## Abstract

The research presented in this paper analytically examines the fire performance of flat plate buildings. The modeling parameters for the mechanical and thermal properties of materials are calibrated from relevant test data to minimize the uncertainties involved in analysis. The calibrated models are then adopted to perform a nonlinear finite element simulation on a flat plate building subjected to fire. The analysis examines the characteristics of slab deflection, in-plane deformation, membrane force, bending moment redistribution, and slab rotational deformation near the supporting columns. The numerical simulation enables the understanding of structural performance of flat plate under elevated temperature and, more importantly, identifies the high risk of punching failure at slab-column connections that may trigger large-scale failure in flat plate structures.

### Keywords

elevated temperature fire flat plate punching shear slab## 1 Introduction

Reinforced concrete flat plate is a type of structural system containing slabs with uniform thickness supported directly on columns without using beams. Flat plates are commonly used in buildings where relatively low gravity loads are applied. A major concern for flat plates is punching failure of slab in the vicinity columns due to high stress concentration. In 2004, a flat plate parking garage collapsed in Gretzenbach, Switzerland after fire was ignited for 90 min inside the garage (Ruiz et al. 2010). The collapse, causing the death of seven firefighters, was triggered by punching failure of slab around one column that immediately propagated over the structure. In spite of this incident, very limited information exists regarding the vulnerability to punching failure of flat plate structures subjected to fire. Moss et al. (2008) numerically studied the fire resistance of a reinforced concrete building consisting of flat plates and conjunctionally used perimeter moment frames. Elevated temperatures simulating fires with and without decay phase were applied below the slab in the entire lowest story. The finite element analyses focused primarily on the vertical deflection, horizontal expansion, bending moment, and in-plane membrane force in the heated slab. However, the risk of slab punching failure was not specifically studied.

A flat plate under severe fire condition experiences significant load redistribution. When the fire load is applied beneath the slab, columns restrain slab flexural deformation induced by thermal gradient, resulting in increased slab negative bending moment near the columns. Moreover, the slab top reinforcement remains relatively cool while bottom reinforcement heats up. The elevated temperature may cause the bottom bars at mid-span to yield at low stress. The load redistribution leads to high negative moment and large inelastic flexural deformation in slab near columns, which likely triggers to a punching failure of the flat plate structure. The study presented herein examines the punching failure potential of slab-column connections in flat plate buildings in the event of uncontrolled fire. For this purpose, nonlinear finite element analyses are performed on a prototype flat plate structure subjected to elevated temperature. The analysis determines the slab local force and deformation demands at columns as well as the degradation of punching shear strength. The research is limited to flat plates where the design of slabs is governed by gravity loads and the slabs are supported on square columns without using any shear reinforcement.

## 2 Description of Finite Element Modeling

### 2.1 General Model and Experiments for Model Calibration

The analyses are performed using Abaqus (Dassault Systèmes Simulia Corporation 2009), a general purpose finite element program. The span-to-thickness ratio of slab in a flat plate structure is normally larger than 30. Four-node thin shell elements with reduced integration are therefore used to simulate reinforced concrete slabs. The mesh size is approximately equal to slab thickness. Simpson’s rule is adopted for integration at a section to evaluate the slab internal forces. Thirteen integration points are defined at a section. The slab-column joint region is taken as rigid. Temperature variation through the thickness of slab is assumed to be piecewise quadratic. Perfect bonding between concrete and reinforcement is assumed. Concrete spalling under elevated temperature is not considered. Siliceous aggregate concrete, more vulnerable to fire-induced damage than carbonate concrete, is assumed for slabs.

To minimize uncertainty involved in analysis, several key modeling parameters for concrete are calibrated from relevant test data. The concrete tensile behavior that greatly affects slab flexural stiffness is calibrated from the tests of two isolated slab-column connections, Specimens B-2 and B-4, tested by Elstner and Hognestad (1956) in ambient temperature. These specimens are chosen because they had slab tensile reinforcement ratios (0.55 and 0.99 %) representative of design practice. The slab of each specimen had identical geometry (1,830 × 1,830 × 152 mm) and the center column stub was 254 mm square. The slab reinforcing bars were uniformly distributed. No shear reinforcement was used. During the tests, the specimens were placed up-side-down and simply supported along slab edges with the slab corners free to lift up. A downward vertical load simulating the effects of concentric gravity loading was applied at the center column stub.

Three furnace tests of two-way reinforced concrete slabs performed by Lim and Wade (2002) are used to examine the definitions of concrete thermal properties specified in ASCE (1992) Manual No. 78 (referred to as ASCE in the following discussions for simplicity) and EC2 (1992). All the specimens (D147, HD12, and 661) were 3.3 m wide, 4.3 m long, and 100 mm thick slabs constructed using siliceous concrete with identical compressive strength (36.6 MPa). The type and amount of flexural reinforcement varied among the specimens. The slabs were simply supported at the four edges in the tests. The corners of Specimens 661 and HD12 were unrestrained from vertical displacement, but the corners of Specimen D147 were clamped down. In each test, constant gravity load of 5.4 kPa was first applied first and then followed by a three-hour standard fire.

### 2.2 Mechanical and Thermal Properties of Reinforcement

The slab flexural reinforcement is modeled as one-dimensional material using *Rebar Layer* available in Abaqus. The reinforcement is assumed to be elastic-perfectly plastic under tension and compression. The modulus of elasticity at higher temperatures is defined following the recommendation by Harmathy (1993) for reinforcing steel. The mass density of reinforcing bars is taken as unchanged at high temperatures. The specific heat and conductivity of reinforcement are not included in the heat transfer analyses because the absence of reinforcing bars in the analyses has negligible effects on the temperature field in slabs (Wang 2004). The thermal expansion of reinforcement is defined based on Eurocode 3 (1995).

### 2.3 Mechanical Properties of Slab Concrete

Slab concrete under triaxial state of stresses is modeled using *Concrete Damaged Plasticity* (Lubliner et al. 1989; Lee and Fenves 1998). The five parameters needed to construct this model are assumed as temperature-independent: (1) Vermeer and De Borst (1984) suggested a non-associated plasticity for concrete where the dilation angle was defined with a value between 0° and 20°. Due to the relative large range of this suggested value, the appropriate definition of dilation angle for use in the analyses of flat plates is calibrated from experiments, as discussed later; (2) a flow potential eccentricity of 0.1 is used because the dilatancy of concrete is known to vanish at high confining pressure (Vermeer and De Borst 1984); (3) based on experimental evidence (Lubliner et al. 1989), the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian is defined as 0.667; (4) The viscosity parameter is taken as zero so that no viscoplastic regulation is enforced; (5) The ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress ranges narrowly from 1.10 to 1.16 (Lubliner et al. 1989). A value of 1.16 is chosen for this ratio.

*f*

_{ c }, after which the material experiences strain hardening. Once the strength

*f*

_{ c }(peak stress) is reached, strain softening is initiated.

For simplicity, a bilinear model is adopted for concrete tensile behavior. The failure stress of concrete in tension, *f*_{
t
}, represents the onset of micro-cracking. Beyond *f*_{
t
}, the stress–strain curve softens to reflect the formation of micro-cracks and further reaches zero stress at *ε*_{
tu
}. Concrete tensile strength vanishes when temperature reaches up to 600 °C. Note that there in no consensus regarding the definition of *f*_{
t
}. Concrete tensile strength was completely neglected in the numerical simulation performed by Lim et al. (2004) and Moss et al. (2008). Ghaffar (2005) define *f*_{
t
} as 0.3*f*_{
r
}, where *f*_{
r
} is the concrete modulus of rupture.

*f*

_{ t }= 0.2

*f*

_{ r }and

*ε*

_{ tu }= 10

*f*

_{ c }/

*E*

_{ c }are calibrated to ensure converged results and to avoid significantly underestimated stiffness of slabs. It is also found that the load-deformation response of slab-column connections is not sensitive to dilation angle, which is thus taken as 15°. Figure 2 compares for the two specimens the measured load-center deflection response with that predicted from analysis using the calibrated modeling parameters. Acceptable agreement is achieved between the simulation and test results. To further validate the finite element simulation, the predicted slab section rotation and rebar force are examined. Figure 3 shows the plan view of slab section rotation about Y-axis in Specimen B-2 when the largest slab center deflection in the test (23 mm) is reached. It is seen that fairly small difference exists in slab rotation at the sections outside the vicinity of column, indicating that slab deforms mainly through rigid body rotation due to the highly localized flexural deformation near the column caused by concrete cracking and reinforcement yielding. Although such a property was not specifically reported by Elstner and Hognestad (1956), it is consistent with the observation made from the similar tests by Guandalini et al. (2009).

### 2.4 Thermal Properties of Slab Concrete

Neglecting the evaporation of free water in concrete, the mass density of concrete is taken constant. Both ASCE (1992) and EC2 (1992) have specified specific heat, thermal conductivity, and thermal expansion for siliceous concrete. The two sources define specific heat and thermal conductivity similarly except that the specific heat defined in ASCE (1992) spikes between 400 to 600 °C due to the assumed presence of quartz and the thermal conductivity defined in EC2 (1992) reduces at a slightly lower rate with increased temperature. However, significant difference exists between ASCE (1992) and EC2 (1992) in defining concrete thermal expansion, a critical parameter for obtaining desirable analysis results of concrete slabs (Huang et al. 1999). To examine the appropriateness of these concrete thermal property models, finite element simulations are performed on the three specimens tested by Lim and Wade (2002). The time–temperature histories measured at slab bottom surface in the tests are applied at this location in the analyses. Temperatures are assumed identical over the slab and therefore vary only through the thickness of slab.

## 3 Analysis of a Flat Plate Building Exposed to Fire

### 3.1 Description of Prototype Structure

^{2}superimposed dead load and 2.39 kN/m

^{2}live load. The slabs are supported on 381 mm square columns without using shear capitals or drop panels. Grade 60 hot-rolled reinforcement (yield strength

*f*

_{ y }= 414 MPa) and normal weight concrete with a cylinder compressive strength of 27.6 MPa are used to construct the slabs and columns. The concrete is made of siliceous aggregates with 9.53 mm maximum size.

Fire is assumed to occur in the center bays on the third floor. The actual temperature in a fire compartment depends on several parameters such as fuel load and ventilation. Estimating the probable fire temperature is beyond the scope of the present study. Thus, the time–temperature history identical to that measured at slab bottom during the furnace testing of Specimen HD12 (Lim and Wade 2002) is applied in analysis at slab bottom surface while the slab top surface has room temperature at the beginning of heating. It is assumed that in the event of fire the prototype structure is subject to a uniformly distributed gravity load of 1.0D + 0.25L, where D and L are the design dead load and live load given previously.

### 3.2 Finite Element Model for Prototype Structure

### 3.3 Analysis Results and Discussions

#### 3.3.1 Results of Heat Transfer Analysis

#### 3.3.2 Slab Vertical Deflection

#### 3.3.3 Slab In-Plane Expansion and Membrane Forces

#### 3.3.4 Bending Moments and Rebar Forces in Slab

#### 3.3.5 Slab Section Rotation

#### 3.3.6 Risk of Punching Failure of Slab-Column Connections

*d*, where θ is the rotation of slab relative to column and

*d*is the effective depth of slab. The punching strength

*V*

_{ R }was then correlated with slab rotation at failure θ

_{ u }as

*b*

_{0}is the perimeter of shear critical section taken as

*d*/2 from the column face,

*f*

_{ c }is concrete compressive strength,

*d*

_{ g }is the maximum size of the aggregate, and

*d*

_{g0}is a reference aggregate size equal to 16 mm.

When Eq. (1) is employed in the present study, the concrete strength *f*_{
c
} is defined as a function of temperature according to EC2 (1992). Given that the temperature is not uniform over slab thickness, the temperature field determined from heat transfer analysis is used to estimate the concrete strength at different locations along the slab depth. It is assumed that the punching strength of a slab-column connection depends primarily on the depth of concrete underneath the inclined crack (*h*_{
c
} shown in Fig. 20) and the average concrete strength within *h*_{
c
} is used to define *f*_{
c
}. Different values of *h*_{
c
}, ranging from 0.1 *h* to 0.9 *h* (*h* is slab depth), are assumed because its exact value is difficult to determine. The 0.9 *h* is chosen herein as an upper bound because it approximates the effective depth of slab.

*V*

_{ R }with different assumed

*h*

_{ c }values. The solid lines in Fig. 21 give the total shear transferred from slab to the center column, which varies slightly over time due to load redistribution. It appears from Fig. 21 that punching shear failure is unlikely to occur within 30 min of fire loading because the shear capacity is always higher than demand. However, this figure also indicates that, at 60 min of heating, punching strength will be less than shear demand if the depth of concrete in compression at the incline crack is assumed less than 70 % of slab thickness (

*h*

_{ c }< 0.7

*h*). Tian et al. (2008) tested three slab-column connections with a slab tensile reinforcement ratio of 0.5 %. The specimens were subjected to different loading histories until failure. It was observed from these tests that the slab inclined crack had deeply extended toward the interface of column and slab bottom surface prior to the ultimate punching failure. The prototype structure analyzed in the present study has a similar slab reinforcement ratio (0.53 %) at columns. It can therefore be assumed that

*h*

_{ c }< 0.7

*h*and, accordingly, punching failure may occur earlier than 60 min of heating. Moreover, regardless of values assumed for

*h*

_{ c }, the shear capacity at 90 min is much less than the shear that the slab-column connection must carry. In brief, the finite element simulation and the use of Eq. (1) indicate that, even if the prototype building is designed with 90 min fire endurance, premature punching shear failure may occur due to the large local deformation of slab caused by combined gravity and thermal loading.

## 4 Conclusions

Nonlinear finite element simulation using calibrated concrete thermal and mechanical properties is carried out on a flat plate building. The analysis indicates that, because the thermal-induced slab rotational deformation is restrained by columns, the slab top reinforcement near the columns yields quickly at around 4 min of heating. Consequently, the heated slab experiences severe bending moment redistribution that changes positive bending moment at the mid-span due to the initial gravity loading into negative moment. However, very little change in bending moment is seen between 30 and 90 min of heating. Due to the restrained thermal expansion, membrane forces in the slab become compressive at all sections after only a short period of thermal loading. Moreover, no collapse mechanism associated with slab flexural yielding is generated at 90 min of fire exposure.

This study reveals serious concern for the risk of punching failure at the interior slab-column connections of a flat plate building subjected to fire. The analysis carried out on the prototype building indicates that, if the depth of concrete underneath the inclined shear crack is less than 70 % of the total slab depth, punching failure may have occurred at 60 min of heating. Even thought 90 % of the slab section is assumed as effective in resisting shear, the shear demand will be much higher than the shear capacity after 90 min of fire loading. It is therefore concluded that, prior to reaching its design fire resistance, the prototype building may have suffered a premature punching shear failure due to the large curvature of slab near column.

## Copyright information

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