$-------------------------------------------------------------------------------
$                       RIGID FORMAT No. 1, Static Analysis
$                 Thermal and Pressure Loads on a Long Pipe Using
$                     Linear Isoparametric Elements (1-13-1)
$                 Thermal and Pressure Loads on a Long Pipe Using
$                    Quadratic Isoparametric Elements (1-13-2)
$                 Thermal and Pressure Loads on a Long Pipe Using
$                      Cubic Isoparametric Elements (1-13-3)
$ 
$ A. Description
$ 
$ These problems demonstrate the use of the linear, quadratic, and cubic
$ isoparametric solid elements, IHEX1, IHEX2, and IHEX3, respectively. A long
$ pipe, assumed to be in a state of plane strain, is subjected to an internal
$ pressure and a thermal gradient in the radial direction.
$ 
$ B. Input
$ 
$ 1. Parameters:
$ 
$    r       = a =  4 in.        (radius to the inner surface)
$     inner
$ 
$    r       = b =  5 in.        (radius to the outer surface)
$     outer
$ 
$                6
$    E = 30. x 10    psi          (Young's Modulus)
$ 
$    v = 0.3                      (Poisson's Ratio)
$ 
$                      -5
$    alpha = 1.428 x 10           (thermal expansion coefficient)
$ 
$                           2
$                  -4 lb-sec
$    p = 7.535 x 10   --------    (mass density)
$                         4
$                       in
$ 
$    p = 10 psi                   (inner surface pressure)
$ 
$    T  = 100.0 deg. F            (inner surface temperature)
$     i
$ 
$    T  = 0.0 deg. F              (outer surface temperature)
$     o
$ 
$ 2. Boundary Conditions:
$ 
$    u sub theta = 0 at all points on the right side
$ 
$    u sub theta = 0 at all points on the left side
$ 
$    u  = 0 at all points on the bottom surface
$     z
$ 
$    u  = 0 at all points on the top surface
$     z
$ 
$ 3. Loads:
$ 
$    Subcase 1,
$ 
$      p = 10 psi    (internal pressure)
$ 
$    Subcase 2,
$ 
$            (T -T
$              i  o)      (b)       100       (5)
$      T  =  --------   ln(-)  =  --------  ln(-)  , where r is any radius.
$       r        (b)      (r)     ln(1.25)    (r)
$              ln(-)
$                (a)
$ 
$ C. Theory
$ 
$ 1. Subcase 1
$ 
$ The normal stresses due to the pressure load (Reference 24) are obtained by
$ 
$                 2 2                2
$                a b      p        pa
$    sigma  =  - -------  --  +  -------                                     (1)
$         r        2  2    2       2  2
$                (b -a )  r      (b -a )
$ 
$                          2 2                2
$                         a b      p        pa
$    sigma sub theta  =   -------  --  +  -------                            (2)
$                           2  2    2       2  2
$                         (b -a )  r      (b -a )
$ 
$ and
$                      2
$                    pa
$    sigma  =  2v  -------                                                   (3)
$         z          2  2
$                  (b -a )
$ 
$ where r is the radius and all shearing stresses are zero.
$ 
$ The displacement in the radial direction is
$ 
$                             2                     2 2
$         (l-2v)(l+v)       pa        (l+v)   l   pa b
$    u  = -----------  r  -------  +  ------  -   -------                    (4)
$     r        E            2  2        E     r     2  2
$                         (b -a )                 (b -a )
$ 
$ and all other displacements are zero.
$ 
$ 2. Subcase 2
$ 
$ The stresses in the radial and tangential directions due to the thermal load
$ (Reference 24) are given by
$ 
$             alphaET      +             2           2        +
$                    i     |    (b)     a           b     (b) |
$    sigma  = -----------  |- ln(-) - -------  (l - --) ln(-) |              (5)
$         r           (b)  |    (r)     2  2         2    (a) |
$             2(l-v)ln(-)  |          (b -a )       r         |
$                     (a)  +                                  +
$ and
$ 
$                  alphaET      +               2           2        +
$                         i     |      (b)     a           b     (b) |
$    sigma       = -----------  |l - ln(-) - -------  (l + --) ln(-) |       (6)
$         theta            (b)  |      (r)     2  2         2    (a) |
$                  2(l-v)ln(-)  |            (b -a )       r         |
$                          (a)  +                                    +
$ 
$ The stress in the axial direction is obtained via the procedure contained in
$ the reference as
$ 
$                  alphaET      +        2                      +
$                         i     |      2a  v     (b)        (b) |
$    sigma       = -----------  |v -  -------  ln(-) - 2  ln(-) |            (7)
$         theta            (b)  |       2  2     (a)        (r) |
$                  2(l-v)ln(-)  |     (b -a )                   |
$                          (a)  +                               +
$ 
$ All shearing stresses are zero.
$ 
$ The displacement in the radial direction is
$ 
$                              ++      +                   +
$                          T   ||      |   2 2             |
$         (l + v)           i  ||   l  |  a b          (b) |
$    u  = -------  alpha ----- || - -  |-----------  ln(-) |
$     r   (l + v)          (b) ||   r  |    2  2       (a) |
$                        ln(-) ||      | 2(b -a )          |
$                          (a) ++      +                   +
$ 
$             +                                            +  ++
$             |                               2            |  ||
$          r  |     (b)              (      2a        (b)) |  ||
$        + -  | 2 ln(-) + l + (l-2v) ( l - -------  ln(-)) |  ||             (8)
$          4  |     (r)              (       2  2     (a)) |  ||
$             |                            (b -a )         |  ||
$             +                                            +  ++
$ 
$ D. Results
$ 
$ Note that five IHEX1 elements were used along the radial thickness, whereas
$ one element was used for each of the IHEX2 and IHEX3 cases. Two values for the
$ stress occur at the boundary of two adjacent IHEX1 elements, resulting in a
$ sawtooth pattern.
$ 
$ APPLICABLE REFERENCES
$ 
$ 24. Timoshenko, S. P. and J. N. Goodier, Theory of Elasticity, McGraw-Hill,
$     Inc., 1961.
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