Article 3.18 presents relationships between a distributed load on a beam, the resulting internal forces and moments, and the corresponding deformations. These relationships provide the key expressions used in the** conjugate-beam method** and the **moment-area method** for computing beam deflections and slopes of the neutral axis under loads. The unit-load method used for this purpose is derived from the principle of virtual work (Art. 3.23).

## Conjugate-Beam Method

## Moment-Area Method

Similar to the conjugate-beam method, the moment-area method is based on Eqs. (3.130a) to (3.130d). It expresses the deviation in the slope and tangential deflection between points A and B on a deflected beam:

Equation (3.131a) indicates that the change in slope of the elastic curve of a beam between any two points equals the area under the M/EI diagram between these points. Similarly, Eq. (3.131b) indicates that the tangential deviation of any point on the elastic curve with respect to the tangent to the elastic curve at a second point equals the moment of the area under the M/EI diagram between the two points taken about the first point.

For this particular example tA = 0, and hence SB = tB.

The moment-area method is particularly useful when a point of zero slope can be identified.

In cases where a point of zero slope cannot be located, deformations may be more readily calculated with the conjugate-beam method. As long as the bending-moment diagram can be defined accurately, both methods can be used to calculate deformations in either statically determinate or indeterminate beams.

## Unit-Load Method

Article 3.23 presents the basic concepts of the unit-load method. Article 3.31 employs this method to compute the deflections of a truss. The method also can be adapted to compute deflections in beams.