**Research article**| 11 Jul 2022

# A model test study on the parameters affecting the cyclic lateral response of monopile foundations for offshore wind turbines embedded in non-cohesive soils

Dennis Frick and Martin Achmus

**Dennis Frick and Martin Achmus**Dennis Frick and Martin Achmus

- Institute for Geotechnical Engineering, Leibniz University Hannover, Hanover, 30167, Germany

- Institute for Geotechnical Engineering, Leibniz University Hannover, Hanover, 30167, Germany

**Correspondence**: Dennis Frick (frick@igth.uni-hannover.de)

**Correspondence**: Dennis Frick (frick@igth.uni-hannover.de)

Received: 01 Nov 2021 – Discussion started: 25 Nov 2021 – Revised: 27 Apr 2022 – Accepted: 17 Jun 2022 – Published: 11 Jul 2022

During their service life, monopiles supporting offshore wind turbines are subjected to a large number of lateral cyclic loads resulting from complex environmental conditions such as wind and waves varying in amplitude, direction, load eccentricity and frequency. The consequential accumulation of displacements and rotations of the foundation structure with cyclic loading is one key concern in the design of monopiles. Nevertheless, the relevant offshore guidelines do not provide suitable procedures for predicting such deformations. Although there are several methods for this purpose in the literature, some of them produce very different or even contradictory results, which prevents a consistent approach to dimensioning. This paper briefly summarizes the current standardization regarding design of monopiles for cyclic lateral loading and provides some examples of possible prediction models from the literature. To highlight the need for further research, the predictions according to different approaches are compared and evaluated by a calculation example and a parameter study. Further, the results of a small-scale 1 g model test campaign on the load-displacement behaviour of monopile foundations subjected to lateral cyclic loading and the influencing parameters are presented, evaluated and compared with the findings of other research groups. In this way the test results can help to support or improve model development and provide insight into key issues relevant to monopile design. The parameters that have been assessed include the cyclic load magnitude, cyclic load ratio, load eccentricity, soil relative density, the grain size distribution of the non-cohesive bedding material and the pile embedment length.

Offshore wind energy is one of the promising solutions for sustainable energy, but for the wind industry to be competitive, it is vital that costs are significantly reduced for future projects. This can be achieved, on the one hand, by introducing new technologies and, on the other hand, by improving existing technologies and design methods. One of the areas where costs can be reduced is the support structure, which accounts typically for about 16 % to 35 % of the total cost of an offshore wind turbine (OWT) and whose cost increase substantially with water depth (Bhattacharya et al., 2021). With regard to continuously increasing water depths of future wind farm sites and projects, an improved design of the foundation structure can therefore make a significant contribution to the competitiveness of offshore wind energy.

Up to now, the prevailing support structure for offshore wind energy
converters at low to medium water depths is the monopile foundation, a
single pile with large diameter (*D*) and a relatively small ratio of
embedment length (*L*) to the diameter ($L/D$) that transfers the predominantly
horizontal loads from the action of wind and waves into the seabed. Its
popularity can be explained by its suitability for mass fabrication,
robustness for most soil conditions, relatively simple design and
therefore cost efficiency. To extend the range of applications of the
monopile and make use of the related benefits, pile diameters have to be
extended (leading to decreasing $L/D$ ratios), and more accurate design
methods, specifically tailored to the offshore wind industry, have to be
developed.

A governing factor in the design of monopile foundations is compliance with serviceability limit state and associated strict tilting tolerances. This means that an accurate prediction of pile displacement and rotation accumulation resulting from cyclic-occurring horizontal loads plays a key role for the final dimensions of the foundation structure and therefore its costs. However, current offshore guidelines do not provide appropriate procedures for the prediction of pile displacement accumulations, which is especially true for monopiles, which due to their large dimensions and low $L/D$ ratios have a significantly different load-displacement behaviour than long and slender piles. For this reason, a variety of new empirical and numerical approaches for the estimation of cyclic deformation behaviour of monopiles have been proposed in the literature. Although these methods were usually developed specifically for monopile foundations, they sometimes provide very different and partly contradictory results with respect to the resulting deformations and the governing parameters.

The article at hand first summarizes the current standards and developments regarding the estimation of permanent deformations of offshore monopiles before selected prediction models are compared with each other on the basis of a calculation example and a parameter study. In order to gain further insight into the deformation behaviour of monopiles due to lateral cyclic loading, results of a comprehensive test campaign of small-scale 1 g model experiments are presented and discussed. The results are used to identify the governing parameters and to evaluate existing empirical approaches. Based on the results, qualitative conclusions can be drawn. The findings can thus contribute to a better understanding of the complex processes associated with the cyclic load-bearing behaviour of piles and to the development of improved calculation approaches.

The design of offshore structures, such as monopile foundations for offshore
wind turbines, is usually based on the latest version of the offshore
guidelines, e.g. DNV GL (2018) or API (2014). These also regulate the
required design checks, which include the proofs for ultimate limit states
(ULSs), serviceability limit states (SLSs), fatigue limit states (FLSs) and
accidental limit states (ALSs). In the case of large-diameter monopiles for OWTs,
the SLS proof for long-term lateral cyclic loading resulting from millions
of wind and wave loads is often decisive for the dimensioning of the
foundation. For this proof, limit values for permanent pile head
displacement or rotation at seabed level are usually specified by turbine
manufacturers or structural designers, whose compliance is to ensure the
safe and smooth operation of the turbine until the end of its planned
service life. As an example, the DNV GL (2018) guideline for this proof
provides the usual limit values of 0.5^{∘} pile head rotation,
including an installation tolerance of 0.25^{∘}, which means that
the additional accumulated rotation due to lateral cyclic loading must be
limited to less than 0.25^{∘} in this case. Both the DNV GL (2018)
and the API (2014) guideline mention the so-called *p*–*y* method as a
possibility to model the pile–soil interaction in horizontal direction and
to predict the load-deformation behaviour of a pile foundation due to
lateral loads. This method models the pile–soil system by discretising the
pile into a number of elastic beam elements, interconnected by nodal points,
and with uncoupled soil support springs laterally attached to these nodal
points. Loads such as horizontal forces or moments are applied directly to
the pile head. The characteristics of the springs (*p*–*y* curves) are hereby
non-linear and describe the relationship between soil's bedding resistance
(*p*) and lateral pile displacement (*y*). Therefore, the reliability of the
calculated prediction of pile deformations by this method strongly depends
on the chosen formulation of the *p*–*y* curves. While API (2014) refers to
API RP 2GEO (2014) for an approach to the construction of *p*–*y* curves, the DNV GL (2018) does not recommend a specific approach but points out that any *p*–*y*
method to be used for piles larger than 1.0 m in diameter should be
validated by means of other methods (e.g. finite element calculations). This
remark results from the fact that most *p*–*y* approaches (including the API RP 2GEO, 2014, method) are largely based on some well-documented field tests on
small-diameter, long and therefore slender piles reported by Reese et al. (1974), O'Neill and Murchison (1984), and others. Since the load transfer
behaviour of such slender and thus more flexible piles with large $L/D$ ratios
differs significantly from that of rigid piles (e.g. typical large-diameter
monopiles), these methods should not be used for this field of application
without further validation. In addition to this general issue regarding the
*p*–*y* method or most of the approaches for the determination of *p*–*y* curves,
the DNV GL (2018) guideline states that the SLS proof of a monopile
requires that it represents the behaviour of the soil under cyclic loading
in such a way that cumulative deformations in the soil are appropriately
calculated as a function of the number of load cycles at each load amplitude
in the applied history of SLS loads. However, no specific procedure for this
purpose is mentioned in DNV GL (2018) either. In contrast, the *p*–*y* method
according to API RP 2GEO (2014) allows for the consideration of cyclic loads by a
simple adjustment of the proposed *p*–*y* curves by an empirical calibration
factor. When being applied, this factor leads to an overall softer
foundation response and a reduced pile capacity without considering
the number of applied load cycles, load magnitude or other parameters of the
load or the pile–soil system. As the calibration factor according to the API RP 2GEO (2014) approach was derived from field tests with less than 100 load
cycles in most cases, this method is widely deemed to be unsuitable for SLS
verifications of monopiles for OWTs, especially when large-diameter piles
are used. In this context, the API (2014) states that the methods referred
to are only intended as recommendation. Therefore, if further detailed
information from advanced soil testing and pile testing in the centrifuge, at
model scale or even at full scale is available, then also other methods
may be justified.

In summary, as can be seen from the above, both offshore guidelines, while
regulating the principles of design of offshore foundations, do not provide
a generally applicable method for pile deformation assessment due to lateral
cyclic loading for SLS verification of large-diameter monopiles. Instead, it
is up to the designer to choose a suitable method for this purpose.
Accordingly, there are several publications on the subject of cyclic
laterally loaded piles in the literature and on how deformations due to such
loads can be predicted. Most of the methods proposed are based either on a
limited number of small-scale model tests at 1 g or tests in the centrifuge, with a
few approaches also based on field experiments. Mostly, these approaches
were derived from best-fit curves, for which it has been found that for a given
load level and type of loading, the ratio of the pile head displacement
accumulated after *N* load cycles (*y*_{N}) and the maximum displacement
reached within the first cycle (*y*_{1}) can most generally be described as
a function of the number of load cycles (*N*) by either a power or a
logarithmic function as shown in Eqs. (1) and (2).

Here, *α* and *t* are referred to as accumulation parameters and may be
defined differently depending on the chosen approach taken from the literature.
It should be noted that according to some methods, also pile head rotations
(*θ*_{N}, *θ*_{1}) are used as deformation variables in
Eqs. (1) or (2) instead of the pile head displacements (*y*_{N}, *y*_{1}).
The maximum deformation reached within the first load cycle (*y*_{1} or
*θ*_{1}) is usually determined from monotonic load-displacement or load-rotation curves, which in turn can be calculated using a suitable method
as, for example, finite element calculations, an appropriate *p*–*y* method (e.g.
Kallehave et al., 2012; Sørensen, 2012; Thieken et al., 2015) or the
PISA method (see, for example, Byrne et al., 2017, 2019; amongst
other). Although both equations (Eqs. 1 and 2) are often considered
to describe the variation in accumulated pile deformations with the number of
load cycles, most studies indicate that for a pile–soil system that behaves
almost rigid, the power function according to Eq. (1) gives more accurate
results, whereas the logarithmic function better fits a flexible pile
behaviour when subjected to cyclic loading (see, for example, Peralta, 2010).

While early publications on the topic of cyclic laterally loaded pile
foundations focused primarily on the behaviour of long and slender piles
with a limited number of mostly one-way load cycles (see, for example, Hettler,
1981; Little and Briaud, 1988; Long and Vanneste, 1994; Lin and Liao,
1999), the interest of the last decade has been mainly in predicting the
behaviour of piles with dimensions and loading conditions typical for
offshore monopile foundations (e.g. rigid pile behaviour, higher number of
load cycles, one- and two-way loading). In order to clearly describe
constant cyclic loading conditions, the two load parameters *ζ*_{b}
and *ζ*_{c} defined by Eqs. (3) and (4) are well established.

In these equations, the reference horizontal force or moment (*H*_{ref} or
*M*_{ref}) is that corresponding to monotonic loading of a pile soil
system at failure or at a reference displacement or rotation (*y*_{ref} or
*θ*_{ref}) at the soil surface. As a geotechnical failure of a rigid,
laterally loaded pile in sand due to monotonic loading can require large
pile deformations, it has become common practice to define *H*_{ref} or
*M*_{ref} not at pile failure but at significantly lower reference values
for *y*_{ref} or *θ*_{ref}. Further, *H*_{min} and *H*_{max} are the
minimum and maximum horizontal forces being applied to the pile within a
load cycle, with associated moments *M*_{min} and *M*_{max} acting on the
pile head at ground level. Therefore, *ζ*_{b} can be interpreted as
the cyclic load magnitude, while *ζ*_{c} is the loading type, with
*H*_{max} and *H*_{min} taking positive and negative values for two-way
loading, respectively.

Since offshore loads are of course not constant cyclic loads, it is common
practice to divide the real and highly variable in situ load series into
load packages with constant mean load and amplitude using various methods
(e.g. rain flow counting). Subsequently, the individual load packages with
corresponding load cycle numbers can be converted into a single load package
with clearly defined load parameters *ζ*_{b} and *ζ*_{c}, as
well as an equivalent number of load cycles (*N*_{equ}), so that this
equivalent load package provides a comparable damage or load to the
structure as the original load series. As a result, even simple approaches,
such as those shown in Eqs. (1) and (2), can be used to predict the
deformation accumulation of a pile under variable load amplitudes. However,
since the determination of equivalent load packages and cycle numbers is a
broad topic, this paper is limited only to the subject of constant loads.
For procedures to determine equivalent load packages from random two-way
lateral loads, reference is made to LeBlanc et al. (2010b) and Jalbi et al. (2020), for example.

In order to investigate the load-bearing behaviour of large-diameter piles
in sand subjected to long-term lateral cyclic loading, Peralta (2010)
conducted a series of 34 scaled 1 g model tests (13 monotonic and 21 cyclic)
on model piles (*D*=60 mm) with $L/D$ ratios within the range of 3.33–8.33
and up to 10 000 load cycles. The tests involved cyclic one-way loading
only, with loads being applied with an eccentricity (*h*) of 240 mm (distance
between load application point and soil surface). Both rigid and flexible
pile–soil systems with different relative soil densities (*D*_{r}) and pile
bending stiffnesses (*E*_{p}*I*_{p}) were investigated. In addition, also
the influence of the cyclic load magnitude *ζ*_{b} was considered. As
a result, it has been found that the measured pile displacement
accumulations of the rigid pile–soil systems followed a power function as
shown in Eq. (1), while a logarithmic trend (Eq. 2) was observed for the
piles with a more flexible behaviour. For the accumulation parameters given
in Eqs. (1) and (2), Peralta (2010) suggests values of *α*_{P}=0.12 and *t*_{P}=0.21 for rigid and flexible pile–soil
systems, respectively, regardless of the soil relative density. An influence
of the load magnitude (*ζ*_{b}) on the accumulation parameters
(*α*_{P} and t_{P}) was also not observed; the load magnitude
(*ζ*_{b}) and soil relative density (*D*_{r}) correlated only with
the value of *y*_{1}.

LeBlanc et al. (2010a) also conducted a series of 21 small-scale model tests
(6 monotonic and 15 cyclic) at 1 g, in which the influence of not only the
load magnitude *ζ*_{b} but also the loading type *ζ*_{c} and
the soil relative density (*D*_{r}) was investigated in more detail. The
rigid model pile had a diameter (*D*) of 80 mm and a $L/D$ ratio of 4.5, which
is typical for large-diameter monopiles. Lateral loads have been applied
with up to 65 370 load cycles (at least 7400) and an eccentricity (*h*) of
430 mm resulting in an $h/D$ ratio of 5.38. In order to take scaling effects
into account and to ensure comparability of the dilatancy and shearing
behaviour of the soil (dry silica sand) between the model and true scale,
the model tests were carried out at relative densities (*D*_{r}) of only
0.04 (very loose) and 0.38 (medium dense). As the shear parameters of the
soil are stress-dependent (at least for very small vertical stresses),
LeBlanc et al. (2010a) provide a graphical relationship between vertical
effective stress with reference stress taken at a depth (*z*) of 0.8 *L*, soil
relative density (*D*_{r}) and peak friction angle (*φ*^{′}), which can
be used to convert the relative densities used in the model tests to a
true-scale monopile. Based on the results of the conducted cyclic tests
LeBlanc et al. (2010a) propose the power function approach given in Eq. (5)
to describe permanent increases in pile head rotation (Δ*θ*)
with load cycle number (*N*).

For the accumulation parameter (*α*_{θ, LB}) they recommend a
value of 0.31. The factors *T*_{b, LB} and *T*_{c, LB} were identified to be
dependent on load characteristics and soil relative density and have been
defined in terms of graphical functions (see LeBlanc et al., 2010a). While
*T*_{b, LB} increases linearly with load magnitude (*ζ*_{b}) and takes
larger values for a higher relative density (*D*_{r}), the
*T*_{c, LB} function indicating the influence of the loading type (*ζ*_{c}) on the pile head rotation accumulation is according to the results
of LeBlanc et al. (2010a) not affected by soil relative density (*D*_{r})
and shows a maximum of approximately 4 at a cyclic load ratio of ${\mathit{\zeta}}_{c}=-\mathrm{0.6}$ (asymmetric two-way loading).

Another approach was proposed by Klinkvort and Hededal (2013), who in their
centrifuge tests (5 monotonic and 12 cyclic) on almost rigid model piles
with diameters (*D*) of 28 and 40 mm, respectively, and a constant $L/D$ ratio
of 6 applied up to 10 000 load cycles (however, most of the tests involved
only 500 load cycles) with a normalized load eccentricity ($h/D$) of 15. The
soil relative density (*D*_{r}) in these tests ranged from 0.79 to 0.96, and
the applied cyclic loads also varied in both their load magnitude (*ζ*_{b}) and cyclic load ratio (*ζ*_{c}). The results regarding the
pile head displacement have been found to follow a power law according to
Eq. (1), but unlike the findings of Peralta (2010), Klinkvort and Hededal (2013) found an influence of the load magnitude (*ζ*_{b}) and cyclic
load ratio (*ζ*_{c}) on the accumulation parameter. An impact of the
soil relative density (*D*_{r}), however, could not be determined. As a
result, Klinkvort and Hededal (2013) defined the accumulation parameter
(*α*) from Eq. (1) as follows:

where *T*_{b,K&H} and *T*_{c, K&H} were defined by two functions
depending on *ζ*_{b} and *ζ*_{c}, respectively. While the
*T*_{b, K&H} function indicates a linear increase with cyclic load
magnitude (*ζ*_{b}), *T*_{c, K&H}(*ζ*_{c}) is given by a
third-order polynomial with a maximum value slightly larger than 1 for
${\mathit{\zeta}}_{c}=-\mathrm{0.01}$ and even negative values of up to −1.95 for
perfect two-way loading (${\mathit{\zeta}}_{c}=-\mathrm{1}$), which means that the
accumulation of displacement is reversed for this loading condition and the
pile moves back to its initial position.

Li et al. (2015) conducted one of the few field test studies, in which four
open-ended steel pipe piles with an outer diameter (*D*) of 340 mm, wall
thickness (*t*) of 14 mm and an embedment length (*L*) of 2200 mm
($L/D=\mathrm{6.47}$) were tested in an over-consolidated fine sand with a relative
density (*D*_{r}) close to 1. All loads have been applied with a normalized
eccentricity ($h/D$) of 1.18. Two pile tests were performed to derive
monotonic load-displacement curves and determine *H*_{ref} (see Eq. 3).
In the other two tests, cyclic one-way loads (*ζ*_{c}=0) were
applied in three load packages of different cyclic load magnitudes,
increasing from small to larger values of *ζ*_{b} and with different
numbers of load cycles ranging from *N*=40 to *N*=4007 for each load
package. After cycling, monotonic tests were performed in order to determine
the post-cyclic load-displacement response of the piles and to see the
effect of the cyclic loading history. The results of the cyclic tests have
been fitted by both power and logarithmic functions as given in Eqs. (1) and (2) and with respect to pile head displacement (*y*), as well as pile head
rotation (*θ*). Further, a Miner-rule-based superposition method with
both models (the logarithmic and power law functions) was applied to the
results to prove the validity of this method to predict the accumulated pile
head response to multi-amplitude lateral cyclic loading. From the evaluation
of the results, Li et al. (2015) propose ${\mathit{\alpha}}_{y,\phantom{\rule{0.125em}{0ex}}L}=\mathrm{0.085}$ and
${\mathit{\alpha}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}L}=\mathrm{0.060}$ as power law accumulation parameters for
displacement (*y*) and rotation (*θ*), respectively. For the
corresponding logarithmic accumulation parameters they suggest ${t}_{y,\phantom{\rule{0.125em}{0ex}}L}=\mathrm{0.125}$ and ${t}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}L}=\mathrm{0.080}$. However, since these values are based
on only two cyclic tests with one-way loading, no conclusions can be drawn
about the influence of varying soil relative density (*D*_{r}) or other
cyclic load ratios (*ζ*_{c}). Regarding the superposition model, Li
et al. (2015) found a very good overall prediction of the model with both
logarithmic and power functions.

A study involving considerably more model tests and different boundary
conditions was conducted by Truong et al. (2019). In this study, 17
centrifuge tests (3 static and 14 cyclic) with different soil relative
densities (*D*_{r}=0.57 to *D*_{r}=0.95), pile slenderness ratios
($L/D=\mathrm{6}$ and $L/D=\mathrm{11.4}$) and load magnitudes (*ζ*_{b}) were
conducted. The model piles had diameters (*D*) of 11 mm and 40 mm. Also
normalized load eccentricity was varied between $h/D=\mathrm{2}$ and $h/D=\mathrm{3}$, and
cyclic loads have been applied with load cycle numbers (*N*) between 50 and
1500 with different cyclic load ratios (*ζ*_{c}). In addition to
these centrifuge tests, Truong et al. (2019) also considered the test
results of Klinkvort and Hededal (2013), Li et al. (2015), and Rosquoët
et al. (2007) to develop a new method for the estimation of pile head
displacement accumulations with load cycle number (*N*). Based on the aforementioned results, they suggest a power law as given in Eq. (1) in
combination with an accumulation parameter (*α*) according to Eq. (7)
to account for different soil relative densities (*D*_{r}*>*0.5)
and cyclic load ratios (*ζ*_{c}).

Following this approach, maximum pile head accumulations result from cyclic
load ratios of about ${\mathit{\zeta}}_{c}=-\mathrm{0.5}$ and lower soil relative
densities (*D*_{r}). Further, Truong et al. (2019) could not confirm a
significant effect of the load magnitude (*ζ*_{b}) or the
eccentricity of applied loads (*h*), although these variables are of course
included in Eq. (1) as the displacement due to initial loading (*y*_{1})
depends on them.

A recent proposal for an approach to calculate the accumulation parameter
(*α*) given in Eq. (1) for pile head displacement from Li et al. (2020) is based on a series of 20 centrifuge tests (2 monotonic and 18
cyclic) on model piles with a diameter (*D*) of 18 mm and an embedment length
(*L*) of 90 mm ($L/D=\mathrm{5}$). In this study, two different soil relative
densities (*D*_{r}=0.5 and *D*_{r}=0.8) have been tested, and cyclic
loads have been applied with load cycle numbers (*N*) ranging from 42 to 153,
a normalized load eccentricity ($h/D$) of 8 and several different load
magnitudes (*ζ*_{b}), as well as cyclic load ratios (*ζ*_{c}).
Together with Eq. (1), for the accumulation parameter (*α*) Li et al. (2020) suggest the formulation of Klinkvort and Hededal (2013) given in Eq. (6) but define a new set of functions to derive *T*_{b} and *T*_{c} (here
referred to as *T*_{b, L} and *T*_{c, L}). In contrast to Klinkvort and
Hededal (2013), Li et al. (2020) found *T*_{b} independent from load
magnitude to be a constant, taking a value of ${T}_{b,\phantom{\rule{0.125em}{0ex}}L}=\mathrm{0.07335}$. The
parameter *T*_{c, L} is given by two equations, each dependent on the cyclic
load ratio (*ζ*_{c}), for the two soil relative densities (*D*_{r})
used in their test series. According to this approach, the largest
accumulation parameter (*α*) and therefore most displacement
accumulation results from asymmetric cyclic two-way loading with ${\mathit{\zeta}}_{c}\approx -\mathrm{0.3}$ and a soil relative density (*D*_{r}) of 0.5.

From the above, it can be seen that a variety of different approaches exists for the estimation of cyclic deformation accumulations. Even if the studies and methods mentioned here represent only part of the approaches to be found in the literature, it is already apparent from this that due to the complexity of the mechanisms driving displacement accumulation and inherent differences in reported tests, disagreements in results of different studies and the approaches derived from them are to be expected. Therefore, careful examination of the assumptions and the applicability of each proposed method is required. To facilitate a comparison of the individual methods mentioned and the most important underlying boundary conditions, these are summarized briefly in Table 1.

^{*} *T*_{b} and *T*_{c} functions fitted based on the graphical representations given in LeBlanc et al. (2010a).

${}^{**}$ Polynomial factors for the determination of ${T}_{c}({\mathit{\zeta}}_{c}\le -\mathrm{0.3})$: *a*=113.33; *b*=288.56; *c*=238.88; *d*=73.48; and *e*=9.94, as well as *T*_{c}(*ζ*_{c}*>*−0.3): *a*=3.06; $b=-\mathrm{6.50}$; *c*=5.22;

$d=-\mathrm{2.76}$; and *e*=0.99.

To allow not only a qualitative but a quantitative comparison of the
different empirical methods for the prediction of monopile deformation
accumulations resulting from lateral cyclic loading given in Table 1, a
calculation example and a parameter study are presented in the following.
Since all the above approaches describe deformation of the pile accumulated
after a certain number of load cycles (*y*_{N} or *θ*_{N}) as a
function of the initial deformation after first loading (*y*_{1} or *θ*_{1}), a monotonic load-displacement or load-rotation curve,
respectively, is the basis for further calculations. Therefore, Fig. 1 shows
the response of a steel pile (${E}_{\mathrm{p}}=\mathrm{21}\times {\mathrm{10}}^{\mathrm{7}}$ kN m^{−2}, *γ*_{s}=68 kN m^{−3}, *ν*_{s}=0.27) with typical monopile
dimensions (*D*=8 m, *t*=0.08 m, *L*=32 m) and an $L/D$ ratio of 4 to
monotonic loading. It was calculated for a load eccentricity (*h*) of 32 m
with the *p*–*y* method proposed by Thieken et al. (2015) using the freely
accessible pile design program IGTHPile V3.1 (Terceros et al., 2015). The
relevant soil parameters for the calculation representative of a homogeneous
and dense sand layer are given in the bottom line of Table 2.

On the one hand, these curves can be used to determine the displacement or
rotation (*y*_{1} or *θ*_{1}) for a given load, and, on the other
hand, they can be used to determine the reference load (*H*_{ref}) for the
definition of the load magnitude (*ζ*_{b}) according to Eq. (3).
However, as there is no single criterion for determining *H*_{ref}, this
value had to be evaluated for each approach according to the specifications
of the respective authors. Relevant deformation criteria for the definition
of the reference load (*H*_{ref}) and corresponding values taken from Fig. 1
are given in Table 2. It should be mentioned that due to the different
specifications regarding the reference load (*H*_{ref}), a direct comparison
of load magnitudes (*ζ*_{b}) between different approaches (see
Table 1) is not possible. In order to be able to make a direct comparison of
the various prediction models within the framework of the parameter study,
the cyclic loads were defined in terms of absolute magnitude (*H*_{max})
rather than their relative load magnitude (*ζ*_{b}). Since the
relative load magnitude (*ζ*_{b}) is nevertheless required as an
input variable for some of the models shown in Table 1, it was determined
and summarized in Table 2 for a bandwidth of horizontal loads (*H*_{max})
depending on the associated value of the reference force (*H*_{ref}) for
each method. Here, it can be seen that both the reference pile capacities
(*H*_{ref}) and therefore also the associated relative load magnitudes
(*ζ*_{b}(*H*_{max})) vary over a wide range depending on the chosen
criterion, even exceeding the value of 1 when *θ*_{ref}=0.5^{∘} is applied as proposed by Truong et al. (2019).

^{*} Calculated with ${\mathit{\gamma}}^{\prime}=\mathrm{10}$ kN m^{−3}; ${\mathit{\phi}}^{\prime}=\mathrm{37.5}$^{∘}; ${E}_{s,\phantom{\rule{0.125em}{0ex}}\mathrm{ref}}=\mathrm{57}\phantom{\rule{0.125em}{0ex}}\mathrm{500}$ kN m^{−2}; ${\mathit{\lambda}}_{{E}_{s}}=\mathrm{0.55}$; ${G}_{o,\phantom{\rule{0.125em}{0ex}}\mathrm{ref}}=\mathrm{71}\phantom{\rule{0.125em}{0ex}}\mathrm{250}$ kN m^{−2}; ${\mathit{\lambda}}_{{G}_{\mathrm{0}}}=\mathrm{0.5}$; and *υ*=0.225 based on Thieken et al. (2015).

Table 1 shows that the majority of the listed approaches refer to the pile
head displacement (*y*) as a deformation variable even if the reference load
(*H*_{ref}) is partly derived from pile head rotation (*θ*). Only
LeBlanc et al. (2010a) and Li et al. (2015) provide methods for calculating
the pile head rotation, whereby Li et al. (2015) propose both accumulation
parameters for pile head displacement and rotation. In order to be able to
compare the individual approaches with each other, the pile head
displacement was chosen as the relevant deformation variable. To enable at
least a qualitative comparison, the approach of LeBlanc et al. (2010a) was
therefore also applied to displacements without changing any of the model
parameters, although this is not actually permissible. Furthermore, it can
be seen from Table 1 that, according to the listed approaches, only the soil
relative density (*D*_{r}), the load magnitude (*ζ*_{b}) and the
cyclic load ratio (*ζ*_{c}) have an influence on the model
parameters. For the parameter study, bandwidths of the mentioned parameters
(*D*_{r}, *ζ*_{b}, *ζ*_{c}) were used, which lie within the
application range of the examined approaches. For the method of LeBlanc et
al. (2010a), the functions for *T*_{b, LB}(*D*_{r}, *ζ*_{b}), which in
model scale apply for soil relative densities (*D*_{r}) of 0.04 and 0.38,
were related to relative densities (*D*_{r}) of 0.08 and 0.75 in true scale
(see Sect. 2). In cases where there are two functions for a parameter
depending on, for example, the soil relative density (see, for example, *T*_{c, L}
according to Li et al., 2020 in Table 1) or another input value, linear
interpolation was performed between the two functions as needed.

The results of the parameter study are given in Fig. 2. Here, the pile head
displacements after a given number of load cycles (*y*_{N}) and the
corresponding normalized pile head displacements (${y}_{N}/{y}_{\mathrm{1}}$)
calculated according to the six methods summarized in Table 1 and the aforementioned assumptions are depicted. In order to assess the influence of the
different input variables separately, only one parameter was varied and
plotted on the *x* axis for each diagram. When evaluating the results
presented in Fig. 2, it must be kept in mind that the approaches of
Peralta (2010) and Li et al. (2015) in particular were derived for one-way
cyclic loading (*ζ*_{c}=0) only. For the sake of completeness,
these are nevertheless shown in Fig. 2d and h, where the absolute and
normalized pile head displacement (*y*_{N} and ${y}_{N}/{y}_{\mathrm{1}}$) is plotted
against the cyclic load ratio (*ζ*_{c}).

For cyclic one-way loading (*ζ*_{c}=0) with *H*_{max}=15 MN
(corresponding to relative load magnitudes (*ζ*_{b}) between 0.18 and
0.77, see Table 2) and a soil relative density (*D*_{r}) of 0.7, it can be
seen from Fig. 2a and e that according to all methods considered an
accumulation of pile head displacement with increasing load cycle number (*N*)
occurs. The lowest accumulation results from the approaches of LeBlanc et al. (2010a) and Li et al. (2015, 2020), in which after 30 000 load
cycles ${y}_{N}/{y}_{\mathrm{1}}$ takes values of approximately 2.2, 2.4 and 2.0,
respectively. However, it should be noted that the approach of LeBlanc et
al. (2010a) will result in slightly higher values (${y}_{N}/{y}_{\mathrm{1}}$) for
further increasing load cycle numbers when compared to Li et al. (2015) and
Li et al. (2020) due to its formulation (see Eq. 5) deviating from Eq. (1)
and the resulting higher accumulation rate. The largest normalized pile head
displacements are predicted when applying the method of Truong et al. (2019),
taking a value of ${y}_{N}/{y}_{\mathrm{1}}=\mathrm{6.1}$ and corresponding to an absolute
pile head displacement (*y*_{N}) of 0.59 m after 30 000 load cycles. The
results according to the approaches of Klinkvort and Hededal (2013) and Peralta (2010) fall between the results of the previously mentioned
methods.

Looking at Fig. 2b showing the influence of the maximum cyclic load
(*H*_{max}) and thus indirectly the influence of the cyclic load magnitude
(*ζ*_{b}) on the pile head displacement (*y*) after 10 000 load cycles
(*N*), it is obvious that in general larger absolute displacements (*y*) occur
for higher cyclic loads (*H*_{max}). When considering the normalized pile
head displacements (${y}_{N}/{y}_{\mathrm{1}}$) in Fig. 2f instead, it becomes clear
that only the results according to LeBlanc et al. (2010a) and Klinkvort and
Hededal (2013) are affected by the maximum cyclic load (*H*_{max}) or the
load magnitude (*ζ*_{b}), respectively. Here, the Klinkvort and
Hededal (2013) approach is much more sensitive to an increase in maximum
cyclic load (*H*_{max}) even though the cyclic load magnitude (*ζ*_{b}) for the chosen loads (10 MN $\le {H}_{\mathrm{max}}\le \mathrm{30}$ MN) and for
the definition of the reference load (*H*_{ref}) according to this approach
is in a moderate range from $\mathrm{0.15}\le {\mathit{\zeta}}_{b}\le \mathrm{0.46}$ (see
Table 2). Also it should be noted that the trends and differences shown with
regard to the results in the accumulation rate (${y}_{N}/{y}_{\mathrm{1}}$) according
to Klinkvort and Hededal (2013) compared to LeBlanc et al. (2010a) increase
further with a larger number of load cycles (*N*). Nevertheless, according to both
Klinkvort and Hededal (2013) and LeBlanc et al. (2010a) the
accumulation rate (${y}_{N}/{y}_{\mathrm{1}}$) generally increases with increasing
cyclic load magnitude (*ζ*_{b}), whereas this is not the case for the
other approaches mentioned. Here, higher absolute displacement (*y*_{N}) in the
case of increasing maximum cyclic loads (*H*_{max}) only results from an
increase in *y*_{1}.

Figure 2c and g show the influence of the soil relative density
(*D*_{r}) on the absolute and normalized pile head displacement (*y*_{N} and
${y}_{N}/{y}_{\mathrm{1}}$). From the plot, it can be seen that relative soil density
(*D*_{r}) only has an impact on the results according to the approaches of
LeBlanc et al. (2010a), Truong et al. (2019), and Li et al. (2020). However,
it must be kept in mind that the soil relative density (*D*_{r}) has an
influence on the soil parameters and thus on the monotonic load-displacement
curve and consequently *y*_{1}, which was not taken into account here for
reasons of comparability. Therefore, lower values would actually be the result for
the absolute displacements (*y*_{N}) with increasing soil density (*D*_{r}).
Nevertheless, according to both Truong et al. (2019) and Li et al. (2020), higher
soil relative density (*D*_{r}) results in lower pile head displacement
accumulation. As already shown in Fig. 2a and e, the approach according
to Truong et al. (2019) here also yields the largest deformations overall.
However, these also decrease the most with increasing soil relative density
(*D*_{r}), although they are still always above the other results. The
results according to LeBlanc et al. (2010a) are somewhat different. Here, a
slight increase in the accumulated displacements with increasing soil
relative density (*D*_{r}) can be observed. Nevertheless, the results
according to LeBlanc et al. (2010a) are in a similar range as those
according to the other approaches with the exception of Truong et al. (2019). This is due to the fact that the soil relative density (*D*_{r})
according to LeBlanc et al. (2010a) seems to have only a minor influence.

In Fig. 2d and h the influence of the cyclic load ratio (*ζ*_{c}) for a maximum cyclic load (*H*_{max}) of 15 MN and a soil relative
density (*D*_{r}) of 0.7 after 10 000 load cycles (*N*) is given. Irrespective
of the fundamental differences in the results according to the approaches
investigated, it follows from these diagrams that a variation in the cyclic
load ratio (*ζ*_{c}) also leads to deviating results with different
overall trends. While for almost all approaches except Peralta (2010) and Li
et al. (2015), who did not provide information on the influence of the
cyclic load ratio (*ζ*_{c}), a rather asymmetric load with ${\mathit{\zeta}}_{c}\le -\mathrm{0.25}$ results in the highest displacements; for the Klinkvort and
Hededal (2013) approach this is true for *ζ*_{c}=0. Furthermore,
it is also apparent that the different approaches are differently sensitive
to the cyclic load ratio (*ζ*_{c}). In particular, the results
according to Li et al. (2020) stand out, in which only a marginal influence
of the cyclic load ratio (*ζ*_{c}) can be recognized. In contrast,
the other approaches show significantly larger differences when *ζ*_{c} differs from zero.

Considering the partly deviating or even contradictory results shown above,
both with respect to the absolute values (*y*_{N}) and the trends shown with
regard to the influence of the different input parameters on the
accumulation rate (${y}_{N}/{y}_{\mathrm{1}}$), it is clear that there is a need for
further research. The inconsistencies shown between the different approaches
lead at best to an over-dimensioning of monopile dimensions and therefore
increasing costs and at worst even to uncertainties with regard to the
long-time deformation behaviour of the foundation that could render the
structure unsuitable for its intended function earlier than planned. One
possible reason for the existing discrepancies could be, for example, the
usually very limited number of tests on which the various approaches are
based.

## 4.1 Objective, test program and experimental set-up

To identify and quantify the influencing parameters affecting the load-displacement behaviour of a rigid pile due to lateral cyclic loading in more
detail, a large campaign of small-scale 1 g model tests has been designed.
The parameters that have been assessed include the cyclic load magnitude
(*ζ*_{b}), cyclic load ratio (*ζ*_{c}), load eccentricity (*h*),
soil relative density (*D*_{r}), the grain size distribution of the
non-cohesive bedding material (soil) and the pile embedment length
(*L*). In total, the entire test program, which is summarized in Table 3,
comprised 15 test series (TSs) with more than 150 individual tests on four
different pile–soil systems in dry sand. A pile–soil system is here defined
as a system with the same embedment length (*L*), soil relative density (*D*_{r}),
pile diameter (*D*) and bedding material (soil). As a model pile a tubular
aluminium pipe with an outer diameter (*D*) of 50 mm and a wall thickness (*t*)
of 3.2 mm was used. Two different embedment lengths (*L*) of 300 mm
($L/D=\mathrm{6}$) and 400 mm ($L/D=\mathrm{8}$), as well as two different bedding
materials (F34 and S40T, see Sect. 4.2) with two relative densities
(*D*_{r}) of 0.4 (medium dense) and 0.6 (dense), are considered by which the
four pile–soil systems (see Table 3) are defined. According to the
non-dimensional stiffness ratio suggested by Poulos and Hull (1989) all four
systems can be characterized to behave almost rigid, similar to a true-scale
monopile. In order to investigate the influence of the load eccentricity (*h*)
or the ratio of horizontal force to overturning moment, the
ratio of load eccentricity (*h*) to embedment length (*L*) was varied in the
range of $h/L=\mathrm{0.6}$ to $h/L=\mathrm{1.0}$ for pile–soil system 1 and $h/L=\mathrm{0.8}$
to $h/L=\mathrm{1.2}$ for pile–soil system 4. For pile–soil systems 2 and 3 the
ratio ($h/L$) was kept constant, taking a value of 1.0.

The loading conditions in the model tests comprised both displacement-controlled monotonic loading tests (*ζ*_{c}=1) and load-controlled cyclic loading (sinusoidal) with different cyclic load magnitudes
(*ζ*_{b}) and cyclic load ratios (*ζ*_{c}) at a constant
frequency of 0.1 Hz with 2500 load cycles (*N*) each. The cyclic load
magnitude (*ζ*_{b}) was defined according to Eq. (3) based on a
reference load (*H*_{ref}) that has been defined for each configuration of
pile–soil system and load eccentricity (*h*) from the determined monotonic
load-displacement curves by application of a pile failure criterion (see
Sect. 4.3.1). To evaluate the influence of different cyclic load ratios
(*ζ*_{c}) as given in Eq. (4), one- and two-way loads with values of
${\mathit{\zeta}}_{c}=-\mathrm{0.75}/-\mathrm{0.50}/-\mathrm{0.25}/\mathrm{0.00}/+\mathrm{0.25}$ have been applied. In
order to increase the significance of the experimental results and to
confirm repeatability, all tests given in Table 3 have been
conducted at least twice.

The small-scale model tests were carried out on a specially designed test
rig, consisting of a sand container, a model pile, an electromechanical
actuator and several sensors. Figure 3 shows the schematic structure of the
experimental set-up, its dimensions and its individual components. For more
detailed information on the test equipment or scaling considerations,
refer to Frick and Achmus (2020). Physical quantities measured in the
*y* direction (see Fig. 3), such as displacements or forces imposed by pulling
with the actuator, are positive in the following.

## 4.2 Soil properties and sand sample preparation

For the model test campaign, two different non-cohesive bedding materials
were chosen to investigate the influence of the grain size distribution to
the pile response due to lateral cyclic loading. Both materials are
commercially available silica sands with the designations F34 and S40T. The
F34 is a fine- to medium-grained sand having a mean effective particle size
(*d*_{50}) of 0.18 mm and a uniformity coefficient (*C*_{U}) of 1.90. In
contrast, the S40T is a coarse sand (*d*_{50}=0.82 mm) with a slightly
narrower graded grain size distribution (*C*_{U}=1.4). For both bedding
materials the sand characteristics and peak friction angles (${\mathit{\phi}}_{\mathrm{peak}}^{\prime}$) determined from standard shear box tests with normal stresses
(*σ*_{v}) of 20, 40 and 80 kN m^{−2} and soil relative
densities (*D*_{r}) of 0.4 are given in Table 4. The grain size
distributions of both soils are depicted in Fig. 4.

Before each small-scale model test a uniform sand sample with one of the two
chosen soil relative densities (*D*_{r}) had to be prepared. This was done
by air pluviation through a series of diffusor meshes and with a defined
drop height. In a series of preliminary tests, appropriate meshes and drop
heights were determined, with whose help the desired soil relative
densities (*D*_{r}) for the two different bedding materials (see Table 3)
could be achieved. Furthermore, it could be shown that the selected
preparation procedure leads to homogeneous and reproducible sand conditions
in a normally consolidated state. To avoid complex stress fields and local
changes in soil density due to model pile installation influencing the test
results, it was decided to omit the pile installation procedure from the
model tests and prepare the sand around the pre-installed pile.
Nevertheless, in order to allow mobilization of shear stresses at the pile
toe, the sand container was first filled to a height of about 5 cm above the
later position of the pile toe before the pile was placed in the container,
slightly pushed into the 5 cm thick sand bed and fixed in position by a
clamping system. The remaining soil preparation was then carried out around
the already installed pile. The soil dry unit weight (*γ*) resulting
from this preparation procedure for the S40T sand with a relative density
(*D*_{r}) of 0.4 is 16.1 kN m^{−3}. For the F34 sand, the soil
dry unit weight (*γ*) is 15.0 and 15.5 kN m^{−3} for relative densities (*D*_{r}) of 0.4 and 0.6,
respectively.

## 4.3 Test results

### 4.3.1 Monotonic test results

In order to be able to apply comparable load conditions in terms of the load
magnitude (*ζ*_{b}) in the cyclic tests despite different pile–soil
systems and lever arms (*h*), first monotonic load tests were carried out for
the determination of load-displacement curves and a reference load
(*H*_{ref}) for each configuration. Figure 5 presents the variations in
normalized monotonic lateral load ($H/(\mathit{\gamma}\cdot {L}^{\mathrm{3}})$) with normalized pile head displacement ($y/L$) at the soil
surface for all four pile–soil systems and in the case of pile–soil systems 1 and
4 for different load eccentricities (*h*) additionally. Here, the pile head
displacements (*y*) calculated from the measured displacements of the two
laser distance transducers (see Fig. 3) are depicted as solid lines. To
ensure the reproducibility of the tests and to check the quality of the sand
sample preparation, each test was conducted at least twice. As some
scattering could be observed especially for pile–soil system 2 ($h/L=\mathrm{1}$)
and pile–soil system 4 with a normalized load eccentricity ($h/L$) of 1.2,
these tests have even been done four times.

For this study, the reference load (*H*_{ref}) should be defined as the
ultimate lateral pile capacity (*H*_{ult}) at total pile failure. The
failure load (*H*_{ult}) that can be resisted by a rigid pile is a function
of the ultimate lateral resistance that can be mobilized by the soil against
the pile. The mobilized soil resistance in the case of a laterally loaded rigid
pile is again characterized by two failure mechanisms. The first occurs at
shallow depths and is due to the formation of a passive wedge in front of
the pile and in the direction of loading. The second is associated with the
plastic flow of soil around the pile in the horizontal plane at greater
depths. For the occurrence of both failure mechanisms and thus the
mobilization of the full soil resistance against the pile, very large
displacements are required. From the monotonic test results depicted in
Fig. 5 it emerges that total pile capacity (*H*_{ult}) defined by full soil plastification and a load-displacement curve approaching a horizontal
tangent has not been reached despite very large displacements. In order to
be able to determine the pile capacities (*H*_{ult}) and therefore the
chosen reference loads (*H*_{ref}) of the individual pile–soil systems, the
method of Manoliu et al. (1985) was applied to the results. This method
assumes that load displacement of a laterally loaded pile can be described
by a hyperbolic function. Therefore, the method allows for the estimation of the
pile failure load (*H*_{ult}) by the extrapolation of measured test data.
Corresponding extrapolation curves derived by application of this approach
are depicted in Fig. 5 as dashed lines.

As some slight scattering could be observed in the monotonic test results
and extrapolations (Fig. 5), mean values for the pile failure load
(*H*_{ult}) or reference load (*H*_{ref}), respectively, were used for each
configuration to define the cyclic load magnitude (*ζ*_{b}) according
to Eq. (3) for each configuration. The normalized and absolute reference
values of the horizontal load (*H*_{ref}) determined for each test
configuration according to the previously mentioned procedure are summarized
in Table 5.

### 4.3.2 Cyclic test results

The cyclic tests summarized in Table 3 have been conducted for cyclic load
magnitudes (*ζ*_{b}) of 0.15, 0.25 and 0.35 and cyclic load
ratios (*ζ*_{c}) ranging from nearly balanced two-way loading (${\mathit{\zeta}}_{c}=-\mathrm{0.75}$) to one-way loading with complete unloading (*ζ*_{c}=0.00) or partial unloading (${\mathit{\zeta}}_{c}=+\mathrm{0.25}$) in each
cycle. Based on the findings of Jalbi et al. (2019), who proposed a
practical method to predict the nature of monopile loading conditions
(*ζ*_{b} and *ζ*_{c}) and evaluated 15 existing wind turbines
in Europe using their method, the load magnitudes (*ζ*_{b}) and
cyclic load ratios (*ζ*_{c}) selected for this study are of
particular interest for practical application. According to Jalbi et al. (2019) typical load magnitudes (*ζ*_{b}) for normal operational
conditions range from 0.1 to 0.2. In extreme wind and wave loading cases
also load magnitudes (*ζ*_{b}) of up to 0.4 may be reached. With
regard to the cyclic load ratio (*ζ*_{c}) they found that loads on
monopiles are mostly one-way (*ζ*_{c}≥0) under normal
operational conditions but may also be two-way (*ζ*_{c}*<*0)
in extreme loading scenarios, especially in deep waters. It should be
mentioned that Jalbi et al. (2019) also assumed the reference load
(*H*_{ref}) for the definition of the load magnitude (*ζ*_{b}) by
back-calculations, in which ultimate pressure of the ground profile was
mobilized (= total ground and therefore pile failure) so that values
given for *ζ*_{b} should be comparable with those of this study (see
Sect. 4.3.1).

In Fig. 6, the results of all cyclic tests are plotted in terms of
normalized pile head displacement (${y}_{N}/{y}_{\mathrm{1}}$) against load cycle
number (*N*) for each of the 15 test series separately. In addition, power
functions according to Eq. (1) have been fitted to the measured test results
and are also shown in Fig. 6. The underlying maximum pile head displacement
after application of the first load cycle (*y*_{1}), as well as the
determined accumulation parameter (*α*) which is dependent on the cyclic
load ratio (*ζ*_{c}) for each individual test, is listed in Table 6
for clarity.

In general, it can be seen from Fig. 6 that the tests can be reproduced well
in most cases. Furthermore, it can be seen that the measured curves can be
described very well by the selected power function (Eq. 1). Only in a few
cases (see, for example, test series 5: ${\mathit{\zeta}}_{c}=-\mathrm{0.25}$) does there seem to be
a minimal overestimation of measured values as the number of load cycles
increases. With respect to the influence of the cyclic load ratio (*ζ*_{c}), a clear trend can be recognized in all test series. Irrespective
of the load eccentricity (*h*) or the pile embedment length (*L*), for pile soil
systems 1 (test series 1–5), 3 (test series 9–11) and 4 (test series 12–15),
the highest displacement accumulation consistently results from an
unbalanced two-way loading with a cyclic load ratio (*ζ*_{c}) of
−0.25. In the case of pile–soil system 3 (test series 9–11) with the S40T sand
as bedding material and a soil relative density (*D*_{r}) of 0.4, however,
the difference between the tests with ${\mathit{\zeta}}_{c}=-\mathrm{0.25}$ and *ζ*_{c}=0.00 is less pronounced. Also for pile–soil system 2 (test
series 6–8) with dense (*D*_{r}= 0.6) F34 sand the largest displacements are the result of an asymmetric two-way loading, however, not with a cyclic load
ratio (*ζ*_{c}) of −0.25 but of −0.5. The lowest displacement
accumulations for all test series result from loading with a cyclic load
ratio (*ζ*_{c}) of −0.75 (nearly balanced two-way loading) or +0.25
(one-way loading without complete unloading in each cycle). A negative
accumulation for loads with large negative cyclic load ratios (*ζ*_{c}) as reported by Klinkvort and Hededal (2013) could not be observed,
although very small accumulations were recorded in some cases for the tests
with ${\mathit{\zeta}}_{c}=-\mathrm{0.75}$ (see, for example, test series 12 and 13). Another
general trend that emerges from the results shown in Fig. 6 is that a large
part of the total deformations due to lateral cyclic loading already takes
place within the first 500 to 1000 load cycles, while subsequently there is
a slowly decreasing accumulation rate (sedation). An exception here is test
series 8 (F34, D_{r}=0.6 and *ζ*_{b}=0.35), in which the test
results with negative cyclic load ratios (*ζ*_{c}) show a strong
increase in displacements even beyond a cycle number (*N*) of 1000. This is
due to the fact that in this configuration with high cyclic loads (large
*H*_{ref} and *ζ*_{b}), especially with alternating loads (*ζ*_{c}*<*0), the pile moved slowly out of the soil while cycling,
resulting in progressive failure. This is also the reason why no results are
shown for cyclic load ratios (*ζ*_{c}) of −0.75 for this test series
as here an even earlier failure occurred. Having this is mind, the results
for pile–soil system 2 (test series 6–8) and especially those of test
series 8 should be treated with caution.

### 4.3.3 Evaluation

In order to evaluate the cyclic test results from Fig. 6 with respect to the
influence of the applied cyclic loading conditions or different parameters
of the pile–soil system on the displacement accumulation, the accumulation
parameters (*α*) for cyclic one-way loading (*ζ*_{c}=0)
from Table 6 were used as a reference value and plotted against the variable
parameters of the investigated pile–soil systems (*D*_{r}, $h/L$, $L/D$) and the
load magnitude (*ζ*_{b}) in Fig. 7. In general it is evident from
Fig. 7 that the determined accumulation parameters for cyclic one-way
loading (*α*(*ζ*_{c}=0)) are subjected to a certain degree
of unsystematic scattering, ranging from a maximum value of 0.1343 to a
minimum value of 0.0983. The mean of all *α* values for cyclic one-way
loading (*ζ*_{c}=0) is 0.1169 (see Table 6). The deviations in
the results for the individual tests with identical boundary conditions
(redundant tests) are generally smaller, but they are nevertheless present
and probably due to experimental scatter.

In Fig. 7a the influence of the cyclic load magnitude (*ζ*_{b})
can be seen. When taking into account only the accumulation parameters
*α*(*ζ*_{c}=0) for pile–soil system 1 (black symbols) or 2
(green symbols), a slight increase in the accumulation parameter (*α*)
with cyclic load magnitude (*ζ*_{b}) can be observed. Nevertheless, the
opposite is true for systems 3 (blue symbols) and 4 (red symbols). When all pile–soil systems are considered in a linear
regression analysis, the aforementioned trends almost cancel each other out,
resulting in only a negligible increase in the accumulation parameter (*α*) with cyclic load magnitude (*ζ*_{b}). Nevertheless, the linear equation
describing the possibly existing dependency of the accumulation parameter
(*α*) on the cyclic load magnitude (*ζ*_{b}) is given in
Fig. 7a for completeness.

An evaluation of the results with respect to the soil relative density
(*D*_{r}) as shown in Fig. 7b also does not allow any clear conclusion to
be drawn. On the one hand, only one system with a higher soil relative
density (*D*_{r}) was investigated, and, on the other hand, the results of
the individual pile–soil systems scatter over such a range that the linear
regression shown in Fig. 7b can only provide an approximation. Similar to
the influence of the cyclic load magnitude (*ζ*_{b}), a trend of a
slightly increasing accumulation parameter (*α*) with soil relative
density can be seen, but this increase is also considered negligible due to
the existing scatter and the minor slope of the regression curve.

In Fig. 7c all accumulation parameters for cyclic one-way loading
(*α*(*ζ*_{c}= 0)) are plotted against the relative load
eccentricity ($h/L$). Taking all values into account a decreasing trend with
increasing load eccentricity (*h*) results from the linear regression
analysis. If only the values for system 1 (black symbols) or system 3 (blue
symbols) are considered, for which the load eccentricity was varied, it
becomes clear that this behaviour is also more likely to be due to
experimental scattering. For system 1, where the normalized lever arm ($h/L$)
was varied in the range of 0.6 to 1.0, the largest accumulation parameter
(*α*) was determined for the mean value of $h/L=\mathrm{1.0}$. For system 3
($\mathrm{0.8}\le h/L\le \mathrm{1.2}$) a similar observation can be made. Since the
lever arm (*h*) only defines the ratio of the horizontal force to the applied
overturning moment, a maximum value of the accumulation parameter (*α*) in the middle of the investigated bandwidth for the normalized load
eccentricity ($h/L$) is not plausible.

Finally, Fig. 7d shows the determined accumulation parameters (*α*)
for *ζ*_{c}=0 as a function of the normalized pile embedment
length ($L/D$). As for the other investigated parameters (*ζ*_{b},
*D*_{r}, $h/L$), there is no appreciable influence of the pile embedment
length (*L*) on the accumulation parameter (*α*) at least for the rigid
piles in the investigated range of normalized embedment length ($L/D$) and for
one-way cyclic loading (*ζ*_{c}=0).

Although no clear trend emerges from any of the graphs in Fig. 7 in view of
the scatter present, a linear regression analysis was performed for each
plot. The resulting equations describing the determined and aforementioned
dependencies are given in the respective diagrams for completeness. Due to
the insignificance of the observed dependencies combined with the existing
variance of the results, it seems that there is no remarkable influence of
the investigated parameters on the accumulation parameter (*α*) at
least for one-way cyclic loading (*ζ*_{c}=0). Nevertheless, it
should be kept in mind that at least the initial displacement (*y*_{1})
depends strongly on the mentioned load or pile–soil system parameters, which
is why the absolute accumulated displacement after a certain number of load
cycles (*y*_{N}) is of course not independent of the mentioned input
variables (see Eq. 1).

Further, Fig. 8 shows the accumulation parameter (*α*) and its
dependency on the cyclic load ratio (*ζ*_{c}) for all four pile–soil
systems. Here, the results for test series 8 with negative cyclic load
ratios (*ζ*_{c}*<*0) have been excluded from evaluation due
to the aforementioned reasons (see Sect. 4.3.2). Furthermore, due to the
previously determined predominant independence of the accumulation
parameter for cyclic one-way loading (*α*(*ζ*_{c}=0)), it
was decided not to normalize the accumulation parameters (*α*(*ζ*_{c})) on the basis of one of the other parameters (e.g. *ζ*_{b}
or *D*_{r}), as suggested, for example, by Klinkvort and Hededal (2013),
Truong et al. (2019), and Li et al. (2020). Instead, the results in Fig. 8 are
enveloped by two functions defining an upper and lower bound of the
accumulation parameter (*α*) for the investigated pile–soil systems
and boundary conditions, illustrating the possible range of *α* values.

In general, it can be seen from Fig. 8 that the largest values for the
accumulation parameter (*α*) result from unbalanced two-way loading
(*ζ*_{c}*<*0) taking a maximum value of approximately 0.17
for pile–soil system 2 at a cyclic load ratio (*ζ*_{c}) of −0.5 and
being more or less independent from cyclic load magnitude (*ζ*_{b}).
On closer examination, it emerges that for all other pile–soil systems (1, 3
and 4) the maximum accumulation parameter (*α*) occurs with a lower
value at a cyclic load ratio (*ζ*_{c}) of −0.25. It could be
concluded that both the maximum of the accumulation parameter (*α*)
and its occurrence with respect to the cyclic load ratio (*ζ*_{c})
depend on the soil relative density (*D*_{r}). In the investigated cases, an
increase in the soil relative density (*D*_{r}) from 0.4 (system 1) to 0.6
(system 2) leads to a slight increase and simultaneous shift in the maximum
accumulation parameter (*α*) towards a more negative value of the
cyclic load ratio (*ζ*_{c}). Nevertheless, as already described in
Sect. 4.3.2, the results for pile–soil system 2 should be handled with
care. When considering only the results for pile–soil systems 1, 3 and 4, a
certain spread of the determined accumulation parameters (*α*) is
still evident, but basically they follow a consistent trend. Within the
above-mentioned range of values for pile–soil systems 1, 3 and 4, the values
for system 1 in particular are at the upper bound, while the accumulation
parameters (*α*) for systems 3 and 4 tend to be below this. Especially
for a cyclic load ratio (*ζ*_{c}) of −0.75 the accumulation
parameters (*α*) for system 4 with a shorter embedment length
partially lie in a very low range. Due to the scattering of the results, a
clear final conclusion cannot be drawn. However, it is evident from the
results that both the embedment length (*L*) of the pile (compare system 1 and
system 4 in Fig. 8) and the grain size distribution (compare system 1 and
system 3 in Fig. 8) appear to have an effect on the accumulation parameter
(*α*).

In this section the findings and results from the conducted experimental 1 g
model test campaign are discussed and compared with those of other research
groups so that a classification of the results is possible. With respect to
the accumulation parameter (*α*) from Eq. (1), the results indicate that
it appears to be largely independent of the cyclic load magnitude (*ζ*_{b}), the soil relative density (*D*_{r}), the load eccentricity (*h*) and
the embedment length of the pile (*L*) for one-way cyclic loading (*ζ*_{c}=0) as long as the pile–soil system is characterized by an
almost rigid load-displacement behaviour. Despite some scattering in the
results for the accumulation parameter (*α*), which could probably be
due to irregularities in the test execution (soil sample preparation, etc.),
a mean value of ${\mathit{\alpha}}_{\mathrm{mean}}({\mathit{\zeta}}_{c}=\mathrm{0})=\mathrm{0.1169}$ (with
${\mathit{\alpha}}_{\mathrm{min}}({\mathit{\zeta}}_{c}=\mathrm{0})=\mathrm{0.0983}$ and ${\mathit{\alpha}}_{\mathrm{max}}({\mathit{\zeta}}_{c}=\mathrm{0})=\mathrm{0.1343}$) could be determined. This mean
value fits quite well with the value of *α*_{P}=0.12 proposed by
Peralta (2010) who also determined it from scaled 1 g model tests on rigid
piles subjected to cyclic one-way loading (*ζ*_{c}=0) only.
Similar to the present study, Peralta (2010) also found the accumulation
parameter (*α*(*ζ*_{c}=0)) to be independent from cyclic
load magnitude (*ζ*_{b}), the soil relative density (*D*_{r}) and the
normalized pile embedment length ($L/D$) as long as the pile behaves almost
rigid.

In contrast, Li et al. (2015) proposed a lower value of ${\mathit{\alpha}}_{y,\phantom{\rule{0.125em}{0ex}}L}=\mathrm{0.085}$ (see Table 1) for cyclic one-way loading (*ζ*_{c}=0)
based on two cyclic laterally loaded field tests on rigid piles. This could
indicate that accumulation parameters (*α*) from small-scale model
tests cannot be easily transferred to true scale due to differences in the
stress state of the surrounding soil and the resulting differences in soil
behaviour (e.g. dilatancy, stiffness). This assumption can be
supported by Richards et al. (2021) who investigated the stress effect on
the response of model monopiles to unidirectional cyclic lateral loading
(*ζ*_{c}=0) in sand by model tests either at 1 g or in the
centrifuge. Although in this study an approximation function according to
Eq. (5) was used to describe the cyclic displacement behaviour, it was found
that the accumulation parameter (*α*) of this equation (not directly
comparable with *α* according to Eq. 1) decreases logarithmically with
stress level under otherwise constant boundary conditions. Qualitatively,
according to Richards et al. (2021) the cyclic responses have been found to
be similar across stress level, anyway. It is therefore obvious that the
present results provide higher accumulation parameters (*α*) than
comparable large-scale experiments or centrifuge tests at higher
stress levels.

This can also be verified using the approach of Li et al. (2020), which is
based on a series of centrifuge tests. According to this approach, the
accumulation parameter (*α*) from Eq. (1) can be calculated by
the multiplication of the two parameters *T*_{b, L} and *T*_{c, L} that describe
the influence of the load magnitude (*ζ*_{b}) and the cyclic
load ratio (*ζ*_{c}), together with the soil relative density
(*D*_{r}), respectively (for corresponding equations see Table 1). Due to the
definition of T_{c,L} according to Eq. (9), together with Eq. (8), for cyclic
one-way loading (*ζ*_{c}=0) the accumulation parameter (*α*) for this approach would have to result in ${T}_{b,\phantom{\rule{0.125em}{0ex}}L}=\mathrm{0.07335}$ when the
proposed functions for *T*_{b, L} and *T*_{c, L} would fit the underlying test
results perfectly.

Since the results of Li et al. (2020), on which the approach and the
functions are based, are also subjected to scattering, the constant value of
*T*_{b, L} is only an approximation of the experimentally determined
accumulation parameters (*α*(*ζ*_{c}=0)), which is why the
proposed *T*_{c, L} functions (see Table 1) do not yield the value of 1 when
the cyclic load ratio (*ζ*_{c}) is 0. Nevertheless, the approach
yields accumulation parameters (*α*) of about ($+/-$) 0.07335 for
cyclic one-way loading (*ζ*_{c}=0), which is slightly less than
the value given by Li et al. (2015) and provides another indication of the
stress dependence of the accumulation parameter (*α*). Further, Li et
al. (2020) confirm the accumulation parameter (*α*) to be independent
from cyclic load magnitude (*ζ*_{b}), similar to the results presented
in the article at hand.

Somehow different are the findings of Klinkvort and Hededal (2013), in which
the accumulation parameter (*α*) depends on the cyclic load magnitude
(*ζ*_{b}) and the cyclic load ratio (*ζ*_{c}). For
cyclic one-way loading (*ζ*_{c}=0) the accumulation parameter
(*α*) according to Klinkvort and Hededal (2013) results directly from
the equation for *T*_{b, K&H} in Table 1, which, for example, yields a value
of 0.231 for a cyclic load magnitude (*ζ*_{b}) of 0.4, linearly
further increasing for higher load magnitudes (*ζ*_{b}). This is
contrary to the findings of most other authors mentioned in Table 1 except
LeBlanc et al. (2010a) whose approach is not directly comparable as it is
not based on Eq. (1). In addition, the Klinkvort and Hededal (2013) approach
seems to provide very high accumulation parameters (*α*(*ζ*_{c}=0)) compared to other methods, at least for load magnitudes
(*ζ*_{b}) larger than 0.2. Here, the definition of the reference load
H_{ref} for the determination of the load magnitude (*ζ*_{b})
according to Klinkvort and Hededal (2013) has to be kept in mind (see
Table 2). Nevertheless, such high accumulation parameters (*α*) from
centrifuge tests are contrary to the findings of Richards et al. (2021) and
the assumption of decreasing accumulation parameters (*α*) with
stress level. Nevertheless, the results of Klinkvort and Hededal (2013)
support the assumption that the accumulation parameter (*α*) is
independent of the soil relative density (*D*_{r}), which has also been
found in the present study, at least for cyclic one-way loading (*ζ*_{c}=0).

According to the approach of Truong et al. (2019), the accumulation
parameter (*α*) for cyclic one-way loading (*ζ*_{c}=0) is
independent from cyclic load magnitude (*ζ*_{b}) as already proposed
by Li et al. (2020) and also found in this study. Nevertheless, it linearly
decreases with soil relative density (*D*_{r}). For relative densities
(*D*_{r}) of 0.4 and 0.6, as used in the experiments presented
above, unidirectional cyclic loading (*ζ*_{c}=0) results in
accumulation parameters of 0.212 and 0.168 using the approach of Truong et
al. (2019). This is significantly higher than the values obtained in the
present study (${\mathit{\alpha}}_{\mathrm{mean}}({\mathit{\zeta}}_{c}=\mathrm{0})=\mathrm{0.1169}$) and
contradicts the assumption of a decreasing accumulation parameter (*α*) with stress level in that the Truong et al. (2019) approach is based on
centrifuge tests. On the other hand, this approach yields an accumulation
parameter (*α*) of 0.113 for a soil relative density (*D*_{r}) of
0.85, which is much closer to the value resulting from this study. Possibly,
a stress-dependent conversion of the soil relative density (*D*_{r}), as
proposed by LeBlanc et al. (2010a), could provide an explanation for the
resulting deviations (see Sect. 2). However, the dependence of the
accumulation parameter (*α*) for unidirectional loading (*ζ*_{c}= 0) on the soil relative density (*D*_{r}) proposed by Truong et
al. (2019) contradicts the results of the present study, as well as those of
Klinkvort and Hededal (2013) and Li et al. (2020).

Further, the influence of a variable cyclic load ratio (*ζ*_{c}) on
the accumulation parameter (*α*) is now discussed. Due to the above-mentioned, partly different dependencies of the estimation approaches for
the accumulation parameter (*α*), however, a direct comparison is not
possible. In order to enable a reliable comparison, the results for the
accumulation parameter (*α*) with cyclic load ratio (*ζ*_{c})
according to the different approaches presented are shown in normalized form
in Fig. 9. By normalizing to *α*(*ζ*_{c}= 0), the previously
mentioned differences of the approaches with respect to the accumulation
parameter (*α*) for one-way loading (*ζ*_{c}=0) are
omitted so that the influence of the cyclic load ratio (*ζ*_{c}) can
be considered in isolation. Only the influence of the soil relative density
(*D*_{r}) according to the approach of Li et al. (2020) cannot be excluded
in this way due to the two proposed non-linear functions for *T*_{c, L}(*ζ*_{c}, *D*_{r}) (see Table 1). For this reason, Fig. 9 shows two curves for
this approach, in which both curves define the limits of applicability of the
Li et al. (2020) method with respect to the soil relative density ($\mathrm{0.5}\le {D}_{\mathrm{r}}\le \mathrm{0.8}$). The results according to Peralta (2010) and Li et al. (2015) are not depicted in Fig. 9 as both methods only propose an
accumulation parameter (*α*) for cyclic one-way loading (*ζ*_{c}=0). To allow a comparison with the results of the present study,
also the lower and upper bound curves for the accumulation parameter
(*α*) determined and proposed in Sect. 4.3.3 are plotted in
normalized form in Fig. 9.

Figure 9 shows that in particular the approach according to Li et al. (2020)
fits well with the results of the current study. For both soil relative
densities (*D*_{r}) of 0.8 and 0.5, the results are within the proposed
limits (lower bound, LB, and upper bound, UB) for two-way loading (*ζ*_{c}*<*0), while
for one-way loading (*ζ*_{c}≥0) the Li et al. (2020) curve for
a soil relative density (*D*_{r}) of 0.8 is increasingly divergent and below
the proposed boundaries (conservative). The Truong et al. (2019) approach
also shows a qualitatively similar shape to the proposed boundary curves
but overall is slightly below the lower bound curve for cyclic two-way
loading (*ζ*_{c}*<*0) and moderately above for one-way
loading (*ζ*_{c}≥0). All the aforementioned curves show a
maximum value between approximately 1.23 and 1.02 for an unbalanced two-way
loading with cyclic load ratio (*ζ*_{c}) in the range of −0.4 to
−0.15. This is generally also in agreement with the findings of LeBlanc et
al. (2010a) who report a maximum accumulation for cyclic two-way loading
with a cyclic load ratio (*ζ*_{c}) of −0.6 but whose approach is not
included in this comparison due to the different formulation of this
approach (see Table 1). Somehow different are the findings of Klinkvort and
Hededal (2013), whose approach provides a maximum accumulation parameter
(*α*) for cyclic one-way loading with complete unloading in each cycle
(*ζ*_{c}=0). For more positive cyclic load ratios (*ζ*_{c}*>*0), the normalized accumulation parameter according to
Klinkvort and Hededal (2013) resembles the values of the Truong et al. (2019) approach that lie slightly above the proposed upper bound curve.

In this paper, a brief summary of current regulations and recommendations
for the serviceability limit state dimensioning of offshore monopile
foundations in sand supporting wind turbines was given. Based on this
summary, it was shown that current offshore guidelines (DNV GL, 2018, and
API, 2014) provide design requirements but do not recommend appropriate
design procedures for predicting deformations for large-diameter piles
subjected to long-term lateral cyclic loading. Instead, a variety of
different methods for the prediction of such deformations can be found in the
literature, some of which have been briefly presented. Based on example
calculations, it was shown that the proposed methods for deriving the cyclic
load-deformation behaviour of monopile foundations yield partly
significantly different results. Furthermore, it could be shown that
depending on the chosen approach, the results exhibit a partly contradictory
trend with regard to the influence of some input variables such as load or
soil parameters. To better understand this outcome, a comprehensive
experimental small-scale model test campaign involving approximately 150
single tests on different pile–soil systems subjected to varying loading
conditions (*ζ*_{b} and *ζ*_{c}) being representative for the
environmental conditions of an offshore monopile foundation (Jalbi et al.,
2019) was conducted and evaluated. Based on the results, it could be
shown that a power function (Eq. 1) is very suitable for representing the
pile head displacement accumulation of rigid piles under different cyclic
one- and two-way loading conditions with constant mean load and amplitude.
For the accumulation parameter (*α*) of the power function, it was
found from the conducted tests that it is almost independent of the cyclic
load magnitude (*ζ*_{b}), the soil relative density (*D*_{r}), the
load eccentricity (*h*) and the pile embedment length (*L*) for cyclic one-way
loading (*ζ*_{c}=0) as long as the pile–soil systems can be
classified as behaving rigidly. Comparison of these findings and the determined
mean value for $\mathit{\alpha}({\mathit{\zeta}}_{c}=\mathrm{0})=\mathrm{0.1169}$ with values
derived from other methods showed that this observation is only shared by
some authors. Furthermore, it was shown that the determined absolute value
of the accumulation parameter for one-way cyclic loading (*ζ*_{c}=0) seems to exhibit a stress dependence. Therefore, the direct
transfer of the presented results to true scale cannot be recommended. With
regard to the influence of different cyclic load ratios (*ζ*_{c}),
the test results of the test campaign conducted showed a relatively clear
trend. Maximum accumulation and therefore accumulation parameters (*α*(*ζ*_{c})) in general result from unbalanced two-way loading
($-\mathrm{0.4}\mathit{<}{\mathit{\zeta}}_{c}\mathit{<}-\mathrm{0.15}$) and lead to an increase in
the accumulation parameter (*α*) by a factor of up to 1.23 compared to
one-way loading (*ζ*_{c}=0). Since the determined accumulation
parameters for variable cyclic load ratios (*α*(*ζ*_{c})), on
the one hand, vary slightly due to experimental scatter and, on the other
hand, seem to be at least slightly influenced by other variables (e.g.
*D*_{r}), two equations for an upper and a lower bound of the accumulation
parameter (*α*(*ζ*_{c})) were proposed. If the proposed limit
curves for the accumulation parameter (*α*(*ζ*_{c})) are
normalized to the accumulation parameter for cyclic one-way loading (*α*(*ζ*_{c}=0)), then a comparison with the results obtained by
other approaches shows relatively good agreement. In order to be able to
make a prediction of the cyclic displacement accumulation of a pile using
the power function according to Eq. (1), the accumulation parameter for
one-way cyclic loading (*α*(*ζ*_{c}=0)) should first be
known as accurately as possible. This can be achieved by site-specific
numerical simulations or centrifuge testing to avoid unwanted stress
effects. Another possibility would be to determine functions for the stress-dependent conversion of the proposed accumulation parameter (*α*) from
the present small-scale model tests (see, for example, Richards et al., 2020). For
deviating load conditions (varying cyclic load ratios (*ζ*_{c})), a
range of possible accumulation parameters (*α*(*ζ*_{c})) can be
estimated using the proposed upper and lower bound curves by normalizing
these curves and multiplying the resulting factor by the pre-determined
site-specific accumulation parameter for one-way loading (*α*(*ζ*_{c}= 0)). Further research should especially focus on the accurate
determination of the accumulation parameter for cyclic one-way lateral
loading (*α*(*ζ*_{c}=0)).

Measurement data from the small-scale 1 g experiments are available upon request from the authors.

DF prepared and carried out the presented test campaign, evaluated the measurement data, and wrote the paper. MA aided in the design of the experimental test campaign, advised on evaluation of the data and supervised the work. Both authors were involved in the peer review of the work.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study was carried out in the scope of the research project “Accumulation of lateral displacements of piles under general cyclic one- and two-way loading” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project no. 393683178. The authors sincerely acknowledge DFG support.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 393683178).

The publication of this article was funded by the open-access fund of Leibniz Universität Hannover.

This paper was edited by Amir R. Nejad and reviewed by Gudmund Reidar Eiksund and Tomas Sabaliauskas.

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- Abstract
- Introduction
- State of the art
- Comparison of different empirical approaches for the estimation of cyclic lateral deformation
- Small-scale model tests
- Discussion
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- State of the art
- Comparison of different empirical approaches for the estimation of cyclic lateral deformation
- Small-scale model tests
- Discussion
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References