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$                      RIGID FORMAT No. 5, Buckling Analysis
$             Buckling of a Tapered Column Fixed at the Base (5-2-1)
$ 
$ A. Description
$ 
$ A buckling analysis of a tapered column fixed at the base is presented. The
$ shallow shell element TRSHL, with membrane and bending stiffness combined, is
$ utilized for modeling the column. (See Reference 31, pp. 190-194). Note that a
$ vertical plane of symmetry is utilized, allowing the model to represent only
$ half the structure.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$                 7            2
$    E =  3.0 x 10  pounds/inch           (Young's modulus)
$ 
$                 7            2
$    G =  1.5 x 10  pounds/inch           (Shear modulus)
$ 
$    L =  3.0 inches                      (Height)
$ 
$    a =  6.056 inches                    (Length)
$ 
$    The area moment of inertia at any cross section is expressed as
$ 
$                   x  4
$       I   =  I  ( - )                                                      (1)
$        x      1   a
$ 
$    I  and I  are the moments of inertia at the top (x=a) and bottom (x=0)
$     1      2
$    of the column respectively and I /I  = 0.2. For this problem 3I  = 2 and
$                                    1  2                           1
$    3I  = 10. The thickness varies linearly from the top (t = 2.0) to the
$      2
$    bottom (t = 3.0) of the column.
$ 
$ 2. Constraints:
$ 
$    theta , theta  = 0          (All grid points)
$         y       z
$ 
$    x, y, z, theta  = 0         (Grids 1, 2, and 3)
$                  x
$    x = 0                       (Grids 4, 7, 10, 13)
$ 
$ 3. Loads:
$ 
$    F  = -166.66                (Grids 13 and 15)
$     y
$ 
$    F  = -666.66                (Grid 14)
$     y
$ 
$ The theoretical solution to this problem is developed on pages 125-130 of
$ Reference 23. The reference defines the buckling factor as
$ 
$                   2
$               P  L
$                cr
$    lambda  =  ------                                                       (2)
$                EI
$                  2
$ 
$ where, for this problem, lambda = 1.505.
$ 
$ D. Results
$ 
$ NASTRAN results for this problem, as modeled with the TRSHL element, are
$ presented below.
$ 
$                        ---------------------------------
$                                                      2
$                                                  P  L
$                                                   cr
$                        Buckling Factor lambda =  ------
$                                                   EI
$                                                     2
$                        ---------------------------------
$                               TRSHL        Theory
$                        ---------------------------------
$                               1.543        1.505
$                        ---------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 23. Timoshenko, S. P., Theory of Elastic Stability, McGraw-Hill, 1961, p 159.
$ 
$ 31. Narayanaswami, R.: Addition of Higher Order Plate and Shell Elements into
$     NASTRAN Computer Program, Technical Report 76-T19, Old Dominion University
$     Research Foundation, Norfolk, Virginia, December, 1976.
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