Structural Engineering and Mechanics

  • 제33권6호
  • /
  • Pages.675-696
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  • 2009
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Solution method for the classical beam theory using differential quadrature

  • Rajasekaran, S. (Department of Civil Engineering, PSG College of Technology) ;
  • Gimena, L. (Department of Projects Engineering, Campus Arrosadia C.P. 31006 Public University of Navarre) ;
  • Gonzaga, P. (Department of Projects Engineering, Campus Arrosadia C.P. 31006 Public University of Navarre) ;
  • Gimena, F.N. (Department of Projects Engineering, Campus Arrosadia C.P. 31006 Public University of Navarre)
  • 투고 : 2008.02.29
  • 심사 : 2009.09.29
  • 발행 : 2009.12.20
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, a unified solution method is presented for the classical beam theory. In Strength of Materials approach, the geometry, material properties and load system are known and related with the unknowns of forces, moments, slopes and deformations by applying a classical differential analysis in addition to equilibrium, constitutive, and kinematic laws. All these relations are expressed in a unified formulation for the classical beam theory. In the special case of simple beams, a system of four linear ordinary differential equations of first order represents the general mechanical behaviour of a straight beam. These equations are solved using the numerical differential quadrature method (DQM). The application of DQM has the advantages of mathematical consistency and conceptual simplicity. The numerical procedure is simple and gives clear understanding. This systematic way of obtaining influence line, bending moment, shear force diagrams and deformed shape for the beams with geometric and load discontinuities has been discussed in this paper. Buckling loads and natural frequencies of any beam prismatic or non-prismatic with any type of support conditions can be evaluated with ease.

키워드

참고문헌

  1. Anonymous (2001), "Three dimensional static and dynamic finite element analysis and design of structures", CSI-SAP 2000NL, Berkeley, California, USA
  2. Bellman, R. and Casti, J. (1971), "Differential quadrature and long term Integration", J. Math. Anal. Appl., 34, 235-238 https://doi.org/10.1016/0022-247X(71)90110-7
  3. Bert, C.W., Wang, X. and Striz, A.G. (1993), "Differential quadrature for static and free vibration analysis of anisotropic plates", Int. J. Solids Struct., 30, 1737-1744 https://doi.org/10.1016/0020-7683(93)90230-5
  4. Bert, C.W., Wang, X. and Striz, A.G. (1994), "Static and free vibration analysis of beams and plates by differential quadrature method", Acta Mech., 102, 11-24 https://doi.org/10.1007/BF01178514
  5. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics", Appl. Mech. Rev., 49(1), 1-28 https://doi.org/10.1115/1.3101882
  6. Gimena, F.N., Gonzaga, P. and Gimena, L. (2003), "New procedure to obtain the stiffness and transfer matrices of an elastic linear element", 16th Engineering Mechanics Conference EM 2003: ASCE. University of Washington. Seattle, WA, USA
  7. Gimena, F.N., Gonzaga, P. and Gimena, L. (2008), "3D-curved beam element with varying cross-sectional area under generalized loads", Eng. Struct., 30(2), 404-411 https://doi.org/10.1016/j.engstruct.2007.04.005
  8. Gimena, L., Gimena, F.N. and Gonzaga, P. (2008), "Structural analysis of a curved beam element defined in global coordinates", Eng. Struct., 30(11), 3355-3364 https://doi.org/10.1016/j.engstruct.2008.05.011
  9. Lam, K.Y. and Li, H. (2000), "Generialized differential quadrature for frequency of rotating multi-layered conical shell", J. Eng. Mech., ASCE, 126(11), 1156-1162 https://doi.org/10.1061/(ASCE)0733-9399(2000)126:11(1156)
  10. Rajasekaran, S. and Padmanabhan, S. (1989), "Equations of curved Beams", J. Eng. Mech., 115(5), 1094-1111 https://doi.org/10.1061/(ASCE)0733-9399(1989)115:5(1094)
  11. Rajasekaran, S. (2007a), "Dynamics of non-prismatic thin-walled hybrid composite members of generic crosssection", Analysis and Design of Plated Structures, Vol. 2, Chapter 3, Dynamics, edited by N.E. Shanmugam and C.M. Wang, Woodhead publishing, U.K
  12. Rajasekaran, S. (2007b), "Symbolic computation and differential quadrature method –a boon to engineering analysis", Struct. Eng. Mech., 27(6), 713-740
  13. Shu, C. (2000), Differential Quadrature and Its Application in Engineering, Berlin, Springer
  14. Striz, A.G., Wang, X. and Bert, C.W. (1995), "Harmonic diferential quadrature method and applications to analysis of structural components", Acta Mech., 111, 85-94 https://doi.org/10.1007/BF01187729
  15. Timoshenko, S.P. (1938), "Strength of materials Part – I Elementary theory and problems", MacMillan and Co., Limited, St. Martin's street, London
  16. Ulker, M. and Civalek, O. (2004), "Application of Harmonic Differential Quadrature (HDQ) to Deflection and bending analysis of beams and plates", F.U. Fen v Muhendislik Bilimleri Dergisi, 16(2), 221-231
  17. Weekzer, J.W. (1987), "Free vibrations of thin-walled bars with open cross sections", J. Eng. Mech., ASCE, 113(10), 1441-1453 https://doi.org/10.1061/(ASCE)0733-9399(1987)113:10(1441)
  18. Weekzer, J.W. (1989), "Vibrational analysis of thin-walled bars with open cross sections", J. Eng. Mech., ASCE, 115(12), 2965-2978 https://doi.org/10.1061/(ASCE)0733-9445(1989)115:12(2965)
  19. Wilson, E.L. (2002), Three Dimensional Static and Dynamic Analysis of Structures, Computers and Structures, Inc, Berkeley, California
  20. Wu, T.Y. and Liu, G.R. (2000), "Axi-symmetric bending solution of shells of revolution by the generalized differential quadrature rule", Int. J. Pres. Ves. Pip., 77, 149-157 https://doi.org/10.1016/S0308-0161(00)00006-5

피인용 문헌

  1. Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials vol.36, pp.7, 2012, https://doi.org/10.1016/j.apm.2011.09.073
  2. Boundary equations in the finite transfer method for solving differential equation systems vol.38, pp.9-10, 2014, https://doi.org/10.1016/j.apm.2013.11.001
  3. Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order vol.49, pp.4, 2014, https://doi.org/10.1007/s11012-013-9847-z
  4. Free vibration analysis of functionally graded beams with variable cross-section by the differential quadrature method based on the nonlocal theory vol.75, pp.6, 2020, https://doi.org/10.12989/sem.2020.75.6.737
  5. Curved beam through matrices associated with support conditions vol.76, pp.3, 2009, https://doi.org/10.12989/sem.2020.76.3.395