The Rotta model is quite simple:
The tire is flat where it contacts the ground, and has a circular cross section where it is not in contact. Each cross section of the tire is treated independently.
Here are tables from which inflation pressure can be computed. The rows correspond to tire widths, in mm. The columns represent rim width, as a ratio to tire width. E.g. the column with 2 at the top is for tire width twice the inner rim width, e.g. a 50 mm wide tire on a 25 mm wide rim.
The tables give the area in square inches of the contact patch. To compute the inflation pressure, divide the load on the wheel by the area of the contact patch. E.g. a 50 mm tire on a 25-559 ETRTO rim will have a 2.06 sq in contact patch when it is squished down 15% of its width. For that contact patch to support a 100 pound load, the inflation pressure should be 100 / 2.06 = 48.5 PSI.
For 622 BSD:
For 559 BSD:
For 406 BSD:
These tables are calculated purely from theory, i.e. no parameters are used to fit them to any experimental data. Do not follow them blindly! They're food for thought & perhaps provide a useful starting point for exploration.
And a 305 BSD table: