ENDURANCE WAVE ANALYSIS

1 1. INTRODUCTION For an offshore structure, wind, wave and gravitational forces are all important sources of loading. The dominant load, however, is normally due to wind-generated random waves. If the structure is located in shallow waters, and in cases that the dominant wave period is obviously higher than the natural period of the structure, wave loads can be applied statically to the structural system. But, a non-linear time history analysis will become necessary to obtain more accurate results for the structure’s response, in particular for deep water structures under extreme loading conditions. It is important to account for the randomness of the loading also. A recently emerged approach called Endurance Time Analysis (ETA) is a novel method for analyzing structures subjected to ground motion excitations. The basic idea of this method was introduced by Estekanchi et al. ETA comes under the category of time domain analysis that in which in the specific times (target times), the input response spectrum will be the same as the design spectra corresponding to the desired levels of risk. In this approach, structure is exposed to an artificial intensifying acceleration time history. Damage indices or any other engineering demand parameters such as base shear, drift, and stress in structural members can be studied through the time variable. ETA method can be applied to any structural system irrespective of the complexity of the structural modeling, and different design criteria can be studied with less cost and computational time. Studies show that this method is more efficient and accurate in the evaluation of structures during the earthquakes previously used methods. Though there are similarities between structures subjected to earthquake and sea wave excitations in basic principles of design and performance, they differ in some aspects. Firstly, time duration of storm wave loading is very large (several hours) compared to earthquake loading occurrence. Secondly, temporal evolution of a stormy sea state may take several hours which covers a sufficient part of the growth and decay phases of the storm. Seismic excitations are almost abrupt and reach to their maximum potential in a short period of time (a few seconds). Here a new approach, called Endurance Wave Analysis (EWA), has been introduced which can be used for non-linear dynamic analysis and assessment of offshore structures subjected to irregular wave forces. This is an extension of the Endurance Time Analysis (ETA) mentioned above for seismic assessment of the structures. In EWA method, the offshore 2 structure is subjected to a predefined Intensifying Wave Train Function (IWTF), which represents different sea states at a specific site. This function is designed so that it characterizes increasing roughness of the sea state over the time which even goes well beyond the design sea state. A non-linear dynamic analysis is conducted by introducing IWTF as the source of external excitations of the offshore structure. From the results of the analysis the structural integrity of an offshore platform can be evaluated. The EWA method has the ability to consider spectral features of the sea state, to incorporate waves of different heights and frequencies in a single dynamic analysis and to take into account the irregularity and randomness of the sea waves. It requires relatively short simulation times than the other time domain analysis procedures. The non-linear behavior of offshore structures can be evaluated. And it is able to consider the different damage indices for evaluating the structural integrity. The method is useful for both design and as well as the assessment and collapse analysis of existing offshore platforms. 2. THE APPROACH OF ENDURANCE WAVE ANALYSIS (EWA) 2.1 The Analogy to the Cardiopulmonary Exercise Test The approach of EWA is owed from the ETA method which was inspired from the standard protocol (exercise test) that is used by cardiologists in order to evaluate the physical condition of the cardiovascular system in the human body. Even though the cardiovascular organization of humankind is one of the most complex systems known in nature, the performance of this system is evaluated with a simple test called as the exercise test. Cardiopulmonary exercise testing is a well-established procedure which is non-invasive, relatively inexpensive, a strong and independent predictor of cardiovascular disease and mortality. The analogy between a cardiopulmonary exercise test and the EWA method is shown in figure 2.1. 2.2 The ETA Method Endurance Time Analysis (ETA) method is a time history based analysis procedure that applies special intensifying acceleration functions for estimating the seismic performance of structures at different excitation levels in each single analysis. The Endurance Time method is a 3 procedure that can be used in both the linear and nonlinear seismic analysis of structures. Its simplicity is Figure 2.1 4 one of the priorities of ET method over response history analysis under actual ground motions. In the ET method, the response of a structure can be monitored against time which is correlated to the intensity of excitation. ET method is evaluated by comparing its results with results of time history analysis under actual records. To simulate excitations, intensifying accelerograms called “ET acceleration functions” (ETAF), are imposed to the structure. These ETAF are generated in such a way that their response spectra increase by the time, hence response of structure under this kind of accelerogram gradually increases with time. A typical ETAF is depicted in Figure 2.2. Figure 2.2 A typical ET Acceleration function 2.3 The EWA Method Endurance Wave Analysis (EWA) method is somewhat similar to the Endurance Time Analysis (ETA). In the EWA method, after a preliminary study on sea state conditions and wave spectrum characteristics of the site, the offshore structure is exposed to an intensifying time history of waves. For this, consecutive time series of stepwisely intensifying constrained irregular waves are joined together to form a single time history of the sea surface. This is called an Intensifying Wave Train Function (IWTF). It is similar to ET Acceleration Function (ETAF) in ETA method. Overall, it represents an artificial gradually deteriorating sea state at the site. This function is a relatively short duration time series of the irregular water surface elevation. It is designed so that it characterizes increasing roughness of the sea state, over the time, which even goes well beyond the design sea state. 5 Similar to a cardiopulmonary exercise test, the non-linear dynamic response of the offshore structure can be evaluated when it is simulated under the IWTF. An arbitrary damage index, for example a drift limit or collapse occurrence, can be defined as a monitoring parameter in the temporal response events. Now, in principle, the sea state which causes the structure to reach the predefined damage target can be easily evaluated by a single dynamic analysis. The dynamic behavior of the structure, from its elastic part to complete failure level, and temporal variations of the desired damage indices can be monitored during an EWA. Using the EWA procedure and considering appropriate target criterion, different levels of performance for the offshore structure can be investigated. The Endurance Wave Analysis (EWA) method provides a powerful and practical approach for the dynamic, time domain and nonlinear analysis of the offshore structures subjected to random sea wave excitations. 2.4 The Hypothetical Shaking Experiment The concept of EWA can be explained with a hypothetical shaking experiment. Three different designs for an offshore jacket type platform with various (say unknown) load bearing capacities, dynamic characteristics and geometrical properties of the structural members are considered. All three structures are considered to be placed in similar offshore locations with identical environments. These structures will be evaluated using an EWA method. For this all three structures become subject to an identical IWTF (Figure 2.3). In the beginning, the wave train and its corresponding wave energy density spectrum are assumed to represent a rough sea state (HS1 and TP1). The EWA analysis results indicate that all three structures exhibit stable responses under this sea environment (Fig. 2.3a). As the time goes on, the sea conditions is intensified (Figure. 2.3b) and the wave train and its corresponding wave energy density spectrum are now assumed to represent a high sea state (HS2 and TP2 , HS2 > HS1 ). This sea state has gone beyond that structure (2) can withstand. As a result a failure occurs in structure (2) (Fig. 2.3b). The failure has been identified by damage indices to exceed a target limit or by unbounded structural responses. Under this sea state, structures (1) and (3) are still performing well and exhibit stable and limited responses. With further intensification of the sea conditions the wave train and its corresponding wave energy density spectrum now, assumedly, represent a phenomenal sea state (HS3 and TP3 , HS3 > HS2). The wave load has now caused structure (3) to fail while structure (1) is still doing well and exhibits stable and limited responses but structure 6 (2) is seriously damaged (Fig. 2.3c). So with a EWA procedure, the non-linear structural dynamic performance of an offshore structure, or its reserve capacity, can be easily evaluated by a single dynamic analysis. Figure. 2.3. Three hypothetical offshore platforms subjected to an Intensifying Wave Train Function (IWTF) 7 Endurance Wave Height (EWH) of an offshore structure refers to the maximum sea state that the structure can withstand under an IWTF. EWH can be defined by monitoring an appropriate damage index when it meets a damage threshold. The EWH corresponds to a time in the analysis when the amount of damage observed in the structure exceeds a target value. As a result, for a particular structure, based on demand levels and the desired performance criteria different EWHs can be identified. The method can also effectively describe the behavior of structure from linear elastic to non-linear plastic zone toward its final collapse by a relatively short duration time domain analysis. 3. THE WAVE TRAIN FUNCTION EMPLOYED 3.1 Intensifying Wave Train Functions (IWTFS) The most important element in the Endurance Wave Analysis (EWA) method is producing appropriate Intensifying Wave Train Functions (IWTFs). Based on the structure type, the water depth, the wave climate condition and the analysis, different wave theories can be used for constructing the IWTF. IWTF may be accordingly selected to be regular or irregular. In practice, ocean and sea waves are both random and irregular and simultaneously contain waves with different shapes, heights, speeds, periods and directions. For engineering purposes, the wave condition may be described either by a deterministic or a stochastic approach using a wave spectrum. The sea state is usually assumed to be a stationary random process. A stationary sea state can be characterized by a set of environmental parameters such as the significant wave height HS and the peak period TP. Three hours has been introduced as a standard time between registrations of sea states when measuring waves, but the period of stationarity can range from 30 min to 10 h. For design applications, regular wave theories such as fifth order Stokes (in deep water) and conidial (in shallow water) are commonly employed to calculate the wave loads on offshore platform. In this approach, it is assumed that all wave energy is concentrated in a fixed frequency. These methods are popular due to their ease of application and low computation time. With dynamically sensitive structures, however, the accuracy of these types of deterministic approach is debatable. 8 The most comprehensive approach in studying the dynamic behavior of offshore structures subjected to ocean waves is stochastic irregular long-term simulation of the sea water level (SWL) and direct method of solving motion equations in a time domain. This is time consuming and computationally expensive, because a time record of the irregular wave input must be provided, while generally, one only has information about the wave spectrum available. Artificial irregular wave records can be regenerated using a wave spectrum but by choosing different random phases a seemingly endless series of time records, all with identical statistical properties, can be generated. It cannot be said which –if any– particular (artificial) time record is correct. Therefore, the time domain analyses need to be repeated many times, using different long duration irregular wave records. This can have a significant influence on the interpretation of the results, especially when extreme values are needed. The interpretation of time domain simulation results forms another difficulty in practice. A designer often needs a “design response” (an extreme dynamic internal load or displacement) with an associated (small) probability that it will be exceeded as “output” from a dynamic analysis. Any wave theory can be used in the EWA procedure. Here CNW model has been used to generate the IWTFs. This is because the irregularity and randomness of the sea surface are sensibly represented by a CNW. A CNW has a relatively short duration (say around 2 min) as compared to typical irregular waves (say 3 h). Other researchers in recent years have shown the CNW model to be a suitable alternative to long-term random time history simulations, and that it would decrease the time and calculation costs. 3.2 Intensifying Constrained NewWave Model (ICNW) Assessment of Gaussian sea state characteristics, near a local maximum, for example highest wave crest, has been studied by Lindgren. In 1981–1982, Quasi-Deterministic (QD) theory was later suggested by Boccotti, into two versions, based on first-order Stokes expansion. The QD theories proved that in a Gaussian sea state if a local wave crest maximum (first QD theory) or a large crest-to-trough (second QD theory) occurs at some fixed time and location, the water surface elevation will be proportional to the autocovariance function. This theory provides necessary and sufficient conditions for simulating a very large irregular wave at a fixed position and time. The first version of QD formulation was similar to NewWave (NW) theory introduced by Tromans et al. The NW theory was based on statistical analysis of irregular sea surface data. 9 They were the first who introduced a practical approach for representing an extreme irregular sea-surface elevation in a short-term history, which was called NewWave. Several specific characteristics included in the model, such as its spectral base and irregular shape, made the NW clearly different from deterministic regular models, for example, 5th-Stokes theory. A single profile of the NW model, however, cannot consider the stochastic nature of the ocean waves. So, this shot-term time history falls short to deliver the randomness of the sea state to an offshore structure when it is modeled under dynamic wave actions. If a NW profile is constrained to a Random Surface Elevation (RSE), an appropriate short-time random time history, with a desired wave crest elevation, can be produced. Accordingly, Taylor et al. proposed the Constrained NewWave (CNW) model. Fig. 3.2.a shows typical shapes of the NW and CNW surface elevations as a function of time. Fig. 3.2a Typical NewWave (left) and Constrained NewWave profiles (right). In EWA method, the CNW model has been used to generate the Intensifying Wave Train Functions (IWTFs). For this, m separate time series of stepwisely intensifying CNWs each with a duration time of td are joined together to form a single artificial time history of the sea surface. The kth CNW profile 𝜂k(t) represents the sea state k (1≤ k ≤ m) which is itself constructed based on the wave energy density spectrum Sk(ω) at a specific site. The kth CNW covers a time period of (k − 1) × td < t < k × td. The roughness of the sea state k and the power of the wave spectrum Sk(ω) stepwisely increases with a linear trend as k increases from 1 to k. The target (say design) wave height HS design and its corresponding energy density spectrum Sdesign(ω) can be located 10 somewhere halfway through the sea states 1 to m. The first generation of IWTF, in which the function growth is linear, can be expressed as follows: 𝜂R1 (t) + ρ1(t) [α1 − 𝜂R1 (tc)] + ̇ 𝜂̇R1 (tc) 0 < t< td,, S1(ω) 𝜂R2 (t) + ρ2(t) [α2 − 𝜂R2 (tc)] + ̇ 𝜂̇R2 (tc) td < t<2 td,, S2(ω) ⋮ 𝜂Rk (t) + ρk(t) [αk − 𝜂Rk (tc)] + ̇ 𝜂̇Rk (tc) (k-1 td < t