CONCEPT OF SHEAR FORCE AND BENDING MOMENT

1.1 Shear force and Bending moment:

When a beam is subjected to a set of loads and reactions, as a result of which the internal forces and moments tend to setup within the beam This internal force is the shear force  and the internal moments are the bending moments. 

For illustration,  consider a beam with a given set of loading supported at two points A &  B giving rise to reactions at the two supports. 

                      (fig.  1.1)

Let's cut the beam into two parts &  let's assume that the resultant of loads (F) and reactions to the left of section xx is vertically upwards. 

                       (fig.  1.2)

For equilibrium of the section, the resultant of loads and reactions to the right of section xx has to be equal to ''F" but vertically downwards. This force is known as shear force. 

Hence SHEAR FORCE is defined as,  the algebraic sum of all the vertical forces either to the right or to the left of a section.


1.2 Sign conventions for shear force:

1). The shear force will be positive if the resultant of forces to the left of the section is upwards, or to the right of the section is downwards.

                            (fig.  1.3)

2). Similarly, if the resultant of forces to the left of the section is downwards or to right of the section is upwards, the shear force will be negative.

                      (fig.  1.4)

Now let's assume,  "M" be the resultant of all the moments about a point just to the left of section xx and let this resultant moment be in clockwise direction.

                      (fig.  1.5)

For equilibrium, the resultant of all the moments about a point just to the right of section xx has to be equal to "M" but the direction of this resultant moment will be counterclockwise.

This resultant moment is known as Bending moment.

Hence Bending Moment is defined as, the Algebraic sum of all the moments either to the right or to the left of a section. 

1.3 Sign conventions for Bending moment:

1). The bending moment at a section is positive if it tends to bend the beam in a manner such that it can retain water on it's curvature(concavity at the top) as shown in Figure. Positive bending moment is also called sagging moment. 

                           (fig.  1.6)

2). Similarly, the bending moment at a section is negative if it tends to bend the beam in a curvature having convexity at the top as shown in Figure. The negative bending moment is also called hogging moment. 

                              (fig.  1.7)

1.4 Shear force and bending moment diagram:

A diagram which shows the variation of shear force along the length of beam is known as shear force diagram and a diagram which shows the variation of bending moment along the length of beam is known as bending moment diagram. 

                      (fig. 1.8)

•  Before drawing SFD &  BMD we must know the different types of beams and types of loads acting on them.

1.5 Types of beams:

1.5.1 Simply supported beam:

Supported or resting freely at supports. 

                           (fig. 1.9)

1.5.2 Overhanging beam:

If the span of the beam is extended beyond it's supports, such type of beam is known as overhanging beam. 

                       (fig. 1.10)

1.5.3 Fixed Beam:

A beam whose both the ends of it's span are fixed or built in walls is known as fixed beam. 

                         (fig. 1.11)

1.5.4 Cantilever Beam:

A beam whose span is fixed at one end and free at the other end is known as a cantilever beam. 

                           (fig. 1.12)

1.5.5 Continuous Beam:

A beam which has more than two supports is known as a continuous beam. 

                        (fig. 1.13)

1.6 Types of loads:

1.6.1 Concentrated or point Load:

A load which is considered to act at a point is known as a concentrated load. 

                        (fig. 1.14)

1.6.2. Uniformly distributed load (UDL):

A load which is spread over a beam in a way that the rate of loading is uniform along the length (i.e, load per unit length is uniform) 

                         (fig. 1.15)

                     

1.6.3. Uniformly varying load (UVL):

A load which is spread over a beam in a way, that the rate of loading varies from point to point along the beam

                     (fig. 1.16)

1.7 Relationship between load, shear force and bending moment: