In my research the 1807.3.2.1 formula is a modified version of formula used by Rutledge, see Pole Building Design by Donald Patterson.
I've taken the approach of using the IBC formula for the embedment depth and then back solving the Rutledge formula elements using that embedment to form the shear and moment diagrams below grade.
Also note the method breaks down for deep/flexible piles would need to swap to something like P-Y Curves (LPile) if you fall into that category.
I went through this recently. I found it surprisingly hard to find good resources on it. I found my structural frame software quite helpful by using a suitable number of spring restraints but I wasn't comfortable relying solely that approach. I eventually found a formula from Tomilinson; Pile Design and Construction Practice. Since I didn't have the derivation of the formula it was more 'black box' to me than using software. But it helped confirm my software approach.
Below is an except from ASI - Design of Portal Frames 5th Edition
Human909, Thank you for sharing this information.
Do these equations (Hu) relate to the internal shear in the individual pile, based on pile length, diameter, pile capacity based on soil, etc..?
Or is it the soil capacity of the pile?
Both. Which may sound a little odd but that is how they have been formulated.
Long piles at least in the definition used here are piles governed by the strength of the pile Mu. Short piles are governed by the capacity of the sub soil.
Using your favourite frame analysis tool is probably easier.
Laterally loaded piles are complicated to hand solve due to them being heavily indeterminate. The interactions between soil stiffness and pile stiffness plays a big role. Hence it being easier just to use a frame analysis tool. For me this was simply a check. And it was pretty accurate with my answer from my frame analysis program. It also should be unit agnostic. (Equation 9.7 seems odd and is possibly a mistake as it isn't needed and it gives LAMBA(s) units.) Simply solve for LAMBA(s) and 7.2a.)
