# New Fuzzy-Heuristic Methodology for Analyzing Compression Load Capacity of Composite Columns

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## Abstract

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## 1. Introduction

## 2. Data Provision

_{y}), and ultimate stress (f

_{u}) of the steel tube, as well as the compressive strength of UHSC (f

_{c}’). The used data are obtained from a study by Tran, Thai, and Nguyen [33]. They created a numerical dataset as the result of an extensive finite element simulation verified with experimental efforts in the literature. The readers are guided to refer to the reference paper [33] for further details of data provision (e.g., material characteristics, assumptions of simulation, parameters, etc.).

_{y}, f

_{u}, and f

_{c}’ fall within (900.0, 4800.0) mm, (300.0, 600.0) mm, (6.0, 30.0) mm, (235.0, 460.0) MPa, (360.0, 540.0) MPa, and (100.0, 200.0) MPa, respectively. An R-value is also calculated for each chart that shows the correlation between the CC and corresponding input. As is seen, the sole meaningful relationship here is between the CC and D with R = 0.90703. Other R values are below 0.5. Moreover, Table 1 gives some statistical details of the used dataset.

_{c}’ are the most important parameters for predicting the CC, while the lowest importance is obtained for the f

_{u}and L.

_{y}, f

_{u}, and f

_{c}’) are referred to as the inputs of the network. The used data consist of 768 records that are divided into two sets after permuting their order. This is performed to achieve a random division. The first set of data, which contains 80% of the whole, is devoted to the training process. In other words, the network goes through these data to learn the mathematical relationship between the CC and inputs. The second set, which contains the remaining 20%, will be later used as non-processed data to evaluate the prediction ability of the models. This process is called the testing phase. In both the training and testing phases, the real values of CC are compared with the products of the models for accuracy evaluation.

## 3. Methodology

#### 3.1. ANFIS

#### 3.2. Metaheuristic Algorithms

## 4. Results and Discussion

_{j}and is calculated as follows:

#### 4.1. Metaheuristic Optimization

#### 4.2. Prediction Results

_{y}, f

_{u}, and f

_{c}’. Apart from the reported RMSEs, the MAEs of 3085.0588, 466.4530, 2296.0564, and 2282.5048 indicate a fine level of learning error. Figure 4 depicts the histogram of the training Err values (see Equation (4)).

#### 4.3. Comparison

_{y}, f

_{u}, and f

_{c}’. The corresponding hybrid tool, i.e., PSO-ANFIS, can be introduced as a reliable and efficient predictive model for this purpose.

## 5. Conclusions

- Metaheuristic algorithms are suitable options for training neuro-fuzzy systems for the mentioned purpose.
- Referring to the correlation values >0.96, all employed fuzzy-metaheuristic models are capable of both comprehending and generalizing the relationship between the CC and input parameters.
- The PSO algorithm emerged as the most suitable optimizer for the ANFIS. This deduction came up due to the highest accuracy, as well as the most time-efficient optimization behavior observed compared to the three other algorithms.
- The PSO-ANFIS could present a finer prediction of extremum CC values.
- In short, the use of the PSO-ANFIS is recommended for practical applications which pursue efficient cost-competitive design of CCFST columns.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

CCFST | Circular concrete-filled steel tube | CC | Compression capacity |

ANN | Artificial neural network | ANFIS | Adaptive neuro-fuzzy inference system |

BART | Bayesian additive regression tree | GA | Genetic algorithm |

ABC | Artificial bee colony | PSO | Particle swarm optimization |

EWA | Earthworm algorithm | SSA | Salp swarm algorithm |

TLBO | Teaching learning-based optimization | f_{c}’ | Compressive strength of UHSC |

L | Length of column | D | Diameter |

t | Thickness | f_{y} | Yield stress |

f_{u} | Ultimate stress of the steel tube | MF | Membership function |

RMSE | Root mean square error | R | Pearson correlation index |

MAPE | Mean absolute percentage error | MAE | Mean absolute error |

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**Figure 1.**The distribution of the CC versus influential factors (

**a**) L, (

**b**) D, (

**c**) t, (

**d**) f

_{y}, (

**e**) f

_{u}, and (

**f**) f

_{c}’ and (

**g**) a schematic view of the column and section.

**Figure 4.**Histogram of the errors obtained for (

**a**) EWA-ANFIS, (

**b**) PSO-ANFIS, (

**c**) SSA-ANFIS, and (

**d**) TLBO-ANFIS.

**Figure 5.**Correlation assessment of the testing data for (

**a**) EWA-ANFIS, (

**b**) PSO-ANFIS, (

**c**) SSA-ANFIS, and (

**d**) TLBO-ANFIS.

Indicator | Factor | ||||||
---|---|---|---|---|---|---|---|

L [mm] | D [mm] | t [mm] | f_{y} [MPa] | f_{u} [MPa] | f_{c}’ [MPa] | CC [kN] | |

Mean | 2475.0 | 450.0 | 15.2 | 331.3 | 460.0 | 150.0 | 30,185.3 |

Std. Error | 47.4 | 4 | 0.2 | 3.1 | 2.5 | 1.2 | 538.3 |

Std. Deviation | 1313.1 | 111.9 | 6.1 | 86 | 70.4 | 34.2 | 14,918.5 |

Sample Variance | 1,724,120 | 12,516.3 | 37.3 | 7401.8 | 4956.5 | 1168.2 | 222,561,708.9 |

Minimum | 900.0 | 300.0 | 6.0 | 235.0 | 360.0 | 100.0 | 8016.3 |

Maximum | 4800.0 | 600.0 | 30.0 | 460.0 | 540.0 | 200.0 | 75,051.6 |

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## Share and Cite

**MDPI and ACS Style**

Karimi Sharafshadeh, B.; Ketabdari, M.J.; Azarsina, F.; Amiri, M.; Nehdi, M.L.
New Fuzzy-Heuristic Methodology for Analyzing Compression Load Capacity of Composite Columns. *Buildings* **2023**, *13*, 125.
https://doi.org/10.3390/buildings13010125

**AMA Style**

Karimi Sharafshadeh B, Ketabdari MJ, Azarsina F, Amiri M, Nehdi ML.
New Fuzzy-Heuristic Methodology for Analyzing Compression Load Capacity of Composite Columns. *Buildings*. 2023; 13(1):125.
https://doi.org/10.3390/buildings13010125

**Chicago/Turabian Style**

Karimi Sharafshadeh, Bizhan, Mohammad Javad Ketabdari, Farhood Azarsina, Mohammad Amiri, and Moncef L. Nehdi.
2023. "New Fuzzy-Heuristic Methodology for Analyzing Compression Load Capacity of Composite Columns" *Buildings* 13, no. 1: 125.
https://doi.org/10.3390/buildings13010125