Numerical simulation of the load-carrying behavior of RC tunnel structures exposed to fire

ARTICLE IN PRESS Finite Elements in Analysis and Design 45 (2009) 958–965

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Finite Elements in Analysis and Design journal homepage: www.elsevier.de/locate/finel

Numerical simulation of the load-carrying behavior of RC tunnel structures exposed to fire ¨ Christian Feist a, Matthias Aschaber b, Gunter Hofstetter b, a b

CENUMERICS—Consultant Engineers for Numerical Simulation, A-6020 Innsbruck, Haspingerstr. 16, Austria Unit for Strength of Materials and Structural Analysis, Institute for Basic Sciences in Civil Engineering, University of Innsbruck, A-6020 Innsbruck, Technikerstr. 13, Austria

a r t i c l e in f o

a b s t r a c t

Article history: Received 12 September 2008 Received in revised form 20 July 2009 Accepted 28 September 2009

A numerical model for the evaluation of the load-carrying behavior of fire exposed reinforced concrete structures is presented. It consists of a thermo-mechanical model based on a phenomenological approach. In the transient thermal and mechanical analysis the temperature dependent thermal and mechanical material properties of concrete and reinforcing steel are taken into account on the basis of test data or—if test data are not available—on the basis of the provisions of the Eurocode 2, part 1–2. For each time step the time-dependent temperature field and the structural behavior are computed in two consecutive steps. A concrete model, based on the combination of plasticity and damage theory, and an elastic–plastic model for the reinforcing steel serve as the basis for the respective material models at high temperatures. The proposed numerical model is validated by the numerical simulation of fire tests on concrete specimens and of a large scale fire test on a reinforced concrete slab. Finally, it is applied to the evaluation of the structural response of a fire exposed tunnel structure. & 2009 Elsevier B.V. All rights reserved.

Keywords: Fire Finite element analysis High temperatures Reinforced concrete Tunnelling

1. Introduction Several major fires in tunnels during the past decade caused severe personal injuries and structural damage to the tunnel lining (cf., e.g. [1]). In addition to preventive safety measures and safety facilities it is of particular importance for personal security to ensure the load-carrying capacity of a structure during a fire. A variety of methods of different complexity for considering the effects of a fire on the load-carrying capacity of a structure is available, ranging from simple design rules for the dimensions of cross-sections to sophisticated multi-phase models for fire exposed concrete. A simple design method for fire exposed structures is given, e.g., in Eurocode 2, part 1–2 [2]. According to the latter, those parts of a cross section of a column, beam or frame with temperatures exceeding 500 3 C are neglected for the load carrying capacity, whereas the remaining parts of the respective cross section are assumed to maintain the original stiffness and strength. In order to make the effects of a fire amenable to commercial computer programs for structural analysis based on linear beam theory, Wageneder [3] proposed to replace the highly nonlinear temperature distributions across the thickness by so-called ‘‘equivalent linear temperature distributions’’.

 Corresponding author.

E-mail addresses: [email protected] (C. Feist), [email protected] (G. Hofstetter). 0168-874X/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2009.09.010

However, this simplified approach suffers from basic inconsistencies, e.g., it does not properly account for the pronounced nonlinear strain distribution across the thickness of fire exposed structures. Within the framework of a phenomenological description of the material behavior of fire exposed reinforced concrete structures a coupled thermo-mechanical approach yields a consistent numerical model. Assuming the temperature distribution in the structure as independent of the structural behavior allows performing the thermal analysis and the mechanical analysis in two consecutive steps. Several thermo-mechanical models were proposed in recent years (cf., e.g. [5–7]). They allow a good representation of the time-dependent overall structural behaviour during a fire and, hence, the assessment of structural safety. In the thermo-mechanical models neither the hygral behavior of heated concrete nor the chemical processes in fire exposed concrete are modelled but the respective effects are only reflected in the phenomenological description of the temperature-dependent material behavior. Hence, comprehensive physically based material models for concrete, subjected to high temperatures, take into account the coupling of heat and moisture transport with phase changes between water, air and vapor [8,9] and the heatinduced dehydration of concrete [10] and the coupling between all of these processes [11]. These multi-field models allow the description of the physical and chemical processes taking place in heated concrete and the effects on the mechanical behavior. They are indispensable, if, e.g., the possibility of spalling of

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concrete is investigated. However, these models are computationally expensive and, because of their complexity, a relatively large number of material parameters is required, of which at least some are not known in the design stage of a structure. Hence, thermo-mechanical models do have their merits for the assessment of the load-carrying capacity of fire exposed structures both in the design stage and if spalling of concrete is prevented, e.g., by adding Polypropylene fibers to the concrete [12]. The proposed thermo-mechanical model for fire exposed concrete structures combines a material model for concrete, formulated within the framework of plasticity theory and enhanced by considering stiffness degradation due to damage. The temperature-dependent material parameters for this model can be obtained from test data or—if test data are not available—they can be taken according to the provisions of the Eurocode 2 [2]. Continuum elements are employed for the discretization of a structure in order to avoid prescribing a linear strain distribution across the thickness, as would be the case for beam-, plate- and shell-elements as a consequence of the underlying kinematic hypothesis. Special attention is paid to the validation of the proposed numerical model by means of fire tests on plate-type concrete specimens and a large-scale fire test on a RC slab, before the numerical model is applied to the numerical simulation of the load-carrying behavior of a RC tunnel structure exposed to fire.

The time-dependent distribution of the temperature T is governed by Fourier’s differential equation for heat conduction @Tðx; tÞ  aDTðx; tÞ ¼ 0: @t

ð1Þ

In (1) x is the position vector of an arbitrary point of the structure under consideration, t denotes time, D is the Laplace operator and a represents the thermal diffusivity. The latter is given as a ¼ l=ðrcÞ with l, r and c denoting the temperature dependent thermal conductivity, density and specific heat capacity of concrete. If no test data are available, the dependence of the thermal diffusivity of concrete on temperature and moisture content can be assumed according to [2]. The solution of the differential equation (1) requires specification of initial conditions at time t ¼ t0 and of boundary conditions for the considered time domain. The heat flux through the surface of a structure can be prescribed by means of Robin’s boundary condition

l

@T ¼  aðT  TÞ  Fem ef 5:67  108 ½ðT þ273Þ4  ðT þ273Þ4 ; @n

finite element method with the unknowns representing the timedependent nodal values of the temperature.

3. Constitutive models 3.1. Concrete model The plastic damage model for concrete, proposed in [18] and implemented, e.g., in the FE-program ABAQUS [17], serves as the basis for the constitutive model for concrete, subjected to high temperatures. The model is formulated within the framework of plasticity theory in combination with damage mechanics. Hence, plastic strains as well as stiffness degradation due to damage are taken into account. Stiffness degradation is described by a simple isotropic damage model in terms of the scalar damage variable d. The relationship between the elasticity tensor Cel 0 of the undamaged concrete and the elasticity tensor Cel of the damaged concrete is given as Cel ¼ ð1  dÞCel 0:

ð2Þ

with n denoting the outward normal direction to the surface of the structure. The right hand side of (2) consists of two parts, the heat flux due to convection and due to radiation. T represents the time-dependent gas temperature of the area on fire and T is the temperature on the surface of the fire exposed structure in degree Celsius, a denotes the heat transfer coefficient of the convective heat flux, F is a configuration parameter and em and ef are the emissivity of the heated surface and of the fire, respectively. Commonly, F ¼ 1, em ¼ 0:8 and ef ¼ 1 can be assumed [13]. For structures with roughly known thermal loads the time-dependent gas temperature T can be prescribed in terms of standardized time-temperature profiles, also known as fire curves (cf., e.g. [14,15]). The numerical simulation of the transient temperature distribution in a fire exposed structure is based on the weak formulation of the heat equation (1) within the framework of the

ð3Þ

The relationship between the nominal stresses r, related to an undamaged surface element, and the effective stresses r , related to the intact part of the respective damaged surface element, is given as

r ¼ ð1  dÞr ¼ ð1  dÞCel0 : ðe  epl Þ ¼ Cel : ðe  epl Þ;

2. Thermal model

959

ð4Þ

with e and epl denoting the total strains and the plastic strains, respectively. The yield function ~ pl f ðr ; e~ pl t ; ec Þ ¼

  1 3 s : s þ ar : I þ b/s max S  g/  s max S  s c ðe~ pl c Þ 1a 2

ð5Þ

is formulated in terms of the effective stresses r and the ~ pl equivalent plastic strains e~ pl t and e c . The latter are determined in a uniaxial tension and compression test. In (5), a is a material parameter, depending on the ratio of the biaxial compressive strength fbc to the uniaxial compressive strength fc , s represents the deviatoric part of the effective stresses, I is the second order unity tensor, s max denotes the largest eigenvalue of the effective stress tensor r and / S is the McAuley bracket (i.e., /xS ¼ x for x Z0 and /xS ¼ 0 for x o 0). The material parameter g is only required for triaxial compressive stress states and the function b is given as ~ pl bðe~ pl t ; ec Þ ¼

s c ðe~ pl c Þ ð1  aÞ  ð1 þ aÞ; s t ðe~ pl t Þ

ð6Þ

~ pl with s c ðe~ pl c Þ and s t ðe t Þ as the effective uniaxial compressive and tensile stress in terms of the equivalent plastic strain in compression and tension, respectively. The plastic strain rate is obtained from a non-associated flow rule. For plane stress states the yield function (5) gives a good approximation of the well known failure curve determined by Kupfer [16]. For undamaged concrete (i.e. for d ¼ 0) the isotropic hardening and softening laws for concrete in compression and tension, s c ðe~ pl and s t ðe~ pl are obtained from the respective c Þ t Þ, seFrelationships for uniaxial compression and uniaxial tension by subtracting the elastic strain. Hence, the hardening and softening laws can be easily determined from the respective seFrelations for uniaxial compression and uniaxial tension, obtained from test data or given in Eurocode 2, part 1–1 and part 1–2. Subsequently, exemplarily the employed approach for the frequently encountered case in engineering practice of test data being not available is described. In this case one can revert to the provisions of the Eurocode 2 [2,19].

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For ambient temperatures the relationship between the uniaxial compressive stress sc and the strain ec is given according to [19] as

sc ¼ fc

k  Z  Z2 ; 1 þ ðk  2Þ  Z

ð7Þ

with Z ¼ ec =ec1 and k ¼ 1; 1  Ec  jec1 j=fc . ec1 is the strain associated with the uniaxial compressive strength fc and Ec denotes the secant modulus of elasticity. For the whole temperature range during a fire the relationship between the temperature-dependent uniaxial compressive strength sc ðTÞ and the strain ec ðTÞ is given according to [2], as

sc ðTÞ ¼ fc ðTÞ

ec ðTÞ  ec1 ðTÞ



3

ec ðTÞ 2þ ec1 ðTÞ

3 :

ð8Þ

The respective temperature-dependent seFcurves for concrete with limestone aggregates are shown in Fig. 1. In this figure the seFcurves for 20 3 C are taken according to (7), whereas the seFcurves at elevated temperatures are taken according to (8). The temperature-dependent seFcurves for concrete with quartzous aggregates differ from the ones of Fig. 1 by a somewhat smaller temperature-dependent compressive strength. The ratio fbc =fc of the biaxial strength to the uniaxial compressive strength increases with increasing temperature, i.e. with increasing temperature the uniaxial compressive strength decreases faster than the biaxial compressive strength [21]. Starting with fbc =fc  1:16 at ambient temperatures this ratio increases to fbc =fc  1:30 at 300 3 C and up to fbc =fc  1:70 at 750 3 C. In uniaxial tension a linear constitutive relationship is assumed up to the tensile strength. According to [2] the latter decreases linearly with increasing temperature from 100 3 C until it is equal to zero at 600 3 C. The post-peak domain in uniaxial tension is described by means of the exponential softening law ! ~ pl e pl t ð9Þ s t ðe~ t Þ ¼ ft exp  pl : e~ t0

related stress σc(T) / σc(T = 20°C)

Within the framework of the employed smeared crack concept represents a fictitious crack the equivalent plastic strain e~ pl t strain, which is obtained from the crack width zn and the equivalent length he of the respective ‘‘cracked’’ finite element ~ pl as e~ pl t ¼ zn =he ; e t0 defines the gradient of the softening law at pl e~ t ¼ 0. In order to ensure objective numerical results of a finite element analysis based on the smeared crack concept independently of the employed element size at least in an approximate

1.0 0.8

20°C 100°C 200°C 300°C 400°C

limestone aggregates

500°C

600°C 700°C

0.6

800°C 900°C 1000°C 1100°C

0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5 3.0 strain ε

3.5

4.0

4.5

5.0 ×10−2

Fig. 1. seFrelations for uniaxial compression of concrete with limestone aggregates at different values of temperature.

manner, e~ pl t0 in (9) is adjusted to the respective element size by means of the specific fracture energy of concrete Gf and the equivalent length he of the respective finite element. The dependence of the specific fracture energy on temperature is taken into account according to test results [20,7]. The value for Gf at 20 3 C increases up to 1:6Gf at 280 3 C and decreases with further increasing temperature until at 600 3 C a value of 0:8Gf is attained. In addition to the strains, given by the constitutive relations (7) and (8), the thermal strains are taken into account according to [2]. 3.2. Steel model The constitutive relations for reinforcing steel consist of a linear-elastic part up to the proportional limit, a nonlinear seFrelation up to the yield strength followed by ideal plastic material behavior and finally a softening branch. The modulus of elasticity, the proportional limit and the yield strength are assumed to depend on temperature according to [2].

4. Numerical study The numerical study consists of two parts. In the first part the numerical model is validated by the numerical simulation of fire tests on plate-type concrete specimens conducted by Ehm [21] and by the numerical simulation of a large-scale fire test on a reinforced concrete slab [4]. In the second part the proposed numerical model is applied to the numerical simulation of the response of a tunnel structure, exposed to fire. 4.1. Fire tests on plate-type specimens Fire tests on plate-type concrete specimens with the dimensions 200  200  50 mm were conducted by Ehm [21] at the Technical University at Brunswick. Since the specimens as well as parts of the loading frame were located within the furnace, both surfaces of the specimens were heated uniformly. The heating rate of the furnace was 2 K/min. The uniaxial compressive strength is given according to [21] as fc ¼ 41 N=mm2 , the modulus of elasticity, the strain associated with the uniaxial compressive strength and the ultimate strain are assumed according to [19] as Ec ¼ 33 590 N=mm2 , ec1 ¼ 0:002213 and ecu ¼ 0:0035. Poisson’s ratio and the ratio of the biaxial over the uniaxial compressive strength are assumed as 0.18 and 1.16, respectively. Here, two of the extensive test series, conducted by Ehm, are simulated numerically. In the first test series the plate-type specimens were heated at constrained deformations in both directions of the specimens midplane and the resulting stresses were measured by load cells placed between the specimens and the hydraulic jacks of the biaxial testing machine [21]. In the second test series the specimens were first loaded by biaxial compressive stresses s1 ¼ s2 ¼ afc , 0 o a o 1, and then the loaded specimens were heated at constant stresses and the resulting strains were measured. Fig. 2 contains a comparison of measured and computed stresses for heated concrete specimens, which are constrained in both directions. The stress–strain relation (7) according to the Eurocode 2, part 1–1, is employed for ambient temperatures whereas the stress–strain relation (8) according to the Eurocode 2, part 1–2, is employed for temperatures exceeding 180 3 C. Within the temperature range from 20 to 180 3 C a smooth transition from the seFrelations (7) to the seFrelations (8) is obtained by linear

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(a) (b)

0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

100

200

300

400 500 600 700 temperature [°C]

800

900 1000

Fig. 2. Comparison of measured and computed stresses for biaxially constrained heated concrete specimens: (a) measurement data, (b) numerical results.

1.0

4.2.1. Thermal analysis The gas temperature is given by the time temperature profile [14]

× 10−2 α = 0.2

T ¼ 20 þ 345 log10 ð8  t þ1Þ;

0.5 α = 0.4

ε1, ε2

0.0 −0.5 −1.0 (a)

−1.5 −2.0

drums. The furnace was heated for 3 h following the ISO 834 standard fire curve [14]. The uniaxial concrete compressive strength is given in [4] as fc ¼ 36 N=mm2 . From the latter, the modulus of elasticity, Poisson’s ratio, the uniaxial tensile strength and the specific fracture energy are estimated as Ec ¼ 32 310 N=mm2 , n ¼ 0:18, ft ¼ 3:0 N=mm2 and Gf ¼ 0:1 N=mm. The slab is reinforced with a mesh of cold drawn reinforcing bars, denoted as type 661, with 7.5 mm diameter and a grid spacing of 150 mm. The yield stress and the ultimate strain are given as 568 N=mm2 and 3.2%, respectively, resulting in a steel content of 295 mm2 =m. The concrete cover of the reinforcement is 25 mm. The experimental program during the fire test contained, e.g., measurements of the temperature across the thickness of the slab, the deflection at the center of the slab and the vertical displacement of the corners of the slab.

α = 0.6

(b) 0

100

200

300

400 500 600 700 temperature [°C]

800

900 1000

Fig. 3. Comparison of measured and computed strains for biaxially loaded concrete specimens: (a) measurement data, (b) numerical results.

interpolation. On the basis of the interpolated stress–strain relations good correspondence between measured and computed stresses is obtained for the whole temperature range. Only for values exceeding 700 3 C the pronounced decrease of stresses is somewhat underestimated by the numerical simulation (Fig. 2, curve (b)). This discrepancy is attributed to the fact that the functional relations, describing the dependence of material parameters on temperature, could not be deduced from test data but were assumed according to the provisions of the Eurocode 2, part 1–2. Fig. 3 shows a comparison of measured and computed strains for biaxially loaded concrete specimens with a ¼ s1 =fc ¼ s2 =fc . It is emphasized that the only material parameter which was available for computing the strains, shown in Fig. 3, was the uniaxial compressive strength. Hence, the correspondence between measured and computed strains seems to be acceptable. 4.2. Large-scale fire test on a RC slab Several large-scale fire tests on RC slabs were carried out at the University of Canterbury in New Zealand [4]. The slab, considered in this paper, had dimensions of 4300  3300 mm and a thickness of 100 mm. It was placed above a furnace with an opening of 4020  3020 mm and was simply supported on all four edges on cylindrical rollers. The corners were free to lift from the supports. In addition to the dead load of the slab, a uniformly distributed load of 3:0 kN=m2 was applied by means of water-filled steel

ð10Þ

with t denoting the fire duration in minutes and T the gas temperature of the area on fire in degree Celsius. With the exception of a 140 mm wide edge strip the bottom surface of the slab was exposed to the fire, resulting in a pronounced nonlinear temperature distribution across the thickness. Fig. 4 shows a comparison of measured and computed temperatures for different distances from the fire exposed bottom surface of the slab. A good correspondence between measured and computed temperatures is achieved. Because of the relatively small thickness of the slab it is essential to take into account the thermal radiation, i.e. the second term on the right hand side of (2), for the upper surface of the slab. If the latter is taken into account, after 180 min of fire the temperature at the upper surface is computed as 280 3 C, whereas neglecting thermal radiation would result in a surface temperature of 380 3 C. The pronounced nonlinear temperature distribution across the thickness for selected time instants during the fire is shown in Fig. 5. 4.2.2. Mechanical analysis The RC slab is discretized by 3D continuum elements. Thus, no restriction with respect to the strain distribution across the thickness is made, as would be the case, if plate elements were employed for the discretization. The reinforcement is modelled by

1200 (a) 1000 temperature [°C]

σ1/ fcm (20°) = σ2/ fcm (20°)

0.7

961

0 mm

(b)

800

25 mm

600

50 mm

400 200 0

75 mm 0

30

60

90 time [min]

120

95 mm 150

180

Fig. 4. (a) Measured and (b) computed temperatures at different distances from the heated surface.

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Fig. 5. Distributions of temperature (left), stress (middle) and strain (right) at the center of the slab for selected time instants in the direction of the longer span.

0 center deflection [mm]

−50 −100 −150 −200 −250 (a) (b) (c)

−300 −350 −400

0

30

60

90 time [min]

120

150

180

Fig. 6. Center deflection of the slab: (a) experimental results, (b) and (c) numerical results without and with consideration of the tensile strength of concrete.

the embedded approach, which is characterized by embedding the reinforcing bars into finite elements of the same type as for the concrete and superposing the concrete elements and the reinforcement elements. Possible lift of the corners from the supports is modelled by contact conditions between the supports and the bottom surface of the slab. Again, within the temperature range from 20 to 180 3 C the seFrelations for concrete are obtained from linear interpolation between the constitutive relations (7) and (8). In addition to the temperature distribution across the thickness of the slab Fig. 5 contains computed stress and strain distributions for selected time instants during the fire. The respective distributions refer to the center of slab in the direction of the longer span. Fig. 5 shows the pronounced change of the stress distribution across the thickness during the fire and the transition of the initially linear strain distribution across the thickness to a nonlinear one shortly after beginning of the fire. Hence, the employed discretization by 3D continuum elements has proven as appropriate. Fig. 6 contains a comparison of (a) the test results for the center deflection of the slab with the computed results (b) neglecting the tensile strength of concrete and (c) considering the temperature-dependent tensile strength and tension softening following crack initiation. As expected, neglecting the tensile strength of concrete results in an overestimation of the deflections

throughout the duration of the fire. Consideration of the temperature-dependent tensile strength results in good correspondence between measured and computed deflections in the initial phase of the fire. In the further course of the fire the computed deflections are somewhat larger than the measured ones. This discrepancy may be attributed to the facts that (i) the computed temperature at the fire exposed surface of the slab is somewhat higher than the measured one, (ii) a phenomenological description of the material behavior is used, not taking into account the coupling of heat and moisture transport with phase changes between water, air and vapor, the heat-induced dehydration of concrete and the coupling between these processes, (iii) the functional relations describing the dependence of the material parameters on temperature were not available from test data but were assumed according to the provisions of the Eurocode 2, part 1–2.

4.3. Numerical simulation of the structural response of a fire exposed tunnel structure In this section the application of the proposed numerical model to the numerical simulation of the response of a shallow RC tunnel structure, exposed to a fire of 90 min duration, is demonstrated. The double-track railway tunnel, put up by the cut and cover method, is characterized by a double box section. The employed concrete quality is C25/30, characterized by mean values of the uniaxial cylindrical compressive strength and of the uniaxial tensile strength of 33 and 2:6 N=mm2 , respectively. The reinforcing steel is of type BSt 550, characterized by the yield strength and tensile strength of 550 and 620 N=mm2 , respectively. The dimensions of the tunnel structure were determined accord¨ NORM B 4700. ing to the Austrian code O Assuming plane strain conditions the cross-section of the tunnel structure is discretized by 2D continuum elements. The dimensions of the cross-section of the tunnel and the FE-mesh are shown in Fig. 7. The reinforcing mesh is modelled by embedded truss elements. Tension stiffening is taken into account in a simplified manner by increasing the specific fracture energy of concrete [22]. The soil is not considered in the FE-mesh, but is taken into account by an elastic foundation of the bottom slab and by the loads exerted on the tunnel structure.

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963

Fig. 7. Cross-section of the tunnel structure (left) and FE-discretization (right).

Fig. 8. Deformations and fictitious crack strains after 10 min (left) and 90 min (right) of fire duration (displacements are magnified by a factor of 50).

The first part of the numerical simulation refers to the thermal analysis yielding the time-dependent temperature distribution in the tunnel structure during the fire. To this end the gas temperature in the tunnel is assumed according to the Hydrocarbon increased ðHC inc Þ time–temperature curve [15] T ðtÞ ¼ 1280ð1  0:325  e0:167t  0:675  e2:5t Þ þ 20:

ð11Þ

It is based on the ambient temperature of 20 C and is characterized by a rapid increase of the gas temperature up to 1300 3 C within a few minutes. The thermal insulation of the bottom slab due to the road bed results in a reduced thermal loading for this part of the tunnel structure. In order to take advantage of symmetry the fire is assumed to simultaneously rage in both parts of the tunnel. In the first step of 3

the mechanical analysis all permanent loads are applied to the tunnel structure. They consist of dead load of the tunnel structure and the road bed, the earth pressure acting on the outer surface of the tunnel structure, and traffic loads. Subsequently, the timedependent temperature changes are applied in the transient part of the mechanical analysis. Fig. 8 shows the deformations of the tunnel structure and the fictitious crack strains for two selected time instants during the fire. From these two figures the increase of the deformations and the increase of damage of the tunnel structure with increasing fire duration can be clearly seen. The distributions of the temperature, the stress and the strain across the thickness of the outer vertical wall of the tunnel structure at a vertical distance of 2 m from the bottom are shown

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Fig. 9. Distributions of temperature (left), stress (middle) and strain (right) across the thickness of the outer vertical wall at a distance of 2 m from the bottom at selected time instants during the fire.

700

fire the stresses are changing from initially compressive stresses to tensile stresses. Contrary to the outer reinforcement, the temperature of the inner reinforcement is increasing during the fire quite strongly. Whereas in the first stage of the fire, the compressive stresses of the inner reinforcement are increasing rapidly, in the following stages they are decreasing due to the decreasing strength caused by the increasing steel temperature.

temperature [°C]

600 500 400

outer vertical reinforcement inner vertical reinforcement

300 200

5. Summary and conclusions

100 0

0

10

20

30

40 50 time [min]

60

70

80

90

100

stress [N/mm2]

0 −100 −200 −300 outer vertical reinforcement inner vertical reinforcement

−400 −500

0

10

20

30

40 50 time [min]

60

70

80

90

Fig. 10. Development of temperature (top) and stresses (bottom) for the inner and outer vertical reinforcement.

for selected time instants during the fire in Fig. 9. The strain distributions in Fig. 9 clearly show the transition from the initially linear strain distribution across the thickness to a pronounced nonlinear one shortly after beginning of the fire. For the same section as in Fig. 9 the evolution of the temperature and of the stresses in the inner and outer vertical reinforcement are plotted in Fig. 10. Expectedly, the outer reinforcement does not experience any temperature increase during the fire. However, due to stress redistribution during the

The presented thermo-mechanical model for fire exposed reinforced concrete structures relies on a phenomenological approach for considering the temperature-dependent material properties of concrete and reinforcing steel. To this end, a plastic damage model for concrete and standard elastic–plastic constitutive relations for the reinforcing steel were extended in order to take into account the impact of elevated temperatures on the material behavior. The extended constitutive relations were implemented into the FE-code ABAQUS [17]. By assuming the temperature distribution in a fire exposed structure as independent of the mechanical behavior, the thermomechanical analysis can be performed in two consecutive steps. The FE-solution of Fourier’s differential equation for heat conduction yields the transient temperature distribution in the structure, which is used for determining the temperaturedependent material properties and the thermal strains in the subsequent transient mechanical FE-analysis. The computational model was validated by the numerical simulation of fire tests on plate-type concrete specimens [21] and of a large-scale fire test on a RC slab [4]. The numerical simulations of the fire tests on the small-scale concrete specimens demonstrate the capability of the numerical model to capture the material behavior of concrete during a fire. The numerical simulation of the large-scale fire test on a RC slab confirms the ability of the numerical model to capture the load-carrying behavior of a RC structure during a fire. In addition, the latter shows the evolution of a pronounced nonlinear strain distribution across the thickness of the slab during the fire. This justifies the discretization by 3D continuum elements in order to avoid prescribing a linear strain distribution across the thickness as would be the case, if shell elements were employed.

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