Imperfection sensitivity of flat plates under combined compression and shear

Abstract

Finite element analysis allows fully non-linear analysis of shells containing geometric imperfections. However, such an analysis requires information on the exact size and shape of the imperfection to be modelled, in order to produce accurate results on which designs can be based. In the absence of such data it is generally recommended that the imperfection be modelled on the first eigenmode with an amplitude selected according to manufacturing procedure. This paper presents the effects of imperfection shape and amplitude on the buckling and postbuckling behaviour of one specific case, to test the accuracy of such recommendations.

Introduction

It has long been accepted that the existence of imperfections in thin-walled structures can substantially reduce their buckling load and affect their postbuckling behaviour. These imperfections can take the form of non-uniformity or local variations in the physical properties of the material, geometric imperfections (deviations in shape, eccentricities, local indentations) and load eccentricities, although the majority of the research carried out to date has been based on geometric imperfections.

Attempts to develop a general theory of buckling and postbuckling theory incorporating imperfection sensitivity began with Koiter [1], [2]. Further contributions were provided by Budiansky and Hutchinson [3], Stein [4], Arbocz [5], and many others.

With the event of finite element analysis the engineer was presented with a tool which allows him to model the buckling and postbuckling of shells under complex load and boundary conditions whilst fully incorporating the effects of imperfections and other non-linearities. However, difficulties still arise in directly using numerical shell buckling analysis in design due to the need to convert numerical buckling loads based on any of several different types of buckling analysis available into a design load for a particular structure. Several approaches have been considered by different researchers and code writing committees [6], [7], [8], [9].

Two main types of approach exist. The first involves the use of a linear elastic bifurcation buckling analysis (often known as an eigenvalue analysis) to determine the bifurcation loads of a structure. Reduction factors are then applied to account for both geometric imperfections and plasticity. This method is attractive in that it is similar to existing techniques based on simple load cases, and a linear bifurcation analysis is relatively quick and simple to perform. However, difficulties exist in determining reduction factors for various loading and boundary conditions. Work has been done in this area by Schmidt and Krysik [6] and Samuelson and Eggwertz [8].

The alternative approach involves the performance of a fully non-linear analysis with geometric imperfections, plasticity and also large deflections being accurately modelled. Although this technique has obvious advantages, the difficulty now lies in deciding upon the amplitude and form of the imperfection to be used. The most accurate method is obviously to base any analysis on real imperfections. Accurate measurement of imperfections in the laboratory and also full-scale shells, have been carried out by Arbocz [10], [11], Arbocz and Babcock [12], Arbocz and Hol [13], Elishakoff et al. [14], Singer [15], Singer and Abromovich [16], and Weller et al. [17]. These and other results have been incorporated into an International Imperfections Data Bank [10], [11].

Unfortunately, detailed information on the amplitude and form of real imperfections in the structure being designed are not always available. In such cases Speicher and Saal [7] have recommended that an equivalent imperfection having the same form at the first bifurcation or eigenmode be used. They have also derived the required magnitude of these imperfections to produce a safe design for the particular case of cylinders, based on existing experimental results. Most commercially available finite element codes recommend a similar approach to designers, who are advised again to base the form of the imperfection on the first bifurcation/eigenmode, and set its maximum amplitude equal to that expected in the component itself, based on a consideration of manufacturing methods used.

This paper presents the results of such an analysis performed for the specific case of a flat plate under complex load and boundary conditions, for which a model previously validated by comparison with experimental results [18] already exists. This was selected specifically to represent part of a component commonly found in aeroengine structures, which are often susceptible to failure by buckling. A simple linear analysis was performed first to determine the eigenmodes of the plate. A series of non-linear analyses based on imperfection in the form of these mode shapes was then carried out to investigate the effect of varying the modal shape used to represent this imperfection, and its amplitude.