FIRE ANALYSIS OF CURVED REINFORCED CONCRETE BEAM

In the present study the fire analysis of a curved reinforced concrete beam exposed to concrete spalling is presented. Due to the complexity of physical and chemical processes in concrete at elevated temperatures, the proposed numerical model is divided into two consecutive mathematically uncoupled phases. In the second phase of the fire analysis a partially coupled numerical model is introduced in order to evaluate the effect of concrete spalling on the behaviour of curved RC beam in fire. In addition, the effect of depth, time development and the length of spalling area on the fire resistance of a curved RC beam is discussed.


INTRODUCTION
Fire safety of the tunnel structures is an important requirement that in general every engineering structure must fulfil.Insufficient level of fire safety can lead into human loss, material damage and pollution of environment.Tunnel structures are usually made of concrete which is heterogeneous material consisting of solid matrix, water and gaseous mixture of dry air and water vapour.When concrete structures are exposed to high temperatures as can be found in fire physical and chemical processes within concrete structure occur.These can lead to unfavourable effect of explosive spalling of concrete, which can cause premature failure of the tunnel structure.The influence of spalling process needs to be taken into account in order to achieve better assessment of the fire resistance.This study presents a novel partially coupled strain-based finite-element numerical model for assessing the fire resistance of curved RC beams exposed to spalling of concrete.The idea of the introduced model is to evaluate the time and the magnitude of concrete spalling on the behaviour of curved RC beam in fire conditions.Due to concrete spalling, which is based on the proper parameters (temperature, pore pressure,...), the cross-section of the observed beam is reduced by the layer of concrete that spalls/peels off.The reduction of the cross-section has effect on the mechanical behaviour and also on the temperature field of the analyzed beam and therefore has to be taken into account.In presented study the effect of depth, time development and length of spalling area on the behaviour of curved RC beam in fire situation is presented.The introduced numerical model can be used for the evaluation of the fire resistance of tunnel lining in natural fire conditions.

PARTIALLY COUPLED NUMERICAL MODEL
Introduced partially coupled numerical model for the fire analysis of curved RC beam is due to the complexity of physical and chemical processes in concrete at elevated temperatures divided into two consecutive mathematically uncoupled phases.The first phase is dedicated to the determination of time and spatial development of temperatures in the fire compartment.Thus hydrocarbon fire curve (EC1, 2004) is used in the study for describing gas temperatures in the fire compartment, which are used as input parameters for the second phase.In the second phase of the fire analysis the heat and mass transfer in concrete and mechanical part (stress-strain state of structure) are partially coupled.The heat and mass transfer in concrete is based on the model proposed by Davie (Davie et al., 2006) and in the mechanical part of the fire analysis curved RC beam is modelled with the kinematically exact planar beam model of Reissner (Reissner, 1972).The system of governing equations is solved numerically using the modified principle of virtual work and a strain-based Galerkin-type of FEM (Planinc, 2001).Further a brief presentation of heat and mass transfer model and mechanical part is given.The algorithm of the partially coupled numerical model for the fire analysis of curved RC beam with concrete spalling involved is described as well.

2.1
Heat and mass transfer model -1 st part of partially coupled numerical model The heat and mass transfer in concrete is described by the system of equations proposed by Davie et al. (2006).Model considers the transfer of free water, water vapour and dry air caused by pressure and concentration gradients and the conversion of energy, and also takes into account the evaporation of water, the liquefaction of water vapour, the dehydration of the chemically bound water, the capillary pressure and the part of free water, coexisting in an adsorbed state.The governing equations of heat mass transfer model are solved numerically with the Galerkin-type of the finite element method employing 4-noded two dimensional (2D) isoparametric elements.The primary unknowns of the problem are temperature T, pressure of gaseous mixture of water vapour and dry air P G and water vapour content V  .For detailed description of the model see (Davie et al., 2006).

Mechanical part of numerical model -2 nd part of partially coupled numerical model
The mechanical model of the curved RC beam is modelled with the kinematically exact planar beam model of Reissner (Reissner, 1972).Within time interval [t i-1 , t i ], i = 1, 2,...,n cr , the stressstrain state at time station t i is calculated iteratively based on the mechanical results at time t i-1 and the results from the hygro-thermal analysis (1 st part of partially coupled numerical model) at time t i .The related governing equations of curved RC beam consist of kinematic, equilibrium and constitutive equations: Kinematic equations: Equilibrium equations: Constitutive equations: In equations (1) the prime ( )  denotes the derivatives with respect to arch length s.Variables u and w are the components of a displacement vector and  is the rotation of a cross-section at the reference axis.Variable and are extensional and shear strains, respectively, while represents the pseudo curvature (flexural deformation) of the beam reference axis.N, Q and M represent equilibrium generalized internal forces, while N C , Q C and M C denote constitutive generalized internal forces.According to the given stress and strain state on the time interval i-1 and temperature on the time interval i, the mechanical strains on the time interval i of any point in the curved beam can be calculated by the equation: ) where i D  is the increment of the total strains (also known as the 'geometrical deformations') in the time interval i. Considering the principle of additivity of strains we end up with the strain increment, i D  , as the sum of the strain increments due to temperature, For a detailed description of each strain increment, see Hozjan et al. (2011) and for detailed description of the proposed model a reader is referred to Ružić et al. (2015).The non-linear stressstrain relations for concrete and reinforcement steel at elevated temperatures and the rules for the reduction of material parameters due to an increased temperature are taken from European building code EC2 (2005).

Algorithm of partially coupled numerical model
The criteria for the commencement of concrete spalling and for the evaluation of the depth of the spalled layer are taken according to Gawin et al. (2006).The conditions are slightly modified, so the spalling process starts, when the pore pressure reaches maximum value and at the same time temperatures reach values between 200 and 250 o C. In general the heat and mass transfer analysis and the mechanical analysis are carried out in parallel.When the algorithm detects that the conditions for the start of explosive spalling (Gawin et al., 2006) are fulfilled, the change of the finite element mesh with corresponding boundary conditions is executed.The process can be repeated several times within the entire fire analysis.However, since the modified geometry of the cross-section, especially in the heat and mass transfer part, presents a numerical 'shock', an appropriate reduction of time step adequate resizing the finite element mesh are needed.The algorithm is simple but necessary for the proper assessment of the fire safety of curved RC beams exposed to concrete spalling.The procedure of the algorithm is shown below; for the sake of transparency the time increment for both parts of the partly coupled numerical model is the same size.
The algorithm of partially coupled numerical model:  1 st phase  The determination of the temperature field in the fire compartment;  2 nd phase  Step 1: time increment [t i-1 , t i ]:  Heat and mass transfer analysis  T, P G ,  V  ;  Mechanical analysis  stress-strain state of the curved RC beam;  The conditions for the start of spalling is checked;  Step 2: Check of conditions for the explosive spalling of curved RC beam:  If YES  re-meshing of the finite element mesh and reduction of the time step t;  If NO  continue to step 3;  Step 3: New time step  t i+1 = t i + t;  Repeat Step 1 and 2 until the failure of curved RC beam or the end of fire; Note.Considering a comparison also uncoupled numerical model is used (Ružić et al., 2015).Important difference compared to the partially coupled numerical model is that the parts (hygrothermal and mechanical part) of the second phase of the fire analysis are performed separately as a continuing process.Therefore the effect of spalling is considered only in the mechanical part of the fire analysis in the form of reduced height of the cross-section.Meaning the temperature state of the cross-section is determined on initial structure and does not change within the analysis.

NUMERICAL EXAMPLE OF CURVED RC BEAM EXPOSED TO SPALLING
In the parametric study curved RC beam exposed to the standard hydrocarbon fire curve as given in EC1 (EC1, 2004) is considered.Geometrical, cross-sectional, material, reinforcement and loading data of the problem are shown in The difference between the distribution of temperatures for partially coupled and uncoupled model is significant, which can be observed in the area of spalling where a huge leap in temperatures for partially coupled model happens.As observed the difference between partially coupled and uncoupled model decreases with the height of the cross-section.Meaning the distribution of temperatures and pore pressure approximately 10 to 15 cm from the exposed edge of the initial section is the same for both models.Proper distribution of temperatures over the cross-section is very important because in the mechanical part of the fire analysis it directly affects the material properties of concrete, and thus the capacity and stability of the arc.Critical times t cr of the curved RC beam exposed to fire for partially coupled numerical model are given in Table 1.Critical times t cr are more pronounced for the asymmetric form of spalling.As anticipated t cr of the curved RC beam decreases with the length of spalling L spal .Though, a little surprising is the re-increase of the critical time for biggest lengths of spalling.This inconsistency is present in symmetric and asymmetric shape of spalling and for both types of soil.The increased stiffness of soil has positive effect on the critical times of arc for both forms of spalling.For instance, in the case of symmetrical spalling for E soil = 30 MPa the beam fails only for L spal ≥ 600 cm, while for E soil = 30 MPa failure occurs for L spal ≥ 300 cm.The deformed configurations of observed beam for the length of spalling L spal = 400 cm are depicted on Fig 3 .The results are presented for critical time t spal = 23 min when spalling occurs.In general the displacements in horizontal direction are larger than in vertical direction, which is caused by the action of springs that simulate the influence of soil.The effect of asymmetric shape is clearly noticed as the whole deformed shape of curved RC beam is shifted towards the direction of the spalling; in this case to the left side of the arc.In the case of less rigid soil (E soil = 10 MPa) the displacements are more pronounced for the asymmetrical case of spalling, while for more rigid soil (E soil = 30 MPa) the direct opposite holds true (Fig. 3).At the end of the parametric study the comparison between partially coupled and uncoupled approach is made.The results for critical time t cr of the curved RC beam for the length of spalling L spal = 600 cm and the depth of spalling d spal = 3 cm are given in Table 2.The difference between critical times for partially coupled and uncoupled model is rather high.For instance in case of less rigid soil and asymmetrical form of spalling the critical time for uncoupled model is almost two times bigger from the critical time for partially coupled model.This indicates that for proper assessment of the fire resistance of RC arcs accurate as possible temperature field is needed.

CONCLUSIONS
A novel partially coupled strain-based finite-element numerical model for assessing the fire resistance of curved RC beams exposed to concrete spalling was presented.The temperature field in concrete was determined with a coupled model of slow transient phenomena involving heat and mass transport and pore pressure increase.Furthermore, the strain-based non-linear beam finiteelement was employed in the mechanical analysis.The parametric study of a curved RC beam in fire indicates that adopting the partially-coupled or fully-coupled models is a must in order to adequately assess the structural safety of concrete structures exposed to fire, especially when properties of the concrete (HPC), thermal field and applied loads favour concrete spalling.
i cr D  , and transient strains of concrete, i tr D  : Fig. 1 Geometrical and material properties of the curved RC beam and the examples of spalling

Fig. 3
Fig. 3 Displacements in reference nodes A and B for spalling case d spal = 3 cm and two types of soil (a) E soil = 30 MPa and (b) E soil = 10 MPa

Table 1
Critical times t cr for different values of L spal for spalling case d spal = 3 cm

Table 2
Critical times t cr for partially coupled and uncoupled model (L spal = 600 cm, d spal = 3 cm)