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1. Introduction
Concrete structures such as dams, bridge foundations, piers, and abutments are often classified as “mass concrete”, (ACI, 2005), (Do, 2014). When a larger volume of fresh concrete is poured, a larger amount of heat releases from the cement hydration process. The temperature at the inside portion of the concrete is increasing while that at the surface is quickly decreasing due to the exposure to the air. This creates a large temperature difference between the core and the outer surface of the concrete, leading to high tensile stresses that may exceed the concrete tensile strength, which will cause thermal cracks (ACI, 2005), (Tian et al., 2012), (Tía, 2013), (Do, 2014) . As long as cracking occurs in the concrete, it begins to affect the regular service and durability of the structure (Maruyama and Lura, 2019). Therefore, it is necessary to analyze the temperature and the corresponding stress fields in mass concrete structures for controlling temperature and preventing crack initiation in the early-age concrete (Nguyen and Luu, 2019), (Nguyen and Bui, 2019).
There are different measures to reduce the temperature difference and thermal tensile stress in mass concrete such as use of lower cement content, use of aggregates (Klemczak et al., 2017) with a low coefficient of thermal expansion, casting concrete at night or in the early morning (Do, 2019), use of ice water, use of liquid nitrogen, pre-cooling of mix constituents, post cooling using embedded pipes (Hong et al., 2017), (Nguyen et al., 2019), and use of insulating formwork (Do, 2020), (Do et al., 2014a), (Do et al., 2013). Recently, laboratory (Zhao et al., 2019), (Do et al., 2019a) and field tests (Sargam et al., 2019) on early-age properties and thermal cracking of concrete have been also conducted to evaluate the cracking risk as well as the efficiencies of thermal control methods to control cracking in mass concrete. Each measure has both advantages and disadvantages depending on construction conditions. Among these measures, the control of concrete placement stages has not been widely used because of its costs. Hence, the aim of this study is to determine the temperature and thermal stress in a concrete footing cast in different placement conditions, thus suggesting the best scenario for reducing cracking risk in the structure.
2. Materials and methods
2.1. Description of the mass concrete footing
In this study, an assumed 8-m×6-m×3-m concrete footing cast on soil is modeled. The modeled soil layer is assumed to have dimensions of 16-m×12-m×4-m. The concrete footing has two planes of symmetry, therefore only one-quarter of the footing is analyzed in order to reduce the computation time. The geometry and dimensions of the footing are shown in (Figure 1).
The ambient temperature can be simulated using (Equation 1) (Léger et al., 1993):
(1)
The temperature of the foundation under the concrete block is considered 25°C and the initial concrete temperature is assumed to be 30°C. The concrete mix proportion is shown in (Table 1) (Tia et al., 2016), (Do, 2016). The adiabatic temperature rise for the concrete mix was tested and is shown in (Figure 2). The thermal and physical characteristics of the concrete and the soil foundation used in the analysis are presented in (Table 2).
w/cm = water/cementitious materials ratio
ACI 318-19 (ACI, 2019) suggests the following expressions (Equation 2) and (Equation 3) for estimating splitting tensile strength and Young’s modulus as functions of the compressive strength:
(2)
(3)
where fsp(t) is the splitting tensile strength (MPa), E(t) is the Young’s modulus (MPa), and fc’ is the compressive strength (MPa).
During the construction of the concrete footing, there are many factors that affect its thermal regime especially construction conditions. In this study, the calculation cases are assumed as follows (Figure 3):
Case 1: no construction joints are used,
Case 2: the concrete is placed in two lifts (1.5 m each) and 5-day time interval for placing each lift,
Case 3: the concrete is placed in three lifts (each lift 1 m) and 3-day time interval for placing each lift.
2.2. FE basis for solving heat transfer problem
The governing equation of a 3D unsteady heat transfer problem is based on the principle of energy conservation and Fourier's law of heat conduction and expressed in (Equation 4) (Cengel, 2014):
(4)
where t is the material temperature (°C), kx, ky, kz are the thermal conductivity coefficients of the material dependent on the temperature in the directions x, y and z, respectively, (W/m-°C), qv is the amount of heat released by internal sources (for example, exothermic heating) to a given moment in time (W/m3), c is specific heat (J/kg-°C), ρ is the density concrete (kg/m3), t is time (s).
Two types of heat transfer boundary conditions are often used to analyze heat problems, which are expressed by (Equation 5) and (Equation 6) (Cengel, 2014):
(5)
(6)
where Tp is the temperature of the surface or foundation (°C), qv is the heat generated per unit volume (J/m3), h is the convective coefficient (W/m2-°C), Ts is the temperature of concrete or foundation (°C), Tf is the ambient temperature of the construction area (°C), lx, ly, and lz are the directional cosine of the surface according to the x, y and x axes, respectively.
The problem of heat transfer is solved using the following matrix equation (Equation 7), (Zienkiewicz and Taylor, 2000):
(7)
In the problem of unstable heat transfer, it is necessary to analyze time into steps Δt as follows (Equation 8):
(8)
(Equation 9) is obtained by combining (Equations 7) and (Equation 8) and expressed as follows:
(9)
where [K] is conductivity operator, [C] is capacity operator, and [Q] is heat load due to heat of hydration, Δt = Δtn-Δtn-1 is the time step.
Solving (Equation 9) gives temperature fields in the concrete block at different time steps.
Thermal stress in the concrete is determined by (Equation 10) (Zienkiewicz and Taylor, 2000):
(10)
where {ơ} is the thermal stress vector, [D] is the elasticity matrix, [B] is the strain-displacement matrix, based on the element shape functions, {u} is the nodal displacement vector, {ɛ}= { ɛx ɛy ɛz ɛxy ɛyz ɛzx} is the strain vector, and {ɛth} is the thermal strain vector. The determination of temperature field in a mass concrete structure is a complex problem because it depends not only on the shape of the structure but also on other factors such as the internal heat generation, construction conditions, and the ambient temperature. In recent years, numerical approaches such as finite difference and FE methods have been widely used to predict temperature and stress fields in mass concrete structures (Nguyen et al., 2019), (Trong et al., 2019), (Aniskin et al., 2018), (Do et al., 2020), (Do et al., 2020a), (Do, 2013). In this study, the Midas/Civil software (MIDAS, 2011) based on the FE method was used to model the early-age behavior of the concrete footing depicted in (Figure 1).
2.3. Crack index for evaluation of early-age cracking risk
The prediction of crack formation in an early-age concrete structure plays an important role in minimizing the cracking risk and/or controlling crack growth. Criteria for evaluating early-age thermal cracking in concrete vary from country to country (Do et al, 2020b). In the United States, the ACI guidelines for assessing thermal cracking are not specified except for a recommended limiting value for the temperature difference between the core and the outer surface of the concrete. In other countries such as Korea and Japan, “crack index” is preferably used as a measure for assessing early-age cracking potential in the concrete. Crack index is determined by (Equation 11) (Kim, 2010), (Japan Concrete Institute, 2017):
(11)
where Ict is the crack index, ft(t) is the maximum tensile stress (MPa), and fsp(t) is the splitting tensile strength of the concrete (MPa).
The tendency of cracking can be evaluated using “crack index” based on engineering experience as introduced in (Table 3) and (Figure 4), (Kim, 2010).
3. Analysis Results and Discussion
The 3-D FE model was created using the Midas/Civil software. The element used in the thermal analysis is a 3-D eight-node thermal solid element, which has one degree of freedom - temperature - at each node. The concrete and foundation in the stress analysis are modeled with a 3-D eight-node structural solid element, which is coupled with the 3-D thermal solid element. The temperature distribution obtained from the thermal analysis is then served as “thermal loading” in the stress calculation. The FE model geometry is depicted in (Figure 5).
The (Figure 6) shows the maximum temperature development at the center of the concrete footing under different construction cases. The maximum temperature decreases as the thickness of each lift decreases during the cement hydration. The maximum temperature value and time of occurrence are listed in (Table 4). The maximum temperatures at the concrete center are 75.52°C, 72.24°C, and 66.74°C in Cases 1, 2, and 3, respectively. It may be noted that in Case 3, the maximum temperature and temperature difference are the smallest compared to Cases 1 and 2. After reaching the peak temperature, it begins to cool down. As predicted, the maximum temperature in the concrete will take a long time to decrease to a stable temperature. The crack index calculated at the surface is also listed in (Table 5).
The (Figure 4) and (Table 4) together show that the probabilities of cracking in the concrete footing are 81%, 50%, and 20%, in Cases 1, 2 and 3, respectively. It can be included that thermal cracking in the concrete in Case 3 is not likely to occur at early ages, thus suggesting the proposed method of concrete pouring is effective. In other hand, other measures should still be taken in Cases 1 and 2 for controlling early-age cracking in the concrete.
4. Conclusions
• The 3-D FE model created in this study has identified the temperature and thermal stress fields in the concrete footing at early ages in different construction scenarios. The results show that the construction schedules significantly affect the temperature development and thermal cracking risk in the concrete.
• When the concrete is cast with a maximum volume of 8-m×6-m×3-m in three lifts (as investigated in this study), thermal cracking will be not likely to occur. Thus, dividing large volume of mass concrete into reasonably smaller lifts can help minimize cracking risk in the concrete.
• For future work, the numerically predicted results (temperature field and crack index) should be compared with experimental results. As long as the research results have been verified, they can be used in the sustainable design and construction of mass concrete structures.
• The developed model can be used to perform thermal and stress analyses, and assess the risk of thermal cracking of other essential concrete members. The research methodology can help engineers/contractors optimize the construction stages and reduce the project schedule.
