# problem parameters for
# 12"-inch schedule 100
b = 323.8/2    # external radius [ mm ]
a = b-21.5     # internal radius [ mm ]
l = 2*(b-a)    # axial length [ mm ]
n = 4          # number of elements through thickness 

E = 200e3      # Young modulus [ MPa ]
nu = 0.3       # Poisson's ratio [ non-dimensional ]
p = 10         # internal pressure [ MPa ]


# definition of analytical solutions for comparison from
# <http://eprints.whiterose.ac.uk/110536/1/art%253A10.1007%252Fs00707-016-1762-7.pdf>
ur(x,y,z) = (p*a**2*sqrt(y**2+z**2))/(E*(b**2-a**2)) * ((1-2*nu)*(1+nu) + (1+nu)*b**2/(y**2+z**2))
sigmal(x,y,z)     = 2*nu*p*a**2/(b**2-a**2)
sigmar(x,y,z)     = p*a**2/(b**2-a**2) * (1 - b**2/(y**2+z**2))
sigmatheta(x,y,z) = p*a**2/(b**2-a**2) * (1 + b**2/(y**2+z**2))