Reliability-based evaluation of ultimate embedment depths of torpedo anchors in spatially variable clay

In this study, a large deformation random finite element (LDRFE) method was used to investigate the dynamic penetration of anchor piles in spatially variable clay. Validation of the rationality of the proposed methodology using data from published studies. The randomness of the undrained shear strength of the soil in the system is considered by three-dimensional random fields to investigate the anchor penetration mechanism and quantitatively assess the ultimate embedment depths (Hran ) of the anchor in spatially variable clay. The strain softening and strain rate effects of the soft clayey soil were considered. The findings indicate that the variability in soil strength significantly affects the flow pattern and failure mechanism of the soil, consequently influencing the Hran of the anchor. Furthermore, according to the computed Hran results, a coefficient of variation of soil shear strength φ can be used to connect the deterministic (Hdet ) and random analysis results. This correlation may facilitate further development of reliability-based designs.


Introduction
The self-penetrating torpedo anchor [1] is widely regarded as a highly promising anchor type owing to its cost-effectiveness, ease of installation, and adaptability to varying water depths (figure 1).However, predicting its embedment depth and subsequent capacity poses challenges [2].Relatively few studies have been conducted to evaluate the depth of torpedo anchors.Current practice relies on the limit equilibrium method, Newton's second law of motion, and considers uniform clay, as in True's method [3].Several studies [1,4] have modified this theory by varying the forces acting on the torpedo anchor.To obtain an accurate penetration mechanism of anchors, several studies [5,6] have simulated the penetration process using various numerical methods (FE, PFC, ALE, and CEL) for torpedo anchors.Among them, the dynamic installation of a torpedo anchor in a homogeneous clayey seabed was modeled by Kim et al. [5] through CEL finite element calculation with simultaneous consideration of the strain rate effect and strain softening effect.A parametric study on the influence of various factors on the embedment depth of the anchor was conducted simultaneously.
However, almost all these deterministic analyses ignore the effects of the inherent spatial variability of marine soils.Similar to terrestrial soils, seabed soils have been attested to exhibit strong spatial variability [7,8].In traditional deterministic geotechnical engineering analyses, the uncertainty of soil properties is often ignored, which may lead to non-conservative solutions and thus endanger structures [6,9].Unfortunately, because of the substantial variety of ground features, it is expensive and difficult to precisely investigate the in-situ ground soil.Therefore, when addressing parameters related to soil mechanical characteristics, a random technique must be used.Some geotechnical design rules have recently shifted toward reliability-based design to consider the variety in soil parameters [10,11].Indeed, in typical clayey seabed soils, where the shear strength increases with depth, increasing the embedment depth results in the anchor being surrounded by stronger soil, thereby enhancing the holding capacity.Therefore, the spatial variability of the seabed soil properties could be shown through the spatial variation of the soil undrained shear strength, which was considered by a normally or log-normally distributed random field characterized by a standard deviation and mean value [12,13].In this study, a three-dimensional large-deformation random finite element analysis was conducted to explore the influence of the spatial variability of soil strength on the penetration mechanism and embedment depth of the torpedo anchor.The dynamic installation of a torpedo anchor in a clayey seabed was modeled using the CEL finite element method with simultaneous consideration of the strain-softening effect, strain-rate effect, and linearly increasing strength with depth.Subsequently, a parametric study on the influences of and on the embedment depth of the anchor was conducted.The results provide a calculation method for the embedment depth of the torpedo anchor under different guarantee rates, which may be helpful for further development of the reliability-based design of torpedo anchors.

Methodological aspects
Large-deformation finite element calculations (using the commercial software Abaqus 6.14) and the random field theory were successfully combined using a Monte-Carlo simulation framework.To create a three-dimensional log-normal random field, an exponential transformation was applied to the three-dimensional Gaussian random field created using Liu's modified linear estimating approach [12].

Random field
Deterministic trends (i.e., mean values) and random residuals (i.e., stochastic fluctuations) are frequently used to describe the randomness of s u in probabilistic geotechnical analysis.The coefficient of variation (COV)su and the autocorrelation length (θ) are two statistical metrics that quantify the random heterogeneity that is reflected in the random residual.According to research [7,14], the COVsu of clayey soil typically ranges from 0.02 to 0.7.It has been reported that the horizontal autocorrelation length of marine soil is two orders of magnitude larger than its vertical corresponding length; among which and are between 0.05~14m and 7~9000m [14], respectively, which may be caused by the geological deposition process.The horizontal and vertical correlation lengths for this paper are taken as 50.7m and 3.8m [6,13,14].However, the diameter of the torpedo anchor is very small relative to the and the range of the mobilized soil during the installation of the anchor is extremely limited in the horizontal direction.Seemingly, there is no need to consider the impact of the horizontal correlation length on the results in the following discussion.In addition, the average value and standard deviation of su are different at different depths [6], which requires the establishment of a nonstationary random field to capture the spatial variation in the intensity, as addressed by Yi [13].
Through the user subroutine VUSDFLD of the Coupled Eulerian-Lagrangian (CEL) finite element program ABAQUS/Explicit V6.14, the subsequent random field generation technique was implemented to enable the subsequent massive deformation finite element analysis.Owing to the significant computational burden of CEL analysis, this study conducted 400 realizations, striking a balance between computational time and solution accuracy.It has been proven that this number of simulations is sufficient to achieve convergence and a stable output.Five series of Monte-Carlo simulations (I-V), each containing 400 possible realizations of the CEL calculations, were performed (see figure 2), as shown in table 1. Figure 2 shows a color contour plot of an example of a nonsmooth stochastic intensity field.

CEL finite element modeling
The large-deformation finite element calculations in this study were performed using the CEL technique available in ABAQUS/Explicit V6.14.Deterministic finite element calculations of soil and anchor parameters were performed using field data reported by Medeiros [1].Dynamic penetration of torpedo anchors in clay involved dynamic large deformations influenced by soil softening and strain rate effects [ 5 ].Therefore, this study adopts an extended Tresca model (equation 1) to account for the effects of strain rate and softening.
where su,m is the mobilized shear strength considering the strain softening and strain rate effects.β and η reflect two strain rate parameters; ξ describes the cumulative plastic shear strain.The variable δrem is the reciprocal of soil's sensitivity, St.In addition, in this contribution, the buoyancy of soil γ'=600kN/m 3 .A uniform stiffness ratio E/su=500 and a constant Poisson's ratio v=0.499 were used to approximate the undrained conditions without causing numerical instability.The other parameters are detailed in figures 3 and 4. It is worth noting that "fully rough" was used to simulate the mechanical behaviors of the anchor-soil interface in this paper because of the uncertainty of the ultimate embedment depth of the torpedo anchor in each realization of the Monte-Carlo simulation.

Model verification against True's results and field test
With the finite element model established in the earlier section, CEL analysis was first undertaken to replicate the field test reported by Medeiros [1] to validate the capability of the CEL model in solving anchor penetration, as shown in figure 3. The results of the CEL simulation calculation showed that the penetration depth of the anchor developed rapidly within the initial 1s; the development gradually slows after 1s, and the penetration depth basically no longer developed and stabilizes at 29.67m after 1.5s.This calculated result is very close to the field test result (29.0m) reported in [1].Several researchers [2,4] improved the True [3] method to predict the penetration depth of torpedo anchors.The essence of this technique was presented to determine the penetration of the anchor into the seafloor by considering Newton's second law of motion and the forces acting on the projectile during penetration, as shown in equation 2.
where su,tip and su,ave are the undrained shear strength at the anchor tip level and the undrained shear strength averaged over the embedded shaft length, respectively, satisfying su,tip=k(L-Ltip)su,ave/2.K represents the strength gradient of the soil.Α is evaluated as the reciprocal of soil sensitivity.CD and v denote the drag coefficient and penetration velocity, respectively.As shown in figure 3, the embedment depth calculated by the improved True theoretical formula was 29.46m, which was also very close to the above-mentioned CEL calculated and field measured value.[1] and Ture [3].

Analyses results and discussion
From this section onwards, the subscripts "det" and "ran" will be added to the various variables to distinguish between the results of deterministic and random calculations.For greater computational performance, a totally rough soil-anchor interface (α=1) and vi =15 m/s were assumed given the substantial number of realizations involved in the random analyses.The values of the remaining parameters were presented in the previous section.
Under the condition of Series I, the embedment depth of the torpedo anchor with velocity in spatially varying soil was explored by Monte-Carlo simulation, as presented in figure 4. As shown in figure 4a, the penetration depth of the anchor increased rapidly and then decreased over time [5].During the acceleration stage, a comparison of the quantitative analysis results showed that the penetration depth of the torpedo anchor was not very sensitive to the spatial variability of the soil.Immediately after entering the second stage (i.e., the deceleration stage), the ultimate penetration depth exhibited obvious disparities.To further illustrate this, based on figure 5a, it can be observed that the size of the plastic flow zone in the soil (as indicated by three realizations) remains approximately the same and exhibits a symmetrical distribution in the shallow region (i.e., at H=1.82m) [5].This indicates that during the acceleration phase, the penetration depth of the torpedo anchor was minimally affected by the randomness in relation to the relatively weaker upper soil in terms of the undrained shear strength.However, during the deceleration phase of the torpedo anchor (H=12m), as shown in figure 5(b), the plastic flow zones in the homogeneous linear gradient soil (deterministic realization) were predominantly symmetrically distributed.While, in No.251 and No.321, cases considering the spatial variability, the soil no longer exhibited a symmetric distribution.In addition, the size of the disturbed zone differed at this stage.The disturbed zone in No.251 realization is smaller than the disturbed zone, whereas the disturbed zone in the 321st case is approximately 1.5 times larger than the deterministic realization.As shown in the figure, at H=12m, the strength of the surrounding soil in No.251 realization was weaker than that in the other two cases.Furthermore, in No.321 realization, the acceleration at t=0.84s in the largest plastic flow zone is approximately twice that of the No.251 realization.This resulted in the most severe attenuation of the penetration velocity of the torpedo anchor in the deceleration phase within the spatially variable soil, along with experiencing the highest resistance and the surrounding soil having the highest strength.Therefore, the size of the plastic zone and variation in the undrained shear strength indirectly reflect the magnitude of the penetration resistance experienced by the torpedo anchor, thereby visually reflecting the final penetration depth.Moreover, the asymmetry in the range of the disturbed zone during the penetration process of the torpedo anchor in spatially variable soil leads to differences in the soil failure mechanisms to a certain extent.Furthermore, the statistics of the ultimate embedment depths, including the mean and coefficient of variation, were calculated for all Monte-Carlo series, and are listed in table 2. The error is within 20% of the quantitative analysis results.The LDFEA results of the other series were similar and are not discussed here.In this study, the results were analyzed using the Kolmogorov-Smirnov test (K-S test), as presented in table 2. Although both the normal and log-normal distribution functions can reasonably fit these IOP Publishing doi:10.1088/1755-1315/1332/1/0120246 embedment depths, the chi-square goodness-of-fit test suggests that the log-normal distributions provide better fitting (see table 2).In addition, a log-normal fit always produces a positive ultimate depth.Therefore, the log-normal distribution function was used to approximate the probability density distribution of the ultimate depth.

Design method of torpedo anchor
According to previous studies, predicting the ultimate embedment depth of a torpedo anchor requires adequate consideration of the influence of a spatially varying clayey seabed.As present in equation 5, su,ave is obtained from the soil strength at half and tip depth of the torpedo anchor, respectively (see in figure 6, φ=0.5).Therefore, this paper proposes a simple and reliable design method for torpedo anchors that considers the selection of su reference point location in the True method.For this purpose, φ is defined as the coefficient of variation of soil shear strength corresponding to su,ave.Based on deterministic analysis, the final penetration of the torpedo anchor can be determined by considering the spatial variation (cov) of the soil using different values of φ given in table 3.For example, the engineer can select the corresponding φ value from table 3 and substitute it into True's method according to the different soil variability parameters at the site.Therefore, a torpedo anchor foundation design method based on Ture's method and considering the spatial variability of the soil was proposed.A flowchart of this method is shown in figure 7.

Conclusion
In this study, in a spatially varying clayey seabed, the penetration depth of a torpedo anchor foundation was primarily determined by the undrained shear strength of the soil during the entire penetration process.During the dynamic penetration process, the spatial variability of the soil in the acceleration phase had little effect, which may have been caused by the resistance provided by the shallow penetration depth of the soil, which was significantly less than the penetration force.However, in the deceleration phase, the spatial variability of the soil had an obvious influence on the embedment depth, and the ultimate embedment depth of the torpedo anchor foundation increased with the cov of soil, indicating that it seems to be somewhat conservative in the traditional design.

Figure 1 .
Figure 1.Schematics of torpedo anchors and typical installation procedure.

Figure 2 .
Figure 2. A three-dimensional random field realization for su.
m and Ws denote the quality and dry weight of the torpedo anchor, respectively.Rf represents the strain rate factor corresponding to the first bracket term of the expanded Tresca model (equation 1).The meanings of the remaining parameters have been mentioned by Kim et al.[ 5 ].Frictional resistance (Fs), bearing resistance (Fb), and inertial drag force (Fd) are expressed as follows:= , ,

Figure 4 .
Figure 4. Comparison between random results and deterministic results in Series Ⅰ in terms of velocity values.

Figure 6 .
Figure 6.Schematic of the selection of reference points for su in Ture's method [4].

Figure 7 .
Figure 7. Flow chart of the torpedo anchor design.

Table 1 .
Statistics of su random fields of Monte-Carlo series

Table 2 .
Ultimate embedment depth statistics of the Monte-Carlo series and the result of the K-S test.