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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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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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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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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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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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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


10/12


11/12


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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and lateral load (P= 0), but slightly underestimating un-

der combined axial and lateral loads.

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Stiffened Plate With Circular Opening