Elsevier

Thin-Walled Structures

Volume 107, October 2016, Pages 1-17
Thin-Walled Structures

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

https://doi.org/10.1016/j.tws.2016.05.003Get rights and content

Highlights

  • Buckling and post-buckling behaviour of stiffened curve plate is investigated.

  • Numerical studies to simulate the nonlinear ultimate bending strength behaviour.

  • Effects of varying geometrical parameters on buckling ultimate strength capacity

  • Proposed design curves for of stiffened curve plate.

Abstract

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.

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], [2], [3], [4], [5]. A brief review follows of recent research related to buckling and ultimate strength behaviour of cylindrically curved 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 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], [12], [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 elasto-plastic large deflection analyses with variations in the curvature, slenderness ratio, aspect ratio, web height, initial imperfection and 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 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], [16], [17], [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 (hw) 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.

Section snippets

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

Eigenvalue buckling analysis and results

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

Buckling and post-buckling behaviour

To investigate the fundamental buckling and post-buckling 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

Influence of slenderness ratio

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).

Web height varied from 150 to 400 mm and flank angle from 5° to 45°. As the flank angle increases, there is a slight

Development of design formula based on FE results

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 thin-walled shells with curvature. In this study, design

Concluding remarks

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

References (28)

  • H.J. Park, S.R. Cho, J.N. Chung, D.B. Lee, Ultimate strength analysis of curved stiffened shell of container bilge...
  • Y.W. Kwen, Y.I. Park, J.K. Paik, J.M. Lee, Buckling and ultimate strength characteristics for ship curved plate...
  • S.R. Cho, H.Z. Park, H.S. Kim, J.S. Seo, Experimental and numerical investigations on the ultimate strength of curved...
  • J.S. Park, K. Yumura, S. Katsura, I. Kazuhiro, Y. Tetsuya, Buckling and post-buckling behaviour of cylindrically curved...
  • Cited by (25)

    • A novel formula for predicting the ultimate compressive strength of the cylindrically curved plates

      2024, International Journal of Naval Architecture and Ocean Engineering
    • A useful guide of effective mesh-size decision in predicting the ultimate strength of flat- and curved plates in compression

      2023, Journal of Ocean Engineering and Science
      Citation Excerpt :

      Until now, we targeted 1.002 of plate slenderness ratio, which presents a thick plate (t = 28mm). Other research groups have widely studied the effect of flank angle on the ULS of the curved plate [34,39,41,42,64,65], so details may not be discussed further here. The effect of mesh-size or the number of elements on the flat- and curved plates defined based on the flank angle in Fig. 5 is investigated in this section.

    • Effects of pre-fatigue damages on ultimate strength of steel columns: From material to structure

      2022, Journal of Constructional Steel Research
      Citation Excerpt :

      These models have been validated by the experiments and they are very effective to simulate the stress-strain behaviors for steels without an apparent yield plateau. Over the past several years, a considerable number of studies have been devoted to the strength evaluation based on the numerical analysis approaches of steel plates [17–25], steel components [26–41] and steel structures [42–47]. To study the effect of constitutive model on the ultimate strength of stiffened steel panels, members, and structures, Tekgoz et al. [17] investigated numerically the effect of residual stress on the ultimate strength of thin rectangular stiffened and unstiffened plates.

    • Stability behaviour of stiffened curved plates subjected to pure compression

      2021, Thin-Walled Structures
      Citation Excerpt :

      Hence, only the influence of the bending and torsional stiffness of the stiffener on the strength of subpanels was investigated. Seo et al. [17] have derived empirical expressions in closed form to predict the compressive strength of curved stiffened plates. The formulae were derived from a database of 150 numerical analysis yet again based on the double span/double bay model, without considering of the overall buckling.

    View all citing articles on Scopus
    View full text