6. Pipe-pipe connection

In order to be able to correctly represent a pipe line where the elbows are not coplanar, it is necessary to choose a common origin of the \(\phi\). Thus, for two elbows belonging to two planes perpendicular to each other, it is necessary to be able to take into account the fact that the displacements in the plane of the first elbow are equal to the movements out of plane of the second in the connecting straight section.

Figure 6-a: Representation of two non-coplanar elbows connected by a straight pipe

In [bib12], this common origin is defined by a continuous generator line along the pipe as shown above. This generator intersects each cross section at one point. The angle between \(Z\) defined on [Figure 2.1.1-a] and the line passing through the center of the cross section and this point is equal to \(\Omega\).

6.1. Construction of a particular generator

For a cross section at the end of the pipe line, a unit origin vector \({z}_{1}\) is defined in the plane of this section. The intersection between the direction of this vector and the mean area of the elbow determines the trace of the generatrix on this section. We call \({x}_{1},{y}_{1},{z}_{1}\) the direct trihedron associated with this section where \({x}_{1}\) is the unit vector perpendicular to the cross section constructed at [Figure2.1.1-a]. For all the other transverse sections, the trihedron \({x}_{k},{y}_{k},{z}_{k}\) is obtained either by rotation of the trihedron \({x}_{k-1},{y}_{k-1},{z}_{k-1}\) in the case of the bent parts, or by translation of the trihedron \({x}_{k-1},{y}_{k-1},{z}_{k-1}\) for the straight parts of the pipe. The intersection between the cross section and the line coming from the center of this section directed by \({z}_{k}\) is the trace of a generatrix shown below.

Figure 6.1-a: Representation of the reference generator

The origin of \(\phi\) common to all elements is defined in relation to the trace of this generator on the cross section. The angle between the generatrix trace and the current position on the cross section is then called \(\psi\).

6.2. Connection from one element to another

The kinematics of [§3.1] is given in the elbow plane. This is determined by the arc generated by the elbow axis. The origin of the angles is the normal to the plane chosen as [§2.1]. Defining the origin from a generator makes it possible to eliminate the problems of continuity of movement from one element to another. In fact, if we postulate that the relative movements of the transverse sections are of the type \(\sum _{p=1}^{M}{u}_{p}^{i}\text{cos}\mathrm{p\psi }+{u}_{p}^{o}\text{sin}\mathrm{p\psi }\) where \(\psi\) is the angle with the trace of the generator on the transverse section, the continuity of the movements is automatically ensured from one element to another.

We note \(Z\) the vector perpendicular to the plane of the elbow corresponding to the origin of the angles chosen so far. Note that the vectors \(Z\) and \({z}_{k}\) are in the plane of section \(k\). \(\phi\) is the angle defined in relation to \(Z\). If we introduce \(\psi\) the angle calculated from the trace of the generatrix on the transverse section (therefore with respect to \({z}_{k}\)) we have the following relationship: \(\psi =\phi -{\Omega }_{k}\) where \({\Omega }_{k}=({Z,z}_{k})\) angle between \(Z\) and \({z}_{k}\) in the plane of the transverse section. So the trips are now of the \(\sum _{p=1}^{M}{u}_{p}^{i}\text{cos}p(\phi -{\Omega }_{k})+{u}_{p}^{o}\text{sin}p(\phi -{\Omega }_{k})\) type. It should be noted that for a given elbow, angle \({\Omega }_{k}\) is identical regardless of the cross section chosen. It is during the transition from one elbow to the other that the value of \({\Omega }_{k}\) changes.

Note:

When the pipe consists of straight collinear elements, we choose arbitrarily \(\Omega =0\) .

6.3. Digital implementation

The pipe line is meshed by straight or curved elements to be ordered. The first element indicates the beginning of the pipe line. For this element, the associated trihedron \({x}_{1},{y}_{1},{z}_{1}\) is determined. If this element is right, we choose \({\Omega }_{1}=0\), otherwise we calculate \({\Omega }_{1}\) as indicated in the previous paragraph. If the first element is straight, the trihedron associated with the first transverse section of the second element \({x}_{2},{y}_{2},{z}_{2}\) is obtained by translating \({x}_{1},{y}_{1},{z}_{1}\). If the first element is curved, the associated trihedron \({x}_{2},{y}_{2},{z}_{2}\) is obtained by rotating \({x}_{1},{y}_{1},{z}_{1}\) in the plane of the elbow. In this case \({\Omega }_{2}=0\) if the second element is straight and \({\Omega }_{2}=({z}^{2},{z}_{2})\) if the second element is curved where \({z}^{2}\) is constructed like the \(z\) of the [Figure 2.1.1-a]. The rest of the construction is easily deduced by recurrence of the previous diagram.