1 Introduction

The effects of mean stress on fatigue strength of welded structure are commonly understood to be not so large due to the existence of large tensile weld residual stresses, and many fatigue test results show this fact. Accordingly, practical fatigue procedures adopted in the design standards for bridges and offshore structures consider the stress range only [1–3] or consider the effect of mean stress conservatively [4, 5]. It is also generally accepted that the effect of material strength on fatigue strength of welded structures is negligibly small even though an effect of material strength on fatigue strength is recognized on a base material. The design standards of welded structures do not consider the difference of material strength [1–5].

Many instances of fatigue damage in the shipping industry occurred in the side longitudinal stiffeners of single-hull very large crude oil carriers (SHVLCCs), and they showed quite distinguishing tendency [6]. The SHVLCC had crude oil tanks (COT) and water ballast tanks (WBT) as side tanks. The occurrence tendency of fatigue damages in side longitudinal stiffeners in both tanks were completely different. The difference of the occurrence tendency of fatigue damages in both tanks were caused by different internal pressures according to the purpose of the tanks. Subsequently, the effect of mean stress on fatigue strength of welded joints of ship structure was investigated; however, there was no effective fatigue test data to verify the effect of mean stress on fatigue strength.

Recently, many experimental studies on the relationship between residual stress and fatigue strength have been carried out. It was revealed that the weld residual stress is easily changed by loading of few times [7–9]. After relaxation of residual stress from overloading, the effect of stress ratio as well as the effect of material strength that were not observed in as-welded joints were recorded [9]. These results indicate that the effect of mean stress on fatigue strength should be considered together with residual stress. Simultaneously, many fatigue tests of HFMI-treated joints and as-welded joints were carried out to investigate the advantage of HFMI-treated joints [10–25] focusing on the relation with residual stress.

This study examines the fatigue evaluation method which considers the mean stress, the residual stress and the material yield strength. Fatigue test data of HFMI-treated joints and as-welded joints were evaluated to confirm the effectiveness of the evaluation method. This method was then applied to the actual fatigue damages in the end connection of side longitudinal stiffeners of midsize double hull tanker in order to confirm the effectiveness.

2 Damage experience

2.1 Report of fatigue damage

Fatigue damages experienced in Japanese SHVLCCs built after 1985 were reported [6]. All of the subject ships were designed and built by Japanese shipbuilders and operated by Japanese shipowners. Therefore, the size and the structural arrangement of SHVLCCs were almost the same. All SHVLCCs were exclusively operated between Japan and the Persian Gulf. Figure 1 shows the typical tank arrangement of SHVLCC. Generally, SHVLCC had four segregated WBTs as illustrated.

Fig. 1
figure 1

Typical tank arrangement of single hull very large crude oil carrier

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According to the report [6], a total of 76 SHVLCCs were checked, and fatigue damage was found in 56 SHVLCCs. The large majority of fatigue damage occurred in the end connections of side longitudinal stiffeners with transverse bulkheads and transverse webs. Figure 2 shows the illustrations of typical fatigue damage, which occurred in the connection between transverse bulkhead and transverse web.

Fig. 2
figure 2

Typical example of fatigue damage in end connection of side longitudinal stiffener

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2.2 Damage statistics and remarks

The dominant load component acting through the side longitudinal stiffeners is wave induced external pressure. Since the distribution of wave pressure along the longitudinal direction is considered to be constant, fatigue damage in six side tanks (two WBTs and one COT per broadside) in the mid region were investigated. Fatigue damage of the side longitudinal stiffeners is influenced by the structural mean stress due to hydrostatic pressure and the distribution of the fluctuated wave external pressure along the ship side. Therefore, the damage statistics were summarized according to the vertical location of the stiffener as shown in Fig. 3. The vertical location is divided into nine regions, that is, three regions between the upper deck and the position at full load water line (FLWL), three regions between the position at FLWL and the position at ballast water line (BLWL) and three regions between the position at BLWL and the bottom of the ship.

Fig. 3
figure 3

Comparison of frequency distribution of fatigue damage in each vertical region of COT and WBT [6]

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Figure 3 shows the average frequencies of fatigue damage per COT or WBT in each vertical region. The amount of fatigue damage in each vertical region in each tank was normalized by the total number of fatigue damage instances in COTs of both sides.

According to the results shown in Fig. 3, the tendencies of fatigue damage occurrence in COT and WBT are completely different although the fluctuated wave external pressure acting on the both tanks are same. In COT, the majority of fatigue damages occurred between 2 and 5 m below FLWL in which the external wave pressure range is the maximum. On the other hand, in WBT, fatigue damage scarcely occurred in this region but occurred in the region below BLWL.

3 Investigation of damage case

Direct load calculations, by which the stress response of each load component was estimated and combined, of a sample ship along the operating route were carried out to understand the stress conditions of side longitudinal stiffeners made from AH36 (rule specified yield strength is 355 MPa). The hot spot stress was obtained by combining the stress components of external pressure, internal pressure and hull girder moments. The long-term probability distribution of hot spot stress range and the static hot spot stress of each connection of side longitudinal stiffener with transverse web were calculated. Figure 4 shows the distribution of expected values in the hot spot stress range, which is obtained by the long-term probability distribution of hot spot stress range, and static hot spot stress along the ship side. The heights of FLWL and BLWL of this sample ship were at 19.48 and 8.16 m, respectively. In these figures, “COT-full” denotes the stress in side longitudinal stiffener in COT at full loaded condition when COT is filled. “COT-ballast” denotes the stress in side longitudinal stiffener in COT at ballast condition when COT is empty. “WBT-full” denotes the stress in side longitudinal stiffener in WBT at full loaded condition when WBT is empty. “WBT-ballast” denotes the stress in side longitudinal stiffener in WBT at ballast condition when WBT is filled. At the end connection of side longitudinal stiffener, external pressure induces compressive stress, and internal pressure induces tensile stress.

Fig. 4
figure 4

Distributions of expected values of hot spot stress range (left side) and static hot spot stress (right side) at end connection of side longitudinal stiffeners along the ship side

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According to the results shown in Fig. 4, large stress ranges are acting on the side longitudinal stiffeners in both COT and WBT in the full loaded condition, and the magnitude of stress range in both tanks are almost same. On the other hand, the static stress conditions of the side longitudinal stiffener in COT and WBT are completely different. In COT, the static stress at the full loaded condition is tension, but, in WBT, the static stress at the full loaded condition is zero or compression.

In order to understand the difference in frequency distributions of fatigue damages between COT and WBT shown in Fig. 3 according to the difference in static stress conditions between COT and WBT shown in Fig. 4, the effect of mean stress should be significant on the fatigue strength of ship structures. For the mean stress to have influence in fatigue strength, it is natural to assume that the residual stress is changing during the operation. Therefore, a fatigue evaluation method considering both the mean stress and the residual stress is necessary.

4 Fatigue evaluation method

4.1 Weld residual stress

It is generally accepted that tensile weld residual stress exists in an as-welded joint. Many studies have been carried out on the relationship between residual stress and fatigue strength [10–15]. In these studies, measurements of residual stress were carried out as well as the fatigue tests. Figure 5 shows the relationship between the weld residual stress and the yield strength of the material of butt and cruciform joints (left side) and longitudinal gusset joints (right side). In these figures, the vertical axis shows the ratio of weld residual stress to the yield strength of the material. Although the amount of data for cruciform joints was limited, the relationship between the weld residual stress and the yield strength of the material is considered to be similar to that of butt joint. On the other hand, the relationship of longitudinal gusset joint differs from those of the butt and cruciform joints.

Fig. 5
figure 5

Relationship between weld residual stress and yield strength of material of butt [7, 10] and cruciform [12] joints (left side) and longitudinal gusset [7, 11, 13–15, 24, 25] joints (right side)

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The results in Fig. 5 show that the weld residual stress increases with the increase in yield strength of the material. Regarding the butt joints and cruciform joints, weld residual stress is not so large in the low yield strength material.

Weld residual stress can easily change after loading of few times [7–9]. It is considered that the relaxation of residual stress occurs when the maximum notch stress exceeds the yield strength [8]. Data of residual stresses before loading and after loading were obtained [7, 10, 11, 13–17, 25]. In this study, the relaxation of residual stress was simply estimated based on the assumption of elastic perfectly plastic solid. Based on the simple assumption, the residual stress after loading can be estimated as below.

$$ {\sigma}_{res, a}=\left\{\begin{array}{ccc}\hfill {\sigma}_y-{\sigma}_a\hfill & \hfill; \hfill & \hfill {\sigma}_y\le {\sigma}_{res, b}+{\sigma}_a\hfill \\ {}\hfill {\sigma}_{res, b}\hfill & \hfill; \hfill & \hfill \mathrm{otherwise}\hfill \\ {}\hfill -{\sigma}_y-{\sigma}_a\hfill & \hfill; \hfill & \hfill {\sigma}_{res, b}+{\sigma}_a\le -{\sigma}_y\hfill \end{array}\right. $$
(1)

where σ res , b and σ res , a are the residual stress before loading and after loading, respectively, σ a is the applied stress, and σ y is the yield strength of the material.

The residual stress relaxation was estimated according to the two scenarios. One is that (a) the notch stress which exceeds the yield strength determines the relaxation, and the other is that (b) the hot spot stress which exceeds the yield strength determines the relaxation. Figure 6 shows the relationship between the measured residual stress after loading and the estimated residual stress after loading based on the notch stress (left side) and the hot spot stress (right side). In these figures, “B,” “C,” and “G” denote butt joint, cruciform joint, and longitudinal gusset joint, respectively. And, “AW” and “UP” denote as-welded joint and HFMI-treated joint, respectively. In the estimation of the relaxation of residual stress, the hot spot stress and the notch stress were estimated as the nominal stress multiplied by the average stress concentration factors [8, 15, 19, 24, 25].

Fig. 6
figure 6

Relationship between measured [7, 10, 11, 13–17, 25] and estimated residual stress after loading

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According to the results shown in Fig. 6, it seems that the estimation by the notch stress overestimates the relaxation of residual stress by loading. As the residual stress is the self-balanced force, stress conditions around the notch affect the re-distribution of residual stress. Stress at the notch only does not seem enough to evaluate the re-distribution of residual stress. As re-distribution of residual stress around the notch is quite a complex phenomenon, the estimation of relaxation by the hot spot stress with the assumption of elastic perfectly plastic solid seems to be effective from a practical point of view. According to this evaluation method, the amount of initial residual stress is referred to judge whether the relaxation of residual stress would occur or not. However, the amount of relaxed residual stress is determined by the relationship between the material yield strength and the applied stress irrespective of the initial residual stress. Although the relaxed residual stress state depends on the initial residual stress state, it is assumed that the relaxed residual stress used in the fatigue evaluation does not depend on the initial residual stress. However, this simple assumption is effective to evaluate the relaxed residual stress as shown in Fig. 6b.

4.2 Evaluation method

Fatigue evaluation method is developed based on the method known as “modified MIL-HDBK-5D” method [26, 27]. The equivalent stress range is defined by the hot spot stress as below. Hereinafter, the equivalent hot spot stress is simply described as the equivalent stress.

$$ \Delta {\sigma}_{eq}=\Delta {\sigma}^{\alpha}{\sigma}_{max}^{1-\alpha} $$
(2)

where α is the material coefficient. Originally, α is determined by the experiments [26]; however, in this study, α = 3/4 is assumed for the simplification to use in design. σ max is the maximum stress which is the sum of residual stress, mean stress, and stress amplitude. It is assumed that the maximum stress does not exceed the yield strength according to the assumption of elastic perfectly plastic solid. Then, in order to keep consistency with the design S-N curve, a correction factor for material yield strength is introduced.

Accordingly, the fatigue evaluation method is expressed as below.

$$ N={C}_D\cdot {\sigma}_{y0}\cdot {\left({f}_y\cdot \Delta {\sigma}_{eq}\right)}^{-4}={C}_D\cdot {\sigma}_{y0}\cdot {f}_y^{-4}\cdot {\sigma}_{max}^{-1}\cdot \Delta {\sigma}^{-3} $$
(3)
C D :

coefficient of S-N curve (N = C D  ⋅ Δσ −3) of welded joints whose structural stress concentration factor is 1

σ y0 :

yield strength of mild steel

f y :

correction factor for material yield strength defined as \( {f}_y={\left(\raisebox{1ex}{${\sigma}_{y0}$}\!\left/ \!\raisebox{-1ex}{${\sigma}_y$}\right.\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} \)

σ y :

yield strength of the material to be evaluated

Δσ eq :

equivalent stress range defined as \( \Delta {\sigma}_{eq}=\Delta {\sigma}^{\alpha}{\sigma}_{max}^{1-\alpha} \)

α :

material coefficient assumed to be 3/4

σ max :

maximum stress defined as \( {\sigma}_{max}= \min \left\{{\sigma}_y, \max \left({\sigma}_{res}+{\sigma}_{mean}+\raisebox{1ex}{$\Delta \sigma $}\!\left/ \!\raisebox{-1ex}{$2$}\right.,-{\sigma}_y+\Delta \sigma \right)\ \right\} \)

In the case of σ max  ≤ 0, fatigue life is assumed to be infinite.

σ res :

residual stress

σ mean :

mean stress

This method was applied to the fatigue data of as-welded joints and HFMI-treated joints obtained from the literature to verify the adequacy of the method. Where numerical data were not indicated, data was retrieved from the data plots. Table 1 shows the summary of obtained fatigue test data.

Table 1 Fatigue test data obtained from the literature
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In the application, the mean S-N curve of the butt joint (D curve, C D  = 1012.6007) presented by UK-HSE [2] was used as the basic S-N curve, where σ y0 = 280 MPa was assumed. The residual stress in as-welded joints was assumed according to the mean lines indicated by the black line in Fig. 5 for each type of joint. In regard to the residual stress of the HFMI-treated joints, the compressive residual stress of −200 MPa was assumed without relation to the joint type and the yield strength of the material, since there was no effective data that suggested any dependency on yield strength and joint type. The hot spot stress concentration factors for butt joint, cruciform joint and longitudinal gusset joint were set as 1.0, 1.20, and 1.47, respectively. As per the rule, a thickness correction factor is introduced to consider the difference in fatigue strength from the fatigue strength by the design S-N curve whose standard thickness is 22 mm in case of UK-HSE’s. Although the minimum thickness of 22 mm was used in the rule for the conservative consideration, the majority of thicknesses of fatigue test specimens were less than 22 mm as shown in Table 1. The thickness effect on fatigue strength was considered according to the correction factor in IACS CSR-H [28].

Figures 7, 8, and 9 show the S-N relation of butt joints, cruciform joints and longitudinal gusset joints respectively. In these Figures, (a) and (b) show the S-N relation according to the nominal stress range and the equivalent stress range, respectively. The red marks and the black marks show the S-N plots of the as-welded joints and the HFMI-treated joints. The mean and the mean minus 2 times standard deviation S-N relations of all data (irrespective of weld toe treatment) are indicated by the solid line and the broken line in these Figures. Table 2 summarizes the obtained S-N relationships.

Fig. 7
figure 7

S-N plots of butt joints according to the nominal stress range and the equivalent (hot spot) stress range

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Fig. 8
figure 8

S-N plots of cruciform joints according to the nominal stress range and the equivalent (hot spot) stress range

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Fig. 9
figure 9

S-N plots of longitudinal gusset joints according to the nominal stress range and the equivalent (hot spot) stress range

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Table 2 S-N relationship (N = CΔS −m) according to the nominal stress range and the equivalent (hot spot) stress range
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As shown in Figs. 7a, 8a, and 9a, according to the nominal stress range, the group of S-N data of as-welded joints and HFMI-treated joints are plotted separately showing the improvement of fatigue strength by the HFMI-treated. This means that the fatigue assessment methods for as-welded joints and HFMI-treated joints should be prepared separately.

On the other hand, as shown in Figs. 7b, 8b, and 9b, according to the equivalent stress range, the group of S-N data of as-welded joints and HFMI-treated joints are plotted closer to coincide with each other. And the scatter of fatigue data due to the differences of residual stress, stress ratio, and material strength becomes small. These results show that the method, which considers the mean stress together with the change in residual stress and the material yield strength, can evaluate fatigue strength sufficiently. Fatigue strength of both as-welded joints and HFMI-treated joints of various mean stresses and material yield strengths can be evaluated simultaneously.

5 Application to the fatigue damage of ship structures

5.1 Consideration of change in residual stress

As shown in the statistics of fatigue damage, which occurred in COT and WBT of SHVLCC, and the evaluated stress conditions at end connection of side longitudinal stiffeners in both tanks, the difference of mean stress should have a significant effect on fatigue strength of ship structures that are subject to the large change of static stress condition according to the loading conditions of the tank. The change in static stress continues alternately during the long life service. Therefore, the stress cycles including the static stress in a first voyage of ballast or full load condition can be considered to be the pre-loading to the continued voyage of the other condition.

Since the residual stress changes according to the loading, the estimation of change in residual stress is necessary in order to consider the effect of mean stress on the fatigue strength. As shown in Fig. 6, the estimation of relaxation of residual stress based on the hot spot stress with the assumption of elastic perfectly plastic solid is a simple and practical procedure. Let the residual stress before loading be σ res , i − 1, and the mean stress and the maximum stress range in the i-th loading condition be σ mean , i and Δσ max , i . The residual stress in the i-th loading condition is estimated as follows. When the maximum stress or the minimum stress exceeds the yield stress in tension side or in compression side, the residual stress after loading is obtained as σ res , i . If the next continuing loading condition exists, σ res , i is to be the residual stress before loading for the next (i + 1)-th loading condition.

$$ {\sigma}_{res, i}=\left\{\begin{array}{ccc}\hfill \min \left\{{\sigma}_{res, i-1},{\sigma}_y-\left({\sigma}_{mean, i}+\raisebox{1ex}{$\Delta {\sigma}_{\mathit{\max}, i}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\ \right\}\hfill & \hfill; \hfill & \hfill {\sigma}_{mean, i}\ge 0\hfill \\ {}\hfill \max \left\{{\sigma}_{res, i-1},-{\sigma}_y-\left({\sigma}_{mean, i}-\raisebox{1ex}{$\Delta {\sigma}_{\mathit{\max}, i}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\ \right\}\hfill & \hfill; \hfill & \hfill {\sigma}_{mean, i}<0\hfill \end{array}\right. $$
(4)

5.2 Numerical example

Fatigue strength of end connections of side longitudinal stiffeners of a damaged ship is evaluated. The subject ship is the midsize double-hull tanker which has seven side wing tanks. According to the operating records, No.5 tank was used as the usual WBT. That is, the tank was filled with ballast water in the ballast condition and was empty in the full load condition. On the other hand, No.7 tank was used as the usual WBT in the half period of service, however this tank was filled with ballast water in the other half period of service even though it was full load condition. Figure 10a shows the frequency distribution of damages at 10 years service along the shipside which is normalized by the number of end connections per longitudinal stiffener in a tank. The tendencies of occurrence of fatigue damages in No.5 tank and No.7 tank are different.

Fig. 10
figure 10

Comparison between frequency distribution of fatigue damage and distribution of calculated cumulative fatigue damage

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Direct load calculations were carried out to obtain the long-term probability distribution of stress range, and the cumulative fatigue damage at 10 years service life was calculated based on the UK-HSE’s mean D curve. Hot spot stress concentration factors were considered for each type of end connection. The side longitudinal stiffeners were made from AH36. The weld residual stress in the joints was assumed to be 250 MPa.

Figure 10b shows distributions of calculated cumulative fatigue damage along the shipside in No.5 tank and No.7 tank. The distribution of calculated cumulative fatigue damage along the shipside shows good correlation with the frequency distribution of the fatigue damage shown in Fig. 10a. By considering the effect of mean stress on fatigue strength according to the equivalent stress range, the fatigue damage that occurred in the ship structure could be well explained.

6 Conclusions

According to the damage statistics regarding the end connection of side longitudinal stiffener of SHVLCCs, the effect of mean stress on fatigue strength seems significant. For ship structures, the static mean stress conditions are usually changing alternatively from tensile side to compressive side depending on the loading conditions during operation. Therefore, it is necessary to consider the effect of mean stress appropriately.

In this paper, the fatigue evaluation method, which considers the mean stress together with the change in residual stress and the material yield strength, is examined. In order to verify this method, it was applied to the existing fatigue test data of as-welded joints and HFMI-treated joints. It was confirmed that the group of S-N data of as-welded joints and HFMI-treated joints were plotted closer to coincide with each other. Also, the scatter of fatigue data due to the differences of residual stress, stress ratio and material strength were reduced. These results show the effectiveness of the method.

The fatigue strength of end connections of side longitudinal stiffeners of midsize double-hull tanker was evaluated considering the past record of usage of the tanks which affected the static mean stress conditions. The results of calculated cumulative fatigue damage for each end connection of side longitudinal stiffener showed good correlation with the tendency of fatigue damage occurrences. The fatigue evaluation method which considered the effect of mean stress was demonstrated in its application to the actual fatigue damages of ship structures subject to the change in mean stress conditions.

However it should be noted that fatigue strength is quite sensitive to the mean stress including residual stress. The evaluation of static mean stress and residual stress should be closer to the actual stress for the effective fatigue evaluation when applying the method based on the equivalent stress.