# Numerical study on mitigating tsunami force on bridges by an SPH model

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## Abstract

This study applies the numerical model of GPUSPH, an implementation of the weakly compressible Smoothed Particle Hydrodynamics method on graphics processing units, to investigate tsunami forces on bridge superstructures and tsunami mitigation on bridges by using a service road bridge and an offshore breakwater. The capability of GPUSPH to predict tsunami forces on bridges is first validated by simulating a laboratory experiment on tsunami impacting a bridge with different configurations of superstructures. To address the uncertainty of tsunami generation with the gate falling method used in the laboratory, this study proposes a new tsunami wave generation method that makes use of the laboratory free-surface measurements to replicate the wave. Furthermore, the tsunami force, in particular, the first impact force, on bridges is reasonably predicted by GPUSPH. Next additional numerical experiments built upon the laboratory work are carried out to examine the efficiency of tsunami mitigation by an upwave service road bridge and an offshore breakwater. It is found that a two-girder service road bridge is effective in reducing tsunami forces on the main bridge. Furthermore, a breakwater can also reduce tsunami forces on a bridge, and there is an optimal distance between the breakwater and the bridge to achieve the best reduction effect. However, the tsunami mitigation structures experience a strong tsunami force, which may lead to the failure of these structures.

## Keywords

Tsunami Bridge Wave–structure interaction Hydrodynamic force Hazard mitigation Smoothed Particle Hydrodynamics## 1 Introduction

Several major tsunamis in the past decades have caused significant damages to bridge structures. Post-disaster surveys have found that scouring of bridge foundations, loss of abutments, uplift due to buoyancy, and bridge superstructure sliding are the major causes of a bridge failure during a tsunami (e.g., Saatcioglu et al. 2006; Kawashima and Buckle 2013). All these failure modes indicate that a bridge failure is closely related to the hydrodynamics of a tsunami, in particular, tsunami forces on bridges. Moreover, countermeasures for mitigating the effect of tsunami forces on bridges should be studied and further implemented in engineering practices. To address the above issues, this work investigates tsunami forces on bridge superstructures, and examines the efficiency of measures used in practice for mitigating tsunami forcing on bridges.

In addition to field surveys, laboratory experiments and numerical simulations have been conducted to study tsunami impact on bridges. A number of experimental studies were carried out after the 2004 Indian Ocean Tsunami, and this study briefly reviews some of them. Arnason et al. (2009) conducted a set of flume experiments on a tsunami impacting vertical columns that resemble bridge piers in real life. Their results show that tsunami forces are influenced by the shapes and orientations of bridge piers. Kosa et al. (2010) investigated tsunami forces on bridge girders by considering the impact of breaking waves and non-breaking waves. They observed that for two waves with approximately the same height, the breaking wave gives a higher horizontal force, while the non-breaking wave results in a larger uplift force. Lau et al. (2011) examined tsunami forces on bridge decks, and further categorized them into four types, i.e., impulsive, slowly varying, uplift, and additional gravity forces. Furthermore, a few experiments also examined tsunami forces on multiple components of a bridge (e.g., Nakao et al. 2013), and even a whole bridge model (e.g., Iemura et al. 2007).

With the rapid development of computer hardware and numerical methods, numerical models are also used to simulate these forces. To mention some of them, Lau et al. (2011) used Flow-3D^{®} to simulate tsunami forces on a bridge deck involving wave impact and overtopping, and they obtained a good agreement with the experimental data for both hydrodynamic pressure and wave force. Hayatdavoodi et al. (2014) applied a two-phase flow solver InterFoam to investigate the force of a tsunami-like solitary wave on bridge deck and girders. They not only compared the experimental force data, but also explored the entrapped air pockets impact on force predictions. Recently, we have applied the GPU-accelerated Smoothed Particle Hydrodynamics (SPH) model, GPUSPH, to investigate tsunami forces on bridge piers, showing that GPUSPH is able to accurately predict tsunami forces on different shapes of bridge piers (Wei et al. 2015).

Furthermore, mitigation of tsunamis was also addressed in several works. For example, coastal structures, such as seawalls and breakwaters, serve as major structural countermeasures to mitigate tsunamis (see e.g., Fujima 2006; Thomas and Cox 2011). Recently, the so-called environment-friendly countermeasures for attenuating tsunamis gain increasing amount of attentions (see e.g., Kathiresan and Rajendran 2005; Tanaka et al. 2007). It is seen that the above countermeasures are proposed to mitigate tsunamis impact on coastal area in a general way, and a very few works have investigated specific countermeasures to mitigate tsunami impact on a bridge (Iemura et al. 2007; Nakao et al. 2013). Considering the massive loss of bridges caused by tsunamis, more effort should be devoted to study mitigating tsunami forcing on bridges. This paper is going to present some preliminary work on this topic by using the numerical model of GPUSPH (Hérault et al. 2010). The model is first validated by the laboratory experiment of Nakao et al. (2013) on tsunami forcing on bridge superstructures, and then additional numerical experiments are conducted to investigate tsunami force mitigation on bridges. The rest of the paper is organized as follows. Section 2 briefly reviews the fundamentals of the numerical model. Section 3 introduces the physical experiment and its corresponding numerical setup. Then the main work in this paper is divided into two parts. The first part validates the capability of GPUSPH to predict tsunami forcing on different bridge superstructures in Sects. 4 and 5. The second part, which is an extension of the first part, examines the efficiency of man-made structures such as a service road bridge and a breakwater for mitigating tsunami forces on bridges in Sect. 6. Finally, conclusions are made in Sect. 7.

## 2 Numerical model

### 2.1 GPUSPH

This study uses the open-source fluid-dynamics SPH code: GPUSPH^{1}, which is an implementation of the weakly compressible SPH (WCSPH) method on graphics processing units (Hérault et al. 2010); the basic equations of GPUSPH follow an early version of SPH code: SPHysics (e.g., Gomez-Gesteira et al. 2012). The theoretical formulations of the WCSPH method and its numerical implementations could be found in many references (e.g., Monaghan 1992, 1994; Dalrymple and Rogers 2006). However, for the completeness of this work, the fundamentals of the SPH method are briefly reviewed and the governing equations of the WCSPH method are presented in this section.

*W*. In this study, a quintic function of Wendland (1995) is used:

*i*and

*j*; and \(\alpha _{D} = 21/(16\pi h_s^{3})\) is used.

*i*is the particle of interest; \(\left\langle f(\mathbf {r}_{i})\right\rangle \) is the approximation for \(f(\mathbf {r})\) at \(\mathbf {r}_{i}\);

*j*is the particle within a radius of \(2h_s\) of the particle

*i*; and

*m*is the particle mass.

*t*is the time; \(\rho \) is the fluid density; \(\mathbf {u}\) is the particle velocity;

*P*is the pressure; \(\mathbf {g}\) is the gravitational acceleration; \(\nu _{0}\) is the laminar kinematic viscosity; and \(\tau \) is the turbulence stress tensor, which is approximated by the sub-particle scale model of Dalrymple and Rogers (2006).

In an SPH model, a numerical boundary condition plays an important role in obtaining correct numerical results. As an SPH particle approaches a wall boundary, its kernel does not have full support domain any more. To address this issue, Dalrymple and Knio (2000) proposed to represent the wall boundary with several layers of dynamic boundary particles, so that the particle of interest inside the computation domain will have a full kernel support near the wall boundary. The smoothing length \(h_\mathrm{s}\) is chosen to be 1.3 times of the particle size \(\Delta p\) in this study, and then only three layers of dynamic boundary particles are needed. These dynamic boundary particles share the same equations of continuity and state as the fluid particles placed inside the domain; however, their positions and velocities remain unchanged in time. In this study, rigid boundaries include bridge structures (e.g., piers, girders, and decks) are represented by dynamic boundary particles, which are also used to measure the hydrodynamic force on structures. Basically, for each dynamic boundary particle, the force exerted by its neighboring fluid particles is first obtained, and then the total hydrodynamic force on the structure is the summation of forces on individual dynamic boundary particles.

### 2.2 Numerical model validation

*h*, and a width of

*b*, the hydrostatic force exerted on its vertical wall is given by

## 3 Physical experiment and numerical model setup

### 3.1 Laboratory experiment

*H*, and the downstream flume has a water depth of

*h*. Although different combinations of

*H*and

*h*were used in the laboratory, this work only simulates cases with \(h= 0.15\) m and a tsunami bore height of \(A= 0.1\) m, as the corresponding experimental data are available to the authors through the US–Japan Tsunami Modeling Workshop held at Oregon State University in December 2014. Considering the physical experiment scale is 1/20, the experiment actually resembled a 2-m-high tsunami propagation over a 3-m deep water in reality.

A bridge structure was placed nearshore, and the distance between the gate and the leading part of the bridge is about 7.6 m. This study considers two bridge models, which are denoted as Type I and Type II bridges hereafter. The Type I bridge consists of a single deck sitting above a bridge pier, a 3D view and a top view of Type I bridge are shown in Fig. 2c, d, respectively. The dimension of the deck of Type I bridge is 0.25 m long in the longitudinal direction of the flume, 0.985 m wide in the transverse direction of the flume, and 0.05 m thick. The round-shaped pier of Type I bridge is 0.305 m long, 0.08 m wide, and 0.18 m high. There are two bearing supports with a dimension (\(0.01\times 0.04\times 0.02\), \(L\times W \times H\), unit m) connecting the pier and the deck, giving the bottom of the deck with a height of 0.2 m above the floor. For Type II bridge, its superstructure was made of a deck and four girders, which were supported by a bridge pier underneath. Similarly, a 3D view and a top view of Type II bridge are shown in Fig. 2e, f, respectively. The dimension of the deck of Type II bridge is 0.5 m long, 0.985 m wide, and 0.03 m thick. Each of the four girders of Type II bridge has a dimension of \(0.02\times 0.985\times 0.07\) (\(L\times W \times H\), unit m), and the distance between girders is about 0.107 m. The pier of Type II bridge also has two round-shaped edges, and it is 0.5 m long, 0.08 m wide, and 0.18 m high. Similar to the setup in Type I bridge, four bearing supports were used to connect girders to the bridge pier in Type II bridge.

In terms of laboratory measurements, time-series free-surface elevations were measured by two wave gages (denoted as 1 and 2), which were located at 2.5 and 1 m offshore of the bridge structure, respectively. The primary concern of the experimental study was to examine tsunami forces on bridge superstructures, and therefore both horizontal and vertical tsunami hydrodynamic forces on bridge superstructures were measured by biaxial load cells located at bearing supports, which were used to connect the bridge pier and the bridge superstructure as mentioned above.

### 3.2 Numerical model setup

For the numerical model setup in GPUSPH, several modifications or adaptations are made with respect to the actual laboratory setup. First of all, as the flume and structures are discretized into a set of particles in the numerical model, tsunami forces on the bridge superstructure are obtained by summarizing the force exerted on individual particles that are used to represent the structure (Wei et al. 2015). Therefore, there is no need for the numerical model to include bearing supports in which biaxial load cells were installed in the laboratory, and then dimensions of the bridges are slightly adjusted. For Type I bridge in the numerical setup, the 0.18-m-high bridge pier is vertically extended by 0.02 m for considering the space occupied by the biaxial load cells in the laboratory; this modification does not change the dimension of the superstructure, i.e., the single deck. With respect to the 0.02-m-high bearing supports in Type II bridge, they are considered as part of the girders in the numerical setup. Since the girder is about 0.985 m wide, this modification slightly increases the frontal area of girders.

*C*is an averaged wave celerity and it is defined by \(C=\sqrt{g(h+A)}\) with the tsunami bore height \(A =\max (\eta (t))\). The variable \(t_{ L}\) is introduced to consider the amount of time needed for the wave propagating a distance (

*L*) between the piston location

*x*(

*t*) and Gage 1, and it is given by

After the above adaptations, the experimental domain is discretized into a collection of particles with a fixed particle size. To evaluate the model convergence, three particle sizes (\(\Delta p= 0.01\), 0.0075 and 0.005 m) are used by GPUSPH. It is noted that \(\Delta p= 0.005\) m was previously used in Wei et al. (2015) to simulate a laboratory experiment with a similar dimension of this work, and it was able to resolve water drops due to tsunami-pier impact. With \(\Delta p= 0.005\) m, the total number of particles for Type I bridge is about 20 million, of which 15 million are fluid particles. The remaining particles are used to represent the boundary and the piston wavemaker. For Type II bridge, there is about 22 million particles, of which 16 million are fluid particles. This study utilizes the multi-GPU version of GPUSPH (Rustico et al. 2012), and the numerical simulations were carried out by running GPUSPH on six NVIDIA Tesla C2050 GPUs. The numerical model simulates 8 s of the physical experiment, and it requires approximately 70 h of computation when the finest particle size of 0.005 m is used.

## 4 Tsunami on Type I bridge

### 4.1 Free-surface evolution

### 4.2 Hydrodynamic forces

## 5 Tsunami on Type II bridge

### 5.1 Free-surface evolution

### 5.2 Hydrodynamic forces

## 6 Tsunami forcing mitigation

In the previous two sections, the capability of GPUSPH to predict tsunami forces on bridges was evaluated by simulating the laboratory experiment of Nakao et al. (2013). Although discrepancies were observed in numerical simulations because a different tsunami generation method has to be used in the single-phase numerical model, GPUSPH with a fine particle size of 0.005 m is able to reasonably predict the first peak of tsunami forces on bridges, which are generally large, and therefore critical for the bridge safety design. In this section, additional numerical experiments based on those presented in previous sections are carried out to examine the efficiency of tsunami mitigation on bridges by two types of man-made structures, i.e., a service road bridge and a breakwater. Although two types of bridges of Nakao et al. (2013) were considered earlier in this work, the following analysis on tsunami force mitigation is only conducted for Type II bridge that has a deck and four girders, because it resembles better a real-life bridge than Type I bridge does. The major activities/steps include: (1) adding a structure offshore of Type II bridge (the bridge hereafter); (2) rerun the tsunami impacting a bridge test as presented in Sect. 5, and (3) comparing the tsunami peak force reduction on the bridge and analyzing the hydrodynamic force on the added structures (i.e., the service road bridge and the breakwater). It should be pointed out that since the followings are pure numerical experiments, it is consistent to analyze the force reduction on the basis of the numerical force computed by GPUSPH in Sect. 5.

### 6.1 Tsunami mitigation by a service road bridge

A service road bridge is a local bridge running parallel to a higher-speed, limited-access highway bridge. During the 2011 Great East Japan Tsunami, superstructures in 12 bridges including service road for pedestrian on national highway route 45 of Japan (main route along the Pacific coast of Tohoku area) were washed away (Hoshikuma and Zhang 2013). In the laboratory work of Nakao et al. (2013), the effect of a service road bridge on tsunami force reduction on bridge superstructures was also investigated. However, their laboratory data and setup are not available to the authors for model-data comparison. Therefore, numerical experiments are conducted to investigate tsunami mitigation by a service road bridge in this study.

### 6.2 Tsunami mitigation by a breakwater

A breakwater is a structure built to reduce the offshore wave impact on harbor and nearshore structures. There is a long history in Japan of using breakwaters for mitigating a tsunami disaster (see e.g., Fukuuchi and Ito (1966); Fujima 2006)). Although it is difficult to collect data and quantify the effect of a breakwater for mitigating a tsunami impacting bridges in real life, laboratory experiment shows that breakwaters placed in front of a bridge are able to reduce tsunami force on a bridge (Iemura et al. 2007). In this study, numerical experiments are conducted to investigate the effect of a breakwater on reducing tsunami force on coastal bridges, in particular, a proper distance between a breakwater and a bridge to achieve a favorable effect.

*D*) between the breakwater and the bridge on reducing tsunami forces on the bridge, and the distance is measured from the center of the breakwater (i.e., the local coordinate \(x= 0\) in Fig. 12b) to the offshore edge of the bridge pier.

## 7 Conclusions

- 1.
This study proposes an alternative tsunami wave generation method that makes use of the laboratory free-surface measurements, and it can be adopted by other numerical models for wave generation;

- 2.
The numerical model of GPUSPH is able to accurately predict forces on structures if the experimental setup is well-designed, and it can also capture the overall trend of tsunami forces on a bridge in a dynamically changing environment;

- 3.
Numerical study in this work confirms the laboratory work finding of Nakao et al. (2013) that a service road bridge can also reduce tsunami forces on a bridge if it survives the tsunami impact, and a two-girder service road bridge is more efficient because of the blockage effect of the girders;

- 4.
An offshore breakwater can also reduce tsunami forces on a nearshore bridge. Based on the specific setup of this study, an optimal distance between the breakwater and the bridge to achieve the best reduction effect is about eight times of the local water depth or 13 times of the incoming tsunami height. However, the breakwater may also fail because of the large tsunami loading and the strong scouring in its lee side.

## Footnotes

- 1.
The code is freely available at: http://www.gpusph.org.

## Notes

### Acknowledgments

The authors acknowledge the support from the Office of Naval Research, Littoral Geosciences, and Optics Program. R.A.D. further acknowledges the ATHOS Consortium and its member organizations for their contributions to the GPUSPH code. The numerical simulations were carried out at the Graphics Processing Lab Cluster of Johns Hopkins University, which is sponsored by the National Science Foundation Grant MRI-0923018.

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