Estimation of Overall Fatigue Life of Jack-up Leg Structure

Jack-up platforms are designed to work in three conditions: Transit, Preloading and Operating. The fatigue life of the jackup platforms in operating condition will be determined to be similar to that of offshore fixed steel structures. In preloading conditions, fatigue damage is usually ignored. Up to now, the fatigue damage of the jack-up leg structure in transit conditions has been calculated at approximately 20% of the total fatigue damage of the jack-up leg structure in two conditions (transit and operating). The approximate calculation method is usually accepted by consultants and register agencies. If the approximation is used, the fatigue life of the jack-up leg structure will be calculated only as for the jacket structure of the fixed offshore structure, with 80% of the allowable fatigue life based on standards. The approximation proved to have many disadvantages: the different travel times of each jack-up rig cannot be mentioned; hot spots that need to be maintained during the transit condition have not been pointed out; it is difficult to guarantee the safety of the jack-up leg structures in the transit condition. In order to overcome the limitations of the approximation method, this paper will propose a method to predict the overall fatigue life of the jack-up leg structure in three main problems. Firstly, we use the analysis method of fatigue of fixed steel offshore structures for jack-up leg structures in operating conditions. Secondly, we suggest a method to analyze the fatigue of the structures in transit conditions. Herein, motion analysis and determination of inertia forces on the leg structure are performed by the Boundary Element Model (BEM) in SACS software. Then the inertia forces are assigned to a Finite Element Model (FEM) in SACS to decide the internal forces of the structures. Hotspot stresses are determined by combining nominal stress from FE analysis results with a concentrated stress factor from the analysis of joint local models in the ANSYS program. Then, fatigue damage and fatigue life of hotspots of the structure are determined in the transit condition. Finally, a formula is suggested to determine total fatigue damage in operating conditions and transit conditions with different cases in relation to different fraction factors. These results are used to predict fatigue life corresponding to the most dangerous cases of structural joints. These new suggestions are applied to fatigue analysis for jack-up Tam Dao 05. Currently, the Tam Dao 05 platform has been operating in the Vietnam East Sea.


Introduction
Jack-up structures include three main parts: a hull, legs, and spudcan structures. Under operating conditions ( Figure  1-a), the platform structures work in the same way as fixed offshore structures. In wet tow conditions (Figure 1-b), the structures work the same way as floating structures. In the dry tow conditions (Figure 1-c), the jack-up rig is a large structure transported on a barge. The strength of jack-up structures is assessed based on ultimate limit state (ULS) in operating conditions, preloading conditions, and transit conditions according to standards [1], DNV-RP-C104 (2012) [2], and American Bureau of Shipping (2014) [3]. Besides, the fatigue life of the structures is evaluated based on Fatigue Limit States (FLS) in operating conditions and transit conditions. In FLS, the fatigue damage in operating conditions is presented by Barltrop & Adams [1] and DNV-RP-C104 (2012) [2], American Bureau of Shipping [4,5]. Cuong & Chinh (2019) and Quang & Vu (2021) clarified a method to evaluate the fatigue life of the structures in transit conditions [6,7].
As per current standards, the method of assessing fatigue life jack-up leg structures can be summarized in steps as follows: Step 1: Assessing fatigue life of jack-up leg structures in the operating conditions.
In the operating conditions, the jack-up rig operates as an offshore fixed steel structure. The fatigue life of the jackup leg structures in the operating conditions, under the influence of sea waves acting directly on the jack-up leg structures, can be determined by the Palmgren-Miner method and has been clearly presented in Barltrop & Adams (1991) [1]. In this paper, the fatigue life of the jack-up leg structures in the operating conditions is denoted as Top.
Step 2: Evaluation of the overall fatigue life of the jack-up leg structures.
Herein, fatigue damage in transit condition is ignored. In return for this situation, the design life of the structures is reduced by 20%. So the fatigue condition can be assessed as below: Safety conditions: Top ≥ 80% [T] (1) whereas: [T] is design life.
Considering that the transit time is equal to 20% of the fatigue life of jack-up leg structures is incorrect. This statement can be clarified for the following reasons:  In fact, the transit time of each jack-up rig is different. There are jack-up rigs that move very little and only inside the oil and gas field. In contrast, many jack-up rigs are used for oil and gas exploration at sea, so the transit time is often very large compared to the total lifetime of the jack-up structure. Thus, the transit time of all jack-up rigs should be considered to be the same and, equal to 20% of the jack-up structural life, as a basis to approximate the total fatigue damage in the transit conditions of the jack-up rig is equal to 20% of total fatigue damage of jack-up structure is incorrect.
 The results of the fatigue analysis will predict the fatigue life of the structure. The fatigue life of the structure is the fatigue life of the hot spots. Pointing out the hot spots of the structure, in addition to the purpose of predicting the structural fatigue life, also gives the locations that need a detailed survey to promptly correct defects or possible problems to ensure safety and prolong the life of the structure. This is an important step in the analysis and reevaluation of the structure as a basis for the re-design of the structure. The approximation method does not show hot spots on the jack-up structure in the transit conditions, it will be difficult to track, ensuring the safety of the jack-up structure in the transit conditions.
 Usually, the location of hot spots of the jack-up foot structure in the transit conditions is different from the position of the hot spots in the operating conditions. Most of the hot spot locations that occur in the operating conditions, in the transit conditions, the location will not be the hot spot. The addition of fatigue life in two conditions (transit and operating) at a location of the structure is possible only when that location is a hotspot in both conditions. This is very unlikely, and this is a limitation of the approximation method. However, the fatigue damage determined in the two conditions can be added. Then the fatigue life of the hot spot in the two conditions will be the inverse of the sum of the fatigue damage of the two conditions. The approximation method does not help us to do this.
 Considering that the fatigue damage of the jack-up rig in the transit conditions accounts for 20% of the total fatigue damage of the jack-up structure, it means that there will not be a hot spot on the jack-up structure in the transit conditions. This is not practical because, for jack-up structures, there is a large travel time and distance, especially when traveling long distances in bad weather, there is a risk of hot spots. In this case, the movement of the jackup structure is not guaranteed.
In order to overcome the shortcomings of the approximate method, this paper proposes a new method to evaluate the overall fatigue life of jack-up leg structures including the fatigue damages in the transit conditions [8]. The new method consists of three steps as follows: Step 1: Assess the fatigue damage of the jack-up leg structures in the operating conditions (Dop).
Step 1 is carried out in the same way as the current method used to evaluate the fatigue life of steel jacket of fixed offshore platforms.
Step 2: Assess the fatigue damage of the jack-up leg structures in the transit conditions (Dts).
Herein, the authors propose a new method to evaluate fatigue damage of the jack-up rig leg structure in the transit conditions, replacing the approximate method currently being implemented in the world. In transit conditions, almost the jack-up leg structures are not directly subjected to wave loads. Fatigue loads are inertia forces generated by motions of the structural systems under the actions of waves.
Step 3: Evaluation of overall fatigue life of the jack-up leg structures takes into account the total fatigue damage of the structure in the operating and transit conditions. Step 3 will, based on the results of step 1 and step 2, propose formulas to evaluate the overall fatigue life of jack-up legs structures.
It is necessary to include fatigue damage in transit conditions to estimate the overall fatigue life of the Jack-up leg structure. The calculation that ignores the fatigue damage in the transit conditions and replaces it by reducing the design life by 20% is incorrect. The three calculation steps proposed in this paper are a new method, aiming to replace the calculation method that is usually accepted by Consultants and Register agencies.

Prediction of Fatigue Life of Jack-up Leg Structures in Operating Conditions
In operating duration, jack-up leg structures are impacted directly by wave loading of different environmental conditions and at different locations. So, in the lifetime, any point of the structures always bears many stress groups Generally, a hotspot of a jack-up structure is assumed to resist m group of stresses ranges Si induced by short-term seastates which specified by significant wave heights Hsi and number of cycles ni (i=1÷m). Whereas, the stress ranges are calculated by multiplication of nominal stresses and stress concentration factors [9,10]. According to Palmgren-Miner rule [1], the fatigue cumulative damage ratio of the hotspot is expressed by formula as below: Commonly, jack-up legs are made of high strength steels with yield limits large than 500MPa, so fatigue limit cycles Ni can be determined based on Si (MPa) by the S-N curve in DNV RP-C203 [11], as below: The fatigue life of a hot spot can be determined by the Equation 4, with a safety factor f. The factor value depends on standards [11,12]: As mentioned above, jack-up platforms must be worked in different locations with various water depths. To analyze fatigue damage in a whole life time, the structures should be simulated by corresponding models. According to basic theories, authors suggested a procedure of fatigue analysis of jack-up structures in operating conditions in Figure 2 as below:

Fatigue Life of Jack-up Leg Structures in Transit Conditions
Generally, fatigue analysis of jack-up legs structures in transit conditions are performed as the same as in operating conditions. However, in transit conditions, the jack-up legs are not directly subjected to wave loads. Fatigue load for fatigue analysis of jack-up rig leg structure in transit conditions is the inertia force, caused by shaking vibrations of jackup rig structure under the action of waves and currents. The following sections will briefly present the calculation of the inertia force and establish the algorithm to evaluate the fatigue life of the jack-up leg structures in transit conditions.

Inertial Load for Jack-up Structural Fatigue Analysis in Transit Conditions
Floating structural motion components are signified in Figure 3. The equation of motion can be expressed in the following Equation: ( where: M is a structural mass matrix in global axis; A is added mass matrix determined from radiation wave potential on the mean wetted body surface of the hull; Damping matrix C is determined from radiation wave potential; Hydrostatic Stiffness Matrix Khys is determined from hydrostatic pressure effect on the wet surface of the hull; ̇,̈ vectors of motion velocities and accelerations; F(t): the vector of wave loads based on incident wave potentials and diffraction wave potentials. Motions at the center of gravity of a jack-up platform are determined by Equation 5 induces inertial forces at i th mass point of the structure. The inertial force includes the components of the translational inertia, the centrifugal inertia and the tangential inertia, which are determined according to DNV-RP-C104 (2012) [2].

Determine the Stress at the Hotspot
Assuming that incident waves are stationary random processes specified by spectrum density functions ( ), the response spectrums of the structures can be expressed in formula as below: whereas, ( ) is Response Amplitude Operator of the structural response u(t). According to DNV-RP-C104 (2012) [2], based on ( ), the inertial force spectrums at mass mi can be determined depending on response accelerations of the structural center of gravity, roll and pitch angles, distances from mi to the center of gravity. The structures will be analyzed by the finite element method, the stress spectrums S() are determined at the hotspot of the structures, including stress concentration factors [5]. Finally, the stress ranges and number of cycles are determined by the formula in [13]. Actually, joints of jack-up legs are complicated, so the hotspot stress concentration factors (SCF) are commonly determined by analysis of local FE models [2,5].

Prediction of fatigue Life of Jack-up Leg Structures in Transit Conditions
The fatigue cumulative damage ratio of the hotspot is expressed by Equation 2. S-N curve in DNV RP-C203 [12] are presented in Equation 3. The fatigue life of a hot spot can be determined by the Equation 4. Herein, the authors establish the general algorithm for fatigue analysis of Jack-up legs structures in transit conditions in Figure 4.

Estimation of Total Fatigue Life of Jack-up Leg Structures
To call fatigue cumulative damage of a hotspot of jack-up leg structures in operating conditions and transit conditions in a year are Dop and Dts, respectively. The damage in operating duration with n of different water depths can be expressed by the formula as below: And the total damage of the hotspot, Dtot, is calculated: whereas, i is a fraction of fatigue life at i th water depth condition (i=1÷n) and β is the average fraction of fatigue life of transit condition in a year. Due to the independent damage and linear relationship, the safety factor is equal to Equations 7 and 8, αi and β are satisfied the equation: The total fatigue life of the hotspot in a year can be calculated as: The suggested formulas will be used to estimate the total fatigue life of a jack-up leg structure in Vietnamese conditions as below.

Input Data
In the case study, we perform fatigue analysis of jack-up platform Tam Dao 05 which is operating in Thien Ung field of Vietnam. Input data are summarized in Table 1: Statistic wave data for fatigue analysis in operating conditions: Wave scattering diagram is from the following report "Metocean criteria and statistics 8 to 9 o N, 107 to 109 o E, Offshore South Vietnam. V1.0 for the Thien Ung Project Feed Design" [14][15][16]. The distribution of waves in accordance with the directions is given below (Table 2, Figure 5). In transit conditions, fatigue wave data for all directions can be assumed: Wave height H = 3m; Period T = 6s.

Structural Modeling
TD-05 jack-up platform structure can be modeled by SACS software [17], including: Mass, Hull and Legs models. There are two models corresponding to operating conditions ( Figure 6) and transit conditions (Figure 7). In operating conditions, hull structures are modeled by an equivalent system with equivalent stiffness according to DNV-RP-C104 (2012) [2]. The structure is analyzed by the Finite Element Method. In transit conditions, hull structures are modeled by an absolute stiffness body with similar geometry to analyze the interaction between waves and large bodies [18][19][20][21]. The structural motions are determined by Boundary Element Method. Then the motions are used as input data to analyze jack-up leg structures by Finite Element Method.

Joint Local Modeling
Joints of jack-up legs are special structures, so there is no formula to determine local stresses in current standards, so stress concentrated factor (SCF) of joints must be found by Local Finite Element Models. In the paper, the joints are modeled by ANSYS software [22]. A typical joint and concentrated stresses are illustrated in Figure 8. According to the analyses, the maximum SCF corresponding to axial force, in-plane bending and out-plane bending are 2.55, 2.43 and 2.33, respectively.

Natural Vibrations
Natural vibrations of jack-up structures are analyzed in relation to 3 water depths in operating conditions as the procedure in figure 3. The first period of the vibrations are 9.13s, 9.32s and 10.42s. The vibration shapes are illustrated in Figures 9, 10 and 11, respectively.

Fatigue Analysis and Results
Fatigue damage of the structural hotspots is analyzed by SACS software. In operating conditions, wave data is given in section 5.1 with Pierson-Moskowitz Spectrum. Joints with the most damage are shown in Figure 15. A typical hotspot stress spectrum and maximum damage of joint 0283 of member 0275-0283 is presented in Figure 16.
In transit conditions, wave data is given in section 5.1 as above with Pierson-Moskowitz Spectrum. Joints with the most damages are shown in Figure 17. A typical hotspot stress spectrum and maximum damage of joint 1009 of member 1009-1031 is presented in Figure 18. Herein, we will choose 6 typical joints to analyze total fatigue life: 3 joints with maximum damage in operating condition and 3 joints with maximum damage in transit condition fatigue life. The results are listed in Tables 3 and 4 as below:

Total Fatigue Life Estimation
Total fatigue lives of the above hotspots of the leg structures will be determined according to the traditional method and suggested method depending on different fraction factor scenarios in six cases as below: Case 0: For the traditional method, the fatigue damage of a hotspot in operating conditions is determined by an average of the damage corresponding to water depths. Besides, the design life when skipping transit time is equal to 80% design life = 25×0.8 = 20 years.

Discussions
Based on the analysis results, we draw diagrams to express the variation of fatigue life of joints in each case ( Figure  19) and the variation of fatigue life of each joint in Case 0 (Tradition Method) and 5 cases with different fraction factors ( Figure 20). Figures 19 and 20, there are some recommendations:  There are some results that are not reasonable when analyzed by the traditional method:

According to results in tables and
o Joints 3014 and 2012 have high fatigue life. It is caused by only considering operating conditions. Besides, these joints are located below, near the spud-cans. In fact, these joints will have significant damage in transit conditions.
o Joints 0283, 0268, and 0253 have fatigue life which is smaller than those in comparison with reality. In fact, these joints have trivial damage in transit conditions, but they are always considered as having a certain damages 5 years when analyzed by an approximate method.
 Based on the results in case of analyzing by suggested method, there are some points that can be seen as below: o Fatigue damage is almost maximum at connecting joints between legs and guides.
o Joints 0283, 0268, 0253 are almost not affected by fatigue in transit conditions. It can be explained that in these conditions, they are on the top level of the legs, so internal forces are small. Whereas, fatigue damages of joints below such as 1009, 3014, 2012 in transit conditions are significant.
o In operating duration, jack-up platforms may be worked in some areas with different water depths, so total fatigue damage is depended on locations and percentage of exploitation at each location. The fatigue damage tends to increase accordingly with water depths.  Algorithms in Figures 2 and 4 have been established into specialized software programs. When using software programs, it is easy to determine the fatigue damage (see Tables 3 and 4). Fatigue life (Tables 6 to 10) is just the application of Equation 10 in specific conditions.
 The proposed formula for assessing the overall fatigue life of the hot spot of the jack-up leg structure was also considered under the condition that the jack-up rig operates in different water depths and each jack-up rig has different transit times. The above factors are mentioned by the coefficients: i is a fraction of fatigue life at i th water depth condition (i=1÷n) and β is the average fraction of fatigue life of transit conditions in a year.
 The theoretical research results of this paper have solved all the limitations of the approximation method that the paper has set out.
 The research results of this paper have been applied in the design and manufacture of the jack-up rig TAMDAO 05 in Vietnam. The jack-up rig TAMDAO 05 has received the certificate of ABS registration. Currently, the jackup rig TAMDAO 05 is operating well, which is a testament to the correctness of the research results presented in this paper.

Recommendations
 Fatigue damage of jack-up structural legs in transit conditions is significant and not the same at different locations. Besides, it is dependent on environmental conditions in the moving area, so fatigue life, taking into account fatigue damage in transit conditions by reducing 20% of design life, is only a reference value in design.
 The fatigue damage in operating conditions and transit conditions usually changes in opposite directions, along with leg structures. Therefore, it is necessary to calculate total fatigue damage at every hotspot of the structures in a total of operating and transit durations. Based on the calculation results, the hotspot with the maximum damage will be found.
 The fraction factors of different working conditions in Equations 7 to 9 have a significant influence on the total fatigue life of jack-up leg structures, so it is necessary to study their sensitivity to make the best operating plans for jack-up platforms in their life cycle next time.
 The research results in this paper will be used as a reference for engineers to design or evaluate the foot structure of jack-up rigs in Vietnam in the future.