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Slope and deflection of beam  
Whenever a beam is loaded with transverse loads, the bending moments are developed which cause the axis of beam to deflect from the original undisturbed position as seen in the following figure.

slope and deflection

In the above figure the undisturbed axis of cantilever beam is AB and the deflected shape is AB'. The deflected shape of the beam is also known as elastic curve.  The deviation of point B to B' is shown as deflection  δB and the change in slope of tangent at B is shown as slope θB .

The differential equation of elastic curve was first given by Euler and written as

          elastic curve equation

where   E = Modulus of elasticity of the beam material

             I = moment of inertia of the beam x-section

             v = deflection along y-axis

             x = distance from the origin to the point of deflection

             M = bending moment at section

Deflection calculations are required for buildings, bridges and machines to satisfy some design criteria and to control vibration.

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There are different methods available to determine slope and deflection of beam. Some of them are given below;

  1. Integration method:

  2. Area-moment method
  3. Conjugate-beam method
  4. Strain-energy method (Castigliano's theorems)
  5. Virtual work method (Unit Load method)
  6. Method of superposition (used to find the slope and deflection for different loading cases by taking the resultant of the values for individual cases)

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Visit Problem Solver  for solved examples on deflection

     
 

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Last updated on Thursday January 31, 2013