Shear and Bending

Shear Stress

If 2 equal and opposite parallel forces Q, not in the same straight line, act on parallel faces of a member then it is said to be loaded in shear.
Shear force in a beam at any section is the force transverse to the beam tending to cause it to shear across the section.
Consider the above Cantilever.

The load P, which is assumed to act at a point is called a concentrated or point load

Consider portion CD
At section C, for balance of forces there must be an upward force Q equal and opposite to the load P at D.

This force is provided by the resistance of the beam to shear at the plane B (this being coincident wit the plane C) Note that it doesn't matter where we make the imaginary cut in the beam.

Now for the portion AB the shear force at the wall (A) must be equal and opposite to the shear force at the 'cut' (B)

The shear force in this beam is constant and = P all along the cantilever beam.
Sign Convention

Positive if right hand side tends to slide downwards.
Shear force diagram

A graph showing the variation of shear force along a beam.
Bending Moment

The bending effect at any section X of a concentrated load P at D is measured by the applied moment Px, where x is the perpindicular distance of the line of action of P from the section X.
Sign Convention

Positive if it tends to make the beam sag
Negative if it tends to make the beam hog (bend upwards)

When more than one load acts on a beam the bending moment at any section is the algebraic sum of the moments due to all the loads on one side of the section. (Doesn't matter which side)
Bending Moment Diagram

The variation of bending moment along the beam is shown in a bending moment diagram.

Using the definition of bending moment above, verify that the bending moment diagram below is correct.

The bending moment at the wall is known as the fixing moment.