COUPLED FEM MODEL FOR CONCRETE EXPOSED TO FIRE LOADING

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1 COUPLED FEM MODEL FOR CONCRETE EXPOSED TO FIRE LOADING Ji í Surovec (1), Petr Kabele (1) and Jan ervenka (2) (1) Czech Technical University in Prague, Czech Republic (2) ervenka Consulting, Czech Republic Abstract A phenomenological material model with ability to describe a behavior of various concrete types during extensive fire load is presented. Our objective was to propose a model sufficiently accurate, yet simple to be easily applicable in wide engineering practice. The state variables are temperature and water vapor density. The presence of liquid water is considered, however, its direct transport is neglected. The liquid water is used only as a container for evaporation or condensation. In the proposed model the evaporation is governed by the phase equilibrium function. The assumption that evaporation does not occur instantaneously, leads to the application of the evaporation function, which determines the evaporation rate dependency on the pore pressure and temperature. Since the model accounts for variable permeability of concrete, which is one of the main factors influencing the humidity transport, it makes it possible to model special types of concretes such as polymeric fiber concrete. 1. INTRODUCTION Several serious accidents in European road tunnels recently caused death of tens of people. Thus the need to satisfactorily predict the behavior of concrete under fire exposure arose. The numerous processes acting in concrete exposed to fire loading are rather complex. Sophisticated numerical models capable to describe numerous physical and chemical processes have been developed. However, the main disadvantage of most of models is the lack of accurate values for material constants and boundary conditions. Usually, only an elasticity modulus, strength in tension and compression, porosity and permeability can be determined. Our aim is to make the analytical model as simple as possible for easy use in engineering practice while maintaining satisfactory accuracy. The main target of the model is to describe the deterioration of load carrying capacity of structures exposed to fire. The model should also 95

2 capture the spalling phenomena, the brittle failure of concrete with most cracks parallel to the heated surface. The proposed material model should be easy to implement in an existing finite element (FE) package. The range of validity of this model must cover the majority of concretes. The material laws must also allow modeling of other materials, such as thermal insulations or reinforcement bars. As a basis for the FE implementation the ATENA FE package developed by ervenka Consulting was selected. By upgrading the well-established ATENA s fracture-plastic model of concrete with the proposed transport model, a strong and sophisticated tool to predict the behavior of concrete during a fire is created. Also, the implementation into an established commercial code will make our proposed model available to a wider engineering community, thus creating a significant contribution to the fire safety. 2. SPALLING PHENOMENA The proposed model should also capture the spalling phenomena. Spalling has two main consequences. It causes the loss of section and the loss of protective cover of reinforcing bars. The cover of reinforcement is crucial for fire resistance of the structure. Moreover, during extinguishing the fire the debris is still susceptible to spalling which endangers the rescue workers. Two hypotheses were considered. First hypothesis states that spalling is caused by restrained thermal dilatation. The temperature in outer layers rapidly rises, causing the expansion. That crates the huge shear stress in between the heated and inner layers, thus inducing the cracks parallel to the surface. The cohesion between the layers decreases. Moreover, the compressive stresses in the heated layers contribute to brittle buckling. Second hypothesis assumes that spalling is caused due to pore pressure build-up. This hypothesis is supported by investigations showing that spalling is closely related to moisture content, concrete permeability and porosity. It is opposed by facts that pre-dried concrete is still susceptible to spalling. Both scenarios may occur simultaneously. Thus both the pore pressure distribution and temperature field is needed for accurate description of concrete behavior. 3. PROPOSED TRANSPORT MODEL DESCRIPTION 3.1 General considerations At high temperatures, the transport of liquid phase is much slower process than the movement of water vapor. Thus, consistently with [1], the liquid humidity transport is neglected and only the transport of gaseous phase is considered. However, evaporation and condensation are taken into account. The liquid water can change its phase to gaseous, change the location and condense back to liquid. To describe these processes we use water vapor density per unit volume ρ and temperature T as the primary state variables. There are two driving forces that influence the humidity transport - the temperature gradient and the humidity gradient. Consider that temperature decreases and humidity increases as we proceed from a heated surface. Then the temperature gradient is pushing the humidity inwards, while the humidity gradient is forcing the humidity towards the heated surface. 96

3 Figure 1: Driving forces influencing the humidity transport 3.2 Governing equations The pore pressure P pore is defined as ρ P pore = R T + Patm pore LQ where R is the gas constant (461.5 J.kg -1.K -1 ); pore is the volume of pores; LQ is the volume of liquid water and P atm is the atmospheric air pressure. Using the nabla operator (2) = ; ; x y z v the velocity of water vapor due to the gradient of pore pressure is given by Darcy s law: K( 1 S ) (3) = Ppore µ where v is the water vapor velocity; K is the specific (intrinsic) permeability of dry concrete; S is the saturation of concrete; µ is the dynamic viscosity of water vapor. Water vapor flux is determined according to Fick s law as follows: K( 1 S ) (4) J = ρ v = ρ Ppore µ (1) where J is the water vapor flux. Total humidity conservation is given as: ρ + J = E t (5) 97

4 where t is time; E is the evaporation rate. Using Equations (1) to (5) we obtain the first governing equation: ρ K( 1 S ) (6) + ρ Ppore = E t µ Energy equilibrium is given by Equation (7): T ρ CC ( K H T ) = Q + λe t where ρ C is the concrete density; C is the specific heat of concrete; K H is the thermal conductivity of concrete; Q is the internal heat source; λ is the specific heat of evaporation or condensation. 3.3 Evaporation For the evaporation rate calculation the ITS-90 formulation, [4], which defines the relationship between saturation vapor pressure over water and temperature, is used. Saturation pressure is the pressure for a corresponding saturation temperature at which a liquid boils into its vapor phase. The ITS-90 formulation is given in Equation (8). 6 i ln es = git 2 + g7 lnt i= 0 where e S is the saturation vapor pressure over water; T is the temperature in Kelvin; g i are coefficients given in [4]. This equation defines the conditions (pressure and temperature) of a steady state at which both phases - liquid and gaseous - may exist simultaneously. If a material point contains both gaseous and liquid water, a change of pore pressure must be accompanied by evaporation or condensation. When the pore pressure at a given temperature is below the ITS-90 curve, the liquid water evaporates, thus creating water vapor that is causing an increase of the pore pressure. On the other hand, when the pressure is above the curve, condensation occurs. The phase change occurs with certain latency, i.e., the steady state is not achieved instantaneously. Therefore, we introduce an evaporation function, which defines the rate of evaporation or condensation dependent on the state in the material point. To this end, the following three hypotheses are taken into consideration. The first hypothesis implies that the evaporation rate is proportional to temperature. The higher the temperature, the higher evaporation rate is observed. The second hypothesis consists in assuming that the evaporation rate decreases proportionally with increasing pore pressure. The third hypothesis assumes that the evaporation rate depends on the proximity of the current state to the steady state defined by Equation (8). We choose the closest distance between the current state and the steady state curve in the temperature-pressure plane as the measure of this proximity. In order to avoid the sensitivity of this measure on the units and scales used, the axes are normalized as follows. The normalized temperature equals zero for T = 0 C and equals unity for temperature at the critical point, T = 374 C. The normalized pore pressure equals zero at saturation vapor pressure over water at saturation temperature (7) (8) 98

5 T = 0 C, i.e. at P pore = 0,6 kpa, and it equals unity for the pressure at the critical point, i.e. at P pore = 24,9 MPa. The evaporation function is then assumed in the following form: E = 2.0 D T 1. 0 P (9) ( )( ) where D is the perpendicular distance between the actual state and normalized evaporation curve in the plane; T is normalized temperature; P is normalized pore pressure as described above. In the numerical algorithm, the evaporation rate is obtained from the previous time step and is kept constant during iterations. This simplification is employed to improve numerical stability. 4. MODEL ALIDATION To validate the proposed model the comparative analysis has been performed on a problem of a tunnel lining segment exposed to fire. The proposed model was compared with results by Tenchev et al. [7]. The test problem represents a 0.5 m thick slab exposed on one side to a standard fire curve (ISO 834) load. The details such as boundary conditions or material parameters are stated in [7]. The results of both the proposed model and the model used by Tenchev et al. [7] are depicted in Figure 2. The pore pressure distributions are shown. The proposed model curves are denoted as PMXX, where XX indicates the time in minutes from beginning of fire loading. Tenchev's model results are labeled TNXX with same notation. Pore pressures 8 7 MPa TN10 TN30 TN60 PM10 PM30 PM distance from heated end [mm] Figure 2: Pore pressure distributions 99

6 The resulting pore pressure distributions show the satisfactory agreement between the two analytical models. 5. NUMERICAL PREDICTION The proposed model was used to predict the behavior of the polymeric fiber concrete exposed to ISO fire load. The aim of this numerical example is to show the difference between a pore pressure occurring in a standard high performance concrete (HPC) and a pore pressure occurring in a polymeric fiber concrete. When the polymeric fiber concrete is exposed to temperatures above the fiber s melting point, i.e. above 160 C, the permeability significantly increases [6]. The melting of fibers evokes the creation of void channels, which allow the humidity to easily escape. That can minimize the spalling phenomena. The chosen numerical example is the same as in Chapter 4. In addition to standard HPC the polymeric fiber concrete with steep increase of permeability above 160 C is used. It should be mentioned that polymeric fibers were employed only to increase the permeability of concrete when heated due to fire. No improvement in mechanical properties due to use of polymeric fibers was taken into account. Relative permeabilities of both standard HPC and a polymeric fiber concrete are presented in Figure 3. The intrinsic permeability of fiber concrete increases between 160 and 170 C six times. This was derived according to Salomão [6]. The value of permeability above 600 C was assumed the same for both standard HPC and fiber concrete. The concrete above 600 C is already significantly damaged, so the influence of voids from melted polymeric fibers is neglected. 120 K [ ] 20 K Temperature [ C] Figure 3: Relative permeability dependence on temperature, continuous line represents HPC, dashed line represents fiber concrete The results of both the HPC concrete and fiber concrete are depicted in Figure 4. The pore pressure distributions are shown. The high-performance concrete is denoted as HPC XX, where XX indicates the time in minutes from beginning of fire loading. Fiber concrete is labeled FC XX with same notation. The resulting pore pressure distributions show the significant decrease of pore pressures when the polymeric fibers are used. The pore pressures in fiber concrete reach only less then 30 % of the pore pressures in HPC. The increased permeability at around 160 C caused by 100

7 melting of the fibres enables faster humidity transport that significantly improves the fire resistance of the concrete. Pore pressures 8 7 MPa FC 10 FC 30 FC 60 HPC 10 HPC 30 HPC distance from heated end [mm] 6. NOTATION Figure 4: Pore pressure distributions For the sake of clarity, the following notation was used in this paper. T temperature [ K] ρ vaporous humidity density [kg/m 3 ] L liquid humidity density [kg/ m 3 ] t time [s] J vapor flux [kg/( m 2.s)] E evaporation rate (liquid vapor) [kg/ m 3 ] λ specific heat of evaporation [J/kg] Q internal source of heat [J/ m 3 ] C specific heat of concrete [J/(kg. K)] K H heat conductivity of concrete [J/( m 3.s. K)] K permeability of concrete [m 2 ] S saturation of concrete voids with liquid [-] pore volume of pores [m 3 ] LQ volume of liquid water [m 3 ] v water vapor velocity [m/s] P pore internal pore pressure [Pa] 101

8 7. CONCLUSIONS P atm atmospheric air pressure [Pa] R gas constant = [J/(kg. K)] µ dynamic viscosity of water vapor [kg/(m.s)] The analytical transport model with only two state variables that is able to describe the behavior of concrete exposed to fire was presented. The evaporation function that governs phase changes of liquid water and water vapor is proposed. The validation by comparison with a more complex model was shown. The importance of permeability on pore pressure and the ability of the model to predict the behavior of various concrete types were presented. It was demonstrated by numerical prediction that the use of polymeric fiber concrete can significantly reduce the pore pressure in concrete. ACKNOWLEDGEMENTS The presented research has been supported by the Czech Science Foundation grant GACR 103/07/1660. REFERENCES [1] Ahmed, G.N. and Hurst, J.P., 'Coupled heat and mass transport phenomena in siliceous aggregate concrete slabs exposed to fire', Fire and Materials 21(1997) [2] Eurocode 2, Design of concrete structures, Part 1-2: General rules - Structural fire design (2004) [3] Gawin, D. and Pesavento, F 'Modelling of hygro-thermal behaviour of concrete at high temperature with thermo-chemical and mechanical material degradation'. Computer methods in applied mechanics and engineering 192 (2003) [4] Hardy, B., 'ITS-90 Formulations for apor Pressure', Proceedings of the Third International Symposium on Humidity & Moisture, Teddington, London, England, April 1998 [5] Janotka, I. and Bagel, L., 'Pore structures, permeabilities, and compressive strengths of concrete at temperatures up to 800 degrees C - Author's closure'. ACI Materials journal 100 (2) (2003) [6] Salomão, R. and Cardoso, F.A Effect of polymeric fibers on refractory castable permeability. In American society ceramic bulletin 4(82): [7] Tenchev, R.T. and Li, L.Y., 'Finite element analysis of coupled heat and moisture transfer in concrete subjected to fire'. Numerical heat transfer part A Applications 39 (7) (2001)