Moment area method

The method for finding deflections in a framed structure by use of the moment area curve.

First Theorem


Theorem 1: The change in slope between any two points on the elastic curve equals the area of the M/EI diagram between these two points.

\theta_{AB} = {\int_A}^B \frac{M}{EI}\;dx

where


 * M moment
 * EI flexural rigidity
 * \theta_{AB} ... change in slope between points A and B
 * A, B ... points on the elastic curve

Second Theorem


Theorem 2: The deviation of the tangent at point B on the elastic curve with respect to the tangent at point A equals the "moment" of the M/EI diagram between points A and B computed about point A (the point on the elastic curve), where the deviation t_{A/B} is to be determined.

t_{A/B} = {\int_A}^B \frac{M}{EI} \bar{x} \;dx

where


 * M moment
 * EI flexural rigidity
 * t_{A/B} ... deviation of tangent at point B with respect to the tangent at point A
 * \bar{x} ... centroid of M/EI diagram measured horizontally from point A
 * A, B ... points on the elastic curve