In this section, we will discuss a simple analysis technique that can predict the dynamic behavior and evaluate the safety of the offshore wind turbine system loaded with the horizontal load discussed above. Poulos [
14] presented a method for calculating the displacement and rotation angle of a pile installed in the ground using a dimensionless influence factor. He assumed the ground to be a homogeneous elastic continuum model; the length-to-diameter ratio (L/D) and the pile flexibility factor, which is defined as a dimensionless measure of the flexibility of the pile relative to soil, are the most influential factors on the behavior of the pile. In addition, Randolph [
15] expanded this model and suggested an algebraic expression that can calculate the displacement and rotation angle of a pile under a horizontal load (F) and moment (M) using fundamental soil properties, as shown in
Figure 6.
3.1. Closed Form Solution for Laterally Loaded Piles
Randolph proposed that the behavior of the pile can be characterized into three cases according to the modulus ratio and the slenderness ratio , which are two dimensionless parameters (hereafter, referred to as influence factors).
: Flexible Pile Region
: Rigid Pile Region
: Intermediate Region
is the effective Young’s modulus of the pile and is defined by Equation (1) when the pile in the ground is actual bending rigidity
.
In addition,
is an equivalent shear modulus that shows the effect of variations in Poisson’s ratio (
) on the deformation of the laterally loaded pile and is described by Equation (2).
In the flexible range, the displacement response of the pile is determined by the modulus ratio
among the aforementioned dimensionless parameters when the applied load is the same, and the horizontal displacement and rotation angle are approximated by Equations (3) and (4). In this case, the applicable range of the equation is
≤
≤
and L/D ≥10.
In contrast, in the rigid range, the displacement response of the pile is determined by the slenderness ratio,
, and John P. Carter proposed the following simple formula through further research [
16].
Its accuracy has been verified only for the following ranges of parameters: 1 ≤ ≤ 10 and ≥ 1. In addition, the displacements in the intermediate range case should be taken as 1.25 times the maximum of either (a) the predicted displacement of a rigid pile with the same slenderness ratio, or (b) the predicted displacement of a flexible pile with the same modulus ratio.
In summary, if the displacement response to the load of the pile is obtained in the form of a closed solution and used in the design of the foundation structure, a quick design decision can be made without time-consuming numerical analysis. In this study, the displacement response of the turbine system is obtained in the form of a closed solution using the basic properties of the suction pile and the ground soil. The effectiveness is verified through a comparison with the Finite Element Method (FEM) analysis result and is then used for a safety review using the quick basic design.
3.2. Influence Factors for Single Suction Pile
First, if the load displacement relations presented in Equations (3) to (6) are applied to a single suction pile, as shown in
Figure 7, the horizontal displacement and rotation angle at the pile head can be obtained as follows using the influence factor.
In the above equations,
,
,
, and
represent dimensionless coefficients of influence, and they are a function of the aspect ratio and stiffness ratio. In each coefficient, the first subscript indicates whether it is related to a horizontal displacement (
u) or an angular displacement (θ), and the second subscript is related to a horizontal load (
F) or a moment load (
M).
represents the vertical force generated when a horizontal load is applied to the multisuction pile, which can also be expressed in the form of an influence factor, as shown in Equation (9).
Next, to obtain the displacement response for a single suction pile in the form of a closed solution, the influence factors for the three pile regions presented in
Section 3.1 will be calculated. First, if the displacement response function in the rigid pile range is defined as
and the function in the flexible range as
, the influence factors of a single suction pile in each area can be obtained as shown in
Table 1 through the results of previous studies.
In this study, an influence factor function that can be applied in the intermediate region is proposed, which is represented by
and
=
at the left boundary (rigid range
) when the approximate function of the displacement response in the intermediate range is
,
=
. This is an exponential function that satisfies the condition. Additionally, at the right boundary, the slope of the function
is zero because it has a constant value regardless of the aspect ratio when the modulus ratio is constant, but the slope between
and
cannot be zero unless exponential function
has an extreme value. However, because the
function is sufficiently attenuated with the
for this range, it can be expressed as a continuous function if appropriate attenuation conditions are applied. Therefore, the analysis was performed by defining the exponential function
, with an attenuation condition of less than 0.01 at the right boundary (flexible range
), as shown in Equation (10).
By synthesizing the above results, the exponential function
of the intermediate range is proposed as Equation (11).
α then requires the following conditions to satisfy Equation (10).
Based on the results obtained above, the coefficient of influence of the rotation angle caused by the moment load over the entire pile range is shown in
Figure 8.
In addition, a simple finite element analysis was performed to understand the characteristics of the influence factor. For the physical properties of the ground soil, a modulus of elasticity of 22.1 MPa and a Poisson’s ratio of 0.25 were applied, in consideration of the soft marine soil. The influence factor was derived using the functions
–
based on the stiffness ratios
of 100, 200, 400, 800, 1600, 3200, and 6400 and an aspect ratio (L/D) of 0.5–8, as shown in
Figure 9.
When observing the characteristics of , , , , the rotation factor () of the moment load has the largest effect on the system (2.3 and 3 times that of and , respectively); this is because the offshore wind turbine has a large horizontal load, which is relatively situated at a higher position than that of the top of the pile, as described above. In addition, for the same pile, the larger the stiffness ratio and the larger the aspect ratio, the smaller the influence factor. This implies that the larger the stiffness of the ground soil, the longer the length of the pile, and the smaller the displacement at the pile head. Specifically, when the stiffness ratio is constant, the influence factor no longer decreases when the slenderness exceeds a certain value, which shows that no matter how deep the pile is driven above a certain depth, the displacement reduction is not affected. Based on the validity of the confirmed influence factor, the analysis was performed by extending the influence factor to the lower support structure.
3.3. Influence Factors for Multisuction Piles
In this section, based on Equations (7) and (8) for the single pile, we derive the load and displacement relations for the multisuction pile. Variables related to the multisuction pile are indicated by subscript
MP, and the analysis section and elements have been presented as shown in
Figure 10. In addition, we assume that each member connecting a single pile is robust and the relative displacement between the piles at the top of the pile is negligibly small. In this assumption, the angular displacement (
) and horizontal displacement (
) of the multipile have the same value at all points of the multipile. The distance from the center point of the multisuction pile to the center of a single suction pile is defined as
.
The lateral force (
) acting on a multipile is simply multiplied by the force acting on a single pile
and the number of single piles
, as shown in Equation (13).
On the other hand, the moment acting on a multipile (
) can be expressed as a combination of a moment acting on a single file (
) and a normal force (
), which is shown in Equation (14).
The vertical force acting on the
i-th suction pile is defined as
and the length of the moment arm due to the vertical force is expressed as
. If
, i.e., when
, the normal force is
.
When substituting Equation (15) into Equation (14) to obtain Equation (16), the moment acting on the multipile by the relational expression
is as shown in Equation (17). It is expressed simply as below.
The relationship between the vertical force applied to a single pile
and the angular displacement and horizontal displacement of a multipile is obtained using Equation (18) in Equation (9) in
Section 3.2.
According to the previous assumption, the displacements of the suction pile (
) are the same as the displacements of the multipile (
and
because the multipile at the top of the pile moves like a rigid body. Therefore, to obtain the angular displacement
and the horizontal displacement
of the multipile, the following equation is derived by substituting the multipile relational Equations (17) and (18) into the influence factor Equations (7) and (8) of a single file.
Substituting Equation (18) to remove the term
from the above equation, the angular displacement and horizontal displacement equation for the multipile can be obtained as follows:
The influence factor of the multipile
is derived from Equations (23) and (24) as follows:
3.4. Tripod and Tower Influence Factors
As shown in the simplified schematic diagram in
Figure 5, we next derive the relationship between force and displacement for the tripod on the top of the suction pile and the tower member. To easily derive the validity and physical concept of the simple analysis, the solution is determined using the static dynamic relationship of force–displacement in the Euler beam to obtain the effective elasticity and effective mass. For the analysis, as shown in
Figure 11, the lower, inner, and upper parts of the tripod member are marked with subscripts 0,
s, and
a, respectively. The relationship between the force applied to the member in the Euler beam
, moment
, angular displacement
, and displacement
can be expressed as Equation (25).
The angular and horizontal displacements
and
at arbitrary positions can be subsequently expressed as follows:
, expressed as an f-factor, which is an influencing coefficient that represents angular displacement θ at an arbitrary point s when the moment acts at the upper end, and other coefficients follow the definition of the aforementioned influence factor.
In the above Equations (26) and (27), the displacement of the tripod is determined by the physical properties of the member as well as by the boundary condition at the lower end (point 0), i.e., the displacement at the upper end of the (single or multi) suction pile. Therefore, by substituting Equations (21) and (22) regarding the influence factor of the suction pile into the displacement at point 0 of Equations (26) and (27), the load–displacement relationship for the entire system can be obtained as follows:
The
c-influence factor of the entire tripod system can be obtained as shown in Equations (30)–(33) below.
The effect of the influence factor of the tripod and tower member can be directly observed in Equations (28) and (29). In this instance, the influence factor
f is a value corresponding to the displacement or angular displacement when the lower end is completely fixed, and if the shape of the beam is expressed as a simple function, a closed form solution may exist.
Table 2 summarizes the influence factors (
f-factors) for a simple beam in which
does not change and the beam changes as a quadratic function. As shown in
Figure 5, the vertical member connecting the tripod and the tower applies a condition with a constant cross-section, and the tripod and the tower perform analysis by assuming the change in cross-section as a quadratic function. The process of deriving a detailed equation for this is detailed in
Appendix A.
To demonstrate the characteristics of the
f-factor expressed in Equations (30)–(33),
Figure 12 shows the factor at
ŝ = 0, i.e., at the top of the member.
is a monopile structure in which the area moment of inertia at the top and bottom of the member is constant. In this case, the values on the left and right sides of
Table 2 are the same. At this time, the ratio of
is 6:3:2, and to compare the three values in the same graph, the values of
are shown. As depicted in
Figure 12, as
n increases, the stiffness of the system increases, and the
f-factor value decreases. For example, in the case of
n = 100, the value of
decreases to 15% of the value at
n = 1, and 2
also sharply decrease to 5% and 2.7%, respectively. In addition,
decreases at the fastest rate as
n increases, and in the case of
n ≫ 100, the values of
become small enough to be negligible compared to
, so the behavior of structure is under the effect of
.
In addition, from the
f-factor graph in
Figure 13, the influence factor increases as the location of the point where the displacement is obtained moves from the bottom to the top. This is because the turbine system has a large horizontal load acting on the top of the tower and dominates the overall behavior as described above. Notably,
do not match; unlike in
Figure 12 for the influence factor at
ŝ = 0, these variables are within the member as shown in
Figure 13, which demonstrates that the symmetry is not secured inside the member.
Figure 13 also shows that the graph of
is convex as the length changes, whereas
changes to a concave shape. Thus, a simple analysis method that can obtain the horizontal displacement and rotation angle among the main design variables in the form of a closed solution has been proposed, and the characteristics of each influence factor at different cases have been examined. In the next section, we will discuss how to determine the first natural frequency of the system, with one of the remaining major design variables.