1

On the Post-buckling of Corroded Steel Plates used in MarineStructures

António F. Mateus and Joel A.WitzDepartment of Mechanical Engineering, University College London

This paper investigates the buckling and post-buckling behaviour of imperfect steel plates used in ship and relatedmarine structures. The analysis is carried out using the non-linear finite element program ABAQUS. The effects ofgeneral corrosion are introduced into the finite element models using the uniform thickness reduction approach and aproposed quasi-random thickness surface model. The results obtained for the corroded plate models show that thereare significant discrepancies in the prediction of plate post-buckling behaviour between the two general corrosionapproaches. The uniform thickness reduction approach for compressive strength predictions produces optimisticresults and therefore the question arises over its adequacy for design purposes.

1. INTRODUCTION

Floating marine structures such as ships, crane barges and offshore floating production units make

extensive use of stiffened flat steel plate in their hull construction. The plates between stiffeners

experience significant compressive loading as a result of the flexing of the hull girder in a seaway

and, therefore, the compressive strength of the steel plates is of primary concern to the designer. As

a consequence the evaluation of plate compressive strength has mainly been achieved through

laboratory experiments in the past few decades. For a realistic assessment of the buckling and post-

buckling behaviour of steel plates large deflection theory must be used. The solution of this

mathematical model for the post-buckling behaviour of plates is extremely complicated and

tedious, and only a few particular approximate solutions have been derived. Many of the

compressive strength models in use today reflect analyses made on the results of the laboratory

experiments. Small deflection theory for plates is well established, see for example Timoshenko

and Woinowsky-Krieger (1959). But unfortunately small deflection plate theory is of limited

practical value for the analysis of ship plates since the deflections experienced by these plates in

severe loading conditions are usually several times the plate thickness which is well beyond the

range of validity of this model. The large deflection mechanics of steel plates is a highly non-linear

problem whose solution relies on the use of numerical techniques such as non-linear finite element

analysis. However, the use of non-linear finite element methods to analyse the buckling and post-

buckling behaviour of plates was not practical until relatively recently because of the computational

resources required by this method.

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Marine structural components have failed at sea due to excessive structural degradation caused by

corrosion, even though some of these structures have met the requirements of the Classification

Societies. This indicates that the evaluation of the compressive strength of corroded structural

components cannot be based simply on a uniform reduction of thickness, but rather in the detailed

study of how each type of corrosion type may affect the strength of the component in question. This

paper addresses this issue.

An investigation into the buckling and post-buckling behaviour of corroded plates used in ships and

other marine structures is presented in this paper. This work first addresses the validation of the non-

linear finite element analysis technique used to perform the simulation of the buckling and post-

buckling behaviour of plates. The non-linear finite element analysis was performed using ABAQUS

V5.4, a finite element analysis software package produced by Hibbitt, Karlsson and Sorensen (1994).

2. STEEL CORROSION IN SHIPS AND MARINE STRUCTURES

Despite the obvious importance of the effects of corrosion on marine structures, its influence on

structural strength has never been fully researched. It is frequently treated by means of simple

empirical deterministic models or relatively generous allowances that are envisaged to cover the level

of uncertainty. The commonest approach these days to corrosion effects on structures is from a

structural reliability perspective using a fully probabilistic analysis as described by Amazigo (1974),

Akita (1987) and Shi (1992). Evidently this method does not enable the actual quantification of

strength reduction of structural components due to corrosion. In essence, it defines the component

strength by a mean value and variance.

Often several types of corrosion manifest their effects simultaneously. These corrosion types are

usually treated as independent variables and their effects are superimposed. Their interdependence is

not yet fully understood and the modelling of such combined situations would be extremely difficult.

The two most relevant types of steel corrosion present in marine structures are general corrosion and

pitting corrosion. The issue of pitting corrosion is not addressed here. Ahammed and Melchers

(1995) give an interesting application of pitting corrosion.

General corrosion is the most common type of corrosion present in steel structures. It is

characterised by the global oxidation of the material surface, either in large areas or in reduced

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patches, but always having more development in area rather than depth. The most common effect of

general corrosion on plates can be seen as producing a random surface contour associated with a

generalised reduction in material thickness. An exception to this general corrosion pattern occurs in

the vicinity of welds where the areas of plate immediately adjacent to the weld are usually corroded

at an higher rate. This relatively small width zone, approximately parallel to the butt weld, suffers a

grooving effect with the corrosion manifesting its effects in depth rather than in extent. The

importance of these grooved areas on the compressive strength of the plate are most certainly

significant, especially if the welds are located at the plate boundaries. This implies a decrease in the

rotational restraint at the plate boundaries, i.e. the plate will tend to behave more like a simply

supported plate as opposed to clamped plate. Some sources, including some Classification Societies

already define this type of corrosion as grooving corrosion. Significant grooving corrosion

invalidates to a certain extent the uniform thickness reduction model.

The extent of general corrosion in marine structures is reflected in the results published by Ohyagi

(1987) who presented the compilation of data gathered by Nippon Kaiji Kyokai (NKK) from

corrosion surveys on ships. Ohyagi gives the mean and maximum values of thickness measured at

each inspection and the associated statistical parameters of significance such as the standard

deviation. According to these results general cargo carriers are most affected by corrosion and the

cargo holds are the most corroded spaces.

For design purposes, two of the major Classification Societies have similar approaches with regard

to the structural effects of corrosion. Det Norske Veritas (DNV) treats the problem of corrosion by

providing a fixed "corrosion addition" to the required strength thickness, varying according to the

type of tank in which the structural members are located. Only structural members located in ballast

tanks and oil cargo tanks are considered for this allowance. Lloyds Register of Shipping (LRS)

allows a corrosion wastage margin which varies according to the type of tank, the slope of the

surface considered within the tank and the required strength thickness of the plating. The wastage

margin is constrained to lie within specified limits. Only tank related structural members are

considered for the allocation of these allowances.

The usual approach to quantify the effects of corrosion is based on the evaluation of material loss

for a given exposure time. The equivalent uniform reduction in thickness is then calculated from

this weight loss. For the case of a plate this may be expressed in numerical terms by

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t t d tW W

abn wn= − = −

−

0 0

0

ρ(Eq.2.1.)

where tn is the equivalent uniform thickness of the corroded plate after n years of exposure, t0 is

the thickness of the uncorroded plate, dw is the uniform reduction in thickness (or mean corrosion

depth) due to corrosion effects after n years of exposure, W0 is the original weight of the

uncorroded steel plate and Wn is the actual weight of the corroded steel plate after n years of

exposure. a is the plate length, b is the plate width, and ρ is the density of steel.

To date the main concern of researchers has been to determine how the uniform reduction in

thickness varies as a function of exposure time. More sophisticated models also try to include the

contributions of other parameters such as location of the structural component, type of steel, etc.

The ability to model the effects of corrosion in structural components is an issue of critical

importance for the integrity assessment after a certain period of time and ultimately the prediction

of the serviceability limit of the structure. As stated by Melchers (1994): “For structural

engineering design applications there is a need to be able to decide how much allowance to make

for material loss due to corrosion over the anticipated lifetime of the structure”.

Ohyagi‘s (1987) uniform thickness reduction general corrosion model has a mean linear model for

maximum corrosion wear rates, following a normal distribution for the probability of exceedance of

wear rate, characterised by

d nw y= 0 34. (Eq.2.2.)

where ny is the number of years of exposure and dw is the uniform reduction in thickness (or

corrosion depth) in millimetres due to corrosion effects after n years of exposure. The standard

deviation associated with the normal distribution is 0.23mm.

In addition to the Ohyagi model there are other general corrosion models available in the literature.

The predictions from several of these corrosion models are presented in Table I. It is possible to

identify immediately the discrepancies between them. If it is assumed that the influence of

measurement errors and other minor possible sources of inaccuracy is negligible then the main

distinction that has to be made between the models is based on the provenience of their respective

data, i.e. steel specimens or steel structural components. The Melchers’ (1994) model is based on

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data from steel specimens while the Ohyagi (1987) and TSCF (1994) models are based on

measurements made on ship structural components. DNV (1991) and LRS (1995) allowances

reflect the conclusions taken from years of surveys and measurements on ship structural

components.

Table I - Thickness reduction of structural components by general corrosion

(All thickness values shown are expressed in millimetres)

ExposureTime

Melchers (1994)

Violette(1994)

Ohyagi(1987)

DNV(1991)

LRS(1995)

(Years) Meannon-linear

Meanpiecewise

linear

Upperbound

piecewiselinear

TSCFpiecewise

linear

NKKlinear

DNV LRS

1 0.083 0.09 0.2 0.005 0.34 --- ---5 0.336 0.266 0.63 0.125 1.7 --- ---10 0.614 0.456 1.13 0.5 3.4 0.5 to

3.00.5 to

4.015 0.873 0.646 1.63 1.27 5.1 --- ---

For an exposure time of 10 years Melchers’ model predicts a uniform reduction in thickness of 1.13

mm while Ohyagi’s model predicts 3.4 mm. This implies that the corrosion level in a ship structural

component after 10 years is expected to be approximately three times that expected in a plate

specimen. This is a significant difference and it requires explanation.

In very basic terms a plate specimen in contact with aerated sea water corrodes. It suffers a

chemical reaction, the oxidation of the iron, which produces iron oxide covering the surface of the

plate specimen. This iron oxide inhibits the contact between the oxygen in the water and the iron

thus reducing the level of aerobic corrosion. The remaining corrosion mechanism will be

predominantly anaerobic and the corrosion rate will decrease moderately or even stabilise at an

approximately constant value. This depends on several parameters such as water temperature,

salinity, chemical composition of the steel specimen, etc. The main point is that the iron oxide layer

inhibits the exponential growth of corrosion rates, which occurs in the initial period of exposure

where there is no significant iron oxide layer covering the plate surface.

For a structural plate member the basic chemical principle of corrosion applies but the interaction

between this and other mechanisms increase dramatically the corrosion rates. A plate, when part of

a ship’s structure, experiences significant cyclic stresses and deflections as a consequence of the

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direct action of loads. The lateral deflections are often large when compared to the plate thickness.

The iron oxide which forms as the plate corrodes is a relatively brittle material and it can be easily

broken off the plate surface as it flexes. This progressive disintegration of the iron oxide layer

leaves the plate surface exposed to the direct action of the corrosion agents, enabling the growth of

the corrosion rate and hence a new reduction in structurally effective plate thickness. The new

reduction in plate thickness will imply that the stresses and strains in the plate will be higher than

before. This also means that the magnitude of the plate’s lateral deflections will tend to increase.

This probably causes the iron oxide layer to break away from the plate’s surface even earlier than

before thus enabling again the growth in corrosion rate.

Since plate panels are welded, the areas adjacent to the welds tend to suffer a grooving corrosion

phenomena which is very likely to be associated with a combination of galvanic corrosion, stress

corrosion, and general corrosion. The galvanic corrosion effect is associated with the inevitable

difference in material galvanic properties between the steel plate and the weld metal. The stress

corrosion will have its origin in the tensile block of weld induced residual stresses which has its

transition from tension to compression slightly out of the weld as confirmed by surveys. These

three types of corrosion do not show their effects individually but what can be seen on in-service

structures is mainly a combination of them. The reduction in effective thickness near the plate

boundaries implies a decrease in the rotational restraint at the plate edges. This not only implies an

immediate decrease in compressive strength (simply supported plates have approximately 30% less

strength than similar clamped plates) but also causes the level of out-of-plane deflections to increase,

thus causing the disintegration of the iron oxide layer to occur more frequently than is the case for

the plate with the original boundary conditions. Also the conditions for the occurrence of stress

corrosion are ideal for a structural plate and these will tend to favour more and more this type of

corrosion as the plate thickness decreases because the level of stress in the plate will increase.

It is the combination of the factors described above that causes the corrosion rates in ship structures

to be significantly higher than the corrosion rates measured on steel specimens. Other factors may

also influence the results such as material properties, location of the surveyed components and local

environmental conditions on board which tend to vary with the type of ship. It can be seen that

corrosion rates in marine structures are significant and their effect on the structural strength needs

to be evaluated.

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3. COMPRESSIVE STRENGTH OF UNCORRODED PLATES

A plate buckles when the applied uniaxial compressive load reaches the critical load, the point at

which the plate becomes unstable. Depending on the plate dimensions the critical load may be due

to elastic or a combination of elastic and plastic effects. As the applied load is increased above the

critical value, the unloaded supported edges provide an increase in plate strength when the centre of

the plate displaces. This produces a redistribution of stresses which are associated with tensile

membrane stresses, stabilising the plate in the new deformed configuration. If the applied load is

continuously increased the edges of the plate will yield thus defining the ultimate load at which

failure will occur. The ratio between ultimate load and critical load increases directly with plate

slenderness. This means that slender plates are more suitable to work in the post-buckling regime

than thicker plates which are more affected by plastic hinge mechanisms after yield occurs. A

commonly used measure of plate slenderness is the slenderness ratio β, given by

β σ= b

t EYP (Eq.3.1.)

where σYP is the yield stress of the plate material, E is Young’s modulus, b is the plate width, and t

is the plate thickness.

Beyond the critical load or stress, the deflections of the plate can no longer be considered small

when compared with its thickness and the membrane stresses at mid-thickness cannot be neglected.

The membrane stresses are generated by the stretching of the plate middle surface due to the

curvature of the plate, ensuring equilibrium of forces and satisfaction of displacement constraints in

accordance with the boundary conditions. In some areas, the plate will have tensile membrane

stresses due to the stretching as a result of bending. These tensile membrane stresses will inhibit

increases in plate deflection caused by the in-plane external compressive load and thus will have a

stabilising effect on the buckled plate. All this is true as long as the unloaded edges of the plate are

constrained to remain plane and prevented from pulling in. These boundary conditions are typical

of those encountered with structural plating. This means that the typical pull-in of the free unloaded

edges due to Poisson’s effect will be prevented and a biaxial stress state will occur due to the fact

that direct stresses perpendicular to the applied load will be present along these edges, especially at

the middle section where the Poisson’s effect is larger. This in-plane displacement restraint, however,

is reasonably difficult to obtain in laboratory tests, a factor present in some of the older series of

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tests. The physical devices available to prevent the edges from pulling in always imply a certain

restraint in rotational displacement which obviously alters the compressive strength of the plate.

The concept of effective width originates from considerations of the typical stress distribution in a

plate in the post-buckling regime where high membrane stresses are concentrated along the

unloaded edges of the plate and the middle strip experiences relatively low stresses. This suggests

the idea that at failure all the load could be assumed to be carried by two bands of plate adjacent

and parallel to the unloaded edges and the remaining central area of the plate would be completely

unloaded. The total width of these two lateral load carrying bands would be equal to the effective

width be. Therefore at failure, the value of the stresses in the load carrying strips would equal the

yield stress. The compressive strength of a plate could be expressed in a non-dimensional form in

terms of its effective width by

b

b

N

bte u

YP

=σ

(Eq.3.2.)

where Nu is the load applied at the edges of the plate when collapse occurs. The maximum average

stress or ultimate stress concept simply takes into account the stress distribution along the width of

the plate when failure occurs. If, for simplicity, in a plate loaded uniaxially in the x-direction,

failure is defined as the point where the edge stress reaches the material yield stress, then the non-

dimensional form of the maximum average stress is given by

( )( )σ

σ

σ

σσ σu

YP

x

b

YPMISES YP

x y dy

bx= =

∫ ,

, ,0 0 (Eq.3.3.)

where σx(x,y) is the function representing the direct stress variation along the width of the plate and

σMISES(x,0) is the Von Mises resultant edge stress. Since both concepts expressed by Eq.3.2 and

Eq.3.3. represent the plate strength it is possible to establish the following important relationship

b

be u

YP

=σσ

(Eq.3.4.)

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The definition of yielding does also depend on the criterion used but for the case of plate theory the

two usual criteria are the maximum shear stress (Tresca) criterion and the maximum distortion

energy (Von Mises) criterion. For the case of ductile materials these two criteria give similar

results. It is the Von Mises criterion that is perhaps the most often used for plates.

The modern foundations for the assessment of compressive plate strength start with Faulkner

(1975) who produced a complete review of all the compressive strength models derived up until

that date. Faulkner derived an effective width model representing the average curve that fitted the

envelope of all the experimental data points considered. The model was specifically dedicated to

simply supported plates where a relatively large set of experimental data was available. An

extension to the model for clamped plates was also produced based on American data collected

from the destructive testing of three decommissioned destroyers. The compressive strength

equation proposed by Faulkner is

b

b

a afore u

YP

= = − ≥σσ β β

β1 22

1, (Eq.3.5.)

where a1, a2 are constants dependent on the plate boundary conditions: a1=2.0, a2=1.0 for simply

supported plates and a1=2.25, a2=1.25 for clamped plates.

More recent work by Guedes Soares (1988) concluded that there was a loss of plate strength

directly related to the level of initial plate imperfection. However, it was argued that the loss of

strength with the increase in initial imperfection was less noticeable for higher values of plate

slenderness. Thus, it was proposed to quantify this strength reduction by means of a multiplicative

reduction factor given by a linear function of plate slenderness and initial deflection. The plate

compressive strength model is then expressed as

b

b

a aB Re u

YPb== = −

σσ β β δ

1 22

(Eq.3.6.)

where Bb is the uncertainty quantification factor and Rδ is the reduction factor due to initial plate

deflection. The above equation is applicable to plates under uniaxial compressive loads, and

represents the perfect plate strength multiplied by the reduction factor Rδ due to initial plate

deflection. Bb is an uncertainty quantification factor whose mean value represents the bias of

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Faulkner’s compressive strength model, given by Eq.3.5. The average values of Bb were determined

from comparison between Faulkner’s strength model predictions and experimental data for

allegedly perfectly flat plates. These values are Bb =1.08 for simply supported plates and Bb =1.03

for clamped plates. From regression analysis of plate experimental data, the reduction factor is

given by

( )Rwtδ β= − −1 0 626 0 121 0. . (Eq.3.7.)

where w0 is the initial plate distortion amplitude.

The presence of initial geometric imperfections in structural components submitted to compressive

loading is of critical importance for their structural behaviour. These imperfections may imply a

decrease in stiffness when compared with a perfectly straight component. In essence, an initial

distortion will have contributions from at least one of the elastic buckling modes implying that the

structural behaviour under compressive loading will no longer reflect a bifurcation problem. The

minimum energy path has already been defined by the existence of the initial imperfection and the

structure will deform continuously until failure occurs. Thus, the structure behaves as if it is already

“slightly” buckled and its compressive strength can no longer be treated in terms of an elastic

critical load (or critical stress) but rather in terms of a maximum load defining the collapse point

and an ultimate load, usually significantly lower than the elastic critical load, defining the limit

state for plate serviceability.

Imperfections are clearly present in ship plating, primarily as a result of distortions originating from

welding and other production processes involved in the manufacture of the grillages. It is possible

to observe the out-of-plane deformations, especially on the shell plating between consecutive

longitudinals and frames. As the ship ages, its plating is submitted to a large number of normal

loads, whether caused by the fall of a weight on a deck or by collision with the quay when going

alongside. These loads tend to cause half-sinusoidal waveform plastic, large amplitude

deformations of the plating which will affect its compressive strength, and hence the grillage

interframe and overall compressive strength.

As described by Smith et al (1987), in the case of square plates, the dominant distortion induced by

welding is dominated by an approximately sinusoidal half-wavelength equal or somewhat less than

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the plate dimension (length or width, depending on the direction of loading). Regression analysis of

data gathered from surveys on ships and welded box girder bridges plating has shown that the most

adequate model for the evaluation of the maximum average out-of-plane initial distortion amplitude

is a function of β, and is given by

w

t0 201= . β (Eq.3.8.)

In basic terms, this approach seems to be consistent with physical reality.

Guedes Soares (1988) argues that the influence of aspect ratio α on the compressive strength of

perfect plates is not very significant. However, when an initial deformation representative of that

encountered with ship plating is present, i.e. a first mode distortion, the overall behaviour is

substantially different. For plates of aspect ratio above 1.5 the effect of this initial distortion is to

strengthen the plate while it degrades the strength in the case of square plates. This means that the

effects of initial distortions are not always a case of downgrading plate strength and each case has

to be assessed independently.

4. FINITE ELEMENT MODEL OF UNCORRODED PLATES

Since this work is aimed at representative plating used in ships, the dimensions, slenderness ratio,

aspect ratio, material properties, definition of boundary conditions and type of loading analysed had

to be consistent with marine structure applications. All the plates analysed here have a width of 0.5

m, which is representative of a typical test plate and all the remaining parameters are scaled relative

to this dimension in accordance with each case. It was decided to analyse unstiffened steel plates

with slenderness ratio values in the range of 1 to 4. As pointed out by Smith et al (1987), this is the

usual range of slenderness ratio encountered with ship and offshore structures applications, although

the majority of these plates have slenderness ratio values between 2 and 3. It was decided to

concentrate the analysis on square plates since for most symmetric modes of failure the behaviour of

a rectangular plate may be fully described by simply analysing one of its square components.

However, the work also includes some analyses of rectangular plates of aspect ratios of 2 and 4.

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In order to assess the influence of material properties, two types of steel were modelled into the

plates analysed: API’s Grade B and E steels, which represent respectively a typical mild and a higher

strength steel used in marine structures. Table II resumes the more relevant mechanical properties of

each of these steels, and Fig.1 illustrates their respective stress-strain curves as implemented in

ABAQUS models.

In terms of definition of boundary conditions, the work was restricted to simply supported plates

since these represent the lower bound of strength in ship plating. In fact, the rotational restraint due

to welding is neither purely clamped nor simply supported. It lies somewhere in between with its

stiffness dependent on parameters such as the butt weld width and depth. Due to the fact that it

became relevant within the scope of this research to investigate the influence of the unloaded edges

restraint on the compressive strength of the plate, a series of square plates with all edges simply

supported but with the unloaded edges free to pull-in were also modelled.

Table II - Mechanical properties of API steels

Property Grade B Grade E(API X52)

Yield Stress, MPa 241 358.5Young’s Modulus, GPa 210 210Poisson Ratio 0.3 0.3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

50

100

150

200

250

300

350

400

450

500Average steel stress-strain curves

API (X-52)Grade E

API Grade B

Strain

Stre

ss (

MPa

)

Fig. 1 - Stress-strain curves for API Grade B and E steels, as implemented in ABAQUS plate models

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The finite element model implemented in ABAQUS for uncorroded square plates consisted on a

mesh of 10 by 10 elements using a eight node quadratic plate element enabling up to five

integration points through the thickness. Convergence of results from this mesh were checked

against results for higher mesh densities such as the one illustrated in Fig.2. The buckled shapes

shown in Fig.2, 3 and 4 represent plate models analysed in ABAQUS. They refer respectively to a

square plate, a plate with an aspect raio of 2, and a plate with an aspect ratio of 4, all with unloaded

edges restrained from pulling-in.

The five series of uncorroded plates analysed were as follows:

• Square plates, Grade B steel, with unloaded edges restrained from in-plane displacement.

• Square plates, Grade E (API X52) steel, with unloaded edges restrained from in-plane

displacement.

• Square plates, Grade B steel, with unloaded edges free to have in-plane displacement.

• Rectangular plates with an aspect ratio of 2, Grade B steel, with unloaded edges restrained from

in-plane displacement.

• Rectangular plates with an aspect ratio of 4, Grade B steel, with unloaded edges restrained from

in-plane displacement.

Fig. 2 - Buckled shape of a square plate uniaxially loaded with the free edges restrainedfrom pulling-in - ABAQUS solution

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The major objective of performing the analysis of uncorroded plates using ABAQUS was to prove

that this type of test could now be carried out accurately, using a numerical model, without the need

to use expensive laboratory tests. In order to perform the validation of the model, it was required to

assess the results of the non-linear finite element analysis against the plate strength experimental

data obtained from several different sources, shown in Fig.5, and the most relevant analytical plate

strength models, shown in Fig.6. As shown in Fig.5, all the data points obtained for the ultimate

strength of the uncorroded plates lie within the envelope defined by the previous experimental tests

Fig. 3 - Buckled shape of a rectangular plate with an aspect ratio of 2 uniaxially loaded withthe free edges restrained from pulling-in - ABAQUS solution

Fig. 4- Buckled shape of a rectangular plate with an aspect ratio of 4 uniaxially loaded withthe free edges restrained from pulling-in - ABAQUS solution

15

considered. The only exception is in the case of the series of rectangular plates with an aspect ratio

of 2, where a higher strength is predicted by the finite element analysis for the plates of higher

slenderness ratio. However, this higher strength is consistent for all the rectangular plates, which

tends to indicate that the combination between boundary conditions, aspect ratio and the type and

magnitude of initial imperfection considered have enhanced the plate strength. The possibility of

this effect had already been discussed by Guedes Soares (1988).

The square plates’ strength values obtained from the non-linear finite element analysis for the more

stocky plates (i.e. slenderness ratio less than 2) fall outside of the experimental data envelope below

its lower bound values. However, it does not reflect analysis error but simply the fact that

experimental tests were not generally performed on square plates of such low slenderness values.

Due to the reasons explained earlier it is expected that square plates have the lowest strength while

most of the laboratory experiments used plates with aspect ratio typical of those encountered in for

ship structures which very rarely is equal to 1.

The hypothesis previously mentioned that the laboratory plate tests boundary conditions would not

represent the case where the unloaded edges are restrained from in-plane displacement seems to be

1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Compressive strength of uncorroded plates - Experimental data & ABAQUS models

AR=1

AR=1, free edge

Plate slenderness ratio

Stre

ngth

(A

vg.S

tr/Y

S)

Fig. 5 - Comparison between ABAQUS curves for ultimate compressive strength ofuncorroded Grade B steel square plates and experimental data

16

confirmed by the fact that the values of ultimate strength for the series of square plates with their

unloaded edges free to pull-in correlates well with the experimental data.

5. RESULTS FOR UNCORRODED PLATES

The load shortening curves for the grade B, shown in Fig.7, and grade E (API X52), shown in

Fig.8, steel square plates with unloaded edges restrained from in-plane displacement present some

interesting results which may be summarised:

• The slopes in the initial linear stress-strain behaviour of the plate increase inversely from the

higher to the lower values of slenderness representing the initial plate stiffness (which is

obviously associated with the magnitude of the initial out-of-plane imperfection).

• While the thicker plate tends to show a more gradual loss of strength in the post-buckling regime

associated with plastic flow of the material, the more slender plates tend to have a more

accentuated and sudden loss of strength after buckling (εε 0

from 0.4 to 0.8, where ε is the end

shortening strain and ε0 is the end shortening yield strain), achieving an almost constant strength

1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Compressive strength of uncorroded plates - Analytical & ABAQUS models

AR=1

AR=1, free edge

Soares

Faulkner

Plate slenderness ratio

Stre

ngth

(A

vg.S

tr/Y

S)

Fig. 6 - Comparison between ABAQUS curves for ultimate compressive strength ofuncorroded square plates and analytical models

17

beyond these values of end shortening. This behaviour is related to the more sudden collapse

mechanism that the slender plates have as a result of elastic effects.

• The magnitude of the remaining nearly constant fully plastic post-failure strength is inversely

proportional to plate slenderness. The value of end shortening at which this stable plastic flow

regime is reached is also inversely proportional to plate slenderness. However, for design

purposes this behaviour is not relevant since it occurs well beyond plate failure.

• The main difference between the Grade B and E steel plates is the relative magnitude of the

ultimate plate strength. Grade E steel plates have higher compressive strength for values of plate

slenderness below 2.5. This is due to the material’s plastic properties which provide these plates

with more stiffness in the post-buckling regime. The more slender plates, say beyond β=2.5,

have their collapse dominated by predominantly elastic mechanisms and therefore the influence

of the material properties is very small.

• The comparative assessment of load shortening curves for Grade B and E steel square plates

showed that the variation in material properties (especially yield stress) influenced significantly

both the initial plate stiffness and the magnitude of edge displacement at which buckling and

failure occurred. This means that the range of edge displacements that Grade E plates can

experience without failure is increased when compared with similar Grade B steel plates. These

results are consistent with the expected performance of a higher yield strength material.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

ABAQUS model- Load shortening curves of Grade B uncorroded plates

B=0.83

B=1.23

B=1.63B=2.04

B=2.45B=2.87B=3.38

Note: "B" represents Beta

End shortening (e/ey)

Avg

.Str

ess/

Yie

ld S

tres

s

Fig. 7 - Set of ABAQUS load shortening curves of uncorroded Grade B square plates withunloaded edges restrained from pull-in

18

The analysis and resultant load shortening curves for the grade B steel square plates with unloaded

edges free in in-plane displacement, shown in Fig.9, confirms what has been said previously for

square plates with the unloaded edges prevented from in-plane displacement. Adding to this, it was

found that:

• For plates of slenderness ratio above 2, the ultimate strength is lower than that for square plates

with the unloaded edges restrained. This discrepancy increases with plate slenderness. This

confirms that for plates where the buckling and post-buckling are primarily dependent on elastic

mechanisms the edge in-plane displacement restraint boundary condition can increase the plate

strength as much as 18% (β=3.38).

• For plates of slenderness ratio below 2, the ultimate strength of this series is slightly higher than

that for square plates with the unloaded edges restrained (the maximum difference is 9% for

β=1.23). This is due to the fact that the unloaded edges are free to pull-in enabling the plate to

deform freely without a significant biaxial stress state at the plate edge. Therefore failure will

occur at higher values of applied load.

The load shortening curves for the Grade B steel rectangular plates with aspect ratios of 2 and 4 and

with unloaded edges restrained from in-plane displacement showed the following:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

ABAQUS model- Load shortening curves of Grade E uncorroded plates

B=1

B=1.5B=2

B=2.5B=3

B=3.5B=4

Note: "B" represents Beta

End shortening (e/ey)

Avg

.Str

ess/

Yie

ld S

tres

s

Fig. 8 - Set of ABAQUS load shortening curves of uncorroded Grade E square plates withunloaded edges restrained from pull-in

19

• The increase in aspect ratio had a direct effect on the increase in ultimate strength. This is due to

the increase in length of the unloaded edges which were forced to remain plane and prevented

from pull-in.

• The comparative assessment of the sets of load shortening curves for square plates and

rectangular plates with an aspect ratio of 2 showed that the range of edge displacements that the

rectangular plate can experience without failure is increased when compared with square plates

but the rectangular plate has a lower initial stiffness. These results are consistent with the

expected performance of a higher aspect ratio plate.

• As shown in Fig.10, the ultimate strength of plates with an aspect ratio of 4 is lower than that for

an aspect ratio of 2. The range of edge displacements that these plates can experience without

failure is approximately five times higher than that for square plates or rectangular plates with an

aspect ratio of 2.

From the results of the uncorroded square plate models an analytical model was derived,

representing the lower bound of ultimate strength for typical ship structural plating. This analytical

model is given by

σσ β β β β βu

YP

= − + − + ≤ ≤0 0226 0 272 11989 2 3407 2 2759 0 83 3 384 3 2. . . . . , . .

(Eq.5.1.)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

ABAQUS model- Load shortening curves of Grade B uncorroded plates

B=0.83

B=1.23

B=1.63B=2.04

B=2.45

B=2.87B=3.38

Note: "B" represents Beta

Plate with unloaded edges free to pull-in

End shortening (e/ey)

Avg

.Str

ess/

Yie

ld S

tres

s

Fig. 9 - Set of ABAQUS load shortening curves of uncorroded Grade B square plates withunloaded edges free to pull-in

20

where the ultimate strength is a fourth order polynomial function of the slenderness ratio.

6. COMPRESSIVE STRENGTH OF CORRODED PLATES

The type of surface present in a corroded plate is far from uniform and in order to evaluate the

compressive strength of a corroded plate a model of the spatial variation of the plate’s thickness is

required. A plate subject to general corrosion has a random distribution of thickness over its area.

The likelihood of these variations in thickness to form plastic hinges that may affect the buckling

and post-buckling behaviour of a corroded plate, and perhaps its ultimate strength, is something

that cannot be discarded without further analysis.

It is reasonable to assume that the corroded surface of a plate with its random thickness variation is

composed of an infinite summation of random coefficients associated with each of the elastic

buckling modes of the plate. The peak amplitude of the random surface is limited to a maximum

value which is governed by the standard deviation of the associated thickness probability

distribution function. The surface roughness will lie in a certain range which will not penalise

1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Ultimate compressive strength of uncorroded plates - ABAQUS models

Grade B, AR=1

Grade E, AR=1

Grade B, AR=2

Grade B, AR=4

Grade B, AR=1, free edge

Plate slenderness ratio

Str

engt

h (A

vg.S

tr/Y

S)

Fig. 10 - ABAQUS solutions for ultimate compressive strength of uncorroded plates(square plates, and aspect ratio 2 and 4)

21

excessively the strength of the plate due to the existence of localised deep grooves but will still

enable the appearance of the localised plastic hinges. The standard deviation of the associated

normal distribution was fixed to a value in accordance with the Ohyagi corrosion model.

In numerical terms the general corrosion model describes the typical surface of a corroded plate as a

random thickness variation, tP, with an average value equal to the original thickness of plate, minus

the corroded equivalent thickness reduction.

( ) ( ) ( )( )t x y t d A f x B g yP w i i k kki

, = − + +=

∞

=

∞

∑∑011

(Eq.6.1.)

where Ai and Bk are the random coefficients associated with mode i in the x-direction and the mode

k in the y-direction, respectively. fi is the ith elastic buckling mode shape in the x-direction, given

by ( )f xi x

ai =

sin.π

and gk is the kth elastic buckling mode shape in the y-direction, given

by ( )g yk y

bk =

sin.π

.

This model assumes that only one of the surfaces of the unpainted plate will be affected by corrosion

and that the perturbation around the uniform reduced thickness is independent of any geometrical

characteristics of the plate (e.g. curvature, proximity to edges, etc.). This means that the lower

uncorroded surface of the plate will only have the geometrical imperfection of the original plate due

to welding and production inaccuracies. This is given by a function h(x,y) which, for typical plates in

ship structures, may be represented by a distortion in the first buckling mode, as suggested by Smith

et al (1987). Defining the average imperfection level as indicated by Smith (1987), in Eq.3.8. , the

Cartesian co-ordinates of an arbitrary point on the plate lower surface may be defined by

( ) ( )z x y h x y tx

ay

aLOWSRF, , . sin

.sin

.= =

0 1 2

0βπ π

(Eq.6.2.)

The upper surface of the plate will reflect a combination of the initial geometrical imperfection and

the random thickness pattern which is consistent with the assumption of a single corroded plate

surface. Thus, the Cartesian co-ordinates of a point located on the upper (corroded) surface of the

plate may be obtained by

22

( ) ( ) ( )z x y t x y h x yUPSRF P, , ,= + =

= − +

+

+

=

∞

=

∞

∑∑t d Ai x

aB

k y

bt

x

a

y

aw i kki

011

2001sin

.sin

.. sin

.sin

.π π β π π

(Eq.6.3.)

Due to the fact that in practical terms it is impossible to consider a summation of infinite number of

terms, a reasonable approximation can be achieved by considering only the first ten elastic buckling

modes. This approximation to the random thickness modelling is called a quasi-random thickness

model of general corrosion.

Two main approaches were adopted to model the effects of corrosion in the ABAQUS finite

element models: the uniform thickness reduction and the quasi-random thickness variation. The

corrosion model assumed was that due to Ohyagi (1987) and the exposure time considered for all

cases was 10 years. The use of a linear model is valid for this exposure time.

The implementation of the uniform thickness reduction approach to corrosion was done by

modelling two series of square plates with the thickness reduced by the amount determined from

Ohyagi’s corrosion model. This gives a thickness reduction of 3.4 mm. For a 10 year exposure

time, the effects of material properties on the compressive strength were investigated by modelling

one series of plates using Grade B steel and the other series using Grade E (API X52) steel. The

finite element model used for all plates was similar to that previously used for uncorroded plates

with the exception of the mesh density. Uniaxial uniform displacements were applied at one plate

edge with the other edges restrained from movement.

Looking at the results shown in Fig.11 obtained from the two series of ABAQUS square plate

models using the uniform thickness reduction approach it is possible to make the following

statements:

• The load shortening curve for the more slender plate with a thickness of 1.6 mm has an unusual

shape which obviously may be associated with the lower capacity of the solution algorithm to

deal with very slender plates. The slenderness ratio for the corroded plate would be equal to

10.58 which obviously shows that errors may be expected to occur in the convergence in the

solution algorithm. Therefore, the results obtained for the very slender plates should be treated

with some scepticism.

23

• Overall, all the load shortening curves have similar general behaviour characteristics as

discussed previously for uncorroded plates. The slopes of all the load shortening curves have

decreased compared with the related curves for uncorroded plates. This shows the direct

dependence of initial plate stiffness not only on initial distortion, but also on plate thickness.

Plate stiffness is the combination of geometric stiffness and flexural stiffness. Geometric

stiffness is dependent on plate curvature and is a linear function of the plate thickness. Flexural

stiffness is directly proportional to the cube power of the thickness. From a quick analysis of the

load shortening curves it may be shown that the ratio between the corroded and the uncorroded

slopes is approximately 0.9, i.e. a relatively small difference, which suggests that the geometric

stiffness of the corroded plates was the component that was most affected by the decrease in

plate thickness.

• The more slender plates, which have their buckling behaviour dominated by elastic mechanisms,

experienced lower net losses in strength compared with the strength losses for the less slender

plates (except for the most slender plate as discussed previously).

In the series of plates modelled in ABAQUS using the quasi-random thickness approach the

thickness distribution along the plate was given by

( ) ( ) ( )( )t x y t A f x B g yP i i k kki

, . .= − + +==

∑∑01

10

1

10

0 0034 0 001 (Eq.6.4.)

where all the thickness values are in metres. The thickness standard deviation was set to

σ tP= 0 2. mm, which is consistent with Ohyagi’s general corrosion model. This general corrosion

model had a maximum thickness amplitude of 0.316 mm, corresponding to a maximum peak to

peak variation of 0.632 mm. A significantly higher mesh density was used for the quasi-random

thickness model.

24

Three ABAQUS plate models with quasi-random thickness variation were subject to analysis.

These analyses are very expensive in terms of computational effort. The level of surface

imperfection was set to a relatively small value in order to ensure that the solution algorithm would

converge. The load shortening curves for these cases are shown in Fig.12 and 13 which show some

interesting results. From the comparison between the load shortening curves shown in Fig.12 and

13, it is possible to highlight the following points:

• The initial stiffness of both corroded plate models are equal, i.e. the slopes of the two load

shortening curves are the same whether the corrosion model used was the uniform thickness

reduction or the quasi-random thickness surface. This shows that the initial pre-buckling elastic

behaviour of the plate is governed by the magnitude of initial imperfections and the mean

thickness of plating.

• Once the applied load exceeds the proportional limit the quasi-random thickness models present

a departure in behaviour from the uniform thickness reduction models. The quasi-random

thickness models present lower maximum and ultimate strength compared with the

corresponding uniform thickness models. The magnitude of this difference between models

increases directly with the thickness of plate, and is almost negligible in the case of the most

slender plate. This suggests that the point of formation of plastic hinge mechanisms and their

location becomes relevant, i.e. once first yield occurs within the plate the quasi-random

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

ABAQUS model- Load shortening curves of Grade B corroded plates

B=0.83

B=1.23

B=1.63

B=2.04

B=2.45

B=3.38

Note: "B" represents Beta

Ohyagi corrosion model, 10 years exposureUniform thickness reduction 3.4mm

End shortening (e/ey)

Avg

.Str

ess/

Yie

ld S

tres

s

Fig. 11 - Set of load shortening curves of corroded square plates with unloaded edgesrestrained from pulling-in, using uniform thickness reduction approach

(β values based on nominal uncorroded scantlings)

25

thickness model will present a substantially different behaviour to that obtained from the

uniform thickness reduction model because several small plastic hinges will tend to form within

the plate according to the thickness variation.

• In the fully plastic post-collapse regime all plates modelled using the quasi-random thickness

have a more sudden and greater drop in strength, based on the formation of small plastic hinges,

presenting a substantially lower remaining strength than that obtained if using the uniform

thickness reduction.

• The load shortening curve for the most slender plate presents similar behaviour to that

previously obtained for the uniform thickness reduction model, raising doubts over its validity

due to its very high slenderness.

• The quantification of the discrepancies between the quasi-random thickness model results and

both the uncorroded and the uniform thickness reduction models are presented in Tables III and

IV. Nevertheless, it seems important to point out that the maximum loss of ultimate strength (for

the thicker plate) did not exceed 8%. This result, however, was obtained for a very small

thickness variation, which seems to indicate that the effects of increasing the severity of the

corrosion grooves merits further investigation. Although not directly relevant for design

purposes, the loss of strength in the fully plastic range presented significantly reduced values of

as much as 68%.

Table III -ABAQUS models: Ultimate compressive strength of uncorroded and corroded Grade B unpainted steel

square plates with unloaded edges restrained from in-plane displacement

β Uncorroded Corroded

Uniform thickness reduction Quasi-random thickness

Strengthσσ

u

YP

Strengthσσ

u

YP

Netstrength

loss

%Strength

loss

Strengthσσ

u

YP

Netstrength

loss

%Strength

loss

0.83 1.0126 0.9551 0.0575 5.7 --- --- ---1.23 0.7626 0.6405 0.1221 16.0 --- --- ---1.63 0.6181 0.5800 0.0381 6.2 0.5250 0.0931 15.12.04 0.5742 0.5314 0.0428 7.5 --- --- ---2.45 0.5543 0.4830 0.0713 12.9 0.4623 0.0920 16.62.87 0.5298 --- --- --- --- --- ---3.38 0.5028 --- --- --- --- --- ---

26

Table IV - ABAQUS models: Compressive strength of uncorroded and corroded Grade B unpainted steel square

plates with unloaded edges restrained from in-plane displacement, in the post-collapse fully plastic regime

β Uncorroded Corroded

Uniform thickness reduction Quasi-random thickness

Strengthσσ

U

YP

Strengthσσ

U

YP

Netstrength

loss

%Strength

loss

Strengthσσ

U

YP

Netstrength

loss

%Strength

loss

1.63 0.4646 0.4246 0.0400 8.6 0.2915 0.1731 37.32.45 0.4211 0.1921 0.2290 54.4 0.1327 0.2884 68.53.38 0.2966 --- --- --- --- --- ---

In summary, strength predictions obtained from the quasi-random thickness general corrosion

model differed significantly from those obtained from the uniform thickness reduction corrosion

model. Essentially, this shows that the uniform thickness approach is easy and convenient but it is

inadequate for establishing the effects of corrosion in the prediction of compressive structural

strength.

An analytical model was derived from the results of the quasi-random thickness corroded square

plate models analyses, representing the ultimate strength for typical ship structural plating with 10

years of in-service life. This analytical model is given by

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ohyagi corrosion model, 10 years exposurea. Quasi-random thickness, avg.t=3.5mm, max.amplt=0.316mmb. Uniform thickness reduction, t=3.5mmc. Uncorroded, t=6.9mm

Random thk.

Uniform thk.

Uncorroded

ABAQUS model- Beta=2.45 Grade B corroded plate load shortening curve

End shortening (e/ey)

Avg

.Str

ess/

Yie

ld S

tres

s

Fig. 12 - Comparison between load shortening curves for 6.9 mm thick uncorroded andcorroded plate (β values based on nominal uncorroded scantlings)

27

σσ β β βu

Y

= − + + ≤ ≤0 1137 0 3873 0 1957 1 63 3 382. . . , . . (Eq.6.1.)

where the ultimate strength is a quadratic function of slenderness ratio. This model should be used

with caution as it is based on a limited set of data.

7. CONCLUSIONS

This paper has shown that is possible to use a fully numerical solution in the form of non-linear

finite element analysis to predict the buckling and post-buckling behaviour of plates. However,

some disadvantages have been identified in the sensitivity of the solution algorithm to particular

plate configurations, especially the most slender plates, where convergence is not always achieved.

The ultimate strength solutions obtained in ABAQUS for uncorroded plates lie within the envelope

defined by the set of experimental data. The major discrepancies that were observed between

ABAQUS and experimental plate test data are essentially due to differences in boundary

conditions. The solutions obtained for ABAQUS plate models showed the dependence of plate

compressive strength on the interaction between aspect ratio, type of initial imperfection, and

boundary conditions.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ohyagi corrosion model, 10 years exposurea. Quasi-random thickness, avg.t=6.95mm, max.amplt=0.316mm, sdev=0.2mmb. Uniform thickness reduction, t=6.95mmc. Uncorroded, t=10.35mm

Random thk.

Uniform thk.

Uncorroded

ABAQUS model- Beta=1.63 Grade B corroded plate load shortening curve

End shortening (e/ey)

Avg

.Str

ess/

Yie

ld S

tres

s

Fig. 13 - Comparison between load shortening curves for 10.35 mm thick uncorroded andcorroded plate (β values based on nominal uncorroded scantlings)

28

The results obtained from the ABAQUS corroded plate models have shown that the usual uniform

thickness reduction approach to account for general corrosion effects is not adequate in the

quantification of the strength of structural components. The proposed quasi-random thickness

surface approach to general corrosion has indicated that in the initial elastic regime the strength of

the plate is primarily determined by the average thickness of the plate. This is verified from the fact

that both uniform thickness reduction and quasi-random thickness plate models have coincident

load shortening curves in this range. This fact obviously confirms the predominant dependence of

the initial plate stiffness on the geometrical stiffness component. Also, once plasticity becomes

relevant in the structural behaviour of a plate, the plastic hinges formed due to plate surface

irregularity will affect significantly the post-buckling behaviour of the plate, decreasing slightly its

ultimate strength when compared with the results obtained using the uniform thickness reduction.

This decrease in ultimate strength was shown to be directly dependent on plate thickness, and the

maximum discrepancy between the two general corrosion models was of the order of 8%. The

formation of the plastic hinges affects dramatically the plate post-collapse fully plastic regime

strength, decreasing these values comparatively with the uniform thickness reduction approach

results for as much as 68%.

LIST OF VARIABLES

α plate aspect ratioa plate lengtha1, a2 constants dependent on the plate boundary conditionsAi , Bk random coefficients associated with mode i in the x-direction and mode k in the y-directionb plate widthBb uncertainty quantification factorbe effective widthdw uniform reduction in thickness (or corrosion depth) due to corrosion effects after n years of exposureε end shortening strainε0 end shortening yield strainE Young’s modulus

fi ith elastic buckling mode shape in the x-direction, given by ( )f xi x

ai =

sin.π

gk kth elastic buckling mode shape in the y-direction, given by ( )g yk y

bk =

sin.π

Nu load applied at the edges of the plate when collapse occursny number of years of exposureρ density of steelRδ reduction factor due to initial plate deflectionσMISES(0) Von Mises resultant edge stressσx(y) function representing the direct stress variation along the width of the plateσYP yield stress of the plate materialσu ultimate compressive stress of the plate

29

σU compressive strength of the plate in the post-collapse fully plastic regimet plate thicknesst0 thickness of the uncorroded platetn equivalent uniform thickness of the corroded plate after n years of exposuretP corroded plate random thicknessw0 initial plate distortion amplitudeW0 original weight of the uncorroded steel plateWn actual weight of the corroded steel plate after n years of exposure

REFERENCES

• Ahammed, M. and Melchers, R.E., (1995), Probabilistic Analysis of Pipelines Subjected to Pitting CorrosionLeaks, Engineering Structures, Volume 17, No.2, pp.74-80

• Akita, Y., (1987), Reliability and Damage of Ship Structures, NK Technical Bulletin, Tokyo• Amazigo, J.C., (1974), Buckling of Stochastically Imperfect Structures, Buckling of Structures, pp.172-182,

IUTAM Symposium, Cambridge, USA• Det Norske Veritas, (1991), Rules for Classification of Steel Ships: Hull Structural Design - Ships With Length

100 Metres and Above, pp.88-95, Part 3, Chapter 1, Section 14, Hovik, Norway• Faulkner, D., (1975), A Review of Effective Plating for Use in the Analysis of Stiffened Plating in Bending and

Compression, Journal of Ship Research, Vol.19, No.1, pp.1-17• Guedes Soares, C., (1988), Design Equation for the Compressive Strength of Unstiffened Plate Elements with

Initial Imperfections, Journal of Constructional Steel Research, Vol.9, pp.287-310• Hibbitt, Karlsson & Sorensen, (1994), ABAQUS Version 5.4, Pawtucket, USA• Lloyds Register of Shipping, (1995), Rules and Regulation for the Classification of Ships: Longitudinal Strength,

pp.8-12, Part 3, Chapter 4, Section 7, London• Melchers, R.E., (1994), Pitting Corrosion in Marine Environments - A Review, Research Report No.101.07.1994,

University of Newcastle, New South Wales, Australia• Melchers, R.E. and Ahammed, M., (1994), Non-linear Modelling of Corrosion of Steel in Marine Environments,

Research Report No.106.09.1994, University of Newcastle, New South Wales, Australia, ISBN 0-7259-0840-8• Ohyagi, M., (1987), Statistical Survey on Wear of Ship’s Structural Members, NK Technical Bulletin, Tokyo• Shi, W.B., (1992), In-service Assessment of Ship Structures: Effects of General Corrosion on Ultimate Strength,

pp.77-91, Transactions RINA, London• Smith, C.S., Davidson, P.C., Chapman, J.C., and Dowling, P.J., (1987), Strength and Stiffness of Ships’ Plating

Under In-plane Compression and Tension, Transactions RINA, Vol.130, London, pp.277-293• Tanker Structures Cooperative Forum, (1993), Guidelines for the Inspection and Condition Assessment of Tanker

Structures• Timoshenko, S.P. and Woinowsky-Krieger, S., (1959), Theory of Plates and Shells, Second Edition, McGraw-Hill

Book Company, London• Violette, F.L.M., (1994), The Effect of Corrosion on Structural Detail Design, Paper No.14, RINA International

Conference on Marine Corrosion Prevention, London