ARCHIVES OF CIVIL ENGINEERING, LIX, 1, 2013 COMPARISON OF ANALYTICAL METHODS FOR ESTIMATION OF EARLY-AGE THERMAL-SHRINKAGE STRESSES IN RC WALLS

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1 DOI: 1.478/ae-13-4 ACIVES O CIVIL ENGINEEING, LIX, 1, 13 COMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES IN C WALLS B. KLEMCZAK 1, A. KNOPPIK-WÓBEL The volume hanges aused by oupled temperature and moisture variations in early-age onrete elements lead to formation of stresses. If a restraint exists along the ontat surfae of mature onrete against whih a new onrete element has been ast, generated stresses are mostly of a restraint origin. In engineering pratie a wide range of externally restrained onrete elements an be distinguished suh as tank walls or bridge abutments ast against an old set foundation, in whih early-age raking may endanger their durability or funtionality. Therefore, for years methods were being developed to predit early-age stresses and raking risk of externally restrained onrete elements subjeted to early-age thermal-moisture effets. The paper presents the omparative study of the most reognised analytial approahes: the method proposed in EC, the method proposed by ACI Committee 7 and the method developed at the Luleå University of Tehnology. Key words: C wall, thermal-shrinkage stresses, degree of restraint, analytial models. 1. INTODUCTION Thermal and shrinkage stresses arise in early-age onrete as a result of volume hanges due to the temperature and moisture variations during hardening proess (ig. 1). The variations of onrete temperature during uring are onneted with an exothermi nature of the hemial reation between ement and water. In strutural elements with thin setions the generated heat dissipates quikly and auses no problem. In thiker setions the internal temperature an reah a signifiant level. urthermore, due to the poor thermal ondutivity of onrete, high temperature gradients may our between the interior and the surfae of thik strutural elements. Conrete uring is also aompanied with moisture exhange with the environment in onditions of variable temperature. The loss of water trough evaporation at the surfae of element results in shrinkage, whih is lassified as an external drying shrinkage. There is also internal drying resulting from the redution in material volume as water is onsumed by hydration, whih is lassified as 1 PhD, Ds., Assoiate Prof., Silesian University of Tehnology, Akademika 5, 44-1 Gliwie, Barbara.Klemzak@polsl.pl Ms., PhD student, Silesian University of Tehnology, Akademika 5, 44-1 Gliwie, Agnieszka. Knoppik-Wrobel@polsl.pl Unauthentiated

2 98 B. KLEMCZAK, A. KNOPPIK-WÓBEL autogenous shrinkage. Additionally, the hemial shrinkage is also distinguished whih ours beause the volume of hydration produts is less than original volume of ement and water. The volume hanges due to the temperature and moisture variations have onsequenes in arising thermal-shrinkage stresses in a onrete element. a) b) ig. 1. Temperature (a) and moisture ontent (b) development in time for an externally restrained onrete wall [7]. Thermal-shrinkage stresses in externally restrained elements result from a oupled ation of self-indued and restraint stresses with a predominant role of restraint stresses. Self-indued stresses our, too, but their influene is muh smaller. This is mainly the result of temperature and moisture onentration distribution within the wall: even though the differene is observed, it is of a relatively small magnitude (KLEMCZAK [1], KNOPPIK-WÓBEL [], ZYC [3]). Self-indued stresses originate from the material itself due to the internal restraint aused by temperature and moisture gradients. In an internally restrained element stress development is haraterised by formation of ompressive stresses in the interior and tensile stresses on the surfae of the element in the heating phase while in the ooling phase stress body inversion in observed. estraint stresses result from external limitation of deformation, usually aused by mature onrete of previously ast layers. Their magnitude depends on a degree of restraint indued by an older part against a newer part of the struture. The degree of restraint is expressed in a form of an restraint fator, γ, whih in any point of the element is defined as a ratio between the stress generated in an unrestrained element σ to the fixation stress σ fix ([4], [5], NILSSON [6], LASON [7]): fix and may take values between at no restraint to 1 at total fixation. It varies throughout the element with the maximum value at the joint in the mid span and dereasing towards Unauthentiated

3 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 99 free edges of the element. The degree of restrain of the element depends predominantly on the length-to-height ratio of the element, L/, but also on the ratio of stiffness of a newly ast (wall) and a mature (foundation) part. estraint stresses have different harater than self-indued stresses: in the heating phase almost the whole volume of the element is subjeted to ompression while in the ooling phase tensile stresses our. The disussed tensile stresses in the ooling phase an reah a signifiant level and raks an be indued in the struture. These raks, espeially deep or through raks, may adversely affet the servieability, lifespan or even bearing apaity of a onrete struture. Some methods were developed to assess the influene of the degree of restraint on the harater and values of tensile stresses ourring in the phase of the element ooling. The methods are based on the onept of the restraint fator. In this paper three most known approahes are presented: the method given in Euroode Part 3 [4], the method proposed by ACI Committee 7 in the eport on Thermal and Volume Change Effets on Craking of Mass Conrete [5] and the method developed at the Luleå University of Tehnology (NILSSON [6], LASON [7]).. EUOCODE APPOAC Euroode Design of onrete strutures in Part 3: Liquid retaining and ontainment strutures [4] presents a simple, engineering method for determination of restraint stresses generated by self-indued strains and resulting risk of raking in externally restrained walls on the example of tanks and silos. The information and requirements provided in Part 3 should be read in onjuntion with the requirements stated in Part 1-1: General rules and rules for buildings [8]..1. TEMPEATUE AND SINKAGE-INDUCED VOLUME CANGES The standard does not provide speifi suggestions for determination of early-age thermal-moisture effets. It is only stated that where onditions during the onstrution phase are onsidered to be signifiant, the heat evolution harateristis for a partiular ement should generally be obtained from tests. The atual heat evolution should be determined taking aount of the expeted onditions during the early life of the member (e.g. uring, ambient onditions). The maximum temperature rise and the time of ourrene after asting should be established from the mix design, the nature of the formwork, the ambient onditions and the boundary onditions. or determination of restraint strains or stresses it is stated that assumption of elasti behaviour of onrete is satisfatorily preise and that the effet of reep an be onsidered by means of effetive (redued) modulus of elastiity. To estimate the magnitude of the restraint strain free thermal and shrinkage strains must be determined and the restraint must be known. The total thermal-shrinkage strain ε tot (t) is a sum of thermal strain ε T (t) and shrinkage strain ε s (t): Unauthentiated

4 1 B. KLEMCZAK, A. KNOPPIK-WÓBEL (.1) t T t t where: tot ε T = α T ΔT thermal strain, ΔT is the temperature differene, C; ε s = ε d + ε a total shrinkage strain: drying shrinkage and autogenous shrinkage strain; α T oeffiient of thermal expansion, 1/ C; the basi value of α T is given as equal to / C but the standard emphasises variability of thermal expansion oeffiient depending on the type of aggregate and moisture ontent in onrete mix. s.. ESTAINT STESSES The stress at any level y in unraked setion due to translational and rotational restraint of the element an be alulated based on the known imposed strain from the equation: y E y y m, eff in whih the atual strain at level y, ε a (y), is given by: where: ax y 1 1 y a ax i m y av en fator defining the degree of external axial restraint provided by elements attahed to the element onsidered. The restraint fators may be alulated on the base of the stiffness of the element onsidered and the members attahed to it. Alternatively, pratial axial restraint fators for ommon situations may be taken from ig. ; m fator defining the degree of moment restraint provided by elements attahed to the element onsidered. In most ommon ases m may be taken as 1.; E m,eff effetive modulus of elastiity of onrete allowing for reep as appropriate; ε iav average imposed strain in the element (i.e. the average strain whih would our if the member was ompletely unrestrained); ε a (y) atual strain at level y; ε i (y) imposed strain at level y; y height from joint plane to setion; y en height from joint plane to setion entroid; 1/r urvature of joint plane (if the restraining element is flexible). i a 1 r Unauthentiated

5 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 11 ig.. Axial restraint fator ax in element restrain along one edge [4]. In the simplified approah when no rotational restraint is onsidered, the problem an be redued to a planar problem (no distribution of strain at the thikness of ross-setion is onsidered), ε i = ε iav. The average imposed strain an be taken as total thermal-shrinkage strain ε tot alulated a. to Eq. (.1). The stress at any level y of entre setion an be then onventionally expressed with use of the restraint fator: (.) γ = ax as: σ = ax E m,eff ε tot The values of material parameters must be defined for the atual age of onrete orreted for temperature effets and influene of reep. The hange of material parameters an be taken aording to [8]. The mean modulus of elastiity development is given by the equation: (.3) E t where: m f f m E m mean 8-day modulus of elastiity of onrete, GPa; f m mean 8-day ompressive strength of onrete, MPa; f m (t) mean ompressive strength of onrete taking into onsideration onrete age, MPa, a. to the equation: f m (t) = β (t) f m ; β (t) oeffiient dependent on the age of onrete. t m.3 E m Unauthentiated

6 1 B. KLEMCZAK, A. KNOPPIK-WÓBEL The effet of reep an be inluded by redution of the modulus of elastiity, however, it is not stated neither in [4] nor in [8] how the effet of reep on stress relaxation in early-age onrete should be onsidered. Nevertheless, it is given that for onrete heated prior to loading, the reep oeffiient may be assumed to inrease with inrease in temperature above normal (assumed as C) by the appropriate fator given [4], hene the effet of self-heating of onrete due to hydration in onstrution stages an be inluded in alulations. 3. ACI COMMITTEE 7 APPOAC The instrutions given in the eport 7.-7 [6] are primarily onerned with ontrol of raking in elements that ours mainly from thermal ontration with restraint, hene, it fouses on evaluation of thermal behaviour of massive onrete strutures. Moisture hanges are also disussed but to a limited extent. Other volume hanges like alkali-aggregate expansion, autogenous shrinkage and hanges due to expansive ement are not onsidered. The report presents a detailed disussion on the effets of heat generation and volume hanges in massive onrete elements, methods to ompute heat dissipation and volume hanges as well as to determine mass and surfae gradient stresses TEMPEATUE AND SINKAGE-INDUCED VOLUME CANGES The maximum effetive temperature hange depends on four basi temperature determinations: the effetive plaing temperature, the final or operating temperature of onrete, the temperature rise of onrete due to hydration and the equivalent temperature hange to ompensate for drying shrinkage. The volume hange that leads to thermal raking results from the temperature differene between the peak temperature of onrete attained during early hydration and the minimum temperature to whih the element will be subjeted under servie onditions. The report assumes a ondition of no initial stress as the initial hydration temperature rise produes little stress in onrete at early age the modulus of elastiity of onrete is so small that ompressive stresses indued by the rise in temperature are insignifiant even in zones of full restraint and, in addition, are relaxed by a high rate of reep. The thermal volume hange ΔV an be omputed as: where: V T f T i Tad Tenv T T f T i final stable temperature of onrete, C; initial plaing temperature of onrete, C; Unauthentiated

7 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 13 T ad adiabati temperature rise of onrete, C; T env temperature hange due to heat added or subtrated from onrete due to environmental onditions, C; α T oeffiient of thermal expansion of onrete, 1/ C. The report proposes methods for determination of eah of the omponents. If no speial treatment was applied, the plaing temperature an be taken as equal to the air temperature at the moment of plaement. The minimum expeted final temperature of a onrete element (final temperature in servie) an be taken as the average minimum exposure temperature ourring during a period of approximately 1 week. Instrutions are given to estimate adiabati temperature rise if no speifi data are provided onerning heat generation in massive onrete elements for a given type and amount of ement ontained. Nevertheless, it is stated that the data should be taken diretly from laboratory tests for speifi ements. Drying shrinkage was expressed in terms of equivalent hange in onrete temperature T DS, aording to the equation: 1V Wu 15 (3.1) T DS 3 S 1 where: W u water ontent of fresh onrete, lb/yd 3, but not less than 5 lb/yd 3 (133 kg/m 3 ); V total volume, in 3 ; S area of the exposed surfae, in. Eq. (3.1) is an empirial equation and the resultant temperature is given in. The approximation assumes equivalent drying shrinkage of and an expansion oeffiient of / C. 3.. ESTAINT STESSES It is defined that the stress at any point in an unraked onrete member is proportional to the strain in onrete. The horizontal stress in a member ontinuously restrained at its base and subjeted to an otherwise uniform horizontal length hange varies from point to point in aordane with the variation in degree of restraint throughout the member. Two restraint fators have been developed to fully model the restraint onditions on a massive struture: the strutural shape restraint fator K and the foundation restraint fator K. Distribution of the K restraint varies with the length-to-height ratio (L / ) of the element. The distribution of the K restraint at entre setion is shown in ig. 3a. The following approximation of K distribution is proposed: Unauthentiated

8 14 B. KLEMCZAK, A. KNOPPIK-WÓBEL K L L L L y y if if L L.5.5 where y signifies loation above the onstrution joint. a) b) ig. 3. a) Strutural shape restraint fator K at entre setion [5], b) Basi resiliene fator s at entre setion [6]. The stresses in onrete due to restraint derease in diret proportion to the derease in stiffness of the restraining foundation material. The restraint fator K has been approximated as: 1 K A E 1 A E where: A gross area of onrete ross-setion, m ; A area of foundation or other element restraining shortening of element, generally taken as a plane surfae at ontat, m ; E modulus of elastiity of wall, GPa; E modulus of elastiity of foundation or restraining element, GPa. The restraint fator is hene given by the equation: res (3.) γ = K K Unauthentiated

9 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 15 and tensile stress at any point on the entreline due to a derease in length ε tot an be alulated as: σ = K K ε tot E m,eff where E m,eff signifies a sustained modulus of elastiity of onrete at the time when thermal-shrinkage strains ε tot ourred and for the duration involved. The value of the modulus of elastiity of onrete an be given taking into aount its development in time related to the onrete temperature development as well as taking into onsideration the influene of reep. Predition of reep is disussed in ACI eport 9, however, it is not designated to early-age onrete. 4. LULEÅ UNIVESITY O TECNOLOGY APPOAC 4.1. TEMPEATUE AND SINKAGE-INDUCED VOLUME CANGES The third approah, derived at the Luleå University of Tehnology, Sweden, is more omplex (NILSSON [6], LASON [7]). The method fouses on estimation of raking risk by means of determination of the restraint fator. It should be emphasised that it has not been larified how the restraint oeffiient shall be applied in a thermal stress analysis. The method does not provide instrutions for determination of temperature and moisture fields. Volume hange is expressed in a form of inelasti strain ε tot (t) originating from a ombination of thermal expansion ε T (t) and autogenous shrinkage ε AD (t): ε tot (t) = ε T (t) + ε AD (t) The disussed approah suggests that parameters required for alulation of thermal and shrinkage strains depend on a onrete mix omposition and should be determined from laboratory tests. 4.. ESTAINT STESSES In this approah the deisive restraint oeffiient is defined as: where:, res, slip the plane-setion restraint oeffiient, whih depends on the geometry of the struture as well as both the rotational rx ry, and translational t boundary restraints from underlying foundation materials; δ res the resiliene fator onsidering the non-linear effets in high walls (typially walls with L / < 5 where plane-setion theory is no longer valid), a. to ig. 3b; Unauthentiated

10 16 B. KLEMCZAK, A. KNOPPIK-WÓBEL δ slip the slip fator whih depits a restraint stresses redution as a result of slip failure in asting joints (usually ourring in walls with L / < 5). The deisive restraint oeffiient distribution at height y in entral setion of a wall an be alulated aording to the following equation: res slip res slip t rx ry or the walls whih are not affeted by the high-walls effets, the formula simplifies to: 1 The oeffiient is alulated based on the geometry of the wall and its foundation: y where: E 1 E B x 1 1 B B en, eff B 1 x en, eff y B en.5 B E E, eff y y y.5 en E E B B, B en B, B eff B, 1 eff eff 1 x en y en x, y oordinates of the analysed point, m; x en, y en oordinates of the entroid of transformed setion (onsidering old and new part of the element), m; height of the wall, m; B thikness of the wall, m; E modulus of elastiity of the wall at the moment of analysis, GPa; depth of foundation, m; B,eff effetive width of foundation, m. The effetive width of foundation is determined to omply with the urvature obtained in E-analysis; E modulus of elastiity of foundation at the moment of analysis, GPa; ω loation fator (defines relative loation of the wall with respet to the entral axis of foundation). When a slip failure in the joint is possible and if the setions do not remain plane under deformation (high walls effet), the restraint in the wall is defined as: Unauthentiated

11 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 17 (4.1) y n n n a ai ai i y en y yen i1 i i i i y i res E B,eff E B,eff 1 y en yen E B E B 1 1 x. B B en 5,eff slip B B B B B,eff E,eff,eff x en x en E B 1 1 where a i are oeffiients of polynomial funtion desribing resiliene fator distribution. Tensile stress an be alulated as: The fixation stress is given by a formula: fix t t t t fix, dtot t where (t,t ) is a relaxation funtion at time t for loading at the age t (in equivalent time determined a. to Arrhenius equation). or a known reep funtion φ(t,t ) it may be expressed as: t,t E t m t,t 1 1 t,t where χ is the ageing oeffiient. The proposed reep funtion was derived from Linear Logarithmi Model, broadly desribed by LASON [7], with parameters to be evaluated from laboratory tests. The development of modulus of elastiity is defined as: E m t E where E m is the 8-day modulus of elastiity of onrete and β E (t) is the relative development expressed as a funtion of equivalent ages, with parameters to be evaluated from laboratory tests. m E t Unauthentiated

12 18 B. KLEMCZAK, A. KNOPPIK-WÓBEL 5. COMPAISON O ANALYTICAL MODELS: ANALYSIS O C WALLS 5.1. MATEIAL, TECNOLOGICAL AND GEOMETICAL DATA In the analysis the distribution of thermal stresses of six walls was investigated. The walls differed in dimensions: walls of thikness 4 and 7 m and length of 8, 16 and 8 m. Geometrial harateristis of the analysed walls are presented in Table 1. Aording to lassifiation proposed by LAGA [9] all the walls are medium-thik elements in whih expeted temperature rise should not exeed C. Moreover, the same value of thermal-moisture influene measure in a form of apparent surfae modulus/equivalent thikness for the walls with the same thikness suggests that the expeted self- -indued part of thermal-shrinkage stress will be of similar magnitude. On the other hand, when analysing the linear restraint expressed in a form of length-to-height ratio of the wall, L /, distribution of total stress (enountering restraint part of stress) should differ between the walls with different L / and higher restraint stresses will be observed in thinner walls due to relatively greater stiffness of restraining foundation. Conrete lass C5/3 was assumed for both the wall and the foundation. or the foundation the material properties were assumed as for 8-day onrete. or the wall, development of material properties in time onsidering influene of temperature and reep was assumed. Detailed material properties, environmental and tehnologial onditions are listed in Table 1, while parameters for numerial, thermal analysis are given in Table. 5.. TEMAL ANALYSIS The analysis foused on strutural behaviour of the walls due to the influene of thermal effets beause this part of the analysis was ommon for all the presented methods and as suh allows of omparison. To ompare the methods, temperature development was determined by means of numerial analysis with use of program TEMWIL developed by KLEMCZAK [1]. It was observed that temperature development and distribution in the entre ross-setion of the wall depends only on the thikness of the wall. igure 4a presents the map of thermal fields at the moment of reahing of the maximum hardening temperature in a hosen wall. It an be observed that onentration of high temperature ours in the entral part of the wall while beause of heat dissipation the values of temperature derease towards the exposed surfaes. igure 4b presents diagrams of temperature development in time in the loation where the maximum temperature is observed in the wall, i.e. in its entral part. It an be notied that the maximum hardening temperatures in the walls were equal to 37.3 C after 1.5 days in 7-m thik walls and 3.9 C after 1.1 days in 4-m thik walls. Unauthentiated

13 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 19 Geometrial, material and tehnologial data for the analysed walls Table 1 L_8,d_.4 L_16,d_.4 L_8,d_.4 L_8,d_.7 L_16,d_.7 L_8,d_.7 GEOMETICAL DATA thikness B [m].4.7 wall length L [m] height [m] 4 depth [m].7 foundation length L [m] width B [m] 4 geometrial harateristis L / A /A MATEIAL DATA onrete lass C5/3 α T.1/ C water 17 l/m 3 ement CEM III 4.5N, 35 kg/m 3 mix omp. 8-days mehanial properties fine aggregate sand, 675 kg/m 3 oarse aggregate (rounded aggregate) dolomite, 1139 kg/m 3 additives & admixtures f m f tm E m 6 kg/m 3 33 MPa.6 MPa 31 GPa TECNOLOGICAL DATA environmental harateristis tehnologial harateristis ambient temperature T a C relative humidity 6% wind veloity v 4 m/s initial temp. of onrete mix T i C formwork 1.8-m plywood on side surfaes, removed after 8 days, top surfae proteted with PE foil Unauthentiated

14 11 B. KLEMCZAK, A. KNOPPIK-WÓBEL Parameters for numerial analysis (temperature fields) Table Coeffiient of thermal ondutivity λ 3.3 W/(mK) Speifi heat b 1. kj/(kgk) Density of onrete ρ 34 kg/m 3 Thermal transfer oeffiient α p without protetion 15. W/(m K) 1.8-m plywood formwork 5.58 W/(m K) PE foil W/(m K) eat of hydration Q(T,t) A. to equation for CEM III 4.5N:.5 e Q T, t Q e at.93 where Q = 469 kj/kg, a = 475.7t e a) b) ig. 4. Distribution of temperature in the wall: a) map of thermal fields (¼ of the L_8,d_.7 wall) at the moment of reahing the maximum hardening temperature, b) diagram of temperature development in time the entre setion for the L_8 wall with.7 m and.4 m thikness TEMAL STESSES The analysis of strutural behaviour of the investigated walls foused on determination of total thermal stress distribution, espeially the influene of restraint stresses, in the entre setion of the wall aording to the presented analytial methods. Tensile stresses were determined from tensile strains resulting from ontration of the wall in the ooling phase and the omputed degree of restraint. Temperature development was taken from the numerial analysis. Thermal strain hange in any loation in the entre setion of the wall above the joint y in a time step Δt when temperature differene ΔT was observed was alulated as: Unauthentiated

15 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 111 and tensile fixation stress in any time t as: fix T t, y T t, yt t y t, y E t, y, T, i m, eff, mean, i i where E m,eff,mean,i signifies mean sustained modulus of elastiity of onrete in a i th time step. Development of the modulus of elastiity in time was alulated a. to Eq. (.3). The effet of inreased temperature was negleted to apply equal values of the moduli. The influene of reep was onsidered by redution of modulus of elastiity of onrete (KIENOŻYCKI [11]): E m, eff Em( t) ( t) 1 t where: β 1 ageing oeffiient, an be taken as.8; φ p t r, t reep oeffiient, an be taken as.6. The total thermal stress at any time t and any loation y on a entreline was alulated as: fix (5.1) t, y t, y t, y where γ is a restraint fator a. to Eq. (.), (3.) and (4.1), for EC, ACI and Luleå approah, respetively. In order to ompare qualitatively the results obtained with different methods the effet of rotational restraint in the definition of the restraint fator was negleted. 1 p r, t Uniform temperature distribution In the first approah, a uniform distribution of temperature in the whole volume of the wall was assumed. Development of temperature in the walls was assumed as given in diagram in ig. 5d. Stresses were determined after 8 days. irstly, in eah ase thermal strain was alulated for total temperature hange during ooling, i.e. ΔT = T max T 8d whih was equal to ΔT = 17. C for 7-m thik and 1.8 C for 4-m thik walls. The resulting thermal strains were equal to ε T = and , respetively. If the mean values of moduli of elastiity were taken E m,eff,mean = GPa for 7-m thik walls and GPa for 4-m thik walls the resulting fixation stresses were equal to σ fix =.96 MPa and.15 MPa. or the final, 8-day values of the moduli (E m,eff,8 =.95 GPa) the fixation stresses were Unauthentiated

16 11 B. KLEMCZAK, A. KNOPPIK-WÓBEL equal to σ fix = 3.56 MPa and.68 MPa. Seondly, thermal strains were numerially integrated in the ooling phase (after the maximum temperature ours) in time steps Δt = hours. In eah step a temperature differene was determined and a resulting strain was omputed. A mean value of the modulus of elastiity was taken in eah step, too. The final values of fixation stresses after 8 days were equal to.3 MPa for 4-m thik walls and.99 MPa for 7-m thik walls. This yields the question of how to assume the mean value of the modulus of elastiity in manual alulations when total strain is onsidered. a) b) ) d) ig. 5. The ase of uniform distribution of temperature: distribution of total thermal stress in entre rosssetion of the walls aording to EC (a), ACI (b) and Luleå () method, d) assumed uniform temperature at the height of the wall (at the moment of reahing the maximum hardening temperature) and fixation stress distribution for the walls with.4 m and.7 m thikness. Unauthentiated

17 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 113 Total stresses were alulated with use of the restrain fator aording to Eq. (5.1), for the final values of fixation stresses after 8 days equal to.3 MPa for 4-m thik walls and.99 MPa for 7-m thik walls. Distribution of stresses at the height of the walls was presented in ig. 5. Table 3 presents the values of the restraint fator and total stress at the bottom and top of the walls. Comparing the results obtained by means of all methods a tendeny an be observed that with the inreasing L / ratio the influene of the restraint at the height dereases upper fibres are allowed to deform more freely. Nevertheless, some differenes may be also notied. Table 3 estraint fator and total thermal stress in hosen loations of the walls a. to different methods EC ACI Luleå γ σ fix γ [MPa] γ σ fix γ [MPa] γ σ fix γ [MPa] L_8,d_.4 L_16,d_.4 L_8,d_.4 L_8,d_.7 L_16,d_.7 L_8,d_.7 y= y= y= y= y= y= y= y= y= y= y= y= The results of the EC show a visible and straightforward differene in the values of stresses reahed in the walls of different thiknesses the maximum value of stress depends only on the thikness of the wall (the fixation stress). This results from the fat that the restraint fator in suh an approah depends only on the L / ratio and the influene of relative foundation stiffness is not onsidered. A similar observation an be made in ase of the stress distribution obtained with use of ACI method, however, the differenes in stresses between the walls of different thiknesses are lower. This an be explained by the fat that although higher fixation stresses an be expeted in thiker walls the restraint exerted by the foundation is in this ase lower so the resultant total stresses differ by a smaller magnitude. In the results obtained with the Luleå approah a strong influene of both the thikness (fixation stresses and relative stiffness of foundation) and length (L / ratio) is Unauthentiated

18 114 B. KLEMCZAK, A. KNOPPIK-WÓBEL visible. It an be summarised that in the walls with lower L / ratio the effet of the restraint onentrates near the joint; in ontrary, in the walls with higher L / ratio the effet of the restraint is more uniform at the whole height of the wall. ene, this approah an be treated as an extension to previous methods Non-uniform distribution of temperature Calulations were made aording to the proedure presented in setion but taking into onsideration real distribution of temperature at the height of the wall. As real thermal fields are non-stationary fields (temperature differs from point to point), a semi-numerial approah was required in whih the wall was divided into a finite number of segments with the same value of temperature assigned. The alulations were made for two walls the L_8,d_.7 wall and the L_16,d_.7 wall. Distribution of maximum temperature arising in the wall and resulting fixation stresses are presented in ig. 6a and ig. 7a. Total thermal stresses along the height of the walls and the restraint fators distribution are shown in ig. 6b, and 7b,. This analysis is foused on determination of the deisive point. Total thermal stresses are a ombination of self-indued stresses, aused by temperature differenes, and restraint stresses, aused by the influene of a linear restraint of foundation. As it an be observed in diagrams in ig. 6 and 7, the restraint oeffiient is a measure that defines inlination of the total stress urve in ross-setion. Therefore, if the value of the restraint oeffiient is more uniform whih is harateristi for the walls with high L / ratio total stress distribution is also more uniform, qualitatively similar to fixation stress. Nevertheless, as fixation stresses derease towards the edges of the wall, the maximum value of the total stress is not observed at the wall foundation joint but at some height above the joint referred to as a deisive point. The loations of the deisive points and the values of stresses in this loations are presented in Table 4. The results obtained with Luleå approah give the highest values of total stresses, espeially onerning the stress in the deisive point, beause of the high values of the restraint assumed. The lowest values are obtained with use of the ACI approah. No diret relationship between the approah and the loation of the deisive point an be made beause the influene of the restraint aused by the relative stiffness of the foundation is inluded in a different manner than in the other methods. In the Luleå approah it is suggested that the deisive point an be assumed one wall thikness above the joint and this is satisfied by the results obtained with use of this method. Nevertheless, the loation of the deisive point varies from ase to ase. In general it an be said that for the walls whih differ only in geometry of the wall the deisive point is loated the losest to the joint in thinner walls with low L / ratio and is being observed higher as the thikness of the wall and L / ratio inrease. That tendeny in visible in the EC and ACI approah. Unauthentiated

19 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 115 a) b) ) ig. 6. The ase of non-uniform distribution of temperature wall L_8,d_.7: a) distribution of temperature at the height of the wall (at the moment of reahing the maximum hardening temperature) and fixation thermal stress, b) restrain fator at the height, ) total thermal stress at the height after 8 days. a) b) ) ig. 7. The ase of non-uniform distribution of temperature wall L_16,d_.7: a) distribution of temperature at the height of the wall (at the moment of reahing the maximum hardening temperature) and fixation thermal stress, b) restrain fator at the height, ) total thermal stress at the height after 8 days Deisive points a. to different methods: loation and the value of total stress Table 4 EC ACI Luleå y [m] σ(y) [MPa] y [m] σ(y) [MPa] y [m] σ(y) [MPa] L_16,d_ L_8,d_ Unauthentiated

20 116 B. KLEMCZAK, A. KNOPPIK-WÓBEL 6. CONCLUSIONS Estimation of raking risk in onrete elements is ruial to ensure their durability and funtionality. In ase of early-age onrete elements this task is espeially diffiult beause of a omplex nature of the material and imposed loading. or years analytial methods were being developed to assess the harater and values of tensile stresses ourring in the phase of onrete element ooling. The derived methods are based on the onept of the restraint fator. In this paper the most known approahes are desribed and analysed: the method proposed in EC [4], the method proposed by ACI Committee 7 [5] and the method developed at the Luleå University of Tehnology [6, 7]. The results obtained for the walls with use of these methods are ompared. The following onlusions an be drawn from the analysis: the proedure proposed by Euroode is a useful method under the assumption that the problem an be redued to a planar problem, the restraining element is indeformable and that total rotational restraint exists. or broader use of the method some parameters, suh as the atual restraint or spatial thermal-moisture fields must be determined whih is possible either numerially or by tests; the proedure proposed by ACI report is a simple approah whih easily provides a designer with estimation of strutural behaviour of an externally restraint onrete element. Nevertheless, the resultant stresses reah relatively low values in omparison to other analytial methods so its quantitative validity should be ontrolled; the proedure proposed by the Luleå team is the most omplex beause it takes into onsideration a great number of fators. In the most simplified form it an be easily applied for manual alulations, however, more extensive appliation requires omputer-aided alulations or tests. The approah in a present form may pose some diffiulties to a potential user but it has a great potential and should be further investigated; definition of the deisive point is vital for sane appliation of analytial approahes. The loation of the deisive point varies from ase to ase and is proposed to be assumed one wall thikness above the joint (NILSSON [6]), however, it hanges not only with the thikness but also with the L / of the element (KLEMCZAK, KNOPPIK- -WÓBEL [1]). Quantitative definition of the deisive point loation is required; the values and the distribution of stresses at the height of the wall depend not only on the applied analytial method but also on the assumed distribution of temperature at the height of the wall. Two ases are disussed in the paper. In the first approah, a uniform distribution of temperature at the height of the wall was assumed suh assumption is ommonly used in analytial methods beause in these methods the temperature rise in the element is also estimated in a simplified way. In the seond approah, real distribution of temperature at the height of the wall was onsidered. It is worth mentioning that stress distributions obtained from the seond approah are in good onformity with the results from the numerial tests (KLEMCZAK, KNOP- PIK-WÓBEL [1]). Suh an assumption enables preise analysis of total stress distri- Unauthentiated

21 C OMPAISON O ANALYTICAL METODS O ESTIMATION O EALY-AGE TEMAL-SINKAGE STESSES 117 bution but the knowledge of temperature distribution yields neessity of numerial alulations; the further researh should fous on verifiation of the results obtained with different analytial methods with laboratory and numerial tests. ACKNOWLEDGEMENTS This paper was done as a part of a researh projet N N Numerial predition of raking risk and methods of its redution in massive and medium-thik onrete strutures funded by Polish National Siene Centre. Co-author of this paper A. Knoppik-Wróbel is a sholar under the projet SWIT POKL /1 o-finaned with the European Union. EEENCES 1. B. KLEMCZAK, A. KNOPPIK-WÓBEL, Analiza naprężeń w śianie żelbetowej poddanej wzesnym wpływom termizno-skurzowym (Analysis of stresses in reinfored onrete wall subjeted to early-age thermal-shrinkage effets). Zeszyty Naukowe Politehniki zeszowskiej. Budownitwo i Inżynieria Środowiska, 59, 3, 85-9, 1.. A. KNOPPIK-WÓBEL, The influene of self-indued and restraint stresses on rak development in reinfored onrete wall subjeted to early-age thermal-shrinkage effets. Proeedings of 14th International Conferene of Postgraduate Students Juniorstav, 16, Brno M. ZYC, Analiza pray śian zbiorników żelbetowyh we wzesnym okresie dojrzewania betonu w aspekie ih wodoszzelnośi (Analysis of work of C tank walls in early ages of onrete uring in the view of their water tightness), PhD Thesis, aulty of Civil Engineering, Craow Tehnial University, EN Euroode Design of onrete strutures Part 3: Liquid retaining and ontainment strutures. 5. ACI Committee 7. eport on Thermal and Volume Change Effets on Craking of Mass Conrete (ACI 7.-7). ACI Manual of Conrete Pratie, Part 1, M. NILSSON, estraint fators and partial oeffi ients for rak risk analyses of early age onrete Strutures. PhD Thesis, Luleå University of Tehnology, M. LASON, Thermal rak estimation in early age onrete: models and methods for pratial appliation. PhD Thesis, Luleå University of Tehnology, EN Euroode Design of onrete strutures Part 1-1: General rules and rules for buildings. 9. K. LAGA, Naprężenia skurzowe i zbrojenie przypowierzhniowe w konstrukjah betonowyh (Shrinkage stresses and surfae reinforement in onrete strutures). Monograph 95, Wydawnitwo Politehniki Krakowskiej, Kraków B. KLEMCZAK, Predition of Coupled eat and Moisture Transfer in Early-Age Massive Conrete Strutures. Numerial eat Transfer. Part A: Appliations, 3, 1-33, W. KIENOŻYCKI, Betonowe konstrukje masywne (Massive onrete strutures). Wydawnitwo Polski Cement, Kraków 3. eeived:.11.1 evised: Unauthentiated