CALIBRATION OF AN ULTIMATE LIMIT STATE FOR MOORING LINES

Structural reliability analysis has been used to calibrate a design equation for mooring lines in their ultimate limit state. The calibration is based on six test cases, for mooring systems in water depths ranging from 70m to 2000m. Three of the cases apply to a turret-positioned ship and three to a semisubmersible. Conventional catenary mooring systems with chain and/or wire components have been studied, whereas taut moorings with tibre rope are not yet included. Environmental conditions from the Norwegian continental shelf and from the Gulf of Mexico have been considered. A design equation format involving two partial safety factors, applied to two tension components is recommended. The two components are: (i) the static tension due to pretension and due to tension induced at the offset position corresponding to the mean environmental forces in an environmental state, and (ii) the dynamic tension component due to time-varying loads; i.e. in this paper defined as the sum of timevarying low-frequency and wave-frequency tensions in the environmental state. The recipes for characteristic values of the tension components and the line capacity are specified, and partial safety factors are given. INTRODUCTION Three criteria should be considered in the structural design of mooring lines for floating offshore structures. Within a structural reliability format it is convenient to formulate these criteria as: (a) an ultimate limit state (ULS) that to ensure that each mooring line is strong enough to withstand the extreme loads it is subjected to, (b) a progressive collapse limit state (PLS) to ensure that the mooring system can withstand the failure of one mooring line due to other causes, and (c) a fatigue limit state (FLS) to ensure that each mooring line has adequate capacity against fatigue. This paper deals with the ULS, while a companion papers deals with the PLS (Mathisen et al. 1998). The results are intended for use in the revision of the Posmoor rules for mooring line design (Sogstad 1998). There is also usually a serviceability requirement in the design of mooring lines, to ensure that the motion of the platform does not exceed limits imposed by attached risers or adjacent structures. This is obviously essential for a satisfactory design, but it is convenient to separate the serviceability requirement from the requirements placed on the strength of the mooring lines. The serviceability is usually adjusted by means of the line pretension, elasticity, weight, or number of lines. After changing any of these parameters it is necessary to check that all limit states are still satisfied. The objective of this work is to calibrate a simplified design method for the ULS, against a detailed structural reliability analysis of the ULS, such that a chosen target reliability level is achieved when the design method is applied. In order to ensure applicability of the rules in very deep water, the test set includes water depths down to 2000 m. The reliability analysis and calibration of the ULS is based on the experience obtained from two preceding joint industry projects: (a) FPS 2000 project 1.8, Reliability of Station Keeping, Systems, Mathisen (1992), and (b) PROMOOR, Reliability-Based Design of Mooring Systems, Mathisen and Horte (1996). The present paper therefore focuses on the calibration results. CALIBRATION Madsen et al. (1986) provide an introduction to calibration. Discussion of calibration can be a little confusing, because two methods of analysing the same problem are involved. The essential difference is that one method is simplified to make it convenient in practical design, while the other method is detailed to carry out the analysis in the best way available within the stateof-the-art. The calibration typically involves adjusting the partial safety factors applied in the design method, so that the resulting designs are close to a chosen target reliability level. The calibration can also be generalised to include other adjustments to the design method, such as changes in the format of the design equation, or in the definitions of the characteristic values that are involved. The calibration process may be considered to be a mathematical optimisation process, to minimise an objective function measuring the distance of the resulting designs away from the target reliability. It is advisable to clearly define the scope of the calibration; i.e. the class of structures that the design method is intended to be applicable to. Here, we intend to encompass mooring systems for floating offshore structures, in water depths from 70 m to 2000 m, using conventional chain and steel wire rope mooring line components. Semisubmersibles and ships are included, while tension leg platforms are excluded. We had hoped to include Spar platforms too, but have not had the resources to include a Spar platform in the test set yet. The results are also intended to be applicable world wide, provided that the recipe for characteristic values including the environmental conditions is followed. Strictly speaking, the calibration should be checked if the design rule is to be applied in locations where the distribution of environmental conditions falls outside the range covered by the two cases that have been used. A set of test structures have been selected to span the scope of the calibration. These include a turret-positioned ship and a semisubmersible, with various mooring systems for water depths of 70 m, 350 m, 1000 m and 2000 m. Environmental conditions for the Norwegian continental shelf and the Gulf of Mexico are included. The calibration is carried out as an iterative process, as indicated in Fig. 1. In practice, there is a need to simplify the iteration process, to avoid excessive computations in each iteration loop. This has been done by assuming that the mooring system response is unaffected by perturbations in the mooring line strength. The reliability analysis can then be carried out beforehand for a few line strengths, and the reliability results needed in the iteration loop can be provided by interpolation. The same assumption is also made in the design analysis, for consistency. RELIABILITY ANALYSIS Limit State Model The present application is concerned with the probabilitv of failure of a single mooring line, under extreme environmental conditions, without prior failure of any of the other mooring lines. This failure mode is referred to as an ultimate limit state (ULS). The ULS is intended to give a sound design of each individual mooring line to withstand the extreme loads it is expected to be exposed to. Mooring system failure is not considered in the ULS, but will be covered by the PLS. The PLS takes account of the possibility that a mooring line may fail due to some exceptional or unknown cause not accounted for in the ULS, and ensures that the mooring system has an ability to withstand such incidents. (Note that the empirical probability of failure of one line is relatively high and needs to be taken into account in the serviceability limit state. This can be done conservatively by assuming that any single line may be missing.) The ULS formulation used was originally developed by Braathen and Mathisen (1991), as has been presented by Mathisen and Mark (1993), and Larsen and Mathisen (1996b). Some additional details of the formulation are included in the description by Mathisen and Horte (1994). Tensile overload failure in any component of the line is included, and the distribution function for the strength of each type of component is required as input. The strengths of individual components are assumed independent. Variation in tension along the length of the line is neglected, based on the results of previous analyses, where it was taken into account, Mathisen and Horte (1994). The probability of failure is calculated by integration over the joint probability distribution of the environmental effects (long term type analysis). Fig. 2 provides an overview of the ULS computation. Only chain link and steel wire rope components from the main body of the line are included, because the design rule is to be calibrated for these types of components for predictable normal conditions. Connecting links and end terminations are omitted in the analysis. Mechanical wear and special load effects at fairlead as well as corrosion are not taken into account. Abnormal conditions are to be covered by PLS. Response Analvsis The main excitation sources for the mooring line are the vessel motions in the mooring line plane at the upper terminal point (fairlead). For a given state of the environmental variables, the low-frequency (LF) motion is the result of vessel low-frequency surge, sway and yaw, while the wave-frequency (WF) motion is defined as the tangential motion transformed to the fairlead point. The tension caused by the pretension, offset due to mean environmental forces and the low-frequency motion may be calculated by quasi-static analysis from the actual line characteristics. The tension due to the tangential WF motions at the fairlead is computed by a dynamic analysis. The total tension at the upper terminal point is the sum of the quasi-static and dynamically calculated tension. Since both the quasi-static top tension due to low-frequency motion and the WF top tension are stochastic processes, some diffrcuhy arises in making this combination. An approach is adopted, based on Turkstra’s hypothesis, which states that the extreme value of the combination is expected to occur when the extreme value of one of the components occurs. Two combinations are considered: CASE A: An extreme value of the quasi-static tension together with a local maximum of the WF tension CASE B: An extreme value of the WF tension together with a random value from the parent distribution of the quasi-static tension Note also that the WF tension is always conditional on the quasi-static tension. The combination is carried out in a short term, stationary en