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Axi s y m m et r i c s h e I I s

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    Axi s y m m et r i c s h e I I s

    The problem of axisymmetric shells is of sufficient practical importance to include
    in this chapter special methods dealing with their solution. While the general
    method described in the previous chapter is obviously applicable here, it will be
    found that considerable simplification can be achieved if account is taken of
    axial symmetry of the structure. In particular, if both the shell and the loading
    are axisymmetric it will be found that the elements become ‘one-dimensional’.
    This is the simplest type of element, to which little attention was given in earlier
    chapters.
    The first approach to the finite element solution of axisymmetric shells was
    presented by Grafton and Strome.’ In this, the elements are simple conical frustra
    and a direct approach via displacement functions is used. Refinements in the
    derivation of the element stiffness are presented in Popov et a1.* and in Jones
    and S t r ~ m e . ~ An extension to the case of unsymmetrical loads, which was
    suggested in Grafton and Strome, is elaborated in Percy et d4 and
    Later, much work was accomplished to extend the process to curved elements and
    indeed to refine the approximations involved. The literature on the subject is
    considerable, no doubt promoted by the interest in aerospace structures, and a
    complete bibliography is here impractical. References 7- 15 show how curvilinear
    coordinates of various kinds can be introduced to the analysis, and references 9
    and 14 discuss the use of additional nodeless degrees of freedom in improving
    accuracy. ‘Mixed’ formulations (Chapter 11 of Volume 1) have found here some
    use.I6 Early work on the subject is reviewed comprehensively by Gallagher’7.18 and
    Stricklin.



    References
    1. P.E. Grafton and D.R. Strome. Analysis of axi-symmetric shells by the direct stiffness
    method. Journal of AZAA, 1, 2342-7, 1963.
    2. E.P. Popov, J. Penzien and Z.A. Liu. Finite element solution for axisymmetric shells. Proc.
    Am. SOC. Civ. Eng., EM5, 119-45, 1964.
    3. R.E. Jones and D.R. Strome. Direct stiffness method of analysis of shells of revolution
    utilizing curved elements. Journal of AIAA, 4, 1519-25, 1966.
    4. J.H. Percy, T.H.H. Pian, S. Klein and D.R. Navaratna. Application of matrix displacement
    method to linear elastic analysis of shells of revolution. Journal of AZAA, 3,
    5. S. Klein. A study of the matrix displacement method as applied to shells of revolution.
    In J.S. Przemienicki, R.M. Bader, W.F. Bozich, J.R. Johnson and W.J. Mykytow (eds),
    Proc. 1st Con$ Matrix Methods in Structural Mechanics, Volume AFFDL-TR-66-80,
    pp. 275-98, Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base,
    OH, October 1966.
    6. R.E. Jones and D.R. Strome. A survey of analysis of shells by the displacement method. In
    J.S. Przemienicki, R.M. Bader, W.F. Bozich, J.R. Johnson and W.J. Mykytow (eds), Proc.
    1st Con$ Matrix Methods in Structural Mechanics, Volume AFFDL-TR-66-80, pp. 205-29,
    Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, OH, October 1966.
    7. J.A. Stricklin, D.R. Navaratna and T.H.H. Pian. Improvements in the analysis of shells of
    revolution by matrix displacement method (curved elements). Journal ofAZAA, 4,2069-72,
    1966.
    8. M. Khojasteh-Bakht. Analysis of elastic-plastic shells of revolution under axi-symmetric
    loading by the finite element method. Technical Report SESM 67-8, University of
    California, Berkeley, CA, 1967.
    9. J.J. Webster. Free vibration of shells of revolu

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