1.Introduction

The experience of building foundations for unique structures [1, 2] shows that the multiplicity of factors determining the quality of massive monolithic foundations determines the urgency of careful study of the totality of issues related to the technology of their concreting, taking into account prescription factors, technical capabilities, and weather conditions [3–5]. To a large extent, the risk of early cracking in the initial period of concrete hardening in massive monolithic reinforced concrete structures is due to temperature and shrinkage deformations of hardening concrete [6–9]. The reliability of the assessment of the level of tensile stresses increases when shrinkage deformations of concrete are taken into account [10, 11], especially when the overlap time of the layers is more than 4 hours and when using concrete mixtures from different plants (on different cements). In some cases, when calculating the stress level in the early period of construction, shrinkage deformations are ignored [11, 12], which is an unjustified simplification. For example, according to [13], in concretes with a compressive strength of more than 50 MPa without taking into account temperature deformations at the age of 3–5 days, the values of autogenous shrinkage-induced (AS-induced) stress may reach 3 MPa with a deformation of AS up to 0.1 mm/m. Obviously, it is impractical to ignore such a factor.

Stress increments in the assessment of the stress-strain state of massive monolithic foundation slabs can be determined by the equation [11, 14]:

                           (1)

where  is the E-modulus of concrete,  is the average increment of total deformation over the thickness of the slab,  is a coefficient of thermal expansion,  is the difference between the temperature at the point at the current and previous time steps,  is the increment of shrinkage deformation,  is the increment of creep deformation,  is the Poisson's ratio of concrete. The multiplier  in the denominator in Equation (1) takes into account the work of concrete under conditions of biaxial tension (compression)  and the value  is determined by the equation:

                                       (2)

where  is the thickness of the foundation plate.

Equations (1, 2) are applicable in the case where the base limits the deformations of the slab along z but does not limit the deformations of the slab along x and y (A soil with a low modulus of elasticity compared to concrete).

Thus, in the early period of concrete hardening, AS poses a certain danger to early cracking of structures [10, 15]. As is well known, AS is caused by a change in the volume of cement paste (stone) during hydration [16, 17]. The share of AS in the total volume change of cement paster during hydration depends on many factors [18]. AS may affect the stress level of massive structures in the early period [19] and increases with a decrease in the W/C ratio, depends on the properties of cement and the presence of chemical admixtures [16, 17], for example, superplasticizers, develops most intensively in the first 5–7 days of hardening [20–22]. AS-induced stress is also influenced by the properties and concentration of aggregates [23].

The multiplicity of factors that influence on the included in Equations (1, 2) values, depending on prescription and technological factors, determines the expediency of using numerical methods in calculating the stress level [24–26]. In this regard, the relevance of obtaining the dependences of the quantities included in Equations (1, 2) on prescription and technological factors in order to model the stress-strain state in the early period of the construction of massive structures is obvious. Self-compacting concrete and concrete mixtures of classes S4, S5 are widely used in the construction of massive monolithic reinforced concrete structures, including foundation slabs. The kinetics of hardening, the formation of the structure and properties of modern concretes with organomineral modifiers has a number of features in comparison with concretes from traditional mixtures [27, 28]. It is well known that to ensure the solidity of massive structures, an important task is to regulate temperature and shrinkage deformations [15, 29–30]. In this regard, in order to predict more precisely the risk of early cracking, it is relevant to assess the role of AS in the formation of a stress-strain state in the early period.

2.Materials and Methods

The research was carried out by modeling the formation of a stress field in a massive block of 20×20×2 m:

at the first stage, studies only of temperature deformations without taking into account stress relaxation; at the second stage of studies both of temperature and shrinkage deformations without stress relaxation were used.

Stress relaxation was not taken into account, which in some cases is accepted in modeling [31], because, firstly, the task was to assess the effect on the stress level of AS, and, secondly, data on stress relaxation in the early period are few and ambiguous [32].

In the simulation, 8 layers of the laid concrete mix of 0.25 m each were assumed with a layer overlap time ("layer birth time") of 4 hours. The simulation uses the results of laboratory studies and the authors' experience in regulating temperature stresses during the construction of foundation slabs with a volume of 1560–1640 m3 in 14–29 hours using concretes grades B25 and B40 from highly mobile concrete mixtures. When modeling, the following provisions are adopted:

the stress increment is calculated according to Equations (1, 2); the calculation of temperature fields is based on the solution of the differential equation of thermal conductivity [7, 12]:

                                               (3)

where  is the coefficient of thermal conductivity;  is the temperature;  is the density of internal heat sources;  is the density of concrete;  is the specific heat of concrete;  is the time.

And since the temperature distribution over the cross section, with the exception of the peripheral zones of the foundation slab, is one-dimensional, Equation (4) is used to determine the function instead of Equation (3) [12, 33]:

                                                              (4)

The boundary conditions in the case of convective heat exchange with the environment have the form:

                                                              (5)

where  is the normal to the surface,  is the heat transfer coefficient, is the temperature of the environment.

Sequential laying of concrete mix layers after 4 hours when calculating the temperature field is taken into account by assigning an abnormally high coefficient of thermal conductivity to each layer before it is laid (1000 W/(m·°C)) and zero heat capacity, which is mathematically equivalent to the absence of a layer [34]. The temperature field was calculated using the finite element method in the program developed by the authors in the MATLAB environment. The influence of temperature and degree of hydration of concrete on the thermal characteristics of the slab was not taken into account. The entry  in Equations (4, 5) means that  changes from an abnormally high value in the actual absence of a layer to a standard value after its "birth."

The kinetics of concrete compressive strength  was given by the authors equation similar equation in EN 1992-1-1:

                                                          (6)

where  was assumed to be 0.2, 0.25, and 0.38, respectively, for rapid (R), moderate (M), and slow (S) hardening concretes;  = 0.25 days;  is the actual curing time, taking into account the "maturity of concrete," days (  = 20 °C).

The function of the strength of concrete under axial tension  was given by the authors equation:

                                                               (7)

The assumption is made that the modulus of elasticity of concrete (E-modulus) is equal under compression and tension. The E-modulus function is given by authors Equation (8) similar equation in EN 1992-1-1 obtained by processing experimental data of 65 pairs of " " values with a range of compressive strength " " from 9.3 to 69.2 MPa and E-modulus from 21.4 to 37.75 GPa of concretes made from self-compacting mixtures and mixtures classes S4, S5:

                                                         (8)

Time of hardening of the concretes ranged from 1 to 28 days. The concretes were made using three Portland cements of class C42.5, of which one is additive-free (CEM I), two with various mineral additives (CEM II). In experimental studies, concrete mixtures were used both without additives and with superplasticizing admixtures PCE or NF type 1 and type 2. The concretes hardened under normal conditions at  = 20...22 °C,  > 95 %, or in the laboratory at  = 22...25 °C,  ≈ 45...55 %, or in the massive volume at a peak temperature of 39...42 °C. The values of " " in real time were recalculated to the actual time according to the "concrete maturity" indicator, taking into account the actual temperature conditions [32, 35].

AS was determined within 7 days using samples made from cement paste with a W/C = 0.27 [21]. Samples after demoulding were isolated from water evaporation. Six cements CEM I and CEM II with three different superplasticizing admixture PCE and NF – type 1 and 2 were used.

The properties of the studied cements are presented in Table 1.

Table 1. Studied cements.

Cement properties

Cements

1

2

3

5

6

7

CEM I 42.5Н1,6

CEM II/А-Ш 42.5Н1,6 slag

CEM I 42.5Н2,6

CEM I 42.5Н3,6

CEM II/А-П 42.5Н СС4,7

CEM I 42.5 Н5,6

Flexural/Compressive strength, N/mm2, 2 days age

4.59/22.4*

4.84/23.4

5.04/28.5

5.82/30.1

5.09/23.5

6.49/32.3

Flexural/Compressive strength, N/mm2, 28 days age

7.81/51.5

8.58/55.3

8.55/50.9

8.96/55.6

9.95/54.6

8.79/63.1

Normal consistency of cement paste, %

22.75

25.0

26.5

26.25

27.25

27.75

Setting time, min

115

150

150

170

150

140

C3S

66.5+0.2

66.7+0.3

64.9+0.4

68.4+0.4

63.8+0.3

58.3+0.3

C2S

12.1+0.3

11.9+0.3

12.1+0.3

10.7+0.3

16.1+0.3

15.3+0.4

C3A

6.8+0.2

6.9+0.3

5.3+0.3

4.96+0.2

4.4+0.2

7.7+0.2

C4AF

12.1+0.2

12.3+0.2

15.1+0.2

13.1+0.3

14.2+0.2

12.3+0.2

SO3

2.37+0.2

2.11+0.2

2.84+0.2

2.78+0.2

2.82+0.2

3.06+0.3

Q7, kJ/kg

340.9

298.1

310.6

321.1

358.8

332.4

1–5 – manufacturers; 6 – GOST 31108-2020; 7 – GOST 22266-2013; * – flexural/compressive strength

Information about the admixture used is presented in Table 2.

Table 2. Admixtures.

No.

Admixture

Standard

Chemical basis

1

"POLIPLAST SP-4"

TS 5745-026-58042865-2007

sopolymers based on naphthalene sulfonic acid (NF type 1)

2

"LINAMIX PC" type 2

TS 5745-033-58042865-2008

polyoxyethylene derivatives of polycarboxylic acids and polyethylene glycol (PCE)

3

PFM-NLK

TS 5745-022-58042865-2007

mixture of sodium salts of polymethylene naphthalene sulfonic acids,

hydrophobizing and air-entraining component (NF type 2)

The kinetics of AS of the studied cements is described by the authors equation [21]:

                                                           (9)

where  = 0.35, 0.45, 0.6;  = 0.6, 0.75, 0.9, respectively, for the moderate (M), slow (S), and very slow (VS) groups;  = 0,25 days;  is the actual curing time, taking into account the "maturity of concrete," days.

At the age of 3 days, the average value of  for the M group was 0.75, this is the upper curve; for the S group, it was 0.6, this is about the average; for the VS group, it was 0.4, this is the lower curve (Fig. 1).

Figure 1. Kinetics of autogenous shrinkage of cements: 0, 1, 2, 3 – without admixtures, with the admixtures of PCE, NF type 1 and type 2, respectively; C1–C7 – cements according to Table 1; M – moderate, S – slow, and VS – very slow groups.

At the age of 3 days, 14 of the 24 formulations had a value of  more than 0.6 and 10 less than 0.6. The AS values of the studied cements with admixture at the age of 7 days were changed from 0.361 to 0.864 mm/m. The predicted AS values of concretes with cement consumption from 260 (C20/25) to 420 kg/m3 (C35/45) at the age of 7 days ranged from 0.004 to 0.15 mm/m.

The AS function, depending on the compressive strength  was given by the authors equation:

                                                          (10)

where:  – relative AS;  – relative compressive strength of concrete;    – the coefficients is equal minus 0.64 and 2.48 for the VS group, 0.24 and 1.4 for the S group, 1.4 and minus 0.12 for the M group, respectively.

According to [17], deformations of AS equal 0.15 mm/m in concretes with a value of W/C = 0.5 occurred after 15 hours of hardening. According to [32], the AS of concretes with a strength of 40...50 MPa at the age of 7 days was equal 0.025...0.07 mm/m. According to [36], already at the age of one day, the AS value of concrete with a strength of 32 MPa was 0.05 mm/m, and concrete with a strength of 100 MPa was 0.1 mm/m. According to [37], at the age of 5 days, concretes with a strength of 54.8...58.1 MPa had an AS value of 0.125...0.175 mm/m. According to [38], at the age of 5 days, concretes with a strength of 41...64 MPa had an AS value of 0...0.195 mm/m. According to [39–41], at the age of 3 days, the AS value of concrete changes from 0.025 to 1 mm/m. An analysis of a not complete database on AS shows that AS values can vary over a very wide range. In our calculations, the AS value of concretes in 7 days age was carefully assumed to be 0.025 mm/m for concretes of class C20/25 and 0.25 mm/m for concretes of class C35/45.

The heat dissipation function  was given by the authors equation:

                                (11)

where    are the coefficients;  – is the duration of the induction period (is a parameter adjusted to take into account different setting times due to different mix temperatures caused by the use of retarding/accelerating admixtures).

The values of the coefficients in Equation (11) are assumed to be  = 0.14, 0.19, 0.24;  = 0.4, 0.51, 0.62, respectively, for concrete R, M, S groups. In Equation (6), the values of the coefficient  are assumed to be 0.2, 0.35, 0.5, respectively. The value of  is assumed according to experimental data from 110 to 190 MJ/m3. The heat transfer coefficient from the upper surface is assumed from 3 to 23 W/(m2 °С). The lower surface of the block is assumed to be in contact with a 3 m thick soil mass with a density of 1800 kg/m3 and a specific heat capacity of 750 J/(kg °C) and a thermal conductivity coefficient of 0.9 W/(m °С).

3.Results and Discussion

According to [9], in a 2 m thick concrete structure of class B35 with a module of surface less than 1.1, the maximum temperature in the center of the structure reached 64.5 °C after 75 hours, and 29 °C on the surface after 52 hours. According to [39], in a 2 m thick concrete structure with a cement consumption of 340 kg/m3, the maximum temperature in the center of the structure reached 63.8 °C after 72 hours, and 28.2 °C on the surface after 48 hours. In our research, when modeling the stress-strain state of a 2 m thick foundation slab made of concrete classes from B25 to B45 with a specific heat dissipation of concrete from 110 to 190 MJ/m3, depending on the kinetics of heat dissipation and the conditions of heat exchange "top surface – ambient," the maximum temperature in the center was from 43.3 to 77.2 °C after a time of 59 to 204 hours. On the top surface, respectively, from 26.1 to 60.2 °C after 24–161 hours. When measured in a real foundation slab with a thickness of 2 m and a volume of 1642 m3 made of rapid-hardening concrete of class B25 with a specific heat dissipation of 130...140 MJ/m3, the maximum temperature in the center after 45 hours was 67.2 °C, on the top surface covered with a tarpaulin, after 25 hours a temperature of 55 °C was recorded at an ambient temperature from 18 to 24 °C.

In structures, the value of maximum tensile stress depends on the concrete properties and structural parameters. For example, according to [40], the excess of the stresses of the tensile strength of concrete was recorded after about 84 hours. According to [6], for concrete with a design strength of 80 MPa, the tensile stress exceeded the tensile strength of 3 MPa after about 54 hours. According to [39], on the top surface of the slab, after 72 hours, at the moment of maximum temperature in the center of the slab, the value of the tensile stresses reached 2.43 MPa. In our studies, in the real foundation slab described above, the tensile stresses on the upper surface reached a value comparable to the ultimate tensile strength of 2.32 MPa after 110 hours with a center-top surface temperature difference of 26 °C.

A significant amount of research is devoted to the role of shrinkage in the formation of the stress-strain state of structures. For example, in the study [41] of the stress-strain state of massive structures as a function of temperature and shrinkage deformations, ACI 209 and CEB-FIP formulas were compared using thick-walled cylinders as example. Tensile stress values from shrinkage to 7 MPa are obtained. According to [4], when modeling with AS according to various standards at a maximum temperature in the center of the structure of 47.7 °C and a maximum temperature difference "center-top" of 28.7 °C, the values of tensile stresses from 1.8 to 4.17 MPa were obtained. According to our data, when modeling the stress-strain state described above foundation slab with a thickness of 2 m, the maximum values of tensile stresses on the upper surface of the foundation slab ranged from 1.59 to 5.4 MPa. However, the authors are not aware of studies that would provide an assessment of the effect of AS, taking into account its magnitude and kinetics on the stress-strain state, which are presented in this paper.

Fig. 2 shows the dependence of the calculated values of the maximum tensile stress level in the section of the massive block under consideration on the class and kinetics of concrete hardening, the magnitude and kinetics of concrete heat dissipation, and the heat transfer coefficient taking into account AS deformations of concrete.

Figure 2. Dependence of the maximum stress level in the section on the kinetics and magnitude of concrete heat dissipation, concrete class, and heat transfer coefficient from the upper surface: 25, 35, 45 – concrete class C20/25, B35 (В35 in accordance with Russian standard (similar C30/37)), C35/45 respectively; R, S – groups according to Equation (6); 3, 23 – heat transfer coefficient.

Fig. 3 shows the dependence of the calculated values of the maximum tensile stress level in the section of the massive block under consideration on the class and kinetics of concrete hardening, the magnitude and kinetics of concrete heat dissipation, and the heat transfer coefficient without taking into account AS deformations of concrete.

Figure 3. Dependence of the maximum stress level in the section on the kinetics and magnitude of concrete heat dissipation, concrete class, and heat transfer coefficient from the upper surface: 25, 35, 45 – concrete class C20/25, B35 (В35 in accordance with Russian standard (similar C30/37)), C35/45 respectively; R, S – groups according to Equation (6); 3, 23 – heat transfer coefficient.

Fig. 4 shows the ratio of the calculated values of the maximum stress level in the section over a 7-day period, taking into account AS deformation and without AS deformation of concrete  depending on the class and kinetics of concrete hardening, magnitude and kinetics heat dissipation of concrete, heat transfer coefficient.

Figure 4. The ratio of the maximum stress levels in the section from the kinetics and magnitude of the heat dissipation of concrete, the concrete class, and the coefficient of heat transfer from the upper surface: 25, 35, 45 – concrete class C20/25, B35 (В35 in accordance with Russian standard (similar C30/37)), C35/45 respectively; R, S – groups according to Equation (6); 3, 23 – heat transfer coefficient.

Analysis of the simulation results shows that:

the effect of a decrease in the heat transfer coefficient from 23 to 3 W/m2·°C is manifested in a decrease in the maximum stress level in the section, depending on the values of the studied factors, by 28–32 % when taking into account only temperature deformations and by 14–27 % when taking into account temperature and AS deformations of concrete; the influence of the kinetics of heat dissipation and kinetic of strength during the transition from the R group to the S group is manifested in a decrease in the maximum stress level in the section, depending on the values of the studied factors, to 6 %, when taking into account only temperature deformations, and in an increase in the maximum stress level in the section by 20–56 % when taking into account for temperature and AS deformations; as a result, when taking into account temperature and AS deformations of concrete together at the "layer birth time" (layer overlap time) of 4 hours, taking into account all the values of the studied factors, manifests itself in a predicted decrease in the stress level to 30 % or the level increase up to 36 %.

Obviously, it is impractical to ignore such a possible influence of AS deformation on the maximum stress level in the section. The ambiguity of the influence of AS deformation of concrete on the formation of the stress field is due to the manifestation of a possible inconsistency between the kinetics of heat dissipation of cement (concrete), as a result of which a temperature field and temperature deformations are formed, and the kinetics of AS deformation of concrete (Fig. 5).

Figure 5. Diagram of the possible development of the resulting deformation depending both on the kinetic of temperature and kinetic of AS deformation of concrete: R, S – rapid, slow groups for the kinetics of heat dissipation and strength of concrete respectively; M, S, VS – medium, slow, very slow groups for AS deformation of concrete, respectively; ε(T), ε(AS) – temperature and AS deformations of concrete, respectively.

Obviously, with a rapid increase in temperature deformations and a slow development of AS deformation, compensation for temperature deformations practically does not occur at the stage of temperature rise. At the stage of cooling, AS deformation increases the total deformations. Such a variant of the influence of AS deformation can be considered as negative. With a slower increase in temperature deformation and a faster development of AS deformation, it partially compensates for temperature deformations at the stage of temperature rise. This variant of the influence of AS, depending on the ratio of the magnitudes of temperature and shrinkage deformations, as well as the values of the E-modulus and the ultimate strength of concrete under axial tension, may be positive. From the results shown in Fig. 6, it follows that for the studied combinations of cements with admixtures, there is no single dependence of the kinetics of AS deformation on the kinetics of heat dissipation

Figure 6. Dependence of relative AS deformation on relative heat dissipation: 1, 3 – at the age  of 1 and 3 days, respectively, C0 – without superplasticizing admixtures.

Table 3 shows the maximum and minimum values of  and  at diurnal age.

Table 3. Maximum and minimum values of Qτ/Q7 and εAS,τ/ εAS,7 at diurnal age.

Indicator

Cement / admixture

max = 0.742

0.181

–

0.244

3 / NF type 2

min = 0.452

0.265

–

0.586

6 / NF type 2

max = 0.43

–

0.659

0.653

6 / NF type 1

min = 0.061

–

0.722

0.084

5 / PCE

From the results presented in Table 3, it follows that, under the condition of a faster development of AS deformation of concrete relative to the kinetics of heat dissipation, preference should be given to C6 cement in combination with an NF type 2 or NF type 1 admixture. The least preferred option is C5 cement in combination with a PCE admixture.

4.Conclusions

It is shown that ignoring the residual deformation in concrete when calculating the stress-strain state of massive monolithic structures during early hardening can lead to an error in the results, depending on the structural parameters and properties of concrete, by more than 30 %, both in terms of reducing and increasing the tensile stress in concrete. For the studied combinations of cements with admixtures, there is no single dependence of the kinetics of AS deformation of concrete on the kinetics of heat dissipation. For concretes intended for the construction of massive monolithic reinforced concrete structures the value  for example, at the age of one day, can be used as one of the criteria for choosing a rational combination of the "cement + admixture." Of course, the values  and  should also be considered as very important criteria, depending on the construction parameters and concrete properties.