Recently, Yang (2020), Kim et al. (2020), and Kim et al. (2021) conducted experimental studies to investigate how the mechanical properties of LWAC-BA differed when the bottom ash fine and/or coarse aggregates were fully or partially replaced with normal-weight aggregates. In the present study, the data related to LWAC-BA in Yang (2020), Kim et al. (2020), and Kim et al. (2021) were collected. Table 1 presents the LWAC-BA mixtures made with partially or fully replaced bottom ash fine aggregate (BAS) and/or bottom ash coarse aggregate (BAC), where each value is the average of three samples. The main parameters observed during the test were the percentage of replaced BAS content (RBAS), the percentage of replaced BAC content (RBAC), and the water-to-cement ratio (W/C), which ranged from 0.3 to 0.45. For example, an RBAS value of 25% indicated that BAS was used as one-fourth of the total sand aggregate. In Table 1, average measures of the mechanical properties at 28 days are given for the following: oven-dried density (ρc,meas), compressive strength (f’c,meas), splitting tensile strength (fsp,meas), elastic modulus (Ec,meas), and bond strength (τb,meas). In the case of LWAC-BA, which consisted of concrete mixed with partial or full bottom ash aggregate, rc,meas ranged from 1730 to 2171 kg/m3, f’c,meas ranged from 23.3 to 52.6 MPa, fsp,meas ranged from 2.34 to 3.95 MPa, Ec,meas ranged from 18.1 to 27.9 MPa, fr,meas ranged from 3.9 to 6 MPa, and tb,meas ranged from 4.3 to 7 MPa. Utilizing LWAC-BA mixtures and measured values as given in Table 1, as well as the NLR analysis performed by Yang et al. (2014a, 2014b)) and Lee et al. (2019a), new straightforward empirical equations for LWAC-BA were derived in the order of ρc, f’c, Ec, ε0, stress–strain, fsp, fr, and τb. Due to the internal number of voids of bottom ash aggregate, bottom ash aggregate generally possessed lower crushing strength and stiffness compared with natural aggregate (Sim & Yang, 2011). Its property affects the compressive strength of concrete, and the compressive strength and weight of the unit volume of bottom ash aggregate are generally inversely proportional to each other (Lee et al., 2021). Therefore, the proposed model presented in this study was more simplified by using the weight of the unit volume of bottom ash aggregate and the compressive strength. The presented model evaluated the mean, standard deviation, and coefficient of variation through comparative analysis with experimental results.

Table 1 Summary of LWAC-BA mixtures and test results (Kim et al., 2020, 2021; Yang, 2020).

Oven-Dried Density

As previously reports by Yang (2020), Kim et al. (2020), Kim et al. (2021), the measured oven-dried density (ρc,meas) of LWAC-BA was affected by W/C, RBAS, and RBAC. Therefore, an equation for oven-dried density should be considered with W/C, RBAS, and RBAC, and two coefficient factors were to be derived. To determine the weight of the effects of BAC, the volume of natural sand (FS) used was fixed. The weight was then calculated from the relationship between RBAC and ρc,meas to wa, where wa is the summation of the absolute unit weight of each ingredient. After that, the weight of the effects of W/C was also calculated from the relationship between W/C and the ratio of ρc,meas to wa. From the weights of the effects of BAC and W/C, the following coefficient factor (α1) pertaining to BAC and W/C was finally derived:

$$alpha_{1} = left( {0.0013left( frac{W}{C} right) – 0.0009} right)R_{text{BAC}} + left( { – 0.3736left( frac{W}{C} right) + 1.1177} right)$$

(1)

By using the same method and procedure, a second coefficient factor (β1) regarding BAS and W/C was also derived:

$$beta_{1} = left( { – 0.0011left( frac{W}{C} right) + 0.0006} right)R_{text{BAS}} + left( {0.3076left( frac{W}{C} right) + 1.0367} right)$$

(2)

Fig. 1 shows the relationship of the measured density (ρc,meas) and the summation of the absolute unit weight of each ingredient (wa) multiplied by the coefficient factors (α1 and β1) for the NLR analysis. By utilizing NLR analysis, the straightforward empirical equation for oven-dried density (ρc) of LWAC-BA can be expressed as

$$rho_{c} = 1.447left( {alpha_{1} beta_{1} w_{a} } right)^{0.93} ,$$

(3)

Fig. 1
figure 1

Relationship of ρc,meas and wa multiplied by coefficient factors.

where ρc is the oven-dried density (in kg/m3) and wa is the summation of the absolute unit weight of each ingredient (in kilograms). The correlation coefficient (R2) was 0.88.

Fig. 2 displays a comparison of ρc,meas and values of predicted oven-dried density (ρc,pred) obtained by using proposed model, ACI 318 (2019), and Lee et al.’s (2019a) equation. The mean value (γm), standard derivation (γsd), and coefficient of variation (γcv) of the measured to predicted density obtained by using the proposed equation are 1.00, 0.03, and 0.034, respectively. Meanwhile, the values of γm of ACI 318 (2019) and the equation of Lee et al. (2019a) are close to 1, while the values of γsd and γcv of ACI 318 (2019) and the equation of Lee et al. (2019a) are slightly higher than those of the proposed equation. However, all values of γcv are 0.03 or less. Overall, the accuracy of the proposed model and the others is similar and acceptable.

Fig. 2
figure 2

Comparison of ρc,meas and ρc,pred.

Compressive Strength

Yang et al. (2014a, 2014b) proposed an equation to predict the compressive strength (f’c) of LWAC. The model was formulated with ρc and C/W (cement-to-water ratio) as the primary parameters, and Lee et al. (2019a) modified the equation so that LWAC-BS would fit. The relationship among compressive strength (f’c), oven-dried density, and C/W of LWAC-BA can be expressed as

$$frac{{f_{c}^{^{prime}} }}{{f_{0} }} = 1.544left[ {alpha_{2} beta_{2} left( {frac{{rho_{c} }}{{rho_{0} }}} right)^{0.8} left( frac{C}{W} right)^{{{}^{1.4}}} } right]^{0.44} ,$$

(4)

where

$$alpha_{2} = left( { – 0.015left( frac{W}{C} right) + 0.002} right)R_{text{BAC}} + left( {0.8left( frac{W}{C} right) + 0.8} right),$$

(5)

$$beta_{2} = left( {0.007left( frac{W}{C} right) – 0.0039} right)R_{text{BAS}} + left( {2.935left( frac{W}{C} right) + 0.283} right).$$

(6)

In aforementioned equations, f’c is the compressive strength of LWAC-BA (in MPa); f0 is the reference compressive strength (= 10 MPa); RBAS is the percentage of replaced content of BAS (= percentage of BAS’s weight to total sand weight); RBAC is the percentage of replaced content of BAC (= percentage of BAC`s weight to total coarse weight); ρc is the oven-dried density (in kg/m3), which can be obtained from Eq. 3; ρ0 is the reference density (2300 kg/m3); and C/W is the cement-to-water ratio.

Values of f’c,meas were also affected by RBAS, RBAC, and W/C, wherein RBAS and RBAC are related to ρc,meas. α2 in Eq. 5 was derived by first determining the relationship between RBAC and f’c,meas and then determining the relationship between W/C and f’c,meas. β2 in Eq. 6 was also derived by first determining the relationship between RBAS and f’c,meas. Following that, the relationship between W/C and f’c,meas was discerned. For NLR analysis, Fig. 3 shows the relationship between f’c,meas and the fundamental form with C/W and ρc,meas multiplied by the coefficient factors, where all individually measured values were used, not the average values from Table 1.

Fig. 3
figure 3

Regression analysis for f’c,meas.

Fig. 4 displays the comparison between f’c,meas and predicted compressive strength (f’c,pred) using the proposed equations (Eqs. (4)–(6)) and Lee et al.’s (2019a) equation, where f’c,pred was calculated with the predicted oven-dried density obtained from Eq. 3. Values of γm, γsd, and γcv of LWAC-BA obtained by using the proposed equation are 1.03, 0.03, and 0.12, respectively. Meanwhile, values of γm, γsd, and γcv of LWAC-BA within Lee et al.’s (2019a) equation are 1.29, 0.22, and 0.17, respectively. Overall, the proposed equation offers better accuracy than Lee et al.’s equation.

Fig. 4
figure 4

Comparison of f’c,meas and f’c,pred.

Elastic Modulus

ACI-318 (2019), MC2010 (2010), and Lee et al.’s (2019a) equation for predicting the elastic modulus of concrete (Ec) are formulated with f’c and ρc; the results indicate that Ec is significantly affected by f’c and ρc. Following the analysis method conducted by Lee et al. (2019a), the relationship between (f_{c,meas}^{^{prime}} rho_{c,meas} /rho_{0}) and the measured elastic modulus (Ec,meas) of LWAC-BA was studied, as shown in Fig. 5. The value of Ec,meas increased as fc,meas and/or ρc,meas increased. From the NLR analysis based on the test results, the elastic modulus Ec, (in MPa) of LWAC-BA can be expressed using f’c, and ρc as

$$E_{c} = 7307left[ {f_{c}^{^{prime}} left( {frac{{rho_{c} }}{{rho_{0} }}} right)} right]^{0.336} ,$$

(7)

Fig. 5
figure 5

Effect of f’c,measρc,meas/ρ0 on Ec,meas and regression analysis.

where f’c is the compressive strength (in MPa), which can be obtained from Eq. 4; ρc is the oven-dried density (in kg/m3), which can be obtained from Eq. 3; and ρ0 is the reference density (2300 kg/m3).

Fig. 6 compares Ec,meas to the predicted concrete modulus (Ec,pred) calculated with the predicted concrete strength and oven-dried density. As observed in Eq. 7 and other existing equations, the values of γm, γsd, and γcv of LWAC-BA obtained by using the proposed equation are 1.00, 0.05, and 0.05, respectively, indicating that the proposed equation is excellent in terms of all indexes. The accuracy of the equation of Lee et al. (2019a) is good when Ec,meas is greater than 22,000 MPa. Meanwhile, the accuracy of MC2010 (2010) is good when Ec,meas is less than 22,000 MPa.

Fig. 6
figure 6

Comparison of Ec,meas and Ec,pred.

Stress–Strain Relationship

Yang et al. (2014a, 2014b) proposed an equation for predicting the stress–strain curve of concrete, including the descending branch covering a wide range of f’c values (from 10 to 180 MPa) and ρc values (from 1200 to 4500 kg/m3). Further, Lee et al. (2019a) presented a modified equation for LWAC-BS by performing the same analysis as that of Yang et al. (2014a, 2014b)) with the test database of LWAC-BS. The two equations have the same fundamental equation (Eq. 8) regarding the corresponding concrete stress (f’c,crs) and specific strain (εc), as well as the equation related to ascending and descending branches being different depending on the properties of the concrete:

$$f_{c,crs}^{^{prime}} = left[ {frac{{left( {beta_{{}} + 1} right)left( {frac{{varepsilon_{c} }}{{varepsilon_{0} }}} right)}}{{left( {frac{{varepsilon_{c} }}{{varepsilon_{0} }}} right)^{{beta_{{}} + 1}} + beta }}} right]f_{c}^{^{prime}} ,$$

(8)

where f’c,crs is the corresponding concrete stress (in MPa) for the specific strain (εc); ε0 is the strain value at peak stress; f’c is the compressive strength (in MPa) of LWAC-BA, respectively; and β is the key parameter determining slopes of the ascending and descending branches of the stress–strain curve.

Yang (2019, 2020) reported that it was difficult to measure a descending branch because of the brittle characteristic of LWAC-BA. Therefore, there are a few data points including a descending branch. For NLR analysis, the relationship of the measured specific strain ((varepsilon_{0,meas})) and (f_{c,meas}^{^{prime}} /E_{c,meas}) was first studied, as shown in Fig. 7. Hence, the equation to predict ε0 at the peak compressive strength of LWAC-BA can be expressed as:

$$varepsilon_{0} = 0.001exp left[ {442left( {frac{{f_{c}^{^{prime}} }}{{E_{c} }}} right)} right].$$

(9)

Fig. 7
figure 7

Effect of f’c,measρc,meas/ρ0 on ε0,meas and regression analysis.

When entering Eq. 9 into Lee et al.’s (2019a) equation, it was found that the slopes of the ascending branch were close to the measured slope, although the slopes of the descending branch were different. Therefore, it was decided that only the equation of the descending branch should be modified, and the constant in the exponential function was changed from 0.58 to 0.3, with the slopes of the descending branch compared with the measured values (Fig. 8). Therefore, the equations for the ascending and descending branches can be expressed as

$$beta = 0.19exp left[ {0.54left( {frac{{f_{c}^{^{prime}} }}{{f_{0} }}} right)left( {frac{{rho_{0} }}{{rho_{c} }}} right)^{1.5} } right]quad {text{for }}varepsilon_{c} le varepsilon_{0} ,$$

(10)

$$beta = 0.32exp left[ {0.3left( {frac{{f_{c}^{^{prime}} }}{{f_{0} }}} right)left( {frac{{rho_{0} }}{{rho_{c} }}} right)^{1.5} } right]quad {text{for }}varepsilon_{c} > varepsilon_{0} .$$

(11)

Fig. 8
figure 8

Stress–strain curves of LWAC-BA.

f’c and ρc are the compressive strength (in MPa) and oven-dried density (in kg/m3) of LWAC-BA, respectively; and (f_{0}) and (rho_{0}) are the 10 MPa and 2300 kg/m3 reference values. Equation 10 is the same equation proposed by Lee et al. (2019a).

Fig. 9 displays the ratios of the measured strain (ε0,meas) to predicted strain (ε0,pred) at peak compressive strength, where values of ε0,pred are calculated with the predicted compressive strength (f’c,pred) and elastic modulus (Ec,pred) of LWAC-BA. All indexes of the proposed equation for reliability are excellent in the overall range. The accuracy of Lee et al.’s (2019a) equation increases as the value of ε0,meas increases.

Fig. 9
figure 9

Comparison of (varepsilon_{0,meas}) and (varepsilon_{0,pred}).

Splitting Tensile Strength, Modulus of Rupture, and Bond Strength

Lee et al. (2019a) also proposed the splitting tensile strength (fsp), modulus of rupture (fr), and bond strength (τb) based on f’c and ρc/ρ0, and the design equations were expressed through the form of {(f’c) n1 (ρc/ρ0) n2}α, where n1, n2, and α as three exponents are the coefficient factors that vary based on mechanical properties. This means that fsp, fr, and τb are strongly affected by f’c and ρc, and the relation of fsp, fr, and τb and{(f’c) n1 (ρc/ρ0) n2}α was also investigated in this study.

Fig. 10 shows the effects of f’c,predρc, pred/ρ0 on the measured splitting tensile strength (fsp,meas), measured modulus of rupture (fr,meas), and measured bond strength (τb,meas) of LWAC-BA, where ρc,pred and f’c,pred are the predicted density and compressive strength obtained from Eqs. 3 and 4, respectively. The values of fsp,meas, fr,meas, and τb,meas of LWAC-BA increased with the rise in f’c,pred and/or ρc,pred. From the LNR analysis in Fig. 10, fsp, fr, and τb of LWAC-BA can be expressed using f’c and ρc as:

$$f_{sp} = 0.5left( {f_{c}^{^{prime}} left( {frac{{rho_{c} }}{{rho_{0} }}} right)} right)^{0.54} ,$$

(12)

$$f_{r} = 1.74left( {f_{c}^{^{prime}} left( {frac{{rho_{c} }}{{rho_{0} }}} right)} right)^{0.32} ,$$

(13)

$$tau_{b} = 0.99left( {f_{c}^{^{prime}} left( {frac{{rho_{c} }}{{rho_{0} }}} right)} right)^{0.51} ,$$

(14)

where fsp, fr, and τb are the predicted splitting tensile strength (in MPa), modulus of rupture (in MPa), and bond strength (in MPa), respectively; f’c is the compressive strength (in MPa); ρc is the oven-dried density (in kg/m3); and ρ0 is the reference density (2300 kg/m3). Here, ρc and f’c can be obtained from Eqs. 3 and 4.

Fig. 11 presents a comparison of the test results and the predicted values iterated by the equation of Lee et al (2019a), ACI-318 (2019), MC 2010 (2010), and the proposed equation. All equations overestimate fsp in fsp,meas range of 2.5 MPa or less, and they exhibit solid accuracy in fsp,meas range of 3 MPa or greater. In the case of fr, the values of γm of the proposed equation and MC2010 (2010) are close to 1.0, while the equation of ACI-318 (2019) underestimates across the entire range. Regarding τb, the values of γm and γcv of LWAC-BA obtained using the proposed equation are 1.01 and 0.08, respectively, which are the best values among all the equations.

Fig. 10
figure 10

Effects of f’c,predρc,pred/ρ0 on measured mechanical properties and regression analysis.

Fig. 11
figure 11

Comparison of test results and prediction values.

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