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# Cosine Curve

## Cosine Curve

(OP)
Long story short,
I am working with a client that has little to no resources when it comes to inspection of their storage tanks. I have been tasked with performing a complete external visual inspection in addition to an external settlement survey.

I am in the process of creating a cosine curve for the settlement survey in accordance with API 653 annex B. I have read through the entire annex and am pretty understanding of it, though cannot seem to understand how to acquire appropriate predicted elevations.

In API 653 it says: Where Elevpred is the elevation predicted by the cosine curve at angle theta. A typical starting point for a computer
best-fit cosine curve is a least-square fit where a, b, and c are chosen to minimize the sum of the square of the
differences between measured and predicted elevations.

What I am getting from this is theta is angular coordinate of the point, and a, b, and c are fitting parameters to be estimated.

However, no where in the code does it specify what a , b, and C values area derived from?

Can anyone help with this.

### RE: Cosine Curve

#### Quote (What I am getting from this is theta is angular coordinate of the point, and a, b, and c are fitting parameters to be estimated.)

-theta is angular coordinate of the point .( correct)..

- a, b, and c are fitting parameters to be estimated .(not correct.. to be calculated rather than estimated )

You may calculate the parameters a, b, and c within a excell spread sheet.

Your post implies you have settlement survey data.. If so, post the survey data together with some tank info ( dia, ht etc..) you may get more useful responds..

### RE: Cosine Curve

Another advantage of an Excel sheet is you can plot the actual elevations vs the best fit. I find this often helps me understand what's happening to the tank.

If the simple rigid body rotation of B.2.2.4 has poor curve fit (R²<0.9) or fails, it is well worth doing the B.2.2.5 review before concluding that releveling of a tank is required.

I've often seen too many survey points taken. This can unnecessarily lower Smax. In this case make use of the note below B.2.2.4.f.

Just a heads up, I see you're in the US so you may need to watch out for having an API 653 Authorized Inspector involved with the inspections, and an experienced Storage Tank Engineer for any evaluation and repair recommendations. See Section 3 for definitions.

### RE: Cosine Curve

I am also having the same problems as original post in finding a solution to the best fit curve models. Can anyone point me in the direction of a resource? I am aiming to build my own excel tool to evaluate settlement. The tank is 44.50' by 36.00'. The rod readings are in feet and taken at every half station (16 in total, spaced at 17.475ft).

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### RE: Cosine Curve

Here's what my spreadsheet looks like with your data. Perhaps this will give you some ideas for developing your own. Since the equation links are specific to the number of survey points I have a separate tabs for different numbers of points.

First I enter the data in the green squares. Then I use Solver to alter the three Curve Parameters to minimize the RMS value.

If I understand correctly the tank is 44.5 ft diameter. Per the Annex B maximum spacing of 32' this would require five survey points, but Annex B also specifies a minimum of 8 points. If 8 survey points are used they are 17.5 feet apart, and your 16 points are only 8.7 feet apart. As I noted previously this unnecessarily reduces the Smax limit.

On the first page I show the output resulting from using all 16 of your data points, and on the second I made a quick alteration to show the results with only 8 data points. While the R2 is below 0.9 it is up to you if it is still acceptable.

PS : Just before posting I noticed I haven't updated to reference 5th Edition, but Annex B didn't change so this is just a typing correction.

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