Thermal fluid-structure interaction and coupled thermal-stress analysis in a cable stayed bridge exposed to fire

Abstract

In this paper, thermal fluid structure-interaction (TFSI) and coupled thermal-stress analysis are utilized to identify the effects of transient and steady-state heat-transfer on the vortex induced vibration and fatigue of a segmental bridge deck due to fire incidents. Numerical simulations of TFSI models of the deck are dedicated to calculate the lift and drag forces in addition to determining the lock-in regions once using fluid-structure interaction (FSI) models and another using TFSI models. Vorticity and thermal convection fields of three fire scenarios are simulated and analyzed. Simiu and Scanlan benchmark is used to validate the TFSI models, where a good agreement was manifested between the two results. Extended finite element method (XFEM) is adopted to create 3D models of the cable stayed bridge to simulate the fatigue of the deck considering three fire scenarios. Choi and Shin benchmark is used to validate the damaged models of the deck in which a good coincide was seen between them. The results revealed that TFSI models and coupled thermal-stress models are significant in detecting earlier vortex induced vibration and lock-in regions in addition to predicting damages and fatigue of the deck due to fire incidents.

Graphical abstract

Keywords

fire scenario / transient heat transfer / TFSI model / coupled thermal-stress / XFEM

1 Introduction

Many environmental thermal effects such as fire, solar radiation and the air temperature have significant impact on bridges. The continuous change of temperatures in their structural elements may result in nonlinear thermal stresses that affect their performance significantly. Generally, fire incidents are caused by smashing of vehicles and combust of fuel on or under the bridges. The variation in the structural temperature and its distribution in bridges lead to displacements, deformations, potentially extreme stresses, cracks and in serious cases of fire might lead to structural collapse. Fires that are originated by gasoline are considered most dangerous than building fires because they are distinguished by a rapid heating rate and very high temperature. Regarding these types of fires, very high temperatures will be acquired during the first few minutes [1–11].

A bridge fire took place in Birmingham, USA in 2002 due to collide of a diesel tanker with one of the Piers. The fire finished after 45 minutes of burning 142,000 liters of diesel approximately. The location of the fire was under the bridge and a part was unexposed. Due to non-regular exposure, the bridge collapsed partially. The MacArthur Maze Bridge in USA in 2007, partially failed due to truck incident with one of the support columns resulted in combustion of 32,600 liters of gasoline for more than two hours [12–14].

Dotreppe et al. [15] used the developed SAFIR code at the University of Liege to implement numerical analysis for the collapse of Vivegnis Bridge. The incident happened when a fire broke out due to explosion of a gas pipe. A room temperature was considered to generate their model so that to validate it with the measurements. They performed a transient heat transfer structural analysis supporting on the hydrocarbon temperature-time data of Eurocode1-1-2. The mode of failure and the time needed in the numerical simulation exhibited a good agreement with the actual data of the bridge collapse.

Kodur et al. [16] discussed the effects of fire incident on a bridge. They stated that the fire effect on bridges girder requires especial modeling as compared to the effect of fire on a building beam. They considered the data for the variation of heat transfer parameters along the depth of the beam. The analysis of the results showed that when the girder depth increases, the web, flange and the slab of the girder are subject to lower radiation effects.

Zhou et al. [9] searched the temperature distribution of the Humber suspension Bridge in UK. They utilized numerical simulation for the box girder in addition to field measurements by considering multiple wind velocity to determine the thermal initial boundary conditions. They performed a transient heat transfer and they investigated the vertical and horizontal temperature differences for the box girder. Then results of the temperature data at different regions and different times were in good agreement with the measured data, where a significant result was detected for the horizontal temperature variation in the box girder.

Modeling of thermal-stress analysis or coupled problems in general [17] poses significant challenges on the computational methods, particularly when coupled with fluid-structure-interaction and fracture. Rabizadeh et al. [18] proposed an adaptive FE analysis for thermoelasticity problems in the context of the finite element method. Efficient approaches to fracture include meshfree methods [19–22] the extended finite element method, phantom node methods [23–25], extended meshfree methods [26–35], efficient remeshing techniques [36–40], screen-Poisson models [41,42], cracking particles methods [43–48] anisotropic softening elements [49], extended IGA [50–53], multiscale methods for fracture [54–60] and peridynamics or dual-horizon peridynamics formulations [61–65], to name a few though IGA and XFEM has also been exploited for other applications [66–78]. Fluid-structure interaction driven fracture has been proposed for instance in [79]. However, a coupled thermal stress analysis involving fluid-structure interaction and fracture has not been carried out so far.

Up to date researches have concentrated on the heat transfer in the structural elements of bridges due to fire incidents and analyzed their failure. Numerical simulations were utilized to model the actual damaged members of the bridges supporting on standard temperature-time curve of Eurocode1. These researches have not covered the effect of transient heat transfer due to fire incidents on the behavior of long span bridges during wind excitations regarding vortex induced vibration aspect. They did not covered how the thermal effect will act in the generation of lock-in phenomena and resonance as a result the failure of the bridge due to serious vibrations, in addition to the role of thermal effect in speeding up the fatigue and failure of the bridge. Based on the above, in this paper numerical simulations of thermal fluid-structure interaction and coupled thermal-stress of three fire scenarios would be adopted to study the effect of thermal effect and transient heat transfer on the behavior of a segmental bridge deck during vortex shedding and lock-in phenomena in addition to the structural deformations and fatigue of the bridge deck during vibrations using ABAQUS CSD and CFD models in addition to XFEM in the analysis.

2 Standard temperature-time fire curve

Fire is one of the dangerous incidents for structures. Concrete is considered to resist the fire fairly compared to other materials such as steel and wood. When a high temperature of fire is exposed to concrete for a long time, mechanical properties of the concrete will be lost. When concrete is not protected, the experiments demonstrate that it would decrease its mechanical properties extremely above 300°C. Standard ISO 834 fire curve is an international temperature-time curve which is used for heat transfer analysis for structures (see Fig. 1). This name is originated due to the document number prepared at the International Standard Organization. The analysis of temperature is set for a specific time duration where the temperature is a function of time, and the temperature rises as a result of the increase in time (see Eq. (1)) [80–83].

(1)T=\mathrm{20}+\mathrm{345}\ast \mathit{log}(\mathrm{8}t+\mathrm{1}),

where T represents the temperature in degree Celsius and t the time in minutes.

3 Heat transfer analysis

The surface of any material element which is exposed to fire is subjected to heat transfer by three modes; conduction, convection and radiation. The conduction is responsible of the internal heat transfer through concrete structural members only. The temperature gradient drives heat transfer from the hot to cold regions, which is called heat flux. Convection heat transfer is due to change in the air density depending on the temperature, where this causes turbulence in the air leading to heat transfer to nearby regions. Usually convection is ignored for the exposed surface of concrete because it is responsible of less than10% of heat transfer at the exposed area, but it is usually considered only for unexposed areas of the concrete surface. The radiation takes place when a structure is exposed directly to a fire, where the surface is heated through heat transfer [84–87].

The convective heat flux q_c is described by Eq. (2) (EC 1991-1-2 2002).

(2){{\displaystyle q}}_c={{\displaystyle h}}_c({{\displaystyle T}}_s-{{\displaystyle T}}_a),

where is convection coefficient, 25 W/m²·K (EC 1991-1-2-2002), T_s is surface temperature [K] and T_a is ambient temperature [K].

Radiation differs from convection, for heat transfer there is no need for a medium. It consists of electromagnetic waves from a region with high energy to a lower energy region. The emissivity factor is the dependable factor for radiation effect where the materials absorb radiated energy. The radiation heat flux q_r is described by Eq. (3) (EC 1991-1-2 2002).

(3){{\displaystyle q}}_r=\varphi \ast {{\displaystyle \epsilon }}_s\ast {{\displaystyle \epsilon }}_a\ast \sigma \ast ({{\displaystyle T}}_s^{\mathrm{4}}-{{\displaystyle T}}_a^{\mathrm{4}}),

where \varphi is shape factor, usually 1.0 when radiation to ambient temperature and no other surface (EC 1991- 1-2-2002), {{\displaystyle \epsilon }}_s is emissivity of surface, {{\displaystyle \epsilon }}_a is emissivity of ambient, fire usually 1.0 (EC 1991-1-2-2002) and \sigma is Stephan-Boltzmann’s constant, 5.67*10^–8 W/m²·K⁴.

The net heat flux is described as the sum of the two Eqs. (2) and (3).

(4){{\displaystyle h}}_{net}={{\displaystyle q}}_c+{{\displaystyle q}}_r.

The convective heat flux and radiation heat flux are depending on the temperature difference between the surface and ambient air. When the temperature is low, the main contribution to the total heat flux is from convection but when the temperature raises, the effect of radiation will govern.

3.1 Thermal analysis of a cable stayed bridge

Generally, the differences in the temperature in the longitudinal direction of a bridge are not considered. Only one segmental section of the deck is appropriate to analyze the distribution of temperature due to fire exposition (see Fig. 2). The process of heat transfer in the segmental deck which is exposed to fire occurs through heat conduction, heat convection and thermal radiation. Where the heat conduction happens in the interior of the deck and it is governed by the Fourier heat transfer equation. While heat convection takes place through energy exchange between the deck surface and the surrounding air which results in the diffusion and motion of the air. Furthermore, the radiation occurs via energy transfer from the fire to the deck and vice versa.

3.2 Heat transfer theory

The temperature of a point in a bridge can be expressed as T=T(x,y,z,t) where x, y, and z are Cartesian coordinates of the point and t = time. Heat-transfer theory is governed by the typical Fourier heat-transfer equation

(5)\rho c{\displaystyle \frac{\partial T}{\partial t}}=k\left({\displaystyle \frac{{\partial }^{\mathrm{2}}T}{\partial x^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}T}{\partial y^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}T}{\partial z^{\mathrm{2}}}}\right),

where k= isotropic thermal conductivity coefficient; \rho = density; and c= specific heat of the material. The temperature field of a structure at a specific time can be obtained by solving the preceding Fourier partial differential equation under initial and boundary conditions.

3.3 Thermal boundary conditions

There are three types of boundary conditions related to the structural thermal analysis. The first is that the structural boundary temperature is exactly known. The second is that the heat flux on the structural boundary is determinate. The third is that the heat flux on the structural boundary is proportional to the temperature difference between the bridge and the air. The boundary conditions associated with Eq. (5) for the thermal analysis of a bridge can be written as a combination of Types 2 and 3:

(6)k{\displaystyle \frac{\partial T}{\partial n}}=h({{\displaystyle T}}_a-{{\displaystyle T}}_v)+q,

where n is normal to the surface and the heat transfer coefficient is: h=hc + hr, which is combining the heat-transfer coefficients of convection c and thermal irradiation r and its unit is W/m²·K.

T_a is air temperature; T_S is structural surface temperature; and q is boundary heat exchange per unit area (heat fluxes, positive for inflow). The heat-transfer coefficient of convection c is related to wind speed. The wind blowing across the bridge surface significantly affects the heat-transfer convection coefficient and consequently influences the accuracy of the thermal analysis results. Previous studies on bridge thermal analysis used a constant value of wind speed for all structural surfaces. The heat-transfer coefficient of thermal radiation hr depends on the structural material, surface temperature, and air temperature [9,88–90].

4 Thermal fluid-structure interaction

Thermal fluid-structure interaction (TFSI) happens in many engineering applications, for example in designing aircraft turbines, vehicle engines and in biomechanical fields. A full set of interaction phenomenon between the fluid and the solid is available in all mentioned applications. The interaction comprises the action of the fluid on the solid, also the solid acts on the fluid and additional heat transfer exists in the whole system. In TFSI, a thermal field is considered in addition to the fluid and the structural fields. Considering the thermal field in the respective domain, flow and solid, describes a volume-coupled problem. The volume coupling to the Thermal field is processed in each domain either with a partitioned approach or with a monolithic scheme, where surface coupling of the thermal field between the two domains is required. Therefore, the interaction of fluid, structure and thermo field leads to many different coupling processes. Reynolds-averaged Navier-Stokes is used to model the interaction phenomenon and the nonlinear Fourier heat conduction equations are used for the fluid and the solid phase, respectively. The thermal fluid-structure interaction investigates the heat exchange between a heated bridge deck and the air flow due to fire [91–96].

4.1 Governing equations

On a domain W₁⊂ ℝ^d the physics is described by a fluid model, whereas on a domain W₂⊂ ℝ^d, a different model describing a structure is used. The two domains are almost disjoint. The part of the interface where the fluid and the structure are supposed to interact is called the coupling interface. On this surface the coupling conditions are the temperature and the normal component of the heat flux which are both continuous across the interface.

4.1.1 Fluid model

The fluid is modeled using the Reynolds-averaged Navier-Stokes equations, which are a second order system of conservation laws (mass, momentum, energy) modeling viscous compressible flow. We consider the two dimensional case, written in conservative variables density r, momentum m = r v and energy per unit volume E.

Herein the following equation represents the viscous shear stress tensor

(7)T=\eta (\nabla v+\nabla v^T).

And the heat flux is:

(8)q=-\lambda \nabla \Theta .

As the equation is dimensionless, the Reynolds number Re and the Prandtl number Pr appear. The equations are closed by the equation of state for the pressure:

(9)p=(\gamma -\mathrm{1})(\rho E-\mathrm{0.5}{\left|v\right|}^{\mathrm{2}}).

Furthermore, a Spalart-Allmaras one-equation model is used for the existing turbulence. The spatial discretization is done by a finite volume method whereas a Runge-Kutta method is used for the time integration.

(10){\displaystyle \frac{\partial \rho }{\partial t}}+\nabla .(\rho v)=\mathrm{0},

(11){\displaystyle \frac{\partial \rho v}{\partial t}}+\nabla .(\rho v\otimes v)+\nabla p={\displaystyle \frac{\mathrm{1}}{\mathrm{Re}}}\nabla .T,

(12){\displaystyle \frac{\partial \rho E}{\partial t}}+\nabla .(v(\rho E+p))={\displaystyle \frac{\mathrm{1}}{\mathrm{Re}}}\left[\nabla .({{\displaystyle T}}_V)-{\displaystyle \frac{\mathrm{1}}{\mathrm{Pr}}}\nabla .q\right].

4.1.2 Solid model

The governing equations within the solid domain W₂ are given by the balance of heat energy and the Fourier law:

(13)\rho .{{\displaystyle c}}_p(\mathrm{\Theta })\dot{\mathrm{\Theta }}(x,t)=-\mathrm{d}\mathrm{i}\mathrm{v}q(x,t,\mathrm{\Theta })-Q(x,t),

(14)q(x,t,\mathrm{\Theta })=-\lambda (\mathrm{\Theta })\nabla \mathrm{\Theta }(x,t),

here, q is the heat flux vector, \mathrm{\Theta } the temperature, \dot{\mathrm{\Theta }} the temperature rate, \rho the density, Q the heating source and \lambda (\mathrm{\Theta }) as well as {{\displaystyle c}}_p(\mathrm{\Theta }) are the temperature dependent conductivity and specific heat capacity. The Neumann boundary conditions are the forced convection \bar{c} of the fluid together with the thermal radiation \epsilon (\mathrm{\Theta })\sigma [{\mathrm{\Theta }}^{\mathrm{4}}-{\mathrm{\Theta }}_{\infty }^{\mathrm{4}}] at the surface with the Stefan Boltzmann constant , the emissivity \epsilon (\mathrm{\Theta }) and the bulk temperature {\mathrm{\Theta }}_{\infty }^{\mathrm{4}}.

5 Finite element model of TFSI

The finite element model of thermal fluid-structure-interaction between the fire-heated segmental deck of a cable stayed bridge and the surrounding air due to heat transfer when there is a wind flow in the same time by considering three fire scenarios is modeled as follows:

The model of the segmental bridge deck is generated in ABAQUS once in CFD and another in CSD. The CFD model is with a dimension 2.6 m height and total width of 22 m as shown in Fig. 3. The thickness is 0.01 m. The flow domain size is 140 m length and 40 m height. Material properties for the CFD model are assigned where the air density is 1.29 kg/m³ and the dynamic viscosity of the air is (1.8E–05) Pa.s. The thermal conductivity of the air is 0.0257 W/m²·K, specific heat of air is 1005.4 J/kg·K and the thermal expansion coefficient of air is 0.00343 K⁻¹.

The CFD model part is meshed using CFD element fluid family with FC3D8: A-8 node linear fluid brick. The deck wall assigned with 0.4 element size and the flow domain with 1 element size.

The CSD model have the following material properties concrete density is assigned 2643 kg/m³, Youngs modulus 200E+08 and the Poissons ratio 0.2. The role of the steel reinforcement is neglected for simplicity because the thickness of the CSD model is too small 0.01m and the reinforcement has no direct contact with the air. The thermal conductivity, specific heat and the thermal expansion coefficient of the concrete are set with amplitudes supporting on temperature dependent data due to Eurocode EN 1994-1-2.The CSD model part is meshed using Heat Transfer family with DC3D8: A-8 node linear heat transfer brick.

A flow step with 100 seconds duration is created for the CFD model by activating the energy equation for temperature to enable the heat transfer in the flow. An initial condition of fluid thermal energy is assigned for the whole model both the deck and the fluid domain to represent the ambient temperature of 293.15 K. Another transient heat transfer step with 100 seconds is created for the CSD model. Also the same initial condition of temperature 293.15 K is assigned for the CSD model. The step time in each model must be the same, in addition to adding interaction with the same name in each model for the surfaces of interaction, also the important issue related to mesh generation is equality of the mesh size in each model.

Four boundary conditions are defined for the CFD model, fluid B.C for the inflow and far fields assigning the air velocity value in the horizontal direction only (zero attack angle). The other two directions with zero magnitudes, fluid B.C for the outflow assigning zero pressure and fluid B.C for the front and back of the flow model with zero velocity magnitude for the third direction perpendicular to the model (z- direction), and no-slip fluid B.C for the wall condition of the deck.

A temperature conduction boundary condition is defined for the surface of the CSD model supporting on three fire scenarios (above, under and beside) the deck by assigning time-temperature data amplitude due to ISO 834 fire curve. A heat flux load of 100,000 W/m² (fire flames) is assigned to the surface of the CSD model three times (three fire scenarios). A convection interaction is assigned (three fire scenarios) with the film coefficient 23.5 W/m²·K and sink temperature 293.15 K. A radiation interaction is assigned to the surfaces of the deck (three fire scenarios) with an emissivity of 0.85 for the concrete and ambient temperature 293.15 K, in addition the absolute zero temperature should be activated and assigned –273.15 and the Stefan Boltzmann constant 5.67E–08. A co-simulation job is generated for both the CFD and the CSD models together to enable transient heat transfer type of thermal coupling between the two models.

5.1 Fire scenarios

Three fire scenarios for 100 seconds duration are being considered to analyze the transient heat transfer due to the difference in temperature between the surrounding air and the segmental deck where the incoming wind is cooler supporting on standard ISO 834 fire time-temperature curve. The first situation of the fire is above the segmental deck, the second scenario is under the segmental deck and the last scenario is beside the deck (see Fig. 4).

5.2 Results of TFSI–models simulations

The numerical simulation of the finite element model of the deck when exposed to three fire scenarios for 100 seconds duration which is considered a transient heat transfer process is simulated in both the vorticity and temperature fields so that to explain the thermal effect of fire on the vortex shedding generated due to wind flow with a velocity of 6 m/s. The effect of the heat transfer occurs due to conduction, convection and radiation methods, which have appreciable effect on the generated lift and drag forces in the deck.

Considering fire scenario 1, in the vorticity field the vortices are shed in a regular pattern from the tail of the deck in the lower region, but in the upper region the vortices tend to connect at the beginning of the shedding due to fire effect, but after a short distance the vortices start to separate generating vortex shedding. The connection of the vortices occurs again and becomes continuous. Regarding the temperature field, the thermal convection vortices are seen with a semi regular pattern with difference in the temperature. The temperature of the vortices is high in the start of the wake region reaching 658.8°C while it decreases after shedding from the tail of the deck regularly till 459.3°C (see Fig. 5(a) and Fig. 5(b)).

Regarding scenario 2, in the vorticity field the vortices are shed in a regular pattern from the tail of the deck in the lower region, but in the upper region the vortices tend to connect at the beginning of the shedding and after a shortly the vortices separate in the same way as scenario1. The connection of the vortices occurs again and becomes continuous. While for the temperature field, the thermal convection vortices are seen with a vanishing pattern with difference in the temperature, which is high in the start of the wake region up to 658.8°C while it decreases after shedding from the tail of the deck linearly till 592.3°C, where the vortices are being cooled slower than of the case in scenario 1 (see Fig. 5(c) and Fig. 5(d)). While in scenario 3, in the vorticity field the vortices are shed in a regular pattern from the tail of the deck in both the lower and upper regions. The vortices shed continuously with a better pattern of vortex shedding. While for the temperature field, the thermal convection vortices are seen with a continuous pattern with difference in the temperature. The temperature of the vortices is high in the start of the wake region up to 658.8°C while it decreases after shedding from the tail of the deck linearly till 426.1°C which means that the vortices suffer from faster cooling during shedding compared to the cases of scenario 1 and scenario 2 (see Fig. 5(e) and Fig. 5(f)). To identify which fire scenario is critical for vortex induced vibration, analysis of generated lift and drag forces is a must, and it is detailed in the next sections.

5.3 Results of lift and drag forces-FSI-models

In order to identify the effect of thermal coupling of fluid-structure interaction due to wind flow for the deck of the cable stayed bridge during fire incident, analysis of the fire scenarios without thermal coupling is a valuable step so that to measure the difference in the behavior of the deck with respect of the vortex induced vibration. The generated lift and drag forces are other criteria of analysis, where the effect of fire is identified. When there is no fire incident and just the fluid-structure interaction is considered, the lift forces amplitudes have a stable and regular frequency with range values (9000–19000) N. The drag forces have the same regular, stable frequency but lower amplitudes ranging between (3000–7000) N. The lift forces values are three times of the drag forces values approximately (see Fig. 6(a)).

Now fire scenarios are added to the numerical simulations to search the effect of transient heat transfer without thermal coupling. In fire scenario 1, the fire region is above the deck. The lift forces values have a stable and regular frequency range of (50–80) N, where these values are much lower than the values of lift forces in the case of no fire scenario. While the drag forces have much lower amplitude values than the values of drag forces in the case of no fire scenario which are between (910–1330) N. The drag forces values are 18 times of the lift forces values approximately (see Fig. 6(b)). This is an indication that fire scenario 1 will decrease the lift and drag force in deck to a high ratio which is very significant in suppressing the vertical and horizontal vibration of the deck.

In fire scenario 2, the fire region is under the deck. The lift forces values have a stable and regular frequency of (50–85) N the same as the scenario1without any change. The drag forces have amplitude values between (910–1330) N the same as scenario 1 without any change (see Fig. 6(c)). This is an indication that fire scenario 2 is very significant like the fire scenario 1 in suppressing the vertical and horizontal vibration of the deck to very high ranges. In fire scenario 3, the fire region is under the deck but just at half of the region in one side only. The lift forces values have a stable and regular frequency of (35–90) N the same as the scenario1approximately. The drag forces have amplitude values between (640–3610) N different from scenario 1 and scenario 2, where the amplitude of the drag forces are much higher (see Fig. 6(d)). This is an indication that fire scenario 3 is more important in increasing the horizontal vibration of the deck, but it decreases the vertical vibration to a range which near to the range of its values in scenario 1 and scenario 2.

When no thermal coupling is considered, there is a nonlinear behavior of the deck under the fire scenarios with respect to both vertical and horizontal vibrations of the deck. So a special consideration should be given to each case supporting on the specifications of a certain design aims when considering transient heat transfer due to fire.

5.4 Results of lift and drag forces-TFSI-models

Thermal coupling in the numerical simulations for the fire scenarios is utilized to search the effect of the of transient heat transfer on vortex induced vibration of the deck and behavior of the deck under the generated lift and drag forces. The case of no fire scenario available in the numerical simulation has been mentioned in the previous section, and here is considered only for comparison purpose so that to see the effect of thermal coupling on the behavior of the deck.

Fire scenarios are added to the numerical simulations to search the effect of transient heat transfer with thermal coupling. In fire scenario 1, the lift forces values have a stable and regular frequency range of (4810–11620) N, where these values are lower than the values of lift forces in the case of no fire scenario up to half of their values approximately. While the drag forces have higher amplitude values than the values of drag forces in the case of no fire scenario but with regular repeated amplitudes relatively, and their values are between (2400–7160) N (see Fig. 7(b)). This is an indication that fire scenario 1 will decrease the lift forces to the half of its value and increase the amplitude of the drag force in deck. This behavior proofs that a significant suppression in the vertical vibration will occur and negatively the horizontal vibration of the deck would be increased.

In fire scenario 2, the lift forces values have a stable and regular frequency of (4810–11620) N the same as the scenario1without any change. The drag forces have amplitude values between (2400–7160) N the same as scenario 1 without any change (see Fig. 7(c)). This is evidence that in fire scenario 2 the deck behaves like its behavior in the case of fire scenario 1 in suppressing the vertical vibration and increasing the horizontal vibration of the deck to a small range.

In fire scenario 3, the lift forces values have a stable and regular frequency of (1010–4420) N with a semi regular and stable frequency but much lower than the lift forces values in scenario 1 and scenario 2. The drag forces have amplitude values between (1210–5270) N which is much lower than its values in scenario 1 and scenario 2 (see Fig. 7(d)). This is an indication that fire scenario 3 is significant among the other scenarios in mitigating both the horizontal and vertical vibrations of the deck

When thermal coupling is adopted in the numerical simulations, there is a nonlinear behavior of the deck under fire scenario1 and scenario 2 with respect to both vertical and horizontal vibrations of the deck. Furthermore, there is a significant behavior of the deck under fire scenario3 where both the horizontal and vertical vibrations are being suppressed to a good extent.

Finally, by comparing the behavior of the deck under fire scenarios in the FSI simulations and the TFSI simulations, a fact arises which is clear that considering TFSI will result in significant suppression of both horizontal and vertical vibrations in scenario 3 where the fire is under the deck at one side. The same suppression in vertical vibration occurs in both scenario 1 and scenario 2 but with a small increase in the horizontal vibration.

5.5 Results of lock-in phenomenon

To analyze the vortex shedding and generation of lock-in phenomenon, TFSI models are utilized supporting on the fire scenario 2 because the critical maximum lift and drag forces are generated in this scenario. Twenty five TFSI models and twenty five FSI models are utilized supporting on different wind flow velocities starting from 1 m/s to 15 m/s this after thorough investigations to identify the first lock-in region in advance before starting analysis. The first lock-in region for TFSI models was found to be between wind velocity (9–11.8) m/s and between (10–11.5) m/s for FSI models (see Table 1). Numerical simulations for all the models are utilized to identify the vortex shedding frequencies where lock-in regions are located (see Fig. 8).

The vortex shedding frequency for TFSI models is starting from 0.122 Hz for the wind velocity 1 m/s and linearly continues till the wind velocity 8 m/s with shedding frequency of 0.979 Hz. After this point the vortex shedding frequency tends to reach a constant point of 1.059 Hz at wind velocity 9 m/s. The vortex shedding will remain constant approximately till wind velocity11.8 m/s at shedding frequency of 1.122 Hz. The wind velocity region from 9 m/s to 11.8 m/s is called lock-in region. While for FSI models the vortex shedding frequency starts at 0.111 Hz where the wind velocity is 1 m/s and the curve continues to rise linearly till shedding frequency of 1.154 Hz for wind velocity 10 m/s. The shedding frequency becomes constant until shedding frequency 1.183 Hz at wind velocity 11.5 m/s. The constant region is the lock-in region for FSI model (see Fig. 8).

In this region the amplitude of vibrations is independent from the shedding frequency despite of the increase in wind velocity, where resonance is expected to occur and serious damage, or may be failure of the structure to take place.

6 Validation of TSFI-models

The results of lock-in phenomenon calculated from the numerical simulations of TFSI models in addition to FSI models are validated with Simiu and Scanlan benchmark which are the numerical results of the lock-in obtained by them. The lock-in region of TFSI models starts from wind velocity 9 m/s and continues till wind velocity of 11.8 m/s, while for FSI models the wind velocity range in lock-in region is (10–11.5) m/s. When comparing the two results with the lock-in region by Simiu and Scanlan it is clear the two models are having the same shape and the same constant region of shedding frequencies approximately which is a very good agreement between al the results (see Fig. 9). In the same time the comparison between the results of lock-in region of TFSI and FSI models, shows that the results of TFSI models has wider lock-in region or wider critical wind velocity range and earlier critical wind velocity is detectable in addition to lower natural frequency of the system for vibration at lock-in region which is very significant in the design of long span bridges regarding vibrations due to wind excitations.

7 Coupled thermal-stress analysis

Coupled thermal-stress in bridges is a heat transfer simulation which induces stresses through thermal expansion of the structure due to elevated temperature of the structure. The thermal effects on the bridges are expansion and deformations of the structure, stresses inducing cracks or spalling of the concrete depending on the duration of the heating source.

7.1 Thermal cracking and spalling of concrete

Excessive differences in temperature within a concrete structure or its surrounding result in thermal cracking. This causes contract of the cool part more than the warmer part and consequently the contraction would be restrained. When the tensile stresses exceed the tensile strength of the concrete, the thermal cracks appear. When extreme temperature differences occur, thermal cracks with a random pattern appear on the concrete structural element. Spalling is another aspect of thermal stresses inducing thermal strains which end with the cracking and spalling of portions of concrete. The high temperature and intensity of the fire causes bursting pressure in the moisture trapped within the microstructure of the concrete. Due to unequal thermal expansion between the aggregates, cementitious paste and the steel bars can increase spalling of the concrete. In addition to material properties of the concrete, geometric parameters like section shape and size contribute to the spalling of concrete. Environmental parameters such as heating rate and profile, thermal restraint and temperature level also affect the occurrence of spalling [97–102].

Heating rates in the range of 20–32°C/min are significant for spalling to occur [103]. The high heating rates occur in the early stages of the fire. Several researchers have identified critical temperature ranges for spalling to start regarding exposed surface to fire. Akhtaruzzaman and Sullivan [104] have stated that exposed surface temperatures in the range of 375°C–425°C for normal weight concretes leads to spalling. Choi and Shin [105] determined theoretically and experimentally the zone of spalling of a reinforced concrete beam exposed to ISO 834 fire where the location of the spalling is at the upper right and left of the external part of the beam. Numerical models of bridges and prediction of the fire response of the bridges have been studied by researchers which are providing modeling guidelines for improving bridge design in the case of fire exposure [106–112].

7.2 Finite element model of coupled thermal-stress

The full model of a cable stayed bridge is created in ABAQUS to undergo thermal stresses due to heat transfer from a fire source for a long duration of 20 minutes supporting on the standard ISO 834 Fire data. Three scenarios (the same as TFSI case) are dedicated to simulate the effect of the fire on the deformations, cracks and spalling of concrete. A cable stayed bridge model is created with 324 m length and 22 m width, the main parts of the bridge is the deck which consists of connected reinforced concrete deck segments with 2.6 m height. Four reinforced concrete pylons with square shapes 4 m×4 m dimensions and 103 m height, and 80 stay cables are connecting the deck to the pylons in a fan shape arrangement, each cable with cross section area 0.00785 m² (see Fig. 10).

The main steel bar diameter is 0.06 m and the diameter of the temperature steel bars in addition to the stirrups are 0.04 m. The boundary condition of the deck is fixed in one side and free for longitudinal translation in the other side. The pylons are fixed at the bottom and each two pylons are connected by six reinforced concrete ties with 4 m×4 m dimensions and 22 m length. The stay cables equivalent Young's modulus of elasticity has been used to approximate the sagging occurrence in the cables because it was modeled as truss elements (see Table 2).

An ambient temperature of 293.15 K is assigned to the whole model, and a static general step is created for duration of 1200 seconds. A predefined field (temperature type) is assigned in to the region where three fire scenarios are considered with amplitude of time dependent temperature (Standard ISO 834 Fire).

To simulate the propagation of the cracks and the spalling in the concrete due to fire action, equivalent thermal stresses are added in the fired regions using eXtended Finite Element Method (XFEM) approach for each fire scenario. Maximum principal stress is assigned in addition to the displacement at failure. The start of damage in the simulations is identified with a maximum principal stress generated at the temperature 400°C where the cracks and spalling in the concrete are prone to occur.

The deck is modeled as (C3D10: A10-quadratic tetrahedron) elements, pylons and ties are modeled as (C3D8R: An 8 node linear brick reduced integration hourglass control) elements, the reinforcing steel bars are modeled as (B31: A 2- node linear beam in space) elements and the stay cables are modeled as (T3D2: A 2- node linear 3D truss) elements.

7.2.1 Extended finite element method (XFEM)

The XFEM is a partition of unity based method in which the classical finite element approximation is enhanced by means of enrichment functions. The enrichment procedure is employed on a local level for the XFEM. Thus, only nodes close to the crack tip, as well as the ones required for the correct localization of the crack, are enriched. In the XFEM, enrichment functions are added to additional nodes, in order to include information about discontinuities and singularities around the crack. These functions are the asymptotic near-tip solutions, which are sensitive to singularities, and the Jump function, which simulates the discontinuity when the crack opens. This evidently entails a tremendous computational advantage. The XFEM method was firstly introduced by Belytsckho and Black in 1999 [113]. Their work, in which a method for enriching finite element approximation in such a way that crack growth problems can be solved with minimal remeshing, represents a milestone in the XFEM history. Later on, much more elegant formulations, including the asymptotic near-tip field and the Heaviside function H(x) in the enrichment scheme, have been proposed [114–116]. The XFEM method, furthermore, has been demonstrated to be well suited for three dimensional crack modelling [117]. In this latter work, geometric issues associated with the representation of the crack and the enrichment of the finite element approximation have been addressed. A major step forward has been then achieved when a generalized methodology for representing discontinuities, located within the domain indipendetely from the mesh grid, has been proposed [114,118]. In such manner, the eXtended Finite Element Method allows to alleviate much of the burden related to the mesh generation, as the finite element mesh is not supposed to conform the crack geometry anymore. This represents certainly, one of the major advantages provided by the XFEM usage. The XFEM capabilities can be extended if employed in conjunction with the Level Set Method (LSM) [119–121]. Such method permits to represent the crack position, as well as the location of crack tips.

7.2.2 Partition of unity method

In the finite element method, a basis function, NI , is associated with a node I. The region of support of a basis, or shape, function is the set of elements that include node I. These shape functions form a partition of unity:

(15){\displaystyle \sum _{I\in N}{{\displaystyle N}}_I}(x)=\mathrm{1.}

It follows that any arbitrary function \gamma (x) may be reproduced exactly by

(16){\displaystyle \sum _{I\in N}{{\displaystyle N}}_I}(x)\gamma (x)=\gamma (x).

It is this ability that forms the basis of the XFEM. By appropriately choosing the function \gamma (x) for each node, a priori knowledge of a model’s behavior may be incorporated while retaining the firm mathematical basis of standard finite element analysis [122].

7.3 Results of deck displacements

Three fire scenarios for the duration of 1200 seconds on the middle part of the cable stayed bridge in the region of (22×20) m is considered and the results of the displacements for each case are calculated for a center point in the middle of the cable stayed bridge model. Considering the maximum horizontal displacements, this maximum value for scenario1 was low which is reached 0.7 cm upward at the end of the 1200 seconds. While for scenario 2, this maximum value was very low which is reached 0.02 cm upward at 172 seconds. Furthermore, the maximum value for scenario3 was high reaching 20.4 cm downward at the end of the 1200 seconds of the fire exposition (see Fig. 11).

Relating to the maximum vertical displacements, all the scenario cases reached teir maximum values at the end of the fire exposition after 1200 seconds. Scenario 1 showed very high maximum value of 44.2 cm upward. This maximum value for scenario 2 was high reached 33.7 cm downward but it is less than of it in the previous scenario. While for scenario3, the maximum vertical displacement reache 18.1 cm downward which is less than all the scenario cases (see Fig. 12). Finally, with respect to the maximum angular displacemnts, all the scenarios reach the maximum value at the end of the 1200 seconds. In scenario 1, the maximum value was 0.038° which is considered very low which is not effective. In the other hand, this value for scenario 2 was 0.064° which is nearly twice the previous value but still not appreciable to induce stresses in the deck. While for scenario 3, the maximum value reached 0.086° which is greater than all the maximum values in scenario 1 and scenario 2 and is three times of the maximum value in scenario 1 approximately and it is not an effective in producing stresses in the deck (see Fig. 13).

This is an indication that the horizontal displacement in scenario 3 where the fire is exposed on one side under the deck is the critical case in generating tension stresses in the fired side and compression stresses in the opposite sides of the deck. The vertical displacement in scenario 1 where the position of the fired region is above the deck is the critical case in inducing tension stresses that are the reason of propagation of cracks and spalling of the concrte above the deck whenever these stresses exceed the tensile strength of the concrete. While the angular displacement in scenario 3 is the critical case but it is too low which is not appreciated in the geneartion and propagation of cracks and spalling of concrete.

7.4 Results of cracking and spalling of the deck

To identify the damaged regions in the cable stayed bridge model, maximum principal stress values calculated in ABAQUS are used for damage detection with the support of XFEM to simulate the propagation of the cracks and occurrence of spalling in the concrete. In all fire scenarios the duration of the fire is 20 minutes supported on standard ISO 834 Fire data.

In fire scenario 1, the fired region is shown (see Fig. 14). Cracks and spalling of concrete can be seen at the middle and external edges of deck (see Fig. 15). The cracks propagated from the inner edges of the flange toward the external edges leading to occurrence of spalling. This is due to high vertical displacement upward and angular displacement of the deck caused by the thermal stresses which induced high tensile stresses in the concrete exceeded the tensile strength of the concrete.

Considering fire scenario 2, the fired region is shown in (Fig. 16). Cracks are obvious in the web of the deck in addition to spalling of concrete at one side of the exterior edges (see Fig. 17). The cracks appear first at the middle and the external edges of the web, then they propagate from the exterior edges toward middle part of the web and consequently spalling occurs. The thermal stresses generated high vertical displacement downward and relative higher angular displacement of the deck with respect to its value in sceanrio1.

While for fire scenario 3, the fired region is shown in (Fig. 18). Cracks with spalling of concrete can be seen at one side of the exterior edges on the web (see Fig. 19). The cracks start first at the fired side and then they propagate toward the middle of the web and as a result, spalling occurs. The thermal stresses in one side of the web created relative high horizontal displacement in the damage side in addition to relative low vertical displacement downward and relative higher angular displacement of the deck with respect to their values in sceanrio1 and scenario 2 resulted in generation of cracks and spalling in the fired region.

Cracks and spalling of concrete occur in all fire scenarios due to high temperature exposure of the deck. Consequently the highest damage in the deck of the cable stayed bridge takes place in the fire scenario1 where the fire is above the deck because larger amounts of cracks and spalling appear in earlier time from the fire incident and the failure of the structure is more prone to occur compared to other two fire scenarios.

8 Validation of coupled thermal- stress models

To validate the models of the spalling and cracking of the deck, a benchmark for the spalling zone of the reinforced concrete beam by (Choi and Shin, 2011) is dedicated, where the spalling occurs in the external parts of the beam at the mid span. The zone of the spalling of concrete for the coupled thermal-stress models are in the external sides of the deck at the mid span in all fire scenarios. Comparing the two results indicates that there is a good agreement between the results of the coupled thermal-stress models and the results of the benchmark model.

9 Conclusions

The obtained conclusions are as follows:

1) When TFSI numerical simulations are adopted, nonlinear behavior is detected for the deck under fire scenario 1 and fire scenario 2 regarding the horizontal and vertical vibrations of the deck which imposes special analysis for each case up to the design needs and specifications of safety against vibrations. While for fire scenario 3 a significant suppression of both horizontal and vertical vibrations to a good level has been identified. This fact can be further utilized to use heating sources to invent new devices for the suppressing vibrations in long span bridges where the lower temperature that enhances that target is lower than 100°C supporting on the results of lift and drag forces for fire scenario 3 and the standard ISO 834 fire curve.

2) An earlier critical wind velocity is discovered for the TFSI simulations in the same time lower vortex shedding frequencies for fire scenario 2 where the fire is under the deck, which enhances the generation of earlier vortex induced vibration and earlier lock-in phenomena which is a serious matter related to the service ability and safety of the cable stayed bridge due to vibrations and failure. As a result a special attention should be dedicated to prevent heating of the deck (above part) especially in hot countries where the source of heating from the environment helps this situation.

3) Using both coupled thermal-stress and XFEM simulations, all fire scenarios generated thermal stresses in the deck due to very high temperature, consequently deformations and damage of the concrete comprising cracks and spalling of the concrete, especially fire scenario 1 where larger amounts of cracks and spalling appear earlier which is considered critical for the safety of the structure compared to ire scenario 2 and fire scenario 3.

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