Beam and Truss Element for Non-Linear Geometric Analysis

Abstract:

The present paper investigates possible improvements to the performances of beam finite element with curvature correction where large slopes as well as large displacement are involved. The displacement field of the beam element is based on simple strain functions satisfying the requirement of exact representation of curvature. The truss element is introduced with the current cross sectional and the current length instead of initial area and initial length in large displacement solution. The finite element method is used in conjunction with linearised incrementation and the Newton-Raphson iterative technique. The two basic formulations to problem involving geometric non-linear, Eulerian and Lagrangian are also discussed. The present elements offer significant advantages over existing stiffness-based elements. Consequently, fewer elements are needed to yield results of comparable accuracy. This is demonstrated with the analysis of several simple example structures by comparing the results to those of stiffness based elements and analytical solution.

• Periodical: International Journal of Engineering Research in Africa (Volume 35)
• Pages: 145-161
• DOI: https://doi.org/10.4028/www.scientific.net/JERA.35.145
• Online since: March 2018
• Authors: Messaoud Bourezane*
• Keywords: Curvature Correction, Eulerian Formulation, Geometric Non-Linear, Lagrangian Formulation, Large Displacement, Linearised Incrementation, Newton-Raphson

References

[1] J.T. Oden, Finite elements of non linear continua, McGraw-Hill New York, (1972).

[2] M.A. Crisfield, Non-Linear finite element analysis of solids and structure, Volume 1, Wiley, New York, (1991).

[3] M. Kleiber, Incremental finite element modeling in non linear solid mechanics, Ellis Horwood limited, John Willey, (1989).

[4] Z.H. Zhong, Finite element procedures for contact impact problems, Oxford University Press, New York, (1993).

[5] O.C. Zienkiewwicz and R.L. Taylor, The finite element method, McGraw-Hill New York, (1991).

[6] K.J. Bathe, Finite element procedures, Prentice Hall, Englewood Cliffs, (1996).

[7] W.F. Chen and E.M. Liu, Stability design of steel frames, CRC Press, Boca Raton, Fla, (1991).

[8] E. Spacone, et al, Mixed formulation of nonlinear beam finite element, Computer and structure, Volume 58, Issue 1, (1996) 71-83.

[9] A. Neuenhofer and F. C. Filippou, Evaluation of nonlinear frame finite element models, Journal of structural engineering, Volume 123, Issue 7, (1997) 958-966.

DOI: 10.1061/(asce)0733-9445(1997)123:7(958)

[10] P. C. Kohnke, Large deflection analysis of frame structures by fictitious forces. International journal of numerical methods in engineering, Volume 12, Issue 8, (1978) 1279-1294.

DOI: 10.1002/nme.1620120806

[11] J. Backlund, Large deflection analysis of elasto-plastic beams and frames. International journal of mechanical sciences, Volume 18, Issue 6, (1976) 269-277.

DOI: 10.1016/0020-7403(76)90028-x

[12] I. Carol, and J. Murcia, Nonlinear time dependent analysis of planar frames using an exact formulation -1: Theory, Computer and structure, Volume 33, Issue 1, (1989) 79-87.

DOI: 10.1016/0045-7949(89)90131-4

[13] S. Kaba and S.A. Mahin, Refined modelling of reinforced concrete column for seisme analysis. EERC report 84/03, Earth quake engineering research center, University of California, Berkeley (1984).

[14] C.A. Zeris and S.A. Mahin, Analysis of reinforced concrete beam-columns under uniaxial excitation. Journal of structural engineering, ASCE, Volume 114, Issue 4, (1988) 804-820.

DOI: 10.1061/(asce)0733-9445(1988)114:4(804)

[15] C.A. Zeris and S.A. Mahin, Behaviour of reinforced concrete structures subjected to biaxial excitation Journal of structural engineering, ASCE, Volume 117, Issue 9, (1991) 2657-2673.

DOI: 10.1061/(asce)0733-9445(1991)117:9(2657)

[16] D.J. Dawe, Finite deflection analysis of shallow circles by the discrete element method. International journal of numerical methods of engineering, Volume 3, Issue 4, (1971) 529-552.

DOI: 10.1002/nme.1620030408

[17] T. Matsui and O.A Matsuoka, new finite element scheme for instability analysis of thin shells. International journal of numerical methods of engineering, Volume 10, Issue 1, (1976) 145-170.

DOI: 10.1002/nme.1620100112

[18] M. Epstein and D. W. Murray, Large deformation in plane analysis of elastic beams. Computer structure, Volume 6, Issue 1, (1976) 1-9.

[19] H. R. Milner, Accurate Finite element analysis of large displacement in skeletal structures, Computer structure, Volume 14, Isues 3-4, (1981) 205-210.

DOI: 10.1016/0045-7949(81)90005-5

[20] J. R. Thomas, Hughes and Belytschko, Ted, Non linear finite element analysis, Volume 1 August (1998) 17-21 in Berlin, Germany.

[21] O.C. Zienkiewicz and G. C. Nayak, A general approach to problems of plasticity and large deformation using isoparametric element, Proc. Conf. Matrix Method Structure Mechanics, Wright -Patterson AFB, Ohio.(1971) (pub 1973).

[22] R.D. Wood, The application of finite element methods to geometrically nonlinear Analysis, P.h.d Thesis University of Wales; Swansea C/Ph/20/73, (1973).

[23] Y.V. Novozhilov, Theory of elasticity, Pergamon Press Oxford, (1961).

[24] Y.C. Fung, Foundation of solid mechanics, Prentice Hall. Englewood Cliff New Jersy, (1965).

[25] G.C. Nayak, Plasticity and large deformation problems by the finite element method, P.h.d Thesis University of Wales, Swansea C/Ph/15/71, (1971).

[26] R.D. Wood and O.C. Zienkiewicz, Geometrically nonlinear finite element analysis of beams, frames, arches and axisymmetric shells, Computer and structures, Volume 7, Issue 6, (1977) 725-735.

DOI: 10.1016/0045-7949(77)90027-x

[27] O.C. Zienkiewicz, The finite element method (3rd edn), Mc-Graw-Hill, New York, (1977).

[28] K.J. Bathe and A.P. Cimento, Some practical procedures for the solution of nonlinear finite element equations, Computer Methods. App. Mech. Engineering, Volume 22, Issue 1, (1980) 59-85.

DOI: 10.1016/0045-7825(80)90051-1

[29] G. A. Mohr and H. R. Milner, Finite element analysis of large displacements in flexural systems. Computer structure, Volume 13, Issue 14, (1981) 533-536.

DOI: 10.1016/0045-7949(81)90048-1

[30] ANSYS, Engineering Analysis Systems, Swanson Analysis Systems Inc, Box 65, Houston, PA, (1987).

[31] S. Timoshenko, Theory of elastic Stability, McGraw Hill book Co, New York, (1936).

[32] S. Timoshenko and J.N. Goudier, Theory of elasticity, McGraw Hill, NY, (1951).

[33] R. Levy and D.R. Brill, Optimisation for Exact Stability of 2-Bar Trusses, Computer utilization in Structural Engineering, (edited by James K. Nelson J.R) American Society of civil engineering New York, (1989) 520-529.

[34] M.A. Biot, Mechanics of Incremental Deformation, John Wiley and Sons, New York, (1965).