Engineering Structures 102 (2015) 369–386

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier .com/ locate /engstruct

Degree of restraint concept in analysis of early-age stresses in concretewalls

http://dx.doi.org/10.1016/j.engstruct.2015.08.0250141-0296/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: agnieszka.knoppik-wrobel@polsl.pl.com (A. Knoppik-Wróbel).

Agnieszka Knoppik-Wróbel ⇑, Barbara KlemczakDepartment of Structural Engineering, Silesian University of Technology, Akademicka 5, 44100 Gliwice, Poland

a r t i c l e i n f o

Article history:Received 6 March 2015Revised 24 July 2015Accepted 18 August 2015Available online 2 September 2015

Keywords:Degree of restraintEarly-age concreteNumerical modellingRestraint stressesSoil–structure interactionWall

a b s t r a c t

The degree of restraint is a useful concept for characterisation of early-age thermal–shrinkage stressesoccurring in externally-restrained concrete elements such as walls. It can be used not only in manual cal-culations, but also in numerical analysis to determine the values and distribution of stresses in walls. Theissues that must be addressed while defining the degree of restraint of the wall include the stiffness of therestraining body (e.g. foundation), translational and rotational restraints, influence of the constructionsequence and support conditions. These issues are discussed in the paper. For the purpose of the studya numerical model is proposed which takes into account sequential casting and interaction betweenearly-age structure and founding soil. The results of the study point out the factors that need be takeninto account when modelling structural behaviour of early-age walls for proper determination of theexpected stresses.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Concrete elements are subjected to early-age volume changesdue to temperature and moisture variations which characterisethe process of concrete hardening. These volume changes inducestresses in concrete elements. In massive concrete elements, suchas foundation slabs or blocks, the stresses are induced mainly bysignificant temperature differences developing between the inte-rior and the surface of the element (self-induced stresses). Inexternally-restrained elements, such as walls, thermal–shrinkagestresses result from a coupled action of self-induced (Fig. 1b) andrestraint stresses (Fig. 1c). The restraint in these elements isexerted by the bond between the new concrete of the elementand the older concrete of the foundation or a previous lift; in a con-crete wall tensile stresses result from the restraint of a potentialcontraction caused by the length changes associated with decreas-ing temperature of the wall. In typical walls restraint stresses playa predominant role because volumetric strains caused by the tem-perature and humidity gradients are relatively small in comparisonto the linear strains caused by the contraction of the element alongthe line of the restraint joint [1,2]. Nevertheless, it must be remem-bered that with the increasing massivity of the wall the share ofthe self-induced stresses increases. Surface tensile stresses occur-ring in thick walls (thermal gradients) and formed by early

formwork removal (both thermal and moisture gradients) maylead to surface cracking which can further develop into throughcracking.

The magnitude of the restraint stresses depends on a degree ofrestraint induced against the early-age part of the structure. Thedegree of restraint can be expressed with the restraint factor, cR– a measure which in any point of the element is defined as a ratiobetween the actual stress generated in the element, r, to the hypo-thetical stress at total restraint, rfix, [4–7]:

cR ¼ rrfix ; ð1Þ

and may take values between 0 at no restraint to 1 at total restraint.It varies throughout the element with the maximum value at thejoint between the wall and the restraining body, decreasing towardsfree edges of the wall.

Fig. 2 presents distribution of stresses in the mid-span cross-section of a base-restrained wall. Stresses generated by the exter-nal restraint (rres) have a major influence on the values and char-acter of the total stresses (rtot). If there were no temperature andhumidity gradients within the element, the stress distributionwould be proportional to the degree of restraint as shown inFig. 2a (there would be no self-induced stresses, rs-ind). Due tothe temperature and humidity gradients at the height of the wall,which generate the self-induced stresses, the maximum stressappears above the joint (Fig. 2b) with the maximum value of thestress above the joint. The temperature and humidity gradientsat the thickness of the wall also cause self-induced stresses, which

Fig. 1. Development of early-age thermal–shrinkage stresses in time in an externally-restrained element [3].

Fig. 2. Distribution of early-age stresses in a cross-section of a wall [3].

370 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

are the reason why the values of the total thermal–shrinkage stres-ses differ in magnitude between the interior and the surface of thewall (Figs. 1d and 2b). If the value of tensile stress in any location ofthe element exceeds the tensile strength of concrete in that

location, a crack is formed (Fig. 1e). When the stress state in cool-ing phase looks like in Fig. 2b, which happens if the wall is kept informwork during the whole process of concrete hardening, devel-opment of cracks initiates from the interior of the wall (internal

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 371

stresses reach higher values than surface stresses). Nevertheless,more often formwork is removed from the wall in cooling phase,and then first cracks appear on the surface of the element. Theinternal stresses can also become of considerable magnitude andconsequently through cracks may develop.

There are multiple examples of externally-restrained concreteelements in which early-age thermal–shrinkage cracking wasobserved or which are susceptible to such cracking: structuralelements of bridges, especially bridge abutments [8,9], tank walls[9–11], walls of nuclear containments and radiation protectionwalls [12–14], retaining walls and tunnel walls cast against hardenedfoundation or cast in stages [7,15]. These elements are charac-terised with different massivity and restraint conditions. Theirthickness can be as little as 40 cm up to 2 m and beyond whilethe length-to-height ratio may range from 1 to even 15 or more.Yet, the observed cracking pattern is similar [3,16]. In the base-restrained walls the occurring cracks are vertical in the central partof the wall and splay towards the ends of the element. A horizontalcrack may occur at the construction joint at the ends of the wall.The greatest height of the crack is observed in the middle of thewall and it declines towards the side edges of the wall. The studypresented in this paper aimed at characterisation of the stressesgenerated in these externally-restrained elements taking intoaccount their restraint conditions, i.e. geometry and dimensionsof the wall and restraining foundation, construction sequenceand support conditions. The concept of the degree of restraintexpressed with the restraint factor was used as a measure.

2. Theoretical background

The early-age stresses in externally-restrained elements resultfrom a coupled action of internal and external restraints. The exter-nal restraint acts against axial deformation and flexural deforma-tion. A free deformation of the concrete element can beseparated into deformation in an axial (X) direction (expansionor contraction) and flexural deformation in a vertical (Z) direction.The total stress consists then of three components: due to theinternal restraint, due to the external restraint against axial defor-mation and due to the external restraint against flexural deforma-tion. The approach based on such assumptions, calledcompensation plane method (CPM), was first proposed in Japan[17] and can be found in Standard Specifications for ConcreteStructures [4]. Other methods for determination of stress state inexternally-restrained concrete elements can be referred whichbase on the same assumptions as the CPM, developed by AmericanConcrete Institute published in ACI Report 207 [5], in Eurocode 2 –Part 3 [18] and enhanced in CIRIA C660 [19], by Nilsson introducedin his Licentiate thesis [6] and developed in further research [11].

Fig. 4. Determination of stresses in a concrete element caused by the externalrestraint according to compensation plane method after JSCE Standard [4].

2.1. Compensation plane method

The stress exerted due to the internal restraint is caused bya differential strain in the cross section resulting from the

Fig. 3. Determination of stresses in a concrete element caused by the internal

temperature and moisture concentration gradients. The incrementof stress due to the internal restraint can be determined from thedifference between the strain value at the point of the compensa-tion line, ecomp, and the thermal–shrinkage strain distributioncurve, e0, (Fig. 3) by the equation:

rint ¼ Ecðe0 � ecompÞ; ð2Þwhere Ec is a modulus of elasticity of the wall, GPa. This approachallows to obtain the increment of free axial strain, �e, and the incre-ment of curvature, �u (Fig. 3).

The internal forces are generated in the element trying to returnthe plane after deformation to the original restrained position –axial force NR and bending moment MR. The restraint stresses arethen caused by the coupled action of the axial force and bendingmoment, as shown in Fig. 4, according to the equation:

rext ¼ NR

AcþMR

Icðz� zcenÞ; ð3Þ

where:

Ac , – cross-section area of the wall, m2;Ic – moment of inertia of the cross-section; m4;ðz� zcenÞ – distance from the centre of gravity of the cross-section, m.

The internal forces can be defined as follows:

NR ¼ RN Ec Ac �e; ð4aÞMR ¼ RM Ec Ic �u ð4bÞ

where the external restraint factors, RN and RM , are introduced torepresent the translational and rotational restraint of the elementby the restraining body. The Eq. (3) gets a form:

rext ¼ RN Ec �eþ RM Ec �u ðz� zcenÞ: ð5Þ

restraint according to compensation plane method after JSCE Standard [4].

372 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

The values of the restraint factors vary within the elementaccording to the degree of restraint. In the base-restrained elementthe greatest degree of restraint occurs at the centreline with themaximum value at the joint between the early-age wall and therestraining body, decreasing towards the top edge of the wall.

2.2. Restraint factor

There exist numerous proposals for determination of the degreeof restraint [4–6,18–20]. All of them state that the degree ofrestraint depends on the linear restraint exerted by the restrainingbody expressed with the ratio of the length and height of the wall,L=Hc , and the ratio of the stiffness of the wall and the restrainingbody:

cR ¼ cRLH;Ac Ec

AF EF

� �; ð6Þ

where Ac and Ec are cross-section area and modulus of elasticity ofthe wall and AF and EF are analogically cross-section area and mod-ulus of elasticity of the restraining body.

The most complete formulation of the restraint factor was pro-posed by Nilsson [6]. This approach, based on CPM, introduces arestraint factor, cR, to determine the restraint stress, r, based onthe stress at total restraint, rfix:

r ¼ cRðc0R; dres; dslipÞrfix; ð7Þwhere:

c0R – plane-section restraint factor, which depends on the geom-etry of the structure as well as the rotational, cryR ; crzR , and trans-lational, ctR, boundary restraints;dres – resilience factor considering the non-linear effects;dslip – slip factor which depicts the reduction of the restraintstresses as a result of slip failure when bond forces betweenthe early-age element and restraining body are broken and hor-izontal crack appears at free ends of the element.

The value of dres changes at the height of the wall and dependson the boundary restraint. It is a product of the basic resilience fac-tor and translational and rotational correction factors,dres ¼ dres d0res; d0trans; d0rot

� �. To simplify, the resilience factor is taken

as equivalent to the basic resilience factor, dres ¼ d0res, and thecorrection to account for the translational and rotational boundaryinfluence is included by introduction of the effective width of the

Fig. 5. Factors accounting for high-walls effect in determi

restraining body, BF;eff , instead of the real width, BF , (see Eq.(10)). Effective width defines a part of the restraining body cross-section which influences deformation of a structure. Its value isdetermined in numerical analysis in such a way that the curvature– measured at the bottom of the wall – obtained with the analyticapproach with the use of restraint factors complies with thenumerically calculated curvature. The resilience factor is analogicalto the structural shape restraint factor, KR, proposed by ACI Report207.2 [5]. The values of d0res are given in diagrams in Fig. 5a or canbe approximated with a polynomial function [11]:

d0res ¼Xni¼1

aizHc

� �i

; ð8Þ

where:

ai – coefficients of a polynomial function describing resiliencefactor distribution;z=Hc – relative location of the analysed point above the joint; zis a location at the height of the wall above the joint, 0 6 z 6 Hc ,and Hc is the height of the wall.

The slip factor depends on the free length, L, the width, Bc , andthe height, Hc , of the casting section. It can be determined experi-mentally or numerically. The values of dslip proposed by Nilsson [6]are given in diagrams in Fig. 5b.

The restraint factor distribution at height z at the mid-span of awall was defined with the following expression:

cRðzÞ ¼ dslip � dresðzÞ � ctRðzÞ þ crzR ðzÞ þ cryR ðzÞ� �� �

; ð9Þ

where ctRðzÞ; crzR ðzÞ and cryR zð Þ are given as follows:

ctRðzÞ ¼Pn

i¼1aiiþ1

1þ EFEc

HFBF;effHcBc

; ð10Þ

crzR ðzÞ ¼zcen � zð Þ zcen

Pni¼1

aiiþ1 � Hc

Pni¼1

aiiþ2

� �Hc

2

12 þ zcen � Hc2

� �2 þ EFEc

HFBF;effHcBc

HF2

12 þ zcen þ HF2

� �2� � ; ð11Þ

cryR ðzÞ ¼ycen �x0:5 BF;eff � Bc

� �� �2Bc2

12 þ ycen �x BF;eff�Bc2

� �2þ EF

Ec

HFBF;effHcBc

BF;eff2

12 þ y2cen� � : ð12Þ

Nilsson emphasised the influence of the subsoil on the valuesof the restraint factor and analysed them thoroughly [6]. He

nation of the restraint factor according to Nilsson [6].

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 373

concluded that a low-friction and non-cohesive foundation mate-rial would pose almost zero translational restrain (restraint againstcontraction) and the restraint would increase with the increasedfrictional and cohesive properties of the restraining body material.Analogically, the stiffness of the foundation material would influ-ence the possibility of bending of the concrete element and as suchwould influence the rotational restraint to this element. Therestraint would increase with the increasing stiffness of therestraining body material and the length of the element.

3. Strategy of analysis

The concept of the restraint factor as a measure of the degree ofrestraint advocated by Nilsson was adopted in numerical analysesof walls by Larson [21], Al-Gburi et al. [17,22] and Hösthagen et al.[15]. In this hybrid approach proposed at the Lule TechnicalUniversity (LTU) the self-induced part of the stress, rfix, is deter-mined numerically for the cross-section of the wall, and the totalstress, r, is calculated by introduction of the restraint factor, cR,analogically as in the analytic approach proposed by Nilsson. Therestraint factor is, however, calculated numerically with thegeneral-purpose software according to either of two methods: lin-ear restraint method (LRM) [17,21] or equivalent restraint method(EQM) [15,17,22].

It is common in the numerical analysis of early-age walls toassume that the foundation is settled firmly on the subgrade – totalrotational restraint of the foundation is then assumed. In the beforementioned LTU approach the influence of the founding soil is takeninto account, but only in the analysis of cross-section (no influenceon the restraint) and limited to thermal analysis (no moisturetransport). The approach in which the founding soil block is takeninto consideration in spatial analysis is used rarely and most oftenit is also limited to thermal analysis [23,24,20]. A few examples offull cooperation can be referred [9,25]. For the purpose of thisstudy the model for thermal–moisture–mechanical analysis ofearly-age walls was proposed which takes into account the soil–structure interaction. An approach similar to the LTU methodwas used, in which the numerically-determined values of stresseswere expressed with the use of the restraint factor. Nevertheless,in the presented approach the whole spatial mechanical analysiswas performed at once. For each wall the stress state was deter-mined for the real restraint conditions (stress r) and under theassumption of total restraint (stress rfix). To clarify the calculationsit was assumed that the temperature and moisture content wereuniform in the volume of the wall, so the obtained stresses werepure restraint stresses. The degree of restraint was in each case cal-culated at the mid-span of the wall according to Eq. (1). Such cal-culated values of the restraint factor were compared to the valuesof the restraint factor calculated with the Nilsson’s approachaccording to Eq. (9).

3.1. Numerical model

The model used in this study was developed based on the pro-posals of Klemczak [26] for thermal–moisture analysis and Majew-ski [27,28] and Klemczak [25,29] for stress and damage analysis. Itis an enhanced form of the model introduced by Klemczak andKnoppik-Wróbel [9]. Formulation of the model is schematicallypresented in Fig. 6.

3.2. Thermal–moisture analysis

3.2.1. Thermal–moisture fieldsThe temperature and moisture fields in early-age concrete were

defined with the coupled equations proposed by Klemczak [26]:

_T ¼ divðaTT gradT þ aTW gradcÞ þ qvðt; TÞcbq

; ð13aÞ_c ¼ divðaWW gradc þ aWT gradTÞ � KH qvðt; TÞ: ð13bÞ

Analogical equations were proposed for soil; only partial cou-pling was assumed (it was suggested by Hillel [30] that the influ-ence of moisture gradients on temperature development do notneed to be taken into account):

_T ¼ divðaTT gradTÞ; ð14aÞ_c ¼ divðaWW gradc þ aWT grad TÞ; ð14bÞwhere:

T – temperature, K;c – moisture concentration by mass (relative humidity), kg=kg;aTT – coefficient of thermal diffusion, m2=s;aWW – coefficient of moisture diffusion, m2=s;aTW – coefficient representing the influence of moisture concen-tration on heat transfer, ðm2 KÞ=s;aWT – coefficient representing the influence of thermal gradienton moisture transport, m2=ðs KÞ;cb – specific heat, kJ=ðkg KÞ;q – density of concrete, kg=m3;KH – water–cement proportionality coefficient describing theamount of water bounded by cement during hydration process,m3=J;qvðt; TÞ – rate of heat generated per unit volume of concrete,W=m3.

As initial conditions initial temperature and humidity of con-crete mix and soil were taken. Boundary conditions were assumed

to be 3rd type boundary conditions of convective type. The coeffi-cients of the heat and moisture exchange were calculated takinginto account the physical properties of materials and coveringmaterial (if applied).

3.2.2. Thermo–physical parametersThe coefficients of thermal diffusion, aTT , both for soil and con-

crete were calculated with the values of thermal conductivity, k,and heat capacity, cv , constant over time. The values of k and cvwere calculated based on the composition of soil and concretemix, respectively. Thermal conductivity of concrete [31–33] andsoil [34] was calculated as, respectively:

k ¼X

pi ki; ð15aÞ

k ¼X

kiffiffiffiffiki

p� �2: ð15bÞ

Heat capacity of concrete [31–33] and soil [35,30] was calculated as,respectively:

cv ¼X

pi qi cb;i; ð16aÞcv ¼

Xki qi cb;i: ð16bÞ

In the equations pi is amass fraction and ki is a volume fraction of theith component: cement, water and aggregate for concrete, and soilparticles and water for soil. The properties of concrete componentswere taken after Kierno _zycki [32] and for soil after Farouki [35].

The coefficient of moisture diffusion, aWW , for concrete was cal-culated after Hancox [36] based on the moisture content byvolume:

DWWðWÞ ¼ 4:6389 � 0:7þ 6Wð Þ2 � 1:0556 � 0:7þ 6Wð Þ þ 0:3055h i� 10�10;

ð17Þ

Fig. 6. Model for analysis of early-age walls with the use of the numerical approach [3].

374 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

by applying the following relationship between the moisture con-tent by volume and by mass:

aWW ¼ DWWqw

q: ð18Þ

The coefficients of the mutual influence of temperature onmoisture fields and vice versa for concrete were taken as constantand equal to aTW ¼ 0:9375 � 10�4 ðm2 KÞ=s after Gawin [37] andaWT ¼ 2:0 � 10�11 m2=ðK sÞ after Wyrwał and Szczesny [38]. Thecoefficient of the water–cement proportionality was taken as equalto KH ¼ 0:3 � 10�8 m3=J (for Portland cement).

Coefficients in moisture analysis for soil, aWW and aWT , were cal-culated based on the isothermal and thermal liquid diffusivitiesdetermined after Clapp and Hornberger [39] and by applying therelationship given in Eq. (18). Because of relatively small changesin moisture content of soil the values of the diffusivities were

assumed as constant and determined under the assumption of fullsaturation (S ¼ 1:0):

DWW ¼ bKh;satwa

n; ð19aÞ

DWT ¼ Kh;satwacT : ð19bÞwhere:

Kh;sat – saturated hydraulic conductivity, m=s;wa – air-entry tension, m;b– fitting parameter;cT – relative change of surface tension with respect to temper-

ature; constant value of cT ¼ 2:09 � 10�3=�C given by Philip and

de Vries [40] can be assumed.

The hydraulic properties of soil (Kh;sat;wa and b) were taken after[39].

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 375

3.2.3. Hydration functionThe rate of heat generated per unit volume of concrete was cal-

culated based on the unit heat rate, qðtÞ (W/g), and the amount ofcement, Cc (kg/m3):

qvðt; TÞ ¼ Ccqðt; TÞ ¼ Cc@Qðt; TÞ

@t: ð20Þ

The unit rate of hydration heat development, qðt; TÞ, was calcu-lated according to Schindler and Folliard [41]:

qðt; TÞ ¼ Q totste

� �b bte

� �aHðteÞe

EKR

1Tref

�1T

� �; ð21Þ

with the parameters calculated for pure Portland cement (no SCMsaddition):

Q tot – total heat of hydration, J=g; calculated based on theknown composition of cement according to the approach pro-posed by Schindler and Folliard [41] after Bogue [42]:

Q tot ¼ 500pC3S þ 260pC2S þ 866pC3A þ 420pC4AF

þ 642pSO3þ 850pMgO; ð22Þ

aHðteÞ – degree of hydration at equivalent age, te, after Rastrup[43]:

aHðteÞ ¼ aHue�steð Þb ; ð23Þ

s; b – hydration time and shape parameters:

s ¼ 66:78p�0:154C3A

p�0:401C3S

Blaine�0:804p�0:758SO3

; ð24aÞb ¼ 181:4p0:146

C3Ap0:227C3S

Blaine�0:535p0:558SO3

; ð24bÞaHu – ultimate degree of hydration (aHu 6 1):

aHu ¼ 1:031w=c0:194þw=c

: ð25Þ

EK – activation energy, J=mol:

EK ¼ 22;100p0:30C3A

p0:25C4AF

Blaine0:35; ð26Þpi – weight ratio of the subsequent cement component withrespect to the total weight of cement.

3.3. Thermal–shrinkage strain

The imposed thermal–shrinkage strains, en, were treated as vol-umetric strains:

den ¼ den;x den;y den;z 0 0 0½ �; ð27Þ

Fig. 7. Boundary surface according to the modified 3-pa

and calculated with the predetermined temperature, T (�C), andmoisture content, W ðm3=m3Þ, change:den;x ¼ den;y ¼ den;z ¼ aTdT þ aWdW; ð28Þwhere aT is the coefficient of thermal dilation and aW is the coeffi-cient of moisture dilation. For concrete the coefficient of thermaldilation was taken after Neville [44] based on the aggregate usedin the mix. The moisture dilation coefficient was taken as equal toaW ¼ 0:002. For soil, the thermal dilation coefficient was taken asequal to aT ¼ 10�5=�C [20] and the moisture dilation coefficientwas taken as equal to aW ¼ 0:001.

3.4. Stress analysis

3.4.1. Early-age concreteFor the stress analysis in early-age concrete viscoelasto–visco

plastic model was used. The applied constitutive formulation ofthe consistent conception of the model was given by Klemczak[25]. The following constitutive equations were defined in the vis-coelastic area and viscoelasto–viscoplastic area, respectively,expressed with the stress and strain rates:

_r ¼ Dve _e� _en � _ec½ �; ð29aÞ_r ¼ Dve _e� _en � _ec � _evp½ �: ð29bÞwhere:

Dve – viscoelasticity matrix;e – strain matrix;en – imposed thermal–shrinkage strain matrix;ec – matrix of strain representing the effect of creep;evp – viscoelastic strain matrix.

Both the yield surface, f, and the boundary surface, F, are rate-dependent and were expressed as functions of the hardeningparameter, j, and its rate, _j:

f ðr;j; _jÞ ¼ 0; ð30aÞFðr;j; _jÞ ¼ 0: ð30bÞ

The creep function, Cðt; sÞ, was assumed according to ModelCode 1990 [45] after Guénot et al. [46]. It was assumed that creepis symmetrical in compression and tension.

The modified 3-parameter Willam–Warnke failure criterion,MWW3, was adopted after Klemczak [29] to describe both theyield surface and the failure surface (see Fig. 7). Final values ofthe mechanical properties (compressive strength, f c;28, tensilestrength, f t;28 and modulus of elasticity, Ec;28) were calculated

rameter Willam–Warnke (MWW3) failure criterion.

376 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

based on the concrete class according to Model Code 2010 [47] asmean values (denoted with index m):

f cm;28 ðMPaÞ ¼ f ck;28 þ 8 ð31aÞf ctm;28 ðMPaÞ ¼ 0:3 f cm;28 � 8

� �2=3; ð31bÞ

Ecim;28 ðGPaÞ ¼ 21:5aEf cm;28

10

� �1=3

; ð31cÞ

where the coefficient aE depends on the type of aggregate. The biax-ial compressive strength was calculated as [48]:

f ccðtÞ ¼ 1:2� f cmðtÞ1000

f cmðtÞ: ð32Þ

Development of the surfaces in time with progressing maturityof concrete was defined by the time-development of the materialproperties, Pðt; TÞ; ageing of concrete was defined as a function ofthe equivalent age of concrete:

Pðt; TÞ ¼ bc½ �nP28; ð33Þwhere:

Pðt; TÞ; P28 – material property (f c; f t or Ec) in time t and at theactual maturity level, and at the age of 28 days, respectively;bc – time development function [47]:

bcðt; TÞ ¼ es 1�

ffiffiffi28te

ph i; ð34Þ

n – exponent dependent on the property to describe and con-crete composition;s – coefficient dependent on the type of cement.

Possibility of cracking and its influence on development ofstresses was taken into account. A smeared cracking image wasused in the model. A possibility that a crack occurs in an analysedpoint was defined with a damage intensity factor given by theequation:

0 6 sl ¼ socts foct

6 1; ð35Þ

where soct is the actual stress state in the analysed point and s foct

represents stress states on the failure surface. The damage intensityfactor equal to 1 is equivalent to the stress reaching the failure sur-face and damage of the element. The character of this damagedepends on the location where the failure surface was reached[28]. In the analysed cases failure surface was always reached inthe range of hydrostatic tensile stresses which was equivalent toformation of the splitting crack in the plane perpendicular to thedirection of the maximum principal stress.

Fig. 8. Scheme of the soft

Yield surface evolves according to the applied hardening lawuntil it reaches failure surface. When failure surface is reached,the material exhibits softening behaviour. Deviatoric and volumet-ric softening was applied. Hardening and softening laws wereadopted after Majewski [28].

3.4.2. SubsoilFor soil the elasto–plastic material model with a modified

Drucker–Prager failure criterion was used. The constitutive equa-tions ware given by Majewski [27]. In the elastic phase the consti-tutive equation has a form:

dr ¼ De de; ð36Þwhere De is elasticity matrix. Beyond the elastic phase the constitu-tive equation was defined:

dr ¼ Depðdee þ depÞ; ð37Þwhere the magnitude of the plastic strain was determined by thelaw of plastic flow and:

Dep ¼ De � Dp: ð38ÞThe plasticity matrix, Dp, was given by the equation:

Dp ¼ De @f@r

� �@f@r

� �T

De @f@j

rT @f@r

þ @f@r

� �T

De @f@r

� �" #�1

: ð39Þ

The yield surface, f, has a form:

f ¼ f 1ðrm;jÞ þ f 2ð�rÞ ¼ 0; ð40Þwhere:

j – hardening parameter;rm – mean stress;�r – stress intensity.

The values of bulk modulus, K, and shear modulus, G, requiredfor definition of the failure surface were assumed in a manner sim-ilar to Duncan and Chang model [49] after Majewski [27]. The bulkmodulus, K, and the initial value of shear modulus, G1, were givenby the following expressions:

K ¼ Koparm

pa

� �0:5

; ð41aÞ

G1 ¼ Goparm

pa

� �0:5

; ð41bÞ

where Ko and Go are material constants, rm is a mean stress and pa

is the atmospheric pressure. For a given stress intensity, �r, the valueof the shear modulus, G, was calculated as:

ware architecture [3].

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 377

G ¼ G1 1� �rrf

rf

� �2

; ð42Þ

where rf is a material parameter and rf is a limit value of stress �rfor a given mean stress, rm.

The values of the cohesion and internal friction angle weretaken as average values according to the literature data.

3.4.3. Contact elementsA layer of contact finite elements was introduced in the mesh

between the concrete element and soil to allow detachment ofthe rotated wall from the soil. Contact elements were assignedwith the material properties of the modified soil. This materialhas no tensile strength and limited ability to transfer shear stres-ses. Such concept was proposed by Majewski [27].

3.5. Implementation

Implementation of the model has a modular architecture(Fig. 8). The mesh is defined with the use of the MAFEM3D modulecreated by Wandzik (see [50]). The same mesh is used in thermal–moisture and stress analysis. The main calculations are performedwith the use of two computational modules. The first module,TEMWIL, is used for thermal–moisture analysis. This moduleimplements the thermal–moisture part of the numerical model.The second module, MAFEM, is used for stress and damage analy-sis. The results are presented with open-source software PARAVIEW.

4. Analysis and results

The factors that have major effects on the early-age stresses inwalls are dimensions and support conditions of the walls[16,25,51]. The presented study was focused on the characterand magnitude of stresses in the view of these factors. The analysiswas performed on 12 walls which dimensions are listed in Table 1.

It has been reported that the stresses differ with respect to L=Hc

ratio of the wall; in this study it was investigated if the effect of therestraint is the same in the walls with equal L=Hc but differentdimensions L and Hc. Dimensions of the walls were chosen in sucha way that wide range of L=Hc was covered (from 1.4 to 10) butwith different walls representing the same L=Hc ratios. AF=Ac ratio

Table 1Geometrical data for the analysed walls.

No. Wall Foundation

L; m Hc ; m Bc ; m Ac ; m2 HF ; m BF ; m AF ; m2 L=Hc

01 15 1.50 0.7 1.05 0.70 1.50 1.05 1002 15 2.14 0.7 1.50 0.70 2.14 1.50 703 15 3.00 0.7 2.10 0.70 3.00 2.10 504 10 1.42 0.7 0.99 0.70 1.42 0.99 705 10 2.00 0.7 1.40 0.70 2.00 1.40 506 10 3.33 0.7 2.33 0.70 3.33 2.33 307 7 1.40 0.7 0.98 0.70 1.40 0.98 508 7 2.33 0.7 1.63 0.70 2.33 1.63 309 7 3.50 0.7 2.45 0.70 3.50 2.45 210 5 1.67 0.7 1.17 0.70 1.67 1.17 311 5 2.50 0.7 1.75 0.70 2.50 1.75 212 5 3.57 0.7 2.50 0.70 3.57 2.50 1.4

Table 2Mineral composition of cement used in the parametric study.

Component C3S C2S C3A C4AF

Amount (%) 64.0 15.0 10.0 8.0

in each case was equal to 1.0, given that HF ¼ Bc and BF ¼ Hc . Themaximum temperature, autogenous shrinkage and temperaturedifference in walls with simple geometry is the same for equalthickness of the wall [52], so all the walls were assigned with thesame thickness.

The ambient temperature was set at Ta ¼ 20 �C and the initialtemperature of concrete at Ti ¼ Ta þ 5 �C ¼ 25 �C (assumption of5 �C temperature increase due to mixing is recommended by JCI[20]). Relative humidity of air was set at 55%. The wall wasassumed to be executed 2 weeks after the foundation. Both ele-ments were kept in formwork during the whole process of curing.The same material was assumed in all the analysed cases, and bothfor the wall and its foundation. The composition of the assumedconcrete was as follows: cement CEM I 42.5N 365 kg=m3, water160 l=m3 (w=c = 0.44), aggregate – sand 641 kg=m3 and granite1289 kg=m3. The design class of concrete was C30/37. Compositionof cement is shown in Table 2. Properties of concrete are listed inTables 3 and 4.

The meshing of the wall and time discretisation of the analysiswere made according to the recommendations of JCI Guideline[20]. Fig. 9 presents a scheme of the assumed model of an exem-plary wall (No. 09) with its finite element mesh. Because the mate-rial model is not regularised, individual finite elements of the meshwere always of the same dimensions. The analysis was performedfor 28 days in total (14 days before execution of the wall and14 days after its casting).

The maximum temperature reached in the core of the wall wasequal to 47:0 �C. The shrinkage strain at the end of the analysis wasequal to 4:80 � 10�5 (this shrinkage can be correlated with autoge-nous shrinkage).

4.1. Influence of construction sequence

Fig. 10a presents development of the temperature in the core ofthe wall with respect to the temperature development in the sup-porting foundation at different depth in the mid-span longitudinalsection of the wall. The temperature rise in the wall led to re-heating of the foundation, which at the moment when the wallwas executed had already cooled down to the temperature of thesurrounding air (Tf ’ Ta). Such a temperature increase in the foun-dation resulted in the strain of the foundation fibres. The strainchange due to re-cooling of the foundation at given depth d wasequal to:

SO3 CaO MgO Blaine; m2=kg

3.3 0.8 0.6 367

Table 3Thermo-physical parameters of concrete used in the parametric study.

Parameter Unit Value

Thermal conductivity, k W=ðm KÞ 2.5Specific heat, cb kJ=ðkg KÞ 0.95Density, q kg=m3 2455

Coefficient of heat exchange, ap W=ðm2 KÞ 3.6 – 1:8 cm plywood0.8 – soil

Coefficient of moisture exchange, bp m=s 0:18 � 10�8 � 1:8 cmplywood

0:01 � 10�8 – soil

Thermal dilation coefficient, aT 1=�C 10 � 10�6

Moisture dilation coefficient, aW – 0.002

Table 4Mechanical parameters of concrete used in the parametric study.

Parameter Unit Value

Compressive strength, f cm MPa 38Tensile strength, f ctm MPa 2.9Modulus of elasticity, Eci GPa 33.0Coefficient s – 0.25Coefficient n for tensile strength – 0.67Coefficient n for modulus of elasticity – 0.50

Fig. 11. Shrinkage strain difference between the wall and the restraining body.

378 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

eT;d ¼ Tmax;d � Tf� �

aT : ð43ÞAt the same time cooling of the wall resulted in the strain

change in the wall:

eT;wall ¼ Tmax;wall � Tf� �

aT : ð44ÞThe strain change that generates the restraint stresses is the dif-

ference between the strain in the wall’s fibres and foundation’sfibres:

r ¼ cR eT;wall � eT;d� �

Ec: ð45ÞThe ratio between the thermal strain during cooling of the wall

and the foundation at the depth of the foundation, shown inFig. 10b, was calculated as:

c ¼ eT;wall � eT;deT;wall

: ð46Þ

A mean value of c ¼ 0:7 can be assumed for this particular case.It means that the linear strain that would produce the restraintstress is about 70% of the strain in the wall itself caused by the

Fig. 9. Model and FE mesh of an exempla

Fig. 10. Relative free thermal stra

temperature difference during cooling. Analogical observationcan be made with respect to the shrinkage strain – the restrainedshrinkage strain that generates stresses in the wall is the differencebetween shrinkage strain in the foundation and in the wall, asshown schematically in Fig. 11. This shows that sequential castingmust be taken into account in the analysis, otherwise the value ofthe stress would be overestimated.

Another issue is the relative stiffness of the restraining body.Stiffness change is caused by development of the modulus of elas-ticity. Assuming that the wall and the foundation are made of thesame material and that the modulus of elasticity development intime can be described with the following function [47]:

EcðteÞ ¼ es 1�ffiffiffi28t

p� � nEc;28; ð47Þ

ry wall used in the parametric study.

in in an exemplary wall [3].

Fig. 12. Relative stiffness of early-age wall in time.

Table 5Polynomial coefficients for calculation of dres used in the parametric study.

L=Hc a0 a1 a2 a3 a4

7 1 �0.185 0.222 �0.253 0.1275 1 �0.387 0.036 0.132 �0.0313 1 �0.912 �0.041 0.189 0.0542 1 �1.238 �0.541 1.158 �0.4411.4 1 �2.362 0.931 1.581 �1.224

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 379

where s and n are material coefficients, than the change of the rel-ative stiffness of the wall, calculated as:

Ec

EF¼ es 1�

ffiffiffi28t

p� �es 1�

ffiffiffiffiffi28tþd

ph i ; ð48Þ

where d is a time between execution of the foundation and the wall,is shown in Fig. 12 for typical values of s (0.20–0.40) and d (7, 14and 28 days).

Fig. 13. Degree of restraint distribution alo

It can be observed that at the moment when cooling begins(usually 1–3 days) the stiffness of the wall can be as little as 20%of the foundation stiffness (for cements with low rate of strengthdevelopment) but should not be less than 50% (after 1 day) to even75% (after 3 days) in concretes with Portland cement. In such con-cretes 90% of the foundation stiffness can be obtained after 7 daysof curing. The increased temperature accelerates the rate of themodulus of elasticity development and Ec=EF is even higher in mas-sive concrete elements. Nevertheless, at the moment when tensilestresses start to develop the ratio between the stiffness of the walland the restraining foundation is non-negligible.

4.2. Influence of dimensions of the wall

In the stress analysis it was assumed that the temperature andmoisture content are uniform in the wall – this allowed to elimi-

ng mid-span section of the walls [3].

Table 6Slip factor dslip used in the parametric study.

Wall no. 01 02 03 04 05 06 07 08 09 10 11 12

dslip 1 1 1 1 1 0.83 0.84 0.77 0.73 0.74 0.68 0.60

Fig. 14. Degree of restraint for walls with equal lengths and different L=Hc ratios [3].

380 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

Fig. 15. Degree of restraint for walls with equal L=Hc ratios and different lengths [3].

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 381

382 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

nate the influence of the self-induced stresses. The temperatureand shrinkage strain in each time step were assumed as equal tothe temperature and shrinkage strain in the core of the element.The influence of the linear restraint, expressed with the L=Hc ratio,on the degree of restraint was analysed.

Fig. 13 presents the results of stress analysis for timet ¼ 16:5 days (age of wall t ¼ 2:5 days). The diagram in Fig. 13apresents distribution of the degree of restraint determined innumerical analysis as a ratio between the stress, r, and the stresscaused by the same strain but under total end restraint,rfix ¼ 1:5 MPa, i.e. cR ¼ r=rfix. For comparison the degree ofrestraint determined with the analytic approach is shown inFig. 13b. The degree of restraint acc. to Nilsson [6] was calculatedtaking into account the resilience and possibility of slip failure;the coefficients of polynomial functions used for definition of theresilience factors, dres, are listed in Table 5 while the values of slipfactors, dslip, are listed in Table 6. No rotation was taken intoaccount (total base restraint). The ratio between the moduli ofelasticity was Ec=EF ¼ 0:9.

Comparing the results obtained with the two calculation meth-ods it can be observed that they comply only to some extent. TheNilsson’s approach reproduced a characteristic that although inthe walls with lower L=Hc the degree of restraint is generally lower,the influence of the restraining body is more pronounced in theclose vicinity of the joint [53]. Nevertheless, when analysing thedegree of restraint obtained with the numerical analysis it is easilynoticed that it is not equal in the walls with the same L=Hc . There-fore, the influence of length, L, and height, Hc , of the wall on thedegree of restraint was further investigated.

Firstly, the diagrams of the degree of restraint distribution forthe walls with the same length were compared to analyse theinfluence of the height of the wall. This comparison is shown inFig. 14. The diagrams on the left show the restraint factor distribu-tion while the diagrams on the right the ‘‘normalised” value of therestraint factor. The normalised restraint factor, cR;1, was calculatedby introduction of the modification factor, M1:

cR;1 ¼ cRM1

: ð49Þ

In the previous study carried out by the authors [53] it wasobserved that when the L=Hc was changing by varying length atconstant height the diagrams of the restraint factor exhibited thefollowing behaviour: the diagrams crossed at the level of 0.25 Hc

and with the increasing L=Hc the value of the restraint factordecreased at the joint and increased at the top of the wall. ‘‘Rota-tion” around the crossing point was observed. Based on that obser-vation the value of M1 was determined so that the diagrams of therestraint factor show the before mentioned characteristic. In this

Fig. 16. Modification factors for dete

case the length was constant while such a behaviour was observedwhen L=Hc was changing due to the change of length. Therefore, therelationship between the value of the modification factor, M1, andthe heights of the walls was investigated. The walls height ratio,fH , was calculated for each length, L, in such a way that the shortestwall was set as a basic wall with the height Hbas and the heights ofthe other walls, Hi, were taken as relative heights, i.e. the wallsheight ratio for the ith wall was calculated as:

fH;i ¼Hbas

Hi; ð50Þ

with Hbas < Hi; for the basic wall fH ¼ 1. Comparing M1 and fH(Fig. 16a) it was observed that with the increasing height of the wallthe magnitude of the restraint decreases. This relationship becomesmore pronounced as the length of the wall increases. 2nd orderpolynomial approximation was proposed to show the trend. It canbe observed that in very long walls (10 and 15 m) the characteristicis identical and it starts to differentiate as the length of the walldecreases. This is most probably the influence of the slip at the endsof the wall.

Analogical analysis was performed on the walls with the sameL=Hc ratios but different dimensions. This comparison is shownin Fig. 15. The diagrams on the left show the actual restraint factordistribution while the diagrams on the right the ‘‘normalised”value of the restraint factor. The normalised restraint factor, cR;2,was calculated by introduction of the modification factor, M2:

cR;2 ¼ cRM2

: ð51Þ

It is suggested by analytic approaches that the restraint factors forthe walls with equal L=Hc are also equal, which means their dia-grams cover. Thus, for each L=Hc group the M2 was determined sothat the diagrams of the restraint factors covered. It was alsoobserved that there is a direct relationship between the value ofthe modification factor, M2, and dimensions of the wall. Since ineach case L=Hc were constant, both length and height of the wallscould have been compared. Thus, for comparison the surface areaswere used (Ai ¼ LiHi). The walls area ratio, fA, was calculated foreach L=Hc ratio in such a way that the longest wall was set as a basicwall with the surface area Abas and the areas of the other walls weretaken as the relative areas, i.e. the walls area ratio for ith wall wascalculated as:

fA;i ¼Abas

Ai; ð52Þ

with Abas > Ai. Comparing M2 and fA (see Fig. 16b) it was observedthat with the increasing area of the wall the magnitude of therestraint decreases. The influence of the walls area increases with

rmination of the restraint factor.

Table 7Thermo-physical parameters of soil used in parametric study.

Parameter Unit Hard soil and soft soil

Thermal conductivity, k W=ðm KÞ 1.5Specific heat, cb kJ=ðkg KÞ 1.0Density, q kg=m3 2600Liquid diffusivity, DWW m2=s 10�5

Liquid diffusivity, DWT m2=ðs KÞ 10�9

Coefficient of heat exchange, ap W=ðm2 KÞ 1.5Coefficient of moisture exchange, bp m=s 0:01 � 10�8

Thermal dilation coefficient, aT 1=�C 10 � 10�6

Moisture dilation coefficient, aW – 0.001

Table 8Mechanical parameters of soil used in parametric study.

Parameter Unit Hard soil Soft soil

Coefficient for bulk modulus, Ko – 900 100Coefficient for shear modulus, Go – 405 45Cohesion, c MPa 0.02 0.05Internal friction angle, u � 40 25

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 383

the increasing L=Hc ratio. Because of a small number of data pointslinear approximation was proposed, which suggests directly pro-portional relationship, but such relationship must be confirmed.

4.3. Influence of support conditions

In the previous analyses it was assumed that the walls had nopossibility to rotate, and to lose contact with the restraining soil,and that the base had infinite stiffness. In this part of the study thesoil block was introduced to simulate the founding soil. Dimensionsof the soil block were chosen based on the recommendations of JCIGuideline [20]. Contact elementswere introduced between the con-crete structure and soil to allow for lifting of walls ends. The influ-ence of the subsoil on the degree of restraint was analysed.

Two walls with distinctly different geometries were chosen – ashort wall (No. 12, L=Hc ¼ 1:4) and a long wall (No. 02, L=Hc ¼ 7).Two types of soil were investigated: soft soil and hard soil.Mechanical parameters of the soils were taken after Majewski[27]. The Model and the finite element mesh are shown inFig. 17 on the example of the long wall. Thermo–physical andmechanical properties of the soils are listed in Tables 7 and 8.The same thermo–physical parameters were assumed for bothtypes of soils to clarify the comparison of stresses. Properties ofconcrete were taken from Tables 3 and 4.

Diagrams in Fig. 18 present distribution of the degree ofrestraint in the walls with different support conditions, calculatedwith the use of numerical and analytic model. In the numerical cal-culations the comparison was made for the degree of restraint

Fig. 17. Model and FE mesh of an exempla

calculated assuming total rotational base restraint as well as takinginto account the real stiffness of the founding soil. Analogical com-parison was made for analytic calculations, however, in case whenrotation of the wall was allowed the founding soil was assumed tobe infinitely stiff.

ry wall with real support conditions.

384 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

It can be observed that both the numerical and the analyticmodel present the same character of the walls behaviour: theeffect of the rotation of the wall on the restraint factor is observedand it is pronounced in the long wall while in the short wall there

Fig. 18. Degree of restraint in the walls w

Fig. 19. Deformation and rxx stresses, MPa, in w

is almost no influence of rotation. The difference in results betweenthe numerical and analytic model is caused by the fact that thelength of the wall itself is not taken into account in determinationof the restraint factor in analytic calculations (which was discussed

ith different support conditions [3].

alls with different support conditions [3].

A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386 385

in the previous section). It can also be observed that the stiffness ofthe soil influences distribution of the restraint factor: when the soilis softer, larger rotation occurs. Also in the long wall it can beobserved that the magnitude of the restraint is smaller when thewall is founded on the soft soil which poses smaller friction.

Deformations of both types of walls are shown in Fig. 19. In thewalls with total restraint the foundation has only possibility ofcontraction; when real founding subsoil is taken into account,the rotation of the foundation occurs. The effect of the rotation isalso visible on the maps of stresses, and is more pronounced onthe soft soil and in the long wall. As it can be deduced from thedegree of restraint diagram, by allowing the possibility of rotationthe value of the tensile stress near the joint increases. Due to thatfact in the long wall founded on the hard soil cracking reachingapprox. 60% of the wall’s height occurred; concentration of stressesin the upper part of the wall results from the consequent redistri-bution of stresses in the cracked wall. In the wall supported on thesoft soil the magnitude of the restraint was generally reduced. As aresult, the value of the stress decreased significantly (with respectto the wall founded on hard soil) and only a part of the wall (60%)was subjected to tensile stresses (with respect to 100% in the wallwithout the possibility of rotation). Nevertheless, the values ofthese stresses were lower than in the wall on the hard soil anddid not lead to cracking.

5. Discussion and conclusions

Dangerous early-age tensile stresses are induced in concretewalls mainly by limitation of the contraction of the wall in thecooling phase (restraint of linear strain). Although walls in variousengineering structures differ in geometry, concrete design or tech-nology of execution, the character of stresses in these elements issimilar. The magnitude of these stresses depends on the degreeof restraint exerted by the restraining body (e.g. foundation) tothe restrained wall. There are multiple proposals for computationof the degree of restraint, which declare that the degree of restraintdepends on the L=Hc ratio of the wall and relative stiffness of therestraining body. These methods are discussed in the paper.

For proper analysis of stresses in early-age concrete elementsspatial numerical analysis is recommended. Nevertheless, whensuch analyses are performed for walls, modelling of the externalrestraint is often not appropriately approached and the obtaineddegree of restraint is incorrect. The most common shortcomingsof the models are: assumption of total rotational restraint of thebase, infinite stiffness of the soil and negligence of different matu-rity development of the early-age element and the restraining con-crete element resulting from the construction sequence.

The presented study focused on the analysis of stresses inearly-age concrete walls in relation to the restraint conditions.The concept of the degree of restraint was used as a measure forcharacterisation of these stresses. A numerical model was pro-posed for simulation of the behaviour of early-age walls includingcasting sequence and soil–structure interaction. This model wasused in further study for determination of stresses in walls withdifferent dimensions and support conditions. The results obtainedwith the use of the numerical model were compared with theresults of the analytic method proposed by Nilsson [6]. The follow-ing conclusions can be drawn from the study:

1. Construction sequence must be taken into account for twomajor reasons: (1) The strain that produces stress in a wall isa differential strain, ediff , resulting from the difference in strainin the wall (due to cooling and autogenous shrinkage), ewall, andfoundation (due to re-heating and autogenous shrinkage), efound,which is less than the strain in the wall itself (ewall > ediff ). (2)

Following layers of concrete in the element are characterisedwith different maturity and stiffness. In typical walls made ofOPC the stiffness of foundation may be even 2 times higher atthe moment when tensile stresses start to develop in the wall,and this ratio may increase when cements with lower rate ofstrength development are used. Consideration of relative stiff-ness was also postulated by Nilsson.

2. The restraint factor can be used as a measure to characterise thestresses induced in the walls, however, the degree of restraintobtained with the numerical and analytic model comply onlyto some extent. The degree of restraint increases with anincreasing L=Hc ratio, which agrees with Nilsson’s approachand other analytic formulations. Nevertheless, in contrary tothem, for the same L=Hc ratios it is not equal but has lower val-ues in walls with larger L and Hc dimensions. Hence, in determi-nation of the degree of restraint not only the L=Hc ratio but alsothe individual dimensions of the wall must be taken intoaccount.

3. Real support conditions of the wall must be simulated to prop-erly determine the degree of restraint. This requires introduc-tion of the soil block to simulate (1) real stiffness and frictionof the restraining body (soil) and (2) possibility of loss of con-tact between the rotating foundation and the soil. When possi-bility of rotation is taken into account, the value of the restraintfactor increases at the joint but decreases at the top edge of thewall; when soil is softer, larger rotation occurs. The influence ofthe rotation of the wall on the degree of restraint increases withan increasing length of the wall and it is practically unnoticed inshort walls. In the long walls it is also visible that the decreaseof frictional properties of soil leads to reduction of the transla-tional restraint. This observation is in agreement with Nilsson’sobservations.

Acknowledgements

The co-author of this paper, Agnieszka Knoppik-Wróbel, was ascholar under the Project ‘‘DoktoRIS”, co-funded by the EuropeanUnion under the European Social Fund.

References

[1] Knoppik-Wróbel A. The influence of self-induced and restraint stresses oncrack development in reinforced concrete wall subjected to early-age thermal-shrinkage effects. In: Proceedings of the 14th international conference ofpostgraduate students juniorstav, Brno, Czech Republic; 2012. p. 162.

[2] Klemczak B, Knoppik-Wróbel A. Analysis of early-age thermal and shrinkagestresses in reinforced concrete walls. ACI Struct J 2014;111(2):313–22. http://dx.doi.org/10.14359/51686523.

[3] Knoppik-Wróbel A. Analysis of early-age thermal–shrinkage stresses inreinforced concrete walls. PhD thesis. Silesian University of Technology,Gliwice, Poland; 2015.

[4] JSCE. Guidelines for concrete no. 15: Standard specifications for concretestructures. Design; 2011.

[5] ACI Committee 207. ACI 207.2R-07: Report on thermal and volume changeeffects on cracking of mass concrete; 2007.

[6] Nilsson M. Thermal cracking of young concrete. Partial coefficients, restrainteffects and influence of casting joints. Master’s thesis. Luleå University ofTechnology, Luleå, Sweden; 2007.

[7] Xiang Y, Zhang Z, He S, Dong G. Thermal–mechanical analysis of a newly castconcrete wall of a subway structure. Tunn Undergr Space Technol2005;20:442–51. http://dx.doi.org/10.1016/j.tust.2005.02.005.

[8] Flaga K, Klemczak B, Knoppik-Wróbel A. Early-age thermal–shrinkage crackformation in bridge abutments – experiences and modelling. In: Proceedings ofthe fib symposium, Tel-Aviv, Israel; 2013. p. 435–8.

[9] Klemczak B, Knoppik-Wróbel A. Reinforced concrete tank walls and bridgeabutments: early-age behaviour, analytic approaches and numerical models.Eng Struct 2015;84:233–51. http://dx.doi.org/10.1016/j.engstruct.2014.11.031.

[10] Seruga A, Zych M. Research on thermal cracking of a rectangular RC tank wallunder construction. I: Case study, J Perform Constr Facil. doi:http://dx.doi.org/10.1061/(ASCE)CF.1943-5509.0000704.

386 A. Knoppik-Wróbel, B. Klemczak / Engineering Structures 102 (2015) 369–386

[11] Nilsson M. Restraint factors and partial coefficients for crack risk analyses ofearly age concrete structures. PhD thesis. Luleå University of Technology,Luleå, Sweden; 2003.

[12] Benboudjema F, Torrenti J-M. Early-age behaviour of concrete nuclearcontainments. Nucl Eng Des 2008;238:2495–506. http://dx.doi.org/10.1016/j.nucengdes.2008.04.009.

[13] Prochaczek M. Technical issues of the design of shielding bunker for 235 MeVcyclotron. In: Proceedings of the 9th central european congress on concreteengineering, Wrocław, Poland; 2013.

[14] Knoppik-Wróbel A. Cracking due to restraint stresses in early-age radiationshielding wall. Arch–Civil Eng–Environ 2014;7(3):49–61.

[15] Hösthagen A, Jonasson J-E, Emborg M, Hedlund H, Wallin K, Stelmarczyk M.Thermal crack risk estimations of concrete tunnel segments – EquivalentRestraint Method correlated to empirical observations. Nordic Concr Res2014;49:127–43.

[16] Klemczak B, Knoppik-Wróbel A. Early-age thermal and shrinkage cracks inconcrete structures – description of the problem. Arch–Civil Eng–Environ2011;4(2):35–48.

[17] Al-Gburi M, Jonasson J-E, Nilsson M, Hedlund H, Hösthagen A. Simplifiedmethods for crack risk analyses of early age concrete. Part 1: Development ofEquivalent Restraint Method. Nordic Concr Res 2012;46:17–38.

[18] PN-EN 1992-3. Eurocode 2 – Design of concrete structures. Part 3: Liquidretaining and containment structures; 2008.

[19] Bamforth PB. CIRIA C660: Early-age thermal crack control in concrete. Techrep. CIRIA, London, UK; 2007.

[20] Japanese Concrete Institute. Guidelines for control of cracking of massconcrete; 2008.

[21] Larson M. Thermal crack estimation in early age concrete. Models andmethods for practical application. PhD thesis. Luleå University of Technology,Luleå, Sweden; 2003.

[22] Al-Gburi M, Jonasson J-E, Yousif ST, Nilsson M. Simplified methods for crackrisk analyses of early age concrete. Part 2: Restraint factors for typical casewall-on-slab. Nordic Concr Res 2012;46:39–56.

[23] Waller V, d’Aloïa L, Cussigh F, Lecrux S. Using the maturity method in concretecracking control at early ages. Cement Concr Compos 2004;26:589–99. http://dx.doi.org/10.1016/S0958-9465(03)00080-5.

[24] Kwak H-G, Ha S-J. Non-structural cracking in RC walls. Part II: Quantitativeprediction model. Cement Concr Res 2006;36:761–75. http://dx.doi.org/10.1016/j.cemconres.2005.12.002.

[25] Klemczak B. Modeling thermal–shrinkage stresses in early age massiveconcrete structures – comparative study of basic models. Arch Civil MechEng 2014;14(4):721–33. http://dx.doi.org/10.1016/j.acme.2014.01.002.

[26] Klemczak B. Prediction of coupled heat and moisture transfer in early-agemassive concrete structures. Numer Heat Transfer, Part A: Appl 2011;60(3):212–33. http://dx.doi.org/10.1080/10407782.2011.594416.

[27] Majewski S. Spre _zysto–plastyczny model współpracujacego układu budynek—podło _ze poddanego wpływom górniczych deformacji terenu. SilesianUniversity of Technology, Gliwice, Poland; 1995.

[28] Majewski S. MWW3 – elasto-plastic model for concrete. Arch Civil Eng2004;50(1):11–43.

[29] Klemczak B. Adapting of the Willam–Warnke failure criteria for youngconcrete. Arch Civil Eng 2007;53(2):323–39.

[30] Hillel D. Fundamentals of soil physics. 1st ed. San Diego (USA): AcademicPress; 1998.

[31] Jendele L, Šmilauer V, Cervenka J. Multiscale hydro-thermo-mechanical modelfor early-age and mature concrete structures. Adv Eng Software2014;72:134–46. http://dx.doi.org/10.1016/j.advengsoft.2013.05.002.

[32] Kierno _zycki W. Betonowe konstrukcje masywne. Polski Cement, Cracow,Poland; 2003.

[33] Zreiki J, Bouchelaghema F, Chaouchea M. Early-age behaviour of concrete inmassive structures – experimentation and modelling. Nucl Eng Des2010;240:2643–54. http://dx.doi.org/10.1016/j.nucengdes.2010.07.010.

[34] Endrizzi S, Quinton WL, Marsh P. Modelling of spatial pattern of ground thawin a small basin in the arctic tundra. Cryosphere Discuss 2011;5:367–400.http://dx.doi.org/10.5194/tcd-5-367-2011.

[35] Farouki OT. Thermal properties of soils. Monograph 81-1. US Army ColdRegions Research and Engineering Laboratory, Hanover, USA; 1981.

[36] Hancox NL. The diffusion of water in concrete. Tech rep. UKAEA, Winfrith, UK;1966.

[37] Gawin D. Numerical solution of coupled heat and moisture transfer problemswith phase changes in porous building materials. Arch Civil Eng 1993;34(4):393–412.

[38] Wyrwał J, Szczesny J. Migracja wilgoci wywołana gradientami temperatury wpłytach betonowych poddanych obróbce termicznej. Arch Civil Eng 1986;32(1):421–32.

[39] Clapp RB, Hornberger GM. Empirical equations for some soil hydraulicproperties. Water Resour Res 1978;14(4):601–4.

[40] Philip JR, de Vries DA. Moisture movement in porous materials undertemperature gradients. Trans Am Geophys Union 1957;38:222–32.

[41] Schindler AK, Folliard KJ. Heat of hydration models for cementitious materials.ACI Mater J 2005;102(1):24–33. http://dx.doi.org/10.14359/14246.

[42] Bogue JH. The chemistry of Portland cement. New York (USA): ReinholdPublishing Corporation; 1947.

[43] Rastrup E. Heat of hydration in concrete. Mag Concr Res 1954;6(17):1–14.[44] Neville AM. Properties of concrete. 5th ed. San Francisco (USA): Prentice Hall;

2012.[45] CEB-FIP fib. Model Code 1990; 1991.[46] Guénot I, Torrenti J-M, Laplante P. Stresses in early age concrete: comparison

of different creep models. ACI Mater J 1996;93(3):254–8. http://dx.doi.org/10.14359/9809.

[47] CEB-FIP fib. Model Code 2010; 2013.[48] CEB-FIP fib. Bulletin 70. State-of-the-art report: code-type models for concrete

behaviour. Background of MC2010. Tech rep; 2013.[49] Duncan JM, Chang CY. Nonlinear analysis of stress and strain in soil. ASCE J Soil

Mech Found Div 1970;96(SM5):1629–53.[50] Majewski S, Wandzik G. MWW3 model for concrete – adjustment of failure

and yield surface for use in computational FEM system MAFEM3D. Arch–CivilEng–Environ 2009;2(2):33–44.

[51] Klemczak B, Knoppik-Wróbel A. Early-age thermal and shrinkage cracks inconcrete structures – influence of geometry and dimensions of a structure.Arch–Civil Eng–Environ 2011;4(3):55–70.

[52] Klemczak B, Knoppik-Wróbel A. Comparison of analytical methods forestimation of early-age thermal–shrinkage stresses in RC walls. Arch CivilEng 2013;59:97–117.

[53] Klemczak B, Knoppik-Wróbel A. Comparative study of methods for analysis ofearly-age thermal–shrinkage stresses in externally restrained concretestructures. In: Proceedings of the 10th international conference on newtrends in statics and dynamics of buildings, Bratislava, Slovakia; 2012. p. 41–4.