Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques
Abstract
:1. Introduction
2. Artificial Intelligence (AI) Techniques
2.1. Genetic Algorithm GA
2.2. Genetic Programming (GP)
2.3. Evolutionary Polynomial Regression EPR
2.4. Artificial Neural Network (ANN)
3. Methodology
3.1. Research Program
3.1.1. Phase 1: Constitutive FEM Models
- Top layer (soil type S1 to soil type S6)
- Bottom layer (soil type S1 to soil type S6)
- Width of strip footing (B) (1.0 m to 5.0 m)
- Top layer thickness (h) (0.5 B to 1.0 B)
- Overburden stress (σ′v) (1.0 m to 3.0 m by the top density γ′t)
3.1.2. Phase 2: Evaluate (a and b) Factors, Generate the Database and Conduct Statistical Analysis
- Cohesion, tangent of friction angle and effective density of top layer (Ct) kN/m2, tan (φt) and (γ′t) kN/m3, respectively.
- Top layer thickness (h) m,
- Cohesion, tangent of friction angle and effective density of bottom layer (Cb) kN/m2, tan (φb) and (γ′b) kN/m3, respectively.
- Strip footing width (B) m,
- Effective over burden stress at foundation depth (σ′v) kN/m2,
- 1000 × hyperbolic factor (a),
- 1000 × hyperbolic factor (b).
3.1.3. Phase 3: Predicting (a and b) Values Using AI Techniques
4. Results and Discussion of the Predictive Models
4.1. Results Presentation
4.1.1. Model (1)—Using (ANN) Technique
4.1.2. Model (2)—Using GP Technique
4.1.3. Model (3)—Using EPR Technique
4.2. Results Discussion
5. Conclusions
- The developed formulas using the GP technique showed a limited accuracy of 50%. All input factors were utilized, except the cohesion of both top and bottom soils (Ct), (Cb).
- EPR technique generated two seven term polynomials out of 1287 possible terms. The accuracy is better than the GP models (65%). In addition, all input factors except the overburden pressure (σ′v) and the cohesion of both the top and bottom soils (Ct), (Cb) were generated.
- Finally, ANN technique presents the best accuracy of 80% and used all the input factors. The relative importance of each factor is indicated by the size of the blocks in Figure 5, and, accordingly, all factors have almost the same effect on the load-settlement curve except (B), tan (φt) and tan (φb), which have a slightly higher effect.
- Both GP and EPR could not capture the influence of soil cohesion on the load-settlement curve, which gives the advantage to the ANN model.
- The developed GP model is not recommended because of its limited accuracy.
- Although the ANN model showed the best accuracy and utilised all input factors, its model is too complicated to be manually handled.
- The developed EPR model could be used for manual calculations, while the ANN model is suitable for computerized calculations
- The developed models should be used within the factor values considered in the study. The prediction accuracy must be verified beyond this range.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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No | Investigator | Year | Suggested Formula |
---|---|---|---|
1 | Winkler | 1867 | |
2 | Biot | 1937 | |
3 | Terzaghi | 1955 | |
4 | Vesic | 1961 | |
5 | Meyerhof and Baike | 1965 | |
6 | Selvadurai | 1984 | |
7 | Bowles | 1988 |
Soil Type | Soil Description | C (kN/m2) | φ (°) | γ (kN/m3) | E (MN/m2) | υ |
---|---|---|---|---|---|---|
S 1 | loose Sand | 0.0 | 29 | 16 | 9.0 | 0.350 |
S 2 | Dense Sand | 0.0 | 38 | 20 | 50.0 | 0.300 |
S 3 | Soft Clay | 25 | 0.0 | 14 | 1.5 | 0.450 |
S 4 | Stiff Clay | 100 | 0.0 | 20 | 10.0 | 0.350 |
S 5 | Soft Silt | 25 | 5 | 18 | 6.0 | 0.400 |
S 6 | Stiff Silt | 100 | 20 | 20 | 30.0 | 0.330 |
Ct | tan (φt) | γ′t | h | Cb | tan (φb) | γ′b | B | σ′v | 1000a | 1000b | |
---|---|---|---|---|---|---|---|---|---|---|---|
kN/m2 | - | kN/m3 | m | kN/m2 | - | kN/m3 | m | kN/m2 | kN/m2 | kN/m2 | |
Training set | |||||||||||
Min. | 0.1 | 0.0 | 14.0 | 0.5 | 0.1 | 0.0 | 14.0 | 1.0 | 18.0 | 0.343 | 0.086 |
Max. | 100 | 1 | 20 | 5 | 100 | 1 | 20 | 5 | 54 | 6.340 | 1.880 |
Avg. | 35.6 | 0.4 | 18.1 | 2.1 | 37.4 | 0.3 | 17.8 | 3.0 | 35.0 | 1.850 | 0.327 |
SD | 46.0 | 0.3 | 2.2 | 1.3 | 41.2 | 0.3 | 2.4 | 1.6 | 14.7 | 1.610 | 0.272 |
VAR | 1.29 | 0.87 | 0.12 | 0.61 | 1.10 | 1.02 | 0.13 | 0.54 | 0.42 | 0.870 | 0.832 |
Validation set | |||||||||||
Min. | 0.1 | 0.0 | 14.0 | 0.5 | 0.1 | 0.0 | 14.0 | 1.0 | 18.0 | 0.372 | 0.094 |
Max. | 100 | 1 | 20 | 5 | 100 | 1 | 20 | 5 | 54 | 6.450 | 1.920 |
Avg. | 40.6 | 0.3 | 18.4 | 1.8 | 39.5 | 0.3 | 17.6 | 2.7 | 38.4 | 1.880 | 0.347 |
SD | 46.9 | 0.3 | 2.0 | 1.2 | 39.9 | 0.3 | 2.5 | 1.5 | 14.5 | 1.540 | 0.263 |
VAR | 1.15 | 0.97 | 0.11 | 0.67 | 1.01 | 1.18 | 0.14 | 0.56 | 0.38 | 0.819 | 0.758 |
Ct | tan (φt) | γ′t | h | Cb | tan (φb) | γ′b | B | σ′v | 1000a | 1000b | |
---|---|---|---|---|---|---|---|---|---|---|---|
Ct | 1.00 | ||||||||||
tan (φt) | −0.66 | 1.00 | |||||||||
γ′t | 0.59 | 0.05 | 1.00 | ||||||||
h | −0.21 | 0.40 | 0.07 | 1.00 | |||||||
Cb | 0.06 | 0.08 | 0.13 | 0.04 | 1.00 | ||||||
tan (φb) | 0.09 | 0.02 | 0.02 | 0.05 | −0.55 | 1.00 | |||||
γ′b | 0.09 | 0.08 | 0.15 | 0.08 | 0.43 | 0.32 | 1.00 | ||||
B | 0.02 | −0.01 | −0.01 | 0.88 | 0.02 | 0.01 | 0.04 | 1.00 | |||
σ′v | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | ||
1000a | −0.12 | −0.50 | −0.55 | −0.30 | −0.20 | −0.29 | −0.38 | −0.07 | −0.18 | 1.00 | |
1000b | −0.15 | −0.08 | −0.30 | 0.41 | −0.19 | −0.25 | −0.47 | 0.56 | 0.01 | 0.13 | 1.00 |
Hidden | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
H1 | H2 | H3 | H4 | H5 | H6 | H7 | H8 | H9 | H10 | |||
Input Layer | (Bias) | −0.96 | 1.31 | −0.01 | −1.42 | −0.34 | 0.29 | −0.39 | 0.17 | −0.59 | −1.54 | |
Ct | −0.27 | 0.63 | 0.75 | −0.07 | 0.12 | 0.22 | −0.15 | −0.41 | 0.00 | −0.1 | ||
tan (φt) | 0.31 | 0.36 | 0.41 | 1.16 | 0.17 | −0.03 | −0.45 | 0 | −0.3 | −0.36 | ||
γ′t | −0.74 | 0.51 | −0.16 | −0.63 | 0.00 | −0.28 | −0.09 | 0.41 | −0.34 | −0.49 | ||
h | 0.15 | −0.65 | 0.23 | 0.02 | −0.04 | −0.22 | −0.11 | 0.4 | −0.35 | −0.15 | ||
Cb | 0.32 | −0.4 | 0.44 | 0.33 | −0.78 | 0.00 | 0.24 | −0.06 | 0.08 | −0.37 | ||
tan (φb) | 1.24 | −0.94 | 0.56 | 0.47 | −0.8 | −0.32 | −0.9 | −0.71 | −0.49 | −0.27 | ||
γ′b | −0.26 | −0.83 | 0.26 | 0.61 | −1.04 | 0.32 | −1.11 | −0.42 | 0.6 | −0.17 | ||
B | 0.1 | 0.25 | −0.15 | 0.15 | 0.37 | 0.5 | 0.02 | −0.55 | 0.5 | −0.01 | ||
σ′v | 0.45 | 0.26 | −0.21 | −0.26 | −0.01 | −0.24 | −0.6 | 0.37 | 0.02 | −0.27 | ||
Hidden | ||||||||||||
H1 | H2 | H3 | H4 | H5 | H6 | H7 | H8 | H9 | H10 | (Bias) | ||
Output | 1000a | −0.81 | −1.33 | −0.51 | −1.28 | −0.16 | 0.15 | 0.12 | 0.09 | 0.03 | 0.92 | −0.49 |
1000b | 0.33 | 0.73 | −0.07 | 0.78 | 0.62 | 0.04 | −0.15 | −0.33 | 0.46 | 0.06 | −0.18 |
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Ebid, A.M.; Onyelowe, K.C.; Salah, M. Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques. Designs 2022, 6, 104. https://doi.org/10.3390/designs6060104
Ebid AM, Onyelowe KC, Salah M. Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques. Designs. 2022; 6(6):104. https://doi.org/10.3390/designs6060104
Chicago/Turabian StyleEbid, Ahmed M., Kennedy C. Onyelowe, and Mohamed Salah. 2022. "Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques" Designs 6, no. 6: 104. https://doi.org/10.3390/designs6060104
APA StyleEbid, A. M., Onyelowe, K. C., & Salah, M. (2022). Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques. Designs, 6(6), 104. https://doi.org/10.3390/designs6060104