Numerical Study on Tsunami Force on Coastal Bridge Decks with Superelevation

Abstract

Many coastal bridges have been destroyed or damaged by tsunami waves. Some studies have been conducted to investigate wave impact on bridge decks, but there is little concerning the effect of bridge superelevation. A three-dimensional (3D) dam break wave model based on OpenFOAM was developed to study tsunami-like wave impacts on bridge decks with superelevation. The Reynolds-averaged Navier–Stokes equations and the k-ɛ turbulence model were used. The numerical model was satisfactorily checked against Stoker’s analytical solution and the published hydrodynamic experiment. The validated model was employed to carry out parametric analyses to investigate the effects of upstream and downstream water depths and the bridge deck’s superelevation. The results show that the tsunami force is proportional to the relative wave height. The dam break wave impact on the bridge deck can be identified as two distinct scenarios according to whether the wave height is higher than the bridge deck top. The trend of the tsunami force is also different in different scenarios. The superelevation will significantly influence the tsunami forces acting on the box girder, with some exceptions.

1. Introduction

The recent occurrences of tsunamis in Indonesia (2004), Chile (2010), Japan (2011), and Tonga (2022) have caused significant devastation to structures and infrastructures in coastal regions. For instance, in the aftermath of the 2011 Japan Earthquake and Tsunami, over 300 bridges were damaged [1]. It is widely believed that the enormous forces generated by tsunami waves can result in the displacement and even the complete unseating or overturning of bridge decks [2]. Therefore, to protect coastal bridges from such disasters, it is necessary to gain a comprehensive understanding of the impact of tsunami waves on bridge decks.

Two distinct wave types have been utilized in previous studies to simulate the im-pact of a tsunami wave on bridge decks: solitary wave [3,4,5,6,7,8,9] and dam break wave [10,11,12,13,14,15,16]. Generally, solitary waves are well suited for simulating the propagation of tsunami waves in the ocean, while dam break waves are ideal for simulating the behavior of tsunami waves once they reach the shore. Leschka and Oumeraci [17] showed that solitary waves and surge forces on coastal structures have different characteristics. Chanson [18] analyzed the surge data observed since the 2004 tsunami disaster and compared them with the results of the analysis of breached waves, showing a reasonable agreement was observed in terms of the surge front celerity and free surface profile. However, there is currently no clear definition for the choice of wave type. In this paper, the dam breaking wave model was utilized to simulate tsunamis.

The factors affecting tsunami loads on bridges are mainly related to wave height, wave speed and girder clearance height, and the effects of these factors have been studied by many scholars [14,19,20,21,22,23,24,25]. These studies show that tsunami forces increase with an increasing wave height and speed and increase with a decreasing girder clearance height. Nakamura et al. [26] investigated the tsunami force characteristics of bridge superstructures from hydraulic model experiments and numerical simulations and derived a tsunami force estimation equation, which can accurately predict horizontal and vertical tsunami forces, except for the short wave period.

For the effect of the shape of the girder on tsunami forces, Winter et al. [23] investigated numerically the variation of the tsunami force in three-girder cases and two-girder cases; the results showed that the peak tsunami force in the three-girder cases was larger than that in the two-girder cases. Yang et al. [19] numerically investigated the tsunami forces for T-girder and box girder bridges, and the results showed that T-girder bridges have higher and more persistent horizontal peak forces. The upward force of the box girder bridge is significantly greater than that of the T-girder bridge. However, when the wave is high enough to impact the entire girder, the effect of the girder shape on the horizontal impact load is negligible. Huang et al. [4] conducted a series of experiments using cnoidal waves and solitary waves to investigate the characteristics of wave forces on box girders and T-beams, and the results showed that the interaction of the waves with these two types of girders differed significantly. Xu et al. [7] conducted a numerical parametric study on the characteristics of solitary-wave-induced force on a typical coastal double-bridge panel.

Farvizi et al. [11] conducted physical modeling experiments to measure the borehole impact forces and pressures for different tsunami borehole strengths on bridge decks with different abutment types (wingwalls and spillways) and different opening and inundation ratios. It was shown that the horizontal and vertical forces increased with an increasing inundation rate for both abutments. The horizontal forces showed a decreasing trend as the opening rate decreased, while the vertical forces initially increased with a decreasing opening rate and then decreased. Lukkunaprasit and Lau [27] investigated the effect of bridge piers on tsunami forces through experiments using isolated waves and showed that for intact bridge structures, the actual component forces generated by hydrodynamic pressures acting on piers alone can be increased by 50% compared to independent piers.

The effect of the air during the tsunami impact is not negligible. Yang et al. [15] numerically investigated the air escape and pressure variation during the submergence of multichamber beams. The results show that when the inundation ratio is small, the air in the chambers is transferred to the adjacent chambers sequentially and finally escapes from the downstream chamber; when the inundation ratio is large, the air in the chambers is drawn into the water or vortex and then escapes directly from the chambers. Issakhov and Borsikbayeva [28] investigated the influence of trapped air on tsunami forces on two common bridge types, and the experimental results show that the air retention had a significant effect on the tsunami loads, wave–structure interactions, and bridge structural response. Bullock et al. [29] conducted drop tests using seawater and freshwater and showed that the maximum impact pressure and rise time were influenced by the level of aeration and impact intensity.

Motley et al. [30] and Farvizi et al. [12] investigated the characteristics of tsunami forces on skewed bridges by numerical simulations and experiments, respectively. The results show that the horizontal and vertical forces, as well as the overturning moment, showed a decreasing trend with the increase in the tilt angle. The lateral force, cross-rolling moment, and cross-swing moment showed an increasing trend with the increase in the tilt angle.

To mitigate damage to bridges from tsunami waves, Xu et al. [6], Hayatdavoodi et al. [3], and Seiffert et al. [31] investigated the effect of air relief openings on tsunami loads. The results show that the air relief venting hole can significantly reduce the vertical tsunami force but increases the horizontal tsunami force. Wei and Dalrymple [32], Nassiraei et al. [33], and Gao et al. [34] investigated the effect of breakwaters on the tsunami force of girders, and the results showed that breakwaters can significantly reduce the tsunami force, and there exists an optimum value for the distance between breakwaters and girders. Nakao et al. [13], Zhang and Hoshikuma [35], and Abukawa et al. [36] investigated the effect of fairings on tsunami forces acting on girders, and the results showed that fairings can significantly reduce the horizontal forces but increase the vertical forces. Nakamura et al. [37] numerically investigated the effect of fairings and fenders on the girder on tsunami forces and showed that both fairings and fenders reduce the horizontal tsunami forces and are independent of the fairing height and fender configuration. In addition, when the fairing surface is configured upward, the downward tsunami force increases and makes the girder more stable.

Bridges are usually built with a superelevation at the curve to ensure the safety of vehicles during turning. Winter et al. [21] investigated the effect of the superelevation on the tsunami force of a T-girder bridge by numerical simulation, and the results showed that the superelevation can significantly affect the tsunami force on the girder. Bricker and Nakayama [38] reached similar conclusions. However, there have been few studies conducted on the effects of tsunami forces on box girder bridges with superelevation. Considering the widespread utilization of box girder bridges in modern construction projects, it is essential to undertake research on the impact of tsunami forces on box girder bridges with superelevation.

This paper established a three-dimensional numerical model to investigate the effect of superelevation on tsunami wave forces acting on box girders. First, a brief introduction of the numerical model built based on OpenFOAM is presented. Then, the numerical model was validated by Stoker’s analytical solution and the published hydrodynamic experiment. The effect of the properties of the incident dam break wave, elevation of the bridge deck, and superelevation of the bridge deck are discussed in Section 3. Finally, several conclusions and findings are summarized.

2. Materials and Methods

2.1. Simulation Methodology

The Open-Source Field Operation and Manipulation software package, generally known as OpenFOAM, was employed in this study. The solver interFoam was used to solve the flow field during the dam break wave impact on bridge decks. The interFoam utilizes the volume of fluid (VOF) method to simulate the flow of two incompressible fluids separated by a free interface. An indicator function, ${\alpha }_1$, was defined, which had a value of 1.0 corresponding to the regions occupied by water (water: $\rho =1000 \mathrm{kg}/{\mathrm{m}}^3$ and $\upsilon =1.0\times {10}^6 {\mathrm{m}}^2/\mathrm{s}$) and a value of 0.0 for air (air: $\rho =1.22 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and $\upsilon =1.48\times {10}^{-5} {\mathrm{m}}^2/\mathrm{s}$); when α is between 0 and 1, this implies that the cell is occupied by a mixture of these two fluids along the free surface, where $\rho$ is the mass density of the fluid, and $\upsilon$ is the kinematic viscosity. The gravitational acceleration was $g=9.81 \mathrm{m}/{\mathrm{s}}^2$. The intermediate values indicate the cells belonging to the free interface. The present numerical model utilized the Reynolds-averaged stress (RAS) and standard $k-\epsilon$ turbulence models, which have been successfully used in previous studies on wave impact structures [21,38,39]. However, it is worth noting that a large-eddy simulation (LES) turbulence model would provide even greater accuracy and precision. Assuming that the fluid is incompressible, the RANS equations can be written as

$$\frac{\partial {\overline{u}}_i}{\partial x_i}=0$$

(1)

$$\rho \frac{\partial {\overline{u}}_i}{\partial t}+\rho {\overline{u}}_j\frac{\partial {\overline{u}}_i}{\partial x_j}=\frac{\partial \overline{p}}{\partial x_i}+\mu \frac{{\partial }^2{\overline{u}}_i}{\partial x_j\partial x_j}-\frac{\partial \rho \overline{{u^{\prime }}_i{u^{\prime }}_j}}{\partial x_j}$$

(2)

where ${\overline{u}}_i$ is the mean velocity in the i-direction, ${u^{\prime }}_i$ is the fluctuating component of the velocity in the i-direction, and $\overline{p}$ is the mean pressure. If $u_i$ is the velocity component in the i-direction, then $u_i={\overline{u}}_i+{u^{\prime }}_i$. The Reynolds stress term in Equation (2) can be computed with

$$-\rho \overline{{u^{\prime }}_i{u^{\prime }}_j}={\upsilon }_t\rho \left[\frac{\partial {\overline{u}}_i}{\partial x_j}+\frac{\partial {\overline{u}}_j}{\partial x_i}\right]-\frac{2}{3}k\rho {\delta }_{ij}$$

(3)

where $k$ is the turbulent kinetic energy. A standard $k-\epsilon$ model was applied to solve for the turbulent kinetic energy, $k$, and dissipation, $\epsilon$,

$$\frac{\partial k}{\partial t}+\frac{\partial k{\overline{u}}_j}{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(\upsilon +\frac{{\upsilon }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+\frac{P_k}{\rho }-\epsilon$$

(4)

$$\frac{\partial \epsilon }{\partial t}+\frac{\partial \epsilon {\overline{u}}_j}{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(\upsilon +\frac{{\upsilon }_t}{{\sigma }_k}\right)\frac{\partial \epsilon }{\partial x_j}\right]+C_{\epsilon 1}\frac{\epsilon }{k\rho }{P}_k-C_{\epsilon 2}\frac{{\epsilon }^2}{k}$$

(5)

where

$$P_k=-\rho \overline{{u^{\prime }}_i{u^{\prime }}_j}\left(\frac{\partial {\overline{u}}_j}{\partial x_i}\right)$$

The turbulence eddy viscosity, ${\upsilon }_t$, is calculated by

$${\upsilon }_t=C_{\mu }\frac{k^2}{\epsilon }$$

The previously mentioned coefficients were applied in this paper as $C_{\epsilon 1}=1.44$, $C_{\epsilon 2}=1.92$, $C_{\mu }=0.09$, ${\sigma }_k=1.0$ and ${\sigma }_{\epsilon }=1.3$ [40].

The initial conditions for the air/water phase indicator (${\alpha }_1$), velocity ($U$), pressure on the boundary minus the hydrostatic pressure ($p_{rgh}$), turbulent kinetic energy ($k$) and turbulent dissipation ($\epsilon$) were specified for all cases, as described in

Table 1. Model boundary conditions.

Boundary	α1	ε	k	prgh	U
Bottom	zeroGradient	epsilonWallFunction	kqRWallFunction	fixedFluxPressure	fixedValue
Left wall	zeroGradient	epsilonWallFunction	kqRWallFunction	fixedFluxPressure	fixedValue
Right wall	zeroGradient	epsilonWallFunction	kqRWallFunction	fixedFluxPressure	fixedValue
Atmosphere	inletOutlet	zeroGradient	zeroGradient	totalPressure	pressureInletOutletVelocity
Bridge model	zeroGradient	epsilonWallFunction	kqRWallFunction	fixedFluxPressure	fixedValue
Front wall	zeroGradient	epsilonWallFunction	kqRWallFunction	fixedFluxPressure	fixedValue
Back wall	zeroGradient	epsilonWallFunction	kqRWallFunction	fixedFluxPressure	fixedValue

.

To ensure the stability of the numerical calculation process, the time step, $\delta t$, is automatically controlled by the solver so that the time step in all cells is less than the user-defined maximum Courant number ($\delta t\left|U\right|/\delta x$) of 0.5. Here, U is the velocity vector through the cell, while $\delta x$ is the cell size in the velocity direction.

2.2. Numerical Model

In the present study, a three-dimensional (3D) numerical model was set up to investigate the wave impact on the bridge deck with superelevation. Figure 1 shows a diagrammatic sketch of the numerical model. A Cartesian coordinate was defined with the origin located at the bottom-left corner of the wave flume. The x-axis was defined along the direction of the dam break wave, while the y-axis yielded a vertical direction. The length of the flume was 30 m in the x-direction in which the reservoir length was 12 m. The bridge deck was located 7.6 m downstream from the gate. The numerical model was 1 m in width along the z-axis, which was perpendicular to the x–y plane, as shown in Figure 1. The convention for positive superelevation is demonstrated in Figure 1. It should be noted that the 3D numerical model was preferred to study the wave impact on the bridge deck, because the two-dimensional (2D) numerical model cannot capture the air behavior along the bridge span. According to Motley et al. [30], air would stagnate among the structural components in the 2D model, resulting in an overestimation of the wave forces acting on the bridge deck.

Three different levels of mesh were generated in the present numerical study, i.e., background, main calculation zone, and local refinement zone around the bridge deck, as shown in Figure 2. The meshes in the main calculation zone, with a 5.0 m length and 0.8 m height, were refined to obtain details of the fluid field during wave impact. The cell lengths were 0.05 m and 0.00625 m for the background and the local refinement zone, respectively. A mesh sensitivity analysis was carried out with four different mesh sizes of the main calculation zone. The cell length, total mesh number, and calculation time (using a computer with a double Inter Xeon Gold 5218R processor (2.1 GHz and 40 M cache) CPU and 128 G internal storage) for four different mesh sizes are listed in

Table 2. Mesh sensitivity testing of the main calculation zone (using a computer with a double Inter Xeon Gold 5218R processor (2.1 GHz and 40 M cache) CPU).

Cell Length (m)	Total Mesh Number	Calculation Time (h)
0.05	526,041	1.32
0.025	691,829	2.05
0.0125	2,074,903	9.47
0.00625	13,431,024	235

. Figure 3 shows the maximum horizontal force (F_x-max) and vertical forces (F_y-max) acting on the bridge deck during the dam break wave impact. It can be seen that the maximum tsunami-induced forces oscillated with a mesh size change from 0.05 m to 0.0125 m. However, further mesh refinement did not significantly improve the numerical results of the forces. A mesh size of 0.0125 m in the main calculation zone was adopted in the following analysis to achieve a balance between the accuracy and computational consumption.

2.3. Validation

All tsunami-like waves used in this study were calibrated in the empty wave flume. Stoker’s theory [41], as shown in Figure 4, was employed to check the wave height and velocity obtained by the present numerical model. According to Stoker’s theory, the free surface profile (h₂) after the dam break is related to the water depth of the upstream reservoir (h₁) and downstream (h₀) at the initial time. Figure 5 shows the calculated time series of the free surface at monitoring locations (x = 18.5, 19.0, 19.5 and 20.0 m) compared with Stoker’s analytical solution for two cases. It can be concluded from Figure 5 that the free surface profiles measured at the monitoring locations after the wave fronts all gradually converged to a stable water level, which is in good agreement with Stoker’s analytical solution. Furthermore, the dimensionless wave height $\left(h_2-h_0\right)/h_0$ and wave velocity $u/c_0$ (where $c_0=\sqrt{g{h}_0}$) were checked against Stoker’s theory, as shown in Figure 6. The results indicate that both the wave height and the wave velocity predicted by the present numerical model were consistent with Stoker’s analytical solution.

The hydrodynamic experiment conducted by Nakao et al. [42] was employed to validate the present numerical model. The investigation was carried out in a wave flume with a size of 30 m in length, 1 m in width and 1 m in height. A 0.1 m thick sluice gate was used to release the water from the 12 m reservoir zone to the 18 m still water zone. The 1/20 scale bridge deck model was set 19.6 m downstream of the sluice gate and 0.2 m high from the flume bottom. A total of five different configurations of bridge models were tested by Nakao et al. [42], in which Model 5 was simulated in this section. Model 5 was a T-girder bridge deck with one slab and four rectangular girders, as shown in Figure 7. The reservoir water level (RWL) was h₁ = 0.617 m, the still water level (SWL) h₀ = 0.1 m, the bridge deck elevation Z_d = 0.2 m and the bridge deck superelevation $\theta =0°$. The tsunami-induced forces were measured by biaxial load cells installed between the bridge model and the supporting pier. It should be noted that the pier was not considered in the present study.

Figure 8 shows the tsunami-induced forces on the T-girder bridge deck. The present numerical model was run under two conditions: with and without a pier, represented by the green, short, and dashed line and the red, dashed line in Figure 8, respectively. In Figure 8, the solid line represents the experimental results measured by Nakao et al. [42], and the dot–dash line represents the numerical results obtained by Motley et al. [30]. It can be seen from Figure 8a that the horizontal force rose sharply when the dam break wave reached the bridge deck. The maximum horizontal force occurred at the impact stage when the wave water started to contact the bridge deck. The horizontal force gradually tended to a relatively stable value after an oscillation decline. It should be noted that the experimental data lag behind the numerical results, with a time difference of approximately 0.4 s. This time difference result of the gate release was not instantaneous during the experiment, which was not simulated in the numerical model. To facilitate a more intuitive comparison of the wave forces, the numerical data presented in Figure 8 were shifted backward. A similar treatment was carried out by Motley et al. [30]. Based on the results presented in Figure 8a, it can be concluded that the present numerical model overestimates the horizontal force by 15–25 percent during the steady state. Nonetheless, from an engineering safety perspective, the horizontal forces obtained using the present numerical model are conservative. As shown in Figure 8b, the present numerical model reasonably predicted the vertical forces acting on the bridge deck with consideration of the pier. To concentrate on the effect of the superelevation, the pier was neglected in the following numerical study.

2.4. Numerical Test Plan

The dam break wave acting on a box girder bridge deck was studied in this paper. Figure 9. shows the configuration of the 1/20 scale box girder bridge deck, which was adopted from Yang et al. [19]. The dam break wave generated by different depths of the upstream reservoir over different depths of the downstream were considered in this study. Figure 10. shows the wave heights and corresponding velocities of all incident dam break waves calculated by Stoker’s theory. In this study, the different elevations (Z_d, as shown in Figure 1) and different superelevations of the bridge deck (distinguished by the rotation angle, as shown in Figure 1) were simulated.

3. Results and Discussion

3.1. Time Histories of the Wave Forces

Figure 11 shows the time series of the forces acting on the box girder bridge for the case h₁ = 0.7 m, Z_d = 0.2 m, h₀ = 0.7 m and $\theta =0°$. It can be seen that the forces acting on the bridge model by the dam break wave can be divided into two stages: impact stage (I) and the flow around stage (II). The impact stage is divided into two substages: rising stage (from Moment A to force peak Moment B) and descending stage (from Moment B to Moment C). Snapshots of the fluid field are displayed in Figure 12.

3.2. Characteristic Tsunami Forces for a Box Girder Bridge Deck without Superelevation

In this section, the effects of different parameters on tsunami forces are studied more comprehensively, with varying upstream reservoir water depths, downstream still water depths and deck clearances. In order to represent the test cases with an upstream reservoir water depth, downstream still water depth and deck clearance in a dimensionless form, the following ratios were calculated

$$R=\frac{Z_d-h_0}{h_0}$$

(6)

$$Q=\frac{h_1-h_0}{h_0}$$

(7)

where R is the relative clearance height, Q is the relative reservoir height, $Z_d$ is the distance from the bottom of the deck to the bottom of the flume, $h_0$ is the downstream still water depth and $h_1$ is the upstream reservoir water depth, as shown in Figure 1.

The tsunami forces on the box girder bridge deck are nondimensionalized without the loss of generality, as follows

$$F_x^{*}=\frac{F_{x,\max }}{0.5\rho u^2{A}_v}$$

(8)

$$F_y^{*}=\frac{F_{y,\max }}{0.5\rho u^2{A}_h}$$

(9)

where $\rho$ is the water density, $u$ is the bore velocity, $A_v$ is the vertical projected area of the bridge deck and $A_h$ is the horizontal projected area of the bridge deck.

The depth of the downstream still water was $h_0=0.1$ m, and the investigation of tsunami force characteristics was conducted by varying the upstream reservoir water depth and deck height. As shown in Figure 13a, when the relative wave height was $h_2/(Z_d+h)\le 1$, the tsunami wave height was below the top of the bridge deck, i.e., non-overtopping stage. As the relative wave height increased, the area of the tsunami wave acting on the bridge girder increased; thus, the horizontal tsunami force also increased. When the relative wave height was $h_2/(Z_d+h)>1$, the tsunami wave height was higher than the top of the bridge deck, i.e., overtopping stage. In the overtopping stage, the area of the tsunami wave acting on the bridge girder was equal to the vertical projection of the girder, and as the relative wave height increased, more water crossed over the girder. Thus, the horizontal tsunami force decreased with an increasing relative wave height. As the relative clearance height, R, increased, the horizontal tsunami force tended to decrease, and it was more obvious in the overtopping stage. As shown in Figure 13b, with the variation in the relative wave height, $h_2/(Z_d+h)$, the vertical tsunami force had the same trend as the horizontal tsunami force. In the overtopping stage, the vertical tsunami force decreased with an increasing relative wave height. In the non-overtopping stage, the vertical tsunami force increased with the relative wave height increase. The tendency of the vertical tsunami force was inversely proportional to the relative clearance height, R.

With a deck height of $Z_d=0.2$ m, an investigation of the tsunami force characteristics was conducted by varying the upstream reservoir water depth and downstream still water depth. Figure 14 shows the trend of the vertical tsunami force, with $Z_d$ remaining constant, which was similar to that of the horizontal tsunami force with $h_0$ remaining constant. However, there was a gentle trend of the tsunami force with the variation of the relative wave height $h_2/(Z_d+h)$. As the relative clearance height, R, increased, the horizontal tsunami force tended to decrease. However, the variation in the tsunami force was not significant at R = 0.67∓1.50. Figure 14b shows that the trend of the vertical tsunami force was the same as that of the horizontal tsunami force, and it was also flatter than that of the horizontal tsunami force, with $h_0$ remaining constant. In addition, the vertical tsunami force tended to decrease with the relative clearance height, R, increases. Moreover, the variation of the vertical tsunami force with R was insignificant. This means that the effect of the variation of the downstream still water depth on the tsunami force was not significant, especially for the vertical tsunami force.

With an upstream reservoir water depth of $h_1=0.6$ m, an investigation of the tsunami force characteristics was conducted by varying the downstream still water depth and deck height. As shown in Figure 15a, the trend of the horizontal tsunami force was also divided into two stages: non-overtopping stage and overtopping stage. In the non-overtopping stage, the horizontal tsunami force was proportional to the relative wave height $h_2/(Z_d+h)$. Meanwhile, it was inversely proportional to $h_2/(Z_d+h)$ in the overtopping stage. As the relative reservoir height, Q, increased, the horizontal tsunami force decreased in the overtopping stage. in Figure 15b shows that the vertical tsunami force increased with the increase in $h_2/(Z_d+h)$ for both stages. As the relative reservoir height, Q, increased, the wave is increased, although the wave height decreased. Furthermore, in the overtopping stage, the change in the vertical tsunami force was not significant. In the non-overtopping stage, the increasing trend of the vertical tsunami force was the result of a larger contact area. In the overtopping stage, although the increase in Q led to more water crossing over the top of the deck, the increase in the wave speed was not negligible. Larger wave velocities lead to larger tsunami forces, and in the case of a constant upstream reservoir depth, the wave velocity becomes the main factor affecting the varying tsunami forces in the overtopping stage. Thus, the vertical tsunami force increased with the increase in Q in the non-overtopping stage and decreased with increase in Q in the overtopping stage.

3.3. Characteristic Tsunami Forces for a Box Girder Bridge Deck with Superelevation

In order to evaluate the effect of the superelevation on box girder bridges, 0°, ±2°, ±4°, ±6°, ±7.5°, ±9°, ±10.5° and ±12° were selected for the analysis.

In the impact stage, the tsunami forces on the box girder bridge deck were nondimensionalized, as follows

$$F_x^{*}=\frac{F_{x,\max }}{F_{0x,\max }}$$

(10)

$$F_y^{*}=\frac{F_{y,\max }}{F_{0y,\max }}$$

(11)

where $F_{0x,\max }$ is the peak horizontal tsunami force at $\theta =0°$, and $F_{0\mathrm{y},\max }$ is the peak vertical tsunami force at $\theta =0°$.

With a deck height of $Z_d=0.2m$ and downstream still water depth of $h_0=0.1$ m, an investigation of the tsunami force characteristics was conducted by varying the upstream reservoir water depth and girder superelevation. Figure 16a displays the trend of the relative horizontal tsunami force with θ and relative wave height of $h_2/(Z_d+h)$. The relative horizontal tsunami force decreased with the increasing superelevation, and this tendency became more pronounced as the relative wave height increased. In addition, the tsunami force caused by the negative superelevation was significantly larger than that of the positive superelevation in the overtopping stage. In the case of positive superelevation, more water flowed through the upper part of the deck, which led to a reduction of the tsunami force. In the negative superelevation case, the bottom surface of the beam was the main contributor to the tsunami force, and the uneven bottom configuration led to a larger tsunami force. However, when the relative wave height was $h_2/(Z_d+h)=0.85$, the relative horizontal tsunami force increased with the increase in the superelevation. At $h_2/(Z_d+h)=0.85$ (i.e., non-overtopping stage), a positive superelevation increased the contact area between the water and the girder side plate, which caused a larger tsunami force. However, in the negative superelevation case, although the contact area increased, the contact area between the water and the side plate was reduced; thus, the tsunami force was reduced. As shown in Figure 16b, the trend of the relative vertical tsunami force was similar to that of the relative horizontal tsunami force, which decreased with the increase in the superelevation. In addition, the effect of the negative superelevation on the tsunami force was greater than the effect of the positive superelevation. Moreover, at $h_2/(Z_d+h)=0.85$ (i.e., non-overtopping stage), the vertical tsunami force decreased and then increased with the superelevation, because the change in the superelevation increased the contact area between the water and the girder, which in turn increased the tsunami force.

In the flow-around stage, the tsunami forces on the box girder bridge deck were nondimensionalized, as follows

$$F_x^{*}=\frac{F_{x,ave}}{F_{0x,ave}}$$

(12)

$$F_y^{*}=\frac{F_{y,ave}}{F_{0y,ave}}$$

(13)

where $F_{x,ave}$ is the average horizontal tsunami force during 5 s–7 s, $F_{y,ave}$ is the average vertical tsunami force during 5 s–7 s, $F_{0x,ave}$ is the average horizontal tsunami force during 5 s–7 s at $\theta =0°$ and $F_{0y,ave}$ is the average vertical tsunami force during 5 s–7 s at $\theta =0°$.

Figure 17a shows the trend of the average horizontal tsunami force with θ and a relative wave height of $h_2/(Z_d+h)$. The average horizontal tsunami force decreased and then increased with the increase in the superelevation, and the tsunami force of the positive superelevation was larger than that of the negative superelevation. The vertical projection area of the girder increased with the superelevation, resulting in a larger contact area between the water and the girder. Therefore, the tsunami force also increased. However, in the case of the negative superelevation, the horizontal component of the gravity of the water canceled part of the horizontal tsunami force, so the average horizontal tsunami force in the negative superelevation was smaller than the average tsunami force in the superelevation. In the overtopping stage, the influence of the variation in the relative wave height on the average horizontal tsunami force was not significant. However, in the non-overtopping stage, the horizontal tsunami force increased with the increase in the superelevation. In the case of the negative superelevation, the lift at the bottom of the girder reduces the contact area between the water and the girder side plates, which in turn led to smaller horizontal tsunami forces. In the case of a positive superelevation, a larger contact area between the water and the girder side plate was achieved, thus leading to a higher horizontal tsunami force. As shown in Figure 17b, the average vertical tsunami force increased with the increase in the superelevation. With the increases in the superelevation, more water acts on the deck, thus enlarging the negative vertical force. However, in the case of a negative superelevation, the waves act mainly on the bottom of the girder, generating a positive vertical force. As the negative superelevation grows, the front side of the bridge is lifted higher and less water crosses the deck, resulting in a decrease in the negative vertical force generated by the water. This explains the generation of positive average vertical tsunami forces at $-12°$. In the overtopping stage, it was not obvious that the average vertical tsunami force was affected by the relative wave height. However, in the non-overtopping stage (i.e., $h_2/(Z_d+h)=0.85$), the average vertical tsunami force was more influenced by the superelevation, and the maximum average vertical tsunami force reached four times that at $\theta =0°$.

4. Conclusions

In this study, a 3D dam break numerical model was developed to simulate the tsunami wave impact on a box girder bridge deck with superelevation, which has seldom been studied in the existing research works. The numerical model was validated by Stoker’s analytical solution and the hydrodynamic experiment conducted by Nakao et al. The effect of the depth of the reservoir, elevation of the bridge deck, downstream still water depth and bridge superelevation on tsunami-like wave generated forces were investigated by the validated numerical model. The following conclusions are drawn from this study:

(1) The two stages of tsunami impact on a box girder were divided into the impact stage and the flow-around stage, where the impact stage can be divided into two substages: rising stage and descending stage. Tsunami waves acting on box beams can be divided into the non-overtopping stage and overtopping stage, and the trend of the tsunami forces varies from case to case.

(2) For box girders without a superelevation, the tsunami force increases with the relative wave height in the non-overtopping stage and decreases with the relative wave height in the overtopping stage. However, in the case of a small relative storage depth, the horizontal tsunami force increases with the increase in the relative wave height in the overtopping stage. The tsunami force decreases with an increasing relative clearance, R, and is more significant in the overtopping stage. The vertical tsunami force increases with increases in Q in the non-overtopping stage and decreases with increases in Q in the overtopping stage.

(3) For box girders with a superelevation, the relative tsunami forces were investigated for the impact stage and the flow-around stage. During the impact stage, the relative tsunami force decreases with an increasing superelevation, but in the non-overtopping stage, the relative tsunami force increases with an increasing superelevation. The relative tsunami force increases with the relative wave height and is more significant in the case of a negative superelevation. During the flow-around stage, the relative average horizontal tsunami force decreases and then increases with an increase in the superelevation, but in the non-overtopping stage, the relative average horizontal tsunami force increases with the increase of superelevation, the relative average vertical tsunami force increases with the increase in the superelevation, and is more significant in the non-overtopping stage. In addition, under certain circumstances, the average vertical tsunami force turns into a lifting force in the case of negative superelevation.