14.1.1 Need for Numerical Analysis for Carrying out the Design

The previous section showed a simplified procedure to estimate SSI. However, this approach is grossly oversimplified and has limitations in the sense that the changes in soil properties due to cyclic loading cannot be considered. Also whether or not the foundation will progressive tilt cannot be studied. While the two simple interaction phenomena (i.e., due to pile bending and to lateral deflection of the pile) are taken into consideration, other aspects, such as interface opening (gap formation) or possible impacts between pile and soils during the dynamic loading cannot be accounted for. Moreover, the nature of dynamic loadings is complicated due to the wind–wave misalignment. These considerations make it necessary to use numerical simulations, which allow one to precisely define the geometry of the problem and can also encompass different types of soils. Furthermore changes in the behavior of the soils surrounding the pile is expected to occur as a consequence of the repetition of cycles. Advanced constitutive models, suitable to reproduce different aspects of cyclic/dynamic behavior, are therefore required for analysis. Classical elastoplastic models cannot estimate the accumulation of plastic volumetric strains due to the repetition of cycles (densification). Therefore advanced constitutive laws capable of simulating the plastic strains in soils not due to the magnitude of the loading, but due to the repetition of cycles are needed [13–15].

14.2.1 Standard Method Based on Beam on Nonlinear Winkler Spring

The American Petroleum Institute (API) code prescribes a methodology known as “p–y” method, which is widely used for design of offshore piles. This is a subgrade modulus method with nonlinear depth dependent load—deformation characteristics representing the pile–soil interaction. This procedure is calibrated to provide the worst case scenario for the foundation behavior under static and cyclic loading. The p–y curve has been adopted in most offshore standards (like API, 2007; GL, 2005; DNV, 2004 or ISO,) as it has been considered, for many years, as the best method available to determine the displacements of the head of the pile under various loading conditions. Fig. 14.6 represents a typical p–y curve and in the figure, from a to b, an increase in the loading carries an increasing rate of degradation. This nonlinear behavior can be determined empirically or using field tests, under either static or monotonic loading conditions. On the other hand, Fig. 14.6B shows the degradation suffered by the soil as the pile deflection increases [19,20,21].

Terzaghi [22] demonstrated that the linear relationship between p and y (from the origin to a in Fig. 14.6A) was only valid for values of p lower than a half of the ultimate bearing capacity of the soil [22]. This criteria is assumed in the construction of the p–y curves by many methodologies. API (2007) [23] takes into account a number of factors for the construction of the curves, which can be summarized as follows:

Some other aspects, such as the effects of pile installations in the soils and the scour, are taken into account, although it is fully recognized that these factors require further analysis and research.

The main limitations of this API (2007) are:

Computational simulations are the alternative to overcome those limitations. Different trends in these methodologies and few worked out examples are presented next, in order to address their suitability and reliability.

14.2.2 Advanced Analysis (Finite Element Analysis and Discrete Element Modeling) to Study Foundation–Soil Interaction

14.2.2.1 Different Soil Models Used in Finite Element Analysis

To account for accumulated strains in the numerical simulations, it is necessary to employ a suitable constitutive model, capable of reproducing realistic soil behavior when subjected to cyclic/dynamic loading conditions. Simple elastoplastic models are usually based on one yield surface without hardening (e.g., the Mohr Coulomb model), with isotropic hardening (e.g., the Cam-Clay model), or on two yield surfaces (isotropic loading and deviatoric loading), e.g., Lade’s. They are ideal and efficient on simulating the soil behavior under monotonic conditions, but not suitable to model cyclic loading. Essentially, in such models, plastic deformations start to appear for a certain magnitude of loading, usually higher that the amplitude of the cycles, and hence, the simulated soil behavior remains elastic under that limit, which is against experimental evidences particularly for granular soils [25]. Fig. 14.7 shows the different phenomenon, which can take place in soil during cyclic loading.

Plastic phenomenon for cyclic loading can either be based on “constant stress amplitude” (so called stress controlled) or “constant strain amplitude” (strain controlled). The phenomenon called “adaptation” refers to less dissipation of energy in each cycle of loading as the number of cycles increases, until convergence to nondissipative elastic cycles. On the other hand “accommodation” refers to change of the dissipation of energy from the beginning with an irreversible cumulative deformation and evolves toward a stabilized cycle. This is experienced in laboratory experiments for drained samples. “Ratcheting” is an irreversible strain accumulation, which keeps the same shape as that of the beginning [25].

Constant strain amplitude cyclic loading phenomenon results in the cyclic hardening or softening. Cyclic hardening occurs when there is an increase in the cyclic stress amplitude with an increase in the number of cycles, e.g., densification during testing of drained soil. On the contrary, cyclic softening occurs when there is a reduction in the cyclic stress amplitude as the number of cycles increases, e.g., an increase in pore pressure during the testing of undrained soil samples (decrease in the effective stress).

In one of the examples presented in this chapter, a kinematic hardening law has been employed by coupling the elastic moduli and a hardening parameter. Kinematic hardening is used to model plastic ratcheting, which is the build-up of plastic strain during cyclic loading, as previously explained. Other hardening behaviors include changes in the shape of the yield surface in which the hardening rule affects only a local region of the yield surface, and softening behavior in which the yield stress decreases with plastic loading. The kinematic hardening model only involves one plastic mechanism with a smooth yield surface. For nonlinear modeling, the material behavior is characterized by an initial elastic response, followed by plastic deformation and unloading from the plastic state. The plasticity is as a result of the microscopic nature of the material particles and includes shear loading that causes particles to move past one another, changes in void or fluid content that result in volumetric plasticity, and exceeding the cohesive forces between the particles or aggregates. The material is defined by model materials subject to loading beyond their elastic limit. See Refs. [25,26] for more details.

14.2.2.2 Discrete Element Model Analysis Basics

Originally proposed by Cundall and Strack [26], Discrete Element Method (DEM) simulates granular materials as assemblies of individual particles, which respond to given load conditions. The interactions between particles are simulated by contact laws, e.g., linear contact model, Hertz-Mindlin contact model [27], and parallel bond model, where the normal and tangential contact forces are dependent on the overlap and relative displacement between two contact particles (Fig. 14.8). The contact forces, accelerations, velocities, and displacements of all particles are updated in each small time step using the central difference time integration method. Stresses and strains are then calculated from the contact forces within a representative volume element (RVE) or along a boundary. The DEM is superior to other numerical methods (e.g., Finite Element Method (FEM)) as it allows the direct monitoring of change in soil stiffness, and more importantly it offers a method to analyze the micromechanics, which underlies the stiffness changes. Fig. 14.9 shows a flowchart of a typical DEM calculation.

14.3.1 Monopile Analysis Using DEM

An analysis of monopile–soil interactions and stiffness variations were carried out using an open-sourced DEM code modified and validated in previous studies (Cui [29], Cui et al. [30], O’Sullivan et al. [31]). More details can be found in Cui and Bhattacharya [32]. A DEM model of a soil tank (100 mm×100 mm×50 mm) was firstly created, which was filled with about 13 000 spherical particles of radii in the range of 1.1–2.2 mm. The particles were deposited under gravity. The pile was 20 mm in diameter and was embedded to a depth of 40 mm by removing particles located in the space, which was to be occupied by the pile. Particles were allowed to settle down again following the installation of the pile. Once the soil particles were settled down in the soil tank, cyclically horizontal movements were assigned to the pile to simulate the cyclic movements of OWT monopile due to the cyclic loadings. Translational movements rather than rotational movements were assigned to the pile at this stage. Three different strain amplitudes, 0.1%, 0.01%, and 0.001%, were chosen to examine the effects of strain levels. For each strain amplitude two types of cyclic loading were applied: symmetric cyclic loading with stains in the ranges of (−0.1%, 0.1%), (−0.01%, 0.01%), and (−0.001%, 0.001%) and asymmetric cyclic loading with strains in the ranges of (0, 0.2%), (0, 0.02%), and (0, 0.002%). As constrained by the computational costs, 500 cycles were simulated for strain amplitude of 0.1% and 1000 cycles were simulated for other strain amplitudes. The simulation parameters are listed in Table 14.1.

The resultant horizontal stress applied on the pile versus the horizontal strain of soil is illustrated in Fig. 14.10. It can be observed that the stress–strain curves forms hysteresis loops, indicating the energy dissipations during the cyclic loading. It is also interesting to observe that, though the stresses in the first half cycle for the asymmetric cyclic loading is positive, it reduced to negative when the strain goes to zero. Following a few cycles, the minimum negative stress approaches the same magnitude as the maximum positive stress. Moreover the magnitude of stresses and the shape of the hysteresis loops for both symmetric cyclic loading and asymmetric cyclic loading are almost identical after many cycles. The system under asymmetric cyclic loading behaves the same as symmetric cyclic loading with the same strain amplitude after many cycles, indicating that the strain amplitude, rather than the maximum strain, dominates the long-term cyclic behavior.

The secant Young’s Modulus of soil in each cycle was calculated by determining the slope of a line connecting the maximum and minimum points of each full loop and is shown in Fig. 14.11. It is evident that the secant Young’s Modulus of soil increased during the cyclic loading. The initial stiffness of asymmetric cyclic loading is lower than that of symmetric cyclic loading with the same strain amplitude due to higher maximum strain applied. However, following a few cycles, the stiffness for both types of cyclic loading approaches the same value. The Young’s Modulus versus horizontal strain in the first half cycle and in the 500th cycle for strain amplitude of 0.1% is shown in Fig. 14.12. The stiffness–strain curve in the first cycle displays the similar “S” shape as expected for the shear modulus–shear strain curve. Following cyclic loadings, all stiffness at different strain levels increases significantly.

To illustrate the ground settlement, plots of incremental soil particle displacements in the symmetric cyclic loading with strain amplitude of 0.1% in the first 50 cycles and in the next 50 cycles are given in Fig. 14.13. Each arrow in the plot starts from the original center of a particle and ends at the new center at the end of a given cycle. It is evident that soil particles surrounding the pile moved downwards, causing ground settlement. Soil densification around pile is the main reason causing the increase of soil stiffness. It also clear that the soil particle displacements are only significant in the first 50 cycles, underlying the significant increase in the soil stiffness in the first 50 cycles. In the current DEM simulations samples with same initial void ratio of 0.539 (medium dense sample) all showed densification behavior and stiffness increase under cyclic loading. It would be interesting to investigate soil behaviors and stiffness evolutions for a wide range of initial void ratios in the future study.

The evolutions of average radial stress in representative cycles are illustrated in Fig. 14.14. Due to the soil densification, the radial stress increased significantly for strain amplitude of 0.1% under cyclic loading. The shapes of the radial stress–strain curves are quite different for different strain amplitude. For symmetric cyclic loading with 0.01% strain amplitude (Fig. 14.14C), the radial stress increases at positive strain values, but decreases slightly at negative strain values. However, for 0.1% strain amplitude (Fig. 14.14A), the radial stress increases at both positive and negative strains, forming a “butterfly” shaped curve. It is also interesting to observe that, for the asymmetric cyclic loading (Fig. 14.14B), the stress–strain response in the first few cycles is different from that in the correlated symmetric cyclic loading, However, it eventually evolves to the same “butterfly” shape after many cycles. It confirms again that the influence of different maximum strain of a cyclic loading can be eliminated after many cycles and the dominant factor for cyclic soil response is strain amplitude.

The evolution of unbalanced horizontal force at the end of each cycle is illustrated in Fig. 14.15. It is evident that the unbalanced horizontal force is more significant with increasing strain amplitude. It can also be seen that the unbalanced force for asymmetric cyclic loading is much larger in magnitude and is oriented in the opposite direction compared with that for symmetric cyclic loading. It is because, for asymmetric cyclic loading pile only moves to the right side and compresses the soils on the right side significantly, therefore the horizontal force on the pile at the end of each cycle is oriented to the left. However, for the symmetric cyclic loading, the pile compressed the soils on both sides to the same strain level. At the end of a full cycle, the residual horizontal force is oriented to the right. In the presented study monopile was driven to move by a predefined constant velocity. It is more realistic to simulate the free motions of monopile under the action of resultant external force/moment, including the interaction force/pressure between the monopile and the soil.

14.3.2 Monopile Analysis Using FEM Using ANSYS Software

This section of the chapter presents the analysis of a monopile embedded 30 m in a cylindrical soil layer (90 m wide, 45 m deep), see Fig. 14.16. The steel pile is open ended with a wall thickness of 90 mm and a diameter of 7.5 m. It is worth noting that monopile diameters of up to 7 m are designed to maintain serviceability of the wind energy converters over several years as a result of harsh environmental conditions offshore. The monopile extended 30 m above the sea bed to the point where it was loaded and driven 30 m into the sea bed where the water depth was 10 m. The software ANSYS 17.0 was used to create the 3D model of wind turbine foundation. Static Structural Analysis was used to analyze static conditions, while the Transient Structural was used for the cyclic load.

The soil–pile model took advantage of the symmetry in geometry and load, hence, only one half of the soil-pile model was modeled, as shown in Fig. 14.17 which means that the transversal displacements at the symmetry plane was set to zero. A fine mesh size was selected for the system as shown in Fig. 14.17 resulting in 50 622 nodes and 36 816 elements. Previous studies demonstrated that the minimum number of elements required for an accurate enough model of monopile wind turbine foundations may be in the range of 30 000–40 000. The interaction between pile and structure was modeled as a frictional interface with friction coefficient of 0.4.

The time step was taken as 0.5 second, which was derived by the integration time step (ITS) method [33]. The force was applied by an idealized system described by Bryne and Houlsby [34] with a generalized loading case for a 3.5 MW turbine in which a 4 MN horizontal load and a 6 MN vertical load applied 30 m above the soil surface was used. This can be represented as:

$F\left(t\right)=F_{\max }\mathit{sin}\left(2\pi \mathit{ft}\right)$ (14.3)

(14.3)

The one-way lateral load was applied with a sinusoidal character of 0.1 Hz (typical frequency ranges in these infrastructures are 0–1 Hz for environmental loads and 5–200 Hz for machine loads [34]). At fixed end time, t of 1000 seconds was used, this gave 100 cycles (frequency=0.1 Hz), which was used in analyzing the results. The input parameters for the pile were calibrated for offshore installation in United Kingdom using typical test results while those obtained after calibration for the soil are shown in Table 14.2. The steel monopile was represented as a linear-elastic, isotropic, homogeneous material. For the soil properties, the Kinematic Hardening law, have been employed.

Table 14.2

Soil Model Material Parameters for Medium Dense Sand (after [19])

Parameter	Symbol	Value
Oedometric stiffness parameter	$\kappa$	600
Parameter for the elastic modulus (eq. 14.4)	$\lambda$	0.55
Poisson’s ratio	Υ	0.25
Unit buoyant weight/KN m−3	γ′	15.5
Internal friction angle/deg	Ø′	35
Dilation angle/deg	ψ	5
Cohesion/KN m−2	c	0.1
Yield surface parameter	α	0.127
Yield surface parameter/kPa	k	1.8
Plastic potential	β	4.05×10−2
Hardening law/MPa	C	22
	D	1200

The elastic modulus of the soil was derived from Janbu (1963) as presented in Eq. (14.2):

$E=60\mathit{MPa}\times \left(\frac{5}{6}\right)\times \left(\sqrt{2}\right)\times {\left(\frac{{\sigma }_y}{{\sigma }_{\mathit{at}}}\right)}^{\lambda }$ (14.4)

(14.4)

where ${\sigma }_{\mathit{y}}$ denotes the soil vertical effective stress, and ${\sigma }_{\mathit{at}}$ is the atmospheric pressure.

The plot of the applied horizontal load versus the pile-head displacement for dynamic loading, p–y (Fig. 14.18) shows a progressive inclination of the hysteretic loops, for every subsequent increase in the number of cycles, resulting in an increasing pile-head displacement. Similar trend can be observed in the result reported in Ref. [13], although that analysis was carried out for a clayey soil under the influence of a two-way lateral loading, which resulted in equal but opposite pile displacements in both directions. This result trend can be referred to as accommodation cyclic behavior (previously presented in this chapter) and is mostly found in laboratory experiments [25].

A closer observation of pile-head displacement (Fig. 14.19) shows an increase in the displacement with time. This can be observed in drained cyclic tests with an increase in density (densification) [25]. However, instabilities caused by alternating degradation and compaction are observed at the earlier cycles, causing negative displacement accumulation at the initial stages of loading. The pile-head displacement at the end of the first cycle equals 1.36 cm and 2.06 cm at the end of 100 cycles, which is the maximum displacement of the sea bed in front of the pile. This gives a total increased displacement of 0.7 cm. The accumulation rate from the 1st to the 100th cycle equals 1.51 cm. The displacement accumulation per cycle reduces drastically for the first 10 cycles, and then tends toward zero as the number of cycles increase, because at deflections larger than that of static loading, the value of soil resistance, p, decreases sharply due to cyclic loading, while afterwards it remains pretty much constant [18]. This is as a result of the kinematic hardening parameter, as also reported in Ref. [25], which stated that the cycles become reversible until a complete stabilization is reached for very high numbers of cycles. Similar results are observed along radial stress paths [35,36].

A comparison of the load-displacement response at the pile head between results from finite element modeling (static and dynamic) and API p–y curve method is provided in Fig. 14.20. The dynamic loading is represented by the curve for 1st, 50th, and 100th cycle. From Fig. 14.20, it is observed that at very low lateral-horizontal forces, the API method is in agreement with numerical simulations for static loading and 1st cycle dynamic loading. This supports the statement that the API method is a pseudo static approach [18]. However, at low lateral-horizontal forces and high number of cycles, the API method underestimates the pile displacement. This can be observed by the difference in pile displacements between the API method and numerical simulations; calling Δ to the displacement obtained when the applied load is 4 MN, we can see that Δ equals 0.3 cm for the 1st cycle, 0.8 cm for the 50th cycle, and 1 cm for the 100th cycle. This underestimation of the pile displacement by the API method at low lateral displacement was also observed by Hearn [37]. This also represents the conclusion presented by Yang et al. [38] who stated that “the cyclic bearing capacity may be significantly lower than the static bearing capacity.” This is true as the displacement increases with an increase in the number of cycles. Thus the displacement as a result of several cycles goes farther away from the displacement from static load/one cycle, since there is an accumulation of volumetric strains in the soil surrounding the pile. The plot also shows that the curve for the static loading and the 1st cycle dynamic loading are similar but the difference in pile displacement at the maximum applied lateral load of 4 MN is 0.3 cm. This gives a percentage difference of 30%. This may be because simulations for transient analysis are based on division of time or steps: therefore, there is a progressive accumulation of residual errors with an increase in the number of time steps [39].