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Change in equation in Roark's Formulas for Stress and Strain

Change in equation in Roark's Formulas for Stress and Strain

(OP)
Hello everyone,

I'm looking at evaluating an increase in max dP of an elephant stool (Outlet collector ring) in a reactor. Evaluating the stiffener ring based on the original calculations from the manufacturer, they reference Roark's Table 35. This reactor was built in 1977 so I'm assuming they used the 5th Edition (1975). The case they pulled references Table 35, Fig. 17 showing a circular plate under uniform compression with the outside edge simply supported, inner edge free. Their equation shows this as a cubed function. Using the 8th Edition of Roark's, on page 747 Table 15.2 Case 12a shows the equation to be a squared function as opposed to a cubed. I'm interested in calculating the max Pressure at the actual thickness since there is the residual difference in THK compared to the t_min to see if we have some room to raise the design dP.

My question is if this is a mistake in the book, mistake in the original calculations, or if there was a change somewhere between the 5th and 8th editions. The 7th edition also shows the same equation as the 8th. Does someone have a 5th edition and can confirm the equation? Or does anyone know a possible reason to why the equation would change? It is going from a more conservative approach to a less, but I don't know if this is based on material properties changing over time, etc.

Thank you all in advance for your time and looking forward to hearing your responses.

RE: Change in equation in Roark's Formulas for Stress and Strain

(OP)
Equation in the original calcs, likely based on the 5th Edition Table 35 Fig. 17:

critical_stress = (KE)/(1-v^2) *(t/a)^3

Equation in the 8th edition, page 747 Table 15.2, case 12a:

critical_stress = (KE)/(1-v^2) *(t/a)^2

So when back solving for t_min @ design stress you're dealing with a cube rooted function as opposed to a square rooted function.

Thanks,
HRS713

RE: Change in equation in Roark's Formulas for Stress and Strain

The seventh edition shows (t/a)^2

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RE: Change in equation in Roark's Formulas for Stress and Strain

My very yellow Fifth Edition (purchased in 1981) has a term (t/l) that is squared, not cubed.
Full formula for "tau-primed" is given as
[E/(1-v^2)] * (t/l)^2 *
[1.27 + sqrt(9.64 + 0.466*H^1.5)]
where H = sqrt(1-v^2) * l^2/tr

This is for the "ends hinged" case.

For the "ends clamped" case the final term has changed to
[-2.39 + sqrt(96.9 + 0.605*H^1.5)]
but the rest of the formula is the same.

RE: Change in equation in Roark's Formulas for Stress and Strain

(OP)
Thank you both for your responses.

@Denial: I apologize for putting Figure 17 in my OP as that may have been misleading. All that is referenced from the original calculations was "Per Roark (Table 35)".
The case seems to be for a "circular plate with a concentric hole under uniform radial compression on outer edge." Is that the case considered for the equations you provided above? Just wondering because the original calcs also use the terms (b/a) to determine "K" and that doesn't seem to be represented by the equations you posted.

Thanks again.

RE: Change in equation in Roark's Formulas for Stress and Strain

That's case 12, not case 17.  And now it looks more like what you had in your OP.

Roark's Fifth has the (t/a) term squared, not cubed.  It also gives a table where you bang in (b/a) and get out "approximate" K.  K ranges from 0.35 for b/a=0 to 0.16 for b/a=1.

RE: Change in equation in Roark's Formulas for Stress and Strain

Nobody mentioned the 6th Edition (1989) so I thought I'd mention it too (to add to the timeline):

Table 35 (Figure 12a), p.688 ("Circular plate with concentric hole under uniform radial compression on outer edge"; "outer edge simply supported, inner edge free")

σ'= K (E/(1-ν2))(t/a)2

Basically the same as the 8th edition.

RE: Change in equation in Roark's Formulas for Stress and Strain

(OP)
Thank you guys greatly for your assistance. It seems as if the equation did not change from the 5th to the 8th edition.

At this point I'm going to assume either the original calculations are incorrect or they took a more conservative approach. Unless editions prior to the 5th had a change and for some reason they calculated it to an earlier edition.

Again, thank you for your time and help!

RE: Change in equation in Roark's Formulas for Stress and Strain

(OP)
Awesome, thanks JAE for the information on the errata.

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