Abstract
An upper bound (UB) and lower bound (LB) finite element limit analysis cooperating with a machine learning method is adopted as a new solution for predicting the bearing capacity of conical foundations embedded in anisotropic and heterogenous clays. The anisotropic and heterogenous clays are simulated by anisotropic undrained strength (AUS) model for capturing the anisotropic strengths of clays. The bearing capacity of the conical foundation is investigated using the dimensionless parameter approach. The bearing capacity factors, as well as the failure mechanisms of conical foundations, are examined through 1296 numerical cases with changing of four input dimensionless parameters, namely cone apex angle, embedded depth ratio, the anisotropic ratio, and the strength gradient ratio. Based on numerical results, a machine learning technique of multivariate adaptive regression splines (MARS) model is used for accessing the sensitivity of each investigated dimensionless parameter and functioning the relationship between input parameters and output bearing capacity factors. The results of the analysis are prepared in charts, design tables, and empirical equations from MARS. The paper can be the theory guidelines for initial design and provide an effective tool for practitioners in determining the bearing capacity of conical foundation embedded in anisotropic and heterogenous clays.
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Data availability statement
All data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.
Abbreviations
- a 0 :
-
The constant in empirical proposed from MARS
- a n :
-
The coefficient added on basic function
- ANFIS:
-
Adaptive neuro fuzzy inference system, a machine learning technique
- ANN:
-
Artificial neural network, a machine learning technique
- AUS:
-
Anisotropic undrained strength, name of soil model
- AVG:
-
Average
- BF:
-
The basic function
- BFs:
-
The basic functions
- CART:
-
Classification and regression trees, a machine leaning technique
- CHFRF:
-
Cone-shaped hollow flexible reinforced concrete foundation
- D :
-
Diameter
- DT:
-
Decision tree, a machine leaning technique
- ELM:
-
Extreme learning machines, a machine leaning technique
- F :
-
Bearing capacity factor
- f(x):
-
Function
- FEA:
-
Finite element analysis
- FELA:
-
Finite element limit analysis
- FEM:
-
Finite element methods
- GCV:
-
Generalized cross-validation
- g n :
-
The nth of the basic functions
- GPR:
-
Gausian process regression, a machine leaning technique
- GRNN:
-
General regression neural network, a machine leaning technique
- h :
-
Is the penalty factor
- H :
-
Embedded depth
- H/D :
-
The embedded depth ratio
- KNEA:
-
Kernel-based nonlinear extension of Arps decline, a machine leaning technique
- LB:
-
Lower bound
- LR:
-
Linear Regression
- M5Tree:
-
Name of a machine leaning technique
- MARS:
-
Multivariate adaptive regression splines model, a machine leaning technique
- max:
-
Maximum
- MLP:
-
Multi-layer perceptron, a machine leaning technique
- MoC:
-
Method of characteristics
- MSE:
-
Mean squared error
- n :
-
Value
- N :
-
Number of the fitted basic functions
- PI:
-
Plasticity index of clay
- q :
-
Uniform pressure
- R :
-
Radius
- ρ :
-
The gradient of linearly increasing strength
- R :
-
The number of data points
- R 2 :
-
The coefficient of determination
- r e = s uDSS/s uTC :
-
An anisotropic strength ratio between suDSS/suTC
- RF:
-
Radio frequency, a machine leaning technique
- RII:
-
Relative importance index
- r s = s uDSS/s uTC :
-
An anisotropic strength ratio between suDSS/suTC
- RMSE:
-
The root mean square error
- SGBT:
-
Stochastic gradient boosting tress, a machine learning technique
- s uDSS :
-
Shear strengths obtained from direct simple shear
- s uDSS0 :
-
Direct simple shear strength at the ground surface
- s uTC :
-
Shear strengths obtained from triaxial compression
- s uTC0 :
-
Triaxial compression shear strength at the ground surface
- s uTE :
-
Shear strengths obtained from triaxial extension
- s uTE0 :
-
Triaxial extension shear strength at the ground surface
- SVM:
-
Support vector machine, a machine leaning technique
- SVR:
-
Support vector regression
- t :
-
Threshold value in MARS model
- UB:
-
Upper bound
- x :
-
A input variable
- X :
-
Variable
- XGBoost:
-
Extreme gradient boosting, a machine leaning technique
- z :
-
The depth starting from the ground surface
- β :
-
The cone apex angle of the conical foundation
- ρD/s uTC0 :
-
The increasing strength gradient ratio
- Σ :
-
Sum
References
Byrne B, Houlsby G (2003) Foundations for offshore wind turbines. Philos Trans R Soc Lond Ser A Math Phys Eng Sci 361(1813):2909–2930
Hu P, **ao Z, Leo C, Liyanapathirana S (2021) Advances in the prediction of spudcan punch-through in double-layered soils. Mar Struct 79:103038
Fan Y, Wang J, Feng S (2021) Effect of spudcan penetration on laterally loaded pile groups. Ocean Eng 221:108505
Mehralizadeh H, Makarchian M (2021) A new method to predict the bearing capacity–penetration curve of spudcans in multi-layered clay soils. Mar Georesour Geotechnol 40:511–522
Li D, Li S, Zhang Y (2019) Cone-shaped hollow flexible reinforced concrete foundation (CHFRF)–Innovative for mountain wind turbines. Soils Found 59(5):1172–1181
Li S, Zhang Y, Li D, Gao M (2022) Lateral bearing capacity of cone-shaped hollow foundation by using limit equilibrium method. Int J Phys Model Geotech 22(3):157–168
Cassidy M, Houlsby G (2002) Vertical bearing capacity factors for conical foundations on sand. Géotechnique 52(9):687–692
Houlsby G, Martin C (2003) Undrained bearing capacity factors for conical foundations on clay. Géotechnique 53(5):513–520
Khatri VN, Kumar J (2009) Bearing capacity factor N for a rough conical foundation. Geomech Eng 1(3):205–218
Chakraborty M, Kumar J (2016) The size effect of a conical foundation on Nγ. Comput Geotech 76:212–221
Chakraborty M, Kumar J (2015) Bearing capacity factors for a conical foundation using lower-and upper-bound finite elements limit analysis. Can Geotech J 52(12):2134–2140
Keawsawasvong S (2021) Bearing capacity of conical foundations on clays considering combined effects of anisotropy and non-homogeneity. Ships Offshore Struct. https://doi.org/10.1080/17445302.2021.1987110
Phuor T, Harahap IS, Ng C-Y (2022) Bearing capacity factors for rough conical foundation by viscoplasticity finite-element analysis. Int J Geomech 22(1):04021266
Casagrande ACN (1944) Shear failure of anisotropic soils. Contributions to Soil Mechanics (BSCE)
Lo KY (1965) Stability of slopes in anisotropic soils. J Soil Mech Found Div 91(4):85–106
Ladd C (1991) Stability analysis during staged construction: J Geotech Engng Div ASCE V117, N4, April 1991, P538–615. In: International journal of rock mechanics and mining sciences & geomechanics abstracts. Pergamon
Ladd CC, DeGroot DJ (2004) Recommended practice for soft ground site characterization: Arthur Casagrande Lecture. Massachusetts Institute of Technology, Cambridge
Grimstad G, Andresen L, Jostad HP (2012) NGI-ADP: anisotropic shear strength model for clay. Int J Numer Anal Meth Geomech 36(4):483–497
Krabbenhøft K, Galindo-Torres SA, Zhang X, Krabbenhøft J (2019) AUS: anisotropic undrained shear strength model for clays. Int J Numer Anal Meth Geomech 43(17):2652–2666
Krabbenhoft K, Lyamin A (2015) Generalised Tresca criterion for undrained total stress analysis. Géotech Lett 5(4):313–317
Ukritchon B, Yoang S, Keawsawasvong S (2018) Bearing capacity of shallow foundations in clay with linear increase in strength and adhesion factor. Mar Georesour Geotechnol 36(4):438–451
Keawsawasvong S (2021) End bearing capacity factor for annular foundations embedded in clay considering the effect of the adhesion factor. Int J Geosynth Ground Eng 7(1):1–10
Lai VQ, Nguyen DK, Banyong R, Keawsawasvong S (2021) Limit analysis solutions for stability factor of unsupported conical slopes in clays with heterogeneity and anisotropy. Int J Comput Mater Sci Eng. https://doi.org/10.1142/S2047684121500305
Huang M, Wang H, Tang Z, Yu J (2021) Basal stability analysis of braced excavations in anisotropic and non-homogeneous undrained clay using streamline velocity fields. Acta Geotech 16(4):1175–1186
Pan Q, Dias D (2016) Face stability analysis for a shield-driven tunnel in anisotropic and nonhomogeneous soils by the kinematical approach. Int J Geomech 16(3):04015076
Keawsawasvong S, Ukritchon B (2021) Design equation for stability of a circular tunnel in anisotropic and heterogeneous clay. Undergr Space 7:76–93
Rao P, Wu J, Mo Z (2020) 3D Limit analysis of the transient stability of slope during pile driving in nonhomogeneous and anisotropic soil. Adv Civ Eng 2020:1–10
Haghsheno H, Arabani M (2021) Seismic bearing capacity of shallow foundations placed on an anisotropic and nonhomogeneous inclined ground. Indian Geotech J 51:1319–1337
Keawsawasvong S, Ukritchon B (2021) Undrained stability of plane strain active trapdoors in anisotropic and non-homogenous clays. Tunn Undergr Space Technol 107:103628
Keawsawasvong S, Shiau J (2021) Stability of active trapdoors in axisymmetry. Undergr Space 7:50–57
Ukritchon B, Keawsawasvong S (2020) Undrained lower bound solutions for end bearing capacity of shallow circular piles in non-homogeneous and anisotropic clays. Int J Numer Anal Meth Geomech 44(5):596–632
Nguyen DK, Nguyen TP, Keawsawasvong S, Lai VQ (2021) Vertical uplift capacity of circular anchors in clay by considering anisotropy and non-homogeneity. Transp Infrastruct Geotechnol. https://doi.org/10.1007/s40515-021-00191-6
Keawsawasvong S, Shiau J, Ngamkhanong C, Qui Lai V, Thongchom C (2021) Undrained stability of ring foundations: axisymmetry, anisotropy, and nonhomogeneity. Int J Geomech 22(1):04021253
Shiau J, Lai VQ, Keawsawasvong S (2022) Multivariate adaptive regression splines analysis for 3D slope stability in anisotropic and heterogenous clay. J Rock Mech Geotech Eng. https://doi.org/10.1016/j.jrmge.2022.05.016
Lai VQ, Banyong R, Keawsawasvong S (2022) Stability of limiting pressure behind soil gaps in contiguous pile walls in anisotropic clays. Eng Fail Anal 134:106049
Lee JK, Jeong S, Lee S (2016) Undrained bearing capacity factors for ring footings in heterogeneous soil. Comput Geotech 75:103–111
Keawsawasvong S, Lai VQ (2021) End bearing capacity factor for annular foundations embedded in clay considering the effect of the adhesion factor. Int J Geosynth Ground Eng 7(1):1–10
Bishop AW (1966) The strength of soils as engineering materials. Geotechnique 16(2):91–130
Likitlersuang S, Plengsiri P, Mase LZ, Tanapalungkorn W (2020) Influence of spatial variability of ground on seismic response analysis: a case study of Bangkok subsoils. Bull Eng Geol Env 79(1):39–51
Misliniyati R, Mase LZ, Irsyam M, Hendriawan H, Sahadewa A (2019) Seismic response validation of simulated soil models to vertical array record during a strong earthquake. J Eng Technol Sci 51(6):772–790
Butterfield R (1999) Dimensional analysis for geotechnical engineers. Geotechnique 49(3):357–366
Lai VQ, Shiau J, Keawsawasvong S, Tran DT (2022) Bearing capacity of ring foundations on anisotropic and heterogenous clays: FEA, NGI-ADP, and MARS. Geotech Geol Eng 40:3929–3941
Brinkgreve R, Vermeer P (2019) PLAXIS 2D reference manual CONNECT edition V20. Delft University, Delft, The Netherlands
OptumCE (2020) OptumG2. Copenhagen, Denmark: Optum Computational Engineering. See https://optumce.com/. Accessed 1 Dec 2020
Griffiths D, Lane P (1999) Slope stability analysis by finite elements. Geotechnique 49(3):387–403
Griffiths D, Marquez R (2007) Three-dimensional slope stability analysis by elasto-plastic finite elements. Geotechnique 57(6):537–546
Moayedi H, Mosallanezhad M, Rashid ASA, Jusoh WAW, Muazu MA (2020) A systematic review and meta-analysis of artificial neural network application in geotechnical engineering: theory and applications. Neural Comput Appl 32(2):495–518
Moayedi H, Rezaei A (2021) The feasibility of PSO–ANFIS in estimating bearing capacity of strip foundations rested on cohesionless slope. Neural Comput Appl 33(9):4165–4177
Singh G, Walia B (2017) Performance evaluation of nature-inspired algorithms for the design of bored pile foundation by artificial neural networks. Neural Comput Appl 28(1):289–298
Abd Elmaboud Y, Abdelsalam SI (2019) DC/AC magnetohydrodynamic-micropump of a generalized Burger’s fluid in an annulus. Phys Scr 94(11):115209
Raza R, Mabood F, Naz R, Abdelsalam SI (2021) Thermal transport of radiative Williamson fluid over stretchable curved surface. Therm Sci Eng Progress 23:100887
Elkoumy S, Barakat E, Abdelsalam S (2013) Hall and transverse magnetic field effects on peristaltic flow of a Maxwell fluid through a porous medium. Glob J Pure Appl Math 9(2):187–203
Abdelsalam SI, Zaher AZ (2021) Leveraging elasticity to uncover the role of rabinowitsch suspension through a wavelike conduit: consolidated blood suspension application. Mathematics 9(16):2008
Eldesoky I, Abdelsalam SI, El-Askary WA, Ahmed M (2020) The integrated thermal effect in conjunction with slip conditions on peristaltically induced particle-fluid transport in a catheterized pipe. J Porous Media 23(7):695–713
Abdelsalam SI, Velasco-Hernández JX, Zaher A (2021) Electro-magnetically modulated self-propulsion of swimming sperms via cervical canal. Biomech Model Mechanobiol 20(3):861–878
Bhatti M, Abdelsalam SI (2021) Bio-inspired peristaltic propulsion of hybrid nanofluid flow with Tantalum (Ta) and Gold (Au) nanoparticles under magnetic effects. Waves Random Complex Media. https://doi.org/10.1080/17455030.2021.1998728
Mekheimer KS, Abo-Elkhair R, Abdelsalam S, Ali KK, Moawad A (2022) Biomedical simulations of nanoparticles drug delivery to blood hemodynamics in diseased organs: synovitis problem. Int Commun Heat Mass Transf 130:105756
Abd Elmaboud Y, Mekheimer KS, Abdelsalam SI (2014) A study of nonlinear variable viscosity in finite-length tube with peristalsis. Appl Bionics Biomech 11(4):197–206
Abumandour RM, Eldesoky IM, Kamel MH, Ahmed MM, Abdelsalam SI (2020) Peristaltic thrusting of a thermal-viscosity nanofluid through a resilient vertical pipe. Z Naturforschung A 75(8):727–738
Bhatti M, Alamri SZ, Ellahi R, Abdelsalam SI (2021) Intra-uterine particle–fluid motion through a compliant asymmetric tapered channel with heat transfer. J Therm Anal Calorim 144(6):2259–2267
Bhatti MM, Marin M, Zeeshan A, Abdelsalam SI (2020) Recent trends in computational fluid dynamics. Front Phys 8:593111
Sloan S (2013) Geotechnical stability analysis. Géotechnique 63(7):531–571
Ukritchon B, Keawsawasvong S (2019) Design equations of uplift capacity of circular piles in sands. Appl Ocean Res 90:101844
Keawsawasvong S, Shiau J (2021) Instability of boreholes with slurry. Int J Geosynth Ground Eng 7(4):1–11
Ukritchon B, Yoang S, Keawsawasvong S (2020) Undrained stability of unsupported rectangular excavations in non-homogeneous clays. Comput Geotech 117:103281
Ukritchon B, Yoang S, Keawsawasvong S (2019) Three-dimensional stability analysis of the collapse pressure on flexible pavements over rectangular trapdoors. Transp Geotech 21:100277
Yodsomjai W, Keawsawasvong S, Senjuntichai T (2021) Undrained stability of unsupported conical slopes in anisotropic clays based on Anisotropic Undrained Shear failure criterion. Transp Infrastruct Geotechnol 8(4):557–568
Keawsawasvong S, Ukritchon B (2020) Design equation for stability of shallow unlined circular tunnels in Hoek–Brown rock masses. Bull Eng Geol Env 79:4167–4190
Keawsawasvong S, Ukritchon B (2022) Design equation for stability of a circular tunnel in an anisotropic and heterogeneous clay. Undergr Space 7(1):76–93
Keawsawasvong S, Ukritchon B (2019) Undrained basal stability of braced circular excavations in non-homogeneous clays with linear increase of strength with depth. Comput Geotech 115:103180
Keawsawasvong S, Thongchom C, Likitlersuang S (2021) Bearing capacity of strip footing on Hoek–Brown rock mass subjected to eccentric and inclined loading. Transp Infrastruct Geotechnol 8:189–200
Keawsawasvong S, Ukritchon B (2019) Undrained stability of a spherical cavity in cohesive soils using finite element limit analysis. J Rock Mech Geotech Eng 11(6):1274–1285
Loukidis D, Bandini P, Salgado R (2003) Stability of seismically loaded slopes using limit analysis. Geotechnique 53(5):463–479
Kumar J, Samui P (2006) Stability determination for layered soil slopes using the upper bound limit analysis. Geotech Geol Eng 24(6):1803–1819
Li A, Merifield R, Lin H, Lyamin A (2014) Trench stability under bentonite pressure in purely cohesive clay. Int J Geomech 14(1):151–157
Shiau J, Hassan MM (2021) Numerical investigation of undrained trapdoors in three dimensions. Int J Geosynth Ground Eng 7(2):1–12
Shiau J, Al-Asadi F (2020) Three-dimensional heading stability of twin circular tunnels. Geotech Geol Eng 38(3):2973–2988
Shiau J, Al-Asadi F (2021) Twin tunnels stability factors Fc, Fs and Fγ. Geotech Geol Eng 39(1):335–345
Yodsomjai W, Keawsawasvong S, Lai VQ (2021) Limit analysis solutions for bearing capacity of ring foundations on rocks using Hoek–Brown failure criterion. Int J Geosynth Ground Eng 7(2):1–10
Ciria H, Peraire J, Bonet J (2008) Mesh adaptive computation of upper and lower bounds in limit analysis. Int J Numer Methods Eng 75(8):899–944
Yodsomjai W, Keawsawasvong S (2021) Limit analysis solutions for bearing capacity of ring foundations on rocks using hoek-brown failure criterion. Int J Geosynth Ground Eng 7(2):1–10
Goudjil K, Arabet L (2021) Assessment of deflection of pile implanted on slope by artificial neural network. Neural Comput Appl 33:1091–1101
Acharyya R, Dey A (2019) Assessment of bearing capacity for strip footing located near slo** surface considering ANN model. Neural Comput Appl 31(11):8087–8100
Suthar M (2019) Applying several machine learning approaches for prediction of unconfined compressive strength of stabilized pond ashes. Neural Comput Appl 32:9019–9028
Cao M, Pan L, Gao Y, Novák D, Ding Z et al (2017) Neural network ensemble-based parameter sensitivity analysis in civil engineering systems. Neural Comput Appl 28(7):1583–1590
Abu Arqub O (2017) Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput Appl 28(7):1591–1610
Abu Arqub O, Rashaideh H (2018) The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs. Neural Comput Appl 30(8):2595–2606
Momani S, Abo-Hammour ZS, Alsmadi OM (2016) Solution of inverse kinematics problem using genetic algorithms. Appl Math Inf Sci 10(1):225
Fei J, Wu Z, Sun X, Su D, Bao X (2021) Research on tunnel engineering monitoring technology based on BPNN neural network and MARS machine learning regression algorithm. Neural Comput Appl 33:239–255
Barzegar R, Sattarpour M, Deo R, Fijani E, Adamowski J (2019) An ensemble tree-based machine learning model for predicting the uniaxial compressive strength of travertine rocks. Neural Comput Appl 32:9065–9080
Moayedi H, Rezaei A (2019) An artificial neural network approach for under-reamed piles subjected to uplift forces in dry sand. Neural Comput Appl 31(2):327–336
Zhang W (2019) MARS applications in geotechnical engineering systems: multi-dimension with big data. Springer, Singapore
Lai F, Zhang N, Liu S, Sun Y, Li Y (2021) Ground movements induced by installation of twin large diameter deeply-buried caissons: 3D numerical modeling. Acta Geotech 16:2933–2961
Zheng G, Yang P, Zhou H, Zeng C, Yang X et al (2019) Evaluation of the earthquake induced uplift displacement of tunnels using multivariate adaptive regression splines. Comput Geotech 113:103099
Raja MNA, Shukla SK (2021) Multivariate adaptive regression splines model for reinforced soil foundations. Geosynth Int 28(4):368–390
Wu L, Fan J (2019) Comparison of neuron-based, kernel-based, tree-based and curve-based machine learning models for predicting daily reference evapotranspiration. PLoS ONE 14(5):e0217520
Zhang W, Zhang R, Wu C, Goh ATC, Lacasse S et al (2020) State-of-the-art review of soft computing applications in underground excavations. Geosci Front 11(4):1095–1106
Friedman JH (1991) Multivariate adaptive regression splines. Ann Stat 19(1):1–67
Steinberg D, Colla P, Martin K (1999) MARS user guide. Salford Systems, San Diego
Gan Y, Duan Q, Gong W, Tong C, Sun Y et al (2014) A comprehensive evaluation of various sensitivity analysis methods: a case study with a hydrological model. Environ Model Softw 51:269–285
Acknowledgements
This work belongs to the project T2022-159 funded by Ho Chi Minh City University of Technology and Education, Vietnam.
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CNV contributed to data curation, software, investigation, methodology, writing–original draft, and project administration. SK contributed to methodology, software, and writing–original draft. DKN contributed to data curation and software. VQL contributed to data curation, software, investigation, methodology, writing–review and editing, and project administration.
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Appendix
Appendix
1.1 Appendix 1: Bearing capacity of conical foundation, H/D = 0
H/D = 0 | |||||||
|---|---|---|---|---|---|---|---|
ρD/suTC0 | r e | β (°) | |||||
30 | 60 | 90 | 120 | 150 | 180 | ||
0 | 0.5 | 6.623 | 5.197 | 4.863 | 4.835 | 4.772 | 4.746 |
0 | 0.6 | 10.929 | 6.864 | 5.895 | 5.579 | 5.469 | 5.389 |
0 | 0.7 | 17.269 | 9.334 | 7.413 | 6.652 | 6.325 | 6.056 |
0 | 0.8 | 27.834 | 13.426 | 9.848 | 8.348 | 7.569 | 6.977 |
0 | 0.9 | 48.934 | 21.507 | 14.603 | 11.542 | 9.820 | 8.490 |
0 | 1 | 70.076 | 29.570 | 19.333 | 14.619 | 11.923 | 9.741 |
1 | 0.5 | 7.164 | 5.581 | 5.222 | 5.141 | 5.107 | 5.095 |
1 | 0.6 | 11.760 | 7.360 | 6.333 | 5.972 | 5.832 | 5.770 |
1 | 0.7 | 18.579 | 9.976 | 7.914 | 7.103 | 6.682 | 6.486 |
1 | 0.8 | 29.936 | 14.322 | 10.477 | 8.889 | 8.081 | 7.499 |
1 | 0.9 | 52.636 | 22.898 | 15.546 | 12.311 | 10.494 | 9.130 |
1 | 1 | 75.334 | 31.486 | 20.564 | 15.633 | 12.769 | 10.523 |
2.5 | 0.5 | 7.594 | 5.848 | 5.443 | 5.374 | 5.344 | 5.336 |
2.5 | 0.6 | 12.419 | 7.720 | 6.585 | 6.233 | 6.105 | 6.043 |
2.5 | 0.7 | 19.633 | 10.469 | 8.241 | 7.402 | 7.033 | 6.831 |
2.5 | 0.8 | 31.632 | 15.004 | 10.933 | 9.276 | 8.434 | 7.908 |
2.5 | 0.9 | 55.610 | 24.023 | 16.223 | 12.863 | 10.991 | 9.658 |
2.5 | 1 | 79.583 | 33.027 | 21.460 | 16.371 | 13.400 | 11.181 |
5 | 0.5 | 7.954 | 6.087 | 5.625 | 5.532 | 5.521 | 5.515 |
5 | 0.6 | 13.060 | 8.017 | 6.840 | 6.444 | 6.312 | 6.267 |
5 | 0.7 | 20.655 | 10.877 | 8.571 | 7.688 | 7.276 | 7.108 |
5 | 0.8 | 33.288 | 15.591 | 11.373 | 9.645 | 8.782 | 8.250 |
5 | 0.9 | 58.526 | 24.967 | 16.876 | 13.392 | 11.467 | 10.120 |
5 | 1 | 83.751 | 34.318 | 22.322 | 17.046 | 13.997 | 11.738 |
10 | 0.5 | 8.282 | 6.279 | 5.779 | 5.674 | 5.671 | 5.666 |
10 | 0.6 | 13.618 | 8.316 | 7.046 | 6.624 | 6.484 | 6.449 |
10 | 0.7 | 21.548 | 11.288 | 8.840 | 7.922 | 7.506 | 7.340 |
10 | 0.8 | 34.727 | 16.182 | 11.737 | 9.952 | 9.063 | 8.553 |
10 | 0.9 | 61.055 | 25.909 | 17.417 | 13.829 | 11.873 | 10.534 |
10 | 1 | 87.384 | 35.607 | 23.041 | 17.614 | 14.507 | 12.241 |
15 | 0.5 | 8.570 | 6.468 | 5.939 | 5.812 | 5.799 | 5.794 |
15 | 0.6 | 14.109 | 8.577 | 7.254 | 6.814 | 6.667 | 6.633 |
15 | 0.7 | 22.331 | 11.647 | 9.108 | 8.160 | 7.739 | 7.573 |
15 | 0.8 | 35.995 | 16.698 | 12.096 | 10.261 | 9.212 | 8.840 |
15 | 0.9 | 63.291 | 26.733 | 17.953 | 14.266 | 12.272 | 10.914 |
15 | 1 | 90.579 | 36.740 | 23.749 | 18.172 | 14.999 | 12.702 |
1.2 Appendix 2: Bearing capacity of conical foundation, H/D = 0.1
H/D = 0.1 | |||||||
|---|---|---|---|---|---|---|---|
ρD/suTC0 | r e | β (°) | |||||
30 | 60 | 90 | 120 | 150 | 180 | ||
0 | 0.5 | 6.796 | 5.541 | 5.286 | 5.309 | 5.299 | 5.276 |
0 | 0.6 | 10.699 | 7.075 | 6.273 | 6.085 | 5.991 | 5.959 |
0 | 0.7 | 15.352 | 8.876 | 7.387 | 6.895 | 6.668 | 6.536 |
0 | 0.8 | 21.025 | 11.057 | 8.700 | 7.826 | 7.490 | 7.124 |
0 | 0.9 | 28.072 | 13.759 | 10.310 | 8.944 | 8.211 | 7.774 |
0 | 1 | 32.292 | 15.356 | 11.260 | 9.580 | 8.814 | 8.094 |
1 | 0.5 | 7.366 | 5.929 | 5.655 | 5.655 | 5.645 | 5.638 |
1 | 0.6 | 11.553 | 7.569 | 6.710 | 6.472 | 6.412 | 6.393 |
1 | 0.7 | 16.554 | 9.496 | 7.893 | 7.347 | 7.136 | 7.039 |
1 | 0.8 | 22.620 | 11.837 | 9.295 | 8.341 | 7.940 | 7.682 |
1 | 0.9 | 30.201 | 14.706 | 10.996 | 9.532 | 8.707 | 8.360 |
1 | 1 | 34.729 | 16.387 | 11.993 | 10.228 | 9.352 | 8.716 |
2.5 | 0.5 | 7.802 | 6.219 | 5.892 | 5.896 | 5.881 | 5.874 |
2.5 | 0.6 | 12.231 | 7.954 | 6.994 | 6.761 | 6.693 | 6.678 |
2.5 | 0.7 | 17.494 | 9.970 | 8.235 | 7.674 | 7.444 | 7.372 |
2.5 | 0.8 | 23.901 | 12.399 | 9.694 | 8.695 | 8.249 | 8.044 |
2.5 | 0.9 | 31.896 | 15.405 | 11.475 | 9.935 | 9.202 | 8.762 |
2.5 | 1 | 36.683 | 17.196 | 12.523 | 10.654 | 9.728 | 9.149 |
5 | 0.5 | 8.202 | 6.451 | 6.114 | 6.080 | 6.080 | 6.075 |
5 | 0.6 | 12.871 | 8.264 | 7.275 | 6.990 | 6.936 | 6.923 |
5 | 0.7 | 18.413 | 10.365 | 8.568 | 7.948 | 7.731 | 7.659 |
5 | 0.8 | 25.155 | 12.889 | 10.088 | 9.040 | 8.581 | 8.379 |
5 | 0.9 | 33.565 | 16.011 | 11.939 | 10.335 | 9.578 | 9.143 |
5 | 1 | 38.602 | 17.873 | 13.032 | 11.086 | 10.130 | 9.554 |
10 | 0.5 | 8.548 | 6.688 | 6.286 | 6.235 | 6.236 | 6.232 |
10 | 0.6 | 13.428 | 8.580 | 7.498 | 7.192 | 7.129 | 7.119 |
10 | 0.7 | 19.209 | 10.762 | 8.839 | 8.190 | 7.964 | 7.892 |
10 | 0.8 | 26.244 | 13.382 | 10.411 | 9.324 | 8.855 | 8.653 |
10 | 0.9 | 35.018 | 16.622 | 12.324 | 10.664 | 9.760 | 9.462 |
10 | 1 | 40.274 | 18.555 | 13.452 | 11.444 | 10.477 | 9.896 |
15 | 0.5 | 8.852 | 6.895 | 6.466 | 6.400 | 6.402 | 6.396 |
15 | 0.6 | 13.918 | 8.855 | 7.725 | 7.402 | 7.331 | 7.320 |
15 | 0.7 | 19.912 | 11.111 | 9.113 | 8.441 | 8.208 | 8.134 |
15 | 0.8 | 27.204 | 13.816 | 10.737 | 9.616 | 9.138 | 8.936 |
15 | 0.9 | 36.296 | 17.159 | 12.708 | 11.001 | 10.181 | 9.785 |
15 | 1 | 41.744 | 19.153 | 13.871 | 11.803 | 10.822 | 10.241 |
1.3 Appendix 3: Bearing capacity of conical foundation, H/D = 0.25
H/D = 0.25 | |||||||
|---|---|---|---|---|---|---|---|
ρD/suTC0 | r e | β (°) | |||||
30 | 60 | 90 | 120 | 150 | 180 | ||
0 | 0.5 | 7.075 | 5.903 | 5.752 | 5.795 | 5.778 | 5.734 |
0 | 0.6 | 10.484 | 7.240 | 6.580 | 6.465 | 6.454 | 6.437 |
0 | 0.7 | 13.625 | 8.451 | 7.333 | 7.014 | 6.933 | 6.873 |
0 | 0.8 | 16.504 | 9.569 | 8.005 | 7.482 | 7.280 | 7.185 |
0 | 0.9 | 19.181 | 10.576 | 8.592 | 7.896 | 7.579 | 7.365 |
0 | 1 | 20.433 | 11.050 | 8.868 | 8.087 | 7.784 | 7.460 |
1 | 0.5 | 7.627 | 6.319 | 6.119 | 6.156 | 6.148 | 6.144 |
1 | 0.6 | 11.290 | 7.773 | 7.052 | 6.908 | 6.877 | 6.856 |
1 | 0.7 | 14.683 | 9.060 | 7.848 | 7.495 | 7.382 | 7.334 |
1 | 0.8 | 17.781 | 10.255 | 8.544 | 7.991 | 7.752 | 7.679 |
1 | 0.9 | 20.650 | 11.325 | 9.178 | 8.427 | 8.170 | 7.950 |
1 | 1 | 22.002 | 11.822 | 9.463 | 8.620 | 8.244 | 8.057 |
2.5 | 0.5 | 8.093 | 6.647 | 6.404 | 6.423 | 6.427 | 6.414 |
2.5 | 0.6 | 11.992 | 8.180 | 7.382 | 7.220 | 7.200 | 7.184 |
2.5 | 0.7 | 15.544 | 9.531 | 8.208 | 7.825 | 7.725 | 7.690 |
2.5 | 0.8 | 18.812 | 10.754 | 8.929 | 8.337 | 8.129 | 8.055 |
2.5 | 0.9 | 21.831 | 11.865 | 9.580 | 8.785 | 8.438 | 8.334 |
2.5 | 1 | 23.263 | 12.386 | 9.876 | 8.986 | 8.590 | 8.448 |
5 | 0.5 | 8.516 | 6.907 | 6.651 | 6.650 | 6.655 | 6.638 |
5 | 0.6 | 12.630 | 8.508 | 7.685 | 7.488 | 7.470 | 7.453 |
5 | 0.7 | 16.371 | 9.913 | 8.546 | 8.131 | 8.018 | 7.987 |
5 | 0.8 | 19.806 | 11.181 | 9.299 | 8.668 | 8.446 | 8.372 |
5 | 0.9 | 22.985 | 12.335 | 9.973 | 9.132 | 8.794 | 8.668 |
5 | 1 | 24.486 | 12.876 | 10.283 | 9.345 | 8.942 | 8.791 |
10 | 0.5 | 8.882 | 7.166 | 6.848 | 6.837 | 6.837 | 6.824 |
10 | 0.6 | 13.185 | 8.840 | 7.928 | 7.709 | 7.688 | 7.670 |
10 | 0.7 | 17.084 | 10.299 | 8.821 | 8.378 | 8.259 | 8.228 |
10 | 0.8 | 20.668 | 11.614 | 9.600 | 8.938 | 8.705 | 8.633 |
10 | 0.9 | 23.982 | 12.812 | 10.294 | 9.420 | 9.071 | 8.944 |
10 | 1 | 25.546 | 13.372 | 10.612 | 9.639 | 9.231 | 9.074 |
15 | 0.5 | 9.202 | 7.392 | 7.046 | 7.018 | 7.023 | 7.008 |
15 | 0.6 | 13.669 | 9.128 | 8.172 | 7.936 | 7.906 | 7.890 |
15 | 0.7 | 17.712 | 10.636 | 9.096 | 8.633 | 8.507 | 8.473 |
15 | 0.8 | 21.426 | 11.993 | 9.900 | 9.215 | 8.977 | 8.899 |
15 | 0.9 | 24.861 | 13.230 | 10.616 | 9.715 | 9.355 | 9.226 |
15 | 1 | 26.482 | 13.806 | 10.947 | 9.940 | 9.525 | 9.364 |
1.4 Appendix 4: Bearing capacity of conical foundation, H/D = 0.5
H/D = 0.5 | |||||||
|---|---|---|---|---|---|---|---|
ρD/suTC0 | r e | β (°) | |||||
30 | 60 | 90 | 120 | 150 | 180 | ||
0 | 0.5 | 7.428 | 6.385 | 6.297 | 6.343 | 6.347 | 6.315 |
0 | 0.6 | 10.255 | 7.451 | 6.934 | 6.911 | 6.887 | 6.865 |
0 | 0.7 | 12.109 | 8.115 | 7.318 | 7.176 | 7.140 | 7.112 |
0 | 0.8 | 13.413 | 8.597 | 7.590 | 7.354 | 7.281 | 7.240 |
0 | 0.9 | 14.383 | 8.951 | 7.777 | 7.469 | 7.082 | 7.321 |
0 | 1 | 14.803 | 9.085 | 7.861 | 7.510 | 7.404 | 7.344 |
1 | 0.5 | 8.017 | 6.828 | 6.708 | 6.772 | 6.766 | 6.756 |
1 | 0.6 | 11.056 | 8.003 | 7.441 | 7.377 | 7.367 | 7.339 |
1 | 0.7 | 13.050 | 8.727 | 7.859 | 7.686 | 7.647 | 7.600 |
1 | 0.8 | 14.458 | 9.231 | 8.148 | 7.868 | 7.795 | 7.734 |
1 | 0.9 | 15.496 | 9.597 | 8.345 | 7.990 | 7.899 | 7.823 |
1 | 1 | 15.930 | 9.744 | 8.413 | 8.026 | 7.921 | 7.848 |
2.5 | 0.5 | 8.498 | 7.197 | 7.037 | 7.079 | 7.084 | 7.065 |
2.5 | 0.6 | 11.716 | 8.425 | 7.803 | 7.721 | 7.722 | 7.701 |
2.5 | 0.7 | 13.815 | 9.187 | 8.235 | 8.025 | 7.999 | 7.974 |
2.5 | 0.8 | 15.298 | 9.710 | 8.521 | 8.217 | 8.158 | 8.120 |
2.5 | 0.9 | 16.397 | 10.089 | 8.724 | 8.337 | 8.272 | 8.203 |
2.5 | 1 | 16.847 | 10.243 | 8.804 | 8.389 | 8.314 | 8.230 |
5 | 0.5 | 8.945 | 7.483 | 7.310 | 7.332 | 7.337 | 7.317 |
5 | 0.6 | 12.347 | 8.768 | 8.119 | 8.006 | 8.009 | 7.989 |
5 | 0.7 | 14.553 | 9.559 | 8.575 | 8.333 | 8.314 | 8.275 |
5 | 0.8 | 16.110 | 10.101 | 8.875 | 8.532 | 8.469 | 8.428 |
5 | 0.9 | 17.268 | 10.493 | 9.086 | 8.665 | 8.584 | 8.521 |
5 | 1 | 17.738 | 10.650 | 9.165 | 8.715 | 8.632 | 8.543 |
10 | 0.5 | 9.332 | 7.765 | 7.529 | 7.541 | 7.545 | 7.524 |
10 | 0.6 | 12.887 | 9.110 | 8.376 | 8.237 | 8.239 | 8.216 |
10 | 0.7 | 15.191 | 9.931 | 8.850 | 8.579 | 8.553 | 8.518 |
10 | 0.8 | 16.814 | 10.493 | 9.161 | 8.790 | 8.725 | 8.677 |
10 | 0.9 | 18.022 | 10.901 | 9.376 | 8.929 | 8.845 | 8.773 |
10 | 1 | 18.516 | 11.062 | 9.460 | 8.979 | 8.892 | 8.801 |
15 | 0.5 | 9.671 | 8.012 | 7.747 | 7.745 | 7.751 | 7.729 |
15 | 0.6 | 13.365 | 9.409 | 8.634 | 8.478 | 8.475 | 8.451 |
15 | 0.7 | 15.751 | 10.256 | 9.128 | 8.840 | 8.809 | 8.768 |
15 | 0.8 | 17.433 | 10.835 | 9.448 | 9.058 | 8.988 | 8.935 |
15 | 0.9 | 18.684 | 11.257 | 9.670 | 9.200 | 9.075 | 9.032 |
15 | 1 | 19.195 | 11.427 | 9.757 | 9.255 | 9.111 | 9.065 |
1.5 Appendix 5: Bearing capacity of conical foundation, H/D = 1
H/D = 1 | |||||||
|---|---|---|---|---|---|---|---|
ρD/suTC0 | r e | β (°) | |||||
30 | 60 | 90 | 120 | 150 | 180 | ||
0 | 0.5 | 7.995 | 7.130 | 7.019 | 6.806 | 6.763 | 6.739 |
0 | 0.6 | 9.998 | 7.757 | 7.450 | 7.344 | 7.225 | 7.116 |
0 | 0.7 | 10.819 | 8.005 | 7.520 | 7.458 | 7.333 | 7.251 |
0 | 0.8 | 11.267 | 8.121 | 7.561 | 7.500 | 7.344 | 7.260 |
0 | 0.9 | 11.547 | 8.191 | 7.592 | 7.512 | 7.374 | 7.287 |
0 | 1 | 11.666 | 8.214 | 7.584 | 7.458 | 7.363 | 7.297 |
1 | 0.5 | 8.648 | 7.633 | 7.539 | 7.502 | 7.469 | 7.434 |
1 | 0.6 | 10.785 | 8.338 | 7.955 | 7.889 | 7.837 | 7.789 |
1 | 0.7 | 11.669 | 8.599 | 8.067 | 7.992 | 7.918 | 7.880 |
1 | 0.8 | 12.160 | 8.724 | 8.111 | 8.009 | 7.940 | 7.906 |
1 | 0.9 | 12.456 | 8.794 | 8.130 | 8.015 | 7.954 | 7.915 |
1 | 1 | 12.570 | 8.825 | 8.132 | 8.020 | 7.929 | 7.893 |
2.5 | 0.5 | 9.190 | 8.045 | 7.937 | 7.938 | 7.944 | 7.919 |
2.5 | 0.6 | 11.426 | 8.781 | 8.346 | 8.301 | 8.288 | 8.248 |
2.5 | 0.7 | 12.356 | 9.041 | 8.451 | 8.371 | 8.343 | 8.318 |
2.5 | 0.8 | 12.867 | 9.176 | 8.494 | 8.386 | 8.375 | 8.331 |
2.5 | 0.9 | 13.184 | 9.248 | 8.508 | 8.399 | 8.366 | 8.331 |
2.5 | 1 | 13.301 | 9.280 | 8.513 | 8.407 | 8.362 | 8.329 |
5 | 0.5 | 9.633 | 8.396 | 8.266 | 8.277 | 8.284 | 8.260 |
5 | 0.6 | 12.032 | 9.153 | 8.681 | 8.627 | 8.626 | 8.593 |
5 | 0.7 | 13.011 | 9.422 | 8.789 | 8.691 | 8.678 | 8.652 |
5 | 0.8 | 13.550 | 9.558 | 8.832 | 8.701 | 8.690 | 8.662 |
5 | 0.9 | 13.884 | 9.638 | 8.846 | 8.696 | 8.683 | 8.656 |
5 | 1 | 14.006 | 9.669 | 8.851 | 8.692 | 8.681 | 8.651 |
10 | 0.5 | 10.045 | 8.682 | 8.529 | 8.542 | 8.548 | 8.523 |
10 | 0.6 | 12.556 | 9.488 | 8.957 | 8.893 | 8.895 | 8.869 |
10 | 0.7 | 13.585 | 9.769 | 9.064 | 8.946 | 8.942 | 8.918 |
10 | 0.8 | 14.141 | 9.912 | 9.106 | 8.955 | 8.946 | 8.921 |
10 | 0.9 | 14.490 | 9.992 | 9.125 | 8.947 | 8.944 | 8.911 |
10 | 1 | 14.618 | 10.023 | 9.130 | 8.946 | 8.935 | 8.904 |
15 | 0.5 | 10.405 | 8.952 | 8.775 | 8.785 | 8.788 | 8.761 |
15 | 0.6 | 13.023 | 9.797 | 9.233 | 9.153 | 9.159 | 9.131 |
15 | 0.7 | 14.084 | 10.089 | 9.345 | 9.212 | 9.211 | 9.182 |
15 | 0.8 | 14.661 | 10.237 | 9.389 | 9.219 | 9.214 | 9.183 |
15 | 0.9 | 15.022 | 10.324 | 9.408 | 9.212 | 9.209 | 9.172 |
15 | 1 | 15.156 | 10.353 | 9.414 | 9.209 | 9.196 | 9.165 |
1.6 Appendix 6: Bearing capacity of conical foundation, H/D = 2.5
H/D = 2.5 | |||||||
|---|---|---|---|---|---|---|---|
ρD/suTC0 | r e | β (°) | |||||
30 | 60 | 90 | 120 | 150 | 180 | ||
0 | 0.5 | 9.016 | 7.115 | 6.815 | 6.736 | 6.731 | 6.698 |
0 | 0.6 | 9.923 | 7.802 | 7.009 | 6.942 | 6.906 | 6.869 |
0 | 0.7 | 10.057 | 7.502 | 6.941 | 6.925 | 6.902 | 6.872 |
0 | 0.8 | 10.115 | 7.977 | 6.938 | 6.937 | 6.913 | 6.881 |
0 | 0.9 | 10.138 | 8.021 | 6.930 | 6.917 | 6.905 | 6.877 |
0 | 1 | 10.143 | 8.031 | 6.935 | 6.859 | 6.822 | 6.874 |
1 | 0.5 | 9.816 | 7.889 | 7.477 | 7.403 | 7.385 | 7.370 |
1 | 0.6 | 10.695 | 8.541 | 7.711 | 7.590 | 7.570 | 7.535 |
1 | 0.7 | 10.838 | 8.638 | 7.694 | 7.585 | 7.561 | 7.531 |
1 | 0.8 | 10.891 | 8.696 | 7.689 | 7.594 | 7.578 | 7.543 |
1 | 0.9 | 10.919 | 8.722 | 7.818 | 7.568 | 7.580 | 7.535 |
1 | 1 | 10.923 | 8.730 | 7.838 | 7.590 | 7.564 | 7.537 |
2.5 | 0.5 | 10.454 | 8.560 | 8.074 | 7.946 | 7.933 | 7.901 |
2.5 | 0.6 | 11.377 | 9.154 | 8.304 | 8.132 | 8.102 | 8.061 |
2.5 | 0.7 | 11.516 | 9.242 | 8.364 | 8.155 | 8.124 | 8.091 |
2.5 | 0.8 | 11.577 | 9.269 | 8.433 | 8.162 | 8.141 | 8.098 |
2.5 | 0.9 | 11.601 | 9.274 | 8.484 | 8.163 | 8.132 | 8.094 |
2.5 | 1 | 11.615 | 9.273 | 8.499 | 8.164 | 8.141 | 8.099 |
5 | 0.5 | 10.971 | 9.173 | 8.599 | 8.449 | 8.435 | 8.398 |
5 | 0.6 | 11.924 | 9.667 | 8.847 | 8.639 | 8.606 | 8.561 |
5 | 0.7 | 12.077 | 9.701 | 8.912 | 8.668 | 8.626 | 8.589 |
5 | 0.8 | 12.134 | 9.712 | 8.966 | 8.678 | 8.646 | 8.600 |
5 | 0.9 | 12.163 | 9.724 | 8.999 | 8.683 | 8.649 | 8.604 |
5 | 1 | 12.172 | 9.703 | 9.006 | 8.689 | 8.648 | 8.604 |
10 | 0.5 | 11.454 | 9.681 | 9.066 | 8.895 | 8.871 | 8.837 |
10 | 0.6 | 12.448 | 10.102 | 9.321 | 9.086 | 9.041 | 9.000 |
10 | 0.7 | 12.606 | 10.112 | 9.381 | 9.120 | 9.081 | 9.030 |
10 | 0.8 | 12.663 | 10.098 | 9.420 | 9.134 | 9.096 | 9.041 |
10 | 0.9 | 12.691 | 10.087 | 9.437 | 9.142 | 9.099 | 9.045 |
10 | 1 | 12.701 | 10.081 | 9.445 | 9.148 | 9.102 | 9.047 |
15 | 0.5 | 11.878 | 10.142 | 9.488 | 9.296 | 9.271 | 9.231 |
15 | 0.6 | 12.907 | 10.482 | 9.745 | 9.490 | 9.440 | 9.393 |
15 | 0.7 | 13.066 | 10.470 | 9.794 | 9.525 | 9.460 | 9.424 |
15 | 0.8 | 13.128 | 10.451 | 9.811 | 9.540 | 9.488 | 9.436 |
15 | 0.9 | 13.159 | 10.435 | 9.817 | 9.549 | 9.492 | 9.443 |
15 | 1 | 13.172 | 10.430 | 9.816 | 9.551 | 9.519 | 9.445 |
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Nguyen Van, C., Keawsawasvong, S., Nguyen, D.K. et al. Machine learning regression approach for analysis of bearing capacity of conical foundations in heterogenous and anisotropic clays. Neural Comput & Applic 35, 3955–3976 (2023). https://doi.org/10.1007/s00521-022-07893-z
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DOI: https://doi.org/10.1007/s00521-022-07893-z