Validation of OpenSees for Tsunami Loading on Bridge Superstructures

In recent years, major tsunamis have taken many lives and caused severe damage to coastal infrastructure. The 2004 tsunami in the Indian Ocean killed 230,000 people, and the 2011 Tohoku tsunami in the Pacific Ocean killed more than 30,000 people and destroyed more than 125,000 buildings. In both of these events, earthquake and tsunami damage was observed for structures of various construction types, including steel, reinforced concrete, and timber (Ghobarah et al. 2006; Saatcioglu et al. 2006; Fraser et al. 2013). In addition to building damage, the Tohoku tsunami washed away more than 300 bridges (Suppasri et al. 2014). The loss of functionality of these bridges severely hindered emergency response and post-event recovery. Aside from the loss of human lives, the Tohoku event was a stark reminder that areas with well-engineered infrastructure designed to resist high seismic loads are not immune to the hydrodynamic loads that can be generated by tsunamis.

The last major tsunami to affect the United States resulted from the magnitude 9.2 Alaskan earthquake in 1964. In addition to damage in Anchorage, tsunami waves reached the Pacific Northwest and northern California, leading to damage and casualties. Hawaii also faces tsunami hazards from events around the Pacific rim; for example, a magnitude 8.3 earthquake in Chile in 1960 caused a tsunami that led to severe damage in the town of Hilo. Coastal communities in the Pacific Northwest face tsunami hazards from the Cascadia Subduction Zone (CSZ), which has a 40% chance of a major earthquake and subsequent tsunami occurring in the next 50 years (Goldfinger et al. 2012). Many bridges constructed during the 1950s and 1960s, with little to no seismic provisions and thus very little resistance to lateral loads, are still in use today along the Oregon and Washington coasts. Some of these bridges serve as the only regional transportation lifeline for coastal communities. Determining tsunami loads is critical to the design and assessment of these, and other, coastal bridges.

Various physical experiments have been conducted to assess tsunami loading on bridges. In response to the 2011 Tohoku event, the Public Works Research Institute (PWRI) in Japan conducted wave flume experiments to determine hydrodynamic loads imparted on bridge superstructures (Hoshikuma et al. 2013). In addition, Hayatdavoodi et al. (2014) and Seiffert et al. (2015) performed experiments and computational simulations for solitary wave loads on deck-girder bridges. The results indicated that trapped air and the number of girders have a significant effect on the forces imparted on elevated deck-girder superstructures. Istrati et al. (2016) conducted experiments to investigate forces on bridge-deck-to-substructure connections during tsunami loading, where it was found that flexible connections can lead to significant reductions in superstructure forces.

With experimental data, numerical simulation models of bridge response to tsunami loading can be validated to explore mitigation strategies and guide the design of additional experiments, as well as to develop guidelines for tsunami-resistant bridge design. Analysis of bridge deck failure resulting from tsunami loading was conducted by Bricker et al. (2012) using Reynolds-averaged Navier-Stokes (RANS) models in the OpenFOAM computational fluid dynamics program. Motley et al. (2016) also used OpenFOAM to investigate the effects of bridge skew on the hydrodynamic forces imparted by tsunami bores. A systematic integration of numerical models for tsunami generation, propagation, and inundation was developed by Azadbakht and Yim (2016), who used site-specific flow height and velocity to compute bridge forces using LS-DYNA. Numerical studies of tsunami loading on bridge piers were carried out by Wei et al. (2015) using smoothed particle hydrodynamics (SPH). In addition, SPH was used to investigate tsunami loads on bridge superstructures with and without protective breakwaters (Wei and Dalrymple 2016).

Simulation of the sequential effects of seismic loading followed by hydrodynamic loading motivates the continued development and validation of OpenSees for fluid–structure interaction (FSI) simulations. OpenSees was developed for the simulation of structural and geotechnical system response to earthquake loading, and it has been used extensively for earthquake engineering simulations of bridge structures (Aygün et al. 2011; Seo and Linzell 2013; Liang et al. 2016). Recently, OpenSees was extended to the particle finite-element method (PFEM) (Zhu and Scott 2014b), a Lagrangian formulation that is able to simulate FSI with the wide range of nonlinear finite-element formulations and constitutive models in OpenSees.

In this article, FSI simulations using the PFEM in OpenSees are validated against wave flume experiments conducted at PWRI on scaled models of various bridge deck configurations (Hoshikuma et al. 2013). After a brief overview of the PFEM approach to FSI, the PWRI wave flume experiments are described. Wave height and velocity in the flume, along with pressures and forces exerted on the bridge decks, are compared with the experimental results. The validated FSI simulations make OpenSees available for a number of wide-ranging applications involving sequential earthquake and tsunami hazards.

Various simulation approaches have been developed for FSI with incompressible Newtonian flows using either Lagrangian, Eulerian, or arbitrary Lagrangian-Eulerian (ALE) formulations (Girault and Raviart 1986; Gunzburger 1989; Baiges and Codina 2010; Radovitzky and Ortiz 1998; Tezduyar et al. 1992). All three formulations are suited to particular classes of problems for the simulation of solids, fluids, FSI, and contact. Important advantages of the Lagrangian formulation with respect to FSI include easy tracking of the fluid free surface and conformity with the Lagrangian formulation of structural mechanics. The PFEM (Oñate et al. 2004; Carbonell et al. 2010; Cremonesi et al. 2011a, b; Oñate et al. 2010; Sabel et al. 2014) is one approach that uses a Lagrangian formulation of fluid response. The motion of fluid and structural particles is tracked at all time steps, and the fluid mesh is updated at every time step to maintain numerical accuracy. With Lagrangian formulations of the fluid and solid response, the PFEM equations of FSI are monolithic, leading to a more efficient implementation than staggered approaches with disparate modules for fluid and structure response.

Initial implementations of the PFEM used a moving-mesh formulation where all meshes of the fluid and structure domains are updated at each time step (Oñate et al. 2004). This moving-mesh approach makes it easy to track the fluid free surface and to have a high-quality fluid mesh at the start of each time step. However, the mesh is not updated during iterations within the time step, which can require small time steps to prevent ill-conditioning of the global system of equations. A further result of the ill-conditioning is a very significant oscillation in the computed pressures and forces on the structure, making it difficult to quantify engineering demand parameters (Cressman 2016). To overcome these numerical difficulties, subsequent versions of the PFEM used a fully fixed mesh with Lagrangian particles (Idelsohn et al. 2014), similar to particle-in-cell methods (Brackbill and Ruppel 1986; Brackbill et al. 1988) and the material point method (Więckowski 2004). The fixed-mesh approach is more computationally efficient than the moving-mesh approach because remeshing is not required. This approach has been used for multifluid flows and FSI (Becker et al. 2015); however, it is limited to specific types of solid elements not typically employed in structural engineering applications.

The approach taken herein to validate OpenSees PFEM simulations of tsunami loading on bridges is the background mesh method, which strikes a balance between the moving-mesh and fixed-mesh formulations (Becker et al. 2015). In this approach, the fluid mesh is fixed, except at the interface of the fluid and structure domains, where a local moving mesh is created at each time step. The structural mesh is not constrained to that of the fixed fluid mesh, and as a result, any line or solid finite element based on a Lagrangian formulation can be used for FSI simulations. Complete details of the PFEM fixed-mesh method are described by Becker et al. (2015); however, a brief overview of the computational steps taken during one time step are described as follows:

With this computational approach, the entire mesh is neither fully fixed nor fully moving because it is only updated in the local area around the structural domain. As a result, the computational cost is significantly reduced compared with the full moving-mesh method. In addition, this approach gives flexibility of the mesh updating in the FSI area because structural nodes are not always on the grid points.

To address bridge failures observed in the 2011 east Japan earthquake and tsunami, PWRI performed a series of experiments in a 1-m-wide, 30-m-long wave flume, as shown in Fig. 2. A gate served as a dam to a 12-m-long reservoir filled to height h_res, and bridge deck models were placed 7.5 m downstream from the gate on a pier 20 cm above the base of the flume. Two wave gauges located 1 and 2.5 m in front of the bridge deck recorded the flow height and velocity of the approaching wave.

The test cases consisted of various bridge deck configurations, standing water levels, and wave heights for which hydrodynamic forces imparted on the bridge decks were recorded (Hoshikuma et al. 2013). The three cases shown in Fig. 3 were selected for validation of the PFEM background mesh implementation in OpenSees. The model bridge deck was developed at 1/20 length scale, with cross sectional dimensions of 50-cm width, 10-cm height, and 5-cm overhang. These dimensions correspond to a prototype bridge deck of 10-m width, 2-m height, and 1-m overhang. Using Froude similitude laws with the 1/20 length scale, the experimental results for time and force were scaled by $\sqrt{20}$ and 20³, respectively, to the prototype bridge. The transverse dimension of the model bridge deck was 0.985 m.

As shown in Fig. 3, the standing water level, h, was 10 cm (2 m at prototype scale), and the clearance was 10 cm (2 m prototype) for Case 1. The reservoir depth, h_res, was 61.7 cm, corresponding to a wave that would be expected to reach the top of the deck. Case 2 differed from Case 1 in its standing water level of 15 cm (3 m prototype), clearance of 5 cm (1 m prototype), and reservoir depth of 51.8 cm, which was also expected lead to the wave reaching the top of the deck. All conditions were the same for Case 3 as in Case 1, except for the addition of a protective fairing on the seaward side of the bridge. It was anticipated that the fairing would divert flow from the deck, thereby reducing hydrodynamic loads on the bridge deck.

Numerical simulations were carried out in OpenSees for the three cases shown in Fig. 3. In the experiments, the dam gate was hinged at its base and released to initiate fluid flow. For the simulations, an instantaneous dam break was used to initiate fluid flow. The initial conditions of fluid flow were identified as important in the numerical simulations of these PWRI experiments (Wei and Dalrymple 2016; Motley et al. 2016), and comparisons of simulated and experimental wave height and velocity at the wave gauge 1 m in front of the bridge were made for each case. In addition, simulated wave pressures at the four locations on the seaward side of the bridge, as shown in Fig. 4, were compared with recorded values. Finally, the simulated total horizontal and vertical force were compared to the forces recorded by biaxial load cells at the four girder-to-pier connections. The simulated forces were obtained by summing the horizontal and vertical reactions, including inertial effects, at the connection locations. Additional details on the experimental setup and recordings can be found in Hoshikuma et al. (2013).

All three cases used a 0.02-m mesh size for the background grid, in which three or four particles were placed in each cell. Physical constants for the fluid were density ρ = 1,000.0 kg/m³ and viscosity μ = 10^–3Pa-s. To represent the deck geometry, the bridge was discretized using 158 structural nodes with 204 triangular plane-stress finite elements of thickness 0.985 m. The bridge deck material had density ρ = 600.0 kg/m³, elastic modulus E = 10 GPa, and Poisson’s ratio ν = 0.3. Because the exact locations of the pressure sensors in Fig. 4 were unknown, nodal pressures were averaged in the area of the approximate sensor location. Horizontal and vertical forces were obtained from the simulation as the dynamic reaction forces at the girder-to-pier connections.

The PFEM simulations in OpenSees use backward Euler time integration with a time step of 0.006 s. Convergence in a time step is achieved when the norms of both the incremental velocity and incremental pressure vectors are less than 10^–6. A single-core Intel Xeon central processing unit (E3-1225 v3) at 3.20 GHz was used for the simulation, and the real time for a simulation time step averaged 4.7 sec. All simulations were performed in two dimensions with single-phase flow, modeling assumptions that will be addressed in the ensuing discussions of simulation results.

Validations of the PFEM in OpenSees for FSI simulations of tsunami loading on bridge decks were presented using data from wave flume experiments conducted at PWRI in Japan. Comparisons of wave height and velocity in the flume show that the OpenSees PFEM implementation is able to simulate tsunami bore flow as it approaches the bridge. In addition, the OpenSees simulations are able to capture pressure at various locations on the bridge deck. The simulations reproduce the peak horizontal forces acting on the bridge decks, including the case where fairing protects the seaward side of the bridge. The simulations, however, tended to underestimate the peak uplift force and overestimate the steady state downward force. These results support the importance of 3D simulation and two-phase flow reported in previous research on simulations of tsunami–bridge interaction.

In future developments of the PFEM in OpenSees, 3D and two-phase simulations will be carried out to capture pressure shocks and the localized air pockets and flows observed between the girders of the bridge superstructure. In addition, for computational efficiency, an approach using a variable mesh size, where a coarse mesh is used for the fluid domain in conjunction with a refined mesh in and around the structural domain, will be addressed in OpenSees. These outcomes have important implications for the development of guidelines and loading equations for tsunami-resistant bridge design and the design of additional experiments to investigate protective strategies. Finally, validated FSI models in OpenSees open this software framework to additional applications for sequential earthquake and tsunami hazards (Carey et al. 2014; Kim et al. 2017; Scott and Mason 2017).

The authors thank Dr. Jun-ichi Hoshikuma and Dr. Hisashi Nakao for sharing data and video from the PWRI wave flume experiments. The research was supported by the Pacific Earthquake Engineering Research (PEER) Center and the Oregon State University (OSU) College of Engineering through the Summer Undergraduate Research Fellowship (SURF) program. Portions of this material are based on work supported by the National Science Foundation under Grant 1344695. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation, PWRI, PEER, or OSU.