Prediction of Thermal Stress in Hardening Mass Concrete Structures Using Temperature Monitoring Data

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1. Introduction A widespread problem for mass concrete and reinforced concrete structures is the formation of temperature cracks during the early stages of hardening. During cement hydration, a large amount of heat is released, causing uneven temperature rise: the core of the structure develops much more heat than the surface. During subsequent cooling, tensile stress that exceeds the strength of early age concrete arises, leading to cracking [1; 2]. This reduces the strength, durability, and aesthetics of structures and increases the risk of corrosion [3]. The most common tool for assessing the risk of early cracking is finite element modeling. It allows to analyse the evolution of temperature distributions and stresses, taking into account changes in the key properties of concrete over time. It is critical to account for the dependence of concrete strength and deformation characteristics on time and hardening conditions, since ignoring this factor leads to an incorrect evaluation of the stress state [4-6]. A limitation of finite element modeling is the inability to account for uncertainties in input data, including ambient and foundation temperatures, wind speed, variations in concrete heat release when using cements from different manufacturers, etc. Variability in input data can lead to significant deviations of the actual temperature and stress values in the structure from the expected. In existing monitoring systems for mass structures (Maturix[2], Giatec SmartRock[3]), only temperature measurements are typically taken, not stress [7]. Stress can be determined indirectly from strain; however, sensors for measuring strain are highly complex and expensive, as they must be capable of compensating their readings with respect to temperature [8]. Therefore, the problem of quickly assessing the actual stress in the structure based on real-time temperature monitoring data is of great relevance. Such an assessment allows for timely measures to be taken to prevent early cracking, including additional thermal insulation of surfaces to reduce temperature gradients when the stress approaches a dangerous level [9]. In [10], a simplified method for estimating thermal stress in mass concrete foundation slabs was proposed. Instead of the traditional approach based on the temperature difference between the center and the surface [11-14], the proposed method uses temperature data at three characteristic points along the slab thickness (bottom, middle, and top). The formulas were verified using numerical simulation and demonstrated high accuracy for slabs up to 2 meters thick. Significant errors for slab thickness values of 2 meters and greater are associated with the assumption of a parabolic distribution of temperature and stress along the slab thickness. This article aims to overcome this limitation using machine learning methods. It should be noted that previously, machine learning methods were used only to predict the heat release of concrete [15], its thermal conductivity coefficient [16], and strength characteristics [17-18], estimating the temperature difference between the center and the surface of the structure [19], determining the maximum stress level for fixed input data [20], and designing an optimal concrete mix composition [21-24]. Unlike previous studies, the aim of this article is to develop and validate a method for predicting thermal stress in mass concrete foundation slabs based on limited temperature monitoring data collected at characteristic points along the slab thickness, using artificial neural networks. To achieve this objective, the following tasks were addressed: 1. Create a representative training dataset based on parametric finite element calculations, covering variations in geometric, strength, and thermophysical parameters. 2. Develop and train three artificial neural network architectures (recurrent, feedforward, and cascade) for stress prediction based on temperature monitoring data. 3. Evaluate the accuracy and comparative performance of the proposed models based on training quality metrics and regression curve analysis. 4. Verify the developed models using experimental data for a real foundation slab. 2. Methods MATLAB R2021a (Neural Network Toolbox) serves as the environment for implementing machine learning models. Seven variables were selected as input parameters for the machine learning models: temperatures at three points along the slab thickness (Tbot at the bottom surface, Tmid in the middle of the slab, and Tup at the top surface), time t in days, concrete compressive strength class B according to the Russian Federal Standard GOST 18105-2018[4], slab thickness h, and hardening rate (rate). The output parameters of the models are three values: stress σbot at the bottom surface, σmid in the middle surface, and σup at the top surface (MPa). Three artificial neural network (ANN) architectures are considered: 1. Layer recurrent neural network (Figure 1); 2. Feedforward neural network (Figure 2); 3. Cascade forward neural network (Figure 3). Figure 1. Recurrent neural network (screenshot from MATLAB) S o u r c e: made by V.S. Tyurina. Figure 2. Feedforward neural network (screenshot from MATLAB) S o u r c e: made by V.S. Tyurina. Figure 3. Cascade forward neural network (screenshot from MATLAB) S o u r c e: made by V.S. Tyurina. For all ANN architectures, a single hidden layer with sixteen neurons was used. When forming the training dataset, the slab thickness varied from 1 to 3 m in 0.5 m increments, the heat transfer coefficient on the upper surface ranged from 2 to 30 W/(m2∙°C) in 4 W/(m2∙°C) increments, and the concrete class ranged from B25 to B45 in 5 MPa increments. The ambient temperature varied from 5 to 35 °C in 5 °C increments. Three integer values were introduced for the hardening rate: 1, 2, and 3, corresponding to high-early-strength, regular, and low-early-strength concrete. The selected ranges of input parameter variations cover the most common parameters for concrete placement and the geometry of mass concrete foundation slabs. The heat release function was defined by equation [25]: (1) where Q28 is the total heat release per 1 m3 of concrete at 28 days, k and x are coefficients determining the kinetics of heat release, and b is the induction period. The heat release parameters are listed in Table 1. For each set of values [B h hup T∞ rate], the temperature distribution and stress state were calculated, with temperature and stress values determined at three characteristic points (bottom, middle, top) for 119 time points ranging from 0.5 to 30 days, with a step size of 0.25 days. The temperature distribution was calculated using the finite element method (FEM) in a simplified one-dimensional formulation according to the method described in [26]. The stress state was calculated using the method presented in [27]. Table 1. Parameters in the heat release equation Parameter k x Q28 MJ/m3 b days High-early-strength (1) 0.14 0.4 130 + 3∙(B - 25) 0.167 Regular (2) 0.19 0.51 Low-early-strength (3) 0.24 0.62 S o u r c e: made by V.S. Tyurina. The thermophysical properties of the soil were assumed to be constant and equal to: thermal conductivity coefficient λg = 0.9 W/(m·°C), specific heat capacity cg = 750 J/(kg·°C), and density ρg = 1800 kg/m³. The following values of thermophysical properties of concrete were assumed: λ = 2.67 W/(m·°C), c = 1000 J/(kg·°C), ρ = 2400 kg/m³. The function describing the change in concrete compressive strength was defined by equation [28]: (2) where R28 = B + 12 is the compressive strength of concrete at 28 days, s is a coefficient dependent on the hardening kinetics of concrete, is the equivalent age of concrete, expressed in terms of its degree of maturity DM: (3) where is the temperature at a point at age . Coefficient s was taken to be 0.2 for high-early-strength concrete, 0.35 for regular concrete, and 0.5 for low-early-strength concrete. The modulus of elasticity of concrete was determined as a function of compressive strength using formula [28]: MPa. (4) The value of R in Equation (4) should be substituted in MPa. Poisson’s ratio of concrete was assumed to be independent of time and hardening temperature (ν = 0.2). Up to an equivalent age of 12 hours, the modulus of elasticity of concrete was assumed to be zero (it was assumed that the concrete was not yet a solid and contained no stress). The layering of the concrete mix was not taken into account in the calculation. The initial soil temperature was assumed to be equal to the ambient temperature. The calculated temperature values Tbot, Tmid, Tup were placed in the input parameter array along with the values [B, h, rate] and time t. Stress values σbot, σmid, σup were placed in the target values array of the output variables. The total volume of the training dataset consisted of 4200 numerical experiments (4200 × 119 = 499800 rows). A sample of the generated training dataset is presented in Table 2. Table 2. A fragment of the training dataset No. Input parameters Output parameters Tbot, °С Tmid, °С Tup, °С t, days B, MPa h, m rate σbot, MPa σmid, MPa σup, MPa 1 23.97 31.98 29.18 0.5 25 1 1 0.000 0.000 0.000 2 27.31 35.90 32.27 0.75 25 1 1 -0.002 -0.088 0.094 3 29.31 37.68 33.69 1 25 1 1 -0.095 -0.116 0.178 4 30.67 38.54 34.33 1.25 25 1 1 -0.248 -0.108 0.256 5 31.63 38.91 34.53 1.5 25 1 1 -0.424 -0.087 0.335 6 32.33 39.01 34.46 1.75 25 1 1 -0.606 -0.062 0.415 7 32.84 38.93 34.24 2 25 1 1 -0.784 -0.036 0.492 8 33.20 38.73 33.91 2.25 25 1 1 -0.954 -0.010 0.565 9 33.44 38.46 33.52 2.5 25 1 1 -1.114 0.014 0.631 10 33.58 38.14 33.10 2.75 25 1 1 -1.262 0.037 0.691 11 33.66 37.77 32.65 3 25 1 1 -1.399 0.059 0.743 12 33.67 37.39 32.20 3.25 25 1 1 -1.525 0.080 0.789 13 33.62 36.98 31.74 3.5 25 1 1 -1.640 0.100 0.828 14 33.54 36.56 31.28 3.75 25 1 1 -1.745 0.118 0.861 15 33.42 36.14 30.83 4 25 1 1 -1.842 0.136 0.889 … … … … … … … … … … … 499786 76.22 70.58 37.56 26.5 45 3 3 -5.600 -1.337 8.696 499787 76.08 70.38 37.54 26.75 45 3 3 -5.609 -1.317 8.631 499788 75.94 70.18 37.52 27 45 3 3 -5.617 -1.297 8.566 499789 75.80 69.99 37.50 27.25 45 3 3 -5.624 -1.278 8.501 499790 75.66 69.79 37.49 27.5 45 3 3 -5.630 -1.258 8.436 499791 75.51 69.60 37.47 27.75 45 3 3 -5.636 -1.239 8.372 499792 75.37 69.41 37.45 28 45 3 3 -5.642 -1.220 8.307 499793 75.23 69.22 37.44 28.25 45 3 3 -5.647 -1.202 8.243 499794 75.09 69.03 37.42 28.5 45 3 3 -5.651 -1.184 8.179 499795 74.94 68.84 37.40 28.75 45 3 3 -5.654 -1.166 8.116 499796 74.80 68.65 37.39 29 45 3 3 -5.657 -1.148 8.053 499797 74.65 68.46 37.37 29.25 45 3 3 -5.660 -1.130 7.990 499798 74.51 68.28 37.36 29.5 45 3 3 -5.662 -1.113 7.927 499799 74.37 68.10 37.34 29.75 45 3 3 -5.663 -1.095 7.864 499800 74.22 67.91 37.32 30 45 3 3 -5.664 -1.078 7.802 S o u r c e: made by V.S. Tyurina. The artificial neural network models were trained using the Levenberg - Marquardt algorithm. The training dataset was randomly divided into three parts: “Training,” “Validation” and “Test,” which were used for training, validation, and testing, respectively. The split ratio was 75:15:15%. The mean squared error (MSE) was used as the training quality metric: (5) where n is the training sample size, are the stress values predicted by the neural network, are the target stress values. 3. Results and Discussion The statistical characteristics of the generated dataset are presented in Table 3. The model operates reliably only within the ranges between min and max for the input parameters, since using artificial neural networks for extrapolation can lead to unpredictable results. Table 3. Statistical characteristics of the training dataset Characteristic Input variables Output variables Tbot Tmid Tup t B h rate σbot σmid σup mean 46.35 46.16 27.78 15.25 35 2 2 -2.98 -0.52 2.42 std. dev. 15.08 16.77 12.71 8.59 7.07 0.71 0.82 2.10 0.97 3.36 min 6.21 6.32 5.23 0.50 25 1 1 -9.21 -4.53 -9.95 25% 35.96 34.20 17.64 7.75 30 1.5 1 -4.41 -1.12 0.27 50% 46.57 45.92 27.39 15.25 35 2 2 -2.94 -0.39 2.45 75% 57.40 58.05 36.59 22.75 40 2.5 3 -1.51 0.18 4.71 max 85.57 99.69 79.89 30.00 45 3 3 3.11 2.27 12.55 S o u r c e: made by V.S. Tyurina. The correlation coefficient matrix is shown in Table 4. Table 4 indicates that a very strong positive correlation (correlation coefficient RXY greater than 0.9) is observed between parameters Tbot and Tmid. A strong positive correlation (0.7 ≤ RXY < 0.9) is also observed between parameters Tup and Tbot, σup and Tbot, σmid and Tbot, σmid and Tmid. Correlation between the temperatures at the bottom surface, in the middle of the slab, and at the top surface is explained by the fact that the heating of the structure due to hydration is accompanied by the rise in temperature at all points. Strong correlation between the temperature and stress parameters also fully corresponds to the physics of the process. A strong negative correlation is observed between parameters σup and σmid. This is consistent with the results of [29], where it was shown that, assuming a parabolic temperature distribution along the thickness and symmetric heat transfer conditions, the following relationship holds between the increments of stresses σup and σmid: (6) Under asymmetric heat exchange conditions (heat exchange with the ground and the atmosphere), this relationship is disrupted, but a strong negative correlation between σup and σmid persists. Moderate positive correlation (0.5 ≤ RXY < 0.7) is observed between parameters h and Tbot, h and Tmid, h and σup. It is evident that as the slab thickness increases, the maximum temperature at the center of the structure also increases. Since most of the heat is dissipated through the top surface of the slab, the temperature at the bottom surface also increases as the thickness increases. Moderate positive correlation (0.3 ≤ RXY < 0.49) is observed between parameters Tup and σup, t and σmid, the hardening rate and the value of σbot. Parameters h and t, B and t, in the training dataset are completely independent. The training process for the recurrent neural network, feedforward neural network, and cascade neural network is shown in Figures 4-6, respectively. The training process was limited to 1000 iterations. The best mean square error value at the 1000th iteration was achieved for the feedforward neural network and amounted to 0.025 MPa². Figures 7-9 show the regression plots for the three machine learning models. The x-axis plots the target stress values T, and the y-axis plots the values Y predicted by the neural networks. The shape of the graphs for the models under consideration is similar; all points lie a short distance from the line Y = T. The correlation coefficients between the target and predicted values are close to one. Table 4. Correlation coefficient matrix Parameter Tbot Tmid Tup t B h rate σbot σmid σup Tbot 1.00 0.96 0.73 -0.18 0.23 0.53 -0.06 -0.21 -0.72 0.79 Tmid 0.96 1.00 0.76 -0.36 0.21 0.55 -0.04 0.02 -0.83 0.84 Tup 0.73 0.76 1.00 -0.20 0.08 0.10 -0.01 0.26 -0.36 0.44 t -0.18 -0.36 -0.20 1.00 0.00 0.00 0.00 -0.46 0.48 -0.34 B 0.23 0.21 0.08 0.00 1.00 0.00 0.00 -0.25 -0.11 0.15 h 0.53 0.55 0.10 0.00 0.00 1.00 0.00 -0.24 -0.68 0.63 rate -0.06 -0.04 -0.01 0.00 0.00 0.00 1.00 0.47 -0.09 0.26 σbot -0.21 0.02 0.26 -0.46 -0.25 -0.24 0.47 1.00 -0.10 0.02 σmid -0.72 -0.83 -0.36 0.48 -0.11 -0.68 -0.09 -0.10 1.00 -0.91 σup 0.79 0.84 0.44 -0.34 0.15 0.63 0.26 0.02 -0.91 1.00 S o u r c e: made by V.S. Tyurina. Figure 4. The training process for the recurrent neural network. S o u r c e: made by V.S. Tyurina. Figure 5. The training process for the feedforward neural network S o u r c e: made by V.S. Tyurina. Prediction Y=T Figure 6. The training process for the cascade neural network. S o u r c e: made by V.S. Tyurina. Figure 7. Regression plot for the recurrent neural network S o u r c e: made by V.S. Tyurina. Prediction Y=T Prediction Y=T Figure 8. Regression plot for the feedforward neural network S o u r c e: made by V.S. Tyurina. Figure 9. Regression plot for the cascade neural network S o u r c e: made by V.S. Tyurina. Figure 10. Graph of temperature changeon the upper surface of the slab before and after smoothing S o u r c e: made by V.S. Tyurina. The trained models were also validated using experimental data for a mass concrete foundation slab 2 m thick, as reported in [30]. Stress pre-dictions were performed based on experimental temperature values. The prediction results were then compared with the field measurements. Since the temperature on the top surface exhibited significant daily fluctuations, preliminary smoothing of the experimental data was applied (Figure 10). For the bottom surface and the middle of the slab, the experimental temperature values are given in Table 5. According to the classification used in this study, the concrete used in the slab is a high-early-strength concrete with a strength class of approximately B22.5. Figure 11 shows a comparison of the predictions made by the artificial neural net-works with experimental data for the center of the foundation slab. For comparison, the results of finite element modeling using relationships (2)-(4) are also presented. The heat release function in the finite element calculation was selected to ensure the best possible agreement between the calculated and experimental temperatures. Table 5. Experimental temperature values in the middle of the thickness and at the lower surface t, days 0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Tmid, °C 23.1 28.8 41.8 44.3 42.6 40 37.3 35.3 33.3 31.5 30.2 29.2 28.3 27.4 26.9 Tbot, °C 23 25.7 32 35 36.3 36.4 35.9 35.1 34.2 33.3 32.4 31.5 30.7 29.9 29.2 t, days 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Tmid, °C 26.1 25.8 25.7 25.8 25.9 25.8 25.7 25.5 25.3 24.4 23.5 23.1 23 23 23.1 Tbot, °C 28.6 28.1 27.6 27.2 26.8 26.6 26.3 26.1 26 25.7 25.4 25.1 24.7 24.4 24.2 S o u r c e: made by V.S. Tyurina. The graphs show that the three machine learning models and the FEM produce roughly the same results, with the exception of the initial time point. The feedforward neural network provides a more accurate prediction at the initial time point, exhibiting the lowest mean square error among all the models considered. At the 16-day mark during the experiment, a crack was observed in the slab, corresponding to a spike in the experimental graph. At this point in time, the neural networks predict the tensile stress value with sufficient accuracy. Deviations of the experimental results from the finite element analysis and the neural network predictions are observed in the time interval from 3 to 11 days. These can be explained by deviation of the actual time-dependent elastic modulus of concrete from that assumed in the model. t, days Figure 11. Comparison of neural network predictions with experimental data and finite element analysis results: 1 - recurrent neural network; 2 - feedforward neural network; 3 - cascade neural network S o u r c e: made by V.S. Tyurina. 4. Conclusion As part of this study, an approach for predicting thermal stress in mass concrete foundation slabs based on temperature monitoring data using machine learning methods was developed and validated. Three artificial neural network architectures were considered as models: a recurrent neural network, a feedforward neural network, and a cascade neural network. Training was conducted using a dataset of 499,800 rows, generated from 4200 numerical experiments and covering a wide range of variations in geometric, strength, and thermophysical parameters. The feedforward neural network model yielded the best results, achieving a mean square error in the stress prediction of 0.025 MPa². All three models demonstrated high prediction accuracy, as confirmed by regression plots with correlation coefficients between the target and predicted values close to one. Verification of the developed ANN models using experimental data for a 2-m-thick slab demonstrated their adequacy and practical applicability. With particular accuracy, the models predicted the emergence of tensile stress corresponding to the crack formation observed in the experiment on the 16th day. The proposed approach overcomes the limitations of existing methods based on the assumption of a parabolic distribution of temperature and stress, and provides accurate predictions for structures thicker than 2 m. The use of ANN significantly reduces computational costs compared to direct finite element modeling, making the method an effective tool for quick analysis of monitoring data in real-time. Thus, the application of machine learning methods to predict thermal stress in hardening mass concrete structures is a promising area of research that helps to improve the accuracy of stress state evaluation and prevent early cracking.

About the authors

Vasilina S. Tyurina

Don State Technical University

Email: vasilina.93@mail.ru
ORCID iD: 0009-0001-6399-401X
SPIN-code: 8808-2687

Candidate of Technical Sciences, Associate Professor of the Department of Structural Mechanics and Theory of Structures

1 Gagarin Sq., Rostov-on-Don, 344003, Russian Federation

Anton S. Chepurnenko

Don State Technical University

Author for correspondence.
Email: anton_chepurnenk@mail.ru
ORCID iD: 0000-0002-9133-8546
SPIN-code: 7149-7981

Doctor of Technical Sciences, Professor of the Department of Structural Mechanics and Theory of Structures

1 Gagarin Sq., Rostov-on-Don, 344003, Russian Federation

Batyr M. Yazyev

Don State Technical University

Email: ps62@yandex.ru
ORCID iD: 0000-0002-5205-1446
SPIN-code: 5970-5350

Doctor of Technical Sciences, Professor of the Department of Structural Mechanics and Theory of Structures

1 Gagarin Sq., Rostov-on-Don, 344003, Russian Federation

References

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