We were discussing basics of shear force and bending moment diagramsÂ and sign conventions for shear force and bending momentÂ 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.Â

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.Â

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.Â

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

### Shear force diagram

Let us assume that FX is shear force and MX 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
FX = Area of triangle BCX
FX = (w*x/L)*x/2
FX = w*x2/2L

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
FB = 0

Similarly, Shear force at fixed end i.e. at point A, x=L
FA = 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.

### Bending moment diagram

Bending moment at section XX will be written asÂ

MX = - (Total load for length of x) * (distance between section XX and point of action of total load for length of x)

MX = - (Total load for length of x) * (distance between section XX and C.G of the triangle )

MX = - (w*x2/2L) * (x/3)

MX = - w*x3/6L

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
MB= 0

Bending moment at fixed end i.e. at A, value of distance x = L
MB= - w.L2/6

Do you have any suggestions? Please write in comment box

### Reference:

Strength of material, By R. K. Bansal