**In this article Learn :** shear force, bending moment, shear force and bending moment,basic concepts of shear force and bending moment,sign convention shear force and bending moment

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**SHEAR FORCE AND BENDING MOMENT**

Consider a simply supported beam AB [Fig. 3.7(a)] having some point loads. If the beam is to be cut in two parts at section X and the right hand portion of the beam is removed, the equilibrium of the left portion will be under the action of the external forces W_{1}, W_{2}, W_{3} and reaction R_{1}, and under the action of internal forces which are distributed over the cross-section X. The internal forces at X represent the acting of the right portion of the beam on the portion.

From the law of parallel forces, the external forces W_{1}, W_{2}, W_{3} and reaction R_{1} can be represented by a single vertical force F acting at x’ from the support. Introducing equal and opposite forces F at the section X [Fig. 3.7(b)] the whole loading is equivalent to an unbalanced vertical force F acting upward and a moment M [Fig. 3.7(c)] in the clockwise direction acting in the plane of X. Similarly if the equilibrium of the right hand portion of the beam is considered, the loading is reduced to an unbalanced vertical forces F, acting downwards and a moment M acting in the anticlockwise direction as shown in Fig. 3.7(d).

Thus, the action of the external loading can be represented by

### (i) a shear force F,

which is defined as the algebraic sum of all vertical forces either to the left or to the right hand side of a section.

### (ii) a bending moment M,

which is defined as the algebraic sum of the moments of all vertical forces either to the left or to the right of a section.

**Sign convention shear force and bending moment :**

For writing the general expressions for B.M. and S.F., we shall be adopting the following sign conventions :

#### (a) *For Shear Force :*

Shear force having an upward direction to the right hand side of a section or downwards to the left of the section will be taken positive. Similarly, a negative S.F. will be one that has a downward direction to the right of the section or upward direction to the left of the section. See Fig. 3.7(e).

#### (b) *For Bending moment :*

A B.M. causing concavity upwards will be taken as negative and called as sagging B.M. Similarly, a B.M. causing convexity upwards will be taken as positive and called hogging B.M. See Fig. 3.7(f).

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