Finite element analysis of long-span rail-cum-road cable-stayed bridge subjected to ship collision

Abstract

Ship collision is rare, yet it leads to serious consequences once it occurs, in particular for long-span bridges. This study investigates dynamic responses of a long-span, rail-cum-road cable-stayed bridge under ship collision through finite element analysis. Three ship tonnages were investigated, which are 3000, 5000, and 8000 t, respectively. The displacement, velocity, and acceleration of the bridge under ship collision are analyzed. The collision process is simulated in two explicit steps to improve the computational efficiency. First, the collision force is determined through a collision simulation of the ship to a rigid body that simulates the massive bridge pier. The collision force is then applied to the bridge to analyze the dynamic responses of the bridge. The simulation results of the collision force are compared with four different design codes. Analysis results from different codes show significant discrepancies, demonstrating lack of reliability of the formula recommended by the codes. The results indicate that the maximum displacement and acceleration occur at the top of the bridge pylon. The bridge’s responses under ship collision decrease as the collision angle increases from 0° to 20°.

Keywords

collision angle dynamic response finite element method long-span bridge rail-cum-road bridge ship collision

Introduction

Bridges over navigable waterways are exposed to the risk of ship collision, which is rare, yet leads to serious consequences once it occurs, in particular for long-span bridges. According to the statistics, at least one serious ship collision accident occurs per year worldwide, and there were 34 major bridge collapses with a total loss of 346 lives during 1960–2007 (Sha and Hao, 2012). In 1993, a barge collided with the CSX Railway Bridge in the United States and caused serious deformation of the railway track, resulting in derailment of a train. The accident caused casualty of 47 passengers and significant capital loss. In china, the Wuhan Yangtze River Bridge in China experienced about 70 ship collision events between 1975 and 1999 (Zhang et al., 2016), and the Nanjing Yangtze River Bridge experienced about 35 collision accidents in the 45 years since its completion in 1968. A more recent example of ship bridge collision was about Jiujiang Yangtze River Bridge; a cargo impacted the bridge pier, causing the collapse of two side spans. The accident caused nine people to lose their lives and significant economic losses and long legal battles (Sha and Hao, 2012). So, it is significant to investigate ship collision resistance of bridges and ensure adequate collision resistance in bridge design.

Minorsky (1959) conducted 26 ship–ship collision tests and derived a relationship between the ship deformation and the dissipated energy through the collision. Based on that research, Woisin (1976) carried out further experimental studies and proposed a new formula to link the ship deformation and the dissipated energy. The experimental studies provided valuable and reliable data of ship–ship collision and improved the understanding of the collision process. As ship collision accidents draw more attention worldwide in the 1980s, design codes were issued and an empirical formula was recommended to consider the collision effect in bridge design (American Association of State Highway and Transportation Officials (AASHTO), 2009; JTG D60-2015, 2015; TB 10002-2017, 2017; Vrouwenvelder, 1998). In the existing methods recommended in the design codes, a collision force is applied to bridge. The collision force is a static force. While the recommended method is convenient to use for engineers, the simplification of the collision process and neglected dynamic effect may lead to remarkable errors and underestimation of the collision effect. In situ test and large-scale model test are effective approaches to investigate the responses of bridges (Gou et al. 2018a, 2018b, 2018c, 2018d, 2018e, 2018f, 2018g). However, ship–bridge collision tests are typically expensive and time-consuming, and are less realistic in bridge design.

As an alternative, the finite element method has been used to investigate ship collision in different scenarios. Since pier is the bridge component that is most probably subjected to ship collision, most of the existing studies were focused on ship collision to bridge pier. Consolazio and Cowan (2003) used ADINA^® to analyze the collision force of barge collision against bridge piers with the consideration of different pier sizes and shapes. Sha and Hao (2012, 2013, 2014) and Sha et al. (2017) conducted comprehensive research on the collision force. A finite element model of barge–pier was developed in LS-DYNA, and the bridge was simplified as a nonlinear single-degree-of-freedom system. Yuan and Harik (2008) studied the time history response of the collision force between bridge pier and ship, and different pier geometries were considered. Gholipour et al. (2018) and Aziz et al. (2017) studied the effects of soil–pile interaction on the responses of bridges subjected to barge collision. The finite element analyses in the literature greatly advance the understanding of ship–pier collision forces. To date, there remains lack of knowledge on the effect of ship collision on the dynamic responses of the bridge. Such knowledge gap creates technical challenges in designing bridges under the risk of ship collision. With an increasing demand of long-span railway bridges in China, there is an urgent need to study the responses of railway bridges under ship collision.

This study investigates the dynamic responses of a long-span, rail-cum-road cable-stayed bridge under ship collision through a finite element analysis. Three ship tonnages were investigated, which are 3000, 5000, and 8000 t. The displacement, velocity, and acceleration of the bridge are analyzed using three-dimensional (3D) finite element models of the bridge and ship. The simulation results are compared with the results determined according to different design standards. This research provides data for improving the design and safety assessment of long-span rail-cum-road cable-stayed bridges subjected to ship collision.

Background

In this study, ship collision analysis is conducted based on the Tianxingzhou Yangtze River Bridge, located in Wuhan, China, as shown in Figure 1(a). The bridge is the world’s longest rail-cum-road cable-stayed bridge and plays an important role in the Beijing–Guangzhou Railway Line and the Wuhan Highway Network. Due to the heavy ship flow in the Yangtze River, the bridge is subjected to high risk of ship collision.

Figure 1.

Tianxingzhou Yangtze River Bridge (unit: m): (a) Tianxingzhou Yangtze River Bridge; (b) elevation view; (c) plan view of the highway level; (d) plan view of the railway level; (e) cross section of section 1-1; and (f) plan view of the bridge pylon.

The bridge has a span layout of 98 m + 196 m + 504 m + 196 m + 98 m, as shown in Figure 1(b). The bridge has two pylons, and there are 3 × 16 stayed cables at each side of a pylon. The cables are anchored to the upper chord of the main girder, which is a steel truss composed of three truss planes with a center-to-center spacing of 15 m in the transverse direction, as depicted in Figure 1(c). The truss members of the center truss have box sections. The truss members of the side trusses have H-shaped sections. Along the longitudinal direction, the trusses have an equal spacing of 14 m.

The width of the bridge accommodates six highway lanes at the top level and four railway lanes at the bottom level, as illustrated in Figure 1. For the top highway level, a steel–concrete composite deck is used at each end measuring 168 m in length, while a steel orthotropic deck is used in the middle length measuring 756 m in length. The design speed is 80 km/h for vehicles on the highway. At the bottom railway level, longitudinal beams and crossing beams with I-shaped cross sections are used to support the railway track. The design speeds are 200 km/h for passenger trains and 120 km/h for freight trains.

The main cables are made of Φ7 galvanized steel wires. The tensile strength, Poisson’s ratio, Young’s modulus, and density of the wire are 1670 MPa, 0.3, 200 GPa, and 8354 kg/m³, respectively. The bridge pylon was made of reinforced concrete, and the structural details of the pylon are shown in Figure 1(f). The design compressive strength of the concrete is 50 MPa. The key properties of the materials used in this study are clarified in the drawings of the bridge design, as listed in Table 1.

Table 1.

Material properties.

Material	Input parameters	Magnitude
Steel (ship and bridge truss)	Mass density	7580 kg/m³
	Young’s modulus	210 GPa
	Poisson’s ratio	0.3
	Yield stress	235 MPa
	Failure strain	0.2
Steel (stayed cables)	Mass density	8354 kg/m³
	Young’s modulus	205 GPa
	Poisson’s ratio	0.3
	Tensile stress	1670 MPa
Concrete	Mass density	2650 kg/m³
	Young’s modulus	46.5 GPa
	Poisson’s ratio	0.2
	Failure strain	0.05
	Compressive strength	50 MPa

Table 2 lists the dimensions of the ships at different tonnage levels. In this study, the ships with the representative bulb bow are considered. The ships are made of a low-carbon steel, as detailed in Table 1.

Table 2.

Dimensions and mass of the ships.

Tonnage (t)	Length (m)	Width (m)	Maximum height (m)	Full-load mass (t)
3000	104	16.5	2.8	5332
5000	110	19.2	4.2	9100
8000	128	18.0	6.5	12,572

Finite element analysis

This section introduces the finite element analysis method adopted in this study. To be strict, in the collision process, the displacements of the ship and the bridge pier are coupled at the collision spot—the vibration of the bridge pier affects the collision process. The computation of fully coupled vibrations of the ship and the bridge needs high capability of the computing hardware and is time-consuming. In this study, a more efficient yet reasonable approach recommended by Sha et al. (2017) is adopted. In general, the deformation of the bridge pier is much smaller than the deformation of the ship, implying that the effect of bridge deformation on the displacement and deformation of the ship is neglected. Thus, a sequentially coupled, two-step analysis is performed to analyze the bridge responses under ship collision for an improved computational efficiency (Sha et al., 2017; Yuan, 2005):

Step 1. To determine the collision force between the ship and bridge pier. The bridge pier is modeled as a rigid body with no deformation. Deformation in the ship is considered.

Step 2. To determine the responses of the bridge. The collision force determined in Step 1 is input to the bridge pier in Step 2. Deformation of the bridge is considered.

The bridge model

The finite element code of the bridge is established using the software ANSYS. The steel orthotropic bridge deck of the bridge’s highway layer is simulated using beam elements with a retained global stiffness for improving the computational efficiency, as elaborated by Li and Qiang (2003). Figure 2 illustrates the arrangement of the beam grid for simulating the upper bridge deck. The truss elements, pylon, and the beam grids are modeled using 3D two-node beam elements (BEAM4). The 168-m steel–concrete composite bridge deck at each end of the bridge is modeled using 3D four-node shell elements (SHELL63). The cables are modeled using 3D two-node tension-only bar elements (LINK10). In total, the bridge model has 15,208 nodes and 24,122 elements. Figure 3 shows the finite element model of the bridge.

Figure 2.

Orthotropic plate simplified calculation model.

Figure 3.

FE model of the Tianxingzhou Yangtze River Bridge.

Considering that materials may undergo inelastic stages during the ship collision process, a concrete damage model that considers the strain rate effect and post-fracture behaviors of concrete is used with the keyword “MAT_CONCRETE_DAMAGE_REL3” (Sha and Hao, 2013) to define the concrete in the bridge pier. As for the cables, the keyword “MAT_PLASTIC_KINEMATIC” is used to consider the inelastic behaviors of the stayed cables.

The geometrical nonlinearity effect is also considered in the bridge model. The cable’s sag effect is analyzed using the Ernst formula through modifying Young’s modulus of the cable material (Gu and Xiang, 2011). An equivalent Young’s modulus is defined

E_{e} = \frac{E_{0}}{1 + \frac{γ^{2} l^{2} E_{0}}{12 σ^{3}}}

(1)

where $E_{e}$ is the equivalent Young’s modulus (unit: MPa); $E_{0}$ is Young’s modulus of the material (unit: MPa); $γ$ is the line severity of the cable (unit: kg/m); l is the horizontal projection length of cables (unit: m); and $σ$ is the cable stress (unit: kN).

The finite element model is verified through two computer tests. In the first test, the simulation results of the bridge deformation under gravity are compared with the test data reported by Wu (2010). Under the most unfavorable loading, the displacement at the center of the middle span was measured to be −0.523 m (negative sign of displacement indicates downward displacement). According to the equivalent load, the deflection ratio under the design live load is 1/889, which is less than 1/500 of the mid-span flexing ratio of the bridge designed by the design department for the specificity of the bridge. Based on the finite element model, under gravity, the displacement at the center of the middle span is predicted to be −0.258 m, and all the bridge components behave within the elastic range; the displacement at the center of the side span is −0.220 m and the displacement at the quarter point of the middle span is −0.013 m. The predicted displacement at the center of the middle span is less than the measured displacement, indicating that the bridge has sufficient stiffness. The predicted displacements are reasonable.

In the second test, the dynamic characteristics of the bridge are analyzed. Modal analysis is performed to determine the vibration characteristics, that is, frequencies and the corresponding mode shapes. Figure 4 shows the first three mode shapes of the bridge under free vibration. The first 10 mode shapes and frequencies are summarized in Table 3. The fundamental transverse and vertical frequencies are 0.269 and 0.408 Hz, respectively, which are much smaller than the frequencies corresponding to the first torsional mode (seventh mode), revealing that the girder has relatively higher torsional stiffness than the transverse and vertical stiffness. This is consistent with the structure of the bridge girder, which is a truss with three truss planes and two layers. The model analysis results are compared with the results reported by Qian et al. (2011), as shown in Table 4. The maximum discrepancy is −7.8%.

Figure 4.

Typical bridge mode shapes: (a) Mode 1 (symmetrical transverse bend); (b) Mode 2 (symmetrical vertical bend); and (c) Mode 3 (symmetrical vertical bend).

Table 3.

Natural frequencies and mode shapes.

Mode no.	Frequency (Hz)	Mode shape
1	0.269	Symmetric transverse bend
2	0.408	Symmetric vertical bend
3	0.576	Asymmetric vertical bend
4	0.658	Asymmetric transverse bend
5	0.705	Asymmetric vertical bend
6	0.955	Asymmetric transverse bend of side span
7	0.973	Symmetric rotation
8	1.008	Second-order rotation
9	1.126	Second-order asymmetric rotation
10	1.140	Second-order asymmetric rotation

Table 4.

Comparison of natural frequency and mode characteristics.

Mode no.	This study		Qian et al. (2011)
Mode no.	Frequency (Hz)	Mode characteristics	Frequency (Hz)	Mode characteristics
1	0.269	Symmetric transverse bend	0.292	Symmetric transverse bend
2	0.408	Symmetric vertical bend	0.381	Symmetric vertical bend
3	0.576	Asymmetric vertical bend	0.571	Asymmetric vertical bend

The above two computer tests demonstrate that the bridge model provides adequate predictions of the static and dynamic responses of the bridge.

The ship model

LS-DYNA is an explicit finite element software which is considered to be effective in solving collision problems. So, it is used for the ship models in this study. As collision takes place, the collision point is at the ship bow in the front of the ship (Sha and Hao, 2013). While the global mesh size is 100 mm, refined (10 mm) meshing is used at the bow locally, as shown in Figure 5. The refined and global mesh sizes are determined through a mesh size convergence study. The ship body is modeled using 3D four-node shell elements (SHELL63). The rear part of the hull away from the bow is modeled as a rigid body.

Figure 5.

FE model of the ship.

Under collision, the materials of the ship and bridge pylon undergo a dynamic loading with a varying strain rate. A material model defined using the keyword “MAT_PLASTIC_KINEMATIC” is used to analyze the inelastic behaviors of the ship. The influence of the strain-hardening effect and strain rate effect on the yield strength is considered (Li and Huang, 2015). The mechanical properties of the steel are listed in Table 1.

During the collision, water around the hull is involved in the hull movement and participates in the collision energy absorption (Zhang et al., 2014). The interaction between the hull and water in the collision process is simulated by additional water mass. The additional water mass depends on multiple factors, such as the characteristics of the ships and the detailed collision process. In this study, the additional water mass is considered 4% of the ship’s mass (Glykas and Das, 2001). In the model, it is simulated by increasing the hull material density by 4%.

A surface-to-surface contact pair is defined to simulate the interaction between the ship and bridge using the keyword “CONTACT_AUTOMATIC_SURFACE_TO_SURFACE” in LS-DYNA. A penalty friction is defined for the tangential behavior with a coefficient of friction of 0.3 (Sha and Hao, 2013). The simulation method for ship–bridge collision was validated by Sha and Hao (2012, 2014).

The investigated parameters

Based on the finite element models of the bridge and ship, a parametric study is conducted to investigate the effects of the tonnage level and the collision angle on the collision force and bridge responses. In this study, the initial velocity of the ship at collision is 4.5 m/s, based on the survey and statistics for the Tianxingzhou Bridge. The velocity is dependent on multiple factors in practical applications (Wang, 2009), such as the distance from the navigation channel centerline to the pier, the length of the ship, and the hydrological conditions of the water channel.

According to the navigation standard of the Yangtze River, three tonnage levels of ships, 3000, 5000, and 8000 t, are investigated. The 5000-t ship represents the standard and majority of ships in the Yangtze River; the 3000- and 8000-t ships, respectively, represent light and heavy ships in comparison with the standard 5000-t ship.

The collision angle refers to the angle between the ship’s traveling direction and the transverse/normal direction of the bridge. The code JTG D60-2015 indicates that an angle greater than 5° may likely cause ship–bridge collision. The collision angle is dependent on the change of river course and river regime, the flow direction, the normal angle between the flow direction and the axes of bridge, the wind and the deflection angle of current pressure at the bridge site, and so on. According to the design of the Tianxingzhou Bridge, the collision angle increases from 0° to 20° at different water levels.

Collision force in design codes

In the existing bridge design codes, the effect of ship collision is considered by applying a static force to the bridge pier. Such simplification fails to consider the effect of the pier geometry, duration of collision, material nonlinearity, and so on (Sha and Hao, 2014). Moreover, since the collision is a dynamic process, the equivalent static analysis may not provide sufficient accuracy in the prediction. The numerical simulation results are compared with the prediction results according to different design codes. The investigated design codes include AASHTO code, International Association for Bridge and Structural Engineering (IABSE) code, Code for Design on Railway Bridge and Culverts (CDRBC; TB 10002-2017), and General Specifications for Design of Highway Bridges and Culverts (GSDHBC; JTG D60-2015).

The AASHTO code

P_{t} = 0.122 {(D W T)}^{1 / 2} V \pm 50 %

(2)

where P_t is the average collision force (unit: MN); DWT is the ship tonnage (unit: t); and V is the initial velocity of the ship at the moment of collision (unit: m/s).

The IABSE code

P = 0.024 {(V D_{max})}^{2 / 3}

(3)

where P is the design collision force (unit: MN); V is the initial velocity of the ship at the moment of collision (unit: m/s); and D_max is the full-load displacement (unit: t).

GSDHBC

F = \frac{WV}{gT}

(4)

where F is the average collision force (unit: kN); W is the drift gravity (unit: kN), determined through surveys; V is the velocity of water (unit: m/s); T is the duration of collision (unit: s), assumed to be 1 s; and g is the gravity acceleration (g = 9.81 m/s²).

CDRBC

F = γ ν \sin α \sqrt{\frac{W}{C_{1} + C_{2}}}

(5)

where F is the collision force (unit: kN); γ is the kinetic energy reduction factor (γ = 0.3 for positive impact); ν is the initial velocity of the ship at the moment of collision (unit: m/s); α is the attack angle (α = 90° in this study); W is the tonnage of the ship (unit: kN); C₁ is the elastic deformation coefficient of the ship; and C₂ is the elastic deformation coefficient of the bridge pier and abutment. In this study, due to lack of data, the sum of C₁ and C₂ is assumed to be 0.0005 m/kN.

Results and discussion

Collision force

The time history curves of the collision forces under different tonnages are shown in Figure 6. For the three tonnages, the collision forces follow a consistent trend. As the ship gets in contact with the bridge pier, the collision force rapidly increases to the peak value and then decreases until zero. In the descending stage, the collision force fluctuates. The fluctuation results from the vibration of the ship and the progressive damage process of the ship. In the collision process, the bow of the ship is in direct contact with the bridge pier, which is represented by the rigid body. The bow of the ship is gradually crushed, accompanied with the change of structural stiffness at the contact location. The maximum collision force increases with the tonnage of the ship. The peak collision forces corresponding to the tonnages of 3000, 5000, and 8000 t are 18, 29, and 38 MN, respectively (Figure 7).

Figure 6.

Time history of collision force: (a) 3000 t, (b) 5000 t, and (c) 8000 t.

Figure 7.

Time history of velocity: (a) 3000 t, (b) 5000 t, and (c) 8000 t.

According to the above formula and the ship dimensions in Table 2, the collision forces can be calculated according to the different design codes, as listed in Table 5. Different design codes provide significantly different results of the collision force, due to the different considerations in the derivation of the formulas in different design codes. It can be seen from Figure 8 that the formula recommended by AASHTO predicts the largest collision force, and the formula recommended by IABSE predicts the collision force consistent with the simulation results. The significant discrepancy generates challenges in bridge design and evaluation, since different codes give different predictions. The different predictions are likely due to the lack of knowledge on the ship collision effect. The currently recommended empirical formulae only consider a limited number of variables, and the dynamic effect due to ship collision is not considered. The formulae recommended by AASHTO and IABSE are based on the experimental data. The effect of the collision velocity on the collision force is considered. The prediction results from the formula recommended by CDRBC are the minimum, possibly attributed to the kinetic energy reduction factor (0.3) in the formula. The prediction results from the formula recommended by GSDHBC significantly vary with the tonnage of the ship. As the collision duration is difficult to estimate, the collision time is assumed to be 1 s. As can been seen from Figure 8 and Table 5, collision forces predicted by the IABSE code and the finite element model are close. Based on the above analysis, the dynamic effect cannot be ignored and the velocity of the ship at the moment of collision and full-load displacement are important for the prediction accuracy of the collision force.

Table 5.

Equivalent static load of codes and maximum collision force of simulation.

Tonnage (t)	AASHTO (MN)	IABSE (MN)	CDRBC (MN)	GSDHBC (MN)	Simulation (MN)
3000	30.1	20.0	21.2	13.9	18
5000	38.8	28.5	36.2	18.2	29
8000	49.1	35.4	50.0	21.4	38

AASHTO: American Association of State Highway and Transportation Officials; IABSE: International Association for Bridge and Structural Engineering; CDRBC: Code for Design on Railway Bridge and Culverts; GSDHBC: General Specifications for Design of Highway Bridges and Culverts.

Figure 8.

Comparison of collision forces predicted by the codes and the FE model.

Bridge responses

A transient analysis is conducted to investigate the bridge responses under the collision force. The time history of the collision force (Figure 6) is applied to the bridge pier. Since the duration of collision is short (about 3–4 s), the interference with the soil–pile interaction is not considered. This is consistent with the recommendations of Sha and Hao, (2012) and Wang et al. (2016). The transverse displacements and accelerations of the bridge at the top of the pylon, the joint between the pylon and the beam, and the middle span of the main girder are obtained from the finite element model under the ship collision. The results are shown in Figures 9 and 10.

Figure 9.

Transverse displacements of (a) pylon top, (b) pylon–girder joint, and (c) mid-span of girder.

Figure 10.

Transverse accelerations of (a) pylon top, (b) pylon–girder joint, and (c) mid-span of girder.

Figure 9 shows the time history curves of the transverse displacements of the bridge. The ship collision force excites vibrations of the bridge. It can be seen from Figure 9 that different tonnages of the ship lead to consistent transverse displacements of the bridge. This is because the frequencies and mode shapes are dependent on the bridge. However, the vibration amplitudes are dependent on the collision force. Taking the 5000-t ship for example, the transverse displacement of the pylon top reaches the peak value (7.5 cm) at t = 0.7 s; the transverse displacement of the joint between the pylon and bridge girder reaches the peak value (0.5 cm) at t = 0.5 s; the transverse displacement of the middle span of the bridge girder reaches the peak displacement (4.5 cm) at t = 5.5 s. Under ship collision, the pylon top exhibits the largest transverse displacement, and the joint between the pylon and bridge girder has the smallest transverse displacement, due to the constraint of the bridge girder.

Figure 10 shows the time history curves of the transverse accelerations at different locations of the bridge. The peak acceleration of the top of the pylon is larger than that of the middle span and the joint between the pylon and beam. Again, this is likely due to the constraint of the bridge girder. The acceleration of the pylon top is more sensitive to the collision and should be paid more attention in bridge design and condition monitoring.

Effect of collision angle

Figures 11 and 12 show the effects of the collision angle on the maximum transverse and longitudinal displacements of the pylon top and the middle span of the bridge girder. The transverse and longitudinal displacements of the pylon top are greater than those of the middle span of the bridge girder under different collision angles. As the collision angle increases from 0° to 20°, the displacement of the pylon top does not change, while the transverse displacements of the middle span are reduced by up to 50%. Therefore, in the anti-collision design of the bridge, in addition to using standard energy absorption devices, increasing the collision angle is a plausible approach to reduce ship collision on the bridge responses.

Figure 11.

Comparison of the transverse maximum displacement of (a) pylon top under different collision angles and (b) the middle span under different collision angles.

Figure 12.

The longitudinal maximum displacement of (a) pylon top under different collision angles and (b) the middle span under different collision angles.

Summary and conclusion

This article investigates the dynamic responses of a long-span, rail-cum-road cable-stayed bridge under ship collision through finite element analysis and presents comparison of the collision forces with the recommendations in different design codes. The formulae in the current design codes give significantly different predictions of the collision force. The finite element method proposed in this study likely provides a reasonable prediction of the collision force to promote bridge design and evaluation. According to the finite element analysis, the velocity of the ship at the moment of collision and full-load displacement are important for the prediction accuracy of the collision force. So, conducting a detailed survey of the tonnage and speed of navigation vessels is strongly suggested in the bridge design, which is also suggested to be taken into account in the design formula in the future study. It can be found by the dynamic response of the bridge that the maximum displacement and acceleration occur at the top of the bridge pylon, which should receive more attention in the bridge design. Properly increasing the collision angle helps reduce the damage severity of the bridge under ship collision. Increasing the width of the navigation channel tends to increase the collision angle. Further research is needed to develop effective approaches to increase the collision angle.

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: The research was funded by the National Natural Science Foundation of China (Grant No. 51878563), the Sichuan Science and Technology Program (Grant Nos 2018JY0294 and 2018JY0549), and the Ministry of Science and Technology of China (Grant No. KY201801005).

ORCID iD

Hongye Gou

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