Dear canercaner,

I think the root of misunderstanding is the fact API/AWWA operates with two models.

The first API/AWWA model assumes a simple thin cylinder with constraints which are continuously distributed over the circumference of the base. In some books of strength-of-materials is given that, in case the cylinder is subject to bending of moment M, the "reaction force" over the base- i.e. a thin walled circle- is 4*M/(PI*D^2)*cos(theta) where "theta" is a polar variable.

The extreme values are 4*M/(PI*D^2) and -4*M/(PI*D^2).

This model is the base of various calculations when we are sure that the contact is over that "thin walled circle", i.e. when there is no uplift because tank is stable on bottom or when thank is kept on foundation by anchors. So the calculation is based on the maximum 4*M/(PI*D^2)+wt that gives the compression stress in shell or on minimum -4*M/(PI*D^2)+wt which (in some circumstances) may be an uplift of 4*M/(PI*D^2)-wt. See for example API 650 improved formulas; E.6.2.2.1-1a, E.6.2.2.2-1a for maximum longitudinal shell compression stress and E.6.2.1.2-1, E.6.2.1.2-2 for the calculated uplift load on the anchors.

Please note that in this model we are not considering any liquid resistance to uplift; we are just sure there is no uplift and calculation is conservative made. For this reason, these formulas cannot be used as a criterion for anchorage!

The warning in AWWA is rather a mathematical warning: what to do if "calculated uplift" for anchors is negative?

The second API/AWWA model is more complex and we need it when we have a partial uplift of shell.

Uplift of the tank shell is resisted by the weight of the shell and supported roof plus a band of liquid adjacent to it. The width of this band of liquid is L (and one main parameter in calculation is wa-Fluid Force resisting uplift in annular region, which is correlated with L). "L" is counted between two seismic plastic hinges that develop under seismic event; L depends on the stiffness (or thickness) of the part of the bottom plate inside the shell, which is called the annular ring. The designer can thicken this ring but there are limitations; it cannot be thicker than the shell and L cannot exceed a certain value. If this is not sufficient, additional restraint in the form of anchors must be provided.

So which is the criterion for anchorage? To understand a little bit more, please have a look in Myers’ book or download

this document - and concentrate on "ß" angle; that ß is a characteristic of the non-lifted shell but API/AWWA does not consider any ß because they are not interested to develop the theory which is behind the results.

When ß is zero, there is a total uplift, when ß=Pi radians we have no partial uplift. The corresponding parameter is " J" that mathematically takes values between PI/4=0.785 (for ß=PI radians) to PI/2=1.57 (for ß=0); for calculation practical reasons API/AWWA considers J=1.54 as "total" uplift. You may see also the graph attached which shows J vs. ß.

For J< 0.785 there is no partial uplift and anchors are not required for this reason, for 0.785< J< 1.54 there is partial uplift and tank may be not-anchored, however the decision is based also considering additional criteria- see API 650/ E.6.2.1.1 Self-Anchored paragraph. In fact, the main concern is the compression in shell because, when shell is partially uplifted, the M seismic bending moment must be compensated by an increased longitudinal stress in shell (due to the reaction force on foundation, only on that part of shell which is not uplifted)- see API650/E.6.2.2.1-2a. Also, the designer may decide to anchor the tank because the attached piping is too sensitive to uplift.

It is true that API 650 says "The maximum width of annulus for determining the resisting force is 3.5% of the tank diameter." However this is not a true criterion, it is just a warning that the decision "tank is not anchored" has been based on the desired algorithm of calculation. In my opinion that sentence can miss there because the same condition is given in various other paragraphs (see E.6.2.1.1) and it is practically impossible to skip it.

I think it is interesting that wa=1.28HDGe (in USC Units) and and L= 0.035D are equivalent by Wozniak & Mitchell model, that;s why API650/ E.6.2.1.1 asks for both.

As limit value, L< 0.035 D appears today as being rather arbitrary; however that limit has been established by Wozniak in 1971 and I guess he had a good reason for it. Today we may understand that the API 650/AWWA algorithm is not an exact one; it is based on some assumptions and has been fine tuned over the years.

My best regards.