Driven Pile (McVay) TZ [McVay et al., 1989]

$$\tag{1} z = \frac{tr}{G_i}[ln(\frac{r_m-\beta}{r-\beta})+\frac{\beta(r_m-r)}{(r_m-\beta)(r-\beta)}]$$

where,

$t$ = skin friction (side friction) $z$ = vertical displacement (settlement) $r$ = pile radius $G_i$ = initial (small-strain) shear modulus of soil $r_m$ = outward radius were the transferred shear stress to soil is negligible $β$ = side resistance parameter

The parameter $β$ is computed as,

$$\tag{2} \beta = \frac{t_r}{t_u}$$

where,

$t_u$ = ultimate side resistance

$r_m$ is assumed to be initially equal to:

$$\tag{3} r_m = \frac{G_{mid}}{G_{tip}}L(1-\nu)$$

where,

$L$ = pile length $ν$ = Poisson's ratio of soil $G_{mid}$ = shear modulus of soil at mid-depth $G_{tip}$ = shear modulus of soil at the pile tip

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[McVay et al., 1989] McVay, M., Townsend, F., Bloomquist, D., O’Brien, M., and Caliendo, J. (1989). Numerical analysis of vertically loaded pile groups. Foundation Engineering, pages 675–690