Bending Moment and Shear Force / #Part - 1 ( PDF note 📒 and video solution)

 Bending Moment and Shear Force

Introducton:

        A structural member which is acted upon by  a system of external loads at right angles to its axis is known as beam. 

We see that whenever a horizontal beam is loaded with vertical loads, sometimes, it bends ( i.e., deflects) due to the action of loads. The amount with which a beam bends, depends upon the amount and type of the loads, length of the beam, elasticity of the beam and type of the beam. The scientific way of studying the deflection or any other effect is to draw and analyse the shear force or bending moment diagram of a beam.

Video solution:

PDF Solution: Click PDF note link 

                  Bending Moment and Shear Force  

Types of Beams and Loadings:

Types of Beams:

       The types of Beams are classified as under:

1. Cantilever beam,

2. Simply support beam,

3. Overhanging beam,

4. Rigidly fixed or built-in -beam and 

4. Continuous beam.

                       Types of Beams

Types of Loadings

        A beam may be subject to either or in combination of the following types of loads:

1. Concentrated or point load,

2. Uniformly distributed load and 

3. Uniformly varying load.

                 Types of Loadings

Shearing Force: The shearing Force at any point along a loaded beam is algebraic sum of all the vertical forces acting to one side of the point. 

Thus for the beam shown in below 👇 (a), the shear force at cross-section x-x as measured from the left hand side is            F1 = R1 - W1 - W2, which is equal to the shear force F2 = WE - RE as measured from the right hand side, i.e. F1 = F2, since the beam is in equilibrium. A thin slice of the beam at section x-x subjected to these shear force is shown in Fig below 👇 (b).

         Shear force Fig. (a) and (b)

       Shear force is assumed to be positive if it produces a clockwise moment and negative if it produces an anti-clockwise moment.

Bending Moment: The bending at any point along a loaded beam is the algebraic sum of the moments of all the vertical forces acting to one side of the point about the point. 

In Fig (a)👆, the clockwise moment at cross-section x-x is:

          M1 = R1 x - W1 (x-a) -W2 (x-b)

Whereas the anti-clockwise moment is:

          M2 = R2 ( l-x) - WE (l - x - c)

For equilibrium of the beam: M1 = ME

       Clockwise moments are assumed to be positive and anti-clockwise negative due to all loads acting to the left of a section.

Shear Force and Bending Moment Diagram:

       The shear force and bending moment can be calculated numerically at any particular section. But sometimes, we are interested to know the manner, in which these values very, along the length of the beam. This can be done by plotting the shear force or the bending moment as ordinate and the position of the cross as abscissa. These diagram are very useful, as they give clear picture of the distribution of shear force and bending moment all along the beam.

Conception of Shear Force and Bending Moment:

         Fig. 1.1 shows a Cantiliver AB whose end A is fixed. Let the cantilever carry a vertical load of 4000 N at C.

         For the equilibrium of the Cantiliver the fixed support at A will provide a vertical reaction vertically upwards, of magnitude Va= 4000N.

         Taking moment about A, we have a clockwise moment of 4000 × 2 = 8000 Nm.
         Hence for the equilibrium of the Cantiliver the fixed support at A must also provide a reacting moment or fixing moment of 8000 Nm of an anti-clockwise order.
         
         Now consider a section D. At this section there is a possibility of failure by shear as shown in Fig. 1.1. If such a failure will occur at section D, the Cantiliver is liable to be sheared off into two parts. It is clear that the force acting normal to the center line of the member on each part equal S = 4000 N. The force acting on the right part of the section D. For the case illustrated above the resultant force normal to the axis of the member on the right part of the section is downwards while the resultant force normal to the axis of the member on the left part of the section is upward. Such a shear force will be regarded as a positive shear force.

     Let us now study another effect of the load applied on the cantilever. The cantilever is liable to bend due to the load on it.

     For instance, the cantilever has a tendency to rotate in clockwise direction about A ( Fig. 1.2). Hence, the fixed support at A has to offer a resistance against this rotation.
    Taking moment about A we find that the applied load of 4000 N has a clockwise moment of 4000 × 2 = 8000 Nm. about A. Hence, for the equilibrium of the cantilever, the fixed support at A will provide a reacting or resisting anti-clockwise moment of 8000 Nm. If the support A is not able to provide such a resisting moment the cantilever will not be in equilibrium and will, therefore, rotate about A in the clockwise order.

     The magnitude of the reacting moment at A depends on (i) the magnitude of the load and (ii) the position of the load. We say that the support A provides the necessary fixing or reacting moment at A, and that at the section A of the beam, there is a bending moment of 4000 × 2 = 8000 Nm.

     Now consider, for instance, the section D. Suppose the part DB was free to rotate about D, obviously the load on the part DB would cause the part DB to rotate in a clockwise order about D. Considering the part DB, taking moment about D, we find that there is a clockwise moment of 4000 × 0.8 = 3200 Nm about D. Hence for the equilibrium of the part DB or restoring anti-clockwise moment of 3200 Nm about D.

      Let us now discuss the equilibrium of the part AD (Fig. 1.2). Taking moment about D, we have following moments about D.

     (i) Va × 4000 × 1.2 = 4800 Nm ( clockwise)

     (ii) Couple = 8000 Nm ( anti-clockwise ) 

     ∆ Net moment about D = 8000 - 4800 = 3200 Nm ( anti-clockwise ).
     Hence, for the equilibrium for the part AD, the part DB should provide a clockwise moment of 3200 Nm.                                           

 

     Hence, we find that at the section D, the part DB provides a clockwise moment of 3200 Nm and the part DA provides an anti-clockwise moment of 3200 Nm. We say that at the section D there is bending moment of 3200 Nm.

    The bending moment at the section D is the algebraic sum of the moment of force and reactions acting on one side of the section about the section.

    In the case illustrated above the tendency of the bending moment at D is to bend it so as to produce convexity above the Center line. Such a Bending moment is called a hogging bending moment. It is quite possible that the bending moment at a section may bend the member at the section so as to produce concavity above the center line. Such a bending moment is called a sagging bending moment.

          Thanku  for watching...

  Follow to me .....

 My own YouTube channel : Please subscribe