Soil Dynamics and Earthquake Earthquake Engin Engineering eering 49 (2013) 165–180
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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildy www.elsevier.com/locate/soildyn n
Dynamic soil–structure interaction of monopile supported wind turbines in cohesive soil Domenico Lombardi a, Subhamoy Bhattacharya b, , David Muir Wood c n
a
Research Student, Department of Civil Engineering, University of Bristol, Bristol, UK Chair in Geomechanics, University of Surrey, Department of Civil Engineering, Guildford, UK c Professor of Geotechnical Engineering, University of Dundee, Dundee, UK b
a r t i c l e
i n f o
Article history: Received 31 July 2012 Received in revised form 20 January 2013 Accepted 23 January 2013 Available Available online 20 March 2013 Keywords: Offshore Wind turbine Monopile Cyclic loading Dynamics Long-term performance Laboratory test Clay
a b s t r a c t
Offshore Offshore wind turbines supported supported on monopile monopile foundations foundations are dynamically sensitive because the overall natural frequencies of these structures are close to the different forcing frequencies imposed upon them. The structures are designed for an intended life of 25 to 30 years, but little is known about their their long long term term behavio behaviour. ur. To study study their their long long term behaviou behaviour, r, a series series of labora laboratory tory tests were conducted conducted in which a scaled model wind turbine supported supported on a monopile in kaolin clay was subjected subjected to between 32,000 and 172,000 cycles of horizontal loading and the changes in natural frequency and dampin damping g of the model model were monitored monitored.. The experim experimenta entall results results are present presented ed using using a nonnondimensional framework based on an interpretation of the governing mechanics. The change in natural frequency was found to be strongly dependent on the shear strain level in the soil next to the pile. Practical guidance for choosing the diameter of monopile is suggested based on element test results using the concept of volumetric threshold shear strain. Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.
1. Introduct Introduction ion
Offshore wind turbines are providing an increasing proportion of wind wind energy energy genera generatio tion n capaci capacity ty becaus because e these these sites sites are characterised by stronger and more stable wind conditions than comp compara arable ble onsho onshore re sites. sites. Offsho Offshore re sites sites also also have have a highe higherr capacity factor (the ratio of the actual amount of power produced over a period of time to the rated turbine power) when compared to equivalent onshore sites. The The design design and and const constru ructi ction on of found foundati ation onss for offsho offshore re turbines turbines are challeng challenging ing because because of the harsh harsh environm environmenta entall conditions and as a result provide a focus of major research in Europe, see for example Achmus et al. [1] [1],, Kuhn [29] [29],, Kuo et al. [30],, Bhattacharya et al. [5] [30] [5].. Different types of foundations have been proposed: including monopile, gravity base, jacket, suction caiss caisson on and floatin floating g system systems. s. Howeve However, r, most most of the the offsho offshore re turbines currently in operation (UK Round 1 development) are supported on driven monopiles. The choice of monopiles results from their simplicity simplicity of installat installation ion and the proven success success of driven piles in supporting offshore oil and gas infrastructures. The available available methods for designing designing monopiles monopiles for offshore offshore wind
n
Corresponding author. E-mail addresses: S.Bhattacharya@surrey.ac.uk S.Bhattacharya@surrey.ac.uk,, subhamoy.bhattacharya@gmail.com (S. Bhattacharya). Bhattacharya).
turbines turbines (e.g. the approach approach suggested suggested by DNV-OS-J DNV-OS-J101 101 [15] or IEC61400-1 [21] IEC61400-1 [21])) are based on the methods originally developed for the offshore oil and gas industry [4] [4].. Fig. 1 shows a typical monopile monopile supported supported wind turbine turbine and a pile supported supported fixed offshore jacket structure. There are, however, obvious differences between those two types of foundations. Piles for offshore structure structuress are typically typically 60–110 m long and 1.8– 2.7 m diameter. By contrast, monopiles monopiles for offshore wind turbines are commonly commonly 30–40 m long and 3.5–6 m diameter. diameter. Degradation Degradation in the upper upper soil soil layers layers result resulting ing from cyclic cyclic loadin loading g is less less severe severe for offshore jacket piles which are significantly restrained from pile head rotation rotation causing lower pile head deflection deflections. s. However, However, the overoverturning moments generated in the jacket superstructure are resisted by pairs of equal and opposite axial resultants in the piles. Such cyclic axia axiall load loadss can can prod produc uce e a loss loss of shaf shaftt capa capaci city ty beca becaus use e of the the development of ‘friction fatigue’ down the piles. Monopiles are freeheade headed d which which encou encourag rages es more more pile pile head head deflect deflection ion.. A design design metho method d usin using g a beam beam on nonnon-li line near ar Wink Winkle lerr spri spring ngss (‘ p method od in API API code code p– y’ meth [4] or [4] or DNV code) may be used to obtain pile head deflection under cyclic loading, but its use is limited for wind turbines because:
(a) the widely widely used API model is calibrat calibrated ed against against respons response e of a few small diameter piles (length to diameter ratio of 30 to 50) subjecte subjected d to small small numbers numbers of cycles cycles (maximum (maximum 200 cycles) cycles) suited for offshore fixed platform applications, e.g. Matlock [38] Matlock [38],,
0267-7261/$0267-7261/$- see front matter matter Crown Copyright Copyright & 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.soildyn.2013.01.015
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D. Lombardi et al. / Soil Dynamics and Earthquake Engineering 49 (2013) 165–180
Nomenclature a CSR D E f f f n G Gmax Gsec I kh K L K R K V L M M N M 1
parameter in the rational function fitting cyclic stress ratio pile diameter Young’s modulus of pile forcing frequency natural frequency shear modulus of soil shear modulus of soil at small strains secant shear modulus of soil second moment of area of pile horizontal coefficient of soil permeability stiffness of transverse spring stiffness of rotational spring stiffness of vertical spring penetration depth of pile external moment acting at the pile head secant modulus of the p – y curve after N th secant modulus of the p – y curve after 1st cycle
O’Neill and Murchison [42], Poulos and Hull [44], Reese et al. [47,48]. In contrast, for a real offshore wind turbine, the length to diameter ratio of piles is of the order of 4 to 8 and 10 7–108 cycles of lateral and moment loading are expected over a lifetime of 20–25 years. (b) It can be shown that the calibrated p– y curves used in the API and DNV codes are based on flexible pile behaviour where the pile is expected to fail by formation of plastic hinges (structural failure of piles). On the other hand, the squat nature of monopiles makes them sufficiently rigid that the formation of plastic hinges is not expected. Rather, a monopile will rotate like a rigid body (potentially including some reverse toe-kick) and the soil next to the pile may fail. (c) under cyclic loading, the API or DNV model always predicts degradation of foundation stiffness in sandy soil. However, recent work by Bhattacharya and Adhikari [7], Cue´llar et al. [13], LeBlanc [32] suggested that the foundation stiffness for a
N P t t w V y
a D
ga gtl gtv d e p es l x
s0v s y
number of load cycles net horizontal load time wall thickness of pile vertical load distance between foundation level and application of P parameter in the logarithm fitting parameter in the rational function fitting average shear strain in the soil linear shear strain threshold volumetric shear strain threshold lateral deflection of pile head strain in pile wall thickness average strain in soil parameter in the rational function fitting damping ratio of model effective vertical on the soil at the same depth as above pile yield stress
monopile in sandy soil will actually increase as a result of densification of the soil next to the pile. (d) The ratio of horizontal load ( P ) to vertical load ( V ) is very high in offshore wind turbines when compared with fixed jacket structures. Therefore, the monopiles experience disproportionately higher moment loading in comparison to a jacket pile. This more extreme loading condition was not taking into account during the calibration of the API and DNV p– y curves.
A similar problem of cyclic degradation of the soil surrounding a relatively short pile (20–30 m) was encountered in designing the floating offshore platforms for the North Sea Alvheim field [8]: Mechanisms included post-holing and possible jetting action due to the one-way cyclic loading on the anchoring piles. These nearsurface effects, Bhattacharya et al. [8] are much more significant for the shorter monopiles, affecting a greater proportion of their length.
Fig. 1. Typical monopile supported wind turbine and a fixed offshore jacket structure supported on piles.
D. Lombardi et al. / Soil Dynamics and Earthquake Engineering 49 (2013) 165–180
The long term performance of offshore wind turbines involves the following two issues:
1. Change (degradation/stiffening) of soil stiffness leads to changes in the natural frequency of the system, with the potential for unplanned system resonances and consequent excessive cyclic displacements. 2. Degradation of soil stiffness may lead to permanent displacement of the turbine which may jeopardise its performance: Wind turbines typically cannot tolerate more than 0.5 degrees tilt. Offshore wind turbines are relatively new structures without any track record of long term performance. Monitoring of a limited number of installed monopile-supported wind turbines has indicated a gradual departure of the overall system dynamics from the design requirements (e.g. the data from Lely wind farm data [28]). Clearly, the long-term performance and the uncertainties related to dynamic response of these dynamically sensitive structures need to be understood. The current codes of practice (for example API, ISO, DNV) for the design of monopile foundations of offshore wind turbines recommend the application of the p–y curves primarily for the evaluation of lateral pile capacity in the ultimate limit state (ULS). The codes provide limited guidance in predicting the change of the foundation stiffness and consequently the change of natural frequency which is an important design driver for serviceability limit state (SLS) requirements. Vibration of a monopile will induce cyclic strains in the soil in its vicinity. Under moderate-to-high amplitudes of cyclic loading most soils change their stiffness and strength. In order to study the changes in soil stiffness due to these cyclic strains, the developing strain in the soil around the shear zone must be taken into consideration. Soils in offshore sites can be fully saturated and therefore pore pressures are likely to develop as a result of these cyclic strains. The pore pressure developed may dissipate to the surrounding soil depending on factors including frequency of loading, permeability of the soil and diameter of the pile. Pore water pressure may also develop in unsaturated soils under cyclic shearing. The dynamic excitations caused by environmental loads (wind and wave) may impose either one-way or two-way cyclic loading on the monopile. It has been observed by Chang and Whitman
167
[12], Little and Briaud [34], Kramer and Heavey [27] that one-way loading develops more soil deformation and consequently more change in foundation stiffness. Research on pile–soil interactions without the effects of superstructure has also been recently carried out by Cue´ llar et al. [13], Li et al. [33], LeBlanc [32], Achmus et al. [1] where it was observed that the pile-soil stiffness changes with cycles of loading. Research considering large numbers of cycles of loading on a soil sample was investigated by Wichtmann et al. [54]. By contrast, the present work is unique in its experimental consideration of the dynamics of the overall system. This paper aims:
(a) to summarise the external dynamic and cyclic loading acting on a typical offshore wind turbine. (b) to present results of tests on scaled model wind turbine supported on a monopile in kaolin clay in order to understand the long term foundation performance. (c) to assess the long term performance of prototype offshore wind turbines using similitude relationships for 1-g testing of models of such turbines. (d) to highlight the practical implications for choice of monopile dimensions.
2. Dynamic and cyclic loading on a wind turbine
Offshore wind turbines are characterised by a unique set of dynamic loading conditions (Fig. 2). The principal external excitations are:
(a) Environmental dynamic loads arising from wind and waves. Fig. 3 shows the plot of power spectral density of wind and wave loading around the UK coastline (particularly in the North Sea). The predominant wave frequency is 0.1 Hz, which corresponds to 10 s wave period. (b) Rotor loading at a frequency which is commonly referred to as 1P . Fig. 3 shows the rotor frequency for a 3.6 MW wind turbine having an operational range between 5 and 13 rpm,
Fig. 2. External loads acting on offshore wind turbine supported on a monopile foundation.
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D. Lombardi et al. / Soil Dynamics and Earthquake Engineering 49 (2013) 165–180
Fig. 3. Simplified power spectral density of the forcing frequencies applied to typical three-bladed 3.6 MW offshore wind turbine with an operational interval in the range of 0.14–0.31 Hz (5–13 rpm).
Table 1 Typical values of forcing and natural frequencies for real offshore wind turbines. Offshore wind farm and the type of turbine
Lely A2 Turbine (Netherlands) Nedwind500KW/41 Zaaijer [57] Irene Vorrink (Netherlands) Nortdtank600/43 turbine Zaaijer [57] Blyth (UK) Vestas V66 2 MW turbine Camp et al. [10] Sheringham Shoal (UK) Siemens SWT-3.6-107 turbine Hamre et al. [18] Kentish Flat (UK) Vestas V90 3 MW turbine. Villalobos (2006) North Hoyle (UK) Vestas V80 2 MW turbine. Carter [11]
n
Foundation and soil data
Forcing frequency ( f f )
Natural f f /f n frequency, f n [Hz]
1 P [Hz]
2 P /3 P [Hz]
Pile passes through soft layer to stiff sandy layer Pile passes through soft layer to stiffer sandy layer Submerged ro cky outcrop
0.53
1.06 (2P )
0.63
0.45
1.35 (3P )
0.56
0.18– 0.4 1
0.54–1 .23 (3P )
0.41
Gravelly sands overlaying firm to stiff sandy clays Soft and stiff clay
0.08–0.22
0.24–0.66 (3P )
0.85–0.96
0.14–0.31
0.42–0.90 (3P )
0.38
The upper seabed layer comprises variations of sand and clay layers. Below is mudstone/sandstone
0.15–0.32
0.45–0.96 (3P )
0.35
nn
nn
n
n
n
n
From 0.84 to 1.68 From 0.80 to 2.41 From 0.44 to 3.00 From 0.09 to 0.69 From 0.37 to 2.37 From 0.43 to 2.74
Design approach
Soft-stiff Soft-stiff Soft-stiff Stiff-stiff Soft-stiff Soft-stiff
Estimated based on Adhikari and Bhattacharya [2]. Measured and reported in the literature.
nn
i.e. 0.14–0.31 Hz. In the power spectral density plot the 1 P frequency appears as a band. (c) The blade passing frequency (3 P or 2P for a three-bladed or two-bladed turbine, respectively) is a forced loading generated from the effect of wind deficiency that occurs as each blade passes through the shadow of the tower. Fig. 3 shows the blade passing frequency for the 3.6 MW wind turbine generator. From Fig. 3 it may be observed that, in order to avoid the resonance of the system, the designed frequency of the overall system must be kept away from the frequency content of applied loads. Specifically, DNV [14] suggests that the natural frequency of the wind turbine should be at least 7 10% away from the 1 P and 2P /3P frequencies. Bearing these considerations in mind, there are three possible ranges in which the natural frequency of the system may lie.
They correspond to three different design approaches namely: softsoft (natural frequency o 1P ), soft-stiff (natural frequency between 1P and 2P or 3P ) and stiff-stiff (natural frequency 4 2P or 3P ). The most common design, used for example in the Round 1 UK development, is ‘‘soft-stiff’’, which implies that the natural frequency lies between 1P and 3P . The design procedure requires an accurate evaluation of the natural frequency, which is dependent on the support condition (i.e. the stiffness of the foundation), which in turn relies on the strength and stiffness of the surrounding soil. Furthermore, it should be ensured that throughout the operational life the natural frequency of the system does not come close to any forcing frequencies: This would lead to amplification of the dynamic response of the turbines, leading to larger tower deflections and/or rotations beyond the 0.5 degrees tilt which can typically be tolerated.
D. Lombardi et al. / Soil Dynamics and Earthquake Engineering 49 (2013) 165–180
Table 1 lists the details of six types of turbines (Nedwind, Nordtank, Seimens, Vestas V66, Vestas V80 and Vestas V90) from six different locations. Dynamic measurements were carried out only on two turbines (Lely and Irene Vorrink wind farm). For the remaining four the assessment of the natural frequency is based on the calibrated mathematical model developed by Adhikari and Bhattacharya [2,3], Bhattacharya and Adhikari [7]. Table 1 shows that the natural frequency of these structures is of the same order as the excitation frequencies imposed by the external dynamic loads, reinforcing the need for dynamic considerations in the design process.
Fig. 4. Physical model of offshore wind turbine.
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3. Experimental apparatus
The experimental investigation was conducted using the BLADE facilities (Bristol Laboratory for Advanced Dynamics Engineering) at the University of Bristol. Tests were carried out on a scale wind turbine model supported on a monopile. The turbine was subjected to up to 172,000 cycles of 3 P loading. The experimental setup is shown in Fig. 4. The model turbine replicates the 3 MW Vestas V90 turbine having a notional scale of 1:100. A homogeneous soil profile of soft speswhite kaolin clay was used for the model tests. The soil was prepared from slurry by mixing kaolin powder with deionised water at a moisture content of about twice the liquid limit. This slurry was then consolidated in a cylindrical concrete tube (diameter 600 mm, height 600 mm). The small strain shear modulus (Gmax) of the clay was assessed by a series of bender element tests placed in the tube and an average value of 6 MPa was measured. The undrained shear strength ( S u) was estimated from the moisture content using correlations and an average value of 14 kPa was estimated. More details are given by Lombardi [35]. The environmental dynamic loads were modelled using an electro-dynamic actuator fixed to the laboratory strong wall and connected to the model wind turbine tower. The force ( P ) applied to the wind turbine could be constantly monitored by a force sensor. An electric motor powered by a DC supply was used to rotate the blades to model the 1P loading and also provided
Fig. 6. Usefulness of scaling law.
Fig. 5. (a) Model set-up and instrumentation used in the experimental investigation. (b) Impedance of the actuator.
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aerodynamic damping to the system. Precise measurements of displacements and accelerations were carried out using LVDTs (linearly varying differential transformers) and piezoelectric accelerometers at two locations (pile head and at the top of the tower). The acceleration responses of the physical model were recorded in two orthogonal directions along the tower. Fig. 5a illustrates the model set-up. In order to investigate the dynamic response of the model, tests were performed at selected excitation frequencies and amplitudes (Table 5). The monopile was installed by controlled jacking into the clay. When a pile is driven in soft clay, the soil around the pile is remoulded and undergoes high shear strains generating positive excess pore pressure around the pile. In order to allow the dissipation of this pore water pressure, initial measurements of the dynamic properties of the wind turbine model were carried out after a pause of 40 min. The assessment of this time required for the dissipation was based on the method suggested by Randolph [46] using the theory of radial consolidation. The initial dynamic properties of the model (i.e. frequency and damping of the overall system) were measured using a free vibration test, also known in the literature as a ‘‘snap back’’ test. The snap back motion was generated by the impact of an impulse force hammer and the response signal was recorded in the time domain by accelerometers. Following the initial measurement of dynamic properties, the electro-dynamic actuator was connected to the model. The model turbine was then subjected to cyclic loading for a chosen time interval (typically 5000 cycles) by spinning the rotor blades (1 P loading) and using the actuator (3 P loading). The dynamic properties were then evaluated at the end of the chosen number of cycles through another snap back test. During these measurements the actuator was detached from the model and the DC motor controlling the 1P loading was also stopped. The actuator spring otherwise provides unwanted impedance to the system (Fig. 5b).
4. Similitude relationships
Derivation of the correct scaling laws constitutes the first step in an experimental study. The similitude relationships are essential for interpretation of the experimental data and for scaling up the results to real prototypes. There are two ways to scale up the model test results as shown in Fig. 6. The first is to use standard tables for scaling and multiply the model observations by the scale factor to predict the prototype response, see for example Muir Wood [39]. The alternative is to study the underlying mechanics/ physics of the problem based on the model tests, recognising that not all the interactions can be scaled accurately in a particular test. Once the mechanics/physics of the problem are understood, the prototype response can be predicted through analytical and/or numerical modelling in which the physics/mechanics discovered will be implemented in a suitable way. The second method is particularly useful for study of the dynamics of offshore wind
turbines: these involve complex dynamic wind–wave–foundation– structure interaction and no physical modelling technique can simultaneously satisfy all the interactions at a single scale. Ideally, a wind tunnel combined with a wave tank on a geotechnical centrifuge would serve the purpose but this is unfortunately not feasible. Special consideration is required when interpreting the test results. As dynamic soil structure interaction of wind turbines are being studied, stiffness of the system is a top priority. Every physical process may be expressed in terms of dimensionless groups and the fundamental aspects of the governing physical processes as encapsulated in these dimensionless groups must be preserved in the design of model tests. Derivations of some aspects of the scaling laws for these experiments can be found in Bhattacharya et al. [6]. In this work the following physical mechanisms and parameters are considered important:
(a) Pile geometry. Because of their low length to diameter ratio (L/D), prototype monopiles are likely to exhibit ‘rigid’ behaviour. (b) Repeated cyclic shear strain. The strains in the soil around a laterally loaded pile control the degradation of soil stiffness. (c) Cyclic stress ratio, (CSR). The strains in the soil depend on the ratio of the shear stress to the vertical effective stress at a particular depth. It is shown in Appendix A and Bhattacharya et al. [6] that CSR can be represented by the non-dimensional group (P /GD2) (see definitions in the footnote of Table 1). (d) Rate of loading. Generation and dissipation of pore pressure are influenced by several parameters. The time ( t ) for significant pore water pressure dissipation will be inversely proportional to the permeability of the soil ( kh) and directly proportional to a characteristic length, for example monopile diameter (D). Time (t ) is inversely proportional to the forcing frequency ( f f ), so that (kh/ f f D) is a simple relevant nondimensional group providing first order similarity. (e) Frequencies of loading and system response. The dynamic response is strongly influenced by the relationship between these frequencies.
Table 2 presents a set of dimensionless groups, along with their physical meaning, considered in this study to model the dynamics of the system. Other dimensionless groups pertinent to study different aspects of the problem can be derived, see for example [6]. The model can be compared with two prototypes: Sheringham Shoal and Kentish Flat. The properties of the foundations are given in Table 3 while the dimensionless groups for model and prototype are presented in Table 4. A few points may be noted:
1. The soil strength ( S u) does not enter in any of the dimensionless groups. This is because of the fact that the present
Table 2 Dimensionless groups considered in this study.
Physical mechanism
Dimensionless group
Strain field in the soil next to the pile, cyclic stress ratio ( CSR) in the soil next to the pile, for details see Bhattacharya et al. [6]
Rate of loading System dynamics (relative spacing of forcing to the natural frequency of the system)
P GD2
kh f f D f f f n
P represents the total equivalent horizontal load acting on the wind turbine at a distance y from the foundation level. Essentially, P is also the net shear acting on the monopile. D is the diameter of the monopile. G, kh are the shear modulus and horizontal coefficient of soil permeability respectively. f f is the forcing frequency, f n is the natural frequency of the system.
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D. Lombardi et al. / Soil Dynamics and Earthquake Engineering 49 (2013) 165–180 Table 3 Properties of monopile and soil profile for physical model and prototypes.
Monopile
Soil profile
n
Property
Model
Prototype (Sheringham Shoal)
Prototype (Kentish Flat)
Material diameter Length wall thickness Young’s modulus Mass Yield stress
Dural alloy 22 mm 500 mm 1.3 mm 70 GPa 61.5 g 250 MPa
Steel 4.7–5.7 m 23–37 m 55 mm 210 GPa 145–283 t 550 MPa
Steel 4.3 m 28 m 45 mm 210 GPa 132 t 550 MPa
Soil type Plasticity Index Shear modulus Horizontal permeability
Soft clay 31% 6 MPa 10 9 m/s
Firm to stiff clay 30% 4–48 MPa 10 9 m/s
nn
nn
London clay 41% (60 MPa) 10 9 m/s
n
n
n
Hamre et al. [18]. Approximate value.
nn
Table 4 Dimensionless groups for model and prototype.
Dimensionless group
P GD2
kh f f D f f f n
Physical model
1.52
10
4
to 3.44
Prototype (Sheringham Shoal)
10
2.3 10 8 ( f f 2 Hz) 2.3 3.6 10 10 ( f f 125 Hz) 0.6–44
¼ ¼
3
10
8.9 9
( f f 20 Hz)
¼
10
3
(P 9.5 MN, D
¼
Prototype (Kentish Flat)
¼ 4.7 m, G ¼ 48 MPa)
Varied between 2.7 10 9 ( f f 0.08 Hz, D 2.7 10 10 ( f f 0.66 Hz, D 5.7 m) from 0.09-0.69
¼
¼
¼
¼ 4.7 m) and
1.05 10 3 (P 1.16 MN, D 4.3 m, G 60 MPa) 1.66 10 9 ( f f 0.14 Hz) 2.58 10 10 ( f f 0.90 Hz) Varied between 0.37 to 2.37
¼
¼
¼
¼ ¼
research focus is on the long term performance i.e. serviceability limit state (SLS) and fatigue limit state (FLS) which is concerned with stiffness and variation of stiffness. There may well be plastic deformations occurring but not significant regions of failure. 2. Fig. 7 shows a simple structural model of the overall system in which the foundation is replaced by springs. The stiffness of the foundation (pile) is represented by rotational stiffness ( K R), lateral stiffness ( K L) and rotational–lateral coupled stiffness (K LR) which are complex functions of soil stiffness, length and bending stiffness of the pile. The natural frequency of the system ( f n) is a function of K R, K L and K LR and therefore pile length (L) and bending stiffness pile are incorporated in the group ( f f / f n). 3. Moment loading will also cause strain in the soil next to the pile. For a specific turbine (e.g. Vestas V80, hub height of 80 m and blade length of 50 m) and under a certain environmental condition (for example, moderate wind and extreme sea state), the net lateral load ( P ) acting on the foundation is a summation of various loads which can be assumed to be relatively constant. Fig. 8(a) shows a schematic diagram showing the main loads (P 1, P 2, P 3) along with their point of application. Fig. 8(b) shows the equivalent load acting on the wind turbine model where the different loads are replaced by a single load (P ) acting at y c .
The moment ( M ) acting on the foundation is M P yc (Fig. 1). Following the similar argument as given in Appendix A, the pile deflection (d) at a depth is a function of the moment ( M ), the shear modulus of the soil ( G) and the pile diameter ( D). The strain in the soil next to the pile is a function of pile deflection and pile diameter D (see Eq. A.1). Therefore, the average strain field in the soil around a pile due to the moment ( M ) can be expressed as a function of three parameters: e s f (M ,D,G)
¼
¼
Fig. 7. Simplified mathematical model of the whole system.
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Table 5 Details of experiments and loading parameters.
Loading conditions
Derived parameters
ID
P [N]
f f [Hz]
N [cycles]
f n-final /f n-initial
C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8 C-9
10 10 10 7 5 2 0.44 1 50
2 20 125 2 2 2 2 2 125
Up to 32,400 Up to 32,400 Up to 32,400 Up to 32,400 Up to 32,400 Up to 32,400 Up to 32,400 Up to 172,800 Up to 32,400
0.63 0.91 0.96 0.79 0.87 0.94 0.98 0.95 0.83
P/GD 2 [%]
f f /f n
0.34 0.34 0.34 0.24 0.17 0.07 0.02 0.03 1.72
0.6–1.0 6.7–7.3 42–44 0.6–0.8 0.6–0.7 0.6 0.6 0.5–0.6 42–44
Remarks (1) Tests C-1, C-2 and C-3 were the initial tests which suggested that drop in frequency is sensitive to 2 Hz i.e. as expected f f /f n close to 1. Therefore the remaining tests focussed on 2 Hz. (2) The measured damping increased by 1.8 during the tests. (3) Results from the test C-9 are not considered in this paper.
f n-final is the final natural frequency of the system following the maximum number of cycles of loading. f f-initial is the initial natural frequency of the system measured 40 min after the installation.
Fig. 8. Lateral loads acting on a wind turbine and load-equivalence for model testing.
A dimensionless group can then be obtained:
M
es ¼ f
½F ½L ½FL2 ½L3
3
GD
ð1Þ
M Oralternatively: P y P Py GD3
¼
c
GD3
¼
c
GD2
D
¼b
GD2
ð2Þ
Which suggests that the strain induced in the soil next to the pile due to the moment loading M can be expressed as a scalar multiplier (b) of ( P /GD2). As the diameter of monopiles ranges between 3 m and 6 m and the hub height above the foundation level can be up to 100 m, the maximum value of ( b) can be 33 but the value depends on many factors, including direction of wind and wave loading, water depth (which will control the force due to current and wave), environmental loading at that point in time (i.e. wind and the sea state), type of turbines, geometry of the tower (uniform of tapered), geometry and orientation of the blades. b will vary throughout the life time of the wind turbine. However, in the model tests, the lateral load was applied at a fixed point i.e. 600 mm above the foundation level giving b E 27.
5. Summary of the test resuls
Table 5 summarises the main characteristics of the tests carried out. Three parameters (i.e. magnitude of the force P , number of
cycles of loading N , and the forcing frequency f f ) were varied during the tests. Tests C-1, C-4, C-5, C-6, C-7 and C-8 were carried out at a constant forcing frequency of 2 Hz, varying the magnitude of strains in the soil by using different P/GD 2 values. These tests applied 32,400 cycles, apart from test C-8 which applied 172,800 cycles. Tests C-2 and C-3 were performed at different forcing frequencies but with the same initial value of P/GD 2 (i.e. 0.34%).
5.1. General features of the vibration
The assessment of natural frequency was carried out in the frequency domain and the fast Fourier transform ( FFT ) was evaluated using the method suggested by Welch [53]. This function estimates the power spectral density ( PSD) of the input (i.e. acceleration record) using Welch’s averaged modified periodogram method of spectral estimation. The first natural frequency of the model corresponds to the frequency having the highest power spectral density value. The damping was assessed in the time domain using the logarithmic decrement method. Figs. 9 and 10 show typical results obtained from the free vibration tests. The acceleration time history response of the physical model installed in the clay sample is shown in Fig. 9 using the continuous line. In the same plot, the frequency response of the fixed base model is shown by a dotted line. The fixed base condition was obtained by clamping the bottom part of the tower to a heavy steel bench.
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(b) From the frequency domain analysis ( Fig. 10), the fixed base natural frequency of the turbine (10.27 Hz) is reduced by about 68% (3.3 Hz) in the presence of the foundation. The foundation provides flexibility to a wind turbine. It may be concluded that the assessment of the dynamic properties of the offshore wind turbines is strongly dependent on the foundation stiffness. 5.2. Change in natural frequency and damping of the system due to cyclic loading
Fig. 9. Time history of acceleration records from free vibration tests carried out on the physical model with or without the foundation.
Table 5 shows the change in natural frequency of the wind turbine system as a function of the number of cycles. The change is represented by the ratio f n-final /f n-initial i.e. the ratio between the natural frequency measured after the maximum number of cycles have been applied and the natural frequency in the first cycle. This ratio provides an indication of the level of degradation caused by the cyclic loading. Fig. 11 plots the change in natural frequency of the overall system with the number of cycles for different values of P/GD 2 which controls the strain level in the soil next to the pile. In all cases the ratio representing the system dynamics, f f /f n, changed between 0.6 and 44. Fig. 12 illustrates the change of natural frequency of the system for different ratios of f f /f n but for a particular strain level P/ GD 2 0.34%. The drop in natural frequency is higher if the forcing frequency is close to the natural frequency of the system, which is basically a resonance type mechanism. The natural frequency of real wind turbines is close to the forcing frequency and therefore the experiments demonstrate the importance of such considerations. Fig. 13 plots the results from the test carried out up to 172,000 numbers of cycles of loading. The test showed that the overall stiffness change is minimal, possibly due to the very low strain level in the soil i.e. P/GD 2. For values of f f /f n close to 1, this change in frequency is strongly dependent on the dimensionless group P/GD 2. On the other hand, for high values of f f /f n (6.7–7.3 and 42– 44 in Fig. 12) relatively high amplitudes of P/GD 2 (0.34%) seem to have little effect on the dynamic response of the model. This observation is consistent with dynamic analysis which indicates that when the excitation frequency is far from the natural frequency of the system the dynamic interaction is negligible and the response becomes inertia dominated. Fig. 14 plots the normalised damping values of the system for four tests. As expected, the damping of the model increased with the number of cycles and higher damping variations are recorded for larger strain amplitudes in the soil.
¼
Fig. 10. Frequency response from free vibration tests carried out on the physical model with or without the foundation.
5.3. Curve fitting of the data based on the dimensionless groups
One of the aims of this paper is to link the change in natural frequency of the wind turbine system based on the dimensionless groups. The test results suggest that the change in natural frequency of the physical model with number of cycles is dependent on the strain level in the soil which is controlled by P/GD 2. The data shown in Fig. 11 can be fitted by a logarithmic regression line of the form:
Fig. 11. Change in frequency of the physical model with number of cycles for different amplitudes of P/GD 2. The fitted line is shown by Eq. (5).
Fig. 8 plots the same data in the frequency domain. Two important observations may be made:
(a) From the pattern of decay of acceleration in Fig. 9, the presence of the soil-foundation system which typically increases damping by a factor of 8 to 12.
f N -cycles f initial
¼ 1a lnðN Þ
ð3Þ
Fig. 15 plots the variation of the coefficient a with P/GD 2: The value of a varies linearly with P/GD 2.
a ¼ 0:0915
P GD2
ð4Þ
A high value of P/GD 2 produces a higher value of a, which corresponds to a much greater change in frequency. A similar form of equation is also used to fit the degradation of the Secant Young 0 s modulus of soil with number of cycles [19,20].
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Fig. 12. Change in natural frequency f N-cycles /f initial for constant P/GD 2 of 0.34%.
Fig. 13. Result from test C-8: 172,800 cycles of loading applied at constant frequency 2 Hz and constant force amplitude 1 N.
Fig. 15. Parameter a as a function of P/GD 2. The line shows the result of linear regression fitting.
Fig. 14. Change in damping with number of cycles (tests C-1, C-5, C-6, C-7).
Using Eq. (3), the degradation is disproportionately higher for the first few cycles, with the subsequent degradation becoming slower as loading continues. This is in line with the observations reported by Long and Vanneste [36] where 34 full-scale cyclic tests on pile foundations were analysed. However, these results do not take into account dynamic considerations, such as the change of the ratio between the frequency of the excitations and the natural frequency of the system ( f f /f n) with number of cycles. As the ratio f f /f n approaches 1, the degradation is amplified. This particular behaviour was observed clearly in test C-1 (solid square data points in Fig. 11). An improved match to the experimental data is
Fig. 16. Experimental results fitted with Eq. (10).
obtained using an alternative fitting Eq. (7) shown in Fig. 16. f N -cycles f initial
¼ ð1DÞ
1
1 ln N =a l
þ½ ð Þ
ð5Þ
D. Lombardi et al. / Soil Dynamics and Earthquake Engineering 49 (2013) 165–180
where the parameters D and l are related to P/GD 2. Their values are plotted in Fig. 17a and b and they may be described by Eqs. (6) and (7). D
l
¼ 11:0371 ¼ 72:083
P GD2
P
ð6Þ ð7Þ
GD2
a is a parameter which seems to lie in the small range from 7.8 to 8.9. An average value of 8.35 has been assumed in Fig. 16. The experimental work suggests that the change in frequency over a wide range of strain level and also number of cycles can be predicted by the following equation where D and l are related to P/GD 2. f N -cycles f initial
¼ ð1DÞ
1 1
þ½lnðN Þ=8:35l
ð8Þ
Comparing Figs. 11 and 16, it may be observed that Eq. (8) gives a better overall fit to the entire range of the data. In particular, Eq. (8) may represent the model behaviour for higher values of P/GD 2 (e.g. the data indicated with the solid square) and for values of f f /f n close to 1. 5.4. Moisture content of the clay next to the pile
Measurements of moisture content of the clay sample next to the foundation were carried out before and after the tests: It was
175
observed to increase considerably due to the cycling. It was also noticed that for the tests performed at higher P/GD 2 values (i.e. 0.34% and 0.24%) and for values of f f /f n in the range 0.6–1.0, the moisture content increased from its initial value of 50% to a maximum of 100%. However, much smaller increments of moisture content were observed for lower values of P/GD 2 and higher values of f f /f n. The increase in moisture content evidently implies reduction of the undrained strength and decrease of the stiffness of the clay. Fig. 18 shows photographs of the monopile following tests in which the entire structure tilted considerably. 5.5. Rigidity of the model pile
It may also be observed from Fig. 18 that the soil surrounding the aluminium alloy model pile failed and there were no evidence of plastic strain in the pile following the tests which ensured that rigid behaviour of the model pile.
6. Discussion: choice of monopile diameter 6.1. Comparison with other published results
Offshore wind turbines are dynamically sensitive structures because the natural frequency of the system is very close to the forcing frequency. The study showed the variation of the normalised frequency of the system with respect to the number of cycles of loading for various strain levels in the soil imposed by the cyclic
Fig. 17. Parameters for fitting the degradation as function of P/GD 2, Eq. (3); (a) parameter D ; (b) parameter l .
Fig. 18. Details of the monopile after the tests.
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loading. As expected, higher strain levels led to higher reductions in natural frequency of the model. For a low value of ( P/ GD 2 0.02%) there is practically no degradation in the natural frequency even after 105 cycles of loading. Based on the trends and in the absence of any other data, these results can be extrapolated to cycles corresponding to a fatigue limit state (FLS) of 107 cycles, which is typically the number of cycles likely to be experienced by an offshore wind turbine. The test results presented in the paper are consistent with the centrifuge tests reported by Jeanjean [23] and Doyle et al. [16] where cyclic loading was applied on pile foundations embedded in kaolin clay and the effects of degradation were monitored. Jeanjean [23] proposed Eq. (9) based on observations made over 1000 cycles.
¼
M N M 1
¼ 0:9 þ 2:5 tan0h:½90:7 logðN Þ
ð9Þ
where M N and M 1 are secant modulus of the p– y curve i.e. factored pile-soil stiffness after N th and 1st cycle respectively. Overconsolidated clays (more representative of those found at locations of the current generation of offshore wind farms) are more difficult to model in a centrifuge, as in-flight consolidation is slow and the clay would need to be prepared off the centrifuge and manipulated to size. Over-consolidated clays respond by predominantly undrained plastic displacement which should be independent of effective stress and capable of study in 1 g experiments which can be used to apply large numbers of cycles in a realistic time frame. It is also easier to isolate the test rig from the effects of external vibration so that parasitic displacements generated by the rotation of the centrifuge itself can be avoided. It is suggested that in the absence of other data, the present data may be used to extrapolate to the long term performance.
soil will reduce. The value of g tl can be estimated from the secant shear modulus reduction curve (schematically shown in Fig. 19) assuming that linear threshold shear strain corresponding to a ratio Gsec /Gmax of 0.99. There exists another value of threshold shear strain, known as volumetric threshold shear strain level [52], denoted by gtv, beyond which permanent microstructural change of the fabric do occur. In other words, beyond gtv the soil is degradable possibly due to pore pressure build up. In the triaxial stress space this state corresponds to the boundary between the ‘recoverable’ and the ‘‘plastic zone’’ [22].gtv can be assessed from the secant shear modulus reduction curve. However two values of Gsec /Gmax ratios (namely 0.85 and 0.60) are often used depending on the application. The value of 0.85 represents a degradation of secant shear modulus beyond which permanent deformation is expected. On the other hand 0.6 is defined on the basis of the accumulation of excess pore pressures. These two values can represent upper and lower bound values of g tv for the problem in hand. The upper bound value will give the highest diameter that may be required. On the other hand, the lower bound will give the minimum diameter necessary. While Fig. 19 shows the schematic representation of threshold strains, Fig. 20 plots typical experimental data obtained from cyclic triaxial tests carried out on 10 samples of Turkish clay having a plasticity index (PI ) ranging from 9 to 40 following Okur and Ansal [41]. Fig. 20 shows an upper and lower bound for gtv which corresponds to Gsec /Gmax ratio of 0.85 and 0.60 respectively. The experimental results clearly show that samples with higher PI tend to
6.2. P/GD 2 and the determination of acceptable values based on soil element tests
Fig. 19 shows the variation of normalised secant stiffness with strain for fully saturated cohesive soil adapted from Vucetic [52]. It seems that there exists a strain level, the threshold linear shear strain, gtl, [22], for which there is no stiffness degradation and the behaviour of the soil is practically linear elastic, so that there is no permanent microstructural changes of the fabric of the soil with cyclic loading. As the strain level is increased beyond g tl, the soil behaves non-linearly and as a result the secant shear modulus of Fig. 20. Experimental secant shear modulus reduction curve for samples of Turkish clay with different plasticity index (after [41]).
Fig. 19. Secant shear modulus reduction curve for fully saturated clay subjected to cyclic undrained loading.
Fig. 21. Volumetric shear strain threshold values for different PI.
D. Lombardi et al. / Soil Dynamics and Earthquake Engineering 49 (2013) 165–180
have a more linear cyclic stress–strain response at small strains and to degrade less at larger strain than soils with lower PI . Therefore clays with higher PI are characterised by higher values of volumetric threshold shear strain. For example, considering the lower bound, gtv increases from 0.025 to 0.15 when PI increases from 9 to 40. Fig. 21 collates the volumetric threshold shear strain value ( gtv) for 18 types of clays having plasticity index ( PI ) ranging from 9 to 100. The experimental data were collected from research published by Kim and Novak [24], [17,26,31,37,41,50,51,56]. In the same figure, two linear correlations for evaluating g tv are also suggested for the investigated range of PI (9 o PI o 100). The experimental data used for deriving the plot in Fig. 21 are given in Appendix B. The experimental results presented in Figs. 11 and 16 indicate that the dimensionless group ( P /GD2) representing the strain level in the soil next to the monopile i.e. the strain in the mobilised shear zone can be mapped to the volumetric threshold strain, gtv which is a fundamental property of a soil and can be obtained from soil testing. The reason being, for ( P /GD2 0.7%) or less, there is less than 5% reduction in natural frequency. For kaolin clay used in this experimental investigation, the value of the threshold volumetric shear strain ( gtv) was estimated by Vucetic [52] to be in the range of 0.080–0.100%, based on the work carried out by Ohara and Matsuda [40]. Vucetic [52] states that the value of g tv increases with the plasticity index ( PI ) of the soil but that the influence of the overconsolidation ratio may be neglected. An attempt has been made to link the value of g tv with the value of (P /GD2) for which negligible change of natural frequency was observed. Figs. 11 and 16 suggest that for ( P /GD2) of 0.07%, the change in natural frequency is within 5% ( f N-cycles /f initial 0.95), and from an engineering point of view, the behaviour can be considered practically non-degradable. If an average value of gtv 0.09% is taken for koalin clay, the ratio of threshold strain and (P /GD2) becomes 1.3, see Eq. (10).
¼
¼
¼
gtv 2
P =GD
¼ 00::09 ¼ 1:3 07
ð10Þ
It is interesting to note that, the average strain in the soil ( gav) around a laterally loaded pile, which can be expressed by Eq. (11), suggested by Klar [25], is of the same order as that given by (10).
d
gav ¼ 2:6
D
6.3. Choosing diameter of monopile for design
The non-dimensional group ( P /GD2) suggests that higher the diameter of the monopile, the lower is the average strain in the surrounding soil and therefore lower is its degradation. For the two prototype turbines considered in this paper (Sheringham Shoal and Kentish Flat, see Table 4), (P /GD2) is in the range of 0.001 to 0.008 which is indeed in the range of gtv for clayey soils, see Fig. 20. For a particular wind turbine (i.e. known P ) to be located at a particular site (representative shear modulus of the soil G is also known) the allowable value of (P /GD2) may be chosen based on volumetric threshold shear strain (gtv) as illustrated earlier. However, if the prediction of change in frequency, based on the value of g tv, may have an adverse effect on the dynamic performance of WTG (Wind turbine Generator), a more conservative approach may assume the linear shear strain (gtl) as threshold value. While valuable insights on the mechanisms that affect the long term performance can be obtained from a relative simple test, more work is necessary if detailed guidelines are to be prepared. The next section shows the applicability of the above concepts for deciding a monopile diameter in clayey soils. 6.4. Example
Consider a stiff monopile of length 20 m, required for vertical bearing capacity to support a 70 m high tower of a 3.5 MW turbine. Based on dynamic considerations, in order to be able to generate power over a wide range of wind speed, it is necessary to be confident that any change in natural system frequency is no greater than 5%. The maximum shear acting in the pile is 2 MN. The soil at the site is stiff overconsolidated high plasticity clay (PI 74) which has a Shear modulus of 100 MPa. Using the fitting equation given in Fig. 21, and for PI of 74, the lower bound volumetric threshold strain can be taken as 0.2%. The allowable (P /GD2) is given by
¼
P 2
GD
where d and D are the displacement and the outer diameter of the pile, respectively. Eqs. (10 and 11) are similar in the sense that the terms ( P /GD2) and (d/D) represent factored average strain in the soil (but not the actual strain) in the deformation mechanism (pile-soil interaction) or in the mobilised strain zone. Further details of derivation of these two terms can be found in Appendix A and Bhattacharya et al. [6]. It is therefore reasonable to make the allowable value of (P /GD2) a multiple of the volumetric threshold strain gtv. While (P /GD2) represents factored average shear strain in the soil (having shear modulus of G ) in the mobilised zone next to the pile (having diameter D) due to a lateral load P , gtv represents the limiting strain level in the same beyond which progressive degradation in the soil will occur. This concept is quite similar to the Mobilisable Strength Design (MSD) concept pioneered by Bolton [9], Osman and Bolton [43] where the average strain in a mechanism (deformed zone or sheared zone) is linked with an element test in a soil. In the absence of more data and in order to ensure a change in natural frequency of the wind turbine within 5%, the ratio expressed in Eq. (10) can be taken equal to 2.5. This provides a factor of safety of about 2.
¼ 2g:tv5 ) 2:5 0:2100 ¼ 0:0008
ð12Þ
and the outer diameter required is D
ð11Þ
177
¼
r ffiffi ffi ffi ffi ffi ffi ffi2ffi ffiMN ffiffiffiffiffiffiffiffiffiffiffi 100 MPa
0:0008 ¼ 5:0 m
ð13Þ
Therefore the minimum diameter required is 5 m. This research suggests that if the diameter is less than 5 m, the strain developed in the soil next to the pile may lead to progressive degradation of the foundation stiffness leading to lower of natural frequency of the overall system.
7. Conclusion
The natural frequency and the long-term performance of a wind turbine model founded on clay soil have been studied using a number of 1-g tests. It has been shown that small scale experimental studies can be carried out to study complex dynamic soil–structure interaction problems. Based on the experimental results reported here, the following conclusions may be drawn:
(a) The dynamic response of the physical model is very sensitive to the flexibility of the foundation. The presence of the foundation provides increased flexibility and increased damping of the system. (b) The natural frequency of a monopile supported wind turbine founded on clayey soil may change with number of cycles of
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repeated loading. For clayey soil, a decrease in natural frequency is expected depending on the strain level in the soil next to the pile and the ratio of system frequency to the forcing frequency. (c) Non-dimensionless groups permit scaling the model test results to a prototype through conceptual understanding and knowledge gained from element tests on soils. It has been shown that the dimensionless group ( P /GD2) (P being is the net shear force in the foundation, D diameter of the monopile and G is the representative shear modulus of the soil) also captures the strain level in the soil due to moment loading and the cyclic stress ratio (CSR) in the soil adjacent to the pile. Tests having a high value of ( P /GD2) showed a higher reduction of natural frequency with long-term cyclic loading. On the other hand, for low values of ( P /GD2) (e.g. 0.02% or lower), the change of natural frequency was negligible. This non-dimensional group also justifies the use of large diameter monopiles as foundations for modern offshore wind turbines. (d) Furthermore, ( P /GD2) is calibrated against a well-known element test parameter threshold volumetric strain ( gtv) of the soil for providing practical design guidelines. (e) Finally, practical guidance for choosing the diameter of monopile foundations has been proposed.
with the strain distribution in the soil and the stresses in the soil must be in equilibrium with the reaction force from the pile. The pile deflection will cause strain in the soil and the strain field in the soil around a pile is very complex, see Fig. A1. A soil element in the front of the pile will have pure compression and a corresponding soil element in the orthogonal direction will have pure shear. Similarly a soil element behind the pile will have pure extension. Any other soil element will have a combination of either ‘shear and compression’ or ‘shear and extension’. However, a spatial average shear strain in whole deformable zone (mobilised) may be obtained. The average shear strain in the soil can be expressed as a function of pile deflection (d) and pile outer diameter ( D) given by Eq. (A.1).
es p
d D
ðA:1Þ
Klar [25] suggested a value of 2.6 for the coefficient of proportionality between the average strain in the soil and the ratio of head deflection and pile diameter. Similar concept is used in API code to relate e50(strain at 50% yield stress) to displacements in the pile. The pile deflection at a particular depth is a function of the external load, P , the shear modulus of the soil, G, and the pile diameter, D. Therefore, the average strain field in the soil around a pile can be expressed as a function of three parameters:
es ¼ f ðP D GÞ ,
,
ðA:2Þ
Appendix A. Derivation of the group CSR (cyclic stress ratio) and ( P /GD2) 8.1. Strain field in the soil around the laterally loaded pile
Repeated shear strain may reduce the stiffness of saturated soils. As discussed in the paper, the changes in soil stiffness may drive the long term performance and as a result, the average strain next to a pile is a governing criterion that must be preserved in order to ensure similar stiffness degradation in both model and prototype. The relevant non-dimensional group can be derived by considering that the average shear strain field around a laterally moving pile. Fig. A1 schematically shows the plane strain idealisation of pile–soil interaction at a particular depth of consideration (represented by section X - X at a depth d ). The deflection of the pile at a depth d is represented by (d). The deflection must be compatible
Fig. A2. Geometry of the characteristic mesh, Randolph and Houslby [45] .
Fig. A1. Schematic diagram showing the average shear strain concept in the soil around the pile .
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The parameters in Eq. (A.2) can be used to obtain a dimensionless group as follows:
P
es ¼ f
2
GD
½F
ðA:3Þ
½FL2 ½L2
Eq. (A.3) describes the non-dimensional group that takes into account a measure of strain field in the soil generated by a lateral loaded pile. Eq. (A.3) shows that the strain in the soil is directly proportional to the horizontal load applied at the pile head, inversely proportional to the soil stiffness and inversely proportional to the square of the pile diameter. 8.2. Cyclic stress ratio (CSR) in the soil in the shear zone
In geotechnical earthquake engineering, it is well established that degradation of a soil due to liquefaction-type failure is a function of cyclic stress ratio ( CSR) which is defined as the ratio of the shear stress to the effective vertical stress at a particular depth, defined by Eq. (A.4) (see, for example, [49]). The cyclic stress ratio (CSR) can be expressed by Eq. (A.4): CSR
¼ tscyc 0
The vertical effective stress can be related to the shear modulus (G) of the soil:
½F ½L2
s0v pG
ðA:7Þ
It is usually found that G is proportional to s0vn where the value of n depends on the type of soil. The value of n varies between 0.435 to 0.765 for sandy soil [55] but a value of 0.5 is commonly used. For clayey soil, the value of n is generally taken as 1. Combining Eqs. (A.4) to (A.7), one can see that the nondimensional group expressed by Eq. (A.3) can also guarantee similarity of cyclic stress ratio. This leads us to a non-dimensional group (Eq. (A.8)) that must be satisfied.
P GD2
mod el
¼
P GD2
It is interesting to note that using two different approaches based on average strain in the soil and on the cyclic stress ratio (CSR) dimensional analysis leads to a unique non-dimensional group given by Eq. (A.8).
ðA:4Þ
v
where tcyc is the cyclic shear stress imposed by the pile on the soil at a particular depth; s 0v is the effective vertical stress on the soil at the same depth.
Appendix B
Fig. 18 in the paper is based on collation of the following data.
Soil name
PI
OCR
Test apparatus
Windsor silty clay Wallaceburg silty clay Hamilton clayey silt Sarnia silty clay Alluvianal clay at Teganuma laboratory-made clays Vallericca clay Pietrafitta clay Todi clay laboratory-made clay Fat clay Not specified Turkish clay Ariake clay Onada clay
30 25 12 14 44, 57, 58, 83, 96 15, 30, 50, 100 31 30 28 22 36 10, 20, 40, 60, 80 9,10,12,15,18,20,25,27,35,40 44 50
2.7 5.1 5.8 1.8 NC NC OC OC OC 3 Not specified Not specified NC NC NC
Resonant Column Resonant Column Resonant Column Resonant Column Cyclic triaxial Not specified Resonant Column Resonant Column Resonant Column Direct simple shear Not specified Not specified Cyclic triaxial Hollow Cylinder Hollow Cylinder
Fig. A2 shows the deformation mechanism for a quarter of a pile, following Randolph and Houslby [45], where D (the angle that defines the characteristics) is a function of roughness of the pile. For a rough pile, D p/2 and therefore angle CQE will be 90 degrees. It can be shown that the area of the deformed zone ( Adef ) is approximately given by Eq. (A.5).
¼
2 p ffiffi D16 ðp3 þ 8 2Þ ¼ 2:644D2
Adef
ðA:5Þ
As the deformed zone is proportional to D 2, cyclic shear stress can be expressed by Eq. (A.6).
tcyc p
P 2
D
ðA:8Þ
prototype
½F ½L2
ðA:6Þ
Reference
[24] [24] [24] [24] Kokusho, 1982 [51] [17] [17] [17] [31] [50] [37]; [41] [56]a [56]a
References [1] Achmus M, Kuo YS, Abdel-Rahman K. Behavior of monopile foundations under cyclic lateral load. Computers and Geotechnics 2009;36(5): 725–35. [2] Adhikari S, Bhattacharya S. Dynamic analysis of wind turbine towers on flexible foundations. Shock and Vibration 2011;19:37–56, http://dx.doi.org/ 10.3233/SAV-2012-0615 IOS press. [3] Adhikari S, Bhattacharya S. Vibrations of wind-turbines considering soilstructure interaction. Wind and Structures 2011;14(2):85–112. [4] API. Recommended practice for planning, designing, and constructing fixed offshore platforms: working stress design. 20th edn. Washington, DC: American Petroleum Institute; 1993 RP2A-WSD. [5] Bhattacharya S, Cox, J, Lombardi, D Muir Wood, D. (2012). Dynamics of offshore wind turbines on two types of foundations. In: Proceedings of ICE—geotechnical engineering. Ahead of print. [6] Bhattacharya S, Lombardi D, Muir Wood D. Similitude relationships of physical modelling of monopile-supported offshore wind turbines. International Journal of Physical Modelling in Geotechnics 2011;11(2):28–68.
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