Moment area method

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

First Theorem

Picture illustrating the 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.

<tex> \theta_{AB} = {\int_A}^B \frac{M}{EI}\;dx </tex>

where

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

Second Theorem

Picture illustrating the 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 <tex>M/EI</tex> diagram between points A and B computed about point A (the point on the elastic curve), where the deviation <tex>t_{A/B}</tex> is to be determined.

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

where

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

References

• Russel C. Hibbeler: Structural Analysis, 3rd Edition, Prentice Hall, 1995, chapter 8, p. 354-569, ISBN 0-02-354041-9

External links

• Colin Caprani: Structural Analysis III The Moment Area Method – Mohr’s Theorems (lecture notes), 2007