Prof.Kodali Srinivas

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). X A   and t   B/A  =  Z  BA   =   1/EI (Area of AB). X B 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.Slop

The deformation of a beam is usually expressed in terms of its deflection from its original unloaded position.  The deflection is measured from the original neutral surface of the beam to the neutral surface of the deformed beam. The configuration assumed by the deformed neutral surface is known as the elastic curve of the beam.  The angle through which the cross-section rotates with respect to the original position is called the angular rotation of the section.  Generalized  Deflection equation Consider a small element dx in deflected beam as shown below. In Cartesian coordinates, the radius of curvature(R) of a curve and deflection y = f(x) is have the relation given below,  The deflection is very small ,the slope of the curve dy/dx is also very small and squaring of this we may get a negligible valve and may neglect in the above equation.  1/R = d2y/dx2 In the derivation of flexure formula, the radius of curvature of a beam is given as  1/R = M /EI  Thus, The differential equation