Weak Rock (Reese) [Reese, 1997]

Ultimate resistance of Rock determined by following equations

$$\tag{1} p_{ur} = \alpha_{r}q_{ur}b(1+1.4\frac{x_r}{b}) \quad for \quad 0 \leq x_r \leq 3b$$

$$\tag{2} p_{ur} = 5.2\alpha_{r}q_{ur}b\quad for \quad x_r > 3b$$

where,

$p_{ur}$ = ultimate rock resistance per unit length $q_{ur}$ = unconfined compressive strength of rock $α_r$ = strength reduction factor $x_r$ = depth below rock surface $b$ = diameter of pile

The initial slope can be determined as,

$$\tag{3} K_{ir} = k_{ir}E_{ir}$$

where,

$K_{ir}$ = initial slope of the p-y curve $E_{ir}$ = initial modulus of rock $k_{ir}$ = dimensionless constant

$k_{ir}$ is calculated by;

$$\tag{4} k_{ir} = (100+\frac{400x_r}{3b})\quad for \quad 0 \leq x_r \leq 3b$$

$$\tag{5} k_{ir} = 500\quad for \quad x_r > 3b$$

The relationship is divided into three portions: initial slope, transition, and ultimate resistance.

$$\tag{6} p = K_{ir}y \quad for \quad y \leq y_A$$

$$\tag{7} p = \frac{p_{ur}}{2}(\frac{y}{y_m})^{0.25} \quad for \quad y \geq y_A \quad and \quad p \leq p_{ur}$$

$$\tag{8} p = p_{ur} \quad for \quad y \geq 16y_m$$

$$\tag{9} y_m = k_{rm}b$$

where,

$y$ = horizontal displacement $p$ = horizontal rock resistance per unit length $y_A$ = horizontal displacement end of linear portion $k_{rm}$ = dimensionless constant, ranging from 0.0005 to 0.00005

The value of the $y_A$ is found solving the following equation,

$$\tag{10} y_A = \biggr[\frac{p_{ur}}{2(y_m)^{0.25}K_{ir}}\biggr]^{1.333}$$

p − y curve for weak rock (Reese) [Reese, 1997]

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[Reese, 1997] Reese, L. C. (1997). Analysis of laterally loaded piles in weak rock. Journal of Geotechnical and Geoenvironmental engineering, 123(11):1010–1017.