Orifice Plate Bending - Mystery Calculation 3

[Plungeman]

Having joined a new company, I've been tasked to "standardise" a batch of old spreadsheets that are frequently reused. Absolutely nothing give references for the calculations embedded and I'm working to change that. I've managed to track down most of them, but have run into a wall relating to deflection of an orifice plate with a horrendous bit of excel that simplifies to this:

[delta]y = Axial deflection of plate inner edge
[delta]P = Differential pressure
Dpipe = Pipe ID, used as the fixed edge of the plate
Dplate = Plate constant/flexural rigidity
dn = Orifice ID

The values for this are essentially the same as using Roark's 7th Ed Table 11.2 Case 2e (for the range where it's practical to real-world circumstances), but I can't find where this much more compact equation comes from - any pointers?

Edit: Define [delta]y

 

[prex]

The original formula can't be correct in general.
Letting a=D[sub]pipe[/sub]/2, b=d[sub]n[/sub]/2 and ρ=b/a, it may be written as [ΔPa[sup]4[/sup]/(64D][sub]plate[/sub])(1-ρ[sup]2[/sup])[sup]2[/sup]. Calling P the total load acting on the plate, P=ΔPπ(a[sup]2[/sup]-b[sup]2[/sup]), that formula may be rearranged as [Pa[sup]2[/sup]/(64D)](0.318-0.318ρ[sup]2[/sup]).
The right parenthesis may be considered as a polynomial expansion of the deflection, but it is far too short.
By calculating the polynomial form of the plate deflection at the inner edge by the minimum elastic energy method, one obtains, for ρ=0.5:
y=[Pa[sup]2[/sup]/(64D)](6.47+69.7ρ+41ρ[sup]2[/sup]-93.3ρ[sup]3[/sup]-23.8ρ[sup]4[/sup]+1.21logρ+34.1ρlogρ+111ρ[sup]2[/sup]logρ+71.4ρ[sup]3[/sup]logρ+6.35ρ[sup]4[/sup]logρ) (see here). The sum of the terms of the expansion, with large terms canceling out, is 0.0669 (and this gives Roark's result), while the original formula gives (0.318-0.318*0.25)=0.239, a quite different result.
However, for smaller values of ρ (say <0.1), the error on deflection is much smaller, so possibly the formula has been obtained for low values of the ratio of the radii.

prex
[URL unfurl="true"]http://www.xcalcs.com[/url] : Online engineering calculations
[URL unfurl="true"]https://www.megamag.it[/url] : Magnetic brakes and launchers for fun rides
[URL unfurl="true"]https://www.levitans.com[/url] : Air bearing pads