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$               RIGID FORMAT No. 4, Differential Stiffness Analysis
$           Differential Stiffness Analysis for a Hanging Cable (4-1-1)
$ 
$ A. Description
$ 
$ NASTRAN provides an iteration procedure for nonlinear differential stiffness
$ (or geometric stiffness) solutions. As described in Section 7 of the NASTRAN
$ Theoretical Manual, the internal loads are recalculated for each iteration.
$ The changes in direction of these internal loads are used to correct the
$ previous solution. External loads retain their original orientation; however,
$ they do travel with the grid point.
$ 
$ A classical nonlinear geometric problem is that of a hanging cable which
$ assumes the shape of a catenary when a uniform gravity load is applied. The
$ model is given a circular shape initially. The resulting displacements of the
$ grid points, when added to their original locations, provide a close
$ approximation to the catenary.
$ 
$ B. Input
$ 
$ The NASTRAN model consists of nine BAR elements connected to ten GRID points
$ evenly spaced on a quarter circle. The bending stiffness of the elements is a
$ nominally small value necessary to provide a non-singular, linear solution.
$ 
$ The axial stiffness of the elements is a sufficiently large value to limit
$ extensional displacements. The basic parameters are
$ 
$    R = 10.0 ft      (initial radius)
$ 
$    w = 1.288 lb/ft  (Height per length)
$ 
$    L = 5 pi
$ 
$ C. Theory
$ 
$ The coordinate positions of the initial circular shape are defined by the
$ equations
$ 
$    x = R cos theta                                                         (1)
$ 
$    y = R sin theta                                                         (2)
$ 
$    s = R theta                                                             (3)
$ 
$ where s is the arc length and is measured in radians. Solving Equation (3) for
$ theta and substituting into Equations (1) and (2), the expressions for the
$ circular shape are
$    _
$    x = R cos (s/R)                                                         (4)
$    _
$    y = R sin (s/R)                                                         (5)
$ 
$ The differential equation for the deformed shape (see Reference 25) is
$ 
$    dy'     w           2  1/2
$    ---  =  - ( 1 + (y')  )                                                 (6)
$    dx      H
$ 
$ where
$ 
$    w is the weight per unit length
$ 
$    H is the tension at x = 0,
$ 
$    y' = dy/dx is the slope of the resulting curve
$ 
$ Dividing both sides of Equation (6) by the radical term and integrating,
$ results in the equation
$ 
$        -1        wx
$    sinh   y'  =  -- + C                                                    (7)
$                  H     1
$ 
$ Since y' = 0 at x = 0 and C  = a, then
$                            1
$ 
$                wx
$    y' = sinh ( -- )                                                        (8)
$                H
$ 
$ Integrating again and applying the known boundary condition y = 0 at x = 0,
$ the equation for the shape is
$ 
$          H         wx
$    y  =  -  [ cosh -- -1 ]                                                 (9)
$          W         H
$ 
$ Since the length of the cable is known but the horizontal force H is unknown,
$ the two may be related by integrating for the arc length L which is
$ 
$                 wx
$          H        o
$    L  =  - sinh ---                                                       (10)
$          w       H
$ 
$ 
$ where x  is one-half the distance between supports. If w, x , and L are given,
$        o                                                   o
$ Equation (10) is solved for H (for x  = 10.0, w/H = .1719266) and Equation (9)
$                                     o
$ is evaluated to obtain the actual shape. However, for a given position s along
$ the cable, the coordinates x and y would be
$ 
$          H     -1   ws
$    x  =  - sinh   ( -- )                                                  (11)
$          w          H
$ 
$          H           ws  2  1/2
$    y  =  - [ ( 1 + ( -- )  )    - 1 ]                                     (12)
$          w           H
$ 
$ The location of points on the initial circular shape are defined in the
$ coordinate system used for the deflected shape using
$           _
$    x   =  x                                                               (13)
$     o
$               _
$    y   =  R - y                                                           (14)
$     o
$ 
$ The deflections of points on the cable are computed with the equations
$ 
$    u   =  x - x                                                           (15)
$     x          o
$ 
$    u   =  y - y                                                           (16)
$     y          o
$ 
$ D. Results
$ 
$ NASTRAN and theoretical results are presented In Table 1 below. Deflections
$ are measured from the initial shape at selected locations.
$ 
$         Table 1. Comparison of NASTRAN Results to Theoretical Predictions
$      ----------------------------------------------------------------------
$                                   u   - Horizontal       u   - Vertical
$      Grid                          x                      y
$      Point       s     theta      Theory      NASTRAN    Theory     NASTRAN
$      ----------------------------------------------------------------------
$      11       13.962     10      -.4856       -.4739    -.1119      -.0408
$ 
$      13       10.472     30      -.8043       -.7666    -.2286      -.1269
$ 
$      15        6.981     50      -.5175       -.4612     .0030       .1470
$ 
$      17        3.491     70      -.1110       -.0877     .5698       .7973
$ 
$      19         .0       90       .0           .0        .9338      1.2167
$      ----------------------------------------------------------------------
$ 
$ APPLICABLE REFERENCES
$ 
$ 25. Spiegel, Murray R.:  Applied Differential Equations. Prentice-Hall, Inc.,
$     1958, pp. 105-108.
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