Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Concerning the tensor-based flexural formulation: Theory

  • Al-Rousan, Rajai Z. (Department of Civil Engineering, Jordan University of Science and Technology) ;
  • Alhassan, Mohammed A. (Department of Civil Engineering, Jordan University of Science and Technology) ;
  • Hejazi, Moheldeen A. (Department of Civil Engineering, Jordan University of Science and Technology)
  • Received : 2018.10.06
  • Accepted : 2019.03.01
  • Published : 2019.05.25
⟨ Previous Next ⟩

Abstract

Since the days of yore, plate's flexural analysis and formulation were dependent on the assumed coordinate system. In uncovering the coordinates-independent flexural interpretation, in this study, the plate bending analysis has been interpreted in terms of the tensor's components of curvatures and bending moments, in accordance with the continuum mechanics. The paper herein presents the theoretical formulations and conceptual perspectives of the Hydrostatic Method of Analysis (HM) that combines the continuum mechanics with the elasticity theory; the graphical statics and analysis; the theory of thin isotropic and orthotropic plates.

Keywords

References

  1. Belkacem, A., Daouadji Tahar, H., Abderrezak, R., Benhenni M.A., Mohamed, Z. and Boussad, A. (2018), "Mechanical buckling Analysis of hybrid laminated composite Plates under different boundary conditions", Struct. Eng. Mech., 66(6), 761-769. https://doi.org/10.12989/sem.2018.66.6.761.
  2. Backus, G. (1966), "Potentials for tangent tensor fields on spheroids", Arch. Ration. Mech. Anal., 22, 210-252. https://doi.org/10.1007/BF00266477.
  3. Rad, A.B., Farzan-Rad, M.R. and Majd, K.M. (2017). "Static Analysis of non-uniform heterogeneous circular Plate with porous material resting on a gradient hybrid foundation involving friction force", Struct. Eng. Mech., 64(5), 591-610. https://doi.org/10.12989/sem.2017.64.5.591.
  4. Bramble, J.H. and Payne, L.E. (1962), "Some uniqueness theorems in the theory of elasticity", Arch. Ration. Mech. Anal. 9, 319-328. https://doi.org/10.1007/BF00253354.
  5. Capaldi, F.M. (2012), Continuum Mechanics: Constitutive Modeling of Structural and Biological Materials, Cambridge University Press, New York, U.S.A.
  6. Cauchy, A. (1900), Oeuvres completes d'Augustin Cauchy, Serie 2, Tome 1, Gauthier-Villars fils, 381-411.
  7. Einstein, A. and Minkowski, H. (1920), "The foundation of the generalised theory of relativity", The University of Calcutta, 89-163.
  8. Euler, L. (1766), "De motu vibratorio tympanorum", Novi. Comment. Acad. Sci. Petropol., 10, 243-260.
  9. Foppl, A. (1900), "Vorlesungen uber Technische Mechanik", Teubner, Leipzig, Germany.
  10. Germain, S. (1821), "Recherches sur la theorie des surfaces elastiques", Mme. Ve. Courcier, Paris, France.
  11. Germain, S. (1826), "Remarques sur la nature, les bornes et l'etendue de la question des surfaces elastiques et equation generale de ces surfaces", Huzard-Courcier, Paris, France.
  12. Gray, M. (1978), "Sophie Germain (1776-1831)", Women of Mathmatics, Greenwood Press, Connecticut, U.S.A.
  13. Harutyunyan, D. (2017), "Gaussian curvature as an identifier of shell rigidity", Arch. Ration. Mech. Anal., 226, 743-766. https://doi.org/10.1007/s00205-017-1143-y.
  14. Kirchhoff, G. (1850), "Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe", J. fur die Reine und Angew Math., 51-88. https://doi.org/10.1515/CRELLE.2008.080
  15. Kirchhoff, G. (1897), Vorlesungen uber mechanic, 4th ed., Teubner, Leipzig, Germany.
  16. Kuhnel, W. (2015), Differential Geometry: Curves-Surfaces-Manifolds, 3rd Ed., American Mathematical Society, U.S.A.
  17. Kurrer, K.E. (2008), "The history of the theory of structures: From arch analysis to computational mechanics", J. Space Struct., 23(3), 193-197. http://doi.org/10.1002/9783433600160.
  18. Lame, G. (1866), Lecons sur la theorie mathematique de l'elasticite des corps solides, 2nd ed., Gauthier-Villars, Paris, France.
  19. Lisle, R.J. and Robinson, J.M. (1995), "The Mohr circle for curvature and its application to fold description", J. Struct. Geol., 17, 739-750. https://doi.org/10.1016/0191-8141(94)00089-I.
  20. Love, A. (1888), "The small free vibrations and deformation of a thin elastic shell", Philos. Trans. R. Soc. London, 179, 491-546. https://doi.org/10.1098/rsta.1888.0016
  21. Mohr, O. (1906), "Abhandlungen aus dem Gebiete der technischen Mechanik mit zahlreichen textabbildungen", Wilhelm Ernst & Sohn, Berlin, Germany.
  22. Murat, A. (2014), "Large deflection Analysis of point supported super-elliptical Plates", Struct. Eng. Mech., 51(2), 333-347. https://doi.org/10.12989/sem.2014.51.2.333.
  23. Navier, C.L.M.H. (1823), "Extrait des recherches sur la flexion des planes elastiques", Bull des Sci Societe Philomath Paris, 10, 92-102.
  24. Navier, C.L.M.H. (1826), "Resume des Lecons donnees a l'Ecole Royale des Ponts et Chaussees sur l'Application de la Mecanique a l'Etablissement des Constructions et des Machines", 1er partie: Lecons sur la resistance des materiaux et sur l'etablissement des constructions en terre, Firmin Didot pere et fils, Paris, France.
  25. Navier, C.L.M.H. (1828), "Remarques sur l'article de M. Poisson, insere dans le cahier d'aout", Annales de chimie et de physique, Landmarks II, Sci. J., 39(1), 145-151.
  26. Norton, J.D. (1993), "General covariance and the foundations of general relativity: Eight decades of dispute", Reports Prog. Phys., 56(7), 791. https://doi.org/10.1088/0034-4885/56/7/001
  27. Parry, R.H.G. (2005), Mohr Circles, Stress Paths and Geothechnics, 2nd ed., CRC Press, U.S.A.
  28. Poisson, S.D. (1829), "Memoire sur l'equilibre et le mouvement des corps elastique", Academie des sciences, Gauthier-Villars, Paris, France.
  29. Quinn, V. and Stubblefield, A. (2012), Continuum and Solid Mechanics: Concepts and Applications, http://doi.org/10.1017/CBO9781107415324.004.
  30. Yahia, S.A., Amar, L.H.H., Belabed, Z. and Tounsi, A. (2018), "Effect of homogenization models on stress analysis of functionally graded plates", Struct. Eng. Mech., 67(5), 527-544. https://doi.org/10.12989/sem.2018.67.5.527.
  31. Szilard, R. (2004), Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods, John Wiley and Sons, U.S.A.
  32. Timoshenko, S.P. (1953), History of Strength of Materials: With a Brief Account of the History of Theory of Elasticity and Theory of Structures, McGraw-Hill, New York, U.S.A.
  33. Tu, T.M., Quoc, T.H. and Long, N.V. (2017), "Bending analysis of functionally graded plates using new eight-unknown higher order shear deformation theory", Struct. Eng. Mech., 62(3), 311-324. https://doi.org/10.12989/sem.2017.62.3.311.
  34. Ugural, A.C. (2010), Stresses in Beams, Plates, and Shells, CRC Press, Florida, U.S.A.
  35. Ugural, A.C. and Fenster, S.K. (2008), Advanced Strength and Applied Elasticity, 4th ed., Prentice Hall, U.S.A.
  36. Ventsel, E. and Krauthammer, T. (2001), Thin Plates and Shells: Theory: Analysis, and Applications, CRC Press, Florida, U.S.A.
  37. Huang, W.H. and Lin, C.C. (1998), "Negatively curved sets on surfaces of constant mean curvature in ${\mathbb{R}}$ 3 are large", Arch. Ration. Mech. Anal., 141, 105-116. https://doi.org/10.1007/s002050050074.