Buckling behaviour of imperfect spherical shells

doi:10.1016/S0020-7462(01)00086-5

International Journal of Non-Linear Mechanics

Volume 37, Issues 4–5, June 2002, Pages 589-604

https://doi.org/10.1016/S0020-7462(01)00086-5 Get rights and content

Abstract

For the design of spherical shells under external pressure relatively few information can be found in corresponding codes and recommendations, e.g. not at all in the new draft of Eurocode 3 ENV 1993-1-6. Under this aspect, new design rules for these shells were developed, which take into account relevant details like boundary conditions, material properties, and imperfections. They are usually based on a large number of systematic numerical simulations to obtain results describing the load carrying behaviour and imperfection sensitivity of thin spherical shells. In addition, previous theoretical and experimental results are discussed. Based on the results, diagrams and design rules have been developed which might be used for new recommendations in the design concept of the Eurocode.

Introduction

The load carrying and buckling behaviour of spherical shells under external pressure has been the subject of many theoretical, experimental, and numerical investigations. Although, the present European Recommendations ECCS from 1988 [1] include some formulas for the design of complete spheres, and national standards [2] give recommendations for spherical caps, in the new draft of Eurocode 3 ENV 1993-1-6 [3] rules for the buckling behaviour of spherical shells are missing. Therefore, it seems necessary to develop new diagrams and formulas for additional recommendations to design thin spherical shells. Previous theoretical and experimental studies are summarized and systematic numerical calculations of different spherical shells are performed to investigate the load carrying behaviour and imperfection sensitivity of spherical shells.

In various kinds of spherical shells under external pressure, in buildings, vehicles, tanks, vessels or silos, the stability of the structural system has to be considered. Most procedures for analysis of the buckling of shells are based on experimental investigations and are available in design codes to a very limited extent only. On the other hand, the improvements of computer technology and finite element methods give the possibility to investigate the behaviour of thin spherical shells by numerical simulations. As non-linear numerical calculations need high professional expertise and are also rather time consuming, it is very helpful for practical applications that new recommendations for shells on the basis of systematic parametrical studies are developed (Fig. 1).

Perfect shells may be considered as optimal structures, in the sense that their load carrying capacity is usually larger than that of shells which show deviations in geometry, material behaviour and boundary conditions. Actual shells exhibit almost more or less pronounced imperfections of geometry or material. Therefore, in numerical simulations both perfect and imperfect structures have to be investigated. A semi-analytic treatment of the field equations governing the non-linear static behaviour of shells of revolution [4], [5] as well as an approach for general shells to construct the relationship between the limit load and amplitudes and shapes of the “worst” imperfections in a direct way [6] are used. Applying these methods different geometrical imperfection shapes of spherical shells are analysed. Furthermore, in the numerical simulations both elastic and elastic–perfectly plastic modelling are considered to show the differences in load carrying behaviour and imperfection sensitivity of spherical shells. Additionally, the paper deals with spheres with different semi-angles φ to demonstrate their small influence on the limit loads. As the load carrying behaviour is much more influenced by boundary conditions than by different semi-angles, shells with various boundary conditions are investigated.

Section snippets

Theoretical developments

The theory of stability of shells is relatively young and very intensive research has been carried out during the last 20 years. Some of the milestones in the development of the shell buckling theory are presented in the following short review.

Euler's formulas, for determining the critical load of a compressed straight bar were published in the middle of the 18th century. The theory was further developed during the next years, and a linear buckling theory for spherical shells was first

Experimental investigations

To confirm theoretical results and to get new knowledge in buckling behaviour of externally pressurized spherical shells, experimental studies were performed by several researchers. However, exact documentations and descriptions of tests are often missing and the specimens as well as the manner in which the experiments are carried out differ from actual service conditions. Nevertheless, experimental tests are essential to confirm theoretical and numerical results.

In this context, a

Scope of numerical simulations

For the development of new design rules it is necessary to investigate a wide range of different shell parameters in addition to previous theoretical, experimental, and numerical results. Therefore, a large number of numerical simulations of perfect and imperfect as well as of elastic and elastic–plastic spherical shells with different geometries and boundary conditions have been performed.

New draft of European recommendations

In addition to the discussion of the numerical results this paper also presents a proposal for generalising the existing rules in the ECCS-Recommendations [1] and in national standards [2]. These rules use the current elastic–plastic buckling interaction rule of ENV 1993-1-6 [3]. The new recommendations are applicable to complete spheres and spherical caps under uniform external pressure with different boundary conditions. The wall thickness of the spheres considered is assumed to be constant,

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Buckling behaviour of imperfect spherical shells

Abstract

Introduction

Section snippets

Theoretical developments

Experimental investigations

Scope of numerical simulations

New draft of European recommendations

Comput. Methods Appl. Mech. Eng.

Thin-Walled Struct

Comput. Methods Appl. Mech. Eng.

A semi-analytical approach to the nonlinear analysis of shells of revolution

Analysis of elastic–plastic buckling and imperfection-sensitivity of shells of revolution