Post-fire nonlinear vehicle–bridge interaction analysis considering hysteretic material behavior

Abstract

Deteriorating bridges and potential vulnerabilities are crucial in the structural assessment of railroad bridges. Among various hazards that a bridge may encounter, evaluating the bridge after a fire is important because it may change the behavior of the bridge. More specifically, the post-fire bridge exhibits structural stiffness degradation, residual stress accumulation, and residual deflection. Due to those residual responses further affecting the dynamics of the bridge, analyzing the post-fire behavior of a bridge subject to normal traffic usage becomes important. Thus, this paper proposes a framework that assesses the dynamic behavior of a post-fire bridge considering the material hysteresis due to thermal loads and vehicle-bridge interaction (VBI). The proposed model is different from previous approaches in that the nonlinear VBI analysis is analyzed in an integrated system equation. So far, most related works exchange the data between separate finite element analysis software and in-house codes, intrinsically facing convergence problems. Instead, by adopting the augmented representation approach, the proposed framework becomes computationally efficient. Herein, nonlinear structural fire analysis is performed to obtain post-fire responses of the bridge. Then, the dynamic analysis is realized by incorporating material hysteretic behavior and VBI systems. The proposed framework is validated against the linear VBI model under gravity load. Subsequently, parametric studies are presented to represent various fire scenarios and long-term serviceability. The results demonstrate that the proposed model provides an effective tool for evaluating post-fire bridge dynamics by including material hysteresis under cyclic traffic loads. Moreover, the proposed framework can easily accommodate various types of hazards as well that cause the nonlinear behavior of the system.

Keywords

assumed modes method Bouc-Wen model finite element method material hysteresis post-fire structures structural dynamics vehicle-bridge interaction (VBI)

Introduction

Bridges present valuable assets for transportation networks, and their serviceability and load-carrying capacity must be ensured. However, these infrastructures are exposed to different hazards during the life cycle, causing gradual or rapid degradation. Thus, to appropriately assess the structure, evaluating the dynamic response of bridges under moving loads is crucial (Frýba, 1999). At the same time, the impact of unexpected hazards, such as fire, which can cause the structure to fail or be partially damaged must be also examined. In this regard, analyzing the post-fire behavior of a bridge subject to vehicle-bridge interaction (VBI) is needed for a comprehensive understanding of a bridge.

To date, the analytical derivation of the VBI problem has been studied extensively since pioneered by Timoshenko (1922). In early efforts, assuming that the moving loads are significantly small compared to the mass of the beam, analytical solutions have been derived by Timoshenko (1922). Intensive studies considering various types of bridge models and the number of loads are carried out by Frýba (1999); the reference has expanded the bridge from a simple beam to a shell and a 3-dimensional solid structure, etc. In addition, Yau et al. (2001) studied VBI problems with elastic bearings and Yang et al. (2004) summarized the effect of vehicle speeds and beam length on structural response of cancelation and resonance conditions. When the mass of the vehicle is significantly large, the inertial force of the mass must be included. In this case, the moving mass assumption yields more complicated analytical solutions than the moving load model. Here, the bridge responses have been evaluated by using a successive approximation method (Jeffcott, 1929) under moving and fluctuating mass. Although analytical derivation can provide an understanding of key parameters affecting VBI, more realistic bridges like multi-span bridges, truss bridges, or suspension bridges, and more complex vehicles with multiple Degrees of Freedom (DOFs) are complicated to solve analytically in VBI problems.

For this reason, numerical approaches have been proposed for analyzing more realistic and thus complex bridges and vehicle systems. In large, numerical approaches to VBI problems including an inertial force of vehicle can be categorized into two types (Chen and Chen, 2014; Deng et al., 2015): (i) Separated representation for each VBI component; (ii) Augmented representation for entire VBI components. Among those two, separated representation first constructs two sets of Equations of Motion (EOM); one system for the bridge and the other system for the vehicle. Then, two systems are coupled using a concept of contact force defined at each time step, and then solved iteratively until satisfying the compatibility and equilibrium conditions. In contrast, the augmented representation forms the EOM for both systems to yield a single and augmented equation. In this approach, the system matrices contain time-varying terms but do not require calculating the two sets of EOMs at each time step.

The separated representation approaches have been studied including linear and non-linear components of VBI systems. In this category, linear vehicle models include a single moving mass and oscillator (Yang and Wu, 2001), half car model (Yang and Fonder, 1996; Wu and Yang, 2003), 3D vehicle model with higher DOFs (Broquet et al., 2004; Liu et al., 2009), and connected multi-vehicles with total DOF of 115 (Xia et al., 2003). Using the finite element (FE) model approach, linear bridge models were expanded from a single-span simple beam (Yang and Wu, 2001), a multi-span bridge (Yang and Fonder, 1996; Wu and Yang, 2003; Broquet et al., 2004), and to a series of simply supported composite-type bridges models (Xia et al., 2003; Liu et al., 2009). Further, the nonlinearities that arose in vehicles (Veletsos and Huang, 1970), contact interfaces (Hutton and Cheung, 1979; Hwang and Nowak Andrzej, 1991; Wang and Huang, 1992), and bridges are also considered. To include bridge nonlinearity, Lu et al. (2020) used FE software, ABAQUS, to simulate the VBI problem; here a penalty contact method is used to estimate and optimize the displacement and force at the contact point. In addition, to examine structural dynamic responses under thermal loads and an accelerating mass, Esen (2019) considered material nonlinearity due to temperature increase for a Timoshenko beam with simple supports. Typically, the frameworks for VBI problems including bridge nonlinearity have been proposed by implementing a coupled system with FE software for bridges and in-house code for vehicle and their interaction (Borjigin et al., 2018; Gong et al., 2020; Stefanidou and Paraskevopoulos, 2022). Then, the responses of each system are exchanged until satisfying equilibrium conditions. When using such frameworks to exchange responses within two software at each time step are time-consuming, and checking equilibrium conditions is numerically inefficient.

When numerically solving separated representations of the total system, the convergence rate problem is intrinsically faced. To resolve this problem, various approaches were developed for coupling the separated system to yield an augmented representation. By combining DOFs at the contact location of a vehicle, VBI analysis including the linear behavior of the system has been carried out (e.g. Yang and Lin (1995) and Kim et al. (2016)). Here, the augmented system not only includes vehicles and a bridge model, but also can be expanded to include track, road roughness, and other excitation loads. Respectively, a wide variety of VBI problems has been solved including a multi-sprung mass on single span beam with an FE model (Cheng et al., 2001; Liu et al., 2021) and a three-dimensional multi-car with higher DOFs (Song et al., 2003; Xia and Zhang, 2005). Also, the complexity of the bridge by exporting the stiffness matrix from ABAQUS (Paraskeva et al., 2017), track system (Cheng et al., 2001), and roughness (Kim et al., 2005) have been included in VBI problem. Furthermore, VBI problems with seismic loads (Kim and Kawatani, 2006), wind loads (Guo et al., 2007), and collision impacts (Xia et al., 2012) are also examined.

The augmented representation approach has been also extended to include nonlinearities of a vehicle (Zhu and Ishitobi, 2006), contact interfaces (Zhai and Cai, 1997; Zhang et al., 2001, 2018; Sun and Dhanasekar, 2002; Dinh et al., 2009; Yang Sin and Hwang Sung, 2016), and a bridge (Au et al., 2001, 2002; Aied et al., 2016). In detail for bridge nonlinearity, the geometric nonlinear behavior of the cable-stayed bridge has been included by Au et al. (2001) and (2002). Moreover, Aied et al. (2016) considered damages on a beam by modeling the material damage behavior at a structure level (i.e., bilinear moment-curvature relationship). In their research, while bridge nonlinearity has been investigated, the nonlinear has not been considered at a material level, and this approach cannot be applied to obtaining stress variation in the depth of the beam. Therefore, studies for VBI analysis considering bridge nonlinearity at a material level using the augmented representation approach have not been investigated.

Especially, when a bridge is subjected to fire, the material nonlinearity can easily be formulated at the structure. Usmani et al. (2001) analytically studied the effects of the various constraint and thermal loads resulting in different residual stresses and material nonlinearity. The material model has been also numerically studied by Franssen (1990) by considering material variation that occurred at increased temperature and cool-down phase. The post–fire mechanical properties of structural steel have been experimentally studied by (Qiang et al., 2012; Wang et al., 2015; Wang and Lui, 2020). Then, simple structures constituted by several trusses (Wang et al., 2008; Lin et al., 2010, 2012) and beams (Iu et al., 2005; Lien et al., 2009) under fire until cool-down also have been investigated to quantitatively assess the residual stress and deflection exist after thermal loads. Moreover, post-fire bridge structures have also been simulated to investigate the residual strength by forcing the bridge until the buckling phase (Tang et al., 2019). To understand the post-fire behavior of a structure appropriately, a material nonlinearity under temperature variation, which also includes the cool-down phase must be considered in the modeling procedure.

To further evaluate the dynamic behavior of a post-fire structure, a few studies have applied dynamic loads after considering material nonlinearity. Seismic performances in various post-fire scenarios have been investigated experimentally (Li et al., 2019a, 2019b; Liu et al., 2021) and numerically (Mo et al., 2004; Ni and Birely, 2018; Cai et al., 2021). These studies showed that post-fire structures under cyclic loads show more rapid stiffness degradation compared to a structure with no fire. Because a bridge subject to a regular traffic load also experiences cyclic stresses, to assess the post-fire residual strength and serviceability of a bridge, VBI after fire needs to be studied, which yet has been lacking.

Thus, this paper proposes a VBI framework that evaluates post-fire bridge nonlinear response accounting hysteretic behavior of material using an augmented representation approach. In the next section, the theoretical backgrounds for the nonlinear behavior of a structure under thermal loads are introduced. Then, the post-fire nonlinear VBI analysis framework is proposed including the formulation of nonlinear dynamic VBI analysis. Subsequently, the proposed nonlinear model is validated with a linear VBI model under the gravity load with no fire scenario. In addition, post-fire structural nonlinear responses such as deflection and stress are observed and compared with a linear model. Using the model various parametric studies are followed: (i) The impact of critical thermal load location on post-fire VBI problem, (ii) The impact of maximum temperature experienced on the residual stress and deflection after cool-down, (iii) the effect of different velocities and the number of moving masses, and (iv) the number of traffic cycles are examined. The results of the proposed study demonstrate that the proposed framework can be an efficient tool for evaluating post-fire behavior under cyclic traffic loads.

Theoretical backgrounds

To understand the nonlinear behavior of a structure, this section performs static nonlinear structural fire analysis and observes the stress variation under thermal loads and various boundary conditions. When a structure is subject to thermal loads and external static loads simultaneously, the governing equilibrium equation (Bathe, 2006) is

R_{s} (u_{s}) = F_{s}

(1)

where

R_{s} (u_{s})

is the internal restoring force which is a function of displacement

u_{s}

, and

F_{s}

is an external force.

F_{s}

includes thermal loads and gravity, where the thermal loads are determined by the material property such as thermal expansion,

α_{T} (T)

, and temperature

T

. A symbol ‘

T

’ with a parenthesis indicates that the variable is a function of temperature. In addition,

R_{s}

is induced by stress resultants,

σ

, of the deformable structure. The distribution of

σ

is determined by the nonlinear constitutive relationship between

σ

and mechanical strain

ϵ_{M}

, where this relationship is controlled by material properties such as elastic modulus

E (T)

and yield stress

σ_{y} (T)

. These material properties vary depending on the temperature, which leads to the nonlinear behavior of the structure. Complying the continuity of a structure, the strain compatibility condition is defined as:

ϵ_{t} = ϵ_{M} + ϵ_{T}

(2)

where

ϵ_{t}

is total strain, related to the displacement of structures, and

ϵ_{T} = α_{T} Δ T

is thermal strain,

Δ T

is temperature increments. By satisfying the strain compatibility condition in equation (2) and the nonlinear constitutive relationship,

u_{s}

can be estimated by solving equation (1).

Figure 1(a) is a simple restrained axial structure prepared to illustrate the behavior of $σ$ and $ϵ_{M}$ with respect to thermal loading due to nonlinear constitutive relationships (Gernay and Gamba, 2018). Note that in this specific example, the magnitude of $ϵ_{M}$ is always the same as $ϵ_{T}$ , because $ϵ_{t} = 0$ (Usmani et al., 2001). To better emphasize the phenomenon, constant initial stress is applied, which the magnitude being half of the yield stress at room temperature 20°C ( $σ_{y 20}$ ). Thermal loads are assumed to be uniform and increase from 20°C to 600°C followed by a cool-down phase, back to 20°C, and examined at a 5°C increment. In the calculation, material properties from Eurocode 3 (CEN, 2005) are adopted and neglected buckling. Figure 1(b) shows the stress variation with respect to $σ_{y 20}$ : In the heating phase (solid line), the column develops compression quickly reaches to $σ_{y 20}$ , at 90°C and then shows nonlinear behavior afterward. At this phase, the stress decreases with the higher temperature due to the reduction of $σ_{y} (T)$ . In addition, at cool-down phase (dotted line), the stress reversely increases resulting in residual tensile stress at room temperature.

Figure 1.

(a) Restrained column; (b) Stress variation with temperature.

In reality, the boundary condition of a structure is determined by the surrounding structures which can be represented with a stiffness exhibiting a value between zero (e.g., fully un-restraint such that $ϵ_{M} = 0$ ) and infinite (e.g., fully restraint such that $ϵ_{t} = 0$ ). To examine the nonlinear variation of $σ$ for a flexural structure with various boundary conditions, a beam shown in Figure 2(a) is prepared. Here, the length of the beam is 10 m, which is subject to distributed loads (gravity) and uniformly distributed local thermal loads at the middle span with a width of 2.5 m. In thermal analysis, the effect of heat transfer is ignored. In addition, the material properties and temperature variation are used as same as the previous one. A constant end spring stiffness is $K_{S}$ , such that the beam becomes pin-roller condition with $K_{S} = 0$ , and pin-pin condition when $K_{S} = \infty$ . Under the condition of $K_{S} = \infty$ , the stress variation at the midspan varies as shown in Figure 2(b). Here, the residual stress also can be observed to be 50.33% of $σ_{y 20}$ , which is lower than the previous restrained column case (i.e., $ϵ_{t} = 0$ ). Figure 2(c) plots the maximum residual stress of beam obtained from different $K_{S}$ , varied from $10^{5}$ to $10^{12} N / m$ . To confirm the convergence of the residual stress, $K_{S} = \infty$ is plotted using dotted line. Therefore, when solving equation (1), the residual behaviors such as residual stress are affected by strain compatibility condition (i.e., equation (2)), and nonlinear constitutive relationship which is controlled by material properties (i.e., $E (T)$ and $σ_{y} (T)$ ).

Figure 2.

(a) Configuration of a beam under gravity and thermal loads; (b) Stress variation versus temperature at midspan (K_S = ∞); (c) Residual stress after the heat- and cool-down phase with various K_S.

Proposed post-fire nonlinear vehicle–bridge interaction analysis framework

General framework for post-fire nonlinear VBI analysis

This section develops a post-fire nonlinear VBI analysis framework for a railroad bridge structure (Figure 3). The proposed framework is composed of two phases: nonlinear structural fire analysis and nonlinear dynamic VBI analysis.

Figure 3.

Proposed post-fire nonlinear VBI analysis framework scheme.

In the step of nonlinear structural fire analysis, to obtain the nonlinear static residual deformation and behavior of a post-fire structure, a bridge under gravity and thermal loads are simulated. Herein, simplified fire loads are assumed, which do not include heat transfer analysis. However, when more complicated fire scenarios are required, the aid of professional software for heat analysis in structures can be adopted for obtaining the structural response after a fire. In this study, this phase is calculated using (ABAQUS, 2020) assuming that single or multiple members on the bridge are subject to fire. Then, the rail system is integrated to satisfy the equilibrium status. As a result, the residual deflection and hysteretic behavior of the bridge are estimated. These outputs are then applied as initial conditions of the structure for the nonlinear dynamic VBI analysis (post-fire phase). Then, in nonlinear dynamic VBI analysis, to include the hysteretic behavior of a material, the EOM of the VBI problem is formulated. Over decades various phenomenological models have been proposed to introduce material nonlinearity during dynamic loading. In this study, the Bouc-Wen model is implemented, a versatile tool that can describe hysteretic behavior with an appropriate choice of model parameters. Finally, the constructed augmented EOM can be solved using a direct time-step integration procedure. The advantage of the proposed framework using the augmented representation approach to include material nonlinearity of the bridge is numerical convergence, simplicity, and accuracy, compared to the separated representation approach.

Post-fire nonlinear VBI formulation for truss railroad bridge

This section applies the proposed framework on a benchmark 2-dimensional truss railroad bridge structure (Figure 4) adopted from Kim et al. (2016). Elements of the bridge and rail are modeled as Euler Bernoulli beams. The rail is assumed as a single continuous and smooth element with a length $l_{R}$ , flexural stiffness $E I$ , density $ρ_{R}$ , area of cross-section $A_{R}$ , and damping coefficient $c_{R}$ . The continuous rail model has the advantage of using less DOF than the FE model for the same accuracy. The bridge is formulated as an FE model and assumed a constant damping coefficient $c_{B}$ and density $ρ_{B}$ for all elements. In addition, discrete elastic springs $k_{s}$ is introduced to link the rail and the bridge to represent sleepers. Two identical lumped masses, $m_{v}$ , moves along the bridge at a constant speed $v$ , and the distance between two masses is $d$ . Moreover, $l_{B i}$ is the length of each member with the subscript $i$ indicating each element, $A_{B i}$ is the cross-section area of element $i$ .

Figure 4.

Configuration of bridge and rail.

Using the augmented representation, the governing EOM of the total structures with the displacement state vector $u = {[\begin{array}{c} u_{B} (t) & q (t) \end{array}]}^{T}$ becomes

M (t) \ddot{u} (t) + C \dot{u} (t) + [\tilde{K} u (t) + \tilde{R} (t)] = F (t)

(3)

where

M

is a mass matrix,

t

is a time variable,

C

is a damping matrix,

R

is an internal restoring force,

\tilde{R} (t) = {[\begin{array}{c} R_{B} (t) & 0 \end{array}]}^{T}

, and

R_{B}

is the restoring force vector of the bridge beam element, and

F

is the external force for both rail and the bridge. In addition, as can be seen from equation (3),

\tilde{K}

consists of linear components (here the rails and sleepers) only, while any system with nonlinear behavior can be embedded in

R_{B}

, which is a function of material nonlinear properties

R_{B} = \sum_{i = 1}^{n_{B}} \int_{0}^{l_{B i}} B \int_{A_{B i}} Y σ_{B} (ϵ) d A d x

(4)

Here, $n_{B}$ is the total number of bridge elements, $B (x)$ is the derivative of the shape function ( $H_{B}$ ), $Y = [\begin{array}{c} 1 & y \end{array}]$ , $y$ is a coordinate of an element, $d A$ , from the neutral axis of the section, and $σ_{B}$ is stress as a function of $ϵ$ . A more detailed formulation procedure is described in Appendix. In the proposed framework, the second-order differential equation Eq. (3) is transferred to an ordinary differential equation such as

{\begin{array}{c} \dot{u} \\ \ddot{u} \end{array}} = [\begin{array}{c} 0 & I \\ 0 & - C M^{- 1} \end{array}] {\begin{array}{c} u \\ \dot{u} \end{array}} + [\begin{array}{c} 0 & 0 \\ M^{- 1} & - M^{- 1} \end{array}] {\begin{array}{c} F \\ R \end{array}}

(5)

where

R (t, u) = [\tilde{K} u (t) + \tilde{R} (t)]

. The initial conditions of the VBI problem equation (5), which are output from the first phase becomes

{[\begin{array}{c} u (t = 0) & \dot{u} (t = 0) \end{array}]}^{T} = {[\begin{array}{c} u_{B} (t = 0) q (t = 0) & {\dot{u}}_{B} (t = 0) & \dot{q} (t = 0) \end{array}]}^{T} = {[\begin{array}{c} d_{0} & 0 \end{array}]}^{T},

and

d_{0}

is the displacement of the rail and the bridge.

When calculating $R_{B}$ , one can apply the Bouc-Wen material model to approximate $σ_{B}$ , by calculating $σ_{B W}$ . In this study, an expression for $σ_{B W}$ , which includes material nonlinearity, strain rate, and a priori stress is chosen as below: (Wen, 1976)

σ_{B W} (t) = α k_{e} ϵ (t) + (1 - α) k_{e} ϵ_{y} z (t)

(6)

Here, $ϵ (t)$ is the strain, $k_{e}$ is the linear elastic modulus, $ϵ_{y}$ is the yield strain, and $α = k_{p} / k_{e}$ is the ratio of the post-to pre-yield stiffness, $k_{p}$ and $k_{e}$ respectively. The evolutionary parameter $z (t)$ models the smooth hysteretic behavior and is governed by the following differential equation

\dot{z} (t) = \frac{1}{x_{y}} [A_{B W} - {| z (t) |}^{n} (β + s g n (\dot{ϵ} (t) z (t)) γ)] \dot{ϵ} (t)

(7)

where

α

γ

β

n

, and

A_{B W}

are the parameters of the Bouc-Wen model governing and controlling the scale and general shape. Throughout the study,

α = 0

γ = 0.45

β = 0.55

n = 4

, and

A_{B W} = 1

are used. The initial condition of the Bouc-Wen model with the following

σ_{B W} (t = 0) = σ_{0}

and

z (t = 0) = ϵ_{0}

. Here

σ_{0}

and

ϵ_{0}

are stress and strain of the bridge.

Using this augmented representation approach, the proposed framework can solve the nonlinear dynamic response of the structure by considering material hysteretic behavior. However, the scope of this paper may be limited to solving the problem with the following assumptions: (1) Buckling is not included. Because of the geometry of the bridge’s truss member such as holes in the member, the local buckling is assumed not to exist. Also, the chosen fire scenarios do not cause global buckling; (2) At nonlinear structural fire analysis, the rail structure is not included because rail is assumed to be a separated system with its movement tied with the bridge due to sleepers; (3) Uniform thermal loads are assumed within each member, due to the aim of the study targeted to observe the fundamental bridge behavior of post-fire VBI problem; (4) Neglected the rail roughness and the rigid body motion of the vehicles to focus on the dynamics of the bridge, as compared to the vehicle dynamics. However, based on the scope of future studies, the aforementioned features can be easily incorporated into the proposed post-fire nonlinear VBI framework.

Model validation

This section validates the proposed framework by comparing the linear behavior with the linear model proposed by Kim et al. (2016). Modal parameters used in two phases are introduced. The primary purpose of this validation is to check whether the augmented representation approach with the Bouc-Wen material model behaves linearly under gravity loads. In addition, the proposed model in material nonlinearity is also evaluated.

Model parameters

The truss bridge used in this study (Figure 4) is assumed to have pin-pin supports. Following the design specifications presented by Kim et al. (2015), the bridge is made of ASTM A36 (ASTM, 2005) steel and the geometry of the cross-section is a combination of W-sections and hollow sections. In the simulation, the number of integration points at each cross-section is 13, and a 2D beam element, B23, is chosen. Modal damping is used for the entire rail-bridge system, with a damping ratio of 2%.

Given the type of steel, the yield strength is 250 MPa and the ultimate tensile is 450 MPa. In addition, the material properties under elevated temperatures are defined following Eurocode 3. Figure 5(a) and 5(b) show temperature-dependent thermal expansion and reduction factors of the effective ultimate strength and the elastic modulus, respectively. Due to the scope of the study, geometric nonlinearity is not considered. Moreover, the mass of each vehicle is set as 20% of the mass of the bridge, and to emphasize the dynamic effects of the mass-selected speed of the vehicle 180 km/h.

Figure 5.

Temperature-dependent material properties: (a) thermal expansion; (b) reduction factor of the effective ultimate strength and the elastic modulus.

Dynamic behavior validation

The performance of the proposed model is compared with a linear model under two simple scenarios: VBI analysis under a material linear range (e.g., applying gravity and masses only), and nonlinear dynamic VBI analysis under a nonlinearity range of material (e.g., subsequently applying gravity, fire and moving masses). Figure 6 shows the simulated deflections at the midspan of the bridge under gravity and moving masses. The solid line shows the results from the proposed framework, compared with the result (the dotted line) from a linear model by Kim et al. (2016). Both models show very similar responses, verifying the proposed framework with the Bouc-Wen model under linear loading scenarios.

Figure 6.

Deflection comparison between the proposed model with Kim et al. (2016)’s model.

Using the proposed framework post-fire nonlinear VBI analysis is also simulated, where the fire scenario leads to material nonlinearity. Here a vertical member (element 20 in Figure 7) at the midspan is heated up to 600°C and cooled down to room temperature with gravity, followed by moving masses. The maximum residual stress of each member is shown in Figure 8(a), where the highest stress of 249 MPa was observed at element 5. More specifically, the stress history with respect to strain is plotted in Figure 8(b), indicating that the member experiences nonlinear behavior. An example of the cross-sectional stress distribution after the moving mass is described in Figure 8(c). As can be seen, the material nonlinear behavior throughout the profile is overserved, with near yield stress at the top fiber in an extreme case. In addition, Figure 9(a) shows the midspan deflection of the proposed model (solid line), in comparison with that under the linear VBI model (dashed line). Note that the non-zero initial condition of the nonlinear dynamic VBI is applied using the results from nonlinear structural fire analysis; whereas zero initial condition is applied for the linear case. Due to the nonlinear behavior, the deflection does not recover to the initial position after the vehicle crosses the bridge, in the post-fire VBI case (Figure 9(b)). This phenomenon is summarized in Table 1; the deflection increased from −5.8461 mm to −5.8955 mm, corresponding to the residual deflection of 0.0494 mm. From the presented study, the proposed model can describe the nonlinear hysteretic strain-stress relationship of a member with non-zero initial conditions at a structure level, resulting the residual deflection after forced excitation.

Figure 7.

Thermal loads in bridge with element ID.

Figure 8.

Results of stress: (a) Maximum stress distribution at each element; (b) Stress history at maximum stress occurred element; (c) Stress distribution at cross-section after VBI.

Figure 9.

Midspan deflection: (a) Entire response; (b) Zoomed results from −0.6 to 0.2 mm.

Table 1.

Residual deflection.

Analysis	Residual deflection (mm)
After nonlinear structural fire analysis	−5.8461
After post-fire nonlinear VBI analysis	−5.8955

Model applications

This section performs the parametric studies using the proposed post-fire nonlinear VBI analysis framework. Four applications are prepared: The first study evaluates the behavior of structures under thermal loads under various locations (elements), followed by the impact of maximum temperatures of the thermal load at a specific location. Then, the effect of moving masses (i.e., number and velocity) is examined, and lastly, the study on the relationship between the number of repetitive loads and residual deflection is discussed.

Thermal load location

To evaluate the critical location under fire, the truss bridge under thermal loads with various locations is simulated. In this study, assuming that the vertical and diagonal elements are less sensitive to boundary conditions, 17 elements (from IDs 12–28) are selected. In each case, the selected element is assumed to be under the thermal load of a maximum of 600°C, followed by passing through two moving masses (20% of the mass of the bridge) at a velocity of 180 km/h. The maximum stresses and residual deflections are compared as summarized in Figure 10. Note that the maximum stress may or may not occur at the same element that was under thermal load. The results show that the cases for elements 16, 20, and 24 under thermal load experienced maximum stresses lower than $σ_{y 20}$ , while other cases exhibit between $σ_{y 20}$ and $σ_{u 20}$ . Figure 10(b) illustrates the maximum residual deflection at a node, compared with Figure 10(a), the maximum residual deflection occurred for vertical members under thermal load (i.e., elements 16, 20, and 24). This phenomenon indicates that the maximum deflection could occur in the case where the temporal maximum stress is not the maximum. From the result of the study, when considering a single-element fire scenario, the vertical member at midspan, i.e., element 20, is chosen for examining the impact of other factors, such as maximum temperature, number of masses and velocity, and repetitions.

Figure 10.

Comparison results under various thermal load scenarios: (a) maximum stress; (b) residual deflection.

Maximum temperatures

To evaluate the effect of a target elevating temperature on residual $σ_{B} (ϵ)$ , bridge responses (residual stress and deflections) under various thermal loads are simulated. Firstly, element 20 in Figure 7 is heated up to target temperatures (i.e., 100, 200, 300, 400, 500, and 600°C) assuming the structure is solely under gravity. The temperature increment from room temperature (20°C) is 5°C. Then, the cool-down phase is followed to 20°C with the same temperature interval. Figure 11(a) and Figure 11(b) show the residual $σ_{B}$ and residual deflection after cool-down phase with respected to various target temperatures, respectively. Due to the symmetry, the maximum residual $σ_{B}$ after cool-down occurs at elements five and 6. As shown in Figure 11(a), the residual $σ_{B}$ increases abruptly beyond target temperature of 300°C. In all cases, the residual $σ_{B}$ does not exceed $σ_{y 20}$ , and at the target temperature of 600°C, the residual stress is nearly 59% of $σ_{y 20}$ . The residual deflections increase in nonlinear manner with the increase of target temperature (Figure 11(b)).

Figure 11.

Results with various temperatures: (a) Residual stress; (b) Residual deflection.

Moving masses number and velocity

To evaluate the impacts of moving masses, post-fire VBI analyses under various numbers of moving masses and velocities are performed. The bridge configuration is the same as the previous example and the target temperature of 600°C is chosen. Vehicle velocity and the number of moving masses are varied from 72 km/h to 252 km/h and from 1 to 4, respectively. Each moving mass is set equal to 20% of the bridge mass. Figure 12(a) and 12(b) show that residual deflection increases with more numbers of moving masses. When velocities are relatively slow (below 180 km/h), the residual deflection tends to increase rather linearly. With the higher speed, the amount of increase shows nonlinear trends. Such a phenomenon may be due to resonance and cancelation on the bridge, which needs to be investigated in more detail in future studies.

Figure 12.

Residual deflection with velocities and the number of moving masses: (a) Contour; (b) Distribution.

Repetitive moving mass

In this section, the impact of repetitive moving mass on the residual deflection is examined. Here, a vehicle, whose mass is about 20% of the bridge crossed the bridge at 180 km/h and repeated for 1,000. Note the wheel distance of a consecutive car is set far enough (476.89 m) such that the dynamic effect of the vehicle on the bridge dissolves. Using a single fire scenario, with the location and the target temperature described in the previous example, the residual deflection respected to the number of loading repetitions is summarized in Figure 13(a). Here, the residual deflection is defined by subtracting the deflection due to the fire; thus, the plot only examines the effect of the repetitive moving mass on the post-fire structure. Figure 13(b) also plots the incremental deflection generated at each moving mass. The residual deflection is sharply increased within 100 repetition moving masses. Then, the increasing rate becomes smoother and is almost decreased to zero, which means that the residual deflection will be linearly increased at a lower rate with repetition number growth. The residual deformation yields residual stress making structures undergo structural nonlinearity with external loads. These results reveal that the repetitive moving vehicles affect the residual deflection of the post-fire structure, and thus must be considered in structural assessment to ensure serviceability.

Figure 13.

Results after repetitive moving masses: (a) Residual deflection; (b) Increased deflection.

Conclusions

The post-fire nonlinear VBI dynamic analysis framework has been proposed in this study for predicting the nonlinear dynamic response of a bridge including the hysteretic behavior of material acted by moving vehicles. Due to the restraint conditions, residual stress accumulates in each structural member during a fire, causing the material to exhibit nonlinear behavior during normal usage. Thus, the primary objective of the presented study is to understand such effects, i.e., post-fire residual stress onto the structural dynamic response under moving masses, in a computationally efficient manner. The developed framework is composed of two parts: a nonlinear structural fire analysis (during the fire phase); and a nonlinear dynamic VBI analysis (post-fire phase). The nonlinear structural fire analysis calculates the post-fire deformation due to an arbitrary member fire. Then, adopting the deformation as the initial conditions, VBI analysis is realized by embedding the Bouc-Wen model in the system equations. The performance of the proposed model has been validated with a linear model to demonstrate that both models match well under linear loading. Beyond the fully linear phase, the model can further accommodate conditions where certain members in the system undergo a nonlinear state consistently, influencing the static components of the bridge response. To illustrate the efficacy of the proposed model for observing the relationship between dynamic response and post-fire residual behavior, examples are provided. From the results of the study, variables such as temperature locations, target temperature, vehicle speed, and the number of passages cause the dynamics of the bridge, while increments of each variable show nonlinear increases. As a result, to ensure the serviceability of a post-fire structure, structural deflection and stress should be assessed accommodating the nonlinear material responses in system analyses. Although the presented framework applied arbitrary fire loads, the model is not restricted to post-fire conditions but can incorporate other types of nonlinear scenarios which can cause nonlinear VBI dynamics such as wind. Also, a fatigue analysis of post-fire bridges under moving vehicles could be developed for future studies. Moreover, quantitatively investigating various thermal loads and more realistic fire scenarios are required to confirm the post-fire effect on structures under extremely severe conditions.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1F1A1051668).

ORCID iD

Robin Eunju Kim

Appendix

This appendix derives EOM for augmented representation of the bridge and the vehicle considering bridge nonlinearity using the principle of virtual work.

By the principle of virtual work, total potential energy by internal virtual work and external virtual work (8)

δ Π = δ U - δ V = 0

where

δ U

is internal virtual work

δ U = δ U_{B} + δ U_{R} + δ U_{S} + δ U_{R D} + δ U_{B D}

, and

δ V

is external virtual work

δ V = δ V_{B I} + δ V_{R I} + δ V_{M I} + δ V_{M}

δ U_{B}

δ U_{R}

δ U_{S}

δ U_{D R}

, and

δ U_{D B}

are internal virtual work done by the bridge’s strain, rail, discrete spring, rail’s damping, and bridge’s damping, respectively.

δ V_{B I}

δ V_{R I}

δ V_{M I}

, and

δ V_{M}

are external virtual work done by the bridge’s inertial force, rail’s inertial force, moving masses’ inertial force, and moving masses. Those are (9)

δ U_{B} = \sum_{i = 1}^{n_{B}} \int_{V_{B i}} δ ϵ σ_{B} (ϵ) d V

(10)

δ U_{R} = \int_{0}^{L} E I u_{R}^{″} (x, t) δ u_{R}^{″} (x, t) d x

(11)

δ U_{S} = \sum_{i = n_{B R}} {\int_{0}^{L} k_{s} [u_{B i} (x, t) - u_{R} (x, t)] δ u_{B i} d x + \int_{0}^{L} k_{s} [u_{R} (x, t) - u_{B i} (x, t)] δ u_{R} d x}

(12)

δ U_{R D} = \int_{0}^{L} c_{R} {\dot{u}}_{R} δ u_{R} d x

(13)

δ U_{B D} = \sum_{i = 1}^{n_{B}} \int_{0}^{l_{B i}} c_{B} {\dot{u}}_{B} δ u_{B} d x

(14)

δ V_{B I} = - \sum_{i = 1}^{n_{B}} \int_{V_{B i}} ρ_{B} {\ddot{u}}_{B} (x, t) δ u_{B} d V

(15)

δ V_{R I} = - \int_{V_{R}} ρ_{R} {\ddot{u}}_{R} (x, t) δ u_{R} d V

(16)

δ V_{M I} = - \int_{0}^{L} m_{v} {\ddot{u}}_{R} (x, t) δ u_{R} d x

(17)

δ V_{M} = \int_{0}^{L} m_{v} g δ u_{R} d x

To establish the EOM of the total system from equation (11), the bridge element and rail element are summarized as follows. The beam element has six DOFs in the global system, i.e., two translational displacements and one rotation at each node. These DOFs are grouped into a vector $u_{B} = {[\begin{array}{c} u_{1} & v_{1} & θ_{1} & u_{2} & v_{2} & θ_{2} \end{array}]}^{T}$ , where $u_{1}$ and $u_{2}$ are axial DOF of beam element, $v_{1}$ and $v_{2}$ are vertical DOF of beam element, $θ_{1}$ and $θ_{2}$ are rotational DOF of the beam element. The displacement component of the beam element along the x-direction (axial displacement) $u_{B u}$ and in y-direction $u_{B v}$ can be interpolated between the element DOFs with shape function (18)

{\begin{array}{c} u_{B u} \\ u_{B v} \end{array}} = H_{B} d = [\begin{array}{c} h_{B 1} & 0 & 0 & h_{B 4} & 0 & 0 \\ 0 & h_{B 2} & h_{B 3} & 0 & h_{B 5} & h_{B 6} \end{array}] u_{B}

where

H_{B}

is shape function, and

h_{B 1} = 1 - x / l_{B i}

h_{B 2} = 1 - 3 {(x / l_{B i})}^{2} + 2 {(x / l_{B i})}^{3}

h_{B 3} = x - 2 L {(x / l_{B i})}^{2} + L {(x / l_{B i})}^{3}

h_{B 4} = x / l_{B i}

h_{B 5} = 3 {(x / l_{B i})}^{2} - 2 {(x / l_{B i})}^{3}

h_{B 6} = - L {(x / l_{B i})}^{2} + L {(x / l_{B i})}^{3}

are an interpolation function related to

x

From the Euler-Bernoulli beam theory, the displacements in terms of displacement components are (19)

u = u_{B u} - y \frac{\partial u_{B v}}{\partial x}

Correspondingly, axial strain is (20)

ϵ = \frac{\partial u_{B u} (x)}{\partial x} - y \frac{\partial^{2} u_{B v} (x)}{\partial x^{2}}

Taking derivatives of the displacement field in equation (18) with respect to x and substituting the results into equation (20), yields in matrix form (21)

ϵ = Y B u_{B} = [\begin{array}{c} 1 & y \end{array}] [\begin{array}{c} b_{1} & 0 & 0 & b_{4} & 0 & 0 \\ 0 & b_{2} & b_{3} & 0 & h_{5} & h_{6} \end{array}] u_{B}

where

Y = [\begin{array}{c} 1 & y \end{array}]

B

is the derivative of

H_{B}

, and

b_{1} = - 1 / l_{B i}

b_{2} = 12 x / {l_{B i}}^{3} - 6 / {l_{B i}}^{2}

b_{3} = 6 x / l_{B i}^{2} - 4 / l_{B i}

b_{4} = 1 / l_{B i}

b_{5} = 6 / {l_{B i}}^{2} - 12 x / l_{B i}^{3}

b_{6} = 6 x / l_{B i}^{2} - 2 / l_{B i}

Rails are modeled as a continuous simple beam using the assumed modes method (22)

u_{R} (x, t) = N_{R} (x) q (t)

where

N_{R}

is a vector with a series the shape functions,

q

is a time-dependent generalized displacement of rails. Finally, by substituting equations (18), (21), and (22) into equations (9)–(17), and combining with equation (8), the EOM of total structures can be obtained with the displacement state vector

u = {[\begin{array}{c} u_{B} (t) & q (t) \end{array}]}^{T}

becomes as equation (3), and can be written in detail as

[\begin{array}{c} M_{B} & 0 \\ 0 & M_{R} + Δ M_{R} (t) \end{array}] {\begin{array}{c} {\ddot{u}}_{B} (t) \\ \ddot{q} (t) \end{array}} + [\begin{array}{c} C_{B} & 0 \\ 0 & C_{R} \end{array}] {\begin{array}{c} {\dot{u}}_{B} (t) \\ \dot{q} (t) \end{array}} + ([\begin{array}{c} K_{B B} & K_{B R} \\ K_{R B} & K_{R R} + K_{R} \end{array}] {\begin{array}{c} u_{B} (t) \\ q (t) \end{array}} + {\begin{array}{c} R_{B} (t) \\ 0 \end{array}}) = {\begin{array}{c} 0 \\ F_{M} (t) \end{array}} (23)

here (24)

M = [\begin{array}{c} \sum_{i = 1}^{n_{B}} \int_{0}^{l_{B i}} ρ_{B} A_{B i} H {(x)}^{T} H (x) d x & 0 \\ 0 & \int_{0}^{l_{R}} ρ_{R} A_{R} N_{R} {(x)}^{T} N_{R} (x) d x + \int_{0}^{l_{R}} m_{v} d x \end{array}]

(25)

C = [\begin{array}{c} \sum_{i = 1}^{n_{B}} c_{B} \int_{0}^{l_{B i}} H_{B} {(x)}^{T} H_{B} (x) d x & 0 \\ 0 & c_{R} \int_{0}^{l_{R}} N_{R} {(x)}^{T} N_{R} (x) d x \end{array}]

(26)

\tilde{K} = [\begin{array}{c} \sum_{i = 1}^{n_{B}} k_{s} \int_{0}^{l_{B i}} H_{B} {(x)}^{T} H_{B} (x) d x & \sum_{i = n_{B R}} - k_{s} \int_{0}^{l_{B i}} N_{R} {(x)}^{T} H_{B} (x) d x \\ \sum_{i = n_{B R}} - k_{s} \int_{0}^{l_{B i}} H_{B} {(x)}^{T} N_{R} (x) d x & k_{s} \int_{0}^{l_{R}} N_{R} {(x)}^{T} N_{R} (x) d x + E I \int_{0}^{l_{R}} {N_{R}}^{″} {(x)}^{T} {N_{R}}^{″} (x) d x \end{array}]

(27)

F_{M} = - m_{v} g \int_{0}^{l_{R}} N_{R} (x) d x

where

M_{B}

is a mass matrix of the bridge,

M_{R}

is a mass matrix of rail,

Δ M_{R}

is a mass matrix of moving vehicles,

C_{B}

is a damping matrix of the bridge,

C_{R}

is a damping matrix of rail,

K_{R}

is stiffness matrix of rail,

n_{B R}

are the elements indicating bridge elements linked with the rail element,

F_{M}

is the force vector of vehicles, and

g

is the gravitational acceleration. Also,

N_{R}

is a vector with a series the shape functions.

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