To rww88
Since you say the PDF example is for a textbook I hope you won't mind a few "review" comments. These comments reflect how I've design API nozzles for the past 30+ years. They may or may not reflect what you want your text to demonstrate.
All API references are for 12th Edition. I'll try to use your nomenclature for variables.
In the initial question you state the pipe physicals are 83% and 85% of the shell, but somehow the allowable is only 64%. Since I can't use Table 5.2 for the pipe allowable I'd use the formulas in clause 5.6.2. Thus I'd estimate that a realistic Sp would be closer to 0.84 x 28,000 = 23,500 psi. You'd still be able to demonstrate the effect of the weaker pipe in your A'2 formula.
Having calculated the required thickness (tr = 1.117") I'd use that for calculating the required area (as permitted by clause 5.7.2.1). The intent of the example seems to be to reduce the repad as much as possible, so using t required instead of t nominal will help.
In step 5 we should be doing the calcs in the corroded condition, so the permitted projection would be 4(tn-c). The available area thus becomes A2 = 8(tn-c)2.
In step 9 you suggest there's an Standard rule requiring the repad to be the same thickness as the shell. While it commonly is, since it's easiest to just order some additional ring 1 plate for the repads, I'm not aware it's an API requirement.
This is almost too picky to even mention. You could be very exact and use the repad ID (10.875" per Table 5.6b column 4) instead of the hole cut in the shell (11") in your formula to calculate do.
Ignoring how my comments above will change the required pad size, I wouldn't be comfortable suggesting 13¼" OD. This is barely over 1" wide. The weld spacing between the shell-to-neck weld and the repad outside fillet is too small, particularly for such a high-strength shell material and large weld sizes. I'd be reluctant to go below 3" width due to both fabrication and weld spacing concerns. If this was a more typical low-strength shell I'd try to use a thinner repad to match this additional width (but this tank is probably stuck with first ring material). The outside fillet size is t, so even though the weld circumference increases for a thinner repad, the weld volume also comes down by t2, so the total weld volume would be smaller (less manhours = less dollars)
The big savings isn't the 100 lbs of material (or whatever the changes above come out to) but the reduced welding manhours from the smaller repad outside circumference. I'd guess the manhours savings might be 10 times the material cost savings. In fact when an additional first ring plate is bought there's no material dollar savings, just more scrap at the end of the job.
I hope you find at least some of these comments helpful.
Good luck with your text book.
Geoff