Influence of Kinematic Analysis Parameters of Drag Anchor Trajectory Prediction Using Yield Envelope Method

Drag anchor is widely applied in offshore engineering for offshore mooring systems. The prediction of the invisible trajectory during its drag-in installation is challenging for anchor design in determining the anchor final position for ensuring sufficient holding capacity. The yield envelope method based on deep anchor failure for kinematic analysis was proposed as a promising trajectory prediction method for drag anchor. However, there is a lack of analysis on the effects of the parameters applied in the kinematic analysis. The current work studies the effects of the yield envelope parameters, anchor line bearing capacity factor and the anchor/soil interface friction. It is found that the accuracy of the yield envelope parameters has large impact on the prediction results based on deep yield envelopes. Analyses of cases with smooth fluke predict deeper embedment depth than that from analyses of cases with rough fluke. The decrease of the capacity factor results in the increase of the anchor embedment depth, the anchor line load, the anchor chain angle and the stable value of the normalized horizontal load component for the same drag length, while the stable value of the normalized vertical load component decreases when the capacity factor decreases. This illustrates the importance in applying reasonable parameters and improving the method for more reliable prediction of the anchor trajectory.


Introduction
For floating offshore structures in oil/gas industry and the floating wind turbines in offshore wind industry, mooring systems are the key supporting structures, which are connected to the seabed soil by different types of anchors. As the increasing need for large offshore structures, anchors such as drag anchors, which include drag embedded anchor (DEA) and vertical loaded anchor (VLA), suction embedded plate anchor (SEPLA), dynamically penetrated anchor (DPA, such as torpedo anchor and OMNI-Max anchor), have been increasingly applied in offshore engineering. Drag anchor is named for its drag-in installation, for which the anchor is dragged into the seabed until the required holding capacity is reached. It is an attractive anchor preference for deep water anchorage due to its low cost of installation and high holding capacity relative to low anchor weight in soft clay condition (Kim, 2005). However, the anchor trajectory during drag-in installation is invisible.
The prediction of the invisible trajectory is challenging but important for anchor design in determining the anchor final position for ensuring sufficient holding capacity, and the field installation operation.
Studies on the trajectory prediction methods of drag anchor have been developed from the traditional empirical method, which determines the drag anchor position after installation based on design charts from the anchor suppliers, such as Vryhof Anchors and Bruce Anchor. Prediction methods such as limit equilibrium method (Stewart, 1992;Neubecker and Randolph, 1995;Dahlberg, 1998;Thorne, 1998), upper bound method (Kim, 2005;Aubeny et al., 2005Aubeny et al., , 2008, yield envelope method (Bransby and O'Neill, 1999;O'Neill et al., 2003;Elkhatib and Randolph, 2005;Elkhatib, 2006;Cassidy et al., 2012;Tian et al., 2014;Wei et al., 2014;Liu et al., 2015Liu et al., , 2016Wei et al., 2015;Han et al., 2018), kinematic model-based analytical method (Liu et al., 2012a(Liu et al., , 2012b(Liu et al., , 2013 and large deformation finite ele-ment method Liu, 2014, 2016) have been presented and applied in the existing literatures. The advantages and limitations of these methods have been discussed in Wu (2016). For the traditional empirical method, the anchor drag length and embedment depth can be easily read from the design chart, but the trajectory related parameters are provided without the detailed trajectory in installation, and the soil condition is not site specific. The limit equilibrium method is the earliest method for anchor trajectory prediction based on geotechnical knowledge. The method is based on the calculation of soil resistance on all the elements using given equations. However, the empirical constants that determined from field and laboratory tests are required. And the parametric study proved that the assumed factors for the resistance calculation influence the accuracy of prediction. The upper bound method is based on the theory of plastic limit analysis, which is more efficient in solving the collapse problem. But the application of the method is complex with requirement of a comprehensive optimization analyses and has limited application in industry. The newly developed kinematic model-based analytical method and method that based on large deformation finite element analysis depend on the accuracy of some mathematical expressions and uncertain parameters. The yield envelope method discussed in the current work requires the yield envelope study, which can be achieved by finite element analyses. In addition, compared with the other existing methods, the prediction flowchart can be easily applied and the prediction can be site specific. By comparing the complexity and accuracy of the yield envelope method with the other existing methods in industry application, yield envelope method has been found as a promising method. It has been applied for the prediction of the trajectory of drag anchor, keying process of SEPLA and OMNI-Max anchor.
Yield envelope method based on deep anchor failure for kinematic analysis proposed by Bransby and O'Neill (1999) served as an alternative for anchor trajectory prediction in industry. However, analyses are insufficient on the effects of the parameters applied for the kinematic analysis and thus it is difficult to evaluate the reliability of the method and in selecting the parameters for the application of the method. For a confident application of this method and the development of the method, the current paper studies the effects of the yield envelope parameters, anchor line bearing capacity factor and the anchor/soil interface friction for evaluating the influence of these parameters on the prediction results.
2 Kinematic analysis using yield envelope method For anchor fluke subjected to combined vertical (V), horizontal (H) and rotational (M) loading during installation in Fig. 1, the behavior of fluke is characterized with yield envelope in VHM space. The yield envelope can be expressed as a function of failure loads V, H and M as . The relative plastic vertical (δv), horizontal (δh) and rotational (δ ) displacement at failure can be determined using the VHM yield envelopes based on the assumptions of associate flow or the condition of normality. For a typical yield envelope plotted in VH space shown in Fig. 2 with M equal to 0 for a simplified rectangular anchor fluke, the plastic potential allows the determination of relative displacement ratio. Finite element study or upper bound limit analysis can be conducted to develop the yield envelopes function. Bransby and O'Neill (1999) suggested the following form from Murff (1994) which can give the best fit of the yield envelope.
where f is the yield locus, V max , M max , and H max are the maximum loads respectively, m n, p, and q are exponents determined by least squares regression scheme. The values of the parameters and V max , M max , and H max are given by O'Neill et al. (2003) from both finite element study and upper bound analysis. The yield envelope can be used to determine the kinematic movement of the anchor in cohesive soil based on the assumption of associated flow in determining the forces  on the fluke and anchor displacements. By ignoring the influence of shank force on the movement and the elastic displacements, the direction of fluke movement is calculated using the following equations after choosing the incremental distance δh: where B is the length of the fluke and t is the fluke thickness in Fig. 2. The suggested steps of kinematic analysis using yield envelope method based on the anchor force system in Fig. 3 are as follows (O'Neill et al., 2003): β (1) Assume the initial anchor position, including the anchor fluke angle and the pad eye depth d a ; (2) Calculate the normal and sliding force on the anchor shank, F 1 and F 2 ; (3) Assume the anchor chain force at the pad eye T a ; θ a (4) Calculate the anchor chain angle according to the anchor chain equation from Neubecker and Randolph (1995) which is dependent on the anchor line force and anchor line angle relationship; (5) Calculate the forces V, H and M on the anchor fluke according to the force equilibrium of the anchor force system; (6) Check if the calculated V, H and M are on the deep yield envelope; β (7) Use normality to determine the displacement ratio and an incremental horizontal displacement δh is assumed for the calculation of the vertical displacement δv and rotation δ based on the determined displacement ratio in the other directions; β β (8) Adjust the anchor position and update the fluke angle and the pad eye depth d a based on the above calculated displacement and repeat Steps (1) to (8) until the anchor reaches a stable state. The anchor fluke is moved with the incremental displacement δh and δv in the local direction of V and H, while rotation of δ is given for the fluke angle change.
The shank resistance has two components, the shank normal force F 1 and sliding force F 2 . The normal force F 1 can be calculated as the product of the shank area perpendicular to the shank, the bearing capacity factor N cs and the soil undrained shear strength. The sliding force F 2 can be calculated as the product of the shank surface area and the soil undrained shear strength, which indicates that the interface of the shank and soil is assumed as rough condition. The bearing capacity factor N cs is assumed as 9 in the studies of Bransby and O'Neill (1999) and O'Neill et al. (2003) while 10.5 is used in Elkhatib (2006) in the kinematic analysis of VLA. The anchor chain equation from Neubecker and Randolph (1995) is used to calculate the anchor line angle from the anchor line force by ignoring the self-weight: where, is the chain angle at pad eye, T a is the chain tension at pad eye, b c is the chain effective width, N cl is the bearing capacity factor for the anchor chain, which is assumed as 9 in Bransby and O'Neill (1999) and O'Neill et al. (2003), d a is the pad eye depth, S u0 is the soil surface undrained shear strength, and S ug is the undrained shear strength gradient. For kinematic analyses, the used yield envelopes defined by the yield envelope parameters are commonly from the finite element study of a deeply embedded anchor with deep localized failure. Large amount of displacement-controlled finite element studies are conducted to get the yield envelopes and the function parameters by curve fitting. More details on the yield envelope study and the kinematic study can refer to O'Neill et al. (2003) and Elkhatib (2006).

Case information and yield envelope parameters
As there is very limited existing work on the prediction of drag anchor, the typical case used in O'Neill et al. (2003) is used in the current study. The kinematic analysis is for a 32 tonne Vryhof Stevpris anchor with a 50° shank. The simplified anchor geometry and soil property are summarized in Table 1. For the kinematic studies, one of the most important parts is the study of yield envelope functions, which is characterized by the yield envelope parameters. Table 2 gives the parameters of yield envelopes for deep anchor failure with rough interface from different researchers (O'Neill et al., 2003;Elkhatib and Randolph, 2005;Elkhatib, 2006;Yang et al., 2010;Andersen et al., 2003;Wu et al., 2017). As for the same simplified strip anchor fluke with length and thickness ratio B/t of 7 and rough interface in uniform clay with undrained shear strength S u , different yield envelope function parameters are given by different researchers, due to the accuracy of the yield envelope study and curve fitting process, as shown in Table 2. The current kinematic analyses applied parameters from different researchers to examine the difference caused by the existing yield envelope parameters. The yield envelope parameters from the authors' previous work (Wu et al., 2017) are also examined and results are compared with those from other researchers.
Since the difficulty in determining the anchor/soil interface friction, the existing yield envelopes from different researchers are all based on the assumption of a rough anchor/soil interface. There is a lack of analysis on the effect of anchor/soil friction on the predicted results. The current study applied the same numerical method and model as that in the authors' previous work in Wu et al. (2017) to give the yield envelope parameters for case with smooth anchor/soil interface. Table 3 gives the parameters of yield envelopes for deep anchor behavior with smooth and rough interface (Wu et al., 2017) from the current study. The kinematic analysis using different yield envelope parameters from rough and smooth cases aims to study the effect of interface friction. Meanwhile, the effect of the shank/soil friction is also studied in the kinematic analysis by considering the case of a smooth shank.

Effect of yield envelope parameters
The validation of the analysis flow using deep yield envelope from Wu et al. (2017) adopted in the current analysis is verified by comparing the results from current study using the same parameters of yield envelope from O'Neill et Then the parameters of yield envelope from other researchers are used to investigate the influence of yield envelopes on the anchor performance. The bearing capacity factor N cl and N cs for the calculation of forces of anchor line and the shank resistance used the same value of 9 in O' Neill et al. (2003). Here the same value is kept in the reproduction and comparison studies. The results of the kinematic analyses are presented and compared in Fig. 4. The anchor efficiency (the holding force over the anchor dry weight, which is 32 t in the current case), anchor trajectory (relationship of anchor drag length X a /B and anchor pad eye depth d a /B), anchor fluke angle , anchor chain angle and normalized load components ( , and ) are shown, respectively. When the fluke reaches a steady state, the kinematic performance parameters are compared as shown in Table 4.
To verify the current prediction procedure, the reproduction of the kinematic analysis using deep anchor behavior in O'Neill et al. (2003) is conducted. This is achieved by using the example case and the yield envelope parameters in O'Neill et al. (2003). The results are shown in Fig. 4 denoting with "O' Neill et al. (2003)" and comparison between the stable values of the kinematic performance parameters from the reproduction using current prediction procedure and the values given in O'Neill et al. (2003) is made in Table 4.    (2003) with a slight difference, resulting from some nonspecific anchor system information in the analysis such as the initial anchor pad eye position, the center of the anchor system and the initial anchor fluke angle. The good agreement of the reproduced results and the values given in O' Neill et al. (2003) verifies that the current prediction procedure using deep yield envelope is correct. Meanwhile, the kinematic process is also used to reproduce the kinematic analysis of the VLA case in Elkhatib (2006) and the results also show good agreement with the results from the kinematic prediction of the case of VLA in Elkhatib (2006). After the verification of the current prediction procedure using deep anchor behavior, the yield envelope parameters in Table 2 from different researchers for the simplified drag anchor with B/t of 7 including those from current study are applied to study the influence of the yield envelopes parameters on the prediction results. The results from kinematic analysis using the existing deep yield envelopes and current deep yield envelopes are shown in Fig. 4 and denoted by the corresponding literatures where the yield envelope parameters for the deep anchor behavior are from. Values of the kinematic performance parameters are compared in Table 4.
The predicted anchor performance using parameters in Table 2 is largely different, as shown in Fig. 4. This is due to the differences in the parameters of the yield envelope function from different researchers. Although there is only a very slight difference of maximum anchor capacity in the three directions, the other parameters in the yield envelope function are determined by curve fitting, the accuracy of which depends on the number of load points obtained from the finite element studies of yield envelopes. It can be seen that the predictions based on current yield envelopes from Wu et al. (2017), Elkhatib (2006) and Andersen et al. (2003) have small differences, while large differences are shown when compared with results of O'Neill et al. (2003) and Yang et al. (2010). Analysis based on yield envelopes from O'Neill et al. (2003) gave the largest anchor efficiency and deepest anchor embedment depth. The anchor fluke normalized load components from the prediction based on the five yield envelopes are almost the same. By comparing the results from using the yield envelopes of API/Deepstar (2003) and O'Neill et al. (2003), which gave two limits of the prediction, the anchor efficiency and anchor embedment depth related to anchor capacity and trajectory increase about 38% and 43%, respectively. The current examination of the influence of yield envelopes illustrates the importance of relatively accurate yield envelope function for prediction using the yield envelope method for anchor kinematic performance. Elkhatib (2006) studied the effect of anchor/soil interface friction on the anchor performance. The interface friction coefficient of 0.4 and 1 (rough interface) were adopted in the studies of the yield envelopes for fluke with uniform thickness and aspect ratio of 7 (drag anchor) and 20 (plate anchor) and the corresponding kinematic analyses for a case of plate anchor. In the current study, the two limits of interface friction condition, smooth ( ) and rough ( ) interface conditions are chosen to examine the effect of interface friction on the kinematic performance of the base case of drag anchor. It should be noted that in the studies of O'Neill et al. (2003), the shank sliding force is calculated as a production of shank's sliding area and the undrained shear strength at the shank midpoint. When the effect of friction coefficient is studied in Elkhatib (2006), the sliding force on the shank is assumed as a production of shank's sliding area, a friction coefficient and the undrained shear strength at the shank midpoint. In the current studies, the shank/soil interface is also assumed smooth when the fluke/soil interface is assumed smooth for investigation of the effect of the fluke interface friction, which means that the corresponding shank/soil interface friction coefficient is assumed 0, which results in the 0 shank sliding force. The kinematic parameters from analyses assuming both the fluke/soil interface and shank/soil interface smooth are shown and compared with those from the prediction using yield envelope for rough fluke/soil interface in Fig. 5. The stable values of the kinematic parameters are summarized in Table 5.

Effect of interface friction
As shown in Fig. 5, for the same drag length, anchor with rough fluke becomes stable earlier than that with smooth fluke. The anchor efficiency of smooth interface is about 14% higher than that of rough interface. This is due to the deeper embedment depth resulting from the smooth interface. As shown in the comparison of anchor trajectory, the embedment depth for smooth interface is about 68% larger than that for rough interface. This large increase of embedment depth results in the anchor entering into stronger layer which increases the anchor line load. For a fairer comparison, the anchor line load factor T a /(AS uH ) is compared (S uH is the soil undrained shear strength at the correspond- Fig. 5. Influence of fluke interface friction on the anchor kinematic performance. ing depth). It is found that the anchor line load factor decreases when the condition of anchor fluke interface changes from rough to smooth, which is contrary to that shown from the comparison of the anchor efficiency. This means that the decreasing fluke interface friction will reduce the required anchor line load for anchor embedding to the same depth. The current finding of the increase of the embedment depth and the decrease of anchor line load factor resulted from smaller interface friction is consistent with the results shown in Elkhatib (2006). Accordingly, the resulted stable fluke angle and chain angle from the two friction conditions are significantly different. The trend of increasing chain angle and decreasing fluke angle over the drag is similar for both smooth and rough interface, which indicates the increasing normalized V load shown in the figure for load components.
Although smooth shank/soil interface is assumed in the above comparison, the influence of the shank roughness is also studied. For the smooth fluke/soil interface, the kinematic analysis is also conducted for the case with rough shank/soil interface. Results compared with those from analysis assuming both the fluke/soil and shank/soil interface as smooth are shown in Table 5. The expected decrease of the embedment depth does not occur when the interface of the shank/soil is changed from smooth to rough. It is found that the influence of the shank friction for the current base case is mainly on the anchor efficiency and the anchor embedment depth, which increase about 13.7% and 6% when the shank is changed from smooth to rough as shown in Table 5. This relatively small difference caused by the shank interface friction change when compared with the influence caused by the fluke interface friction change, especially on the embedment depth ratio, illustrates the importance of fluke interface friction in anchor installation performance. Because of the slight influence of the shank friction, the shank/soil interface will be kept smooth for the calculation of the shank force in the following studies when the fluke/soil interface is assumed smooth.

Effect of bearing capacity factor
For the existing application of yield envelope method using deep yield envelopes, the bearing capacity factors N cl and N cs for the anchor line equation and the shank force calculation are usually kept the same and there is no discussion on the selection of these parameters. In the previous studies, the bearing capacity factor of 9 is chosen to be consistent with O'Neill et al. (2003) for the current study. However, considering the possible decrease of bearing capacity factor when the shallow anchor behavior is integrated, the influence of the bearing capacity factor is studied first here for the analyses using deep yield envelopes and will also be discussed in the following studies when the effect of shallow anchor behavior is introduced in the kinematic analyses. Degenkamp and Dutta (1989) applied the formula given by Skempton (1951) on calculating the soil resistance of the strip footing to calculate the anchor chain bearing resistance. Degenkamp and Dutta (1989) recommended that the bearing capacity factor for the anchor chain factor N cl is from 5.1 at the seabed to 7.6 at the chain depth of 6 times the chain diameter and the maximum value of N cl is 7.6. House (2002) suggested that the chain with bearing behavior in the phase of tensioning can be represented by an embedded cylinder, for which the bearing resistance factor N cl value is between 10 and 11. While the chain with a taut configuration can be represented by an embedded strip footing due to the similar failure mechanism, for which the bearing resistance factor N cl value is between 7 and 8. DNV (2012) suggested a default N cl of 11.5, and the lower bound N cl of 9 and upper bound N cl of 14 for the anchor chain or wire. Liu et al. (2010) summarized that the value of N cl is in the range from 7.6 to 14 based on the existing literatures. O'Neill (1999) andO'Neill et al. (2003) assumed the same value of 9 for the shank bearing capacity factor N cs and the anchor chain bearing capacity factor N cl , which is 10.5 for Elkhatib (2006) in the kinematic analysis of VLA. The two bearing capacity factors for the calculation of anchor chain force and force on the shank are also kept the same N c in the current analysis, to reduce the influence parameters on the anchor kinematic performance.
Based on the above literature review on the value of the bearing capacity factor for anchor line equation, a value of 7 at the embedment ratio of 2 for strip footing in Skempton (1951) is selected to study the influence of bearing capacity factors. This value is also chosen based on the conclusion that the difference in the anchor behavior under unidirectional and combined loading is mainly for the embedment depth ratio between 1 and 3 for anchor fluke in uniform clay and the horizontal anchor fluke in clay with linearly increasing shear strength. With the possible decrease of bearing capacity factors in the shallow embedment depth for the integration of anchor shallow behavior, kinematic analyses using deep yield envelope applying bearing capacity factor of 7 is also conducted for both conditions with rough and smooth fluke/soil interface. The changes of the kinematic parameters with different N c for conditions of rough and smooth interface are shown in Fig. 6 and Fig. 7, respectively. For cases with N c = 9, the stable state is reached for the studied drag length of 50B for both two interface conditions while the kinetic parameters except normalized load components are still varying for N c of 7. The kinematic parameters at the stable state or the drag length of 50B are summarized in Table 6.
For both two interface conditions, the decrease of N c results in the increase of the anchor embedment depth, the anchor line load, the anchor chain angle and the stable value of the normalized horizontal load component for the same drag length, while the stable value of the normalized vertical load component increases when N c decreases from 9 to 7. It should be noted that the influence of N c on the change of the fluke angle is different for the drag length smaller or lar- Fig. 6. Influence of N c on anchor kinematic performance for rough interface. Fig. 7. Influence of N c on anchor kinematic performance for smooth interface.

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WU Xiao-ni et al. China Ocean Eng., 2020, Vol. 34, No. 2, P. 257-266 ger than a critical drag length, which is about 10B for the rough interface and 20B for the smooth interface. For the smooth interface, due to the large rotation of the anchor fluke, the normalized rotational load component decreases with the drag length of the anchor while the normalized rotational load component is around 0 for the rough interface. The changing is not compared in Table 6.

Conclusions
Kinematic analyses using yield envelope method based on deep anchor behavior are conducted for the example case. The effects of the yield envelope parameter, anchor/ soil interface friction and anchor line bearing capacity factors are examined. The current work provided a deep understanding on the influence of kinematic analysis parameters on the trajectory prediction of drag anchor. The main conclusions are summarized as follows.
(1) The accuracy of the yield envelope parameters has large impact on the prediction results based on deep yield envelopes. The maximum predicted anchor efficiency and anchor embedment depth using the existing yield envelope parameters in literatures can be about 38% and 43% larger than the minimum prediction results, respectively. The current examination of the influence of yield envelopes illustrates the importance of relatively accurate yield envelope function for prediction using the yield envelope method for anchor kinematic performance.
(2) Analyses based on deep yield envelopes from smooth fluke predict greater embedment depth than that from analyses based on deep yield envelopes from rough fluke. The decrease of the fluke interface friction will reduce the required anchor line load for anchor embedding to the same depth. The influence of the shank friction for the current base case is mainly on the anchor efficiency and the anchor embedment depth, which increase about 13.7% and 6% when the shank is changed from smooth to rough.
(3) The decrease of the capacity factor results in the increase of the anchor embedment depth, the anchor line load, the anchor chain angle and the stable value of the normalized horizontal load component for the same drag length, while the stable value of the normalized vertical load component decreases when the capacity factor decreases from 9 to 7.
The kinematic analysis for trajectory prediction using yield envelope method is largely dependent on the accuracy of yield envelope parameters, anchor/soil interface friction condition and the bearing capacity factors applied for the anchor line. It is necessary to improve the yield envelope method in future work to ensure a reliable prediction and reasonable selection of the other impacting parameters in practical application.