How does one calculate the depth and diameter of laterally loaded cast-in-place concrete piles.  Planning a conc pile cap on 2 or more piles to carry the structural steel base of WF vertical columns forming a closed sign 12' wide and 25' high. Not fussy about embedding the WF columns. Obviously it's going to be a function of certain properties of the soil. We are confident of the appropriate wind load we have researched.

I did a large sign like you have described not so long ago (45' x 8') which had two 4' diameter piles embedded 14' into the ground. The best thing to do would be to speak with a geotechnical engineer (has a geotechnical engineer been engaged for the job) and ask them how much passive resistance the soil is capable of. I had a geotech recently give me advice that the passive resistance that the soil can tolerate is 9*Cu where Cu is the undrained shear strength of the soil.

I am unsure on what the soil material was (I will dig up the design tomorrow at work) but you can back-calculate what soil strength you need to resist the overturning moment and give this to the geotech and see what they recommend.

Personally, I don't think a pile cap will be necessary. You could just have the WF columns bear on the pile head (with a thick size baseplate, typically 1-1/2"). I would also embed a end-plate welded to the hold-down bolts.

I can attach a sketch if it were to be more helpful.

Look at some standard drawings for highway sign structures from your local DOT, they should be a good indication on what the design looks like.

Thank you kindly ASIXTH.  Yes It would be helpful if you could send a sketch and I appreciate your offer. Can you do that here? We will be working in medium to low plastic clay