In our previous topics, we have seen some important
concepts such as deflection
and slope of a simply supported beam with point load, deflection
and slope of a simply supported beam carrying uniformly distributed load, deflection
and slope of a cantilever beam with point load at free end and
deflection and slope of a cantilever beam loaded with uniformly distributed load in
our previous post.

Now we will start here, in this post, another
important topic i.e. Macaulay’s method to determine the deflection and slope of
a loaded beam with the help of this post.

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 discuss the Macaulay’s method and
its significance 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.

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.

Differential
equation for elastic curve of a beam will also be used in Macaulay’s
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*

*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.

As we have seen above that 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.

Let us consider one case of a simply supported beam with
multiple loading and let us write here the complete steps one by one in order
to understand the complete process of Macaulay’s method.

Let us consider a beam AB of length L is
simply supported at A and B as displayed in following figure,
Let us consider that there are three loads W

_{1}, W_{2}and W_{3}are acting on the beam AB at point C, D and E respectively and we have displayed these loads in following figure.
We have following information from above figure,

W

_{1}, W_{2}and W_{3}= Loads acting on beam AB
a

_{1}, a_{2}and a_{3}= Distance of point load W_{1}, W_{2}and W_{3}respectively from support A
AB = Position of the beam before loading

AFB = Position of the beam after loading

θ

_{A}= Slope at support A
θ

_{B}= Slope at support B
y

_{C}, y_{D}and y_{E }= Deflection at point C, D and E respectively###
*Boundary
condition*** **

*Boundary condition*

We must be aware with the boundary
conditions applicable in such a problem where beam will be simply supported and
loaded with multiple point loads.

Deflection at end supports i.e. at
support A and at support B will be zero, while slope will be maximum.

####
*Step: 1*

*Step: 1*

First of all we will have to determine
the value of reaction forces. Here in this case, we need to secure the value of
R

_{A}and R_{B}.####
*Step: 2*

*Step: 2*

Now we will have to assume one section
XX at a distance x from the left hand support. Here in this case, we will
assume one section XX extreme for away from support A and let us consider that
section XX is having x distance from support A.

####
*Step: 3*

*Step: 3*

Now we will have to secure the moment of
all the forces about section XX and we can write the moment equation as
mentioned here.

M

_{X}= R_{A}. x – W_{1}(x- a_{1}) – W_{2}(x- a_{2}) – W_{3}(x- a_{3})We have taken the concept of sign convention to provide the suitable 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”.

####
*Step: 4*

*Step: 4*

Now we will have to consider Differential
equation for elastic curve of a beam and bending moment determined
earlier about the section XX and we will have to insert the expression of bending
moment in above equation. We will have following equation as displayed here in
following figure.

####
*
Step: 5*

*Step: 5*

We will now integrate this equation. 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.

####
*Step: 6*

*Step: 6*

We will apply the boundary conditions in order to
secure the values of constant of integration i.e. C

_{1}and C_{2}.
At x
= 0, Deflection (y) = 0

At x
= L, Deflection (y) = 0

####
*Step: 7*

*Step: 7*

We
will now insert the value of C

_{1}and C_{2}in slope equation and in deflection equation too in order to secure the final equation for slope and deflection at any section of the loaded beam.####
*Step: 8*

*Step: 8*

We will
use the value of x for a considered point and we can easily determine the
values of deflection and slope of the beam AB at that respective point.

We will see another topic in our next post. Please
comment your feedback and suggestions in comment box provided at the end of
this post.

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

###
**Also
read**

*Keep reading............*

Thanks, this is really helpful. Please continue the explanations.

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