Study on Temperature Field Massive Concrete in Early Age Based on Temperature Influence Factor

In order to solve the problem of insufficient accuracy of early temperature field caused by the change of hydration rate under different temperatures, the theoretical formula of finite element calculation based on temperature influence factor is put forward and then the theory is tested. On this basis of this theory, the early temperature field of a RCC dam is numerically simulated and the variation law of concrete hydration rate under different temperatures is studied. .e numerical simulation results are compared with the results without considering the temperature effect and the measured temperature data. .e results show that the theoretical results are in agreement with the measured temperature data, and the accuracy and applicability of the theoretical formula are proved.


Introduction
Due to great temperature difference, the temperature stresses are caused by heat of hydration of cement which leads to temperature cracks during mass concrete construction processing. Temperature cracks seriously affect the durability of concrete structures and reduce the service life of the project [1][2][3]. Cement hydration of mass concrete produces vast quantities of heat. However, concrete as a poor conductor of heat leads to its slow heat dissipation and high internal temperature. e heat of hydration generation rate of cement varies with different temperatures, and in the first 2 days, the heat of hydration of cement is 40-80% of the total heat [4,5]. erefore, it is more significant to simulate accurately the early-age temperature field of mass concrete considering temperature effect.
Considering the temperature influence on the mass temperature field, Saul [6] and Rastrup [7] firstly proposed the concept of equivalent age. Based on the concept of equivalent age, the domestic and foreign scholars have carried out a lot of research. For example, the Jin et al. [8] calculated equivalent age based on the equivalent quantity of cement and determined the concrete temperature field equation; Zhang et al. [9]. studied the influence of equivalent age on the adiabatic temperature rise of concrete; Schindler [10] studied the temperature effect on hydration heat of concrete based on equivalent age; Schutter [11] studied the heat of hydration from concrete based on the degree of hydration, which is consistent with the concept of equivalent age. Dong and Li [12] considered the chemical reaction rate of cement at different curing temperatures and deduced the hydration-heat released model of cement based on equivalent age. In order to directly reflect the change of hydration rate during the actual age, Zhang et al. [4]. proposed the temperature influence factor to accurately simulate the temperature field of PCCP (Prestressed Concrete Cylinder Pipe) on the high temperature curing stage. On this basis, this study presents a finite element theoretical formula based on the temperature influence factor and studies the early-age temperature field variation law of mass concrete considering the influence of temperature. Furthermore, compared with the measured temperature data, the theoretical formula is verified to be the correction and a basis for accurately simulating the temperature field of mass concrete is provided.

Hydration Rate Based on Temperature
Influence Factor 2.1. Temperature Influence Factor. Arrhenius proposed the empirical formula for the rate of chemical reactions [13][14][15][16][17][18]: where A is the pre-exponential factor and E a is the reaction activation energy; Ordinary Portland cement E a is 33500/ (J·mol −1 ), atmospheric constant R � 8.314 (J·mol −1 ); T is temperature (k). Integral expression of equation (1) is shown as Making temperature influence factor c T i � (v i /v r ), c T i is the temperature influence factor at time i, which is the ratio of the reaction rate corresponding to the temperature at time i to the reaction rate at reference temperature. Formula (3) of temperature influence factor is obtained: where T r is the reference temperature and T i is the concrete temperature at a certain moment i.

e Equation of Hydration Rate Based on the Temperature
Influence Factor. Assuming any point coordinate (x, y, z) in the concrete, the heat conduction equation of the transient temperature field [18] is shown as where c is the specific heat capacity; ρ is the density; T is the temperature; and t is the actual age.
In the temperature field simulation of mass concrete, the temperature influence factor c T is introduced to the heat conduction equation (4) considering the temperature effect.
e modified formula is shown as e heat conduction equation (5) is to consider the temperature effect in transient temperature field, and the innovation is the introduction of temperature influence factorc T .
Concrete hydration heat is generated and produced, in essence, by the compounds of cement hydration reaction with water hydration heat. Cement hydration heat is not fully released in the hardening process, but gradually released as time [19][20][21][22]. e hydration heat of cement changes with time: where t is the actual age; Q ∞ is the final hydration heatt ⟶ ∞; and m and n are the constant coefficients. e hydration rate equation considering the temperature effect for any element in the finite element model at time t i is shown as In formula (7), the temperature influence item is i ; moreover, this item declines with the increase of age and changes in (0, 1).
When the initial time (t � 1 d) and pouring temperature is T 0 , the initial hydration heat rate value is q 1 . If the initial value of the hydration heat rate is known, the temperature T i at a certain unit time t i can be obtained by program language programming. us, the temperature influence factor c T i at time t i can be obtained from the heat conduction equation (3). e temperature field of mass concrete affected by temperature is calculated by the heat conduction equation (5).

Verify the Theory
In order to study the temperature field variation law of mass concrete under the influence of different temperature, the concrete finite element model of 1m thickness, 5 m long, and 5 m wide is used to verify the theory. e temperature field of the center point in the model is studied under the different ambient temperatures. ree calculation conditions: the 1st condition is cold environment, 5°C ambient temperature and 10°C pouring temperature; the 2nd condition is high temperature environment, 35°C ambient temperature and 10°C pouring temperature; the 3rd condition is high temperature environment, 40°C ambient temperature and 20°C pouring temperature. Under the different conditions, comparison analysis of temperature field in the center point M considering and not considering the temperature effect is studied. e results are shown in Figure 1.
In Figure 1, we can see that (1) In the 1st condition, the slope of the point on L 1 curve is obviously smaller than that of curve L 01 after considering temp-effect. Temperature change rate decreases and the peak temperature is decreased from 20.4°C to 17.2°C. After reaching the peak temperature on the 10th day, the slope difference between the curve L 1 and L 01 becomes weaker and weaker gradually. e reason is that the temperature influence factor c T i < 1 and the heat of hydration rate are lower than that of without considering. It leads to the decrease of the temperature change rate and the temperature field changes obviously. With the increase of age, the age effect term t n−1 i e mt n i is gradually reduced to zero. us, the temperature change rate is getting smaller and smaller.
(2) In the 2nd condition, when the temperature is between 10 -20°C, the slope of L 2 curve is smaller than curve L 02 , the temperature change rate decreases.
When the temperature is higher than 20°C, the slope of the point on the L 2 curve is obviously larger than that of L 02 and the peak temperature increases from 36.8°C to 39.5°C. After reaching the peak temperature on the 50th day, the slope difference between the curve L 2 and L 02 becomes weaker and weaker gradually. e reason is that the temperature influence factor c T i > 1 and the heat of hydration rate is higher than that of without considering temperature influence. It leads to the increase of the temperature change rate and the temperature field changes obviously. With the increase of age, the age effect term t n−1 i e mt n i is gradually reduced to zero. us, the temperature change rate is getting smaller and smaller.
(3) In the 3rd condition, the slope of curve L 3 is obviously larger than L 03 in early age. e temperature change rate increases, and the peak temperature increases from 42.3°C to 51.9°C. When after reaching the peak temperature on the 35th day, the slope difference between the curves L 3 and L 03 becomes weaker and weaker gradually. After considering the temperature effect, the temperature difference between inside and outside increases from 2.3°C to 11.9°C. (4) e temperature effect is weakened with the increase of age. In order to save the finite element calculation time, the temperature effect is taken into consideration only in early age, which is within two months after pouring concrete. Because after two months, the both temperature changes tend to be consistent. Beyond the time, the temperature effect is not considered. Considering the influence of temperature, the higher the temperature, the faster the hydration rate and the more obvious the change of temperature. It is more unfavorable to the concrete temperature control during the construction period, and vice versa. e calculation results under the different conditions are shown in Table 1. In Table 1, it can be seen that, in hightemperature areas, the temperature difference has increased by five times after considering the temperature influence factors; however, in cold areas, the temperature difference does not change significantly. For the mass concrete engineering in the high temperature region and the high temperature curing, it is necessary to consider the temperature effect especially in the early temperature field simulation, which can improve the simulation accuracy of the temperature field and better construction temperature control.

Material Parameters and Hydration Heat Calculation
(1) Material parameters and cement compositions: material parameters and cement components used in the finite element calculation for the RCC dam are shown in Tables 2 and 3, respectively. (2) m and n parameters: Borg analyzed a large number of cement hydration heat test data, using the least squares method to obtain the empirical formula of multiple regression hydration heat [14,15]: In formula (8), Q (t) is cement hydration heat on the age of t, kJ/kg; P a , P b , P c , and P d are C 3 S, C 2 S, C 3 A, and C 4 AF percent content, respectively; A t , B t , C t , and D t corresponding to unit mass C 3 S, C 2 S, C 3 A, and C 4 AF hydration heat value in t-age, kJ/kg. e heating quantity of various hydration substances in cement clinker at different ages at 20°C is shown in Table 4 [18][19][20]. e empirical formula for the hydration heat of mixed cement with admixture [20][21][22][23][24]: where Q p is the hydration heat of the cement mixed with admixture; Q 0 is the hydration heat of the cement without admixture; p is the percentage of the admixture; the maximum value of p is 60%; and k is empirical coefficient 0.55. When the content of admixture is greater than 60%, the empirical coefficient k value is 0.5.

Advances in Civil Engineering
According to different cement mineral compositions and fly ash contents, the heat of hydration value can be calculated in different ages of the project through formulas (8) and (9). And the heat of hydration formula was fitted with the double exponential formula (6) by the hydration heat value of different ages. e fitting results of m and n coefficients are shown in Table 5.

Temperature Field Simulation.
e ambient temperature of the engineering site varies with time according to the cosine curve of T �14.8-13.3cos [3.14 (t−1.1)/6] [24][25][26]. e mean annual temperature is 14.8°C. Roller-compacted concrete dam is compacted layer by layer, and each layer pouring temperature and time are different. e dam's temperature field is simulated using new heat conduction equation (formula (5)). Because the influence of temperature effect is obvious only in the early age, in order to save the calculation time, the calculation example only takes into account the influence of temperature in 30 days. Point T 1 is chosen to study the temperature field of the RCC dam on the early age. e position of T 1 is at the 82.5 m elevation and shown in Figure 1. e temperature history curve and the monitoring temperature data of position T 1 are shown in Figure 3.     In Figure 3, it can be seen that (1) the pouring temperature is 18°C, and the heating rate is slower at the beginning of a few days after considering the influence of temperature. When the temperature is higher than 20°C, the heating rate becomes faster. Maximum temperature increased by 1.2°C. (2) e monitoring data is more consistent with the finite element calculation result considering the temperature effect, which shows that the numerical simulation results are more accurate after considering the temperature effect. (3) e ambient temperature is closer to   Advances in Civil Engineering the reference temperature of 20°C; thus, the temperature range of change is small. It is assumed that the RCC dam is constructed in a high temperature region with an average temperature of 28°C and the ambient temperature varies with time according to the cosine curve T � 28 + 10.8 cos[3.14 (t − 66.8)/6]. e position of point T 1 and point T 2 is at 82.5 m and 90.5 m elevation, respectively, as shown in Figure 1. e temperature history curves of point T 1 and point T 2 are shown in Figure 4.
In Figure 4, it can be seen that (1) concrete temperature rising speed increases in high-temperature area after considering temp-effect. Point T 1 reached the peak temperature on 100th day. e peak temperature increased from 36.3°C to 39.9°C, and maximum temperature increased by 3.6°C. Point T 2 reached the peak temperature on 110th day. e peak temperature increased from 36.9°C to 43°C, and maximum temperature increased by 6.2°C. (2) When the ambient temperature drops sharply, the larger temperature difference will lead to higher thermal stress and concrete temperature cracks. erefore, better temperature control is necessary during the summer construction period in high-temperature areas such as pouring temperature control and take concrete cooling measures.

Conclusions
e heat conduction equation and the finite element theory formula based on the temperature influence factor are proposed. e theory is verified and the variation law of temperature field is analyzed. e temperature field of the RCC dam is simulated by an engineering example. e finite element simulation results are in agreement with the measured temperature data after considering the temperature effect, which proves that the theory formula is correct.
Temperature has a great influence on the early temperature field. In order to save the finite element calculation time, the temperature effect is considered only in one to two months after the pouring time.
Especially in the high-temperature region, early ages' accurate simulation of mass concrete temperature field, adopting the reasonable temperature control measures, control right into the molding temperature, and break and intermittent layer thickness is important, such as the choice of which can effectively guide the engineering construction rapidly.

Data Availability
e data used to support the findings of this study are included in the article.

Conflicts of Interest
e authors declare no conflicts of interest.