Shear Force and Bending Moment

Types of Beam

Simply Supported Beam

If the ends of a beam are made to rest freely on supports it is called a simply (freely) supported beam.

Fixed Beam

If a beam is fixed at both ends it is called fixed beam its another name is encastre or built-in beam.

Cantilever Beam

If a beam is fixed at one end while other end is free, it is called cantilever beam.

Continuous Beam

If more than two supports are provided to beam, it is called continuous beam.

Shear Force

It is the internal resistance developed at any section to maintain free body equilibrium of either left or right part of the section.

Sign Convention

Shear force having an upward direction to the left-hand side of the section or downward direction to the right-hand side of section will be taken positive and vice-versa.
It may be a horizontal or vertical Shear force at any section is the algebraic sum of all transverse forces either from the left or right of that section.

Bending Moment

Bending moment at any section is the internal reaction due to all the transverse force either from the left side or from the right side of that section.

It is equal to the algebraic sum of moments in that section either from the left or from the right side of that section.
The bending moment is different from the twisting moment.

Sign convention of Bending moment

A bending moment causing concavity upward will be taken as positive and called sagging bending moment. 
A bending moment causing convexity upward will be taken as negative and will be called a hogging bending moment.

Relationship Between Bending Moment (M), Shear Force (S) and Loading Rate (w)

The rate of change of shear force is equal to load. 
w = Load per unit length
The negative slope represents downward loading.

The rate of change of bending moment along the length of the beam is equal to shear force.
At hinge, the bending moment will be zero.
The bending moment is maximum or minimum when the shear force is zero or changes sign at a section.
If the degree of loading curve= n then
degree of shear force curve= n+1
and degree of bending moment curve= n+2
The point of contra-flexure/inflexion is that point where bending moment changes its sign.

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