Drilled Shaft Sand TZ (Reese and O'Neill) [O’Neil et al., 1999]

$$\tag{1} \beta=1.5-0.135z^{0.5} \quad for \quad 0.25 \leq \beta \leq 1.2$$

$$\tag{2} t_u = K\sigma'_{z}tan\varphi=\beta\sigma'_{z} \leq 2.1 \quad psi$$

The non-linear t-z relationship can be obtained using the following equations;

$$\tag{3} t = t_u(-2.16R^{4}+6.34R^{3}-7.36R^{2}+4.15R) \quad for \quad Z \leq 0.9083$$

$$\tag{4} t = 0.9781t_u \quad for \quad Z > 0.9093$$

$$\tag{5} Z = \frac{z}{D}*100$$

where,

$t$ = skin friction $Z$ = normalized settlement $D$ = pile diameter

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[O’Neil et al., 1999] O’Neil, M. W., Reese, L. C., et al. (1999). Drilled shafts: Construction procedures and design methods. Technical report, United States. Federal Highway Administration. Office of Infrastructure.