Deflection of Beams

when there is the vertical displacement at any point on the loaded beam, it is said to be deflection of beams. The maximum deflection of beams occurs where slope is zero.

Slope of the beam is defined as the angle between the deflected beam to the actual beam at the same point.

The general and standard equations for the deflection of beams is given below :

 

 

Where,
M = Bending Moment,
E = Young’s Modulus,
I = Moment of Inertia.

The product of E.I is known as flexural rigidity.

There are many types of beams and for these different types of beams or cases the formula will not be the same. It has to be modified according to the case or the type of the beam. Now let us see the following cases.

Different Types of Cases for the Deflection of Beams.

1. Simply Supported Beam With a Central Point Load :

A simply supported beam AB of length l is carrying a point load at the center of the beam at C. The deflection at the point C will be :

2. Simply Supported Beam With an Eccentric Point Load :

A simply supported beam AB of length l is carrying an eccentric point load at C as shown in the fig. The deflection of the beam is given as follows :

Since b > a, therefore maximum deflection occurs in CB and its distance from B is given by :

and maximum deflection is given by :

3. Simply Supported Beam With Uniformly Distributed Load :

A simply supported beam AB with a uniformly distributed load w/unit length is shown in figure,

The maximum deflection occurs at the mid point C and is given by :

4. Simply Supported Beam With Gradually Varying Load :

A simply supported beam of AB of length l carrying a gradually varying load from zero at B to w/unit length at A, is shown in fig below,

The maximum deflection of beam occurs when  x = 0.519 l and its value is given by :

5. Cantilever Beam With the Point Load at Free End :

A cantilever beam AB of length l carrying a point load at the free end is shown in fig. The deflection at any section X at a distance x from the free end is given by :

The maximum deflection occurs at the free end (when x = 0) and its value is given by

6. Cantilever Beam With a Uniformly Distributed Load :

A cantilever beam AB of length l carrying a uniformly distributed load w/unit length is shown in fig. The deflection at any section X at a distance x from B is given by

The maximum deflection occurs at the free end (when x=0) and its value is given by

                                                                                                                                         

When a cantilever is partially loaded as shown in the fig, then the deflection at point C (at a distance from the fixed end) is given by :

and the maximum deflection occurs at B whose value is given by

7. Cantilever Beam With a Gradually Varying Load :

A cantilever beam  AB of length l carrying a gradually varying load from zero at B to w/unit length at A is shown in fig. The deflection at any section X at a distance x from B is given by

The maximum deflection occurs at the free end (when x = 0) and its value is given by

8. Fixed Beam carrying a central point load :

A fixed beam AB of length l carrying a point load at the center of the beam C as shown in fig. The maximum deflection of beam occurs at C and its value is given by

9. Fixed Beam Carrying an Eccentric Load :

A fixed beam AB of length l carrying an eccentric point load at C as shown in fig. The deflection at any section X at a distance x from A is given by

The maximum distance occurs when,

Hence Maximum deflection of beam,

and deflection under the load at C,

10. Fixed Beam Carrying a Uniformly Distributed Load :

A fixed beam AB of length l carrying a uniformly distributed load of w/unit length as shown in fig. The deflection at any section X at a distance x from A is given by

The maximum deflection of beam occurs at the centre of the beam and its value is given by

                                                                                                                                    ….

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