Elsevier

Thin-Walled Structures

Volume 39, Issue 3, March 2001, Pages 215-246
Thin-Walled Structures

Large deflection orthotropic plate approach to develop ultimate strength formulations for stiffened panels under combined biaxial compression/tension and lateral pressure

https://doi.org/10.1016/S0263-8231(00)00059-8Get rights and content

Abstract

This paper uses the large deflection orthotropic plate approach to develop the ultimate strength formulations for steel stiffened panels under combined biaxial compression/tension and lateral pressure loads, considering the overall (grillage) buckling collapse mode. The object panel has a number of one-sided small stiffeners in either one or both orthogonal directions. The stiffened panel is then modeled as an equivalent orthotropic plate, for which the various elastic constants characterizing structural orthotropy are determined in a consistent systematic manner using classical theory of elasticity. The panel edges are considered to be simply supported. The influence of initial deflections is taken into account. The membrane stress distribution inside the panel under combined uniaxial loading (in either longitudinal or transverse direction) and lateral pressure is analyzed by solving the nonlinear governing differential equations of large deflection orthotropic plate theory. It is presumed that the panel collapses when the most highly stressed boundary location yields, resulting in closed-form expressions for the ultimate strength of the stiffened panel. Based on the insights previously developed through numerical studies, the panel ultimate strength interaction formulation between biaxial loads, with lateral pressure regarded as a secondary load component is then proposed as a relevant combination of the two sets of panel ultimate strength formulations, i.e. one for combined longitudinal axial load and lateral pressure and the other for combined transverse axial load and lateral pressure. The validity of the proposed ultimate strength formulations is verified by a comparison with nonlinear finite element and other numerical solutions.

Introduction

For convenience, the primary modes of overall failure for a stiffened panel subject to predominantly compressive loads will be categorized into the following six types, namely

  • Mode I: Overall (grillage) buckling collapse,

  • Mode II: Yielding along the plate-stiffener intersection,

  • Mode III: Column or beam-column type collapse of the plate-stiffener combination as representative of the stiffened panel,

  • Mode IV: Local buckling of stiffener web,

  • Mode V: Tripping of stiffener, and

  • Mode VI: Gross yielding

Mode I typically represents the collapse pattern when the stiffeners are relatively weak. In this case, the stiffeners can buckle together with plating, the overall (grillage) buckling behavior remaining elastic. The stiffened panel can normally sustain further loading even after overall (grillage) buckling in the elastic regime occurs, and the ultimate strength is eventually reached by formation of a large yield region inside the panel and/or along the panel edges. In Mode I, the stiffened panel behaves as an “orthotropic plate”.

The other groups (i.e. Modes II to VI) normally take place when the stiffeners are relatively strong so that the stiffeners remain straight until the plating between stiffeners buckles or even collapses locally. The stiffened panel will eventually reach the ultimate limit state by failure of stiffeners together with some associated plating. It is noted that the stiffened panel with weak stiffeners where failure of stiffeners occurs prior to buckling of plating normally follows Mode I, i.e. failure after overall (grillage) buckling occurs in the elastic regime.

Mode II typically represents the collapse pattern wherein the panel collapses by yielding along the plate-stiffener intersection, which is normally termed a plate-induced failure. This type of collapse can also occur in some cases when the panel is predominantly subjected to biaxial compressive loads. Mode III indicates a failure pattern in which the ultimate strength is reached by column or beam column type collapse of the plate-stiffener combination with the associated effective (reduced) plating. Modes IV and V failures typically arise when the ratio of stiffener web height to stiffener web thickness is too large and/or when the type of the stiffener flange is inadequate to remain straight so that the stiffener web buckles or twists sideways. Mode IV represents a failure pattern in which the panel collapses by local buckling of stiffener web, while Mode V can occur when the ultimate strength is reached subsequent to tripping of stiffeners.

Mode VI typically takes place when the panel slenderness is very small (i.e. the panel is very stocky or thick) and/or when the panel is predominantly subjected to the axial tensile loading so that neither local nor overall (grillage) buckling occurs until the panel cross section yields entirely.

Calculation of the ultimate strength of the stiffened panel under combined loads, taking into account all of the possible failure modes noted above, is not straightforward, because of the interplay of the various factors previously such as geometric/material properties, loading, post-weld initial imperfections (i.e. initial deflection and welding induced residual stresses) and boundary conditions. As an approximation, it is normally considered that the collapse of stiffened panels occurs at the lowest value among the various ultimate loads calculated for each of the above collapse patterns. This leads to the easier alternative wherein one calculates the ultimate strengths for all collapse modes mentioned above separately and then compares them to find the minimum value which is then taken to correspond to the real panel ultimate strength.

In this regard, it is normally of interest to calculate the panel ultimate strength for each of the above collapse patterns. This paper is concerned with the ultimate strength of stiffened panels subject to Mode I failure. When the panel has a number of relatively small stiffeners, the stiffened panel can normally be replaced by an “orthotropic plate”. In this case, the stiffeners are smeared into the plating. The orthotropic plate theory taking into account large deflection effects can then be used to calculate the membrane stress distribution of a stiffened panel whose failure follows Mode I.

A large number of studies on stiffened panel behavior have of course been previously undertaken. Some of the relevant results have been reviewed by Technical Committee III.1 of ISSC (International Ships and Offshore Structures Congress). In this paper, previous studies related to stiffened panel behavior primarily applying orthotropic plate approach are briefly reviewed. Mansour [1] presented the nonlinear theory to analyze the elastic behavior of orthotropic plates. He used this theory to develop some design charts (curves) for post-buckling behaviour of stiffened panels under combined in-plane and lateral pressure loads and with small initial deflections [2]. Aalami and Chapman [3] employed the large deflection orthotropic plate theory to analyze the post-buckling behavior of ship stiffened panels under combined in-plane and lateral pressure loads. Mansour [4] examined the characteristics of existing methods to predict the ultimate strength of ship gross panels under combined biaxial and lateral pressure loads. Christodoulides and de Oliveira [5] calculated the plastic collapse loads of orthotropic plates. Shen [6] studied the post-buckling behavior of orthotropic plates on an elastic foundation and under uniaxial loads, applying large deflection orthotropic theory. Banka and Jianshenga [7] presented buckling strength solutions for various types of orthotropic panels under uniaxial compression. Mikami and Niwa [8] presented an approximate method to predict the ultimate strength of orthogonally stiffened panels under uniaxial compression. Pavlovic et al. [9], [10] derived closed-form equations of the effective breadth for orthotropic plates under edge shear. Bedaira [11], [12] performed a useful study on stability of stiffened panels under unaixial compression, including an extensive literature review and development of buckling load formulations applying orthotropic plate theory. Smith and colleagues [13], [14], [15], [16], [17], [18], [19] provided a noteworthy contribution to the advanced design of ship stiffened panels, in part using the orthotropic plate approach as part of their studies.

While some of the previous studies related to the orthotropic plate approach were concerned with post-buckling and ultimate strength behavior taking into account large deflection effects, others have been limited to linear small deflection analysis. Also, while some studies do deal with combined loading under two or more load components, most of them are limited to a single load component such as uniaxial compression alone. The stiffened panel in steel plated structures is generally subjected to combined in-plane and lateral pressure loads. These loads are not always applied simultaneously, but more than one load component can normally exist and interact. Thus it is of interest to better understand the ultimate limit state characteristics of stiffened panels under combined loads.

In this paper, the large deflection orthotropic plate approach is used to develop the ultimate strength formulations of stiffened steel panels under combined biaxial compression/tension and lateral pressure loads, showing Mode I (i.e. overall grillage) collapse mode noted above, where the stiffened panel is replaced by an equivalent orthotropic plate. The panel edges are considered to be simply supported. The post-weld initial deflection is included in the strength calculations as a parameter of influence. The membrane stress distribution inside the panel under combined uniaxial compression/tension and lateral pressure is calculated by solving the nonlinear governing differential equations of large deflection orthotropic plate theory, where the elastic constants for material orthotropy of the panel are determined in a consistent theoretical manner. It is considered that the panel collapses if the most highly stressed panel edge yields, resulting in closed form expressions for the ultimate strength of the panel under combined axial compression/tension in either longitudinal or transverse direction and with lateral pressure applied. Based on insights previously developed by nonlinear finite element analysis, the panel ultimate strength interaction relationship between biaxial compression/tension and lateral pressure loads are suggested as a relevant combination of the two sets of the strength formulations, i.e. one for combined longitudinal axial compression/tension and lateral pressure and the other for combined transverse axial compression/tension and lateral pressure. The developed strength formulations are compared with nonlinear finite element analysis and other numerical solutions.

Section snippets

Basic idealizations

This paper is solely concerned with a stiffened panel with a number of relatively small stiffeners so that the stiffeners can buckle together with the plating, i.e. following Mode I, i.e. overall (grillage) collapse mode. In this case, the stiffened panel is modeled as an equivalent “orthotropic plate” by smearing stiffeners into the plating. In the following, other basic idealizations made for analysis of the panel are described.

Governing nonlinear differential equations of the orthotropic plate

To calculate the distribution of membrane stresses inside the idealized orthotropic plate, this paper attempts to analytically solve the nonlinear governing differential equations of large deflection orthotropic plate. As previously reviewed, this problem has of course been studied before, by a number of investigators. The reliability of orthotropic plate analysis depends significantly on various elastic constants that need to be determined when a stiffened panel is replaced by an equivalent

Ultimate strength formulation under longitudinal axial load and lateral pressure

The ultimate longitudinal axial strength formulation for the panel is now derived when it is subjected to combined σxav (either compressive or tensile) and p, the latter being regarded as a secondary load component. To analytically solve the nonlinear governing differential , , it is desirable to use the relevant simplified initial and added deflection functions which involve only one deflection term. For a panel under predominantly longitudinal compressive axial loads, the deflection term

Ultimate strength formulation under transverse axial load and lateral pressure

The ultimate transverse axial strength formulation is now derived when the panel is subjected to combined σyav (either compressive or tensile) and p, the latter being regarded as a secondary load component. The dominant term of the initial and added deflection functions of the panel under σyav and p can be assumed by considering only the buckling mode, as followswo=AonsinπxLsinnπyBw=AnsinπxLsinnπyBwhere An is the unknown amplitude of the added deflection function, while Aon=woplBon since the

Ultimate strength formulation under combined biaxial compression/tension and lateral pressure

So far, two sets of the ultimate longitudinal or transverse axial strength formulations have been developed by taking into account the influence of p as the secondary load component. By combining these two sets of the strength formulations, a complete set of the ultimate strength formulation of the panel under three load components (i.e. σxav, σyav and p) is now derived when the panel follows Mode I collapse.

To investigate the characteristics of the ultimate strength interaction relationship of

Illustrative examples

To test the validity of the proposed ultimate strength formulation, i.e. Eq. (40), illustrative examples are now considered. In these examples, the proposed strength formulations are compared with either the nonlinear finite element analyses using ANSYS [28] or the semi-analytical (incremental Galerkin) method using SPINE [29]. These programs compute the elastic-plastic large deflection behavior of the stiffened panel modeled as an assembly of both plating and stiffeners until the ultimate

Concluding remarks

The aim of the present study has been to develop ultimate strength formulations for stiffened steel panels under combined biaxial loads and lateral pressure when the stiffened panel behaves as an “orthotropic plate”. For this purpose, a large deflection orthotropic plate approach is primarily employed, where elastic constants for orthotropic plates are determined in a consistent manner using classical theory of elasticity. The support condition for the panel is assumed to be simply supported

Acknowledgements

The present research study was undertaken with support from the Korea Research Foundation for Faculty Research abroad and the Brain Korea 21 project who are thanked for their support.

References (29)

  • M. Kmiecik et al.

    Statistics of ship plating distortions

    Marine Structures

    (1995)
  • J.K. Paik et al.

    An analytical method for the ultimate compressive strength and effective plating of stiffened panels

    J. of Constructional Steel Research

    (1999)
  • C. Guedes Soares et al.

    Compressive strength of rectangular plates under biaxial load and lateral pressure

    Thin-Walled Structures

    (1996)
  • A.E. Mansour

    On the nonlinear-theory of orthotropic plates

    J. of Ship Research

    (1971)
  • A.E. Mansour

    Post-buckling behaviour of stiffened plates with small initial curvature under combined loads

    International Shipbuilding Progress

    (1971)
  • B. Aalami et al.

    Large deflection behaviour of ship plate panels under normal pressure and in-plane loading

    Trans. Royal Institution of Naval Architects

    (1972)
  • Mansour AE. Gross panel strength under combined loading. Ship Structures Committee, SSC-270,...
  • J.C. Christodoulides et al.

    Plastic collapse of orthotropic plates

    J. of Ship Research

    (1982)
  • H.S. Shen

    Post-buckling of orthotropic plates on two-parameter elastic foundation

    J. of Engineering Mechanics

    (1995)
  • L.C. Banka et al.

    Buckling of orthotropic plates with free and rotationally restrained unloaded edges

    Thin-Walled Structures

    (1996)
  • I. Mikami et al.

    Ultimate compressive strength of orthogonally stiffened steel plates

    J. of Structural Engineering

    (1996)
  • M.N. Pavlovic et al.

    Shear lag and effective breadth in rectangular plates with material orthotropy, Part 1: Analytical formulation

    Thin-Walled Structures

    (1998)
  • M.N. Pavlovic et al.

    Shear lag and effective breadth in rectangular plates with material orthotropy, Part 2: Typical results of parametric studies

    Thin-Walled Structures

    (1998)
  • O.K. Bedaira

    The elastic behaviour of multi-stiffened plates under uniform compression

    Thin-Walled Structures

    (1997)
  • Cited by (0)

    View full text