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

![p − y curve for sand (Reese) \[Reese and Van Impe, 2011\]](https://openbrim.atlassian.net/wiki/download/attachments/3049717819/SandReesePY.jpg?api=v2)

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

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 \[Reese and Van Impe, 2011\]](https://openbrim.atlassian.net/wiki/download/attachments/3049717819/SandReeseA.jpg?api=v2)

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 \[Reese and Van Impe, 2011\]](https://openbrim.atlassian.net/wiki/download/attachments/3049717819/SandReeseB.jpg?api=v2)

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,

$$
\tag{10}
m = \frac{p\_u-p\_m}{y\_u-y\_m}
$$

$$
\tag{11}
n = \frac{p\_m}{my\_m}
$$

$$
\tag{12}
\bar{C} = \frac{p\_m}{y\_m^{1/n}}
$$

$$
\tag{13}
y\_k = \biggl(\frac{\bar{C}}{k\_{py}x}\biggl)^{n/n-1}
$$

***

\[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.


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