In our previous topics, we have seen some important
concepts such as

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*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.

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**
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.

- Double integration method
- Moment area method
- 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.

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**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,

R

_{A}= Reaction force at support A = W/2
R

_{B}= Reaction force at support B = W/2
θ

_{A}= Slope at support A
θ

_{B}= 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.

- Deflection at end supports i.e. at support A and at support B will be zero, while slope will be maximum.
- Deflection will be maximum at the center of the loaded beam
- 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.

M

_{x}= + R_{A}. x
M

_{x}= (W/2) x
M

_{x}= 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.

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**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.

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**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

y

_{A}= y_{B}= 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, y

_{c}could be secured by using the value of x = L/2 in deflection equation as displayed here.
Please
comment your feedback and suggestions in comment box provided at the end of
this post.

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**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

Nice illustration, thanks.

ReplyDeleteWhat if l/2<x<l

ReplyDeleteLoad is acting at l/2 from the supports so section is taken as x distance from any support which should be less than l/2 not greater

Delete