Analytical model for evaluation of thermal–shrinkage strains and stresses in RC wall-on-slab structures

archives of civil and mechanical engineering 17 (2017) 75–95

Available online at www.sciencedirect.com

ScienceDirect journal homepage: http://www.elsevier.com/locate/acme

Original Research Article

Analytical model for evaluation of thermal–shrinkage strains and stresses in RC wall-on-slab structures Barbara Klemczak a,*, Kazimierz Flaga b, Agnieszka Knoppik-Wróbel a a b

Faculty of Civil Engineering, Silesian University of Technology, Akademicka 5, 44-100 Gliwice, Poland Faculty of Civil Engineering, Cracow University of Technology, Warszawska 24, 31-155 Kraków, Poland

article info

abstract

Article history:

Recent experiences have shown that thermal–shrinkage cracks in reinforced concrete walls

Received 1 April 2016

are a common phenomenon. The cracks appear above the joint between the wall and the

Accepted 31 August 2016

foundation at the construction phase of these structural members. This problem affects,

Available online

among the others, bridge abutments, retaining walls, tank walls and radiation protection shields, in which cracking is highly undesirable due to tightness requirements. Prediction of

Keywords:

the early-age thermal and shrinkage effects is not an easy task because of the complexity of

Early-age concrete

the issue and a large number of contributing technological and material factors deciding

Wall-on-slab structure

about the magnitude and character of early-age volume changes.

Temperature Shrinkage Restraint stresses

This paper presents a complete analytical model for determination of hardening temperature, shrinkage deformations and thermal–shrinkage stresses in early stages of hardening of reinforced concrete walls cast against previously executed foundation. As a basis for the model development, numerical analysis of 39 walls was performed in which the analysed walls had various dimensions and were made of concretes with different types of cements and aggregates. A calculation method for determination of stresses proposed by Eurocode 2 is also referred. # 2016 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.

1.

Introduction

In most cases construction of a wall-on-slab structure starts with casting of the foundation (phase I), shallow or deep, after 1–3 months followed by execution of the rest of a loadbearing system in a form of a wall. At the moment of execution of wall concrete the foundation concrete is already cooled down (hydration heat has been released to the

environment) and a part of shrinkage strains has occurred. The temperature in the wall increases for the first 2–3 days (50–80 h), after which the temperature starts to decrease slowly and after, on average, 1–2 weeks the wall reaches the temperature of the surrounding air (Fig. 1a). At the same time, in the wall there occur moisture content changes which result from water transfer to the environment; part of the water is also bound in the process of cement hydration (Fig. 1c).

* Corresponding author. E-mail address: [email protected] (B. Klemczak). http://dx.doi.org/10.1016/j.acme.2016.08.006 1644-9665/# 2016 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.

76

archives of civil and mechanical engineering 17 (2017) 75–95

Fig. 1 – Time development during hardening of reinforced concrete wall: (a) hardening temperature, (b) restraint thermal stresses, (c) moisture content in hardening concrete, (d) restraint shrinkage stresses.

Walls in the heating phase undergo significant self-heating due to the so-called thermal shock (first 50–80 h of concrete hardening), which may lead to formation of considerable thermal strains in these members. Displacement of the wall with respect to the foundation is restrained by transverse reinforcement in the joint and developing in time bond forces between the concretes of the wall and the foundation. In the first phase of this bond development (heating phase) compression of the concrete of the wall can be observed due to thermal expansion of the wall concrete (Fig. 1b). After the maximum temperature is reached, cooling phase begins in which inversion of stresses occurs. In the cooling phase, when heated concrete of the wall cools down and shrinks, compressive stresses decrease rapidly and tensile stresses appear due to significant restraint of the previously cast foundation (Fig. 1b). At this stage bond at the joint between the two concretes develops considerably. Tensile force formed in this zone causes eccentric tension of the wall and eccentric compression of the foundation. Diagrams in Fig. 1 depict the processes of time-development of temperature, moisture content and thermal–shrinkage stresses in the discussed walls. In reality the change of temperature and moisture content is not uniform in the cross-section of the wall due to

heat and moisture exchange with the environment. As a result, the values of generated stresses differ in different locations in the wall due to different values of temperature and moisture content, as well as due to variable degree of restrain of the wall at its length. Nevertheless, the described changes exhibit a character similar to the one shown in Fig. 1 in most areas of the wall. For example, spatial stress distribution in the heating and cooling phase in the wall is presented in Fig. 2. More detailed discussion of the character and distribution of thermal–shrinkage stresses can be found in the works [1–7]. The discussed thermal–shrinkage stresses often reach considerable values and may lead to cracking of a structure, which is of high interest from durability point of view. In general, it can be said that the crack is formed when tensile stress in a given location in the structure exceeds tensile strength of concrete in this location. In the discussed structures crucial are restraint tensile stresses induced by a linear external restraint formed at the joint between the wall and the previously cast foundation [5–7]. Thus, a potential crack can be formed in the cooling phase in the wall (Fig. 3a). Experiences of realisations of the walls show that the first crack appears not directly at the joint between the wall and the foundation but at some height above this joint. Also the greatest width of these cracks is observed above the joint (Fig. 3b). It can be explained by distribution of stresses at the height of the wall – the highest value of tensile stress occurs above the joint (Fig. 4), which has been proven in the works [4–7]. Assessment of the risk of thermal–shrinkage crack formation in wall-on-slab structures is a difficult task because of a large number of contributing factors deciding about the magnitude and character of early-age volumetric changes. It is also an untypical task – the source of the loads is the material of which the structure is made. It must be also remembered that these changes occur in the material which mechanical properties change in time. Experiences gathered during realisations of massive concrete structure are of importance here, so is the ability to predict the magnitude and character of early-age thermal and shrinkage effects occurring during hardening of concrete at the design phase of new structures. Therefore, determination of the final effect of thermal–shrinkage loads action, which is potential cracking of a structure, requires calculation of thermal–shrinkage strains and then of distribution and magnitude of resulting stresses. For assessment of the discussed influences analytical or numerical methods can be used [7]. Numerical methods allow for precise recognition of thermal–moisture fields and generated stresses, however, their use requires specific software [8–13]. That is why analytical models are useful for that purpose, which aid preliminary evaluation of the magnitude of thermal–shrinkage stresses and cracking risk as well as answer a frequent designers' question: would thermal– moisture effects have an important impact on the structure that I am designing and may they lead to its cracking? Several authors have contributed to the field of analytical modelling of stresses induced in wall-on-slab structure. The distribution of the thermal–shrinkage stresses at the height of the wall depending on its slenderness has been investigated by Schleeh [14], Rüsch and Jungwirth [15]. Rostásy and Henning [16] also provide equations allowing the determination of the

archives of civil and mechanical engineering 17 (2017) 75–95

77

Fig. 2 – Maps of an exemplary reinforced concrete wall: (a) distribution of temperature after 2 days of concrete hardening, (b) distribution of moisture content after 14 days of concrete hardening, (c) map of stresses in heating phase, (d) map of stresses in cooling phase.

forces in the wall-on-slab, depending on the relative stiffness of the wall and foundation. Another proposals of analytical models can be found in, among the other, ACI Committee 207 Report [17,18], JSCE Standard [19], JCI Guideline [20], CIRIA C660 [21], Eurocode 2-3 [22], and other works [23,24]. Nevertheless, most of the analytical models concentrate on determination of restraint thermal–shrinkage stresses, referring in a limited range or not referring at all to the methods of determination of temperature of concrete resulting from the exothermal process of cement hydration (e.g. EC2 standard provides no such recommendations). The paper presents a simple analytical method for determination of early-age thermal–moisture effects in reinforced concrete walls cast against previously executed foundation. The proposed method covers:

– determination of the maximum and mean hardening temperature of concrete and thermal strains, – determination of shrinkage strains, – determination of thermal–shrinkage stresses. It should be mentioned that the analytical procedure related to the stress calculation, presented in next chapters, can be considered as similar to the aforementioned approaches [14–16]. Numerical validation of the analytical model is also presented in the paper. Analytical and numerical calculations were performed on 39 walls with different dimensions and concrete mix composition. The results of these calculations were compared regarding distribution of temperature in the

78

archives of civil and mechanical engineering 17 (2017) 75–95

foundation. As it has already been mentioned, in such types of walls restraint tensile stresses have a dominant role, that is why the proposed procedure relates to determination of the restraint stresses in the concrete cooling phase, the heating phase is not considered. This is a simplified approach, because the previously described, two-phase character of the stresses induced in the wall is omitted. Nevertheless, the good compliance between the results from the proposed procedure and full two-phase, 3D numerical analysis have been obtained. For calculation of stresses mean thermal–shrinkage strains, and hence mean values of temperature and shrinkage of the wall, are used. For the description of the concrete behaviour the viscoelastic material model is assumed, both in the analytical procedure and numerical model. It is problematic in simplified analytical calculations to assume appropriate time for calculations, relating to:

Fig. 3 – (a) Time-development of thermal–shrinkage stresses during hardening and crack formation in reinforced concrete walls. (b) Typical crack pattern in reinforced concrete walls [4].

cross-section of the wall and distribution of stresses in chosen vertical sections at the length of the wall. All the numerical calculations were performed with the use of 3D FEM model of the wall with foundation and fixed supports under the foundation, i.e. compliant with the model of the wall assumed in the analytical calculations. The applied original numerical model was described in previous works [6,7,25]. Stresses determined in the analysis were also compared with the results obtained with the Eurocode 2-3 method [22].

2.

Analytical procedure

In reinforced concrete walls cast against previously executed foundation potential cracking may occur in the phase of temperature decrease in the zone near the joint with the

– Determination of the maximum hardening temperature and related amount of the released hydration heat, which depends on a number of technological and material factors [26], as well as thickness of the member [27]. Experiences and numerical analyses show here that it is usually a period of 2 to 5 days after casting of the wall, depending on the thickness of the wall and the type of cement used. – Determination of the duration of cooling phase, which depends – among the others – on the thickness of the wall, type of cement used and time of formwork removal. Experiences and numerical analyses show that it is usually a period of 7 to 14 days after casting of the wall, however, in case of the walls of considerable thickness (e.g. 1.5 m) kept in formwork for a significant time it can be even 20 days and beyond [26]. – Determination of shrinkage strains accompanying thermal strains. – Assumption of the age of concrete for determination of the modulus of elasticity of concrete, which value increases during maturing of concrete, and which is necessary for determination of stresses. A similar problem occurs when calculating tensile strength of concrete which is used for evaluation of cracking risk. In the proposed model, based on the experiences and extensive numerical analyses presented in Section 3 of the paper, it is proposed:

Fig. 4 – Distribution of hardening temperature, moisture content change as well as self-induced and restraint stresses in vertical central section of the reinforced concrete wall in cooling phase.

79

archives of civil and mechanical engineering 17 (2017) 75–95

– For calculations of the released hydration heat and hardening temperature to assume the time t = 7 days as a mean time of the maximum temperature occurrence. This time is greater than the actual time of the maximum temperature occurrence (2–5 days) because it takes into consideration accelerated maturing of concrete in the conditions of increased temperature, which happens in massive structures. The influence of the thickness of the member is taken into account by introduction of an appropriate coefficient. – In the procedure for temperature calculation only the time of the maximum temperature occurrence and the final temperature difference DT between the analysed wall and its foundation are considered without precise indication of the end of cooling phase. Calculations of shrinkage strains are recommended to be performer for the time corresponding to the moment of cooling down of the wall – it can be assumed, depending on the thickness of the wall and curing conditions: approx. 7 days for relatively thin walls made with low-heat cements, approx. 10–12 days for thin and medium-thick walls, approx. 20 days for thick walls. Assumption of inexact time for calculation of shrinkage strain does not significantly influence calculation of stresses because of the predominant role of thermal strain – the share of shrinkage strains in total restraint strains are estimated to be on the level of 12–14% [6]. – For calculations of the modulus of elasticity and tensile strength of concrete it is recommended to assume time t = 7 days, as mean age of concrete in the analysed life phase of the wall. The time equal to 7 days is used for modulus of elasticity as an average value from the cooling phase, also because the most significant decrease of temperature occurs after the maximum temperature when the modulus of elasticity is relatively small. – It should be mentioned that in later period the stresses can increase due to the developing shrinkage strains and the possible influence of the seasonal temperature. For estimation of stresses in later ages the bigger value of the modulus of elasticity should be taken.

2.1. Calculation of temperature distribution at the thickness of the wall A. Calculation of hardening temperature of concrete in adiabatic conditions (no heat exchange with the environment) DTadiab: DTadiab ¼

CQ 7 cb rb

(1)

where C – amount of cement in 1 m3 of concrete, kg, cb – specific heat of concrete, kJ/(kg K), rb – volumetric mass density of concrete, kg/m3, Q7 – hydration heat of cement after 7 days of hardening, kJ/kg. It has already been mentioned that it is recommended in the presented procedure to assume for calculations time t = 7 days as mean time of the maximum temperature occurrence. This time is greater than the actual time when the maximum temperature occurs (which is usually 2–5 days) because it takes into account accelerated hardening of concrete in the conditions of the increased temperature, which happens in massive concrete members. This heat can be assumed based on the laboratory tests (Q7) or can be calculated based on the mineralogical composition of cement (Q1). Table 1 presents approximate values of Q7 heat for chosen cements after 7 days of hardening and a7 coefficient for determination of this heat based on the total hydration heat of cement (in such case Q7 = a7Q1). In case of using Portland cement CEM I together with mineral additions of ground-granulated blast furnace slag and/or siliceous fly ash added directly to the concrete a7 coefficient can be taken according to the content ratio of individual components in the binder. Total heat of hydration can be calculated based on the model proposed by Schindler and Folliard [29]. The equation for the total heat of hydration of cements with different compositions considers the content of slag, fly ash and silica fume: cem þ 461pslag þ 1800pFA pFA-CaO þ 330pSF Q 1 ¼ pcem Q1

(2)

cem with Q1 quantified on the basis of Portland cement composition:

Table 1 – Values of Q7 heat for chosen cements after 7 days of hardening and a7 coefficient (predicted on the basis of experimental tests [28] and the model proposed by Schindler and Folliard [29]). Cement type

CEM I 42.5R CEM II/B-V 32.5R CEM II/B-S 32.5R CEM III/A 32.5N-LH/HSR/NA CEM V/A (S-V) 32.5R-LH VLH V/B (S-V) 22.5 a

Q1 a

Component, % Portland clinker

Slag (S)

Siliceous fly ash (V)

kJ/kg

95.7 67.3 68.3 41.1 62.2 32.3

– – 27.1 58.9 18.2 34.4

– 29.1 – – 19.6 33.3

501 410 490 498 430 362

a7

Q7 kJ/kg

0.65 0.50 0.60 0.52 0.58 0.51

325 205 294 258 249 185

Note: – Q1 heat calculated based on the mineralogical composition of cement is not the heat after 28 days of hardening but the heat released assuming total hydration, – in case when 28-day heat Q28 is known (it is usually 0.8Q1 for cements CEM I and CEM II, and 0.7Q1 for cements CEM III, CEM V and VLH) for calculation of the heat after 7 days of hardening coefficient a07 ¼ a7 =0:8 or a07 ¼ a7 =0:7 should be taken.

80

archives of civil and mechanical engineering 17 (2017) 75–95

Table 2 – Coefficients in Eq. (4). Component

Table 3 – Suggested values of x coefficient for reinforced concrete walls with typical plywood formwork (predicted on the basis of numerical tests).

fci

Water Cement Sand Basalt Dolomite Granite Quartz Riolite

0.0418 0.0056 0.0074 0.0077 0.0082 0.0047 0.0072 0.0078

x coefficient

Wall thickness, m Portland cement CEM Ia

Other cements (with mineral additions)

0.6 0.7 0.8

0.45 0.57 0.7

0.5 1.0 1.5 a

cem Q1 ¼ 500pC3 S þ 260pC2 S þ 866pC3 A þ 420pC4 AF þ 624pSO3

þ 1186pfree

CaO

þ 850pMgO

(3)

where pi – weight ratio of ith component in Portland cement, pcem – Portland cement weight ratio, pslag – slag weight ratio, pFA – fly ash weight ratio, pFA-CaO – CaO weight ratio in terms of total fly ash content, pSF – silica fume weight ratio. Specific heat of concrete cb can be taken from the laboratory test or can be calculated based on the composition of concrete mix according to equation [30]: n X Gi f ci (4) cb ¼ i¼1

where Gi means mass % content of subsequent components in 1 m3 of concrete, fci is a value for the subsequent components of the mix (Table 2). It is proposed to assume, based on the conducted own laboratory tests, for concrete with gravel aggregate cb = 0.84 kJ/ (kg K), for concrete with basalt aggregate cb = 0.80 kJ/(kg K), for concrete with granite aggregate cb = 0.88 kJ/(kg K) and for concrete with limestone aggregate cb = 0.80 kJ/(kg K). B. Calculation of hardening temperature with the reduction coefficient taking into account heat exchange with the environment and non-adiabatic conditions within the element: red ¼ DTadiab x DTadiab

(5)

Coefficient x takes into account heat release from the core of the member due to its exchange at the surface of the member with the cooler environment. This coefficient is x = 1 for adiabatic conditions; in other cases x < 1. For reinforced concrete wall the value of the coefficient is related to the thickness of the wall and type of cement – the proposed values, determined in the numerical analyses, are collectively presented in Table 3. Intermediate values can be interpolated with linear regression. C. Calculation of temperature inside the element Tint: red Tint ¼ Tbo þ DTadiab

The bigger coefficients have been noticed during the numerical validation of the proposed model for Portland cement. It is probably connected with the rate of heat released in concrete with Portland cement. It means that in case of Portland cement the rate of the heat evolved is faster than the heat dissipation through the surfaces, especially when the influence of the curing temperature on the amount of heat released during the hydration process is considered.

Hence, assuming 2nd order parabolic distribution of temperature at the thickness of the wall, the following equation can be used: Tp ¼ Tint þ

Ta Tint b  ðb=2Þ þ 2ðlb =ap Þ 2

(8)

where Ta is the ambient temperature, b is the thickness of the wall. The value of thermal conductivity coefficient of concrete lb can be calculated with equation [30]: lb ¼

n X Gi f li

(9)

i¼1

where Gi means mass % content of subsequent components in 1 m3 of concrete, fli is a value for the subsequent components of the mix (Table 4). It is also proposed to assume, based on the conducted own laboratory tests, for concrete with gravel aggregate lb = 2.96 W/ (m K), for concrete with basalt aggregate lb = 2.04 W/(m K), for concrete with granite aggregate lb = 2.41 W/(m K) and for concrete with limestone aggregate lb = 2.48 W/(m K).

(6)

where Tbo is the initial temperature of concrete. D. Calculation of temperature at the surface of the wall Tp: Temperature at the surface of the wall can be determined considering surface gradient of temperature dTðtÞ dx jp , coefficient of thermal conductivity of concrete lb, heat exchange coefficient ap and by using 3rd type boundary condition (Fig. 5): ap dTðtÞ j ¼  ðTp Ta Þ dx p lb where Ta is the ambient temperature.

(7) Fig. 5 – Distribution of temperature in the cross-section of the member.

archives of civil and mechanical engineering 17 (2017) 75–95

Table 4 – Coefficients in Eq. (9).

2.3.

Component

fli

Water Cement Sand Basalt Dolomite Granite Quartz Riolite

0.0060 0.0128 0.0308 0.0191 0.0432 0.0294 0.0460 0.0188

Coefficient of heat exchange on the surface of a concrete member ap has an important influence on development of temperature in surface areas. Exchange of heat between the concrete and the ambient environment can occur due to convection or radiation. Heat release from the surface by convection can be free (due to sole temperature difference between the surface of the member and surrounding) or forced (due to wind action). Convection heat exchange coefficient due to joint free and forced convection can be taken from Table 5 depending on the wind speed [30]. Thermal insulation on the surface can be taken into account by introduction of the substitute coefficient apz in Eq. (8) instead of ap [30]: apz ¼

li ap li þ di ap

(10)

where li – thermal conductivity coefficient of the insulation layer, W/(m K), di – thickness of the insulation layer, m. E. Calculation of mean temperature at the thickness of the wall: 1 Tm ¼ Tint  ðTint Tp Þ 3

2.2.

(11)

81

Calculation of shrinkage strains

Calculations can be performer according to the equations given in the standard Eurocode 2-1-1 [31]. Total shrinkage strain ecs is calculated as a sum of drying shrinkage strain ecd and autogenous shrinkage strain eca: ecs ¼ ecd þ eca

(14)

Drying shrinkage strain development in time is given by equation: ecd ðtÞ ¼ bds ðt; ts Þkh ecd;0

(15)

in which: bds ðt; ts Þ ¼

tts

qffiffiffiffiffi tts þ 0:04 h30

(16)

and t – age of concrete at the analysed moment, days, ts – age of concrete at the beginning of drying process; usually it is the time when curing is finished, days, h0 – notional size of the member equal to 2Ac/u, m, Ac – cross-section area of concrete member, m2, u – perimeter of the cross-section subjected to drying, m; kh – coefficient dependent on the notional size h0, ecd,0 – ultimate value of strain. Autogenous shrinkage strain development is given by equation: eca ðtÞ ¼ bas ðtÞeca;1

(17)

in which eca;1 ¼ 2:5ðf ck 10Þ106

(18)

bas ðtÞ ¼ 1expð0:2t0:5 Þ

(19)

Shrinkage strains must be determined independently for the foundation and the wall considering appropriate time periods. Hence, the following must be calculated:

Calculation of thermal strains

Mean thermal strain of the wall: DeT ¼ aT DT

(12)

where aT is the coefficient of thermal expansion of concrete. Temperature difference DT between the analysed wall and its foundation is equal to: DT ¼ g F ðTm Ta Þ

(13)

where Ta is the ambient temperature and at the same time temperature of the foundation. Such an assumption can be made because, in general, at the moment when the wall is cast the foundation has been cooled down to the temperature of the surrounding air. Reduction factor gF takes into account reheating of the part of the foundation during concreting of the wall. Suggested value of this coefficient is gF = 0.9.

Table 5 – Coefficient of convection heat exchange depending on the wind speed. v [m/s] 2

ap [W/(m K)]

0

1

2

3

4

5

6

6.0

10.4

14.5

18.6

22.6

26.7

34.5

1. Shrinkage strain in the foundation (element I) right before execution of the wall (element II) – time tI – according to the equations: – drying shrinkage strain at time tI and time ts (formwork removal from the foundation) according to Eq. (15); – autogenous shrinkage strain at time tI according to Eq. (17); 2. Shrinkage strain in the foundation (element I) for the analysed time tI + tII; tII = 7 days after execution of the wall: – drying shrinkage strain at time tI + tII and time ts (formwork removal from the foundation) according to Eq. (15); – autogenous shrinkage strain at time tI + tII according to Eq. (17); 3. Shrinkage strain in the wall (element II) for the analysed time tII = 7 days after execution of the wall: – drying shrinkage strain at time tII and time ts (formwork removal from the wall) according to Eq. (15); this strain should be calculated only if the formwork was removed from the wall earlier than in 7 days after concreting of the wall; – autogenous shrinkage strain at time tII according to Eq. (17).

82

archives of civil and mechanical engineering 17 (2017) 75–95

2.6.1. s H¼0 res

Standard method of Eurocode 2-3 [22]

¼ RH¼0 ax Ecm;eff De

(26)

H¼h s H¼h res ¼ Rax Ecm;eff De

(27)

H¼0 Rax ; RH¼h ax

Fig. 6 – Difference in shrinkage strains of the wall and its foundation.

Then a difference between shrinkage strains in the foundation and the wall must be determined Decs (Fig. 6): Decs ¼ ecs;II ðtII Þ½ecs;I ðtI þ tII Þecs;I ðtI Þ

(20)

where ecs,II(tII) – total shrinkage strain (autogenous and drying shrinkage strain) of element II at time tII, ecs,I(tI + tII) – total shrinkage strain (autogenous and drying shrinkage strain) of element I at time tI + tII, ecs,I(tI) – total shrinkage strain (autogenous and drying shrinkage strain) of element I at time tI.

2.4.

Calculation of total strains

Mean strain of the wall in cooling phase: De ¼ DeT þ Decs

2.5.

(21)

Calculation of the modulus of elasticity development

For further calculations a change of the modulus of elasticity during hardening needs to be known. According to the recommendations of Model Code 90 [32] it can be calculated according to the following equation: Ecm ðtÞ ¼ bE ðtÞEcm

(22)

in which ( "   #) 28 1=2 bcc ðtÞ ¼ exp s 1 t bE ðtÞ ¼ ½bcc ðtÞ0:5

(23) (24)

where – restraint factor according to Eurocode 2-3 (2008), at the wall–foundation joint (H = 0) and at the top edge of the wall (H = h), respectively – Fig. 7; Ecm,eff – effective modulus of elasticity of the wall concrete at the age of t = 7 days, MPa, De – mean free thermal–shrinkage strain, H – height H¼h of the wall; s H¼0 res ; s res – restraint stresses, at the wall–foundation joint (H = 0) and at the top edge of the wall (H = h), respectively. The advantage of the standard Eurocode 2-3 [22] method is its applicability to all types of restraint conditions of the walls and simplicity of determination of stresses. The drawback is pre-determined, constant distribution of stresses in the central part of the wall. In the following section the paper presents a proposal of extension of the EC2-3 method for the base-restrained walls. The extension covers consideration of the foundation stiffness and bending of the wall, which allows for determination of stresses in any section of the wall. The model introduces 3 coefficients g1, g2, g3, determined in extensive numerical analyses of the walls with different dimensions and L/H ratios. There was a good compliance between the results of the numerical analysis and calculations made with the use of the proposed method. These results are presented in Section 3 of the paper.

2.6.2.

Proposed analytical method M2 y0 M1 y0  Ix Ix

(28)

M2 yh M1 yh þ Ix Ix

(29)

s H¼0 res ¼ s tension þ s H¼h res ¼ s tension  in which s fix ¼ DeEcm;eff

(30)

s tension ¼ g 1 s fix

(31)

V ¼ 0:5ðg 2 s fix Þðg 2 lz Þb

(32)

lz ¼ 0:5L

(33)

where t is time in days, Ecm is a mean value of the modulus of elasticity of concrete after 28 days in MPa, s is a coefficient dependent on the type of cement. The effects of creep can be taken into account using the effective modulus of elasticity: Ecm;eff ðtÞ ¼

Ecm ðtÞ 1 þ fðt; t0 Þ

(25)

where f(t, t0) is a creep coefficient; it is recommended to take f (t, t0) = 1.1.

2.6.

Calculation of stresses

Normal horizontal restraint stresses along the axis of symmetry of the wall can be calculated with one of the two methods.

Fig. 7 – Restraint factors according to EC2 [22].

archives of civil and mechanical engineering 17 (2017) 75–95

83

Fig. 8 – Set of forces acting in the wall.

  1 M1 ¼ V ðlz xÞ ðg 2 lz Þ 3

(34)

T2 ¼ ADeEcm;eff

(35)

M2 ¼ T2 y0

(36)

where A – cross-section area of the wall, m2, Ecm,eff – effective modulus of elasticity of the wall concrete at the age of t = 7 days, MPa, De – mean free thermal–shrinkage strain, x – distance of the calculated cross-section of the wall to its axis of symmetry (Fig. 8), m, b – thickness of the wall, m, Ix – moment of inertia of the cross-section of the wall and foundation; for determination of the moment of inertia the ratio of Ecm,eff (wall concrete) to Ecm,eff (foundation concrete) equal to 0.8 should be introduced, which takes into account the ratio of both the ages of wall and foundation concrete as well the effect of creep; this ratio, representing the relation between wall stiffness and foundation stiffness, equals to 0.8 has been assumed on the basis of the suggested value in CIRIA C660 [21] as well as on the basis of the numerical results, m4, yh, y0 – distances according to Fig. 8, y0 = g3  H, m. Model coefficients g1, g2, g3 were determined based on numerical calculations of 44 base-restrained walls with different dimensions. The results of these calculations were discussed in Section 3 of the paper. Coefficient g1 reduces the share of the tensile stresses in the final restraint stresses according to the existing not full restraint of the wall. Coefficient g1 depends on the L/H ratio and can be calculated with equation:

g 1 ¼ 0:0592ðL=HÞ þ 0:3059a1 g 1 ¼ 0:6019a1 g 1 ¼ 0:65a1

for walls with L=H5 for walls with 5 < L=H7 for walls with L=H > 7

cross-section located at the distance x = 0.75lz it is 0.1. In other cross-sections its value can be taken from Fig. 9b. Coefficient g2 describes the relation between the vertical stresses sz at the end of the wall and horizontal stress sfix as well as the range of the vertical stresses (Fig. 8). Coefficient g2 depends on the length of the wall and can be taken from Fig. 9c or calculated with equation: g2 ¼

0:01 L a2

(38)

where a2 is equal to 1 m. Coefficient g3 has been estimated during the calibration of the model on the basis of the results of numerical tests. Coefficient g3 also depends on the length of the wall and can be taken from Fig. 9d. It is worth noticing that consideration of bending moment M1 from sz stresses influences a change of the values of stresses in different sections of the wall. The stresses calculated with the use of the proposed procedure are the horizontal stresses. These stresses reach the highest value in the mid-span cross section and decrease towards the ends of the wall. In the zones near the wall ends the vertical stresses contribute to the main stresses and there the horizontal stresses do not reach the same value as main stresses, as it is in the central part of the wall. Thus, the predicted horizontal stresses can indicate the zone of the vertical cracks in the wall – on the basis of EC2-3 the magnitude of stresses is constant in the whole middle zone of the wall and does not show the area of the wall with vertical cracks. Simultaneously, in the zones near the wall ends the main stresses should be considered in the cracking risk assessment.

(37)

The value of g1 coefficient for a cross-section located in the axis of symmetry is shown in Fig. 9a. Coefficient a1 takes into account a change of g1 coefficient at the length of the wall. In the cross-section in the axis of symmetry of the wall a1 is therefore equal to 0, in the cross-section located at the distance x = 0.5lz from the axis it is equal to 0.05, and in the

3.

Numerical validation

3.1.

Range and data for analysis

Validation of the analytical model covered determination of hardening temperature of concrete and thermal–shrinkage restraint stresses in reinforced concrete wall. This validation

84

archives of civil and mechanical engineering 17 (2017) 75–95

Fig. 9 – Model coefficients.

was performed by comparison of the results of calculations performed according to the procedure described in Section 2 and numerical calculations with the use of TEMWIL and MAFEM software described in previous works [6,7,25,33]. It should be added that in MAFEM software viscoelasto– viscoplastic material model is originally implemented [25]. However, to validate the analytical model, which applies

viscoelastic material model of concrete, viscoelastic model is also used in numerical calculations. Validation of the method for determination of hardening temperature was performed for the walls with thicknesses of 0.5 m and 1.5 m. In all the cases the same concrete mix composition was assumed, changing only the type of cement and aggregate. Concrete mix composition was as follows:

Table 6 – Thermo-mechanical properties of concretes used in calculations. Concrete symbol

CEM CEM CEM CEM

I_o I_b I_g I_l

CEM II/BS_o CEM II/BV_o

Cement

Aggregate

lb

cb

rb

Ecm

fcm

fctm

W/m K

kJ/(kg K)

kg/m 3

MPa

MPa

MPa

CEM I 42.5R

Rounded Basalt Granite Lime

2.960 2.040 2.412 2.484

0.84 0.80 0.88 0.80

2370 2573 2363 2400

36,400 43,500 37,500 41,500

52.0 63.6 56.2 59.9

3.36 4.05 4.04 3.85

CEM II/B-S 32.5R CEM II/B-V 32.5R

Rounded Rounded

2.96 2.96

0.84 0.84

2370 2366

33,900 26,200

47.1 37.1

3.30 2.94

CEM CEM CEM CEM

III_o III_b III_g III_l

CEM III/A 32.5N-LH/HSR/NA

Rounded Basalt Granite Lime

2.96 2.040 2.412 2.484

0.84 0.80 0.88 0.80

2343 2576 2370 2433

32,100 39,000 33,600 38,700

44.3 47.9 45.7 45.6

3.10 3.72 3.32 3.82

CEM CEM CEM CEM

V_o V_b V_g V_l

CEM V/A (S-V) 32.5R-LH

Rounded Basalt Granite Lime

2.96 2.040 2.412 2.484

0.84 0.80 0.88 0.80

2366 2560 2316 2420

28,800 37,600 32,900 40,100

40.4 48.0 45.2 50.6

3.08 3.81 3.39 3.75

VLH V/B (S-V) 22.5

Rounded

2.96

0.84

2330

25,700

36.0

2.90

CEM VLH_o

lb – coefficient of thermal conductivity, cb – specific heat, rb – density, Ecm – modulus of elasticity, fcm – mean compressive strength, fctm – mean tensile strength.

85

archives of civil and mechanical engineering 17 (2017) 75–95

300 kg/m3 of cement, 150 l/m3 of water, 583 kg/m3 of 0–2 mm sand, 427 kg/m3 of 2–8 mm aggregate, 389 kg/m3 of 8–16 mm aggregate and finally 544 kg/m3 of 16–31.5 mm aggregate. Six types of cement were assumed in the concrete mix with different content of Portland clinker and mineral additions (ground-granulated blast furnace slag and siliceous fly ash). Compositions of these cements were given in Section 2.1. For concretes made with CEM I 42.5R and CEM III/A 32.5N-LH/HSR/ NA cements, 4 types of aggregates were additionally assumed:

gravel, basalt, granite and limestone. In other cases CEM I 42.5R cement and rounded aggregate were assumed. Detailed data about concrete mix composition as well as its thermal and mechanical properties are presented in Table 6. These data were obtained from laboratory tests [25]. Analyses presented in the work [34] have shown that distribution of stresses in the discussed walls depends not only on the L/H ratio but also on the length and height themselves. Thus, the stress analysis was performed for the

Table 7 – Dimensions and denotations of the analysed walls. Description

No

Walla

Denotation of wall and concrete symbol b, m

L, m

H, m

Foundationa L/H

bF, m

L F, m

H F, m

1.5 m thick wall 20 m length Influence of cement type

1 2 3 4 5 6

20_4_1.5_CEM I_o 20_4_1.5_ CEM II/BS_o 20_4_1.5_CEM II/BV_o 20_4_1.5_CEM III_o 20_4_1.5_CEM V_o 20_4_1.5_CEM VLH_o

1.5 1.5 1.5 1.5 1.5 1.5

20 20 20 20 20 20

4.0 4.0 4.0 4.0 4.0 4.0

5 5 5 5 5 5

5.5 5.5 5.5 5.5 5.5 5.5

20 20 20 20 20 20

1.0 1.0 1.0 1.0 1.0 1.0

0.5 m thick wall 20 m length Influence of cement type

7 8 9 10 11 12

20_4_0.5_CEM I_o 20_4_0.5_ CEM II/BS_o 20_4_0.5_CEM II/BV_o 20_4_0.5_CEM III_o 20_4_0.5_CEM V_o 20_4_0.5_CEM VLH_o

0.5 0.5 0.5 0.5 0.5 0.5

20 20 20 20 20 20

4.0 4.0 4.0 4.0 4.0 4.0

5 5 5 5 5 5

4.5 4.5 4.5 4.5 4.5 4.5

20 20 20 20 20 20

0.7 0.7 0.7 0.7 0.7 0.7

0.5 m thick wall 20 m length Influence of aggregate type

13 14 15 16 17 18

20_4_0.5_CEM 20_4_0.5_CEM 20_4_0.5_CEM 20_4_0.5_CEM 20_4_0.5_CEM 20_4_0.5_CEM

I_b I_g I_l III_b III_g III_l

0.5 0.5 0.5 0.5 0.5 0.5

20 20 20 20 20 20

4.0 4.0 4.0 4.0 4.0 4.0

5 5 5 5 5 5

4.5 4.5 4.5 4.5 4.5 4.5

20 20 20 20 20 20

0.7 0.7 0.7 0.7 0.7 0.7

0.5 m thick wall 20 m length Influence of L/H

19 20 21 22 23 24 25

20_2.5_0.5_CEM I_o 20_4_0.5_CEM I_o 20_5_0.5_CEM I_o 20_6_0.5_CEM I_o 20_8_0.5_CEM I_o 20_10_0.5_CEM I_o 20_12_0.5_CEM I_o

0.5 0.5 0.5 0.5 0.5 0.5 0.5

20 20 20 20 20 20 20

2.5 4.0 5.0 6.0 8.0 10.0 12.0

8 5 4 3.33 2.5 2 1.67

4.5 4.5 4.5 4.5 4.5 4.5 4.5

20 20 20 20 20 20 20

1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.5 m thick wall 15 m length Influence of L/H

26 27 28 29 30 31 32

15_2.25_0.5_CEM I_o 15_3_0.5_CEM I_o 15_3.75_0.5_CEM I_o 15_4.5_0.5_CEM I_o 15_6_0.5_CEM I_o 15_7.5_0.5_CEM I_o 15_8.95_0.5_CEM I_o

0.5 0.5 0.5 0.5 0.5 0.5 0.5

15 15 15 15 15 15 15

2.25 3.0 3.75 4.5 6.0 7.5 8.95

6.67 5 4 3.33 2.5 2 1.67

4.5 4.5 4.5 4.5 4.5 4.5 4.5

20 20 20 20 20 20 20

1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.5 m thick wall 10 m length Influence of L/H

33 34 35 36 37 38 39

10_1.5_0.5_CEM I_o 10_2_0.5_CEM I_o 10_2.5_0.5_CEM I_o 10_3_0.5_CEM I_o 10_4_0.5_CEM I_o 10_5_0.5_CEM I_o 10_6_0.5_CEM I_o

0.5 0.5 0.5 0.5 0.5 0.5 0.5

10 10 10 10 10 10 10

1.5 2.0 2.5 3.0 4.0 5.0 6.0

6.67 5 4 3.33 2.5 2 1.67

4.5 4.5 4.5 4.5 4.5 4.5 4.5

20 20 20 20 20 20 20

1.0 1.0 1.0 1.0 1.0 1.0 1.0

a

Dimensions according to the graph:

86

archives of civil and mechanical engineering 17 (2017) 75–95

Table 8 – Coefficients used in numerical and analytical calculations. Coefficient

Denotation

Unit

Value

Comment

Initial temperature Ambient temperature Initial moisture

Tbo Ta Wbo

8C 8C m3/m 3

20 20 0.15

Ambient humidity Ambient moisture Coefficient of thermal diffusion

RH Wa aTT

% m3/m 3 m2/s

70 0.035 –

Coefficient of moisture diffusion

aWW

m2/s

–

Coefficient representing the influence of moisture concentration on heat transfer Thermal coefficient of moisture diffusion Rate of heat generated by cement hydration per unit volume of concrete Time of shuttering removal Time interval between foundation and wall Thermal transfer coefficient

aTW

(m2 K)/s

0.9375  104

– – There is a following relationship between mass concentration c (kg/kg) and volumetric moisture W (m3/ m3): W ¼ rc=r0w , with density of water: r0w ¼ mw =V w , kg/m 3 – Wa = 0.0005RH Depends on the type of concrete; according to the equation: aTT = lb/(cb  rb) According to the equation proposed by Hancox [35]: aWW(W) = aW12 + bW1 + c with a = 4.6389  1010 m2/s b = 1.0556  1010 m2/s c = 0.3055  1010 m2/s W1 = 0.7 + 6 W –

aWT

m2/(s K)

2.0  1011

–

qv

W/m 3

–

Depends on the type of cement, on the basis of Table 1

ts tI

days days

3 28

– –

ap

W/m2 K

Moisture transfer coefficient

bp

m/s

10.4 5.8 4.0 2.78  108 2.78  108 0.15  108

Coefficient of thermal deformability

aT

1/8C

– after formwork removal – top surface before formwork removal – side surfaces before formwork removal – after formwork removal – top surface before formwork removal – side surfaces before formwork removal Depends on the type of aggregate: – gravel – basalt – granite – limestone

Coefficient of moisture deformability Coefficient of mechanical development

aW s

– –

0.00001 0.000008 0.000008 0.0000072 0.001 0.2 0.25 0.25 0.38 0.25 0.38

walls with different L/H and variable both height and width of the walls. Thermal–shrinkage stresses were determined in the walls with the lengths of 20 m, 15 m and 10 m, and with the thickness of 0.5 m, which height was applied so that the L/H ratio ranged from 8 to 1.67. Calculations were also performed for: the walls of 20 m length, 4 m height and 0.5 m as well as 1.5 m thickness with different types of cement used; the walls of 20 m length, 4 m height and 0.5 m thickness with different types of aggregate used. Dimensions of the walls and their denotations are collectively presented in Table 7. The remaining data for both models are presented in Table 8. Fig. 10 shows an exemplary finite element mesh and support conditions for ¼ of the analysed wall-on-slab structure. In numerical calculations of the walls presented in Table 7 subsoil was neglected because the soil–structure interaction is not considered in most of the analytical models, as it was not considered in the proposed analytical model presented in

Depends on the type of cement: – CEM I 42.5R – CEM II/B-S 32.5R – CEM II/B-V 32.5R – CEM III/A 32.5N-LH/HSR/NA – CEM V/A (S-V) 32.5R-LH – VLH V/B (S-V) 22.5

Fig. 10 – Exemplary finite element mesh – 20_6_0.5_CEM I_o wall.

archives of civil and mechanical engineering 17 (2017) 75–95

87

Fig. 11 – Comparison of results of numerical and analytical calculations – 1.5 m thick wall – 20 m length–influence of cement type.

88

archives of civil and mechanical engineering 17 (2017) 75–95

Fig. 12 – Comparison of results of numerical and analytical calculations – 0.5 m thick wall – 20 m length – influence of cement type.

archives of civil and mechanical engineering 17 (2017) 75–95

Fig. 13 – Comparison of results of numerical and analytical calculations – 0.5 m thick wall – 20 m length – influence of aggregate type.

89

90

archives of civil and mechanical engineering 17 (2017) 75–95

Fig. 14 – Comparison of results of numerical and analytical calculations – 0.5 m thick wall – 20 m length – influence of L/H ratio.

archives of civil and mechanical engineering 17 (2017) 75–95

91

Fig. 15 – Comparison of results of numerical and analytical calculations – 0.5 m thick wall – 15 m length – influence of L/H ratio.

92

archives of civil and mechanical engineering 17 (2017) 75–95

Fig. 16 – Comparison of results of numerical and analytical calculations – 0.5 m thick wall – 10 m length – influence of L/H ratio.

archives of civil and mechanical engineering 17 (2017) 75–95

Section 2. Nevertheless, it has been proven in the work [34] that consideration of the subsoil in modelling of wall-on-slab structures has an influence on distribution of the obtained thermal–shrinkage stresses. This influence depends most of all on the stiffness of the subsoil and length of the wall, and can be summarised as follows: subsoil of low stiffness allows for larger rotation of the wall, which results in a decrease of stresses at the joint between the wall and foundation; this effect is more visible in longer walls. Hence, it can be assumed that the values of stresses determined without consideration of the subsoil give safe estimation.

3.2.

Results

The results of numerical and analytical calculations of all the walls defined in Table 7 are shown in Figs. 11–16. Diagrams of temperature distribution refer to horizontal cross-section of the wall (marked in a scheme in Table 7) at the moment when its maximum value was reached in the core of the wall. Diagrams of thermal–shrinkage stresses were made for a vertical section of the wall in its axis of symmetry (Figs. 11–16) and additionally in two sections located at the distance x from this axis (Figs. 14–16). These diagrams present stress distribution at the moment of complete cool-down of the wall. In the analysed walls it was on average 12 days after casting of the wall. Figs. 14–16 present also diagrams of stresses determined with the use of Eurocode 2-3 method. In all cases good agreement was achieved for temperature distribution at the moment when the maximum temperature was reached in the core of the wall, determined with the use of numerical and analytical model. This was true for the walls made of concrete with different types of cements and aggregates. This agreement proves proper assumption of the model parameters in thermal analysis. It also confirms that this part of the model can be used independently for determination of hardening temperature even if stress analysis is performed with the use of different analytical model. In numerical calculations the greatest values of thermal stresses were obtained not at the joint between the wall and foundation but at some height above this joint. Such distribution was obtained because realistic, non-linear distribution of temperature was assumed and resulting selfinduced stresses were taken into account in this analysis. In analytical calculations the assumed temperature was a mean temperature, uniform over the volume of the wall. Thus, the obtained distribution of stresses was linear and the greatest value of stress occurred at the joint with the foundation. The best agreement between the numerical and analytical model was obtained for the section located in the axis of symmetry. This conclusion was true for all the analysed walls. The only difference occurred in the vicinity of the joint with the foundation and resulted from the before mentioned linear distribution of stresses in the analytical model. In other sections, located at the distance x from the axis of symmetry, the agreement between the results obtained with the two models was satisfactory. The worst agreement was obtained in the section located the furthest from the axis of symmetry of the wall (x = 3/8L), especially in the walls with L/H ratio less than 2.5. It must be emphasised, though, that this discrepancy

93

refers to distribution of stresses, which was highly non-linear at the height of the wall in this section, not the value of the maximum tensile stress; in this regard the agreement of the discussed results was good. As far as the Eurocode 2 model is concerned, the best agreement of stresses was obtained for the walls with low L/H ratio (below 2.5). This conclusion refers to the section located in the axis of symmetry of the wall and the section located at the distance x = ¼L; for the walls with higher values of the L/H ratio stresses determined with the EC2 model had proper distribution (inclination) but their values were underestimated. The level of this underestimation increased with an increase of L/H and the length of the wall. Moreover, comparing EC2 model results with the results of numerical calculations it can be concluded that the value of the restraint factor should be dependent on the L/H ratio of the wall; the performed calculations suggest that this value should increase with an increase of L/H ratio. Currently, it is advised to be taken as 0.5 irrespectively of the L/H ratio. Nevertheless, EC2 method is the simplest of the available analytical methods and it seems that introduction of a slight correction to the restraint factor at the joint with the foundation would significantly improve its reliability. In such an event, the analytical model proposed in this paper could be used for determination of temperature, shrinkage and mechanical properties while stress analysis could be performed with the use of the modified EC2 method, which is objectively simpler than the proposed method. It has to be noted that apart from proper estimation of hardening temperature and thermal–shrinkage strain in earlyage concrete it is crucial to properly determine the value of the modulus of elasticity and creep coefficient of concrete, which reduces the value of the elastic modulus. In the proposed model it was assumed that the value of the modulus of elasticity is calculated after 7 days and that the value of creep coefficient is equal to 1.1 (irrespectively of the concrete composition). Good agreement was achieved between the results obtained with the analytical model with such assumptions and numerical model, in which time-development of mechanical properties and creep were assumed. Assumption of different values of these properties (smaller or larger) would directly influence the obtained values of stresses. Thus, in authors' opinion, each analytical model should be complete, i.e. refer not only to determination of restraint factors and stresses, but also provide precise recommendations for determination of hardening temperature, strains and modulus of elasticity. Only such a model can be reliable and useful for evaluation of stresses.

4.

Conclusions

The paper presents a proposal of a calculation algorithm which allows for relatively simple determination of thermal–shrinkage strains and stresses in walls in the early phases of concrete hardening. This model can be regarded as an extension to the recommendations of Eurocode 2-3 (2008) covering determination of hardening temperature and more precise determination of stress distribution in early-age base-restrained wall. The advantage of this method is its completeness – it provides a

94

archives of civil and mechanical engineering 17 (2017) 75–95

complete set of recommendations to determine temperature, shrinkage, modulus of elasticity and stresses. The proposed analytical model was developed based on the results of numerical analyses covering walls with different dimensions and L/H ratios. It should be mentioned that the proposed method for calculation of hardening temperature of concrete can be also used for walls in other restraint conditions because it also relates to distribution of temperature at the thickness of the wall. Restraint factors proposed in EC2 can be then used for determination of the resulting stresses. The proposed procedure can be helpful in preliminary assessment of the level of thermal–shrinkage stresses and cracking risk in walls. Despite good compliance between the results obtained with the use of the numerical and analytical model it must be remembered that evaluation of the discussed thermal–shrinkage effects in the walls with the use of analytical methods is always approximate. This results from a series of simplifications that must be made as it is not possible to take into account complex impact of all material and technological factors, which are additionally variable in time. It must be also remembered that the discussed wall-onslab structures are always cast on a subgrade which interacts with the structure. The analyses performed in works [34] have shown that the subgrade has an important influence on distribution of stresses in the walls. At the same time, based on the referred works it can be said that assessment of stresses neglecting the soil–structure interaction is a safe assessment as it gives higher values of stresses in the zone of the joint between the wall and the foundation. To conclude, precise analysis of the discussed phenomena is possible only with the use of numerical methods and specialised computer software which allow for recognition of thermal–moisture fields and generated stresses in respective time steps corresponding to maturing of concrete, taking into account various technological and material conditions and subgrade. The proposed model allows for determination of hardening temperature distribution in the cross-section of the wall, resulting thermal and shrinkage strains and finally distribution of restraint stresses in various areas of the wall. It was proven that the use of this complete procedure provides safe yet economical estimation of restraint stresses, especially in central section of the wall where the maximum values of stresses and initiation of cracking are expected. The results of analyses show also that parts of the procedure for determination of strains and stresses can be used independently from one another and that stress analysis can be performed with other analytical methods. In this regards the model proposed by Eurocode 2-3 is shown to be a promising, simpler alternative given that slight modifications in determination of the restraint factor are introduced.

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15] [16]

[17]

[18]

[19]

references

[20] [21]

[1] M. Al-Gburi, J.E. Jonasson, M. Nilsson, H. Hedlund, A. Hosthagen, Simplified methods for crack risk analyses of early age concrete, part 1: development of equivalent

[22]

restraint method, Nordic Concrete Research Publication 46 (2) (2012) 17–38. M. Al-Gburi, J.E. Jonasson, S.T. Yousif, M.M. Nilsson, Simplified methods for crack risk analyses of early age concrete, part 2: restraint factors for typical case wallon-slab, Nordic Concrete Research Publication 46 (2) (2012) 39–56. F. Benboudjema, J.M. Torrenti, Early-age behaviour of concrete nuclear containments, Nuclear Engineering and Design 238 (10) (2008) 2495–2506. K. Flaga, K. Furtak, Problem of thermal and shrinkage cracking in tanks vertical walls and retaining walls near their contact with solid foundation slabs, Architecture–Civil Engineering–Environment 2 (2) (2009) 23–30. B. Klemczak, A. Knoppik-Wróbel, Early age thermal and shrinkage cracks in concrete structures – description of the problem, Architecture–Civil Engineering–Environment 4 (2) (2011) 35–48. B. Klemczak, A. Knoppik-Wróbel, Analysis of early-age thermal and shrinkage stresses in RC wall, ACI Structural Journal 111 (2) (2014) 313–322. B. Klemczak, A. Knoppik-Wróbel, Reinforced concrete tank walls and bridge abutments: early-age behaviour, analytic approaches and numerical models, Engineering Structures 84 (2015) 233–251. M. Aurich, A.C. Filho, T.N. Bittencourt, S.P. Shah, Finite element modeling of concrete behaviour at early age, Revista IBRACON de Estruturas e Materiais 2 (1) (2009) 37–58. M. Azenha, C. Sousa, R. Faria, A. Neves, Thermo-hygromechanical modelling of self-induced stresses during the service life of RC structures, Engineering Structures 33 (12) (2011) 3442–3453. M. Briffaut, F. Benboudjema, J.M. Torrenti, G. Nahas, Effects of the early age thermal behaviour on long term damage risks in massive concrete structures, European Journal of Environmental and Civil Engineering 16 (5) (2012) 589–605. H.G. Kwak, S.J. Ha, Non-structural cracking in RC walls. Part II: quantitative prediction model, Cement and Concrete Research 36 (2006) 761–775. H.G. Kwak, S.J. Ha, J.K. Kim, Non-structural cracking in RC walls. Part I: finite element formulation, Cement and Concrete Research 36 (2006) 749–760. J. Zreiki, F. Bouchelaghema, M. Chaouchea, Early-age behaviour of concrete in massive structures – experimentation and modelling, Nuclear Engineering and Design 240 (2010) 2643–2654. W. Schleeh, Die Zwängspannungen in einseitig festgehaltenen Wandscheiben, Beton- und Stahlbetonbau 57 (3) (1962) 64–72. H. Rüsch, D. Jungwirth, Stahlbeton-Spannbeton, Band 2, Werner-Verlag, Dusseldorf, 1976. F.S. Rostásy, W. Henning, Zwang in Stahlbetonwänden auf Fundamenten, Beton- ind Stahlbetonbau 84 (8) (1989) 208–214. ACI Committee No 207.2R, Effect of restraint, volume change, and reinforcement on cracking of mass concrete, ACI Materials Journal 87 (3) (1990) 271–295. ACI Committee 207, Report on Thermal and Volume Change Effects on Cracking of Mass Concrete (ACI 207.2R-07). ACI Manual of Concrete Practice, Part 1, 2011. JSCE, Guidelines for Concrete. No. 15: Standard Specifications for Concrete Structures. Design, 2011. Japanese Concrete Institute, Guidelines for Control of Cracking of Mass Concrete, 2008. P.B. Bamforth, CIRIA C660. Early-age Thermal Crack Control in Concrete, CIRIA, Classic House London, 2007. EN 1992-3, Eurocode 2 – Design of concrete structures – Part 3: Liquid retaining and containment structures, 2008.

archives of civil and mechanical engineering 17 (2017) 75–95

[23] M. Nilsson, Restraint factors and partial coefficients for crack risk analyses of early age concrete structures, (PhD thesis), Luleå University of Technology, 2003. [24] M. Larsson, Thermal crack estimation in early age concrete: models and methods for practical application, (PhD thesis), Luleå University of Technology, 2003. [25] B. Klemczak, Modelling thermal–shrinkage stresses in early age massive concrete structures – comparative study of basic models, Archives of Civil and Mechanical Engineering 14 (4) (2014) 721–733. [26] B. Klemczak, A. Knoppik-Wróbel, Early age thermal and shrinkage cracks in concrete structures – influence of curing conditions, Architecture–Civil Engineering–Environment 4 (4) (2011) 47–58. [27] B. Klemczak, A. Knoppik-Wróbel, Early age thermal and shrinkage cracks in concrete structures – influence of geometry and dimension of a structure, Architecture–Civil Engineering–Environment 4 (3) (2011) 55–70. [28] B. Klemczak, M. Batog, Heat of hydration of low-clinker cements. Pt. 1. Semi-adiabatic and isothermal tests at

[29]

[30] [31] [32] [33]

[34]

[35]

95

different temperature, Journal of Thermal Analysis and Calorimetry 123 (2) (2016) 1351–1360. A.K. Schindler, K.J. Folliard, Heat of hydration models for cementitious materials, ACI Materials Journal 102 (1) (2005) 24–33. W. Kiernożycki, Massive Concrete Structures, Polski Cement, Kraków, 2003 (in Polish). EN 1992-1-1, Eurocode 2 – Design of concrete structures – Part 1-1: General rules and rules for buildings, 2008. CEB-FIP FIB Model Code 1990, Thomas Telford, London, 1991. B. Klemczak, Prediction of coupled heat and moisture transfer in early-age massive concrete structures, Numerical Heat Transfer. Part A: Applications 60 (3) (2011) 212–233. A. Knoppik-Wróbel, B. Klemczak, Degree of restraint concept in analysis of early-age stresses in concrete walls, Engineering Structures 102 (2015) 369–386. N.L. Hancox, The Diffusion of Water in Concrete, UKAEA, Winfrith, 1966.