SEATECH

KNOWLEDGE LIMITS IN CUMULATIVE FATIGUE ASSESSMENT OF MARINE STRUCTURES

Michel Huther, Guy Parmentier, S. Mahérault

Bureau Veritas, 17 bis Place des Reflets, La Défense 2, 92077 Paris La Défense Cedex

Abstract

Due to the merchant and navy fleets ageing and the extensive use of computer calculations with scantling reductions, fatigue cracking has become the major failure mode of the ship structures. The aim of this paper is to analyse the reality of the used fatigue assessment methods pointing out the limits of the knowledge.

The basic procedure of the fatigue verification of a structural detail are presented and the necessary data for the assessments are identified. Then all steps and the data are reviewed and analysed giving the present State of the Art and the existing uncertainties. Both approaches, Miner sum and crack propagation are considered.

In conclusion, it is reminded that, if all methods are today based on well developed theories, one shall never forget that they are only models of the reality which correspond to the knowledge level at a given time, that they are neither the exact image of the reality but include numerous simplifications and uncertainties. As a consequence, the engineer shall neither have a blind confidence in calculation results but shall have to remain critical and neither loss the physical sense of the things.

Presented at SEATECH – Fatigue of Marine Structures – 19th October 2004 – Brest (France)

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1. INTRODUCTION

The beginning of the use of computers in the years 60s leads to the development of direct calculations of the structures. The enormous progresses in this area allow today to compute ship behaviour on regular and irregular waves, and the resulting static and dynamic loads. The finite elements then allow to compute the stresses in the selected details by the designer.

Considering the tools and their capacities at his disposal, the engineer today has tendency to consider that, with the help of the computer, every thing is possible and allowed. He has also tendency to forget that the tools at his disposal have been developed by other engineers or scientists and that they are, through models which may be not free of errors, the mere translation of the knowledge of the moment on the treated subject.

Numerous incidents or accidents unfortunately recall that the models are only a simplified representation of the reality, but neither the reality itself.

Return experience in marine structures shows that the fatigue failure mode (structural component cracking) is now the main failure mode. It is why we select this example to illustrate the uncertainties which can exist in a model although considered today, by many people, perfectly dealt with.

2. METHODOLOGY

Before the analysis of the state of the knowledge, simplifications and approximations, uncertainties of the today practices, we shall remind the used procedures for the fatigue verification of a welded structural detail.

As for any structural strength verification, three items have to be known: loads and stresses, strength limit state, failure criteria.

2.1. Loads and stresses

The fatigue strength verification of the ship structures requires the determination of the stress ranges at the more loaded points of the ship hull. Two approaches are today possible, a so called spectral analysis, and a so called conventional analysis.

The spectral analysis involves the computation of the ship behaviour on regular waves in various possible loading conditions, wave incidences, ship speeds. It is so possible to determine the sine and cosine components of the transfer functions of the load variations on the different parts of the ship. Using a finite element model it is then possible to calculate the stress range transfer functions at the selected points for the fatigue verification [1-2].

Knowing a transfer function, the sea states in the ocean areas where the ship is planned to operate (significant heights, associated apparent mean periods), it is selected an energy density spectrum formula (Pierson-Moskowitz, Jonswap) and it is calculated the short term and long term distributions of the stress ranges [3-4].

The so called conventional analysis is based on the ship rules stress calculation practices. Loading cases (light ship and loading masses , draughts , accelerations) corresponding to the on wave crest and trough extremes, head and beam seas, are determined and applied to a finite element model. The loads are determined at a conventional probability level to be exceed during the ship life time, 10-5 for a life time of 20 years in the Bureau Veritas steel

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ship classification rules. So, the maximum and minimum stresses are calculated at the selected points for the fatigue verification.

The stress range at the fixed probability level is obtained by the difference between the maximum and minimum stress, the mean stress being the average. The long term histogram is then determined using a conventional probability law, a two parameters Weibull law [5-6].

Fatigue failure being a very local phenomenon, cracks generally appear at weld toes (figure 1), and the stress to be determined is a local stress taking into account the geometrical shapes of the structural details, called hot spot stress. But this hot spot stress does not take into account the weld toe shape (angle and radius) nor its quality.

Figure 1: Fatigue crack example

To integrate these parameters in the fatigue calculation formula, Bureau Veritas decided to use the so called notch stress which takes into account the local shape of the weld toe through the weld angle and the quality through a factor function of the weld type (but weld, fillet weld, etc…).

However the three stress type, nominal, hot spot, notch, are used in codes and standards. In the three approaches the final result allowing the fatigue resistance calculation at a particular point of the structure is a cumulated long term histogram of the stress range giving the number of cycles which exceed a series of stress range levels (figures 2 and 4).

2.2. Strength characteristic

The fatigue phenomenon of a welded component corresponds to the initiation of cracks followed by a rupture when the component is submitted to cyclic loads during a long time, until reaching millions of cycles.

The fatigue life time of a structural detail is determined by tests on samples which represent the considered detail. The loads are applied with various stress ranges ∆S and the number of cycles N at failure is counted.

Generally, for welded details, they are submitted to tension-compression constant stress range sinusoidal cycles, more commonly zero-tensile (0-Smax) cycles to prevent gap problems and unclamping in the testing rig jaws.

The results are presented in form of ∆S-N curves called Whoeler or S-N curves. The curves in a log-log scale diagram are straight lines giving the following formula:

∆Sm N = K (1)

with m = 3 for welded steels.

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The determination of the curve is performed by a linear regression and statistical treatment of log(N) versus log(∆S) which provides the mean curve and the log(K) standard deviation.

As the mean curve corresponds to a probability of failure of 50%, it is not considered acceptable for a project verification, the risk of failure is too important. For ship structures, the practice is to consider acceptable a probability of failure of 2.5% which correspond to a so called design S-N curve at minus 2 standard deviations below the mean curve.

The K value is function of the structural detail and of the used stress, nominal, hot spot or notch stress. The K values used in the classification rules, codes and standards are based on the gathering of numerous test results of different origins for the nominal stress, some few test results and finite element calculations for the hot spot stress, and finite element analysis with correlation with the return experience for the notch stress.

2.3. Failure criteria

Two methods exit to determine the failure risk of a structural component.: the Miner sum and the crack propagation.

2.3.1. Miner sum

The fatigue verification consists to calculate from the stress range histogram and the design S-N curve a scalar which value allows to determine if there is a risk of failure of the considered detail or not.

The world wide used scalar for design of marine structure is the Miner sum. This sum can be obtained by representing the long term histogram by a stair curve, each step giving the number of cycles of the corresponding stress range. The damage, following the Miner rule, is assumed proportional to the ratio of the numbers of cycles ni of the step and Ni of the S-N curve at the same stress range level. The total damage is the sum of the damages at each step (figure 2 and equation 2).

D = nN

i

i∑ (2)

log N

log∆SS-N curve

histogram

∆Si

Ni

ni

D = Σ ni/Ni

Figure 2: Long term histogram and S-N curve

It is commonly considered that there is no failure risk when D ≤ 1 and risk of failure and rupture when D > 1.

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2.3.2. Crack propagation

The determination of the risk of failure is performed by calculation of the propagation of a small crack assumed always existing at the weld toes.

The propagation of a crack is characterised by the increase of its size versus the applied stress cycles. The figure 3 illustrates a propagation in a stiffener flange at a bracket weld toe.

The analysis of laboratory tests on samples have shown that the propagation can be represented by a relationship between the crack length increase ratio da/dN and the stress intensity factor range ∆K (N number of cycles at constant ∆K level).

The commonly used formula is the Paris law (equation 3):

da/dN = C (∆K)m (3)

∆K = ∆S πa f(geometry) (4)

a

Figure 3: Through thickness crack propagation

Using the same histogram than for the Miner sum, it is possible with the equation (4) to integrate the equation (3) and to calculate the final crack length.

The failure criteria is then, either the crack going through the plate inducing a leak, or the rupture of a stiffener flange, or a brittle fracture when K ≥ KC , critical value of the stress intensity factor.

The values of C and KC are material characteristics of same nature than yield or ultimate strength, and are determined by specific tests.

3. STATE OF THE ART AND LIMITS

The different means and parameters necessary to verify the risk of fatigue cracking and failure being given, we shall now see the state of our knowledge at their subject.

3.1. Loads

The loads which induce the stresses are at the origin of the fatigue accumulation and of the risk of cracking are the result of the ship motions on waves and resulting effects.

These loads are therefore, function of the ship characteristics and her loading cases. If the ship characteristics are well defined, the loading cases are less. Effectively, the loading cases are defined by the loading manual, but the frequencies of use, of which the long term

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histogram (figure 2) depends, are not known in advance, they need assumptions for the project development.

For a given loading case, the load long term histogram and therefore of the stress ranges, is function of the encountered sea states during the ship life, so of the navigation zones and seasons during which she will be in these zones [4-7]. It is therefore clear that during the project such knowledge is not available.

For the fatigue verification it has been fixed a conventional sea state statistic the use of which is only justified by the return experience since various tenth of years. The used statistic until recently corresponded to the known North Atlantic observations. This conventional character is shown by the decision some years ago to use a less severe statistic for ships designed for a navigation spread all over the world oceans, although the analysis of the sea states observations of the North Pacific, for example, leads to conclude that they are not significantly different than those of the North Atlantic.

This conventional character is also illustrated by the determined long term distributions of two sister ships of the French Navy after a navigation time of about 5 years (figure 4) [7] which show that at the end of the same number of years the stress histograms are clearly different.

1

10

100

1000

1,0E+0 1,0E+1 1,0E+2 1,0E+3 1,0E+4 1,0E+5 1,0E+6 1,0E+7 1,0E+8

N

S

Figure 4: Two navy ships of same age stress range histograms

The structural load distribution can be evaluated by direct ship behaviour on wave computations with data including ship speeds and wave incidences. To allow the calculations it has been necessary to select a limited number of speeds and wave incidences, although these parameters are continuous functions. Also they depends of the Master decisions versus sea states, a factor which is not possible to take into account in the today models.

Another approximation corresponds to the fact that the computation models of behaviour on waves are based on the spectral approach (transfer function) and so are implicitly linear. But, even for medium wave heights, the ship behaviour becomes non-linear.

Also these calculations are rather heavy and cannot be performed for each ship project. So the applicable rules and standards are based on results obtained on typical ships and provide long term distributions function of some selected ship parameters, distributions which are considered realistic but are not real.

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3.2. Stresses

The local stress calculations from the loads require the used of finite element models, type of calculations which is actually commonly used by the design offices.

The determination of a local stress long term distribution by the spectral approach requires, from real cases computed some months ago, some 400 finite element calculations with refined mesh. It is clear that this method is not practicable for a ship design.

The applied method is therefore an adaptation of the regulatory calculations of the conventional approach. The finite element model is loaded with combinations of static wave pressures, masses of the structural components and loadings with acceleration effects representing the extreme cases, maximum and minimum. The stress range, necessary for the fatigue verification, is simply obtained by the difference between the maximum and minimum corresponding to two or three loading cases and two wave incidences, head sea and beam sea.

To remain in the elastic calculation domain, the level of the loads is so that they correspond on the long term distribution curve to a probability to be exceeded of an order of magnitude of 10-5, an order of magnitude, as there is no exhaustive verification to justify this value. But the whole profession agrees to accept this hypothesis, and the obtained results on existing ships do not deny this assumption.

The stress long term distribution is then obtained by using a 2 parameters Weibull law which passes by the above defined point, with a total number of cycles and a shape factor determined by conventional standard formula.

Those approximations are not the only ones. The finite element models provide local stresses which integrate the effect of the local geometry of the connection, but not the weld shape and less the weld toe quality effects, although they have a great influence on the fatigue life time of the welded joints, as cracks start at this location.

To solve this problem, Bureau Veritas for example, uses the notch stress, nominal stress multiplied by the geometrical stress concentration factor Kg which can be determined by finite elements (hot spot stress), and a notch factor Kf which integrates the weld quality factor (formula 5).

∆Snotch = Kg Kf ∆Snom (5)

with

Kf = λ θ30

(6)

θ weld toe angle λ weld type factor (~ quality)

Considering the lack of knowledge in the above mentioned transformations, the today formula don't allow to take into account the building quality factor which, however, can be very different from a fabrication to another, for example, between manual and automatic welding, in workshop or in open air yard, etc …

To end this review we shall also remind that the finite element calculation results are very sensitive to the modelling, type and size of the elements. As there is no modelling standards, two calculations of the same detail performed by two different engineers will provide two different sets of results (figure 5).

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Figure 5: Results of 3 finite models of the same cantilever beam in bending

3.3. Strength

The difficulties follow. The S-N curves are, at the origin, known in nominal stress range when finite elements provide local stresses which integrate the local geometrical effect of the connections, and the notch stress integrates the weld toe quality effect.

To be homogeneous with the provided stresses by the calculations and formula, the S-N curves will have to be expresses in notch stress ranges.

As it is not possible to obtain the necessary data for finite element calculations of the samples used to determined the S-N curves, (mixture of different results which origin has been lost), the transformations from nominal stress range to notch stress range have been performed using theoretical arguments and expert advises.

As it has been mentioned, the actual S-N curves correspond to linear regressions in log-log of gathered test results providing a mean curve at 50% probability of failure and the standard deviation of the log of the constant K of the equation (1).

When looking back, the practices show that the gatherings were done without verification of the validity by statistical tests, although gathering data with different statistical distributions seriously modifies the linear regression results, and more deeply the standard deviation of log(K) values [8].

Now, as it has been said, the design S-N curve is obtained from the mean curve minus 2 standard deviations, and so this design curve is doubly affected by the gathering practice uncertainties.

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This fact is clearly shown when fatigue tests are performed with perfectly identified fabrications. For good quality welds, the results provide systematically smaller standard deviations that those considered in rules, codes and standards.

3.4. Criteria

The last point of the fatigue verification is the failure criteria, limit of what is acceptable or not.

For the fatigue tests used to determined the S-N curves, the tests being done with small samples, the failure criteria is clearly defined by the sample rupture.

For a large size complex structures, the failure criteria definition is much more difficult and can be different versus the considered component or detail.

In practice there is no real definition of a ship structure failure by fatigue cracking. The practice is to consider that there is failure when the Miner sum calculated with the S-N curves derived from small sample test results reaches the value of 1, without any possibility to say to what crack size corresponds this rule (figure 1 and 6).

That a crack appears when the Miner sum equal 1 is only a hypothesis. In fact, tests performed with similar samples with random loads presenting different irregularity factors show that the failure is observed for Miner sum which can be smaller or greater than 1.

Tests performed in Sweden on welded joints submitted to loads which simulated stress range histograms corresponding to sea states with large, medium and narrow band signals provide Miner sum at failure varying from 0.5 to 2.1 [9-10].

Taking into account the previous remarks, there is a tendency to consider that the use of the crack propagation approach is better and more precise than the Miner sum.

Figure 6: Cracked ship deck

If theoretically the idea is correct, in practice it is not so. In fact the crack propagation calculations require the definition of an initial crack from which the propagation is calculated.

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Studies performed on laboratory test results on welded tubular nodes [11] show that the initial crack size is not a physical parameter which can be measured but a fictitious parameter linked to the used propagation model.

In practice the initial crack size can only be determined using S-N curves results. So at the end, the use of this approach cannot be more precise than the Miner sum.

Another difficulty also exists with the crack propagation approach. The Paris law (equation 3) integration shows that the crack length increase is not linear versus the number of cycles. So the order of application of the stress range histogram cycles greatly influences the final result. For example, studies performed at the Osaka university (Japan), using various possible sequences corresponding to a same ship long term stress range histogram of a ship operating in North Pacific provide the results of the figure 7 [12].

0

2

4

6

8

10

12

0 5 10 15 20

Vayage t ime (year)

Cra

ck

dept

h (

mm

)

Analysis m odel (1)

Analysis m odel (2)

Analysis m odel (3)

Figure 7: Crack propagation curves at an OBO transversal bulkhead bottom

As it can be seen, the calculated life time varies from 9 to 20 years.

Today there is no standard to define how to apply the stress long term histogram cycles and this approach cannot, therefore, be considered applicable for a ship project fatigue verification.

4. CONCLUSION

If the analysis of the existing calculation methods for ship structure fatigue verifications appears very negative, it must be recognized that it is thanks to them that numerous progresses have been possible since the first 300,000 tdw tankers and the large container ship of the end of the 60s.

It is thanks to these methods that the use of the high tensile steels has been developed, use which allows lighter hulls with a same safety level than those of the ordinary steel, that it has been possible to safely design the 500,000 tdw tankers which fatigue failure risk was increased by the springing phenomenon [4-13], and that the oil reserves exploitation at sea has been developed thanks the offshore platforms.

These methods are in fact today at their higher level of knowledge and the given remarks are not critics but aim to remind all the hypothesis and simplifications they have required.

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They allow to remind to the user that he must neither forget that our calculation methods are only simplified representations of the reality and that only the comparison with the return experience can validate them, and neither pure theoretical considerations, as it is too often seen.

In addition, it appears that performing more complex calculations not provides always an improvement in the result precision when the basic hypothesis or data remain the same. It is so the case for the crack propagation approach for which the initial crack size is obtained from S-N curves or the direct calculations on wave for which the encountered sea states by the ship during her future life cannot be known.

The evolution of the methods belongs to the R&D which has to precise the basic data and hypothesis, decrease the uncertainties, improve the models and develop more efficient new methods.

So the engineer must neither have a blind confidence in calculation results, he must always remain critical, neither loss the physical sense of the things and neither reject the remarks he may receive from other engineers without questioning himself. In one word, he must remain humble in front of the extend of our lack of knowledge of the reality.

REFERENCES

[1] J.M. Planeix – Etude dynamique du navire sur houle simple – Bulletin Technique du Bureau Veritas – Aout, Septembre, Octobre 1968

[2] J.M. Planeix, F. Tournan – Efforts tranchants, moments fléchissants, accélérations verticales et mouvements relatifs du navire par rapport à la surface sur houle simple, debout ou oblique – Bulletin Technique du Bureau Veritas – Janvier 1970

[3] M. Huther, J.M. Planeix - Etude des phénomènes dynamiques à la mer – Bulletin Technique du Bureau Veritas – Septembre 1972

[4] Ships, the Sea and the Computer – Bureau Veritas Technical note – Intec Press Ltd - 1975

[5] Vérification en fatigue des détails de structures - Partie B Chap 7 Sect 4, Règlement Bureau Veritas pour la Classification des Navires en Acier – NR 468 1 DTM R00 E/F - juin 2000

[6] Fatigue Strength of Welded Ship Structures – Note d'information Bureau Veritas – NI 393 DSM R01 E – July 1998

[7] M. Bousseau – Différences d'endommagements subis en service par deux bâtiments de surface – IIS/IIW Commission XIII, Doc XIII-1648-96 - 1996

[8] M. Huther – Fatigue testing and Evaluation of Data for Design – IIS/IIW Commission XIII symposium – Université de Tokyo – Avril 2002

[9] R.I. Petersen, H. Agerskov, L. Lopez Martinez, V. Askegaard – Fatigue Life of High-Strength Steel Plate Elements under Stochastic Loading – Technical University of Denmark report – Serie R, No 320 – 1965

[10] R.I. Petersen, H. Agerskov, L. Lopez Martinez – Fatigue Life of High-Strength Steel Offshore Tubular Joints – Technical University of Denmark report – Serie R, No 1 - 1996

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[11] Guides pratiques sur les ouvrages en mer – Assemblages tubulaires soudés - ARSEM, Editions Technip - 1985

[12] Y. Tomita, K. Hashimoto, N. Osawa, K. Terai – Fatigue Strength Evaluation of Ship Structural Members based on Fatigue Crack Propagation Analysis – NAV 2000 conference - Venise - Septembre 2000

[13] B. d'Hautefeuille, M. Huther, M. Baudin, J. Osouf – Calcul de springing par équations intégrales – Nouveautés Techniques Maritimes - 1977