Discussion of Some Important Parameters in Fatigue Loading Calculation for Ship Structural Design

YANG Peng,GU Xue-kang

(China Ship Scientific Research Center,Wuxi 214082,China)

1 Introduction

After developing and investigating for many years,IACS has formulated CSR-OT[1]rules for oil tankers and CSR-BC[2]rules for bulk carriers,respectively.However,there are many differences between them and it is confused and inconvenient for users to apply them in ship research and design practice.Recently,IACS has been developing a general rule called harmonized common structural rules(HCSR)for bulk carriers and oil tankers.Many classification societies and institutes have been involved in this project to make effort to give more rational solutions in the new rules.A vast number of theoretical and numerical analysis,comparative study and empirical estimation have been made to eliminate differences between CSR-OT and CSRBC.And some new processes,including fatigue damage assessment approach which is claimed more reasonable,are presented in the first version of HCSR.

For example,both CSR-OT and CSR-BC assume that long-term characteristics of stress range abided by Weibull distribution and representative value of the stress is corresponding to exceeding probability of 10-4.The Weibull shape parameter given in CSR-OT depends on location of structural component and ship length,whereas the shape parameter in CSR-BC is constantly equal to 1.0.But in HCSR,based on primary investigation of some members,the representative stress range has been set as the value corresponding to exceeding probability of 10-2and the shape parameter is equal to 1.0.To investigate the rationality of the new parameters given in fatigue loading assessment of HCSR,theoretical analysis and numerical simulation for fatigue life estimation are carried out.The exceeding probability of the stress range that has maximum contribution to fatigue damage in long term extent,the Weibull shape parameter that has minimum influence on the exceeding probability and their sensitivities to reverse slope of S-N curve,have been deeply discussed.Some different conclusions to HCSR have been made.

2 Stress range for maximum fatigue damage contribution

2.1 Long-term stress distribution

In general,a two-parameter Weibull distribution[3]is used to describe the long-term stress response range in marine structures.This distribution reads

where k and ξ are scale and shape parameters.Its corresponding probability density function(pdf)is:

Nolte and Hansford[4]derived a closed-form expression for the fatigue damage under the assumption of two-parameter Weibull distribution long-term stress ranges and S-N data given by Ns=AS-m:

where S represents the long-term stress range with pdf f（s）,Nsis the number of cycles corresponding to stress range S,A and m are the material parameters,N is the total number of cycles.Fdif（s）is defined fatigue damage intensity factor as a function of stress range s by

From expression(3),it shows that the peak of Fdif（s）is corresponding to stress range s which has the maximum contribution to structural fatigue damage.

Fig.1 shows the distribution of fatigue damage intensity factor for semi-submersible in different exceeding probability levels.In Fig.1(a)and(b),S-N inverse slope m are 3 and 5 respectively,and the scale parameter k is 50.Fig.1 reveals that the exceeding probability of stress range corresponding to maximum distribution of structural fatigue damage does not lie in 10-2(but,HCSR recognizes as 10-2);meanwhile,maximum contribution has relation to Weibull shape parameter.

Fig.1 Contribution of stress range corresponding to various exceeding probabilities

Mathematically,when setting the derivate of Fdif（s）equal to zero,the stress range corresponding to the maximum contribution to fatigue damage can be determined.The solution is

Substituting Eq.(5)into expression of exceeding probability,which becomes

When m=3,Eq.(6)could become into

Fig.2 shows the relationship between stress range smaxcorresponding to maximum contribution to structural fatigue damage and Weibull shape parameter ξ.While ξ=1.0,the exceeding probability corresponding to maximum contribution to fatigue damage is 10-1.3.If Weibull shape parameter varies from 0.5 to 2.0,then exceeding probability varies from 10-2.17to 10-0.89.Especially,the exceeding probability is 10-2corresponding to shape parameter 0.55.In addition,Moan et al[5]calculated this kind of exceeding probability for FPSO assuming m equals to 3 under stress range distribution in gamma distribution,which revealed exceeding probability was 0.02.

Fig.2 Exceeding probability corresponding to stress range of maximum contribution(m=3)

Fig.3 Exceeding probability corresponding to stress range of maximum contribution(m=5)

When m=5,Eq.(6)could become into

Fig.3 shows the relationship between stress range smaxcorresponding to maximum contribution to structural fatigue damage and Weibull shape parameter ξ.While ξ=1.0,the exceeding probability corresponding to maximum contribution to fatigue damage is 10-2.17.If Weibull shape parameter varies from 0.5 to 2.0,then exceeding probability varies from 10-3.91to 10-1.30.Especially,the exceeding possibility is 10-2corresponding to shape parameter 1.11.

Above all,when Weibull shape parameter is 1.0,if m=3,then exceeding probability is 10-1.3corresponding to stress range which has maximum contribution to structural fatigue damage.If m=5,the exceeding probability is 10-2.17.Thus,conclusion given by HCSR that exceeding probability is 10-2corresponding to stress range which has maximum contribution to structural fatigue damage does not have universal meaning.

Alternating stress ranges are composed of low stress range and high stress range during ship service life,which depend on ship style,length of ship,loading status,routine,wind and wave,component location,nodal style,etc.It has comparative difference about stress range level and in actual ship structure,E.g.CSR-OT supposes the Weibull shape parameter range is 0.6～1.1 when length of ship varies from 150 m to 500 m.

Furthermore,the stress range level given by HCSR corresponding to exceeding probability 10-2is equal to the maximum wave load encountering in every fifteen minutes during shipping.That is to say,this characteristic wave loads which in low sea states make maximum contribution to fatigue damage.However,lots of study show wave loads in moderate sea states make maximum contribution to fatigue damage,so standpoint of HCSR does not have proper physical meaning.

3 Weibull shape parameter influence to fatigue life

3.1 Theoretical analysis

Fatigue stress cycle characteristicwhich is exceeded with probabilityduring ship service life is defined as

From Eqs.(1)and(9),expression of scale parameter k can be obtained as follows:

Structural fatigue life using S-N curve method in Weibull distribution model is

Substituting expression N=fL·T into Eq.(11),the equation of structural fatigue life is

where fLand Δ are mean frequence of stress cycles and design fatigue cumulative damage.Non-dimensional fatigue lifecan be obtained by dividing T by fatigue life T0which is the value when shape parameter equals to ξ0.It reads

Eq.(13)shows effects of shape parameter to fatigue life only depending on NLand m,in other words,only depending on stress range exceeding probability,reverse slope of S-N curve and actual shape parameter.

3.2 Shape parameter effect

Munse et al[6]compared practical stress range distribution with Weibull distribution,and,found Weibull shape parameter was between 0.7 and 1.3.Soares and Moan[7]calculated wave bending moment for lots of ships with linear theory using North Atlantic sea state statistical data,they had obtained shape parameters after fitting Weibull distribution.Soares and Moan gave the expression of shape parameter as follows:

where L is ship length.The expression given by DNV[8]is

The expression given by Cui[9]is

If m=3.0,and Weibull shape parameters of stress range long-term distribution are 0.6,0.8,1.0,1.2,respectively.Fig.4 shows the relationship betweenand ξ,which can be calculated by Eq.(13).When ξ equals tois 1.0 from Eq.(13),which means different curves of various exceeding probabilities have intersection.Fig.4(a～d)and Tab.1 show various exceeding probabilities when Weibull shape parameter takes minimum influence to fatigue life,respectively.At the same time,exceeding probability varies larger synchronously following shape parameter,but rate will turn down.

Fig.4 Effects of Weibull shape parameter on fatigue life(m=3)

Tab.1 Exceeding probabilities with various shape parameters(m=3)

3.3 Influence of S-N curve reverse slope

While assessing structural fatigue life with S-N curve method,m equals to 5.0 under low stress range level,m equals to 3.0 under high stress range level.Thus,it is necessary to study effect of Weibull shape parameter on fatigue life with different value of m.Assuming m=5.0,and Weibull shape parameters of stress range long-term distribution are 0.6,0.8,1.0 and 1.2,respectively.Fig.5 shows the relationship betweenand ξ,which can be calculated by Eq.(13).Fig.5(a～d)and Tab.2 show various exceeding probabilities when Weibull shape parameter has minimum influence to fatigue life,respectively.Comparing the results of m equals to 3.0 and m equals to 5.0,it indicates the exceeding probability is smaller when m becomes larger.

Tab.2 Exceeding probabilities with various shape parameters(m=5)

Fig.5 Effects of Weibull shape parameter on fatigue life(m=5)

3.4 Influence of dual linear format of S-N curve

Assuming Q is the point of intersection of the two lines with corresponding stress range SQand fatigue life NQ,the expression of dual linear of S-N curve is

where,m and m′are reverse slopes,Nsis cycle number of stress range,A and A′are constants.The formula of fatigue cumulative damage with this S-N curve reads

where fLand Δ are mean frequence of stress cycles and design fatigue cumulative damage.Non-dimensional fatigue lifecan be obtained by dividing T by fatigue lifewhich is the value when shape parameter equals to ξ0.It reads

Eq.(20)shows the effects of shape parameter on fatigue life depending on SL,NL,m,m′,SQand actual shape parameter.

Since value of stress range SLrepresents the long-term characteristic of the stress range distribution and the parameters of S-N curve are different at high or low stress range level,it is necessary to investigate the effect of shape parameter on fatigue life under various distributions of stress range.If ξ0=1.0,and the parameters of S-N curve are chosen from the D curve of DNV(2005)[10]（SQ=52.63 MPa,m=3,m′=5）,Fig.6 shows the effects of shape parameter on fatigue life with various exceeding probabilities and long-term distributions of stress range.Tab.3 shows the exceeding probability corresponding to minimum effect is decreasing when SLbecomes smaller.The results coincide with the results in single line format of S-N curve.Eq.(13)shows the effects of shape parameter are independent of SL.Meanwhile,with the single line format of S-N curve,when m equals to 3,the exceeding probability is 10-1.5;m equals to 5,the exceeding probability is 10-2.2.When SLis comparatively large the long-term stress range mainly lies on high stress zone,the exceeding probability corresponding to minimum effect of shape parameter on fatigue life approaches the results with m=3.When SLis comparatively small the long-term stress range mainly lies on low stress zone,the exceeding probability corresponding to minimum effect of shape parameter on fatigue life approaches the results with m=5.

Tab.3 Exceeding probabilities with various SL

Above all,exceeding probability level corresponding to minimum effect of shape parameter on fatigue life mainly depends on reverse slope of S-N curve,shape parameter and the long-term distribution of stress range in practice.Meanwhile,the commendation given by HCSR,i.e.while exceeding probability level is 10-2the variance of shape parameter variant has a minimum effect on fatigue life,does not have general meanings.

Fig.6 Effects of shape parameter on fatigue life with different SL

4 Conclusions

Based on theoretical and numerical investigations of parameters in fatigue loading estimation process carried above,some conclusions can be achieved:

(1)When Weibull shape parameter and reverse slope of S-N curve are 1.0 and 3,respectively,the exceeding probability corresponding to stress range which has maximum contribution to structural fatigue damage is 10-1.3,whereas for reverse slope of 5,the corresponding exceeding probability is 10-2.17.The exceeding probability is related to the stress level experienced by the structure concerned.

(2)When Exceeding probability and reverse slope of S-N curve are 10-1.5and 3,respectively,fatigue life estimated is not sensitive to Weibull shape parameter,whereas for reverse slope of 5,the corresponding exceeding probability changes to 10-2.2.The choice of the Weibull shape parameter is also related to the stress level experienced by the structure concerned.

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