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$                       RIGID FORMAT No. 1, Static Analysis
$            Static Analysis of a Beam Using General Elements (1-14-1)
$ 
$ A. Description
$ 
$ This problem demonstrates the use of general GENEL elements having various
$ types of input format in the static analysis of a cantilever beam subjected to
$ tension and bending. The beam consists of five GENEL elements and one BAR
$ element.
$ 
$ The GENEL elements are constructed as follows:
$ 
$      GENEL Element   Approach      Matrix Size  {u }   [S]
$                                                   d
$ 
$            1         Flexibility        3        No     No
$            2         Stiffness          6        No     No
$            3         Stiffness          3        Yes    Yes
$            4         Stiffness          3        Yes    No
$            5         Flexibility        3        Yes    No
$ 
$ B. Input
$ 
$ 1. Parameters
$ 
$    l = 6.0 m (length)
$ 
$               2
$    E = 6.0 N/m  (modulus of elasticity)
$ 
$    V = 0.3 (Poisson's ratio)
$ 
$             2
$    A = 1.0 m  (cross-sectional area)
$ 
$              4
$    I = .083 m  (bending moment of inertia)
$ 
$    F  = 1.0 N (axial load)
$     x
$ 
$    P  = 1.0 N (transverse load)
$     y
$ 
$ C. Theory
$ 
$ The stiffness matrix for the element in its general form is given in section 8
$ of the NASTRAN Programmer's Manual.
$ 
$ Define [Z] as the matrix of deflection influence coefficients (flexibility
$ matrix) whose terms are {u } when {u } is rigidly constrained,
$                           i         d
$ [K] as the stiffness matrix,
$ [S] as a rigid body matrix whose terms are {u } due to unit motions of {u }
$                                              i                           d
$ when all {f } = 0,
$            i
$ {f } as the vector of forces applied to the element at {u },
$   i                                                      i
$ and {f } as the vector of forces applied to the element at {u }. They are
$       d                                                      d
$ assumed to be statically related to the {f } forces, i.e., they constitute a
$                                           i
$ nonredundant set of reactions for the element. If transverse shear is neglected
$ and the beam is confined to motion in the X-Y plane, then
$ 
$      {f }  =  [K] {u }
$        i            i
$ 
$ where
$             +   +
$             | F |              +        +
$             | V |              | deltax |
$      {f } = |  2|       {u } = | deltay |
$        i    | M |         i    | deltaz |
$             |  1|              +        +
$             +   +
$            +                    +
$            | AE                 |     +               +
$            | --     0       0   |     | 6     0     0 |
$            | l                  |     |               |
$            |       12EI    6EI  |     |               |
$      [K] = | 0     ----    ---- |  =  | 0     6     3 |
$            |        3       2   |     |               |
$            |       l       l    |     |               |
$            |                    |     |               |
$            |       6EI     4EI  |     | 0     3     2 |
$            | 0     ----    ---- |     +               +
$            |        2       l   |
$            |       l            |
$            +                    +
$                   +              +
$                   | 1            |
$                   | -    0    0  |
$                   | 6            |
$               -1  |      2       |
$      [F] = [K]    | 0    -    -1 |
$                   |      3       |
$                   |              |
$                   | 0    -1   2  |
$                   +              +
$ 
$ and
$             +              +   +              +
$             |1    0    ^u  |   | 1    0    0  |
$             |            y |   |              |
$             |              |   |              |
$      [S] =  |0    1    ^u  | = | 0    1    -1 |
$             |            x |   |              |
$             |              |   |              |
$             |0    0    1   |   | 0    0    1  |
$             +              +   +              +
$ 
$ where ^u = u  - u  ,i.e., the difference between the dependent displacement
$             d    i
$ degree of freedom {u } and the independent displacement degree of freedom
$                     d
$ {u }.
$   i  
$ 
$ D. Results
$ 
$ The theoretical maximum deflections of the cantilever beam subjected to
$ tension and bending (for the input values) are
$ 
$                 F
$                  l
$      deltax  =  --  =  1.0 m (tension)
$                 AE
$ 
$ and
$ 
$                   3
$                 Pl
$      deltay  =  ---  =  144.0 m (bending)
$                 3EI
$ 
$ These results are obtained by NASTRAN.
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