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 during our previous
posts.

Today we will see here the concept to draw shear
force and bending moment diagrams for a cantilever beam with a uniformly
distributed 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 uniformly distributed load.

Let us consider that cantilever beam AB is loaded
with uniformly distributed load of w per unit length over the entire cantilever
beam 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.

###
**Shear
force diagram**

Recall the concept of different
types of loads on beam, during determination of the total load, total
uniformly distributed load will be converted in to point load by multiplying
the rate of loading i.e. w (N/m) with the span of load distribution i.e. L and
will be acting over the midpoint of the length of the uniformly load
distribution.

Let us assume that F

We will recall here the sign conventions for shear force and bending moment and we can conclude here that resultant force acting to the right portion of the section will be w*x and it will be positive.

_{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 can conclude here that resultant force acting to the right portion of the section will be w*x and it will be positive.

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

F

_{X}= w*x
As we can see here from above equation that shear
force at section XX will follow the linear 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
F

_{A}= W
Where, W = w*L

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 the linear equation and hence we can draw shear force
diagram i.e. SFD as displayed in following figure.

###
**Bending
moment diagram**

Bending moment at section XX will be written as

M

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

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

_{X}= - w*x^{2}/2Above equation indicates that bending moment at section XX will follow here the parabolic 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 square 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}/2
M

_{B}= - W.L/2
Where, W = w*L

Do you have any suggestions? Please write in comment
box

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

We will see another important topic i.e. concept to draw shear force and bending moment diagrams for a cantilever beam carrying a gradually varying load in the
category of strength of material.