Simplified modeling of rubberized concrete properties using multivariable regression analysis

2.1. Collection of rubberized concrete properties

The final dataset, Table 1, was based on about 1000 experimental findings from 28 papers that discussed the performance of rubberized concrete. Accordingly, the content of natural fine and coarse aggregates, rubber replacement method, rubber content, and rubber replacement ratio were collected from each paper in addition to the reported mechanical and dynamic properties of the produced concrete. In general, the concrete mixtures that were considered in this study represented conventional concrete or concrete with silica fume and/or fly ash, while those with fibers were ignored. Moreover, during the database generation stage, issues such as the size of the test specimen, concrete mixture’s age at testing, and repetition of the same findings in other papers/reports were considered carefully to prevent any problem during the analysis.

2.2. Prediction strategy

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 single-graded rubber particles while other utilized well-graded 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.

2.3. Empirical formulation and regression analysis

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 [1] (4949. Achen, C.H. (1982) Interpreting and using regression, Vol. 29, Sage.
):

where y represent the list observations on the dependent variable; x₁ , …,x _k are the independent variables; β₀ is an intercept term; β₁ , …, β_x are the coefficients to be estimated; and ε is a list of random errors, also known as the residual.

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 case-by-case 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 least-squares method (5050. Cui, W.; Mansour, A.E. (1998). Effects of welding distortions and residual stresses on the ultimate strength of long rectangular plates under uniaxial compression.Mar. Struct. 11. 251-269
), nonlinear least-squares procedure (5151. Hu, B.; Cui, A.; Cui, K.; Liu, Y.; Li, J. (2021) A novel nonlinear creep model based on damage characteristics of mudstone strength parameters. Plos one. 16, e0253711. https://doi.org/10.1371/journal.pone.0253711.
), and sequential least squares programming algorithm (5252. Kraft, D. Algorithm 733: (1994) TOMP-Fortran modules for optimal control calculations. ACM Trans. Math. Softw. 20, 262-281.
, 5353. Alibrahim, B.; Uygar, E. (2021) Nonlinear calculation method for one-dimensional compression of soils. Arab. J. Sci. Eng. 47, 4865-4877. https://doi.org/10.1007/s13369-021-06270-7.
).

In this study, the nonlinear least-squares procedure was adopted to achieve the optimal solution for each of the developed models.

Let’s consider an arbitrary empirical equation given in Equation [2]:

where a, b, and c are the parameters to be fitted and x₁ and x₂ are the input variables.

In a matrix form, this equation can be written as follows:

where y is the list of observations on the dependent variable; X are the independent variables; β is an intercept term to represent the list of coefficients to be optimized, and ε is a list of residuals.

The nonlinear least-squares procedure tries to solve the objective function in Equation [4] to minimize the sum squared residuals.

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 i^th observation is defined in Equation [5]:

Additionally, the goodness-of-fit or proportion of the variance of each developed model can be represented by the coefficient of determination (Equation [6]):

where y_i is the actual value, ${\widehat{y}}_i$ is the predicted one, and $\stackrel{-}{y}$ is the mean of the actual values.

2.4. Error analysis

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, root-mean-square error (RMSE), Equation [7], and mean absolute percentage error (MAPE), Equation [8], will be used to evaluate the estimation models.

where y_i is the actual value, ${\widehat{y}}_i$ is the predicted one, and n is the number of observations.

3.1. Prediction of hardened density

the descriptive statistics of the data used for proposing the hardened density model are shown in Table 2. The constructed mathematical expression in this section is generally shown in Equation [9], and its performance is investigated in Figure 1. Generally, the R² value of this model is equal to 0.898, which gives a good sense of accuracy in prediction.

This is also indicated in the prediction model’s performance against the measured values, Figure 1-a, and the difference between the measured values and the predicted ones (residual), Figure 1-b.

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 Figure 1-c. It can be noticed that the model provided very high accuracy in predicting the density of RBC up to 2200 kg/m³, whereas a lower accuracy is faced when the density goes below that reaching a residual value of almost 150 kg/m³. This point can also be observed from the box plots provided in Figure 1-c, in which the high side of the predicted box has better matching to the measured values compared to the lower one. This reduced capabilities in the model when it comes to lightweight concrete can be attributed to the lesser data available on RBC density with a considerably high amount of rubber that can result in a significant density reduction.

The analysis of the error in this model can be seen in Table 3 in which the standard deviation of the residuals is almost 100 kg/m³ and the MAPE was 3.37%. Thus, this model can be reliably used for predicting the hardened density of RBC.

3.2. Prediction of compressive strength

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 Table 4.

The mathematical expression of the estimation model is given in Equation [10], and the R² value of this model was 0.766, whereas its fitting performance was analyzed in detail, as shown in Figure 2:

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 Figure 2-a and Figure 2-c. On the other hand, the residuals of the estimated values, Figure 2-b, show that the error in the model’s prediction is not dependent on the specimen type or size. Furthermore, the box plots of the measured and predicted values, Figure 2-c, depict that this model could estimate the general trends and statistical properties of the dataset.

The error analysis in this section is shown in Table 5. It can be seen that the MAPE value was about 20.28% which represents that this model can be reliably used for predicting the compressive strength of RBC mixtures.

3.3. Prediction of static modulus of elasticity

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 (5454. ACI318. (2019) ACI 318-19: Building code requirements for structural concrete and commentary; American Concrete Institute: Farmington Hills, USA.
), Equation [11], and ACI 363 (5555. ACI363. (2010) ACI 363R-10 Report on high-strength concrete; American Concrete Institute: Farmington Hills, USA.
), Equation [12], methods in the case of RBC will be investigated, and enhanced ones will be proposed. The aim behind using different ACI models is that the collected dataset ranges between normal and high strength concrete, which means there might be a lack of predictability in one of the models. Unlike previous models with multiple inputs, the proposed equation in this section for the modulus of elasticity of RBC is developed in a similar form to the ACI standard to provide an alternative fit-for-purpose method as given in the US codes practice but with better performance for RBC mixtures. Moreover, the descriptive statistics of the datasets used in this analysis are shown in Table 6.

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 (5656. ASTM. (2019) ASTM C192 Standard practice for making and curing concrete test specimens in the laboratory.
) requirements. Generally, ACI 318 (5454. ACI318. (2019) ACI 318-19: Building code requirements for structural concrete and commentary; American Concrete Institute: Farmington Hills, USA.
) and ACI 363 (5555. ACI363. (2010) ACI 363R-10 Report on high-strength concrete; American Concrete Institute: Farmington Hills, USA.
) explicitly permit the compressive strength test to be performed on cylinders of either 100 mm diameter or 200 mm diameter. Accordingly, these two sizes were defined as one category in developing the database of this section so that the prediction models in the design codes can be applied correctly. Therefore, three different models for predicting the static models of elasticity of concrete based on the size of the compressive strength test were used herein, which are the 100 mm cubic specimen, 150 mm cubic one, and cylinders both of 100 mm and 150 mm diameters where the later one was compared to the code-based estimation equations.

ACI 318 (5454. ACI318. (2019) ACI 318-19: Building code requirements for structural concrete and commentary; American Concrete Institute: Farmington Hills, USA.
)

ACI 363 (5555. ACI363. (2010) ACI 363R-10 Report on high-strength concrete; American Concrete Institute: Farmington Hills, USA.
)

The prediction models are given in Equation [13], and their performances are investigated in Figure 3. In general, similar estimation capabilities are expected when using any of the proposed models since their R² values are close to each other, especially those with cubic specimens. A comparison between the measured and predicted values could be seen in Figure 3-b and Figure 3-c. Furthermore, the residuals plot is provided in Figure 3-d. It can be seen that the prediction models provide acceptable accuracy. Moreover, the regression line in Figure 3-b and the box plot, Figure 3-e, compares the capability of the prediction models in terms of the entire dataset with mixed sizes. It can be seen that the models have achieved an excellent matching to the reference dataset and were able to estimate its behavior reliably.

where E_s is the modulus of elasticity of concrete, f_c28 is the 28 days compressive strength of concrete, and a is a coefficient related to the size of the compressive strength specimen as shown in Table 7.

A comparison between the code-based prediction approaches and the proposed one for cylinder specimens can be seen in Figure 4. Generally, it can be observed that the ACI 363 (5555. ACI363. (2010) ACI 363R-10 Report on high-strength concrete; American Concrete Institute: Farmington Hills, USA.
) method provided the least capability in predicting the static modulus of elasticity of the concrete mixtures. On the other hand, both the proposed model and the ACI 318 (5454. ACI318. (2019) ACI 318-19: Building code requirements for structural concrete and commentary; American Concrete Institute: Farmington Hills, USA.
) give a similar behavior, with the proposed one having slightly better performance and lower residual values. This can be further highlighted from scatter plots in Figure 4-e the box plots, Figure 4-f, where the proposed approach provided the best matching as compared to the measured values.

The error analysis of the investigated models is given in Table 8. On the other hand, the proposed model provided the lowest errors as compared to ACI 318 and ACI 363 except for the MAPE, where the ACI 318 had a slightly lower value. This can be attributed to the fact that the MAPE measures the average error which means that even if most of the predicted values have small residuals, an error in a certain measurement would influence the MAPE value significantly due to the lack of weight-based computation in this method.

3.4. Prediction of flexural strength

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 Table 9.

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/m³ 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 [14] to Equation [16]. The R² values were calculated as 0.712, 0.947, and 0.817 for the 100 cube, 150 cube, and cylinders equations. Moreover, the performances of these estimation formulas are investigated in Figure 5.

It can be seen from Figure 5-a, Figure 5-b, and Figure 5-c that the proposed models provide high performance with generally small residual values regardless of the size of the specimen being tested in compression.

Furthermore, the error analysis presented in Table 10 indicates that the models can be used reliably to estimate the flexural strength of RBC concrete.

3.5. Relation between compressive strength and dynamic modulus of elasticity

In fact, concrete testing approaches are be divided into destructive and non-destructive ones. Generally, destructive tests provide more reliable results as compared to non-destructive ones. However, the required procedure is rather long and expensive and will slightly or entirely damage the concrete member (5757. Jones, R. (1949) The non-destructive testing of concrete. Mag. Concr. Res. 1, 67-78.
). Thus, in certain cases of practical applications, engineers tend to utilize nondestructive methods for fast measurements. Previously, several researchers have discussed the prediction of compressive strength of concrete based on the dynamic modulus of elasticity. Moreover, Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
) have proposed a mathematical model, Equation [17], during the early investigations on RBC. In this section, the capability of this model will be investigated and compared against the proposed one based on the collected findings from the literature with the descriptive statistics shown in Table 11.

Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
)

In fact, to produce good non-destructive 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 [18] to Equation [20]. In general, substituting zero instead of the R_c term means that the estimation is being conducted for normal concrete.

The R² 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 Figure 6. Generally, it can be seen that the models are highly capable of predicting the compressive strength of concrete with normal to high strength capacity. However, they show some deficiency in estimating values above 60 MPa, as shown in Figure 6-a and Figure 6-c, due to the lack of more experimental data to improve the prediction model. On the other hand, the residual plot in Figure 6-b depicts that the 150 cube and cylinders equations provide lower error values as compared to the 100 cube one due to the existence of a larger dataset with various rubber content in the latter case, which resulted in the more generalized model but will lower fitting rate.

A comparison between the proposed model and the one suggested by Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
) is provided in Figure 7. In general, the proposed one provides significantly better matching capabilities, Figure 7-c, performance compared to the measured values, Figure 7-d, and reduced errors, Figure 7-e, compared to the Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
) approach. This can be attributed to utilizing a larger dataset and introducing the R_c term in the prediction equation, which has improved the fitting performance significantly.

The error analysis of these models is given in Table 12. It can be seen that the percentage of error represented by MAPE value reached its highest of 16% in the case of 100 cube which means that the proposed models can be used reliably in estimating the compressive strength of RBC mixtures. Furthermore, the errors in the proposed method were considerably lower than these of the Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
) model.

3.6. Relation between dynamic and static moduli of elasticity

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 (5959. Lydon, F.D.; Balendran, R.V. (1986) Some observations on elastic properties of plain concrete. Cem. Concr. Res. 16, 314-324. https://doi.org/10.1016/0008-8846(86)90106-7.
, 6060. BSI. (1995) BS 8110-2 Structural use of concrete-Part 2: code of practice for special circumstance; British Standard Institute: London.
) and RBC (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
). Furthermore, Habib et al. (1010. Habib, A.; Yildirim, U.; Eren, O. (2020) Mechanical and dynamic properties of high strength concrete with well graded coarse and fine tire rubber. Constr. Build. Mater. 246, 118502. https://doi.org/10.1016/j.conbuildmat.2020.118502.
) observed that there is a good correlation between the static and dynamic moduli of elasticity of RBC mixtures. In this section, the applicability of previously proposed models, Equation [21] to Equation [23], to RBC will be investigated based on the dataset with the descriptive statistics shown in Table 13, and an enhanced model will be suggested.

Lydon & Balendran (5959. Lydon, F.D.; Balendran, R.V. (1986) Some observations on elastic properties of plain concrete. Cem. Concr. Res. 16, 314-324. https://doi.org/10.1016/0008-8846(86)90106-7.
)

BS 8110-2 (6060. BSI. (1995) BS 8110-2 Structural use of concrete-Part 2: code of practice for special circumstance; British Standard Institute: London.
)

Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
)

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 [24]. Generally, the R² value of this model was computed as 0.77. Furthermore, the performance of the model is shown in Figure 8-a.

A comparison between the capability of the proposed model and the previously introduced ones are shown in Figure 8. In general, the estimation model suggested in this study provides significantly better performance than the other models. Furthermore, the residuals plot, Figure 8-e, depicts that there are very high errors in the case of BS 8110-2 (6060. BSI. (1995) BS 8110-2 Structural use of concrete-Part 2: code of practice for special circumstance; British Standard Institute: London.
) model and Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
) one as compared to the Lydon & Baledran (5959. Lydon, F.D.; Balendran, R.V. (1986) Some observations on elastic properties of plain concrete. Cem. Concr. Res. 16, 314-324. https://doi.org/10.1016/0008-8846(86)90106-7.
) model, whereas, the proposed expression gives the lowest errors among other equations. Moreover, the suggested model achieved the best matching to the reference dataset compared to others, as seen in Figure 8-f and Figure 8-g.

The error analysis, Table 14, proves that the proposed model gives the best results among others, while the errors in Lydon & Baledran (5959. Lydon, F.D.; Balendran, R.V. (1986) Some observations on elastic properties of plain concrete. Cem. Concr. Res. 16, 314-324. https://doi.org/10.1016/0008-8846(86)90106-7.
) were lower than both Goulias & Ali (5858. Goulias, D.G.; Ali, A.H. (1998) Evaluation of rubber-filled concrete and correlation between destructive and nondestructive testing results. Cem. Concr. Agg. 20, 140-144. https://doi.org/10.1520/CCA10447J.
) model and the BS 8110-2 (6060. BSI. (1995) BS 8110-2 Structural use of concrete-Part 2: code of practice for special circumstance; British Standard Institute: London.
) model. On the other hand, the highest error was observed in the case of BS 8110-2 (6060. BSI. (1995) BS 8110-2 Structural use of concrete-Part 2: code of practice for special circumstance; British Standard Institute: London.
).

3.7. Relation between dynamic modulus of elasticity and flexural strength

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 Table 15.

The rubber content and the dynamic modulus of elasticity were used in the estimation model given in Equation [25]. In fact, the model has an R² value of 0.6 and a performance, as shown in Figure 9. It can be seen from Figure 9-c that the model provides an acceptable accuracy in the prediction.

Moreover, the error analysis, Table 16, indicates that the model provides good results since its MAPE value is about 15.5%.