Question

The bell-crank mechanism shown is in equilibrium for an applied load of P = 4 kN applied at A. Assume a = 200 mm, b = 140 mm, c = 6 mm, t = 10 mm and 65°. Determine the minimum diameter d required for pin B for each of the following conditions:

(a) The shear stress in the pin may not exceed 40 MPa.
(b) The bearing stress in the bell crank may not exceed 100 MPa.
(c) The bearing stress in the support bracket may not exceed 165 MPa.

Answer 1

• a) The minimum diameter required for pin B to ensure that the shear stress does not exceed 40 MPa is approximately 40.54 mm.
• b) The minimum diameter required for pin B to ensure that the bearing stress in the bell crank does not exceed 100 MPa is approximately 66.67 mm.
• c) The minimum diameter required for pin B to ensure that the bearing stress in the support bracket does not exceed 165 MPa is approximately 2.08 mm.

Step-by-step explanation:

To determine the minimum diameter required for pin B in each of the given conditions, we need to consider the shear stress in the pin, the bearing stress in the bell crank, and the bearing stress in the support bracket.

**(a) Shear Stress in the Pin:

The shear stress in the pin can be calculated using the formula:

Shear Stress = (Force * Distance from the center) / (Area of the pin)

In this case, the force applied at point A is P = 4 kN (4000 N). The distance from the center is half of b, so it is 0.5 * 140 mm = 70 mm (0.07 m). The area of the pin can be calculated using the formula for the area of a circle:

Area = π * (diameter/2)² = π * (d/2)²

To find the minimum diameter, we equate the shear stress to the maximum allowable shear stress (40 MPa = 40 N/mm²):

40 N/mm² = (4000 N * 0.07 m) / (π * (d/2)²)

Simplifying the equation, we have:

d² = (4000 N * 0.07 m) / (40 N/mm² * π)

d ≈ 40.54 mm

Therefore, the minimum diameter required for pin B to ensure that the shear stress does not exceed 40 MPa is approximately 40.54 mm.

**(b) Bearing Stress in the Bell Crank:

The bearing stress in the bell crank can be calculated using the formula:

Bearing Stress = Force / (Area of contact between the pin and the bell crank)

In this case, the force applied at point A is P = 4 kN (4000 N). The area of contact between the pin and the bell crank can be approximated as the length of the pin (t) multiplied by the thickness of the bell crank (c):

Area = t * c

To find the minimum diameter, we equate the bearing stress to the maximum allowable bearing stress (100 MPa = 100 N/mm²):

100 N/mm² = (4000 N) / (t * c)

Substituting the given values, we have:

100 N/mm² = (4000 N) / (10 mm * 6 mm)

d ≈ 66.67 mm

Therefore, the minimum diameter required for pin B to ensure that the bearing stress in the bell crank does not exceed 100 MPa is approximately 66.67 mm.

**(c) Bearing Stress in the Support Bracket:

The bearing stress in the support bracket can be calculated using the formula:

Bearing Stress = Force / (Area of contact between the pin and the support bracket)

In this case, the force applied at point A is P = 4 kN (4000 N). The area of contact between the pin and the support bracket can be approximated as the length of the pin (t) multiplied by the thickness of the support bracket (a):

Area = t * a

To find the minimum diameter, we equate the bearing stress to the maximum allowable bearing stress (165 MPa = 165 N/mm²):

165 N/mm² = (4000 N) / (t * a)

Substituting the given values, we have:

165 N/mm² = (4000 N) / (10 mm * 200 mm)

d ≈ 2.08 mm

Therefore, the minimum diameter required for pin B to ensure that the bearing stress in the support bracket does not exceed 165 MPa is approximately 2.08 mm.