$-------------------------------------------------------------------------------
$                     RIGID FORMAT No. 1, Static Analysis
$               Delta Wing with Biconvex Cross Section (1-1-1)
$ Delta Wing, Biconvex Cross Section Using QDMEM1 and QDMEM2 Elements (1-1-2)
$      Delta Wing, Biconvex Cross Section Using QDMEM1 Elements (1-1-3)
$      Delta Wing, Biconvex Cross Section Using QDMEM2 Elements (1-1-4)
$
$ A. Description
$
$ This series illustrates the use of various NASTRAN elements in the solution of
$ an actual structural problem. The delta wing model is composed of membrane,
$ shear panel, and rod elements. Due to the existence of symmetry or
$ antisymmetry in the structure and loading conditions, only one-quarter of the
$ wing needs to be modeled. The midplane of the wing (the plane dividing the
$ wing into upper and lower halves) is a plane of symmetry, as is the center
$ plane (the plane that divides the wing into left and right halves). The
$ loading conditions are antisymmetrical with respect to the midplane of the
$ wing and symmetric with respect to the center plane.
$
$ B. Input
$
$ The surface skin of the wing is modeled with membrane elements while the ribs
$ and spars are modeled with a combination of shear panels and rods. The shear
$ load carrying capability of ribs and span is represented by shear panels. The
$ bending stiffness of the ribs and spars is modeled with rod elements placed in
$ the plane of the skin surface.
$
$ Since a quarter model is used, the loading conditions require that an
$ antisymmetric boundary be provided on the midplane and a symmetric boundary
$ must be provided on the center plane. These boundary conditions are provided
$ by constraining all grid points on the midplane in the x and y directions and
$ all grid points on the center plane in the x direction. Supports for the
$ structure are provided by constraining grid points 13, 33, 53, 73, and 93 in
$ the z direction. Since no rotational rigidity is provided by the elements used
$ in the model, all rotational degrees of freedom have been removed by the use
$ of the GRDSET card.
$
$ The problem is first modeled (Problem 1-1-1) with a load on the trailing
$ edge and a checkpoint is requested. The modified restart (Problem 1-1-1A)
$ capability is used to perform the analysis associated with the leading edge
$ loading condition. The ability of NASTRAN to change rigid formats on a restart
$ is demonstrated by the third case (Problem 1-1-1B). The natural modes of the
$ structure are extracted using the Inverse Power method. Since the symmetric
$ boundary conditions are used, only the modes with symmetric motion about the
$ center line will be extracted. If the unsymmetric modes were required, a
$ separate run with the appropriate boundary conditions could be submitted.
$
$ A second variation (Problem 1-1-2) of the basic problem is obtained by
$ replacing the quadrilateral membrane elements (QDMEM) with the QDMEM1 and
$ QDMEM2 elements. This modification demonstrates the ability to reproduce
$ previously derived theoretical results. The SORT2 format of the printed output
$ allows the results obtained with a leading and trailing load to be compared. A
$ third case (Problem 1-1-3) is modeled with all QDMEM elements replaced by
$ QDMEM1 (Reference 26) elements. A grid point force balance is requested to
$ verify that the static equilibrium of forces at a grid point (due to the load,
$ constraints, and element forces) is zero. A fourth modeling of the wing
$ (Problem 1-1-4) uses QDMEM2 elements in place of the QDMEM elements. In this
$ case, element strain energy is requested to exhibit the energy transmitted by
$ each of the elements due to the load and resultant deflections.
$
$ 1. Parameters
$
$                6      2
$ E   = 10.4 x 10  lb/in           (modulus of elasticity)
$
$                6      2
$ G   =  4.0 x 10  lb/in           (shear modulus)
$
$                  -4       2   4
$ p   =  2.523 x 10   lb sec /in   (density)
$
$ 2. Constraints
$
$ theta sub x  = theta sub y  = theta sub z   = 0.0      All grid points
$
$ U   =  0.0               Grids 13, 33, 53, 73, and 93
$  z
$
$ U   =  0.0               Grids 11, 31, 51, 71, and 91
$  x
$
$ U   =  U   =  0.0        Grids 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 41,
$  x      y                      42, 43, 44, 45, 61, 62, 63, 64, 81, 82, and 83
$
$ 3. Loads
$
$ Problems 1-1-1, 1-1-2, 1-1-3, 1-1-4
$
$     Grid 16   F   = -500.0  (trailing edge)
$                z
$ Problem 1-1-2
$
$     Grid 36   F   = -500.0  (leading edge)
$                z
$
$ 4. Eigenvalue extraction data
$
$ Method:  Inverse Power
$
$ Region of interest:  30.0 <= f <= 160.0
$
$ Number of desired roots:  3
$
$ Number of estimated roots:  1
$
$ C. Results
$
$ No closed-form or theoretical solution exists for this problem. However, a
$ passive analog computer simulation (Reference 1) and a laboratory test
$ (Reference 2) have been performed for this structural model. The displacements
$ calculated by NASTRAN and the experimentally measured and simulated
$ displacements are shown in Tables 1 and 2. The natural frequencies and modal
$ displacements are shown in Tables 3 and 4. Table 5 presents the displacements
$ for the static loading conditions when elements 1 through 9 are CQDMEM1
$ elements and the other quadrilaterals are CQDMEM2 elements.
$
$ Table 1. NASTRAN and Experimental Deflections - Concentrated Load on Outboard
$          Trailing Edge
$            --------------------------------------------------------
$                                      Z DISPLACEMENT
$             GRID         ------------------------------------------
$            NUMBER        NASTRAN        EXPERIMENTAL         ANALOG
$            --------------------------------------------------------
$               14           -.082           -.08               -.080
$               15           -.221           -.22               -.210
$               16           -.424           -.39               -.400
$               34           -.063           -.07               -.061
$               35           -.162           -.16               -.157
$               36           -.293           -.28               -.286
$               54           -.043           -.05               -.044
$               55           -.104           -.12               -.144
$               74           -.025           -.03               -.030
$            --------------------------------------------------------
$ 
$ Table 2. NASTRAN and Experimental Deflections - Concentrated Load on Outboard
$          Leading Edge.
$            --------------------------------------------------------
$                                      Z DISPLACEMENT
$             GRID         ------------------------------------------
$            NUMBER        NASTRAN        EXPERIMENTAL         ANALOG
$            --------------------------------------------------------
$               14           -.063           -.06               -.060
$               15           -.163           -.15               -.157
$               16           -.293           -.28               -.286
$               34           -.057           -.06               -.057
$               35           -.148           -.15               -.150
$               36           -.280           -.30               -.290
$               54           -.046           -.05               -.048
$               55           -.118           -.13               -.127
$               74           -.030           -.04               -.035
$            --------------------------------------------------------
$ 
$            Table 3. NASTRAN and Analog Computer Analysis Eigenvalues
$                     -----------------------------------------
$                     Mode No.   NASTRAN (cps.)   ANALOG (cps.)
$                     -----------------------------------------
$                        1             40.9           41.3
$                        2            115.3          131.0
$                        3            156.2          167.0
$                     -----------------------------------------
$ 
$                   Table 4. Mode Displacements For First Mode
$                            -------------------------
$                                      Z DISPLACEMENT
$                             GRID    ----------------
$                            NUMBER   NASTRAN   ANALOG
$                            -------------------------
$                               14      .250     .273
$                               15      .601     .630
$                               16     1.000    1.000
$                               34      .210     .239
$                               35      .504     .558
$                               36      .854     .902
$                               54      .162     .192
$                               55      .391     .462
$                               74      .112     .148
$                            -------------------------
$ 
$                     Table 5. Comparison of Z Displacements
$          ---------------------------------------------------------------
$                     Trailing Edge Load             Leading Edge Load
$                    --------------------------   ------------------------
$                                   CQDMEM1 and                CQDMEM1 and
$          Grid      CQDMEM         CQDMEM2       CQDMEM       CQDMEM2
$          Point     Elements       Elements      Elements     Elements
$          ---------------------------------------------------------------
$           14        -.082           -.082         -.063        -.064
$           15        -.221           -.224         -.163        -.167
$           16        -.424           -.433         -.293        -.300
$           34        -.063           -.064         -.057        -.059
$           35        -.162           -.166         -.148        -.154
$           36        -.293           -.300         -.280        -.294
$           54        -.043           -.044         -.046        -.047
$           55        -.104           -.108         -.118        -.123
$           74        -.025           -.026         -.030        -.031
$          ---------------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$
$ 1. Richard H. MacNeal and Stanley U. Benscoter, "Analysis of Multicell Delta
$    Wings on Cal-Tech Analog Computer", NACA TN 3114, 1953.
$
$ 2. George W. Zender, "Comparison of Theoretical Stresses and Deflections of
$    Multicell Wings with Experimental Results Obtained from Plastic Models",
$    NACA TN 3913.
$
$ 26. Adelman, Howard E.; Walz, Joseph E.; and Rogers, James L., Jr.:  "An
$     Isoparametric Quadrilateral Membrane Element for NASTRAN", NASA TN X-2637,
$     September, 1972, pp. 315-336. 
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