Sand (Reese) [Reese et al., 1974]

p − y curve for sand (Reese) [Reese and Van Impe, 2011]

$$\tag{1} \alpha = \frac{\phi}{2};\quad \beta = 45 + \frac{\phi}{2}; \quad K_a = \tan^2(45-\frac{\phi}{2})$$

where,

$ϕ$ = internal friction angle $K_a$ = coefficient of active earth pressure

The earth pressure at rest, $K_0$ is assumed as constant and $K_0$ = 0.4

The unfactored horizontal ultimate sand resistance per unit length of pile, ps is defined as the minimum of the following two equations,

$$\tag{2} p_s = min(p_{st},p_{sd})$$

$$\tag{3} \begin{aligned} p_{st} = \gamma z\Bigl[\frac{K_0\tan(\phi)\sin(\beta)}{\tan(\beta-\alpha)\cos(\alpha)}+\frac{\tan(\beta)}{\tan(\beta-\alpha)}(b+z\tan(\beta)\tan(\alpha)) \\ +K_0z\tan(\beta)(\tan(\phi)\sin(\beta)-\tan(\alpha))-K_ab\Bigl] \end{aligned}$$

$$\tag{4} p_{sd} = K_ab\gamma z (\tan^8(\beta)-1)+K_0b\gamma z \tan(\phi)\tan^4(\beta)$$

where,

$p_s$ = unfactored horizontal ultimate sand resistance per unit length of pile $b$ = pile diameter $z$ = depth $γ$ = sand unit weight

Compute $p_{ult}$ by the following equation:

$$\tag{5} p_{ult} = \bar{A_s}p_s \quad or \quad p_{ult} = \bar{A_c}p_s$$

use the appropriate value of $\bar{A_s}$ or $\bar{A_c}$ for static and cyclic loading case respectively from the following figure.

Values of coefficients [Reese and Van Impe, 2011]

Compute $pm$ by the following equation:

$$\tag{6} p_m = \bar{B_s}p_s \quad or \quad p_m=\bar{B_c}p_s$$

use the appropriate value of $\bar{B_s}$ or $\bar{B_c}$ for static and cyclic loading case respectively from the following figure.

Values of coefficients [Reese and Van Impe, 2011]

The two straight-line portions of the p-y curve can now be established. Establish following two definitions:

$$\tag{7} y_u = 3b/80 \quad and \quad y_m = b/60$$

Establish the initial straight-line portion of the p-y curve.

$$\tag{8} p = (k_{py}z)y$$

Use the appropriate value for $k_{py}$ from following tables

Representative values of $k_{py}$ for submerged sand

Relative density	Loose	Medium	Dense
Recommended $k_{py}$ (MN/m3)	5.4	16.3	34

Representative values of $k_{py}$ for sand above water table

Relative density	Loose	Medium	Dense
Recommended $k_{py}$ (MN/m3)	6.8	24.4	61

Establish the parabolic section of the p-y curve,

$$\tag{9} p = \bar{C}y^{1/n}$$

Fit the parabola between points $k$ and $m$ as follows,

$$\tag{10} m = \frac{p_u-p_m}{y_u-y_m}$$

$$\tag{11} n = \frac{p_m}{my_m}$$

$$\tag{12} \bar{C} = \frac{p_m}{y_m^{1/n}}$$

$$\tag{13} y_k = \biggl(\frac{\bar{C}}{k_{py}x}\biggl)^{n/n-1}$$

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[Reese et al., 1974] Reese, L. C., Cox, W. R., and Koop, F. D. (1974). Analysis of laterally loaded piles in sand. In Offshore Technology Conference, pages OTC–2080. OTC.

[Reese and Van Impe, 2011] Reese, L. C. and Van Impe, W. F. (2011). Single piles and pile groups under lateral loading. CRC Press/Balkema.