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Comparison of different structural stress approaches for fatigue assessment of welded ship structures
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Marine Structures 18 (2005) 473–488 www.elsevier.com/locate/marstruc
Comparison of different structural stress approaches for fatigue assessment of welded ship structures Wolfgang Fricke, Adrian Kahl Hamburg University of Technology, Ship Structural Design and Analysis, 3-06, 21073 Hamburg, Germany Received 31 October 2005; received in revised form 3 February 2006; accepted 7 February 2006
Abstract The structural stress approach, which considers the stress increase due to the structural configuration, allows the fatigue strength assessment of welded ship structures with various geometries on the basis of an S–N curve depending only on the type of weld. However, a unique definition and the numerical calculation of the structural stresses are problematic, which has resulted in the development of different variants of the approach. These are discussed and compared with each other in the present paper. The application to three examples shows the variation and differences in the analysed stresses and predicted fatigue lives, which are compared with those derived from fatigue tests. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fatigue strength; Welded joint; Ship structural detail; Structural stress; Hot spot stress
1. Introduction The initiation and early propagation of cracks at welded joints under fatigue loading is primarily determined by the local stress distribution. According to Radaj et al. [1], first attempts to correlate the fatigue strength with a local stress were made already in the 1960s by several researchers, including Peterson, Manson and Haibach, who related the fatigue strength to a local stress or strain measured at a certain point close to the weld toe, for example at a distance of 2 mm [2].
Corresponding author. Tel.: +49 40 428 786089; fax: +49 40 428 786090.
E-mail address: [email protected] (W. Fricke). 0951-8339/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2006.02.001
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The stress or strain determined in this way depends, however, of the component size or plate thickness, which has led in the 1970s to the development of the well-known hot spot stress approach with reference points for the stress evaluation and extrapolation, which are located at distances from the hot spot (weld toe) depending of the plate thickness. This allows hot spot stress concentration factors to be derived in relation to dimensionless geometry parameters. The approach was at first successfully applied to the fatigue assessment of tubular joints in offshore platforms [3,4]. Radaj [4] summarized these and other investigations and defined the structural stress at the hot spot (weld toe) as the surface stress which can be calculated in accordance with structural theories used in engineering. He demonstrated that the structural stress can be analysed either by surface stress extrapolation or by linearization, e.g. through the wall thickness. Both procedures exclude the local non-linear stress peak caused by the weld toe, whose effect is considered in the S–N curve. In the early 1990s, Petershagen et al. [5] derived a generalized hot-spot stress approach for plate structures using Radaj’s effective notch stress approach [4] and applied it to complex welded structures [6]. Detailed recommendations concerning stress determination for fatigue analysis of welded components were given by Niemi [7], Huther et al. [8] and Fricke [9], among others. A comprehensive guidance by the International Institute of Welding (IIW) is presently under preparation [10,11], where the relevant stress is termed ‘structural hot-spot stress’ to avoid confusion created by the different terms used previously (structural stress; hot-spot stress; geometric stress). In the recent years, further variants of the structural stress approach were developed. From these, particularly the approaches by Dong et al. [12,13] and Xiao and Yamada [14] are remarkable and will be described and discussed in the following chapter together with the ‘conventional’ structural hot-spot stress approach. Subsequently, the approaches will be applied to three typical structural details, where fatigue test results are available so that not only stresses, but also predicted fatigue lives can be compared. It should be noted that the approaches described here are applicable only to the fatigue strength assessment of weld toes, whereas weld roots, particularly at non-fused root faces, have to be assessed with modified or other approaches. 2. Structural stress approaches In this chapter, three approaches are briefly described and discussed in more detail. Emphasis is placed on welded plate structures being typical for ships, although they are also well-suited for welded joints in shell structures such as tubular joints. A more comprehensive description of the approaches can be found in [1]. 2.1. Structural hot-spot stress approach according to the IIW The traditional approach to derive the structural hot-spot stress is the linear or quadratic extrapolation of strains measured at two or three reference points in front of the weld toe. In the recommendations of the International Institute of Welding (IIW) [11], distances of the reference points from the weld toe of 0.4t/1.0t or 0.4t/0.9t/1.4t are recommended, where t is the plate thickness. Here it is assumed that the local stress increase due to the notch at the weld toe disappears within 0.4t. At plate edges, quadratic extrapolation over reference points at fixed distances from the weld toe (4/8/12 mm) is
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recommended, as plate thickness is not considered as a suitable parameter to define the location of the reference points at plate edges. The surface extrapolation of stresses can accordingly be applied to f.e. analyses, Fig. 1a. Alternatively, stress linearisation over the thickness leads also to the exclusion of the local stress peak in plate or shell structures, Fig. 1b. In the case of solid models, the arrangement of three or more elements over the thickness is recommended, because the stresses in the section directly below the weld toe are disturbed by the notch singularity, which affects the linearized structural stress considerably in case of only one or two elements. Systematic variation of stress analyses has shown that detailed rules for finite element modelling and stress evaluation are necessary to avoid large scatter and uncertainties particularly in connection with surface stress extrapolation [9,10,15]. Fig. 2 shows examples for the extrapolation of stresses from different kinds of models. The left part contains the application of the above mentioned reference points to relatively fine f.e. meshes, whereas the right part shows the stress extrapolation for relatively coarse models as recommended by some classification societies. It should be noted that further mesh refinement, e.g. in case a), should be performed in both directions (see example in Fig. 1a) to avoid over-estimation of stresses. The associated design S–N curves were defined on the basis of extensive evaluation of fatigue tests [16]. As stated in [11], fatigue class FAT 100 ( ¼ fatigue strength reference value in [N/mm2] at two million cycles) is recommended in normal cases for welded joints in steel structures. Exceptions are longer attachments (4100 mm) at plate edges as well as load-carrying fillet welds (due to the additional local stress concentration at the weld toe, which is not captured by the structural stress defined above), for which FAT 90 applies as demonstrated by the open symbols in Fig. 3. An alternative procedure to capture the increased stresses in load-carrying fillet welds by a bilinear stress distribution has been proposed in [17]. For welded joints at plate thickness t larger than t0 ¼ 25 mm, the well-known thickness correction on fatigue strength has to be considered with an exponent on the thickness ratio t/t0 varying from n ¼ 0.1 for welds at plate edges over n ¼ 0.2 for butt joints to n ¼ 0.3 for other joints [11]. A special problem is fabrication-related axial and angular misalignments. As measured structural hot-spot stresses are the basis for the design S–N curves, which already contain the effects of possible misalignment, these have to be explicitly taken into account in the
Fig. 1. Evaluation of structural stress at weld toe by surface stress extrapolation (a), linearisation over plate thickness (b) and equilibrium with stresses at distance d (c).
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Fig. 2. Examples of different rules for modelling and stress extrapolation.
Fig. 3. Fatigue test results for non-load and load-carrying fillet welds in terms of measured structural hot-spot stress [16].
structural hot-spot stress in the considered case. In the nominal stress approach, the effects are implicitly taken into account by the design S–N curves up to a certain amount. Usually, stresses are computed with perfectly aligned f.e. models which do not contain any pre-deformations. Their effects on the structural hot-spot stress have to be considered in plate structures particularly at butt- and cruciform joints with non-continuous loaded plates (due to possible axial misalignment) and at one-sided, transverse fillet welds (due to
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possible angular misalignment). If no information about misalignment is available, IIW [11] recommends to multiply the axial (membrane) component of the plate stress with km factors, which contain the effects of axial misalignment of 5–15% of the plate thickness (km ¼ 1.1–1.4). With these factors, the fatigue classes of the nominal and structural hotspot stress approach become compatible. 2.2. Structural stress approach according to Dong The approach with linearisation of the stress over the plate thickness was adopted by Dong and extended such that particularly the effect of the stress gradient along the anticipated crack path is taken into account using fracture mechanics [12,13]. The stress linearisation over the thickness t of a plate with one-sided weld is illustrated in Fig. 4a. In certain cases, the linearisation up to a depth t1 ot is recommended, Fig. 4b, e.g. for welds at plate edges, where t1 corresponds to the final crack length. In case of two-sided welds with symmetrical geometry and loading, a linearisation over half the plate thickness (t1 ¼ t/2) is proposed, Fig. 4c, which means a different structural stress definition compared to the approach mentioned before. Generally, the linearisation according to Dong is performed only over a monotonic decreasing stress distribution. Dong et al. propose special procedures for the computation of the structural stress, which are considered to be rather mesh-insensitive. As element stresses depend on the mesh fineness and are affected by the notch singularity at the weld toe, they should be evaluated in a distance d from the weld toe, Fig. 1c. Using equilibrium conditions, the membrane and bending portion of the stress and thus the linear stress distribution in the through-thickness section at the weld toe can be determined from the normal and shear stresses acting in the distance d. However, this procedure neglects the shear stresses at the other element faces, which causes errors in case of high local stress concentrations [15]. If the stresses are linearized over the depth t1, the stress components acting at the lower edge of the area d t1 have to be included in the equilibrium equations. As an alternative, Dong proposes to determine the structural stress from the internal nodal forces in the through-thickness section at the weld toe, as these generally satisfy equilibrium conditions. This approach is particularly well-suited for shell models, where work-equivalent line forces and moments can be computed from the nodal forces and moments along the weld toe line using the element displacement functions. The line forces and moments yield directly the structural stresses. Partial linearisation over the depth t1 is, of course, not possible here.
Fig. 4. Definition of the structural stress according to Dong [12].
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The endurable stresses or load cycles are determined from a master S–N curve using an equivalent structural stress parameter DSs, which results from the structural stress range Dss as follows [13]: m2
1
DSs ¼ Dss t 2m IðrÞm
(1)
with plate thickness t [mm], the exponent m of the Paris crack propagation law (with m ¼ 3.6 according to Dong) and the integral I(r), which depends of the ratio r between the bending portion and the total structural stress and also of the boundary conditions during crack propagation (load- or displacement-controlled). Typical curves are given for I(r) in [18,19]. The plate thickness is considered in Eq. (1) by an exponent of 0.22. The master S–N curve shown in Fig. 5 has been derived from a large number of fatigue tests, for which the structural stress according to Fig. 4 and the equivalent structural stress parameter in Eq. (1) have been analysed. Misalignment has not been considered explicitly, i.e. it affects the master S–N curve to an extent as it has been present in the tests. 2.3. Structural stress approach according to Xiao and Yamada In view of more powerful soft- and hardware, which allow the generation of finer meshes without high expenditure, Xiao and Yamada [14] have recently proposed a new structural stress approach which assumes the computed stress at a point in a depth of 1 mm below the weld toe in the direction of the expected crack path as relevant parameter for the fatigue strength. The selection of this point is verified by a reference detail, a 10 mm thick plate with transverse stiffeners on both sides, Fig. 6. Finite element computations have shown that the local stress at the weld toe of this detail decreases much faster in thickness direction than along the surface. In the latter, the local stress increase disappears in a distance of 2.5 mm, while the nominal stress is already reached in a depth of approximately 1 mm, irrespectively from the local shape of the weld toe (radius and flank angle varied in Fig. 6). Insofar, similarities exist with the approach by Haibach [2]. However, additional justification is given in [14] by showing that the 1-mm-stress is a representative load parameter for the early crack propagation phase.
Fig. 5. Master S–N curve according to Dong [20] with scatter band for two probabilities of survival Ps.
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Fig. 6. Stress distribution at the reference detail: (a) along the plate surface and (b) in thickness direction [14].
Fig. 7. Evaluation of test results for relatively thick specimens [14].
Finite element analyses require a mesh, which yields the 1-mm-stress with sufficient accuracy. It is stated that the element length should not exceed 1 mm, which will be discussed further in connection with the examples below. The approach has been applied to several types of welded joints, where the geometry is similar to that of the reference detail, i.e. longitudinal and transverse attachments on continuous plates. Fatigue test results, if plotted against the calculated 1-mm-stress, show a fairly small scatter with a lower boundary according to FAT 100. Furthermore, it has been shown that the 1-mm-stress considers the thickness effect very well, see Fig. 7. The scatter of the results is smaller than for the conventional structural hot-spot stress approach and for Dong’s structural stress approach.
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The applicability of the approach to other welded joints (e.g. cruciform joints with loadcarrying fillet welds or one-sided longitudinal and transverse attachments) and load cases (e.g. with pronounced bending) needs still to be verified. 3. Selected structural details Three structural details have been selected for the present investigation. They will be described in the following together with the computed structural stress concentration factors taken mainly from previous investigations, which are summarized in Table 1. Regarding the surface stress extrapolation, only results are considered which follow the recommendations given above. 3.1. Detail 1: one-sided doubler plate The first detail is the specimen shown in Fig. 8 with a round doubler plate on one side, subjected to tensile forces. The specimen was investigated in a Japanese research project [21]. The fatigue critical position is the weld toe on the plate oriented transversely to the loading. The circular doubler plate causes a non-uniform stress distribution in the transverse direction. In a separate unpublished study, the longitudinal stresses were measured in front of the critical weld toe and linearly extrapolated over 0.4t/1.0t, yielding a structural hot-spot stress concentration factor Khs of 1.62 based on the nominal stress in the plate. It was noticed that the specimen had an angular distortion due to the one-sided fillet welding, from which a stress magnification factor of about km ¼ 1.2 was derived. The detail was investigated also in the round-robin analysis described in [9], where different techniques of modelling the one-sided doubling plate were applied. Fig. 9 shows two typical f.e. models with 20-noded solid elements based on the meshing according to Fig. 2a and b [15]. The results were included in the evaluation together with two results from [22], comprising in total eight solid models. Also shell element models were created in [9], where different types of weld modelling by vertical or inclined shell elements or rigid links caused some additional scatter. However, the scatter of the structural stress
Table 1 Structural stress concentration factors obtained for the three details by different methods and f.e. models Structural detail
Structural stress concentration factor Ks at hot spot Khs from measurement
1. Doubler plate 2. Edge gusset 3. Stiffener on T-bar a
Khs acc. to IIW
Ks acc. to Dong
Ks for 1-mm-stress
Shell models
Solid models
Shell models
Solid models
Shell models
Solid model
1.46–1.59a 1.92–2.08 1.77–1.85
1.54–1.58a 1.49–1.56b 1.85–1.90c
1.35–1.59a – –
– 1.84 1.64–1.95
1.48a – –
1.62 (0.4t/1.0t) 1.52–1.66a 2.38 (4/8/12 mm) 1.89–2.04 1.85 (4/8/12 mm) 1.64–2.18
Incl. km ¼ 1.2 for angular misalignment, For t1 ¼ 30 mm, c for t1 ¼ 20 mm. b
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Fig. 8. One-sided doubler plate investigated by Yagi et al. [21].
Fig. 9. Different finite element meshes for modelling the one-sided doubler plate [15] (half model).
concentration factor Khs derived by surface stress extrapolation from the different f.e. models is fairly small, see 3rd and 4th column in Table 1. Dong’s method was applied using seven solid and three shell f.e. models [15,22], yielding the stress concentration factors Ks given also in Table 1 for through-thickness linearisation. The slightly larger variation of the results from solid models is due to element sizes ranging from 0.4t to 2.0t in front of the weld toe and the stress evaluation according to Fig. 1c [15], where the shear stresses at the side faces of the elements are neglected. The approach of Xiao and Yamada was applied to a model with 8-node solid elements having a size of 1 mm in the critical region. Therefore, only one value representing the stress concentration at the node 1 mm below the weld toe is given in Table 1. 3.2. Detail 2: edge gusset The second detail is the specimen shown in Fig. 10, having two 150 mm long and 10 mm thick edge gussets shown in Fig. 10. It was investigated in a larger test program [23] with different types of specimens representing typical ship structural details such as face bars with tapered ends. For longitudinal loading of the specimen, the fatigue critical position is the weld toe on the plate edge at the end of the gusset. The measured stresses at the plate
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Fig. 10. Specimen with edge gussets investigated in [23].
edge, which showed some scatter between the specimens, were quadratically extrapolated to the weld toe, yielding a mean structural stress concentration factor Khs ¼ 2.38 referring to the nominal stress in the plate. The detail was also investigated in the round-robin analysis [9]. Four solid and five shell element models were created. The structural stress concentration factors derived from surface stress extrapolation according to Fig. 2c and d are given in Table 1. The values are smaller than the measured one and show a relatively small scatter. Dong’s method applied to four shell models [22] yields smaller stress concentration factors Ks, which is obviously due to the stress linearisation over half the plate breadth according to Fig. 4c, i.e. t1 ¼ 30 mm. The approach of Xiao and Yamada was also applied to a shell model with 4-noded elements, where the weld region was simply modelled by increasing the thickness of the shell elements in way of the fillet weld from 10 to 15 mm. The Ks-value resulting from the 1-mm-stress is slightly smaller than that obtained from surface stress extrapolation. 3.3. Detail 3: loaded stiffener on T-bar The third detail is again a specimen from the test program [23], representing a stiffener connection being subjected to shear and bending, see Fig. 11a. The fatigue critical position is the weld toe on the upper plate edge of the flat bar. The measured stresses at the plate edge showed again some scatter between the specimens. A quadratic extrapolation to the weld toe yields a mean structural stress concentration factor Khs of 1.85 referred to the nominal bending stress at this point in the stiffener. Also this detail was investigated in the round-robin analysis [9] with three solid and five shell element models. Fig. 11b shows a simplified shell model from an additional analysis by the authors (two models), where the fillet weld is again represented by elements with increased thickness. The structural stress concentration factors derived from surface stress
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Fig. 11. Loaded stiffener on a T-bar and shell modelling of the critical area around the weld.
extrapolation according to Fig. 2c and d are given in Table 1. The values show a relatively large scatter. Apart from the notch singularity, which affects the stresses close to the hot spot, different types of weld modeling play a role. Particularly in cases where the weld is omitted, the structural hot-spot stress is increased when extrapolated to the intersection point of the plates. Again, Dong’s method was only applied to shell models according to Fig. 11b with three different element lengths ‘ ranging from 2 to 10 mm. The stress linearisation was performed over a depth t1 ¼ 20 mm, which corresponds to the final crack length in the tests. The derived stress concentration factors Ks show very small scatter, because the weld modelling was not further varied. The approach of Xiao and Yamada was also applied to the above mentioned shell model after mesh refinement, disregarding the discontinuity due to the non-fused weld root face. In the investigation, a variation of the f.e. model was studied. The element length was varied from ‘ ¼ 0:5 to 1 mm, and 4-node as well as 8-node shell elements were used, having linear and quadratic displacement functions, respectively. The 1-mm-stress shows a surprisingly large scatter, see Table 1. Particularly the 8-node shell elements with 1 mm length give remarkable smaller stresses. The reason is seen in the notch singularity at the weld toe, which causes increased stresses at the plate edge and correspondingly decreased stresses at the lower edge of the element in the depth of 1 mm. Therefore, only elements without midside nodes are recommended if element sizes of 1 mm are chosen, which has been the case for the other two details.
4. Fatigue life prediction for the three details In this chapter, the fatigue life is predicted for the three different structural details described before, assuming a constant nominal stress range of Dsn ¼ 150 MPa for details 1 and 3 and Dsn ¼ 80 MPa for detail 2. The predicted lives are compared with test results, from which a lower bound S–N curve is derived having the same probability of survival Ps as the design S–N curves, i.e. two standard deviations below the mean line (Ps ¼ 97.7%).
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4.1. Test results The test results for the first detail (doubler plate) are published in [21]. The six results for series 15A were statistically evaluated, yielding FAT 71 for Ps ¼ 97.7%, if the slope exponent of the S–N curve is fixed to three. The test results for the other two details are given in [23]. Several test series were investigated with different mean stresses and pre-loads. From these, the results for the highest mean load were chosen. The failure criterion was specimen fracture for detail 2 (edge gusset) and a 20 mm long crack for detail 3 (stiffener on T-bar). The statistical evaluation of the test results is given in [24], using again a fixed slope exponent of the S–N curve. For the lower-bound S–N curve (Ps ¼ 97.7%) in the following form, log N ¼ log a m log Dsn
(2)
the constants are summarized in Table 2. The resulting lower-bound fatigue lives for the nominal stress ranges Dsn assumed above for the three details are given in the third column of Table 3. 4.2. Life prediction with the structural stress approaches The structural stress ranges are calculated from the nominal stress ranges and the stress concentration factors given in Table 1, including variations due to modelling effects. For the structural hot-spot stress approach based on the IIW recommendations, the design S–N curve FAT 100 was assumed for the first detail, whereas FAT 90 was assumed for detail 2 (due to edge attachment longer that 100 mm) and for detail 3 (due to loadcarrying fillet welds). The resulting design fatigue lives are given in the 4th and 5th column of Table 3. In the structural stress approach according to Dong, the equivalent structural stress parameter DSs in Eq. (1) was calculated using the integral value I(r) containing the degree of bending r of the stress linearized over t or t1. It was assumed in all cases that the cracks propagated under load-controlled conditions. Values for I(r) are given for semi-elliptical cracks in [18], which apply to detail 1, for which a bending ratio r ¼ 0.32 was found. For the edge cracks in details 2 and 3, corresponding values for I(r) can be found in [19]. The linearized stress over t1 gave again a bending ratio r ¼ 0.32 for detail 2 and r ¼ 0.41 for detail 3. The fatigue lives in the 5th and 6th column of Table 3 were obtained from the lower bound master S–N curve in Fig. 5. With the approach by Xiao and Yamada [14], the fatigue life was predicted using the 1-mm-stress as well as the FAT 100 S–N curve. The results are given in the last columns of Table 3.
Table 2 Parameters of the lower bound S–N curves (Ps ¼ 97.7%) derived from test results
a m
1. Doubler plate
2. Edge gusset
3. Stiffener on T-bar
11.855 3
11.436 3
12.118 3
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Table 3 Fatigue lives derived from the fatigue tests and predicted with different methods based on the results from various f.e. models Structural detail
Dsn
Fatigue lives [cycles] for Ps ¼ 97.7%
[MPa]
Fatigue test
1. Doubler plate 2. Edge gusset
150
209,000
80
533,000
3. Stiffener on T-bar
150
389,000a
a
IIW recommendations
Dong’s master curve
Xiao/Yamada
Shell models
Solid models
Shell models
Solid models
Shell models
Solid model
130,000– 169,000 337,000– 423,000 41,700– 97,900
150,000– 190,000 318,000– 399,000 67,900– 77,500
97,400– 106,000 315,000– 364,000 44,800– 46,400
94,800– 158,000 –
–
183,000
630,000
–
79,500– 134,000
–
–
For 20 mm crack; long endurance due to compressive residual stresses verified by measurements.
Fig. 12. Fatigue lives for the doubler plate (detail 1) derived from fatigue tests and predicted with the different structural stress approaches.
The fatigue lives are compared with each other for the three details in Figs. 12–14. The variation in the results is shown by the hatched part of the bars. In some cases it is quite large, such as for the shell models of detail 3 evaluated according to the IIW recommendations (Fig. 14). As mentioned before, the different kinds of modelling the weld play a significant role here. But also the variation in the approach by Xiao and Yamada is large for this detail—due to the small structural stress computed with 8-node shell elements. In other cases, where only the element size was varied, the scatter in results is rather small, such as in several analyses using Dong’s approach. Although the differences in the stress concentration factors in Table 1 are partly large, the life predictions based on the three approaches are not too distant from each other
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Fig. 13. Fatigue lives for the edge gusset (detail 2) derived from fatigue tests and predicted with the different structural stress approaches.
Fig. 14. Fatigue lives for the stiffener on T-bar (detail 3) derived from fatigue tests and predicted with the different structural stress approaches.
and—with one exception—conservative when compared with the results of the fatigue tests. For detail 3 (stiffener on T-bar), the fatigue life predictions are over-conservative. The reason for the long fatigue life observed in the tests was found by beneficial residual stresses, which were shown by measurement to be compressive on the upper side of the stiffener [23]. Compressive residual stresses were found also in similar situations [25], which were shown to be due to longitudinal shrinkage of the fillet welds, causing in-plane-bending of the stiffener, which is constrained and thus leads to compressive stresses at the upper and lower edge of the stiffener. Insofar the over-conservatism in fatigue life prediction is not due to the structural stress approaches, but to the neglect of a significant influence factor.
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5. Conclusions Three different structural stress approaches to the fatigue strength assessment of welded structures were reviewed and applied to three structural details. The results of several stress analyses from previous investigations were supplemented by few additional stress analyses. Fatigue lives were predicted using the design S–N curves recommended within the different approaches and compared with the results of fatigue tests evaluated for a corresponding probability of survival. From the results, the following conclusions are drawn:
Uncertainties in the computed structural stresses and predicted fatigue lives are mainly due to the element properties and sizes, to the stress evaluation and in particular to the weld modelling in shell element models; In spite of different structural stress definitions, the fatigue lives predicted with the three approaches are not too distant from each other; Also, the fatigue life predictions are—with one exception—conservative compared with fatigue tests; The results of fatigue tests are partly affected by beneficial residual stresses, which were verified by measurements and cause large differences between tests and predictions.
Acknowledgements The major part of this work has been performed within the project ‘‘Network of Excellence on Marine Structures’’ (MARSTRUCT). The participants are Instituto Superior Te´cnico, Universities of Glasgow & Strathclyde, Universite´ de Liege, Technical University of Varna, Technical University of Denmark, Helsinki University of Technology, VTT Industrial Systems, Kvaerner Masa-Yards, Bureau Veritas, Principia Marine, Sirehna, Germanischer Lloyd, Hamburg University of Technology, Flensburger Schiffbau-Gesellschaft mbH & Co KG, CMT—Center of Maritime Technologies e.V., National Technical University of Athens, CETENA—Centro Tecnico Navale, Universita` di Genova, TNO—Netherlands Institute for Applied Scientific Research, Schelde Naval Shipbuilding, Norwegian University of Science and Technology, Det Norske Veritas, Technical University of Szczecin, CTO—Centrum Techniki Okretowej, Lisnave Estaleiros Navais SA, Estaleiros Navais de Viana do Castelo, University ‘‘Dunarea de Jos’’ of Galati, IZAR Construcciones Navales S.A., Chalmers University of Technology, Technical University of Istanbul, University of Newcastle, University of Southampton and The Welding Institute. The work has been partially funded by the European Union through the Growth programme under contract TNE3-CT-2003-506141. References [1] Radaj D, Sonsino CM, Fricke W. Fatigue assessment of welded joints by local approaches. Cambridge: Abington Publ; 2006 [2nd edition under preparation]. [2] Haibach E. Fatigue strength of welded joints from viewpoint of local strain measurement. Report FB-77, Fraunhofer-Institut fu¨r Betriebsfestigkeit (LBF), Darmstadt; 1968 [in German]. [3] van Wingerde AM, Packer JA, Wardenier J. Criteria for the fatigue assessment of hollow structural section connections. J Construct Steel Res 1995;35:71–115. [4] Radaj D. Design and analysis of fatigue-resistant welded structures. Cambridge: Abington Publ; 1990.
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[5] Petershagen H, Fricke W, Massel T. Application of the local approach to the fatigue strength assessment of welded structures in ships. IIW Doc. XIII-1409-91, International Institute of Welding, 1991. [6] Fricke W, Petershagen H. Detail design of welded ship structures based on hot spot stresses. In: Caldwell JB, Ward G, editors. Practical design of ships and mobile units. Amsterdam: Elsevier Science; 1992. [7] Niemi E., editor. Recommendations concerning stress determination for fatigue analysis of welded components. Cambridge: Abington Publ; 1995. [8] Huther I, Gorski S, Lieurade HP, Laborde S, Recho N. Longitudinal non loaded welded joints-geometrical stress approach. Welding in the World 1999;43(3):20–6. [9] Fricke W. Recommended hot spot analysis procedure for structural details of ships and FPSOs based on round-robin FE analyses. Int J Offshore Polar Eng 2002;12(1):40–7. [10] Niemi E, Fricke W, Maddox S. Structural stress approach to fatigue analysis of welded components— designer’s guide. IIW-Doc. XIII-1819-00/XV-1090-01, International Institute of Welding, Final draft 2004. [11] Hobbacher A., editor. Recommendations for fatigue design of welded joints and components. IIW-Doc. XIII-1965-03/XV-1127-03, International Institute of Welding, Final draft 2005. [12] Dong P. A structural stress definition and numerical implementation for fatigue analyses. Int J Fatigue 2001;23(10):865–76. [13] Dong P, Hong JK, Cao Z. Structural stress based master S–N curve for welded joints, IIW Doc XIII-193002/XV-1119-02, International Institute of Welding, 2002. [14] Xiao Z-G, Yamada K. A method of determining geometric stress for fatigue strength evaluation of steel welded joints. Int J Fatigue 2004;26:1277–93. [15] Doerk O, Fricke W, Weissenborn C. Comparison of different calculation methods for structural stresses at welded joints. Int J Fatigue 2003;25:359–69. [16] Maddox SJ. Hot-spot stress design curves for fatigue assessment of welded structures. IIW-Doc. XIII-1999a00, International Institute of Welding 2000. Int J Offshore Polar Eng 2002;12:134–41. [17] Pouitiainen I, Marquis G. A single-point structural stress assessment procedure for load-carrying fillet joints. IIW Doc XIII-2012-04/XV-1174-04, International Institute of Welding; 2004. [18] Dong P, Hong JK. Recommended structural stress procedure for fatigue assessment. Proceedings of OMAE Specialty Conference on FPSO systems, OMAE-FPSO’04-0029, Houston, TX, ASME International Petroleum Technical Institute; 2004. [19] Dong P. The mesh-insensitive structural stress and master S–N curve method for ship structures. Proceedings of OMAE Specialty Conference on FPSO Systems, OMAE-FPSO’04-0021, Houston, TX, ASME International Petroleum Technical Institute; 2004. [20] Dong P. Private communication with the authors; 2005. [21] Yagi J, Machida S, Tomita Y, Matoba M, Kawasaki T. Definition of hot spot stresses in welded plate type structure for fatigue assessment. IIW-Document XIII-1414-91, International Institute of Welding; 1991. [22] Dong P, Hong JK. Hot spot stress and structural stress analyses of FPSO fatigue details. Proceedings of OMAE Specialty Conference on FPSO systems, OMAE-FPSO’04-0023, Houston TX, ASME International Petroleum Technical Institute; 2004. [23] Kim WS, Lotsberg I. Fatigue test data for welded connections in ship-shaped structures. Proceedings of OMAE Specialty Conference on FPSO systems, OMAE-FPSO’04-0018, Houston, TX, ASME International Petroleum Technical Institute; 2004. [24] Lotsberg I, Sigurdsson G. Hot spot stress S–N curve for fatigue analysis of plated structures. Proceedings of OMAE Specialty Conference on FPSO systems, OMAE-FPSO’04-0014, Houston, TX, ASME International Petroleum Technical Institute; 2004. [25] Fricke W, Doerk O, Gru¨nitz L. Fatigue strength investigation and assessment of fillet welds around stiffener and bracket toes. Proceedings of the OMAE Specialty Conference on FPSO systems, OMAE-FPSO’04-0010, Houston, TX, ASME International Petroleum Technical Institute; 2004.
Marine Structures 18 (2005) 473–488 www.elsevier.com/locate/marstruc
Comparison of different structural stress approaches for fatigue assessment of welded ship structures Wolfgang Fricke, Adrian Kahl Hamburg University of Technology, Ship Structural Design and Analysis, 3-06, 21073 Hamburg, Germany Received 31 October 2005; received in revised form 3 February 2006; accepted 7 February 2006
Abstract The structural stress approach, which considers the stress increase due to the structural configuration, allows the fatigue strength assessment of welded ship structures with various geometries on the basis of an S–N curve depending only on the type of weld. However, a unique definition and the numerical calculation of the structural stresses are problematic, which has resulted in the development of different variants of the approach. These are discussed and compared with each other in the present paper. The application to three examples shows the variation and differences in the analysed stresses and predicted fatigue lives, which are compared with those derived from fatigue tests. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fatigue strength; Welded joint; Ship structural detail; Structural stress; Hot spot stress
1. Introduction The initiation and early propagation of cracks at welded joints under fatigue loading is primarily determined by the local stress distribution. According to Radaj et al. [1], first attempts to correlate the fatigue strength with a local stress were made already in the 1960s by several researchers, including Peterson, Manson and Haibach, who related the fatigue strength to a local stress or strain measured at a certain point close to the weld toe, for example at a distance of 2 mm [2].
Corresponding author. Tel.: +49 40 428 786089; fax: +49 40 428 786090.
E-mail address: [email protected] (W. Fricke). 0951-8339/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2006.02.001
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The stress or strain determined in this way depends, however, of the component size or plate thickness, which has led in the 1970s to the development of the well-known hot spot stress approach with reference points for the stress evaluation and extrapolation, which are located at distances from the hot spot (weld toe) depending of the plate thickness. This allows hot spot stress concentration factors to be derived in relation to dimensionless geometry parameters. The approach was at first successfully applied to the fatigue assessment of tubular joints in offshore platforms [3,4]. Radaj [4] summarized these and other investigations and defined the structural stress at the hot spot (weld toe) as the surface stress which can be calculated in accordance with structural theories used in engineering. He demonstrated that the structural stress can be analysed either by surface stress extrapolation or by linearization, e.g. through the wall thickness. Both procedures exclude the local non-linear stress peak caused by the weld toe, whose effect is considered in the S–N curve. In the early 1990s, Petershagen et al. [5] derived a generalized hot-spot stress approach for plate structures using Radaj’s effective notch stress approach [4] and applied it to complex welded structures [6]. Detailed recommendations concerning stress determination for fatigue analysis of welded components were given by Niemi [7], Huther et al. [8] and Fricke [9], among others. A comprehensive guidance by the International Institute of Welding (IIW) is presently under preparation [10,11], where the relevant stress is termed ‘structural hot-spot stress’ to avoid confusion created by the different terms used previously (structural stress; hot-spot stress; geometric stress). In the recent years, further variants of the structural stress approach were developed. From these, particularly the approaches by Dong et al. [12,13] and Xiao and Yamada [14] are remarkable and will be described and discussed in the following chapter together with the ‘conventional’ structural hot-spot stress approach. Subsequently, the approaches will be applied to three typical structural details, where fatigue test results are available so that not only stresses, but also predicted fatigue lives can be compared. It should be noted that the approaches described here are applicable only to the fatigue strength assessment of weld toes, whereas weld roots, particularly at non-fused root faces, have to be assessed with modified or other approaches. 2. Structural stress approaches In this chapter, three approaches are briefly described and discussed in more detail. Emphasis is placed on welded plate structures being typical for ships, although they are also well-suited for welded joints in shell structures such as tubular joints. A more comprehensive description of the approaches can be found in [1]. 2.1. Structural hot-spot stress approach according to the IIW The traditional approach to derive the structural hot-spot stress is the linear or quadratic extrapolation of strains measured at two or three reference points in front of the weld toe. In the recommendations of the International Institute of Welding (IIW) [11], distances of the reference points from the weld toe of 0.4t/1.0t or 0.4t/0.9t/1.4t are recommended, where t is the plate thickness. Here it is assumed that the local stress increase due to the notch at the weld toe disappears within 0.4t. At plate edges, quadratic extrapolation over reference points at fixed distances from the weld toe (4/8/12 mm) is
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recommended, as plate thickness is not considered as a suitable parameter to define the location of the reference points at plate edges. The surface extrapolation of stresses can accordingly be applied to f.e. analyses, Fig. 1a. Alternatively, stress linearisation over the thickness leads also to the exclusion of the local stress peak in plate or shell structures, Fig. 1b. In the case of solid models, the arrangement of three or more elements over the thickness is recommended, because the stresses in the section directly below the weld toe are disturbed by the notch singularity, which affects the linearized structural stress considerably in case of only one or two elements. Systematic variation of stress analyses has shown that detailed rules for finite element modelling and stress evaluation are necessary to avoid large scatter and uncertainties particularly in connection with surface stress extrapolation [9,10,15]. Fig. 2 shows examples for the extrapolation of stresses from different kinds of models. The left part contains the application of the above mentioned reference points to relatively fine f.e. meshes, whereas the right part shows the stress extrapolation for relatively coarse models as recommended by some classification societies. It should be noted that further mesh refinement, e.g. in case a), should be performed in both directions (see example in Fig. 1a) to avoid over-estimation of stresses. The associated design S–N curves were defined on the basis of extensive evaluation of fatigue tests [16]. As stated in [11], fatigue class FAT 100 ( ¼ fatigue strength reference value in [N/mm2] at two million cycles) is recommended in normal cases for welded joints in steel structures. Exceptions are longer attachments (4100 mm) at plate edges as well as load-carrying fillet welds (due to the additional local stress concentration at the weld toe, which is not captured by the structural stress defined above), for which FAT 90 applies as demonstrated by the open symbols in Fig. 3. An alternative procedure to capture the increased stresses in load-carrying fillet welds by a bilinear stress distribution has been proposed in [17]. For welded joints at plate thickness t larger than t0 ¼ 25 mm, the well-known thickness correction on fatigue strength has to be considered with an exponent on the thickness ratio t/t0 varying from n ¼ 0.1 for welds at plate edges over n ¼ 0.2 for butt joints to n ¼ 0.3 for other joints [11]. A special problem is fabrication-related axial and angular misalignments. As measured structural hot-spot stresses are the basis for the design S–N curves, which already contain the effects of possible misalignment, these have to be explicitly taken into account in the
Fig. 1. Evaluation of structural stress at weld toe by surface stress extrapolation (a), linearisation over plate thickness (b) and equilibrium with stresses at distance d (c).
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Fig. 2. Examples of different rules for modelling and stress extrapolation.
Fig. 3. Fatigue test results for non-load and load-carrying fillet welds in terms of measured structural hot-spot stress [16].
structural hot-spot stress in the considered case. In the nominal stress approach, the effects are implicitly taken into account by the design S–N curves up to a certain amount. Usually, stresses are computed with perfectly aligned f.e. models which do not contain any pre-deformations. Their effects on the structural hot-spot stress have to be considered in plate structures particularly at butt- and cruciform joints with non-continuous loaded plates (due to possible axial misalignment) and at one-sided, transverse fillet welds (due to
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possible angular misalignment). If no information about misalignment is available, IIW [11] recommends to multiply the axial (membrane) component of the plate stress with km factors, which contain the effects of axial misalignment of 5–15% of the plate thickness (km ¼ 1.1–1.4). With these factors, the fatigue classes of the nominal and structural hotspot stress approach become compatible. 2.2. Structural stress approach according to Dong The approach with linearisation of the stress over the plate thickness was adopted by Dong and extended such that particularly the effect of the stress gradient along the anticipated crack path is taken into account using fracture mechanics [12,13]. The stress linearisation over the thickness t of a plate with one-sided weld is illustrated in Fig. 4a. In certain cases, the linearisation up to a depth t1 ot is recommended, Fig. 4b, e.g. for welds at plate edges, where t1 corresponds to the final crack length. In case of two-sided welds with symmetrical geometry and loading, a linearisation over half the plate thickness (t1 ¼ t/2) is proposed, Fig. 4c, which means a different structural stress definition compared to the approach mentioned before. Generally, the linearisation according to Dong is performed only over a monotonic decreasing stress distribution. Dong et al. propose special procedures for the computation of the structural stress, which are considered to be rather mesh-insensitive. As element stresses depend on the mesh fineness and are affected by the notch singularity at the weld toe, they should be evaluated in a distance d from the weld toe, Fig. 1c. Using equilibrium conditions, the membrane and bending portion of the stress and thus the linear stress distribution in the through-thickness section at the weld toe can be determined from the normal and shear stresses acting in the distance d. However, this procedure neglects the shear stresses at the other element faces, which causes errors in case of high local stress concentrations [15]. If the stresses are linearized over the depth t1, the stress components acting at the lower edge of the area d t1 have to be included in the equilibrium equations. As an alternative, Dong proposes to determine the structural stress from the internal nodal forces in the through-thickness section at the weld toe, as these generally satisfy equilibrium conditions. This approach is particularly well-suited for shell models, where work-equivalent line forces and moments can be computed from the nodal forces and moments along the weld toe line using the element displacement functions. The line forces and moments yield directly the structural stresses. Partial linearisation over the depth t1 is, of course, not possible here.
Fig. 4. Definition of the structural stress according to Dong [12].
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The endurable stresses or load cycles are determined from a master S–N curve using an equivalent structural stress parameter DSs, which results from the structural stress range Dss as follows [13]: m2
1
DSs ¼ Dss t 2m IðrÞm
(1)
with plate thickness t [mm], the exponent m of the Paris crack propagation law (with m ¼ 3.6 according to Dong) and the integral I(r), which depends of the ratio r between the bending portion and the total structural stress and also of the boundary conditions during crack propagation (load- or displacement-controlled). Typical curves are given for I(r) in [18,19]. The plate thickness is considered in Eq. (1) by an exponent of 0.22. The master S–N curve shown in Fig. 5 has been derived from a large number of fatigue tests, for which the structural stress according to Fig. 4 and the equivalent structural stress parameter in Eq. (1) have been analysed. Misalignment has not been considered explicitly, i.e. it affects the master S–N curve to an extent as it has been present in the tests. 2.3. Structural stress approach according to Xiao and Yamada In view of more powerful soft- and hardware, which allow the generation of finer meshes without high expenditure, Xiao and Yamada [14] have recently proposed a new structural stress approach which assumes the computed stress at a point in a depth of 1 mm below the weld toe in the direction of the expected crack path as relevant parameter for the fatigue strength. The selection of this point is verified by a reference detail, a 10 mm thick plate with transverse stiffeners on both sides, Fig. 6. Finite element computations have shown that the local stress at the weld toe of this detail decreases much faster in thickness direction than along the surface. In the latter, the local stress increase disappears in a distance of 2.5 mm, while the nominal stress is already reached in a depth of approximately 1 mm, irrespectively from the local shape of the weld toe (radius and flank angle varied in Fig. 6). Insofar, similarities exist with the approach by Haibach [2]. However, additional justification is given in [14] by showing that the 1-mm-stress is a representative load parameter for the early crack propagation phase.
Fig. 5. Master S–N curve according to Dong [20] with scatter band for two probabilities of survival Ps.
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Fig. 6. Stress distribution at the reference detail: (a) along the plate surface and (b) in thickness direction [14].
Fig. 7. Evaluation of test results for relatively thick specimens [14].
Finite element analyses require a mesh, which yields the 1-mm-stress with sufficient accuracy. It is stated that the element length should not exceed 1 mm, which will be discussed further in connection with the examples below. The approach has been applied to several types of welded joints, where the geometry is similar to that of the reference detail, i.e. longitudinal and transverse attachments on continuous plates. Fatigue test results, if plotted against the calculated 1-mm-stress, show a fairly small scatter with a lower boundary according to FAT 100. Furthermore, it has been shown that the 1-mm-stress considers the thickness effect very well, see Fig. 7. The scatter of the results is smaller than for the conventional structural hot-spot stress approach and for Dong’s structural stress approach.
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The applicability of the approach to other welded joints (e.g. cruciform joints with loadcarrying fillet welds or one-sided longitudinal and transverse attachments) and load cases (e.g. with pronounced bending) needs still to be verified. 3. Selected structural details Three structural details have been selected for the present investigation. They will be described in the following together with the computed structural stress concentration factors taken mainly from previous investigations, which are summarized in Table 1. Regarding the surface stress extrapolation, only results are considered which follow the recommendations given above. 3.1. Detail 1: one-sided doubler plate The first detail is the specimen shown in Fig. 8 with a round doubler plate on one side, subjected to tensile forces. The specimen was investigated in a Japanese research project [21]. The fatigue critical position is the weld toe on the plate oriented transversely to the loading. The circular doubler plate causes a non-uniform stress distribution in the transverse direction. In a separate unpublished study, the longitudinal stresses were measured in front of the critical weld toe and linearly extrapolated over 0.4t/1.0t, yielding a structural hot-spot stress concentration factor Khs of 1.62 based on the nominal stress in the plate. It was noticed that the specimen had an angular distortion due to the one-sided fillet welding, from which a stress magnification factor of about km ¼ 1.2 was derived. The detail was investigated also in the round-robin analysis described in [9], where different techniques of modelling the one-sided doubling plate were applied. Fig. 9 shows two typical f.e. models with 20-noded solid elements based on the meshing according to Fig. 2a and b [15]. The results were included in the evaluation together with two results from [22], comprising in total eight solid models. Also shell element models were created in [9], where different types of weld modelling by vertical or inclined shell elements or rigid links caused some additional scatter. However, the scatter of the structural stress
Table 1 Structural stress concentration factors obtained for the three details by different methods and f.e. models Structural detail
Structural stress concentration factor Ks at hot spot Khs from measurement
1. Doubler plate 2. Edge gusset 3. Stiffener on T-bar a
Khs acc. to IIW
Ks acc. to Dong
Ks for 1-mm-stress
Shell models
Solid models
Shell models
Solid models
Shell models
Solid model
1.46–1.59a 1.92–2.08 1.77–1.85
1.54–1.58a 1.49–1.56b 1.85–1.90c
1.35–1.59a – –
– 1.84 1.64–1.95
1.48a – –
1.62 (0.4t/1.0t) 1.52–1.66a 2.38 (4/8/12 mm) 1.89–2.04 1.85 (4/8/12 mm) 1.64–2.18
Incl. km ¼ 1.2 for angular misalignment, For t1 ¼ 30 mm, c for t1 ¼ 20 mm. b
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Fig. 8. One-sided doubler plate investigated by Yagi et al. [21].
Fig. 9. Different finite element meshes for modelling the one-sided doubler plate [15] (half model).
concentration factor Khs derived by surface stress extrapolation from the different f.e. models is fairly small, see 3rd and 4th column in Table 1. Dong’s method was applied using seven solid and three shell f.e. models [15,22], yielding the stress concentration factors Ks given also in Table 1 for through-thickness linearisation. The slightly larger variation of the results from solid models is due to element sizes ranging from 0.4t to 2.0t in front of the weld toe and the stress evaluation according to Fig. 1c [15], where the shear stresses at the side faces of the elements are neglected. The approach of Xiao and Yamada was applied to a model with 8-node solid elements having a size of 1 mm in the critical region. Therefore, only one value representing the stress concentration at the node 1 mm below the weld toe is given in Table 1. 3.2. Detail 2: edge gusset The second detail is the specimen shown in Fig. 10, having two 150 mm long and 10 mm thick edge gussets shown in Fig. 10. It was investigated in a larger test program [23] with different types of specimens representing typical ship structural details such as face bars with tapered ends. For longitudinal loading of the specimen, the fatigue critical position is the weld toe on the plate edge at the end of the gusset. The measured stresses at the plate
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Fig. 10. Specimen with edge gussets investigated in [23].
edge, which showed some scatter between the specimens, were quadratically extrapolated to the weld toe, yielding a mean structural stress concentration factor Khs ¼ 2.38 referring to the nominal stress in the plate. The detail was also investigated in the round-robin analysis [9]. Four solid and five shell element models were created. The structural stress concentration factors derived from surface stress extrapolation according to Fig. 2c and d are given in Table 1. The values are smaller than the measured one and show a relatively small scatter. Dong’s method applied to four shell models [22] yields smaller stress concentration factors Ks, which is obviously due to the stress linearisation over half the plate breadth according to Fig. 4c, i.e. t1 ¼ 30 mm. The approach of Xiao and Yamada was also applied to a shell model with 4-noded elements, where the weld region was simply modelled by increasing the thickness of the shell elements in way of the fillet weld from 10 to 15 mm. The Ks-value resulting from the 1-mm-stress is slightly smaller than that obtained from surface stress extrapolation. 3.3. Detail 3: loaded stiffener on T-bar The third detail is again a specimen from the test program [23], representing a stiffener connection being subjected to shear and bending, see Fig. 11a. The fatigue critical position is the weld toe on the upper plate edge of the flat bar. The measured stresses at the plate edge showed again some scatter between the specimens. A quadratic extrapolation to the weld toe yields a mean structural stress concentration factor Khs of 1.85 referred to the nominal bending stress at this point in the stiffener. Also this detail was investigated in the round-robin analysis [9] with three solid and five shell element models. Fig. 11b shows a simplified shell model from an additional analysis by the authors (two models), where the fillet weld is again represented by elements with increased thickness. The structural stress concentration factors derived from surface stress
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Fig. 11. Loaded stiffener on a T-bar and shell modelling of the critical area around the weld.
extrapolation according to Fig. 2c and d are given in Table 1. The values show a relatively large scatter. Apart from the notch singularity, which affects the stresses close to the hot spot, different types of weld modeling play a role. Particularly in cases where the weld is omitted, the structural hot-spot stress is increased when extrapolated to the intersection point of the plates. Again, Dong’s method was only applied to shell models according to Fig. 11b with three different element lengths ‘ ranging from 2 to 10 mm. The stress linearisation was performed over a depth t1 ¼ 20 mm, which corresponds to the final crack length in the tests. The derived stress concentration factors Ks show very small scatter, because the weld modelling was not further varied. The approach of Xiao and Yamada was also applied to the above mentioned shell model after mesh refinement, disregarding the discontinuity due to the non-fused weld root face. In the investigation, a variation of the f.e. model was studied. The element length was varied from ‘ ¼ 0:5 to 1 mm, and 4-node as well as 8-node shell elements were used, having linear and quadratic displacement functions, respectively. The 1-mm-stress shows a surprisingly large scatter, see Table 1. Particularly the 8-node shell elements with 1 mm length give remarkable smaller stresses. The reason is seen in the notch singularity at the weld toe, which causes increased stresses at the plate edge and correspondingly decreased stresses at the lower edge of the element in the depth of 1 mm. Therefore, only elements without midside nodes are recommended if element sizes of 1 mm are chosen, which has been the case for the other two details.
4. Fatigue life prediction for the three details In this chapter, the fatigue life is predicted for the three different structural details described before, assuming a constant nominal stress range of Dsn ¼ 150 MPa for details 1 and 3 and Dsn ¼ 80 MPa for detail 2. The predicted lives are compared with test results, from which a lower bound S–N curve is derived having the same probability of survival Ps as the design S–N curves, i.e. two standard deviations below the mean line (Ps ¼ 97.7%).
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4.1. Test results The test results for the first detail (doubler plate) are published in [21]. The six results for series 15A were statistically evaluated, yielding FAT 71 for Ps ¼ 97.7%, if the slope exponent of the S–N curve is fixed to three. The test results for the other two details are given in [23]. Several test series were investigated with different mean stresses and pre-loads. From these, the results for the highest mean load were chosen. The failure criterion was specimen fracture for detail 2 (edge gusset) and a 20 mm long crack for detail 3 (stiffener on T-bar). The statistical evaluation of the test results is given in [24], using again a fixed slope exponent of the S–N curve. For the lower-bound S–N curve (Ps ¼ 97.7%) in the following form, log N ¼ log a m log Dsn
(2)
the constants are summarized in Table 2. The resulting lower-bound fatigue lives for the nominal stress ranges Dsn assumed above for the three details are given in the third column of Table 3. 4.2. Life prediction with the structural stress approaches The structural stress ranges are calculated from the nominal stress ranges and the stress concentration factors given in Table 1, including variations due to modelling effects. For the structural hot-spot stress approach based on the IIW recommendations, the design S–N curve FAT 100 was assumed for the first detail, whereas FAT 90 was assumed for detail 2 (due to edge attachment longer that 100 mm) and for detail 3 (due to loadcarrying fillet welds). The resulting design fatigue lives are given in the 4th and 5th column of Table 3. In the structural stress approach according to Dong, the equivalent structural stress parameter DSs in Eq. (1) was calculated using the integral value I(r) containing the degree of bending r of the stress linearized over t or t1. It was assumed in all cases that the cracks propagated under load-controlled conditions. Values for I(r) are given for semi-elliptical cracks in [18], which apply to detail 1, for which a bending ratio r ¼ 0.32 was found. For the edge cracks in details 2 and 3, corresponding values for I(r) can be found in [19]. The linearized stress over t1 gave again a bending ratio r ¼ 0.32 for detail 2 and r ¼ 0.41 for detail 3. The fatigue lives in the 5th and 6th column of Table 3 were obtained from the lower bound master S–N curve in Fig. 5. With the approach by Xiao and Yamada [14], the fatigue life was predicted using the 1-mm-stress as well as the FAT 100 S–N curve. The results are given in the last columns of Table 3.
Table 2 Parameters of the lower bound S–N curves (Ps ¼ 97.7%) derived from test results
a m
1. Doubler plate
2. Edge gusset
3. Stiffener on T-bar
11.855 3
11.436 3
12.118 3
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Table 3 Fatigue lives derived from the fatigue tests and predicted with different methods based on the results from various f.e. models Structural detail
Dsn
Fatigue lives [cycles] for Ps ¼ 97.7%
[MPa]
Fatigue test
1. Doubler plate 2. Edge gusset
150
209,000
80
533,000
3. Stiffener on T-bar
150
389,000a
a
IIW recommendations
Dong’s master curve
Xiao/Yamada
Shell models
Solid models
Shell models
Solid models
Shell models
Solid model
130,000– 169,000 337,000– 423,000 41,700– 97,900
150,000– 190,000 318,000– 399,000 67,900– 77,500
97,400– 106,000 315,000– 364,000 44,800– 46,400
94,800– 158,000 –
–
183,000
630,000
–
79,500– 134,000
–
–
For 20 mm crack; long endurance due to compressive residual stresses verified by measurements.
Fig. 12. Fatigue lives for the doubler plate (detail 1) derived from fatigue tests and predicted with the different structural stress approaches.
The fatigue lives are compared with each other for the three details in Figs. 12–14. The variation in the results is shown by the hatched part of the bars. In some cases it is quite large, such as for the shell models of detail 3 evaluated according to the IIW recommendations (Fig. 14). As mentioned before, the different kinds of modelling the weld play a significant role here. But also the variation in the approach by Xiao and Yamada is large for this detail—due to the small structural stress computed with 8-node shell elements. In other cases, where only the element size was varied, the scatter in results is rather small, such as in several analyses using Dong’s approach. Although the differences in the stress concentration factors in Table 1 are partly large, the life predictions based on the three approaches are not too distant from each other
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Fig. 13. Fatigue lives for the edge gusset (detail 2) derived from fatigue tests and predicted with the different structural stress approaches.
Fig. 14. Fatigue lives for the stiffener on T-bar (detail 3) derived from fatigue tests and predicted with the different structural stress approaches.
and—with one exception—conservative when compared with the results of the fatigue tests. For detail 3 (stiffener on T-bar), the fatigue life predictions are over-conservative. The reason for the long fatigue life observed in the tests was found by beneficial residual stresses, which were shown by measurement to be compressive on the upper side of the stiffener [23]. Compressive residual stresses were found also in similar situations [25], which were shown to be due to longitudinal shrinkage of the fillet welds, causing in-plane-bending of the stiffener, which is constrained and thus leads to compressive stresses at the upper and lower edge of the stiffener. Insofar the over-conservatism in fatigue life prediction is not due to the structural stress approaches, but to the neglect of a significant influence factor.
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5. Conclusions Three different structural stress approaches to the fatigue strength assessment of welded structures were reviewed and applied to three structural details. The results of several stress analyses from previous investigations were supplemented by few additional stress analyses. Fatigue lives were predicted using the design S–N curves recommended within the different approaches and compared with the results of fatigue tests evaluated for a corresponding probability of survival. From the results, the following conclusions are drawn:
Uncertainties in the computed structural stresses and predicted fatigue lives are mainly due to the element properties and sizes, to the stress evaluation and in particular to the weld modelling in shell element models; In spite of different structural stress definitions, the fatigue lives predicted with the three approaches are not too distant from each other; Also, the fatigue life predictions are—with one exception—conservative compared with fatigue tests; The results of fatigue tests are partly affected by beneficial residual stresses, which were verified by measurements and cause large differences between tests and predictions.
Acknowledgements The major part of this work has been performed within the project ‘‘Network of Excellence on Marine Structures’’ (MARSTRUCT). The participants are Instituto Superior Te´cnico, Universities of Glasgow & Strathclyde, Universite´ de Liege, Technical University of Varna, Technical University of Denmark, Helsinki University of Technology, VTT Industrial Systems, Kvaerner Masa-Yards, Bureau Veritas, Principia Marine, Sirehna, Germanischer Lloyd, Hamburg University of Technology, Flensburger Schiffbau-Gesellschaft mbH & Co KG, CMT—Center of Maritime Technologies e.V., National Technical University of Athens, CETENA—Centro Tecnico Navale, Universita` di Genova, TNO—Netherlands Institute for Applied Scientific Research, Schelde Naval Shipbuilding, Norwegian University of Science and Technology, Det Norske Veritas, Technical University of Szczecin, CTO—Centrum Techniki Okretowej, Lisnave Estaleiros Navais SA, Estaleiros Navais de Viana do Castelo, University ‘‘Dunarea de Jos’’ of Galati, IZAR Construcciones Navales S.A., Chalmers University of Technology, Technical University of Istanbul, University of Newcastle, University of Southampton and The Welding Institute. The work has been partially funded by the European Union through the Growth programme under contract TNE3-CT-2003-506141. References [1] Radaj D, Sonsino CM, Fricke W. Fatigue assessment of welded joints by local approaches. Cambridge: Abington Publ; 2006 [2nd edition under preparation]. [2] Haibach E. Fatigue strength of welded joints from viewpoint of local strain measurement. Report FB-77, Fraunhofer-Institut fu¨r Betriebsfestigkeit (LBF), Darmstadt; 1968 [in German]. [3] van Wingerde AM, Packer JA, Wardenier J. Criteria for the fatigue assessment of hollow structural section connections. J Construct Steel Res 1995;35:71–115. [4] Radaj D. Design and analysis of fatigue-resistant welded structures. Cambridge: Abington Publ; 1990.
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