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Cosine Integral Transform Method for Solving the Westergaard Problem in Elasticity of the Half-Space | ||
| Civil Engineering Infrastructures Journal | ||
| دوره 53، شماره 2، اسفند 2020، صفحه 313-339 اصل مقاله (931.81 K) | ||
| نوع مقاله: Research Papers | ||
| شناسه دیجیتال (DOI): 10.22059/ceij.2020.285125.1596 | ||
| نویسنده | ||
| Charles Chinwuba Ike* | ||
| Member, Nigerian Society of Engineers. COREN REGISTERED ENGINEER. | ||
| چکیده | ||
| The cosine integral transform method is applied to find the expressions for spatial variations of displacements and stresses in the Westergaard continuum under vertical concentrated loading, and distributed loadings acting over lines and geometric areas on the surface. The half-space is considered to be horizontally inextensible and the displacement field reduces to the vertical displacement component. The paper derives a displacement formulation of the equation of equilibrium in the vertical direction. Cosine integral transformation is applied to the formulated equation and the Boundary Value Problem (BVP) is found to simplify to Ordinary Differential Equation (ODE). The general solution of the ODE is obtained in the cosine integral transform space. The requirement of bounded solutions is used to obtain one integration constant. Inversion of the bounded solution gave the solution in the real problem domain space. The stress fields are obtained using the stress-displacement equations. The requirement of equilibrium of the vertical stress fields and the vertical point loading at the origin is used to determine the remaining integration constant, and thus the vertical deflections and the stresses. The solutions obtained are kernel functions employed to derive the expressions for solutions for line, and uniformly distributed loads applied over given geometric areas such as rectangular and circular areas. The vertical stresses are expressed in terms of dimensionless vertical stress influence factors and tabulated. The vertical displacements and stresses obtained are identical with Westergaard solutions obtained by stress function method. The solutions agree with results obtained by Ike using Hankel transform method. | ||
| کلیدواژهها | ||
| Cosine Integral Transform Method؛ Elastic Half-Space؛ Inverse Cosine Integral Transform؛ Stress Fields؛ Westergaard Problem | ||
| مراجع | ||
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Abeyaratne, R. (2012). Continuum mechanics, Volume 11 of Lecture notes on the Mechanics of Elastic Solids. Cambridge. http//web.mit.edu/abeyaratne/lecture notes/html 11th May 2012. Updated 6th May, 2015.Retrieved 8th April 2016.
Andrews, L.C. and Shivamoggi, B.K. (1999). Integral transforms for engineers. SPIE Optical Engineering Press. A publication of SPIE, The International Society for Optical Engineering, Bellingham Washington USA.
Anyaegbunam, A.J., Osadebe, N.N. and Eze-Uzoamaka, O.J. (2011). “Non-existence of solution for horizontally rigid half-space”, Journal of Geotechnical and Geoenvironmental Engineering, 137(4), 431-434.
Apostol, B.F. (2017). “Elastic displacement in half space under the action of a tension force, General solution for the half space with point forces”, Journal of Elasticity, 126, 231-244.
Ardeshri-Behrestaghi, A., Eskandari-Ghadi, M. and Vaseghi-Amiri, J. (2013). “Analytical solution for a two-layer transversely isotropic half-space affected by an arbitrary shape dynamic surface load”, Civil Engineering Infrastructures Journal, 46(1), 1-14.
Barbar, J.R. (2010). Elasticity, Third Revised Edition. Springer Science and Bussiness Media. Dordrecht, The Netherlands.
Barden, L. (1963). “Stresses and displacements in a cross-anisotropic soil”, Geotechnique, 13, 198-210.
Bowles, J.E. (1997). Foundation analysis and design, International Edition. McGraw Hill International Book Company, Tokyo.
Chau, K.T. (2013). Analytical methods in Geomechanics, CRC Press, Taylor and Francis Group, New York.
Davis, R.O. and Selvadurai, A.P.S. (1996). Elasticity and goemechanics, Cambridge University Press.
Dobrovolsky, I.P. (2015). “Unidimensional inhomogeneous isotropic elastic half-space”, Open Access Library Journal, 2(7), 1-6.
Eskandari-Ghadi, M., Charmhini, A.H. and Ardeshi-Behrestaghi, A. (2014). “A method of function space for vertical impedance function of a circular rigid foundation on a transversely isotropic ground”, Civil Engineering Infrastructures Journal, 47(1), 13-27.
Fadum, R.E. (1948). “Influence values for estimating stresses in elastic foundations”, Proceedings Second International Conference on Soil Mechanics and Foundation Engineering, 3, 77-84.
Green A.E. and Zerna W. (1954). Theoretical elasticity, Oxford University Press, London.
Godara, Y., Sahrawat, R.K. and Singh, M. (2017). “Static elastic deformation in an orthotropic half-space with rigid boundary model due to non-uniform long strike slip fault”, Journal of Earth System Science, 126(97), 1-10.
Hayati, Y., Eskandari-Ghadi, M., Raoofian, M., Rahimian, M. and Ardalan, A.A. (2013). “Dynamic Green’s functions of an axisymmetric thermoelastic half-space by a method of potentials”, Journal of Engineering Mechanics,139(9), 1166-1177.
Hazel, A. (2015). MATH 35021 elasticity, University of Manchester, Manchester. www.maths.manchester.ac.lc/_ahazel/MATH/35021/MATH 35021.html, November 30.
Ike, C.C. (2017). “First principles derivation of a stress function for axially symmetric elasticity problems, and application to Boussinesq problem”, Nigerian Journal of Technology, 36(3), 767-772.
Ike, C.C. (2018a). “Hankel transform method for solving axisymmetric elasticity problems of circular foundation on semi-infinite soils”, International Journal of Engineering and Technology, 10(2), 549-564.
Ike, C.C. (2018b). “General solutions for axisymmetric elasticity problems of elastic half space using Hankel transform method”, International Journal of Engineering and Technology, 10(2), 565-580.
Ike, C.C. (2018c). “On Maxwell’s stress functions for solving three-dimensional elasticity problems in the theory of elasticity”, Journal of Computational Applied Mechanics, 49(2), 342-350.
Ike, C.C. (2019a). “Hankel transformation method for solving the Westergaard problem for point, line and distributed loads on elastic half-space”, Latin American Journal of Solids and Structures, 16(1), 1-19.
Ike, C.C. (2019b). “Solution of elasticity problems in two dimensional polar coordinates using Mellin transform”, Journal of Computational Applied Mechanics, 50(1), 174-181.
Ike, C.C. (2020a). “Elzaki transform method for solving elasticity problems in two-dimensional polar coordinates”, Journal of Computational Applied Mechanics, Article in Press: DOI:10.22059/JCAMECH:2020.296012.472.
Ike, C.C. (2020b). “Fourier cosine transform method for solving the elasticity problem of point load on an elastic half-plane”, International Journal of Scientific and Technology Research, 9(4), 1850-1856.
Ike, C.C., Onah H.N. and Nwoji C.U. (2017a). “Bessel functions for axisymmetric elasticity problems of the elastic half space soil, a potential function method”, Nigerian Journal of Technology, 36(3), 773-781.
Ike, C.C., Mama, B.O., Onah, H.N. and Nwoji, C.U. (2017b). “Trefftz Harmonic function method for solving Boussinesq problem”, Electronic Journal of Geotechnical Engineering, 22(12), 4589-4601.
Kachanov, M.L., Shafiro, B. and Tsukrov, I. (2003). Handbook of elasticity solutions, Springer Science and Bussiness Media, Kluwer Academic Publishers, Dordrecht, The Netherlands.
Liao, J.J. and Wang, C.D. (1998). “Elastic solutions for a transversely isotropic half-space subjected to a point load”, International Journal for Numerical and Analytical Methods in Geomechanics, 22, 425- 447.
Lurie, S.A. and Vasilev, V.V. (1995). The biharmonic problem in the theory of elasticity, Gordon and Breach Publishers, United States.
Miroshnikov, V.Yu. (2018). “First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities”, Journal of Mechanical Engineering, 21(2), 12-18.
Miroshnikov, V.Yu (2017). “On computation of the stress-strain state of a space weakened by a system of parallel circular cylindrical cavities with different edge conditions”, Science and Practice: A New Level of Integration in the Modern World, 4th Conference Proceedings Scope Academic House, Sheffield, UK, pp. 77-83.
Naeeni, M.R. and Eskandari-Ghadi, M. (2016). “A potential method for body and surface wave propagation in transversely isotropic half – and full – spaces”, Civil Engineering Infrastructures Journal, 49(2), 263-288.
Naeeni, M.R., Campagna, R., Eskandari-Ghadi, M. and Ardalan, A.A. (2015). “Performance comparison of numerical inversion methods for Laplace and Hankel integral transforms in engineering problems”, Applied Mathematics and Computation, 250, 759-775.
Nwoji, C.U., Onah, H.N., Mama, B.O. and Ike, C.C. (2017a). “Solution of elastic half space problem using Boussinesq displacement potential functions”, Asian Journal of Applied Sciences, 5(5), 1100-1106.
Nwoji, C.U., Onah, H.N., Mama, B.O. and Ike, C.C. (2017b). “Solution of the Boussinesq problem using Green and Zerna displacement potential function method”, Electronic Journal of Geotechnical Engineering, (22.11), 4305-4314.
Ojedokun, O.Y. and Olutoge, F.A. (2012). “Application of Boussinesq’s and Westergaard’s formulae in analysing foundation stress distribution for a failed telecommunication mast”, African Journal of Mathematics and Computer Science Research, 5(4), 71-77.
Palaniappan, D. (2011). “A general solution of equations of equilibrium in linear elasticity”, Applied Mathematical Modelling, 35, 5494-5499.
Piessens, R. (2000). “The Hankel transform” in “The transforms and applications handbook, Second Edition, Alexander D. and Poularikas Boca Raton (eds.), Raton CRC Press.
Podio-Guidugli, P. and Favata, A. (2014). Elasticity for geotechnicians, A modern exposition of Kelvin, Boussinesq, Flammant, Cerrutti, Melan and Mindlin problems, Solid mechanics and its applications, Springer.
Rocscience (2018). Westergaard stress solution method, https//www.rocscience.com/ documents/../Westergaard_Stress_Solution_Method pdf. Retrieved 02/09/2018.
Sadd, M.H. (2014). Elasticity theory, application and numerics, Third Edition, University of Rhode Island, Elsevier Academic Press, Amsterdam.
Sitharam, T.G. and GovindaRaju, L. (2017). 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.
Sneddon, I.N. (2010). Fourier transforms, Dover Publication.
Sneddon, I.N. (1992). “Fourier transform solution of a Boussinesq problem for a hexagonally aeolotropic elastic half-space”, The Quarterly Journal of Mechanics and Applied Mathematics, 45(4), 607-616.
Tarn, J.Q. and Wang, Y.M. (1987). “A fundamental solution for a transversely isotropic elastic space”, Journal of the Chinese Institute of Engineers, 10(1), 13-21.
Teodorescu, P.P. (2013). Treatise on classical elasticity theory and related problems, Mathematical and analytical techniques with applications to engineering, Springer Dordrecht.
Tuteja, R., Joloree, S. and Goyal, A. (2014). “Application of Hankel transform of I-function of one variable for solving axisymmetric Dirichlet potential problem”, Global Journal of Science Frontier Research: Mathematics and Decision Sciences, 14(4), 11-16.
Westergaard, H.M. (1938). A problem of elasticity suggested by a problem in soil mechanics: soft soil reinforced by numerous strong horizontal sheets, Contributions to the mechanics of solids, Stephen Timoshenko 60th Birthday Anniversary Volume, Macmillan, New York.
Voegtle, I.C. (2017). The Bessel function, the Hankel transform and an application to differential equations, Electronic Thesis and Dissertations, Jack N. Anerity, College of Graduate Studies, Georgia Southern University.
VSUT (2018). Stress distribution in soil, www.vsut.ac.in/lecture_notes/lecture 1424085471.pdf, Retrieved 01/11/2018.
Yokoyama, T. (2014). The Hankel transform, www.stat.phys.titech.ac.jp/~Yokayama/memo4.pdf, Accessed 1/7/2018.
Zhou, S. and Gao, X-L. (2013). “Solutions of half-space and half-plane contact problems based on surface elasticity”, Zeitschrift für angewandte Mathematik und Physik, 64, 145-166. | ||
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