Safety Assessment of Coastal Bridge Superstructures with Box Girders under Potential Landslide Tsunamis

Abstract

The superstructure of a coastal bridge is prone to overturn or unseating under a catastrophic tsunami, which seriously affects the post-disaster emergency rescue. In this paper, we establish a safety assessment framework for the superstructure of a bridge with a box girder under a potential landslide tsunami, and apply it to an in-service box girder and Baiyun Slide Complex on the southeast coast of China. First, a meshless numerical approach called Tsunami Squares (TS) is used to predict the movement of landslides and tsunamis. Additionally, we introduce the velocity-weakening basal friction effect in the model to optimize the landslide dynamics. Second, the maximum lateral and vertical wave loads on a box girder can be estimated using the time series of the wave height and velocity in the TS model. Third, we construct a safety evaluation method for the superstructure using the reaction of the bearing as the critical index. The results indicate that the framework developed here provides instructive guidance for evaluating the safety of coastal bridge superstructures during tsunami disasters, and we discuss the influence of the basal friction effect, bridge elevation, and support type on the structural safety.

1. Introduction

The frequent occurrence of extreme events such as debris flows, storm surges, landslide tsunamis, hurricanes, and earthquake tsunamis [1,2] pose a severe threat to coastal structures, such as floating docks [3,4,5,6] and bridge superstructures [7,8]. Among these hazards, the risk of catastrophic landslide tsunamis should be include as they occur suddenly and are extremely destructive. In 2011, a devastating tsunami, triggered by the seismic forces of the Great East Japan Earthquake and subsequent submarine landslides, destroyed more than 300 bridges by damaging their bearings, the transverse offset of the deck, and overturning [9]. In 2018, the flank collapse of the Anak Krakatau volcano generated tsunami waves that caused over 430 fatalities and catastrophic damage in the coastal regions [10]. The failures of coastal bridge superstructures may further influence post-tsunami relief programs and induce more casualties during tsunami events [11]. Hence, there is an urgent need to improve our understanding of the safety of coastal bridge superstructures under potential landslide tsunamis.

The numerical simulation of landslide tsunamis is based on various models and assumptions, such as Smoothed Particle Hydrodynamics, Discontinuous Deformation Analysis, Particle Flow Code, Tsunami Ball, and Tsunami Squares, which is related to landslide geometric and material parameters [12]. Among these models, the basal friction at the bottom of the moving mass is an essential factor affecting the accuracy of landslide simulation. This parameter describing energy dissipation is generally simply replaced by a static friction coefficient, although this assumption does not always hold, especially for large and fast-moving landslides [13]. Lucas et al. [14] gave a consistent estimate of the effective friction coefficient of natural landslides across the Solar System demonstrating that friction decreases with increasing volume. Various mechanical and thermodynamic processes are conducive to the instability of shear resistance upon fast shearing, possibly explaining the effective friction coefficient change in a sliding movement [15,16]. Several high-velocity rotary shear experiments have analyzed the evolution of the frictional coefficient with shear displacement, and the results showed that the frictional coefficient reached its peak value only with a small shear displacement, significantly declining after the peak and approaching a steady state value with increasing shear displacement [17,18]. The above research shows that the conventional static basal friction coefficient cannot accurately reflect the energy loss of large-scale landslides interacting with the bottom surface. Landslide modeling should consider the time-varying properties of basal friction and its performance in specific scenarios should be examined.

In 2004 and 2005, hurricanes Ivan and Katrina devastated several coastal bridges in the U.S., which marked a turning point in the study of extreme wave loads on coastal bridge superstructures [19]. The earlier research mainly focused on the wave forces on flat plates or T-type girders [20,21,22,23,24]. During the past decade, more researchers have started paying attention to the box girder [25,26,27], which is widely used in coastal bridges along the Western Pacific region due to its strong integrity, high torsional stiffness, and excellent adaptability [28,29]. Based on theories regarding T-type girders [30], Huang et al. [11] proposed an improved empirical equation to estimate the solitary wave force on the box girder. Gao et al. [31] established a model to analyze the hydrodynamic wave forces on the box girder behind a trench or breakwater. Istrati [32] investigated the tsunami impact on prototype box-girder bridges located in Oregon U.S. by both 2D and more realistic 3D CFD modeling. Huang et al. [33] presented a fragility framework to assess the failure probability of a box girder with different connection types under an extreme random wave. Chen et al. [27] investigated the effect of lateral restraining stiffness on its solitary wave force. However, most studies focus on the behavior of box girders under simplified extreme waves, such as regular and solitary waves. Few have considered its safety under a real extreme wave, let alone a landslide tsunami [34].

In this study, we develop and illustrate a suitable modeling framework to assess the safety of a box-girder superstructure under a potential landslide tsunami. We first introduce three primary components of this framework: the landslide tsunami simulation modeling, the wave load calculation theory, and the safety assessment method of the box girder. In landslide motion, a velocity-weakening friction law is implemented to consider the time-varying characteristics of the basal friction coefficient. Then, we carry out a case study on an in-service coastal bridge with a box girder subjected to a potential landslide in the South China Sea. In this framework, we investigate the dynamic characteristics of landslide tsunamis with real physical mechanisms and the law of wave loads and bridge safety in different directions. A series of quantitative analysis data and conclusions can effectively guide the design, construction, and maintenance of coastal bridges.

2. Methodology

To evaluate the safety of the box-girder superstructure under the potential landslide tsunami waves, we established an analytical framework as shown in Figure 1. First, we analyze and identify landslides that may pose a threat to a bridge superstructure based on the geographical location of the bridge. Second, we update the Tsunami Squares (TS) method by introducing velocity-weakening basal friction to optimize the simulation of landslides and tsunamis and obtain a more reasonable tsunami wave height and velocity around the bridge. Third, the force of a wave, if tsunami waves impact the box girder, can be calculated using the summarized empirical algorithms. Finally, we establish a finite element model of the coastal bridge and compute the reaction of the bearing under tsunami waves. After a comprehensive comparison of the support’s horizontal and vertical security redundancy, the safety assessment conclusions of the box girder can be determined based on whether vertical disengagement or lateral displacement occurs.

2.1. Tsunami Squares Method

TS, a two-dimension simulation method, can resolve the process of fluid or solid motion, such as tsunamis, floods, landslides, and storm surges. This depth-averaged method can simultaneously fulfill volume and momentum conservation. It was initially proposed by Xiao et al. [35] and Wang et al. [36], based on the previous work ‘Tsunami Balls’ [37,38,39]. TS establishes an innovative dual-grid partition method, namely, fixed grid and ghost grid, which significantly improves the calculation efficiency. The fixed grid stores information on the sliding/water squares at each time moment. The ghost grid is an intermediation that updates the data for the next time step of the fixed grid. A series of experiments and cases showed that TS can be suitable for large-scale landslide tsunami research and can ensure simulation accuracy [40,41,42].

TS splits the landslide tsunami into two steps: landslide motion and tsunami motion. As shown in Equation (1), for landslide modeling, material squares speed up under the influence of gravity and decelerate due to basal and dynamic frictions. For tsunami modeling, fluid squares are controlled by accelerations from gravity, dynamic friction, ‘lift up’ (LU), ‘push ahead’ (PA), and ‘drag along’ (DA) effects. The lift up effect considers the impact of the thickness of the moving slide on the surface elevation of the water but transfers no momentum to the water. A detailed description of the numerical scheme of TS can be found in Xiao et al. [35] and Wang et al. [40].

$$\left\{\begin{array}{l}{a}_{\mathrm{l}}\left(r,t\right)=a_{\mathrm{g}}\left(r,t\right)+a_{\mathrm{b}}\left(r,t\right)+a_{\mathrm{d}}\left(r,t\right)\\ a_{\mathrm{t}}\left(r,t\right)=a_{\mathrm{g}}\left(r,t\right)+a_{\mathrm{d}}\left(r,t\right)+a_{\mathrm{DA}}\left(r,t\right)+a_{\mathrm{PA}}\left(r,t\right)\end{array}\right.$$

(1)

where, $a_{\mathrm{l}}\left(r,t\right)$ and $a_{\mathrm{t}}\left(r,t\right)$ is the total acceleration of the landslide and tsunami, respectively. $a_{\mathrm{g}}\left(r,t\right)$ is the gravitational acceleration. $a_{\mathrm{b}}\left(r,t\right)$ is the basal friction acceleration, which depends on the material type, solid fraction, and bed roughness. $a_{\mathrm{d}}\left(r,t\right)$ is the dynamic friction acceleration. $a_{\mathrm{DA}}\left(r,t\right)$ and $a_{\mathrm{PA}}\left(r,t\right)$ is the ‘drag along’ acceleration and ‘push ahead’ acceleration of a landslide on water, respectively.

$$a_{\mathrm{g}}\left(r,t\right)=-g{\nabla }_{\mathrm{h}}\zeta \left(r,t\right)$$

(2)

$$a_{\mathrm{b}}\left(r,t\right)=-{\mu }_{\mathrm{b}}g{v}_{\mathrm{s}}\left(r,t\right)/\left|v_{\mathrm{s}}\left(r,t\right)\right|$$

(3)

$$a_{\mathrm{d}}\left(r,t\right)=\left\{\begin{array}{l}-{\mu }_{\mathrm{d}}{v}_{\mathrm{s}}\left(r,t\right)\left|v_{\mathrm{s}}\left(r,t\right)\right|/H_{\mathrm{s}}\left(r,t\right) \mathrm{Landslide} \mathrm{motion}\\ -{\mu }_{\mathrm{d}}{v}_{\mathrm{w}}\left(r,t\right)\left|v_{\mathrm{w}}\left(r,t\right)\right|/H_{\mathrm{w}}\left(r,t\right) \mathrm{Tsunami} \mathrm{motion}\end{array}\right.$$

(4)

$$a_{\mathrm{DA}}\left(r,t\right)=C_{\mathrm{DA}}\left[\left(v_{\mathrm{s}}\left(r,t\right)-v_{\mathrm{w}}\left(r,t\right)\right)\cdot {\widehat{v}}_{\mathrm{s}}\left(r,t\right)\right]\frac{v_{\mathrm{s}}\left(r,t\right)}{H_{\mathrm{w}}\left(r,t\right)}$$

(5)

$$a_{\mathrm{PA}}\left(r,t\right)=C_{\mathrm{PA}}\left[\left(v_{\mathrm{s}}\left(r,t\right)-v_{\mathrm{w}}\left(r,t\right)\right)\cdot {\widehat{v}}_{\mathrm{s}}\left(r,t\right)\right]\frac{\left|{\mathrm{dT}}_{\mathrm{s}}\left(r,t\right)\right|v_{\mathrm{s}}\left(r,t\right)}{\Delta t\left|v_{\mathrm{s}}\left(r,t\right)\right|H_{\mathrm{w}}\left(r,t\right)}$$

(6)

Here, $g$ is the acceleration of gravity. ${\nabla }_{\mathrm{h}}$ is the surface gradient. $\zeta \left(r,t\right)$ is the upper surface elevation of the square position of r and time t. ${\mu }_{\mathrm{b}}$ and ${\mu }_{\mathrm{d}}$ are the basal and dynamic friction coefficients. $v_{\mathrm{s}}\left(r,t\right)$ and $v_{\mathrm{w}}\left(r,t\right)$ are the velocities of the slide and water square. ${\widehat{v}}_{\mathrm{s}}\left(r,t\right)$ is the unit vector in the velocity direction of the slide. $H_{\mathrm{s}}\left(r,t\right)$ and $H_{\mathrm{w}}\left(r,t\right)$ are the thickness of the slide and water square, respectively. ${\mathrm{dT}}_{\mathrm{s}}\left(r,t\right)$ is the variation in slide thickness over the time interval $\Delta t$. $C_{\mathrm{DA}}$ and $C_{\mathrm{PA}}$ are the ‘drag along’ and ‘push ahead’ unitless coefficients.

Among these effects that control landslide motion, basal frictional mechanisms can be analyzed by high-velocity rotary shear experiments. Mizoguchi et al. [43] developed an exponential-decay equation for the friction coefficient of a fault gouge by fitting the experimental data. Togo et al. [44] collected soil samples from the accident scene and measured their frictional properties by experiment. Yang et al. [45] gathered data on the friction characteristics of the shale powder and fault gouge from the Tsaoling landslide site and presented a velocity-displacement dependent friction law that can account for most of the experimental data. According to the above research, the following velocity-weakening friction coefficient can describe the phenomenon that the steady-state friction coefficient decreases with increasing velocity.

$${\mu }_{\mathrm{b}}={\mu }_{ss}+\left({\mu }_{\mathrm{p}}-{\mu }_{\mathrm{ss}}\right)\exp \left(-v_{\mathrm{s}}/V_{\mathrm{c}}\right)$$

(7)

Here, ${\mu }_{\mathrm{ss}}$ is the steady-state friction coefficient when the velocity approaches infinity. ${\mu }_{\mathrm{p}}$ is the peak frictional coefficient and ${\mu }_{\mathrm{p}}=\tan \left({\varphi }_{\mathrm{bed}}\right)$, where ${\varphi }_{\mathrm{bed}}$ is the basal angle. $v_{\mathrm{s}}$ is the velocity of the slide square. $V_{\mathrm{c}}$ is the material’s constant. Therefore, we introduce this velocity-dependent friction model in TS and calculate the basal friction acceleration using Equations (3) and (7).

2.2. Wave Force on a Box Girder

The safety evaluation of coastal bridges with box girders under landslide tsunamis relies on an accurate calculation of the tsunami load. After conducting a series of experiments and numerical simulation studies, Huang et al. [11] proposed an empirical method of the maximum wave load on a box girder, which can apply to most wave conditions. However, when the tsunami does not go over the top of the deck, the simplified calculation method proposed by Li et al. [46] is more suitable for structural buoyancy. In light of this, integrating the methods of Huang et al. [11] and Li et al. [46], the correlations between the wave features and maximum wave forces are quantified as follows:

$$\left\{\begin{array}{l}{f}_{\mathrm{H}}=f_{\mathrm{Hs}\_\mathrm{front}}-f_{\mathrm{Hs}\_\mathrm{back}}+f_{\mathrm{D}}\\ f_{\mathrm{V}}=f_{\mathrm{HV}}-f_{\mathrm{w}}+f_{\mathrm{l}}+f_{\mathrm{HC}}\end{array}\right.$$

(8)

Here, the horizontal wave forces per unit length ($f_{\mathrm{H}}$) consist of the hydrostatic constituents ($f_{\mathrm{Hs}\_\mathrm{front}}$ and $f_{\mathrm{Hs}\_\mathrm{back}}$) and the hydrodynamic constituent ($f_{\mathrm{D}}$). The vertical wave forces per unit length ($f_{\mathrm{V}}$) include the hydrostatic members ($f_{\mathrm{HV}}$ and $f_{\mathrm{w}}$), the hydrodynamic component ($f_{\mathrm{l}}$), and the contribution of the horizontal load ($f_{\mathrm{HC}}$), which is dependent on the web inclination.

$$f_{\mathrm{Hs}\_\mathrm{front}}=0.5\alpha C_w{C}_d\rho g\left(H-H_{\mathrm{bottom}}\right)d_{\mathrm{girder}}$$

(9)

$$\left\{\begin{array}{ll}{f}_{\mathrm{Hs}\_\mathrm{back}}=0.5\alpha \rho g\left(0.2H-H_{\mathrm{bottom}}\right)d_{\mathrm{girder}}& 0.2H>H_{\mathrm{bottom}}\\ f_{\mathrm{Hs}\_\mathrm{back}}=0& 0.2H\le H_{\mathrm{bottom}}\end{array}\right.$$

(10)

$$f_{\mathrm{D}}=0.5\alpha C_D{C}_I\rho u_h^2{d}_{\mathrm{girder}}$$

(11)

If the tsunamis go over the top of the box girder, $H>H_{\mathrm{deck}}$

$$f_{\mathrm{HV}}=C_w{C}_d\rho g{W}_{\mathrm{girder}}\left(H-H_{\mathrm{bottom}}\right)$$

(12)

$$f_{\mathrm{w}}=0.5\rho g{W}_{\mathrm{girder}}\left(H-H_{\mathrm{flange}}\right)$$

(13)

If there are no overtopping tsunamis on the top of the girder, $H\le H_{\mathrm{deck}}$

$$f_{\mathrm{HV}}=\rho g\left(H-H_{\mathrm{bottom}}\right)\left[W_{\mathrm{bottom}}+\left(1-\alpha \right)\left(H-H_{\mathrm{bottom}}\right)\right]$$

(14)

$$f_{\mathrm{w}}=0$$

(15)

$$f_{\mathrm{l}}=0.5C_l{C}_I\rho u_v^2{W}_{\mathrm{girder}}$$

(16)

$$f_{\mathrm{HC}}=\left(1-\alpha \right)\left(f_{\mathrm{Hs}\_\mathrm{front}}+f_{\mathrm{Hs}\_\mathrm{back}}+f_{\mathrm{D}}\right)$$

(17)

where α is the inclination coefficient and α = 1 − tanφ, where φ is the angle between the girder web and the vertical direction. $C_w$ and $C_d$ are the effective coefficients of the girder width and deck thickness, which are all conservatively set equal to 1.0. ρ is the density of the water. $g$ is the acceleration of gravity. As seen in Figure 2, H is the tsunami wave height. $H_{\mathrm{bottom}}$ and $H_{\mathrm{flange}}$ are the distance from the static water level (SWL) to the bottom of the girder and flange. d is the water depth. $d_{\mathrm{girder}}$ is the height of the girder. $H_{\mathrm{girder}}$ is the distance from the SWL to the top of the box girder. $W_{\mathrm{girder}}$ is the width of the girder. $W_{\mathrm{flange}}$ is the width of the flange plate. $W_{\mathrm{bottom}}$ represents the width at the bottom of the girder. $C_D$ and $C_l$ are the drag and lift force coefficients, respectively, which are both equal to 1.0. $C_I$ is the impact coefficient corresponding to a wave breaking. Here, we do not consider the impulse forces caused by a wave breaking, and $C_I=1.0$. $u_h$ represents the maximum lateral velocity of water at the center position of the girder. $u_v$ is the maximum vertical velocity of water at the bottom of the girder. The fluid is simplified to a two-dimensional model without considering the height direction in the TS method. Here, $u_h$ takes the velocity of the fluid square and $u_v$ is the variation of the water surface elevation in unit time.

For bridges whose elevation varies with the longitudinal position, the resultant wave forces can be calculated by the $L_{\mathrm{bridge}}$ integral:

$$\left\{\begin{array}{l}{F}_H={\displaystyle \int f_{H}d{L}_{\mathrm{bridge}}}\\ F_V={\displaystyle \int f_{V}d{L}_{\mathrm{bridge}}}\end{array}\right.$$

(18)

Here $F_H$ and $F_V$ are the resultant horizontal and vertical wave forces. $L_{\mathrm{bridge}}$ represents the span of the whole bridge.

2.3. Safety Assessment

According to the post-disaster investigations and safety evaluations of coastal bridges subjected to disaster events [47,48], a structure failure due to the wave force is defined as the deck being moved either vertically or horizontally. A horizontal failure of the girder is equivalent to the fact that the bearing cannot provide a lateral restraint, and the vertical displacement of the girder means that the bearing is separated from the girder. If it is a vertical tensile bearing, the vertical restraint has failed. Therefore, the unseating failure of the girder is determined as long as the maximum wave force exceeds the bearing’s capacity, either vertically or horizontally.

$$\left\{\begin{array}{l}{f}_{\mathrm{hmax}}>R_{\mathrm{limit}}^{\mathrm{h}}\\ f_{\mathrm{vmax}}>R_{\mathrm{limit}}^{\mathrm{v}}+f_{\mathrm{gravity}}^{\mathrm{v}}\end{array}\right.$$

(19)

where $f_{\mathrm{hmax}}$ and $f_{\mathrm{vmax}}$ represent the horizontal and vertical maximum bearing reaction under the tsunami. $R_{\mathrm{limit}}^{\mathrm{h}}$ and $R_{\mathrm{limit}}^{\mathrm{v}}$ are the horizontal and vertical resistance of the bearing. If there is no vertical tensile bearing, $R_{\mathrm{limit}}^{\mathrm{v}}=0$. $f_{\mathrm{gravity}}^{\mathrm{v}}$ is the reaction of the bearing of the deck under its gravity load. Here, static analysis, a time-independent method, is adopted to calculate the most unfavorable bearing reaction.

$$\left[K\right]\left\{u\right\}=\left\{F_{wave}\right\}$$

(20)

Here, $\left[K\right]$ is the stiffness matrix of the structure. $\left\{u\right\}$ is the displacement vector. $\left\{F_{wave}\right\}$ is the nodal force vector excited by tsunami waves, including the horizontal and vertical force.

3. Case Study

3.1. Prototype Bridge

As shown in Figure 3 and Figure 4, a continuous box girder bridge with five spans (5 × 50 m) in southeast China is selected for safety assessment. A box girder with a twin chamber gradually narrows from 20.3 m to 16.3 m in width. The height of the box girder is 3.0 m, and the flange plate is 0.24 m high and 3.5 m wide. Figure 4c displays the structural details of the two cross sections. The bottom elevation of the girder gradually climbs from 4.51 m to 10.55 m. Above each pier (P1–P6), there are two identical hyperbolic spherical bearings, including fixed and longitudinally moveable bearings. All bearings do not provide vertical tension, and the horizontal ultimate bearing capacity of B1-1–B6-1 is 1250, 2000, 2000, 2000, 1750, and 1000 kN.

To obtain the reaction of the bearing under tsunami waves, a finite element model of the coastal bridge using ANSYS is established. Both the girder and restraint are simulated in the model.

Table 1. The bearing reaction under the gravity of a box girder.

Number	Bearing Reaction (kN)	Number	Bearing Reaction (kN)	Number	Bearing Reaction (kN)
B1-1	5359.8	B2-1	14,124.5	B3-1	12,357.8
B1-2	5357.0	B2-2	14,123.2	B3-2	12,399.6
B4-1	12,136.2	B5-1	12,817.5	B6-1	4536.1
B4-2	12,559.7	B5-2	13,044.5	B6-2	4376.1

illustrates the bearing reaction under the gravity of the box girder. We additionally set the vertical restraint of the bearing to analyze the vertical bearing failure quantitatively, and it still retains a lateral restraint when the bearing reaction exceeds its lateral capacity.

3.2. Baiyun Slide Complex

The Baiyun Slide Complex (BSC), a large-scale study of a slope collapse, is located in the middle and lower landscape region of the Pearl River Mouth Basin (Figure 3). The BSC covers an area of ~5500 km² and has a maximum length of 200 km. The water depth is 1100–3100 m, and the average slope is ~0.65° [49]. Based on high-quality seismic reflections and multi-beam bathymetry data, Li et al. [50] pointed out that the BSC is a hotspot for mass movement due to the excess pore pressure aided by volcanic activity, tectonic overstepping, and gas hydrate dissociation. Additionally, Sun et al. [51] suggested that the lateral migration of free gas is likely to trigger a multi-stage slope failure. However, whether such a massive landslide occurred in a single significant incident or multiple incidents is still controversial. Wang et al. [52] divided it into four regions utilizing the maximum likelihood classification. Li et al. [50] identified a conceptual four-phase model analyzing its sedimentary and seismic characteristics. After studying a large amount of seismic data and multi-beam bathymetrical data, Sun et al. [53] proposed six events of Mass Transport Deposits (MTDa–MTDd, MTDo1, and MTDo2) and supplied detailed information on their age, volume, geographical area, and thickness.

In this study, to analyze the tsunami formed by landslides in various regions and its hazard to the safety of a bridge deck, we constructed three probable landslide scenarios (BSC-1–BSC-3) based on the geological and depositional data provided by Sun et al. [53]. As shown in Figure 3b and

Table 2. Maximum thickness, area, and volume of three landslide events.

Parameter	BSC-1	BSC-2	BSC-3
Maximum thickness/m	188.7	296.8	359.1
Area/km2	4076.1	4155.9	4836.6
Volume/km3	324.3	405.6	838.7

, these three scenarios’ maximum thickness, area, and volume gradually increase. Among them, BSC-1 is composed of MTDa (~0.19 Ma), MTDb (~0.54 Ma), and MTDc (~0.79 Ma). Based on BSC-1, BSC-2 adds MTDd (~1.59 Ma) with an earlier formation time. As the largest area and volume condition, BSC-3 consists of MTDb and MTDo2. It should be noted that these scenarios represent possible landslide conditions in the study area rather than actual landslide events.

3.3. Landslide and Tsunami Simulation

Based on the geography of the coastal bridge and BSC, we select 14° N–26° N to 104° E–122° E as the research area. The elevation data comes from the General Bathymetric Chart of the Ocean 2022 digital bathymetric model [54]. Most of the surrounding terrain has been incorporated into this region, providing more realistic tsunami dynamics such as superposition, diffraction, and refraction. The simulation model comprises 4320 × 2880 squares each with a dimension of 15 arc-second. The time interval is set to 0.5 s to ensure that each step’s displacement is less than the square dimension.

As shown in Equation (21), a velocity-weakening friction law [14,17,43,45,55] was incorporated into the TS model to describe the time-varying characteristics of the basal friction effect.

$${\mu }_{\mathrm{b}}=0.006+0.004\exp \left(-v_{\mathrm{s}}/3.0\right)$$

(21)

To analyze the impact of the friction law on landslide tsunamis, we also set the static basal friction coefficient ${\mu }_{\mathrm{b}}=0.01$. The dynamic friction coefficient is set at 0.02, which slows the landslide motion. For tsunami propagation, the dynamic friction coefficient is 0.001 only when the tsunami submerges the land and 0 otherwise. We set the PA coefficient C_PA = 0.25 and the drag along coefficient C_DA = 0.12 in all landslide events [41,42].

3.3.1. Landslide Motion

The landslide magnitude and dynamics have a significant impact on tsunami generation. Figure 5a–d show the snapshots of BSC-1 with velocity-weakening basal friction (BSC-1V) at 30 min, 1 h, 2 h, and 3 h. The soil slips from northwest to southeast and finally forms a fan deposit in the southeast. For the BSC-1 with the static basal friction model (BSC-1S, Figure 5g), the last deposit and average velocity for three hours are slightly smaller than those of BSC-1V. The snapshots of BSC-2 and BSC-3 in Figure 5e,f,h,i show a similar law to BSC-1. Figure 6 shows that the coverage area and average velocity considering the velocity-weakening basal friction model are always greater than the static basal friction model, and the difference in coverage area gradually increases over time. Close to 3 h under all conditions, the coverage area and average velocity have tended to be a stable value, and the maximum average speed is only 0.40 m/s. Therefore, the total duration of landslide simulation is 3 h, ensuring that most of the energy of the soil is transferred to the water.

To quantitatively analyze the influence of two basal frictions on landslide movement,

Table 3. Maximum coverage area and velocity of landslide considering velocity-weakening and static basal friction.

Case	Maximum Velocity (m/s)			Maximum Coverage Area (km2)
	BSC-1	BSC-2	BSC-3	BSC-1	BSC-2	BSC-3
Velocity-weakening friction model	14.55	15.24	23.87	5894	6739	10,524
Static friction model	14.05	14.71	23.29	5648	6505	9969
Relative difference	3.47%	3.44%	2.44%	4.17%	3.47%	5.27%

shows the landslide’s maximum coverage area and velocity. The relative difference coefficient $R_{\mathrm{VS}}$ is used to analyze the difference between the crucial landslide and tsunami parameters under two basal friction coefficients:

$$R_{\mathrm{VS}}=\left(P_{\mathrm{V}}-P_{\mathrm{S}}\right)/P_{\mathrm{V}}\times 100\%$$

(22)

Here, $P_{\mathrm{V}}$ and $P_{\mathrm{S}}$ represent the values of some parameters, such as the maximum velocity, under velocity-weakening and the static basal friction coefficient, respectively.

The maximum coverage area and average velocity of the velocity-weakening basal friction model are larger than those of the static basal friction, indicating that this friction can characterize the superfluidity of landslides. The relative difference in maximum velocity steadily declines when the landslide scenarios (BSC-1 to BSC-3) vary. However, the relative difference in the coverage area does not follow a similar pattern. This may result from variable soil distribution under different landslide scenarios.

3.3.2. Tsunami Motion

To investigate the propagation characteristics of the tsunami, Figure 7 shows the tsunami snapshots of BSC-1V at four moments. Half an hour later, a nearly circular tsunami-affected area is formed centered on the sliding site, and the principal propagation direction is from northwest to southeast (Figure 7a). After 1 h, the tsunami affects the coastal areas of the Philippines, and part of the tsunami continues to spread northwest (Figure 7b). About 3 h later, the tsunami strikes the southeastern coast of China and extends perpendicularly to the coastline (Figure 7c). After that, the tsunami wave reflection and refraction last for more than 4 h (Figure 7d).

Landslide motions with different friction models will cause different tsunamis, and the height and velocity of waves are the most typical tsunami parameters. Figure 8 presents the tsunami wave height and velocity at two positions (A₁ and A₂). A₁ is located southeast of the landslide area (Figure 7), which is the main direction of tsunami propagation. A₂ represents the geographical location of the coastal bridge in Section 3.1. It can be seen from Figure 8a,b that in the main propagation direction of the tsunami, the first wave peak under the velocity-weakening basal friction model is higher than that of the static basal friction, and the wave velocity is always larger within 3 h. As shown in Figure 8c–d, the wave height at the bridge site gradually increases and fluctuates. The wave height under the two friction models alternately peaks. The wave velocity under BSC-1 and BSC-2 decays rapidly after reaching the ridge and then continues to rise for about 1 h. However, after a brief decline after the first peak, the wave velocity of BSC-3 rises rapidly and continuously to form a relatively stable peak interval (about 4.9 h–5.2 h).

Table 4. Maximum tsunami wave height and velocity at A₁ and A₂.

Landslide Motion	Maximum Wave Height (m)		Maximum Wave Velocity (m/s)
	A1	A2	A1	A2
BSC-1V	5.02	3.30	1.14	6.04
BSC-1S	4.82	3.06	0.99	5.83
RVS	3.97%	7.28%	13.14%	3.45%
BSC-2V	6.44	3.49	1.46	5.93
BSC-2S	6.15	3.16	1.31	5.72
RVS	4.53%	9.19%	10.26%	3.58%
BSC-3V	20.85	6.45	3.45	8.05
BSC-3S	20.11	6.46	3.12	7.83
RVS	3.57%	−0.06%	9.63%	2.77%

compares the maximum wave height and velocity at A₁ and A₂ under different wave conditions. Under the velocity-weakening friction model, the top wave height and velocity at A₁ are higher than those under the static friction model, and the relative wave velocity difference is greater than the wave height. However, the maximum wave height of the velocity-weakening basal friction model at A₂ is not always greater than that of the static basal friction model, which might be the result of the shallow water topography at A₂ leading to more complex tsunami motion patterns.

3.4. Safety Assessment

3.4.1. Tsunami Wave Force

The tsunami wave force is the basis of assessing bridge safety. Only the tsunami with velocity-weakening basal friction is utilized as the load input since it more closely matches the superfluidity of the landslide movement. It can be seen from Figure 4b that the lowest elevation of the bridge is 4.51 m, and the maximum wave heights for BSC-1V–BSC-3V are 3.30, 3.49, and 6.45 m, respectively. Therefore, only the tsunami under BSC-3V can interact with the box girder.

Figure 8c and Figure 9 show the evolution of wave height and velocity with time under BSC-3V. The vertical load is mainly affected by the wave height and vertical speed. Among them, the vertical velocity has a low temporal variability, and the vertical load’s velocity component (Equation (16)) is the second-order function of vertical velocity. Therefore, the vertical load will be mainly affected by the wave height. The lateral wave loads are primarily controlled by the wave height and horizontal velocity (Equations (9)–(11)), both of which have a high value. Thus, the wave height and horizontal velocity are the main parameters affecting the vertical and horizontal wave force. As shown in Figure 8c, the tsunami of BSC-3V is composed of a series of tsunami waves, and its peak time is 5.51 h. Moreover, the horizontal velocity peaks at 4.90 h. Therefore, we select the peak time of the wave height and horizontal velocity, respectively, and compute the wave force by the method shown in Section 2.2.

The tsunami under BSC-3V will act on the first (S1) and second (S2) spans.

Table 5. Tsunami wave force on box girder under BSC-3V.

Time (h)	Vertical Force/kN			Horizontal Force/kN			Ratio of Total Vertical to Horizontal Force
	S1	S2	Total	S1	S2	Total
4.90	6440.5	148.9	6589.4	3633.1	324.1	3957.2	1.665
5.51	8049.1	663.3	8712.5	970.1	172.0	1142.0	7.629

shows the horizontal and vertical loads of S1 and S2 at two moments. The vertical and horizontal force of S1 are higher than those of S2, which is due to the greater range of the wave impact on S1 with a lower elevation. The resultant vertical force is higher than the transverse load, consistent with previous tests or numerical simulations [26,56,57], and the ratios are 1.665 and 7.629 at 4.90 h and 5.51 h, respectively.

3.4.2. Safety Evaluation

We can obtain the bearing reaction ($f_{\mathrm{hmax}}$ and $f_{\mathrm{vmax}}$) by applying the tsunami wave force to the finite element model of the coastal bridge. Based on Equation (19), we define the safety coefficient $C_{\mathrm{S}}$ to evaluate the bearing:

$$C_{\mathrm{S}}=\left\{\begin{array}{ll}1-f_{\mathrm{hmax}}/R_{\mathrm{limit}}^{\mathrm{h}}& \mathrm{for} \mathrm{horizontal}\\ 1-f_{\mathrm{vmax}}/f_{\mathrm{gravity}}^{\mathrm{v}}& \mathrm{for} \mathrm{vertical}\end{array}\right.$$

(23)

Here, $C_{\mathrm{S}}$ represents the safety coefficient of the bearing. If $C_{\mathrm{S}}>0$, the bearing is safe for vertically or horizontally directed waves, otherwise it is a failure. A vertical failure means that the box girder is separated from the support, and a horizontal failure represents the lateral constraint failure of the bearing.

Figure 10 shows the horizontal and vertical safety coefficients of bearings (B1-1(2)–B3-1(2) in Figure 4a) under BSC-3V. The safety coefficients of all bearings are greater than 0; so there will be no lateral displacement or vertical separation, and the bridge remains safe. As the bearing position moves from the side span to the middle span, its horizontal and vertical safety coefficients gradually increase. The horizontal safety coefficient of bearings in the same row is equal, while the vertical safety coefficient on the side of the tsunami impact is lower than on the other side. The vertical safety coefficient of the third row is greater than 1.0 since this support will bear positive pressure when S1 is mainly exposed to the vertical impact of waves. Therefore, we choose the first and second rows of the bearing on the wave impact side (B1-1 and B2-1) to evaluate its safety in the subsequent analysis. Moreover, the horizontal or vertical safety coefficient is the maximum of two moments (4.90 h and 5.51 h).

3.5. Parametric Analysis of Bridge Elevation and Support Type

According to the calculation results of the safety coefficient, the coastal bridge has a high design redundancy to withstand the impact of tsunamis. Furthermore, this section examines the influence of the design parameters such as the elevation and support type on the superstructure’s safety, serving as a guide for the engineering design of coastal bridges under the impact of a severe tsunami.

3.5.1. Bridge Elevation

The minimum designed elevation of the bridge is 4.51 m (Figure 4b) and it gradually increases from east to west to 10.55 m. Under the premise of the same longitudinal slope, we analyze its safety coefficient under BSC-3V after the deck is uniformly raised or lowered. The maximum lifting or lowering height is 1.0 m with an interval of 0.1 m. Figure 11 shows the bearing reaction and safety coefficient for different relative elevations.

With a decrease in the deck elevation, the vertical bearing reaction gradually increases. The vertical bearing reaction of B2-1 gradually exceeds that of B1-1; this is because B2-1 is more sensitive to S2, whose greater range will be impacted by the tsunami with the decline in elevation. However, the bearing reaction of B2-1 under gravity is far higher than that of B1-1 (Table 1), and its safety coefficient is always greater than that of B1-1.

The horizontal bearing reaction is always lower than the vertical one. The lateral safety coefficient significantly reduces with the decreased bridge elevation. However, in contrast to the bearing whose vertical safety factor is always high than zero, B1-1 will have a lateral failure when the falling height exceeds 0.705 m, indicating that the lateral redundancy of the bridge against tsunami is lower than the vertical one. The fact that the wave forces on superstructures are not fully considered in the design of coastal bridges may be the main reason for this unbalanced lateral and vertical redundancy. Therefore, a design elevation with a larger bottom clearance can ensure the bridge’s safety and avoid the need to evaluate the transverse and vertical wave force under a complex and changeable tsunami.

3.5.2. Support Type

This coastal bridge adopts the hyperbolic spherical bearing (HSB) to connect the superstructure and substructure. To study the lateral safety of the deck under different connection modes, we supplement two common forms of simple support, rubber bearing (RB) and direct contact (DC), which both limit the lateral displacement of the girder by friction. Based on the studies of Lum [58] and Hayatdavoodi et al. [59], the friction coefficient $\mu$ is equal to 0.1 for RB and 0.8 for DC. Equation (24) shows the calculation method of the lateral safety coefficient, and the results can be found in Figure 12.

$$C_{\mathrm{S}}=1-f_{\mathrm{hmax}}/\left[\mu \left(f_{\mathrm{gravity}}^{\mathrm{v}}-f_{\mathrm{vmax}}\right)\right]$$

(24)

With a decreased deck elevation, the bearing’s vertical compressive stress under the combined action of the deck gravity and tsunami gradually decreases. The lateral safety coefficient under a RB and DC shows a nonlinear downward trend, which is more evident for the support with lower elevation in the bridge side span (B1-1). The B1-1(RB) cannot provide sufficient lateral restraints at all the relative elevations. The horizontal safety coefficient of B2-1(RB) reached zero when the height decreased by about 0.47 m. At the designed elevation of this coastal bridge, the horizontal safety of DC is higher than that of HSB. When the elevation decreases by 1.0 m, the safety coefficient of B1-1(DC) will be lower than 0 and less than B1-1(HSB). Therefore, the DC and HSB can provide relatively reliable lateral protection, and the RB is not recommended in a tsunami-prone region. The bridge designers need to combine the design requirements of the coastal bridge and the potential tsunami hazard level to demonstrate and determine more suitable bearings.

4. Conclusions

In this study, we establish a safety assessment framework of coastal bridge superstructures with box girders taking potential landslide tsunami forces into consideration. A continuous box girder and Baiyun landslide on the southeast coast of China are selected as the safety assessment objects and disaster sources, respectively. After analyzing the effects of different basal friction models on landslides and tsunamis, we evaluate the safety of bridges under tsunamis and discuss the impact of the bridge design elevation and support forms on their safety. We find that this framework can provide instructive guidance for the safety evaluation of coastal bridge superstructures.

(1) Landslide dynamics: The average sliding velocity of the Baiyun Slide Complex will rapidly increase to the peak value and then gradually decay; this process will continue for about 3 h. Compared with the static basal friction model, the average velocity and coverage area considering velocity-weakening friction are greater (2.44%–5.27%), and this sliding process can better represent its superfluidity.

(2) Tsunami dynamics: About 3 h after a submarine landslide, the tsunami spread to the bridge and could not be simplified into a single wave. Only the BSC-3 could interact with the box girder, and its maximum wave height and velocity were 6.45 m and 3.45 m/s, respectively. In the main propagation direction of the tsunami, the wave height and velocity under the velocity-weakening basal friction model are always higher than that of the static basal friction. However, the wave height at the bridge does not exhibit the above stable pattern, which may be due to the shallow water topography leading to more complex tsunami motion patterns.

(3) Safety assessment: The maximum vertical force is higher than the horizontal load, and the maximum ratio is 7.629, but the peak time is different. The bridge has no lateral displacement or vertical separation from the bearing under the worst tsunami. As the bearing position moves from the side span to the middle span, its horizontal and vertical safety coefficients gradually increase. The vertical safety coefficient on the side of tsunami impact is less than on the other side.

(4) Parametric analysis: With the decrease of deck elevation, the vertical bearing reaction of the middle span gradually exceeds that of the side span, but its safety coefficient is always greater. The horizontal bearing reaction is always lower than the vertical one. The hyperbolic spherical bearing and direct contact can provide relatively reliable lateral protection, and a rubber bearing is not recommended in a tsunami-prone region.

In future related studies, we will explore the impact of more varieties of vertical and lateral bearings on the security of bridge superstructures. Continued exploration is necessary to fully understand how debris flows [60] and other extreme events can impact the safety of bridges. TS still has the potential for further exploration in simulating the superfluidity characteristics of large landslides.