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Area Moment Method

Consider 2 points A and B on the beam, as shown in the figure below:

Area Moment Diagram

Theorem 1: The change in slope between the tangents drawn to the elastic curve at any 2 points A and B, $\theta_{BA}$, is equal to the product $1/EI$ multiplied by the area of the moment diagram between A and B. That is,

\[ \theta_{BA} = \frac{1}{EI} \left( \text{Area of moment diagram from A to B} \right) \]

Theorem 2: The deviation of any point B relative to a tangent drawn to the elastic curve at any other point A, $t_{B/A}$, is equal to the product $1/EI$ multiplied by the moment of an area about B of that part of the moment diagram between points A and B. That is,

\[ t_{B/A} = \frac{1}{EI} \left( \text{Area of moment diagram from A to B} \right) \left( \bar{x}_B \right) \]

Sign Convention:
The rotation, $\theta$, is positive if counterclockwise from the left tangent and negative if clockwise from the left tangent. The deviation, $t$, is positive if the point is above the tangent, and negative if the point is below the tangent.

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