Abstract
This article investigated the effect of structural flexibility on a coastal highway bridge subjected to Stokes waves through a three-dimensional numerical model. Wave-bridge interaction modeling was performed by an FSI model with the coupling of finite element and finite volume methods. An experimental model validated the FSI numerical analysis. Eventually, the overall results of hydrodynamic and structural analyses are presented and discussed. The results illustrate that the structural flexibility significantly increases the initial shock of the wave force on the flexible bridge. In contrast, the fixed bridge tolerates the least forces in the initial shock of the wave force. Then, by adding a wedge-shaped part to the bridge structure, an attempt was made to reduce the initial shock of the wave force to the structure. The results showed the wedge-shaped part with an angle of 30° reduces the initial shock of wave forces down to 50% for horizontal force and 43% for vertical force on the flexible structure.
1. Introduction
Extreme wave-induced force by coastal flooding, hurricanes, storm surges, and tsunamis impacts the coastal highway bridges (CHBs). To prevent future events damage over CHBs, the extent of wave forces on these structures must be specified with sensible accuracy. Worldwide, one of the most critical difficulties of coastal areas is the resistance of structures to natural hazards. Given that the bridges play an essential role in emergency services access when coastal accidents occur, the high cost of building a bridge increases the need for careful study of coastal bridges against hazards [1].
In recent years, tsunamis and hurricanes have created structural damages to several CHBs around the world. Some of these events are Tohoku (2011) and Indian Ocean (2004) tsunamis, Queensland cyclones (2012), hurricanes Ivan (2004), Katrina (2005), Ike (2008), Sandy (2012), Haiyan (2013), Maria, Irma, and Harvey (2017), and Florence (2018) which have caused severe damage to coastal bridges. Therefore, it is necessary to identify the factors that cause damage to bridges when these accidents occur [2]. In the great east Japan earthquake, the bridges that remained intact against the 9-magnitude earthquake were entirely damaged by the tsunami [3]. Hence, it can be mentioned that one of the main factors of damage to CHBs is the extreme waves due to occur hurricanes, storm surges, and tsunamis. The countless numbers of CHBs damaged during these events have attracted the consideration of many researchers to analyze the extreme wave loads on CHBs. Figure 1 shows a photograph of damage to the I-10 Bridge across Escambia Bay, Florida, as a result of the Hurricane.
Although the production sources of the storm surge and tsunami are different, the physical characteristics of their nonlinear transformation (shallow water), wave propagation (deep water), and runup (land inundation) are similar [4]. In general, in two-dimensional analyses, wave loads on the superstructure are divided into horizontal and vertical. As shown in Figure 2(a), in this study, the total horizontal forces on the center of gravity of the superstructure are defined as the horizontal force , the total vertical forces on the center of gravity of the superstructure are defined as the vertical force , and the total moment on the center of gravity of the superstructure is defined as . The magnitude and performance of these forces depend on the inundation depth, wave parameters, and bridge topologies. As shown in Figure 2(b), coastal bridges have three main sections, foundation, substructure, and superstructure, so that the substructure is connected to the foundation, and the superstructure is connected to the substructure. The bridge geometrical characteristics that are effective in wave force include the deck support, air vents, deck slope, girder spacing, girder height, girder type, and railing. The main wave parameters combine wave height , wavelength , water depth , wave period , and clearance that are shown in Figure 2(a). The best way to accurately predict the wave loads on a CHB is the modes that consider the most influential parameters simultaneously. The inundation ratio is a nondimensional parameter that considers wave and structure specifications simultaneously and is defined by , where is the distance between the underside of the girders and seabed. The height of the superstructure is [5].
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Many researchers have worked on the parameters that are effective on the wave forces on bridges. In most of the damage caused to CHBs over extreme waves, the superstructure is detached from the substructure. This subject has led researchers to pay more attention to determining the value of wave loads on the deck. Here, there is a summary of the work done on the parameters that affect the determination of the wave forces of coastal bridges.
As the water level increases to below the deck, the empty spaces between the deck girder (air pocket) generate the entrapped air force. A numerical simulation was done by Kerenyi et al. They analyzed different depths of AASHTO girders of a deck under the storm wave and concluded that increases with a rise in the depth of girders [5]. Hayatdavoodi et al. analyzed a two-dimensional model of a deck in the different number of girders under the solitary wave force. They concluded that the number of girders had not affected the loads [6]. Sheppard et al. showed that increasing the distance between the girders decreases the trapped air force, and reducing the number of girders reduces the slamming force [7]. As a computational fluid dynamics (CFD) study, Seiffert et al. investigated the wave loading on a superstructure in various water depths. They realized that creating the air vent between the girder causes a significant decrease in the lift and slamming force, but it does not affect the drag force [8]. Qu et al. (2017) presented a two-dimensional numerical simulation on superstructure with air vents under solitary waves. They proved that setting an air vent with an area around 0.06 of the deck’s area can decrease the lift force up to , the drag force up to 39%, and the slamming force up to [9]. Xu et al. did a numerical simulation of a superstructure with six different venting ratios under wave loading and realized that the deck air vent up to 0.05 caused a significant decrease in the lift, drag, and slamming forces, but the air vent with more area does not have much effect on decreasing the forces [10]. Therefore, it can be concluded that the air pocket of the deck causes a significant rise in the wave load, and in this study, the effect of the air involved in modeling is considered.
Several studies have shown that increasing the wave height increases the deck’s vertical force [11–14]. Also, high amounts of illustrate the enormous value of wave amplitudes at high elevations and low values at low wave amplitudes at down elevations [6, 15]. The wave period is also a parameter that affects the magnitude of forces. An increase in the wave period causes increases to [16]. Also, the largest happens at comparatively low wave periods [12].
The water level plays an essential role in the waves loads and is usually analyzed by parameters inundation and clearance in the coastal bridge studies. Azadbakht et al. examined five California coastal bridges under tsunami loads. They analyzed and as the water level is in the initial impact, overlapping, and full inundation stages. They showed that changes to a downward force when the free surface attains the top of the bridge deck, and is essentially caused by the pressure of water at high submergence. The buoyant force also shows a high value at high submergence levels, and the slamming force and weight of water generate a downward force on the deck at large submergence levels. Huang et al. showed that with the increase of , the lift force increases up to deck self-weight, and after that, the lift force does not increase much [17]. Marin et al. showed that the maximum lift force happens at coefficients of , , and for AASHTO II, IV, and VI girders, respectively [18]. Studies show that when the deck is submerged, the slamming and weight of the water forces made a downward force on the superstructure, and the maximum slamming force occurs in , and in larger clearance , slamming force is small [18, 19].
In the wave-bridge interaction phenomenon, the bridge moves and deforms under the wave force. The movement and deformation of the bridge can vary depending on the structural flexibility of the bridge. In general, structural flexibility depends on the specifications of the materials, sections, members, and connections. The substructure stiffness depends on the structural stiffness of the piles and piers, the soil condition, etc. The deck stiffness depends on the substructure and superstructure connections, such as shear keys, bearing types, shape memory alloys, and restraining cables [20, 21]. There are two approaches in wave-bridge interaction studies. Some studies have neglected the effect of bridge flexibility, and some have considered it in their calculations in different ways. Many studies have been done on the bridge superstructure with rigid support under wave loads [6, 15, 17, 22–24]. The bridge deck with flexible support is a problem that has not been adequately studied yet. Today, powerful software packages are made by computers that can analyze fluid-solid interaction (FSI) problems accurately, but using them to analyze any problem requires verification with similar laboratory models [23]. As an experimental work that considered deck flexibility, we can mention Bradner et al. They made one span of the I-10 Bridge over Escambia Bay with a scale in the Oregon State University wave flume. This experiment was the basis for validating most numerical studies that consider deck flexibility. This study showed that the bridge with a flexible substructure resists larger than the rigid substructure, for all the measured wave heights [11]. Xu et al. analyzed the influence of lateral stiffness in the wave-bridge interaction. They used solitary wave theory for wave loading. They concluded that raising the horizontal direction of structural flexibilities results in larger horizontal forces on the connections of the superstructure and the substructure [25]. The studies of Xu et al. and Bradner et al. indicated that structural flexibility has less effect on than . Cai et al. verified their CFD model by the experimental work of Bradner et al. and then modeled a deck with flexibility in both horizontal and vertical directions under the wave force. They showed that deck flexibility in the vertical direction did not have a significant effect on increasing wave force [26]. Istrati and Buckle carried an FSI analysis in LS-DYNA utilizing a 2D bridge model with a flexible state, which indicated that the dynamic properties of the bridge could influence the created wave load on the bridge [27]. This study exposed the importance of bridge dynamics and FSI; nevertheless, because of the simplified 2D bridge model, it was observed that additional 3D FSI investigations should be performed.
The studies reviewed show that coastal bridges under extreme waves are new and do not have much precedent and need more investigation. Researchers have analyzed this problem in various ways in their numerical, analytical, and experimental work. A study that would consider more realities of the wave force event on coastal bridges is more accurate in predicting wave forces. Some studies have considered the bridge structure to be rigid and fixed and have passed up the structure-fluid interaction [6, 9, 15, 17, 24, 28–30]. This simplification significantly reduces the cost of calculations. It was observed that ignoring the effect of structural flexibility leads to inaccurate results in predicting wave forces on the bridge. In all reviewed studies, the bridge deck has been analyzed alone, and the effect of the substructure and the superstructure has not been analyzed simultaneously under the wave forces. Also, in none of these studies, the three-dimensional display of wave impact on the structure has not been shown. The studies that modeled deck flexibility released the deck in one direction. Most studies examining bridge flexibility concluded that deck flexibility does not help reduce wave forces and suggested deck fixing [11, 25, 26]. However, all factors of structural flexibility cannot be completely solidified. The flexibility of the joints between the deck and the substructure may be significantly reduced, but some factors, such as the flexibility of reinforced concrete, are unavoidable. This study intends to provide a numerical model to supplement some of the shortcomings of previous studies. It is almost impossible to consider all the factors influencing deck flexibility in numerical modeling. However, modeling the structure with natural materials can allow the structure to move in all directions, similar to the natural state.
In this study, a 3D model of the superstructure and substructure of a CHB under the extreme wave loads is presented to determine the effect of all different structural flexibility directions in the wave forces. An experimental study of the substructure and superstructure of the bridge requires a lot of cost and time. Therefore, using a reliable numerical method will be very useful. In the analysis of engineering problems, different methods are used. Each of these methods has its own advantages and disadvantages that make them used in their own problem. Sometimes, it is needed to utilize more than one method to solve the problem to use the appropriate characteristics of each method in its problem and avoid the disadvantages of each method. The FEM is usable for describing vibrations in elastic structures, and the FVM for describing vibrations in the fluid, which is equally suitable. In this case, FEM (structure)/FVM (fluid) coupling can be used to solve the coupled problems. Until a few years ago, FSI problems were done in the form of computational code written by the researcher. The fluid software did not have the ability to move the boundary and especially to calculate the stress in the solid area, and on the other hand, the solid software was not suitable for fluid flow analysis. Today, some reputable engineering software types can do coupling between solid and fluid solvents. ANSYS software has an environment called Workbench that has the ability to link the software in it together. This software uses a module called a system coupling for coupling. In the fluid part, it can use Fluent software, which is software based on FVM. In the solid part, it can use the Mechanical solver, which is software based on the FEM. In Section 2, the numerical method of the FSI model and governing equations are described to simulate the 3D wave-bridge interaction. In Section 3, to validate the FSI numerical model, the results of comparing numerical simulations with an experimental study are presented and compared. In Section 4, to consider the effect of substructure and natural material of the bridge (reinforced concrete) in wave forces, a bridge with substructure and superstructure is modeled of reinforced concrete, and the connection between them is considered fixed and rigid, which means that the flexibility of the structure is due to its material nature. Finally, the results of the structural and hydrodynamic analysis of the simulated model are presented and investigated.
2. Governing Equation and Numerical Method
Investigating the structural dynamic response of the bridge under wave loads is a multiphysics problem. This study uses a structural solver to analyze the dynamic response of the bridge and a fluid solver to analyze the wave hydrodynamics. To achieve this goal, an FSI model is needed. This section provides precise information on the governing equations for the structural solver and fluid solver. In the present study, commercial software ANSYS has been used for FSI simulation. The FSI model requires a coupling process between the structural and fluid solvers. The coupling among the two tools works in both directions at their surface of contact: deformations in the structure resulting from the forces applied by the fluid flow modify the state of the fluid-structure interface; this changes the flow conditions of the fluid, which causes a variation in the forces exerted on the structure at the interface, thus bringing the interaction cycle to a close. In this study, ANSYS performs coupling analysis by the load transfer method. In this method, the unidirectional or bidirectional FSI component analyzes external Mechanical or fluid data systems. In two-way coupling modeling, the displacement of structure is transmitted to the fluid solver, but in one-way coupling modeling, only the fluid pressure on the structure is transmitted to the structural solver. In general, the two-way coupling solution is more reliable. Therefore, this study uses two-way FSI coupling. Figure 3 shows the chart of the two-way FSI scheme. The transient structural module is the structure solver, and the Fluent module is the fluid solver. FSI problems are ordinarily complex. The discretization of the mathematical model in time and space, with time integration for both fluid flow and structure domain, creates an algebraic equation system. In this instance, there is a transition of loads from structure and fluid at the interface and vice versa. Therefore, the boundary conditions change, and the mesh must be updated at each step of the solution. The conservation of mass equation is the same in the fluid, and the general momentum equation cannot be utilized in the transient analysis as the solution domain keeps changing every time. Hence, the mesh has to get updated to the changed flow boundary. The real fluid velocity with respect to a constant mesh is replaced by a relative velocity relating the real fluid velocity to the mesh velocity. Therefore, the momentum equation is changed so that the mesh gets updated every time [31]. ANSYS Fluent uses the dynamic mesh model to simulate flows that the form of the domain is varying with time due to motion on the domain boundaries. The dynamic mesh to model flows when the domain changes with time because of motion on the boundaries of the solid domain. According to dynamic meshes, the integral form of the conservation equation for a general scalar, , on an arbitrary control volume, , where the boundary is moving can be defined aswhere is the flow velocity vector, is the fluid density, is the mesh velocity of the moving mesh, is the diffusion coefficient, is the source term of , and is used to represent the boundary of the control volume, .
2.1. Solid Domain Modeling
Transient dynamic analysis is used to specify the behavior of the bridge. This is a numerical technique employed to specify the structural dynamic response under the action of time-dependent forces. The fundamental equation of motion solved by a transient dynamic analysis iswhere is the mass matrix, is the damping matrix, is the stiffness matrix, is the load vector, is the nodal acceleration vector, is the nodal velocity vector, and is the nodal displacement vector.
The bridge is modeled by reinforced concrete material in the ANSYS Mechanical package. The concrete density is 2400 , and the corresponding poison ratio is 0.18. This concrete is modeled by SOLID186 solid elements, which is a higher-order 3D 20-node solid element. This element shows second-class displacement behavior, and this is defined by 20 nodes by three degrees of freedom per node. The element supports hyperelasticity, plasticity, creep, large deflection, large strain capabilities, and stress stiffening.
2.2. Fluid Domain Modeling
ANSYS Fluent solves conservation equations of mass and momentum for all flows. The conservation equation of mass and continuity equation can be defined as
Equation (3) is the total form of the mass conservation equation and is reliable for incompressible and compressible flows. The source is the added mass to the constant phase from the diffused second phase, and in 2D axisymmetric geometries, the continuity equation is defined aswhere is the radial coordinate, is the axial coordinate, is the radial velocity, and is the axial velocity.
Conservation of momentum in the inertial reference frame (nonaccelerating) is defined as [32] where is the static pressure, is the stress tensor (described below), and and are the gravitational body force and external body forces, respectively. The stress tensor is defined aswhere is the molecular viscosity, is the unit tensor, and the second term on the right-hand side is the effect of volume dilation. Fluent uses the FVM to solve the governing equations for numerical modeling. In this method, the domain of computational is divided into small zones called control volumes, and the governing equations are discretized and determined iteratively for each control volume. The governing equations are combined over each control volume, such that the momentum and mass are conserved in a discrete sense for each control volume. Then, each variable’s approximated values, such as pressures and velocities, are determined at each point throughout the domain.
The fluid domain includes two phases, air with a density of 1.225 and water with a density of 1025. The water is assumed as a viscous, incompressible fluid, and the Volume of Fluid (VOF) method is used to prescribe the dynamic free surface [33]. The VOF model was used to model the fluid. The VOF can make two or more immiscible fluids models using a single set of momentum equations and exploring the volume fraction of each of the fluids all over the domain. The exploration of the interface is accomplished by the continuity equation solution for the volume fraction of one or more phases between phases. For the phase, this equation is defined as follows:where is the mass transfer from phase to phase and is the mass transfer from phase to phase . is zero, but it can be a constant for each phase. For the primary phase, the volume fraction equation does not solve. In the two-phase models, notes that the mesh cell is completely occupied with the phase that is the primary phase, and means that the other phase fluid, the secondary phase, occupies the mesh cell. Nevertheless, shows that the mesh cell is given by both fluids with their respective occupation ratio. In this study, for CHB-wave interaction, the air is introduced as the primary phase and water is the secondary phase, and a is defined as
In order to reduce computational costs, the laminar model is adopted. Laminar flow is ruled by the unsteady Navier–Stokes equations. Xu et al. solved their problem with the turbulence model and the laminar model [25]. Then, they compared their result with the results of McPherson (2008). They showed that there is no notable difference between results. The difference between the results of turbulent flow and laminar flow is seen to be for most utmost conditions.
These wave models can be categorized into two common types: one is a periodical wave model with finite wavelength and wave period, and the other is the solitary wave model with theoretically infinite wavelength and wave period. The wave forces on a bridge by applying different wave models should have some variations. For instance, given the same wave height, the wave forces of a cnoidal wave with a large wavelength are in conventional accordance with those of a solitary wave; nevertheless, there are differences between the results of these two wave models when the cnoidal wave has a comparably smaller wavelength. This is because one single wave in the cnoidal wave series with a large wavelength is similar to a solitary wave in a general wave shape. Nevertheless, the cnoidal wave is beneath the SWL, but a solitary wave is fully over the SWL. Therefore, the results of solitary waves may not be right utilized for those bridges under periodical waves with relatively small wavelengths. The structural oscillation caused by periodical waves may be more incited by the following incident waves, while the oscillation induced by solitary waves decreases all the way to zero. Consequently, according to previous studies considering solitary wave effects on bridges, the present study particularly studies the bridge subjected to one common periodical wave (Stokes 2nd order) [10]. Generally, the wave profiles for second- to fifth-order Stokes theories are defined as follows [34]:and the velocity potential for shallow/intermediate waves is defined aswhere is the wave speed , is the wavenumber , is the wave frequency, is the wavelength, is the wave amplitude, and , , and are functions of liquid height and wavelength for shallow and intermediate waves and permanent amounts for short gravity waves.
3. Validation of the Numerical Model
Bradner et al. investigated the I-10 Bridge over Escambia Bay (Florida) with a 1 : 5 scale as an experimental study. This bridge was damaged in Hurricane Katrina. The capability of the FSI model described previously has been validated by this experimental work. The experiment was done in the large wave flume at Oregon State University. This flume has 104 m long, 3.66 m wide, and 4.57 m deep. The experiment conditions and position of the Gauges are shown in Figure 4. Six scaled AASHTO type III girders were evenly set under the deck slab, and as shown in Figure 5, in this study, all girders were simplified by rectangles. This simplification is usually accepted in past studies [6, 16, 17, 26, 35]. The dimensions shown in Figure 5 are the actual dimensions of the bridge deck, which converted to a scale of 1 : 5 in the experimental study. The deck is supported by two bents, which are fixed with load cells, and a soft dynamic setup is mounted into the reaction frame to examine the effect of structural flexibility on the response of the deck. The natural period of the superstructure is 0.95 s, and the damping effect has been neglected. The fixed and flexible deck was investigated under the wave forces by considering the effect of wave period, wave height, and water level. Other than the shape of the girders, all other parameters considered in the numerical modeling are exactly equal to those used in the experimental work, as shown in Table 1.
In order to simulate this experiment in ANSYS, a 3D model was simulated with experiment specifications. Figure 6(a) shows a two-dimensional view of the sketch of dimensions and boundary conditions considered in the modeling. Wave is generated at the inlet boundary surface and propagates into the outlet boundary surface. The outlet and top surface are managed as a pressure outlet that cannot entirely prevent wave reflection into the domain. The left and right boundary surface is defined as a symmetry boundary condition. The bottom boundary surface is defined as a wall boundary condition. In order to exert the structural flexibility effect, two linear spring elements with the horizontal stiffness were selected and joined to the deck. The experiment does not consider into account the vertical motion and connection failure, so a collection of joints was installed at the bottom of each beam, and to model the rigid state, the deck was fixed from the bottom. These joints free the structure only in the X direction and allow it to displace. The dimensions of the deck in ANSYS were chosen exactly like the experiment so that the span length is , width is , girder height is , girder spacing (CL to CL) is , deck thickness is , and overall height is .
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One of the most important parts of FSI modeling is the meshing process of the solution domain. In solving coupling problems, when two computational methods solve a problem, equilibrium and compatibility conditions in the solution algorithm must be satisfied at the boundary between the two areas where the coupling is done. Therefore, at each point on the common boundary, the displacements in the two regions are equal. To converge equations and satisfy these conditions, the elements of the two domains must be equal at the common boundary. Fluid domain meshing was performed in Fluent, and solid domain meshing was performed in the Mechanical module. Meshing in the fluid domain (Fluent) and the deck (Mechanical) was done with the equal element , and then, the meshing of the fluid and structure was equal parts in the Z direction (). This is because there is less change in the direction perpendicular to the propagation wave. Larger elements were chosen due to small changes in the direction perpendicular to the wave propagation direction. The elements of the fluid domain have larger dimensions as they move away from the deck. Figures 6(b) and 6(c) show the mesh style of fluid and structure domains. The simulation time is with a time step of . To obtain these values, six different resolutions of the space and time in which the smallest mesh sizes range from 0.006 to 0.02 m with time steps 0.01 and 0.02 are adopted for analysis. Figure 7 compares the horizontal and vertical wave forces obtained from the different mesh solutions. As shown in this figure, there is not much difference between the results of , with , , and , . To validate the numerical model, a Stokes wave (the reason for choosing this wave was explained in Section 2) impacted the deck with , height, and period . Figure 8 shows the time histories of the total horizontal and vertical wave forces on the fixed and flexible bridge deck. Figure 9 shows a comparison of the numerical computations and the experimental measurements of the maximum vertical and horizontal wave loads on the bridge deck model for different wave heights at and . As obtained in Figures 8 and 9, the numerical model calculations of both the vertical and horizontal forces have a good agreement with experimental data.
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4. Numerical Simulation and Results Analysis
4.1. A Numerical Model of a CHB under Wave Loads
In this part of the study, a superstructure and substructure of a bridge under wave load are modeled to determine the structural flexibility effect on the initial wave shock. Therefore, according to the purpose of the study, there is no need to accurately model a bridge because the purpose is not to estimate the amount of wave load entering a specific bridge. In this case, two bridge models in equal conditions are required except for flexibility conditions. Given that the modeling is to compare the flexible and fixed structures, the dimensions and problem conditions are considered equal for both modes. Therefore, in this study, it was assumed that the deck is fully integrated with the substructure (rigid connections), and the flexibility of the bridge is due only to the materials of the bridge (reinforced concrete). Therefore, in order to investigate the effect of this flexibility, two bridges in the fixed and flexible modes were modeled in ANSYS. Fixed structure modeling was performed in Fluent software, and then, the flexible structure modeling was performed by Fluent and Mechanical coupling (the solution process is described in Section 2). The problem domain dimensions and boundary conditions are shown in Figures 10 and 11. In this simulation, a 3D numerical domain with a size of 62.1 m in length, 4.2 m in wide, and 5.7 m in height is used, where the bridge model is placed at the 40 m of domain start (see Figures 10 and 11(a)). Figure 11(b) shows the bridge model for comparing fixed (rigid) and flexible modes. According to the purpose of this study, an idea has been proposed to reduce the initial shock of the wave force. The idea is to add a wedge-shaped part to the deck to absorb the wave forces. As shown in Figure 11(c), this wave shock absorber (WSA) compares with three angles (β = 15°, 30°, and 45°). The dimensions of the bridge deck are close to the dimensions selected in the validation section. So, the span length is 3 m, width is 2.1 m, girder height is 0.3 m, girder spacing (CL to CL) is 0.3 m, deck thickness is 0.1 m, and overall height is 0.4 m. Reasonable dimensions are considered for the pier (0.3 × 0.3 × 2 m) and pier-cap (0.3 × 0.3 × 2.1 m) of the bridge. As described in the introduction, FSI modeling is a very complex process. Some techniques can simplify the modeling and calculation process, in which regular meshing is one of them. In this study, meshing was done in such a way where all elements of the structure and the part of the fluid around the structure are equal (∆X = ∆Y = 0.02 m), and then, the meshing of the fluid and structure was 28 and 20 parts in the Z direction (∆Z), respectively. Because in this direction, as shown in Figure 10, the fluid is 1.2 m more than the structure in width. This mode allows the coupling process between solid and fluid solvers to be performed correctly. The elements of the fluid domain have larger dimensions as they move away from the bridge. All elements of the solid and fluid domain are rectangular cubes. The simulation time is 15 s with a time step of 0.01 s. All the characteristics of the wave used and the structure and conditions of the problem are described in Section 2. Second-order Stokes’s wave theory is employed to model cyclic waves by entering the inlet boundary condition as an open channel wave boundary condition. This is coded by Fluent. The water particle velocities u, , and of the second-order Stokes wave theory and free surface profile are described in Section 2. The wave height and wavelength used are similar to those used in the validation section (Stokes second-order wave with 0.5 m in height and 8.6 m in length). This wave has been confirmed as a tsunami wave [11, 26]. Because some studies have shown that the maximum wave force is created in d∗ = 0 [18, 19], in this study, the deck bottom is set at the water level, which means the elevation of the deck bottom is 2.3 m, and for all calculations and comparisons, the wave force impacts the bridge in d∗ = 0. The type of setting set in the Fluent is such that the wave impacts the structure simultaneously as the simulation begins.
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4.2. Hydrodynamic Analysis
In order to compare the wave forces on the fixed and flexible bridges, the results are presented. Figure 12 shows the snapshots of the volume of fluid (VOF) profiles in the wave structure interaction for the flexible bridge. and on the bridge with flexible and fixed modes are shown as a function of time in Figure 13. The simulation is set so that the wave force affects the structure when time starts. The most noticeable difference between flexible and fixed structures occurred in early times. This event lasted for up to in horizontal force and up to in vertical force. The fixed structure tolerates the least forces in the early times. However, this phenomenon is reversed for the flexible structure, and the structure tolerates the most significant forces and moments in the early times. This phenomenon is the initial shock of the wave to the structure that significantly affects the flexible structure. At this moment, the vertical force is upwards (positive), indicating a vast applied moment due to the applied extreme wave loads. In the time of 1 to of the simulation, wave forces become negative after the initial shock, and then, it undulates from negative to positive for the horizontal force. In this event, for vertical force, there is another increase. Then, it was shown that the wave forces have an almost constant trend in to . In general, the results that can be inferred from Figure 13 are that the structure flexibility increases the forces when the wave hits the structure, and this increase will continue for the vertical force but has less effect on the horizontal force. The results show that the initial shock has the most significant impact on the bridge. Since there would be this amount of flexibility in the bridge due to the nature of the bridge materials, in any case, a way had to be devised to prevent its damage. Therefore, the idea of adding a wedge-shaped part to the side of the bridge was considered to counteract the initial wave shock. Three angles (β = 15°, 30°, and 45°) of WSA are considered for this part. As shown in Figure 13, WSA with the angle of has had the most wave force reduction on the flexible structure. When the wave hits the flexible structure, a huge horizontal negative force is created in the structure, which is well controlled when the WSA is added, and it reduces the initial shock of wave forces up to for horizontal force and for vertical force.
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4.3. Structural Analysis
This part presents the structural analysis results of the flexible bridge under wave force in the 3D contour and chart form. These results include deformations and stresses of the flexible structure under wave force. The deformation of each structural element is calculated by Equation (12), where is the total deformation and the three-component deformations , , and are directional deformations.
Structural stresses are presented as von Mises stress, which is part of the maximum equivalent stress failure theory that would determine yielding in a ductile material. Equivalent stress allows any arbitrary 3D stress state to be represented as a single positive stress value. Equivalent (von Mises) stress is related to the principal stresses by where is the equivalent stress and , , and are principal stresses.
Figures 14 and 15 show the maximum deformation and equivalent stress of the flexible structure in 15 s of simulation time. Figures 16 and 17 are 3D snapshots of these deformations and stresses at three times of simulation. The rest of the deformations and stresses can be detected according to the diagrams. Figure 14 shows the maximum deformation of the flexible bridge for the total situation and directions X, Y, and Z. As shown in Figure 16(a), most deformation has occurred in the deck, which is normal according to the wave level. Although the wave forces are greater in the vertical direction (direction Y), most deformation occurred in the direction of wave propagation (direction X). However, due to the direction of these forces, the vertical force causes more damage due to the creation of larger moments. As in many bridge failures, the deck is detached due to large moments. According to the deformation mode of the deck, it can be recognized that the slamming force has the greatest impact in the early times, which is the main factor of the initial shock in the wave-bridge interaction. What is clear from all the results is that most of the stress occurred in the early times. Figure 17 provides exciting results on bridge stresses. In 0.5 s of simulation, the critical points of the bridge with maximum stress are clearly shown. The connections of the piers to the cap, the cap to the deck, and the middle part of the girders are the critical areas of the bridge that have maximum stresses. For better display, the deformations of the flexible structure are shown with a very large deformation scale factor (10000).
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5. Conclusion
In the present study, a 3D numerical model of a CHB under the wave loads is simulated to determine the structural flexibility effect in the initial shock of wave forces. The capabilities of the provided FSI model are validated by an experimental model. The results showed the FSI model is reliable in predicting wave-bridge interaction. Since previous studies concluded that, by giving different types of stiffness to the deck, the flexibility of the bridge deck does not reduce the wave force, and the stability of the deck decreases the wave forces, in this study, the connection between the deck and substructure is considered fixed and rigid, and the flexibility of the bridge is due to its material nature (reinforced concrete). In huge structures such as coastal bridges and construction of the bridge with any rigidity, there is still this flexibility. The structural material is modeled to match the real conditions fully. This material is modeled by SOLID186 solid elements of ANSYS, which is a higher-order 3D 20-node solid element that shows second-class displacement behavior. The element supports plasticity, hyperelasticity, creep, stress stiffening, large deflection, and large strain capabilities. Based on the numerical computations, the following results are obtained:(1)The flexibility of the bridge significantly increases the wave forces so that the initial shock of the wave on the flexible bridge creates the most remarkable forces, while the fixed structure tolerates the least forces in the initial wave shock. This increase in forces has a more significant effect due to the structure flexibility in the vertical force.(2)Adding a wedge-shaped part to the side of the bridge with an angle reduces the initial shock of wave forces down to for horizontal force and for vertical force in the flexible structure.(3)The connections between the piers and pier-cap, the pier-cap and deck, and the middle part of the girders of the deck are the critical areas of the bridge that have maximum stresses and deformations.(4)In general, this study concluded that considering the actual conditions of the wave-bridge interaction model, such as the effect of structural materials, has a significant influence on the results, and their simplification cannot provide realistic results.
As a work in the future, we suggest simulating an FSI model to compare the effect of structural flexibility on (a) applied loads from the integration of pressures in the fluid solver, for both the fixed and the flexible component and (b) the reaction forces from the structural solver.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Meysam Rajabi contributed to conceptualization, methodology, investigation, data curation, writing of the original draft, formal analysis, visualization, and validation; Fahimeh Heydari reviewed and edited the paper; Hassan Ghassemi involved in supervision and reviewing and editing of the paper; Mohammad Javad Ketabdari and Hamidreza Ghafari helped in reviewing and editing of the paper.
Acknowledgments
The High-Performance Computing Research Center (HPCRC) of the Amirkabir University of Technology provided the HPC resources necessary to run the advanced numerical analyses under contract ISI-DCE-DOD-Cloud-700101-4423.