Note 1.1 Introduction to fatigue design

April 2009/ John Wægter Note 1.1 Introduction to fatigue design General...2 The S-N curve...2 Fatigue crack propagation...3 Definition of basic S-N curves...6 Tubular joints...9 Influence of the parent material...11 Influence of plate thickness...11 Influence of residual stresses...13 Cumulative damage...14 References...17 1

Introduction to fatigue design General Fatigue may be defined as a mechanism of failure based on the formation and growth of cracks under the action of repeated stresses. Normally, small cracks will not cause failure, but if the design is insufficient in relation to fatigue, the cracks may propagate to such an extent that failure of the considered detail occurs. Fatigue is associated with stresses that vary with time often in a repeated manner. There are many possible sources of time varying stresses e.g. Fluctuating live loads Acceleration forces in moving structures Pressure changes Temperature fluctuations Mechanical vibrations Environmental loading (wind, waves and current) Fatigue failure is therefore likely to occur in many types of structures, but in the context of this course the main emphasis will be on welded offshore steel structures. Further, only fatigue failure from many stress cycles (high cycle fatigue) will be considered. Fatigue is the most common cause of structural failures, it is frequently claimed that at least 80 % of all structural failures are associated with fatigue. It should however be noticed, that in most cases where fatigue was identified as responsible for a structural failure, it was not because the fatigue design was inadequate, but rather because no attempt was made to design against fatigue and give attention to proper joint selection and detailing. With todays design approaches it is possible to perform a fatigue design that can almost eliminate the risk of failure. In theory it can be based on fracture mechanics techniques, and significant advances have been made in this field. But as such calculations inevitably must be based on certain assumptions (e.g. initial crack/flaw size) they are mainly used for comparative analyses where the results finally can be calibrated to test results. Most fatigue design in the industry is therefore closely based on fatigue strength data (S-N curves) that have been obtained experimentally. The S-N curve Fatigue cracks initiate at some form of stress concentration. The stress concentration may be caused by either a gross change of shape in the structure or by a local change of shape e.g. a bolt hole. In welded structures fatigue cracks typically initiate at a weld toe or from a defect at the root of the weld. For these reasons fatigue cracks are very commonly associated with joints. 2

Data on fatigue strength are usually presented in the form of S-N curves, which show, for a particular type of specimen, the number of stress cycles (N) required to cause failure under a given repeated constant stress range (S). S is the difference between the maximum and the minimum stress in the cycle. To produce such an S-N curve fatigue tests are carried out at different stress ranges on a number of identical specimens representing the detail under consideration. The results are plotted on the basis of log S versus log N, and a typical relationship is shown in Figure 1. Figure 1 Typical S-N curve for welded joints It can be seen that, as the applied stress range is decreased, the number of cycles required to cause failure increases until ultimately a stress is reached below which failure does not occur. This limiting stress is referred to as the fatigue limit (the endurance limit). The two branches of the S-N curve represent an important different behaviour. For stress ranges above the fatigue limit, cracks initiate and propagate to final rupture, while below the fatigue limit crack are initiated, but they do not propagate. In both cases the existence of cracks suggests that fracture mechanics may be a useful tool for analysing the behaviour. Fatigue crack propagation For welded joints it has been found that crack propagation is the essential parameter for the fatigue life. The rate of propagation depends on the actual stress or strain condition at the crack tip at any moment, which in linear fracture mechanics (LFM) can be expressed in terms of the stress intensity factor, K. Conventionally, studies of fatigue crack propagation aim to relate the rate of propagation to the range of the stress intensity factor, or Δ K = Kmax Kmin, where K max and K min are the values of K 3

corresponding to the upper and lower values of the stress cycle, Smax and Smin. This approach is logical under wholly tensile loading, but it may not be so obviously correct when the stress cycle alternates between tension and compression, since at some stage the crack will close and at least part of the compressive part of the cycle will have no effect on the crack. The first investigators to suggest that the rate of fatigue crack propagation was related to Δ K were Paris and Erdogan. They found that the rate of propagation depended on the stress condition adjacent to the tip of the crack, and that it could be expressed in terms of Δ K. They expressed the rate of propagation as da C( K) m dn = Δ (1) where C and m are material constants, a is the crack size and n is the number of stress cycles. It has since been shown that this relationship is only valid for a certain range of for da is plotted against the value of dn K 2 is obtained, where the Paris law is a straight line in the middle region. Δ K. If the equation Δ on a log log basis, a curve of the shape shown in Figure Figure 2 Typical relationship between da dn and Δ K 4

The lower value of Δ K represents a threshold value, Δ Kth, that must be exceeded before any propagation can occur at all. The existence of this threshold value explains why some cracks do not propagate under fatigue loading. K C represents a critical value of Δ K triggering fast failure. For steels it has been found, Ref. /1/, that there is a simple empirical relationship between m and C, in the Paris law region which is 4 1.315 10 C = (2) 895.4 m for da in mm/cycle and dn Δ K in Nmm-3/2. Furthermore, for normal weldable structural steels virtually all the results lie in the range m = 2.4 3.6 with an average value of 3.0. Given that the values of m and C are known and that an initial crack size is assumed, it is possible to integrate Equation (1) to calculate the expected fatigue life. Although this in principle is simple, it has not been possible to determine values of m and C that relate to the full interval of Δ K for a given steel quality in a particular structure. In general using linear fracture mechanics, the values of weld subjected to stress range Δ σ can be written as Δ K = F( a) Δ σ πa (3) Δ K for a crack depth a, at the toe of the where F is a function of the crack size and the joint geometry. As previously indicated, the stress intensity factor K describes the stress field at the crack tip as indicated in Figure 3. Figure 3 Stress field around crack tip 5

Combining Equations (1) and (3) gives da dn ( ( ) σ πa) = C F a Δ (4) m which on integration gives da a f m m 2 = C Δ σ π N a m (5) i ( Fa ( ) a) where N is the number of cycles for the crack to grow from an initial size ai to a final size a f. Examination of Equation (5) shows, that for a given type of joint (i.e. a known value of F(a)) and constant values of a and a, the value of the integral is a constant, so that Equation (5) can be written as i f m ( σ ) N constant = Δ = a (6a) log N = log a m log Δ σ (6b) which is the equation of an S-N curve. The constant a in Equation (6) above should not be confused with the crack size a used in fracture mechanics. The implication of Equation (6) is that if the fatigue life of a specimen consists wholly of crack propagation, then the S-N curve would be a straight line with slope -1/m when plotted on the basis of log Δ σ versus logn. Many welded joints effectively fall into this category, since it has been found that fatigue cracks normally initiate at small, slag intrusions at the weld toe. It should, however be emphasised that not all fatigue cracks initiate from the weld toe. When F, C and m are known, Equation (5) generally provides the key equation for fatigue calculations since it allows the calculation of any one of the four parameters ai, a f, Δ σ and N given the three others. Definition of basic S-N curves It is not possible to determine S-N curves for welded joints for all joint geometries entirely based on fracture mechanics. This would require knowledge of the stress intensity factor for each type of joint together with details of the assumed pre-existing defect (flaw) and the weld shape in the vicinity of those defects. Such information does not, in general exist. It is therefore inevitable that the basic S-N curves for design must be based on experimental results obtained under constant amplitude loading. 6

The fatigue strength S of a welded joint depends on the stress concentration it produces. It is the difference between the degrees of stress concentration that distinguish the good solution from the bad one. Considering the joints shown in Figure 4, it can be seen that the butt weld will produce less disturbance of the stress flow than any of the other solutions. The stress concentration at the weld toe will be lower than in any of the last three solutions based on fillet welds, thus the butt weld solution has the highest fatigue strength. It is important to notice that features that may increase the static strength of a welded joint, such as the presence of butt weld overfill or welded attachments or stiffeners inevitably reduce the fatigue strength. Figure 4 Various methods of welded joints and lines of stress flow A summary of some typical fatigue strengths (at 2 x 10 6 cycles) for welded joints together with an indication of the point of origin of fatigue cracks in them is shown in Table 1. While the strength quoted in different design standards vary somewhat, they are usually not very different from the values given in the table which is based upon analysis of a large mass of experimental data. 7

Table 1 Fatigue strengths and cracking patterns in simple welded joints For these simple joints, the design stress range given ( σ local ) is the nominal value as determined in beam stress analysis, away from the stress concentration. When calculating the allowable design stress range, the stress concentration effect of the joint itself is thus included in the quoted fatigue strengths. However, if e.g. a hole exists adjacent to the joint, the extra stress concentration from the 8

hole (SCF) must be taken into consideration when determining the local stresses σ = SCF σ ), see Figure 5. ( local nominal Figure 5 Determination of local stresses The fatigue strength in Table 1 refers to simple joints, typically 13 mm thick, tested in air. In some cases these allowable stress ranges may need to be modified If there is a larger eccentricity than assumed in the test If the plate is thicker If the joint is placed in a corrosive environment Similarly, if the point of crack initiation is affected by a large geometric discontinuity in the structure, e.g. a cut-out in a plate as shown in Figure 5, it will be necessary to consider this stress concentration when determining the design stress as already mentioned above. Based on the above it can be concluded, that the general rule in designing against fatigue is to keep all stress flow paths smooth by eliminating as many stress concentrations as possible. Tubular joints In more complicated joints, such as tubular joints as used in the offshore industry, it is not possible to relate the fatigue strength directly to nominal stresses (beam stresses) as for the simple cases considered in Table 1. However, a reasonable correlation with fatigue test data can be obtained by including the stress concentration of the joint itself in the stresses considered, using a special design S-N curve and stresses in terms of the hot spot stress range. 9

A tubular joint consists of a chord and one or more braces, see Figure 6. The joint is analysed by considering the hot spot stress range in both the brace and in the chord based on the nominal stress ranges in the brace and appropriate stress concentration factors (SCFs). Nominal stresses arise due to the tubes behaving as beam-columns, and may be calculated by frame analysis of the structure. However, the total stress picture in a tubular joint is generally caused by three main influences 1. The basic structural response of the joint to the applied loads in terms of beam stresses (nominal stresses) 2. The stress concentration caused by the actual shape of the tubular joint (geometric stresses) 3. Highly localised stresses caused by the shape of the weld in part of the tube and the weld near the brace-chord intersection (notch stresses) Figure 6 Determination of hot spot stresses in a tubular joint To capture the different types of stresses different types of FE analysis are required. As already mentioned nominal stresses may be determined by frame analysis, geometric stresses can be found using shell elements while the determination of local stresses in the vicinity of the weld requires solid elements. Under cyclic loading fatigue cracks will appear first in the point or region of highest stress range. This point is called the hot spot, and the stress in the hot spot is the hot spot stress. The hot spot stress incorporates the effects of the overall tube geometry (brace and chord). It disregards the stress concentration from the weld itself, but this stress concentration is included in the appropriate S-N curve to be used for tubular joint design. 10

The determination of hot spot stresses can be done very reliably using FE analysis, or for simple joints where parametric formulae exist, based on nominal stresses and (parametric) stress concentration factors. For unusual and complex details an experimental determination of the fatigue strength is sometimes carried out. Influence of the parent material For unwelded components the fatigue strength increases with material tensile strength. For example, plain polished steel specimens, with or without notches, display an increase of the fatigue strength with the tensile strength. However, this is not the case with welded joints, in which fatigue failure occurs by crack propagation from initial flaws. For such welded joints the fatigue strength is independent of the static strength, see Figure 7. Figure 7 Fatigue strength versus static strength of parent material An element in the explanation of this difference is that for the unwelded component, a fatigue crack first has to be initiated and then propagate, while in the welded joint it is only necessary for the crack to propagate. Influence of plate thickness It has been found experimentally that the fatigue strength of test specimens is size-dependent. This is true for both unwelded and welded specimens, see Figure 8. 11

B Figure 8 Size effect on fatigue specimens For the same limiting stress a thin plate experiences a steeper stress gradient than a thick plate under pure bending. Such a steep stress gradient has been found to be less damaging than a small or no gradient. It has also been found that a combination of bending stresses and axial stresses is less damaging than a pure axial stress of the same value. In practical design it has always been assumed that the use of fatigue strengths obtained under axial loading would be safe, even if conservative, under bending stresses. In consequence, fatigue design rules for welded joints have traditionally been based upon results obtained under axial loading. With the issue of the British Department of Energy Guidance Notes (1984), Ref. /3/, it became usual to quantify a thickness correction factor. They based their recommendations on newer test normalised to a thickness of 32 mm as shown in Figure 9. Based on these results a thickness correction factor was introduced of the form S 1 4 tb = SB t (7) where S is the fatigue strength of the joint under consideration, t is the thickness, S B is the fatigue strength of the joint using the basic S-N curve and t is the thickness corresponding to the basic S-N BB curve. Newer design codes often use the term t ref instead of t B. B The practical implication of Figure 9 being, that thicker plates have lower fatigue strength than thin plates. 12

B Figure 9 Thickness corrections, summary of tests For design purposes t B B = 32 mm should be used for tubular joints, while t = 22 mm should be used BB for other types of joints. The choice of reference thickness (t B = 32 mm) for tubular joints was consistent with the data actually used for tubular joints, while for other types of joints the chosen value of t = 22 mm was less obvious since the results were bases on specimen of approximately BB 12.7 mm on average. - Newer design codes represent a further development of the above basic considerations in relation to thickness correction. Influence of residual stresses The majority of welded structures are not stress relieved, so that it is realistic to assume that high tensile residual stresses of yield stress magnitude will exist in some places, both in weld metal and in the adjacent parent material. Since residual stresses are not normally modelled in the structural analysis, this means that the stress range determined coexist with an unknown level of residual stresses. In other words, the actual stress range may be determined as the nominal stress range found in the structural analysis, but the mean stress level would be changed in the presence of residual stresses. In Figure 10 a test specimen with a non-load carrying attachment has been tested under a pulsating tension (R = 0), one series being as-welded and one series being stress relieved. 13

partly tensile/partly compressive pulsating tension Figure 10 Influence of stress ratio and stress relief on fatigue strength It can be seen that the fatigue strength is increased when the specimen is stress relieved, although not so much. In addition a stress relieved specimen with partly tensile and partly compressive stresses (R = -1) was tested. The increase in allowable stress range is significant, and the results confirms that if a joint contains low residual stresses and is subjected to partly compressive stresses, then the fatigue strength is increased. As can be seen in Figure 10, for stress relieved joints there is a significant increase in fatigue strength when subjected to partially compressive stresses as opposed to pulsating tension. It would therefore theoretically be safe to increase the allowable stress range for such joints, and Gurney, Ref. /1/, gives the following advice: A possible method of doing that would be to design for a stress range equal to the tensile component of stress, plus 60 % of the compressive component. However, it is emphasised that a necessary assumption for using the increased fatigue strength in practical design work is that the residual stresses have been reduced effectively. Based on these considerations, the conclusion for practical design work normally is that the fatigue strength of welded joints should be considered independent of the mean stress and be based on the stress range alone. Cumulative damage Most fatigue tests have been carried out under constant amplitude loading and design rules for welded joints have been based on these tests, see Figure 11. 14

popular testing area Figure 11 Definition of terms for S-N testing The parameters that characterize the loading is the stress amplitude S a or stress range S r, and the mean stress S m. S r = S max S min (8) R S S min = (9) max S m Sr 1+ R = 2 1 R (10) In real life many structures experience loads of variable amplitude. For such structures the fatigue design is normally carried out based on Miner s linear damage rule, which states that the fatigue damage D can be found as n n n n = + +... = 1.0 (11) 1 2 i i D N 1 N 2 N i i N i where n 1, n 2 etc. are the numbers of stress cycles of stress range S 1, S 2 etc. expected during the life of the structure, and N 1, N 2 etc. are the corresponding numbers of cycles to failure under constant amplitude loading at those stress ranges, see Figure 12. - The values of N 1, N 2 etc. should be derived from the relevant design S-N curve for the joint in question. The application of Miner s rule requires knowledge of the expected loading/response history because a fatigue design is to be based on the long term distribution of stress ranges for the joint considered. In some instances the loading assumed for design purposes will be specified in the relevant application standard/design basis. In cases where such information is not available, the designer has to make reasonable (conservative) assumptions as to the stress range history, e.g. based on information from similar structures, or from loading/response readings obtained from continuous monitoring. 15

Figure 12 Miner s rule When long term stress histories are known, the corresponding long term stress range history may be established using an appropriate standard cycle counting technique such as Rainflow counting. The stress history can then be expressed in terms of coexisting stress ranges and number of stress cycles. As may be noted, Miner s rule does not account for the sequence of the stress ranges, only for the number of stress ranges. However, experimental data suggest that the sequence of stress ranges does actually influence the fatigue strength of a welded joint. At present no simple and reliable remedy is available to account for this sequence effect. For design purposed a pragmatic correction is typically carried out by reducing the limiting damage to a value less than unity in combination with a modified S-N curve. The modified S-N curve is normally modified to account for the stresses lower than the constant amplitude fatigue limit because in a variable loading history the higher stress ranges tend to propagate small defects and fatigue cracks to such an extent, that they will become capable of being propagated further by even the lowest stress ranges. Nevertheless, the lower stress ranges can usually be assumed to be less damaging so that in many standards it is assumed that the S-N curve is bent from a m = 3 to a shallower slope m = 5 at 10 7 cycles. Experimental data suggest that the design S-N curve should be determined as the mean minus two standard deviations, and this is used by most design standards i.e. log N = log a 2 s m log Δσ (12) where s is the standard deviation of log N. The design S-N curve is therefore often written as log N = log a m log Δ σ (13) where log a = log a 2 s. 16

References Ref. /1/ Gurney, T. R.: Chapter 5.4 Fatigue Design. Constructional Steel Design. An International Guide. Elsevier Applied Science, 1992. Ref. /2/ Fatigue Handbook. Offshore Steel Structures. Edited by A. Almar-Næss. Tapir 1985. Ref. /3/ Ref. /4/ Department of Energy. Offshore Installations: Guidance on design, construction and certification. London: HMSO, 1984. DNV-RP-C203: Fatigue Design of Offshore Steel Structures. October 2008. Det Norske Veritas. 17