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$                      RIGID FORMAT No. 5, Buckling Analysis
$                    Symmetric Buckling of a Cylinder (5-1-1)
$ 
$ A. Description
$ 
$ This problem demonstrates the use of buckling analysis to extract the critical
$ loads and the resulting displacements of a cylinder under axial loads. The
$ Buckling Analysis rigid format solves the statics problem to obtain the
$ internal loads in the elements. The internal loads define the differential
$                    d
$ stiffness matrix [K ], which is proportional to the applied load. The load
$ factors, lambda , which cause buckling are defined by the equation:
$                i
$ 
$               d
$    [lambda  [K ] + [K]] {u }  =  0                                         (1)
$           i               i
$ 
$ where [K] is the linear stiffness matrix. This equation is solved by the real
$ eigenvalue analysis methods for positive values of lambda . The vectors {u }
$                                                          i                i
$ are treated in the same manner as in real eigenvalue analysis.
$ 
$ The problem consists of a short, large radius cylinder under a purely axial
$ compression load. A section of arc of 6 degrees is used to model the
$ axisymmetric motions of the whole cylinder.
$ 
$ All three types of structure plots are requested: undeformed, static,  and
$ modal deformed. The undeformed perspective plot is fully labeled for checkout
$ of the problem. The modal orthographic plots specify a range of vectors {u }
$                                                                           i
$ which includes all roots. A longitudinal edge view of the model is also
$ plotted for easy identification of mode shapes.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    R  =  80           (Radius)
$ 
$    h  =  50           (Height)
$ 
$                  4
$    E  =  1.0 x 10     (Modulus of elasticity)
$ 
$    v  =  0.0          (Poisson's ratio)
$ 
$    t  =  2.5          (Thickness)
$ 
$    I  =  1.30208      (Bending inertia)
$     b
$ 
$ 2. Loads:
$                         3
$    p  =  1.89745 x 10 /3 deg. ARC
$ 
$ 3. Constraints:
$ 
$    a) The center point (17) is constrained in u .
$                                                z
$ 
$    c) All points are constrained in u     , theta , and theta .
$                                      theta       r           z
$ 
$    d) The top and bottom edges are constrained in u .
$                                                    r
$ 
$ 4. Eigenvalue Extraction Data:
$ 
$    a) Method: Unsymmetrical Determinant
$ 
$    b) Region of Interest:  .10 < lambda < 2.5
$ 
$    c) Number of estimated roots = 4
$ 
$    d) Number of desired roots =  4
$ 
$    e) Normalization: Maximum deflection
$ 
$ C.  Results
$ 
$ The solution to this problem is derived In Reference 9, p. 439. For
$ axisymmetric buckling, the number of half-waves which occur when the shell
$ buckles at minimum load are:
$ 
$                                   2
$       ~  h                 12 (1-v )
$    m  =  --  4th root of ( --------- )                                     (2)
$          pi                   2 2
$                              R t
$ 
$ where m is the closest integer to the right-hand values.
$ 
$ The corresponding critical stress is:
$ 
$                  2 2  2           2
$                Et m pi          Eh
$    sigma    =  ---------   +  -------                                      (3)
$         cr        2    2       2 2  2
$                12h (1-v )     R m pi
$ 
$ Using the values given, the lowest buckling mode consists of a full sine wave.
$ The NASTRAN results and the theoretical solutions for the critical load for
$ each buckling mode are listed below:
$ 
$                    ---------------------------------
$                    Number of
$                    Half Waves
$                        m       NASTRAN    ANALYTICAL
$                    ---------------------------------
$                        1       2.2889      2.2978
$                        2        .99424     1.0
$                        3       1.2744      1.26402
$                        4       2.0070      1.86420
$                    ---------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 9. S. Timoshenko, THEORY OF ELASTIC STABILITY. MGraw-Hi11, 1936.
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