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.
Slope of a Beam: Slope of a beam is the angle between deflected beam to the actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on a loaded beam. There are many methods to find out the slope and deflection at a section in a loaded beam.
Differential Equation of the Deflection Curve of Beam
Numerous methods are available for the determination of beam deflections. These methods include:
1. Double Integration MethodIn the derivation of flexure formula, the radius of curvature of a beam is
ρ = EI / M
Deflection of beams is so small, such that the slope of the elastic curve dy/dx is very small, and squaring this expression the value becomes practically negligible, hence:
If EI is constant, the equation may be written as:
EIy′′ = M
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
The product EI is called the flexural rigidity of the beam.
Integrating one time:
The first integration y'(dy/dx) yields the Slope of the Elastic Curve.
Second Integration:
EI y =  ∫∫M
The second integration y gives the Deflection of the Beam at any distance x.
(a) Theorems of AreaMoment Method
The method of superposition, in which the applied loading is represented as a series of simple loads for which deflection formulas are available. Then the desired deflection is computed by adding the contributions of the component loads(principle of superposition).
(a) Deflection for Common Loadings
(i) Concentrated load at the free end of cantilever beam (origin at A)
(ii) Concentrated load at any point on the span of cantilever beam
(iii) Uniformly distributed load over the entire length of cantilever beam
(iv) Triangular load, full at the fixed end and zero at the free end
(v) Moment load at the free end of cantilever beam
(vi) Concentrated load at the midspan of simple beam
(vii) Uniformly distributed load over the entire span of simple beam
(viii) Triangle load with zero at one support and full at the other support of simple beam
(ix) Triangular load with zero at each support and full at the midspan of simple beam
Rule.1: The slope at any point of a real beam, relative to the original axis of the beam, is equal to the shear force at the corresponding point of the conjugate beam.
Rule.2: The deflection at any point of a real beam, relative to the original axis of the beam is equal to the bending moment at the corresponding point of the conjugate beam.
Support conditions for the real and conjugate beam:
Deflections by Castigliano's Theorem
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1. What is deflection in beams? 
2. How is beam deflection calculated? 
3. What factors affect beam deflection? 
4. How does beam deflection affect structural integrity? 
5. What are some practical applications of beam deflection analysis? 

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