Bending Moment and Shear Force / #Part - 2 ( With PDF note 📒 and video solution)

 Definition of shear force and bending moment:

Shear Force:  

           The shearing force at any point a loaded beam is the algebraic sum of all the vertical forces acting to one side of the point. 
Thus for the beam shown in Fig. 1.1 (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 - R2 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. 1.1 (b).

Bending Moment:

         The bending moment at any point along a loaded beam is the algebraic sum of the moments of all the vertical force acting to one side of the point about the point.

   In Fig. 1.1 (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 = M2

Video solution:

>                                                                                                                                                                       PDF Solution:Click to download PDF note 📒##################################

Units for Shear Force and Bending Moment:

1. Units of Shear Force = N, Kn.

2. Units of Bending Moment = K Nm, Nm.

Sign Conventions for Shear Force and Bending Moment:

1. Shear Force: We know that as the shear force is the unbalanced vertical force, therefore it tends to slide one portion of the beam, upwards or downwards with respect to the other. We take shear force at a section as positive, when the left hand portion tends to slide upwards or the right hand portion tends to slide downwards shown in Fig. 1.2 (a).

          Similarly, we take shear force at a section as negative, when the left hand portion tends to slide downwards or the right hand portion tends to slide upwards as shown in Fig. 1.2 ( b).

2. Bending Moment: We take the bending moment at a section as positive, ij it tends to bend the beam at that point to a curvature having concavity at the top, shown in Fig. 1.3 (c).

   On the other hand, we take the bending moment at a section as negative, if it tends to bends the beam at that point to a curvature having convexity at the top , as shown in Fig. 1.3 (d).

         We often call the positive bending moment as sagging moment and negative bending moment as hogging moment.

        Another way of assigning the sign convention to the bending moment is by the direction in which it acts at a section. We take the bending moment at a section as positive, when it is acting in clockwise direction to the left or in anti-clockwise direction to the right.

Note - While calculating bending moment or shear force, at a section the beam will be assumed to be weightless.

#We find different sign convention in differents website and books, regarding shear force and bending moment at a section. But in this content the following sign conventions will be used, which are widely followed and international recognised.

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