How to find the diameter of bolts in a flanged joint?

Question

So I'm given this question with the following diagram:

Two sections of a pipe are joined in a flanged joint as shown in the Figure below. The difference in pressure from interior to exterior is 2 MPa.

The joint is connected with 12 off bolts, equally pitched around a pitch circle diameter of 350 mm. If the maximum allowable tensile stress on the bolts is 300 MPa, what diameter do the bolts need to be?

My working out

First I find the area of the section that the bolt's diameter takes up:

$A = \pi (0.175+r)^2 - \pi(0.175-r)^2 $

$A = 0.7\pi r\ $

Then I find out the force acting on that area of section and divide by twelve to obtain the force for a section that has one bolt.

$F_{12} = (2\times 10^6)(0.7\pi r) = 14 \times 10^5 \pi r$

$F_1 = 366519.1429r$

I then sub this into the stress equations since I know the maximum tensile stress for 1 bolt:

$A = {F_1 \over \sigma_{max}}$

$\pi r^2 = {366519.1429r \over 300 \times 10^6}$

$Diameter = 0.039 m$

But the answer I'm given says it is supposed to be 5.9mm. Is there anything wrong in my thinking?

Answer 1

Your calculation for the maximum forces to the bolt appears flawed.

The forces resisted by the bolts in tension is equal to the total force of the area on either side of the tube.

You have the tube diameter and thickness, you can calculate the interior area.

Multiple the pressure by this area to find the total force trying to pull these two sides apart.

Divide that by 12 and you have your force in one bolt. Then divide by the allowable bolt stress to get the minimum required area and then diameter.

When I do this I get 5.8mm this is the root area (doesn't include the threads).