Recently, at work, there was some work where someone needed to break out ASME B31.3 and figure out the allowable pressure for a given length of pipe -- back calculating from what was there to what was "OK".

This is neither here nor there, but since I was around I read through what they were doing. I have only vague memories of the intro mechanical engineering courses required for chemical engineers so I was interested in seeing what that entailed and where those equations came from.

In B31.3 is the basic minimum thickness formula: $$ t = \frac{ p D}{2 \left( SEW + pY \right)} $$

Or in terms of stress $$ SEW = \sigma = p \left( \frac{D}{2t} - Y \right) $$

Where:

*t*is the thickness of the pipe,*p*is pressure,*D*is the outside diameter of the pipe,*S*is the allowable stress, with*E*and*W*efficiency factors for the joint and weld, I am just going to lump it all together into $\sigma$,*Y*is the "Y factor"... okay...

I asked around at work but none of the people knew what the physical significance of the Y factor was, the best answer I got was that it was a fitting parameter to make the equation line up with empirical data.

Well, let's rewind and do some math. Suppose we have a pipe with internal pressure *p* at inner radius *r _{i}* and zero pressure at the outer radius

*r*. First consider a small element of a cylindrical shell within this pipe. If we do a force balance:

_{o}We end up with: $$ \sigma_c - \sigma_r = r \frac{d \sigma_r}{dr} $$

Where *σ _{c}* is the stress in the circumferential direction, i.e. the hoop stress, and

*σ*is the stress in the radial direction. This is the setup for the Lamé equations.

_{r}If we assume that the stress in the longitudinal direction is zero, we have a system of plane stress, and the stress invariant is: $$ \sigma_r + \sigma_c = \mbox{a constant} = C_1 $$

Which gives us: $$ C_1 - 2 \sigma_r = r \frac{d \sigma_r}{dr} $$

Separating and integrating: $$ \ln r = -\frac{1}{2} \ln \left( C_1 - 2 \sigma_r \right) + C_2 $$

Where *C _{2}* is a constant of integration, if we suppose $ C_2 = -\frac{1}{2} \ln C_3$ we can clean things up quite a bit:

$$ -2 \ln r = \ln \left( C_3 \left( C_1 - 2 \sigma_r \right) \right) $$

$$ \sigma_r = \frac{1}{2} \left( \frac{1}{C_3 r^2} - C_1 \right) $$

At the outer boundary the radial stress is zero (from the boundary conditions) $$ \sigma_r = 0 = \frac{1}{2 C_3 r_o^2} - \frac{C_1}{2} $$ $$ C_1 = \frac{1}{C_3 r_o^2} $$

At the inner boundary the radial stress equals *-p* (from the boundary conditions)
$$ \sigma_r = -p = \frac{1}{2 C_3} \left( \frac{1}{r_i^2} - \frac{1}{r_o^2} \right) $$
$$ C_3 = - \frac{1}{2p} \left( \frac{r_o^2 - r_i^2}{r_o^2 r_i^2} \right) $$

Finally, after some algebra: $$ \sigma_r = \frac{p r_i^2}{r_o^2 - r_i^2} \left(1 - \frac{r_o^2}{r^2} \right) $$ and $$ \sigma_c = \frac{p r_i^2}{r_o^2 - r_i^2} \left(1 + \frac{r_o^2}{r^2} \right) $$

Now the maximum shear stress at any given point in the pipe wall is (if you don't trust me look at Mohr's circle for this situation): $$ \tau_{max} = \frac{\sigma_c - \sigma_r}{2} $$

Assuming the pipe is a ductile material, Tresca's failure criterion is appropriate, which is: $$ \tau_{max} \le \frac{\sigma_{failure}}{2} $$ $$ \sigma = \sigma_c - \sigma_r \le \sigma_{failure} $$

Note that the shear stress is at a maximum at the inside wall of the pipe. Consider at $r_i$:

$$ \sigma = p \frac{ r_o^2 + r_i^2 }{r_o^2 - r_i^2} + p $$ $$ \sigma = p \frac{ 2 r_o^2 }{r_o^2 - r_i^2} $$

Letting $r_o = \frac{D}{2}$ and $r_i = \frac{D}{2} - t$:

$$ \sigma = p \frac{ D^2 }{2t \left( D - t \right)} $$

We can further simplify to: $$ \sigma = p \left( \frac{ D }{2t} - \frac{D}{2 \left( D - t \right)} \right) $$

If we assume $ t \ll D$ then $\frac{D}{2 \left( D-t \right)} \approx \frac{D}{2D} = \frac{1}{2}$ $$ \sigma = p \left( \frac{ D }{2t} - \frac{1}{2} \right) = p \left( \frac{ D }{2t} - Y \right) $$

Where *Y = 0.5*.

Apparently, according to the internets, the ASME task group for this decided to use *Y = 0.4* as a safety factor in 1943. Later editions of B31.3 and B31.1 replaced this with a generic fudge factor, *Y*, which is based on burst test results. The Y factor is now a function of temperature as the previous version of the equation lead to excessively thick pipes in high temperature steam service, with all the associated difficulties in that.