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$                  RIGID FORMAT No. 6, Piecewise Linear Analysis
$              Piecewise Linear Analysis of a Cracked Plate (6-1-1)
$ 
$ A. Description
$ 
$ This problem illustrates elastic-plastic deformation of a thin plate
$ uniaxially loaded across a crack at the center of the plate. The same problem
$ was solved by J. L. Swedlow (Reference 10).
$ 
$ Piecewise Linear Analysis involves loading the plate in increments and
$ recalculating the material properties for each element as a function of the
$ element stresses for the last load increment.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    L  =   9.0 inch (Total length of plate)
$ 
$    W  =   6.0 inch (Total width of plate)
$ 
$    w  =   2.0 inch (Total width of crack)
$ 
$    t    = 1.0 inch (thickness)
$ 
$                    6      2
$    E    = 10.8 x 10  lb/in  (Modulus of elasticity at zero strain)
$     o
$ 
$    v    = .3333 (Poisson's Ratio at zero strain)
$     o
$            _____
$ 2. Loads:  sigma is the applied load.
$ 
$                  _____      2                      _____      2
$    Load Factor   sigma lb/in       Load Factor     sigma lb/in
$ 
$         1          2,300                14           7,000
$         2          2,500                15           7,500
$         3          2,800                I6           8,000
$         4          3,100                17           8,500
$         5          3,400                18           9,000
$         6          3,700                19           9,500
$         7          4,000                20          10,000
$         8          4,400                21          10,500
$         9          4,800                22          11,000
$        10          5,200                23          11,500
$        11          5,600                24          12,000
$        12          6,000                25          12,500
$        13          6,500                26          13,000
$ 
$ 3. Constraints:
$ 
$    a) All grid points are constrained in u , theta , theta , and theta .
$                                           z       x       y           z
$ 
$    b) Grid points along the Y-axis are constrained in the u  direction.
$                                                            x
$ 
$    c) Grid points along the X-axis from the crack tip (x = 1.0) to the edge
$       (x = 3.0) are constrained in the u  direction.
$                                         y
$ 
$ C. Theory
$ 
$ The finite element model utilizes two planes of symmetry, so only one quarter
$ of the structure (the first quadrant) is modeled. All membrane elements use
$ stress-dependent materials, duplicating the model in Reference 10.
$ 
$ The octahedral stress used in the determination of the material properties was
$ defined in Reference 10 as:
$ 
$           sqrt(2)            2                        2          2
$    tau  = ------- sqrt( sigma  - sigma  sigma  + sigma  + 3 sigma   )      (1)
$       o      3               x        x      y        y          xy
$ 
$ The octahedral strain was defined by:
$ 
$                +
$                | tau (1+v )/E                 tau <= tau
$                |    o    o   o                   o      limit
$    epsilon   = |                                                           (2)
$           o    | tau (1+v )/E + epsilon       tau > tau
$                |    o    o   o         p         o     limit
$                +
$ 
$ where
$ 
$                          -3               -1 1/0.3964
$    epsilon   = 9.716 x 10   (tau /tau       )
$           p                     o    limit
$ 
$    tau       = (sqrt(2/3)) sigma
$       limit                     limit
$ 
$                            2
$    sigma     = 11,500 lb/in
$         limit
$ 
$ NASTRAN uses an equivalent uniaxia1 stress-strain curve defined by
$ 
$    sigma  =  3/sqrt(2) tau                                                 (3)
$                           o
$ 
$ and
$ 
$    epsilon   =  sigma/E + sqrt(2) epsilon p                                (4)
$ 
$ A complete discussion of the equations may be found in Reference 10.
$ 
$ D. Results
$ 
$ In the NASTRAN analysis, the octahedral stress is calculated for each load
$ factor as a function of the current values of the stresses. In Reference 10
$ the current value of the octahedral stress is obtained by accumulating
$ incremental values of the octahedral stress. This procedure results in a
$ generally more flexible model. The resulting differences in calculated
$ stresses are particularly noticeable at the higher load levels.
$ 
$ APPLICABLE REFERENCES
$ 
$ 10. J. L. Swedlow, "The Thickness Effect and Plastic Flow in Cracked Plates",
$     Office of Aerospace Research, USAF, APL 65-216.
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