Nonlinear t-z curves for driven piles into sand can be based on the recommendation from Mosher (1984) where a hyperbolic representation of t-z curve was proposed.
The initial slope Kf of the curve is given in the following table as a function of internal friction angle.
This nonlinear t-z curve as recommended by Mosher (1984) has been adopted by PileAXL program for driven pile into cohesionless materials. Refer to the figure below for a typical t-z curve based on Mosher (1984)'s approach.
Single Piles and Pile Groups under General Loading