We were
discussing basics
of shear force and bending moment diagrams and sign
conventions for shear force and bending moment in our recent posts. We
have also discussed the concept to draw shear
force and bending moment diagrams for a cantilever beam with a point load and
shear force and bending moment diagrams for a cantilever beam with a uniformly distributed load during our previous posts.

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Today we
will see here the concept to draw shear force and bending moment diagrams for a
cantilever beam carrying a gradually varying load with the help of this post.

Let us consider one beam AB of length L as displayed in following figure. As we can
see here that one end of the beam AB is fixed at one end i.e. at A and other
end is free i.e. at end B. Therefore, we can say that we have one cantilever
beam here and we will have to find the method to draw shear force and bending
moment diagram when cantilever beam will be loaded with gradually varying load.

Let us
consider that cantilever beam AB is loaded with a gradually varying load from
zero at free end and w per unit length at the fixed end as displayed in
following figure.

Let us
consider one section XX at a distance x from free end i.e. from end B. Now we
will have two portion of the beam AB i.e. left portion and right portion. Let
us deal with the right portion here; you can also go with left portion of the
beam in order to draw shear force and bending moment diagram.

Let us
understand here first about the rate of loading. Rate of loading at free end will
be zero as displayed in figure and rate of loading is w per unit length at the
fixed end and it is also displayed in figure.

Now we
will see the rate of loading or intensity of the load at section XX and it can
be written as (w/L)*x or w*x/L

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**Shear
force diagram**

Let us
assume that F

_{X}is shear force and M_{X}is the bending moment at section XX. Shear force at section XX will be equal to the resultant force acting to the right portion of the section. We will recall here the sign conventions for shear force and bending moment and we will determine here the resultant force acting to the right portion of the section.
Recall the
concept of different
types of loads on beam, during determination of the total
load in case of beam loaded with gradually varying load, we will determine the
area of the triangle and the result i.e. area of the triangle will be the total
load and this total load will be assumed to act at the C.G of the triangle.

Therefore,
shear force at section XX will be written as mentioned here

F

_{X}= Area of triangle BCX
F

_{X}= (w*x/L)*x/2
F

_{X}= w*x^{2}/2L
As we can
see from above equation that shear force at section XX will follow here the
parabolic equation and on the basis of value of x we can conclude value of
shear force at critical points i.e. at point A and at point B.

Shear
force at free end i.e. at point B, x=0

F

_{B}= 0
Similarly,
Shear force at fixed end i.e. at point A, x=L

F

_{A}= w*L/2
Now we have data for shear force at critical points i.e. at point A and at point B and
as we have stated above that shear force at section XX will follow here the
parabolic equation and hence we can draw shear force diagram i.e. SFD as
displayed in following figure.

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**Bending
moment diagram**

Bending
moment at section XX will be written as

M

_{X}= - (Total load for length of x) * (distance between section XX and point of action of total load for length of x)
M

_{X}= - (Total load for length of x) * (distance between section XX and C.G of the triangle )
M

_{X}= - (w*x^{2}/2L) * (x/3)
M

_{X}= - w*x^{3}/6L
Above
equation indicates that bending moment at section XX will follow here the cubic
equation.

Again if
we will recall sign
conventions for shear force and bending moment, we will conclude here that
bending moment at section XX will be negative.

We can
also conclude here that bending moment will be directionally proportional with
the cube of distance x and we can secure here the value of bending moment at
critical points i.e. at point A and at point B.

Bending
moment at free end i.e. at B, value of distance x = 0

M

_{B}= 0
Bending moment
at fixed end i.e. at A, value of distance x = L

M

_{B}= - w.L^{2}/6
Do you
have any suggestions? Please write in comment box

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**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

We will
see another important topic i.e. in the category of strength of material.

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