Optimization of Curing Regimes for Precast Prestressed Members with Early-Strength Concrete

Compressive tests of ESC were performed for three different \(f_{{{\text{cd}}}}^{\prime}\) s of 30, 40 and 50 MPa with target slump of 200 ± 20 mm and air content of 3 ± 1 %. Table 1 summarizes mix proportions for the ESC. Crushed gravels with 20 mm nominal maximum size, specific gravity of 2.6 g/cm³, and fineness modulus of 6.8 were used. River sand was used as fine aggregates, which had a specific gravity of 2.6 g/cm³, and fineness modulus of 2.2. Superplasticizer (SP) was used to increase workability.

Mixing and specimen preparation were carried out at room temperature and humidity of 20 ± 2 °C and 50 ± 5 %, respectively. After completing mixing with a 120-liter-capacity concrete pan mixer, the slump and air content of the fresh concrete were measured according to ASTM C143/C143M-10 (2010) and ASTM C231/C231M-14 (2014). From the same batch, 14 cylinders of 100 × 200 mm were cast. Before submerging them into water for curing, each mold was enclosed in a waterproof plastic bag and the top surface was covered with a plastic sheet to prevent evaporation of the water in the mold. The cylindrical molds were then put into the water in order to cure the ESC according to the pre-determined curing regime (Fig. 1). Portable and manually controllable coiled water heaters were used to adjust water temperature. Thermocouples were used to measure the temperatures at the mid-depth of ESC in the cylindrical mold and water. From the preliminary tests, it was found that the ESC developed 70 and 100 % of its \(f_{{{\text{cd}}}}^{\prime}\) value at 36 and 168 h (7 days) after casting, respectively, at ambient temperature of 20 °C. Accordingly, seven different ages of 9, 12, 18, 24, 36, 100 and 168 h at ambient temperature were selected for the measurement of compressive strength.

For each \(f_{{{\text{cd}}}}^{\prime}\), five different maximum temperatures (T _max) of 20, 30, 40, 50 and 60 °C were considered. As the maximum temperature of 60 °C has been commonly applied to steam curing fabrication of architectural precast prestressed units in practice, the maximum value of T _max was chosen to be 60 °C in this study. At temperatures other than 20 °C, equivalent ages (t _eq in hour) were estimated corresponding to the seven different ages of measurement. The concept of t _eq given in Eq. (1), which was suggested by Freiesleben and Pederson (1977) and also recommended by ASTM C 1074-04 (2004), was used.where, Δt is the time interval (day), E _eq is the activation energy (=33,500 J/mol) used in the estimation of t _eq, R is the universal gas constant (=8.314 J/mol/K) and T _c = temperature of concrete (°C).

Figure 2a illustrates typical curing regimes adopted in this study for the ESC subjected to different T _max. A 3-h delay period was assigned for initial setting at the beginning of each curing regime. Then temperature was increased gradually up to T _max and maintained until a t _eq of 36 h, the equivalent age at which the compressive strength of ESC reaches 70 % of \( f_{{{\text{cd}}}}^{\prime}\). Then the temperature was lowered from T _max to 20 °C and maintained at that temperature until the total elapsed t _eq became 168 h. For all cases, the rates of temperature increase and decrease of the concrete were chosen to be 10 °C/h and −10 °C/h, respectively. In Fig. 2b, the relationships between the t _eq and real elapsed time (t) at strength measurement are presented for each curing regime for a corresponding T _max.

At each age of measurement, a pair of replica ESC cylinders from the same batch and with the same curing condition were tested in compression. Top surfaces were ground shortly before they were tested in compression with a hydraulic servo-controlled compressive testing machine of 1000 kN capacity. The rate of loading was within the range of 0.25 ± 0.05 MPa/s. The results of 210 compression tests are given in Table 2. Test specimens were designated in Table 2 as CSn, where C stands for cylinder, S for concrete compressive strength (S = L, M and H for \( f_{{{\text{cd}}}}^{\prime } = 30,40 \) and 50 MPa, respectively) and n for the two-digit T _max in °C.

Figure 3a illustrates the effects of T _max on the development of compressive strength of ESC for the cases with T _max = 20, 40 and 60 °C. Strength development of the ESC was characterized by a rapid increase during early-stage curing followed by a gradual increase and then asymptotic approach to ultimate strength. Figure 3a also indicates that higher curing temperature accelerates the rate of hydration of ESC in early-age of curing but results in crossover effect in later-stage. The crossover effect is known to occur due to a retarding effect on the diffusion of the hydrate in the secondary reaction of cement hydration by built-up of hydrate shells of low-permeability hydration products around the cement grains (McIntosh 1956; Alexander and Taplin 1962; Carino and Lew 1981; Kjellsen et al. 1990; Yi et al. 2005). Formation of microcracks and/or coalesce by tensile stress resulting from the difference of volume expansion between water and voids is also known to cause a decrease in strength with an increase in initial curing temperature (Verbeck and Helmuth 1968; Carino and Lew 1981; Kjellsen and Detwiler 1993).

The rate of strength development is presented for different \( f_{{{\text{cd}}}}^{\prime}{s}\) in Fig. 3b. A more rapid increase in strength at early-stage curing was observed with higher \( f_{{{\text{cd}}}}^{\prime}\) under the same T _max. At later-stage curing, however, similar rate of strength increase was observed for all \( f_{{{\text{cd}}}}^{\prime} {s}\).

In Fig. 5a, a typical 3-6-3 curing regime is illustrated with a 24-h turnover period. It consists of a 3-h preliminary work period for preparation and concrete casting followed by a 3-h delay period before the initiation of steam curing, a 12-h steam curing period, a 3-h cooling period, and a 3-h posterior work period for cutting, demolding and transportation. The 12-h steam curing period comprises a 3-h temperature increase at a rate of 13.3 °C/h, a 6-h constant T _max equal to 60 °C, and a 3-h temperature decrease at a rate of 13.3 °C/h. When prestress is transferred at the end of cooling period, the concrete is expected to have developed 70 % of \( f_{{{\text{cd}}}}^{\prime}\) value. This provides for sufficient bond strength to transfer the prestress to the surrounding concrete while ensuring that stresses in the concrete induced by the transferred prestress do not exceed the allowable stresses specified in the code provisions (Erdogdu and Kurbetci 1998; ACI318-11 2011).

The objective of this study is to find the optimized curing regime, which enables ESC to develop 70 % of \( f_{{{\text{cd}}}}^{\prime}\) value with minimum consumption of fuel within a given 24-h turnover period. For the purpose of measuring fuel consumption, the energy index (E _idx in  °C/hr) was calculated for each curing regime (Ramezanianpour et al. 2013). E _idx is defined as the total area under the curing time versus temperature curve during the elevated temperatures above T _r (20 °C in this study) (see Fig. 5a). According to this definition, the value of E _idx for a typical 3-6-3 curing regime (E _idx ^t ) is estimated to be 360  °C/h.

In optimization of curing regime, all candidate regimes consist of a series of sub-periods within the 24-h turnover period: a 3-h preparation period followed by a 3-h delay period at ambient temperature of 20 °C (from t _o to t _de in Fig. 5b), a temperature ascending period (t _de to t _cs), a constant T _max period (t _cs to t _ce), a temperature descending period (t _ce to t _e), and a cooling period (t _e to t _f). Since a rapid increase or decrease of the temperature could have a detrimental effect on the concrete material properties such as undesirable porosity or cracking (Alexanderson 1972), constraints were imposed against the rates of temperature increase (k _a) and decrease (k _d) so that 0 °C/h ≤ k _a ≤ 20 °C/h and −20 °C/h ≤ k _d ≤ °C/h, respectively, according to code recommendations (AASHTO 2004; PCA 2006).

It has been reported that a desirable T _max in steam curing for concrete made of Type I cement is in the range of 40 to 85 °C. At temperatures above 85 °C, long-term development of compressive strength and durability could be adversely affected (Türkel and Alabas 2005; Hwang et al. 2012). In this study, different upper bounds on T _max between 40 and 60 °C were considered and the corresponding values of E _idx were evaluated for the optimization process. In optimization, the value of S developed during 18-h curing period from the beginning of delay period (t _o) to the end of cooling period (t _f) is required to be greater than or equal to 0.7 \( f_{{{\text{cd}}}}^{\prime}\).

In addition to the constraints given to the values of \( {S}, {k}_{\text{a}} , {k}_{\text{d}} \, {\text{and}}\, T_{ \hbox{max} } \), an additional constraint was imposed to prevent the occurrence of an incompatible curing regime. An incompatible curing regime could occur by an improper combination of design variables, for which the value of temperature obtained at the intersecting point of extending temperature increase and decrease lines (T _its) becomes less than the value of T _max as shown in Fig. 5c. If this happens, then T _its needs to replace T _max for compatible curing regime. The objective function and all the constraints are presented in the following constrained optimization formulation:

Subject to: 0 °C/h ≤ k _a ≤ 20 °C/h; and 20 °C/h ≤ k _d ≤ 0 °C/h. where, T _max = 40, 50 or 60 °C and \( {T}_{\text{its}} = 20 + {k}_{\text{a}} \cdot \left( {\frac{{3 \cdot {k}_{\text{a}} - {k}_{\text{d}} \cdot {t}_{\text{e}} }}{{{k}_{\text{a}} - {k}_{\text{d}} }} - 3} \right) \).

Optimum curing regimes were found for ESC having \( f_{{{\text{cd}}}}^{\prime}\) the values of 30, 40 and 50 MPa. For each value of \( f_{{{\text{cd}}}}^{\prime}\), the values of four design variables k _a, T _M, k _d and t _e that minimize the value of E _idx in Eq. (7) were found for each of the t _f values 12, 14, 16 and 18 h. For each of the previous conditions, three different cases were considered with the temperature T bounded between 20 °C and the smaller value of T _its and T _max values. Therefore, a total of 36 cases of candidate steam curing regimes were considered resulting from combinations of three values of \( f_{{{\text{cd}}}}^{\prime}\), four values of t _f and three values of T _max (=40, 50 and 60 °C).

In the preliminary search, a step-by-step method was used to find a set of discrete values of design variables leading to a near local minimum value of E _idx in Eq. (7). In implementing Eq. (7), four loops were used, where the value of a particular design variable was increased by a finite increment from its lowest value to highest value step-by-step. The values of 0.5, −0.5 and 0.5 °C/h were assigned as incremental values for k _a, k _d and t _e, respectively. An extensive number of searches were made in the domain of design variables for a specific optimization case with pre-assigned values of \( f_{{{\text{cd}}}}^{\prime}\), t _f and T _max. These values leading to the near local minimum value of E _idx were then saved.

In the subsequent search, values of design variables found from the preliminary search were further refined by using a nonlinear searching technique. For this, Eq. (7) was transformed to an unconstrained optimization formulation using the penalty terms as given in Eq. (8).where, W _E and W _i are the weights given to r _idx and the i-th penalty, respectively (\( {i} = 1, 2,\ldots, 6) \), r _idx = E _idx/E _idx ^t , \( \left\langle v \right\rangle = \left\{ {\begin{array}{*{20}c} {0\, {\text{if}}\, v \le 0} \\ {v \, {\text{if }}\,v > 0} \\ \end{array} } \right. \) and x _i,\( {x}_{i}^{L} \),\( {x}_{i}^{U} \) and \( {x}_{i}^{m} \) are the i-th design variable, its lower and upper bounds and its median values, respectively, as tabulated in Table 4.

The Powell’s algorithm was applied to Eq. (8), which converges to a global minimum in n number of searches for n-dimensional quadratic functions (Powell 1964). The Powell’s method is preferred in minimizing Eq. (8) as it has quadratic terms in its formulation, and also because the method does not require gradients of a given function. The design variable values obtained from the first level of searches were used as initial values in the Powell’s algorithm. After refinement by the subsequent search, the values of E _idx in Eq. (7) from the preliminary searches were further reduced only by 2.0 % or less.

From preliminary and subsequent searches, it was consistently observed that the constraints on T and k _a became near active or active at their upper bound values for all cases with differing values of \( f_{{{\text{cd}}}}^{\prime }\), T _max and t _f. This was because ESC reaches higher compressive strength more effectively under the rapidly increased higher temperature during the early stage of curing. Based on this observation, the values of upper bound for T and k _a were fixed to their maximum values (T = T _max and k _a = 20 °C/h), and the optimized design values of k _d and t _e were searched again from Eq. (8) using the Powell’s method. This further refined the values of k _d and t _e but resulted in the marginally reduced value of E _idx.

As the difference was marginal in the optimized values of E _idx by the discrete step-by-step search and by the Powell’s method, discrete design values from the step-by-step search were presented in Table 5 for practical purpose. In Table 5, the results of optimization obtained for different values of \( f_{{{\text{cd}}}}^{\prime }\), T _max and t _f were listed. For each case, values of optimized design variables as well as the corresponding values of S in terms of S/0.7\( f_{{{\text{cd}}}}^{\prime }\) and r _idx were also tabulated. The values of r _idx in Table 5 show that for all values of \( f_{{{\text{cd}}}}^{\prime }\), the optimized values of E _idx for ESC are 23 to 57 % less than those from the typical 3-6-3 curing regime for concrete made of Type I cement.

More reduction in Eidx was realized for ESC with higher \( f_{{{\text{cd}}}}^{\prime}\) than with lower \( f_{{{\text{cd}}}}^{\prime}\) under the same T _max and t _f values. The largest reduction in E _idx was realized for all \( f_{{{\text{cd}}}}^{\prime}\) with T _max = 60  °C and \( {t}_{\text{f}} = 18\,{\text{h}} \). The values of r _idx are 0.59, 0.52 and 0.43 for \( f_{{{\text{cd}}}}^{\prime } = 30, 40\, {\text{and }}50 \, {\text{MPa}} \), respectively.

The values of r _idx from optimized regimes in Table 5 indicate that lower fuel consumption can be realized by allowing a longer cooling period with a greater value of t _f. This was because an additional increase in concrete strength can be attained during an extended cooling period without fuel consumption. As the upper bound of T was lowered from T _max = 60 C to 40 °C, a longer period with a larger value of t _f was needed. For this case, however, a feasible design with sufficient concrete strength was available only with the longest period of t _f = 18 h for all \( f_{{{\text{cd}}}}^{\prime} {s} \).

In Fig. 6(a), the optimum curing regimes with the lowest value of E _idx obtained from the condition of T _max = 60  C and t _f = 18 h are presented for each \( {f{^{\prime}}}_{\text{cd}} \) value with the corresponding development of compressive strength of ESC based on Eq. (6). Because a higher increase rate of k _a = 20  C/hr was adopted in the optimized regime, a rather rapid increase in compressive strength in the ESC was observed from 3 to 5 h after increasing temperature. After 8 h of curing, more than 50 % of 0.7\( f_{{{\text{cd}}}}^{\prime }\) was developed by the optimized regimes for all \( f_{{{\text{cd}}}}^{\prime }\) values. The rate of strength increase was then decreased and the strength gradually approached the target strength of 0.7\( f_{{{\text{cd}}}}^{\prime }\) during the temperature decrease and cooling periods after a constant T _max period. The optimized values of E _idx (r _idx) were 211.5 (0.59), 188.4 (0.52) and 155.6  C/h (0.43) for \( f_{{{\text{cd}}}}^{\prime }\) equal to 30, 40 and 50 MPa, respectively.

Figure 7a illustrates the effects of t _f on E _idx when T _max = 60. As can be seen in Fig. 7a, for all values of \( f_{{{\text{cd}}}}^{\prime }\), the values of E _idx are shown to decrease almost linearly with the increase in the values of t _f. As explained earlier, it was mainly due to the additional strength gain during an extended cooling period without fuel consumption at ambient temperature of 20 °C after early-stage rapid strength increase at elevated temperature.

Figure 7a also shows that more reduction of E _idx can be realized for a greater \( f_{{{\text{cd}}}}^{\prime }\) value at the same values of t _f. This is attributed to a more rapid increase in strength development for ESC with higher compressive strength at the same T _max as shown in Figs. 3 and 6.

Figure 7b shows the effects of limiting the maximum value of T (T _max) on the E _idx value. Given a value of t _f = 18 h, the curing regime with a higher T _max became more efficient, resulting in a smaller value of E _idx for all. As can be seen in Table 5, a larger amount of E _idx was reduced at lower T _max as the value of t _f increased for all \( f_{{{\text{cd}}}}^{\prime }\). This is because a longer period with an elevated temperature is needed for ESC to sufficiently develop its early strength with a relatively lower T _max than a higher T _max so that its strength can reach 0.7\( f_{{{\text{cd}}}}^{\prime }\) with an additional strength gain during the subsequent cooling period.

Additional compression tests were performed for three sets of ESC with \( f_{{{\text{cd}}}}^{\prime }\) equal to 30, 40 and 50 MPa to verify the proper development of compressive strength of the ESC when the optimized curing regimes as illustrated in Fig. 6a were applied. The same curing method and mix proportion were used as described for the 210 compression tests in previous sections. Two cylinders of 100 × 200 mm from the same batch were tested in compression for each \( f_{{{\text{cd}}}}^{\prime }\) value. The results are shown in Fig. 6a. The averages of the ratio of S/0.7\( f_{{{\text{cd}}}}^{\prime }\) were less than 1.0, showing 0.92, 0.94 and 0.96 for \( f_{{{\text{cd}}}}^{\prime }\) equal to 30, 40 and 50 MPa, respectively. This might be due to statistical variety related to μ (=0.99) and σ (=0.20) for the ratios of model predictions to experimental findings (see Case III in Table 3). Modified optimum curing regimes were then suggested by increasing the values of the optimized t _e by 3 h. Increase in t _e was preferred than increasing the curing period related to T _max because it may need a less fuel consumption for the same amount of additional increase in E _idx. In Fig. 6b, modified curing regimes were illustrated. Predicted strength ratios of S/0.7\( f_{{{\text{cd}}}}^{\prime }\) were 1.07, 1.06, 1.08 for \( f_{{{\text{cd}}}}^{\prime }\) equal to 30, 40 and 50 MPa, respectively. From the additional compression tests performed on two cylinders for each \( f_{{{\text{cd}}}}^{\prime }\) cured under the modified regimes, the averages of the ratio of S/0.7\( f_{{{\text{cd}}}}^{\prime }\) were observed to be 1.02, 1.03 and 1.03 for \( f_{{{\text{cd}}}}^{\prime }\) equal to 30, 40 and 50 MPa, respectively. The values of E _idx (r _idx) from the modified optimum curing regimes were 271.5 (0.75), 248.4 (0.69) and 215.6  C/h (0.60) for f _c ^’ equal to 30, 40 and 50 MPa, respectively.

In Table 5, the values of design variables for the modified curing regime and the corresponding values of relative strength development (S/0.7\( f_{{{\text{cd}}}}^{\prime }\)), E _idx and r _idx are presented. The modified curing regimes were provided for \( f_{{{\text{cd}}}}^{\prime }\) values in the range of 30 to 50 MPa and the upper bound of T between T _max = 50^oC and 60  C. The cases either with T _max = 40 C or t _f ≤ 14 h were excluded as their E _idx values were found to be relatively large compared with those from higher T _max and t _f values (see Table 5). Different value of t _f with 16 h or 18 h can be applied depending on the given value of T _max.

Although the modified optimum regimes became less effective in E _idx than the optimum ones at the price of conservatism, they still save energy by 25, 31 and 40 % for \( f_{{{\text{cd}}}}^{\prime }\) = 30, 40 and 50 MPa, respectively, for 24-turnover period (or t _f = 18 h) compared with those by typical 3-6-3 curing regimes (Table 5).