Slope and Deflection of a 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.

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  δ_B and the change in slope of tangent at B is shown as slope θ_B .

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

         

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.

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

1. Double Integration method/Macaulay's Method
2. Moment-Area method
3. Conjugate-beam method
4. Strain-energy method (Castigliano's theorems)
5. Virtual work method (Unit Load method)

Use our Deflection Calculator Easy to use calculator for different loads on beams

Moment Distribution Calculator^Newfor solving Indeterminate Beams with different loads