Numerical Evaluation of the Upright Columns with Partial Reinforcement along with the Utilisation of Neural Networks with Combining Feature-Selection Method to Predict the Load and Displacement
<p>The reinforcement system and the constituent elements: (<b>a</b>) graphical section detail and (<b>b</b>) upright column in tests.</p> "> Figure 1 Cont.
<p>The reinforcement system and the constituent elements: (<b>a</b>) graphical section detail and (<b>b</b>) upright column in tests.</p> "> Figure 2
<p>Upright configuration details (dimensions are in millimeters).</p> "> Figure 3
<p>(<b>a</b>) Schematic of compressive test on uprights; (<b>b</b>) testing rig.</p> "> Figure 4
<p>(<b>a</b>) Ball bearing; (<b>b</b>) cap plates.</p> "> Figure 5
<p>Interaction and connection properties of a typical model.</p> "> Figure 6
<p>A typical model with a mesh matrix view on the polygon and circular perforations.</p> "> Figure 7
<p>Designation of models.</p> "> Figure 8
<p>Comparison of the experimental and numerical results of normalised load for (<b>a</b>) 3600L-1.6T, (<b>b</b>)3600L-2.5T.</p> "> Figure 9
<p>Comparison of experimental and numerical results of normalised load for (<b>a</b>) 3000L-1.6T, (<b>b</b>) 3000L-2.5T.</p> "> Figure 10
<p>Comparison of experimental and numerical results of normalised load for (<b>a</b>) 2400L-1.6T, (<b>b</b>) 2400L-2.5T.</p> "> Figure 11
<p>Comparison of experimental and numerical results of normalised load for (<b>a</b>) 1800L-1.6T, (<b>b</b>) 1800L-2.5T.</p> "> Figure 12
<p>Normalised load-displacement diagrams of the FE results for: (<b>a</b>) 3600L-1.6T, (<b>b</b>) 3600L-2.0T, (<b>c</b>) 3600L-2.5T, and (<b>d</b>) 3600L-3.0T models.</p> "> Figure 13
<p>Normalised load-displacement diagrams of the FE results for: (<b>a</b>) 3000L-1.6T, (<b>b</b>) 3000L-2.0T, (<b>c</b>) 3000L-2.5T, and (<b>d</b>) 3000L-3.0T models.</p> "> Figure 14
<p>Normalised load-displacement diagrams of the FE results for: (<b>a</b>) 2400L-1.6T, (<b>b</b>) 2400L-2.0T, (<b>c</b>) 2400L-2.5T, and (<b>d</b>) 2400L-3.0T models.</p> "> Figure 15
<p>Normalised load-displacement diagrams of the FE results for: (<b>a</b>) 1800L-1.6T, (<b>b</b>) 1800L-2.0T, (<b>c</b>) 1800L-2.5T, and (<b>d</b>) 1800L-3.0T models.</p> "> Figure 16
<p>Normalised load-displacement diagrams of the FE results for: (<b>a</b>) 3600L-50B, (<b>b)</b> 3000L-50B, (<b>c</b>) 2400L-50B, and (<b>d</b>) 1800L-50B models.</p> "> Figure 16 Cont.
<p>Normalised load-displacement diagrams of the FE results for: (<b>a</b>) 3600L-50B, (<b>b)</b> 3000L-50B, (<b>c</b>) 2400L-50B, and (<b>d</b>) 1800L-50B models.</p> "> Figure 17
<p>Ultimate load capacities based on thickness and reinforcement spacing for (<b>a</b>) 1800 mm models and (<b>b</b>) 2400 mm models.</p> "> Figure 18
<p>Ultimate load capacities based on thickness and reinforcement spacing for (<b>a</b>) 3000 mm models and (<b>b</b>) 3600 mm models.</p> "> Figure 19
<p>Schematic representation of MLP neuron.</p> "> Figure 20
<p>Flowchart of typical single line hidden layer MLP for identifying a problem.</p> "> Figure 21
<p>PSO sequential flowchart.</p> "> Figure 22
<p>Feature selection technique steps.</p> "> Figure 23
<p>The flowchart of the sequential combination of hybrid MLP-PSO-FS algorithm.</p> "> Figure 24
<p>Regression of the training (above charts) and testing (below charts) phase results with measured values of displacement for (<b>a</b>) one input, (<b>b</b>) two inputs, (<b>c</b>) three inputs, (<b>d</b>) four inputs, (<b>e</b>) five inputs.</p> "> Figure 25
<p>Tolerance diagram of the displacement prediction corresponding to the MPF model with five inputs: (<b>above</b>) training phase, and (<b>below</b>) testing phase.</p> "> Figure 26
<p>Error histograms for displacement prediction by the MPF model with five inputs: (<b>above</b>) training phase, and (<b>below</b>) testing phase.</p> "> Figure 27
<p>Regression of the training (above charts) and testing (below charts) phase results with measured values of normalised load for (<b>a</b>) one input, (<b>b</b>) two inputs, (<b>c</b>) three inputs, (<b>d</b>) four inputs, (<b>e</b>) five inputs.</p> "> Figure 28
<p>The MPF (five inputs) prediction vs experimental diagram for ultimate load: (<b>above</b>) training phase, (<b>below</b>) testing phase.</p> "> Figure 29
<p>The MPF (five inputs) error histograms for ultimate load prediction: (<b>above</b>) training phase and (<b>bellow</b>) testing phase.</p> ">
Abstract
:1. Introduction
2. Finite Element (FE) Modelling
2.1. Material Properties
- σ and ε are = stresses and strains derived from the coupon tests.
- Poisson ratio = 0.3.
- Module of elasticity = 200 GPa.
2.2. Connections and Interactions
2.3. Boundary Conditions, Loading and Mesh
2.4. FE Model Verification
2.5. Result and Discussion of the FE Analysis
3. Artificial Intelligence Prediction
3.1. Algorithm Methodology
3.1.1. Multi-Layer Perceptron (MLP)
3.1.2. Particle Swarm Optimisation (PSO)
3.1.3. Feature Selection (FS) Technique
3.1.4. MLP-PSO-FS Architecture
- si⇀ = particle’s position;
- vi⇀ = particle’s velocity;
- pi⇀ = most appropriate position;
- w = inertia weight;
- c1 and c2 = acceleration coefficients;
- ∅1⇀ and ∅2⇀ = uniformly-distributed random vectors in [0,1].
3.1.5. Performance Evaluation
3.2. Algorithm Results and Discussion
4. Displacement Prediction
5. Ultimate Load Prediction
6. Conclusions
- According to the FE results, using reinforcement at closer distances increased the ultimate load capacity compared to other models, especially in models with 1.6 mm thickness. By comparing the thicknesses, the model with 2.5 mm thickness presented the most load capacity increment among other thicknesses. The model with 1800 mm length and 50 mm spacing showed the most capacity compared to other models. Models with 3600 mm length indicated more ductile behaviour in comparison with other models.
- MPF network was successfully developed and the predicted results was stasifying, that indicated proficiency of MLP. Neural network prediction revealed a harmonious relation between load and displacement. FS technique was employed to select the best possible input arrangements from introduced parameters. Results of the MPF algorithms in the displacement prediction phase represented that the model’s prediction with 45 iterations and 250 populations is better than others. The model with five inputs represented the optimum parameter composition to predict the displacement, which its performance parameters were RMSE = 0.001, r = 1.000, R2 = 0.999, NS = 1.000, MAE = 0 and WI = 1.000. In the case of load prediction, the model with 150 iterations, 250 populations and five input combinations predicted the most accurate values along best precision values including RMSE = 1.678, r = 0.800, R2 = 0.907, NS = 0.435, MAE = 1.203 and WI = 0.882.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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3600 mm | NoB | 400B | 350B | 300B | 250B | 200B | 150B | 100B | 50B |
---|---|---|---|---|---|---|---|---|---|
1.6T | 0.320355 | 0.324957 | 0.329694 | 0.334311 | 0.338452 | 0.345572 | 0.345014 | 0.34748 | 0.349518 |
2.0T | 0.267865 | 0.274581 | 0.283207 | 0.286614 | 0.288806 | 0.291152 | 0.291867 | 0.292715 | 0.293467 |
2.5T | 0.266615 | 0.290798 | 0.29735 | 0.302896 | 0.30518 | 0.311646 | 0.314002 | 0.315819 | 0.317829 |
3.0T | 0.244069 | 0.266756 | 0.273336 | 0.284606 | 0.292642 | 0.299102 | 0.304017 | 0.305707 | 0.30739 |
3000 mm | NoB | 400B | 350B | 300B | 250B | 200B | 150B | 100B | 50B |
---|---|---|---|---|---|---|---|---|---|
1.6T | 0.466192 | 0.476169 | 0.488906 | 0.496421 | 0.503579 | 0.51175 | 0.511099 | 0.514937 | 0.518591 |
2.0T | 0.397264 | 0.410294 | 0.423495 | 0.429079 | 0.43454 | 0.438375 | 0.441986 | 0.443907 | 0.445772 |
2.5T | 0.367905 | 0.404657 | 0.411269 | 0.419328 | 0.42485 | 0.429913 | 0.435515 | 0.438243 | 0.440754 |
3.0T | 0.364267 | 0.407849 | 0.416635 | 0.431377 | 0.442595 | 0.450041 | 0.457685 | 0.461408 | 0.464957 |
2400 mm | NoB | 400B | 350B | 300B | 250B | 200B | 150B | 100B | 50B |
---|---|---|---|---|---|---|---|---|---|
1.6T | 0.44809 | 0.47159 | 0.484062 | 0.491135 | 0.498348 | 0.505794 | 0.512634 | 0.520127 | 0.527014 |
2.0T | 0.394546 | 0.426041 | 0.434678 | 0.44741 | 0.458877 | 0.468928 | 0.473545 | 0.480774 | 0.483447 |
2.5T | 0.452161 | 0.498086 | 0.508837 | 0.519291 | 0.529506 | 0.537428 | 0.545083 | 0.550086 | 0.552796 |
3.0T | 0.382484 | 0.454062 | 0.472006 | 0.48701 | 0.502558 | 0.516978 | 0.529511 | 0.538297 | 0.543857 |
1800 mm | NoB | 400B | 350B | 300B | 250B | 200B | 150B | 100B | 50B |
---|---|---|---|---|---|---|---|---|---|
1.6T | 0.446135 | 0.498162 | 0.531249 | 0.557774 | 0.580285 | 0.594211 | 0.607148 | 0.615757 | 0.624738 |
2.0T | 0.469598 | 0.533296 | 0.555708 | 0.589511 | 0.619443 | 0.645763 | 0.657006 | 0.67562 | 0.682954 |
2.5T | 0.457849 | 0.549937 | 0.585468 | 0.599585 | 0.608311 | 0.624156 | 0.656714 | 0.671063 | 0.682757 |
3.0T | 0.428473 | 0.523704 | 0.547287 | 0.566293 | 0.587712 | 0.606822 | 0.6235 | 0.635513 | 0.642487 |
Phase | Network Result | |||||
---|---|---|---|---|---|---|
R2 | R | NS | RMSE | MAE | WI | |
Test | 0.999 | 1.000 | 1.000 | 0.001 | 0.000 | 1.000 |
Train | 1.000 | 1.000 | 1.000 | 0.000 | 0.000 | 1.000 |
Phase | Network Result | |||||
---|---|---|---|---|---|---|
R2 | R | NS | RMSE | MAE | WI | |
Test | 0.907 | 0.800 | 0.435 | 1.678 | 1.203 | 0.882 |
Train | 0.847 | 0.820 | 0.511 | 1.590 | 1.137 | 0.895 |
FIS Clusters | Population Size | Iterations | Inertia Weight | Damping Ratio | Learning Coefficient | |
---|---|---|---|---|---|---|
Personal | Global | |||||
10 | 150~350 | 45~100 | 1 | 0.98 | 2 | 3 |
Parameter characteristics used for the MLP | |
Hidden Layers | Training Function |
10 | Levenberg–Marquardt back-propagation (LMBP) |
Parameter characteristics used for FS | |
Number of runs | Number of functions(nf) |
3 | 1~5 |
Feature | Number of Inputs | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
Length | X | X | |||
Bolt | X | X | X | ||
Thickness | X | X | X | X | |
X | |||||
Axial load | X | X | X | X | X |
Train | ||||||||
The MPF Network | ||||||||
Iteration | Population | nf | R2 | r | NS | RMSE | MAE | WI |
45 | 250 | 1 | 0.885 | 0.941 | 0.870 | 0.035 | 0.026 | 0.969 |
45 | 250 | 2 | 0.971 | 0.985 | 0.970 | 0.018 | 0.003 | 0.993 |
45 | 250 | 3 | 0.825 | 0.996 | 0.992 | 7.244 | 5.270 | 0.998 |
45 | 250 | 4 | 0.999 | 1.000 | 1.000 | 0.001 | 0.000 | 1.000 |
45 | 250 | 5 * | 1.000 | 1.000 | 1.000 | 0.000 | 0.000 | 1.000 |
Test | ||||||||
The MPF Network | ||||||||
Iteration | Population | nf | R2 | r | NS | RMSE | MAE | WI |
45 | 250 | 1 | 0.992 | 0.935 | 0.854 | 0.038 | 0.028 | 0.965 |
45 | 250 | 2 | 0.997 | 0.978 | 0.955 | 0.022 | 0.004 | 0.989 |
45 | 250 | 3 | 0.921 | 0.907 | 0.798 | 7.199 | 5.246 | 0.951 |
45 | 250 | 4 | 0.999 | 1.000 | 1.000 | 0.001 | 0.000 | 1.000 |
45 | 250 | 5 * | 0.999 | 1.000 | 1.000 | 0.001 | 0.000 | 1.000 |
Feature | Number of Inputs | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
Length | X | X | X | ||
Bolt | X | X | X | X | |
Thickness | X | X | X | ||
X | X | ||||
Displacement | X | X | X |
Train | ||||||||
The MPF Network | ||||||||
Iteration | Population | nf | R2 | r | NS | RMSE | MAE | WI |
150 | 250 | 1 | 0.524 | 0.709 | 0.011 | 1.966 | 1.386 | 0.816 |
150 | 250 | 2 | 0.650 | 0.796 | 0.422 | 1.672 | 1.121 | 0.879 |
150 | 250 | 3 | 0.674 | 0.812 | 0.484 | 1.620 | 1.148 | 0.890 |
150 | 250 | 4 | 0.683 | 0.818 | 0.503 | 1.575 | 1.125 | 0.894 |
150 | 250 | 5 * | 0.847 | 0.820 | 0.511 | 1.590 | 1.137 | 0.895 |
Test | ||||||||
The MPF Network | ||||||||
Iteration | Population | nf | R2 | r | NS | RMSE | MAE | WI |
150 | 250 | 1 | 0.785 | 0.697 | −0.034 | 1.984 | 1.400 | 0.809 |
150 | 250 | 2 | 0.835 | 0.782 | 0.357 | 1.762 | 1.189 | 0.869 |
150 | 250 | 3 | 0.853 | 0.806 | 0.464 | 1.655 | 1.159 | 0.886 |
150 | 250 | 4 | 0.812 | 0.822 | 0.489 | 1.639 | 1.163 | 0.894 |
150 | 250 | 5 * | 0.907 | 0.800 | 0.435 | 1.678 | 1.203 | 0.882 |
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Taheri, E.; Mehrabi, P.; Rafiei, S.; Samali, B. Numerical Evaluation of the Upright Columns with Partial Reinforcement along with the Utilisation of Neural Networks with Combining Feature-Selection Method to Predict the Load and Displacement. Appl. Sci. 2021, 11, 11056. https://doi.org/10.3390/app112211056
Taheri E, Mehrabi P, Rafiei S, Samali B. Numerical Evaluation of the Upright Columns with Partial Reinforcement along with the Utilisation of Neural Networks with Combining Feature-Selection Method to Predict the Load and Displacement. Applied Sciences. 2021; 11(22):11056. https://doi.org/10.3390/app112211056
Chicago/Turabian StyleTaheri, Ehsan, Peyman Mehrabi, Shervin Rafiei, and Bijan Samali. 2021. "Numerical Evaluation of the Upright Columns with Partial Reinforcement along with the Utilisation of Neural Networks with Combining Feature-Selection Method to Predict the Load and Displacement" Applied Sciences 11, no. 22: 11056. https://doi.org/10.3390/app112211056