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. deflection and slope of a cantilever beam loaded with gradually
varying load 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 calculate the deflection and slope
of a cantilever beam loaded with gradually varying load with the help of this
post.

Cantilever beam is basically defined as a beam where
one end of beam will be fixed and other end of beam will be free.

Uniformly varying load or gradually
varying load is the load which will be distributed
over the length of the beam in such a way that rate of loading will not be
uniform but also vary from point to point throughout the distribution length of
the beam.

Uniformly varying load is also termed as
triangular load. Let us see the following figure, a beam AB of length L is
loaded with uniformly varying load or we can also say gradually
varying load.

###
**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 cantilever beam which is loaded with gradually
varying load.

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*

*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 cantilever beam which is loaded with gradually
varying load.

Let us consider a cantilever beam AB of length L
which is fixed at the support A and free at point B and loaded with gradually
varying load in such a way that load is varying from 0 at B to w per unit run
at A, as displayed in following figure.

We have following information from above figure,

w = Rate of gradually varying loading in N/m

AB = Position of the cantilever beam before loading

AB’ = Position of the cantilever beam after loading

Î¸

_{A}= Slope at support A
Î¸

_{B}= Slope at support B
y

_{B }= Deflection at free end B###
*Boundary
condition*** **

*Boundary condition*

We must be aware with the boundary conditions
applicable in such a problem where beam will be a cantilever
beam with gradually
varying load. We have following boundary condition as mentioned here.

At point A, Deflection will be zero

At point A, Slope will be zero

At point B, Deflection will be maximum

At point B, Slope will also be maximum

Let us consider one section XX at a distance x from
end support A, let us calculate the bending moment about this section. Let us
consider the triangle XCB and we can easily determine the length XC i.e. load
at section XX.

XC = w (L-x)/L

BX = (L-x)

Force or load acting due to the gradual loading up
to section XX

F

_{X}= (XC * BX)/2
F

_{X}= w (L-x)^{ 2 }/2L
Distance of point of action of load F

_{X }from section XX = (L-x)/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”.

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 expressions for slope as
well as deflection at a section of the beam and we can write the equations for
slope and deflection for loaded beam as displayed here.

Where, C

_{1}and C_{2}are the constant of integration and we can secure the value of these constant C_{1}and C_{2}by considering and applying the boundary condition.
Let us use the boundary condition as we have seen
above.

At point A i.e. x = 0, Slope will be zero i.e. dy/dx
=0

At point A i.e. x = 0, deflection will be zero i.e.
y = 0

After applying the boundary conditions in above
equations of slope and deflection of beam, we will have following values of
constant C

_{1}and C_{2}as mentioned here.
C

_{1 }= - wL^{3}/24
C

_{2 }= wL^{4}/120
Let us insert the values of C

_{1}and C_{2}in slope equation and in deflection equation too and we will have the final equation of slope and also equation of deflection at any section of the loaded beam. We can see the slope equation and deflection equation in following figure.###
**Slope
at the free end**

*At x = L, Î¸*_{B}= Slope at end B
Let us use the slope equation and insert the value
of x = L, we will have value of slope at support B i.e. Î¸

_{B}
Î¸

_{B}= - w.L^{3}/24EI
Negative sign represents that tangent at end B makes
an angle with beam axis AB in anti-clockwise direction.

###
**Maximum
deflection**

At point B i.e. x = L, Deflection will be maximum

Let us use the deflection equation and insert the
value of x = L in deflection equation, we will have value of deflection at
point B.

y

_{B}= - wL^{4}/30EI
Negative sign represents here that deflection in the
loaded beam will be in downward direction.

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

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