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On the quasi-static granular convective flow and sand densification

around pile foundations under cyclic lateral loading

Pablo Cuéllar, Steven Georgi, Matthias Baeßler, Werner Rücker

BAM Bundesanstalt für Materialforschung und -prüfung (Federal Institute for Materials Research and Testing) - Division 7.2 “Buildings and Structures“. Unter den Eichen 87, 12205 Berlin, Germany

TEL. +49 / (0)30 / 81043888

FAX. +49 / (0)30 / 81041727

pablo.cuellar@bam.de

www.bam.de

Abstract The saturated sand surrounding an offshore pile foundation under quasi-static cyclic lateral load can show the physical phenomena of macromechanical densification and convective granular flow. Based on the results from physical model tests at different geometrical scales, this paper provides a certain quantification of such phenomena and discusses their causes and consequences. The progressive sand densification leads to subsidence of the soil surface and a significant stiffening of the pile behaviour. Conversely, the ratcheting convective motion of two closed cells of soil beneath the pile-head is responsible for an endless grain migration at the soil surface, the inverse grading of the convected material and a direct shear of the sand at the distinct boundary of the revolving soil domain. In this respect, and from a macromechanical perspective considering the soil as a continuum, it appears that the convecting material tends to follow gradient lines of shear stress during its ratcheting motion. Concluding the paper, the practical relevance of these phenomena and their extrapolation to other conditions are briefly discussed.

Keywords: Offshore pile foundation, cyclic loading, ratcheting, granular convection, densification.

1. Introduction

In the context of the offshore wind power generation, the installation of large monopiles of up to 8 metres in diameter and about 30 metres of embedded length is currently being considered feasible, and in some cases even necessary, for a safe foundation of the wind turbines on the sea-bed [1,2]. The reason for such extensive dimensions of the foundation stems from the need to withstand the harsh loading conditions imposed by the marine environment, which are characterised by large bending moments and relatively high ratios of lateral to vertical loads (see Figure 1) and typically involve large numbers of load cycles caused by the wind and sea waves (in the order of 109 cycles during the turbine’s service lifetime). However, there are still many open questions concerning the behaviour of such foundations and their cyclic interaction with the

surrounding soil (mainly sandy soils in the case of the North Sea). The main purpose of the experimental investigations reported here was to study the structural stability of laterally loaded offshore piles embedded in sand after the application of several millions of load cycles, and specifically to assess whether a brittle failure as in the case of axially loaded piles could take place or not. However, the quasi-static experiments have also shown some singular phenomena within the embedding soil, such as an endless granular flow and the formation of closed convective cells, which resemble fluid-like behaviour and hint towards the special properties of granular materials, often considered a fourth state of matter apart from solid, liquid and gaseous [3]. This paper quantifies to some extent such phenomena and discusses their origin, consequences and practical relevance for the offshore foundation.

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Fig. 1 Approximate dimensions of a 5-MW offshore wind turbine with monopile foundation and order of magnitude of the maximum loads during the 30-years return period storm in the North Sea

2. Motivation and scope

The effects of cyclic loading are only seldom taken into account explicitly during pile design, but it is well known and widely accepted that cyclic axial loading of piles can cause a so-called friction fatigue [4,5], which eventually can lead to a sudden failure of the pile. The origin of such phenomenon appears to lie on the cyclic densification (and hence contraction) of the soil at the pile-soil interface, which leads to a reduction of the radial stress exerted by the far field on the pile shaft and thus to a reduction in pile bearing capacity. This can have particularly serious consequences due to the potentially abrupt nature of such failure, where the progressive decrease of radial stress at the interface might remain unnoticed while the brittle failure might only happen after the application of several tens of thousands of load

cycles with a seemingly stable behaviour of the pile (see for instance [6] or [7]). In contrast, it is generally assumed that the cyclic lateral loading of piles in sand normally involves a rather benign behaviour of the pile: an attenuation of the cyclic displacements and only potential serviceability problems rather than an eventual loss of mechanical equilibrium (see the decreasing rate of cyclic permanent displacements shown for instance in [8-10]). However, such assumption is based mainly on experimental tests with a low number of load cycles (from the few tens or hundreds of cycles in the early investigations to the tens of thousands of cycles in recent works). Since the pile foundations in the offshore environment are exposed to millions of load cycles during their service life-time, the absence in the literature of empirical studies of lateral pile behaviour in such range of number of cycles prompted the performance of the physical model tests in reduced scale presented in this paper. The loading schedules, each of them including millions of cycles, were therefore aimed at elucidating whether an abrupt failure analogous to those shown by the cyclic axially loaded piles could also take place at some point in the long-term when the cyclic loading is applied laterally. In the course of the experiments, the authors observed consistently some progressive subsidence of the soil around the pile and a particular grain migration following convective patterns, which were presented and discussed in [11]. There, the origin of these phenomena were attributed respectively to two consecutive phases of soil deformation, the first one characterised by the densification of the sand under the effects of cyclic solicitation, and a second phase marked by the formation of convective ratcheting cells of soil around the pile-head, whereby the existence of two clearly distinct soil domains, namely the convected and static soil domains, and a limiting direct shear surface between them were proposed (see Figure 2). In this paper, the results from further experiments confirming these phenomena are presented and several open issues are

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addressed. Among others, a comparison of physical tests at different geometrical scales (1:100 and 1:30) is put forward and the limits of the densification phase and subsidence cone as well as the extents of the convected soil domain are discussed.

Fig. 2 Convected and static soil domains within the saturated sand surrounding a cyclic laterally loaded pile and close-up details of the sheared material at the transition surface [11] Furthermore, topographic measurements of the soil surface before and after the tests show that cyclic densification and soil improvement do effectively occur at some stage during the tests, whereby the differential soil volume between the original and final states indicates the averaged degree of densification. Finally, the shape of the direct shear surface at the transition between the static and convected soil domains is here investigated and discussed in the light of results from simulations with a numerical model of the pile-soil system.

3. Physical evidence

3.1. Physical testing in 1-g conditions

Physical testing in a reduced scale implies that, in order to be able to relate the results with the corresponding variables of the prototype system in real size, there are certain conditions that must be met so that the kinematic and

constitutive similarity between the systems is preserved. However, the satisfaction of such scaling conditions for similarity in granular systems is in general not a trivial matter (see e.g. [12,13]). Among other technical difficulties of physical tests with geomaterials, the direct scaling of the grain dimensions is particularly problematic, since it may introduce undesired cohesive forces into play. Moreover, the pressure dependence of the soil’s constitutive nature implies that, under 1-g (natural gravity) conditions, homologous points of the soil in the model and prototype will show different deformational behaviour since they are subject to different stress conditions. In order to preserve the constitutive similarity across the scales, special techniques need to be adopted so that the stress levels at homologous points in model and prototype are equal (e.g. the centrifuge testing, the fluid-gradient method or the pressure-vessel confinement, as shown for instance in [14,15]). Given the practical difficulty to fulfill simultaneously all the scaling laws and assuming a certain constitutive dissimilarity between model and prototype, the physical tests presented here were performed in 1-g conditions for the sake of simplicity and repeatability. The procedure to obtain the similarity relationships for physical tests is normally based on Buckingham’s Π-Theorem [16,17] and basically consists on identifying the quantities relevant for the physical relation under consideration and deriving suitable dimensionless combinations of variables Π that need to be equal in both scales (see for instance [18], [19] or the recent [20]). Based on the two independent ratios between model and prototype to be kept in the 1-g tests (ratio of geometrical lengths λ and ratio of gravity accelerations κ=1), the conservation of the combinations Π across the scales leads to the scaling rules for the rest of physical quantities, which are summarised in Table 1, extracted from [21] and [12]. The interested reader may find the specific dimensionless combinations Π used for these investigations in [22] and [23].

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Table 1 Summary of scaling laws for 1-g physical tests in relation to the geometrical scaling factor λ. Subscripts “M” and “P” denote the model and prototype scales respectively

Physical quantity Scaling law Dimensions

Length, L LM = LP / λ L [m]

Force, F FM = FP / λ3 F [N]

Distributed line load, q qM = qP / λ2 F / L [N/m]

Stress (or Pressure), σ σM = σP / λ F / L2 [Pa]

Unit weight, γ γM = γP F / L3 [N/m3]

Moment, M MM = MP / λ4 F x L [N m]

Bending stiffness, EI (EI)M = (EI)P / λ5 F x L2 [N m2]

Time, t tM = tP / λ1/2 T [s]

Frequency, f fM = fP / λ-1/2 1 / T [Hz]

3.2. Experimental setups and testing schedules

With the main goal of a qualitative assessment of the long-term behaviour of the foundation rather than the quantification of the complex phenomena associated to the cyclic pile-soil interaction, the authors have been carrying out physical tests at geometrical scales of 1:100 and 1:30 of a monopile prototype with 7.5 metres in diameter and about 30 metres of embedded length. The main magnitudes in the tests compared to those of the selected prototype are shown in Table 2, while the experimental setups are illustrated in Figure 3. The soil used for the tests was the Berliner sand, which is a narrowly graded sand with rounded grains whose granulometric properties are described in [24]. The sand was placed in a dense state (for all the tests approximately between 93% and 97% of the ASTM Proctor density) and then flooded with water until full saturation. For the open-ended hollow model piles, different materials were employed (PVC and Aluminium for the 1:100 and 1:30 models, respectively). The flexural stiffness of these piles was slightly higher than what it should have been according to the scaling laws presented before, but neither a softer material with the appropriate dimensions nor piles with

thinner walls were commercially available within reasonable cost margins. Furthermore, in view of the elevated loads required by the hydraulic jacks for the pile installation (up to 90 KN of axial loading) a strict scaling of the pile wall thickness would have also caused driveability problems (pile buckling). Finally, in order to investigate the characteristical grain migration towards the pile, several patterns of bands (each 2 cm thick) with coloured sand grains were placed on the soil surface, as reported and described in [11,24]. The average size of the coloured grains was about 2 mm. Concerning the cyclic pile solicitation, harmonic loading schedules with different amplitudes and average loads were applied, including one-way and two-way asymmetric sinusoidal loads at frequencies of 1 Hz and 0.5 Hz and maximum loads of up to 27% and 20% of the static ultimate capacity of the piles, respectively for the tests at 1:100 and 1:30 scale. The static capacity was eventually measured to range around 150 N at a pile-head displacement of 0.1 D for the model scale 1:100 and around 4.8 kN for the system at scale 1:30. Additionally, a force lever arm with respect to the pile-head level (i.e. the mudline) equal to the embedded length L was kept for all the testing programs. With reference to the scaling laws for 1g model tests presented in

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Table 1, such load magnitudes represent approximately the extremal lateral loading that an offshore prototype in the North Sea might suffer with a return period of 50 years (see for instance [1,2]). The loading frequencies were chosen for practical purposes aiming to avoid inertia effects and pore pressure variations, and incidentally correspond approximately to a prototype loading frequency of 0.1 Hz, or conversely to a wave period of 10 seconds, typical of the offshore environment. Given the relatively high permeability of these clean sands and the extremely short drainage paths to the surface in the models, these frequencies and the load magnitudes were considered to be low enough to disregard the possibility of any significant pore pressure accumulation within the soil. Being out of the scope of this paper, such extent has been investigated numerically for the instance of a full-size prototype in [25,24]. The different loading schedules for the 1:100 scaled model consisted in the application of 5 million load cycles each, which means that, with the frequency of 1 Hz, every test lasted for about 2 months. On the other hand, the model at the 1:30 scale was only subject to around a

million load cycles due to the elevated logistical expense associated to each test in such a scale.

3.3. Observed phenomena

Considering that the focus of this paper lies mainly on the physical phenomena taking place within the soil, this section is devoted just to the observed soil subsidence, grain migration and convective granular flow. Concerning the observed behaviour of the pile itself, here it may suffice to point out that the pile lateral displacement and inclination did consistently increase for all tests following an attenuating pattern (i.e. with an ever-decreasing but never vanishing rate of accumulation) while the amplitude of cyclic displacements always showed a certain decaying character. The interested reader may find further details about the evolution of pile displacements in [24], where a general empirical form for the accumulation of permanent displacements is proposed, including a pseudo-logarithmic-linear function of the number of load cycles featuring a term for the decay of the displacement rate in the long-term.

Table 2 Comparison of main dimensions and magnitudes between the foundation prototype and the physical models in reduced scales

Physical quantity Prototype Model tests 1:100 Model tests 1:30

PIL

E

Pile diameter, D 7.5 m 0.075 m 0.25 m

Pile’s embedded length, L 30 m 0.3 m 0.94 m

Pile’s bending stiffness, EI 2.5 x 1012 N m2 2 x 103 N m2 1.9 x 106 N m2

SO

IL

Sand grain size, d50 0.5 mm 0.5 mm 0.5 mm

Sand grain unit weight, γS 26.5 kN/m3 26.5 kN/m3 26.5 kN/m3

Sand permeability, k 2.5 x 10-4 m/s 2.5 x 10-4 m/s 2.5 x 10-4 m/s

LO

AD

S Horizontal load, H 15 MN -10 N to 40 N -0.4 kN to 1.0 kN

Lever arm, h 30 m 0.3 m 0.94 m

Loading frequency, f 0.1 Hz 1 Hz 0.5 Hz

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Fig. 3 Sketches of the experimental setups for the 1-g physical tests in reduced scale of a pile foundation under cyclic horizontal loading H and embedded in saturated sand. Prototype replication in geometrical scales of 1:100 (a) and 1:30 (b) respectively

3.3.1. Soil subsidence and densification

As advanced in the introduction to this paper, the cyclic lateral loading was observed to cause a pronounced subsidence in the immediate vicinities of the pile, while in the outskirts of the subsided cone the sand would typically emerge, forming two small hills with elongated crescent-moon shape, similar to those reported by Brown et al. in their field tests [26]. The cause of the progressive subsidence can be attributed to the grain rearrangement and, in general, densification of the sand triggered by the pile’s cyclic lateral displacements, whose existence and extent were verified by performing topographic measurements of the soil surface before and after the tests with a 3D structured-light scanner. In this respect, an averaged measure of the densification of the whole system accumulated during the cyclic loading was estimated as the difference between the heaved and subsided volumes, taking as a reference the original soil surface mapped before the test. An example of the scanned topography of the soil around the pile after the application of the cyclic loading is shown in Figure 4 for both test scales, where the colour field shows the

measured elevations relative to the position of the original soil surface. A plot of the surface profile along the loading axis and perpendicular to it illustrates some geometrical details of the subsided cone, as shown exemplarily in Figure 5 for the scale 1:100. There, it can be seen that the base of the subsided cone (excluding the emerged part) is an almost perfect circle, with a diameter of 3D in both directions. Such circular regularity of the cone is a remarkable feature, since both the lateral loading and the soil heave did only take place along a single direction and in an asymmetric way. That suggests the existence of out-of-plane grain displacements within the soil, which was later confirmed by the use of sand markers (see next section). Interestingly, the dimensions and geometry of the subsided cone were almost the same for all the tests, with the mere exception of the one-way loading test, which produced an elongation of the subsided zone along of the loading axis (i.e. an ovalization) with a maximum diameter of 7/3 D. This means that the extension of the densified region is significantly smaller for the one-way test as for the two-way counterparts, which would be consistent with the observations made by Brown et al. [26].

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Fig. 4 Subsidence and heave surveyed with a topographic scanner after the tests. Colour field shows measured height difference relative to the initial topography before the tests. Physical test in 1:100 (a) and 1:30 (b) geometrical scales respectively In order to quantify somehow the sand densification, a spatial integration of the surface elevations with respect to a given reference provides an estimate of the absolute volume of the soil for both the initial and final states. In none of the tests did the heaved volume add up to account for the whole subsided one, so the cyclic densification of such very dense sand did indeed take place (at least in an average sense) for all the tests. For the tests in the 1:100 scale, the absolute difference between the subsided and heaved volumes was measured to range between 550 cm3 and 620 cm3. As an order of magnitude, consideration can be made that, since the model pile had 7.5 cm in diameter and 30 cm in embedded length, 620 cm3 represents around 47% of the soil volume enclosed by the pile.

Finally, and concerning the temporal evolution of the subsidence, the fact that all of the tests tended to reach quickly a steady form and magnitude of the subsided cone, consistently within the first hundred thousand of cycles, reinforces the idea that the observed subsidence is mainly due to the cyclic densification of the surrounding soil, which after a certain number of load cycles reaches its maximum density, sooner or later depending on the magnitude of the loads. Afterwards, the soil’s void ratio in that zone would be practically unaffected by further cycles, regardless of their number and magnitude, and no more significant plastic volumetric strains would take place around the pile.

Fig. 5 Photogrammetric scan of subsided zone and measured surface profiles along the loading axis (B-B’) and perpendicular to it (A-A’)

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Fig. 6 Hardening effects of soil densification on the pile behaviour. (a): Evolution of amplitude of pile-head cyclic displacements for different tests at 1:100 scale. (b): Evolution of pile bending moments at half the embedded depth, compared with measurements from control strain gauges outside the soil This also agrees with the experimental fact that the amplitude of pile-head cyclic displacements tends to reduce until a certain point, after which it remains rather stable (generally at some point between N=104 and N=105 cycles, see Figure 6-a). That point would mark the moment where the soil around the pile has reached its maximum density, since then no more increments of soil stiffness would take place and hence no further reductions of displacement amplitude would be caused. This very same argument of an initial rapid densification of the soil around the pile and its consequent stiffening of the pile’s lateral behaviour is also supported by the obtained measurements of pile bending moments, which were recorded with strain gauges at a depth half-way between the soil surface and the pile-base (see Figure 6-b). The measured data shows a progressive reduction of bending moments at that depth (and exclusively there), which hints towards the increasing rigidisation of the upper layers of soil. The other two strain gauges, which were installed for calibration and control during the test and in theory should not be affected by the soil densification, show as expected a fairly constant level of bending moments, hence excluding an electrical drift of

the signal as explanation for the reduction in the measured moments.

3.3.2. Superficial migration

Noteworthily, a continuous grain migration towards the pile was consistently observed during all of the tests at both scales, taking place along with the local subsidence discussed in the previous section. Therefore, the tests were devised incorporating bands of coloured markers on the soil surface to analyse such “steady-state” grain migration and trace back its causes. The recorded images (see some instances in Figure 7) show that the laterally loaded pile acts effectively as a permanent attractor (or sink) at the soil surface and that the inflow of sand grains at the pile-soil interface starts immediately with the beginning of the cyclic loading. This way, with an initial migrating speed around a millimetre per hour, the first grain marker band (the red markers) would normally disappear within the first day of loading, while the rate of motion always tended to decrease progressively and eventually stabilize at an estimated average rate of a couple of millimeters per day (i.e. approx. 20 nanometres per load cycle).

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Fig. 7 Progression of the granular migration at the soil surface. State of the coloured sand markers at three different moments of a physical test in a 1:100 scale. Pictures taken from underneath the pneumatic actuator (i.e. the perspective line is coincident with the loading axis) By the end of the tests after 5 million load cycles, the particle migration was in all cases still active and at a rather constant speed, with markers from the previously swallowed coloured bands appearing back into the soil surface at some distance from the pile, and then being mixed and drawn back to the pile along with the rest of sand particles affected by the grain migration, clearly indicating the closed convective nature of the migration (see [11]). Interestingly, such endless pattern of grain movement towards the pile was always observed to happen radially in every direction of the soil’s surface plane, even along the orthogonal to the loading direction. The event horizon of the granular migration (i.e. the limit within which any particle at the soil surface would be attracted towards the pile) was observed at a distance of around 6 cm from the pile wall for the tests at model scale 1:100, and around 11 cm at the 1:30 scale. Sand particles beyond these limits did in

general not experience any substantial migration.

3.3.3. Convective granular flow

The drainage and excavation of the soil after the tests revealed in all cases the existence of two distinct domains within the soil that suggest a closed-cell convective pattern of grain migration, and, notably, also a dark transition band at the limit between the two domains (see Figure 8). The convected domain, located always beside the pile-head and right underneath the soil depression, featured in all cases a heterogeneous mixture of sand grains and coloured particles from the different marker bands. For the two-way tests at the 1:100 scale, it reached a depth of about a third of the embedded length along the pile-soil interface, while the one-way testing with the same loading amplitude seemed to produce a convective cell of smaller dimensions.

Fig. 8 Convective cells observed in physical tests at different geometrical scales. Soil excavation along the loading axis, after testing at the geometrical scales 1:100 (a) and 1:30 (b) respectively

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The very same phenomena were also observed in the larger 1:30 scale tests, where, incidentally, the convected domain reached the same proportional depth as in the smaller scale tests, namely a third of the pile’s embedded length (see Figure 8). The extents of the convected domain for the different tests are summarised in Table 3. Concerning the second soil domain, the static one, it encompassed the rest of the soil, from the far field soil right up to the limits of the transition band, and it was always characterised by a complete absence of coloured sand markers. Noteworthily, the transition surface separating the convected and static domains featured the presence of very fine dark particles which were generated by the direct shearing taking place between the moving and the static soil masses, whereby the abrasion occurring at the contact between sliding grains did wear off the paint-coating of the sand markers (see [11]). As regards to the three-dimensional geometry of the convected domain, it can be observed that the migrated markers and the dark transition surface were also present in vertical planes along directions other than the loading line, although only reaching shallower depths and narrower extensions. This is depicted by Figure 9 which shows several soil cuts along different vertical planes and the final

excavation of the convected soil domain right up to the transitional direct-shear surface. A remarkable feature is the absence of any convected material within the vertical plane transversal to the loading direction. There, the sand grains at the soil surface first migrate radially towards the pile, that is, moving within their vertical plane containing the pile. But once they reach the pile and penetrate into the soil at the pile-soil interface, they seem to experience an out-of-plane convection that deviates their flow towards the loading axis. The three-dimensional shape of the convected soil volume interpolated out of the different soil cuts is sketched in Figure 10. As a final observation, the sand grains within the convected domain presented consistently a notable size segregation and inverse grading, with the coarser particles being typically found in the upper parts of the domain (see Figure 11). This phenomenon of upward coarsening, also known as the “Brazil nut” effect, appears to be one characteristical consequence of the convective motion of granular matter (see e.g. [27-29]), which may be explained by the higher mobility of the smaller particles which are able to reach deeper regions and also owing to the fact that once the bigger particles reach the surface it is more difficult for them to enter again into the bulk of the soil.

Table 3 Extents of convective cells measured after different physical tests

Physical test Loading range Depth of convection Width of convection

1:100 ; Two-way -10 N to 20 N 9.5 cm ~ 6 cm (*)

1:100 ; Two-way -10 N to 30 N 10.5 cm 6 cm

1:100 ; One-way 0 N to 30 N 7 cm 5 cm

1:30 ; Two-way -400 N to 1000 N 30 cm (**) 11 cm

NOTES: Depths as measured below subsided soil surface; Widths as measured from pile surface.

(*): Estimated after static lateral capacity test. (**): The two convective cells were in this case asymmetric. The one on the pile’s side under higher loads (1 KN) featured a smaller cell, reaching a depth of about 20 cm (see Figure 8-b).

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4. Discussion

4.1. Mechanical interpretation

4.1.1. On the ratcheting motion

The convective flow of granular matter when it is subject to dynamic excitation (vibrations and relatively high accelerations) has been studied and described profusely since its first report in the 19th century by Michael Faraday, e.g. in [30-33,3]. However, the convective flow presented here has instead a quasi-static "ratcheting" nature, whereby, due to the low loading frequency and small pile-head displacements, the induced accelerations of the soil particles must have been negligible compared to the earth’s gravity. For the interested reader, a discussion on the

micromechanical origin of the granular ratcheting and its implications for the shakedown of cohesionless soils can be found for instance in [34] and [35]. The actual mechanism responsible for the ratcheting convection may be explained in the following manner: during the cyclic lateral loading of the pile, and every time that the pile moves back after a loading peak, a small gap opens at the pile-soil interface allowing the sand grains adjacent to the pile-head to move downwards along the interface. Once they reach a critical depth where the gap is not big enough and they cannot move further down, the sand particles would then be pressed into the soil by the advancing pile, moving a little bit further with every load cycle, in a ratchet-like fashion.

Fig. 9 Tomographic excavation of the convection cell through vertical planes along different radial directions and final excavation of the distinct transition surface between the convected and the static soil domains

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Fig. 10 Sketch of the ortographic projections of the convected soil volume reconstructed from the tomographic excavations of the soil A similar ratcheting motion of granular matter coupled with a certain convective vorticity has already been described in recent numerical studies at a micromechanical scale (see e.g. [36]), where it was also shown that, even under very small loading amplitudes, a large number of particle contacts can reach the sliding condition and produce irreversible deformations. Furthermore, they also provided evidence supporting that ratcheting “is a purely quasistatic effect”, independent of any accelerations or damping in the material. On the other hand, the idea of ratcheting behaviour has also been used by several authors in the field of soil mechanics, for instance in [37-39], to explain the progressive accumulation of plastic deformations of the soil under cyclic loads. Concerning the role of the water saturation in the observed phenomena, the presence of water

probably did enhance the soil migration and convective flow, since the effective stresses within the submerged soil matrix are lower than those within a dry material where there would be no buoyancy. This, in turn, permits a larger number of contacts between particles reaching the sliding condition, favouring thus the ratcheting displacements. On the other hand, it appears reasonable to expect these phenomena also to happen in dry sands, although probably to a minor extent, and perhaps even in partially saturated sands.

4.1.2. On the general shape of the convective trajectories

Considering the micromechanical particle slips and reorientations which occur under increasing shear stress as the main source of plastic deformation and ratcheting flow, the

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examination of the gradients of shear stress within the soil viewed as a continuum can provide some clues about the trajectories of the convected material. As explained before, the convective motion would start as a collapse of the loose sand grains at the soil surface down the open pile-soil interface. Then, driven by gradients of shear stress and towards areas of lower confining stresses, i.e. upwards, the migrating grains would move forward pushed by the following grains, setting in motion a

whole ratcheting convective cell within the pile-head vicinities. At the limit where the pile-soil interface remains “closed” and there is no inflow of new material, i.e. where the sand grains remain rather “static”, a direct shear would take place between the convecting and the static soil masses, since there the migrating grains would have to override the standing particles in order to move forward.

Fig. 11 Inverse grading (upward coarsening) of the convected material. Soil excavation after physical test at 1:30 scale

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Such interpretation of the convective mechanism seems to be somehow confirmed by the shape of the limiting direct shearing surface observed experimentally and by inspecting the isolines of shear stress within the soil during a loading peak, which can be obtained with a numerical continuum model (more details on the elastoplastic finite element model employed here may be found in [24]). As shown in Figure 12, the normals to the isolines of shear stress (i.e. the gradients of shear stress) reproduce fairly well the shape of the convected domain and even the

characteristical return towards the pile near the soil surface.

4.1.3. On the gap opening as triggering factor

Along with the cohesionless nature of the saturated sand, it is likely that the recurring opening of a gap at the pile-soil interface is the key element for the appearance of such grain migration and quasi-static convective cell, because it permits the downwards movement of the grains adjacent to the pile-head.

Fig. 12 Gradients of shear stress as mechanical explanation of convected geometry. (a): Isolines of shear stress obtained by FE analysis. (b): Magnification of soil region near the pile-head and comparison to experimental evidence (c)

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Therefore, the depth of the convection cell is probably determined by the magnitude of the opening gap and the relative size of the migrating grains. As proposed before, the sand grains would only migrate downwards at the interface just until reaching the depth where the gap is not big enough to let them through. This is also supported by the three-dimensional shape of the convected domain and particularly by the soil cuts performed at 90° off the loading axis. The decreasing depth of the convection cell within the vertical planes as they depart from the loading axis and the total absence of convected material in the plane perpendicular to the loading direction seem to agree with the idea of a strong (and perhaps even necessary) coupling between the ratcheting convection and the open interface, since the pile lateral loading in a single direction should cause little or no gap opening in the plane normal to it. At this point it is important to note that here the terms “gap” and “interface opening” do not imply an actual empty space between the pile and the soil, which in general should only take place in the case of cohesive or partially saturated soils, but rather a “relaxation” at the interface which is immediately filled by the cohesionless grains accompanying the receding pile. Such relationship between granular convection and open interface appears also to imply that, for a same level of maximum loading, a two-way cyclic lateral load should produce a grain migration of greater magnitude and a deeper convective cell than a one-way cyclic loading, since the two-way loading normally features a higher amplitude of cyclic displacement and hence causes a greater aperture of the pile-soil interface. The observations from the single one-way test performed here seem to indicate so (see Table 3). Similarly, some numerical simulations presented in [40], which were performed with a hypoplastic constitutive law, have shown how the two-way loading can produce some hardening of the surrounding soil and thus reduce the pile-head displacements, in contrast to a one-way cyclic lateral load causing mainly a progressive foundation inclination. The

authors of the simulations interpreted such hardening effect under two-way cycling as being conditioned by the slippage of soil at the interface after each loading peak, so again the triggering effect of the “opening” interface can be highlighted.

4.1.4. On the phases of soil deformation. Densification and convection

The empirical evidence discussed so far suggests a two-phase scenario of pile-soil interaction, where the saturated sand surrounding a flexible pile foundation, subject to a cyclic horizontal loading on the pile-head, undergoes two main distinct phases of deformation and grain displacement: an initial densification-dominated phase and a subsequent convection-dominated phase. The first phase, the densification phase, starts immediately after the first cycle of loading and is characterised by a progressive subsidence of the soil surface surrounding the pile. During this phase, the cyclic compaction of the soil due to the pile displacements causes a grain rearrangement and, in general, a reduction of inter-granular voids until the soil reaches its maximal density. This phase would also be characterised by a progressive reduction of the amplitude of cyclic pile displacement as a consequence of the hardening of the soil, and in general by a logarithmic accumulation of pile-head permanent displacements, as described for instance in [41] or [42]. The duration of this initial phase would be mainly influenced by the magnitude of the pile loading and particularly by the initial relative density of the soil. Although it may not be possible to define a clear temporal limit of this phase, for the presented tests it appears to have lasted for a number of cycles between N=104 and N=105. The same densifying explanation has been proposed by Gudehus for both the subsidence and grain migration near a cyclically axially loaded pile or wall, where the cyclic shearing would cause a net contraction of the soil near the structure, thereby producing a flow of material towards the latter and downwards “until the maximal density of the soil is

16

reached” [43]. After most of the densification has taken place, and once the soil depression reaches a rather constant depth, a second phase starts, namely the convection-dominated phase. During this endless phase, rather than producing further densification of the soil, the cyclic lateral movements of the pile would mainly cause a convective ratcheting displacement of the sand particles. Such a convective flow explains why despite the permanent inflow of material into the pile-soil interface no more significant densification of the soil is taking place. This second phase would also be characterised by a fairly constant amplitude of cyclic pile displacements and in general by an over-logarithmic accumulation of permanent displacements, leading eventually to an incremental collapse of the pile in the sense of an ever-decreasing but never vanishing rate of accumulation of permanent displacement. It appears reasonable to assume that in most cases these two phenomena, densification and convection, are not completely decoupled. Some convective grain migration probably takes place simultaneously with the densification during the first loading cycles and, reciprocally, some degree of further densification might also occur during the convection-dominated phase, as the densified soil would approach its maximum density asymptotically. The existence of two such kinds of deformation regimes in cohesionless soil samples subjected to cyclic loading has been confirmed both theoretically and experimentally before and seems to suggest that such phenomena are inherent to the behaviour of granular materials rather than just related to this specific soil-structure system. The numerical micromechanical investigations of discrete granular packings by Alonso-Marroquin et al. did indeed show short-time regimes featuring a fast accumulation of plastic deformation and long-time ratcheting regimes with slow rates of plastic deformation, along with the formation of ratcheting convecting vortices [36]. A clear analogy between such micro-mechanical effects and the two stages of macro-mechanical

pile-soil interaction proposed here may be drawn. On the empirical side, the experimental results presented by Bobryakov et al. [44] also showed two stages of soil deformation in their particular system (a retaining wall and a raking slope of confined sand in plane strain conditions), where they also report an initial non-stationary phase with densifying character during the first few load cycles and a subsequent stationary stage “with steady state material density and unchanging specimen surface”.

4.2. Practical relevance and extrapolation to other conditions

A particularly striking conclusion, if yet unconfirmed, is that if the results of these tests were to be up-scaled back to a prototype-size, it would mean that the granular convective flow could reach a depth of up to 10 meters. However, the fact that so far the convected domain seemed to hold the scale between the physical tests at different scales has to be regarded with caution, since these model tests have been carried out under very particular 1g conditions and not all the appropriate scaling laws have been strictly followed. Moreover, the loading schedules applied to the piles during the tests included large numbers of load cycles of very high amplitude, proper of extreme conditions, which do not represent the variable loading that a real offshore foundation might be normally exposed to. Therefore, the extent to which these phenomena take place around real-size offshore prototypes remains unclear. In any case, considering the apparent direct dependence of the convective flow with the cyclically opening interface, it may be possible to relate the depth and extent of the convected soil domain to the different design parameters of the pile foundation (flexural stiffness, service loads, grain size distribution, etc...) so that an extrapolation to real pile dimensions can be made and a clearer picture of the pile bedding emerges. The main tasks in this respect would involve the determination of the opening depth at the pile-soil interface and the

17

assessment of the gradient path of shear stress at that depth. Relative to the possibility of soil softening or even liquefaction within the convective sand around the offshore pile, the effects of any excess pore water pressure have been disregarded in the investigations so far, but they might not be negligible for the case of real-scale large-diameter monopiles, as shown in [24]. In such cases, the lengthy drainage paths imposed by the large pile can lead to a progressive accumulation of the transient excesses of pore pressure caused by the cyclic loading, and thus to a reduction of the effective stresses within the soil matrix. This, in turn, would certainly enhance the mobility of the single grains and any convective motion, since the contacts between the grains could then reach the sliding condition easier. Another practical consequence might concern the scouring protection systems for offshore operations. Even though the phenomena reported here are not related to the scour that often occurs in under-water foundations, they might interfere with any counter-measures against it, since the soil subsidence and grain migration will probably still occur due to the densification and gap-openings caused by the cyclic loading. Therefore, a re-examination of the current scouring-protection systems (see e.g. [45,46]) taking into account these issues might help to assess their suitability and improve their efficiency. Finally, it is important to remark the possible impact of the discussed progressive densification on the dynamic behaviour of the turbines. The shift of the foundation's eigenfrequencies in the long-term due to the densification and hardening of the soil may have significant consequences for the turbine’s operational thresholds if resonance phenomena are to be avoided.

5. Conclusions and outlook

The physical phenomena of macromechanical densification (i.e. an overall reduction of intergranular voids) and convective granular flow have been shown to take place in the

saturated sand surrounding a pile foundation under quasistatic cyclic lateral loading. A certain quantification of these phenomena, which were consistently observed in several model tests at different geometrical scales under 1-g conditions, has been provided here. Relative to the progressive sand densification around the pile, the three-dimensional shape and magnitude of the subsidence cone as well as the overall volume loss have been measured with a topographic scanner, while its consequential hardening effect on the foundation’s stiffness has been shown by the amplitude reduction of the cyclic pile displacements and its concurrent abatement of bending moments in the embedded pile. This progressive stiffening effect of the soil around the pile may lead to a shift of the pile’s eigenfrequencies and produce a significant change in the foundation’s susceptibility to dynamic resonance under the external loading, as already discussed in [47]. A persistent migration of the sand grains at the soil surface was observed taking place concomitant with the sand densification and subsidence, but lasting beyond them, with an initially decreasing speed of migration which eventually stabilised in all cases to a rather constant rate of motion. The grain movements were driven in all cases towards the pile and in every radial direction, with the pile acting as a permanent attractor with an event horizon contained within the subsidence cone. Such endless migration happens to be just the visible effect at the soil surface of a couple of convective cells of sand that surround the pile-head down to a certain depth. The quasi-static ratcheting nature of the convective motion originates from the periodical gap-opening (or, strictly speaking, stress relaxation) and material inflow at the pile-soil interface every time the pile recedes after a load peak, the downwards movement of the sand grains along the “relaxed” interface and finally their eventual push-through into the soil mass once they cannot move further down. The convective cells present a distinct limiting surface where a direct shearing of the convecting material takes place and whose shape can be approximated

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reasonably well by the gradient paths of shear stress. The nature and implications of these phenomena may be further clarified with the help of meshless numerical models or in the frame of a discrete element (DEM) approach, so that a simulation of the grain migration and soil convection can be performed. That way, an analysis of the results employing appropriate discrete averaging techniques could also provide a better insight into the shear and volumetric strain fields, and hence to the degree of densification and the extents of soil subject to it. On the other hand, it should also be possible to characterise more accurately the migration at the surface or in the vertical symmetry plane by means of PIV techniques (particle image velocimetry, see e.g. [48]), which also provides a basis for the experimental determination of the strain fields, as done for instance in [49]. The significance of these phenomenological investigations can be highlighted by the fact that the upper parts of the pile foundations are the most relevant for their lateral bearing capacity. A better understanding of the physical processes inside the soil and of the pile-soil interaction at shallow depths might help improve the design and safety of future foundations.

Acknowledgements The German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU) has kindly provided the funding for these investigations, which were approved by the Project Management Organisation Jülich (PTJ) and carried out in the frame of the RAVE (research at alpha ventus) research program. Special thanks are also due to Fred Ziegler and Wilfried Wuttke for their helpful assistance and technical expertise with the physical tests.

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