$-------------------------------------------------------------------------------
$                       RIGID FORMAT No. 1, Static Analysis
$         Bending of a Beam Fabricated from HEXA1 Solid Elements (1-8-1)
$ 
$ A. Description
$ 
$ The properties of solid bodies may be modeled with the NASTRAN tetrahedra,
$ wedge, or hexahedron finite elements. This problem demonstrates the analysis
$ of a solid fabricated from the six-sided HEXA1 solid elements. The problem
$ consists of a rectangular parallelopiped subdivided into forty cubic
$ subelements.
$ 
$ The loads were chosen to approximate the stress distribution due to a moment
$ on one end of a beam; the other end is constrained to resist the moment. Two
$ planes of symmetry were used to simulate an actual problem having twice the
$ width and twice the height.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    l = 20.0 (length)
$ 
$    w = 4.0 (width of full section)
$ 
$    h = 16.0 (height of full section)
$ 
$                6
$    E = 3.0 x 10  (modulus of elasticity)
$ 
$    v = 0.2 (Poisson's ratio)
$ 
$ 2. Boundary Constraints:
$ 
$    on y = theta plane, u  = u  = theta (antisymmetry)
$                         x    z
$ 
$    on z = theta plane, u  = theta (symmetry)
$                         z
$ 
$    on x = theta plane, u  = theta (symmetry)
$                         x
$ 
$ 3. Loads:
$                                  3
$    Total Moment:  M  = 2.048 x 10
$                    y
$ 
$ This moment will produce bending about the z axis. It is modeled by a set of
$ axial loads at x = l which, in turn, represent an axial stress distribution:
$ 
$ 
$    sigma   = 1.5y                                                          (1)
$         xx
$ 
$ C. Theory
$ 
$ A prismatic beam with an axial stress which varies linearly over the cross
$ section has an exact solution. The theoretical stress distribution is
$ 
$               M
$    sigma   = --- y                                                         (2)
$         xx    I
$ 
$ and
$ 
$    sigma   = sigma   = tau   = tau   = 0
$         yz        zz      xy      yz
$ 
$            1    3
$ where I = --- wh .
$           12
$ 
$ The displacements are:
$ 
$                M
$       u  = -  ---  xy                                                      (4)
$        x      EI
$ 
$              M     2     2    2
$       u  =  ---  (x  - vy  -vz )                                           (5)
$        y    2EI
$ 
$ and
$               M
$       u  = v ---  yz                                                       (6)
$        z     EI
$ 
$ D. Results
$ 
$ Tables 1 and 2 are comparisons of displacements and stresses for the
$ theoretical case and the NASTRAN model.
$ 
$                      Table 1. Comparisons of Displacement
$                     ---------------------------------------
$                                                         -4
$                                        DISPLACEMENT x 10
$                                       ---------------------
$                     POINT/DIRECTION    THEORY      NASTRAN
$                     ---------------------------------------
$                         21/y           .0400         .0417
$                         41/y           .1600         .1607
$                         61/y           .360          .366
$                         81/y           .640          .651
$                        101/y          1.000         1.016
$                        109/x          0.800         0.844
$                        110/z           .016         0.007
$                     ---------------------------------------
$ 
$                         Table 2. Comparisons of Stress
$           ------------------------------------------------------------
$                             THEORY                     NASTRAN
$           ELEMENT    -----------------------    ----------------------
$                      sigma    sigma    tau       sigma   sigma   tau
$                           xx       yy     xy          xx      yy    xy
$           ------------------------------------------------------------
$              1       -1.5       0       0       -1.56     .02   -.01
$ 
$              2       -4.5       0       0       -4.53     .036  -.05
$ 
$              3       -7.5       0       0       -7.39     .06   -.06
$ 
$              4      -10.5       0       0       -9.95    -.11    .12
$           ------------------------------------------------------------
$            NOTE:  NASTRAN stresses are average; theoretIcal stresses
$            are calculated at the center of the element.
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