Prediction of the Yield Strength of RC Columns Using a PSO-LSSVM Model
Abstract
:1. Introduction
2. Pseudo-Static Cyclic Test and Yield Point
2.1. Pseudo-Static Cyclic Test
2.2. Yield Point
- For the Geometric Graphic Method shown in Figure 2a, the yield displacement and yield point are defined as follows: Draw the tangent line to the curve at ‘O’. The line is extended to the intersection with a horizontal line through ‘D’ at ‘A’, where the ‘D’ corresponds to the maximum applied shear Pmax shown in Figure 2a. The perpendicular of line ‘OA’ intersects the curve at ‘B’. Connect ‘O’ and ‘B’ and extend line ‘OB’ to meet the horizontal line ‘DA’ at ‘C’, and then project onto the horizontal axis to obtain the yield displacement ∆y and the yield point ‘E’ on the curve, where it corresponds to the yield applied shear Py.
- For R. Park Method shown in Figure 2b [48], the yield displacement and yield point are defined as follows: A secant ‘OB’ is drawn to intersect the lateral load-displacement relation at a certain proportion of the maximum applied shear, i.e., the ‘B’ on the curve corresponding to αPmax shown in Figure 2b. Similarly, this extension of line ‘OB’ and the horizontal line corresponding to the maximum applied shear Pmax intersect at ‘A’, and then projects onto the horizontal axis to obtain the yield displacement ∆y. The intersection point ‘C’ of the vertical and curve is defined as the yield point, which corresponds to the yield applied shear Py.
- For Equivalent Elasto-Plastic Energy Method shown in Figure 2c, the yield displacement and yield point are defined as follows: Determine a point ‘B’ on the curve and draw secant ‘OB’ to intersect the curve for satisfying the principle that the energy absorbed by the ideal elastoplastic structure is equal to that absorbed by the actual structure, i.e., the areas of shaded area ‘OAB’ and ‘BCD’ are equal shown in Figure 2c. Similar to the R. Park Method in Figure 2b, this extension of line ‘OB’ and the horizontal line corresponding to the maximum applied shear Pmax intersect at ‘C’, and is then projected onto the horizontal axis to obtain the yield displacement ∆y. The intersection point ‘E’ of the vertical and curve is defined as the yield point, which corresponds to the yield applied shear Py.
3. Data Collection and Pre-Processing
4. Model Development and Verification
4.1. Least Squares Support Vector Machine
Algorithm 1: Pseudo-code of LSSVM algorithm | |
1 | Initialize LSSVM parameters |
2 | Normalize data using Equation (8) |
3 | whilestopping condition is not met do |
4 | Train LSSVM model with parameters (σ and γ) using each training data point |
5 | end while |
6 | Predict testing data using trained LSSVM model |
7 | returnaccuracy |
4.2. Particle Swarm Optimization
Algorithm 2: Pseudo-code of LSSVM algorithm | |
1 | Initialize population of particles and derive local and global best particles (pbesti and gbest) |
2 | for k = 1 to maximum number of iterations do |
3 | for i = 1 to population size do |
4 | Update the velocity of the ith particle (vi) using Equation (6) |
5 | Update the location of the ith particle (ui) using Equation (7) |
6 | if F(ui) < F(pbesti) then |
7 | pbesti = ui. |
8 | if F(pbesti) < F(gbest) then |
9 | gbest = pbesti |
10 | end if |
11 | end if |
12 | end for |
13 | end for |
14 | returngbest |
- (i)
- Initialize the parameters in PSO algorithm.
- (ii)
- Calculate the fitness value of each particle, and evaluate their fitness with Equation (8), i.e., F(u) = F(σ, γ) = RMSE.
- (iii)
- Update the location and velocity of the ith particle with Equations (6) and (7) and compare the current fitness value of each particle F(ui) with the individual best fitness value F(pbesti), if satisfying F(ui) < F(pbesti), pbesti = ui.
- (iv)
- Compare the current fitness values of all particles in the swarm F(ui) with the fitness value of the best location of the swarm F(gbest), if satisfying F(ui) < F(gbest), the global optimal solution gbest = ui.
- (v)
- Check whether the termination condition is met. If the error accuracy is satisfied or the maximum number of swarm evolution is reached, then the process of searching for the optimal solution (σ, γ) ends and the optimal values of σ and γ are outputted. Otherwise, proceed to the next step from step (ii) to continue the process of parameter optimization.
- (vi)
- Substitute the optimal parameters (σ, γ) into Equation (1) for predicting the yield strength and displacement of RC columns.
4.3. Implementation of the Yield Strength and Displacement Prediction
4.4. Evaluation of the PSO-LSSVM Prediction Performance
4.4.1. Performance Evaluation Indicators
4.4.2. Evaluation of the PSO-LSSVM Prediction Results
4.4.3. Comparative Evaluation of Model Prediction Performance
5. Conclusions
- (1)
- The determination coefficient R2 is 0.96, if 80% of the whole dataset is used for training the PSO-LSSVM model, which means a low prediction error. Meanwhile, the RMSE and MAE are 31.45 and 23.25, respectively, which indicates that the prediction model has a low prediction deviation.
- (2)
- The PSO adopted for parameter optimization of σ and γ that are embedded in the LSSVM model, can quickly find the optimal parameters to effectively assist the LSSVM model in prediction work.
- (3)
- The proposed PSO-LSSVM model can predict the yield strength and displacement of RC columns with high efficiency and accuracy by comparing them with the LSSVM models optimized by other metaheuristic algorithms.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Study | Model | Issues or Objectives |
---|---|---|
Xu et al. [18] | LSSVM | Predict the strength of radiation-shielding concrete |
Vu and Hoang [19] | LSSVM | Predict the ultimate punching shear capacity of FRP-RC slabs. |
Luo and Paal [20] | ML-BCV | Predict backbone curves of RC columns |
Mangalathu and Jeon [21] | MLs | Predict shear strength for RC beam-column joints |
Ning et al. [22] | DE-ANN | Predict the hysteresis loop of RC columns |
Influencing Factors | Parameters | Description |
---|---|---|
Geometries | Dc | Section size (mm) |
ρr = L/Dc | Span-to-depth ratio (%) | |
SecT | Section type | |
Materials | fc | Strength of concrete (MPa) |
fyl | Yield strength of longitudinal bars (MPa) | |
fyv | Yield strength of transverse bars (MPa) | |
Reinforcement | ρl | Longitudinal reinforcement ratio (%) |
ρv | Transverse reinforcement ratio (%) | |
Loading | ρn = P/(Ag·fc) | Axial load ratio |
Variables | Min. | Max. | Mean | Std. |
---|---|---|---|---|
Dc | 80 | 1520 | 345.85 | 149.98 |
ρr | 1 | 10 | 3.44 | 1.65 |
fc | 16 | 118 | 46.54 | 25.98 |
fyl | 0 | 587.1 | 422.84 | 72.41 |
fyv | 0 | 1424 | 454.52 | 206.60 |
ρl | 0.0046 | 0.0603 | 0.03 | 0.01 |
ρv | 0 | 4.27 | 0.40 | 0.69 |
ρn | −0.099 | 0.9 | 0.22 | 0.18 |
Fy | 16.27 | 2654.11 | 202.49 | 204.25 |
Δy | 0.54 | 114.70 | 12.64 | 14.55 |
σ | γ | |
---|---|---|
Fy | 0.47 | 195.45 |
Δy | 0.43 | 49.93 |
Model | Training | Testing | |||||||
---|---|---|---|---|---|---|---|---|---|
RMSE | MAE | R2 | EV | RMSE | MAE | R2 | EV | ||
Fy | LSSVM | 25.41 | 15.65 | 0.9859 | 98.5933 | 43.29 | 26.58 | 0.9248 | 92.6090 |
GA-LSSVM | 20.61 | 13.41 | 0.9907 | 99.0746 | 31.92 | 23.93 | 0.9591 | 95.9857 | |
ABC-LSSVM | 20.54 | 13.25 | 0.9908 | 99.0802 | 31.60 | 23.38 | 0.9599 | 96.0527 | |
DE-LSSVM | 19.90 | 12.94 | 0.9913 | 99.1371 | 31.42 | 23.17 | 0.9603 | 96.1449 | |
PSO-LSSVM | 20.53 | 13.22 | 0.9907 | 99.0821 | 31.45 | 23.25 | 0.9601 | 96.0898 | |
Δy | LSSVM | 2.067 | 1.282 | 0.9775 | 99.7536 | 7.870 | 2.895 | 0.7848 | 78.5773 |
GA-LSSVM | 1.800 | 1.108 | 0.9830 | 98.2953 | 4.647 | 2.87 | 0.9249 | 92.5010 | |
ABC-LSSVM | 1.876 | 1.168 | 0.9815 | 98.1484 | 4.943 | 2.674 | 0.9151 | 91.5142 | |
DE-LSSVM | 1.831 | 1.139 | 0.9823 | 98.2360 | 4.975 | 2.686 | 0.9140 | 91.4036 | |
PSO-LSSVM | 1.795 | 1.140 | 0.9832 | 98.3245 | 2.798 | 1.822 | 0.9728 | 97.2999 |
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Wang, B.; Gong, W.; Wang, Y.; Li, Z.; Liu, H. Prediction of the Yield Strength of RC Columns Using a PSO-LSSVM Model. Appl. Sci. 2022, 12, 10911. https://doi.org/10.3390/app122110911
Wang B, Gong W, Wang Y, Li Z, Liu H. Prediction of the Yield Strength of RC Columns Using a PSO-LSSVM Model. Applied Sciences. 2022; 12(21):10911. https://doi.org/10.3390/app122110911
Chicago/Turabian StyleWang, Bochen, Weiming Gong, Yang Wang, Zele Li, and Hongyuan Liu. 2022. "Prediction of the Yield Strength of RC Columns Using a PSO-LSSVM Model" Applied Sciences 12, no. 21: 10911. https://doi.org/10.3390/app122110911
APA StyleWang, B., Gong, W., Wang, Y., Li, Z., & Liu, H. (2022). Prediction of the Yield Strength of RC Columns Using a PSO-LSSVM Model. Applied Sciences, 12(21), 10911. https://doi.org/10.3390/app122110911