Journal of the Korean Geotechnical Society (한국지반공학회논문집)

Korean Geotechnical Society (한국지반공학회)
권호 표지
DOI QR코드

DOI QR Code

Numerical Formulation of Consolidation Based on Finite Strain Analysis

대변형 압밀방정식의 수식화

  • 신호성 (울산대학교 건설환경공학부) ;
  • 이승래 (한국과학기술원 건설및환경공학과)
  • Received : 2013.06.11
  • Accepted : 2013.06.24
  • Published : 2013.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

Embankments on soft ground experience significant deformation during time-dependent consolidation settlement, as well as an initial undrained settlement. Since infinitesimal strain theory assumes no configuration change and minute strain during deformation, finite strain analysis is required for better prediction of geotechnical problems involving large strain and geometric change induced by imposed loadings. Updated Lagrangian formulation is developed for time-dependent consolidation combining both force equilibrium and mass conservation of fluid, and mechanical constitutive equation is written in Janumann stress rate. Numerical convergence during Newton's iteration in large deformation analysis is improved by Nagtegaal's approach of considering the effect of rotation in mechanical constitutive relationship. Numerical simulations are conducted to discuss numerical reliability and applicability of developed numerical code: deformation of cantilever beam, two-dimensional consolidation. The numerical results show that developed formulation can efficiently describe large deformation problems. Proposed formulation is expected to facilitate the upgrading of a numerical code based on infinitesimal strain theory to that based on finite strain analysis.

연약지반위에 성토를 할 경우 초기 비배수 상태 뿐만 아니라 압밀 과정 중에도 큰 변형이 발생한다. 기존의 미소변형률 이론은 변형률이 작고 초기의 기하학적인 형상이 변형과정 동안 변하지 않는다고 가정하므로 큰 변형이 유발되는 지반공학 문제들을 해석하기 위해서는 대변형 해석을 수행하여야 한다. 힘평형 방정식과 유체 연속방정식이 결합된 압밀지배 방정식을 Updated-Lagrangian 형태로 수식화하고, Jaumann stress rate을 이용하여 역학적 구성관계를 표현하였다. 그리고 Nagtegaal이 제안한 회전을 고려한 구성관계를 적용하여 Newton의 반복과정을 통한 해의 수렴성과 정확도를 향상시켰다. 개발된 대변형 압밀해석 프로그램을 검증하기 위하여 켄틸레버보와 이차원 압밀문제를 해석하였다. 수치해석 결과는 큰 변형률과 기하학적 회전을 포함하는 대변형 문제를 효과적으로 묘사할 수 있음을 보여주었다. 기존의 미소변형이론에 근거한 유한요소 프로그램은 제안한 방법을 통하여 대변형 해석 프로그램으로 용이하게 전환될 수 있다.

Keywords

References

  1. Biot, M. A. (1941), "General theory of three dimensional consolidation", J. Appl. Phys., 12, 155-169. https://doi.org/10.1063/1.1712886
  2. Carter, J. P. (1977), "A theory of finite elastic consolidation", Int. J.Solids Structure, 13, 467-478. https://doi.org/10.1016/0020-7683(77)90041-5
  3. Carter, J. P. (1979), "The analysis of finite elasto-plastic consolidation", Int. J. Num. Anal. Meth. Geom., 3(2), 107-129. https://doi.org/10.1002/nag.1610030202
  4. Gere, J. M. and Timoshenko, S. P. (1997), "Mechanics of materials", Pws-kent publishing company.
  5. Chakrabarty, J. (2010), "Applied Plasticity", Springer, p.758.
  6. Gibson, R. E., England, G. L., and Hussey, M. J. L. (1967), "The theory of one dimensional consolidation of clays", Geotechnique, 17, 261-273. https://doi.org/10.1680/geot.1967.17.3.261
  7. Cryer, C. W. (1963). "A comparison of three dimensional theories of Biot and Terzeghi", Quarterly Journal of Mechanics and Applied Mathematics, 16, 72-81.
  8. Hibbit, H. D., Marcal, P. V., and Rice, J. R. (1970), "A finite element formulation for problems of large strain and large displacement", Int. J. Solids Structure, 6, 1069-1086. https://doi.org/10.1016/0020-7683(70)90048-X
  9. Hughes, Thomas. J. R. (1984), "Numerical implementation of constitutive models: rate- independent deviatoric plasticity", Martinus Nijhoff Publishers.
  10. Irgens, F. (2008), "Continuum mechanics", Springer, p.679.
  11. Mandel, J. (1953), "Consolidation des sols (etude mathematique)", Geotechnique, 3, 287-299. https://doi.org/10.1680/geot.1953.3.7.287
  12. Mcmeeking, R. M. and Rice, J. R. (1975), "Finite-element formulation for problems of large elastic-plastic deformation", Int. J. Solids Structure, 11, 601-616. https://doi.org/10.1016/0020-7683(75)90033-5
  13. Nagtegaal, J.C. (1988), "On the development of a general purpose finite element program for analysis of forming process", Int. J. Numer. Meth. Eng., 25, 113-131. https://doi.org/10.1002/nme.1620250111
  14. Nazem, M., Sheng, D.C., and Carter, J.Pl. (2006), "Stress integration and mesh refinement for large deformation in geomechanics", Int. J. Num. Anal. Meth. Geom., 65(7), 1002-1027. https://doi.org/10.1002/nme.1470
  15. Prevost, J. H. (1981), "Consolidation of an elastic porous media", Proc. ASCE EMI, 169-186.
  16. Terzaghi, K. (1925), "Erdbaumechanik auf bodenphysikalischer Grundlage", F. Devtike, Viena.
  17. Zienkiewica, O. C. and Shiomi, T. (1984), "Dynamic behavior of saturated porous media: the generalized Biot formulation and its numerical solution", Int. J. Numer. Meth. Eng., 8, 71-96. https://doi.org/10.1002/nag.1610080106