$-------------------------------------------------------------------------------
$                       RIGID FORMAT No. 1, Static Analysis
$        Fully Stressed Design of a Plate with a Reinforced Hole (1-16-1)
$ 
$ A. Description
$ 
$ A flat plate with a reinforced hole in the center is optimized for stresses
$ due to a uniform end load. Restrictions on the minimum thickness are
$ maintained. This problem has been investigated by G. G. Pope (Reference 21).
$ 
$ Due to symmetry, only one quadrant is modeled. Due to the membrane load all
$ rotations and normal displacements are constrained. The QDMEM and TRMEM
$ elements are used for the plate and ROD elements for the reinforcement around
$ the hole.
$ 
$ The problem demonstrates several features unique to fully stressed design
$ capability in NASTRAN. These features are:
$ 
$ 1. Elements with no limits on the range of the property change, i.e., the ROD
$    has no PLIMIT data.
$ 
$ 2. Elements with a lower limit on the property optimization card. All membrane
$    elements are required to have a resultant thickness which must not be less
$    than a minimum thickness. This minimum is determined from the thickness
$    obtained when the plate without a hole is subjected to an end load at a
$    prescribed stress limit.
$ 
$ 3. Elements whose stress is not inspected but being in an area of nearly
$    uniform stress have their properties changed due to another element's
$    stress. Element 7 has no stress request but does have the same property
$    identification number as element 17. This type of optimization can save
$    computer time at the expense of a design that may not be truly optimized.
$ 
$ 4. A property whose value depends on the maximum stress of elements. Elements
$    5 and 15 have the same property card. This option may be necessary if
$    insufficient core is allocated.
$ 
$ 5. Temperature dependent stress limits for material 3.
$ 
$ 6. Using one stress limit only. The membrane elements use the maximum
$    principle shear only. This is 1/2 the major principle stress allowed. This
$    stress limit was chosen to better model the octahedral limit in Reference
$    21. The rod elements use only the tension and compression stress
$    appropriate to the given property, namely area.
$ 
$ 7. An additional load case that was not included in the fully stressed design
$    because a stress request was not made. The second subcase may be considered
$    a displacement verification of this load case.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    l  = 30.0 in      (total length)
$    w  = 20.0 in      (total width)
$    d  = 10.0 in      (hole diameter)
$    t  =  3.348  in   (initial plate thickness)
$     o             2
$    A  =  1.674  in   (initial rod cross sectional area)
$     o         6
$    E  = 30.x10   psi (modulus of elasticity)
$    v  =  0.3         (Poisson's ratio)
$    t  = 1.0 in       (lower limit for plate thickness corresponding to a
$     e                       3
$                      25.0x10  maximum principle stress)
$ 
$ 2. Boundary conditions:
$ 
$    on y = 0 plane, u  = 0 (symmetry)
$                     y
$    on x = 0 plane, u  = 0 (symmetry)
$                     x
$    all points u  = theta  = theta  = theta  = 0 (permanent constraints)
$                z        x        y        z
$ 3. Loads:
$                                               3
$    First subcase:  uniform load, F   = 25.0x10  lb/in
$                                   10
$    Second subcase:  at grid points 69 and 79, F   = -1000.0 lb
$                                                12
$                     at grid point 78, F   = -2000.0 lb
$                                        12
$              (contact load on rim of hole - displacement check only)
$ 
$ C. Theory
$ 
$ The theoretical approach developed for the property optimization technique in
$ NASTRAN is contained in the NASTRAN Theoretical Manual, Section 4.4. This
$ technique is a fully stressed design approach. A mathematical programming
$ technique is used in reference 21, from which the example problem was taken.
$ 
$ The two techniques might be expected to give similar results when the same
$ model is used. However, reference 21 employs elements which allow varying
$ properties and stresses, while NASTRAN elements allow only constant properties
$ and constant stresses. Somewhat different geometry is used in the NASTRAN
$ model, i.e., the use of quadrilateral elements for illustration. Additional
$ features of the NASTRAN model are discussed in items 3, 4, and 5 of Part A.
$ 
$ D. Results
$ 
$ The optimization process in this problem is terminated at 5 iteratlons. The
$ initial weight to final weight ratio is 2.70 compared to Pope's results of
$ 2.63. Tables 1 and 2 show the optimized nondimensional properties of the
$ elements around the arch. Note that the results from reference 2l are averaged
$ to provide an equivalent constant property element for comparison.
$ 
$             Table 1. Optimized Nondimensional Thickness Comparisons
$               ----------------------------------------------------
$                             Original     Reference 21     NASTRAN
$                               t/t        Average t/t       t/t
$                Element           e                  e         e
$               ----------------------------------------------------
$                  37          3.348          1.24          1.00
$                  38          3.348          1.00          1.04
$                  39          3.348          1.00          1.00
$                  46          3.348          2.10          1.14
$                  47          3.348          1.34          2.00
$                  57          3.348          3.32          1.34
$                  59          3.348          3.19          4.40
$                  67          3.348          4.58          5.47
$                  68          3.348          3.26          1.00
$                  69          3.348          4.52          5.49
$               ----------------------------------------------------
$ 
$                Table 2. Optimized Nondimensional Area Comparisons
$             ---------------------------------------------------------
$                           Original     Reference 21     NASTRAN
$                             A/dt       Average A/dt      A/dt
$              Element            e                  e         e
$             ---------------------------------------------------------
$                101        .1674           .0249         .00716
$                102        .1674           .0238        0.0 effective
$                103        .1674           .0636         .05019
$                104        .1674           .1880         .1839
$                105        .1674           .3540         .3287
$             ---------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 21. Pope, G. G., "Optimum Design of Stressed Skin Structures", AIAA Journal,
$     Vol. 11, No. 11, pp 1545-1552, November 1973.
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