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$                     RIGID FORMAT No. 1, Static Analysis
$                Spherical Shell with Pressure Loading (1-2-1)
$ 
$ A. Description
$ 
$ This problem demonstrates the finite element approach to the modeling of a
$ uniform spherical shell (Problem 1-2-1). A spherical coordinate system is
$ chosen to describe the location and displacement degrees of freedom at each
$ grid point. Triangular plate elements are chosen to provide a nearly uniform
$ pattern. Two symmetric boundaries are used to analyze the structure with a
$ symmetric pressure load.
$ 
$ Two different sets of boundary conditions are used on the outside edge to
$ demonstrate the ability of NASTRAN to restart (Problem 1-2-1A) with different
$ constraint sets by simply changing the case control request. The membrane
$ support, under a uniform inward pressure load, results in uniform in-plane
$ compression in two directions. The clamped support produces bending moments in
$ addition to in-plane stresses.
$ 
$ The grid point numbering sequence used minimizes the computer time required to
$ perform the triangular decomposition of the constrained stiffness matrix. This
$ numbering sequence results in a partially banded matrix with all terms outside
$ the band located in a single column. The grid points are arranged to form five
$ rings; the center point is sequenced last.
$ 
$ Orthographlc and perspective plots of the deformed and undeformed structure
$ are requested. For the orthographlc projections the plots are fully labeled to
$ aid in checking the model. The perspective projection uses the symmetric
$ plotting capability to plot all four quadrants of the shell. A region request
$ is used to find an origin location that will allow all quadrants to be
$ plotted. The deformed plot uses plot elements to simplify the presentation.
$ Umderlays of the undeformed structure are also shown for both projections.
$ 
$ B. Input
$ 
$ 1. Parameters
$ 
$      r = 90.0 in.           (radius)
$ 
$      t = 3.0 in.            (thickness)
$                   6      2
$      E = 3.0  x 10  lb/in   (modulus of elasticity)
$ 
$      v = .1666              (Poisson's ratio)
$ 
$ 2. Constraints
$ 
$    Problem 1-2-1
$ 
$      a)  Grids at phi = theta degrees and phi = 90 degrees are constrained
$          u sub phi = theta sub r  = 0.0
$ 
$      b)  Grids at theta = 35 degrees are constrained u sub theta  = 0.0 only
$ 
$    Problem 1-2-1A
$ 
$      a)  Grids at phi = 0 degrees and phi = 90 degrees are constrained
$          u sub phi = theta sub r  = 0.0
$ 
$      b)  Grids at theta = 35 degrees are constrained u sub r = u sub phi =
$          u sub theta = theta sub r = theta sub phi = theta sub theta = 0.0
$ 
$ 3. Loads
$ 
$                                   2
$ A uniform pressure load of 1 lb/in  is applied in the -R direction (acting
$ inward).
$ 
$ C. Theory
$ 
$ Theoretical solutions for the continuum shell were obtained from Reference 4
$ using the first 20 terms of the series shown in Equation (j) of Section 94.
$ 
$ D. Results
$ 
$ The slight differences between theoretical and computed answers are due to the
$ combined effects of the finite element theory and the structural behavior in
$ the region of the clamped boundary. In the region of the clamped boundary, in-
$ plane stresses and bending moments are predicted to have large variations.
$ However, the elements used in the model assume a constant in-plane stress and
$ linearly varying bending moment and do not accurately represent the structural
$ response. In addition, the irregularities of the finite element model cause
$ extra coupling between bending and membrane action. Since the elements are
$ planar, the curvature is modeled, in effect, by the dihedral angles between
$ elements. Since the elements are different sizes and shapes, these dihedral
$ angles vary, which results in slight differences In curvature that cause small
$ errors.
$ 
$ APPLICABLE REFERENCES
$ 
$ 4. S. Timoshemko, THEORY OF PLATES AND SHELLS. McGraw Hill, 1940.
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