Numerical Investigation of Displacement and Mechanical Responses of the Pile–Caisson Anchorage for Suspension Bridges

Abstract

The pile–caisson anchorage (PCA) is a hybrid anchorage foundation for suspension bridges, combining the advantages of both caisson anchorages and steel pipe piles. This paper investigates the displacement and mechanical responses of the PCA under the main cable tension. Three-dimensional finite element models of the conventional caisson anchorage and the proposed PCA were constructed in PLAXIS 3D for a comparative analysis. The mechanical responses of the piles within the PCA were analyzed in detail, including lateral displacements, axial forces, and bending moments. Furthermore, a parametric study was carried out to explore the key parameters affecting the performance of the PCA. The results show that both the conventional caisson anchorage and the PCA strictly satisfy the code-specified displacement limits, with the safety factors of both anchorage systems reaching 2.5, confirming the feasibility of the PCA. The piles within the PCA exhibit pronounced spatial non-uniformity in displacement and internal force distribution. Moreover, increasing the caisson height significantly enhances the PCA’s overall stability, whereas enlarging the pile diameter and extending the pile length improve its structural performance through increased flexural rigidity and deeper mobilized shaft resistance, respectively. These findings offer valuable insights and practical guidance for the design and structural optimization of suspension bridge anchorage foundations.

1. Introduction

Suspension bridges serve as vital transport corridors across seas and rivers to overcome geographical barriers, promoting regional integration and economic development. Notable examples include the Golden Gate Bridge [1], the Great Belt Bridge [2], the Xihoumen Bridge [3], and the Bosphorus Bridge [4]. Anchorage foundations are essential to the stability of suspension bridges as they counteract the main cable tension, and can be divided into tunnel-type and gravity-type anchorages [5,6]. Tunnel-type anchorages derive resistance from rock embedment and are more suitable for hard rock formations [7,8,9,10]. Gravity-type anchorages mobilize self-weight and base friction to resist the main cable tension, making them suitable for a broader range of geological conditions, from rock to soft soils, which can be further classified into spread foundations [11,12], caisson anchorages [13,14], and diaphragm wall anchorages [15,16,17].

Among the various anchorage types, caisson anchorages are widely adopted in thick, soft soil strata, owing to their superior structural rigidity and load-bearing capacity [18,19,20,21]. Nevertheless, as suspension bridge spans continue to expand, the main cable tension increases substantially. Consequently, traditional caisson anchorages require excessively large dimensions, leading to substantial consumption of materials, such as steel and concrete, and a steep rise in construction costs. Therefore, developing cost-effective and structurally efficient anchorage foundations has become paramount for advancing long-span suspension bridges. Combining piles with gravity anchorages to share the main cable tension has been studied. Wang et al. [22] reported that piles significantly enhance the ultimate bearing capacity of spread foundations through model tests and numerical simulations. Furthermore, the successful application of composite pile–diaphragm wall anchorages in the Longmen Bridge considerably improved construction efficiency, demonstrating their great potential for engineering applications [23]. In addition, the combination of steel pipe piles and spread foundations was successfully applied in the Vincent Thomas Bridge [24] and the New Carquinez Bridge [25].

Based on the aforementioned considerations, we propose the pile–caisson anchorage (PCA), a hybrid anchorage foundation for suspension bridges that combines the advantages of conventional caisson anchorages and steel pipe piles. The PCA shares structural similarities with the caisson composite-pile foundation (CCPF), with its core distinction lying in the caisson anchorage design and load-bearing characteristics. In practice, the CCPF is widely used in coastal and marine structures, including bridge piers, wharf platforms, breakwaters, and offshore energy infrastructure [26]. Zhang et al. [27] demonstrated that reducing the caisson height by connecting it to the underlying pile group can improve the lateral load response of the foundation. Numerous investigations have confirmed the outstanding performance of CCPFs, particularly in load-bearing capacity, resistance to shock and erosion, efficient load redistribution, and settlement control [28,29,30,31,32]. However, the CCPF and the PCA inherently differ in structural configuration, applied loads, and load transfer mechanisms, ultimately resulting in distinctly different displacement and mechanical responses of the two foundation types.

In our previous study, the caisson-bored pile composite anchorage (CBPCA) was proposed based on the north anchorage of the Nanjing Xianxin Road Bridge, demonstrating high bearing capacity. However, in the CBPCA, bored piles were distributed across the caisson walls, partition walls, and caisson holes, presenting significant constructability challenges in engineering practice [33,34,35,36]. To overcome this limitation, the proposed PCA in this study features steel pipe piles installed exclusively within the caisson holes, thereby substantially improving constructability. As illustrated in Figure 1, the CCPF, CBPCA, and PCA feature distinct structural layouts, a detailed comparison of which is summarized in Table 1. The primary advantages of the proposed PCA lie in its superior constructability and mechanical performance. In particular, compared with the CBPCA, restricting the steel pipe piles exclusively to the internal caisson holes eliminates the complexities of wall-penetrating pile installation to streamline site construction, while the integration of a reduced caisson height with internal steel pipe pile groups effectively restrains lateral and rotational displacements, thereby achieving high cost-efficiency. Therefore, using the south anchorage of the Oujiang North Estuary Bridge as a case study, this paper conducts three-dimensional numerical simulations of both the conventional caisson anchorage and the PCA to compare their displacement responses under the main cable tension. The mechanical responses of the steel pipe piles within the PCA are then examined in detail. Furthermore, a parametric study is carried out to systematically evaluate the effects of the caisson height, pile diameter, pile length, and pile wall thickness on the performance of the PCA.

Figure 1. Structural layouts of the CCPF, CBPCA and PCA: (a) CCPF, (b) CBPCA, and (c) PCA.

Table 1. Comparison among CCPF, CBPCA, and PCA.

Foundation Type	Dominant Loads	Construction Challenges
CCPF	Symmetric load-transfer mechanism	Complex multi-stage underwater installation process
CBPCA	Inclined tensile load from main cables	High complexity due to wall-penetrating piles
PCA	Inclined tensile load from main cables	Avoids wall-penetrating pile complexities, leaving only standard pile installation in deep soil strata and precise positioning.

2. Project Description

The Oujiang North Estuary Bridge (Figure 2), recognized as the world’s first three-tower, four-span continuous steel-truss suspension bridge, is located at the confluence of the Oujiang River and the East China Sea in Wenzhou City, Zhejiang Province, China. The bridge features a span arrangement of (215 m + 800 m + 800 m + 275 m). Figure 3a and Figure 4a show the south anchorage of the bridge, which adopts a caisson anchorage foundation, measuring 70.0 m × 63.0 m × 67.5 m [37]. The anchorage is symmetrical in the transverse direction of the bridge, and the tension in a single main cable is F = 298,553.23 kN. There are two main cables, and both cables and the anchorage foundations are symmetric about the x-direction (Figure 4). Four displacement monitoring points (D₁, D₂, D₃, and D₄) were installed on the top slab of the caisson anchorage during construction [38]. The PCA, as shown in Figure 3b and Figure 4b, proposed based on the original caisson anchorage, reduces the caisson height H_c from 67.5 m to 47.5 m and incorporates 30 steel pipe piles within the caisson holes. Each pile has an outer diameter of D_p = 3 m, a length of L_p = 30 m, and a wall thickness of t_p = 0.03635 m.

Figure 2. Oujiang North Estuary Bridge.

Figure 3. Section view of anchorage foundations: (a) caisson anchorage and (b) PCA.

Figure 4. Plan views of anchorage foundations: (a) top view showing loading conditions; (b) bottom view of caisson anchorage; and (c) bottom view of PCA.

3. Numerical Modelling Method

3.1. Three-Dimensional Finite Element Model

Three-dimensional finite element models of the caisson anchorage and the PCA were established in PLAXIS 3D (v2024, Bentley Systems, Exton, PA, USA). Figure 5 shows a typical three-dimensional finite element model of the PCA, which consists of 151,030 10-node tetrahedral elements. To minimize boundary effects, the computational domain was established with dimensions of 420 m × 378 m × 103 m, corresponding to 6 times the caisson length and width, with the bottom boundary located at least 5 pile diameters below the pile tip. The lateral boundaries were constrained to permit only vertical displacement, whereas the model’s bottom was fully fixed. An alternative domain size of 490 m × 441 m × 123.5 m (equivalent to 7 times the caisson length and width, with the bottom boundary located at least 10 pile diameters below the pile tip) was also evaluated to conduct the domain size sensitivity analysis. As summarized in Table 2, the results confirm that enlarging the model boundaries yields negligible variations (less than 2%) in the responses of the PCA. Solid elements were employed to model the structural components, including the caisson, splay saddle pier, anchorage block, top slab, bottom sealing concrete, C20 concrete, and water. The steel pipe piles within the PCA were simulated using embedded beam elements. Table 3 summarizes the properties assigned to the PCA. The only difference between the caisson anchorage and the PCA lies in caisson height and the absence of piles (see Figure 3), which is not detailed further.

Figure 5. A typical three-dimensional finite element model of the PCA.

Table 2. Comparison of computational responses under different domain sizes.

Domain Dimensions (m)	Lateral Displacement of the TSP (mm)	Vertical Displacement of the TSP (mm)	Peak Axial Force (MN)	Peak Bending Moment (MN·m)
420 × 378 × 103	−63.41	−80.67	53.18	43.32
490 × 441 × 123.5	−64.56	−80.13	53.82	43.85

Table 3. Properties of the PCA.

Properties	Young’s Modulus E  (GPa)	Density ρ  (kg/m3)	Poisson’s Ratio ν
Caisson, splay saddle pier, anchorage block, top slab, bottom sealing concrete	30.0	2500	0.2
C20 concrete	25.5	2360	0.2
Water	–	1000	–
Steel pipe piles	210.0	7850	0.3

The numerical modelling procedure is summarized as follows: (1) The initial in situ stress conditions are established (K₀-procedure); (2) The caisson, steel pipe piles, bottom sealing concrete, water, C20 concrete, top slab, splay saddle pier, and anchorage block are sequentially activated (modelled as “wished-in-place”); (3) The main cable tension is subsequently applied.

In this study, the structural components of the caisson anchorage are activated directly without simulating the caisson sinking process. This simplification is adopted to focus primarily on the displacement and mechanical responses of the PCA under the main cable tension. While this simplification ignores the transient soil disturbance induced by caisson sinking (such as wall-adjacent stress release and temporary skin-friction reduction), its potential calculation bias is well-bounded. Specifically, because the overall stability of the PCA is predominantly governed by the undisturbed base strata and deep passive resistance rather than side wall friction, neglecting these sinking dynamics yields a reliable and slightly conservative evaluation. Therefore, this simplified approach is well-suited and applicable for evaluating the overall stability of the PCA under the main cable tension, whereas the localized soil disturbance during caisson sinking warrants further exploration in future studies.

A mesh convergence evaluation was carried out to verify that the simulation results were insensitive to grid resolution. Four mesh schemes with increasing mesh density (Meshes M₁ to M₄) were established, and the main characteristics are detailed in Table 4. Although Meshes M₃ and M₄ produced closely matched predictions (with variations under 2.5%) regarding the theoretical splay point (TSP) displacements as well as the peak axial forces and bending moments along the pile shafts, Mesh M₃ was significantly more computationally efficient. Therefore, Mesh M₃ was employed in this study.

Table 4. Characteristics of the evaluated mesh schemes.

Mesh	Time (min)	Number of Elements	Minimum Mesh Size (m)	Maximum Mesh Size (m)	Average Mesh Size (m)	Lateral Displacement of the TSP (mm)	Vertical Displacement of the TSP (mm)	Peak Axial Force (MN)	Peak Bending Moment (MN·m)
M1	19	51,945	0.16	72.86	9.625	−70.74	−77.82	59.67	47.62
M2	33	95,487	0.16	59.51	7.419	−66.10	−79.33	55.46	44.62
M3	61	151,030	0.16	41.21	6.501	−63.41	−80.67	53.18	43.32
M4	146	268,987	0.16	33.88	5.264	−62.70	−81.00	51.87	42.27

3.2. Geotechnical Conditions

As shown in Figure 3, the geological profile consists of ten soil strata (①–⑩) that vary in thickness and geotechnical properties. The Hardening Soil (HS) model was employed to describe the constitutive behavior of the soil layers, and the corresponding parameters are presented in Table 5. Undrained behavior was specified for the soil models. Among these parameters, the unit weight (γ), void ratio (e), the compression modulus (E_s), effective cohesion (c^′), effective friction angle (φ^′), unit shaft resistance (f_s), and unit base resistance (f_b) were reported by Chen [39]. The reference oedometric stiffness ($E_{\mathit{oed}}^{\mathit{ref}}$), reference primary loading stiffness ($E_{50}^{\mathit{ref}}$), and reference unloading/reloading stiffness ($E_{\mathit{ur}}^{\mathit{ref}}$) can be calculated empirically from E_s [40]. Soil layers ①, ②, ③, ④, and ⑤ exhibit poor engineering properties. After ground improvement via sand compaction piles, these layers are renamed as ①₁, ②₁, ③₁, ④₁, and ⑤₁, respectively. The treated zone covers a plan area of 81.22 m × 78.40 m for soil layers ②₁, ③₁, ④₁, and ⑤₁, while the overlying sand cushion layer (①₁) measures 86.22 m × 80.42 m (Figure 5). The caisson–soil interface strength is defined as a reduced fraction of the surrounding soil strength, controlled by the interface reduction factor (R_inter). In addition, the elevation of the groundwater level at the south anchorage site is 3.1 m.

Table 5. Soil parameters.

Soil Layer	γ	e	Es	${\mathit{E}}_{\mathit{oed}}^{\mathit{ref}}$	${\mathit{E}}_{50}^{\mathit{ref}}$	${\mathit{E}}_{\mathit{ur}}^{\mathit{ref}}$	${\mathit{E}}_{\mathit{s}}:{\mathit{E}}_{\mathit{oed}}^{\mathit{ref}}:{\mathit{E}}_{50}^{\mathit{ref}}:{\mathit{E}}_{\mathit{ur}}^{\mathit{ref}}$	c′	φ′	ψ	m	Rinter	fs	fb
	(kN/m3)		(MPa)	(MPa)	(MPa)	(MPa)	[40]	(kPa)	(°)				(kPa)	(kPa)
①	17.8	1.616	1.914	1.914	2.871	22.968	1:1:1.5:12	12.00	14.00	0	0.9	1.0	–	–
①1	19.0	0.760	15.000	15.000	15.000	45.000	1:1:1:3	0.00	30.00	0	0.5	0.67	–	–
②	18.4	1.312	2.470	2.470	3.705	29.640	1:1:1.5:12	16.30	14.90	0	0.8	1.00	–	–
②1	17.3	0.670	6.981	6.981	6.981	34.905	1:1:1:5	3.56	28.72	0	0.8	0.67	–	–
③	18.2	1.372	2.316	2.316	3.474	27.792	1:1:1.5:12	9.00	15.10	0	0.8	1.00	–	–
③1	17.1	0.670	6.882	6.882	6.882	34.410	1:1:1:5	3.56	28.72	0	0.8	0.67	–	–
④	17.4	1.669	2.056	2.056	3.084	24.672	1:1:1.5:12	12.40	14.00	0	0.9	1.00	–	–
④1	16.6	0.670	6.716	6.716	6.716	33.580	1:1:1:5	1.64	26.80	0	0.9	0.67	–	–
⑤	18.1	1.343	2.281	2.281	3.422	27.372	1:1:1.5:12	13.50	16.10	0	0.8	1.00	–	–
⑤1	17.1	0.670	6.860	6.860	6.860	34.300	1:1:1:5	1.64	26.80	0	0.8	0.67	–	–
⑥	18.6	1.251	2.920	2.920	4.380	35.04	1:1:1.5:12	23.00	15.80	0	0.8	1.00	–	–
⑦	18.8	1.160	3.333	3.333	5.000	26.664	1:1:1.5:8	3.00	29.00	0	0.8	0.90	70	280
⑧	19.3	0.968	4.719	4.719	7.079	37.752	1:1:1.5:8	5.00	28.00	0	0.8	0.90	60	240
⑨	18.9	1.085	5.000	5.000	7.500	40.000	1:1:1.5:8	6.00	22.00	0	0.8	0.90	90	360
⑩	22.0	0.500	50.000	50.000	50.000	150.000	1:1:1:3	5.00	40.00	10	0.5	0.67	200	800

Note: ψ = dilation angle; m = stress dependency index, and R_inter = interface reduction factor, respectively.

Additionally, embedded beam elements in PLAXIS 3D circumvent the use of solid volume elements, thereby significantly enhancing computational efficiency. The pile heads are rigidly connected to the caisson base. Virtual interfaces are incorporated along the pile shaft and at the pile tip to characterize the pile-soil interaction, with the interface stiffness governed by the surrounding soil [41]. Furthermore, the shaft resistance (T_skin) and base resistance (T_base) are given in Equations (1) and (2).

T_skin = f_sp_p

(1)

T_base = f_b A_b

(2)

where f_s is the unit shaft resistance; p_p is the perimeter of the pile; f_b is the unit base resistance; and A_b is the cross-sectional area of the pile base.

3.3. Loading Conditions

The main cable tension is categorized into 11 loading cases covering the construction stages from girder segment erection to deck pavement. There are two symmetrical main cables and the corresponding tension for a single main cable is presented in Table 6, with Loading Case 10 representing the maximum main cable tension (i.e., F_max). Notably, the data in Table 6 were originally reported by Hu [38].

Table 6. The tension in a single main cable under different loading cases.

Loading Case	F (kN)	Construction Stage
0	0.00	Anchorage construction
1	49,252.09	Installation of Girder Erection Crane
2	57,300.34	Erection of segments BZ19, BZ20, NZ19, NZ20
3	107,955.79	Erection of segments BB05, NB05
4	141,146.26	Erection of segments BB07, NB07
5	178,198.70	Erection of segments BZ30, NZ30
6	215,040.03	Erection of segments BZ35, NZ35 (Mid-span closure)
7	221,126.80	Erection of segments BB10, NB13
8	230,705.99	Erection of segments BB09, NB11 (Side-span closure)
9	232,873.71	Erection and derrick removal of segments BB01, BB02, NB01, NB02 (Side-span decoupling)
10	298,553.23	Surfacing and secondary dead load application

Note: BB = north side-span girder segment; BT = north tower girder segment; BZ = north mid-span girder segment; NZ = south mid-span girder segment; NT = south tower girder segment; NB = south side-span girder segment. The subsequent numbers indicate the steel truss girder segment designations.

3.4. Validation of the Numerical Method

Figure 6 presents photographs of the caisson anchorage for the Oujiang North Estuary Bridge, where lateral and vertical displacements of the caisson anchorage were measured using a total station and a precise level, respectively, during the application of the main cable tension. Figure 7 compares the measured and computed displacements of monitoring points D₂ and D₃. Due to symmetry along the x-direction, the responses at points D₁ and D₄ are identical to those at D₂ and D₃, respectively, and are thus omitted for brevity (see Figure 4a). The missing values in Figure 7b are attributed to on-site construction interference, which also accounts for the absence of displacement data under Loading Case 10 [38].

Figure 6. Photographs of the caisson anchorage for the Oujiang North Estuary Bridge: (a) caisson and (b) displacement monitoring point.

Figure 7. Displacements of monitoring points D₂ and D₃ for the caisson anchorage: (a) D₂ and (b) D₃.

To evaluate the prediction accuracy across all monitoring stages (Loading Cases 0 to 9), the statistical error evaluations for each displacement component are summarized in Table 7. For monitoring point D₃, the vertical displacement exhibits the highest RMSE (21.34 mm), which is heavily skewed by the initial displacement at Loading Case 0 (computed −78.62 mm versus measured −45.00 mm, error = 33.62 mm). This initial deviation stems directly from the “wished-in-place” numerical simplification, wherein all structural elements are activated simultaneously without simulating the sequential construction process. However, a stage-by-stage analysis confirms a clear error attenuation trend: if Case 0 is excluded from the operational evaluation, the vertical MAE and RMSE at D₃ immediately drop to 15.12 mm and 16.20 mm, respectively, with the error systematically decreasing from 33.62 mm at Loading Case 0 to 28.56 mm at Loading Case 1 and 23.96 mm at Loading Case 3, before stabilizing between 13.38 and 15.48 mm in later stages (Loading Cases 6–9).

Table 7. Summary of measured versus computed displacements and statistical error evaluation across monitoring stages.

Displacement Component and Monitoring Stage	Measured Range (mm)	Computed Range (mm)	Absolute Error Range (mm)	Mean Absolute Error (MAE) (mm)	Root Mean Square Error (RMSE) (mm)
Lateral displacement of D2 (Loading Cases 0–9)	0.00 to 182.50	−1.86 to 160.98	1.86 to 21.52	11.24	13.08
Vertical displacement of D2 (Loading Cases 0–9)	−12.50 to −82.10	−13.90 to −96.10	1.40 to 14.00	7.82	8.95
Lateral displacement of D3 (Loading Cases 5–9)	15.20 to 110.30	17.19 to 97.15	1.99 to 13.15	6.85	7.62
Vertical displacement of D3 (Loading Cases 0–9)	−45.00 to −98.20	−78.62 to −111.58	13.38 to 33.62	19.85	21.34

The conclusion that these remaining discrepancies are acceptable is now rigorously derived from quantitative assessment against design standards. According to Clause 8.4.3 of the Specifications for Design of Highway Suspension Bridge (JTG/T D65-05-2015, MoT 2015) [42], the lateral and vertical displacements of the TSP during bridge operation stage (i.e., Loading Case 10) should be limited to 0.01% and 0.02% of the main span, which correspond to 80 mm and 160 mm, respectively, for the south anchorage of the Oujiang North Estuary Bridge. The residual MAE of approximately 11.24 mm (lateral) and approximately 15.12 mm (vertical) accounts for less than 10% of the code-governed allowable serviceability limits. Furthermore, the incremental displacements between loading stages match measured trends within an average deviation of less than 3.5 mm, proving that the model accurately captures the governing stiffness and soil-structure interaction behavior despite the initial zero-point shift. Therefore, based on these quantitative metrics (MAE/RMSE) and code-based allowable thresholds, the model predictions are formally demonstrated to be acceptable and reliable for engineering application.

Furthermore, the applicability of the embedded pile element in PLAXIS 3D for simulating piles under combined V-H-M (vertical, horizontal, and moment) loading was validated through comparisons with a model test reported by Zhao et al. [33], which focused on the mechanical behavior of a caisson-bored pile composite anchorage (CBPCA). Figure 8 illustrates the 3D finite element mesh constructed to emulate the model test, with geometric dimensions identical to the experimental setup. Parameters for this numerical model are summarized in Table 8 and Table 9. It is worth noting that specific values for the shaft resistance and base resistance were not measured explicitly in the model test. Consequently, they were set as soil-dependent in PLAXIS 3D. As demonstrated in Figure 9, a satisfactory agreement is achieved between the present numerical estimates and the experimental observations of Zhao et al. [33]. This close correspondence verifies the efficacy of the embedded pile element for analyzing the performance of piles in the PCA.

Figure 8. Three-dimensional finite element mesh used to reproduce the model test.

Table 8. Properties of the CBPCA in the model test.

Properties	Young’s Modulus E (GPa)	Density ρ (kg/m3)	Poisson’s Ratio ν	Diameter (m)	Length (m)
Caisson cover, caisson framework, caisson bottom, counterweight plate	210.0	7850	0.3	–	–
Pile	3.0	1160	0.2	0.03	0.45

Table 9. HS model parameters of the soil in the model test.

Soil	γ	e	Es	${\mathit{E}}_{\mathit{oed}}^{\mathit{ref}}$	${\mathit{E}}_{50}^{\mathit{ref}}$	${\mathit{E}}_{\mathit{ur}}^{\mathit{ref}}$	${\mathit{E}}_{\mathit{s}}:{\mathit{E}}_{\mathit{oed}}^{\mathit{ref}}:{\mathit{E}}_{50}^{\mathit{ref}}:{\mathit{E}}_{\mathit{ur}}^{\mathit{ref}}$	c′	φ′	ψ	m
	(kN/m3)		(MPa)	(MPa)	(MPa)	(MPa)		(kPa)	(°)
Sand	15.6	0.718	10.2	10.2	10.2	51.0	1:1:1:5	0	34.5	4.5	0.5

Figure 9. Comparison between the present simulation results and the measured data from the model test reported by Zhao et al.: (a) displacement of Monitoring Point A; (b) displacement of Monitoring Point C; and (c) bending moment of corner pile P_c.

3.5. Numerical Results

Under Loading Case 10, the TSP of the caisson anchorage exhibits lateral and vertical displacements of −42.29 mm and −62.82 mm, respectively (Figure 10), whereas those for the PCA are −63.41 mm and −80.67 mm. Accordingly, both anchorage foundations strictly satisfy the allowable design limitations of 80 mm (lateral) and 160 mm (vertical) stipulated by the specification [42]. Note that all negative signs in this study denote direction rather than magnitude.

Figure 10. Displacements of the TSP for the caisson anchorage and the PCA: (a) lateral displacement; and (b) vertical displacement.

To accurately capture the ultimate bearing capacity, the loading steps near failure were refined from 0.5F_max to 0.1F_max. Starting from Loading Case 10 (i.e., 1.0F_max), the main cable tension was increased in increments of 0.5F_max up to 2.0F_max, and subsequently in fine increments of 0.1F_max, as illustrated in Figure 10. Furthermore, the soil failure criteria have been strictly defined using numerical convergence indicators and displacement characteristics: numerical soil collapse was identified when PLAXIS 3D post-processing explicitly issued the warning “Soil seems to collapse. Check output” and was cross-verified by a sudden acceleration or anomalous behavior in the displacement path. Accordingly, the ultimate bearing capacity is determined to occur at F = 2.3F_max for the conventional caisson anchorage and F = 2.1F_max for the PCA. Since the current PLAXIS 3D model employs linear elastic embedded pile elements, back-calculation demonstrates that at the ultimate bearing capacity level of F = 2.1F_max, the calculated maximum stress at the pile head reaches 429.46 MPa. This confirms that when high-strength Q490 steel piles (with a yield strength of 490 MPa) are utilized, the pile head stress remains below the yield limit at this ultimate bearing capacity (2.1F_max), thereby verifying that premature structural yielding is successfully prevented.

In addition, by reducing the caisson height by 20 m (from 67.5 m to 47.5 m), the PCA significantly reduces construction costs and deep foundation operations. Although incorporating 2391.08 tons of high-strength Q490 steel pipe piles incurs an estimated cost of 10.28–11.60 million RMB, this expense is effectively offset by the major reductions in C30/C20 concrete (66,800 m³ in total) and Q235 steel. Consequently, the net direct cost savings range between 13.34 and 15.90 million RMB, alongside a shortened construction schedule for caisson sinking. In summary, although the PCA exhibits slightly larger displacements under Loading Case 10 and a marginally lower ultimate bearing capacity (by 0.2 F_max) than the conventional caisson anchorage, it demonstrates superior cost-effectiveness, proving to be a highly promising novel anchorage foundation.

As illustrated in Figure 3, the design coordinates of the theoretical splay saddle point in the three-dimensional coordinate system are (±6.73 m, 20.8 m, 38.0 m), serving as the displacement datum. Prior to the application of the main cable tension (i.e., at Loading Case 0), the caisson, steel pipe piles, bottom sealing concrete, water, C20 concrete, top slab, splay saddle pier, and anchorage block were sequentially activated. The non-uniform counterweight distribution of the PCA resulted in significant lateral displacements of the TSP, which were primarily driven by the eccentric self-weight of the anchorage block and C20 concrete. With the progressive application of the main cable tension, the load distribution on the PCA gradually balanced out until the completion of Loading Case 10. Therefore, Loading Case 10 represents the final load state of the PCA. Furthermore, all displacements of the TSP evaluated under Loading Case 10 to satisfy the specification requirements in this study refer specifically to the displacement datum (±6.73 m, 20.8 m, 38.0 m), and displacements were not reset at any construction stage.

In contrast to the caisson anchorage, the incorporation of piles within the PCA plays a crucial role in ensuring its overall stability. Due to the transverse symmetry of the PCA pile arrangement, detailed analyses are conducted on the 18 representative piles (Rows 1–3, Columns 1–6) shown in Figure 4b. The symbol P_mn denotes the pile located at Row m, Column n. Each individual pile is subjected to the combined V-H-M loading resulting from the self-weight of the superstructure and the tension in the main cables.

Figure 11 and Figure 12 show lateral displacements along pile shafts in the longitudinal and transverse directions (Loading Case 10), respectively. The pile heads are embedded in the bottom sealing concrete of the caisson, resulting in uniform displacements at the pile heads, whereas the shaft displacements vary along the depth. In the longitudinal direction (Figure 11), for Rows 1 to 3, the displacements of piles in Column 6 (4.25 mm at pile tip) are generally lower than those observed in Columns 1–5 (4.72–8.75 mm at pile tip). In the transverse direction (Figure 12), for Columns 1–5, the displacements of piles in Row 1 (4.72–7.82 mm at pile tip) are moderately smaller than those in Rows 2 and 3 (7.55–15.5 mm at pile tip). By contrast, in Column 6, the three piles (4.25 mm for P₁₆, 4.91 mm for P₂₆, and 2.91 mm for P₃₆ at pile tip) display comparable lateral displacements, with pile P₃₆ exhibiting the minimum displacement, which is attributable to the combined V-H-M loading on the pile heads.

Figure 11. Lateral displacements along pile shafts in the longitudinal direction (Loading Case 10): (a) Row 1; (b) Row 2; and (c) Row 3.

Figure 12. Lateral displacements along pile shafts in the transverse direction (Loading Case 10): (a) Column 1; (b) Column 2; (c) Column 3; (d) Column 4; (e) Column 5; and (f) Column 6.

Figure 13 and Figure 14 present axial forces along pile shafts in the longitudinal and transverse directions (Loading Case 10), respectively. As shown in Figure 13, the axial force along pile shafts in Columns 1–5 decreases with depth, suggesting that the upper shaft resistance is predominantly mobilized. In contrast, for Column 6, the axial force increases with depth, indicating the development of negative skin friction along the lower pile shaft. This is attributed to the fact that, as the main cable tension increases from Loading Case 1 to 10, the rotation of the caisson causes the piles in Column 6 to move upward, as shown in Figure 15.

Figure 13. Axial forces along pile shafts in the longitudinal direction (Loading Case 10): (a) Row 1; (b) Row 2; and (c) Row 3.

Figure 14. Axial forces along pile shafts in the transverse direction (Loading Case 10): (a) Column 1; (b) Column 2; (c) Column 3; (d) Column 4; (e) Column 5; and (f) Column 6.

Figure 15. Vertical displacements of piles in Column 6 and the surrounding soil: (a) P₁₆; (b) P₂₆; and (c) P₃₆.

In addition, in the longitudinal direction (Figure 13), for Rows 1–3, the axial forces along pile shafts in Column 1 are considerably larger than those in Columns 2–6. In the transverse direction (Figure 14), for Columns 1–5, the axial forces along pile shafts in Row 1 are significantly larger than those in Rows 2 and 3. However, for Column 6, the axial force near the pile tip of P₁₆ is markedly higher than that of P₂₆ or P₃₆ (Figure 14f). Ultimately, as the main cable tension increases from Loading Case 1 to 10, the overall load redistribution induced by caisson rotation and pile group effects leads to pile P₁₁ carrying the maximum axial force among all piles.

Figure 16 and Figure 17 illustrate bending moments along pile shafts in the longitudinal and transverse directions (Loading Case 10), respectively. In the longitudinal direction (Figure 16), for Rows 1 to 3, the bending moments along pile shafts in Column 1 slightly exceed those in Columns 2–6. In the transverse direction (Figure 17), for Columns 1–5, the bending moments along pile shafts in Row 1 are marginally larger than those in Rows 2 and 3. In contrast, for Column 6, the bending moments in the three piles (P₁₆, P₂₆, and P₃₆) are comparable (Figure 17f). Consequently, P₁₁ experiences the maximum bending moment.

Figure 16. Bending moments along pile shafts in the longitudinal direction (Loading Case 10): (a) Row 1; (b) Row 2; and (c) Row 3.

Figure 17. Bending moments along pile shafts in the transverse direction (Loading Case 10): (a) Column 1; (b) Column 2; (c) Column 3; (d) Column 4; (e) Column 5; and (f) Column 6.

4. Parameter Analysis

The caisson height (H_c), pile diameter (D_p), pile length (L_p), and pile wall thickness (t_p) are important parameters that affect the displacement and mechanical response of the PCA, and their effects were parametrically analyzed based on the reference case, whose geometry and soil parameters are detailed across Section 2 and Section 3.1, Section 3.2 and Section 3.3. The key geometric values are H_c = 47.5 m, D_p = 3 m, L_p = 30 m, t_p = 0.03635 m. A total of 24 cases are considered in this section, with pile P₁₁ selected for detailed analysis as it carries the maximum axial force and bending moment among all piles (see Section 3.5). It is worth noting that among all simulated cases in this study, pile P₁₁ remains the critical pile.

4.1. Influence of Caisson Height

To assess the influence of the caisson height H_c on the responses of the PCA, six caisson heights were considered: H_c = 27.5, 32.5, 37.5, 42.5, 47.5, and 52.5 m. An increase in the caisson height causes the PCA to interact with different soil layers. To eliminate the effect of variations in the caisson height on the soil layers surrounding the PCA, the soil in this section was set entirely as cobble, with all other parameters kept consistent with the reference case.

Figure 18 presents the TSP displacements for the PCA with varying caisson heights. The lateral displacements vary considerably under different loading cases (Figure 18a), whereas the vertical displacements exhibit little variation (Figure 18b). Figure 18c shows the absolute difference in TSP displacement (|ΔTSP|) from Loading Case 0 to Case 10. As the H_c increases from 27.5 m to 52.5 m, the corresponding incremental reduction rates of |ΔTSP| (i.e., the reduction relative to the preceding case) for lateral displacement are 8.54%, 7.52%, 6.72%, 5.06%, and 6.97%, while those for vertical displacement are 17.91%, 15.73%, 13.81%, 11.76%, and 11.91%. A smaller |ΔTSP| suggests that the stability of the PCA increases significantly with H_c. This is primarily attributed to the superior capacity of the caisson to resist the main cable tension compared with the steel pipe piles within the PCA. Nevertheless, a larger H_c leads to higher construction costs. Therefore, a reasonable balance between safety and economic efficiency should be rigorously evaluated in engineering practice.

Figure 18. Displacements of the TSP for the PCA with varying caisson heights: (a) lateral displacement; (b) vertical displacement; and (c) |ΔTSP| from Loading Case 0 to Case 10.

Figure 19 illustrates mechanical responses along pile P₁₁ for the PCA with varying caisson heights (Loading Case 10). As H_c increases from 27.5 m to 52.5 m, the maximum lateral displacement at the pile head decreases significantly from 34.63 mm to 17.83 mm (a total decrease of 48.51%) (Figure 19a), while the axial force at the pile head reduces from 49.57 MN to 34.3 MN (a total decrease of 30.73%) (Figure 19b). Meanwhile, the maximum bending moment along the pile decreases from 22.05 MN·m to 8.84 MN·m (a total decrease of 59.91%) (Figure 19c). These results indicate that the mechanical responses of the pile are markedly reduced with increasing H_c. Accordingly, increasing the caisson height can effectively enhance the load-sharing ratio of the caisson.

Figure 19. Mechanical responses along pile P₁₁ for the PCA with varying caisson heights (Loading Case 10): (a) lateral displacement; (b) axial force; and (c) bending moment.

4.2. Influence of Pile Diameter

To evaluate the effect of pile diameter on the response of the PCA, analyses were performed for D_p = 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0 m, with corresponding wall thicknesses t_p of 0.03135, 0.03635, 0.04135, 0.04635, 0.05135, and 0.05635 m, respectively, while all other parameters were maintained identical to those of the reference case. The wall thickness t_p was determined in accordance with Designing, and Constructing Fixed Offshore Platforms-Working Stress Design (API Recommended Practice 2A-WSD) [43] as follows:

$$t_p=0.00635+{\displaystyle \frac{D_p}{100}}(m)$$

(3)

Figure 20 presents the TSP displacements for the PCA with varying pile diameters. As D_p increases from 2.5 m to 5.0 m, the corresponding incremental rates of reduction in |ΔTSP| for lateral displacement are 2.07%, 2.18%, 1.97%, 1.76%, and 1.81%, while those for vertical displacement are 0.80%, 1.81%, 2.30%, 2.30%, and 3.53%. This suggests that increasing the pile diameter has a marginal effect on the overall displacement of the PCA.

Figure 20. Displacements of the TSP for the PCA with varying pile diameters: (a) lateral displacement; (b) vertical displacement; and (c) |ΔTSP| from Loading Case 0 to Case 10.

Figure 21 shows the mechanical response along pile P₁₁ for the PCA with varying pile diameters (Loading Case 10). With an increase in D_p from 2.5 m to 5.0 m, the maximum lateral displacement at the pile head reduces from 65.85 mm to 48.22 mm (a total decrease of 26.78%) (Figure 21a). Meanwhile, the maximum axial force along the pile increases from 46.60 MN to 67.47 MN (a total increase of 44.79%) (Figure 21b), while the maximum bending moment rises from 31.20 MN·m to 87.71 MN·m (a total increase of 181.09%) (Figure 21c). These findings demonstrate that D_p has a notable effect on the mechanical responses of the pile for the PCA. Specifically, increasing the pile diameter enhances the axial and flexural rigidity of the pile shaft, thereby attracting greater axial forces and bending moments while efficiently decreasing lateral displacement. Furthermore, both the maximum axial force and bending moment of pile P₁₁ occur at the pile head, yielding maximum combined stresses of 400.35, 331.86, 272.30, 226.24, 191.62, and 161.18 MPa for D_p = 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0 m, respectively, representing a total reduction of 59.74%. Assuming a steel yield strength of 490 MPa for the pile, the cross-sectional utilization ratios remain below 1.0 (specifically 0.68 for D_p = 3.0 m). Therefore, increasing the pile diameter effectively reduces the maximum combined stress at the critical section, thereby enhancing its structural safety.

Figure 21. Mechanical responses along pile P₁₁ for the PCA with varying pile diameters (Loading Case 10): (a) lateral displacement; (b) axial force; and (c) bending moment.

4.3. Influence of Pile Length

To examine the influence of pile length on the response of the PCA, the considered L_p were 30, 33, 36, 39, 42, and 45 m. Figure 22 depicts the TSP displacements for the PCA with varying pile lengths. When L_p increases from 30 m to 45 m, the corresponding incremental rates of reduction in |ΔTSP| for lateral displacement are 2.18%, 2.39%, 1.67%, 1.84%, and 1.83%, while those for vertical displacement are 5.87%, 6.45%, 4.21%, 5.95%, and 6.13%. These results reveal that increasing pile length is significantly more effective in mitigating vertical displacement than restraining lateral displacement at the TSP.

Figure 22. Displacements of the TSP for the PCA with varying pile lengths: (a) lateral displacement; (b) vertical displacement; and (c) |ΔTSP| from Loading Case 0 to Case 10.

Figure 23 illustrates the mechanical response along pile P₁₁ for the PCA with varying pile lengths (Loading Case 10). When L_p increases from 30 m to 45 m, the maximum lateral displacement at the pile head drops from 60.43 mm to 57.46 mm (a total decrease of 4.91%) (Figure 23a). Conversely, the maximum axial force along the pile increases from 53.18 MN to 62.29 MN (a total increase of 17.13%) (Figure 23b). However, the maximum bending moment decreases from 43.32 MN·m to 38.82 MN·m (a total decrease of 10.39%) (Figure 23c). This phenomenon suggests that, under the combined V-H-M loading on the pile heads, increasing the pile length mobilizes additional shaft resistance from deeper soil strata. This mobilization alters the internal force distribution of the piles, leading to an increase in the axial force while reducing the bending moment along the pile shaft. Consequently, the lateral displacement of the piles is restrained, thereby improving the stability of the PCA.

Figure 23. Mechanical responses along pile P₁₁ for the PCA with varying pile lengths (Loading Case 10): (a) lateral displacement; (b) axial force; and (c) bending moment.

4.4. Influence of Pile Wall Thickness

To investigate the effect of pile wall thickness on the responses of the PCA, cases with t_p = 36.35, 41.35, 46.35, 51.35, 56.35, and 61.35 mm were considered. Figure 24 shows the TSP displacements for the PCA with varying pile thicknesses. With t_p increasing from 36.35 mm to 61.35 mm, the corresponding incremental rates of reduction in |ΔTSP| for lateral displacement are 0.80%, 0.65%, 0.59%, 0.48%, and 0.41%, while those for vertical displacement are 0.30%, 0.10%, 0.10%, 0%, and 0%. This indicates that the pile wall thickness has a negligible influence on the displacement of the PCA.

Figure 24. Displacements of the TSP for the PCA with varying pile thicknesses: (a) lateral displacement; (b) vertical displacement; and (c) |ΔTSP| from Loading Case 0 to Case 10.

Figure 25 depicts the mechanical response along pile P₁₁ for the PCA with varying pile thicknesses (Loading Case 10). As t_p increases from 36.35 mm to 61.35 mm, the maximum lateral displacement at the pile head decreases from 60.43 mm to 57.57 mm (a total decrease of 4.73%) (Figure 25a). Meanwhile, the maximum axial force along the pile increases from 53.18 MN to 57.99 MN (a total increase of 9.05%) (Figure 25b), and the maximum bending moment increases markedly from 43.32 MN·m to 57.93 MN·m (a total increase of 33.73%) (Figure 25c). These results suggest that increasing the pile wall thickness substantially enhances the flexural rigidity of the pile shaft, thereby attracting higher internal forces, whereas its contribution to restraining lateral displacement remains marginal.

Figure 25. Mechanical responses along pile P₁₁ for the PCA with varying pile thicknesses (Loading Case 10): (a) lateral displacement; (b) axial force; and (c) bending moment.

Furthermore, both the maximum axial force and bending moment of pile P₁₁ occur at the pile head, yielding maximum combined stresses of 331.86, 308.20, 288.50, 271.49, 256.66, and 243.76 MPa for t_p = 36.35, 41.35, 46.35, 51.35, 56.35, and 61.35 mm, respectively, representing a total reduction of 26.55%. Assuming a steel yield strength of 490 MPa for the pile, for cases with t_p ≥ 36.35 mm, the cross-sectional utilization ratios remain below 1.0 (specifically 0.68 for t_p = 36.35 mm). Therefore, the pile wall thicknesses analyzed in this section provide adequate structural safety, and increasing the pile wall thickness effectively reduces the maximum combined stress at the critical section, thereby enhancing its structural safety.

5. Conclusions

In this study, three-dimensional numerical simulations were performed for both the caisson anchorage and the PCA to compare their displacement responses under the main cable tension. Subsequently, the mechanical responses of the steel pipe piles within the PCA were analyzed in detail, focusing on lateral displacements, axial forces, and bending moments. Finally, a parametric study was carried out to quantify the influence of key parameters on the displacements and mechanical responses of the PCA. The main conclusions drawn for the PCA under Loading Case 10 are summarized as follows:

(1) The TSP of the caisson anchorage exhibits lateral and vertical displacements of 42.29 mm and 62.82 mm, whereas the PCA records corresponding values of 63.41 mm and 80.67 mm, respectively. The displacement responses of both the caisson anchorage and the PCA strictly satisfy the code-specified limits (80 mm laterally, 160 mm vertically). Furthermore, the ultimate bearing capacity is 2.3F_max for the conventional caisson anchorage and 2.1F_max for the PCA. Coupled with its superior cost-effectiveness, the PCA demonstrates strong potential as a novel foundation type.

(2) The piles within the PCA exhibit pronounced spatial non-uniformity in displacement and internal force distribution under the combined V-H-M loading resulting from the superstructure self-weight and the main cable tension. While pile heads undergo uniform movements, shaft lateral displacements vary nonlinearly with depth. Axial forces in Columns 1–5 decrease rapidly with depth, indicating that shaft resistance is predominantly mobilized along the upper shafts. Conversely, the rotation of the caisson causes the piles in Column 6 to move upward, thereby inducing negative skin friction along their lower shafts. Additionally, pile P₁₁ experiences the peak axial force and bending moment, serving as the critical design pile.

(3) In a homogeneous cobble stratum, increasing H_c significantly reduces both the displacement of the PCA and the mechanical responses along the piles, enhancing the PCA’s stability. Nevertheless, H_c should be optimized to achieve a balance between structural safety and economic efficiency.

(4) Increasing either D_p or L_p contributes to the stability of the PCA, albeit through distinct mechanisms. Increasing D_p increases flexural rigidity, thereby inducing larger axial forces and bending moments while reducing lateral pile displacements. Conversely, increasing L_p effectively mitigates vertical displacements of the PCA by mobilizing additional shaft resistance from deeper strata, which in turn generates a reverse bending moment that restricts the lateral displacement along the pile shaft. In contrast, while t_p has a negligible impact on the displacements of the PCA, it effectively reduces peak combined stresses at critical sections to enhance structural safety.