A mathematical trick to identify the new spoolpiece lifting points!

In the offshore construction industry, the connection between the newly installed pipeline and the riser is accomplished via a series of ‘spoolpieces’ (or spools). The spool is fabricated by welding pipe joints to form an L-shaped, Z-shaped, or possibly a straight pipe. The figure below shows that two L-spools connect the riser flange to the pipeline flange.

The lifting analysis of a subsea spoolpiece typically yields the specifications of the lifting slings and the lifting configuration, which represents the location of the pick-up (rigging) points. In addition, the analysis checks the stresses in the spools resulting from the lifting forces. However, the pipe stresses would be of concern in limited cases where we have to use a small lifting angle (with the horizontal), and the bend angle is approximately 90 degrees, coupled with a small pipe (e.g., 8 inches or less). Otherwise, lifting the spool does not usually cause stresses exceeding the allowable limits.

In all cases, we must obtain the lifting configuration; thus the tension in the wires, to correctly choose the suitable slings. That means we must find/purchase the slings matching the analysis results, which may cost much. However, the main problem is the potential delay in providing the slings if we have a tight schedule, e.g., when changing the lifting configuration of the closing tie-in spoolpiece.

What if the lengths of the available slings become an input in the analysis procedure? In other words, the question becomes, “Where can I install a sling of a particular length on the spool, with a particular hook height?” So if we could change the way we do the analysis, we would be using slings available in the stock, provided they have reasonable sizes and lengths, saving costs in many ways. Is that possible?

The answer is yes. The basic idea of obtaining the location of the pick-up points is that they represent the intersection between a sphere and a straight line, but how?

The sphere is centered at the hooking point, and its radius is the sling length. That means that the sphere surface represents all the possible points in space around the hooking point that are at an equal distance of the sling length from it. These points, of course, include the pick-up points we are looking for; thus, our job now is to find the intersection points between the sphere and the spool straight lines. See the below illustrating sketch.

Mathematically speaking, the sphere equation is defined by the center (hooking point) and the radius (sling length). And of course, the hooking point should be at a suitable vertical distance (hooking height) from the center of gravity, which should be calculated as well. Of course, that hook height should be reasonably shorter than the shortest sling. On the other hand, the spool equation is identified by the spool pipes’ lengths and bends’ angles. Consequently, the two equations must be solved simultaneously, yielding, ideally, two quadratic root(s), representing the coordinates of the pick-up points of one particular sling length on a specific spool pipe.

Furthermore, the tension in the slings can be found by simple equilibrium equations of rigid body mechanics. So we have six equilibrium equations: three for summing forces in the three coordinate directions and three for summing moments. In that case, if we imagine that the max number of slings is six (which is practically too many), we can use as many equations as the number of slings we wish to use. Ultimately, we solve these equations simultaneously, get the tension, and subsequently choose the proper slings or probably check if the tension matches the slings we have in stock.

This procedure can be done by Excel VBA. A program has actually been developed that performs the whole process from identifying the center of gravity, getting the pick-up points, and obtaining the tension. And, of course, the user can redistribute the slings and get new tension values if he wishes. Obtaining positive tension values is a sign that the slings distribution is physically possible and stable.

The following video explains the concept graphically.