Abstract
Shallow foundations are typically the first option for the foundation engineer due to its lesser construction costs, unless they are deemed inadequate. Determining the bearing capacity of a strip footing under eccentrically inclined loading is crucial in designing foundations. In the design of shallow foundation, machine learning (ML) models have been broadly used to predict the reduction factor (the ratio of ultimate bearing capacity of strip footing under an eccentrically inclined load to the ultimate bearing capacity of strip footing under a centric vertical load) for strip footing resting over granular soil subjected to eccentrically inclined load. Convolutional neural networks (CNN), recurrent neural networks (RNN), and long short-term memory (LSTM) are utilized in this study to predict reduction factor (RF), which will be used to calculate the ultimate bearing capacity of an eccentrically inclined loaded strip footing. By taking into account three crucial inputs (e/B, α/φ and D/B) for predicting reduction factor, these three ML models are applied to 140 datasets. Various performance parameters (R2, VAF, WI, LMI, RMSE, EAE, MAE and U95) are used to evaluate how well the established ML models are being used. Using performance parameters, the results reveal that CNN had the best predictive performance among all three proposed ML models, with the highest value of coefficient of determination (R2) = 0.998 and the lowest value of root mean square error (RMSE) = 0.009 in the training phase and R2 = 0.996 and RMSE = 0.016 in the testing phase. Additionally, rank analysis, regression curve, error matrix, objective function criterion, Akaike information criterion, and performance strength criterion are used to analyze the model’s performance. Seven second-order reliability method (SORM) formulas are used to compute the probability of failure and reliability index and are compared with the failure probability and reliability index computed by first-order reliability method (FORM). An uncertainty study is performed to check the proposed ML models are capable of accurately predicting the outcomes and to evaluate the model’s robustness, external validation is performed. A sensitivity study is also performed to determine the influence of each input parameters on the output. The research finding have a big impact on geotechnical engineering and give academics and engineers new knowledge about how CNN models can be used to determine bearing capacity of strip footings under inclined loading.
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Abbreviations
- ML :
-
Machine learning
- CNN :
-
Convolutional neural network
- RNN :
-
Recurrent neural network
- LSTM :
-
Long short-term memory
- GMDH :
-
Group method of data handling
- RF :
-
Reduction factor
- e :
-
Load eccentricity
- D :
-
Depth of footing
- φ :
-
Angle of shearing resistance
- α :
-
Load inclination
- B :
-
Width of footing
- e/B :
-
Eccentricity ratio
- D/B :
-
Embedment ratio
- α/ φ :
-
Inclination ratio
- FORM :
-
First-order reliability method
- SORM :
-
Second-order reliability method
- BC :
-
Bearing capacity
- R 2 :
-
Coefficient of determination
- R :
-
Coefficient of correlation
- VAF :
-
Variance account factor
- WI :
-
Willmott’s index of agreement
- LMI :
-
Legate and McCabe’s index
- RMSE :
-
Root mean square error
- EAE :
-
Maximum absolute error
- MAE :
-
Mean absolute error
- U 95 :
-
Expanded uncertainty
- MSE :
-
Mean squared error
- E :
-
Coefficient of efficiency
- PI :
-
Performance index
- TMP :
-
Trend measuring parameters
- EMP :
-
Error measuring parameters
- TR :
-
Training
- TS :
-
Testing
- β :
-
Reliability index
- P f :
-
Probability of failure
- ANFIS :
-
Adaptive neuro fuzzy inference system
- FIS :
-
Fuzzy inference system
- ANN :
-
Artificial neural network
- PSO :
-
Particle swarm optimization
- GOA :
-
Grasshopper optimization algorithm
- SSA :
-
Salp swarm algorithm
- ACO :
-
Ant colony optimization
- LCA :
-
League champion optimization
- WOA :
-
Whale optimization algorithm
- MFO :
-
Moth-flame optimization
- OBJ :
-
Objective function criterion
- AIC :
-
Akaike information criterion
- PSC :
-
Performance strength criterion
- SD :
-
Standard deviation
- ME :
-
Mean error
- MOE :
-
Margin of error
- UBW :
-
Uncertainty band width
- LB :
-
Lower bound
- UB :
-
Upper bound
- \({\mathrm{\upepsilon }}_{S}\) :
-
Standard error
- AME :
-
Absolute mean error
- SOR :
-
Strength of relation
- MBE :
-
Mean bias error
- GPR :
-
Gaussian process regression
- MCS :
-
Monte Carlo simulation
- UBC :
-
Ultimate bearing capacity
- ELM :
-
Extreme learning machine
- MPMR :
-
Minimax probability machine regression
- MLR :
-
Multiple linear regression
- CV :
-
Cross validation
- SVM :
-
Support vector machine
- RVM :
-
Relevance vector machine
- MARS :
-
Multivariate adaptive regression splines
- KNN :
-
K-nearest neighbor
- XGBoost :
-
Extreme gradient boosting
- DT :
-
Decision tree
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Rashid Mustafa: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing-original draft, Writing-review & editing; Pijush Samui: Supervision, Validation; Sunita Kumari: Supervision; Danial Jahed Armaghani: Supervision.
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Mustafa, R., Samui, P., Kumari, S. et al. Appraisal of numerous machine learning techniques for the prediction of bearing capacity of strip footings subjected to inclined loading. Model. Earth Syst. Environ. (2024). https://doi.org/10.1007/s40808-024-02008-0
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DOI: https://doi.org/10.1007/s40808-024-02008-0