# A 400-lb vertical force is applied at D to a gear attached to the solid 1-in. diameter shaft AB. Determine the principal stresses and the maximum shearing stress at point H located as shown on top of the shaft.

Question-AnswerCategory: Strength of MaterialsA 400-lb vertical force is applied at D to a gear attached to the solid 1-in. diameter shaft AB. Determine the principal stresses and the maximum shearing stress at point H located as shown on top of the shaft.

A 400-lb vertical force is applied at D to a gear attached to the solid 1-in. diameter shaft AB. Determine the principal stresses and the maximum shearing stress at point H located as shown on top of the shaft. Step No: 1

Consider a section at H.
Calculate the moment at H using the formula,
Moment Here, F is the force and l is the perpendicular distance of the line of action of force from H.
Substitute for F and for l. Step No: 2

Calculate the torque at H using the formula,
Torque Here, is the radius of the gear.
Substitute for F and for . Step No: 3

Calculate the polar moment of inertia of the shaft using the formula, Here, d is the diameter of the shaft.
Substitute for d. Calculate the moment of inertia of the shaft using the relation, Substitute for J. Step No: 4

Calculate the shear stress using the relation, Here, r is the radius of the shaft.
Substitute for  for , and for r.  Step No: 5

Calculate the bending stress using the relation, Here, y is the half of the diameter of the shaft.
Substitute for  for , and for y.  Step No: 6

Transverse shear stress at point H is zero.
Write the resultant stress. Calculate the average stress as follows: Substitute for and 0 for . Step No: 7

Calculate the maximum shear stress as follows: Substitute for , 0 for , and for . Therefore the maximum shear stress is .

Step No: 8

Calculate the maximum principal stress using the formula, Substitute for and for . Therefore, the maximum principal stress is .

Step No: 9

Calculate the minimum principal stress using the formula, Substitute for and for . Therefore, the minimum principal stress is .