Ike, C. (2023). Solving Westergaard Half-Space Problems Using Potential Theory. , 42(5), 446-459. doi: 10.30684/etj.2023.141182.1486
Charles Chinwuba Ike. "Solving Westergaard Half-Space Problems Using Potential Theory". , 42, 5, 2023, 446-459. doi: 10.30684/etj.2023.141182.1486
Ike, C. (2023). 'Solving Westergaard Half-Space Problems Using Potential Theory', , 42(5), pp. 446-459. doi: 10.30684/etj.2023.141182.1486
Ike, C. Solving Westergaard Half-Space Problems Using Potential Theory. , 2023; 42(5): 446-459. doi: 10.30684/etj.2023.141182.1486

Solving Westergaard Half-Space Problems Using Potential Theory

Engineering and Technology Journal
Volume 42, Issue 5, May 2024, Pages 446-459    PDF (1.15 M)
Document Type: Research Paper
DOI: 10.30684/etj.2023.141182.1486
Author
Charles Chinwuba Ike*
Civil Engineering Dept., Enugu State University of Science & Technology, Agbani, Enugu State, Nigeria.
Abstract
The Westergaard half-space problem has been solved using the potential theory in this work. It is a classical theme in elasticity theory that seeks to find the displacements and stresses in the half-space caused by known boundary loads. It has important applications in analyzing stresses and displacement fields in soil due to applied points and distributed loads on the boundary caused by structures placed on the soil. It is governed by stress–strain, strain-displacement, and equilibrium equations. Horizontal inextensibility is assumed in developing the problem, simplifying the displacement formulation to a three-dimensional (3D) Laplace equation. The potential theory is applied to find the vertical displacement. Stress–displacement equations obtained from the simultaneous use of the kinematic and stress-strain equations are used to obtain the stress fields. The specific problem of point load at the origin was considered and solved. The equilibrium of internal vertical stresses and the external vertical load is used to find the integration constant. Hence, vertical displacements were found. The stress fields were found from the stress–displacement equations. The expressions for the vertical displacements and stresses were found to be exact within the framework of the theory used, as they satisfied all the governing equations of the problem. However, the solutions become unbounded at the origin due to the singularity of the vertical displacement and stresses. The obtained solutions are identical to previously obtained solutions.

Highlights
  • The governing differential equation is derived for an elastic half-space problem with horizontal inextensibility.
  • The potential theory is applied to solve the Westergaard problem for a point load on the boundary.
  • The approach adopts first principles to derive the governing equations, highlighting limitations and scope.
  • Results are validated by comparison with literature sources.
Keywords
Displacement formulation; Horizontal inextensibility; Laplacian; Potential theory; Navier-Cauchy equations; Westergaard half-space
References
  1. U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike, Solution of the Boussinesq Problem using Green and Zerna Displacement Potential Function Method, Electron. J. Geotech. Eng., 22 (2017) 4305 – 4314.
  2. C. Ike, B.O. Mama, H.N. Onah, C.U. Nwoji, Trefftz Harmonic Function Method for Solving Boussinesq Problem, Electron. J. Geotech. Eng., 22 (2017) 4589 – 4601.
  3. U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike, Solution of Elastic Half-Space Problem using Boussinesq Displacement Potential Functions, Asian J. Appl. Sci., 5 (2017) 1100 – 1106.
  4. N. Onah, M.E. Onyia, B.O. Mama, C.U. Nwoji, C.C. Ike, First Principles Derivation of Displacement and Stress Functions for Three-Dimensional Elastostatic Problems, and Application to the Flexural Analysis of Thick Circular Plates, J. Comput. Appl. Mech., 51 (2020) 184 – 198. https://doi.org/10.22059/jcamech.2020.295989.471
  5. A. Lubarda, M.V. Lubarda On the Kelvin , Boussinesq and Mindlin Problems. Acta Mechanica., 231 (2020) 155 – 178. https://doi.org/10.1007/S00707-019-02539-z
  6. C. Ike, Fourier-Bessel Transform Method for Finding Vertical Stress Fields in Axisymmetric Elasticity Problems of Elastic Half-Space involving Circular Foundation Areas, Adv. Model. Anal. A, 55 (2018) 207 – 216. https://doi.org/10.18250/ama_a.550405
  7. C. Ike, On Maxwell’s Stress Functions for Solving Three-Dimensional Elasticity Problems in the Theory of Elasticity, J. Comput. Appl. Mech., 49 (2018) 342 – 350. https://doi.org/10.22059/Jcamech.2018.266787.330
  8. C. Ike, Hankel Transform Method for Solving Axisymmetric Elasticity Problems of Circular Foundation on Semi-Infinite Soils, Int. J. Eng. Technol., 10 (2018) 549 – 564. https://doi.org/10.21817/ijet/2018/v10i2/181002111
  9. C. Ike, General Solutions for Axisymmetric Elasticity Problems of Elastic Half-Space Using Hankel Transform Method, Int. J. Eng. Technol., 10 (2018) 564 – 580. https://doi.org/10.21817/ijet/2018/v10i2/181002112
  10. C. Ike, First Principles Derivation of a Stress Function for Axially Symmetric Elasticity Problems and Application to Boussinesq Problem, Nig. J. Technol., 36 (2017) 767 – 772. https://doi.org/10.4314/njt.v36i3.15
  11. G. Taylor, J.H. Chung, Application of low-order potential solutions to higher-order vertical traction boundary problems in an elastic half-space, R. Soc. Open Sci., 5 (2018) 180203. https://doi.org/10.1098/rsos.180203
  12. M. Becker, M. Bevis, Love’s Problem, Geophy. J., 156 (2004) 171 – 178. https://doi.org/10.1111/j.1365-246X.2003.02150.x
  13. P.S. Selvadurai, On Boussinesq’s Problem for a Cracked Half-Space, J. Eng. Math., 107 (2017) 269 – 282. http://doi.org/10.1007/s10665-017-9934-6
  14. P.S. Selvadurai, On Boussinesq’s Problem, Int. J. Eng. Sci., 39 (2001) 317 – 322. https://doi.org/10.1016/S0020-7225(00)00043-4
  15. P.S. Selvadurai, On Frohlich’s Solution for Boussinesq Problem, Int. J. Numer. Anal. Methods Geomech., 38 (2014) 925 – 934. https://doi.org/10.1002/nag.2240
  16. F. Apostol, Elastic Displacement in Half Space under the Action of a Tension Force, General Solution for the Half Space with Point Forces, J. Elast., 126 (2017) 231 – 244. https://doi.org/10.1007/s10659-016-9592-3
  17. F. Apostol, Elastic Equilibrium of the Half-Space Revisited, Mindlin and Boussinesq Problems, J. Elast., 125 (2016) 139 – 148. https://doi.org/10.1007/s10659-016-9574-5
  18. P.S. Selvadurai, The Analytical Method in Geomechanics, Appl. Mech. Rev., 60 (2007) 87 – 107. https://doi.org/10.1115/1.2730845
  19. Palaniappan, A General Solution of Equations of Equilibrium in Linear Elasticity, Appl. Math. Model., 35 (2011) 5494 – 5499. https://doi.org/10.1016/j.apm.2011.01.041
  20. Chau, T. Analytic Methods in Geomechanics, CRC Press Taylor and Francis Group, New York, 2013.
  21. Ferretti, Satisfying Boundary Conditions in Homogeneous, Linear-Elastic and Isotropic Half-Spaces Subjected to Loads Perpendicular to the Surface: Distributed Loads on Adjacent Contact Areas, Curved Layer. Struct., 6 (2019) 11 – 29. https://doi.org/10.1515/cls-2019-0002
  22. Sadd, H. Elasticity Theory Application and Numerics, Third Edition, University of Rhode Island, Elsevier Academic Press, Amsterdam, 2020.
  23. Bowles, J.E. Foundation Analysis and Design, International Edition, McGraw Hill International Book Company, Tokyo,1997.
  24. Podio-Guidguli, and A. Favata, Elasticity for Geotechnicians: A Modern Exposition of Kelvin, Boussinesq, Flammant, Cerruttti, Melan and Mindlin Problems, Solid Mechanics and its Applications, Springer, New York, 2014. https://doi.org/10.1007/978-3-319-01258-2
  25. G. Sitharam, and L. Govindaraju, Applied Elasticity for Engineers Module: Elastic Solutions with Applications in Geomechanics 14.139.172.204/nptel/1/CSE/web/105108070/module 8/lecture 17.pdf, 2017. Retrieved 12th February 2019.
  26. Teodorescu, P. Treatise on Classical Elasticity Theory and Related Problems. Mathematical and Analytical Techniques with Applications to Engineering, Springer, Dordrecht, 2013. https://doi.org/10.1007/978-94-007-2616-1
  27. Davis, O. and Selvadurai, A.P.S. Elasticity and Geomechanics, Cambridge University Press, Cambridge, 1996.
  28. Abeyartne, Continuum Mechanics, Volume 11 of Lecture notes on the mechanics of elastic solids, Cambridge, 2012. http://web.mit.edu/abeyaratne/lecture
  29. Zhou, X. L. Gao‏, Solutions of Half-Space and Half-Plane Contact Problems Based on Surface Elasticity, Z. Angew. Math. Phys., 64 (2013) 145 – 166. https://doi.org/10.1007/s00033-012-0205-0
  30. S. Khapilova, S.V. Zaletov, The Exact Solution of the Problem on a Concentrated Force Action on the Isotropic Half-Space with the Boundary Fixed Elastically, St Petersbg. Polytech. Uni. J.: Phys. Math., 1 (2015) 287 – 292. https://doi.org/10.1016/j.spjpm.2015.11.004
  31. Morshedifard, and M. Eskandari-Ghadi, Coupled BE-FE Scheme for Three-Dimensional Dynamic Interaction of a Transversely Isotropic Half-space with a Flexible Structure, Civ. Eng. Infrastruct. J., 50 (2017) 95-118. https://doi.org/10.7508/CEIJ.2017.01.006
  32. A. Tekinsoy, T. Taskiran, C. Kayadelen, T. Baran, An approximation to the stress distribution analysis for anisotropic clayey soil, J. Sci. Res. Essay, 4 (2009) 078 – 087.
  33. J. Anyaegbunam, N.N. Osadebe, O.J. Eze-Uzoamaka, Non-Existence of Solution for Horizontally Rigid Half-space, ASCE J. Geotech. Geoenvironmental Eng., 137 (2011) 431 – 434. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000444
  34. Y. Ojedokun, F. A. Olutoge, Application of Boussinesq’s and Westergaard’s Formulae in Analysing Foundation Stress Distribution for a Failed Telecommunication Mast, Afr. J. Math. Comput. Sci. Res., 5 (2012) 71 – 77. https://doi.org/10.48550/arXiv.1207.5403
  35. Westergaard M., A problem of elasticity suggested by a problem in soil mechanics: soft soft reinforced by numerous strong horizontal sheets, Contributions to the Mechanics of Solids. Stephen Timoshenko 60th Birthday Anniversary Volume Macmillan, New York, 1938.
  36. C. Ike, Elzaki Transform Method for Finding Solutions to Two-Dimensional Elasticity Problems in Polar Coordinates using Airy Stress Functions, J. Comput. Appl. Mech., 51 (2020) 302 – 310. https://doi.org/10.22059/jcamech.2020.296012.472
  37. C. Ike, Fourier Integral Transformation Method for Two-Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions, Math. Model. Eng. Prob., 8 (2021) 333 – 346. https://doi.org/10.18280/mmep.080302
  38. C. Ike, Hankel Transformation Method for Solving the Westergaard Problem for Point, Line and Distributed Loads on Elastic Half-Space, Lat Am J. Solids Struct., 16 (2019) 1 – 19. https://dx.doi.org/10.1590/1679-78255313
  39. C. Ike, Fourier Sine Transform Method for Solving the Cerrutti Problem of Elastic Half Plane in Plane Strain, Math. Model. Civ. Eng., 14 (2018) 1 – 11. https://doi.org/10.2478/mmce-2018-0001
  40. C. Ike, Solution of Elasticity Problems in Two Dimensional Polar Coordinates using Mellin Transform, J. Comput. Appl. Mech., 50 (2019) 174 – 181. https://doi.org/10.22059/jcamech.2019.27
  41. C. Ike, Cosine Integral Transformation Method for Solving the Westergaard Problem in Elasticity of the Half-Space, Civ. Eng. Infrastruct. J., 53 (2020) 313 – 339. https://doi.org/10.22059/ceij.2020.285125.1596
  42. C. Ike, Fourier Cosine Transform Method for Solving the Elasticity Problem of Point Load on an Elastic Half-Plane, Int. J. Sci. Technol. Res., 9 (2020) 1850 – 1856.
  43. C. Ike, Exponential Fourier transform method for stress analysis of boundary load on soil, Math. Model. Eng. Prob., 5 (2018) 33 –39. https://doi.org/10.18280/mmep.050105
  44. C. Ike, Closed Form Solutions of the Navier’s Equations for Axisymmetric Elasticity Problems of the Elastic Half-Space, J. Comput. Appl. Mech., 52 (2021) 588 – 618. https://doi.org/10.22059/JCAMECH.2021.329433.646
  45. C. Ike, B.O. Mama, H.N. Onah, C.U. Nwoji, Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems, Civ. Eng. Archit., 9 (2021) 559 – 583. https://doi.org/10.13189/cea.2021.090301
  46. C. Ike, H.N. Onah, C.U. Nwoji, Bessel Functions for Axisymmetric Elasticity Problems of the Elastic Half-Space Soil, A Potential Function Method, Niger. J. Technol., 36 (2017) 773 – 781. https://doi.org/10.4314/njt.v36i3.16
  47. N. Onah, B.O. Mama, C.U. Nwoji, C.C. Ike, Boussinesq displacement potential functions method for finding vertical stresses and displacement fields due to distributed loads on elastic half space, Electron. J. Geotech. Eng., 22 (2017) 5687 – 5709.
  48. E. Gibson, Some results concerning displacements and stresses in a non-homogeneous elastic half-space, Geotechnique, 17 (1967) 58 – 67. https://doi.org/10.1680/geot.1967.17.1.58
  49. D. Cawier, J.J. Christian “Analysis of an inhomogeneous elastic half-space.” J. Soil Mech. Found. Div., 99 (3), 301 – 306, 1975.
  50. Ter-Martirosyan, Z. G. Soil Mechanics, Moscow, 2009.
  51. Florin, V.A. Fundamentals of Soil Mechanics, Vol. 1 Leningrad State Publishing House of Literature on Construction, Architecture and Building Materials, 1959.
  52. Sitharam, T.G., Govindaraju, L. Theory of Elasticity, Springer, 2021.
  53. Taylor, D.W. Fundamentals of Soil Mechanics. John Wiley and Sons Inc., New York, 1948.
  54. Bhushan, S.C. Haley, Stress distribution for heavy embedded structures, ASCE J. Geotech. Geoenvironmental Eng., 102 (1976) 807 – 810. https://doi.org/10.1061/AJGEB6.0000300
  55. Sadek, J. Shahrour, Use of the Boussinesq solution in geotechnical and road engineering: influence of plasticityUtilisation de la solution de Boussinesq en géotechnique et dans le domaine des chaussées : influence de la plasticité, C.R. Mec., 335 (2007) 516 – 520. https://doi.org/10.1016/j.crme.2007.08.007
  56. C. Ike, Love Stress Function Method for Solving Axisymmetric Elasticity Problems of the Elastic Half-Space, Electron. J. Geotech. Eng., 24 (2019) 663 – 706.
  57. Kachanov, M.L. , Shapira, B. , Tsukrov, I. Handbook of Elasticity Solutions. Springer Science and Business Media Kluwer Academic Publishers Dordrencht, The Netherlands, 2003.
  58. Love, A.E.H. Mathematical Theory of Elasticity, 4th Edition Dover Publications Inc New York ,1944.
  59. Favata, A. On the Kelvin problem, Mathematical Physics (math-ph) 2012. https://doi.org/10.48550/arXiv.1202.1719
  60. Green, A.E. , Zarna, W. Theoretical Elasticity. Oxford University Press, London,1992.
  61. Fujiyoshi, H. Hasegawa, A Boussinesq’s problem of an elastic half-space with an elastic spherical inclusion, Trans. Jpn Soc. Mech. Engrs. Ser. A, 71 (2005) 852 – 857. http://dx.doi.org/10.1299/kikaia.71.852
  62. Hasegawa, K. Watanabe, Green's Functions for Axisymmetric Surface Force Problems of an Elastic Half-Space with Transverse Isotropy, Trans. Jpn Soc. Mech. Engrs, Ser. A, 62 (1996) 1736 – 1740. https://doi.org/10.1299/kikaia.62.1736
  63. Hazel, A. MATH 35021 Elasticity University of Manchester, Manchester. math.manchester.ac.uk/~a.hazel/MATH.3502k html. Accessed 30/11/2015.
  64. Helms, L.L. Potential Theory, Springer London, 2014. https://doi.org/10.1007/978-1-4471-6422-7
  65. Strauss, W.A. Partial Differential Equations: An Introduction Second Edition, John Wiley and Son’s Ltd, 2008.
  66. Kreyszig, E., Kreyszig, H., Norminton, E. J. Advanced Engineering Mathematics 10th Edition, John Wiley and Sons Inc., USA, 2011.
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