Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models

ACME-183; No. of Pages 13 archives of civil and mechanical engineering xxx (2014) xxx–xxx

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Original Research Article

Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models B.A. Klemczak * Silesian University of Technology, Akademicka 5, 44-100 Gliwice, Poland

article info

abstract

Article history:

The hydration of cement generates heat and can subject structural elements to temperature

Received 30 January 2013

variations that can be significant, particularly in massive concrete structures. Considerable

Accepted 1 January 2014

attention has been focused on this problem dating back to the 1930s during construction of

Available online xxx

concrete dams in North America. Thermal cracking in massive concrete as well as cracking resulting from drying and autogenous shrinkage, not only forms mechanical weaknesses

Keywords:

and cracking but also causes a reduction in durability. Therefore, prediction of the thermal-

Early age concrete

shrinkage stresses and the risk of cracking in massive concrete structures is the important

Thermal-shrinkage stresses

engineering task. The basic models, which can be implemented for the evaluation of the

Massive structures

thermal-shrinkage stresses, are briefly described and compared in the paper. The results of

Modeling

numerical analysis presented in the paper showed differences in values and distribution of

Numerical analysis

thermal-shrinkage stresses predicted with different models. The analyses were performed for two types of structures: the massive foundation block as the example of internally restrained structure and the reinforced concrete wall as the example of externally restrained structure. # 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved.

1.

Introduction

Massive concrete elements are a special type of structure – in the phase of their erection loads, originating from the material of which a structure is made, play a significant role. These loads, caused by temperature and humidity changes of hardening concrete, are defined as indirect interactions. Temperature changes in massive concrete structure are related to the exothermic nature of cement hydration. The concrete temperature increases as a result of heat released in this process. The cooling of surface layers of the structure and

relatively low value of concrete thermal conductivity result in temperature diversification between the surface layers and the inside of the structure. Concrete curing is also accompanied by a loss of moisture. The moisture content decreases due to moisture diffusion after the concrete is exposed to ambient air in conditions of variable temperatures. There is also internal consumption of water during cement hydration. The originating non-linear and non-stationary coupled thermal-humidity fields generate self-induced stresses in the structure (related with internal constraints of the structure, resulting from inhomogeneous distribution of thermal-humidity fields) and restraint stresses (related to limitation of structure

* Tel.: +48 32 237 20 37. E-mail address: [email protected]. 1644-9665/$ – see front matter # 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved. http://dx.doi.org/10.1016/j.acme.2014.01.002 Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

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deformations freedom). During the phase of temperature increase tensile stresses originate in surface layers of the massive element and compressive stresses inside the element. An inversion of the stress body occurs during the cooling phase: inside we observe tensile stresses, in the surface layers – compressive stresses. It is necessary to add that the discussed stresses originate in a material with not fully developed structure, which is subject to relatively quick changes of mechanical properties. The character of the early age thermal and shrinkage stresses as well as possible cracking in structural elements has been presented inter alia by Klemczak and Knoppik-Wróbel [1]. The discussed problems were also investigated in the papers [2,3] that were focused on the influence of structural and technological factors on development of temperature, moisture and finally induced stresses. Taking into account concrete structure and its changes during curing, two possibilities to model an early age concrete appear: a structural approach and a phenomenological approach. The starting point for structural models is the description of internal concrete structure and of thermal–moisture–mechanical effects occurring in this structure. Appropriate constitutive equations are written for the solid, liquid and gaseous phase of the medium and then averaged for a multi-phase medium. These models enable a precise analysis of physical phenomena and of the internal material structure influence on these phenomena. Such approach for porous media has been presented inter alia by Bažant and Thonguthai [4] and Gawin et al. [5,6]. In phenomenological models concrete is treated as a continuous medium. A detailed analysis of physical processes related to phase transitions and of chemical processes occurring in a hardening concrete is neglected in those models and a macroscopic description of thermal–moisture–mechanical phenomena is used. Both in structural and in phenomenological models it is necessary to assume appropriate material model of the concrete, as the basis to define the stress state and possible damage of massive structure. Determination of thermalshrinkage stresses in a maturing concrete is considered a complex issue. Apart from generally known problems of stress modeling in concrete, such as taking into account cracking, anisotropy after cracking or concrete softening, in a curing concrete imposed strains (variable in time and space, originating from temperature and humidity changes), variability of concrete mechanical properties (related to its aging) as well as viscous effects should be additionally considered. So the consideration of the above specific nature of early age concrete in a massive structure requires considering a spatial stress state because of imposed strains spatial variability and using an incremental algorithm due to the variability in time of imposed strains and variability of mechanical properties. Also viscous effects cannot be neglected in the model, both due to a long-term nature of thermal-shrinkage loads and also due to properties of an early age concrete, which shows features of this type much stronger than a mature concrete [7–10]. So the necessity to consider viscous effects in early age concrete modeling disqualifies the models, which do not describe these properties. Viscous effects may be considered in the material model in the following way:


assuming that viscous effects are related only to elastic strains – this leads to viscoelastic or viscoelasto-plastic models, assuming that viscous effects are related only to plastic strains – elasto-viscoplastic models, assuming that viscous effects occur in the whole range of strains – viscoelasto-viscoplastic models.

2.

Review of basic models

2.1.

Viscoelastic model

The first attempts to describe mechanical fields in massive structures used primarily an elastic model of concrete. Now, to evaluate thermal-shrinkage stresses a viscoelastic model is most often used [11–16]. A model consisting of an elastic element and Kelvin body is most often assumed here. Concrete aging has been considered by expressing the model parameters as functions of time, temperature and time or degree of hydration [17,18]. The relationship between stresses and strains in a viscoelastic model can be written in the following way: Dsðtiþ1 Þ ¼ Dve ðtiþ1 Þ½Deðtiþ1 Þ  Den ðtiþ1 Þ  Dec ðtiþ1 Þ

(1)

where denotation Dec ðtiþ1 Þ is given by formulae: "Z Z tiþ1 ti @Cðtiþ1 ; tÞ @Cðtiþ1 ; tÞ 1 c sðtÞdt þ sðti Þ dt De ðtiþ1 Þ ¼ D @t @t 0 ti # Z ti @Cðti ; tÞ sðtÞdt (2)  @t 0 In formula (1) Dve ðtiþ1 Þ is a viscoelasticity matrix given by the formula: De ðtiþ1 Þ Dve ðtiþ1 Þ ¼¼ (3) 1 þ 0:5Eðtiþ1 Þ½ð1=Eðti ÞÞ  ð1=Eðtiþ1 ÞÞ þ Hðtiþ1 ; ti Þ where Z Hðtiþ1 ; ti Þ ¼

tiþ1

ti

@Cðtiþ1 ; tÞ dt @t

(4)

Imposed strains en are treated as volumetric strains:  n den ¼ dex

deny

denz

0 0 0

T

(5)

and calculated based on predetermined temperature and humidity changes [19], according to the equation: denx ¼ deny ¼ denz ¼ aT dT þ aW dW

(6)

where aT is the coefficient of thermal deformability, aW is the coefficient of moisture deformability, T is the temperature, W is the moisture. Determination of stresses based on Eq. (1) requires assuming a creep function C(t,t) and a function describing the modulus of elasticity varying during concrete curing E(t).

2.2.

Viscoelasto-plastic model

The supplementing of a viscoelastic model by a plastic element results in a viscoelasto-plastic model, in which

Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

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viscous effects are taken into account only for the elastic phase of material operation [20]. Total strain in the viscoelastoplastic model is equal to: de ¼ dee þ deve þ de p þ den

(7)

The magnitude of plastic strain is determined by the law of plastic flow: de p ¼ dl

A constitutive relationship may be written in the form of:   Dve mnT Dve ðde  den  dec Þ ds ¼ Dve  h þ nT Dve m

@g @s

(8)



Dve mð@ f =@tÞdt h þ nT Dve m

(18)

After introduction of additional notations:   Dve mnT Dve Dvep ¼ Dve  ve T hþn D m

(19)

A1 ¼

  Dve mnT Dve Dve  dec h þ nT Dve m

(20)

Determination of constitutive relationships for a plastic material requires also assuming yield surface f. In the case of curing concrete the yield surface evolution in the stress space depends on the plastic parameter k = k(ep) and on concrete age t:

A2 ¼

Dve mð@ f =@tÞdt h þ nT Dve m

(21)

f ðs; k; tÞ ¼ 0

Eq. (22) has been derived for an unassociated plasticity law, hence for the plastic potential surface g defined independently of the yield surface f. The defining of the potential surface is not an easy task, so an assumption is frequently made that the potential surface is identical with the yield surface:

where dl is a scalar proportionality coefficient, g is the surface of plastic potential, defined in the stress space as: g ¼ gðs; k; tÞ

(9)

(10)

The derivative of plasticity function f against the stress vector determines a vector normal to the yield surface n, in a similar way the vector m is a vector normal to the plastic potential surface. @f ; n¼ @s

@g m¼ @s

(11)

The yield surface f, irrespective of the assumed function describing this surface, represents areas of real physical states. So the postulate of point representing the current state of stress in the stress space remaining on the yield surface must be realized. This means that each load must correspond to such a change of hardening parameter that the point representing the state of stress is situated on the yield surface. So this consistence condition may be written in the form of: d f ðs; k; tÞ ¼ 0

(12)

Assuming the hardening modulus in the form of: h¼

@ f dk @k dl

(13)

the consistence condition assumes the form of: nT s  h dl þ

@f dt ¼ 0 @t

(14)

Having considered (7) it is possible to write: ds ¼ Dve ½de  den  dec  de p 

(15)

After substitution of Eq. (15) to the consistence condition (14) we obtain: nT Dve de  nT Dve den  nT Dve dec  nT Dve dlm  hdl þ ¼0

@f dt @t (16)

Hence parameter dl is equal to: dl ¼

nT Dve ðde  den  dec Þ ð@ f =@tÞdt þ h þ nT Dve m h þ nT Dve m

(17)

finally it is possible to write ds ¼ Dvep ðde  den Þ  A1  A2

g  f;

(22)

mn

(23) vep

The symmetry of viscoelasto-plastic matrix D is an additional advantage of assuming an associated plasticity law. A full description of viscoelasto-plastic model requires also defining the failure criterion, which in the stress space is represented by the failure surface with the consistence condition: Fðs; tÞ ¼ 0

(24)

dFðs; tÞ ¼ 0

(25)

2.3.

Models considering viscoplasticity

A set of viscoplastic models equations depends inter alia on the adopted viscoplasticity concept, i.e. the way of viscoplastic strains determination. The concept of overstress is one of proposals in this area. Its basis consists of the assumption that a viscoplastic strain originates only once a stress path goes outside the yield surface. In above concept the viscoelastic strain rate is proportional to the excess stress as against a plastic state. The Perzyna proposal is more general in this field, due to assuming a function of the excess stress f( f), while in the Duvaut–Lions proposal the viscoelastic strain rate is directly proportional to the vector of stress increase above a plastic state (s  s) [21,22]. It is important that both proposals allow a stress path to go outside the yield surface and thereby the consistence condition df = 0 is not fulfilled. The postulate that the point representing the current state of stress in the stress space remains on the yield surface is fulfilled in a consistent concept. A viscoplastic strain is calculated according to the equation: @f e_vp ¼ l_ @s

(26)

Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

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where l_ is a positive scalar, called a consistent parameter [22,23]. Consistent approach viscoelasto-viscoplastic model equations, taking into consideration imposed strains, may be written as: e

ve

e_ ¼ e_ þ e_

vp

n

þ e_

þ e_

(27)

Having used relationships (26), (27) and having assumed an associated law of plasticity the relationship for stresses may be written in the form of: _ s_ ¼ Dve ðe_ e_n  e_c  lnÞ

(28)


the value €ln is determined for step n €ln ¼  bn l_ n  cn an an

(37)


in step n + 1 according to the Euler method the value l_ nþ1 is equal to: l_ nþ1 ¼ l_ n þ €ln Dt

(38)

Having assumed numerical solution of Eq. (33) the constitutive relationship remains in the form of (28).

In the consistent concept the yield surface is assumed as: f ðs; k; k_Þ ¼ 0

(29)

So the yield surface is expressed as a function of hardening parameter and its rate. The description adopted in this way is determined by respecting the consistence condition, hence having the point representing the state of stress in the stress space remaining on the yield surface. The yield surface position depends on the rate, with which the simulated process proceeds. Simultaneously the time (concrete age) in this case may not occur openly in the surface description, but may be considered in the adopted law of hardening k = k(evp,t). The consistence condition, assuming an associated law of flow, takes the form of: nT s_ þ

@f @f €k ¼ 0 k_þ @k @k_

(30)

Assuming the relationship: k_ ¼ hðs; kÞ l_

(31)

Having considered relation (28) the consistence condition (30) assumes the form of:

3.

Comparative analysis of basic models

3.1.

Scope and the method of analysis

The following models, described in the previous chapter were compared: viscoelastic model, viscoelasto-plastic and viscoelasto-viscoplastic model. Additionally, the elastic and elastoplastic model were considered, mainly in order to show the importance of viscous effects in an early age concrete. In elastic model equation (1) was used with neglecting viscous strains. In elastic model as well as in all analyzed models the development of modulus of elasticity Eb(t) in time was assumed according to CEB FIP Model Code 1990 [24], which provide here the following relationships: "

Eb ðtÞ ¼ bE ðtÞEb ;

( "



28 bE ðtÞ ¼ exp s 1  t=t1

1=2 #)#0:5 (39)

in which t1 = 1 day, s stands for a coefficient depending on the cement type. The influence of elevated curing temperatures on concrete mechanical properties is considered by introducing equivalent time te instead of time t: Z t eðEK =RÞðð1=TÞð1=To ÞÞ dt (40) te ¼ 0

nT Dve ðe_ e_n  e_c Þ  nT Dve D l_ þ

@f _ @f _ _ @f € hlþ hl þ hl ¼ 0 @k @k_ @k_

(32)

After ordering, the equation providing the basis to determine parameter l_ may be written as: a €l þ d l_ þ k ¼ 0

(33)

where a¼

@f h @k_

d ¼ nT Dve n þ

(34) @f @f _ hþ h @k @k_

k ¼ nT Dve ðe_  e_n  e_c Þ

(35) (36)

Differential equation (33) may be resolved with the use of numerical methods. The Euler method is very convenient and simple in application, being a special case of both differential methods as well as of Runge–Kutty type methods. So Eq. (33) is resolved as follows:

where T stands for temperature, K; To is the reference temperature, K; R stands for the gas constant, EK is the activation energy. In viscoelastic model equation (1) was used. The creep function suggested by the CEB FIP MC90 [24] was applied: !  0:3 1 16:8 1 tt Cðt; tÞ ¼ (41) Eb ðtÞ f c ðtÞ0:5 0:1 þ t 0:2 1500 þ t  t Elasto-plastic, viscoelasto-plastic and viscoelastoviscoplastic model used in presented analysis are based on the elasto-plastic model proposed by Majewski [25]. In viscoelasto-viscoplastic model, the viscoelastic and viscoplastic areas were distinguished. These two areas are separated by the initial location of the yield surface, the formula of which is defined by the boundary surface multiplied by the coefficient less than one. The coefficient can be classified as the viscoelasticity limit. Its value depends on the concrete strength in uniaxial compression, as in the equation: elim ¼ 1  e0:02 f c ðte Þ

(42)

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The following constitutive equation was assumed in the viscoelastic area: s_ ¼ Dve ðe_  e_n  e_c Þ

(43)

where s_ ¼

ds ; dt

e_ ¼

de ; dt

e_n ¼

den ; dt

e_ c ¼

dec dt

(44)

In the viscoelasto-viscoplastic area the constitutive equation has a form: s_ ¼ Dve ðe_  e_n  e_ c  e_vp Þ

(45)

where e_vp ¼ l_

@f @s

(46) where

In the consistent concept of the viscoplastic strains description, both the yield surface f and the failure surface F are rate-dependent surfaces: f ðs; k; k_Þ ¼ 0

(47)

Fðs; k; k_Þ ¼ 0

(48)

The failure surface is described as a modified 3-parameter Willam–Warnke failure criterion [26]. The failure criterion involves all stress invariants in the form of non-dimensional values of the octahedral stress sm ¼ s m = f c ðte Þ, t0 ¼ tokt = f c ðte Þ and the angle of similarity u. The failure surface for young concrete was assumed as a fixed surface in the proposed coordinate system. The meridians are straight lines and in the low-compression and tension regime the caps described as the second-order parabolas were introduced. In the deviatoric plane the failure surface has a noncircular cross-section, described according to the Willam–Warnke conception as a part of an elliptic curve (Fig. 1). Entering of the stress path on the failure surface means material failure. The failure is connected with material structural changes, which in material model is taken into account by introducing the law of softening. If the entrance takes place in the tensile mean stress area, failure manifests as a splitting crack. In such circumstances a substantial anisotropy of the concrete appears. Then, when describing material after failure the law of softening and material anisotropy is considered. In the model, the associated, 2-parameter anisotropic law of softening is assumed. The parameters are plastic part of volumetric strain and plastic part of deviatoric strain calculated as a square root of the second invariant of the strain deviator. A smeared crack image was assumed in the model. The possibility of crack occurrence is defined based on the location of the point representing a stress state with respect to the failure surface. This location can be described by the formula (Fig. 2): sl ¼

tokt f

tokt

Reinforcement is modeled as bar elements connected with concrete elements in the nodes. Bond forces between reinforcement and adjoining concrete are not considered, so the assumed model is simplified. Elastic–perfectly-plastic material model with Huber–von Mises–Hencky failure surface is assumed for reinforcement. It should also pointed that before the start of stress analysis non-linear and non-stationary thermal-humidity fields must be determined. The coupled temperature and moisture fields in early age concrete were described by the following equations [27]: 1 q T_ ¼ divðaTT grad T þ aTW gradcÞ þ cb r v (50) c_ ¼ divðaWW grad c þ aWT grad TÞ  Kqv

(49)

where sl is referred to as the damage intensity factor. The damage intensity factor equal to 1 is equivalent to the stress reaching the failure surface and signifies failure of the element. Character of this failure depends on the location where the failure surface is reached.

T – temperature, K c – moisture concentration, kg/kg T_ ¼ @T @t – time derivative of temperature c_ ¼ @c @t – time derivative of moisture concentration aTT – coefficient of thermal diffusion, m2/s aWW – coefficient of moisture diffusion, m2/s aTW – coefficient representing the influence of moisture concentration on heat transfer, (m2 K)/s aWT – thermal coefficient of moisture diffusion, m2/(s K) cb – specific heat, kJ/(kg K) r – density of concrete, kg/m3 K – coefficient of water–cement proportionality, which describes the amount of water bounded by cement during hydration process with the rate of heat generated by cement hydration per unit volume of concrete, m3/J qv – rate of heat generated by cement hydration per unit volume of concrete, W/m3 The rate of heat generated per unit volume of concrete qv can be determined as shown in Eq. (51): qv ðT; tÞ ¼ cc qðT; tÞ

(51) 3

where cc is the amount of cement in 1 m of concrete mix and: qðT; tÞ ¼

dQðT; tÞ dt

(52)

In Eq. (52) Q(T,t) is the cement heat of hydration in concrete generated in temperature T and time t. In the next chapter results for moisture distribution were presented with the use of the volumetric moisture content W (m3/m3), which is introduced in place of the mass concentration c (kg/kg). There is the following relation between mass concentration and volumetric moisture: rc ¼ r0w W

(53)

with r0w ¼

mw Vw

(54)

where mw – mass of water in concrete, kg Vw – volume of water in concrete, m3

Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

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Fig. 1 – (a) Development of the failure surface of young concrete to the maturing process. (b) Tensile and compressive meridians of the 3-parameter Willam–Warnke model [26].

With respect to the engineering application of the theoretical models, the computer programs were also developed: TEMWIL for the purpose of thermal–moisture analysis and MAFEM_VEVP for the stress analysis. The computer program TEMWIL was developed on the base of Eq. (50) with

Fig. 2 – Illustration of the damage intensity factor calculation method.

the use of the compiling program Fortran Power Station. The program can be used to solve three-dimensional problems with variable boundary and initial conditions to investigate temperature and moisture transfer process. More details connected with the applied incremental algorithm applied in the program are given by Klemczak [27]. To determine stresses in spatial reinforced concrete elements at early ages the MAFEM program, originally developed by Majewski [25], was modified. The elasto-plastic model of concrete, which was originally used in the MAFEM program, was replaced by viscoelasto-viscoplastic model (VEVP). The model was also supplemented with elements specific to maturing concrete, as the variation of the mechanical properties in time of curing. More details of the program MAFEM_VEVP, describing, inter alia, the applied incremental-iterative algorithm to solve nonlinear problems can be found in the paper [25] and the monograph [19]. The soil can be also considered in the analysis

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both in thermal–moisture and stresses problems. For soil the elasto-plastic material model proposed by Majewski [28] is implemented. Both in the TEMWIL program as well as in MAFEM_VEVP program the spatial concrete elements are modeled as simple cubic complexes with nodes in vertexes, whereas reinforcement is modeled in a form of rod elements. The current program version enables the analysis of structures with the number of nodes not exceeding 33,000. The thermal–moisture and stress problems use one and the same mesh, though employ different numerical models and computer programs. Therefore, the complex analysis of a structure with the use of presented programs consists of three steps. The first step is related to determination of temperature and moisture development, in the second step thermal-shrinkage strains are calculated and these strains are used as an input for computation of stress development in the last step. The whole time period assumed in the analysis, is divided into smaller intervals so that the moments of subdomains joining to the domain coincide with the beginning of the intervals. The solution is proceeded successively by intervals starting from the first one. After each step the results are saved on disk in the form of files including information on temperature, moisture and displacements of nodes as well as the stress components. For presentation of results the open-source application PARAVIEW was adopted. Undoubtedly, validation of the numerical results with experimentally-obtained data is necessary to confirm reliability of the theoretical model. Because the proposed model includes two issues connected with determination of thermal– moisture fields and stress and strain states, validation of the model should be two-phase as well. It should be noted that this is untypical task in which firstly the loads to which the structure is subjected (thermal–moisture fields) are validated, and then the effects of these loads – stress state – is validated. This results from a specific nature of massive concrete structures, where the material of the structure itself is the source of the load. In validation of thermal–moisture fields author's own experimental results were used. In case of stress state, because of the assumed complex material model, validation was initially performed with simple numerical tests, which allowed of qualitative confirmation of the proposed material model correctness. Firstly, creep test simulation in heavily stressed conditions was performed. Then, the results of numerical simulations of simple stress states in concrete samples were

7

presented, i.e. uniaxially compressed and tensiled samples and cylindrical samples in split test. In the second phase the thermal-shrinkage stresses were analyzed. The results of the author's own as well as other experiments were used. Because of their volume, these validation are only mentioned here and can be found in the monograph [19].

3.2.

The objects of analysis

Thermal-shrinkage stresses are investigated in two types of structures. One of them was a massive foundation block with the base dimensions 10 m  10 m and thickness 3 m. The finite element mesh of the analyzed block was shown in Fig. 3. Because of the symmetry only the quarter of the block was modeled. Essential elements of the block used in presentation of calculation results were marked in bold in Fig. 3. The foundation block was assumed to be reinforced with a 20 cm  20 cm mesh at the top, bottom and side surfaces. Steel class RB400 and f12 bars were assumed for calculations. The second object was a reinforced concrete wall cast against an old set foundation. The analyzed wall was assumed to have 20 m of length, 4 m of height and 40 cm of thickness, supported on a 4 m wide and 70 cm deep continuous foundation of the same length. The wall with the assumed mesh for finite element analysis is presented in Fig. 4. Because of the symmetry only the quarter of the structure was modeled. The wall and the foundation of the wall were assumed to be reinforced with a near-surface reinforcing net of f16 bars. The wall was reinforced at both surfaces with horizontal spacing of 20 cm and vertical spacing of 15 cm. The foundation of the wall was reinforced with a 20 cm  20 cm mesh at the top and bottom surface. Steel class RB400 was also assumed for the wall. It was assumed that the analyzed block and wall were made of the following concrete mix: cement CEM II/BS 42.5N 350 kg/m3, water 170 l/m3, sand 665 kg/m3, granite 1230 kg/m3. Environmental and technological conditions were taken as: ambient temperature 20 8C; initial temperature of concrete mix was assumed as equal to the ambient temperature. The initial volumetric moisture content W (m3/m3) was 0.170 m3/m3; the initial mass concentration c (kg/kg) was c = 7.04  10–5 kg/kg. Both for the massive foundation block and RC wall a wooden formwork of 1.8 cm plywood on side surfaces and the protection of top surface with foil were assumed in analyses. Additionally, it was assumed that formwork is removed in 28 days after concrete casting in all analyzed cases. The final values for 28-day

Fig. 3 – Dimensions of ¼ of the massive foundation block with the assumed finite element mesh. Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

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Fig. 4 – Dimensions of ¼ of the reinforced concrete wall with the assumed finite element mesh.

concrete were assumed on the base of the experimental tests: the compressive strength fcm = 35.0 MPa, the tensile strength fctm = 3.0 MPa and the modulus of elasticity Eb = 32 GPa. The soil is also considered in the performed analysis. For soil the following data is assumed: cohesion c = 25 kPa, friction angle f = 208, density % = 1800 kg/m3, coefficient of thermal diffusion aTT = 8.82  107 m2/s, humidity of soil 11%. The initial temperature of soil was assumed as 15 8C – it is average temperature of surface and deeper layers of soil [29]. Thermal and moisture coefficients necessary for calculations were set in Table 1. The values of coefficients were assumed according to suggestions given in literature [9,28]. Additional layer (wooden formwork or foil) on the concrete

surface were considered by reducing thermal transfer coefficient according to the formula [29]: a pz ¼

li a p li þ di a p

(55)

where: li – coefficient of thermal conductivity of the additional layer, W/(m K) di – thickness of the additional layer, m. The moisture transfer coefficient was reduced with the use of the same method [29].

Table 1 – Thermal and moisture coefficients. Thermal fields Coefficient of thermal conductivitya Specific heata Density of concrete Coefficient of thermal diffusion Coefficient representing the influence of the moisture concentration on the heat transfer Thermal transfer coefficient

l, W/m K cb, kJ/kg K r, kg/m 3 aTT, m2/s aTW, m2 K/s

2.57 0.82 2415 10.641 T 107 9.375 T 105

ap, W/m2 K

6.00 (without protection) 3.58 (plywood) 5.80 (foil) Acc. to equation: 0:5 a QðT; tÞ ¼ Q 1 eate ; a ¼ a1 te 2 with Q1 = 466 kJ/kg, a1 = 480.51 and a2 = 0.115

Heat of hydrationb

Q(T,t)

Moisture fields Coefficient of the water–cement proportionality Coefficient of moisture diffusion

K, m3/J aWW, m2/s

Thermal coefficient of moisture diffusion Moisture transfer coefficient

aWT, m2/s K bp, m/s

a b

0.3 T 109 Acc. to equation: aWW ðWÞ ¼ aW 1 2 þ bW 1 þ c a ¼ 4:6389  1010 m2 =s, b ¼ 1:0556  1010 m2 =s, c ¼ 0:3055  1010 m2 =s, W1 ¼ 0:7 þ 6W 2  1011 2.78  108 (without protection) 0.18  108 (plywood) 0.10  108 (foil)

Coefficients are taken on the base of concrete mix compositions. Coefficients are taken on the base of experimentally determined development of heat of hydration for the analyzed cement CEM II/BS 42.5N.

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ACME-183; No. of Pages 13 archives of civil and mechanical engineering xxx (2014) xxx–xxx

4.

Results of comparative analysis

4.1.

Thermal and moisture fields

Firstly, the temperature and moisture development were determined in analyzed structures. The results are shown in Figs. 5 and 6. A significant difference in temperatures of the interior and the top surface of the block is observed, for the assumed curing conditions it is nearly 20 8C (Fig. 5a). The temperature of the bottom of the block is higher compared to the top surface, because of the existing soil under the block, which is also warmed up. A difference in temperatures of the center and the bottom of the wall (Fig. 6a) is smaller compared to the massive foundation block. The moisture loss from structures occurs slowly and there are no significant differences in moisture loss of the interior and the surface because the surfaces are protected with formwork. The loss of moisture from the bottom of the block is more smaller because of the foil layer assumed between block and the soil.

4.2.

Thermal-shrinkage stresses

For the known thermal–moisture fields and imposed thermalshrinkage strains, stress state was determined. In the block, the originating non-linear and non-stationary coupled thermal-humidity fields generate mainly self-induced stresses,

a

related to the internal restraints of the structure resulting from inhomogeneous distribution of thermal-humidity fields. During the phase of temperature increase the tensile stresses are induced in surface layers of the block and compressive stresses inside the element. It should be mentioned that during the cooling phase an inversion of the stress body may occur. In such case the tensile stresses are observed inside and compressive stresses in the surface layers. The development of induced stresses in the essential points of the massive foundation block, determined with the use of different models is shown in Fig. 7. In case of medium-thick structures deprived of the possibility of deformation, such as a analyzed wall cast against an old set foundation, the development of stresses is different. Stresses developing in the wall result mainly from restraint stresses, connected with restraint of deformation of the wall. There are also self-induced stresses resulting from non-uniform temperature and moisture distribution in the wall. However, in massive blocks the self-induced stresses reach comparatively higher values and, as a consequence, are predominant impacts, their influence in walls is much lower. This is mainly the result of the hardening temperatures distribution within the wall. Even though a difference in temperatures of the interior and the bottom of the wall is observed, the difference is small in magnitude (Fig. 6a). Therefore, in the first phase the wall extends being opposed by the weakly bonded foundation, which results in occurrence of compressive stresses (Fig. 8). These are usually the first

60

Temperature, oC

50 40 30

20

center

top

bottom

10 0

2

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Moisture x 100, m3/m3

Time, days

b

18 16 14 12

center

top

14

16

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10 0

2

4

6

8

9

10

12

18

20

Time, days Fig. 5 – Temperature (a) and moisture (b) development in essential elements of the analyzed massive foundation block.

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a

60 wall_center

wall_bottom

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Moisture x 100, m3/m3

Time, days

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wall_center

wall_bottom

16 14 12 10 0

2

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16

Time, day Fig. 6 – Temperature (a) and moisture (b) development in essential elements of the analyzed wall. 1–3 days. As soon as the maximum self-heating temperature is reached, the wall starts to cool down, which takes another few days, restrained by a cooled foundation. This leads to development of tensile stresses in the wall (Fig. 8). The described development of stresses is observed both in the center and in the bottom layers of the wall. The results related to the stress distribution determined with the use of different models can be summarized as follows:
as it was expected maximum values of stresses were obtained from the elastic model. It is also interesting to note that in the analysis of the wall the compressive stresses arising in the heating phase are greater than in other models but simultaneously the inversion of the stress body occurs later. In this case the tensile stresses in the cooling phase occur later and reach lower values. It can be the reason for the underestimation of the cracking risk in such structures,
the elasto-plastic model and elastic model provide almost the same results in the case of wall where the failure does not exist for the assumed curing conditions. In the case of massive foundation block this similarity is visible only in the first period of analysis until the failure was appeared in the top surface element,
the results obtained from the viscoelastic and viscoelastoplastic model is very close both for the massive foundation block and the wall,
the main fact observed in the stress analysis performed with the use of viscoelasto-viscoplastic model is the faster inversion of the stress body. It is perfectly visible in the

stress development in the wall (Fig. 8) where the tensile stresses develop relatively early and achieve greater values as compared with the others models. This fact can be important for the assessment of the cracking risk in structures subjected to early age deformations of the thermal and shrinkage origin..

5.

Summary

A mathematical model of phenomena proceeding in a early age concrete is related to determination of thermal-humidity and mechanical fields. Assuming that mechanical fields affect the phenomena of heat and moisture diffusion to a small extent, the issue boils down to determination of massive structure loads in the form of imposed thermal-shrinkage strains and to determination of the stress and strain state, which results from the action of thermal-shrinkage distortions. At the determination of thermal-shrinkage stresses in massive concretes it is significant to assume a proper material model of young concrete. The presented results of comparative analyses showed mainly the importance of viscous effects in an early age concrete, which are much stronger in young concrete than a mature concrete. Finally, it should be noted that the main aim of this paper is to show the nature of differences in thermal-shrinkage stresses predicted with different material models of early age concrete. These differences were showed on the exemplary foundation block and RC wall. The aim of the analysis was not to determine

Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

ACME-183; No. of Pages 13 archives of civil and mechanical engineering xxx (2014) xxx–xxx

a

4

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ELASTIC_center VISCOELASTIC_center ELASTO-PLASTIC_center VISCOELASTO-PLASTIC_center VISCOELASTO-VISCOPLASTIC_center

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Stress σx , MPa

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Stress σx , MPa

2

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0 0

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24

-1

-2

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Time, days

Fig. 7 – Stress development in essential elements (see Fig. 3) of the analyzed massive foundation block: (a) center, (b) top, and (c) bottom.

Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

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Stress σx , MPa

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-1 ELASTIC_center VISCOELASTIC_center ELASTO-PLASTIC_center VISCOELASTO-PLASTIC_center VISCOELASTO-VISCOPLASTIC_center

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-1 ELASTIC_bottom VISCOELASTIC_bottom ELASTO-PLASTIC_bottom VISCOELASTO-PLASTIC_bottom VISCOELASTO-VISCOPLASTIC_bottom

-2 Time, days

Fig. 8 – Stress development in essential elements (see Fig. 4) of the analyzed wall: (a) center and (b) bottom.

the risk of cracking for the specific case. Therefore, some technological and material conditions was assumed only for purpose of a comparative study. Obviously, assuming the other conditions different quantitative results would be obtained.

Acknowledgement This paper was done as a part of a research project N N506 043440 entitled Numerical prediction of cracking risk and methods of its reduction in massive concrete structures, funded by Polish National Science Centre.

references

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Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002

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Please cite this article in press as: B.A. Klemczak, Modeling thermal-shrinkage stresses in early age massive concrete structures – Comparative study of basic models, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.002