MODIFICATION OF THE API P-Y FORMULATION OF INITIAL STIFFNESS OF SAND D Kallehave, C LeBlanc Thilsted and MA Liingaard DONG Energy A/S, Fredericia, Denmark
Abstract
Monopiles are currently the preferred concept of support structures for offshore wind turbines. However, ex periences from operating operat ing offshore wind farms indicate that the current design guidelines (e.g. American Petroleum Institute (API)) under-predict the soil stiffness for large-diameter monopiles. Due to the structural dynamic of a wind turbine, it is unconservative to both over-predict and under-predict the soil stiffness. Only an exact prediction is conservative. The objective with this paper is to introduce an approximate method for determining the soil stiffness of sand regarding large-diameter monopiles by modifying the initial stiffness of the API p-y formulation. The modification introduces both a stress level and a strain level correction derived on basis of sound theoretical considerations without introducing new empirical parameters. It has been shown by benchmarking with full-scale measurements from Walney offshore wind farm that the t he modified approach provides a more accurate determination of the total soil stiffness, although it is still under-predicted. shore wind turbines located in sand profiles in the Walney offshore wind farm (see Figure 1) reveal an under-prediction of the wind turbine structures’ fundamental frequency of ~5–7%. Figure 1 shows the relative frequency deviation ( f Rel ) being the difference between the measured and predicted frequency relative to the predicted frequency. Each dot represents a 10min average measured value in the period 15 August 2011 to 26 September 2011. The measured frequencies are seen to drop slightly at high wind speeds, which is most likely due to an increased load cycle amplitude at such wind speeds.
1. Introduction
The p-y curves evolved primarily from research in the oil and gas industry, as the demand for large pile-supported offshore structures increased during the 1970s and 1980s. Research has included testing of full-sized piles in sand under both static and cyclic loading conditions. The p-y curves for piles in sand described by Reese et al. (1974) and O’Neill and Murchison (1983) led to recommendations in the American Petroleum Institute (API) standards for oil and gas installations (2011). In 2004 these recommendations were adopted in the Det Norske Veritas (DNV) standard (2004), which represents the current state of the art for design of monopiles in the offshore wind industry. The p-y curves for piles in sand were developed based on full-scale load tests on long, slender and flexible piles with a diameter of 0.61m (Reese et al., 1974). In addition, they have been widely applied to relatively shorter and stiffer piles with diameters di ameters up to 6.0m in the offshore wind turbine industry.
Under most circumstances and particularly for static structures, under-predicting the stiffness is conservative. However, because of the structural dynamic of a wind turbine, both over-predicting and under predicting the soil stiffness is unconservative. Only an exact prediction is conservative. It must be em phasised that the p-y curves have never been developed with the objective to accurately predict the soil stiffness for large-diameter piles.
The impacts of applying p-y curves empirically developed outside the verified range can now be observed. Nacelle measurements from DONG Energy’s offshore wind turbines show that the fundamental frequencies are much higher than bestestimate predictions using the API p-y formulation for piles in sand. This may be due to an under prediction of the soil stiffness. For example, fullscale measurements for three randomly chosen off-
The objective of this paper is therefore to present modifications to the current p-y curves that result in a better prediction of the soil stiffness. From a commercial point of view, under-predicting the soil stiffness increases uncertainties, adds additional but unnecessary costs to the industry and decreases the feasibility of the monopile foundation. Moreover, in the worst case it could have a negative influence on the structural lifetime.
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where A = 0.9 is a factor to account for cyclic loading; k is is the initial modulus of sub-grade reaction according to Reese et al. (1974); z is is the depth; and pu is the modified ultimate soil resistance, according to Bogard and Matlock (1980). It is given by:
(2)
where c1, c2 and c3 are factors depending on the internal friction angle of the sand; D is the diameter of the pile; and is the effective soil weight. In the current formulation, the initial slope of the p-y curve is assumed to be:
(3)
The tangent hyperbolic shape of the p-y formulation was first suggested by Parker and Reese (1970) as the best fit of p-y curves to be described as a continuous function providing a smooth transition between the initial slope of the curve and the ultimate soil resistance. This shape then assumes that the degradation of stiffness as a function of increasing deformation can be represented by the tangent hyper bolic function. In addition, it can be assumed that the shape of the curve is scaled mainly through the ultimate soil resistance. O’Neill and Murchison (1983) show that the currently applied method is a best estimate after reviewing a database of full-scale load tests with diameters in the range of 51mm to 1.22m. They concluded that it could not reasonably be expected to under-predict static pile-head displacement by more than about 70%, or static maximum moment by more than about 20%, although over-predictions in both cases can be higher. Over-predicting the pile-head dis placement is equivalent to under-predicting the soil stiffness and would result in an under-prediction of the fundamental frequency of the wind turbine structure, as shown in Figure 1. The API p-y curves for two different pile diameters is shown in Figure 2. They are D = 0.61m, which is equivalent to the pile diameter in the Mustang Island tests reported by Reese et al. (1974), and D = 6m, which is a typical pile diameter for offshore wind turbine monopile foundations. The p-y curves are shown plotted for a depth of 8m and for an internal friction angle of 39° equivalent to the Mustang Island test site. The considerations shown in Equations 2 and 3 are directly observed in the figure.
Figure 1: Relative frequency frequency deviations from three three offshore wind turbines in the Walney offshore wind farm
2. Review of the API p-y formulation
The API p-y formulation for piles in sand (API, 2011), are based upon the recommendations from O’Neill and Murchison (1983) and Reese et al. (1974). The lateral soil resistance ( p) as a function of the lateral deflection ( y) are then assumed as:
2.1 Governing parameters
A review of input parameters and the chosen shape of the API p-y curve is needed to identify short-comings that could potentially explain why a larger soil stiffness is observed for large diameter piles, as shown in Figure 1. The governing parameters are the ultimate soil resistance, the initial modulus of subgrade reac-
(1)
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tion, the initial stiffness distribution with depth and finally the representation of the strain level.
proximately two times steeper for a 6m-diameter pile than a 0.61m diameter pile for a deformation level of 1mm. It is easily observed form Figure 2 that the API p-y formulation do not properly account for changes in strain levels in the soil as a result of diameter variations.
Figure 2: The p-y curves for piles piles in sand: (solid line) line) D = 0.61m and (dashed line) line) D = 6m, with with z = 8m
2.1.1 Ultimate soil resistance Considering the ultimate soil resistance obtained from Equation 2, the expression represents the failure mode of shallow and deep soil layers. It is also a slight simplification of the ultimate soil resistance by Reese et al. (1974), which used large efforts to derive a rigorous theoretical basis for ultimate soil resistance. This basis accounts for the pile diameter, and it is currently assumed to be equally applicable to large-diameter piles. The formulations were adopted by API and have long, proven track records from the oil and gas industry. Thus, in respect of ultimate soil resistance, the formulation by Reese et al. (1974) is considered adequate in this paper for the design of large-diameter piles, although they might turn out to be over-conservative. 2.1.2 Effects of strain level First, a lateral depth related to the diameter of the pile, in which strains are mobilised as, for example, the concept in Terzaghi (1955) of bulb of pressure, is assumed. The average shear strain ( ) mobilised for a lateral pile movement ( y y) in the soil around the pile are given by Kagawa and Kraft (1980), after Matlock (1970):
Figure 3: Degradation of shear modulus as a function of shear strain: (hollow markers) D = 0.61m; (solid markers) D = 6m; ( ) y = 0.02mm; ( ) y = 1mm; ( ) y = 5mm
The concept of considering the strain level in the soil is supported by the diameter effects included in the p-y formulation for piles in clay (Stevens and Audibert, 1979). Stevens and Audibert recasted existing p-y formulations for piles in clay with a dependency on pile diameter. After reviewing a broader database of load tests, including large-diameter piles, they were able to derive an expression for the initial p-y slope increasing with the square root of pile diameter. Later research has revealed that a gradual transition between the modulus reduction behaviour of sand and clay (e.g. Ishibashi and Zhang (1993) sup porting that a formulation similar to that of Stevens and Audibert) could potentially be equally applica ble for piles in sand. 2.1.3 Initial sub-grade reaction modulus The applied initial sub-grade reaction modulus ( k ) is obtained from Meyer and Reese (1979) and is directly evaluated from Reese et al. (1974). The initial sub-grade reaction modulus is defined from the theory of linear elasticity (Terzaghi, 1955). Therefore, when considering the behaviour of sand, it is applicable for very small strain levels (<10 –5). This suggests that in general the initial sub-grade reaction modulus should be applied when determining the initial slope of the p-y curve. As shown in Figure 3, the soil modulus is already affected for deformations larger than 0.02mm for the Mustang Island test setup and reduced to ~50% for a deflection of 1mm. A linearisation of this part of the curve would therefore seem logical, although it would result in k being a
(4) where is Poisson’s ratio. By keeping y constant, it
then follows that the shear strain decreases as the diameter increases. Decreased shear strains yield an increase of soil shear modulus, which will effectively increase the p-y stiffness as the diameter increases. This is shown in Figure 3 following the stiffness degradation approach given in Khouri (1984). The markers illustrate the concept for lateral deformations for a 0.61m and a 6m diameter pile, respectively. Following the results of Figure 3, the slope of the p-y curve should theoretically be ap-
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function of a specific strain level. In this paper, the initial sub-grade reaction modulus is not investigated even though it could be subjected to variations not accounted for in the API approach. In general an in situ evaluation of k would yield more accurate results.
It is the opinion of the authors that k should should be considered as the initial modulus of sub-grade reaction and hence a constant. However, the soil modulus should be evaluated based on the correct strain level. It is therefore questionable if the API p-y formulation is the best estimate for evaluating deformations of large-diameter piles when the choice of the shape of the curve and n depends on a particular deflected shape of the pile.
2.1.4 Initial stiffness profile The current p-y formulation assumes the initial stiffness to increase linearly with depth, however, as is well recognised for sand, the response is governed by the isotropic stress level. A common attempt to account for the influence of isotropic stress level can be made by expressing the soil modulus ( E E ) as:
3. Suggestions for Modifications
Based on this review of the governing parameters, the API p-y formulation seems to provide a poor representation of the small strain stiffness variation with depth and the rate of stiffness degradation with increasing shear strain. Considering the behaviour of other geotechnical structures, Clayton (2011) analysed the deformation of a retaining wall by a range of constitutive models. He concluded that high initial stiffness, coupled with stiffness degradation with increasing strain, is needed to mimic the pattern of observed ground surface movements for structures that take the soil to intermediate strain levels. Clayton also states that predicted displacement patterns are sensitive to most parameters, including very small strain stiffness, rate of stiffness degradation, and anisotropy.
(5) where is the effective stress level; the value of is a reference is a reference soil modulus; and effective stress level for which . This formu-
lation is equally applicable to the shear modulus of the sand (G).
Much research has been carried out to determine the variation of the small strain soil moduli E and and G as a function of the confining pressure. Hertz (1881) found that n = 0.33 for uniform spheres by applying contact theory, and Goddard (1990) reported n = 0.5 for conical asperities. Based on measurements on real soil Hardin and Richart (1963), Hardin and Black (1968) and Drnevich and Richart (1970) all suggested applying n = 0.5 as a representative value. Considering the range of values reported in the literature, Hryciw and Thomann (1993) reported values of n ranging from 0.39 to 0.72 based on bender element tests on various sands. Wichtmann and Triantafyllidis (2009) carried out 163 resonant column tests on 25 different grain size distributions of quartz sand with subangular grain shape. They found n ranging from 0.41 to 0.58, increasing with decreasing uniformity of the sand grains. Furthermore, Wroth et al. (1979) re ported values of n to vary from 0.435 at very small strains, to 0.765 at very large strains.
A need for modification of particularly the small strain stiffness variation with depth and the rate of stiffness degradation with increasing shear strain could be justified to obtain a more accurate determination of the soil stiffness for large-diameter piles. This justification could be based on the strain levels observed in Figure 3 and the fact that the API ap proach has been fitted to a particular deformation shape (Parker and Reese, 1970). As a matter of choice, it is proposed to maintain the overall format of the API p-y formulation, but include an isotropic stress level correction for the small strain soil modulus. In addition, a strain level correction is to be included to account for the rate of stiffness degradation with increasing strain in the formulation of the initial stiffness. The proposed modifications take their set-point from the Mustang Island tests (Reese et al., 1974). 3.1 Depth effects
Parker and Reese (1970) discussed the representation of the initial stiffness of the p-y curve in a format equivalent to Equation 5. They concluded that for a realistic problem, k and and n may not be constants, but may be functions of a number of parameters, one of which is the deflection of the pile. Since the variation in soil modulus with depth may be approximated by a straight line for a particular deflected shape, the use of Equation 3 as a computational technique is valid.
On basis of fundamental behaviour of sand, and using that is approximately proportional to z , it is reasonable to assume that the formulation could be extended to account for the effective stress level by:
(6) where is a reference depth identified from the
original formulation and n is a site specific parameter expected to be in the range of 0.4–0.7.
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The deflections in the Mustang Island tests (Reese et al., 1974) were primarily in the upper 5m (200 inches) of the soil, and the moment peak was ~2.5m (100 inches) below the seabed. Thus, if = 2.5m is used, then the modified formulation should remain consistent with the Mustang Island tests. This modification causes a slightly higher stiffness above z = 2.5m and a slightly lower stiffness below z = = 2.5m. The curves fitted for the Mustang Island tests in Reese et al. (1974) are reproduced in Figures 4 and 5. It is seen that the measured moment peaks were actually above the predicted moment peaks. This supports the conclusion that the soil stiffness must have been higher above = 2.5m and lower below z = 2.5m. While this correction may have a small effect on the Mustang Island results, the effective stress correction is important when scaling to larger and stiffer piles, where layers of higher stress level are mobilised.
where D0 is a reference diameter that is equal to the pile diameter used to derive the original formulation D0 = 0.61m; is the initial soil stiffness; and m is a diameter exponent. Stevens and Audibert suggest applying m = 0.5 for clay. Tabulated values for m(y) have been included in Table 1 for realistic values of y after the stiffness degradation approach by Khouri (1984) and for a pile diameter of 6m.
Table 1: Diameter exponent for varying lateral displacement y [mm] [mm] M
1.0
2.5
5.0
7.5
10.0
0.29
0.41
0.52
0.58
0.63
Due to the nonlinear formulation of the stiffness degradation, m is a function of the deformation shape of the pile. Therefore, the highest values of m are applied for the top soil layers in which the pile deformations are largest, and conversely, the smaller values are applied for lower soil layers. However, as a first approximation a constant value of m is assumed. By choosing m = 0.5, the stiffness of the top soil layers will be under-predicted while the stiffness of the lower layers will be over-predicted. It is therefore estimated that this will be a representative value when evaluating the total soil stiffness. 3.3 Modified formulation
A modified formulation of the initial stiffness of the p-y curve therefore applies. Combining Equations 6 and 7 gives:
Figure 4: Static Mustang Island test (Reese et al., 1974)
(8)
where D0 = 0.61m and z 0 = 2.5m to ensure that the formulation is consistent with the Mustang Island tests. The modified formulation accounting both for strain and stress level dependency is illustrated in Figure 6 for a pile with D = 6m, together with the individual contributions to the modified stiffness. The figure is plotted for m = 0.6, as this value has been used in the benchmark study in section 4.
Figure 5: Cyclic Mustang Mustang Island test (Reese et al., 1974)
3.2 Strain level effects
The approach by Stevens and Audibert (1979) suggest the diameter to be included in the formulation as:
(7)
For completeness, the unmodified API p-y curve illustrated in Figure 2 is reproduced and plotted together with the modified p-y curves in Figure 7. They are plotted in a depth of 8m, which is below the reference depth. Hence the initial stiffness of the p-y curve for the D = 0.61m pile decreases, whereas the initial stiffness of the D = 6m pile increases. Ap plying Equation Equati on 8 instead of Equation 3 is therefore expected to give a more accurate evaluation of the soil stiffness, and hence the fundamental frequency of wind turbine structures supported by monopile foundations.
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cal parameters. It has been found that the proposed modifications provide a better prediction of the fundamental frequency, although they have only been verified for structures within one wind farm.
Figure 6: Modified p-y stiffness with D = 6m: (Solid thin thin line) Equation 3; (dotted-dashed line) line) Equation 6; (dashed line) line) Equation 7; (thick solid solid line) Equation 8, m = 0.6.
Figure 7: The p-y curves for piles piles in sand: (solid lines) lines) D = 0.61m; (dashed lines) D = 6m and X= 8m; (thin lines) unmodi fied API p-y curve; (thick lines) lines) modified API p-y curve
3.4 Benchmark study of full-scale measurements
To validate Equation 8, the modification of the API p-y formulation has been benchmarked against fullscale measurements from the same three wind tur bines in i n the Walney offshore wind farm (see Figure 1). The results are shown in Figure 8 assuming m = 0.6. In general, the modified p-y formulation gives a better prediction of the fundamental frequency and hence the total soil stiffness, although it still underestimates the fundamental frequency with ~2–4%. Based on the current level of knowledge, it is unwise to assign the evaluated effects in the modified ap proach higher importance. More detailed analysis and in situ measurements are needed to determine and understand the complexity in the results and the effects causing the remaining stiffness.
Figure 8: Benchmark of modified p-y formulation formulation using Equation 8 with with full-scale measurements measurements
A site-specific evaluation of the governing parameters k , m and n might therefore be required when considering other structures. The current tuning has only been made considering the total soil stiffness and a constant strain level correction has been as-
3. Discussion and Future Works
The proposed modifications to the API p-y formulation have been derived on basis of theoretical considerations without the introduction of new empiri470
sumed. However, it should be noted that a strain level correction depending on the deflection of the pile would be more accurate. The resultant p-y curves have not been compared to experimental determined p-y curves for large-diameter piles. This is necessary to increase robustness of the proposed modification. Therefore it is only recommended to extend the formulation to other sites if more detailed investigations and site-specific parameters are taken into account. The results presented in Figures 1 and 8 only covers a period of one and a half months within the first years of the Walney offshore wind farm lifetime. It could therefore easily be argued that the fundamental frequency of the wind turbine structures would decrease over time due to cyclic degradation of the soil. This is why application of a stiffer soil response should only be done with great caution. Although based on a five-month long record of the fundamental frequency for more wind turbines and corresponding wind speeds, it has been found that variations in the fundamental frequency are due to variations in the wind speed, as illustrated in Figure 9.
During periods with high mean wind speed, a decrease of the fundamental frequency is observed. As soon as the wind speed decreases, the fundamental frequency increases to the same value as previously observed for similar wind speed. This is expected to be directly related to the variations of soil stiffness. Based on this consideration and the current level of knowledge, degradation of soil stiffness has not been observed. Therefore, the proposed modifications of the API p y formulation provide a more accurate yet conservative approach in the determination of the total soil stiffness than the unmodified approach. It is also believed that the soil stiffness will not degrade with time. This hypothesis will need to be verified by longer data records. 4. Conclusion
The current API p-y formulation was found to significantly underestimate the stiffness of sand. An attempt was made to derive corrections to the initial stiffness of the API p-y curve by adding both a stress-level and a diameter correction. These corrections were derived on basis of sound theoretical considerations without the introduction of new empirical parameters. The modified p-y formulations were benchmarked against three randomly chosen wind turbine structures on the Walney offshore wind farm. Measurements of the fundamental frequency showed that they provide a better estimate of the total soil stiffness, although the best-estimate fundamental frequency still under-predicts the actual frequency of the structures. Benchmarking with full-scale measurements for large-diameter piles is found to be the best approach for a possible reformulation of the currently applied API p-y formulation. However, the proposed modifications must be benchmarked with full-scale measurements from structures within more offshore wind farms. In addition, the theoretical p-y curves must be compared with experimental p-y curves obtained from large-diameter piles before a complete and rigorous p-y formulation can be obtained. Nevertheless, it is a fact that the actual soil stiffness is under-predicted by the actual approach and that the modified formulation provides a better prediction of the measured fundamental frequency of the structures.
Figure 9: (Top) Evaluation Evaluation of long-term variation variation of the relative 1hr average fundamental frequency for one offshore wind turbine; (bottom) corresponding wind speed
Finally, long-term effects were considered based on the currently available data covering a five-month period. It was concluded that the t he soil stiffness is not expected to degrade over time, although longer timeseries is needed to verify this.
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