The analytical models for predicting age-dependent mechanical properties (i.e., compressive strength, splitting tensile strength, and elastic modulus) of RCA, BA, and their combination are needed to minimize the construction time and assess the health of the existing structures. The results obtained from experiments are affected by the recycled coarse aggregate, bottom ash, and their combination. The reason may be the poor bonding between new and old materials and variation in recycled coarse aggregate and bottom ash properties. Therefore, in the present study, new analytical solutions for predicting mechanical properties (i.e., compressive, splitting tensile strength, and static elastic modulus) of recycled coarse aggregate and bottom ash concrete at any time have been proposed using crushing strength, UPV and RHN.
4.1 Age-dependent models for compressive and tensile strength of recycled aggregate + bottom ash concrete
The multivariable regression analysis has been carried out on experimental results for all concrete mixes. The proposed models are shown in Eqs. (1), (2), and (3) are formed on the 28-day crushing compressive strength of conventional concrete, UPV, and RHN values. It means that the models are given in Eqs. (2) and (3) are applicable at the onsite inspection to assess the structural health of the buildings in terms of their residual strength capacity at any age and replacement of recycled coarse aggregate and bottom ash. Because during structural health monitoring at different intervals of the age of the structure, it is not possible to determine the strength of concrete based on a crushing test by taking a core from the existing structure. Therefore, the proposed models given in Eqs. (2) and (3) will be helpful to determine the strength of RCA based concrete structures containing bottom ash by using UPV and RHN at any age, respectively, without knowing the laboratory based crushing strength of concrete.
$${\left({f}_{c}^{\text{'}}\right)}_{t}=\frac{{exp}\left({C}_{3}{p}_{rca}+{C}_{4}{p}_{ba}\right)t{\left({f}_{c}^{\text{'}}\right)}_{28,plain}}{{C}_{1}+{C}_{2}t}$$
1
$${\left({f}_{c}^{\text{'}}\right)}_{t,UPV}=\frac{{C}_{10}{exp}\left({C}_{7}{p}_{rca}+{C}_{8}{p}_{ba}+{C}_{9}UPV\right)t}{{C}_{5}+{C}_{6}t}$$
2
$${\left({f}_{c}^{\text{'}}\right)}_{t,RHN}=\frac{{C}_{16}{exp}\left({C}_{13}{p}_{rca}+{C}_{14}{p}_{ba}+{C}_{15}RHN\right)t}{{C}_{11}+{C}_{12}t}$$
3
Where, \({\left({f}_{c}^{{\prime }}\right)}_{t}\) = concrete compressive strength at age ‘t’ on cylindrical specimen (MPa); t = concrete age in days; \({\left({f}_{c}^{{\prime }}\right)}_{28,plain}\) = 28-day conventional concrete compressive strength in MPa on cylindrical specimen; \({p}_{rca}\) = recycled coarse aggregate percentage in fraction; \({p}_{ba}\) = percentage of bottom ash in fraction; \({\left({f}_{c}^{{\prime }}\right)}_{t,UPV}\) = concrete compressive strength at age ‘t’ using UPV in MPa; \({\left({f}_{c}^{{\prime }}\right)}_{t,RHN}\) = concrete compressive strength at age ‘t’ using RHN in MPa; C1 to C16 = model parameters having values 2.675, 0.947, -0.056, -0.412, 0.591, 0.918, 0.003, -0.266, 0.315, 5.921, 0.034, 0.118, 0.006, -0.290, 0.034 and 1.215 respectively.
Figure 9 (a) to (c) shows a comparison of the experimental and predicted compressive strength of RCA and BA based concrete using \({\left({f}_{c}^{{\prime }}\right)}_{28,plain}\), UPV and RHN (i.e., using Eqs. 1 to 3) respectively with age. It has been observed that 100%, 87.5%, and 94.0% data points fall within an error band of ± 5% for the strength prediction using Eq. 1, 2, and 3, respectively. The acceptability of these models can also be justified with R2 values, i.e., 0.94, 0.98, and 0.98 for Eq. 1, 2, and 3, respectively. From the above model prediction analysis, it can be suggested that the concrete compressive strength can be predicted by using Eqs. (2) and (3) based on UPV and RHN, respectively, for any percent replacement of RCA and BA. These models are also applicable to the existing concrete buildings to assess strength capacity without knowing the 28-day compressive strength in the laboratory based on the crushing test.
The analysis of experimental results and the empirical relationships in design codes of different countries revealed that compressive strength could play an essential role in predicting concrete splitting tensile strength (fspt). However, an age-dependent model for predicting fspt of concrete prepared with RCA and BA is still needed considering compressive strength at 28-day, UPV, and RHN values. Thus, based on the multivariable regression analysis and by using models given in Eq. (1), (2), and (3), the following models presented in Eq. (4) to (6) have been proposed for predicting the splitting tensile strength of RCA, BA and their combination.
$${\left({f}_{spt}\right)}_{t}={k}_{1}{\left[{\left({f}_{c}^{\text{'}}\right)}_{t}\right]}^{{K}_{2}}$$
4
$${\left({f}_{spt}\right)}_{t,UPV}={k}_{3}{\left[{\left({f}_{c}^{\text{'}}\right)}_{t,UPV}\right]}^{{K}_{4}}$$
5
$${\left({f}_{spt}\right)}_{t,RHN}={k}_{5}{\left[{\left({f}_{c}^{\text{'}}\right)}_{t,RHN}\right]}^{{K}_{6}}$$
6
Where, \({\left({f}_{spt}\right)}_{t}\) = concrete splitting tensile strength at age ‘t’ in GPa; \({\left({f}_{spt}\right)}_{t,UPV}\) = concrete splitting tensile strength at age ‘t’ using UPV in GPa; \({\left({f}_{spt}\right)}_{t,RHN}\) = concrete splitting tensile strength at age ‘t’ using RHN in GPa; k1 to k6 = model parameters having values 0.021, 1.661, 0.026, 1.594, 0.028, and 1.568, respectively.
The comparative analysis between predicted splitting tensile strength and the measured values for all RCA and BA concrete mixes and at all ages using the proposed models as given in Eqs. (4) to (6) are presented in Figs. 10 (a) to (c) respectively. From Figs. 10 (a) to (c), it seems that 100% of data lie within the error band ± 5%, ± 10%, and ± 10% in the prediction of fspt using the proposed models given in Eqs. 4 to 6, respectively. Thus, it can be said that the proposed models for predicting fspt of RCA + BA based concrete are in acceptable agreement well with the measured data. Hence the models proposed in Eqs. 5 and 6 will be helpful in the prediction of the splitting tensile strength of existing concrete buildings at the age ‘t’ and any percentage of RCA, BA, and their combination.
4.2 Age-dependent models for elastic modulus of RCA + BA concrete
The current state of the art on the concrete static elastic modulus (Ec) indicates the need for a new model to predict age-dependent Ec of RCA, BA, and RCA + BA based concrete. Moreover, age-dependent formulae for Ec of concrete proposed by researchers and empirical relationships proposed by design codes of different countries are silent on the effect of RCA and BA and prediction based on UPV and RHN values. Therefore, the multivariable regression analysis has been carried out and proposed new age-dependent models for predicting Ec of concrete for any percentage replacement of natural crushed rock and river bed aggregate with RCA and BA, respectively. A model is shown in Eq. 7 to predict the Ec of RCA and BA based concrete at any age by using 28-day conventional concrete compressive strength. Similarly, the models are shown in Eqs. 8 and 9 have been proposed for predicting the Ec of RCA and BA based concrete at any age by using UPV and RHN values, respectively. Thus, it can be said that the models present in Eqs. 8 and 9 are more rational than the model given in Eq. 7 as these models are not dependent on the 28-day conventional concrete strength. Hence, the models proposed in Eqs. 8 and 9 can estimate the Ec of concrete buildings containing RCA, BA, and their combination by using UPV and RHN values at any age without knowing the conventional concrete compressive strength in the laboratory at 28 days.
\({\left({E}_{c}\right)}_{t}={J}_{1}{\left[{\left({f}_{c}^{\text{'}}\right)}_{t}\right]}^{{J}_{2}}\) (7)
\({\left({E}_{c}\right)}_{t,UPV}={J}_{3}{\left[{\left({f}_{c}^{\text{'}}\right)}_{t,UPV}\right]}^{{J}_{4}}\) (8)
\({\left({E}_{c}\right)}_{t,RHN}={J}_{5}{\left[{\left({f}_{c}^{\text{'}}\right)}_{t,RHN}\right]}^{{J}_{6}}\) (9)
Where, \({\left({E}_{c}\right)}_{t}\) = concrete elastic modulus at age ‘t’ in GPa; \({\left({E}_{c}\right)}_{t,UPV}\) = concrete elastic modulus of at age ‘t’ using UPV in GPa; \({\left({E}_{c}\right)}_{t,RHN}\) = concrete elastic modulus at age ‘t’ using RHN in GPa; J1 to J6 = model parameters having values 3.739, 0.564, 3.660, 0.571, 3.665, and 0.571, respectively.
The comparison between age-dependent predicted Es using proposed models using Eqs. 7 to 9 and the measured results for all the RCA and BA concrete mixes are shown in Figs. 11 (a) to (c), respectively. Using these models, the error band of ± 5% has also been plotted in these figures by using these models. From these Figs., it is observed that 93%, 100%, and 93% of data lie within these error bands of ± 5%, respectively. This indicates that the proposed models for predicting Es of RCA and BA based concrete are in acceptable agreement with the experimental results and can be applied in the design and construction of concrete buildings made with RCA, BA, and their combination.