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 
.