1 On the Post-buckling of Corroded Steel Plates used in Marine Structures António F. Mateus and Joel A.Witz Department of Mechanical Engineering, University College London This paper investigates the buckling and post-buckling behaviour of imperfect steel plates used in ship and related marine structures. The analysis is carried out using the non-linear finite element program ABAQUS. The effects of general corrosion are introduced into the finite element models using the uniform thickness reduction approach and a proposed quasi-random thickness surface model. The results obtained for the corroded plate models show that there are significant discrepancies in the prediction of plate post-buckling behaviour between the two general corrosion approaches. The uniform thickness reduction approach for compressive strength predictions produces optimistic results 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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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.
2
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
3
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
4
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
5
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)
Fig. 1 - Stress-strain curves for API Grade B and E steels, as implemented in ABAQUS plate models
13
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
14
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
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
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
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