There are many different possible calculations.

Generally for a sheet pile wall it would be considered flexible. Most geotechnical textbooks have examples for determine the applied soil stresses etc on the sheet pile.

I'm not sure exactly what you are after since this is a very vague question.

You need to know how to design a sheet pile wall. You need to determine the soil properties and lateral loads due to the soils and applicable surcharges. You need to know the water levels behind and in front of the wall. Do the water levels fluctuate with tide? If it is an existing wall, you need to know the specifics of the wall - the sheet pile size, length, steel grade, location of any tie rods or tieback anchors. You need to know if there are any other surcharge loads applied to the wall, such as ship mooring loads.

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Perhaps you can assume a sheet pile section and length and then backcalculate the surcharge load considering that the applied stresses in the sheet pile are equal to the allowable stresses...