Mathematical Problems in Engineering

Volume 2015, Article ID 514267, 10 pages

http://dx.doi.org/10.1155/2015/514267

## An Analytical Insight into the Buckling Paradox for Circular Cylindrical Shells under Axial and Lateral Loading

^{1}Department of Mechanical, Aerospace and Civil Engineering, Brunel University, Uxbridge UB8 3PH, UK^{2}Department of Structural Engineering, University of Naples “Federico II”, Via Claudio 21, 80125 Naples, Italy

Received 23 December 2014; Accepted 28 May 2015

Academic Editor: Xin-Lin Gao

Copyright © 2015 Rabee Shamass et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A large number of authors in the past have concluded that the flow theory of plasticity tends to overestimate significantly the buckling load for many problems of plates and shells in the plastic range, while the deformation theory generally provides much more accurate predictions and is consequently used in practical applications. Following previous numerical studies by the same authors focused on axially compressed cylinders, the present work presents an analytical investigation which comprises the broader and different case of nonproportional loading. The analytical results are discussed and compared with experimental and numerical findings and the reason for the apparent discrepancy on the basis of the so-called “buckling paradox” appears once again to lay in the overconstrained kinematics on the basis of the analytical and numerical approaches present in the literature.

#### 1. Introduction

Plastic buckling generally takes place in the case of moderately thick cylinders subjected to axial compression, external pressure, or torsion, alone or in combination, and has been largely investigated.

The plasticity models that have been proposed for metals in the strain hardening range for the study of plastic buckling can be divided into two groups: the “deformation theory” of plasticity and the “flow theory” of plasticity. In both of these theories the plastic deformation is a function of the second invariant of the deviatoric part of the stress tensor, given that volume changes in the plastic range are not permitted. The difference lies in the fact that in the deformation theory of plasticity it is assumed that the state of stress is uniquely determined by the state of strain as it would happen in any path-independent nonlinear elastic constitutive law. Therefore, after a strain reversal, instead of recovering the initial elastic stiffness, the initial loading curve is followed. This behaviour is in contrast with the usual findings from experimental tests. The flow theory of plasticity, on the contrary, assumes that the increment of strain/stress is uniquely determined by the existing stress/strain and its increment. This assumption gives origin to a path-dependent relationship and the current stress depends not only on the value of the actual total strain but also on how this value has been reached.

As a consequence, notwithstanding its mathematical advantages, the deformation theory of plasticity is considered to lack somehow physical rigour in comparison to the flow theory [1, 2]. Surprisingly, the employment of the deformation theory has been reported to predict buckling loads that are smaller than those obtained with the flow theory and in better accordance with the experimental tests. This fact has been often designated as the “plastic buckling paradox.”

Recently the present authors have started to investigate the plastic buckling paradox by conducting accurate finite-element modelling of buckling of cylindrical shells using both the flow theory and the deformation theory of plasticity [3, 4]. Contrary to the common belief, they showed that, by using an accurate and carefully validated geometrically nonlinear finite element modelling, a very good agreement between numerical and experimental results can be obtained also in the case of the physically sound flow theory of plasticity. Consequently, according to the performed numerical investigations in the case of axially loaded cylinders, it can be affirmed that no plastic buckling paradox actually exists. Additionally, the flow theory of plasticity, which provides a physically sound description of the behaviour of metals, can even lead to predictions of the buckling stress which are in better agreement with the corresponding test results than those provided by use of the deformation theory, in contrast with the widely accepted belief that the flow theory leads to a significant overestimation of the buckling stress while the deformation theory leads to much more accurate predictions and, therefore, is the recommended choice for use in practical applications. On the basis of these numerical investigations, it was suggested that the roots of the discrepancy lie in the simplifying assumptions which have been regularly made with respect to the buckling modes and that the adoption of the deformation theory of plasticity simply results in counterbalancing the greater stiffness induced by kinematically constraining the cylinders to follow predefined buckling modes.

However, the case of axially loaded cylinders is relatively limited in order to draw general conclusions since in this case material points are generally subjected to proportional loading in the elastic range, and this remains relatively true also in the initial phase of plastic buckling. In fact, even more significant discrepancies are reported in the literature between the results of the flow and the deformation theories in the case of nonproportional loading.

Therefore the present investigation extends the analysis to the much more general case of nonproportional loading and, at the same time, makes use of an analytical treatment of the problem instead of the numerical one employed for the case of proportional loading. This makes it possible to analyse in detail the shape of the buckling modes both in the cases of the flow and of the deformation theory of plasticity.

The obtained analytical results are compared with the experimental and numerical results obtained in [5, 6] using the code BOSOR5 [7] and, for the purpose of validation, also with the numerical ones by the present authors [8].

Blachut et al. [5] conducted experimental and numerical analyses for 30 mild-steel machined cylinders, of different dimensions, subject to axial tension and increasing external pressure. Using the code BOSOR5 [7] for their numerical analyses they reported that the agreement between the two plasticity theories appeared strongly dependent on the diameter, , and the length, , of the cylindrical shell. For short cylinders (), the plastic-buckling pressure results predicted by the flow and deformation theories coincided only when the tensile axial load vanished [5]. By increasing the axial tensile load, the plastic buckling pressures predicted by the flow theory started to diverge quickly from those predicted by the deformation theory. Additionally, the flow theory failed to predict any buckling for high axial tensile load while tests confirmed the buckling occurrence. For specimens with length-to-diameter ratio ranging from 1.5 to 2.0 the results predicted by both theories were identical for a certain range of combined loading. However, for high values of applied tensile load, the predictions of the flow theory began to deviate from those of the deformation theory and became unrealistic in correspondence with large plastic strains.

Giezen et al. [6] conducted experiments and numerical analyses on two sets of tubes made of aluminium alloy 6061-T4 and subjected to combined axial tension and external pressure in order to highlight the difference in buckling predictions of both the flow theory and the deformation theory again using the code BOSOR5. These tubes have ratios equal to one. In their test two different loading paths were considered. In the first one the axial tensile load was held constant and the external pressure was increased; in the second one, the external pressure was held constant and the axial tensile load was increased. Their numerical studies showed that the buckling pressure based on the flow theory increases with increasing applied tensile load while the experimental test revealed a reduction in buckling resistance with increasing axial tension. Thus as axial tension increased the discrepancy between test results and numerical results predicted by the flow theory also significantly increased. On the other hand, results predicted by the deformation theory displayed the same trend as in the test results. However, the deformation theory significantly underpredicted the buckling pressure observed experimentally for some loading paths. Therefore, Giezen [9] concluded that, generally speaking, both plasticity theories were unsuccessful in predicting buckling load.

For the case of cylinders subjected to axial tensile load and external pressure, Blachut et al. [5] and Giezen et al. [6] concluded that the flow theory significantly overpredicts the plastic strains and buckling loads for high tensile loads while deformation theory leads to acceptable plastic strains and buckling loads that are more in line with experimental observations in most cases. This moved researchers to attempt a revised deformation theory by including unloading [10] or propose a total deformation theory applicable for nonproportional loading defined as a sequence of linear loadings [11].

The analytical approach employed in this work moves from the formulation presented by Chakrabarty [12] and encompasses, differently from the original formulation of Chakrabarty, both the flow and deformation theories.

It is found that the plastic buckling results calculated analytically using both the flow and deformation theories closely match those, when available, obtained numerically by using the code BOSOR5 [5, 7]. The analytical results thus confirm that the flow theory seems to overpredict buckling pressures for high values of applied tensile load while the deformation theory predictions appear to be in better agreement with experimental results.

However, going more in depth by means of the proposed analytical approach, it is possible to focus the attention on the buckling shapes and confirm, in such a way, that the root of the discrepancy lays in the assumed harmonic buckling modes along the circumference at the bifurcation.

In fact it is found that in the case of nonproportional loading the analytical and numerical approaches based on a certain class of harmonic buckling functions tend to overestimate the buckling loads when using the flow theory on account of a shift of the buckling modes. The deformation theory, on the contrary, tends to point to buckling shapes of lower order and thus counterbalances the excessive stiffness of the cylinder caused by the kinematic overconstraining.

In conclusion, when the buckling modes are the same, and in the case of nonproportional loading, the flow and deformation theory of plasticity provide the same buckling loads. It is worth recalling that the numerical FE approach [8], which is not kinematically overconstrained by a choice of predefined harmonic buckling modes, provides results which are in line with the experimental ones in the case of proportional loading.

As such, the conclusion of the present investigation is that also in the case of cylinders subjected to the more general and technically relevant case of nonproportional loading actually there seems to be no plastic buckling paradox.

#### 2. Experimental Data

Blachut et al. [5] conducted tests on 30 machined cylinders made of mild steel with outer diameter mm and length-diameter ratio () of 1.0, 1.5, and 2.0. In the experimental setting, one flange of the specimen was rigidly attached to the end flange of the pressure chamber and the other flange was bolted to a coupling device which in turn was bolted to the load cell; see Figure 1.