momentarea

Moment-area, in structural engineering, refers to the moment-area method (also called area-moment method), a classical technique for finding slopes and deflections of beams under bending. The method leverages the relationship between bending moments, beam stiffness, and the elastic curve to compute displacements without solving the full differential equation of the beam.

The core idea is that the bending curvature of a beam is proportional to the bending moment divided by the flexural rigidity EI. The method uses the bending moment diagram M(x) along the beam and the constant EI to form an M/(EI) diagram. Two key results, known as the moment-area theorems, relate areas under this diagram to changes in slope and to deflections.

Theorem I states that the change in slope between two points on the beam equals the area under the M/(EI) diagram between those points. In practical terms, if φ is the slope, then φ(B) − φ(A) = (1/EI) ∫ from A to B M(x) dx.

Theorem II provides a relation for deflection: the deflection at a point can be found from a corresponding area relation, often approached via a conjugate-beam construction or by taking the first moment of the M/(EI) diagram about the point of interest. This enables the determination of vertical displacement without directly integrating the beam’s deflection equation.

The method is applicable to prismatic beams (constant E and I) under small-deflection, linear-elastic behavior and can handle statically determinate and certain indeterminate cases when used with superposition. It is commonly taught as part of classical beam theory alongside other methods such as the unit-load and slope-deflection methods.