Derive the formula for shear stress in the beam.

• The vertical shear force at the section of beam results in shear stress that varies along the depth of the beam. Variation of shear stress along the depth of the beam is of significant importance and is analyzed below.

• Consider two section xx and yy in the loaded beam as shown in the figure. Let the distance between the two sections be 'dx'.

• Let M and M+dM be the bending moments at sections xx and yy respectively.

• Let us consider plane EF at a distance from N.A.

Consider an elemental area 'da' at the distance 'y' from N.A If is the bending stress on the elemental area on the section xx and is the bending stress on the elemental area on the section yy.

Then and

• The thrust on elemental area 'da' on the face AA'E'E of the section xx is = and on the face CC'F'F on the section yy is
• The thrust on face AA'E'E of section xx is and on face CC'F'F of section yy is

The resultant thrust experienced by the portion of the beam between two sections xx and yy and the planes AA'C'C and EE'F'F is (P2-P1) from right to left. This resultant thrust causes shearing of the portion of the beam at the plane EE'F'F. This shearing force is resisted by shear stress 'q' generated on the surface EE'F'F.

Therefore, where b is the width of beam i.e FF'.

or

• SInce is the shear force and is the moment of area of X-section of the beam above the plane EE'F'F about N.A

i.e is Shear Stress Formula

Where A is the area of X-section of the beam above EF or FF' and is the distance of the centroid of this area from N.A.