Penetration Depth of Torpedo Anchor in Two-Layered Cohesive Soil Bed by Free Fall

The penetration depth of torpedo anchor in two-layered soil bed was experimentally investigated. A total of 177 experimental data were obtained in laboratory by varying the undrained shear strength of the two-layered soil and the thickness of the top soil layer. The geometric parameters of the anchor and the soil properties (the liquid limit, plastic limit, specific gravity, undrained shear strength, density, and water content) were measured. Based on the energy analysis and present test data, an empirical formula to predict the penetration depth of torpedo anchor in two-layered soil bed was proposed. The proposed formula was extensively validated by laboratory and field data of previous researchers. The results were in good agreement with those obtained for two-layered and single-layered soil bed. Finally, a sensitivity analysis on the parameters in the formula was performed.


Introduction
Torpedo anchor is a kind of anchorage equipment with fast and convenient installation (O'Loughlin et al., 2004;Wang et al., 2016). It has been used for deep-water mooring facilities since 2002. The holding capacity, which is dependent on the anchor penetration depth, is an important parameter to evaluate the anchorage performance and safety of a torpedo anchor . Thus, it is of practical significance to study the penetration depth of torpedo anchor into the actual seabed.
Two-layered soil bed is common in both shallow and deep ocean floors, as it is found in the West coast of Africa, the Sunda Shelf, offshore Malaysia, Australia's Bass Strait and North-West Shelf (Colliat et al., 2011;Sturm et al., 2011;Kuo and Bolton, 2013;Hossain et al., 2014b). The seabed sediments and natural soil comprise of discrete layers of different thicknesses and properties (Lee et al., 2016). The two-layered soil bed can be classified based on the undrained shear strength of the top and bottom layers (Lee et al., 2016;Kim et al., 2018).
Thus far, many experimental studies have been conducted on the penetration depth of torpedo anchor into singlelayer soil bed using conventional laboratory 1-g tests (Medeiros, 2002;O'Loughlin et al., 2004O'Loughlin et al., , 2009O'Loughlin et al., , 2013Audibert et al., 2006;Richardson, 2008;Lieng et al., 2010;Hossain et al., 2014aHossain et al., , 2015Wang et al., 2016), centrifuge lab tests (O'Loughlin et al., 2004(O'Loughlin et al., , 2013Audibert et al., 2006;Richardson, 2008) and field tests (Medeiros, 2002;Lieng et al., 2010;O'Loughlin et al., 2013). Analytically, a prediction model based on Newton's second law was used by O'Loughlin et al. (2004O'Loughlin et al. ( , 2013, Richardson (2008), Hossain et al. (2015) and Kim et al. (2015aKim et al. ( , 2015b. While, a simple method based on the total energy was proposed by O' Loughlin et al. (2013). However, the focus of these studies was on the anchor penetration depth into single-layered soil bed. The prediction methods mentioned above cannot be directly applied to torpedo anchor penetrating into twolayered soil bed, as Kim et al. (2018) found that the penetration process of torpedo anchor into two-layered soil bed is different from that in the case of single-layered soil bed.
However, limited efforts were contributed to explore the penetration depth of torpedo anchor in two-layered soil bed. Few field experiments were carried out by Sturm et al. (2011) at the Troll Field in the North Sea, offshore Norway. Similar field tests were carried out by  in Lower Lough Erne. Kim et al. (2018) proposed a modified empirical total energy method to predict the penetration depth of torpedo anchors in two-layered soil bed based on the total energy method used for single-layer soil beds. The proposed method was mainly verified using numerical China Ocean Eng., 2018, Vol. 32, No. 6, P. 706-717 DOI: https://doi.org/10.1007/s13344-018-0072-3, ISSN 0890-5487 http://www.chinaoceanengin.cn/ E-mail: coe@nhri.cn data. However, the available test data are not sufficient to validate the numerical simulations and develop the formulae to calculate the penetration depth. Therefore, it is necessary to conduct further experimental tests on the penetration depth of torpedo anchor in two-layered soil bed, in order to propose a reliable prediction formula to calculate the penetration depth of torpedo anchor in natural soil seabed.
This study is to investigate the effect of the two-layered soil bed parameters on the penetration depth of torpedo anchor, and propose a reliable formula for predicting the penetration depth of torpedo anchor in two-layered soil bed. Thus, extensive lab tests on the penetration depth of torpedo anchor were carefully conducted. Energy analysis was performed for the anchor during the penetration process at different soil properties. A formula to calculate the penetration depth of torpedo anchor in two-layered soil bed was proposed based on the experimental data and theoretical analysis. The developed model was validated by the collected data of previous researches, including the data from Sturm et al. (2011 and Kim et al. (2018) as well as the data for single-layered soil bed (Medeiros, 2002;Richardson, 2008;Richardson et al., 2009;O'Loughlin et al., 2009;Hossain et al., 2014a;Kim et al., 2015b). A sensitivity analysis was performed on the parameters used in the formula. The advantages of the developed formula were compared with those of the modified energy method proposed by Kim et al. (2018) and bearing resistance model.

Experimental setup
The experimental setup is schematically illustrated in Fig. 1. It was comprised of two fixed pulleys, a nylon string of 2 mm in diameter, a 5 m scaffold, steel tape, stainless steel cylindrical bucket with well-stirred saturated soft soil, a torpedo anchor and a cantilever bar. A bucket with the diameter of 35 cm and depth of 60 cm was used. One end of the cantilever bar was mounted on the scaffold. The right pulley was fixed on the cantilever bar to ensure that the pulley was positioned directly above the center of the bucket.

Torpedo anchor
Seven different stainless steel torpedo anchors with solid conical tip were manufactured for the tests as shown in Fig. 2. The cavity of the tested anchor shaft was firmly filled with lead not only ensure the falling direction stability of torpedo anchor, but also increase its penetration depth. The torpedo anchor had a taper angle of 30°. The length of the anchor tip was equal to 1.87 times the diameter (d) of the anchor. The width (W f ) of the fin was equal to the diameter of the anchor. The thickness (t f ) of the fin was 1.2 mm. Table 1 lists the geometric dimensions of the anchors. For Anchor A30-C2s listed in Table 1, the first characters A, B, and C denote three anchor diameters (d = 1.9 cm, 2.5 cm, and 3.3 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, 2 and s indicate three ratios of the fin length (L f ) to the anchor length (L) (i.e., 0, 1/3, 2/3, and 4/9), respectively.

Sedimentary bed preparation and soil characteristics
Two different types of cohesive soil collected from Shanghai Jinshan Port and Shanghai Huangpu River in China were used to constitute saturated cohesive soil for the experiment. The median particle diameters (d 50 ) of the two types of soil were 19.49 μm and 29.31 μm, respectively.
The sedimentary bed in the test bucket was elaborately molded in the experimental bucket. The soil bed consisted of two layers with different thicknesses and undrained shear strengths. First, to obtain the designed thickness of the top soil layer, certain ratios of soil and water were added to a separate bucket with a thin sheet of geo-fabric mats laid at the bottom. Then, the mixture was fully stirred by a cement vibrator to prepare the top layer of the soil bed. On the other hand, the thickness of the bottom soil layer could be as thick as possible to avoid the anchor touching the bottom of the bucket. Hence, certain ratios of soil and water were added to the experimental bucket, and then fully stirred by the cement vibrator to prepare the bottom layer of the soil bed. Afterwards, the top layer was carefully and gently added on the top of the soil surface previously prepared in the experi-  WANG Cheng et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 706-717 707 mental bucket. Then, the sheet of geo-fabric mats was gently peeled off to obtain two soil layers (consolidation at 1-g) molded in the experimental bucket. The geological properties of the soil bed in the test bucket were carefully measured. The liquid limits of the soil collected from Shanghai Jinshan Port and Shanghai Huangpu River in China were 57% and 52%, and their plastic limits were 28% and 24%, respectively. The specific gravity (G s ) of the two types of soil was 2.65 and 2.62, respectively. The undrained shear strength (S u ) of the soil was measured by a portable vane shear apparatus, i.e., SZB-1.0. The shear ρ ρ δ rate of SZB-1.0 was 0.1 s -1 . Table 2 lists the geological characteristics of the experimental cohesive soil, where S ut is the undrained shear strength at the mudline; k 1 is the undrained shear strength gradient for the top soil layer; S ub is the undrained shear strength at the top-bottom soil layer interface; k 2 is the undrained shear strength gradient for the bottom soil layer; s1 is the average density of the top soil layer; s2 is the average density of the bottom soil layer; S t1 is the average soil sensitivity of the top soil layer; S t2 is the average soil sensitivity of the bottom soil layer and is the side adhesion factor. 2.4 Test procedure In order to simplify the experimental conditions and change soil properties conveniently, all tests were conducted in air. The entire penetration process was as follows: First, some connections and adjustments were completed before the torpedo anchor was free-fall released. The cantilever bar was adjusted up and down to ensure a constant vertical height (D) from the torpedo anchor tip to the soil surface. Then, the cantilever bar was fixed on the scaffold. As shown in Fig. 1, one end of the nylon rope was connected to the torpedo anchor while the other end went through the pulley being held by hand. The initial orientation of the torpedo anchor was set downwards restrictively.
Next, the free fall drop of the torpedo anchor was carried out by releasing the hand-held rope, and the torpedo anchor freely fell vertically into the soil bed in the experimental bucket. The free-falling anchor test began 10 min after stirring.

√ 2gD
During the anchor falling, the friction between the nylon rope and pulley could be ignored as the nylon rope used in this experiment was thin and light. The friction between the anchor and air has also been ignored. Thus, the impact velocity v i could be calculated by . Finally, after the penetration stopped, the penetration depth of the anchor tip was measured by a steel tape. Afterwards, the undrained shear strength, soil sensitivity, density and water content of the soil were immediately measured. Each experimental test was repeated 3 times in order to eliminate the single measurement errors. The averaged penetration depth was regarded as the final penetration depth.

Experimental data
A total of 177 sets of laboratory tests were conducted. The thickness ratio (H/L) of the top soil layer varied from 0.2 to 3.2. Thirty-one different shear strength profiles of the soil bed were tested. The impact velocity of the torpedo anchor was 7.1 m/s, and the experimental penetration depth of the torpedo anchor was approximately (1.04-4)L. The experimental data are summarized in Table 3, where H is the thickness of the top soil layer and d e, t is the measured tip penetration depth.

Effect of top layer thickness ratio
As shown in Fig. 3, the top layer thickness ratio significantly affects the penetration depth of the torpedo anchor in the two-layered soil bed. When the shear strength of the top layer is smaller than that of the bottom layer, the penetration depth ratio increases with the increase of the top layer thickness ratio. Besides, when the shear strength of the top layer is larger than that of the bottom layer, the penetration depth ratio increases as the top layer thickness ratio decreases. For instance, for the soft-over-stiff soil bed (S ut =2.096 kPa, S ub =4.139 kPa), d e, t /L of Anchor B12 increases by 43.1% as H/L increases from 0.2 to 1.8. However, d e, t /L decreases by 26.8% as H/L increases from 0.2 to 1.5 for the stiff-over-soft soil bed (S ut =4.177 kPa, S ub =1.412 kPa).

Discussion
4.1 Empirical formula to predict the anchor penetration depth Many forces act on the torpedo anchor during the dynamic penetration of the anchor in two-layered soil bed, which inevitably include the anchor weight, side-adhesion resistance between the anchor surface and soil, end-bearing resistance on the anchor tip, buoyancy force, drag resistance, and the reverse end-bearing resistance at the upper end of the anchor fins.
The side-adhesion and end-bearing resistances can be expressed as follows (Terzaghi and Peck, 1967;True, 1976): where F AD is the side-adhesion resistance; S t is the average soil sensitivity; A S is the penetrator side area; F BE is the endbearing resistance; N c is the bearing capacity factor and A F is the penetrator frontal area. The real side-adhesion and end-bearing resistances could be affected by the high shear strain rate. It is noted that the high shear strain rate would cause the real shear strength higher than that at the reference strain rate (Kim et al., 2015a(Kim et al., , 2015b. Thus, both Eq. (1) and Eq. (2) must be modified by two strain rate functions (R f1 , R f2 ) (O'Loughlin et al., 2013;. These two strain rate functions were presented in either a logarithmic, inverse sin-hyperbolic or power-law equation (Steiner et al., 2014). In this study, the modified forms are assumed based on the power-law equation as follows: where v is the penetrating velocity; is the strain rate parameter and is the reference shear strain rate. In general, a 1 varies from 0 to 1, and a 2 is larger than 1.
Therefore, the modified side-adhesion and end-bearing resistances can be respectively expressed as: If a small penetration displacement of the torpedo anchor ΔZ is assumed, the energy losses by the side-adhesion and end-bearing resistances may be expressed as: where W AD is the energy consumptions of the side-adhesion and W BE is the energy consumption of the end-bearing resistances.
The buoyancy force and drag resistance can be expressed as follows: ρ where s is the density of soil; V is the volume of the soil displaced by the anchor; g is the gravitational acceleration; and C d is the drag coefficient.
A schematic diagram of the penetration process of the torpedo anchor in two-layered soil bed is shown in Fig. 4. When the torpedo anchor moves a minute displacement (ΔZ) from position j to j+1, the energy losses caused by the buoyancy force and end drag resistance may be described as: where W b is the energy loss by the buoyancy force and W d is the energy loss by the end drag resistance. Correspondingly, the initial mechanical energy E j (E j = mv j 2 /2+W g ) of the anchor is converted into the energy losses caused by the resistance and residual kinetic energy EK j+1 (EK j+1 =mv j+1 2 /2). Hence, according to the energy conservation law, it is obtained that Then, when the anchor tip arrives at Position j+1, the residual kinetic energy can be written as: The residual kinetic energy can be rewritten as: By considering the dynamic penetration velocity in Eq. (15) as v j , the residual kinetic energy can be estimated as follows: The penetration depth can be obtained by solving Eq. (16) using iterative procedure. The penetration depth of the anchor in Positions j and j+1 are Z j and Z j+1 (Z j+1 =Z j +ΔZ), respectively. The assigned value of ΔZ would affect the prediction accuracy. Practically, the value of ΔZ can be assigned to be H/1000 according to present numerical tests. When the anchor penetrates from Position j to j+1, it is necessary to determine whether the residual kinetic energy Ek j+1 is 0. If Ek j+1 is larger than 0, the anchor would continue to penetrate. After the penetration process ends, the anchor position should be determined; whether it is located in the top layer (Z j+1 <H), partially in the bottom layer (H<Z j+1 <H+L), or completely in the bottom layer (Z j+1 >H+L). At each step of iteration, the energy losses caused by the side-adhesion resistance, end-bearing resistance and etc., largely depend on the position of the anchor. For instance, in the anchor position shown in Fig. 4d, the energy loss caused by the reverse end-bearing resistance does not exist; while the anchor would be subjected to two different side-adhesion resistances when it is partially in the top and bottom soil layers owing to the different soil shear strengths and anchor-soil contact areas. Otherwise, the anchor would stop penetrating when it reaches its ultimate penetration depth d t (d t =Z j /2+Z j+1 /2). β Eq. (16) was used to calculate the penetration depths of the torpedo anchor in the present study based on the following parameters: N c was considered being 12 . The reverse end-bearing resistance at the upper end of the anchor fins was calculated using N cf =7.5 (Skempton, 1951). C d =0.63 was estimated for the ellipsoidal-tipped, four-fin anchor, while C d =0.24 was estimated for the ellipsoidal-tipped, finless anchor (Richardson, 2008;. =0.07 was adopted (O'Loughlin et al., 2013;. The values of a 1 and a 2 determined by the back-calculations are 0.8 and 1, respectively. Fig. 5 presents the comparison between the experimental and calculated penetration depth ratios (d t /L). It is shown that the present method can provide reasonable predictions as 94.4% of the 177 data points calculated by the proposed method falls within the relative error range of ±20%.

Data obtained for single-layered soil bed
To verify the reliability of Eq. (16) for a single-layered soil bed, 218 sets of data of other researchers are used. As shown in Fig. 6, the penetration depths (d tc ) calculated by Eq. (16) are consistent with the experimental data. All data fall within a relative error range of smaller than 20%. The values of N c and N cf used in Eq. (16) are 12 and 7.5, respectively. The other input parameters are listed in Table 4. Hence, Eq. (16) can be used to predict the penetration depth of the torpedo anchor in single-layered soil bed. Sturm et al. (2011) and To verify the reliability of Eq. (16) for two-layered soil   β bed, the field data of Sturm et al. (2011) and are used. Sturm et al. (2011) conducted the field tests at Troll Field in the North Sea, offshore Norway using a 1:3 reduced scale anchor. The density of the soil was 1.5 g/cm 3 . The seabed undrained shear strength measured at a reference shear strain rate of 0.56 s -1 could be simplified as S u =5 kPa for Z=0-2.5 m, and S u =5+2.69Z kPa for Z=2.5-15 m. The sensitivity of the soil bed was S t =5.5 (Lunne et al., 2006). Recently, O'Beirne et al. (2017) have conducted field tests using a 1:20 scale model anchor. The tests were performed in Lower Lough Erne, a glacial lake located in Northern Ireland. The density of the soil was 1.1 g/cm 3 . The undrained shear strength measured at a reference shear strain rate of 0.18 s -1 could be conveniently idealized by S u =1.5Z for Z=0-1.5 m, and S u =2.25+0.8Z for Z=1.5-3 m. The input parameters used in Eq. (16) are as follows: N c =12, N cf =7.5, C d =0.7 (for Sturm cases) or 0.67 (for O'Beirne cases), and =0.07. The values of a 1 and a 2 from the back-calculations, respectively, were 1 and 2.6 for Sturm et al. (2011) data, while they were 1 and 1.8 for O'Beirne (2017) data. Fig. 7 shows that all data fall within a relative error range of smaller than 10%. It can be seen that the penetration depths (d tc ) calculated by Eq. (16) are consistent with the values measured by Sturm et al. (2011) and O'Beirne (2017). Kim et al. (2018) Eq. (16) developed to predict the penetration depth was further validated by the numerical data obtained by Kim et al. (2018). In their study, the thickness ratio of the top soil layer was smaller than 1. The soil sensitivity was 3. The values of a 1 and a 2 from back-calculations were 0.78 and 1, respectively. Fig. 8 illustrates that the penetration depths calculated by Eq. (16) are consistent with the values simulated by Kim et al. (2018) as 87.5% of all data falls within a relative error range of smaller than 20%. Although 6 groups specially marked groups of data points show larger deviations, it may be acceptable as the soil sensitivity varies when the soil has significantly different shear strength (Wang et al., 2016;. According to Wang et al. (2016), the penetration depth prediction is sensitive to the soil sensitivity.

Numerical data obtained by
Typical sensitivity values for marine soil bed might range from 2 to 5 (Kim et al., 2015a). If a soil sensitivity of 5 was adopted for the crust soil layer, 95% of all data would fall within a relative error range of smaller than 20%.

Sensitivity analysis β
Sensitivity analysis was performed to evaluate the effects of the tip bearing capacity factor, drag coefficient, shear-thinning index, shear strength, shear strength gradient and soil sensitivity on the calculated penetration depth. Herein, it was assumed that N c =12, C d =0.63 (four-fin anchor) or 0.24 (finless anchor), =0.07, and the values obtained by the measured experiment were used for other parameters (S ut , S ub , k 1 , k 2 , S t1 , S t2 ). The values of N c were assumed to range from 8 to 20 (Lunne et al., 1997). Fig. 9 indicates that the differences of ±50% were yielded in the calculated results. Thus, the penetration depth is significantly sensitive to N c as the relative change in the prediction results was equivalent to that of N c . The drag coefficient C d may vary from 0 to 0.7 (True, 1976). Similarly, Fig. 10 reveals that the differences of at most 20% in the calculated results were yielded when C d =0, and only 8% differences were achieved when C d =0.7. It can be seen that the penetration depth is not significantly sensitive to C d as the relative change in the prediction results was approximately half of     Fig. 11 indicates that the penetration depth is significantly sensitive to as the relative change in the prediction results was equivalent to that of . For the values of other parameters (S ut , S ub , k 1 , k 2 , S t1 , and S t2 ), it was assumed that the tests errors were ±10%. The relative errors of the calculated penetration depth using Eq. (16) are shown in Figs. 12-17, respectively. Therefore, the formula for calculating the penetration depth of the torpedo anchor is significantly sensitive to the tip bearing capacity factor, shearthinning index, undrained shear strength of two soil layers and soil sensitivity; however, it is less sensitive to the drag coefficient and shear strength gradient. The sensitivity degrees of the parameters may influence their contribution to the energy losses. The variations in these parameters directly change the energy losses caused by the side-adhesion resistance, end-bearing resistance, inertia resistance coefficient and buoyancy force. The contribution of various forces to the amount of the energy losses is shown in Fig. 18. Relatively large proportions of the energy losses are caused by the side-adhesion and end-bearing res-      istances. Hence, the proposed formula to calculate the penetration depth of the torpedo anchors is significantly sensitive to the bearing capacity factor, shear-thinning index, undrained shear strength, and soil sensitivity; however, it is insensitive to the inertia resistance coefficient. This can be mainly attributed to the fact that the energy losses caused by the inertia resistance comprise a small proportion of the total energy consumption.

Comparison with the total energy method
Based on the total energy method, an empirical model to predict the penetration depth of the torpedo anchor in single-layered soil bed was developed by O'Loughlin et al.
(2013), Kim et al. (2015aKim et al. ( , 2015b and Hossain et al. (2015). Lately, the model was further advanced by Kim et al. (2018) to predict the penetration depth of the torpedo anchor in two-layered soil bed and the following formula was presented: where p and q are the exponent and coefficient of the energy models, respectively; E top is the mechanical energy of the torpedo anchor in the top soil layer; E bot is the mechanical energy of the torpedo anchor in the bottom soil layer; A tot is the total surface area of the anchor; k eff1 (k eff1 =S ut /H+k 1 ) is the effective shear strength gradient for the top layer; k eff2 (k eff2 =S ub /(d t -H)+k 2 ) is the effective shear strength gradient for the bottom layer; is the anchor effective mass for the top soil layer and is the anchor effective mass for the bottom soil layer.
A nonlinear curve fitting method is used to determine the values of p and q, which are 4 and 0.28, respectively. Fig. 19 shows the comparison between the results calculated by Eq. (17) and the measured values. 72.9% of the 177 data points calculated by the total energy method fall within the band with a relative error of ±20%, and 89.8% of the points calculated by this method fall within the band with a relative error of ±30%. Comparatively, the accuracy of Eq. (17) is smaller than that of Eq. (16) with 94.4% of the 177 data points falling within a relative error range of ±20%. Furthermore, the penetration depth calculated iteratively by Eq. (17) is sensitive to the initial value d tc0 , which is investigated by a set of simulated data provided by Kim et al. (2018) as listed in Table 5. The relative error of the calculated d t is 7.5% when the initial value d tc0 is 1, while the relative error of the calculated d t is as high as 172.5% when the initial value d tc0 is 5. In addition, O'Beirne et al. (2017) noted that Eq. (17) did not account for variations in the interface friction along the anchor-soil interface.

Comparison with the bearing resistance model
The penetration depth was also calculated using a bearing resistance model (True, 1976;O'Loughlin et al., 2013;Hossain et al., 2015;Kim et al., 2015aKim et al., , 2015b Fig. 20 shows the comparison between the results calculated by Eq. (18) and the measured values. 82% of the 177 data points calculated by the bearing resistance model fall within the band with a relative error of ±20%, and 99.4% of the points calculated by this method fall within the band with a relative error of ±30%. Comparatively, the accuracy of the bearing resistance model is smaller than that of Eq. (16) with 94.4% of the 177 data points falling within a relative error range of ±20%. Besides, compared with Eq. (16), most of the data points (i.e. 87% of points) calculated by Eq. (18) are underestimated. The difference of the point distribution calculated by Eq. (16) and Eq. (18) may be due to the different distribution of the side-adhesion and end-bearing resistances in these two methods, which can be verified by the curves of the anchor velocity vs. tip penetration depth, as shown in Fig. 21. The velocity calculated by Eq. (18) is smaller than that calculated by Eq. (16), which leads to a smaller penetration depth.

Conclusions
In this study, the penetration depth of the torpedo anchor that penetrated into two-layered soil beds was investig-ated. A total of 177 experimental data were obtained in laboratory by varying the undrained shear strength of the two-layered soil bed and the thickness of the top soil layer. The conclusions drawn from this study are summarized as follows.
A formula to predict the penetration depth of the torpedo anchor in two-layered soil beds was proposed based on the analysis of the energy loss during the anchor penetration and the obtained experimental data. This formula was validated by comparing the obtained results with the experimental data by Sturm et al. (2011, and Kim et al. (2018) as well as the data obtained for singlelayered soil beds reported by Medeiros (2002), Richardson (2008), Richardson et al. (2009), Hossain et al., (2014a, and Kim et al. (2015b).
The sensitivity analysis on the parameters used in the proposed formula showed that the tip bearing capacity factor, shear-thinning index, undrained shear strength of two soil layers and soil sensitivity are the key parameters in Eq. (16). Variations in these parameters would change the energy losses caused by the side-adhesion and end-bearing resistances, which represent a relatively large proportion of the total energy losses.
In addition to the influence of the soil parameters on the anchor penetration, many other factors such as the tilting angle, impact velocity and experimental environment (in water, not in air) may also affect the penetration depth of the torpedo anchor into two-layered soil. These factors on the anchor penetration in two-layered soil bed may be paid attention to in the future study.