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A Four-node General Shell Element with Drilling DOFs

면내회전자유도를 갖는 4절점 곡면 쉘요소

  • 정근영 (전남대학교, 산학협력단) ;
  • 김재민 (전남대학교, 해양토목공학과) ;
  • 이은행 (전남대학교, 해양토목공학과)
  • Received : 2012.04.26
  • Accepted : 2012.07.19
  • Published : 2012.08.31

Abstract

In this study, a new 4-node general shell element with 6 DOFs per node is presented. Drilling rotational degrees of freedom are introduced by the variational principle with an independent rotation field. In formulation of the element, substitute transverse shear strain fields are used to avoid shear locking, while four nonconforming modes are applied in the in-plane displacement fields as a remedy for membrane locking. In addition, a direct modification method for nonconforming modes is employed in the numerical implementation of nonconforming modes to represent constant strain states. A 9-points integration rule is adopted for volume integration in the computation of the element stiffness matrix. With the combined use of these techniques, the developed shell element has no spurious zero energy modes, and can represent a constant strain state. Several numerical tests are carried out to evaluate the performance of the new element developed. The test results show that the behavior of the elements is satisfactory.

이 연구에서는 감절점쉘요소의 개념에 근거한 새로운 4절점 곡면 쉘요소를 제시하였다. 회전장이 독립변수로 도입된 범함수에 의하여 면내회전자유도를 도입함으로써 개발된 쉘요소에서는 절점당 6자유도를 갖도록 하였다. 아울러 쉘요소의 면내거동 개선을 위하여 4개의 비적합변위형에 의한 비적합변위를 면내방향의 변위성분에 추가하였으며, 면외거동 개선을 위하여 대체전단변형률장이 적용되었다. 이 연구에서의 비적합변위형의 수치적 구현에 있어서 일정한 변형률상태를 표현할 수 있도록 하기 위하여 비적합변위형의 직접 수정법이 적용되었다. 이렇게 정식화된 쉘요소 강성행렬의 수치적분에 있어서는 부피적분을 위하여 9점 적분법이 사용되었다. 개발된 쉘요소는 바람직하지 못한 영에너지모드를 갖지 않으며, 일정한 변형률 상태를 표현할 수 있음을 확인하였다. 개발된 4절점 곡면 쉘 요소에 대한 다양한 수치예제를 통한 검증 결과, 전반적으로 양호한 거동을 보여주고 있음을 확인하였다.

Keywords

References

  1. 최창근, 정근영, 이태열, "회전자유도를 갖는 비적합 8-절점 입체요소의 개선," 한국전산구조공학회 논문집, 제13권, 제4 호, 475-484, 2000.
  2. 백종균, 대체 변형률장에 의한 효율적인 4절점 쉘 유한요소의 개발, 박사학위논문, KAIST, 1994.
  3. 이태열, 직접 수정된 비적합 변위모드를 가진 평면쉘요소의 개발 및 쉘 구조물 해석, 박사학위논문, KAIST, 2002.
  4. Hughes, T.J.R. and Brezzi, F., "On drilling degrees of freedom," Comput. Methods Appl. Mech. Engrg., Vol. 72, 105-121, 1989. https://doi.org/10.1016/0045-7825(89)90124-2
  5. Iura, M. and Atluri, S.N., "Formulation of a membrane finite element with drilling degrees of freedom," Computational Mechanics, Vol. 9, 417-428, 1992. https://doi.org/10.1007/BF00364007
  6. Cook, R.D., Malkus, D.S., and Plesha, M.E., Concepts and applications of finite element analysis, 3rd edition, John Wiley & Sons, 1989.
  7. Choi, C.K., Chung, K.Y., and Lee, T.Y., "A direct modification method for strains due to non-conforming modes," Structural Enigineering and Mechanics, Vol. 11, No. 3, 325-340, 2001. https://doi.org/10.12989/sem.2001.11.3.325
  8. Choi, C.K., Lee, T.Y., and Chung, K.Y., "Direct modification for non-conforming elements with drilling DOF," Int. J. for Numerical Methods in Engineering, Vol. 55, 1463-1476, 2002. https://doi.org/10.1002/nme.550
  9. 최창근, 이태열, 정근영, "개선된 비적합 변위형과 대체전단 변형율장을 이용한 4절점 평판 휨 요소," 대한토목학회 논문집, Vol. 21, No. 2-A, 279-286, 2001.
  10. Bathe, K.J. and Dvorkin E.N., "A continuum mechanics based four-node shell element for general nonlinear analysis," Engineering Computation, Vol. 1, 77-88, 1984. https://doi.org/10.1108/eb023562
  11. MacNeal, R.H. and Harder, R.L., "A proposed standard set of problems to test finite element accuracy," Finite Elements in Analysis and Design, Vol. 1, 3-20, 1985. https://doi.org/10.1016/0168-874X(85)90003-4
  12. Hinton, E. and Huang, H.C., "A family of quadrilateral Mindlin plate elements with substitute shear strain fields," Computers and Structures, Vol. 23, 409-431, 1986. https://doi.org/10.1016/0045-7949(86)90232-4
  13. Groenwold, A.A. and Slader, N., "An efficient 4-node 24 DOF thick shell finite element with 5-point quadrature," Engrg. Compt. Vol. 12, 723-747, 1995. https://doi.org/10.1108/02644409510104686
  14. Taylor, R.L., "Finite elment analysis of linear shell problem," in Whiteman, J.R.(ed.), Proc. of Mathmatics of Finite Elements and Application VI, Academic Press, New York, 191-203, 1987.
  15. Simo, J.C., Fox, D.D., and Rifai, M.S., "On a stress resultant geometrically exact shell model. Part II: The linear theory; Computational aspect," Comput. Methods Appl. Mech. Engrg., Vol. 73, 53-92, 1989. https://doi.org/10.1016/0045-7825(89)90098-4
  16. Choi, C.K. and Lee, T.Y., "Efficient remedy for membrane locking of 4-node flat shell elements by non-conforming modes," Comput. Methods Appl. Mech. Engrg., Vol. 192, 1961-1971, 2003. https://doi.org/10.1016/S0045-7825(03)00203-2
  17. Belytchko, T. and Leviathan, I., "Physical stabilization of the 4-node shell element with one point quadrature," Comput. Methods Appl. Mech. Engrg., Vol. 113, 321-350, 1994. https://doi.org/10.1016/0045-7825(94)90052-3
  18. U.S. Nuclear Regulatory Commission, Standard Review Plan 3.7.2 Revision 3, NUREG-0800, 2007.
  19. Ryu, J.S., Seo, C.G., Kim, J.M. and Yun, C.B., "Seismic response analysis of soil-structure interactive system using a coupled three-dimensional FE-IE method," Nuclear Engineering and Design, Vol. 240, 1949-1966, 2010. https://doi.org/10.1016/j.nucengdes.2010.03.028
  20. Lysmer, J., Tabatabaie-Raissi, M., Tajirain, F., Vandahi, S. and Ostadan, F., "A System for Analysis of Soil-Structure Interaction User's Manual," University of California, Berkeley, 1988.