# Prediction of Bidirectional Shear Strength of Rectangular RC Columns Subjected to Multidirectional Earthquake Actions for Collapse Prevention

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. MEP

_{1}, Z

_{2}, Z

_{3}, Z

_{4}}. The combination of chromosomes in the example is as follows [22]:

_{1}

_{2}

_{3}

_{4}

_{1}, Z

_{2}, Z

_{3}and Z

_{4}, respectively [22]. The expressions for these genes are shown in Equation (1).

_{2}) shows the operator + on the operands located at the position 0 and 1 on the chromosomes. Similarly, Gene 4 (G

_{4}) and Gene 6 (G

_{6}) represent the operator × and ^ on the operands located at positions 2, 3 and 4, 5, respectively [22]. The expressions encoded by these genes are shown in Equations (2)–(4).

#### 1.2. ANFIS

**Layer 1**is an adaptive layer as the parameters involved need to be adjusted in the training process. In this layer, every node “i” represents an adaptive node with a node membership function as shown in Equations (5) and (6) below [24]:

_{i}

^{1}= μ

_{Ai}(x

_{1}), i = 1, 2, …, n

_{i}

^{1}= μ

_{Bi}

_{−2}(x

_{2}), i = 3, 4, …, n

**Layer 2**calculates the firepower of the predetermined fuzzy rules via ∏ operator. It is a non-adaptive layer [24]. For the ANFIS network shown in Figure 3, we have:

_{i}

^{2}= w

_{i}= μ

_{Ai}(x

_{1}) × μ

_{Bi}(x

_{2}), i = 1, 2, …, n

**Layer 3**is also a non-adoptive layer, which calculates the firepower of a rule from layer 2. Every node in this layer is a fixed node labelled as ‘N’ [24]. The outputs are the normalized firing power of the rules expressed, as given in Equation (8) below:

**Layer 4**is an adaptive layer in which each node represents a consequent part of the fuzzy rule. The outputs are the products of the weights (normalized) into the node’s function, which can be represented as given below [24]:

**Layer 5**is a non-adaptive layer whose output is the final output of the ANFIS network. For the network shown in Figure 3 with one output, defuzzification is performed by the node (∑), which is shown below [24]:

## 2. Methodology

#### 2.1. Fitting Parameters

#### 2.1.1. MEP Parameters

#### 2.1.2. ANFIS Parameters

#### 2.2. Performance Evaluation of Models

^{th}experimental, predicted, mean experimental, and mean predicted values, respectively, while $n$ is the total number of data points used for modelling. The subscripts ${n}_{L}$ and ${n}_{T}$ represent the number of learning (training and validation) and testing datapoints, respectively. The ${\rho}_{L}$ and ${\rho}_{T}$ denote the performance index of learning and testing sets, respectively.

#### 2.3. Modelling

^{2})

## 3. Results Analysis and Discussion

#### 3.1. ANFIS Modeling Results

#### 3.2. MEP Modeling Results

## 4. Parametric Analysis of MEP-Based Models

_{w}, and d) used in the other models in Figure 8. From the figure, it can be noted that V increased with an increase in all the input parameters for the four models. From Figure 8a,b, we can see that V increased linearly with an increase in ${f}_{c}^{\prime}$ and $N$, respectively. An increase in ${f}_{c}^{\prime}$ and $N$ proved to be beneficial to the shear strength of the RC columns, and the observations are valid from an engineering viewpoint and conforms with the conclusions obtained by [4,46]. Similarly, V increased due to an increase in the percentage of both longitudinal and transverse reinforcement in the specimens (vid. Figure 8d,e). This can be attributed to the fact that an increasing ${\rho}_{l}$ allows the specimens to sustain greater damage in the core, while an increasing ${\rho}_{w}$ confines the longitudinal reinforcement and prevents it from buckling [47,48]. Lastly, the biaxial shear strength increased due to increasing ${A}_{g}$ and ${f}_{y}$, which is also correct from an engineering viewpoint. Hence, the results of the parametric analyses of the models show that the MEP algorithm accurately presents the system under consideration. The results of the parametric study of the common parameters in this study agree well with the results of Murad et al. [12].

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

RC | Reinforced Concrete |

MEP | Multi Expression Programming |

V | Biaxial Shear Strength |

GEP | Gene Expression Programming |

GA | Genetic Algorithm |

GP | Genetic Programming |

G_{2} | Gene 2 |

G_{4} | Gene 4 |

G_{6} | Gene 6 |

x_{1} & x_{2} | Sample inputs in ANFIS |

μ_{Ai} & μ_{Bi}_{−2} | Weights obtained while connecting fuzzy membership functions |

${\overline{w}}_{i}$ | Firing strength |

f_{i} | Linear function |

p_{k}, q_{k} & r_{k} | Linear function parameters for particular rule ‘k’ |

OF | Objective Function |

ρ | Performance Index |

${e}_{i}$ | ith Experimental |

${m}_{i}$ | ith Predicted |

${\overline{e}}_{i}$ | ith Mean Experimental |

${\overline{m}}_{i}$ | ith Mean Predicted |

${n}_{L}$ | Number of learning (training and validation) data |

${n}_{T}$ | Number of testing dataset |

${\rho}_{L}$ | Performance index of learning dataset |

${\rho}_{T}$ | Performance index of testing dataset |

R | Correlation coefficient |

MAE | Mean Absolute Error |

RMSE | Root Mean Square Error |

RRMSE | Relative Root Mean Square Error |

${f}_{c}^{\prime}$ | Concrete Compressive Strength |

${A}_{g}$ | Gross Sectional Area of Column |

${\rho}_{l}$ | Longitudinal Reinforcement Percentage |

${\rho}_{w}$ | Shear Reinforcement Percentage |

${f}_{y}$ | Yield Strength of Longitudinal Reinforcement |

$N$ | Axial Load Of Column |

${b}_{w}$ | Width Of Column Web |

$d$ | Depth Of Column |

$H$ | Column Height |

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**Figure 3.**Sample architecture of an ANFIS network with two inputs and one output [24].

**Figure 4.**Comparison between experimental and various ANFIS models results: (

**a**) M1-ANFIS vs. experimental results, (

**b**) M2-ANFIS vs. experimental results, (

**c**) M3-ANFIS vs. experimental results, (

**d**) M4-ANFIS vs. experimental results.

**Figure 5.**Comparison between experimental and various MEP models results: (

**a**) M1-MEP vs. experimental results, (

**b**) M2-MEP vs. experimental results, (

**c**) M3-MEP vs. experimental results, (

**d**) M4-MEP vs. experimental results.

**Figure 8.**Results of parametric analysis. (

**a**) Compressive Strength of Concrete, (

**b**) Axial Load (

**c**) Column Cross-sectional Area, (

**d**) Longitudinal Reinforcement Percentage, (

**e**) Shear Reinforcement Percentage, (

**f**) Yield Strength Longitudinal Reinforcement, (

**g**) Column Height, (

**h**) Width of Column Web (mm), (

**i**) Depth of Column (mm).

Parameters | Settings | |||
---|---|---|---|---|

M1-MEP | M2-MEP | M3-MEP | M4-MEP | |

Number of sub-population | 30 | 30 | 30 | 30 |

Size of subpopulation | 200 | 200 | 200 | 200 |

Code length | 45 | 35 | 30 | 35 |

Crossover probability | 0.9 | 0.9 | 0.9 | 0.9 |

Mathematical operators | +, −, ×, ÷, √ | +, −, ×, ÷, √ | +, −, ×, ÷, √ | +, −, ×, ÷, √ |

Mutation probability | 0.01 | 0.01 | 0.01 | 0.01 |

Tournament size | 4 | 4 | 2 | 2 |

Operators | 0.5 | 0.5 | 0.5 | 0.5 |

Mutation probability | 0.5 | 0.5 | 0.5 | 0.5 |

Number of generations | 1000 | 2000 | 5000 | 3000 |

Parameters | Settings | |||
---|---|---|---|---|

M1-ANFIS | M2-ANFIS | M3-ANFIS | M4-ANFIS | |

Linear parameters | 729 | 243 | 243 | 729 |

Non-linear parameters | 54 | 45 | 45 | 54 |

Fuzzy rules | 729 | 243 | 243 | 729 |

Nodes | 1503 | 524 | 524 | 1503 |

Epochs | 50 | 50 | 50 | 50 |

Error goal | 0 | 0 | 0 | 0 |

Type of MF | Trimf | Trimf | Trimf | Trimf |

Structure of fuzzy | Sugeno | Sugeno | Sugeno | Sugeno |

Type of FIS | Grid Partition | Grid Partition | Grid Partition | Grid Partition |

Optimization technique | Backpropagation and least square | Backpropagation and least square | Backpropagation and least square | Backpropagation and least square |

Type of output function | Linear | Linear | Linear | Linear |

Parameter | Expression | Criteria |
---|---|---|

Correlation coefficient (R) | $\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({e}_{i}-{\overline{e}}_{i}\right)\left({m}_{i}-{\overline{m}}_{i}\right)}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left({e}_{i}-{\overline{e}}_{i}\right)}^{2}{{\displaystyle \sum}}_{i=1}^{n}{\left({m}_{i}-{\overline{m}}_{i}\right)}^{2}}}$ | >0.8 [40] |

Mean absolute error (MAE) | $\frac{{{\displaystyle \sum}}_{i=1}^{n}\left|{e}_{i}-{m}_{i}\right|}{n}$ | Minimum [41] |

Root mean square error (RMSE) | $\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({e}_{i}-{m}_{i}\right)}^{2}}{n}}$ | Minimum |

Relative root mean square error (RRMSE) | $\frac{1}{\left|\overline{e}\right|}\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({e}_{i}-{m}_{i}\right)}^{2}}{n}}$ | 0–0.1 (Excellent) or 0.11–0.2 (Good) [42] |

$\mathrm{Performance\; index}(\rho )$ | $\rho =\frac{\mathrm{RRMSE}}{1+R}$ | <0.2 [34] |

Objective function (OF) | $\left(\frac{{n}_{L}-{n}_{T}}{n}\right){\rho}_{L}+2\left(\frac{{n}_{T}}{n}\right){\rho}_{T}$ | Close to zero [25] |

Parameter | ${\mathit{f}}_{\mathit{c}}^{\prime}\left(\mathbf{MPa}\right)$ | $\mathit{N}$ (kN) | ${\mathit{b}}_{\mathit{w}}\left(\mathbf{mm}\right)$ | $\mathit{d}$ (mm) | ${\mathit{A}}_{\mathit{g}}\left({\mathbf{mm}}^{2}\right)$ | ${\mathit{\rho}}_{\mathit{l}}(\%)$ | ${\mathit{f}}_{\mathit{y}}\left(\mathbf{MPa}\right)$ | ${\mathit{\rho}}_{\mathit{w}}(\%)$ | $\mathit{H}$ (mm) |
---|---|---|---|---|---|---|---|---|---|

Mean | 19.23 | 8.44 × 10^{5} | 1.85 | 28.05 | 1.16 × 10^{5} | 1.568 | 440.93 | 0.25 | 1493.86 |

Standard Error | 0.694 | 8.02 × 10^{4} | 0.070 | 0.693 | 6.77 × 10^{3} | 0.115 | 10.82 | 0.03 | 79.38 |

Median | 23 | 1.00 × 10^{6} | 1.755 | 27 | 1.20 × 10^{5} | 1.056 | 400.00 | 0.13 | 1700.00 |

Mode | 23 | 1.00 × 10^{6} | 2.06 | 38.2 | 9.00 × 10^{4} | 2.547 | 400.00 | 0.11 | 1700.00 |

Standard Deviation | 6.13 | 7.08 × 10^{5} | 0.615 | 6.12 | 4.34 × 10^{4} | 0.739 | 69.30 | 0.18 | 508.28 |

Sample Variance | 37.60 | 5.02 × 10^{11} | 0.378 | 37.51 | 1.88 × 10^{9} | 0.546 | 4801.83 | 0.03 | 258,343.76 |

Kurtosis | −1.1036 | 1.30 × 10^{1} | −0.556 | −0.572 | 5.57 | −1.312 | 0.45 | −0.58 | −0.69 |

Skewness | −0.710 | 2.83 | 0.379 | 0.120 | 1.37 | 0.724 | 0.08 | 0.94 | −0.62 |

Minimum | 8.85 | 1.00 × 10^{5} | 0.7 | 15.92 | 4.00 × 10^{4} | 0.848 | 276.00 | 0.09 | 570.00 |

Maximum | 25.7 | 4.29 × 10^{6} | 3.35 | 38.2 | 2.92 × 10^{5} | 2.777 | 575.60 | 0.63 | 2438.40 |

Parameter | ${\mathit{f}}_{\mathit{c}}^{\prime}\left(\mathbf{MPa}\right)$ | $\mathit{N}$ (kN) | ${\mathit{b}}_{\mathit{w}}\left(\mathbf{mm}\right)$ | $\mathit{d}$ (mm) | ${\mathit{A}}_{\mathit{g}}\left({\mathbf{mm}}^{2}\right)$ | ${\mathit{\rho}}_{\mathit{l}}(\%)$ | ${\mathit{f}}_{\mathit{y}}\left(\mathbf{MPa}\right)$ | ${\mathit{\rho}}_{\mathit{w}}(\%)$ | $\mathit{H}$ (mm) |
---|---|---|---|---|---|---|---|---|---|

${f}_{c}^{\prime}$ (MPa) | 1.00 | 0.22 | −0.32 | 0.21 | −0.11 | −0.30 | −0.07 | −0.20 | 0.03 |

$N$ (kN) | 0.22 | 1.00 | 0.10 | 0.17 | 0.14 | 0.24 | 0.29 | −0.04 | 0.49 |

${b}_{w}$ (mm) | −0.32 | 0.10 | 1.00 | 0.16 | 0.76 | 0.05 | 0.51 | −0.28 | 0.36 |

$d$ (mm) | 0.21 | 0.17 | 0.16 | 1.00 | 0.73 | −0.26 | −0.32 | −0.52 | 0.49 |

${A}_{g}$ (mm^{2}) | −0.11 | 0.14 | 0.76 | 0.73 | 1.00 | −0.11 | 0.11 | −0.43 | 0.51 |

${\rho}_{l}$ (%) | −0.30 | 0.24 | 0.05 | −0.26 | −0.11 | 1.00 | 0.34 | 0.67 | −0.22 |

${f}_{y}$ (MPa) | −0.07 | 0.29 | 0.51 | −0.32 | 0.11 | 0.34 | 1.00 | 0.14 | 0.25 |

${\rho}_{w}$ (%) | −0.20 | −0.04 | −0.28 | −0.52 | −0.43 | 0.67 | 0.14 | 1.00 | −0.47 |

$H$ (mm) | 0.03 | 0.49 | 0.36 | 0.49 | 0.51 | −0.22 | 0.25 | −0.47 | 1.00 |

Model | Dataset | R | MAE | RMSE | RSE | RRMSE | ρ | OF |
---|---|---|---|---|---|---|---|---|

M1-ANFIS | Training | 0.992 | 3.94 | 11.50 | 0.01 | 0.07 | 0.03 | 0.026 |

Validation | 0.999 | 4.83 | 9.46 | 0.011 | 0.05 | 0.02 | ||

Testing | 0.994 | 3.16 | 5.83 | 0.016 | 0.05 | 0.02 | ||

M2-ANFIS | Training | 0.983 | 11.55 | 25.25 | 0.03 | 0.12 | 0.06 | 0.015 |

Validation | 0.996 | 3.45 | 4.89 | 0.009 | 0.04 | 0.02 | ||

Testing | 0.999 | 1.01 | 1.75 | 0.001 | 0.01 | 0.01 | ||

M3-ANFIS | Training | 0.961 | 13.34 | 26.48 | 0.08 | 0.15 | 0.08 | 0.03 |

Validation | 0.997 | 2.54 | 3.97 | 0.006 | 0.03 | 0.01 | ||

Testing | 0.999 | 0.83 | 1.33 | 0.0005 | 0.01 | 0.01 | ||

M4-ANFIS | Training | 0.991 | 8.27 | 15.90 | 0.03 | 0.09 | 0.05 | 0.04 |

Validation | 0.999 | 1.88 | 3.08 | 0.0003 | 0.01 | 0.01 | ||

Testing | 0.989 | 12.25 | 28.21 | 0.027 | 0.11 | 0.06 |

Model | Dataset | R | MAE | RMSE | RSE | RRMSE | ρ | OF |
---|---|---|---|---|---|---|---|---|

Biaxial shear strength | Training | 0.993 | 9.20 | 12.49 | 0.01 | 0.06 | 0.03 | 0.059 |

Validation | 0.995 | 8.76 | 10.95 | 0.014 | 0.06 | 0.03 | ||

Testing | 0.977 | 8.35 | 10.64 | 0.054 | 0.09 | 0.04 |

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Pang, Y.; Azim, I.; Rauf, M.; Iqbal, M.F.; Ge, X.; Ashraf, M.; Tariq, M.A.U.R.; Ng, A.W.M. Prediction of Bidirectional Shear Strength of Rectangular RC Columns Subjected to Multidirectional Earthquake Actions for Collapse Prevention. *Sustainability* **2022**, *14*, 6801.
https://doi.org/10.3390/su14116801

**AMA Style**

Pang Y, Azim I, Rauf M, Iqbal MF, Ge X, Ashraf M, Tariq MAUR, Ng AWM. Prediction of Bidirectional Shear Strength of Rectangular RC Columns Subjected to Multidirectional Earthquake Actions for Collapse Prevention. *Sustainability*. 2022; 14(11):6801.
https://doi.org/10.3390/su14116801

**Chicago/Turabian Style**

Pang, Yingbo, Iftikhar Azim, Momina Rauf, Muhammad Farjad Iqbal, Xinguang Ge, Muhammad Ashraf, Muhammad Atiq Ur Rahman Tariq, and Anne W. M. Ng. 2022. "Prediction of Bidirectional Shear Strength of Rectangular RC Columns Subjected to Multidirectional Earthquake Actions for Collapse Prevention" *Sustainability* 14, no. 11: 6801.
https://doi.org/10.3390/su14116801