Fourier series method for finding displacements and stress fields in hyperbolic shear deformable thick beams subjected to distributed transverse loads

Document Type : Research Paper

Author

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

Abstract

This paper presents a systematic formulation of the hyperbolic shear deformation theory for bending problems of thick beams; and the Fourier series method for solving the resulting system of coupled differential equations and ultimately finding the displacements and stress fields. Hyperbolic sine and cosine functions are used in formulating the displacement field components such that transverse shear stress free conditions are achieved at the top and bottom surfaces of the beam, thus obviating the shear correction factors of the first order shear deformation theories. The vanishing of the first variation of the total potential energy functional is used to obtain the system of coupled differential equations for the domain and the boundary conditions. The domain equations are solved using Fourier series method for simply supported ends for linearly distributed and uniformly distributed loads. The solutions are found as infinite series with good convergence. Solutions obtained for the axial and transverse displacements, and normal and shear stresses at critical points on the beam agree remarkably well with previous solutions, and for normal stresses, the errors of the present method are less than 0.5% for aspect ratio of 4 and less than 1.9% for aspect ratio of 10.

Keywords

[1]     C.C. Ike, Fourier sine transform method for the free vibration of Euler-Bernoulli beam resting on Winkler foundation, International Journal of Darshan Institute on Engineering Research and Emerging Technologies, Vol. 7 No.1, pp 1 – 6, 2018. DOI: 10.32692/IJDI-ERET/7.1.2018.1801.
[2]     C.C. Ike, Point collocation method for the analysis of Euler-Bernoulli beam on Winkler foundation, International Journal of Darshan Institute on Engineering Research and Emerging Technologies, Vol. 7, No.2, pp 1 – 7, 2018.
[3]     C.C. Ike,  E.U. Ikwueze, Ritz method for the analysis of statically indeterminate Euler-Bernoulli beams, Saudi Journal of Engineering and Technology, Vol.3, No 3, pp 133 – 140, 2018. DOI:10.21276/sjeat.2018.3.3.3
[4]     C.C. Ike, E.U. Ikwueze, Fifth degree Hermittian polynomial shape functions for the finite element analysis of clamped simply supported Euler-Bernoulli beam, American Journal of Engineering Research, Vol.7 No.4, pp 99 – 105, 2018.
[5]     B.O. Mama, O.A. Oguaghamba, C.C. Ike, Quintic polynomial shape functions for the finite element analysis of elastic buckling loads of Euler-Bernoulli beam resting on Winkler foundation, Proceedings, 2nd NIEEE Nsukka Chapter Conference on Sustainable Infrastructure Development in Developing Nations, pp 122-128, 2020.
[6]     S.P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine and Journal of Science, Vol. 41, No 6, pp 744 – 746, 1921. https://doi.org/10.1080/14786442108636264.
[7]     K.S. Pakhare, M. Mitra, R.P. Shimpi,  Development of single variable new first order shear deformation theories for plates and beams, Master of Technology Dissertation, Department of Aerospace Engineering, Indian Institute of Technology, Bombay, 2016.
[8]     I. Senjanovic, N. Vladimir, N. Hadzik, M. Tomic,   New first order shear deformation beam theory with in-plane shear influence” Engineering Structures, Vol. 110, No. 1, pp. 169 – 183, 2016. https://doi.org/10.1016/j.engstructures.2015.11.032
[9]     C.C. Ike, Timoshenko beam theory for the flexural analysis of moderately thick beams – variational formulation and closed form solutions. Tecnica Italiana – Italian Journal of Engineering Science, Vol.63, No.1, pp. 34–45, 2019. https://doi.org/10.18280/ti-ijes.630105
[10]  C.C. Ike, C.U. Nwoji, H.N. Onah, B.O. Mama, M.E. Onyia, Modified single finite Fourier cosine integral transform method for finding the critical buckling loads of first order shear deformable beams with fixed ends. Revue des Composites et des Materiaux Avances, Vol. 29, No. 6, pp. 357–362, 2019. https://doi.org/10.18280/rcma.290603
[11]  C.C. Ike, C.U. Nwoji, B.O. Mama, H.N. Onah, M.E. Onyia, Laplace transform method for the elastic buckling analysis of moderately thick beams, International Journal of Engineering Research and Technology, Vol. 12, No.10, pp. 1626 – 1638, 2019.
[12]  H.N. Onah, C.U. Nwoji, M.E. Onyia, B.O. Mama, C.C. Ike, Exact solutions for the elastic buckling problems of moderately thick beams, Revue des Composites et des Materiaux Avances, Vol.30, No.2, pp. 83 – 93, 2020. https://doi.org/10.18280/rcma.300205
[13]  R.P. Shimpi, P.J. Guruprasad, K.S. Pakhare,  Simple two variable refined theory for shear deformable isotropic rectangular beams, Journal of Applied and Computational Mechanics, Vol. 6, No.3, pp  394 – 415, 2020. DOI:10.22055/JACM.2019.29555.1615.
[14]  R.P. Shimpi, R.A. Shetty, A. Guha, A simple single variable shear deformation theory for a rectangular beam. Proceedings of the Institution of Mechanical Engineers Part C. Journal of Mechanical Engineering Science, Vol. 231, No.24, pp. 4576 – 4591, 2017.  https://doi.org/10.1177/0954406216670682
[15]  M. Levinson, A new rectangular beam theory, Journal of Sound and Vibration, Vol. 74, No.1, pp. 81 – 87, 1981. https://doi.org/10.1016/0022-460X(81)90493-4
[16]  Y. Gao, M. Wang, The refined theory of deep rectangular beams based on general solutions of elasticity, Science in China: Series G Pysics, Mechanics and Astronomy, Vol. 49, No 3, pp 291 – 303, 2006. https://doi.org/10.1007/s11433-006-0291-0
[17]  G. Shi, G. Z. Voyiadjis, A sixth order theory of shear deformable beams with variational consistent boundary conditions, Journal of Applied Mechanics, Vol. 78, No.2, pp 021019, (11pgs), 2011. https://doi.org/10.1115/1.4002594
[18]  C.C. Ike, O.A. Oguaghamba, Trigonometric shear deformation theory for bending analysis of thick beams: Fourier series method, Proceedings, Conference on Engineering Research, Technology Innovation and Research, 2020.
[19]  C.C. Ike, Ritz variational method for the flexural analysis of third order shear deformable beams, Proceedings, Conference on Engineering Research, Technology Innovation and Practice, pp. 64, 2020.
[20]  Y. Ghugal, V. Nakhate, Flexure of thick beams using trigonometric shear deformation theory, The Bridge and Structural Engineer – The Quarterly Journal of the Indian National Group of International Association for Bridge and Structural Engineering, Vol. 39, No.4, pp 1 – 17, 2010.
[21]  Y. M. Ghugal, R.P. Shimpi, A trigonometric shear deformation theory for flexure and free vibration of isotropic thick beams, Proceeding of Structural Engineering Convention (SEC 2000), IIT Bombay Mumbai India, pp 255 – 263, 2020.
[22]  Y. Ghugal, R. Shimpi, A review of refined shear deformation theories for isotropic and anisotropic laminated beams, Journal of Reinforced Plastics and Composites, Vol. 20, No 3, pp 255 – 272, 2001.
[23]  Y. Ghugal, A. Dahake,  Flexural analysis of deep beam subjected to parabolic load using refined shear deformation theory, Applied and Computational Mechanics, Vol 6, pp 163 – 172, 2012.
[24]  A. Pote Rohit, U.S. Ansari, A.S. Sayyad, Refined beam theory for flexural analysis of composite beam, International Journal of Recent Scientific Research, Vol 7, No 7, pp 12382 – 12385, 2016.
[25]  K.S. Pakhare, R.P. Shimpi, P.J. Guruprasad, Buckling analysis of thick isotropic shear deformable beams, Proceedings of ICTACEM 2017 International Conference on Theoretical, Applied, Computational and Experimental Mechanics, IIT Kharagpur, India, pp 1 – 7, 2017.
[26]  A.S. Sayyad, Y.M. Ghugal, Single variable refined beam theories for the bending, buckling and free vibration of homogeneous beams. Applied and Computational Mechanics, Vol 10, No 2, pp 123 – 138, 2016.
[27]  Y.M. Ghugal, A new refined bending theory for thick beam including transverse shear and transverse normal strain effects, Departmental Report Applied Mechanics Department, Government College of Engineering, Aurangabad India, pp 1 – 96, 2006.
[28]  Y.M. Ghugal, A single variable parabolic shear deformation theory for flexure and flexural vibration of thick isotropic beams. Proceedings of Third International Conference on Structural Engineering Mechanics and Computation (SEMC-2007), Cape Town, South Africa, pp 77 – 78. 10 – 12 September, 2007.
[29]  P.G. Darak, M.N. Bajad, Static flexural analysis of thick beam by hyperbolic shear deformation theory, International Journal of Science, Engineering and Technology Research, Vol 6, No 6, pp 977 – 983, 2017.
[30]  Y. Ghugal, R. Sharma, Hyperbolic shear deformation theory for flexure and vibration of thick isotropic beams, International Journal of Computational Methods, Vol 6, No 4 pp 585 – 604, 2009.
[31]  A. Sayyad, Y. Ghugal, Flexure of thick beams using new hyperbolic shear deformation theory, International Journal of Mechanics, Vol 5, No3, pp 113 – 122, 2011.
[32]  S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, 3rd International Edition, McGraw Hill, Singapore, 1970.
[33]  Y. Ghugal, A two-dimensional exact elasticity solution of thick beam, Departmental Report-1 Department of Applied Mechanics Government Engineering College, Aurangabad India, pp 1 – 96, 2006.
[34]  A.V. Krishna Murthy, Towards a consistent beam theory. AIAA Journal, Vol 22: 811 – 816, 1984.
[35]  Y.M. Ghugal, Flexure and vibration of thick beams using trigonometric shear deformation theory, Journal of Experimental and Applied Mechanics, Vol 1, No 1, pp 1 – 27, 2010.
[36]  J.N. Reddy, Energy and Variational Methods in Applied Mechanics, Wiley, New York, 1984.
[37]  J. N. Reddy, Simple higher order theory for laminated composite plates, ASME Journal of Applied Mechanics, Vol 51, No 4, pp 745 – 752.1984
[38]  J. N. Reddy, Canonical relationships between bending solutions of classical and shear deformation beam and plate theories, Annals of Solid and Structural Mechanics, Vol 1, No 1, pp 9 – 27, 2010.
[39]  U.P. Naik, A.S. Sayyad, P. N. Shinde, Refined beam theory for bending of thick beams subjected to various loading. Elixir Applied Mathematics,Vol  43, pp 7004 – 7015, 2012.
[40]  F.G. Canales and J.L. Mantari Boundary discontinuous Fourier analysis of thick beams with clamped and simply supported edges via CUF. Chinese Journal of Aeronautics Vol 30, No 5, pp 1708 – 1718. http://dx.doi.org/10.1016/j.cja.
[41]  A Barati, A. Hadi, M.Z. Nejad, R.Noroozi, On the vibration of bi-directional functionally graded nanobeams under magnetic field, Mechanics Based Design of Structures and Machines, Vol 50, No 2 pp 468 - 485 2022. Doi: 10.1080/15397734.2020.1719507.
[42]  M.Z. Nejad, A. Hadi, Eringen’s non-local elasticity theory for bending analysis of bi-directional functionally graded  Euler-Bernoulli nanobeams, International Journal of Engineering Science Vol 106, pp 1 – 9, 2016.
[43]  M.Z. Nejad, A. Hadi, Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli beams, International Journal of Engineering Science Vol 105, pp 1 – 11, 2016.
[44]  S.M. Ghumare, A.S. Sayyad, A new fifth order shear and normal deformation theory for static bending and elastic buckling of P-FGM beams, Latin American Journal of Solids and Structures, Vol 14, pp 1893 – 1911, 2017.
[45]  M. Zidi, M.S.A. Houari, A. Tounsi, A. Bessaim, S.R. Mahmoud, A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams, Structural Engineering and Mechanics, Vol 64, No 2, pp 145 – 153, 2017.
[46]  E.M.M. Fonseca, F.J.M.Q. de Melo, Numerical solution of curved pipes submitted to in-plane loading conditions, Thin Walled Structures Vol 48, No 2, pp 103 – 109, doi:10.1016/j.tws.2009.09.004.
[47]  E.M.M. Fonseca, F.J.M.Q. de Melo, C.A.M. Oliveira, Trigonometric function used to formulate a multi-nodal finite tubular element Journal Mechanics Research Communications Vol 34, No 1, pp 54 – 62, 2007. Doi:10.1016/j.mechrescom.2006.06.008.
[48]  A. Karamanli, Analytical solutions for buckling behaviour of two directional functionally graded beams using a third order shear deformable beam theory, Academic Platform Journal of Engineering and Science, Vol 6, No 2, pp 164 – 178, 2018.
[49]  A.S. Sayyad, Y.M. Ghugal, “On the buckling analysis of functionally graded sandwich beams using a unified beam theory,” Journal of Computational Applied Mechanics Vol 51, No 2, December 2020, pp 443 – 453. Doi: 10.22059/jcamech.2020.310180.557.
[50]  E.M.M. Fonseca, F.J. Melo, C.A.M. Oliveira, “Numerical analysis of piping elbows for in-plane bending and internal pressure” Thin-Walled Structures Vol 44, No 4, pp 393 – 398, 2006. Doi: 10.1016/j.tws.2006.04.005.
[51]  E.M.M. Fonseca, F.J. Melo, C.A.M. Oliveira, Determination of flexibility factors in curved pipes with end restraints using a semi-analytic formulation, International Journal of Pressure Vessels and Piping Vol 19, No 12, pp 829 – 840, December 2002. Doi: 10.1016/S0308-0161(02)00102-3.
[52]  A. Hadi, M.Z. Nejad, M. Hosseini Vibrations of thick-dimensionally graded nanobeams. International Journal of Engineering Science Vol 128, pp 12 – 23, 2018.
[53]  A. Hadi, M.Z. Nejad, A. Rastgoo, M. Hosseini Buckling analysis of FGM Euler – Bernoulli nano-beams with 3D – varying properties based on consistent couple – stress theory. Steel and Composite Structures, An International Journal Vol 26 No 6, pp 663 – 672, 2018.
[54]  A. Hadi, A. Rastgoo, A.R. Daneshmehr, F. Ehsani Stress and strain analysis of functionally graded rectangular plate with exponentially varying properties. Indian Journal of Material Science, Vol 2013. 206239, http://dx.doi.org/10.1155/2013/206239.
[55]  A. Hadi, A.R. Daneshmehr, S.M.N Mehrian, M. Hosseini, F. Ehsani Elastic analysis of functionally graded Timoshenko beam subjected to transverse loading. Technical Journal of Engineering and Applied Sciences Vol 3 No 13, pp 1246 - 1254, 2013.
[56]  H. Hosseini, M. Shishesaz, K.N. Tahan, A. Hadi Stress Analysis of rotating nano-disks of variable thickness made of functionally graded materials. International Journal of Engineering Materials, Vol 109, pp 29 – 53, 2016.
[57]  M. Hosseini, M. Shishesaz, A. Hadi Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness. Thin Walled Structures Vol 134, pp 508 – 523, 2019.
[58]  M.Z. Nejad, A. Hadi, A. Rastgoo Buckling analysis of arbitrary two-dimensional functionally graded Euler-Bernoulli nano-beams based on nonlocal elastic theory. International Journal of Engineering Science Vol 103, pp 1 – 10, 2016.
[59]  M.Z. Nejad, A. Rastgoo, A. Hadi Exact elasto-plastic analysis of rotating disks made of functionally graded materials. International Journal of Engineering Science, Vol 85, pp 47 – 57, 2014.
[60]  M.Z. Nejad, A. Hadi, A. Farajpour Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials. Structural Engineering and Mechanics, An International Journal Vol 63 No 2, pp 161 – 169, 2017.
[61]  M.Z. Nejad, N. Alamzadeh, A. Hadi Thermoelastic plastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic fully plastic condition. Composites Part B: Engineering Vol 154 pp 410 – 422, 2018.
[62]  M.Z. Nejad, A. Hadi, A. Omidvari, A. Rastgoo Bending analysis of bi-directional functionally graded Euler-Bernoulli nanobeams using integral form of Eringen’s non-local elasticity theory. Structural Engineering and Mechanics, An International Journal Vol 67, No 4 pp 417 – 425, 2018.
[63]  M. Shishesaz, M. Hosseini, K.N. Tahan, A. Hadi Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory. Acta Mechanica Vol 228 No 12, pp 4141 – 4168, 2017.
[64]  M. Shishesaz, M. Hosseini Mechanical behavior of functionally graded cylinders under radial pressure based on strain gradient theory. Journal of Mechanics, Vol 35, No 4, pp 441 – 454, 2019.
[65]  M. Mohammadi, M. Shishesaz, A. Hadi 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, pp 103793, 2019.
[66]  H.H. Gorgani, M.M. Adeli, M. Hosseini Pull-in behavior of functionally graded micro/nanobeams for MEMs and MEMs switches. Microsystem Technologies, Vol 25, No 8, pp 3165 – 3173, 2019.
[67]  M.M. Khoram, M. Hosseini, A. Hadi, M. Shishesaz Bending analysis of bidirectional FGM Timoshenko nanobeam subjected to mechanical and magnetic forces and resting on Winkler – Pasternak foundation. International Journal of Applied Mechanics, Vol 12 No 8 pp 2050093, 2020.
[68]  M. Mousavi Khoram, M. Hosseini, M. Shishesaz A concise review of nanoplates. Journal of Computational Applied Mechanics Vol 50 No 2, pp 420 – 429, 2019.
[69]  A. Daneshmehr, A. Rajabpoor, 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, pp 23 – 35, 2015.
[70]  Z. Mazarei, M.Z. Nejad, A. Hadi Thermo-elastic-plastic analysis of thick-walled spherical pressure vessels made of functionally graded materials. International Journal of Applied Mechanics Vol 8 No 04 pp 1650054, 2016.
[71]  M. Gharibi, M. Z. Nejad, Elastic analysis of functionally graded rotating thick cylindrical pressure vessels with exponentially-varying properties using power series method of Frobenius. Journal of Computational Applied Mechanics Vol 48, No 1, pp 89 – 98, 2017.
[72]  R. Noroozi, A. Barati, A. Kazemi, S. Norouzi, 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, pp 13 – 24, 2020.
[73]  A. Barati, M.M. Adeli, A. Hadi Static torsion of bi-directional functionally graded microtube based on the couple stress theory under magnetic field. International Journal of Applied Mechanics Vol 12 No 2, pp 2050021, 2020.
Volume 53, Issue 1
March 2022
Pages 126-141
  • Receive Date: 21 October 2021
  • Revise Date: 12 March 2022
  • Accept Date: 18 March 2022
  • First Publish Date: 18 March 2022