Effect of Stress Level on Response of Model Monopile to Cyclic Lateral Loading in Sand
Publication: Journal of Geotechnical and Geoenvironmental Engineering
Volume 147, Issue 3
Abstract
Monopile foundations supporting offshore wind turbines are exposed to cyclic lateral loading, which can cause accumulated pile displacement or rotation and evolution of the dynamic response. To inform the development of improved design methods, the monopile’s response to cyclic lateral loading has been explored through small-scale physical modeling at and in the centrifuge, as well as at large-scale in the field. There are advantages and disadvantages to each physical modeling technique, and the response may be most efficiently explored through a combination of modeling techniques. However, stress levels vary significantly between these techniques, and only centrifuge testing can simulate full-scale stress levels. This paper explores the effect of stress level on the response of a monopile foundation in dry sand to monotonic, unidirectional cyclic and multidirectional cyclic lateral loading with small-scale tests at and in the centrifuge at and . With an identical setup at each , stress-level effects were isolated. Qualitatively, the responses are similar across the stress levels, but some important quantitative differences are revealed. In particular, the rate of accumulation of pile displacement and the rate of change of secant stiffness under cyclic loading are found to reduce with increasing stress level. The results highlight the need to simulate full-scale stress levels to thoroughly understand foundation behavior, but also demonstrate the qualitative insight that can be gained through physical modeling. The data and trends presented in this paper provide input for the modeling of monopile responses at different stress levels.
Introduction
Monopiles are the principal foundation solution for offshore wind turbines (OWTs) in shallow to medium water depths and currently support 87% of OWTs in Europe (Wind Europe 2018). The response of these large-diameter hollow steel piles has been the subject of much recent research because their geometry and load regime differ from piles traditionally used for offshore oil and gas and onshore applications. Monopiles are subjected to long-term multidirectional cyclic lateral loading caused by a combination of wind, waves, and current loads, which has been shown in model tests to cause accumulated displacement or rotation of the foundation (e.g., Li et al. 2010; Truong et al. 2019) and evolution of foundation stiffness (Leblanc et al. 2010) and energy dissipation (Abadie et al. 2019). Accumulated rotation of the foundation is of particular concern because turbine manufacturers specify strict rotation limits, and changes to foundation stiffness and energy dissipation impact the dynamic response of the foundation and can enhance dynamic amplification of loads and increase fatigue damage of the OWT structure (Bhattacharya 2014).
Given that monopiles may be up to 8 m in diameter (Sørensen et al. 2017), with the latest designs being even larger, it is neither practical nor economic to test monopiles at full scale. However, physical modeling—at in the centrifuge and at large (but reduced) scale in the field—has been used extensively to explore the response of a monopile. Because cyclic tests often involve many thousands of cycles and need to be performed under load control, many cyclic campaigns have been performed at where full-scale stress levels are not simulated (e.g., Leblanc et al. 2010; Nicolai and Ibsen 2014; Arshad and O’Kelly 2017). Some studies have explored a monopile’s cyclic response in the centrifuge, simulating full-scale stress levels, although the number of cycles is limited (e.g., Li et al. 2010; Klinkvort and Hededal 2013; Truong et al. 2019). Large-scale field testing has also been performed to understand a monopile’s response (principally under monotonic loading) (Byrne et al. 2017). These tests provide valuable insight with stress level, grain size, and installation method closer to that at full scale; however, large-scale tests are very costly and there is intrinsic residual uncertainty about the soil conditions at each pile location.
Understanding a monopile’s response to realistic multidirectional cyclic lateral loading is necessary for the development of new design methods for the next generation of monopiles. Given the complexity of the problem, it may be most efficiently explored through a combination of tests, centrifuge tests, and large-scale field tests. However, stress levels vary significantly among these test types, and only centrifuge tests are able to simulate the full-scale stress regime. Understanding the impact of stress level on a monopile’s cyclic response is therefore an important step in bringing together observations from various test types. Other parameters such as grain size and installation method will also vary among these test types, but this work focuses exclusively on stress level.
The fundamental behavior of sand depends on the stress level. For example, dilatancy and peak friction angle increase approximately logarithmically with reducing stress level (Bolton 1986), whereas the maximum shear modulus increases as a power function of stress level, with the exponent typically 0.5 [first reported for angular grains by Hardin (1965), with a more recent review by Oztoprak and Bolton (2013)]. This stress-level-dependent behavior is reflected in the global foundation response, as observed by Ovesen (1975) for footings and Klinkvort (2012) for monotonically laterally loaded monopiles.
Few studies have explored the impact of stress level on a monopile’s response to cyclic lateral loading despite the prevalence of testing in this area. Rudolph et al. (2014a, b) reported a decrease in accumulated displacement for centrifuge tests compared with tests, and Nicolai et al. (2017) compared postcyclic behavior between and centrifuge tests. However, the setup and sand type differed between and centrifuge tests in these studies, and so stress-level effects were not completely isolated.
This paper extends the work of Richards et al. (2019) and specifically addresses the important issue of stress level, presenting the response of a monopile in dry sand to monotonic, unidirectional cyclic and multidirectional cyclic loading at three -levels. Tests were performed in the beam centrifuge at the University of Western Australia (UWA) at and , with corresponding tests performed on the laboratory floor. By varying the -level, effective unit weight and stress level were varied; with an identical setup at each -level, stress-level effects were isolated from other phenomena.
Experimental Setup
Sample Preparation
UWA superfine (SF) silica sand obtained from Sibelco, Perth, Australia was used for these tests, with properties as summarized in Table 1. Samples were prepared by air pluviation into a square strongbox with base dimensions of to a sample height of 376 mm. Three dense samples were prepared with an average unit weight (average relative density ). The samples were prepared dry to ensure a fully drained response during pile loading.
Property | Notation | Value | Unit |
---|---|---|---|
Specific gravity | 2.67 | — | |
Particle size | 0.12, 0.18, 0.19 | mm | |
Minimum effective unit weight | 14.69 | ||
Maximum effective unit weight | 17.40 | ||
Maximum void ratio | 0.78 | — | |
Minimum void ratio | 0.51 | — | |
Critical state friction angle (triaxial testing) | 31.9 | Degrees |
Source: Data from Chow et al. (2018).
The piles were installed wished-in-place during the sand raining process. Although it is acknowledged that this installation method is not representative of full-scale monopile installation and that the pile installation method can affect the lateral stiffness and bearing capacity (e.g., Klinkvort 2012; Fan et al. 2019), this method avoids the introduction of complex stress fields and local density changes through in-flight (or ) installation, which may scale differently from the postinstallation response investigated here.
Sand was first rained to the depth of the pile tip before hanging the piles in position and raining around them. The sand surface was then vacuumed to achieve an average pile embedment . Nine piles were installed per strongbox, with a minimum center-to-center spacing of . This layout was chosen to avoid significant boundary effects while maximizing the number of tests per strongbox sample; the center-to-center spacing is in line with previous studies (e.g., Leblanc et al. 2010; Abadie 2015; Albiker et al. 2017).
Cone penetration tests (CPTs) were performed using a cone to characterize each soil sample. For Sample S1, used for testing, two CPTs were performed at each stage: (a) before testing, and (b) after testing. For Samples S2 and S3, used for testing at and , respectively, one CPT was performed at each stage: (a) at before spinning, (b) at before testing, (c) at after testing, and (d) at after spinning down. Fig. 1 presents the CPT profiles measured at each -level. At , the S1 CPTs and prespinning S2 and S3 CPTs show good consistency, in line with the small variation in global unit weight measured across the samples. The consistency of the pretesting and posttesting S2 and S3 CPTs also gives confidence in the homogeneity of the sample and insensitivity to repeated spin cycles associated with each monopile test. The postspinning S2 and S3 CPTs show increased CPT resistance, which may be attributed to overconsolidation effects given that these samples previously experienced higher stress levels (Gui et al. 1998; Roy et al. 2019). However, no monopile tests were performed on these overconsolidated samples.
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Model Pile, Loading, and Instrumentation
Tests were performed on sandblasted mild steel piles with properties summarized in Table 2. The small ratio is representative of modern full-scale monopiles (Sørensen et al. 2017; Schroeder et al. 2015), whereas the large ratio is expected to be representative of operational loading conditions (Richards 2019) and ensures a rotation-dominated response. The surface is classified as smooth on the basis of the normalized roughness (e.g., Hu and Pu 2004), in contrast to typically rough full-scale monopiles. However, roughness is likely to have only a small impact on the pile’s lateral response (Klinkvort 2012). The pile has a closed end, which is appropriate given the wished-in-place installation method. Although full-scale monopiles are open-ended, the contribution of base shear and base moment is typically small (Burd et al. 2020). The pile wall thickness is selected to ensure rigid behavior at all three stress levels; such rigid pile behavior is expected to be representative for modern monopiles (e.g., Abadie 2015).
Property | Notation | Value | Unit |
---|---|---|---|
Outside diameter | 42 | mm | |
Target embedded length | 168 | mm | |
Length to diameter ratio | 4 | — | |
Wall thickness | 3.2 | mm | |
Eccentricity | 424 | mm | |
Eccentricity to diameter ratio | 10 | — | |
Mean surface roughness | 3 | ||
Normalized roughness | 0.016 | — |
Fig. 2 shows the loading and instrumentation apparatus, which was modified from that designed by Herduin (2019) for multidirectional loading of anchors. Actuators sit perpendicular to Platform A and apply load to the pile via cables, which travel around pulleys on Platform C and attach to in-line load cells at the base of the pile stickup. Pile displacement was measured with six string potentiometers. Three were positioned 253 mm above the load application point (on Platform B), and three were positioned 30 mm above the load application point (on Platform C). Each triplet of string potentiometers was arranged in a 120° star to attempt to eliminate the net load applied to the pile by these sensors because each applies a tensile load of in the direction of action. The pile stick-up allowed straightforward connection of the string potentiometers and load lines to the pile. The stick-up consists of an insert that is fixed inside the pile head and a threaded rod to which the potentiometers and load lines were attached.
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The setup shown in Fig. 2 is for multidirectional loading, with three actuators and associated load lines positioned 120° apart. For unidirectional loading, only two actuators were used, positioned 180° apart, to simplify the setup. All cyclic tests were performed under load control with sinusoidal waveforms, whereas initial monotonic tests were performed under displacement control. Under load control, all load lines were kept in tension, with one or two lines typically holding constant load and one line applying cyclic loading.
To resolve pile position, vertical displacements were neglected and the planar position of the pile at the level of each triplet of string potentiometers was calculated. Under unidirectional loading, pile displacement was obtained directly from the string potentiometer aligned with the loading direction. Under multidirectional loading, measurements from all three potentiometers were used, and to account for the measurement redundancy, the error between actual and measured length was assumed to be equal for all three string potentiometers. Net loads were obtained directly under unidirectional loading, whereas under multidirectional loading, they were resolved to account for the current pile position and load line angle.
Methodology
In this study, the stress level was controlled by varying the model -level using the centrifuge. With the same setup at each -level, stress-level effects were isolated from other variables. The were chosen to vary logarithmically in line with the expected logarithmic variation of dilatancy with stress level.
The vertical effective stress at 70% pile embedment was used as a reference effective stress level () and as a proxy for the mean effective stress . For the model and sand sample () used, Table 3 presents the reference effective stress level and normalized effective stress level () corresponding to each . Table 3 also presents the prototype monopile diameter simulated at each , assuming a dense saturated sand at prototype-scale (). Fig. 3 shows how the model tests translate to prototype scale. Although modern monopiles are even larger () than that simulated by the tests, the range of stress levels is sufficiently large to observe relevant stress-level effects.
, | (kPa) | (m) | |
---|---|---|---|
1 | 2 | 0.02 | 0.071 |
9 | 18 | 0.18 | 0.64 |
80 | 162 | 1.60 | 5.7 |
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Modeling of models campaigns (e.g., Ovesen 1975; Dewoolkar et al. 1999; Klinkvort et al. 2013) are more common than stress-level investigations and involve tests at various -levels and model sizes to simulate the same prototype response to explore scale effects or verify centrifuge modeling techniques. An example modeling of models campaign performed by Klinkvort et al. (2013) is shown in Fig. 3 for context.
Monotonic Response
Fig. 4 presents the response for the monotonic tests alongside the responses for the initial loading portion of the unidirectional cyclic tests. Fig. 4 highlights the significant variation in response as the stress level is varied, as well as the consistency of behavior at each stress level.
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Data are presented in terms of applied horizontal load and pile displacement at the load application point (both at model scale). The imperfectly rigid connection between the pile and pile stickup introduces some error into the resolved pile rotation and prevents presentation of the response in terms of applied moment and pile rotation . However, conclusions on stress-level effects are not expected to be affected by the choice of either work-conjugate pair.
Normalization Approaches
Casting the monopile’s response in dimensionless form facilitates comparison of tests conducted across this large range of stress levels and can help provide insight into stress-level effects. Three different normalization approaches are considered following (1) Klinkvort et al. (2013), (2) Klinkvort (2012), and (3) Leblanc et al. (2010), as summarized in Table 4.
Parameter | Normalization 1(Klinkvort et al. 2013) | Normalization 2(Klinkvort 2012) | Normalization 3 (Leblanc et al. 2010; Abadie 2015) |
---|---|---|---|
Load | |||
Displacement | |||
Fig. 5(a) shows the result of applying Normalization 1, which accounts for variation of (but not for changes in stiffness or angle of friction) to the monotonic responses. This normalization therefore highlights stress-level effects: the decrease in normalized initial stiffness and inferred normalized capacity with increasing stress-level is indicative of stress dependent stiffness and dilatant behavior.
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When exploring stress level effects, Klinkvort (2012) proposed a further approach (Normalization 2), which introduces Rankine’s passive earth pressure coefficient to attempt to account for stress dependent changes in peak angle of friction . Klinkvort (2012) used this normalization to successfully collapse the response of a centrifuge model monopile at stress levels from to , with values obtained from complementary triaxial testing. Complementary triaxial tests are not available for this study, and moreover, obtaining definitive values at stress levels corresponding to the tests (2 kPa) is likely to be very challenging. A number of relationships exist that allow to be estimated for a given stress level and relative density (Bolton 1986, 1987; Chakraborty and Salgado 2010), but there is disagreement at low stress levels.
The relationship proposed by Bolton (1987), which limits dilatancy at low stress levels, is employed in this work. This relationship is supported by the work of e.g., Tatsuoka et al. (1986) and the recent investigation into the behavior of Leighton Buzzard sand by White (2020), which found that although the shear strength properties (e.g., ) tend to increase with decreasing effective confining stress, this effect becomes increasingly minor for confining stresses less than approximately . However, other studies (Ponce and Bell 1971; Quinteros et al. 2017) have presented conflicting results. Employing the relationship of Bolton (1987) to estimate for Normalization 2, changes negligibly over the stress range of concern here, and the plot resembles Fig. 5(a).
In contrast to the approach of Klinkvort (2012), Leblanc et al. (2010) incorporated stress dependent stiffness into Normalization 3, but did not consider dilatancy (i.e., stress-level-dependent changes in angle of friction). Stress dependent stiffness is incorporated by assuming . The exponent is chosen as , which is generally accepted at small strains () i.e., (e.g., Hardin 1965). At larger strains, (Oztoprak and Bolton 2013). Fig. 5(b) shows the results of applying Normalization 3. The normalization successfully collapses the results toward a single curve, particularly at the small displacements that are relevant to the cyclic tests and which are expected to be controlled by small strain stiffness.
Together, the normalizations indicate the presence of stress-level-dependent stiffness and possibly stress-level-dependent dilatant behavior, although it is not possible to decouple these phenomena in this study. The normalization approach of Leblanc et al. (2010) is adopted here for comparison of responses across stress levels.
Maximum Stiffness
The monopile’s maximum tangential stiffness is expected to vary with stress level in the same way as the soil’s maximum shear modulus ; indeed, this assumption is incorporated into the normalization approach of Leblanc et al. (2010).
The initial loading stiffness is the obvious choice for estimating , but there is significant variability in this value, probably due to bedding-in effects associated with the wished-in-place installation method. Maximum tangential stiffness values obtained from the first unloading portion (available for all cyclic tests) are more consistent and are therefore used here, although they may be affected by densification during initial (albeit small-amplitude) loading.
Fig. 6 presents the variation of with stress level. Here, is obtained by manual fitting to all relevant tests, and the range and mean values of are indicated. A power function fits the variation of mean with stress level well, as shown by the dotted line in Fig. 6. However, the exponent obtained from least-squares fitting is , somewhat lower than the for that is implicit in the Leblanc et al. (2010) normalization. The cause of this discrepancy is unclear, but might be explained by experimental variability, the impact of stiffening during the initial loading portion, or three-dimensional effects when considering the monopile system. Using in place of in Normalization 3 produces a poorer result.
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Cyclic Definitions
Loading
Constant-amplitude cyclic loading can be characterized by two parameters (Leblanc et al. 2010) as follows:where and are the maximum and minimum loads applied in each cycle, respectively; characterizes the load symmetry ( for one-way loading, for symmetric two-way loading, and for constant loading); and and together describe the load amplitude, relative to an arbitrary reference load , where is defined at a reference pile displacement value at the load application point. This reference displacement approximately corresponds to 2° pile rotation () and pile displacement at the mudline, which may represent an ultimate failure criteria; similar reference values have been used in studies by Abadie et al. (2015) and Arshad and O’Kelly (2017). Table 5 summarizes the monopile’s reference load values at each -level obtained from initial monotonic tests.
(1)
(2)
, | (N) |
---|---|
1 | 15.7 |
9 | 99.5 |
80 | 597 |
The average load amplitude and cyclic load amplitude are also useful parameters that can be used to describe the cyclic response.
Cyclic Response
The monopile’s cyclic response in dry sand is characterized in terms of permanent accumulation of displacement (ratcheting), evolution of secant stiffness, and evolution of energy dissipation per cycle.
Ratcheting is interpreted in terms of accumulated mean displacement at the load application point (), following the approach of Richards et al. (2019), but with displacement instead of rotation as the strain variable.
Secant stiffness is defined at the center of the cycle (shown in Fig. 7 for Cycle 1) to minimize conflation of stiffness change with ratcheting. This stiffness is the inverse of the average of the loading and unloading flexibilities and may be expressed in terms of the secant loading stiffness and secant unloading stiffness , (each indicated in Fig. 7) as follows:
(3)
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The energy dissipation is interpreted in terms of an energy loss factor proportional to the ratio of hysteretic energy loss to maximum stored energy per cycle (Inman 2014) as follows:where the maximum stored energy per cycle can be related to the secant stiffness with:
(4)
(5)
For nonclosing hysteresis loops (due to ratcheting), the hysteretic energy loss across a cycle is defined as follows:where the reversal () and maximum () points are indicated in Fig. 7. The energy loss factor becomes:
(6)
(7)
Unidirectional Cyclic Response
Test Program
Table 6 summarizes the unidirectional cyclic tests conducted as part of this campaign. One-way () and two-way () tests were conducted, and the majority of tests involved 1,000 initial loading cycles followed by a reload to to explore the postcyclic reloading response. The tests are characterized in terms of nominal and applied values of and . The nominal values are those demanded of the control system, and the applied loads are obtained from the in-line load measurements; the discrepancy is small and reduces with increasing load level as the relative size of temperature effects and the load cell signal-to-noise ratio reduces.
Test | , | No. of cycles, | Reload | ||||
---|---|---|---|---|---|---|---|
Nominal | Applied | Nominal | Applied | ||||
U.1.A | 1 | 0.4 | 0.38–0.40 | 0 | 1000 | Yes | |
U.1.B | 1 | 0.3 | 0.28–0.30 | 0 | 770a | Yes | |
U.1.C | 1 | 0.2 | 0.18–0.17 | 0 | 1,000 | Yes | |
U.1.D | 1 | 0.4 | 0.40–0.42 | 500b | No | ||
U.9.A | 9 | 0.4 | 0.39–0.40 | 0 | 1,000 | Yes | |
U.9.B | 9 | 0.3 | 0.29–0.30 | 0 | 1,000 | Yes | |
U.9.C | 9 | 0.2 | 0.19–0.20 | 0 | 1,000 | Yes | |
U.9.D | 9 | 0.4 | 0.39–0.40 | 1,000 | Yes | ||
U.80.A | 80 | 0.4 | 0.39–0.40 | 0 | 1,000 | Yes | |
U.80.C | 80 | 0.2 | 0.20–0.20 | 0 | 1,000 | Yes | |
U.80.D | 80 | 0.4 | 0.40–0.40 | 1,000 | Partial |
a
Final 230 cycles of Test U.1.B excluded as they appear to be anomalous, probably due to temperature effects, which were significant at .
b
Only 500 cycles completed for Test U.1.D due to problems with load control.
Load-Displacement Response
Fig. 8 presents the load-displacement (H-u) responses for the tests summarized in Table 6, alongside the corresponding monotonic responses (). The full responses are shown for one-way tests , , and , but only the first five cycles of two-way tests are shown to highlight the loop shape. The responses are normalized by the load and displacement reference values ( and ) to present data from all three stress levels on comparable axes. The results show good repeatability at each stress level and qualitatively similar behavior across the three stress levels, although it is clear that the responses become more linear with increasing stress level. This variation in linearity is linked to stress dependent stiffness and dilatant behavior, as previously discussed, as well as the relative load amplitude. To better match the linearity of the cyclic responses across the stress levels, the reference pile displacement —which determines and therefore the cyclic amplitude —may be adjusted with stress level, perhaps employing the dimensionless framework of Leblanc et al. (2010). However, the variation in backbone linearity is a stress-level effect, and it is not clear whether it is appropriate to eliminate this effect by modifying .
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Fig. 8(b) also reveals evidence of gapping-like behavior in tests U..D at and to a lesser extent at . The inflexion in the load-displacement response around zero load is indicative of a gapping-type response because the tangent stiffness would reduce significantly when a pile traverses a gap. Similar load-displacement responses were recorded during field testing in both sand and clay as part of the pile soil analysis (PISA) project, where gapping was also observed on site (Beuckelaers 2017). Gapping was not observed during these tests, but the setup was not designed to record gap formation. A gap of only a fraction of a millimeter would be sufficient to account for the observed behavior at .
For these tests, the gapping-like behavior reduces with increasing stress level. Some cohesion is necessary for gap formation, and so it is not generally expected in dry sand. However, the tendency for sand particles to move into a gap will reduce with reducing stress levels. At very low stress levels, electrostatic effects or ambient moisture may then provide enough cohesion for gap formation. The wished-in-place installation will also lead to very low horizontal stresses adjacent to the pile, increasing the likelihood of gap formation.
Ratcheting Response
Fig. 9(a) presents evolution of normalized accumulated displacement () with cycle number () for the one-way unidirectional cyclic tests; a power law [Eq. (8)] is also fitted to each test response and shown as a dashed line. Various studies have shown ratcheting to evolve approximately as a power law with cycle number (Leblanc et al. 2010; Klinkvort 2012; Abadie 2015; Albiker et al. 2017; Truong et al. 2019), and a power law is also suitable here at all stress levels:
(8)
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The power-law coefficient has been found to vary with and (Leblanc et al. 2010; Nicolai and Ibsen 2014), and the power-law exponent has often been reported as a constant within individual studies (Leblanc et al. 2010; Nicolai and Ibsen 2014; Albiker et al. 2017). However, Truong et al. (2019) suggested that varies with and . This paper investigates only a single value of and .
No dependence of on stress level is observed, although Fig. 9(b) shows how varies with as a power law, in line with Abadie (2015) as follows:
(9)
Conversely, no dependence of on is observed (in agreement with previous studies), but there is a trend for decreasing with increasing stress level, as shown in Fig. 9(c). To facilitate prediction of behavior at other stress levels, the ratcheting exponent is presented in Fig. 9(c) against normalized reference stress level rather than -level.
A logarithmic trend line is fitted to the data in Fig. 9(c) to quantify the variation of with stress level and is shown as a dashed line. The line is defined by:
(10)
This trend line suggests that for a full-size monopile in dense saturated sand (, , and ), the ratcheting exponent may be as low as 0.11, approximately half of the value in the model tests.
The value of appears to vary with the chosen strain variable, and it is therefore difficult to compare values from independent tests campaigns to build confidence in this conclusion on stress dependency; for example, Albiker et al. (2017) found with pile rotation as the strain variable and with pile displacement (some distance above mudline) as the strain variable. However, the results presented here appear to be consistent with the increase in normalized pile displacement for test compared with centrifuge tests reported by Rudolph et al. (2014a, b).
Secant Stiffness Response
Fig. 10(a) presents evolution of the monopile’s secant stiffness with cycle number for all unidirectional cyclic tests. Stiffness is normalized by the maximum stiffness at each , obtained from the first-cycle unloading portion. An increase in secant stiffness with cycle number is observed for all tests, with plateauing at high cycle number in all tests except U.1.D.
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The initial normalized secant stiffness [] is plotted in Fig. 10(b). There is no strong dependence of general initial normalized secant stiffness values on stress level; however, the spread of initial stiffness values decreases with increasing stress level, consistent with the increasing linearity of the responses. At each -level, the initial stiffness values generally decrease with cyclic amplitude , as expected, given the nonlinear load-displacement response.
Evolution of secant stiffness with cycle number has previously been described by logarithmic functions (Klinkvort and Hededal 2013; Leblanc et al. 2010; Abadie 2015). Although a logarithmic function captures these data reasonably well, a power law [Eq. (11)] fits equally well, and is preferred here for consistency of interpretation with ratcheting. However, neither function captures the response particularly well for . Power-law fits are shown by a dashed line in Fig. 10(a) following:
(11)
The impact of stress level on the evolution of stiffness is assessed by plotting the power-law exponent () against stress level in Fig. 10(c). The exponents for tests U.1.D () and U.9.D () lie off this plot, being considerably greater than the other exponents. These high values of may be linked to the gapping-like behavior observed for these tests. If a gap is opening, the possibility of grain migration, and therefore densification close to the pile, is increased.
A logarithmic trend line is fitted to the variation of with stress level, neglecting the two-way tests. The dashed trend line in Fig. 10(c) has the following equation:
(12)
This trend line suggests that for a full-size monopile, the one-way stiffness exponent may be as low as 0.011, or 25% of the stiffness exponent value in the model tests.
The power-law function for [Eq. (11)] is not bounded as approaches infinity. However, given the small magnitude of at the stress levels of interest, reasonable values for secant stiffness can be expected for all practical cycle numbers.
Energy Loss Factor Response
Fig. 11(a) shows the evolution of unidirectional energy loss factor with cycle number. Abadie (2015) also explored the evolution of an energy loss factor with cycle number and found that a power law adequately captured the evolution. A power law [Eq. (13)] is also found to provide a good fit to the energy loss factor evolution here at all stress levels, and the resulting fits are shown by a dashed line in Fig. 11(a):
(13)
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For ratcheting and stiffness evolution, the variation of the energy loss factor exponent with stress level is plotted in Fig. 11(c). However, no dependence of on stress level is observed. Instead, the initial energy loss factor is found to decrease with increasing stress level, as shown in Fig. 11(b). Similar behavior would be observed by plotting coefficient against stress level. The anomalously low value of initial energy loss factor for Test U.1.D is linked to the significant gapping-like behavior observed, which reduces the hysteretic energy loss compared with no gapping.
A logarithmic function is fitted to the variation of initial energy loss factor with stress level. The following equation defines this trend line:
(14)
This trend line suggests an initial energy loss factor of around 0.12 for an equivalent full-size monopile, which is 16% of the value in the model tests.
As for secant stiffness, the power-law function for [Eq. (13)] is not bounded as approaches infinity. However, given the small magnitude of at the stress levels of interest, reasonable (nonnegligible) values of energy loss factor are expected for all practical cycle numbers.
Reloading Response
Postcyclic reloading was performed after 1,000 loading cycles for the majority of the unidirectional cyclic tests. Fig. 8(a) shows how the reloading response tends to rejoin the backbone curve following one-way cyclic loading at all stress levels. It is understood that ratcheting and stiffening mechanisms act in competition, often resulting in no significant net effect on the displacement response at large loads (Abadie et al. 2019). The reloading response is also presented in Fig. 12 with the displacement rezeroed (from displacement at start of reload ). This plot shows how the reloading response is significantly stiffer than the initial loading response for , consistent with the observed increase in secant stiffness with cycling. In general, when compared with the corresponding backbone curves, the reloading responses are qualitatively similar at all stress levels.
![](/cms/10.1061/(ASCE)GT.1943-5606.0002447/asset/691dc88d-00f9-42bc-8c45-7d56d0a92921/assets/images/large/figure12.jpg)
Discussion
Qualitatively, the unidirectional cyclic response is similar across the three stress levels investigated, with similar reloading responses and with power-law expressions approximately capturing the evolution of ratcheting, stiffness, and energy loss factor at all stress levels. Quantitatively, however, the results reveal some important stress-level effects.
The decrease of initial energy loss factor with stress level can be directly linked to the linearity of the load-displacement response. The response linearity increases with stress level for a given normalized load amplitude and is linked to stress dependent stiffness and dilatant behavior. However, the logarithmic decrease in ratcheting and stiffness exponent with stress level [Figs. 9(c) and 10(c)] appears to be independent of load amplitude and therefore of linearity of the load-displacement response.
Cuéllar et al. (2012) and Nicolai (2017) explored the mechanism driving ratcheting and stiffening behavior for cyclically loaded monopiles in sand. Cuéllar et al. (2012) used colored sand grains at , and Nicolai (2017) used particle image velocimetry in the centrifuge. Both authors observed local sand densification and convective particle movements during cyclic loading, and Cuéllar et al. (2012) identified three-dimensional (3D) convective cells. Cui and Bhattacharya (2016) conducted distinct element method (DEM) simulations to explore the same problem and also observed convective particle movement and densification. Cui and Bhattacharya (2016) also suggested that the convective region is the shape of an inverted cone [also observed by Cuéllar (2011)] because increasing stress level with depth imposes increasing constraint on particle movement.
The increase in the monopile’s secant stiffness is probably caused by a local densification mechanism, and the ratcheting behavior is probably caused by both a three-dimensional convective mechanism and any asymmetry in local densification. Given that both local densification and convective mechanisms require particle rearrangement, it is explicable that these mechanisms, and associated ratcheting and stiffening behavior, will be affected by increasing stress level, where—as highlighted by Cui and Bhattacharya (2016)—there is increasing constraint on particle rearrangement.
Multidirectional Cyclic Response
Test Program and Loading Definition
Table 7 summarizes the multidirectional cyclic tests performed as part of this campaign. Two multidirectional tests were performed at each -level: a T-shaped test and an L-shaped test. These tests explore the impact of cyclic load and load bias direction and are intended to be compared with the one-way unidirectional tests at the same cyclic load amplitude and average load amplitude .
Test | -level, | -direction | II-direction | No. of cycles, | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Nominal | Applied | Nominal | Applied | Nominal | Applied | Nominal | Applied | |||
M.1.T | 1 | 0.35 | 0.37–0.39 | 1 | 0.80–1.05 | 0.2 | 0.19–0.22 | 250a | ||
M.1.L | 1 | 0.35 | 0.36–0.39 | 1 | 0.98–1.08 | 0.4 | 0.37–0.38 | 0 | 153a | |
M.9.T | 9 | 0.2 | 0.20–0.23 | 1 | 0.95–0.99 | 0.2 | 0.19–0.20 | 1,000 | ||
M.9.L | 9 | 0.2 | 0.21–0.24 | 1 | 0.93–0.97 | 0.4 | 0.39–0.38 | 0 | 1,000 | |
M.80.T | 80 | 0.2 | 0.20–0.22 | 1 | 0.95–1.00 | 0.2 | 0.20–0.20 | 1,000 | ||
M.80.L | 80 | 0.2 | 0.20–0.22 | 1 | 0.95–1.00 | 0.4 | 0.39–0.39 | 0 | 1,000 |
a
Fewer than 1,000 cycles completed for these tests due to load control problem.
Fig. 13 shows the loading pattern for each test type: the selected unidirectional tests involve one-way cycling (), T-shaped tests involve symmetric cycling () perpendicular to a constant-load bias, and L-shaped tests involve one-way cycling () perpendicular to a constant-load bias. Unidirectional, T-shaped, and L-shaped tests therefore have average load biases in the -direction, -direction, and at 45° to the axes respectively, following the axes in Fig. 13. The T- and L-shaped tests may represent misaligned wave and wind loading on an OWT, where cyclic loading approximates wave loading and the load bias approximates the more slowly varying wind loading.
![](/cms/10.1061/(ASCE)GT.1943-5606.0002447/asset/18857089-2994-4e9e-8b75-c79963e047bd/assets/images/large/figure13.jpg)
Table 7 characterizes the tests in terms of nominal and applied values of and . The nominal values are those demanded of the control system during testing without considering load adjustment due to pile movement. The applied values are resolved from the load cell measurements posttest, accounting for the current pile position and load line angle. The multidirectional tests at and correspond to unidirectional tests U..A conducted at the same cyclic load amplitude and average load amplitude . The multidirectional tests have a nominal load bias in the -direction 75% larger than U.1.A, and comparisons at should therefore be made with caution.
Displacement and Ratcheting Response
Fig. 14 presents the displacement response for the multidirectional tests summarized in Table 7 alongside the corresponding unidirectional tests. The displacements at cycles 1, 10, 100, (and 1,000) are marked. Fig. 14 shows how the monopile moves broadly in the direction of the load bias at all stress levels, though there is some deviation in M.9.T and M.80.T. Fig. 14 also highlights the increase in amplitude of cyclic displacement with stress level, which is in line with the changing linearity of the load-displacement response with stress level.
![](/cms/10.1061/(ASCE)GT.1943-5606.0002447/asset/508d9e5d-e39e-449b-822b-f850c9885d6e/assets/images/large/figure14.jpg)
Fig. 15(a) presents the accumulated displacement response for the multidirectional tests alongside the corresponding unidirectional tests. Multidirectional behavior is presented in terms of orthogonal - and II-direction components. For the T-shaped tests, significant ratcheting only occurs and is only reported in the -direction. For the L-shaped tests ratcheting is reported in both the - and II-directions. A power law [Eq. (8)] is fitted to the evolution of with cycle number and shown as a dashed line in Fig. 15(a). The power law generally captures the evolution of ratcheting well for , but tends to overpredict ratcheting for for the T-shaped test. The response of test M.80.L(I), particularly for , may be anomalous.
![](/cms/10.1061/(ASCE)GT.1943-5606.0002447/asset/3892fbf1-ecad-4142-ad81-d9542d578fd7/assets/images/large/figure15.jpg)
Fig. 15(b) presents the variation of ratcheting coefficient with test type and stress level. No clear dependence of on either test type or stress level is observed. The ratcheting exponent is plotted against stress level in Fig. 15(c), accompanied by the dashed trend line obtained for the unidirectional tests [Eq. (10)]. This trend line also fits the multidirectional data well. In general, Fig. 15 shows no clear dependency of ratcheting behavior on test type (at least for ).
The greater scatter in Figs. 15(b and c) for the tests is explained by the inconsistency in applied load bias.
Secant Stiffness Response
Fig. 16(a) presents the evolution of secant stiffness for the multidirectional tests and corresponding unidirectional cyclic tests. For multidirectional loading, evolution of secant stiffness is only relevant in the direction of cycling. For , the response is well described with a power law [Eq. (11)], which is fitted and shown as a dashed line in Fig. 16(a).
![](/cms/10.1061/(ASCE)GT.1943-5606.0002447/asset/eab96462-353a-4cd6-abca-032de423ce2d/assets/images/large/figure16.jpg)
The variation of initial normalized secant stiffness [] is shown in Fig. 16(b). As for the unidirectional tests, there is no dependence on stress level; there is also no dependence on test type. The variation in stiffness exponent is plotted in Fig. 16(c), with the trend line obtained for unidirectional loading [Eq. (12)] shown as a dashed line. The values of for the multidirectional tests at depart from the trend line (although this may be related to the inconsistency in applied load bias), whereas the multidirectional tests at and are aligned with the unidirectional trend.
Energy Loss Factor Response
The evolution of energy loss factor with cycle number is shown in Fig. 17(a), with a power law [Eq. (13)] fitted and shown as a dashed line. The multidirectional behavior is consistent with that observed for the unidirectional tests: there is no clear dependence of the energy loss factor exponent on stress level [Fig. 17(c)], but the initial energy loss factor decreases logarithmically with stress level [Fig. 17(b)]. The trend line obtained for unidirectional loading [Eq. (14)] also fits this data well, as shown by the dashed line in Fig. 17(b). There is no clear variation in initial energy loss factor or exponent with test type.
![](/cms/10.1061/(ASCE)GT.1943-5606.0002447/asset/24ca009b-e4aa-43de-ae08-3d2627ae5b86/assets/images/large/figure17.jpg)
Discussion
In general, the stress-level effects observed in the unidirectional tests are also observed in the multidirectional tests. The trend lines obtained from the unidirectional results for variation of ratcheting exponent, stiffness exponent, and initial energy loss factor with stress level are also appropriate for the multidirectional results.
For , the T-shaped tests exhibit less accumulated displacement than the corresponding L-shaped and unidirectional tests. However, in general, there is little variation in magnitude and evolution of ratcheting, stiffness, and energy loss factor with test type at all stress levels. This implies that for a given mean load and cyclic load amplitude , the direction of cyclic loading relative to the mean load has an insignificant impact on the cyclic response. Ratcheting is also seen to occur in the direction of the mean load at all stress levels, regardless of the cyclic loading direction, in agreement with Richards et al. (2019). These observations have important modeling implications and suggest that misaligned wind and wave loading may be as damaging as aligned wind and wave loading.
Conclusions
Lateral monotonic, unidirectional cyclic and multidirectional cyclic loading tests have been performed on a model monopile in dry dense sand at three different -levels to investigate stress-level effects. Stress-level effects were isolated experimentally by using the same experimental setup at each -level.
The monotonic responses exhibit stress-level effects that appear to be mostly explained by stress dependent stiffness, with some contribution from stress dependent dilatancy, although decoupling these phenomena is difficult. The monotonic responses support the use of the normalization approach proposed by Leblanc et al. (2010), particularly at small displacements. However, a lower exponent than that implied in the Leblanc et al. (2010) normalization was obtained when exploring the stress dependency of maximum (small-displacement) foundation stiffness.
Qualitatively, the cyclic responses are similar across the stress levels, with similar postcyclic reloading behavior and with power-law expressions approximately capturing the evolution of ratcheting, stiffness, and energy loss factor with cycle number. Quantitatively, however, the results reveal some important stress-level effects.
The ratcheting and stiffness exponents, which control the rate of change of these parameters with cycle number, both decrease logarithmically with increasing stress level. The trends suggest that at full scale, the ratcheting exponent may be around half of the value obtained in model tests, and the stiffness exponent is projected to reduce to 25% of the model test value at full scale. This behavior is explained by the increase in confinement with stress level, which inhibits the particle movements that are thought to cause ratcheting and stiffness changes with cycling.
In this study, the model reference pile displacement —which determines the reference load amplitude —was maintained constant (in absolute terms) with stress level. As a result, the linearity of the monotonic and cyclic responses (at a given ) increases with increasing stress level; correspondingly, the initial energy loss factors reduce with increasing stress level. Future studies into the effect of stress level on the cyclic response might change or minimize the effect of the backbone linearity by setting the reference pile displacement in a different way (e.g., by employing a constant dimensionless value).
The multidirectional tests exhibit stress-level effects consistent with the unidirectional tests and provide insight into the impact of cyclic load direction. For a given mean load and cyclic load amplitude, the cyclic direction has no significant impact on the cyclic response, and ratcheting occurs in the direction of the mean load.
This test campaign provides new insight into the impact of stress level on the response of a monopile, with the monotonic response and evolution of ratcheting and stiffness exhibiting significant stress level dependency. The stress level effects highlight the need to simulate full-scale stress levels to understand foundation behavior thoroughly, while the qualitative similarities in response demonstrate the insight that can be gained from testing. The trends presented in this paper can help compare monopile behavior observed at different stress levels using different physical modeling approaches to inform better the development of new design methods for the next generation of monopile foundations.
Notation
The following symbols are used in this paper:
- , ,
- coefficients;
- pile outside diameter;
- relative density;
- particle sizes;
- void ratio;
- maximum void ratio;
- minimum void ratio;
- shear modulus;
- maximum (small strain) shear modulus;
- specific gravity;
- acceleration due to gravity;
- horizontal load;
- hysteretic energy loss;
- average load amplitude;
- cyclic load amplitude;
- maximum load (during cyclic loading);
- minimum load (during cyclic loading);
- reference load;
- pile loading eccentricity;
- relative density index;
- passive earth pressure coefficient;
- maximum (small displacement) stiffness;
- secant stiffness;
- secant stiffness on loading;
- secant stiffness on unloading;
- pile embedded length;
- maximum point on cycle;
- cycle number;
- -level;
- mean effective stress;
- atmospheric pressure;
- general variable;
- CPT cone resistance;
- general variable;
- roughness;
- normalized roughness;
- reversal point on cycle;
- pile wall thickness;
- displacement at load application point;
- reference displacement at load application point;
- , ,
- general exponents;
- effective unit weight;
- minimum effective unit weight;
- maximum effective unit weight;
- accumulated mean pile displacement;
- parameter characterizing cyclic load amplitude;
- parameter characterizing cyclic load symmetry;
- energy loss factor, general exponent;
- friction angle;
- critical state friction angle;
- peak friction angle;
- reference effective stress level;
- vertical effective stress; and
- initial vertical effective stress.
Data Availability Statement
Data that support the findings of this study (pile load-displacement responses) are available from the corresponding author upon reasonable request.
Acknowledgments
Centrifuge tests were undertaken with the support of the technical team at the National Geotechnical Centrifuge Facilities at UWA, whose contribution is gratefully acknowledged. The authors would also like to thank Manuel Herduin for assistance when using a modified version of his apparatus. The second author is supported as the Fugro Chair in Geotechnics. This work was supported by Grant No. EP/L016303/1 for Cranfield University, the University of Oxford, and Strathclyde University, Centre for Doctoral Training in Renewable Energy Marine Structures (REMS, http://www.rems-cdt.ac.uk/) from the UK Engineering and Physical Sciences Research Council (EPSRC).
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Received: Mar 24, 2020
Accepted: Sep 9, 2020
Published online: Jan 4, 2021
Published in print: Mar 1, 2021
Discussion open until: Jun 4, 2021
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