Bending behaviour of steel–concrete composite beam with partial shear interface using MCS and ANN

Abstract

A one-dimensional C^o finite element model (FEM) based on cubic-order beam theory in conjunction with the artificial neural network–genetic algorithm (ANN–GA) and artificial neural network–grey wolf optimization (ANN–GWO) metamodels has been adopted for the bending analysis of the pervious composite beam. The axial displacement field is a third-order equation and a function of the variation of the thickness of the beam. The parabolic variation of the shear stress is assumed; therefore, the shear stress value is zero at the extreme surface of the beam. The validation of the presented model has been done with the published research work and found suitable for further analysis. The Monte Carlo simulation–finite element method (MCS–FEM) has been developed and coded in the FORTRAN environment. The dataset obtained from MCS–FEM is used for the training and testing of the machine learning model. The ANN–GA and ANN–GWO metamodels are coded in the MATLAB environment. The various statistical parameters, regression analysis, and rank analysis have been done to find out the accuracy and efficiency of the proposed soft computation metamodels.

Appendix

$$\Gamma_{i} \, = \,1 - \left( {\frac{2}{\pi }\sqrt {1 - e_{ui} } - \frac{2}{\pi } + 1} \right)^{2}$$

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Statistical parameters:

$$R^{2} = 1 - \left( {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {d_{i} - y_{i} } \right)^{2} } }}{{\sum\nolimits_{i = 1}^{n} {\left( {d_{i} - d_{mean} } \right)^{2} } }}} \right)$$

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$${\text{Adj}}.R^{2} = 1 - \left\{ {\frac{{\left( {n - 1} \right)}}{{\left( {n - p - 1} \right)}}\left( {1 - R^{2} } \right)} \right\}$$

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$${\text{WMAPE}} = \frac{{\sum\nolimits_{i = 1}^{n} {\left| {\frac{{\left( {d_{i} - y_{i} } \right)}}{{d_{i} }}} \right|\, \times d_{i} } }}{{\sum\nolimits_{i = 1}^{n} {d_{i} } }}$$

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$${\text{NASH}} = \,1 - \frac{{\sum\nolimits_{i = 1}^{n} {\left( {y_{i} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y}_{i} } \right)^{2} } }}{{\sum\nolimits_{i = 1}^{n} {\left( {y_{i} - y_{{{\text{mean}}}} } \right)^{2} } }}$$

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$${\text{RMSE}} = \sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {d_{i} - y_{i} } \right)^{2} } }$$

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$${\text{VAF}} = \left( {1 - \frac{{{\text{var}} \left( {d_{i} - y_{i} } \right)}}{{{\text{var}} \left( {d_{i} } \right)}}} \right) \times 100$$

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$${\text{PI}} = \,{\text{Adj}}.R^{2} + \left( {0.01 \times {\text{VAF}}} \right) - {\text{RMSE}}$$

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$${\text{RSR}} = \frac{{{\text{RMSE}}}}{{\sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left( {d_{i} - d_{{{\text{mean}}}} } \right)^{2} } } }}$$

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$$WI = 1 - \left[ {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {d_{i} - y_{i} } \right)^{2} } }}{{\sum\nolimits_{i = 1}^{n} {\left\{ {\left| {\left( {y_{i} - d_{{{\text{mean}}}} } \right)} \right| + \left| {\left( {d_{i} - d_{{{\text{mean}}}} } \right)} \right|} \right\}^{2} } }}} \right]$$

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$${\text{MAE}} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left| {\left( {y_{i} - d_{i} } \right)} \right|}$$

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