# Prediction of Bond-Slip Behavior of Circular/Squared Concrete-Filled Steel Tubes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Database

_{i}), the cubic strength of the concrete mixture (${f}_{cu}$), and finally the age of the concrete mixture.

## 3. Analysis of Variance (ANOVA)

_{i}) less than 1200 mm and the concrete age (T) of not more than one year. The equations proposed for circular specimens are

^{2}of (${\tau}_{uExp.}/{\tau}_{uPred.})$ are presented in Table 1. It is found that the ANOVA results compare well with the experimental results and the mechanical-based discussion is presented in Section 4. ANOVA is an efficient parametric method for analyzing experiment data since it is practical and adaptable [32]. However, it is a considerably complex and subtle method to use as there are numerous ANOVA variations, each of which corresponds to a specific experimental situation. As such, it is possible to use the wrong type of ANOVA for a given experimental situation and draw incorrect conclusions from the data [33].

## 4. Artificial Neural Network (ANN)

_{max}and ANN

_{min}values are chosen to be used in scaling the inputs and outputs so the scaled values are between 0.1 and 0.9 as shown in Equations (12) and (13). However, ANN models’ prediction is restricted to be within the actual maximum and minimum values. Equations (12) and (13) are utilized in normalizing and denormalizing the data respectively.

^{2}) are calculated and listed. The testing datasets are used to test the accuracy of the trained networks. The trained network with the lowest ASE value of the testing datasets is defined as the network with the optimum number of hidden nodes. After the optimization, both training and testing datasets are combined into one train-all datasets group and utilizedtoretrain the optimum network to get the connection links weights of the train-all model. Combining these two dataset groups allows the network to use more datasets in model training and capturing the relations between inputs and outputs which helps in increasing the model accuracy and reducing the error. The validation datasets (which were not used in training or testing the model) are utilized to check and validate the model by comparing them with the model’s outputs.

- ANN1 (for circular column): 157 datasets of circular columns are utilized. The datasets are divided into 117 training datasets, 30 testing datasets, and 10 validation datasets. ANN models’ ASE
_{training}, ASE_{testing}, ASE_{train-all}, and ASE_{validation}values are listed in Figure 5a. Based on the values shown in Figure 5a, the optimum number of hidden nodes for ANN1 is found to be 9; because it has the lowest ASE_{testing}value. Hence, ANN1 is denoted by its architecture as 5-9-4. - ANN2 (for square column): 105 datasets of circular columns are utilized. The datasets are divided into 75 training datasets, 20 testing datasets, and 10 validation datasets. As shown in Figure 5b, the optimum number of hidden nodes for ANN2 is found to be 7. So, ANN2 is denoted by its architecture as 5-7-4.

## 5. Comparison and Discussion of Results

#### 5.1. Comparison of Bond Strength $({\tau}_{u})$ Predicted by ANOVA and ANN with Existing Models

#### 5.2. Comparisons of ${\tau}_{u}$, ${S}_{u}$, $\overline{\beta}$ and α Predicted by ANOVA and ANN for Both Circular and Squared CFSTs

_{u}plus standard deviation.

## 6. Finite Element Modeling of CFST’s Columns

#### 6.1. Experimental Data for Calibration

#### 6.2. Finite Element Discretization

#### 6.3. Constitutive Models of Materials

#### 6.4. Tube-Core Interface

#### 6.5. Finite Element Results

#### 6.6. Sensitivity Analysis

## 7. Conclusions

^{2}values are found ranging from 0.674 to 0.876 for circular CFSTs and from 0.548 to 0.837 for squared CFSTs, which are high compared to ANOVA results. Consequently, it can be concluded that ANN models are appropriate to anticipate the full behavior of circular and square CFSTs and are better than ANOVA.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Idealized response curves of concrete-filled steel tubes (CFSTs) under push-out loading test to represent; (

**a**) the overall behavior and (

**b**) the normalized post-peak behaviour.

**Figure 8.**Experimental versus predicted interfacial behavior using ANN and ANOVA analysis of (

**a**) circular and (

**b**) squared specimens.

**Table 1.**Comparison between mean value, standard deviation, and R

^{2}of the ratio of experimental values to estimated values using the proposed ANOVA and ANN.

Shape | Output | ANN Results | ANOVA Results | ||||
---|---|---|---|---|---|---|---|

Mean Value, μ | Standard Deviation, σ | R^{2} | Mean Value, μ | Standard Deviation, σ | R^{2} | ||

Circular | ${\tau}_{u}$ (MPa) | 0.999 | 0.166 | 0.876 | 1.001 | 0.277 | 0.328 |

S_{u} (mm) | 1.023 | 0.462 | 0.674 | 1.006 | 0.611 | 0.140 | |

β | 1.000 | 0.052 | 0.742 | 1.000 | 0.062 | 0.394 | |

α | 1.090 | 0.713 | 0.695 | 0.935 | 0.945 | 0.409 | |

Square | ${\tau}_{u}$ (MPa) | 1.004 | 0.188 | 0.837 | 1.067 | 0.596 | 0.623 |

S_{u} (mm) | 1.160 | 0.658 | 0.809 | 0.338 | 2.932 | 0.442 | |

β | 1.000 | 0.096 | 0.570 | 1.000 | 0.117 | 0.231 | |

α | 1.175 | 0.985 | 0.548 | 0.907 | 1.922 | 0.358 |

Limit | f_{cu} (MPa) | Age (days) | L_{i} (mm) | T (mm) | B (mm) | D (mm) | ${\mathit{\tau}}_{\mathit{u}}\left(\mathbf{MPa}\right)$ | ${\mathit{S}}_{\mathit{u}}\left(\mathbf{mm}\right)$ | $\overline{\mathit{\beta}}$ | α | |
---|---|---|---|---|---|---|---|---|---|---|---|

ANN1 | Max | 96.43 | 365 | 1095 | 6 | - | 219 | 2.55 | 6.83 | 1.04 | −1 × 10^{−3} |

Min | 9.11 | 28 | 190 | 2.5 | - | 107.7 | 0.61 | 0.23 | 0.69 | −0.31 | |

ANN_{max} | 120 | 450 | 1300 | 7 | - | 250 | 3 | 8 | 1.1 | 0 | |

ANN_{min} | 0 | 0 | 0 | 1.5 | - | 75 | 0 | 0 | 0.6 | −0.35 | |

ANN2 | Max | 58.31 | 365 | 1498.6 | 6.6 | 254 | - | 1.74 | 10.42 | 1.06 | −4 × 10^{−4} |

Min | 9.11 | 28 | 190 | 3 | 90.85 | - | 0.17 | 0.13 | 0.5 | −0.62 | |

ANN_{max} | 70 | 450 | 1750 | 7.5 | 310 | - | 2 | 12.5 | 1.25 | 0 | |

ANN_{min} | 0 | 0 | 0 | 1.75 | 30 | - | 0 | 0 | 0.3 | −0.75 |

Input_{j} | Connection Links Weights between Inputs and Hidden Nodes (Input_{j}–HN_{i}Connection) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

HN_{1} | HN_{2} | HN_{3} | HN_{4} | HN_{5} | HN_{6} | HN_{7} | HN_{8} | HN_{9} | ||

Input_{1}D(mm) | −11.17 | −18.56 | −20.95 | −11.36 | −8.69 | −11.29 | −2.26 | −3.64 | −2.72 | |

Input_{2}T(mm) | 11.47 | 0.96 | 30.81 | −8.25 | −18.52 | −7.72 | −4.56 | 5.32 | 1.46 | |

Input_{3}L_{i}(mm) | −6.08 | −15.98 | 0.44 | −5.25 | −4.93 | −12.12 | −4.70 | −2.71 | 0.74 | |

Input_{4}f_{cu}(MPa) | −15.35 | −15.39 | 1.55 | −3.66 | 20.37 | 1.73 | 1.55 | −5.55 | 0.90 | |

Input_{5}A(days) | −16.51 | −0.93 | −6.22 | 9.25 | −0.62 | −6.42 | −2.36 | −3.37 | −3.67 | |

Bias_{1} | 5.26 | 11.98 | −8.71 | 4.71 | 7.26 | 6.24 | 2.34 | −0.84 | −0.69 | |

Output_{m} | Connection Links Weights between Hidden Nodes and Outputs (HN_{i}–Output_{m}Connection) | |||||||||

HN_{1} | HN_{2} | HN_{3} | HN_{4} | HN_{5} | HN_{6} | HN_{7} | HN_{8} | HN_{9} | Bias_{2} | |

Output_{1}τ_{u}(MPa) | 4.28 | −2.75 | −3.64 | 1.46 | 0.65 | 6.97 | −6.84 | 2.22 | 0.36 | 0.72 |

Output_{2}S _{u}(mm) | −5.90 | −0.84 | 7.19 | 3.65 | 3.37 | 2.21 | −0.46 | 5.25 | 1.49 | −5.70 |

Output_{3}$\overline{\mathbf{\beta}}$ | −3.21 | 0.31 | 4.45 | −0.11 | 1.36 | 2.51 | 0.19 | −1.23 | −3.11 | 0.20 |

Output_{4}α | −5.13 | 3.49 | 3.70 | −5.49 | 1.50 | 0.01 | −1.90 | −0.06 | −0.06 | 2.44 |

Input_{j} | Connection Links Weights between Inputs and Hidden Nodes (Input_{j}–HN_{i}Connection) | |||||||
---|---|---|---|---|---|---|---|---|

HN_{1} | HN_{2} | HN_{3} | HN_{4} | HN_{5} | HN_{6} | HN_{7} | ||

Input_{1}B(mm) | 3.49 | −24.58 | 5.49 | −7.53 | −2.29 | −1.42 | −0.37 | |

Input_{2}T(mm) | −11.94 | 10.40 | −7.95 | 0.68 | 1.88 | −2.02 | −2.41 | |

Input_{3}L_{i}(mm) | 10.68 | 2.55 | 14.40 | −0.92 | −1.03 | −1.43 | −1.16 | |

Input_{4}f_{cu}(MPa) | 19.57 | 9.37 | −0.57 | 8.37 | −2.67 | −6.08 | 0.00 | |

Input_{5}A(days) | 20.11 | −6.96 | 14.91 | 0.27 | 5.37 | −4.21 | −0.42 | |

Bias_{1} | −11.30 | −5.28 | −4.33 | −2.57 | −1.66 | 1.15 | −0.39 | |

Output_{m} | Connection Links Weights between Hidden Nodes and Outputs (HN_{i}–Output_{m}Connection) | |||||||

H1 | H2 | H3 | H4 | H5 | H6 | H7 | Bias_{2} | |

Output_{1}τ_{u}(N/mm^{2}) | −4.20 | 4.89 | 1.95 | 2.85 | 1.65 | −0.75 | −1.34 | −0.19 |

Output_{2}S_{u}(mm) | 5.75 | 8.42 | −6.37 | −0.78 | 1.96 | 0.86 | −0.81 | −3.70 |

Output_{3}$\overline{\mathbf{\beta}}$ | 1.39 | −7.36 | −1.31 | 0.30 | 0.50 | 1.96 | 1.40 | 0.05 |

Output_{4}α | 12.72 | 16.91 | −9.67 | −1.06 | −1.81 | 4.44 | 0.07 | 0.41 |

Model | 5-9-4 ANN1 (Circular Column Model) | 5-7-4 ANN2 (Square Column Model) |
---|---|---|

ASE_{training} | 0.00724 | 0.007836 |

ASE_{testing} | 0.006465 | 0.01313 |

ASE_{train-all} | 0.007167 | 0.006721 |

ASE_{validation} | 0.007048 | 0.007940 |

MARE_{training} | 33. 648 | 41.586 |

MARE_{testing} | 33.177 | 44.317 |

MARE_{train-all} | 33. 229 | 39.400 |

MARE_{validation} | 33.064 | 42.304 |

R^{2}_{training} | 0.67767 | 0.59295 |

R^{2}_{testing} | 0.54837 | 0.44174 |

R^{2}_{train-all} | 0.74671 | 0.691078 |

R^{2}_{validation} | 0.75730 | 0.631577 |

**Table 6.**Comparison among mean value, standard deviation, and R

^{2}regarding the ultimate bond strength.

References | Prediction Model | Mean Value, μ | Standard Deviation, σ |
---|---|---|---|

Circular CFST | |||

Roeder et al. [7] | ${\tau}_{u}=2.109-0.026(D/t)$ | 1.197 | 0.524 |

Xue and Cai [20] | ${\tau}_{u}=0.1{\left({f}_{cu}\right)}^{0.4}$ | 4.316 | 1.809 |

Lyu and Han [21] | ${\tau}_{u}=0.071+4900\left(t/{D}^{2}\right)$ | 1.212 | 0.549 |

Chen et al. [19] | ${\tau}_{u}=[0.0336+0.0141\delta -0.0028({L}_{e}/d)]{f}_{cu}$ | 2.487 | 1.414 |

Proposed ANOVA | 1.001 | 0.277 | |

Proposed ANN | 0.999 | 0.1660 | |

Squared CFST | |||

Parsley et al. [18] | ${\tau}_{u}=0.013+1751\left(t/{b}^{2}\right)$ | 1.417 | 0.679 |

Lyu and Han [21] | ${\tau}_{u}=0.043+1100\left(t/{B}^{2}\right)$ | 2.019 | 0.937 |

Xue and Cai [20] | ${\tau}_{u}=0.1{\left({f}_{cu}\right)}^{0.4}$ | 2.544 | 1.255 |

Proposed ANOVA | 1.067 | 0.596 | |

Proposed ANN | 1.004 | 0.188 |

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**MDPI and ACS Style**

Allouzi, R.A.; Almasaeid, H.H.; Salman, D.G.; Abendeh, R.M.; Rabayah, H.S.
Prediction of Bond-Slip Behavior of Circular/Squared Concrete-Filled Steel Tubes. *Buildings* **2022**, *12*, 456.
https://doi.org/10.3390/buildings12040456

**AMA Style**

Allouzi RA, Almasaeid HH, Salman DG, Abendeh RM, Rabayah HS.
Prediction of Bond-Slip Behavior of Circular/Squared Concrete-Filled Steel Tubes. *Buildings*. 2022; 12(4):456.
https://doi.org/10.3390/buildings12040456

**Chicago/Turabian Style**

Allouzi, Rabab A., Hatem H. Almasaeid, Donia G. Salman, Raed M. Abendeh, and Hesham S. Rabayah.
2022. "Prediction of Bond-Slip Behavior of Circular/Squared Concrete-Filled Steel Tubes" *Buildings* 12, no. 4: 456.
https://doi.org/10.3390/buildings12040456