Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models
Abstract
:1. Introduction
- First, the influence of using steel fibers on the SFRC’s basic mechanical properties and the reinforced SFRC beam’s shear bearing capacity are briefly reviewed. In addition, the database of SFRC beams containing no stirrups established by Lantsoght [28] is succinctly analyzed.
- A gray relational analysis is then performed to identify the parameter importance.
- AI models, including back-propagation artificial neural network (BPANN), random forest (RF) and multi-gene genetic programming (MGGP), are developed to simulate the shear strength of the reinforced SFRC beam without stirrups.
- A parametric study is finally carried out to validate and explain the AI models.
2. Literature Review, Experimental Database and Typical Prediction Models
2.1. Literature Review
2.2. Experimental Database for Shear Testing of SFRC Beams without Stirrups
2.3. Assessing Existing Prediction Models for Shear Capacity of SFRC Beams without Stirrups
3. Parameter Sensitivity Evaluation Using GRA
3.1. Gray Relational Analysis Principle
3.2. Assessment of Parameter Sensitivity
- (a)
- The values of the gray relational factor (λ) for concrete strength fc and maximum aggregate size smax are 87.79% and 82.61%, respectively. The value for the fiber-related parameter F is 84.23%. These results suggest that, compared with steel fibers, concrete has a greater bearing on the shear strength. This is probably because concrete can resist external loads throughout the entire loading course [49]. Meanwhile, with the widening of the critical crack, the role of steel fiber will be weakened [30].
- (b)
- The shear contribution provided by the longitudinal reinforcement cannot be ignored. This can be stressed by assigning ρs with secondary importance (λ = 86.69%).
- (c)
- Additionally, the parameters d/b and a/d have notable effects on the shear capacity of SFRC beams containing no stirrups, where they are registered with λ values of 85.69% and 81.96%, respectively. This phenomenon is consistent with that observed in [30].
4. Shear Capacity Prediction Using Artificial Intelligence Models
4.1. Back-Propagation Artificial Neural Network (BPANN)
4.2. Random Forest (RF)
4.3. Multi-Gene Genetic Programming (GP)
4.4. Prediction Results and Discussion
5. Parametric Study
5.1. Influence of the SFRC’s Compressive Strength
5.2. Influence of the Shear Span-to-Effective Depth Ratio
5.3. Effect of the Area Ratio of Longitudinal Reinforcement
5.4. Effect of the Maximum Aggregate Size
5.5. Effect of the Fiber Factor
5.6. Effect of the Sectional Effective Depth-to-Width Ratio
6. Conclusions
- (1)
- The empirical strength models evaluated in this paper cannot predict with desirable accuracy the shear bearing capacity of SFRC beams without stirrups. There are a number of reasons for this, including the inadequate account of the role of steel fibers (such as the effective fiber distributed area along the critical diagonal shear crack, the total amount of fibers in this area, the fiber type and orientation, and the individual fiber pull-out load–slip relationship) and the limited database that the models were derived from.
- (2)
- The GRA results indicate that the shear strength of the beams depends crucially on the following parameters: the material properties of concrete, the amount of longitudinal reinforcement, the attributes of steel fibers, and the geometrical and loading characteristics of SFRC beams in shear. The λ values of these parameters are all greater than 80%, indicating that all of these parameters should be considered for a more rational prediction of beam shear strength. Unfortunately, none of the empirical models evaluated take these parameters into full account.
- (3)
- The three AI models—BPANN, RF and MGGP—are effective in predicting the shear capacity of SFRC beams without stirrups. Their predictive performance is excellent, with all R2 values higher than 0.95. By contrast, RF slightly surpasses BPANN and MGGP (mainly because RF is an ensemble learning method, which combines the results of multiple weak learners), while MGGP provides an unambiguous design expression. The AI models fit the experimental data in both the training and testing sets, showing good generalization capacity within the range of the data collected.
- (4)
- The AI models were used to perform a parametric study to strengthen support for experimental trends. The results show that these models reveal the potential effects of all of the important factors affecting the shear capacity, and these effects can be reasonably explained. Therefore, the developed AI models can be used as fast, accurate and simulation-free tools for designing SFRC beams without stirrups.
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Parameter | d/b | a/d | fc (MPa) | smax (mm) | ρs (%) | F (%) | vu (MPa) |
---|---|---|---|---|---|---|---|
Maximum | 4.90 | 6.00 | 215.0 | 22.0 | 5.72 | 285.75 | 13.96 |
Mean | 1.81 | 2.92 | 49.0 | 10.7 | 2.46 | 53.95 | 3.64 |
Minimum | 0.42 | 0.46 | 9.8 | 0.4 | 0.37 | 7.50 | 0.60 |
Standard deviation | 0.77 | 0.98 | 25.2 | 5.1 | 1.01 | 35.96 | 2.14 |
Standard error | 0.03 | 0.04 | 1.1 | 0.2 | 0.05 | 1.63 | 0.10 |
Median | 1.57 | 3.00 | 40.3 | 10.0 | 2.54 | 48.75 | 3.01 |
Mode | 1.26 | 2.00 | 33.2 | 10.0 | 3.09 | 60.00 | 2.61 |
Kurtosis | 3.99 | 0.37 | 7.8 | −0.1 | 0.98 | 7.14 | 4.72 |
Skewness | 1.94 | 0.00 | 2.2 | 0.0 | 0.77 | 2.02 | 2.06 |
Reference | Equation |
---|---|
CECS38-2004 [41] | |
DAfStB-2012 [42] | |
Fib-2010 [43] | |
Greenough and Nehdi [44] | |
Imam et al. [45] | |
Kuntia et al. [46] | |
Sharma [47] | |
Yakoub [48] |
Parameter Definition | Setting |
---|---|
Population size | 1000 |
Number of generations | 1000 |
Max number of genes | 10 |
Max genes’ tree depth | 6 |
Function set | plus, minus, times, divide, sqrt, square, cube, sin |
Tournament size | 20 |
Elitism | 5% of population |
Probability of crossover event | 0.85 |
Probability of mutation event | 0.10 |
Probability of reproduction event | 0.05 |
Term | Value |
---|---|
Bias | 53.4 |
Gene 1 | |
Gene 2 | |
Gene 3 | |
Gene 4 | |
Gene 5 | |
Gene 6 | |
Gene 7 | |
Gene 8 | |
Gene 9 | |
Gene 10 |
Group | Influence Parameters | vu Predicted by Different Methods | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
No. | [d/b, a/d, fc, smax, ρs, F] | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
I | [2, 3.0, 30, 10, 2.5%, 50%] | 1.55 | 1.44 | 1.22 | 2.18 | 2.14 | 1.60 | 2.22 | 1.27 | 2.29 | 2.49 | 2.74 |
[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |
[2, 3.0, 70, 10, 2.5%, 50%] | 2.89 | 2.10 | 1.64 | 2.41 | 2.60 | 2.44 | 3.39 | 1.74 | 2.92 | 3.43 | 3.64 | |
[2, 3.0, 100, 10, 2.5%, 50%] | 3.34 | 2.45 | 1.89 | 2.52 | 2.85 | 2.92 | 4.05 | 2.01 | 2.93 | 3.67 | 3.68 | |
II | [2, 0.5, 50, 10, 2.5%, 50%] | 3.81 | 1.81 | 1.45 | 3.93 | 100.1 | 2.06 | 4.48 | 35.05 | 12.80 | 11.49 | 14.59 |
[2, 1.0, 50, 10, 2.5%, 50%] | 3.81 | 1.81 | 1.45 | 3.17 | 18.73 | 2.06 | 3.77 | 7.21 | 8.40 | 7.26 | 9.11 | |
[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |
[2, 6.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 1.94 | 1.47 | 2.06 | 2.41 | 1.22 | 2.55 | 2.62 | 3.22 | |
III | [2, 3.0, 50, 10, 0.5%, 50%] | 2.38 | 1.41 | 0.85 | 1.58 | 1.04 | 2.06 | 2.86 | 0.77 | 1.77 | 1.65 | 1.67 |
[2, 3.0, 50, 10, 1.5%, 50%] | 2.38 | 1.66 | 1.22 | 2.03 | 1.81 | 2.06 | 2.86 | 1.22 | 2.20 | 2.72 | 2.61 | |
[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |
[2, 3.0, 50, 10, 5.0%, 50%] | 2.38 | 2.07 | 1.83 | 2.81 | 3.60 | 2.06 | 2.86 | 2.13 | 3.66 | 4.29 | 4.44 | |
IV | [2, 3.0, 50, 2.5, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 1.42 | 2.06 | 2.86 | 0.91 | 3.26 | 3.38 | 3.72 |
[2, 3.0, 50, 5.0, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 1.88 | 2.06 | 2.86 | 1.20 | 3.00 | 3.18 | 3.47 | |
[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 2.85 | 3.37 | |
[2, 3.0, 50, 20, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.88 | 2.06 | 2.86 | 1.84 | 2.47 | 2.63 | 3.30 | |
V | [2, 3.0, 50, 10, 2.5%, 0%] | 1.59 | 1.65 | 1.38 | 1.32 | 1.83 | 1.18 | 2.86 | 1.44 | 1.69 | 2.66 | 1.98 |
[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |
[2, 3.0, 50, 10, 2.5%, 100%] | 3.17 | 1.98 | 1.51 | 3.27 | 2.54 | 2.95 | 2.86 | 1.62 | 3.41 | 3.71 | 3.86 | |
[2, 3.0, 50, 10, 2.5%, 200%] | 4.76 | 2.30 | 1.62 | 5.10 | 2.72 | 4.72 | 2.86 | 1.79 | 3.98 | 3.94 | 4.27 | |
VI | [0.5, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 2.39 | 2.04 | 3.09 | 3.26 | 2.06 | 2.86 | 2.09 | 3.51 | 3.69 | 4.74 |
[1.0, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 2.03 | 1.69 | 2.63 | 2.88 | 2.06 | 2.86 | 1.84 | 3.21 | 3.29 | 3.59 | |
[2.0, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 2.85 | 3.37 | |
[5.0, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.67 | 1.25 | 2.03 | 1.73 | 2.06 | 2.86 | 1.10 | 2.08 | 2.40 | 2.93 |
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Yu, Y.; Zhao, X.-Y.; Xu, J.-J.; Wang, S.-C.; Xie, T.-Y. Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models. Materials 2022, 15, 2407. https://doi.org/10.3390/ma15072407
Yu Y, Zhao X-Y, Xu J-J, Wang S-C, Xie T-Y. Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models. Materials. 2022; 15(7):2407. https://doi.org/10.3390/ma15072407
Chicago/Turabian StyleYu, Yong, Xin-Yu Zhao, Jin-Jun Xu, Shao-Chun Wang, and Tian-Yu Xie. 2022. "Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models" Materials 15, no. 7: 2407. https://doi.org/10.3390/ma15072407
APA StyleYu, Y., Zhao, X. -Y., Xu, J. -J., Wang, S. -C., & Xie, T. -Y. (2022). Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models. Materials, 15(7), 2407. https://doi.org/10.3390/ma15072407