As an exercise, I have been trying to correlate SWSim results with the Lamé Equations for stresses in a thick walled cylinder subjected to internal/external pressure. My issues come from trying to determine which value of P1/P2/P3/von Mises stress should match with my hand calc for Hoop Stress.
My model is a quarter-cylinder, 5in long with an internal radius of 0.6in and an external radius of 0.8315in. I have applied a symmetry restraint to three faces (each of the longitudinal faces, and one end face). Normal pressure is applied on the internal face and the external face (as two separate loads, to allow a differential across the two). Mesh element size is nominally 0.05in, giving me 5 elements through the wall thickness.
The formula I have used for hoop stress is:
I have set r = ri to get the peak stress on the inner wall.
The following table shows the pressure values used and the results generated for hoop stress from the above equation (Pi, Po and Calc values all in psi):
Negative values indicate compressive stress at inner surface.
Each case was then run in SWSim (2011 SP5) as a Static Study. Results were then taken from four stress plots; P1, P2, P3 and von Mises. The stress values for each of these were then compared with the hand-calc value by dividing through to generate a ratio – obviously, the closer to 1 the value, the better the match between SW and the hand-calc. These are tabulated below:
Values with more than a 2% discrepancy and shaded grey as these are deemed to be outside of acceptable limits. Values shaded orange are outside 2%, but are the best option available out of the four. For ratio calculation, all values are normalised to positive to achieve the magnitude of the discrepancy, as SW provides only the magnitude value for von Mises.
As the data shows, P1 provides adequate results for pressure ratios (Pi / Po) above 1.5. Below this, selection of stress criteria becomes more difficult – von Mises often provides the best approximation, but this still is a poor answer for some cases, as far as 12.5% off. I’d have expected P1 or P2 to have provided the correct Hoop Stress value for the whole range, depending on how the software defines which is which – is this reasonable?
The data is represented graphically below. X-axis in each instance is the pressure ratio between the internal and external wall. Y-axis is discrepancy ratio:
So my question is, is there a definitive answer for which stress plot to use when determining hoop stress in a thick walled cylinder? And why is there so much deviation in what is usable either side of a 1:1 pressure ratio?
Hopefully, I’ve just mis-understood the use of the Lamé equations – they’re something that I never covered in my degree and so I’ve had to pick up the information from elsewhere. Otherwise, is there a chance that a bit of code somewhere in SWSim is hashing up and generating the aforementioned results?
I haven’t yet tried with different geometry, that’s this afternoons job.