Nonlinear structural behaviour and design formulae for calculating the ultimate strength of stiffened curved plates under axial compression

Thin-Walled Structures 107 (2016) 1–17

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Nonlinear structural behaviour and design formulae for calculating the ultimate strength of stiffened curved plates under axial compression Jung Kwan Seo a, Chan Hee Song c, Joo Shin Park b,n, Jeom Kee Paik a,c a The Korea Ship and Offshore Research Institute (The Lloyd's Register Foundation Research Centre of Excellence), Pusan National University, Busan, Republic of Korea b Central Research Institute, Samsung Heavy Industries Co., Ltd., Geoje, Republic of Korea c Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan, Republic of Korea

art ic l e i nf o

a b s t r a c t

Article history: Received 21 January 2016 Received in revised form 26 April 2016 Accepted 6 May 2016

Cylindrically curved and stiffened plates are often used in ship and offshore structures. For example, they can be found in the cambered decks, fore and aft side shells and circular bilge parts of ships. A number of studies have investigated curved plates in which the buckling/ultimate strength is increased according to the curvature under various loading scenarios and design formulas. However, information regarding the nonlinear structural behaviour and design formulas for calculating the ultimate strength of the stiffened curved plates is currently limited. In this paper, a series of finite element analyses are performed on stiffened curved plates with varying geometric parameters. The existing curvatures are also analysed to clarify the effects of these parameters on the buckling/post-buckling characteristics and collapse behaviour under axial compression. The results are used to derive closed-form expressions to predict the ultimate compressive strength of curved stiffened plates for marine applications. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Cylindrically curved plate Stiffened curved plate Axial compression Ultimate strength Design formula

1. Introduction Thin-walled cylindrical shells are widely used as structural elements, such as for oil and gas storage, offshore structures, cooling towers and ship hulls. It is therefore important to clarify the elastic and plastic stability of cylindrical shells under various loading conditions. In particular, cylindrically curved plates are used extensively in ship structures, such as for cambered deck plating, side shell plating at fore and aft and for circular bilge parts. The structural modelling and investigation of unstiffened and stiffened curved plates can be fundamentally treated as part of a cylinder. To understand the structural behaviour and strength of these curved plates, they should first be subjected to axial compression loading conditions. Then, they can be observed under combined loading conditions (biaxial compression with lateral pressure) specific to certain ships. Studies on the buckling theory of curved panels are not numerous due to the complexity of the problem and to its late application in marine structures compared with land-based structures [1–5]. A brief review follows of recent research related to buckling and ultimate strength behaviour of cylindrically curved n

Corresponding author. E-mail address: [email protected] (J.S. Park).

http://dx.doi.org/10.1016/j.tws.2016.05.003 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

plates and stiffened plates used in design formulae for marine and/ or ship structures. Maeno et al. [6] performed a series of large deflection elastoplastic analyses to investigate the buckling and plastic collapse behaviour of ship bilge strakes, which are unstiffened curved plates subjected to axial compression. Based on the results, a simple formula was derived to calculate buckling and ultimate strength and to simulate the average stress-average strain relationship of the bilge structure under axial compression. It was found that a bilge structure of conventional shape and size when reaching ultimate strength will yield before buckling. Therefore, hard corner elements can be used for bilge parts in the ultimate hull girder strength evaluation applying Smith's method, and the effects of buckling of bilge parts should be accounted for in addition to ultimate strength to provide comparative estimates in the post-ultimate strength range. Yumura et al. [7] investigated the buckling and plastic collapse behaviour of cylindrically curved plates under axial loading. They performed a series of elastic eigenvalue analyses while changing the curvature of the plate to clarify the fundamentals of its elastic buckling behaviour. Park et al. [8] performed non-linear finite element model (FEM) analyses using a commercial program for the actual stiffened curved plates of a container ship while varying curvature and stiffener spacing. In the analysis, initial shape imperfections and residual stresses were considered and combined axial compression and hydrostatic pressure loads were applied. Kwen et al. [9] performed non-linear FEM analyses using a

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Table 1 Finite element analysis meshing and number of elements. Part

Number of elements

Plate 10 Stiffener Web 6 Stiffener Flange 4

Table 2 Material properties for stiffened curved plate. Material

High Tensile Steel

Elastic Modulus (E) 205.8 GPa Poisson’ Ratio (v) 0.3 Yield Stress ( σy ) 352.8 MPa

Fig. 1. FEM of a stiffened curved plate (Double-span & double-bay).

commercial program for unstiffened curved plates, varying the aspect ratio, slenderness ratio and curvature according to load conditions such as longitudinal compression, transverse compression and shear load. Based on the analysis, a simple formula was proposed to predict ultimate strength and then the calculated results were compared with those predicted by the DNV buckling formulae with plasticity correction. Cho et al. [10] performed both ultimate strength tests and nonlinear finite element analyses on six stiffened curved plates under axial compression. The numerical predictions were compared with the results, and experimental and numerical information regarding curvature effects and collapse patterns under axial compression were given. Recently, Park et al. [11–13] clarified the buckling, post-buckling and collapse behaviour of unstiffened and stiffened cylindrically curved plates compared with that of a circular cylinder under axial loading. They performed a series of elastic and elastoplastic large deflection analyses with variations in the curvature, slenderness ratio, aspect ratio, web height, initial imperfection and

Fig. 3. Elastic buckling strength of stiffened curved plates subjected to axial compression (tee-bar stiffener).

stiffener type. A simple formula incorporating the effects of several parameters was proposed to predict ultimate strength, which was then compared with the predictions of the classification society buckling formulae [14]. In addition, they investigated secondary buckling behaviour for all cases using the arc-length method. A large container ship has a greater variety of degrees of curvature in the bottom bilge strake than other ship types, because a sharp hull form is required for higher speeds at sea. Therefore, the curvatures and the slenderness ratio should both be changed with

Fig. 2. Typical shapes of curved plate and stiffener combinations.

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Fig. 4. Typical buckling modes of stiffened curved plates.

a new hull form. Hence, calculation of their capacity as a strength member compared with other ship types should be considered. At present, the classification society has guidelines for evaluating the buckling strength of a ship's curved plate but these do not seem to reflect the effects of curvature. In contrast to design formulae for predicting the ultimate strength of stiffened and unstiffened cylindrically curved plates, related studies on buckling and ultimate strength behaviour are limited, at least for marine loading intensities and geometries, although some useful formulations for the ultimate compressive strength of unstiffened plates and stiffened panels can be found in the literature [15–18]. In the present paper, to clarify and examine the fundamental buckling and post-buckling behaviour of stiffened curved plates, a series of elastic and elasto-plastic large deflection analyses and elastic eigenvalue analyses are performed. On the basis of the calculated results, the influences of curvature (θ ), slenderness ratio ( β ), web height ( h w ) and stiffener type (flat, angle and tee bar) on the buckling characteristics and ultimate strength behaviour of stiffened curved plates under axial compression are discussed. The aim of the present study is to derive closed-form expressions to predict the ultimate compressive strength of stiffened curved plates used in marine applications.

2. Stiffened curved plate model and method of analysis 2.1. FEM analysis model Fig. 1 shows the finite element model of a continuous stiffened curved plate in a typical large container ship. As shown by the dotted lines in Fig. 1, a FEM covering a half span on both sides of the transverse frame and a half bay on both sides of a longitudinal stiffener is used. This model is called a double-span and double-bay model [19]. Along each edge of the dotted line, symmetry conditions are imposed and the in-plane displacement perpendicular to the edge is set to be uniform to account for the continuity of plating. The transverse frame is assumed to be flexurally rigid but torsionally weak and therefore simply supported boundary conditions are imposed. The geometry of stiffeners is classified into the three types indicated in Fig. 2. The height and thickness of the stiffener web are h w and t w , respectively. The width and thickness of the stiffener flange are bf and t f , respectively. The adequacy of the finite element mesh adopted in the model was investigated using variations in the element size [13]. Good simulation results were obtained with the use of the following approximate element numbers: 10 for the plate, 6 for the web and 4 for the flange elements (Table 1).

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Fig. 5. Comparison of the average stress and average strain curves at varying flank angles of a curved plate with flat-bar stiffening (hw ¼ 400 mm) under axial compression.

The material is assumed to be elastic perfectly plastic, and the strain hardening is set at zero. The isotropic hardening law is assumed, using von Mises's yield condition. The mechanical properties of the material are shown in Table 2. 2.2. Initial imperfections To fabricate the stiffened cylindrically curved plates for ship structures, fillet welding is usually performed to attach stiffeners to the plating, and thus post-weld initial imperfections such as initial deflections and residual stresses develop in the plating and stiffeners. In advanced ship structural design, load-carrying capacity calculations of ship plating are performed to consider the influences of post-weld initial deflections [20,21]. The characteristics of the post-weld initial imperfections are not completely clear, however, as it is difficult to define the real magnitude and shape of the initial imperfection [22,23]. In the present paper, the magnitude of the initial imperfection proposed by Smith [24] is used.

Smith suggested the following maximum values of representative initial deflections wopl and wosx for stiffened flat plates in merchant vessel structures.

Plate :

wopl t

Stiffener:

⎧ 0.025β 2 for slight level⎫ ⎪ ⎪ ⎪ ⎪ = ⎨ 0.1β 2 for average level ⎬ ⎪ ⎪ 2 ⎪ ⎩ 0.3β for severe level ⎪ ⎭

⎧ 0.00025 for slight level ⎫ ⎪ wosx ⎪ = ⎨ 0.0015 for average level⎬ a ⎪ ⎪ ⎩ 0.0046 for severe level ⎭

(1)

(2)

where t and a are the thickness and the length of the panel, respectively, and β = b /t (σy/E ) is the slenderness ratio of the local panel with a yielding stress of σy . Using wopl and wosx in (Eqs. (1) and 2), initial deflection of the following form is assumed.

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The initial deflection of the stiffener is assumed to be

wosx = sin

πx a

(4)

which corresponds to the slight level in Eq. (2). The influence of welding residual stress is not considered in the present paper. For the elastic and elastoplastic large deflection analyses, the FEM code ANSYS [25] is used with the shell-181 element, which is quite suitable for analysing thin to moderately thick shell structures. The element is a four-node Reissner-Mindlin shell element with six degrees of freedom at each node, and is well suited for large strain nonlinear applications. The arc-length method is applied in conjunction with the modified NewtonRaphson method in both standard and modified forms.

3. Eigenvalue buckling analysis and results

Fig. 6. Change of deflection mode of the stiffened curved plate with a tee bar in flank angle 45° (hw¼ 400 mm) under axial compression.

wopl =

∑ ∑ A omn sin m

n

mπx nπ y sin a b

(3)

where m and n represent the number of half-waves in the x and y direction, respectively. The shape of the initial deflection in the local panel is assumed to follow a buckling mode. Magnitudes of Aomn are determined based on the condition that the maximum magnitude of the initial deflection is equal to wopl = 0.025β 2t corresponding to the slight level in Eq. (1).

A series of eigenvalue buckling analyses are carried out to evaluate buckling strength and examine the significant buckling mode [26]. The latter result is used to produce initial deflection. Fig. 3 shows the influence of the web height of the stiffener on elastic buckling strength. The web height varies from 50 to 400 mm in 50 mm increments, and flank angles of 5°, 10°, 20°, 30° and 45° are considered. When the flank angle increases, elastic buckling stress occurs and varies according to the web height; however, where the flank angle is below 10°, there are no effects on elastic buckling stress as web height changes. Fig. 4 indicates the buckling modes for several significant cases. Fig. 4(a) shows the web buckling mode, which results in stiffener induced failure. This type of failure generally occurs when the web is high. Fig. 4(b) shows local panel buckling mode, the most common buckling collapse mode when actual structures are subjected to axial loading. The buckling shown in Fig. 4(c) typically occurs when the stiffeners are relatively weak, and that in Fig. 4 (d) when the stiffeners have high torsional rigidity.

4. Buckling and post-buckling behaviour To investigate the fundamental buckling and post-buckling

Fig. 7. Comparison of the average stress and slenderness ratio of a curved plate with angle and tee-bar stiffening under axial compression.

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Fig. 8. Comparison of the average stress and slenderness ratio of a curved plate with angle-bar stiffening varying in web height from 200 to 400 mm under axial compression.

behaviour of stiffened curved plates under an axial compressive load, the plate thickness was varied from 12 to 26 mm, the web height was 150–400 mm per 50 mm, the flank angles were 5°, 10°, 20°, 30° and 45° and the stiffener types were flat, angle and tee bar. The plate thickness was set at 15 mm, the web thickness at 12 mm and the flange thickness at 15 mm. Fig. 5(a) shows the elastic large deflection behaviour of the stiffened curved plate with flat-bar stiffening and a varying flank angle under axial compression. As the flank angle increases, the elastically calculated buckling strength increases, and it exceeds the yielding stress of the material when the flank angle is 20°. With a flank angle of 45°, almost linear behaviour is observed in the analyses under axial compression. Fig. 5(b) and (c) show the results of analyses for a curved plate stiffened with an angle bar and a tee-bar, respectively, with a flank angle varying from 5° to 45°. With the increase in the flank angle, the elastically calculated buckling strength also increases relative to the previous results. In particular, when the flank angle is 20°, very complicated behaviour is observed to be induced by secondary buckling, accompanied by a change of mode deflection, as shown in Fig. 5(b). The mode generally changes to a higher order mode as shown in Fig. 6. The unloading path along which both average stress and average strain decreases is an unstable equilibrium and physically does not exist. The actual behaviour is dynamic, and a snapthrough takes place from the stress pick point to the following stable equilibrium path. In all cases, except with a flank angle of 5°, secondary buckling takes place accompanied by unstable behaviour, as shown in Fig. 5(c).

5. Collapse behaviour 5.1. Influence of slenderness ratio

Fig. 9. Comparison of overall buckling collapse mode and local panel collapse mode of a curved plate with tee-bar stiffening under axial compression (amplification factor 25).

A series of elasto-plastic large deflection analyses were performed to examine the influence of the slenderness ratio, flank angle, stiffener height and stiffener type on the ultimate compressive strength of stiffened curved plates. The calculated ultimate strength for angle-bar and tee-bar stiffeners, respectively, are plotted against the slenderness ratio in Fig. 7(a) and (b).

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Fig. 10. Comparison of average stress and average strain curves of a curved plate with angle-bar stiffening and varying slenderness ratios under axial compression.

Web height varied from 150 to 400 mm and flank angle from 5° to 45°. As the flank angle increases, there is a slight increase in ultimate strength in the case of a slenderness ratio below 2.0. However, with an increased slenderness ratio, a big difference is observed between flank angles of 5° and 45 °. Load-carrying capacity also increases slightly with an increase in web height. In general, reducing the slenderness ratio is expected to increase ultimate strength and buckling strength. However, with a web height of 400 mm and a flank angle of 30° and 45°, ultimate strength is reduced with a change in slenderness ratio, as shown in Fig. 7(b). This was also observed in some exceptional cases, such as a slenderness ratio of 3.5. A higher slenderness ratio with a 20° flank angle tends to increase the capacity, which is close to the behaviour of a cylindrical column due to its small web height. In cases with a large web height, the capacity slightly decreases as a result of the plate behaviour. In these results, the distribution of the ultimate strength is sharply divided into two domains by a flank angle below 10° and a flank angle above 30°. A flank angle of 20° is in the middle of the domain, and it can resemble the plate and cylinder behaviour

depending on the web height and slenderness ratio. Fig. 8(a) shows the relationship between the average stress and slenderness ratio with an angle-bar stiffener and a web height of 150 mm, where flank angle varies from 5° to 45°. When the slenderness ratio is 2.3, all cases show severely reduced ultimate strength. Because buckling collapse in these cases induces simultaneous stiffener failure, there is a rapid reduction of in-plane rigidity. Fig. 8(b) shows the elasto-plastic large deflection behaviour of a stiffened curved plate with angle-bar stiffening, two slenderness ratios and a varying flank angle. When buckling collapse occurs it induces stiffener failure on the top, abruptly decreasing the load-carrying capacity with an increase in panel deflection. Fig. 9 shows a comparison of overall buckling collapse mode with local panel collapse mode of a stiffened curved plate with tee-bar stiffening, a web height of 150 mm and a flank angle between 5° and 10° under axial compression. At a slenderness ratio of 1.59 and a flank angle of 10°, ultimate strength is lower compared with 5°. This can be explained by different collapse modes,

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Fig. 11. Comparison of average stress and average strain curves of a curved plate with flat-bar stiffening (hw ¼150 mm, 250 mm) and varying slenderness ratios under axial compression.

Fig. 12. Average stress and slenderness ratios of a curved plate with tee-bar stiffening and a local curved plate under axial compression.

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as shown in Fig. 9. When an axial force is applied to the stiffener, overall collapse is induced by stiffener failure, and load-carrying capacity is simultaneously reduced despite the increased flank angle. Fig. 10(a)–(d) shows average stress and average strain relationships for curved plates with angle-bar stiffening at a web height of 300 mm. For all slenderness ratios, the average stress and average strain curves are smooth and show a simple path compared to those attained by elastic large deflection analyses. This is because deflection does not follow yielding of the elastoplastic behaviour, thus avoiding the mode change observed in elastic large deflection behaviour. In almost all cases, except with a flank angle of 45°, the curved panel undergoes local buckling in three half-waves in the loading direction in one panel. In many cases with larger flank angles, large torsional deformation is observed. In these cases, rapid reduction of ultimate capacity is observed in the post-ultimate strength range. Collapse behaviour is observed in most cases with various combinations of slenderness ratio, flank angle and stiffener type. The results of behaviour were expected to show meaningful behaviour with slenderness ratio, angle and stiffener type. However, in two of them (slenderness ratios of 1.97 and 3.45), overall buckling induced stiffener failure. Fig. 11 (a) shows the case when overall buckling takes place in the stiffened curved plate with a

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Fig. 14. Comparison of average stress and slenderness ratios with varying web height at a flank angle of 5° under axial compression.

Fig. 13. Average stress and slenderness ratios of a curved plate with flat-bar stiffening and a local curved plate under axial compression.

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Fig. 15. Average stress and slenderness ratios of a curved plate with flat-bar stiffening at varying web heights under axial compression.

flat-bar stiffener with a height of 150 mm. When the flank angle is 5°, 10° and 20°, the primary buckling in overall collapse mode is columnar and the local panel does not undergo local buckling. In such cases, both average stress and average strain decrease beyond the ultimate strength, which is the buckling strength. This decreasing path is an unstable equilibrium path which does not exist physically. In actual behaviour, snap-through is expected at the maximum average stress point. When the flank angle is 30°, yielding occurs first, and then overall buckling, at which point overall load-carrying capacity decreases abruptly. Fig. 11 (b) indicates another exception in the form of the results for a stiffened curved plate with a flat-bar stiffener of hw ¼250 mm and a slenderness ratio of 3.45. It can be explained by observing the mode shape and buckling half-wave number before and after distribution of stresses and deformation shapes in the FE analysis results. When the flank angle is 10°, the collapse mode is five halfwaves. In this case, the local buckling of three half-waves takes place, and the secondary buckling of five half-waves occurs near the maximum stress point.

5.2. Influence of stiffeners Fig. 12 shows the average stress and slenderness ratios of a curved plate with tee-bar stiffening to clarify the influence of the stiffeners. Symbols indicate flank angles ranging from 5° to 45°, while dotted and solid lines show finite element analysis results for the local and stiffened curved plate, respectively. It is seen that with a tee-bar stiffener ultimate strength is distributed in two distinct domains: flank angles below 10° and above 20°. At a flank angle below 10°, the estimated ultimate strength of the stiffened curved plate is below that of a local curved plate 18–21 mm thick. However, with a flank angle of over 20°, ultimate strength is 20% higher. The ultimate strength of an unstiffened curved plate increases uniformly as the slenderness ratio decreases, but local curved plates with stiffeners do not appear to have this characteristic, because tee-bar stiffeners can create greater resistance than flatbar stiffeners to longitudinal compressive stress. In general, stiffened plates have increased stiffness compared to local plate structure when used as part of the deck, side shell or bottom of a

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compression it shows greatly reduced ultimate strength compared to other collapse modes. The lowest ultimate strength among the stiffened models is shown at a web height of 250 mm and a flank angle of 5°. One explanation for the above result is that the flat-bar stiffener has relatively little torsional rigidity and is thus susceptible to tripping or local buckling; when it starts to yield, overall buckling collapse finally occurs. This collapse mode is called SIF (stiffener induced failure). In such cases, one should be careful to design the stiffened curved plate such that tripping or local web buckling of the stiffener does not occur prior to bucking of the plating between stiffeners. Changing the stiffener shape clearly produces very different elasto-plastic behaviour. One major difference is that the flange resists rotation of the web in the case of angle bars under axial compression. In particular, with a flat-bar flank angle of 10° snap-back behaviour takes place and in-plane rigidity decrease abruptly after ultimate strength. With a flank angle below 10°, the ultimate strength is higher for a curved plate without stiffening compared to flat-bar stiffening, but in the other cases flat-bar stiffening increases ultimate strength. With flat-bar stiffening, web buckling and tripping easily occur on the web plate under axial compression, and the decision is mainly about ultimate strength and collapse pattern. Fig. 16. Average stress and average strain of a curved plate with varying slenderness ratios and flat-bar stiffening (hw ¼400 mm) at a flank angle of 20°.

Fig. 17. Change in deformation shape and spread of equivalent stress of a curved plate with flat-bar stiffening (CASE-3, hw ¼ 400 mm).

ship. Fig. 13 shows the average stress and average strain for a curved plate with a flank angle ranging from 5° to 45°; solid lines indicate flat-bar stiffening; dotted lines indicate no stiffening. At a plate thickness of 18 mm the curved plate with flat-bar stiffening shows the lowest ultimate strength compared to other conditions. The characteristic of collapse mode appears as a single web collapse induced by buckling. If a stiffener collapses by buckling under axial

5.3. Influence of web height To examine the influence of changing the section modulus of the web, elasto-plastic large deflection analysis was carried out for a stiffened curved plate of varying slenderness ratios under axial compression. Fig. 14 shows average stress and slenderness ratios of a curved plate with flat-bar stiffening at varying web heights and a flank angle of 5°. At a slenderness ratio of 1.95, overall buckling collapse due to stiffener induced failure (SIF) occurs in all cases except at a web height of 300 mm. At a web height of 150 mm, overall buckling accompanied by SIF [20] occurs simultaneously, abruptly reducing load-carrying capacity. Fig. 15 shows the average stress and slenderness ratios of a curved plate with flat-bar stiffening and varying flank angles. At a web height of 150 mm and a flank angle of 10°, overall buckling collapse occurs as a result of SIF at slenderness ratios from 1.59 to 2.75, and the ultimate strength decreases more rapidly than under other conditions. With a flank angle of 20°, overall buckling collapse occurs as a result of SIF at slenderness ratios from 2.3 to 2.7. A flank angle between 20° and 30° with a web height of 250 mm gives the highest ultimate strength across the slenderness ratios. Despite an increase in the web height, the ultimate strength is lower at the 300-mm web height. In these results, the distribution of ultimate strength is also divided into two domains by a flank angle below 10° and a flank angle above 30°. A flank angle of 20° is the middle of the domain, and it sometimes resembles the plate and cylinder behaviour depending on the web height and slenderness ratio. A flank angle of less than 20° yielded plate behaviour with increasing web height. According to previous results, with a web height of 400 mm and a flank angle of 20°, the ultimate strength is mostly lower at various slenderness ratios. To understand this behaviour, the phenomenon is investigated using a detailed FE simulation, as shown in Fig. 16. In general, this behaviour is seen because the web member buckles with increasing lateral deflection and the plate member does not deflect the buckling phenomenon. At the structural design stage, therefore, the use of flat-bar stiffening requires an examination of the collapse mode characteristics according to changes in web height under axial compression. Figs. 16 and 17 show that the ultimate strength is slightly better for CASE3 and CASE-4 than for CASE-5. The ultimate strength of these

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Fig. 18. Average stress and slenderness ratios of a stiffened curved plate with varying flank angles under axial compression.

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5.4. Influence of curvature Influence of curvature of the ultimate strength of a stiffened curved plate is investigated by varying the web height from 250 to 400 mm, flank angle from 5° to 45° and slenderness ratio from 1.59 to 3.45 as shown in Fig. 18. At a flank angle of 5°, ultimate strength at various slenderness ratios is slightly lower than with no flank angle, while ultimate strength remains similar with increasing web height. In previous research, we confirmed that a curved plate with a flank angle of 5° increased ultimate strength by about 10% compared with a flat plate [8]. At a flank angle above 20°, the increase in ultimate strength is conspicuous as the slenderness ratio increases. This range of flank angle is used extensively in the bilge structures of bulk carriers and tankers. Comparisons show that increasing the curvature of a stiffened curved plate does not always increase ultimate strength and buckling strength. However, a flank angle of less than 10° appears to consistently improve the ultimate strength of a stiffened plate.

6. Development of design formula based on FE results Fig. 19. Accuracy of the empirical formulae for curved plate structures stiffened with tee bars.

models gradually increases with increased plate thickness; however, this increase does not occur as a result of snap-back behaviour induced by secondary buckling accompanied by a change in the deflection mode. Fig. 17 shows the deflection shape and spread of equivalent stress at ultimate strength at a slenderness ratio of 2.3. As shown by the average stress and average strain curves, the 1st and 2nd deflection is mainly induced by buckling of the web; therefore, there is an abrupt increase in lateral deflection and the yielding area is extended. As secondary buckling behaviour occurs, the deflection mode is changed by tripping collapse on the top edge of the web, and this post-ultimate strength behaviour rapidly reduces load-carrying capacity.

Most design rules of classification societies calculate the approximate inelastic buckling strength of curved plates by applying a correction to the elastic buckling strength using the so called Johnson-Ostenfeld formula [27]. This approach tends to underestimate the buckling strength for uni-axial compression. Therefore, a new rational formula must be developed to predict the buckling and ultimate strength for the characteristics of thinwalled shells with curvature. In this study, design formulae were proposed to estimate the ultimate strength of a stiffened curved plate under a compressive axial load. The formulae are derived from a numerical database of 150 cases (see Appendix) and the applied curved plate slenderness ratios are as follows. The slenderness ratio β is the square root of the ratio of yield strength σy to elastic buckling strength σE , as found using the following equation [28]:

β=

b σy t E

Fig. 20. The mean, bias and COV (coefficient of variation) for the empirical formulae in (Eqs. (13) and 14) for curved plate structures stiffened with tee bars.

(5)

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σY σEflat

7. Concluding remarks

12(1 − ν 2) kπ 2

=β

(6)

This gives the definition of the slenderness parameter β′ as

β′ = β

σEflat σECurved

(7)

The elastic buckling strength of a curved plate under axial compressive load σECurved , as found from the following equation [18], is

σECurved = k

CB =

2

⎛t⎞ π 2E ⎜ ⎟ × C B 12(1 − ν 2) ⎝ b ⎠

(1 − ν 2) × θ ×

β σY /E

(8)

× CR

(9)

⎛ R ⎞−0.751 CR = 0.0323⎜ ⎟ × CJ ⎝t⎠

(10)

⎧ 1.0 for β < 2.0 at0≤ θ ≤ 10 CJ = ⎨ ⎩ 0.88 for β ≥ 2.0

(11)

⎧ 1.0 for β < 2.0 CJ = ⎨ at 10 ≤ θ ≤ 30 ⎩ 1.1 for β ≥ 2.0

(12)

The design formulae for stiffened curved plates are derived by the least squares method based on the FE results. The accuracy of the present formulae plotted against the curved plate slenderness ratio was checked by comparison with FE results for a flank angle of 5° and a range of 10–45° (Fig. 19). The design formulae for a stiffened curved plate are:

σu = σYeq σu = σYeq

1 0.0683 + 0.994β′ + 0.0065β′2

(θ ≤ 5°)

1 1.1339 − 0.1291β′ + 0.0392β′2

(13) (θ > 5°) (14)

Fig. 20 indicates the mean, bias and COV (coefficient of variation) for the design formulae in (Eqs. (13) and 14). The mean, bias and COV in the empirical formula of a curved stiffened plate subjected to longitudinal compression when compared to the finite element analyses were 0.002, 0.012 and 0.057 for under 5° and 0.001, 0.006 and 0.118 for over 5°, respectively.

As a first step towards achieving and understating of ultimate strength of stiffened curved plates, a dimensional analysis was performed to identify the parameters that characterise the behaviour and strength of stiffened curved plates. The selected parameters were slenderness ratio, curvature, web height and stiffener shapes. The selected parameter set was found to predict the behaviour and strength of stiffened curved plates for different scales of the model. A series of elastic and elastoplastic large deflection analyses were then carried out on over 400 cases of stiffened curved plates under axial compression. The following conclusions can be drawn from the numerical investigation. 1) Stiffener induced failure mode is a potentially severe failure mode in stiffened curved plate structures, resulting in a decrease in ultimate strength and an abrupt loss of in-plane rigidity. 2) An increase in flank angle increases the elastic buckling strength, and at 20 degrees it exceeds the yielding stress of the material. 3) In general, the type of stiffener does not affect the ultimate strength in a stiffened curved plate, but it causes the collapse pattern to occur in different modes. With flat-bar stiffeners, failure is induced by the stiffener, whereas with angle-bars and tee-bars, failure occurs from buckling of the plate and local stiffener, respectively. However, in some cases with a flank angle of 20 degrees, complex behaviour was observed near the plate and/or cylinder depending on web height and slenderness ratio. Therefore, the distribution of the ultimate strength is sharply divided into two domains by the flank angle: below 10 degrees and above 30 degrees. 4) A stiffened curved plate attached to a tee-bar stiffener at a flank angle of 5 degrees has slightly lower ultimate strength at various slenderness ratios compared to no flank angle, and the ultimate strength remains similar with as the web height increases. When the flange is considered, the same distribution pattern of ultimate strength appears at varying web heights. 5) A simple formula developed for curved plates could reasonably be used to estimate the ultimate strength under longitudinal loading conditions. Good correlation with the ultimate strength is observed with the application of the proposed empirical formula.

Appendix See Appendix Table A1.

Table A1 Structural dimensions, properties and FEA solutions. Id

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stiffener type

Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee

bar bar bar bar bar bar bar bar bar bar bar bar bar bar

θ

0 0 0 0 0 0 0 0 0 0 0 0 0 0

a

b

t

hw

tw

bf

tf

σYp

σYs

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

(MPa)

(MPa)

3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

26 21 18 15 12 26 21 18 15 12 26 21 18 15

150 150 150 150 150 200 200 200 200 200 250 250 250 250

12 12 12 12 12 12 12 12 12 12 12 12 12 12

100 100 100 100 100 100 100 100 100 100 100 100 100 100

15 15 15 15 15 15 15 15 15 15 15 15 15 15

352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8

352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8

λ

β

β′

FEA

(σu/σYeq) 0.900 0.848 0.811 0.770 0.725 0.681 0.639 0.612 0.581 0.548 0.543 0.509 0.488 0.464

1.590 1.971 2.300 2.761 3.452 1.591 1.969 2.292 2.764 3.456 1.591 1.973 2.293 2.756

1.590 1.971 2.300 2.761 3.452 1.591 1.969 2.292 2.764 3.456 1.591 1.973 2.293 2.756

0.958 0.890 0.799 0.720 0.589 0.986 0.898 0.811 0.712 0.681 0.984 0.899 0.829 0.733

J.K. Seo et al. / Thin-Walled Structures 107 (2016) 1–17

15

Table A1 (continued ) Id

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

Stiffener type

Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee

bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar

θ

0 0 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 20 20 20

a

b

t

hw

tw

bf

tf

σYp

σYs

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

(MPa)

(MPa)

3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18

250 300 300 300 300 300 400 400 400 400 400 150 150 150 150 150 200 200 200 200 200 250 250 250 250 250 300 300 300 300 300 400 400 400 400 400 150 150 150 150 150 200 200 200 200 200 250 250 250 250 250 300 300 300 300 300 400 400 400 400 400 150 150 150 150 150 200 200 200 200 200 250 250 250

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8

352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8

λ

β

β′

FEA

(σu/σYeq) 0.439 0.449 0.422 0.404 0.385 0.366 0.330 0.311 0.299 0.286 0.273 0.900 0.848 0.811 0.770 0.725 0.681 0.639 0.612 0.581 0.548 0.543 0.509 0.488 0.464 0.439 0.449 0.422 0.404 0.385 0.366 0.330 0.311 0.299 0.286 0.273 0.900 0.848 0.811 0.770 0.725 0.681 0.639 0.612 0.581 0.548 0.543 0.509 0.488 0.464 0.439 0.449 0.422 0.404 0.385 0.366 0.330 0.311 0.299 0.286 0.273 0.900 0.848 0.811 0.770 0.725 0.681 0.639 0.612 0.581 0.548 0.543 0.509 0.488

3.453 1.586 1.969 2.303 2.763 3.450 1.586 1.969 2.303 2.763 3.450 1.590 1.971 2.300 2.761 3.452 1.591 1.969 2.292 2.764 3.456 1.591 1.973 2.293 2.756 3.453 1.586 1.969 2.303 2.763 3.450 1.586 1.969 2.303 2.763 3.450 1.590 1.971 2.300 2.761 3.452 1.591 1.969 2.292 2.764 3.456 1.591 1.973 2.293 2.756 3.453 1.586 1.969 2.303 2.763 3.450 1.586 1.969 2.303 2.763 3.450 1.590 1.971 2.300 2.761 3.452 1.591 1.969 2.292 2.764 3.456 1.591 1.973 2.293

3.453 1.586 1.969 2.303 2.763 3.450 1.586 1.969 2.303 2.763 3.450 6.852 8.269 8.879 10.420 12.668 6.855 8.260 8.853 10.429 12.683 6.855 8.277 8.857 10.401 12.674 6.838 8.260 8.891 10.425 12.662 6.838 8.260 8.891 10.425 12.662 3.735 4.507 4.840 5.680 6.905 3.736 4.502 4.825 5.685 6.913 3.737 4.512 4.828 5.669 6.908 3.727 4.502 4.846 5.682 6.902 3.727 4.502 4.846 5.682 6.902 1.821 2.197 2.638 3.096 3.764 1.822 2.195 2.630 3.099 3.768 1.822 2.200 2.631

0.620 0.992 0.901 0.828 0.759 0.579 0.996 0.889 0.826 0.754 0.579 0.955 0.880 0.784 0.702 0.630 0.974 0.884 0.790 0.703 0.646 0.953 0.933 0.814 0.786 0.654 0.977 0.902 0.810 0.745 0.661 0.973 0.902 0.819 0.814 0.662 0.931 0.926 0.844 0.737 0.658 0.992 0.927 0.846 0.743 0.702 0.968 0.935 0.856 0.823 0.677 0.988 0.938 0.828 0.768 0.684 0.996 0.964 0.845 0.793 0.688 0.965 0.962 0.950 0.905 0.822 0.995 0.978 0.951 0.907 0.876 0.985 0.976 0.952

16

J.K. Seo et al. / Thin-Walled Structures 107 (2016) 1–17

Table A1 (continued ) Id

89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

Stiffener type

Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee Tee

bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar bar

θ

20 20 20 20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45

a

b

t

hw

tw

bf

tf

σYp

σYs

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

(MPa)

(MPa)

3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12 26 21 18 15 12

250 250 300 300 300 300 300 400 400 400 400 400 150 150 150 150 150 200 200 200 200 200 250 250 250 250 250 300 300 300 300 300 400 400 400 400 400 150 150 150 150 150 200 200 200 200 200 250 250 250 250 250 300 300 300 300 300 400 400 400 400 400

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15

352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8

352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8 352.8

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λ

β

β′

FEA

(σu/σYeq) 0.464 0.439 0.449 0.422 0.404 0.385 0.366 0.330 0.311 0.299 0.286 0.273 0.900 0.848 0.811 0.770 0.725 0.681 0.639 0.612 0.581 0.548 0.543 0.509 0.488 0.464 0.439 0.449 0.422 0.404 0.385 0.366 0.330 0.311 0.299 0.286 0.273 0.900 0.848 0.811 0.770 0.725 0.681 0.639 0.612 0.581 0.548 0.543 0.509 0.488 0.464 0.439 0.449 0.422 0.404 0.385 0.366 0.330 0.311 0.299 0.286 0.273

2.756 3.453 1.586 1.969 2.303 2.763 3.450 1.586 1.969 2.303 2.763 3.450 1.590 1.968 2.300 2.761 3.452 1.591 1.969 2.292 2.764 3.456 1.591 1.973 2.293 2.756 3.453 1.586 1.969 2.303 2.763 3.450 1.586 1.969 2.303 2.763 3.450 1.590 1.971 2.300 2.761 3.452 1.591 1.969 2.292 2.764 3.456 1.591 1.973 2.293 2.756 3.453 1.586 1.969 2.303 2.763 3.450 1.586 1.969 2.303 2.763 3.450

3.090 3.765 1.817 2.195 2.641 3.097 3.762 1.817 2.195 2.641 3.097 3.762 1.277 1.538 1.850 2.171 2.639 1.277 1.539 1.844 2.173 2.642 1.277 1.542 1.845 2.167 2.640 1.274 1.539 1.852 2.172 2.638 1.274 1.539 1.852 2.172 2.638 0.895 1.080 1.297 1.522 1.850 0.896 1.079 1.293 1.523 1.853 0.896 1.081 1.294 1.519 1.851 0.893 1.079 1.299 1.523 1.850 0.893 1.079 1.299 1.523 1.850

0.921 0.860 0.995 0.980 0.967 0.929 0.860 0.998 0.984 0.960 0.920 0.851 0.971 0.976 0.965 0.956 0.934 0.988 0.990 0.979 0.958 0.943 0.992 0.990 0.981 0.961 0.935 0.996 0.991 0.984 0.968 0.946 0.998 0.996 0.990 0.967 0.980 0.977 0.981 0.978 0.980 0.949 1.000 0.998 0.993 0.981 0.956 1.000 1.000 1.000 0.998 0.980 1.000 1.000 1.000 0.983 0.991 1.000 1.000 1.000 0.992 0.990

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