Study on Temperature Control and Cracking Risk of Mass Concrete Sidewalls with a Cooling-Pipe System

Abstract

Hydration heat of early-age sidewalls can cause cracks owing to thermal stress, reducing the durability of underground space structures. The heat can be removed by the flowing water in the cooling pipe system. However, the cooling pipe may cause thermal stress due to the temperature gradient in the region adjacent to the cooling pipe, resulting in concrete cracking. To minimize the temperature peak of sidewalls and cracking risks in the region adjacent to the cooling pipe, the crack-distribution characteristics, temperature, and strain evolution of an early-age sidewall with a cooling pipe system are analyzed by concrete temperature and strain tests. Furthermore, a model that accounts for the early-age behavior of concrete and cooling-pipe effects is developed and solved. Finally, the effects of cooling-pipe parameters and ambient temperature on the sidewall’s temperature field and cracking risk are analyzed. The results indicate that the cracks emerge in the first two weeks after concrete pouring; most are vertical, and a few oblique cracks emerge in the wall corner. The tensile stress in the region adjacent to the cooling pipe gradually decreases along the flow direction. Reducing the water temperature and increasing the flow rate reduces the sidewall’s temperature peak and cooling rate. However, they increase the cracking risk in the region adjacent to the cooling pipe. When the flow rate exceeds 0.6 m³/h, further increasing the flow rate does not significantly affect the temperature field. Reducing the distance between cooling pipes reduces the temperature peak, cooling rate, and cracking risk in the region adjacent to the cooling pipe. In high-temperature environments, the cracking risk in the region adjacent to the cooling pipe increases significantly.

1. Introduction

Due to the influence of groundwater, strict requirements must be observed to maintain the underground space structure’s waterproofing performance and durability. As a typical mass concrete structure, the sidewall plays a crucial role in the antiseepage and anticracking performance of the underground space structure [1]. At an early stage, the hydration heat causes the concrete temperature to rise and fall sharply, accompanied by thermal deformation. The influence of the ambient environment results in a large temperature gradient in the thickness direction of the early-age concrete, so the structure is compressed internally and tensile externally [2]. For sidewalls with thin thickness and large specific surface area, even though the drying shrinkage caused by the hydration reaction is completed quickly, the dry environment will also cause drying shrinkage of the concrete. Additionally, the constraint effect of old concrete on the shrinkage deformation of early-age concrete is also an essential reason for forming concrete cracks [3]. When the constraint stress exceeds the concrete strength, the sidewall will crack, which significantly reduces the durability of the underground space structure [4]. Therefore, it is necessary to take measures to reduce the cracking risk of early-age sidewalls.

Additives and admixtures such as fiber, mineral admixtures, temperature inhibitors, and microexpansion agents have been widely used in concrete engineering, significantly improving concrete strength and durability [5,6,7,8,9,10,11]. However, concrete structures exhibit numerous cracks at an early age. The American Concrete Institute proposed that measures must be taken to cope with the heat generated by the hydration reaction and the subsequent volume changes to minimize cracks in mass concrete structures [12]. It is important to ensure that the temperature peak of early-age concrete does not exceed 70 °C and that the temperature difference between the surface and interior of concrete remains below 20 °C. Achieving these two conditions can effectively minimize the formation of cracks owing to thermal stress. The conditions can be achieved by reducing the cement content in concrete and adopting reasonable construction measures. Based on a numerical simulation analysis of the temperature field of an early-age box girder, Cai et al. [13] found that as the concrete section size, pouring temperature, and cement content increased, the temperature peak of concrete and the temperature difference between the surface and interior of concrete rose significantly. With the on-site temperature monitoring and numerical simulation analysis, Zhang et al. [14] presented that reducing the concrete pouring temperature can increase the cooling rate of mass concrete, and the ambient temperature dramatically influences the temperature difference between the surface and interior of concrete. Using a shrinkage test for early-age concrete, Zhang et al. [15] discovered that the exposure of the cement paste to a windy environment significantly increases shrinkage deformation. In civil engineering, the monitoring system can reflect the health status of the structure in time, which has become an essential means of obtaining structural state data [16]. The curing regimen for early-age concrete devised by Yang et al. [17] relied on the monitoring of hydration reactions as well as real-time tracking of temperature and humidity. This approach ensured that the concrete humidity and temperature were meticulously regulated to prevent the formation of cracks due to inappropriate artificial maintenance practices. In addition, many prediction models were proposed to guide mass concrete construction to reduce the crack risk, which can predict the temperature and stress state of mass concrete structures [18,19,20].

In 1931, the first use of a post-cooling-pipe system, in the Owyhee Dam in Oregon, achieved good construction results [21]. Since that time, the post-cooling system has become increasingly popular in mass concrete engineering. Wang et al. [22] designed the cooling system of the bridge deck and determined the effective flow-rate range of the cooling water through field temperature monitoring and numerical simulation. Chen et al. [23] performed artificial cooling tests on cast-in-place piles in the Beilu River of the Qinghai–Tibet Plateau to address thermal-disturbance issues during construction in permafrost regions. It was found that the cooling-pipe system demonstrated a notable cooling effect, leading to a rapid and significant improvement in the bearing capacity of the pile foundation. Lu et al. [24] embedded cooling pipes in the surface layer of the concrete structure and connected the cold and heat sources to achieve temperature control of the dam concrete. Wang et al. [25] used numerical simulation analysis to determine the optimal pipe spacing of the cooling pipe system that was embedded in the concrete water tower, which controlled the temperature and thermal stress of the early-age water tower. Additionally, many studies have optimized the cooling pipe system according to the working principle of the cooling pipe to improve the cooling efficiency of the cooling pipe system. Zhou et al. [26] proposed an annular finned cooling tube, which improves the cooling efficiency of the cooling pipe by increasing the contact area between the cooling pipe and the concrete. Li et al. [27] developed a new automatic air-cooling system on the basis of the cooling-pipe principle. It was discovered that the air-cooling pipe is capable of significantly reducing concrete cracks when the gas flow rate reaches 12 m/s. Based on the working principle of cooling pipes, Zhang et al. [28] proposed a novel longitudinal circular opening structure for tunnel lining to reduce the temperature peak of early-age tunnel lining and, thus, reduce its thermal stress. Kheradmand et al. [29] proposed a flexible cooling pipe system, which allows the cooling pipe to be recycled after the concrete hardening to realize the reuse of the cooling pipe. In addition, the numerical simulation method of the cooling pipe system was improved in many studies, which significantly improves the computational efficiency and accuracy of the numerical simulation and provides a method for the optimal design of the cooling system [30,31,32].

Notably, although the cooling pipe can effectively reduce the temperature peak of early-age concrete, the temperature gradient in the region adjacent to the cooling pipe can cause concrete cracks [33]. The temperature gradient can be effectively reduced by reasonable water temperature, pipe spacing, pipe material, and pipe arrangement [34,35]. According to the above, the cooling pipe system has been widely used in mass concrete engineering, such as bridge decks, cast-in-place piles, dams, water towers, and tunnel linings. However, the cooling pipe system was not applied in sidewall engineering. Although Wang et al. [36] found that the cooling pipe system can effectively reduce the temperature peak and cracking risk of sidewall by the temperature and strain test of early-age concrete, the thermal stress in the region adjacent to the cooling pipe was not further analyzed. In addition, compared with other mass concrete structures, owing to the thinner thickness and larger surface area of the sidewall, the temperature field of the sidewall is more sensitive to the changing ambient temperature. The cooling pipe parameters of other engineering cannot provide a reference for it.

This study aims to guide the application of the cooling pipe system in mass concrete sidewall engineering to reduce the temperature peak and thermal stress of sidewalls and avoid cracking in the region adjacent to the cooling pipe. Examining the sidewall of a subway station, this study investigates the crack distribution characteristics, temperature, and strain evolution of early-age sidewalls with a cooling-pipe system by conducting temperature and strain tests on early-age concrete. To describe the complex multifield-coupling relationship of early-age sidewalls, a chemo-thermo-hygro-mechanical coupling model that accounts for the behavior of concrete at an early age and the action of cooling pipes is developed. The secondary development function of finite element software is used to solve the multifield-coupling model, and the temperature, humidity, and stress fields of the early-age sidewall with a cooling-pipe system are analyzed. In addition, the effects of water temperature, flow rate, and pipe-spacing distance on the temperature field and cracking risk of the early-age sidewall, as well as the applicability of the cooling pipe system in sidewall engineering at different ambient temperatures, are discussed.

2. Early-Age-Sidewall Temperature and Strain Tests

2.1. Engineering Situations

The Xingbin Metro station in Xiamen Metro Line 6 adopts a composite wall structure; the diaphragm-wall thickness is 800 mm, and the sidewall thickness is 400 mm. The composite-wall structure is poured with C40 concrete. The mix proportion of concrete is shown in

Table 1. Concrete mix proportions.

Cement Type	Water (kg/m3)	Cement (kg/m3)	Sand (kg/m3)	Aggregate (kg/m3)	Water Reducing Admixture (kg/m3)	Fly Ash (kg/m3)	Blast Furnace Slag (kg/m3)
P.O42.5	159	249	729	1051	7.51	99	47

, and the main performance parameters of concrete are shown in

Table 2. Main performance parameters.

Air Content (%)	Porosity (%)	Compressive Strength for Fresh Concrete (MPa)	Compressive Strength for Hardened Concrete (MPa)
2.6	12	42.6	43.5

. The sidewall is poured at a length of 12 m and a height of 4.8 m. The sidewall is constructed with wood formwork, and the formwork is removed at two days of age. The cooling-pipe system adopts a steel pipe with an inner diameter of 30 mm and a wall thickness of 1.5 mm. The water flow rate is 1.5 m³/h, the water temperature is 25 °C, and the spacing distance between cooling pipes is 1 m. Figure 1a presents a schematic diagram of the cooling pipe.

2.2. Experimental Description

As shown in Figure 1, there was a temperature and strain monitoring section at a sidewall height of 2.4 m. In the monitoring section, the measuring points were distributed along the thickness direction of the sidewall, located at the surface, center, and bottom of the sidewall. A total of six temperature sensors were installed in the sidewall, named T11~T13 and T21~T23; six strain sensors were installed along the length direction of the sidewall, named S11~S13 and S21~S23. The temperature and strain of the sidewall were measured by a temperature sensor and vibrating wire strain sensor. The measurement ranges of the temperature sensor and strain sensor were −30~70 °C and 0~4000 με, respectively, and the precision values were 0.1 °C and 0.1 με, respectively.

In order to completely record the temperature and strain evolution of the early-age sidewall and consider the hydration process of the concrete, the monitoring period was 28 days, in which the data were recorded every 2 h from 0 days to 7 days, every 4 h from 7 days to 14 days, every 6 h from 14 days to 21 days, and every 8 h from 21 days to 28 days. During the monitoring period, the development of cracks on the sidewall surface was observed and recorded, and the crack width was recorded by the crack width gauge. The precision value of the crack width gauge was 0.01 mm. In addition, the temperature and humidity of the environment were recorded by a temperature and humidity meter. The experimental procedure is shown in Figure 2.

2.3. Results

2.3.1. Crack Distribution of the Early-Age Sidewall

Figure 3 shows the crack distribution of the early-age sidewall. At 4~6 days of age, there were five vertical cracks and one oblique crack in the sidewall, and none of the cracks had seeped. At 9 days of age, cracks 7 and 8 appeared in the sidewall, accompanied by water seepage. This is because the shrinkage deformation of the sidewall concrete was constrained by the old concrete, resulting in cracks 7 and 8 developing outward from the inside of the sidewall and water seepage. At this time, cracks 5 and 8 had extended to the construction joint at the top of the sidewall, and crack 1 began to seep. In addition, as the age gradually approached 14 days, the temperature and shrinkage deformation of the sidewall concrete gradually decreased, which caused the overall development speed of the sidewall crack to slow down gradually. At 13 days of age, the length of cracks 1–3 and 6 were slowly growing, and cracks 2 and 3 began to seep. Moreover, crack 7 stopped seeping, and the crack contained calcium carbonate crystals. At 16 days of age, crack 2 had extended to the top of the sidewall, but the development of other cracks was slow. At 17~28 days of age, the length of most cracks did not grow significantly, and cracks 1–6 and 8 exhibited seepage accompanied by a small amount of calcium carbonate crystals. It indicated that the sidewall cracks mainly appeared within 14 days after concrete pouring. Notably, most of the cracks extended upward from the construction joints at the bottom of the sidewall, and most of them were vertical cracks; a few oblique cracks were located in the corner of the wall. This occurred because the shrinkage deformation of the early-age sidewall was constrained by the old concrete, and the principal shrinkage stress was along the horizontal direction. The constraint effect on the end of the sidewall was small, and the principal stress direction was inconsistent with the sidewall’s length direction.

Figure 4 depicts the crack-width evolution of cracks 1 and 6. As the figure shows, the width at both ends of the crack was smaller than the width in the middle, and the shape resembled a jujube core. At the time of crack initiation, the widths of cracks 1 and 6 were 0.15 and 0.11 mm, respectively. The width of crack 1 underwent its fastest growth rate at 4~9 days after concrete pouring, and the crack seeped when its width was 0.26 mm. The width of crack 6 underwent its fastest growth rate at 6~13 days after concrete pouring, and water seepage occurred when the crack width reached 0.20 mm. This analysis demonstrated that the sidewall crack width underwent its fastest growth rate in the first two weeks after concrete pouring and that there was a risk of water seepage in the sidewall when the crack width exceeded 0.20 mm.

2.3.2. Evolution of Temperature and Strain

Figure 5 shows the temperature and strain evolution of the early-age sidewall. As the figure shows, the temperature of the sidewall rose and fell sharply, along with evident expansion and shrinkage, in the first eight days after concrete pouring. At 0.78~0.83 days of age, the temperature of measuring points reached the peak. The temperature peaks of bottom points T13 and T23 were 49.1 and 49.7 °C, respectively, and center points T12 and T22 were 48.6 and 49.4 °C, respectively. Compared with the temperature peak of the center points, the temperature peak of surface points T11 and T21 decreased by 4.73% and 4.86%, respectively. This decrease enabled the heat of the sidewall to be diffused into the environment across the formwork, which caused the temperature of the sidewall surface to fall below that of the inside of the sidewall. The temperatures of points T21, T22, and T23 were higher than those of points T11, T12, and T13, respectively, because the water temperature of the cooling pipe gradually rose along the flow direction. After the formwork was removed, the coefficient of the heat dissipation of the sidewall surface increased significantly, and the temperature of the sidewall underwent an apparent fluctuation. Moreover, as the figure shows, the strain peaks of center point S12 and S22 were −83.7 and −84.2 με, respectively, and the strain peaks of bottom points S13 and S23 were −87.1 and −93.9 με, respectively. When comparing the strain at the top of the sidewall with the strain at the center and bottom of the sidewall, the strain peaks of surface point S11 decreased by 14.10% and 17.45%, respectively, while the strain peaks of the surface point S21 decreased by 21.14% and 29.29%, respectively. In addition, at 5 days of age, the strain curve of the surface point S11 exhibited a strain jump of 154.0 με, indicating that the surface had cracked at point S11.

3. Numerical Model of the Early-Age Sidewall

3.1. Chemo-Thermo-Hygro-Mechanical Coupling Model

3.1.1. Cement Hydration

At an early age, the heat released by the hydration reaction is the main reason for the increase in concrete temperature. The hydration degree formula, based on the Arrhenius reaction rate formula and material activation energy, was applied to describe the hydration process of cement in this paper. The heat released by the hydration reaction at a certain time under adiabatic conditions is expressed as follows [37]:

$$Q\left(t\right)=\alpha (t)Q_{\max },$$

(1)

where Q_max is the final heat release of the cement and $\alpha (t)$ is the hydration degree of the cement. This study adopted the following hydration degree calculation model based on equivalent age:

$$\alpha \left(t_{\mathrm{e}}\right)={\alpha }_{\infty }(1-\exp (-a{t_{\mathrm{e}}}^b)),$$

(2)

where t_e is equivalent age [37], ${\alpha }_{\infty }$ is the final hydration degree, and a and b are the model parameters controlling the evolution of the hydration reaction.

3.1.2. Governing Equation for Temperature Evolution

For the mass concrete structure with a cooling pipe system, the temperature field is mainly related to hydration heat, ambient temperature, and cooling pipe. Considering the influence of the hydration reaction, ambient temperature, and cooling pipe system, the heat balance equation for the sidewall is as follows:

$$C_{\mathrm{c}}{\rho }_{\mathrm{c}}\frac{\mathit{\partial }T}{\mathit{\partial }t}=\nabla \left({\lambda }_c\nabla T\right)+Q+Q_{\mathrm{ext}},$$

(3)

where C_c is the specific heat of the concrete, ${\rho }_{\mathrm{c}}$ is the density of the concrete, q is the convective heat flux, Q is the hydration heat of the concrete per unit of time, Q_ext is the external heat source accounting for the effect of the cooling pipe, and ${\lambda }_c$ is the thermal conductivity of the concrete.

The thermal properties of early-age concrete are related to the hydration degree, showing significant time characteristics. Accounting for the influence of the hydration degree, Van Breugel [38] and De Schinder and Taerwe [39] proposed the equation for specific heat and thermal conductivity. The equation of specific heat is as follows:

$$\{\begin{array}{ll}{C}_{\mathrm{c}}\hfill & =\left(W_{\mathrm{c}}\alpha C_{\mathrm{cef}}+W_{\mathrm{c}}\left(1-\alpha \right)C_{\mathrm{cem}}+W_{\mathrm{a}}{C}_{\mathrm{a}}+W_{\mathrm{w}}{C}_{\mathrm{w}}\right)/{\rho }_{\mathrm{c}}\hfill \\ C_{\mathrm{cef}}\hfill & =8.4T+339\hfill \end{array},$$

(4)

where W_c, W_a, and W_w are the quality of cement, aggregate, and water per cubic meter of concrete, respectively; C_cem, C_a, and C_w are the specific heat of the cement, aggregate, and water, respectively; C_cef is the assumed heat of the cement.

The equation of thermal conductivity is as follows:

$${\lambda }_{\mathrm{c}}\left(\alpha \right)={\lambda }_{\mathrm{p}}\cdot \left(1.33-0.33\alpha \right),$$

(5)

where ${\lambda }_{\mathrm{c}}\left(\alpha \right)$ is the thermal conductivity of the concrete based on hydration degree, and ${\lambda }_{\mathrm{p}}$ is the final thermal conductivity of the concrete.

For surface contact with air, the boundary conditions are as follows:

$$-{\lambda }_{\mathrm{c}}(\frac{\mathit{\partial }T}{\mathit{\partial }n})={\beta }_{\mathrm{c}}(T-T_{\mathrm{ext}}),$$

(6)

where T_ext is ambient temperature and ${\beta }_{\mathrm{c}}$ is the coefficient of the heat release of the concrete surface.

Since the surface heat release coefficient of the concrete template is smaller than the surface heat release coefficient of the concrete, the concrete template has a thermal insulation function. Therefore, when solving the temperature field of concrete, the influence of concrete formwork on boundary conditions must be considered. When the formwork is not removed, the coefficient of the heat release of the concrete surface is as follows:

$${\beta }_{\mathrm{e}}=\frac{1}{1/{\beta }_{\mathrm{w}}+h_{\mathrm{w}}/{\lambda }_{\mathrm{w}}},$$

(7)

where ${\beta }_{\mathrm{e}}$ is the equivalent heat dissipation coefficient, ${\beta }_{\mathrm{w}}$ is the coefficient of the heat release of the formwork surface, h_w is the thickness of the formwork, and ${\lambda }_{\mathrm{w}}$ is the thermal conductivity of the formwork.

When the cooling pipe works, the equation for the fluid’s heat balance is as follows:

$${\rho }_{\mathrm{w}}A{C}_{\mathrm{w}}\frac{\mathit{\partial }T}{\mathit{\partial }t}+{\rho }_{\mathrm{w}}A{C}_{\mathrm{w}}{u}_1•\nabla T_1=\nabla •\left(Ak\nabla T_1\right)+f_D\frac{\rho A}{2d_{\mathrm{h}}}{\left|u_1\right|}^3+Q+Q_{\mathrm{wall}},$$

(8)

where ${\rho }_{\mathrm{w}}$ is the fluid density, A is the area of the pipe’s cross section, u₁ is the velocity field, k is the thermal conductivity of fluid, T₁ is the fluid temperature, f_D is the Darcy resistance coefficient, d_h is the average hydraulic diameter, Q is the general heat source, and Q_wall is external heat exchange through the pipe wall.

3.1.3. Governing Equation for Moisture Transport

At an early age, the moisture consumption of concrete is caused by moisture diffusion and the hydration reaction [40]. The governing equation for the humidity field is obtained by Fick law, as the following equation shows:

$$\frac{\mathit{\partial }h}{\mathit{\partial }t}=D_{\mathrm{h}}{\nabla }^{2}h+\frac{\mathit{\partial }{h}_{\mathrm{s}}}{\mathit{\partial }t},$$

(9)

where h is the relative humidity of the concrete, D_h is the coefficient of the concrete’s moisture diffusion, and $\mathit{\partial }{h}_{\mathrm{s}}/\mathit{\partial }t$ is the rate of the humidity loss caused by the hydration reaction. The equation for the two-stage moisture-consumption model established in the literature is as follows [41]:

$$h_{\mathrm{s}}=\{\begin{array}{l}\begin{array}{cc}0& (\alpha \le {\alpha }_{\mathrm{c}})\end{array}\hfill \\ \begin{array}{cc}(h_{\mathrm{su}}-1){(\frac{\alpha -{\alpha }_{\mathrm{c}}}{{\alpha }_{\infty }-{\alpha }_{\mathrm{c}}})}^{k_{\mathrm{s}}}+1& (\alpha >{\alpha }_{\mathrm{c}})\end{array}\hfill \end{array},$$

(10)

where h_su is the relative humidity corresponding to the final hydration degree of the cement during self-drying (h_su = 0.84), ${\alpha }_{\mathrm{c}}$ is the critical hydration degree (${\alpha }_{\mathrm{c}}=0.52$), and k_s is the model parameter (k_s = 1.5).

The boundary conditions for surface contact with the air are as follows:

$$-D_{\mathrm{h}}(\frac{\mathit{\partial }h}{\mathit{\partial }n})={\beta }_{\mathrm{h}}(h-h_{\mathrm{ext}}),$$

(11)

where h_ext is the humidity of the environment and ${\beta }_{\mathrm{h}}$ is the coefficient of the moisture exchange of the concrete surface. This paper adopted the coefficient of the moisture exchange of the concrete surface proposed by Wong [42], that is,

$${\beta }_{\mathrm{h}}=a_{\mathrm{h}}(\frac{w}{c})-b_{\mathrm{h}},$$

(12)

where w/c is the water–cement ratio, a_h = 2.32 × 10⁻³, and b_h = 0.9 × 10⁻⁴.

At an early age, temperature has a strong influence on the moisture diffusion rate of concrete. The moisture diffusion coefficient that accounts for the influence of temperature is as follows [43]:

$$\{\begin{array}{c}{D}_{\mathrm{h}}(h)=D_1\left\{\frac{D_0}{D_1}+\frac{1-D_0/D_1}{1+{\left[(1-h)/(1-h_{\mathrm{c}})\right]}^n}\right\}•f(t_{\mathrm{e}})\hfill \\ f(t_{\mathrm{e}})=\exp [\frac{E_{\mathrm{ad}}}{R}(\frac{1}{273+T}-\frac{1}{273+T_0})]\hfill \end{array},$$

(13)

where D₀ is the coefficient of the concrete’s moisture diffusion for h = 0%, D₁ is that for h = 100%, n is the fitting coefficient of the nonlinear humidity diffusion equation, which represents the rate of decrease in the moisture diffusion coefficient, h_c is the relative pore humidity, $f(t_{\mathrm{e}})$ is the function related to equivalent age, E_ad is the activation energy for moisture diffusion, and R is the universal gas constant. This paper employs D₀/D₁ = 0.05, n = 15, h_c = 0.8, E_ad = 35 kJ/mol, and R = 8.314 J/(mol·°C).

3.1.4. Deformations of Early-Age Concrete

The deformation of early-age concrete mainly consists of instantaneous elastic deformation, shrinkage deformation, thermal deformation, and creep deformation. The strain of early-age concrete can be decomposed as follows:

$$\epsilon ={\epsilon }_{\mathrm{e}}+{\epsilon }_{\mathrm{c}}+{\epsilon }_{\mathrm{th}}+{\epsilon }_{\mathrm{sh}},$$

(14)

where ${\epsilon }_{\mathrm{e}}$ is the instantaneous strain, ${\epsilon }_{\mathrm{c}}$ is the creep strain, ${\epsilon }_{\mathrm{th}}$ is the shrinkage strain, and ${\epsilon }_{\mathrm{sh}}$ is the thermal strain.

The microprestress–solidification (MPS) theory has been widely used to calculate the concrete creep and has produced reasonably accurate calculations [44,45,46,47]. The MPS theory consists of the solidification and microprestress theories. The solidification theory holds that the gradual hydration reaction of a large number of cement particles without hydration reaction in concrete causes the decrease in the short-term creep rate. The microprestress theory holds that at the microscale, concrete exhibits a self-balanced microstress and that the creep rate decreases with the release of microstress during the long-term loading process. Inspired by molecular dynamics, Rahimi-Aghdam et al. [48] developed an extension of the MPS theory that considers the inherent mechanism of drying creep. This paper uses the extended microprestress–solidification theory to calculate the creep deformation of concrete.

The elastic strain, which is the instantaneous strain of concrete specimens after applying uniaxial stress $\sigma$, is age-independent. The elastic strain equation is as follows:

$${\epsilon }_{\mathrm{e}}=\frac{p_1}{E_{28}}\sigma ,$$

(15)

where E₂₈ is the elastic modulus at 28 days of age and p₁ is an empirical parameter that transforms the E₂₈ into the initial instantaneous elastic modulus E₀.

According to the MPS theory, the creep strain of concrete is the combination of viscoelastic strain and viscous flow strain. The viscoelastic strain rate is expressed as follows:

$${\dot{\epsilon }}_v\left(t\right)=\frac{\dot{\gamma }(t)}{v(t)},$$

(16)

where $\dot{\gamma }(t)$ is the viscoelastic strain rate and $v(t)$ is the volume growth function of the solidified product during the hydration process.

Referring to the B3 model and the B4 model [49,50], the expression of viscoelastic strain is as follows:

$${\epsilon }_{\mathrm{v}}=\sigma \{q_{2}Q(t,t\prime )+q_3\ln [1+{(t-t\prime )}^n]\},$$

(17)

where q₂ and q₃ are model parameters related to the cement type, t is the current time, $t\prime$ is the loading time, and $Q(t,t\prime )$ is a function related to the equivalent age θ and the loading time $t-t^{\prime }$. The expressions $Q(t,t\prime )$ is as follows:

$$\{\begin{array}{l}\frac{dQ(t,t^{\prime })}{dt}={(\frac{{\lambda }_0}{\theta (\alpha )})}^m\frac{n{(t-t\prime )}^{n-1}}{{\lambda }_0[1+{(t-t\prime )}^n]}\hfill \\ \theta (\alpha )={\left[\frac{0.28}{w/c}(\frac{{\alpha }_{\infty }}{\alpha }-1)\right]}^{-4/3}\hfill \end{array},$$

(18)

where ${\lambda }_0$, m, and n are fixed, as follows: n = 0.1, ${\lambda }_0=1\mathrm{day}$, and m = 0.5. Notably, as the temperature increases, the aging rate of the concrete accelerates, resulting in a gradual reduction in the creep rate. Accounting for the influence of temperature on the viscoelastic strain, the equivalent time t_e is substituted for the actual time t in Equation (18).

The flow strain ${\epsilon }_{\mathrm{f}}$ represents the fully irrecoverable creep strain, and the rate of viscous flow strain is as follows:

$${\dot{\epsilon }}_{\mathrm{f}}(t)=\frac{\sigma (t)}{{\eta }_M},$$

(19)

where ${\eta }_M$ is the viscosity, which is expressed as a function related to the microprestress S, temperature T, and relative humidity h. The viscosity ${\eta }_M$ is expressed as follows:

$$\frac{1}{{\eta }_M}=(eS+f\left|\dot{S}\right|){\beta }_{\mathrm{\eta }}(T,h),$$

(20)

where $|\cdot |$ is the absolute value. The model parameters e and f reflect the increase in effective viscosity with age, which depends on the degree of hydration. The function for e and f is as follows:

$$e=e_0\frac{{\alpha }_{\infty }}{\alpha }, f=f_0\frac{{\alpha }_{\infty }}{\alpha },$$

(21)

where e₀ and f₀ are model constants.

The change in microprestress obeys the Maxwell rheological model, and the equation for its evolution is as follows:

$$\dot{S}+\frac{C_S{\beta }_{\eta }(T,h)}{e{\beta }_{C_{\mathrm{s}}}(T)}{S}^2=c_1(\frac{T\mathit{\partial }h}{h\mathit{\partial }t}+\frac{T\mathit{\partial }h\mathit{\partial }T}{h\mathit{\partial }T\mathit{\partial }t}+\dot{T}\ln (h(t,T))+c_h{h}^3\dot{T}),$$

(22)

where C_s is the spring stiffness coefficient in the Maxwell series and c₁ and c_h are model parameters. In the initial hydration stage, the microprestress S depends on the hydration degree, and the expression for S is as follows:

$$\begin{array}{ll}S=S_0=c_0{q}_4\hfill & (\alpha <0.6{\alpha }_{\infty })\hfill \end{array},$$

(23)

where S₀ is the initial microprestress, c₀ is the model parameter, and q₄ is the creep parameter, referring to the B4 model.

The variables ${\beta }_{\mathrm{\eta }}\left(t\right)$ and ${\beta }_{C_s}(t)$, which are related to temperature in Equation (22), are expressed as follows:

$${\beta }_{\mathrm{\eta }}\left(t\right)=\exp [\frac{Q_{\mathrm{\eta }}}{R}(\frac{1}{T_0}-\frac{1}{T(t)})][p_0+\frac{1-p_0}{1+{(1-h/1-h^{*})}^{n_{\mathrm{h}}}}],$$

(24)

$${\beta }_{C_{\mathrm{s}}}(t)=\exp [\frac{Q_{C_{\mathrm{S}}}}{R}(\frac{1}{T_0}-\frac{1}{T(t)})],$$

(25)

where $Q_{\mathrm{\eta }}$ and $Q_{C_{\mathrm{S}}}$ are the activation energy of viscosity change and C_s, respectively, and p₀, h*, and n_h are model constants, where p₀ = 5, h* = 0.75, and n_h = 2.

The expression of thermal strain at an early age is as follows:

$${\epsilon }_{\mathrm{th}}=k_T(T-T_0),$$

(26)

where $k_{\mathrm{T}}$ is the thermal expansion coefficient and T₀ is the initial temperature of concrete.

At an early age, the moisture in the concrete will participate in the hydration reaction, reducing the concrete moisture. In addition, when the ambient moisture is less than the concrete moisture, the moisture in the concrete will diffuse into the environment. The above two reasons cause the drying shrinkage of concrete. Using the relative humidity test of early-age concrete, Zhang et al. [51] proposed a two-stage model of shrinkage deformation in concrete. The expression is as follows:

$${\epsilon }_{\mathrm{sh}}=h_{\mathrm{w}}+k_{\mathrm{h}}(h_0-h),$$

(27)

where h_w = 2.82 × 10⁻⁶, which describes the shrinkage deformation of the concrete at stage I, k_h is the coefficient of the drying shrinkage of the concrete, and h₀ is the initial humidity of the concrete.

The mechanical parameters of early-age concrete depend on the hydration degree, which has evident aging characteristics. Many studies have proposed models for the elastic modulus and strength of early-age concrete. This paper adopts the models of elastic modulus and strength based on hydration degree [52,53], and the equations are as follows:

$$E(t_{\mathrm{e}})=E_{28}\cdot [e^{s(1-\sqrt{\frac{672}{t_{\mathrm{e}}-t_0}})}{]}^{n_{\mathrm{E}}},$$

(28)

$$F_t(t_{\mathrm{e}})=f_{\mathrm{t}28}\cdot [e^{s(1-\sqrt{\frac{672}{t_{\mathrm{e}}-t_0}})}{]}^{n_{\mathrm{t}}},$$

(29)

where E₂₈ is the elastic modulus of concrete at 28 days of age, f_t28 is the tensile strength of concrete at 28 days of age, t₀ is the initial time for the development of concrete strength (where t₀ = 10 h), and s, n_E, and n_t are the model parameters.

3.2. Finite Element Model

3.2.1. Geometric Model and Element Discretization

The geometric model was established according to the dimensions of the composite wall in the actual subway station, and the element size of the composite wall was determined according to the computational efficiency of the model. Figure 6 depicts the geometric model and mesh elements. In this study, the cooling pipe was built by a line element to reduce the computational cost.

3.2.2. Initial Value and Boundary Conditions

Figure 7 illustrates the boundary conditions of the finite element model. The initial humidity and temperature of the old concrete, which was composed of the diaphragm wall, low wall, old sidewall, and foundation slap, were 85% and 28 °C, respectively. The initial humidity and temperature of the new sidewall were 100% and 28 °C, respectively. In this study, the third boundary condition was used to calculate the temperature field and humidity field of the sidewall. Before the formwork was removed, the wall surface was regarded as a dehumidification condition, and the equivalent heat dissipation coefficient was used to calculate the convective heat transfer. In addition, the normal displacement of the wall surface was constrained. The composite wall’s low heat exchange and humidity diffusion from the surrounding environment were ignored. The zero heat flux was defined by $\mathit{q}\cdot \mathit{n}=0$ (q was the heat flux, n was the surface normal vector), and the zero humidity transfer was defined by $\mathit{F}\cdot \mathit{n}=0$ (F is the humidity diffusion rate). According to the on-site ambient humidity, the ambient humidity of the model was 50%. The ambient temperature of the model is shown in Equation (30), which was obtained by fitting the on-site ambient temperature curve.

$$T_{\mathrm{ext}}=28.4+4.6·\cos [\frac{\pi }{345}·(t+38)],$$

(30)

where t is the age of the sidewall.

3.2.3. Model Parameters

Table 3. Chemo-thermo-hygro model parameters.

Type	Parameter	Value	Unit	Type	Parameter	Value	Unit
Thermal parameters	${\rho }_{\mathrm{c}}$	2400	Kg/m3	Thermal parameters	Qc	277.53	kJ/kg
	${\beta }_{\mathrm{c}}$	40	kJ/(m2·h·°C)		a	0.34	-
	${\beta }_{\mathrm{e}}$	27.73	kJ/(m2·h·°C)		b	0.86	-
	Ccem	0.86	kJ/(kg·°C)	Humidity parameters	D1	3.13 × 10−10	m2/s
	Ca	0.76	kJ/(kg·°C)
	Cw	4.2	kJ/(kg·°C)
	${\lambda }_{\mathrm{p}}$	8.3	kJ/(m·h·°C)
	${\rho }_{\mathrm{w}}$	103	Kg/m3
	k	52	W/m·°C

presents the parameters of the chemo-thermo-hygro-mechanical coupling model, which was used to calculate the hydration reaction, temperature, and humidity fields of the early-age sidewall.

Table 4. Mechanical model parameters.

Type	Parameter	Value	Unit	Type	Parameter	Value	Unit
Thermal strain and drying shrinkage	kT	1.0 × 10−5	1/K	Creep deformation	q4	8.39	10−6 MPa−1
	kh	1.11 × 10−5	-		e0	0.05·q4	-
Elastic modulus and strength model	E28	3.1 × 104	MPa		f0	0.0125·q4	-
	ft28	3.1	MPa		Cs	1.6/q4	-
	t0	10	days		c1	22.5·q4	-
	s	0.21	-		ch	0.035	-
	ne	0.34	-		c0	2.38	10−3 MPa−1
	nt	0.62	-		$Q_{\mathrm{\eta }}/R$	3000	K
Creep deformation	p1	0.7	-		$Q_{C_{\mathrm{s}}}/R$	1500	K
	q2	1.98	10−4 MPa−1		${\displaystyle \prod _{i=1}^2}K$	1.7	-
	q3	4.46	10−6 MPa−1

presents the mechanical model parameters. This study selected the basic parameters of creep deformation on the basis of the B4 model. Additionally, to account for the influence of fly ash (FA) and blast furnace slag (BFS) on creep deformation, the creep correction factor (${\displaystyle \prod _{i=1}^n}K$) was introduced using the experimental data in the literature [54].

3.3. Numerical-Simulation Results

3.3.1. Model Verification

Figure 8 shows the tested and simulated strain values of points S12 and S22. It was found that the difference between the tested and simulated strain values was small, indicating that the numerical simulation results were reliable.

3.3.2. Temperature Field and Humidity Field

Figure 9 depicts the temperature evolution of points T11, T12, and T13 and the temperature curve between the cooling pipes at the sidewall height of 2.4 m for several time periods obtained by numerical simulation. The figure shows that the temperature evolution obtained by the numerical simulation followed the same principle as the experimental temperature curve and that there was a significant temperature difference around the cooling pipe. Moreover, the temperature difference increased initially and then decreased as age increased, reaching its maximum at 0.81 days of age. The maximum temperature difference was 22.8 °C. In addition, Figure 10 shows the temperature contour between the cooling pipes for several time periods. The picture shows that the temperature of the cooling pipe in the direction of the water flow gradually increased, and the temperature of the concrete around the cooling pipe also gradually increased. Notably, because the left and right ends of the sidewall could transfer heat to the old sidewall and the environment, respectively, the temperature difference at the end of the sidewall was significantly smaller than that between the cooling pipes.

Figure 11 illustrates the humidity evolution at points H11, H12, and H13 and the humidity curve between the cooling pipes for several time periods. Points H11, H12, and H13 correspond to the relative humidity of points T11, T12, and T13, respectively. As the figure shows, when the age exceeded two days, the relative humidity of the sidewall surface gradually exceeded the relative humidity of the center and bottom of the sidewall. This was the reason for the formwork’s removal, and the drying rate of the sidewall surface increased significantly. When the age exceeded 14 days, the relative humidity of the sidewall decreased slowly with increasing age. As in the temperature field, the cooling pipes exhibited a humidity difference, and the relative humidity at the end of the sidewall dropped sharply. Notably, although the hydration reaction consumed some of the moisture, the drying of the early-age sidewall was a slow process, and the humidity difference between the cooling pipes was small.

3.3.3. Stress Field

Figure 12 illustrates the distribution of maximum principal stress in the central section of the sidewall for several time periods. At 0.5 days of age, there was a tensile stress around the cooling pipe that exhibited a maximum stress of 1.27 MPa. Because the temperature of the water gradually increased along the flow direction, the tensile stress around the cooling pipe gradually decreased along the flow direction. Notably, the tensile stress peak was located around the inlet of the cooling pipe. Additionally, due to the heat exchange with the environment and the interaction between the cooling pipes, the tensile stress around the transverse cooling pipe at the top of the sidewall was higher than in other areas. At 0.81 days of age, the tensile stress around the cooling pipe reached its maximum value, which was 49.61% higher than the maximum tensile stress at 0.5 days of age. At 7 days of age, the temperature difference around the cooling pipe gradually decreased as the sidewall temperature decreased. Because the shrinkage deformation of the concrete was constrained by the old concrete, there was a tensile stress around the construction joint; its maximum value was 0.91 MPa.

4. Effect of the Cooling Pipe Parameters and Ambient Temperature

The analysis presented in the previous section demonstrated that when the cooling-pipe system was applied to the sidewall engineering, there was a significant temperature difference around the cooling pipe, which was not conducive to minimizing the cracking risk of the sidewall. Therefore, it is necessary to further discuss the influence of cooling-pipe parameters, such as water temperature, flow rate, and pipe-spacing distance, on the temperature field and cracking risk of the early-age sidewall as well as the suitability of using a cooling pipe in sidewall engineering at different ambient temperatures. The expression of the cracking risk according to the strength criterion is as follows:

$$\eta =\frac{\sigma (t)}{f_{\mathrm{t}}(t)},$$

(31)

where $\sigma (t)$ is the maximum tensile stress of concrete and $f_{\mathrm{t}}(t)$ is the tensile strength of concrete.

4.1. Water Temperature

Figure 13 depicts the temperature evolution of point T12 and the cracking risk around the inlet at different water temperatures. At a water temperature of 30 °C, the temperature peak at the sidewall center was 50.2 °C, and the maximum cooling rate was 9.0 °C/d. When the water temperature decreased from 30 to 10 °C, the temperature peak and maximum cooling rate of the sidewall center decreased by 15.94% and 24.44%, respectively, and the cracking risk around the inlet increased from 0.85 to 5. Reducing the water temperature reduced the temperature peak and cooling rate of the early-age sidewall. However, because reducing the water temperature will significantly increase the cracking risk of the sidewall, it is necessary to strictly control the water temperature of the cooling-pipe system.

4.2. Flow Rate

Figure 14 depicts the temperature evolution of point T12 and the cracking risk around the inlet at different flow rates. The figure shows that when the flow rate was 0.3 m³/h, the temperature peak and the maximum cooling rate of the side wall were 49.5 °C and 9.1 °C/d, respectively, and the cracking risk around the inlet was 1.30. Compared with a flow rate of 0.3 m³/h, with a flow rate of 1.5 m³/h, the temperature peak and maximum cooling rate of the sidewall were reduced by 2.42% and 4.39%, respectively, and the cracking risk was increased by 1.54%. When the flow rate exceeded 0.6 m³/h, there was no significant impact on the temperature peak and cracking risk of the sidewall, even with an increase in the flow rate.

4.3. Cooling Pipe Spacing Distance

Figure 15 illustrates the temperature evolution of point T12 and the cracking risk around the inlet at different cooling-pipe-spacing distances. The figure shows that when the spacing distance was 100 cm, the temperature peak and maximum cooling rate of the sidewall were 48.2 °C and 8.7 °C/d, respectively, and the cracking risk around the inlet was 1.32. Compared with a spacing distance of 100 cm, when the spacing distance was 40 cm, the temperature peak and maximum cooling rate of the side wall were reduced by 24.69% and 59.14%, respectively. Nevertheless, the cracking risk around the inlet was reduced by 35.61%. It was shown that reducing the cooling-pipe-spacing distance can prolong the length of the cooling pipe and increase the scope and the heat absorption of the cooling pipe. Consequently, the heat absorption rate of the cooling pipe was greater than the heat release rate of hydration.

4.4. Ambient Temperature

Figure 16 illustrates the temperature evolution of point T12 and the cracking risk around the inlet at different ambient temperatures. As the figure shows, at an ambient temperature of 5 °C, the temperature peak and maximum cooling rate of the sidewall were 34 °C and 7.6 °C/d, respectively, and the cracking risk around the inlet was 0.94. As the ambient temperature increased from 5 to 45 °C, the temperature peak and maximum cooling rate of the sidewall increased by 63.24% and 17.11%, respectively, but the cracking risk around the inlet increased by 144.68%. Thus, increasing the ambient temperature accelerated the hydration reaction of the concrete; consequently, the heat release rate of the concrete was greater than the heat absorption rate of the cooling pipe, and the temperature peak, cooling rate, and temperature difference around the cooling pipe increased.

5. Discussion

The thermal stress in the region adjacent to the cooling pipe has always been a hot problem for engineers. This study aims to guide the application of the cooling pipe system in mass concrete sidewall engineering to reduce the temperature peak and thermal stress of sidewalls and avoid cracking in the region adjacent to the cooling pipe. This study analyzed the crack distribution characteristics, temperature, and strain evolution of early-age sidewalls by the temperature and strain monitoring of early-age sidewall. Furthermore, the temperature, humidity, and stress fields of the early-age sidewall with the cooling pipe system were analyzed using numerical simulation. Finally, the effects of water temperature, flow rate, pipe spacing, and ambient temperature on the temperature field of the early-age sidewall and the cracking risk in the region adjacent to the cooling pipe were analyzed. The results show that the temperature of the early-age sidewall experiences a sharp increase and decrease, accompanied by apparent thermal deformation. This process is completed in about 1 week, and the sidewall temperature will reach the peak value in about 1 day. Compared with other mass concrete engineering (e.g., concrete dam), the temperature change rate of the early-age sidewall is greater, and the temperature field is easily affected by the ambient temperature. The numerical simulation results indicate that the cooling pipe system can effectively reduce the temperature peak of the sidewall. Consistent with other mass concrete engineering, the region adjacent to the cooling pipe has a temperature gradient. In addition, the influence of parameters such as water temperature, flow rate, and pipe spacing on the temperature field and the temperature gradient in the region adjacent to the cooling pipe is consistent with the literature [14,34,35]. Reducing the water temperature will increase the thermal stress in the region adjacent to the cooling pipe. Notably, reducing the distance between cooling pipes is the most effective method to reduce the temperature peak of the sidewall and the thermal stress in the region adjacent to the cooling pipe, which is consistent with the literature [25]. Additionally, the temperature gradient shows noticeable sensitivity to the change in ambient temperature. The temperature gradient in the region adjacent to cooling pipes will increase significantly in the low-temperature environment. Therefore, according to the ambient temperature conditions, adjusting the parameters of the cooling pipe can reduce the temperature peak of the sidewall and the temperature gradient around the cooling pipe. Moreover, the effects of cooling pipe material, cooling pipe diameter, and cooling pipe working time on the temperature field of the sidewall need to be further analyzed in subsequent research.

6. Conclusions

This study used temperature and strain tests of early-age concrete to determine the crack-distribution characteristics, temperature evolution, and strain evolution of early-age sidewalls with a cooling-pipe system, and it developed a chemo-thermo-hygro-mechanical coupling model that accounts for the early-age behavior of the concrete and the action of the cooling pipes. On this basis, the temperature, humidity, and stress fields of the early-age sidewall were obtained using the differential-equation-modeling function of the finite element software. The effects of water temperature, flow rate, pipe spacing, and ambient temperature on the temperature field and cracking risk of the early-age sidewall were determined. Four conclusions can be drawn:

• The early-age sidewall’s cracks appear in the first two weeks after concrete pouring. The cracks extend upward from the bottom construction joints. Most of the cracks are vertical, and a few oblique cracks are located near the corner of the wall.
• There is a tensile stress around the cooling pipe that gradually decreases along the flow direction and reaches its maximum around the inlet.
• As the temperature of the water drops from 30 to 10 °C, the temperature peak and cooling rate of the sidewall decrease by 15.94% and 24.44%, respectively, and the cracking risk around the inlet increases by 488%. When the flow rate is between 0.3 m³/h and 1.5 m³/h, the temperature peak and cooling rate of the sidewall decrease by 2.42% and 4.39%, respectively, and the cracking risk around the inlet increases by 1.54%. Notably, when the water flow rate exceeds 0.6 m³/h, increasing the flow rate does not significantly improve the cooling efficiency of the cooling pipe. Compared with a spacing distance of 100 cm, when the spacing distance is 40 cm, the temperature peak, cooling rate, and cracking risk around the inlet decrease by 24.69%, 59.14%, and 35.61%, respectively.
• In a high-temperature environment, the cracking risk in the region adjacent to the cooling pipe will increase significantly.
• During the sidewall construction, reducing the distance between cooling pipes and strictly controlling the water temperature are effective methods to reduce the temperature peak of the early-age sidewall while avoiding a large temperature gradient in the area adjacent to the cooling pipe. In addition, in a high-temperature environment, the water temperature of cooling pipes should not be too high.

In future research, the effects of cooling pipe material, cooling pipe diameter, and cooling pipe working time on the temperature field of the sidewall need to be further analyzed. Furthermore, it is recommended that the cooling pipe system, which can automatically adjust the cooling pipe parameters, is studied to improve work efficiency and avoid thermal stress in the region adjacent to the cooling pipe.