Reliability study of suction caissons for catenary and taut-leg mooring systems

Structural Safety 45 (2013) 59–70

Contents lists available at ScienceDirect

Structural Safety journal homepage: www.elsevier.com/locate/strusafe

Reliability study of suction caissons for catenary and taut-leg mooring systems F. Silva-González a,⇑, E. Heredia-Zavoni a, C. Valle-Molina a, J. Sánchez-Moreno a, Robert B. Gilbert b a b

Instituto Mexicano del Petróleo, Programa de Explotación de Campos en Aguas Profundas, Eje Central Lázaro Cárdenas Norte 152, San Bartolo Atepehuacán, México DF 07730, Mexico Department of Civil Engineering, The University of Texas at Austin, Austin, TX 78712, United States of America

a r t i c l e

i n f o

Article history: Received 31 December 2011 Received in revised form 8 March 2013 Accepted 28 August 2013 Available online 13 November 2013 Keywords: Reliability analysis Partial safety factors Plastic limit models Response surfaces Mooring systems

a b s t r a c t A detailed study on the reliability of suction caissons for moorings of floating systems is presented. Reliability analyses are performed considering mooring line tensions and suction caisson capacity at the mudline. The probabilistic modeling of caisson capacities is based on simulations using a plastic limit model calibrated to numerical computations using finite element modeling. Loading for catenary and taut-leg mooring systems were considered for the reliability analyses; loads were expressed as functions of uncertain metocean variables by means of response surfaces. Reliability analyses were carried using FORM. Linear relationships were established between the logarithm of failure probability and caisson length for both cases of mooring systems. Partial safety factors for capacity, and mean and dynamic line tensions were calibrated separately for caissons in connection to catenary and taut-leg moorings. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Suction caissons are a critical component of mooring systems. Their main function in floating systems is to hold the mooring lines in place under severe and normal environmental conditions. They are thus subject to mooring lines loading acting on the caissons padeye, as depicted in Fig. 1. For many years suction caissons were used for catenary moorings in moderate water depths exposed to predominantly horizontal loads. However, as the development of oil fields has moved into deeper waters, taut-leg moorings have become a preferred solution. Caissons for taut-leg moorings are subject to greater vertical loading, which makes them an even more critical component of the mooring system than before. It is thus necessary to focus on the reliability of these mooring components. Reliability of suction caissons has been studied relatively recently. Clukey et al. [1] used linear response surfaces expressing the lateral resistance of suction caissons as a function of shear strength and an empirical model of mooring line tensions in terms of significant wave height. Reliability was estimated considering a simple limit state function in terms of load and capacity for lateral and axial loading failure representing catenary and taut-leg moorings. In connection with a study on system reliability of moorings, Choi [2] analyzed the reliability of suction caissons. Capacity was

⇑ Corresponding author. Tel.: +52 55 91758238. E-mail address: fl[email protected] (F. Silva-González). 0167-4730/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.strusafe.2013.08.011

evaluated at the padeye using the upper bound plastic limit formulation proposed by Aubeny et al. [3–5] and then examined at mudline using the Neubecker and Randolph [6] formulation to account for soil–chain interaction effects. Comparing results of laboratory tests against the analytical capacity, biases and coefficients of variation of caisson capacity were estimated. Based on line tensions from a SPAR model, loading was characterized considering that dynamic tensions were Gaussian and using distributions for hurricane and loop current sea-state conditions in the Gulf of Mexico. Assuming lognormal distributions for capacity and loading, reliability was assessed in terms of median safety factors, and biases and coefficients of variation for loads and capacity. More recently Valle-Molina et al. [7] used response surfaces obtained from catenary mooring lines of a Floating Production, Storage and Offloading (FPSO) system to model the mean and the expected maximum dynamic tensions as functions of the environmental variables describing extreme sea states. Weibull distributions were fitted to samples of total line tensions generated using the response surfaces and simulations of the environmental variables. The caissons capacities were characterized using the plastic limit formulation proposed by Aubeny et al. [3–5] and Monte Carlo simulation was used to evaluate caisson reliabilities. From these studies it is concluded that a proper characterization of the caisson capacity under both lateral and axial loading is a challenge to be addressed and that reliability assessment needs to be further studied, particularly considering the significant differences in terms of loading and capacity for catenary and taut-leg moorings.

60

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

Fig. 2. Finite element model with optimal mesh developed in ANSYSÒ.

Fig. 1. (a) Mooring system; and (b) Soil–caisson system subjected to inclined loading at padeye.

Regarding code calibration, safety factors have been proposed for design of suction caissons in normally consolidated clays. In DNV-RP-E303 [8], partial safety factors are calibrated for the capacity only while safety factors on the mean and dynamic components of line tensions are given as in DNV-OS-E301 [9] for mooring design. Thus, no distinction is made between catenary and taut-leg moorings. Safety factors have also been calibrated to design caissons for Tension-Leg Platforms (TLP) where the caisson uplift capacity has been characterized based on the upper bound plastic limit method and on finite element modeling [10]. In this study, reliability analyses are performed considering mooring line tensions and capacity of the caisson at mudline. A 3D finite element model (FEM) is developed using ANSYSÒ to evaluate the caisson capacity. The plastic limit model (PLM) developed by Aubeny et al. [3–5] is calibrated to the FEM results. Samples of caisson capacity are generated at mudline and at the padeye using the calibrated PLM and simulations of the undrained shear strength gradient, the side shear factor, the reverse-end bearing factor and the line tension angle at the padeye. The probability densities of caisson capacities take into account a lower-bound axial capacity defined by the shear strength of remolded soil. The mean and the expected maximum dynamic tensions from catenary and taut-leg moorings of an FPSO model are expressed as functions of uncertain metocean variables by means of response surfaces. Reliability analyses are carried using First Order Reliability Method (FORM) and calibration of partial safety factors is performed for caissons in connection to catenary and taut-leg moorings.

2. Suction caisson modeling A three-dimensional finite element model (FEM) of the soil– caisson system subjected to inclined loading was developed using ANSYSÒ. The caisson and the soil were modeled using ANSYSÒ SOLID45 eight-node brick and prism elements. Point-to-surface elements were used to model the exterior and interior soil–caisson interfaces at the caisson shaft, and surface-to-surface elements

Fig. 3. Interaction diagrams obtained with FE models and PLM capacity estimates.

were used for the head and tip of the caisson. The interface elements along the wall of the caisson reproduced the frictional shaft capacity by means of the adherence or side shear factor, a. The Von Mises yield surface was used in the model with no dilation and undrained conditions. Results of the FEM developed here were compared to those presented in Andersen et al. [11] for a caisson with 5 m diameter and 25 m long, as shown in Fig. 2. The loading was applied at the padeye located at approximately 2/3 of the caisson length and considering a soil profile of normally consolidated clay with side shear factor a = 0.65. In this case, the model considers an anisotropic undrained shear strength taken from triaxial extension testing results for the passive wedge, triaxial compression shear strength for the active wedge, and direct simple shear strength for the flow at the base of the caisson. Capacity of the caisson was estimated varying the load inclinations at padeye in the range from 0° to 90°. For loading inclinations around 20° and less, the hyperbolic Kondner model was fitted to the horizontal load versus horizontal displacement curves. For loading inclinations between 20° and 35° the hyperbolic Kondner model was fitted to the inclined load versus inclined displacement curves. In both cases the capacity was defined based on the asymptotic behavior of the fitted model at large displacements. For greater load inclinations, capacity was obtained from the asymptotic behavior of the vertical load versus vertical displacement curves at large displacement levels of about 0.5–1.0 m. Fig. 3 shows the interaction diagrams of horizontal and vertical components of capacity obtained with the FEM developed here and those given in Andersen et al. [11]. The horizontal component of capacity varies from 0 to about 22,000 kN, while the vertical component ranges from 0 to 11,800 kN. It is seen that the results obtained here are in very good agreement with those given in Andersen et al. [11].

61

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

A plastic limit model (PLM) was used in this study as an efficient analytical tool for intensive computations of the inclined capacity of suction caissons for the purpose of probabilistic characterization of capacity as discussed below. The plastic limit analytical model is an upper bound plasticity solution [12] originally developed for pure lateral loading of caissons [13] and then extended for the case of inclined loading by Aubeny et al. [3–5]. It holds for normally consolidated clay deposits with undrained shear strength Su varying linearly with depth. Comparisons of results obtained with analytical plastic limit formulations and numerical FE models have been performed widely for suction caissons and discrepancies have been found to fall in the range from 10% to 20% [4,5,11]. The PLM used here was calibrated to the numerical results obtained with the FEM by means of varying the side shear factor, a, and the reverse-end bearing factor, Nab. In Fig. 3 capacity estimates using PLM are shown along the FEM interaction diagrams for different combinations of a and Nab. It was found that using a = 0.7 and Nab = 9, considering suction on the back side of the caisson and a mean profile of undrained shear strength from direct simple shear (DSS) tests, the PLM results matched reasonably well with FE element results. Suction caisson models were designed according to DNV-RPE303 [8] for consequence class 2 and target failure probability of 105. The caissons were designed considering line tensions from catenary and taut-leg mooring systems of an FPSO at 630 m and 1565 m water depth, respectively, for two sites in the Bay of Campeche, Gulf of Mexico [14]. A characteristic linear profile of Su given by a value of 0.97 kPa at mudline and a gradient of 1.60 kPa/m was employed; this profile can be considered representative of clayey soils in the Bay of Campeche area. Table 1 lists the characteristic mean tension, Tmean-C, and characteristic mean maximum dynamic tension, lTdyn-C, for 100 year return period sea states, as well as the design line tensions at the mudline, Td, and the corresponding line angles at mudline, ho. Suction caisson models were designed for length-diameter ratios Lf/D = 4, 5 and 6; the padeye was located at 2/3 the caisson length. The capacity of suction caissons was calculated with plastic limit analysis using a = 0.8 and Nab = 9 as characteristic values. In engineering design practice, typical characteristic a values range from 0.5 to 1.0. For normally consolidated clays, an alpha value close to 1.0 is generally used in the design of offshore piles. From tests in 1-g physical models in kaolinite, El-Sherbiny [15] found average alpha values of 0.78 at failure load. Moreover, results from 1-g physical models of offshore piles for fixed platforms using clayey soil from the Bay of Campeche indicated alpha values equal to 0.83 [16]. Thus, considering the soil conditions at sites in the Bay of Campeche area, a characteristic value of a = 0.8 is deemed to be

appropriate for this study. The design criteria in DNV-RP-E303 [8] establishes that the capacity be compared to design line tensions applied at the padeye. The loss in line tension between the mudline and the padeye was assessed using the formulation developed by Neubecker and Randolph [6]. Table 2 shows the lengths, diameters, design tensions at the padeye, Td,a, design resistances at the padeye, Rd,a, and the corresponding line tension angles at the padeye, ha. For the design conditions considered, Table 2 indicates that caisson diameters are about 3–4 m for the catenary lines and 4–5 m for the taut-leg system. Line tension angles at the padeye are 15–17° for the catenary mooring and 38–40° for the taut-leg one. We consider realistic to use caisson models of such dimensions for the reliability analysis. Thus, three groups of caisson models with ratios Lf/D = 4, 5, 6 for each of the mooring systems were used for reliability analysis considering diameters varying from 3 m to 6 m. 3. Reliability analysis The net load that is transferred to the suction caisson is the resultant force of the mooring line tension at the mudline and the forces due to chain–soil interaction along the chain segment of mooring line from the dip down point to the caisson’s padeye. The chain–soil interaction effects depend on the undrained shear strength of the soil and hence the net load transferred to the caisson’s padeye also depends on the soil shear strength. If the limit state function is formulated at the padeye, the statistical dependence between the applied load and the caisson capacity would have to be considered, thus increasing significantly the complexity of the reliability analysis. An alternative approach proposed by Choi [2] is to formulate the limit state function at the mudline. In such case, the chain–soil interaction effects need to be taken into account only in the assessment of the loading capacity. It is then reasonable to consider that at the mudline the line tensions and the caisson capacity are statistically independent variables. In this work the limit state function is set in terms of the caisson capacity and the line tensions at the mudline,

    M R; Hs ; T p ; U 10 ¼ R  T mean Hs ; T p ; U 10   lTdyn;max Hs ; T p ; U 10

ð1Þ

where R is the caisson capacity at the mudline, Tmean is the mean line tension due to pretension and mean environmental loads, and lTdyn,max = E[Tdyn,max] is the expected maximum dynamic line tension during an extreme sea state induced by low-frequency and wave-frequency motions; both Tmean and lTdyn,max are expressed at the mudline as functions of the significant wave height (Hs), peak period (Tp), and wind velocity (U10) which describe an

Table 1 Characteristic and design tensions for the mooring systems.

Catenary Taut-leg

Hs

Tp

U10

Tmean-C (kN)

lTdyn-C (kN)

Td (kN)

ho (°)

10.8 11.1

14.2 14.4

21.4 22.1

2871 6200

2277 2129

8801 13152

1 36

Table 2 Suction caisson models. Mooring system

Caisson model

Lf/D

Lf (m)

D (m)

Td,a (kN)

Rd,a (kN)

ha (°)

Catenary

C-4 C-5 C-6 T-4 T-5 T-6

4 5 6 4 5 6

15.65 16.80 17.95 19.10 21.65 23.87

3.91 3.36 2.99 4.78 4.33 3.98

7904.78 7834.27 7772.56 12860.19 12783.18 12708.74

7994.51 7950.75 7950.30 12920.42 12870.58 12787.57

14.88 16.04 17.06 38.68 39.41 40.11

Taut-Leg

62

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

Fig. 5. Ratio of: (a) capacity at padeye to capacity at mudline; and (b) load angle at padeye to load angle at mudline.

Fig. 4. Capacity interaction diagrams at the padeye and mudline: (a) D = 3 m, Lf/D = 6, (b) D = 5 m, Lf/D = 4, and (c) D = 4 m, Lf/D = 6.

extreme sea-state. The caisson capacity and load tensions are evaluated in the direction of the mooring line at the dip-down point, i.e. at an angle ho with the horizontal. The probability of failure pF during a given extreme sea state is:

    pF ¼ P M R; Hs ; T p ; U 10  0

ð2Þ

Assuming a Poisson model for the occurrence of extreme sea states, the relationship between the probability of failure pF and the annual probability of failure pFa is:

pFa ¼ 1  exp ðkpF Þ

ð3Þ

where k is the mean annual rate of extreme sea states; pFa  kpF for small values of kpF. 3.1. Probabilistic modeling of caisson capacity Capacity at the padeye is evaluated using the PLM and then the formulation developed by Neubecker and Randolph [6] is applied to assess the chain–soil interaction effect and obtain the capacity at mudline. Fig. 4 shows capacity interaction diagrams at the padeye and the mudline considering the characteristic Su profile and a = 0.7, Nab = 9. Fig. 5a shows the ratio of load capacity at

the padeye (Ra) to capacity at the mudline (R) as a function of the loading angle at mudline (ho). For purely horizontal loading, the capacity at the padeye is about 90% that at mudline. For loading angles less than 20°, the capacity at padeye maybe up to 95% that at mudline. For larger loading inclinations, say angles greater than 60°, the difference in capacity is less than 3%. Fig. 5b shows the ratio of load angle at the padeye (ha) to load angle at the mudline (ho) as a function of ho. It is observed that ratio ha/ho increases quite rapidly as ho approaches zero; ha/ho varies from 1.25 to 3 as ho varies from 20° to 5°. For loading inclination greater than 30° the angles at padeye and mudline are similar, and for ho > 50° the ratio ha/ho is close to 1. Clearly, capacity estimates at mudline and at the padeye differ in a more significant way for small loading inclinations or for purely horizontal loading. Samples of caisson capacity were obtained using the PLM and simulations of the undrained shear strength gradient (Su1), the side shear factor (a), the reverse-end bearing factor (Nab), and the line tension angle at the padeye (ha). A lognormal distribution was assumed for Su1, with mean value 1.70 kPa/m and coefficient of variation dSU1 = 0.20 [17,18]; this random variable accounts for uncertainty due to systematic test variations as well as spatial variability in the soil properties. At mudline it is considered that the undrained shear strength is Suo = 1.02 kPa [17,18]. Nab and a are modeled using a bivariate lognormal distribution, with mean values la = 0.7 and lNab = 9, which calibrated the PLM, coefficients of variation da = 0.2 and dN ab = 0.25, and correlation coefficient q = 0.8 [15,19,20]. Since a given load capacity can be explained by different combinations of a and Nab, and considering that capacity increases with any of them, hence a and Nab are negatively correlated. The load angle at the padeye was modeled as ha = lha + Dha, where Dha is a zero-mean normal random variable with standard deviation equal to 2° [21], and the mean value lha was estimated as the load angle obtained from the design of the caisson models.

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

63

Fig. 6. Histograms of simulated capacities (left) and fitted/sample distribution functions of capacity at mudline (right); caisson models for catenary mooring and undisturbed soil. Fig. 7. Mean and standard deviation of load capacity versus caisson length; Lf/D = 6.

Samples of capacity were obtained at the padeye and mudline for the caisson models based on ensembles of 5000 simulations of Su1, a, Nab and ha, considering both undisturbed and remolded undrained shear strengths. Using maximum likelihood estimators, lognormal probability density functions were fitted to the simulated samples of caisson capacity. Fig. 6 shows histograms of simulated capacities at mudline and the fitted distribution functions. The mean (l), standard deviation (r), median value (m), and coefficient of variation (d) of capacities at the padeye and mudline were computed for all caisson models; results for some of the models are listed in Table 3. The results show that mean capacity at mudline is 4–10% greater than at the padeye in case of caisson models for the catenary mooring; it is 2–4% greater in case of caisson models for the taut-leg mooring. The differences in capacity at mudline and at the padeye decrease slightly as the diameter and length increase. The same conclusions can be drawn in terms of the median capacities at mudline and at the padeye. There is no significant difference between coefficients of variation of capacity at mudline and at the padeye; they are about 19% and 20% for caisson capacities of the catenary and taut-leg moorings, respectively. The mean and median capacity of suction caissons for the catenary system are about 20–55% greater than the ones for the taut-leg system, depending on the Lf/D ratio.

Estimates of the lower bound of the axial capacity component can be obtained through capacity analyses using the strength of remolded soil [20]. Simulations were performed considering a remolded shear strength profile defined by a lognormal gradient with mean value 0.45 kPa/m and coefficient of variation dSU1 = 0.05. At mudline it was considered that Suo = 0.95 kPa. The corresponding mean, standard deviation, median value and coefficient of variation of capacities for some caisson models are given in Table 4. The results showed that the mean capacity at mudline is about 6–7% greater than at the padeye, in case of caissons for the catenary systems, and 2–4% greater in case of caissons for the taut-leg moorings. The same conclusions can be drawn in terms of the median capacities. The coefficients of variation of capacity at mudline and at the padeye are practically the same; they are 6% and 8% for caissons of the catenary and taut-leg moorings, respectively. A lognormal distribution truncated at the lower bound capacity was used. This was defined as the caisson capacity corresponding to the remolded shear strength. A finite mass probability was added to the truncated distribution to account for the probability of capacity being less than its lower bound [20]. Table 5 shows that the ratio of the lower-bound capacity to the median capacity at mudline is equal to 0.3 and 0.35–0.40 for caisson models of the catenary and taut-leg moorings, respectively.

64

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

Table 3 Mean, standard deviation, median value, and coefficient of variation of capacity at padeye and at mudline. Undisturbed soil. Mooring system

Models

Lf/D

Lf (m)

Catenary

C-4

4

C-6

6

T-4

4

T-6

6

12 24 18 36 12 24 18 36

Taut-Leg

Padeye

lR/lRa

Mudline

lRa (kN)

rRa (kN)

dRa

mRa (kN)

lR (kN)

rR (kN)

dR

mR (kN)

4599.4 35339.5 9973.3 77727.8 3933.0 30498.1 6643.7 51953.6

846.5 6717.3 1860.7 14815.9 760.0 6016.9 1301.4 10327.2

0.184 0.190 0.187 0.191 0.193 0.197 0.196 0.199

4523.4 34717.9 9804.1 76353.1 3861.6 29921.4 6519.8 50956.6

5079.3 37064.2 10991.5 80662.2 4065.9 30981.0 6929.2 52990.8

937.6 7039.3 2053.7 15387.7 781.7 6100.7 1352.5 10520.1

0.185 0.190 0.187 0.191 0.192 0.197 0.195 0.199

4994.9 36413.3 10804.5 79233.4 3992.8 30397.2 6800.8 51976.4

1.10 1.05 1.10 1.04 1.03 1.02 1.04 1.02

Table 4 Mean, standard deviation, median value, and coefficient of variation of capacity at padeye and at mudline. Remolded soil. Mooring system

Catenary

Taut-Leg

Models

Lf/D

C-4

4

C-6

6

T-4

4

T-6

6

Lf (m)

Padeye

12 24 18 36 12 24 18 36

lR /lRa

Mudline

lRa (kN)

rRa (kN)

dRa

lR (kN)

rR (kN)

dR

1463.3 10256.2 3054.7 22228.2 1466.5 10714.3 2451.6 18470.4

74.5 552.0 158.5 1204.2 91.0 619.4 194.5 1487.6

0.051 0.054 0.052 0.054 0.062 0.058 0.079 0.081

1555.3 10890.4 3271.8 23724.6 1515.2 10871.8 2550.3 18799.0

84.6 612.6 180.4 1317.3 92.8 625.6 199.0 1503.3

0.054 0.056 0.055 0.056 0.061 0.058 0.078 0.080

1.06 1.06 1.07 1.07 1.03 1.01 1.04 1.02

Table 5 Ratio of lower-bound capacity to median capacity at mudline. Lf (m)

C-4

Lf (m)

C-5

Lf (m)

C-6

Lf (m)

T-4

Lf (m)

T-5

Lf (m)

T-6

12 16 20 24

0.31 0.31 0.30 0.30

15 20 25 30

0.30 0.30 0.30 0.30

18 24 30 36

0.30 0.30 0.30 0.30

12 16 20 24

0.38 0.37 0.36 0.36

15 20 25 30

0.38 0.37 0.37 0.36

18 24 30 36

0.37 0.36 0.36 0.35

To develop models of capacity distribution, square polynomials were fitted for the mean and standard deviations of capacity as a function of caisson length for each group of models. Fig. 7 shows the variation of mean and standard deviations of capacity at padeye and at mudline for caissons of the catenary system with Lf/D = 6. Similar fits were made for capacities associated to the remolded shear strength. In this way, probability distributions of load capacity were established as a function of caisson length for varying Lf/D ratios. 3.2. Probabilistic modeling of mooring line tensions Assume that the dynamic tension in a mooring line can be modeled as a Gaussian process [2,14,22]. The expected maximum dynamic line tension during an extreme sea state of duration Dt, can be expressed as [23]:

hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 lnðmH Dt=2Þ E T dyn;max H  lTdyn;max ¼ # 0:5772 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rT;H 2 lnðmH Dt=2Þ

ð4Þ

where H denotes a random vector of r uncertain environmental variables that characterize the extreme sea state, and rT,H = r(H), mH = m(H), are the standard deviation and the mean crossing rate of the dynamic tension, respectively. Response surfaces were developed to express the mean line tension Tmean and the expected maximum dynamic line tension at mudline lTdyn,max in terms of H using second order polynomial expansions of the type:

YðHÞ ¼ c þ aT H þ HT bH

ð5Þ

Fig. 8. Response surfaces for Tmean and lTdyn,max as functions of wave height (Hs) and peak period (Tp): (a) Catenary mooring; and (b) Taut-leg mooring; U10 = 25 m/s, Uc = 1.75 m/s, xw = p, dww = 0, dwc = 0.

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

65

where, Y(H) is the response of interest, H is the vector of environmental variables of order r 1 and c, a (r 1) and b (r r) are the unknown coefficients to be determined from response analysis. Response surfaces were computed using response data of the mooring systems at the two sites in the Bay of Campeche, for a set of 8100 combinations of values of r = 7 environmental variables: significant wave height (Hs), peak period (Tp), wind velocity (U10), surface current velocity (Uc), wave direction (xw), and wind and current directions relative to wave direction (dww and dwc). Response data for the most loaded section at mudline among all mooring lines was used for estimation of the response surfaces. Fig. 8 shows response surfaces for Tmean, and lTdyn,max in terms of significant wave height and peak period. Extreme sea states during a storm event are defined as those for which Hs  Hth S provided Hs at time tpeak is a peak value within time window tpeak + DTcluster and tpeak  DTcluster, where Hth S is a threshold value and DTcluster is a de-clustering time window. The values of all other environmental variables are taken as those occurring at time tpeak. Data to characterize the extreme values of environmental variables are taken from time series for one-hour sea states available from hindcast studies for tropical (hurricanes) and extra-tropical storms (northers) over a 41 year observation period for the two sites in the Bay of Campeche [24]. Marginal probability distributions of H = [HS, TP, U10]T have been estimated using the Peaks over Threshold (POT) method based on a set of 54 extreme seastate events in the hindcast time series at site 1 and 58 events at site 2. Maximum likelihood estimates of the mean annual rate k are 54/41 = 1.32 per year for site 1 and 58/41 = 1.41 per year for site 2. More detail on the estimation of the marginal and joint probability distributions of the environmental variables for these sites, using Nataf model, can be found in [14]. Fitted marginal distributions, maximum likelihood estimates of the distribution parameters, and estimated correlation coefficients are given in Tables 6 and 7. 3.3. Results Reliability analyses were performed using FORM. To avoid convergence problems in FORM, the finite probability (spike) concentrated at the lower-bound capacity was modeled as a very narrow Gaussian distribution appropriately weighted to ensure that the total probability under the mixed capacity density equals one [25]. Fig. 9 shows the variation of the annual reliability index as a function of caisson length. For reliability indices between 3.7 and 4.2, which correspond to target failure probabilities between 104 and 105 typically used for the ultimate limit state design of several offshore systems [8,9,26,27], dimensions of the suction caissons range from 15.5 m to 17 m for Lf/D = 4, from 17 m to 18 m for Lf/D = 5, and from 18 m to 19 m for Lf/D = 6 in case of the catenary moorings; they range from 19 m to 20 m for Lf/D = 4, from 21.5 m to 23 m for Lf/D = 5, and from 24 m to 25 m for Lf/D = 6 in case of the taut-leg moorings. It can be seen that increasing the diameter has a greater effect on the reliability of the caisson models for the taut-leg mooring. Take for instance a length of 19 m: if Lf/D varies from 5 to 4, the reliability index increases from 2.3 to 3.7, thus reducing the failure probability by two orders of magnitude from 1.07  102 to 1.08  104. In case of the catenary mooring, if Lf/D varies from 5 to 4, the reliability index increases from 4.44 to 4.95 and the failure probability is reduced by one order of magnitude from 4.5  106 to 3.7  107. These results indicate that increasing the diameter has a greater influence on the capacity of caissons for the taut-leg moorings. The taut leg capacity is more sensitive to diameter because it is governed by the axial capacity of the caisson, with the end bearing being proportional to the square of the diameter, as compared to lateral capacity which is proportional to the diameter. Fig. 9 also shows that

Fig. 9. Annual reliability index versus caisson length: (a) catenary mooring; and (b) taut-leg mooring.

Fig. 10. Comparison of annual reliability indexes for catenary and taut-leg moorings; Lf/D = 4.

a reliability level can be achieved by different caisson dimensions. For instance, for a reliability index bannual = 4.1, caisson lengths may be 20 m for Lf/D = 4, 22.5 m for Lf/D = 5 and 25.1 m for Lf/D = 6, in case of caissons for the taut-leg moorings. If the sole criterion for optimal design was to minimize the caisson volume, the largest

66

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

Table 6 Distribution parameters of environmental variables. Variables

Probability distribution

Distribution parameters

Site 1 (630 m)

Site 2 (1565 m)

Hs

3-parameter Weibull

Scale Shape Location

Tp

Lognormal

U10

Lognormal

llnT rlnT llnU rlnU

2.4587 1.4696 4 2.4337 0.1189 2.8028 0.1629

2.0806 1.2052 4 2.4090 0.1178 2.7860 0.1597

p p 10 10

Table 7 Correlation coefficients for environmental variables. Site 1

HS TP U10

Site 2

HS

TP

U10

HS

TP

U10

1.00 0.6714 0.7732

0.6714 1.00 0.1446

0.7732 0.1446 1.00

1.00 0.7485 0.7910

0.75 1.00 0.3990

0.7910 0.3990 1.00

Lf/D ratio would be the optimal solution: in this case Lf/D = 6, length 25.1 m and diameter 4.18 m. Using the results in Fig. 9, the relation between caisson length and the logarithm of failure probability, log(pFa), was obtained. Figs. 10–12 show the variation of caisson length versus log(pFa) as a function of Lf/D for both cases of mooring systems. It is seen that a linear relationship can be established between log(pFa) and

caisson length. The slope of such linear relations defines the increment in caisson length necessary to achieve a reduction of the failure probability by a factor of 10, i.e. one order of magnitude. This is useful input for life-cycle cost benefit analysis, where modeling initial costs as a function of failure probability is needed. The increase of initial cost can be associated to increase of material volume, and therefore of caisson length. Figs. 10–12 show that slopes are higher in case of the caissons for catenary than for taut-leg mooring. For Lf/D = 4, the required length increase is 1.14 m for taut-leg and 1.35 m for catenary. For Lf/D = 5 it is 1.28 m for taut-leg and 1.49 m for catenary. For Lf/D = 6, the corresponding length increases are 1.42 m and 1.88 m. Thus, considering the lengths indicated above for reliabilities between 3.7 and 4.2, the necessary length increase to reduce by 10 the failure probability are in the order of 6% for the taut-leg mooring and 9% for the catenary one. These would result in increments of caisson volume of the order of 20–30% for the taut-leg and catenary moorings, respectively. It is also seen that for each mooring system, the larger the ratio Lf/D the greater the length increase necessary to reduce by 10 the failure probability. 4. Calibration of partial safety factors Consider the characteristic capacity at mudline Rc; the design equation is then

RC

cR

Fig. 11. Comparison of annual reliability indexes for catenary and taut-leg moorings; Lf/D = 5.

 cmean T meanC þ cdyn lTdynC

where cR, cmean and cdyn are the partial safety factors on the capacity, mean tension, and expected maximum dynamic tension, respectively, to be calibrated for a given target reliability. Herein, RC is defined through a characteristic linear profile of Su, and characteristic values of a and Nab as was described in Section 4. The characteristic values of tensions (Tmean-C and lTdyn-C,) are defined from 100 year return period environmental contours for which the total characteristic tension is maximum. Both the characteristic capacity and characteristic tensions are evaluated in the direction of the mooring line at the dip-down point, i.e. at an angle ho with the horizontal. We used the method proposed by Ditlevsen and Madsen [28] where design cases are generated so that they meet the target reliability index and then a set of partial safety factors is obtained for each design case. Consider m representative design cases; the annual reliability for each design case is approximated as bj = b0(aj  d), j = 1, 2, . . ., m, where b0 is the design target reliability index, aj is the vector of direction cosines for the most probable failure point of design case j in standard normal space, and vector d is obtained from the following optimization problem:

min d

Fig. 12. Comparison of annual reliability indexes for catenary and taut-leg moorings; Lf/D = 6.

ð6Þ

m X



xj b0 ðaj  dÞ  b0

2

ð7Þ

j¼1

where xj is a weight factor for design case ‘‘j‘‘. The representative design point for all cases, in standard normal space, is calculated as uD = b0d. The philosophy behind this approach is to identify a vector d which defines a unique design point for all design cases

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

such that the expected loss (penalty) in Eq. (7) becomes minimal. The next step is to transform this design point to the physical space, e.g. through the Nataf or the Rosenblatt transformations. Let RD,j, Tmean-D, j and lTdyn-D, j be the values in physical space for design case ‘‘j’’ which correspond to the design point uD in the standard normal space. For each design case ‘‘j’’, the safety factors are then obtained based on the characteristic values as follows: cR, j = RC, j/RD, j, cmean, j = Tmean-D, j/Tmean-C, j, cdyn, j = lTdyn-D, j / lTdyn-C, j. Safety factors were calibrated for a target annual reliability index b0 = 4.1 corresponding to an acceptable annual failure probability of 2  105. This target reliability was proposed by Rendón et al. [29] as a result of a life-cycle cost benefit analysis applying the framework for risk assessment and failure consequence modeling formulated by Faber et al. [30] and Heredia-Zavoni et al. [31] for the purpose of developing design criteria of floating systems. A model for estimating the expected monetary value of caisson failure consequences was developed employing a Bayesian probability network. Using a formulation of total expected costs, which includes the failure consequences and a simple model that relates initial costs to failure probability, the target reliability was established as that which minimizes the total expected costs over required life service of the suction caissons to be designed. The reader is referred to [29–31] for detailed information on these models and the formulation employed. Here, the calibration of safety factors is carried out separately for the caissons for each type of mooring. This is considered to be most appropriate taking into account that: (1) the contribution of the dynamic tension is much larger, compared to the mean tension, in a catenary than in a taut-leg system; (2) the caisson capacity may be governed by different failure mechanisms: lateral failure in a catenary system and axial failure in a taut-leg system; and (3) the lower-bound capacity of the caissons for the taut-leg system is greater because of the influence of the caisson’s net weight. The design cases shown in Table 8 were produced for the calibration process using the results of the reliability analyses in Fig. 9. The design points (RD, Hs-D, Tp-D, U10-D) were determined from reliability analyses using FORM. The corresponding design point values of the mean tension Tmean-D and the expected maximum dynamic tension lTdyn-D were determined using the design values Hs-D, Tp-D, U10-D and the response surfaces. The design and characteristic values of capacity, mean tension and expected maximum dynamic tension, as well as the resulting partial safety factors are given in Table 9. It is seen that partial safety factors for loads are the same for all of the design cases for each mooring system; only safety factors on the resistance differ slightly among design cases. Two criteria could be used to define the resistance partial safety factors for design standardization purposes: (1) to take the average safety factors; thus cR = 1.38, cmean = 1.10 and cdyn = 2.55 for the catenary mooring caissons, and cR = 1.65, cmean = 1.05 and cdyn = 1.87 for the taut-leg system caissons; (2) to take the maximum safety factors: then for the catenary system caissons: cR = 1.40, cmean = 1.10 and cdyn = 2.55, and for the taut-leg system: cR = 1.67, cmean = 1.05 and cdyn = 1.87. Since both sets of partial safety factors are similar, the maximum safety factors are established as the optimum load safety factors. The results show that the safety factor for capacity is 20% greater in case of caissons for taut-leg systems. It is observed that capacity at the design point, RD, is greater for taut leg systems than for catenary system; this is due to the fact that there is more uncertainty in the capacity of the first type of suction caissons than in the second ones, as shown in Section 3.1. On the other hand, the characteristic capacity of caissons for the taut-leg system is greater than the one for catenary systems since their design tension is greater than for the latter ones. The safety factor on the mean tension component is slightly greater for the catenary mooring. The safety factor on the dynamic component is 35% greater in case of caissons for the catenary mooring, which is related to the dynamic

67

Fig. 13. Partial safety factors versus target reliability index: (a) for suction caisson capacity; (b) for mean line tension; and (c) for mean maximum dynamic line tension.

tension having a greater contribution to the total loading in these systems. These results account for the differences in terms of the relative contributions of the mean and dynamic tension components and their uncertainties to the total loading. Sensitivity analyses of the partial safety factors to variations in the target reliability were performed. Fig. 13 shows values of partial safety factors for target reliabilities from 3.1 to 4.1, which correspond to target failure probabilities from 1  103 to 2  105. For the whole range of target reliabilities it is also seen that the safety factor on capacity is greater for the taut-leg mooring and the safety factors on the line tensions are greater for the catenary

68

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

Table 8 Design cases and characteristic capacity at mudline. Mooring system

Design case

Lf/D

D (m)

Lf (m)

ha (°)

RC (kN)

Catenary

1 2 3 1 2 3

4 5 6 4 5 6

4.17 3.60 3.18 4.97 4.50 4.16

16.67 18.02 19.10 19.86 22.50 24.93

14.45 15.47 16.54 38.58 39.23 39.90

12659.76 12916.48 12695.23 17718.90 17808.14 18074.65

Taut-leg

Table 9 Set of partial safety factors associated to the design cases. Mooring system

Design case

RC (kN)

RD (kN)

cR

Tmean-D (kN)

cmean

lTdyn-D (kN)

cdyn

Catenary

1 2 3 1 2 3

12659.76 12916.48 12695.23 17718.90 17808.14 18074.65

9230.23 9227.51 9237.00 10806.98 10824.22 10813.81

1.37 1.40 1.37 1.64 1.65 1.67

3145.44 3145.44 3145.44 6537.49 6537.49 6537.49

1.10 1.10 1.10 1.05 1.05 1.05

5805.65 5805.65 5805.65 3989.02 3989.02 3989.02

2.55 2.55 2.55 1.87 1.87 1.87

Taut-leg

Table 10 Partial safety factors validation. Mooring system

Design

Lf (m)

D (m)

Lf/D

bannual

pFa

Catenary

A B C A B C

16.60 18.00 19.00 19.80 22.40 24.70

4.15 3.60 3.17 4.95 4.48 4.12

4 5 6 4 5 6

4.08 4.10 4.08 4.07 4.06 4.02

2.29 2.11 2.29 2.32 2.41 2.96

Taut-leg

mooring, as discussed above for b0 = 4.1. All three safety factors increase when the target reliability increases. The safety factors on capacity and mean tension (Fig. 13a and b) are significantly less sensitive to variations in the target reliability than the safety factor for dynamic tension (Fig 13c). The average rates of change of the safety factor for caisson capacity with target reliability are 0.09 (catenary mooring) and 0.11 (taut-leg mooring), whereas for the safety factor on the mean tension are 0.07 (catenary mooring) and 0.06 (taut-leg mooring). In contrast, the safety factor on the dynamic tension exhibits average rates of change of 1.17 (catenary mooring) and 0.89 (taut-leg mooring). In general, the average rate of change of partial safety factors with the target reliability depends on several factors such as the magnitude of the coefficients of variations of loads and capacity, the relative magnitude of the standard deviations of loads to that of capacity, and the magnitude of the characteristic values. In this application, the coefficients of variation of caisson capacity at mudline are about 19% (catenary mooring) and 20% (taut-leg mooring). The coefficients of variation of the mean tension at mudline are about 2% and 1% for the catenary and taut-leg moorings, respectively. The coefficients of variation of the dynamic tension at mudline are about 23% and 13% or the catenary and taut-leg moorings, respectively. These values would suggest that the safety factor of capacity should be as sensitive to changes in target reliability as the safety factor of the dynamic tension. However, this does not occur because the characteristic capacity of the caisson also increases with the reliability index since the design cases are generated to meet the target reliability. On the contrary, the characteristic values of line tensions are independent of the target reliability since they are defined in terms of the 100 year return period environmental contours. The calibrated safety factors for b0 = 4.1 were tested applying them to the design of suction caissons. The partial safety factors

     

105 105 105 105 105 105

for catenary and taut-leg moorings were used in the design equation at the mudline, as formulated in this paper. Table 10 lists the dimensions of the designed suction caissons and their computed annual reliability indexes. As seen, for both kind of mooring systems the reliability of the designs is very close to the target b0 = 4.1. The good agreement between the obtained reliability indexes and the target one indicates the robustness of the calibration process and reliability assessment. It should be noticed that the quality of the calibration process basically depends on the penalty function employed and the design cases selected. Alternative penalty functions could be used in order to penalize more the reliability indexes (of design cases) that are less than the target (e.g. Lind [32]). On the other hand, the design cases should be well distributed over the class of possible caissons on which the partial safety factors should apply and the number of design cases should be as large as feasible in order to achieve the desired accuracy.

5. Concluding remarks A 3D finite element model of suction caissons was developed using ANSYSÒ to evaluate the loading capacity. Failure criteria were established for three different loading inclination ranges associated with failure under pure translation, pure pullout, and combined translation and pullout. Interaction diagrams of horizontal and vertical capacity at the padeye produced using the finite element model (FEM) were in very good agreement compared to other interaction diagrams published in the literature. The capacity of suction caissons was also assessed at the mudline accounting for soil–chain interaction effects. A plastic limit model (PLM) was calibrated by varying the side shear factor and the reverse-end bearing factor in order to match the FEM results. The calibrated PLM was used for characterizing capacities for the reliability analysis.

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

Reliability analyses were performed considering both mooring line tensions and capacity of the caisson at the mudline, where they are statistically independent. Mean and dynamic tensions from catenary and taut-leg moorings of an FPSO designed for the Bay of Campeche, Gulf of Mexico, at 630 m and 1565 m water depths were used. Suction caisson models with length–diameter ratios 4, 5 and 6, for each type of mooring system, were used for the reliability analysis. Samples of caisson capacity were generated at mudline and at the padeye using the PLM and simulations of the significant variables such as the undrained shear strength gradient, the side shear factor, the reverse-end bearing factor and the line tension angle at the padeye. Probability densities of capacity were obtained as function of length and length-diameter ratio by fitting lognormal distributions to the simulated capacities; these distributions were truncated at a lower bound defined by the mean value capacity obtained with remolded soil and completed with a finite probability concentrated at such lower bound. Reliability analyses were carried using FORM and safety factors were calibrated for a target reliability index equal to 4.1. Suction caissons were designed using the calibrated safety factors; it was verified that reliability of the designs was close to the target. The main findings of this study are: 1. The mean and median capacities at mudline were about 4–10% greater than at padeye for the catenary mooring, and 2–4% greater in case of the taut-leg. Coefficients of variation of capacity at mudline and at the padeye were similar; they are about 19% and 20% in case of the catenary and taut-leg moorings, respectively. 2. The mean and median capacities of caissons for the catenary system were greater than the ones for the taut-leg system; differences in capacity were of the order of 20–55% depending on the caisson’s length–diameter ratio. 3. The ratio of the lower-bound to median capacity was equal to 0.3 for caisson models of the catenary system, and 0.35–0.4 for caisson models of the taut-leg system. In this study the lower-bound capacity had essentially no effect on the reliability index. 4. Reliability of caissons for taut-leg moorings is more sensitive to caisson diameter than that of caissons for catenary moorings. Capacity of caissons for taut-leg moorings is governed by the axial contribution with the end bearing capacity being proportional to the square of the diameter whereas lateral capacity is proportional to the caisson diameter. 5. Failure probabilities (in log scale) were expressed as a linear function of caisson length. Length increments of about 6% and 9% reduced by 10 the failure probability of caissons for the taut-leg and catenary moorings, respectively. These result in 20–30% volume increase, which is a reasonable basis for estimating initial cost increments to reduce failure probabilities by one order of magnitude. 6. The safety factor for capacity is 20% greater in case of caissons for taut-leg systems, cR = 1.67, than for catenary ones, cR = 1.40. They account for uncertainty in soil–chain interaction, and could be expected to be greater than a safety factor on capacity at the padeye. 7. The safety factor on the dynamic component is 35% greater in case of caissons for catenary than for taut-leg moorings. On the mean tension component the safety factor is a little greater for catenary systems as well. These results account for the differences in terms of the relative contributions of the mean and dynamic tension components. Anchoring of moored floating structures used in offshore oil and gas production facilities is a critical role to be performed by suction caissons. Design criteria for suction caissons should be established

69

within a risk management framework. Current trends in the design philosophy of modern codes for offshore facilities establish safety classes defined in terms of risks indicators such as life safety, economic consequences of failure, and damages to the environment. Target reliabilities or probabilities of failure are specified for each safety class and limit state, and the corresponding safety factors to be used in a design equation are given. This design philosophy has been adopted by several standards such as DNV-RP-E302 [33], DNV-RP-E303 [8], and DNV-OS-E301 [9] dealing with structural or geotechnical design. ISO standards have also been developed under the philosophy that classifies systems into exposure levels which depend on life safety categories and failure consequences, see e.g. ISO 19904-1:2006 [34]. These standards have been adopted as national standards by regulatory bodies such as BSI, see e.g. BS ISO 19901-7:2005 [35]. In the Mexican oil industry, risk-based standards have also been developed for design and reassessment of fixed jacket platforms and submarine pipelines [36,37]. The formulation advanced in this paper can be used to support safety factors for code development in order to achieve target reliabilities. The formulation can be applied also to assess compliance of existing facilities with requirements of new codes. As given here, the formulation is an improvement compared to previous studies provided it considers a more rigorous treatment of components of line tensions, accounts for soil–chain interaction effects, and treats metocean and geotechnical uncertain variables explicitly. Parameter uncertainties and their effect on the reliability of suction caissons have not been considered in this work. These are parameters involved in the probability distributions and the numerical or physical models used for the reliability assessment, estimated on the basis of some statistical inference processes that introduce a degree of uncertainty in their estimates. In the case of suction caissons, parameter uncertainty may arise from the modeling of suction caisson capacity, including uncertainties in the finite element modeling and in the calibrated parameters of the PLM, as well as from the coefficients and error residuals of the response surfaces used to characterize the loading. Some geotechnical data is available for assessing some of the modeling uncertainties. The mean value and coefficient of variation of the ratio of measured to predicted uplift capacity using PLM have been estimated for suction caissons in normally consolidated clay based on data from labscale model tests (1-g modeling), centrifuge tests and full scale field tests [39]. Furthermore, capacities in the complete range of loading inclinations from 0 to 90° have been evaluated using labscale suction caisson models and the results have been compared with the numerical predictions of the PLM model [15]. Parameter uncertainties may also result from the probabilistic modeling of the tension loads in the mooring lines. For instance, if the maximum dynamic line tension was considered in the limit state function, the reliability analysis would have to consider that the distribution parameters are functions of the uncertain metocean variables that characterize the extreme sea states. Hence, uncertain parameters in the probabilistic modeling would be introduced. In this study we used the expected maximum dynamic line tension based on formulations of previous works in order to simplify the reliability analyses. Although it simplifies the reliability analysis, it cannot account for random variations of the dynamic tension within the duration of a sea state. A measure of reliability that takes into consideration parameter uncertainties is the so-called predictive reliability index [38]. In principle, it can be assessed using well established methods of structural reliability such as FORM, SORM or Monte Carlo simulation. An alternative approach would be to use a nested reliability formulation based on auxiliary limit state functions. Under the assumption of a normal distribution for the reliability index, Der Kiureghian [38] derived a simple formulation for the predictive reliability which can make use of

70

F. Silva-González et al. / Structural Safety 45 (2013) 59–70

first order approximations for the mean and variance of the reliability index in terms of the mean values of the uncertain parameters. This is an option that could be explored for developing a formulation to assess the uncertainty in the reliability index of suction caissons which results from parameter uncertainties. The assessment of the uncertainty in reliability index accounting for parameter uncertainties is beyond the scope of this work and should be addressed in future studies. Acknowledgement Funding for this research was provided by the Mexican Petroleum Institute under project D.00703 ‘‘Análisis de riesgo y confiabilidad estructural para el diseño de pilotes de succión de sistemas flotantes’’. References [1] Clukey EC, Banon H, Kulhawy FH. Reliability assessment of deep water suction caissons. In: Proceedings, 32nd offshore technology conference, OTC 12192. Houston, Texas. 2000, pp. 777–85. [2] Choi YJ. Reliability assessment of foundations for offshore mooring systems under extreme environments [dissertation]. Austin, TX: The University of Texas at Austin; 2007. [3] Aubeny CP, Han S, Murff J. Refined model for inclined load capacity of suction caissons. In: Proceedings, 22nd intl. conf. offshore mechanics and arctic engineering, Cancun, Mexico. 2003, pp. 883–7. [4] Aubeny CP, Han S, Murff J. Inclined load capacity of suction caissons. Int J Numer Anal Meth Geomech 2003;27:1235–54. [5] Aubeny CP, Han S, Murff J. Suction caisson capacity in anisotropic purely cohesive soil. Int J Geomech 2003;3:225–35. [6] Neubecker SR, Randolph MF. Performance of embedded anchor chains and consequences for anchor design. In: Proceedings, offshore technology conference, OTC 7712, Houston, Texas. 1995, pp. 191–200. [7] Valle-Molina C, Heredia-Zavoni E, Silva-González FL. Reliability analysis of suction caisson for FPSO systems. In: Proceedings, 27th intl. conf. offshore mechanics and arctic engineering, Estoril, Portugal. 2008. [8] Det Norske Veritas. Geotechnical design and installation of suction anchors in clay. Recommended Practice DNV-RP-E303. Norway: Det Norske Veritas; 2005. [9] Det Norske Veritas. Position mooring. Offshore Standard DNV-OS-E301. Norway: Det Norske Veritas; 2008. [10] Sánchez-Moreno J. Proposta de metodologia para critério de projeto de fundações de estacas de sucção para TLP [dissertation]. Rio de Janeiro, Brazil: COPPE-UFRJ; 2005. [11] Andersen K, Murff J, Randolph M, Clukey EC, Erbrich C, Jostad HC. Suction anchors for deep waters applications. In: Proceedings, intl. symposium on frontiers in offshore geotechnics, Perth, Australia. 2005. [12] Chen WF, Liu XL. Limit analysis in soil mechanics. Amsterdam, The Netherlands: Elsevier Science; 1990. [13] Murff JD, Hamilton JM. P-ultimate for undrained analysis of laterally loaded piles. J Geotech Eng Div, ASCE 1993;119(1):91–107. [14] Montes-Iturrizaga R, Heredia-Zavoni E, Silva-González F, Straub D. Nested reliability analysis of mooring lines for floating systems. Appl Ocean Res 2012;34:107–15. [15] El-Sherbiny RM. Performance of suction caisson anchors in normally consolidated clay [dissertation]. Austin, TX: The University of Texas at Austin; 2005. [16] Rufiar M, Mendoza MJ, Ibarra E. Axial load capacity for an instrumented pile model: a review of alpha and beta methods. In: Proceedings, Pan American CGS geotechnical conference, Toronto, Canada. 2011.

[17] Silva-González F, Heredia-Zavoni E, Sánchez-Moreno J, Valle-Molina C. Análisis de confiabilidad y calibración de factores de seguridad para el diseño de pilotes de succión. Instituto Mexicano del Petróleo; 2012 INDA Registry N°: 03-2012110810402800-01. [18] Sánchez-Moreno J, Valle-Molina C, Silva-González F, Gilbert RB, HerediaZavoni E. Caracterización de la capacidad de carga de pilotes de succión en arcillas normalmente consolidadas. Instituto Mexicano del Petróleo; 2012 INDA Registry [forthcoming]. [19] Luke AM, Rauch AF, Olson RE, Meacham EC. Components of suction caisson capacity measured in axial pullout tests. In: Proceedings, intl. symposium on deepwater mooring systems, concepts, design, analysis, and materials. OTRC; 2003. pp. 1–12. [20] Najjar S. The importance of lower-bound capacities in geotechnical reliability assessments [dissertation]. Austin, TX: The University of Texas at Austin; 2005. [21] Gilbert RB. FORM Analysis for suction caisson anchor. University of Texas at Austin; 2010 Sep. Attachment G In: Technical Report N°: 1. Purchase Order N°: 2010000253: UTA09–000748. Sponsored by Instituto Mexicano del Petróleo. [22] Sarkar A, Eatock Taylor R. Effects of mooring line drag damping on response statistics of vessels excited by first and second-order wave forces. Ocean Eng 2000;27(6):667–86. [23] Davenport AG. Note on the distribution of the largest value of a random function with applications to gust loading. Proceedings of the Institution of Civil Engineers, London, England 1964;28(2):187–96. [24] Oceanweather Inc. Update of metocean design data for Zona Norte and Sonda de Campeche. Oceanweather Inc.; 2006 [Final Report. Purchase Order N°: 2006000748. Sponsored by Instituto Mexicano del Petróleo]. [25] Melchers RE, Ahammed M, Middleton C. FORM for discontinuous and truncated probability density functions. Struct Saf 2003;25:305–13. [26] Det Norske Veritas. Dynamic risers. Offshore Standard DNV-OS-F201, Norway: Det Norske Veritas; 2010. [27] Det Norske Veritas. Submarine pipeline systems. Offshore Standard DNV OSF101, Norway: Det Norske Veritas; 2010. [28] Ditlevsen O, Madsen HO. Structural reliability methods. New York, NY: John Wiley and Sons; 1996. [29] Rendón C,Heredia-Zavoni E,Silva-González F. Evaluación de riesgo de falla de pilotes de succión. Instituto Mexicano del Petróleo; 2012 INDA Registry N°: 03–2012–121811211700–01. [30] Faber M, Straub D, Heredia-Zavoni E, Montes-Iturrizaga R. Risk assessment for structural design criteria of FPSO systems Part I: generic models and acceptance criteria. Mar Struct 2012;28(1):120–33. [31] Heredia-Zavoni E, Montes-Iturrizaga R, Faber M, Straub D. Risk assessment for structural design criteria of FPSO systems. Part II: consequence models and applications to determination of target reliabilities. Mar Struct 2012;28(1):50–66. [32] Lind NC. Reliability-based structural codes. Practical calibration. In: Holand, et al., editors. Safety of structures under dynamic loading, vol. 1. Trondheim: Tapir; 1978; 149–61. [33] Det Norske Veritas. Design and installation of plate anchors in clay. Recommended Practice DNV RP-E302, Norway: Det Norske Veritas; 2010. [34] International Organization for Standardization. Petroleum and natural gas industries. Floating offshore structures. Part 1: monohulls, semi-submersibles and spars. ISO 19904-1; 2006. [35] British Standards Institution. Petroleum and natural gas industries. Specific requirements for offshore structures. Part 7: stationkeeping systems for floating offshore structures and mobile offshore units. BS EN ISO 19901-7; 2005. [36] PEMEX. Diseño y evaluación de plataformas marinas fijas en el Golfo de México. NRF-003-PEMEX-2007; 2008. [37] PEMEX. Diseño de líneas submarinas en el Golfo de México.NRF-013-PEMEX2009; 2009. [38] Der Kiureghian A. Analysis of structural reliability under parameter uncertainties. Probab Eng Mech 2008;23:351–8. [39] Gilbert RB, Choi YJ, Dangayach S, Najjar SS. Reliability-based design considerations for deepwater mooring system foundations. In: Proceedings, intl. symposium frontiers in offshore geotechnics, Perth, Australia. 2005.