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Numerical Heat Transfer, Part A: Applications: AnInternational Journal of Computation and MethodologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/unht20

HEAT, MOISTURE TRANSPORT, AND INDUCED STRESSESIN POROUS MATERIALS UNDER RAPID HEATINGPradip Majumdar a & A. Marchertas aa Department of Mechanical Engineering , Northern Illinois University , DeKalb, Illinois,60115, USAPublished online: 12 Mar 2007.

To cite this article: Pradip Majumdar & A. Marchertas (1997) HEAT, MOISTURE TRANSPORT, AND INDUCED STRESSES IN POROUSMATERIALS UNDER RAPID HEATING, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation andMethodology, 32:2, 111-130

To link to this article: http://dx.doi.org/10.1080/10407789708913883

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HEAT. MOISTURE TRANSPORT, AND INDUCEDSTRESSES IN POROUS MATERIALS UNDERRAPID HEATING

Pradip Majumdar and A. MarchertasDepartment ofMechanical Engineering, Northern Illinois University, DeKalb,Illinois 60//5, USA

High temperalures and other severe thermal conditions cause releaseofsignificant amountsof water in porous materials, and induce pore pressure and temperature gradients underwhich water is transported through pores. A moJhernatiJ:aJ nwdel thoi simuloJes the coupledheoJ and mass transfer in heated porous media as weU as the resalling stress are discussed.A finite elemeai analysis for the souaion ofsuch a model is developed and used to stwJy thetemperature and pore pressure distribution and the resalling stresses. A comparison ofpredicted temperature distribution with experimental data is made, and sensitivity of thecoejJicientof shrinkage on the induced stress is investigated.

INTRODUCTION

Considerable interest is being shown in the problems of moisture propagationand resulting stress in heated porous media. Some investigators are concerned withthe structural integrity of the nuclear reactor containment structure typically madeof concrete when subjected to severe physical phenomena during accidents. Othersare concerned with failure of material due to the formation of cracks or explosivespalling of structural material during rapid heating. Many materials contain ex­tremely fine pores with large amounts of water in various forms, such as free orcapillary water and adsorbed and chemically bound water. High temperatures orother severe thermal boundary conditions may cause release and evaporation of asignificant amount of water and induce a pressure gradient under which water istransported toward the surface through pores. These induced pore pressures, thethermal stress due to the temperature gradient, and shrinkage stress due to therelease of water may cause internal stresses above maximum allowable tensilestress of the material and may cause cracks and failure of the structure. Cracks docreate additional passageways for leakage and facilitate additional moisture migra­tion and heat dissipation.

Bazant and Thonguthai U] studied the magnitude of pore pressure and loss ofmoisture caused by heating of concrete using finite element techniques. Dayan andGluekler [2] studied heat and moisture transport in surface heated concrete slabs,

Received 1 March 1997; accepted 15 April 1997.Part of the work described in this article was supported by the SHIMIZU Corporation, Nuclear

Reactor Division, through a subcontract of Argonne National Laboratory.Address correspondence to Dr. P. Majumdar, Department of Mechanical Engineering, Northern

Illinois University, DeKalb, IL 60115-2854, USA. E-mail: majumdar@ceet.niu.edu

Numerical Heat Transfer, Part A, 32:111-130,1997Copyright © 1997 Taylor & Francis

1040-7782/97 $12.00 + .00 111

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112 P. MAJUMDAR AND A. MARCHERTAS

NOMENCLATURE

a permeability R body force in r directiona, thermal moisture flux S surface0ww, aWT • QTW transport coefficients t time[BJ straindisplacement matrix T temperatureC specific isobaric heat capacity To temperature in environment

of concrete u displacement in r directionC. heal of sorption of free water V volumeC.. heat capacity of isobaric water w displacement in Z directionID] material matrix W free waterE modulus of elasticity w.. dehydration waterlJ) element force vector W.. change in dehydration waterIF) global force vector Z coordinates in axialdirectiong acceleration due to gravity Z body force in axial directionhT convectiveheat transfer Q coefficient of thermal

coefficient expansionJ moistureflux ~ coefficient of shrinkagek thermal conductivity d incremental changelk] element stiffness matrix ET thermal strain[KJ global stiffness matrix EW strain due to dehydrationK. mass transfer coefficient E porosityn outward normal to the surface p density of concreteN shape function P.. density of water

P pressure CT stressPo pressurein environment }; summation symbolq heat flux 9 circumferential directionr coordinatesin radial direction

'"relative humidity

in the context of structural integrity of nuclear reactor containment structures. Aheat and mass transfer model, based on an evaporation and recondensationmechanism, excluding the release of nonevaporable water, was solved numerically.Results were presented for water release, temperature, and pore pressure distribu­tion.

Kikuchi et aI. [3] reported results of heating tests of a thick reinforcedconcrete wall under high temperatures. Temperatures of the liner plate and heatedtest specimen were measured and presented. Wu and Ouyang [4] presented a finiteelement technique for computation of shrinkage stress in prestressed concretecontainment due to both internal and external restraints. They presented aconstitutive model for drying creep and shrinkage stress. A maximum shrinkagestress close to 90% of the tensile strength of concrete was observed in a very slowdrying process under normal temperature conditions. Numao and Mihashi [5]reported a relationship between drying shrinkage and water migration. Experimen­tal results of drying shrinkage, caused by water loss in a thin-wall cylindricalspecimen, were presented. Temperatures were maintained at constant values(below 90°C) to remove the influence of thermal expansion on the drying shrink­age. Chapman and England [6] conducted experiments to study the effect ofmoisture migration on shrinkage in a cylindrical concrete specimen heated to atemperature above 150°C. Temperature and pressure as well as transverse and

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HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS 113

longitudinal shrinkages were measured. Their results showed that shrinkage tendedto occur whenever the induced pore pressures were sufficient to cause moisturemigration and in the region where moisture is lost. Thermal and drying shrinkagestress analysis using the finite element method was carried out by Takiguchi andHotta [7] to predict the occurrence of cracks in a concrete specimen. Largestresses, which are almost equal to the tensile strength of the material, wereobserved in their low-temperature heating conditions. Majumdar et al. [8] used thefinite element formulation to estimate stresses due to thermal pore pressureloading and the shrinkage during moisture migration due to heating of concretewalls. The locations of impending cracks at different times were presented. Theeffects of different heating rates or time-dependent temperature boundary condi­tions on the induced stresses were further investigated [9].

The objective of this study is to simulate the coupled heat and mass transferin heated porous media as well as to estimate the resulting stress. A mathematicalmodel that simulates the coupled heat and mass transfer in heated porous media aswell as the resulting stress is discussed. A finite element analysis for the solution ofsuch a model is presented and used to study the temperature and pore pressuredistribution and the rate of moisture propagation through the material. Results arealso presented for internal stress caused by the presence of temperature gradients,pore pressure, and release of bound water. Sensitivity of parameters such as thecoefficient of shrinkage is also investigated.

HEAT AND MASS TRANSPORT ANALYSIS

When a high temperature is applied to such porous material, water pressurewithin its pores increases, and moisture movement is induced within the body.Treatment of the dependent variables within the concrete body is facilitated by theequation of state for water and conservation equations. The relationship betweentemperature, pore pressure, and moisture content within concrete is provided bytwo conservation equations of energy and mass as a function of time.

The calculated pressures, temperatures, and shrinkage due to water lossinduce strains inside concrete. Stress analysis formulations are provided that yieldinformation on resulting deformations and stresses. Analytical formulations areexpressed in axisymmetric geometry as shown in Figure 1.

State of Water in Concrete

The state of water in concrete is categorized as free or capillary water oradsorbed or chemically bound water. Figure 2 shows the schematic view of thestate of water in a pore system. Water in chemically bound form strongly adheresto surfaces and is considered an integral part of the solid. Adsorbed water covers apore surface in several monolayers. The amount of adsorbed water or the numberof monolayers of adsorbed water decreases with the decreased relative humidity inthe pore system. At the pore wall the innermost adsorbed water is considered to beless accessible for motion. The rest of the pore space is assumed to be filled by freewater, which is removable by heating. The capillary water is thus considered to bepart free water and part adsorbed water. A distinction is also made in terms of

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114 P. MAJUMDAR AND A. MARCHERTAS

Moisture now

--Sted liner

(a)

(b)

Figure 1. (a) Physical model of a slab. (b) Discretizationof the analytical model.

evaporable and nonevaporable water. Part of the evaporable water is the same asfree water, and the rest includes the outer layer of adsorbed water.

When the concrete structures are subjected to high temperatures or heating,a significant amount of bound water, also referred to as dehydration water, isreleased, and pressure is induced in the pores. This causes transport of capillarywater through the pores. Heat and mass transport within heated concrete struc­tures are therefore coupled and described by energy and mass conservationequations.

Figure 2. State of water in pore system.

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HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS 115

Heat and Mass Transfer Analysis

Moisture in either liquid or vapor form or in combination migrates throughthe pore structure as a result of diffusion, capillary action, pressure, and tempera­ture gradients. Moisture migration may be caused by a temperature gradient inaddition to a moisture concentration and/or a pressure gradient in vapor phases aswell as in liquid phases, which in turn, result from a temperature gradient in themedia. Such mutual interactions require multiple transport properties to model thephysical process mathematically. Luikov [10] proposed a heat and mass transfermodel that requires nine transport coefficients. Lei et al. [11] developed a modelbased on multiphase permeability and vapor phase diffusion, which require abso­lute permeability of porous media as well as relative permeability values for liquidwater and gas (air-vapor mixture). However, the mutual interaction betweenmoisture transfer and heat transfer makes it difficult to obtain these transportproperties experimentally for different porous structures. In this study, a simplifiedmodel is used based on the following assumptions. The diffusion mass flux due to aconcentration gradient is given by Fick's law, whereas the mass flux due to capillaryaction and pressure gradient is given by Darcy's equation. In the liquid phase, massflux is caused primarily by capillary forces, and in the gas phase, mass flux is causedprimarily by diffusion when the pressure gradient is low. However, even at lowpressure gradients, the order of magnitude of diffusion of water vapor in the airbecomes small compared to that given by capillary action for smaller pore sizes.Also, for smaller pore sizes, a few monolayers of adsorbed liquid water almost fillup the pore space. An effective permeability coefficient then could be used toexpress moisture flux due to capillary forces and pressure gradient as expressed byDarcy's equation without making any distinction between the liquid and vaporphase migration.

With such understanding and assumptions, the mass transport analysis in thisstudy is simplified in comparison with the conventional permeability calculations. Ittakes into account low porosity of concrete and thus neglects the effect of diffusionof moisture in the air within concrete. This assumption appears to be good for thistype of material. It also has two other major advantages of practical value. First, itmakes the analytical formulation considerably simpler than it would be otherwise,and second, the required material properties for the implementation of the analysisbecome tractable. Determination of the permeability coefficient as an effectivevalue simplifies the experimental collection of data.

Transfer Rate Equations

The flux of moisture in the porous media due the gradient of free waterconcentration and gradient of temperature is expressed as

J = -aww grad W - aW T grad T (1)

As free water content W is a function of pore pressure p and temperature T, asgiven by the equation of state [W = W(p/g, n], the gradient of which may be

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116

written as

P. MAJUMDAR AND A. MARCHERTAS

law awgradW= --gradp + -gradT

g sp or

Substituting Eq. (2) into Eq. (1) and upon further simplification, we get the transferrate equation for moisture as

aJ= --gradp-a gradTg t

where a = aww aw/ap, defined as the coefficient of permeability, and at =a ww aW/aT + aWT'

Heat Transfer Rate Equation

Heat flux due to a temperature gradient (according to the Fourier law ofconduction) and moisture gradient (due to Dufour effect) is expressed as

q = -aTW grad W - K grad T

Neglecting the Dufour effect, the heat transfer rate equation is given by

q = -KgradT

(4)

(5)

The conservation of mass and energy give the following set of coupled governingequations:

Governing Equations

The conservation of mass is represented by

awat

a~-divJ + -­at

(6)

The conversation of energy is represented by

or awPc- = c - + C JgradT- divqat a at W

Boundary Conditions

The boundary conditions for heat and moisture transfer at the surface are

n . J = Ks(p - Po)

n . q = hT(T - To) + CwK,(p - Po)

(8)

(9)

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HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS 117

where n is the unit outward normal at the surface. These boundary conditions maybe classified, under limiting cases, as follows:

• Moisture/pressureSealed case: no moisture flux at the boundary, i.e., J = 0 at the surface.Unsealed case: perfect moisture transfer at the boundary, i.e., P = Po.

• TemperaturePerfect heat transfer from the surface, i.e., T = To.Time-dependent specified temperature, i.e., T = Tit).

Equation of State

Governing Eqs. (1)-(4) are complemented by the equation of state relatingfree water contents W, pressure p, and temperature T as shown in Figure 3, andbound water content of concrete as shown in Figure 4. For analytical purposes thetreatment for estimating free water content was subdivided into three parts,depending on the magnitude of water pressure p with respect to the saturationpressure Ps' or .p = pips' Estimation of free water content is based on assumingthree regions: unsaturated, saturated, and transition. In the unsaturated region (for.p ~ 0.96), the state of water is given in the form of sorption isotherms, whichdepend on the type of concrete used.

In the saturated region (for .p ~ 1.04) the equation of state is based onthermodynamic properties of water as given in the steam tables. The amount of

0.4 -,-------------,

z;'"~ 0.3

c

1'1..()

- 0.2:;;;;~....It

0.1

0.0 1.0

RBlalive Vepor Pre..ura, P/Ps(T)

2.0

Figure 3. Sorption isotherms.

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118 P. MAJUMDAR AND A. MARCHERTAS

1000

'"I of

~IOO.a....e.."-.5..g' 10...cU...!:.!i..a:

.1 +--'----r-----,-------,----I

o 100

rempe,etu,e, C

200Figure 4. Dependence of permeability ontemperature and humidity.

(10)

water is given by

available pore space eW=-------

specific volume v

The pore space available for such water is estimated by taking into account theincreases in pore space due to elastic volume expansion, decrease in adsorbablewater, and/or partial dehydration.

Due to the unreliable and poorly reproduced data from references, Bazantand Thonguthai [1] represented the transition region (for 0.46 S cP s 1.04) artifi­cially by a linear straight line between two limits with a lower limit given byunsaturated region expression and an upper limit given by saturation regionexpression. Inflection points at the two limits of these assumptions may causecomputation difficulty or instability. In order to avoid such computational difficul­ties or instabilities during the run of the program, a more accurate and smoothertransition region is developed using a so-called S-shape equation, which matchesthe coordinates and slopes of the transition limits at cPt = 0.96 and cPz = 1.04 [9].

Permeability Coefficient

The permeability coefficient used in the mathematical model needs to beobtained experimentally for the concrete. One such correlation for permeability isgiven by Bazant [1], obtained by fitting experimental data for concrete:

(11)

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HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS 119

and

(12)

where ao is the permeability at reference temperature To = 25°C and fl' f2' and f3are empirical functions. Figure 5 shows a graphical representation of the coeffi­cient of permeability used in this study.

FINITE ELEMENT FORMULATION

The governing differential equations, Eqs. (1-4), when expressed in terms ofthe cylindrical coordinate system are given as follows:

1[ a (a ap)] 1[ a ( aT)] awap ow or aWd (13)-; ar ""iTa; + -; aT alTa; - ap at - aT at + ---at = 0

1 [ a (aT)] sw s» (aw ) et al¥.t- - kr- + C - - + C - - pC - + C --T aT ar a ap at a aT at d at

_ C (~ ap + a aT) aT = 0W g ar 1 aT ar (14)

The functions of p, W, and T, which makes Eqs. (13) and (14) nonlinear, are put

0.11 -,--------------,

0.04

0.07

0.08

0.05

0.03

0.08

0.01

0.09

0.10

0.02

o 100 200 300 400 500 800 700 800 900

Tempe'.lu,e C

Figure S. Release of bound water.

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120 P. MAJUMDAR AND A. MARCHERTAS

into the form of coefficients and expressed as

1[ a (a ap)] 1[ a (aT)] sp er- - -r- + - - a r- + A - + A - + A = 0r ar g ar r ar I ar 1 at 2 at 3

l[a( aT)] ap et et- - kr- + A - + A - + A + A - = 0r ar ar 4 at 5 at 6 7 ar

where

(15)

(16)

awA =C--pC

5 a aT

awA =--

I ap

awA =C-

4 a ap

awA =--

2 aTau-;,

A -­3 - at

(17)

A = -C (~ ap + a aT)7 W g ar 1 ar

By making use of the Galerkin-integration procedure, the above governing differ­ential equations are transformed into the following matrix equations:

and

where {pI and {T} are column matrices of nodal values; {F I } and {F2} are columnmatrices; and [QI]' [Q2]"'" and [Q7] are square matrices. Their components forindividual elements are the volume integral expressions:

Q1 =AI!.[Nf[N]dVv

Q = _~!. (a[Nf a[N] + a[Nf a[N]) dV -K,3 g V ar sr az az

Q4 = -al!. (a[Nf a[N] + a[Nf a[N]) dV (20)v ar ar az az

Q6 =As!.[Nf[N]dVv

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and

HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS

T [u(ap ap) (aT aT)]F2 = A 6f [N] dV + C; - - + - + u1 - + - + hTTov g ar az ar az

121

(21)

For the quadrilateral element, the shape functions [NY in terms of the localcoordinates ~ and 71 have the following form:

(22)

The Crank-Nicolson type implicit algorithm is employed to solve the matrixdifferential equations, Eqs. (18) and (19). Time derivatives are approximated usinga central difference scheme resulting in a set of quasi-linear algebraic equations.

STRESS ANALYSIS

In this section, analysis for computing stresses resulting from various effectscaused by heating of the concrete is presented. Since heating of concrete causesexpansion, pore pressure, and shrinkage resulting from the release of bound water,the stresses resulting from these three effects are computed together, and the totaleffect is observed.

Stress analysis formulations are superimposed over existing heat and masstransfer solutions, so that the calculated nodal temperatures and pore pressurevalues are used as input to perform stress analysis. A detailed description of thestress model is given in our previous paper [9].

The analysis of mass transport in concrete displayed a very large volumestrain due to the loss of bound water [9]. In order to assess the relative effect ofthis component of strain, as compared to that due to pore pressure or temperature,an elastic analysis was undertaken. This analysis disclosed that under conventionalthermal gradients, the loss of moisture affecting concrete cracking was significantlylarger than that due to thermal effects. The effect of pore pressure was even lesssignificant. Even though the actual material behavior is nonlinear, and a nonlinearconstitutive model is required to predict stress-strain behavior accurately, suchelastic analysis yields relative merits of the individual components qualitatively.

Next, a brief description of the finite element model for stress is given. Usingthe principle of minimum potential energy, we have the global system of discretizedequations:

[K]{U} - {F} = {O} (23)

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122 P. MAJUMDAR AND A. MARCHERTAS

The contribution to the global stiffness matrix [K] and the force vector {F} byeach element is given by

[k] = f [Bf[D][B] dVv

(24)

(25)

where the individual component terms of the force vector {f} correspond to thechanges in temperature, moisture content, body force, and pressure force. Theforce due to pressure is calculated by summing contributions from all four sides ofan element as shown in Figure 6. The average pressure of an element is expressedas

i = 1,2,3,4 (26)

where P; are the pressures at the nodes. Pressure is converted to nodal forces bysumming contributions from all four sides. Along one representative side ij, thecorresponding rand z components of force are

(2r; + rj)P,

= 7TLij (2ri + r)Pz

3 (r; + 2rj)P,

(ri + 2r)Pz

(2ri + rj)(zj - z)

7TPav (2ri + rj)(ri - rj)=--

3 (ri + 2rj)(zj - z)

tr, + 2rj)(ri - r)

(27)

This contribution is summed for all four sides. The remaining symbols in theforce vector of Eq. (25) are the rand Z components of the body force (R, Z),which are incorporated in the developed computer' program. However, for theproblem under consideration, the body force component is neglected.

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HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS 123

For axisymmetric geometry, the stress and strain components are {uV =[u,uzu,zOO] and {"y = [e, Ez Erz Eo]' The thermal strain vector {Eh = a~T[l 1 01] and the change in nodal temperature values (~T) are utilized to formulate theelement volumetric thermal strain components. Similarly, the strain vector result­ing from the change in volume because of moisture loss is {dT = Ew[1 1 0 1].

The strain resulting from dehydration Ew is estimated as the change in thevolume of concrete per unit volume of concrete:

(28)

where f3 is defined as the coefficient of shrinkage, representing the ratio of thechange in concrete volume to the volume of water released by dehydration. Also, E

is the porosity, Ww is the change in adsorbed or dehydration water used in the masstransport calculation, and Pw is the density of water as a function of temperature.

RESULTS AND DISCUSSION

The input parameters for the problem under consideration are given in Table1. A parametric study was made to conduct an analysis of mesh size refinement anderrors. Since the pore pressure moves through the medium with time, a uniformmesh size distribution was assumed. The number of elements was varied from 100to 600, and results for pore pressure distribution at different times are shown inFigure 7. Results show convergence of pore pressure distribution with an increasein the number of elements but at a slow rate. Even though the number of elementswas increased by an order of 100, the number of elements in the narrow band ofthe pore pressure distribution was increased by a few, causing this slow rate ofconvergence. Percent relative error for peak pore pressure is estimated andpresented in Table 2 and Figure 8. Results show improvement in the convergedvalue of peak pore pressure for mesh sizes of 400 or higher. Percent relative errorshowed a faster convergence for longer time values and a slower rate withoscillation at shorter time values.

The strain and stress due to the loss of moisture materialize because of thedifference of volume change between the adjacent elements. As water migratesthrough the specimen, it undergoes nonuniform shrinkage, which induces shrink-

L. Figure 6. Schematic for pressure distri­bution in an element.

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124 P. MAJUMDAR AND A. MARCHERTAS

Table I. Input parameters

Parameter

Age of concreteInitial relative humidity of concreteRelative humidity of environmentInitial temperatureThickness of slab or cylinderSaturation watercontentMaximum dehydration waterMassof cement per unit concrete volumeDensity of concreteWaterlcement ratioVolume of specimenInnerradiusThermal conductivityPermeability

Value

40 days0.950.7020'C2m100 kg/m3

0.1 g/cm3

300kg/m3

2400kg/m3

0.5210m3

lOoom1.674 Jim s·C10- 11 mls

age stresses due to the internal restraints. If a uniform contraction were allowed toexist in all elements, no strain or stress would result. This component of strain isassociated with the loss of bound water and migration of free water in concrete. Amaterial property of concrete, the coefficient of shrinkage designed by f3, must bedetermined experimentally before this strain can be evaluated. A rough estimate ofthis coefficient was made from some experiments by Chapman and England [6],resulting in f3 = 0.06. For this study, results are computed assuming the value ofthe coefficient of shrinkage to be in the range between 0.05 and 0.08.

Figure 9 shows a comparison of temperatures as determined in a tempera­ture-induced moisture propagation through a specimen of a concrete wall and theanalytical predictions. This figure shows a comparison of predicted temperaturedistributions at different locations in the media with experimental data reproducedfrom the study made by Kikuchi et al. [3]. The specimen tested by Kikuchi et al.had a crack line measuring 3 mm in width. The heating surface was covered with athick steel plate and was heated up to 800°C. Near the surface, results showed goodagreement except a little temperature rise in the experiment in the area of cracksaround 100°C. However, greater discrepancy is noticed at a location of 20 em andhigher that could be caused by additional cracks formed due to induced stresses.While some discrepancies may be explained by assuming nonuniform propertiesthroughout the concrete, some rather abrupt' changes cannot. A fair amount ofthese, however, may be attributed to material discontinuities or cracks.

Because of the very close temperature values at 2.5 em to those at thesurface, it seems that an original opening or a crack extending from the surface tothe 2.S-cm temperature probe would be the cause. The same original openingwould also explain the nearly universal increase of experimental temperaturesthroughout the concrete specimen: as if the same surface temperature werepresent at some distance inside the specimen. The presence of the cracks causedenhanced moisture migration and hence a higher heat transfer rate at locationsnear the cracks. This situation of measured higher temperatures appears to exist

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Table 2. Percent relative error for peak pore pressure

5 Hours 10 Hours 15 Hours 20 Hours

Number of p, Error, p, Error, p, Error, p, Error,elements kPa % kPa % kPa % kPa %

100 195.78 8.82 396.41 8.70 563.07 19.7 763.58 5.08200 191.58 2.14 435.63 9.89 600.99 6.70 752.54 1.44300 203.50 6.22 446.92 2.59 615.87 2.48 744.94 1.01400 208.08 2.25 451.18 0.95 620.53 0.76 743.73 0.16500 217.14 4.35 451.69 0.10 621.80 0.20 746.04 0.30600 219.82 1.23 451.43 0.05 623.96 0.34 748.38 0.30

throughout the experiment at the range below 100°C. Since cracking is discrete,and some cracks may be closer to some temperature probes than to some others,deviation from a general trend would be expected.

At temperatures somewhat above 100°C, and inside the surface of thespecimen, new cracking due to the loss of bound water was predicted by prelimi­nary stress analysis. It is interesting to note that there is considerable deviation ofexperimental temperatures from the linear behavior expected in thermal conduc­tion through isotropic material. Cracking again may affect the experimental valuesdifferently, depending on how far cracking exists from the particular temperatureprobe. The important factor to note, however, is that substantial new cracking ispredicted inside the specimen in the range llO-120°C.

Figure 8. Percent relative errors for peakpore pressure using different mesh sizedistributions.

600500300 400

Number of Elements

200100

20

- 5 hrs-- 10 hrs

15 15 hrs

1!- .. 20 brs

W...l:..G 10a:E..~..

11.

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HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS 127

500

400

§:e

300~

;;:0D.E.....

200

100

a

a 10 20

Time (hr)

30

Figure 9. Comparison of temperaturedistributions in the specimen predictedby finite element model with experimen­tal data [3].

A preliminary elastic stress analysis has revealed that the loss of moisturewithin concrete due to an imposed temperature at the surface would be signifi­cantly large to cause cracking. This factor is related to the analytical model, whichsimulates moisture propagation through the concrete. The variation of temperatureand free water content in the specimen at different times for this example isdisplayed in Figures 10 and 11, respectively. The constant cross section of thespecimen indicates the presence of moisture at different times. An approximateamount of total water pushed forward and expelled through the inside surface maybe inferred from Figure 11 if the moisture flux through the outside wall isneglected.

Elastic stress analysis, a very simplistic approach to evaluating stress inconcrete at high temperatures, has nevertheless revealed certain important factors.The relative magnitude of stress caused by moisture loss is significantly larger thanthat caused by thermal effects or pore pressure. This is especially noticeable incomparing the plots for f3 = 0.05 and f3 = 0.08 in Figures 120 and 12b, respec­tively. At f3 = 0.05 only stresses due to pore pressure and thermal effects exist, butfor f3 = 0.08 the stresses due to the loss of bound water also come into the picture.The stress due to moisture loss is about 30 times larger. While the strength ofconcrete in tension is in the neighborhood of 3 X 106 Pa, the appearance of thisstress would be coincident with the new cracking of concrete. This being the case,the importance of knowing the actual value of f3 becomes significant. The specificvalue of f3 would then indicate how far from the surface inside the specimen thecracking would begin.

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500

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€..'00i!e

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200

100

o

P. MAJUMDAR AND A. MARCHERTAS

Dloton.o 1m)

0.0

200

0.2 0.4 0.6 O.B 1.0Figure 10. Temperature distributions inspecimen with distance at different times.

~ 150 -ue~

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=c.. 100 ­u

=..•..~...

50 -

o

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; . ,I . i,II,,IIIII . ILJ

I

0.2

-5 hro- - 10 hro

··15 hro-'-20 hro

I I' I

0.4 0.6 O.B

Dloton.o 1m)

1.0 Figure 11. Free water content in thespecimen at different times.

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HEAT, MOISTURE, AND STRESSES IN POROUS MATERIALS 129

8E+5 .---------------, 3E+7 ,-----------,

1.0

~aO.08

-5 hrs--·10 hrs--,--15 hrs-,-20 hra

I I

0.6 0.80.0

«i IE+?"0....

..a.:; 2E+7 -..e;;;

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Distanca (m)

(a)

Distance (m)

(b)

Figure 12. Variation of total tensile stress in the specimen for different values of coefficient ofshrinkage: (a) fJ = 0.05 and (b) fJ = 0.08.

CONCLUSIONS AND RECOMMENDATIONS

Induced pore pressures, thermal stress due to the temperature gradient, andshrinkage stress due to the release and migration of water may cause tensile stressabove the maximum tensile stress of the materials. However, results seem to bevery sensitive to the value of the coefficient of shrinkage, which has to be estimatedaccurately based on experimental data. Even though the assessment yieldedrelative merits of the individual strain components, an accurate contribution ofmoisture loss to the cracking of concrete would need certain relationships betweenthe loss of bound water with the actual decrease in volume. Assuming that thisinformation can be experimentally determined, the strain contributions may nowbe implemented into a nonlinear constitutive model. The contribution of the effectof moisture in the nonlinear context would then be much more meaningful.

REFERENCES

1. Z. P. Bazant and W. Thonguthai, Pore Pressure and Drying of Concrete at HighTemperature, J. Eng. Mech., vol. EMS, pp. 1059-1079, 1978.

2. A. Dayan and E. L. Gluekler, Heat and Mass Transfer Within an Intensely HeatedConcrete Slab, Int. J. Heat Mass Transfer, vol. 25, no. 10, pp. 1461-1467, 1982.

3. R. Kikuchi, T. Kume, M. Hiramoto,M. Yamazaki, T. Hasegawa, and K. Hirakawa, Studyof Heat Transfer Properties of Reinforced Concrete Members Under High Tempera­ture, SMiRT-12 Trans., 12th Int. Conf. Structures Nucl. Reactors, vol. Q, pp. 97-102,1993.

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130 P. MAJUMDAR AND A. MARCHERTAS

4. R. F. Wu and H. Ouyang, Computation of Shrinkage Stresses in Prestressed ConcreteContainments, SMiRT-IO Trans., 10th Int. Conf. Structures Nucl. Reactors, vol. Q., pp.47-51,1989.

5. T. Numao and H. Mihashi, Moisture Migration and Shrinkage of Hardened CementPaste at Elevated Temperatures, SMiRT-11 Trans., 11th Int. Con]. Structures Nucl.Reactors, vol. H, pp. 37-42, 1991.

6. D. A. Chapman and G. L. England, Effects of Moisture Migration on Shrinkage, PorePressure and Other Concrete Properties, Trans. 4th Int. Conf. Structures Nucl. Reactors,vol, H, pp. 1-14, 1977.

7. K. Takiguchi and H. Hotta, A Numerical Analysis Method on Thermal and ShrinkageStress of Concrete, SMiRT-11 Trans., 11th Int. Conf. Structures Nucl. Reactors, vol. H,pp.49-54,1991.

8. P. Majumdar, A. Gupta, and A. Marchertas, Stress Analysis of Heated Concrete UsingFinite Elements, Nucl. Eng. Des., vol. 147, pp. 287-298, 1994.

9. P. Majumdar, A. Gupta, and A. Marchertas, Moisture Propagation and Resulting Stressin Heated Porous Media, Adu. Heat TransferASME Publ., PO-vol. 64-1, pp. 81-91, 1994.

10. A. V. Luikov, Systems of Differential Equations of Heat and Mass Transfer inCapillary-Porous Bodies, Int. J. Heat Mass Transfer, vol. 18, pp. 1-14, 1975.

11. S.-Y. Lei, B.-x. Wung, T.-I. Lu, and G.-M. Xiang, Investigation of the Transient Heatand Mass Transfer Process in Unsaturated Porous Media Around a Buried Pipe, ASMEPublication 94-WA/HT-13, 1994.

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