Sand (API) [API, 2010]

To develop the p-y curve for API Sand the friction angle and initial modulus of subgrade reaction is required. Hyperbolic p-y relationship for sand for both short-term static and cyclic loading conditions are recommended by [API, 2010].

At a given depth, $p_u$ is taken as the lesser of the calculated $p_{us}$ and $p_{ud}$.

$$\tag{1} p_u = min(p_{us},p_{ud})$$

$$\tag{2} p_{us} = (C_1z+C_2b)\gamma z$$

$$\tag{3} p_{ud} = C_3b\gamma z$$

where,

$b$ = pile diameter $γ$ = sand unit weight $p_{us}$ = horizontal ultimate resistance per unit length at shallow depths $p_{ud}$ = horizontal ultimate resistance per unit length at greater depths

and

$$\tag{4} C_1 = \tan{\beta}\Bigl\{K_p\tan{\alpha}+K_0\Bigr[\tan{\phi}\sin{\beta}(\frac{1}{\cos{\alpha}}+1)-\tan{\alpha}\Bigr]\Bigl\}$$

$$\tag{5} C_2 = K_p - K_a$$

$$\tag{6} C_3 = K_p^{2}(K_p+K_0\tan{\phi})-K_a$$

$$\tag{7} K_0 = 0.4, \quad K_a = \tan^2{(\frac{\pi}{4}-\frac{\phi}{2})}, \quad and \quad K_p = \frac{1}{K_a}$$

where,

$ϕ$ = internal friction angle $α$ = $ϕ$/2 $β$ = π/4 + $ϕ$/2 $K_0$ = coefficient of earth pressure at rest $K_a$ = coefficient of Rankine's active earth pressure $K_p$ = coefficient of Rankine's passive earth pressure

The p-y curve is computed on $p_u$ based on:

$$\tag{8} p = Ap_u\tanh{\frac{kz}{Ap_u}y}$$

where,

$k$ = initial modulus of subgrade reaction $z$ = depth $y$ = horizontal displacement $p$ = horizontal resistance per unit length $p_u$ = horizontal ultimate resistance per unit length $A$ = factor depending on loading type: 0.9 for cyclic loading; for static loading

$$\tag{9} A = 3 - 0.8\frac{z}{b} \geq 0.9$$

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[API, 2010] American Petroleum Institute (2010). Recommended Practice for Planning, Designing, and Constructing Fixed Offshore Platforms-Working Stress Design. API RP 2A.