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Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
©2013: The Royal Institution of Naval Architects A-9
FATIGUE ASSESSMENT OF CORRODED DECK LONGITUDINALS OF TANKERS
J Parunov, I Gledić, University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Croatia.
Y Garbatov and C Guedes Soares, Centre for Marine Technology and Engineering (CENTEC), Instituto Superior
Técnico, Technical University of Lisbon, Portugal.
SUMMARY
Fatigue life of deck longitudinals of oil tankers is analysed based on linear elastic fracture mechanics. A parametric
formulation for the estimation of stress intensity factors and the Paris-Erdogan law are applied. Long-term effects of
corrosion are modelled based on regression equations fitted to thickness measurements made during inspections of two
tankers. Parametric studies are performed in order to investigate the importance of the governing parameters of crack
propagation. A comparison of the fatigue analyses performed by linear fracture mechanics and S-N approaches is
presented.
NOMENCLATURE
a Crack depth, m
a0 Initial crack size, m
a/2c Crack aspect ratio
c half of surface crack length, m
C Material parameter (crack growth parameter);
C* Regression constant for corrosion progression
d stress gradient correction factor decay
coefficient
da/dN Crack growth rate
fΔσ Probability density function of stress range
F Joint geometry function
FE Basic crack shape factor
FG Stress gradient factor
FS Front face factor
FT Back face factor (finite thickness factor)
HGSM Hull girder section modulus, m3
I Regression constant for corrosion progression
K Stress intensity factor, MPa m0.5
KC Fracture toughness, MPa m0.5
ΔK Stress intensity factor range, MPa m0.5
ΔKeff Effective stress intensity factor range, MPa m0.5
Kg Hot-spot stress concentration factor
ΔKth Threshold stress intensity factor range, MPa m0.5
Kw Notch factor
m Material parameter
n0 Total number of load cycles
q Stress gradient correction factor decay exponent
r(t) reduction of the midship section modulus
R Stress ratio (R=
min/
max)
SCF Total stress concentration factor at notch
SIF Abbreviation for stress intensity factor
t0 coating life, years
U Crack closure ratio
Greek symbols
α non-dimensioned crack length
β load interaction coefficient
λ Material constant
θ Scale parameter of the Weibull distribution
ξ Shape parameter of the Weibull distribution
Δσ Nominal stress range, N/mm2
Δσeq Equivalent constant amplitude stress range,
N/mm2
Δσ0 Reference stress range, N/mm2
1. INTRODUCTION
Fatigue life predictions of ship structural details have
traditionally been carried out using S-N approach and the
Palmgren-Miner’s rule. The principal objective of such
approach is to estimate the time to failure in order to
ensure a satisfactory design lifetime of ship structural
components.
For fatigue life assessments different procedures have
been developed based on databases of the fatigue
behaviour of welded structural components as a result of
both tests and theoretical investigations [1] [2]. A fatigue
analysis includes the wave induced loading [3], the stress
distribution in the structure [4], the model of fatigue
damage (S–N approach) or fracture mechanics approach
[5] and the probabilistic evaluation of the different steps
to arrive at a safety index or time dependent reliability
[6].
The analysis of stresses is a complex task due to the
complexity of a ship structure. The most accepted
methods are the hot spot stress approach [7] and the
effective notch stress approach [8]. Recently, new
structural approaches were developed in [9] and [10].
Fracture mechanics has been applied to determine the
fatigue life of cracked components [11]. A finite element
analysis procedure for the development of a probabilistic
crack growth model for a stiffened panel has been
proposed in [12], allowing for the existence of multiple
cracks both in the stiffeners and in the plate and
accounting for the correlation between them. The
developed probabilistic model may be used for fatigue
crack growth analysis and is suitable for reliability
assessment of a stiffened panel subjected to correlated
crack growth [13]. Recent developments in fatigue
assessment methods are provided by the ISSC,
Committee on Fatigue and Fracture [14].
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
A-10 ©2013: The Royal Institution of Naval Architects
Welded structures are not perfect and their behaviour
depends on a variety of influential factors, namely
geometric and material properties, loadings, initial or
post built imperfections, deterioration, crack propagation,
denting etc. The imperfections change permanently the
structural capacity of welded structures that initially have
been designed to resist loadings, keeping a certain level
of safety.
Corrosion can also impact the fatigue strength, the
geometry of the structural component changes with time
and as a result, the nominal stress and the stress levels
around the hot spot increase with different intensities
(e.g. Garbatov, et al. [15], Moan and Ayala-Uraga [16]).
The present paper aims to analyse the fatigue life of main
deck structural components of ageing oil tankers using
linear elastic fracture mechanics and a parametric
formulation to estimate stress intensity factors. The
presence of cracks in the deck structure of oil tankers is
an important concern for the marine industry because the
crack propagation will reduce the collapse strength of the
deck-stiffened panels and consequently decrease the
ultimate hull girder capacity of ship structures.
Furthermore, the effect of crack growth can be even
more critical as a result of imperfections that exist in ship
structures as demonstrated in [17]. System reliability
approach of ship hull structures under corrosion and
fatigue is proposed in [18] and [19]. An evidence of the
importance of fatigue for global ship strength is the
“fatigue section modulus”, newly introduced into
Common Structural Rules for Double-hull Oil Tankers,
which is a simplified fatigue control measure against the
fatigue hull girder stresses in the longitudinal deck
structure [20].
In the first section of the paper, two single hull tankers
built in the nineteen-eighties are described as well as
details of the connection of main deck longitudinals with
the stiffener of deck transverse, where a crack is assumed
to initiate. In the following sections, a procedure is
proposed to estimate the fatigue life of main deck
longitudinals based on the Paris-Erdogan law [5]. The
intention is to use a procedure that is of comparable
simplicity as the conventional S-N approach.
The corrosion, represented as a time dependent function,
decreases the plate thickness and affects the midship
section modulus, resulting in an increase in stresses. The
midship section modulus, as a function of time, is
defined by a regression analysis based on corrosion
thickness measurements, taken during regular surveys
[21]. After that, the case study of crack propagation in
deck longitudinals is presented. Parametric studies are
performed in order to investigate the importance of the
governing parameters of crack propagation. Finally, a
comparison of fatigue analyses performed by linear
fracture mechanics and S-N approaches is presented.
2. SHIPS STRUCTURAL DESCRIPTION
The structural detail considered in the present analysis is
located at the intersection between a deck transverse and
deck longitudinal. It is assumed that the crack is initiated
at the vicinity of the weld toe as a two-dimensional
surface crack (see Figure 1 and 2). The structural detail is
subjected to fatigue loading because of the vertical wave
induced bending moment. Two single-hull tankers built
in the early nineteen-eighties with principal particulars
shown in Table 1 are studied here. The first ship, Ship I,
is a tanker whose deck structure is built of high tensile
steel, with the yield stress of 315 MPa, with a T profile
of deck longitudinals. The second ship, Ship II, is built of
mild steel with the yield stress of 235 MPa with a flat bar
profile of deck longitudinals.
In the case of a T profile longitudinal stiffener, the crack
propagation is assessed based on the ratio a/h, where a is
the crack depth and h the flange thickness. The flange
thickness, h=0.019 m is assumed to be an indicator for
achieving the critical crack depth. When the crack depth
reaches the flange thickness, the crack propagates further
as a trough-thickness crack. The through-thickness crack
propagates rather fast in the case of a T profile
longitudinal stiffener as has been confirmed by the
experimental and numerical investigations reported by
Jang, et al. [22].
Table 1: Principal particulars of ships
Ship I Ship II
Length, L [m] 237.00 203.00
Breadth, B [m] 42.00 48.00
Depth, D [m] 20.50 18.00
Block coefficient, Cb [-] 0.80 0.77
Midship cross-section area, A [m2] 4.00 4.32
Moment of inertia of the mid-ship section, I [m4] 269.08 260.33
Position of neutral axis, zT [m] 9.44 8.88
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
©2013: The Royal Institution of Naval Architects A-11
Figure 1: Ship structural detail and crack location, Ship I
Figure 2: Ship structural detail and crack location, Ship II
In the case of a flat bar longitudinal stiffener, the crack
propagation is assessed according to the ratio 2c/h of a
crack width of 2c and a thickness of h of the flat bar
longitudinal stiffener. The thickness of h=0.028 m of the
flat bar stiffener is assumed to be an indicator for
achieving the critical crack depth.
3. CRACK GROWTH ASSESSMENT
The fatigue life may generally be subdivided in a crack
initiation period and a crack growth period, ending in
failure. The assumption adopted in the present paper is
that the crack initiation period in welded joints, that are
not stress relieved, occupies a small part of the total life
and may be neglected [23]. This assumption is supported
by full scale experiments performed by Lotsberg and
Salama [24] on details of FPSOs, which are similar to
those from Fig. 1 and 2. They concluded that the number
of cycles for crack growth is several times (more than 7
times) larger than the number of cycles to initiate 1mm
crack.
The crack growth rate is defined as a function of the
stress intensity factor at the crack tip, assuming that the
yielded area around the crack tip is relatively small.
Based on the Paris and Erdogan [5] law, the crack growth
rate da dN is given as:
m
da CK
dN , ΔK > ΔKth (1)
where the crack growth rate varies from 10-3 to 10-6
mm/cycle for marine structures [23]. C, m are material
parameters, ΔK is the stress intensity factor range at the
crack tip and ΔKth is the threshold stress intensity factor
range.
The Paris-Erdogan equation, as can be seen from Figure
3 [18], remains conservative in region A, and non-
conservative in the region C, which leads to a conclusion
that most of the fatigue life is consumed when the crack
propagates in regions A and B. When a crack enters
region C, i.e. when the stress intensity factor reaches the
fracture toughness Kc, not much fatigue life is left.
The stress range intensity factor range is described by the
following equation:
a .aKF
(2)
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
A-12 ©2013: The Royal Institution of Naval Architects
where
is the nominal stress range, a is the depth of a
surface crack and F(a) is the joint geometry function
accounting for the crack size, shape, and stress gradient
at the crack tip.
If Eq. (2) is inserted in Eq. (1), the Paris-Erdogan law
may be presented in the form suitable for numerical
integration:
2
12
1()
a
m
a
da
N
CaFa
(3)
where N1-2 represents the number of cycles necessary for
the crack to propagate from the depth a1 to a2. Since the
joint geometry function F(a) is a function of a crack size
and the nominal stress range
is a nonlinear function
of time because of the corrosion degradation, an
analytical solution of Eq. (3) is not possible.
Figure 3: Schematic of crack growth as a function of ΔK
Two alternatives are suggested in BS7910 [21] for
numerical integration of the Paris-Erdogan law. The first
option is to use simple, a one-stage crack growth
relationship, whereas the second alternative proposes a
two-stage relationship. Only the former approach, i.e. the
simplified but a conservative one-stage fatigue crack
growth law is used in the present study.
The material parameters C and m for steels, operating in
air or other non-aggressive environments are defined
based on BS7910 [25] and, Lassen and Recko [26].
BS7910 recommends m=3 and the upper bound of C as
1.64·10-11. Lassen and Recko [26] proposed also 5.79·10-
12 and 9.49·10-12 as the mean value and the mean value
plus two standard deviations of C, respectively. The
material parameter C, as defined here, should be used for
stresses in MPa and the crack depth in meters, while for
the units in MPa and mm the material parameter C
should be divided by 31.62. For steel operating in marine
environments, C =7.27·10-11 is recommended [25].
Based on the results of the experimental tests of mild and
high tensile steel, the following expression is suggested
in [23]:
7
1.315 10
28.31m
C
(4)
that agrees favourable with the mean value proposed in
[26].
The stress intensity factor K is the principal governing
parameter for a crack growth in the linear fracture
mechanics approach. It incorporates the effects of the
stresses and the crack size within the crack tip zone. The
stress field at the crack tip may be determined if the
stress intensity factor is known. The stress intensity
factor depends on load and crack geometry (size and
shape) providing a link between a very specific localized
tip response and more global structural conditions. In a
crack propagation analysis, the stress range intensity
factor,
K represents the difference between the
maximum and the minimum stress intensity factors.
The joint geometry function is defined as [23]:
ESTG
F
FFFF
(5)
where FE is the basic crack shape factor that accounts for
the effect of the crack shape:
0.5
1.65
14.5945 2
E
a
Fc
(6)
FS is the front face factor that accounts for a free surface
at the “mouth” of the crack:
',10,16
22
SS SS
aa
FF x f F cc
(7)
where a uniform crack stress distribution FS
’=1 for θ=45°
of the weld angle is considered here.
FT is the back face or a finite thickness factor that
accounts for the effect of a finite plate thickness:
2.454 2 1.005
; 1 0.008 0.0534
2
aa
Fx y yx yx
Tch
(8)
The crack aspect ratio a/2c is assumed from the equation
[23]:
26.712.58ca
(9)
It should be noted that BS7910 enables a more accurate
calculation of the crack aspect ratio as the geometry
function is given at various points along the crack front.
However, in the present study, the growth is simplified
using the one-directional approach by forcing the aspect
ratio in accordance to e.g. Eq. (9). Lassen and Recko [26]
proposed following expression for the crack shape in a
fillet weld joint:
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
©2013: The Royal Institution of Naval Architects A-13
2 3.83 2.92ca (10)
Another option for the crack shape that may be
considered in the present study is the one experimentally
obtained by Jang, et al. [22]:
29.812.84ca (11)
The aspect ratio development, according to different
studies and loading modes is presented by Moan and
Ayala-Uraga [16].
The stress gradient factor FG takes into account the non-
uniform crack opening stresses, i.e. the gradient stress
field at the crack locus. In the case of a stiffener welded
to the flange of the longitudinal stiffener, the stress
gradient factor is calculated as:
Gq
SCF
F
1d
(12)
where α represents a non-dimensioned crack length
defined as ah , q is the stress gradient correction
factor given as 0.6051, d is the stress gradient correction
factor decay coefficient taken as 1.158 and SCF is the
stress concentration factor at the notch, which is defined
according to DnV [27] as:
2.1
gw
SCF K K (13)
where Kg is the hot spot stress concentration factor due to
the gross geometry of the detail considered and Kw is the
notch stress concentration factor due to the weld
geometry.
Variable amplitude fatigue loading, which is inherent to
ship structures because of the random wave-induced
loads, adds complexity to predicting the crack
propagation. The sequence and interaction of loading
events may have a major influences on fatigue life. The
crack growth depends not only on the number of cycles
but also on the exact details of the load history. For
example, even a small tensile overload can produce a
plastic zone at the tip of the crack, forming compressive
residual stresses and crack closure that may retard the
growth of the crack. Compressive overloads have a
different effect as they can form tensile residual stresses
at the crack tip and thus accelerate the growth of cracks.
To account for these effects, detailed step-by-step
analysis would be required.
Such details of the load history, however, for random
wave loading of ship structures are entirely
unpredictable. Consequently, there may be limited gains
from the detailed analysis and for that reason often
simplified models are preferred [28].
One approach to the problem is to define an equivalent
constant amplitude stress range, which should cause the
same amount of fatigue damage, i.e. a crack growth as
the variable amplitude fatigue loading. The equivalent
deterministic stress range is defined by adjusting
empirically the constant β in the following expression:
1/
0
d
eq f
(14)
where
f
is the probability density function of
the stress range
and β is the interaction coefficient,
which can be assumed as β=m=3 in the case of the
Palmgren-Miner’s fatigue damage approach.
4. CASE STUDY
The risk of failure of ships at sea has risen due to
corrosion deterioration of ship hull structures. Numerous
statistical data suggest that corrosion is the primary cause
of sea accidents, especially for aged ships. Severe
corrosion may cause cracks on deck along the ship’s
service life, which consequently may result in the loss of
the entire ship. The hull girder section modulus (HGSM)
is the basic measure of ship’s longitudinal strength,
which is decreasing as a function of time because of
corrosion deterioration. Ship Classification Rules assume
a constant loss of HGSM during the ship’s service life.
Although such approach may be practical, but it is not a
realistic one since HGSM is a time-dependant non-linear
function [21].
The approach adopted here is to consider a corrosion
model that provides the trend that is derived from the
corrosion mechanism and then it fit to the field data.
Corrosion thickness measurements for the two single hull
tankers were performed during periodic dry-dockings
and regular inspections of ships in service after 10, 15
and 20 years. Based on corrosion thickness
measurements, HGSM has been determined accounting
for the coating life. The reduction of the midship section
modulus, r(t) as a result of a structural deterioration,
calculated as a function of HGSM, is defined as [21]:
10
H
GSM t
rt HGSM
(15)
where HGSM(0) is the hull girder section modulus as
built and HGSM(t) is the hull girder section modulus at
any year t during the service life.
Eq. 15, which is directly based on the real corroded plate
thickness measurement, is fitted to the following
regression equation:
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
A-14 ©2013: The Royal Institution of Naval Architects
Table 2: Crack propagation descriptors
Features T profile (Ship I) FB profile (Ship II)
Initial crack size, a0 [m] 0.001 0.001
Interaction coefficient, β 3 3
Shape Weibull parameter, ξ 0.94 0.98
Stress range corresponding to 10-5 exceeding probability, Δσ0 [MPa] 167.3 131.7
Crack growth parameter, m 3 3
Crack growth parameter, C 5.79 x 10-12 5.79 x 10-12
Number of stress cycles in 1 year NL 2.8 x 10-6 2.9 x 10-6
*0
I
rt C t t (16)
where t0 is the equivalent coating life of the entire
structure, C* and I are regression constants, which are
defined as 5 years, 0.6 and 0.58 for the Ship I and 6
years, 0.6 and 0.86 for the Ship II respectively. The
HGSM reduction as a function of time, as shown in
Figure 4, demonstrates that the loss is still well below
10%, which is the permissible reduction of HGSM
according to the IMO regulations [21, 29].
It must be emphasized that the present analysis is
considering only the longitudinal stresses because of
wave-induced vertical bending moments, which are the
dominating in the main deck structure of oil tankers. The
structural components in the ship side and bottom will be
subjected to lateral pressure inducing other fatigue
cracks, propagating in different directions and having
thus a different effect on longitudinal strength is
expected.
Figure 4: HGSM reduction as a function of time
A crack propagation analysis is performed according to
the parameters specified in Table 2. The initial crack size
and the interaction coefficient are assumed according to
Almar-Naess [23] and Lassen and Recko [26]. The
vertical wave induced bending moments as well as the
shape parameter ξ of the Weibull distribution are taken
from IACS [20]. Applying Eq. (14), the equivalent stress
range is calculated. Furthermore, the stress range is
increasing in time because of the HGSM loss. The crack
propagations in deck longitudinals are presented in
Figure 5. Clearly, the crack propagates more rapidly in
Ship I than in Ship II. This may be explained with the
fact of using high tensile steel for Ship I, so the wave-
induced stresses are higher for Ship I, as has been shown
in Table 2. However, details of the geometric function
used for the calculation of stress intensity factor, as the
finite with correction factor (Eqn. (8)), may also cause
this difference in the crack propagation rate. Geometry
functions for the two examples are shown in Figure 6,
being in remarkable agreement with the geometry
function for weld toe crack as presented in [24].
The fatigue life of the deck longitudinal of Ship I is about
16 years, which is lower than the normal design lifetime
of ship structural details. This may be due to the fact that
mean compressive stresses are not taken into account in
the calculation of the crack propagation in Figure 5. The
effect of mean compressive stresses may increase the
fatigue life considerably, as discussed in Section 5.
Figure 5: Crack propagation in deck longitudinals
The parametric analysis is carried out to study how the
crack propagation is affected by the input data. The
parameters considered are the initial crack size a0, the
crack shape 2ca, the stress ratio R, the interaction
coefficient
, the crack growth parameter C, the stress
concentration factor SCF, the threshold stress intensity
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 5 10 15 20 25
r (t)
Years
Ship II
Ship I
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2
c/h
a/h
Year s
Ship I
Ship II
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
©2013: The Royal Institution of Naval Architects A-15
factor Kth , the crack closure parameter U and residual
stresses.
Figure 6: Geometry functions for two ships
In the design phase, the initial crack size is generally
unknown. Three different initial crack sizes are assumed
in the present study: 0.5 mm, 1 mm and 5 mm. The crack
propagation is presented in Figure 7 for Ship I and Ship
II respectively. Obviously, the variation of the initial
crack size may largely influence the fatigue life. It is to
be mentioned that DNV-RP-C203 [30] recommends a
crack depth of 0.5 mm if other documented information
about the crack depth is not available.
Figure 7: Crack propagation as a function of the initial
crack size for Ship I (up) and Ship II (down)
The crack shape influences
K (Eqns. 6-8) significantly
and thus the fatigue life. In principle, the aspect ratio
should be calculated by a consistent two-dimensional
crack growth model [25]. For practical purposes,
however, the crack growth is simplified by the one
directional approach, using the forcing function of the
crack aspect ratio. Most often, a semi-elliptical shape is
assumed described by the aspect ratio 2ca. The
prediction of the 2ca evolution during the crack growth
is usually empirically done based on the measurements
of the fractured surface of welded joints.
In the present study, three different forcing functions for
the crack aspect ratio, given by Eqns. 9 to 11, are
analysed. The results of the analysis are presented in
Figure 8. Here, the different behaviour of the T
longitudinal profile and the flat bar longitudinal is
evident. For the T-profile longitudinal (Ship I), the
assumed crack shape influence both, the crack
propagation rate and the final crack size, while for the
flat bar (Ship II), the crack propagation rate is not
substantially affected.
Figure 8: Crack propagation as a function of the assumed
forcing function of the crack aspect ratio for Ship I (up)
and Ship II (down)
The analysis of the effect of the stress ratio R=
min/
max
may be performed by the modified Paris-Erdogan law,
which also accounts for the fracture toughness, Kc [23]:
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
Geometry function (F)
a/h (Ship I) 2c/h (Ship II)
Ship I
Ship II
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
a/h
Year s
a0=0.001
a0=0.0005
a0=0.005
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2c/h
Year s
a0=0.001
a0=0.0005
a0=0.005
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
a/h
Year s
2c=6.71+2.58a
2c=3.83+2.92a
2c=9.81+2.84a
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2c/h
Year s
2c=6.71+2.58a
2c=3.83+2.92a
2c=9.81+2.84a
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
A-16 ©2013: The Royal Institution of Naval Architects
1
m
c
da C K
dN R K K
(17)
The fracture toughness Kc is the upper bound of stress
intensity factor and represents the critical value of the
stress intensity factor when the crack propagates in a
rapid unstable manner. For an accurate definition of Kc,
knowledge of the precise metallurgic and chemical
characteristics of the material is required [22]. Eqn. (17)
is also known as the Forman equation [22] and its
drawback is that the growth parameters C and m are
different from the ones governing the original Paris-
Erdogan law, Eqn. (2). Another commonly used
expression to account for the stress ratio 0R is the
Walker relationship [28].
1
1
m
m
da C K
dN R
(18)
where C and m are the Paris coefficient and the slope for
R=0, respectively, and
is a material constant with a
typical value around 0.5 [28]. The influence of the
variation of the stress ratio R on the crack propagation is
presented in Figure 9, using Walker Eqn. (18). The rate
of the crack propagation clearly increases with increasing
the stress ratio.
Figure 9: Crack propagation as a function of the ratio R
for Ship I (up) and Ship II (down)
The variable amplitude of loading adds complexity to the
problem of predicting fatigue life. The variable stress
history is transformed into an equivalent constant stress
range that causes the same fatigue damage, i.e. the same
crack growth. If the stress range is fitted to the Weibull
distribution, then the equivalent constant amplitude stress
range may be approximately expressed as [23]:
1
1.6 1.26
0.0076 x
e
eq
(19)
where 1x
,
is the shape parameter and
is the
scale parameter of the Weibull distribution. The
interaction coefficient β takes into account the effects of
the interaction between stress cycles. When the
interaction coefficient is β=3, which in the present case
equals the exponent m, the interaction between the stress
cycles is not taken into consideration, i.e. it corresponds
to Palmgren-Miner approach. When β=2, it corresponds
to the equivalent stress range calculated by the root-
mean-square rule (RMS). The scale parameter
is
defined as:
0
1
ln 0
n
(20)
where Δσo is the reference stress range, which is defined
as the one exceeded once out of no cycles and no is the
total number of cycles associated with the fatigue service
life.
The effect of these two interaction coefficients to the
crack propagation is compared in Figure 10 up and down
for Ship I and II, respectively. Obviously, β=2 leads to a
considerably slower crack propagation for both ships.
Whereas RMS has a defined physical meaning related to
energy in many applications, it has no physical
significance in fatigue. Hence, it is not likely that such
approach would be a rational choice for calculating the
equivalent stress range. It would be more appropriate in
many cases, β=3 as such the equivalent stress results in
the same damage in S-N approach as corresponding
irregular load history. As has been discussed in [23],
however, for many offshore structures the interaction
effects are so important that failure may occur for the
Palmgren-Miner approach [31] much less than unity.
With that respect, even using β=3 should be taken with
caution.
Although the described approach for calculating
equivalent stress range has no sound theoretical basis, it
is a practical empirical approach and hence
recommended by some authors like Hughes and Paik
[23].
The crack growth parameter C considered in this study is
specified in [25, 26]. The mean value, mean value + 2
SD and the upper bound of C, applicable for steels
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
a/h
Yea rs
R=0
R=0.3
R=0,5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2c/t
Yea r s
R=0
R=0.3
R=0.5
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
©2013: The Royal Institution of Naval Architects A-17
operating in the air are compared and the results are
presented in Figure 11. As may be seen in Figure 11, the
variation of the parameter C has a significant influence
on the crack propagation rate. Expectedly, when C equals
to the upper bound defined by BS7910 [25], the crack
propagation is the fastest. It also appears from the results
that for Ship II, that the choice of parameter C could
have a decisive impact on the conclusions of the crack
propagation assessment.
For the structural detail considered here, the SCF is 2.1
according to DnV [27]. The SCF may be reduced either
by completely removing the web stiffener or by
providing a stiffener with a soft bracket. It should be
pointed out that the former solution is employed
occasionally in the deck area of oil tankers, while the
latter solution is usually used for the side shell
longitudinals. The comparison of the crack propagation
for the two different SCFs is presented in Figure 12 for
Ship I and II, respectively. The lower SCF in Figure 12
corresponds to the case of the flat bar stiffener with a soft
bracket.
Figure 10: Crack propagation as a function of interaction
coefficient β for Ship I (up) and Ship II (down)
The effect of the threshold stress intensity factor, ΔKth is
also analysed. The assumed threshold value reads
ΔKth=2MPa m0.5 [32]. The stress range intensity factor is
above 2.6 for Ship I while the whole range of the stress
intensity factors is between 1.16 and 1.75 for Ship II.
Consequently, the threshold stress intensity factor does
not influence the crack propagation of Ship I, while the
crack propagation in Ship II would not start at all if the
threshold was taken into account.
Figure 11: Crack propagation as a function of crack
growth parameter C for Ship I (up) and Ship II (down)
One of the factors that affects the crack growth rate is the
crack closure effect. Accounting for this effect, fatigue
cracks remain closed during a part of the loading cycle
under constant and variable amplitude loading. The
effective stress intensity factor range, including the crack
closure effect, is defined in Eqn. 21 using the crack
closure ratio U:
max
max
1
;1
op
eff op
K
K
KK KUKU R
(21)
where Kop is the SIF at which the crack opens (or closes).
According to Lassen and Recko [26], for welded steel
joints it may be assumed that for full alternating loading
(R= -1), Kop/Kmax=0.2 implying U=0.4, while for R=0,
Kop/Kmax=0.25 and U=0.75. The crack closure effect to
the crack propagation is shown in Figure 13 for Ship I
and II, respectively. It appears that the crack closure
effect may slow down the crack propagation
considerably.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
a/h
Yea r s
β=2
β=3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2c/h
Yea r s
β=2
β=3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
a/h
Years
C=5.7957E-12
C=9.49E-12
C=1.64e-11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2c/h
Yea rs
C=5.7957E-12
C=9.49E-12
C=1.64e-11
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
A-18 ©2013: The Royal Institution of Naval Architects
Figure 12: Crack propagation as a function of SCF for
Ship I (up) and Ship II (down)
However, it should be emphasised that there is a large
uncertainty regarding the validity of the mentioned crack
closure ratios and for that reason, the crack closure is
seldom used in practical calculations. Although the crack
closure effect is very important for the shallow cracks, it
is often neglected because the closure effect in a surface
crack is generally less significant than the one observed
in through-thickness crack geometries [22].
The presence of residual stresses is another important
source of uncertainty of the fatigue crack growth near
welded areas. For a crack starting from the weld toe end,
the stress ratio depends on the residual stress distribution
near the toe end. The stress ratio can be adjusted as:
min
max
res
T
res
KK
RKK
(22)
In the case of tensile residual stresses, RT>R and thus the
residual stresses have a detrimental effect on the crack
propagation. Compressive residual stresses may cause a
cracking closure and thus Eq. (21) should be considered.
The residual stress intensity factor depends on the
residual stress distribution and on the geometry of the
welded joints. The quantitative assessment of residual
stresses at the web stiffener weld toe end has not yet been
reported [22].
For details, similar as those studied in the present paper,
Lotsberg and Salama [24] measured tensile residual
stresses. However, they also explained that shake-down
of residual stresses is expected in the first year of ship
service as a consequence of maximum loading. Shake-
down effect changes residual stresses from tensile to
compressive causing crack closure and reduced fatigue
damage. To allow for the influence of residual stresses, a
simple and conservative low for R > 0.5 is proposed by
BS7910 [25].
Figure 13: Crack propagation as a function of the crack
closure ratio U for Ship I (up) and Ship II (down)
5. COMPARATIVE ANALYSIS
The S-N approach often covers the total fatigue life from
the crack initiation to final failure. Coupled with a Paris-
Erdogan law [5] to determine the fatigue life consumed
during the crack propagation, the S-N approach then
enables an approximate estimate of the fatigue life
corresponding to the fatigue crack initiation [14].
However, as discussed in Section 3, the crack initiation
period is neglected. Therefore, the fatigue life calculated
by the S-N approach is directly comparable to the one
calculated by the crack propagation law.
The S-N approach used in the present study has been
defined based on IACS [20] for the F-class of S-N curve.
The only difference, comparing to CSR, is that in the
present study, the stresses increase due to the corrosion
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
a/h
Year s
SCF=2.1
SCF=1.88 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2c/h
Year s
SCF=2.1
SCF=1.883
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
a/h
Year s
U=1
U=0.75
U=0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
2c/h
Year s
U=1
U=0.75
U=0.4
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
©2013: The Royal Institution of Naval Architects A-19
wastage of the hull girder section modulus is determined
for each year during ship service life. Thus, the stress
range increases as the section modulus is reduced
because of the corrosion wastage. The accumulated
damage, according to the Palmgren-Miner’s rule is
calculated for each year of the ship's lifetime and is
added to the accumulated damage from previous years as
has been described by Frost, et al. [33].
The parameters used in the crack propagation assessment
are those indicated in Table 4, except that the crack
growth parameter C is taken as 9.49·10-12, as
recommended by Lassen and Recko [26] for mean plus
two standard deviations. This is consistent with the fact
that in the S-N approach, the design S-N curves specified
in CSR are used, rather than the mean S-N curves [20].
The comparison between the S-N and the crack
propagation approaches is shown in Figure 14 up and
down for Ship I and II, respectively. In the case of Ship I
(see Figure 14, up), the accumulated damage closely
follows the crack propagation. For both approaches, the
predicted fatigue failure occurs much before than the end
of the ship’s service life. It should be pointed out that
such short calculated fatigue life may be explained by the
effect of the mean compressive stresses that are not taken
into account. Actually, fully loaded oil tankers are in the
sagging condition in still water, leading to mean
compressive stresses in the main deck structure [34].
Consequently, the fatigue life will increase considerably,
as the compressive stresses have a favourable influence
reducing to the effective range of stresses [20]. A
comparison of S-N and crack propagation approaches,
taking into account the correction for the mean stresses is
presented in Figure 15. Mean compressive stresses allow
a reduction of the effective stress range by about 40%
[20].
The mean stress effect has to be accounted for in the
fatigue damage assessment as has been recommended in
DnV CN 30.7 [27] and DnV-RP-C203 [30] and for
tankers as has been postulated in IACS [20]. However, in
the first two documents, beneficial influence of the mean
compressive stresses is considered only for base material,
which is not significantly affected by residual stresses
due to welding. Lotsberg and Salama [24] presented
evidences from in-service experience and full scale
measurements on FPSOs that the fatigue capacity of the
hot spots exposed to mean compressive stress are well
above those for tensile load cycling.
With Ship II (see Figure 14, down), the crack
propagation rate is lower than the rate of damage
accumulation and the agreement of the two approaches is
less favourable compared to Ship I. A fatigue life bigger
than 25 years indicates that the crack does not propagate
sufficiently to reduce the load-carrying capacity of the
deck longitudinal.
Figure 14: Crack propagation and accumulated damage
for Ship I (up) and Ship II (down)
Figure 15 Crack propagation and accumulated damage
for Ship I with correction for mean stresses
6. CONCLUSIONS
The work presented here describes a practical application
of fracture mechanics approach to the fatigue assessment
of main deck longitudinals of an oil tanker. The crack
propagation at the connection of the deck longitudinal
and the web stiffener of the transverse deck girder in two
single hull oil tankers is studied. The proposed procedure
takes into account the stress increase due to the corrosion
degradation of the midship section modulus. The
parametric analysis showed in which way and to what
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
DM
i
a/h
Year s
Crack propagation C=9.49e-12
DMi for Ship I
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
DM
i
Year s
2c/h
Crack propagation C=9.49e-12
DMi for Ship II
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
DM
i
a/h
Year s
Crack propagation C=9.49e-12
DMi for Ship I
Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
A-20 ©2013: The Royal Institution of Naval Architects
extent any individual input parameter affects the crack
propagation.
The initial crack depth, crack aspect ratio, the crack
growth parameter C, the stress concentration factor
(SCF), the interaction coefficient,
, the stress ratio R and
the crack closure effect strongly affect the rate of the
crack propagation. The only parameter that may readily
be controlled during the design is the SCF, as a better
structural design resulting in a reduction of SCF and
consequently a significant reduction of the crack
propagation rate.
Regarding the comparison of the results obtained from S-
N and fracture mechanics approaches, it can be
concluded that the two approaches agree satisfactory.
The ship that did not meet the requirements of fatigue
life implied by the S-N approach also had a crack
propagation rate outside acceptable limits. The ship that
met the requirements of the fatigue assessment implied
by the S-N approach also produced satisfactory results
with fracture mechanics, as the crack propagation rate
was slow.
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Trans RINA, Vol 155, Part A1, Intl J Maritime Eng, Jan-Mar 2013
A-22 ©2013: The Royal Institution of Naval Architects
