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Interaction curves for stiffened panel with circularopening under axial and lateral loadsM. Suneel Kumar a , P. Alagusundaramoorthy b & R. Sundaravadivelu ca Department of Civil Engineering, College of Engineering , Sri Venkateswara University ,Tirupati, 517 502, Indiab Department of Civil Engineering , IITMadras , Chennai, 600 036, Indiac Department of Ocean Engineering , IITMadras , Chennai, 600 036, IndiaPublished online: 08 May 2009.

To cite this article: M. Suneel Kumar , P. Alagusundaramoorthy & R. Sundaravadivelu (2009) Interaction curves forstiffened panel with circular opening under axial and lateral loads, Ships and Offshore Structures, 4:2, 133-143, DOI:10.1080/17445300902746420

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Ships and Offshore StructuresVol. 4, No. 2, 2009, 133–143

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 1.0–4.5 and 0.32–1.00,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 curvesfor 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 stiffenersd Diameter of openingE Young’s modulus of elasticityL Length of platePsq Squash loadPu, Pu0 Axial ultimate load (Q = 0)PuQ Axial ultimate load (for Q = 1/3 qut, Q = 2/3

qut, and Q = qut)Q Lateral loadQn Normalised lateral loadqult Lateral ultimate loadr Radius of gyrationt Thickness of Platew Lateral deflection at the centre of plateβ Plate slenderness ratioυ Poisson’s ratioλ Column slenderness ratioσu Ultimate Stress of PlateσF , σy Yield Stress of plate and stiffenerσavg Average axial stress in the plateηp Maximum allowable usage factor (usually

taken as 1.0)

∗Corresponding author. Email: suneel78@rediffmail.com

1. Introduction

Longitudinal stiffened panels in ship decks and offshorestructures are subjected to axial compression due to sag-ging and hogging moments. Cargo loading and buoyancyexert 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 providedin the decks for access, inspection and maintenance. Theseopenings are provided in the centre of the panel for theminimal reduction in elasto-plastic buckling stress (KhaledEl-Sawy et al. 2004). The presence of these openings inplates causes a redistribution of membrane stresses accom-panied by a significant change in the buckling and ultimatestrength 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 ofthe opening decrease ultimate strength significantly and arecritical 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 canbe performed at (1) grillage level, (2) stiffened panel

ISSN: 1744-5302 print / 1754-212X onlineCopyright C© 2009 Taylor & FrancisDOI: 10.1080/17445300902746420http://www.informaworld.com

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Figure 1. Ship deck (a) with circular opening and (b) typical loads acting.

level between two adjacent transverses and (3) bare plateelement level. Openings in the deck plates are provided formaintenance, access and piping. Figure 1(a) shows thelocation and provision of openings in ship decks. Constantcargo load acts on the plate surrounding the opening(Figure 1b) and the same situation is considered in thepresent study. The size of opening under any combinationof loads is critical if it is extended over the full widthbetween stiffeners (Gendy and Saleeb 1995; Alagusun-

daramoorthy et al. 1999). Because of the symmetry in thestiffened panel, only one panel consisting of a portion ofthe plate of width bp with a stiffener centred on the platestrip is considered in the present study (shaded panel inFigure 1a) as it was found to give a better understanding ofthe complex behaviour of a stiffened panel with T stiffenersunder combined load and can be used reasonably for goodprediction 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 openingunder compression and bending (Sheikh et al., 2003).

of a unit panel is adopted in this paper with the centralcircular opening extending for the full width betweenthe stiffeners for determining the ultimate strength underaxial, lateral and combined axial and lateral loads usingnon-linear elasto-plastic FEA.

The normal range of lateral loads acting on ship deckplates due to cargo varies in the range of 30–40 kN/m2. Asper clause 3.3.5 of Det Norske Veritas (DnV) Classifica-tion Notes 30.1 (1995), the design pressure (pd ) for platessubjected to constant load is given as

pd ≤ ηp ∗ 4 ∗ σF ∗(

t

b

)2

. (1)

The present study considers constant lateral load acting ondeck plates and not variable hydrostatic loading, which actson ship-side plates and curved shells. The present studyis only to assess the reduction in ultimate load carryingcapacity and to determine the interactive behaviour due toopenings. On the basis of the reduction in ultimate load,insert plates can be provided to compensate for the loss ofarea which is not in the scope of the present study. Thehigh stress concentration around the hole observed at theearly stages of loading practically disappears at loads inexcess of the critical load due to the development of largebuckles (Ritchie and Rhodes 1975). Hence, at ultimate load,stress concentration is not of a concern as redistributionof the stresses for the surrounding area takes place andthe present study is done up to ultimate load. The presentstudy focuses on the interactive behaviour between axialand lateral loads. The interaction equations (IEs) withoutmuch computational effort for the purpose of designershave also been developed.

Figure 3. Unit panel.

2. Literature review

Azizian and Roberts (1983) have presented a FEA for theelastic buckling and geometrically non-linear elasto-plasticcollapse of perforated plates subjected to inclined loading.Triangular elements with three bending and two membranedegrees of freedom at each corner node are used to modelthe plates. Sabir and Chow (1983) employed the methodof FEA to determine the elastic critical buckling loadsof flat square panels containing circular and square holes.In-plane loads such as uniaxial, biaxial or shear distributeduniformly along the straight edges of the panels were con-sidered. Narayanan and der Avanessian (1984) dealt withelastic shear buckling of simply supported and clampedplates with circular and rectangular openings. Roberts and

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136 M. Suneel Kumar et al.

Table 1. Details of parametric study.

Plate Slenderness ratio

Sl. no. b (mm) t (mm) b/t β Section

ColumnSlenderness

ratio (λ) Specimen Pu(kN) q (kN/m2)

1 170 6 28 1 ISA 5030 6 0.92 B1C1 374.86 0.00267.95 61.88167.60 123.77

0.00 185.65ISA 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 31.30204.80 62.61

0.00 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 22.43170.80 44.86

0.00 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 axialload.

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 touniaxial compression, biaxial compression and pure shear.For uniaxial compression, the buckling load was almostindependent of the size of the hole for d/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 inthe ultimate load was most significant for low plate slender-ness values. Narayanan and Chow (1984) developed designcharts based on ultimate capacity of uniaxially compressedperforated plates with square and circular openings for therange of d/b of 0 to 0.5. Narayanan and Chan (1985) pre-sented design charts based on ultimate strength of platescontaining circular holes for the range of d/b of 0 to 0.5under linearly varying edge displacements. A Marginal de-crease in ultimate load was observed for many practicalplates (b/t < 50) with unloaded edges free to wave in orkept straight. Also, ultimate load is not much affected forplates with holes located away from the centre. Shanmugamet al. (1986) presented an approximate method for the anal-ysis of stiffened flange plates containing circular or squareopenings loaded axially. The failure is considered to oc-cur in the plate first. The loss in the stiffness of plates wastaken into account by means of effective width. Reportsof the experiments carried out on small-scale steel modelsof perforated stiffened plates were presented. A compar-ison of ultimate loads with the proposed method was ingood comparison with experimental results. Brown et al.(1987) determined the buckling coefficients for plates withrectangular 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 Kumar et al.

all four edges subjected to combined action of axial and lat-eral loads. The test specimens were analysed by using theelasto-plastic FE package ABAQUS to determine the be-haviour and ultimate load capacity of stiffened panels. Thecomparison 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 openingsfor the range of d/b of 0 to 0.5 in plate elements subjected touniaxial compression, biaxial compression and pure shearin stiffened plates, shear webs and cold formed steel sec-tions. Grondin et al. (1998) investigated experimentally andnumerically the effect of unloaded edges boundary restraint,combination of axial and lateral loads, and imposed platedamage on the buckling capacity of stiffened steel plates.Shanmugam et al. (1999) presented the design formula foraxially compressed perforated plates with circular openingsunder axial compression for simply supported and clampedboundary conditions. The ultimate load carrying capacity ofperforated plates decreased significantly with the increasein hole size and slenderness ratio. The strength of perforatedplates with simply supported edges was lower as comparedto that of plates with clamped edges. The plates with cir-cular holes generally had higher ultimate loads comparedto the square perforated plates. Paik et al. (2001) presentedultimate strength formulations for ship plating with plateslenderness ratio in the range of 1.0 to 5.0 under combinedbiaxial compression/tension, edge shear and lateral load.The study was focused at bare plate level and the validityof the proposed interaction formulae was verified. Sheikhet al. (2003) studied the behaviour of steel plates stiffenedwith T stiffeners subjected to axial load with and withoutbending moments by considering unit panel. The variousmodes 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) andAmerican Petroleum Institute (API) codes is made. It wasfound that the predicted ultimate strengths using DnV andAPI for stiffener tripping mode were unreliable. SuneelKumar et al. (2007) studied the ultimate strength of shipdeck plating with a centrally located increasing rectangularopening along the width of the plate. A design equation wasproposed for a square plate with rectangular opening underaxial compression.

A comprehensive procedure for the computation ofbuckling strength of stiffened panels under axial and lat-eral loads exists in API and DnV codes. The interactivebuckling phenomenon is not mentioned in either of thesetwo codes. Design guidelines for a stiffened panel with cir-cular opening extending full width between stiffeners underaxial 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 studiesare conducted on the effect of circular opening in the rangeof d/b of 0 to 0.9 on the buckling and ultimate strength ofunstiffened plates only. Till now researchers have focusedon the effect of circular openings on the ultimate strength atthe bare element level only. This methodology is only validif the stiffener is stocky and the failure is by plate buckling.In case of low stiffener rigidity, the interaction betweenplate and stiffener has to be considered. Hence, it is desir-able to analyse at panel level to account for the rigidity ofthe stiffener in the analysis. It may be observed that thereis not much information available on the ultimate strengthof a stiffened panel with a circular opening extending fullwidth between stiffeners under axial and lateral loads in theliterature. An attempt is made in this paper to determine theinteraction 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 axialand 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 withcircular opening in the centre of the panel is as shown inFigure 3. As it is symmetrical with respect to the unit panel,only a half portion of the circular opening is considered inthe analysis (shaded portion of panel shown in Figure 1).The length of panel and thickness of the plate are takenas 1500 mm and 6 mm, respectively. The widths of plateconsidered in the present study are 170 mm, 340 mm, 510mm 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 denotesan unequal Indian standard angle of flange width 30 mm,with overall web depth of 50 mm and uniform thickness ofsection 6 mm. The practical range of plate slenderness ratio

(β = bt

√σy

E) and column slenderness ratio λ = L

rπ

√σy

Eof

the stiffener associated with plating used in ship construc-tion of 1.0–4.5 and 0.15–0.90, respectively (Smith 1975)are adopted in the study. The selected plate and columnslenderness 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 foundto be 8.82, 4.41, 2.94 and 1.96 covering the entire range ofslender plates used in frigates and tankers (Guedes Soares1988). The depth to width of circular opening (d/b) is keptconstant at 1.0 throughout the study. A combination of ev-ery plate slenderness ratio with all four column slendernessratios is accounted. The study comprises a total of sixteenspecimens with four plate and four column slendernessratios. Each specimen is subjected to four load cases asdetailed in the subsequent section. Thus the entire studyresults in FEA of sixty-four load cases. The details of theparametric study are given in Table 1. The yield strength ofplate (σy) is assumed as 250 N/mm2 with Young’s modulusof elasticity (E) as 2 × 105 N/mm2 and Poisson’s ratio (υ)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 withcircular opening under axial and lateral loads. Discretestiffened plate approach is adopted for modelling of thestiffened panels. This approach enables the prediction ofoverall 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 Kumar et 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 inboth in-plane directions. This element is well suitable foranalysing 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 andUZ , and three rotations θX, θY and θZ . Square shaped el-ements are used for meshing regular area while triangularmesh is adopted around the circular opening (Khaled El-Sawy and Aly Nazmy 2001). Both geometric and materialnon-linearities are considered in the analysis. Geometricnon-linearity involves large displacement static analysiswith stress stiffening. The model invokes large displace-ment using an updated Lagrangian formulation. Bilinearisotropic rate independent hardening with von Mises yieldcriteria is used in material non-linearity. Simply supportedboundary conditions along the loading and reactive edgeare considered. The rotation about the longitudinal edgeis arrested at all the nodes along the unloaded edges. The

displacement along the same nodes is allowed to freelywave in. This condition is to generate the actual situationof continuity between individual stiffened panels. A Con-straint equation is applied along the loading edge on axialdeformation degree of freedom to attain uniform compres-sion of the loading edge. Incremental load is applied up toand beyond collapse. Newton–Raphson iterative procedurein the initial stage of loading and then arc length methodis used to trace the post peak axial load/axial deformationbehaviour.

A stiffened plate of size 500 mm × 500 mm × 5 mmsimply supported on all four sides, subjected to axial loadis considered for validation (Ueda and Yao 1983). The sizeof flat stiffener is 50 mm × 5 mm. The yield stress, Young’smodulus and Poisson’s ratio of plate and stiffener are 250N/mm2, 2 × 105 N/mm2 and 0.3, respectively. Based onthe convergence study, it is found that the mesh size of 25mm × 25 mm is found to be satisfactory for ultimate loadand is used for the analysis of all specimens in this study.The load/deflection curve shown in Figure 4 obtained usingFEA is in good comparison with analytical and numericalstudies conducted by Ueda and Yao (1983). For triangu-lar mesh around circular opening, the size of the elementadopted is least of b/50 or πd/40 (Khaled El-Sawy andAly Nazmy 2001). The FE model of typical unit panel withcircular opening is shown in Figure 5. Each specimen issubjected 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 toconstant lateral load (Q = 1/3 qu), and (4) axial load (tillfailure, PuQ) corresponding to constant lateral load (Q =2/3 qu). Ultimate load of the specimens is determined fromthe 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 axialload and lateral load Q = 1/3 qut, Q = 2/3 qut, and Q =qut, respectively) are determined for varied plate and col-umn slenderness ratios. The values of ultimate strength forall 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 inFigure 7.

5. Results and discussion

The axial and lateral ultimate load obtained from the presentstudy (Table 1) are as (1) normalised axial ultimate loadas Pu0/Psq(for Q = 0), Psqbeing the squash load; (2) nor-malised axial ultimate load under combined load as PuQ/Pu0

and (3) normalised lateral load (Qn) as QE

(σy )2 (Guedes Soaresand Gordo 1996) in which Q = 1/3qut, Q = 2/3qut, andQ = qut. Squash load (Psq) is defined as the summation of

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Ships and Offshore Structures 141

Figure 9. Interaction curves for stiffened panel with circular opening under axial and lateral load.

product of cross-sectional area of plate with yield stress ofplate and cross-sectional area of stiffener with yield stressof stiffener (Alagusundaramoorthy et al. 1999). Using thegeometrical parameters (β and λ), interaction curves aredeveloped for varied column slenderness ratios as (1) nor-malised axial load (Pu0/Psq) vs. plate slenderness ratio (β),(2) normalised lateral load (Qn) vs. plate slenderness ratio(β) and (3) normalised axial load (PuQ/Pu0) vs. normalisedlateral load (Qn). The effect of plate and column slender-ness ratios on the normalised ultimate strength for all thespecimens subjected to axial load (with Q = 0) and lateralload (with P = 0) are shown in Figure 8a and 8b, respec-tively. It is observed from Figure 8a that the entire range ofλ (0.32–1.00), the normalised axial ultimate load decreaseslinearly with increase in β for the range 1.0–4.5. From Fig-ure 8b it can be ascertained that normalised lateral loadcarrying capacity decreases drastically for β in the range of1.0–2.0 beyond which the decrease is gradual and remainslinear for the entire range of λ (0.32–1.00). The interactioncurves for panels with circular opening for varied plate and

column slenderness ratios are shown in Figure 9. The in-teraction behaviour is non-linear for plate slenderness ratioonly up to 2.0 beyond which it is linear for the entire rangeof column slenderness ratio (0.3–1.0) for all the specimensunder axial and lateral load.

Shanmugam et al. (1999) used best fit regression analy-sis in developing the design formula to predict the ultimateload of square plates with centrally placed square or circu-lar openings subjected to uniaxial and biaxial compression.Based on the present study, design equations are developedusing regression analysis to determine the ultimate loadcarrying capacity of stiffened panels with circular opening(d/b = 1.0). The below mentioned interaction in Equations(2) and (3) correspond to stiffened panels under axial load(with Q = 0) and lateral load (P = 0), respectively.

Pu0

Psq= −0.092β + 0.027λ + 0.657β−0.025λ−0.202 (2)

Qn = 0.031β − 0.091λ + 0.643β−0.828λ−1.149. (3)

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142 M. Suneel Kumar et 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 observedthat relationship between plate slenderness ratio (β), col-umn slenderness ratio (λ), and normalised lateral load (Qn)on the normalised axial load (PuQ/Pu0) varies non-linearly.The interaction of these parameters is important and thereis 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 regressioncan estimate models with an arbitrary relationship betweenthe dependent variable (PuQ/Pu0) and a set of independentvariables (β, λ and Qn). The proposed IE for the designof stiffened panels with circular opening under axial andlateral loads is given below:

Pu0

Psq= 0.001β2.831 + 1.067λ0.061 + 0.595Qn − 0.041βλ

−1.606λQn − 0.095βQn − 0.765βλQn. (4)

It is found that for the above mentioned proposed inter-action in Equations (2), (3) and (4), the R-squared value isfound to be 0.992, 0. 999 and 0.985, respectively and hencefits well in the data obtained using non-linear FEA. Thus thedeveloped formula is simple, reliable and can be used forthe estimation of ultimate strength by practical engineers.The mean (x), standard deviation (σ ) and coefficient of vari-ation (cov) are found to be 1.001, 0.032 and 0.032 for theresults obtained in Equation (2), 1.002, 0.040 and 0.040 forEquation (3), and 0.970, 0.073 and 0.076 for Equation (4),respectively. It can be observed that the ultimate strength ofpanels 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 onFEA vs. normalised ultimate load obtained using proposedIE for the specimens considered in the present study. Theultimate load can be predicted using the proposed Equa-tions (2), (3) and (4) with an error of ±4 %, +5% to −10%and ±2 % correspondingly for the practical range of plateand column slenderness ratios used in ship construction of1.0–4.5 and 0.15–0.90, respectively.

6. Summary and conclusions

The interaction behaviour of a unit stiffened panel withcircular central opening on the ultimate strength under ax-ial, lateral and combined load was studied. The effect ofplate and column slenderness ratios on ultimate strengthwas also evaluated. Equations for the developed interac-tion curves for stiffened panels with circular opening underaxial, lateral and combined load were developed based onnon-linear regression analysis. The error analysis for thedeveloped equations indicates to yield satisfactory results.The proposed equations were found to predict accuratelythe 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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