According to the current trend of larger size container ship, the effect of hull-girder vibrations such as springing and whipping on the fatigue strength becomes the main concern of classification societies as well as ship structural designers. This is because the wave-induced hull-girder vibrations are likely to occur due to relative decrease of flexural rigidity of larger ships, and this fact has raised fears that the high frequency stress components of vibration increase the fatigue damages. In spite of the fear, however, serious damages attributed to such hull-girder vibrations were rarely reported in well-maintained properlyoperated ships up to now.

The fatigue strength of a ship is usually evaluated by using Miner’s rule. Miner’s rule is based on the linear cumulative law and the fatigue damage is considered to be proportional to the ratio of the number of applied stress cycle to that to cause failure at any stress range. This means that the small amplitude components are still effective on the fatigue damage, particularly in the case where the number of occurrence is large enough.

The fatigue strength has well been estimated by Miner’s rule in a practical sense so far. This is because the calibration of the method using accumulated experiences and past data was rightly done. However, the applicability of Miner’s rule to the case where the smaller amplitude higher frequency stress components are superposed on the larger amplitude lower frequency stress components is not clear at the moment. Actually, through fatigue tests and numerical simulations, it was shown recently that the smaller amplitude higher frequency stress component does not affect so much as is expected (Matsuda et al., 2011; Gotoh et al., 2012; Kitamura et al., 2012; Sumi, 2012).

In the present paper, the effects of hull-girder vibration upon fatigue strength of a Post-Panamax container ship are investtigated. Nonlinear time-domain simulations of ship response are performed in short-term sea states, and the obtained stress histories are processed by means of the rainflow counting method. The cumulative fatigue damage factors are then evaluated by using Miner’s rule disaggregated by short-term sea state. Through the calculated results, the characteristics of fatigue damage factor estimated by using Miner’s rule is discussed in the case where the smaller amplitude higher frequency stresses are superposed on the larger amplitude lower frequency stresses.

Assuming that the lifetime of a ship is 25 years and the mean duration of a short-term sea state is roughly 3 hours, the occurrence probability of a short-term sea state is given by (Fukasawa et al., 2007).

Eq. (1) means that the total number of short-term sea states which a ship encounters in her lifetime can be considered to be the order of 10⁵. The occurrence probability of a short-term sea state is given in a wave scatter diagram such as the IACS North Atlantic wave scatter diagram.

In order to categorize the short-term sea state according to the occurrence probability, the following variable K is introduced.

where Q is the occurrence probability of a short-term sea state. K is to be rounded to unit for convenience sake. As the order of occurrence probability of a short-term sea state is written as 10^-K, K can be used as an index of the occurrence probability of the short-term sea state. By using the IACS North Atlantic wave scatter diagram, K in each short-term sea state was calculated and is shown in Table 1 up to K = 5 because the total number of short-term sea states which a ship encounters in her lifetime can be considered to be the order of 10⁵. The fatigue damage factors of a ship in the short-term sea states of K = 1, 2, 3, 4, 5 in Table 1 are investigated in the present paper.

The effect of hull-girder vibrations upon fatigue damage of a ship will be investigated in the present paper. A Post-Panamax type container ship is used in the study, of which particulars are L × B × D - d = 280.0m × 42.8m × 24.5m - 14.20m. The ship speed is taken to be V = 15.27knots (Fn = 0.15) taking account of the speed reduction in rough seas.

Ship responses in waves are more or less nonlinear and the strongly nonlinear phenomenon such as slamming occasionally occurs in rough seas. Moreover, hull-girder vibrations such as springing and whipping come to be significant with the decrease of flexural rigidity caused by scale-up of the ship. A computational code TSLAM, therefore, is used in the present paper in the calculation of nonlinear ship response in irregular waves because of its robustness in strong nonlinear phenomena. TSLAM is based on the time-domain nonlinear strip theory and the nonlinearities with respect to slamming impact and resulting whipping vibrations can be taken into account (Yamamoto et al., 1983). The hull-girder of the ship is considered to be both a rigid body and a flexible body to clarify the effect of vibratory stress component of the ship upon fatigue damages. As for the structural damping, the logarithmic decrement of vibration was taken as 0.05 for the flexible ship by reference to measured results in actual ships.

Numerical simulations are performed to calculate the stress due to longitudinal vertical bending moment at midship in irregular waves. The calculated vertical bending moment is converted to the hull-girder hot-spot stress with the stress concentration factor of 1.5. Numerical simulations are carried out over 3 hours in each short-term sea state, because the mean duration of a short-term sea was assumed to be 3 hours.

A ship in service experiences complex load histories and the resulting stress amplitude is not constant, which bears little resemblance to the constant amplitude fatigue test. In order to apply Miner’s rule in the assessment of the fatigue life of a ship, the complex stress time history have to be broken down into recognizable stress cycles by using a ‘cycle counting method’. The cycle counting is a procedure which is carried out to convert a complex stress history into a sequence of identifiable stress cycles; the complex stresses are divided into blocks of constant amplitude, and the service stresses are expressed in terms of numbers of complete cycles at particular stress levels.

The rainflow counting method is the most popular counting technique in the analysis of fatigue data, where a stress spectrum is reduced into a set of simple stress reversals (Matsuishi et al., 1968). The rainflow algorithm can be explained by an analogy to the flow of rainwater over a series of rooftops. The water flows down from each turning point and continues until either it is interrupted by flow from above or it reaches a turning point which is larger than the one that started. By using the rainflow counting method, the stress range against number of cycles occurring at that range can be obtained.

The fatigue life of a structural member subjected to the stress sequence is then estimated by summing up the calculated fatigue damage arising from each block of cycles using a cumulative damage rule.

The damage rule which is most widely used nowadays is Palmgren-Miner linear cumulative damage rule, or simply called as Miner’s rule. This rule states that the fatigue damage from any particular stress range is proportional to the ratio of the number of cycles applied at that stress to the number of cycles which is required to cause failure. The cumulative fatigue damage factor D, therefore, is defined as the sum of all the ratios, that is,

where n_i is the number of cycles applied at the i-th stress range and N_i is the number of cycles to failure at the i-th stress range. It is assumed that the failure occurs under a sequence of different stresses when the fatigue damage factor equals unity.

The relationship between nominal stress range and number of cycles to failure for any particular type of loading can be established by a series test under the constant amplitude loading conditions of identical specimens in the laboratory, and the S-N curve is produced from the test results. The fatigue endurance of a structural member subjected to a particular cyclic stress can be estimated by using the S-N curve for that geometry of the structural member. The general equation relating the applied stress range and fatigue life is given by

where Δσ_i is the i-th nominal stress range and N_i is the number of cycles to failure in the i-th stress range. The parameters m and c are the slope and the constant which refers to the S-N curve respectively. These parameters are taken as m = 3 and c = 1.513 × 10¹² in the present paper according to the classification society’s rule provided by ClassNK (Nippon Kaiji Kyokai, 2003).

Miner’s rule proves to be reasonably accurate if the applied stress spectrum resembles narrow-band random loading (Maddox, 1991). The significant characteristic of such a spectrum is that the stress range changes gradually, with the results ‘crack growth retardation’ is not significant. In the case of a ship, the wave-induced slowly-varying stress spectrum can be considered to be narrow-banded in a short-term sea state. The fatigue strength due to such stresses has well been estimated by Miner’s rule in a practical sense, with the aid of the calibration of the method using accumulated experiences and past data. However, as a ship is supposed to encounter a lot of different short-term sea states, the stress spectrum is unlikely to be narrow-banded in the long-term sense. Moreover, the stress spectrum is no more narrow-banded when the hull-girder vibrations such as whipping or springing occur. The applicability of Miner’s rule has not stringently been verified in this case.

The stress history of the ship in each short-term sea state was firstly calculated by means of nonlinear simulations of ship response. The calculated stress peaks were processed by the rainflow counting method, and the stress ranges against number of cycles occurring at that ranges were obtained. The fatigue damages were then estimated by applying Miner’s rule in each shortterm sea state.

Firstly, the effect of the hull-girder vibrations to the fatigue damage factor is investigated in the disaggregated short-term sea state. For this purpose, the calculated results assuming the ship’s hull to be a rigid body are compared with the results assuming that to be a flexible body. The rigid ship is the rigid limit of the hull and there is no vibration expected.

Table 2 shows the rounded ratio of the fatigue damage factor in flexible body assumption to that in the rigid body assumption, where the ship is travelling in each short-term sea state for 3 hours. It can be seen from the table that the ratio is large around the sea state of Hs = 7.5m and Tz = 6.5s, which is comparatively shorter wave period region. These sea states are in the category of K = 4 or 5 defined by Eq. (2). The larger fatigue damage factor might have been caused by significant hull-girder vibrations, which may be attributed to severe whipping vibrations due to bow-flare slamming. It should be noted here that the occurrence probability of these sea states is very low, although the ratio of fatigue damage factors is large.

As was mentioned before, the total number of occurrence of short-term sea states in the lifetime of a ship can be considered to be the order of 10⁵. In the present paper, therefore, the total number of short-term sea states a ship encounters is assumed to be N_total = 100,000. In order to clarify the contribution rate of each short-term sea state of occurrence probability of 10^-K, the cumulative fatigue damage factors are calculated in the case where the ship encounters the short-term sea state 100,000 times in total according to the occurrence probability given by the wave scatter diagram.

Firstly, in the case where the ship’s hull is assumed to be a rigid body, the fatigue damage factor in each short-term sea state is shown in Table 3. The largest fatigue damage can be seen in the short-term sea state of Hs = 5.5m and Tz = 9.5s, where the fatigue damage factor is 0.044 to be exact. As the total fatigue damage factor in 100,000 sea states is 0.968 in this case, it seems that the fatigue damage factor can well be estimated by Miner’s rule in the case where the ship’s hull is assumed to be a rigid body.

On the other hand, the fatigue damage factor in each short-term sea state is shown in Table 4, where the ship’s hull is assumed to be a flexible body. Comparing Table 4 with Table 2, the wave period of the sea state where the fatigue damage factor becomes larger is shifted to the longer wave period side due to the occurrence probability of the sea state, and the largest fatigue damage factor appears nearly in the same short-term sea state as that of the rigid body assumption case shown in Table 3. However, the fatigue damage factors are increased due to the flexibility of the hull; that is, the largest fatigue damage factor is 0.098 to be exact in the sea state of Hs = 6.5m and Tz = 9.5s, and the total fatigue damage factor in 100,000 sea states is 2.172 in this case. Comparing Table 4 with Table 3, the fatigue damage factor of flexible ship nearly doubles that of rigid ship almost evenly in each short-term sea state. This may be caused by the effect of hull-girder vibrations, which can be confirmed in Fig. 1, where the time histories of hull-girder stress are shown in the case of both rigid body assumption (top figure) and flexible hull assumption (bottom figure) in the short-term sea state of Hs = 6.5m and Tz = 9.5s. The whipping vibrations due to bow-flare slamming are significant in the flexible ship.

The cumulative fatigue damage factors in each short-term sea state category of the occurrence probability of 10^-K are shown in Fig. 2 for both the rigid body assumption case and the flexible body assumption case. It can be seen from the figure that the short-term sea state of the occurrence probability of 10^-2 contributes largely to the fatigue damage factor among the categorized short-term sea states. It is also seen that the fatigue damage factor is largely affected by the hull-girder vibration almost evenly in each short-term sea state category.

The fatigue damage factor is larger in the short-term sea state category of occurrence probability of 10^-2 than that of other sea state categories as was shown in Fig. 2. The reason of this result will be considered in the followings.

As the S-N curve falls to a straight line by using logarithmic scales, the S-N curves are usually plotted in a double logarithmic chart of stress range and fatigue life. The long-term prediction results of the applied stress, on the other hand, are usually plotted in a semi-logarithmic scale. For a better understanding of the relationship between the long-term prediction result and the S-N curve, it would be convenient that the S-N curve is plotted in the same semi-logarithmic chart as the long-term prediction result. Fig. 3 shows the comparison of these curves, where the abscissa is the wave-induced linear stress range and the ordinate is the logarithm of the number of stress cycles. In the fatigue strength estimation, the estimated number of stress cycles is compared to the number of cycle to failure given by the S-N curve. This leads to that the stress range where the long-term prediction curve comes closer to the S-N curve has a profound effect on the fatigue damages. According to Fig. 3, the stress range around 100MPa has a significant effect on the fatigue damage in this context in the case of this container ship. The occurrence probability of this stress range is about 10^-2 according to the long-term prediction results of wave-induced stress range. This fact was reported in the following papers (Derbanne et al., 2011; Fukasawa, 2012). This means that the stress range of small amplitude, which occurs with comparatively greater frequency, is dominant in the fatigue damage if Miner’s rule is applied.

In the next place, the contribution rate of the stress range to the fatigue damage factor is investigated. The histograms of the hull-girder stress range and the corresponding fatigue damage factors in the short-term sea states of significant wave height Hs = 6.5m and average zero up-crossing period Tz = 9.5s are shown in Fig. 4. The generated stress range is mostly below 100 MPa in this case, and the fatigue damage factor is increasing with the increase of the stress range. On the other hand, the histograms of the hull-girder stress range and the corresponding fatigue damage factors in the short-term sea states of significant wave height Hs = 13.5m and average zero up-crossing period Tz = 9.5s are shown in Fig. 5. The stress range over 100MPa is frequently observed in this case, and the fatigue damage factor has a peak around the stress range of 100 MPa. This is interprettable as the effects of the abovementioned stress range of 100MPa shown in Fig. 3; that is, the fatigue damage factor is increasing with the stress range in case the stress range is below 100MPa, while it is decreasing with the stress range in case the stress range is over 100MPa. It can also be seen in Figs. 4 and 5 that the fatigue damage factors are quite small in the small stress range although the number of occurrence of stress peak is large. To summarize this point, if the stress range around 100MPa is generated by the hull-girder vibration, the fatigue damage factor increases significantly. This is an inherent characteristics of Miner’s rule, but it is very important to confirm whether this phenomenon is observed or not in the actual fatigue damage of a ship.

The fatigue crack propagation behaviors under variable amplitude loading with different frequency components are investigated recently by means of fatigue testing and numerical simulations (Matsuda et al., 2011; Gotoh et al., 2012; Kitamura et al., 2012; Sumi, 2012). According to the results of such research works, the smaller amplitude component of higher frequency superposed on the larger amplitude component of lower frequency has not significant effect on the fatigue damage, but only the enlargement effect of the total amplitudes of the superposed stress is effective. This means that the smaller amplitude component of stress range does not affect the fatigue strength so much even if the number of cycles is large, but the enlargement effect in stress range by the smaller amplitude component of higher frequency is important in the fatigue strength. This effect can roughly be demonstrated by counting zero crossing peaks, rather than local peaks, in the rainflow counting method.

The fatigue damage factor in each short-term sea state obtained by using the zero crossing peak counting method is shown in Table 5, where the ship’s hull is assumed to be a flexible body. The largest fatigue damage factor is 0.094 to be exact in the sea state of Hs = 6.5m and Tz = 9.5s, and the total fatigue damage factor in 100,000 sea states is 1.995. Comparing the results in Table 5 with that in Table 4, the fatigue damage factors obtained by using the zero crossing peak counting method are not so much different from that obtained by using the local peak counting method.

In order to confirm this result, the ratio of the fatigue damage factors obtained by the local peak counting method to that obtained by the zero crossing peak counting method in each short-term sea state is shown in Table 6. It can be seen from the table that the ratio is almost 1 except for the lower significant wave height region.

It would appear that the increase of the number of cycle of fluctuating stress due to hull-girder vibration affects the fatigue damage only in the region of lower significant wave height. In the region of moderate or higher significant wave height, the enlargement effect of the total amplitudes of superposed stress is dominant in the fatigue damage factor, and the number of cycle of the small amplitude component has insignificant effect. This finding is consistent with the fatigue test results and the crack propagation simulation results, therefore, it can be concluded that the fatigue damage factor is reasonably evaluated by the rainflow counting method even in the case where the smaller amplitude component of higher frequency vibration superposed on the larger amplitude component of lower frequency vibration.

The cumulative fatigue damage factors in the short-term sea state category of the occurrence probability of 10^-K are shown in Fig. 6 for both the results obtained by using the local peak counting method and the zero crossing peak counting method. Significant differences caused by the cycle counting method are only seen in the short-term sea state category of higher occurrence probability; that is, K = 1 or 2. The conclusion drawn before can also be confirmed in this figure.

The crack initiation and propagation are nonlinear phenomena and there may be some limitation in the application of Miner’s rule to estimate the fatigue damage precisely, because Miner’s rule is based on the linear cumulative law. Including the behavior of fatigue damage factor around 100MPa stress range, further investigation would be necessary to clarify the effect of hull-girder vibration upon the fatigue strength of a ship by taking account of stress histories which is effective in the crack initiation and propagation. Hasty conclusions obtained by Miner’s rule by using only the load histories leads sometimes to misunderstandings on the fatigue strength of a ship.

The effect of hull-girder vibrations on the fatigue damage factor is investigated in the present paper for a Post-Panamax container ship by applying the rainflow counting method and Miner’s rule to the fatigue strength estimation. The conclusions obtained are as follows:

(1) If the ship is assumed to encounter 100,000 sea states according to the occurrence probability of the short-term sea state given in the IACS North Atlantic wave scatter diagram, the total cumulative fatigue damage factor is almost unity in the case where the ship’s hull is assumed to be rigid. In this context, Miner’s rule can be considered to be reasonably applicable to the fatigue strength design of a ship.(2) However, if the ship’s hull is assumed to be flexible, the total cumulative fatigue damage factor becomes almost double as many as that in the case of rigid body ship.(3) In the above two cases, the short-term sea state of the occurrence probability of 10-2 contributes largely to the fatigue damage factor among the categorized short-term sea states of the occurrence probability of 10-K.(4) The stress range around 100MPa has a significant effect on the fatigue damage in the present ship, and a peak can be seen in the fatigue damage factor around the stress range of 100MPa. It is important to confirm whether this phenomenon can be observed or not in the actual fatigue damage of a ship taking account of the nonlinearity in crack initiation/propagation behavior.(5) In the case where the smaller amplitude component of higher frequency vibration superposed on the larger amplitude component of lower frequency vibration, the enlargement effect of the total amplitudes of superposed stress is dominant in the fatigue damage factor, and the number of cycle of the small amplitude component has insignificant effect.(6) The fatigue damage factor is reasonably evaluated by the rainflow counting method even in the case where the smaller amplitude component of higher frequency vibration superposed on the larger amplitude component of lower frequency vibration.(7) The crack initiation and propagation behaviors are nonlinear phenomena and there may be some limitation in the application of Miner’s rule to estimate the fatigue damage precisely. Further investigation would be necessary to clarify the effect of hull-girder vibration upon the fatigue strength of a ship by taking account of stress histories which is effective in the crack initiation and propagation as well as the mean stress effect due to still water bending moment.