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$                       RIGID FORMAT No. 1, Static Analysis
$            Nonsymmetric Bending of a Cylinder of Revolution (1-5-1)
$ 
$ A. Description
$ 
$ This problem illustrates the application of the conical shell element and its
$ related special data. This element uses the Fourier components of displacement
$ around an axisymmetric structure as the solution coordinates. The geometry of
$ the structure is defined by rings instead of grid points. Its constraints must
$ be defined by the particular Fourier harmonics, and the loads must be defined
$ either with special data or in a harmonic form. This element may not be used
$ in conjunction with any of the other structural elements.
$ 
$ The structure to be solved is described in Reference 6. It consists of a
$ short, wide cylinder with a moderate thickness ratio. The applied loads and
$ the output stresses are pure uncoupled harmonics. The basic purpose of this
$ problem is to check the harmonic deflections, element stresses, and forces.
$ 
$ B. Input
$ 
$ The Fourier coefficients of the applied moment per length are:
$ 
$ m   = cos(n theta)                                                         (1)
$  n
$ 
$ The applied input loads are defined as:
$ 
$ M   = integrate o to 2 pi (m  cos (n theta) R d theta)                     (2)
$  n                          n
$ 
$ The values of applied moment on the MOMAX cards are:
$ 
$ M      = piR      n > 0                                                    (3)
$  n phi
$ 
$ and
$ 
$ M      =  2piR    n = 0                                                    (4)
$  o phi
$ 
$ The bending moments in the elements are defined as:
$ 
$ M   =  Moment about u                                                      (5)
$  v                   phi
$ 
$ and
$ 
$ M   =  Moment about u                                                      (6)
$  u                   z
$ 
$ Positive bending moments indicate compression on the outer side.
$ 
$ 1. Parameters:
$ 
$  R  =  50    Radius
$ 
$  s  =  50    Height
$ 
$  t  =  1.0   Thickness
$ 
$  E  =  91.0  Modulus of Elasticity
$ 
$  v  =  0.3   Poisson's Ratio
$ 
$ 2. Loads:
$ 
$  M (100) =  157.0796    Force x Length
$   n
$ 
$  M (50)  = -157.0796    Force x Length
$   n
$ 
$ 3. Single Point Constraints:
$ 
$  Ring ID  Harmonic   Coordinates
$ 
$    50       all      u ,u   ,u    Radial, tangential and axial translations
$                       r  phi  z
$   100       all      u ,u   ,u    Radial, tangential and axial translations
$                       r  phi  z
$   all       all      theta        Rotation normal to surface
$                           r
$ 
$ The AXISYM = COSINE statement in case control defines the motions to be
$ symmetric with respect to the x-z plane.
$ 
$ C. Results
$ 
$ Notice that for higher harmonics the effect of the load is limited to the
$ edges. A smaller element size at the edges and a relatively large size in the
$ center would have given the same accuracy with fewer degrees of freedom.
$  
$ APPLICABLE REFERENCES
$ 
$ 6. B. Budiansky and P. P. Radkowski, "Numerical Analysis of Unsymmetrlc
$    Bending of Shells of Revolution", AIAA Journal, August, 1963.
$ 
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