$-------------------------------------------------------------------------------
$                       RIGID FORMAT No. 1, Static Analysis
$            Thermal and Applied Loads on HEXA2 Solid Elements (1-9-1)
$          Thermal and Applied Loads on TRIM6 Membrane Elements (1-9-2)
$ 
$ A. Description
$ 
$ This problem demonstrates a static analysis of a cantilevered beam under two
$ loading conditions: axial stress and thermal expansion. The analysis is
$ performed twice, once with a model consisting of HEXA2 solid hexahedron
$ elements (Problem 1-9-1) and once with a model built using the TRIM6 higher
$ order triangular membrane element (Problem 1-9-2).
$ 
$ Forty HEXA2 elements are used to model a symmetric quarter of the 4 x 4 x 20
$ beam. Symmetric boundary conditions are used on both the vertical and the
$ horizontal planes of symmetry.
$ 
$ Ten TRIM6 elements are used to model one half of the 4 x 4 x 20 beam. Symmetry
$ boundary conditions are used on the vertical plane of symmetry (see Reference
$ 31, pp. 168-172).
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    l =  20.0          (length)
$ 
$    w = 4.0            (width)
$ 
$    h = 4.0            (height)
$ 
$                6
$    E = 3.0 x 10       (modulus of elasticity)
$ 
$    v = 0.2            (Poisson's ratio)
$ 
$    alpha = .001       (thermal expansion coefficient)
$ 
$    T  = 10 deg.       (reference temperature)
$     o
$ 
$ 2. Support Boundary Constraints:
$ 
$    HEXA2 Model                  TRIM6 Model
$ 
$    u  = 0 at x = 0              u  = u  = 0 at x = 0
$     x                            x    y
$ 
$    u  = 0 at y = 0              u  = 0      at y = 0
$     y                            y
$ 
$    u  = 0 at z = 0              u  = 0      at all grid points
$     z                            z
$ 
$ 3. Loads
$ 
$    Subcase 1 (HEXA2 Model):
$ 
$    An axial force F  distributed for uniform pressure over the end of the
$                    x
$    beam where
$ 
$                    3
$        F  = 24 x 10  (total axial force)
$         x
$ 
$    Subcase 1 (TRIM6 Model):
$ 
$                    3
$        F  = 24 x 10  (total axial force)
$         x
$ 
$        Total force on symmetric part = 24/2 = 12.
$ 
$                +----->1
$                |
$                +----->4      Force divided into the ratio of
$                |                           1 x 12    2 x 12    1 x 12
$          - - - +----->1      1:4:1, i.e.,  ------- , ------- , ------
$                                              6         6         6
$ 
$    Subcase 2 (Both Models):
$ 
$        T = 60 deg. (uniform temperature field)
$ 
$        T  = 10 deg. (reference temperature)
$         o
$ 
$    Subcase 3 (TRIM6 Model Only):
$ 
$                +----->1
$                |      
$                +----->2      Force divided into the ratio of
$                |                           1 x 12    2 x 12    1 x 12
$          - - - +----->1      1:2:1, i.e.,  ------- , ------- , ------
$                                              4         4         4
$ 
$ 
$ C. Theory
$ 
$ 1. Subcase 1 and Subcase 3
$ 
$ The distributed axial load is equivalent to a stress field of:
$ 
$                  3
$   å    = 1.5 x 10                                                          (1)
$    xx
$ 
$ and
$ 
$   sigma   = sigma   = tai   = tai   = tai   = 0                            (2)
$        yy        zz      xy      xz      yz
$ 
$ The displacement field is
$ 
$        sigma
$             xx              -3
$   u  = -------  x = 0.5 x 10  x                                            (3)
$    x    E
$ 
$        -vsigma
$               xx               -3
$   u  = ---------  y = -0.1 x 10  y                                         (4)
$    y     E
$ 
$ and
$ 
$        -vsigma
$               xx               -3
$   u  = ---------  z = -0.1 x 10  z                                         (5)
$    z     E
$ 
$ 2. Subcase 2
$ 
$ The uniform expansion due to temperature will not cause any stress. The
$ strains, however, are uniform and equal. Therefore, the displacements are
$ 
$   u  =  sigma(T-T )x = .05x                                                (6)
$    x             o
$ 
$   u  =  sigma(T-T )y = .05y                                                (7)
$    y             o
$ 
$   and
$ 
$   u  =  sigma(T-T )z = .05z                                                (8)
$    z             o
$ 
$ where T is the uniform temperature and T  is the reference temperature.
$                                         o
$ 
$ D. Results
$ 
$ The results of both subcases are exact to the single precision limits of the
$ particular computer used. Table 1 presents the theoretical solutions and the
$ results of the TRIM6 finite element model analysis.
$ 
$                    Table 1. TRIM6 and Theoretical Solutions
$ --------------------------------------------------------------------
$                  Pressure Load                  Temperature Load
$      -------------------------------------    ---------------------
$                   Subcase       Subcase
$      Exact Sol.      1             3
$          -3          -3            -3                      Subcase
$   X   (10  )      (10  )        (10  )        Exact Sol.     2
$ --------------------------------------------------------------------
$   0        0           0            0              0         0.
$ 
$   2        1           0.98         0.98           0.1       0.109
$ 
$   4        2           1.98         1.98           0.2       0.2093
$ 
$   6        3           2.98         2.981          0.3       0.3093
$ 
$   8        4           3.98         3.98           0.4       0.4093
$ 
$  10        5           4.98         4.981          0.5       0.5093
$ 
$  12        6           5.98         5.981          0.6       0.6093
$ 
$  14        7           6.98         6.98           0.7       0.7093
$ 
$  16        8           7.98         7.98           0.8       0.8093
$ 
$  18        9           8.98         8.99           0.9       0.9093
$ 
$  20       10           9.98        10.026          1.0       1.00937
$ --------------------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 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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