Prediction of Behaviour of Cementitious CompositesUnder Fire Conditions

JIRI VALABrno University of TechnologyFaculty of Civil Engineering

Veverı 95, 602 00 BrnoCZECH REPUBLICvala.j@fce.vutbr.cz

ANNA KU CEROVABrno University of TechnologyFaculty of Civil Engineering

Veverı 95, 602 00 BrnoCZECH REPUBLIC

kucerova.a@fce.vutbr.cz

PETRA ROZEHNALOVABrno University of TechnologyFaculty of Civil Engineering

Veverı 95, 602 00 BrnoCZECH REPUBLIC

rozehnalova.p@fce.vutbr.cz

Abstract:Prediction of non-stationary behaviour of cementitious composites as crucial materials of concrete build-ing structures, at high temperatures, namely under fire conditions contains still a lot of difficulties, related toi) multiphysical analysis, dealing with the coupled heat and (liquid and vapour) mass transfer and related me-chanical effects, ii) open questions of existence and uniqueness of formal mathematical solutions, including theconvergence of numerical algorithms, iii) design and identification of suitable material characteristics, supportableby rather inexpensive experiments: the macroscopic mass, momentum and energy conservation equations need toexploit available information from micro-structural considerations. This paper discusses common traditional andadvanced computational modelling and simulation approaches and shows a possible practical compromise betweenthe model complexity and the reasonable possibility of identification of material characteristics of all componentsand their changes. An illustrative example confronts computational results from such a simplified model with ex-perimental ones, performed in the specialized laboratory at the Brno University of Technology.

Key–Words:Cementitious composites, modelling of heat and mass transfer, fire simulation.

1 Introduction

Problems concerning cementitious composites, es-pecially concrete both in its regular and more ad-vanced forms, as high-strength, stamped, (ultra-)high-performance, self-consolidating, pervious or vacuumconcrete, limecrete, shotcrete, etc., exposed to ele-vated and high temperatures are significant and wideranging. The increasing interest of designers of build-ing structures in this field in several last decades hasbeen driven by the exploitation of advanced materi-als, structures and technologies, reducing the exter-nal energy consumption, whose thermal, mechanical,etc. time-dependent behaviour cannot be predicted us-ing the simplified semi-empirical formulae from mostvalid European and national technical standards. Suchmodels as that of one-dimensional heat conductionwith (nearly) constant thermal conductivity and ca-pacity, give no practically relevant results for refrac-tory or phase change materials, thermal storage equip-ments, etc. – cf. [17]. Fortunately, in such casesthe correlation between the predicted and measuredquantities, e. g. of the annual energy consumption ofa building, can be available, to help to optimize thedesign of both new and reconstructed building struc-tures. This is not true for high temperatures leading topartial or total collapse of a structure, as those caused

by thermal radiation during a fire.

A lot of historical remarks and relevant referencesfor the analysis of structures exposed to fire can befound in [7]. The physical background for the evalua-tion of simultaneous temperature and moisture redis-tributions under such conditions is based on the bal-ance law of classical thermodynamics. However, it isnot trivial to supply a resulting system of evolutionby reasonable effective constitutive relations, valid atthe microstructural level, despite of the complicated(typically porous) material structures. Most classi-cal approaches to evaluate temperature and moistureredistributions rely on the (semi-)empirical relationsfrom [1]. The attempts to improve them can be seenin various directions, with still open problems and dif-ficulties everywhere. The physical and/or mathemati-cal homogenization approaches applied to a referencevolume element seem to be useful. The process ofcement hydration, distinguishing between anhydrouscement scale, cement-paste scale, mortar scale andmacro-scale, handled by specific physical and math-ematical formulations, with a posteriori scale bridg-ing applying least squares arguments, has been de-scribed in details, evaluating the hydration of particu-lar clinker phases, in [21]. However, the thermal gainsand losses from chemical reactions cannot be con-

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sidered as deterministic ones, moreover correspond-ing computations just in the case of dehydration mustsuffer from the lack of relevant data at some scales.The formal mathematical two-scale or similar homog-enization, introduced in [5], needs to remove assump-tions of periodicity using stochastic or abstract de-terministic homogenization structures; this leads tophysically non-transparent and mathematically verycomplicated formulations, whose applicability to theconstruction of practical algorithms for the analysisof engineering problems is not clear. From such pointof view, the most promising approach of the last yearsseems to be that of [13] and (in a substantially gener-alized version) of [12], replacing the proper homoge-nization by some arguments from the mixture theory;this will be the principal idea even in our followingconsiderations in this short paper.

2 Definitions of Function Spaces andNotation

Let us describe physical processes, which occur inconcrete during fire. Concrete is non-combustible ma-terial with low thermal conductivity. Although con-crete does not contribute to fire load of the structuressignificant changes occur in its structure during a fireexposure. Besides reduction of mechanical, deforma-tion and material properties also chemical composi-tion of concrete is varied during heating [2].

Concrete, as a porous material, contains a largeamount of pores, which can be filled fully (satu-rated concrete) or just partially with water. The wa-ter occurred in the pores is evaporable water andstarts to evaporate at early beginning of the fire. Thefirst changes of concrete structure arise at 105C asstated in [16], when chemically bounded water is re-leased from cement gel to the pores. Some smallmicro-cracks start to appear as the capillary porosityarises. The peak of the dehydration process is reachedaround 270C. The color of concrete is changed anda slight decrease of strength, modulus of elasticity andchanges in material properties like thermal conduc-tivity can be noted. Temperature of 300C is theextreme temperature beside which the concrete struc-ture is irreversibly damaged [15]. In range of 400C -600 C calcium hydroxide decomposes into calciumoxide plus water (rise of amount of free water) andtransition of and quartz, accompanied by increase inits volume, induces another creation of severe cracksin concrete.

Not only high temperature can affect adverselythe load bearing capacity of concrete structures. Si-multaneously with a temperature profile along a con-crete member should be investigated also the change

of mass of free water (mostly vapour) in concreteduring fire and distributions of pore pressure. Thepore pressure is one of the main reasons of concretespalling (Figure 1 and 2), which happened at the be-ginning of heating (10 - 30 minutes) and is acciden-tal. Small or grater (1 m wide and 1 m long) areasof concrete cover can be broken and cross section ofmember is reduced then. Furthermore in most casesthe reinforcement is exposed directly to the fire andthe member is heated faster, which can lead to loss ofloadbearing capacity.

3 Mathematical considerationsFollowing [12] (unlike [21]), for the quantification ofthe dehydration process we shall work with the hy-dration degreeΓ, a number between 0 and 1, as thepart of hydrated (chemically combined) watermw inits standard mass, constituted usually during the early-age treatment of a cementitious composite. Althoughthe evaluation ofΓ from a simple algebraic formula

Figure 1: Areal spalling of concrete.

Figure 2: Local spalling of concrete.

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is not available because it must take into account thechemicalaffinity and the fact that the accessibility ofwater for chemical reactions is controlled by the watercontentηw inside the pores under certain temperatureT , we shall applyΓ as a known function of such (orslightly transformed) variables.

The multiphase medium at the macroscopic levelcan be considered as the superposition of 4 phases:solid material, liquid water, water vapour and dryair, identified by their indicesε ∈ s, w, v, a.In the Lagrangian description of motion, following[24], the deformation tensorF s can be derived usingthe derivatives of displacements of particular pointswith respect to Cartesian coordinate systemx =(x1, x2, x3) in the 3-dimensional Euclidean space. Ifωε is a source corresponding to certain scalar quan-tity φε then the conservation of such quantity can beexpressed by [4], p. 4, and [10], p. 9, as

φε + (φεvεi ),i = ωε ; (1)

this formula contains the dot notation for the deriva-tive with respect to any positive timet, (. . .),i meansthe derivative with respect toxi wherei ∈ 1, 2, 3,vεi = uεi , with uεi referring to displacements re-lated to the initial geometrical configurationx0 =(x01, x02, x03) (for t = 0) and later alsoaεi = uεi ;the Einstein summation is applied toi and j from1, 2, 3 everywhere. For the brevity,φε will be more-over used instead ofφεηε whereηε(n, S) is the vol-ume fraction of the phaseε, a function of the mate-rial porosityn and the saturationS. ClearlydetF s =(1−n)/(1−n0) with n0 corresponding tox0. The sat-urationS is an experimentally identified function ofthe absolute temperatureT and of the capillary pres-surepc, needed later; the assumption of local thermalequilibrium yields the same values ofT for all phases.

In addition toT , we have 4 a priori unknownmaterial densitiesρε and 12 velocity componentsvεi .Assuming that vapour and dry air are perfect gases,we are able to evaluate their pressurespv(ρv, T ) andpa(ρa, T ) from the Clapeyron law; the capillary pres-sure is thenpc = pv + pa − pw, with the liquid waterpressurepw, or alternatively just withpc, as an addi-tional unknown variable.

In the deterioration of a composite structure 2 cru-cial quantities occur: the massmw of liquid waterlost from the skeleton and the vapour massmv causedby evaporation and desorption. The time evolution ofmassmw can be determined from the formally simpleformulamw = Γmw

0, with m0 related tot = 0. The

vapour fractionζ remains to be calculated from thesystem of balance equations of the type (1), suppliedby appropriate constitutive relations.

The mass balance works with

φε = ρε ,

ωs = −mw , ωw = mw − mv ,

ωv = mv , ωa = 0

in (1). No additional algebraic relations are necessary.Since all phases are considered as microscopi-

cally nonpolar, the angular momentum balance forcesonly the symmetry of the partial Cauchy stress tensorτ , i. e. τij = τji for such stress components. Theformulation of the linear momentum balance in 3 di-rection is more delicate. Introducingwε

i = ρεvεi and

choosing (for particulari)

φε = wεi ,

we can evaluate, using the Kronecker symbolδ, thetotal Cauchy stressσ in the formσij = τijδ

sε andfinally set

ωε = σεij,j + ρε(gi − aεi + tεi )

in (1) wheregi denotes the gravity accelerations andtεi the additional accelerations caused by interactionswith other phases, whose evaluation is possible fromthe Darcy law, as explained lower. The constitutiverelationships for the solid phase, e. g. those betweenτandus, vs, etc., need the multiplicative decompositioninto a finite numberm of matrix componentsF s =F s1 . . . F sm (elasticity, creep, damage, etc. ones) toexpressτ(F s1 . . . F sm, F s1 . . . F sm, . . .). Forε 6= s,introducing the dynamical viscosityµε and the per-meability matrixKε

ij , depending onρε again, we canformulate the Darcy law as

µερε(vεi − vsi ) = Kε

ij(ρε(gj − aεj + tεj)− pε,j) . (2)

The energy balance inserts

φε =1

2wεi v

εi + ρεκ

ε

with κε usuallydefined ascεT , using the thermal ca-pacitiescε, in general functions ofT andpc again, andsome internal heat fluxesqεi , and also

ωε = (σεij,j + qεi ),j + (gi − aεi + tεi ) +ε ,

s = −mwhw , w = mwhw − mvhv ,

v = mvhv , a = 0

into (1); two new characteristics here are the specificenthalpies of cement dehydrationhw and evaporationhv. The internal heat fluxesqεi come from the consti-tutive relation

qεi = −λεijT,j − ξεijp

c,j ; (3)

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the first (typically dominant) additive term corre-spondsto the well-known Fourier law of thermal con-duction, the second one to the Dufour effect, withsome material characteristicsλε

ij andξεij dependent onT andpc.

Similarly to (3), it is possible derive the diffusivefluxesrεi = ρε(v

εi − vsi ) with i ∈ v, w in the form

rεi = −ςεijT,j − γεijpc,j , (4)

due to the Fick law (the second additive term), re-specting the Soret effect (the first one), with some ma-terial characteristicsςεij andγεij dependent onT andpc; for more detailed analysis of Dufour and Soret ef-fects cf. [19].

Now we have 4 mass balance equations, 3×4=12momentum ones and 4 energy ones, in total 20 par-tial differential equations of evolution for 3 groups of

Figure 3: Fire simulation at Brno University of Tech-nology: laboratory setting.

Figure 4: Fire simulation at Brno University of Tech-nology: detail of real experiment.

variables:

R = (ρs, ρw, ρv, ρa) ,

V = (vs1, vs2, v

s3, v

w1 , v

w2 , v

w3 , v

v1 , v

v2 , v

v3 , v

a1 , v

a2 , v

a3),

T = (T, pc,mv, ζ) ,

supplied by appropriate initial and boundary condi-tions, e. g. for a priori known values of all variablesfor t = 0 of the Dirichlet, Cauchy, Neumann, Robin,etc. types, for local unit boundary outward normalsn(x) = (n1(x), n2(x), n3(x)) in particular

• σijnj = ti with imposed tractionsti,

• (ρa(vai − vsi ) + rai )ni = ra with imposed air

fluxesra,

• (ρw(vwi − vsi ) + rwi + ρv(v

vi − vsi ) + rvi )ni =

rw + rv + β(ρv) with imposed liquid water andvapour fluxesrw, rv and some mass exchangefunctionβ,

• (ρw(vwi − vsi )hv − λijT,j − ξεijp

c,j)ni = q +

α(T ) with imposed heat fluxes and some heatexchange functionα, e. g. by the Stefan -Boltzmann law, proportional toT 4.

This seems to be a correct and complete formulationfor the analysis of time development ofR, V andT .

4 Computational analysisUnfortunately, a lot of serious difficulties is hidden inthe above presented formulation, e. g. the still miss-ing “Millenium Prize” existence result on the solv-ability of Navier - Stokes equations (cf. the “mysteri-ously difficult problem” of [23], p. 257), the physicaland mathematical incompatibilities like [11], as wellthe absence of sufficiently robust, efficient and reli-able numerical algorithms based on the intuitive time-discretized computational scheme:

1. setR,V andT by the initial conditions fort = 0,

2. go to the next time step, preservingR, V andT ,

3. solve some linearized version of (1) with themass balance choice, evaluate and perform thecorrectionǫR,

4. solve some linearized version of (1) with the mo-mentum balance choice, evaluate and performthe correctionǫV ,

5. solve some linearized version of (1) with the en-ergy balance choice, evaluate and perform thecorrectionǫT ,

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6. if ǫR, ǫV andǫT aresufficiently small, return to3,

7. stop the computation if the final time is reached,otherwise return to 2.

Consequently all practical computational toolsmake use of strong simplifications. The rather generalapproach of [12] introduces the additive linearizedstrain decomposition instead of the multiplicative fi-nite strain one, ignores some less important terms, asthe kinetic energy (the first additive term inφε) inthe energy balance interpretation of (1), as well asthe Dufour and Soret effects in (3) and (4), and re-duces the number of variables, comparing (2) withthe differencesvεi − vsi with ε ∈ v, w, a from themomentum balance interpretation of (1), and presentsa computational scheme of the above sketched type,applying the Galerkin formulation and the finite ele-ment technique. Nevertheless, most engineering ap-proaches, as [3] or [18], endeavour to obtain a sys-tem of 2 equations of evolution forT andpc (or someequivalent quantity) only, pre-eliminating or neglect-ing all remaining ones, using arguments of variouslevels: from micro-mechanically motivated physicalconsiderations to formal simplification tricks.

The reliability of computational results dependson the quality and precision of input data, includingthe design and identification of all material character-istics. Even the reasonable setting of basic materialcharacteristics, just at elevated and high temperatures,for separate non-stationary heat transfer is not triv-ial; the relevant least-squares approach is discussed in[25]. Probably all authors take some of them from theliterature, not from their extensive original experimen-tal work; the variability of forms of such characteris-tics and generated values, accenting those from (2),including the formulae from [12], has been discussedin [7]. Nevertheless, the proper analysis of uncertaintyand significance of particular physical processes leadto even more complicated formulations, analyzed (formuch easier direct model problems) in [26] using thespectral stochastic finite element technique, or in [14]the Sobol sensitivity indices, in both cases togetherwith the Monte Carlo simulations.

Unlike most experiments in civil and material en-gineering in laboratories and in situ, nearly no relevantresults for advanced numerical simulations are avail-able from real unexpected fires, as the most dangerousconditions for building structures, crucial for the reli-able prediction of behaviour of concrete and similarstructures. Since also laboratory experiments with ce-mentitious composites under the conditions close toa real fire, as that documented on Fig. 1 and Fig. 2,performed at the Brno University of Technology, sim-ilar to the building structure considered in [22], are

expensive, producing incomplete and uncertain quan-titative data anyway, the reasonable goal of numeri-cal simulation is to test simple numerical models withacceptable correlation with real data, as the first stepfor the development of more advanced multi-physicalmodels.

5 ExampleLet us present rather simple mathematical modelbased on previous considerations. The model is com-patible with the slightly modified and revisited ap-proach of [3].

Let us considerx ∈ Ω, whereΩ is a simple do-main and represents the part of space occupied by con-crete. Further, letΓ be its boundary, which consist oftwo non-intersecting parts:ΓN andΓR and it holdsΓN ∪ ΓR = Γ. The partΓR represents piece of theboundary, which is exposed to the fire andΓN denotesthe part exposed to the atmosphere.

Model is two-dimensional, i.e.x = (x1, x2), andit neglects mechanical loads, strains and stresses, i.e.

usi = 0, vsi = 0, asi = 0, (5)

andτ = 0, σ = 0, F s = 0.

We consider only two phase - solid phase and freeevaporable water. Let us denote byρ = ρ(x, t) den-sity of the free water, i.e.

ρ = ρw + ρv,

and its velocity byv = (v1, v2).The mass balance equation for free water reads:

∂ρ

∂t+∇ · (ρ v) =

∂mw

∂tin Ω× (0,∞), (6)

wheremw = Γmw0

. Hydration degreeΓ is empiricalfunction specified for example in [8].

In the model, we suppose the concrete be anisotropic material, therefore all material characteris-tic are not represent by the matrix but by a function.For that reason let us denote byK = K(T, pc) thepermeability of concrete and byλ = λ(T ) thermalconductivity of concrete.

Diffusive flux r = ρ v we approximate only byFick law since Soret effect is appeared to be rathersmall [2], i.e.

r = −K

g∇pc,

in which g is gravity acceleration (included for thereasons of dimensionality).

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Since (5) is assumed, the mass balance for solidphaseis trivial.

Summing up the enthalpy balances for free waterand solid phase, we obtain

ρsCs∂T

∂t+∇ · q = (7)

Cwρ v · ∇T + hw∂mw

∂t− hv

∂ρ

∂tin Ω× (0,∞) .

The last term on right-hand side approximateshvmv.Heat fluxq is considered only in form of Fourier law

q = −λ∇T,

since Dufour effect is negligible [2].To keep the model simple we do not consider mo-

mentum balance equations at all.Now, we have three variablesρ, pc, T and only

two equations (6), (7). For that reason we add empiri-cal state equation

ρ = Φ(pc, T ).

FunctionΦ can be found in [6], p. 530.The model is completed by boundary and initial

conditions. The boundary conditions are:

−r · n = β(pc − p∞) onΓ× (0,∞) ,

−q ·n = α(T−T∞) onΓN×(0,∞) ,

−q ·n = α(T−Ten)+eσ(T 4−T 4

en) onΓR×(0,∞) .

By n = (n1, n2) we denote an outer unit normal,p∞andT∞ is pressure and temperature of the outer envi-ronment. Ten represents temperature caused by fire.We usedTen given by ISO curve [9].e denotes emis-sivity of concrete andσ is Stefan- Boltzmann con-stant.

Initial conditions are

pc(x, 0) = p0 for x ∈ Ω,

T (x, 0) = T0 for x ∈ Ω,

wherep0 andT0 are pressure and temperature fort =0.

Fig. 5, Fig. 6 and Fig. 7 are the outputs fromthe model described above. Solver combines the fi-nite element techniques together with the iterated timediscretization scheme (the numerical construction ofRothe sequences), implemented in the MATLAB en-vironment. Fig. 6 demonstrates the time developmentof temperature, Fig. 5 that of pressure and Fig. 7that of moisture content. The fire is considered asthe boundary thermal radiation on the left and upperedges of the rectangle. Surprisingly some phenom-ena observed in situ can be explained from the deeperanalysis of results of such seemingly simple calcula-tions; the deeper analysis of this preliminary results,directed just to the practical application to fire protec-tion in civil engineering, is in preparation.

6 Conclusion

Development of reasonable models and computa-tional algorithms for the prediction of thermal, me-chanical, etc. behaviour of cementitious compositesand whole building structures, is strongly required bythe applied research in civil engineering, which can-not be ignored, although the formal mathematical ver-ification of such models is not available and the prac-tical validation suffers from the lack of data from ob-servations in situ, as well as of large databases fromrelevant (very expensive) experiments. The first steps,both in the sufficiently general formulation of theproblem, containing most practical computational ap-proaches as special ones, as well as the methodologyof computational and experimental work, sketched inthis paper, should be a motivation for further extensiveresearch.

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Acknowledgements: The authors wish to thank tothe financial support from the research project FAST-S-14-2490 at Brno University of Technology.

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Figure 7: Time development of moisture contentρw+ρv [kg m−3].

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WSEAS TRANSACTIONS on HEAT and MASS TRANSFER Jiri Vala, Anna Kucerova, Petra Rozehnalova

E-ISSN: 2224-3461 8 Volume 10, 2015