width: 2.4 kPa of floor load on joists at 400 mm centres is 2.4 × 0.4 = 0.96 kN/m per joist, and forgetting the tributary width is the most common error in the whole calculation. Second, this is the simply supported case; a fixed-fixed beam peaks at wL²/12 over the supports and only wL²/24 at midspan, and a propped cantilever gives wL²/8 at the fixed end. Continuous multi-span beams are somewhere in between, which is why they use less steel.