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Interaction curves for stiffened panel with circular opening under axial andlateral loadsM. Suneel Kumar a; P. Alagusundaramoorthy b; R. Sundaravadivelu caDepartment of Civil Engineering, College of Engineering, Sri Venkateswara University, Tirupati, India b
Department of Civil Engineering, IITMadras, Chennai, India cDepartment of Ocean Engineering, IITMadras,Chennai, India
First Published:June2009
To cite this ArticleKumar, M. Suneel, Alagusundaramoorthy, P. and Sundaravadivelu, R.(2009)'Interaction curves for stiffened panelwith circular opening under axial and lateral loads',Ships and Offshore Structures,4:2,133 143
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Ships and Offshore Structures
Vol. 4, No. 2, 2009, 133143
Interaction curves for stiffened panel with circular opening under axial and lateral loads
M. Suneel Kumara, P. Alagusundaramoorthyb and R. Sundaravadiveluc
aDepartment of Civil Engineering, College of Engineering, Sri Venkateswara University, Tirupati 517 502, India; bDepartment of CivilEngineering, IITMadras, Chennai 600 036, India; c Department of Ocean Engineering, IITMadras, Chennai 600 036, India
(Received 22 October 2008; final version received 14 January 2009)
Stiffened panels in ships and offshore oil platforms are provided with circular openings for repair, access and maintenance.This paper presents the numerical study carried out on the ultimate strength of stiffened panel with central circular openingsubjected to axial load, lateral load and a combination of axial and lateral loads. Ultimate strength of the panel was evaluatedconsidering both geometric and material non-linearities using FEA software ANSYS. Plates of varied widths and opensection unequal angle stiffeners covering plate and column slenderness ratios in the practical range of 1.04.5 and 0.321.00,respectively, keeping the opening ratio equal to 1.0 are the parameters considered in this study. On the basis of the study,interaction curves were developed for normalised axial load and normalised lateral load. The developed interaction curves
for stiffened panels with angle stiffeners and circular opening were found to be non-linear for lower plate slenderness ratioup to 2.0 and for the range of column slenderness ratio covered in the present study. Interaction equations were also proposedbased on non-linear regression analysis for determining the ultimate strength of stiffened panel under axial, lateral and alsounder combined axial and lateral loads.
Keywords: axial load; circular opening; interaction curves; interaction equations; lateral load; stiffened panel; ultimatestrength
Notation
b Width of Plate between stiffeners
d Diameter of opening
E Youngs modulus of elasticity
L Length of plate
Psq Squash load
Pu, Pu0 Axial ultimate load (Q = 0)
PuQ Axial ultimate load (for Q = 1/3 qut, Q = 2/3
qut, and Q = qut)
Q Lateral load
Qn Normalised lateral load
qult Lateral ultimate load
r Radius of gyration
t Thickness of Plate
w Lateral deflection at the centre of plate
Plate slenderness ratio
Poissons ratio
Column slenderness ratio
u Ultimate Stress of PlateF,y Yield Stress of plate and stiffener
avg Average axial stress in the plate
p Maximum allowable usage factor (usually
taken as 1.0)
Corresponding author. Email: [email protected]
1. Introduction
Longitudinal stiffened panels in ship decks and offshore
structures are subjected to axial compression due to sag-
ging and hogging moments. Cargo loading and buoyancy
exert lateral pressure over these panels. The failure in pan-
els subjected to axial compression occurs due to instability.
The presence of constant lateral load further reduces the ax-
ial load carrying capacity. Circular openings are provided
in the decks for access, inspection and maintenance. These
openings are provided in the centre of the panel for the
minimal reduction in elasto-plastic buckling stress (Khaled
El-Sawy et al. 2004). The presence of these openings in
plates causes a redistribution of membrane stresses accom-
panied by a significant change in the buckling and ultimate
strength characteristics. The increase in the size of opening
(d/b) increases the buckling strength up to a certain ex-
tent beyond which it decreases. But increases in the size of
the opening decrease ultimate strength significantly and are
critical between stiffeners with extending full width in be-tween them (Gendy and Saleeb 1995; Alagusundaramoor-
thy et al. 1999).
The analysis of a typical stiffened deck in a ship can
be performed at (1) grillage level, (2) stiffened panel
ISSN: 1744-5302 print/ 1754-212X online
Copyright C 2009 Taylor & Francis
DOI: 10.1080/17445300902746420http://www.informaworld.com
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134 M. Suneel Kumaret al.
Figure 1. Ship deck (a) with circular opening and (b) typical loads acting.
level between two adjacent transverses and (3) bare plate
element level. Openings in the deck plates are provided for
maintenance, access and piping. Figure 1(a) shows the
location and provision of openings in ship decks. Constant
cargo load acts on the plate surrounding the opening
(Figure 1b) and the same situation is considered in the
present study. The size of opening under any combination
of loads is critical if it is extended over the full width
between stiffeners (Gendy and Saleeb 1995; Alagusun-
daramoorthy et al. 1999). Because of the symmetry in the
stiffened panel, only one panel consisting of a portion of
the plate of width bp with a stiffener centred on the plate
strip is considered in the present study (shaded panel in
Figure 1a) as it was found to give a better understanding of
the complex behaviour of a stiffened panel with T stiffeners
under combined load and can be used reasonably for good
prediction of ultimate strength using finite element analysis
(FEA) as stated by Sheikh et al. (2003). The same concept
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Ships and Offshore Structures 135
Figure 2. Modes of failure of stiffened plate without opening
under compression and bending (Sheikh et al., 2003).
of a unit panel is adopted in this paper with the central
circular opening extending for the full width between
the stiffeners for determining the ultimate strength under
axial, lateral and combined axial and lateral loads using
non-linear elasto-plastic FEA.
The normal range of lateral loads acting on ship deck
plates due to cargo varies in the range of 3040 kN/m2. As
per clause 3.3.5 of Det Norske Veritas (DnV) Classifica-
tion Notes 30.1 (1995), the design pressure (pd) for plates
subjected to constant load is given as
pd p 4 F
t
b
2. (1)
The present study considers constant lateral load acting on
deck plates and not variable hydrostatic loading, which acts
on ship-side plates and curved shells. The present study
is only to assess the reduction in ultimate load carrying
capacity and to determine the interactive behaviour due to
openings. On the basis of the reduction in ultimate load,
insert plates can be provided to compensate for the loss of
area which is not in the scope of the present study. The
high stress concentration around the hole observed at the
early stages of loading practically disappears at loads in
excess of the critical load due to the development of large
buckles (Ritchie and Rhodes 1975). Hence, at ultimate load,
stress concentration is not of a concern as redistribution
of the stresses for the surrounding area takes place and
the present study is done up to ultimate load. The present
study focuses on the interactive behaviour between axial
and lateral loads. The interaction equations (IEs) without
much computational effort for the purpose of designers
have also been developed.
Figure 3. Unit panel.
2. Literature review
Azizian and Roberts (1983) have presented a FEA for the
elastic buckling and geometrically non-linear elasto-plastic
collapse of perforated plates subjected to inclined loading.
Triangular elements with three bending and two membrane
degrees of freedom at each corner node are used to model
the plates. Sabir and Chow (1983) employed the method
of FEA to determine the elastic critical buckling loads
of flat square panels containing circular and square holes.
In-plane loads such as uniaxial, biaxial or shear distributed
uniformly along the straight edges of the panels were con-
sidered. Narayanan and der Avanessian (1984) dealt with
elastic shear buckling of simply supported and clamped
plates with circular and rectangular openings. Roberts and
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136 M. Suneel Kumaret al.
Table 1. Details of parametric study.
Plate Slenderness ratio
Sl. no. b (mm) t (mm) b/t Section
ColumnSlenderness
ratio () Specimen Pu(kN) q (kN/m2)
1 170 6 28 1 ISA 5030 6 0.92 B1C1 374.86 0.00267.95 61.88
167.60 123.770.00 185.65ISA 7045 6 0.60 B1C2 427.43 0.00
322.90 114.04216.95 228.08
0.00 342.12ISA 10065 6 0.40 B1C3 502.50 0.00
384.45 189.54247.80 379.08
0.00 568.62ISA 12595 6 0.31 B1C4 584.98 0.00
457.95 254.82308.90 509.65
0.00 764.472 340 6 57 2 ISA 5030 6 1.00 B2C1 628.39 0.00
402.80 31.30
204.80 62.610.00 93.91ISA 7045 6 0.64 B2C2 681.34 0.00
522.00 57.22334.80 114.45
0.00 171.67ISA 10065 6 0.42 B2C3 755.66 0.00
590.50 95.36369.00 190.72
0.00 286.08ISA 12595 6 0.32 B2C4 838.00 0.00
672.70 127.24438.70 254.49
0.00 381.733 510 6 85 3 ISA 5030 6 1.03 B3C1 564.67 0.00
397.50 22.43
170.80 44.860.00 67.29ISA 7045 6 0.66 B3C2 621.47 0.00
492.70 38.39304.90 76.79
0.00 115.18ISA 10065 6 0.43 B3C3 692.76 0.00
540.00 67.35271.80 134.70
0.00 202.05ISA 12595 6 0.32 B3C4 791.89 0.00
584.70 93.01300.40 186.01
0.00 279.024 765 6 128 4.5 ISA 5030 6 1.05 B4C1 618.51 0.00
362.30 15.72129.70 31.43
0.00 47.15ISA 7045 6 0.67 B4C2 695.40 0.00
480.00 28.03222.40 56.07
0.00 84.10ISA 10065 6 0.43 B4C3 747.85 0.00
519.70 48.54248.30 97.09
0.00 145.63ISA 12595 6 0.32 B4C4 846.08 0.00
593.50 65.68306.30 131.36
0.00 197.04
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Ships and Offshore Structures 137
Figure 4. Load/deflection curves for stiffened plate under axial
load.
Azizian (1984) generated interaction curves for the ulti-
mate strength of square plates with central square and cir-
cular holes for the range of d/b of 0 to 0.5 subjected to
uniaxial compression, biaxial compression and pure shear.
For uniaxial compression, the buckling load was almost
independent of the size of the hole ford/b ranging from
0 to 0.5. The ultimate load of all the plates studied de-
creased with increase in size of the hole. The reduction in
the ultimate load was most significant for low plate slender-
ness values. Narayanan and Chow (1984) developed design
charts based on ultimate capacity of uniaxially compressed
perforated plates with square and circular openings for the
range ofd/b of 0 to 0.5. Narayanan and Chan (1985) pre-sented design charts based on ultimate strength of plates
containing circular holes for the range ofd/b of 0 to 0.5
under linearly varying edge displacements. A Marginal de-
crease in ultimate load was observed for many practical
plates (b/t < 50) with unloaded edges free to wave in or
kept straight. Also, ultimate load is not much affected for
plates with holes located away from the centre. Shanmugam
et al. (1986) presented an approximate method for the anal-
ysis of stiffened flange plates containing circular or square
openings loaded axially. The failure is considered to oc-
cur in the plate first. The loss in the stiffness of plates was
taken into account by means of effective width. Reports
of the experiments carried out on small-scale steel models
of perforated stiffened plates were presented. A compar-
ison of ultimate loads with the proposed method was in
good comparison with experimental results. Brown et al.
(1987) determined the buckling coefficients for plates with
rectangular opening.
Shanmugam and Arockiasamy (1996) conducted ulti-
mate strength tests on stiffened plates simply supported on
Figure 5. Finite element mesh.
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138 M. Suneel Kumaret al.
all four edges subjected to combined action of axial and lat-
eral loads. The test specimens were analysed by using the
elasto-plastic FE package ABAQUS to determine the be-
haviour and ultimate load capacity of stiffened panels. The
comparison of FE results with those obtained experimen-
tally established the accuracy of the FE modelling. Motok
(1997) carried out stress concentration studies on the con-tour of a plate opening of an arbitrary corner radius of cur-
vature. Shanmugam (1997) reviewed the effects of openings
for the range ofd/b of 0 to 0.5 in plate elements subjected to
uniaxial compression, biaxial compression and pure shear
in stiffened plates, shear webs and cold formed steel sec-
tions. Grondin et al. (1998) investigated experimentally and
numerically the effect of unloaded edgesboundary restraint,
combination of axial and lateral loads, and imposed plate
damage on the buckling capacity of stiffened steel plates.
Shanmugam et al. (1999) presented the design formula for
axially compressed perforated plates with circular openings
under axial compression for simply supported and clamped
boundary conditions. The ultimate load carrying capacity of
perforated plates decreased significantly with the increase
in hole size and slenderness ratio. The strength of perforated
plates with simply supported edges was lower as compared
to that of plates with clamped edges. The plates with cir-
cular holes generally had higher ultimate loads compared
to the square perforated plates. Paik et al. (2001) presented
ultimate strength formulations for ship plating with plate
slenderness ratio in the range of 1.0 to 5.0 under combined
biaxial compression/tension, edge shear and lateral load.
The study was focused at bare plate level and the validity
of the proposed interaction formulae was verified. Sheikh
et al. (2003) studied the behaviour of steel plates stiffenedwith T stiffeners subjected to axial load with and without
bending moments by considering unit panel. The various
modes of failure (Figure 2) observed were categorised as (1)
plate-induced overall buckling, (2) stiffener-induced over-
all buckling, (3) plate buckling and (4) stiffener tripping.
It is observed that both the behaviour and strength are pre-
dominantly affected by plate and column slenderness ratio.
A comparison of ultimate strength with DnV (1995) and
American Petroleum Institute (API) codes is made. It was
found that the predicted ultimate strengths using DnV and
API for stiffener tripping mode were unreliable. Suneel
Kumar et al. (2007) studied the ultimate strength of ship
deck plating with a centrally located increasing rectangular
opening along the width of the plate. A design equation was
proposed for a square plate with rectangular opening under
axial compression.
A comprehensive procedure for the computation of
buckling strength of stiffened panels under axial and lat-
eral loads exists in API and DnV codes. The interactive
buckling phenomenon is not mentioned in either of these
two codes. Design guidelines for a stiffened panel with cir-
cular opening extending full width between stiffeners under
axial and lateral loads are not available in the DnV and API
Figure 6. Axial load/axial deformation curves for (a) case(i), (b) case (ii), and (c) case (iii).
codes. A critical review of literature indicates that studies
are conducted on the effect of circular opening in the range
ofd/bof 0 to 0.9 on the buckling and ultimate strength of
unstiffened plates only. Till now researchers have focused
on the effect of circular openings on the ultimate strength at
the bare element level only. This methodology is only valid
if the stiffener is stocky and the failure is by plate buckling.
In case of low stiffener rigidity, the interaction between
plate and stiffener has to be considered. Hence, it is desir-
able to analyse at panel level to account for the rigidity of
the stiffener in the analysis. It may be observed that there
is not much information available on the ultimate strength
of a stiffened panel with a circular opening extending full
width between stiffeners under axial and lateral loads in the
literature. An attempt is made in this paper to determine the
interaction behaviour between axial and lateral loads for a
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Ships and Offshore Structures 139
Figure 7. Deformation pattern plate buckling overall (Vector mode).
stiffened panel with a circular opening subjected to axial
and lateral loads.
3. Numerical study
A unit panel consisting of a portion of the plate of width
b with a stiffener centred on the plate strip provided with
circular opening in the centre of the panel is as shown in
Figure 3. As it is symmetrical with respect to the unit panel,
only a half portion of the circular opening is considered in
the analysis (shaded portion of panel shown in Figure 1).
The length of panel and thickness of the plate are taken
as 1500 mm and 6 mm, respectively. The widths of plate
considered in the present study are 170 mm, 340 mm, 510
mm and 765 mm. Unequal Indian Standard Angles (ISA)
such as ISA 5030 6, ISA 7045 6, ISA 10065 6 and ISA12595 6 are used as stiffeners. Typical ISA 5030 6 denotes
an unequal Indian standard angle of flange width 30 mm,
with overall web depth of 50 mm and uniform thickness of
section 6 mm. The practical range of plate slenderness ratio
( = bt
y
E) and column slenderness ratio = L
r
y
E of
the stiffener associated with plating used in ship construc-
tion of 1.04.5 and 0.150.90, respectively (Smith 1975)
are adopted in the study. The selected plate and column
slenderness ratios for the present study considered are 1,
2, 3, and 4.5, and 0.9, 0.6, 0.4 and 0.3, respectively. The
corresponding plate aspect ratios (A/B) adopted are found
to be 8.82, 4.41, 2.94 and 1.96 covering the entire range of
slender plates used in frigates and tankers (Guedes Soares
1988). The depth to width of circular opening (d/b) is kept
constant at 1.0 throughout the study. A combination of ev-
ery plate slenderness ratio with all four column slenderness
ratios is accounted. The study comprises a total of sixteen
specimens with four plate and four column slenderness
ratios. Each specimen is subjected to four load cases as
detailed in the subsequent section. Thus the entire study
results in FEA of sixty-four load cases. The details of the
parametric study are given in Table 1. The yield strength of
plate (y ) is assumed as 250 N/mm2 with Youngs modulus
of elasticity (E) as 2 105 N/mm2 and Poissons ratio ()
of 0.3.
4. Non-linear FEA
A general purpose FE software ANSYS is used for mod-
elling, analysis and post processing of stiffened plate with
circular opening under axial and lateral loads. Discrete
stiffened plate approach is adopted for modelling of the
stiffened panels. This approach enables the prediction of
overall buckling and local buckling of the stiffened plates,
and interaction between the plate and stiffener and lo-
cal yielding. A four noded quadrilateral isoparametric lin-
ear shell element (SHELL181) available in the ANSYS
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140 M. Suneel Kumaret al.
Figure 8. Effect of plate and column slenderness ratios on (a)normalised axial load (Q = 0) and (b) normalised lateral load(P = 0).
element library is used for modelling the plate and stiff-
ener. The deformation shapes of the element are linear in
both in-plane directions. This element is well suitable for
analysing the linear, large rotation, and/large strain non-
linear applications. The element has six degrees of free-
dom at each node viz. three translations in UX, UY and
UZ, and three rotations X, Y andZ. Square shaped el-
ements are used for meshing regular area while triangular
mesh is adopted around the circular opening (Khaled El-
Sawy and Aly Nazmy 2001). Both geometric and material
non-linearities are considered in the analysis. Geometric
non-linearity involves large displacement static analysis
with stress stiffening. The model invokes large displace-
ment using an updated Lagrangian formulation. Bilinear
isotropic rate independent hardening with von Mises yield
criteria is used in material non-linearity. Simply supported
boundary conditions along the loading and reactive edge
are considered. The rotation about the longitudinal edge
is arrested at all the nodes along the unloaded edges. The
displacement along the same nodes is allowed to freely
wave in. This condition is to generate the actual situation
of continuity between individual stiffened panels. A Con-
straint equation is applied along the loading edge on axial
deformation degree of freedom to attain uniform compres-
sion of the loading edge. Incremental load is applied up to
and beyond collapse. NewtonRaphson iterative procedurein the initial stage of loading and then arc length method
is used to trace the post peak axial load/axial deformation
behaviour.
A stiffened plate of size 500 mm 500 mm 5 mm
simply supported on all four sides, subjected to axial load
is considered for validation (Ueda and Yao 1983). The size
of flat stiffener is 50 mm 5 mm. The yield stress, Youngs
modulus and Poissons ratio of plate and stiffener are 250
N/mm2, 2 105 N/mm2 and 0.3, respectively. Based on
the convergence study, it is found that the mesh size of 25
mm 25 mm is found to be satisfactory for ultimate load
and is used for the analysis of all specimens in this study.
The load/deflection curve shown in Figure 4 obtained using
FEA is in good comparison with analytical and numerical
studies conducted by Ueda and Yao (1983). For triangu-
lar mesh around circular opening, the size of the element
adopted is least of b/50 or d/40 (Khaled El-Sawy and
Aly Nazmy 2001). The FE model of typical unit panel with
circular opening is shown in Figure 5. Each specimen is
subjected to and analysed for four different load cases as;
(1) axial load (till failure, Pu0) with no lateral load (Q =
0), (2) no axial load (P= 0) but lateral load (till fail-
ure, qu), (3) axial load (till failure, PuQ) corresponding to
constant lateral load (Q = 1/3 qu), and (4) axial load (till
failure, PuQ) corresponding to constant lateral load (Q =2/3qu). Ultimate load of the specimens is determined from
the peak of axial load/axial deformation plots (Figure 6).
The axial ultimate load, Pu0 (the specimen is under ax-
ial load and lateral load Q = 0) and axial ultimate load,
PuQ (the specimen is under the combined action of axial
load and lateral loadQ = 1/3 qut, Q = 2/3 qut, andQ =
qut, respectively) are determined for varied plate and col-
umn slenderness ratios. The values of ultimate strength for
all the specimens with varied plate and column slender-
ness ratios are given in Table 1. A typical lateral deforma-
tion pattern for overall plate buckling mode is shown in
Figure 7.
5. Results and discussion
The axial and lateral ultimate load obtained from the present
study (Table 1) are as (1) normalised axial ultimate load
as Pu0/Psq(forQ = 0), Psqbeing the squash load; (2) nor-
malised axial ultimate load under combined load as PuQ/Pu0and (3) normalised lateral load (Qn) as
QE
(y )2(Guedes Soares
and Gordo 1996) in which Q = 1/3qut, Q = 2/3qut, and
Q = qut. Squash load (Psq) is defined as the summation of
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142 M. Suneel Kumaret al.
Figure 10. Comparison of normalised ultimate load for panelsunder (a) axial load (Q = 0), (b) axial and lateral load, and (c)lateral load (P = 0).
From the interaction curves (Figure 9), it may be observed
that relationship between plate slenderness ratio (), col-
umn slenderness ratio (), and normalised lateral load (Qn)
on the normalised axial load (PuQ/Pu0) varies non-linearly.
The interaction of these parameters is important and there
is a need for simple formulae for the design. Hence non-
linear regression analysis is adopted to predict the rela-tionship among these variables. A non-linear regression
can estimate models with an arbitrary relationship between
the dependent variable (PuQ/Pu0) and a set of independent
variables (, andQn). The proposed IE for the design
of stiffened panels with circular opening under axial and
lateral loads is given below:
Pu0
Psq= 0.0012.831 + 1.0670.061 + 0.595Qn 0.041
1.606Qn 0.095Qn 0.765Qn. (4)
It is found that for the above mentioned proposed inter-action in Equations (2), (3) and (4), the R-squared value is
found to be 0.992, 0. 999 and 0.985, respectively and hence
fits well in the data obtained using non-linear FEA. Thus the
developed formula is simple, reliable and can be used for
the estimation of ultimate strength by practical engineers.
The mean(x), standard deviation () andcoefficientof vari-
ation (cov) are found to be 1.001, 0.032 and 0.032 for the
results obtained in Equation (2), 1.002, 0.040 and 0.040 for
Equation (3), and 0.970, 0.073 and 0.076 for Equation (4),
respectively. It can be observed that the ultimate strength of
panels obtained using the proposed Equations (2) and (3)
are slightly higher while Equation (4) underestimates andhence conservative for the purpose of design. A plot (Fig-
ure 10) is drawn for the normalised ultimate load based on
FEA vs. normalised ultimate load obtained using proposed
IE for the specimens considered in the present study. The
ultimate load can be predicted using the proposed Equa-
tions (2), (3) and (4) with an error of4 %, +5% to10%
and2 % correspondingly for the practical range of plate
and column slenderness ratios used in ship construction of
1.04.5 and 0.150.90, respectively.
6. Summary and conclusions
The interaction behaviour of a unit stiffened panel with
circular central opening on the ultimate strength under ax-
ial, lateral and combined load was studied. The effect of
plate and column slenderness ratios on ultimate strength
was also evaluated. Equations for the developed interac-
tion curves for stiffened panels with circular opening under
axial, lateral and combined load were developed based on
non-linear regression analysis. The error analysis for the
developed equations indicates to yield satisfactory results.
The proposed equations were found to predict accurately
the ultimate strength of panels under axial load (Q = 0)
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Ships and Offshore Structures 143
and lateral load (P= 0), but slightly underestimating un-
der combined axial and lateral loads.
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