Numerical Study on the Keying of Suction Embedded Plate Anchors with Chain Effects

Abstract

Suction embedded plate anchors are widely used in deepwater mooring systems, which can withstand significant vertical loading. During the installation, the mooring chain is tensioned and causes the anchor to rotate, which is known as keying. With a large deformation finite element approach of the coupled Eulerian–Lagrangian method, the chain effects are incorporated into the keying of suction embedded plate anchors. The effectiveness of the proposed method is verified by numerical results and centrifuge tests. The numerical study reveals that the installation angle of the chain has a significant effect on the loss of embedment, especially combined with the effects of load eccentricity and soil strength. The losses of embedment are 0.024~0.273 and 0.217~1.755 anchor width for the installation angles of 15° and 90°, respectively. The ultimate bearing capacity factor decreases with the increasing of load eccentricity and soil strength, because a cavity is formed at the anchor back. Empirical formulae are finally developed for engineers to rapidly estimate the embedment loss and ultimate pullout capacity of suction embedded plate anchors.

1. Introduction

The exploration of oil and gas resources in deep and ultra-deep waters poses significant challenges for floating platforms [1], where mooring systems play a critical role in ensuring structural positioning and stability. Suction embedded plate anchors (SEPLAs) are regarded as deep and ultra-deep water mooring solutions because of their precise positioning and high bearing capacity [2,3,4]. The SEPLA is embedded to the targeted depth assisted with the suction caisson [5], as shown in Figure 1. The SEPLA is vertically located in the seabed when the suction caisson is retrieved. The mooring chain is then tensioned, causing the SEPLA to rotate to an inclination that is approximately normal to the chain, which is known as the keying of SEPLA. The keying of SEPLA will induce a loss of embedment Δz, resulting in the decrease in anchor-bearing capacity.

The keying of SEPLAs involves the interactions among mooring chain, anchor and soil. Previous studies paid more attention on the effect of anchor–soil interaction on the loss of embedment, in which a vertical load or velocity was applied at the padeye until the SEPLA rotated horizontally without the chain effect. It was demonstrated that the loss of embedment is highly sensitive to anchor geometry (Figure 2), such as load eccentricity [7,8], padeye offset [9,10,11,12,13,14], flap [12,13,14,15,16] and shank [17]. Centrifuge tests by O’Loughlin [7] found that the loss of embedment increased from 0.1B to 1.5B with load eccentricity decreasing from 1.0 to 0.17, in which B is the anchor width. Large deformation finite element (LDFE) analysis by Yu et al. [8] found that the loss of embedment increased from 0.21B to 1.01B, when the load eccentricity decreased from 1.0 to 0.25 for square anchors. Tian et al. [9,10,11,12,13] paid attention to the effect of padeye offset on the “diving” performance of SEPLAs. Results indicated that the padeye offset enhanced the ability of the anchor to dive and hence the loss of the embedment was reduced. Centrifuge tests reported that the flap did not create any reduction in embedment loss, because the flap hardly rotates during keying [16]. However, new designs of the flap could reduce the loss of embedment, such as the inward flap by Tian et al. [13] and restrained flap by Liu et al. [15]. The embedment loss reduced with taking the shank into account [17]. For the pull of installation angle θ_e of 40°, the embedment loss was reduced from 0.318B to 0.164B (nearly twice) when the shank was considered. Furthermore, some papers [6,18,19,20,21,22] have investigated the influence of soil properties on the keying of SEPLAs, including the effects of soil remodeling [6,21].

During the keying of SEPLAs, the mooring chain forms a reverse catenary shape in the soil, interacting with the anchor at the padeye, as shown in Figure 1. It is reported by centrifuge tests that the mooring chain also has a significant effect on the loss of embedment. Gaudin et al. [23] found that the losses of embedment were 0.25B and 1.15B in the centrifuge test with the installation angle θ_e of 45°and 90°, respectively. Centrifuge tests conducted by Song et al. [18] reported that the losses of embedment were 0.33B and 0.65B with the installation angle θ_e of 60°and 90°, respectively. In the theoretical analysis, the effect of mooring chain was considered by a chain equation, such as the plasticity model [17,24,25] and the mechanical model [26]. The plasticity model described the keying of SEPLAs as a yield envelope that fitted by finite element (FE) analysis. Combined with the chain equation, the loss of embedment was obtained by incremental calculations. The retrospective simulation of anchor keying was performed by the plasticity model, where the simulation results agreed well with the centrifuge test data under the installation angle θ_e of 40° [17]. The mechanical model was developed on the limit equilibrium method, which proved that the loss of embedment ranged from 0.19B to 1.13B for the installation angle θ_e between 0°and 90° [26].

For FE analysis, there were two solutions to incorporate the chain effect into the keying of SEPLAs, i.e., combining with the chain equation and constructing the chain by FE method. Song et al. [27] incorporated the chain equation into the LDFE analysis with the remeshing and interpolation technique with small strain (RITSS). It was found by the RITSS analysis that the loss of embedment ranged from 0.24B to 0.51B for the installation angle θ_e of 30°~90°. Zhao et al. [28] adopted the coupled Eulerian–Lagrangian (CEL) method to simulate the keying of SEPLAs, where the mooring chain was constructed by connecting cylindrical elements with LINK connectors. The effectiveness of the CEL analysis was validated by the centrifuge test data of Song et al. [27] under the installation angle θ_e of 60°. The diameter of the chain is much smaller, while the length of the chain is much longer than the anchor width. This results in massive soil elements, and hence a time-consuming computation. To improve the computational efficiency, the chain equation was incorporated into the CEL analysis by Zhao et al. [29], using a user subroutine VUAMP written in FORTRAN. It was demonstrated that the computation time by VUAMP was 36.7~41.6% smaller than that by LINK connectors. However, Zhao et al. [29] focused on the investigation of drag anchor installation, while giving little attention to the keying of SEPLAs.

Overall, most of the previous studies focused on the keying of SEPLAs without considering the chain effects. The keying of SEPLAs is a coupled process where the anchor, chain and seabed interact with each other. It was proved by model tests [18,23] that the chain plays a significant role in the keying of SEPLAs. Nevertheless, it is a challenge to effectively incorporate chain effects into the numerical analysis. Although some studies have addressed this issue in the FE analysis [27,28,29], almost no one has systematically investigated the influence of chain effects on the keying of SEPLAs.

In the present study, the CEL model was established to analyze the keying of SEPLAs, incorporating chain effects by a user subroutine VUAMP. A sensitivity analysis was first performed to determine the parameters introduced in the subroutine. The CEL results were then compared with those of RITSS and CASPA and model test to prove its validity. With chain effects, the influences of installation angle, load eccentricity, soil strength and mechanical parameters of the chain on the keying of the SEPLAs were investigated. Finally, empirical formulae were established for the embedment loss and ultimate bearing capacity of SEPLAs.

2. Numerical Modeling

2.1. CEL Method

For large deformation problems like the keying of SEPLAs, the traditional FE method is unsuitable. As the soil deformation grows, soil meshes may severely distort, causing convergence issues. The CEL method, a LDFE approach, was widely used in analyzing geotechnical problems [20,28,29,30,31,32]. The CEL method divides the FE model into two parts, i.e., Eulerian meshes for the soil where material flows in fixed meshes, and Lagrangian meshes for the anchor that moves with the material. During the Lagrangian phase, nodes are assumed to be temporarily fixed within the material, and elements deform with the material. A tolerance is used to determine which elements are significantly deformed at the end of the Lagrangian phase. During the Eulerian phase, the deformation is suspended, and elements with significant deformation are automatically remeshed. The convergence issues can be effectively solved by the Eulerian algorithm [33]. The CEL method uses an explicit time integration scheme [28]. The unknown solution in the next time step can be directly calculated by the solution of the previous time step without any iteration. A critical increment is introduced at each time step to ensure numerical stability. The CEL method tracks soil–structure contact changes via structural boundaries, and calculates the Eulerian volume fraction (EVF) for each element to track soil deformation. EVF = 1 indicates the element is full of soil, while EVF = 0 means no soil is present.

2.2. CEL Model

2.2.1. Model Establishment

The numerical model was established by the CEL method in the software ABAQUS 6.14, as illustrated in Figure 3. To enhance the calculation efficiency, a symmetric model was constructed due to the symmetry of the problem. S, W and H are the length, width and height of the soil domain; L, B and t are the length, width and thickness of the SEPLA; e is the padeye eccentricity, which is the distance from the padeye to the plane of the anchor. The drag force T_a was applied at a drag angle θ_ah with the chain equation, which will be described in Section 2.2.2.

The soil was modeled as the undrained saturated clay, governed by an ideal elastoplastic constitutive law with the Tresca yield criterion [34,35,36,37]. The SEPLA was modeled as a rigid body. For the soil–anchor interaction, a general contact was defined in the normal direction, while the tangential contact followed the Coulomb friction theory. The frictional coefficient between the anchor and the soil μ_c = 0.3 was chosen. The soil was meshed by EC3D8R elements, allowing it to flow freely within Eulerian meshes. The EC3D8R element is an eight-node reduced integration Eulerian mesh used for explicit analysis in the software ABAQUS. An air layer was set above the soil to simulate the deformation of seabed surface. The bottom of the soil is fixed, and a symmetric boundary condition is applied on the symmetric plane. Other soil boundaries are constrained in the direction normal to their surfaces.

2.2.2. Incorporating Chain Effects into the CEL Model

In the software ABAQUS, the changes in the drag force T_a and the drag angle θ_ah at the padeye (Figure 1) cannot be directly implemented by the CAE interface or input file. In the present study, the user subroutine VUAMP was used to reflect chain effects, in which the chain equation of Neubecker and Randolph [38] was adopted:

$$-{\displaystyle \frac{T_{\mathrm{a}}}{1+{\mu }^2}}{\left[e^{\mu \left({\theta }_{\mathrm{ah}}-\theta \right)}\left(\cos \theta +\mu \sin \theta \right)\right]}_{{\theta }_{\mathrm{e}}}^{{\theta }_{\mathrm{ah}}}={\displaystyle {\int }_0^{z_{\mathrm{a}}}{N}_{\mathrm{cl}}{E}_{\mathrm{n}}d{s}_{\mathrm{u}}dz}$$

(1)

where μ is the frictional coefficient between soil and chain, θ is the angle formed by the chain to the horizontal at the embedment depth of z, θ_ah is the drag anchor at the padeye, θ_e is the drag angle at the embedment point which is also called the installation angle (Figure 1), z_a is the embedment depth of the padeye, N_cl is the pullout capacity factor for the chain, ranging from 7.6 to 14, E_n is the multiplier to give the effective width in the normal direction, d is the chain diameter, and s_u is the undrained shear strength of the soil.

The user subroutine VUAMP enables user-defined amplitude curves for load conditions, boundary conditions and predefined fields, describing their variations with time and displacement. The VUAMP subroutine was employed to define the time-dependent amplitude curves of Ta via Equation (1), reflecting the chain effects. In the VUAMP subroutine, three operations are required: firstly, declare history output variables used in Equation (1), including the coordinates of the padeye (x_a and z_a), which are designated as sensors for the subroutine VUAMP. Secondly, create the horizontal and vertical components of the drag force. The drag force T_a was decomposed into horizontal and vertical components, i.e., T_ax = T_acosθ_ah and T_az = T_asinθ_ah, and their amplitude curves are calculated dynamically by the VUAMP. Finally, prior to each increment step, retrieve amplitude values from the VUAMP to update T_a and θ_ah.

Figure 4 illustrates the flowchart to incorporate chain effects with the VUAMP. The procedure for analyzing the keying of SEPLAs with chain effects can be summarized as

• Give the initial values of T_a and θ_ah at the padeye, i.e., T_a = T_a0 and θ_ah = θ_ah0.
• Calculate T_ax = T_acosθ_ah and T_az = T_asinθ_ah acted on the padeye, and conduct an incremental CEL analysis.
• If the time interval Δt is equal or greater than the control time interval t_c, update the padeye location (x_a, z_a) and calculate the average rotational velocity ur_ave = Δur/Δt. ur is the rotation angle of the anchor. If Δt < t_c, execute step b.
• Compare the values of ur_ave and the control rotational angular velocity ur_c. If ur_ave > ur_c, apply a new drag force T_a = T_a − ΔT, otherwise T_a= T_a + ΔT, where ΔT is a load increment.
• Update the value θ_ah via Equation (1), with the updated T_a and z_a.
• Repeat steps b–e until the predefined inclination of the SEPLA is achieved.

2.3. Sensitivity of FE Parameters

2.3.1. Sensitivity of Mesh and Domain Sizes

Preliminary calculations were performed to study the sensitivity to the soil mesh size and the far-field boundary effect. The model parameters were adopted the same as RITSS [19], as listed in

Table 1. Model parameters.

Type	Model Parameters	RITSS [19]	CASPA [17]	Centrifuge Test [27]
Soil	S (m)	19.7	27.8	25.8
	W (m)	4	7.9	4
	H (m)	16	25.9	16
	su (kPa)	0.7z	1 + 1.25z	18
	Poisson’s ratio	0.49	0.49	0.49
	γ′soil (kN/m3)	6.5	6.5	9.2
	E/su	500	500	500
Anchor	L (m)	3	7.92	4
	B (m)	3	4.64	4
	t (m)	0.2	0.16	0.2
	e/B	0.17, 0.5 *, **, 1.0	0.558	0.625
	Poisson’s ratio	0.3	0.3	0.3
	γ’anchor (kN/m3)	67	65	67.8
	Hi/B	3	4.47	3
Chain	Ncl, En, μ	7.6, 1, 0.1	7.6, 1, 0.1	7.6, 1, 0.1
	d (m)	0.1	0.41	0.1
	θe (°)	30 **, 45, 60, 75, 90 *	40	60

* Parameters used in Section 2.3.1. ** Parameters used in Section 2.3.2.

, where γ′_soil and γ′_anchor are the submerged unit weight of the soil and the anchor, respectively; E is the Young’s modulus. Figure 5a,b present the relationship between the embedment loss and the anchor inclination under three soil mesh sizes and soil domains, respectively. The parameters and results for different soil mesh size and soil domain are summarized in

Table 2. Parameters and results under different mesh and domain sizes.

Case	Minimum Mesh Size	Length of Soil Domain (m)	Number of Elements	Embedment Loss Δz/B
Mesh 1	B/10	19.7	32,256	0.58
Mesh 2	B/20	19.7	152,520	0.51
Mesh 3	B/30	19.7	406,000	0.49
Domain 1	B/20	10.7	83,640	0.51
Domain 2	B/20	19.7	152,520	0.51
Domain 3	B/20	37.7	250,920	0.52

. It was found that Mesh 2 and Domain 2 are considered sufficiently fine in terms of accuracy and adopted for all subsequent analyses.

2.3.2. Sensitivity of Parameters of VUAMP

In the VUAMP, five parameters were introduced by the present study, i.e., the initial drag force T_a0 and drag angle θ_ah0, the controlled rotational angular velocity ur_c, the controlled time interval t_c and the load increment ΔT. These parameters may influence the behaviors of SEPLAs during keying. At the initial stage, θ_ah0 was set as 90°.

Table 3. Cases assessing the dependency of parameters in VUAMP.

Cases	Ta0 (kN)	urc (°/s)	tc (s)	ΔT (kN)
1	28 57 171 285 570	1.0	0.001	0.5
2	114	0.5 1.0 2.0	0.001	0.5
3	114	1.0	0.1 0.01 0.001	0.5
4	114	1.0	0.001	0.25 0.5 1.0

lists the cases for assessing the dependency of T_a0, ur_c, t_c and ΔT. The model parameters were adopted the same as RITSS [19], as listed in Table 1. Five values of T_a0 were set as 0.5, 1.0, 3.0, 5.0 and 10.0 times the value of s_uiA, where s_ui is the soil strength at the initial embedment depth of the anchor, and A is the fluke area. Figure 6a indicates that the keying of SEPLAs is not affected by T_a0 of 28 kN~171 kN. As the value of T_a0 continually increases, the drag force accelerates the keying of SEPLAs and leads to an excessive embedment loss, especially for T_a0 = 570 kN. In addition, the calculation time decreases with an increasing value of T_a0. Hence, it is suggested that T_a0 = s_uiA~3s_uiA is adopted for the keying of SEPLAs. Figure 6b–d shows that ur_c, t_c and ΔT have little effect on the embedded loss. However, the calculation time increases with a decreasing value of ur_c. The value of t_c has a significant effect on the average rotational velocity of the anchor ur_ave. The values of ur_ave are 0.52°/s, 0.91°/s and 1.08°/s at t_c of 0.1 s, 0.01 s and 0.001 s, respectively. The average rotational velocity ur_ave approximates the control rotational angular velocity ur_c at t_c = 0.001 s. In the following analysis, T_a0, ur_c, t_c and ΔT are adopted as 2s_uiA, 1.0°/s, 0.001 s and 0.5 kN, respectively.

2.4. Validation of the CEL Model

The CEL model was compared with the results of the RITSS method by Wang et al. [19], CASPA method by Wei et al. [17] and centrifuge test by Song et al. [27]. The model parameters are listed in Table 1 for validations with RITSS, CASPA and the centrifuge test. Figure 7a gives the relationships between the embedment loss Δz/B and the anchor inclination β for both the CEL and RITSS methods at vertical loading (θ_e = 90°). For load eccentricities e/B = 0.17, 0.5 and 1.0, the embedment losses calculated by the CEL are 1.75B, 0.51B and 0.21B, respectively, compared to 1.6B, 0.5B and 0.2B by the RITSS [19]. The deviations range from 2% to 9.4%, indicating minor overestimations by the CEL model. To further validate the method to incorporate chain effects (Figure 4), the embedment losses of five installation angles, i.e., θ_e = 30°, 45°, 60°, 75° and 90°, were compared with the RITSS of Wang et al. [19]. Figure 7b demonstrates that the embedment losses calculated by the CEL agree well with those by the RITSS, with a maximum deviation less than 5%. Figure 7c shows the comparison of embedment loss between the CEL and CASPA methods at θ_e = 40°. The CASPA method reported an embedment loss of Δz = 0.318B and a final anchor inclination of β_f = 34.5°, while the CEL method yielded Δz = 0.287B and β_f = 35.6°, representing a 9.7% reduction in the embedment loss. Figure 7d shows the comparison between the CEL and centrifuge test [27] at θ_e = 60°. The embedment loss of Δz and the final anchor inclination of β_f calculated by the CEL are 0.25B and 33.3°, respectively, while the centrifuge test shows Δz = 0.32B and β_f = 38°. These comparisons confirm the effectiveness of the CEL model.

3. Parametric Study

Based on the established CEL model, a parametric study was performed to quantify the effects of installation angle, load eccentricity, soil strength and mechanical parameters of the chain on the keying of the SEPLAs. The model parameters were adopted the same as RITSS [19] (Table 1), while the cases of parametric study are listed in

Table 4. Cases of parametric study.

Case	Cf (m)	e/B	su (kPa)	θe (°)
1	1.9	0.17	0.7z	15, 30, 45, 60, 75, 90
2	1.9	0.5	0.7z	15, 30, 45, 60, 75, 90
3	1.9	0.5	1.4z	15, 30, 45, 60, 75, 90
4	1.9	0.5	2.8z	15, 30, 45, 60, 75, 90
5	1.9	1.0	0.7z	15, 30, 45, 60, 75, 90
6	3.8	0.5	0.7z	15, 30, 45, 60, 75, 90
7	5.7	0.5	0.7z	15, 30, 45, 60, 75, 90

.

3.1. Installation Angle

As shown in Equation (1), different installation angles of θ_e induce different drag angles at the padeye θ_ah, affecting the keying of SEPLAs. Numerical simulations were conducted for θ_e = 15°, 30°, 45°, 60° and 75°, and compared with the results of vertical uplift of θ_e = 90° (Case 2 in Table 4). Figure 8a illustrates the relationship between the θ_ah and Δz/B. As the embedment loss Δz/B increases, the drag angle at the padeye θ_ah decreases. The value of θ_ah is always greater than that of θ_e during the keying, confirming that the chain presents a reverse catenary shape in the soil. The anchor trajectory is shown in Figure 8b, presenting the non-dimensional curves of the horizontal displacement (Δx/B) and vertical displacement (Δz/B) of the anchor. It is evident that a larger installation angle requires the anchor to rotate more to reach the normal bearing state, resulting in greater vertical displacement and embedment loss. Each trajectory has a distinct inflection point at different installation angles, after which the anchor displaces along a fixed direction.

Figure 8c shows that the embedment loss increases by 0.06B~0.1B for every 15° increase in the installation angle. The rotational velocities are nearly the same for each installation angle, arriving at final anchor inclination β_f. The values of β_f are 75.64°, 60.37°, 45.12°, 30.31°, 14.96° and −0.28° for θ_e = 15°, 30°, 45°, 60°, 75° and 90°, respectively. It is found that the final anchor inclination is approximately complement of the installation angle, i.e., β_f + θ_e ≈ 90°. Figure 8d presents the relationship between the pullout capacity factor N_c and embedment loss Δz/B, where N_c = T_a/s_ucA, where s_uc is the soil strength of the anchor center during the keying. It is shown that the ultimate pullout capacity factors N_cu (N_cu = max(N_c)) are 14.18, 14.62, 15.15, 15.62, 16.22 and 16.97 for θ_e = 15°, 30°, 45°, 60°, 75° and 90°, respectively.

The soil flow mechanism is illustrated in Figure 9, which proves that the soil failure mechanisms for θ_e = 30°~90° are nearly the same, with a local symmetrical failure. Hence, the value of N_cu decreases only by 3.10% to 4.62% for every 15° decrease in the installation angle θ_e. The effect of θ_e on N_cu is only reflected by the final anchor inclination β_f.

3.2. Load Eccentricity

Three cases with load eccentricities e/B = 0.17, 0.5 and 1.0 were designed to investigate the influence of e/B on the keying of SEPLAs (Cases 1, 2 and 5 in Table 4). Figure 10 illustrates the relationship between installation angle θ_e and embedment loss Δz/B as well as ultimate pullout capacity factor N_cu under different load eccentricities. It is proved that both embedment loss and ultimate pullout capacity of the anchor are negatively correlated with the load eccentricity. For instance, when the installation angle is 15°, the embedment loss decreases by 91.14% and the ultimate pullout capacity factor by 6.41%, as the value of e/B increases from 0.17 to 1.0. Similarly, during vertical uplift of θ_e = 90°, the embedment loss decreases by 87.67% and the ultimate pullout capacity factor by 29.74% with the increasing of e/B from 0.17 to 1.0. The embedment loss shows an approximate linear relationship with the installation angle. When e/B is small (e.g., e/B = 0.17), the installation angle significantly affects the embedment loss and the ultimate pullout capacity factor. When e/B is between 0.5 and 1.0, the installation angle θ_e has little effect on the ultimate bearing capacity factor N_cu, with maximum differences of 5.65% for every 15° increase in the installation angle θ_e.

Figure 11 illustrates the soil flow mechanisms at θ_e = 60° with load eccentricity of e/B = 0.17, 0.5 and 1.0. For e/B = 0.17, the anchor rotates about the anchor top, which induces an intense unsymmetrical soil disturbance with the soil flowing back to the bottom of the anchor. The rotation around the anchor top leads to an amplified embedment loss, while the intense unsymmetrical soil disturbance increases the ultimate bearing capacity factor. The soil flow mechanisms are nearly the same for e/B = 0.5 and 1.0, while there is a cavity on the back of the anchor for e/B = 1.0, as shown in Figure 11. The cavity makes a small reduction in the ultimate bearing capacity factor (Figure 10b).

3.3. Soil Strength

In the numerical study by Wang et al. [19], the SEPLA was vertically pulled (θ_e = 90°) in the soil with a strength of s_u = 6.3 kPa and s_u = 0.7z, where the initial embedment depth of the anchor H_i was 9 m (yielding the soil strength at the initial embedment depth s_ui = 6.3 kPa). It was demonstrated by Wang et al. [19] that the embedment loss only depends on the value of s_ui, independent of the soil strength gradient. In the present study, two cases of s_u = 25.2 kPa and s_u = 7.2 + 2z kPa were designed with the anchor initial embedment depth of H_i = 9 m, in which the soil strengths at the initial embedment depth were both s_ui = 25.2 kPa. Figure 12a presents the relationship between Δz/B and θ_e. The maximum deviation of embedment losses is only 0.05B for s_u = 25.2 kPa and s_u = 7.2 + 2z kPa with θ_e = 15°~90°. That means the embedment loss only depends on the soil strength at the initial embedment depth, even at different values of θ_e. However, the final anchor inclination β_f is affected by the soil strength with a decreasing value of θ_e. The maximum difference is 10° at θ_e = 15°, as shown in Figure 12b.

Based on the aforementioned findings, three soil strengths, i.e., s_u = 0.7z, 1.4z and 2.8z, were designed to evaluate the effects of soil strength on the keying of SEPLAs under different installation angles (Cases 2–4 in Table 4), as shown in Figure 13. Figure 13a demonstrates that the embedment loss increases with increasing soil strength. When the soil strength increases from s_u = 0.7z to s_u = 2.8z, the embedment loss increases from 0.09B to 0.27B, 0.17B to 0.48B, 0.27B to 0.69B, 0.35B to 0.82B, 0.42B to 0.96B and 0.52B to 1.28B for θ_e = 15°, 30°, 45°, 60°, 75° and 90°, respectively. Figure 13b shows that the ultimate bearing capacity factor decreases with increasing soil strength.

Figure 14 illustrates the soil flow mechanisms at θ_e = 60° with a soil strength of s_u = 0.7z, 1.4z and 2.8z. For s_u = 0.7z, the symmetrical circular flow is formed around the anchor sides, and the soil sticks the anchor back, which induces a higher ultimate bearing capacity factor. There is a cavity at the anchor back both for s_u = 1.4z and 2.8z, decreasing the ultimate bearing capacity factor with increasing soil strength.

3.4. Mechanical Parameters of the Chain

There are three mechanical parameters in Equation (1), including the chain diameter d, the bearing capacity factor of the chain N_cl and the multiplier to give the effective width of the chain E_n. C_f denotes the product of d, N_cl and E_n, i.e., C_f = dN_clE_n (unit: m). Three cases of C_f = 1.9 m, 3.8 m and 5.7 m were designed to evaluate their effects on the keying of SEPLAs (Cases 2, 6 and 7 in Table 4), as shown in Figure 15. The embedment loss increases with the increase of C_f for 15° ≤ θ_e ≤ 45°, while the value of C_f has no effect on the embedment loss for 45° < θ_e ≤90°. The smaller θ_e can form a reverse catenary shape of the chain in the soil, influencing the drag angle at the padeye θ_ah and hence the anchor embedment loss. Figure 15b shows the relationship between the ultimate bearing capacity factor N_cu and the installation angle θ_e. It is demonstrated that the value of C_f has little effect on N_cu at different installation angles, with the maximum deviation of 1.9%.

4. Empirical Formulae

Considering that the CEL analysis is very time-consuming, empirical formulae will be established for embedment loss and ultimate bearing capacity factor in the present study, which is a benefit to engineering practice and easy to use by engineers. It is noted that the chain effects are incorporated into the empirical formula, which is hardly considered by previous researchers.

4.1. Embedment Loss

Based on the parametric study in Section 3 and dimensional analysis, the empirical formula for anchor embedment loss was developed:

$${\displaystyle \frac{\Delta Z}{B}}={\displaystyle \frac{{\left[x_1\sin {\theta }_{\mathrm{e}}+x_2{\left(\frac{{kH}_{\mathrm{i}}}{s_{\mathrm{u}0}}\right)}^{x_3}{\left(\frac{C_{\mathrm{f}}}{B}\right)}^{x_4}\right]}^{x_5}}{{\left(\frac{e}{B}\right)}^{x_6}}}$$

(2)

where x₁, x₂, x₃, x₄, x₅ and x₆ are the fitting coefficients for the empirical formula of Equation (2). It is noted that s_u0 adopts 0.001 kPa when s_u0 = 0 kPa. With CEL calculations in

Table 5. Verification of Equation (2).

θe (°)	e/B	su (kPa)	Cf (m)	CEL Analysis	Equation (2)	Deviation
20°	0.3	1.0z	2.85	0.300	0.272	9.33%
40°	0.6	2.0z	4.05	0.375	0.391	4.27%
90°	0.3	1.0z	2.85	1.041	1.105	6.15%
90°	0.6	2.0z	4.05	0.783	0.785	0.26%

, x₁, x₂, x₃, x₄, x₅ and x₆ are fitted as 0.183, 0.316, 0.085, 0.01, 9.383 and 1.167, respectively. Figure 16 illustrates the comparison between the results of Equation (2) and the CEL analysis, with R² = 0.988.

To verify the empirical formula for embedment loss, four validation cases were designed in Table 5 and calculated by CEL analysis and Equation (2). The deviations between the CEL analysis and Equation (2) are 0.26~9.33%, validating the efficiency of Equation (2).

4.2. Ultimate Pullout Capacity Factor

Based on the parametric study in Section 3 and dimensional analysis, the empirical formula for ultimate pullout capacity factor was also developed:

$$N_{\mathrm{cu}}={\displaystyle \frac{\left[x_7{\left(\frac{{kH}_{\mathrm{i}}}{S_{\mathrm{u}0}}\right)}^{x_8}{N}_{\mathrm{c},{\theta }_{\mathrm{e}}={90}^{\circ }}+x_9\right]\cdot {\left(\frac{\theta \mathrm{e}}{{90}^{\circ }}\right)}^{x_{10}}\cdot {\left(\frac{C_{\mathrm{f}}}{B}\right)}^{x_{11}}}{{\left(\frac{e}{B}\right)}^{x_{12}}}}$$

(3)

where x₇, x₈, x₉, x₁₀, x₁₁, and x₁₂ are the fitting coefficients for the empirical formula of Equation (3). It is noted that s_u0 adopts 0.001 kPa when s_u0 = 0 kPa. With CEL calculations in Table 4, x₇, x₈, x₉, x₁₀, x₁₁, and x₁₂ are fitted as 0.963, −0.027, 4.183, 0.069, −0.090 and 0.012. Figure 17 illustrates the comparison between the results of Equation (3) and the CEL analysis, with R² = 0.954.

To verify the empirical formula for ultimate pullout capacity factor, two validation cases were designed in

Table 6. Verification of Equation (3).

θe (°)	e/B	su (kPa)	Cf	${\mathit{N}}_{\mathbf{c},{\mathit{\theta }}_{\mathbf{e}}={90}^{\circ }}$	CEL Analysis	Equation (3)	Deviation
20°	0.3	1.0z	2.85	14.279	13.865	13.640	1.63%
40°	0.6	2.0z	4.05	9.89	10.278	10.666	3.78%

and calculated by CEL analysis and Equation (3). The deviations between the CEL analysis and Equation (3) are 1.63~3.78%, validating the efficiency of Equation (3).

5. Conclusions

In the present study, the CEL method was employed to establish an FE model for analyzing the keying of SEPLAs, incorporating chain effects via the user subroutine VUAMP. Based on the sensitivity analysis for the introduced parameters in VUAMP, it is suggested that the initial drag force T_a0, the controlled rotational angular velocity ur_c, the controlled time interval t_c and the load increment ΔT are adopted as 1~3s_uiA, 1.0°/s, 0.001 s and 0.5 kN, respectively. The effectiveness of the CEL model was demonstrated by validations against RITSS, CASPA and the centrifuge test.

With chain effects, the influences of installation angle, load eccentricity, soil strength and mechanical parameters of the chain on the keying of the SEPLAs were investigated. The results show the following:

(1) The final anchor inclination β_f is approximately complement of the installation angle θ_e at the completion of the keying. The loss of embedment Δz and ultimate pullout capacity factor N_cu increase with increasing θ_e. Each 15° increase in θ_e elevates Δz by 0.06B~0.1B and N_cu by 3.10~4.62%.

(2) The values of Δz and N_cu decrease with increasing load eccentricity e. Increasing e/B from 0.17 to 1.0 reduces Δz/B by up to 94.14% and N_cu by 29.74%. Lower e/B (e.g., e/B = 0.17) amplifies the effects of θ_e on Δz and N_cu, because the anchor rotates around the anchor top which induces an intense unsymmetrical soil disturbance with the soil flowing back to the bottom of the anchor.

(3) The loss of embedment only depends on the soil strength at the initial embedment depth of SEPLAs. The value of Δz increases and that of N_cu decreases with increasing value of s_u. When the soil strength increases, there is a cavity at the anchor back, impairing the bearing capacity factor N_cu.

(4) The value of Δz increases with increasing chain parameter C_f for 15° ≤ θ_e ≤ 45°, while the value of C_f has no effect on Δz for 45° < θ_e ≤90°. The smaller θ_e forms a reverse catenary shape of the chain in the soil, influencing the drag angle at the padeye θ_ah and hence the loss of embedment Δz. The value of C_f has little effect on the ultimate bearing capacity factor N_cu at different installation angles.

Based on the parametric study, empirical formulae were established for the embedment loss and ultimate bearing capacity of SEPLAs, yielding average deviations of 9.69% and 3.24%, respectively. By inputting the parameters of the anchor, chain and soil, the embedment loss and ultimate bearing capacity of SEPLAs can be easily calculated by Equations (2) and (3), respectively. The empirical formulae are easy to use by engineers and can reduce design cycles.

The chain effect was implemented by the chain equation (Equation (1)), which cannot reflect the effects of chain length. In addition, the installation angle was assumed by a constant during the keying of SEPLAs. These limitations also restrict the application of the empirical formulae. In the future, the researchers can pay more attention to the development of the chain equation or find a more accurate and effective FE method to simulate the chain.