# Stiff Clay with Free Water (Reese)\[Reese et al., 1975]

### Static Loading Condition

![Figure 1. p − y curve for static loading in stiff clay with free water\[Reese and Van Impe, 2011\]](https://openbrim.atlassian.net/wiki/download/attachments/3052634117/ReeseClayWFreeWater-20250402-093012.jpg?api=v2)

Obtain undrained shear strength $$c\_u$$, soil submerged unit weight $$γ′$$ and compute the average undrained shear strength $$c\_a$$ over the depth $$z$$ and value of $$c\_u$$ at depth $$z$$.

$$p\_u$$, the ultimate soil resistance per unit length of pile defined as:

$$
\tag{1}
p\_u = min(p\_{ct},{p\_cd})
$$

$$
\tag{2}
p\_{ct} = 2c\_ab+\gamma'bz+2.83c\_az
$$

$$
\tag{3}
p\_{cd} = 11c\_ub
$$

Initial straight-line portion of the *p-y* curve,

$$
\tag{4}
p = (k\_{py}z)y
$$

Following values are suggested for $$k\_{py}$$ over-consolidated clay by \[Reese and Van Impe, 2011],

| $$c\_a(kPA)$$       | $$50-100$$ | $$100-200$$ | $$300-400$$ |
| ------------------- | ---------- | ----------- | ----------- |
| kpy (static)(MN/m3) | 135        | 270         | 540         |
| kpy (cyclic)(MN/m3) | 55         | 110         | 540         |

Define $$y\_{50}$$ as,

$$
\tag{5}
y\_{50} = \varepsilon\_{50}b
$$

Following values are suggested for $$ε\_{50}$$ over-consolidated clay by \[Reese and Van Impe, 2011].

| $$ca(kPA)$$ | $$50-100$$ | $$100-200$$ | $$300-400$$ |
| ----------- | ---------- | ----------- | ----------- |
| $$ε\_{50}$$ | 0.007      | 0.005       | 0.004       |

Obtaion As from following figure,

![Figure 2. Constants for As and Ac \[Reese and Van Impe, 2011\]](https://openbrim.atlassian.net/wiki/download/attachments/3052634117/ReeseClayWFreeWaterA.jpg?api=v2)

First parabolic portion of the *p-y* curve is given by the Eqn. (6). This portion extends from $$y$$ is equal to intersection of Eqn. (4) and Eqn. (6) to $$y$$ is equal to $$A\_sy\_{50}$$.

$$
\tag{6}
p = 0.5p\_u(\frac{y}{y\_{50}})^{0.5}
$$

Second parabolic portion of the *p-y* curve is given by the Eqn. (7). This portion extends from $$y$$ is equal to $$A\_sy\_{50}$$ to $$y$$ is equal to $$6A\_sy\_{50}$$.

$$
\tag{7}
p = 0.5p\_u(\frac{y}{y\_{50}})^{0.5} - 0.055p\_u(\frac{y-A\_sy\_{50}}{A\_sy\_{50}})^{1.25}
$$

Next straight line portion of the *p-y* curve is given by the Eqn. (8). This portion extends from $$y$$ is equal to $$6A\_sy\_{50}$$ to $$y$$ is equal to $$18A\_sy\_{50}$$.

$$
\tag{8}
p = 0.5p\_u(6A\_s)^{0.5}-0.411p\_u-\frac{0.0625}{y\_{50}}p\_u(y-6A\_sy\_{50})
$$

Establish the final straight-line portion of the *p-y* curve,

$$
\tag{9}
p = 0.5p\_u(6A\_s)^{0.5}-0.411p\_u-0.75p\_uA\_s \quad
$$

or

$$
\tag{10}
p= p\_u (1.225\sqrt{A\_s}-0.75-0.411)
$$

\[Reese and Van Impe, 2011] states that there might be no intersection between Eqn. (6) with any of the other equations given to define portions of the *p-y* curve. Eqn. (6) defines the *p-y* curve until it intersects with one of the other equations. However, if no intersection occurs, the *p-y* curve is defined by Eqn. (6).

### Cyclic Loading Condition

Straight portion of the curve is same as static loading condition.

Choose appropriate value for $$A\_c$$ from the Figure 2. and compute the following value:

$$
\tag{11}
y\_p = 4.1A\_cy\_{50}
$$

First parabolic portion of the *p-y* curve is given by the Eqn. (12). This portion extends from $$y$$ is equal to intersection of Eqn. (4) and Eqn. (12) to $$y$$ is equal to $$0.6yp$$.

$$
\tag{12}
p = A\_cp\_u\Bigg\[1- \Bigg|\frac{y-0.45y\_p}{0.45y\_p}\Bigg|^{0.25}\Bigg]
$$

The next straight portion of the *p-y* curve is given by the Eqn. (13). This straight line lies in between from $$y$$ is equal to $$0.6yp$$ to $$y$$ is equal to $$1.8yp$$.

$$
\tag{13}
p = 0.936A\_cp\_u-\frac{0.085}{y\_{50}}p\_u(y-0.6y\_p)
$$

Lastly, final straight portion of the curve is given by the following,

$$
\tag{14}
p = 0.936A\_cp\_u-\frac{0.102}{y\_{50}}p\_uy\_p
$$

***

\[Reese and Van Impe, 2011] states that there might be no intersection between Eqn. (12) with any of the other equations given to define portions of the *p-y* curve. Eqn. (12) defines the *p-y* curve until it intersects with one of the other equations. However, if there is no intersection occurs, *p-y* curve is defined by smallest value of $$p$$ for any value of $$y$$.


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