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]

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]

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,

$$\tag{14} p = 0.936A_cp_u-\frac{0.102}{y_{50}}p_uy_p$$

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[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 $y$.