Application of Moment - Area Method for Simply supported Beams

Moment - Area Method - Mohr's Theorems 

Otto Mohr (1835-1918)
 German Civil Engineer

In this method, to determining the slopes and deflections in beams involves the Area and Moment of the 'Bending moment diagram'. 

( B.M.D )

First Theorem :

The angle between tangents drawn at any two points on deflected curve, is equal to area of the M/EI diagram between the two points.

The angle between the tangents from points A and B 

O AB =  1/EI (Area of AB)

Second Theorem :

The intercept on a vertical line made by two tangents drawn at the two points on the deflected curve is equal to the moment of the M/EI diagram between two points about the vertical line.

The intercepts at point A,and B are,

t A/B = Z AB  =1/EI (Area of AB). XA

  and

t B/A = Z BA  = 1/EI (Area of AB). XB

Sign Convention :

For +ve B.M. ,the area of  M/EI diagram is considered +ve , and for -ve B.M.,the area of M/EI diagram is considered as -ve  

1. 1.Slope :

1. Measured from left tangent, if θ is anti-clockwise, the change of slope is positive, negative if θ is clockwise.

1. 2. Deflection

1. 1. The intercept  at any point is positive if it lies above the tangent, negative if the it is below the tangent.
   2. 
      
   3. Application of Moment - Area Method for Cantilever Beams
      

      1. 1.  

            

             

            From the figure above, the deflection at B denoted as δ_B is equal to the deviation of B from a tangent line through A denoted as t_B/A. This is because the tangent line through A lies with the neutral axis of the beam.

            Similarly the angle between the tangents will equal to slope of the point. 

         2. Application of Moment - Area Method for Simply supported Beams

            
            

            The deflection δ at some point B of a simply supported beam can be obtained by the following steps:
            

            1.  

            

            
            

            1. Compute the vertical intercept between Cand A  = ZCA

            2. Compute the vertical intercept between B and C  = ZBA

            3. Slope at support A = O A = t CA /L = ZCA / L

            4. Slope at support C = O C = tCA/ L = ZCA / L
   4.  Generally, the tangential deviation t is not equal to the beam deflection. In cantilever beams, however, the tangent drawn to the elastic curve at the wall is horizontal and coincidence therefore with the neutral axis of the beam. The tangential deviation or vertical intercept  in this case is equal to the deflection of the beam as shown below.