Monday, 29 May 2017

DEFLECTION AND SLOPE OF SIMPLY SUPPORTED BEAM WITH POINT LOAD AT CENTER

DEFLECTION AND SLOPE OF SIMPLY SUPPORTED BEAM WITH POINT LOAD AT CENTER

In our previous topics, we have seen some important concepts such as Deflection of beams and its various terms, Concepts of direct and bending stresses, shear stress distribution diagram and basic concept of shear force and bending moment in our previous posts.

Now we will start here, in this post, another important topic i.e. deflection and slope of a simply supported beam carrying a point load at the midpoint of the beam. 

We have already seen terminologies and various terms used in deflection of beam with the help of recent posts and now we will be interested here to calculate the deflection and slope of a simply supported beam carrying a point load at the midpoint of the beam with the help of this post.

 

Basic concepts

There are basically three important methods by which we can easily determine the deflection and slope at any section of a loaded beam.
  1. Double integration method
  2. Moment area method
  3. Macaulay’s method
Double integration method and Moment area method are basically used to determine deflection and slope at any section of a loaded beam when beam will be loaded with a single load. 

While Macaulay’s method is basically used to determine deflection and slope at any section of a loaded beam when beam will be loaded with multiple loads.

We will use double integration method here to determine the deflection and slope of a simply supported beam carrying a point load at the midpoint of the beam.

Differential equation for elastic curve of a beam will be used in double integration method to determine the deflection and slope of the loaded beam and hence we must have to recall here the differential equation for elastic curve of a beam.

Differential equation for elastic curve of a beam
After first integration of differential equation, we will have value of slope i.e. dy/dx. Similarly after second integration of differential equation, we will have value of deflection i.e. y.

Let us come to the main subject i.e. determination of deflection and slope of a simply supported beam carrying a point load at the midpoint of the beam

Let us consider a beam AB of length L is simply supported at A and B as displayed in following figure. Let us think that one load W is acting at the midpoint of the beam. 
We have following information from above figure,
RA = Reaction force at support A = W/2
RB = Reaction force at support B = W/2
θA = Slope at support A
θB = Slope at support B

Boundary condition

 
We must be aware with the boundary conditions applicable in such a problem where beam will be simply supported and loaded with a load at the center and we have following boundary condition as mentioned here.
  1. Deflection at end supports i.e. at support A and at support B will be zero, while slope will be maximum.
  2. Deflection will be maximum at the center of the loaded beam
  3. Slope will be zero at the center of the loaded beam
Let us consider one section XX at a distance x from end support A, let us calculate the bending moment about this section.
Mx = + RA. x
Mx = (W/2) x
Mx = W. x/2
We have taken positive sign for above calculated bending moment about section XX. For more detailed information about the sign convention used for bending moment, we request you to please find the post “Sign conventions for bending moment and shear force”.

Let us recall the differential equation of elastic curve of a beam and we can write the expression for bending moment at any section of beam as mentioned here in following figure.
Let us consider the bending moment determined earlier about the section XX and bending moment expression at any section of beam. We will have following equation as displayed here in following figure.
We will now integrate this equation and also we will apply the boundary conditions in order to secure the expression for slope at a section of the beam and we can write the equation for slope for loaded beam as displayed here.

Let us determine the value of slope at end supports i.e. at A and B

 
Slope at support A could be determined by using x = 0 in above slope equation and similarly we can also secure the value of slope at support B by using x = L
Let us integrate the slope equation and considering the boundary condition, we will have equation for deflection at a section of beam.
Above expression indicates the equation of deflection at any section of the loaded beam. We will find here the value of deflection at each support and also at center of the beam.

Let us determine the value of deflection at end supports i.e. at A and B and also at the center of the beam

Deflection at end supports A and B, x = 0
yA = yB = 0

Deflection will be maximum at the center of the loaded beam i.e. at x = L/2. Deflection at the center of the beam, yc could be secured by using the value of x = L/2 in deflection equation as displayed here.
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Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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