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.

$\theta _{AB}={\int _{A}}^{B}{\frac {M}{EI}}\;dx$

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

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 $M/EI$ 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

References

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

External links

• Deflection of Cantilever Beam by Area Moment Method
• Deflection of Simply Supported Beam by Area Moment Method
• Colin Caprani: Structural Analysis III The Moment Area Method – Mohr’s Theorems (lecture notes), 2007