Compressive strength evaluation of concrete confined with spiral stirrups by using adaptive neuro-fuzzy inference system (ANFIS)

The compressive strength of concrete confined with spiral stirrups was an important parameter to evaluate the load-bearing capacity of concrete columns. The confinement provided by spiral stirrups let concrete under the triaxial compression state and improved the compressive strength of concrete. However, the relationships between concrete and stirrups were complex and the existing prediction models for evaluating the compressive strength of confined concrete were various. In this paper, an adaptive neural-fuzzy inference system (ANFIS) model was developed to evaluate the compressive strength of concrete confined with stirrups. A set of 231 experimental results of concrete confined with spiral stirrups were collected from the previous studies to establish a reliable database. The investigated parameters included the aspect ratio of specimens, the diameter, spacing, yield strength, and volumetric ratio of stirrups, the ratio of longitudinal reinforcement, and the compressive strength of concrete. The results showed that the ANFIS model predicted the compressive strength of confined concrete accurately. By comparing with existing models, the proposed ANFIS model had high applicable and reliability. The effects of the investigated parameters on the compressive strength of concrete were analyzed based on the proposed ANFIS model.


Introduction
The compressive strength of concrete was a crucial parameter to evaluate the load-bearing capacity of concrete columns. The compressive strength of concrete would be enhanced when concrete was confined with stirrups, which let concrete under triaxial compression (Nemecek et al. 2005). It was important to evaluate the compressive strength of confined concrete. The compressive strength of confined concrete was close related to the mechanical properties of stirrups (Saatcioglu and Razvi 1998). However, the relationships between concrete and stirrups were complex, which can be attributed to the tensile strength of stirrups may not reach the yield strength at the compressive strength of confined concrete (Légeron and Paultre 2003). It was not suitable for evaluating the compressive strength of confined concrete by using the tensile strength; on the other hand, the existing models for evaluating the compressive strength of confined concrete were proposed by the limited experimental datasets, which should be reevaluated when the new test data were introduced. Thus, it was necessary to establish a reliable prediction model to estimate the compressive strength of concrete confined with spiral stirrups.
The compression behavior of concrete confined with spiral stirrups has been studied over one century, and many typical prediction models for evaluating the compressive strength of confined concrete were proposed. Richard et al. explored the compression behavior of confined concrete and found that the compressive strength of confined concrete was improved by lateral confinement of stirrups (Richart et al. 1929). Besides, they firstly proposed the prediction models for evaluating the compressive strength of confined concrete (Richart et al. 1928). Mander et al. indicated that longitudinal reinforcement and spiral stirrups both had obvious effects on the compressive strength of concrete based on the experimental results and theoretical analysis (Mander et al. 1988a) and the analytical model for predicting the compressive strength of confined concrete by considering the effects of longitudinal reinforcement and stirrups based on the ''Mohr-Coulomb criteria'' (Mander et al. 1988b). Cusson proposed the concept of ''confinement index,'' which was the ratio between the lateral confining stress and compressive stress of unconfined concrete, to evaluate the compressive strength of confined concrete (Cusson and Paultre 1994). Legeron proposed a prediction model for evaluating the compressive strength of confined concrete based on the theoretical analysis and a large experimental results (Légeron and Paultre 2003). Bing found that the high strength of spiral stirrups can significantly improve the compressive strength of confined concrete (Bing and Park 2001). Saatcioglu and Razvi indicated that the lateral confining stress decreased exponentially with the increase of concrete compressive strength (Saatcioglu and Razvi 1992). Sharma thought that the load-bearing capacity of confined concrete columns would be decreased with the increase of the compressive strength of concrete, while the volumetric ratio and spacing of stirrups had obvious influence on the compressive behavior of confined concrete (Sharma et al. 2005). Wang found that the compressive strength of confined concrete increased with the increase of the yield strength of stirrups, while the volumetric ratio of stirrups also had the same effects (Wang 2014). Wei demonstrated that the high strength steel wire can effectively improve the compressive strength of confined concrete (Wei and Wu 2014a). Cao found that the high strength stirrups had insignificant effects on the load-bearing capacity of confined concrete (Li et al. 2018). Deng et al. found that high strength stirrups had superiority in improving the compressive strength of confined concrete compared with normal strength stirrups (Deng and Yao 2020). Based on the previous studies, the effects of the influential parameters on the compressive strength of confined concrete only considered the yield strength and the volumetric ratio of stirrups and the compressive strength of concrete. Moreover, the existing prediction models were established based on a specific set of experimental results; when new test results were introduced, the applicability and reliability of models should be re-evaluated. Moreover, some existing models with numerous variables and complex computational processes were difficult to utilize in practical civil engineering design.
In recent years, adaptive neural-fuzzy inference system (ANFIS) combined with both learning and reasoning capability of artificial neural network and fuzzy logic has been developed for solving the complex problems (Yalpir and Ozkan 2018). Akbarpour proposed an ANFIS model to predict the punching shear strength of two-way slabs based on 189 experimental results and found that the proposed model can evaluate the punching load with an acceptable error (Akbarpour and Akbarpour 2017). Khademi et al. (2016) and Bilgehan (2011) predicted the 28-day compressive strength of recycled aggregate concrete by using ANFIS with 14 different input parameters and indicated that the ANFIS model was suggested to be used in the mix design optimization and be utilized for preliminary mix design of concrete, respectively. Vakhshouri designed ANFIS models to establish the relationships between the compressive strength of self-compacting concrete and mixture proportions and slump flow and indicated the proposed models gave the best prediction of the compressive strength (Vakhshouri and Nejadi 2018). Based on the previous studies, the ANFIS had high prediction performance and good reliable for predicting the mechanical properties of concrete. Thus, ANFIS was suitable for evaluating the compressive strength of concrete confined with stirrups.
The objective of this paper was to evaluate the compressive strength of concrete confined with spiral stirrups. To achieve this purpose, 231-group experimental results of concrete confined with spiral stirrups were collected from previous studies to establish a reliable database. Based on the database, an ANFIS model was developed to evaluate the compressive strength of confined concrete. Furthermore, the effects of the influential parameters on the compressive strength of confined concrete were analyzed.

Database preparation
To establish the ANFIS model for evaluating the compressive strength of concrete confined with spiral stirrups, a reliable database consisting of 231 group of experimental results of concrete columns confined with spiral stirrups gathered from the previous studies was established, listed in Table 1 attached in appendix (Mander et al. 1988a;Sakai 2001;Sakai et al. 2000;Antonius. 2014;Assa et al. 2001;Bing 1993;Yang et al. 2016;Cusson et al. 1998;Montgomery 1996;TokIucu 1992;Sheikh and Toklucu 1993;Silva 2000;Wang et al. 2013Wang et al. , 2017Razvi 1995;Wei and Wu 2014b;Marvel et al. 2014). To ensure the reliable database, the collected data should obey the following criteria: (1) all specimens had one-layer spiral stirrup; (2) the aspect ratio of all specimens was no more than 8 to avoid the bucking failure; (3) all specimens were tested under monotonically concentric loads.
The compressive strength of concrete confined with spiral stirrups was related to the aspect ratio of specimens, the diameter, spacing, volumetric ratio, and yield strength of spiral stirrups, the ratio of longitudinal reinforcement, and the compressive strength of concrete. The range of influential parameters is listed in Table 2, and the distribution of the influential parameters is shown in Fig. 1. 3 Adaptive neuro-fuzzy inference system (ANFIS)

Conception
Adaptive neuro-fuzzy inference system (ANFIS) integrated the advantages of both neural networks and fuzzy logic systems with high self-adaptability and self-learning ability to be identified as a universal estimator for responding to complex problems (Yalpir and Ozkan 2018). The ANFIS was a class of adaptive, multi-layer, and feed-forward networks which was comprised of input-output variables and a fuzzy rule base of the Takagi-Sugeno type (Jiang 1993). The ANFIS model incorporated the human-like reasoning style of fuzzy inference system through the use of input-output sets and a linguistic model consisting of a set of IF_THEN fuzzy rules, which was expressed as following: Rule 1 IF x was A 1 and y was B 1 , THEN f 1 = p 1 x ? q 1 y ? r 1 ; Rule 2 IF x was A 2 and y was B 2 , THEN f 2 = p 2 x ? q 2 y ? r 2 ; The principle structures of ANFIS model were consisted of fiver layers, including input layer, input membership function layer, rule layer, output membership function layer, and output layer (Assa et al. 2001). The framework of different layers was different with each other, while the nodes of the same layer performed similarly to each other. The principle structures of ANFIS are shown in Fig. 2.
Input layer: the influential parameters determined the number of nodes of the input layer. If the number of input variables was N, the number of input layer nodes was N.
Layer1 the membership function in this layer can fuzzy the input variables. Every node i in this layer was a square node with a membership function. The membership function is expressed in Eq. (1).
in which x was the input variables, A i was the fuzzy sets; O 1 i was the subordinative function values of A i , which represented the degree belonging to A i ; l A i x ð Þ was the membership function.
Layer2 the regular strength release layer. The nodes in this layer were responsibility for multiplying the input signals from the previous layer; meanwhile, the outputs of each node represented the credibility of the rules. The outputs of Layer2 are described in Eq. (2).
where x i was the outputs of the Layer2.
Layer3 the normalization layer of rules. The i th node calculates the ratio of the i th rule firing strength to the sum of all rule firing strength. The outputs of Layer3 are shown in Eq. (3).
where x i was the outputs of the Layer3 which is also called normalized firing strength.
Layer4 calculating the outputs of fuzzy rules. Each node was an adaptive node and the output is given in Eq. (4). where L=D was the aspect ratio of specimens, L was the height of specimens, and D was the diameter of specimens; d was the diameter of stirrups; s was the spacing of stirrups; f y was the yield strength of specimens; q sv was the volumetric ratio of specimens; q s was the ratio of longitudinal reinforcement; f c was the compressive strength of concrete; f cc was the compressive strength of confined concrete. where O 4 i was the output of Layer4; x i was the output of Layer3; p i ; q i ; r i f gwere parameter sets of nodes in Layer4. Layer 5 only had one fixed node to calculate the sum of total input signals. The total output of Layer5 is shown in Eq. (5).
The O 5 i also can be described as a linear combination of parameter sets of nodes in Layer4.
Output layer: the target values were obtained in this layer.

ANFIS model establishment
In this section, the ANFIS model for evaluating the compressive strength of concrete confined with spiral stirrups was established. The proposed ANFIS model was consisted  of seven input parameters including the aspect ratio of specimens, the diameter, spacing, yield strength, and volumetric ratio of stirrups, the ratio of longitudinal reinforcement, and the compressive strength of concrete and one output variable (the compressive strength of confined concrete). The four membership functions including the triangular, trapezoidal, Gaussian, and P-shape were selected to construct the proposed ANFIS model, named as ANFISI, ANFISII, ANFISIII, ANFISIV, to obtain the suitable membership function, while the membership function in output layer was constant. The ANFIS mode was trained by 185-group data sets and tested by 46-group data sets, which were selected randomly. Up to 100 epochs were specified for the training process to obtain the minimum error tolerance. Furthermore, hybrid learning procedure which combined back-propagation gradient descent and least squares method for identification of premise and consequent parameters was adopted to establish the ANFIS model.
The performance of ANFIS model was examined by RMSE and R2 which are listed in Table 3. In Table 3, all ANFIS models had acceptable prediction performance. Among those models, ANFISIII constructed with Gaussian member function exhibited the best prediction performance, which the RMSE was 0.9986 and 0.9026 and the R 2 was 0.9986 and 0.9026 for training and testing phases. Thus, the ANFIS model constructed with Gaussian membership function was determined and the structures of the proposed ANFIS model are shown in Fig. 3. Figure 4 presents the comparison between the predicted results of the proposed ANFIS model and experimental results. In Fig. 4, the prediction results from the proposed ANFIS model were well matched the experimental results in training and testing, which meant that the proposed models had good reliable and high performance for evaluating the compressive strength of concrete confined with stirrups.

Results assessment criteria
A successfully trained ANFIS model should give an accurate output prediction, not only for training process but also for new testing data. In this study, four assessment indicators were applied to evaluate the prediction performance of the proposed ANFIS model, which were the rootmean-square error (RMSE), coefficient of determination (R 2 ), integral absolute error (IAE), and a20 À index [44], which are expressed in Eqs. (6-9).

RMSE
where n was the total number of samples; t k was the experimental result of k th data; o k was the output value of Légeron and Paultre (2003) f where f cc was the compressive strength of confined concrete; f c was the compressive strength of concrete; q sv was the volumetric ratio of stirrups; f yv was the yield strength of stirrups; k e was the effective confinement coefficient; s was the spacing of stirrups; D was the diameter of specimens; q s was the ratio of longitudinal reinforcement; k 1 was the parameter related to the yield strength and the volumetric ratio of stirrups.
k th data; n20 was the number of samples which the values of the f cc;predicted =f cc;experiment were in the range of 0.8-1.2. The value of R 2 was applied to evaluate the variation between predicted results and experimental results. The value of RMSE and IAE was applied to evaluate the errors between predicted results and experimental results. The value of a20 À index was applied to evaluate the number of the predicted results falling in a deviation of compared with experimental results. Theoretically, the higher R 2 and lowest RMSE and IAE indicated the good prediction performance of proposed models, while a20 À index was expected to 1 in the perfect prediction models.

Existing prediction models for compressive strength of confined concrete
In this section, four existing models proposed by Richart et al. (1928), Mander et al. (1988b), Saatcioglu ( 1992), and Legeron ( 2003) for predicting the compressive strength of confined concrete were reviewed and are listed in Table 4. In Table 4, the existing models for evaluating the compressive strength of concrete confined with stirrups were proposed by considering the effects of the compressive strength of concrete, the volumetric ratio, and the yield strength of stirrups, which ignored the effects of the aspect ratio of specimens, the spacing and diameter of spiral stirrups, and the ratio of longitudinal reinforcements. On the other hand, those existing models were proposed based on a set of specific experimental results; when the new test data sets were introduced, the models may not performed well. Figure 5 shows the comparison of compressive strength of concrete confined with stirrups of existing and proposed prediction models (listed in Table 4) and experimental results. The 45 degree line indicated the perfect predicted results, equaling the experimental results. It was obvious that the compressive strength of concrete confined with stirrups calculated from the all existing models was lower than the experiment results, which can be attributed to the effects of concrete strength on the shear strength of concrete that were underestimated and the ultimate shear strength of concrete fluctuated heavily; the upper limited value of the compressive strength of concrete confined with stirrups was derived as the lower limited value of experimental results in existing models. The predicted results from the proposed models were also compared with experimental results. In Fig. 5, the scatter of proposed models in this study approximated to the experimental results. Moreover, Fig. 6 shows the histograms of the proposed models, demonstrating the good distribution with the mean values of unity. Figure 7 shows the box plot of f cc;predicted f cc;exp eriment with different prediction models, and Table 5 lists the performance indicators of different models. It was highlighted that the proposed models were suitable for evaluating the compressive strength of concrete confined with stirrups due to the lowest RMSE, IAE, and SD and the unity of mean values.  Compressive strength evaluation of concrete confined with spiral stirrups by using adaptive… 11879

Parameter analysis
In this study, the effects of the investigated parameters including the aspect ratio of specimens, the diameter, spacing, yield strength, and volumetric ratio of stirrups, the ratio of longitudinal reinforcement, and the compressive strength of concrete were performed based on the proposed ANFIS models. The mean values (listed in Table 2) of the investigated parameters were set as the basic values, while the certain parameter was varied from the minimum value to the maximum value. The outputs were obtained from the proposed ANFIS model.  Figure 8 shows the relationships between the compressive strength of confined concrete and the aspect ratio of specimens. The compressive strength of confined concrete increased firstly and then decreased with the increase of the aspect ratio of specimens.  Figure 10 presents that the compressive strength of confined concrete varied with the spacing of stirrups. The increase of the spacing of stirrups declined the effective confinement area of specimens and the longitudinal reinforcement tended to be bucking failure, which decreased the compressive strength of confined concrete.

Yield strength of stirrups
In Fig. 11, the increase of the yield strength of stirrups enhanced the compressive strength of confined concrete. However, when the yield strength of stirrups exceeded 700 MPa, the compressive strength of confined concrete increased slowly.

Volumetric ratio of stirrups
In Fig. 12, the increment of the volumetric ratio of stirrups improved the compressive strength of confined concrete. The increment of the volumetric ratio of stirrups enhanced the lateral confinement of stirrups, which improved the compressive strength of confined concrete. Figure 13 shows that the compressive strength of confined concrete varied with the ratio of longitudinal reinforcement. When the ratio of longitudinal reinforcement was lower than 2%, the ratio of longitudinal reinforcement improved the compressive strength of confined concrete. When the ratio of longitudinal reinforcement was over 2%, the ratio of longitudinal reinforcement had the negative effects on the compressive strength of confined concrete.

Conclusions
The aim of this paper was to evaluate the compressive strength of concrete confined with spiral stirrups. A reliable database consisting of 231-group experimental results collected from previous studies was established, and the ANFIS model for evaluating the compressive strength of confined concrete was developed. The parametric analysis was performed based on the ANFIS model. The conclusions were drawn as following: Fig. 6 Histograms of the proposed models Fig. 7 Box plot of f cc;predicted f cc;exp eriment with different prediction models     The ANFIS model was adopted to predict the compressive strength of concrete confined with spiral stirrups, which has high prediction performance for both training and testing data sets. The ANFIS model proposed in this study has high reliability and high applicable in predicting the Based on the proposed ANFIS model, the effects of influential parameters including the aspect ratio of specimens, the diameter, spacing, yield strength, and volumetric ratio of stirrups, the ratio of longitudinal reinforcement, and the compressive strength of concrete on the compressive strength of confined concrete were analyzed.