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$          RIGID FORMAT No. 13 Normal Modes with Differential Stiffness
$      Normal Modes of a 100-Ce1l Beam with Differential Stiffness (13-1-1)
$ 
$ A. Description
$ 
$ This problem illustrates the effects of differential stiffness on the solution
$ for the normal modes of a beam under axial compression.
$ 
$ The natural frequencies of the beam are affected by this load as shown in
$ Reference 23. The loading specified here is one half of the Euler value for
$ compression buckling, which decreases the unloaded natural frequency, w,
$ proportional to
$ 
$      +            +
$      |   2        | 1/2
$      | pi EI      |
$      | -----  - F |
$      |   2        |
$      |  l         |
$      +            +
$ 
$ where F is the applied load.
$ 
$ The structural model is a uniform 100 cell beam hinged at both ends.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    A = 2.0         (cross sectional area)
$ 
$    I = 0.667       (bending inertia)
$ 
$                 6
$    E = 10.4 x 10   (modulus of elasticity)
$ 
$    l = 100.0       (length)
$                -4
$    p = 2.0 x 10    (mass density)
$ 
$ 2. Constraints:
$ 
$    u  = theta  = 0  = 0     (all points)
$     z        x    y
$ 
$    u  = 0                    (point 101)
$     y
$ 
$    u  = u  = 0               (point 1)
$     x    y
$ 
$ 3. Loads:
$ 
$    F      = 3423.17
$     101,x
$ 
$    B      = 1.0       (default load factor)
$ 
$ C. Theory
$ 
$ The theoretical natural frequency for the first mode is given by
$ 
$         +                      +
$         |             2        | 1/2
$         |    l      pi EI      |
$    f  = |  ------  (----- - F) |                                           (1)
$         |       2      2       |
$         |  4pA l      l        |
$         +                      +
$ 
$ For this loading of one half the Euler buckling value, the theoretical value
$ is 14.6269 Hertz for the bending mode.
$ 
$ D. Results
$ 
$ The natural frequency computed using NASTRAN is 14.62325 Hertz.
$ 
$ APPLICABLE REFERENCES
$ 
$ 23. Timoshenko, S. P., Theory of Elastic Stability, McGraw-Hill, 1961, p 159.
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