Cellular Finite Beam Element for Nonlinear Analysis of Concrete...

Cellular Finite Beam Element for Nonlinear Analysis ofConcrete Structures under Fire

Fabio Biondini, M. ASCE1; and Andrea Nero2

Abstract: A novel approach to nonlinear finite-element analysis of concrete structures exposed to fire is presented. The proposed formu-lation refers to frame systems, but it can be extended to other types of structures. The main novelty of this formulation is the use of a specialclass of evolutionary algorithms, known as cellular automata, to describe the heat transfer process induced by fire and to create an effectivelink between the simulation of the thermal process and the structural analysis. The heat transfer process is reproduced by considering heatconduction, heat convection, and thermal radiation. The temperature effects on the structural performance are taken into account by means oftemperature-dependent thermal and mechanical properties of concrete and steel. In this way, the general criteria for nonlinear finite-elementanalysis of concrete structures are applied to formulate a cellular reinforced concrete beam element with temperature-dependent character-istics. The effectiveness and applicability in engineering practice of the proposed formulation is demonstrated through applications. Theresults prove the accuracy of the proposed procedure and show that, for statically indeterminate structures, fire safety needs to be evaluatedat the global level by taking into account the actual role played by the structural scheme. DOI: 10.1061/(ASCE)ST.1943-541X.0000307.© 2011 American Society of Civil Engineers.

CE Database subject headings: Concrete structures; Finite element method; Fire resistance; Heat transfer; Nonlinear analysis.

Introduction

In structural design, approximate methods are applied to assessthe structural performance of reinforced concrete cross sectionsor members under fire [Hertz 1985; European Committee forStandardization (CEN) 2004; American Concrete Institute(ACI)/Minerals, Metals and Materials Society (TMS) 2007]. Thisapproach can provide useful information at a preliminary designstage, and can be easily applied for statically determinate struc-tures. However, particularly for statically indeterminate systems,accurate fire simulations by means of time-variant nonlinear analy-sis procedures are often required to properly address the choicesinvolved in the design of new structures, and in the rehabilitationof the existing ones, to achieve the required level of fire perfor-mance and safety.

Several fire-dedicated computational tools based on finite-element or finite-difference methods have been proposed in theliterature, including computer programs such as FIRE-RC (Beckerand Bresler 1974), FIRES-T3 (Iding et al. 1977), CONFIRE(Forsén 1982), TASEF (Sterner and Wickström 1990), TEMP-CALC (Anderberg 1991), and SAFIR (Franssen et al. 2000).The standard computational approach is to perform a thermal fireanalysis and use the temperature maps in a separate finite-elementprogram to evaluate the temperature-dependent structural perfor-mance. Finite-element procedures have also been proposed toinvestigate the effects of fire-induced spalling that may occur for

low permeability concrete materials, such as high-strength concreteor shotcrete (Caner et al. 2005; Pichler et al. 2006; Kodur andDwaikat 2008).

The aim of this paper is to present a novel integrated approach tothermomechanical nonlinear finite-element analysis of concretestructures exposed to fire (Biondini and Nero 2006). The attentionis focused on normal strength concrete frame systems, but theproposed formulation can be specialized to other types of concretematerials and structures. The main novelty of this formulation is theuse of a special class of evolutionary algorithms, known as cellularautomata, to describe the heat transfer process induced by fire andto create an effective link between the simulation of the thermalprocess and the structural analysis. This link makes the proposedapproach more general than other procedures in which thermal andstructural analyses are uncoupled. In fact, although the thermalproperties of concrete are generally assumed to be independentfrom the stress state (CEN 2004), in some cases the occurrenceof stress-related damage phenomena, such as severe cracking,may modify the temperature distribution and lead to a couplingbetween heat transfer and structural response.

Recently, cellular automata have been successfully applied tosimulate the diffusion processes involved in durability analysisof concrete structures (Biondini et al. 2004). The algorithm pro-posed in this paper starts from similar criteria. However, its formu-lation has been extended to accurately reproduce not only theinternal diffusive flow associated with heat conduction, but alsothe external thermal flow due to heat convection and thermalradiation.

On the basis of this evolutionary model, temperature effectson the structural performance are evaluated by introducing temper-ature-dependent laws for both the thermal and mechanical proper-ties of the component materials, concrete and steel, and by takinginto account the additional deformative effects induced by heattransfer. In this way, the coupling between thermal process andstructural behavior leads to the cellular automata formulation ofa deteriorating reinforced concrete finite beam element for thetime-variant nonlinear structural analysis under fire.

1Dept. of Structural Engineering, Politecnico di Milano Piazza L. daVinci, 32 20133 Milan, Italy (corresponding author). E-mail: [email protected]

2Tecnomare S.p.AVia Enrico Caviglia, 11 20139 Milan, Italy. E-mail:[email protected]

Note. This manuscript was submitted on December 18, 2008; approvedon August 23, 2010; published online on August 25, 2010. Discussion per-iod open until October 1, 2011; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Structural Engineer-ing, Vol. 137, No. 5, May 1, 2011. ©ASCE, ISSN 0733-9445/2011/5-543–558/$25.00.

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MAY 2011 / 543

The effectiveness and applicability in engineering practice of theproposed methodology is demonstrated through applications(Biondini and Nero 2006, 2007). The obtained results prove theaccuracy of the proposed procedure and show that for staticallyindeterminate structures, fire safety needs to be evaluated at theglobal level by taking into account the actual role played by thestructural scheme.

Modeling of Heat Transfer

Heat transfer is the energy transmission process that involvesobjects having different temperatures. There are primarily threedistinct ways of heat transfer: heat conduction, heat convection,and thermal radiation. In fire structural engineering, all such trans-mission methods can play an important role (Buchanan 2001). Forthis reason, an accurate modeling of heat transfer must accountfor conduction into solid bodies and for convection and radiationbetween the solid body and the outer heating sources.

Heat Conduction

The conduction of heat in solids is usually described by the Fouriermodel, which, for isotropic media and in absence of internal sour-ces of heat, is represented by the following second order partialdifferential nonlinear equation (Incropera and De Witt 2002):

∇ðλ∇TÞ ¼ cρ∂T∂t ð1Þ

where T ¼ Tðx; tÞ = temperature at point x ¼ ðx; y; zÞ and time t;λ ¼ λðx; TÞ = thermal conductivity; c ¼ cðx;TÞ = specific heat;ρ ¼ ρðx; TÞ = material density; and ∇T ¼ gradT . If the thermalconductivity λ does not depend on position x, the previous differ-ential equation simplifies in the following form:

D∇2T ¼ ∂T∂t ð2Þ

where D ¼ Dðx;TÞ ¼ λ=ðcρÞ = thermal diffusivity; and ∇2 ¼∇ ·∇. If the thermal diffusivity D does not depend on temperatureT , the previous differential equation is linear. However, the ther-mal properties of concrete strongly depend on temperature, asshown in Fig. 1 (CEN 2004), where for the conductivity λ thelower limit is usually assumed. Therefore, the heat conductionprocess in reinforced concrete structures exposed to fire is gener-ally nonlinear.

The thermal properties shown in Fig. 1 refer to normal strengthconcrete. The effects of temperature on thermal properties of high-strength concrete are investigated in Kodur and Sultan (2003). Thethermal properties of steel are not considered in the proposedapproach because the effects of reinforcement are usually negli-gible in the heat transfer process.

Heat Convection and Thermal Radiation

The heat flow q ¼ qðxs; tÞ prescribed at the surface point xs of thesolid body may be decomposed into two contributions, qc ¼qcðxs; tÞ and qr ¼ qrðxs; tÞ, owing to heat convection and thermalradiation, respectively:

q ¼ qc þ qr ð3Þ

Heat convection is a thermal phenomenon in which heat flow isgenerated in consequence of macroscopic movements of a fluid

system. In general, the analytical description of a convection pro-cess is very complex. For this reason, heat convection is usuallymodeled by using empirical formulations. Newton’s law of cooling(Incropera and De Witt 2002) is adopted in this study:

qc ¼ hðTc � TÞ ð4Þ

where Tc ¼ TcðtÞ = temperature of the convective medium; T ¼Tðxs; tÞ = temperature of the solid surface; and h ¼ hðxs;TÞ =convection coefficient.

Thermal radiation is the transmission of heat flow from anobject in virtue of its temperature, without any mass transfer. Basedon the Stefan-Boltzmann law, the radiative heat flow between twogray surfaces can be evaluated as follows (Incropera and De Witt2002):

qr ¼ bðT4r � T4Þ ð5Þ

where Tr ¼ TrðtÞ = temperature of the radiative body; b ¼bðxs; tÞ ¼ σnεrφ = radiation coefficient; σn = Boltzmann constant;εr ¼ εrðxs; TÞ = emissivity coefficient; and φ ¼ φðxsÞ = viewfactor.

0.0

0.5

1.0

1.5

2.0

Temperature T [°C]

Co

nd

uct

ivit

y λ

[W/m

K] Upper limit

Lower limit

800

1000

1200

1400

1600

Temperature T [°C]

Sp

ecif

ic H

eat

c [J

/kg

K]

2000

2100

2200

2300

2400

0 200 400 600 800 1000 1200

0 200 400 600 800 1000 1200

0 200 400 600 800 1000 1200Temperature T [°C]

Den

sity

ρ [

kg/m

3]

Fig. 1. Thermal properties of concrete (moisture content of 1.5%)

544 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MAY 2011

Cellular Automata Solution of the Heat TransferEquations

Cellular automata represent simple mathematical idealizations ofphysical systems in which space and time are discrete, and physicalquantities take on a finite set of discrete values (Wolfram 1994). Inits basic form, a cellular automaton consists of a regular uniformgrid of sites or cells, theoretically having infinite extension, with adiscrete variable in each cell that can take on a finite number ofstates. The state at time t of the cellular automaton is then com-pletely defined by the values of the variable si ¼ siðtÞ at each celli. During time, cellular automata evolve in discrete time stepsaccording to a parallel state transition determined by a set of localrules: the variables skþ1

i ¼ siðtkþ1Þ at each site i at time tkþ1 areupdated synchronously based on the values of the variables sknin their “neighborhood” n at the preceding time instant tk .The neighborhood n of a cell i is typically taken to be the cellitself and a set of adjacent cells within a given radius r, orði� rÞ ≤ n ≤ ðiþ rÞ. Thus, the dynamics of a cellular automatoncan be formally represented as

skþ1i ¼ ϕðsknÞ; i� r ≤ n ≤ iþ r ð6Þ

where ϕ = evolutionary rule of the automaton. Clearly, a properchoice of the neighborhood plays a crucial role in determiningthe effectiveness of such rule. Fig. 2 shows an example of typicalneighborhoods for one- and two-dimensional cellular automata, butpatterns of higher complexity can also be proposed. Special atten-tion is due to neighborhoods along the sides of the finite grid, forwhich the evolutionary rule should be defined according to theimposed boundary conditions.

In general, any physical system satisfying differential equationsmay be modeled as a cellular automaton. In particular, it can beshown that the heat transfer process in d-dimensions (d ¼ 1, 2,3) can be effectively simulated by adopting a von Neumann neigh-borhood with radius r ¼ 1 and the following evolutionary rule(Biondini and Nero 2006):

Tkþ1i ¼ ϕi;0Tk

i þXdj¼1

ðϕ�i;jT

ki�1;j þ ϕþ

i;jTkiþ1;jÞ ð7Þ

where the discrete variable ski ¼ Tki ¼ Tðxi; tkÞ represents the tem-

perature of the cell i at time tk , and the value of the evolutionarycoefficients ϕi;0 and ϕ�

i;j depends on the nature of the neighborhoodcells (i� 1; j).

If the cells (i� 1; j) are associated with a fire source, representedby convective fluids and radiative bodies, the corresponding evolu-tionary coefficients ϕ�

i;j can be set as follows:

ϕ�i;j ¼ FiðB�

i;j þ R�i;jÞ ð8Þ

where the dimensionless factors Fi (Fourier number), B�i;j (Biot

number), and R�i;j (Radiation number), represent the contribution

of heat conduction, heat convection, and thermal radiation,respectively. For a cellular automaton defined by a grid dimen-sion Δx and a time step Δt, such contributions are related to thethermal properties of the cell i and to both the convective andradiative characteristics of the cells (i� 1; j), in the followingway:

Fi ¼λiΔt

ciρiΔx2¼ DiΔt

Δx2ð9Þ

B�i;j ¼

hi�1;jΔx

λið10Þ

R�i;j ¼

bi�1;jΔx

λiTk3i�1;j ð11Þ

Conversely, if the cells (i� 1; j) are not associated with a firesource, the evolutionary coefficients ϕ�

i;j simplify as follows:

ϕ�i;j ¼ F�

i;j ¼λ�i;jΔt

ciρiΔx2¼ D�

i;jΔt

Δx2ð12Þ

where the following nonlocal definition of equivalent thermalconductivities λ�

i;j of the cell i is introduced:

λ�i;j ¼ 2

�1λi

þ 1λi�1;j

��1ð13Þ

Finally, based on the continuity principle, the central evolution-ary coefficient ϕi;0 is obtained:

ϕi;0 ¼ 1��ψþ

Xdj¼1

ðϕ�i;j þ ϕþ

i;jÞ�

ð14Þ

where the coefficient ψ represents the eventual nonlinear supplyof the radiative thermal flow:

ψ ¼Xdj¼1

½Δϕ�i;j þΔϕþ

i;j� ð15Þ

with

Δϕ�i;j ¼ ΔR�

i;j ¼bi�1;jΔx

λiðTk3

i � Tk3i�1;jÞ ð16Þ

when the cells (i� 1; j) are associated with a radiative source,and Δϕ�

i;j ¼ 0 otherwise.Clearly, a proper balance between grid dimension Δx and time

step Δt must be achieved consistently with the required level ofaccuracy. To this aim, a suitable discretization in space and timemay be chosen with reference to the case of homogeneous andisotropic solid medium, which usually characterizes the problem

r = 1

von Neumann (r = 1)

Radius r = 2

Moore (r = 1)

Radius

Fig. 2. Typical neighborhoods for one- and two-dimensional cellularautomata


at the initial time t ¼ 0. In such cases, the symmetry conditionϕ�i;j ¼ ϕþ

i;j ¼ ϕi;1 ≡ ϕ1ðj ¼ 1;…; dÞ must hold for each cell i, withϕi0 ≡ ϕ0 ¼ ð1� 2dϕ1Þ. Therefore, after the initial value of theevolutionary coefficients ϕ0 and ϕ1 is properly fixed, a consistentvalue of both grid dimension Δx and time step Δt is chosento regulate the process according to a given reference value ofthe thermal diffusivity D:

D ¼ 1� ϕ0

2dΔx2

Δt¼ ϕ1

Δx2

Δtð17Þ

Suitable initial values of the evolutionary coefficients are ϕ0 ¼ 1=2and ϕ1 ¼ 1=ð4dÞ.

Validation of the Cellular Automata Solution

The cellular automata solution of the heat transfer equations hasbeen successfully validated for one-, two-, and three-dimensionalproblems (Nero 2006). Temperature maps for cross sectionsexposed to fire are available in CEN (2004) and ACI/TMS(2007). As an example, Fig. 3 shows the time-evolution of thetemperature maps for a 300 × 300 mm concrete cross sectionsubjected to a standardized fire ISO 834 (ISO 1975) along theboundary surface. The comparison presented in Fig. 3 betweenthe results of a cellular automata simulation and those provided

in CEN (2004) proves the high accuracy of the proposed numericalmodel.

Constitutive Laws of Materials

The effects of high temperatures on material behavior usuallyinvolve remarkable strength reductions and the onset of a temper-ature-induced strain εT in addition to the stress-induced strain εσ, orε ¼ εσ þ εT , for both concrete and steel.

Concrete Behavior

The instantaneous stress-induced strain εσ in compression ismodeled by using the Schneider’s stress-strain constitutive law(CEN 2004; Schneider 1988):

σf c;T

¼ nðεσ=εc1;TÞðn� 1Þ þ ðεσ=εc1;TÞn

; εσ ≤ εcu;T ð18Þ

where εc1;T = peak strain associated with the compression strengthf c;T at the temperature T; εcu;T = corresponding ultimate strain; andn ¼ 3. The thermal degradation of concrete properties depends onthe stress level (ACI/TMS 2007). This dependency is particularly

Fig. 3. Time-evolution of the temperature maps for a 300 × 300 mm concrete cross section subjected to a standardized fire ISO 834 (ISO 1975) alongthe boundary surface (thermal properties of Fig. 1, with h ¼ 25 kW=m2 and b ¼ 0:7σn), comparison between the grayscale results provided by acellular automata simulation (Δx ¼ 1:874 mm; Δt ¼ 0:01 min) and the contour maps given in CEN (2004)


relevant for structures loaded in the elastic range with a stressratio σ=f c;20 ≤ 0:3 (Schneider et al. 2008), where f c;20 is the com-pression strength at the initial temperature T ¼ 20°C. Althoughthe dependency of thermal degradation on the loading historycan easily be implemented in the proposed formulation accordingto CEN (2004), a constitutive model independent from thestress level is considered in this study. Figs. 4(a) and 4(b) showthe evolution with temperature of the compression stress-strainlaw for concrete with siliceous and calcareous aggregates, respec-tively. The values of compression strength f c;T and ultimate strainεcu;T are assumed consistently with CEN (2004). Peak strain is in-stead evaluated according to the following relationship (Franssen1987):

εc1;T ¼ 2:5 × 10�3 þ 4:1 × 10�6ðT � 20Þþ 5:5 × 10�9ðT � 20Þ2 ≤ 10�2 ð19Þ

In fact, the values of εc1;T provided in CEN (2004) are overesti-mated to implicitly include, in an approximated way, the transientstrain component of the temperature-induced strain εT in thestress-induced strain εσ [Fig. 4(c)]. For the stress-induced strainεσ in tension, the bilinear model shown in Fig. 4(d) is adopted(Rigberth 2000).

The temperature-induced strain εT is due to the contribution ofthree terms: thermal strain εth, transient strain εtr , and creep strainεcr , or εT ¼ εth þ εtr þ εcr. According to CEN (2004), the thermalstrain εth is given by the following relationships, for both siliceousaggregates:

εth ¼ �1:8 × 10�4 þ 9 × 10�6T þ 2:3 × 10�11T3;

for 20°C ≤ T ≤ 700°Cð20a Þ

εth ¼ 14 × 10�3; for 700°C < T ≤ 1200°C ð20b Þ

and calcareous aggregates:

εth ¼ �1:2 × 10�4 þ 6 × 10�6T þ 1:4 × 10�11T3;

for 20°C ≤ T ≤ 805°Cð21a Þ

εth ¼ 12 × 10�3; for 805°C < T ≤ 1200°C ð21b Þ

Transient strain εtr and creep strain εcr are evaluated according tothe formulations proposed by Anderberg and Thelandersson(1976). The transient strain εtr is considered to be proportionalto the thermal strain εth and to the ratio between actual stress σand compression strength f c;20 at the initial temperature T ¼ 20°C:

εtr ¼ �2:35σ

f c;20εth ð22Þ

The contribution Δεcr to creep strain εcr depends instead on theratio between actual stress σ and compression strength f c;T attemperature T > 20°C:

Δεcr ¼ �530 × 10�6 σf c;T

ffiffiffiffiffiffiΔt3

re3:04ðT�20Þ=1000 ð23Þ

where Δt (h) = reference time interval. Such contribution is par-ticularly important with sustained high temperatures and in theevaluation of the time-dependent deflections during the coolingphase (Hertz 1985).

(a) (b)

(c) (d)

Fig. 4. Temperature-dependent concrete behavior. Stress-strain relationships in compression for (a) siliceous aggregates and (b) calcareousaggregates; (c) peak and ultimate strains in compression; (d) strength and elastic modulus in tension


Steel Behavior

The instantaneous stress-induced strain εσ is modeled by using thestress-strain constitutive laws shown in Fig. 5, for both hot-rolledand cold-worked steel (CEN 2004).

The temperature-induced strain εT is attributable to the contri-bution of two terms: thermal strain εth, and creep strain εcr , orεT ¼ εth þ εcr . According to CEN (2004), the thermal strain εthis computed by using the following relationships:

εth ¼ �2:416 × 10�4 þ 1:2 × 10�5T þ 0:4 × 10�8T2;

for 20°C ≤ T ≤ 750°Cð24a Þ

εth ¼ 11 × 10�3; for 750°C < T ≤ 860°C ð24b Þ

εth ¼ �6:2 × 10�3 þ 2 × 10�5T; for 860°C < T ≤ 1200°C

ð24c Þ

The contribution of the creep strain εcr becomes significant whenvery sustained high temperatures occur (Rigberth 2000). Suchcontribution is implicitly included, in an approximate way, in thestress-strain constitutive laws shown in Fig. 5. The accuracy of thisapproach is generally adequate for structures with relatively lowpercentage of steel reinforcement (Anderberg 1988).

Time-Variant Structural Analysis under Fire

On the basis of general criteria for nonlinear finite-element analysisof concrete structures (Bontempi et al. 1995; Malerba 1998;Biondini 2004), in the following a finite-element formulation fortime-variant structural analysis of framed systems under fire isdeveloped (Biondini and Nero 2006).

Nonlinear Analysis of Deteriorating ReinforcedConcrete Cross Sections

The formulation assumes the linearity of concrete strain field andneglects shear failures and bond-slip of reinforcement (Fig. 6). Forthe cross section x, the vectors of the stress resultants r ¼ rðx; tÞ ¼½N Mz My�T (axial force N and bending moments Mz and My)and of the global strains e ¼ eðx; tÞ ¼ ½ ε0 χz χy�T (axial strainε0 and bending curvatures χz and χy) are then related, at each timeinstant t, as follows:

rðx; tÞ ¼ r0ðx; tÞ þ rTðx; tÞ ¼ Hðx; tÞeðx; tÞ ð25Þ

where H ¼ Hðx; tÞ = 3 × 3 stiffness matrix of the cross section;r0 ¼ r0ðx; tÞ ¼ ½N0 Mz0 My0�T = vector of the applied stressresultants; and rT ¼ rTðx; tÞ ¼ ½NT MzT MyT �T = vector ofthe stress resultants equivalent to the temperature-induced effects.

On the basis of the principle of virtual displacements, the stiff-ness matrix H ¼ Hðx; tÞ and the thermal stress vector rT ¼ rTðx; tÞare derived by integration over the area of the composite crosssection, or by assembling the contributions of both concrete andsteel as follows:

(a) (b)

Fig. 5. Temperature-dependent reinforcing steel behavior, stress-strain relationships for (a) hot-rolled steel and (b) cold-worked steel

Fig. 6. Main parameters, local reference system, and sign conventions for the finite beam element


Hðx; tÞ ¼ Hcðx; tÞ þHsðx; tÞ ð26a Þ

Hcðx; tÞ ¼ZAcðxÞ

Ecðx; y; z; tÞbðy; zÞTbðy; zÞdA ð26b Þ

Hsðx; tÞ ¼Xm

Esmðx; tÞbTmbmAsm ð26c Þ

rTðx; tÞ ¼ rcTðx; tÞ þ rsTðx; tÞ ð27a Þ

rcTðx; tÞ ¼ZAcðxÞ

Ecðx; y; z; tÞεcTðx; y; z; tÞbðy; zÞTdA ð27b Þ

rsTðx; tÞ ¼Xm

Esmðx; tÞεsTmðx; tÞbTmAsm ð27c Þ

where the symbol “m” refers to the mth reinforcement bar locatedat ðym; zmÞ; Ec ¼ Ecðx; y; z; tÞ and Esm ¼ Esmðx; tÞ = generalizedmoduli of the materials; and bðy; zÞ ¼ ½ 1 �y z� . The vectorsr and e have to be considered as total or incremental quantitiesdepending on the nature of the stiffness matrix H, which dependson the adopted formulation (i.e., secant or tangent) for the gener-alized moduli of the materials.

Formulation of a Deteriorating Reinforced ConcreteFinite Beam Element

The previous cross-sectional formulation is extended to define thecharacteristics of a reinforced concrete finite beam element fornonlinear analysis of framed structures under fire. The beamelement shown in Fig. 6 is considered. The torsional degrees offreedomare not included because torsion is assumed to be uncoupledfrom axial and bending deformations. The vector of displacementsu ¼ uðx; tÞ ¼ ½ uTa j uTb �T ¼ ½ u0 j v0 w0 �T and the vectorof generalized strains e ¼ eðx; tÞ ¼ ½ ε0 χz χy �T can be related tothe vector of nodal displacements s0 ¼ s0ðtÞ ¼ ½ sTa j sTb �T ¼½ u1 u2 j v1 φz1 w1 φy1 v2 φz2 w2 φy2 �T asfollows:

uðx; tÞ ¼ NðxÞs0ðtÞ ð28Þ

eðx; tÞ ¼ BðxÞs0ðtÞ ð29Þwhere

NðxÞ ¼ NaðxÞ 00 NbðxÞ

� �ð30Þ

BðxÞ ¼ ∂NaðxÞ=∂x 00 ∂2NbðxÞ=∂x2

� �ð31Þ

The axial Na ¼ NaðxÞ and bending Nb ¼ NbðxÞ displacementfunctions of a linear elastic beam element having uniformcross-sectional stiffness and loaded only at its ends are adopted.On the basis of this assumption, the 10 × 10 element stiffnessmatrix K0 ¼ K0ðtÞ and the nodal force vectors f 0p ¼ f 0pðtÞand f 0T ¼ f 0TðtÞ equivalent to the applied loads p ¼ pðx; tÞ ¼½ p0x p0y p0z �T and to the temperature-induced effects rT ¼rTðx; tÞ, respectively, are derived by applying the principle ofvirtual displacements and then evaluated at each time instant t byintegration over the length l of the beam:

K0ðtÞ ¼ K0MðtÞ þK0

GðtÞ

¼Z

l

0BðxÞTHðx; tÞBðxÞdxþ

Zl

0Nðx; tÞGðxÞTGðxÞdx ð32Þ

f 0pðtÞ ¼Z

l

0NðxÞTpðx; tÞdx ð33Þ

f 0TðtÞ ¼Z

l

0BðxÞTrTðx; tÞdx ð34Þ

where K0M ¼ K0

MðtÞ and K0G ¼ K0

GðtÞ = material and geometricalcontributions to the element stiffness matrix K0, respectively;N ¼ Nðx; tÞ = axial force; and G ¼ GðxÞ = corresponding com-patibility matrix:

GðxÞ ¼ 0 ∂NbðxÞ=∂x½ � ð35ÞThe previous quantities need to be evaluated at each time instant

by numerical integration. To this aim, without loss of generalityand for sake of simplicity, the integration can be effectively basedon the three-dimensional grid of the cellular automaton adoptedto describe the heat transfer process. To this regard, the gridsassociated with adjacent beam elements must be properly intercon-nected to assure the continuity of the thermal flow at theirboundaries.

Finally, by assembling the stiffness matrix K ¼ KðtÞ andthe vectors of the nodal forces f ¼ fðtÞ in a global referencesystem, the equilibrium of the whole structure can be expressedas follows:

KðtÞsðtÞ ¼ fðtÞ ð36Þ

where s ¼ sðtÞ = global vector of the nodal displacements. Asat the cross-sectional level, the vectors f and s have to be consideredas total or incremental quantities, depending on the nature ofthe global stiffness matrix K, which depends on the adoptedformulation (secant or tangent) for the cross-sectional stiffnessmatrix H.

Applications

The effectiveness of the proposed methodology is demonstratedby means of applications: two simply supported beams tested inlaboratory, a precast beam with elastic restraints, a continuousbeam under different fire scenarios, and a frame structure undercompartmented fire are analyzed. The time-variant nonlinear analy-ses are based on the previously introduced thermal and mechanicalcharacteristics of concrete with siliceous aggregates and hotrolled reinforcing steel (CEN 2004). The heat transfer processis described by three-dimensional cellular automata with griddimension Δx ¼ 13:3 mm and time step Δt ¼ 0:5 min.

Simply Supported Beams

The simply supported beams shown in Fig. 7(a) have been exposedto a standardized fire ASTM E119 (ASTM 1976) along their wholelength. The beams are subdivided into four finite elements. Thenumerical results obtained from a time-variant nonlinear analysisare compared with the experimental results of laboratory tests(Hertz 1985). For the beam with rectangular cross section, theresults of the numerical analysis are also compared withthe results provided by an approximated method formulated by


Bar 1

Bar 2

Bar 2

Bar 1

(a)

(b)

(c)

(d)

Fig. 7. (a) Simply supported beams with rectangular cross section (left side) and T cross section (right side). Comparison between numerical andexperimental results in terms of time evolution of (b) temperature in two reinforcing steel bars; (c) maximum deflection; and (d) resistant bendingmoment versus the maximum acting bending moment


Fig. 9. Precast beam. Time evolution of the structural response in terms of maximum deflection vmax, axial force N, and bending moment at the endsM� and middle span Mþ

Fig. 8. Precast beam: (a) material properties, geometrical dimensions of the structure; (b) characteristics of the cross sections and fire scenario


Hertz (Hertz 1985; CEN 2004). These comparisons prove theprimacy of the proposed procedure over the approximated methodand highlight its effectiveness in reproducing, with high accuracy,the evolution of both the thermal process [Fig. 7(b)] and the struc-tural response [Fig. 7(c)], and in predicting the structural lifetimeagainst fire collapse [Fig. 7(d)].

Precast Beam

The precast beam shown in Fig. 8 is exposed to a standardized fireISO 834 (ISO 1975) along its whole length. The role of the axialstiffness κ and rotational stiffness μ of the elastic restraints isinvestigated by considering three separate cases: (1) simplysupported beam (κ ¼ μ ¼ 0); (2) beam with elastic restraintswith stiffness κ ¼ 8:95 MN=m, and μ ¼ 3κ ¼ 26:85 MNm;(3) clamped beam (κ ¼ μ ¼ ∞). The beam is subdivided into fourfinite elements. The results of the time-variant nonlinear analysesare summarized in the diagrams of Figs. 9 and 10. The followingremarks can be made:• The beam deflection increases over time for all the investigated

cases, and its maximum value occurs for the simply supportedbeam during the whole fire event. The deflection of the beamwith elastic restraints is initially higher than the deflection ofthe clamped beam, whereas a reversed tendency arises after25–30 min (Fig. 9).

• The simply supported beam is axially unloaded. The beam withelastic restraints and the clamped beam are subjected to an axialforce which starts from the low initial value induced by the

nonlinear coupling with bending behavior and evolves duringtime according to the corresponding thermal effects (Fig. 9).

• The bending moment diagram of the simply supported beamis constant over time. The time evolution of the bending mo-ment diagram for the beam with elastic restraints and theclamped beam is characterized by remarkable redistribution ef-fects (Fig. 9).

• The most critical cross section is located at the ends for the beamwith elastic restraints, and at the middle span for both the simplysupported beam and the clamped beam (Fig. 10). For the threeinvestigated cases (1), (2), and (3), the first cross-sectionalfailure is reached after 45, 40, and 20 min, respectively.

• For the simply supported beam, the local failure of the middlecross section leads to the collapse of the structure. On thecontrary, the beam with elastic restraints and the clamped beamcan still carry the load after their first local failure, as shown bythe comparison between the time evolution of the axial force-bending moment resistance domains and the correspondingloading path in the critical cross sections (Fig. 10). The collapseof these beams is reached after 55 and 32.5 min, respectively.

Continuous Beam

The structural performance of the continuous beam in Fig. 11 isstudied with reference to the five scenarios of standardized fireISO 834 (ISO 1975) described in Fig. 12. The beam is subdivided,in each span, into three finite elements. The results of the time-variant nonlinear analyses are summarized in the diagrams ofFigs. 13 and 14. The following remarks can be made:

Fig. 10. Precast beam. Comparison between the time evolution of the axial force versus bending moment resistance domains (Δt ¼ 2:5 min) and thecorresponding loading paths


• The axial force starts from the low initial value induced by thenonlinear coupling with bending behavior and evolves duringtime in accordance with the corresponding evolution of the axialthermal effects. The maximum value, which clearly depends onthe amount of surface exposed to fire, is reached in eachscenario after approximately 70–80 min (Fig. 13).

• The time evolution of the bending moment diagram is charac-terized by remarkable redistribution effects, particularly loca-lized in the zones exposed to fire. In such zones, the bendingmoment initially tends to increase at the ends and to decreasewithin the spans, whereas a reversed tendency appears after thefirst 40–50 min (Fig. 13).

• For Scenarios 1, 3, and 4, the failure of the most critical crosssection is reached after 130, 180, and 280 min, respectively. ForScenarios 2 and 5, no failure appears during the first 280 min(Figs. 13 and 14).

• The failed cross section is located at the ends of the central spanfor Scenario 1, at approximately 15 cm from the left end of thecentral span for Scenario 3, and at the middle of the central spanfor Scenario 4 (Fig. 14).

• The comparison between the time evolution of the axial force-bending moment resistance domains and the correspondingloading path in the most critical cross sections shows that, as ex-pected, the margin of safety initially decreases owing to both thedecreasing structural resistance and increasing thermal-inducedstress contributions. A change in this tendency, however, arisesafter the first 70–80 min, when the reduction of the axial forcebegins. As a consequence, in some cross sections, the margin ofsafety may unexpectedly increase during time depending on thecorresponding rate of structural damage (Fig. 14).The previous comments clearly show that, for redundant

structures, the evaluation of fire performance cannot be afforded

FIRE FIRE

FIRE

20°C

20°C

20°C

20°C

20°C

SCENARIO 1

SCENARIO 3

SCENARIO 4

EXPOSEDEXPOSED NOT EXPOSED

EXPOSED

EXPOSEDEXPOSED

EXPOSED

20°C

20°C

20°C

20°C

20°C

20°C

NOT EXPOSED

NOT EXPOSED

NOT EXPOSED NOT EXPOSED

NOT EXPOSED

EXPOSEDEXPOSED

EXPOSED

EXPOSEDSCENARIO 520°C20°C

SCENARIO 220°C20°CNOT EXPOSED

Fig. 12. Fire scenarios for the continuous beam

Fig. 11. Continuous beam. Mechanical and thermal properties of the materials, main geometrical dimensions of the structure, distribution andcharacteristics of the cross sections


-2.00

-1.60

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Axi

al F

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e [

MN

]

Scenario 1Scenario 2Scenario 3Scenario 4Scenario 5

End 1-0.20

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M+ m

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m]


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ax [

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Span 2

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ax [

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Span 3

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End 4-0.20

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0 30 60 90 120 150 180 210 240 270 300 0 30 60 90 120 150 180 210 240 270 300

0 30 60 90 120 150 180 210 240 270 300 0 30 60 90 120 150 180 210 240 270 300

0 30 60 90 120 150 180 210 240 270 300 0 30 60 90 120 150 180 210 240 270 300

0 30 60 90 120 150 180 210 240 270 300 0 30 60 90 120 150 180 210 240 270 300

Time [min]

Ben

din

g M

om

ent

M− m

ax [

MN

m]


Fig. 13. Continuous beam. Time evolution of the axial force and of the maximum bending moment within the spans (Mþmax) and at the beam ends

(M�max) for the five investigated fire scenarios


SCENARIO 1

-0.5

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T=20°C

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SCENARIO 3

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SCENARIO 4

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-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0

Axial Force N [MN]

Ben

din

g M

om

ent

M [

MN

m]

T=20°C

First Failure

(b)(a)

Fig. 14. Continuous beam. Comparison of the time evolution of the axial force versus bending moment resistance domains (Δt ¼ 10 min)and the corresponding loading paths for Scenarios 1 (tfail ¼ 130 min), 3 (tfail ¼ 180 min), and 4 (tfail ¼ 280 min): (a) cross section at the leftend of the central span (Scenario 1), and at approximately 15 cm from such end (Scenarios 3 and 4); (b) cross section at the middle of the centralspan


by simply considering the evolution of the thermal-induced damageat the cross-sectional level, but needs to be developed at thestructural level by taking into account the actual role played bythe structural scheme in the time-variant stress redistribution pro-cess. With this regard, the fire performance is strongly dependenton both mechanical and thermal loading history, and the mostdamaged structural configuration at the end of fire exposition isgenerally not the most critical.

Frame Structure

The structural performance of the frame structure shown in Fig. 15is studied with reference to the three scenarios of standardized fireISO 834 (ISO 1975) described in Fig. 16. The beams subject to firefrom the top are exposed on the top face only. The beams subject tofire from the bottom are exposed on the bottom face and the lateralfaces. The columns under fire are exposed on the internal face andon the two lateral faces. Each beam and each column are subdi-vided into four finite elements. The results of the time-variant non-linear analysis are resumed for all scenarios in the diagrams ofFig. 17. Such diagrams show that the time evolution of the bendingmoments in the monitored cross sections A–F (Fig. 17) is charac-terized by remarkable redistribution effects, and the margin ofsafety strongly depends on the fire scenario. With this regard,for some cross sections, the compartmented fire in Scenarios 1and 2 is more critical than Scenario 3, characterized by a higherthermal load.

Conclusions

A novel approach to the time-variant nonlinear finite-elementanalysis of concrete structures exposed to fire has been presented.The main novelty of the proposed formulation is the use of cellularautomata to describe the heat conduction, convection, and radiationinduced by fire, and to create an effective link between thesimulation of the thermal process and the finite-element structuralanalysis. In this study, the thermal properties of concrete have beenassumed to be independent from the stress state, as proposed inCEN (2004). However, it is emphasized that the possibility ofaccounting for a thermomechanical coupling makes the proposedapproach more general than other procedures in which the finite-element analysis is based on temperature maps obtained from aseparate thermal fire analysis. In fact, in some cases, the occurrenceof stress-related damage phenomena, such as severe cracking, maymodify the temperature distribution and lead to a coupling betweenheat transfer and structural response.

The proposed methodology has been validated through applica-tions: two simply supported beams tested in laboratory, a precastbeam with elastic restraints, a continuous beam under different firescenarios, and a frame structure under compartmented fire, havebeen analyzed. The results have proven the accuracy of the pro-posed procedure in reproducing the evolution of both the thermalprocess and structural response, and in predicting the structural life-time against fire collapse. Moreover, the results clearly showedthat, for redundant structures, the evaluation of fire performance

3.0m

3.0m

55 kN/m

40 kN/m

A1A1 B1

A2A2 B2

C

C

C

C+ Dead Load

5.0m

50cm

30cm

Ø18

4cm

4cm

Ø22

50cm

30cm

Ø18

4cm

4cm

30cm

30cm

8x Ø22

4cm

4cm

SECTION A1 SECTION B1 SECTION C

50cm

30cm

Ø18

4cm

4cm

50cm

30cm

Ø18

4cm

4cm

SECTION A2 SECTION B2

(exposed)kW/m25

(not exposed)kW/m9

056.

430 MPa

3 MPa

30 MPa

2

2

=

=

=

===

h

h

b

f

f

f

n

sy

ct

c

σ

Fire: ISO-834

(1.00 m) (1.00 m)

(1.50 m) (1.50 m)

Fig. 15. Frame structure. Material properties, geometrical dimensions of the structure, and characteristics of the cross sections

FIRE

20°C FIRE

20°C FIRE

FIRE

A

B

C

F G

ED

F G

C D E B

A

Scenario 1 Scenario 2 Scenario 3

Fig. 16. Fire scenarios for the frame structure


cannot be afforded by simply considering the evolution of thethermal-induced damage at the cross-sectional level, but needsto be developed at the structural level by taking into account theactual role played by the structural scheme in the time-variant stressredistribution process.

With this regard, it is outlined that the margin of safety dependson the prescribed fire scenario, and that the most critical scenariomay not be associated with the maximum thermal load. Moreover,for a prescribed fire scenario, the structural performances are

strongly dependent on both mechanical and thermal loading his-tory, and the most damaged structural configuration at the endof fire exposition is generally not the most critical. As a conse-quence, a reliable assessment of fire safety can be carried out onlyby means of a time-variant nonlinear analysis that is able to followthe complex loading paths associated with thermal effects. All theseaspects can be consistently taken into account by the proposedmethodology, which has been proven to be an effective engineeringtool in fire analysis of concrete structures.

Acknowledgments

This study has been partially supported by research fundsPRIN2005 (prot. 2005082490), Italian Ministry of Universityand Research—Department of Structural Engineering, Politecnicodi Milano.

References

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-0.20

-0.12

-0.04

0.04

0.12

0.200 30 60 90 120 150 180

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din

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M [

MN

m]

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G

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0.200 30 60 90 120 150 180

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G

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0.200 30 60 90 120 150 180

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g M

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M [

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m]

A

B

C

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F

G

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Scenario 2

Scenario 3

Time [min]

Time [min]

Fig. 17. Frame structure. Time evolution of the bending moment in themonitored cross sections A–F (see Fig. 16) for the three investigatedfire scenarios


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