Nonlinear dynamic response of sea-crossing bridges to 3D correlated wind and wave loads

Abstract

Long span sea-crossing bridges are often slender and sensitive to wind and wave loads. Nonlinear dynamic response analysis of the bridges under three-dimensional (3D) correlated wind and wave loads is performed in this study in consideration of both geometric and aerodynamic nonlinearities. An optimized C-vine copula is first used to construct a 3D joint probability distribution and environmental contour of mean wind speed, significant wave height and peak wave period. Multi-point fluctuating wind loads with Davenport coherence function and random wave loads with pile group effect are then determined using wind and wave spectra respectively. The nonlinear wind-wave-bridge system considering geometric and aerodynamic nonlinearities is solved by the Newmark-β method with the 3D correlated wind and wave parameters as an input. The proposed approach is finally applied to a real sea-crossing cable-stayed bridge with the measured wind and wave data. The results show that the nonlinear response of the bridge is higher than its linear response with the same input. The bridge response is significantly reduced if the 3D correlated wind and wave loads other than conventional uncorrelated wind and wave loads are considered.

Keywords

Sea-crossing bridge wind and wave environmental contour aerodynamic nonlinearity nonlinear dynamic response

Introduction

In recent years, more and more long-span sea-crossing bridges are designed and constructed in offshore or deep sea to meet the need of modern transportation system. Major environmental loads acting on these bridges are wind and wave loads, which affect the service performance of or even cause the observed damage to the bridges (Padgett et al., 2008; Robertson et al., 2007). The determination of wind and wave induced dynamic responses is thus important for the performance evaluation of the bridges.

Most efforts spent on the dynamic response analysis of marine structures under wind and wave loads focus on small-scale structures, such as offshore wind turbines (Wei et al., 2017a; Yue et al., 2020), oil platforms (Zaheer and Islam 2012, 2017) and single towers (Guo et al., 2016; Wei et al., 2017b). Only recently, with advance of computation resource and measurement data, the dynamic response analysis of long-span bridges under wind and wave loads attracts attention. Zhu et al. (2018) calculated the dynamic response of a cable-stayed bridge under different wind and wave loading cases. Meng et al. (2018) studied stochastic buffeting response of a cable-stayed bridge under correlated wind and waves using Frank copula. Fang et al. (2020a, 2020b) investigated wind and wave induced response of a sea-crossing bridge using different surrogate models. Bai et al. (2022) analyzed the extreme response of a sea-crossing bridge under joint action of wind, wave and seafloor earthquakes, and they found that the bridge response under correlated wind and waves is approximately equal to that under 0.2 g offshore ground motion. In the above studies, only the bivariate correlation between wind speed and wave height are considered. Because wind speed, wave height and wave period are all important factors affecting wind and wave loads (Ti et al., 2019), the trivariate correlation among wind and wave parameters is necessary to be considered. Moreover, only the geometric nonlinearity of the cable sag effect and the large deformation effect is considered in the above studies, and the aerodynamic nonlinearity, which includes wind incidence angle effect on self-excited forces, is neglected, but it should be considered due to the large deformation of the bridge under design wind and wave loads

Regarding the trivariate correlation among wind and wave parameters, some researchers simulated the correlation among the three variables by using high-dimensional parametric copulas: Plackett copulas, vine copulas and multivariable copulas (Cheng et al., 2020; Li et al., 2018; Luo and Huang 2017; Luo et al., 2021; Montes-Iturrizaga and Heredia-Zavoni 2016; Xiao et al., 2022; Wang et al., 2021; Zhang et al., 2020). However, they focused on the joint distribution characteristics of three variables and the simulation accuracy only, and little information is available for the multivariable environmental contours. Fang et al. (2022) proposed an optimized C-vine copula model and derived the multivariable environmental contour for the sea-crossing bridge. The optimized C-vine copula and the corresponding environmental contours figured out by Fang et al. (2022) will be adopted in this study because the optimized C-vine copula provides more accurate joint distribution and more reasonable decomposition method for the selected key variable.

Furthermore, the existing buffeting response analyses of long-span bridges often assume that wind incidence angle effect is negligible (Bai et al., 2022; Meng et al., 2018; Sun et al., 1999; Xu et al., 2000). However, the design wind load and wave load with a high return period determined by the environmental contours could induce a large deformation of the bridge. The wind angle of attack along a long-span bridge deck could change with bridge vibration and initial wind angle of attack. Since wind loads depend not only on time-varying wind speed but also on time-varying wind angle of attack (Chen and Kareem 2001), the aerodynamic nonlinearity cannot be neglected in this study.

This study therefore proposes an analytical method for calculating nonlinear dynamic response of sea-crossing bridges to three dimensional (3D) correlated wind and wave loads in consideration of both geometric and aerodynamic nonlinearities. Firstly, the optimized C-vine copula and the corresponding environmental contour are briefly introduced, and the 3D correlated wind and wave loads are determined. Secondly, the nonlinear wind-wave-bridge (NWWB) system is established and the solution method is interpreted. Thirdly, a real long-span sea-crossing bridge is selected as a case study, in which the 3D correlated wind speed, wave height and wave period are simulated based on the measurement data, the trivariate environmental contours are drawn to obtain the correlated wind and wave parameters at different return periods, and the wind-wave load and the corresponding aerodynamic coefficients are elaborated. Finally, the dynamic responses of the bridge under wind and wave loads in different return periods, in linear or nonlinear cases, and in univariate or trivariate cases, are computed and compared.

Methodology

This section presents a general analytical method for calculating the nonlinear dynamic response of sea-crossing bridges in consideration of 3D correlated wind and wave loads and both geometric and aerodynamic nonlinearities.

3D correlated wind speed, wave height and wave period

As an effective simulation method of high-dimensional joint probability distribution, C-vine copula is used to decompose multiple random variables from a high-dimensional copula into several two-dimensional copulas (pair-copulas). The joint probability density function (JPDF), $f (U, H, T)$ , of wind speed U, wave height H and wave period T is given by

f (U, H, T) = f (U) \cdot f (H) \cdot f (T) \cdot c (F_{U} (U), F_{H} (H), F_{T} (T))

(1)

where

f (U) \cdot f (H) \cdot f (T)

represents the product of the marginal probability density functions (MPDFs) of wind speed, wave height and wave period. These MPDFs can be estimated by the five commonly-used marginal density functions, which include Lognormal, General Extreme Value (GEV), Gumbel, Weibull and Gamma distribution functions, in terms of the measurement data.

c (F_{U} (U), F_{H} (H), F_{T} (T))

is a three-dimensional copula density function;

F_{U} (U), F_{H} (H) a n d F_{T} (T)

are the marginal cumulative distribution function (MCDFs) of wind speed, wave height and wave period.

Define ${u_{1} = F}_{U} (U)$ , ${u_{2} = F}_{H} (H)$ and $u_{3} = F_{T} (T)$ . The three-dimensional copula density function $c (u_{1}, u_{2}, u_{3})$ can be decomposed into three pair-copulas of $c_{12}, c_{13}$ and $c_{23 | 1}$ and expressed as follows

c (u_{1}, u_{2}, u_{3}) = c_{12} (u_{1}, u_{2}) \cdot c_{13} (u_{1}, u_{3}) c_{23 | 1} (h (u_{2} | u_{1}), h (u_{3} | u_{1}))

(2)

h (u_{i} | u_{j}) = F (x_{i} | x_{j}) = \frac{\partial C_{i j} (u_{i}, u_{j})}{\partial u_{j}}

(3)

where

c_{12}

and

c_{13}

are the copula density functions for the variables (

u_{1}, u_{2}

) and (

u_{1}, u_{3}

), respectively;

c_{23 | 1}

is the conditional copula density as an intermediate variable independent of

u_{1}

;

h (\cdot | \cdot)

and

F (\cdot | \cdot)

are the conditional probability function.

The four commonly-used pair-copulas, that is, Gumbel copula, Clayton copula, Gaussian copula and Frank copula, can be used to simulate the bivariate correlation. Copula functions are usually divided into two families: Archimedean copulas and Elliptical copulas. The Gaussian, Gumbel, Clayton and Frank copulas used in this study are very representative of Copula functions. The Gumbel copula and Clayton copula belong to the asymmetric Archimedean copulas, the Frank copula belongs to the symmetric Archimedean copula, and the Gaussian copula belongs to the Elliptical copulas. The four pair-copulas used here cover almost all the characteristics of the copula function in terms of lower-tail and upper-tail dependences. Specifically, the Clayton copula has more probability concentrated in lower-tail. The Gumbel copula has more probability concentrated in upper-tail. The Gaussian copula is neither lower-tail nor upper-tail dependent, which is also approximated as the Nataf transformation. These four pair-copulas have been widely used in wind and ocean engineering.

The optimized C-vine copula proposed by Fang et al. (2022) are used to estimate $f (U, H, T)$ . The mean wind speed is first determined as a key variable through the multivariable correlation analysis of the measurement data of wind speed, wave height and wave period. The one-step optimization method is then used to best fit the correlations of the three random variables among the five commonly-used MPDFs and the four commonly-used pair-copulas. The JPDF, $f (U, H, T),$ of the wind speed, wave height and wave period, is finally established using the respective optimal pair-copulas and optimal marginal distribution functions obtained from the one-step optimization. Details of the one-step optimization method can be found in Fang et al. (2022).

Trivariate environmental contours

The environmental contour is the basis for determining the design values of the three environmental variables with a given return period. Once the JPDF of the three environmental variables is obtained, the environmental contours could be obtained by the Rosenblatt transformation (Rosenblatt, 1952). The Rosenblatt transformation maps the environmental variables ( $x_{1}, x_{2}, x_{3}$ ) in the physical space to the independent standard normal variables Z $= (Z_{1}, Z_{2}, Z_{3})$ in a reduced space with the radius of the hypersphere defined by $| Z | = β$ (Montes-Iturrizaga and Heredia-Zavoni 2016).

By assuming that the occurrence of extreme event is regarded as a Poisson process with a return period $T_{R}$ and a mean annual rate $λ_{E}$ , the radius of the hypersphere $β$ is given as

β = - Φ^{- 1} [- λ_{E}^{- 1} \ln (1 - 1 / T_{R})]

(4)

Once the radius β and Z $= (Z_{1}, Z_{2}, Z_{3})$ are determined in the uncorrelated standard normal space, the environmental variables x = ( $x_{1}, x_{2}, x_{3}$ ) can be derived using the optimal copula as follows.

Letting x₁=U, x₂=H, x₃=T, ${u_{1} = F}_{U} (U)$ , ${u_{2} = F}_{H} (H)$ and $u_{3} = F_{T} (T)$ , the inverse Rosenblatt transformation yields

u_{1} = Φ (Z_{1})

u_{2} = C^{- 1} (Φ (Z_{2}) | u_{1})

(5)

u_{3} = C^{- 1} (Φ (Z_{3}) | Φ (Z_{1}), u_{2})

where

C^{- 1}

is the inverse transformation of conditional distribution

C

; and

Φ (\cdot)

represents the standard normal distribution function. Once the values of

(u_{1}, u_{2}, u_{3})

are obtained by equation (5), the corresponding design values of the three environmental variables can be calculated by

{U = F_{1}^{- 1} (u}_{1})

{H = F_{2}^{- 1} (u}_{2})

and

{T = F_{3}^{- 1} (u}_{3})

Therefore, with a given return period, a specific combination of the design values $(U, H, T)$ can be obtained in terms of their environmental contours. For the long span sea-crossing bridge concerned, the maximum value of the dominant variable $U$ and the corresponding values of $H$ and T are selected as the design values as the most unfavorable combination.

Wind loads

In the boundary layer wind, the real wind velocity at a point can be decomposed as a mean wind speed plus the three fluctuating wind speeds in the three orthogonal directions. The mean wind speed profile along the height z above the design water level (DWL) can be described by

U_{z} (z) = U_{10} [\ln (z / z_{0}) / \ln (10 / z_{0})]

(6)

where

U_{10}

is the 10-min mean wind speed at a 10 m height above the DWL;

z_{0}

is the sea surface roughness which is approximately equal to 0.01 m for a rough sea surface.

For a long-span sea-crossing bridge, the wind field of the bridge consists of wind velocities at multiple points along the bridge deck and towers. The means wind speeds at the multiple points along the bridge deck are often taken as a constant value and those along the bridge towers are determined by equation (6). The fluctuating wind speed at a point in one direction can be treated as a uniformly modulated process and simulated by the spectral representation method (SRM). The Kaimal spectrum and the Lumley-Panofsky spectrum are selected as the lateral and vertical wind spectrum to generate the fluctuating wind speed on the bridge deck and tower, respectively.

\frac{n S_{u} (f)}{u_{*}^{2}} = \frac{200 f}{{(1 + 50 f)}^{5 / 3}}

(7)

\frac{n S_{w} (f)}{u_{*}^{2}} = \frac{3.36 f}{{(1 + 10 f)}^{5 / 3}}

(8)

where

f = n z / U (z)

is the dimensionless frequency; n is the frequency in Hz; and

u_{*} = K U (z) / \ln (z / z_{0})

is the friction velocity, where K is the von Kármán constant of 0.4.

The correlation between the fluctuating wind speeds at multiple points in one direction is considered in terms of the Davenport coherence function. The wind loads acting on the bridge deck are accordingly decomposed as the static wind forces, buffeting wind forces, and self-excited wind forces with each wind force consisting of the three components: drag, lift, and moment. As a typical blunt structure, the wind loads acting on the bridge tower only consider the static wind forces and the buffeting wind forces, and the self-excited wind forces are negligible.

The static wind forces resulting from the mean wind speeds and acting on the bridge deck can be expressed by

F_{s t}^{D} (α) = 0.5 ρ U_{d}^{2} C_{d} (α) H_{b}; F_{s t}^{L} (α) = 0.5 ρ U_{d}^{2} C_{l} (α) B; F_{s t}^{M} (α) = 0.5 ρ U_{d}^{2} C_{m} (α) B^{2}

(9)

where

F_{s t}^{D} (α), F_{s t}^{L} (α), F_{s t}^{M} (α)

are respectively the static wind drag, lift and moment forces per unit length, which are the function of wind angle

α

of attack;

C_{d} (α), C_{l} (α), C_{m} (α)

are respectively the drag, lift, and moment coefficients at wind angle

α

of attack;

ρ

is the air density;

U_{d}

is the mean wind speed at the height of the bridge deck; H_b and B are the height and the width of the bridge deck, respectively.

The buffeting wind forces are caused by fluctuating winds and the self-excited wind forces are generated by the interaction between wind and bridge. For the sake of concise, only the buffeting lift force and self-excited lift force per unit length are given below.

F_{b u}^{L} (t, α) = 0.5 ρ U_{d}^{2} B [2 C_{l} (α) \frac{v (t)}{U_{d}} + (C_{l}^{'} (α) + \frac{H_{b}}{B} C_{d} (α)) \frac{w (t)}{U_{d}}]

(10)

F_{s e}^{L} (t, α) = 0.5 ρ U_{d}^{2} \int_{- \infty}^{t} [f_{L p} (α, t - τ) p (τ) + f_{L h} (α, t - τ) h (τ) + f_{L θ} (α, t - τ) θ (τ)] d τ

(11)

where

F_{b u}^{L} (t, α)

and

F_{s e}^{L} (t, α)

are the buffeting lift force and the self-excited lift force, respectively;

v (t) a n d w (t)

are respectively the lateral and vertical fluctuation wind speed at t time;

C_{l}^{'} (α)

is the first derivative of the lift coefficient to wind angle

α

of attack;

p (τ), h (τ), θ (τ)

are the lateral, vertical, and torsional displacement responses of the bridge deck, respectively;

f_{L j} (j = p, h, θ)

are the impulse functions determined by flutter derivatives based on the rational approximation approach.

As a result, the total wind-induced load $F_{w i n d}$ acting on the sectional length $L_{s}$ of the bridge deck is given by

F_{w i n d} (t, α) = [\begin{array}{c} \int_{0}^{L_{s}} F_{D} (t, α) d z \\ \int_{0}^{L_{s}} F_{L} (t, α) d z \\ \int_{0}^{L_{s}} F_{M} (t, α) d z \end{array}] = [\begin{array}{c} \int_{0}^{L_{s}} {(F}_{s t}^{D} (α) + F_{b u}^{D} (t, α) + F_{s e}^{D} (t, α)) d z \\ \int_{0}^{L_{s}} {(F}_{s t}^{L} (α) + F_{b u}^{L} (t, α) + F_{s e}^{L} (t, α)) d z \\ \int_{0}^{L_{s}} {(F}_{s t}^{M} (α) + F_{b u}^{M} (t, α) + F_{s e}^{M} (t, α)) d z \end{array}]

(12)

where

F_{D} (t, α), F_{L} (t, α)

and

F_{M} (t, α)

are the drag, the lift, and the moment with respect to the time t and the wind angle

α

of attack, respectively; and

L_{s}

is the sectional length of the bridge deck. Detailed derivation of the mean wind forces, buffeting wind forces and self-excited wind forces on the bridge deck and the bridge towers can be found in (Chen and Kareem 2001).

Wave loads

The random wave load depends on water velocity and acceleration. The horizontal water velocity ${\dot{u}}_{w}$ and acceleration ${\ddot{u}}_{w}$ are expressed by Chakrabarti (1971) as follows, which are only suitable for linear waves.

{\dot{u}}_{w} (z, t) = \frac{H}{2} w \frac{\cosh k_{i} z}{\sinh (k_{i} d + φ (t))} \cos (k_{i} x_{j} - w t)

(13)

{\ddot{u}}_{w} (z, t) = \frac{H}{2} w^{2} \frac{\cosh k_{i} z}{\sinh (k_{i} d + φ (t))} \sin (k_{i} x_{j} - w t)

(14)

where H is the wave height;

x_{j}

is the horizontal length from the initial position to the j^th pile foundation of the bridge along the wave travel direction; z is the height from the calculation point to the pile bottom;

w = 2 π / T

is the wave frequency; d is the DWL;

k_{i}

is the i^th component of wave number satisfying the linear dispersion relationship such that

w^{2} = g k \tanh (k d)

;

φ (t)

is the wave surface elevation, which can be generated from the random wave spectrum by using the wave superposition method. The JONSWAP spectrum is selected in this study as the random wave spectrum expressed by

S (w) = \frac{1}{2 π} β_{j} H_{s}^{2} T_{p}^{- 4} {(\frac{w}{2 π})}^{- 5} \exp [- \frac{5}{4} {(T_{p} \frac{w}{2 π})}^{- 4}] \cdot γ^{\exp [- {(\frac{w}{w_{p}} - 1)}^{2} / 2 σ^{2}]}

(15)

where

β_{j} = \frac{0.06238}{0.230 + 0.0336 γ - 0.185 {(1.9 + γ)}^{- 1}} \cdot [1.094 - 0.01915 \ln γ]

;

T_{p} = \frac{T_{s}}{1 - 0.132 {(γ + 0.2)}^{- 0.559}}

;

H_{s}, T_{s}

are the significant wave height and the associated wave period, respectively;

T_{p}, w_{p}

are the peak wave period and the corresponding circular frequency, respectively;

σ = 0.07

when

w \leq w_{p}

, or

σ = 0.09

when

w > w_{p}

; and

γ

is the peak enhancement factor of 3.3.

The wave load acting on the pile foundations is calculated by the Morison equation owing to satisfying the condition of $\frac{D_{p}}{L} < 0.2$ , where $D_{p}$ is the pile diameter and L is the wave length. The horizontal wave force per unit height can be given by

F_{w} (z, t) = 0.5 ρ_{w} C_{D} D_{p} ({\dot{u}}_{w} (z, t) - {\dot{u}}_{b} (z, t)) | ({\dot{u}}_{w} (z, t) - {\dot{u}}_{b} (z, t)) |

+ ρ_{w} A {\ddot{u}}_{w} (z, t) + (C_{M} - 1) ρ_{w} A ({\ddot{u}}_{w} (z, t) - {\ddot{u}}_{b} (z, t))

(16)

where

{\dot{u}}_{b} (z, t), {\ddot{u}}_{b} (z, t)

are the velocity and acceleration of a pile with respect to the height z from the pile bottom and time t;

ρ_{w}

is the water density; A is the cross-sectional area of the pile;

C_{D}

C_{M}

are the drag coefficient and the inertia coefficient.

In general, wave load on a single pile is significantly affected by the neighboring piles, which calls pile group effect. Therefore, for the pile group fixed on a common bridge foundation as used in this study, the pile group effect should be considered for such densely arranged piles. The interference coefficient $K_{G}$ is thus adopted in terms of the laboratory test results (Bonakdar et al., 2015), and $K_{G}$ is taken as 1.0 for the side piles and $1.265 - 0.225 \ln (a / D)$ for the middle piles, where $a$ is the distance between two neighboring piles.

As a result, the total wave-induced load $F_{w a v e}$ acting on the sectional height $L_{p}$ of the bridge pile in water is given by

F_{w a v e} (t) = K_{G} \int_{0}^{L_{p}} F_{w} (z, t) d z

(17)

where

L_{p}

is the sectional height of the pile in water.

Equation of motion of nonlinear wind-wave-bridge system

The nonlinear wind-wave-bridge (NWWB) system includes three major components: wind field, wave field and the bridge itself, as shown in Figure 1. The coupling effects among the three components are identified as: (1) the aerodynamic coupling effect between wind and bridge, accounting for wind loads acting on the bridge deck and towers; (2) the hydrodynamic coupling effect between wave and bridge, accounting for wave loads acting on the group piles of the bridge; and (3) the coupling effect between wind and wave, accounting for 3D correlated wind and wave parameters.

Figure 1.

Schematic diagram of nonlinear wind-wave-bridge system.

The governing equation of motion of the NWWB system can be written as

M_{b} {\ddot{u}}_{b} + C_{b} {\dot{u}}_{b} + K_{b} u_{b} = F_{s b} (α) + F_{b u b} (α_{e}) + F_{s e b} (α_{e}) + F_{w b}

(18)

where

M_{b}, C_{b}, K_{b}

are the mass, damping and stiffness matrixes of the bridge, respectively;

{\ddot{u}}_{b}, {\dot{u}}_{b}, u_{b}

are the acceleration, velocity and displacement response vectors of the bridge, respectively;

F_{s b} (α)

is the static wind forces with respect to wind attack angle

α

;

F_{w b}

is the interaction forces between wave and bridge;

F_{b u b} (α_{e})

and

F_{s e b} (α_{e})

are the buffeting forces and self-excited forces with respect to the effective wind angle

α_{e}

. The effective wind angle

α_{e} = α + α_{s} + α_{w b}

, where

α_{s}

is the torsional angle of the bridge deck caused by the static wind forces, and

α_{w b}

is the time-varying torsional angle of the bridge caused by the interaction between the fluctuating wind and the bridge. Because of the time-varying torsional angle, equation (18) should be solved iteratively within each time step.

Computing procedure

The procedure of computing the nonlinear dynamic response of sea-crossing bridge under 3D correlated wind and wave loads is illustrated in Figure 2 and explained as follows.

Step 1: Establish the 3D joint probability distribution of the three environmental variates $(U_{10}, H_{s}, T_{p})$ using the optimized C-vine copula.

Step 2: Derive trivariate environmental contours using the Rosenblatt transformation and determine the correlated design values of $(U_{10}, H_{s}, T_{p})$ for a given return period.

Step 3: Simulate the fluctuating wind field by SRM and the random wave field by the wave superposition method using the correlated design values of $(U_{10}, H_{s}, T_{p})$ .

Step 4: Establish the finite element model of the bridge considering the geometric nonlinearity due to stay cables and/or large deformation.

Step 5: Compute the static deformation of the bridge due to the static wind forces and take it as the initial position of the bridge for nonlinear dynamic response analysis.

Step 6: Calculate the effective wind angle and the corresponding aerodynamic coefficient and flutter derivative at t time step.

Step 7: Calculate the time-varying wind loads of $F_{b u b}, F_{s e b}$ and wave load $F_{w b}$ .

Step 8: Solve the nonlinear dynamic responses of the bridge until to meet the convergence criteria at time t. Since all the buffeting forces, self-excited forces and wave forces are motion-dependent, the nonlinear dynamic responses at all nodes of the bridge deck and the bridge piles should meet the convergence criteria. Specifically, the Newmark-β method is used to solve the equation of motion and reach the prescribed convergence. The stiffness matrixes are also updated during iteration because of changes in structural internal forces.

Step 9: Repeat steps 6–7 until nonlinear response time histories at all the nodes of the bridge are completed.

Figure 2.

Computing procedure for nonlinear wind-wave-bridge system.

Case study

Bridge and bridge site

A sea-crossing cable-stayed bridge with a span arrangement of (90 + 145 + 560 + 145 + 90) m, as shown in Figure 3, is taken as a case study to demonstrate the proposed method for determining the nonlinear dynamic responses of the bridge to 3D correlated winds and waves with a given return period. The sea-crossing bridge is located in the East China Sea, connecting the Fuzhou city to the Pingtan island. As a typical island sea area, the bridge is surrounded by more than 20 islands and reefs. The measurement data of winds and waves are collected to develop the JPDF and environmental contours of wind speed, wave height and wave period. The measurement point is about 5 km offshore with 30.5 m water depth, as shown in Figure 3.

Figure 3.

Schematic diagram of the sea-crossing cable–stayed bridge (unit: cm).

The sea-crossing bridge consists of two towers, one truss deck (girder), four auxiliary piers and 108 stay cables. Each tower with a height of 206 m consists of two tower columns, one cap above the DWL and 38 piles below the DWL. The cap with 81.2 m × 33.2 m × 6 m (length × width × height) are supported by the group piles marked from P1 to P38. All piles and auxiliary piers are embedded in the seabed rock layer. In the finite element modelling of the bridge, the steel truss girder with a width of 22 m and a height of 6 m is divided into 117 nodes marked from G1 to G117. The reinforced concrete tower columns are divided into 21 nodes marked from T1 to T21. Each bored pile with a radius of 3.4 m is divided into several sections with a sectional height of 5 m.

The finite element model of the full bridge is built by ANSYS with a total of 5622° of freedom. The bridge deck, the towers, and the piers are simulated by 3D beam elements(BEAM4). The cables are simulated by uniaxial tension-compression link elements (LINK11) to consider cable sag effect. The added masses for considering the secondary dead loads are simulated by mass elements (MASS21) which are then added on the corresponding nodes of the bridge. The modal analysis shows that the first natural frequency of the bridge in the lateral direction (z-axis) is 0.210 Hz while the first natural frequency of the bridge in the vertical direction (y-axis) is 0.278 Hz and the first torsional frequency (about x-axis) is 0.435 Hz. The damping ratio of 2% is adopted according to JTG-T (2018). The Rayleigh structural damping matrix is then figured out by using the mass coefficient of 0.0057 and the stiffness coefficient of 0.0619.

JPDF of wind and wave parameters

A total of 10 years of measurement data from 2007 to 2016 are collected, which contains the maximum 10-min mean wind speed U₁₀, the maximum 30-min significant wave height H_s and the associated peak wave period T_p within 1 hour. Before analyzing the wind and wave data to construct the JPDF and environmental contours with different return periods, the extreme values of three variables need to be selected to form the database of extreme values. To make sure that the selected samples of extreme events are independent and identically distributed (IID), all the selected samples of (U₁₀, H_s, T_p) should be spaced more than 72 h apart, and the typhoon events are also removed from the data set. As a result, a total of 60 triplets of (U₁₀, H_s, T_p) are identified using the R Largest Order Statistics method (Fang et al., 2022).

The optimized C-vine copula proposed by (Fang et al., 2022) is used in the present work to obtain the JPDF of the extreme values of the three variables. Firstly, the mean wind speed is decided as the key variable based on the multivariate correlation analysis of the selected samples of extreme events. Secondly, the one-step optimization method is adopted to obtain the optimal marginal distributions and the optimal pair-copulas, in which the maximum log-likelihood (MLogL) values and the root mean square error are used to evaluate the fitting accuracy of the marginal distributions and pair-copulas, respectively. As shown in Figure 4, the optimal pair copula of (U₁₀-H_s) obeys the Gumbel copula and the marginal distributions are the Weibull distribution for U₁₀ and the Gamma distribution for H_s. The optimal pair copula of (U₁₀-T_p) is the Frank copula and the marginal distribution is the GEV distribution for both U₁₀ and T_p. Once the optimal pair copulas and marginal distributions are obtained, the trivariate JPDF can be achieved according to equations (1)–(3).

Figure 4.

Bivariate joint probability density function contours of (U₁₀-H_s) and (U₁₀-T_p) and the corresponding marginal distributions.

Trivariate environmental contours

The 3D environmental contours can be drawn in terms of the obtained JPDF of (U₁₀-H_s-T_p) and by using equations (4) and (5). In consideration of only 10 years measurement data available and assuming that the materials of all the bridge structural components are linear and elastic, the considered return periods are set as 20 years, 30 years and 50 years respectively.

The values of $β$ are accordingly calculated as 2.107, 2.385 and 2.533 with a mean annual rate of $λ_{E} =$ 6. The 3D environmental contours of (U₁₀-H_s-T_p) at the three different return periods are figured out and presented in Figure 5. It can be seen that the 3D environmental contour is an ellipsoid, which represents all possible combinations of the design values of the three variables (U₁₀-H_s-T_p). It can be observed that the maximum values of the mean wind speed, significant wave height and peak wave period do not occur at the same time. Because wind dominates the nonlinear dynamic response of the bridge in this study, the maximum value of U₁₀ and the corresponding values of H_s and T_p are used and marked in Figure 5. Obviously, the design values of the wind and wave parameters increase with the increase of return period.

Figure 5.

3D environmental contours of (U₁₀-H_s-T_p) at different return periods: (a) 20-year; (b) 30-year; (c) 50-year.

Table 1 lists the design values of U₁₀, H_s and T_p obtained by using either trivariate or univariate distribution for different return periods. By using univariate distributions, the correlation between wind speed, wave height and wave period is ignored and each variable is independent. In trivariate cases, the design values of the mean wind speed are 19.23 m/s, 20.72 m/s, and 22.29 m/s at 20-year, 30-year and 50-year return periods. They are actually the maximum values of U₁₀ in the 3D environmental contours, which are close to the design values obtained by the Inverse First Order Reliability Method (IFORM) based on the univariate distribution. However, the design values of H_s and T_p obtained based on the trivariate and univariate distributions show significant differences. The design values of H_s using the trivariate distribution are about 77.3%–79.6% of the design values of H_s using the univariate distribution. The design values of T_p using the trivariate distribution are about 83.6%–84.0% of the design values of T_p using the univariate distribution. Clearly, the design values of H_s and T_p without considering their correlations with U₁₀ are conservative.

Table 1.

The design values of U₁₀-H_s-T_p using the trivariate and univariate distributions.

Return periods	Trivariate			Univariate			Reduction rate
Return periods	U₁₀/m/s	Hs/m	T_p/s	U₁₀/m/s	Hs/m	T_p/s	Hs (%)	T_p (%)
20-year	19.23	4.50	7.85	19.30	5.65	9.35	79.6	84.0
30-year	20.72	4.69	7.97	20.77	6.00	9.50	78.2	83.9
50-year	22.29	4.91	8.11	22.35	6.35	9.70	77.3	83.6

3D correlated wind and wave loads

In order to assess the effects of 3D correlated environment and aerodynamic nonlinearity on the dynamic response of the sea-crossing bridge, a total of four load types are considered: trivariate distribution, univariate distribution, wind load only and wave load only. Each load type considers the three different return periods so that there are a total of 12 computation cases, as listed in Table 2. In Table 2, U_d represents the 10-min mean wind speed at the bridge deck level of 46.5 m above DWL.

Table 2.

Four load cases and 12 computation cases.

Type	Return period	Case	U_d/m/s	Hs/m	T_p/s	Type	Case	U_d/m/s	Hs/m	T_p/s
Trivariate	20-year	1	23.51	4.50	7.85	Wind only	7	23.51	—	—
	30-year	2	25.33	4.69	7.97		8	25.33	—	—
	50-year	3	27.25	4.91	8.11		9	27.25	—	—
Univariate	20-year	4	23.59	5.65	9.35	Wave only	10	—	4.50	7.85
	30-year	5	25.39	6.00	9.50		11	—	4.69	7.97
	50-year	6	27.32	6.35	9.70		12	—	4.91	8.11

Once the input wind and wave parameters (mean wind speed, wave height and wave period) are decided, the wind loads acting on the bridge can be determined according to equation (12) while the wave loads acting on the bridge can be determined according to equation (16). It should be noted that all the buffeting forces, self-excited forces and wave-excited forces are motion-related so that the computation should be iterated at each time step.

When determining the buffeting forces and the wave forces, the lateral and vertical wind fields and the wave field are simulated by the Kaimal, the Lumley-Panofsky and the JONSWAP spectra, respectively. The simulated time histories are then converted to the simulated spectrum to compare with the target spectrum to make sure the simulated time histories are accurate enough. As an example, the targeted spectra and the simulated spectra for Case 1 are shown in Figure 6. As observed, the simulated spectrum matches the target spectrum very well for both wind and wave.

Figure 6.

Comparison between target and simulated spectra: (a) lateral wind; (b) vertical wind; (c) wave.

When determining the buffeting forces and the self-excited forces, the aerodynamic coefficients and the flutter derivatives are needed. The drag, lift, and moment coefficients of the bridge deck at wind attack angles from −12° to 12° are shown in Figure 7. The flutter derivatives at the 0° wind attack angle are listed in Table 3. All aerodynamic coefficients and flutter derivatives of the bridge deck are obtained by wind tunnel sectional model tests. In considering that the hydrodynamic coefficients are independent of wind attack angle, the drag coefficient and the inertia coefficient are taken as 1.2 and 2.0, respectively, according to the Chinese Code JTS (2015).

Figure 7.

Sectional model and aerodynamic coefficients of bridge deck: (a) segment model; (b) aerodynamic coefficients.

Table 3.

The flutter derivatives of the bridge deck at 0° wind attack angle.

U _d /fB	H ₁ ^*	A ₁ ^*	H ₄ ^*	A ₄ ^*	U _d /fB	H ₂ ^*	A ₂ ^*	H ₃ ^*	A ₃ ^*
1.332	−0.271	0.029	0.220	0.150	0.786	0.154	−0.011	0.523	0.043
1.956	0.181	0.007	−0.463	0.118	1.152	0.883	−0.005	0.564	−0.023
2.631	0.858	0.030	−0.715	0.258	1.568	−2.601	−0.199	4.747	−0.039
3.912	−2.047	0.157	−2.042	0.228	2.380	4.405	−0.089	1.625	0.164
4.966	−4.247	0.310	−5.148	0.200	3.210	1.274	−0.025	0.296	0.218
6.575	−1.820	−0.023	−0.917	0.222	4.078	−0.508	−0.113	4.591	0.323
6.955	1.982	0.143	−3.757	0.212	4.523	0.074	−0.097	5.999	0.387

Results and discussion

Without loss of generality, the vertical, lateral and torsional static wind displacements of the bridge deck for Case 3 are presented in Figure 8. As observed, the three maximum static wind responses all occur at the middle section of the main span. It is noteworthy that only the dynamic effects of the wind and wave are considered in the subsequent study and the static wind response will not be discussed further. The dynamic responses of the sea-crossing bridge are computed for the four types of loading and the 12 computation cases. The results are presented and discussed as follows.

Figure 8.

Static wind responses along the bridge deck (Case 3): (a) lateral response; (b) vertical response; and (c) torsional response.

Comparison of bridge responses with and without aerodynamic nonlinearities

The dynamic responses of the bridge from Cases 1, 2 and 3 are calculated based on the NWWB system while those from Cases 4, 5 and 6 are computed based on the WWB system. The WWB system only considers the initial geometric nonlinearity caused by the dead weight of the bridge and the initial tension forces in the cables, but the aerodynamic nonlinearity and geometric nonlinearity in the iteration process are ignored compared with the NWWB system. Regarding the computation cost, for one computation case with a time history of 600 s and a time step of 0.25 s using a workstation equipped with an Intel Xeon E5-2650v5 3.40 GHz CPU and 16 GB memory size, the WWB system needs about 1 h while the NWWB system spends 8.5 h.

Figure 9 shows the time histories of lateral, vertical and torsional responses of the bridge deck at the middle section of the main span for Cases 1 and 4. The linear dynamic responses (without considering aerodynamic nonlinearity, Case 4) are represented by the black line while the nonlinear dynamic responses (with aerodynamic nonlinearity, Case 1) are represented by the red line in Figure 9. It can be observed from Figure 9 that nonlinear lateral, vertical and torsional displacement responses are slightly higher than the linear lateral, vertical and torsional displacement responses. The similar results are also found for Cases 2 and 5 with the 30 years return period and for Cases 3 and 6 with the 50-year period. Nevertheless, both the linear and nonlinear dynamic responses of the bridge increase with the increase of return period.

Figure 9.

Dynamic responses of the middle section of the main span (Case 3): (a) lateral response; (b) vertical response; and (c) torsional response.

The standard deviations of the lateral displacement response

L_{H}

, vertical displacement response

L_{V}

and torsional displacement response

θ_{t}

at the midspan of the bridge are listed in Table 4. With the increase of the three environmental variables from Case 1 to Case 3, the bridge responses increase from the 20-year return period to the 50-year return period. The SD values of the torsional displacement responses for the three nonlinear cases increase by 19.0%–21.2% compared with those for the three linear cases. The SD values of the vertical displacement responses increase by 15.3%–15.4% for the three nonlinear cases compared with those for the three linear cases. The SD values of the lateral displacement responses for the three nonlinear cases increase by 10.2%–12.7% in comparison with those for the three linear cases. The SD values of the lateral displacement response

L_{H}

, longitudinal displacement response

L_{L}

and torsional angle response

θ_{t}

at the top tower are listed in Table 5. Different from the bridge responses of the bridge deck, the maximum increment of the nonlinear dynamic responses compared with the linear dynamic responses is less than 4.0% for all the three cases (three return periods).

Table 4.

Standard deviation values of linear and nonlinear responses at the middle section of the bridge deck.

	Linear			Nonlinear			Increment
	$θ_{t}$ /rad	$L_{V}$ /m	$L_{H}$ /m	$θ_{t}$ /rad	$L_{V}$ /m	$L_{H}$ /m	$θ_{t}$ (%)	$L_{V}$ (%)	$L_{H}$ (%)
Case 1	9.14 × 10⁻⁵	2.41 × 10⁻²	1.22 × 10⁻²	1.11 × 10⁻⁴	2.78 × 10⁻²	1.36 × 10⁻²	21.0	15.3	11.9
Case 2	1.07 × 10⁻⁴	2.59 × 10⁻²	1.24 × 10⁻²	1.28 × 10⁻⁴	2.99 × 10⁻²	1.37 × 10⁻²	19.0	15.3	10.2
Case 3	1.10 × 10⁻⁴	2.78 × 10⁻²	1.37 × 10⁻²	1.33 × 10⁻⁴	3.21 × 10⁻²	1.54 × 10⁻²	21.2	15.4	12.7

Table 5.

Standard deviation values of linear and nonlinear responses at top tower.

	Linear			Nonlinear			Increment
	$θ_{t}$ /rad	$L_{H}$ /m	$L_{L}$ /m	$θ_{t}$ /rad	$L_{H}$ /m	$L_{L}$ /m	$θ_{t}$ (%)	$L_{H}$ (%)	$L_{L}$ (%)
Case 1	3.07 × 10⁻⁵	1.19 × 10⁻²	1.38 × 10⁻²	3.15 × 10⁻⁵	1.22 × 10⁻²	1.38 × 10⁻²	2.3	2.7	0.3
Case 2	3.25 × 10⁻⁵	1.27 × 10⁻²	1.50 × 10⁻²	3.31 × 10⁻⁵	1.31 × 10⁻²	1.50 × 10⁻²	1.7	3.4	0.5
Case 3	3.39 × 10⁻⁵	1.36 × 10⁻²	1.55 × 10⁻²	3.52 × 10⁻⁵	1.41 × 10⁻²	1.56 × 10⁻²	4.0	3.4	0.8

Figure 10 shows the SD values of the linear and nonlinear dynamic responses along the bridge deck for Case 3 and Case 6 with a return period of 50 years. As observed, the nonlinear responses are larger than the linear response in general. The largest difference between linear and nonlinear responses occurs at the midspan of the bridge, and the difference gradually decreases from the midspan to both side spans. The maximum standard values of the lateral response, vertical response and torsional response are 1.54 cm, 3.21 cm and 1.33 × 10⁻⁰⁴ rad, respectively. This shows that the bridge is safe enough at 0° wind attack angle to resist wind and waves within a 50-year return period.

Figure 10.

Standard deviation values of linear and nonlinear responses along the bridge deck (Case 3 vs Case 6): (a) lateral response; (b) vertical response; and (c) torsional response.

Comparison of bridge responses among only wind load, only wave load and wind-wave load

According to Table 2, Cases 7–9 consider only wind load cases while Cases 10–12 consider only wave load cases and Cases 1–3 are the wind-wave load cases. The SD values of only wind, only wave, and wind-wave induced responses along the bridge deck for the 50-year return period are shown in Figure 11. It can be observed that the wind load dominates the lateral displacement response, vertical displacement response and torsional angle response of the bridge deck. For the lateral displacement response (see Figure 11(a)), the only wave-induced response is just 7% of the wind-wave-induced displacement response. For the torsional angle response (see Figure 11(c)), the only wave-induced response is about 11% of the wind-wave-induced response. Since the wave load is in the lateral direction (z-axis) only, the wave load hardly causes any vertical response of the bridge deck as shown in Figure 11(b).

Figure 11.

Standard deviation values of only wind, only wave, and wind-wave load induced nonlinear displacement responses along the bridge deck: (a) lateral response; (b) vertical response; and (c) torsional response.

The SD values of only wind, only wave, and wind-wave load induced responses at the top tower are summarized in Table 6. For the lateral displacement response of the tower, the only wind load induced and only wave load induced displacement responses account for 66% and 34% of the total wind-wave load induced displacement response, respectively, implicating that wind and wave loads jointly control the lateral displacement response of the bridge tower. For the longitudinal displacement response of the bridge tower, the only wind load induced displacement response accounts for about 94% of the total wind-wave load induced displacement response, indicating that wind load is a dominant load for the longitudinal displacement response of the bridge tower.

Table 6.

Standard deviation values of only wind, only wave, and wind-wave load induced response at top tower.

	Wind + Wave		Wind		Wave
	$L_{H}$ /m	$L_{L}$ /m	$L_{H}$ /m	$L_{L}$ /m	$L_{H}$ /m	$L_{L}$ /m
Cases 1,7,10	1.22 × 10⁻²	1.38 × 10⁻²	7.92 × 10⁻³	1.29 × 10⁻²	4.27 × 10⁻³	6.61 × 10⁻⁴
Cases 2,8,11	1.31 × 10⁻²	1.50 × 10⁻²	8.64 × 10⁻³	1.42 × 10⁻²	4.45 × 10⁻³	7.95 × 10⁻⁴
Cases 3,9,12	1.41 × 10⁻²	1.56 × 10⁻²	9.44 × 10⁻³	1.47 × 10⁻²	4.65 × 10⁻³	7.41 × 10⁻⁴

Comparison of bridge responses between univariate and trivariate cases

In order to assess the effect of 3D correlated wind-wave loads on bridge response, the nonlinear dynamic responses of the sea-crossing bridge are computed for both the trivariate cases 1–3 and the univariate cases 4–6. The SD values of bridge nonlinear responses along the bridge deck for a 50-year return period are shown in Figure 12. As observed from Figure 12(a), the lateral displacement response of the bridge deck in the univariate case is greater than those in the trivariate case in general. The largest difference of the lateral displacement response between the univariate and trivariate cases is 7.0% at the midspan. As observed from Figure 12(c), the maximum SD value of the torsional angle response is increased by 7.1% in the univariate case, compared with that in the trivariate case. Since the wave load has almost no effect on the vertical displacement response of the bridge deck, the SD value of the vertical displacement response is almost the same for both cases, as seen from Figure 12(b).

Figure 12.

Standard deviation values of nonlinear displacement responses in the trivariate Case 3 and univariate Case 6: (a) lateral response; (b) vertical response; and (c) torsional response.

The SD values of nonlinear displacement responses of the top tower in the trivariate and univariate cases are listed in Table 7. The torsional angle responses in the trivariate cases are reduced by 7.2%–13.2% from the 20-year return period to the 50-year return period compared to those in the univariate cases. The lateral displacement responses are reduced by 17.0%–17.4% in the trivariate cases from the 20-year return period to the 50-year return period compared to those in the univariate cases. This is because the influence of wave load on the tower response is more significant than on the deck response, and larger differences between the trivariate and univariate cases are thus found in the top tower response.

Table 7.

Standard deviation values of nonlinear displacement responses of top tower in trivariate and univariate cases.

	Univariate			Trivariate			Increment
	$θ_{t}$ /rad	$L_{H}$ /m	$L_{L}$ /m	$θ_{t}$ /rad	$L_{H}$ /m	$L_{L}$ /m	$θ_{t}$ (%)	$L_{H}$ (%)	$L_{L}$ (%)
Cases 1,4	3.39 × 10⁻⁵	1.47 × 10⁻²	1.40 × 10⁻²	3.15 × 10⁻⁵	1.22 × 10⁻²	1.38 × 10⁻²	−7.2	−17.0	−1.3
Cases 2,5	3.69 × 10⁻⁵	1.58 × 10⁻²	1.53 × 10⁻²	3.31 × 10⁻⁵	1.31 × 10⁻²	1.50 × 10⁻²	−10.4	−17.2	−2.0
Cases 3,6	4.06 × 10⁻⁵	1.70 × 10⁻²	1.60 × 10⁻²	3.52 × 10⁻⁵	1.41 × 10⁻²	1.56 × 10⁻²	−13.2	−17.2	−2.0

Conclusions

A framework for determining the nonlinear dynamic responses of a sea-crossing bridge under 3D correlated wind and wave loads has been proposed in this study. The optimal C-vine copula is adopted to select the most appropriate fitting parameters and construct the joint probability distribution function of wind and wave loads. The 3D environmental contours under different return periods are derived and used to determine the design values of correlated wind-wave loads for a given return period. The nonlinear wind-wave-bridge (NWWB) system considering both geometric nonlinearity and aerodynamic nonlinearity is established to compute the nonlinear dynamic response of the bridge by Newmark-β method. A case study using a real sea-crossing cable-stayed bridge has been performed and the effects of aerodynamic nonlinearity and 3D correlated wind-wave loads on bridge responses have been investigated. The main conclusions from this study can be summarized as follows:

1. The obtained dynamic responses of the bridge using the NWWB system are higher than those using the linear WWB system. For the dynamic responses of the bridge deck at midspan, the lateral and vertical displacement responses and the torsional angle response using the NWWB system increase by about 10.2%–21.2% from the 20-year return period to the 50-year return period compared with those using the linear WWB system.

2. Compared with univariate cases without considering the correlation of wind and wave, the design values of significant wave height and peak wave period obtained for the trivariate cases are reduced by 22.7% and 16.4% in the 50-year return period. As a result, the lateral displacement responses of the bridge deck at the midspan and the top tower in the trivariate cases are reduced by 7.0% and 17.2%, respectively, compared with those in the univariate cases. The bridge response will be thus overestimated if the correlation between wind and waves is not considered.

3. Wind load is found to be a dominating load for the bridge response in this study by comparing only wind load, only wave load, and wind-wave load induced responses. The wave load shows only a slight amplification effect on the lateral displacement and torsion angle responses of the bridge deck but 34% of the lateral displacement response of the top tower is contributed by wave load, indicating that wave load has a significant impact on the bridge tower.

It is noted that the present study considered wind and wave loads of a return period less than or equal to 50 years and assumed that the deformation of bridge members is still within the linear elastic range and that only aerodynamic nonlinearity and geometric nonlinearity are considered. Nevertheless, the full nonlinearity including material nonlinearity of the bridge under the extreme wind and wave loads of 100-year return period needs further investigation.

Footnotes

Acknowledgements

The works described in this paper are financially supported by the Changjiang Scholars Program of the Ministry of Education of China to which the authors are most grateful.

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 study is supported by Changjiang Scholar Program of Chinese Ministry of Education (YH1199911012201) and the National Natural Science Foundation of China (Grants No. 52208504).

Disclaimer

Any opinions and conclusions presented in this paper are entirely those of the authors.

ORCID iDs

Chen Fang

You-Lin Xu

References

Bai

Jiang

Song

, et al. (2022) Extreme responses of sea-crossing bridges subjected to offshore ground motion and correlated extreme wind and wave. Ocean Engineering 247: 110710.

Bonakdar

Oumeraci

Etemad-Shahidi

(2015) Wave load formulae for prediction of wave-induced forces on a slender pile within pile groups. Coastal Engineering 102: 49–68.

Chakrabarti

(1971) Discussion of nondeterministic analysis of offshore structures. Journal of the Engineering Mechanics Division 97(3): 1028–1029.

Chen

Kareem

(2001) Nonlinear response analysis of long-span bridges under turbulent winds. Journal of Wind Engineering and Industrial Aerodynamics 89(14): 1335–1350.

Cheng

(2020) Multivariate joint probability function of earthquake ground motion prediction equations based on vine copula approach. Mathematical Problems in Engineering 2020: e1697352.

Fang

Tang

, et al. (2020a) Effects of random winds and waves on a long-span cross-sea bridge using Bayesian regularized back propagation neural network. Advances in Structural Engineering 23(4): 733–748.

Fang

Tang

, et al. (2020b) Stochastic response of a cable-stayed bridge under non-stationary winds and waves using different surrogate models. Ocean Engineering 199: 106967.

Fang

Y-L

(2022) Optimized C-vine copula and environmental contour of joint wind-wave environment for sea-crossing bridges. Journal of Wind Engineering and Industrial Aerodynamics 225: 104989.

Guo

Liu

Chen

, et al. (2016) Experimental study on the dynamic responses of a freestanding bridge tower subjected to coupled actions of wind and wave loads. Journal of Wind Engineering and Industrial Aerodynamics 159: 36–47.

10.

JTG-T 3360-01-2018 (2018) Wind-resistant design specification for highway bridges. Beijing, China: Ministry of Transport of the People’s Republic of China.

11.

JTS-145-2015 (2015) Chinese code of the hydrology for harbor and waterway. Beijing, China: Ministry of Transport of the People’s Republic of China.

12.

Zhou

Liu

(2018) Statistical modelling of extreme storms using copulas: a comparison study. Coastal Engineering 142: 52–61.

13.

Luo

Huang

(2017) Characterizing dependence of extreme wind pressures. Journal of Structural Engineering, ASCE 143(4): 04016208.

14.

Luo

Huang

Han

, et al. (2021) Evaluation of full-order method for extreme wind effect estimation considering directionality. Wind and Structures 32(3): 193–204.

15.

Meng

Ding

Zhu

(2018) Stochastic response of a coastal cable-stayed bridge subjected to correlated wind and waves. Journal of Bridge Engineering 23(12): 04018091.

16.

Montes-Iturrizaga

Heredia-Zavoni

(2016) Multivariate environmental contours using C-vine copulas. Ocean Engineering 118: 68–82.

17.

Padgett

Reginald

Bryant

, et al. (eds) (2008) Bridge damage and repair costs from hurricane katrina. Journal of Bridge Engineering 13(1): 6–14.

18.

Robertson

Riggs

Yim

, et al. (2007) Lessons from hurricane katrina storm surge on bridges and buildings. Journal of Waterway, Port, Coastal, and Ocean Engineering 133(6): 463–483.

19.

Rosenblatt

(1952) Remarks on a multivariate transformation. The Annals of Mathematical Statistics 23(3): 470–472.

20.

Sun

, et al. (1999) Fully-coupled buffeting analysis of long span cable-supported bridges: formulation. Journal of Sound and Vibration 228(3): 569–588.

21.

Zhang

, et al. (2019) Numerical study on the stochastic response of a long-span sea-crossing bridge subjected to extreme nonlinear wave loads. Engineering Structures 196: 109287.

22.

Wang

Zhang

, et al. (2021) Circular-linear-linear probabilistic model based on vine copulas: an application to the joint distribution of wind direction, wind speed, and air temperature. Journal of Wind Engineering and Industrial Aerodynamics 215: 104704.

23.

Wei

Zhou

(2017b) Experimental study of the hydrodynamic responses of a bridge tower to waves and wave currents. Journal of Waterway, Port, Coastal, and Ocean Engineering 143(3): 04017002.

24.

Wei

Myers

Arwade

(2017a) Dynamic effects in the response of offshore wind turbines supported by jackets under wave loading. Engineering Structures 142: 36–45.

25.

Xiao

Liu

Zhou

, et al. (2022) Reliability analysis of bridge girders based on regular vine Gaussian copula model and monitored data. Structures 39: 1063–1073.

26.

Sun

, et al. (2000) Fully coupled buffeting analysis of Tsing Ma suspension bridge. Journal of Wind Engineering and Industrial Aerodynamics 85(1): 97–117.

27.

Yue

Liu

, et al. (2020) Effects of heave plate on dynamic response of floating wind turbine Spar platform under the coupling effect of wind and wave. Ocean Engineering 201: 107103.

28.

Zaheer

Islam

(2012) Stochastic response of a double hinged articulated leg platform under wind and waves. Journal of Wind Engineering and Industrial Aerodynamics 111: 53–60.

29.

Zaheer

Islam

(2017) Dynamic response of articulated towers under correlated wind and waves. Ocean Engineering 132: 114–125.

30.

Zhang

Wang

Liu

(2020) Joint distribution of wind speed, wind direction, and air temperature actions on long-span bridges derived via trivariate metaelliptical and Plackett copulas. Journal of Bridge Engineering 25(9): 04020069.

31.

Zhu

Zhang

(2018) Coupled dynamic analysis of the vehicle-bridge-wind-wave system. Journal of Bridge Engineering 23(8): 04018054.