Tuan Anh Pham This email address is being protected from spambots. You need JavaScript enabled to view it.1, Huong-Lan Thi Vu1, and Hong-Anh Thi Duong1

1University of Transport Technology, Hanoi 100000, Vietnam


Received: March 9, 2021
Accepted: August 16, 2021
Publication Date: September 10, 2021

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||https://doi.org/10.6180/jase.202204_25(2).0012  


Ultimate bearing capacity is one of the most important parameters in designing shallow foundations. This study focused on developing a hybrid model using Random Search (RS) technique and Deep Neural Network (DNN) to predict the maximum bearing capacity of shallow foundations in sandy soil. The data included 97 load tests on the steps that were used to train and test the model. This data is divided into two parts of the training data set (7%) and the testing set (30%) to build and validate the corresponding models. The performance of the final DNN model is comprehensively assessed with a random hyper-parameters DNN model developed independently using the same data. The values of performance evaluation measures such as R-squared (R2), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE) and the Variance Accounted For (VAF) are used to determine to get the performance of the DNN model in predicting the ultimate bearing capacity of shallow foundations. In addition, a parallel coordinate plot is utilized to show and evaluate the effect of hyperparameters combination on the performance of DNN model. Besides, a global sensitivity analysis technique was deployed to detect the most important input variables in predicting the ultimate bearing capacity of shallow foundations. This study can provide an effective tool to identify the ultimate bearing capacity of shallow foundations.

Keywords: Deep Neural Network; hyperparameters tuning; shallow foundations; sensitive analysis


  1. [1] H. A. Taiebat and J. P. Carter, (2000) “Numerical studies of the bearing capacity of shallow foundations on cohesive soil subjected to combined loading" Géotechnique 50(4): 409–418.
  2. [2] D. Raj and M. Bharathi, (2013) “Bearing Capacity of Shallow Foundation on Slope: A Review" Proc. GGWUIP India: Ludhiana:
  3. [3] H. T. Eid, (2013) “Bearing capacity and settlement of skirted shallow foundations on sand" International Journal of Geomechanics 13(5): 645–652.
  4. [4] P. Samui and T. G. Sitharam, (2008) “Least-square support vector machine applied to settlement of shallow foundations on cohesionless soils" International Journal for Numerical and Analytical Methods in Geomechanics 32(17): 2033–2043.
  5. [5] S. S. Tezcan, A. Keceli, and Z. Ozdemir, (2006) “Allowable bearing capacity of shallow foundations based on shear wave velocity" Geotechnical & Geological Engineering 24(1): 203–218.
  6. [6] K. Terazaghi, (1965) “Theoretical soil mechanics" John Wiley and Sons:
  7. [7] A. Vesic, (1975) “Bearing Capacity of Shallow Foundations, Foundation Engineering Handbook, ed.Winterkorn, FS and Fand, HY":
  8. [8] B. Hansen, (1961) “A general formula for bearing capacity" Danish Geotechnical Institute, Bulletin 11:38–46.
  9. [9] G. G. Meyerhof, (1963) “Some recent research on the bearing capacity of foundations" Canadian Geotechnical Journal 1(1): 16–26.
  10. [10] S. M. Gourvenec, C. Vulpe, and T. G. MURTHY, (2014) “A method for predicting the consolidated undrained bearing capacity of shallow foundations" Géotechnique 64(3): 215–225.
  11. [11] M. Ghavami, M. M. Tamizdoust, and O. Ghasemi-Fare, (2019) “Determination of allowable bearing capacity of shallow foundation using modified hyperbolic stressstrain model" Journal of Applied Geophysics 166: 1–9.
  12. [12] M. M. Nujid and M. R. Taha, (2014) “A review of bearing capacity of shallow foundation on clay layered soils using numerical method" Electronic Journal Geotechnical Engineering 19: 811–825.
  13. [13] G. Gandhi. “Study of bearing capacity factors developed from lab. Experiments on shallow footings on cohesionless soils". (phdthesis). 2003.
  14. [14] A. Azami, S. Pietruszczak, and P. Guo, (2010) “Bearing capacity of shallow foundations in transversely isotropic granular media" International Journal for Numerical and Analytical Methods in Geomechanics 34(8): 771–793.
  15. [15] S. D. Nielsen, L. B. Ibsen, and B. N. Nielsen, (2016) “Advanced laboratory setup for testing offshore foundations" Geotechnical Testing Journal 39(4): 543–556.
  16. [16] T. TATSUOKA, (1991) “Progressive failure and particle size effect in bearing capacity of footing on sand" ASCE Geotechnical Special Publication 27: 788–802.
  17. [17] Y. M. Chew, K. S. Ng, and S. F. Ng, (2015) “The effect of soil variability on the ultimate bearing capacity of shallow foundation" Journal of Engineering Science and Technology 10: 1–13.
  18. [18] S. A. Ziaee, E. Sadrossadat, A. H. Alavi, and D. M. Shadmehri, (2015) “Explicit formulation of bearing capacity of shallow foundations on rock masses using artificial neural networks: application and supplementary studies" Environmental earth sciences 73(7): 3417–3431.
  19. [19] M. Omar, K. Hamad, M. Al Suwaidi, and A. Shanableh, (2018) “Developing artificial neural network models to predict allowable bearing capacity and elastic settlement of shallow foundation in Sharjah, United Arab Emirates" Arabian Journal of Geosciences 11(16): 464.
  20. [20] V. R. Kohestani, M. Vosoghi, M. Hassanlourad, and M. Fallahnia, (2017) “Bearing capacity of shallow foundations on cohesionless soils: A random forest based approach" Civil Engineering Infrastructures Journal 50(1): 35–49.
  21. [21] A. H. Alavi and E. Sadrossadat, (2016) “New design equations for estimation of ultimate bearing capacity of shallow foundations resting on rock masses" Geoscience Frontiers 7(1): 91–99.
  22. [22] M. A. Shahin, M. B. Jaksa, and H. R. Maier, (2001) “Artificial neural network applications in geotechnical engineering" Australian geomechanics 36(1): 49–62.
  23. [23] M. A. Shahin, H. R. Maier, and M. B. Jaksa, (2002) “Predicting settlement of shallow foundations using neural networks" Journal of Geotechnical and Geoenvironmental Engineering 128(9): 785–793.
  24. [24] H. Shahnazari, M. A. Shahin, and M. A. Tutunchian, (2014) “Evolutionary-based approaches for settlement prediction of shallow foundations on cohesionless soils" International journal of civil engineering 12(1): 55–64.
  25. [25] R. Mohanty and S. K. Das, (2018) “Settlement of shallow foundations on cohesionless soils based on SPT value using multi-objective feature selection" Geotechnical and Geological Engineering 36(6): 3499–3509.
  26. [26] M. Bagi´nska and P. E. Srokosz, (2019) “The optimal ANN Model for predicting bearing capacity of shallow foundations trained on scarce data" KSCE Journal of Civil Engineering 23(1): 130–137.
  27. [27] L. Liu, H. Moayedi, A. S. A. Rashid, S. S. A. Rahman, and H. Nguyen, (2020) “Optimizing an ANN model with genetic algorithm (GA) predicting load-settlement behaviours of eco-friendly raft-pile foundation (ERP) system" Engineering with Computers 36(1): 421–433.
  28. [28] A. H. Gandomi, S. M. Tabatabaei, M. H. Moradian, A. Radfar, and A. H. Alavi, (2011) “A new prediction model for the load capacity of castellated steel beams" Journal of Constructional Steel Research 67(7): 1096–1105.
  29. [29] A. Saha, S. Nama, and S. Ghosh, (2019) “Application of HSOS algorithm on pseudo-dynamic bearing capacity of shallow strip footing along with numerical analysis" International Journal of Geotechnical Engineering: 1–14.
  30. [30] A. Soleimanbeigi and N. Hataf, (2005) “Predicting ultimate bearing capacity of shallow foundations on reinforced cohesionless soils using artificial neural networks" Geosynthetics International 12(6): 321–332.
  31. [31] E. Sadrossadat, F. Soltani, S. M. Mousavi, S. M. Marandi, and A. H. Alavi, (2013) “A new design equation for prediction of ultimate bearing capacity of shallow foundation on granular soils" Journal of Civil Engineering and Management 19(sup1): S78–S90.
  32. [32] P. Pakdel, R. Jamshidi Chenari, and M. Veiskarami, (2019) “An estimate of the bearing capacity of shallow foundations on anisotropic soil by limit equilibrium and soft computing technique" Geomechanics and Geoengineering 14(3): 202–217.
  33. [33] H. Rezaei, R. Nazir, and E. Momeni, (2016) “Bearing capacity of thin-walled shallow foundations: an experimental and artificial intelligence-based study" Journal of Zhejiang University-SCIENCE A 17(4): 273–285.
  34. [34] O. E. David and I. Greental. “Genetic algorithms for evolving deep neural networks”. en. In: [Online; accessed 2020-03-17]. Vancouver, BC, Canada: ACM Press, 2014, 1451–1452. DOI: 10.1145/2598394.2602287.
  35. [35] R. Zemouri, N. Omri, F. Fnaiech, N. Zerhouni, and N. Fnaiech, (2019) “A new growing pruning deep learning neural network algorithm (GP-DLNN)" Neural Computing and Applications: 1–17.
  36. [36] T. A. Pham, V. Q. Tran, H.-L. T. Vu, and H.-B. Ly, (2020) “Design deep neural network architecture using a genetic algorithm for estimation of pile bearing capacity" PLOS ONE 15(12): e0243030. DOI: 10.1371/journal. pone.0243030.
  37. [37] T. A. Pham, V. Q. Tran, and H.-L. T. Vu, (2021) “Evolution of Deep Neural Network Architecture Using Particle Swarm Optimization to Improve the Performance in Determining the Friction Angle of Soil" Mathematical Problems in Engineering 2021: 1–17. DOI: 10.1155/2021/5570945.
  38. [38] W. Liu, Z. Wang, X. Liu, N. Zeng, Y. Liu, and F. E. Alsaadi, (2017) “A survey of deep neural network architectures and their applications" Neurocomputing 234:11–26.
  39. [39] K. Aljanabi, (2008) “Prediction of Ultimate Bearing Capacity of Shallow Foundations on Cohesionless Soils Using Back Propagation Neural Networks (BPNN)" Iraqi Journal of Civil Engineering IJCE 8th year: 162–176.
  40. [40] H. Guo, J. Zhou, M. Koopialipoor, D. Jahed Armaghani, and M. M. Tahir, (2021) “Deep neural network and whale optimization algorithm to assess flyrock induced by blasting" Engineering with Computers 37(1): 173–186. DOI: 10.1007/s00366-019-00816-y.
  41. [41] W. Yong, J. Zhou, D. Jahed Armaghani, M. M. Tahir, R. Tarinejad, B. T. Pham, and V. Van Huynh, (2020) “A new hybrid simulated annealing-based genetic programming technique to predict the ultimate bearing capacity of piles" Engineering with Computers: DOI: 10.1007/s00366-019-00932-9.
  42. [42] J. Zhou, Y. Qiu, S. Zhu, D. J. Armaghani, C. Li, H. Nguyen, and S. Yagiz, (2021) “Optimization of support vector machine through the use of metaheuristic algorithms in forecasting TBM advance rate" Engineering Applications of Artificial Intelligence 97: 104015. DOI: 10.1016/j.engappai.2020.104015.
  43. [43] Y. Bengio, (2009) “Learning Deep Architectures for AI" Found. Trends Mach. Learn. 2(1): 1–127. DOI: 10.1561/2200000006.
  44. [44] A. T. C. Goh, (1995) “Back-propagation neural networks for modeling complex systems" Artificial Intelligence in Engineering 9(3): 143–151. DOI: 10.1016/0954-1810(94)00011-S.
  45. [45] K. Hornik, (1991) “Approximation capabilities of multilayer feedforward networks" Neural Networks 4(2): 251–257. DOI: 10.1016/0893-6080(91)90009-T.
  46. [46] J. Bergstra and Y. Bengio, (2012) “Random Search for Hyper-Parameter Optimization" Journal of Machine Learning Research 13(10): 281–305.
  47. [47] P. Liashchynskyi and P. Liashchynskyi, (2019) “Grid Search, Random Search, Genetic Algorithm: A Big Comparison for NAS" arXiv:1912.06059 [cs, stat]:
  48. [48] T. Chai and R. R. Draxler, (2014) “Root mean square error (RMSE) or mean absolute error (MAE)? – Arguments against avoiding RMSE in the literature" Geoscientific Model Development 7(3): 1247–1250. DOI:10.5194/gmd-7-1247-2014.
  49. [49] A. Colin Cameron and F. A. G. Windmeijer, (1997) “An R-squared measure of goodness of fit for some common nonlinear regression models" Journal of Econometrics 77(2): 329–342. DOI: 10.1016/S0304-4076(96)01818-0.
  50. [50] R. A. Peterson, (2000) “A Meta-Analysis of Variance Accounted for and Factor Loadings in Exploratory Factor Analysis" Marketing Letters 11(3): 261–275. DOI: 10.1023/A:1008191211004.
  51. [51] H. Muhs and K. Weiß. Untersuchung von Grenztragfähigkeit und Setzungsverhalten Flachgegründeter Einzelfundamente im ungleichförmigen nichtbindigen Boden. Berlin: Ernst W. + Sohn Verlag, 1, 1982.
  52. [52] K. Weiß. Der Einfluß der Fundamentform auf die Grenztragfähigkeit flachgegründeter Fundamente, Untersuchungen ausgef.... von Klaus Weiß: mit 14 Zahlentaf. Ernst, 1970.
  53. [53] H. Muhs, R. Elmiger, and K. Weiß. Sohlreibung und Grenztragfähigkeit unter lotrecht und schräg belasteten Einzelfundamenten; mit 128 Bildern und 13 Zahlentafeln. Ernst, 1969.
  54. [54] M. H and W. K, (1973) “Inclined load tests on shallow strip footings" Proceedings of the 8th international conference on soil mechanism and foundation engineering II: 173–9.
  55. [55] J.-L. Briaud and R. Gibbens, (1999) “Behavior of Five Large Spread Footings in Sand" Journal of Geotechnical and Geoenvironmental Engineering 125(9):787–796. DOI: 10.1061/(ASCE)1090-0241(1999)125:9(787).
  56. [56] M. Maccarini, (1993) “A comparison of direct shear box tests with triaxial compression tests for a residual soil" Geotechnical and Geological Engineering 11(2): 69–80. DOI: 10.1007/BF00423336.
  57. [57] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: A Simple Way to Prevent Neural Networks from Overfitting": 30.
  58. [58] I. M. Sobol, (2001) “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates" Mathematics and Computers in Simulation 55(1): 271–280. DOI: 10.1016/S0378-4754(00)00270-6.
  59. [59] A. Saltelli, (2002) “Making best use of model evaluations to compute sensitivity indices" Computer Physics Communications 145(2): 280–297. DOI: 10.1016/S0010-4655(02)00280-1.


33rd percentile
Powered by  Scopus

SCImago Journal & Country Rank

Enter your name and email below to receive latest published articles in Journal of Applied Science and Engineering.