Maximum Force of Inclined Pullout of A Torpedo Anchor in Cohesive Beds

Torpedo anchors have been used in mooring systems for deep-water oil and gas projects owing to their prominent advantages, such as low cost and easy installation. The maximum force of torpedo anchors is crucial not only to the safety and stability of vessels and other marine facilities, but also for an economical design. It is necessary to develop reliable formula for fast predicting their maximum inclined force of a torpedo anchor in cohesive beds. In this study, the maximum inclined force of a torpedo anchor vertically embedded in cohesive beds was extensively investigated. 316 sets of inclined pullout laboratory tests were carried out for 9 differently shaped torpedo anchors which were vertically embedded in different cohesive beds. The loading curves were automatically acquisitioned and their characteristics were analyzed. The load angle relative to the horizontal varied from 20° to 90°. A new formula for fast calculating the maximum inclined force of the torpedo anchor vertically embedded in cohesive beds was obtained based on force analysis and a nonlinear regression on the data from the present and other studies. Effect aspects on the tests are discussed and further studies are highlighted.


Introduction
An accurate estimation of the maximum inclined force of an anchor is crucial not only to ensure the safety and stability of vessels and other marine facilities, but also to realize an economical design. In order to anchor the platforms and ships, a variety of anchorage systems have been designed using stockless anchors, screw anchors, anchor plates, suction anchors, vertically loaded anchors (i.e., VLAs), or torpedo anchors. The geotechnical focus of the offshore oil and gas industry has recently shifted to the development of cost-effective anchoring systems that meet both the geotechnical and economic demands associated with hydrocarbon exploration and extraction in deep water Liu et al., 2016). Torpedo anchors are regarded as a type of the cost-effective anchoring system owing to their short installation duration and convenient anchoring equipment (Kim et al., 2018;Wang W.K. et al., 2018). A torpedo anchor can usually be divided into four parts: a cylindrical anchor shaft, fins, a tip segment and a padeye segment (O'Beirne et al., 2015). The padeye segment is used to connect the anchor and the anchor chain, which is often hinged at the padeye at an angle (load angle) of 30°-45° to the horizontal (Fu et al., 2017). Since its first application in 2002, several different forms of torpedo anchors have been successfully used in ocean engineering projects. T-98 was used to anchor the PETROBRAS FPSO P-50 in 2003 (de Araujo et al., 2004). Two full-scale 80 t torpedo anchors were installed at the Gjøa field in the North Sea in August 2009; the embedment depths of these anchors were up to 1.85 to 2.38 times their length into the seabed, respectively (Lieng et al., 2010). The maximum inclined force of a torpedo anchor is a crucial index for a mooring system. Therefore, it is of practical significance to study the maximum inclined force of a torpedo anchor vertically embedded in cohesive beds.
Numerous studies, including physical laboratory, field tests and numerical analyses, have been conducted on the pullout characteristics of torpedo anchors. The physical laboratory experiments can be classified into 1-g tests (which were conducted under one-gravity acceleration conditions) and centrifuge tests (which were completed under higher-gravity acceleration conditions). The 1-g tests were mainly conducted by Gilbert et al. (2008) and Wang W.K. et al. (2018), while the centrifuge tests were principally conducted by Richardson et al. (2009), Hossain et al. (2015), and Fu et al. (2017). With respect to the field tests, Medeir-os (2002), Brandão et al. (2006), and Lieng et al. (2010) conducted full-scale tests. Recently, O'Beirne et al. (2015) carried out reduced-scale field tests using one 1:20 scale model anchor with four wide fins. From the above literature review, it can be found that most of the tests were conducted to investigate the vertical pullout characteristic of torpedo anchor. Besides, the effects of anchor aspect ratio, slenderness ratio and fins on the maximum inclined pullout force of the torpedo anchor in cohesive bed were also limited. Therefore, it is necessary to conduct additional physical tests to extend the scope of the tests and investigate the soil failure characteristics.
Numerical models were used to calculate the maximum inclined force of a torpedo anchor (de Araujo et al., 2004;Brandão et al., 2006;de Sousa et al., 2010;Hossain et al., 2013;O'Beirne et al., 2015;Fu et al., 2017). These numerical models are generally regarded as effective tools for analyzing the maximum inclined force of a torpedo anchor which is vertically embedded in saturated cohesive beds. Assuming the soil as an ideal elastoplastic material and adopting the Drucker-Prager model, de Sousa et al. (2010) used ANSYS to explore the effects of the load angle, number of fins, soil properties, and width of fins on the maximum inclined force of torpedo anchors. Assuming the soil as an elastic perfectly plastic material obeying a Tresca yield criterion, Kim and Hossain (2016) used Abaqus to investigate the effects of the installation method, impact velocity, anchor geometry, padeye position, pullout angle on the maximum inclined force of torpedo anchors. Nevertheless, it is very difficult to correctly formulate the module for considering the soil-anchor interaction and complex rheological properties of the soil. These numerical models have to be extensively validated to obtain the correct simulation results using sufficient data of physical tests. Moreover, for the engineering design of a torpedo anchor, it is time-consuming to calculate the maximum inclined force by these numerical models. The API (2014) method has been recommended for design O'Beirne et al., 2015). In the API method, the ultimate axial pile capacity is considered the sum of the shaft friction capacity and end bearing capacity (API, 2014).The maximum vertical force calculated by API method (2014) was relatively sensitive to the values of friction ratio ( ) (Fu et al., 2017). However, the value of in API method (2014) varies within a certain range; there are still divergences in determining its exact value (Fu et al., 2017;Wang W.K. et al., 2018). Fu et al. (2017) stressed that API method is restricted to vertical pullout and not applicable for inclined pullout. Furthermore, the empirical factor N c is assumed to be 9 (API, 2014), whereas O'Loughlin et al. (2009) and Gilbert et al. (2008) regarded this bearing ca-pacity factor N c as 12 and 17 for the ellipsoidal tip, respectively. It can be observed that the bearing capacity factor depends on the local soil properties at the anchor position, which requires further investigation. O'Beirne et al. (2015) proposed a design procedure for a dynamically installed anchor (DIA) under inclined loading, based on 1:20 scale field tests and finite element analysis. However, this procedure is strictly applicable only to the specific anchor geometry in their study (Fu et al., 2017). Fu et al. (2017) proposed a method for calculating the maximum inclined force of a torpedo anchor based on the API (2014), Broms (1964), and linear fitting method. Nevertheless, the method in Fu et al. (2017) is complicated to calculate the maximum inclined force of torpedo anchor and requires program or spreadsheet. Thus, it is necessary to develop a new formula for fast estimation of maximum inclined force. Wang C. et al. (2018) have investigated the penetration characteristics of a torpedo anchor in cohesive bed. This study was designed to investigate the maximum inclined force of a torpedo anchor. By conducting comprehensive experimental tests in the laboratory, the effects of the load angle, embedment ratio, anchor geometry, and soil shear strength on the maximum inclined force of a torpedo anchor were studied. Finally, based on the force analysis and experimental data, a formula was proposed to fast estimate the maximum inclined force of a torpedo anchor in cohesive beds. The reliability of the formula was validated by the data of other researchers.

Experimental setup
The experimental setup for anchor pullout tests is schematically illustrated in Fig. 1, comprising of an I-beam, two pulleys, a steel string of 1 mm in diameter, a stepper motor with a frequency converter, a barrel (with 120 cm and 120 cm of diameter and depth, respectively) filled with well-stirred soft soil (with 116 cm of depth), a load cell, a computer, a data acquisition card, and anchors. In order to minimize boundary effects from the barrel, the diameter of the barrel was 120 cm allowing an enough distance of 55.2 cm (5.75 times the anchor diameter) between the anchor and the wall of the barrel. The I-beam was erected over the barrel, which was attached to two pulleys. The left pulley, which was fixed at a predetermined position on the I-beam, was installed above the stepper motor. Another pulley was fixed by a clamp near the top of the torpedo anchor, and it could be moved to a designated position on the I-beam to obtain the required load angle ( =arctan(H 1 /S), where S and H 1 are defined in Fig. 1) between the mooring line and mudline. One end of the steel string was passed over the pulleys and connected to the anchor padeye, while the other end of the steel string was attached to one end of the load cell (measuring range of 50 kg). The other end of the cell was connected to the motor via another string. A 380 V stepper motor provided sufficient torque and pullout speed. By adjusting the frequency of the motor, the required pullout speed was obtained. The load cell was connected to the data acquisition card.

Torpedo anchors
Nine different full-torpedo anchors with conical tips (30°) were manufactured for the anchor pullout tests. The torpedo anchors were fabricated according to the geometric shape of the full-scaled T-98 torpedo anchor with dimensional scales from 1:65 to 1:180, which were identical to those used by Wang et al. (2016) and Wang W.K. et al. (2018). The T-98 torpedo anchor had 98 tons dry mass (m), 17 m in length (L), 1.07 m in diameter (d), and four fins of dimensions 0.9 m × 10 m (width × length). The torpedo anchors were made of stainless steel. The anchor shaft was hollow, and the conical tip was solid. The tested model torpedo anchors had three different shaft diameters, seven different shaft lengths, and three different slenderness ratios. The thickness of each fin was 1.2 mm. The distance from the upper end of the torpedo anchor to the upper end of the fin was 6 mm. The purpose of choosing these shapes and sizes was to explore the effect of aspect ratio, slenderness ratio and fins on the maximum inclined pullout force of the torpedo anchor in cohesive bed. Table 1 lists the geometric dimensions of the anchors. For the tested model anchor A12-C22 listed in Table 1, the first characters A, B, and C denote three anchor diameters (d=1.9 cm, 2.5 cm, and 3.2 cm, respectively); the second characters 1, 2, and 3 represent the three slenderness ratios of the anchor (i.e., 5, 8, and 11), respectively; the third characters 0, 1, and 2 indicate three ratios of fin length to anchor length (i.e., 0, 1/3, and 2/3), respectively.

Characteristics of cohesive beds
Two types of cohesive silt were used to constitute the cohesive beds for the full-anchor pullout tests. The sediments were collected from the seabed of Shanghai Jinshan Port and the riverside of the outlet of Shanghai Huangpu River, respectively. The median particle diameters (d 50 ) of the two sedimentary beds were 19.49 μm and 29.31 μm. The properties of the prepared test cohesive bed were measured. The liquid limits of the two sedimentary beds were 57% and 52%, respectively. The plastic limits of the two sedimentary beds were 28% and 24%, respectively. The undrained shear strength (S u ) of the sedimentary beds was measured by the vane shear test. The properties of the experimental cohesive sediments are listed in Table 2, where Z is the depth below the bed surface.
In order to derive a formula for the maximum inclined force of a torpedo anchor, the representative undrained shear strength was used. This study used S u, ave as the representative undrained shear strength, which is the average undrained shear strength over the length of the anchor at its final embedment depth ).

Test procedure
The whole process of a pullout test of the torpedo anchor could be divided into two stages: installation and pul- lout. The sampling rate of torpedo anchor pullout force was 4 Hz. In order to simplify the experimental conditions, all tests were conducted in air and only a thin layer of water covered the sedimentary bed surface. During the pullout process, the stepper motor was turned on 10 min after the torpedo anchor installation to pull the torpedo anchor out at the designated speed (v=2 mm/s). The friction between the mooring line and pulleys and the weight of the mooring line could be neglected in the present tests. Simultaneously, the load cell with the acquisition software began to record the pullout load. The pullout test proceeded until the torpedo anchor was removed completely from the bed. After that, the motor was switched off.

Results and analysis
A total of 316 tests were conducted using nine different torpedo anchors embedded in cohesive beds to investigate the maximum inclined force of the torpedo anchors. The load angle varied from 20° to 90°.

Loading curves
The loading curves of torpedo anchor A32 in two different beds are shown in Fig. 2. These curves can be divided into three stages. In the first stage, the pullout force rapidly increased with pullout time until it reached a peak (regarded as the maximum inclined force F PU ). In the two test beds, the displacement of the torpedo anchors was different when the peak value was reached. In the second stage, the load decreased gradually until the top of the torpedo anchor emerged from the bed. In the last stage, after the top of the torpedo anchor reached the bed, the load dropped quickly because the torpedo anchor did not have any remaining soil weight. It was noteworthy that the loading curve had sever-al fluctuations, which might have been caused primarily by the collapse of the wall of the cavity beneath the torpedo anchor. For identical pullout speed and torpedo anchor, the fluctuations of the curve in Bed A were larger than those in Bed B; this difference resulted from the different undrained shear strengths of the beds. The higher undrained shear strength of the wall of the cavity had a stronger ability to resist its collapse, so that the cavity could endure higher negative pressures. Similar characteristics of pullout process were observed in the other tests using different torpedo anchors.
3.2 Effect of load angle on the maximum inclined force of torpedo anchor 3 shows the maximum total inclined force, its horizontal (maximum inclined force times ) and vertical components (maximum inclined force times ) versus load angle in Bed A. For large load angles larger than 80°, the maximum inclined force remains almost constant. When the load angle decreases from 80° to 50° (or less), the maximum inclined force increases obviously. This increase in the maximum inclined force can be attributed to an increase in its horizontal component, as a simultaneous increase in the total and horizontal components can be observed. Besides, the corresponding vertical component remains roughly unchanged in the range of 80° to 50°. This is consistent with the phenomena described in Fig. 4, which presents the normalized vertical component (F PU, V0 /F PU, V ) versus the load angle, where F PU, V0 is the vertical component of the maximum inclined force, F PU, V is the maximum vertical force, and is the submerged unit weight of the soil. This behavior indicates that, for load angles larger than 50°, anchor failure is controlled predominantly by the maximum vertical force and the failure mode is likely to be ver-  tical. For load angles smaller than 50°, both the horizontal and vertical components undergo great changes, and the failure mode is not vertical.
for anchor A32. That is, the anchor with a lower value of corresponded to a higher maximum inclined force, as high as 1.46 times that of the vertical pullout with a higher load angle. Evidently, a larger load angle makes it easier to pull out the anchor. Furthermore, the data are fitted with a power functional relation, as , and the fitted curves are plotted in Fig. 5 as dashed lines.
The values of C 0 and C 1 , and the statistics of the fitted curves are listed in Table 3. The results of the fitting show a good power functional relation between F PU /F PU, V and /90°. The values of C 1 were 0. The values of C 0 decreased with the increasing embedment ratio, i.e., the values of C 0 decreased from -0.126 to -0.179 when the embedment ratio increased from 1.48 to 3.39 for anchor A32 embedded in Bed B. Similar phenomena were found for other tests using different anchors. Hence, the maximum inclined force depends on the load angle and embedment ratio. The non-dimensional factor basically represents the ratio of the undrained shear strength over the pressure at the location of the torpedo anchor. As shown in Fig. 5, a positive correlation was found between F PU /F PU, V and a non-dimensional factor . That is, the value of F PU /F PU, V increased from 1.17 to 1.47 for the embedment ratio of 1.96 and the load angle of 22 when the value of increased from 0 to 0.45. Furthermore, Fig. 5 and Table 3 also depict that the values of C 0 decreased with increasing . The values of C 0 decreased from -0.134 to -0.313 for the embedment ratio of 1.96 when increased from 0 to 0.45. Hence, the maximum inclined force depends on the value of .

Force analysis
Referring to the force analysis of other anchor structures performed by other researchers Bridge et al., 2004;Singh and Ramaswamy, 2008), the forces acting on a torpedo anchor vertically embedded in a cohesive bed are as follows: a maximum inclined force (F PU ), suction force (F SU ), side adhesion force (F AD ), additional shear force (F AS ), the submerged (or buoyant) weight of the torpedo anchor in the soil (W AS ), and the weight of the upward soil (W s ), as shown in Fig. 6.
For vertical pullout with a load angle of 90°, the suction force formula for torpedo anchors was obtained by referring to the formula used by marine facilities. The vertical suction force (F SU, V ) proposed by Das et al. (1994) for the plate anchors is described as: where A F and H are the projected area of the anchor and distance from the anchor tail to the bed surface, respectively. Eq.
(1) shows that the suction force is negatively related to H/d and positively related to S u, ave . However, other researchers Bridge et al., 2004;Singh and Ramaswamy, 2008) considered that the suction force should be positively related with these two parameters. Hence, a general form of the vertical suction force for torpedo anchors may be expressed as: where f is the function of H/d. As stated above, the equation is linear, and it can be presented as , where and are constants.  2017) introduced a friction ratio and expressed it as a general form: where A S is the lateral area of the anchor.

ξ ξH
The soil on the top of the anchor, which was compressed during the pullout process, induced an additional shear force on the anchor. In order to estimate the vertical additional shear force (F AS, V ), the soil failure surface was assumed to be cylindrical, which was in line with the assumptions made by Majer (1955) and Wang and O'Loughlin (2014). Hence, the height of the cylindrical soil failure surface may be calculated by introducing a coefficient owing to the complexity of the torpedo anchor, i.e., . The vertical additional shear force can be presented as where C F is the cross-section perimeter of the anchor. The vertical weight (W S, V ) of the upward soil can be presented as (Singh and Ramaswamy, 2008;Singh et al., 2017): The vertical submerged (or buoyant) weight (W AS, V ) of the anchor in the bed can be written as follows: where m s is the mass of the soil that would occupy the volume of the anchor and m is the mass of the anchor. Thus, a general form of the maximum vertical force during the pullout process can be expressed as: During the pullout process, the force terms on the right side of Eq. (7) kept changing, except the forth term W AS, V , but did not achieve their maxima in synchronization. In other words, the fourth term did not change and the first three terms kept decreasing, while the suction force increased to form a maximum cavity, and then fluctuated, which may be due to the dynamic change of the cavity caused by the soil mobilization. The result of such non-synchronized variations in the five forces was the maximum force obtained after a certain displacement of the anchor. 4.2 Empirical formula for maximum inclined force of a torpedo anchor By substituting Eq. (1)-Eq. (6) into Eq. (7), a general form of the maximum vertical force can be presented as: The normalized maximum vertical force F N, V , i.e., F PU, V /(A F S u, ave ), is presented as: Based on the results of the present experiment and the data of Medeiros (2002), O'Beirne et al. (2015), and Fu et al. (2017), the normalized maximum vertical force of the torpedo anchor can be expressed by the following relationship using multiple linear regression analysis: Thus, the maximum vertical force of the torpedo anchor can be described as: Fig. 7 presents the comparison between the measured and calculated F N, V . Of the 112 test data points, 98.2% fell within the band with a relative error of ±20%. The value of R 2 =0.91 further indicated that Eq. (11) has acceptable accuracy for calculating the maximum vertical force. θ 0 For the case of an inclined pullout, because F PU /F PU, V corresponded to a power functional function of /90°, as described in Section 3.2, F PU /F PU, V can be presented as  C0 . As stated above, as a shearing interface with dynamic position and variable shape exists between the adhered and the stagnant soil, the adhesion soil induces an additional shear force on the anchor. The additional shear force is a function of the integration of the adhesion force on this interface, which depends not only on the mean undrained shear strength, but also on the value of . Again, the additional shear is a component of the maximum inclined force. Hence, the maximum inclined force depends on the value of , which is consistent with the results in Section 3.3. Furthermore, referring to the research on the plate anchor carried out by Merifield et al. (2005) and Singh et al. (2017) as well as the dynamically embedded plate anchor conducted by Wang and O'Loughlin (2014), the maximum inclined force depends on the value of D/L. Hence, the normalized maximum inclined force (F PU /F PU, V ) can be assumed as where a 1 , b, and c are coefficients. Based on the present experimental data and the data of de Sousa et al. (2010), O'Beirne et al. (2015, Hossain (2016), andFu et al. (2017), the normalized maximum inclined force of the torpedo anchor can be expressed by the following relationship using multiple nonlinear regression analysis: Thus, the maximum inclined force of the torpedo anchor can be described as: Fig. 8 presents the comparison between the experimental and calculated values of the normalized maximum inclined force F N , i.e., F PU /(A F S u, ave ). Of the 441 test data points, 94.8% fell within the band with a relative error of ±20%. The value of R 2 = 0.95 further indicated that Eq. (14) has satisfactory accuracy for calculating the maximum inclined force. It should be noted that Eq. (14) is probably not suitable for those cases in which the embedment depth is smaller than one anchor length and load angle is smaller than 20°.
The influence of the major parameters (undrained shear strength, undrained shear strength gradient, and load angle) on the maximum inclined force calculated by Eq. (14) was assessed by sensitivity analysis. In general, it is impossible to avoid the test error of these parameters. Supposing that the tests errors are ±10% for the undrained shear strength, undrained shear strength gradient, and load angle, all calculated errors of the maximum inclined force are smaller than ±10%. These major parameters that affect the calculation accuracy of the maximum inclined force in Eq. (14) are ranked in terms of the sensitivity as follows: undrained shear strength, undrained shear strength gradient, and load angle. In practice, these major parameters should be determined as accurately as possible before using Eq. (14).
In the present test, it was found that the shaft resistance and bearing resistance decreased with the increase of load angle. For instance, as shown in Fig. 9, for the cohesive soil Bed A (S u, ave =3.297 kPa) and the anchor tip embedment ratio (D/L=2.44), when the load angle increased from 25° to 90° for anchor A32, the values of shaft resistance and bearing resistance decreased from 72.3 N and 20.1 N to 54.6 N and 12.4 N, respectively.

API method
The maximum vertical force of the torpedo anchors can be calculated by the API (2014) method. In the method, the bearing capacity factor N c may have different values for different parts of the anchor O'Beirne et al., 2015;Fu et al., 2017). For anchor fins, N c =7.5 (Skempton, 1951) while N c =12 for the tip , and N c =9 for the padeye (Skempton, 1951).  47 tests. 66% of the points calculated by the API (2014) method fell within the band with a relative error of ±20%, and 93.6% of the points calculated by the API (2014) method fell within the band with a relative error of ±30%. The value of R 2 is 0.81. Although its accuracy is lower than Eq. (14), it is acceptable for present data in terms of engineering practice. Since Fu et al. (2017) stressed that API method is restricted to vertical pullout and is not applicable for inclined pullout, whether the API method is applicable for the inclined one requires further validation. Fu et al. (2017) Fu et al. (2017 proposed a complicated method to calculate the maximum inclined force. Fig. 11 shows a comparison of the maximum inclined force calculated by Fu et al. (2017) method with those measured in the present tests. 63.2% of the points calculated by the Fu et al. (2017) method fell within ±20%, which was lower than the 94.3% of the present method. Thus, the reliability of the method devised by Fu et al. (2017) seems to require further validation.

Effect of installation
The tilt of the torpedo anchor in the bed would affect the maximum inclined force of a torpedo anchor. The maximum inclined force of a slant anchor would be higher than that of a vertical anchor. In present tests, the verticality of the torpedo anchor in the bed was guaranteed. The posture of the slender stainless steel rod was checked to be vertical using a level ruler after the anchor was inserted into the designated experimental depth. If the slender stainless steel rod was vertical, then the torpedo anchor would be vertical, as the slender stainless steel rod had the same direction as the attached anchor.
In engineering practice, the typical torpedo anchor is installed by free-fall under gravity. The velocity of the torpedo anchor changes with the depth in a parabolic nature, with initially reaching a large velocity, which then decreases to zero. The high penetration velocities would lead to a high strain rate in the vicinity of the torpedo anchor. The high strain rate leads to increase in undrained shear strength of the soil (DeJong et al., 2011;Kim and Hossain, 2016). Furthermore, the soil in the vicinity of the torpedo anchor would also undergo softening caused by the soil remoulding (Randolph, 2004;Lunne et al., 2011;Gaudin et al., 2014). Overall, it was found by Kim and Hossain (2016) that the soil strength increased sharply as the torpedo anchor tip approached the designated depth due to the effect of higher strain rate, and then dropped abruptly as the torpedo anchor tip passed through the depth, which was associated with the domination of remoulding or softening. Before the pullout test, the undrained shear strength would gradually increase due to the combined effects of thixotropy and consolidation (Richardson, 2008;Hossain et al., 2015).
In the present tests, the torpedo anchor was very slowly pushed into the designated depth, which was different from the installation process in engineering practice. The shear strength near the torpedo anchor installed at very slow velocity was higher than that by free-fall due to a lower soil disturbance adjacent to the slow installed torpedo anchor (Kim and Hossain, 2016). Then, the maximum inclined force of the torpedo anchor installed at very slow velocity was also higher than that of the torpedo anchor installed by free-fall. However, if the suitable waiting time from the end of the installation to the start of the pullout exists, the difference of the shear strength and maximum inclined force under these two different installation methods could be ignored Kim and Hossain, 2016). For the present tests, the installation effect was neglected. Furthermore, Eq. (14) for estimating the maximum inclined force of torpedo anchor is valid for the condition that the installation effect can be ignored.

Effect of mooring line
The curvature of the mooring line would influence the maximum inclined force. If the mooring line is coarse and heavy, and the bed is stiff, the mooring line may be curved during the pullout process. In such cases, the mooring line  angle at the padeye is higher than the mooring line angle at the mudline as stated by O'Beirne et al. (2015). In the present tests, according to the chain solution proposed by Neubecker and Randolph (1995), the difference between the mooring line angle at the padeye and the mooring line angle at the mudline was smaller than 3° and could be neglected (O'Beirne et al., 2015). Furthermore, the mooring line was assumed to be straight in the study because the mooring line used in the present tests was light and thin, and the bed was soft (Fu et al., 2017). Therefore, the influence of the mooring line curvature on the maximum inclined force was ignored in the present tests.

Effect of pullout speed
It is noteworthy to highlight that the pullout speed may affect the maximum inclined force of torpedo anchor. The maximum inclined force would be larger if the pullout speed increased significantly, as the pullout speed or the shear rate caused changes in the soil shear strength. The effect of shear rate on shear strength for Bed B is shown in Fig. 12. Clearly, the shear strength increases with the increasing shear rate, which is validated by Eq. (15) (Liu et al., 2016;Kim and Hossain, 2016;Kim et al., 2018). In general, the pullout speeds were sufficiently small in previous tests conducted by Hossain et al. (2015), O'Beirne et al. (2015) and Fu et al. (2017), so that the effect of pullout speed on maximum inclined force could be ignored. However, due to the importance effect of the pullout speed on the maximum inclined force of torpedo anchor, it is necessary to conduct further research.
where is the viscous property, is the soil shear strain rate, is the reference shear strain rate, is the shearthinning index, is the fully remoulded ratio and equal to 1/St, is the cumulative plastic shear strain, is the cumulative plastic shear strain required for 95% remoulding, and S u, ref is the undrained shear strength at the reference strain rate.

Effect of S u measurement
The undrained shear strength used in Eq. (14) can represent that in the vicinity of the torpedo anchor. However, it was impossible to measure the undrained shear strength in the vicinity of the torpedo anchor. Practically, there are two methods: (1) measure it before the anchor installation, then modify it with the friction ratio (Gilbert et al., 2008;Richardson et al., 2009;O'Beirne et al., 2015;Hossain et al., 2015;Fu et al., 2017); (2) wait for the effect of the installation on the pullout restored to acceptable level and then measure it away from the torpedo anchor. In present tests, the second method was used. For this method, suitable waiting time from the end of the installation to the start of the pullout was required to ensure the effect of installation acceptable. According to Kimura and Saitoh (1983) and Yang et al. (2014), the waiting time is likely dependent on the disturbance rate (or impact rate; V i /d) of the torpedo anchor and restoration speed of the bed sediments. In the present tests, the waiting time was determined by trial tests (Kimura and Saitoh, 1983).The results of the trial tests for Bed B are shown in Fig. 13. When the waiting time was longer than 10 min for the bed (sheared 1 min at the disturbance rate of 1 -s 1 ) of present tests, the difference of the shear stress between the sheared bed and the un-sheared bed was smaller than 10%. According to the previous sensitivity analysis, the corresponding difference of the maximum inclined force calculated using the undrained shear strength of the sheared bed (corresponding to the undrained shear strength in the vicinity of the torpedo anchor in the present tests) and the undrained shear strength of the un-sheared bed (corresponding to the undrained shear strength away from the torpedo anchor) would be smaller than 10%. Hence, the waiting time from the end of the installation to the start of the pullout should be at least 10 min in the present tests. Thereafter, the effect of the installation was assumed to be restored to an acceptable level, and the soil shear strength which is measured away from the torpedo anchor and subsequently used in Eq. (14) can represent that in the vicinity of the torpedo anchor.  WANG Cheng et al. China Ocean Eng., 2019, Vol. 33, No. 3, P. 333-343 341 The effects of the consolidation on S u during the pullout process were assumed to be negligible in the study. In general, the value of S u would increase with time as the bed consolidation (Kamei et al., 1987). In present tests, the time from the start of the pullout to the moment when the peak pullout force (i.e., the maximum inclined force) was reached is less than 5 min. To evaluate the changes in S u because of the bed consolidation during this period, the rheological curves were measured using a RheolabQC rheometer for the soil Bed B at different consolidation time, as shown in Fig. 14. The rheological curves corresponding to the consolidate time of 5, 10 and 20 min were quite close. Thus, the effects of the consolidation on S u during the pullout process were assumed to be negligible in the study.
This study only focuses on the maximum inclined force of torpedo anchors vertically embedded in un-stratified cohesive soils subject to a constant pullout speed. In practice, the bed may be composed of several layers with different kinds of sediments, such as clay, silt, and sand. Moreover, an anchor may be embedded in the soil bed in a tilted manner, rather than a vertical manner. Further investigation on the maximum inclined force of torpedo anchors embedded at an angle in stratified beds is necessary. Another issue to highlight is that the anchor pullout performance may vary when the anchor is subjected to a vibrational or an intermittent pullout rather than a constant pullout (Richardson, 2008). The soils around the anchor may be liquefied under a vibrational or an intermittent pullout force (Yang et al., 2014). Once the soil around the anchor is liquefied, its undrained shear strength would be significantly reduced, and the anchor would provide a lower maximum inclined force (Richardson, 2008). Hence, further investigation on the maximum inclined force of torpedo anchors subjected to the vibrational or an intermittent pullout force is necessary.

Conclusions
The maximum inclined force of a torpedo anchor vertically embedded in cohesive beds was extensively investigated in this study. The effects of the soil, anchor shape, embedment ratio, and load angle on the maximum inclined force were investigated by using nine different types of torpedo anchors and two types of silts. The cohesive bed properties were measured by using precise instruments, and the maximum inclined force were recorded by a high-speed data acquisition system. The load angle varied from 20° to 90°. A total of 316 sets of experimental data were collected in the laboratory. Effect aspects on the tests are discussed and further studies are highlighted. The main conclusions of this study can be summarized as follows.
Eq. (14) was proposed for fast calculating the maximum force of inclined pullout of a torpedo anchor in cohesive beds. This formula was validated by comparing the obtained results with the numerical simulated, laboratory and field data by Medeiros (2002), de Sousa et al. (2010), O'Beirne et al. (2015, Kim and Hossain (2016), and Fu et al. (2017).
The sensitivity of major parameters in Eq. (14) reduces as follows: the undrained shear strength, undrained shear strength gradient, and load angle. The calculated errors of the maximum inclined force of the torpedo anchor are smaller than ±10% when the tests errors in the undrained shear strength, undrained shear strength gradient, and load angle were ±10%.
The proposed formula is suitable for the torpedo anchor without a fin or with four fins, but has not been verified by other anchors such as plate anchor, OMNI-Max and dynamically embedded plate anchor (DEPLA). Whether the formula is suitable for other types of anchors needs further investigation. Owing to the critical importance of large-scale mooring system, the difficulty of seabed detection and the limitation of formulae, a trial test may be necessary in future engineering practice.
In addition to the influence of the soil parameters, anchor parameters and load angle on the inclined pullout force of torpedo anchor, many other factors such as pullout speed, mooring line, vibrational or an intermittent pullout force and layered soil may also affect the inclined pullout force of torpedo anchor in cohesive soil bed. These factors may be paid attention to in future study.