OPEN ACCESS
In this work, the Fourier-Bessel transformation method was used to determine the vertical stress fields in axisymmetric elasticity problems of elastic half space involving circular foundation areas subject to uniformly distributed loads. A stress-based formulation of the elasticity problem was adopted. The biharmonic stress compatibility equation was solved using the variable separable technique to obtain a general solution for the bounded stress-functions as Fourier-Bessel integrals. Egorov expressions for the vertical stress fields defined in terms of harmonic functions were used to obtain the vertical stress fields. The load distribution was similarly transformed by the Fourier-Bessel transformation. Enforcement of the boundary condition of the equilibrium of the internal vertical stress at the z = 0 plane and the applied load yielded the unknown parameter of the bounded Fourier-Bessel transform integral, and thus, the full determination of the bounded stress function $\Omega(r, z)$. The vertical stress fields $\sigma_{z z}(r, z)$ were obtained from the bounded stress potential function using Egorov expressions for $\sigma_{z z}(r, z)$ Evaluation of the integration problem yielded mathematical expressions for the vertical stresses at any point in the elastic half space. The vertical stresses at any point under the center of the circular foundation were also determined, and tabulated. The mathematical expressions for vertical stresses obtained using Fourier-Bessel transform method were identical with solutions in the technical literature.
Fourier-Bessel transform method, axisymmetric elasticity problem, circular foundation, elastic half space, vertical stress field, stress potential function
[1] Ike CC. (2006) Principles of Soil Mechanics. De-Adroit Innovation, Enugu.
[2] Chau KT. (2013) Analytic methods in geomechanics. CRC Press Taylor and Francis Group, New York.
[3] Sadd MH. (2014) Elasticity theory, applications and numerics. Third Edition, Elsevier Academic Press, Amsterdam.
[4] Padio-Guidugli P, Favata A. (2014) Elasticity for geotechnicians: A modern exposition of Kelvin, Boussinesq, Flammant, Cerrutti, Melan and Mindlin problems. Solid Mechanics and Its Applications, Springer.
[5] Hazel Andrew (2015) MATH 35021 Elasticity. University of Manchester, Manchester. www.maths.manchester.ac.uk/ahazel/MATH35021/ MATH35021.html.
[6] Palaniappan D. (2011) A general solution of equations of equilibrium in linear elasticity. Applied Mathematical Modelling 35: 5494-5499, Elsevier.
[7] Barber JR. (2010) Elasticity, 3rd Revised Edition. Springer Science and Business Media, Dordrecht, the Netherlands.
[8] Kachanov ML, Shafiro B, Tsukrov I. (2003) Handbook of elasticity solutions. Springer Science and Business Media Kluwer Academic Publishers Dordrecht, the Netherlands.
[9] Sitharam TG, Govinda-Reju L. Applied Elasticity for Engineers Module: Elastic Solutions and Applications in Geomechanics. 14.139.172.204/ nptel/1/CSE/web/ 105108070/module 8/lecture 17.pdf
[10] Abeyaratne R. (2012) Continuum Mechanics, Volume II of lecture notes on the Mechanics of Elastic Solids Cambridge http//web.mit.edu/abeyartne/ lecture_notes.html, 11th May, 2012. Updated 6th May, 2015. ISBN – 13:978-0-9791865-0-9. ISBN – 10:0-9791865-0-1.
[11] Timoshenko SP, Goodier JN. (1970) Theory of Elasticity, Third Edition. McGraw Hill, New York.
[12] Sokolnikoff IS (1956) Mathematical Theory of Elasticity, Second Edition. Data McGraw Hill Publishing Company Ltd, Bombay New Delhi.
[13] Lurie SA, Vasilev VV. (1995) The Biharmonic Problem in the Theory of Elasticity. Gorden and Breach Publishers, United States.
[14] Onah HN, Mama BO, Nwoji CU, Ike CC. (2017) Boussinesq displacement potential functions method for finding vertical stresses and displacement fields due to distributed load on elastic half space. Electronic Journal of Geotechnical Engineering (EJGE) 22(14): 5687-5709.
[15] Ike CC, Mama BO, Onah HN, Nwoji CU. (2017) Trefftz harmonic function method for solving Boussinesq problem. Electronic Journal of Geotechnical Engineering (EJGE) 22(12): 4589-4601.
[16] Nwoji CU, Onah HN, Mama BO, Ike CC. (2017) Solution of the Boussinesq problem of half space using Green and Zerna displacement potential function method. The Electronic Journal of Geotechnical Engineering (EJGE) 22(11): 4305-4314.
[17] Ike CC, Onah HN, Nwoji CU. (2017) Bessel functions for axisymmetric elasticity problems of the elastic half space soil: A potential function method. Nigerian Journal of Technology (NIJOTECH) 36(3): 773-781. http://dx.doi.org/w4314/nijtv36i3.16
[18] Onah HN, Osadebe NN, Ike CC, Nwoji CU. (2016) Determination of stresses caused by infinitely long line loads on semi-infinite elastic soils using Fourier transform method. Nigerian Journal of Technology (NIJOTECH) 35(4): 726-731. https://doi.org/10.4314/nijt/v35i4.7
[19] Nwoji CU, Onah HN, Mama BO, Ike CC. (2017) Solution of elastic half space problem using Boussinesq displacement potential functions. Asian Journal of Applied Sciences (AJAS 05(05): 1100-1106.
[20] Onah HN, Ike CC, Nwoji CU, Mama BO. (2017) Theory of elasticity solution for stress fields in semi-infinite linear elastic soil due to distributed load on the boundary using the Fourier transform method. Electronic Journal of Geotechnical Engineering (EJGE) 22(13): 4945-4962.
[21] Ike CC. (2017) First principles derivation of a stress function for axially symmetric elasticity problems, and application to Boussinesq problem. Nigerian Journal of Technology (NIJOTECH) 36(3): 767-772.
[22] Ike CC. (2018) General solutions for axisymmetric elasticity problems of elastic half space using Hankel transform method. International Journal of Engineering and Technology (IJET) 10(2): 565-580. https://10.21817/ijet/2018/ v10i2/181002112
[23] Ike CC. (2018) Hankel transform method for solving axisymmetric elasticity problems of circular foundation on semi-infinite soils. International Journal of Engineering and Technology (IJET) 10(2): 549-564. https://10.21817/ijet/2018/v10i2/181002111