PAPUSHA A. N., SINYAK R. A.
MURMANSK STATE TECHNICAL UNIVERSITY, Russia, FSUE 'Arktikmomeftegazrazvedka", Russia

Until now computer methods of analysis based on the symbolic computing and exact calculations concerning deflected mode of a slightly raised and sagging part of a submarine pipeline are not developed or absent in most cases.

A slightly raised and sagging part of a pipeline is known to be under the complex action of different external stresses as it is bend and stress-strain simultaneously. There are no calculating methods giving complete and mechanically justified submission about deflected mode of the slightly raised and sagging part of a submarine pipeline.

Mechanical justification is explained as the representation of deflected mode of a slightly raised and sagging part of a tube, which is based on the strict mathematical specification statement and definition of the deflected mode, and would be based on the accurate solutions of the appropriate edge tasks.

Mathematical modeling experience demonstrates that appropriate symbolic solutions allow not only to solute task of stability of a pipe while laying it by a pipelaying vessel, but also to give the practical recommendations to the engineer how to use the nonconventional character solutions for others traditional (basically for FEM) and other numerical realizations distinct from finite element method.

Up to date facilities of a submarine pipeline designing based on the mathematical modeling of deflected mode are usually based on numerical accounts using a finite element method which doesn't allow to represent the strained of state of the pipe correctly.

As a rule this fact shows that designer has no correct representation about bending pipe. It is necessary to notice, that using methods of computer algebra allows to find the exact solutions for deflected mode analysis of the submarine pipeline.

Thus, formulating tasks and their solutions at all stages, the mathematical expressions-beginning from function of a bending flexure of a tube and finishing the formulas for a tension on its surface are developed using the methods of computer algebra.

These results are based on the exact solutions of appropriate edge tasks. Such solutions don't lose their meaning even if there is strong and well organized numerical realization of finite element method (FEM), as they are fundamental and main mathematical landmark for numerical solutions with the use of other methods, as a rule, checking, authenticity and reality of numerical design ascertains on exact solutions of linear edge tasks. And it runs out of mathematical design experience.

Thus, major reason of a scientific research is a need of computer method organization of calculation and design of submarine pipelines of large diameter being laid in deep water.

There computer solutions are ready for a designer and technologist as they allow to lay submarine pipeline of large diameter in deep water.
In the world practice in laying submarine pipeline two main methods of laying are used. They are S-laying method and J-laying method
Both methods installation took their names after the shape of the deformed part of a pipe under water in a sagging position, i. e. a part of a pipeline not adjoining with the seabed.

Till now the exact mathematical formula depicting the shape of a sagging deformed part of a pipe was not obtained.

The mathematical formula will be found by methods of computer algebra applying the software of Wolfram Research - "Mathematica".

Talking about the technology of laying a submarine pipeline we'd like to point out that both methods of laying are characterized by the possibility of strain force regulation by a power unit, installed on a pipelaying vessel.

Evidently, changing a pipe strain as a free mechanical parameter in a task will simultaneously cause changing its deflected mode. Studying an influencing of a pipe strain on deflected mode of submarine pipeline is an important problem requiring its solving.

In spite of this, the ability of using both methods and constructing technology in a sea is determined firstly by strain deformed position of the slightly raised and sagging part of a pipeline, because only exceeding pipe strain leads to breaking geometry of a pipe profile.

From another side reaching limit strain of crumpling in a pipe wall in the moment of laying leads to crashing the pipe shape and consequently to breaking the construction technology, that is inadmissible during the exploitation of a pipeline.

That is why the task of analysis of deflected mode of a slightly raised and sagging part of a pipe with two absolutely different methods of pipeline laving: J-laying method and S-laying method is a new fundamental problem for continuum mechanics, which has an important applied significance for marine oil and gas techniques, concerning marine large diameter pipeline installation in deep water.

Opposite to traditional methods of a deflected mode of lengthy marine constructions calculating saluted by continuum mechanics methods, we shall apply computer (mostly symbolic) techniques of calculating based on up to date computer techniques and solution methods of appropriate edge tasks of mechanics of rigid body. The software of Wolfram Research – “Mathematica” will be applied for symbolic computation.

To solve the task of analyzing the deflected mode of any sagging pipe part a certain sequence is used:

1. Application of computer algebra compiling of the applicable boundary value problem:
2. For task solution standard codes of the software "Mathematica" are applied.
3. At first all solutions must be found in symbolic form and after that on the basis of analysis of the symbolic computation the numerical and plot dependences are developed. They give a designer a visual notion of the deflected mode of any sagging part of a submarine pipeline.
4. On the basis of the symbolic and numerical solutions a designer makes a conclusion about the rational and possible method of laying a pipeline, and technical means of laying a pipeline in deep water.
5. The analysis by computer technique and symbolic computation is possible only after weighted estimation and fighting force of a pipeline calculation.

Thus a new computer technique will be developed to solute a task of a deflected mode application and designing a sagging part of a submarine pipeline, being under the interaction of different power stresses, common for marine oil & gas technology of laying large diameter submarine pipelines in deep water.

New tasks of calculating a deflected mode of a sagging part of a submarine pipeline by computer algebra methods are formulated and solved in our research work.

The obtained symbolic solutions of calculating the strain of a sagging pipeline part are suitable for submarine pipeline designing in the case of using large diameter pipes in deep water.

All of design problems are solved without any complementary limitation of the sea depths.

We should also mention that symbolic solutions having been offered are applied for both methods of pipe laying, what gives solutions of new practically important tasks of Mechanics of rigid body which have an important applicable notion for marine oil & gas technology.

In conclusion we'd like to mention, that all normative documents regulating submarine pipeline design say the following:
1. Up to now there is no common theoretical basis of deep water pipeline design.
2. There is no whole calculating theory of a deflected mode of a pipe while laying and using.
3. There are no true experimental quantities of the deflected mode of deep water pipes while laying submarine pipelines.

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ARCTIC SHELF OIL AND GAS CONFERENCE 2004