Buckling strength of steel plating with elastically restrained edges

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Abstract

The twin aims of the present study are to investigate the buckling strength characteristics of steel plating elastically restrained at their edges and also to develop simple design formulations for buckling strength as a function of the torsional rigidity of support members that provide the rotational restraints along either one set of edges or all (four) edges. The characteristic equation for the buckling strength of steel plating that is elastically restrained along either long or short edges while the other edges are simply supported was derived by an analytical method. Using the computed results obtained by directly solving the buckling characteristic equation, closed-form expressions of the buckling strength of the plating with one set of edges elastically restrained while the other set of edges is simply supported are derived empirically by curve fitting. Based on the insights developed in the present study, approximate equations for the buckling strength for plating with all edges elastically restrained are proposed as a function of a relevant combination of the three simpler edge condition cases (i.e., long edges elastically restrained/short edges simply supported, long edges simply supported/short edges elastically restrained, and all edges simply supported). The effect of distortion of support members before the plating buckles is also approximately accounted for. The validity of the proposed closed-form buckling strength design formulations is studied by a comparison with theoretical and numerical solutions.

Introduction

Ship steel plating can buckle in compression if the applied compressive stresses reach a critical value which is influenced by a variety of parameters including the plate dimensions and boundary conditions. Stiffened plating in ships is supported by various types of members along the edges, which have a finite value of the torsional rigidity. This is in contrast to the idealized simply supported boundary conditions often assumed for design purposes. Depending on the torsional rigidity of support members, the rotation along the plate edges will to some extent be restrained. When the rotational restraints are zero, the edge condition corresponds to a simply supported case, while the edge condition becomes clamped when the rotational restraints are infinite.

Most current practical design guidelines from classification societies for the buckling and ultimate strength of ship plating are based on boundary conditions in which all (four) edges are simply supported. In real ship plating, idealized edge conditions as simply supported or clamped never occur because of finite rotational restraints.

According to the study of Paik et al. [1] who investigated the bending and torsional rigidities of support members for deck, side and bottom plating in merchant ships, the amount of the torsional rigidity of support members at long edges (parallel to the ship length direction) normalized by the bending rigidity of attached plating, denoted by GJLbD (G = shear modulus, JL = torsion constant of support member at long edges, b = plate breadth, D=Et312(1−ν2), E = Young's modulus, t = plate thickness, ν = Poisson's ratio), is normally in the range of 0.05 to 3.0 (and usually not exceeding 5.0) while the amount at short edges (normal to the ship length direction), denoted by GJSbD (JS = torsion constant of support member at short edges), is normally in the range of 0.1 to 8.0 (and usually not exceeding 13.0). This implies that there is of course no case with zero or infinite torsional rigidities in practice as long as support members exist at their edges, and the amount of the torsional rigidities at one set of long or short edges is normally different from each other as well. It was also found from the investigation of Paik et al. [1] that the bending rigidity of support members are sufficient enough so that the relative lateral deflection of typical members providing the support to plating at edges is small.

For more advanced design of steel plating against buckling, it is hence important to better understand the buckling strength characteristics of plating as a function of the torsional rigidity of support members along the edges.

This problem has of course been studied before, by a number of investigators. Lundquist and Stowell [2] studied the effect of the edge condition on the buckling strength of rectangular plates subject to uniaxial compressive loads where the support along the unloaded edges was intermediate between simply supported and clamped. Bleich [3] and Timoshenko and Gere [4] discussed the buckling strength of plates with various boundary conditions that one set of edges is elastically restrained while the other set of edges is either simply supported or clamped. Gerard and Becker [5] surveyed literature and results for the buckling of rectangular plates under various combinations of two or three types of loading and a number of edge conditions. Evans [6] carried out an extensive experimental study for the buckling strength of wide plates that the loaded (long) edges are elastically restrained while the unloaded (short) edges are simply supported. Based on the experimental results, he derived a closed-form expression of the compressive buckling strength of wide plates taking into account the effect of rotational restraints along the loaded edges. McKenzie [7] studied the buckling strength of plating under biaxial compression, bending and edge shear that is simply supported along short edges (at which bending is applied) and elastically restrained along long edges. These and other previous studies are quite useful for the buckling strength design of plating considering the rotational restraint effect along the edges. To the authors' knowledge, systematic investigations on the buckling strength of plating with elastically restrained along both long and short edges appear to be difficult to come by and are needed.

The aims of the present study are:

  • to investigate the buckling strength characteristics of plating with the boundary conditions which are elastically restrained along the edges, and also

  • to develop simple buckling design formulations of plating taking into account the torsional rigidity of support members along either one set of edges or all (four) edges, see Fig. 1, Fig. 3, Fig. 11 or Fig. 12.

In a sense, the present study is a generalization of such past efforts, in that it attempts to consider both long and wide edge cases in an integrated way in order to make possible solutions of wider applicability. It is also important in that it provides potentially useful design formulations that are obtained by curve fitting, and validated against the more exact theoretical and numerical (finite element) results.

The characteristic equation for the buckling of plating with elastic restraint along either long or short edges while the other edges are simply supported is derived analytically. By solving the buckling characteristic equation, the buckling strength characteristics of plating are investigated varying the plate aspect ratio and the torsional rigidity of support members. Based on the computed results, closed-form expressions of the plate buckling strength are derived empirically by curve fitting. Simplified buckling design formulations for plating with all edges elastically restrained are also derived using the insights developed in the present study. The validity of the buckling design formulations developed in the present study is confirmed by a comparison with the more exact theoretical solutions and also numerical results obtained by finite element analysis.

Section snippets

Buckling strength of steel plating with elastically restrained boundary conditions and uniaxial compressive loading

The elastic buckling strength of steel plating with boundary conditions which depend on the torsional rigidity of support members at edges can be calculated as a characteristic value problem. For simplicity, the following basic hypotheses are made:

  • The x axis is taken in the long direction and the y axis is taken in the short direction, so that the plate aspect ratio a/b is always greater than 1.0.

  • The geometric and material properties of support members are the same in the same direction.

  • The

Effect of distortion of support members

So far we have assumed that support members remain straight until the plating buckles. This assumption is normally appropriate to apply for practical use, but the support members may in some cases distort due to axial compression before the inception of plate buckling so that the support members will not fully contribute to the rotational restraints along the edges.

It is considered that the effectiveness of support members depends on the relative torsional rigidities of support member to the

Buckling strength of steel plating subject to biaxial compression

So far the buckling strength formulations for plating under unaxial compressive loading were derived taking into account the effect of the torsional rigidity of support members along the plate edges. The buckling strength interaction equation of plating subject to biaxial compressive loads is now studied, considering the torsional rigidity effect of support members. Ueda et al. [8] proposed and investigated the following elastic buckling strength interaction equation for plating simply

Verification examples using finite element analysis

The validity of the empirical formulations for the plate buckling strength so far developed is now confirmed by a comparison with applicable finite element solutions as obtained by eigen-value analysis. As a representative study case, the buckling of a continuously flat bar stiffened panel subject to either uniaxial or biaxial compressive loads such as that shown in Fig. 11 is analyzed. Because of symmetry, the central part of the panel with the stiffeners in the cruciform is taken as the

Concluding remarks

The twin aims of the present study have been to investigate the buckling strength characteristics of steel plating elastically restrained at their edges and also to develop related simplified design formulations for predicting the buckling strength as a function of the torsional rigidity of support members along either one set of edges or both long and short edges.

The buckling strength characteristic equation for steel plating which is elastically restrained along either long or short edges

Acknowledgements

The present study was undertaken under support from the Brain Korea 21 Project, the American Bureau of Shipping and the Hyundai Heavy Industries who are thanked for this support. The second author was with the American Bureau of Shipping when this work was carried out. The views expressed in this paper are those of the authors and are not necessarily those of the institutions the authors are affiliated with.

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