Offshore Pipe Laying Analysis: General Definitions

The theory

The pipe-laying stress analysis can be understood by considering the pipeline as a slender beam that is fixed at both ends and subjected to large deflection. By large deflection, I mean (among other things) that my calculations deal with the true locations of the forces at time t, not the original locations.

During laying, the pipeline is subjected to cyclic loads and thus oscillation movements. There are a number of ways to formulate the differential equation of motion; at the end of the day, however, we will reach a matrix equation of the following general form:

[M]{u?

} + [C]{u?} +[K]{u} = {F}

This equation is nothing more than Newton equation of motion, F=ma; here, however, u?

represents acceleration expressed as the second derivative of the displacement u and {M} is the mass matrix. The second and the third terms of the left-hand side (LHS) represent the damping and restoring forces, respectively, where, C is the damping coefficient, and K is the stiffness coefficient. The velocity is represented by u?. Rearranging the above equation, we can bring the second and third terms to the RHS with minus signs so that the LHS represents the inertia force, ma, and the RHS represents all the total force, including those forces causing movement, F and Ku, and those resisting movements, Cu?.

In a finite-element model, a static analysis is performed by solving the above differential equation with respect to the curve length, with the time being constant. As a result, we get the static configuration, which is used later in the dynamic analysis.

To perform a dynamic analysis, we solve the differential equation in time. The non-linearity of the large displacement is overcome by discretizing the pipeline into finite element beams, each one of which has a local set of axes. At each timestep, we get a new pipeline configuration, with the local set of axes rotating and translating to the new location, where it will be used to calculate the new configuration in the next timestep, and so on and so forth. See figure 1.

Figure 1: Local coordinates

A global coordinate system is fixed somewhere according to the procedure used. The changing position of each origin of the local coordinates is identified with respect to the global coordinates; thereby we have a different profile (elastic curve) at each timestep that is plotted against the global coordinates.

Loads

The hydrostatic load is acting only on the wetted surfaces, normal to the pipe axis. According to the Gaussian theorem, this surface force can be converted into a body force (buoyancy) acting in the vertical direction. Then, however, as we are working on finite elements, the cross-sections of the element beams cannot be regarded as wetted surfaces, so we compensate for these compressive hydrostatic forces (implicitly acting upon the cross sections) by axial tension forces that are added to the other internal tensile forces.

Due to wave and current, the hydrodynamic loads are acting in the unsupported span and are composed of inertia and drag force components (Morison equation).

Beyond the touchdown point (TDP), the pipeline is treated differently according to the seabed interaction model used. In OFFPIPE, the vertical soil reaction is calculated by assuming that the soil is linearly elastic, whereas, in Orcaflex, a more sophisticated nonlinear model is used whereby the pipeline embedment can be predicted to some degree of accuracy (but is, to the best of my knowledge, still experimental).

Above water, the dynamic load comes primarily from the wave-induced vessel motions, which are calculated by using the vessel’s characteristic transfer functions, known as Response Amplitude Operators (RAOs).

Waves

Wave motion is regarded as one form of energy transmission. For example, the energy from the sun is transmitted by waves in the atmosphere. Also, when a musical instrument is being played, sound waves spread throughout the room.

When ocean waves interact with floating or submerged bodies, they cause them to move if the wave frequency is high enough.

In order to study the effects of waves on offshore structures, the wave’s motion and amplitude must be represented mathematically. There are a number of wave theories that assume that the wave has a single frequency (known as regular waves), which is good enough during the conceptual design phase. Nothing in the real world, however, has a single frequency, and we typically deal with a spectrum of wave frequencies known as a wave spectrum.

As can be seen in figure 2, the wave spectrum is a relationship between the frequency and the energy density. 

Figure 2: Wave spectra

These graphs are created from site measurements and are designed so that when the frequency is multiplied by its corresponding energy density, it gives the wave energy, the amount of which depends on the wave’s amplitude. Now, if we discretize this graph into a number of frequencies (harmonics), we can get the wave elevations as a sum of sinusoid functions coupled with a random phase, thereby creating a time trace of wave elevations and thus a time trace of loads, strains, stresses, etc.

Summary

To put all this in perspective, analyzing the stresses in the pipeline during laying is like any design problem, where you need to identify five essential elements: the system, the loads, the material, the configuration, and the boundary conditions.

Before calculating the dynamic stresses, we must first identify the configuration or the elastic curve in the static equilibrium condition. This is done via performing static analysis, which also tells us the (static) stresses along the pipeline from the line-up station to the touchdown point.

This means that we first have to calculate the shape that the pipeline will take, given the pipe geometry, material, water depth, stinger angle, and barge tension, without considering the dynamic loads at this stage.

The stresses must not exceed the allowable limits identified in the engineering codes. If the static analysis fails, it will be repeated over and over, changing the input parameters (like tension, rollers’ height, etc.) until the resulting stresses are acceptable according to the adopted criteria.

Having identified the configuration via the static analysis, we perform the dynamic analysis given the obtained elastic curve and the wave parameters to check the extreme stresses, hence identifying the maximum workable sea state (typically the wave height). Perhaps in some cases, it is necessary to estimate the fatigue damage.

So, in the dynamic analysis, we can say that we perform a stress analysis to many fixed-fixed beams (that represent the pipeline elements) at each timestep, See Figure 3.

Figure 3: Pipeline loadings during laying

About the Author

Mohamed Hermas is a chartered offshore installation engineer, with 14 years’ experience in the offshore construction industry. He worked on a variety of projects, including pipelines, platforms, risers & spools, PLEM & SPM, hook up and more. He has always had his own style and approach in writing installation procedures and method statements. During working in VMGL, he has written many spectacular documents that constitute benchmarks for the company for years to come.

Mohamed Hermas has a working knowledge of a wide range of disciplines, like planning, pipelay analysis, pipeline design, AutoCAD and Adobe Illustrator. This combination of skills enables him, not only to be an excellent team player but also to play different roles if the need arises.

Mohamed's overall expertise qualifies him best to handle EPC offshore projects from the FEED phase all the way to the mechanical completion and pre-commissioning. He has unparalleled ability to do all the coordination work necessary to deliver the project's documents, ensuring coherency and consistency between different interdependent contents. Mohamed is able to be at the center of any job and to take critical decisions as and when required.