The conversion of energy supply to the extensive utilization of renewable energies can only be realized by further expansion of offshore wind energy. In this regard, it is crucial to minimize the costs for offshore wind energy exploitation. Significant cost savings are possible by optimizing the foundation elements of offshore wind energy converters.

In the last years, the monopile foundation, consisting of a circular steel pipe of very large diameter (Fig. 1), was proven to be an economic and robust foundation solution for water depths of up to around 40 m. For the time being, projects are being realized with monopiles of up to 10 m outer diameter. However, there are still basic uncertainties in the design of such monopiles. In practice, usually the p–y method is applied in the calculation of the monopile behaviour. This is a special subgrade reaction method, which utilizes nonlinear and depth-dependent spring characteristics, the p–y curves. Here p is the resultant soil resistance (horizontal bedding stress times pile diameter) at a certain depth, and y is the corresponding horizontal pile displacement. In offshore guidelines, for example, API (2014) approaches to derive p–y curves dependent on the parameters of the present soil are given. For piles in sand, the only parameters required are the buoyant unit weight γ′ and the angle of internal friction ϕ′. However, several investigations in recent years have shown that these approaches, which have been calibrated by field test results of slender, flexible piles, cannot be used without modification for the large-diameter and almost rigid monopiles. This applies both to the p–y approaches for static (monotonic) and cyclic loads.

In the classic approach for piles in sand according to API (2014), a tangent hyperbolic function is used to describe the dependence of bedding resistance p on horizontal displacement y at a certain depth below sea bottom z: 1p=A⋅pu⋅tanh⁡k⋅zA⋅pu⋅y. Here pu is the theoretical maximum value of bedding resistance, which is basically the passive earth pressure times pile diameter D. This value is calculated from Eq. (2) as a minimum of two values for shallow and deep failure modes pus and pud, which depend on buoyant unit weight γ′ and three coefficients C1, C2 and C3 correlated with the angle of internal friction ϕ' (Fig. 2 left). 2pu=min⁡pus=C1⋅z+C2⋅D⋅γ′⋅zpud=C3⋅D⋅γ′⋅z

The k value in Eq. (1) is a stiffness parameter and is also given in API (2014) dependent on the angle of internal friction (Fig. 2 right).

For the factor A in Eq. (1), the API approach distinguishes static and cyclic loading conditions. For static loading, Astat=3.0-0.8⋅z/D≥0.9 applies, whereas for cycling loading the constant factor (independent of depth) Acyc=0.9 shall be used.

Several investigations have shown that for large-diameter piles the pile–soil stiffness obtained with the static p–y method is overestimated under extreme loads and underestimated under smaller operational loads (cf. e.g. Thieken et al., 2015a). Therefore, new static p–y approaches for monopiles in sand have been proposed in recent years. Sørensen (2012) replaced the k value of the API guidelines by a k value depending on depth z, the soil's oedometric stiffness Es and the pile diameter D, which for large-diameter piles leads to greater displacements under service loads than the API approach. Kallehave et al. (2012) also suggested a modified initial stiffness formulation, but their target was to avoid an underestimation of stiffness under small operational loads, which is usually applied for the determination of the natural frequency of the whole structure. Here, the parameter k is formulated dependent on depth z and pile diameter D. This modified p–y formulation results in a considerably “stiffer” behaviour than the API formulation. Thieken et al. (2015b) proposed a more sophisticated p–y curve approach, in which also the soil's greater stiffness for small strains is considered and which accounts for the effect of the pile deformation on the p–y curves by an iterative procedure. They showed that this approach gives reasonable results both for small operational and large service loads. Recently, Byrne et al. (2017) also proposed a new p–y approach especially developed for monopiles in sand soil (see also Byrne et al., 2015; Burd et al., 2020). Herein, besides p–y springs, also rotational springs and a pile tip spring are introduced to the beam-spring model. However, the spring characteristics are to be determined by calibration with a numerical model, which makes the approach not straightaway applicable to a certain system.

Cyclic loading of a pile can lead to both an increase of pile deformation with the number of load cycles and a reduction of the pile capacity, whereby in sand soils the effect on deformation usually dominates the design. Cyclic loading effects are considered in design by applying cyclic p–y curves in the calculation. As shown above, in the approach of API (2014) cyclic loading is considered by a reduction of the parameter A and hence the bedding resistance p down to a depth of z= 2.625 D. However, since the parameter A also affects the argument of the tanh function, also displacements are affected, which means that the API consideration for cyclic loads is a mixture of p- and y-multiplier methods. The approach is based on field tests, in which for the determination of cyclic loading effects only limited load cycle numbers (at maximum around 200) were realized (Cox et al., 1974). It is generally assumed that the approach represents the pile behaviour due to around 100 load cycles.

Apart from the fact that the cyclic API approach is not validated for large-diameter monopiles, the consideration of just one cyclic p–y curve valid for 100 load cycles is not sufficient for the design of piles for offshore wind foundations. These foundations are subject to intense cyclic loading induced by wind and wave loading. The design checks for cyclically accumulated deformations and in particular rotations of the foundation structure is quite important for wind energy structures. In most wind farm projects, a maximum permanent rotation of the tower of 0.5∘ is required. Thus, an accurate prediction of the accumulated pile rotation of the monopile to be expected over the whole lifetime of the structure is necessary. This means that the deformations must be calculated under consideration of the actual number of load cycles.

Dührkop (2009) conducted model tests with almost rigid monopiles in sand under cyclic loading. Based on the results, he proposed a modification of Eq. (1) for the case of cyclic loading as follows: 3p=Ãcyc⋅pu⋅tanh⁡k⋅z0.9⋅pu⋅y, with Ãcyc=rA⋅3-1.143z/D+0.343z/D.

The factor rA is dependent on the number of load cycles. With rA= 1, the API approach for monotonic loading and with rA= 0.3 for cyclic loading is obtained. Thus, rA= 0.3 can be used for a load cycle number of N= 100. For greater load cycle numbers, rA decreases. For N > 105, rA shall be set to zero (Dührkop, 2009). With this approach, the monopile deflection can be calculated dependent on the actual number of load cycles. However, the given Acyc function is valid only for the model test boundary conditions realized by Dührkop. For a certain system with different boundary conditions, model tests or numerical investigations are necessary.

Besides from local approaches modifying the p–y curves, also global approaches predicting the increase of pile head deflection or rotation can be applied. In general, the increase of head deflection due to one-way loading with full unloading can be described by the following equation: 4yN=y1⋅fN(N).

Here yN and y1 are the horizontal pile head deflections after N load cycles and after 1 load cycle (monotonic loading), respectively. fN(N) is a function which describes the increase of deflections with the number of load cycles. As long as the cyclic load amplitude is well below the ultimate pile capacity, sedation behaviour can be expected, which means that the deflection rate decreases with increasing number of load cycles. The most common functions of displacement of structures under cyclic loading that are found in the literature are of the exponential type such as Eq. (5) (e.g. Little and Briaud, 1988) and of logarithmic type such as Eq. (6) (e.g. Hettler, 1981): 5fN=Nm6fN=1+t⋅ln⁡N. Here m and t are empirical accumulation parameters. Assuming that these parameters are constants, Eqs. (5) and (6) imply that the function of load cycle number is independent of the load amplitude.

A few investigations regarding the above accumulation parameters exist. In addition to the above-mentioned works, Lin and Liao (1999) and Long and Vanneste (1994) should be mentioned. However, it is not clear how pile geometry (in particular, pile rigidity) and soil conditions affect the accumulation parameters. Based on model tests, LeBlanc et al. (2010) showed that the rate of deformation accumulation also depends on the load level, i.e. the ratio of maximum cyclic load to the ultimate pile capacity. In contrast, model test results of Peralta and Achmus (2010) did not show a significant effect of the load level.

For practical design, a simple-to-use approach for the derivation of cyclic p–y curves would be highly desirable, which just modifies a chosen static p–y curve by p or y multipliers depending on the number of load cycles and other relevant parameters. Such an overlay model describes just the change of static p–y curves with increasing load cycle numbers and could be applied to arbitrary static p–y curve approaches.

A calculation approach termed “stiffness degradation method” (SDM) combining numerical simulations and cyclic triaxial tests has been developed at the Institute for Geotechnical Engineering of Leibniz University Hanover (Achmus et al., 2009). This method allows the calculation of pile deflection lines dependent on the actual number of cycles of a given load and with that also the derivation of p–y curves for a given number of load cycles. This method is applied in the paper at hand for the development of a cyclic p–y overlay model. Therefore, the SDM is briefly described in the following section.

The SDM is described in detail in Kuo (2008) and in other publications, e.g. Achmus et al. (2009) and Kuo et al. (2012), and thus here the method shall be outlined just briefly. A 3-dimensional finite element model is used to calculate the pile deformation behaviour. The soil behaviour under static load is modelled by an elastoplastic material law with the Mohr–Coulomb failure criterion and stress-dependent stiffness. The stiffness modulus, i.e. the oedometric stiffness determined under constrained lateral strain, is defined as follows: 7Es=κσatσmσatλ. Herein σat= 100 kN m-2 is a reference (atmospheric) stress, σm is the current mean principal stress in the considered soil element, and κ and λ are soil stiffness parameters.

For describing the increase of the plastic strain of soil with the number of load cycles in element tests (cyclic triaxial tests), an approach by Huurman (1996) is used. The decrease of the (secant) stiffness modulus of the soil with the number of cycles can be approximated by the following equation: 8Es,NEs,1=N-b1Xb2. Herein N is the number of cycles, b1 and b2 are regression parameters and X=σ1,cyc/σ1,f is the cyclic stress ratio, whereby σ1,cyc is the maximum of the principal stress in a cycle and σ1,f is the main principal stress at failure (in a static test). For each element of the finite element model, from the consideration of the initial stress state and the stress state after applying the horizontal load, a cyclic stress ratio quantifying the intensity of cyclic loading is derived. With the reduced stiffness values obtained from Eq. (8), the system's behaviour under the lateral load is calculated again. The result represents the increased pile deformation after the considered number of cycles. An illustration of the procedure is given in Fig. 3.

For cyclic one-way loading and drained conditions, the SDM allows the evaluation of the pile deformation behaviour under consideration of the site-specific soil conditions as well as the loading conditions and the number of cycles. The application requires the definition of six material parameters accounting for the soil behaviour under static loading, for which comprehensive experiences exist, and two parameters b1 and b2 describing the stiffness degradation under cyclic loading.

The method has been validated by back-calculation of various series of model tests in medium-dense as well as in dense sand (see Albiker, 2016; Albiker and Achmus, 2018). From these back-calculations, it could be concluded that for comparable relative densities of the soil also consistent sets of values for b1 and b2 have to be chosen. Value sets for medium-dense and dense sand were determined by comparison of calculations and experimental measurements, and the ranges of values were also verified by regression values derived from cyclic triaxial tests (Albiker, 2016).

The derived cyclic overlay model is an easy-to-use approach to predict the translocation of p–y curves due to lateral cyclic load for any static p–y approach. In the following, the overlay model is applied to the static p–y approach stated by API (2014) (see Sect. 2, Eqs. 1 and 2). The resulting deflections, bending moments and p–y curves can be compared to the results of the cyclic API approach, which is usually assumed to represent approximately 100 load cycles.

A monopile with a diameter of D = 5 m and an embedded length of L= 25 m (L/D= 5) in very dense sand (ϕ′= 40∘, γ′= 10.31 kN m-3; cf. Table 1) is considered. A constant wall thickness of t= 7 cm and a load eccentricity e/L= 0.6 are assumed.

Figure 14 shows deflection lines and bending moments for a horizontal load of H= 10 MN, calculated with the static API approach, the cyclic API approach and the cyclic overlay model for load cycle number of N= 100, 1000 and 10 000. The cyclic API approach predicts an increase of the pile head deflection (Fig. 14 left) with respect to the static approach of 30.5 %, which is in rather good agreement with the result of the cyclic overlay model for N= 100 (22.1 %). However, the overlay model also predicts pile head deflections for other load cycle numbers. For N= 1000 and 10 000, the head displacement increases by 35.6 % and 51.1 %, respectively.

Figure 14 right shows that the cyclic API approach overestimates the maximum bending moment of the monopile. Compared to the cyclic overlay model with N= 100, a 5.6 % greater maximum bending moment is gained. Also the bending moments of the cyclic overlay model with N= 1000 and 10 000 are considerably smaller than for the cyclic API approach.

Figure 15 shows the reason for that. Here, the p–y curves at four distinct depths along the monopile are depicted. The cyclic API approach results in a severe degradation of bedding resistance and stiffness in shallow depth. In contrast, in greater depth (in this example below z/L= 0.525) no modification of the p–y curves is to be considered. For a stiff or almost rigid pile, this is of course unrealistic. It should be noted that the cyclic API approach was calibrated on lateral load tests on small and flexible piles. According to the cyclic overlay model, the p–y curves are subject to softening along the whole pile length. Therefore, at shallow depths much stiffer p–y curves apply, which results in considerably smaller maximum bending moments.

By comparison of numerical simulations of monopile behaviour once under monotonic and once under cyclic loading with a defined number of load cycles (utilizing the SDM), a cyclic p–y overlay model for monopiles in sand soils could be developed. Applying the derived equations for y multipliers, arbitrary p–y curves for monopile behaviour under monotonic loading can be adapted to a given number of load cycles. The y multipliers are formulated as a product of a term valid for constant horizontal deflection and a correction term accounting for the actual deflection line and loading conditions.

It was found that the first term could be formulated dependent only on the load cycle number and the angle of internal friction of the sand. In contrast, the second term was found to depend on load cycle number, monopile geometry (length-to-diameter ratio), load eccentricity and relative depth.

The new cyclic overlay model gives plausible results and can be applied to any monotonic p–y approach for monopiles in sand. However, an experimental validation of the model is still missing. It is planned to do this with cyclic large-scale pile load tests in an ongoing research project.