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P-y models: Modelling details |
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P-y models: Modelling details |
A P-y model is essentially a relationship between the lateral deflection $y$ of a vertical-axis cylinder which is at least partially-submerged in soil, and the corresponding soil resistance $p$ to that deflection. OrcaFlex offers a number of different types of relationship, or model types. Further, different model types (or the same model type with different data) can be defined for different depths below the soil surface.
You apply a P-y model in OrcaFlex by associating it with a line (which is not required to be vertical): OrcaFlex then implements that model at each sub-surface node on that line, taking into account the nominal depth of the node in choosing which specific depth range applies. At any time, the node's deflection is calculated as the difference between its instantaneous position and its initial, nominal position. Having calculated the node's lateral deflection $y$, we look up the the corresponding soil resistance $p$ in the appropriate P-y data set, and apply the corresponding resistive force at the node. We now explain these steps in more detail.
Before the statics calculation, each node on a line with an associated P-y model is assigned a nominal position, $\vec{q}_0$, as follows
| Figure: | P-y modelling |
| Note: | It is expected that the P-y curve depth data will start at the seabed, i.e. depth from will increase from zero. However, OrcaFlex does not enforce this and will not attach P-y models to any nodes that are above this initial depth, even if they are below the seabed. |
The nominal depth of each node, i.e. the depth at its nominal position, is used to determine which set of P-y curve data is used in each case. The nominal horizontal position of each is used in calculating the instantaneous lateral deflection $y$, as follows.
Each node's deflection is obtained by subtracting the node's instantaneous position, $\vec{q}$, from its nominal position, $\vec{q}_0$, and projecting onto the horizontal plane; the deflection $y$ is then the magnitude of this horizontal vector. If we write $\vec{q}-\vec{q}_0 = \vec{d}$, then in component form $\vec{d} = (d_x,d_y,d_z)$ and $y = (d_x^2 + d_y^2)^{1/2}$. From the lateral deflection value $y$, the soil resistance $p$ is calculated, according to the model type, as described below. The magnitude of the resistive force is then $p d l$ where $d$ and $l$ are the node's outer contact diameter and associated length, respectively. This force is applied at the node, towards the projection of $\vec{q}_0$ in the horizontal plane. The deflection and resistance at each node are available as results.
| Note: | The line properties report contains detailed information on how the P-y model data are interpreted and applied in the OrcaFlex model. |
When a P-y model is active a number of modifications are made to the modelling of the line. In particular, all normal seabed reaction forces are suppressed for the line. Other lines in the model which do not use P-y models are still subject to normal seabed reaction forces. Nodes that have P-y models attached are further modified as follows:
Buoyancy and pressure calculations are not affected by the presence of a P-y model. You may extend the sea density profile beneath the seabed if you want to vary the fluid density used to calculate these effects.
The use of P-y models also influences the interfaces to SHEAR7 and VIVA.
The load-deflection formulation is based on the ultimate unit lateral bearing capacity of the soil, $\pu$. The API code provides two equations for this \begin{align} (6.8.2\text{-}1) \qquad \pu &= 3 c + \gamma X + J\frac{cX}{D} \\ (6.8.2\text{-}2) \qquad \pu &= 9c \end{align} where $c$ is the undrained shear strength, $\gamma$ is the effective unit weight of soil, $X$ is the depth below soil surface, $J$ is a dimensionless empirical constant, and $D$ is the pile diameter.
The first equation, 6.8.2-1 aims at capturing the reduced lateral bearing capacity of the soil close to the surface. The intent of the code is that $\pu$ is defined by whichever of 6.8.2-1 and 6.8.2-2 gives the smaller value. The code approaches this by defining $X_\mathrm{R}$, the depth to which the reduced resistance zone extends, and applying 6.8.2-1 for $X \lt X_\mathrm{R}$ and 6.8.2-2 for $X \ge X_\mathrm{R}$.
This approach using $X_\mathrm{R}$ breaks down when the soil properties vary with depth, so we must treat these equations slightly differently. Starting at zero depth, OrcaFlex processes each node with a soft clay P-y model in turn. The two prospective values of $\pu$ are calculated using 6.8.2-1 and 6.8.2-2. So long as 6.8.2-1 produces the smaller value, that value is used for $\pu$. Once a node is reached for which 6.8.2-1 gives a greater value of $\pu$ than 6.8.2-2, then 6.8.2-2 is used for that node and all subsequent nodes.
Once the value of $\pu$ has been determined, $p$ is calculated by linear interpolation of the following table:
| $p/\pu$ | $y/y_\mathrm{c}$ |
|---|---|
| 0.00 | 0.0 |
| 0.23 | 0.1 |
| 0.33 | 0.3 |
| 0.50 | 1.0 |
| 0.72 | 3.0 |
| 1.00 | 8.0 |
| 1.00 | $\infty$ |
where $y_\mathrm{c} = 2.5\ \epsilon_\mathrm{c} D$.
| Note: | The table above is taken from API RP 2A supplement 3 and differs from earlier versions of the API code. OrcaFlex 9.5 used the version of the table that was presented in earlier versions of the API code. OrcaFlex 9.6 and later use the version of the table presented above. |
The load-deflection formulation is based on the ultimate bearing capacity of the soil, $\pu$. The code provides two equations for this \begin{align} (6.8.6\text{-}1) \qquad \pu &= (C_1 H + C_2 D) \gamma H \\ (6.8.6\text{-}2) \qquad \pu &= C_3 D \gamma H \end{align} where H is the depth below soil surface, D is the pile diameter, and ɣ is the effective unit weight of soil.
Both equations 6.8.6-1 and 6.8.6-2 should be evaluated and $\pu$ taken to be the smaller value. The resistance $p$ is then defined to be \begin{equation} p = \frac{A \pu}{D}\ \tanh\left(\frac{kH}{A \pu}y\right) \end{equation} where $A = \max\left(0.9, 3-0.8\frac{H}{D}\right)$.
The input data define a table of deflection and resistance values. The deflection values must include zero and must be given in increasing order. Linear interpolation is used within this table. For values of deflection exceeding the largest given value, the resistance corresponding to that largest deflection value is used.