# 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

***

\[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


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