A model for the numerical analysis of hygro-thermal and mechanical behaviour
of concrete at high temperatures including transport, reaction and deformation
processes is presented. For the description of transport and material behaviour the model
comprises the balance equations for mass and enthalpy and the constitutive equations
for concrete at high temperatures. The degradation of the cement minerals C-S-H and
portlandite is described by two chemical reactions. The consideration of thermo-chemical
damage in the classical non-local damage theory is discussed. Finally, the finite element
formulation for solving the coupled balance equations regarding the primary variables
displacement, temperature and relative humidity is presented and numerical results of a
3-D frame corner are discussed

Cramer, F.

Dinkler, D.

Kowalsky, U.

Ostermann, L.

English

UPCommons. Portal del coneixement obert de la UPC

1342IV International Conference on Computational Methods for Coupled Problems in Science and EngineeringCOUPLED PROBLEMS 2011M. Papadrakakis, E. Oñate and B. Schrefler (Eds)COUPLED ANALYSIS OF TRANSPORT PROCESSES ANDMECHANICAL BEHAVIOUR OF CONCRETE AT HIGHTEMPERATURESF. CRAMER∗, L. OSTERMANN∗, U. KOWALSKY∗ D. DINKLER∗∗Institute for Structural AnalysisTU Braunschweig, 38106 Braunschweig, Germanye-mail: f.cramer@tu-bs.de, www.statik.tu-bs.deKey words: Concrete, High Temperatures, Thermo-chemical Damage, Mechanical Be-haviour, Non-local Damage, Structural AnalysisAbstract. A model for the numerical analysis of hygro-thermal and mechanical be-haviour of concrete at high temperatures including transport, reaction and deformationprocesses is presented. For the description of transport and material behaviour the modelcomprises the balance equations for mass and enthalpy and the constitutive equationsfor concrete at high temperatures. The degradation of the cement minerals C-S-H andportlandite is described by two chemical reactions. The consideration of thermo-chemicaldamage in the classical non-local damage theory is discussed. Finally, the finite elementformulation for solving the coupled balance equations regarding the primary variablesdisplacement, temperature and relative humidity is presented and numerical results of a3-D frame corner are discussed.1 INTRODUCTIONSeveral serious fires in multi-story buildings and tunnels have been leading to increasedresearch interest in material behaviour under extremely high temperatures. Especially inbuildings made of concrete, the most frequently reported building material, fire may effectthe strength of construction and reduces the load-carrying capacity.Concrete is a composite material and consists of several phases. The solid phasecomprises of aggregates and cement paste. Here, the cement paste is composed ofCalcium-Silicate-Hydrates (C-S-H) and portlandite (CH) and the aggregates are madeup of siliceous sand and gravel. The liquid phase consists of capillary water and physi-cally bounded water, dry air and vapour provide the gas phase. High temperatures risingup to 1000 K in the case of fire lead to evaporation of capillaries and absorbed waterwith concurrent increase of the pore pressure in areas close to surfaces. Above the crit-ical temperature of water, Tcr = 647.3 K, water only exists in form of gas, i.e. vapour.11343F. Cramer, L. Ostermann, U. Kowalsky and D. DinklerAdditionally a chemical reaction in the cement paste leads to degradation of the cementminerals, vapour is released and the porosity of the cement stone increases. As the re-sult of evaporation and dehydration the rising pore pressure and thermal strains lead todamage which often manifests in the form of spalling.2 BALANCE EQUATIONFor the description of transport and reaction phenomena the balance equations ofvapour and water and the conservation of energy are taken into account. In the proposedmodel the effect of dry air is neglected, thence for the gas phase only a vapour balanceis necessary. The following balance equations are formulated for the volume average onmacroscopic level∂∂t(Sg n ρv) + ∇ · qv = sv↔w + sv,deh . (1)The first term of the vapour balance covers the local mass and depends on the porosityn, the saturation of the gas phase Sg and the density of vapour ρv. The source termssv↔w and sv,deh represent sources from phase changes between liquid and gas and thefreed vapour from the dehydration reaction. The water balance equation is formulatedequivalently with saturation of the liquid phase Sw, density of water ρw and the phasechange sw↔v. The second term in equation (1) represents the pressure driven flow. Here aDarcy law is used for water and vapour. With the assumption of an instant phase changeof water and vapour, the summation of water and vapour balances leads to the generalwater balance equation∂∂t(Sg n ρv) + ∇ · qv + ∂∂t(Sw n ρw) + ∇ · qw = sv,deh . (2)In context of modelling concrete under high temperatures the formulation of the conser-vation equation of energy leads to the conservation of heat which depends on the heatcapacities Ci of each phase, flows, the source terms for evaporation or condensation andthe heat of dehydration(∑Ciρi) ∂T∂t+ ∇ · qT = sv↔w · ∆Hv↔w + ∆Hdeh . (3)The first term of Equation (3), the local change of heat, depends on temperature T anddensity of the components related to the concrete volume ρi. The next term describesthe heat flow which is divided in a convective and a conductive flow following the Fourierlaw. The convective part of heat flow is described by the product of the respective flowvelocity, density and heat capacity. The heat of evaporation and condensation is theproduct of change of pore water and latent heat of water which depends on temperature.The last term gives the heat of chemical reaction and may be determined as the productof dehydration rate multiplied by the specific reaction heat. For coupling heat and mass21344F. Cramer, L. Ostermann, U. Kowalsky and D. Dinklertransfer with mechanical behaviour in addition the balance of linear momentum∇ · σ + ρc g = 0 (4)with Cauchy stress σ, density of concrete ρc and gravitation g is formulated. The densityof concrete ensues from the density of the components in consideration of their volumecontents.3 CONSTITUTIVE EQUATIONSIn addition to the balance equations constitutive models for transport, chemical re-actions and the mechanical behaviour are necessary which relate the primary variablesrelative humidity ϕ, temperature T and displacement vector u with other variables. Theyare assembled in the next section.3.1 Heat and mass transportThe transport processes in general may be divided into convection and diffusion. Theconvective transport is described by Darcy’s law and reads for the liquid phase asqw = ρw · k · krwµw· (−∇pw + ρw g) . (5)The water pressure pw which can be expressed in terms of the gas pressure and the capillarypressure pw = pg − pc is the driving force for the transport in the liquid phase, thencethe last term is significantly smaller. Further parameters are the intrinsic permeabilitytensor k, which is depending on the degree of dehydration and the grade of damage, therelative permeability krw and the dynamic viscosity of water µw which is a function oftemperature. Analogously the vapour flow is determined by respective parameters. Therelative permeabilities krw and krg depend only on saturation [1] and are given in Figure 1.As a result of evaporation of pore water and dehydration vapour is generated, what locallyleads to an increase of vapour pressure and therewith a higher vapour concentration. Theresulting diffusion is described by the diffusion model after Fickqv = −D · ∇ρv. (6)The diffusion coefficient tensor D incorporates the influence of pore structure and dependson the saturation, temperature and pressure as well.The saturation of water described by pc-Sw-relation is received from experimental re-sults [1]. Furthermore, saturation depends on temperature which influences the surfacetension of water. Above the critical temperature only vapour exists which is described bythe ideal gas law.31345F. Cramer, L. Ostermann, U. Kowalsky and D. Dinkler0.0 0.2 0.4 0.6 0.8 1.0Sw [−]0.00.20.40.60.81.0k rw,krg[−]Füllerkrwkr gFigure 1: Relative permeability of liquid and gas phase3.2 Chemical reactionBesides the transport dehydration is a determining process at high temperatures. Dur-ing dehydration water, which the concrete contains in form of pore water or chemicallybounded water in the crystals, is released and the cement paste is decomposed. Figure2 shows the degree of dehydration. Here, degradation of portlandite and C-S-H phases400 600 800 1000 1200T [K]0.00.20.40.60.81.0ξ dehyd[−]400 600 800 1000 1200T [K]0.100.110.120.130.14n[−]Figure 2: Degree of dehydration and porositymay be considered by two chemical reactions, which are leading to additional sources inthe heat balance and in the vapour balance [6]. Due to the decomposed cement paste theporosity n increases, see Figure 2, which leads to a significant change in permeability. Theenormous ascent of the curves in the temperature range 800-850 K is due to the reactionof the portlandite phase.After the dehydration process the micro structure is not destructed completely, there-fore the scalar chemical damage Dchem is defined with the linear relationship Dchem =0.5 · ξdehyd.41346F. Cramer, L. Ostermann, U. Kowalsky and D. Dinkler3.3 Mechanical behaviourThe stress state at a material point is bounded by a failure surface consisting of aRankine criterion for tensile principal stresses and of a Drucker-Prager criterion for com-pressive stresses. Due to the composite failure surface, Koiter’s flow rule for multi-surfaceplasticity [5] with the internal variables λR and λDP is used to compute the inelastic straintensor for stress states at the surface.Figure 3: Segmented failure surface for tension and compressionDue to the fact that in classical damage models the results depends on the discretisationparameters the non-local damage theory is used. The isotropic damage parameters DRand DDP are described with an exponential approach by Peerlings [7] and are affected bythe non-local variables λR for tension and λDP for compression.The total mechanical damage which is defined byDmech = 1 − (1 − DR) · (1 − DDP ) (7)allows an individual damage evolution for concrete in uniaxial and multiaxial stress states.3.4 Coupling of heat and mass transport with the mechanical behaviourFor the coupling of heat and mass transport with the mechanical behaviour it is essen-tial to consider the pore pressure. Often an effective stress tensorσ′ = σ + b ppore I (8)is employed, which is the sum of the Cauchy stress tensor and the product of effectivepore pressure time the Biot number b times the unit tenor [4]. Baroghel-Bouny et al.define the effective pore pressure as the of the sum of the pressures of liquid and gas51347F. Cramer, L. Ostermann, U. Kowalsky and D. Dinklerphase [1]. Against this Gawin et al. propose a definition of the effective pore pressureas the difference of gas pressure and the capillary pressure [2]. On the other hand porepressure can be considered by using an equivalent strainε = εel + εin + εhom . (9)The elastic strain εel and the inelastic strain εin result from external load or displacementconstraints, whereas the homogenised strain εhom is effected by pore pressure within thecement paste, thermal strain of aggregates and cement, and chemical and hygral shrinkageof the cement paste.Furthermore a combination of mechanical damage Dmech and thermo-chemical damageDchem for the description of the total damage D is necessary. Therefore the productapproachD = 1 − (1 − Dchem) · (1 − Dmech) (10)is used. The total damage D is included by the effective stress concept in the materialmodel.4 STRUCTURAL ANALYSISThe balance equations are discretised in space by a finite element formulation. Theprimary variables relative humidity, temperature and displacement are expressed by theirnodal values. The discretised form of the model equations reads as the non-linear coupledequations systemCij(x)∂x∂t+ Kij(x)x = fi(x). (11)The non-linear coefficient matrix C describes the local changes of variables with respectto time, K the fluxes. For the discretisation in time an implicit Euler scheme is used.The system of non-linear equations is solved by the Newton-Raphson procedure.For numerical analysis of the process of heating concrete, a 3-D frame corner with heatimpact at three surfaces is considered taking the symmetry of the structure into account.The dimension of the cross-section area is 30x30 cm2. The exposure of the structure tofire is modelled by means of the volume averaged flux boundary condition as describedin [3]. The initial conditions of temperature and relative humidity are T0 = 293.15 K andϕ0 = 0.6, the temperature of the fire T∞ follows the Standard-Fire characteristics.Figure 4 depicts results of numerical analysis after 10 min. for temperature, degree ofdehydration, relative humidity and vapour pressure. During the heat impact the temper-ature increases and diffuses into the interior. The highest temperature and therefore thegreatest damage occurs between the claimed surfaces. At the same time, water evaporatesrapidly which induces enormous pressure increase in the gas phase and causes a moistureclog which can be identified by local increase in relative humidity. The highest vapourpressure from evaporation and dehydration appears behind the edges near to the surfaces.Here spalling occurs first.61348F. Cramer, L. Ostermann, U. Kowalsky and D. DinklerT [K ] ξdehyd [−]φ[−] pv [Pa]Figure 4: Numerical analysis of a 3-D frame corner - distributions of temperature, degree of dehydration,relative humidity and vapour pressure after 10 min.5 CONCLUSIONSA mathematical model for the numerical analysis of hygro-thermal behaviour of con-crete structures subjected to high temperatures, including the developing of temperaturesand vapour pressure and the damage evolution is presented. The macroscopic balanceequations for the conservation of heat, mass of water and vapour and the linear momentumare formulated and important constitutive relations are proposed. The thermo-chemicaldamage in concrete under high temperatures is considered by non-local damage theory.Finally results of numerical analysis of a 3-D frame corner are discussed. The presentedmodel allows a more detailed understanding of material degradation of concrete at hightemperatures. The decrease of load-carrying capacity after fire events may be predictedmore realistically.71349F. Cramer, L. Ostermann, U. Kowalsky and D. DinklerREFERENCES[1] V. Baroghel-Bouny, M. Mainguy, T. Lassabatere and O. Coussy, Characterizationand identification of equilibrium and transfer moisture properties for ordinary andhigh-performance cementitious materials. Cement and Concrete Research (1999)29:1225–1238.[2] D. Gawin, C.E. Majorana and B.A. Schrefler, Numerical analysis of hygro-thermal be-haviour and damage of concrete at high temperature. Mechanics of cohesive-frictionalmaterials (1999) 4:37–74.[3] D. Gawin, F. Pesavento, and B.A. Schrefler, Modelling of hygro-thermal behaviour ofconcrete at high temperature with thermo-chemical and mechanical material degra-dation. Comp. Meth. Appl. Mech. Engrg. (2003) 192:1731–1771 .[4] W.G. Gray and B.A. Schrefler, Thermodynamic approach to effective stress inpartially saturated porous media. European Journal of Mechanics-A/Solids (2001)20:521–538.[5] W.T. Koiter, General theorems for elastic-plastic solids. Progress in Solid Mechanics(1960) 167–221.[6] L. Ostermann, and D. Dinkler, Numerical Simulation of the transport and the me-chanical behaviour of concrete at high temperatures. Proceedings EURO:TUN 2007,Vienna, Austria (2007).[7] R.H.J. Peerlings, R. De Borst, W.A.M. Brekelmans and M.G.D. Geers, Gradient-enhanced damage modelling of concrete fracture. Mechanics of Cohesive-FrictionalMaterials (1998) 3:323–342.8

Coupled analysis of transport processes and mechanical behaviour of concrete at high temperatures

https://upcommons.upc.edu/bitstream/2117/328229/1/Coupled-2011-121_Coupled%20analysis%20of%20transport.pdf

Coupled analysis of transport processes and mechanical behaviour of concrete at high temperatures

Abstract

Similar works

Full text

Available Versions

UPCommons. Portal del coneixement obert de la UPC