# Lateral Soil Methods¶

There are various methods available that can be used to calculate PY curves for further lateral analysis:

 Method Required Input Additional Required Input 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 24 SANDAPI X X O CLAYAPI X X X X CLAYAPI Factor, Limit su CLAYREESE X X X O O Reese K Static, Reese K Cyclic DYSON X X X X X WESSELINK X X X x0 ROCKREESE97 X X X Alpha, Eir, Krm, Top depth of rock layer EVANS X X X X O O O O FRAGIO X X ABBS X X X X X O O Reese K Static DUNNAVANT X X X X x0, number of cycles NOVELLO99 X X X Gamma C, number of cycles KALLEHAVE X X O SORENSEN X O CLAYJEANJEAN X X X X X CLAYJJ2017 X X X CUSTOM PY X

List of soil parameters used in the lateral methods:

No.

Symbol

Unit

Soil Parameter

1

$$s_{u,top}$$

(kPa)

undrained shear strength at the top of the layer

2

$$s_{u,bottom}$$

(kPa)

undrained shear strength at the bottom of the layer

3

$$J$$

(-)

empirical constant with values ranging from 0.25 to 0.5 having been determined by field testing

4

$$e_{50}$$

(-)

strain at one half the maximum deviator stress in laboratory undrained compression tests

5

$$\phi$$

(deg)

angle of internal friction in sand

6

$$K$$

(kPa/m)

rate of increase with depth of the modulus of subgrade reaction

7

$$p_{lim}$$

(kPa)

Limit for the ultimate lateral resistance

8

$$q_{c,top}$$

(kPa)

CPT cone resistance at the top of the layer

9

$$q_{c,bottom}$$

(kPa)

CPT cone resistance at the bottom of the layer

10

$$Dyson R$$

(-)

R parameter for Dyson method

11

$$Dyson N$$

(-)

N parameter for Dyson method

12

$$Dyson M$$

(-)

M parameter for Dyson method

13

$$Wess R$$

(-)

14

$$Wess N$$

(-)

15

$$Gamma$$

(-)

16

$$UCS$$

(kPa)

Unconfined compressvie strength

17

$$Dunn K_{r}$$

(-)

K_{r} parameter for Dunnavant method

18

$$Reese K Static$$

(kPa/m)

Static K parameter for Reese method

19

$$Reese K Cyclic$$

(kPa/m)

Cyclic K parameter for Reese method

20

$$Reese \alpha$$

(-)

alpha parameter for Reese method

21

$$Reese E_{ir}$$

(kPa)

E_{ir} parameter for Reese method

22

$$Novello \gamma_{C}$$

(-)

gamma C parameter for Novello method

23

$$G_{max,A}$$

(kPa)

Small strain shear modulus at reference depth

24

$$G_{max,B}$$

(-)

Exponent for Gmax variation as function of depth

For SANDAPI, ABBS, KALLEHAVE and Sorensen Plim is an optional input.

Note

Where possible the notation used in the explanation of each method is the same as used in the original source paper, rather than translate many different methods and styles of notation to a common one to be used in OPILE. A full list of different notations and descriptions is given within this help file.

Attention

It is always advised to consult the original reference sources of these methods if the user is not familiar with the implementation of a particular method. The references cited within the OPILE documentation are not intended to be exhaustive. Note that easy unit conversions can be made when inputting data such as dimensions and shear strengths, see units convention section for details.

## SANDAPI¶

OPILE has one implementation of the SANDAPI formulation even if API2000 and API RP2GEO formulate the methodology using slightly different equations. The equations as presented in API RP2GEO have been implemented:

According to API RP2GEO (2011) the ultimate lateral bearing capacity for sand has been found to vary from a value at shallow depths determined by:

$p_{us} = (C_{1} \cdot z + C_{2} \cdot D)C_{1} \cdot \gamma' \cdot z$

To a value at large depths determined by:

$p_{ud} = C_{3} \cdot D \cdot \gamma' \cdot z$

At any given depth the smallest value of pu should be used as the ultimate bearing capacity.

Where:

 $$\gamma'$$ is the submerged soil unit weight $$z$$ is the depth below the original seafloor. $$D$$ is the pile outside diameter $$C_{1}$$ coefficient calculated as: $$C_{1} = \frac {(\tan{\beta})^2 \cdot \tan{(\alpha)}}{\tan{(\beta-\phi')}} \cdot K_{0} \cdot \left[\frac {\tan{(\phi')} \cdot \sin{(\beta)}}{\cos{(\alpha)} \cdot \tan{(\beta - \phi')}} + \tan{\beta} \cdot (\tan{\phi'} \cdot \sin{\beta}-\tan{\alpha})\right]$$ $$C_{2}$$ coefficient calculated as: $$C_{2} = \frac {\tan{(\beta)}}{\tan{(\beta - \phi')}} - K_{a}$$ $$C_{3}$$ coefficient calculated as: $$C_{3} = K_{a}((\tan{\beta})^8 - 1 )+ K_{0} \cdot \tan{\phi'} \cdot (\tan{\beta})^4$$ $$\alpha$$ $$\alpha = \frac {\phi'}{2}$$ $$\beta$$ $$\beta = 45 + \frac {\phi'}{2}$$ $$K_{0}$$ $$0.4$$ $$K_{a}$$ $$\frac {1-sin(\phi')}{1+sin(\phi')}$$

The lateral soil resistance-deflection (p-y) relationships for sand are non-linear and may be approximate at a specific depth z by the following expression:

$p = A \cdot p_{u} \cdot \tanh{\left(\frac{kz}{Ap_{u}} \right)}$

Where:

 $$A$$ is a factor to account for cyclic or static loading condition, evaluated by: $$A= (3.0-0.8\frac{z}{D}) \geq 0.9$$ for static loading and $$0.9$$ for cyclic loading. $$k$$ is the rate of increase with depth of initial modulus of subgrade reaction $$[kN/m^{3}]$$ - see table below $$p_{u}$$ is the ultimate lateral resistance at depth z

For SANDAPI there is an optional parameter $$P_{lim}$$ which limits the lateral pressure, if necessary. If this optional parameter is left blank then it will not be used, otherwise the limit will be applied.

Optional entry of a limiting lateral ulitmate pressure

Coefficients as a function of internal friction angle, ref API RP2GEO (2011)

The the rate of increase with depth of initial modulus of subgrade reaction can be derived from the table below as a function of $$\phi'$$.

 $$\phi'$$ $$k$$ $$MN/m^3$$ $$lb/in^3$$ 25 5.4 20 30 11 40 35 22 80 40 45 165

## CLAYAPI¶

According to API (2000) for static lateral loads the ultimate unit lateral bearing capacity pu for a pile embedded in CLAYS is calculated according to:

$\begin{split}p_{u} = \left\| \begin{eqnarray} 3 \cdot c + \sigma'_{v} + J \cdot \frac{c . X}{D} & \quad \text{if } X<X_{R} \\ 9 \cdot c & \quad \text{if } X \geq X_{R} \end{eqnarray}\right.\end{split}$

Where:

 $$p_{u}$$ is the ultimate lateral resistance (kPa) $$c$$ is the undrained shear strength for undisturbed clay soil samples. $$D$$ is the pile outside diameter $$\sigma'_{v}$$ is the effective overburden pressure at the depth in question (kPa) $$J$$ is a dimensionless empirical constant with values ranging from 0.25 to 0.5. $$X$$ is depth below soil surface $$X_{R}$$ is depth below soil surface to bottom of reduced resistance zone. Given by: $$X_{R} = \frac {6 \cdot D}{\frac {\gamma' \cdot D}{c}+ J}$$

The curve for STATIC loading is generated from the following co-ordinate points:

 $$p/p_{u}$$ $$y/y_{c}$$ 0.00 0 0.23 0.1 0.33 0.3 0.50 1.0 0.72 3.0 1.00 8 1.00 infinity

where:

 $$p$$ is the acutal lateral resistance (kPa) $$y$$ is the actual lateral deflection [m] $$y_{c}$$ $$y_{c} = 2.5 \cdot \epsilon_{50} \cdot D$$ $$\epsilon_{50}$$ is the strain which occurs at one-half the maximum stress on laboratory undrained compression tests of undisturbed samples. Within OPILE it is referred to as “e50”

For the case where equilibrium has been reached under CYCLIC loading the PY curves may be generated from the following table:

 $$X > X_{R}$$ $$X < X_{R}$$ $$p/p_{u}$$ $$y/y_{c}$$ $$p/p_{u}$$ $$y/y_{c}$$ 0.00 0 0.00 0 0.23 0.1 0.23 0.1 0.33 0.3 0.33 0.3 0.50 1.0 0.50 1.0 0.72 3.0 0.72 3.0 0.72 infinity 0.72 $$\frac{X}{X_{R}}$$ 15 0.72 $$\frac{X}{X_{R}}$$ infinity

In addition some account of the recommendation within API for the treatment of stiff clays is made. For stiff clay (c > 96 kPa) which might undergo rapid deterioration under cyclic loadings the ultimate resistance will be reduced to something considerably less than that for soft clays. While stiff clays also have non-linear stress-strain relationships, they are generally more brittle than soft clays. In developing stress-strain curves and subsequent PY curves for cyclic loads, good judgment should reflect the rapid deterioration of load capacity at large deflections for stiff clays (according to API WSD (2000)). This is accounted for in OPILE by the application of a factor to the ultimate resistance, where the undrained shear strength is above a specified shear strength limit (usually 96kPa):

CLAYAPI other lateral parameters

The “Stiff CLAY API Factor” is applied to the calculated ultimate lateral resistance where the shear strength is above the specified limit. The application of this factor is only made if the STATIC/CYCLIC option is set to CYCLIC. The initial points on the PY curve remain the same, after a larger displacement the ultimate pressure is reduced. The single shear strength value (as recommended by API WSD (2000) implements a cut-off rather than a transition. If stiff clays are to be analysed the use of the Dunnavant method could also be considered.

Specifically for CLAYAPI, the parameter Xr can be manually set by the user. When the tickbox ‘Manually define Xr’ is checked, a text box for defining Xr will appear. When checked, OPILE will overwrite Xr by the user inputted value.

CLAYAPI Manually define Xr

A modification to Yc for large diameter piles, according to Stevens and Audibert (1979) can be performed by ticking the tickbox ‘Modify Y for large diameter piles’. The modification performed in OPILE is:

$Y_{c} = Y_{c} \cdot \left(12.75 \cdot \frac{0.0254}{D} \right)^{0.5}$

## CLAYREESE¶

Reese et al (1975) developed a PY curve suitable for use in stiff clays. The PY curve has a complex shape and can be applied for static and cyclic loading. For both the static and cyclic curves an initial linear section is defined using an initial stiffness which is assumed to vary linearly with depth, from zero at the surface. The linear section is defined by an initial stiffness gradient k. Typical values for k are shown in the table below:

 $$k (kPa/m)$$ undrained shear strength (kPa) 50 - 100 100 - 200 200 - 400 static loading 135 000 270 000 540 000 cyclic loading 55 000 110 000 220 000

The PY curve for STATIC conditions is then calculated using the following logic:

$\begin{split}p_{u} = \left\| \begin{eqnarray} \textrm{min} \left(k \cdot H \cdot y, \frac{p_{u}}{2} \cdot \sqrt{\frac{y}{y_{c}}} \right) & \quad \text{if } y \leq A \cdot y_{c} \\ \frac{p_{u}}{2} \cdot \sqrt{\frac{y}{y_{c}}} - 0.055 \cdot p_{u} \cdot \left(\frac{y - A \cdot y_{c}}{A \cdot y_{c}} \right)^{1.25} & \quad \text{if } A \cdot y_{c} < y \leq 6 \cdot A \cdot y_{c} \\ 0.5 \cdot p_{u} \cdot \sqrt{6 \cdot A} - 0.411 \cdot p_{u} - \frac{p_{u}}{16 \cdot y_{c}} \cdot (y - 6 \cdot A \cdot y_{c}) & \quad \text{if } 6 \cdot A \cdot y_{c} < y \leq 18 \cdot A \cdot y_{c} \\ 0.5 \cdot p_{u} \cdot \sqrt{6 \cdot A} - 0.411 \cdot p_{u} - 0.75 \cdot p_{u} \cdot A & \quad \text{if } y > 18 \cdot A \cdot y_{c} \end{eqnarray}\right.\end{split}$

The PY curve for CYCLIC conditions is then calculated using the following logic:

$\begin{split}p_{u} = \left\| \begin{eqnarray} \textrm{min} \left(k \cdot H \cdot y, B \cdot p_{u} \left(1-\left(\frac{y - 0.45 \cdot y_{p}}{0.45 \cdot y_{p}}\right)^{2.5}\right) \right) & \quad \text{if } y \leq 0.6 \cdot y_{p} \\ 0.936 \cdot B \cdot p_{u} - 0.085 \cdot p_{u} \cdot \frac{y - 0.6 \cdot y_{p}}{y_{c}} & \quad \text{if } 0.6 \cdot y_{p} < y \leq 1.8 \cdot y_{p} \\ 0.936 \cdot B \cdot p_{u} - \frac{0.102}{y_{c}} \cdot p_{u} \cdot y_{p} & \quad \text{if } y > 1.8 \cdot y_{p} \end{eqnarray}\right.\end{split}$

Where:

 $$p_{u}$$ $$\textrm{min} \left(b \cdot (2 \cdot s_{uav} + \sigma'_{v} + 2.83 \cdot s_{uav} \cdot \frac{H}{b}), b \cdot 11 \cdot s_{u}\right)$$ $$s_{u}$$ su is the undrained shear strength at the depth in question $$s_{uav}$$ is the undrained shear strength averaged from the current depth to the seabed (or to the depth of the layer above with a different lateral method). $$H$$ is the depth $$b$$ is the pile diameter $$\sigma'_{v}$$ is the effective overburden pressure $$y_{c}$$ $$y_{c} = \epsilon_{50} \cdot b$$ $$y_{p}$$ $$y_{p} = 4.1 \cdot A \cdot y_{c}$$ $$\epsilon_{50}$$ is the strain at one half the maximum undrained shear strength in a triaxial test. $$A$$ is an empirical adjustment factor determined for STATIC loading: $$\textrm{min} \left(-0.05 \cdot (\frac{H}{b})^{2} + 0.29 \cdot \frac{H}{b} + 0.2, 0.6\right)$$ $$A$$ is an empirical adjustment factor determined for CYCLIC loading: $$\textrm{min} \left(-0.055 \cdot (\frac{H}{b})^{2} + 0.15 \cdot \frac{H}{b} + 0.2, 0.3\right)$$

CLAYREESE STATIC Curve

CLAYREESE CYCLIC Curve

## DYSON¶

The DYSON p-y curve method is intended for use in calcareous soils and is described by Dyson & Randolph (2001)

$p = \gamma' \cdot D \cdot R \cdot \left(\frac{q_{c}}{\gamma' \cdot D} \right)^{n} \cdot \left(\frac{y}{D}\right)^{m}$

where:

 $$p$$ lateral resistance (kPa) $$R$$ is a constant for curve stiffness (-) $$q_{c}$$ is cone tip resistance at the specified depth in [kPa], which in OPILE is interpolated between cone resistances specified for the top and bottom of a layer. $$n$$ is a constant $$m$$ is a constant controlling the amount of curvature in the p-y relation. $$\gamma'$$ is the soil submerged unit weight (kN/m3) $$D$$ is the pile diameter (m).

For a calcareous soil near the Goodwyn A platform Dyson and Randolph (2001) found R to be between 2.56 and 2.84, n=0.72 and m is between 0.52 and 0.64.

## ROCKREESE97¶

The ROCKREESE97 method has been taken from Reese, L. & Van Impe, W.F. (2001). The PY curve has a complex shape and can be applied for static and cyclic loading.

The PY curve is calculated using the following logic:

$\begin{split}p = \left\| \begin{eqnarray} \textrm{min}\left[\frac{p_{ur}}{2} \cdot \left(\frac{y}{y_{rm}} \right)^{0.25} , K_{ir} \cdot y \right] & \quad \text{if } y \leq 16 \cdot y_{rm} \\ p_{ur} & \quad \text{otherwise } \end{eqnarray}\right.\end{split}$

The ultimate lateral resistance of the rock $$p_{ur}$$ in kN/m can be calculated using the following logic:

$\begin{split}p_{ur} = \left\| \begin{eqnarray} \alpha_{r} \cdot q_{u} \cdot B \cdot (1 + 1.4 \cdot \frac{x_{r}}{B}) & \quad \text{if } 0 \lt x_{r} \lt 3 \cdot B \\ 5.2 \cdot \alpha_{r} \cdot q_{u} \cdot B & \quad \text{otherwise } \end{eqnarray}\right.\end{split}$

Where:

 $$y$$ is the displacement $$q_{u}$$ is the compressive strength of the rock $$x_{r}$$ is the depth below the rock surface $$B$$ is the pile diameter $$y_{rm}$$ $$y_{rm} = k_{rm} \cdot B$$ $$k_{rm}$$ is a constant ranging from 0.0005 to 0.00005, for more information see Reese, L. & Van Impe, W.F. (2001) $$K_{ir}$$ $$K_{ir} = k_{ir} \cdot E_{ir}$$ $$k_{ir}$$ $\begin{split}k_{ir} = \left\| \begin{array} 1100 + 400 \cdot \frac{x_{r}}{3 . B} & \quad \text{if } 0 < x_{r} < 3B \\ 500 & \quad \text{otherwise } \end{array}\right.\end{split}$ $$E_{ir}$$ is the initial reaction modulus of the rock (user input) in kPa $$\alpha_{r}$$ is a strength reduction factor

Characteristic shape of p-y curves for weak rock (after Reese & vam Impe, 2001)

## EVANS¶

Evans and Duncan (1982) developed PY curves for c-$$phi$$ soils for the application of short-term static loading and for cyclic loading. Similar as the CLAYREESE method, for both the static and cyclic curves an initial linear section is defined using an initial stiffness which is assumed to vary linearly with depth, from zero at the surface. The linear section is defined by an initial stiffness gradient k. Typical values for k are shown in the table below:

 $$k (kPa/m)$$ undrained shear strength (kPa) 50 - 100 100 - 200 200 - 400 static loading 135 000 270 000 540 000 cyclic loading 55 000 110 000 220 000

The PY curve for STATIC and CYCLIC conditions is calculated using the following logic:

$\begin{split}p_{static}(y) = \left\| \begin{eqnarray} k \cdot x & \quad \text{if } y \leq y_{k} \\ C \cdot y^{1/n} & \quad \text{if } y_{k} < y \leq \frac{b}{60} \\ p_{m} - \frac{p_{m} - p_{y}}{\frac{3 \cdot b}{80} - \frac{b}{60}} \cdot (y - \frac{b}{60}) & \quad \text{if } \frac{b}{60} < y \leq \frac{3 b}{80} \\ p_{u} & \quad \text{if } y > \frac{3 b}{80} \end{eqnarray}\right.\end{split}$

Where:

 $$p_{u}$$ $$p_{u} = A' \cdot p_{u\phi} + p_{uc}$$ $$A'$$ taken from the figure below $$p_{u\phi}$$ $\begin{split}p_{u\phi} = Min \left\| \begin{array} \gamma \gamma \cdot X \left(\frac{K_{0} \cdot x \cdot \tan{(\phi)} \cdot \sin{(\beta)}}{\tan{(\beta - \phi)} \cdot \cos{(\alpha)}} + \frac{\tan{(\beta)}}{\tan{(\beta - \phi)}} \cdot (b + x \cdot \tan{(\beta)} \cdot \tan{(\alpha)}) \cdot K_{0} \cdot x \cdot \tan{(\beta)} * (\tan{(\phi)} \cdot \sin{(\beta)} - \tan{(\alpha)}) - K_{A} \cdot b \right) \\ K_{A} \cdot b \cdot \gamma \cdot x \cdot \left(\tan{(\beta)}^{8}-1 \right) + K_{0} \cdot b \cdot x \cdot \tan{(\alpha)} \cdot \tan{(\beta)}^{4} \end{array}\right.\end{split}$ $$p_{uc}$$ $\begin{split}p_{uc} = Min \left\| \begin{array} ( \left(3 + \frac{\gamma'}{c} . x + \frac{J}{b} . x \right) . c . b \\ 9 . c . b \end{array}\right.\end{split}$ $$\alpha$$ $$\frac{\phi}{2}$$ $$\beta$$ $$45 + \alpha$$ $$c$$ is the undrained shear strength at the depth in question (cohesion) $$\phi$$ is the angle of internal friction $$x$$ is the depth at which the passive resistance is considered $$b$$ is the pile diameter $$\gamma$$ is the unit weight of the soil $$K_{0}$$ taken as 0.4 $$K_{A}$$ $$K_{A} = tan(45 - \frac{\phi}{2})^{2}$$ $$m$$ $$m = \frac{p_{u} - p_{m}}{y_{u} - y_{m}}$$ $$n$$ $$n = \frac{p_{m}}{m \cdot y_{m}}$$ $$C$$ $$C = \frac{p_{m}}{y_{m}^{1/n}}$$ $$y_{k}$$ $$y_{k} = (\frac{C}{k \cdot x})^{\frac{n}{n-1}}$$ $$A'$$ is an empirical adjustment factor determined for STATIC loading:

Values of coefficients A’ for the STATIC and the CYCLIC condition

Characteristic shape of p-y curves proposed for :math:c-phi soils

Note

Note that it can occur that $$y_{k} > \frac{b}{60}$$, in this case the exponential part of the curve is not present.

## FRAGIO¶

Fragio et al (1985) developed a PY curve criterion for weak calcareous claystone. It was developed from pile load tests in calcareous claystone with strengths of between 9 and 36MPa. The response is intended to represent brittle failure near the surface. The initial linear response is influenced by the rock mass stiffness and the peak stress is reached at a displacement yu. The peak stress is given by:

$\begin{split}p_{u} = \left\| \begin{eqnarray} 3s & \quad \text{at the surface } \\ 9s & \quad \text{at depth } \end{eqnarray}\right.\end{split}$

Where:

 $$s$$ is the rock mass shear strength [kPa]

Fragio et al (1985) found that by fitting the test data a value of s of 10% (which can be changed by the user in OPILE) of the unconfined compressive strength of the intact rock gave a good match with the measured response. After the initial response the curve remains at pu until y exceeds 3yu where it drops to 0.5pu as shown in the figure below.

The reduction only occurs at the surface where it is expected that a brittle failure may occur, the reduction depth is typically 6 pile diameters below mudline and can be varied in OPILE. At some transition depth the curve takes on the shape of the deep failure curve, with no reduction from the peak stress. A linear interpolation is used between the surface and the transition depth to determine the response at displacements where the reduction takes place.

Typical shape of PY curves for Zumaya claystone, ref Fragio et al (1985)

## ABBS¶

The ABBS method was designed for application in soft rock and is described by Abbs (1983). It was developed for carbonate rocks having strengths in the range of 0.5 to 5MPa. The first part of the curve is given by the CLAYREESE method for static loading in stiff clay, up until the end of the second parabolic section (i.e. $$y \leq 6Ayc$$). After that the pressure undergoes a rapid change to the residual pressure given by the SANDAPI method for cyclic loading.

The Abbs (1983) method essentially assumes that the behaviour is elastic up to the peak in the parabolic section. Once this point is passed, inter-particle bonding is assumed to be destroyed and the resistance is assumed to be represented by the “residual frictional resistance” curve. The slope of the drop from peak resistance to the residual frictional resistance curve is arbitrary. On first loading the slope would probably be more shallow than shown but as the structures are subject to cyclic loading the acutal slope is not critical. On subsequent load cycles the residual frictional curve is assumed to apply. The fall from the peak to the residual curve is therefore assumed to occur during an additional 10% displacement after the point of peak resistance. This slope is convenient from a computational point of view and represents the most dramatic fall that is anticipated from the more brittle materials encountered in the field.

The PY curve is generated using the following logic:

$\begin{split}p_{abbs}(y) = \left\| \begin{eqnarray} \textrm{min} \left(k \cdot H \cdot y , \frac{p_{u}}{2} \cdot \sqrt{\frac{y}{y_{c}}} \right) & \quad \text{if } y \leq A \cdot y_{c} \\ \frac{p_{u}}{2} \cdot \sqrt{\frac{y}{y_{c}}} - 0.055 \cdot p_{u} \cdot \left(\frac{y - A \cdot y_{c}}{a \cdot y_{c}} \right)^{1.25} & \quad \text{if } A \cdot y_{c} < y \leq 6 \cdot A \cdot y_{c} \\ b \cdot Sand_{A} \cdot p_{sand} \cdot \tanh{\left(\frac{Sand_{K} \cdot H \cdot y}{Sand_{A} \cdot p_{sand} \cdot b}\right)} & \quad \text{if } y > 6 \cdot A \cdot y_{c} \end{eqnarray}\right.\end{split}$

Where:

 $$p_{u}$$ $$p_{u} = \textrm{min} \left( b \cdot (2 \cdot s_{uav} + \sigma'_{v} + 2.83 \cdot s_{uav} \cdot \frac{H}{b}), b \cdot 11 \cdot s_{u} \right)$$ in kN/m $$s_{u}$$ is the undrained shear strength at the depth in question $$s_{uav}$$ is the undrained shear strength averaged from the current depth to the seabed (or to the depth of the layer above with a different lateral method). $$H$$ is the depth $$b$$ is the pile diameter $$\sigma'{v}$$ is the effective overburden depth $$y_{c}$$ $$y_{c} = \epsilon_{50} \cdot b$$ $$\epsilon_{50}$$ is the strain at one half the maximum undrained shear strength in a triaxial test $$A$$ $\begin{split}A = \left\| \begin{eqnarray} 0.6 & \quad \text{if } \frac{H}{b} \gt 3 \\ \textrm{min} \left(-0.05 \cdot \frac{H}{b}^{2} + 0.29 \cdot \frac{H}{b} + 0.2 , 0.6 \right) & \quad \text{if } 0 \gt \frac{H}{b} \geq 3 \\ 0.2 & \quad \text{if } \frac{H}{b} \leq 0 \end{eqnarray}\right.\end{split}$ $$Sand_{A}$$ is a factor to account for CYCLIC loading - taken as 0.9. $$p_{Sand}$$ is the ultimate pressure (i.e. at large displacements) calculated using the SANDAPI method. $$Sand_{K}$$ is the subgrade modulus

Caution

Note, it is possible that the SANDAPI method returns a higher residual pressure than that calculated by the CLAYREESE method. This results in an unusual and unlikely looking PY curve. OPILE does not automatically check and warn the used this has hapenned. The user therefore remains responsible for checking the input parameters and the PY curves returned.

Typical curves for the ABBS method are shown below:

Example of ABBS Py curves

## DUNNAVANT¶

The Dunnavant PY curve method for STATIC conditions is calculated by:

$p = 1.02 \cdot p_{u} \cdot \tanh{\left(0.537 \cdot \left(\frac{y}{y_{50}} \right)^{0.7} \right)}$

Where:

 $$p_{u}$$ $$p_{u} = N_{p} \cdot s_{u} \cdot B$$ $$N_{p}$$ $$N_{p} = \textrm{min}(2 + \frac{\sigma'{v}}{s_{uav}} + 0.4 \cdot \frac{x}{B} , 9)$$ $$s_{u}$$ su is the undrained shear strength at the depth in question $$s_{uav}$$ is the undrained shear strength averaged from the current depth to the seabed (or to the depth of the layer above with a different lateral method). $$x$$ is the depth $$B$$ is the pile diameter $$\sigma'{v}$$ is the effective overburden pressure $$y_{50}$$ $$y_{50} = 0.0063 \cdot \epsilon_{50} \cdot B \cdot K_{R}^{-0.875}$$ $$\epsilon_{50}$$ is the strain at one half the maximum undrained shear strength in a triaxial test $$K_{R}$$ is the relative soil-pile stiffness and is included as a parameter to account for elastic coupling of the p-y curves. $$K_{R}$$ might typically be 0.001.

Example of a static Dunnavant Py curve

For CYCLIC conditions there are a number of steps that need to be followed. The initial shape of the PY curve follows that for STATIC conditions, at a certain displacement the CYCLIC curve leaves the STATIC curve and it is necessary to determine the displacement at which this occurs. The peak lateral pressure for CYCLIC conditions is given by:

$p_{cm} = N_{cm} . s_{u} . B$

Where:

 $$N_{cm}$$ $$N_{cm} = N_{p} \cdot \textrm{min}\left(1 - (0.45 - 0.18 \cdot \frac{x}{x_{0}}) \cdot \log{(N)} , 1 \right)$$ $$N$$ is the number of cycles, typically 100 to 200.

The displacement at the peak cyclic pressure is then given by rearranging the STATIC PY curve equation to give the $$y_{peak}$$ Y value at $$p_{cm}$$, thus:

$y_{peak} = \textrm{1m} \cdot 10^{\left[ \log{\left[\left(\frac{y_{50}}{1m}\right)^{0.7} \right]} \cdot \frac{\textrm{atanh}\left(\frac{0.98 \cdot p_{cm}}{p_{u}}\right)}{0.537} \cdot \frac{1}{0.7} \right] }$

The residual of the PY curve is then calculated using:

$p_{r} = p_{cm} \cdot \textrm{min} \left( 1 - \Big(0.25 - 0.07 \cdot \frac{x}{x_{0}}\Big) \cdot log(N) , 1 \right)$

The residual pressure is mobilised at a displacement of 12y50,with the overall shape of the curve as shown below:

Example of a CYCLIC Dunnavant Py curve

The shape of the CYCLIC curve makes a transition towards the STATIC curve as the depth increases.

## NOVELLO99¶

NOVELLO99 is a PY curve method which is intended for use in uncemented/very weakly cemented calcareous sediments and is described by Novello (1999).

The STATIC curve is given by:

$p = 2 \cdot D \cdot (\sigma'_{v})^{0.33} \cdot q_{c}^{0.67} \cdot \left(\frac{y}{D} \right)^{0.5} < p_{limit}$

Where:

 $$D$$ is the outer diameter (m) $$\sigma'_{v}$$ is the effective overburden pressure (kPa) $$q_{c}$$ is the cone tip resistance (kPa) $$P_{limit}$$ is the limiting plastic stress

The CYCLIC curve is given by:

$p = 2 \cdot D \cdot (1 - 0.6 \cdot U^{*}) \cdot (\sigma'_{v})^{0.33} \cdot q_{c}^{0.67} \cdot \left(\frac{y}{D}\right)^{0.5} < (1 - 0.6 \cdot U^{*}) . p_{limit}$

Where:

 $$U^{*}$$ is the excess pore pressure coefficient

The excess pore pressure coefficient is calculated according to Dobry (1988):

$U^{*} = \frac{(N \cdot g(\gamma_{c}))^{0.37}}{1 + (N \cdot g(\gamma_{c}))^{0.37}}$

Where:

 $$N$$ is the number of cycles $$\gamma_{c}$$ cyclic shear strain [-] $$g(\gamma_{c})$$ $\begin{split}g(\gamma_{c}) = \left\| \begin{eqnarray} \textrm{Exp}(5.47 - \frac{2.08}{\gamma_{c} \cdot 100 + 0.11}) & \quad \text{if } \gamma_{c} \gt 0.01\% \\ 0 & \quad \text{otherwise } \end{eqnarray}\right.\end{split}$

## KALLEHAVE¶

The KALLEHAVE method is presented in Kallehave et. al. (2012) for application on large diameter piles for offshore wind turbines in sand, where the API recommendations for sand resulted in under-prediction of soil stiffness. The method is based on the API RP2GEO sand method presented in API RP2GEO (2011), but uses a modification on the subgrade reaction modulus $$(k*z)$$:

$E_{mod}^{py} = k \cdot z_{0} \cdot \left(\frac{z}{z_{0}} \right)^{m} \cdot \left(\frac{D}{D_{0}} \right)^{0.5}$

Where:

 $$k$$ subgrade reaction modulus (Sand K) [kPa/m] $$z_{0}$$ Reference depth = 2.5 [m] $$D_{0}$$ Reference diameter = 0.61 [m] $$m$$ 0.6

## Sorensen¶

The SORENSEN method is based on Sorensen et. al. (2010) and provides p-y curves for monopile foundations with diameters of 4m to 6m in sand. The method uses a similar approach as KALLEHAVE and is calibrated for dense sand with friction angle 40 degrees.

The SORENSEN method is based on the API RP2GEO sand method presented in API RP2GEO (2011), but uses a modification on the SAND subgrade reaction modulus $$(k*z)$$:

$E_{mod}^{py} = a \cdot \left(\frac{z}{z_{ref}} \right)^{b} \cdot \left(\frac{D}{D_{ref}} \right)^{c} \cdot \phi^{d}$

Where:

 $$\phi_{d}$$ Friction angle (deg) $$z_{ref}$$ Reference depth = 1.0 [m] $$D_{ref}$$ Reference diameter = 1.0 [m]

The parameters a, b, c and d are defined in Table 1 of Sorensen et. al. (2010):

a

b

c

d

[kN/m^2]

[-]

[-]

[-]

50 000

0.6

0.5

3.6

## CLAY JEANJEAN¶

The CLAYJEANJEAN method is based on Jeanjean (2009) and provides p-y curves for soft clays subjected to cyclic loads. The PY curves were derived from Finite Element Analysis of cyclic loaded conductors and are defined by the following equation:

$\frac{p}{p_{max}} = \tanh{\left(\frac{G_{max}}{f_{Su} \cdot s_{u}} \cdot \left(\frac{y}{D}\right)^{0.5}\right)}$
$p_{max} = s_{u} \cdot \left(N_{p0} - N_{pd} \cdot e^{\frac{- \xi \cdot z}{D}}\right)$
$G_{max} = A \cdot \left(\frac{z}{z_{0}} \right)^{B}$

Where:

 $$p$$ Soil resistance [kPa] $$p_{max}$$ Maximum soil resistance [kPa] $$G_{max}$$ Maximum shear modulus [kPa] defined with parameters A [kPa] and B [-] from soil input table $$z_{0}$$ Depth reference for Gmax [m] $$N_{p0}$$ Lateral soil resistance factor [-] $$N_{pd}$$ Lateral depth dependent soil resistance factor [-] $$f_{Su}$$ Clay maximum stiffness to strength ratio [-] $$s_{u}$$ Undrained shear strength [kPa] $$D$$ Pile diameter [m] $$\xi$$ empirical factor [-]

The depth reference $$z_{0}$$ for the maximum shear modulus can be overwritten by checking the “z0 for Gmax” checbox in the Lateral Soil tab:

Overwrite the Gmax reference depth

The factor $$\xi$$ is defined as:

$\begin{split}\xi = \left\| \begin{eqnarray} 0.25 + 0.05 . \lambda & \quad \text{if } \lambda \lt 6 \\ 0.55 & \quad \text{otherwise } \end{eqnarray}\right.\end{split}$

where:

 $$\lambda$$ $$\frac{s_{u0}}{s_{u1} \cdot D}$$ $$s_{u0}$$ Shear strength intercept at top of layer [kPa] $$s_{u1}$$ Rate of increase of shear strength with depth [kPa/m]

The rate of increase of shear strength with depth is defined as:

$s_{u1} = \frac{s_{u,bottom} - s_{u,top}}{z_{bottom} - z_{top}}$

The parameters $$N_{p0}$$, $$N_{pd}$$, $$f_{Su}$$ and $$d$$ are defined in Jeanjean (2009):

$$N_{p0}$$

$$N_{pd}$$

$$f_{Su}$$

[kN/m^2]

[-]

[-]

12

4

100

## CLAYJJ2017¶

The CLAYJJ2017 method is based on Jeanjean et al. (2017). In this approach a normalised PY curve is derived either from site specific DSS stress-strain curves or, in the absence thereof, on the tabulated normalised curves as repeated below. The tabulated values are based on database of DSS test results as evaluted by Jeanjean. OPILE only implements the tabulated normalised PY curves.

 $$s_{u} \leq 100kPa$$ $$s_{u} > 100kPa$$ $$\frac{p}{p_{u}}$$ $$\frac{y}{D}$$ $$\frac{y}{D}$$ 0 0 0 0.05 0.0003 0.0004 0.2 0.003 0.0035 0.3 0.0053 0.007 0.4 0.009 0.0125 0.5 0.0145 0.02 0.6 0.022 0.03 0.7 0.032 0.045 0.8 0.05 0.07 0.9 0.082 0.114 0.975 0.15 0.16 1.0 0.25 0.25

The tabulated normalised PY curves are then de-noramlised into a py curve for a given pile diameter, $$D$$, and with a value of the ultimate lateral pressure, $$p_{u}$$, calculated as:

$P_{u} = N_{p} \cdot s_{u}$
$\begin{split}N_{p} = \left\| \begin{eqnarray} N_{p0} + \frac{\gamma z}{s_{u0}+s_{u1}z} \leq N_{pd} & \quad \text{if gap assumed} \\ 2 \cdot N_{p0} \leq N_{pd} & \quad \text{if no gap assumed } \end{eqnarray}\right.\end{split}$

Where:

 $$N_{p0}$$ $$N_{1} - (1 - \alpha) - (N_{1} - N_{2}) \left[1- \left(\frac{z}{dD} \right)^{0.6} \right]^{1.35} \leq N_{pd}$$ $$N_{1}$$ 12 $$N_{2}$$ 3.22 $$d$$ $$16.8 - 2.3 \textrm{log_{10}}(\lambda) \geq 14.5$$ $$\lambda$$ $$s_{u0} / (s_{u1}D)$$ $$s_{u}$$ $$s_{u0}+s_{u1}z$$ , as determined in DSS testing $$N_{pd}$$ 9 + 3 $$\alpha$$ $$\alpha$$ the soil pile adhesion factor

For gapping conditions, the value of $$N_{p0}$$ in the shallow wedge is multiplied by the correction factor $$C$$ to account for strength anisotropy.

$C = 1 + \left(\frac{s_{uTE}}{s_{uDSS}}-1\right) \left(\frac{N_{pd}-N_{p}}{N_{pd}-N_{p|z=0}} \right)$

where $$N_{p|z=0}$$ is the value of $$N_{p}$$ at ground level (z=0). The ratio $$\frac{s_{uTE}}{s_{uDSS}}$$ is, in OPILE, taken by default as 0.9, but can be specified by the user. For non gapping conditions, the value of $$p_{u}$$ is not modified.

## Custom lateral methods¶

The custom lateral methods “CUST_PY” numbers 1 to 3 which are available under the lateral soil input allow the generation of custom PY curve shapes. The custom curves are used in conjunction with the limiting lateral pressure Plim and the Yc normalisation parameter for the lateral displacement. The custom py curves are thus defined normalised in terms of P/Pult (a noramlised lateral pressure) and Y/Yc (a normalised lateral displacement). Both normalisation factors, Yc and Pult, are defined on the lateral soil input.

Illustration of custom lateral method selection

The normalised PY curves can be defined on the Custom Input tab under the Custom Response Curves tab. The table to be completed will be highlighted yellow as soon as a custom PY curve method is selected on the Lateral Soil input.

Definition of custom lateral reponse