Two shafts are connected by means of a flange coupling to transmit a torque of 30 N-m. The flange of the coupling are fastened by four bolts of same material at the radius of 30 mm . What will be the core diameter of bolts if the allowable shear stress of bolt material is 30 MPa?

Concept:

We use torque transmission and shear stress principles in a flange coupling to determine the required core diameter of the bolts.

Given:

• Torque, \( T = 30 \, \text{N-m} = 30 \times 10^3 \, \text{N-mm} \)
• Number of bolts, \( n = 4 \)
• Radius of bolt circle, \( R = 30 \, \text{mm} \)
• Allowable shear stress, \( \tau = 30 \, \text{MPa} = 30 \, \text{N/mm}^2 \)

Step 1: Calculate shear force per bolt

The torque is transmitted through shear forces on the bolts. The shear force \( F \) on each bolt is:

\( T = F \times R \times n \)

\( F = \frac{T}{R \times n} = \frac{30 \times 10^3}{30 \times 4} = 250 \, \text{N} \)

Step 2: Relate shear force to shear stress

The shear stress \( \tau \) on each bolt is given by:

\( \tau = \frac{F}{A} \)

where \( A = \frac{\pi d^2}{4} \) is the cross-sectional area of the bolt core (diameter \( d \)).

Substituting values:

\( 30 = \frac{250}{\frac{\pi d^2}{4}} \)

\( 30 = \frac{1000}{\pi d^2} \)

Step 3: Solve for core diameter

Rearranging for \( d^2 \):

\( d^2 = \frac{1000}{30 \pi} = \frac{100}{3 \pi} \)

Taking the square root:

\( d = \left( \frac{100}{3 \pi} \right)^{1/2} \, \text{mm} \)