buckling of stiffened plates

Journal of Constructional Steel Research 50 (1999) 151–175

Buckling of stiffened steel plates—a parametricstudy

G.Y. Grondin*, A.E. Elwi, J.J.R. ChengDepartment of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta,

Canada, T6G 2G7

Received 18 August 1997; received in revised form 28 July 1998; accepted 2 August 1998

Abstract

The stability of plates stiffened with tee-shape stiffeners was investigated using a finiteelement model. Four series of stiffened plate panels were modeled using a finite strain four-node shell element. The model was validated using the results of tests on full-size stiffenedplate specimens and was subsequently used to perform the study of various parameterspresented in this paper. The parameters investigated are: the shape and magnitude of initialimperfections in the plate; residual stress magnitude and direction of applied uniform bending;plate slenderness ratio; plate aspect ratio; and plate to stiffener cross-sectional area ratio. Theeffect of the investigated parameters on the axial load carrying capacity and the mode offailure of stiffened plates is investigated both in the elastic and inelastic ranges. A comparisonof these results with design guidelines formulated by Det norske Veritas and the AmericanPetroleum Institute indicates that the guidelines are generally conservative for cases whereinitial imperfection magnitudes do not exceed the guidelines’ prescribed maximum. 1999Elsevier Science Ltd. All rights reserved.

Keywords:Steel plate; Buckling; Effective width; Finite element; Initial imperfections; Residual stresses;Tee stiffener; Tripping

NOMENCLATURE

Ap area of the plate (B 3 t)Ast area of the stiffener

* Corresponding author. Tel:1 1-403-492-2794; fax:1 1-403-492-0249; e-mail: [email protected]

0143-974X/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0143 -974X(98)00242-9

152 G.Y. Grondin et al. /Journal of Constructional Steel Research 50 (1999) 151–175

B longitudinal stiffeners spacing, also taken as the width of a stiffenedplate panel

Be plate panel effective widthbf stiffener flange widthE modulus of elasticityh height of the stiffener stemL length of a stiffened plate panelnp number of half sine waves in the deformed shape of the plate along

the length of the plater radius of gyration of stiffenert plate thicknesstf stiffener flange thicknessW stiffener stem thicknessw0 magnitude of the maximum out-of-plane deflectionwp magnitude of out-of-plane deflectionx distance along the length of the platey distance across the width of the plateb plate slenderness ratio defined by Eq. (2)sY yield strength

1. Introduction

Thin steel plates that are stabilized in one direction by stiffeners are used exten-sively for plating of ship decks and hulls, components of offshore structures, box-girder bridges, bridge decks and other structures in which a high strength-to-weightratio is important. Flexure of the entire hull of a ship or box girder of a bridgewill induce longitudinal compressive stresses in the stiffened panels that form theseelements. This may be coupled with local bending moments arising from transverseloads acting directly on the stiffened panels, e.g. wheel loads acting on a bridgedeck. Because of the presence of compressive axial forces and bending moments,stiffened plates are susceptible to failure by instability. Instability failure can takevarious forms: buckling of the plate between stiffeners; lateral torsional buckling ofthe stiffeners, also called tripping; overall buckling of the stiffened panel as a column,referred to in the following as Euler buckling, or; local buckling of the stiffener.This last mode of failure is not considered in this study.

Test results [1,2] have indicated that failure by tripping of the stiffener is morecritical than failure by buckling of the plate because it is associated with a suddencollapse. Although plates stiffened on one side have considerable ability to carrytransverse loads that put the flange of the stiffener in tension, tripping of the stiffenermust also be considered in the analysis where the structure is such that bending willcause compression to develop in the flange of the stiffener. For example, initialimperfections favoring tripping in the lower deck of a box-girder cannot be avoided.Similarly, wheel loads acting on alternate panels of the upper deck will induce nega-tive moments (which put the flange of the stiffener in compression) in the unloadedpanels, again predisposing the stiffeners to tripping failure.

153G.Y. Grondin et al. /Journal of Constructional Steel Research 50 (1999) 151–175

Collapse analysis of stiffened plates subjected to axial compression has receivedconsiderable attention [2–5]. A common simplified approach to predict the capacityof stiffened plates has been to treat one stiffener with the associated plate width asa simply supported beam-column [6,7]. Assuming that the ultimate load is reached asthe outer fibers reach yield under compression and bending, a simple Perry-Robertsonformula can be derived. Using this approach, Euler buckling is obtained using acolumn formed of one stiffener and an attached plate width equal to the stiffenerspacing. Similarly, the plate buckling mode of failure is taken into account by reduc-ing the width of the plate associated with each stiffener, from the stiffener spacingto an effective width. The effective width of the plate has been determined eitheranalytically or experimentally [7]. The main advantage of this method is its sim-plicity. This model, however, is not normally suitable for tripping of stiffeners,though it could if satisfactory predictions could be obtained for the limiting stressin the stiffeners. Since the model cannot properly account for interaction betweenthe stiffener and plate in the pre- and post-buckling ranges it has limited applicability.

With current analysis tools and computing power, however, more precise modelingof stiffened steel plates can be achieved. Factors such as residual stresses, initialimperfections and yielding of significant parts of the cross-section can be explicitlyincorporated into numerical models. The finite difference method used by Smith andco-workers [8] included material yielding, plate and stiffener imperfections and localbuckling. That model, however, assumed that tripping of the stiffeners is suppressedby suitable proportioning of the stiffeners. The authors of the current work haveobtained excellent correlation [1] between results of tests on full-size stiffened platesand a numerical model using the finite strain four node shell element S4R fromABAQUS [9]. The same finite element model was used to perform an extensivestudy of the behaviour and strength of stiffened plates under axial in-plane loadingand bending moment. This paper presents the results of the numerical investigation.The parameters that have been investigated are the effects of the magnitude andshape of initial imperfections, residual stresses, plate slenderness, interaction betweenaxial load and out-of-plane bending, plate aspect ratio, and stiffener to plate arearatio.

2. Finite element model

A stiffened plate is typically fabricated from a flat plate with equally spaced longi-tudinal stiffeners and may span between girders. A typical cross-section of the typeof stiffened plate panel considered in this investigation is shown in Fig. 1. Becauseof the symmetry of the stiffened plate, only one panel, namely a portion of the plateof width B with a stiffener centered on the plate strip, was modeled. The stiffenedplate panel was modeled and analyzed using the commercial finite element codeABAQUS. A total of 576 plate bending S4R elements were used to model the stiff-ened panel. The flange and stem of the stiffener were each modeled with 96 elementsand the plate was modeled with 384 elements. This mesh size was found to yieldsatisfactory convergence. The S4R element is a four node shell element that allows


Fig. 1. Stiffened plate panel.

for changes in the thickness as well as finite membrane strains. The model invokedlarge displacement using a Total Lagrangian formulation. The plate material behav-iour was modeled by an elastic-plastic constitutive model incorporating a von Misesyield surface and an isotropic strain-hardening flow rule. The rotation about the longi-tudinal axis was suppressed at all the nodes along the unloaded edges to simulatefull continuity. The loaded ends of the plate panel were constrained so that the planeend sections would remain plane. This type of boundary conditions simulates theframing of the stiffeners into transverse beams or bulkheads.

In all cases studied the residual stresses were first introduced using a procedureto be described in detail later. If required, an initial bending moment was then intro-duced. Finally, in-plane axial compressive forces were applied and increased gradu-ally to failure.

In order to model the full range of behaviour of the panel, including both the pre-buckling and the post-buckling regimes, the solution strategy started with a loadcontrol standard Newton-Raphson iterative procedure in the initial stage of loading,then shifted to a modified Riks procedure as the ultimate load was approached. Thisprocedure permits tracing the behaviour in the softening post-buckling regime [10].

The cross-sectional parameters outlined in Fig. 1 can all vary independently ofeach other over a certain range. In order to keep the various possible combinationsof dimensions to a manageable number, simplifying assumptions had to be made.The panel width,B, the stiffener flange width,bf, and the overall depth of the stiff-ened panel (t 1 h 1 tf) were taken as 500 mm, 100 mm and 125 mm, respectively.The ratio of the stiffener stem thickness to flange thickness was taken as 0.75. Forall the cases investigated, except where indicated otherwise, the ratio of plate areato stiffener area, (Ap/Ast), was taken as 3.0, the length to width aspect ratio,L/B,was taken as 4.0, the compressive residual stress in the plate was taken as 15% ofthe yield strength,sy, and the initial imperfection pattern was taken to be a threehalf wave sinusoidal pattern of average magnitude. Factors investigated such asAp/Ast, residual stresses, initial imperfection and transverse loads will be discussed indetail later. That leaves the plate width to thickness ratioB/t as a primary parameter.


3. Modes of failure of stiffened plates

Stiffened plates in real structures can be loaded under a combination of in-planeand out-of-plane loading. In-plane loads can include axial or biaxial compressionand in-plane shear. Out-of-plane loading would include lateral pressure or bendingabout the transverse and longitudinal axes of the stiffened plate. Although a stiffenedplate panel can be loaded in a number of ways, the work presented in the followingis limited to two different loading conditions. In one, uniaxial compression only wasintroduced in the direction of the stiffeners. In the other, an initial bending aboutthe transverse axis was introduced followed by uniaxial compression. When axialloading is superimposed on bending, an initial bending moment of either 30 percentor 60 percent of the plastic bending moment capacity of the stiffened plate panelcross-section is applied following the application of the residual stresses. These load-ing conditions are common in several civil engineering structures such as the bottomflange of box girders and bridge decks.

Instability of stiffened plates under such loading conditions can take one of threeforms: buckling of the plate between the stiffeners (Fig. 2(a)); lateral torsional buck-ling of the stiffener with rotation of the stiffener about the junction between the plateand the stiffener (this mode of failure is also referred to as tripping of the stiffener)(Fig. 2(b)) and; buckling of the plate between the stiffeners followed by an overall,or “Euler”, buckling (Fig. 2(c)).

For all the cases investigated a tripping failure mode was triggered only when abending moment was applied to induce initial compressive stresses in the flange ofthe stiffener. Tripping of the stiffener was not observed with other loading conditions.Danielson and co-workers [11] demonstrated that tripping of the stiffener is depen-dent on the torsional stiffness of the stiffener. The torsional stiffness of the stiffenerrelative to the stiffness of the plate in bending is believed to be a determining factorin deciding whether, under axial compression, the failure mode will take the formof plate buckling or stiffener tripping. In the work presented here the ratio of stiffenertorsional stiffness to plate bending stiffness is probably too high to make stiffenertripping the governing buckling mode, except when the compression in the plate wasdecreased by the application of bending moment.

The load deformation response of stiffened plates is strongly affected by the modein which the stiffened plate fails. The load versus plate shortening response for stiff-ened plates failing by various failure modes are displayed in Fig. 3. The three stiff-ened plate specimens failing by plate buckling and Euler buckling were loaded inaxial compression whereas the plate that failed by stiffener tripping was loaded undercombined end moments and axial compression. Each of the stiffened plates illustratedin Fig. 3 had a different slenderness,B/t, ratio so that different failure modes couldbe triggered. A comparison of the load deformation behaviour for the modes offailure observed in the analysis indicates that failure by Euler buckling is the mostfavorable mode of failure since it has a more stable post-buckling behaviour. On theother hand, failure by tripping of the stiffener is the least desirable since it is charac-terized by a sudden drop in load capacity just after the peak load is reached.

Although the average applied peak stress in several of the cases investigated was


Fig. 2. Failure modes of stiffened plates.


Fig. 3. Typical load vs. deformation behaviour of stiffened steel plates.

well below the yield stress of the material, the analysis showed that in all cases therewas always some degree of yielding in the cross-section before the peak load wasreached. Localized high strains were reached in slender plates as a result of thesuperposition of residual stresses, applied axial stresses, and local bending stressesintroduced by the axial load acting on the initially deformed shape. Details of themagnitude and distribution of the residual stresses and initial imperfections arepresented in the following sections.

The effective width concept has frequently been used in the design of stiffenedplates to account for the post-buckling strength reduction. After buckling of the platebetween the stiffeners, a portion of the load in the plate shifts towards the stiffener.As a result, a non-uniform stress distribution is formed in the plate with the maximumstress magnitude present in the plate strip immediately adjacent to the stiffener. Atthe time of failure it is assumed that the load is resisted by a strip of material oneither side of the stiffener stressed to the yield level, whereas a central portion ofthe plate between the stiffeners takes no load. This concept has been shown to workwell when applied to plate failure alone. However, in the case of stiffener failurethe behaviour is more complicated because tripping failure is essentially torsionalin nature [11] and will result in interaction between the stiffener and the plate.

The membrane stress distribution across the width of a stiffened plate panel atthe third point along the length of the plate is plotted in Fig. 4 at various stages ofloading. Although residual stresses were included in the model, only the appliedstresses are shown here. In the following the term “applied membrane stresses” willbe used to reflect the removal of residual stresses from the stress picture. Fig. 4illustrates the case where the plate was loaded under combined axial load and aninitial bending moment of 0.3 Mp that placed the flange of the stiffener initially incompression. One observes that the applied bending stresses are initially almost uni-


Fig. 4. Stress distribution in the plate (failure by stiffener tripping).

form across the width of the plate. As the compressive load is increased, the appliedmembrane stresses become gradually non-uniform but remain symmetric about thestiffener. At the peak load the stresses are significantly higher in the vicinity of thestiffener. This phenomenon results from shear lag in the plate and from the partialloss of plate stiffness as the out-of-plane deformations increase. In the post bucklingrange, at an axial shortening of about 1.0 percent of the plate length, the appliedmembrane stresses in the plate are reversed and skewed. This is caused by the suddendrop in axial load at tripping accompanied by large lateral displacements as well astwisting of the stiffener. The flexural component becomes dominant causing stressreversal and the twist introduces warping stresses in the system.

Fig. 5 illustrates the applied membrane stress distribution at various stages ofloading of a panel failing by plate buckling. At the initial stage of loading (designatedas the pre-buckling stage in Fig. 5) the applied stresses are uniform across the platepanel. As the peak load level is approached, however, the portion of the plate nearthe stiffener carries more load. Since the stiffener in this type of failure retains itseffectiveness, as evidenced by the stress distribution in the post-buckling range, thepost-buckling behavior after plate buckling is much more ductile than in the case oftripping failure. The stress distribution in the plate, therefore, does not change charac-ter in the post-buckling range.


Fig. 5. Stress distribution in the plate (failure by plate buckling).

4. Parametric study

4.1. Effect of initial imperfections

Irrespective of the degree of sophistication of the numerical method used, the maindrawback of any procedure attempting to use numerical results as a basis for derivingdesign data remains the uncertainty related to the magnitudes and distributions ofthe initial deformations and residual stresses. Initial imperfections in structures thathave not been subjected to damage usually result from the fabrication process. Alarge number of measurements of post-welding distortions have been reported in theliterature [12]. However, the majority are made on panels which were specificallyproduced for experiments. Most of the panels were not made to full scale, and it isdoubtful whether the welding conditions for the stiffeners resemble those in pro-duction workshops. Although a number of studies on plate distortion were made onactual ship hulls and decks only in very few cases have the distortions been referredto the dimensions of the plates. Furthermore, often only the maximum or centraldeflections have been reported.

In this study the effect of initial imperfections and of residual stresses are con-sidered in turn. Only one type of initial out-of-straightness is accounted for, namely,out-of-plane imperfections in the plate. The effect of stiffener imperfections is notconsidered in this study.

From measurements on 196 plates, Carlsen and Czujko [12] determined that the


deformed shape of welded stiffened plates used in ship structures can be expressedby a double trigonometric series. For simplification in this study, only one term ofthe series is used as follows:

wp(x,y) 5 w0 sin (nppx/L)sin(py/B) (1)

wherex andy are distances along the length and across the width of a plate, respect-ively, L is the length of the plate,B is the width of the plate,w0 is the magnitudeof the maximum out-of-plane deflection, andnp is the number of half sine waves inthe deformed shape of the plate along its length. Carlsen and Czujko [12] found thatthe most critical shape was obtained when only one term of the series was used withnp 5 3 for a plate with an aspect ratio of 3.0.

Using the results of extensive surveys of actual ship deformations, Smith et al.[13] definedw0 as slight (w0 , 0.025b2t), average (w0 , 0.10b2t), and severe (w0

, 0.30b2t) whereb is the plate slenderness ratio:

b 5Bt !

sY

E(2)

In this study the initial imperfections are defined according to Eq. (1) withB, thestiffener spacing, taken as 500 mm. The analysis is performed for a range of slender-ness ratios (b 5 0.63, 1.67 and 2.72) and for initial imperfection magnitudes equal tothe upper limits defined by Smith et al. for slight, average and severe imperfections.

The constant termnp in Eq. (1), the number of half waves along the length of theplate, is varied from 1 to 12. The deformed shape of a stiffened plate panel fornp

equal to 10 is shown in Fig. 6(a). A limited number of other initial imperfectionswere also investigated. In two models (b 5 1.67 and 2.72) randomly distributed out-of-plane ripples were introduced using a random number generator but constrainedto zero at the plate to stiffener intersection as shown in Fig. 6(b). In two other cases(b 5 1.67 and 2.72) the initial imperfection pattern consisted of one single waveplaced diagonally across the plate as shown in Fig. 6(c). The crest of the sine wavefollows a straight line intersecting the stiffener at zero imperfection.

The magnitude of the compressive residual stresses in the plate was taken as 15%of the yield strength. The distribution of residual stresses in the specimen and themethod used to incorporate both the residual stresses and the initial imperfectionsin the model are discussed in the following section. It should be noted also that allthe models investigated above were loaded only with an axial force. Therefore, thetripping mode was excluded.

Given the very large number of analyses performed, only some of those resultswill be discussed in detail. The effect of the magnitude of initial imperfections forone and four half sinusoidal waves along the length of the stiffened plate is presentedin Figs. 7 and 8, respectively, for different plate slenderness ratios. Fig. 7 indicatesthat the magnitude of the initial imperfections has little effect on the stiffened platecapacity when initial imperfections take the form of one half wave. Fig. 8, however,shows that the magnitude of initial imperfections has a very significant effect on the


Fig. 6. Initial imperfections.


Fig. 7. Effect of the magnitude of initial imperfections (one half sine wave).

Fig. 8. Effect of the magnitude of initial imperfections (four half sine waves).

stiffened plate ability to carry load when the initial imperfections take the form offour half waves. A conservative estimate of plate capacities for initial imperfectionsconsisting of two and three half waves can be obtained by linear interpolationbetween the results obtained for one half wave and four half waves.

The mode of failure changes from Euler buckling (Fig. 2(c)) to plate buckling


(Fig. 2(a)) as the plate slenderness increases. The plate slenderness ratio where thischange in mode of failure is observed is indicated in Figs. 7 and 8.

Examination of these figures indicates that the load carrying capacity of the stiff-ened plate decreases as the number of half-waves in the initial imperfection patternincreases. This is observed more readily in Fig. 9 where the ratio of peak load toyield load is plotted as a function of the number of half waves along the length ofa plate panel. Three curves are presented: one for a plate slenderness ratio of 2.72(failure by buckling of the plate with little yielding), one for a plate slenderness ratioof 1.67 (failure by bucking of the plate with significant yielding of the section), anda curve corresponding to a plate slenderness ratio of 0.63 (failure by Euler buckling).Average magnitude initial imperfections were assumed in all cases. Fig. 9 indicatesthat, for plates with a slenderness ratio of 2.72 and 1.67 the critical load is stronglydependent on the shape of the initial imperfections. The minimum critical load isreached when eight half-waves are used for the configuration of the initial imperfec-tions. A reduction of capacity of about 25 percent is observed as the number of half-waves in the initial shapes increases from one to eight. For initial imperfections withone to 12 half-waves the analysis showed that the initial imperfections are amplifieduntil the peak load is reached. After the peak load is reached, typically only twohalf waves amplify while the other half-waves gradually become smaller as the loaddecreases in the post-buckling range. Fig. 2(a) shows the final configuration of astiffened plate that started out with 10 half-waves in its initial configuration. Theeffect of initial imperfections was found to be negligible for a stiffened plate thatfails by Euler buckling (this is the case for the plate with a slenderness ratio,b,of 0.63).

The peak loads obtained for the random initial imperfections and the skewed wave

Fig. 9. Effect of the number of half waves on the load carrying capacity of a stiffened plate.


were found to be somewhere between the peak loads obtained for a plate with fourhalf-waves and a plate with two half waves. It is apparent from Fig. 9 that theprediction of plate capacity based on initial imperfections consisting of three to fourhalf sine waves may be sufficiently conservative. Measurements of post-weld distor-tions of plates of different aspect ratios have indicated that the deformed shape con-sists of two to three half waves along the length of a panel [12]. As the number of halfwaves in the initial deflected shape increases from four to eight a further reduction ofcapacity of as much as 15% is observed. For all the initially deformed shapes investi-gated in this study the post-buckling configuration of the plate consists of two half-waves located near the centre portion of the plate as shown in Fig. 2(a).

In general, this study clearly shows the sensitivity of stiffened plates to initialimperfections. Measurements of initial imperfections by others have indicated thattwo to three half waves along the length of a plate are quite possible. As a conserva-tive estimate, if the actual shape of the initial imperfections is not known, four halfsine waves can be assumed for the shape of the initial imperfections. One shouldkeep in mind that eight half sine waves is highly unlikely and indicate a very severecondition. As a less conservative assumption for the initial imperfections, the follow-ing study uses three half waves to describe the shape of the initial imperfections.The reader can accurately make the necessary corrections for other numbers of halfwaves using Figs. 7–9. When using these figures to estimate the capacity of a stiff-ened plate with a number of half waves other than one or four, Figs. 7 and 8 areused to obtain the capacity for one and four half sine waves, respectively. Thesevalues can then be plotted in Fig. 9 where, using the trend established in the figurefor plate slenderness ratios of 0.63, 1.67, and 2.72, the buckling capacity for thegiven number of half waves can be readily obtained. Alternatively, lines similar tothose shown in Figs. 7 and 8 can be reconstructed from data points obtained alonga vertical line drawn in Fig. 9. The reconstructed curve can be used to estimate thefailure load for slenderness ratios,b, other than those presented in Fig. 9.

4.2. Effect of residual stresses

The presence of residual stresses in stiffened plates is mainly attributable to thewelding of stiffening members to the plate. The residual stresses in the weld and inthe plate and stiffener material in the vicinity of the weld are close to the yield limitas a result of the contraction of the welds. Residual stress measurements in stiffenedplates have shown that the tension zone around the weld extends three to six timesthe plate thickness [1,14,15].

The magnitude and distribution of the residual stresses are governed by weldingparameters such as heat input and cooling rate; both are governed by the weldingprocedure adopted for fabrication. Most test specimens would be fabricated usingmanual welding, such as shielded metal arc welding. In production workshops, how-ever, semi-automatic or automatic welding techniques are more likely to be used.Hence the information on residual stresses in real structures is scarce. For this reason,measurements of longitudinal residual stresses were performed by Grondin et al. [1]on test specimens fabricated using automatic submerged arc welding to weld the


stiffener to the plate simultaneously on both sides of the stem. This welding techniqueis representative of large scale fabrication processes.

A typical residual stress pattern used for the analysis is shown in Fig. 10. Thetension existing at the plate to stiffener junction is balanced by residual compressionwhich exists largely in the plate. It has been reported that the magnitude of thecompressive residual stresses increases as the plate slenderness becomes smaller.Values of up to 75 percent of the yield strength have been measured, although 25percent of the yield strength is believed to be more common for typical ship struc-tures [15]. Because of the relatively high magnitude of residual stresses in stiffenedplates, it is important that residual stresses be included in strength calculations.

The residual stress picture of Fig. 10 was introduced in the finite element modelby imposing initial strains in the form of a temperature distribution. Only longitudinalresidual stresses were investigated in this study. In order to introduce initial strainsonly in the longitudinal direction, an orthotropic temperature material property wasused that had zero thermal expansion coefficients in directions transverse to the axisof the stiffened plate. The initial strains introduce initial stresses, upon which aniteration is carried out in order to establish equilibrium. A complication arisesbecause the introduction of residual stresses in the model creates a distortion of thespecimen which must be considered when investigating the effect of initial imperfec-tions. The application of residual stresses was therefore performed in two stages.First, a set of strains, equal in magnitude but opposite to the desired residual strains,was introduced in the model with the desired initial imperfections. The resultingdisplacements were superimposed on the initial imperfections and a new deformed,stress-free, mesh was thus generated. Initial strains corresponding to the residualstrains were then applied on the newly generated model. At this stage the modelincorporates the desired magnitude and distribution of the initial imperfections andresidual stresses.

The magnitude of the residual stress at the plate to stiffener junction (Fig. 10)was taken as the yield strength of the material in tension. The residual stresses inthe stiffener were assumed to be similar to those measured in the full-size specimenreported by Grondin et al. [1]. The exact distribution, however, varied slightly,depending on the changes required during an equilibrium iteration. As mentioned

Fig. 10. Typical residual stress pattern in a stiffened plate.


before, the compressive residual stresses in the plate were taken to be 15 percent ofthe yield strength of the plate material in the models used to investigate the effectof initial imperfections, out of plane loading, plate aspect ratio and plate to stiffenerarea ratio. The 15 percent level is consistent with the level measured in a full-sizepanel fabricated using automatic submerged dual arc welding with the fillets on eitherside of the stiffener stem laid simultaneously [1].

To investigate the effect of the level of residual stresses, three cases were exam-ined, namely, residual stresses in the plate of zero, 15 and 30 percent of the yieldstrength. Fig. 11 shows the effect of residual stresses on the capacity of stiffenedplates. At plate slenderness ratios larger than about 1.7 the residual stresses in theplate decreases the strength of stiffened plates roughly in direct proportion to themagnitude of the compressive residual stresses in the plate. As the plate slendernessratio where the mode of failure changes from plate buckling to Euler buckling isapproached, the capacity does not decrease proportionally with the magnitude of theresidual stresses in the plate. This is due to a number of reasons. For cases wherefailure is by plate buckling, the uniform residual compressive stresses, present overmost of the plate, govern the behaviour of the stiffened plate. For this later condition,the buckling stress in the presence of residual stresses can be approximated as thedifference between the critical stress when the plate is free of residual stresses andthe magnitude of residual stresses. When Euler buckling is the governing mode offailure, the residual stresses in the plate become less influential and the overall distri-bution rather than the residual stresses in the plate only becomes important. It shouldalso be noted that at plate slenderness ratios,b, smaller than about 1.7, the combinedeffect of the average buckling stress and the residual stress in the plate causes theplate to yield before buckling.

Fig. 11. Effect of residual stresses.


4.3. Effect of out-of-plane loading

The superposition of out-of-plane loads to in-plane loads was investigated bysuperposing equal end moments to the in-plane axial load. Five loading combinationswere investigated, namely, no applied end moments, end moments of 30 percent ofthe plastic moment capacity of the cross-section to place the stiffener initially eitherin tension or in compression, and end moments of 60 percent of the plastic momentcapacity placing the stiffener initially either in tension or in compression. For eachloading condition the plate area to stiffener area ratio,Ap/Ast, was taken as 3.0, theresidual compressive residual stresses in the plate were fixed at 15 percent of theyield strength, and initial imperfections of average amplitude with three half sinewaves along the length of the plate were assumed. The plate slenderness ratio,B/t√sY/E, was varied from about 0.6 to about 2.8. Fig. 12 shows that superimposedbending has two distinct effects, namely, to reduce the load carrying capacity ofstiffened plates and to change the failure mode from one of plate buckling when thebending moment places the flange of the stiffener initially in tension, to stiffenertripping when the bending moment places the flange of the stiffener initially in com-pression.

As might be expected, when tripping is the mode of failure, plate slenderness haslittle to no effect on the load carrying capacity of the stiffened plate. However, whenplate buckling is the mode of failure, the slenderness of the plate has obviously astrong effect on the capacity of the stiffened plate. As the failure mode becomesEuler buckling, the effect of plate slenderness on the ratio of buckling load to yieldload becomes insignificant.

When stiffened plates are subjected to a bending moment that places the flangeof the stiffener in compression, the analysis shows that an increase in the bending

Fig. 12. Effect of initial bending moment and plate aspect ratio.


moment from 0.3 Mp to 0.6 Mp, results in a decrease in capacity of about 0.3 PY.A comparison of the buckling curve for stiffened plates with no bending momentwith that for plates subjected to 0.3 Mp with the stiffener initially in compressionindicates a similar reduction in capacity of 0.3 PY when Euler buckling is the govern-ing failure mode (these are plates with small slenderness ratios). However, whenfailure occurs by plate buckling the change in load carrying capacity as the appliedmoment is increased from zero to 0.3 Mp varies with the plate slenderness ratio andbecomes very small as the plate slenderness approaches 3.0.

Fig. 12 also indicates that stiffened plates are not as sensitive to bending momentswhen the applied bending moment places the flange of the stiffener into tension. Anincrease in moment of 0.3 Mp generally results in a decrease in capacity of 0.12 PY.This strength reduction is observed for all values of plate slenderness.

4.4. Effect of plate aspect ratio

Plates of two aspect ratios,L/B, were investigated, namely, 3.0 and 4.0. The aspectratio was varied by changing the length of the plate. The plates with an aspect ratioof 3.0 were investigated with a superimposed bending moment of 0.3 Mp and 0.6Mp applied to cause initial compression or initial tension in the flange of the stiffener.The initial imperfection shape was taken as three half waves of average magnitude.The results for plate slenderness,b, from 0.6 to 2.8 are presented in Fig. 12. Thefigure indicates only a small change in capacity as the plate aspect ratio changesfrom 4.0 to 3.0. The figure also indicates that, in general, a decrease of plate aspectratio increases the plate capacity when the failure mode is either Euler buckling orstiffener tripping. Insignificant change in capacity is observed when failure occursby plate buckling.

4.5. Effect of plate to stiffener area ratio

The parametric study presented in the preceding sections was performed with aplate to stiffener area ratio of 3.0. Additional cases were investigated where the plateto stiffener area ratio was varied by changing the stiffener area while keeping theoverall depth of the section and its slenderness ratio,L/r, constant. Two load caseswere investigated, namely one where the stiffened plate is subjected to axial loadingand the other where a bending moment is superimposed to the axial load. The super-imposed moment corresponds to 30 percent of the plastic moment capacity andcauses compression in the flange of the stiffener. The plate slenderness ratio,b, usedfor this investigation is 2.09. The initial imperfections are taken as three half wavesof average magnitude. The residual compressive stresses in the plate are assumedto be 15 percent of the yield strength.

Fig. 13 presents a plot of the ratio of peak to yield load versus plate to stiffenerarea ratio. Fig. 13 indicates that the mode of failure of stiffened plates under pureaxial load is plate buckling for all the plate to stiffener area ratios investigated. Inaddition, it is seen that the area ratio has little effect on the load carrying capacityfor area ratios less than about 1.5 and greater than about 4.0. However, for area


Fig. 13. Effect of plate to stiffener area ratio.

ratios between 1.5 and 4.0, the effect of plate to stiffener area ratio is significantand a reduction in capacity of 20 percent is observed over that range.

When stiffened plates are subjected to a bending moment that places the flangeof the stiffener in compression, the mode of failure is either tripping of the stiffener(for plate to stiffener area ratios greater than about 2.5) or Euler buckling. The loadcarrying capacity is found to decrease almost linearly with increasing plate to stiff-ener area ratio. The sensitivity of the stiffened plates to the area ratio is more signifi-cant for this loading case than for the case where no moment is applied to thestiffened plate. This is likely a reflection of the fact that the area ratio was variedby varying the area of the stiffener while keeping the area of the plate constant.Since the tripping and Euler buckling failure modes are controlled by the stiffener,the effect of varying the stiffener area is probably more influential on these twomodes of failure than it is when plate buckling is the governing failure mode.

The above observations appear at first to be in contrast to the observations madeby Carlsen [16]. In the work of Carlsen the plate to stiffener cross-sectional arearatio was varied by changing the area of the plate while keeping its slenderness ratioconstant. Two failure conditions were detected, namely, a plate induced failure anda stiffener induced failure. When the failure mode was stiffener tripping, little effectof the area ratio was detected. This is likely because the area of the stiffener waskept constant. Carlsen also found that the plate to stiffener area ratio was not influen-tial when the mode of failure was plate buckling. This, again is expected since theplate slenderness ratio was kept constant when its area was changed. Consequently,Carlsen concluded that the plate to stiffener cross-sectional area ratio was not sig-nificant.

The work performed by the authors and the work of Carlsen are reconciled if itcan be stated that the cross-sectional area of the stiffened plate component that causes


instability of the stiffened plate is the important parameter rather than the ratio ofthe area of the plate to the area of the stiffener.

5. Comparison with design practice

A large number of numerical analyses were carried out to compare with widelyaccepted design guidelines [17,18]. The numerical investigation included the effectof initial imperfections and residual stresses, which are usually not specificallyincluded in the design procedures. The effect of residual stresses and initial imperfec-tions are normally accounted for through the selection of a plate effective width,obtained from examination of test results. Furthermore, the bulk of the research con-ducted on the behaviour and strength of stiffened steel plates has concentrated onthe elastic behaviour. The numerical analysis presented here has considered bothelastic and inelastic behaviours. An assessment of current design practice is thereforeperformed in light of this new body of data.

The two design guidelines selected for this assessment are Classification note No.30.1 by Det norske Veritas [17] and the American Petroleum Institute Bulletin 2V[18]. Both guidelines present a comprehensive procedure for computation of thebuckling strength of stiffened steel plates whereby the stiffened plate capacity isevaluated based on the various failure modes identified in this paper.

The procedure adopted in the two guidelines uses significantly different plate paneleffective widths. The plate panel effective width adopted by DnV [17] depends onwhether failure of the stiffened plate in induced by the stiffener or by the plate. APIRP 2V [18] does not make this distinction. It does reduce, however, the plate effec-tive width significantly when stiffener failure occurs before the plate material aroundthe stiffeners reaches yield. This reduction was supported by test results and wasrecommended by Faulkner [14]. Fig. 14 illustrates the difference in the ratio of plateeffective width,Be, to full width, B, predicted using API RP 2V and DnV designapproaches. It can be observed that API RP 2V is significantly more conservativethan DnV when stiffener failure occurs before the plate panel yields near the stiff-eners.

A comparison between API, DnV and the numerical analysis results is presentedin Fig. 15. The results of the numerical analysis are presented for two differentshapes and two different magnitudes of initial imperfection, namely, one half andfour half sine waves along the length of the plate and for small and severe magnitudesof initial imperfections. The cross-sectional parameters outlined in Section 2 of thispaper were used. The plate length and width used here are 2000 mm and 500 mm,respectively. Since the current design methods [17,18] are semi-empirical in nature,they account for some level of geometrical imperfections and residual stresses. Thepredictions of stiffened plate capacity all follow the same general trend across theplate slenderness range investigated. As expected from the comparison of plate paneleffective width presented in Fig. 14, API RP 2V provides a more conservative esti-mate of the plate buckling capacity than DnV. In fact, when compared to the resultsof the numerical analysis, API RP 2V provides a conservative strength prediction


Fig. 14. Plate panel effective width.

Fig. 15. Effect of plate slenderness—assessment of design practice.


for the entire range of plate slenderness investigated. On the other hand, strengthpredictions based on DnV exceed those obtained using the numerical analysis in therange of slenderness from 1.1 to 2.5 for plates with severe imperfections and fourhalf waves. It should be noted, however, that the maximum initial imperfections forwhich DnV guidelines are applicable fall in the range of small to medium amplitudeimperfections as defined earlier.

Fig. 16 presents a comparison of the predicted plate capacity under combined axialload and uniform bending. Initial imperfections are assumed to consist of three halfwaves of average magnitude. The uniform bending moments used for this assessmentare 0.3 and 0.6 times the plastic moment capacity of the cross-section. The formu-lation presented in DnV limits the maximum strain in the cross-section to the yieldstrain. API RP 2V allows the section to reach its ultimate capacity, namely the plasticmoment capacity of the section. This has the effect of reducing somewhat the gapin predicted capacity between DnV and API that was observed above. Fig. 16(a)shows the predicted axial load carrying capacity for the case where the bendingmoment is applied to place the plate into compression. A very good agreement isobserved between the DnV guidelines and the numerical analysis results. On theother hand, API RP 2V still shows a very conservative strength prediction at bothmoment levels.

Fig. 16(b) shows the predicted strength when the applied moment places the platepanel initially in tension. For this case, the DnV guidelines are found to be moreconservative than the API guidelines, especially at higher applied moment. It canbe seen that, with an applied moment of 60 percent of the plastic moment capacity,the DnV design procedure would allow little to no axial load. For this case, however,API RP 2V is in excellent agreement with the finite element predictions.

6. Summary and conclusions

The capacity of stiffened plates subjected to combinations of axial compressionand bending was investigated using a finite element model which was demonstratedto be able to predict accurately the capacity of full-size stiffened plates tested undervarious loading conditions. A large displacement formulation with an elastic-plasticisotropic strain-hardening material model was used for the analysis. The parametersthat were investigated are initial imperfections in the plate, residual stresses, endbending moments and plate to stiffener cross-sectional area ratio. The stiffener spac-ing, B, the stiffener flange width,bf, and the overall depth of the stiffened panel (t1 h 1 tf) were taken as 500 mm, 100 mm and 125 mm, respectively. This wasdone to keep the number of variables in the investigation to a reasonable size. Theratio of the stiffener stem thickness to flange thickness was taken as 0.75. For allthe cases investigated, except where indicated otherwise, the ratio of plate area tostiffener area, (Ap/Ast), was taken as 3.0, the plate aspect ratio,L/B, was taken as4.0, the compressive residual stress in the plate was taken as 15% of the yieldstrength, and the initial imperfection pattern was taken to be a three half sine wavesof average magnitude. The plate width to thickness ratioB/t was taken as the primaryparameter and its value was varied to include elastic as well as inelastic behaviour.


Fig. 16. Effect of initial bending—assessment of design practice.


The load versus deformation response of stiffened plates in the post-buckling rangewas found to be strongly affected by the mode of failure of the stiffened plate, withoverall buckling being the most desirable mode of failure and stiffener tripping theleast desirable due to its abrupt reduction in capacity past the peak load.

Both the magnitude and the shape of the initial imperfections were shown to havea significant influence on the capacity of stiffened plates failing by plate buckling.

Although a minimum plate buckling capacity is reached when the initial imperfec-tion configuration consists of eight half sine waves along the length of a panel, it isrecommended that three or four half sine waves be assumed when the exact shapeof the initial imperfections is unknown. Initial imperfections in the plate were foundto have negligible effect on the capacity of plates failure by overall Euler buckling.

Three levels of residual stresses, namely, compressive residual stresses in the plateof zero, 15 and 30 percent of the yield strength were investigated. At plate slender-ness ratios,b, larger than about 1.7 the residual stresses in the plate decreases thestrength roughly in direct proportion to the magnitude of the compressive residualstresses in the plate. However, when yielding sets in before buckling, the effect ofresidual stresses is diminished. When failure of stiffened plates is by overall Eulerbuckling the effect of the residual stresses in the plate is less pronounced.

Out-of-plane loading, applied here in the form of end moments, has a significantimpact on the mode of failure. For the stiffened plate proportions used in this investi-gation it was found that out-of-plane loading causing compression in the flange ofthe stiffener was necessary to trigger stiffener tripping. The sensitivity of stiffenedplates to out-of-plane loading was found to be twice as severe when the out-of-planeloading causes compression in the flange of the stiffener, compared to tension in thestiffener flange.

An investigation of the plate to stiffener area ratio indicates this parameter is notinfluential. Rather, the cross-sectional area of the stiffened plate element that initiatesinstability of the stiffened plate is the important parameter. This observation is con-sistent with earlier observations made by Carlsen [16].

An assessment of current design practice outlined in DnV [17] and API RP 2V [18]indicated that current practice is generally conservative. Although a large number ofcases were investigated in this study, further work is necessary to investigate abroader range of parameters. More specifically, the effect of initial imperfections inthe stiffener and its influence on the tripping behaviour of stiffened plates need tobe investigated.

References

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[2] Murray NW. Buckling of stiffened panels loaded axially and in bending. The Structural Engineer1973;51(8):285–301.

[3] Timoshenko SP, Gere JM. Theory of elastic stability, McGraw-Hill, 1961.[4] Sherbourne AN, Liaw CY, Marsh C. Stiffened plates in uniaxial compression. IABSE 1971;31:145.[5] Balaz I, Murray NW. A comparison of some design rules with results from tests on longitudinally

stiffened deck plates of box-girders. Journal of Constructional Steel Research 1992;23:31–54.


[6] Bonello MA, Chryssanthopoulos MK, Dowling PJ. Ultimate strength design of stiffened plates underaxial compression and bending. Marine Structures 1993;6:533–52.

[7] Carlsen CA. Simplified collapse analysis of stiffened plates. Norwegian Maritime Research1977;4:20–36.

[8] Smith CS. Strength of stiffened plating under combined compression and lateral pressure. Trans-actions of the Royal Institution of Naval Architects 1991;133:131–47.

[9] Hibbit et al. ABAQUS/Standard. Version 5.4, Hibbitt, Karlsson and Sorensen, Inc: Pawtucket, RI,1994.

[10] Riks E. An incremental approach to the solution of snapping and buckling problems. InternationalJournal of Solids and Structures 1979;15:529–51.

[11] Danielson DA, Kihl DP, Hodges DH. Tripping of thin-walled plating stiffeners in axial compression.Thin-Walled Structures 1990;10:121–42.

[12] Carlsen CA, Czujko J. The specification of post-welding distortion tolerances for stiffened plates incompression. The Structural Engineer 1978;56(5):133–41.

[13] Smith CS, et al. Strength and stiffness of ship plating under in-plane compression and tension. TheRoyal Institute of Naval Architects, 1987.

[14] Faulkner D. A review of effective plating for use in the analysis of stiffened plating in bending andcompression. Journal of Ship Research 1975;19(1):1–17.

[15] Faulkner D, Adamchak JC, Snyder GJ, Vetter MF. Synthesis of welded grillages to withstand com-pression and normal loads. Computers and Structures 1973;3:221–46.

[16] Carlsen CA. A parametric study of collapse of stiffened plates in compression. The Structural Engin-eer 1980;58(2):33–40.

[17] Det norske Veritas. Buckling strength analysis of mobile offshore units. Classification Note 30.1,October, 1987.

[18] American Petroleum Institute. Bulletin on design of flat plate structures. API Bulleting 2V, 1st Ed.,May 1, 1987.

buckling of stiffened plates

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