Journal of the Computational Structural Engineering Institute of Korea (한국전산구조공학회논문집)

Computational Structural Engineering Institute of Korea (한국전산구조공학회)
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Improved Method Evaluating the Stiffness Matrices of Thin-walled Beam on Elastic Foundations

탄성지반위에 놓인 박벽보의 강성행렬산정을 위한 개선된 해석기법

  • 김남일 (명지대학교 토목환경공학과 BK21) ;
  • 정성엽 (평화엔지니어링 구조부) ;
  • 이준석 (성균관대학교 토목환경공학과) ;
  • 김문영 (성균관대학교 건설환경시스템공학과)
  • Published : 2007.04.30
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Abstract

Improved numerical method to obtain the exact stiffness matrices is newly proposed to perform the spatially coupled elastic and stability analyses of non-symmetric and open/closed thin-walled beam on elastic foundation. This method overcomes drawbacks of the previous method to evaluate the exact stiffness matrix for the spatially coupled stability analysis of thin-walled beam-column This numerical technique is accomplished via a generalized eigenproblem associated with 14 displacement parameters by transforming equilibrium equations to a set of first order simultaneous ordinary differential equations. Next polynomial expressions as trial solutions are assumed for displacement parameters corresponding to zero eigenvalues and the eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition. And then the exact displacement functions are constructed by combining eigensolutions and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently an exact stiffness matrix is evaluated by applying the member force-deformation relationships to these displacement functions. In order to illustrate the accuracy and the practical usefulness of this study, the numerical solutions are compared with results obtained from the thin-walled beam and shell elements.

탄성지반 위의 비대칭 개/폐단면의 박벽보에 대한 탄성해석 및 안정성해석을 수행하기 위해 엄밀한 강성행렬을 계산하기 위한 개선된 수치해석 기법을 새롭게 제시한다. 본 연구에서 제시한 수치해석기법은 박벽보의 안정성 해석을 위한 엄밀한 강성행렬을 산정하는 선행된 수치해석기법의 결점을 보완하고 있다. 본 연구에서 제시한 기법은 일반화된 고유치 문제에 관한 해를 얻는 것으로서 일반화된 14개의 변위에 대한 고유치 문제를 평형방정식에 관한 1차의 연립상미분 방정식으로 변환함으로써 얻어진다. '0'의 고유치에 대응되는 변위파라미터에 대해 다항식이 가정되며 항등조건으로부터 '0'의 고유치의 수와 동일한 미결정된 파라미터를 포함하는 고유 모우드가 결정되고 이로부터 'non-zero'의 고유치와 다항식의 해를 조합함으로써 엄밀한 변위함수가 결정된다. 이후 부재력-변위의 관계를 이용하여 엄밀한 강성행렬을 산정하게 된다. 본 연구에서 개발한 수치해석 기법의 타당성을 검증하기 위해서 본 연구에서 제시한 이론에 의한 해를 제시하고 보요소 및 쉘요소을 사용한 유한요소해와 비교 검토한다.

Keywords

References

  1. 김문영, 윤희택, 곽태영 (2002) 균일하게 탄성지지된 보-기둥 요소의 엄밀한 동적강성행렬 유도, 한국전산구조공학회 논문집, 15(3), pp.463-469
  2. ABAQUS (2003). Standard user's manual, Ver. 6.1, Hibbit, Kalsson & Sorensen Inc
  3. Aydogan, M. (1995) Stiffness-matrix formulation of beams with shear effect on elastic foundation. Journal of Structural Engineering, 121, pp.1265-1270 https://doi.org/10.1061/(ASCE)0733-9445(1995)121:9(1265)
  4. Banerjee, J.R., Williams, F.W. (1996) Exact dynamic stiffness matrix for composite Timoshenko beams with applications. Journal of Sound and Vibration, 194. pp.573-585 https://doi.org/10.1006/jsvi.1996.0378
  5. Banerjee, J.R. (1998) Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method. Computers & Structures, 69. pp.197-208 https://doi.org/10.1016/S0045-7949(98)00114-X
  6. Chen, C.N. (1998) Solution of beam on elastic foundation by DQEM. Journal of Engineering Mechanics, 124. pp.1381-1384 https://doi.org/10.1061/(ASCE)0733-9399(1998)124:12(1381)
  7. Dube, G.P., Dumir, P.C. (1996) Tapered thin open section beams on elastic foundation I. buckling analysis. Computers & Structures, 61, pp.845-857 https://doi.org/10.1016/0045-7949(96)00112-5
  8. Gendy, A.S., Saleeb, A.F. (1999) Effective modeling of beams with shear deformations on elastic foundation, Structural Engineering and Mechanics, 8. pp.607-622 https://doi.org/10.12989/sem.1999.8.6.607
  9. Guo, Y.J., Weitsman, Y.J. (2002) Solution method for beams on nonuniform elastic foundations. Journal of Engineering Mechanics, 128. pp.592-594 https://doi.org/10.1061/(ASCE)0733-9399(2002)128:5(592)
  10. Hetenyi, M. (1946) Beams on elastic foundations. Scientific Series, vol. XVI. Ann Arbor: The University of Michigan Press. University of Michigan Studies
  11. Hosur, V, Bhavikatti, S.S. (1996) Influence lines for bending moments in beams on elastic foundations. Computers & Structures, 58. pp.1225-1231 https://doi.org/10.1016/0045-7949(95)00219-7
  12. Kim, M.Y., Chang, S.P., Kim, S.B. (1996) Spatial stability analysis of thin-walled space frames. Int. J. Num. Meth. Engng., 39. pp.499-525 https://doi.org/10.1002/(SICI)1097-0207(19960215)39:3<499::AID-NME867>3.0.CO;2-Z
  13. Kim, M.Y., Yun, H.T., Kim, N.I. (2003) Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns. Computers & Structures, 81, pp.1425-1448 https://doi.org/10.1016/S0045-7949(03)00082-8
  14. Kim, S.B., Kim, M.Y. (2000) Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frames. Engineering Structures, 22, pp.446-458 https://doi.org/10.1016/S0141-0296(98)00140-0
  15. Kuo, Y.H., Lee, S.Y. (1994) Deflection of nonuniform beams resting on a nonlinear elastic foundation. Computers & Structures, 51, pp.513-519 https://doi.org/10.1016/0045-7949(94)90058-2
  16. Lee, S.Y., Ke, H.Y., Kuo, Y.H. (1990) Exact static deflection of a non-uniform Bernoulli-Euler beam with general elastic end restraints, Computers & Structures, 36, pp.91-97 https://doi.org/10.1016/0045-7949(90)90178-5
  17. Leung, A.Y.T. (1992) Dynamic stiffness for lateral buckling, Computers & Structures, 42, pp.321-325 https://doi.org/10.1016/0045-7949(92)90028-X
  18. Leung, A.Y.T., Zeng, S.P. (1994) Analytical formulation of dynamic stiffness, Journal of Sound and Vibration, 177(4), pp.555-564 https://doi.org/10.1006/jsvi.1994.1451
  19. Microsoft IMSL Library (1995) Microsoft Corporation
  20. Morfidis, K., Avramidis, I.E. (2002) Formulation of a generalized beam element on a two-parameter elastic foundation with semi-rigid connections and rigid offsets, Computers & Structures, 80, pp.1919-1934 https://doi.org/10.1016/S0045-7949(02)00226-2
  21. Onu, G. (2000) Shear effect in beam finite element on two-parameter elastic foundation, Journal of Structural Engineering, 126, pp.1104-1107 https://doi.org/10.1061/(ASCE)0733-9445(2000)126:9(1104)
  22. Pazaqpur, G.A., Shah, K.R. (1991) Exact analysis of beams on two-parameter elastic foundation, International Journal of Solids and Structures, 27, pp.435-454 https://doi.org/10.1016/0020-7683(91)90133-Z
  23. SAP2000 NonLinear Version 6.11 (1995), Integrated finite element analysis and design of structures, Berkeley, California, USA: Computers and Structures, Inc
  24. Shirima, L.M., Giger, M.W. (1992) Timoshenko beam element resting on two-parameter elastic foundation, Journal of Engineering Mechanics, 118, pp.280-295 https://doi.org/10.1061/(ASCE)0733-9399(1992)118:2(280)
  25. Tong, P., Rossettos, J.N. (1977) Finite element method: Basic techniques and implementation. Cambrdge, Mass: Massachusetts Institute of Technology Press
  26. Vallabhan, C.V.G., Das, Y.C. (1991) Modified Vlasov model for beams on elastic foundations, Journal of Geotechnical Engineering, 117, pp.956-966 https://doi.org/10.1061/(ASCE)0733-9410(1991)117:6(956)