### Least squares weighted residual method for finding the elastic stress fields in rectangular plates under uniaxial parabolically distributed edge loads

Document Type : Research Paper

Authors

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

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

Abstract

In this work, the least squares weighted residual method is used to solve the two-dimensional (2D) elasticity problem of a rectangular plate of in-plane dimensions 2a 2b subjected to parabolic edge tensile loads applied at the two edges x = a. The problem is expressed using Beltrami–Michell stress formulation. Airy’s stress function method is applied to the stress compatibility equation, and the problem is expressed as a boundary value problem (BVP) represented by a non-homogeneous biharmonic equation. Airy’s stress functions are chosen in terms of one and three unknown parameters and coordinate functions that satisfy both the domain equations and the boundary conditions on the loaded edges. Least squares weighted residual integral formulations of the problems are solved to determine the unknown parameters and thus the Airy stress function. The normal and shear stress fields are determined for the one-parameter and the three-parameter coordinate functions. The solutions for the stress fields are found to satisfy the stress boundary conditions as well as the domain equation. The presented solutions for the Airy stress function and the normal stresses and shear stress fields are identical with solutions obtained by using variational Ritz methods, Bubnov–Galerkin methods and agree with results obtained by Timoshenko and Goodier.

Keywords

[1]  S. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw Hill Book Company Inc. 1951.
[2]  R. Richards, Principles of Solid Mechanics, CRC Press, Washington DC, 2001. https/doi.org/10.1201/9781420042207.
[3]  Yu-hua Yang and Xin-Wei Wang, Stress analysis of thin rectangular plates under non-linearly distributed edge loads, Engineering Mechanics, Vol. 28, Issue 1, p. 37 – 42, 2011.
[4]  B.O. Mama, C.U. Nwoji, H.N. Onah, and C.C. Ike, Bubnov–Galerkin method for the elastic stress analysis of rectangular plates under uniaxial parabolic distributed edge loads, International Journal of Engineering and Technology (IJET) Vol 9 No 6 Dec 2017 – Jan 2018), p. 4323 – 4332, 2017 DOI: 10.21817/ijet/2017/v.9i6/170906060.
[5]  C.U. Nwoji, C.C. Ike, H.N. Onah and B.O. Mama, Variational Ritz method for the elastic stress analysis of plates under uniaxial parabolic distributed loads, IOSR Journal of Mechanical and Civil Engineering, (IOSR JMCE) Volume 14, Issue 2, Version 2 p. 60 – 71, March – April 2017. https/doi.org/10.9790/1684-1402026071 [Cross Ref].
[6]  J.R. Barber, The solution of elasticity problems for the half-space by the method of Green and Collins, Applied Scientific Research,Volume 40, Issue 2, Martinivas Nijhoff Publishers, The Hague, Netherlands p 135 – 157, 1983 https//doi.org/10.1007/bf00386216 [Cross Ref].
[7]  J. Blaauwendraad, Theory of elasticity Ct 5141 Direct Methods,  Faculty of Civil Engineering and Geosciences, Delft University of Technology, 2004.
[8]  T. Kimura, Studies on stress distribution in pavements subjected to surface shear forces, Proceedings of the Japan Academy Series B, Volume 90,No 02, p 47 – 55, 2014. http//doi.org/10.2183/pjab.90.47 [Cross Ref].
[9]  R.A. Patil, Complete stress analysis for two dimensional inclusion problem using complex variables,  MSc Mechanical Engineering Thesis, Faculty of Graduate School, The University of Texas at Arlington, August 2007.
[10] K.K. Davarakonda, Buckling and flexural vibration of rectangular plates subjected to half-sinusoidal load on two opposite edges, PhD Dissertation School of Aerospace and Mechanical Engineering Graduate Faculty, University of Oklahoma Graduate School, p 139, 2004.
[11] M.Z. Nejad and A. Hadi, Eringin’s non-local elasticity theory for bending analysis of bi-drectional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 106, p.1 – 9, 2016.
[12] M.Z. Nejad and A. Hadi, Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams, International Journal of Engineering Science, Vol. 105, p.1 – 11, 2016.
[13] M.Z. Nejad, A. Hadi and A. Rastgoo. Buckling Analysis of arbitrary two-dimensional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 103, p.1 – 10, 2016.
[14] M.Z. Nejad, A. Hadi and A. Farajpour, Consistent couple-stress theory for free vibration analysis of Euler –Bernoulli nano-beams made of bi-directional functionally graded material,  Structural Engineering and Mechanics,Vol. 63 , No.2, p.161 – 169, 2017.
[15] M.Z. Nejad, A. Hadi, A. Omidvari, and A. Rastgoo, Bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams using integral form of Eringen’s non-local elasticity theory, Structural Engineering and Mechanics, Vol. 67, No.4, p. 417 – 425, 2018.
[16] M.Z. Nejad, M. Jabbari and A.Hadi, A review of functionally graded thick cylindrical and conical shells, Journal of Computational Applied Mechanics, Vol. 48, 2, p.357 – 370, 2017.
[17] A. Daneshmehr, A. Rajalipoor and A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with higher order theories, International Journal of Engineering Science, Vol. 95, p. 23 – 35, 2015.
[18] A. Hadi, M.Z. Nejad and M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, p. 12 – 23, 2018.
[19] A. Hadi A, M.Z. Nejad, A. Rastgoo and M. Husseini, Buckling analysis of FGM Euler–Bernoulli nano-beams with 3D-varying properties based on consistent couple – stress theory, Steel and Composite Structures,Vol. 26 No.6, p. 663 – 672.
[20] K. Dohshahri, M.Z. Nejad, S. Ziace, A. Nikrejad and A. Hadi, Free vibrations analysis of arbitary three dimensionally FGM nanoplates, Advances in Nano Research, Vol. 8  No.2, p. 115 – 134, 2020.
[21] A. Barati, A. Hadi, M.Z. Nejad, and R. Noroozi, On vibration of bi-directional functionally graded nanobeams under magnetic field, Mechanics Based Design of Structures and Machines, p. 1 – 18, 2020.
[22] R. Noroozi, A. Barati, A. Kazemi, S. Nourouzi and A. Hadi, Torsional vibration analysis of bi-directional FG nano-cone with arbitrary cross-section based on nonlocal strain gradient elasticity, Advances in Nano Research, Vol. 8 No.1,p. 13 – 24.
[23] E. Zarezadeh, V. Hosseini and A. Hadi, Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory, Mechanics Based Design of Structures and Machines, p. 1 – 16, 2019.
###### Volume 51, Issue 1June 2020Pages 107-121
• Receive Date: 17 February 2020
• Revise Date: 23 May 2020
• Accept Date: 31 May 2020