# 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.
Beer Johnston asked 12 months ago

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.

Mazurek Gravity answered 12 months ago

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 .