5.2 Shear and bending moment diagrams
Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of a structural element such as a beam. These diagrams can be used to easily determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure. Another application of shear and moment diagrams is that the deflection of a beam can be easily determined using either the moment area method or the conjugate beam method. For the determination of maximum absolute values of shear and of the bending moment in a beam V and M plotted against x. This plotting method is important for the determination of the deflection of a beam.
A
A
EXAMPLE
P
1
W
C
P
2
B x
A
P
1
W
C
V
M
R
A
M’
V’
C
B
R
B
Figure 5.2.1: Determination of V and M

Free-body diagrams
As a simple starting example, consider a beam clamped (cantilevered) at one end and subjected to a load P at the free end. A free body diagram of a section cut transversely at position x shows that a shear force V and a moment M must exist on the cut section to maintain equilibrium. As usual, we will consider section areas whose normal point in the +x direction to be positive; then shear forces pointing in the +y position on the +x faces will be considered positive. Moments whose vector direction as given by the right-hand rule are in the +z direction will be positive when acting on +x faces. Another way to recognize positive bending moments is that they cause the bending shape to be concave upward.
∑F
Y
= 0 = V + P ; V = constant = − P
∑M
0
= 0 = − M + Px ; M = M (x) = Px
Note that the moment increases with distance from the loaded end, so the magnitude of the maximum values of M compared with V increases as the beam becomes longer. This is true of most beams, so shear effects are usually more important in beams with small length-to-height ratios.
V
-P (-)
M (+)
PL
Figure 5.2.2: Shear and Bending Moment Diagram
Summarizing the sign convections we have presented, we state:
The shear V and the bending moment M at a given point of a beam are said to be positive when the internal forces and couples acting on each portion of the beam are directed as shown. These convections easily remembered by remembering:
1.
The shear at any point is positive when the external forces shearing off the beam as
Figure 5.6 (b)

2.
The bending moment is positive when the external forces acting on the beam as Figure
5.6 (c).
M’
V’
M a)Internal forces(All +ve)
V b) Effect of external forces(+ve shear) c)Effect of external forces(+ve bending moment)
Figure 5.2.3: Sign convection for shear and bending moment
As stated earlier, the stresses and deflections will be shown to be functions of V and M, so it is important to be able to compute how these quantities vary along the beam’s length. Plots of V(x) and M(x) are known as shear and bending moment diagrams and it is necessary to obtain them before the stresses can be determined.
We must note these important things while doing this shear and bending-moment diagram:
1. First draw the free-body-diagram of the beam with sufficient room under it for the shear and moment diagrams (if needed, solve for support reactions first).
2. Draw the shear diagram under the free-body-diagram. The distributed load is the slope of the shear diagram and each point load represents a jump in the shear diagram. Label all the loads on the shear diagram
3. Draw the moment diagram below the shear diagram. The shear load is the slope of the moment and point moments result in jumps in the moment diagram. The area under the shear diagram equals the change in moment over the segment considered (up to any jumps due to point moments). Label the value of the moment at all important points on the moment diagram

Real Life Application
Discuss the shear forces and bending moment set up on the bookshelf when there is a constant load. Draw the shear and bending-moment diagram based on this application.
Figure 2.5.3: Book Shelf