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 and deflection and slope of a cantilever beam with point load at free end 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
uniformly distributed load throughout the length of the 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 calculate the deflection and slope of a cantilever beam loaded
with uniformly distributed load throughout the length of the beam 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 distributed load is the load
which will be distributed over the length of the beam in such a way that rate
of loading will be uniform throughout the distribution length of the beam
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
uniformly distributed load throughout the length 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 cantilever beam which is loaded with uniformly
distributed load throughout the length of the beam.
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 uniformly
distributed load as displayed in following figure.
We have following information from above figure,
w = Rate of loading in N/m
AB = Position of the cantilever beam before loading
AB’ = Position of the cantilever beam after loading
θA = Slope at support A
θB = Slope at support B
yB = Deflection at free end B
Total load due to udl = W = w.L
Boundary condition
We
must be aware with the boundary conditions applicable in such a problem where beam
will be a cantilever beam with udl. 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.
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, C1 and C2
are the constant of integration and we can secure the value of these constant C1
and C2 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 C1 and C2 as mentioned here.
C1 = - wL3/6
C2 = wL4/24
Let us insert the values of C1
and C2 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, θB =
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. θB
θB = - w.L3/6EI
θB = -W.L2/6EI
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.
yB = - wL4/8EI
yB = - WL3/8EI
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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