Empirical Equation for Mechanical Properties of Lightweight Concrete Developed Using Bottom Ash Aggregates

The mechanical properties of lightweight aggregate concrete developed with the use of bottom ash aggregate (LWAC-BA) as a partial or full replacement of lightweight aggregate differ from those of general lightweight concrete made using natural fine and/or coarse aggregates. The mechanical properties of LWAC-BA are difficult to predict using the existing equations proposed by codes or researchers. Therefore, in this study, empirical equations using nonlinear regression analysis are proposed to predict the mechanical properties of lightweight concrete mixed with bottom ash aggregate, based on the collected measured values from other studies (Yang "Development of replacement technology for ready mixed concrete with bottom ash aggregates", 2020; Kim et al. Appl Sci, 10: e8016, 2020; Constr Build Mater 273: e121998, 2021). The collected data include density, compressive strength, elastic modulus, modulus of rupture, splitting tensile strength, and stress–strain relation of LWAC-BA featuring varying amounts of bottom ash fine aggregate and/or coarse aggregate. The proposed empirical equations for each mechanical characteristic are developed considering the replacement volume of bottom ash fine/coarse aggregates. The mean values of the ratios of the measured to predicted values obtained using the proposed equation range from 1.00 to 1.05, with a standard deviation ranging from 0.002 to 0.013, indicating a reasonably positive agreement.


Introduction
Many researchers continue to struggle to identify new materials for replacing conventional ingredients for concrete mixtures. This is because the natural resources used in concrete are becoming increasingly scarce. In particular, the by-products and waste materials are net positive, with examples such as fly ash, blast-furnace slag, and bottom ash. These materials satisfy the research objectives as they are economical and preserve nature by recycling resources. Among by-products, bottom ash is an incombustible by-product collected from the bottom furnace of thermal power stations. Many researchers reported that bottom ash aggregate has irregular rough surface and porous structure (Kim et al., 2020Lee, 2018;Lee et al., 2021;Nisnevich et al., 1999). Due to its porous structure, bottom ash aggregate has a dry density of about 40-70% compared with normal-weight aggregate, while its moisture content is approximately 5-20%, which represents a factor of 3-13 times higher than that of natural aggregate (Lee et al., 2021). The density of aggregate is an important factor that in turn affects the density and quality of concrete (Lee et al., 2019b). As constituents of bottom ash, SiO 2 and Al 2 O 3 account for more than 60% of the total composition, Fe 2 O 3 accounts for approximately 15%, and CaO accounts for about 10%. Bottom ash aggregate was effective at improving the long-term strength and durability of concrete, as insoluble and stable calcium silicate which was produced by pozzolanic reactivity between the bottom ash aggregate and calcium hydroxide (Kim, 2015). Kim et al. (2021) conducted an experimental study on the effects of concrete unit weight on the mechanical properties of concrete containing bottom ash and determined that density was an important factor in determining mechanical properties. Kim et al. (2020) also investigated the workability and mechanical properties of concrete produced with bottom ash aggregates in relation to three water-to-cement ratios and the replaced ratio of bottom ash aggregates. The slump was seen to decline regardless of the water-to-cement ratio. Bottom ash coarse aggregates had a relatively larger effect on compressive strength than fine aggregate, and the tensile and shear friction strength rose as the density of concrete increased. Lee et al. (2019b) investigated the various mechanical properties of LWAC mixed with expanded bottom ash and dredged soil-based artificial lightweight aggregates and novel formulas were proposed to anticipate early-age and long-term strength for that. The research revealed that the density of LWAC mixed with expanded bottom ash and dredged soil-based artificial lightweight aggregates was a key factor for determining compressive strength. To examine the feasibility of applying pre-cast concrete panels, Yang et al. (2019) evaluated the consistency and mechanical properties of LWAC mixed with bottom ash with a pre-formed foam volume ratio of less than or equal to 25%. In concrete mixture, ordinary Portland cement was partially replaced with 50% groundgranulated blast-furnace slag and 20% fly ash, while natural fine and coarse aggregates were fully replaced with bottom ash aggregates. As observed in the results, the splitting tensile strength and modulus of rupture declined as foam volume fraction increased. Lee et al. (2019a) examined the mechanical properties of lightweight aggregate concrete made with expanded bottom ash and dredge soil granules (LWAC-BS), proposing an equation to predict compressive strength, elastic modulus, tensile strength, shear friction, bond strength and also to determine the relationship between compressive strength and strain. Yang (2019) conducted an experiment to investigate the effect of the water-tocement ratio (W/C) and replacement ratio of bottom ash aggregate on the mechanical properties of LWAC-BA. The value of measured compressive strength (f' c,meas ) of LWAC-BA increased with lower W/C and higher density, which was likely a tendency of general LWAC. The value of the elastic modulus divided by the square root of f' c,meas increased smoothly as the density of LWAC-BA was increased. The value of the splitting tensile strength of LWAC-BA was lower than that of general LWAC, and the value of the modulus fracture divided by the square root of f' c,meas of LWAC-BA declined slightly as the density of LWAC-BA increased. The bond strength (τ b ) between LWAC-BA and the reinforcing steel-bar was considered weak, because the value of τ b divided by the square root of f' c,meas of LWAC-BA was lower than that of LWAC-BS.
As described previously, concrete mixed with partial or full bottom ash aggregate possesses mechanical properties that differ from those of conventional LWAC. Therefore, this study aimed to develop empirical equations for mechanical properties such as density (ρ c ), compressive strength (f' c ), elastic modulus (E c ), stress-strain relationship, splitting tensile strength (f sp ,) modulus of rupture (f r ), and bond strength (τ b ) of concrete in consideration of the replacement volume of bottom ash fine and coarse aggregates based on nonlinear regression (NLR) analysis and collected experimental data. The proposed empirical equations were compared with the existing design equations, such as ACI 318, fib Model Code (2010) (hereafter MC2010), and Lee et al., (2019aLee et al., ( , 2019b.

Development of Equation
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 (R BAS ), the percentage of replaced BAC content (R BAC ), and the water-to-cement ratio (W/C), which ranged from 0.3 to 0.45. For example, an R BAS 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 (ρ c,meas ), compressive strength (f ' c,meas ), splitting tensile strength (f sp,meas ), elastic modulus (E c,meas ), and bond strength (τ b,meas ). In the case of LWAC-BA, which consisted of concrete mixed with partial or full bottom ash aggregate, r c,meas ranged from 1730 to 2171 kg/m 3 , f ' c,meas ranged from 23.3 to 52.6 MPa, f sp,meas ranged from 2.34 to 3.95 MPa, E c,meas ranged from 18.1 to 27.9 MPa, f r,meas ranged from 3.9 to 6 MPa, and t b,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. (2014aYang et al. ( , 2014b) and Lee et al. (2019a), new straightforward empirical equations for LWAC-BA were derived in the order of ρ c , f ' c , E c , ε 0 , stress-strain, f sp , f r , and τ b . 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.

Oven-Dried Density
As previously reports by Yang (2020), Kim et al. (2020), Kim et al. (2021), the measured oven-dried density (ρ c,meas ) of LWAC-BA was affected by W/C, R BAS , and R BAC . Therefore, an equation for oven-dried density should be considered with W/C, R BAS , and R BAC , and two coefficient factors were to be derived. To determine the weight of the effects of BAC, the volume of natural sand (F S ) used was fixed. The weight was then calculated from the relationship between R BAC and ρ c,meas to w a , where w a 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 ρ c,meas to w a . From the weights of the effects of BAC and W/C, the following coefficient factor (α 1 ) pertaining to BAC and W/C was finally derived:  (Kim et al., 2020Yang, 2020).
R BAS is the percentage of replaced content of BAS (= 100 × BAS's weight to total sand weight); R BAC is the percentage of replaced content of BAC aggregate (= 100 × BAC's weight to total coarse weight); W/C is the water-to-cement ratio; S/a is the fine aggregate ratio; W is the water volume; C is the cement; F s and C G are the natural sand and coarse aggregates, respectively; BAS and BAC are the bottom ash fine and coarse aggregate, respectively; A c is the air content; and ρ c,meas , f' c,meas , f sp,meas , f r,meas , E c,meas , and τ b,meas are the measured density, compressive strength, splitting tensile, modulus of rupture, elastic modulus, and bond strength at 28 days, respectively. By using the same method and procedure, a second coefficient factor (β 1 ) regarding BAS and W/C was also derived: Fig. 1 shows the relationship of the measured density (ρ c,meas ) and the summation of the absolute unit weight of each ingredient (w a ) multiplied by the coefficient factors (α 1 and β 1 ) for the NLR analysis. By utilizing NLR analysis, the straightforward empirical equation for oven-dried density (ρ c ) of LWAC-BA can be expressed as where ρ c is the oven-dried density (in kg/m 3 ) and w a is the summation of the absolute unit weight of each ingredient (in kilograms). The correlation coefficient (R 2 ) was 0.88. (1)

Specimens W/C R BAS (%) R BAC (%) S/a (%) Unit volume weight (kg/m 3 ) A c ρ c,meas f' c,meas f sp,meas f r,meas E c,meas τ b,meas
(2) (3) ρ c = 1.447(α 1 β 1 w a ) 0.93 , are slightly higher than those of the proposed equation. However, all values of γ cv are 0.03 or less. Overall, the accuracy of the proposed model and the others is similar and acceptable. Yang et al. (2014aYang et al. ( , 2014b proposed an equation to predict the compressive strength (f' c ) of LWAC. The model was formulated with ρ c 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 where In aforementioned equations, f' c is the compressive strength of LWAC-BA (in MPa); f 0 is the reference compressive strength (= 10 MPa); R BAS is the percentage of replaced content of BAS (= percentage of BAS's weight to total sand weight); R BAC is the percentage of replaced content of BAC (= percentage of BAC`s weight to total coarse weight); ρ c is the oven-dried density (in kg/m 3 ), which can be obtained from Eq. 3; ρ 0 is the reference Fig. 1 Relationship of ρ c,meas and w a multiplied by coefficient factors. Fig. 2 Comparison of ρ c,meas and ρ c,pred .
Values of f' c,meas were also affected by R BAS , R BAC , and W/C, wherein R BAS and R BAC are related to ρ c,meas . α 2 in Eq. 5 was derived by first determining the relationship between R BAC and f' c,meas and then determining the relationship between W/C and f' c,meas . β 2 in Eq. 6 was also derived by first determining the relationship between R BAS 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 ρ c,meas multiplied by the coefficient factors, where all individually measured values were used, not the average values from Table 1.  Lee et al. (2019a), the relationship between f ′ c,meas ρ c,meas /ρ 0 and the measured elastic modulus (E c,meas ) of LWAC-BA was studied, as shown in Fig. 5. The value of E c,meas increased as f c,meas and/or ρ c,meas increased. From the NLR analysis based on the test results, the elastic modulus E c, (in MPa) of LWAC-BA can be expressed using f' c , and ρ c as where f' c is the compressive strength (in MPa), which can be obtained from Eq. 4; ρ c is the oven-dried density (in kg/m 3 ), which can be obtained from Eq. 3; and ρ 0 is the reference density (2300 kg/m 3 ). Fig. 6 compares E c,meas to the predicted concrete modulus (E c,pred ) calculated with the predicted concrete strength and oven-dried density . As observed in Eq. 7 and other existing equations, the values of γ m , γ sd , and γ cv 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 E c,meas is greater than 22,000 MPa. Meanwhile, the accuracy of MC2010 (2010) is good when E c,meas is less than 22,000 MPa. Fig. 3 Regression analysis for f' c,meas .
where n 1 , n 2 , and α as three exponents are the coefficient factors that vary based on mechanical properties. This means that f sp , f r , and τ b are strongly affected by f' c and ρ c , and the relation of f sp , f r , and τ b and{(f' c ) n1 (ρ c/ ρ 0 ) n2 } α was also investigated in this study. Fig. 10 shows the effects of f' c,pred ρ c, pred /ρ 0 on the measured splitting tensile strength (f sp,meas ), measured modulus of rupture (f r,meas ), and measured bond strength (τ b,meas ) of LWAC-BA, where ρ c,pred and f' c,pred are the predicted density and compressive strength obtained from Eqs. 3 and 4, respectively. The values of f sp,meas , f r,meas , and τ b,meas of LWAC-BA increased with the rise in f' c,pred and/ or ρ c,pred . From the LNR analysis in Fig. 10, f sp , f r , and τ b of LWAC-BA can be expressed using f' c and ρ c as: where f sp , f r , and τ b 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); ρ c is the oven-dried density (in kg/ m 3 ); and ρ 0 is the reference density (2300 kg/m 3 ). Here, ρ c and f' c can be obtained from Eqs. 3 and 4.

Conclusions
In this study, empirical equations were derived from the experimental results for oven-dried density (ρ c ), compressive strength (f' c ), splitting strength (f sp ), bond strength (τ b ), elastic modulus (E c ), and stress-strain curve of lightweight concrete made with bottom ash fine  Page 8 of 10 Lee et al. Int J Concr Struct Mater (2022) 16:23 and/or coarse aggregates, which was suitable material for lightweight aggregate concrete because of its low density.
The following conclusions could be made: 1. The density and compressive strength were comprehensively affected by the combination of the waterto-cement (W/C) ratio and replacement ratios of bottom ash fine and/or coarse aggregates. The proposed