Shear Force and Bending Moment Diagrams in Cantilever Beams

Q.1 Draw the shear force and bending moment diagrams of a cantilever beam carrying a point load W at its free end as shown in figure. 

Solution:

For cantilevered beams we need not find support reactions, if we considering the free-end of the beam as the initial starting point of the analysis.

Let us consider one section XX at a distance x from free end i.e. from end B. 

Now we will have two portions of the beam AB i.e. left portion and right portion. 

Let us deal with the right portion and draw the free body diagram.

(you can also go with left portion of the beam, you required to find the support reactions ) 

The section is valid in between B and A, becouse of there is no load in between the points.

Let us assume that Vx 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. 

Vx = + W ---(1)

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 and it will be positive.

When x=0 ,the shear force at free end B is 

VB = +W

When x=L ,the shear force at fixed end A is

VA = +W 

Bending moment at section XX will be written as

Mx = - W. x   -- (2)

Above equation indicates that bending moment will follow here the linear 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.

Bending moment will be directionally proportional with the distance x and we can secure here the value of bending moment at point A an at point B as mentioned here.

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

MA= - W.L

After calculating shear force and bending moments values at critical points the SFD and BMD are drawn below.

(The negative values are plotted to scale below a horizontal reference line and positive values are plotted above the line.)

Shear Force and Bending Moment Diagrams

Q.2 Draw the shear force and bending moment diagrams of a cantilever beam carrying  point loads as shown in figure. 

Solution:

For cantilevered beams we need not find support reactions, if we considering the free-end of the beam as the initial starting point of the analysis.

Segment AB

Let us consider one section XX in between A and B,at a distance x from free end A.

The limits of x is zero to 1 meter.

The S.F. equation Vx = +10 KN

The B.M. equation Mx = - 10.X

section XX in between A and B

When x = 0,

Va = 10 KN, and Ma = 0

When x = 1m,

Vb = 10 KN. and Mb = -10 KN-m

Segment BC

Now we will consider another section XX in between B and C,at a distance x from free end A, and consider right side portion.

The limits of x is 1 meter to 2 meters.

The S.F. equation 

Vx = (+10 + 20 )KN = + 30 KN

The B.M. equation Mx = - 10.X - 20 (x- 1) 

section XX in between B and C

When x= 1m,

Vb = 30 KN, and Ma = - 10

When x = 2m,

Vb = 30 KN. and Mb = - 40 KN-m

Segment CD

Now we will consider anther section XX in between C and D,at a distance x from free end A.

The limits of x is 2 meter to 3 meters.

The S.F. equation 

Vx = (+10 + 20 +20 )KN = + 50 KN

The B.M equations 

Mx = - 10.X - 20 (x- 1) - 20 (x- 2)

section XX in between C and D

When x= 2m,

Vb = +50 KN, and Ma = - 40 KN-m

When x = 3m,

Vb = +50 KN. and Mb = - 90 KN-m

Plot the SFD and BMD by using the above values.

Loading Diagram

Shear Force Diagram

Bending Moment Diagram

Q.3 Draw the shear force and bending moment diagrams of a cantilever beam carrying uniformly distributed load w (N/m) as shown in figure.

Solution :

For cantilevered beams we need not find support reactions, if we considering the free-end of the beam as the initial starting point of the analysis.
Let us consider one section XX in between A and B,at a distance x from free end B and consider right side portion.

The limits of x is zero to L.
The S.F. equation at the section,Vx = +wx
The B.M. equation at the section, Mx = - wx. {x/2}

The above B.M. equation is represented parabola equation, 

When x= 0,

The shear force at B is Vb = 0, 

and The bending moment Mb = 0 

When x = L,

The shear force at B is Va = +wL. 

and The bending moment Ma = - w.L. L/2

Plot the SFD and BMD by using the above values.

Assignment :

Q1.1. Draw the shear force and bending moment diagrams of a cantilever beam carrying  point loads as shown in figure. 

Q1.2. Draw the shear force and bending moment diagrams of a cantilever beam carrying  a point load as shown in figure.

Q1.1. Draw the shear force and bending moment diagrams of a cantilever beam carrying  uniformly distributed load as shown in figure.