Reliability assessment of drag embedment anchors in sand and the effect of idealized anchor geometry
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Abstract
In this study, the reliability of drag embedment anchors in the sand was assessed and the effect of anchor geometrical idealization on reliability indices was investigated as an inherited characteristic of analytical approaches. The anchor holding capacity was obtained by performing a series of iterative limit state analyses and a probabilistic model was developed for the selected anchor families. The tensions of the mooring lines connected to a semisubmersible platform were obtained by performing a series of time-domain dynamic mooring analyses using the OrcaFlex software. The uncertainties in environmental loads, metocean variables, and stress distribution along the catenary mooring lines were incorporated into the line tensions through the response surfaces. An iterative procedure was performed by adopting the first-order reliability method (FORM) to calculate the comparative failure probabilities in sand and clay. The study showed significant dependence of the anchoring system reliability on geometrical configuration of anchors, the seabed soil properties, and the environmental loads. It was observed that the implementation of the reliability-based design into the existing in-filed trial procedures could significantly improve the efficiency and cost-effectiveness of the design practice.
Keywords
Reliability analysis Drag embedment anchor Catenary mooring Response surface Numerical method Sand seabedNomenclature
- A_{s}
area of shank.
- d
nominal chain diameter.
- D
pad-eye embedment depth.
- d_{f}
fluke thickness.
- d_{s}
average depth of the shank.
- du_{a}
the absolute displacement of the anchor.
- du_{s}
soil wedge displacement.
- du_{sa}
displacement of the soil relative to the anchor.
- d_{w}
wave direction.
- d_{wc}
current direction relative to wave.
- d_{ww}
wind direction relative to wave.
- E_{n}
normal circumference parameter.
- E_{n}
tangential circumference parameter.
- f
form factor (Neubecker and Randolph, 1996a).
- F
friction force.
- F_{f}
the fluke force.
- F_{fb}
the force on the back of the fluke.
- F_{s}
the shank force.
- h
back edge of the fluke.
- H
depth of fluke tips.
- H_{s}
significant wave height.
- L_{f}
fluke length.
- L_{f}
caisson length (Silva-González et al., 2013)
- L_{s}
shank length.
- N_{q}
standard bearing capacity factor.
- N_{qs}
shank bearing factor.
- p_{F}
probability of failure.
- p_{Fa}
annual probability of failure.
- q
bearing pressure.
- Q
normal soil reaction on chain segment.
- \( \overline{\mathrm{Q}} \)
average bearing resistance per unit length of chain over embedment depth.
- R
anchor capacity at mudline.
- R
soil reaction.
- R_{a}
anchor capacity at pad-eye.
- R_{d}
design anchor capacity at mudline.
- R_{d,a}
design resistances at the pad-eye.
- r_{i}
distance between point i and anchor shackle.
- s
length of chain.
- SF
side friction.
- T
line tension.
- T_{a}
line tension at the pad-eye.
- T_{d}
design line tension at mudline.
- T_{d,a}
design tensions at the pad-eye.
- T_{dyn,max}
mean maximum dynamic line tension.
- T_{dyn,max-C}
characteristic mean maximum dynamic tension.
- T_{mean}
mean line tension.
- T_{mean-C}
characteristic mean line tension.
- T_{o}
Chain tension at mudline.
- T_{p}
spectral peak period.
- T*
normalized tension.
- ∆t
extreme sea state duration
U_{10}
wind velocity.
- U_{c}
surface current velocity direction.
- w
chain self-weight per unit length.
- W_{a}
anchor dry weight.
- W_{s}
the mobilized soil mass.
- x_{a}
anchor horizontal displacement.
- X
absolute displacement of point i.
- x*
horizontal distance normalised by D
- ∆x
absolute penetration increment of the origin Y
absolute displacement of point i
- ∆y
absolute penetration increment of the origin z
depth below mudline
- z*
depth normalised by D
- β
inclination of fluke
- β
reliability index
- β_{annual}
annual reliability index
- ∅^{′}
soil friction angle
- ∅_{p}
sand peak friction angle
- γ′
effective unit weight of soil
- γ_{dyn}
partial safety factor on dynamic line tension
- γ_{mean}
partial safety factor on mean line tension
- λ
failure wedge angle
- λ
mean annual rate of extreme sea states
- η_{a}
anchor efficiency
- μ
chain-soil friction coefficient
- Θ
vector of environmental variables
- θ
line tension angle
- θ_{a}
line tension angle at the pad-eye
- θ_{i}
polar coordinate angle of point i
- θ_{fs}
fluke-shank angle
- θ_{o}
line tension angle at mudline
- ∆θ
rotation increment of the origin
- ψ
dilation angle
Introduction
Drag embedment anchors are widely used as a cost-effective solution for temporary and permeant station keeping of floating structures. By growing offshore exploration and productions, the number of incidents in floating facilities induced by the failure of mooring system has been increased, subsequently (Wang et al., 2010; Ma et al., 2013). This has caused the industry to further emphasize on reliability assessment of the mooring systems and their key components in various types of seabed sediments. Drag embedment anchors are amongst the crucial components of the mooring systems that are used with catenary and taut leg mooring systems.
There are several studies in the literature that have considered the reliability assessment of various anchor families such as suction anchors (Choi, 2007; Valle-molina et al., 2008; Clukey et al., 2013; Silva-González et al., 2013; Montes-Iturrizaga and Heredia-Zavoni, 2016; Rendón-Conde and Heredia-Zavoni, 2016). The high level of certainties in the evaluation of the holding capacity of suction anchors and the promising results obtained in aforementioned reliability studies has provided confidence about no need for field trials in assessment of holding capacities of these anchors. However, the situation is totally different in drag embedment anchors. Difficulties in collecting in-field holding capacity databases, the complicated interaction between the anchor and the seabed, the unknown ultimate depth and location of the anchor, and the need for the extensive amount of costly computational analyses have resulted in limitations to assess the reliability of these important anchor families. Therefore, the current design practice includes performing costly filed trials (e.g., API RP 2SK, 2008). Moharrami and Shiri (2018) published the first study on the reliability of drag embedment anchors and initiated a reliability assessment approach that could potentially result in elimination or mitigation of the filed trial expenses in future. However, the study was limited to clay and used plastic yield loci to be obtained from a series of time-consuming finite element analyses to characterize the fluke-soil interaction and failure states. Also, the authors did not investigate the impact of anchor geometrical idealization of the reliability indices, which is an inherited consequence of the numerical and analytical approaches.
In this study, the reliability of the drag embedment anchors was assessed in sand that has not been investigated in the past. A limit state approach, with no need to time-consuming finite element analysis was adopted to characterize the anchor failure state. The influence of using idealized anchor geometry on reliability indices was also examined and comparative studies were conducted between the sand and clay to obtain the effect of different uncertainties in shear strength parameters on reliability indices.
The holding capacity of anchors was calculated by developing an Excel spreadsheet and incorporation of the limit state analysis proposed by Neubecker and Randolph (1996a). There are several studies on the prediction of drag anchors capacity by analytical and empirical solutions (Neubecker and Randolph, 1996a; Thorne, 2002; O’Neill et al., 2003; Aubeny and Chi, 2010). However, the adopted solution (Neubecker and Randolph, 1996a) benefits from several advantages such as simplified prediction of the anchor capacity and trajectory, incorporation of chain-sand interaction, and comprehensive validation against the experimental studies (Neubecker and Randolph, 1996a; Neubecker and Randolph, 1996b; O’Neill et al., 1997). This model has been widely used in several studies in the literature (Neubecker and Randolph, 1996b; Neubecker and Randolph, 1996c; O’Neill et al., 2003) and recommended by design codes (e.g., API RP 2SK, 2008). The mooring line tensions were obtained by performing dynamic mooring analysis using OrcaFelx software and a generic semisubmersible platform. Reliability assessment was performed by using the first-order reliability method (FORM) through developing a probability model for anchor holding capacities.
The study further prepared the ground for improvement of the anchor design codes (e.g., API RP 2SK, 2008), where the effect of different reliability indices on proposing an optimized filed trials is currently neglected, and an identical holding capacity evaluation procedure is recommended in both sand and clay.
Methodology
The reliability analysis was conducted by calculation of the anchor capacity against the mooring line tensions. The model proposed by Neubecker and Randolph (1996a) was used to analyze chain-soil and anchor-soil interactions in the sand and predict the anchor capacity at the mudline and shank pad-eye. The anchor model was programmed in an Excel spreadsheet macro using Visual Basic Application (VBA) (see Appendix 1 (Table 17)). OrcaFlex software package was employed to model a generic semisubmersible platform in the Caspian Sea to obtain the characteristic mean and maximum dynamic line tensions for a 100 years return period sea states. Various key parameters were incorporated in the estimation of anchor capacities including peak friction at the seabed, dilation angle, soil density, fluke and shank bearing capacity factors, anchor geometrical configurations, line tension angle at mudline, and side friction factor. The response surfaces were used to determine the mean and expected maximum dynamic line tensions. First-order reliability method (FORM) was used to assess the reliability of anchors connected to the catenary mooring line. The DNV design code (DNV-RP-E301, 2012) was used to define the partial design factors on the mean and maximum dynamic line tensions and capacities.
Anchor-seabed interaction
The anchor system mobilizing the ultimate holding capacity comprises of the anchor and the connected chain, both of which were modeled in this study. The Stevpris MK5 and MK6 anchors were used, as the most popular choices in the industry.
Frictional capacity of chain
The parameter T is the line tension; θ is the inclination from the horizontal; F is the friction force, and Q is the typical soil reaction on the chain segment.
Anchor holding capacity
Main dimensions for 12 t anchors (Vryhof Anchors, 2010)
Dimension | Mk5 (L_{f}/d_{f} = 6.67) | Mk6 (L_{f}/d_{f} = 3.09) |
---|---|---|
A (mm) | 5908 | 5593 |
B (mm) | 6368 | 6171 |
C (L_{f}), (mm) | 3624 | 3961 |
E (mm) | 3010 | 2642 |
F (d_{f}), (mm) | 543 | 1282 |
H (mm) | 2460 | 2394 |
S (mm) | 150 | 140 |
Fluke-shank angle(θ_{fs}), (°) | 32.00 | 32 |
Properties of the modeled drag anchors and the corresponding resistance and line angles
Anchor type | L_{f}/d_{f} | L_{f}(mm) | d_{f}(mm) | R_{d,a}(kN) | θ_{a}(°) |
---|---|---|---|---|---|
Mk5 | 6.67 | 4297 | 644 | 2275 | 13.0 |
Mk6 | 3.09 | 4534 | 1468 | 2267 | 12.9 |
The selection of these anchor families facilitated making comparisons between the current study and the results obtained in earlier investigations in clay.
Anchor kinematics
The minimum work approach was applied and the penetration ∆y and rotation ∆θ were considered to obtain the incremental anchor displacements.
Developing iterative macro for prediction of anchor performance
The static limit state and kinematic models were coded into an Excel spreadsheet using VBA macros to calculate the ultimate holding capacity of the anchor-chain system and the anchor trajectory. The developed spreadsheet performed a series of iterative analyses with the calculation procedure outlined in Fig. 5.
Soil and anchor input parameters in the current analysis
Parameter | Value |
---|---|
Anchor dry weight, W_{a} (kN) | 98.06 |
Fluke length, L_{f} (m) | 3.41 |
Fluke width, b_{f} (m) | 5.99 |
Fluke thickness, d_{f} (m) | 0.51 |
Shank length, L_{s} (m) | 5.55 |
Shank width, b_{s} (m) | 2.31 |
Fluke-Shank angle, θ_{fs} (°) | 32 |
Effective chain width, b_{c} (m) | 0.24 |
Chain self-weight, w_{c} (kN/m) | 2 |
Chain soil friction coefficient, μ | 0.4 |
Peak friction angle, ϕ_{p} (°) | 35 |
Residual friction angle, ϕ_{r} (°) | 25 |
Dilation angle, ψ (°) | 8.5 |
Effective unit weight, γ^{′} (kN/m3) | 10 |
Figure 6 shows a perfect agreement between the developed VBA macro and the results published by the developer of the original limit state anchor solution.
Time-domain mooring analysis of semisubmersible platform
Performing a three hours’ time-domain simulation, the most critically loaded line was detected for the environmental loads with a 100 years return period (i.e., H_{s} = 9.5 m, T_{P} = 12.8 s, and U_{10} = 29 m/s). A similar head sea response amplitude operator (RAO) of the platform published by Moharrami and Shiri (2018) was adopted to facilitate comparison of the results.
Catenary mooring system characteristic
H_{s}(m) | T_{P}(s) | U_{10}(m/s) | T_{mean-C}(kN) | T_{dyn,max-C}(kN) | T_{d}(kN) | θ_{o}(°) |
---|---|---|---|---|---|---|
9.5 | 12.8 | 29 | 846 | 623 | 2493 | 1.3 |
The main output of the analysis includes the parameters T_{d} (design line tension), θ_{o} (line angle at mudline), T_{mean-C} (characteristic mean tension), and T_{dyn,max-C} (characteristic mean maximum dynamic tension) that will be used for reliability assessment in the next section.
First-order reliability analysis
First-order reliability method (FORM) was adopted through an iterative procedure to obtain the probabilistic results by incorporation of uncertainties in seabed soil properties and environmental loads. The probabilistic modeling of anchor capacity was conducted by using the limit equilibrium method. The embedment profile and the frictional capacity of the chain were also accounted for in the calculation of ultimate holding capacities. The response surface approach and appropriate probability density functions were used to take into consideration the uncertainties of the environmental loads and metocean variables including significant wave height, spectral peak period, wind velocity, and consequently the stress distribution throughout the catenary lines. A target failure probability of 10E-5 was set assuming a consequence class of 2 as per recommendations made by DNV-RP-E301 (2012). Further details are provided in the coming sections.
Limit state function
Probabilistic modelling of anchor capacity
The crucial factors that were used to construct the anchor capacities database were including the peak friction angle (ϕ_{p}), the dilation angle (ψ), and the soil density (γ′). The mean value of peak friction angle (\( {\upmu}_{\varnothing_{\mathrm{p}}} \)) for lognormal distribution was set to 35° with a coefficient of variation (\( {\updelta}_{\varnothing_{\mathrm{p}}} \)) equal to 0.05 to take into consideration the uncertainty due to systematic test variations and spatial variations of the soil properties (Basha and Babu, 2008; Anchor manual, 2010). A normal distribution with a mean value (μ_{ψ}) of 8.49° and a coefficient variance (δ_{ψ}) of 0.28 was adopted for the sand dilation angle (ψ) that was calculated by using Bolton’s empirical equation for sand (Bolton, 1986; Phoon, 1999; Simoni and Houlsby, 2006). The soil density was represented by a normal distribution with a mean value (\( {\upmu}_{\gamma^{\prime }} \)) of 10.07 and a coefficient variance (\( {\updelta}_{\gamma^{\prime }} \)) of 0.02 (Neubecker, 1995; Phoon, 1999; Simoni and Houlsby, 2006). To construct the capacity database, 5000 simulations were conducted by adopting different values of ϕ_{p}, ψ and γ′.
Statistical properties of anchor capacity at pad-eye and mudline
Model | L_{f}/d_{f} | L_{f}(m) | Padeye | Mudline | μ_{Ra}/μ_{R} | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
μ_{Ra}(KN) | σ_{Ra} (KN) | δ_{Ra} | m_{Ra}(KN) | μ_{R} (KN) | σ_{R} (KN) | δ_{R} | m_{Ra} (KN) | ||||
MK5 | 6.67 | 2.707 | 2283.1 | 506.6 | 0.222 | 2419.0 | 2650.5 | 618.8 | 0.233 | 2801.0 | 0.86 |
MK5 | 6.67 | 3.166 | 3754.2 | 978.8 | 0.260 | 4001.0 | 4314.8 | 1174.1 | 0.272 | 4620.0 | 0.87 |
MK5 | 6.67 | 3.410 | 4874.1 | 1273.2 | 0.261 | 5183.5 | 5590.7 | 1522.7 | 0.272 | 5970.0 | 0.87 |
MK5 | 6.67 | 3.624 | 6093.6 | 1636.5 | 0.268 | 6506.0 | 6978.8 | 1949.2 | 0.279 | 7485.0 | 0.87 |
MK6 | 3.09 | 2.958 | 2876.4 | 429.2 | 0.150 | 2917.0 | 3357.1 | 524.4 | 0.156 | 3411.5 | 0.86 |
MK6 | 3.09 | 3.460 | 5149.2 | 885.5 | 0.172 | 5246.0 | 5983.4 | 1070.7 | 0.178 | 6095.0 | 0.86 |
MK6 | 3.09 | 3.728 | 6702.5 | 1101.0 | 0.164 | 6822.2 | 7768.5 | 1327.9 | 0.170 | 7434.0 | 0.90 |
MK6 | 3.09 | 3.961 | 8451.4 | 1505.3 | 0.178 | 8588.0 | 9779.6 | 1805.6 | 0.184 | 9958.5 | 0.86 |
The mean capacity at mudline is 10 - 14% higher than the mean capacity at pad-eye. Commonly in all anchor models, when the fluke length and fluke thickness increase, the differences between capacity at the pad-eye and mudline increase. The same conclusion can be driven for differences between median capacities at the pad-eye and the mudline, but in some anchor models (MK6 with L_{f} = 3.961 m) the difference between median at the mudline and pad-eye decreases by an increment of fluke length and thickness. The coefficients of variation of the capacity at pad-eye and mudline are about 23-27% for all MK5 anchor families and are about 16-18% for all MK6 anchor families.
Probabilistic modelling of line tension
Distribution parameters of environmental variables
Variable | Probability distribution | Distribution parameters | |
---|---|---|---|
H_{s} (m) | Weibull | Scale | 9.5351 |
Shape | 10.1552 | ||
T_{p} (s) | Lognormal | μ_{lnTp} | 2.4966 |
σ_{lnTp} | 0.1196 | ||
U_{10} (m/s) | Lognormal | μ_{lnU10} | 3.4827 |
σ_{lnU10} | 0.1095 |
Estimated correlation coefficients
H_{s}(m) | T_{p}(s) | U_{10}(m/s) | |
---|---|---|---|
H_{s}(m) | 1.0 | 0.9728 | 0.9905 |
T_{p}(s) | 0.9728 | 1.0 | 0.9935 |
U_{10}(m/s) | 0.9905 | 0.9935 | 1.0 |
Results of reliability analysis
As shown in Fig. 11, to achieve specified target reliability, different anchor families with different fluke lengths and weights are available. For example, for desired reliability index of β_{annual} = 2, either MK5 with L_{f} = 3.46 m and W_{a} = 8 t (point 2) or MK6 with L_{f} = 2.95 m and W_{a} = 5 t (point 5) could be used. Figure 11 shows that for the anchors with the close magnitude of fluke length but different weights (from MK5 and MK6), the corresponding reliability levels are remarkably different. For instance, the 10 tones MK5 and 8 tones MK6 anchors with L_{f} values of 3.41 m (point 3) and 3.46 m (point 6) have a reliability index of 3.2 and 5.08, respectively. This shows that despite the clay (Moharrami and Shiri, 2018) the anchor weight is less influential in the sand, so a lighter MK6 anchor gives higher holding capacity, higher reliability index, and consequently lower failure probability (reduces from 0.0119 to 1.5 × 10^{−7}) compared with a heavier MK5 anchor. These ranges of failure probability are commonly used for ultimate limit state design in offshore systems (DNV-OS-E-301, 2010; DNV-OS-F201, 2010; DNV-OS-F101, 2013). As observed in Fig. 11, the fluke length has a significant effect on reliability indexes in both anchor families; the larger fluke length, the higher holding capacity, the higher reliability index, and the lower probability of failure.
As shown in Fig. 12, in MK6 anchor families, beyond an anchor weight of 8 t and fluke length of 3.46 m the annual reliability index and the logarithm of failure probability approaches the infinity that is shown by dashed lines.
Equivalent reliability study in sand and clay
Equivalency map of anchor classes in sand and clay with close reliability indices
Clay | Anchor Classes in Clay | |||||
---|---|---|---|---|---|---|
Sand | 15 t | 18 t | 22 t | 25 t | 30 t | |
Anchor Classes in Sand | 5 t | MK6 | MK6 | |||
8 t | MK5 | |||||
10 t | MK5 | |||||
12 t | MK5 | |||||
15 t | MK5 |
Further studies can be conducted to determine the equivalency map between the different anchor groups, e.g., MK3, MK5, MK6, and even other anchors in the market. This kind of information can provide better insight for operators and designers to select the required anchors with the desired level of reliability that may vary depending on project conditions. As instance, the results presented in Table 8 indicates that for mobile platforms with existing classes of anchors, the number of anchoring field trial in sand seabed to achieve the target reliability can be reduced in comparison with clay and this can result in a considerable saving in the resources.
Idealization of anchor geometry
Fluke and shank equivalent dimensions for Case 1,2
Case #1 | Case #2 | |||||||
---|---|---|---|---|---|---|---|---|
Anchor weight (ton) | 5 | 8 | 10 | 12 | 5 | 8 | 10 | 12 |
Shank length (m) | 4.41 | 5.16 | 5.55 | 5.90 | 4.41 | 5.16 | 5.55 | 5.90 |
Shank effective width (m) | 0.67 | 0.78 | 0.85 | 0.90 | 0.71 | 0.84 | 0.90 | 0.96 |
Fluke length (m) | 2.70 | 3.16 | 3.41 | 3.62 | 2.70 | 3.16 | 3.41 | 3.62 |
Fluke effective width (m) | 2.63 | 3.07 | 3.31 | 3.51 | 2.99 | 3.50 | 3.77 | 4.01 |
These results show that the geometrical idealization procedure needs particular attention to improve the accuracy of the anchor holding capacity and consequently the results of reliability studies. This can be achieved by adjusting the idealization approach through comparisons with the existing manufacturer and field trials data in every specific anchoring project. It would be highly beneficial if the obtained results in the current study were compared to a benchmark design of the anchor geometry without idealization. However, it is not feasible at the moment due to having no analytical solution to model the real anchor geometry. In addition, using advanced large deformation finite element (LDFE) analysis or remeshing and interpolation technique with small strain (RITSS) analysis to run thousands of extremely time-consuming LDFE analysis for reliability studies is not currently feasible. These limitations need to be resolved by developing improved analytical and numerical solutions.
Conclusions
The costly in-field testing procedure recommended by design codes for estimation of the anchor capacities are identical for all of the anchor families, seabed soil types, environmental loads, and operation conditions. This approach neglects the reliability effects associated with various uncertainties and results in less cost-effective solutions. The current study showed that anchoring design procedures could be improved by performing reliability studies using analytical and numerical models.
The existing theoretical anchor solutions are developed based on simplified anchor geometries. The study showed that idealization of anchor geometry affects the reliability indices. Further investigations and improved numerical and analytical solutions are required to determine the best practice for the idealization of anchor geometry and approach the results of field trials.
A target reliability index for a given anchor family in the sand can be achieved by a lighter anchor compared to the clay. It is challenging to determine a corresponding set of soil parameters in clay and sand to result in an identical reliability index. However, further studies in this area can be beneficial in proposing a more cost-effective infield testing procedure depending on the seabed soil properties.
The geometrical configuration of the anchors, particularly the fluke length, is the most influential parameters in determining the reliability indices. The anchor weight has a beneficial contribution to achieving a higher level of reliability but to a less extent. A well-designed anchor geometry can significantly dominate the weight effect. For instance, some lighter MK6 anchors result in a higher reliability index compared to heavier MK5 models due to their superior geometrical design.
It is worth mentioning that the reliability models for assessing the anchor capacities can be significantly improved by having access to the in-field test databases and corresponding seabed soil properties, and the statistics of failures. These kinds of information are mandatory for obtaining absolute reliability indices or failure probabilities. However, the closed-form solution is highly beneficial for performing comparative studies and improving the recommended practices. Also, the current study is limited to one anchor group and a specific geographical location. Further investigations using different anchors and a range of environmental loads, and system configuration would provide much better insight into this challenging area of engineering.
Notes
Acknowledgments
The authors gratefully acknowledge the financial support of this research by Memorial University of Newfoundland through VP start-up fund and school of graduate studies (SGS). The technical advice of Mr. Mohammad Javad Moharrami is also kindly acknowledged.
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