O R I G I N A L A R T I C L E
Probabilistic modelling of the ultimate strength of ship plates
with non-uniform corrosion
A . P . Teixeira • C. Guedes Soares • G . W a n g
Received: 11 November 2010/Accepted: 14 August 2012/Published onhne: 28 September 2012 © J A S N A O E 2012
Abstract This paper presents the results of a parametric study o f probabilistic modelhng o f the ultimate strength o f ship plates w i t h non-uniform corrosion represented by ran-dom fields. The load-shortening behaviour of the plates w i t h non-uniform reduction o f thickness due to coiTOsion under longitudinal compression is obtained using a general-putpose nonlinear finite element analysis program. A n o n l i n e a r time-dependent c o i T o s i o n model is used to define the probabifistic characteristics o f the random fields based on corrosion data measured i n plate elements at different locations o f b u l k carriers. Based on the probabifistic models derived by Monte Carlo simulation, equations to predict the mean and the 5 % characteristic value o f the ultimate strength o f plates w i t h non-uniform corrosion ai^e developed. Finally a regression equation is proposed to take into account the effect o f non-u n i f o r m corrosion patterns i n the predictions o f the non-ultimate strength o f plates w i t h u n i f o r m coiTosion.
K e y w o r d s N o n - u n i f o r m corrosion • Plates • Ultimate strength • M o n t e Carlo simulation • Ship structures
A . P. Teixeira • C. Guedes Soares ((Sl)
Centre f o r Maiine Technology and Engineering (CENTEC), Institute Superior T é c n i c o , Technical University of Lisbon, A v . Rovisco Pais, 1049-001 Lisbon, Portugal
e-mail: guedess@raar.ist,utl.pt A . P. Teixeira
e-mail: teixeira@mar.ist.utl.pt G. Wang
American Bureau of Shipping, Singapore, Singapore e-mail: gwang@eagle.org
1 Introduction
The random characteristics of several parameters that influence the ultimate strength o f plates have m o t i v a t e d many authors to propose probabilistic models f o r the plate strength, as discussed b y Guedes Soares [ 1 ] and more recentiy b y Teixeira and Guedes Soares [2]. These models describe the effect o f the variability of the parameters and the uncertainty o f the models i n the assessment o f the ultimate strength o f the structures and can be used as tools to derive probabilistically based design rules as demon-strated by Guedes Soares [ 3 ] . These are essential elements that conttibute to the time dependent r e l i a b i l i t y o f ship stiaictures [4, 5 ] .
I n these studies, the basic structural parameters were assumed to be discrete, allowing them to. be represented b y single-valued random variables. This assumption is v a l i d f o r quantities that are concentrated at discrete points or u n i f o r m l y distributed i n space. However, most parameters i n a structure are distributed i n space. Examples o f such parameters are distributed loads, material and geometric properties and the reduction of thickness due to corrosion that varies over the length o f a beam or the area o f a plate. Such quantities cannot be expressed as single r a n d o m variables, but only as a collection o f spatially correlated random variables or, more appropriately, as r a n d o m fields.
Random fields o f f e r an important mathematical f r a m e -w o r k to represent parameters -w i t h continuous spatial fluctuation over a given domain. Its relevance f o r engineering applications deiives f r o m the possibility o f s i m u -lating spatial distribution o f a particular parameter f r o m a l i m i t e d number o f samples available at a f e w locations.
A c o m m o n application o f random fields i n stochastic analysis o f structures is on the m o d e l l i n g o f geometrical imperfections, w h i c h are interpreted as r a n d o m spatially
distributed deviations f r o m a perfect geometrical shape [ 6 ] . M o s t et al. [7] have performed a stability analysis o f a cylindrical panel w i t h stochastic loading and geometrical imperfections represented by conditional random fields, since the location o f the supports was assumed to be deterministic [8, 9 ] . Its parameters were determined via a stochastic interpolation scheme, w h i c h is based on the m a x i m u m l i k e l i h o o d principle [ 8 ] . Kolanek and Jendo [10] have derived a method f o r defining random field models o f geometrical imperfections o f rectangular plates satisfying clamped boundary conditions that consists o f a develop-ment o f the method considered by M o s t et al. [ 7 ] .
The loss o f material due to coiTosion is also an i m p o r -tant factor that affects the l i f e o f metal structures. Corro-sion is i n essence a process o f an uncertain nature governed by many variables and, therefore, only probabilistic models can describe the coiTosion process itself and its effect o n the strength o f structural components.
A l t h o u g h the probabilistic characteristics o f the corro-sion process have been recognized, there are not many studies o f probabifistic m o d e l l i n g o f the strength o f cor-roded structural elements [11]. The first studies have assumed a constant corrosion rate, leading to a linear relationship between the material loss and t i m e , and a u n i f o r m reduction o f plate thickness due to c o i T o s i o n . This approach has been adopted by several authors f o r proba-bilistic m o d e l l i n g of the ultimate strength o f corroded plates and f o r the refiabifity analysis o f plates [ 1 2 - 1 6 ] . However, i n addition to the general wastage that is reflected i n the generalized decrease o f plate thickness, the iTucroscopic variations on the surface o f the metal tend to cause different forms of c o i T o s i o n and also variations i n the corrosion rate over wide or small areas [ 1 7 ] , w h i c h can be represented by random fields.
A d i f f e r e n t problem is the extremely localized types o f corrosion, as p i t t i n g coiTosion, w h i c h are not considered i n the present paper. This problem has been studied by several authors and includes the definition o f the shape o f the corrosion pits, the characterization o f the actual state o f p i t t i n g coiTOsion and the assessment o f its effect on the strength o f the structural elements (e.g. [ 1 8 - 2 1 ] ) . However, f r o m a theoretical point of view, configurations o f p i t t i n g con-osion can also be represented by stochastic simulation o f random fields. Since pits often f o r m i n groups at discrete locations and somedmes as isolated pits, several random field models, each one w i t h a particular correlation struc-ture, must be combined to obtain a representative pattern o f the general and localised corrosion.
The use o f random fields f o r coiTOsion m o d e l l i n g was first suggested by Orisamolu [ 2 2 ] . I n this w o r k a stochastic finite element methodology has been employed f o r the probabilistic response estimation, reliabifity evaluation and residual strength assessment. M o r e recently, Teixeira and
Guedes Soares [23] have studied the ultimate strength of c o i T o d e d plates w i t h spatial distribution o f corrosion also represented by random fields, w h i c h were discretized using the Expansion Optimal Linear Estimation method proposed by L i and Der Kiureghian [24]. This preliminary study has indicated that the strength o f plates w i t h spatial distribution o f c o i T o d e d thickness is usually l o w e r than the one obtained f o r u n i f o r m corrosion and, therefore, i t is expec-ted that this better representation o f the coiTOsion patterns w o u l d influence the probabilistic models o f the residual strength o f ship plates under in-plane compression, as w i l l be demonstrated.
I n the present paper, a nonlinear time-dependent corro-sion model proposed b y Guedes Soares and Garbatov [25] is fitted to corrosion data measured i n plate elements at d i f f e r e n t locations o f b u l k carriers. The model has been fitted and validated w i t h operational data f r o m tankers [26] and f r o m b u l k carriers [27] and has been extended to take into account the effect o f the environment [ 2 8 - 3 0 ] . This model is used to define the time variation o f the probabi-listic characteristics o f the random fields o f coiTosion. However, f o r any point i n time i n w h i c h the average thickness wastage is given b y t h a t model, this study con-siders that the thickness reduction is not u n i f o r m but instead varies spatially i n a random manner. Based on the probabilistic models derived by M o n t e Carlo simulation, equations to predict the mean and the 5 % characteristic value o f the ultimate strength o f plates w i t h n o n - u n i f o r m coiTosion are developed. F i n a l l y , a coiTection equation is proposed to take into account the effect o f n o n - u n i f o r m coiTosion patterns i n predictions of the u l t i m a t e strength o f plates based o n u n i f o r m coiTOsion.
2 T i m e dependent probabilistic modelling of the corrosion i n ship plates
2.1 Corrosion requirements and inspections
CoiTOsion is one o f the m a j o r issues f o r the structural condition assessment o f aged ships. The c o n d i t i o n of the steel structure o f ships is subject to requirements o f clas-sification societies, as stipulated i n the rules f o r classifi-cation, and to international requirements being controlled b y statutory regulations specified i n several I M O (Inter-national M a r i t i m e Organization) resolutions. I n particular, the Enhanced Survey Programme (ESP) Guidelines [ 3 1 ] , Resolution A . 7 4 4 (18), define the requirements f o r plan-ning, execution and reporting f o r h u l l surveys o f o i l / chemical tankers, obo and b u l k carriers. T h e ESP survey schedules and extent o f surveys are based o n the under-standing that the real condition can only be revealed w i t h detailed close-up inspections a n d extensive thickness
measurements, and that the deterioration process happens s l o w l y over time. ESP is based o n two principal criteria: the condition o f the coating and the extent o f structural coiTosion. O f primary importance is when a coating has been f o u n d to be i n a "less than good" condition or when a structure has been f o u n d to be substantially coiToded, i.e., a wastage between 75 and 100 % o f the allowable d i m i n u -tion f o r the structural member i n ques-tion or a measured thickness between /„et + 0.5 m m and ?net for vessels b u i l t under the l A C S Common Structural Rules (CSR) [32].
For inspections to be carried out effectively, suitable means o f access to the vessel's structure are required. I M O has recently addressed this issue w i t h the adoption o f res-olutions M S C . 151(78) and M S C . 158(78) into S O L A S Regulation II-1/3-6 on "Access to and w i t h i n spaces i n the cargo area o f o i l tankers and b u l k caniers". O i l tankers and b u l k earners constructed after 1 January 2006 are required to c o m p l y these new requirements that w i l l ensure that vessels can be properly inspected throughout their lifespan.
M o r e recentiy, a Performance Standard f o r Protective Coatings (PSPC) has been adopted b y I M O (Resolution MSC.215(82)) that became mandatory on 1 July 2008 through an amendment o f the S O L A S 11-1/3-2 regulation. The I M O PSPC is intended to reduce the corrosion encountered i n steel ships, and applies specifically to the protective coatings used f o r dedicated seawater baUast tanks i n a l l types o f ships, and also i n the double-side skin spaces o f bulk carriers. I n addition, a new PSPC require-ment, stemming f r o m the recent adoption o f I M O Reso-l u t i o n MSC.288(87), w i U become e f f e c t i v e 1 January 2013 f o r the crude o i l tanks o f o i l tankers. The a i m o f both PSPC standards is to ensure a u s e f u l coating l i f e o f 15 years, w h i c h w i l l influence considerably the corrosion levels o f the o l d ships i n the future.
I n addition to mandatory inspections, several industry driven inspections have been created such as vetting inspections (e.g. G D I , S I R E and Rightship) performed on o i l tankers, chemical tankers, gas earners and b u l k carriers on behalf o f o i l majors or other cargo owners or on behalf o f the ship owner. The G D I (Chemical Industry Institute) inspections are p r i m a r i l y p e r f o r m e d on chemical tankers whereas SIRE (Ship Inspection Report Program) inspec-tions are perfonned by o i l companies o f the O C I M F ( O i l Companies International M a r i n e F o r u m ) and cover o i l tankers. B o t h C D I and S I R E inspections are based on a standardized questionnaire covering statutory requirements (based on the international conventions), and other requirements introduced b y C D I participants or by some o i l majors during the SIRE inspections. Rightship is a ranking system w h i c h combines i n f o r m a t i o n obtained thi^ough vetting inspections, port state control, casualties, ship particular information and shipowner i n f o r m a t i o n . The R i g h t s h i p service includes vessel inspection along w i t h the
data analysis. The inspections cover tankers and b u l k car-riers but are p r i m a r i l y f o r dry b u l k earners. Particularly concerned w i t h the aging bulker fieet, RightShip also promotes a standard Condition Assessment Program ( C A P ) covering the ship f r o m deck fittings to h u l l structure.
This b r i e f overview o f the recent actions taken by the maritime industry clearly shows that coixosion and the condition assessment o f ships are important issues, and that recent requirements and instruments are expected to have a significant long-term impact on the coiTosion levels o f the ships.
2.2 Thickness measurements and corrosion m o n i t o r i n g
Traditionally, the classification societies are the entities responsible f o r the largest amount o f coiTosion measure-ments obtained during the structural condition assessment of aged ships. Corrosion monitoring o f ship structures is typically p e r f o r m e d thi'ough ultrasonic thickness measure-ments ( U T M ) carried out by qualified operators using specialized measurement equipment. Procedures f o r U T M are w e l l established and mostly governed i n general terms by l A C S requirements [ 3 3 - 3 5 ] and by the i n d i v i d u a l classification societies at a more detailed level.
The purpose o f thickness measurement is to establish i n conjunction w i t h a visual examination that the condition o f the existing structure is, or w i l l be after the required repairs, fit f o r continued service during the subsequent survey interval. The gauging requirements include mea-surements w h i c h are used to v e r i f y remaining longitudinal strength, transverse sections, as w e l l as measurements o f areas k n o w n to be potential problem areas, m a i n deck plates and w i n g and water strakes.
A c c o r d i n g to the current practice i n the execution o f the thickness measurements on board, the surveyor w i l l direct the gauging operation by selecting locations such that readings taken represent, on average, the condition o f the structure f o r that area. W h e n a single reading is not rep-resentative o f the coiTosion i n a particular area o f plating, additional readings are taken and assessed together w i t h close visual examination by the surveyor f o r determination of the extent o f the corrosion pattern. I n this case, the surveyor assesses the average condition based on obser-vations o f the structure f r o m visual examination and the gauged readings.
A key question concerns w h i c h parameters are appropriate f o r evaluating coiTosion: average thickness, m i n i -m u -m thickness, p i t intensity, etc. Class societies generally require average thickness, p i t m a x i m u m depth and p i t intensity (as a percentage o f the plate surface).
There is a trend towards a more quantitative d e f i n i t i o n of corrosion intensity, as the current practice o f gauging, even i f c a n i e d out according to the rule requirements, may
not represent the reality [36]. Teixeira and Guedes Soares [37] have investigated h o w the number o f thickness mea-surements and their location influence the correct repre-sentation o f the corrosion patterns and the correct assessment o f the ultimate strength o f the c o i T o d e d plates. For this purpose the n o n - u n i f o r m coiTosion patterns were represented by random fields, as suggested by Teixeira and Guedes Soares [ 2 3 ] . They showed that taking the average o f the measurements to represent the c o i T o d e d plate thickness, i n alternative to a more correct representation o f the corrosion patterns, can lead to optimistic assessment o f strength o f the stractural elements.
I n spite o f these requirements, there are not many studies presenting i n a systematic and coherent way statistical information about measurements o f thickness reduction due to corrosion i n maritime structures. Questions related to data confidentiaUty and difficulties i n transfonning dispersed data sets into a sti-uctured w o r k have contributed to the fact that only a smafi part o f the o w n e d data is available. Despite these problems, some r e l e v a n t studies w i t h statistical i n f o r -mation about the eoiTOsion i n m a i i t i m e structures can be identified, as reviewed b y Melchers [17].
2.3 Corrosion m o d e l l i n g
T y p i c a l l y , coiTosion models assume a constant coiTosion rate corresponding to a linear relationship between material loss and time [12, 14, 38]. However, experimental w o r k and systematic measurements o f corrosion i n structures i n operation done by several authors have shown that there is a nonlinear dependency between the coixosion rate and time [25, 3 9 - 4 2 ] .
Yamamoto and I k e g a m i [ 4 2 ] , have seen that corrosion i n structural members is consequence o f an extremely large number o f pits g r o w i n g progressively and i n d i v i d u a l l y and have proposed a corrosion model w i t h three phases: period when the anticorrosive paint coating is effective (including the period o f generation o f active p i t t i n g points), period when p i t t i n g points are progressed and period when cor-rosion process stops and c o i T o s i o n rate becomes zero.
Guedes Soares and Garbatov [25] have also suggested a model that characterizes the coiTOsion process by three phases: durability or l i f e o f the coating, a transition period, and a nonlinear c o i T o s i o n progression. Paik et al. [41] considered also three phases but allowed the corrosion progression to be represented by linear, convex or concave type curves. Melchers [40] proposed a nonlinear corrosion model that divides the corrosion process into f o u r stages: initial c o i T o s i o n , oxygen d i f f u s i o n controlled by corrosion products and nficro-organic g r o w t h , l i m i t a t i o n on f o o d supply f o r aerobic activity and anaerobic activity.
The model developed by Guedes Soares and Garbatov [25], w h i c h is adopted here, is able to describe the gradual
reduction o f the corrosion rate w i t h time u n t i l i t reaches zero, when the oxidation on the surface o f the plate inhibits the evolution o f corrosion itself. A c c o r d i n g to this model, the reduction o f thickness o f the plate due to corrosion (d) is given i n f u n c t i o n o f time T by:
d{T) = 0 r < Tc
where is the l i f e t i m e o f the anti-corrosion protection o f the plate and TJ is the transition t i m e o f the corrosion process, defined as the period o f time after the beginning o f the c o i T o s i o n process up to 63 % o f the m a x i m u m thick-ness (doo). The parameters that define the nonlinear cor-rosion model, namely T C , t t and d^o, have been estimated f r o m data on con'osion i n plate elements o f several b u l k earners i n operation collected by Paik et al. [43], A total o f 2973 measurements o f thickness reduction due to coiTOsion were reported f r o m plates and stiffeners at d i f f e r e n t loca-tions o f the cross section o f 44 b u l k earners. Table 1 illustrates the type o f data f o r m a t reported by Paik et al. [43] f o r the particular case o f b o t t o m plates, w h i c h basi-cally consists i n the number o f measurements o f corrosion levels ranging f r o m 0 to 3 m m as f u n c t i o n o f time.
T h e data have been transformed i n t o piecewise time averages d{T) (see last c o l u m n o f T a b l e 1) and then fitted to the nonlinear time-dependent m o d e l proposed b y Guedes Soares and Garbatov [ 2 5 ] , b y estimating the m o d e l parameters w i t h a least-squares approach f o r each data set o f a particular l o c a t i o n . Figure 1 shows the nonlinear fitting o f the corrosion m o d e l to plates at d i f f e r e n t locations and summarizes the m o d e l parameters f o r each case.
One can see that the evolution o f the c o i x o s i o n process w i t h time depends strongly on the location o f the plates. For inner b o t t o m plates ( I B P ) , the c o i T o s i o n process is clearly nonlinear and the con'osion model fits adequately the annual averages o f thickness reduction due to coiTO-sion. However, f o r submerged plates, namely b o t t o m plates (BP), lower w i n g tank side shells ( L W T S S ) and side shells (SS), the corrosion rate is approximately constant and, therefore, the relationship between thickness reduction and ship age is close to linear. F i n a l l y upper deck plates ( U D P ) and upper w i n g tank side shells ( U W T S S ) demonstrate a heavy thickness loss due to c o n o s i o n i n a short period o f time, c o i T C s p o n d i n g to the l i f e t i m e o f the anti-coiTOsion protection, after w h i c h c o n o s i o n remains approximately constant. I t should be noted that wastage o f inner b o t t o m plates o f b u l k carriers is a combination o f corrosion and mechanical abrasion, w h i c h results f r o m cargo handling. However, although the physics is not necessarily the same, the non-linear model can s t i f i be used, as illustrated i n F i g . 1.
T a b l e 1 Wear o f thickness due to conosion in bottom plates (673 measurements) [43]
Time, T (year) Depth o f coiTosion, d (mm) Time average, d{T)
Time, T (year) ~ 0 - 0 . 5 - 1 . 0 ~ 1 . 5 - 2 . 0 - 2 . 5 - 3 . 0 Time average, d{T) 4.75-5.00 16 7 0.15 6.0-6.25 17 10 0.19 6.25-6.5 2 26 I 0.48 6.5-6.75 14 13 4 0.47 7.25-7.5 19 7 1 0.17 7.75-8.0 10 4 0.14 8.0-8.25 2 11 9 5 0.81 10-10.25 20 17 10 3 1 0.51 11-11.25 0 4 8 3 0.97 11.75-12 0 9 1 0.55 13-13.25 10 55 1 0.43 13.75-14 0 14 0.50 14-14.25 1 7 2 3 3 2 1.17 14.25-14.5 4 22 0.42 14.5-14.75 0 2 10 8 1.15 14.75-15 14 25 0.32 15-15.25 14 70 6 0.46 16-16.25 5 15 13 0.62 16.25-16.5 3 14 4 4 6 14 1.42 18-18.25 4 3 3 0.95 20-20.25 5 10 14 14 15 11 1.41
F r o m a strength point o f v i e w one o f the important differences between the plates i n deck and bottom w i l l be the lateral pressure [ 4 4 ] .
T h e statistical analysis o f the corrosion measurements also allows the estimation o f the average variance o f thickness reduction due to corrosion at the d i f f e r e n t loca-tions, w h i c h has been f o u n d to be constant (i.e. not time dependent). A s illustrated i n the last c o l u m n o f Table 2 the variability o f the thickness reduction {d) is very large. I n average terms, f o r each set o f plates, the coefficient o f variation o f the thickness reduction ranges f r o m 0.42 to 0.81 f o r double b o t t o m and bilge plates, respectively. I t should be noted that the lower value o f the coefficient o f variation (0.42) occurs f o r the set o f plates at the inner bottom, w h i c h have the larger mean value o f thickness reduction due to corrosion.
3 Discretization of random fields
I n probabilistic analyses, i t is convenient to represent the random fields i n terms o f a discrete set o f random v a r i -ables. This is k n o w n as discretization o f random fields. Since these random variables are obtained f r o m the same random field, there is statistical coiTelation among them
and the mathematical understanding o f the correlation relationships i n a random field is therefore essential i n order to coiTectly discretize the field.
Several methods have been proposed f o r discretization o f random fields, i n particular, f o r use i n finite element r e l i a b i l i t y analysis o f structures [ 4 5 ] , as reviewed b y L i and D e r K i u r e g h i a n [ 2 4 ] , Ditievsen [ 4 6 ] , S c h u ë l l e r [ 4 7 ] , M a t -thies et al. [48] and Chen and Guedes Soares [ 4 9 ] .
I n this study, the random field o f coiTosion is discretized using the Expansion O p t i m a l Linear Estimation method ( E O L E ) proposed by L i and Der K i u r e g h i a n [ 2 4 ] . The E O L E discretization method assumes that a Gaussian random field H{x) can be defined as a linear f u n c t i o n o f a vector h = {H{x\),.. .,H{XN)} o f N nodal values H(xi) o f the original random field given by:
H{x) = «(.r) + b^ix)H{xi) = a{x) + b^{x)h (2)
;=1
where a(x) is a scalar f u n c t i o n o f x, b{x) is a vector f u n c t i o n o f x and i n the number o f nodal points i n the domain.
A s s u m i n g that the vector h o f random variables can be expressed i n terms o f its spectral decomposition, i t is possible to determine the functions a(x) and b(x) that m i n i m i z e the variance o f the error V A R [ 7 Ï ( . Ï ) - i ï ( . ï ) ] .
1.80 1.60 t '•''^ é 1-00 0.80 .2 0.60 g 0.40 U 0.20 3.00 ^ 2.50 2.00 1.50 u p o 0.50 0 Data ottom plates Non-lin ar model fil E
o ottom plates O ^ 0 O 0 0 ) 5.0 10 0 15 0 20 0 25
Exposure time in years (t)
0.00 0.0
0 Data
Non- inear mode! fit
Lower s oping plates^
0
° Z '
/ °
5.0 10.0 15.0 Exposure time in years (t)
5.0 10.0 15.0 Exposure time in years (t) 1.60 n 1.40 -'s 1.20 •a 1.00 --S c u 0.80 --o _o 0.60 -SO. U 0.40 -O u 0.20 0.00 -° Data
Non-li Tear model fit Upper sl apling plates—
°° /
P o
0.0 10.0 15.0
Exposure time in years (t)
20.0 25.0 3.00 2.50 •X3 2.00 _ C S, 1.50 c O 1.00 O
fa
O U 0.50 0.00 0.0 1.60 1.40 1.20 1.00 0.80 0.60 0.40° Dala Inner Bottom plates " Non-1 near model fit
1 °
5.0 10.0 15.0 20.0 Exposure time in years (r)
25.0
CJ 0.20 0.00
- — 0 Dala - — 0 Dala
Non-linear model fit
0
° '
L o w e r wing taiiK siae siietjs
1
0.0 5.0 10.0 15.0 20.0 Exposure time in years (t)
25.0 1.60 1.40
?
g 1.20-'— -3 1.00-é 0) 0.80-C O 0.60-Ofa
0.40 -O U 0.20- 0.00-o DalaNon-lii ear model fit
- upper wing t ank side
shells-0 o
°
J
1 °0.0 5.0 10.0 15.0 20.0 Exposure time in years (t)
25.0 3.00 2.50 •a 2.00 g. 1.50 O o 0.50 0.00 0.0 1 O Data 1
Uoner deck olates
1 ' o
°
5.0 10.0 15.0 20.0 Exposure time in years (t)
25.0
F i g . 1 Nonhnear coixosion model f o r different plate locations
subjected to H{x) being an unbiased estimator o f the H{x) i n the mean, i.e., E[H{X) - H{x)] = 0. Hence, according to the called Expansion O p t i m a l Linear Estimation method ( E O L E ) , the approximated Gaussian random f i e l d is given by [ 2 4 ] :
ff(.v)=M(.v) +
E4k'^rCH(.v)/,
(3)where //(.v) is the mean f u n c t i o n o f the random f i e l d , 0,-, 0,- are the eigenvalues and eigenvectors o f the
T a b l e 2 Parameters f o r the nonlinear corrosion model f o r different plate locations
Plate location No. o f
measurements
COV(rf)
1-Bottom plates (BP) 673 5.0 1.65 14.37 0.740 2-Inner bottom plates (IBP) 556 5.0 1.91 2.53 0.420 3-Lower sloping plates (LSPCITE) 220 5.0 1.54 3.18 0.629 4-Lower w i n g tank side shells 152 5.0 10.11 119.69 0.814
(LWTSS)
5-Side shells (SS) 383 5.0 1.10 6.29 0.706
6-Upper w i n g tank side shells 201 5.0 0.73 1.53 0.729 (UWTSS)
7-Upper sloping plates (USP) 432 5.0 0.82 3.25 0.638 8-Upper deck plates (UDP) 361 5.0 1.23 0.07 0.674
A l l plates 2978 5.0 1.07 1.64 0.809
- — - 4 - L W T S S
covariance matrix C/,;, o f h, {C,-; / = 1. • •>'} is a set o f r independent standard normal distributions (zero mean, u n i t variance and zero correlation) and Cu{x)ii is a /• x 1 vector containing the covariances o f H{x) w i t h the ele-ments of h.
Therefore, this method requires first the d e f i n i t i o n o f a g r i d o f points w i t h coordinates {xi = xi,.. .,XN} that defines the dimension N x N of the covariance m a t r i x Chi, whose terms are given f r o m the autoconelation f u n c -tion p and the standard devia-tion a o f the random field (i.e. C,,„ = {pix,x') • a{x) • (j{x'); x,x' = A ' I , . . .,XN}). C/,,, defines the eigenvalue p r o b l e m {Chh<j>i = di<i>i) ^^'^ number o f terms r w i t h the largest eigenvalues that corre-spond to the number o f random variables ({(,•; / = 1 - . • ' ' } ) that are used to represent the random field. Then the value o f the discretized random field H at .v {H{x)) is given b y E q . 3, where the vector Cfj{x)ii contains the covariances between x and the grid points .r,- (i.e. {CH{x)h = p{x,Xi) • a{x) • a{xi); i = 1.. .)•}) (see L i and Der K i u -reghian [24] f o r more details o f the E O L E method).
T h e variance o f the Expansion O p t i m a l L i n e a r Estima-tion method is given by:
YAR[H{X) - H { X )
YAR[H{x)]=J2iU'^C -H{x)h (4)
Thus, the variance en-or i n the E O L E method can be estimated by comparing the variance o f random field {a^{x)) w i t h the variance o f the represented field, i.e..
w i t h ' V A R [ H ( . r ) ]
VAR[7^(.v) - H(x)\ = L \ x ) - wi^'^HixV.
(5) The high-level efficiency o f this approach i n the sense that i t requires a smaU number o f random variables to represent the random field within a given level o f accuracy makes this model paiticulaiiy useful for stochastic analysis and, therefore, i t w i l l be used on further calculations. For the coixelation structure used i n the present study, i t was found that only 15 random vaiiables ({C,-; / = ! . . . ; • = 1 5 } ) are sufficient to assure less than 5 % eiTor i n the discretization (i.e. eiT(.v) < 5 % ) .
I n the present study the reduction o f plate thickness due to coiTOsion is described by a homogeneous l o g n o r m a l random field H\„{x), i.e., both the mean value and the standard deviation o f the random field are constant over the plate surface and equal to / i j ^ and trin, respectively. T h e lognormal random field H\n{x) is defined by a transfor-mation o f the Gaussian field H{x) as:
Hi„{x)=exp[H{x)] (6) I n this case, the mean value fi{x) = p and standard
deviation a{x) = a o f the underlying Gaussian homogenous random field H{x) must be first calculated f r o m the mean value p^^ and standard deviation a\n of the lognormal r a n d o m field H\n {x) by:
Fig. 2 Realizations o f the lognormal random field o f thickness reduction due to coiTosion (plate dimensions: a = b = 1000 mm, Lognormal random field: = 1.02 m m , a,„ = 0.88 m m , = 03a)
^ = \ A ^ f l + 4 ) a n d
^, =ln(/4)-^
(7)V \ Ihn/ ^
The probabiHstic characteristics o f the homogeneous lognormal random field that describes the random patterns o f reduction o f plate thickness due to corrosion are established on the basis o f the analysis o f the coiTosion data presented i n Sect. 2.3. Therefore, f o r a particular plate, the mean value is g i v e n as a f u n c t i o n o f time by the non-linear c o i T O s i o n model (Eq. 1) (i.e. / t ] n ( r ) = d{T)) w i t h the corresponding parameters presented i n Table 2. The standard deviation is also defined f o r each plate location f r o m the coefficient o f variation o f the thickness reduction due to c o i T o s i o n , presented i n the last c o l u m n o f Table 2 (i.e. ai„{T)=d(T)-CÖy{d)).
For the numerical analysis presented i n this paper, the f o l l o w i n g autocorrelation f u n c t i o n has been considered f o r the Gaussian random field {H{x)), w h i c h is converted f r o m the o r i g i n a l l o g - n o r m a l random field {Hia{x)) b y E q . 6:
p{x,x')=e 'i (8) i n w h i c h the parameter Zc is a measure o f the rate o f
fluc-tuation o f the random filed, c o m m o n l y k n o w n as the cor-relation length. This parameter can be estimated by investigating the dependence between the measurements at d i f f e r e n t distances f r o m each other. However, the precise locations o f the corrosion measurements have not been registered i n the available coiTOsion data. Therefore a representative value f o r the correlation length o f 0.3 m , corresponding to 30 % o f the plate length, was chosen i n the present study, w h i c h allows the study o f the e f f e c t o f the spatial representation o f the coiTosion patterns on the ultimate strength o f the plates. Figure 2 illustrates three realizations o f the random field o f reduction o f plate thiclcness due to the corrosion process.
4 Ultimate strength of steel plates with non-uniform corrosion patterns
4.1 Characteristics o f the models
The ultimate strength calculations were carried out f o r simply supported square plates w i t h length (a) and w i d t h
/
b T,. = T J = Rj, = R j = 0 b b N T J = R, = R j = 0 b ^ T , = T J = R , = R J = 0 b r a XF i g . 3 Imposed displacement (S) and boundary conditions o f the plate mode (T indicates translational and R rotational constraints)
0.0 0.2 0.4 0.5 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Axial displacement/Yield displacement
F i g . 4 Longitudinal normalized stress-displacement curves o f the non-con-oded plates
(£>) o f 1000 m m (a/b = 1), and slenderness (b/t) f r o m 40 to 60 (/ f r o m 25 m m to 16.7 mm) w i t h l o n g i t u d i n a l edges restrained against transverse displacement. I n fact reahstic b/t ratios o f double b o t t o m plates o f actual ships range f r o m around 35 to 55, and therefore the value o f b/t — 60 is considered as a l i m i t i n g value.
The results are represented by the normalized ultimate strength o f the plate ( c f ) = ffu/ffo) that coiTesponds to the m a x i m u m value (cr,,) o f the l o n g i t u d i n a l stressdisplacement curve o f the plate under inplane l o n g i t u d i n a l c o m -pression obtained b y nonlinear finite element analysis using the A N S Y S software d i v i d e d by the y i e l d stress o f the plate (ag). The load is a u n i f o r m prescribed longitu-dinal displacement { ö ) applied along the transverse edge o f the plate, as illustrated i n F i g . 3. A n i n i t i a l geometric
imperfection shape (wz) represented by only one compo-nent o f a Fourier series was considered i n this study:
. Tlx ny
M'z = Wo sm — sm — (9) a b
where a and b are the plate dimensions and Wq is the average value o f the amphtude o f the shape o f the i n i t i a l imperfection given as f u n c t i o n of the plate slenderness { f i ) by (e.g. [50, 5 1 ] ) :
VFoA = 0 . 1 ^ w i t h p = b/tZZJË (10)
where t is the i n i t i a l plate thickness, and and E are the y i e l d stress and Young's modulus, respectively. Figure 4 illustrates the normalized longitudinal stress-displacement curves o f the noncorroded plates under inplane l o n g i t u -dinal compression.
4.2 Nonlinear FE modelling o f steel plates w i t h n o n - u n i f o r m con-osion
A preUminaiy study of the effect o f the finite element (FE) modelhng of the reduction of plate thickness due to coixosion on the ultimate strength of the plate was canied out to decide which model should be used i n the f o l l o w i n g calculations. The decision was made based on an evaluation o f the e f f i -ciency o f each model (results difference versus computational effort). T w o different models were tested: one uses two-dimensional shell elements and the other one uses three-dimensional sofid elements. I n both cases, one-side conosion (asymmetric conosion), has been modelled assuming a linear-vaiiation o f the thickness between the nodes of each element.
I n case o f shell elements, i n order to guarantee one-side corrosion, an eccentricity to the middle surface o f the elements has been applied so that the i n f e r i o r surface o f plate remains straight, as shown i n Fig. 5a. The eccentricity is applied by displacing the nodes; this displacement is calculated using expression (11), where n is the total number o f nodes i n the model. Figure 5b illustrates the modelled middle surface o f the plate w i t h conosion using a scale factor o f 100:1 i n the vertical axis.
Ti Ti - Ci Ti - Ti - f Ci Ci . ch=^ ^ = 2 = 2 ' ' = 1 ' - - - ' "
(11)
A n alternative to m o d e l l i n g w i t h shell elements is to use solid elements to model the field o f the corrosion. I n this case, one layer o f twenty nodes solid elements has been adopted. The nodes i n the m i d d l e surface o f this solid element facilitate the analysis, namely, the application o f the boundary conditions and o f the load (Fig. 6a). The thickness reduction is applied i n the eight nodes at the upper surface of the element allowing an even better representation o f the corrosion random field (Fig. 6b).
However, there are two issues related to the used solid elements: the application o f the load on the plate and an increase i n flexibility o f the m o d e l because o f the existence o f cross-thickness strain. The second issue is not i n fact a problem; i t arises f r o m the theoretical difference between solid and shell elements. Theory tells us that cross-thick-ness strains can be neglected i n the shell elements due to their small impact on the results, and this influence shall decrease further as the plate thickness decreases. The issue related to the loading comes f r o m the fact that one has to apply loads i n all the section o f the plate a l l o w i n g , at the same time, the rotation o f the section ai-ound its central axis. The obvious solution w o u l d be to use a distributed force (pressure). However, i n the A N S Y S , as i n other F E codes, the pressures remain perpendicular to the surface when the plate starts to d e f o r m . So, a more conect option is to apply a displacement i n the section o f plate a l l o w i n g i t , at the same time, to rotate around its central axis. T o achieve this condition, one uses shell elements w i t h a m u c h higher r i g i d i t y than the solid elements (about two hundred times higher). This binds the nodes at the top sections o f the plate, where the loads are applied. This way, r i g i d body rotation o f the plate section at the boundaries can be sim-ulated. The boundary conditions are applied at the middle surfaces o f the plate.
Table 3 summarizes the results obtained f o r five d i f -ferent realizations of the random field o f conosion f o r a plate o f bit = 40 and 80. I n the table, ^2D-V and < / ) 3 d are the normalized ultimate strength o f the models w i t h shell and solid elements, respectively. A mesh o f 400 shell and solid elements has been considered f o r both models.
I t can be seen that the difference between the 2 D and 3 D models is o f the order o f 0.5 % f o r the t h i c k plate and o f 1.7 % f o r the slender plate. I t is clear that m o d e l l i n g using
F i g . 6 a Scheme o f a solid element w i t h twenty nodes (eight i n each face), b Detail o f the corrosion field
Table 3 Comparison between the normalized ultimate strength 4> = a, j/<To o f the different models (plate dimensions: a = b = 1000 m m , Lognormal random field: / i i ^ = 1.02 m m , cr|„ = 0.88 m m , = 0.3a)
Realization o f random field bit = 40 (f = 25 mm) bit = 8( ) (f = 12.5 m m )
< ^ 2 D - 1 ' 4>m - • ^ 2 0 - 1 ' ( % ) ' P 2 D - 1 ' 4>3D - i>2D-V (%) 1 0.897 0.897 0.000 0.608 0.614 0.987 2 0.899 0.905 0.667 0.615 0.623 1.301 3 0.899 0.901 0.222 0.623 0.616 1.124 4 ^ 0.898 0.902 0.445 0.614 0.624 1.629 5 0.902 0.903 0,111 0.632 0.621 1.741
shell elements, linear variation o f the thickness and eccentricity is the most efficient option since the models o f sohd elements take a considerable longer time to r u n and there is only a small variation on the results.
4.3 Importance o f representing c o i T o s i o n by random fields
A s previously stated, the corrosion process i n metallic structures is sufficientiy complex and uncertain. The corro-sion develops i n random patterns that normally correspond to the overlapping o f some typical forms o f c o i T o s i o n . F i g -ure 7 clearly shows the importance o f the spatial represen-tation o f the c o t T o s i o n as compared to the traditional approach based on a u n i f o r m reduction o f plate thickness. The figure shows the normalized ultiinate strength o f a typical double bottom plate w i t h b = 1000 m m o f bit — 50 f o r various reafizations o f the random field o f corrosion obtained f r o m Monte Carlo simulation versus the one obtained assuming a u n i f o r m reduction o f plate thickness w i t h equivalent reduction o f volume. I t can be seen f r o m Fig. 7 that the strength o f the plate w i t h spatial distribution o f corroded thickness represented by random fields is i n most o f the cases lower than the one obtained f o r u n i f o r m con-osion. I n average, this difference is small, but i t increases f o r larger levels o f coiTOsion. This trend is more evident f o r plates o f bit = 60, as shown i n F i g . 8, f o r which a Ifigher dispersion o f the results is observed. This situation illustrates that the assimrption o f u n i f o r m reduction o f plate
Normalized ultimate strength of the plate f) (Random fields of corrosion)
F i g . 7 Normalized ultimate strength (p o f square plates o f bit = 50 w i t h corrosion represented by random fields and u n i f o r m reduction o f thickness (b = 1000 mm, t = 20 mm) (Lognormal random field:
/ i i n = 1.02 m m , (Tin = 0.88 m m , 1^ = 0.3a)
thickness can i n many situations overestimate the real u l t i -mate strength o f the coiToded plate.
5 Probabilistic modelling of the ultimate strength of corroded steel plates
The probabilistic m o d e l l i n g o f the ultimate strength o f coiToded plates is caixied out f o r bottom (BP) and inner bottom square plates ( I B P ) w i t h w i d t h b = 1000 m m . These t w o cases have been selected as they reflect t w o
different coiTOsion processes previously observed i n the analysis o f the con-osion data at the different locations o f the cross section o f the b u l k canies. I n fact, i t has been shown that the coi-rosion rate i n bottom plates is almost constant and, consequently, the reduction of thickness w i t h time is almost linear whereas conosion i n inner bottom plates demonstrates a clear non-linear relationship w i t h time. Furthermore, although the reduction o f the plate thickness due to corrosion is larger f o r inner bottom plates, the coefficient o f variation is larger f o r bottom plates (see Table 2) and, therefore, both cases are relevant.
The characteristics (mean and coefficient o f variation) o f the random field o f thickness reduction due to corrosion f o r the two cases are obtained directly f r o m the nonfinear time-dependent coiTOsion model based on the analysis o f the coiTOsion data, as described at the end of chapter 3. A total
0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 Normalized ultimate strength of the plate (p
(Random fields of corrosion)
F i g . 8 Normalized ultimate strength (f> of square plates o f bit = 60 w i t h corrosion represented by random fields and u n i f o r m reduction o f thickness {b = 1000 m m , t = 16.7 mm) (Lognormal random field; rtn = 1-02 m m , cTin = 0.88 m m , /c = 0.3n)
of 1000 simulations f o r each case have been perfoimed so that the enor on the average ultimate strength o f the cor-roded plates at a 95 % confidence level is always less than 0.5 %. Figure 9 illustrates the vai-iability on the stress-dis-placement curves of bottom and inner bottom plates o f bit = 5Q{t — 20 m m ) w i t h n o n - u n i f o i m corrosion obtained by Monte Carlo simulation and non-lineai- finite element analysis. Figure 10 shows the coiTesponding statistics, i.e., the mean value and the coefficient o f variation o f the plate ultimate strength f o r increasing levels o f axial displacement. I n particular f o r T = 10 years, i t can be seen that i n average the ultimate strength o f corroded inner bottom plates reduces i n about 12 % and the variability on plate ultimate sttength induced by the random patterns o f coixosion is characterized by a coefficient o f variation ( C O V ) o f 2.6 % .
A three-parameter lognormal distribution has been adopted to describe the probabilistic characteristics o f the ultimate strength o f the plates w i t h n o n - u n i f o r m coixosion represented by random fields, as proposed by Sadovsky and Pales [ 5 2 ] . The three-parameter l o g n o r m a l distribution o f normalized ultimate strength (j) w i t h a positive skewness {sk^ > 0) is related to the normal distribution U b y :
U = log((/) - Ö) w i t h (/) G [5, +oo[ (12)
where ö denotes the smaller value o f ( p .
I n the case o f a negative skewness (skti,<0), the three-parameter lognormal distribution o f (f> is related to the normal distribution U by:
[ / = - l o g ( , 5 - ( / ) ) w i t h (/) g ] - o o , < 5 ] (13)
where ö is n o w the larger value o f the normalized ultimate strength <^ o f the plate.
Using these definitions, the f r a c t i l e (t>p o f ( f ) can be calculated by:
1.25 1.6
Fig. 9 Normalized stress-displacement curves f o r a bottom and b inner bottom plates with non-uniform conosion {bit = 50, T = 10 years)
0.9 1 T , , , , , , r 4.5
Non comjded plale (b/t=50)
Axial displacement / Yield displacement
F i g . 10 Statistics o f the stress-displacement curves o f bottom (BP) and inner bottom (IBP) plates w i t h non-uniform corrosion {b/t = 50, r = 10 years)
Inner bottom Plates-Lognormal 3 par. - Bottom pJates-Lognormal 3 par.
0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 Normalized collapse strength
F i g . 11 Probabihty density function of the normalized ultimate strength o f corroded plates {b/t = 50, T = 10 years)
(l>p = ö + exp{i.iu + au^~\p)}, whenslc^>Q (14) or
(l>p = S ~exp{-pu-(^u'i'~Hp)}, w h e n 5 / : ^ < 0 (15) where O is the standardized normal p r o b a b i l i t y distribution A^(0,1), and ay and py are the mean value and the standard deviation o f the normal variable U, given by:
cTa = {\og[l + a l / { ö - p ^ Y y ^
1/2
and
/lu = ^og{p^ - <5) - ö " ü / 2 , when slc^ > 0 or
/^u •^og{ö- p^)+al/2, wh&ns/(^<0
(16)
(17)
(18)
where the third parameter Ö o f the distribution is obtained b y using the sample skewness o f the values o f the plate normalized ultimate strength as point estimator o f the skewness o f the tlu-ee-parameter l o g n o r m a l distribution given by [52]:
• + •
{ / I , - 5 ) { p , - ö f
(19)
Figure 11 shows the 3-parameters lognormal probabihty density functions o f the normalized ultimate strength o f plates o f b/t — 50 obtained b y M o n t e Carlo simulation o f the random field o f coiTosion. As one can see, the average ultimate strength o f inner bottom plates is smaller than the one o f bottom plates, which is a direct consequence of the higher level o f c o i T o s i o n i n plates at this particular location.
It can also be seen that the probability density f u n c t i o n f o r the corroded bottom plates is obviously truncated at the ultimate strength o f the uncorroded plate and that the dis-tribution is h i g h l y asymmetric. However, F i g . 11 clearly
shows that the 3-parameters lognormal distribution is able to capture this asymmetric characteristic o f the distribution and provides a good fitting at its l o w e r t a i l , w h i c h is especially important i n the calculation o f the characteristic values, 4>^% and o f the strength o f corroded plates. The 0.1 % f r a c t i l e o f the plate strength is considered as the design value coiTesponding to the target reliability o f 3.8, w h i l e the 5 % fractile coiTesponds to the characteristic value o f the resistance n o r m a l l y adopted i n the structural design [ 5 3 ] .
Tables 4 and 5 summarize the probability characteristics o f the normalized ultimate strength o f corroded plates as a f u n c t i o n o f t i m e f o r both b o t t o m and inner b o t t o m plates, respectively. I t can be seen that the average value o f plate strength f o l l o w s , i n both cases, the trend o f the reduction o f thickness due to c o i T o s i o n w i t h the ship age, i.e., linear f o r bottom plates (Fig. 12a) and nonlinear f o r inner b o t t o m plates (Fig. 13a). I t is also seen that the c o e f f i c i e n t o f variation o f the strength o f corroded b o t t o m plates tends to increase w i t h time whereas the one o f inner b o t t o m plates takes the m a x i m u m value at T = 13. This fact influences the value o f the strength fractiles, w h i c h are m i n i m u m s f o r the same exposure time {T = 13) f o r inner b o t t o m plates ( F i g . 13b), and tend to decrease linearly f o r b o t t o m plates (Fig. 12b).
Tables 6 and 7 summarize the results o f a parametric study o f the probabilistic m o d e l l i n g o f the normalized ultimate strength o f bottom and inner bottom square plates o f slendernesses ranging f r o m b/t — 40 to b/t = 60 w i t h random patterns o f coiTOsion.
A l t h o u g h the mean value {p^^ and standard deviation (ffin) o f the homogeneous lognormal random field o f cor-rosion have been estimated f r o m the analysis o f c o i T O s i o n data, its correlation structure has been assumed to be described by the exponential f u n c t i o n w i t h a correlation length lo o f 0.3 m (i.e. 30 % o f the plate length and w i d t h ) . I t should be noted that the correlation functions and the
T a b l e 4 Probabilistic model of the normalized ultimate strength
cj) o f coiToded plates o f b/t = 50
(BP-Bottom plates)
BP-Bottom plates (b/t = 50) / = 20 m m , i = 1 m, = 0.3b
Exposure time i n years, T BP-Bottom plates (b/t = 50) / = 20 m m , i = 1 m, = 0.3b 7.5 10 12.5 15 20 Mean value 0.787 0.773 0.762 0.752 _ 0.737 Standard deviation 0.005 0.009 0.013 0.016 0.022 Coef. o f variation 0.006 0.012 0.016 0.021 0.030 Skewness - 0 . 7 6 4 - 1 . 1 6 5 - 1 . 0 1 0 - 1 . 2 8 1 - 1 . 2 5 4 5 % Fractile (cpg,,,) 0.778 0.756 0.739 0.722 0.695 0.1 % Fractile (i^o.i%) 0.767 0.728 0.703 0.670 0.624 T a b l e 5 Probabilistic model o f the normalized ultimate strength (() o f corroded plates o f b/t = 50 (IBP-Inner bottom plates)
IBP-Inner bottom plates (b/t = 50) / = 20 m m , b = lm,I^ = 0.3b
Exposure time i n years, T IBP-Inner bottom plates (b/t = 50)
/ = 20 m m , b = lm,I^ = 0.3b 7.5 10 12.5 15 20 Mean value 0.731 0.706 0.696 0.694 0.691 Standard deviation 0.014 0.018 0.020 0.019 0.017 Coef. o f variation 0.019 0.026 0.028 0.028 0.024 Skewness - 0 . 6 0 9 - 0 . 5 2 3 - 0 . 6 2 9 - 0 . 4 0 4 - 0 . 4 4 5 5 % Fractile ((ps^J 0.706 0.673 0.660 0.660 0.661 0.1 % Fractile ((po.m) 0.674 0.656 0.615 0.622 0.627 0.04 0.03 0.02 10 12.5 15 17.5 20 Exposure time T (year)
F i g . 12 Normalized ultimate strength <p o f coiroded plates o f b/t = variation, b characteristic values o f the ultimate strength
appropriate values o f correlation length are important f o r the results, and w o r k is still required to i d e n t i f y the most appropriate ones w i t h field data.
6 Prediction of the ultimate strength of non-uniform corroded plates
Several semi empirical formulas have been proposed to predict the ultimate strength o f plates subjected to predominantly compressive inplane loads. These f o r m u l a -tions have been reviewed by Guedes Soares and Gordo [54]
10 12.5 15 Exposure time T (year)
17.5 20
50, t = 20 m m (BP bottom plates), a Mean value and coefficient of
w h o compared them w i t h experimental and numerical results, proposing a f o r m u l a t i o n to be adopted f o r predic-tion o f the biaxial strength o f steel plates.
The existing formulations are able to predict the effect o f several factors on the plate strength, such as the i n f l u -ence o f i n i t i a l distortions, residual stresses and transverse loading. The effect o f c o i T o s i o n is usually treated i n these formulations as a u n i f o r m reduction o f plate thickness. This approach may be appropriate f o r estimating the mean value o f the ultimate strength o f coixoded plates. M o r e o v e r , due to the uncertain nature o f the corrosion process, the design o f structures should rely on characteristic values o f strength
ti) 3 (a) ('•74 0.73 0.72 0.71 0.70 0.69 > 0.68
I
0.67 § 0.66 0.65 0.64 0.63 0.62 1 — Ö — Mean value oef. of variat on 0.04 0.03 >8
0.01 0.00 (b) 0.-% 0.70 s 0.65 •3 =5 0.60 ( ( —e—5%F --^--0.1% ractile Fractile —e—5%F --^--0.1% ractile Fractile ?.5 10 12.5 15 17.5 Exposure time T (year)20 10 12.5 15
Exposure time T (year)
F i g . 13 Normalized ultimate strength ^ of corroded plates o f bit = 50, Ï = 20 m m (IBP inner bottom plates), i i Mean value and coefficient o f variation, b characteristic values o f the ultimate strength
T a b l e 6 Probabilistic model o f the normalized ultimate strength <p o f con'oded bottom plates Plate slenderness Probabilistic characteristics of (j> Exposure time i n years, T
7.5 10 12.5 15 20
bit = 40 Mean value 0.901 0.889 0.878 0.869 0,854
Coef. o f vaiiation 0.004 0.009 0.013 0.016 0,022
Skewness - 0 . 7 4 1 - 1 . 1 1 4 - 1 . 0 2 5 - 1 , 1 7 2 - 1 , 0 2 6
5 % Fractile (tps^) 0.894 0.874 0.858 0.843 0,819
0.1 % Fractile (<^o.i%) 0.884 0.850 0,826 0.799 0,764
bit = 50 Mean value 0.787 0.773 0.762 0.752 0,737
Coef. o f vaiiation 0.006 0.012 0.016 0.021 0,030
Skewness - 0 . 7 6 4 - 1 . 1 6 5 - 1 . 0 1 0 - 1 , 2 8 1 - 1 , 2 5 4
5 % Fractile (^5%) 0.778 0.756 0,739 0,722 0,695
0.1 % Fractile ((^0.1%) 0.767 0.745 0.703 0.670 0.624
bit = 60 Mean value 0.711 0.697 0.685 0.675 0.658
Coef. o f variation 0.007 0.015 0.020 0.027 0.037
Skewness - 0 . 8 1 9 - 1 . 2 7 2 - 1 , 0 3 7 - 1 , 3 7 9 - 1 , 2 9 2
5 % Fractile ((^5%) 0.702 0.678 0,659 0,641 0,612
0.1 % Fractile ((/)o.i%) 0.689 0.645 0.619 0.580 0.532
that depend on the probability characteristics o f the strength o f corroded plates.
Figure 14 shows the reduction o f the ultimate strength A,^ o f plates o f slenderness ratio bit f r o m 40 to 60 as a f u n c t i o n o f the volume reduction due to coiTOsion A y , calculated based on the lognormal random field o f c o i T o -sion referent to r = 10 years (i.e. /<in = 1.02 m m ,
= 0,88 m m , = 0,3 m ) . B o t h , A,^ and A y are defined i n percentage o f the strength (j)^ and volume Vo o f the non-coiToded plate, respectively, by:
= ^'t>o-<Po) ^ jy^j ^ j ^ ^
A y =
7
X 100 [%] (20)where is the ultimate strength and Vc the v o l u m e o f the corroded plate.
Figure 14 clearly shows an almost linear relationship between the strength degradation and the loss o f plate volume. This relationship is demonstrated by observing the exponent parameter close to 1 o f the nonlinear f u n c t i o n obtained b y nonlinear regression analysis o f the values plotted i n the figure,
A ^ = 1.70 X A,';"' (21)
Using this relation and the definitions o f A,^ and A y given i n Eq. 20, the ultimate sttength o f the corroded plate ( ( p ^ ) can be obtained f r o m the strength o f the non-corroded plate
{(f>g) and the volume reduction due to c o i T o s i o n A y , by:
Table 7 Probabilistic model o f the normalized ultimate strength < /) o f corroded inner bottom plates Plate slenderness Probabilistic characteristics o f tj) Exposure time i n years, T Plate slenderness Probabilistic characteristics o f tj)
7.5 10 12.5 15 20
bit = 40 Mean value 0.849 0.826 0.816 0.814 0.812
Coef. o f variation 0.015 0.020 0.022 0.021 0.019
Skewness - 0 . 5 3 6 - 0 . 5 0 8 - 0 . 6 0 2 - 0 . 4 0 1 - 0 . 4 3 1
5 % Fractile (1^5%) 0.827 0.797 0.785 0.784 0.785
0.1 % Fractile (0o.i%) 0.801 0.762 0.746 0.750 0.755
bit = 50 Mean value 0.731 0.706 0.696 0.694 0.691
C o e f o f variation 0.019 0.026 0.028 0.028 0.024
Skewness - 0 . 6 0 9 - 0 . 5 2 3 - 0 . 6 2 9 - 0 . 4 0 4 - 0 . 4 4 5
5 % Fractile ((^5%) 0.706 0.673 0.660 0.660 0.661
0.1 % Fractile ((^0.1%) 0.674 0.656 0.615 0.622 0.627
bit = 60 Mean value 0.652 0.626 0.615 0.614 0.611
Coef. o f variation 0.024 0.032 0.035 0.034 0.030
Skewness - 0 . 6 4 9 - 0 . 5 0 1 - 0 . 6 5 5 - 0 . 3 9 3 - 0 . 4 6 2
5 % Fractile (055,,) 0.624 0.590 0.576 0.577 0.578
0.1 % Fractile (0o.i%) 0.588 0.548 0.526 0.536 0.540
^ 40.0
Volume reduction due to corrosion, Av [%]
F i g . 14 Reduction o f the ultimate strength o f the plate ( A ^ ) as f u n c t i o n o f the volume reduction (Ay) [ % ] {T = 10 years, " A l l plates", Lognormal random field: /(!„ = 1.02 m m , uin = 0.88 m m , /e = 0.3 m)
l - 0 . 0 1 7 [ A y ] ' ° ' (22)
Table 8 and F i g . 15 illustrate the adequacy o f E q . 22 i n predicting the mean ultimate strength o f c o i T o d e d plates w i t h d i f f e r e n t slenderness ratios i n terms o f the average v o l u m e reduction o f each group o f plates ( A y ) .
Based on the values o f Table 8, a similar equation can be proposed to predict the 5 % f r a c t i l e o f the p r o b a b i l i t y distribution o f the strength o f plates w i t h n o n - u n i f o r m c o i T o s i o n (</)5%) b y i n c l u d i n g also the v a r i a b i l i t y on the average v o l u m e reduction ( A y ) f o r each group o f plates characterized by the c o e f f i c i e n t o f variation C O V ( A y ) ,
Table 8 Predicted normalized ultimate strength (j) of corroded plates
{T = 10 years. Bottom plates)
bit 4 0 5 0 6 0 0 . 0 2 5 0 . 0 2 0 0 . 0 1 7 A ^ ( % ) 1 . 7 0 2 . 1 3 2 . 5 5 C 0 V ( A 7 ) 0 . 3 1 0 . 3 1 0 . 3 1 4>o 0 . 9 1 5 0 . 8 0 4 0 . 7 2 8 0 . 8 8 9 0 . 7 7 3 0 . 6 9 7 0 . 8 7 4 0 . 7 5 6 0 . 6 7 8 <l>J4>o 0 . 9 7 1 0 . 9 6 2 0 . 9 5 8 <l>5%l'l>o 0 . 9 5 5 0 . 9 4 1 0 . 9 3 2 4:J4>, (Eq. 2 2 ) 0 . 9 7 1 0 . 9 6 4 0 . 9 5 6
</>5%/0o (Eq. 2 3 )
0 . 9 5 3 0 . 9 4 1 0 . 9 2 9 1 -<pj Ip, ( E q . 2 2 ) ( S 5 5 / 0 „ ( E q , 2 3 ) 35 40 45 50 55 60 65 b/tF i g . 15 Predicted normalized ultimate strength (f> o f corroded bottom plates ( 7 = 1 0 years)
•§ 0.84 ^ 0.82
J 0.80 I :
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 Volume reduction due to corrosion, Ay [%]
Fig. 16 Ratio o f the normalized ultimate strengths o f plates o f b/t f r o m 40 to 60 w i t h corrosion represented by random fields (cp^) and by u n i f o r m reduction o f thiclcness ((^„c) (T = 10 year's, " A l l plates", Lognormal random field: ;ii„ = 1.02 m m , <7in = 0.88 m m , = 0.3 m)
^ = 1 - 0 . 0 1 7 [ A 7 ( 1 + 2 - C 0 V ( A ; ; ) ) ] ^ - ° ' ( 2 3 )
The results o f this equation are also presented i n Table 8 (and F i g . 1 5 ) , showing that the deviation o f the predicted values is less than 0 . 4 % i n the range o f plate slenderness between b/t = 4 0 a n d b/t = 6 0 . I t should be noted that the effect o f the n o n - u n i f o r m corrosion depends o n t h e i n i t i a l plate thickness and, therefore, Eqs. 2 2 and 2 3 a r e only v a l i d f o r the range o f thicknesses considered i n the analysis (i.e. between 1 6 . 7 and 2 5 m m ) .
A n alternative equation c a n be proposed by analysing t h e ratios between t h e ultimate strength o f plates w i t h c o i T O s i o n represented by random fields (</);.ƒ) a n d by equivalent u n i f o r m reduction o f thickness ( 0 , j c ) a f u n c t i o n o f t h e volume reduction d u e to corrosion Ay, presented i n F i g . 1 6 .
I n this case, o n e may calculate t h e strength degradation d u e to corrosion b y using the existing f o n n u l a t i o n s to predict the ultimate strength o f plates assuming a u n i f o r m reduction o f the plate thickness i(f)uc)- f l o w e v e r , this value has to be coiTCcted to take into account t h e e f f e c t o f non-u n i f o r m coiTOsion patterns on the non-ultimate strength o f the plate.
A nonfinear regression analysis o f the ratio ( i ^ r f / ' ^ u c )
between the ultimate strength o f plates w i t h c o i T O s i o n represented b y random fields (cp^f) and b y equivalent u n i -f o r m reduction o -f thickness {(puc) -foi" the entire range o -f plate slenderness, illustrated i n F i g . 1 6 , h a s led to the f o l -l o w i n g correction equation:
^ = 1 - 5 . 1 A^ ( 2 4 ) V'uc
I t c a n be seen f r o m Fig. 1 6 t h a t t h e considered random field o f corrosion applied to plates w i t h i n i t i a l thickness f r o m 2 5 to 1 6 . 7 m m (i.e. b/t f r o m 4 0 to 6 0 ) result i n
volume reductions Ay that range f r o m 1 to 1 5 % w i t h an average value o f 5 % . I t can also be seen that most o f realizations o f corrosion patterns lead to the small differences between the ultimate strength o f plates w i t h corrosion represented by random fields (1^^) and b y equivalent u n i f o r m reduction o f thickness(0uj,). I n fact the results obtained indicate that the average value o f the ratio ( p ^ / c p a c is around 0 . 9 8 . However, when the reduction o f the plate volume due to coiTosion increases, the ultimate strengths o f the plates w i t h coiTosion represented by random fields and u n i f o r m reduction o f thickness deviate, as shown i n F i g . 1 6 . I n this case a large variability o f the values o f </>rf/</'uc ean also be seen, as slender plates w i t h patterns o f h i g h levels o f c o i T O s i o n may have an ultimate strength ((/),.f) considerable d i f f e r e n t f r o m the one calculated assuming an equivalent u n i f o r m reduction o f
thickness
(t^uc)-I t should be r e f e i r e d that after the introduction o f the net scantling approach i n l A C S C o m m o n Structural Rules f o r o i l tankers and b u l k carriers, constant allowable margins o f corrosion have been clearly defined f o r all structural members. This means that the m a x i m u m volume reduction due to c o i T o s i o n is e x p l i c i t l y defined and more strictiy controlled. For example, the m a x i m u m allowable wastage o f b o t t o m plates w o u l d correspond to around 1 5 % i n volume reduction. F r o m the nonlinear fit illustrated i n F i g . 1 6 i t can be seen that f o r this l i m i t i n g v o l u m e reduc-tion the ratio 0rf/(/>uc is around 0 . 9 .
7 Conclusions
A parametric study o f probabilistic m o d e l l i n g o f the u l t i -mate strength o f steel plates w i t h spatial distribution o f c o i T o d e d plate thicknesses represented by random fields was presented.
A nonlinear c o i T o s i o n m o d e l was fitted i n d i v i d u a l l y f o r data sets o f particular locations, showing that its parameters depend strongly on the location o f the plates. For sub-merged plates, namely b o t t o m plates, bilge plates and side plates, the corrosion rate is approximately constant and, therefore, the relationship between thickness reduction and ship age is linear. However, f o r inner tank plates (bilge tanks and upper side tanks) the data measurements showed that wear o f thickness w i t h t i m e is nonlinear and can be w e l l described by the nonlinear corrosion m o d e l adopted.
A preliminary study has been performed to assess the effect o f the finite element (FE) m o d e l l i n g o f the non-u n i f o r m rednon-uction o f plate thickness dnon-ue to corrosion on the ultimate strength o f the plate. The results showed that the use o f 2 D shell elements w i t h a linear variation o f the thickness between the nodes o f each element can represent correctly the patterns o f corrosion and were adopted, i n
