Dynamic analysis of the wind-wave-current-bridge system considering pile-soil interaction

Abstract

Coastal bridges are normally confronted with a complex marine environment including wind, wave and current. Present researches rarely take the current action and pile-soil interaction into consideration. It is thus essential to conduct the dynamic analysis of the cable-stayed bridge under wind-wave-current action considering pile-soil interaction. A coupled wind-wave-current-bridge (WWCB) system is proposed in this study to help understand the bridge dynamic performance. In the WWCB system, the bridge model is established through the finite element method, and the pile-soil interaction is considered through the pile-soil solid contact (PSSC) model. The wave-current action is obtained through a numerical wave tank, while the wind is simulated as the stochastic random process and generated with spectrums. The initial geostress, mass and density distribution are also considered in the WWCB system. To illustrate the bridge dynamic characteristics, dynamic responses under different load combinations are presented. Furthermore, the natural frequency and spectral density function comparison between the consolidation model, PSSC model and soil-spring model are provided to investigate the influence of the pile-soil interaction. The results show that different from previous conclusions, the wave-current action plays the dominant role instead of the wind action in some cases when considering the pile-soil interaction. Additionally, the consideration of the pile-soil interaction would decrease the structural stiffness, resulting in lower natural frequencies and wider frequency distribution. Consequently, the influence of the wave-current action and pile-soil interaction should be noted in the practical engineering.

Keywords

coastal bridges wind-wave-current action dynamic response pile-soil interaction

Introduction

With a growing demand on the transportation in coastal areas, massive sea-crossing bridges have been built, such as Øresund Bridge (Ejermo et al., 2021), Hong Kong-Zhuhai-Macao Bridge (Li et al., 2014) and Yokohama Bay Bridge (Wada et al., 1991). These coastal bridges are normally subjected to the combined action of wind, wave and current.

A great number of previous studies focus on the bridge structure dynamic responses under combined wind and wave action. Some scholars investigated the dynamic responses of the bridge tower under combined wind and wave action experimentally (Guo et al., 2016) and numerically (Fang et al., 2019a). The bridge pylon mechanism under wind-wave action was also studied through frequency domain analysis (Li et al., 2019). Moreover, the numerical floating bridge model (Sha et al., 2018) was established to figure out their structural responses under separate wind and wave action. These studies neglect the influence of the current, while the results considering the current action are more practical. It is thus essential to take the current action into account.

Another concern that this study makes efforts to address is the influence of the pile-soil interaction on the bridge dynamic responses. Several scholars (Fang et al., 2020; Fang et al., 2019b; Meng et al., 2018; Zhu et al., 2018) have conducted studies related to the cable-stayed bridge responses numerically. Zhu et al. (2018) built a cable-stayed bridge model with the finite element method to investigate the bridge dynamics under wind-wave action, where the vehicle was also complemented. Meng et al. (2018) adopted the Frank copula function to integrate wind and wave action, and then applied these actions to a coastal cable-stayed bridge to study its stochastic buffeting response through power spectrum analysis. Fang et al. (2019b, 2020) established a wind-wave-vehicle-bridge model, whose governing equations are solved by Newmark-β method. This process was regarded as a time-consuming issue, where the machine-learning based methods were utilized. Similar conclusions were drawn through these researches that the wind load was the predominate load on the girder displacement responses and the wave load controlled the foundation base shear response. Contrary to these conclusions, the wave load was found to dominate the girder vertical displacement responses in the present study, indicating the necessity of considering the influence of the pile-soil interaction.

It could be concluded that some issues remain in the influence of current and pile-soil interaction on the cable-stayed bridge dynamic responses. The present study aims at tackling these issues. The Wind-Wave-Current-Bridge (WWCB) model established in this study is introduced in WWCB system modeling , including the description of the wind velocity field, regular wave-current field, the dynamic interaction, and WWCB framework. Bridge dynamic responses provides the displacement and internal force response results under different load combinations, to ensure their roles in the combined wind, wave and current action. To investigate the influence of pile-soli interaction, the dynamic characteristics analysis and frequency domain analysis are conducted and compared between the consolidation model, soil-spring model as well as pile-soil solid contact (PSSC) model in influence of pile-soil interaction. The conclusions are drawn in conclusions.

WWCB system modeling

Wind velocity field

The wind velocity field at a point is normally decomposed into the mean wind u_m and fluctuating wind u(t), v(t), w(t). The fluctuating wind components u(t), v(t), w(t) are at lateral, longitudinal and vertical directions, respectively. In the consideration of the cable-stayed bridge model, the longitudinal fluctuating wind at the girder could be neglected. Due to the significantly large dimension along the vertical direction of the tower and foundation, their vertical fluctuating winds could be not taken into consideration. The instantaneous wind velocity at a point of the girder U_g(t), tower U_t(t) and foundation U_f(t) could be thus defined as:

U_{g} (t) = [\begin{array}{l} u (t) \\ w (t) \end{array}] + [\begin{array}{l} u_{m} \\ 0 \end{array}]

(1)

U_{t} (t) = U_{f} (t) = [\begin{array}{l} u (t) \\ v (t) \end{array}] + [\begin{array}{l} u_{m} \\ 0 \end{array}]

(2)

Each one dimensional component of the fluctuating wind is normally described by the cross-spectral density matrix (Deodatis, 1996), presented as:

S^{0} (ω) = [\begin{matrix} S_{11}^{0} (ω) & S_{12}^{0} (ω) & S_{13}^{0} (ω) \\ S_{21}^{0} (ω) & S_{22}^{0} (ω) & S_{23}^{0} (ω) \\ S_{31}^{0} (ω) & S_{32}^{0} (ω) & S_{33}^{0} (ω) \end{matrix}]

(3)

For the wind velocity spectrum, the Simiu spectrum (Simiu et al., 1981) is applied to the lateral and longitudinal wind velocity spectrum while the vertical velocity spectrum adopts the Lumley-Panofsky spectrum (Lumley and Panofsky, 1965). The Davenport coherence function (Davenport, 1962) should be taken into consideration in the cross-power spectral density functions between nodes, described as:

Co h_{j k} (ω) = \exp (- χ \frac{n r_{j k}}{u_{m}})

(4)

Where χ is dimensionless attenuation factor, χ = 7.0; r_jk is the distance between node j and node k; u_m is the mean velocity at the height z.

To validate the wind fields adopted, the turbulence intensity between target values and numerical solutions at various heights (z = 42 m, 81 m, 136 m, 163 m and 190.5 m) are compared and provided in Figure 1. The turbulence intensity is defined as the ratio of the root mean square value of fluctuating wind velocity and the mean wind velocity. Target values of the turbulence intensity is ensured by the code (CCC Highway Consultants CO., 2004). It could be found that the target values and numerical solutions of wind field turbulence intensity vary slightly from each other, illustrating the reliability of the wind field utilized.

Figure 1.

The comparison of the turbulence intensity between target values and numerical solutions. (a) lateral turbulence intensity, I_u. (b) longitudinal turbulence intensity, I_v.

Figure 2(a) and (b) present the wind velocity fields at a point of the bridge tower and girder respectively. The fields are obtained through the method depicted above and applied in the following study.

Figure 2.

Wind velocity fields. (a) tower, z = 81 m. (b) girder, z = 73 m.

Regular wave-current field

In the regular wave-current field simulation, a numerical wave tank is established. The current is considered through setting the initial current velocity in the inlet boundary. The Fenton’s Stokes fifth-order theory (Fenton, 1985) is applied on the left side of the tank to generate regular waves. The velocity potential, wave surface and velocity components are presented as follow:

φ = \frac{c}{k} \sum_{i = 1}^{5} φ_{i} \cosh i (z + d) \sin i (k x - ω t)

(5)

η = \frac{1}{k} \sum_{j = 1}^{5} η_{j} \cos j (k x - ω t)

(6)

u_{x} = \frac{\partial φ}{\partial x}

(7)

u_{z} = \frac{\partial φ}{\partial z}

(8)

Where H, T, k and d represent wave height, wave period, wave number as well as still water level respectively.

z + d

means the distance from the bottom to the free wave surface;

φ_{i}

is the velocity potential coefficient,

φ_{1} = α (A_{11} + α^{2} A_{13} + α^{4} A_{15})

φ_{2} = α^{2} (A_{22} + α^{2} A_{24})

φ_{3} = α^{3} (A_{33} + α^{2} A_{35})

φ_{4} = α^{4} A_{44}

φ_{5} = α^{5} A_{55}

η_{j}

is a coefficient related to the wave form,

η_{1} = α

η_{2} = α^{2} B_{22} + α^{4} B_{24}

η_{3} = α^{3} B_{33} + α^{5} B_{35}

n_{1} = α^{4} B_{41}

η_{5} = α^{5} B_{55}

. α is a parameter associated with H, T, k, d, B_ij and C_i, could be solved by equations as below:

H / (2 d) = \frac{[α + α^{3} B_{33} + α^{5} (B_{35} + B_{55})]}{k d}

(9)

\frac{4 π^{2} d}{(g T^{2})} = k d (1 + α^{2} C_{1} + α^{4} C_{2}) \tanh (k d)

(10)

Where A_ij, B_ij and C_i could be referred to the research by Fenton (1985).

Notably, due to the weak correlation between wind and wave-current at the bridge site during the normal weather, the correlation of their actions on the cable-stayed bridge is not taken into consideration.

The cable-stayed bridge modeling

A numerical cable-stayed bridge model is established in ABAQUS. As shown in Figure 3, the total length of the bridge is 1188 m. Four subsidiary foundations (N01, N02, N05, N06) and two pylon foundations (N03, N04) are set 0 m, 132 m, 1056 m, 1188 m, 328 m as well as 860 m away from the left side of the bridge model, respectively. The combined action of wind, wave and current is applied to the bridge as Figure 3(a) shows. It is noteworthy that the still water level is 22 m, leading to the employment of the wave-current action on the foundations, including piles and bearing platforms. The wind action on the bridge cables is neglected, mainly applied to the bridge girder and towers. In the present study, the wave height, period, current velocity and mean wind velocity are set to be 5 m, 6.6 s, 0.6 m/s and 44.7 m/s individually.

Figure 3.

Numerical cable-stayed bridge model setup (unit: m). (a) bridge elevation. (b) typical cross section of girder. (c) cross section of pile foundations.

Wind-bridge interaction

The wind action applied on the bridge consists of three components: static wind forces, buffeting forces and self-excited forces. The static wind forces on the bridge girder are defined as:

D_{s t} = 0.5 ρ_{a} u_{m}^{2} H_{g} L_{g} C_{D}, L_{s t} = 0.5 ρ_{a} u_{m}^{2} B_{g} L_{g} C_{L}, M_{s t} = 0.5 ρ_{a} u_{m}^{2} B_{g}^{2} L_{g} C_{M}

(11)

Where ρ_a is air density; u_m is the mean velocity; H_g, B_g, L_g denote the height, width and length of the girder respectively; lift force coefficient C_L, drag force coefficient C_D and moment coefficient C_M corresponds to the static lift force L_st, drag force D_st as well torsional moment M_st, individually.

The buffeting forces are defined through the quasi-steady aerodynamic formula, proposed by Scanlan (1978), described as:

{\begin{cases} D_{b} (t) = \frac{1}{2} ρ_{a} u_{m}^{2} B_{g} [2 C_{D 1} (\frac{u (t)}{u_{m}}) + C_{D 2} \frac{w (t)}{u_{m}}] \\ L_{b} (t) = \frac{1}{2} ρ_{a} u_{m}^{2} B_{g} [2 C_{L 1} (\frac{u (t)}{u_{m}}) + (C_{L 2} + C_{D 1}) \cdot \frac{w (t)}{u_{m}}] \\ M_{b} (t) = \frac{1}{2} ρ_{a} u_{m}^{2} B_{g} [2 C_{M 1} (\frac{u (t)}{u_{m}}) + C_{M 2} (α_{0}) \cdot \frac{w (t)}{u_{m}}] \end{cases}

(12)

Where u(t) and w(t) are the wind velocity in the lateral and vertical direction at time t. α₀ is the bridge equilibrium angle of the bridge under the mean wind velocity u_m. D_b, L_b and M_b are buffeting drag force, lift force as well as moment respectively.

Compared with bridge girders, the stiffness of the cable-stayed bridge tower is significantly large. The self-excited wind forces on the tower are thus ignored. For the bridge girders, the neglection of the self-excited wind forces could bring about greater results, beneficial to the structural safety consideration. Hence the self-excited wind forces on the bridge girders and towers are both not taken into consideration here.

Wave-current-bridge interaction

As depicted in regular wave-current field, to investigate the wave-current-bridge interaction, a numerical wave tank is established in FLOW-3D, as Figure 4 shows. The tank is set 900 m(length) × 75 m(width) × 30 m(height). The bridge foundation model is positioned 360 m away from the inlet boundary and the still water level is 22 m. To eliminate the wave reflection, a damping zone is applied to the outlet boundary.

Figure 4.

Numerical wave tank setup.

The model is governed by Reynolds-averaged Navier-Stokes (RANS) equations, where the fluid is supposed to be incompressible and viscous. The continuity and momentum equations of RANS equations are given by:

\nabla \cdot \vec{U} = 0

(13)

\frac{\partial ρ_{w} \vec{U}}{\partial t} + \nabla \cdot (ρ_{w} \vec{U} {\vec{U}}^{T}) = - \nabla \cdot p + \nabla \cdot (μ \nabla \vec{U} + ρ_{w} τ)

(14)

Where

\vec{U}

is the velocity vector; ρ_w denotes the water density; p, t, μ, τ represents the pressure, time, dynamic viscosity and Reynold stress tensor, respectively.

The wave-current forces on the bridge foundations are solved by the renormalization group (RNG) κ-ε turbulent model (Yakhot and Orszag, 1986). The boundary conditions described in regular wave-current field are applied to the model. And the results are calculated and extracted in FLOW-3D. Considering the relatively large stiffness of bridge foundations, the foundation structures are regarded as rigid bodies and the fluid-structure interaction is neglected. To trace the free surface of the fluid, the Volume of Fluid (VOF) method is utilized, proposed by Hirt and Nichols (1981).

The foundation of the cable-stayed bridge investigated is the composite foundation, comprising the cylinder structure and dumbbell structure. The wave forces on these structures are verified through comparing numerical solutions with experimental results (Zhu et al., 2019) respectively. The comparison is presented in Figure 5, showing good agreement, illustrating the reliability of the wave-current action adopted.

Figure 5.

The comparison of total horizontal wave force on different structures. (a) cylinder structure, H = 0.1 m, T = 1.4 s, d = 0.2 m. (b) dumbbell structure, H = 0.08 m, T = 1.0 s, d = 0.2 m.

The wave-current action applied in this study is illustrated in Figure 6. It could be found that the total horizontal wave force patterns on the bridge foundations are regular when the time history profiles reach stable, while the total vertical wave force and the longitudinal moment still fluctuate slightly. This is probably due to the wave force phase difference induced by the distribution of pile groups and the wave buoyance on the composite foundation.

Figure 6.

Time history curves of the wind-current action.

Pile-soil interaction

The pile-soil interaction is studied through different models, including the pile-soil solid contact (PSSC) model, soil spring model as well as consolidation model. The PSSC model takes the nonlinearity of the pile-soil interface, such as contact and friction into consideration. The properties of soil layers in PSSC are provided in Table 1. When setting the contact type between the soil and pile, hard contact type is utilized at the direction normal to the interface and sliding friction is applied at the tangential direction. To consider the geostress, the geostatic step is set before the wind action and wave-current action is applied.

Table 1.

Properties of soil layers.

Category	Elevation (m)	Density (kg/m³)	Poisson’s ratio	Elasticity (MPa)	Internal friction angle (°)
Medium dense silt	−23.0	1800	0.30	16	12
Dense silty sand	−26.1	1950	0.28	28	16
Dense medium sand	−38.7	2020	0.25	40	10
Decomposed granite	−58.2	2100	0.25	50	6

In the soil spring model, a series of stiffness-damping soil springs are adopted to replace the soil restraint on piles. The stiffness of springs is calculated according to the Code for Design on Subsoil and Foundation of Railway Bridge and Culvert (China Railway Design CO., 2017), where the soil layer properties and thickness applied could be found in Table 1. Notably, the stiffness of soil springs on the pile sides is obtained through Matlock method, neglecting the damping coefficient. The springs connecting pile bottoms and the ground are constrained in the lateral and longitudinal directions.

To investigate the influence of the pile-soil interaction, the consolidation model is also used to conduct comparative analysis, which fixes the pile bottoms. The soil mechanical properties are obtained through Mohr-Coulomb theory, described as below:

τ_{f} = c + σ_{f} \tan θ

(15)

Where τ_f and σ_f represent shear stress and normal stress respectively; c means the cohesion force; θ is the soil friction angle.

WWCB analysis framework

The entire procedure to investigate bridge dynamic responses under wind-wave-current action is summarized as Wind-Wave-Current-Bridge (WWCB) model in Figure 7. The model involves two main components: the loads and finite element bridge model. A wave-current boundary, containing parameters like the wave height, period and current velocity, is utilized in a numerical wave tank as the inlet boundary condition to obtain wave-current loads. With wind velocity field ensured, wind loads could be calculated through quasi-steady aerodynamic theory. A finite element bridge model is then established, taking the pile-soil interface nonlinearity, initial geostress as well as hydrostatic pressure into consideration. The loads computed above are applied to the various bridge sections afterwards and their dynamic responses are analyzed. Based on the displacement and internal force response results, the performance of WWCB model could be assessed.

Figure 7.

Flowchart of WWCB analysis procedure.

To validate the WWCB analysis framework proposed, a finite element model of the bridge tower is established in ABAQUS, referring to the model investigated by Guo et al. (2016) through experiments. The prototype of the bridge tower is freestanding, with a height of 233.9 m. Its foundation is composed of 16 piles, whose size is 3.0 m in diameter and 23.0 m in length. To connect the piles, a pile cap with a height of 7.0 m is utilized, supporting the superstructure of the bridge tower. Figure 8 presents the finite element model built and the natural frequencies are compared in Table 2. It could be found that the errors between the numerical solutions and experimental results are relatively small, indicating the reliability of the numerical model.

Figure 8.

The finite element model of the bridge tower.

Table 2.

The comparison of natural frequencies.

Direction	Design frequency (Hz)	Numerical frequency (Hz)	Error (%)
Longitudinal	1.278	1.255	1.80
Lateral	2.678	2.698	0.75

The displacements with various wave heights under combined wind and wave action at the tower top are also compared to further verify the WWCB system, as shown in Figure 9. It could be observed that the numerical solutions vary slightly from experimental results, illustrating the accuracy of the WWCB system.

Figure 9.

The comparison of the displacements under various wave heights at the tower top, U = 1.73 m/s.

Bridge dynamic responses

Displacement responses

To investigate the bridge displacement responses under wind-wave-current action, three loading combinations are adopted: (1) wind only, (2) combined wave and current, (3) combined wind, wave and current. Figure 10 depicts the time history curves of the lateral, vertical and torsional displacement at the midpoint of the girder. To understand the displacement responses intuitively, the root mean square (RMS) values along the girder at various directions are provided in Figure 12. It could be observed that for the lateral and torsional displacement, it is the wind action that makes the major contribution to the bridge response. This could also be found in Figure 12(a) and (c).

Figure 10.

Displacement time history curves at the midpoint of the girder (PSSC model).

Figure 12.

Root mean square (RMS) values of displacement along the girder at different directions. (a) lateral displacement (m). (b) vertical displacement (m). (c) torsional displacement (rad).

For the vertical displacement of the girder when considering the pile-soil interaction, the combined action of wave and current plays an important role in the responses, as shown in Figures 10 and 12(b). This is due to that the pile bottoms are not fixed directly in the PSSC model in Figure 10. For the soil model in the PSSC model, its bottoms are fixed while only the lateral displacements of the sides are constrained. Pile bottoms are tied with soil bottoms to make a connection with longitudinal and vertical degrees of freedom. The wave-current action applied to the foundations causes the foundation vibration, leading to the large amplitude responses of stay cables which are transmitted through bridge towers. The stay cables drive the girder vibrating in the vertical direction, resulting in greater vertical displacement responses.

In the consolidation model, the wave-current action has limited effect on the bridge foundations, resulting in that the wind action plays the dominant role in the vertical displacement responses of the girder, as presented in Figure 11. Additionally, it is noted that the displacements at the midpoint of the girder when the pile bottoms are fixed are smaller than that when considering pile-soil interaction, further indicating the necessity for considering pile-soil interaction.

Figure 11.

Displacement time history curves at the midpoint of the girder (consolidation model).

The time history curves of the tower lateral displacement under various loading combinations with PSSC model and consolidation model are exhibited in Figure 13. It could be found that the wave-current action makes a slight difference in the lateral displacement responses of the tower under wind-wave-current action. This conclusion is consistent regardless of whether taking the pile-soil interaction into the consideration, which could be attributed to the great stiffness of the tower.

Figure 13.

Lateral displacement time history curves of the top of the tower N03.

To better investigate the influence of the wind-wave-current action on the lateral displacement responses of the tower, its lateral RMS values and extreme values are presented in Figure 14. From Figure 14(a), it could be concluded that the wind action plays a dominant role in the joint action of wind, wave and current. However, the extreme displacement caused by wave-current action is twice as large as that caused by wind action at the lower part of the tower, as shown in Figure 14(b). This indicates the significance of taking the wave-current action into account while designing the lower part of the tower.

Figure 14.

Lateral displacement responses of the tower N03. (a) RMS values (mm). (b) extreme values (mm).

Internal force responses

Figure 15 provides the RMS values of the girder internal force under wind action, wave-current action as well as wind-wave-current action. Notably, only the force results at key sections are extracted, resulting in the cusps in the curves. It could be found that the combined wave and current action has little influence on the internal force responses of the girder in general. This is due to the influence of the wave-current action which needs to be transmitted through towers, auxiliary piers and stayed cables, while the wind action is applied to the girder directly. On the other hand, the static wind action is considered in the present study, leading to the great effect of the wind action on the girder. Notably, the influence of the wave-current action could not be neglected on the axial force response, especially at the midspan.

Figure 15.

RMS values of internal force responses along the girder. (a) axial force (kN). (b) lateral force (kN). (c) vertical force (kN). (d) torque (kN·m)

Influence of pile-soil interaction

Dynamic characteristics analysis

To figure out the influence of pile-soil interaction on the dynamic characteristics, the first 10 natural frequencies and corresponding bridge modes with the PSSC model, soil spring model and consolidation model are listed in Table 3. Figure 16 provides the comparison graphic of natural frequencies between various models.

Table 3.

First 10 structural natural frequencies and mode shapes with various models.

Mode number	Consolidation model	Soil spring model	PSSC model	Mode shape description
1	0.08611	0.084184	0.085182	Longitudinal floating, vertical bending (girder)
2	0.19948	0.19713	0.19714	1st symmetric lateral bending (girder)
3	0.33055	0.32251	0.32457	1st symmetric vertical bending (girder)
4	0.36885	0.35144	0.35765	1st opposite lateral bending (tower)
5	0.41226	0.38144	0.39530	1st homolateral bending (tower)
6	0.46542	0.43438	0.45135	2nd asymmetric lateral bending (girder)
7	0.48058	0.45816	0.46028	2nd asymmetric vertical bending (girder)
8	0.48675	0.45607	0.46459	3rd torsion (girder)
9	0.50074	0.47021	0.47515	3rd asymmetric lateral bending (girder)
10	0.54889	0.51623	0.52250	3rd symmetric lateral bending (girder)

Figure 16.

Natural frequency comparison between different models.

It could be observed that the natural frequencies with different models vary slightly from each other in the low-frequency range. This variation is magnified gradually with the frequency increasing. The consolidation model that consolidates the pile bottoms possesses the maximum structural stiffness, leading to the largest natural frequencies. Compared with the consolidation model, the utilization of soil springs in the soil spring model raises the structural flexibility, resulting in the minimum natural frequencies. As a result of considering the initial geostress, mass and density distribution as well as interface behavior definition in the PSSC model, its natural frequencies range in between as shown in Figure 16.

Figure 17(a)–(c) present the displacement responses with different models along the girder. As depicted above, the stiffness of the consolidation model is largest, causing the minimum displacement responses. The displacement responses between the PSSC model and soil-spring model are pretty close, due to the similar structural stiffness. The same conclusion could also be drawn for the upper tower displacement, as shown in Figure 18. It is noted that the displacement at the lower part of the tower with the consolidation model is the smallest. This result could be ascribed to the decreasing impact of pile-soil interaction on the bridge tower displacement along the tower height.

Figure 17.

Extreme displacement responses with different models along the girder. (a) lateral displacement (m). (b) vertical displacement (m). (c) torsional displacement (m).

Figure 18.

RMS values of displacement responses with different models along the tower height.

Frequency domain analysis

To further investigate the influence of the pile-soil interaction, the frequency domain analysis with the consolidation model and PSSC model is conducted, as presented in Figures 19 and 20. The frequencies where peak values locate correspond to the structure natural frequencies of each order. Figure 19 provides the spectral density functions of longitudinal and lateral displacement responses at the tower top. It could be found that with the PSSC model, the displacement response frequency distributes more widely in the frequency domain, while its peak values are smaller than that with the consolidation model. This illustrates that taking the pile-soil interaction into consideration would reduce the tower stiffness, leading to a decrease in the natural frequencies.

Figure 19.

Spectral density functions of longitudinal and lateral displacement responses at the top of tower N03.

Figure 20.

Spectral density functions of lateral, vertical and torsional displacement responses at midspan.

Figure 20 describes the spectral density functions of lateral, vertical and torsional displacement responses at midspan. Different from the frequency distribution at the tower top, results at midspan with the consolidation model as well as PSSC model vary slightly, especially in the low-frequency range. It is noteworthy that for the vertical displacement responses, the frequency obtained from the PSSC model has a wider distribution and the peak value occurs at the lower frequency. As depicted above, the influence of the pile-soil interaction on the girder needs to be transmitted through towers, axillary piers and stayed cables. Thus, under low-frequency excitation, the girder is scarcely affected by the pile-soil interaction.

Conclusions

A Wind-Wave-Current-Bridge (WWCB) system is introduced in the present study to investigate the cable-stayed bridge dynamic responses under wind-wave-current action, taking the pile-soil interaction into consideration. To ensure the combined wave and current action, a numerical wave tank is established, where Fenton’s Stokes fifth-order theory and initial current velocity are adopted. The mean wind and fluctuating wind are utilized to simulate wind field and the wind action is calculated on this basis. To compose the WWCB model, a three-dimensional numerical cable-stayed bridge model is built, where the actions obtained above are applied. The bridge dynamic responses of displacement and internal force are then studied respectively. To figure out the influence of the pile-soil interaction, the bridge dynamic characteristic analysis and frequency domain analysis are conducted. The main conclusions drawn in this study are listed as follows:

For the bridge girder, it is the wave-current action that plays the crucial role in the vertical displacement response under the combined action of wind, wave and current. Additionally, the influence of the wave-current action on the axial force response should be noted, especially at the midpoint of the girder. In other cases, the wind action is dominant.

For the bridge tower, the wave-current action would bring about greater lateral deformation than the wind action at the lower part, concluded from the extreme lateral displacement response. Consequently, the effect of the combined wave and current action should not be dismissed in the bridge dynamic responses under wind-wave-current action, no matter for the girder or the tower.

Compared with the consolidation model, the natural frequencies with PSSC model and soil-spring model are lower, as a result of taking pile-soil interaction into account. The natural frequencies with PSSC model are greater than that with the soil-spring model, due to considering the initial geostress, mass and density distribution as well as interface behavior definition. The PSSC model results are thus more practical.

In the frequency domain, the frequency of the PSSC model distributes more widely than that of the consolidation model at the tower top. At midspan, the frequencies of both vary slightly from each other under low-frequency excitation. For the vertical displacement responses, the PSSC model possesses wider distribution and its peak value occurs at a lower frequency than the consolidation model. This further illustrates that the PSSC model would cause a decrease in the structural stiffness.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Open Project of the Key Laboratory for Comprehensive Simulation of Earthquake Engineering and Urban-Rural Seismic Toughness of China Seismological Bureau (EESSR 19-XX) and National Natural Science Foundation of China - China railway corporation high-speed railway basic research joint fund project (U1834207).

Data availability statement

The data supporting this article is available from the corresponding author upon reasonable request.

ORCID iD

Bing Zhu

References

CCC Highway Consultants CO., L (ed) (2004) Wind-resistent Design Specification for Highway Bridges (JTG/T D60-). Beijing, P.R.China: Ministry of Transport of the People’s Republic of China.

China Railway Design CO., L (ed) (2017) Code for Design on Subsoil and Foundation of Railway Bridge and Culvert (TB10093-20th ed.). Beijing, P.R.China: China Railway Publishing House.

Davenport

(1962) Buffeting of suspension bridge by storm winds. Journal of the Structural Division 88(3): 233–268.

Deodatis

(1996) Simulation of ergodic multivariate stochastic processes. Journal of Engineering Mechanics 122: 778–787.

Ejermo

Hussinger

Kalash

, et al. (2021) Innovation in Malmö after the Öresund bridge. Journal of Regional Science 62: 5–20. DOI: 10.1111/JORS.12543

Fang

Chen

, et al. (2019a) Extreme response of a sea-crossing bridge tower under correlated wind and waves. Journal of Aerospace Engineering 32(6): 05019003. DOI: 10.1061/(asce)as.1943-5525.0001083

Fang