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$                       RIGID FORMAT No. 1, Static Analysis
$  Axisymmetric Cylindrical Thick Shell Subjected to Asymmetric Pressure Loading
$                                    (1-15-1)
$ 
$ A. Description
$ 
$ This problem demonstrates the use of elements TRAPAX and TRIAAX in the
$ analysis of asymmetrically loaded solids of revolution. The structure consists
$ of a circular cylindrical shell loaded with a uniform external pressure over a
$ small square area.
$ 
$ The cylindrical shell wall is assumed to be simply supported, i.e., the radial
$ and circumferential deflections and the bending moments are zero at the ends.
$ 
$ Trapezoidal elements having small and large dimensions are used in the
$ vicinity of the load and away from the load, respectively. A transition area,
$ between the two trapezoidal configurations, is modeled with triangular
$ elements to illustrate their use.
$ 
$ The loads and deflections, not required to be axisymmetric, are expanded in
$ Fourier series with respect to the azimuthal coordinate. Due to the one plane
$ of symmetry of this problem with respect to the phi = 0 plane, the deflections
$ are represented by a cosine series selected by the AXISYM Case Control card.
$ The highest harmonic used, 10, is defined on the AXIC bulk data card. The
$ pressure load is defined using PRESAX bulk data cards.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    r   =  15 in.         (Average radius)
$     a
$ 
$    t      1 in.          (Thickness)
$ 
$    l   =  45 in.         (Length)
$ 
$    2c  =  3.75 in.       (Load Length)
$ 
$    beta =  0.125 radians (Load Arc (beta = c/r ))
$                                               a
$ 
$    E   =  66666.7 psi    (Modulus of Elasticity)
$ 
$    v   =  0.3            (Poisson's ratio)
$ 
$    n   =  10             (Harmonics)
$ 
$ 2. Loads:
$ 
$    p = 7.11111 psi       (Pressure)
$ 
$                 2                             2
$    A = 14.063 in         (Area of Load (A = 4c ))
$ 
$ 3. Supports
$ 
$    Simply supported at the ends:   u   =  0, u     =  0
$                                     r         phi
$ 
$    Symmetry at the midplane:       u   =  0
$                                        z
$ 
$ C. Theory
$ 
$ Theoretical results for this problem are taken from Reference 20, p. 568.
$ 
$ The following theoretical values occur at the center of the load
$      l
$ (z = -, phi = 0):
$      2
$ 
$              pA
$   u   =  272 ---  =  0.0272 in.       (Radial Deflection (inward))
$    r         Er
$                a
$ 
$   M     = 0.1324 pA = 13.24 in-lb/in  (Circumferential Bending Moment)
$    phi
$ 
$   M   = 0.1057 pA = 10.57 in-lb/in    (Longitudinal Bending Moment)
$    z
$ 
$                   pA
$   F     = -2.6125 ---  = -17.42 lb/in (Circumferential Membrane Force)
$    phi            r
$                    a
$ 
$                pA
$   F   = -2.320 ---  =  -15.47 lb/in   (Longitudinal Membrane Force)
$    z           r
$                 a
$ 
$ Theoretical stresses on the inside and outside walls at this point
$      l
$ (z = -, phi = 0) are computed as follows:
$      2
$ 
$            F        6M
$             z         z     47.95 psi (Inside Wall Longitudinal Stress)
$   sigma  = --  +/-  ---  =
$        z   t         2     -78.89 psi (Outside Wall Longitudinal Stress)
$                     t
$ 
$              F          6M
$               phi         phi   62.02 psi (Inside Wall Circumferential Stress)
$   sigma  = --    +/-  ---    =
$        phi   t           2    -96.86 psi (Outside Wall Circumferential Stress)
$                         t
$ 
$ D. Results
$ 
$ The solution is near convergence with ten harmonics.
$ 
$ Ten harmonics shows very good convergence to nearly the theoretical values
$ computed above. However, seven harmonics would result in relatively poor
$ convergence. Thus, displacement convergence alone may be an invalid indicator
$ of an adequate solution.
$ 
$ APPLICABLE REFERENCES
$ 
$ 20. Biljaard, P. P., ASME "Pressure Vessel and Piping Design", Welding Journal
$     Research Supplement. 1954, pp 567-575.
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