Max Moment — Simple Beam, Off-Centre Point Load

Put a single load anywhere on a simply supported span and the moment diagram is two straight lines meeting under the load, peaking at $M = Pab/L$ where $a$ and $b$ are the distances to each support. A 40 kN load at 2 m of a 6 m span gives $40{,}000 \times 2 \times 4/6 = 53.3$ kN·m. Check it independently: the near reaction is $Pb/L = 26.7$ kN, and walking out to the load gives $26.7 \times 2 = 53.3$ kN·m. Same number.

The useful shape of this result is that it is a parabola in $a$, maximised at midspan where it becomes the familiar $PL/4$. Our 53.3 kN·m is only 89% of the 60 kN·m a midspan load would produce, and moving the load from the third point out to the quarter point drops it further to 75%. So off-centre loads are always kinder than centred ones, which cuts two ways: it means a lifting beam checked at midspan is conservative wherever the hook actually is, and it means a moment measured in the field cannot tell you where the load was without more information — two positions symmetric about midspan give exactly the same peak.

Two traps. First, the maximum moment is under the load, but the maximum deflection is not — it sits nearer midspan, at $\sqrt{(L^2-b^2)/3}$ from the far support, and for a badly off-centre load the two locations are far apart. Second, this is one load in isolation. Add a second point load and neither result transfers; you have to build the shear diagram and find where it crosses zero, because that crossing, not the load position, is what locates the peak moment.