Consider a train moving on a bridge. Here, an unsymmetrical load is applied to a supported beam where the maximum deflection doesn't usually occur in the middle.

The maximum deflection of the beam is calculated by identifying the point O on the curve, where the beam's tangent is horizontal.

The slope of the tangent at point X is determined by calculating the tangential deviation between the supports and dividing it by their distance.

Since the slope at point O is zero, the slope between O and X equals the negative slope at X.

The first moment-area theorem is used to locate point O by measuring an area under the M/EI diagram equal to the negative slope at support X.

The maximum deflection equals the tangential deviation of support X about point O. This value can be obtained by calculating the first moment relating to the vertical axis through X of the area between X and O.