16.2.1 Simplified Method (Closed-Form Solutions)

Closed-form solutions have been proposed by Shadlou and Bhattacharya [13]. Values of $K_{\mathrm{L}}$, $K_{\mathrm{R}},$ and $K_{\mathit{LR}}$ are summarized in Tables 16.1 and 16.2 for rigid and flexible piles for three types of ground profile: homogenous, linear, and parabolic, see Fig. 16.4. Homogeneous soils are soils which have a constant stiffness with depth such as over-consolidated clays. On the other hand, a linear profile in typical for normally consolidated clays and parabolic behavior can be used for sandy soils. It may be observed from Table 16.1 that the stiffness term is a function of the aspect ratio (L/D_p) of the pile and soil stiffness (E_S0) and this is a characteristic of rigid behavior. Table 16.2 is a function of relative pile–soil stiffness (E_P/E_S). Further details of classification of rigid and flexible piles in relation to monopile application can be found by Shadlou and Bhattacharya [13].

Table 16.1

Formulas for Stiffness of Monopiles Exhibiting Rigid Behavior

Ground Profile (See Fig. 16.4)	KL	KLR	KR
Homogeneous	$3.2{\left(\frac{L}{D_{\mathrm{P}}}\right)}^{0.62}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}$	$-1.8{\left(\frac{L}{D_{\mathrm{P}}}\right)}^{1.56}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D_{\mathrm{P}}}^2$	$1.65{\left(\frac{L}{D_{\mathrm{P}}}\right)}^{2.5}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D_{\mathrm{P}}}^3$
Parabolic	$2.65{\left(\frac{L}{D_{\mathrm{P}}}\right)}^{1.07}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}$	$-1.8{\left(\frac{L}{D_{\mathrm{P}}}\right)}^2{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D_{\mathrm{P}}}^2$	$1.63{\left(\frac{L}{D_{\mathrm{P}}}\right)}^3{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D_{\mathrm{P}}}^3$
Linear	$2.35{\left(\frac{L}{D_{\mathrm{P}}}\right)}^{1.53}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}$	$-1.8{\left(\frac{L}{D_{\mathrm{P}}}\right)}^{2.5}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D_{\mathrm{P}}}^2$	$1.59{\left(\frac{L}{D_{\mathrm{P}}}\right)}^{3.45}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D_{\mathrm{P}}}^3$

$*f_{(\mathrm{vs})}=1+0.6\left|{\upsilon }_{\mathrm{s}}-0.25\right|$

Table 16.2

Tables for Monopiles Exhibiting Flexible Behavior

Ground Profile (See Fig. 16.4)	KL	KLR	KR
Homogeneous	$1.45{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.186}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}$	$-0.3{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.5}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}^2$	$0.19{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.73}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}^3$
Parabolic	$1.015{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.27}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}$	$-0.29{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.52}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D_{\mathrm{P}}}^2$	$0.18{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.76}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}^3$
Linear	$0.79{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.34}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}$	$-0.27{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.567}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}^2$	$0.17{\left(\frac{E_{\mathrm{P}}}{E_{\mathrm{SO}}}\right)}^{0.78}{f}_{(\mathrm{vs})}{E}_{\mathrm{SO}}{D}_{\mathrm{P}}^3$

$*f_{(\mathrm{vs})}=1+0.6\left|{\upsilon }_{\mathrm{s}}-0.25\right|$

16.2.2 Standard Method

Traditionally in the offshore industry, a nonlinear p–y method is employed to find out pile head deformations (deflection and rotation) and foundation stiffness. The approach can be found in API [14] and also suggested in DNV [15]. Originally it was developed by Matlock [16]; Reese et al. [17]; O’Neill and Murchinson [18], and the basis of this methodology is the Winkler approach [2] whereby the soil is modeled as independent springs along the length of the pile, see Fig. 16.5. The p–y approach uses nonlinear springs and produces reliable results for cases in which it was developed, i.e., small diameter piles and for few cycles of loading. The method is not validated for large diameter piles, in fact, using this method, underprediction of foundation stiffness has been reported by Kallehave and Thilsted [19], who also proposed an updated p–y formulation. Many researchers have recently worked on developing design methodologies for the large diameter more stocky monopiles with length to diameter ratios typically in the range between 4 and 10.

16.2.3 Advanced Method

In this method, soil–structure interaction is carried out using 2D and 3D FEA (finite element analysis) and different sophisticated soils models can be used. This requires advanced soil testing to feed the soil parameters. It is important to note that such models usually require not only knowledge but also expertise and experience. It is not advisable to use this method during the preliminary and optimization stages due to the high cost and the time required for calculation. This method best serves as a final check to a few optimized cases as results obtained are usually of high accuracy. Fig. 16.6 shows a typical section through a finite element model with various degrees of freedom showing the soil mass, the pile, the pile–soil interface, and the boundary conditions.

16.3.1 Example Problem (Monopile for Horns Rev 1)

The case study of Horns Rev 1 is used to show the application of the methodology. Various data on the case study can be found by Augustensen et al. [20]. The monopile foundation supports a 60 m long tower carrying a Vestas V80 2 MW wind turbine. The pile has a diameter of 4.0 m and varying wall thickness (WT in Fig. 16.8). The reported ultimate loads on the pile head were H=4.6 MN and M=95 MN.m. Table 16.3 provides a detailed description of the soil layers which was obtained through an extensive test program including geotechnical borings, cone penetration tests (CPTs), and triaxial tests given by Augustensen et al. [21]. ALP (Oasys) has been used for the p–y analysis of the monopile which can compute deflections, rotations, and bending moments along the pile. Several other software packages such as PYGM and LPILE are available for this type of analysis, or can be solved numerically through a MATLAB program. In this study, p–y curves along different depths are modeled in two ways: (1) elastic–plastic springs by taking the soil properties given in Table 16.3; (2) API recommended p–y springs.

The API code provides the following formulations for sandy soils:

$p=A{p}_{\mathrm{u}}\tanh \left(\frac{kz}{A{p}_{\mathrm{u}}}y\right)$ (16.16)

(16.16)

where A is a factor that depends on the type of loading and p_u depends on the depth and angle of internal friction ϕ′.

However for the elastic–plastic model used in the study, the following equations were used to construct the p–y curves.

$p=kyk=(E_{\mathrm{s}}h)$ (16.17)

(16.17)

where h is the midpoint of the elements immediately above and below the spring under consideration.

The plastic phase of the curve is given by:

$F_{\mathrm{p}}=(K_{\mathrm{q}}\sigma {\text{'}}_{\mathrm{v}}+c{K}_{\mathrm{c}})h{D}_{\mathrm{P}}$ (16.18)

(16.18)

where K_q and K_c are factors that depend on depth and ϕ′ [22].

Fig. 16.9 shows the results obtained from the analysis.

It is evident from the Fig. 16.9 that the ULS loads lie within the linear range which means the deflections and rotations arising from such forces can be estimated using K_L, K_R, and K_LR. Table 16.4 summarizes the results for the analysis where two load cases have been applied (H=0.2 MN and M=0), and (H=0 and M=0.2 MN m) where the major steps are also shown. It may also be noted that Fig. 16.9B plots the ultimate capacity of the pile based on two considerations: (1) soil fails and the pile fails as rigid body failure and (2) pile fails by forming plastic hinge.

Using Eq. (16.13) and applying to the results obtained and shown in Table 16.4, stiffness terms as predicted by elastic–plastic model is given in Eq. (16.19).

Elastic–plastic formulation

$\begin{array}{l}I=\left[\begin{array}{cc}0.00215& 0.00021\\ 0.00021& 0.000042\end{array}\right]\Rightarrow I^{-1}=K=\left[\begin{array}{cc}{K}_{\mathrm{L}}& K_{\mathrm{LR}}\\ K_{\mathrm{RL}}& K_{\mathrm{R}}\end{array}\right]=\left[\begin{array}{cc}894.1& -4451.3\\ -4451.3& 46252.1\end{array}\right]\hfill \\ \begin{array}{lll}{K}_{\mathrm{L}}=894.1\frac{\mathrm{MN}}{\mathrm{m}}\hfill & K_{\mathrm{LR}}=-4451.3\mathrm{MN}\hfill & K_{\mathrm{R}}=46252.1\frac{\mathrm{MN}\mathrm{m}}{\mathrm{rad}}\hfill \end{array}\hfill \end{array}$ (16.19)

(16.19)

The same approach has been used for the API model and the results are shown in Eq. (16.20).

API formulation

$\begin{array}{l}I=\left[\begin{array}{cc}0.00075& 0.00011\\ 0.00011& 0.00003\end{array}\right]\Rightarrow I^{-1}=K=\left[\begin{array}{cc}{K}_{\mathrm{L}}& K_{\mathrm{LR}}\\ K_{\mathrm{RL}}& K_{\mathrm{R}}\end{array}\right]=\left[\begin{array}{cc}3102& -11823.7\\ -11823.7& 78275.5\end{array}\right]\hfill \\ \begin{array}{lll}{K}_{\mathrm{L}}=3102\frac{\mathrm{MN}}{\mathrm{m}}\hfill & K_{\mathrm{LR}}=-11823.7\mathrm{MN}\hfill & K_{\mathrm{R}}=78275.5\frac{\mathrm{MN}\mathrm{m}}{\mathrm{rad}}\hfill \end{array}\hfill \end{array}$ (16.20)

(16.20)

It may be noted that the stiffness terms are quite different in the two formulations given in Eqs. (16.19) and (16.20). This is because the API formulation is empirical and is calibrated against small diameter piles and the extrapolation to large diameter piles is not validated or verified.

Finite element software package PLAXIS 3D was also used to evaluate the deflection and rotation due to the pile head loads. To save computational efforts space, half the pile was modeled (Figs. 16.10 and 16.11). A classical Mohr–Coulomb material model was set for the soil with the same stiffness and strength properties provided in Table 16.3. For the pile material, elastic perfectly plastic model was used. Ten node tetrahedral elements were assigned to the soil volume and six node triangular plate elements were used for the pile. The interface between the soil and the pile was modeled with double noded elements. The soil extents were set as 20D_p, and a medium fine mesh was used. The pile was extended by 21 m above the ground to simulate the effect of the applied moment. Similar plots such as that shown in Fig. 16.9 were obtained. Fig. 16.12 plots the deflection along the length of the pile obtained for H=4.6 MN and M=95 MN m.

16.4.1 Pile Head Deflections and Rotations

Eq. (16.5) can be used to predict pile head deflections and rotations as shown in Eq. (16.21). It may be noted from Fig. 16.9 that the ultimate loads were within the linear range.

$\begin{array}{l}Elastic–plasticformulation\hfill \\ \left[\begin{array}{c}\rho \\ \theta \end{array}\right]={\left[\begin{array}{cc}894.1& -4451.3\\ -4451.3& 46252.1\end{array}\right]}^{-1}\left[\begin{array}{c}4.6\\ 9.5\end{array}\right]=\left[\begin{array}{c}0.03\\ 4.8E-03\end{array}\right]\hfill \\ APIformulation\hfill \\ \left[\begin{array}{c}\rho \\ \theta \end{array}\right]={\left[\begin{array}{cc}3102& -11823.7\\ -11823.7& 78275.5\end{array}\right]}^{-1}\left[\begin{array}{c}4.6\\ 9.5\end{array}\right]=\left[\begin{array}{c}0.014\\ 3.5E-03\end{array}\right]\hfill \end{array}$ (16.21)

(16.21)

The results predict a deflection of 30 mm and a rotation of 0.280° (degree) using elastic–plastic model. On the other hand, the results predict 14 mm of deflection and 0.20° using API model. The advanced model based on PLAXIS 3D shown in Fig. 16.12 predicts a deflection of 42.5 mm and a rotation of 0.3°.

16.4.2 Prediction of the Natural Frequency

These foundation stiffness values can be used to predict natural frequency following the method proposed by Arany et al. [1]. The steps are also shown as follows.

Step 1: Fixed based natural frequency of the tower
Calculate the bending stiffness ratio of tower to the pile:

$\chi =\frac{E_{\mathrm{T}}{I}_{\mathrm{T}}}{E_{\mathrm{P}}{I}_{\mathrm{P}}}$ (16.22)

(16.22)

Calculate the platform/tower length ratio:

$\psi =\frac{L_{\mathrm{s}}}{L_{\mathrm{T}}}$ (16.23)

(16.23)

Calculate the substructure flexibility coefficient to account for the enhanced stiffness of the transition piece:

$C_{\mathrm{MP}}=\sqrt{\frac{1}{1+(1+{\psi }^3)\chi -\chi }}$ (16.24)

(16.24)

Obtain the fixed base natural frequency of the tower:

$f_{\mathrm{FB}}=\frac{1}{2\pi }{C}_{\mathrm{MP}}\sqrt{\frac{3E_{\mathrm{T}}{I}_{\mathrm{T}}}{\left(m_{RNA}+\frac{33m_T}{140}\right){L_{\mathrm{T}}}^3}}$ (16.25)

(16.25)

where the cross-sectional properties of the tower can be calculated as:

$\begin{array}{ll}{D}_{\mathrm{T}}=\frac{D_{\mathrm{b}}+D_{\mathrm{t}}}{2}\hfill & I_{\mathrm{T}}=\frac{1}{8}{\left(D_{\mathrm{T}}-t_{\mathrm{T}}\right)}^3{t}_{\mathrm{T}}\pi \hfill \end{array}$ (16.26)

(16.26)

Step 2: Calculate the nondimensional foundation stiffness parameters
The equivalent bending stiffness of tower needed for this step:

$\begin{array}{lll}q=\frac{D_{\mathrm{b}}}{D_{\mathrm{T}}}\hfill & f(q)=\frac{1}{3}\times \frac{2q^2{(q-1)}^3}{2q^2\ln q-3q^2+4q-1}\hfill & E{I}_{\mathrm{\eta }}=E{I}_{\mathrm{top}}\times f(q)\hfill \end{array}$ (16.27)

(16.27)

where I_top is the second moment of area of the top section of the tower.

${\eta }_{\mathrm{L}}=\frac{K_{\mathrm{L}}{L}^3}{E{I}_{\mathrm{\eta }}}$ (16.28)

(16.28)

${\eta }_{\mathrm{LR}}=\frac{K_{\mathrm{LR}}{L}^2}{E{I}_{\mathrm{\eta }}}$ (16.29)

(16.29)

${\eta }_{\mathrm{R}}=\frac{K_{\mathrm{R}}L}{E{I}_{\mathrm{\eta }}}$ (16.30)

(16.30)

Step 3: Calculate the foundation flexibility factors

$C_{\mathrm{R}}({\eta }_{\mathrm{L}},{\eta }_{\mathrm{LR}},{\eta }_{\mathrm{R}})=1-\frac{1}{1+0.6\left({\eta }_{\mathrm{R}}-\frac{{{\eta }^2}_{\mathrm{LR}}}{{\eta }_{\mathrm{L}}}\right)}$ (16.31)

(16.31)

$C_{\mathrm{L}}({\eta }_{\mathrm{L}},{\eta }_{\mathrm{LR}},{\eta }_{\mathrm{R}})=1-\frac{1}{1+0.5\left({\eta }_{\mathrm{L}}-\frac{{{\eta }^2}_{\mathrm{LR}}}{{\eta }_{\mathrm{R}}}\right)}$ 16.32)

16.32)

Step 4: Calculate the flexible natural frequency of the offshore wind turbine structure (OWT) system

$f_0=C_{\mathrm{L}}{C}_{\mathrm{R}}{f}_{\mathrm{FB}}$ (16.33)

(16.33)

Due to lack of the some aspects of tower data, a similar 2 MW wind turbine from North Hoyle has been used. Table 16.5 summarizes the tower properties used for the calculations.

For simplicity the substructure flexibility coefficient (C_MP) is considered to be 1. Therefore, following Eqs. (16.22)– (16.33), we obtain:

$\begin{array}{l}\begin{array}{ll}{D}_{\mathrm{T}}=\frac{D_{\mathrm{b}}+D_{\mathrm{t}}}{2}\Rightarrow \frac{4+2.3}{2}=3.15\hfill & I_{\mathrm{T}}=\frac{1}{8}{\left(3.15-0.035\right)}^3\times 0.035\times \pi =0.415m^4\hfill \end{array}\hfill \\ f_{\mathrm{FB}}=\frac{1}{2\pi }\sqrt{\frac{3\times 210E9\times 0.415}{\left(100000+\frac{33\times 130000}{140}\right){70}^3}}=0.385\mathrm{Hz}\hfill \end{array}$

The fixed base frequency is therefore 0.385 Hz.

$\begin{array}{l}\begin{array}{ll}q=\frac{4}{2.3}=1.74\hfill & f(q)=\frac{1}{3}\times \frac{2\times {1.74}^2{(1.74-1)}^3}{2\times {1.74}^2\ln 1.74-3\times {1.74}^2+4\times 1.74-1}=3.56\hfill \end{array}\hfill \\ E{I}_{\mathrm{\eta }}=210\times \frac{1}{8}{(2.3-0.035)}^3\times 0.035\times \pi \times 3.56=119.4\mathrm{GPa}\hfill \end{array}$

The nondimensional groups are:

$\begin{array}{lll}{\eta }_{\mathrm{L}}=\frac{0.8941\times {70}^3}{119.4}=2568\hfill & {\eta }_{\mathrm{LR}}=\frac{-4.45\times {70}^2}{119.4}=-182.6\hfill & {\eta }_{\mathrm{R}}=\frac{46.25\times 70}{119.4}=27.1\hfill \end{array}$

The foundation flexibility coefficients are given as follows:

$\begin{array}{ll}{C}_{\mathrm{R}}\left({\eta }_{\mathrm{L}},{\eta }_{\mathrm{LR}},{\eta }_{\mathrm{R}}\right)\hfill & =1-\frac{1}{1+0.6\left(27.1-\frac{-{182.6}^2}{2568}\right)}=0.894\\ C_{\mathrm{L}}\left({\eta }_{\mathrm{L}},{\eta }_{\mathrm{LR}},{\eta }_{\mathrm{R}}\right)\hfill & =1-\frac{1}{1+0.5\left(2568-\frac{-{182.6}^2}{27.1}\right)}=0.999\end{array}$

The natural frequency is therefore given by:

$f_0=0.894\times 0.999\times 0.385=0.344\mathrm{Hz}$

16.4.3 Comparison With SAP 2000 Analysis

The system was modeled using SAP 2000 and a modal analysis is carried out to obtain the natural frequency. Nonprismatic beam elements of varying diameter were assigned to the tower while the soil–structure interaction was represented by discrete linear Winkler springs. The spring stiffness values were taken from the soil properties (Table 16.3) and are a function of modulus of elasticity at the location of the spring and the spacing between two adjacent springs. A lumped mass was assigned at the tower head to model the dead mass of the RNA (rotor nacelle assembly). The first natural frequency recorded was 0.351 Hz and Fig. 16.14 shows the first mode of vibration.