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Deflection of Beams

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 of elastic curve is EI d2y / dx2 = M 
The product EI is called the flexural rigidity of the beam, If EI is constant,the General equation may be written as:
 equ. [1]
Note :The downward deflection will consider here as -ve , 
where x and y are the coordinates shown in the figure of the elastic curve of the beam under load, y is the deflection of the beam at any distance x. E is the modulus of elasticity of the beam, 
I -represent the moment of inertia about the neutral axis, and M represents the bending moment at a distance x from the end of the beam. 

Direct Integration Method or Double Integration Method 

The double integration method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.
The first integration of equation of (1) yields the slope of the elastic curve and the second integration equation (1) gives the deflection of the beam at any distance x. 
The resulting solution must contain two constants of integration since EI y" = M is of second order. 
These two Integral constants must be evaluated from known conditions concerning the slope deflection at certain points of the beam. 

BOUNDARY CONDITIONS: 

1.A simply supported beam with rigid supports, 
at x = 0 and x = L, the deflection y = 0, and in locating the point of maximum deflection, we simply set the slope of the elastic curve y' to zero. 
2.A Fixed support, 
at fixed end, the deflection y is zero and slope dy/dx ( ø)is zero. 
Macaulay's Method For Beam Deflections 
For complex loading, specially where the span is partially loaded or loaded with number of concentrated loads,this Macaulay's method is more useful. 
Note : 
1. While using the Macaulay's method, the section 'x' is to be taken in the last portion of the beam. 
2. the quantity with in brackets ( ), should be integrated as a whole. 
3.The expression for Bending Moment (Mx) equation can be used at any point, provided the term within the brackets becomes negative is omitted.
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