Connection Between Pipes and Structures
This section describes the theory and assumption behind the Structure-Pipe Connection multiphysics coupling. The coupling is an extension of the transition type couplings in Solid-Beam Connection and Shell-Beam Connection to also account for radial deformation of the pipe caused by the fluid pressure and the temperature distribution over the cross section. For a more general background to the coupling of beam type elements to solids and shells, see Connection Between Shells and Solids and Connection Between Shells and Beams in the Shell documentation.
When connecting a pipe to a solid domain it is assumed that the pipe cross section is circular, and that no warping occurs. The connection thus, on the intersecting boundaries of the solid domain, adds the following constraint equation
(11-3)
where us is the displacement of the solid, νs is Poisson’s ratio of the solid, up is the displacement of the pipe, θp is the rotation of the pipe, r the distance from the center of the pipe, and eyl and ezl are the base vectors of the local yl-axis and zl-axis. The first four terms in Equation 11-3 are identical to the transition type coupling in the Solid-Beam Connection, while the last one is added to account for the fluid pressure and the temperature difference in the pipe. The radial displacement uradial included in Equation 11-3 is computed from the radial deformation ur of the pipe as
where T is a matrix the describes the transformation from the local coordinates of the pipe to global coordinates. Above, dy and dz are the distances, in local coordinates, and R is the radial distance from the center of the pipe. The radial displacement is given by
(11-4)
where Es is Young’s modulus of the solid, αp is its coefficient of thermal expansion, Ri and Ro are its inner and outer radii, and Tref is its volume reference temperature. The stresses in the pipe due to the temperature difference between the inside and outside temperatures Tin and Tout, and the fluid pressure p on the inside surface of the pipe are given by the following analytical expressions
When the structure comes from a Shell interface, additional constraints are added for the rotational degrees of freedom as is done for the Shell-Beam connection, so that
where a is the shell normal displacement, and t1 and t1 are the shell tangents. Also, the expression for the radial deformation ur in Equation 11-4 is simplified, and it is for a shell connection given by
Moreover, the expressions for stresses in the pipe simplifies to the thin pressure vessel approximation, so that
Here sp and sT are scaling factors, which are necessary to avoid abrupt changes in uradial.