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Characteristic soil strength for axial pile capacity and its estimation with confidence for offshore applications
Structural Safety 63 (2016) 81–89
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Characteristic soil strength for axial pile capacity and its estimation with confidence for offshore applications Knut O. Ronold DNV GL, Veritasveien 1, N-1322 Høvik, Norway
a r t i c l e
i n f o
Article history: Received 29 June 2015 Received in revised form 28 August 2016 Accepted 28 August 2016
Keywords: Soil strength Pile capacity Model uncertainty Probabilistic analysis Code calibration Confidence estimation
a b s t r a c t Many offshore design codes require the characteristic soil strength for axial pile capacity to be estimated with caution or conservatism, i.e. – in statistical terms – it has to be estimated with confidence. This study is concerned with a reliability-based calibration of the necessary minimum confidence for such estimation. The study demonstrates how to estimate characteristic soil strength for axial pile capacity with confidence and provides an approach to such estimation that will render an incentive to obtain more soil strength data and thereby give credit to the geotechnical designer who opts to test more. The study capitalizes on existing load and resistance models for an offshore example pile from the literature and refers to a number of relevant offshore foundation design codes. In order to accomplish the purpose, a number of prerequisites for a successful study are dealt with before the confidence calibration itself is presented. A suitable approach to represent model uncertainty associated with axial pile capacity predictions is presented and implemented in the reliability analyses that are used for the calibrations. The approach implies that the standard deviation of the ratio between true and predicted capacity is represented as a function of the pile length rather than as a constant. This approach is supported both by theory and by data. Based on stochastic models for load and resistance in conjunction with a method for prediction of axial pile capacity, a second-order reliability analysis of the example pile is carried out. With an assumption of perfect knowledge of the soil strength, with characteristic soil strength defined as the mean value and with prescribed load factors for permanent and environmental loads, the necessary requirement for the material factor on the characteristic soil strength is calibrated by tuning the reliability analysis to meet a prescribed target failure probability. With the calibrated material factor kept unchanged, the reliability analysis is repeated, now with the stochastic model for soil strength altered to include statistical uncertainty owing to limited soil data. The value of the characteristic soil strength is adjusted downward until the result of the reliability analysis again meets the prescribed target failure probability. The resulting value of the characteristic soil strength is interpreted as ‘‘the conservatively assessed value” to be used in design when soil data are limited and this value is subsequently used to find the corresponding minimum confidence for characteristic value estimation by capitalizing on the properties of the Student’s t distribution. Results are presented for target annual failure probabilities in the range 105–104. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction For geotechnical design of axially loaded offshore piles, the characteristic value of the soil strength as a function of depth is used in conjunction with a capacity prediction model and a material factor to determine the design capacity. A number of different capacity prediction models exist; see for example Lehane et al. [17] and Lacasse et al. [14].
E-mail address: [email protected] http://dx.doi.org/10.1016/j.strusafe.2016.08.004 0167-4730/Ó 2016 Elsevier Ltd. All rights reserved.
The characteristic value of the soil strength is usually defined in the design standard which is used for the geotechnical design of the pile, and the definition is usually some objective measure in the probability distribution of the strength, for example the mean value or some lower-tail quantile. Sometimes the design standard also provides requirements for the estimation of the characteristic soil strength with caution and conservatism, i.e. – in statistical terms – estimation with confidence, such as when the estimation is to be based on statistical methods and limited data. Requirements for the material factor to be used in the geotechnical pile design are also given in the design standard and these require-
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K.O. Ronold / Structural Safety 63 (2016) 81–89
ments are specific for the specific definition of characteristic soil strength employed by the standard. The offshore standard NORSOK G-001 [20] specifies that the characteristic value of soil strength to be selected for use in design shall be a ‘‘conservatively assessed mean value”. This implies that the definition of the characteristic value is the mean value and that the assessment of this characteristic value shall be conservative, i.e. in statistical terms that it has to be estimated with confidence. The offshore standard NORSOK N-001 [21] has similar wording for characteristic soil strength, but neither NORSOK G-001 nor NORSOK N-001 specifies any particular requirement regarding which confidence level shall be used for the conservative estimation. DNVGL-OS-C101 [5] is a standard for design of offshore steel structures and their foundations. For cases where a large soil volume is involved, such as for geotechnical design of a long offshore pile, DNVGL-OS-C101 specifies that the characteristic soil strength is defined as the mean value, thereby reflecting that fluctuations of the soil strength from point to point within the soil volume tend to average out over the length of the pile. DNVGL-OS-C101 requires this characteristic soil strength estimated with caution in design and – when statistical methods are used for the estimation – recommends that the estimation be carried out with at least 95% confidence. When the mean value has to be estimated from limited data, the resulting central (unbiased) estimate – the sample mean – will be encumbered with uncertainty. There will be about equal probability for the true but unknown mean value of being greater than respectively smaller than this central estimate. It is not attractive to use this estimate in design when there is such a large probability – about 50% – that the true but unknown mean value is smaller. As exemplified above, the design standards therefore often require that a smaller estimate than the central estimate be used in design, i.e. an estimate with confidence. The confidence is the probability that the true but unknown mean value is greater than the smaller estimate used in design. The smaller the estimate used, the larger is the confidence. Statistical methods exist that specify how much smaller than the central estimate a smaller estimate needs to be in order to meet a specified confidence level. Rather than referring to confidence, some design standards refer to ‘‘the probability of a worse value”, which is the complement of the confidence; for example 5% probability of a worse value corresponds to 95% confidence. Because of the uncertainty in the central estimate of the mean value when data are limited, use of this estimate in design will imply that the safety level which is achieved under an assumption of perfect knowledge, including known mean value, cannot be met. This can be remedied by using a smaller estimate in design, i.e. an estimate with confidence. The smaller the estimate used, i.e. the larger the confidence applied, the larger the safety level achieved in design will be. Because perfect knowledge is usually assumed when design codes are calibrated and because it is a goal to maintain the safety level achieved under perfect knowledge when carrying out designs based on limited data, the requirement for minimum confidence can be established as the confidence level which is such that the safety level achieved in design is equal to the safety level achieved under perfect knowledge. This paper presents the probability calculations necessary to determine which confidence level should be required as a minimum for the estimation of characteristic soil strength for axial pile capacity prediction when data are limited and when the characteristic soil strength is defined as the mean value. A site-specific example case referring to an offshore jacket pile is used for this purpose. The resulting requirement for minimum confidence represents what should be a lower bound for the confidence level from a safety perspective. The determination of the requirement for
minimum confidence serves as support for deciding which confidence level should be used in design. The calculations are based on structural reliability methods applied to a case study which capitalizes on axial pile capacity predictions by means of a recognized method. The case study considered is a study of a jacket pile foundation in clay reported by Lacasse et al. [14] and the capacity prediction method considered is the so-called NGI-05 method described by Karlsrud et al. [13]. To accomplish the calculations, an initial reliability analysis of the pile in question is carried out, leading to calibration of a case-specific set of partial safety factors that will provide a design meeting a prescribed target safety level. This requires an adequate stochastic model for the model uncertainty associated with the chosen method for axial pile capacity prediction. A new such model, which accounts properly for the prediction uncertainty’s dependency on the pile length, is developed for this purpose and is presented first. This is followed by a presentation of the particular example case together with probabilistic and deterministic modelling for the initial reliability analysis. This probabilistic analysis is executed under the assumption of perfect knowledge of the stochastic soil strength, i.e. no statistical uncertainty in the soil strength properties is included. This is a standard approach in code theory as outlined in Madsen et al. [18]. The characteristic soil strength is defined as the mean value which leads to a characteristic axial pile capacity equal to the mean value of the capacity. The results of the probabilistic analysis are presented in terms of the required material factor on capacity as a function of the target annual failure probability. A modification of the probabilistic model is subsequently introduced by which the assumption of perfect knowledge is omitted and replaced by a stochastic model for soil strength which reflects the effect of limited data. This is done by including statistical uncertainty in the reliability analysis by means of probability distributions of the statistically uncertain soil strength parameters. The reliability analysis is repeated based on this revised model and the numerical value of the characteristic soil strength is adjusted downward until the reliability analysis with this model provides the same annual failure probability as the one achieved under the assumption of perfect knowledge. The thus adjusted value of the characteristic soil strength corresponds to a characteristic strength estimated with confidence. This confidence can now be calibrated by means of statistical formulas for estimation of mean values with confidence. The paper shows how this is done and presents the resulting calibrated necessary minimum confidence. The paper may serve as a contribution to ongoing discussions in the geotechnical community regarding the characterization of soil properties. The paper may thereby supplement other recently published material relevant to this issue of characterization and dealing with the same topics that are addressed in the paper. DNV-RP-C207 [2] addresses on a general basis the issue of definition of characteristic values and how to estimate them with confidence. Schneider and Schneider [31] deal with an interpretation of the current definition of characteristic soil strength in Eurocode 7 (EN 1997-1) [6] for onshore foundations and present a simplified way to account for the variance reduction that comes with spatial averaging of soil properties. Orr [22] reviews how important aspects of risk and reliability in geotechnical engineering are addressed in Eurocode 7. ISO 2394:2015 [10] has a wider scope than Eurocode 7 and provides state-of-the-art guidelines for how to carry out reliability-based safety factor calibrations. Fenton et al. [8] provides a review of current practice in various standards regarding definition and estimation of characteristic soil strength. Phoon and Kulhawy [23,24] deal with statistical characterization
K.O. Ronold / Structural Safety 63 (2016) 81–89
of soil properties and Ching et al. [1] brings this one step further by addressing the statistical uncertainty in estimates of parameters used to characterize soils.
2. Stochastic representation of model uncertainty in axial pile capacity predictions The model uncertainty associated with predictions of axial pile capacity by means of a particular capacity prediction model can be represented by a random model uncertainty factor, see Ronold and Bjerager [28]. The random model uncertainty factor is defined as the ratio between the true axial pile capacity and the predicted axial pile capacity as predicted by the prediction model in question. The random model uncertainty factor is a random, or stochastic, factor because the ratio between the true and predicted axial pile capacities exhibits variability from case to case, i.e. from one pile to another. It is a common and accepted approach to calibrate the random model uncertainty factor – its mean value, standard deviation and probability distribution – to results of full scale tests where the true capacity is represented by the capacity measured in the tests, see Ditlevsen [3] and ISO 2394:2015, Annex D, Clause D4.1 [10]. It has been common in the past to represent the standard deviation of the random model uncertainty factor for axial pile capacity as a constant independent of the pile length, see for example Ronold et al. [27] and Lacasse et al. [14]. It is, however, not obvious that this is an adequate representation. The true pile capacity is a random variable. Ronold [25] showed that when the soil strength adheres to the quadratic exponential decay model for vertical autocorrelation, q(r) = exp(r2/R2), then the variance of the axial pile capacity of a friction pile in clay is proportional to
" ! #! pffiffiffi pffiffiffiffi L 1 1 2 L2 2 þ R exp 2 1 k ¼ 2 LR p U R 2 2 L R 2
ð1Þ
lim kL ¼ R
L=R!1
pffiffiffiffi
p
3. Probabilistic and deterministic modelling 3.1. Example case A four-legged offshore jacket structure supported by a group of four tubular steel piles at each leg is considered. The piles are driven 90 m into a layered soil deposit consisting mainly of a series of clay layers. The water depth at the location is 119 m. The pile diameter is 2.438 m. Failure is defined to occur when the average load per pile in the most heavily loaded pile group of four piles, after full redistribution of loads between the pile groups and between the piles within each group, exceeds the capacity of one pile. This example appears as Case Study A in Lacasse et al. [14] from which useful information about loading, soil strength and axial pile resistance is extracted. In the hypothetical case that the four pile capacities in the most heavily loaded pile group are perfectly correlated, the failure criterion implies full utilization of the piles in this pile group and the foundation system has no reserve capacity. In the realistic case that the four pile capacities are only partially correlated, the failure criterion implies that the foundation system has some reserve capacity. 3.2. Loading The average axial load per pile in the most heavily loaded pile group of four piles is considered. This load is taken as one-fourth of the annual maximum total compressive axial load for this pile group, since there are four piles in a pile group and since failure is defined to occur when the average load per pile exceeds the capacity of one pile in the most heavily loaded pile group. The average annual maximum axial compressive load of one pile in the most heavily loaded pile group is represented as the sum of two components
S ¼ X1 þ X2 X3
where k is referred to as a variance reduction factor and expresses the ratio between the capacity variances with and without spatial average effects included. L denotes the pile length, R denotes the vertical correlation length in the quadratic exponential decay model for the autocorrelation function for the undrained shear strength field along the pile, and U denotes the standard Gaussian cumulative distribution function. As the ratio L/R grows large, kL will approach the following limit
ð2Þ
which is a constant when the vertical correlation length R is roughly the same from site to site, which is not an unreasonable assumption. The convergence is fast. Actually the limit for kL for large values of L is a constant regardless of the choice of autocorrelation function model, see Vanmarcke [32]. This observation regarding the limit for kL serves as inspiration to suggest that the standard deviation of the model uncertainty factor for total axial pile capacity predicted by the NGI-05 method (and any other prediction method for that matter) may reasonably well be assumed to be proportional to the inverse of the square root of the pile length. This is assumed in the following and can be considered reasonable as long the capacity is not too dominated by tip resistance. Although maybe not a perfect model for the standard deviation of the model uncertainty factor when there is soil stratification and substantial tip resistance, it still honours available pile test data considerably better than the model with a constant standard deviation independent of the pile length.
83
ð3Þ
where X1 = 25.8 MN is a fixed term due to deadweight and X2X3 is a stochastic term. Reference is made to Lacasse et al. [14]. The variable X3 has a Gumbel distribution with mean value 11.17 MN and standard deviation 4.57 MN, see Lacasse et al. [14]. The 99% quantile (‘‘the 100-year value”) in this distribution is 25.5 MN and is used as the characteristic environmental load, X3char. The variable X2 is a random factor meant to account for statistical uncertainty in the fitted Gumbel distribution as well as model uncertainty associated with adopting the Gumbel distribution as the distribution model. The distribution of X2 is taken as a normal distribution with mean value 1.0 and standard deviation 0.05. The characteristic deadweight load X1char is set equal to X1. With a load factor cfD for deadweight load and a load factor cfE for environmental load, the total design load becomes
Sd ¼ cfD X 1 þ cfE X 3char
ð4Þ
It is noted that the applied loading leading to the average annual maximum axial load S on one pile consists of dynamic wave loading superimposing a static permanent deadweight load. The pile is at the same time subject to a shear force and bending moment caused by the same wave loading; however, this shear force and bending moment and the lateral capacity of the pile to withstand it are outside the scope of this study and not further dealt with here. 3.3. Soil strength and axial pile capacity The soil profile consists primarily of a sequence of clay layers with only two thin sand layers interbedded at and near the seabed.
K.O. Ronold / Structural Safety 63 (2016) 81–89
The undrained shear strength in the clay layers has been sampled by a number of UU tests in the laboratory. The soil stratification appears from Table 1 together with an indication of the number of strength observations in each clay layer. The soil stratification is also indicated in Fig. 1, which is based on Lacasse et al. [14], together with estimates of mean value and standard deviation of the undrained shear strength in each layer. The shear strength of soil comes about as the sum of the strengths of frictional resistance bonds between soil particles over a small unit area; see Lambe and Whitman [15]. Under the central limit theorem, this implies that the shear strength will asymptotically be normally distributed. This observation is also acknowledged by Vanmarcke [33] who refers to a number of supporting studies. Within each identified clay layer in Fig. 1, the undrained shear strength is therefore modelled as a normally distributed variable with mean value and standard deviation as specified in Fig. 1. For deterministic design, the characteristic undrained shear strength is defined as the mean value of the undrained shear strength. This definition is consistent with the definition used in NORSOK G-001 [20] and DNVGL-OS-C101 [5] for the characteristic soil strength for axial pile capacity. It is noted that while the undrained soil strength is assumed to be normally distributed as justified above, a lognormal distribution is often used in practice in order to avoid unrealistic realizations of negative shear strengths when the standard deviation is large, see for example Fenton and Griffiths [7] and Schneider and Schneider [31]. Other distribution models could be applied with this goal in mind, such as the quadratic Weibull distribution described by Lange and Winterstein [16] – a generalized and ‘‘distorted” Weibull distribution which, in contrast to the lognormal distribution, is capable of retaining not only the mean value and standard deviation, but also the skewness of the undrained shear strength. For the present case, the standard deviations are of a magnitude that the issue of achieving unphysical realizations of negative shear strengths is immaterial, but this may not necessarily be the general case. Based on the undrained shear strengths in the soil layers penetrated by the pile, the axial pile capacity is predicted by means of the so-called NGI-05 method; see Karlsrud et al. [13]. The mean value of the predicted axial pile capacity is E[Q] = 118.6 MN, see Lacasse et al. [14]. The coefficient of variation of the predicted axial pile capacity can be determined to be COV[Q]=0.0854 based on information given in Lacasse et al. [14]. With approximately linear relationships between soil strength and segment capacity in each layer and between soil strength and tip resistance in the bottom layer, the distribution of the axial pile capacity becomes a normal distribution when the soil strength in each layer is normally distributed. This is adopted in the following. The characteristic axial capacity for use in deterministic design becomes
Q char ¼ E½Q ¼ 118:6 MN
ð5Þ
Table 1 Soil stratification. Soil layer No.
Depth range (m)
Soil type
1 2 3 4 5 6 7 8 9 10
0–1 1–3.5 3.5–18.5 18.5–24 24–32.2 32.2–38.5 38.5–50 50–56.2 56.2–69.5 69.5–92
Sand Clay Clay Sand Clay Clay Clay Clay Clay Clay
No. of observations of sUU u 3 18 7 6 6 5 7 16
Undrained shear strength, suUU (kPa) 0
100
200
300
400
500
600
0 Mean value 10
Mean-one st.dev.
20
30
40
Depth (m)
84
50
60
70
80
90
100 Fig. 1. Undrained shear strength vs. depth: Mean value and mean minus one standard deviation as estimated from data. Soil layer interfaces at discontinuities and knee points. (Based on Lacasse et al. [14].).
and with a material factor cm for axial pile capacity, the design capacity becomes
Qd ¼
Q char
ð6Þ
cm
Let li denote the expected pile segment capacity in the ith soil layer and let ri denote the standard deviation of the pile segment capacity in the ith soil layer when no spatial averaging is accounted for. It is unrealistic not to account for spatial averaging of soil strength along each pile segment. It is not unreasonable to assume that the autocorrelation function for the undrained shear strength follows the quadratic exponential decay model, q(r) = exp(r2/R2), also known as the Gaussian autocorrelation model. Vanmarcke [32] uses this autocorrelation model for soil strength. Isaaks and Srivastava [12] state that this model is often used to model extremely continuous phenomena. Soil strength may be assumed to fall in this category when considering the geological processes that created the associated soil deposits. Based on the assumption of the quadratic exponential decay model for the autocorrelation function, a variance reduction factor ki can be applied on the capacity variance r2i for the ith pile segment
ki ¼
2 L2Si
" ! #! pffiffiffi LSi 1 1 2 L2Si þ R exp 2 1 LSi R p U 2 2 2 R R pffiffiffiffi
ð7Þ
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K.O. Ronold / Structural Safety 63 (2016) 81–89
where LSi is the segment length in the ith soil layer and R is the vertical correlation length in the quadratic exponential decay model for the autocorrelation function for soil strength, assumed to be identical in all soil layers. Reference is made to Ronold [25]. Ignoring the minor capacity contributions from the two thin sand layers at the seabed and at 20 m depth, respectively, and utilizing that the major part of the axial pile capacity comes from frictional capacity in the clay layers as confirmed by Lacasse et al. [14], the coefficient of variation of the capacity Q can be approximated by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u N u k i r2 i u u COV Q ðRÞ COV½Q u i¼1N u X t r2
ð8Þ
i
i¼1
where the sums are over N clay layers and COV[Q] = COVQ(1) = 0.0854 is the coefficient of variation of the capacity Q when no spatial averaging is accounted for. Note that Eq. (8) is derived by assuming that the pile segment capacities in different layers are statistically independent. Table 2 summarizes the mean value and coefficient of variation of the pile capacity for a number of relevant values for the vertical correlation length R. Vanmarcke [32,33] reports vertical scales of fluctuation dV in the range between 1 and 5 m for soil properties, the higher value being for an example consisting of the undrained shear strength of a clay deposit. This corresponds to vertical correlation lengths R in the range between approximately 0.5 and 3 m, referring to pffiffiffiffi the relationship dV ¼ R p. For vertical correlation lengths R in this range, Table 2 shows a considerable effect of spatial averaging relative to the case without any spatial averaging (R = 1), whereas there is hardly any such effect if R had been as large as 10 metres.
justified above. A nonlinear regression analysis of the data leads to the following estimates for the mean value and the standard deviation of the model uncertainty factor
^ mod ¼ 1:533 L0:0966 E½X
ð10Þ
^ mod ¼ 0:917 pffiffiffi D½X L
ð11Þ
in which the pile length L is given in units of metres. The distribution of the model uncertainty factor conditioned on L is assumed to be a lognormal distribution. This representation of the model uncertainty factor is derived from compressive tests on axially loaded piles in clay and is applied as a good approximation in the present case where the layered foundation soils consist primarily of clay layers with only a couple of relatively thin sand layers interbedded near the seabed. If the model uncertainty factor for total axial capacity of a 90 m long pile in sand has the same mean, standard deviation and distribution type as the model uncertainty factor for total axial capacity of a 90 m long pile in clay, then this representation of Xmod would be exact for the present example pile. If the mean and standard deviation for a pile in sand deviate somewhat from those for the pile in clay, then this representation would only be approximate; however, it would be a good approximation, because the sand layers are relatively thin and the major part of the axial pile capacity comes from frictional capacity in the clay layers as confirmed by Lacasse et al. [14]. Note that what is being characterized as model uncertainty here, in addition to true model uncertainty, implicitly contains other sources of uncertainty (spatial variability and statistical uncertainty) associated with the pile capacity predictions in the database. 3.5. Failure criterion and design rule
3.4. Model uncertainty The model uncertainty factor Xmod for total axial pile capacity is defined as the ratio Qtrue/Qpredicted in which Qtrue denotes the true capacity and Qpredicted denotes the predicted capacity, here specifically the capacity predicted by the NGI-05 method. When data from full scale pile tests to failure are available, measured capacities – one per test – are used as measures of the true capacity, and one realization of Xmod can be established per test. This allows for interpretation of the empirical distribution of Xmod. Sixteen observations of the model uncertainty factor (i.e. of the ratio Qtrue/Qpredicted) are available from compressive axial load tests on piles in clay. This data set stems from the Norwegian Geotechnical Institute’s ‘‘super piles” database. The mean value exhibits a slight variation with the pile length L. This variation can reasonably well be represented by
E½X mod ¼ c1 Lc2
ð9Þ
over the range of pile lengths covered by the tests. This range includes the example pile length of 90 m. The standard deviation is assumed to be inversely proportional to the square root of L as
Table 2 Axial pile capacity properties as function of vertical correlation length. Vertical correlation length, R (m)
Mean capacity, E [Q] (MN)
Coefficient of variation of capacity, COVQ
0.5 1 3 5 10 1
118.6 118.6 118.6 118.6 118.6 118.6
0.0265 0.0366 0.0575 0.0677 0.0784 0.0854
Failure is defined above to occur when the average load per pile in the most heavily loaded pile group of four piles, after full redistribution of loads between the pile groups and between the piles within each group, exceeds the capacity of one pile. This implies that the failure criterion can be written as
S P X mod Q
ð12Þ
where S denotes the average annual maximum axial compressive load per pile in the most heavily loaded pile group, Q denotes the predicted axial capacity of one pile, and Xmod is the model uncertainty factor used to account for the model uncertainty associated with this prediction, such that the product XmodQ represents the true capacity. As stated in 3.1, this failure criterion implies some reserve capacity in the foundation system when the four pile capacities in the most heavily loaded pile group are not fully correlated. The corresponding deterministic design rule is
Sd 6 Q d
ð13Þ
in which Sd is the design load and Qd is the design capacity. Introducing characteristic load and resistance values and corresponding load and material factors, the design rule can be rewritten as
cfD X 1char þ cfE X 3char 6
Q char
ð14Þ
cm
The design equation comes about when the design rule is turned into an equality,
cfD X 1char þ cfE X 3char ¼
Q char
cm
:
ð15Þ
86
K.O. Ronold / Structural Safety 63 (2016) 81–89
3.6. Limit state function and failure probability The failure criterion, normalized by the design equation, can be used to formulate the following limit state function
gðXÞ ¼
X mod Q cm X1 þ X2 X3 Q char cfD X 1char þ cfE X 3char
ð16Þ
in which X denotes the vector of stochastic variables. The probability of failure is
PF ¼ P½gðXÞ 6 0
ð17Þ
and can be calculated by a second-order reliability method (SORM) as described in Madsen et al. [18]. 4. Reliability analysis A second-order reliability method as described in Madsen et al. [18] is used to carry out reliability analyses and calculate the probability of failure. The load factor cfE for environmental load is prescribed equal to 1.3 in compliance with the requirements set forth in NORSOK [20,21] and DNVGL-OS-C101 [5]. Likewise, the load factor for deadweight load is prescribed equal to 1.0. This leaves the value of the material factor cm as a free parameter to be adjusted until the reliability analysis produces a probability of failure which is equal to a specified target failure probability. The value of cm which is thus calibrated to meet the specified target reliability is thus the value of the material factor which is necessary to meet this safety level in design. Results are presented in Table 3 for three values of the target annual failure probability, viz. 105, 5105 and 104, and for a range of values for the vertical correlation length R for the soil strength. Table 3 indicates that for the specific 90 m long pile studied and assuming the characteristic pile capacity Qchar is defined as the mean capacity, see Eq. (5), then a material factor requirement of cm = 1.3 is sensible for a realistic range of vertical correlation lengths between 0.5 and 3 m. The results in Table 3 indicate some sensitivity in the calibrated requirement for the material factor cm to the target annual failure probability and some sensitivity to the value of the vertical correlation length R for the soil strength. In particular, this implies that ignoring the effect of spatial averaging by assuming infinite vertical correlation length R will lead to a somewhat conservative calibrated requirement for cm. For the expected range of realistic R values, i.e. 0.5–3 m, it is reassuring to find that the requirement for cm comes out relatively close to the requirement 1.3 set forth in NORSOK and DNVGL-OSC101. The calibrated requirement is somewhat higher, approximately 1.45–1.50, if one follows the recommendation of a target annual failure probability of 105 for high safety class according to NKB [19] for a structure which has some redundancy at failure. However, it is almost ‘‘spot on”, in the neighbourhood of 1.3, if the target annual failure probability can be set according to accepted past practice: For offshore structures appropriately designed to meet recognized codes, Fjeld [9] reports that the annual failure
probability is of the order of 105 to 104 with higher values for steel structures – such as the present example pile – than for concrete structures. 5. Calibration of necessary minimum confidence level For each vertical correlation length R considered, the calibrated material factor cm is kept unchanged and thereby also the specified target failure probability. Let again li denote the expected pile segment capacity in the ith soil layer and let ri denote the standard deviation of the pile segment capacity in the ith soil layer when no spatial averaging is accounted for. When statistical uncertainty is accounted for in the reliability analysis, the predicted stochastic total pile capacity can be expressed as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q¼ ðli þ ri ki ðRÞ þ T i ðni ranki ÞÞ ni i¼1 N X
ð18Þ
in which T i ðni ranki Þ is a Student’s t distributed variable with ni ranki degrees of freedom, ni is the number of soil strength observations in the ith soil layer, ranki = 1 when the mean soil strength is constant in the ith soil layer, and ranki = 2 when the mean soil strength is linear with depth in the ith soil layer. N is the number of pile segments and equals the number of soil layers, li is the expected pile segment capacity of the ith segment, and ri is the standard deviation of the ith segment capacity when no spatial averaging is accounted for. Contributions from the two thin sand layers near the top of the pile are ignored as insignificant. It is noted that Eq. (18) is correct given the assumptions being made that the actual mean value of the ith summand follows the distribution of the sample mean and the actual variance of the ith summand follows the distribution of the sample variance. The rationale behind Eq. (18) is that when the soil strength sui in the ith soil layer is assumed to be normally distributed, and data for estimation of its mean value E½sui and standard deviation D½sui are limited and cause the estimates of these distribution parameters to be uncertain, then sui can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sui ¼ E½sui þ D½sui 1 þ 1=ni T i , where Ti is Student’s t distributed as described above, see Ditlevsen and Madsen [4]. When this shear strength is integrated over the circumference and height of the ith pile segment (in accordance with the NGI-05 method), spatial averaging comes into play. When the spatial averaging is taken into account, the expression for the ith segment capacity in the summand in Eq. (18) results. Statistical uncertainty in the estimates of the undrained shear strength has been retained through all steps and is thus reflected in the resulting expression for the capacity Q in Eq. (18). From Eq. (7), the variance reduction factor for pile segment capacity in the ith soil layer reads
ki ¼
2 L2Si
" ! #! pffiffiffi LSi 1 1 2 L2Si þ R exp 2 1 LSi R p U 2 2 2 R R pffiffiffiffi
Table 3 Results of reliability analysis: Calibrated requirement for the material factor cm. Vertical correlation length R (m)
0.5 1 3 5 10 1
ð19Þ
Target annual failure probability 105
5105
104
1.45 1.46 1.49 1.51 1.54 1.56
1.33 1.34 1.36 1.38 1.40 1.42
1.28 1.28 1.31 1.33 1.35 1.36
where LSi is the segment length in the ith soil layer and R is the vertical correlation length in the quadratic exponential decay model for the autocorrelation function for soil strength, assumed to be identical in all soil layers. The segment length equals the soil layer thickness in each soil layer, except in the lowermost layer penetrated by the pile where the pile does not penetrate all the way to the bottom of the layer.
K.O. Ronold / Structural Safety 63 (2016) 81–89
The characteristic value of the total capacity, estimated based on soil strength estimates with confidence 1 a, can be expressed as
^ char ¼ Q
N X r^ i ^ i pffiffiffiffi ðl t ni ranki ð1 aÞÞ ni i¼1
ð20Þ
where tni-ranki(1 a) is the 1 a quantile in the Student’s t distri^ i is the central estimate bution with ni ranki degrees of freedom, l ^ i is of the expected pile segment capacity of the ith segment, and r the central estimate of the standard deviation of the ith segment capacity when no spatial averaging is accounted for. It is here utilized that the confidence 1a is defined as the probability that the true but unknown mean soil strength in the ith layer is greater than the characteristic soil strength estimated with this confidence. It is noted that for given value of 1 a, the quantile tni-ranki(1 a) is ^ char will increase a decreasing function as ni increases, such that Q with increasing number of soil strength observations ni, i = 1,. . .N. ^ char , necessary to maintain the reliability Once the value of Q when statistical uncertainty in the soil strengths is included in the reliability analysis, has been determined, Eq. (20) can be solved with respect to 1 a, the resulting value of which can then be interpreted as the necessary minimum confidence level for cautious estimation of the characteristic value of soil strength in design. The results of this calibration of the necessary minimum confidence level required to maintain the specified target reliability when the amount of soil strength data are limited to the extent available for the present example case are presented in Table 4. The results are presented as a function of the vertical correlation length R and of the target failure probability. Each tabulated value for the confidence level 1 a in Table 4 is calibrated based on the corresponding value of the material factor cm tabulated in Table 3 under preservation of the value of the target annual failure probability. It appears from Table 4 that the requirement for confidence in the estimation of characteristic soil strength is rather sensitive to the value of the vertical correlation length R. The larger the correlation length, the higher is the requirement for confidence level. The sensitivity to the specified target annual probability of failure is considerably smaller, as long as realistic correlation lengths in the range 0.5–3 m are considered. For the expected range of vertical correlation lengths up to about 3 m, a confidence of no more than 70% appears to suffice. The results presented in Table 4 are valid for the actual pile length of 90 m investigated. For shorter piles for which less effect of spatial averaging of soil strength can be expected and for which the standard deviation of the model uncertainty factor will be larger, higher requirements for the confidence level can be expected. Also for piles with less variability in the annual maximum load, higher requirements for the confidence level can be expected.
Table 4 Results of reliability analysis: Calibrated requirement for minimum confidence level 1a. Vertical correlation length R (m)
0.5 1 3 5 10 1
Target annual failure probability 105
5105
104
0.583 0.600 0.663 0.719 0.825 0.994
0.576 0.591 0.645 0.691 0.761 0.840
0.574 0.587 0.639 0.680 0.740 0.800
87
6. Sensitivity to number of strength observations The results presented for the necessary confidence in Table 4 are based on the actual number of test data in each of the soil layers identified in Table 1 and Fig. 1. It is of interest to study the sensitivity in the calibrated confidence to a change in the number of test data. A major contribution to the axial pile capacity comes from the lowermost layer along the pile, from 69.5 m depth downwards. There are 16 observations of the undrained shear strength in this layer. The effect of halving this number to 8 is studied. For a vertical correlation length R = 3 m, it appears that the necessary confidence will increase slightly from 0.66 to 0.69 when the target failure probability is 105, and from 0.64 to 0.66 when the target failure probability is 104. In other words, there is only a moderate dependency of the required confidence level on the number of available soil strength observations. This is much in agreement with the findings by Ronold and Lotsberg [29] for a different limit state (fatigue failure rather than ultimate failure) and a different definition of characteristic strength (the 2.3% quantile rather than the mean value), for which the required confidence level also came out with only a moderate dependency on the number of strength data available.
7. Discussion It appears from the calibration that for vertical correlation lengths up to an expected maximum of no more than 5 m, a confidence of up to 72% will be sufficient for estimation of characteristic soil strengths. It is noted that this result is obtained for a 90 m long pile for target failure probabilities in the range 105 to 104 and that it remains to be verified if this result can be generalized to other pile lengths. Selection of characteristic soil strengths with a confidence larger than the calibrated minimum required for estimation presented in Table 4, for use in design, can be done in order to account for other uncertainties than statistical uncertainty not accounted for in the calibration of the material factor, for example the uncertainty associated with the fact that the vertical correlation length R is usually unknown. Measurement uncertainty associated with the level of quality of the soil investigation program, the soil sampling procedures and the laboratory testing of soil samples forms another uncertainty source that one may want to account for by selecting characteristic values with a larger confidence than the minimum required to account for the effect of limited data. Note that in case of for example onshore piles where the loading is less variable than for offshore piles and more uncertainty importance therefore is on soil strength, higher requirements for confidence for cautious estimation of characteristic soil strengths would result than those reported in Table 4. This can be illustrated by looking at the sensitivity for an increased coefficient of variation COVQ of the predicted axial pile capacity. Such an increase would correspond to a situation with reduced uncertainty importance on load and increased uncertainty importance on capacity. If COVQ is increased by 10% from 0.0575 to 0.0633 for R = 3 m, the calibrated confidence requirement for estimation of soil strength increases from 0.66 to 0.71 when the target failure probability is 105, and from 0.64 to 0.67 when the target failure probability is 104. This also illustrates the importance of having as correct a model for model uncertainty representation as possible in the reliability analysis, thereby to get a correct split of uncertainty importance between load and strength variables before the confidence requirement is calibrated.
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Lacasse et al. [14] recommends that the characteristic value of soil strength for use in design of axially loaded piles should be selected as the sample mean minus one half of the sample standard deviation. Schneider [30] proposes the same recommendation for characteristic soil strength in general, not only for design of axially loaded piles. Calculating the corresponding characteristic axial pile capacity – which then becomes the ‘‘selected characteristic pile capacity” for use in design – and substituting this biased characteristic capacity estimate on the left hand side of Eq. (20), Eq. (20) can be solved with respect to the confidence 1a. The result of this exercise is an associated confidence 1a which on average over the soil layers is about 90%. This result is specific to the particular amount of soil strength data available for the present example case. The 90% is well above the confidence levels which have been calibrated as a necessary minimum for a realistic range of the vertical correlation length R for the soil strength and which are presented in Table 4, and it is somewhat smaller than the confidence level of 95% recommended by DNVGL-OS-C101 [5]. The recommendation by Schneider [30] and Lacasse et al. [14] to select the characteristic soil strength as the sample mean minus one half of the sample standard deviation thus appears to be on the conservative side, at least for the pile length of 90 m investigated here; however, this is a recommendation which is independent of the amount of soil test data and which therefore unfortunately does not offer any incentive for the geotechnical designer to test more.
8. Application to ISO 19902 ISO 19902 [11] is the international standard for fixed offshore steel structures for the oil and gas industry. This standard also covers geotechnical design of jacket piles against axial loading. It is therefore of interest to look into the characteristic soil strength used in ISO 19902 and its estimation with confidence. Unfortunately, ISO 19902 does not disclose to the user of the standard which definition it applies for the characteristic soil strength to be used for prediction of axial pile capacity. Yet, it does specify a material factor requirement which is cm = 1.25 and which is to be used in conjunction with the characteristic soil strength. Assume for now that the characteristic soil strength in ISO 19902 is defined as the mean value just as it is in NORSOK G-001, NORSOK N-001 and DNVGL-OS-C101. The same physical reasoning for definition of characteristic soil strength may well have been applied for ISO 19902, namely that fluctuations of the soil strength about the mean tend to average out over the length of the pile. Based on this assumption in conjunction with ISO 19902’s requirements for load factors which are cfD = 1.1 for deadweight load and cfE = 1.40 for environmental load for North Sea conditions, a calibration of the necessary material factor requirement has been carried out a described in Sections 3 and 4. The calibration is carried out to meet an annual probability of failure of 3105, which is ISO 19902’s specified target for exposure level L1 which is the relevant exposure level for the present jacket pile. The results of this calibration are presented in Table 5 for the expected range of the vertical correlation length R of the soil strength.
Table 5 Results of reliability analysis: Calibrated requirement for the material factor cm for ISO 19902 based on characteristic soil strength defined as the mean value. Vertical correlation length R (m)
Material factor cm
0.5 1 3
1.258 1.266 1.290
It appears from Table 5 that regardless of the value of R within the considered expected range, the requirement for the material factor comes out very close to the current requirement of cm = 1.25 in ISO 19902. This implies that it is more than likely that the undisclosed definition of the characteristic soil strength for axial pile capacity in ISO 19902 actually is the mean value. Moreover, with the definition thus in place that the characteristic soil strength indeed is the mean value, Eq. (20) can then be used to estimate the characteristic axial pile capacity with confidence when ISO 19902 is to be applied for geotechnical pile design. It is noted in this context that another reason for defining the characteristic strength as the mean value is that the mean value provides a physically relevant representation of how the foundation is expected to perform. If a small-percentile value for the strength is used in deterministic design, then the performance predicted by the design calculations may be unlikely and even not physically possible in extreme cases. For example, a 5th percentile value of the undrained shear strength could be below the remoulded undrained shear strength of the soil.
9. Future work The 90 m offshore pile analysed above represents one design case. Future work will have to cover similar analyses for a range of design cases, considered representative for the future demand and covering a range of pile lengths, a range of soil profiles with various representative soil stratifications, and a range of load situations representative for the expected future combinations of superstructures, water depths and loading environments. For a prescribed set of load factors, the material factor requirement will then have to be determined by minimizing some penalty function that penalizes deviations of the achieved safety level from the target over the range of design cases, for example in manner demonstrated by Ronold [26]. Once the material factor requirement has been established by such optimization, a reliability-based calibration of the necessary confidence for estimation of characteristic soil strength under limited data can be carried out for each representative design case as described above, and a common minimum confidence requirement can be selected to cover all cases, perhaps expressed as a function of the pile length and perhaps also expressed in dependence on whether the loading is characterized by small or large variability.
10. Conclusions A case-specific study has been carried out to establish the minimum confidence level required for estimation of the characteristic value of soil strength for axial pile capacity predictions based on limited soil strength data. This has been done by requiring that the safety level under limited data is kept equal to the safety level which is achieved under an assumption of perfect knowledge. As a basis for the study, a refined model for representation of model uncertainty in axial pile capacity predictions has been developed and is presented. The model consists of a random model uncertainty factor whose mean value and standard deviation are both represented as functions of the pile length rather than as constants, thereby to honour available pile test data in an appropriate manner. With this refined model uncertainty representation in place, a case-specific reliability-based calibration of the necessary material factor requirement for geotechnical design of a sample jacket pile against failure in axial loading has been carried out. This has been done based on the characteristic soil strength defined as the mean value and under the assumption of perfect knowledge about the soil strength.
K.O. Ronold / Structural Safety 63 (2016) 81–89
For vertical correlation lengths of the soil strength in the range 0.5–3 m, material factors in the range 1.28–1.49 resulted from this exercise when target annual probabilities of failure between 104 and 105 were prescribed. The larger the vertical correlation length or the smaller the target annual probability of failure, the higher is the calibrated requirement for the material factor. The reliability analysis was subsequently repeated with the assumption of perfect knowledge replaced by soil properties based on limited data and as such encumbered with statistical uncertainty which was then included in the probability distributions for the soil strength. For this analysis the material factors were kept equal to those calibrated under the assumption of perfect knowledge. On this basis the repetition of the reliability analysis was carried out by adjusting the value of the characteristic soil strength downwards until the target failure probability was achieved again. The properties of the Student’s t distribution were subsequently used to convert the adjusted reduced value of the characteristic soil strength to a requirement for minimum confidence. Results for a 90 m long offshore example pile in clay have been presented. For realistic vertical correlation lengths for soil strength, relatively moderate requirements for minimum confidence of up to approximately 70% for estimation of characteristic soil strength were calibrated, reflecting that for this long pile the variability in the loading is by far the most important uncertainty source. Higher confidence requirements would result for a shorter pile for which less spatial averaging of the soil strength would imply a shift of uncertainty importance from load to capacity. Likewise, higher confidence requirements would also result for a pile subject to loading with less variability, such as in the case of onshore piles carrying the static self-weight of a building. The calibrated requirements for minimum confidence represent what should be lower bounds for the confidence level from a safety perspective. The requirements for minimum confidence serve as support for deciding which confidence level should be used in design. Some sensitivity in the calibrated requirements for minimum confidence level to the vertical correlation length of the soil strength was found. Relatively little sensitivity in the calibrated requirements for minimum confidence level to the number of available soil strength observations for estimation was found. The conclusions presented for the investigated 90 m long offshore example pile regarding requirements for minimum confidence level for estimation of characteristic soil strength can not necessarily be generalized to onshore piles. Higher requirements for the minimum confidence level can be expected for onshore piles. The major reason for this is that onshore piles are usually meant to carry static self-weight of buildings, at least in seismically non-active regions, whereas offshore piles are meant to carry wind, wave and current loads that are of a dynamic and more variable nature. Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The Norwegian Geotechnical Institute kindly placed data from full scale pile tests in clay at disposal for estimation of model uncertainty characteristics of the NGI-05 method for prediction of axial pile capacity. This contribution is gratefully acknowledged. References [1] Ching J, Wu SS, Phoon KK. Statistical characterization of random field parameters using frequentist and Bayesian approaches. Can Geotech J 2016;53(2):285–98.
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[2] Det Norske Veritas (DNV), Statistical Representation of Soil Data, DNV-RPC207, Høvik, Norway; 2012. [3] Ditlevsen O. Model uncertainty in structural reliability. J Struct Saf 1982;1 (1):73–86. [4] Ditlevsen O, Madsen HO. Structural reliability methods. Chichester, West Sussex, U.K.: John Wiley & Sons; 1996. [5] DNV GL, Design of Offshore Steel Structures, General (LRFD Method), DNVGLOS-C101, Høvik, Norway; 2016. [6] EN 1997-1:2004. Eurocode 7: Geotechnical design Part 1: General rules. Brussels, Belgium: European Committee for Standardization (CEN); 2004. [7] Fenton GA, Griffiths DV. Risk assessment in geotechnical engineering. New York, N.Y.: John Wiley & Sons; 2008. [8] Fenton GA, Naghibi F, Dundas D, Bathurst RJ, Griffiths DV. Reliability-based geotechnical design in the 2014 Canadian Highway Bridge Design Code. Can Geotech J 2016;53(2):236–51. [9] Fjeld S. Reliability of offshore structures. J Pet Technol 1978:1486–96. [10] International Organization for Standardization (ISO), General principles on reliability for structures, ISO 2394; 2015. [11] International Organization for Standardization (ISO), Petroleum and natural gas industries – Fixed steel offshore structures, ISO 19902; 2007. [12] Isaaks EH, Srivastava RM. Applied Geostatistics. New York, N.Y.: Oxford University Press; 1989. [13] Karlsrud K, Clausen CJF, Aas PM. Bearing capacity of driven piles in clay, the NGI approach. Proceedings, International Symposium on Frontiers in Offshore Geotechnics. Perth, Australia: A.A. Balkema Publishers; 2005. [14] Lacasse S, Nadim F, Andersen KH, Knudsen S, Eidsvig U, Yetginer G, Guttormsen TR, Eide A. Reliability of API, NGI, ICP and Fugro Axial Pile Capacity Calculation Methods, In: Proceedings, Offshore Technology Conference, Paper No. OTC 24063, Houston, Texas; 2013. [15] Lambe TW, Whitman RV. Soil mechanics. New York, N.Y.: John Wiley and Sons; 1969. [16] Lange CH, Winterstein SR. Fatigue design of wind turbine blades: load and resistance factors from limited data. Proceedings, 15th ASME Wind Energy Symposium, Houston, Texas. [17] Lehane BM, Schneider JA, Xu X. A review of design methods for offshore driven piles in siliceous sand. Perth, Australia: University of Western Australia; 2005. UWA Report GEO 05358. [18] Madsen HO, Krenk S, Lind NC. Methods of structural safety. Englewood Cliffs, N.J.: Prentice-Hall Inc.; 1986. [19] NKB (The Nordic Committee on Building Regulations), Recommendation for Loading and Safety Regulations for Structural Design, NKB Report No. 36, Copenhagen, Denmark; 1978. [20] NORSOK, Marine Soil Investigations, NORSOK Standard G-001, Rev. 2; 2004. [21] NORSOK, Integrity of Offshore Structures, NORSOK Standard N-001, Rev. 7; 2010. [22] Orr TLL. Managing risk and achieving reliable geotechnical designs using Eurocode 7. In: Phoon KK, Ching J, editors. Chapter 10 in Risk and Reliability in Geotechnical Engineering. Boca Raton, Fla.: CRC Press, Taylor & Francis Group; 2015. p. 395–433. [23] Phoon KK, Kulhawy FH. Characterization of geotechnical variability. Can Geotech J 1999;36(4):612–24. [24] Phoon KK, Kulhawy FH. Evaluation of geotechnical property variability. Can Geotech J 1999;36(4):625–39. [25] Ronold KO. Random Field Modeling of Foundation Failure Modes. J Geotech Eng, ASCE 1990;116(4):554–70. [26] Ronold KO. Reliability-based optimization of design code for tension piles. J Geotech Geoenviron Eng, ASCE 1999;125(8):690–5. [27] Ronold KO, Bergan J, Arnesen K, Guttormsen TR. Reliability of axially loaded jacket pile. Proceedings 31st International Conference on Ocean, Offshore and Arctic Engineering, Rio de Janeiro, Brazil, Vol. 2, 2012, ISBN 978-0-7918-4489-2. Paper No. OMAE2012-84162. [28] Ronold KO, Bjerager P. Model uncertainty representation in geotechnical reliability analyses. J Geotech Eng, ASCE 1992;118(3):363–76. [29] Ronold KO, Lotsberg I. On the estimation of characteristic SN curves with confidence. J Mar Struct 2012;27:29–44. [30] Schneider HR. Definition and characterization of soil properties. Proceedings, XIV ICSMGE, Hamburg, Germany, Balkema Publishers, Rotterdam, The Netherlands. [31] Schneider HR, Schneider MA. Dealing with uncertainties in EC7 with emphasis on determination of characteristic soil properties. In: Arnold P, Fenton GA, Hicks MA, Schweckendiek T, Simpson B, editors. Modern Geotechnical Design Codes of Practice. Implementation, Application and Development. Amsterdam, The Netherlands: IOS Press; 2013. p. 87–101. [32] Vanmarcke E. Probabilistic modeling of soil profiles. J Geotech Eng Div, ASCE 1977;103(GT11):1227–46. [33] Vanmarcke E. Reliability of earth slopes. J Geotech Eng Div, ASCE 1977;103 (GT11):1247–65.
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Structural Safety journal homepage: www.elsevier.com/locate/strusafe
Characteristic soil strength for axial pile capacity and its estimation with confidence for offshore applications Knut O. Ronold DNV GL, Veritasveien 1, N-1322 Høvik, Norway
a r t i c l e
i n f o
Article history: Received 29 June 2015 Received in revised form 28 August 2016 Accepted 28 August 2016
Keywords: Soil strength Pile capacity Model uncertainty Probabilistic analysis Code calibration Confidence estimation
a b s t r a c t Many offshore design codes require the characteristic soil strength for axial pile capacity to be estimated with caution or conservatism, i.e. – in statistical terms – it has to be estimated with confidence. This study is concerned with a reliability-based calibration of the necessary minimum confidence for such estimation. The study demonstrates how to estimate characteristic soil strength for axial pile capacity with confidence and provides an approach to such estimation that will render an incentive to obtain more soil strength data and thereby give credit to the geotechnical designer who opts to test more. The study capitalizes on existing load and resistance models for an offshore example pile from the literature and refers to a number of relevant offshore foundation design codes. In order to accomplish the purpose, a number of prerequisites for a successful study are dealt with before the confidence calibration itself is presented. A suitable approach to represent model uncertainty associated with axial pile capacity predictions is presented and implemented in the reliability analyses that are used for the calibrations. The approach implies that the standard deviation of the ratio between true and predicted capacity is represented as a function of the pile length rather than as a constant. This approach is supported both by theory and by data. Based on stochastic models for load and resistance in conjunction with a method for prediction of axial pile capacity, a second-order reliability analysis of the example pile is carried out. With an assumption of perfect knowledge of the soil strength, with characteristic soil strength defined as the mean value and with prescribed load factors for permanent and environmental loads, the necessary requirement for the material factor on the characteristic soil strength is calibrated by tuning the reliability analysis to meet a prescribed target failure probability. With the calibrated material factor kept unchanged, the reliability analysis is repeated, now with the stochastic model for soil strength altered to include statistical uncertainty owing to limited soil data. The value of the characteristic soil strength is adjusted downward until the result of the reliability analysis again meets the prescribed target failure probability. The resulting value of the characteristic soil strength is interpreted as ‘‘the conservatively assessed value” to be used in design when soil data are limited and this value is subsequently used to find the corresponding minimum confidence for characteristic value estimation by capitalizing on the properties of the Student’s t distribution. Results are presented for target annual failure probabilities in the range 105–104. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction For geotechnical design of axially loaded offshore piles, the characteristic value of the soil strength as a function of depth is used in conjunction with a capacity prediction model and a material factor to determine the design capacity. A number of different capacity prediction models exist; see for example Lehane et al. [17] and Lacasse et al. [14].
E-mail address: [email protected] http://dx.doi.org/10.1016/j.strusafe.2016.08.004 0167-4730/Ó 2016 Elsevier Ltd. All rights reserved.
The characteristic value of the soil strength is usually defined in the design standard which is used for the geotechnical design of the pile, and the definition is usually some objective measure in the probability distribution of the strength, for example the mean value or some lower-tail quantile. Sometimes the design standard also provides requirements for the estimation of the characteristic soil strength with caution and conservatism, i.e. – in statistical terms – estimation with confidence, such as when the estimation is to be based on statistical methods and limited data. Requirements for the material factor to be used in the geotechnical pile design are also given in the design standard and these require-
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ments are specific for the specific definition of characteristic soil strength employed by the standard. The offshore standard NORSOK G-001 [20] specifies that the characteristic value of soil strength to be selected for use in design shall be a ‘‘conservatively assessed mean value”. This implies that the definition of the characteristic value is the mean value and that the assessment of this characteristic value shall be conservative, i.e. in statistical terms that it has to be estimated with confidence. The offshore standard NORSOK N-001 [21] has similar wording for characteristic soil strength, but neither NORSOK G-001 nor NORSOK N-001 specifies any particular requirement regarding which confidence level shall be used for the conservative estimation. DNVGL-OS-C101 [5] is a standard for design of offshore steel structures and their foundations. For cases where a large soil volume is involved, such as for geotechnical design of a long offshore pile, DNVGL-OS-C101 specifies that the characteristic soil strength is defined as the mean value, thereby reflecting that fluctuations of the soil strength from point to point within the soil volume tend to average out over the length of the pile. DNVGL-OS-C101 requires this characteristic soil strength estimated with caution in design and – when statistical methods are used for the estimation – recommends that the estimation be carried out with at least 95% confidence. When the mean value has to be estimated from limited data, the resulting central (unbiased) estimate – the sample mean – will be encumbered with uncertainty. There will be about equal probability for the true but unknown mean value of being greater than respectively smaller than this central estimate. It is not attractive to use this estimate in design when there is such a large probability – about 50% – that the true but unknown mean value is smaller. As exemplified above, the design standards therefore often require that a smaller estimate than the central estimate be used in design, i.e. an estimate with confidence. The confidence is the probability that the true but unknown mean value is greater than the smaller estimate used in design. The smaller the estimate used, the larger is the confidence. Statistical methods exist that specify how much smaller than the central estimate a smaller estimate needs to be in order to meet a specified confidence level. Rather than referring to confidence, some design standards refer to ‘‘the probability of a worse value”, which is the complement of the confidence; for example 5% probability of a worse value corresponds to 95% confidence. Because of the uncertainty in the central estimate of the mean value when data are limited, use of this estimate in design will imply that the safety level which is achieved under an assumption of perfect knowledge, including known mean value, cannot be met. This can be remedied by using a smaller estimate in design, i.e. an estimate with confidence. The smaller the estimate used, i.e. the larger the confidence applied, the larger the safety level achieved in design will be. Because perfect knowledge is usually assumed when design codes are calibrated and because it is a goal to maintain the safety level achieved under perfect knowledge when carrying out designs based on limited data, the requirement for minimum confidence can be established as the confidence level which is such that the safety level achieved in design is equal to the safety level achieved under perfect knowledge. This paper presents the probability calculations necessary to determine which confidence level should be required as a minimum for the estimation of characteristic soil strength for axial pile capacity prediction when data are limited and when the characteristic soil strength is defined as the mean value. A site-specific example case referring to an offshore jacket pile is used for this purpose. The resulting requirement for minimum confidence represents what should be a lower bound for the confidence level from a safety perspective. The determination of the requirement for
minimum confidence serves as support for deciding which confidence level should be used in design. The calculations are based on structural reliability methods applied to a case study which capitalizes on axial pile capacity predictions by means of a recognized method. The case study considered is a study of a jacket pile foundation in clay reported by Lacasse et al. [14] and the capacity prediction method considered is the so-called NGI-05 method described by Karlsrud et al. [13]. To accomplish the calculations, an initial reliability analysis of the pile in question is carried out, leading to calibration of a case-specific set of partial safety factors that will provide a design meeting a prescribed target safety level. This requires an adequate stochastic model for the model uncertainty associated with the chosen method for axial pile capacity prediction. A new such model, which accounts properly for the prediction uncertainty’s dependency on the pile length, is developed for this purpose and is presented first. This is followed by a presentation of the particular example case together with probabilistic and deterministic modelling for the initial reliability analysis. This probabilistic analysis is executed under the assumption of perfect knowledge of the stochastic soil strength, i.e. no statistical uncertainty in the soil strength properties is included. This is a standard approach in code theory as outlined in Madsen et al. [18]. The characteristic soil strength is defined as the mean value which leads to a characteristic axial pile capacity equal to the mean value of the capacity. The results of the probabilistic analysis are presented in terms of the required material factor on capacity as a function of the target annual failure probability. A modification of the probabilistic model is subsequently introduced by which the assumption of perfect knowledge is omitted and replaced by a stochastic model for soil strength which reflects the effect of limited data. This is done by including statistical uncertainty in the reliability analysis by means of probability distributions of the statistically uncertain soil strength parameters. The reliability analysis is repeated based on this revised model and the numerical value of the characteristic soil strength is adjusted downward until the reliability analysis with this model provides the same annual failure probability as the one achieved under the assumption of perfect knowledge. The thus adjusted value of the characteristic soil strength corresponds to a characteristic strength estimated with confidence. This confidence can now be calibrated by means of statistical formulas for estimation of mean values with confidence. The paper shows how this is done and presents the resulting calibrated necessary minimum confidence. The paper may serve as a contribution to ongoing discussions in the geotechnical community regarding the characterization of soil properties. The paper may thereby supplement other recently published material relevant to this issue of characterization and dealing with the same topics that are addressed in the paper. DNV-RP-C207 [2] addresses on a general basis the issue of definition of characteristic values and how to estimate them with confidence. Schneider and Schneider [31] deal with an interpretation of the current definition of characteristic soil strength in Eurocode 7 (EN 1997-1) [6] for onshore foundations and present a simplified way to account for the variance reduction that comes with spatial averaging of soil properties. Orr [22] reviews how important aspects of risk and reliability in geotechnical engineering are addressed in Eurocode 7. ISO 2394:2015 [10] has a wider scope than Eurocode 7 and provides state-of-the-art guidelines for how to carry out reliability-based safety factor calibrations. Fenton et al. [8] provides a review of current practice in various standards regarding definition and estimation of characteristic soil strength. Phoon and Kulhawy [23,24] deal with statistical characterization
K.O. Ronold / Structural Safety 63 (2016) 81–89
of soil properties and Ching et al. [1] brings this one step further by addressing the statistical uncertainty in estimates of parameters used to characterize soils.
2. Stochastic representation of model uncertainty in axial pile capacity predictions The model uncertainty associated with predictions of axial pile capacity by means of a particular capacity prediction model can be represented by a random model uncertainty factor, see Ronold and Bjerager [28]. The random model uncertainty factor is defined as the ratio between the true axial pile capacity and the predicted axial pile capacity as predicted by the prediction model in question. The random model uncertainty factor is a random, or stochastic, factor because the ratio between the true and predicted axial pile capacities exhibits variability from case to case, i.e. from one pile to another. It is a common and accepted approach to calibrate the random model uncertainty factor – its mean value, standard deviation and probability distribution – to results of full scale tests where the true capacity is represented by the capacity measured in the tests, see Ditlevsen [3] and ISO 2394:2015, Annex D, Clause D4.1 [10]. It has been common in the past to represent the standard deviation of the random model uncertainty factor for axial pile capacity as a constant independent of the pile length, see for example Ronold et al. [27] and Lacasse et al. [14]. It is, however, not obvious that this is an adequate representation. The true pile capacity is a random variable. Ronold [25] showed that when the soil strength adheres to the quadratic exponential decay model for vertical autocorrelation, q(r) = exp(r2/R2), then the variance of the axial pile capacity of a friction pile in clay is proportional to
" ! #! pffiffiffi pffiffiffiffi L 1 1 2 L2 2 þ R exp 2 1 k ¼ 2 LR p U R 2 2 L R 2
ð1Þ
lim kL ¼ R
L=R!1
pffiffiffiffi
p
3. Probabilistic and deterministic modelling 3.1. Example case A four-legged offshore jacket structure supported by a group of four tubular steel piles at each leg is considered. The piles are driven 90 m into a layered soil deposit consisting mainly of a series of clay layers. The water depth at the location is 119 m. The pile diameter is 2.438 m. Failure is defined to occur when the average load per pile in the most heavily loaded pile group of four piles, after full redistribution of loads between the pile groups and between the piles within each group, exceeds the capacity of one pile. This example appears as Case Study A in Lacasse et al. [14] from which useful information about loading, soil strength and axial pile resistance is extracted. In the hypothetical case that the four pile capacities in the most heavily loaded pile group are perfectly correlated, the failure criterion implies full utilization of the piles in this pile group and the foundation system has no reserve capacity. In the realistic case that the four pile capacities are only partially correlated, the failure criterion implies that the foundation system has some reserve capacity. 3.2. Loading The average axial load per pile in the most heavily loaded pile group of four piles is considered. This load is taken as one-fourth of the annual maximum total compressive axial load for this pile group, since there are four piles in a pile group and since failure is defined to occur when the average load per pile exceeds the capacity of one pile in the most heavily loaded pile group. The average annual maximum axial compressive load of one pile in the most heavily loaded pile group is represented as the sum of two components
S ¼ X1 þ X2 X3
where k is referred to as a variance reduction factor and expresses the ratio between the capacity variances with and without spatial average effects included. L denotes the pile length, R denotes the vertical correlation length in the quadratic exponential decay model for the autocorrelation function for the undrained shear strength field along the pile, and U denotes the standard Gaussian cumulative distribution function. As the ratio L/R grows large, kL will approach the following limit
ð2Þ
which is a constant when the vertical correlation length R is roughly the same from site to site, which is not an unreasonable assumption. The convergence is fast. Actually the limit for kL for large values of L is a constant regardless of the choice of autocorrelation function model, see Vanmarcke [32]. This observation regarding the limit for kL serves as inspiration to suggest that the standard deviation of the model uncertainty factor for total axial pile capacity predicted by the NGI-05 method (and any other prediction method for that matter) may reasonably well be assumed to be proportional to the inverse of the square root of the pile length. This is assumed in the following and can be considered reasonable as long the capacity is not too dominated by tip resistance. Although maybe not a perfect model for the standard deviation of the model uncertainty factor when there is soil stratification and substantial tip resistance, it still honours available pile test data considerably better than the model with a constant standard deviation independent of the pile length.
83
ð3Þ
where X1 = 25.8 MN is a fixed term due to deadweight and X2X3 is a stochastic term. Reference is made to Lacasse et al. [14]. The variable X3 has a Gumbel distribution with mean value 11.17 MN and standard deviation 4.57 MN, see Lacasse et al. [14]. The 99% quantile (‘‘the 100-year value”) in this distribution is 25.5 MN and is used as the characteristic environmental load, X3char. The variable X2 is a random factor meant to account for statistical uncertainty in the fitted Gumbel distribution as well as model uncertainty associated with adopting the Gumbel distribution as the distribution model. The distribution of X2 is taken as a normal distribution with mean value 1.0 and standard deviation 0.05. The characteristic deadweight load X1char is set equal to X1. With a load factor cfD for deadweight load and a load factor cfE for environmental load, the total design load becomes
Sd ¼ cfD X 1 þ cfE X 3char
ð4Þ
It is noted that the applied loading leading to the average annual maximum axial load S on one pile consists of dynamic wave loading superimposing a static permanent deadweight load. The pile is at the same time subject to a shear force and bending moment caused by the same wave loading; however, this shear force and bending moment and the lateral capacity of the pile to withstand it are outside the scope of this study and not further dealt with here. 3.3. Soil strength and axial pile capacity The soil profile consists primarily of a sequence of clay layers with only two thin sand layers interbedded at and near the seabed.
K.O. Ronold / Structural Safety 63 (2016) 81–89
The undrained shear strength in the clay layers has been sampled by a number of UU tests in the laboratory. The soil stratification appears from Table 1 together with an indication of the number of strength observations in each clay layer. The soil stratification is also indicated in Fig. 1, which is based on Lacasse et al. [14], together with estimates of mean value and standard deviation of the undrained shear strength in each layer. The shear strength of soil comes about as the sum of the strengths of frictional resistance bonds between soil particles over a small unit area; see Lambe and Whitman [15]. Under the central limit theorem, this implies that the shear strength will asymptotically be normally distributed. This observation is also acknowledged by Vanmarcke [33] who refers to a number of supporting studies. Within each identified clay layer in Fig. 1, the undrained shear strength is therefore modelled as a normally distributed variable with mean value and standard deviation as specified in Fig. 1. For deterministic design, the characteristic undrained shear strength is defined as the mean value of the undrained shear strength. This definition is consistent with the definition used in NORSOK G-001 [20] and DNVGL-OS-C101 [5] for the characteristic soil strength for axial pile capacity. It is noted that while the undrained soil strength is assumed to be normally distributed as justified above, a lognormal distribution is often used in practice in order to avoid unrealistic realizations of negative shear strengths when the standard deviation is large, see for example Fenton and Griffiths [7] and Schneider and Schneider [31]. Other distribution models could be applied with this goal in mind, such as the quadratic Weibull distribution described by Lange and Winterstein [16] – a generalized and ‘‘distorted” Weibull distribution which, in contrast to the lognormal distribution, is capable of retaining not only the mean value and standard deviation, but also the skewness of the undrained shear strength. For the present case, the standard deviations are of a magnitude that the issue of achieving unphysical realizations of negative shear strengths is immaterial, but this may not necessarily be the general case. Based on the undrained shear strengths in the soil layers penetrated by the pile, the axial pile capacity is predicted by means of the so-called NGI-05 method; see Karlsrud et al. [13]. The mean value of the predicted axial pile capacity is E[Q] = 118.6 MN, see Lacasse et al. [14]. The coefficient of variation of the predicted axial pile capacity can be determined to be COV[Q]=0.0854 based on information given in Lacasse et al. [14]. With approximately linear relationships between soil strength and segment capacity in each layer and between soil strength and tip resistance in the bottom layer, the distribution of the axial pile capacity becomes a normal distribution when the soil strength in each layer is normally distributed. This is adopted in the following. The characteristic axial capacity for use in deterministic design becomes
Q char ¼ E½Q ¼ 118:6 MN
ð5Þ
Table 1 Soil stratification. Soil layer No.
Depth range (m)
Soil type
1 2 3 4 5 6 7 8 9 10
0–1 1–3.5 3.5–18.5 18.5–24 24–32.2 32.2–38.5 38.5–50 50–56.2 56.2–69.5 69.5–92
Sand Clay Clay Sand Clay Clay Clay Clay Clay Clay
No. of observations of sUU u 3 18 7 6 6 5 7 16
Undrained shear strength, suUU (kPa) 0
100
200
300
400
500
600
0 Mean value 10
Mean-one st.dev.
20
30
40
Depth (m)
84
50
60
70
80
90
100 Fig. 1. Undrained shear strength vs. depth: Mean value and mean minus one standard deviation as estimated from data. Soil layer interfaces at discontinuities and knee points. (Based on Lacasse et al. [14].).
and with a material factor cm for axial pile capacity, the design capacity becomes
Qd ¼
Q char
ð6Þ
cm
Let li denote the expected pile segment capacity in the ith soil layer and let ri denote the standard deviation of the pile segment capacity in the ith soil layer when no spatial averaging is accounted for. It is unrealistic not to account for spatial averaging of soil strength along each pile segment. It is not unreasonable to assume that the autocorrelation function for the undrained shear strength follows the quadratic exponential decay model, q(r) = exp(r2/R2), also known as the Gaussian autocorrelation model. Vanmarcke [32] uses this autocorrelation model for soil strength. Isaaks and Srivastava [12] state that this model is often used to model extremely continuous phenomena. Soil strength may be assumed to fall in this category when considering the geological processes that created the associated soil deposits. Based on the assumption of the quadratic exponential decay model for the autocorrelation function, a variance reduction factor ki can be applied on the capacity variance r2i for the ith pile segment
ki ¼
2 L2Si
" ! #! pffiffiffi LSi 1 1 2 L2Si þ R exp 2 1 LSi R p U 2 2 2 R R pffiffiffiffi
ð7Þ
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where LSi is the segment length in the ith soil layer and R is the vertical correlation length in the quadratic exponential decay model for the autocorrelation function for soil strength, assumed to be identical in all soil layers. Reference is made to Ronold [25]. Ignoring the minor capacity contributions from the two thin sand layers at the seabed and at 20 m depth, respectively, and utilizing that the major part of the axial pile capacity comes from frictional capacity in the clay layers as confirmed by Lacasse et al. [14], the coefficient of variation of the capacity Q can be approximated by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u N u k i r2 i u u COV Q ðRÞ COV½Q u i¼1N u X t r2
ð8Þ
i
i¼1
where the sums are over N clay layers and COV[Q] = COVQ(1) = 0.0854 is the coefficient of variation of the capacity Q when no spatial averaging is accounted for. Note that Eq. (8) is derived by assuming that the pile segment capacities in different layers are statistically independent. Table 2 summarizes the mean value and coefficient of variation of the pile capacity for a number of relevant values for the vertical correlation length R. Vanmarcke [32,33] reports vertical scales of fluctuation dV in the range between 1 and 5 m for soil properties, the higher value being for an example consisting of the undrained shear strength of a clay deposit. This corresponds to vertical correlation lengths R in the range between approximately 0.5 and 3 m, referring to pffiffiffiffi the relationship dV ¼ R p. For vertical correlation lengths R in this range, Table 2 shows a considerable effect of spatial averaging relative to the case without any spatial averaging (R = 1), whereas there is hardly any such effect if R had been as large as 10 metres.
justified above. A nonlinear regression analysis of the data leads to the following estimates for the mean value and the standard deviation of the model uncertainty factor
^ mod ¼ 1:533 L0:0966 E½X
ð10Þ
^ mod ¼ 0:917 pffiffiffi D½X L
ð11Þ
in which the pile length L is given in units of metres. The distribution of the model uncertainty factor conditioned on L is assumed to be a lognormal distribution. This representation of the model uncertainty factor is derived from compressive tests on axially loaded piles in clay and is applied as a good approximation in the present case where the layered foundation soils consist primarily of clay layers with only a couple of relatively thin sand layers interbedded near the seabed. If the model uncertainty factor for total axial capacity of a 90 m long pile in sand has the same mean, standard deviation and distribution type as the model uncertainty factor for total axial capacity of a 90 m long pile in clay, then this representation of Xmod would be exact for the present example pile. If the mean and standard deviation for a pile in sand deviate somewhat from those for the pile in clay, then this representation would only be approximate; however, it would be a good approximation, because the sand layers are relatively thin and the major part of the axial pile capacity comes from frictional capacity in the clay layers as confirmed by Lacasse et al. [14]. Note that what is being characterized as model uncertainty here, in addition to true model uncertainty, implicitly contains other sources of uncertainty (spatial variability and statistical uncertainty) associated with the pile capacity predictions in the database. 3.5. Failure criterion and design rule
3.4. Model uncertainty The model uncertainty factor Xmod for total axial pile capacity is defined as the ratio Qtrue/Qpredicted in which Qtrue denotes the true capacity and Qpredicted denotes the predicted capacity, here specifically the capacity predicted by the NGI-05 method. When data from full scale pile tests to failure are available, measured capacities – one per test – are used as measures of the true capacity, and one realization of Xmod can be established per test. This allows for interpretation of the empirical distribution of Xmod. Sixteen observations of the model uncertainty factor (i.e. of the ratio Qtrue/Qpredicted) are available from compressive axial load tests on piles in clay. This data set stems from the Norwegian Geotechnical Institute’s ‘‘super piles” database. The mean value exhibits a slight variation with the pile length L. This variation can reasonably well be represented by
E½X mod ¼ c1 Lc2
ð9Þ
over the range of pile lengths covered by the tests. This range includes the example pile length of 90 m. The standard deviation is assumed to be inversely proportional to the square root of L as
Table 2 Axial pile capacity properties as function of vertical correlation length. Vertical correlation length, R (m)
Mean capacity, E [Q] (MN)
Coefficient of variation of capacity, COVQ
0.5 1 3 5 10 1
118.6 118.6 118.6 118.6 118.6 118.6
0.0265 0.0366 0.0575 0.0677 0.0784 0.0854
Failure is defined above to occur when the average load per pile in the most heavily loaded pile group of four piles, after full redistribution of loads between the pile groups and between the piles within each group, exceeds the capacity of one pile. This implies that the failure criterion can be written as
S P X mod Q
ð12Þ
where S denotes the average annual maximum axial compressive load per pile in the most heavily loaded pile group, Q denotes the predicted axial capacity of one pile, and Xmod is the model uncertainty factor used to account for the model uncertainty associated with this prediction, such that the product XmodQ represents the true capacity. As stated in 3.1, this failure criterion implies some reserve capacity in the foundation system when the four pile capacities in the most heavily loaded pile group are not fully correlated. The corresponding deterministic design rule is
Sd 6 Q d
ð13Þ
in which Sd is the design load and Qd is the design capacity. Introducing characteristic load and resistance values and corresponding load and material factors, the design rule can be rewritten as
cfD X 1char þ cfE X 3char 6
Q char
ð14Þ
cm
The design equation comes about when the design rule is turned into an equality,
cfD X 1char þ cfE X 3char ¼
Q char
cm
:
ð15Þ
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3.6. Limit state function and failure probability The failure criterion, normalized by the design equation, can be used to formulate the following limit state function
gðXÞ ¼
X mod Q cm X1 þ X2 X3 Q char cfD X 1char þ cfE X 3char
ð16Þ
in which X denotes the vector of stochastic variables. The probability of failure is
PF ¼ P½gðXÞ 6 0
ð17Þ
and can be calculated by a second-order reliability method (SORM) as described in Madsen et al. [18]. 4. Reliability analysis A second-order reliability method as described in Madsen et al. [18] is used to carry out reliability analyses and calculate the probability of failure. The load factor cfE for environmental load is prescribed equal to 1.3 in compliance with the requirements set forth in NORSOK [20,21] and DNVGL-OS-C101 [5]. Likewise, the load factor for deadweight load is prescribed equal to 1.0. This leaves the value of the material factor cm as a free parameter to be adjusted until the reliability analysis produces a probability of failure which is equal to a specified target failure probability. The value of cm which is thus calibrated to meet the specified target reliability is thus the value of the material factor which is necessary to meet this safety level in design. Results are presented in Table 3 for three values of the target annual failure probability, viz. 105, 5105 and 104, and for a range of values for the vertical correlation length R for the soil strength. Table 3 indicates that for the specific 90 m long pile studied and assuming the characteristic pile capacity Qchar is defined as the mean capacity, see Eq. (5), then a material factor requirement of cm = 1.3 is sensible for a realistic range of vertical correlation lengths between 0.5 and 3 m. The results in Table 3 indicate some sensitivity in the calibrated requirement for the material factor cm to the target annual failure probability and some sensitivity to the value of the vertical correlation length R for the soil strength. In particular, this implies that ignoring the effect of spatial averaging by assuming infinite vertical correlation length R will lead to a somewhat conservative calibrated requirement for cm. For the expected range of realistic R values, i.e. 0.5–3 m, it is reassuring to find that the requirement for cm comes out relatively close to the requirement 1.3 set forth in NORSOK and DNVGL-OSC101. The calibrated requirement is somewhat higher, approximately 1.45–1.50, if one follows the recommendation of a target annual failure probability of 105 for high safety class according to NKB [19] for a structure which has some redundancy at failure. However, it is almost ‘‘spot on”, in the neighbourhood of 1.3, if the target annual failure probability can be set according to accepted past practice: For offshore structures appropriately designed to meet recognized codes, Fjeld [9] reports that the annual failure
probability is of the order of 105 to 104 with higher values for steel structures – such as the present example pile – than for concrete structures. 5. Calibration of necessary minimum confidence level For each vertical correlation length R considered, the calibrated material factor cm is kept unchanged and thereby also the specified target failure probability. Let again li denote the expected pile segment capacity in the ith soil layer and let ri denote the standard deviation of the pile segment capacity in the ith soil layer when no spatial averaging is accounted for. When statistical uncertainty is accounted for in the reliability analysis, the predicted stochastic total pile capacity can be expressed as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q¼ ðli þ ri ki ðRÞ þ T i ðni ranki ÞÞ ni i¼1 N X
ð18Þ
in which T i ðni ranki Þ is a Student’s t distributed variable with ni ranki degrees of freedom, ni is the number of soil strength observations in the ith soil layer, ranki = 1 when the mean soil strength is constant in the ith soil layer, and ranki = 2 when the mean soil strength is linear with depth in the ith soil layer. N is the number of pile segments and equals the number of soil layers, li is the expected pile segment capacity of the ith segment, and ri is the standard deviation of the ith segment capacity when no spatial averaging is accounted for. Contributions from the two thin sand layers near the top of the pile are ignored as insignificant. It is noted that Eq. (18) is correct given the assumptions being made that the actual mean value of the ith summand follows the distribution of the sample mean and the actual variance of the ith summand follows the distribution of the sample variance. The rationale behind Eq. (18) is that when the soil strength sui in the ith soil layer is assumed to be normally distributed, and data for estimation of its mean value E½sui and standard deviation D½sui are limited and cause the estimates of these distribution parameters to be uncertain, then sui can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sui ¼ E½sui þ D½sui 1 þ 1=ni T i , where Ti is Student’s t distributed as described above, see Ditlevsen and Madsen [4]. When this shear strength is integrated over the circumference and height of the ith pile segment (in accordance with the NGI-05 method), spatial averaging comes into play. When the spatial averaging is taken into account, the expression for the ith segment capacity in the summand in Eq. (18) results. Statistical uncertainty in the estimates of the undrained shear strength has been retained through all steps and is thus reflected in the resulting expression for the capacity Q in Eq. (18). From Eq. (7), the variance reduction factor for pile segment capacity in the ith soil layer reads
ki ¼
2 L2Si
" ! #! pffiffiffi LSi 1 1 2 L2Si þ R exp 2 1 LSi R p U 2 2 2 R R pffiffiffiffi
Table 3 Results of reliability analysis: Calibrated requirement for the material factor cm. Vertical correlation length R (m)
0.5 1 3 5 10 1
ð19Þ
Target annual failure probability 105
5105
104
1.45 1.46 1.49 1.51 1.54 1.56
1.33 1.34 1.36 1.38 1.40 1.42
1.28 1.28 1.31 1.33 1.35 1.36
where LSi is the segment length in the ith soil layer and R is the vertical correlation length in the quadratic exponential decay model for the autocorrelation function for soil strength, assumed to be identical in all soil layers. The segment length equals the soil layer thickness in each soil layer, except in the lowermost layer penetrated by the pile where the pile does not penetrate all the way to the bottom of the layer.
K.O. Ronold / Structural Safety 63 (2016) 81–89
The characteristic value of the total capacity, estimated based on soil strength estimates with confidence 1 a, can be expressed as
^ char ¼ Q
N X r^ i ^ i pffiffiffiffi ðl t ni ranki ð1 aÞÞ ni i¼1
ð20Þ
where tni-ranki(1 a) is the 1 a quantile in the Student’s t distri^ i is the central estimate bution with ni ranki degrees of freedom, l ^ i is of the expected pile segment capacity of the ith segment, and r the central estimate of the standard deviation of the ith segment capacity when no spatial averaging is accounted for. It is here utilized that the confidence 1a is defined as the probability that the true but unknown mean soil strength in the ith layer is greater than the characteristic soil strength estimated with this confidence. It is noted that for given value of 1 a, the quantile tni-ranki(1 a) is ^ char will increase a decreasing function as ni increases, such that Q with increasing number of soil strength observations ni, i = 1,. . .N. ^ char , necessary to maintain the reliability Once the value of Q when statistical uncertainty in the soil strengths is included in the reliability analysis, has been determined, Eq. (20) can be solved with respect to 1 a, the resulting value of which can then be interpreted as the necessary minimum confidence level for cautious estimation of the characteristic value of soil strength in design. The results of this calibration of the necessary minimum confidence level required to maintain the specified target reliability when the amount of soil strength data are limited to the extent available for the present example case are presented in Table 4. The results are presented as a function of the vertical correlation length R and of the target failure probability. Each tabulated value for the confidence level 1 a in Table 4 is calibrated based on the corresponding value of the material factor cm tabulated in Table 3 under preservation of the value of the target annual failure probability. It appears from Table 4 that the requirement for confidence in the estimation of characteristic soil strength is rather sensitive to the value of the vertical correlation length R. The larger the correlation length, the higher is the requirement for confidence level. The sensitivity to the specified target annual probability of failure is considerably smaller, as long as realistic correlation lengths in the range 0.5–3 m are considered. For the expected range of vertical correlation lengths up to about 3 m, a confidence of no more than 70% appears to suffice. The results presented in Table 4 are valid for the actual pile length of 90 m investigated. For shorter piles for which less effect of spatial averaging of soil strength can be expected and for which the standard deviation of the model uncertainty factor will be larger, higher requirements for the confidence level can be expected. Also for piles with less variability in the annual maximum load, higher requirements for the confidence level can be expected.
Table 4 Results of reliability analysis: Calibrated requirement for minimum confidence level 1a. Vertical correlation length R (m)
0.5 1 3 5 10 1
Target annual failure probability 105
5105
104
0.583 0.600 0.663 0.719 0.825 0.994
0.576 0.591 0.645 0.691 0.761 0.840
0.574 0.587 0.639 0.680 0.740 0.800
87
6. Sensitivity to number of strength observations The results presented for the necessary confidence in Table 4 are based on the actual number of test data in each of the soil layers identified in Table 1 and Fig. 1. It is of interest to study the sensitivity in the calibrated confidence to a change in the number of test data. A major contribution to the axial pile capacity comes from the lowermost layer along the pile, from 69.5 m depth downwards. There are 16 observations of the undrained shear strength in this layer. The effect of halving this number to 8 is studied. For a vertical correlation length R = 3 m, it appears that the necessary confidence will increase slightly from 0.66 to 0.69 when the target failure probability is 105, and from 0.64 to 0.66 when the target failure probability is 104. In other words, there is only a moderate dependency of the required confidence level on the number of available soil strength observations. This is much in agreement with the findings by Ronold and Lotsberg [29] for a different limit state (fatigue failure rather than ultimate failure) and a different definition of characteristic strength (the 2.3% quantile rather than the mean value), for which the required confidence level also came out with only a moderate dependency on the number of strength data available.
7. Discussion It appears from the calibration that for vertical correlation lengths up to an expected maximum of no more than 5 m, a confidence of up to 72% will be sufficient for estimation of characteristic soil strengths. It is noted that this result is obtained for a 90 m long pile for target failure probabilities in the range 105 to 104 and that it remains to be verified if this result can be generalized to other pile lengths. Selection of characteristic soil strengths with a confidence larger than the calibrated minimum required for estimation presented in Table 4, for use in design, can be done in order to account for other uncertainties than statistical uncertainty not accounted for in the calibration of the material factor, for example the uncertainty associated with the fact that the vertical correlation length R is usually unknown. Measurement uncertainty associated with the level of quality of the soil investigation program, the soil sampling procedures and the laboratory testing of soil samples forms another uncertainty source that one may want to account for by selecting characteristic values with a larger confidence than the minimum required to account for the effect of limited data. Note that in case of for example onshore piles where the loading is less variable than for offshore piles and more uncertainty importance therefore is on soil strength, higher requirements for confidence for cautious estimation of characteristic soil strengths would result than those reported in Table 4. This can be illustrated by looking at the sensitivity for an increased coefficient of variation COVQ of the predicted axial pile capacity. Such an increase would correspond to a situation with reduced uncertainty importance on load and increased uncertainty importance on capacity. If COVQ is increased by 10% from 0.0575 to 0.0633 for R = 3 m, the calibrated confidence requirement for estimation of soil strength increases from 0.66 to 0.71 when the target failure probability is 105, and from 0.64 to 0.67 when the target failure probability is 104. This also illustrates the importance of having as correct a model for model uncertainty representation as possible in the reliability analysis, thereby to get a correct split of uncertainty importance between load and strength variables before the confidence requirement is calibrated.
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Lacasse et al. [14] recommends that the characteristic value of soil strength for use in design of axially loaded piles should be selected as the sample mean minus one half of the sample standard deviation. Schneider [30] proposes the same recommendation for characteristic soil strength in general, not only for design of axially loaded piles. Calculating the corresponding characteristic axial pile capacity – which then becomes the ‘‘selected characteristic pile capacity” for use in design – and substituting this biased characteristic capacity estimate on the left hand side of Eq. (20), Eq. (20) can be solved with respect to the confidence 1a. The result of this exercise is an associated confidence 1a which on average over the soil layers is about 90%. This result is specific to the particular amount of soil strength data available for the present example case. The 90% is well above the confidence levels which have been calibrated as a necessary minimum for a realistic range of the vertical correlation length R for the soil strength and which are presented in Table 4, and it is somewhat smaller than the confidence level of 95% recommended by DNVGL-OS-C101 [5]. The recommendation by Schneider [30] and Lacasse et al. [14] to select the characteristic soil strength as the sample mean minus one half of the sample standard deviation thus appears to be on the conservative side, at least for the pile length of 90 m investigated here; however, this is a recommendation which is independent of the amount of soil test data and which therefore unfortunately does not offer any incentive for the geotechnical designer to test more.
8. Application to ISO 19902 ISO 19902 [11] is the international standard for fixed offshore steel structures for the oil and gas industry. This standard also covers geotechnical design of jacket piles against axial loading. It is therefore of interest to look into the characteristic soil strength used in ISO 19902 and its estimation with confidence. Unfortunately, ISO 19902 does not disclose to the user of the standard which definition it applies for the characteristic soil strength to be used for prediction of axial pile capacity. Yet, it does specify a material factor requirement which is cm = 1.25 and which is to be used in conjunction with the characteristic soil strength. Assume for now that the characteristic soil strength in ISO 19902 is defined as the mean value just as it is in NORSOK G-001, NORSOK N-001 and DNVGL-OS-C101. The same physical reasoning for definition of characteristic soil strength may well have been applied for ISO 19902, namely that fluctuations of the soil strength about the mean tend to average out over the length of the pile. Based on this assumption in conjunction with ISO 19902’s requirements for load factors which are cfD = 1.1 for deadweight load and cfE = 1.40 for environmental load for North Sea conditions, a calibration of the necessary material factor requirement has been carried out a described in Sections 3 and 4. The calibration is carried out to meet an annual probability of failure of 3105, which is ISO 19902’s specified target for exposure level L1 which is the relevant exposure level for the present jacket pile. The results of this calibration are presented in Table 5 for the expected range of the vertical correlation length R of the soil strength.
Table 5 Results of reliability analysis: Calibrated requirement for the material factor cm for ISO 19902 based on characteristic soil strength defined as the mean value. Vertical correlation length R (m)
Material factor cm
0.5 1 3
1.258 1.266 1.290
It appears from Table 5 that regardless of the value of R within the considered expected range, the requirement for the material factor comes out very close to the current requirement of cm = 1.25 in ISO 19902. This implies that it is more than likely that the undisclosed definition of the characteristic soil strength for axial pile capacity in ISO 19902 actually is the mean value. Moreover, with the definition thus in place that the characteristic soil strength indeed is the mean value, Eq. (20) can then be used to estimate the characteristic axial pile capacity with confidence when ISO 19902 is to be applied for geotechnical pile design. It is noted in this context that another reason for defining the characteristic strength as the mean value is that the mean value provides a physically relevant representation of how the foundation is expected to perform. If a small-percentile value for the strength is used in deterministic design, then the performance predicted by the design calculations may be unlikely and even not physically possible in extreme cases. For example, a 5th percentile value of the undrained shear strength could be below the remoulded undrained shear strength of the soil.
9. Future work The 90 m offshore pile analysed above represents one design case. Future work will have to cover similar analyses for a range of design cases, considered representative for the future demand and covering a range of pile lengths, a range of soil profiles with various representative soil stratifications, and a range of load situations representative for the expected future combinations of superstructures, water depths and loading environments. For a prescribed set of load factors, the material factor requirement will then have to be determined by minimizing some penalty function that penalizes deviations of the achieved safety level from the target over the range of design cases, for example in manner demonstrated by Ronold [26]. Once the material factor requirement has been established by such optimization, a reliability-based calibration of the necessary confidence for estimation of characteristic soil strength under limited data can be carried out for each representative design case as described above, and a common minimum confidence requirement can be selected to cover all cases, perhaps expressed as a function of the pile length and perhaps also expressed in dependence on whether the loading is characterized by small or large variability.
10. Conclusions A case-specific study has been carried out to establish the minimum confidence level required for estimation of the characteristic value of soil strength for axial pile capacity predictions based on limited soil strength data. This has been done by requiring that the safety level under limited data is kept equal to the safety level which is achieved under an assumption of perfect knowledge. As a basis for the study, a refined model for representation of model uncertainty in axial pile capacity predictions has been developed and is presented. The model consists of a random model uncertainty factor whose mean value and standard deviation are both represented as functions of the pile length rather than as constants, thereby to honour available pile test data in an appropriate manner. With this refined model uncertainty representation in place, a case-specific reliability-based calibration of the necessary material factor requirement for geotechnical design of a sample jacket pile against failure in axial loading has been carried out. This has been done based on the characteristic soil strength defined as the mean value and under the assumption of perfect knowledge about the soil strength.
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For vertical correlation lengths of the soil strength in the range 0.5–3 m, material factors in the range 1.28–1.49 resulted from this exercise when target annual probabilities of failure between 104 and 105 were prescribed. The larger the vertical correlation length or the smaller the target annual probability of failure, the higher is the calibrated requirement for the material factor. The reliability analysis was subsequently repeated with the assumption of perfect knowledge replaced by soil properties based on limited data and as such encumbered with statistical uncertainty which was then included in the probability distributions for the soil strength. For this analysis the material factors were kept equal to those calibrated under the assumption of perfect knowledge. On this basis the repetition of the reliability analysis was carried out by adjusting the value of the characteristic soil strength downwards until the target failure probability was achieved again. The properties of the Student’s t distribution were subsequently used to convert the adjusted reduced value of the characteristic soil strength to a requirement for minimum confidence. Results for a 90 m long offshore example pile in clay have been presented. For realistic vertical correlation lengths for soil strength, relatively moderate requirements for minimum confidence of up to approximately 70% for estimation of characteristic soil strength were calibrated, reflecting that for this long pile the variability in the loading is by far the most important uncertainty source. Higher confidence requirements would result for a shorter pile for which less spatial averaging of the soil strength would imply a shift of uncertainty importance from load to capacity. Likewise, higher confidence requirements would also result for a pile subject to loading with less variability, such as in the case of onshore piles carrying the static self-weight of a building. The calibrated requirements for minimum confidence represent what should be lower bounds for the confidence level from a safety perspective. The requirements for minimum confidence serve as support for deciding which confidence level should be used in design. Some sensitivity in the calibrated requirements for minimum confidence level to the vertical correlation length of the soil strength was found. Relatively little sensitivity in the calibrated requirements for minimum confidence level to the number of available soil strength observations for estimation was found. The conclusions presented for the investigated 90 m long offshore example pile regarding requirements for minimum confidence level for estimation of characteristic soil strength can not necessarily be generalized to onshore piles. Higher requirements for the minimum confidence level can be expected for onshore piles. The major reason for this is that onshore piles are usually meant to carry static self-weight of buildings, at least in seismically non-active regions, whereas offshore piles are meant to carry wind, wave and current loads that are of a dynamic and more variable nature. Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The Norwegian Geotechnical Institute kindly placed data from full scale pile tests in clay at disposal for estimation of model uncertainty characteristics of the NGI-05 method for prediction of axial pile capacity. This contribution is gratefully acknowledged. References [1] Ching J, Wu SS, Phoon KK. Statistical characterization of random field parameters using frequentist and Bayesian approaches. Can Geotech J 2016;53(2):285–98.
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