# 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

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

***

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