The two models that are central to all questions of the design or assessment of offshore structures are the load model and the structural model, i.e. models for blocks 2 and 3 in figure 3.4. A description of the environmental loads is so closely interrelated with a description of the environment itself that it is commonplace to refer to a combination of blocks 1 and 2 as the environmental load model. This practice will be followed here too. In applications all loads should be considered; however, for the present discussion we will focus on the environmental loads and more particularly on the wave and current loads. The second model, the structural model, relates to all system properties, i.e. mass, spring and damping characteristics, as well as to the system’s boundary conditions (e.g. its foundation or anchoring system).
Model attributes
Each of the two models may be identified by three main characteristics or attributes, i.e. they are deterministic or probabilistic (D or P); they are linear or non-linear (L or NL); and they are time dependent or time independent (TD or TI). When we refer to a structural model as time dependent we mean that it includes mass and damping properties, either as a rigid body or as an elastic system, enabling it to exhibit dynamic response. This terminology is perhaps somewhat unusual, but by introducing it we are able to make common distinctions for both models.
Combining 2 choices for each of 3 model attributes leads to 8 different possibilities for each model, as illustrated in figure 3.5. Hence there are 8 x 8 = 64 different combinations in a joint application of the two models. A number of these combinations are in everyday use, other ones only find application in special cases, while some of them are rather hypothetical; see the typical applications that are also indicated in figure 3.5. However, all 64 possibilities do exist and the most appropriate choice for each and every application should be consciously made, keeping the state-of-the-art of the technology and the (im)practicality of the particular model(s) for the intended purpose firmly in mind.
Examples of choices to suit applications
Let us by way of example consider a few applications. A fixed offshore structure is normally designed using a design wave procedure, i.e. an extreme single, periodic wave is selected and ‘frozen’ at different times in its passing through the structure. The load model usually chosen for this event is deterministic, non-linear and time independent (D,NL,TI). The wave is selected by the designer and any possible variation of this selection is treated as a separate load case; hence the model is deterministic. For a single periodic wave non-linear features can easily be retained so that it presents no difficulties to use a non-linear formulation for the wave and the associated wave loading. Finally, although several wave positions may be considered in succession, each of these is independent of time. The usual choice for the structural model is either (D,L,TI) or (D,NL,TI), mainly depending on the properties of the foundation and whether structure and foundation are integrally or separately modelled.
For a fatigue or a dynamic analysis of a structure the time varying nature of the loading is an essential feature and must be retained. As long as the structural response may be treated as quasi-static the structural model does not need to include time dependent properties to enable it to exhibit dynamic response. However, when the response is truly dynamic time dependency is an important element and must be incorporated. For fatigue as well as for
Chapter 3 – Models and modelling 3-12 dynamic problems it is generally necessary to include the random nature of the environment in a realistic manner. If a spectral approach is used the loading must be linearised and the load model is chosen as (P,L,TD). In case of time domain simulations the non-linearity may be retained and the load model can consequently be (P,NL,TD). The structural model normally remains deterministic and satisfies (D,L or NL, TI or TD). The choice of L/NL depends on the use of a spectral or a time domain simulation approach, while the choice TI/TD depends on the quasi-static or dynamic response of the structure as discussed before.
As a final example, the vertical motions of a semi-submersible due to heave, pitch and roll can usually be determined while ignoring the anchoring system. As these vertical motions are time dependent, the random nature of the environment must be retained and a spectral approach has become the norm since many years. Consequently, the load model is chosen as (P,L,TD) and the structural model as (D,L,TD). Applications concern e.g. the assessment of downtime of floating drilling rigs. For the horizontal motions of anchored floaters low frequency drift forces play an important role. These forces are non-linear with the wave environment; therefore time domain simulation would seem to be called for, but in various cases reasonable results have been obtained by second order spectral methods in the frequency domain. The load model would correspondingly be chosen to be (P,NL,TD) while the structural model remains (D,L,TD).
A more systematic overview of combinations used in practical applications is shown in figures C.1 to C.3 in Annex C. These illustrate the current state of affairs and the future development required. Each of these figures is a two-dimensional representation of a really three-dimensional picture.
Probabilistic v. deterministic models
All science and engineering has probably begun by developing deterministic models. In marine and offshore technology the prime source of randomness is the ocean environment, as is illustrated by a saying of Lord Rayleigh towards the end of the last century: “The basic law of the seaway is the apparent lack of any law”. The advent of probabilistic models in oceanography dates back to the days of World War II when attempts at modelling the ocean surface more realistically than by periodic waves began to have success and a modern era of oceanography emerged. Probabilistic environmental loading models combined with a deterministic model of the structure have been used since the early fifties for studying ship motions. They were introduced jointly by a naval architect (St Denis) and an oceanographer (Pierson) in a famous paper for the Society of Naval Architects and Marine Engineers in New York [3.4]. They introduced their work under the heading “In a broad sense the laws of nature are Gaussian”. A very valid and useful statement. The fact that it is not strictly true and that in may instances we are now looking for refinements and improvements on it is only an indication of the degree of progress we have made over the last forty plus years. Since then spectral analysis techniques and statistical treatment of problems based on Gaussian random processes have been widely used. An example of a discussion of some basic aspects of the methodology and some applications in wind and water may be found in ref. [3.5].
In the offshore arena analogous applications of probabilistic loading and deterministic structural models have gradually become common practice for assessing most aspects related to the motion behaviour of floating offshore craft. Examples are motion downtime
analyses for the then new generation of northern North Sea structures (see e.g. [3.6]). Such methods are now fairly routinely applied to large and important structures, especially if dynamic behaviour is to be incorporated. An extension of this situation by also introducing probabilistic structural models is the ultimate challenge of structural reliability analysis. These probabilistic structural models should then account for such things as tolerances on dimensions, geometrical or material imperfections, variations in yield stress and distribution of residual stresses. However, this is largely still beyond the present state-of-the-art.
Generally speaking, the offshore industry has been somewhat slow in adopting random analysis procedures. This is probably due to a lack of familiarity with these techniques as well as a failure to recognise the essential probabilistic nature of the environment (and hence of the environmental loading) amongst most engineers in the early days of offshore developments. However, this has now been largely overcome. What remains and forms a serious complication for the load model is the problem of loading non-linearities. Such non-linearities are e.g. due to the wave kinematics around the free water surface (which cannot be predicted well by linear wave theory), drag forces of viscous origin, and time dependent lengths of structural members that are subjected to wave forces. However, this difficulty cannot be solved by adopting a model that includes non-linearities at the expense of reducing wave loading to a deterministic and periodic phenomenon. To avoid misunderstanding, a deterministic and periodic load model is generally quite suitable as an abstraction of reality for designing fixed structures; it has proven its worth for this purpose. However, it is so great an over-simplification that it loses realism and is inadequate for a rational debate of the occurrences in nature.
The use of probabilistic models is nearly always more suited to analysis than to design. The design process is generally based on simplified deterministic load cases, drawing heavily on the experience of the design engineer. The use of such simplified procedures is likely to remain the norm for practising engineers in the everyday routine of design applications. There is nothing wrong with this, provided the engineer is aware of the simplifications involved and can revert to other approaches when circumstances demand it.
Chapter 3 – Models and modelling 3-14
3.4 References
3.1 Chakrabarti, S.K., Offshore Structures Modelling; series on Advances in Ocean Engineering, volume 9; World Scientific Publishing, 1994, Singapore