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

Computational Structural Engineering Institute of Korea (한국전산구조공학회)
권호 표지

Derivation of Exact Dynamic Stiffness Matrix of a Beam-Column Element on Elastic Foundation

균일하게 탄성지지된 보-기둥요소의 엄밀한 동적강성행렬 유도

  • 김문영 (성균관대학교 토목환경공학과) ;
  • 윤희택 (한국철도기술연구원) ;
  • 곽태영 (성균관대학교 토목환경공학과)
  • Published : 2002.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

The governing equation and force-displacement rotations of a beam-column element on elastic foundation we derived based on variational approach of total potential energy. An exact static and dynamic 4×4 element stiffness matrix of the beam-column element is established via a generalized lineal-eigenvalue problem by introducing 4 displacement parameters and a system of linear algebraic equations with complex matrices. The structure stiffness matrix is established by the conventional direct stiffness method. In addition the F. E. procedure is presented by using Hermitian polynomials as shape function and evaluating the corresponding elastic and geometric stiffness and the mass matrix. In order to verify the efficiency and accuracy of the beam-column element using exact dynamic stiffness matrix, buckling loads and natural frequencies are calculated for the continuous beam structures and the results are compared with F E. solutions.

탄성지반 위에 놓인 보-기둥 요소의 총포텐셜 에너지로부터 변분원리를 적용하여 지배방정식과 힘-변위 관계식을 유도하였다. 4계 상미분방정식 형태의 지배방정식을 4개의 변위 파라메타를 도입하여 1계 연립미분방정식 형태의 선형 고유치 문제로 전환하고, 힘-변위 관계식을 적용하여 엄밀한 정적, 동적 요소강성행렬을 유도하였다. 직접강성법을 이용하여 구조물 강성행렬을 구하고, 2차원 보-기둥구조의 엄밀한 좌굴하중과 고유진동수를 구하고, 결과를 유한요소해와 비교함으로써 본 연구의 타당성을 검증하였다. 이러한 엄밀한 해석방법은 Hermitian 다항식을 형상함수로 도입하여 요소의 강성행렬을 산정하는 유한요소법과 비교할 때, 요소의 수를 대폭 줄일 수 있는 장점이 있다.

Keywords

References

  1. Vlasov, V. Z., Thin Walled Elastic Beams, 2nd ed., Islael Program for Scientific Transactions, Jerusalem, 1961
  2. Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, 2nd ed., McGraw-Hill, New York, 1961
  3. Barsoum, R. W. and Gallagher, R. H. 'Finite element analysis of torsional and torsional flexural stability problems', Int. J. Num. Meth. Eng. Vol. 2, 1970, pp.335-52 https://doi.org/10.1002/nme.1620020304
  4. Chen, W. F. and Atsuta, T., Theory of Beam Columns, Vol. 2, Space Behavior and Design, McGraw-Hill, New York, 1977
  5. Kim, S. B. and Kim, M. Y., 'Improved formulation for spatial stability and free vibration of thin walled tapered beams and space frames', Engng. Struct. Vol. 22, No. 5, 2000, pp.446-458 https://doi.org/10.1016/S0141-0296(98)00140-0
  6. Friberg, P. O., 'Beam element matrices derived from Vlasov's theory of open thin- walled elastic beams', Int. J. Numer. Methods Eng. Vol. 21, 1985, pp.1205-1228 https://doi.org/10.1002/nme.1620210704
  7. Banerjee, J. R. and Williams, F. W., 'Coupled bending-torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element', Int. J. Solids Structures, Vol. 31, No. 6, 1994, pp.749-762 https://doi.org/10.1016/0020-7683(94)90075-2
  8. Banerjee, J. R., Guo, S., and Howson, W. P., 'Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping', Computers and Structures, Vol. 59, No. 4, 1996, pp.613-621 https://doi.org/10.1016/0045-7949(95)00307-X
  9. IMSL. Library, Problem-solving software system for mathematical and statistical FORTRAN programming, IMSL Inc., Houston, 1984 https://doi.org/10.1016/0045-7949(95)00307-X