Study on Foundation Constraint Modeling of a Sea-Crossing Cable-Stayed Bridge Under Combined Wind–Wave Actions

Abstract

Foundation constraints are commonly defined according to the deformation characteristics of the supporting system; however, structural deformation is also strongly affected by external loads. Compared with inland bridges, sea-crossing bridges experience much larger horizontal loads under combined wind–wave actions, and whether foundations in hard-soil conditions can be simplified as rigidly fixed still requires verification. In this study, the m-method is used to determine the equivalent spring stiffness of each soil layer from soil parameters, and a spring-based soil–foundation interaction model is established. This spring-based model is taken as the reference to evaluate the applicability of the rigidly fixed foundation assumption. Using the Qiongzhou Strait highway–railway combined cable-stayed bridge as the engineering background, both rigidly fixed and spring-based foundation models are developed to simulate foundation constraints. The dynamic responses of a single bridge tower and of the entire bridge system under combined wind–wave loading are computed. The influences of foundation constraints on tower-top displacement, foundation reaction forces, and bending moments are investigated. The maximum discrepancy between the two approaches reaches 7.83%, providing a rational basis for selecting foundation constraint conditions in dynamic analysis and design of sea-crossing bridges.

1. Introduction

Bridge boundary conditions are considered to play a decisive role in determining the constraint stiffness of the structure, and the structural response—particularly the dynamic behavior—is consequently influenced. In engineering practice, bridge foundations are generally characterized by relatively high stiffness, large embedment depth, and small structural deformation. Therefore, the restraining effect of the surrounding soil on the structure is often simplified as a rigidly fixed boundary [1,2] to facilitate analysis. When the stiffness of the supporting soil is sufficiently large, the interaction between the foundation and the surrounding soil can be reasonably represented by the rigidly fixed model, and reliable predictions of the structural dynamic response under external loads can thereby be obtained. However, when the foundation is embedded in relatively soft soil layers [3], non-negligible computational errors may be introduced by such a simplified assumption.

To more reasonably account for the influence of soil–foundation interaction on the response of the superstructure, several foundation constraint models have been proposed in previous studies, mainly including the rigidly fixed model, the equivalent fixed model, and the spring model. Among these, the rigidly fixed model [4,5,6,7] is established based on the assumption that the stiffness of the soil is much greater than that of the superstructure, and the foundation is therefore treated as completely fixed. This approach is characterized by simple modeling and high computational efficiency and is considered suitable for bridge structures in which soil deformation can be neglected. Under such circumstances—for example, when bridges are mainly subjected to quasi-static loads such as gravity, vehicle loads, and wind loads—the overall stiffness characteristics of the structural system can be reasonably represented by the rigid assumption.

By contrast, in the equivalent fixed model [8,9,10,11], the restraining effect of the soil is simulated by shifting the fixed depth downward. As a compromise between the rigid model and the elastic model, relatively high computational efficiency can be maintained while soil deformation effects are reflected to a certain extent. Therefore, this model is commonly applied to bridges founded on soils with moderate stiffness or to structures subjected to periodic wind loads or minor lateral disturbances. Furthermore, in the spring constraint model [12,13,14,15], horizontal, vertical, and rotational springs are introduced at the foundation base so that pile–soil interaction and its potential nonlinear characteristics can be simulated more accurately. Such models are particularly suitable for bridges founded on soft soil layers or for sea-crossing bridges subjected to the combined actions of wind, waves, and ocean currents [16,17,18,19].

Unlike inland bridges, which are mainly subjected to conventional environmental loads such as gravity, traffic, and wind, sea-crossing bridges are not only exposed to stronger wind actions but are also subjected to significant dynamic effects induced by waves and ocean currents [20,21]. To ensure the overall stability and safety of the pile–soil system, larger foundation embedment depths are usually adopted in the design of such bridges. Under these complex marine environmental conditions—characterized by large horizontal loads, deep foundations, and pronounced soil–structure interaction—the deformation effects of soft soils are often more significant. Numerous studies have indicated that, for sea-crossing bridges, soil–foundation interaction generally needs to be described using equivalent fixed models or spring models rather than a simple rigidly fixed assumption [22,23,24,25,26]. However, existing studies on foundation modeling for sea-crossing bridges have mainly focused on pile–soil interaction problems under soft soil conditions, with emphasis placed on the applicability of equivalent fixed models and spring models in structural dynamic response analyses. In contrast, relatively limited attention has been paid to the modeling of foundation constraints for sea-crossing bridges founded in relatively stiff or hard soil layers. For bridges founded on hard soils, although large foundation embedment depths and significant horizontal loads may still exist, if the rigidly fixed model can accurately predict the structural dynamic response, the analysis and design process can be greatly simplified and computational efficiency can be improved. Therefore, the applicability of the rigid boundary assumption for sea-crossing bridge foundations under hard soil conditions needs to be systematically evaluated.

Based on the above background, the Qiongzhou Strait rail–road dual-use cable-stayed bridge is taken as an engineering example in this study to investigate the influence of foundation constraint models on the dynamic response of sea-crossing bridges founded on hard soils. First, the equivalent spring stiffness of each soil layer is derived from geotechnical parameters using the m-method, and a spring-type soil–foundation interaction model is thereby established. Meanwhile, a rigidly fixed model is constructed to represent the condition of a completely fixed foundation. On this basis, both a single-tower model and a full-bridge model are established, and dynamic response analyses under combined wind–wave actions are carried out. By comparing the calculation results obtained from the two foundation constraint models and analyzing the underlying mechanisms, the applicability of the rigid boundary assumption under hard soil conditions is evaluated. The results of this study are expected to provide a theoretical basis and engineering reference for the rational selection of foundation constraint models in the dynamic analysis of sea-crossing bridges.

2. Simulation of the Interaction Between Circular End-Shaped Bridge Foundation and Foundation Soil

Foundation Restraint Model

In bridge dynamic analysis, the choice of foundation constraint model directly determines the appropriateness of the structural boundary conditions and the reliability of the computed results. The Code for Design of Highway Bridges and Culverts-Foundations (JTG/TD63-2021) [27] sets out principled requirements for the interaction between bridge foundations and supporting soils. In accordance with the relevant codes and engineering practice, two representative constraint models are adopted for comparative analysis in this study: the rigid embedment model and a multi-spring model based on the m-method [28].

(1)

Rigidly embedded restraint model

This method assumes a rigid connection between the bridge foundation and the soil, where all degrees of freedom at the foundation nodes on the ground surface are fully constrained, equivalent to treating the soil as an undeformable rigid body. This approach greatly simplifies the calculation and is considered reasonable when the foundation embedment is deep, the structural stiffness is large, and soil deformations are relatively small.

(2)

Multi-spring model based on the m-method

To more accurately simulate soil–structure interaction under wind and wave loads, the widely used “m-method” [29] is introduced. This method is founded on the Winkler elastic foundation model, in which the soil is represented by a series of spring elements whose stiffness varies with depth. In this way, while maintaining computational simplicity, the method effectively reflects the stratification characteristics of the soil and the distribution of horizontal resistance.

According to the relevant Standard [27,30], the following assumptions are made regarding the interaction between the bridge foundation and the soil:

• The interaction between the foundation and the soil is assumed to be linear elastic, and spring elements with stiffness varying linearly with depth are used to simulate the interaction.
• Within the same soil layer, the equivalent spring stiffness values are assumed to be equal and constant.
• Friction between the foundation and the soil layers is neglected.

The equivalent stiffness of the multi-spring model is calculated as follows:

$${K_z=b}_{1}mz{h}_z$$

(1)

$${b_1=kk}_f\left(d+1\right)$$

(2)

$$k_f=\left(1-0.1{\displaystyle \frac{a}{d}}\right)$$

(3)

In Equations (1)–(3), $K_z$ represents the spring stiffness at depth $z$; $b_1$ is the effective foundation width for calculation; $m$ is the subgrade reaction coefficient, the value of which is determined with reference to [30]; $h_z$ is the thickness of the soil layer at depth $z$; $k$ is the pile interaction coefficient, which is taken as $k=1$ when the interaction parallel to the horizontal load direction is considered; $k_f$ is the shape conversion factor for round-ended foundations; $d$ is the pile diameter or the foundation width perpendicular to the horizontal load direction; and $a$ is the round-end diameter.

3. Comparative Analysis of the Effects of Foundation Restraint Models on the Structural Dynamic Effects of Cross-Sea Bridges

3.1. Introduction of Actual Engineering

A large-scale highway–railway cable-stayed bridge spanning the sea is the focus of this study. The bridge comprises two towers, three cable planes, and three spans, as illustrated in Figure 1. The total length of the cable-stayed bridge is 2296 m, and the main span is 1120 m. The span between each side pier and its adjacent intermediate pier is 140 m, while the span between each intermediate pier and the adjacent main tower is 448 m. Cable stays are symmetrically arranged on both sides of the main towers at a spacing of 14 m. A total of 444 cable stays are used throughout the bridge.

Two intermediate piers and two end piers are provided along the bridge. The side and intermediate piers are twin-column piers with V-shaped gravity foundations, whereas the main towers have diamond-shaped cross sections and are founded on circular caisson foundations. The height from the top of each main tower to the top of its foundation is 389 m. The circular caisson foundation has plan dimensions of 80 m by 60 m and is embedded 40 m into the seabed; the water depth at the site is 50 m. Figure 2 shows the overall layout of the cross-sea cable-stayed bridge.

The main girder adopts a two-tier structure with highway and railway decks arranged on separate levels Figure 3. The upper deck serves as a two-way, six-lane highway deck, whereas the lower deck serves as a two-way, two-track railway deck. The cross section of the main girder is 16 m high and 35 m wide. Six bearings are installed at the top of each pier to support the main girder.

3.2. Wind and Wave Design Parameters

According to the hydrological and meteorological conditions near the Qiongzhou Strait Bridge [31], the designed wind parameter corresponds to a 100-year return period, with a 10-min mean reference wind speed of 44.2 m/s at a height of 10 m. In the present study, the wind velocity is decomposed into a mean component and a fluctuating component. Based on the reference mean wind speed, a stochastic fluctuating wind field was simulated, and the instantaneous wind velocity time histories were obtained by superimposing the fluctuating component onto the mean wind component. The resulting wind velocity time series were subsequently used to derive the wind load histories for the dynamic analysis.

3.3. Finite Element Modeling and Verification of Bridge Structures

3.3.1. Finite Element Model of Bridge Structure

In this study, a finite element (FE) model of the cable-stayed bridge was established using the general-purpose FE software ANSYS 2022 R2, as shown in Figure 4. The FE model of the entire bridge includes the main girder, bridge towers, piers, stay cables, and bearings, and is composed of 1326 elements and 881 nodes. The main girder, bridge towers, and foundations are modeled using BEAM188 beam elements. BEAM188 is defined as a two-node, three-dimensional linear beam element suitable for the analysis of slender beam structures. Both linear analyses and nonlinear analyses involving large deflections and large strains can be effectively performed using this element [32]. In addition, predefined cross-sectional types provided in the element library can be selected, and arbitrary cross-sectional shapes can also be defined through user-specified section types and sectional parameters, providing considerable flexibility in structural modeling.

The main girder is simplified using an equivalent fishbone beam model, in which a central longitudinal girder and 165 rigid transverse fishbone beams are included, with a spacing of 14 m. Owing to the complex cross-sectional configuration of the main girder, its sectional properties cannot be directly determined. Therefore, a cantilever beam model is established, and loads are applied at the free end so that the equivalent sectional area, bending stiffness, and torsional stiffness of the main girder can be evaluated. The mass of the main girder is subsequently determined based on the equivalent sectional area and the material density. The stay cables are modeled using LINK10 elements. To account for the influence of cable sag induced by self-weight on the axial stiffness, the equivalent elastic modulus of the stay cables is calculated using the Ernst formula [18]. The material parameters adopted for the different components of the cable-stayed bridge FE model are listed in

Table 1. Material property table of each part of cable-stayed bridge.

Structural Parts	E (N/m2)	ν	Density (kg/m3)
caisson foundation	3.00 × 1010	0.2	2.5 × 103
Cushion cap	3.00 × 1015	0.2	2.5 × 103
Lower tower	3.55 × 1010	0.2	2.5 × 103
Cross beam for tower	3.55 × 1010	0.2	2.5 × 103
Middle tower	3.55 × 1010	0.2	2.5 × 103
Upper tower	3.55 × 1015	0.2	2.1 × 103
main girder	3.55 × 1010	0.2	2.0 × 103
Fishbone crossbeam	2.10 × 1011	0.3	2.6 × 104
Lower gravity foundation	2.10 × 1016	0.3	1.0 × 100
Upper gravity foundation	3.55 × 1010	0.2	2.5 × 103
Pier column for gravity foundation	3.55 × 1010	0.2	2.5 × 103

.

The seabed soil distribution in the Qiongzhou Strait is characterized by an upper layer of sediment primarily consisting of silty clay, clayey fine calcareous silty sand, and gravelly sand, while the underlying strata are mainly composed of silty clay, clayey chalk, chalk, and sandy clay, with local interlayers of clayey sand [33]. In Equation (1), the horizontal soil resistance coefficients are determined using the parameters $m$ and $m_0$ specified for non-rock soils in the General Specifications for Design of Highway Bridges and Culverts. The reference values adopted are based on

Table 2. m value and m₀ value of non-rocky soil.

Soil Type	$\mathit{m}$ and ${\mathit{m}}_0$ (kN/m4)	Soil Type	$\mathit{m}$ and ${\mathit{m}}_0$ (kN/m4)
Flow-plastic clay, IL > 1.0, soft plastic cohesive soil 1.0 ≥ IL > 0.75, mud	3000–5000	Hard or stiff to semi-stiff clay IL ≤ 0, coarse sand; dense silty soil	20,000–30,000
Plastic clay, 0.75 ≥ IL > 0.25, silty clay; marine mud	5000–10,000	Gravel, angular gravel, rounded gravel, crushed stone, cobbles	30,000–80,000
Hard-plastic clay, 0.25 ≥ IL ≥ 0, fine sand, medium sand, medium-dense silty soil	10,000–20,000	Dense gravelly sand; dense pebbles; cobbles	80,000–120,000

[30].

As the boundary restraint conditions at the base of the bridge change, the dynamic characteristics of both the bridge tower and the entire bridge also change accordingly. Therefore, a brief analysis is conducted below on the natural frequencies and mode shapes under the two different restraint conditions, so as to provide a basis for the subsequent calculations and analysis. According to the finite element model, the natural frequencies of the whole bridge can be obtained, as listed in

Table 3. Dynamic characteristics of structures with different constraints at the bottom of the foundation.

Constraint Type	Natural Frequency (Hz)
Mode Order	1st	2nd	3rd	4th
Rigidly Fixed	0.045	0.058	0.127	0.158
Spring	0.045	0.058	0.126	0.150

. The corresponding mode shapes of the whole bridge are longitudinal drift of the main girder, first-order lateral bending of the main girder, first-order vertical bending of the main girder, and first-order symmetric bending of the bridge tower.

3.3.2. Model Constraints and Boundary Conditions

To reasonably reflect the actual stress state of the bridge structure, the connections between the main girder and the towers were modeled as simply supported, that is, six bearings were arranged at the top of each pier to support the girder. The bearings were constrained only in the vertical displacement, while the other degrees of freedom were released to allow for thermal expansion of the girder along the bridge axis, which is consistent with the structural configuration of the main girder in practical cross-sea cable-stayed bridges. For the foundation boundary conditions, both the rigidly embedded model and the multi-spring model based on the “m-method” were employed.

3.3.3. Node Configuration of the m-Method Foundation Spring Model

To establish the soil–foundation interaction model of the bridge tower foundation based on the “m-method,” multiple key nodes were arranged along the depth of the surrounding soil in ANSYS, to which equivalent horizontal spring elements were applied. As shown in

Table 4. The m value and stiffness value of the seabed foundation soil layer.

Node Number	b (m)	m (kN/m4)	z (m)	h (m)	k (kN/m)
2	72.9	3.00 × 104	40	4	3.50 × 108
3	72.9	3.00 × 104	36	4	3.15 × 108
4	72.9	1.00 × 104	32	4	9.33 × 107
5	72.9	1.00 × 104	28	4	8.16 × 107
6	72.9	9.00 × 103	24	4	6.30 × 107
7	72.9	9.00 × 103	20	4	5.25 × 107
8	72.9	7.00 × 103	16	4	3.27 × 107
9	72.9	7.00 × 103	12	4	2.45 × 107
10	72.9	6.00 × 103	8	4	1.40 × 107
11	72.9	5.00 × 103	4	4	5.83 106

, the node numbers correspond to the soil nodes beneath the seabed surface of the bridge tower foundation, distributed from 4 m to 40 m below the seabed with an interval of 4 m. The node numbering increased with depth, consistent with the discretization scheme of the soil layers in the m-method. These nodes were connected to the corresponding nodes on the foundation edges, forming a multi-layer horizontal spring system to simulate the lateral restraint provided by the soil along the depth direction. For each soil layer, spring elements were applied in both the longitudinal and transverse directions of the bridge, thereby establishing a multi-layer spring system. Based on the geotechnical survey data and Equation (1), the equivalent m-values and stiffness parameters of the soil layers were calculated, as summarized in Table 4.

3.3.4. Finite Element Modeling and Validation of the Bridge Structure

To verify the rationality of the finite element modeling approach developed in this study, the results obtained from the FE model of a single bridge tower were compared with the experimental results reported in Ref. [34]. To ensure consistency with the boundary conditions of the physical test model in Ref. [34], the base of the tower was constrained in the same manner, that is, by applying a fixed-end restraint at the seabed surface. The first two natural frequencies of the tower structure obtained from the FE model, together with the corresponding experimental results, are presented in

Table 5. The first–second order mode shapes and natural frequencies of bridge tower structures.

Order	Mode Shape	Calculated Frequency (Hz)	Test Frequency (Hz)
1	First order bending along the bridge	0.099	0.094
2	First order bending of cross-bridge	0.179	0.189

.

To verify the consistency of the finite element model with experimental results under wave loading, the regular wave condition adopted in the test (wave height of 9.2 m and period of 9.9 s) was selected for the numerical simulation. The tower-top displacement obtained from the numerical analysis was compared with the experimental results, as shown in Figure 5. The comparison indicates that the numerical results of this study agree well with the experimental data reported in Ref. [1], further validating the accuracy of the finite element modeling approach for the bridge structure.

3.4. Wind and Wave Load Calculation Method

3.4.1. Wind Load Calculation

Under wind action, the cable-stayed bridge is subjected to static loads induced by the mean wind. The calculation method for the static wind load on the bridge tower is expressed as follows:

$$F_x=0.5\rho U^2{C}_x{S}_x$$

(4)

$$F_y=0.5\rho U^2{C}_y{S}_y$$

(5)

where $F_x$ and $F_y$ are the longitudinal and transverse forces, respectively; $S_x$ and $S_y$ are the projected areas of the bridge tower in the longitudinal and transverse directions, respectively; $C_x$ and $C_y$ are the wind drag coefficients in the longitudinal and transverse directions, respectively; $U$ is the mean wind speed; and $\rho$ is the air density. The drag coefficients of the bridge tower were obtained from wind tunnel tests conducted by Ref. [35].

Under wind loading, the main girder of the cable-stayed bridge is subjected to drag force $F_H$, lift force $F_V$, and torsional moment $M$, which can be calculated as follows:

$$F_H={\displaystyle \frac{1}{2}}{C}_H\rho U^{2}H$$

(6)

$$F_V={\displaystyle \frac{1}{2}}{C}_V\rho U^{2}B$$

(7)

$$M={\displaystyle \frac{1}{2}}{C}_M\rho U^2{B}^2$$

(8)

where $\rho$ is the air density; $U$ is the mean wind speed; $H$ and $B$ are the structural height and width, respectively; and $C_H$, $C_V$, and $C_M$ are the drag, lift, and moment coefficients, respectively. The values of these coefficients were obtained from wind tunnel tests reported by Ref. [36].

The Kaimal wind spectrum [37] was employed to simulate the turbulent wind field. The buffeting forces induced by turbulence were considered for the bridge tower structure. As the vertical stiffness of the tower is relatively large, the vertical wind load was neglected. The longitudinal buffeting force $F_{bx}\left(t\right)$ and the transverse buffeting force $F_{by}\left(t\right)$ were calculated as follows:

$$F_{bx}\left(t\right)={\displaystyle \frac{1}{2}}{C}_x\rho A_x\left[u^2\left(t\right)+2Vu\left(t\right)\right]$$

(9)

$$F_{by}\left(t\right)={\displaystyle \frac{1}{2}}{C}_y\rho A_y\left[u^2\left(t\right)+2Vu\left(t\right)\right]$$

(10)

where $u\left(t\right)$ is the fluctuating wind velocity time history at the node, and $V$ is the mean wind speed at the node.

The self-excited aerodynamic forces acting on the main girder of the cable-stayed bridge can be expressed as follows:

$$L_{ae}={\displaystyle \frac{1}{2}}\rho V^{2}B\left[K{H}_1^{\ast }{\displaystyle \frac{\dot{h}}{V}}+K{H}_2^{\ast }{\displaystyle \frac{B\dot{\phi }}{V}}+K^2{H}_3^{\ast }\phi +K^2{H}_4^{\ast }{\displaystyle \frac{h}{B}}\right]$$

(11)

$$M_{ae}={\displaystyle \frac{1}{2}}\rho V^2{B}^2\left[K{A}_1^{\ast }{\displaystyle \frac{\dot{h}}{V}}+K{A}_2^{\ast }{\displaystyle \frac{B\dot{\phi }}{V}}+K^2{A}_3^{\ast }\phi +K^2{A}_4^{\ast }{\displaystyle \frac{h}{B}}\right]$$

(12)

$$D_{ae}={\displaystyle \frac{1}{2}}\rho V^{2}B\left[K{P}_1^{\ast }{\displaystyle \frac{\dot{p}}{V}}+K{P}_2^{\ast }{\displaystyle \frac{B\dot{\phi }}{V}}+K^2{P}_3^{\ast }\phi +K^2{P}_4^{\ast }{\displaystyle \frac{p}{B}}\right]$$

(13)

where $L_{ae}$, $M_{ae}$, and $D_{ae}$ denote the vertical, torsional, and lateral self-excited forces of the main girder, respectively; $K$ is the reduced frequency defined as $K={\displaystyle \frac{B\omega }{U}}$, where $\omega$ is the structural vibration frequency; and $h$, $\phi$, and $p$ represent the vertical displacement, torsional angle, and lateral displacement of the girder, respectively. $H_i^{\ast }$, $A_i^{\ast }$, and $P_i^{\ast }(i=1\text{–}4)$ are the flutter aerodynamic derivatives of the main girder. The values of $D_{ae}$, $H_i^{\ast }$, and $A_i^{\ast }$ were obtained from wind tunnel tests reported by Ref. [36].

The buffeting forces of the main girder were calculated using the Davenport buffeting force model, as expressed by the following equations:

$$D_b\left(t\right)={\displaystyle \frac{1}{2}}\rho V^{2}B\left[2C_D{\displaystyle \frac{u\left(t\right)}{V}}+C_D^{\prime }{\displaystyle \frac{w\left(t\right)}{V}}\right]$$

(14)

$$L_b\left(t\right)={\displaystyle \frac{1}{2}}\rho V^{2}B\left[2C_L{\displaystyle \frac{u\left(t\right)}{V}}+\left(C_L^{\prime }+C_D\right){\displaystyle \frac{w\left(t\right)}{V}}\right]$$

(15)

$$M_b\left(t\right)={\displaystyle \frac{1}{2}}\rho V^2{B}^2\left[2C_M{\displaystyle \frac{u\left(t\right)}{V}}+C_M^{\prime }{\displaystyle \frac{w\left(t\right)}{V}}\right]$$

(16)

where $D_b$, $L_b$, and $M_b$ are the drag force, lift force, and torsional moment of the main girder, respectively; $B$ is the transverse width of the girder; $C_D$, $C_L$, and $C_M$ are the drag, lift, and moment coefficients, respectively; $C_D^{\prime }$, $C_L^{\prime }$, and $C_M^{\prime }$ are the derivatives of these coefficients with respect to the wind attack angle α; $V$ is the mean wind speed; and $u$ and $w$ represent the longitudinal and vertical fluctuating wind velocity time histories, respectively. The coefficients $C_D$, $C_L$, and $C_M$ were obtained from wind tunnel tests reported by Ref. [36].

3.4.2. Wave Load Calculation

This section introduces the calculation method of wave force time histories on the round-ended foundation, as shown in Figure 6, under both regular and irregular wave conditions. An improved JONSWAP spectrum [38], $S_w\left(\omega \right)$, was employed to compute the wave force spectrum $S_F\left(\omega \right)$ at each node of the foundation elements. Subsequently, the wave force time histories at the two end nodes of the foundation elements were obtained using the harmonic superposition method [39].

Following the simplified solution formulas for wave forces on round-ended caisson foundations derived by Ref. [34], the wave force per unit height of the foundation is calculated as:

$$f_{re}\left(z\right)=C_{mre}{\displaystyle \frac{\gamma \pi D^{2}kH}{8}}{\displaystyle \frac{\mathit{cosh} k z}{\mathit{cosh} k d}}\mathit{cos} \omega t$$

(17)

$$C_{mre}=4{\displaystyle \frac{L^2{f}_A}{{\pi }^3{D}^2}}+0.39{\displaystyle \frac{bL}{D^2}}$$

(18)

$$f_A={\displaystyle \frac{1}{\sqrt{{\left[J_1^{\prime }\left(\frac{\pi D}{L}\right)\right]}^2+{\left[Y_1^{\prime }\left(\frac{\pi D}{L}\right)\right]}^2}}}$$

(19)

By integrating Equation (17) over each foundation element, the concentrated nodal wave force $F_W$ under regular wave excitation is obtained and is expressed as:

$$F_w={\int }_{z_1}^{z_2}{f}_{re}\left(z\right)dz=C_{mre}{\displaystyle \frac{\gamma \pi D^{2}H}{8}}{\displaystyle \frac{(\mathit{sinh} k z_2-\mathit{sinh} k z_1)}{\mathit{cosh} k d}}\mathit{cos} \omega t$$

(20)

Let $z_1$ and $z_2$ denote the elevations of the lower and upper nodes of an element, respectively; $\gamma$ is the unit weight of water; $H$, $L$, $\omega$, and $k$ represent wave height, wavelength, wave frequency, and wave number, respectively; $D$ is the diameter of the caisson foundation; $b$ denotes the length of the rectangular section of the caisson foundation; $J_1^{\prime }$ and $Y_1^{\prime }$ are the derivatives of the first-order Bessel functions of the first and second kinds, respectively; and $C_{mre}$ is the inertia force coefficient.

Under irregular wave action, the wave force spectrum at each node of the foundation is derived, and the transfer function between the wave force spectrum and the incident wave spectrum is formulated as follows:

$$T\left(\omega \right)=C_{mre}{\displaystyle \frac{\gamma \pi D^2}{4}}{\displaystyle \frac{(\mathit{sinh} k z_2-\mathit{sinh} k z_1)}{\mathit{cosh} k d}}$$

(21)

The wave force spectrum at each node of the foundation is obtained as follows:

$$S_F\left(\omega \right)={\left[T\left(\omega \right)\right]}^2{S}_w\left(\omega \right)$$

(22)

The frequency range of the wave spectrum, ${\omega }_L$ to ${\omega }_H$, is divided into $N$ equal intervals, with a frequency increment of $\Delta \omega =\frac{\left({\omega }_H-{\omega }_L\right)}{N}$. The representative frequency within each interval is given by:

$${\omega }_i={\displaystyle \frac{\left[\Delta \omega \left(i-1\right)+\Delta \omega i\right]}{2}}, \left(i=1,\cdots ,N\right)$$

(23)

The irregular wave load at each node of the foundation is obtained by superimposing the cosine waves within the $N$ intervals using the harmonic superposition method:

$$F_{rw}\left(t\right)={\sum }_{i=1}^M\sqrt{2S_F\left({\omega }_i\right)\Delta \omega }\mathit{cos}\left({\omega }_{i}t+{\alpha }_i\right), \left(i=\mathrm{1,2},3,\cdots ,N\right)$$

(24)

By using Equation (24), the wave force time histories at each node of the main tower foundation under irregular wave loading can be determined, providing a basis for the subsequent calculation of the dynamic response of the cable-stayed bridge.

3.5. Comparative Analysis of the Dynamic Response of the Cross-Sea Bridge

Based on the designed wind speed, wind velocity time histories were generated using the harmonic superposition method [40], and the corresponding wind load time histories required for structural response analysis were calculated according to the windward area of the structure. Similarly, the wave surface time histories were generated using the harmonic superposition method based on the design significant wave height and spectral peak period. Combined with the characteristic dimensions of the foundation and the wave force calculation formulas, the wave load time histories required for evaluating structural response were obtained. The resulting wind and wave load time histories were applied to the finite element model of the structure. Using the elasto-plastic time-history analysis capability in ANSYS 2022 R2, the time-history responses of the structural dynamic effects were determined.

3.5.1. Comparative Analysis of the Dynamic Response of Bridge Towers

In analyzing the dynamic response of the bridge towers under wind and wave loading, the longitudinal windward area of the towers is considered, as it is the largest along the bridge axis, and the wave-exposed area is also maximal. Consequently, the corresponding wind and wave loads reach their peak values. Therefore, the wind and wave loads along the bridge axis were selected for the analysis in this study. When calculating and comparing the dynamic response of the towers under different loading conditions, only the responses at critical nodes were considered. Specifically, the shear forces and bending moments at the foundation nodes located on the seabed, as well as the displacements at the tower top nodes, were extracted for comparative analysis.

The wind load time histories at the tower top nodes and the wave load time histories at the nodes on the water surface are presented in Figure 7 and Figure 8, respectively.

Using the elasto-plastic time-history analysis method, the dynamic responses of the bridge towers were calculated under different restraint conditions for three loading scenarios: wave loading alone, wind loading alone, and combined wind and wave loading along the same direction. The results are presented in

Table 6. Structural effects and their error statistics under different restraint conditions under wave loads.

Restraint Type	Top Tower Displacement (m)			Base Shear (N)			Base Moment (N·m)
	Mean	Std	Maximum	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	3.89 × 10−1	1.60 × 10−1	8.09 × 10−1	8.56 × 107	4.61 × 107	2.19 × 108	4.28 × 109	1.92 × 109	9.92 × 109
Spring model	4.08 × 10−1	1.68 × 10−1	8.74 × 10−1	8.61 × 107	4.64 × 107	2.27 × 108	4.31 × 109	1.93 × 109	1.06 × 1010
Error	4.66%	5.76%	7.44%	0.58%	0.65%	3.52%	0.70%	0.52%	6.42%

,

Table 7. Structural effects and their error statistics under different restraint conditions under wind load.

Restraint Type	Top Tower Displacement (m)			Base Shear (N)			Base Moment (N·m)
	Mean	Std	Maximum	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	1.25 × 100	2.18 × 10−1	1.83 × 100	3.60 × 107	6.30 × 106	5.17 × 107	1.01 × 1010	1.77 × 109	1.42 × 1010
Spring model	1.28 × 100	2.22 × 10−1	1.84 × 100	3.61 × 107	6.36 × 106	5.18 × 107	1.03 × 1010	1.79 × 109	1.43 × 1010
Error	2.34%	1.80%	0.54%	0.28%	0.94%	0.19%	1.94%	1.12%	0.70%

and

Table 8. Structural effects and their error statistics under wind and wave loads.

Restraint Type	Top Tower Displacement (m)			Base Shear (N)			Base Moment (N·m)
	Mean	Std	Maximum	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	1.39 × 100	2.39 × 10−1	2.00 × 100	1.13 × 108	4.72 × 107	2.42 × 108	1.25 × 1010	2.41 × 109	1.86 × 1010
Spring model	1.47 × 100	2.45 × 10−1	2.09 × 100	1.14 × 108	4.75 × 107	2.51 × 108	1.27 × 1010	2.48 × 109	1.96 × 1010
Error	5.44%	2.50%	4.31%	0.88%	0.63%	3.59%	1.57%	2.82%	5.10%

, respectively. The dynamic responses of the bridge towers under combined wind and wave loading for different restraint conditions are illustrated in Figure 9. The error values presented in the tables were calculated using the following formula:

Error = |S_p − E_m|/S_p × 100%

(25)

In the equation, $S_p$ denotes the structural response obtained using the spring restraint model, while $E_m$ represents the structural response obtained using the rigidly embedded restraint model.

As shown in Table 6, Table 7 and Table 8 and Figure 9, both the spring-based restraint model and the rigid-fixity model exert certain effects on the structural dynamic responses, though the overall differences remain small and can be considered negligible. The maximum discrepancy in the mean dynamic response is 5.44%, while the maximum discrepancy in peak responses reaches 6.43%. This phenomenon is primarily attributed to the relatively stiff soil conditions at the bridge site, under which the rigid-fixity assumption is generally reasonable. However, cross-sea bridges are subjected to substantial horizontal loads, and even stiff soils undergo noticeable deformation under such loading. The rigid-fixity model is inherently unable to capture these soil deformations, whereas the spring-based model can represent this behavior more accurately. Furthermore, the structural displacement responses under the spring-based restraint are consistently larger than those under rigid fixity. This indicates that simplifying the soil to a completely rigid boundary neglects the portion of displacement contributed by soil deformation, leading to an underestimation of the tower-top displacement response.

From Table 6, Table 7 and Table 8, it can also be observed that the influence of the two restraint models is smallest for the foundation shear force, moderate for the foundation bending moment, and most pronounced for the tower-top displacement. This can be explained as follows: The foundation shear force is primarily governed by the applied horizontal loads and is largely independent of the restraint model; therefore, both models yield nearly identical shear responses. The difference in foundation bending moments is greater than that in shear forces because bending moments are influenced not only by shear but also by the effective lever arm. Different restraint models lead to different deformation patterns, resulting in variations in the location where the same horizontal load is applied, and consequently different moment values. The difference is most significant for tower-top displacement, as this response is jointly controlled by foundation shear, foundation moment, and the boundary restraint conditions. Since the restraint model directly affects the overall structural stiffness—and displacement is highly sensitive to stiffness—tower-top displacement shows the largest discrepancy between the two models. According to Table 6, Table 7 and Table 8, although the shear force induced by waves is 1.38 times larger than that induced by wind, the bending moment produced by wind is 1.39 times greater than that produced by waves. Because the tower behaves primarily as a bending-dominated structure, its top displacement is mainly governed by bending moments. Consequently, the tower-top displacement is dominated by the contribution from wind loads, whereas the influence of wave loads is comparatively minor.

3.5.2. Comparative Analysis of the Dynamic Response of the Entire Bridge

As the windward area of the main girder of the entire bridge is relatively large under wind loading, the corresponding dynamic response is more pronounced. Therefore, this study focuses on the dynamic response of the entire bridge under the combined action of wind and wave loads along the transverse across-bridge direction. When calculating and analyzing the dynamic response of the entire bridge under different restraint conditions and loading scenarios, only the responses at critical nodes were considered. These include the shear forces and bending moments at the bridge foundation nodes on the seabed, as well as the displacements at the tower top nodes and the midspan nodes of the main girder, for the purpose of comparative analysis. The wind load time histories at the tower top nodes, the wind load time histories at the main girder midspan nodes, and the wave load time histories at the nodes on the water surface are presented in Figure 10, Figure 11 and Figure 12, respectively.

Using the elasto-plastic time-history analysis method, the dynamic responses of the entire bridge were calculated under different restraint conditions for three loading scenarios: wave loading alone, wind loading alone, and combined wind and wave loading along the same direction. The results are presented in

Table 9. Load effects and their error statistics under different restraint conditions under wave loads.

Restraint Type	Base Shear (N)			Base Moment (N·m)
	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	6.02 × 107	2.94 × 107	1.26 × 108	2.35 × 109	1.09 × 109	4.85 × 109
Spring model	6.04 × 107	3.02 × 107	1.27 × 108	2.37 × 109	1.11 × 109	4.87 × 109
Error	0.33%	2.65%	0.79%	0.84%	1.80%	0.41%

,

Table 10. Structural responses and their error statistics under wave loads.

Restraint Type	Top Tower Displacement (m)			Mid-Span Displacement (m)
	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	5.10 × 10−2	2.32 × 10−2	9.64 × 10−2	1.87 × 10−2	1.08 × 10−2	4.53 × 10−2
Spring model	5.21 × 10−2	2.36 × 10−2	1.03 × 10−1	1.91 × 10−2	1.12 × 10−2	4.80 × 10−2
Error	2.11%	1.70%	6.41%	2.09%	3.57%	5.63%

,

Table 11. Load effects and their error statistics under different restraint conditions under wind loads.

Restraint Type	Base Shear (N)			Base Moment (N·m)
	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	6.19 × 107	7.99 × 106	8.46 × 107	1.19 × 1010	1.32 × 109	1.59 × 1010
Spring model	6.20 × 107	8.03 × 106	8.49 × 107	1.22 × 1010	1.38 × 109	1.63 × 1010
Error	0.16%	0.50%	0.35%	2.46%	4.35%	2.45%

,

Table 12. Structural responses and their error statistics under wind loads.

Restraint Type	Top Tower Displacement (m)			Mid-Span Displacement (m)
	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	3.63 × 10−1	4.25 × 10−2	4.90 × 10−1	8.14 × 100	1.15 × 100	9.78 × 100
Spring model	3.75 × 10−1	4.32 × 10−2	4.95 × 10−1	8.14 × 100	1.15 × 100	9.89 × 100
Error	3.20%	1.62%	1.01%	0.09%	0.00%	1.11%

,

Table 13. Load effects and their error statistics under wind loads and wave loads.

Restraint Type	Base Shear (N)			Base Moment (N·m)
	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	1.12 × 108	2.78 × 107	1.73 × 108	1.33 × 1010	1.76 × 109	1.76 × 1010
Spring model	1.14 × 108	2.82 × 107	1.73 × 108	1.37 × 1010	1.80 × 109	1.77 × 1010
Error	1.75%	1.42%	0.00%	2.92%	2.22%	0.57%

and

Table 14. Structural responses and their error statistics under wind loads and wave loads.

Restraint Type	Top Tower Displacement (m)			Mid-Span Displacement (m)
	Mean	Std	Maximum	Mean	Std	Maximum
Embedded model	3.65 × 10−1	5.25 × 10−2	5.08 × 10−1	8.13 × 100	1.15 × 100	9.79 × 100
Spring model	3.96 × 10−1	5.52 × 10−2	5.29 × 10−1	8.14 × 100	1.16 × 100	9.89 × 100
Error	7.83%	4.89%	3.97%	0.12%	0.86%	1.01%

. The dynamic responses of the entire bridge under combined wind and wave loading for different restraint conditions are illustrated in Figure 13.

Based on the results presented in Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13 and Table 14 and Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, the dynamic responses of both the bridge tower and the full bridge obtained using the spring-based restraint model and the rigid-fixity model remain generally consistent, although the spring model yields slightly larger response amplitudes. The mean tower-top displacement of the full-bridge system is more sensitive to the choice of restraint model than that of the single-tower analysis, with the maximum discrepancy reaching 7.83%. This difference can largely be attributed to the fact that, compared with the isolated tower, the full-bridge system is subjected to substantially greater wind-induced loads and corresponding bending moments under combined wind–wave action. Specifically, the total wind load acting on the full bridge is approximately 75% higher than that on the single tower. Overall, the spring-based restraint model produces slightly larger dynamic responses than the rigid-fixity model. When horizontal loads become significant, the rigid-fixity model tends to underestimate structural displacements.

From the results in Figure 9 and Figure 13, it can also be observed that the mechanism governing the mid-span displacement of the main girder differs from that of the foundation shear force, foundation bending moment, and tower-top displacement. Figure 13 shows that under wave loading, the influence of the two restraint models on mid-span displacement is minimal, whereas under wind loading, the influence is pronounced for both models. Although the external actions include both wind and wave loads, the static effects generated by wave loads cannot be directly transferred to the girder; they can only be transmitted through the dynamic interactions among the tower, girder, and stay cables. However, the dynamic component of the wave load is not significant, and therefore its contribution to mid-span displacement is negligible. As a result, the mid-span displacement of the girder is governed primarily by wind loading. The secondary restraint system of the girder consists mainly of the tower-girder connections and the stay-cable support, whereas the overall bridge system is dominated by soil–foundation restraints. Due to the fundamental differences between these constraint systems, the girder and the full bridge exhibit distinct response mechanisms under external and internal actions. By contrast, the two restraint models exhibit similar influence patterns on the foundation shear force, foundation bending moment, and tower-top displacement of both the single tower and the full bridge. This consistency arises from the fact that the boundary conditions and load variation characteristics of the tower and the full-bridge system are largely comparable.

These findings indicate that when the foundation soil is relatively stiff, the combined wind–wave environment still induces substantial horizontal loads, causing the elastic and rigid-fixity restraint models to produce noticeable differences in the dynamic responses of both the tower and the full-bridge structure. The maximum discrepancy in the mean dynamic response reaches 5.44% for the bridge tower and 7.83% for the full bridge. Therefore, during the preliminary design stage of cross-sea bridges—when the required accuracy is moderate—the soil restraint may be reasonably simplified as a rigidly fixed boundary, as the overall computational deviation remains within 8%, while the modeling and computation efficiency is significantly improved. However, in the detailed design stage where higher accuracy is required, the spring-based restraint model should be adopted to appropriately account for the influence of soil–foundation interaction on the structural response.

4. Conclusions

Taking the Qiongzhou Strait cross-sea highway–railway cable-stayed bridge as the engineering background, this study conducts a time-history dynamic analysis to investigate the structural responses of the bridge under combined wind–wave loading using two different foundation restraint models: the rigid-fixity model and the spring-based model. The main conclusions are as follows:

(1)

The spring-based and rigid-fixity restraint models both exhibit measurable influences on the dynamic responses of the bridge tower and the full-bridge system. The maximum discrepancy in the mean dynamic response reaches 5.44% for the tower and 7.83% for the full bridge. Although the foundation soil at the project site is relatively stiff, the substantial horizontal loads induced by the wind–wave environment necessitate a more realistic representation of the force-deformation behavior at the foundation–superstructure interface. The spring-based model captures this interaction more accurately, whereas the rigid-fixity model neglects the actual soil deformation.

(2)

For both the bridge tower and the full-bridge system, the structural displacement responses obtained under the spring-based restraint are significantly larger than those computed using the rigid-fixity model. This indicates that simplifying the soil restraint as a fully rigid boundary neglects the portion of deformation contributed by soil flexibility, thereby leading to an underestimation of the displacement response at the top of the structure.

(3)

When the foundation soil is relatively stiff, the rigid-fixity model may be adopted during the preliminary design stage of cross-sea bridges to achieve higher computational efficiency while meeting general accuracy requirements. However, in the detailed design stage where higher precision is required, the spring-based restraint model should be employed to simulate soil–foundation interaction. Although this approach involves greater computational complexity, it provides a more realistic representation of foundation forces and deformations, thereby enhancing the reliability of dynamic response analysis and structural safety assessment.