1On Advanced Buckling and UltimateStrength Design of Ship Plating
Jeom Kee Paik (M), Pusan National University, PusanAnil Kumar Thayamballi (LM), Chevron Shipping, San RamonGe Wang (M), American Bureau of Shipping, HoustonBong Ju Kim (SM), Pusan National University, Pusan
ABSTRACT
This paper is a summary of recent research and development in areas related to advanced bucklingand ultimate strength design of ship plating, jointly undertaken by the American Bureau of Shipping andthe Pusan National University. The behavior of ship plating normally depends on a variety of influentialfactors, namely geometric/material properties, loading characteristics, initial imperfections, boundaryconditions and deterioration arising from corrosion, fatigue cracking and accidental dents. In achieving amore advanced buckling and ultimate strength design of ship plating, we are still confronted with a numberof problem areas to be more completely solved and these would need methods that are more sophisticatedthan most existing simplified approaches. In this regard, this paper focuses on the following five subjectswhich have been studied by the authors theoretically, numerically and experimentally: mathematical mod-eling of fabrication related imperfections (i.e., initial deflections and residual stresses), characteristics ofthe plate buckling with elastically restrained edge conditions, capacity equations based on buckling andultimate strength under combined loads including biaxial loads, edge shear and lateral pressure, collapsestrength characteristics under axial compressive dynamic loads, and design equation of the plate capacityunder impact lateral pressure loading. Useful results, important insights and conclusions developed fromthe studies are summarized and recommendations are made with respect to both technologically improveddesign procedures, and also needed future research.
NOMENCLATURE
a = plate lengthb = plate breadth
fxb , fyb = flange breadth of longitudinals or transverses
D = ( )2
3
112 ν−Et
E = Young’s modulus
wxh , wyh = web height of longitudinals or transverses
p = average net lateral pressure
t = plate thickness
fxt , fyt = flange thickness of longitudinals or transverses
wxt , wyt = web thickness of longitudinals or transverses
ββββ =Et
b oσ
νννν = Poisson’s ratio
oσσσσ = yield stress
rcxσσσσ , rcyσσσσ = compressive (negative) residual stress in the
x or y direction
xavσσσσ , yavσσσσ = average longitudinal or transverse axial
stress (compression: negative, tension:positive)
xEσσσσ , yEσσσσ = elastic longitudinal or transverse compressive
buckling stress
xBσσσσ , yBσσσσ = buckling based capacity for xavσ or yavσ
xuσσσσ , yuσσσσ = ultimate strength based capacity for xavσ or
yavσ
avττττ = average edge shear
oττττ =3oσ
Bττττ = buckling based capacity for avτ
uττττ = ultimate strength based capacity for avτ
Paper Number 13 2
INTRODUCTION
The overall failure of a ship hull girder is normallygoverned by buckling and plastic collapse of the deck, bot-tom or sometimes the side shell stiffened panels. Therefore,the relatively accurate calculation of buckling and plasticcollapse strength of stiffened plating of the deck, bottomand side shells is a basic requirement for the safety assess-ment of ship structures. In stiffened panels, local bucklingand collapse of plating between stiffeners is a primary fail-ure mode, and thus it would also be important to evaluatethe buckling and collapse strength interactions of platingbetween stiffeners under combined loading.
The behavior of ship plating normally depends on a va-riety of influential factors, namely geometric/material prop-erties, loading characteristics, initial imperfections (i.e., ini-tial deflections and residual stresses), boundary conditionsand existing local damage related to corrosion, fatigue crackand denting.
The geometry of plating found in ship and offshorestructures is normally rectangular and the material used isusually mild or high tensile steel (Note that the use of alu-minum alloys is increasing in the design and fabrication ofhigh speed vessel structures). The boundary condition forthe rectangular plate elements making up steel plated struc-tures is normally assumed to be simply supported or some-times clamped for practical purposes of analysis. In real shipplating, however, such ideal edge conditions may never oc-cur due to rotational restraint by support members along theplate edges.
The ship plating is generally subjected to combined in-plane and lateral pressure loads. In-plane loads include biaxialcompression/tension and edge shear, which are mainly in-duced by overall hull girder bending and/or torsion of thevessel. Lateral pressure loads are due to water pressure and/orcargo. The extrema of such load components may not occursimultaneously, and more than one load component maynormally exist and interact. Hence, for more advanced designof ship structures, it is of crucial importance to better under-stand the characteristics of the buckling and ultimate strengthfor ship plating under combined loads.
Since the post-weld initial imperfections in the form ofinitial deflections and residual stresses exist in ship steelplating and can affect significantly the strength, such weld-ing induced initial imperfections should be included in thestrength calculations as parameters of influence.
When a perfectly flat plate (i.e., without initial imper-fections) is subjected to predominantly compressive loads,buckling (bifurcation) can occur if the applied compressivestress reaches a critical bifurcation stress, see Figure 1.However, the in-plane stiffness of plating with initial imper-fections decreases from the very beginning as the compres-sive loads increase. In this more general case, it is not possi-ble to define a bifurcation point for buckling.
The phenomenon of buckling may be categorized byplasticity considerations into three classes, namely elastic
buckling, elastic-plastic buckling and plastic buckling, thelast two being called inelastic buckling. The first class (i.e.,elastic buckling) typically indicates that buckling occurssolely in the elastic regime. This class of buckling is oftenseen in very thin steel plates. The second (i.e., elastic-plasticbuckling) normally represents the case wherein buckling oc-curs after plastification has occurred in a local region in theplate. The third (i.e., plastic buckling) indicates that buck-ling occurs in the regime of gross yielding, i.e., after theplate has yielded over large areas. Relatively thick platingmay exhibit either elastic-plastic or plastic buckling.
Unlike columns, plating can normally sustain additionalapplied loads even after elastic buckling occurs since mem-brane tension develops along the plate edges resists anyabrupt increase in lateral deflection. A plate buckled in theelastic regime will eventually collapse by a rapid decreaseof in-plane stiffness (or an abrupt increase of lateral deflec-tion) as the yield zone inside the plate is expanded. On theother hand, if buckling occurs in the elastic-plastic or plasticregime the plating normally immediately reaches the ulti-mate limit state.
Figure 1. A schematic of the collapse behavior of steelplating under predominantly compressive loads
From the view point of a structural designer, it can besaid with reasonable certainty that the buckling and ultimatestrength problem for ship plating under a single load appli-cation and common idealized edge conditions (e.g., simplysupported along four edges) has been almost completelysolved. In the more general case, however, we are still con-fronted with a number of problem areas that remain un-solved due to the various influential factors previouslymentioned. In the following, a literature review of selectedstudies related to the buckling and ultimate strength of plat-ing is now made.
Depending on the rotational restraints and torsional ri-gidity of support members along the plate edges, the com-mon ideal edge conditions (i.e., the assumption that the plate
Paper Number 13 3
edges are simply supported or clamped) may or may not beappropriate to apply (Bleich 1952, Timoshenko & Gere1963). The plate element is normally subjected to combinedloads and the buckling mode depends on the interaction ofthese load components. Therefore, the plate bucklingstrength should in principle be evaluated by taking into ac-count the effects of boundary condition and load componentinteractions among other factors.
Williams (1976) investigated the buckling strengthcharacteristics of plate elements varying torsional rigidity ofsupport members along their edges. Paik et al. (1993) sur-veyed the bending and torsional rigidities of support mem-bers for plate elements in merchant vessel structures. Basedon the survey results, they concluded that due to the rota-tional restraint by support members at plate edges the plateedge condition would be in an intermediate situation, i.e.,between a simply supported and a clamped condition. Mostrecently, Paik & Thayamballi (2000) investigated the buck-ling strength characteristics of steel plating elastically re-strained at their edges and developed simple design formu-lations for buckling strength as function of the torsionalrigidity of support members that provide the rotational re-straints along either one set of edges or all (four) edges.
Mansour (1976) developed charts for predicting thebuckling and post-buckling behavior of simply supportedplates under combined in-plane and lateral pressure loads.Steen & Valsgard (1984) developed a simplified buckling andultimate strength equation for plates under biaxial compres-sion and lateral pressure loads. They define a pseudo-buckling (non-bifurcation) strength for initially deflectedplating. Ueda et al. (1985) developed elastic buckling interac-tion equations for simply supported plates subject to five loadcomponents, namely biaxial compression, biaxial in-planebending and edge shear. Paik et al. (1992a) developed theelastic buckling interaction equation for simply supportedplates under biaxial compression, edge shear and lateral pres-sure loads. The post-weld residual stresses were also later in-corporated in the plate buckling design formula (Paik et al.1992b). To appropriately include the effects of post-weld ini-tial imperfections in the strength calculations, an idealizedmodel representing the distribution of the post-weld initialimperfections is used. Mazzolani et al. (1998) studied the ef-fect of welding on the local buckling of aluminum thin plates.The influence of welding induced initial deflection and resid-ual stresses on the buckling and ultimate strength of platingunder uniaxial compression and lateral pressure was studiedby Yao et al. (1998).
Most design rules of classification societies approxi-mately calculate the inelastic buckling strength of plate ele-ments by a correction for plasticity applied to the elasticbuckling strength, using the so-called Johnson-Ostenfeldformula. This approach normally tends to underestimate thebuckling strength for one single stress component loading,but in some cases for combined loading it can overestimatethe buckling strength. Paik et al. (1992b) and Fujikubo et al.(1997) have derived newer empirical formulations of the
plasticity correction by curve fitting based on nonlinear fi-nite element solutions.
Following von Karman et al. (1932), the concept of ef-fective width has been recognized as an efficient device forcharacterizing the post-buckling strength behavior of a platein compression. For collapse strength prediction of steelplates, the effective width concept has also been widely used(Faulkner 1975). For such use, the reduction of in-plane stiff-ness of a buckled plate is evaluated by using the effectivewidth concept, and it is assumed that the plate reaches the ul-timate limit state if the normal stress components within theplate field satisfy certain predefined ultimate strength criteria.
An extensive review of a number of studies for thederivation of the effective width formulae for plates, under-taken until the early 80s, has been made by Rhodes (1984).Since then, Ueda et al. (1986a) derived the effective widthformula for a plate under combined biaxial compression andedge shear taking into account the effects of initial deflec-tions and welding induced residual stresses. Usami (1993)studied the effective width of plates buckled in compressionand in-plane bending.
While the concept of effective width is aimed at theevaluation of in-plane stiffness of plate elements buckled incompression, Paik (1995) suggested a new concept of the ef-fective shear modulus to evaluate the effectiveness of plateelements buckled in edge shear. The effective shear modulusconcept is useful for computation of the post-buckling be-havior of plate girders under predominant shear forces.
Regarding the ultimate strength interaction equationsfor plate elements under combined loads, a number of stud-ies have also been undertaken in the past, e.g., for uniaxialcompression and shear (Fujita et al. 1979), for in-planecompression and tension (Smith et al. 1987), for uniaxialcompression and lateral pressure (Aalami & Chapman 1972,Aalami et al. 1972, Okada et al. 1979, Paik & Kim 1988),for biaxial compression (Dier & Dowling 1983, Ohtsubo &Yoshida 1985), for biaxial compression and lateral pressure(Dowling & Dier 1978, Soreide & Czujko 1983, Steen &Valsgard 1984, Davidson et al. 1991, Soares & Gordo 1996,Wang & Moan 1997), for biaxial compression and shear(Ueda et al. 1984, 1995, Davidson et al. 1989), for biaxialcompression, shear and lateral pressure (Ueda et al. 1986b),among others. Some of the methods mentioned above ap-proximately accommodate post-weld initial imperfections,but others neglect them.
For safety assessment of aging ship structures, it is nec-essary to better understand the influence of local damagerelated to corrosion, fatigue cracking and dents on thestrength. Smith & Dow (1981) review structural damage ina ship’s bottom or side shell as may be caused by collisions,grounding, hydrodynamic impact or explosions, with par-ticular reference to the influence of such damage on hullgirder bending strength. Paik et al. (1998a) proposed a prob-abilistic corrosion rate estimation model of ship plating.They also studied the ultimate strength reliability of shipstructures related to corrosion damage (Paik et al. 1998b,
Paper Number 13 4
1998c). Mateus & Witz (1997, 1998) studied the bucklingand post-buckling behavior of corroded steel plates usingthe nonlinear finite element method.
The response of ship plating is dynamic in principlesince ships are subjected to both low and high frequencydynamic loads induced by waves. It is recognized that thestrength characteristics of structural elements under dy-namic loading can be quite different from those under astatic or quasi-static loading situation (Jones 1989). An ex-tensive survey of the studies related to dynamic plastic be-havior of marine structures that have been published overthe last two decades was made by Jones (1997). Accordingto his review, most previous studies related to the dynamicplastic behavior of structural members are limited to eitherbeam members or plates under lateral pressure loading, andthe studies for plate elements subject to dynamic in-planeloads are limited to crushing and are not seen for dynamicultimate strength. Also, the ship bottom plating can be sub-jected to impact pressure loads due to slamming and cancollapse (Jones 1973). For advanced strength design of shipstructures, it would therefore be important to better under-stand the collapse strength characteristics of ship platingunder dynamic/impact in-plane or lateral pressure loads.
Based on the literature surveys mentioned above, it isevident that some of the issues that need to be studied fur-ther for facilitating more refined buckling and ultimatestrength of ship plating are as follows
• Modeling of the fabrication related initial im-perfections (i.e., initial deflections and residualstresses) and their effect,
• Effects of rotational restraints and torsional ri-gidity of support members,
• Ultimate strength interaction characteristics un-der any combination of potential load compo-nents, including biaxial compression/tension,edge shear and lateral pressure loads,
• Effects of structural deterioration such as dueto corrosion, fatigue cracking and local dents.
This paper is a summary of recent research and devel-opment in the above areas, jointly undertaken by the PusanNational University and the American Bureau of Shipping.The paper focuses on the following five subjects which arestudied theoretically, numerically and experimentally: mod-eling of post-weld initial imperfections (i.e., initial deflec-tions and residual stresses) and their effects, influence ofrotational restraints and torsional rigidity of support mem-bers on the plate buckling strength, ultimate strength designequations under combined loads including biaxial compres-sion/tension, edge shear and lateral pressure loads, and dy-namic collapse strength characteristics under dynamic axial
compressive loads or slamming induced impact lateral pres-sure loading. Selected useful results and insights developedare summarized, and recommendations are made with re-spect to related enhancements in the advanced ship struc-tural design and also needed future research.
BUCKLING/ULTIMATE STRENGTHDESIGN PROCEDURE
Figure 2 shows a schematic of the typical steel platedstructure. The response of such a structure can be classifiedinto three levels, namely the bare plate element level, thestiffened panel level and the entire plated structure level.This paper is concerned with the design for the first level(i.e., the plating between longitudinals and transverses). Insuch a case, the structure is to be designed so that the ca-pacity (resistance) with allowable usage factor should not beless than the corresponding applied loads. To prevent thestructure from failure (instability) under applied loading,therefore, the following criterion is to be satisfied:
ησσ ≤
c
(1)
where σ = applied load (stress), cσ = structural capacity
(stress), and η = allowable usage factor which is the inverse
of the conventional factor of safety.
Stiffeners Plate field
Stiffened panel
Heavy longitudinals and transverses
Figure 2. A typical stiffened plate structure in a ship
The applied load (stress) components are to be deter-mined using any acceptable method such as the finite ele-ment approach. The structural capacity is normally deter-mined based on either buckling or ultimate strength. Thispaper focuses on the advanced design equations for the ca-pacity based on both buckling and ultimate strength.
GEOMETRIC AND MATERIAL PROPERTIES
The length and breadth of plating are a and b , respec-tively. The long direction is taken as the x axis and theshort direction is taken as the y direction, that is, 1/ ≥ba .
Paper Number 13 5
The thickness of plating is t . The Young modulus and Pois-son ratio are E and ν , respectively. The yield stress ofmaterial is oσ . The plating is supported by longitudinals
and transverses. Figure 3 shows a typical geometry of thesupporting members in the x and y directions. The rota-
tional restraint parameters for the boundary longitudinalsand transverses are defined as follows
bD
GJ LL =ζ ,
aD
GJ SS =ζ (2)
where Lζ , Sζ = rotational restraint parameters for the
longitudinals and transverses, respectively, with
( )ν+=
12
EG ,
6
33fxfxwxwx
L
tbthJ
+= , ( )2
3
112 ν−= Et
D ,
6
33fyfywywy
S
tbthJ
+= .
For a simply supported condition, Lζ and Sζ are set to
be zero, while their values will become infinity for aclamped edge condition. For practical purposes, the value ofthe rotational restraint parameter for clamped edges may beconsidered to be 20.
(b) y-stiffener
bfy
tfy
hwy
zoy
a
N. A.twy
t
(a) x-stiffener
bfx
tfx
hwx
zox
b
N. A.twx
t
z z
y x
Figure 3. Typical geometry for the longitudinals and trans-verses
b
a
σx1
σx2
σy1 σy2
p1 p2
τ
x
y
b
a
σx1
σx2
σy1 σy2
p1 p2
τ
x
y
Figure 4. The plating under a general pattern of combinedexternal loads
LOAD (STRESS) APPLICATION
Figure 4 shows a general loading condition on theplating between longitudinals and transverses. For the platecapacity calculations, the distribution of applied loads is of-ten idealized by their average values, similar to that shownin Figure 5. The compressive stress is taken as negative andthe tensile stress is taken as positive. The average values ofthe applied stresses (loads) are defined as follows
2
21 xxxav
σσσ
+= ,
221 yy
yav
σσσ
+= , ττ =av ,
221 pp
p+
= (3)
where xavσ = average axial stress in the x direction, yavσ =
average axial stress in the y direction, avτ = average edge
shear stress, and p = average net lateral pressure.
τav
y
x
p
a
b
τav
σxav
σyav
τav
y
x
p
a
b
τav
σxav
σyav
Figure 5. Idealized load application for the plating underuniform biaxial, edge shear and lateral pressure loads
The effect of in-plane bending stress in the x or y di-
rection is included in the buckling based capacity analysis.The in-plane bending stresses are defined as follows (Forthe symbols, see Figure 4)
=σσ
=σ
σ−=σσ=σ
−≠φ≠σσσ
=φσφ+φ−
=
σ−σ=σ−σ=σ
02
101
1
12
1221
11
2
21
xx
xav
xxxx
xxx
xxxav
x
x
xavxxavxxb
if
if
,if,
(4.a)
Paper Number 13 6
=σσ
=σ
σ−=σσ=σ
−≠φ≠σσσ
=φσφ+φ−
=
σ−σ=σ−σ=σ
02
101
1
12
1221
11
2
21
yy
yav
yyyy
yyy
yyyav
y
y
yavyyavyyb
if
if
,if,
(4.b)
where xbσ , ybσ = in-plane bending stress in the x or y
direction, respectively.For safety evaluation using equation (1), the measure of
the applied stresses can be defined for combined loading, asfollows
222avyavxav τσσσ ++= (5)
MODELING OF FABRICATIONRELATED IMPERFECTIONS
To fabricate the ship stiffened plate structure, welding isnormally used and thus the post-weld initial imperfections(initial deflections and residual stresses) develop in thestructure. In advanced ship structural design, capacity cal-culations of ship plating should accommodate post-weldinitial imperfections as parameters of influence. The char-acteristics of the post-weld initial imperfections are uncer-tain, and an idealized model is used.
Figure 6 shows a schematic of the post-weld initial de-flections in ship stiffened plate structure. The measurementsof welding induced initial deflection for plating in merchantship structures reveal a complex multi-wave shape in thelong direction and one half wave is found in the short direc-tion (Carlsen & Czujko 1978, Antoniou 1980, Kmiecik et al.1995). In this case, the plate initial deflection can approxi-mately be expressed by
∑=
=M
ioi
opl
o
b
y
a
xiB
w
w
1
sinsinππ
(6)
where oplw = relative maximum initial deflection of the
plating between stiffeners, and oiB = initial deflection am-
plitudes normalized by oplw .
B
Ly
xwosx
wopl
wopl
wosy
b
b
b
b
a aaa
Figure 6. Fabrication related initial deflections in steelstiffened panels
0
1
wo
/ wop
l
a/2 a(a) Initial deflection shape #1
0
1
wo
/ wop
l
a/2 a(b) Initial deflection shape #2
0
1
wo
/ wop
l
a/2 a
(c) Initial deflection shape #3
a/2 a
wo
/ wop
l
0
1
(d) Initial deflection shape # 4Figure 7. Some typical patterns of welding induced initialdeflection in ship plating
Table 1. Initial deflection amplitudes for various initial deflection shapes indicated in Figure 7Initial
Paik & Pedersen (1996) examined 33 sets of measure-ments and showed that equation (6) with 11=M could rea-sonably model the measured initial deflections. For theshapes of initial deflection in ship plating shown in Figure7, for instance, the coefficients oiB are given as those indi-
cated in Table 1. Smith et al. (1987) suggest the followingmaximum values of representative initial deflections forplating in merchant vessel structures which may be used toapproximate oplw in equation (6):
=
levelseriousfor
levelaveragefor
levelslightfor
t
wopl
2
2
2
3.0
1.0
025.0
β
β
β
(7)
The welding induced residual stress distributions can beidealized to be composed of tensile and compressive stressblocks, as shown in Figure 8. Along the welding line, tensile(positive) residual stresses are usually developed with mag-nitude rtxσ in the x direction and rtyσ in the y direction
since welding is normally performed in both x and y di-
rections. In order to obtain equilibrium, corresponding com-pressive (negative) residual stresses with magnitude rcxσ in
the x direction and rcyσ in the y direction are developed
in the middle part of plating. The breadths of the related ten-sile residual stress blocks in the x and y directions can be
shown to be as follows:
rtxrcx
rcxt
b
b
σσσ
−=
2,
rtyrcy
rcyt
a
a
σσσ
−=
2 (8)
where the tensile residual stress normally reaches the yieldstress of material for mild steel plating (e.g.,
ortyrtx σσσ ≈= ), while it is usually somewhat less (ap-
proximately 80% of the material yield stress) for high ten-sile steel plating (e.g., ortyrtx σσσ 8.0≈= ).
rcyσ
rtyσ
rcxσrtxσ
Comp.
Tens.
Tens.
x
y
at ata −2at
b−2 b
tb t
b t
Figure 8. Idealization of welding induced residual stressdistribution inside plating in the x and y directions
Once the magnitudes of the compressive and tensile re-sidual stresses are known, breadths of the tensile residualstress blocks can be determined from equation (8). The re-sidual stress distributions in the x and y directions may be
approximated by
≤≤−−<≤
<≤=
bybbfor
bbybfor
byfor
trtx
ttrcx
trtx
rx
σσσ
σ0
(9.a)
≤≤−
−<≤
<≤
=
axaafor
aaxafor
axfor
trty
ttrcy
trty
ry
σ
σ
σ
σ
0
(9.b)
Smith et al. (1987) also suggest the following repre-sentative values of welding induced compressive residualstress in the longitudinal ( x ) direction:
−−−
=levelseriousfor
levelaveragefor
levelslightfor
o
rcx
3.0
15.0
05.0
σσ
(10)
The magnitude of welding induced residual stresses inthe longer direction will normally be larger because the weldlength is longer. Therefore, the transverse (plate breadth di-rection) residual stresses may be approximated as follows:
rcxrcy a
b σσ = (11)
By substituting equation (7) with Table 1 into equation(6) or equations (8), (10) and (11) into equation (9), thepost-weld initial deflection and residual stress distributioncan reasonably be defined for practical design purposes.
BUCKLING BASED CAPACITY
Design EquationsThe basis of the plate capacity nominally adopted by
most classification societies is buckling. For one singlestress component loading, the buckling based capacity Bσ(i.e., xBσ for xavσ , yBσ for yavσ and Bτ for avτ ) is de-
fined using the so-called Johnson-Ostenfeld equation (orsometimes called Bleich-Ostenfeld equation) to account forthe effect of plasticity, as follows
( )
>
−−
≤
=5.0
411
5.0
k
E
E
kk
k
EE
B
if
if
σσ
σσσ
σσ
σ
σ (12)
Paper Number 13 8
where Eσ = elastic buckling stress for one single stress
component, (i.e., xEσ = as defined in eq. (26) for compres-
sive xavσ , yEσ = as defined in eq. (27) for compressive
yavσ and Eτ = as defined in eq. (28) for avτ ), ok σσ = for
either xavσ or yavσ , and 3/ook στσ == for avτ . It is
taken as oxB σ=σ for tensile xavσ and oyB σ=σ for ten-
sile yavσ .
The elastic buckling stress equations suggested in ourstudy accommodate the in-plane bending, lateral pressure,residual stress, and rotational restraints as necessary, but theeffect of initial deflection is not included since clear bifur-cation buckling is not defined for the initially deflectedplating.
For combined stress component loading, the buckling
based capacity component *Bσ is obtained as a solution of
the following equations (comp.:-, tens.:+)(a) When both xavσ and yavσ are compressive:
1222
=
+
+
B
av
yB
yav
xB
xav
ττ
σσ
σσ
(13.a)
(b) When either xavσ , yavσ or both are tensile:
1222
=
+
+
−
B
av
yB
yav
yB
yav
xB
xav
xB
xav
ττ
σσ
σσ
σσ
σσ
(13.b)
By taking xavσ as the reference (non-zero) stress com-
ponent, for instance, the solution of equation (13) with re-gard to xavσ is given by
(a) When both xavσ and yavσ are compressive:
( )222
2222
122
* 1
yBxBBxBByB
ByBxBxB
CC σστστσ
τσσσ
++−= (14.a)
(b) When either xavσ , yavσ or both are tensile:
( )222
2222
12
122
*
yBxBBxBByBxBByB
ByBxBxB
CCCs
σστστσστσ
τσσσ
++−=
(14.b)
where 1−=s if xavσ is compressive and 1=s if xavσ is
tensile. xav
yavC
σσ
=1 , xav
avCστ
=2 .
For safety evaluation using equation (1), the bucklingbased capacity measure cBσ of the plating under combined
loading are therefore given by holding the loading ratio con-stant, as follows
22
21
* 1 CCxBcB ++= σσ (15)
A similar method can be applied to calculate the buck-ling based capacity measures for the cases in which either
yavσ or avτ is taken as the reference stress.
Validity of the Johnson-Ostenfeld EquationTo account for the influence of plasticity on the buck-
ling based capacity equation, the Johnson-Ostenfeld formulais used, as defined in equation (12). Figures 9 to 11 showthe validity of the Johnson-Ostenfeld equation by compar-ing with the nonlinear finite element inelastic buckling (ul-timate strength) solutions for the plating under one singlestress component, varying the edge condition and the aspectratio. It is seen that the Johnson-Ostenfeld equation gener-ally has the tendency to underestimate the inelastic bucklingstrength.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0σxE/σo
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
σ xu/ σ
o
a/b = 3.0All edges remain straight (SE)
Johnson - Ostenfeld equation
: All edges simply supported (SS) : Simply supported alone longitudinal edges & clamped alone transverse edges (SC) : Clamped along longitudinal edges & simply supported along transverse edges (CS) : All edges clamped (CC)
Figure 9. The ultimate capacity versus the elasticbifurcation buckling stress of plating under longitudinalcompression alone, 0.3/ =ba (symbol: FEA)
Paper Number 13 9
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
σyE/σo
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2σ y
u/ σo
a/b = 3.0All edges remain straight (SE)
Johnson - Ostenfeld equation
: All edges simply supported (SS) : Simply supported alone longitudinal edges & clamped alone transverse edges (SC) : Clamped along longitudinal edges & simply supported along transverse edges (CS) : All edges clamped (CC)
Figure 10. The ultimate capacity versus the elastic bifurca-tion buckling stress of plating under transverse compressionalong, 3/ =ba (symbol: FEA)
: All edges simply supported (SS) : Simply supported alone longitudinal edges & clamped alone transverse edges (SC) : Clamped along longitudinal edges & simply supported along transverse edges (CS) : All edges clamped (CC)
Figure 11. The ultimate capacity versus the elastic bifurca-tion buckling stress of plating under edge shear alone,
3/ =ba (symbol: FEA)
Effect of Rotational RestraintsThe rotational restraints of the support members are in-
cluded in the elastic buckling equations for both xavσ and
yavσ as parameters of influence. Ship plating is supported
by various types of members along the edges, which have afinite value of the torsional rigidity. This is in contrast to theidealized simply supported boundary conditions often as-sumed for design purposes. Depending on the torsional ri-gidity of support members, the rotation along the plateedges will to some extent be restrained. When the rotationalrestraints are zero, the edge condition corresponds to a sim-ply supported case, while the edge condition becomesclamped when the rotational restraints are infinite.
Most current practical design guidelines from classifica-tion societies for the buckling and ultimate strength of shipplating are based on boundary conditions in which all (four)edges are simply supported. In real ship plating, idealizededge conditions such as simply supported or clamped how-ever may never occur because of finite rotational restraints.
According to the study of Paik et al. (1993) who inves-tigated the bending and torsional rigidities of support mem-bers for deck, side and bottom plating in merchant ships, themagnitude of the rotational restraint parameter Lζ at longedges (ship’s longitudinal direction) is normally in the rangeof 0.05 to 3.0 (and usually not exceeding 5.0) while theamount Sζ at the short edges (normal to the ship longitudi-nal direction) is normally in the range of 0.1 to 8.0 (andusually not exceeding 13.0). Thus, there is of course no casewith zero or infinite rotational restraints in practice as longas support members exist at their edges, and the amount ofthe rotational restraints at one set of long or short edges isnormally different from each other as well. It was also foundfrom the same investigation that the bending rigidities ofsupport members are usually sufficient enough so that therelative lateral deflection of typical members providing thesupport to plating at edges can be taken to be small.
For advanced design of ship plating against buckling, itis hence important to better understand the buckling strengthcharacteristics of plating as a function of the rotational re-straints of support members along the edges. This problemhas of course been studied before, by a number of investiga-tors. Lundquist & Stowell (1942) studied the effect of theedge condition on the buckling strength of rectangular platessubject to uniaxial compressive loads where the support alongthe unloaded edges was intermediate between simply sup-ported and clamped. Bleich (1952) and Timoshenko & Gere(1963) discussed the buckling strength of plates with variousboundary conditions that one set of edges is elastically re-strained while the other set of edges is either simply sup-ported or clamped. Gerard & Becker (1954) surveyed litera-ture for the buckling of rectangular plates under variouscombinations of two or three types of loading under a numberof edge conditions. Evans (1960) carried out an extensive ex-perimental study on the buckling strength of wide plates withthe loaded (long) edges elastically restrained while the un-loaded (short) edges are simply supported. Based on the ex-perimental results, he derived a closed-form expression of thecompressive strength of wide plates taking into account theeffect of rotational restraints along the loaded edges. McKen-zie (1963) studied the buckling strength of plating under bi-axial compression, bending and edge shear that is simplysupported along short edges (at which bending is applied) andelastically restrained along long edges.
These various previous studies are quite useful for thebuckling strength design of plating considering the rota-tional restraint effect along the edges. To the authors’knowledge, however, systematic investigations on thebuckling strength of plating which is elastically restrainedalong both long and short edges appear to be difficult tocome by and were thus needed. The aims of our study re-lated to this issue (Paik & Thayamballi 2000) were to
• investigate the buckling strength characteristicsof plating with the boundary conditions whichare elastically restrained along the edges, and to
Paper Number 13 10
• develop simple buckling design formulationsof plating taking into account the rotational re-straints of support members along either oneset of edges or all (four) edges.
The simplified formulations referred to are based onmore exact solutions as obtained by directly solving thebuckling characteristic equations for a variety of the tor-sional rigidities of support members and the plate aspect ra-tio. The characteristic equation for the buckling of platingwith elastic restraints along either long or short edges whilethe other edges are simply supported is derived analytically.By solving the characteristic equation, the buckling strengthcharacteristics of plating are investigated varying the plateaspect ratio and the torsional rigidity of support members.Based on the computed results, closed-form expressions ofthe plate buckling strength are obtained empirically bycurve fitting. Simplified buckling design formulations forplating with all edges elastically restrained are also derived.
Figures 12 to 15 show some selected sets of the bucklingcoefficients as obtained by directly solving the theoreticalcharacteristic buckling equation plotted against the plate as-pect ratio and the torsional rigidity of support members alongthe plate edges. The accuracy of the proposed simplifiedequations obtained by curve fitting the more exact results maybe verified by comparison with the exact theoretical solutions,see Figures 12 and 16 to 18. The curve-fit design equations
121 ,, yxx kkk and 2yk are given in Appendix 1.
One of the useful insights developed herein is that thebuckling coefficient for the plating elastically restrained atboth long and short edges can be expressed by a relevantcombination of the following three edge conditions, namely(a) elastically restrained at long edges and simply supportedat short edges, (b) simply supported at long edges and elas-tically restrained at short edges, and (c) simply supported atall edges. Specifically it was noted that the following heldapproximately:
xoxxx kkkk −+= 21 , yoyyy kkkk −+= 21 (16)
where xk = buckling coefficient of plating elastically re-
strained at both long and short edges for longitudinal com-pression, yk = buckling coefficient of plating elastically re-
strained at both long and short edges for transversecompression, xok = buckling coefficient of plating simply
supported at all edges for longitudinal compression whichmay be taken as 0.4≅xok , and yok = buckling coefficient
of plating simply supported at all edges which may be taken
as { }22)/(1 abk yo += . xok , 1xk , 2xk , yok , 1yk , 2yk = as
defined in Appendix 1.It was also found that the buckling interaction equation
of the plating elastically restrained along all edges and undercombined loading can approximately take the same relation-
ship as that with simply supported conditions at all edges,but by replacing the buckling stress components of theplating simply supported at all edges with the correspondingones for the elastically restrained plating. As two specificcases of plating under combined biaxial compression orcombined axial compression and edge shear, where the plateedges are all clamped, i.e., with infinite rotational restraints,Figures 19 and 20 show the elastic buckling interaction re-lations varying the plate aspect ratio where the theoreticalpredictions were obtained by the formulae for simply sup-ported plates as given in Appendices 2 and 3 while FE solu-tions were calculated for clamped plates.
0 1 2 3 4 5 6 7 8 9 10 110
2
4
6
8
10
a/b
k x1
ExactApproximate
GJL/bD = 20.0
GJL/bD = 2.0
GJL/bD = 0.3
GJL/bD = 0.0
Figure 12. Buckling coefficient 1xk for a plate under lon-
gitudinal compression, elastically restrained at the longedges and simply supported at the short edges as obtainedby directly solving the buckling characteristic equation andby the proposed approximate equation
0 1 2 3 4 5 63
4
5
6
7
8
9
k x2
a/b
GJS/aD = 20.0
GJS/aD = 0.4
GJS/aD = 0.2 GJS/aD = 0.1 GJS/aD = 0.0
Figure 13. Buckling coefficient 2xk for a plate under lon-
gitudinal compression, elastically restrained at the shortedges and simply supported at the long edges as obtained bydirectly solving the buckling characteristic equation
Paper Number 13 11
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00
1
2
3
4
5
6
7
b/a
k y1 = 500.0
= 20.0
= 10.0
= 4.0
= 2.0= 1.0
= 0.0
GJL/bD = oo
Figure 14. Buckling coefficient 1yk for a plate under trans-
verse compression, elastically restrained at the long edgesand simply supported at the short edges as obtained by di-rectly solving the buckling characteristic equation
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
1
2
3
4
5
6
7
8
k y2
b/a
= 10.0
GJS/aD=oo
= 2.0
= 1.0= 0.5
= 0.2
= 0.0
Figure 15. Buckling coefficient 2yk for a plate under
transverse compression, elastically restrained at the shortedges and simply supported at the long edges as obtained bydirectly solving the buckling characteristic equation
0 5 10 15 20 25
0
1
2
3
4
5
6
7
8
kx2
GJS/aD
ExactApproximate
a/b = 1.0
a/b = 1.5a/b = 2.0a/b = 3.0a/b = 5.0
Figure 16. Accuracy of the design equation for the buck-ling coefficient 2xk
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
9
GJL/bD
k y1
a/b = 1.0
a/b = 0.8
a/b = 0.5
a/b = 0.2
a/b = 0.0
ExactApproximate
Figure 17. Accuracy of the design equation for the buck-ling coefficient 1yk
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
9
10
ky2
GJS/aD
b/a = 1.0
= 0.9
= 0.8
= 0.5
= 0.0
ExactApproximate
Figure 18. Accuracy of the design equation for the buck-ling coefficient 2yk
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
FEM (ANSYS) : a/b=2: a/b=1
: a/b=3
a/b=1
a/b=2
a/b=3
xE
xav
σσ
yE
yav
σσ
Figure 19. Elastic buckling interaction relationships forplating under combined biaxial compression (symbol: eigenvalue finite element solutions for plating clamped at alledges, line: design equation for plating simply supported atall edges)
Paper Number 13 12
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
a/b=1a/b=2
a/b=3
FEM (ANSYS) : a/b=2: a/b=1
: a/b=3
E
av
ττ
xE
xav
σσ
Figure 20. Elastic buckling interaction relationships ofplating under combined axial compression and edge shear(symbol: eigen value finite element solutions for platingclamped at all edges, line: design equation for plating sim-ply supported at all edges)
Effect of Residual StressesThe welding induced residual stress (compression: -,
tension: +) is included in the elastic buckling equations forboth xavσ and yavσ as a parameter of influence. The elas-
tic buckling stress of simply supported plating under uni-form axial compression (i.e., without in-plane bending) isgiven by (Paik et al. 2000) (For the symbols unless specifiedbelow, refer to the section on “modeling of fabrication re-lated imperfections”)
reyrexxEbm
a
mb
a
a
mb
tb
D σσπσ22
22
2
2
−−
+−= (17)
where ( )
−−+=
b
bbb
bt
trcxrtxrcxrexπ
πσσσσ 2
sin2
2,
( )
−−+=
a
am
m
aa
at
trcyrtyrcyreyπ
πσσσσ 2
sin2
2
The second and third terms of the right hand side ofequation (17) reflect the effect of welding induced residualstresses on the plate compressive buckling stress. m is thebuckling half wave number which is determined as a mini-mum integer satisfying the following equation
reybm
a
mb
a
a
mb
tb
D σπ22
22
2
2
+
+
( ) rey
bm
a
bm
a
a
bm
tb
D σπ22
22
2
2
1)1(
)1(
++
+
++≤ (18)
Without the post-weld residual stresses, i.e.,0== reyrex σσ , equation (18) simplifies to the well-known
condition
( )1** +≤ mmb
a (19)
where *m is the buckling half wave number when the resid-ual stresses do not exist.
In the similar way, the elastic buckling stress yEσ of
the simply supported plating subject to axial compression inthe y direction can be given by
reyrexyEa
b
a
b
tb
D σσπσ −−
+−=
2
22
2
2
2
2
1 (20)
where rexσ and reyσ are defined as those in equation (17)
but replacing by 1=m . The second and third terms of theright hand side of equation (20) reflect the effect of weldinginduced residual stresses.
Figure 21 shows the influence of welding induced re-sidual stress on the compressive buckling stress for the hightensile steel plating with the yield stress of MPao 352=σ .
In the calculations indicated in Figure 21, the level of resid-ual stresses and the plate slenderness ratio (i.e., tb / ratio)are varied. Two types of welding induced residual stressesin the y direction are presumed, namely one with zero re-
sidual stresses and the other with rcxrcy a
b σσ = . It is in the
analysis assumed that the magnitude of the tensile residualstresses is 80% of the yield stress, that is,
ortyrtx σσσ 8.0== . It is evident from Figure 21 that the
welding residual stresses can significantly reduce the com-pressive buckling stress of the plating. The reduction ten-dency of the buckling stress for thin plating is faster thanthat for thick plating, as expected. It is also noted from Fig-ures 21c and 21d that the residual stresses in the y direction
may change the longitudinal buckling half wave number ofthe plating.
Paper Number 13 13
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
05.0/ =orcx σσ
15.0/ =orcx σσ
30.0/ =orcx σσ
50/ =tb0.0/ =orcy σσ
*/
xExE
σσ
ba /
m/m* =1/1
m/m* =2/2
m/m* =3/3
m/m* =4/4
m/m* =5/5
Figure 21a. Variation of the elastic compressive bucklingstress (normalized by the elastic buckling compressive stresswithout residual stresses) as a function of the welding in-duced residual stress and the plate aspect ratio, 0=rcyσ ,
Figure 21b. Variation of the elastic compressive bucklingstress (normalized by the elastic compressive buckling stresswithout residual stresses) as a function of the welding in-duced residual stress and the plate aspect ratio, 0=rcyσ ,
Figure 21c. Variation of the elastic compressive bucklingstress (normalized by the elastic compressive buckling stresswithout residual stresses) as a function of the welding in-duced residual stress and the plate aspect ratio,
abrcxrcy /σσ = , 50/ =tb , 07.2=β , MPao 352=σ ,
ortx σσ 8.0= ( *xEσ = buckling stress without residual
Figure 21d. Variation of the elastic compressive bucklingstress (normalized by the elastic compressive buckling stresswithout residual stresses) as a function of the welding in-duced residual stress and the plate aspect ratio,
abrcxrcy /σσ = , 100/ =tb , 14.4=β , MPao 352=σ
ortx σσ 8.0= ( *xEσ = buckling stress without residual
stress)
Effect of Cut-OutsIn ship plating, cut-outs are often located to make a way
of access or to lighten the structure. These will reduce thecapacity of the plating. The opening is included in the elasticbuckling equations for xavσ (compression), yavσ (compres-
sion) and avτ as a parameter of influence. For a circular type
of opening, the buckling reduction factors are in our study
Paper Number 13 14
suggested by curve fitting based on the eigen value finiteelement solutions, as follows
for the effect of cut-outs under xavσ , yavσ and avτ , re-
spectively, and cd = diameter of circular opening .
Figures 22 and 23 show the validity of equation (21) bycomparing with the eigen value finite element solutions,varying the diameter of opening. It is seen that with anopening of which diameter is 50% of the plate breadth, theplate compressive buckling becomes 70% of the originalstrength. As the diameter of opening increases, the bucklingreduction tendency for edge shear loading is slightly fasterthan that for axial compressive loading.
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0d c /b
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
Rcx
( ) 0.1/57.0 +−= bdR ccx
dc b
a/b=1.0
a
: FEM (ANSYS)
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0d c /b
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
Rcx
( ) 0.1/57.0 +−= bdR ccx
dc b
a/b=1.0
a
dc b
a/b=1.0
dc b
a/b=1.0
a
: FEM (ANSYS)
Figure 22. Buckling reduction factor accounting for the ef-fect of cut-outs under axial compression
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0d c /b
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
Rcs
dc b
a/b=1.0
a
( ) 0.1/68.0 +−= bdR ccs
: FEM (ANSYS)
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0d c /b
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
Rcs
dc b
a/b=1.0
a
dc b
a/b=1.0
a
( ) 0.1/68.0 +−= bdR ccs
: FEM (ANSYS)
Figure 23. Buckling reduction factor accounting for the ef-fect of cut-outs under edge shear
Effect of Lateral PressureThe lateral pressure is included in the elastic buckling
equations for both xavσ and yavσ as a parameter of influ-
ence. When the buckling half wave corresponds to the platedeflection pattern caused by lateral pressure alone, e.g., in anearly square plate with 0.1/ ≈ba or a long plate underpredominantly transverse compression ( yavσ ), lateral pres-
sure reduces the compressive buckling stress. However, lat-eral pressure loads increase the compressive buckling stressfor a long plate under predominantly longitudinal compres-sive loads ( xavσ )since lateral pressure disturbs the occur-
rence of buckling in this case. The buckling correction fac-tors accounting for the effect of lateral pressure are given by(Paik & Kim 1988)
>
+
≤≤
−
=
20.1
21056.00.1
2/1
4
4
4
4
b
afor
Et
pbC
b
afor
Et
pb
R
p
px
−=
4
4
056.00.1Et
pbRpy
(22)
where pxR , pyR = buckling correction factor accounting for
the effect of lateral pressure under xavσ and yavσ , respec-
tively, with
>
≤≤+
−
=
414.0
42360.0155.0025.02
b
afor
b
afor
b
a
b
a
C p
Figure 24 shows the variation of the elastic longitudinalcompressive buckling plotted against the magnitude of lateralpressure loads. It is seen from Figure 24 that for a squareplate lateral pressure reduces the buckling, while bucklingstrength of a long plate under axial compressive loads in thelongitudinal direction is increased by lateral pressure.
Effect of In-Plane BendingAs shown in Figure 4, ship plating is often subjected to
in-plane bending which affects the buckling capacity. Thein-plane bending stresses are included in the elastic com-pressive buckling equations as parameters of influence. Ingeneral, the in-plane bending stresses can be given in termsof average axial stresses, as defined in equation (4). Theelastic buckling interaction equations between xavσ and
xbσ or between yavσ and ybσ may be approximated by
12
*=
+
xbE
xb
xE
xav
σσ
σσ , 1
2
*=
+
ybE
yb
yE
yav
σσ
σ
σ (23)
where xbEσ , ybEσ = as defined in Appendix 4
Paper Number 13 15
2.0 4.0 6.0 8.0 P
0.2
0.4
0.6
0.8
1.0
σxE
σxE*
a/b = 1
P =pb4
Et4
0
σxE = Elastic buckling under axialcompression and lateral pressure loads
σxE*
= Elastic bucklingcompressive loads alone
stress
under axialstress
2.0 4.0 6.0 8.0 P
0.2
0.4
0.6
0.8
1.0
σxE
σxE*
a/b = 1
P =pb4
Et4
0
σxE = Elastic buckling under axialcompression and lateral pressure loads
σxE*
= Elastic bucklingcompressive loads alone
stress
under axialstress
Figure 24a. Variation of the elastic compressive bucklingfor a square plate against the magnitude of lateral pressureloads
10 20 30 40 50 60 P
0.5
1.0
1.5
2.0
2.5
3.0
3.5
σxE
σxE*
a /b = 3
P =pb4
Et4
0
σxE = Elastic buckling stress under axialcompression and lateral pressure loads
σxE*
= Elastic bucklingcompressive loads alone
stress under axial
Elastic buckling for clamped plating
Design equation
10 20 30 40 50 60 P
0.5
1.0
1.5
2.0
2.5
3.0
3.5
σxE
σxE*
a /b = 3
P =pb4
Et4
0
σxE = Elastic buckling stress under axialcompression and lateral pressure loads
σxE*
= Elastic bucklingcompressive loads alone
stress under axial
Elastic buckling for clamped plating
Design equation
Figure 24b. Variation of the compressive buckling for arectangular plate of a/b=3 against the magnitude of lateralpressure loads
In the above equation, *xEσ is the elastic buckling stress
under uniform longitudinal axial compression (i.e., withoutin-plane bending) which is given considering the effects ofrotational restraints, residual stress, lateral pressure and cut-outs and is defined, as follows
( ) cxpxreyrexxxE RRmb
a
b
tEk
−−
−−= σσ
νπσ
22
2
2*
112
(24)
where xk = as defined in Appendix 1, rexσ , reyσ , m = as
defined in equation (17), pxR = as defined in equation (22),
cxR = as defined in equation (21)*yEσ in equation (23) is the elastic buckling stress under
uniform transverse axial compression (i.e., without in-planebending) which is given considering the effects of rotationalrestraints, residual stress, lateral pressure and cut-outs and isdefined, as follows
( ) cypyreyrexyyE RRa
b
b
tEk
−
−
−−= σσ
νπσ
22
2
2*
112
(25)
where yk = as defined in Appendix 1, rexσ , reyσ = as de-
fined in equation (20), pyR = as defined in equation (22),
cyR = as defined in equation (21)
Considering the relationship between xavσ and xbσ as
defined in equation (23), the axial compressive bucklingstress xEσ taking into account the effect of in-plane bending
stresses are given, as follows(a) When 21 xx σσ = :
*xExE σσ = (26.a)
(b) When 21 xx σσ ≠ :
1
12
22
2
4
F
FFFxE
+−−=σ (26.b)
(c) When 21 xx σσ −= :
( ) xbExE σσ 1−= (26.c)
where 2
2
1xbE
CF
σ= ,
*21
xE
Fσ
= ,
−≠=
−≠≠=+−
=
211
2111
2
,00.1
,0,1
1
xxx
xxxx
xx
x
x
if
ifC
σσσ
σσσσσφ
φφ
In the similar way, the axial compressive bucklingstress yEσ taking into account the influence of in-plane
bending is calculated from considering the relation between
yavσ and ybσ defined in equation (23), as follows
(a) When 21 yy σσ = :
*yEyE σσ = (27.a)
Paper Number 13 16
(b) When 21 yy σσ ≠ :
1
1222
2
4
G
GGGyE
+−−=σ (27.b)
(c) When 21 yy σσ −= :
( ) ybEyE σσ 1−= (27.c)
where 2
2
1ybE
CG
σ= ,
*21
yE
Gσ
= ,
−≠=
−≠≠=+−
=
211
2111
2
,00.1
,0,1
1
yyy
yyyy
yy
y
y
if
ifC
σσσ
σσσσσ
φφφ
Elastic Edge Shear BucklingThe elastic shear buckling stress taking into account the
influence of cut-outs is defined as follows
( ) cssE Rb
tEk
2
2
2
112
−=
νπτ (28)
where sk = as defined in Appendix 5, csR = as defined in
equation (21)
ULTIMATE STRENGTH BASED CAPACITY
Most classification society criteria and procedures forship structural design are based on the first yield of hullstructures together with buckling checks for structural com-ponents. While service proven, the traditional design criteriaand associated linear elastic stress calculations do not neces-sarily define the true ultimate limit state which is the limit-ing condition beyond which a ship hull will fail to performits function. Neither do such procedures help understand thelikely sequence of local failure prior to reaching the ultimatelimit state. It is of course important to determine the true ul-timate strength if one is to obtain consistent measures ofsafety which can form a fairer basis for comparisons of ves-sels of different sizes and types. An ability to better assessthe true margin of safety should also inevitably lead to im-provements in related regulations and design requirements.
In the case of plate elements which constitute a signifi-cant portion of the hull and thus affect its weight and otherdesign characteristics, it is now known that a single set ofultimate strength interaction equations will not successfullyrepresent the ultimate limit state of ship plating under com-bined loads since collapse patterns significantly depend onthe types and relative magnitudes of primary load compo-nent involved. The strength interaction relationship would
thus be different depending on which load component ispredominant. In this regard, the present authors have devel-oped three sets of such equations considering each primaryload component, namely longitudinal axial load, transverseaxial load and edge shear, while lateral pressure is regardedas secondary. The ultimate strength interaction equation un-der all of the load components is derived by a relevant com-bination of the individual strength formulae (Paik et al.1999a). In the following, the plate ultimate strength equa-tions for plating under combined in-plane and lateral pres-sure loads are presented.
Ultimate Strength Equation for CombinedLongitudinal Axial Load and Lateral Pressure
Figure 25 shows a typical example of the axial mem-brane stress distribution inside a plate element under pre-dominantly longitudinal compressive loading, before andafter buckling occurs. It is noted that the membrane stress
x
y
∫=b
xxav dyb 0
1 σσ
σxav
a
b
(a) Before buckling
σxmin
σxav σxmaxx
y a
b
∫=b
xxav dyb 0
1 σσ
(b) After buckling, unloaded edges move freely in plane
σxmin
σxmax
σxav
σyminσymax
x
b
y a
ò=
b
xxav dyb 0
1 σσ
(c) After buckling, unloaded edges keep straight
Figure 25. Membrane stress distribution inside the plateelement under longitudinal compressive loads
Paper Number 13 17
distribution in the loading ( x ) direction can become non-uniform as the plate element deflects (or buckles). Themembrane stress distribution in the y direction also be-
comes non-uniform if the unloaded plate edges remainstraight, while no membrane stresses will develop in the y
direction if the unloaded plate edges move freely in plane.The maximum compressive membrane stresses are devel-oped around the plate edges that remain straight, while theminimum (tensile) membrane stresses occur in the middle ofthe plate element where a membrane tension field is formedby the plate deflection since the plate edges remain straight.
With increase in the plate deflection, the upper and/orlower fibers in the mid-region of the plate element will ini-tially yield by the action of bending. However, as long as itis possible to redistribute the applied loads to the straightplate boundaries by the membrane action, the plate elementwill not collapse. Collapse will then occur when the moststressed boundary locations yield, since the plating can notkeep the boundaries straight any further, resulting in a rapidincrease of lateral plate deflection (Paik & Pedersen 1995).
Hence the ultimate strength formulation for ship platingsubject to uniaxial compression/tension and lateral pressureloads is in the present study derived under the somewhatpessimistic assumption that the plating collapses when ini-tial plastic yield at the plate edges occurs.
The occurrence of yielding can be assessed by using thevon Mises yield criterion. For longitudinal axial load andlateral pressure, the most probable yield locations will befound at longitudinal mid-edges where the maximum com-pressive stress in the x direction and the minimum tensilestress in the y direction develop, as shown in Figure 26a.
The resulting yield criterion is in this case expressed by (Forthe stress related symbols, see Figure 25c)
012
minminmax2
max =−
+
−
=Γ
o
y
o
y
o
x
o
xu σ
σσ
σσ
σσ
σ
(29)
where maxxσ and minyσ are given in terms of xavσ , p and
post-weld initial imperfections (initial deflection and resid-ual stresses), as defined in Appendix 6.
The ultimate strength based capacity xuσ for longitudi-
nal axial load is obtained as the solution of equation (29)with regard to xavσ . As an approximation, xuσ is taken as
the initial (minimum) value at 0≥Γu by increasing xavσwith the increment of 1% yield stress, i.e.,
oxav σσ 01.0−=∆ for compressive xavσ and
oxav σσ 01.0=∆ for tensile xavσ . Figure 27 shows the
variation of the ultimate longitudinal compressive stressesplotted against the plate reduced slenderness ratio, as ob-tained from equation (29) and from nonlinear FEA. Thecompressive stresses at the initial yielding are also plotted.
Figure 26. Possible locations for the initial plastic yield atthe plate edges under combined uniaxial load and pressure
0 .0 1 .0 2 .0 3 .0 4 .0 5 .0
β = b / t , √σo /E
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
1 .2
σ xu /
σ o
t= 2 5 m m
t= 1 5 m m
: E la s tic b u ck lin g s tre n g th : E la s tic b u ck lin g s tre n g th w ith p la s tic ity c o rre c tio n
F E M (F u jik u b o e t a l. 1 9 9 7 ) : U l tim a te s tre n g th (σxu /σo) : In itia l y ie ld in g (σx /σo)
P re se n t d e s ig n fo rm u la : U l tim a te s tre n g th (σxu /σo)
Figure 27. Variation of the ultimate longitudinal compres-sive strength of a long plating shown as a function of the re-duced slenderness ratio, 3/ =ba
It is evident that equation (29) agrees very well with themore refined nonlinear FEA. Equation (29) shows meanbias = 0.978 and COV = 0.065 against FEA. Figure 28shows the variation of the ultimate axial compressive stressplotted against the initial deflection with the shape #1 inFigure 7. Figure 29 shows the ultimate strength interaction
Paper Number 13 18
relationship between longitudinal axial compression and lat-eral pressure. In these calculations, the plate edges are sim-ply supported and kept straight.
0 0 .1 0 .2 0 .3 0 .4
wo p l
(×β2t)
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
σ xu / σ
o
F E M (A N S Y S ) : U ltim a te s tre n g th (a /b = 1 ) : U ltim a te s tre n g th (a /b = 3 )P resen t d e sig n fo rm u la : U ltim a te s tre n g th (a /b = 1 ) : U ltim a te s tre n g th (a /b = 3 )b×t = 1 ,0 0 0×1 5 m mσo = 2 3 5 .2 M P a , E = 2 0 5 .8 G P a
Figure 28. Effect of initial deflection on the plate ultimatecompressive strength
0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7p ( N /m m 2 )
0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
σ xu
/σo
E x p e rim e n t : O k a d a e t a l. [1 ] (1 98 0 ) : O k a d a e t a l. [2 ] (1 98 0 )P re se n t d e sig n fo rm u la : [1 ] : [2 ]
a×b = 9 9 4×33 0 m m , E = 2 0 0 .9 G P aw o p l = 0 .0 , σr c x = 0 .0
[1 ] t = 3 .2 2 m m , σo = 3 17 .5 2 M P a
[2 ] t = 4 .4 8 m m , σo = 3 03 .8 M P a
Figure 29. The ultimate strength interaction of platingbetween axial compression and lateral pressure
Ultimate Strength Equation for CombinedTransverse Axial Load and Lateral Pressure
In this case, the most probable yield location is found atthe transverse mid-edges where the maximum compressivestress in the y direction and the minimum tensile stress in
the x direction develop, as shown in Figure 26b. The re-sulting yield criterion is then given by (For the stress relatedsymbols, see Figure 25c)
012
maxmaxmin2
min =−
+
−
=Γ
o
y
o
y
o
x
o
xu σ
σσ
σσ
σσ
σ
(30)
where minxσ and maxyσ are obtained in terms of yavσ , p
and post-weld initial imperfections (initial deflection andresidual stresses), as defined in Appendix 7.
The ultimate strength based capacity yuσ for transverse
axial load is obtained as the solution of equation (30) withregard to yavσ . As an approximation, yuσ is taken as the
initial (minimum) value at 0≥Γu by increasing yavσ with
the increment of oyav . σ−=σ∆ 010 for compressive yavσ
and oyav σσ 01.0=∆ for tensile yavσ . Figure 30 shows the
variation of the ultimate transverse compressive stress plot-ted against the plate slenderness ratio, as obtained fromequation (30) and from nonlinear FEA. Equation (30) showsmean bias = 0.997 and COV = 0.078 against FEA.
0.0 1 .0 2 .0 3 .0 4 .0 5 .0
β = b /t , √σo /E
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
1 .2
σ yu /
σo
t= 1 5 m m
t= 2 5m m
: E lastic b u cklin g s treng th : E lastic b u cklin g s treng th w ith p lastic ity co rrec tio n
F E M (F u jiku b o e t a l . 1 9 9 7) : U ltim a te s treng th (σyu /σo) : In itia l y ie ld in g (σy /σo)
P resen t d esig n form ula : U ltim a te s treng th (σyu /σo)
Figure 30. Variation of the ultimate transverse compressivestrength of a long plating shown as a function of the reducedslenderness ratio, 3/ =ba
Ultimate Strength Equation for Edge ShearBased on a series of the nonlinear finite element calcu-
lations for simply supported plating varying the plate ge-ometry and the aspect ratio, Paik (1999) derived the fol-lowing empirical formula for the ultimate edge shearstrength of plating, namely
>ττ
≤ττ
<+
ττ
+
ττ
−
ττ
≤ττ
<
ττ
=ττ
029560
02503880
676027400390
5003241
23
.for.
..for.
...
.for.
o
E
o
E
o
E
o
E
o
E
o
E
o
E
o
u (31)
Paper Number 13 19
where Eτ is the elastic shear buckling stress of the plating
with 1/ =ba , which is taken as
( )2
2
2
11234.9
−=
b
tEE ν
πτ .
Figure 31 shows the variation of the ultimate shearstrength of ship plating against the elastic shear bucklingstress. The nonlinear finite element solutions varying themagnitude of post-weld initial deflections are compared.The dotted line represents the elastic shear bucklingstrengths with plasticity correction made by the Johnson-Ostenfeld formula. It is noted that the influence of lateralpressure on the ultimate shear strength is normally small,and equation (31) can approximately be applied for theplating under combined avτ and p as well. Equation (31)
shows mean bias = 0.931 and COV = 0.075 against FEA.
E las tic b u ck lin g s tre n g thw ith p la s tic ity c o rrec tio n
τu /τo = 1 .3 2 4(τE /τo ) . .. . .. if 0 < τE /τo ≤ 0.5F E M (w op l = 0 .1β2 t) : a /b = 1 : a /b = 3 : a /b = 5F E M (w op l = 0 .0 5 t) : a /b = 1 : a /b = 2 : a /b = 3
Figure 31. The ultimate strength versus the elastic bifurca-tion buckling stress of plating under edge shear
Ultimate Strength Equation for Combined BiaxialLoad, Edge Shear and Lateral Pressure
Based on the insights developed (Paik 1999), our stud-ies suggest the following ultimate strength equations ofplating under combined biaxial load, edge shear and lateralpressure (comp.:-, tens.:+)
(a) When both xavσ and yavσ are compressive:
1222
=
+
+
u
av
yu
yav
xu
xav
ττ
σσ
σσ
(32.a)
(b) When either xavσ , yavσ or both are tensile:
1222
=
+
+
−
u
av
yu
yav
yu
yav
xu
xav
xu
xav
ττ
σσ
σσ
σσ
σσ
(32.b)
where xuσ , yuσ = solution of equations (29) and (30), re-
spectively, and uτ = as defined in equation (31).
By taking xavσ as the reference (non-zero) stress com-
ponent, for instance, the solution of equation (32) with re-gard to xavσ is given by
(a) When both xavσ and yavσ are compressive:
( )222
2222
122
1
yuxuuxuuyu
uyuxu*xu
CC σσ+τσ+τσ
τσσ−=σ (33.a)
(b) When either xavσ , yavσ or both are tensile:
( )222
2222
12
122
yuxuuxuuyuxuuyu
uyuxu*xu
CCCs
σσ+τσ+τσσ−τσ
τσσ=σ
(33.b)
where xav
yavCσσ
=1 , xav
avCστ=2 , s = as defined in equation
(14.b)For safety evaluation based on the ultimate strength
using equation (1), the plate capacity measure cuσ is then
given by holding the loading ratio constant, as follows
22
21
* 1 CCxucu ++= σσ (34)
Figures 32 and 33 show the validity of equation (32) for theplating under combined biaxial compression by a compari-son with the conventional nonlinear finite element solutions,varying the aspect ratio, the plate thickness and the level ofthe post-weld initial imperfections. For both FEA and de-sign formula predictions, the shape #1 of initial deflectionas indicated in Figure 7 is presumed. All edges are simplysupported and kept straight. Figure 34 shows the plate ulti-mate strength interaction between axial compression andedge shear, as those obtained by the present design formulaand the FEA.
COMPARISON BETWEEN BUCKLING ANDULTIMATE STRENGTH BASED CAPACITIES
Figures 35 and 36 compare the plate ultimate strengthinteractions between biaxial compression or tension, asthose obtained by the FEA, the buckling or ultimate strengthbased capacity equations, varying the aspect ratio and theplate thickness. For FEA, average level of initial deflectionis considered with the shape #1 as indicated in Figure 7, butno residual stresses are presumed. For the buckling formulapredictions, no initial deflection is considered while a slightlevel of residual stresses is assumed. For the ultimatestrength formula predictions, both initial deflection and re-sidual stresses are presumed.
It is seen that for thin plating which buckles in the elasticregime the formula prediction based on the buckling is too
Paper Number 13 20
pessimistic when compared against the ultimate limit state,while for relatively thick plating that buckles in the inelasticregime it provides good measures for the structural capacity.The capacity formula based on the ultimate strength gives ex-cellent indications for both thin and thick plating.
a×b ×t= 2 ,5 5 0 ×8 5 0 ×1 3 m mσo = 3 5 2 .8 M P aE = 2 0 5 .8 G P a
β = b /t√σo /E = 2 .7 0 7
w op l = 0 .1 ×β2t = 9 .5 2 6 m mσr cx = - 0 .1 ×σo
σr cy = b /a×σrcx
F E M (A N S Y S ) : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s sP re s e n t d e s ig n fo rm u la : W ith o u t in iti a l im p e r fe c tio n s : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s s : W ith b o th in itia l d e fle c tio n a n d r e s id u a l s tr e s s
a×b ×t= 2 ,5 5 0 ×8 5 0 ×2 1 m mσo = 3 5 2 .8 M P aE = 2 0 5 .8 G P a
β = b /t√σo /E = 1 .6 7 6
w op l = 0 .1 ×β2t = 5 .8 9 9 m mσr cx = - 0 .1 ×σo
σr cy = b /a×σrcx
F E M (A N S Y S ) : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s sP re s e n t d e s ig n fo rm u la : W ith o u t in iti a l im p e r fe c tio n s : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s s : W ith b o th in itia l d e fle c tio n a n d r e s id u a l s tr e s s
a×b ×t= 5 ,1 0 0 ×8 5 0 ×1 3 m mσo = 3 5 2 .8 M P aE = 2 0 5 .8 G P a
β = b /t√σo /E = 2 .7 0 7
w op l = 0 .1 ×β2t = 9 .5 2 6 m mσr cx = - 0 .1 ×σo
σr cy = b /a×σrcx
F E M (A N S Y S ) : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s sP re s e n t d e s ig n fo rm u la : W ith o u t in iti a l im p e r fe c tio n s : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s s : W ith b o th in itia l d e fle c tio n a n d r e s id u a l s tr e s s
a×b ×t= 5 ,1 0 0 ×8 5 0 ×21m mσo = 3 5 2 .8 M P aE = 2 0 5 .8 G P a
β = b /t√σo /E = 1 .6 7 6
w op l = 0 .1 ×β2t = 5 .8 9 9 m mσr cx = - 0 .1 ×σo
σr cy = b /a×σrcx
F E M (A N S Y S ) : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s sP re s e n t d e s ig n fo rm u la : W ith o u t in iti a l im p e r fe c tio n s : W ith in itia l d e f le c tio n a n d w ith o u t r e s id u a l s tr e s s : W ith b o th in itia l d e fle c tio n a n d r e s id u a l s tr e s s
: F E M (A N S Y S ) w ith in itia l d e f lec t io n an d w ith o u t re sid u a l s tre s s
a×b×t = 5 ,1 0 0×8 5 0×2 1 m mσo = 3 5 2 .8 M P a , E = 2 0 5 .8 G P a
β = b /t√σo / E = 1 .6 7 6 , w opl = 0 .1 β2t
σrcx = -0 .0 5 .σo , σrcy = b /a .σrcx
v o n M ise s ' e llip se
σyu / σo , σycr / σo
U ltim a te s tren g th b a sed cap ac i tyw ith b o th in it ia l d e f lec tio nan d re s id u a l s tre s s
B u ck lin g b a sed ca p ac ityw ith o u t in i tia l d e f lec tio nan d w ith re s id u a l s tre ss
σxu / σo , σxcr / σo
Figure 36b. Plate capacity interactions between biaxialcompression as those obtained by FEA, buckling andultimate strength based capacity formulae,
,6/ =ba mmt 21= , initial deflection shape #1
The loading speed was varied from 0.05 to 400 mm/sec.Based on the test results, the effect of the loading speed onthe ultimate compressive strength of mild steel plates is in-vestigated. Also, a simple formula for predicting the buck-ling collapse strength of steel plates taking into account thestrain rate effect is empirically derived by curve fitting theexperimental results. A more detailed description of thestudy may be found in Paik et al. (1999b).
Figure 37 shows a selected set of the test results indi-cating the applied axial compressive loads versus lateral de-flection relations for the plate test specimen. It is seen from
Figure 37 that with increase in the speed of axial compres-sive loading the stiffness and ultimate compressive strengthof steel plates both increase. This is because the strain ratefor the material involved increases as the speed of loadingincreases. However, it should be noted that the value of ax-ial displacements at the ultimate limit state is also increasedas the speed of loading increases. Further, it appears that thelateral deflections at the ultimate limit state also increasewith increase in the loading speed. The unloading pattern inthe post-ultimate strength regime tends to be more rapid asthe speed of loading increases, meaning that the tendencyfor unstable plate behavior is greater as the speed of axialcompressive loading becomes faster.
0 2 4 6 8 1 0D eflec tio n (m m )
0
1 0
2 0
3 0
4 0
5 0
6 0
Loa
d (k
N)
: V o= 0 .0 5 m m /se c: V o= 1 0 0 m m /sec: V o= 3 0 0 m m /sec: V o= 4 0 0 m m /sec
E x perim en ts :
U S P -SU S P -1 0 0U S P -3 0 0U S P -4 0 0
Figure 37. Dynamic compressive load versus deflectionbehavior at center of the specimen varying the loading speed(USP stands for un-stiffened plate specimen)
Based on such test results, relevant useful formulationsfor assessing the dynamic collapse strength characteristicsof plates are empirically derived as a function the strain rateby curve fitting, resulting in
21.2
1
41.50.1
+= ε
σσ
u
ud (35)
where uσ , udσ = ultimate compressive stress of plate ele-
ments under a quasi-static or dynamic condition, respec-
tively. ε = strain rate which may be taken as aVo /=ε ,
where oV = loading speed in .sec/m and a = plate length.
In a manner similar to the derivation of the bucklingcollapse strength formula, our studies indicated that the end-shortening and maximum deflection at the ultimate limitstate of plating under dynamic axial compressive loads maybe approximately predicted by
16.3
1
74.00.1
+= ε
u
ud
U
U (36.a)
Paper Number 13 23
43.2
1
21.10.1
+= ε
u
ud
W
W (36.b)
where udu UU , = end-shortening at the ultimate limit state
of plate elements under a quasi-static or dynamic loadingcondition, respectively, and udu WW , = maximum deflec-
tion at the ultimate limit state of plate elements under aquasi-static or dynamic loading condition, respectively. Theaccuracy of equations (35) and (36a) is seen in Figure 38 bycomparison with the plate test data.
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0S tra in ra te ε
0 .0
0 .5
1 .0
1 .5
2 .0
σu d
σu
: E x p e rim en ts
E m p irica l fo rm u la
Figure 38a. Variation of the normalized ultimate compres-sive stress against the strain rate
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0S tra in ra te ε
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
U ud
U u
: E xp erim e n ts
E m piric a l fo rm u la
Figure 38b. Variation of the normalized end-shorting at theultimate limit state against the strain rate
ADVANCED ULTIMATE STRENGTH DESIGN OFSHIP PLATING UNDER IMPACT LATERALPRESSURE LOADS
Ship structures can be subjected to impact loads inducedby waves. The magnitude of such loads is likely to becomelarger as vessel speeds increase. In vessels carrying liquidcargo, structural members are internally subjected to sloshinginduced impact loads resulting from roll or pitch motion ofthe vessel. The plating of ship’s bow and flare parts may beexternally subjected to slamming induced impact loads in
rough sea state. Such hydrodynamic impact loading can infact occasionally result in serious structural damage.
According to the ISSC (1991), more than 10% ofstructural damages for conventional vessel structures arecaused by hydrodynamic impact loading, e.g., by slamming.High speed vessel structures fabricated of aluminum alloys,composites or high tensile steels may in theory experienceeven larger impact loads compared to the conventional ves-sels, and thus the amount and extent of structural damagethey may be subjected to can potentially be more serious.
As previously noted, structural response in an impactloading condition is quite different from that in a static orquasi-static condition. For practical purposes, ship structuraldesign for an impact load is usually carried out based onequivalently defined values of effective quasi-static loadsinstead of direct application of the impact loads themselves,the equivalence being based on certain characteristics of thelikely actual response. Since it lacks a strict theoretical ba-sis, it is possible that such an “equivalent quasi-static loadapproach” may in some cases lead to either unsafe or tooheavy structures. For more rational design of modern shipstructures, therefore, it is of importance to better understandthe strength and response characteristics of ship plating un-der dynamic/impact loading.
The literature review made by Jones (1997) includes anumber of useful studies related to the strength of structuralmembers under impact loading. Selected literature related tothe strength of plating under impact pressure loading is nowreviewed. Chuang (1966) experimentally investigated thecharacteristics of the pressure distribution on ship platingsubject to slamming induced impact loads. Jones (1973) de-rived the maximum permanent deflection equations for shipbottom plating under slamming. Mori (1976) investigatedthe collapse strength characteristics of aluminum alloy shipbottom plating under slamming induced impact pressures,both theoretically and experimentally. Recently, Caridis &Stefanou (1997) performed a nonlinear elastic/visco-plasticnumerical simulation to investigate the strength characteris-tics of plating subject to wave impact loads.
The previously mentioned studies provide useful infor-mation. However, there are still a number of problem areaswhere the state of the art can be potentially improved. Inparticular, to more reasonably and precisely accommodatethe impact load effects in the preliminary design stage ofship structures, we need to have better understanding of (a)hydrodynamic wave impact loading characteristics and (b)collapse strength characteristics of ship plating under impactpressure, among others.
The present authors (Paik et al. 1999c) aim to providesome new contributions to the design technology for shipstructures considering the dynamic/impact load effects. Forthis purpose, the idealization of impact pressure distributionat the ship bottom plating subject to slamming induced im-pact loading is studied. Using a nonlinear finite elementprogram that applies the dynamic approach, a series of col-lapse strength analyses for ship plating under lateral pres-
Paper Number 13 24
sure loading are carried out varying the plate dimensions,aspect ratio and loading speed. Based on the computed re-sults, the collapse strength characteristics of ship bottomplating subject to impact pressure loading are investigated.A simplified theoretical formula for predicting the collapsestrength of ship plating under impact pressure loads is thendeveloped by including the strain rate sensitivity within astrength formula that is derived based on the rigid-plastictheory for plating under a quasi-static pressure loading.
Idealization of Pressure Distribution on Ship Platingunder Hydrodynamic Wave Impact Loads
To analyze the impact load related response characteris-tics (i.e., magnitude and pattern of the response as a functionof time) for structural members, it is important to first definethe characteristics of the impact load versus time history. Thecharacteristics of dynamic/impact loading for ship structuralmembers can be quite complex. The developments presentedin this study are, for reasons of convenience and efficiency,based on certain simplifications of typical load time histories.The nature of these simplifications and their relationship tothe response under the actual (non-idealized ) load versustime histories are studied. According to the experiments un-dertaken by Chan & Melville (1987), the pressure rise time isone order of magnitude smaller than the decay period and it ismore relevant to idealize that the pressure increases instanta-neously to the peak value (i.e., with zero rise time) and thendecays following the exponential law.
Figure 39 indicates possible idealizations of the pressureimpulse history on ship plating, where the curve “e” is themost realistic. As shown in the figure, in the limit, two typesof idealizations, one being of a rectangular pattern and theother being of a triangular pattern, can be possible. Holdingthe amount of impulse the same, the former type (i.e., rectan-gular pattern) is depicted by curve “a” and the latter (i.e., tri-angular pattern) is given by curve “d”. For comparison pur-poses, curves denoted “b” and “c”, the former with twotimes impulse of curve “a” and the latter with a half impulseof curve “c”, are also shown in the figure.
P (T )
P o
τ0 2τ T im e (se c )
a b
c
d e
Figure 39. Idealized types of pressure pulse history
Figure 40 shows the variations of lateral deflection withtime for an example rectangular steel plate subject to impact
lateral pressure loads varying the pressure impulse historiesas indicated above. The results of Figure 40 in the presentstudy are obtained by using a nonlinear finiteelement (FE)program based on a dynamic formulation, STARDYNE(1996). For these analyses, the boundary conditions of theplating are modeled as clamped at all (four) edges. Figure40 also shows relevant experimental data for the maximumpermanent deflection for the same case, as obtained byJones et al. (1970) using a pressure versus time history ofthe more realistic “e” type.
a, b
de
c
0.00 0.05 0.10 0.15 0.20 0.25Time (sec)
0
1
2
3
4
5
Wo
t
Po=5.5 N/mm2
Wo
: Experiment (Jones et al. 1970): FEM
128.6mm
76.2
mm E=205,800N/mm2
t=2.5mmσ
o=233.2N/mm2
Figure 40. Deflection-time history at center of the all edgedclamped plate subject to various types of pressure pulse
It is of interest to note that when using the rectangulartype of pressure impulse, as in either curve “a” or “b” ofFigure 39, the responses obtained are almost same in eithercase even if the duration of impact impulse is different (al-though the maximum permanent deflection is overestimatedcompared to the experimental results). As evident from Fig-ure 40, curves “a” and “b” have the same initial maximumimpact pressure (and the same impulse history at the earlyimpact stage). That is, the rectangular type of impulse mayin such cases give almost same maximum structural re-sponse regardless of duration of the pressure impulse (or theimpulse amount) as long as the initial maximum impactpressure is the same.
Unlike the rectangular type of pressure impulse, it isseen from Figure 40 that the triangular type may producedifferent results if the duration of impact pulse is different(although the initial maximum impact pressure is the same).In some cases (e.g., if the duration of impact pulse is as-sumed to be small), the calculation using the triangular typeof impulse history will underestimate the permanent deflec-tion of plating as well. The differences in the permanentplate deflection between curves “c” and “d” using the trian-gular type of impact pulse history is due to the different du-ration of impulse even though the initial maximum impactpressure is the same in either case.
An advantage of using the rectangular type of impulsehistory for simplified analysis or design is that the related cal-culations result in similar impact responses with only the ini-
Paper Number 13 25
tial maximum pressure known, i.e., regardless of the durationof pulse (or impulse amount). For simplicity, therefore, thepresent theoretical calculation idealizes the impulse historyfor ship plating under impact pressure loads as a rectangulartype with the same initial maximum impulse as the actual andmore complicated impact load history. Of course, if oneknows the values of both duration of pulse (or impulseamount) and initial maximum pressure, an idealization byway of the triangular type of impulse history would be desir-able in the sense of accuracy. For illustrative examples(where both duration of pulse and initial maximum pressureare known), therefore, the various finite element numericalanalyses of this study are carried out using a triangular type ofimpact pressure pulse history which is assumed to representthe same amount of impulse as in the actual more compli-cated impact load history which in reality may take an expo-nential form in its latter phase. The numerical finite elementresults are then compared with the simplified theoretical so-lutions, i.e., using the rectangular type of pressure pulse withonly the maximum impact pressure known.
It is known that duration of impact pulse for ship bot-tom plating subject to slamming induced impact loads is inthe range of 0.025~0.25 seconds (Ochi 1967, Wheaton et al.1970). Hence the duration of an impact pressure pulse maybe taken as 0.1 second for the interests of illustrative exam-ples. The study methodology can of course be applied to anyother duration value equally well.
Prediction of the Initial Maximum Impact PressureIt is necessary to predict the initial maximum impact
pressure to analyze the behavior of ship plating subject topressure impact. When ship bottom plating strikes watersurface, e.g., in slamming event of vessels, air cushioningbetween bottom plating and water surface makes the timehistory of impact pressure pulse more complex. Since it isnot an easy task to theoretically study the influence of aircushioning on the behavior of plating under hydrodynamicwave impact loads, most previous studies related to thisproblem were undertaken experimentally. According to suchprevious studies (for instance, Chuang 1966, Lewison &Maclean 1968, Verhagen 1967), it is known that the mag-nitude of the initial maximum impact pressure felt by thestructure becomes smaller if there exists more of an aircushioning effect, because air particles between plating andwater surface which are more compressible than water ab-sorb the impact energy to a greater extent.
Based on his own experimental results, Chuang (1966)suggested an empirical formula for predicting the initialmaximum impact pressure of plating which strikes thewater surface at a right angle with the initial impact speedof oV , taking into account the influence of air cushioning,
as follows:
oafo VCe
P ρπ
++= − 0.1
4.1
32
14.1
22
(37)
where oP = peak impact pressure on the plating, fρ = den-
sity of sea water (= 1,025 3/ mkg ), aC = velocity of sound
in the air (=342.9 sec/m ), and oV = initial impact velocity
between plate and water surface.For simplicity, the present study uses equation (37) to
calculate the magnitude of the initial maximum impact pres-sure for the various theoretical and numerical analysis ofship bottom plating under slamming induced impact loads.Actual values from realistic cases may also be readily usedfor the same purposes where available.
Ultimate Strength Design Formulation ofShip Plating under Impact Pressure Loads
Based on rigid-plastic theory, a simplified formula forpredicting the collapse strength of ship plating under impactpressure loads is now theoretically derived. For conven-ience, the process of the formula derivation is split into twosteps, namely the quasi-static loading step and the dy-namic/impact loading step. The accuracy of the formula soderived is then verified by comparing with the correspond-ing nonlinear FE solutions. The boundary conditions for theplate are assumed to be clamped at all (four) edges. Theplastic collapse mode of the plate under equilibrium condi-tion can be assumed as shown in Figure 41. In the figure,the dotted lines indicate the plastic hinge lines which are ina gross yielding condition. Except for the plastic hinge lines,each plate region is assumed to behave as a rigid body. Thecollapse load cp of the plating under static pressure is
given as follows
( )2
22 /3/
48
−+=
ababb
Mp p
c (38)
where 4
2tM o
pσ
= .
a
b
Wo
Wo
Pt
a-btanφ
I
II
Plastic hinge lines φ
b2tan φ b
2tan φ
Figure 41. Plastic hinge mechanism for the all edgesclamped plate subject to uniformly distributed static pres-sure loads
Paper Number 13 26
The relationship between the applied static pressure p
and the maximum (permanent) deflection oW for the plating
can be shown as follows
( )o
ooo
c t
W
p
p
ξξξ
−−+
+=
3
23
3
11
22
if 1≤t
Wo (39.a)
( )
−
−−
+=o
ooo
o
c
W
t
t
W
p
p
ξ
ξξ
3
312
12 2
2
if 1>t
Wo (39.b)
where cp = as defined in equation (38),2
2
3
−+
=
a
b
a
b
a
boξ .
The structural response in the dynamic/impact condi-tion is in principle different from that in the quasi-staticcondition. Three related aspects of a dynamic/impact load-ing situation are possibly relevant, namely material strainrate sensitivity, inertia effects and dynamic frictional effects.For steels, with increase in the strain rate, the yield strengthof the material can increase and rupture strain can decrease.Due to inertia effects, deformation patterns may be varied aswell. It is also known that as the speed of dynamic loadingincreases, the coefficient of friction becomes lower.
However, in most practical marine cases of interestwhere dynamic/impact loads are applied, the structural re-sponse is mainly affected by the material strain rate sensi-tivity (Jones 1989, Paik & Wierzbicki 1997). Therefore, thepresent study approximately develops the dynamic collapsestrength formula by including the strain rate effect aloneinto the static collapse strength formula. The validity of thisapproximation is to some extent verified by comparing withappropriate numerical and experimental results.
As previously noted, the pressure versus time historyduring hydrodynamic wave impact loading can be simpli-fied using the rectangular type of pressure pulse. If the du-ration of impact pulse is relatively short, the relationshipbetween initial velocity oV and maximum pressure pulse oP
can then be given by
TPV oo =µ (40)
where T = duration of pressure pulse, µ = mass per unit
area of the impacted plate, oP = peak impact pressure on the
plate, and oV = initial impact velocity between plate and
water surface.
Perrone & Bhadra (1984) formulated the strain rate εof plate elements under impact pressure loading in terms of
initial impact velocity oV and maximum permanent deflec-
tion oW as follows
≈
2
2
23
2
b
VW ooε (41)
Using the initial impact velocity oV calculated from
equation (40), the strain rate ε can then be expressed byequation (41) in terms of the plate dimensions and themaximum deflection. To estimate the dynamic yieldstrength of the material, odσ from the static yield stress oσ ,
with ε known, the Cowper & Symonds equation has beenwidely used, namely
q
o
od
H
/1
0.1
+= ε
σσ
(42)
where H and q are coefficients to be determined based on
test data. For example, the parameters for the initial yield of
mild steel under dynamic loading are 1sec4.40 −=H and
5=q (Cowper & Symonds 1957). Recently, Paik et al.
(1999d) found that equation (42) is also applicable to hightensile steel material, but with coefficients that are differentfrom those of mild steel. Based on existing test data for hightensile steel materials, they determined a sample set of the
Cowper-Symonds coefficients as 1sec3200 −=H and
5=q . For aluminum alloys, Jones (1974) suggests that the
relevant coefficients can be 11288000 −= secH and 4=q .
Substituting equation (41) into equation (42), the dy-namic yield stress can be given by
qoo
o
od
Hb
WV/1
22120.1
⋅
+=σσ
(43)
Therefore, it is proposed that the collapse behavior ofplating under impact pressure loading can be assessed by theformula developed for a quasi-static condition, i.e., equation(39), but with the use of the dynamic yield stress odσ de-
fined in equation (43) in place of its static counterpart oσ .
Figure 42 shows a selected set of the illustrative exam-ples representing the variation of the maximum permanentdeflection for the all edges clamped steel plate under impactpressure as a function of the loading speed, as obtained by theabove developed theoretical formula, and also FE calculationsand experiments. The theoretical solutions are compared withthe nonlinear FE results (with 5% of the strain hardening) andcorresponding test data as obtained by Jones (1970). It is seenfrom Figure 42 that both theoretical and numerical solutionsagree well with the experimental data.
Paper Number 13 27
0 50 100 150 200 250 300ρVo
2a2
4σot2
0
1
2
3
4
5
6
7
8
Wo
t
Experiment (Jones et al. 1970)FEM (Strain hardening included)Rigid perfectly plastic theory
a=128.6mmb=76.2mmt=1.63mmσo=246.96N/mm2
Figure 42. Effects of loading velocity on collapse behaviorof the all edges clamped rectangular plate subject to impactuniform pressure load
Figure 43 shows the influence of the plate aspect ratioand slenderness ratio of the maximum lateral deflection.With a constant speed of loading, the influence of the plateaspect ratio can be ignored for large aspect ratios, e.g., herewhen 2/ >ba . With increase in the plate slenderness (orwith decrease in the plate thickness) the maximum deflec-tion is however seen to increase significantly.
0 1 2 3 4a/b
0
1
2
3
4
5
6
7
8
Wo
t
Rigid perfectly plastic theoryFEM(Vo=25m/sec)
Etb oσσσσββββ ====
82.2====ββββ
69.1====ββββ
13.1====ββββ
Figure 43. Effects of aspect and slenderness ratio on themaximum deflection of clamped plating under impact lateralpressure loads, as those obtained by FEA and the presentdesign formula
Based on the insights developed above, it may be con-cluded that the collapse behavior of steel plating under impactpressure loading is mostly affected by the loading speed andthe plate slenderness ratio, and to much less extent by theplate aspect ratio. In addition, some important insights devel-oped in the present study are to be noted, as follows: Thestrain hardening effect of material may vary with the magni-tude of plate deflection and it can not be neglected in the re-gime of large deflection that produces membrane effects.
Further investigations are needed in this regard. Also, it isevident that as the loading speed increases the plate maximumdeflection increases remarkably. It is also evident that thetheoretical formula developed in the present study is usefulfor assessment of collapse strength characteristics for shipplating under impact pressure loading.
CONCLUDING REMARKS
The behavior of ship plating normally depends on a vari-ety of influential factors, namely geometric/material proper-ties, loading characteristics, initial imperfections, boundaryconditions and also local deterioration related to corrosion,fatigue cracking and dents. To achieve the advanced bucklingand ultimate strength design of ship plating, we would needmore sophisticated methods than existing simplified ap-proaches. The aim of the present study has been to developmore advanced buckling and ultimate strength design tech-nology for ship plating. The present paper focuses on the fol-lowing five areas which were studied theoretically, numeri-cally and experimentally as appropriate:
• Mathematical modeling for fabrication relatedimperfections (i.e., initial deflections and re-sidual stresses),
• Characteristics of the plate buckling with elas-tically restrained edge conditions,
• Strength equations for ship plating under com-bined static loads including biaxial compres-sion/tension, edge shear and lateral pressure,
• Characteristics of the plate capacity under in-plane dynamic loads, and
• Characteristics of the plate capacity underslamming induced lateral impact pressureloads.
Some collected results and conclusions developed inthe present study are as follows:
(1) During fabrication of ship structures, the initialimperfections (initial deflection and residualstresses) inevitably develop and can signifi-cantly affect the structural capacity. The char-acteristics of the fabrication related imperfec-tions are uncertain, and an idealized modelwas proposed in the present study.
(2) The buckling strength of plating can be af-fected significantly by the torsional rigidity ofsupport members as well as the plate dimen-sions. The proposed buckling based capacityformula accommodates the torsional rigidity ofsupport members as a parameter of influence.
(3) Ship plating is generally subjected to combinedin-plane and lateral pressure loads and the platecapacity should be evaluated taking into ac-count the effect of combined loads. Two types
Paper Number 13 28
of the capacity formulations, namely one basedon the buckling and the other based on the ulti-mate strength, were proposed by accommodat-ing the combined loads (i.e., biaxial compres-sion/tension, edge shear and lateral pressure)and fabrication related imperfections. For thinplating which buckles in the elastic regime, thecapacity formula based on the buckling is toopessimistic against the ultimate limit state,while it provides a good measure of the struc-tural capacity for relatively thick plating whichbuckles in the inelastic regime. The capacityformula based on the ultimate strength providesa good indication for both thin and thick platingagainst collapse.
(4) The in-plane stiffness and ultimate compres-sive strength of steel plating both tend to in-crease with increase in the speed of loading.This is due to the fact that the strain rate of thematerial involved increases as the loadingspeed increases. However, it is also noted fromthe tests that the values of axial displacementsand lateral deflections until the plate reachesthe ultimate limit state increases rapidly as theloading speed increases. Also, the unloadingpattern in the post-ultimate strength regime ismore rapid with increase in the loading speed.This would mean that as the speed of loadingincreases the response of steel plates will be-come more unstable after the ultimate strengthis reached. For structural design of ship plat-ing, while it is recommended that the increaseof the ultimate strength due to the dynamicloading effect should not be overestimated, itis also recommended that the potential for dy-namic instability of steel plates in the post-ultimate strength regime should not be under-estimated. The related phenomena depend ofcourse on the rates of loading involved, as thedata presented herein clearly indicate.
(5) The collapse behavior of steel plating underimpact pressure loading is in principle affectedsignificantly by the impact pressure versustime history. Starting some time after thestructure is initially impacted, the impact pulseusually decreases in a way of the exponentialfunction with time. When only the initialmaximum impact pressure is known, the rec-tangular type of idealization of an impact pulsehistory would be useful for practical purposes.The maximum permanent deflection for shipplating under impact pressure loading in-creases with increase in the loading speed. Fora constant loading speed, the maximum de-flection for a rectangular plate is larger thanthat of a square plate, but the effect of the plate
aspect ratio appears to become smaller as as-pect ratios get to be greater than 2. Also, themaximum deflection of ship plating increasessignificantly as the plate slenderness ratio in-creases, as expected. To prevent ship platingunder impact pressure loading from dynamicbuckling collapse, one consideration of im-portance is of course to limit the slendernessratio under a critical value.
ACKNOWLEDGEMENTS
The present research study was undertaken with supportfrom the American Bureau of Shipping, the Korea ResearchFoundation (for Faculty Research abroad), the Brain Korea21 project and the Pusan National University who arethanked for this support.
REFERENCES
Aalami, B. and Chapman, J.C. (1972). Large deflectionbehavior of ship plate panels under normal pressure and in-plane loading, Trans. RINA, Vol.114, March.
Aalami, B., Moukhtarade, A. and Mahmudi-Saati, P.(1972). On the strength design of ship plates subjected toin-plane and transverse loads, Trans. RINA, Vol.114, No-vember.
Antoniou, A.C. (1980). On the maximum deflection ofplating in newly built ships, J. of Ship Research, Vol.24,No.1, pp.31-39.
Bleich, F. (1952). Buckling strength of metal structures,McGraw-Hill Book Co., New York.
Caridis, P.A. and Stefanou, M. (1997). Dynamic elas-tic/visco-plastic response of hull plating subjected to hydro-dynamic wave impact, J. of Ship Research, Vol.41, No.2,pp.130~146.
Carlsen, C.A. and Czujko, J. (1978). The specificationof post-welding distortion tolerance for stiffened plates incompression, The Structural Engineer, Vol.56A, No.5,pp.133~141.
Chuang, S.L. (1966). Experiments on flat-bottomslamming, J. of Ship Research, Vol.10, No.1, pp.10~17.
Cowper, G.R. and Symonds, P.S. (1957). Strain hard-ening and strain rate effects in the impact loading of cantile-ver beams, Brown University Division of Applied Mathe-matics, Report No.28, September.
Davidson, P.C., Chapman, J.C., Smith, C.S. and Dowl-ing, P.J. (1989). The design of plate panels subjected to in-plane shear and biaxial compression, Trans. RINA, Vol.132,pp.267-286.
Davidson, P.C., Chapman, J.C., Smith, C.S. and Dowl-ing, P.J. (1991). The design of plate panels subjected to bi-axial compression and lateral pressure, Trans. RINA,Vol.134, pp.149-160.
Dier, A.F. and Dowling, P.J. (1983). The strength ofplates subjected to biaxial forces, Proc. of Conference on
Paper Number 13 29
Behavior of Thin-Walled Structures, University of Strath-clyde, Glasgow, pp.329-353.
Dowling, P.J. and Dier, A.F. (1978). Strength of ship’splating under combined lateral loading and biaxial pressure,CESLIC Report SP6, Imperial College, London, September.
Evans, J.H. (1960). Strength of wide plates under uni-form edge compression, Trans. SNAME, Vol.68, pp.585-621.
Faulkner, D. (1975). A review of effective plating foruse in the analysis of stiffened plating in bending and com-pression, J. of Ship Research, Vol.19, No.1, March, pp.1-17.
Fujikubo, M., Yao, T. and Varghese, B. (1997). Buck-ling and ultimate strength of plates subjected to combinedloads, Proc. of the 7th International Offshore and Polar En-gineering Conference, Vol. IV, Honolulu, pp.380-387.
Fujita, Y., Nomoto, T. and Niho, O. (1979). Ultimatestrength of rectangular plates subjected to combined loading(1st report) –Square plates under compression and shear, J.of the Society of Naval Architects of Japan, Vol.145,pp.194-202 (in Japanese).
Gerard, G. and Becker, H. (1954). Handbook of struc-tural stability, Part I. Buckling of flat plates, NACA Techni-cal Note, No.3781.
ISSC (1991). Dynamic load effects, Report of Com-mittee II.2, The 11th International Ship & Offshore Struc-tures Congress, Wuxi, China, pp.239~318.
Jones, N. (1973). Slamming damage, J. of Ship Re-search, Vol.17, No.2, pp.80-86.
Jones, N. (1974). Some remarks on the strain rate sen-sitive behavior of shells, Problems of Plasticity, Vol.2,pp.403~407.
Jones, N. (1989). Structural Impact, Cambridge Univer-sity Press.
Jones, N. (1997). Dynamic plastic behavior of ship andocean structures, Trans. RINA, Vol.139, Part A, pp.65~97.
Jones, N., Uran, T.O. and Tekin, S.A. (1970). The dy-namic plastic behavior of fully clamped rectangular plates,Int. J. of Solids and Structures, Vol.6, pp.1499~1512.
Kmiecik, M., Jastrzebski, T. and Kuzniar, J. (1995).Statistics of ship plating distortions, Marine Structures,Vol.8, pp.119-132.
Lewison, G. and Maclean, W.M. (1968). On the cush-ioning of water impact by entrapped air, J. of Ship Research,Vol.12, No.2, pp.116~130.
Lundquist, E. and Stowell, E.Z. (1942). Critical com-pressive stress for flat rectangular plates – Elastically re-strained, NACA Technical Note, No.733.
Mansour, A.E. (1976). Charts for the buckling and post-buckling analyses of stiffened plates under combined load-ing, Technical and Research Bulletin, No.2-22, SNAME,July.
Mazzolani, F.M., Landolfo, R. and De Matteis, G.(1998). Influence of welding on the stability of aluminumthin plates, Stability and Ductility of Steel Structures, El-sevier Science, pp.225-232.
Mateus, A.F. and Witz, J.A. (1997). Post-buckling ofcorroded steel plates: an assessment of the design codes,
Proc. of the 8th International Conference on the Behavior ofOffshore Structures (BOSS’97), Elsevier Science, Delft,July.
Mateus, A.F. and Witz, J.A. (1998). On the post-buckling of corroded steel plates used in marine structures,Trans. RINA, Vol.140, Part C, pp.165-183.
McKenzie, K.I. (1963). The buckling of a rectangularplate under combined biaxial compression, bending andshear, The Aeronautical Quarterly, August.
Mori, K. (1976). Response of the bottom plate of highspeed crafts under implusive water pressure, J. of the Soci-ety of Naval Architects of Japan, Vol.142, pp.297~305.
Ochi, M.K. (1967). Ship slamming hydrodynamic im-pact between waves and ship bottom forward, Fluid-SolidInteraction, ASME, Edited by J.E. Greenspon, pp.223~240.
Ohtsubo, H. and Yoshida, J. (1985). Ultimate strengthof rectangular plates under combination of loads (part 2) –Interaction of compressive and shear stresses, J. of theSociety of Naval Architects of Japan, Vol.158, pp.368-375(in Japanese).
Okada, H., Oshima, K. and Fukumoto, Y. (1979). Com-pressive strength of long rectangular plates under hydro-static pressure, J. of the Society of Naval Architects of Ja-pan, Vol.146, pp.270-280 (in Japanese).
Paik, J.K. and Kim, C.Y. (1988). Simple formulae forbuckling and ultimate strength estimation of plates subjectedto water pressure and uniaxial compression, J. of the Societyof Naval Architects of Korea, Vol.25, No.4, pp.69-80 (inKorean).
Paik, J.K., Ham, J.H. and Kim, U.N. (1992a). A newplate buckling design formula, J. of the Society of NavalArchitects of Japan, Vol.171, pp.559-566.
Paik, J.K., Ham, J.H. and Ko, J.H. (1992b). A new platebuckling design formula (2nd report) –On the plasticity cor-rection, J. of the Society of Naval Architects of Japan,Vol.172, pp.417-425.
Paik, J.K., Kim, J.Y., Kim, W.S. and Lee, B.H. (1993).Development of an efficient and accurate buckling designsystem for ship structures (II), Final Report to the HyundaiHeavy Industries, Ulsan, by the Department of Naval Ar-chitecture and Ocean Engineering, Pusan National Univer-sity, Pusan, August.
Paik, J.K. (1995). A new concept of the effective shearmodulus for a plate buckled in shear, J. of Ship Research,Vol.39, No.1, pp.70-75.
Paik, J.K. and Pedersen, P.T. (1995). Ultimate andcrushing strength of plated structures, J. of Ship Research,Vol.39, No.3, Sept., pp.250-261.
Paik, J.K. and Pedersen, P.T. (1996). A simplifiedmethod for predicting the ultimate compressive strength ofship panels, International Shipbuilding Progress, Vol.43,No.434, pp.139-157.
Paik, J.K. and Wierzbicki, T. (1997). A benchmarkstudy on crushing and cutting of plated structures, J. of ShipResearch, Vol.41, No.2, pp.147-160.
Paper Number 13 30
Paik, J.K., Kim, S.K. and Lee, S.K. (1998a). A prob-abilistic corrosion rate estimation model for longitudinalstrength members of bulk carriers, Ocean Engineering,Vol.25, No.10, pp.837-860.
Paik, J.K., Thayamballi, A.K., Kim, S.K. and Yang,S.H. (1998b). Ultimate strength reliability of corroded shiphulls, Trans. RINA, Vol.140, pp.1-18.
Paik, J.K., Thayamballi, A.K., Kim, S.K. and Yang,S.H. (1998c). Ship hull ultimate strength reliability consid-ering corrosion, J. of Ship Research, Vol.42, No.2, June,pp.154-165.
Paik, J.K. (1999). Buckling and ultimate strength designequations of plates and stiffened panels under combinedloads, Final Report to ABS New York.
Paik, J.K., Thayamballi, A.K. and Kim, B.J. (1999a).Advanced ultimate strength design equations for ship plat-ing subject to combined biaxial compression/tension, edgeshear and lateral pressure loads, to be published.
Paik, J.K., Thayamballi, A.K. and Chung, J.Y. (1999b).Experimental investigation of the dynamic collapse strengthcharacteristics for ship plating under axial compressiveloads, to be published.
Paik, J.K., Chung, J.Y. and Paik, Y.M. (1999c). Oncollapse strength characteristics of ship bottom plating sub-ject to slamming induced impact pressure loads, J. of theSociety of Naval Architects of Korea, Vol.36, No.2,pp.77~93 (in Korean).
Paik, J.K., Chung, J.Y. and Paik, Y.M. (1999d). On dy-namic/impact tensile strength characteristics of thin hightensile steel materials for automobiles, Trans. the Korea So-ciety of Automotive Engineers, Vol.7, No.2, SAE No.99370060, pp.268~278 (in Korean).
Paik, J.K. and Thayamballi, A.K. (2000). Bucklingstrength of steel plating with elastically restrained edges,Thin-Walled Structures, Vol.37, Issue 1, pp.27~55.
Paik, J.K., Thayamballi, A.K. and Kim, B.J. (2000).Ultimate strength and effective width equations for shipplating under combined axial load, edge shear and lateralpressure, to be published.
Perrone, N. and Bhadra, P. (1984). Simplified large de-flection mode solution for impulsively loaded, viscoplastic,circular membranes, J. of Applied Mechanics, Vol.51,pp.505~509.
Rhodes, J. (1984). Effective widths in plate buckling,Chapter 4, Developments in Thin-Walled Structures, Editedby J. Rhodes and A.C. Walker, Applied Science Publishers,London, pp.119-158.
Smith, C.S., Davidson, P.C., Chapman, J.C. and Dowl-ing, P.J. (1987). Strength and stiffness of ships’ plating un-der in-plane compression and tension, Trans. RINA,Vol.129, pp.277-296.
Smith, C.S. and Dow, R.S. (1981). Residual strength ofdamaged steel ships and offshore structures, J. of Construc-tional Steel Research, Vol.1, No.4, pp.2-15.
Soares, C.G. and Gordo, J.M. (1996). Compressivestrength of rectangular plates under biaxial load and lateralpressure, Thin-Walled Structures, Vol.24, pp.231-259.
Soreide, T.H. and Czujko, J. (1983). Load carrying ca-pacities of plates under combined lateral load and ax-ial/biaxial compression, Proc. of the Second InternationalSymposium on Practical Design in Shipbuilding(PRADS’83), Tokyo and Seoul, October.
STARDYNE (1996). User’s manual (version 4.4), Re-search Engineers, Inc.
Steen, E. and Valsgard, S. (1984). Simplified bucklingstrength criteria for plates subjected to biaxial compressionand lateral pressure, Proc. of the Ship Structure Symposium,SNAME, Arlington, October, pp.257-272.
Timoshenko, S.P. and Gere, J.M. (1963). Theory ofelastic stability, McGraw-Hill Book Co., London.
Ueda, Y., Rashed, S.M.H. and Paik, J.K. (1984). Buck-ling and ultimate strength interactions of plates and stiffenedplates under combined loads –In-plane biaxial and shearingforces, J. of the Society of Naval Architects of Japan,Vol.156, pp.377-387 (in Japanese).
Ueda, Y., Rashed, S.M.H. and Paik, J.K. (1985). Newinteraction equation for plate buckling, Trans. JWRI,Vol.14, No.2, Welding Research Institute of Osaka Univer-sity, pp.159-173.
Ueda, Y., Rashed, S.M.H. and Paik, J.K. (1986a). Ef-fective width of rectangular plates subjected to combinedloads, J. of the Society of Naval Architects of Japan,Vol.159, pp.269-281 (in Japanese).
Ueda, Y., Rashed, S.M.H. and Paik, J.K. (1986b). Plateand stiffened plate units of the idealized structural unitmethod (2nd report) –Under in-plane and lateral loading con-sidering initial deflection and residual stress, J. of the Soci-ety of Naval Architects of Japan, Vol.160, pp.321-339 (inJapanese).
Ueda, Y., Rashed, S.M.H. and Paik, J.K. (1995). Buck-ling and ultimate strength interaction in plates and stiffenedpanels under combined inplane biaxial and shearing forces,Marine Structures, Vol.8, pp.1-36.
Usami, T. (1993). Effective width of locally buckledplates in compression and bending, J. of Structural Engi-neering, ASCE, Vol.119, No.5.
Verhagen, J.H.G. (1967). The impact of a flat plate on awater surface, J. of Ship Research, Vol.11, No.4,pp.211~223.
von Karman, T., Sechler, E.E. and Donnell, L.H.(1932). Strength of thin plates in compression, Trans.ASME, Vol.54, No.5.
Wang, X. and Moan, T. (1997). Ultimate strengthanalysis of stiffened panels in ships subjected to biaxial andlateral loading, Int. J. of Offshore and Polar Engineering,Vol.7, No.1, pp.22-29.
Paper Number 13 31
Wheaton, J.W., Kano, C.H., Diamant, P.T. and Bailey,F.C. (1970). Analysis of slamming data from the S.S. Wol-verine State, Ship Structure Committee, Report No. SSC-210.
Williams, D.G. (1976). The influence of the torsionalrigidity of plate stiffeners on plate effectiveness, Interna-tional Shipbuilding Progress, Vol.23, No.268, pp.355-360.
Yao, T., Fujikubo, M., Yanagihara, D., Varghese, B.and Niho, O. (1998). Influences of welding imperfections onbuckling/ultimate strength of ship bottom plating subjectedto combined biaxial thrust and lateral pressure, Thin-WalledStructures, Research and Development, 2nd InternationalConference on Thin-Walled Structures, pp.425-432.
