External pressure - Linking FEA and EN codes?

[dimisor]

Hi everybody,

I am currently performing a finite element analysis of a tank for the transport of dangerous goods according to EN 13094. As these kinds of FEA (where you have to adhere to a code) are not my daily routine I am having some trouble interpreting the EN 13094 code. I can follow most of the text up to the point where the effect of pressure on the convex face of an end section is evaluated. In the code one calculates an elastic modulus that should be less than the actual modulus of the material used. I can understand this in terms of buckling. The numbers used in the formula however confuses me. It reads:

E_calculated = (100 x R^2 x 2.2 x P)/(36.6 x t^2)

This is obviously related to the classical buckling load of a sphere:

P_critical = (2 x E_material) / sqrt(3 x (1 - mu^2)) x t^2 / R^2

Assuming mu is constant (say 0.3) these 2 formulas are proportional to each other so I would assume that a safety factor was introduced in the EN 13094 formula. However, in order to compare the safety factor obtained from a finite element buckling analysis with the cited formula I need to make sense of the numbers 100, 2.2 and 36.6. The code does not give any clue whatsoever.

Perhaps anyone of you can shed some light on this?

Thanks,

Dimitri

 

[doct9960]

I am not familiar with EN 13094, but the term 36.6/100 (or 0.366) was probably derived from the term 0.365 in the formula below.

From Roark's Formulas for Stress and Strain,
for a thin sphere under uniform external pressure,
the probable actual minimum external pressure (q') at which elastic buckling occurs...

q' = 0.365 E t2/r2

Also check "The Buckling of Spherical Shells by External Pressure" by von Karman and Tsien. This was the reference of the Roark book for the above formula.

 

[prex]

I agree with doct9960, 36.6/100 should be 0.365, so 2.2 becomes the factor of safety. However all this is for a full sphere, while you are dealing with a spherical end cap: also for this case the Roark gives a formula (also cited by Timoshenko and Gere), but this one gives smaller values and strongly depends on geometry. It is valid for a R/t ratio between 400 and 2000: you could be out of this interval.
I'm also afraid that your tentative of justifying the code formula by FEM can't succeed. In fact Roark's formula with the factor 0.365 gives a practical prediction of the buckling pressure based on experiments, and is more than 3 times smaller that the theoretical value that Timoshenko and Gere derive with a great deal of analytical effort. Depending on how you perform your analysis, you could get the theoretical value or anything between it and the empirical one.
This situation is clearly explained here:

Experiments with thin spherical shells subjected to uniform external pressure show that buckling occurs at pressures much smaller than that given by Eq. (11-31) and that the collapse of the buckled shell occurs suddenly...

prex

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[dimisor]

Thanks doct9960 and prex for your comments.

I have to agree that the FEA will do no good unless I introduce a safety factor of at least 2.2/0.365 = 6 since a linear buckling analysis will approximately yield the same results as the classical formula. Delving into experimental literature I have found that deviations from this formula by a factor 5 can easily be obtained for the dimensions I'm working with. So I will reconsider using FEA for these kinds of analysis.

Thanks

Dimitri