When bending is produced in a rod, some filaments of the rod are extended, some are contracted, and the others remain unchanged. So longitudinal stresses are developed in the rod due to the elastic reaction against extension and contraction of the filaments. As a result of the presence of the longitudinal stress and strain, Young’s modulus comes into play.
When the thickness of the rod is very small as compared to the length of the rod, then the shearing stress over any section of the rod is negligible for small bending. So the modulus of rigidity is not involved and it does not occur in the expression of depression.