Area-Moment Method | Beam Deflections

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Another method of determining the slopes and deflections in beams is the area-moment method, which involves the area of the moment diagram.
 

Deviation and Slope of Beam by Area-Moment Method

 

Theorems of Area-Moment Method
Theorem I
The change in slope between the tangents drawn to the elastic curve at any two points A and B is equal to the product of 1/EI multiplied by the area of the moment diagram between these two points.
 

$ \theta_{AB} = \dfrac{1}{EI}(Area_{AB}) $

 

Theorem II
The deviation of any point B relative to the tangent drawn to the elastic curve at any other point A, in a direction perpendicular to the original position of the beam, is equal to the product of 1/EI multiplied by the moment of an area about B of that part of the moment diagram between points A and B.
 

$ t_{B/A} = \dfrac{1}{EI}(Area_{AB}) \cdot \bar X_B $

 

and
 

$ t_{A/B} = \dfrac{1}{EI}(Area_{AB}) \cdot \bar X_A $

 

Rules of Sign

area-moment-rules-of-sign.jpg

 

  1. The deviation at any point is positive if the point lies above the tangent, negative if the point is below the tangent.
  2. Measured from left tangent, if θ is counterclockwise, the change of slope is positive, negative if θ is clockwise.

 

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