The studies on rubberized concrete have increased dramatically over the last few years due to being an environmentally friendly material with enhanced vibration behavior and energy dissipation capabilities. Nevertheless, multiple resources in the literature have reported reductions in its mechanical properties directly proportional to the rubber content. Over the last few years, various mathematical models have been proposed to estimate rubberized concrete properties using artificial intelligence, machine learning, and fuzzy logicbased methods. However, these models are relatively complicated and require higher computation efforts than multivariable regression ones when it comes to the daily usage of practicing engineers. Additionally, most of the study has mainly focused on the compressive strength of rubberized concrete and rarely went into more details considering other properties and sample sizes. Therefore, this study focuses on developing simple yet accurate rubberized concrete multivariable regression models that can be generalized for various mixtures of rubberized concrete considering different sample sizes.
Los estudios sobre hormigón incorporando caucho han aumentado drásticamente en los últimos años debido a que es un material ecológico con un comportamiento de vibración mejorado y capacidades de disipación de energía. Sin embargo, múltiples trabajos en la literatura han indicado reducciones en sus propiedades mecánicas directamente proporcionales al contenido de caucho. En los últimos años se han propuesto varios modelos matemáticos para estimar las propiedades del hormigón con caucho utilizando inteligencia artificial, aprendizaje automático y métodos basados en lógica difusa. Sin embargo, estos modelos son relativamente complicados y requieren mayores esfuerzos de cálculo que los de regresión multivariable en el día a día de los ingenieros. Además, la mayor parte de los estudios se han centrado principalmente en la resistencia a la compresión del hormigón con caucho y rara vez entran en más detalles considerando otras propiedades y tamaños de muestra. Por lo tanto, este estudio se centra en el desarrollo de modelos de regresión multivariable de hormigón con caucho, simples pero precisos, que se pueden generalizar para varias mezclas de hormigón de este tipo, considerando diferentes tamaños de muestra.
Symbol  Unit  Description 

RBC    Rubberized concrete 
RMSE    rootmeansquare error 
MAPE    mean absolute percentage error 
R_{m}    Rubber replacement method 
R_{c}  (kg)  Rubber content 
R_{p}  (%)  Rubber replacement percentage 
ρ  kg/m^{3}  The hardened density of concrete 
ρ_{c}  kg/m^{3}  The hardened density of control concrete 
ρ_{r}  kg/m^{3}  The hardened density of RBC 

(MPa)  Compressive strength of concrete at 28 days 

(MPa)  Compressive strength of control concrete at 28 days 

(MPa)  Compressive strength of RBC at 28 days 

(MPa)  Flexural strength of concrete 
E_{s}  (GPa)  Static modulus of elasticity 
E_{d}  (GPa)  Dynamic modulus of elasticity 
Nowadays, the disposal of old rubber tires into landfills represents a severe environmental hazard worldwide (
The final dataset,
Author/s  Rubber  Concrete Properties  

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indeed, several obstacles can be encountered when it comes to proposing a generalized regression model for RBC mixtures with fairly controlled errors due to the variation in the type of binding materials being used, the concrete mixture compositions, and inconsistency in using a specific rubber replacement method. In fact, the effects of the cement type and the concrete mixture composition can be directly seen on the control specimen’s properties when comparing two types of concrete together. Thus, to take the influence of these factors into account, this study uses a parameter that can reflect the variation of the mix design on the properties of the concrete as an input to the regression model.
Accordingly, the main aim of using the property of the control concrete as input to the regression model is to account for the influence of the concrete mixture properties (i.e., water/cement ratio, cement type, the addition of chemical and mineral admixtures, density of natural aggregates as well as its particle size distribution). This technique is expected to improve the outcomes of the prediction and allow the generalization of the developed models. In this regard, the models would be used to design rubberized concrete mixture and optimize the rubber content into a given concrete mix proportion. On the other hand, to overcome the problem of the inconsistency in the rubber replacement method where some studies replaced the natural stone particles by volume while others did it by weight, this study uses the rubber content represented by the weight of the rubber in kg the mixture as input to the proposed models. Moreover, this study does not consider the influence of the rubber particle size distribution when developing the regression equations. The main aim behind this comes from the inconsistency of the literature in that matter. For instance, some papers have used singlegraded rubber particles while other utilized wellgraded ones. Additionally, some used a range of 2 mm to 4 mm, while others had 6 mm to 12 mm in a single concrete mixture. This issue has caused a huge difficulty in numerically incorporating the influence of the rubber particle size on strength reduction. On the other hand, to overcome this issue and prevent any limitation in the model to specific rubber particle size distribution, it is suggested to use data for fine and coarse rubber aggregates in which the proposed models can be generalized for a wide range of cases at the cost of for slightly higher distortion in the results.
in general, multivariable linear regression analysis is a statistical method used to model the relationship between multiple independent variables an output response. The mathematical model used in this approach is given in Equation [
where y represent the list observations on the dependent variable;
On the other hand, the performance of the multivariable linear regression can result in an underfitting of the data to its nature of simplicity. For such purposes, engineers and scientists tend to use nonlinear empirical formulations that contain multiple coefficients in line with the input variables to reach a high predictability level. These equations are not necessarily systematic such as the multivariable linear regression ones but rather are prepared on a casebycase routine to fit the given dataset. Indeed, various types of empirical formulation are available in the literature where they include polynomial functions while others were exponential and even logarithmic.
To optimize the constants in the function, multiple methods were adopted in the literature, such as the leastsquares method (
In this study, the nonlinear leastsquares procedure was adopted to achieve the optimal solution for each of the developed models.
Let’s consider an arbitrary empirical equation given in Equation [
where a, b, and c are the parameters to be fitted and
In a matrix form, this
where
The nonlinear leastsquares procedure tries to solve the objective function in Equation [
Once the empirical models are developed, it is important to estimate errors. This is usually done by performing a residual analysis. The residuals of an
Additionally, the goodnessoffit or proportion of the variance of each developed model can be represented by the coefficient of determination (Equation [
where
in fact, measuring the accuracy of a particular prediction modeling is not an easy task. Currently, several methods for calculating the error are available. In this study, rootmeansquare error (RMSE), Equation [
where
the descriptive statistics of the data used for proposing the hardened density model are shown in
Statistics  R_{c} 



Sample size  102  102  102 
Mean  111.8  2412.4  2215.2 
Standard error  12.9  11.1  31.1 
Standard deviation  129.9  112.5  314.1 
Minimum  7  2213.7  1086.9 
First quartile  36.2  2274.8  2161.5 
Median  63.5  2421.3  2324.8 
Third quartile  129.7  2508.9  2407.7 
Maximum  609.5  2546.7  2524 
This is also indicated in the prediction model’s performance against the measured values,
In fact, due to the significant difference in the rubber aggregates’ weight compared to the natural ones, RBC can have the density of normal concrete or lightweight ones based on the amount of natural aggregates being replaced. Thus, the capability of this model in estimating both normal and lightweight RBC mixtures can be seen in
The analysis of the error in this model can be seen in
Error Type  Model Performance 

RMSE  99.67 
MAPE (%)  3.37 
in this section, a simplified model for predicting the compressive strength of concrete based on a very wide range of rubber content and different types of rubber replacement methods, including replacement by fine, coarse, or total natural aggregates. Moreover, the model can be used to predict the compressive strength of concrete with different specimen sizes. The sample size of the dataset used was 174 RBC results, with the descriptive statistics shown in
Statistics  R_{c} 



Sample size  174  174  174 
Mean  82.35  54.14  33.3 
Standard error  6.43  1.23  1.04 
Standard deviation  84.86  16.28  13.72 
Minimum  6.95  27.62  6.94 
First quartile  35.04  39  22.51 
Median  55.4  56.1  31.5 
Third quartile  80.57  63.2  42.38 
Maximum  490.3  104.8  70.5 
The mathematical expression of the estimation model is given in Equation [
In general, it can be seen that the proposed regression model provides reasonably high performance in predicting the compressive strength of rubberized concrete with different specimen sizes, as presented in
The error analysis in this section is shown in
Error Type  Model Performance 

RMSE  6.63 
MAPE  20.28 
the static modulus of elasticity is indeed a critical parameter for the analysis and design of reinforced concrete structures. Currently, several relationships for estimating this parameter based on the 28 days compressive strength of concrete are available in design codes. In this section, the capability of ACI 318 (
Statistics  100 Cube  150 Cube  Cylinders  

Input  Output  Input  Output  Input  Output  
Sample size  55  55  31  31  55  55 
Mean  38.33  23.831  37.58  24.87  30.68  26.89 
Standard error  1.88  0.534  3.36  1.62  2.19  1.21 
Standard deviation  13.91  3.958  18.68  9.02  16.21  8.97 
Minimum  16.4  14.5  15.32  11.15  6.94  10.58 
First quartile  28  20.9  22.49  18.9  20.6  21.14 
Median  35.85  23.6  34.95  24.32  29.7  25.37 
Third quartile  47  26.82  48.58  28.67  38.11  32.68 
Maximum  75.3  32.4  104.8  56.86  98.25  56.86 
The collected static modulus of elasticity findings in this study were all from tests on either 100 mm and 150 mm cylinders so that these specimens follow ASTM C192 (
ACI 318 (
ACI 363 (
The prediction models are given in Equation [
where
Specimen Size  α 

100 Cube  3.889 
150 Cube  4.245 
Cylinders  5.075 
A comparison between the codebased prediction approaches and the proposed one for cylinder specimens can be seen in
The error analysis of the investigated models is given in
Specimen Size  Error Type  Proposed Model  ACI 318  ACI 363 

100 Cube  RMSE  1.91     
MAPE  6.72      
150 Cube  RMSE  4.49     
MAPE  14.27      
Cylinders  RMSE  3.44  4.02  5.32 
MAPE  10.68  10.28  13.92 
A prediction model for the flexural capacity of RBC based on its compressive strength is investigated in this section. In similar to the previous section, three mathematical models were developed based on the specimen size of the compressive strength test. The descriptive statistics of the utilized datasets for fitting the models can be seen in
Statistics  100 Cube  150 Cube  Cylinders  

R_{c} 


R_{c} 


R_{c} 



Sample size  52  52  52  21  21  21  15  15  15 
Mean  38.8  37.5  5.48  38.9  37.7  3.49  175  28  2.72 
Standard error  4.48  2.09  0.17  6.21  2.9  0.14  45.8  3.56  0.27 
Standard deviation  32.3  15.1  1.21  28.5  13.3  0.62  177  13.8  1.04 
Minimum  0  17  3.7  0  15.3  2.59  0  6.94  1.06 
First quartile  14.7  25  4.49  19.9  26.4  2.89  39.5  19  1.76 
Median  36.8  33.6  5.24  39.8  37.5  3.46  123  27.9  2.57 
Third quartile  50.9  46.6  6.22  59.7  49.5  4.06  362  36.2  3.47 
Maximum  132  71  8.42  79.6  62.2  4.39  490  61  4.5 
In fact, ACI 318 design code provides an equation for estimating the flexural strength of concrete. This equation requires defining a coefficient to take the density of normal weight concrete or the equilibrium density of the lightweight one into account. In section 3.1, it was shown that RBC density can vary significantly based on the utilized rubber content and in some cases, RBC mixtures reach as low as almost 1100 kg/m^{3} even though the control concrete has a normal weight. Thus, to take the influence of rubber aggregates on the unit weight of concrete into consideration, this study suggests using the rubber content as an input in the prediction equation. The proposed models are given in Equation [
It can be seen from
Furthermore, the error analysis presented in
Specimen Size  Error Type  Model Performance 

100 Cube  RMSE  0.64 
MAPE  7.56  
150 Cube  RMSE  0.14 
MAPE  3.02  
Cylinders  RMSE  0.43 
MAPE  15.99 
In fact, concrete testing approaches are be divided into destructive and nondestructive ones. Generally, destructive tests provide more reliable results as compared to nondestructive ones. However, the required procedure is rather long and expensive and will slightly or entirely damage the concrete member (
Goulias & Ali (
Statistics  R_{c}  E_{d} 


Sample size  99  99  99 
Mean  58.41  28.092  36.19 
Standard error  5.02  0.848  1.44 
Standard deviation  49.95  8.438  14.32 
Minimum  0  11.3  10 
First quartile  15.9  21.7  25.24 
Median  47.5  28.5  33.85 
Third quartile  80.1  34.65  45.75 
Maximum  227.3  44.6  75.3 
In fact, to produce good nondestructive models for RBC, it is suggested to use both the dynamic modulus of elasticity and the mixture’s rubber content as inputs to the prediction equations. The proposed models for each specimen size are shown in Equation [
The R^{2} values are 0.797, 0.948, and 0.923 for the 100 cube, 150 cube, and cylinders models respectively. In addition, the performance of the models is shown in
A comparison between the proposed model and the one suggested by Goulias & Ali (
The error analysis of these models is given in
Specimen Size  Error Type  Model Performance  Goulias & Ali 

100 Cube  RMSE  6.55   
MAPE  15.93    
150 Cube  RMSE  2.40   
MAPE  9.45    
Cylinders  RMSE  1.98  9.49 
MAPE  6.48  29.55 
another common mathematical expression in nondestructive testing is the one between dynamic and static moduli of elasticity. Such an approach was previously discussed in the literature for the case of conventional concrete (
Lydon & Balendran (
BS 81102 (
Goulias & Ali (
Statistics  R_{c}  E_{d}  E_{s} 

Sample size  78  78  78 
Mean  48.14  27.644  24.218 
Standard error  4.89  0.931  0.449 
Standard deviation  43.18  8.219  3.966 
Minimum  0  11.54  17.9 
First quartile  15.9  21.7  21.3 
Median  47.5  27.1  23.8 
Third quartile  65.97  34.387  26.863 
Maximum  227.3  44.6  33.2 
Similar to the previous section, both the dynamic modulus of elasticity and the rubber content were used as inputs to the proposed prediction model, Equation [
A comparison between the capability of the proposed model and the previously introduced ones are shown in
The error analysis,
Model Performance  BS 81102  Lydon & Balendran  Goulias & Ali  

RMSE  1.89  11.41  4.49  9.89 
MAPE  6.14  41.42  17.05  36.24 
this section is intended to propose a relationship to estimate the flexural strength of RBC using its dynamic modulus of elasticity. The descriptive statistics of the dataset are shown in
Statistics  R_{c}  E_{d} 


Sample size  15  15  15 
Mean  65.6  34.98  5.68 
Standard error  11.2  1.57  0.473 
Standard deviation  43.3  6.07  1.831 
Minimum  0  24.1  3.185 
First quartile  44.1  30  3.47 
Median  50.3  34.11  6.115 
Third quartile  88.1  39.84  7.45 
Maximum  132.2  44.6  8.42 
The rubber content and the dynamic modulus of elasticity were used in the estimation model given in Equation [
Moreover, the error analysis,
Error Type  Model Performance 

RMSE  1.12 
MAPE  15.54 
This study has focused on investigating the effect of rubber aggregates including its hardened density, compressive and flexural strengths, static and dynamic moduli, damping ratio, and natural frequency of RBC using a high number of experimental observations, observe the correlation between the reduction in the compressive strength of concrete and change in the other mechanical and dynamic properties of RBC, and to propose some prediction models for this type of concrete. Overall, about 1000 experimental observations were used in the numerical investigations. This study proposed several prediction models, including some relationships for nondestructive testing. It was shown that these equations could be used reliably to predict the hardened density, mechanical, and dynamic properties of RBC. Moreover, based on the comparative study that was conducted in some sections to evaluate the capabilities of the proposed models against the available ones, it was clear that the suggested estimation methods provide better performance in the case of RBC. Further research is still needed in the field of RBC to investigate the influence of mixing different types of recycled aggerates with rubber particles green concrete with an enhanced vibration behavior and provide a solid understanding of the performance of this material when used in structural applications especially in earthquakeprone counties.
Conceptualization: A. Habib, U. Yildirm. Methodology: A. Habib, U. Yildirm. Formal analysis: A. Habib. Writing, original draft: A. Habib. Writing, review & editing: U. Yildirm.Visualization: A. Habib.