study notes on deflection of beam

 

Deflection of Beams is an important topic of Strength of Materials (SOM).

This article will provide a basic overview of Deflection of beams along with the solved examples.

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What is Deflection of Beams?

The deflection is usually measured in terms of the deformation of the beam from its original unloaded position.The deflection is usually calculated from the original neutral surface to the neutral surface of the deformed beam.

The configuration attained by the deformed neutral surface is known as the Elastic Curve of the beam.

Factors Affecting Deflection of Beam

The Deflection of a spring beam is affected by the following factors:

  • The length of the beam
  • The cross-sectional shape of the beam
  • Nature of the Material where the deflecting force is applied
  • The way the beam is supported

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Slope of a Beam

It is the angle between the deflected and the actual beam at the same point.

Methods of Determining Beam Deflection

Double-Integration Method

This method is a very powerful tool for calculating the deflection and slope of the beam at any point since we are able to obtain the equation of the elastic curve.

                                                            

                                                                                                                                          

                Where

                M denotes the Bending Moment

                I denotes the Moment of Inertia of the beam section

                y/v denotes the Deflection of the beam

                E denotes the Modulus of elasticity of beam material

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 Area-Moment Method

This is an alternate method for determining the slope and deflection in beams based on the area of the moment diagram.

This method is a semi graphical procedure which computes the slope and deflections utilizing the area under the bending moment diagram.

 This method computes the deflection at a specific location quickly if the bending moment diagram has a simple shape.

Method of Superposition

In this method, the applied load is represented as a series of simple loads. Then the desired deflection is calculated by summing up the contributions of the component loads.

 

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Solved Examples

Q1.      Calculate the value of the Moment M for the beam loaded so that the moment of area about A of the M diagram between A and B is zero. Also, highlight the physical significance of the result?

                                                               

Solution               Ʃ MA = 0                                   

                                                            

                               

4R2 + M = 100 (4) (2)

                R2 = 200 – ¼ M

                ƩMB = 0

                4R1 = 100 (4) (2) + M

                R1 = 200 + ¼ M

                (Area)AB X̅̅A = 0

                ½ (4) (800 - M)(4/3) – 1/3 (4) (800) (1) = 0

                8/3 (800 - M)= 3200/3

                Answer: M = 400 lb.ft

               

The segment AB will deflect downward due to uniform load over span AB.  The moment load applied at the free end will make the slope through B horizontal. Further, this will make the deviation of A from the tangent through B equal to zero.

The moment load will counter the downward deflection due to the uniform load.

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Beam Deflection Formula:

  • Cantilever Beams:

 

Simply supported Beams:

 

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