### First Principles Derivation of Displacement and Stress Function for Three-Dimensional Elastostatic Problems, and Application to the Flexural Analysis of Thick Circular Plates

Document Type: Research Paper

Authors

1 Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State, Nigeria

2 Department of Civil Engineering, Enugu State University of Science and Technology, Enugu State, Nigeria

Abstract

In this study, stress and displacement functions of the three-dimensional theory of elasticity for homogeneous isotropic bodies are derived from first principles from the differential equations of equilibrium, the generalized stress – strain laws and the geometric relations of strain and displacement. It is found that the stress and displacement functions must be biharmonic functions. The derived functions are used to solve the elasticity problem of finding stresses and displacement fields in a thick circular plate with clamped edges for the case of uniformly distributed transverse load over the plate domain. Superposition of second to sixth order Legendre polynomials which are biharmonic functions are used in the thick circular plate problem as the stress function with the unknown constants as the parameters to be determined. Use of the stresses and displacement fields derived in terms of the stress and displacement function yielded the stress fields and displacement fields in terms of the unknown constants of the biharmonic stress function. Enforcement of the boundary conditions yielded the unknown constants, leading to a complete determination of the stress and displacement function for the stress fields and the displacement fields. The solutions obtained are comparable to solutions in the technical literature.

Keywords

[1]   M. Kashtalyan, J.J. Rushchitsky. Revisiting displacement functions in three – dimensional elasticity of inhomogeneous media. International Journal of Solids and Structures, Elsevier Vol 46, pp 3463 – 3470, 2009. journal homepage www.elsevier.com/locate/ijolstr. doi:10.1016/j.ijolstr.2009.06.001.
[2]   M. Sutti. Elastic Theory of Plates. www.unigeich/math/folks/sutti/Elastic_Theory of _Plates.pdf. Accessed on 12/03/2019, 2015.
[3]   M. Amirpour, R. Das, E.I. Saavedra Flores. Analytical solutions for elastic deformation of functionally graded thick plates with in-plane stiffness variation using higher order shear reformation theory. Composites part B: Engineering, Vol 94, pp 109 – 121, 2016. https//doi.org/10.1016/j.composites b.2016.03040.
[4]   P.C. Chou, N.J. Pagano. Elasticity. Von Nostrand Princeton, 1967.
[5]   G.Z. Voyiadjis, P.I. Kattan. Bending of thick plates on elastic foundation. Advances in the theory of plates and shells. Edited by G.Z. Voyiadjis and D. Karamanlidis. Elsevier Science Publishers B.V. Amsterdam The Netherlands, 1990.
[6]   M. Kashtalyan. Three dimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of Mechanics A/Solids 23(5) pp 853 – 864, 2004.
[7]   C.C. Ike. First principles derivation of a stress function for axially symmetric elasticity problems, and application to Boussinesq problem. Nigerian Journal of Technology (NIJOTECH), Vol 36 No 3, pp. 767 – 772,  July 2017. www.nijotech.com. http://dx.doi.org/10.4314/nijt.v36i3.15  [Cross Ref].
[8]   C.C. Ike, H.N. Onah, C.U. Nwoji. Bessel functions for axisymmetric elasticity problems of the elastic half space soil, a potential function method. Nigerian Journal of Technology (NIJOTECH), Vol 36, No 3, pp 773 – 781, July 2017. www.nijotech.com http//dx.doi.org/10.4314/nijtv36i3.16. [Cross Ref].
[9]   C.U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike. Solution of the Boussinesq problem of half-space using Green and Zerna displacement potential function method. The Electronic Journal of Geotechnical Engineering (EJGE) Vol. 22 Bundle 11, (22.11) pp 4304 – 4314, 2017. Available at www.ejge.com.
[10] C.C. Ike, B.O. Mama, H.N. Onah, C.U. Nwoji. Trefftz harmonic function method for solving Boussinesq problem. Electronic Journal of Geotechnical Engineering (EJGE), (22.12) pp 4589-4601, 2017. Available at www.ejge.com.
[11] C.U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike. Solution of elastic half space problem using Boussinesq displacement potential functions. Asian Journal of Applied Sciences (AJAS), Vol 5, Issue 5, pp. 1100 – 1106. October 2017. ISSN: 2321-0893.
[12] C.C. Ike. Hankel transform method for solving axisymmetric elasticity problems of circular foundation on semi-infinite soils. International Journal of Engineering and Technology (IJET), Vol 10, No 2, pp. 549 – 564, Apr – May 2018. DoI:10.21817/ijet/2018/v10.i.2/181002111. eISSN: 09754024, pISSN: 23198613.
[13] C.C. Ike. General solutions for axisymmetric elasticity problems of elastic half space using Hankel transform method. International Journal of Engineering and Technology (IJET), Vol 10, No 2, pp. 565 – 580, Apr – May 2018. DoI: 10.21817/ijet/2018/v10i2/181002112.
[14] C.C. Ike. Fourier – Bessel transform method for finding vertical stress fields in axisymmetric elasticity problems of elastic half space involving circular foundation areas. Advances in Modelling and Analysis A (AMA A) Vol 55, No 4, pp. 207 – 216, Dec. 2018. https//doi.org/10.18208/ama_a550405.
[15] C.C. Ike. 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 (LAJSS), Vol 16, No 1, pp 1 – 19, 2019. eISSN: 1679-7825, pISSN: 1679-7817. DoI: http:/dx.doi.org/10.1590/1679-78255313.
[16] H.N. Onah, C.C. Ike, C.U. Nwoji, B.O. Mama. 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), pp. 4945 – 4962, 2017. Available at www.ejge.com.
[17] H.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 load on elastic half space. Electronic Journal of Geotechnical Engineering (EJGE), (22.15), pp. 5687 – 5709, 2017. Available at www.ejge.com.
[18] C.C. Ike. Fourier sine transform method for solving the Cerrutti problem of the elastic half plane in plane strain. Mathematical Modelling in Civil Engineering (MMCE) Vol 14, No 1, pp. 1 – 11, 2018. Doi:10.2478/mmce-2018-0001.  [Cross Ref].
[19] C.C. Ike. Kantorovich – Euler Lagrange – Galerkin method for bending analysis of thin plates. Nigerian Journal of Technology (NIJOTECH), Vol 36, No 2, pp. 351 – 360, April 2017. www.nijotech.com http//dx.doi.org/10.4314/nijt/v36i2.5  [Cross Ref].
[20] C.C. Ike. On Maxwell’s stress function for solving three dimensional elasticity problems in the theory of elasticity. Journal of Computational Applied Mechanics (JCAMECH), Vol 49, Issue 2, pp. 342 – 350, Dec 2018. DoI: 10.22059/jcamech.2018.266787.330.
[21] C.C. Ike. Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates. Mathematical Models in Engineering, Vol 4, Issue 2, pp. 50 – 72, June 2018. DOI: https//doi.org/10.21595/mme.2018.19825 [Cross Ref].
[22] C.C. Ike. Equilibrium approach in the derivation of differential equations for homogeneous isotropic Mindlin plates. Nigerian Journal of Technology (NIJOTECH), Vol 36, No 2, pp 346 – 350, April 2017. www.nijotech.com http://dx.doi.org/10.4314/nijtv36i2.4  [Cross Ref].
[23] C.C. Ike. Variational formulation of the Mindlin plate on Winkler foundation problem. Electronic Journal of Geotechnical Engineering (EJGE), 22.12, pp 4829- 4846, 2017. Available at www.ejge.com.
[24] C.C. Ike. Flexural analysis of Kirchhoff paltes on Winkler foundations using finite Fourier sine integral transform method. Mathematical Modelling of Engineering Problems (MMEP), Vol 4, No 4, pp 145 – 154. Dec 2017. DoI: 10.18280/mmep.040402. eISSN: 2369-0747 pISSN 2369-0739.
[25] C.C. Ike, C.U. Nwoji, E.U. Ikwueze, I.O. Ofondu. Bending analysis of simply supported rectangular Kirchhoff plates under linearly distributed transverse load. Explorematics Journal of Innovative Engineering and Technology (EJIET), Vol 01, No 01, pp 28 – 36, September, 2017.
[26] C.C. Ike, C.U. Nwoji, I.O. Ofondu. Variational formulation of Mindlin plate equation and solution for deflections of clamped Mindlin plates. International Journal for Research in Applied Science and Engineering (IJRASET), Vol 5, Issue 1, pp. 340 – 353, January 2017.
[27] C.C. Ike, B.O. Mama. Kantorovich variational method for the flexural analysis of CSCS Kirchhoff – Love plates. Journal of Mathematical Models in Engineering (JVEMME), Vol 4, Issue 1, pp 29 – 41, 2018. eISSN: 2424-4627 pISSN: 2351-5279. DoI: https//doi.org/10.21595/mme.2018.19750.
[28] C.C. Ike, C.U. Nwoji. Kantorovich method for the determination of eigen frequencies of thin rectangular plates. Explorematics Journal of Innovative engineering and Technology (EJIET), Vol 01, No 01, pp. 20 – 27, September 2017.
[29] C.U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike. Theory of elasticity formulation of the Mindlin plate equations. International Journal of Engineering and Technology (IJET), Vol 9, No 6, Dec 2017 – Jan 2018 pp 4344 – 4352, 2017. DoI: 10.21817/ijet/2017/v9i6/170906074.
[30] C.U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike. Ritz variational method for bending of rectangular Kirchhoff – Love plates under transverse hydrostatic load distribution. Mathematical Modelling of Engineering Problems (MMEP), Vol 5, No 1, pp. 1 – 10, March 2018. https//doi.org/10.18280/mmep.050101.
[31] C.U. Nwoji, H.N. Onah, B.O. Mama, C.C. Ike, E.U. Ikwueze. Elastic buckling analysis of simply supported thin plates using double finite Fourier sine integral transform method. Explorematics Journal of Innovative Engineering and Technology (EJIET), Vol 01, No 01, pp. 37 – 47, September 2017.
[32] C.U. Nwoji, B.O. Mama, H.N. Onah, C.C. Ike. Kantorovich – Vlasov method for simply supported plates under uniformly distributed loads. International Journal of Civil, Mechanical and Energy Science (IJCMES), Vol 3, Issue 2, pp. 69 – 77, March – April 2017. http//dx.doi.org/1024001/ijcmes.3.2.1. ISSN: 2455-5304 [Cross-Ref].
[33] C.U. Nwoji, B.O. Mama, H.N. Onah, C.C. Ike. Flexural analysis of simply supported rectangular Mindlin plates under bisinusoidal transverse load. ARPN Journal of Engineering and Applied Sciences, Vol 13, No 15, pp. 4480 – 4488, August 2018. ISSN: 1819-6608.
[34] C.U. Nwoji, B.O. Mama, C.C. Ike, H.N. Onah. Galerkin-Vlasov method for the flexural analysis of rectangular Kirchhoff plates with clamped and simply supported edges. IOSR Journal of Mechanical and Civil Engineering (IOSRJMCE), Vol 14, Issue 2, Version 1, pp. 61 – 74, March – April, 2017. DOI:10-9790/1684-1402016174 eISSN: 2278-5728 www.iosrjournals.org. [Cross Ref].
[35] E. Reissner. On the theory of bending of elastic plates. Journal of Mathematics and Physics, Vol 23, pp. 184 – 191, 1944.
[36] E. Reissner. The effect of shear deformation on the bending of elastic plates. Journal of Applied Mechanics, Vol 12, pp. 69 – 75, 1945.
[37] S.G. Lekhnitskii. Anisotropic Plate. Gordon and Breach, New York, 1968.
[38] P.S. Gujar, K.B. Ladhane. Bending analysis of simply supported and clamped circular plates. SSRG International Journal of Civil Engineering (SSRG IJCE), Vol 2, Issue 5, pp. 44 – 51, May 2015. ISSN: 2348-8352. www.internationaljournalsssrg.org.
[39] N.N. Osadebe, C.C. Ike, H. Onah, C.U. Nwoji, F.O. Okafor. Application of the Galerkin – Vlasov method to the flexural analysis of simply supported rectangular Kirchhoff plates under uniform loads. Nigerian Journal of Technology (NIJOTECH), Vol 35, No 4, pp. 732 – 738, October 2016. http//dx.doi.org/10.4314/nijtv35i4.6 [Cross Ref].
[40] B.O. Mama, C.C. Ike, H.N. Onah, C.U. Nwoji. Analysis of rectangular Kirchhoff plate on Winkler foundation using finite Fourier sine transform method. IOSR Journal of Mathematics (IOSRJM), Vol 13, Issue 1, Version VI, pp. 58 – 66, January – February 2017. DOI:10.9790/5728-1301065866 www.iosrjournals.org [Cross Ref].
[41] B.O. Mama, C.U. Nwoji, C.C. Ike, H.N. Onah. Analysis of simply supported rectangular Kirchhoff plates by the finite Fourier sine transform method. International Journal of Advanced Engineering Research and Science (IJAERS), Vol 4, Issue 3, pp. 285 – 291, March 2017. https//dx.doi.org/10.22161/ijars.4.3.44 [Cross Ref].
[42] H.N. Onah, B.O. Mama, C.C. Ike, C.U. Nwoji. Kantorovich – Vlasov method for the flexural analysis of Kirchhoff plates with opposite edges clamped and simply supported (CSCS plates). International Journal of Engineering and Technology (IJET), Vol 9, No 6, Dec 2017 – Jan 2018 pp 4333 – 4343, 2017.  DoI: 10.21817/ijet/2017/v9i6/170906073.
[43] S.G. Lekhitskiy. Theory of anisotropic thick plates. ANSSSR Izvestiya Mekhanika 1. Mashinostroyeniye (Academy of Sciences of the USSR News. Mechanics and Machine Building) No 2 1959, pp. 142 – 145, 1959. Translated by D. Koolbeck/TDBRS-3 Translation Division Foreign Technology Division WP-AFB. Ohio.
[44] S.P. Timoshenko, J.N. Goodier. Theory of Elasticity. (Third Edition) McGraw Hill New York, 1970.
[45] H.J. Ding, R.Q. Xu, F.L. Guo. Exact axisymmetric solution of laminated transversely isotropic piezoelectric circular plates (II) – Exact solution for elastic circular plates and numerical results. Science in China (Series E) Vol 42, pp. 470 – 478, 1999.
[46] J.Z. Luo, T.G. Liu, T. Zhang. Three dimensional linear analysis for composite axially symmetrical circular plate. International Journal of Solids and Structures, Vol 41 pp. 3689 – 3706, 2004.
[47] H.J. Ding, D.J. Huang, H.M. Wang. Analytical solution for fixed end beam subjected to uniform load. Journal of Zhejiang University (SCIENCE), 6A(8), pp. 779-783, 2005.
[48] H.J. Ding, X.Y. Lee, W.Q. Chen. Analytical solutions for a uniformly loaded circular plate with clamped edges. Journal of Zhejiang University SCIENCE, Vol 6A, Issue 10, pp. 1163-1168, 2005. ISSN: 1009-3095 http//www.zju.edu.cn/jzus.
[49] H. Ding, W. Chen, L. Zhang. Elasticity of transversely isotropic materials. Springer Dordrecht, 2006.
[50] S.P. Timoshenko, S. Woinowsky – Krieger. Theory of Plates and Shells, (Second Edition). McGraw Hill New York, 1959.
[51] W.D. Tseng, J.Q. Tarn. Exact Elasticity solution for axisymmetric deformation of circular plates. Journal of Mechanics, Vol 31, Issue 6, pp. 617 – 629, December 2015. https//doi.org/10.1017/jmech2015.37.
[52] K.T. Sundara Raja Iyengar, K. Chandrashekhara, V.K. Sebastian. A higher order theory for circular plates using higher order approximations. https//repository.lib.ucsu.edu/bitstream/handle/1840.20/28839/MS-3.pdf …1, 1973.
[53] K.T. Sundara Raja Iyengar, K. Chandrashekhara, V.K. Sabastain. Method of Initial Functions in the Analysis of Thick Circular Plates. Nuclear Engineering and Design. Vol 36, Issue 3, pp. 341 – 354, March 1976. Science Direct ISSN 0029-5493. https/doi.org/10.1016/0029-5493(76)90027-3.
[54] X.Y. Li, H.J. Ding, W.Q. Chen. Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk. International Journal of Solids and Structures, Vol 45, (2008), pp. 191 – 210, 2008. doi:10.1016/j.ijsolstr.2007.07.023. www.elsevier.com/locate/ijsolstr.
[55] K. Chandrashekhara. Theory of Plates. Universities Press (India) Limited ISBN 8173712530, 9788173712531 410pp, 2001.
[56] H.A. Elliot. Three dimensional stress distributions in hexagonal aeolotroic crystals. Proceedings of Cambridge Philosophical Society, 44 pp. 522 – 533, 1948.
[57] H. Hu. On the three – dimensional problems of the theory of elasticity of a transversely isotropic body. Acta Scientia Sinica 2(2), pp. 145 – 151, 1953.
[58] R.P. Shimpi. Refined plate theory and its variants. AIAA Journal, Vol 40, pp 137 – 146, 2002.
[59] M. Danesh, A. Farajpour, M. Mohammadi. Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method” Mechanics Research Communications, Vol 39 No 1, pp 23-27, January 2012. DOI:10.1016/j.mechrescom.2011.09.004.
[60] M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. Rastgoo. Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads. European Journal of Mechanics – A / Solids, Vol 77, Sept – Oct 2019, 103793, https://doi.org/10.1016/j.euromechsol.2019.05.008.
[61] M. Mohammadi, A. Rastgoo. Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core. Structural Engineering and Mechanics, Vol 69 No 2, pp 131-143, 2019. DOI:10.12989/sem.2019.69.2.131
[62] M. Mohammadi, A Rastgoo. Primary and secondary resonance analysis of FG / lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium. Mechanics of Advanced Materials and Structures, pp 1-22, Dec 2018. https://doi.org/10.1080/15376494.2018.1525453
[63] M. Safarabadi, M. Mohammadi, A. Farajpour, M. Goodarzi. Effect of surface energy on the vibration analysis of rotating nanobeam. Journal of Solid Mechanics, Vol 7 No 3, pp 299-311, 2015.
[64] S.R. Aseni, M. Mohammadi, A. Farajpour. A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory. Latin American Journal of Solids and Structures, Vol 11 No 9, pp 1541-1564, 2014.
[65] M. Mohammadi, A. Farajpour, M. Goodarzi, H. Mohammadi. Temperature effect on the vibration analysis of annular graphene sheet embedded on visco-Pasternak foundation. Journal of Solid Mechanics, Vol 5 No 3, pp 305-323, 2013.
[66]         M. Goodazi, M.N. Bahrami, V. Tavat. Refined plate theory for free vibration analysis of FG nanoplates using the nonlocal continuum plate model. Journal of Computational Applied Mechanics, Vol 48 No 1, pp 123-136, June 2017. DOI: 10.2205/JCAMECH.2017.236217.155
###### Volume 51, Issue 1June 2020Pages 184-198
• Receive Date: 15 January 2020
• Revise Date: 03 February 2020
• Accept Date: 04 February 2020