S C I E N C E

RESEARCH

D E V E L O P M E N T

E U R O P E A N

COMMISSION

technical steel research

Properties and ¡η­service performance

Enhancement

of ECCS design

recommendations

and development

of Eurocode 3 parts

related to shell

buckling

h Report

m EU R 1 8 4 6 0 EN STEEL RESEARCH

EUROPEAN COMMISSION

Edith CRESSON, Member of the Commission responsible for research, innovation, education, training and youth

DG XII/C.2 — RTD actions: Industrial and materials technologies — Materials and steel

Contact: Mr H. J.-L. Martin Address: European Commission, rue de la Loi 200 (MO 75 1/10), B-1049 Brussels — Tel. (32-2) 29-53453; fax (32-2) 29-65987

European Commission

technical steel research Properties and in-service performance

Enhancement of ECCS design recommendations and development

of Eurocode 3 parts related to shell buckling

R. Saikin European Convention for Constructional'Steelwork

Avenue des Ombrages 32/36 bte 20 B-1200 Brussels

Contract No 7210-SA/208 1 April 1991 to 30 September 1995

Final report

Directorate-General Science, Research and Development

1998 EUR 18460 EN

LEGAL NOTICE

Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of the following information.

A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server (http://europa.eu.int).

Cataloguing data can be found at the end of this publication.

Luxembourg: Office for Official Publications of the European Communities, 1998

ISBN 92-828-4414-5

© European Communities, 1998

Reproduction is authorised provided the source is acknowledged.

Printed in Luxembourg

PRINTED ON WHITE CHLORINE-FREE PAPER

List of contents page

Summary 9

1 Overall introduction 11

1.1 Scope of research 11 1.2 The five subprojects 12

2 Final reports of the five subprojects

A Stability and strength of stiffened conical shells 15

Al Introduction 17 A 1.1 Summary of previous work 17 Al .2 Objectives of current investigation 18 A2 Experimental tests 19 A2.1 Model geometries . 19 A2.2 Manufacturing of spacimens 19 A2.3 Measurement of material properties 20

A2.3.1 Yield stress 20 A2.3.2 Modulus of elasticity 21

A2.4 Test rig and measurind devices 21 A2.4.1 Arrangement of logging equipment 21 A2.4.2 Mesurement of end shortening 21

A2.5 Testprocedure 22 A2.6 Summary oftest results 22

A2.6.1 Geometric imperfections 23 A2.6.2 Load-end shortening 24 A2.6.3 Strains and stresses 24 A2.5.4 Lateral displacements 25

A2.7 Conclusions 25 Figures for Chapter A2 27

A3 Validation of finite element modelling 41 A3.1 Selection of FE models 41

A3.1.1 Unstiffened conical shells 41 A3.1.2 Stiffened conical shells 42

A3.2 Modelling aspects and convergence study 42 A3.2.1 Axisymmetric model 43 A3.2.2 Conical panel model 43 A3.2.3 Complete cone model 43 A3.2.4 Model comparisons 43

A3.3 Comparison of FE results with tests 43 A3.3.1 Analysis of imperfect geometries 43 A3.3.2 Unstiffened cones 44 A3.3.3 Stringer-stiffened cones 45

A3.4 Conclusions 46 Figures for Chapter A3 47

A4 Design implications 51 A4.1 Review of current ECCS rules on unstiffened cones 51 A4.2 Development of design proposal for stiffened cones 52 A4.3 Validation of design proposal for stiffened cones 53

A4.3.1 Critical buckling behaviour 53 A4.3.2 Imperfection reduction factors 5 3

A4.4 Concluding remarks 55 Figures for Chapter A4 57 Annex to Chapter A4: design proposals 61

A5 Conclusions 69

Notation 70

References 71

Β Local loads in cylindrical structures 73 Bl Introduction 77

B2 Survey of the pertinent literature 78

B3 Scope of work 79

B4 Experimentation and numerical results 80 B4.1 Fabrication of cylindrical steel models and test set-up 80

B4.1.1 Fabrication 80 B4.1.2 Test set-up 82

B4.1.2.1 Load application and measurement of reactions 82 B4.1.2.2 Measurment of imperfections 84

B4.2 Unstiffened cylinders 85 B4.2.1 Experimental results 85 B4.2.2 Comparison with numerical computations 89 B4.2.3 Comments 92 B4.2.4 Parametric study and development of a design rule 93

B4.2.4.1 Introduction 93 B4.2.4.2 Summary of numerical results 94 B4.2.4.3 Compact representation of the numerical results by

analytical formulae 96 B4.2.5 Design rule 104 B4.2.6 Formulation of an alternative design rule 107 B4.2.7 Comparison with an existing design proposal and rigid support

conditions 111 B4.2.8 Examples 112

B4.2.8.1 Example 1 113 B4.2.8.2 Example 2 114 B4.2.8.3 Example 3 114 B4.2.8.4 Example 4 114

B4.2.8.5 Comparison with other design rules 114 B4.2.9 Development of a design rule, which covers also the higher steel

grades Fe 430 and Fe 510 115 B4.2.9.1 Design rule 115 B4.2.9.2 Examples 115

B4.2.10 Other effects 116 B4.2.10.1 Effect of edge-ring-stiffeners & flexible support plates 116 B4.2.10.2 Interactive effect of internal pressure 118

B4.3 Stiffened cylinders with stepped wall-thickness 119 B4.3.1 Experimental results 119 B4.3.2 Comparison with finite element computations 123 B4.3.3 Development of a design rule 124

B4.3.3.1 Introduction 124 B4.3.3.2 Summary of numerical results and design rule 125

B5 Conclusions 127

C Shells of revolution with arbitrary meridional shapes -buckling design by use of computer analysis 131

CI Introduction 133

C2 Experimental investigations I34 C2.1 Testprogram I34

C2.1.1 Parameter "action causing buckling" I34 C2.1.2 Parameter ' 'meridional break" 134 C2.1.3 Parameter "shell geometry" 134 C2.1.4 Parameter "shell slenderness" , 135

C2.2 Test specimens 135 C2.2.1 Material properties 135 C2.2.2 Manufacturing 137 C2.2.3 Geometrical dimensions 137 C2.2.4 Shape imperfections 138

C2.3 Axial load tests 138 C2.3.1 Test set-up 138 C2.3.2 Test procedure 138 C2.3.3 Test results 138

C2.4 External pressure tests 140 C2.4.1 Test set-up 140 C2.4.2 Test procedure 140 C2.4.3 Test results 140

C3 Numerical investigations 142 C3.1 General 142 C3.2 Benchmarking 142

C3.2.1 Buckling analysis on GN level 142 C3.2.2 Collapse analysis on GMNA level 142

C3.3 Comparative numerical calculations for the axial load specimens 143 C3.3.1 MA level 143

C3.3.2 LA and GN A level 1 4 4 C3.3.3GMNA level 1 4 4 C3.3.4 Discussion of the results ¡45

C3.4 Comparative numerical calculations for the external pressure specimens 146 C3.4. IMA level 1 4 6 C3.4.2 LA and GNA level 146 C3.4.3 GMNA level 146 C3.4.4 Discussion of the results 147

C4 Evaluation of design procedure 148 C4.1 Generalized reduction factor approach (RFA) for shell buckling design 148 C4.2 Application of RFA to test specimens and comparison with test results 149

C4.2.1 Axial load specimens 149 C4.2.2 External pressure specimens 150

C5 Conclusions 151

C6 References 152

Tables for part C 153

Figures for part C 161

D Thin-walled shells under wind loading ¡73 Dl Aim of the investigations and state of knowledge ¡75 D 1.1 Background I75 D 1.2 State of knowledge ¡75 D1.3 General lay-out of the present investigation 176

D2 Experimental procedures 177 D2.1 Testing set-up and flow conditions 177 D2.2 Pressure test models 180 D2.3 Buckling test models 182 D2.4 Testing procedures 183

D2.4.1 Presssure measurements 183 D2.4.2 Buckling tests 184

D2.5 Testing programme 185

D3 Test results 186 D3.1 Presssure distribuition 186 D3.2 Buckling patterns 192

D4 Interpretation and conclusion 198 D4.1 Buckling under static wind load 198 D4.2 The effect of wind turbulence 200

D5 Summary 201

D6 References 202

£ Effects of cut-outs and openings in shells 207

El Introduction and literature revue 209

E2 Methodology 210

E2.1 Definition of shells 210

E2.2 Test cases of shell with opening 211

E2.3 Material and manufacturing 211

E2.4 Experimental set-up 212

E3 Experimental and numerical results 212

E3.1 Effect of shape of a cut-out 212

E3.1.1 Square openings 212

E3.1.2 Rectangular openings 213

E3.1.3 Circular openings 214

E3.2 Effect of opening angle 214

E3.3 Effect of length to radius ratio 215

E3.4 Effect of geometrical imperfections 215

E4 Proposed rules ■ 216

E4.1 General concept 216

E4.2 Definition of the parameters 217

E4.3 Justification of the method 218

E4.3.1 Transition between coupling and no-coupling range 218

E4.3.2 Determination of the slope 221

E4.4 Limits of validity of the proposed rules 222

Figures for part E 223

E5 References . . 241

3 Overall concluding remarks 243

Summary

For thin-walled plated steel structures of curved shape, e.g. tanks, silos, chimneys, towers, pipelines etc., shell buckling is an important design aspect. Relevant design rules are available in the European Recommendations on Buckling of Steel Shells, edited by ECCS. However, these Recommendations do not yet cover every shell buckling case and need continuous enhancement. Five essential deficiencies were prioritized where the existing rules were to be introduced into Eurocode 3 parts related to shell buckling. Collective efforts to eliminate these deficiencies were panelled through the present ECSC Steel Research Contract consisting of five subprojects. The five subprojects are: A - Stability and strength of stiffened conical shells; Β - Local loads on cylindrical structures; C - Shells of revolution with arbitrary meridional shape - buckling design by use of

computer analysis; D - Thin-walled shells under wind loading; E - Effects of cut-outs and openings in shells.

The common aim of the five subprojects was to gain deeper knowledge about the buckling behaviour of the particular structure under the particular loading and to develop from this knowledge simplified design rules which may be introduced into the ECCS Recommendations and into relevant Eurocode 3 parts. The coordination was achieved by the Technical Working Group TWG 8.4 of ECCS.

The research work included in all five subprojects experimental investigations: Axial load tests on unstiffened and stiffened cones in subproject A; axial load tests on locally-supported cylinders in subproject Β; axial load and external pressure tests on cone/cone and cylinder/cone assemblies in subproject C; wind tunnel tests on open vertical cylinders in subproject D; and axial load tests on cylinders with unreinforced openings in subproject E.

In four of the five subprojects comprehensive comparative numerical calculations were carried out in order to improve the understanding of the load carrying and failure characteristics of the tested structures. The validated numerical models were in three subprojects used to extend the covered parametric range by means of additional numerical studies. Experimental and numerical findings were in all five subprojects condensed into recommendations on how to proceed in practical design cases.

The results of this research will enable design engineers to come up with more economic, but still sufficiently safe steel shell structures.

1 Overall introduction

1.1 Scope of the research In 1980 the first edition of the ECCS European Recommendations on Buckling of Steel Shells was published. It had been worked out by the Technical Working Group 8.4 "Shells" (TWG 8.4) of the Technical Committee No. 8 "Structural Stability" (TC8) of the European Convention for Constructional Steelwork (ECCS). It was the first attempt to develop a recommendation-type document covering simplified design rules for buckling-endangered steel shell structures, without being restricted to specific application fields or being related to specific national safety philosophies. Until then, only some isolated buckling design specifications could be found in national standards for specific application fields, e.g. steel chimneys or vertical steel welded storage tanks for the petroleum industry.

The Recommendations have, within short time, gained a worlwide reputation. The latest edition (the fourth one) was published in 1988. It covers several basic shell buckling cases which are common to various application fields. Among them are unstiffened circular cylinders and cones under various loading types, stringer-stiffened cylinders under axial compression and ring-stiffened cylinders under external pressure. However, many questions are still unanswered and many practical problems are left to the designer. He will base his design in such cases either on rough approximations - which are necessarily overconservative, or on tests - which are expensive, or on comprehensive numerical buckling calculations -which are not only expensive but also problematic if not properly conducted, i.e. if not based on a profound personal kowledge of the complex stability behaviour of shells.

It has been (and is still) the strong opinion of TWG 8.4 that the Recommendations - besides continuously being improved - should be extended to further practical shell buckling problems which are not yet addressed, neither in its own present edition nor in any other code or design guide in the world. Parallel to these general enhancement efforts, TWG 8.4 offered assistance to Eurocode 3 Project Teams developing EC3 Parts related to shell buckling.

The scheme of Eurocode 3 "Design of Shell Structures" includes several parts which are, more or less, related to steel shell structures. These parts are: - Part 3 : Steel Towers, Masts and Chimneys, - Part 4: Steel Silos, Tanks and Pipelines, - Part 7: Marine and Maritime Steel Structures, - Part 8: Agricultural Steel Structures. Of these parts, the first two are presently being drafted. The existing design rules on buckling cases of cylindrical and conical shells are being encountered in the drafting of these parts. However, as stated above, not every information that would be needed is available.

The members of TWG 8.4 identified 1991 five outstanding shell buckling problems for which design rules were needed in the context with drafting further parts of EC3, but for which no appropriate source material was available. They gave priority to these topics in their general efforts to enhance the ECCS Shell Buckling Recommendations and proposed to address these design oriented problems in an ECSC-Steel Research Project. The resulting Contract No. 7210-SA/208 between ECSC and ECCS was signed at the end of 1991. The work started 1992 and ended 1995.

11

According to the mentioned five topics, the research was divided into five subprojects A to E

which were conducted in different institutions by five different research teams. The common

aim of all five subprojects was to gain deeper kowledge about the buckling behaviour of the

particular structure under the particular loading and to develop from this knowledge

simplified design recommendations which may be introduced into the ECCS

Recommendations and also into relevant Eurocode 3 Parts. The coordination was achieved by

TWG 8.4. The five subprojects are briefly described in the following subsection, including

their particular aims and the contributing researchers. The detailed reports are then given

separately in five similarly structured chapters A to E.

1.2 The five subprojects

The titles, places and researchers are listed in table 1.

Table 1: Subprojects of Contract No. 7210­SA/208

No.

A

Β

C

D

E

Topic

Stability and strength of stiffened

conical shells

Local loads on cylindrical structures

Shells of revolution with arbitrary

meridional shapes ­ buckling design

by use of computer analysis

Thin­walled shells under wind loading

Effects of cut­outs and openings in

shells

Institution

Imperial College

London

Politecnico di Milano

Universiteit Gent

'Techn. Universität

Graz

Universität Essen

Ruhr­Universität

Bochum

INSA Lyon

Researchers

P. J. Dowling

M. Chryssanthopoulos

C. Poggi

J. Rathé

G. Lagae

R. v. Impe

F. Dhanens

R. Greiner

W. Guggenberger

H. Schmidt

P. Swadlo

H. J. Niemann

V. Gornandt

M. Kasperski

J. F. Jullien

The practical implications and the particular aims of the subprojects may be summarized as

follows:

A ­ Stability and strength of stiffened conical shells

Conical shells are frequently used as truncated cones in tubular members to accommodate the

transition between different diameters. Typical applications include the legs of compliant

offshore structures, the towers of wind generators, vertical process engineering components

and pipelines or ducts. Another practical application of conical shells are roofs of silos and

storage tanks.

In the current version of the ECCS Recommendations design guidance is only given for

unstiffened cones under well­defined elementary boundary conditions. Stringer­stiffening as a

design tool for optimizing the structure (as often realized in cylindrical components) is not at

12

the designer's easy disposal because relèvent guidelines are missing. Furthermore, in many

applications the boundary conditions of the truncated cone deviate considerably from the

presumed elementary ones. (With regard to the latter point see also subproject C.)

Research A was aimed at filling these deficiencies in the Recommendations. First the effects

of various factors on the buckling behaviour of unstiffened and stiffened truncated cones were

to be quantified. Based on this knowledge, the existing design guidelines for unstiffened

conical shells were to be validated and improved, and they were finally to be extended into

stringer­stiffened conical shells.

Β ­ Local loads on cylindrical structures

Many vertical steel structures of cylindrical shape under axial loading are ­ contrary to the

presumptions of the buckling design rules in the Recommendations ­ not loaded and/or

supported uniformly around the circumference, but concentratedly along short parts of the

circumference. Typical applications include point supports of large elevated silos and storage

tanks (representing local loads at the lower cylinder edge) and rafter connections of large silo

or tank roofs (representing local loads at the upper cylinder edge).

Currently no design guidance is provided which would yield economically optimized

structural solutions for these locally loaded parts of the cylinder. The designer has to refer to

extremely simplified approaches in order to prove that he is on the conservative side.

Research Β was aimed at developing simplified and economic buckling design rules for these

cases. The real buckling and load carrying behaviour of axially loaded circular cylindrical

steel shells on point supports was to be carefully analysed, both experimentally and

numerically. The investigations were to cover a wide range of geometric parameters,

including cylinders which are stiffened by a 50% thicker bottom course of their wall. The

results were to be condensed into a design procedure.

C ­ Shells of revolution with arbitrary meridional shapes ­ buckling design by use of

computer analysis

Fundamental meridional shapes of shells of revolution include single cylinders and single

cones both supplied with well­defined boundary conditions. One of the more frequent non­

fundamental shell configurations, which the design engineer will come across in structural

applications, are shells of revolution of which the meridional shapes are arbitrary

combinations of cylinders and cones. Such cylinder/cone or cone/cone assemblies may

represent transition zones between different cross­sections in tubular members or in tower­

like structures or in pipelines (see also subproject A), or they may be special structural

solutions for process engineering purposes.

On the basis of current knowledge the designer will provide strong stiffening rings at the

meridional breaks in order to supply the cylindrical and conical sections with rigid radial

restraint conditions at their junctions. This makes them edge­supported fundamental shell

buckling cases for which the ECCS Recommendations apply. If the ring stiffeners shall be

omitted ­ because of cost­minimizing or any other reasons ­, the designer has to rely on

computer buckling analysis comprising the whole shell configuration. However, no guidance

is at present available how to handle and to interpret the results of this analysis in terms of a

safe and economic design.

13

Research C was aimed at developing such guidance. A set of cone/cone and cone/cylinder assemblies were to be tested experimentally under the two basic loading types, i.e. axial compression and external pressure. By comparing the test results with parallel numerical results, the interrelations were to be evaluated and finally to be processed into recommendations for the design engineer.

D - Thin-walled shells under wind loading

Wind loading produces a shell buckling problem in any cylindrical wall exposed to free atmospheric conditions. Among practical cases are tanks or silos when empty, free-standing chimneys and pipelines above ground.

Though no specific wind buckling rules are given in the ECCS Recommendations, sufficient source material is available in national codes and published research results to draft design rules for the relevant EC3 Parts which would yield wind-buckling-resistant cylindrical walls. However, there is one important economic aspect in connection with thin-walled open steel tanks which is not yet satisfyingly covered by reliable design rules: If the cylinder's top edge is strongly stiffened, its wall exhibits good-natured postbuckling behaviour. That means that the tank wall may be designed considerably thinner than necessary for full wind buckling resistance.

Research D was aimed at verifying this good-natured postbuckling behaviour experimentally under realistic wind tunnel conditions. The understanding of this complex phenomenon was to be enhanced, in order to create a reliable basis for shell buckling designs which take temporary buckles in empty steel tanks under strong wind deliberately into account.

E - Effects of cut-outs and openings in shells

The situation concerning cut-outs and openings in cylindrical shells is very similar to the one concerning local loads and supports (see subproject Β). Contrary to the presumptions of the buckling design rules, the shell walls are often equipped with openings of various shapes, sizes and locations. The openings will in most cases be reinforced by adequate stiffening or increased wall thickness around them. Typical applications include flue inlets in chimneys, nozzle openings in tanks or silos and maintenance doors in wind generator towers.

Currently only scarse design guidance is provided which would help to design the disturbed parts around the openings economically with specific regard to shell buckling. The designer has - similarly to the local load aspect - to refer to very much simplified approaches which often lead to overconservative structural solutions. Above all, there is still a remarkable lack of knowledge about the axial load carrying capacity (beyond the initial buckling resistance) of cylindrical walls with unreinforced openings.

Research E was aimed at filling this gap. The real buckling and load carrying behaviour of axially loaded circular cylindrical steel shells with openings was to be carefully analysed, both experimentally and numerically. The investigations were to cover a wide range of geometric parameters (shape, size and position of the openings with respect to the edges of the cylinder). The resulte were to be condensed into a design procedure.

14

ECSC Contract No. 721 O-S A/208

Enhancement of ECCS Design Recommendations and Development of Eurocode 3 Parts Related to Shell Buckling

Part A

STABILITY AND STRENGTH OF STIFFENED CONICAL SHELLS

Final Report

M K Chryssanthopoulos C Poggi Department of Civil Engineering Dipartimento di Ingegneria Strutturale

Imperial College Politecnico di Milano

A.l Introduction Conical frustra are often used in shell structures, for example, as transition elements between cylinders of different diameter in chimneys and marine structures, as hoppers in cylindrical silos or as end closures in tanks and pressure vessels. As with many other shells, buckling behaviour is an important design criterion, especiallly for thin-walled applications. Considering the possible load cases that arise in the above mentioned applications, it is evident that axial compression is one of the important conditions. Together with cylinders and spheres, conical shells may be regarded as elementary shell geometries, and as such it might be expected that their design, including buckling criteria, is well covered in present codes of practice. This is not the case, particularly when geometries in the intermediate slenderness are considered, which are affected by both material and geometric non-linearities. In fact, compared with their cylindrical counterparts, conical shells have received much more limited attention. The behaviour of stiffened cones (whether ring- or stringer-stiffened) is even less well researched than unstiffened cones and validated design rules are practically non-existent. It could be argued that stiffened cones are not widely used but, of course, this hrnited use could well be the result of scarse design information. At least insofar as transition elements are concerned, matching the overall geomerty of the conical shell to the top/bottom cylinder would offer advantages in load transfer and manufacturing. Moreover, stiffening along'a meridional direction should be an effective arrangement in resisting axial loads. In many modern limit state codes dealing with shell buckling problems, e.g. [1], a practical stability-based approach is followed. Generally, the elastic critical buckling stress is first given by analytical or semi-analytical expressions derived for a specific geometry, load type and boundary conditions. The so-called imperfection reduction factors (a factors) are then specified, largely based on lower bound curves using test data and/or some additional conservative assumptions. Finally, the interaction between elastic buckling and squashing is dealt with, which, in the case of steel structures, involves the relevant material property (in most cases the uniaxial tensile yield stress). The most difficult part in developing guidelines based on this approach is the appropriate specification of the imperfection reduction factors, especially in cases where the experimental data are limited or are not representative of full scale production. For unstiffened cones under compression, a formula for the elastic critical load does exist, subject to some assumptions. The α factors are also specified, with the help of cylinder results. However, as will be seen in the latter parts of this report, the entire procedure needs to be developed for stringer-stiffened cones.

A.l.l Summary of Previous Work The buckling behaviour of unstiffened cones under compression has been the subject of some early analytical studies based on linear theory, e.g. Seide's reference paper [2] in which axisymmetric elastic buckling is investigated. The classical solution given in this work is

_ 2 π £ ί 2 cos2p

V3 (1 - v2) from which it can be seen that, in comparison to a cylinder, the critical buckling load of a cone is also affected by the semi-vertex angle, p. Subsequent studies, e.g. [3], dealt with the asymmetric buckling problem and found that the above expression is also valid for this type of buckling, for some of the commonly assumed stability boundary conditions. Recently, this problem has been revisited, in order to quantify the critical buckling mode and to look at the effect of clamping [4].

17

Chang and Katz [5] give a concise review of research on cones up to 1980 before moving on to present a study on the effect of edge constraints on the elastic stability of unstiffened cones. More recently, the elasto-plastic response of axially compressed unstiffened steel cones, both as isolated elements [6] and as part of larger shell assemblies [7] has been investigated. In spite of these studies, it is fair to say that the behaviour of conical shells remains, to a significant degree, unexplored. This is also evident from the fact that current design procedures [1] treat the conical shell as an equivalent cylinder, even though important differences are generally acknowledged in research studies. For example, the imperfection sensitivity of axially compressed cones is not thought to be identical to that of cylinders but its exact nature has not been quantified. Equally important from a design point of view, any procedure that treats the cone as an equivalent cylinder must specify a limiting semi-vertex angle beyond which it no longer applies. There seems to be a lack of studies which concentrate on validating the range of applicability of current design approaches. Stiffened conical shells have been the subject of very few studies [8, 9] mainly dealing with a smeared approach applied to ring stiffened cones. If, however, the cone acts as a transition element in an axially compressed shell assembly, the presence of stringers is likely to be a more effective stiffening arrangement. Furthermore, in certain applications, such as offshore structures, the spacing of the stringers is fairly wide and, as a result, the assumptions of smeared theory become invalid. No studies on widely spaced stringer-stiffened cones under compression have been found but an important piece of work for the present investigation is a publication by Samuelson [10], where approximate methods for the design of stiffened cones are discussed.

A detailed review of previous work is not intended within this summary report. The above overview points to both historical and recent studies, which can be consulted in order to trace a more complete set of references on the buckling behaviour of conical shells.

A.1.2 Objectives of Current Investigation The latest version of the ECCS Recommendations [1] contains design guidance for unstiffened conical shells under various loads (meridional compression, uniform hydrostatic lateral pressure, torsion and/or shear, liquid-filled cones). However, no guidance is given for stiffened conical shells and, moreover, the rules for unstiffened cones need to be further validated ans expanded. The present research programme is aimed at quantifying the effect of various factors (e.g. boundary conditions, initial imperfections) on the buckling behaviour of unstiffened and stiffened conical shells and, more importantly, at producing design guidelines for stiffened conical shells, which could be included in the next edition of the ECCS recommendations or in the corresponding shell buckling parts of EC3. It is neither possible nor cost-effective to study all the factors influencing buckling response using solely experimental methods. Hence, a combination of experimental work and numerical studies was undertaken summarized in the following chapters. Simplified formulations suitable for design recommendations were developed and validated using numerically derived design data. The project was jointly carried out by Imperial College and Politecnico di Milano. Full reports on the studies undertaken are given in the reports/theses referred to in the ensuing chapters. However, important results and main conclusions, together with the design implications, are presented herein.

18

A.2 Experimental Tests As demonstrated in many other shell buckling studies, experimental tests play an important role in obtaining a better understanding of the structural behaviour through observation and measurement of the primary factors which influence the response. In addition, if properly instrumented, tests provide data on which the validation of numerical/analytical tools can be carried out. The present study includes tests on unstiffened and stringer-stiffened conical shells under axial compression. The selection of specimen geometries was guided by the need to obtain elasto-plastic buckling failures, relevant to the validation of analysis tools and the development of design formulations for steel shells. It also took into account previous tests (in order to provide meaningful comparisons) and some restrictions imposed by experimental facilities. Full experimental results are contained in [11] and [12] for unstiffened and stringer-stiffened cones respectively; a summary is presented below.

A.2.1 Model Geometries The geometric properties of the unstiffened cones are shown in Table A.2.la. The dimensions are similar to those adopted by Krysik & Schmidt in their testing programme [13]. However, the present specimens are made somewhat more slender by using thinner material, since the objective is to study geometries that come closer to the transition range (between elastic and elasto-plastic buckling). This is generally considered the most challenging range in order to test analytical/numerical predictions, because of the interaction between material and geometric non-linearity. It is also the slenderness range in which many steel shells are designed. In Table A.2.la, the dimensions refer to clear distances between end rings, which are fitted to the models after manufacturing in order to create well defined end conditions. Figure A.2.1 shows the models, before and after fitting of end rings.

Table A.2.1a: Model dimensions - Unstiffened Cones Model

Reference

UC01 UC02

UC03 UC04

Tapering Angle

Ρ (°) 15 15

30 30

SmaU Radius

Rl (mm)

100 100

100 100

Large Radius

Rl (mm)

225 225

225 225

Slant Length

L (mm)

482.9 482.9

250.0 250.0

Shell Thickness

t (mm)

0.9 0.7

0.9 0.7

In selecting model geometries for stiffened cones, a wide range of possibilities was faced, since no previous tests have been performed. In order to allow a comparison with unstiffened counterparts, it was decided to have the same overall geometric parameters as UC01 and UC02, but to vary the number of stiffeners. The final geometries of the stringer-stiffened models (Table A.2. lb) are such that they belong to the family of sparsely-stiffened shells, for which local buckling in the elastic-plastic range is likely to occur.

A.2.2 Manufacturing of Specimens The unstiffened specimens were fabricated from steel sheets by cold rolling and seam-welding along a single longitudinal axis. In order to remove high weld induced residual stresses, the models were stress-relieved in a heat treatment oven. Thus, each model was fabricated from a single sheet of material reducing the effect of material property variations.

19

Boundary conditions were provided through heavy accurately machined steel rings. These were attached to the upper and lower end of the models using a mixture of araldite and sand which filled the gap between two concentric rings (Figure A.2.2).

Table A.2.1b: Model dimensions - Stiffened Cones

Model Ref.

SCOIA

SCOIB

SC01B

Tapering Angle

Ρ (°) 15

15

15

Top Radius

Rl (mm)

94.6

94.6

94.6

Bottom Radius

Rl (mm)

230.4

230.4

230.4

Slant Length

L (mm)

524.3

524.3

524.3

Shell thickness

t (mm)

0.8

0.8

0.8

Stiff. No.

8

8

16

Stiffener thickness

tw (mm)

1.7

0.8

0.8

Stiffener depth

hw (mm)

10

10

10

The fabrication of the stiffened cones was significantly more complex, mainly because of the much more extensive welding required, which is always problematical in small-scale models. In model SCOIA, the stiffeners were first positioned by means of small indentations on the shell and corresponding notches on the stiffeners. A staggered TIG weld (placed alternatively on the two sides of the stiffener) was then created along the shell/stringer junction. However, this process resulted in relatively high distortions on both shell and stringers. There was also significant addition of weld material, which implied an effective stringer thickness higher than the nominal value of the sheet.

Better quality was obtained in models SC01B and SC02B through the use of a purpose built copper jig. This enabled accurate positioning and locking of each stiffener on the shell prior to welding, which improved weld induced distortions. The copper jig was also able to absorb a large part of the weld induced heat. Finally, the welding process was modified to allow a virtually continuous spot TIG weld to be produced, with very small weld material deposited along the weld. In keeping with common practice, all three models were stress relieved after welding.

A.2.3 Measurement of Material Properties A.2.3.1 Yield Stress According to the specification, the material category is CR4 (cold rolled on wide mills to the final thickness) which according to BS 1449 (1983) has a specified 0.2 % yield stress of 140 N/mm2. From each sheet used in the manufacturing of a single cone specimen, a number of 400mm χ 40mm coupons in different orientations were prepared for tensile testing. Figure A.2.3 (upper part) shows a typical coupon arrangement for one of the models.

In general, the tests revealed that the material remains elastic up to approximately 180 N/mm2 and then starts to exhibit plastic deformation, reaching the ultimate strength at a strain of about 25%. As can be seen from Figure A.2.3 (lower part), the material is ductile without a flat plateau following initial yielding. It is, thus, reasonable to calculate a yield stress using the 0.2% proof value. In terms of the influence of orientation, there are no clearly established trends, and it was decided to treat the material as homogeneous and isotropic. Since the shell models are tested in axial compression, compression coupon testing would have been appropriate. Because of the thinness of the material (as low as 0.7mm) this was not considered feasible. This should be noted in connection with calculated squash loads, since some previous studies suggest that compressive yield stress can be about 10% higher than its tensile counterpart.

20

A.2.3.2 Modulus of Elasticity In order to obtain a more accurate measurement of the elastic modulus, an extensiometer was used with an accuracy of 10 με over a gauge length of 50 mm. On average, from 36 data points, a value of 201.2 kN/mm2 was obtained for E with a coefficient of variation of about 5% [11]. This value is somewhat larger than what is normally expected for the variation of E (typically 2-4% cov's have been found in previous studies on structural steel).

A.2.4 Test Rig and Measuring Devices The models were tested in a circular rig shown schematically in Figure A.2.4. Axial compression is applied by a screw jack (100 ton capacity), driven by a variable speed electric motor through a low ratio gearbox, thus enabling a very slow loading rate to be achieved. The force created by the jack is transmitted to the model as a uniform axial displacement through rigid circular plates sitting on ball bearings.

A.2.4.1 Arrangement of Logging Equipment A large number of imperfection and deflection measurements is needed in shell buckling tests to enable analytical/numerical modelling to be undertaken but also in order to understand the pattern of deformation under load. In small-scale tests, data acquisition and processing becomes a difficult task because of the required accuracy in measurements, as well as due to the precision required in the manufacturing of the measuring equipment. Forty and twenty transducers, for the long (UC01, UC02, all SC models) and short cones (UC03, UC04) respectively, were used to measure the initial imperfections and the displacements under load. The number of transducers was decided based on the elastic buckling modes which consisted of 21 and 11 axial half-wave for unstiffened long and short cones respectively. The circumferential spacing was determined in order to maintain approximately a square mesh. The two transducer arms were connected to a small circular frame which can be rotated (and locked at predetermined positions through pegs) about the central vertical axis of the rig (see Figure A.2.4). Hence, a complete scan of the model could be achieved by rotating the frame through a full circle. In addition to transducers which measure out-of-plane profiles, another set was used to monitor axial end shortening (see next section). In order to obtain a reference surface for zero displacement, two trapezoidal plates were accurately machined to represent the perfect geometry of the models. The readings taken on these plates provided the datum from which subsequent transducer measurements on the models (imperfections or deflections) could be evaluated. In dealing with stringer-stiffened models, measurement of stringer imperfections/deflections is also necessary. This could not be performed inside the loading rig but was recorded on a separate piece of equipment. Thus, only stringer imperfections and post-buckling permanent deformations have been recorded. The detailed strain gauge layout for a typical long cone (UC01) is shown in Figure A.2.5. In general, more strain gauges were attached close to the small-radius end as the response of the shell in that area was expected to be important with respect to the predicted failure mode. In the stiffened models, meridional strains along the stiffener length were also monitored by positioning gauges close to the stiffener tip.

A.2.4.2 Measurement of End Shortening A set of three transducers, placed at equidistant radial positions from the axis of the cone, give a complete scan of the end shortening pattern of the model. In all models, the end shortening was measured between end plates (see Fig. A.2.6, end shortening denoted as 'B'). In addition, end shortening was measured between end-rings for one unstiffened and one

21

stiffened model (see Fig. A.2.6, end shortening denoted as 'A'). The reason for this alternative measuring system, introduced in the latter part of the study, is because of discrepancies found in pre-buckling stiffnesses when experimental results were compared with FE calculations. Clearly, end shortening 'A' is confined to the clear model length, whereas end shortening 'B' includes the part which is embedded in the araldite-sand mixture and the 2mm clearance between end-rings and loading plates (Fig. A.2.2).

A.2.5 TestProcedure Before mounting each model into the rig, the reference surface for displacement measurements was obtained. This was achieved by positioning the trapezoidal plate representing the 'perfect' cone into the rig and recording initial readings at a number of circumferential positions. The plate was then replaced by the actual model. Elastic tests were first carried out to check concentricity of applied load and the correctness of the test set-up. Once the set-up was confirmed to work properly and repeatability of measurements was established, the position of the model in the rig was fixed and imperfection measurements were carried out. Table A.2.2 contains a summary of extreme imperfection values measured on each model (inward/outward). The full imperfection data have been processed using a 'best-fit' procedure and Fourier analysis (see section A.2.6.1 below) but it is worth bearing in mind that the measured imperfections on the models are in excess of the tolerance values specified in the ECCS recommendations [1]. This is not surprising given the small scale of the models but should be taken into account in correlating the experimental results with the design procedure adopted in the ECCS recommendations [1].

The failure test was then carried out by incrementing the applied axial displacement in small steps. All the models were loaded far beyond the peak load to obtain information on the post­buckling response.

Table A.2.2: Summary of Extreme Imperfection Values

Model Ref.

Max. Imp. In / Out (mm)

UC01

1.8/1.7

UC02

1.7/1.4

UC03

1.1/0.8

UC04

1.1/1.4

SCOIA

1.8/1.9

SC01B

1.6/1.0

SC02A

3.4/1.7

Note: These are the values obtained after the "best-fit' procedure.

A2.6 Summary of Test Results Table A.2.3a presents a summary of the experimental results on unstiffened cones in terms of peak loads [11]. The peak loads of the thicker models (UC01 and UC03) were close to the squash load (about 93%) whereas for the thinner models (UC02 and UC04) the peak load was about 77% of squash. Similarly Table A.2.3b summarizes the results for the stiffened cone models [12]. In the unstiffened cones, the collapse mode was confined to the small-radius end, with a fairly regular axisymmetric bulge forming very close to the end ring. This is the so-called 'elephant foot' mode, see Figure A.2.7a. Only model UC02 showed some evidence of non-axisymmetric behaviour, which was possibly due to overall bending taking place after the peak load had been exceeded. The stringer-stiffened models exhibited collapse modes contained within the shell panels between stringers, commonly referred to as local panel buckling. Clearly, the collapse modes are asymmetric as a result of discrete stiffening. In models SC01B and SC02B, the largest deformations developed once more close to the small-radius end, whereas in model SCOIA collapse took place at mid-length. Considering that SCOIA and SC01B are nominally identical but produced by different manufacturing methods, this result shows how

22

manufacturing distortions can influence the collapse mode. Figure A.2.7b shows a view of model SCOIA after collapse.

Table A.2.3a: Summary of test results ­ Unstiffened cones

Model Ref

UC01 UC02

UC03 UC04

Elastic budding

load

Per (kN)

578.2 349.8

464.8 281.2

Squash Load

Po (kN)

104.3 94.2

93.5 84.5

PerlPo

5.54 3.71

4.97 3.33

Test ultimate load

( failure mode)

97 (axisym.) 73 (axisym.)

87 (axisym.) 66 (axisym.)

Pu (exp)IPo

0.93 0.77

0.93 0.78

Notes: Elastic buckling load calculated from Seide's formula (see Chapter A.l) Squash load calculated using 0.2% proof stress values from tensile tests

Table A.2.3b: Summary of test results ­ Stiffened cones

Model Ref

SCOIA SCOIB SC02B

Squash Load

Po (kN)

115.2 105.7 109.3

Test ultimate load

( failure mode)

'¿sr' 110.3 (local) 107.3 (local) 121.1 (local)

Pu (exp)'Po

0.96 1.01 1.11

Note: Squash load calculated using 0.2% tensile proof stress

A.2.6.1 Geometric Imperfections

Initial geometric imperfections were recorded on all specimens using the procedure described earlier. In order to render these measurements useful for comparative studies and numerical modelling, the imperfections were subjected to the following data processing techniques:

(i) best­fit analysis (ii) two­dimensional Fourier analysis

The first procedure is necessary in order to remove the influence of possible misalignments and rigid body movements in the rig. Fourier analysis is then performed on the best­fit data, in order to characterize the entire imperfection surface through a set of coefficients of simple harmonic modes. This enables the identification of dominant modes, and facilitates comparisons of imperfections with critical buckling modes. Figure A.2.8 shows a typical imperfection surface after best­fit analysis for one of the stringer­stiffened models (SCOIA). The influence of stringer spacing on the dominant circumferential imperfection wavelengths can be clearly seen. This is also shown in terms of the dominant Fourier modes in Figure A.2.9. A two­dimensional Fourier sine expansion has been used in this case, i.e.

wo (χ, Θ) = 2­1 ξ# sin^p­ sin (/Θ + (Sfa )

/=0

23

Note the small number of dominant modes that are present in the imperfection surface, a feature common to all the stringer-stiffened models. For the unstiffened models, there is a higher spread of dominant modes, although long wavelength modes are still governing. Full plots of raw imperfection data followed by the best-fit data plots and those arising from the ensuing Fourier analysis are given in [11] and [12]. This systematic approach to imperfection measurement and further processing is essential in the validation of numerical tools, as will be seen in the next chapter. Having the full imperfection surface described concisely via a set of Fourier coefficients facilitates considerably the input to FE models, enables mesh selection to be made independently of measurement considerations and renders the experimental results accessible to wider use.

A.2.6.2 Load-End Shortening The load-end shortening plots, as an indication of the overall response, are shown in Figure A.2.10 for the unstiffened models and in Figure A.2.11 for the stiffened models. In general, a linear behaviour is exhibited until the peak load is reached, although in the stiffened models (SC02B in particular) some non-linearity is evident prior to that point (this is due to the higher stiffening ratio as a result of closer stiffener spacing in this model). A sudden drop in stiffness occurs at the peak load and the post-buckling response is characterized by negative slope (unstable post-buckling) with the unstiffened models clearly undergoing steeper unloading than their stiffened counterparts. This can be attributed to the higher degree of stress redistribution possible in the stiffened models (related to material non-linearity) and to their lower imperfection sensitivity (related to geometric non-linearity).

As far as the pre-buckling stiffness is concerned, due to the differences observed between experimental and numerical values (as well as analytical 'membrane solutions', which are available for unstiffened cones), a second set of readings (see Fig. A.2.6, end shortening denoted as 'A') was performed in some of the models. Figure A.2.12 compares the results obtained for one unstiffened and one stiffened model. Note that Model UC05 has the same overall dimensions as UC01 and UC02 but its thickness lies in between the other two values (0.8 mm for UC05 compared to 0.9mm for UC01 and 0.7mm for UC02). As can be seen ,the difference between end shortening 'A' and 'B' is very significant, with 'A' being in much closer agreement with analytical/numerical values, as discussed in the following chapter.

A.2.6.3 Strains and Stresses Typical strain gauge results for UC02, together with a schematic diagram showing their location on the model, are shown in Figure A.2.13. At low load levels, the strains are distributed fairly evenly around the circumference. Furthermore, strain increments are linear with respect to load increments. However, as the load increases, the distribution starts losing both linearity and uniformity. In general, the circumferential position 07360° exhibits lower strains whereas the diametrically opposite position is the more strained area. The single longitudinal weld located on the 0°/360° Une seems to be the reason for this non-uniformity. In fact, it is worth noting that an approximately diametrically opposite position seems to the one at which the collapse mode is initiated. This observation is valid for the other models as well [11]. As expected, strains are generally higher near the small-radius end.

The variation of strain with increasing load for UC02 is shown in Figure A.2.14. For comparison, the value obtained by assuming linear membrane pre-buckling is also included and, as can be seen, the agreement between theoretical and experimental results is generally satisfactory. Finally, the stresses in UC02 are plotted in Figure A.2.15. As expected, the hoop stresses are generally small, apart from region close to small-radius end, where, as it might be expected, the assumption of membrane pre-buckling is not wholly appropriate. The hyperbolic shape of the meridional stress distribution along the length is also important, being consistent with the theoretical distribution predicted by membrane pre-buckling.

Similar plots and comparisons have been undertaken for all the models within the present test series and the results are extensively discussed in [11, 12]. Conclusions are broadly in line

24

with the above remarks. Since conical shells, are not as widely investigated as their cylindrical counterparts, these results should help in forming a well documented database for future reference and use.

A.2.6.4 Lateral Displacements The availability of full imperfection and deflection scans enables observations to be made about the growth of displacements under load. Relative pre-buckling deflections with respect to the imperfections for model SCOIA are depicted in Figure A.2.16a (cf. Fig. A.2.8 which shows the imperfect surface of the same specimen). It can be seen that deflection growth takes place primarily in the middle part of the cone, where five inward lobes in five different panels develop, triggered by the presence of high initial imperfections in the same areas. These initial bulges grow further, together with other modes that develop close to the small-radius end, as can be seen in Figure A.2.16b, which is the final scan in the post-ultimate range. The load levels at which these scans were made can be found in Fig. A.2.11. It should be noted that Fourier analysis of the pre-buckling deflections just prior to collapse is generally not accurate due to the localised deflection growth, which inevitably makes harmonic decomposition problematical. Detailed comparisons can be found in [12].

Finally, in [11] and [12] a comparison between imperfections and deflections under load is also made by plotting a whole range of circumferential profiles for each model (typically 20 or 40 profiles per specimen for short and tall cones respectively). These profiles enable-a quantitative comparison to be made, in addition to the overall picture obtained from the figures presented in this report.

A.2.7 Concluding Remarks Fabrication and testing of small-scale models have been undertaken to examine the buckling behaviour of unstiffened and stiffened conical shells. Manufacturing of the latter has proved to be a very demanding process requiring extensive jigging to avoid unacceptable weld distortions. In all models, small-scale manufacturing has produced relatively high imperfection values, which cannot be considered typical of full-scale structures. However, since the full imperfection surface has been recorded, the test results can be used to validate numerical (finite element) or other models. The behaviour of the models is summarised in Table A.2.3 (peak loads) and Figures A.2.7 (failure modes), A.2.10 and A.2.11 (load-end shortening). Representative results of stress, strain and displacement plots and their potential use in further comparisons have been presented for typical models. Since, in all cases, the squash load is well below the elastic critical load, the failure mode was of the axi symmetric 'elephant foot' type in the unstiffened models and local elasto-plastic shell buckling in the stiffened models. The ratio of experimental failure load to squash load {PulPo) is approximately equal to 0.77 and 0.93 for thin and thick unstiffened cones respectively, regardless of the variation in the tapering angle. These values indicate that the thinner cones are influenced to a greater degree by initial imperfections and boundary conditions. The same ratio {PJPo) is much closer to unity for stiffened cones, indicating the beneficial effect of stiffeners in the slenderness range considered in this study. Moreover, the stiffeners play a role in the post-buckling characteristics by reducing the negative slope of the post-buckling path.

25

ro

Fig. A.2.1: Unstiffened conical shell specimens prior to testing

Fig. A.2.2: Schematic diagram of end conditions

28

<

/ \ 45

°

1

2

/

4 3 specimen

1000

400

300

200

Ä- 10°

E .E ζ

w M t) u *J

CO

400

300

200

100

Strain, ε

Strain scale

0% 5% 10% 15% 20% 25% 30% 35% 40%

Fig. A.2.3: Measurement of yield stress

29

LEGEND

fi) Cross Beam

(T) Load Cells

( s ) Model

Ç*) Array of Transducers

(V) Screw Jack

(7) Ball Bearing

Fig. A.2.4: Schematic diagram of the test rig

30

120°

Fig. A.2.5: Strain gauge instrumentation on UC01

End ring:

Cross beam

Moving bottom plate

A - End shortening measured between end rings (specimens SC02B & UC05)

Β - End shortening measured between end plates (all specimens)

Fig. A.2.6: Measurement of end shortening

31

OJ K )

Fig. A.2.7: Collapse modes (a) unstiffened (b) stiffened

Fig. A.2.8: Typical imperfection surface (SCOIA)

(1.3)

Fig. A.2.9: Amplitude of Fourier imperfection modes (SCOIA)

33

a, ■β «

2

120

100

80

60

40

20

120

100

80

60

40

20

0.5 1 1.5 End­shortening, Δ^ [mm]

2.5

Fig. A.2.10: Load­end shortening plots (unstiffened cones)

34

2 "O (O O

"δ

mu -

120 -

100 -

80 -

60 -

40 -

20 -

0 -

+ Scan positions _

jfe 'JT !**""-**. ί ^ ^ · * ^ _

! Ù* f \ ; ^XJ

Jf ¡ ^ ^ ^ \ jl·

fj i \ / / /

ƒ ƒ ! / 7—ji" ff \ \' ι ι ff \ A ι /

ff \ ι \ i / / . . 1 i / i/ / / ι

7 ¿ /

/ i / / !· r 1 ' ■ ■ i ι

0.0 0.5 1.0 1.5 2.0

end­shortening [mm] 2.5

SC01A

SC01Β

SC02B

Fig. A.2.11: Load­end shortening plots (stiffened cones)

140

ζ ¿^ •a to o

n

0.0 0.5 1.0 1.5 2.0

end­shortening [mm] 2.5

Fig. A.2.12: Comparison of alternative end­shortening measurements

35

0.0 0.033

0.1 ÍS

Circumferential Position ['

Fig. A.2.13: Strain distribution in specimen UC02

36

Load,P|kN|

LO

51 qq

> to

CL

<

k e. 3

S" 45 8

3

C

9 to

(J­J,)/L ■ 0.967 (s­s^tL­ 0.303 (J­J,)/L« 0.155 (J­J,)/L ­ 0.033

07360°

Circumferential positions

120° 240°

LO 0 0

8

a ζ

D"

δ 0)

ì o

•c

a a

0*

! (0

α

S se

1 ton

2 tons

3 tons

4 tons

5 tons

6 tons

O 0.2 0.4 0.6 0.8 I Axial Coordinate, ( s-s. ) / L

0.2 0.4 0.6 0.Θ

Fig. A.2.15: Stress distribution in specimen UC02

Fig. A.2.16: Deflection growth in specimen SCOIA (a) pre-buckling (b) post-buckling

39

A.3 Validation of Finite Element Modelling A.3.1 Selection of FE Models The first action undertaken in the numerical activities was the calibration of the numerical tool. The optimum finite element model to analyse the structural problem under consideration was determined simulating the elastic-plastic buckling tests on conical shells described in the preceding chapter. An extensive numerical investigation was performed to test the full range of available finite element models. In fact, shell buckling problems are known to be among the most difficult to tackle via numerical analysis, both owing to unstable post-buckling behaviour and because of high imperfection sensitivity typically encountered in these structural elements. All the FE calculations have been performed using the general purpose finite element program ABAQUS [14].

A.3.1.1 Unstiffened Conical Shells Three different geometrical models have been examined: i) axisymmetric models ii) conical panel models with varying width iii) complete cone models In addition, different finite elements (in terms of element formulation) from the extensive library available in the package used have been tested. The advantages and disadvantages of the three models are reported in the following. Model (i) Advantages

The geometry is described by modelling only the meridian. The resulting number of equations is limited and therefore this model is the most convenient in terms of required computer time.

Disadvantages The model cannot describe an asymmetric'buckling mode (which is likely in conical shells, at least insofar as elastic buckling is concerned). It is not possible to include asymmetric imperfections, which are invariably present in any realistic application.

Model (ii) Advantages - Disadvantages

Asymmetric imperfections can be modelled but only part of the entire imperfection surface may be considered in any one model. The boundary conditions along the straight edges of the panel (along meridians) can be specified as either symmetric or antisymmetric but, in either case, this implies that some restrictions are imposed on the buckling mode shape allowed to develop. In terms of computer time this model is much more demending than model (i) but still manageable, even for the purposes of parametric studies.

Model (iii) Advantages

The geometric imperfections can be modelled in detail (i.e. both axisymmetric and asymmetric modes). The response in terms of displacements, strains and stresses can be compared in detail with any available test results.

Disadvantages The model is the most burdensome in terms of computer time and because of the high number of equations (degrees of freedom) it becomes difficult to accommodate even on dedicated workstations.

41

A.3.1.2 Stiffened Conical Shells The simulation of the experiments on stringer-stiffened cones was carried out by means of a model of the complete cone but other simplified models were also used. In particular, some stiffened panel models, exploiting the symmetry conditions, were studied. Panels with different width, defined by a pair of successive mid-panel meridians (and, therefore, with different numbers of stringers) were analysed [12,15,16]. The finite element models used shell elements for both the shell wall and the stringers. The particular element adopted is a nine-node Lagrange doubly-curved shell element using a reduced 2x2 Gauss integration scheme and five degrees of freedom per node. Five integration points across the thickness were considered to account for non-linear material behaviour. The element formulation is based on large displacements and small strains.

A.3.2 Modelling Aspects and Convergence Study In all the models the following conditions were assumed: Loading conditions The study is devoted to axially compressed conical shells and in all the experiments performed only vertical loads were imposed. During the tests the load was applied as imposed shortening by means of heavy plates placed at the two ends of the model. These plates were in full contact with both the shell and the stringers around the entire specimen circumference through an adequate clearance at the ends, see Chapter A.2. Similarly, in the numerical models the load was applied at the top end (small-radius end) and was uniformly distributed. If present, the stringers were also loaded in the same manner. The effect of the stiff end-plates, which is to impose uniform end shortening, was simulated in the numerical model through the use of constraint equations, which link the displacement(s) of a set of nodes in a pre-defined manner (in this case, imposing equal axial deflections to all the nodes of the small-radius end).

Boundary conditions With reference to Figure A.3.1 and the adopted symbols, the boundary conditions are the following: top circle (small-radius end): v = 0

u sinp + w cosp = 0 u.s = 0

bottom circle (large-radius end): fully clamped Therefore, the conical shell is effectively clamped at both ends, and the shortening at the ends is constrained to take place in the axial direction only (thus simulating the presence of heavy rings or bulkheads at the two ends).

Material properties The material properties used in the numerical models are based on the results obtained in the experiments (coupon testing). In particular, Young modulus = 201200 MPa ; Poisson ratio = 0.3 The full stress-strain relationships obtained from the experiments were used (see chapter A.2). As mentioned previously, the yield stress for each conical shell specimen was defined as the average value determined from several coupon tests. This was the best option in the absence of any clearly identifiable trends realted to the orientation of coupons. Furthermore, since no compression coupon testing has been undertaken, it was decided not to make any changes to the tension coupon test results. As is well known, some studies advocate the use of higher yield stress in compression than in tension. The Von Mises yield criterion is used in the FE analyses for general stress conditions.

42

A.3.2.1 Axisymmetric Model The axisymmetric finite element SAX2 has been used. Each element has 3 nodes with five integration points over the thickness and two along the length. The resulting finite element mesh and the first eigenmode are reported in Figure A.3.2. A convergence procedure was adopted to evaluate the appropriate number of elements to model the particular buckling problem in hand. The number of elements was varied from 50 to 200 although the model should also conform with the number of imperfection and deflection readings taken. Thus, to simulate the experiments one has to adopt a model with only 21 elements so as to introduce the geometric imperfections in the exact location where readings were taken (i.e. where the LVDTs were positioned).

A.3.2.2 Conical Panel Model This model was studied mainly in view of its potential use in the analysis of stringer-stiffened shells. The convergence procedure was based on the simulation of a corresponding unstiffened cone. The conical panel width is equal to 1/8,1/16 and 1/32 of the circumference and the FE models are based on 16x40, 8x40 and 4x40 (circumferential χ axial) meshes respectively [15,17].

A.3.2.3 Complete Cone Model A model comprising 25 elements in the axial direction and 72 elements in the circumferential direction was used. It was demonstrated that this number of elements was sufficient to evaluate correctly the first eigenmodes which contain up to 23 axial half-waves. The first eigenmode for a typical unstiffened cone is reported in Figure A.3.3 [15, 17]. Different considerations prevail in the case of stiffened cones, which are described in more detail below (section A.3.3.3).

A.3.2.4 Model Comparisons For any particular unstiffened cone geometry, the eigenvalues obtained with the models outlined above were all very close, even though the first model is only able to evaluate axisymmetric buckling modes. This observation indicates that the linear buckling behaviour of an axially compressed conical shell is similar to the behaviour of its cylindrical counterpart, which, as is well known, is characterized by a number of almost coincident buckling loads, each admitting a different buckling mode shape. The analysis of the shape of the eigenmodes is straight forward in the axisymmetric models, since the number of waves in the axial direction can be easily counted. However, in the other models (panel and, more importantly, complete) it was necessary to perform a Fourier analysis of the FE eigenmode results in order to determine dominant harmonics. Since the buckling modes cannot be readily represented in analytical form (as is the case for cylinders) three different ways of performing a Fourier analysis were tested [15]. The eigenmodes of the panel model are comparable with those of the complete model, provided the width analysed can conform with the circumferential buckling wavelength. A common feature of all FE eigenmodes is that, for the unstiffened cone geometries analysed (which correspond to the specimens tested experimentally), damped modes are found in the axial direction (i.e. the amplitude of the waves decreases as the distance from the small-radius end increases).

A.3.3 Comparison of FE Results with Tests A.3.3.1 Analysis of Imperfect Geometries The simulation of shell buckling experiments by finite elements must involve modelling of the geometric imperfections recorded on the test specimen. As reported in chapter A.2, the

43

real imperfections were processed through best-fit and Fourier analysis. Hence, analytical expressions of the imperfection surfaces were available and a selection of relevant modes for inclusion in the numerical model could be made. As mentioned before, the sole use of axisymmetric Fourier modes of the imperfect surface which can be introduced in an axisymmetric FE model - model (i)- does not describe adequately the actual imperfection surface and, in most cases, produces a response which is different from that observed experimentally. Even the use of a model based on a single imperfect panel cannot fully simulate the experiments undertaken. Once more in this case, the selected sector of the actual imperfect surface which is inputted in FE model cannot capture the full response. Furthermore, the choice of the most appropriate imperfect sector to be considered is not easy, since it cannot be based on simple criteria such as the maximum deviation from the ideal geometry. In the case of stringer stiffened cones, the stringer imperfections have also been considered. However, the experimental data refer only to the out-of-plane initial deflections along the stiffener outstand, and a linear interpolation is used to complete the stiffener profile. It is worth noting that the use of sector (or single panel) models has a stronger justification in this case, especially if the geometry is such that local buckling between stiffeners (as opposed to overall buckling) governs the response. This is further discussed below.

A.3.3.2 Unstiffened Cones A most significant comparison can be made in terms of load-end shortening curves since these describe the overall response of the structural element under consideration. A typical comparison is reported in Figure A.3.4. As described in Chapter A.2 and reported in [18], the elastic stiffness of the specimens measured experimentally is close both to the membrane solution and to the FE solution if the elastic shortening is measured between the two rings i.e. if the measurements refer to the clear part of the model. In fact, it was noted that the elastic stiffness evaluated on the basis of end shortening measurements for the complete cone is significantly lower. Various FE models concentrating on the end details of the tested models have been studied in [18] in order to quantify these differences. On the post-buckling path the FE models exhibit a slope similar to that recorded experimentally and the agreement is considered satisfactory. The collapse mode observed in the experiments is also very similar to that obtained with FE analysis. For the four unstiffened models tested, it consists primarily of an axisymmetric bulge near the top end, i.e. the small-radius end. This is the so-called 'elephant foot' collapse mode.

In terms of peak loads, the FE prediction is very satisfactory (see Table A.3.1), although marginally better for long cones where the discrepancy is confined to within 2%. The introduction of the full geometric imperfections seems to be necessary, even though the final collapse mode is axisymmetric and the collapse load is not far off from the squash load.

Table A.3.1: Comparison of FE and experimental results for unstiffened cones

Model Ref.

UCOl UC02

UC03 UC04

Slenderness parameter

(ECCS defn.)

0.56/0.80 0.71 /1.01

0.61/0.86 0.77 1.09

Test ultimate load over FE prediction perfect geometry

0.90 0.76

0.88 0.75

Test ultimate load over FE prediction imperfect geometry

0.98 1.00

0.96 0.96

Note: Two values for the ECCS slenderness parameter are given: - the first based on an α factor applicable to imperfections less than the tolerances - the second based on a reduced value (a /2) due to the high imperfections in the specimens.

44

A.3.3.3 Stringer-Stiffened Cones A complete model was used to perform both eigenvalue analysis and fully non-linear step-by-step analysis. The eigenvalue analysis is not directly comparable with experimental results but it is important for evaluating the slenderness of the model (P0/Pcr) and the reduction factor (PJPcr)· It is also relevant since it allows the determination of the eigenmode which needs to be compared with the Fourier imperfection shape to detect the presence/absence of the critical mode in the actual imperfection. Note that, contrary to many other cases (including the axially compressed unstiffened cones) there are no analytical expressions with which the FE eigenvalue predictions can be compared. In this respect, the validation of numerical tools for stringer-stiffened cones can only be based on comparisons with the experimental results which are all in the elasto-plastic regime. Thus, the ultimate load Pu is expected to be affected by pre-buckling non-linearities (if present), the effects of geometric imperfections and material non-linearities, with the latter having perhaps the most important influence. The first eigenmodes for the three experimental models (SCOI, SCOIB and SC02B) are depicted in Figure A.3.5 [12]. It should be noted that, in all cases, local panel buckling is predominant. Significant stringer out-of-plane deflections are evident for model SCOIB, where elastic buckling occurs close to the smaller radius, but they are present also in the other models. These out-of-plane deflections develop so as to maintain the stringers perpendicular to the buckled shell wall. Since local panel buckling is dominant, eigenvalue analysis can also be performed using an FE panel model with very satisfactory results. Further comments and considerations about the eigenvalue analysis can be found in [12,16].

It was significant, as in other elastic-plastic buckling problems, to evaluate the elastic-plastic behaviour of the perfect shells to determine the interaction between plasticity and pre-buckling non-linearities (Pu IP0). The geometries tested were characterized by a ratio Pu IP0 close to unity mainly because of the low slenderness involved (the ratio P0 IPcr being close to 0.22). Nevertheless it should be noted that the squash load P0 is underestimated since it is evaluated under the hypothesis of perfectly plastic material, while in both tests and FE analyses, the stress-strain curve presents a distinct stress hardening behaviour.

The most significant comparisons between FE models and test results are obtained when both the effects of geometric imperfections and of plasticity are included. However, as previously commented in discussing model (iii), non-linear step-by-step analyses based on the complete stringer-stiffened cone can only be performed for a few experimental comparisons since they are extremely demanding in terms of computer time. Comparisons have been made in terms of load vs. axial displacement curves, out of plane displacements and strains [12]. A typical load vs. axial displacement curve is reported in Figure A.3.6 while the results of all the analyses in terms of collapse load are summarized in Table A.3.2, together with the FE predictions made using a singe panel model. In general, good agreement with the experimental results is evident even though the FE models predict in most cases somewhat lower collapse loads. The underestimation, which is within 10-12 %, could be due to the stress-strain model used in the FE analysis obtained from tensile specimens, although in this case the yield stress in compression is more relevant and this could be somewhat higher.

Table A.3.2: Comparison of FE and experimental results for stringer-stiffened cones

Model

SCOIA SCOIB SC02B

FEperf.

Per (kN)

524 490 521

(complete)

PolPer

0.22 0.22 0.21

FE

Pu (kN)

110 100 106

imperi.

Pu/Po

0.95 0.94 0.97

(complete)

Po/Per

0.21 0.20 0.20

FE

Pu (kN)

112 106 113

perfect

Pu/Po

0.97 1.00 1.03

(panel)

Pu/Per

0.21 0.22 0.22

Test

Pu (kN)

110 107 121

Results

Pu/Po

0.96 1.01 1.11

45

As described in chapter A.2 and in the preceding section on unstiffened cones, the experimental elastic stiffness evaluated between the end-rings compares well with FE results [12, 16]. Once more, the comparison is not as good when the end-parts are included. Finally, the comparison between FE results and experiments is also very good in terms of post-buckling slope. This is shown to be less steep for the models having higher stiffening ratio both experimentally and via FE analysis.

A.3.4 Conclusions The numerical tools to be adopted in the parametric studies have first been calibrated by simulating a series of elastic-plastic buckling tests on conical shells. This activity has also allowed a more general evaluation of various FE models on shell buckling problems, which, as is well known are among the most difficult to tackle via numerical analysis. An important aspect in the present study has been the modelling of the initial geometric imperfections recorded on the test specimens. These were processed through best-fit and Fourier analysis and, on this basis, a selection of relevant modes was made for inclusion in the numerical models. It has been demonstrated that modelling of the full imperfection surface is of primary importance in evaluating the ultimate load even in cases where the final collapse mode is axisymmetric. The comparison of FE with test results on unstiffened and stringer-stiffened cones has been satisfactory both in terms of ultimate loads and collapse modes. The comparison has also been very good in terms of stiffness and post-buckling slope; however, it was shown that considerable care is needed in measuring end-shortening values experimentally and that the conical shells are very sensitive to the degree of radial edge constraint.

46

Fig A.3.1: Definition of conical shell geometry

47

Fig. A.3.2: Axisymmetric model - First eigenmode

Fig. A.3.3: Complete cone model - First eigenmode

48

120.0

8 0 . 0 -

Τ3 Λ Ο

40.0-

UC01 unstiffened cone tf· Exp. between plates

O FE model

— Q — Exp. between rings

0.0 —ι 1 1 r— 0.00 0.50 1.00 1.50 2.00 2.50

End shortening (mm)

Fig A.3.4: Comparison of load-end shortening results for UC01

120.00

80.00

'S o 40.00-

-i 1 1 i r

0.00 1.00 2.00 3.00 End shortening (mm)

Fig A.3.6: Comparison of load-end shortening results for SC01B

49

O

(a) (b) (c)

Fig. A.3.5: Critical eigenmodes for SCOIA, SCOIB and SC02B

A.4 Design Implications Following the validation of the numerical tools, several investigations were carried out with the main objective of improving existing rules for unstiffened cones and of developing guidance for stringer-stiffened cones; a summary is given herein.

A.4.1 Review of current ECCS rules on unstiffened cones In Figure A.4.1 a comparison of tests with existing ECCS rules is undertaken [11]. In addition to the present series, the conical specimens tested by Kry sik and Schmidt [13] are included. All test results lie beyond the ECCS curve, even though the imperfections in these small-scale specimens are generally much higher than the ECCS tolerances (the ECCS curve shown is based on 'severe' imperfection reduction factors, which are half the 'normal' value, but without further allowance to account for imperfections higher than that upper limit). The degree of conservatism varies but is in the range of 15-45%. In the light of these results, either higher imperfection reduction factors or higher tolerance values may be accepted; it might be preferable to concentrate on the latter, since the measuring of imperfections of cones is not an easy procedure. In any case, it would appear that the current ECCS mie which stipulates use of the small radius in calculating imperfection reduction factors is correct, since it leads to the lowest possible slenderness (hence, the highest imperfection reduction factor) being considered.

In view of the fact that the failure mode in the present specimens (and in those tested by Krysik and Schmidt) is axisymmetric ('elephant foot'), it was thought appropriate to compare test results with the axisymmetric plastic mechanism formulation developed by Poggi [6]. This approach is based on minimising the energy dissipated during the formation of an axisymmetric bulge consisting of two straight line segments. The results are summarised in Table A.4.1, indicating that, for the stockier models, the mechanism approach gives very good predictions of failure loads and could be used as an alternative design approach [11]. As might be expected, the mechanism predictions become unconservative as the slenderness increases, since they are derived from an upper bound solution.

Table A.4.1: Comparison of tests with plastic mechanism

Model Ref.

UCOl UC02

UC03 UC04

Slenderness parameter

(ECCS defn.)

0.56 0.71

0.61 0.77

Test ultimate load over plastic mechanism prediction

0.99 0.80

0.97 0.84

An area of concern in designing conical shells using 'equivalent cylinder' approaches, such as the one given in the ECCS recommendations, is to establish their range of applicability, by setting an upper limit on the tapering (semi-vertex) angle. As is well known, the stability nature changes from a bifurcation to a limit point response as the tapering angle increases from 0° (cylinder) towards the physical upper bound of 90° (annular plate). A FE parametric study was undertaken to investigate further this aspect. Linear and non-linear analyses were performed on conical shells with different angles. A range of imperfection amplitudes for each geometry was examined, in all cases assuming that the imperfection shape is similar to the first eigenmode. The results are reported in Figure A.4.2 in terms of imperfection sensitivity curves. A corresponding limiting curve derived from the current ECCS rules is

51

also shown. By considering the results for w/t = 0 (perfect cone) it may be concluded that the ECCS procedure covers conservatively cone angles up to 70° (the current limit is 65°). In terms of imperfection sensitivity slope, the same conclusion may be made but some additional work is needed in terms of alternative imperfection shapes. In the current ECCS recommendations, the procedure for cones is based on the calculation of a critical buckling stress. Considering that the critical buckling load (Seide's classical solution) is given by (note that it is independent of radii)

_ 2 π E t2 cos2p

V3 (1 - v2) and that the cross-sectional area of the cone changes along the meridional co-ordinate, i.e.

A - 2nR t cos ρ because the radius R changes, it means that the critical buckling stress is also a function of the meridional location, i.e.

E t coso Oer =

V3 (1 - v2) R The acting (applied) stress on the cone due to an axial load P, which could be evaluated by considering the membrane solution (generally this is sufficiently accurate away from boundaries), i.e.

σχ = Ρ /A=P I InRt cos ρ also varies along the meridional co-ordinate. Thus, the buckling check should, in principle, be made at different points to ensure that the acting stress remains below the design buckling stress (which is not equal to acr , since this theoretical value is modified for imperfections and plasticity, but nevertheless remains a function of the meridional co-ordinate). In order to avoid possible misinterpretation of stress based rules with regard to the appropriate location for carrying out the checks, it is believed that using a buckling load approach is preferable. In this case, all the design parameters (elastic critical buckling load, squash load, acting or applied load) become unambiguous. For this reason, the present ECCS rules have been recast in a load based formulation and are presented in the Annex to this Chapter. In addition, the rules presented in the Annex adopt the Eurocode format in order to render the formulas readily available for EC3 use.

A.4.2 Development of design proposal for stiffened cones As already mentioned, currently there are no design rules for stringer-stiffened cones. However, a short study by Samuelson [10] has indicated that it might be possible to adapt the ECCS stringer-stiffened cylinders rules for this purpose. Furthermore, there are no analytical solutions, similar to Seide's formula for unstiffened cones given above, on elastic critical buckling loads of stringer-stiffened cones.

Thus, the starting point of the present investigation was to establish, through an equivalent cylinder concept, a procedure which would lead to an estimate of the critical buckling load of the stiffened shell, bearing in mind that either local shell buckling or stiffened panel buckling may govern, depending on geometric parameters. Since R is a function of the meridional co-ordinate, there are many different equivalent cylinders that can be defined for any given cone. In the absence of analytical formulations to help in selecting the appropriate value, it was decided to look for a clear and simple definition, which could then be validated (and, if necessary, calibrated) against FE results.

52

Figure A.4.3 shows typical results obtained for a stiffened cone of dimensions similar to specimen SCOIA by considering the full range of possible equivalent cylinders [16]. The results are presented both in terms of stress and load (the latter being equal to the critical stress calculated for any given location multiplied by the cross-sectional area appropriate to that location) and, once more, it becomes clear that an unambiguous definition is essential. Considering that, in some cases, the variation along the length is not monotonie (see elastic stiffened panel buckling load line in Fig. A.4.3) and in the absence of analytical results to support the choice of one particular radius value, it was decided to specify that the full distribution of elastic critical loads should be calculated for each mode and that the two minimum values (one from considering the local mode and the second from the stiffened panel mode) should be taken as design estimates.

However, once the elastic buckling loads are determined in this way, it was felt that the remaining two steps of the design procedure (imperfection and plasticity reduction factors for the two possible buckling modes) should be linked to the elastic buckling loads in a unique manner. The most appropriate parameters for an equivalent cylinder would be obtained by calibrating against FE results, but also taking into account that, insofar as the local mode is concerned, the proposals should be compatible with the rules for unstiffened cones.

A.4.3 Validation of design proposal for stiffened cones Validation of the proposed procedure was undertaken using a wide range of FE models [12]. This was necessary in view of the different objectives of each step in this validation process and also because of the different buckling modes involved. For example, in generating results for local shell mode, a single panel model (defined by a pair of successive mid-panel meridians) was primarily used, whereas in investigating the stiffened panel mode, complete or multi-panel models were used. Furthermore, in the latter case, in order to arrive at imperfection reduction factors that can be compared with the ECCS cylinder values for stiffened cylinders, 'smeared' models were used, where the stringer stiffness is distributed along the shell. Finally, some discretely stiffened multi-panel models (Fig. A.4.4) were also generated in order to check the above mentioned simplified models.

A.4.3.1 Critical buckling behaviour Results from a typical subset of the eigenvalue analyses undertaken in order to check the elastic critical buckling load predictions of the proposed design approach are shown in Figure A.4.5. The top graph shows the variation of Pcr with increasing number of stringers for a typical stiffened cone geometry (p=15°, /?;/i=200, R]/L=l, £/=0.30). As can be seen, both FE results and the design formulae predict a changeover of govering mode in roughly the same range and, furthermore, the design predictions are reasonable, especially for the critical buckling load of the governing mode. The lower graph in Figure A.4.5 shows the results from an additional comparison examining the variations as a function of the stiffening ratio (p=15°, Rj/t=200, Rj/L-l, ns =32) with broadly similar conclusions. Further comparisons have been undertaken to validate the design predictions for a range of semi-vertex angles (up to p=60°) and also for different Rjlt and RjlL values. The results were satisfactory, except when the ratio RjlL took on extreme values; tentatively, predictions for geometries in the range between 0.5 <RjlL < 2.0 are believed to be acceptable.

A.4.3.2 Imperfection reduction factors Use of FE to generate appropriate design data for imperfection reduction factors is a promising development but the procedure is sensitive to the assumptions made in selecting the FE models, in deciding appropriate input values for imperfection parameters (amplitude and shape) and in interpreting the results. However, it is not realistic to expect that physical tests can be used on their own for developing design rules, considering the size and

53

homogeneity of the database required. This is especially valid for structures like the stiffened cones and other non-standard geometries. In the current study, geometrically non-linear analyses of stiffened cone geometries were undertaken using the eigenmode as the appropriate imperfection shape (this is acceptable provided that the non-linear behaviour of the perfect geometry does not result in significantly different pre-buckling shapes). In order to specify an imperfection amplitude that can be used in a rational comparison of a wide range of geometries, it was decided to use the ECCS tolerances as limiting values. Thus, the maximum imperfection of the selected pattern affine to the buckling mode should stay within the envelope of ECCS tolerances (it is an envelope rather than a single value because tolerance values are specified as a fixed fraction of a gauge length, which varies depending on the location measured). Typical results for imperfection reduction factors (ratio of limit load to critical load) obtained in this manner from geometrically non-linear FE analyses are presented in Tables A.4.2a and A.4.2b for local and stiffened panel buckling respectively. Further cases have been examined and are reported in [12].

Table A.4.2a: Imperfection reduction factors - local shell buckling (p = 15° Rj/t = 200 RjlL =1 £> = 0.30)

ns

8 16 32 44

«FE

0.55 0.59 0.89 0.92

Table A.4.2b: Imperfection reduction factors - stiffened panel buckling (Rj/t = 200 Rj/L = 1)

Ρ

0°

15°

30°

45°

60°

ns

32 64 32 64 32 64 32 64 32 64

OFE

0.10

0.85 0.76

ζι 0.30 0.87 0.83 0.86 0.83 0.89 0.86 0.88 0.85 0.82 0.81

0.50

0.95 0.96

In order to discuss the significance of these results, it is appropriate to examine the ECCS philosophy behind the imperfection factors in stiffened-cylinders, which is illustrated in Figure A.4.6. Thus, it can be seen that reduction factors change as a function of the stiffening ratio (in the case of any given cylinder this parameter defined as C=As/bt has a unique value). For low values (f<0.06) the unstiffened cylinder factor is used (a0), whereas for high values (C>0.2) a value of asp =0.65 is specified, with a linear interpolation to cover the range in between. A similar philosophy is considered appropriate for stiffened cones but the influence of an additional parameter, the semi-vertex angle, should also considered. In addition, it is important to note that the stiffening ratio in any given stringer-stiffened cone is not constant since the panel width varies with respect to the meridional co-ordinate.

54

On the basis of the results obtained for stiffened cones through FE analysis, and considering also the need to arrive at relatively simple design values, it is proposed to use the same factors for stringer-stiffened cones as for stringer-stiffened cylinders. The appropriate slenderness value for local shell buckling is to be determined using the smaller radius (in line with the design rule for unstiffened cones) and the appropriate stiffening ratio for stiffened panel buckling is to be based on an average stiffening ratio. There is some evidence (see the Tables above) to suggest that these values are conservative in certain cases (especially for local shell buckling) but further numerical studies in this area would be advisable before higher factors can be introduced. On stiffened panel buckling, the results have not revealed any substantial trend with respect to the semi-vertex angle and, in general, the cone factors are of the same order as FE derived cylinder factors. They are both found to be somewhat higher than the 0.65 value in the current ECCS recommendations (by about 20%) but numerical factors do not take account of any additional imperfections (other than the geometric pattern inputted to the model), such as loading eccentricities etc, and, for this reason, it is not considered prudent to modify the above value. The entire design proposal for stiffened cones is set out in detail in the Annex.

A.4.4 Concluding remarks Results from a range Of design orientated activities are summarised in this chapter. The design of both unstiffened and stiffened cones is considered and proposals for improvement or further development of the ECCS recommendations are made. Most of the proposals are based on FE results, but it is believed that sufficient confidence on the use of the particular tools and models has been created from comparisons with the tests, as outlined in the preceding chapter. The design proposals are presented in a self-contained document, which is the Annex to this chapter. It can be readily used both for the next ECCS edition and for the relevant EC3 parts currently under development. Naturally, it can be updated in the future pending further checks, numerical parametric studies and any new tests.

55

1.2

Λ

0.8

0.6

0.4

0.2

KS6III

• KS3III

KS3I

• ι ^Vv AUC01

• \ Ä

UC03

Χ ι X KS6I

\ ·

•

aUC02

■ DnV(1987)

Λ Present Experiment

• Kryslk & Schmidt (1990)

0.5 1.5 2.5

Figure A.4.1: Comparison of tests with ECCS rules ­ Unstiffened cones

1.00

Pu/Pcr

0.80

Imperfection Kuit iv i ty curves

Axisymmetric model

&— S­v. *»l»­20 (POT­330Æ KN)

i.m¡eMr­iO<Pa­li4J¡7KN) I

■». Mgta­70 (FO­43.S9 KN)

0.60

0.40

0.20 —

0.00

Figure A.4.2: Imperfection sensitivity analysis of unstiffened cones

57

Local Panel Buckling Mode

1000·

900­

800­

~ 70

°· 2 ecu­

's 500­

£ 400­

300­

200­

100­

0­

100 125 150 175 200 225

R[mm]

600

550

500

450

„400

|[ 350

•o 300

S 250

""200

150·

100·

50·

0·

— — - - _ _ , ***~ "^ *

100 125 150 175 200 225

R[mm]

Stiffened Panel Buckling Mode

1000

900

800

~ 700

1 600

8 500

£ 400 CO

300

200

100

0

■ ^ * «OB ....■— ■ ■■— ■ ■■'-

100 125 150 175 200 225

R[mm]

Critical elastic stress

Ultimate elastic stress

Collapse plastic stress

600

550

500

450

„400

1 350

^"300

o 250

"* 200

150

100

50

0

■ ■"■ «κ ^ ; ■ ■ ■ '

100 125 150 175 200 225

R[mm]

Critical elastic load

— — — Ultimate elastic load

Collapse plaste load

Figure A.4.3: Development of design strength predictions ­ Stiffened cones

58

DISPLACEMENT ΚΑΟίΙΠΟΤΙΟΚ nCTOR - I C I

UJOUS TOtSION: S. 3-1 DATI: 21-OCT-»« Tt«Œ: 21: HZ*

STI/ 1 ZMCUfŒVT î

Figure A.4.4: Eigenmode of multi­panel discretely stiffened FE model

2500

2000

£ 1500

g,

Γ0 1000

500

o

\

▲

< > ' ♦

A

•*¡f

A

10 20 30 40 50 60 70

η

500Í1­

2" 1500­

a.b 1000­

sno­

0­

L

Á l ^ X "

'

<

▲

♦

Design local buckling

Design Stiffened Buckling

ABAQUS stiffened Buckling

ABAQUS local buckling

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

ζ,

Figure A.4.5: Comparison of FE results with design predictions ­ Stiffened cones

59

1.00 -ι

0.80 -

0.65

0.60

o.

á0

0.40 -

0.20 -

r/t=40

0.00

0.2 0.3 As/(bt)

—τ ' 1

0.4 0.5

Figure A.4.6: Imperfection reduction factors for stiffened cylinders in ECCS rules

60

ANNEX TO CHAPTER A.4

PROPOSED DESIGN RULES FOR UNSTIFFENED CONES IN COMPRESSION

Definition of Geometry. Range of Applicability

The dimensions of the cone and the coordinates are reported in the figure below. The length L denotes either the total length of an unstiffened conical shell or the length of a ring-stiffened conical shell between two rings.

The rules apply only if the cone semi-vertex (tapering) angle is less than 65 °.

Buckling Modes

The following modes of instability may occur in unstiffened cones under axial compression:

(a) axisymmetric buckling (b) asymmetric buckling.

Both modes are covered by the rules below. Provided that the limitation L < 0.95 Re

V2 (where Re is the minimum radius of an equivalent cylinder) is fulfilled, no column buckling strength has to be considered:

61

Boundary Conditions

The rules apply only if the boundary condtions are such that

v=0 and u sinp - w cosp=0

along both edges of the cone. This is in agreement with the applications where the curved edges of the cone are attached to heavy circular rings constraining the radial displacement of the shell. If these conditions are not met, alternative design rules need to be considered.

Limitation of the Imperfections

The specifications defined for circular cylinders shall be applied. At each specific location on the shell the length of the rod and of the circulare template shall be defined on the basis of the principal radius of curvature of the cone at the midpoint of the rod or template. This means that the length depends on where the measurements are taken. During measurements the circular template shall be placed in a plane perpendicular to the axis of revolution, and measurements shall be taken normal to the shell.

Axially Compressed Conical Shells

The design value of the acting compressive load PxSd shall satisfy the following relations:

Px.Sd — Ρχ,Κά where

Px,Rd = Px,Rk ! ΊMx (!)

PxRd is the design buckling resistance, and PxRk is the characteristic buckling resistance that may be obtained following the procedure below.

yMx is the shell safety factor for buckling.

Elastic Buckling Load

The critical buckling load may be derived from

ρ = 2 π £r2 cos2p ,?, x.cr —. = T vW

The imperfection reduction factor ax is given by

62

Oi, 0.83 for R1/t<2í2

a =

Rì 1. + 0.01-i t

0.70 for RJt > 212

0.1 + 0.01 ί ί t

Rl being the minimum radius of the cone.

Ultimate Buckling Load

The ultimate buckling load is taken as:

'x,Rk X* "o

where P0 — fy2^Ri t cosp

and the factor χχ is determined from

a. Xx = when Xx > \J2ax

(3a)

(3b)

(4)

(5a)

χχ = 1.0-0.25 X;

a. when J.x < \j2ax

where \ . == V {P0IP„) is the shell buckling slenderness

Shell Safety Factors

They may be taken as:

7 t o = 1-33 yMI when \ > ^ o ~

lMx = (1.0 + 0.33 Xx / fio") 7A/J wÄen λ, < ^2o~

where 7 ^ is the basic safety factor for buckling.

(5b)

(6a)

(6b)

63

PROPOSED DESIGN RULES FOR STRINGER­STIFFENED CONES IN COMPRESSION

Definition of Geometry

The cone is considered as a cylinder with parameters (ns,t,hw,tw) but with a height equal to the slant length L of the cone and a radius equal to its radius of curvature Re (Re(x)=R(x)/cosp) see the figure below. Cylinders with different radii Re

have to be considered as Re is a function of the axial co­ordinate, x.

The width between two successive stringers is the panel width be. Note that since be=be(x), the stiffening ratio and the local panel slenderness change along the length of the conical shell.

Detail of Stringer

shell wall

¿¿hw

Buckling Modes

The following modes of instability may occur in stringer­stiffened cones:

(a) Local stiffener buckling, i.e. flexural or torsional buckling of the stringer;

(b) Local shell buckling, i.e. buckling of shell panel between stiffeners (subscript / for stress and load notation);

(c) Stiffened panel buckling, i.e instability of the stiffened shell including the stringers which participate in the buckling mode (subscript ρ for stress and load notation).

64

Provided that the limitation L < 0.95Æ/'2 is fulfilled, no column buckling strength has to be considered. The minimum Re is to be used in the above.

Buckling mode (a) is assumed not to be critical since stringer geometry limitations are given to avoid stringer buckling. For a flat bar stiffener with t^-t the following limitation should be fulfilled:

h I t < 0.35 fy

Boundary Conditions

The rules apply only if the boundary conditions are such that v=0 and u sinp -wcosp=0, i.e. circumferential and radial displacements are constrained along the curved edges. This is in agreement with applications where the curved edges of the cone are attached to heavy circular rings constraining the radial displacements of the shell. If these conditions are not met, alternative design rules need to be considered.

Imperfections

The shell wall imperfections w normal to the surface should not exceed the following limitation:

" < 0.01

where Lr=4VRet, but not greater that 95 % of the distance between successive circular welds or meridional welds. Note that the gauge length is a function of the axial co­ordinate.

The imperfection limitations of the stringers refer to the inwards or outwards lack of straightness of the stringers and to the lateral misalignment of the stiffener attached to the shell as well as to the initial tilt of the stringer.

Thus,

w < 0.0015/G

where lc — min(L, L„) with Lm being the critical axial buckling half-wavelength.

For low stiffening ratios, additional tolerances may be specified.

Lateral misalignment shall be limited to the tolerance values given for stringer-stiffened cylinders.

65

Elastic Critical Buckling Load

For each equivalent cylinder, the following formulae must be evaluated to

obtain local panel and stiffened panel elastic critical buckling stresses.

For buckling mode (b), the critical elastic stress acrl is given either by the

expression for a complete cylinder or by that for a flat plate whichever is higher,

depending on the ratio be /VRj:

'cr,l

V/3(1 ­ V2) Re

= > 2.44 complete cylinder (7)

cr.l

4x2£

12(1 ­ v2)

t

T L­< 2.44 flat plate (8)

R.t

Since the stringers mainly act in constraining the radial displacement of the shell

along the shell­stringer junction, the formula for flat plate buckling is obtained

assuming simple supported boundary conditions along these lines (the torsional stiffness

of the stringer is neglected).

For buckling mode (c), the critical elastic stress σ is evaluated by minimising

the following expression with regard to the number of longitudinal and circumferential

waves (m, ri):

Al2A23 Al2A22 Λ33 — Λ13

Α.,A­­. A 12

A12A13 AnA23

AUA22 A '23

12

cr.p

mir

(9)

According to the ECCS Recommendations, in minimising (9) the cases n = 1 and

η=2,3 are dropped since the former does not represent shell buckling and the latter

gives unsafe results.

The coefficients A¡j are functions of the number of waves, the geometric

parameters of the equivalent cylinder and the orthotropic elastic constants. The latter

are functions of the spacing, cross­sectional area and eccentricity of the stringers. The

elastic constants are also functions of the panel width: according to plate theory, as the

buckling load is reached, the stress distribution is no longer uniform and an effective

width be must be considered.

Since the effective width be js a function of the oCTllacr ratio, an iterative

procedure is required. Starting with be = be, σ is calculated by minimising (9). A cr,p

66

new value of be is then calculated from the new acr¡/acrp ratio, and so on until the solution has converged.

Since equation (9) considers the stringers as 'smeared' along the shell wall, the stiffeners must be closely spaced so that ns <3.5n where n is the minimised number of circumferential waves.

The critical buckling stresses defined by equations (7), (8) and (9) must be evaluated for the full range of equivalent cylinders corresponding to a particular cone. Then, the corresponding elastic critical buckling loads should be calculated taking into account the variation in the cross­sectional area of the cone. For all critical stresses mentioned above:

Ρ = A ocr = (2irRt+nsAs) ocr cosp (10)

The minimum value of equation (10) gives the approximate elastic buckling load of the cone. The minimum values should be found independently for local and stiffened panel modes.

Imperfection Reduction Factors

The base reduction factor a0, which takes into account the imperfection sensitivity of the shell is given by:

a0 = ° · 8 3 for RJt<2\2

1 + 0.01—î t

a0 = ° · 7 0 — forRJt>2\2

(Ha)

D

0.1 + 0.01—ï t

(Hb)

From the basic factor a0, a reduction factor a¡ is proposed for shell panel buckling:

α;=α0 for w/Lr<0.01

Only the value of Re corresponding to the small radius 7?j is needed in calculating a¡.

For stiffened panel buckling, the reduction factor (a ) is given by:

67

<*^=0.65 for AJb;>0.2 (12a)

αψ=α0 for As/bet<0.06 or Re/t<60 (12b)

Since AJbet varies along the length of the cone (depending on the axial co­ordinate) an average value based on upper and lower be values (be at R=R} and be at R=R2) is proposed.

A linear interpolation between 0.65 and a0 is taken for 0.06<As/bet<0.2.

The ultimate elastic buckling load is calculated as:

pu.d = m i n

aP. a Ρ Γ cr J sp i

sp cr,p

7 7

(13)

where y is an additional shell safety factor (γ=4/3 apart from the flat plate local panel buckling mode where 7=1).

Elasto­Plastic Effects

In the elasto­plastic range the influence of plasticity is considered by calculating a plasticity reduction factor. For local panel buckling if λ < V2ocl

Pu,pi = (xh)P0 complete cylinder (14a)

where χ = 1­ 0.25 X2/a¡ and y = (1.0 + 0.33X /V2a¡)

Κ.Ρ,=χΡ0 flat plate (14b)

where χ = 1­ 0.25 λ2

For stiffened panel buckling mode if λ < V2oLsp:

PU,P, = (Χ/7) Λ, <15>

where χ and 7 are as given above in equation (14a)

and λ = V(P0/Pcr) and P0=(2irRet+nsAs)fycosp is the squash load calculated at minimum Re. The appropriate Pcr must be used, depending on which mode (local or stiffened.panel) is being checked.

68

A.5 Conclusions The project comprised experimental, numerical and design-orientated activities. Preceding chapters have outlined the methodology followed in each of these areas and presented typical results. The following is a summary of the main conclusions reached: (1) Tests on unstiffened cones buckling elasto-plastically have confirmed that the current ECCS formulation gives conservative predictions, even though the specimen imperfections were in excess of ECCS tolerances; conservatism is thought to be attributed to the ECCS imperfection reduction factors. (2) For the stockier set of test specimens (ECCS slenderness < 0.6) a plastic mechanism approach can deliver accurate predictions for design purposes; for higher slenderness values (0.7-0.8), the plastic mechanism approach becomes non-conservative by about 15-20%. (3) Comparison of test results for unstiffened cones with a wide range of FE models indicates that the closest agreement between FE and tests is obtained when full imperfection information is included in the numerical models; even for the unstiffened cone specimens, whose final collapse mode is largely axisymmetric, the effect of non-axisymmetric imperfections is not insignificant. Hence, full imperfection scans in shell buckling experiments are thought to be essential when comparison with (or validation of) numerical tools is undertaken. (4) Tests on widely-spaced stringer-stiffened cones, and comparison with their unstiffened counterparts, have confirmed the efficiency of this stiffening arrangement in axially compressed cones. (5) Comparisons of test results for stiffened cones with FE calculations have shown that numerical models which can incorporate the full test imperfection perform better than others, but single panel models have also a useful role to play, particularly when conservative assumptions regarding the imperfection pattern are introduced (e.g. as done in parametric studies). (6) Specification of end conditions for conical shells is a crucial part of strength analysis; both tests and FE calculations have shown the sensitivity of stiffness and strength predictions to this aspect; the exact conditions prevailing on the radial constraint at the ends are the most important element of the boundary conditions. (7) Some proposals for improvement of current ECCS formulation for unstiffened cones have been made, namely: - using a critical buckling load (rather than a stress) parameter avoids potential confusion with the definition of the carrying capacity of the cone - using only the lowest radius-thickness value in deriving the imperfection reduction factor will make the design approach less conservative - by investigating the nature of stability in cones as a function of the tapering (semi-vertex) angle, the limiting value of 65° was found to be acceptable. (8) A design method for stringer-stiffened cones compatible with the ECCS format has been proposed and checked through a series of FE calculations; it is based on the existing ECCS stringer-stiffened cylinder formulation but with the following provisos: - it uses the concept of critical buckling load (instead of stress) - it proposes the use of small radius slenderness for local buckling reduction factors (in line with the conclusion reached earlier on unstiffened cones) - it proposes the use of an average stiffening ratio for panel buckling reduction factors - it retains tentatively the ECCS elasto-plastic reduction equation until further validation.

69

Notation b(x) conical panel width fy yield stress hw stringer depth k number of half-waves in meridional direction / number of full waves in circumferential direction ns number of stringers in stiffened cone t shell thickness tw stringer thickness u,v,w meridional, circumferential and normal coordinates u],u2,u3 radial, circumferential and axial coordinates χ co-ordinate in meridional direction (0<x<L) As stringer area (= hw tw) E Young's modulus L slant length of cone Ρ applied load Per elastic critical load (classical solution or linear eigenvalue analysis) Pi limit load (elastic geometrically non-linear analysis) Ρ o squash load Ρ n ultimate load (test or fully non-linear analysis) R1P2 small and large radii of cone a imperfection reduction factor ζ stiffening ratio (= As I b(x) t) θ circumferential co-ordinate λ slenderness parameter ξ/cl amplitude of Fourier mode (k,l) ρ tapering or semi-vertex angle of cone v Poisson ratio Φΐ>Φ2·Φ3 rotations about uj,u2,u3 directions φ/cl phase angle of Fourier mode (k,l)

70

References

1. European Convention for Constructional Steelwork, Buckling of Steel Shells, European Recommendations, 4th edn, Brussells, 1988.

2. Seide, P., Axisymmetrical buckling of circular cones under axial compression, J. Appi. Mech., 23 (4), 1956, 625­8.

3. Baruch M., Harari, O. & Singer J., Low buckling loads of axailly compressed conical shells, J. Appi. Mech., 37, 1970, 586­71.

4. Pariatmono, N. & Chryssanthopoulos, M. K., Asymmetric elastic buckling of axially compressed conical shells with various end conditions, AJAA J., 33 (11), 1995,2218­27.

5. Chang, C. H. & Katz, L., Buckling of axially compressed conical shells, / . Eng. Mech. Div., ASCE, 106 (3), 1980,501­16.

6. Poggi, C , The collapse of ring­stiffened cones under axial compression and external pressure, in Proc. ECCS Colloquium on Stability of Plate and Shell Structures, Ghent, 1987, 405­10.

7. Schmidt, H. & Krysik R., Static strength of transition cones in tubular members under axial compression and internal pressure, in Proc. 6th Int. Symp. Tubular Structures, Melbourne, Australia, 1994.

8. Singer J., The influence of stiffener geometry and spacing on the buckling of axially compressed cylindrical and conical shells, in Proc. IUTAM Symp. Theory of Thin Elastic Shells, ed. F. I. Niordson, Berlin, 1969, 234­63.

9. Tong, L., Tabarrok, B. & Wang, T.K., Simple solutions for buckling of orthotropic conical shells, Int. J. Solids and Struct., 29 (8), 1992,933­46.

10. Samuelson, L.A., Stiffened conical shells; approximate methods of analysis. In Proc. ECCS Colloquium on Stability of Plate and Shell Structures, Ghent, 1987,411­6.

11. Pariatmono, The Collapse of Axially Compressed Conical Shells, PhD Thesis, Imperial College, University of London, 1994.

12. Spagnoli, Α., Buckling Behaviour and Design of Stringer­stiffened Cones in Compression, PhD Thesis, Imperial College, University of London, to be submitted 1996.

13. Krysik, R. and Schmidt. H., Beulversuche an langsnahtgeschweissten stählernen Kreiszylinder und Kegelstumpf schalen im elastisch­plastischen Bereich unter Meridiandruck und innerer Manteldruckbelastung, Technical Report 51, University of Essen, September 1990.

14. Hibbit, Karlsson & Sorensen, ABAQUS Version 5.4: Theory and User's Manual, Providence, Rhode Island, 1994.

15. Lucchinetti, E., Analisi numerica e sperimentale dell'influenza delle imperfezioni geometriche sull'instabilità elasto­plastica di gusci assialsimmetrici, Diploma Thesis, Dipartimento di Ingegneria Strutturale, Politecnico di Milano, 1994.

16. Spagnoli, A. & Chryssanthopoulos, M.K., Experimental and modelling techniques for buckling analysis of stiffened cones, CESLIC Report SS5, Imperial College, 1995.

17. Donatelli, R., Analisi numerica degli effetti delle imperfezioni geometriche sull'instabilità elasto­plastica di gusci tronco­conici compressi assialmente, Diploma Thesis, Dipartimento di Ingegneria Strutturale, Politecnico di Milano, 1993.

18. Poggi, C , Numerical simulation of tests on conical shells ­ the effects of radial constraints, Technical Report, Dipartimento di Ingegneria Strutturale, Politecnico di Milano, May 1995.

71

ECSC Contract No. 7210­SA/208

Enhancement of ECCS Design Recommendations and

Development of Eurocode 3 Parts Related to Shell Buckling

Part Β

LOCAL LOADS IN CYLINDRICAL STRUCTURES

Final Report

Ghent University

Laboratory for Model Research

Prof. Dr. ir. J. Rathe

Technical University of Graz

Institut für Stahlbau, Holzbau und Flächentragwerke

Prof. Dr. ir. R. Greiner

TABLE OF SYMBOLS

b d frj,2 fu fy h h, lref

limp

n r,R t ti

half-width of local support width of local support 0,2 % proof stress ultimate tensile stress yield stress overall-height of the cylinder model height of reinforced bottom course reference length, size of potential buckles for a cylinder under uniform axial compression axial length of imperfection number of supports radius of cylinder thickness wall-thickness of reinforced bottom course

B

E F Fer ** max Fref Fyield,l

GNL GNLI GMNLI

8 th of circular circumference

Χ = · ' cr,GNLI

modulus of elasticity axial load in mathematical analysis model buckling load for mathematical analysis cut-out collapse load for mathematical analysis cut-out model reference axial load in mathematical analysis model axial load corresponding with squash yielding of analysis cut-out = 27rrtfy/8 geometrically nonlinear elastic analysis geometrically nonlinear analysis of imperfect shell fully nonlinear (geometrical and material nonlinear) analysis of imperfect shell

dimensionless elastic local buckling stress

Y = cr,GMNL1 dimensionless elastoplastic local buckling stress

75

ßy δ

<|>u

η

Κ2

K2.LOCAL

K2,L0CAL,RIGID

K2.UNIF0RM

μ

λ

λο

λο,βΜΝίΙ

λο,σΝίΐ

yield stress imperfection amplitude

O'er

0"cr,GMNLl

0"cr,GNLI

0"cri

σ exp

steel grade factor = 235 fy expressed in MPa

outer diameter support opening angle = d/R reduction factor = acr/fy

reduction factor for local buckling reduction factor for local buckling, rigid boundary conditions reduction factor, according to DIN 18800/4 for cylinders under uniform axial compression

supported part of circumference =

dimensionless load factor = A/Ayieid,i. 100 Arj/Ayieid, ι. 100 dimensionless load parameter dimensionless load parameter, GMNLI analysis dimensionless load parameter, GNLI analysis

slenderness parameter = É

VCTeri

slenderness parameter, based upon elastic critical stress of the imperfect

shell; λ = I y— V ücr,GNLI

critical or buckling stress

critical, local buckling stress from GMNLI analysis = elastoplastic buckling

stress for the imperfect shell

critical, local buckling stress from GNLI analysis = elastic buckling stress

for the imperfect shell

classical elastic buckling stress for a cylinder under uniform axial

compression

experimental, axial compressive buckling stress

Λ

ΔΛο

Δλ0

Δλο,υΜΝΟ

AXo,GNLI

Λο

­/».max

Λ yield

dimensionless load factor = F/Fref

inclination of the fitting line in a Λ­μ diagram

inclination of fitting line = Δλο/Λγκω,ι.ΙΟΟ

inclination of fitting line for XO,GMNLI analysis

inclination of fitting line for XO,GNLI analysis

dimensionless collapse load factor Fmsx./FTe{ for μ = 0 (extrapolation)

maximum dimensionless load factor Λ

dimensionless load factor for squash yielding : Ayieid = Fyieid,i/Fref

76

Bl Introduction The design of thin-walled steel shell structures with respect to buckling is covered

by national and international standards for idealized shell elements (cylinders, cones, torisphe-res) subjected to uniform loading conditions (e.g. [B.l] ). These design regulations are the fruits of years or even decades of both experimental and theoretical investigations. On the other hand, the topic of local force introduction in cylinders - for instance, large elevated cylindrical silos and tanks, which are generally supported on a number of columns where high, meridional compressive stresses arise above the column tenninations - has received relatively little attention in the past and growing interest in this practical problem started about a decade ago. Nevertheless, the designer often finds himself lost by the lack of design regulations or ex­perimental evidence, and the success of the project not seldomly bears heavily for a great deal on the engineer's intuition and his bom feeling for common sense.

The present research programme tackles the problem of local axial loads on cylin­ders and it should cover the field of unstiffened as well as of stiffened structures. In the present context, the significance of the word 'stiffened' may not be misunderstood : stiffening the thin shell wall against local axial forces means that certain structural elements are employed in order to reduce the high stress concentrations at the supports and to provide a way for introducing the concentrated loads smoothly into the shell wall (fig. B.l).

The research activity, sponsored by the European Community, should solve the problem of stability and load-bearing capacity of cylindrical steel structures subjected to such lo­calized loadings. Its purpose is to develop simple design recommendations which can be intro­duced into the ECCS rules and structural Eurocodes in order to assist designers in the steel in­dustry. The present report deals with situations, illustrated in figure B.l as a and b.

thicker lower portion

extended columns

two rings + extended columns

1

I . a /

r !

1 \

Η

b /

1 \

c /

1 1 1 J

1

d /

ί 1 unstiffened cylinder practical solutions for vertical load introduction

Figure B.1 : Local force introduction

The scientific research comprises an experimental part, which has been carried out at the Laboratory for Model Research of the Ghent University (Belgium) and a numerical part, which was ordered by the Laboratory for Research on Models and carried out by the Technical University of Graz (Austria).

77

B2 Survey of the pertinent Literature An overview of the relatively scarce literature available for locally supported cylin­

ders is given below, where a classification is done according to whether the studies concentrate on linear elastic behaviour, on linear bifurcation analysis, or on nonlinear buckling behaviour.

Linear elastic behaviour has been studied by a number of researchers. It appears that the semi-membrane theory, adopted by Greiner [B.2, B.3] and Öry et al.. [B.4] offers an eco­nomical analysis tool. Greiner's variant leads to a model that is similar to the beam-on-elastic-foundation problem. Öry et al., who use the matrix transfer method, claim that their simplified approach, derived by neglecting the longitudinal bending stiffness and torsional stiffness of the cylinder wall, which results in a much easier method of solving the differential equations for the cylindrical shell, yields sufficiently accurate results. It is also worthwhile to mention the merits of Flugge's contribution [B.5], in which he develops a numerical technique in order to solve the differential equations for a semi-infinite cylinder, and that of Bodarski et al. [B.6, B.7] who examine damage in a steel silo with a bottom ring resting on four supports.

Linear bifurcation analysis of perfect shells under circumferentially varying axial loads is dealt with in only a few studies, (e.g. [B.6-B.12] ) despite the extensive research efforts on shell buckling over the last few decades. In all these cases the lowest eigenvalues are only slightly larger than the classical one, which corresponds with uniformly distributed compressive axial stresses, except for small support widths for which higher buckling stresses are found. Mo­reover, these studies deal with perfect shells, even though it is well-known that shell buckling under axial compression is normally very much imperfection sensitive. Öry and Reimerdes (e.g. [B.13] ) suggested that column-supported cylinders might be 20-30% less imperfection sensitive than uniformly supported cylinders. We surmise that the degree of imperfection sensitivity is the same, provided that the structural elements for axial load introduction are fully adequate, i.e. are of such a nature that the stress distribution in the shell is very similar to the one prevailing in a cylinder under uniform compression. On the other hand, Rotter and Teng (e.g. [B.14] ) have a point when they state that the imperfection sensitivity should steadily decrease for unstiffened cylinders as the stress state becomes more localized, leading to vanishing sensitivity for very lo­cal stresses. Yet, the outcome might be unclear for the extremely unusual but potentially dange­rous situation where shape imperfections are concentrated in the immediate vicinity of the local supports.

To date, some publications (e.g. [B.14-B.17] ) about nonlinear buckling behaviour of unstiffened cylindrical shells subjected to localized loads are available, most if not all of them being based upon the finite element technique. Rotter and Teng [B.l6] seem to be the first to have examined, on a theoretical basis, the behaviour of unstiffened, perfect and imperfect cylin­der shells on discrete column supports in the elastic range, where the shell wall is exposed to uniformly distributed meridional traction which represents the frictional force imposed on the silo wall by a bulk solid. Their studies, in which a number of parameters are varied in a syste­matic way, incorporate geometrical nonlinearity, and the effect of a local axisymmetric imper­fection at a specific height above the support is taken into account. The number of supports is varied between 4 and 20.

Samuelson (e.g. [B.17] ) conducted a limited number of computations for one parti­cular elastic silo model with different types ofring and stringer arrangements, comparable with figs B.l (a), B.l(c) and B.l(d) respectively.

At the time the present investigation started, the Laboratory for Model Research got a copy of W. Guggenberger's doctoral dissertation [B.l8] in which the basic problem of local force introduction in unstiffened and stiffened cylinders is dealt with numerically.

78

B3 Scope of Work The project was logically subdivided into several parts. The investigations were

focussing on so-called Unstiffened Cylinders and on Cylinders with Reinforced Wall thickness of The Bottom Course.

The test programme was accompanied by a theoretical-numerical programme of detailed re-analysis of selected test cases by finite element computations. Thereby different ty­pes of material behaviour up to fairly large strain levels (obtained by uni-axial tension tests) had to be taken into account : material models Ml exhibiting a pronounced hardening behavi­our and material models M2 exhibiting a marked yield plateau with subsequent hardening. The first test cylinders were nominally perfect with manufacture-induced imperfections (cutting, rolling and soldering). A second series of tests was carried out with additional artifical imper­fections above the supports that were produced by manual cold-forming with a suitably shaped device.

The second part of the project was devoted to numerical parametric studies for the purpose of developing design recommendations. Thereby, contrary to the testing conditions, idealized conditions were assumed with respect to material behaviour and support conditions. Ideal Mises-elastoplasticity without hardening was assumed (simulating ordinary mild con­struction steel) and the support conditions were assumed completely flexible. This means that the axial support force is introduced into the shell wall by uniform line loads over the suppor­ted portion of the circumference, because the stiffness of the potential support plate is comple­tely neglected. This assumption produces results for the buckling strength which are on the safe side.

The test cylinders are 350 mm in radius and 700 mm in height. The sheet thicknes­ses are t = 0.7 , 1.0 and 1.5 mm for material Ml and t = 0.6 , 1.0 and 1.5 mm for material M2. The test cylinders are terminated by a flat-bar ring (t = 0.7 mm) at the lower edge and are

supported on n = 4 equidistant support plates of variable widths. Radial imperfection measu­rements were carried out for all specimens along 8 generatrices along the central support me­ridians and the 45deg-meridians in-between, except for a few cases in which the local regions above the supports were covered by a fine grid of measurement points. The buckling tests we­re carried out in a simple way by pure load-control, which had the natural effect that the pro­cess of instability after reaching the load maximum was in most cases of highly explosive natu­re.

The following modelling assumptions were adopted throughout for the analysis models of the re-analysis of tests : *Rigid support conditions, i.e. the support plates were simulated by built-in conditions with total displacement constraints for all deformation components.

*The out-of-plane bending stiffness of the lower edge-ring, which has the shape of a flat bar, was either neglected (simply supported conditions) or the edge-ring was taken into account as a curved elastoplastic beam.

*Different theoretical models were applied concerning the representation of the elastoplastic material behaviour of the test specimens which was measured by common tension tests for each sheet thickness. Ramberg-Osgood deformation plasticity, incremental small-strain Mises-elastoplasticity with hardening and large-membrane-Log-strain Mises-elastoplasticity were considered.

*Manufacture-induced imperfections were not taken into account with their actually measured values but by simple representative shapes and mean amplitudes. These shapes were intended to represent the most obvious characteristics of the recorded imperfections and miss the

79

tiny details. It turned out that the main characteristics of manufacture-induced imperfections consisted in outward-directed axisymmetric trumpet-shaped imperfections at the lower edge of the cylinders. Also the artifical imperfections were not taken into account with their ac­tual distributions but with their theoretically intended shapes. The amplitudes were chosen as representative mean values.

The second part of the project was devoted to parametric finite element studies for the development of design rules. For this purpose, commonly used but far-reaching assump­tions were adopted which are more or less in contrast to the properties of the 'real' test mo­dels. * Flexible support conditions * Classical simply-supported boundary conditions at the upper and lower edge * Ideal Mises-elastoplasticity without hardening representing mild construction steel of

common grades Fe 360, Fe 430 and Fe 510. * Theoretical local imperfections above the supports are adopted.

These parametric studies were performed for Unstiffened cylinders and for Cylin­ders with Reinforced Wall thickness and the standard steel grade Fe 360. In addition, the ef­fect of higher steel grades Fe 430 and Fe 510 was systematically investigated. Finally, further important effects were studied, concerning rigid support conditions, edge-ring stiffeners and support plates and the interaction with internal pressure.

On the basis of the results of the foregoing numerical analyses it was possible to derive design recommendations for all investigated aspects of the problem so that these basic cases may be viewed as solved. However, there remain several open points, especially con­cerning the effects of edge-ring stiffeners and supports plates which, at the present state, could be covered only in a very simplified way.

B4 Experimentation and Numerical Results

B4.1 Fabrication of Cylindrical Steel Models and Test Set-up [B.20]

Fabrication

Photo B.1 : Sheet shearing machine

The overall dimen­sions of the steel mo­dels are : height = 700 mm and radius = 350 mm. They are suppor­ted at four discrete points. The width of the support plates can be varied : during the present investigation support widths of 13,7 mm ; 27,5 mm ; 41,2 mm and 55 mm res­pectively have been used. These values

80

correspond with a supported portion of the lower rim that is equal to 2,5% ; 5% ; 7,5% and 10% of the entire circumference.

Photo B.2 : Bending machine

The cylinder models are made out of rectangular steel plates. First of all, the rectan­gular plates are brought to the desired size by means of a cutting operation with a sheet shearing machine (photo B.l).

Subsequently, the flat plate is rolled into a cylindrical shape with a bending machine (photo B.2). After having accomplished this rolling operation, the seam between the neighbou­ring longitudinal free edges of the cylinder is welded. Because the thickness of the parent mate­rial is very small, a special welding technique is adopted (photo B.3). As a matter of fact, the T.LG. welding process is applied in such a way that the adjacent edges are melted locally into each other along the seam. In this way, the application of a welding rod is avoided, which re­sults in a fine, smooth weld (photo B.4)

Photo B.3 : Welding process

It is evident that the fabricated cylinder possesses little transverse stiffness, i.e. in a direction which is perpendicular to the generatri­ces. In order to maintain the circu­lar shape, a ring-stiffener is solde­red to the lower rim of the cylin­der. The steel ring is cut out of the same rectangular steel plates that are used to manufacture the cylinder models. In order not to waste a lot of material, it is made by cutting two semi-circular strips which are welded together to form a complete circle. Ring and cylin­der are connected together by sol­dering (photo B.5).

81

The finished cylindrical steel model might give the impression that the circularity of the upper edge will not be maintained during the tests. This, however, is not true : the upper rim is supported by a rigid circular disk, mounted at the top of the cylinder and which is used to introduce the applied axial force smoothly into the cylinder wall. (cfr. B4.1.2 : Test Set­up).

Photo B.4 : Longitudinal weld

B4.1.2

Photo B.5 : Soldering ring­stiffener to cylinder

Test Set­up (figure B.2)

B4.1.2.1 Load Application and Measurement of Reactions (photo B.6, B.7)

• r ­7

■¿fei

I__ ­

p2¡ ~~­r

.t'—r

'r-1

hfy' ­

~τΒ ΓΠ«

H w&m t t r r

\¡\ ­ISl

κ BM"\H?P4C

HB

¡Atti ¡ilHK κ jggigga β Β — χ

i

! î ■ J

1 ­ui

Photo B.6 :Test set­up

The cylindrical model is mounted on a thick and rigid horizontal circular bottom plate ((j>u = 1175 mm ; t = 62 mm), which, in turn is sup­ported by four relatively short channel sections. A circle with a radius of 350 mm has been drawn on the upper surface of the bottom plate and the four dynamo­meters are installed along the circumference of this circle at angles of revolution which are equal to 0° , 90° , 180° and 270° respectively.

82

In fact, the cylinder models are supported at the lower rim by these four equidistant dynamometers (see photo B.7 for a detail); i.e. the angle of revo­lution formed by the radii that join the centre of the circular bottom plate with the position of two neighbouring dynamometers is equal to a quarter of a circle.

In this way, the magnitude of the individual sup­port reactions can be recorded during the test. It is, quite evidently, an illusion to think that these reac­tions will be exactly the same because small but inevitable imperfections of the cylindrical model and of the test set-up will make these support reactions differ from one another. Details of the dynamome­ters, their measuring range and the method of gauging them are given in [B.20].

The test cylinder is placed on the support plates and positioned very carefully in order to keep the

Photo B.7 : Detail of support eccentricity of the vertical cylinder axis with respect to the vertical centreline of the circular, thick bottom plate as small as is practically possible. A thick circular plate covers the top of the cylinder. The outer diameter of the upper part of this disk (corresponding with approximately half its thickness) is 730 mm, whereas this diameter is reduced stepwise to 700 mm minus a couple of tenths of a millimeter for the lower part of the disk. In this manner, the upper edge of the cylindrical model is supported along a perfect circu­lar boundary and forced to maintain this circular geometry during the load application.

ring

anchoring

inductive def lectoneter

ring s t i f fener

=H= bottom plate

s t r u t s hannel sections

SECTIDN« | E>VIEW

Figure B.2 : Test set-up

83

A flat hard PVC ring is interposed between the upper rim of the cylindrical shell and the top plate in order to smoothen small irregularities between the contact surfaces.

The axial force is applied by means of a hydraulic jack and a tie rod. The upper end of the cable is anchored at the centre of the top plate. The cable passes through the central hole in the bottom plate and ends up in the fixed head of the jack. The movable head of the jack pushes a pressure cell against the lower part of the thick bottom plate and brings the steel model under compression. At any particular time of the test, the sum of the readings of the four dyna­mometers must obviously be equal to the force, recorded at the pressure cell plus the dead weight of the jack, cable, top plate and cylindrical model.

The test itself is fairly simple : the force exerted by the jack onto the model is in­creased slowly and the signals from the pressure cell and from the four dynamometers are captu­red by a data-logger which transfers them to a personal computer where they are stored on disk. The scan interval of one second is the smallest one we could use, taking into account the speed of the processor of the data-logger, the transfer rate to the PC and the writing operation to disk, but from a practical point of view it is largely sufficient to meet our needs. The increase of the axial force is hold on until instability of the shell wall occurs.

Β 4.2.1.2 Measurement of Imperfections

Photo B.8 : Recording shape imperfections of cylinder model

A measuring instrument is needed to record the shape or geometric imperfections of our cylindrical models. It appeared to be a little difficult to perform such measurements at the interior of the cylinder. Therefore, we chose to measure imperfections of shape at the exterior face of the model by means of a rotating arm (photo B.8), on which an inductive deflectometer can move up and down.

The deflectometer records small irregularities, perpendicular to the shell wall, of any particular meridian of the cylinder (photo B.9).

Further details about the measuring technique and calibration are given in [B.20].

84

( Photo B.9 : Detail of inductive displacement recording apparatus

B.4.2 Unstiffened Cylinders

B4.2.1 Experimental Results

Photo B.10 : Cylinder COIU after testing Photo B.11 : Cylinder C04U after testing

Every test proceeded as folows : the load, exerted by the hydraulic jack was increa­sed slowly and the reactions of the supports and of the pressure cell were recorded electronically at regular time intervals. By meridional buckling stress we mean the quantity obtained by divi­ding the support reaction by the width of the support and the thickness of the shell. It is the

85

average contact stress at the supported lower rim. Due to inevitable imperfections of the model

and the set­up, the reactions at the four bearings are not exactly the same at a particular time of

the test, although they are approximately equal at the start by adjusting the vertical position of

the bearings [B.20]. This is the reason why buckling does not occur simultaneously at the four

supports but is initiated instead at one particular support.

Photo B.12 : Detail of postbuckling pattern, obser­ved on model COIU

Photo B.13 : Detail of postbuckling pattern, obser­ved on model C04TJ

The nature of buckling depends on the support width. For smaller support widths

the appearance of a dent occurs quite swiftly but not in a flush. The rather "slow" appearance of

buckles must obviously be attributed to plastification. Buckling of a rather explosive nature,

where failure is almost instantaneoulsy, occurs when the support width is increased.

Some buckling patterns are shown on photographs B.10 ­ B.13. They always consist of one

single, isolated inward dent in the area above the support.

The shape of the buckle which

originates at the supported edge

has a sort of elliptic contour (see

figure B.3). The lower part of the

elliptic region seems to be cut

away by the ring stiffener, though.

The unstiffened cylinders tested

at the beginning of the research

project were nominally perfect,

that is, they did not possess inten­

tional imperfections

buckling

/ ^ * /

V­¿—===■—

ring stiff ener

— A

' ' . ' ■ ' . * % . · ­ . ' ­ ­ ':■■.'/;■■

inward

shell wall

) t ­i=—­ ; * /

—A Sectie m AA

Figure B.3 : Typical post­buckling pattern

86

©

artificial imperfections above supports size = 4 frf

The tests which were carried out during the first

half of 1994 focussed on the buckling behaviour of

unstiffened cylinders with artificial imperfections

(fig. B.4). Two series of cylinders have been exami­

ned, one with a ratio of r/t equal to 500 and a second

one with r/t = 233. Imperfections are applied right

above the supports and they are characterised by the

size = 4 Vrt and the depth of the inward dent δ/t = 1.

The variation of the buckling stress as a function of

the width of the support has been studied.

Figure B.4 : unstiffened cylinder

Meridional Shape Imperfections (C25Ub­10%)

« W r ø O T g S S S3 3 3 2 5 2

^ ^

C O

j ì>^mymrrmmmmm^^

«g 0.5

c E o e. re .π ­0.5

­1

­1.5

n ι n n n 11 ί ι n iJXEtrcrxDxiiTx Π Π Ι Π Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Η ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ κ nnnnmnmni mnmmnnnn nnnm

røg

^ ^ ^ π π π π π π π ρ

o

χ

Δ

a

4

1

2

3

0 50 100 150 200 250 300 350 400

Distance along generatrix [mm]

Figure B.5 : Imperfections for a nominally perfect model

A special device has been fabricated in order to create the artificial dents into the

shell wall. Figures B.5 and B.6 show the typical shape of the meridians above the supports for a

cylinder model without artificial imperfections and for a model where imperfections have been

applied intentionally. Models without artificial imperfections show a trumpet shaped generatrix

near the lower (and the upper) rim which is obviously due to the fabrication process ; these sha­

pe irregularities are more or less axi­symmetric.

87

Meridional Shape Imperfections (C25Ui-10%)

E E

o

1.5

ς 0.5

I o

« -0.5 .c <J1

■1.5

50 lOU 150 200 250 300 350

Distance along generatrix [mm]

Figure B.6 : Intentionally applied imperfections

400

o

X

Δ

D

4

1

2

3

MODEL

COIU

C02U

C03U

C04U

COlUa

C02Ua

C03Ua

C04Ua

C03Ub

C02Ub

C18Ub

C20Ub

C23Ub

C25Ub

t

[mm]

0,7

0,7

0,7

0,7

0,7

0,7

0,7

0,7

0,7

0,7

1,5

1,5

1,5

1,5

r/t

Η

500

500

500

500

500

500

500

500

500

233

233

233

233

233

μ

[%]

2,5

5,0

7,5

10,0

2,5

5,0

7,5

10,0

7,5

5,0

2,5

5,0

7,5

10,0

σ„ ε χ ρ

[MPa]

298,4[4]

196,0[4]

180,7[1]

189,7[2]

189,6[3]

155,8[2]

147,4[3]

304,4[2]

163,9[3]

153,2[3]

170,3[4]

130,1[4]

154,8[1]

163,7[2]

237,6[1]

492[2]

481 [3]

345[3]

277[1]

250[2]

273[3]

ÍM

[MPa]

173,2

170,9

170,1

167,3

163,3

162,7

163,0

165,0

165,0

170

203

203

203

203

fu

Η

301,1

293,5

297,6

294,5

300

300

300

300

300

300

350

350

350

350

Table B.1 : Experimental buckling stresses for nominally perfect cylinders [B.20, B.21, B.22]

88

Tables B.l and B.2 collect the experimental buckling stress obtained on nominally

perfect models and imperfect models. Figures between square brackets in the column of these

tables indicate the number of the support where local buckling occurred. The support reac­

tions were recorded almost continuously (time interval = 1 second) during the test. Neverthe­

less, because of the rapid nature of the buckling phenomenon, it was not always possible to

indicate at which support buckling occurred first and to determine which was the value of the

corresponding buckling stress, μ is the supported part of the lower rim, expressed as a certain

percentage of the circumference. The value of the imperfection amplitude for imperfect mo­

dels, δ , which was measured after a particular cylinder had been made, is indicated as well.

The given amplitude corresponds with the support where failure occurred.

MODEL

COlUi

C14UÌ

C15UÌ

C16UÌ

C17UÌ

C26Ui

C27UÌ

C36UÍ

C18UÌ

C19UÌ

C20UÌ

C21Ui

C22UÌ

C23UÌ

C24Ui

C25UÌ

t

[mm]

0,7

0,7

0,7

0,7

0,7

0,7

0,7

1,0

1,5

1,5

1,5

1,5

1,5

1,5

1,5

1,5

r/t

Η

500

500

500

500

500

500

500

350

233

233

233

233

233

233

233

233

μ

[%]

2,5

5,0

5,0

10,0

10,0

7,5

7,5

2,5

10

2,5

2,5

5,0

5,0

7,5

7,5

10,0

10,0

Ou""

[MPa]

343[1]

163,6[2]

165[1]

175[2]

132[3]

122[1]

146[1]

144[1]

141 [4]

186[3]

464,5[2]

447[1]

500[4]

318[3]

304[4]

.257[3]

259[3]

254[4]

230[3]

222[2]

4» [MPa]

170

170

170

170

170

170

170

175

203

203

203

203

203

203

203

203

fu

[MPa]

300

300

300

300

300

300

300

320

350

350

350

350

350

350

350

350

δ

[mm]

0,64

0,88

0,87

0,94

0,67 .

0,63

0,51

0,62

0,70

0,70

1,52

1,47

1,50

1,39

1,54

1,48

1,51 1,51

Table B.2 : Experimental buckling stresses for cylinders with artificial imperfections [B.22]

B4.2.2 Comparison with Numerical Computations [B.23, B24] The Laboratory for Research on Models of the Ghent Univeristy requested Prof

Greiner and Dr. W. Guggenberger of the Technical University of Graz (Austria) to perform

numerical computations in order to establish a connection as close as possible between experi­

mental and theoretical load bearing capacities by introducing all important details of the test

situation into the analysis model. At the same time the mathematical model could be calibrated

to perform a parametric study afterwards.

The numerical studies of the present investigation are based on finite element dis­

cretized nonlinear shell models and were carried out with the program system ABAQUS by

Hibbit, Karlsson and Sorensen.

The analysis model is shown in figure B.7.The geometrical proportions of the ana­

lysis model and the test model are equal (r/t = 500, h/r = 2). Since only local buckling effects

are investigated (general buckling excluded) and effects of unsymmetric imperfections are ne­

glected, a 45­degree analysis cut­out is sufficient (dark­shaded region in figure B.7). The me­

ridional boundaries of the analysis cut­out have symmetry boundary conditions in circumferen­

89

R/t = 500 L/R = 2

tial direction. As a simplification, symmetry in axial direction has been assumed as well. Ac­

cording to Greiner and his staff, the systematic error in limit load values, introduced by this as­

sumption, remains below the value of 2 % for the case of the largest investigated support width

μ = 0,125 of the computer simulation. Moreover, the error lies on the safe side and is even

smaller for the other support widths.

The axial loads are

introduced at 4 equidis­

tant rigid supports. The

points on the shell wall

are rigidly connected to

the support points along

the supported portions

of the circumference.

The remainder of the

circular boundary is

classically supported

and corresponds to a

reinforcement by an

ideal ring which is rigid

in its plane but has zero

stiffness perpendicular

to that plane.

Figure B.7 : Analysis model

The material model has been adapted to meet the experimentally observed stress­

strain relation : smaller yield limit, pronounced hardening, Ramberg­Osgood fit for small strain

conditions. Special attention is given to the effect of large elasto­plastic compressive strains,

which, as the test results show, are likely to occur for the test models with small support widths

(μ = 2,5 %) where the values of the compressive strain amount to 30 % approximately.

Support Construction: classical hinged boundary conditions

Support width μ=ο/Β> 0 4 equidistant rigid column supports

Material Data: Ε­Modulus = 20600 kN/cm2

Poisson Ratio ν = 0.3

Deformation Plasticity Model: Ramberg­Osgood fit £¡>L.ID = α/Ε·(ΐοΐ/σο)»η·Γσ σο = 17.0kN/cm2, n=11.65, α=2.405

Loading: symmetrical local loading at both edges

Reference Load Magnitude: F=q.b=150kN

Actual Load Magnitude = LPF (Load Proportionality Factor) ' F

variation of μ = b/B

inward inward inward . inward inward

standard 12 shifted 13 test 15 axisymmetric imperfections

II (0.25t) Π (0,50t)

Figure B.8 : Imperfections The real geometric imperfections which have been measured at eight generatrices of

each test specimen have been simplified for analytical purposes. Summarizing all the observed imperfect shapes of the nominally perfect cylinder generatrices of the test models a representati­ve imperfection pattern in the form of an axi-symmetric trumpet-shaped outward deflection of the lower rim has been chosen (fig. B.7). The axial length of the imperfection (limp = 25t and 50t) is in proportion with the imperfection amplitude δ/t = 0,25 and 0,50. These approximate values cover the range of imperfections observed in the tests. For the tested cylinders with arti­ficial imperfections, several approaches have been followed in the modellization of the shape of the meridians (fig. B.8).

90

Figure B.9 collects the experimental and theoretical values for r/t = 500 and figure B.10 shows the correlation between theory and experiment for r/t = 233.

350

2.5 5 7.5 10

μ [%]

Figure B.9 : Unstiffened cylinders, t = 0,7 mm

12.5

500

400

2 ■ * ­

1 ­** B O

υ W 300 IM

α»

4¡

200

< i

\ •

/

* ►

f \ V

i

ι <

•

>

♦ test_nom.perfect

X test_imperfect

­ ­ O ­ Perfect

O I5­M3

! ­ ­

> ► *

* {

* * Τ

-^4 2.5 5 7.5

μ [%] Figure B.10 : Unstiffened cylinders, t = 1,5 mm

10

91

As a general conclusion, it can be stated that the agreement between the numerical approach and the experiments is quite satisfactory considering the idealizations still involved in the analytical modelling : idealized imperfections, material model and boundary conditions.

B4.2.3 Comments Writing a design recommendation requires that a pool or database of so-called 'de­

sign buckling loads' be available. What is a 'design buckling load', how can such 'design buck­ling loads' be obtained ? First of all, it has to be decided which type of structural geometry, in­cluding support and boundary conditions, which type of material behaviour and which type of loading (environmental and service conditions) should be covered by the intended design rule.

Imperfections The lifetime of a real structure consists of several successive time spans each of

which may be the source of imperfections that finally add up. Firstly, the process of produc­tion of the steel members themselves (rolling of plates, heat treatment, cold forming etc.) and secondly, the process of manufacturing and erecting the complex structural components ge­nerally leads to residual stresses and plastic strains as well as deviations of the intended struc­tural shapes {geometric and structural imperfections). Thirdly, during ordinary operation of the designed structural component, the purpose for which it is intended, it may generally be exposed to load patterns that are not contained in the idealized load patterns for which the structure is designed {load imperfections). Examples of such load perturbations may be the unavoidable inertia effects of mainly statically loaded structures, e.g. vibrations, complex inter­action forces of other media, e.g. moving air, water or soils, granular contents of silos, service operations, e.g. filling/emptying of silos, relatively light accidental impacts (heavy impacts should be explicitely included in the design). Fourthly, as an exceptional case, the designed structure may suffer considerable damage in a working accident (causing catastrophic imper­fections) and the question may arise whether or not the structure will continue to sustain its designed loads with a prescribed factor of safety.

As a conclusion, what we have to do is to investigate characteristic geometric, structural and load imperfectons (with respect to amplitude, shape and spatial position) relating to real large-scale structures. These imperfections thus become known and may be termed systematic imperfections. We may expect that more detailed knowledge of systematic imper­fections of specific structures is likely to be available in the near future due to extensive measu­rements. Now let us ask : if we were able to introduce all these known imperfections into the test or analysis models, would this be enough ? Perhaps it may come as a surprise, but the answer is 'no' due to the possible imperfection sensitivity of the imperfect structure. That me­ans, some specified percentage of total imperfections should be viewed as random imperfec­tions (in geometry and/or loading) in order to exclude pathological cases and to obtain buck­ling load values of desired robustness. The term 'random' implies that these particular imper­fections may have the worst possible effects. However, reasonable restrictions could be im­posed on random imperfections if it can be justified. Consequently, these random imperfec­tions are also called "worst' design imperfections. They do not necessarily relate to imperfec­tions that occur in the actual structures and in this sense they are artificial and only used to generate design buckling loads. The attribute 'design' relates to the fact that the generated buckling loads should be robust, in other words insensitive to additional imperfections.

It is therefore advisable to work exclusively with random imperfections or worst design imperfections (100 percent) of a certain magnitude as long as there is not enough

92

knowledge about systematic imperfections of real structures. One has to stay safe and besides, this is the approach adopted in the study of this section.

Support Conditions Another important aspect we have to face is the variation of the support condi­

tions. Because of the very local nature of buckling above the supports it is clear that support conditions and imperfections in this local area directly affect the buckling strength. The as­sumption of rigid supports is well justified for the present test configurations. However in many practical situations it may turn out that such extreme conditions are not applicable. The­refore we are interested in the possible decrease of buckling strength when the support condi­tions are released. For a particular situation it turns out that the worst condition occurs when classical boundary conditions are applied.

B4.2.4 Parametric Study and Development of a Design Rule B4.2.4.1 Introduction

The buckling strength of cylinders with constant wall thickness under local axial loads has been systematically investigated and is documented in [B.25]. Shell slenderness ra­tios r/t = 200 , 300 , 500 and 750 and dimensionless support widths μ = 0.025 , 0.05 , 0.075 ,. 0.1 and 0.2 were considered. The height of the cylindrical shell was h/r = 2 and the number of equidistant supports at the lower edge was always η = 4 (fig. B.l). These parameter values were chosen to cover the relevant range occuring in practical silo design.

It should be noted that the local supports were assumed completely flexible in axial direction, i.e. the support forces were introduced by constant Une loads across the widths of the local supports. This case represents a limiting case lying on the safe side since the supports themselves have no stiffness.

The complementary limiting case, which was also investigated and is reported about in [B.26], is represented by the assumption of completely rigid local supports.

The strengthening effects of support plates and ring-stiffeners at the lower edge were investigated, assuming realistic dimensions of these members, and are reported about in [B.27]. The according results serve as a starting point for the judgement whether actual sup­port designs correspond more closely to the assumption of flexible supports or that of rigid supports. Moreover, the main effect which is responsible for the increased buckling resistance of the rigid-support case, opposite to the flexible-support case, should be clearly worked out on the basis of the results obtained so far {interactive effects of axial warping and meridional bending constraints).

Elastoplastic material behaviour was taken into account in the underlying analyses representing the behaviour of mild construction steel of the common grade Te 360'. Addi­tionally, the effects of local geometric imperfections, i.e. inward-dents above the supports, we­re considered.

Summarizing, the underlying analysis represent the highest possible degree of mo­delling concerning realistic nonlinear structural behaviour. Therefore, the results of these ana­lyses may serve a basis for the development of a design rule. The first step in developing such a design rule requires to cast the analysis results into a com­pact form, that means to describe them by more or less simply shaped analytic curve-fitting formulae. In a second step these ultimate-load-formulae have to be supplemented by further conditions which, on the one hand, limit the range of validity of these formulae and which, on the other hand, make also statements about limiting conditions relating to additional relevant failure modes.

93

B4.2.4.2 Summary of Numerical Results

The results of the Geometrically (and Materially) Non­Lineai /mperfect analyses

{GNLI and GMNLI) are summarized in the form of critical mean meridional support stresses,

which are retrieved from [B.25], in tables B.3 and B.4 below and converted into dimensionless

form in tables B.5 and B.6. A non­dimensionalized imperfection amplitude of 5/t=1.0 was

used throughout for all r/t­values 200, 300, 500 and 750. According to the fabrication tole­

rances for cylinders under uniform compression the applicable imperfection amplitude for r/t =

200 and 300 is only 6/t=0.65 which is about 2/3 of the applied value 5/t=1.0. Some additional

calculations with the reduced imperfection amplitude give a small increase of the buckling

stresses. The design rules are based on a fixed imperfection amplitude of 6/t=1.0, and are

slightly conservative [B.44]. For completeness, these results are also graphically represented

once again in figs. B.l 1 to B.14 below. The results within braces for r/t = 500 were reported

in the doctoral thesis [B.18].

Ultimate stresses are also converted into dimensionless form, for convenient later

reference, in the following way. Firstly, the elastoplastic imperfect buckling stresses are refer­

red to the yield stress (value Y) and then the elastic imperfect buckling stresses are referred to

the elastoplastic imperfect buckling stresses (actually the inverse values of these fractions are

used, values X). The value of the yield stress used in the foregoing analyses is fy = 235 N/mm2.

r/t

100

200

300

500

750

1000

μ = b/B = nd/(2nt) = (2/π) .d/r

0.025

(0.021)

:292:

331

(252;8)

182

0.05

(0.042)

245

188

140 (157.3)

105

0.075

(0.083)

(103.5)

U>9

0.1

170

134 :■■:■■

97.3

78,2

(0.125)

(92.0)

0.2

145

123 1

(0.25)

(96.5)

Table B.3 : Maximum elastoplastic buckling stress acr,GMNu [N/mm2]

r/t

100

200

300

500

750

1000

μ = b/B = nd/(2nc) = (2/π). d/r

0.025

(0.021)

1689

963

(523.8)

256

0.05

(0.042)

875

489

228

(2655)

129

0.075

(0.083)

(142.6)

96.8

0.1

446

264

126

82.5

(0.125)

(110.9)

0.2

274

192

(0.25)

(96.0)

Table B.4 : Maximum elastic buckling stress CS„,GNU [N/mm2]

94

r/t

100 ί 200 ! 300

500

ί ; 750 '■;

1000

μ = b/B = nd/(2nt) = (2/π). d/r

0.025

(0.021)

1.243

.1 .409 ·'

(1:076)7

0.774 "

0.05

(0.042)

■ ■ ■ ■ ' ■ ' . ,

1.043

; 0.800

0.596

(0.669)

:¡:0;447 i

0.075

(0.083)

•

(0.440)

0.361

0.1

0.723

0.57Ó Γ

0.414

0.333

(0.125)

(0.391)

0.2

0.617

:0.523:

(0.25)

(0411

Table B.5 : Maximum elastoplastic buckling stress referred to the yield stress

Y = K

2,LOCAL =

^c^GMNLl/fy

r/t

100

200

300

500

750

1000

0.025

(0.021)

0.173:

0.344

(0.483)

0.711

0.05

(0.042)

0.286

0.384:

0.614­

(0;592)

^ 8 1 4

μ = ο/Β =

0.075

(0.083)

(0.726)

0;877¿

= nd/(2nt) =

0.1

0.381

0.508

0.772

0:948­:

= (2/π). d/r

(0.125)

(0.829)

0.2

0:529

0.641

(0.25)

(1.005)

Table B.6 : Maximum elastoplastic buckling stress referred to the maximum elastic buckling stress

Χ = σ, /σ,, cr.GMNLI' °cr,GNLI

0,05 0,1 0,15

relative support width μ = b/B [­]

0,05 0,1 0,15

relative support width μ = b/B [­]

Figure B.11 : Dimensionless load factors Λ = F/Fref

for shell slenderness r/t =200 (Analysis Series B) Figure B.12 : Dimensionless load factors Λ = F/Fref for shell slenderness r/t =300 (Analysis Series B)

95

<

o ro

LL

■α ro o

0 0,05 0,1 0,15

relative support width μ = b/B [­]

Figure B.13 : Dimensionless load factors Λ = F/Fref

for shell slenderness r/t =500 (Analysis Series B)

0.05 0,1 0,15

relative support width μ = b/B [­

0,2

Figure B.14 : Dimensionless load factors Λ = F/Fref

for shell slenderness r/t =750 (Analysis Series B)

B4.2.4.3 Compact Representation of the Numerical Results by Analytical Formulae

B4.2.4.3.1 Preliminary Remarks concerning the Selection of Imperfections There exist several possibilities to obtain an analytic representation of the numerical

analysis results (maximum loads or mean support stresses) by final curve­fitting formulae which depend mainly on the kind of philosophy by the aid of which the imperfections are introduced into these analyses.

Probabilistic Approach If a large number of analysis results is available a probabilistic approach could be

adopted to establish a design rule. In this case the design curve is obtained as a lower bound (e.g. 5­percent­fractile) data fit of the maximum­load result points or only part of these points which correspond to the characteristic failure mode. Thereby it is assumed that the feature of probability is introduced into the analysis primarily by the uncertainty of geometric imperfec­tions with respect to amplitude, shape and location. Of course, this purely probabilistic approach is extremely analysis­intensive and resembles in many features that of a pure laboratory testing approach and, apart from other reasons, it was not adopted in the present study.

Deterministic Approach

In the present study the geometric imperfections were prescribed in a deterministic

manner with fixed amplitudes of about one percent of the reference length lref = 4 /r/t This re­

ference length is used in shell buckling codes [B.l, B.28] as the length of the base line for mea­

suring the radial geometric fabrication tolerances of the cylindrical shell wall. Shape and location of these initial dents were chosen with the heuristic idea in mind

to render them geometrically affine to the postbuckling mode of the perfect structure. Finally it has been attempted to approximate this postbuckling mode by a simple analytic spline surface with overall in­surface dimensions according to the aforementioned reference length. This pro­cedure has been chosen for the purpose of easier specification and better reproduceability of the imperfection shape compared to the actual postbuckling mode which depends on the load level.

96

By such a postbuckling mode is meant a deformation pattern which results as the scaled difference of a postbuckling deformation and the corresponding prebuckling deforma­tion at equal load levels (P, P i , P2 in fig. B.l5). Obviously, this definition also includes snap-through eigenmodes and bifurcation eigenmodes as limiting cases (L, B). In these cases the reference load levels coincide with the snap-through load (or limit load) and the bifurcation load respectively. For very small imperfection amplitudes the snap-through or bifurcation ei­genmodes must be considered to represent the 'worst' or 'most unfavourable' imperfections with respect to the reduction of the perfect structure's load carrying capacity. This latter statement is supported by the results of Koiter's asymptotic approach to imperfection sensitivi­ty analysis [B.29].

For larger imperfection amplitudes the imperfection shape, which is the most critical one, generally tends to change to the postbuck­ling mode of the perfect structure obtained at the re­duced maximum load level-of the imperfect structure. The imperfect structure progressively loses memory what happens to its perfect counterpart, far up at the perfect maximum load level, i.e. the larger the imperfec­tion amplitude becomes and the larger the load reduc­tion is compared to the

classical buckling eigenmode

nonlinear Bifurcation eigenmode (=zero-eigenvalue)

Snap-through eigenmode

Postbuckling deformations

various posibilities)

inear "small" displacements or any prebuckling

deformation

characteristic displacement component

Figure B.15 : Use of actual deformations or eigenmode shapes of the perfect structure as theoretical imperfection patterns

perfect case. However, when the imperfection amplitudes become very large, we have essen­tially to deal with a new structure which may exhibit a completely different load bearing beha­viour. In this extreme case the 'perfect postbuckling mode'-approach to geometric imperfec­tions, as described above, may become increasingly inappropriate.

Summing up, the adopted approach of selection of postbuckling modes of the per­fect structure as theoretical imperfection patterns is based on mechanical features of the actual buckling behaviour of the perfect structure. Therefore this approach proves itself more ratio­nal, mechanically better founded and versatile in general situations compared to the classical-buckling-eigenmode approach which relates to the approximative, and often unrealistic case of a linearized local stability condition about the initial undeformed state (C).

B4.2.4.3.2 Curve-fitting of Numerical Results A comparatively small number of analyses has been performed, as mentioned in the

introduction, applying 'worst' geometric imperfections - in the restricted sense of approximate postbuckling modes - in a deterministic way. Essentially about 16 maximum-load results {GNLI and GMNLI) are available (4r/t-values times 4 μ-values) which are now ready to be fit­ted by lower-bound curves.

The main idea adopted in curve-fitting of these maximum-load results bears on the following three basic observations :

97

* For very narrow supports the axial load­carrying capacity of the local support is dictated by the

2D Von­Mises yield condition (plastic yielding) and therefore the maximum load is in direct

proportion to the support width.

* For wide supports the axial carrying capacity of the support is approximately equal to the axial

carrying capacity of the uniformly loaded cylinder, expressed in critical buckling stresses σ0Ι =

K2.fy (K2 ... reduction factor).

*In an intermediate range of support widths the axial buckling load which can be resisted by the

local supports is nearly independent of the actual support width. Expressed more precisely, the

axial buckling load shows a typical increase with increasing support width, which is less pro­

nounced for slender cylinders. The key point for understanding the following procedure is

such that these variations of the maximum load factors Amæi with the dimensionless support

width μ are geometrically approximated by lower­bound straight lines , i.e. of all four investi­

gated shell slenderness ratios r/t (fig. B.l6). These lower­bound fitting lines were originally

derived from the maximum load diagrams figs. B.l 1 to B.l4 above in which they appear as

thick­dashed lines.

9.60

1.50 •Í­

R/t = 20fl

1.155 · ßyieid/ /

/ /

/ ^y

"""G^LI /

/

'ÍGMNLI

11.30 R/t = 300,

Λ R/t = 500

/ , / l

/1.155 f ßyjeid

3.6 D

μ^0.2

μ­Ξ=0.2

/ R/t = 750

/ 1 . 1 5 5 · ßyiekj

ΪΠ=0.2

Figure B.16 : Linear approximations of Λ^-μ curves for an intermediate range of support widths for slen­derness ratios r/t = 200,300,500 and 750.

The relevant information contained in the foregoing diagrams (fig. B.16) is summa­rized in tabular form (tab. B.7). The dimensionless load factors Amax = Fmax/Fref refer to a fixed reference load of Fref = 150kN acting on one eighth of the cylinder circumference in axial direc­tion (45° analysis cutout). As can be seen from fig. B.16 the solid straight fitting lines are conti­nued to the μ = 0-ordinate on the left of the diagrams. They may be uniquely described by the extrapolated load factor A^a = Λο of the left μ^β = 0-ordinate and by the inclinations ΔΛο of the fitting lines which are computed as follows (μπ^ί = 0.175 or 0.200) :

ΔΛο = (A right Λο^/μ right (1)

98

r/t

100 200

300 500

750 1000

r [cm]

300.

300.

500.

750.

t [cm]

L5

1 1.

1.'

μ right

0.2

0.2 0.175

0,2

Λο - Fmax / Fref

GLNI

9.60

3.50

2.70

2.20

GMNLI

1.50

0.80

1.10

1.50

ΔΛο

GNLI

8.500

6.000

5.143

5.750

GMNLI

23.750 !

12.500

14 286

9.250

Ayield, 1 -

Fyield,l /

Fref

55.37

36.91

61.52

92.28

Table B.7 : Parameters of straight-line-fits of maximum load values

In the table B.8 the parameters A0 and ΔΛ0 have been converted into parameters λο and Δλο. This conversion is accomplished by a change of the reference basis from the reference load Fref, which is mainly useful in the numerical analyses, to the physically more meaningful value of the plastic yield load Fyiei<j,i = r7rtfy/4 of each shell configuration (fy = 23.5 kN/cm2 for Fe 360). The following relation holds for the dimensionless critical load factor λ :

λ = ? yield, 1 yield, 1

ΐ{π/4)ίσ„μ τ{π/4){Λ μ. κ 2.LOCAL (2)

r/t

100 200 300 500 750 1000

λο = Λο /Ayieid.i - 100

GLNI

17.34 9.48 4.39 2.38

GMLNI

2.709 2.167 1.788 1.625

Δλο=ΔΛο/Λ^Ι(1,ι. 100

GNLI

15.35 16.26 8.360 6.231

GMNLI

42.89 33.87 23.22 10.02

Table B.8 : Converted parameters of straight-line-fits of maximum load values

For reference and comparison purposes with the case of the cylinder under uniform axial loading conditions the reduction fac­tors κ2 of the german shell buckling code DIN 18800/part 4 [B.28] is used. This co­de is formulated in such a way that the re­duction factor K2 = aCI/fy , which is the 'characteristic' buckling stress referred to the yield stress, is related to a shell slen­derness parameter , which is a generaliza­tion of the common beam slenderness pa­rameter occuring in the Euler buckling formulas for beams and plates.

I = \t,-CTcri

1 r, _ A|0.605Et/r A

(Ten

1 23.5 10.605. 20600 '

1 r 530.3 ' t

i- 1 23.03 '

ι? Vt

(3)

The reduction factor κ2 for uniform axial loading conditions is defined in DIN 18800/part 4 according to the following equation (eq. 4). Reduction factors are summarized in tab. B.9 below for the r/t-ratios used in the numerical analyses underlying the present study.

99

r/t

100

200

300

500

750

1000

r

[cm]

300.

500.

750.

t

[cm]

1.5

1

1

1 '.

0.605Et/r

124.63

6232

41.54

24.93

16.62

12.46

* = ^ / σ ­

0.434

0.614

0.752

0.971

1.189

1.373

^ 2 K2juniform

0.828

0.660

0.531

0.327

0.178

0.116

Table B.9 : Reduction factors κ2 according to the german code DIN 18800/4

K2 ~ ^.uniform

1

1.233­0.933.1

0.3 IV

0.2 /J2

for

I < 0.25

025 < λ < 1

1 < λ < 1.5

1.5 < Χ

(4)

Curve­fitting of dimensionless Load Factors λ0 and Δλ0 over r/t

dimensionless load parameter λο

The dimensionless load parameter λο is approximated by a. power fit over r/t or the

shell slenderness parameter respectively, for both the elastic (GNLI) and the elastoplastic

(GMNLI) case resulting in the following fitting formulae :

/IO.GMNLI — 20.193 . [ —

Λ0,' GNLI = 50000.

■0.385

1.503

1.804. X 0.77

4.018. J 3.006

(5a)

(5b)

49.31

Õ

ra

(lar E

, ■

log

3.429

1 ­

Ν

Ν.

V. Ν.

Ν.

Ν.

­ * ­ lambda,0­GMNLI

­ * ­ lambda,0­GNLl

\ G N L I

GMNLI ^ s _

1 ' Γ ' Ι ι , . . . , ι ■ f . — f —

1.549

1.413

100 log (RA)

1000

Figure B.17 : Dimensionless load pa­rameter λο ­ r/t diagram in double lo­garithmic scale (dashed lines repre­

sent power fitting curves)

100

20

15

ro

Ειο-ε ro

I ι I

­ ­~~^

\ v\ \\ \\ \\ \\ \\ \\

\ X \ \ \ \ \ ■

■

1 1 1 ­ 4 ­ lambda.O­GMNLI

­ * ­ lambda,0­GNLI

■.^v

^=>= = s = ­ a^

^ ­ ­ ­

100 200 300 400 500 600 700 800

R/t

Figure B.18 : Dimensionless load pa­rameter λο ­ r/t diagram in linear scale (dashed lines represent power fitting

curves)

A graphical representation of

these power fitting curves is given

in fig. B.18. These curves appear

as straight lines when plotting is

performed in double logarithmic

scale (fig. B.l7) and lead to the

following formulae :

logCkoMNLi) = l­305­0.385.log [­J = 0.256­0.77.1οΕλ

logOkGNLi) = 4.700­1.503.1og[ ­ j ­ 0.604­3.006.1ogX

(6.a)

(6.b)

dimensionless load parameter Δλο

The dimensionless load parameter Δλο is approximated by a linear fit over the shell

slenderness parameter for both the elastic (GNLI) and the elastoplastic (GMNLI) case resulting

in the following fitting formulae. A graphical representation of these linear fitting curves is

given in fig. B.l9.

UU­i

75­

0­

\

__

1 1 1 1 1 ­ * ­ Dlambda.O­GMNLI

­ * ­ Dlamb

GM

Gf

NLI

vILI I I I I

da,0­

TTTT­

3NLI

I I I I Ι Ι f l

Figure B.19 : Dimensionless load parameter Δλο

­λ diagram (λ = ­Jí^lããi)

(dashed lines represent linear fitting curves)

0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4

Lambda­quer

Δλο,οΜΜ., = 76.91 ­ 56.05. λ

Δλο,οΝίι - 27.93 - 18.58. λ

(7.a)

(7.b)

101

Summary of fitting Formulae for the Maximum Loads

Xo.GMNLI

Xo.GNLI

A Xo.GMNLI

A Xo.GNLI

=

=

=

=

1.8/I077

4/X3

77-56. Χ 28 - 19. Χ

(8)

Final form of the Approximate Formulae for the Maximum Loads First we recall the basic assumption of this subsection on 'Curve-fitting of Numeri­

cal Results', namely that the dimensionless maximum loads λ are linearly approximated over the dimensionless support width μ :

X[in%] = Xapprox[in%] = Xo(X) + μ. Δχ0(Χ) (9)

Now we are prepared to write the final approximation formulae for the maximum loads λ , referred to the yield of the uniformly loaded cylinder, in dependence of the parameters of the local support width (μ) and the shell slenderness (X or r/t respectively) as follows :

XGMNLi[in%] = 1.8/Χ077 + μ. (77 - 56. Χ) XGNLI[in%] = 4 / χ 3 + μ. (28 - 19. Χ)

(10)

Β.4.2.4.3.3 Engineering Representation of the Approximate Formulae

Parameter Changes In this sub-section the approximate formulae (eq. 10) are converted into a form

which is more appropriate for the purposes of practical application in engineering design. Consequently, two parameter changes are introduced into these formulae as follows : * Instead of using dimensionless maximum loads λ, which has proved to be advantageous in

deriving the approximate formulae, we use maximum mean support stresses σ0Γ, according to eq. 2 :

λ = μ. μ. κ 2.LOCAL (H)

* Instead of using the dimensionless support width μ , which depends on the number η of co­lumns (n = 4 in the present study) we use the parameter η = d/r (support opening angle ; d denotes circumferential width of the local support), which is independent of the number of co-lumns. The choice of η is physically motivated and reflects the fact that the local load be­aring capacity actually depends on the circumferential opening angle of the local supports as long as these local supports are located at a sufficient distance from each other so that inter­action effects do not play any role [B.30, B.31] :

d.n n d n μ = = — . — = — . 7 7 - »

2Γ7Γ 2π r 2π

μ = —.η = 0.63Ίη for n = 4 π

(12)

102

Introducing these parameter changes into the general formula eq. 9 we arrive at the following conversion formula for the critical stress :

^ = κ _ λ _ λ[ΐη %] 2.LOCAL

λο Δλο 0.9 λο Δλο ■ τ — τ ■

μ ΙΟΟμ 100_2 100 rtfdeg] 100

π

or alternatively (13)

κ 2.LOCAL = 1.571λο + Δλο

100

Final Approximate Formulae

When the approximate expressions given by eq. 10 are introduced into the conver­

sion formula eq. 13 the final representation of the approximate formula of the elastoplastic case

(elastoplastic reduction factor under local support conditions) is :

κ _ Q"cr,GMNLI 2.83

2.LOCAL

λα77η[%] + (0.77 ­ 0.56. λ) (14.a)

A graphical representation of this approximate formula is given in fig. B.20 below (r/t is curve parameter).

1,2

z: Έ (3 evi

0,8

0,6

< O o _ι 0,4

0,2

-

-

-

-

L _

\ \ \ \

, 1 . . . . . .

R/t=100

■—_____¿uu

—___300

4nn

— — _ 5 0 0

_ _ _ _ 6 0 0

700

r——MÖU

_ _ _ _ 9 0 0

~R/t=100Ö" I I I ,

10 15 20

η = d/R [%]

25 30

Figure B.20 : Fitting curves of elastoplastic critical stress according to eq. 14.a

103

Similarly, we arrive at the approximate expression of the elastic critical stress which

is also referred to the yield stress fy. Then the approximate formula of the elastic case reads as

follows :

Ccr.GNLI 6.28

λ3 ■ η[%] + (0.28 ­ 0.19 . λ) (14.b)

We obtain an alternative representation of eq. 14.b in terms of r/t when the parame­

ter λ is converted back according to eq. 3. In addition, the elastic critical stress is now referred

to the ideal elastic bifurcation stress aCú and not to the yield stress fy . This conversion is again

carried out with the help of eq. 3.

O" cr.GNLI Q"cr,GNLI η­2

­a

1.45 +

r / t

1000 .(0.53­0.0156^/r7t)

VrTt.ri

(imperfection reduction factor) LOCAL

(14.C)

B4.2.5 Design Rule

One of the basic ideas of the formulation of the design rule for axially loaded cylin­

ders on local supports was to establish a directly visible transition to the cylinder under uniform

axial loading. This case is already well treated by design codes and, in our opinion, serves as an

appropriate basis of reference from the engineering point of view. One of the main features of

the buckling behaviour under local axial support conditions is given by the fact that it is never

less favourable than under uniform loading conditions. As soon as the support width becomes

sufficiently narrow the critical axial buckling stress increases, compared to the case of uniform

loading, until it is ultimately limited by the axial yield stress. It turned out to be relatively easily

possible to create such an integrated representation which contains both the new local loading

case as well as the already well­known uniform loading case. The characteristic increase in load

carrying capacity becomes directly visible in the chosen representation of the final design dia­

gram which is eventually used by the practical design engineer.

For this purpose we utilized the german shell buckling code DIN 18800/ part 4 as a

starting point which represents the buckling loads of cylindrical shells under uniform axial loa­

ding in the form of reduction factors K2. These reduction factors are applicable to the yield stress

fy (eq. 15.a below) and depend on a dimensionless slenderness parameter λ = ­Jíy/om (eq. 3).

Therefore, we have to express the local buckling stress with reference to the yield stress, i.e. to

define a local reduction factor K2,LOCAL This local reduction factor depends on the dimension­

less support width parameter η = d/r and the shell slenderness parameter r/t which may be easily

converted into λ . This local reduction factor represents the final result of the numerical study of

this part of the project in condensed form and has already been worked out in eq. 14.a above.

A disadvantage of this formulation ­ as one may claim ­ is that the yield stress also

occurs in the right­hand­side of the formula for λ and thus lets one believe that it is valid for

any value of the yield stress. The computations which serve as the basis of the approximation

formula eq. 14.a­c were performed for the steel grade Fe 360 only. Therefore, it remains to futu­

re investigations to show that similar approximation formulae are also valid for other commonly

used steel grades, e.g. Fe 430 and Fe 510.

104

Final Design Formula The final design formula may be written in the form of 3 conditions as follows :

K 2.LOCAL _ »J cr _ 2.83

<1 or 1.155

+ (0.77-0.561;

>κ ,

0) (2)

(3)

(15.a)

* The second one of these conditions describes the local reduction factor in dependency of the dimensionless support width η = d/r , the support opening angle expressed in percent, and the dimensionless shell slenderness parameter λ = Jfy/acri , where acri is the ideal elastic bifurca­tion stress of the uniformly loaded cylinder acri= 0.605Et/r.

* The first one of these conditions describes the plastic limit condition which becomes relevant for very narrow supports. If two-dimensional Von-Mises plasticity is taken into account at the supports, the higher value applies (1.155) due to fully restrained straining conditons in circum­ferential direction. Such conditions could be imagined to be established by very stiff edge rings. However, in order to stay on the safe side and to cover all possibilities of practical sup­port constructions (flexible edge rings or even no edge rings at all) one should not allow for a value greater than the uniaxial yield stress.

* The third one of these conditions describes the limit of the uniform loading condition which applies to wider supports. When the support widths increase beyond certain limits there is no additional gain in load carrying capacity of the locally loaded case compared to the uniformly loaded case. Or put the other way round, the critical buckling stress of the locally loaded case never falls below the value of the critical buckling stress of the uniformly loaded case.

Further Limiting Conditions * Finally, a fourth conditon of technical nature has to be added. This condition is intended to

prevent the case that a cylinder on many narrow, closely spaced, supports yields a higher buckling load than the according uniformly loaded cylinder (eq. 15.b).

K2.LOCAL — K2 with n d

μ - — . — 2π r

(15.D)

μι-2

M-effective

Figure B.21 : Limiting condition for several (> 2) closely spaced narrow supports

That means that it should be ex­cluded to obtain higher buckling loads by a purely formal redistri­bution of axial loads over the cir­cumference, when the underlying loading condition is actually more or less uniform over a wide range of the circumference.

* A further limiting condition of technical nature may become ne­cessary which limits the minimum distance between neighbouring supports or which determines when neighbouring supports have to be treated as one single support

105

25 30 10 15 20 η = d/R [%]

Figure B.22 : Graphical representation of the design formula (Eq. 15.a) · elastoplastic critical stress as a function of η (r/t is curve parameter)

for the purpose of buck­ling design calculations. This condition is similar to the foregoing condition with the difference that it also applies to distinct groups of a small number (> 2) of narrow closely spaced local supports. This situation is explained qualitatively by fig. B.21 alongside. Further thin­king is required to formu­late an adequate transition rule which states when closely spaced local sup­ports are to be treated as independent of each other and when an effective support width has to be applied covering interac­tion effects in an ap­proximative, conservative way.

λ= β Yield'

Figure B.23 : Graphical representation of the design formula (Eq. 15.a) · elastoplastic critical stress as a function of λ (η is curve parameter)

Graphical Representations of the Design Formula

Graphical representations of the design formula are shown in figs. B.22 and B.23, the first one using η as the abscissa and r/t as curve parameter and the second one using λ as the abscissa and η as curve parameter. A 3D-surface representation is given in fig. B.24 below.

The graphical representa­tion of fig. B.23 is proposed to be used as the final de­sign diagram. This diagram is parametrized in the same way as the diagram for uni­form axial loading condi­tions in the german code DLN 18800/part 4 [B.28].

106

This representation allows a direct comparison of the load bearing capacities for local and uniform axial loading conditions in one and the same diagram, for given geometry parameters r/t and η = d/r , which ma­kes it well suited for appli­cation in practical engineer­ing design.

Figure B.24 :3D graphical re­presentation of the design for­mula (Eq. 15.a) ­ elastoplastic critical stress as a function of

dimensionless shell slenderness parameter λ and support

width parameter η = d/r

B4.2.6 Formulation of an Alternative Design Rule In this section an alternative formulation of a design rule is presented based on the

numerical results of section B4.2.4.2 as before. The philosophy, underlying this design rule, cor­responds to the ECCS approach to shell buckling design [B.l] and was originally utilized by Rotter et al. in 'Proposed Design Rule for Buckling Strength Assessment of Cylindrical Shells under Local Axial Loads', submitted to ECCS TWG 8.4 Buckling of Shells [B.31]. A descrip­tion of this design rule will be given in the next section.

The main idea of this alternative design approach is to split the buckling design pro­cess into two distinct steps. This is opposite to the one­step­procedure of the design procedure presented in the preceding section where the elastoplastic imperfect critical stress acr¡;GMNU is directly calculated from the geometry and material parameters according to eq. 15. Thereby the shell slenderness parameter r/t is hidden in the dimensionless parameter which depends on the classical bifurcation stress of the cylinder under uniform axial loading conditions.

* The first step consists in calculating the elastic imperfect critical stress acrjoNLi from an ap­proximate formula (eq. 14.b or c) in dependence of the geometry parameters (r/t and η) of the problem. This approximative formula is obtained by a fitting procedure as demonstrated in the preceding section

* The second step consists in calculating the elastoplastic imperfect critical stress acr>GMNu by utilizing an elastic­plastic interaction relation (figs. B.25­B.27).

107

0,9

0,8

0,7

0,6

0,5

Y 0,4

0,3

0,2

0,1

" " " % ■

;

;

\ Y

; y

\ *

= t

\ *

JU

\ *

"PU

S τ ­

1 \ *

.. I

tan α = I

1 * 1 1 I

\ *

NPUT

\

% \

À\

0 0,1 0,2 0,3 0,4 0.5 0,6 0,7 0,8 0,9 1

X

Figure B.25 : Alternative design rule ­ linear elastic­plastic interaction relation

The analytic representation of the elas­

tic­plastic interaction relation, which is

assumed linear in the present case, is

computed as the intersection Ρ of the two

straight lines in fig. B.25, the inclined

dashed interaction line and the solid line,

emanating from the origin, which is incli­

ned by the angle α (eq. 16). The vertical

part of the dashed interaction polygon at

the right side corresponds to purely elas­

tic behaviour (no interaction). The upper

horizontal part of the interaction polygon

corresponds to purely plastic yielding.

The parameter on the abscissa is defined

as the fraction X = acr,GMNu/acr,GNu ,

which contains the unknown elastoplastic

critical stress. The parameter on the ver­

tical ordinate is defined as Y = acr,GMNLi/fy

, which is the unknown elastoplastic criti­

cal stress related to the yield stress and

corresponds to the reduction factor

K2,LOCAL (fig. B.25).

5 7,5

X = tan(alfa)

1

0,9

0,8

0,7

0,6

0,5

ι 0.4

0,3

0,2

0.1

0 ELASTIC !

! PLASTIC

Ί YIELDING

0,01 0,1 10 100 X = tan(alfa)

Figure B.26 : Alternative design rule - direct para- Figure B.27 : Alternative design rule in linear-log metrization as a function of scale - direct parametrization as a function of

X = tan α = aCr,GNLi/fy X = tan α = acr>GNLl/fy

Y = (Tcr.GMNLI

1 +

1 + a 1 + a 1

f f cr.GNLI f y / 1 + 1

(16)

tana

Remark that 'a' is shown in figure B.25

108

The practical application of the elastic­plastic interaction works in the following way

(fig. B.25). First compute the fraction a^oNu/fy = tan α from an existing fitting formula {INPUT

parameter). Then draw a straight line from the origin, which is inclined under the angle α , and

obtain the intersection with the inclined interaction line. The vertical coordinate of this intersec­

tion point represents the required value Y = a^oMNi/fy {OUTPUTparameter).

Another equivalent representation of the aforementioned interaction relation is

shown in fig. B.26. This curve is obtained by directly using the INPUT parameter acr,GNLi/fy =

tan α = X as the new parameter on the abscissa. In this case the OUTPUT parameter Y is obtai­

ned directly and no auxiliary geometrical construction is needed as before. Drawing the abscissa

axis in a logarithmic scale logX = log(tan a) results in the converted shape of the interaction

curve of fig. B.27.

A second equivalent representation of the interaction relation is obtained (eq. 18 and

fig. B.28 below) by introducing a transformation of the abscissa parameter :

λ = 1

tana 'CTcr.GNLI

­ » 0"cr,GNLI 1

(17)

O" cr.GMNLI

1 plastic yielding

(1 + a ) / ( l + %2) if elastic ­ plastic interaction

i/X2 elastic

(18)

0,9

0,8­

0,7­

0,6­

Y 0,5

0,4­

0,3­

0,2­

0,1

;

CD Ζ ­Ω _ l

■UJ­

> o

H

< CL

E ­ASTIC ¡—>

X = lambda

Figure B.28 : Alternative design rule ­ direct parametrization

as a function of X = ­^/l/tana = Jfy /σ CT GNLI = λ

This final representation of the elas­

tic­plastic interaction relation of fig.

B.28, which is equivalent to the primary

representation of fig. B.25, allows se­

veral comparative observations :

* The structure of the abscissa parameter

λ is similar to the parameter λ used

in the original design rule (section

B4.2.5) However, these parameters

have different mechanical meaning.

* The parameter λ" depends on the im­

perfect elastic critical stress under lo­

cal support conditions, which in turn

depends on the two geometry parame­

ters r/t (shell slenderness) and η

(support width).

* The parameter λ , which depends on

the classical bifurcation stress under

uniform axial loading conditions and

thus depends on the shell slenderness

parameter r/t only. Consequently se­

parate interaction curves have to be

109

1.155

1

ω 0,9 >. rà E 0,8 gi ω

0,7

CD 0,6

E en

in II

< O O

0,5

0,4

0,3

0,2

0,1

;

;

;

;

­

Ò.O750S

t§;É ,

r* \ ♦

\

*

χ \ X w

•V

^¿\

­ ­m­ MUE ­ 0,025

— · ­ MUE = 0,05

­ * ­ MUE = 0,1

—♦— MUE = 0,2

■■■ I — I — I — I ■ ■ Γ Τ ­

rX V

\ fe

^ %

'V ♦

X

S: ■*■

'V ■V I 1

­> I I1

0.15

0.075

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Sigma­cr,GMNLI/Sigma­cr,GNLI

Figure B.29 : Alternative design rule and numerical analysis re­

sults (μ is curve parameter and r/t varies along the curves)

1.155

2 ω >­ró E σι W

3 0,7

0,9

0,8

5 O

CO

II

< O o

0,6

0,5

0,4

0,3

0,2

0,1

0

;

;

;

;

;

;

;

;

­

0.075 0.15 ■ \

" " " • i

V M

\

V *

X \ 1

NS V

­ ­m­ R/T = 200

­ · ­ R/T = 300

­ * ­ R/T = 500

—Φ— R/T = 750

■ ■ . . I . . . . I M M I . M .

> &

I I I I

\

\

\

\

. χ fc^ X

X V

1 1 I !

rv V

ι ι ι I

X T L » .

S: %

X X

\ ; \

1 ! 1 1

X X

' \ >

I I

0.15

0.075

plotted for different η­values

(support width).

* The two design approaches are es­

sentially equivalent concerning

their potential of representing the

final results. Their main difference

lies in the fact that the original de­

sign approach is a direct one­step­

procedure (leading to elastic­plastic

critical stress) whereas the latter

one is an indirect two­step­

procedure (elastic critical stress +

elastic­plastic interaction).

However, it is interesting to note

that, if similar parametrizations are

used (λ and λ) , similarly shaped

elastoplastic load reduction curves

are obtained in both cases. Histori­

cally, the first approach is used in

the german shell buckling code

DIN 18800/4 [B.28] whereas the

latter alternative approach is used

in the ECCS Recommendations

[B.l].

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Sigma­cr,GMNLI/Sigma­cr,GNLI

Figure B.30 : Alternative design rule and

numerical analysis results (r/t is curve pa­

rameter and μ varies along the curves)

Formulation of the Alternative Design Rule

The actual formulation of the alternative design rule is given by eq. 14.b (section

B4.2.4.3.3) for the elastic critical stress on the one hand and by the elastic­plastic interaction rule

eqs. 17 and 18 on the other hand. The interaction polygon represents a lower bound to all avai­

110

lable numerical results and it is determined by the value of the parameter a = 0.075 (see fig. B.25). The following two figures show different representations of the numerical results, taken from tables B.5 and B.6 above (section B4.2.4.2), which are bounded from below by the interac­tion polygon (thick dashed lines in figs. B.29 and B.30).

B4.2.7 Comparison with an existing Design Proposal and Rigid Support

Conditions

The preceeding design rules are compared with an existing design proposal 'Pro­posed Design rule for Buckling Strength Assessment of Cylindrical Shells under Local Axial Loads' by Rotter et al., which was submitted to ECCS TWG 8.4 Buckling of Shells [B.31]. The database of numerical results underlying this proposal consists of comprehensive results provi­ded by the Edinburgh group (Prof. Rotter) on cylinders with rigid local supports and on a limited number of results provided by the Gent/Graz group (Prof. Rathe and Prof. Greiner) on cylinders ■wiih flexible supports (for shell slenderness ratio r/t = 500 only). The above design proposal served as the starting point of the study of the foregoing section and is represented by fig.B.31 and the formulas of eqs. 19 and 20 below. The position of the straight interaction line is defined by the value of the parameter a = 0.15. This straight interaction line is plotted as an inclined thin dashed line in figs. B.29 and B.30 for comparison purposes.

0.15

a = 0.15

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Sigma­cr.GMNLI/Sigma­cr.GNLI

Figure B.31 : Design rule of Rotter et al.

O" cr,GMNLI

1 1 + a

1 + l/fo­cr.GNLl/fy^

O" cr.GNLI / f y

plastic yielding

if elastic ­ plastic interaction

elastic

(19)

The elastic imperfect critical stress, which is needed as input for the computation of the elastoplastic imperfect critical stress in eq. 19 is defined in [B.31] and is given here for completeness :

111

O" cr.GNLI ffcri

fv 0"cr,GNLI 0.605 .

r / t 0.01 + 0.06

+ 65 r / t

(20)

This formula corresponds to eqs. 14.b and c for the fitted elastic imperfect critical stress underlying the alternative design rule (section B4.2.4.3.3). Again, the parameter η = d/r is the dimensionless support width and σ0Γί = 0.605 .Et/r is the classical bifurcation stress of the cy­linder under uniform axial loading conditons.

Modification of the Design Proposal On the wider basis of numerical results on locally supported cylinders on flexible

supports (Gent/Graz) and on rigid supports (Edinburgh), which are now available, it becomes possible to refine the design proposal of Rotter et al., i.e. to provide separate design formulas corresponding to the lower limiting case oí flexible supports and the upper limiting case of rigid

supports (fig. B.32).

Figure B.32 : Suggested modification of the design proposal of Rotter et al.

,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Siqma­cr.GMNLI/Siqma­cr.GNLI

The design equations for rigid support

conditions have the same structure as

those for flexible supports and differ

only in the coefficients. Details of their

derivation are given in [B.33].

K2,LOCAL,RIGID O'er

fy

< 1 or 1.155

5.0

Xh*.r][%] + (0.68­0.50.λ)

^ K2

(a)

00

(c)

(21)

The question which support condition is applicable with confidence in practical si­

tuations cannot be answered so far and further investigations are necessary. Until this question

is clarified it is recommended to use the design equation for flexible support conditions.

B4.2.8 Examples In the subsequent examples the following material data for constructional steel grade

Fe 360 are used : E = 20600 kN/cm2, ν = 0.3 , fy = 23.5 kN/cm2.

112

The reduction factor K2 for highly imperfection sensitive shells according to DIN

18800/part 4 applies to cylindrical shells under uniform axial loading and is defined by the code

as follows :

K2

— K2,unifonn

1

1.233­0.933. Jt

0.3/λ3

0.2/λ2

for

λ < 0.25

0.235<λ<1

1 < λ < 1.5

1.5 < λ

(4)

Β4.2.8.1 Example 1 : r = 300 cm, t= 1 cm , d = 35 cm , η = 4 columns.

η = d/r = 35/300 = 0.11667 = 11.67 % ­> μ = (η/2π).η = 0.0743, r/t = 300.

Uniform compression : σα\ = 0.605.Et/r = 41.54 kN/cm2,

­> λ = Jti<ycri = 0.752 and κ2 = 1.233 ­ 0.933 λ = 0.531

Local support condition :

K2,iocai = 2.83/(λ °·77. η[%]) + (0.77 ­ 0.56λ ) = 0.651 > κ2 (+ 23%).

σ0Γ = K2)iocai.fy = 0.651 .23.5 = 15.30 kN/cm2 ­> F cr = n.acr.d.t = 2144 kN.

For this shell of moderate slenderness and medium support width there is a notable gain of 23 %

in the critical axial buckling stress due to the local support condition, compared to the case of

uniform loading.

B4.2.8.2 Example 2 : r = 450 cm, t = 0.6 cm , d = 52.5 cm , n = 4 columns.

η = d/r = 52.5/450 = 0.11667 = 11.67 % ­» μ = (η/2π).η = 0.0743 , r/t = 750 .

Uniform compression : σ^ = 0.605.Et/r = 16.62 kN/cm ,

­ > λ = / Γ / σ ~ = 1.189 and κ? = 0.3/λ3 = 0.178 . y en

Local support condition :

K2)iocai = 2.83/(λ °­77. η[%]) + (0.77 ­ 0.56 λ ) = 0.316 > κ2 (+ 76 %)

σ0Γ = K2,iocai.fy = 0.316.23.5 = 7.43 kN/cm 2 ­>F c r = n.acr.d.t = 936kN.

For this shell of pronounced slenderness and medium support width there is now a pronounced

gain of 76 % in the critical axial buckling stress due to the local support condition, compared to

the case of uniform loading.

B4.2.8.3 Example 3 : r = 300 cm, t = 1.5 cm, d = 70 cm, n = 4 columns.

η = d/r = 35/300 = 0.23333 = 23.33 % ­> μ = (η/2π). η = 0.1485 , r/t = 200.

Uniform compression : acr¡ = 0.605 . Et/r = 62.32 kN/cm ,

λ = Λ / ξ 7 σ ~ = 0.614 andΚ2= 1.233­0.933 λ = 0.660.

Local support condition :

giocai = 2.83/(λ ° ·7 7 . η[%]) + (0.77 ­ 0.56 λ ) = 0.603 < κ2

113

­ > K2,iocal = κ 2 = 0.660 (± 0 % )

σ0Γ = giocai ■ fy = 0.660 .23 .5 = 15.51 k N / c m 2 ­ > F c r = η . σ „ . d . t = 6 5 1 6 k N .

For this shell of moderate thickness and wide supports there is NO gain in the critical axial

buckling stress due to the local support condition, compared to the case of uniform loading.

Since the local buckling stress is never smaller than the buckling stress under uniform loading,

the present case is governed by the uniform loading condition.

B4.2.8.4 Example 4 : r = 500 cm,t = l c m , d = 15cm,n = 4 columns.

η = d/r = 15/500 = 0.03 = 3 % ­> μ = (π/2π). η = 0.0191, r/t = 500.

Uniform compression : σ<ή = 0.605 . Et/r = 24.93 kN/cm2 , -> λ = A / Ç 7 o ~ = 0.971 and κ2= 1.233­0.933 λ = 0.327.

Local support condition : 0.77.

K2,iocai = 2.83/( λ υ".η[%]) + (0.77 ­ 0.56 λ ) = U91 > 1

­>K2,local=L0(+206%).

σ α = K2>ioeai. fy = 1.0. fy = 23.5 kN/cm2 ­> Fcr = Fy¡eid n.fy.d.t=1412kN

For this slender shell on narrow supports there is an extreme gain in the critical local axial buck­

ling stress, compared to the case of uniform loading. Fully plastic conditions occur at the sup­

ports.

B4.2.8.5 Comparison with other design rules

The results of the foregoing design examples, using the developed design rule

(section B4.2.5), are compared with the alternative design rule (section B4.2.6) and with the

existing design proposal of Rotter et al. (section B4.2.7). The results are compared on the basis

of local reduction factors K2ILOCAL = σα/fy in the following table B.10. The computed strength

gains K2,LOCAL / κ2 in % always refer to the reduction factor for uniform axial loading.

EXAMP

No.

1

2

3

4

r/t

300

750

200

500

d/r

[%]

11.67

11.67

23.33

3.00

K2

0.531

0.178

0.660

0.327

K21ocal

design

rule

01651

0.316

(0.603) 0.660

(1.191) 1.000

alternat,

rule

0.628

0.293

(0.613)

0.660

0.757

Rotter et

al.

0.652

0.347

0.703

0.798

strength gaios'in[%]

idesign

rule

­ 2 3 % ;;N

76%

■ ':.r—­S

206%

alternat

rule

18%

65%

—

132%

Rotter et

al.

23%

95%

7%

144%

Table B.10 : Local reduction factors K2,LOCAL obtained by different design rules

Discussion

* Compared to the developed design rule the alternative design rule yields strength gains which

are formally up to 15 to 20% lower (examples 1 to 3). This means that the latter formulations

seems to be less suited to provide a tight lower­bound fit to the numerical results in a large

range of parameters.

* Compared to the developed design rule the design proposal of Rotter et al. yields strength

gains which result up to 25% higher (examples 1 to 3).

114

* For wide supports (example 3) there is practically no strength gain which is predicted by all three design formulations in nearly the same way.

* For narrow supports (example 4) plastic yielding occurs at the supports which is well predic­ted by the developed design rule and results in an extreme strength gain. However, both of the competing two design formulations are not able to reflect this situation and therefore re­sult in much lesser strength gains.

B4.2.9 Development of a Design Rule, which covers also the Higher Steel Grades Fe 430 and Fe 510 The design rule given for the buckling strength of unstiffened cylinders (with con­

stant wall thickness), made of mild construction steel of the common grade Fe 360, is given by formulas 15.a and 15.b. This design rule for steel grade Fe 360 has been extended in a straightforward manner to a refined design rule which covers also higher steel grades Fe 430 and Fe 510 [B.32].

B4.2.9.1 Design Rule This refined design formula may be written in the form of 3 conditions as follows

κ 2.LOCAL <7cr

fv

2.83 . fc

I077fE . 77[%] 2.83 0.77

lFe360

< 1 or 1.155

+ (0.19-1.04.log!)

+ (0.77-0.56. IFe360;

> χ·,

(a) (b-1)

(b-2)

00

(22)

The first condition refers to the limit of plastic yielding of narrow supports whereas the last condition refers to wide supports which buckle under the same conditions as for uni­form axial loading. These conditions are valid for any steel grade. The second condition re­presents the local reduction factor in explicit dependence of the support width, the shell slen­derness and the steel grade (eq. 22.b-l). The third condition (eq. 22.b-2) may alternatively be used for steel grade Fe 360.

Thereby the modification factors fc and ίΈ , the steel grade parameter ε and the slenderness parameter are computed according to the following formula :

fc fE

λ

5ε - (0.43 + 3.57¿r2)

6.38¿r ­ (0.87 + 4.5If2)

AFe360 = W ε = ^235/fy

<7CRI —

(23)

B4.2.9.2 Examples

We refer to the 4 examples presented in B4.2.8 about the design rule for unstiffe­

ned cylinders of steel grade Fe 360 and extend them to the higher steel grades Fe 430 and Fe

510.

115

Local reduction factors K2;LOCAL are computed according to the new design rule (Eq. 22) and

compared with the reduction factors for uniform axial loading K2;UNIFORM to evaluate the strength

gains due to the local loading conditons.

Tables B.l 1 and B.12 below contain auxiliary parameters which are needed for the

evaluation of the local reduction factors and the uniform reduction factors for comparison pur­

poses. The local reduction factors and the strength gains referred to the uniform reduction fac­

tors are presented in table B.13.

ε

fc

fE

steel grade

Fe 360

1

Fe 430

0.92

1.148

1.183

Fe 510

0.81

1.278

1.339

Table Β. 11 : Steel grade parame­

ter & modification factors

EXAMP

No.

1

2

3

4

r/t

300

750

200

500

d/r

[%]

(*)

λ

Fe

360

0.752

1.189

0.614

0.971

Fe

430

0.817

1.292

0.667

1.055

Fe

510

0.928

1.468

0.758

1.199

K2,UNIFORM

Fe

360

0.531

0.178

0.660

0.327

Fe

430

0.474

0.141

0.613

0.259

Fe

510

0.371

0.096

0.529

0.177

Table B.12 : Uniform reduction factors K^OCAL for higher steel grades

(*) = not révélant

EXAMP

No.

1

2

3

4

r/t

300

750

200

500

d/r

[%]

11.67

11.67

23.33

3.00

K2,LOCAL

Fe 360

0.621

0.324

(0.587) 0.660

(1.168) 1.000

Fe 430

0.616

0.295

(0.574) 0.613

(1.197) 1.000

Fe 510

0.558

0.225

0.530

(1.108) 1.000

strength gains in [%]

Fe 360

17%

82%

­

206%

Fe 430

30%

109%

­

286%

Fe 510

50%

134%

­

465%

Table B.13 : Local reduction factors K2,LOCAL for higher steel grades

Discussion

* Similar trends are observed for all steel grades concerning effects of support width and shell

slenderness compared to steel grade Fe 360 (see section. B4.2.8).

* The local reduction factors and strength gains for steel grade Fe 360 differ slightly compared

to the values of table B.10, section B4.2.8.5. The reason is, that the unified design equation

(eq. 22.b­l) was used in the present calculations whereas the design equation for steel grade Fe

360 (eq. 22.b­2) was used in B.4.2.8.5

* The striking effect of the steel grade manifests itself in much higher strength gains with increa­

sing value of the yield strength, as can be seen from the following table.

B.4.2.10 Other effects

B.4.2.10.1 Effect of Edge­Ring­Stiffeners & Flexible Support Plates

Practical constructions of local supports of upright cylindrical silos may themsel­

ves be complicated structures with specific stiffness properties that may affect the axial buck­

ling resistance of the shell wall above the supports. In order to circumvent the difficulty of

dealing with that situation in a systematic manner, the reasonable limiting assumptions of

116

completely flexible supports and of completely rigid supports were adopted within the scope of the present project (fig. B.33.a and B.33.Í). 'Flexible' means that the uniform local support forces are directly introduced into the shell wall without any accompanying deformation con­straint against meridional displacements and rotations (U = <J>X = free). On the other hand, 'rigid' means that all deformation degrees of freedom at the local supports are constrained to zero (in special U = φχ = 0). In this case the highly uneven distribution of axial support forces develops in accordance with these geometric constraints, with singularities at the corners of the rigid support block. Design rules were worked out for these cases, based on the results of nonlinear numerical Finite element studies, which are reported in [B.42, B.43]. It turned out that there is a marked increase in buckling strength for rigid supports compared to flexible ones, especially for intermediate support widths and shells of low slenderness ratios (about r/t < 500). As a consequence, there remains the important question, how the majority of real practical supports behaves concerning their stiffness properties, i.e. whether their behaviour resembles more closely that of flexible supports or that of rigid ones or if their characteristic behaviour lies somewhere in­between these extreme limiting cases.

For this purpose a numerical study was carried out [B.40, B.41]. However, the scope of this study was limited, compared to the wide variety of possible practical support constructions and must therefore be considered as a first step only. Firstly, the strengthening effect of an edge­ring­stiffener was investigated in an exemplary manner, assuming fixed di­mensions of the ring, shell slenderness r/t = 500 and support widths μ = 0.05 and 0.1 (fig. B.33.d). Secondly, the effect of the meridional rotational constraint was investigated in detail for the case of rigid supports (fig. B.33.Í). Axial warping had been prevented throughout (U = 0) but the meridional rotational constraint was released (φ = free). An overview of selected idealized local supports is presented in fig. B.33.b­e below.

t ttt f ­o |d |o­

a. flexible support

t f ¥ t ■­o|d

b. support plate c. extended support pi.

4imm|mmm pmmimmm pmmjmum­

d. edge ring e. support plate & edge ring f. rigid support

Figure B.33 : Practical support conditions between the limiting cases of flexible and rigid supports

117

The results of the aforementioned investigations may be summarized as follows : * Firstly, provision of lower edge-rings resulted in an increase of buckling strength of about

27-35 % fig. B.33.d) compared to the flexibly supported case (fig. B.33-a). However, this increase amounts to a fraction of only about 1/3 - 2/3 of the increase which is effected by rigid support conditions (fig. B.33-e).

* Secondly, the release of the meridional rotation constraint in the case of rigid support pla­tes resulted in a decrease of buckling strength on only about 6-7 % compared to fully rigid support conditions (fig. B.33-e). As a side effect, the axial warping constraint also pre­vents rigid-body-rotation of the rigid support plates.

The present investigations give some first insight into the effect of edge-rings and support plates on the buckling strength of axially locally supported cylinders. However, much further study is needed to gain a complete picture of the situation which would permit to draw general quantitative conclusions that may be cast into the form of a detailed design rule. This is not possible at the present time due to the limited scope of the available results.

Yet, as a first step, a simplified design recommendation may be provided which accounts for the effects of edge rings and extended support plates (figs. B.33.c-e) in an ap­proximate manner. An effective support width is introduced : deff = d + 2.trjng , which is the original support width plus two times the thickness of the edge-ring (fig. B.34)

ω T3

o weld seam

x.Local •'x.Local

actual support width

Figure B.34 : Design rule accounting for the strengthening effects of edge-rings and extended support plates

B.4.2.10.2 Interactive Effect of Internal Pressure [B.42] Based on the results, given in [B.42] for support widths μ = 0.05 and 0.1 over the

complete lateral-pressure-range it is possible to derive a design recommendation for the buck­ling strength of axially loaded unstiffened cylinders on local supports, taking into account the additional effect of internal pressure. It is assumed that the internal pressure is exerted onto the inner shell wall by the filling material and no further gas pressure is present.

Design Rule * The axial buckling strength above local supports is, as a simplification, not affected by

additional internal pressure caused by the filling material. The increase of buckling strength is neglected.

118

* The design formulas for local buckling without internal pressure apply for unstiffened cy­linders as wall as cylinders with reinforced wall thickness of the bottom course for the common steel grades Fe 360 to Fe 510.

* The biaxial plastic Mises-interaction does not become relevant except for very large pres­sure values and therefore it need not be taken into account for the practical range of pressu­re values. This is due to the pronounced load reduction caused by the effect of imperfec­tions of the elastoplastic structure

B.4.3 Stiffened cylinders with Stepped Wall-Thickness

B.4.3.1 Experimental Results Experimental research also focussed on the behaviour of stiffened cylinders. Stif­

fening of the cylinders was achieved by choosing a larger wall-thickness for the bottom course of the models, that is, where the local loads are applied (figure B.35).

Figure B.35 : Stiffened cylinder

Several failure models are possible a priori :

local support local support local support

Figure B.36 : Possible failure modes - appearance of buckles in the thicker bottom course, the thinner shell segment remaining intact

(figure B.36-a), - buckling deformations spreading over a vaster area which affects also parts of the thinner seg­

ment (figure B.36-b), - buckles appearing just above the bordering parallel circle which separates the two segments

(figure B.36-C). In order to have a fairly broad investigation program the dimensions of our test cylinders have been chosen as follows :

119

The radius and the overall-height of the models are r = 350 mm , h - 700 mm res­pectively. The support width is equal to 5 % and 10 %. The height of the bottom hi course ranges from 35 mm to 175 mm , which means that the range of the dimensionless parameter η = hi/r is from 0,1 to 0,5 . Several combinations t/ti have been selected : 0,7/1,0 ; 0,7/1,5 ; 1,0/1,5 ; 0,6/1,0. These correspond with the dimensionless combinations r/t (n/t) 500(350) -500(233) - 350(233) and 583(350) .

Stiffened cylinders are fabricated as described below [B.21]. First of all, two rectangular plates were removed from steel plates with thicknesses of 0,7 mm and ti respectively by means of a cutting operation with a sheet shearing machine. The length of the plates is 2199,1 mm and their width is 700 - hi (mm) and hi (mm). These plates are welded together along one of the 2199,1 mm long edges.

Subsequently the composite rectangular 2199,1 mm χ 700 mm plate is rolled into a cylindrical shape with a rolling machine. After having accomplished this rolling operation, the seam between the neighbouring longitudinal free edges of the cylinder is welded. The fabrica­ted cylinder possesses little transverse stiffness, i.e. in a direction which is perpendicular to the generatrices. In order to maintain the circular shape in the support area, a ring stiffener is sol­dered to the lower rim of the test model.

The circularity of the upper rim is guaranteed by the 15 mm circular top plate co­vering the cylinder and where the axial force is introduced at the centre.

Some of the tested cylinders are nominally perfect, but because real cylindrical tanks used in industry will certainly possess relatively larger imperfections it is imperative to test cylinders where shape imperfections are introduced in an artificial way, by means of a punch and die with appropriate shape. These shape imperfections are applied right above the supports : the intended size of the imperfections is lref = 4 -N/rt7 and the intended depth is 0,01W. [B.21, B.22 , B.34]

Two different post-buckling patterns have been observed : a pattern, restricted to the stiffening bottom course, the shape of which is very similar to that of the buckles in unstif­fened cylinders (photo B.14) and a second one with a more complex shape that extends into the upper portion of the cylinder, (photos B. 15 , B. 16 and B. 17 and figure B.37).

The experimental buckling stresses are given in table B.14. Note that auexp = sup­

port reaction/bti where b denotes the support width. The actual depth, δ , of the imperfections is given in the table as well.

Photo B.14 : Buckling shape restricted to the stif­fening bottom course

Photo B.1S : Buckling shape extending into the up­per portion of the cylinder

120

Photo B.16 : Buckling shape extending into the up­per portion of the cylinder

Photo B.17 : Buckling shape extending into the up­per portion of the cylinder

Figure B.37 : Buckling shape extending into the upper portion of the cylinder

121

cylnr.

c28s[3] c08s[l] c08s[4] c07s[4] c07sb[4] c05s[3] c05sb[4] c05sc[4]

c29s[3] clls[3] cllsb[2] cllsb[3] c09s[l] c06s[4] c06sb[4] clOs[I]

c35si[l] c35si[4] c34si[l] c33si[4] c33si[4] c32si[3] c32si[4] c31si[l]

c30s[2] cl2s[3] cl2s[2] cl3s[3]

c45si[3] c45si[4] c44si[2] c44si[3] c43si[l] c43si[4] c42si[l]

c41si[l] c40si[l] c40si[3] c39si[2] c39si[3] c38si[l] c38si[4]

c49si[2] c49si[3] c48si[2] c48si[3] c46si[2] c46si[3] c47si[3]

c52si[4] c53si[3] c53si[4] c50si[2] c50si[3] c51si[2] c51si[3]

t> [mm]

1.5 1.5 1.5 1.5

1.5 1.5 1.5 1.5 1.5 1.5 1.5

1.5 1.5 1.5 1.5 1.5 1.5 1.5

r/t, H

350 350 350 350 350 350 350 350

350 350 350 350 350 350 350 350

350 350 350 350 350 350 350 350

233.33 233.33 233.33 233.33

233.33 233.33 233.33 233.33 233.33 233.33 233.33

233.33 233.33 233.33 233.33 233.33 223.33 233.33

350 350 350 350 350 350 350

350 350 350 350 350 350 350

t [mm]

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

0.7 0.7 0.7 0.7

1 1 1 1 1 1 1

1 1 1 I 1 1 1

0.6 0.6 0.6 0.6 0.6 0.6 0.6

0.6 0.6 0.6 0.6 0.6 0.6 0.6

r/t H

500 500 500 500 500 500 500 500

500 500 500 500 500 500 500 500

500 500 500 500 500 500 500 500

500 500 500 500

350 350 350 350 350 350 350

350 350 350 350 350 350 350

583.33 583.33 583.33 583.33 583.33 583.33 583.33

583.33 583.33 583.33 583.33 583.33 583.33 583.33

h, [mm]

35 70 70 105 105 140 140 140

35 70 70 70 105 140 140 175

35 35 70 105 105 140 140 175

35 70 70 105

105 105 105 105 140 140 140

105 105 105 140 140 140 140

90 90 90 90 140 140 140

90 90 90 140 140 140 140

h|/r (-]

0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.4

0.1 0.2 0.2 0.2 0.3 0.4 0.4 0.5

0.1 0.1 0.2 0.3 0.3 0.4 0.4 0.5

0.1 0.2 0.2 0.3

0.3 0.3 0.3 0.3 0.4 0.4 0.4

0.3 0.3 0.3 0.4 0.4 0.4 0.4

0.2571 0.2571 0.2571 0.2571

0.4 0.4 0.4

0.2571 0.2571 0.2571

0.4 0.4 0.4 0.4

m [%]

5 5 5 5 5 5 5 5

10 IO 10 10 10 10 IO 10

10 10 10 10 10 10 10 10

5 5 5 5

5 5 5 5 5 5 5

10 10 10 10 10 IO 10

10 10 10 10 10 10 10

5 5 5 5 5 5 5

σ."" [MPa]

220 225 247 245 266

280.4 265 242

131 156.7 133.5 164 167 184 163 165

116.5 123.6 136

168.7 166.6 184 181 161

186.2 243 220 272

334 349 345 344 308 320 341

218 218 224 219 218 221 227

128 116 129 118 167 173 149

250 254 242 275 280 313 283

HUJ [MPa]

175 173 173 173 173 179 173 173

175 173 173 173 173 179 179 173

175 175 175 175 175 175 175 175

205 205 205 205

201 201 206 202 208 202 210

204 204 204 213 213 213 213

250 251 246 247 250 245 248

245 245 245 255 252 255 255

tu [MPa]

166 166 166 166 166 163 165 165

165 166 166 166 165

164.7 164 165

165 165 165 165 165 165 165 165

165 165 165 165

239 239 239 239 239 239 239

239 239 239 239 239 239 239

215 215 215 215 215 213 213

213 216 216 214 214 216 216

«... [MPa]

320 320 320 320 320 320 320 320

320 320 320 320 320 320 320 320

320 320 320 320 320 320 320 320

350 350 350 350

319 319 320 320 320 320 320

318 318 318 328 328 328 328

347 347 347 348 350 347 350

351 351 351 354 352 352 352

C [MPa]

300 300 300 300 300 300 300 300

300 300 300 300 300 300 300 300

300 300 300 300 300 300 300 300

300 300 300 300

350 350 350 350 350 350 350

350 350 350 350 350 350 350

338 338 336 336 336 336 337

336 339 339 336 336 338 338

δ

n.p. n.p. n.p. n.p. n.p. n.p. n.p. n.p.

n.p. n.p. n.p. n.p. n.p. n.p. n.p. n.p.

0.93 0.84 0.71 0.65 0.76 0.63 0.84 0.84

n.p. n.p. n.p. n.p.

0.67 0.95 0.98 0.95 1.12 1.09 1.08

1.07 0.85 0.98 0.96 0.98 1.02 1.01

0.77 0.56 0.89 0.82 0.75 0.88 0.71

0.89 0.65 0.66 0.79 0.62 0.67 0.61

C* [MPa]

200

247

129

180

198

314

Table B.14 : Cylinders with reinforced wall-thickness

122

B.4.3.2 Comparison with Finite Element Computations A selected number of nominally perfect and imperfect test specimens have been cal­

culated by GREINER et al [B.35, B.36] for comparison purposes and adaptation of the finite element model.

The numerical study examines and takes account of the special features of the test specimens : geometry, loading, support and boundary conditions, material model, eccentricity in the joint between upper and lower part of the cylinder and geometric imperfections. Numerical buckling loads were obtained by load­controlled analyses which yield accurate values of the load maxima (limit points). However, with this simple method no information can be obtained con­cerning post­buckling deformations because any load­controlled analysis inherently breaks down at the limit point. Consequently, each case had to be analyzed twice, the second time by apply­ing an arc­length­controlled analysis.

Comparative buckling stresses for nominally perfect specimens are given in table B. 14 for a selected number of specimens.

350

­ 300

250

200

Analyses Series C2 ­

R/t = 233 (350)

μ = b/B = 0.05 & 0.10

ÉääS

TEST­

* ' ■

'alues

1.15£ •ßüLT = 370

Sss^ S&£ '.·■"

jflfSje^tft^iiS^ B GMNLl­h ­μ=0.05

O GMNIi7­h ­μ=0.05

O GMNL­h · μ=0.05

O GMNL ­ μ=0.05

Β GMNLI­h ­μ=0.10

□ GMNU7­h ­μ=0.10

Η GMNL­h ­ μ=0.10

G GMNL ­ μ=0.10

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

dimensionless length of wall reinforcement η = r^/R [­]

Analyses Series C2 ­

R/t = 350 (583)

μ = b/B = 0.05 & 0.10 1.155 · β υ ο · = 404

_ 300

— 250

100

0 0,1 0,2 0,3 0,4 0.5 0,6 0,7 0,8 0,9 1

dimensionless length of wall reinforcement η = h./R [­]

Figure B38 : Maximum mean support stress of artifl­ Figure B.39 : Maximum mean support stress of arti­cially imperfected reinforced cylinders made of the ficially im perfected reinforced cylinders made of the "new" material. Project part C2/1994 : r/t=233(350) "new" material. Project part C2/1994 : r/t=350(583)

Summaries of results for imperfect models are presented in figures B.38 and B.39 for the investigated slenderness ratios r/ti = 233 and 350 of the reinforced bottom course. The maximum mean meridional buckling stress acr at the rigid supports is plotted against the dimen­sionless length η of the wall reinforcement, η is the ratio : length of bottom course/radius. In order to give an impression of the effect of material plasticity on the computed buckling stress, the plastic hardening range for the sheet used for the bottom course, is indicated by light grey shading in these figures.The ranges oftest results are indicated by dark grey shading on the ac­

123

companying figures.lt may be noted that the agreement with the numerical results is satisfactory

in general.

B.4.3.3 Development of a Design Rule [B.37, B.38]

B.4.3.3.1 Introduction

Practical silo structures are very often designed with stepped wall thickness in verti­

cal direction for the primary purpose to provide the necessary resistance against the increasing

radial pressure of the filling material in an economic manner. The obvious complexity of the

problem had to be reasonably idealized within the framework of the present project. This was

done by adopting several assumptions in the following way :

* The cylindrical shell consists of only two regions of different wall thickness.

The wall thickness of the lower part ti is 50 % larger than that of the upper one.

* The support forces are introduced by \ocdl flexible supports.

Based on these assumptions the problem is reduced to the following question. Given the

buckling strengths for the two slenderness ratios r/t and r/ti of the upper part and the lower part

(with 50 % increased wall thickness) assuming for flexible support conditions : What is the di­

mensionless minimum required height hi/R of the reinforced bottom course of the cylinder so

that the cylinder with one­stepped wall thickness reaches, within certain limits, the buckling

strength of the cylinder with uniform (i.e non­stepped) wall thickness, equal to that of the rein­

forced lower part ? That means we have essentially to deal with a so­called 'Mindeststeifig­

keitsproblem' (problem of nuhimum­required stiffness) which, however, may be defined with

respect to various levels of analytical modelling. The buckling loads could be computed utili­

zing any combination of geometric non/linearity and material non/linearity with/without imper­

fections. Since we have to deal with mild construction steel, which exhibits a pronounced plas­

tic limit behaviour, a geometrically and materially nonlinear approach seems reasonable any­

way (GMNL analyses). But the question remains wether or not imperfections should be inclu­

ded in these analyses which are primarily aimed at determing the minimum­required stiffness of

the structure, expressed by the value of ΙΙΙ,ΜΓΝΛ· (fig· Β .40). The general approach, adopted in the

present study, was to include the effect of geometric imperfections (GMNLI analyses) as will be

discussed below.

Imperfections Jjj Buckling modes ïïf

Figure B.40 : Assumed local imperfections and characteristic buckling modes of cylinders with reinforced wall thickness of the bottom course

The buckling behaviour of the geometrically perfect system {GMNL analyses) may be characte­

rized as follows (right part of fig. B.40) :

* If the height of the wall reinforcement is small the shell always buckles immediately above the

reinforced part in the thinner part of the shell.

124

* If the height of the wall reinforcement is greater than the critical height ΙΜ,ΜΓΝ the shell always buckles immediately above the supports.

In the presence of local geometric imperfections either above the reinforced part or above the supports (left part of fig. B.40) these tendencies do not alter but they are even ampli­fied {GMNLI analyses). An exception occurs for the case of small values of the height of the wall reinforcement and local imperfections directly above the supports. In this case a pro­nounced interaction between the imperfection shape and the buckling mode shape may occur re­sulting in a 'mixed' buckling mode shape, which is partly located within the reinforced and partly within the thin upper part. The clean separation of modes of the perfect system is destroyed.

An overview how this situation reflects from the viewpoint of buckling loads in de­pendence of the height of the wall reinforcement is presented in the schematic diagram below (fig. B.41). The left ordinate corresponds to the limiting case of zero wall reinforcement (hi/r = 0) and the smallest possible buckling loads apply. The horizontal lines correspond to the com­plementary limiting case of total wall reinforcement (hi^/r^hi /r^l) and the largest possible buckling loads apply. The inclined lines correspond to intermediate values of wall reinforce­ment and show an almost linear increase of buckling strength with increasing height of wall rein­forcement. Ultimately, these lines intersect the horizontal lines for total wall reinforcement at values of the critical height ΙΙΙ,ΜΙΝ/Γ and the buckling loads cannot be further increased (points U,R and V in fig. B.41). Three alternative situations are presented, the behaviour of the perfect system {GMNL) and of systems with local imperfections located directly above the supports {GMNLIJjot) or directly above the reinforced part in the thinner part of the wall {GMNLIJop).

10

o CO

LL T3 CO O

0 0,1 design rule

Figure B.41 : Buckling strength in dependence of the axial length of the wall reinforcement for different imperfection assumptions - schematic diagram

for r/t=500(750), μ=0.05

B.4.3.3.2 Summary of Numerical Results and Design Rule Based on the numerical results for the rninimum-required heights ΙΙΙ,ΜΓΝ/Γ , presented

in the table B.l5 below a design rule is developed which is valid for 50 % increased wall thickness of the bottom course or greater (ti > 1.5t).

125

ni,MiN ' r ­

' 0.2 + 0.53 (1 ­ ε)

0.1 + 2μ + 0.53 (1 ­ ε)

4/Vr7t

μ < 0.05

for 0 .05<μ< 0.1

all μ

(24)

. „ , , , . , . . . , . ,

!­; hj.MiN/i"

; (GMNLÎ):Î

μ =0.05

μ =0.1

r/t = 200

Fe 360

W$ßi ; 0.13

Fe 430

,' 0.11

0.14

Fe510

0.11

02

r/t = 500

Fe 360

0.11

0.14

Fe 430

0.11 .'■;■'·,

0.17

Fe510

\ 0.14

0:2

Table B.15 : values of minimum required heights of the wall reinforcement in dependence of slenderness R/t,

steel grade & support width

.^DESIGN:

hijvHN/r ­

e=^235/f;

μ = 0.05

μ = 0.1

steel grade

Fe 360

1

0.2

spilli

Fe 430

0.92

0.24

;,; 0.34V,

Fe510

0.81

: üü , 0.41

Table B.16 : Proposed design values of minimum required height of the wall reinforcement in dependence of

steel grade & support width

*The convention has been adopted to double the values of critical heights of the numerical ana­

lyses for design purposes. This leads to the values of tab. B.16 above. It is noticeable that the

shell slenderness r/t does no longer appear because, by simplification, the respectively larger

values were adopted.

* Linear interpolation of these design values, within the range of investigated support widths and

steel grades, leads to the design formula eq. 24. For narrow support widths μ < 0.05 the design

values for μ = 0.05 apply throughout. Graphical representations of this design equation are

presented in fig. B.42.

rr

ε= (235/ßy)

Figure B.42 : Design diagrams for estimating the minimum required height of the

wall reinforcement

126

* A further purely pragmatic condition was additionally provided which states that the height

of the wall reinforcement should be larger than the approximate length of the local buckles of

uniformly compressed cylinders (last formula of eq. 24). Thereby the shell slenderness reap­

pears in the overall design equation.

♦Finally, a parameter change remains to be applied introducing the dimensionless support

width η = d/r [B.32]. This leads to the final form of the design equation (eq.25), a graphical

representation of which is given in fig. B.43 below.

hUÆN /

r ­

0.2 + 0.53(1 ­ε) η< 0.075

0.1 + 1.3377 + 0.53(1 ­ ε) for 0.075 < η < 0.15

4 /Vr7t all 77

(25)

aí

1.00

ε= J 235/ßy

(0,0) (0,05) , (0,10)

0,0 0,075 0,015

b»"=â­â η = d/R

Figure B.43 : 3D­representation of the final design diagram

B.5 Conclusions A design rule for axially loaded upright cylinders on discrete flexible supports has

been derived by applying curve­fitting procedures. Flexible means that the supports themsel­

ves have no stiffness against axial warping and meridional bending in particular. The equidis­

tant support forces are directly introduced into the shell wall by uniform Une loads over the

proportion of the circumference which is supported. The derivation of this design rule is ex­

clusively based on results of nonlinear numerical buckling analyses taking into account geome­

trically and materially nonlinear behaviour as well as the effect of initial geometric imperfec­

tions in a local region above the supports. The geometrical parameter values cover the rele­

vant range occuring in practical silo design. The investigated shell slenderness parameter

ranges from r/t = 200 to 750 and the dimensionless support width covers the range up to fairly

wide supports η = d/r = 0.3. Elastoplastic material behaviour was taken into account repre­

senting the behaviour of mild construction steel of steel grades Fe 360, Fe 430 and Fe 510.

The format of the design rule has been chosen in a way that allows a direct compa­

rison of the characteristic increase of buckling capacity of the locally supported case with the

127

uniformly loaded case. This has been accomplished with the help of the german shell buckling

code DIN 18800/ part 4, which defines an overall reduction factor κ2 = K2 {λ ) = aCI/fy appli­

cable to the uniaxial yield stress and which depends on the dimensionless shell slenderness pa­

rameter λ = . / ζ / crcri . Thereby acri = 0.605.Et/r is the classical elastic bifurcation stress of

a perfect cylinder under uniform axial compression (ideal critical stress). Analogously, in the

present case a 'local' overall reduction factor K2¿OCAL was defined, which additionally depends

on the dimensionless support width η = d/r. Accordingly, since the abscissa parameter λ is

kept unaltered in the locally supported case compared to the uniformly loaded case, a family of

curves is obtained which are situated above the K2­design curves. These curves are bounded

by the simple condition κ2 < K2,LOCAL < 1 , i.e. they are bounded by the reduction factor for

uniform loading condition from below and by the condition of purely plastic yielding from

above. By this way, the characteristic strength gain of the locally supported case compared to

the uniformly loaded case becomes directly apparent to the design engineer.

In addition, an equivalent alternative formulation of the design rule has been wor­

ked out, based on the same numerical data, which follows the ECCS design philosophy. The­

reby a well­known two­step­procedure is adopted, determining the elastic imperfection reduc­

tion factor α = OGNLi/Ocn in a first step and applying a linear elasto­plastic interaction rule in a

second step.

Finally, the prediction capability of the design rules was tested by 4 examples and compared

with an existing design proposal published by Rotter et al. (1993) [B.31].

The effect of higher steel grades was introduced into the design equations by the

following concept : The structure of the already derived design equation for the reference steel

grade Fe 360 was left intact and so­called modification factors were provided which apply to

the coefficients of the design equation for Fe 360. Therefore the new design equation may be

used for the whole range of investigated steel grades by simple interpolation of the modifica­

tion factors.

Finally, the effect of internal pressure combined with local axial force introduction

and the effect of edge ring stiffeners and flexible support plates has been examined and simpli­

fied design rules are given.

A lot of both theoretical and experimental research has been devoted to the problem of

cylinders on local supports with a reinforced, thicker bottom course. The experimental inves­

tigation covers support widths equal to 5 % and 10 % , and the height of the reinforcement,

expressed as a partion of the radius of the cylinder ranges from 0,1 r to 0,5 r. Several combi­

nations of the thicknesses of the upper portion and the lower reinforced part were tested : t/ti

= 0.7/1.0 ; 0.7/1.5; 1.0/1.5 and 0.6/1.0 .

Numerical simulation by means of finite element techniques led to very satisfactory

results taking into account the features of the test specimens : geometry, loading, support and

boundary conditions, material model, eccentricity of the joint between upper and lower part of

the cylinder and geometric imperfections.

Within the framework of the present project, the development of a design rule had

to be simplified due to the complexity of the problem. The idealizations are : the cylindrical

shell consists of only two regions of different wall thickness ; the wall thickness of the lower

part 11 is 50 % larger than that of the upper one ; the support forces are introduced by flexible

supports. Under these conditions a design rule has been derived giving the minimal required

height of the wall reinforcement in order not to lower the strength of a cylinder with uniform

thickness ti on local supports. The effect of the steel grade is incorporated in the design pro­

posal.

128

REFERENCES

[B.l] European Convention for Constructional Steelwork, Technical Committee 8 ­ Structural Stability,

TWG 8.4 ­ Stability of Shells, Buckling of Steel Shells ­ European Recommendations, 4th edn, No

56, ECCS, General Secretariate, Brussels, 1988.

[B.2]. Greiner, R , Zur Längskrafteinleitung in stehende zylindrische Behälter aus Stahl. Der Stahlbau,

7(1984)210­5.

[B.3] Greiner, R , Sonderprobleme des Konstruktiven Ingenieursbaus, Heft 3. Institut für Stahlbau,

Holzbau und Flächentragwerke, Fakultät für Bauingenieurwesen, Technische Universität Graz,

Austria, 1983.

[B.4] Öry, H., Reimerdes, H.­G. & Tritsen W., Beitrag zur Bemessung der Schalen von Metallsilos. Der

Stahlbau, 8 (1984) 243­8.

[B.5] Flügge, W., Stresses in Shells, 2nd edn Springer Verlag, Berlin, 1973.

[B.6] Bodarski, Z., Hotala, E. & Pasternak, H., Zum Einfluß der Biegedrillsteifigkeit des Fußrings auf den

Störungsbereich im Mantel von Metallsilos. Der Bauingenieur, 57 (1982) 423­7.

[B.7] Bodarski, Z., Hotala, E. & Paternak, H., Zur Buerteilung der Tragfähigkeit von Metallsilos. Der

Bauingenieur, 60 (1985) 49.

[B.8] Bijlaard, P. & Gallagher, RH., Elastic instability of a cylinderical shell under arbitrary

circumferential variation of axial stresses. Journal of the Aerospace Sciences, 27(11) (1959) 854­8.

[B.9] Hoff, N.J. Chao, C.C. & Madsen, W.A., Buckling of a thin­walled circular cylindrical shell heated

along an axial strip. Journal of Applied Mechanics, ASME, 31 (1964) 253­8.

[B.10] Libai, A. & Durban, D., A method for approximate stability analysis and its application to circular

cylindrical shells under circumferentially varying edge loads. Journal of Applied Mechanics, ASME,

40 (1973) 971­6.

[Β. 11] Libai, Α. & Durban, D., Buckling of cylindrical shells subjected to nonuniform axial loads. Journal

of Applied Mechanics, ASME, 44 (1977) 714­20.

[Β. 12] Gould, P. L., Sen, S. K., Wang, R. S. C. & Lowrey, D., Column supported cylindrical conical tanks.

Journal of the Structural Division, ASCE, 102 (ST2) (1976) 429­47.

[B.13] Öry, Η. & Reimerdes, H.­G., Stresses in and stability of thin­walled shells under non­ideal load

distribution. Proceedings of the International Colloquium on the Stability of Plate and Shell

Structures, ed. P. Dubas & D. Vandepitte. Gent University, Gent, Belgium, 1987, pp. 555­61.

[B.14] Rotter, J. M. & Teng, J. G, A study of buckling in column supported cylinders. Proceedings of the

IUTAM Symposium on contact Loading and Local Effects in Thin­walled Plated and Shell

Structures, ed. V. Krupka & M. Drdacky. Prague, Czechoslovakia, 1990, p. 52­61.

[B.15] Li, H. Y., Rotter, J. M. & Teng, J. G, Buckling in thin elastic cylinders on column supports. In

Buckling of Shell Structures, on Land, in the Sea and in the Air, ed. J.F. Jullien, Elsevier Applied

Science, London, pp. 334­343.

[B.16] Rotter, J. M. & Teng, J. G, Linear bifurcation of perfect column­supported cylinders : support

modelling and boundary conditions. Thin­Walled Structures, 13 (1992), 241­63.

[B.17] Samuelson, L. Α., Buckling of cylindrical shells under axial compression and subjected to localized

loads. Proceedings of the EuroMech Colloquium on Post­Buckling of Elastic Structures,

Elsevier/Akadémai Kiado, Mtrafüred, Hungary, 1985.

[B.18] Guggenberger, W., Nichtlineares Beulverhalten von Kreiszylinderschalen unter lokalen

Axialbelasting. PhD Dissertation, Technische Universität Graz, Austria, 1991.

[B. 19] Vandepitte, D., Berekening van Constructies. Vol. I. E. Story­Scientia P.V.B.A. Gent, Belgium,

1979, pp.411­28.

[B.20] Dhanens F. & Van Impe, R, Local Buckling of Unstiffened Steel Cylinders, report LMO ­ 93 ­

1903 , 124 pp., Ghent University, 1993

[B.21] Dhanens, F. & Van Impe, R, Local Buckling in Cylindrical Structures, report LMO ­ 94 ­ 1703 ,

57 pp. , Laboratory for Model Research, Ghent University, Belgium.

[B.22] Dhanens, F. & Van Impe, R , Local Loads in Cylindrical Structures, report LMO ­ 95 ­ 2003 , 43

pp., Laboratory for Model Research, Ghent University, Belgium.

[B.23] Guggenberger, W. & Greiner, R, Local Buckling of Unstiffened Steel Cylinders ­ numerical results

and comparison with test results, report G/G N°. 1.1/1993 Institute for Steel & Shell Structures,

Technical University of Graz, Austria.

129

[B.24] Guggenberger, W. & Greiner, R., Numerical Analysis of Unstiffened Cylinders with Artificial Imperfections & Comparison with Test Results, report G/G. N°. 2A/1994, Institute for Steel & Shell Structures, Technical University of Graz, Austria.

[B.25] Guggenberger, W. & Greiner, R, Numerical Analysis of Unstiffened Cylinders - Development of a Design rule, report G/G N°. 2B/1994, Instittute for Steel & Shell Structures, Technical University of Graz, Austria.

[B.26] Guggenberger, W. & Greiner, R, Effect of rigid Support Conditions on the Buckling Strength of Axially Loaded Unstiffened Cylinders on Local Supports. Report G/G. N°. 3G/1995, Institute for Steel & Shell Structures, Technical University of Graz, Austria.

[B.27] Guggenberger, W. & Greiner, R, Effect of the Edge-Ring-Stiffeners and Support Plates on the Buckling Strength of Axiais Loaded Unstiffened Cylinders on Local Supports , Report G/G. N°. 3H/1995, Institute for Steel & Shell Structures, Technical University of Graz, Austria.

[B.28] DIN 18 800/part 4, Steel Structures, Buckling of Shells, November 1990. [B.29] Koiter, T.W., The Stability of Elastic Equilibrium, English translation Technical report AFFDL-TR-

70-25, Wright-Patterson Air-Force-Base, Ohio, Feb. 1970. [B.30] Guggenberger, W., Greiner, R Rotter, J.M., The Behaviour of Locally Supported Cylindrical Shells :

Unstiffened Shells, submitted for publication, 1995. [B.31] Rotter, J.M., Greiner, R, Guggenberger, W., Li, H.Y., She, K.M., Proposed Design Rule for

Buckling Strength Assessment of Cylindrical Shells under Local Axial Load, submission to ECCS TWG 8.4, Buckling of Shells, Sept. 1993.

[B.32] Guggenberger, W., Greiner, R, Effect of Steel Grade on the Buckling Strength of Axially Loaded Unstiffened Cylinders on Local Supports, Development of a Design Rule, Report G/TG N°. 4E/1995, Technical University of Graz, Austria.

[B.33] Guggenberger, W., Greiner, R, Effect of rigid Support conditions on the Buckling Strength of Axially Loaded Cylinders with Reinforced Wall Thickness on Local Supports, Report G/G N°. 4.G/1995, Technical University of Graz, Austria.

[B.34] Dhanens, F., & Van Impe, R, Local Loads in Cylindrical Structures, Report nr. LMO-95-1509, Laboratory for Model Research, Ghent University, Belgium.

[B.35] Guggenberger, W., Greiner, R., Numerical Analysis of Nominally Perfect Cylinders with Reinforced Wall Thickness and Comparison with Test Results, Report G/G N°. 2C1/1994, Technical University of Graz, Austria.

[B.36] Guggenberger, W., Greiner, R, Numerical Analysis of Artificially Imperfect Cylinders with Reinforced Wall Thickness & Comparison with Test Results, Report G/G N°. 2C2/1994, Technical University of Graz, Austria.

[B.37] Guggenberger, W., Greiner, R, Buckling Strength of Axially Loaded Cylinders with Reinforced Wall Thickness on Local Supports - Development of a Design Rule, report G/G N°. 4DF/1995, Technical University of Graz, Austria.

[B.38] Guggenberger, W., Greiner, R., Numerical Analysis of Cylinders with Reinforced Wall Thickness -Development of a Design Rule, Report G/G N°. 2D/1994, Technical University of Graz, Austria.

[B.39] Guggenberger, W., Greiner, R., Buckling Strength of Unstiffened Cylinders under Axial Loads -Development of a Design rule, Report G/G N°. 4B/1995, Technical University of Graz, Austria.

[B.40] Guggenberger, W., Greiner, R, Effect of Edge-ring Stiffeners and Flexible Support Plates -Development of a Design rule, Reports G/G N°. 44/1995 & 3H/1995, Technical University of Graz, Austria.

[B.41] Guggenberger, W., Greiner, R, Effect of Internal Pressure on the Buckling Strength of Unstiffened Cylinders under Local Axial Loads - Development of a Design Rule, Report G/G N°. 4J/1995, Technical University of Graz, Austria.

[B.42] Guggenberger, W. & Greiner, R, Buckling Strength of Unstiffened Cylinders under Local Axial Loads - Development of a Design Rule, Report G/G. N°. 4G/1995, Gent/Graz Analysis Project Part 4.B/1994, Technical University of Graz, Austria.

[B.43] Guggenberger, W. & Greiner, R, Effect of rigid Support Conditions on the Buckling Strength of Axially Loaded Unstiffened Cylinders on Local Supports - Development of a Design Rule, Report G/G. N°. 4G/1995, Gent/Graz Analysis Project Part 4.B/1994, Technical University of Graz, Austria.

[B.44] Guggenberger, W.& Greiner, R, Corrections & Additions to Reports : G/G. N°. 2.B/1994, G/G N°. 2.C2/1994, G/G N°. 3.E/1995.

130

ECSC Contract No. 7210-SA/208

Enhancement of ECCS Design Recommendations and Development of Eurocode 3 Parts Related to Shell Buckling

PartC

SHELLS OF REVOLUTION WITH ARBITRARY MERIDIONAL SHAPES

-BUCKLING DESIGN BY USE OF COMPUTER ANALYSIS-

Final Report

Universität GH Essen FB Bauwesen · Stahlbau Prof. Dr.-Ing. H. Schmidt

Dipl.-Ing. P. Swadlo

May 1996

Cl Introduction The European Recommendations on "Buckling of Steel Shells" [Cl], as well as other codes of practice or design recommendations dealing with the stability of thin-walled metal shell structures (e.g. [C2, C3]), contain design rules for unstiffened or stiffened fundamental types of shells of revolution under fundamental loads. Fundamental types of shells are for instance cylinders, cones or spheres; fundamental loads are for instance uniform axial compression, uniform external pressure or uniform torsional shear. The rules are, explicitly or implicitly, based on the critical buckling resistance of the idealized shell in combination with appropriate reduction factors taking care of the imperfection-induced and (if relevant) of the plasticity-induced decreases down to the characteristic buckling resistance of the "real" shell.

In most cases, approximate formulas are given for "critical buckling stresses" representing the critical buckling resistance. The reduction factors are of empirical character and are, for each particular fundamental buckling case, experimentally calibrated.

No simple design rules are available how to handle other than the fundamental shells of revolution and/or other than the fundamental loads. On the other hand, powerful computer programs are available to analyse either the idealized shell configuration, i.e. the geometrically perfect and purely elastic shell, or even a discreticized FE-model of the simulatedly imperfect shell configuration. The problem is not the numerical or algorithmic capability of the computer package, but the proper input data and, even more, the proper use of the numerical output data in terms of a balanced safe and economic design. The task of formulating such general guidance is under discussion in several working groups, among others in ECCS-TWG 8.4[C8].

One of the more frequent non-fundamental shell configurations, which the design engineer will come across in structural applications, are shells of revolution of which the meridional shapes are combinations of the fundamental types, e.g. polygonal cylinder/cone assemblies. It may be expected of him to solve - by means of appropriate numerical tools - the eigenvalue problem of the inherent idealized elastic shell configuration. However, if he applies the same reduction factors as specified for the fundamental shell types the result might be overconservative. On the other hand, if he applies no imperfection reduction at all, the result would certainly be unconservative. No guidance is at present available how to transfer the numerical eigenvalue result into a reasonable design buckling resistance [C7, CIO].

The present research subproject C aims at providing the design engineer with recommendations how to handle the buckling design of non-fundamental shells of revolution. This was to be achieved by investigating comprehensively a set of cone/cone and cone/cylinder assemblies as typical and frequently used examples of non-fundamental shells. The investigations were to be carried out on a comparative experimental and numerical basis. The recommendations were to be drawn from these comparisons.

The experimental and numerical investigations are published in full documentary evidence in the research report [CI 1]. This chapter C represents a short version of [CI 1] with either typical or summarizing results, however with the full set of conclusions. The full report [Cll] is available on request.

133

C2 Experimental investigations

C2.1 Test program The test program is shown in table CI. The ideas behind the test program are as follows.

C2.1.1 Parameter "action causing buckling"

Of the three buckling relevant membrane forces in shells, the shear forces are less important in shell structures with polygonal meridional shapes. The other two membrane forces, i.e. meridional and circumferential compressive forces, are in like manner important. However, the buckling behaviour and imperfection sensitivity are principally different. It was therefore necessary to cover both in an as comparable manner as possible. This was achieved by a systematically dual set-up of the program: Each tested shell geometry is represented by two nominally identical specimens (e.g. KK-V50 and KK-V51, see table CI), of which one is tested under central axial compression and the other one is tested under hydrostatic external pressure.

C2.1.2 Parameter "meridional break"

The carrying capacity of a shell of revolution that is composed of cylindrical and conical portions may be considerably lower than predicted by the elementary buckling stresses of the individual cylinders and cones assumed to be edge-supported at the meridional breaks. The governing parameters for the resistance deficit are the angle and the direction of the meridional break (convex or concave). With growing angle, the resistance deficit will increase for axial compression because of deviating the meridional membrane forces, but will decrease for external pressure because of the stiffening effect of the break. Deviating the meridional membrane forces at concave breaks causes inward bending combined with circumferential compressive forces, at convex breaks outward bending combined with circumferential tensile forces.

The semi-vertex angle of all conical portions of the test specimens has been chosen as 20° (see table CI). This is a typical value in practical structures. Using this basic cone angle, four types of meridional breaks are possible: cone/cone-convex, cone/cone-concave, cylinder/cone-convex, cylinder/cone-concave. All of them are covered by the test program (see table CI).

C2.1.3 Parameter "shell geometry"

The logical shell configuration in order to test the behaviour of cone/cone meridional brakes are double cone assemblies, as shown in the first two columns of table CI.

For the cylinder/cone meridional breaks it seemed necessary to check additionally the influence of the boundary conditions at the far ends of the two shell portions forming the cylinder/cone meridional break. Thus, four triple shell configurations have been developed in which each of the two meridional break types is combined with three different boundary conditions (see columns 3 to 6 of table CI).

134

The largest diameter has, in regard to the space of the testing machine, been chosen as d = 450 mm. The other diameters follow from the condition that the conical portions should be long enough to represent "medium-long" shells in terms of buckling theory.

C2.1.4 Parameter "shell slenderness"

Combined shells of revolution without rings or diaphragms at the intersections, as investigated in this research project, are to be found in practical structures with small or medium r/t-ratios in the range of r/t = 100 + 500. Therefore, two wall thicknesses t = 0,5 mm and 1,0 mm have been chosen for the axial load tests, yielding r/t-ratios of 225 and 450. For the external pressure tests only t = 0,5 mm was chosen, because an extrapolation to thicker walls was, considering the smaller imperfection sensitivity of the external pressure buckling case, thought to be feasible.

C2.2 Test specimens C2.2.1 Material properties

C2.2.1.1 Used sheet material

The specimens were manufactured from thin cold rolled sheets (3000 χ 1500 mm) of mild unalloyed steel Stl2 (Material-No. 1.0330) ace. to DIN 1623, which corresponds to sheet category Fe POI ace. to EN 10130. This thin sheet material is produced for cold forming operations rather than for structural purposes. Its specified material properties are

yield stress R^ or Rpo 2 <, 280 MPa, tensile strength R ^ 270 -s- 410 MPa, percentage elongation after fracture Ag0 £ 28%.

The advantage of this material for the present research purposes is - besides its availability on the steel market - its high ductility which is important for simulating the good-natured plastic deformation behaviour of unalloyed structural steels. The disadvantage is that its yield stress is not specified against a minimum value but against a maximum value. That entails a relatively large scatter of yield stress values from sheet to sheet and even within one sheet. Furthermore, the sheets exhibit two different types of stress-strain-behaviour, obviously depending on the particular "skin-passing" procedures of the different steel producers or production heats respectively.

In order to overcome these disadvantages, a number of tension and compression coupons respectively have been taken in each sheet from around the positions of the shell specimens' wall pieces. The cutting plans of all sheets are given in [CI 1].

C2.2.1.2 Tension coupons

The tension coupons have been tested in two phases with different degrees of measurement precision. In the first (strain-controlled) phase a high-precision extensiometer was used which produced exact stress-strain-curves up to e = 4,5%, including an unloading/reloading loop for determining the modulus of elasticity E and three relaxation halts for determining the "tensile static yield stress" R ^ (see fig. CI). The second phase was standard, i.e. piston-displacement-controlled; it delivered a stress-displacement-curve from which the tensile strength Rm and an

135

approximate value for the percentage elongation eg before local reduction of cross-sectional area can be determined. Thus, it will be possible to construct, for numerical purposes, full range stress-strain-curves which are very precise in the important small-strain range and sufficiently good for the rest. After the tension test the percentage elongation after fracture A50 has been determined.

All tension coupon diagrams and the material property values derived from them are given in [CI 1]. The coupons exhibit the abovementioned different types of stress-strain-behavior (see fig. CI): Type (a) has a distinct upper yield stress with the stress-strain-curve being ideally elastic-plastic with a horizontal plateau up to ev ~ 2,5%, before strain-hardening with an approximate slope Ev begins, whereas type (b) has no upper yield stress and an approximately bilinear stress-strain-curve with a round transition from elastic to plastic range and with the plastic range being strain-hardening with an approximate slope Ev from the very beginning.

Table C2 shows how the tension coupon results from around particular specimen wall pieces are collected in order to generate average material property values for the numerical analyses.

C2.2.1.3 Compression coupons

Although it is the compressive stress-strain-behaviour (including the compressive yield stress) which controls the load-carrying capacity of buckling-endangered structures, it is common practice to assume, in thin-walled structures, that the results of tension coupons are applicable to compressive behaviour just by changing the sign. The reason is that it is rather difficult and expensive to produce experimentally true (i.e. uniaxial) compressive stress-strain-curves for thin sheet material. In the present research project an economically feasible compromise has been chosen: Very short compression coupons which do not buckle before excessive yielding (h = 6 mm for t = 1 mm) are used to determine approximately the compressive yield stress of the 1 mm-material; however, the shape of the stress-strain-curve including E has to be taken from the tension coupon tests. For the 0,5 mm-material such compression coupon tests are not feasible. Fortunately the problem of correct compressive yield stress is not as important for the thinner specimens.

Figure C2 shows typical compression coupon diagrams. As can be seen, the compressive behaviour is also different: Type (a) has a short horizontal plateau, whereas type (b) shows bilinear curves. As mentioned above, these diagrams may not be used to derive "compressive Ε-values" or compressive stress-strain-curves! However, they deliver usable compressive yield stress values R,,SiC.

C2.2.1.4 Material input data for comparative numerical calculations

In tables C3 and C4 all average material property values for all test specimens are collacted. Additionally the type of stress-strain-behaviour (a or b ace. to fig. CI and C2) is indicated. From table C4 it may be read that for the 1 mm (a)-material the well-known identity of R,.s , and R,.sc is being confirmed , whereas for the 1 mm (b)-material R ^ i s 15 + 20 % higher than Res,. Also, there is a 20 % difference in the Rest-values of the 1 mm (a)-and 1 mm (b)-material, the latter being lower. The same ReSt difference may be read from table C3 for the 0,5 mm (a) and (b) materials. Thus, it is near at hand to estimate the R^-values of the 0,5 mm (a)-material as identical with R^t and the ones of the 0,5 mm (b)-material as R^sc ~ 1,175 R,.s,t.

136

For numerical calculations a compressive stress-strain-curve is needed. It may be taken as shown in the relevant subfigure of fig. CI, using the tensile values E, e v and Ey from tables C3 and C4, but as yield stress fy the R^-values as explained before.

C2.2.2 Manufacturing

The specimens were manufactured in the University's mechanical workshop in a four-step-procedure.

Step 1: Cutting

The shell wall pieces had to be cut from the sheets as precisely as possible, in order to get exact circumferential lengths along their end circles. Otherwise it would not have been possible to manage the circumferential butt welding along the cone/cone or cone/cylinder intersections, above all for the 0,5 mm-specimens. It was necessary to use computer-aided oxylaser-cutting for producing the wall pieces.

Step 2: Rolling

The wall pieces were cold rolled into their cylindrical and conical shapes respectively. Step 3: Longitudinal welding

The cylindrical and conical shell portions were then completed by longitudinal butt welding. The TIG welding method without filler metal has been used. For the 0,5 mm-specimens it was necessary to improve the heat flow by welding against a steel bed with copper cover. Step 4: Circumferential welding

The circumferential butt welding had turned out to be rather difficult, above all for the 0,5 mm-sheet material. After intensive discussions with experts from the welding industry and pilot welding with several methods, it was decided to use also the TIG welding method without filler metal, but against circular welding devices with machined copper covers. Through that, and because of the precisely cut circumferential lengths, the edges fit sufficiently well together for producing a proper butt weld.

Figure C3 shows photographs of two specimens before testing. As can be seen, the circumferential welds were reasonable. With regard to photographs of all other specimens, see [CU]-

C2.2.3 Geometrical dimensions

The height and diameter values of all specimens have been controlled; they were in sufficient accordance with the nominal values ace. to table CI. The wall thicknesses have been measured by means of ultrasonics at a selected number of points. The scatter is relatively small. A typical set of results is plotted in figure C4. As can be seen, the thicknesses are rather constant around the circumferences and along the meridians, so that it would be reasonable to use average values for numerical calculations. These average values are given for all specimens in table C5.

137

C2.2.4 Shape imperfections

The shape imperfections of all specimens have been measured by means of a special measuring device which had been manufactured on occasion of this particular research project (fig.C5). Specimens with up to 1 m maximum diameter and 2 m height may be handled on this device. A group of horizontally installed transducers scans the surface of the specimen while being rotated. The vertical position of the transducer group is varied by means of a vertical sliding carriage. Both movements (the rotational and the vertical one) are automatically controlled by a computer which also takes the measurement data of the transducers. In order to calibrate the imperfection measuring device, a highly precise calibration cylinder (0,4 m χ 2,00 m) has been machined from a thick steel tube. By scanning it and evaluating its measured surface values and comparing the results with the known dimensions, specific corrections could be derived taking care of the unavoidable inaccuracies of the device. By introducing them into the imperfection evaluation analysis it is possible to define the "best fit shell" for each specimen and to calculate the deviations from it. For the purposes of this project, it proved to be sufficient to extract only the radial deviations from the circular shape on each measuring plane. Fig. C6 shows a typical plot of these imperfections for one of the specimens. Similar plots for all specimens are given in [CI 1]. The imperfection data sets have been evaluated with regard to the imperfection tolerance values for axially compressed cylinders and cones prescribed in [CI]. This has been done by numerically simulating the procedure of holding a straight or a circular template of reference length lr = 4 \/rt against any meridian and any parallel circle respectively. The results of this imperfection evaluation are presented in [Cll]. They may be used for classifying the specimens with regard to their imperfection level.

In general, it may be stated that all specimens were more or less within the prescribed tolerances, or that exceedings (if present) were of a type which had no correlation with the observed load carrying behaviour. That means that it is possible to draw direct conclusions from the test results for design rule purposes.

C2.3 Axial load tests C2.3.1 Test set-up

The specimen ends were machined in order to guarantee an equally distributed introduction of the testing machine axial load. The specimens were then positioned between thick machined steel plates having circular grooves in which their edges were fixed against radial displacements by means of a high-strength resin. In case of conical ends, that one of the two groove sides against which the horizontal component of the meridional end force would act has been exactly machined fitting to the cone end diameter. By this means the cone's meridional end force was taken directly by contact steel-steel instead of partly steel-resin which would have caused imprecise boundary conditions. A total of 24 strain gages were attached to each specimen: 2 circular planes χ 3 circumferential positions (every 120°) χ 2 directions (meridional, circumferential) χ 2 wall sides (internal, external). This is the needed minimum in order to control the prebuckling stress state with regard to its global correctness. The two circular planes were situated near the meridional

138

breaks. By this way the local shell bending caused by the deviated meridional forces could be monitored experimentally. The unit "specimen plus two end plates" was positioned between the thick loading plates of a testing machine. Within the specimen, in its axis, a transducer was installed for measuring its axial shortening. The transducer's displacement signals were manually controlled from outside by means of two dial gauges. In [Cll] photographs of some specimens ready for testing are presented.

C2.3.2 Test procedure

At first, preliminary loadings were applied to check concentricity of applied axial load and correctness of the elastic prebuckling state. The loading procedure of the following ultimate load test was shortening-controlled by means of a servohydraulic system using the axial transducer signal as controlling signal. The shortening was incremented in steps which were chosen increasingly smaller when approaching the supposed failure load. Quasi-static equilibrium states were obtained, once plastic deformations had started, by keeping the shortening constant for at least 10 min, allowing the material to yield or to relax respectively. That is why all experimental load-deformation- and load-strain-curves given in this report ought to be taken as "quasi-static" curves. Hence, comparison analyses should be performed using the static yield stress values rather than the dynamic ones. All specimens were shortened far beyond the peak load until the plastic postbuckling pattern could clearly be identified. In figure CIO photographs of two axial load test specimens after testing are presented; photographs of all other specimens see [Cll].

C2.3.3 Test results

C2.3.3.1 Load-deformation-curves

The load-shortening-curves of the 12 axial load tests are, in a non-dimensional manner, plotted in figures C7 and C8. The reference load Fp,M for the vertical axis and the reference shortening ΔΗρΙ

Μ for the horizontal axis represent the elementary theoretical state with meridional membrane yielding at the small radius end of the cone with 2r = 304 mm, see table CI. The values are given in table C6. It becomes evident from fig. C7 and C8 that the elementary membrane theory is certainly not an adequate tool to describe the load carrying behaviour of shell configurations with meridional breaks; see later on in this report. Load-strain-curves for all axial load tests are given in [Cll]. The measured surface strains have been converted into membrane strains and bending strains. The membrane strains show that the elastic prebuckling states of all specimens were properly uniform.

C2.3.3.2 Failure modes For all 1 mm-specimens (r/t <. 225) the failure mode was dominated by the axisymmetric bending at the meridional breaks. Under the peak loads, pure combined two-dimensional yielding on account of meridional bending moments and both direction membrane forces was responsible for the following decrease of load. The yielding was, for the same r/t-ratio, more pronounced at convex meridional breaks (see fig. CIO, specimen KK-X10) than at concave ones; that is logical from the viewpoint of yield criterion. It was only beyond the peak loads that non-axisymmetric deformations from secondary plastic buckling showed up.

139

For all 0,5 mm-specimens (r/t > 300) the failure mode was more or less an interaction between axisymmetric yielding and non-axisymmetric buckling. The buckles were, for concave breaks, situated directly at the break deforming the junction circle into a polygon - because of the compressive circumferential forces, for convex breaks on both sides of the break leaving the junction circle undeformed - because of the tensile circumferential forces (see fig. CIO, specimen KZK-W 50). The circumferential mode length were clearly shorter than for the 1 mm-specimens. The different load-carrying behaviour may also be recognized from the load-shortening-curves on fig. C7 and C8. Some thinner specimens showed a more rapid load decrease beyond ultimate load than their thicker counterparts (compare ZKZ-XV50 with ZKZ-XV10 and KK-V50 with KK-V10), a clear evidence for buckling interaction.

The ultimate axial load values Fu are, together with a short description of the failure mode, given in table C6.

C2.4 External pressure tests C2.4.1 Test set-up The specimen ends were prepared and connected to thick machined steel plates in a similar way as the axial load specimens (see C2.3.1). The unit "specimen plus two end plates" was set upon the rotary-table of the imperfection measuring device described in section C2.2.4. By this means, it rendered possible to rotate the specimen under load in order to monitor the developing of the geometrical shape during testing.

Similarly to the axial load tests, the shortening was measured by a transducer and plotted versus the pressure during testing.

C2.4.2 Test pocedure The external pressure was applied through underpressure from inside. This was achieved by evacuating the interior of the specimen by means of a vacuum pump from outside. The underpressure was incremented in steps which were chosen increasingly smaller when approaching the supposed failure pressure. Between the loading steps, the pressure was kept constant, and the circumferential shapes were traced at selected measuring planes. Load-deformation-curves may be extracted from these measurements by plotting selected radial displacements versus the underpressure.

Because of the physical characteristics of air pressure, it was not possible to achieve anything like a deformation-controlled loading pocedure. It proved unavoidable that the specimens, reaching their buckling pressure, either snapped through into a stable postbuckling state and failed later on completely (see fig.Cll), or failed completely without a stable intermediate state.

C2.4.3 Test results C2.4.3.1 Load-deformation-curves

Pressure-displacement-curves of five external pressure tests are plotted in fig. C9. (For the sixth one the measurement equipment failed during testing.) The displacement on the

140

horizontal axis is the axial shortening. Because the test procedure was not deformation controlled (see above), no unloading path could be monitored.

C2.4.3.2 Failure modes For all specimens the failure mode was a more or less distinct two-step buckling failure of snap-through type. Local buckling of either a conical or a cylindrical shell section, with the meridional breaks acting as eigenmode node circles and with the postbuckling pattern being virtually stable, was followed by overall buckling, with the junction circles at the meridional breaks collapsing.

Fig. Cll shows specimen KK-V51 during its quasistable postbuckling state and after collapse. The described two-step buckling failure may well be recognized on the two photographs. It is interesting that from local buckling to overall buckling no pressure increase could be achieved, although the local postbuckling state (fig. CI la) proved to be stable during a complete unloading/reloading loop (see fig. C9).

The ultimate pressure values pu are, together with the number nu of buckles around the circum­ference, given in table C7.

141

C3 Numerical investigations

C3.1 General Four types of comparative numerical calculations have been performed for all 18 specimens: (a) Membrane analysis (MA) according to elementary membrane theory, applied to the

cylindrical and conical partial shells, neglecting the mechanical interaction at their junctions. These theoretical values are used as basic references, e.g. for the presentation of experimental results (see C2.3.3.1).

(b) Linear shell analysis (LA), applied to the geometrically perfect shell assembly, including eigenvalue search.

(c) Geometrically nonlinear elastic shell analysis (GNA), applied to the geometrically perfect shell assembly, including eigenvalue search.

(d) Geometrically and materially nonlinear shell analysis (GMNA), applied to the geometrically perfect shell assembly, including eigenvalue search.

For the calculations on levels (b) to (d) the computer program F04 B08 has been used. It was specificly developed by M. ESSLINGER et. al. [C4, C5, C6] for shells of revolution and is based on an axisymmetric discrete modelling of the shell configuration. For each ring element the shell differential equations are satisfied by means of the collocation method, delivering a transfer matrix. The transfer matrices of all ring elements (including stiffening annular plates if relevant) are assembled into the global set of equations for the complete shell. This set of equations is basicly nonlinear - geometrically nonlinear for level (c) and additionally materially nonlinear for level (d). The solution procedure is iterative via linearized intermediate stages. Level (b) represents virtually the first linearized stage. On all of the three levels deformations and stresses may be calculated (called "prebuckling state"), as well as eigenvalues and eigenmodes may be determined (called "buckling analysis").

F04 B08 has been used for shell buckling research at Essen University since many years [C9]; its proper handling is therefore well established. Nevertheless, some benchmarking checks were felt to be necessary in order to prove its suitability for the present purposes.

C3.2 Benchmarking C3.2.1 Buckling analysis on GNA level The algorithm for searching the lowest bifurcation eigenvalue is a basic numerical element of any shell buckling computer program. The one included in F04 B08 has been checked for a selected number of shell geometries against a conventional linear FE-program. The results were reasonably identical.

C3.2.2 Collapse analysis on GMNA level The highest demands in terms of nonlinear shell analysis within the present subproject are requested by the geometrically and materially nonlinear analysis up to (or beyond) the axisymmetric collapse load (limit load). A benchmarking effort involving six independent research teams using five différant computer programs has been dedicated to an axially loaded

142

shell with the geometry of specimen ZKZ-XV 10 (called "milk can shell"). The calculated load-shortening-curves and axial limit loads Flim

GMN are presented in fig. C12. It should be mentioned that these numerical results may not be compared directly with the factual test results of specimen ZKZ-XV 10, because - at the time of the benchmarking - the material properties had only been estimated.

Of the five used computer programs, three (F04 B08, BOSOR 5, HORUS) are based on a ring type discretization, but apply rather different theoretical approaches. It is interesting that they deliver practically identical limit loads (fig. C12). The two other computer programs (ABAQUS with two different users, INCA with two alternative element types) are based on a "complete" discretization, but using different shell elements. They deliver limit loads which are from 3,5% to 8% higher than the before mentioned value. The loading paths are practically identical for all of the seven numerical models (fig. C12), whereas of the unloading paths the F04 B08 one is significantly steeper than the ABAQUS and INCA ones (BOSOR 5 and HORUS do not include unloading paths). Thus, the result of the GMNA benchmarking with regard to the present subproject is as follows:

• F04 B08 is a reliable "numerical tool" for handling the geometrically and materially nonlinear axisymmetric behaviour of complex shells of revolution including the limit load (collapse load).

• However, F04 B08 tends to underestimate the ductility of failed shells in the post-limit range.

C3.3 Comparative numerical calculations for the axial load specimens

C3.3.1 MA level The following elementary membrane theory values were calculated:

Plastic reference force:

FpiM = R« · rcdt · cos 20°

with d = 304 mm (including KZK-VX in spite of its smaller cone), because in a cone the small radius end is the critical one in terms of meridional stress, and Res= fy = R ^ .

Plastic reference shortening (sum of partial shells):

ΔΗρΙΜ = (Fpl

M/27cEt) · [Ecy ,(H/r) + 1/sin 20° · Σ ^ ΐ η ^ / η ) ]

with E,t = average values for the whole specimen, r2,r, = large/small end radii of a cone.

Critical buckling force:

FCTM = 0,605 · E · 2TCÍ2 · (cos 20°)2.

It is remarkable that this value from elementary linear shell buckling theory does not depend on the radius.

If the two cones of a specimen have different values for R^ and t, that one yielding the smaller reference value is introduced into the calculation. The results are given in table C6.

143

C3.3.2 LA and GNA level

The following values have been extracted from linear and geometrically nonlinear shell analysis:

Plastic reference force Fp,L or Fp,GN:

This is the axial load F at which the v. MISES effective stress in the midsurface of the geometrically perfect shell - i.e. the effective membrane stress oeffM - firstly reaches at any point of the shell the yield stress:

°efr,M = [ K M ) 2 - (°xM>(°yM) + K M ) 2 ] 0 , 5 = KS~+ FplL or Fp l

G N

Critical buckling force FcrL or Fc

GI' cr

This is the lowest eigenvalue of F, at which the elastically calculated prebuckling state of the geometrically perfect shell bifurcates.

The GNA level is the theoretically exact level, presumed that the whole shell stays purely elastic until its first bifurcation. The definition of the plastic reference force Fp

GN and the critical buckling force Fcr

GN is analogous to that one on level LA (see above). However, the results are somewhat different, because the deviation of the meridional membrane forces at the meridional breaks causes nonlinear effects on the elastic stress and deformation state. The differences are not dramatic; therefore only the GNA results are included in this report. Full evidence is given in [CI 1].

As an example for typical GNA results, the ones for specimen KZK-VX 10 are illustrated in fig. C13. The strong influence of the meridional break on both, the plastic reference force and the critical buckling force, may be realized from the peak effective membrane stress caused by the circumferential tensile force at the convex break (fig. C 13c) and from the critical buckling mode having its maximum amplitude near the same break (fig. CI 3d). The latter is contrary to an end-supported fundamental conical shell whose critical buckling mode has its maximum amplitude near the small end.

The GNA results of all axial load specimens are given in table C6.

C3.3.3 GMNA level

The basic numerical result delivered by a GMNA calculation is the axisymmetric limit (collapse) force FiimGMN defined by the peak of the load-shortening-curve (see fig. C12). It describes correctly the load-carrying capacity of the geometrically perfect shell, except a bifurcation happens before reaching the peak. If this is the case, the bifurcation load is called the critical buckling force Fcr

GMN.

As a third significant force value for characterizing the GMNA behaviour of a shell, the plastic reference force Fp

GMN has been extracted from the analysis. Its definition is similar to the ones on LA and GNA level: When the v. MISES effective stress in the middle of the wall thickness firstly reaches the yield stress at any point of the shell.

GMNA results are inevitably more siginificantly influenced by the material input data than the elastic calculated results. In fig. C14 three calculated load-shortening-curves for two of the 1 mm-specimens made of type (a) and (b) sheet material respectively (see C2.2.1.4) are compared. The following stress-strain-curves have been used alternatively:

144

• Ideally elastic-plastic with fy = R^t, • ideally elastic-plastic with fy = R^,., • strain hardening with fp02 = ReS,c, but Ev from the tension coupon test.

As can be seen, even the latter approach approximates the observed experimental behaviour rather poorly. The reason for this discrepancy is probably a weld-hardening effect in close vicinity to the circumferential welds at the junctions of the partial shells. Fig. CI 3 shows that the through-thickness yielding which causes numerically the collapse occurs locally at these junctions. In [Cll] some attempts to simulate this localized weld-hardening effect are discussed.

The GMNA results given in Table C6, are those which belong to ideally elastic-plastic stress-strain-curves. In practical GMNA analyses only this approach would be feasible. The numerical strain-hardening values would strongly depend on the input parameters; they are discussed in detail in [Cll]. As yield stress values fy the compressive values R.^ as explained in C2.2.1.4 have been introduced.

C3.3.4 Discussion of the results

A complete set of numerical results for one specimen is, in its relation to the experimental load-shortening-curve, illustrated in fig. C15.

It is obvious, that the membrane solution for the critical buckling force FcrM is of no account

for shell assemblies with meridional breaks; this is confirmed by all other specimens (see table C6). In fact, this could be expected, because the assumed pure membrane state is not an equilibrium state. The real elastic state incorporates very high circumferential membrane forces and meridional bending moments at the junctions (see fig. C13 a, b, c), which trigger the bifurcation buckling at considerably lower loads Fcr

GN.

That means that elastic shell theory (LA or -better - GNA level) for the overall shell configuration is obligatory if a critical buckling resistance is needed for a design procedure (see section C4.2 of this report).

The situation is different when the plastic reference forces are considered. Again the elastic values F ^ delivered by shell theory are much lower than the membrane solutions Fp,M(see fig. CI5 and table C6), but now it seems that the latter are more suitable if a plastic reference resistance is needed for a design procedure. The mechanical background for the extremely low values Fpl

GNbecomes clear from fig. C13: The value Fp,GN describes virtually a highly localized circular plastic hinge mechanism rather than a membrane phenomenon which may cause instability. This question will be reverted to in section C4.2.

The fact that the GMNA limit loads FiimGMN are too low compared with the experimental ultimate loads Fu - especially for the stubber 1 mm-specimens, where strain-hardening has a greater influence - has already been discussed. For the evaluations in section C4.2 the numerical values FHm

GMN are taken as conservative estimates of the axial collapse loads.

145

C3.4 Comparative numerical calculations for the external

pressure specimens

C3.4.1 MA level

In analogy to the axial load specimens, the following elementary membrane theory values were

calculated:

Plastic reference pressure:

pplM = 1,155­ ReS ­(t/r)· cos 20°

with r = 225 mm (large radius end of conical partial shell).

The factor 1,155 results from v. MISES' yield criterion because hydrostatic pressure

produces circumferential plus meridional compressive membrane stresses.

Critical buckling presssure:

pcrM = 0,92 · E · (r*/l*) ■ (t/r*)2·5

with either r*, 1* being the effective radius and length values of the partial conical

shell, ace. to DIN 18800/4,

or r*, 1* being the true radius and length values of the partial cylindrical

shell, whichever yields the smaller value pcrM.

The results are given in table C7.

C3.4.2 LA and GNA level

In analogy to the axial load specimens, the following values have been extracted from linear

and geometrically nonlinear shell analysis:

Plastic reference pressure pp,L or pp

GN:

This is the hydrostatic pressure ρ at which the v. MISES effective membrane stress aeñM in

the geometrically perfect shell firstly reaches at any point of the shell the yield stress.

Critical buckling pressure pcrL or pcr

GN:

This is the lowest eigenvalue of p, at which the elastic prebuckling state bifurcates.

Fig. CI 6 shows a set of typical GNA results for one of the external pressure test specimens.

All results for GNA level are given in table C7. The results for LA level are nearly identical

and are omitted here.

C3.4.3 GMNA level

In analogy to the axial load specimens, the following values have been extracted from

geometrically and materially nonlinear shell analysis:

146

Plastic reference pressure ppGMN:

This is the hydrostatic pressure when the v. MISES effective stress in the middle of the wall thickness firstly reaches the yield stress at any point of the shell.

Axisymmetric limit (collapse ) pressure PiimGMN:

This is the hydrostatic pressure at the peak of the pressure-deformation-curve. It would be the exact solution for the pressure-carrying capacity of the geometrically perfect shell, except a bifurcation would happen before reaching the peak.

Critical buckling pressure pcrGMN:

This is the bifurcation pressure mentioned before.

The results are given in table C7.

C3.4.4 Discussion of the results

From table C7 it may be noted that the load-carrying behaviour of the present shell assemblies under external pressure (more precisely: hydrostatic external pressure) is - from the numerical point of view - rather unproblematic compared to the axial load case. Obviously the meridional break angle of 20° is large enough to enforce circular node lines of the eigenmodes at the junctions. Therefore the critical buckling pressures and critical buckling modes delivered by membrane and shell theories respectively are practically identical. On top ofthat, the buckling happens under such low stresses that the GMN analysis also delivers the same buckling pressures.

Between numerical and experimental results satisfying agreement may be stated: The test buckling pressures pu amount to 60 -*- 95 % of the critical buckling pressures. This corresponds to the well-known imperfection influence range of externally pressurized cylinders and cones: Consequently the results of the external pressure tests may be directly utilized for validating design recommendations (see section C4.2).

147

C4 Evaluation of design procedure

C4.1 Generalized reduction factor approach (RFA) for shell

buckling design

The classic reduction factor approach (RFA) for fundamental shell buckling cases is well­

known. It is used throughout the world in most of the codes of practice and design

recommendations, among others in the ECCS Recommendations [CI]. One of the aims of the

present research was to check if the approach may be generalized to non­fundamental shells of

revolution and ­ if so ­ to propose suitable reduction factors.

The following generalized buckling design procedure has been headed for by the authors:

(a) Calculate the elastic stress state of the geometrically perfect shell under the considered

combination of actions. (Indications for the needed theoretical level are to be formulated.)

(b) Derive the plastic reference resistance Fvp, by applying the v. MISES yield criterion to the

membrane forces of (a).

(c) Derive the elastic critical buckling resistance R,., by applying an eigenvalue analysis to (a).

(d) Determine the shell slenderness parameter

λ = (Rp/RJ0·5. (CI)

(e) Choose the elastic imperfection reduction factor a. If no specific value is known, use the

a0­values of the fundamental edge­supported cylinder. According to [CI], this would

mean:

(el) For meridional compression:

α = a0 = 0,83 / [1 + 0,00605 · λ2 · (E/fy)]0·5

when X¿ 18,7/ (E/fy)0·5, (C2a)

α = a0 = 0,70 / [1 + 0,00605 · λ2 ■ (E/fy)]0·5

when λ > 18,7 / (E/fy)0·5. (C2b)

(e2) For external pressure:

α = α0 = 0,5 (independent of λ). (C3)

(f) If plasticity is involved, determine the plasticity reduction factor κ as function of λ.

According to [CI], this would mean:

κ = 1 ­ 0,25 ­λ11 a when λ ^ (2a)0·5. (C4)

This formula is known to be somewhat unconservative for medium­thick cylinders on the one

hand, but somewhat overconservative and unwieldy for thick cylinders on the other hand. A

better agreement with cylinder tests in the medium­thick range and an easier wielding for thick

cylinders is achieved by a formula that has been proposed by the first author for the shell

buckling rules of Eurocode 3 Part 3 and 4. It reads as follows:

148

κ = 1 when λ ¿ λ0 , (C5a)

0,6

κ = 1 (λ ­ λ0) when λ0 < λ <; (2,5α)0·5, (C5b)

(2,5α)0·5­λ, ο

with λ0 = 0.2 for meridional compression, (C6)

λ0 = 0.4 for external pressure. (C7)

(g) Determine the characteristic value Rk of the buckling resistance from

Rk = α · Rcr if no plasticity is involved, i.e. λ > (2,5 α )0·5, (C8)

Rk = κ · R,,, if plasticity is involved, i.e. λ ¿ (2,5 α )0·5. (C9)

C4.2 Application of RFA to test specimens and comparison

with test results

C4.2.1 Axial load specimens

In table C8 the above outlined RFA procedure for meridional compression has been applied to

the axial load specimens. Three different numerical "resistance pairs " have been coupled into

a shell slenderness parameter λ ace. to eqn. (CI) :

(I) Pure MA level ­> FplM and Fcr

M ;

(II) Combined MA/GNA level ­> FplM and Fcr

GN ;

(III) Pure GNA level -► FplGN and Fcr

GN .

The obtained characteristic resistance values Fk for the axial load must be compared, as explained at the end of subsection C3.3.4, to the numerical limit loads F,imGMN. These are conservative estimates of the axial collapse loads neglecting material hardening effects. From table C8 the following findings may be drawn:

A RFA design based on pure membrane theory (column I) would be extremely unsafe. Of course, this was supposed from the beginning; it has now been substantiated. The mechanic reason is, as already indicated in C3.3.4, that the bifurcation buckling behaviour is significantly influenced by the equilibrium disturbances at the meridional breaks.

A RFA design based on pure elastic shell theory (column III) would be overconservative. The mechanic reason is, as also indicated in C3.3.4, that the plastic reference force Fp]

GN describes a limit state which has virtually nothing to do with shell buckling. This limit state is characterized by plastic expending (or crushing respectively) of the junction circles at the meridional breaks because of interactive through-thickness yielding caused by both direction membrane forces and meridional bending moments. To apply an additional imperfection reduction factor to this limit state, does not make any sense.

The only reasonable RFA design seems to be the one based on a combination of membrane and shell theory (column II in table C8). It yields a little too high Fk-values for the 1 mm-specimens, but lower Fk-values for the 0,5 mm-specimens, both compared to F |im GMN. That appears logical considering that in the 0,5 mm-tests a more distinct buckling influence had been observed than in the 1 mm-tests ( see C2.3.3.2). The combined approach may be associated with the following mechanical idea: Bifurcation buckling is a phenomenon which

149

affects a certain area of the shell midsurface. Consequently it has - in terms of yielding as the limiting case of buckling - to be coupled to a membrane compressive stress state which affects the whole buckling area, instead of only a narrow hoop.

Based on the foregoing considerations the authors recommend to calculate both:

• the characteristic buckling resistance Fk using the reduction factor approach II • and the plastic resistance Fp

GN without additional reduction

and to use the smaller one of the two values for the design. This recommendation is validated by the present test series (see table C8) for r/t-ratios larger than ca. 150.

4.2.2 External pressure specimens Table C9 contains the results of the RFA application to the external pressure specimens. The interpretation of these results is rather straightforward: The validity of an RFA design for externally pressurized cone/cone and cone/cylinder assemblies of arbitrary shape may be taken as proved by the present test series for r/t-ratios larger than ca. 150.

However, one may not conclude from the fact of practically identical pk-predictions in columns I and II in table C9, that it would be sufficient to calculate the critical buckling pressure for the fundamental partial shells only . This is true for the present cone angle of 20°, but certainly not for smaller cone angles. Therefore it is unrenouncable to draw the critical buckling pressure j>„ from a shell theory analysis (LA level is sufficient) of the whole shell configuration.

As to the plastic reference pressure ppl, the same recommendation as given above for meridionally compressed shells should be considered here: Use ppl

M from membrane theory for the RFA check; carry out an alternative check using pp,L or pp

GN from shell theory analysis for the whole shell configuration, but without buckling reduction factor.

A comment on the elastic imperfection factor α for external pressure should be added. As can be seen from table C9, the ECCS-value 0,5 is conservative, whereas the DIN-value 0,65 [C2] is very close to experimental evidence. Since the latter value is calibrated against end-supported fundamental cylinders under pure external pressure (not hydrostatic pressure), it is strongly recommended to use α = 0,5 ace. to eqn. (C3) for general shells of revolution under external pressure of any type.

150

C5 Conclusions (1) Shells of revolution being an arbitrary assembly of cylinders and cones and having no ring

stiffeners at the junctions may - with regard to shell buckling - not be designed as if the partial shells would be edge-supported at the junctions.

(2) Under axial compression, for semi-vertex angles 20° -*- 40° at the meridional breaks and r/t-ratios of up to 200, the failure modes are dominated by axisymmetric interactive yielding under both direction membrane forces and meridional bending moments in narrow bands along the circular junctions. This failure mode is not significantly influenced by imperfections and may therefore be covered by numerically calculating the nonlinear axisymmetric elastic-plastic collapse load or - as a replacement - the plastic reference load from nonlinear elastic shell theory, both without any imperfection reduction.

(3) Under axial compression, for r/t-ratios larger than 200, the axisymmetric local yielding at the breaks is increasingly interacted by non-axisymmetric buckling. This interactive failure mode may be covered by a generalized reduction factor approach (RFA), coupling the plastic reference load from membrane theory (MA) and the critical buckling load from geometrically nonlinear elastic shell theory (GNA) into a slenderness parameter, and applying appropriate imperfection factors α and plasticity reduction factors κ.

(4) Using the cylinder imperfection reduction factor α = α0 for meridional compression as specified in the ECCS Recommendations is conservative.

(5) Under external pressure, for r/t-ratios of more than 150, the failure mode is a non-axisymmetric buckling. This failure mode may be covered by the same generalized RFA as explained in (3).

(6) Using the cylinder imperfection factor α - α0 for external pressure as specified in the ECCS Recommendations is conservative.

(7) For the plasticity reduction factor κ, a new formula is proposed which defines κ as a function of α and of the general stability slenderness parameter λ = (R^/R,.,.)0·5, where R,,, and Rer are the plastic reference and critical buckling resistances respectively.

151

C 6 References

[Cl] Buckling of Steel Shells ­ European Recommendations, 4th edition. Brussels: ECCS

1988.

[C2] DIN 18800 Part 4: Stahlbauten ­ Stabilitätsfalle, Schalenbeulen. Berlin: Beuth­

Verlag 1990.

[C3] Samuelson, L.A./Eggwertz, S.: Shell Stability Handbook. London, New York:

Elsevier 1992.

[C4] Esslinger, M./Kerkhoff, H./Melzer, H.W./Taelmann, E.W.: Berechnuung der

Beullasten von dünnwandigen Rotationsschalen unter axialsymmetrischer Belastung

im elastischen Bereich. Report KfK­CAD 176. Karlsruhe: Kernforschungszentrum

1981.

[C5] Esslinger, M./Geier, B./Wendt, U : Berechnung der Spannungen und Deformationen

von Rotionsschalen im elasto­plastischen Bereich. Stahlbau 53 (1984), ρ 17 ­ 25.

[C6] Esslinger, M./v. Impe, R.: Theoretical Buckling Loads of Conical Shells. In: Dubas,

P./Vandepitte, D. (ed.): Stability of Plate and Shell Structures ­ Proc. Int. Coll., ρ

387 ­ 395. Ghent: University/ECCS 1987.

[C7] Knödel, P.: Cylinder­Cone­Cylinder Intersections under Axial Compression. In:

Jullien, J.F. (ed.): Buckling of Shell Structures on Land, in the Sea and in the Air, ρ

296 ­ 303. London, New York: Elsevier 1991.

[C8] Schmidt, H./Krysik, R.: Towards Recommendations for Shell Stability Design by

Means of Numerically Detennined Buckling Loads. In: (as [C7]), ρ 508 ­ 519.

[C9] Krysik, R.: Stabilität stählerner Kegelstumpf­und Kreiszylinderschalen unter Axial­

und Innendruck. Dr.­Ing thesis, University Essen, 1994.

[C10] Schmidt, H./Krysik, R.: Static Strength of Transition Cones in Tubular Members

under Axial Compression and Internal Pressure. In: Grundy, P. /Holgate, A.I Wong,

B.: Tubular Structures VI, ρ 163 ­ 168. Rotterdam: A.A. Balkema 1994.

[Cll] Swadlo, P./Schmidt, H.: Experimental and Numerical Investigations on Cone/Cone

and Cone/Cylinder Shell Assemblies under Axial Compression and External

Pressure. Research Rep. No. 68, University Essen, Dep. Civil Engineering, 1996.

152

L/i

H m O

H fD i - f

i-I O

00

δ' 3 on O

oo Ό Π) Ο

Π)

ρ 00

η

ο

ο CD W 00

3 3

1.0

0.5

G 3 CD

0.5

meridional breaks cone/cone cone-cone

convex

Ι-τ-Η

1.50

KK - Χ 10

KK - Χ 50

KK - Χ 51

cone-cone concave

KK - V 10

K K - V 50

KK - V 51

cylindercone-cylinder convex - concave

301.

A- ι V

ZKZ- XV 10

ZKZ - XV 50

ZKZ- XV 51

meridional breaks cylinder/cone

convex - convex

ï — - ï

/"JO·

\

')

/

u=±=u-KZK-XX 10

KZK - XX 50

KZK-XX 51

cone-cylinder-cone concave · convex

KZK - VX 10

KZK - VX 50

KZK-VX 51

concave - concave

KZK - VV 10

KZK - VV 50

KZK - VV 51

specimen ZKZ - XV 50

X

X

X

V[%]

type

part A

E ΊΟ' 3

[MPal

Res.t

[MPal

meridiona

222.0

234.2

228.1

252

247

249

Rm

[MPa]

372

376

374

circumferential

209.1

226.4

-

217.8

222.9

4.7

251

231

-

241

245

3.9

375

360

-

367

371

2.1

a

part Β

E I O ' 3

[MPa]

Res.!

[MPa]

meridiona

194.0

-

194.0

198

-

198

Rm [MPa]

337

-

337

circumferential

185.1

199.5

194.3

193.0

193.2

3.1

210

176

184

190

192

7.8

329

318

321

323

326

2.6

b

part C

E IO'3

[MPa]

Res.t

[MPa]

meridiona

227.0

222.6

224.8

240

243

242

Rm

[MPa]

359

372

365

circumferential

194.4

226.6

-

210.5

217.7

7.2

222

244

-

233

237

4.3

348

371

-

360

362

3.1

a

total

E I O 3

[MPa]

211.3

8.0

R|S.l

[MPa]

225

11.9

Rm

[MPa]

353

6.2

(a)

specimen ZKZ - XV 10

X

ι

X

V[%]

type

part A

E I O ' 3

[MPa]

Res.l

[MPa]

Rm

[MPa]

R|*.C

[MPa]

meridional

208.0

-

-

208.0

215

-

-

215

331

-

-

331

circumferentia

194.9

199.0

192.5

195.5

198.6

3.4

199

179

203

194

199

7.6

317

290

314

307

313

5.4

237

-

-

237

193

201

223

206

213

9.3

a

part Β

E IO ' 3

[MPa]

RB,t

[MPa]

Rm

[MPa]

Re« [MPa]

meridional

188.9

200.3

213.7

201.0

228

213

213

218

328

317

298

314

circumferentia

200.8

198.8

207.0

202.2

201.6

4.1

212

227

210

216

217

3.8

316

347

323

329

321

5.0

211

213

218

214

-

-

218

218

215

1.6

a

part C

E IO 3

[MPa]

Res.t

[MPa]

Rm

[MPa]

Re«

[MPa]

meridional

206.7

211.2

206.8

208.2

208

205

205

206

295

308

316

306

circumferentia

202.4

193.5

190.4

195.4

201.8

4.1

209

203

189

200

203

3.6

325

321

297

314

310

4.0

232

217

239

229

212

216

230

219

224

4.9

a

total

E I O 3

[MPa]

200.9

3.8

Res.t

[MPa]

207

5.9

Rm

[MPa]

315

4.7

"esx

[MPa]

219

5.9

(b) Table C2. Typical tension coupon test results appointed to buckling test specimens

(a) ZKZ-XV 50 (b) ZKZ-XV 10

154

specimen no.

KK-X50

KK-X51

KK-V50

KK-V51

ZKZ-XV50

ZKZ - XV 51

KZK · XX 50

KZK-XX 51

KZK - VX 50

KZK - VX 51

KZK - VV 50

KZK - VV 51

1 — co OL

A Β Σ A Β Σ A Β Σ A Β Σ Α Β C Σ Α Β C Σ Α Β C Σ Α Β C Σ Α Β C Σ Α Β C Σ Α Β C Σ Α Β C Σ

tension

Ε ΊΟ'3

[MPa]

204 209 207 184 203 194 203 206 205 194 206 200 223 193 218 211 212 197 212 207 214 229 207 217 199 231 197 209 204 203 211 206 207 201 211 206 202 207 201 203 203 201 198 201

"es.t

[MPa]

187 191 189 183 183 183 190 188 189 192 185 189 245 192 237 225 243 186 243 224 187 246 191 208 194 242 180 205 194 181 192 189 187 177 182 182 186 180 190 185 177 187 186 183

" m

[MPa]

321 325 323 321 320 321 330 324 327 328 331 330 371 326 362 353 369 319 369 352 324 373 330 342

' 336 364 314 338 333 322 336 330 325 319 330 325 325 315 325 322 317 321 327 322

[%] ------------1.9 -1.9 -

2.1 -

2.0 --

2.1 ---

2.2 ------------------

Ε, Ί Ο 3

[MPa]

1.27 1.47 1.37 1.43 1.82 1.63 1.58 1.51 1.54 1.46 1.95 1.70 1.56 1.64 1.57 1.59 1.61 1.66 1.58 1.62 1.70 1.53 1.70 1.64 1.80 1.55 1.74 1.70 1.80 1.85 1.96 1.87 1.82 1.79 1.94 1.85 1.89 1.64 1.84 1.79 1.78 1.81 1.89

1.83

type

b b

b b

b b

b b

a b a

a b a

b a b

b a b

b b b

b b b

b b b

b b b

TableC3. Average material property values for all 0,5 mm specimens

155

specimen no.

KK-X10

KK-V10

ZKZ-XV 10

KZK-XX 10

KZK-VX 10

KZK-VV10

ι ­οί EL

A Β Σ A Β Σ A Β C Σ A Β C Σ A Β C Σ A Β C Σ

tension

Ε ΊΟ 3

[MPa]

193 193 193 184 191 188 199 202 202 201 208 198 201 202 212 205 201 206 197 187 193 192

Res,t

[MPa]

157 154 156 171 170 171 199 217 203 206 220 212 213 215 214 213 213 213 164 165 166 165

Rm

[MPa]

295 288 292 331 335 333 313 321 310 315 331 327 329 329 330 327 329 329 328 317 328 324

ε»

[%]

------

2.3 2.2 2.3 2.3 2.5 2.3 2.4 2.4 2.5 2.5 2.4 2.5 ----

E/103

[MPa]

1.41 1.40 1.41 2.03 1.88 1.96 1.48 1.51 1.40 1.46 1.46 1.47 1.54 1.49 1.73 1.46 1.54 1.58 1.92 1.83 1.92 1.89

compression

"es.c

[MPa]

182 180 181 197 196 197 213 215 224 217 220 217 218 218 207 202 218 209 198 204 192 198

type

b b

b b

a a a

a a a

a a a

b b b

Table C4. Average material property values for all 1,0 mm specimens

wall thickness [mm]

cu E Ό cu ta. co

part A

partB

parte

Σ

o > χ Χ

1.021

1.023

-

1.022

o

X

ie: i ¿

1.014

1.022

­

1.018

nom ™

> >

i ¿ I N I χ

1.035

1.033

1.035

1.034

1,0 mm

CS

Χ

> Χ INI χ

1.028

1.032

1.028

1.029

o

Χ χ

rvj i d

1.020

1.020

1.017

1.019

o

> Χ

INJ

rsj

1.031

1.026

1.027

1.028

o LR

> Χ Χ

0.515

0.517

­

0.516

i n

> χ X

0.525

0.524

­

0.525

o LO X

X X

0.513

0.514

­

0.514

m

X

X X

0.517

0.518

­

0.518

o I D > >

X rsi X

0.516

0.512

0.516

0.515

nom ™

L O

> >

X INI X

0.513

0.522

0.514

0.516

0,5 mm

o m X

> X INI X

0.520

0.513

0.519

0.517

1X9

X

> X INI X

0.532

0.525

0.528

0.528

o m X X

X ΓΝΙ

χ

0.522

0.523

0.524

0.523

I f )

χ χ

χ INI χ

0.518

0.522

0.517

0.519

ο LT)

> Χ

Γν4 Χ INI

0.515

0.512

0.516

0.514

i n

> χ

ÍNI Χ INI

0.514

0.512

0.513

0.513

Table C5. Average thickness values for all specimens

156

^ 1

H ta jr

Π Os

*-· EL o

CO Κ W χ

Ό

S 3 rt I s C L

3

3. o

3 c co

specimen no.

KK-Χ 10

KK-V 10

ZKZ-XV 10

KZK XX 10

KZK-VX 10

KZK-VV 10

KK - X 50

KK - V 50

ZKZ - XV 50

KZK - XX 50

KZK - VX 50

KZK - VV 50

ASY PB

Fu

[kN]

51.96

45.36

93.00

97.20

77.70

87.90

19.40

17.40

31.20

30.00

28.20

26.91

- axisyn - postul

AHU

[mm]

0.9

1.1

0.9

1.0

0.8

0.4

1.2

0.6

0.7

1.0

0.7

0.7

imetric timate b

test failure mode

ASY at break

ASY at break, PB at break with nu-6

ASY at both breaks, PB at concave break with nu-5 and at the convex break with nu-4 ASY at both breaks, PB at one break withnu-7 ASY at the convex break, PB at the convex break withnu-5

ASY at both breaks, PB with n„-7

IASY and NSB at the break with nu-10

I ASY and NSB at the break with n„-12

IASY and NSB at the concave break with nu-11

IASY and NSB at one break with nu-9

IASY and NSB at convex break with nu-4

IASY and NSB at both breaks with n„-8

yielding uckling

comparative analyses membrane

CM Γ pi

[kN]

166.0

181.1

201.0

200.8

196.8

184.0

102.1

102.5

103.5

104.2

102.7

100.0

IASY NSB

ΔΗΜρ 1

[mm]

0.31

0.34

0.82

0.55

0.74

0.36

0.30

0.30

0.68

0.45

0.66

0.47

F M 1 cr

[kN]

676.6

645.6

712.3

729.7

714.3

709.4

181.0

182.1

170.1

190.5

184.1

179.8

geometrically nonlinear

r-GN Γ pi

[kN]

35.5

37.4

76.0

76.2

59.1

77.6

15.8

14.4

31.0

30.1

23.1

27.4

pGN ' CI

[kN]

249.9

353.4

335.0

350.0

369.0

537.7

59.9

79.7

74.9

83.2

87.0

121.3

"cr

[ ·]

10

22

11

11

9

20

13

33

14

14

12

11

- interaktive axisymmetric yielding - non-axisymmetric buckling

ΔΗ„

[mm]

2.0

2.9

2.4

2.2

2.5

2.6

1.1

1.4

1.1

1.1

1.3

2.0

geometrically--materially nonlinear

C6MN ■" pi

[kN]

31.0

31.8

64.0

68.6

51.7

58.8

13.7

12.8

26.6

26.9

20.2

23.0

pGMN r lim

[kN]

38.7

39.1

77.2

81.1

63.5

71.4

16.3

14.2

30.3

31.2

23.3

26.0

ΔΗηηι

[mm]

0.6

0.4

0.6

0.7

0.5

0.3

0.5

0.4

0.5

0.5

0.4

0.4

co

H

È CD

Π m

Π)

3 Β.

Ό ►J πι c/5 m

η ö­Π) C/5 r­+

en

i? Π>

EL

Cu ¡3

Π>

D. o P3

CT) co

specimen no.

KK­X51

KK­V 51

ZKZ ­ XV 51

KZK­XX 51

KZK·VX 51

KZK­W51

test

Pu

[MPa]

0.050

0.059

0.014

0.027

0.042

0.041

failure

mode

n u

[ ­ ]

10

10

8

7

7

7

membrane

M Ρ pi

[MPa]

0.456

0.478

0.553

0.515

0.463

0.453

PM

cr

[MPa]

0.054

0.056

0.022

0.039

0.066

0.058

M " cr

[ ­ ]

12

12

8

11

12

12

comparative analyses

geometrically

nonlinear

GN

Ρ PI

[MPa]

0.095

0.131

0.250

0.233

0.215

0.214

PGN

cr

[MPa]

0.056

0.062

0.022

0.041

0.070

0.061

GN 1 ■ cr

11

11

11

8

10

11

11

geometrically­

­materially

nonlinear

_.GMN Ρ pi

[MPa]

0.080

0.125

0.185

0.170

0.215

0.209

GMN Ρ cr

[MPa]

0.056

0.062

0.022

0.041

0.070

0.061

GMN i ' cr

[­]

11

11

8

10

11

11

_GMN Ρ lim

[MPa]

0.098

0.176

0.205

0.389

0.242

0.236

Pu/pGMN

cr

[­]

0.89

0.95

0.64

0.66

0.60

0.67

H Ρ çr et

o co

>

g.

δ S3

Ο »­ι (Τ)

ë­ο

δ'

8 •β Τ3

Ο

Ο

tr

ο Ci­en

Ό Π) Ο

Οι

en

specimen no.

KK-Χ 10

KK-V 10

ZKZ-XV 10

KZK-XX 10

KZK-VX 10

KZK-VV 10

KK - X 50

KK · V 50

ZKZ - XV 50

KZK - XX 50

KZK - VX 50

KZK - VV 50

(I) RFA on MA level

( FMpl and FM

cr )

λ

(-1 0.495

0.530

0.531

0.525

0.525

0.509

0.751

0.750

0.780

0.740

0.744

0.744

α

[ · ] 0.517

0.513

0.517

0.520

0.514

0.523

0.342

0.343

0.331

0.343

0.345

0.344

κ

[■ ]

0.811

0.788

0.788

0.793

0.791

0.803

0.544

0.545

0.510

0.554

0.552

0.834

FK

[kNl

134.6

142.6

158.4

159.2

155.7

147.8

55.4

55.9

52.6

57.7

56.3

83.1

(II) RFA on MA/GNA level

(FM

plandFGN

cr)

λ

[■ ]

0.815

0.716

0.775

0.757

0.730

0.585

1.306

1.133

1.174

1.119

1.082

0.906

α

[■ ]

0.304

0.352

0.335

0.341

0.345

0.479

0.214

0.244

0.235

0.244

0.254

0.294

κ

[ · ]

0.451

0.581

0.518

0.538

0.564

0.742

-

-

-

-

-

-

FK

[kN]

74.8

105.1

104.0

108.0

110.9

136.5

12.8

19.4

17.6

20.3

22.1

35.7

(III) RFA on GNA level

(FGN

pl and FGN

cr )

λ

[ · ]

0.377

0.325

0.476

0.467

0.400

0.380

0.514

0.425

0.643

0.601

0.515

0.475

α

ί · ] 0.599

0.654

0.551

0.557

0.597

0.611

0.525

0.585

0.382

0.398

0.525

0.549

κ

[·1 0.896

0.930

0.830

0.837

0.883

0.896

0.801

0.866

0.658

0.698

0.800

0.830

FK

[kN]

31.8

34.8

63.1

63.7

52.2

69.5

12.7

12.5

20.4

21.0

18.4

22.7

rGN Γ pi

[kN]

35.5

37.4

76.0

76.2

59.1

77.6

15.8

14.4

31.0

30.1

23.0

27.4

rGMN ·" lim

[kN]

38.7

39.1

77.2

81.1

63.5

71.4

16.3

14.2

30.3

31.2

23.3

26.0

Fu

[kN]

52.0

45.4

93.0

97.2

77.7

87.9

19.4

17.4

31.2

30.0

28.2

26.9

H Ρ σ­

ο o

Π

> -α TSL δ' Ά ο Ο *"+5

Π! ο* c ο Γ+ κ- ■

Ο

O »ι

i­I Ο Ρ

ο cr Π)

Χ r­t­

EL Ό M Π) m en

η en

Ό Πι O

Β Π) 3 en

specimen no.

KK-Χ 51

KK-V51

ZKZ-XV 51

KZK-XX 51

KZK-VX 51

KZK-VV 51

(I) RFA on MA level

(pMplandpM

cr)

λ

[ · ]

2.91

2.92

5.01

3.63

2.65

2.79

EC

α

[ · ]

0.50

0.50

0.50

0.50

0.50

0.50

CS

Pk

[MPa]

0.027

0.028

0.011

0.020

0.033

0.028

D α

[■ ]

0.65

0.65

0.65

0.65

0.65

0.65

Ν

Pk

[MPa]

0.035

0.036

0.014

0.025

0.043

0.038

(II) RFA on MA/GNA level

/nM *nJn

GN 1

(P pi and ρ cr )

λ

[­1

2.86

2.80

5.01

3.54

2.57

2.73

ECCS

α pk

[­1

0.50

0.50

0.50

0.50

0.50

0.50

[MPa]

0.028

0.031

0.011

0.021

0.035

0.031

DIN

α pk

[■ ]

0.65

0.65

0.65

0.65

0.65

0.65

[MPa]

0.036

0.040

0.014

0.027

0.046

0.040

(III) RFA on GNA level

(ρ „, and ρ cr )

λ

[ · ]

1.30

1.45

3.37

2.38

1.75

1.87

ECCS

α pk

[ · ]

0.50

0.50

0.50

0.50

0.50

0.50

[MPa]

0.028

0.031

0.011

0.021

0.035

0.031

DIN

α pk

[ · ]

0.65

0.65

0.65

0.65

0.65

0.65

[MPa]

0.036

0.040

0.014

0.027

0.046

0.040

Pu

[MPa]

0.050

0.059

0.014

0.027

0.042

0.041

α)

measured diagram

R m

r: ^ \ ι arctan E

idealized | stress-strain- curve

V

0,5 1,0 1,5 ε ν k^t measured

diagram

E[%] —►

0,5 1,0 1,5

Figure Cl. Typical measured tension coupon diagrams and idealized stress-strain-curves for numerical comparative analyses

Ahlmm] , .

Figure C2. Typical measured compression coupons diagrams

161

Figure C3. Test specimens ZKZ-XV 10 and KZK-W 50 before testing

1.050

1.040

1.030

r 1020

1.010

1.000

— · — 0 o - meridian

- · — SO" - meridian

-A—180° · meridian

- * — 270° - meridian

— — average part C

- - average part Β

— - average pari A

" " " * average average [mm]

1.028 I

0 100 200 300 400 500 600 700 800 900 1000

axial coordinate [mm]

Figure C4. Typical results from thickness measurements

162

Figure C5. Specimen ... on the imperfection measuring device

specimen KK - Χ 10

E E

ι——ι

5 <

tn i -(Μ tn

T - r * N r s e i c o f o » * · o c N L o r - o c s L o r ^ o « - « - v - r - I M P J N N

circumferential angle [ °]

CM in co co co

Figure C6. Typical imperfection plot

163

2

►n σο

§

O

o

§ co

S. δ' Ρ" Ρ

I'S* cT Ρ­en co

tr tr CD O

ï= ft co fcj·

Β °? CT Ο

rt co

Π) co

O i­h

B­C l

o P­l ­ f Π) co

α

Ξ 11.

α χ υ

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

'Λ\/%

Μ Υ Χ

ƒ Λ χ

/ / / Ν \ /// V af ƒ Λ* *> /// Λ1· ν

[///■' \ 's·

Ι / ■' \ *

ι ν ■ \

Ι.// '­¡Ψ ■ iii ·

ΜΙ ■ IJl ■'

­ ff'

F.'

r Ι ι I I

Ν, \

Ν S

ν s X. s

X^ s

.s s V v­

N. **· ■

1 , , ,

Ν

s

•a

ι 1 1 1

«•>s_

" ^ , .

■a

" * " — ­ ÍS " ­ ­ ­

1 ι — π 1

­ · ­ . .

• Ma

* «aa

­

ZKZ­XV 10

KZK­XX 10

— ­ KZK­VX 10

K Z K ­ V V 1 0

ZKZ­XV 5 0

KZK ­XX 5 0

KZK ­ VX 5 0

KZK ­ VV 5 0

­ ­ ­ „

τ 1 1 ; 1 1 ; 1

6 8

ΔΗβ χ ρ

/ΔΗΜ

ρ Ι

10 12 14

u i

►η

TO

πι

π 00

ο Ρ

Π)

Er Ρ ­η Γ!

cr η

co tr (D μ—'

fS CO

CD

3 cr h­ ·

n> ro

O

ta Ρ r——· O Ρ CL. ι

(Λ Ρ

4

η

ft Π)

2. C3

0 0

ο

■ ^

cu co

O i­h

£ Π)

g iE. t — ■

o Ρ P . Γ­+­

re CO

LL

0.35

0.30

0.25

■EL 0.20

L

LL 0.15

0.10 ­

0.05

0.00

/

/

/ /

/ /

—ψ/7

Ψ Ψ lì

'Γ

ψ

* * f

—

**.

" " " ­

*»%

" χ .

> ν .

^ ­ ^

" " " " ^ ^ " " " " ■

ι τ — ι — ι — |

Ι Γ

KK­V10

ΚΚ­Χ10

­ ­ — ^

" 1 — ι — ι ι

ΚΚ ­ V 50

ΚΚ­Χ50

" ­ ­ ­

— ι — ι — ι — ι —

■ — , _ _

* * ■ — a a . _ ,

τ—ι 1 1—Ι

8 10 12

ΔΗβ χ ρ

/ΔΗΜ

ρ Ι

14 16 18 20

0\

y. era'

H η

η "tí i - l

Π) co co

co

HL ρ" o Hi

Π)

3 ι

ο rt co O ►-+> Ι­+­

ρ­rt rt Χ rt Ρ ,

■ρ

Π) co co

Π) co

puKKV51

= 0.059 [MPa]

LB (n) ­ local buckling with η buckles

. local buckling

4B» collapse

2 3

displacement ΔΗ [mm]

(a) (b)

Figure CIO. Axial load specimens KK-X 10 (a) and KZK-XX 50 (b) after testing

Figure C l l . External pressure specimen KK-V 51 during (a) and after (b) testing

167

0 0

TI era

rt

n ι — '

tf10

ρ ο co tr" Ρ Ί Α t i . co >— r t

3- BL ¡3­ co <T> K '

<δ °

g Ρ rt <f

O >

""*> sr co 5 rt £L o t r

g' co n 3 o

n

N

I

o et Ρ ¡3· ¡3 θ ' CL, Ρ

2 — o Ρ p .

co Ρ" Ό Ά

Β. Ρ ο Ρ . Ρ- Π) • Ρ 5'

era ι ο

Fl¡m ­ 83,2 [kN] by INCA (el. COEP), Lyon

F04 B08, Essen

Calculations on GMNA level

0.0 1.0

C1­C1 by F04 B08

­­•by HORUS

­χ­ ­ S1­S1 / C1­C1 by ABAQUS

­^r­by INCA (element COQUE)

+ ­ b y INCA (element COEP)

by INCA (el.COQUE), Lyon

80,3 [kN] by ABAQUS, Graz

79,8 [kN] by ABAQUS, Milano

77.0 [kN] by HORUS, Munchen

77,0 [kN] by BOSOR, Liverpool

Introduced stress­

strain­curve

fw­ 216 MPa

V 7

E = 205 10JMPa

2.0 3.0

ΔΗ in [mm]

4.0 5.0 6.0

benchmark test on milk can type shell

with geometry of ZKZ­XV 10

rt CO

meridonial stresses circumferential stresses effective stresses buckling mode

VO

ρ" ¡? ^ 05' S * w rt

ço UJ ¡t · rt H CO j rt Ό co - · S- EL ^ O II 2 !

htj > ^ H-,

S rt Ζ ryi

X £. r i . r *

ö CO

g* c?

E g Ρ Ρ*

era —* 3 o" o t

23

a ρ· α» %

« 8 era a

rt 3 Ρ rt

a f P­ 'S

^ <3 n χ

ζ · ·

E

ε, cu

­t—<

ro α o CJ

"ro Χ ro

inner surface

midsurface

outer surface

oszillating with

n = 9

in circumferential

direction

(d)

a)

X¡ ro o

X

CD

100

80

60

m 40

20

0

/ / 1/

í i

!

specimen KZK

experiment

M m

*

..ideally

with R,

\

Na,

/

­ X X 10

I I I

with R e s c =

·»

/

" *s

* ­ * " ­

Å plastic

ÌSjC = 218 [MPa]

I . I

■■ 218 [MPa]

• « „

/

­ ideal ly plastic

w i t h R e s t =215 [MPa]

l , ' i , ι... , 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

displacement ΔΗ [mm]

b)

■F. ­s«:

U­

•o ca o

ro

Χ ra

50

45

40

35

3U

25

20

15

10

f 1 hi­ll 1

.—y ι

1/ = ' /

1/ I ƒ

ì 7

*

«»»

ideall· with

specimen KK - V 1 0

experiment strain hardening

* - »

*«, «

«■

rZ v 1 plastic

"^Sr

■a

v*"* /

V t =1 7 1

>M P

W l

■" """S"

«*

;h Res>c = 197 [MPa]

»* **

* *

" — —.

~z

~5—=

y ideally plastic

a] u· "es.c

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

displacement ΔΗ [mm]

Figure C14. Typical GMNA results for axial load specimens

170

F „ - 712,3 [kN]

specimen ZKZ - XV 10

FLcr = 363 [kN]

Ό CO

_o 75 X (O

-GMN

0 1 2 3 4 5 6 7

displacement ΔΗ [mm]

Figure C15. Typical set of numerical results compared to experimental behaviour: Axial load specimen ZKZ-XV 10

171

ro

ο ­ρ­ό -~s ££ Ρ

era SK

ero' H ΠΙ

η ι » O N

Ο EL

S > rt co Ρ

se 6- co η

era 55 Ρ

g S EL

ρ-Ό H T J i» El "sT β "Ö z N '

Ό O Ζ

co

rt o Β η 3 co Ν

meridional stresses circumferential stresses effective stresses buckling mode 1000

inner surface

midsurface

outer surface

o ¿=.

■ inner surface

­midsurface

­ outer surface

­inner surface

­midsurface

• outer surface

k. (b) (c)

oszillating with

n = 8

in circumferential

direction

(d)

U l

ECSC Contract no.7210-SA/208

Enhancement of ECCS Design Recommendations and Development of Eurocode 3 Parts Related to Shell Buckling

Part D

THIN-WALLED SHELLS SUBJECTED TO WIND LOADING

Final Report

Ruhr-Universität Bochum Institut für Konstruktiven Ingenieurbau

Arbeitsgruppe Aerodynamik im Bauwesen Prof. Dr.-Ing. H.-J. Niemann

Dr.-Ing. M. Kasperski Dipl.-Phys. V. Görnandt

D 1 Aim of the Investigations and State of Knowledge

D 1.1 Background The research project is focused on steel containments of circular cylindrical shape with low aspect ratios (height/radius ratio : h/r = below or equal to 1). The wall thickness ratio (wall thickness to radius : t/r) is typically in the order of 1/3000. Overall dimensions are in the range of some meters up to 20 meters with wall thickness of a few millimetres. The shell structures are open at the top or equipped with one or more stiffening rings. Such buildings are used as tanks and silos for storing fluids or bulk goods in the industrial or agricultural field. In the partially filled or empty state there is little or no support from internal resistance of the contents and the shell is sensitive to external forces especially wind loads. In strong winds damages have occurred where shell buckling was identified as a dominant failure mode. On the other hand, theoretical investigations indicate that the shell can resist a further increase of the wind load after buckling has been initiated. The post-buckling resistance is particularly high if the upper edge of the shell is equipped with a stiff ring beam. For an economic and safe design it is worthwhile to include this additional loadbearing capacity. The main objective of the study is to investigate experimentally the principal features of the post-buckling behaviour of such structures.

D 1.2 State of Knowledge The design for buckling under radial compressive forces at present follows the guideline [D5], in which the design load effects are compared to the buckling stresses, eventually divided by an appropriate partial safety factor. Buckling is to be avoided. The wind load is approximated by an equivalent pressure uniformly distributed over the shell surface. This approach has been based on extensive experimental and analytical investiga­tions, see e.g. ref.s [D18], [D19], [D23], and [D34] to [D36]. One specific result is that for low aspect ratios (height to diameter below 0,75), the equivalent uniform pressure is equal to the local maximum of the wind pressure. For open tanks, the pressure is composed of the external and internal pressures which are strongly non-uniform along both the height and the circumference. The meridional non-uniformity is amplified in natural wind flow by the presence of a wind profile, whereas the refered tests were performed in uniform flows. However, as mentioned earlier the initiation of buckling is dominated by the maximum pressure occurring at the windward face of the cylinder in the region of half of its height, which justifies the use of the uniform pressure concept.

The natural wind flow is in addition time dependent due to its strong turbulence. Little is known about the buckling behaviour under fluctuating load. Its effect is accounted for by applying an equivalent gust velocity pressure averaging the gust wind speed over some seconds. It is open to question which averaging time is appropriate for the buckling problem. The post buckling behaviour has been investigated in some theoretical studies, see e.g. refs. [D41] to [D45]. Experimental verifications have not come to the attention of the authors.

175

D 1.3 General Lay-Out of the Present Investigation The present investigation aims at an experimental verification of the post-buckling behaviour in order to provide a better understanding of the phenomena and an information on the possible margins of safety inherent in the classical design procedure. It consists of two main parts and an addition. In the first part, the aerodynamic forces are investigated which produce buckling. The flow is as uniform as possible with a low degree of turbulence. In the second part, elastic models are used in identical flow conditions which buckle at low flow speeds and permit to observe the development of the buckling patterns over a large range of speeds beyond the critical. The model shapes have a low aspect ratio, their heights ranging from 0,5 to 1,0 of the shell radius. The flow field is therefore strongly three-dimensional with separations occurring both at the top and at the sides of the circumference. Probably, the top separation dominates the flow field, and the sensitivity to Reynolds number typical to rounded shapes is less pronounced. However, in view of the main focus it was sufficient to ensure a pressure distribution ideally independent of the flow velocity within the buckling and post-buckling range. This includes also independence of the deformations of the shell surface, once buckling has been initiated. The latter requirement could not be verified experimentally. In earlier tests on inflated semi-cylinders [D12] it could be shown, that even large deforma­tions of 20% of the radius did not change the pressures considerably. This result was considered as a sufficient indication that the effect of the buckling deformations may be neglected.

The additional test was aimed at the effect of load fluctuations on the critical load and the post-critical behaviour. In this case, the model was completely immersed in a simulated atmospheric boundary layer flow, non- uniform with regard to the mean flow and highly turbulent.

176

D 2 Experimental Procedures

D 2.1 Testing Set­Up and Flow Conditions

All experiments were performed in the Boundary Layer Wind Tunnel ­BLWT­ of the Arbeitsgruppe Aerodynamik im Bauwesen ­Affi­ at the Ruhr­Universität Bochum ­RUB­, which is a common Eiffel type (fig. D 2.1.1). It has a closed working section approximately 10 m long, 1.8 m wide and 1.6 high with an adjustable ceiling. The aerodynamic circuit is a open­return air flow through the testing room where the BLWT is situated.

. end dlffuaor a 2.23 / a 2.80

β 2.228

compensator ζ 2.23

transitional diffusor

laser-light intersection

faten with variable ceiling

1.80 X 1.60 - 1.S0

drive 110 KW^

nozzle

intake chamber with honeycomb stralgbtener.

2.80 χ 2.40

Fig. D 2.1.1 The Boundary Layer Wind Tunnel schematic view, dimensions in m

The principal testing area, the measurement chamber, is equipped with a turntable (0=1.7 m), supply adaptors for several kinds of sensors and the laser equipment. The fan is situated behind the working section (suction type) which, compared to the blower type, diminishes the disturbances in the flow induced by the fan. Maximum empty­tunnel flow speed is 30 m/s.

The boundary layer evolves naturally over the artificially roughened floor (cubes of different sizes as roughness elements) attaining a height of 0,4 m at the measurement section. It can be thickened up to a height of 1,2 m using Counihan's vortex generating method, with a floor barrier and spires at the end of the inlet nozzle. The geometric model scales thus defined are in the range of 1/1000 to 1/200. All roughness generating devices can be removed to produce a uniform flow with low turbulence.

Testing set­up (smooth, uniform flow experiments)

At the position of the measurement chamber a false­floor of 1.75 m length was installed (ref. to fig. D 2.1.2 and D 2.1.3), spanning through the full tunnel width. Upside of the plate the test cylinder (Pressure Model : PM... , Buckling Model BM...) was mounted. The other side of the base plate bears the equivalent counterpart, which is a variable height dummy cylinder for the rigid models. For the elastic models the respective rigid pressure model is used as

177

counterpart. The minimum distance between the axis of the cylinders and the front edge of the false floor was 0.90 m Ξ 2 D of the cylinder model. The respective value for the leeward side was 0.85m = l,9D. For the setup of the pressure measurements an optional extension piece of 1 m length for the back end of the false floor was used, moving the test assembly about this distance upstream from the midpoint of the turntable. For the buckling experiments this long version of the false floor was not used, since such model position is outside the operating range of the laser equipment.

This design was chosen to give a symmetrical flow and to avoid interferences between the

two sides of the plate.

400

Π \ ζ

V 5­©

1750 7615

,Λη stralghtener

and

honeycomb

Fig. D 2.1.2 Test assembly for the smooth, uniform flow experiments, cross schematic view of the wind tunnel, not to scale, dimensions in mm

ΑΑΑΑΑΑΑΑΑΑΛΛ^νΑΑΑΑ^ΑΑΑΑΑΑ^^ΑΑΑΑΑΑΑΑΑΑΑ^νΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΛΑΑ^

ceiling

I

false floor

test Wæms model j

•t

I rigid 1 dummy

455.00 mm

prandtl­tube

75.00 mm

­i \­3 ^

850.00 mm 900.00 mm

S

floor

ΙΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑΑ^,­.,Ν,>.,χ^ν,Λ,ν,χ,χ„Ν„

Fig. D2.1.3 Test assembly for the smooth, uniform flow experiments cross schematic view of the measurement chamber

The inevitable boundary layer of the false floor has a vertical extension of about 0.1 m at the position of the cylinder models (fig. D 2.1.4).

178

E E

500 ­

400 ­

300 ­

200 ­

100 ­

0,0

J 1,0

q / q„ 1

M SO

M 75

M 1 00

2,0 3,0

Fig. D2.1.4 Boundary layer above the false floor, normalised velocity pressure profile compared to the heights of the cylinder models

Testing set­up (turbulent flow experiments)

For the preliminary test under turbulent flow the boundary layer was modelled to natural wind conditions for rural terrain with an exponent of α = 0.18 for the power­law model of the mean velocity profile:

,oc

u ζ _

"ref VZref, u = mean velocity at height z,

The complete set of the characteristic properties for this flow is described in [D8]. This single test just led to a first „qualitative" insight into the post­buckling behaviour under turbulent flow condition.

The test model BM 75 (2) was mounted on the turntable in the ground floor of the wind tunnel. For the magnification factor values for the mean velocity for sample heights ref. to tab. D.2.1.1.

ζ / m m

z / h

ü z / ü r e f

5

0,03

0,46

20

0,12

0,54

60

0,35

0,59

90

0,53

0,63

120

0,70

0,65

180

1,05

0,72

Tab. D 2.1.1 Magnification factors for the mean velocity for sample heights

The effect of blockage

The wind tunnel boundaries impose constraints on the flow around the model, so that the measured parameters differ from the free flow values. In a first order approximation the influence depends on the geometrical blockage ratio ε :

S 8 =

X ' S = model area cross section, A = area of the wind tunnel cross section.

For the actual test setup (with the twin cylinder arrangement mounted on the false floor and the single cylinder arrangement installed on the turntable at the wind tunnel floor) the resulting ε are given in tab. D 2.1.2. For a blockage ratio ε below 1% effects can be neglected, an ε in the region of 5% is acceptable in wind tunnel experiments. For an ε of over 10% appreciable corrections to the measured results are required in order to extrapolate to unconstrained flow conditions. In the experiments, the range of blockage ratios was 2% ­ 7%,

179

and no correction was applied for the reasons discussed in section D 1.3. Some correction would be needed to apply the pressure distributions to full­scale conditions.

wind tunnel

PM100

PM75

PM50

BM100

BM75

BM50

BM 75 (turbulent)

D / m

1,8

0,457

0,457

0,457

0,445

0,445

0,445

0,445

h / m

1,6

0,228

0,172

0,114

0,228

0,171j

0,114

0,171

Ae f f /m^

2,88

0,208

0,157

0,104

0,203

0,152

0,101

0,076

ε

0,072

0,055

0,036

0,070

0,053

0,034

0,026

Tab. D2.1.2 Blockage ratio for the test models

The effect of Reynolds number

The flow field depends on the Reynolds number Re defined as:

Re = L· — , L = characteristic length, ïï = mean velocity, ν

ν = viscosity of the fluid (=1.5· 10 " m/s for air at room temperature).

The range of Re varied by a factor of 25 ranging from 15.000 to 380.000 when refered to the height, see tab. D 2.1.3. These values are in the range of Re­independence for sharp­edged bodies with a fixed separation, which applies here for the top side separation. This was confirmed by a preliminary test in which Re was varied in the test range.

model

BM 50 ­h­

BM 75 ­h­

BM 100 ­h­

BM 50 ­D­

BM 75 ­D­

BM 100 ­D­

PM 50 ­h­

PM 75 ­h­

PM 100 ­h­

PM (all) ­D­

L /m

0,114

0,171

0,228

0,455

0,455

0,455

0,114

0,171

0,228

qmin / P a

10

5

3

10

5

3

0,457

qmax / P a

45

25

15

45

25

15

Vmin / ms"1

4,0

2,8

2,2

4,0

2,8

2,2

2

2

2

2

Vmax / ms"1

8,5

6,3

4,9

8,5

6,3

4,9

25

25

25

25

Remin/104

3,0

3,2

3,3

12,1

8,6

6,6

1,5

2,3

3,0

6,1

Remax/104

6,4

7,2

7,4

25,7

19,2

14,9

19,0

28,5

38,0

76,2

Tab. D 2.1.3 Reynolds number for the experimental parameter range ­h­, ­D­ denotes height, respective diameter of the model

D 2.2 Pressure Test Models

The main body of the pressure test models (fig. D 2.2.1) consists of a prefabricated plexiglas (acryle) tube (thickness 8 mm) which was screw fixed with a ground plate of the same material (thickness 5 mm). The overall diameter was 457 mm.

180

A (detail) : pressure tap

tube

8 mm

• ♦i κ— 0.86 mm ■H w­ 1.27 mm

model PM 100

h/r = 1.0

«— t = 8 mm

t = 6 mm

D = 457 mm

model PM 75

h/r = 0.75

t = 8 mm

t = 6 mm

J Î

D = 457 mm

model PM 50

h/r = 0.50

Fig. D 2.2.1 Pressure Models, PM 100, PM 75 and PM 50

The pressure taps were arranged along one meridian at seven levels listed in table D 2.2.1. At each level, both inside and outside pressure borings were drilled. The tubes were installed along the meridian inside the wall in order to avoid any flow disturbance, see fig. D 2.2.1. Electrical pressure transducers were used monitoring the pressure time histories simultane­ously from the 12 pressure holes. The pneumatic connection from the taps to the pressure transducers used a particular structure of optimised tubes composed of two sections :

First stage : length = 490 mm, 0 ^ = 0.86 mm

Second stage : length =115 mm, 0 ^ = 0.5 mm.

181

model

PM 50

PM 75

PM 100

h /mm

114

171

228

z / h 0,15

17,1

25,7

25,7

0,30

34,2

51,3

51,3

0,50

57,0

85,5

85,5

0,65

74,1

111,2

111,2

0,80

91,2

136,8

136,8

0,90

102,6

153,9

153,9

0,97

110,6

165,9

165,9

Tab. D 2.2.1 Elevation heights of the pressure taps for the pressure models

­in mm from ground­

D 2.3 Buckling Test Models

The buckling test models (fig. D 2.3.1) are of modular design with a circular base plate

(aluminium) of 455,0 mm diameter and 10,0 mm thickness and an upper stiffening ring

(aluminium) of 455,0 mm external diameter, 5 mm width and 2 mm thickness.

BUCKLING MODELS

t =0.1 mm

t =10.0 mm

Mw**t*HœSBB

D = 455.0 mm

\

A (detail) : ring stiffner >·■■·""' 5 mm

w w &666­>>>>3S

aluminium

2 mm

t = 0 . 1 mm

model BM 100

h/r = 1.0

aluminium

t = 0.1 mm

t =10.0 mm

1 D = 455.0 mm

f i ve

e re­

model BM 75

h/r = 0.75

t = 0.1 mm

t = 10.0 mm

D = 455.0 mm

ΙΛ

r­

model BM 50

h/r = 0.50

Fig. D2.3.1 Buckling Models, BM 100, BM 75 and BM 50

182

The wall of the cylinders is made from an elastic Mylar foil of 0,1 mm thickness which is cut

to the appropriate height of the buckling models. The foil is glued to the plate and the ring.

Similarly the closing of the circle is done by gluing the overlapping parts (with an approxi­

mate overlapping length of 10 mm), this overlapping part being always positioned at the

leeward side of the model.

The weight of the ring introduces a constant meridional compressive stress into the shell,

which may have an effect on the buckling. The influence is calculated with a density of

aluminium of 27 kN/m :

• ,· i G hi>

meridional stress σ7 = = γ_ · ­ ^ · r · z Ut ÏR t

ri . „ η l f t3XT__­2 1—ï­ = 2.70­10JNm~

(G : Force of Gravity, U : periphery, yR : density, t : thickness of shell= 0.1 mm, hR : height of ring= 2mm, r : outer radius = 227.5 mm, r¡ : inner radius = 222.5 mm)

critical stress [D 34] o z c r i t = 0.605·Es·f­)■ , = 1.26· 105Nm"2

v +ioo­t

(Es : Young's modulus of the shell = 2.3 ­109 N/m

2)

The acting stress is thus only 2% of the critical stress and its effect on buckling can be

neglected.

The rigidity of the ring compared to the shell is expressed by the stiffness parameten χ :

ER­IZ­Í­V?) stiffness parameter χ = ­¡ ¿­

Es­t4 · (ER : Young's modulus of the aluminium = 7­10 N/m , Iz : moment of Inertia)

„. _ l iR­(r­r¡) moment of inertia IR = K 12

(vs : Poisson's ratio of the shell)

The stiffness parameter is therefore 5.7 ­10 , which is in practice an unlimited rigid ring, leading to fixed boundary condition at the upper edge of the shell.

D 2.4 Testing Procedures

D 2.4.1 Pressure Measurements

The monitored pressure time histories were after amplification processed to the computer

system for evaluation of the mean and rms values. Only the mean pressures are considered for further evaluations. The pressures are refered to the mean velocity pressure measured at the

Prandtl tube to give pressure coefficients.

183

D 2.4.2 Buckling Tests The aim of the buckling tests was to evaluate the shape of the initial buckling patterns and its development with increasing load, to measure quantitatively the depth and width of the buckles, and to determine the wind load, at which eventually the collapse of the model occurs. A square grid of 20 mm was drawn on the model surface, which allowed to sketch the buckling patterns by visual inspection with reasonable accuracy. This was possible since the buckling pattern at a constant load intensity was very stable and, for one and the same model also reproducible. The border picture could not be recorded completely on photos. For the fixed viewing angle of the camera only a small area on the surface provide the right illumina­tion conditions for the borders to become visible. The buckling deformations were visualised using a horizontal laser light sheet, positioned subsequently at several elevations within the model height. The illuminated deformed shell wall was clearly visible in its main parts and could be photographed from above the wind tunnel ceiling through a transparent window from a position right over the cylinder axis of the models. An Argon-Ion laser with a power of 100 mW was used. The laser light sheet optics consists of a scanning (galvanometer type) mirror working at a frequency of 200 Hz. The recorded pictures (standard KODAK black and white 35 mm film material, 400 ASA, t=l/10s at F=2.8) were digitised using a Microtec slide scanner with an optical resolution of 1870 pixel/inch. The resulting output were 256 step grey-level images with 1021*1021 respectively 1201*1201 pixel in TIF format. The geometrical resolution of the raw images, as defined by the later evaluated circle radius in the bitmap system (in pixel), compared to the real radius of the cylinder models, ranged from 2.13 pixel/mm (BM 50, z/h = 0.27) to 1.49 pixel/mm (BM 100 , z/h = 0.8). A standard­ised value of 2 pixel/mm as the final working resolution was used after transformation. The corresponding angular resolution was 10 pixel/0, which is equivalent to 2,5 pixel/mm at radius position. The raw TIF-files were transformed to a Bit-mapped format BMP, using a program for exchanging graphic file formats (alchemy version 1.5.1). A simple FORTRAN program provide the transformation step from this BMP format to the IMG working format, which is a low level implementation of the FITS format, widely used for graphic data exchange. For the complete set of the file formats specifications ref. to [D47]. Further image processing steps used a program library for co-ordinate transformation and picture processing, developed for astronomical purposes [D49], with some adaptation to the present problem. The evaluation of the deformation profile proceeded in the following steps : 1. Find the centre of the circle 2. Find the radius of the circle 3. Polar co-ordinate transform (circle to line) 4. Find the shape defining pattern and its position Since the position defining marks on the turntable were found not to be precise enough to fix the absolute position with the appropriate resolution, an internal relative reference system was used. For finding the centre of the circle and its radius, the picture was scanned in two

184

orthogonal directions and the positions of the found pixels were fitted to a circle (two stage Hough transform [D50], for position and radius). Even for images with a strongly deformed circle this approach provided good results.

Using the obtained values, a co­ordinate transform was applied, to treat the polar (r,õ) description of the circle as an orthogonal axis system for the further processing steps [D46]. For generating required grey level values for resampling in transformation steps, a bilinear interpolation was used [D48]. Each pixel line, representing one angular co­ordinate step, was scanned to find the position of the profile defining pixels.

To distinguish shell defining pixel from background and from single noise pixels a three level threshold system was applied. Pixel candidates must match : 1. the typical grey level interval of the illuminated shell, 2. the typical linear thickness interval of the illumination, 3. the typical geometrical position of the deformation profile. In a calibration stage these parameters were adjusted using sample images.

The output of the automatic processing was usually a 80% ­ 90% covering of the full circle.

D 2.5 Testing Programme

The pressure tests started with an investigation of the Re­number effect. The model PM 50 was used and the velocity range was varied from 1,8 m/s to 25 m/s. Since the pressure distributions remained rather insensitive to the variation of the wind velocity, the main tests were performed at a velocity of 25 m/s with the models PM 50, PM 75, and PM 100.

The buckling tests in uniform flow were performed subsequently, individually varying the velocity according to the development of the buckles in the following steps (tab. D 2.5.1) :

(C CL

ω σ> as

­·—■

<ο

co ο

ΒΜ100

light

sheet

0

3

6

9

12

15

buckling

pattern

9,5

11,5

15,5

Β

light

sheet

5

10

15

20

25

M 75

buckling

pattern

15

20

25

Β

light

sheet

10

15

20

25

30

35

40

M 50

buckling

pattern

20

35

48

Tab. D 2.5.1 Test programme : load stages

185

Small deviations of the actual from the nominal velocity pressures occurred. The BM 75 test was repeated with a new model BM 75 (2) fabricated in the same manner in order to control the reproducibility of the buckling patterns. For the laser light sheet levels see tab. D 2.5.2 :

h/mm

BM100

227,50

ζ / m m

183

137 114 92

45

z / h

0,80

0,60 0,50 0,40

0,20

BM75

170,63

z / mm

126

86

43

z / h

0,74

0,50

0,25

BM50

113,75

ζ / m m

84

59

31

z / h

0,74

0,52

0,27

Tab. D 2.5.2 Test programme : Laser light sheet levels In the third campaign, the buckling model BM 75 (2) was placed on the base of the wind tunnel and tested under turbulent flow conditions, equivalent to natural wind. In this case, the buckling pattern only could be monitored due to the instationary nature of the buckling process.

D 3 Test Results

D 3.1 Pressure Distribution The pressure measurements in the velocity range from 2 to 25 m/s showed no major changes in the pressure distribution around the cylinders. It can be stated that the flow field is insensitive to the effect of Reynolds number in the velocity range required for the buckling tests. The following pressure measurements were then made at high speed (25 m/s) to get high useful signals for optimal signal to noise ratio. Fig. D 3.1.1 shows as an example for model PM 100 the pressure distributions for the outside wall (a), the inside wall (b). The difference of (a) and (b) which corresponds to the time averaged wind-load shows Fig. D 3.1.2a. The external pressure distribution is typical to circular cylinders with positive pressures at the stagnation line decreasing gradually further downstream. At an angle of 37° it crosses the zero level, the minimum pressure occurs at the flanks and is followed by a pressure increase until separation takes place. Due to the 3-D nature of the flow field, the point of separation shifts from 110° at the base to 130° at the top, the position of the minimum pressure similarly from 80° to 90°. The pressure at the stagnation line is disturbed in the upper third of the height. The pressure coefficients on the internal face of the wall are distributed more uniformly although not fairly constant as might have been expected: they are negative everywhere with maxima at 0° and 180° and a minimum at 100°. The resulting net pressure coefficients show that the positive sign, i.e. compressive load prevails over most of the shell surface, only a small area at the flanks is subjected to negative pressure.

186

PM 100

160 angle /

PM 100

160

Fig. D 3.1.1 Mean Pressure coefficients of Model PM 100 a) External wall surface b) Internal wall surface

The results for the other models are principally the same. Fig. D. 3.1.2 presents the complete set of the net pressure distributions for all three models. Tab. D. 3.1.1 to Tab. D. 3.1.3 contain the complete set of pressure coefficients. The results are rather similar to the pressure distributions measured by Holroyd [D24].

187

Fig. D 3.1.2 Combined Pressure distribution a) Model PM 100 b) Model PM 75 c) Model PM 50

188

PM 100

External

Internal

Combined

z/h angle / °

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0,15

0,977 0,934 0,692 0,333

-0,121 -0,614 -1,085 -1,412 -1,518 -1,379 -0,652 -0,431 -0,365 -0,388 -0,423 -0,471 -0,460 -0,368 -0,308

-0,760 -0,743 -0,725 -0,760 -0,825 -0,845 -0,876 -0,906 -0,939 -0,959 -0,974 -0,976 -0,963 -0,932 -0,880 -0,795 -0,677 -0,576 -0,514

1,737 1,677 1,417 1,093 0,704 0,231

-0,209 -0,506 -0,579 -0,420 0,322 0,545 0,598 0,544 0,457 0,324 0,217 0,208 0,206

0,30

0,999 0,950 0,711 0,360

-0,120 -0,618 -1,103 -1,446 -1,589 -1,445 -1,052 -0,423 -0,366 -0,386 -0,453 -0,484 -0,459 -0,365 -0,316

-0,758 -0,729 -0,716 -0,764 -0,862 -0,898 -0,930 -0,957 -0,991 -1,005 -1,021 -1,006 -1,001 -0,965 -0,916 -0,858 -0,739 -0,647 -0,607

1,757 1,679 1,427 1,124 0,742 0,280

-0,173 -0,489 -0,598 -0,440 -0,031 0,583 0,635 0,579 0,463 0,374 0,280 0,282 0,291

0,50

1,032 0,961 0,725 0,379

-0,086 -0,585 -1,064 -1,421 -1,597 -1,531 -1,360 -0,635 -0,360 -0,406 -0,461 -0,498 -0,467 -0,387 -0,330

-0,831 -0,851 -0,866 -0,901 -0,958 -0,967 -0,989 -1,025 -1,062 -1,087 -1,092 -1,068 -1,037

. -1,011 -0,993 -0,954 -0,869 -0,775 -0,732

1,863 1,812 1,591 1,280 0,872 0,382

-0,075 -0,396 -0,535 -0,444 -0,268 0,433 0,677 0,605 0,532 0,456 0,402 0,388 0,402

0,65

1,033 0,946 0,710 0,383

-0,064 -0,546 -1,013 -1,367 -1,558 -1,560 -1,352 -0,837 -0,418 -0,380 -0,440 -0,475 -0,441 -0,373 -0,312

-0,930 -0,930 -0,934 -0,965 -1,008 -1,008 -1,047 -1,103 -1,158 -1,181 -1,182 -1,152 -1,110 -1,079 -1,051 -1,010 -0,964 -0,874 -0,829

1,963 1,876 1,644 1,348 0,944 0,462 0,034

-0,264 -0,400 -0,379 -0,170 0,315 0,692 0,699 0,611 0,535 0,523 0,501 0,517

0,80

0,779 0,765 0,588 0,271

-0,140 -0,578 -1,011 -1,352 -1,563 -1,587 -1,439 -1,009 -0,548 -0,424 -0,466 -0,516 -0,530 -0,416 -0,345

-0,929 -0,923 -0,927 -0,955 -0,989 -1,001 -1,061 -1,135 -1,203 -1,238 -1,245 -1,209 -1,156 -1,099 -1,039 -0,979 -0,947 -0,886 -0,854

1,708 1,688 1,515 1,226 0,849 0,423 0,050

-0,217 -0,360 -0,349 -0,194 0,200 0,608 0,675 0,573 0,463 0,417 0,470 0,509

0,90

0,667 0,633 0,465 0,194

-0,168 -0,553 -0,937 -1,239 -1,435 -1,492 -1,372 -1,051 -0,654 -0,424 -0,406 -0,447 -0,450 -0,357 -0,303

-0,843 -0,856 -0,873 -0,896 -0,932 -0,957 -1,029 -1,113 -1,192 -1,242 -1,255 -1,217 -1,155 -1,079 -0,994 -0,914 -0,870 -0,852 -0,846

1,51 1,489 1,338 1,090 0,764 0,404 0,092

-0,126 -0,243 -0,250 -0,117 0,166 0,501 0,655 0,588 0,467 0,420 0,495 0,543

0,97

0,227 0,232 0,129

-0,082 -0,352 -0,635 -0,926 -1,161 -1,343 -1,415 -1,362 -1,102 -0,747 -0,501 -0,431 -0,463 -0,507 -0,406 -0,356

-0,814 -0,819 -0,832 -0,861 -0,900 -0,937 -1,016 -1,109 -1,201 -1,266 -1,282 -1,242 -1,218 -1,209 -1,120 -0,996 -0,932 -0,969 -0,984

1,041 1,051 0,961 0,779 0,548 0,302 0,090

-0,052 -0,142 -0,149 -0,080 0,140 0,471 0,708 0,689 0,533 0,425 0,563 0,628

Tab. D.3.1.1 Pressure coefficients of the mean pressure referred to the Prantl tube velocity pressure for PM 100

189

PM 75

External

Internal

Combined

z/h angle / °

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0,15

0,944 0,900 0,709 0,380

-0,030 -0,463 -0,876 -1,183 -1,345 -1,279 -0,777 -0,330 -0,324 -0,281 -0,288 -0,293 -0,276 -0,214 -0,176

-0,709 -0,691 -0,704 -0,712 -0,707 -0,718 -0,731 -0,758 -0,781 -0,797 -0,801 -0,808 -0,802 -0,776 -0,713 -0,628 -0,526 -0,457 -0,408

1,653 1,591 1,413 1,092 0,677 0,255

-0,145 -0,425 -0,564 -0,482 0,024 0,478 0,478 0,495 0,425 0,335 0,250 0,243 0,232

0,30

0,980 0,931 0,734 0,401

-0,025 -0,471 -0,896 -1,223 -1,392 -1,319 -0,969 -0,339 -0,290 -0,268 -0,286 -0,302 -0,302 -0,277 -0,254

-0,679 -0,688 -0,705 -0,698 -0,710 -0,737 -0,756 -0,796 -0,827 -0,850 -0,849 -0,854 -0,856 -0,827 -0,771 -0,677 -0,575 -0,516 -0,479

1,659 1,619 1,439 1,099 0,685 0,266

-0,140 -0,427 -0,565 -0,469 -0,120 0,515 0,566 0,559 0,485 0,375 0,273 0,239 0,225

0,50

1,020 0,970 0,779 0,447 0,027

-0,417 -0,853 -1,192 -1,374 -1,326 -1,183 -0,473 -0,240 -0,239 -0,264 -0,285 -0,286 -0,278 -0,261

-0,677 -0,673 -0,689 -0,700 -0,721 -0,749 -0,779 -0,816 -0,843 -0,851 -0,857 -0,855 -0,852 -0,852 -0,812 -0,735 -0,643 -0,571 -0,535

1,697 1,643 1,468 1,147 0,748 0,332

-0,074 -0,376 -0,531 -0,475 -0,326 0,382 0,612 0,613 0,548 0,450 0,357 0,293 0,274

0,65

1,002 0,954 0,765 0,444 0,041

-0,381 -0,805 -1,137 -1,321 -1,302 -1,174 -0,602 -0,226 -0,227 -0,251 -0,273 -0,273 -0,257 -0,232

-0,729 -0,726 -0,741 -0,755 -0,777 -0,806 -0,833 -0,878 -0,920 -0,938 -0,934 -0,939 -0,942 -0,932 -0,878 -0,802 -0,726 -0,659 -0,624

1,731 1,680 1,506 1,199 0,818 0,425 0,028

-0,259 -0,401 -0,364 -0,240 0,337 0,716 0,705 0,627 0,529 0,453 0,402 0,392

0,80

0,834 0,779 0,616 0,328

-0,036 -0,438 -0,813 -1,123 -1,328 -1,387 -1,253 -0,873 -0,463 -0,336 -0,336 -0,354 -0,364 -0,341 -0,311

-0,738 -0,730 -0,744 -0,759 -0,777 -0,807 -0,843 -0,900 -0,954 -0,981 -0,981 -0,987 -0,995 -0,976 -0,902 -0,818 -0,742 -0,686 -0,656

1,572 1,509 1,360 1,087 0,741 0,369 0,030

-0,223 -0,374 -0,406 -0,272 0,114 0,532 0,640 0,566 0,464 0,378 0,345 0,345

0,90

0,690 0,650 0,502 0,248

-0,070 -0,415 -0,749 -1,024 -1,211 -1,269 -1,186 -0,922 -0,557 -0,357 -0,325 -0,338 -0,347 -0,324 -0,287

-0,707 -0,693 -0,709 -0,724 -0,743 -0,769 -0,817 -0,885 -0,941 -0,973 -0,987 -0,986 -0,984 -0,988 -0,932 -0,834 -0,735 -0,683 -0,658

1,397 1,343 1,211 0,972 0,673 0,354 0,068

-0,139 -0,270 -0,296 -0,199 0,064 0,427 0,631 0,607 0,496 0,388 0,359 0,371

0,97

0,295 0,259 0,150

-0,036 -0,269 -0,530 -0,764 -0,981 -1,161 -1,302 -1,285 -1,037 -0,688 -0,468 -0,402 -0,406 -0,416 -0,381 -0,344

-0,693 -0,683 -0,698 -0,713 -0,734 -0,765 -0,817 -0,896 -0,976 -1,044 -1,062 -1,031 -1,036 -1,107 -1,085 -0,994 -0,872 -0,812 -0,772

0,988 0,942 0,848 0,677 0,465 0,235 0,053

-0,085 -0,185 -0,258 -0,223 -0,006 0,348 0,639 0,683 0,588 0,456 0,431 0,428

Tab. D.3.1.2 Pressure coefficients of the mean pressure referred to the Prantl tube velocity pressure for PM 75

190

PM 50

External

Internal

Combined

z/h angle / °

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180

0,15

0,858 0,822 0,675 0,423 0,076

-0,280 -0,623 -0,889 -1,037 -1,046 -0,856 -0,347 -0,190 -0,184 -0,146 -0,161 -0,115 -0,072 -0,033

-0,728 -0,741 -0,768 -0,793 -0,815 -0,796 -0,790 -0,796 -0,802 -0,814 -0,827 -0,881 -0,871 -0,802 -0,699 -0,591 -0,518 -0,483 -0,466

1,586 1,563 1,443 1,216 0,891 0,516 0,167

-0,093 -0,235 -0,232 -0,029 0,534 0,681 0,618 0,553 0,430 0,403 0,411 0,433

0,30

0,849 0,813 0,659 0,393 0,053

-0,308 -0,647 -0,924 -1,081 -1,087 -0,903 -0,337 -0,197 -0,168 -0,154 -0,156 -0,133 -0,101 -0,079

-0,581 -0,597 -0,625 -0,658 -0,709 -0,701 -0,688 -0,688 -0,701 -0,722 -0,754 -0,815 -0,797 -0,697 -0,559 -0,439 -0,341 -0,288 -0,277

1,430 1,410 1,284 1,051 0,762 0,393 0,041

-0,236 -0,380 -0,365 -0,149 0,478 0,600 0,529 0,405 0,283 0,208 0,187 0,198

0,50

0,994 0,958 0,793 0,514 0,146

-0,238 -0,605 -0,909 -1,083 -1,088 -0,941 -0,399 -0,157 -0,111 -0,123 -0,134 -0,120 -0,105 -0,101

-0,646 -0,658 -0,691 -0,727 -0,756 -0,751 -0,741 -0,755 -0,774 -0,803 -0,838 -0,875 -0,859

. -0,795 -0,706 -0,621 -0,562 -0,538 -0,525

1,640 1,616 1,484 1,241 0,902 0,513 0,136

-0,154 -0,309 -0,285 -0,103 0,476 0,702 0,684 0,583 0,487 0,442 0,433 0,424

0,65

0,712 0,680 0,532 0,294

-0,011 -0,327 -0,633 -0,892 -1,042 -1,065 -0,951 -0,573 -0,312 -0,255 -0,261 -0,270 -0,257 -0,245 -0,252

-0,498 -0,526 -0,571 -0,619 -0,662 -0,659 -0,650 -0,675 -0,712 -0,759 -0,801 -0,845 -0,824 -0,742 -0,637 -0,536 -0,459 -0,404 -0,377

1,210 1,206 1,103 0,913 0,651 0,332 0,017

-0,217 -0,330 -0,306 -0,150 0,272 0,512 0,487 0,376 0,266 0,202 0,159 0,125

0,80

0,692 0,650 0,516 0,291

-0,020 -0,340 -0,643 -0,889 -1,050 -1,095 -0,993 -0,716 -0,404 -0,266 -0,257 -0,259 -0,238 -0,220 -0,220

-0,523 -0,547 -0,586 -0,630 -0,668 -0,665 -0,656 -0,684 -0,731 -0,784 -0,819 -0,857 -0,836 -0,772 -0,671 -0,551 -0,466 -0,426 -0,406

1,215 1,197 1,102 0,921 0,648 0,325 0,013

-0,205 -0,319 -0,311 -0,174 0,141 0,432 0,506 0,414 0,292 0,228 0,206 0,186

0,90

0,730 0,695 0,561 0,341 0,049

-0,248 -0,532 -0,770 -0,930 -0,990 -0,940 -0,697 -0,369 -0,206 -0,187 -0,190 -0,159 -0,122 -0,120

-0,507 -0,527 -0,571 -0,617 -0,651 -0,645 -0,636 -0,668 -0,720 -0,781 -0,810 -0,839 -0,840 -0,815 -0,713 -0,556 -0,444 -0,411 -0,398

1,237 1,222 1,132 0,958 0,700 0,397 0,104

-0,102 -0,210 -0,209 -0,130 0,142 0,471 0,609 0,526 0,366 0,285 0,289 0,278

0,97

0,304 0,267 0,166 0,007

-0,210 -0,421 -0,609 -0,775 -0,930 -1,067 -1,049 -0,801 -0,478 -0,329 -0,302 -0,293 -0,252 -0,206 -0,192

-0,500 -0,523 -0,567 -0,613 -0,649 -0,646 -0,640 -0,678 -0,747 -0,829 -0,849 -0,865 -0,917 -0,910 -0,829 -0,643 -0,488 -0,467 -0,454

0,804 0,790 0,733 0,620 0,439 0,225 0,031

-0,097 -0,183 -0,238 -0,200 0,064 0,439 0,581 0,527 0,350 0,236 0,261 0,262

Tab. D.3.1.3 Pressure coefficients of the mean pressure referred to the Prantl tube velocity pressure for PM 75

191

D 3.2 Buckling Patterns The buckling pattern on the surface of the cylinder models have a strong 3-dimensional structure. Fig. D 3.2.1 shows perspective view of a buckled model developed from a mapping of the buckling pattern (BM 75 at highest load intensity).

Fig. D 3.2.1 3-dimensional perspective view of a buckling pattern

The evolution of the buckling pattern with increasing load intensity proceeded in several, well defined steps, which were specific to each of the models. After unloading and repetition of the loading process these steps were reproducible in every detail. The following ranges of buckling reaction could be identified:

1. No visible reaction of the membrane;

2. Small, non stable movements of the surface;

3. Initial buckling starts;

4. Evolution of the buckles from the initial buckling pattern to the upper and lower edges of the cylinder, appearance of the next modes of primary buckles and secondary distortions in addition to the primary buckles.

The onset of the second level is not sharply defined.

The first visible buckling pattern on the surface (fig. D 3.2.2 and fig. D 3.2.3a) of all cylinder models shows a clear and stable pattern of primary buckles.

192

Β M 1 00 q = 9 .5 Pa

2 2

0

— "

­

_^_ _

­ I . . . I I . _

" \ — ""

­ mmmm

­

?Λ V\

^£^3 • s

^—

—

■"—"» Λ «—

• S s/

— — J ^ ­— —f —

. ^ _

— · — ■

^^_

!0 ­ 1 6 ­ 1 2 ­8 ­4 0 4

c m

1 2 1 6 2 0 2 4

E o

BM 75

1fi

14

19

m η

R

Λ

9

0

_ _

_ _ .

KJ"

/K / \

f-\ \

— ï /

L__ / V 7 ν /y

/ V Λ ' \ >

/ \

AN \j V

q = 15Pa

r~~ ­ — — ■

­24 ­20 ­16 ­12 ­ 8 ­ 4 0 4

cm

8 12 16 20 24

BM 50

q = 20 Pa

0

D

3

Λ

η

0

/ ' ν > ( ¿- \ Γ­ ­|Λ —|γ—

νν _ —

ε ο

­24 ­ 2 0 ­ 1 6 ­ 1 2 ­ 8 ­4 0 4 8 12 16 20 24

cm

Fig. D 3.2.2 Initial buckling pattern a) BM 100, b)BM 75, c)BM 50,

as seen from inside of the cylinder; the 0 meridian represents the stagnation line.

193

The further development of the buckling pattern is shown in fig. D 3.2.3. With increasing load, the buckles grow in area and in depth. Near the stagnation line all primary buckles tend to extend to the edges of the cylinder. Although the buckling pattern could always be reproduced for each individual model, different patterns developed when the test was repeated with a second, nominally identical model fabricated separately. As an example consider the differences from the model BM 75 and BM 75 (2) in fig D 3.2.2b and fig D 3.2.3a. The quantitative deformations obtained from the light sheets images are presented in fig. D 3.2.4 and fig. D 3.2.5 in terms of deformation profiles for the complete load range of BM 75 at z/h = 0.5. Buckling pattern from the buckling patterns map can be related directly to the corresponding deformation profiles. The peak to peak amplitude of the visible initial buckling pattern was always identified as 4 - 5 mm, see for example fig. D 3.2.2b and fig. D 3.2.5a. Contrarily buckling could be detected in the deformation profiles at smaller intensities due to the higher resolution, see fig. D 3.2.4c as an example. Moreover the deformation profiles show that at higher load intensities buckles occur even at the leeward side of the cylinder, see fig. D 3.2.5b and fig. D 3.2.5c. This observation corresponds to the fact that the pressure distribution provide in this range a net compressive load. The wave widths identified from the deformation profiles can be compared to theoretical values of the wave number, i%, which for the discussion in section D 4 are calculated from [D34]:

m * = 2 J 4 TÍ' i t ' where c accounts for the boundary conditions of the shell, and a value of c=1.5 is applied for both ends fixed.

194

Β M 75(2)

q = 1 4.5 Pa

B M 75(2) q = 20.0 P¡

ε

1 6

1 4

1 2

1 0

8

6

4

2

0

/

f

A ' \

\

/

/

f

\

\

\

/ \

\

\

ì

J /

/

y

\

\

\

\

\

/

/

/

/

/

/

/

/

|

\

\

\

k

y \

\

/

/

/

/

-24 -20 -16 -12 - 8 - 4 0 4 12 16 20 24

Β M 75(2) q = 25.0 Pa

1 6

I 8

0

\

Λ

25Γ­/ v^

*

■ ­ ­

' " ­ ­ ­

_.

­ ­ '

/

\ " " ­

'*' 'A

# \ .

C^ V

­ \f

, . ­ ' '

, '

.­"'

Æ~

Xr

f /

"­S/

_­1

/ J / /

-24 -20 -16 -12 - 8 - 4 0 4 8 12 16 20 24

Fig. D 3.2.3 Development of post­buckling pattern ­ model BM 75 (2) a) q = 14.5 Pa, b) q = 20.0 Pa, c) q = 25.0 Pa,

as seen from inside of the cylinder; the 0 meridian represents the stagnation line.

195

2

E ° E

■ E

s ­« α M

* ­6

­8

■ 10

B M 7 5

■ Ιιί.1 L.I.

ΠτΠ nyp I 0 Paj

-90 -60 -30 0 30 ΘΟ 90 120 150 180 210 240 270

angle / "

BM 75

.iLiáJrí MWr'' I 5-3 Pa|

-90 -60 -30 0 30 60 90 120 150 180 210 240 270 angl. / °

4

2

¡ ° Έ "2 α • ¡¡ ­< α m * ­β

­β

­10

ûÀnJték ■PIT "W M rv H#

. | 10.1 Pa|

-90 -60 -30 0 30 60 90 120 150 160 210 240 270

angle / °

Fig. D 3.2.4 Bückling displacement with increasing load at midheigth of BM 75

a) q = 0 Pa, b)q= 5.3 Pa, c)q=10.1Pa.

The 0° meridian represents the stagnation line

196

E E

i E ■ O

a ■ =5

β

0

■β

­Β

­10

­12

­9

nua

0 ­β

m ΓΠ

ii N

0 ­3

\Λ v\ Τ li

¡Α­Π r ι nr '

'\

0 t

Ι

3

fa ΓΨ

ΒΜ 75

0 60 90 120

angle / °

150 180 210 240

| 14.6 Pa |

270

ΐ -J

*m¥ W^ ι *\k ΛΛ πι

L 1 Junik ñ

\ > i

Λ . Ν

J

\*n éjÈMÚ UI

™<1 ài L Aww

ι

'y^itoitøm"

-90 -60 -30 0 30 60 90 · 120 150 180 210 240 270

angle / °

BM 75

E " E

ï ­2

I s < M

•s _.

lør^ L Jn

\Λ V' Λ

1, ft

' '

1 / V' V

_ | /

Ι, Λ I * '

ï

i U _

Hh»» 1 J.I .1.

KUK.TÆI I'll Λ 5 Ε Μ F

I ­ , I 24.8 7ä]

-90 -60 -30 0 30 60 90 120 150 180 210 240 270

angle / *

Fig. D 3.2.5 Buckling displacement with increasing load at midheigth of BM 75 a) q = 14.6 Pa, b) q = 20.2 Pa, c) q = 24.8 Pa.

The 0° meridian represents the stagnation line

197

D 4 Interpretation and Conclusion

D 4.1 Buckling under Static Wind Load

The theoretical values of the critical buckling pressure ρ„γ^ and wave number m^ were calculated assuming a constant pressure over the height and the circumference with boundary conditions clamped at the upper and lower ends of the shell. The effect of imperfections on the critical pressure was not included. It is estimated in general by a reduction factor of 0,7 [D34]. The experimental values of the depth and the width of the buckles in the stagnation region were determined as means from two buckles forming on both sides of the stagnation line. These are related to the local maxima of the pressure difference between external and internal pressure as found in the pressure tests, i.e. the maximum net wind pressure max pw. The following net pressure coefficients apply: 1,96 for model BM 100; 1.73 for model BM 75; 1.64 for model BM 50. The buckling depth is depicted in fig. D 4.1.1.

buckling depth at z/h = 0.5

25

20

15 E

ε 10

o^——xo­—

f

/

CÁ

s y

o

—χ—

χ

ο­­·

­ΒΜ50

ΒΜ 50­0

­ΒΜ75

ΒΜ75­0

­ΒΜ100

ΒΜ 100­0

Fig. D 4.1.1

0 1 2 3

max ρ*,/ρ««,»

Development of the maximum buckle depth with respect to the maximum net wind pressure

For a perfect shell, instability would be expected to occur near p^a/max pw = 1. Imperfections are either caused by pre­deformations or by initial stresses introduced by the process of manufacturing. The pre­deformations of the three models as measured were in the order of 0,46% of the radius for model BM 100; 0,39% for the model BM 75, and 0,44% for BM 50. The order of magnitude is the same, nevertheless the models exhibit different degrees of sensitivity to the imperfections: for the small height of 0,5 h, a distinct onset of buckling occurs between 0,65 and 0,94 of pcr th m conformity with the factor of 0.7 mentioned before. For the large height of 1,0 h, buckles start already at a very small velocity pressure of 3 Pa corresponding to 0,5 of the critical pressure. In this case, the buckles could not be seen but only be identified from the laser sheet inspection. A clear instability could not be detected but rather a problem of large deformations, as is typical for shells with strong sensitivity to imperfections.

198

The initial regular pattern of 4 to 5 well separated buckles develops with increasing pressure into overlapping of buckles and formation of secondary buckles. At the highest load levels, folds and additional buckles appear within the initial buckles, and small regular buckles are found around most of the circumference. No collapse of the system was observed in all tests, i.e. up to a limit of 2,5 of the critical pressure. Repetitions of the tests with a different model of the same configuration BM 75 (2) revealed that the pattern could not be reproduced identically. However, the principal observations were the same. The width of the buckles was always larger than the theoretical value estimated from the theoretical wave number; it increased with increasing load, see fig D 4.1.2. In summarising it is concluded, that open, thin walled shells of the type considered do not collapse by instability up to 2.5 of the critical pressure, determined for uniform load and perfect geometry, provided that the upper edge is sufficiently stiffened by a ring beam. The formation of sharp edged folds will however lead to local overstressing beyond a limit of 1.5 times the critical pressure.

buckling width at z/h = 0.5

—O—BM50 Ο BM 50-th

— χ — BM75 X BM 75-th

— ο — BM100 o BM 100-th

Fig. D 4.1.2 Development of the maximum buckle width with respect to the maximum net wind pressure

199

D 4.2 The Effect of Wind Turbulence An indication of the effect of the natural wind flow on the buckling behaviour was obtained from an additional test in a simulated boundary layer flow. The profile of the mean velocity pressure, q, referred to the reference pressure at the Prandtl tube, qp,,,, is

where ζ = height above wind tunnel floor in mm. The intensity of turbulence in the range of the upper part of the model is 18%, corresponding to a typical rural terrain. The new model BM 75 (2) was tested in both flow conditions. The maximum net pressure at 0,65 of the height can be considered as dominant to the initiation of buckling. Generally speaking, the pressure on the external shell surface at the windward side varies according to the velocity pressure, apart from the upper region with its edge disturbance. The pressure on the internal surface is approximately proportional to the velocity pressure at the model top. The maximum net pressure is approximated assuming that the pressure coefficients of the uniform flow apply namely 1,0 at the external and -0,73 at the internal surface :

max pw - (0,43 + 0,73 0,5) q^ = 0,79 q^ comparable to 1,73 q ^ in uniform flow. Visible buckling was observed at a reference pressure q ^ of 20 Pa corresponding to a mean (with respect to time) maximum pressure of 15,8 Pa. In the smooth, uniform flow, the visible onset of buckling occurred at max pw= 1,73-14,5 = 25,1 Pa. Due to the pressure fluctuations induced by gustiness, the critical value of the mean wind pressure in natural wind is smaller by a factor of 1,58. This factor defines a gust velocity pressure appropriate in this case for the buckling design. It is smaller than usual gust factors which means, that the design could have been based on a rather long gust averaging time, i.e. on a gust wind pressure clearly smaller than the 3 to 5 sec gust specified in some codes. This observation indicates an additional source of conservatism. Under turbulent flow conditions, the buckling pattern is no longer constant in time but shows a permanent oscillatory character. Fatigue may therefore become a problem if frequent appearance of buckling is allowed in the design.

200

D 5 Summary In the Boundary Layer Wind Tunnel -BLWT- of the Arbeitsgruppe Aerodynamik im Bauwesen -Affi- at the Ruhr-Universität Bochum -RUB- two series of cylindrical models were investigated under one respectively two different flow conditions. One set of models was manufactured as rigid acrylic models for pressure measurements the second set were elastic models for investigating the buckling pattern on the surface of the cylinders and the development of the buckles under increasing wind load, i.e. the post­buckling behaviour. Each model set consists of three species with different heights (height/radius ratio : h/r = 0.5, h/r = 0.75, h/r = 1.0; with a common radius of 0.225 m). The elastic models are designed as modular elements with a circular base plate, an exchangeable elastic Mylar foil forms the surface of the cylinders (giving a resulting wall thickness ratio of about t/r « 1/2225) and an upper stiffening ring. Pressure measurements were performed under smooth uniform flow condition in the velocity range of 2 m/s to 25 m/s. At maximum speed of 25 m/s the pressure coefficients were determined at 7 heights for the inner and outer side of the cylinders with a rotatory resolution step of 10 degrees. Similarly the buckling tests were performed in smooth uniform flow in the velocity range adjusted for the individual model (equivalent pressure range : 0 - 50 Pa, 0 - 25 Pa, 0 - 15 Pa). Different approaches for recording the three-dimensional buckling pattern were tested. For practical use only two methods were carried out. In order to obtain a qualitative information, the surface was marked with a grid. The visually seen borders of the individual buckles were copied onto corresponding grid paper. The technique to get quantitative distortion data was to „cut" the surface of the cylinder with a laser light sheet and evaluate the absolute deformation from the original circular shape with no wind load. This is done with a computer based semi­automatic procedure using the recorded pictures of a standard 35 mm photocamera. The achieved resolution was about ±1,5 mm absolute and +0.5 mm relative displacement and 0.1 degree in angular direction at radius position.

The lowest level of an identified buckling wave profile from the background was about 2 - 3 mm peak to peak amplitude. The first visible surface pattern occurrence shows 4 - 5 mm peak to peak amplitude with this method. Extrapolation of the measured amplitude and wavelength of the buckles to zero amplitude fits with the theoretical values for the critical wind load. The first visible buckling pattern starts at a wind load of 2 times over this limit. In the tests (with no permanent surface damage) the maximum reached velocity pressures were always 2 -3 times greater than the theoretical values for the start of buckling. A preliminary test in turbulent flow was done with one of the elastic models, but it was not possible to get quantitative results with the methods developed. No stable pattern on the surface evolved. The first visible flipping buckles start at a equivalent velocity (mean value of the velocity profile at the cylinder height) of about 0.6 the value as in uniform flow.

201

D 6 References WINDLOADS ON CIRCULAR CYLINDERS [D 1] Basu, RL; Aerodynamic forces on structures of circular cross-section . Part 2. The

influence of turbulence and three-dimensional effects; Journal of Wind Engineering and Industrial Aerodynamics, Vol 24 (1986) S. 33-59

[D 2] Briassoulis, D. und Pecknold, D.A., M. ASCE; Anchorage requirements for wind-loaded empty silos; Journal of Structural Engineering, Vol. 112, No. 2, February, 1986, S. 308-325

[D 3] Cheung, J.C.K. und Melbourne, W.H.; Turbulence effects on some aerodynamic parameters of circular cylinder at supercritical Reynolds numbers; Journal of Wind Engineering and Industrial Aerodynamics, Vol. 14 (1983) S. 399-410

[D 4] Esslinger, M., Ahmed, S. R, Schroeder, H. H.; Stationäre Windbelastung offener und geschlossener kreiszylindrischer Silos; Der Stahlbau 40 (1971), No. 12, S. 361-368

[D 5] ECCS Recommendations No. 56, 'Buckling of Steel Shells', 4th Ed. 1988 [D 6] Gopalacharyulu, S. and John, D.S.; Cantilever shells under assumed wind pressure;

Journal of Engineering Mechanics Division, Vol. 99, No. EM5, 1973, S. 943-956 [D 7] Hoelscher, N.; Ein multivariater Ansatz für die aerodynamische Übertragungs­

funktion der Winddrücke in atmosphärischer Grenzschichtströmung; Technisch­wissenschaftliche Mitteilungen Nr. 93-3, Institut für Konstruktiven Ingenieurbau, Ruhr-Universität Bochum, 1993

[D 8] Kunieda, H., Yokoyama, Y. und Arakawa M.; Cylindrical membrane structures subject to wind; Journal of Engineering Mechanics, Vol. 107, No.EM5, Oktober 1981, S. 851-867

[D 9] Kwok, K.S.C; Turbulence effect on flow around circular cylinder; Journal of Engineering Mechanics, Vol. 112, No. 11, November 1986, S. 1181-1197

[D 10] Niemann, H.-J.; The boundary layer wind tunnel: an experimental tool in building aerodynamics and enviromental engineering; Journal of Wind Engineering and Industrial Aerodynamics, Vol 48 (1993) S. 145-161

[Dll] Niemann, H.-J., Hölscher,N; A review of recent experiments on the flow past circular cylinders; Journal of Wind Engineering and Industrial Aerodynamics, Vol 33 (1990) S. 197-209

[D 12] Niemann, H.-J.; Wind tunnel experiments on aerolastic models of air supported structures : results and conclusions; Proceedings of the International Symposium on Pneumatic Structures; Delft University of Technology, Stichting P.D.O.B., The Netherlands, 1972, 5.11

[D 13] Purdy, D. M., Maher, F. J. and Frederick, D.; Model studies of wind loads on flat-top cylinders; Journal of Structural Division, Vol. 93 (1967), No. St 2 (April), S. 379-395

202

[D 14] Uchiyama, K., Uematsu, Y. und Katsura, S.; Aerodynamic behavior of cylindrical shells; Journal of Engineering Mechanics, Vol. 112, No. 12, December 1986, S. 1346-1362

[D 15] Uchiyama, K, Uematsu, Y. und Orimo, T.; Experiments on the deflection and buckling behavior of Ring-Stiffened cylindrical shells under wind pressure; Journal of Wind Engineering and Industrial Aerodynamics, Vol. 26 (1987) S. 195-211

[D 16] Uematsu, Y.; Aeroelastic behavior of a pair of thin circular cylindrical shells in staggered arrangement; Journal of Wind Engineering and Industrial Aerodynamics, Vol. 22 (1986) S. 23-41

[D 17] Uematsu, Y., Uchiyama, K; An experimental investigation of wind-induced ovalling oscillations of thin, circular cylindrical shells; Journal of Wind Engineering and Industrial Aerodynamics, Vol. 18 (1985) S. 229-243

BUCKLING OF CIRCULAR CYLINDERS

[D 18] Almroth, B.O.; Buckling of a cylindrical shell subjected to nonuniform external pressure; Journal of applied Mechanics, Vol. 29, 1962, S. 675-682

[D 19] Brendel, Β., Ramm, E., Fischer, D.F. und Rammerstorfer, F.G.; Linear and nonlinear stability analysis of thin cylindrical shells under wind loads; Journal of Structural Mechanics, Vol. 9, No. 1; 1981, S. 91-113

[D 20] Brush, D.O. and Almroth, B.O.; Buckling of bars, plates and shells; McGraw-Hill, New-York, 1975

[D 21] De Witt, J.; How to calculate the stability of empty storage tanks; Oil and Gas International, Vol. 11, No. 8, 1971, S.74-80

[D22] Ebner, Η.; Theoretische und experimentelle Untersuchungen über das Einbeulen zylindrischer Tanks durch Unterdruck; Der Stahlbau 21 (1952) No. 9, S. 153-159

[D 23] Holownia, B.P.; Buckling of cylindrical shells under wind loading; Proceedings of the Symposium on Wind Effects on Buildings and Structures; Loughborough University of Technology, England, 1968, S. 35.1-35.9

[D 24] Holroyd, R. J.; On the behaviour of open-topped oil storage tanks in high winds. Part 1. Aerodynamic aspects; Journal of Wind Engineering and Industrial Aerodynamics, 12 (1983) S. 329-352

[D 25] Holroyd, R. J.; On the behaviour of open-topped oil storage tanks in high winds. Part 2. Structural aspects; Journal of Wind Engineering and Industrial Aerodynamics, 18 (1985) S. 53-73

[D 26] Holroyd, R. J.; On the behaviour of open-topped oil storage tanks in high winds. Part 3. A structural dynamic instability mechanism? ; Journal of Wind Engineering and Industrial Aerodynamics, 21 (1985) S. 339-341

[D 27] Jerath, S.,M. ASCE und Sadid H.; Buckling of orthotropic cylinders due to wind load; Journal of Engineering Mechanics, Vol. I l l , No. 5, May 1985, S. 610-622

\D 28] Kapania, R. K.und Yang,T.Y.; Stability of wind-loaded cylindical shells; Journal of Wind Engineering and Industrial Aerodynamics, Vol. 28 (1988) S. 281-290

203

[D 29] Langhaar, H.L. und Miller, RE.; Buckling of an elastic, isotropic cylindrical shell subjected to wind pressure; Proceedings of the Symposium of Shells to Honor L. H. Donnei, University of Housten, Housten, Texas, 1967, S.404­429

[D 30] Maderspach, V., Gaunt, J.T. and Swort, J.H.; Buckling of cylindrical shells due to wind loading; Rep. VPI­E­71­25, College of Engineering, Virginia Polytechnic Institut and State University, 1971

fD 31] Maderspach, V. and Kamat, M.P.; Buckling of open cylindrical tanks due to wind loading; Der Stahlbau, H. 2, Feb. 1979, S. 53­56

[D 32] Mungan, I.; Buckling stress of cylindrical shells; Journal of Structural Division, Vol. 100, No. ST11, November 1974

[D 33] Prabhu, S.K. and Gopalacharyulu, S. and John, D.S.; Stability of cantilever shells under wind loads; Journal of Engineering Mechanics Division, Vol. 101, No. EM5, 1975, S. 513­530

[D 34] Resinger, F. and Greiner, R; Buckling of wind loaded cylindrical shells ­ application to unstiffened and ring­stiffened tanks; Buckling of shells ­ state of the art coloquium, E. Ramm, Ed. Springer­Verlag, Berlin, 1982, S. 305­331

[D 35] Resinger, F. und Greiner, R; Praktische Beulberechnung oberirdischer zylindrischer Tankbauwerke für Unterdruck; Der Stahlbau 45 (1976) S. 10­15

[D 36] Resinger, F. and Greiner, R; Zylinderschalen unter Winddruck ­ Anwendung auf die Berechnung oberirdischer Tankbauwerke; Der Stahlbau 50 (1981) H3 S. 65­72

[D 37] Wang, Y.­S. and Billington, D.P.; Bückling of cylindrical shells by wind pressure; Journal of Engineering Mechanics Division, Vol. 100, No. EM5, 1974, S. 1005­1023

[D 38] Yamaki, N ; Buckling of circular cylindrical shells under external pressure; Rep. Inst. High Speed Mech. Tohoku Univ., 20(1968/69) 35­54

[D 39] Yamaki, N.; Elastic stability of circular cylindrical shells; North Holland, Amsterdam, 1984, S.306­347

[D40] Zintilis, G.M. und Croll, J.G.A.; Combined axial and pressure buckling of end supported shells of revolution; Engineering Structures, 1983, Vol. 5, S. 199­206

POSTBUCKLING BEHAVIOUR

[D 41] Eßlinger, M. und Geier, Β.; Buckling and postbuckling behavior of thin­walled circular cylinders; Deutsche Luft­ und Raumfahrt­Forschungsbericht 1969, S. 69­99

[D 42] Sridharan, S. und Graves­Smith, Tom R; Postbuckling analyses with finite strips; Journal of Engineering Mechanics, Vol. 107, No.EM5, Oktober 1981, S. 869­887

[D 43] Thielemann, W. und Eßlinger, M.; Beul und Nachbeulverhalten isotroper Zylinder unter Außendruck; Der Stahlbau 36 (1967), No. 6, S. 161­175

[D 44] Uematsu, Y., Uchiyama, Κ.; Deflection and buckling behavior of thin, circular cylindrical shells under windload; Journal of Wind Engineering and Industrial Aerodynamics, Vol. 18 (1985) S. 245­261

204

[D 45] Yamaki, N. und Otomo, K.; Experiments on postbuckling behavior of circular cylindrical shells under hydrostatic pressure; Exp. Mech., 13 (1973) 299-304

PICTURE PROCESSING [D46] Fischer, N.I., Lewis, T., Embleton, B.J.J.; Statistical analysis of spherical data;

Cambridge University Press, Cambridge, 1987 [D 47] Encyclopedia of Graphics File Formats; Ed. Murray. J.D., vanRyper, W.; O'Reily,

Sebastopol, USA, 1994, [D 48] Göpfert, W.; Raumbezogene Informationssysteme; Wichmann, Karlsruhe, 1987 [D 49] Gömandt, V; Untersuchungen zur Methodik der Flächenphotometrie für Kugel­

spiegelkameras mit großem Bildwinkel; Diplomarbeit Ruhr-Universität Bochum, Bochum, 1991

[D 50] Kimme, C, Ballard, D., Sklansky, J.; Finding Circdes by an Array of Accumulators; Short Comm. Graphics and Image Processing, Comm. of the Assoc, of Computer Machinery, 18 (1975) 120-122

[D 51] Serra, J.; Introduction to Mathematical Morphology; Comput. Vision Gaphics Image Process., 35 (1986) 283-305

205

ECSC Contract No. 7210-SA/208

Enhancement of ECCS Design Recommendations and Development of Eurocode 3 Parts Related to Shell Buckling

Part E

EFFECT OF CUT-OUTS AND OPENINGS IN SHELLS

Final Report

INSA de LYON France URGC-Structures

J.F. JULLIEN

El Introduction and literature revue

In many industrial applications shells are equipped with openings of various shapes, sizes and locations within the lateral surface. The objective of the present research is to gain an understanding of the effect of cut-outs on the critical buckling load of thin cylindrical shells and on that basis develop new design rules. The shells are made of mild steel and are subjected to uniaxial compressive load..

No reinforcement along the periphery of the cut-out were considered in the present study. In general, there are many ways of making up for the weakenning effect of the opening. Openings can be reinforced by a set of two rings and stringers of a certain lenght. Alternativelly, an additional reinforcement can be provided by an increased thickness along the edges of the opening. Also, a smaller diameter welded tube acts as a perfect stiffener.

The result of reinforcement has a strong effect on the buckling load and is also important from the point of view of stress concentration and fatigue. However, these factors have not been considered in the present study. Nor has the influence of non-uniform axial stress induced by external bending been included in this work.

Because cylindrical shells subjected to axial loads are imperfection-sensitive, it is important to know as to what extent this effect persists in the case of shells with opening. The present report gives a clear answer on the above question.

Little work has been done in the litterature on the subject. Early work dates back to Tennyson [1] who performed an experimental study on the effect of small circular openings on shells which buckle in the elastic range. This work was followed by Starnes [2] and Toda [3] who tested shells with larger range of a diameter of the opening.

Knödel and Schulz [4] were first to perform tests on steel cylinders with a large spectrum of imperfections (geometry, loading, material). This gave a possibility for a statistical evaluation of the buckling strength of shells. At the same time no physical understanding of the phenomenon was gained.

The first numerical calculations on cylindrical shells with cut-outs were reported by Almroth and Holmes [5]. Based on the available results, Samuelson [6] proposed a simple analytical description of the effect of hole diameter on the buckling strength of shells.

All of the above mentioned research have not provided sufficient information to develop a reliable physical model of the considered problem and to deduce a general rule.

In order to achieve this goal, a new series of tests have been undertaken with a different shapes (square, rectangular, circular), sizes and position of openings with respect to the boundaries of the shell. Furthermore, all tests were of high quality with imperfections representative to industrial applications. Parallel to physical testing, numerical calculations were run and an extensive parametric study was performed.

209

E2 Methodology

E2.1 Definitions of shells

The notation used is defined in Figure El. The shell is of the length i, radius of midsurface r and thickness t. The upper and lower cross-section (A and B) are assumed to remain plain and are refrained from warping and/or ovalisation.

The upper cross-section is free to translate and rotate in three directions while the lower cross-section is considered fixed. The loading is applied at the center of the upper cross-section and a constant rate of displacement uz (compression) is applied.

In the circumferentiel direction, the opening is defined by the central angle Θ, or the curvilinear lenght 2c = rQ or by the dimensionless length r

47t

In the axial direction, the hight of square and rectangular cut-outs is defined by h0. The distance from the lower edge of the opening to the lower cross-section of the shell Β is denoted by hb.

The presence of an opening introduces an asymmetry in the stress distribution so that an internal bending moment is developed. Therefore, the shell response will depend on the length of the shell. For longer shells, the second order terms (Finite rotation) and the effect of ovalisation becomes more important than for shorter shells. The effect of the length of the shell on the critical buckling load obtained by Finite element calculations is shown in Figure El7. The experimental study and the ensuing proposition of the revised rules is limited to relatively short shells with approximately i/r = 2.

The study was done with shells with aspect ratio r/t = 280 which could buckle in the elastic or elasto-plastic range, with multi-modal post-critical behaviour.

210

E2.2 Test cases of shell with opening

Over one hundred specimens were manufactured with the diameter 2r = 99 mm thickness / = 0.175 mm and length £ = 104 mm. Different types of square, rectangular and circular openings were made, as shown in Figure E 2. For comparison, a series of shell (case 1) was made without the cut-out.

Characteristic dimensions of openings and their positions are defined in the tables below. For single square openings, tests (case 2 and 3) were run on five different central opening (Table El) and four meridional positions (Table E2). In addition, four combinations of multiple equi-distanced opening (case 6) were considered, Table E3. Similarly, five different highs (case 4) and five widths (case 5) were studied for a single rectangular opening (Tables E4, E5). For comparison, seven different radii of circular openings (case 7) were considered, (Table E6).

The entire stress-strain characteristic of the material is defined in Figure E 3, and the material parameters are given in Table E7.

With the above material parameters, the classical Donnell buckling stress of a complete

shell (without openings) is acr = 0,605 E - =458 MPa. According to the current ECCS r

recommendation, the reduction factor for r/t = 283 is a0 = 0.409. This gives an approximate buckling stress of the imperfect shell α σ σ = 257 MPa. The application of Eq (5) in Ref [7] gives the value of the parameter λ = 1.567.

Since this value is greater than v2 , one expects elastic buckling to occur first (provided the material is elastic-perfectly plastic). In the case of the present (strain-hardening) material and ranges of geometrical parameter buckling occured in the elastic-plastic range.

E2.3 Material and manufacturing

Cylindrical shells are made from sheet metal by means of rolling and electric welding along a generator. These operations are automated to ensure that the resulting geometric imperfections are the same across all specimens which is a necessary condition for the present comparative study.

The opening is always situated on the opposite side of the weldement. Great care is given so that the cutting process of the opening does not introduce any additional imperfections, as shown in Figure E4.

211

E2.4 Experimental set-up

For each test piece the geometric imperfections are measured by means of a contactless sensor at 25000 points. For that purpose a special automated scanning system is mounted on the testing machine.

The geometric imperfections are located mainly in the vicinity of the welded generator. The imperfections are characterised in the circumferential direction by the wavelength corresponding to n = 7 or 8 with a maximum amplitude of 2w, ·. = 2 χ 157 pm. The

W s \

corresponding dimensionless amplitude is — (n) = 0,9. These were no imperfections in the

meridional direction {m = 0) except along the welded generator which shows two half waves {m = 2) with an amplitude 2 w(ni)= 2 χ 98 pm — [m) = 0,56

v t J

The boundary conditions were introduced in the test pieces by means of two rigid plattens which were connected to cylinders by a sobering technique, (Figure E5).

E3 Experimental and numerical results

The results oftest showing the effect of cut-outs in presented first. This is followed by the numerical analysis of perfect and imperfect shells. The results for different cases of openings are confronted with those corresponding to shells without cut-outs.

The reproducibility of the present test methodology is verified by comparing results for eight specimens without opening. The mean value of buckling load was Po = 11,78 KN with a standard deviation of 58 N which corresponds to 5 % of Po. The corresponding stress is <Jcr = 216 MPa representing 47 % of the theoretical buckling stress Ocr.

E3.1 Effect of shape of a cut-out

E3.1.1 Square openings

For a given dimensions of the considered square and centred opening, the force -displacement characteristic of the shell remains linear up to the first local buckling (L.B.) which occurs in the vicinity of a contour of the opening. After this point the behaviour is linear and stable with a reduced slope until a global buckling (collapse buckling, C.B.) takes place which involves the entire shell, see Figure E7.

In the pre-buckling range, before the first local buckling, large radial displacements develop near the edges of the openings, Figure E8. These observations are visualised in Figure

212

E 9 which shows radial displacements for two types of square openings at 50 % and 80 % of the local buckling, at the first local buckling and finally at the collapse buckling.

A square cut-out in a cylindrical shell subjected to the axial compression reduces the first local buckling load and the global collapse buckling load as compared to the reference shell without a hole. The drop of critical loads with the dimensionless geometrical parameter of the opening r is described by a linear function in the studied range 1,47 < r < 8,81. The ratio between these two critical loads is a function of r, with a mean value of approximately 1.1 (Figure E10). Beyond the point r = 10 approximately, the first local buckling does not develop.

Multiple and equi-distanced openings over the circumferential direction produce a similar type of instabilities (first local buckling followed by collapse buckling) with a corresponding loss of rigidity. In the case of multiple openings the reduction in the critical load is smaller when the comparison is made taking the sum of opening angle, as shown in Figure E l l . Because of symmetry, the appearance of critical loads is not accompanied by the development of an internal» bending. The above dependence can be converted to a different co-ordinate system in which the abscise is not the sum of openings but the width of a single opening. It is interesting to note that in the above co-ordinate system, the first local buckling load is same for shell with single and multiple openings.

For a chosen dimensions of the square cut-out r = 4.40 the position of the opening along the meridional direction (axis of a cylinder) with respect to cross-sections, A or Β does not change the critical load. This interesting result indicates that the critical load is not linked to the direct distribution of stresses above or below the hole. The above conclusion is valid as long as cross-sections A and Β remain plane but could rotate. In practice this boundary condition could be satisfied by placing reinforcing rings of sufficient rigidity at these cross-sections, Figure E12.

£3.1.2 Rectangular openings

A cylindrical shell with a rectangular opening carries the same critical buckling load as an identical shell with the square opening provided the cut-outs are of an identical length in the circumferential direction, Figure El2. The hight of the opening does not intervene in the studied range of parameters. At the same time this conclusion can not be extended to cut-outs in the form of a slot without a separate study.

213

E3.1.3 Circular openings

A cylindrical shell with a circular opening of the diameter equal to the width of the square opening develops the same collapse buckling load. At the same time the first load buckling is not observed in this case. The above result indicates that the occurrence of the first local buckling is preceded by a redistribution of stresses due to stress concentration at corners of square or rectangular cut-outs.

For this type of openings the effect of small hole dimension r = 0.15 ; 0.28 ; has also been analysed. It was found that very small holes do not cause any appreciable change in the critical load.

E3.2 Effect of opening angle

The present experimental study have determined the effect of the shape of the opening on the critical load of cylinders subjected to axial compression. It was shown that the key parameter controlling the buckling strength of the shell is the dimension of a cut-out in the circumferential direction. In the case of multiple openings positioned equidistantly around the circumference, the control parameter is the sum of hole dimensions of all openings.

The above experimental results are confirmed by Finite element calculations. The calculations were run using the three dimensional code CASTEM 2000, developed by the French Atomic Energy Commission. The code is capable of dealing with various types of singularities. Half of the shell was modelled by 3500 triangular elements DKT (3 displacements and 3 rotations per node), as shown in Figure E6. Contrary to previous analyses on buckling of shells, the linear analysis (L.A.) were found to give a smaller critical load than the geometrically non-linear analysis (GNA), as shown in Figure E14. This can be explained by an inability of the geometrically linear model to account for large radial displacements around the opening.

In fact, already at 20 % of the critical load, radial displacements exceed approximately two times the displacements predicted by linear theory (Figure E16). Further refinement of the model by introducing the non-linear material behaviour (GMNA) brings the level of stresses even lower.

The result of the linear (L.A.) and non-linear analyses (GMNA) mentioned above corresponds to the loading case (a) when the cross-sections A and Β remain parallel. This condition does not fully reflect a reality because a single opening produces an asymmetry and the so-called internal bending. At the same time, the results of calculations with the geometrical and material non-linearities referenced in Figure El4 and El 5 as the case (b) correspond to the boundary condition in which the cross-section A is free to displace and rotate in all three directions.

It should be noted that for a design purpose one has to consider rather conservatively only the first critical buckling load even though there could be a stable post-critical range until

214

collapse buckling occurs. Recall that for circular holes there is no first local buckling.

Therefore the design should be made on the basis of the global collapse buckling.

E3.3 Effect of length to radius ratio

As it was mentioned in section E2.1 all the results presented so far are valid only for

shells with £/r= 2. The amount of bending deformation introduced by the presence of

unsymmetrically positioned openings will depend on a distance between two cross­sections

taking into account the second order effect and greater susceptibility for shape distortion in

the critical cross­section in the case of longer cylinders. Figure El7 indicates that there is an

asymptotic limit for the length dependence. A more detailed analysis is necessary to quantify

the above dependence. In the mean time, the existing results can be used for shells reinforced

by two heavy rings at cross­sections A and Β distanced £ = 2r apart. This will ensure that

cross­sections remain plane without warping and ovalisation.

E3.4 Effect of geometrical imperfections

The experimental results for different forms of openings correspond to metal cylinders

characterised by reproducible geometric imperfections encountered in industrial practice.

These results were compared with earlier results reported by STARNES [2], see Figure El 8.

While the shell material is different (Mylar versus steel) and the geometrical imperfections in

STARNES experiments are much smaller compared to the present ones, the overall behaviour

of both types of shells is identical in a certain range the parameter r . The range of interest

corresponds to angular openings larger than a limiting value denoted by r¿.

In the first approximation, the geometrical imperfections are seen not to have any

significant effect as compared to the effect of opening beyond the critical limit re .

It is possible to identify a reduction function for a given opening over the range r ) r¿.

From Figure El 8 it transpires that for small openings up to this limiting value, the effect of

imperfections must be coupled with that of the cut­out. At the same time for very small

openings, their size is unimportant as compared to a significant imperfection sensitivity of

cylinders subjected to compressive loads.

The above conclusions are reinforced by three­dimensional numerical calculations in

conjunction with the model of representing measured imperfections developed at INS A [ ].

As shown in Figure El9 a comparison between calculations and experimental results is in full

agreement.

The obtained results are limited to the aperture size r < 6,61 (i.e. θ = 45°) and

212 < r/t < 400. These limits could be extended in the continuation of this research.

215

In Figure El 9, limits of two distinct zones are clearly identified. In the case of small openings, it is necessary to include in the analysis a coupling between the geometrical imperfection and presence of an opening. In the range of moderately large openings, the existence of geometrical imperfections of any sizes can be neglected.

E4 Proposed rules

E4.1 General concept

The proposed rule uses a reduction (knock-down) factor to the critical Donnell stress σ cr in the same way as it is done for a cylindrical shell without an opening. This rule is developed for the most unfavourable case, i.e. for cut-outs with sharp corners (squares, rectangles). For the purpose of this report, the same notation is used for the reduction factor a, as in the classical buckling analysis of imperfect shells. The reduction factor α is a linear piece-wise function of the geometrical parameter of the openings r , referee to Figure E20. It is proposed to distinguish in the design rule three distinct regimes, denoted respectively by ®, © and ®. The regime Φ corresponds to very small openings in which the size of the hole does not have practically any effect on the critical load. The shell response is dominated by the geometrical imperfections. The reduction factor is constant with respect to the opening parameter r and its magnitude is equal to the reduction factor of an imperfect cylinders without an opening subjected to axial compression.

Response of shells with relatively large openings are dominated by the presence of a cut­outs rather them geometrical imperfections. This function is described by a linear dependence given by segment ® in Figure E20. For intermediate hole sizes there is a strong coupling effect between the opening parameter r and geometrical imperfections. The reduction factor in this range is approximated by a straight line, designated in Figure E20 as the segment ©.

In order to unequally position the above three lines on the α - f plane it is necessary to define six parameters .

216

E4.2 Definition of the parameters

Line © is defined by the intercept with the vertical axis (point A) and the slope β. The

co­ordinates of the point A are :

r = 0

α = α0

where a0 is the reduction factor for pure axial compressive load taking into account the effect

of imperfections defined by formula 13 in the reference[7].

The slope of line Φ is ßj = 0.

Line <D is defined by giving co­ordinates of the end points Β and C.

The co­ordinates of the point Β are :

5

α = α0

The co­ordinates of the point C are :

_ _ 2,3 r r = r.

8(rtrlr/£)(r/t)°­5Y

and the ordinate α is defined by line ® at r = re.

Line ® is defined by the intercept D with the vertical axis and the slope β3.

The co­ordinates of the point D are

r = 0

α = αΗ

The slope of the line ® is β3 =

­ 0,02 f° r a single opening

­ 0,01 for multiple openings

From the above values it is easy to determine the following equations for the three

straight lines :

ψ Line © : range 0 < r < —

α = αη

217

Line © : range rtl5< r < rt

for a single opening

a, a = a 0 + —£ + 0,005 r, a

- r,

fi. + M

for multiple openings :

a , a = a0 + —V- + 0,005 rt 4 a „ , 0.05

l r <

Line © : range rt< r < rmax

. for a single opening

for multiple openings

α = αΗ - 0,01 r

α = αΗ - 0,02 r

Ε4.3 Justification of the method

E4.3.1 Transition between coupling and no-coupling range

Figure E21 shows examples of buckling patterns for different cases of opening geometries. A careful examination of post-critical geometry for all cases considered indicates that the modal geometry is not affected by the presence of openings. This statement is valid for different dimensions and shapes of tested shells. One can conclude that the mode numbers η and m are «invariants» because they depend on the geometry of a shell rather than that of an opening.

For the clarity of presentation it will be helpful to recall few facts from the classical buckling analysis of shells without cut-outs. A numerical analysis in the non-linear formulation indicates that there exists a weak minimum of the eigenvalue (dimensionless critical load) with respect to the circumferential number of waves η and this minimum occurs at η = 14. The corresponding axial mode is found to be m = 6. The above result compares only partially with the mode number found experimentally. In particular, measurements show that η = 8 and m = 6. The question on how can one predict correct buckling modes without reference to experiments is a key point in the development of the present rule. While the number of waves in the axial direction can be determined from the numerical solution, as shown above, our task is to develop simple formulas for the buckle wavelengths.

Our research have shown that buckles develop in the meridional direction during the pre­buckling phase over the region of the shell affected by geometrical imperfections. The onset

218

of buckling corresponds to a sudden propagation of axial buckles in the circumferential

directions, [11]. It should be mentioned that the wavelength in the circumferential direction is

largely influenced by certain harmonics of the Fourier decomposition of geometrical

imperfections.

The circumferential imperfections are imposed on a shell by a manufacturing method

which usually. Involves rolling, welding, connecting with other structural members, etc.

Experience shows that the nature of these imperfections is similar to buckling modes develop

by the external pressure. This can be explained by the fact that in both cases the shell wall is

deformed by circumferential contraction. Furthermore, the corresponding circumferential

wavelength of the diamond buckling pattern is equal to the axial wavelength of the identical

shell subjected to axial loads.

It should be recalled that the critical load has an asymptote with increasing amplitude of

imperfections. It is important to note that the dependence of the critical load on the hole size

has a similar form as the known imperfection sensitivity. The reduction factor reachs a

maximum (asymptotic) value where the opening size corresponds to the half­length of the

buckling wave, Figure E22. It is shown in Figure El9 that geometrical imperfection have no

longer any effect on acr for f > 2,94 i.e. for the opening equal or larger than the half wave of

the Yoshimura pattern.

It follows from the geometry of a single buckle, shown in Figure E22 that one half of the

length of the square opening, inscribed into the diamond shape is equal to

2·94

7 A7 r = = 1,47 .

It should be noted that for r = 1,47, there is some effect of imperfections, which

disappears for f = 2,94, see Figure El 9. The above construction constitutes a conservative

approach to the description of the reduction factor, see Figure E26.

In view of the above discussion, the circumferential and axial wavelengths are taken to be

the same, £n = £m Using the well known expression for the half wavelength £n for a shell

under an external pressure we have

2.3 r Í. =

tørøJT Therefore, using a definition of r the limiting value between the coupling and no­

coupling response is

t. 2.3 r

r'"^~~ „vr [me It is proposed to define the limit of applicability of the regime Φ (no effect of opening) as

219

5 '

The empiricai constant - comes from the analysis of experimental results shown in

Figure El3 and El 8.

In order to define ordinate of point D, it is hypothesised that because imperfections and opening effect is in a certain sense equivalent, the validity of the straight line segment © can be extended all the way to f = 0. The ordinate of the point D, represent the maximum reduction factor corresponding to the lower limit of the force-shortening curve. Two approaches are possible, the first with the most unfavourable geometrical imperfections, the second with a medium imperfection.

On Figure E24 results are presented on the experimental critical load of a cylinder with most unfavourable geometrical imperfections. The most unfavourable form of imperfection is that of the Yoshimura type. In our study, tests was run in two phases. In the first phase shells were brought into the post-buckling stage and subsequently unloaded. Then the same pre-deformed shells where subjected to axial load. It was found that the new critical buckling load is equal to

(o- ) = 0.2 σcr. V cry new cr

It should be pointed out that experimental points corresponding to shells with or without openings fall on the same horizontal line. The interesting results confirm once again the equivalence between openings and maximum amplitude geometrical imperfections.

The imperfection reduction factor for the studied geometry of the shell obtained from the current rules, [7] is α 0 = 0.409.

It is seen that the reduction factor 0,2 of the extreme case of Yoshimura imperfection is half of the reduction factor obtained for the « industrial » imperfection using the rules. It is proposed to generalise this observation and assume that the ordinate of the point D is aH — 0.5 α 0. Note that in view of the results shown in Figure E25, the above proposition is very conservative.

The point of departure of the second approach is again the equivalence of the sensitivity to larger amplitude geometrical imperfection and the effect of the size of holes greater than re. The above similarity permits one to make use of the concept of a «lower limit» for the critical load of shells without cut-outs. The present hypothesis is confirmed by two independent observations :

- the critical load relationship a ( r ), extrapolated to f = 0 gives the same value 0.37 acr

for both single and multiple openings.

- the critical load found by numerical calculations for a geometrically imperfect shell gives the lower limit equal 0.37 acr.

220

With this hypothesis the value of the critical load is

( " ­ L , « 0,37o­cr

oraH = 0,9 a 0

The second approach is valid when the «best» representations of the real behaviour of the

shell with opening is available. The use of this representations considers only the effect of

geometrical imperfections and the internal bending but does not consider the imperfection of

loading.

E4.3.2 Determination of the slope

The preceding results as applied to the shell with equi­distanced openings indicates that

the effect of the opening angle is proportional to the loss of cross­section produced by the

openings. Indeed, the broken line corresponding to the real weakened section is parallel to the

experimental curve (full line). This means that slopes of both curves are the same and equal to

03 ­ Χ ­ ­ °°' whereby and^40 denote the weakened and original cross­sections, respectively.

This observation holds also for a single opening with a correction factor that includes the

moment of inertia of the section (because of internal bending). It can be shown that the

corrected value of the slope is

where I0 and Ij denote, respectively, the moment of inertia of the weakened and initial cross­

sections

Similarly, v¡ and v0 denote positions of extreme fibers from the bending axes of the cross­

section. This proposition is confirmed by Figure E23 which shows that the slopes of the lines

corresponding to a single openings are the same. Using the proceeding equation, the slope for

the case of one opening is ß3 = ­ 0.02.

221

E4.4 Limits of validity of the proposed rules

The rules described in this chapter are developed on the basis of limited number of tests and thus can not cover a wide range of cylindrical shells and cut-outs encountered in practical applications. It is useful to recall limits of applicability and restrictions on various geometrical parameters of the problem. The ranges of validity of various parameters and assumptions are :

r

where £ a distance between two cross-sections restraint from warping and ovalisation.

2) - > 212 t

3) 0 < f < fm for square and circular openings

4) rt < r < rm for rectangular opening

where rm = 6.61 (corresponding to the opening angle θ = 45°).

5) - Axial loading only. No external bending applied. Note that in the presence of a single opening, an internal bending and relative rotations between cross-sections A and Β may develop.

6) - Existence of a single opening or multiple, equi-distanced openings in the circumfe­rential direction.

7) α H = 0,9 a 0 for a structures of quality, in particular as far as the loading is concerned

aH = 0,5 α 0 for a classical structures

Figure E26 shows the application of the rule for two classes of quality of the shells

- classical structure - quality structures

as compared with the observed behaviour. The value a0 is actually the same for the above two classes.

The same concept of constructing the design rule can be extended beyond the above limits, in a continuation of this research.

222

θ (degree)

2c (mm)

f

10

8.64

1.47

20

17.28

2.94

30

25.92

4.40

45

38.88

6.61

60

51.84

8.81

Table El - Dimensions of centrally positionned square opening

r

f

4.40

1

4.40

0.75

4.40

0.50

4.40

0.25

4.40

0

Table E2 - Distance of the lower edge of the square opening from cross section Β

nb

1.47

2

4.40

2

4.40

3

8.81

2

Table E3 - Number of equi-distanced square openings

r

ho/1

2.94

0.17

2.94

0.25

2.94

0.37

2.94

0.50

2.94

0.75

Table E4 - Higth of centrally positionned rectangular opening

ho/1

f

0.17

2.94

0.17

4.40

0.17

6.61

0.17

8.81

0.17

13.23

Table E5 - Dimensionless curvilinear width of centrally positionned rectangular opening

223

r 0.15 0.28 0.73 1.46 2.88 4.34 5.75

Table E6 - Dimensionless diameter of centrally positioned circular opening

E

ν

fl

fr

213960 MPa

0,3

197 MPa

630 MPa

Young modulus

Poisson ratio

proportionality limit

conventional yield stress σ0>2

Table E7 - Materiel constants

224

Figure £1 - Section of a cylindrical shell and notation

case 1 case 2 case 3 case

case 5 case 6 case 7

Figure E2 - Shapes and positions of openings

225

1 * 8 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Strain %

Figure E3 - Material characteristic of a coupon cut-out in the meridional direction of a shell

226

100 μιτι Amplitude (inward)

Ζ [mm]

àio Angle [degree]

100 μm Inward

Ζ [mm] eo

Angle [degree]

100 μπι f Inward

Angle [degree]

Figure E4 - Measured profiles of geometrical imperfections for

three different cases of openings

227

Figure E5 ­ Boundary and loading conditions

■ ^ ■ ^ ■ ^ ■ ^ ■ j

HHjjjH^ iiii ! ' i f røo*SS?S*f

^ÍAs­S >í ili |tatftå%% « « « « « « Μ Α

1 11 IIII f» Mte «n

IM ■

■^^^■■i Τ ι , in ι '1 i l 1 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ B

»« ­" & ■ O »Ί O k* ■

¿^,«^.­.­'«.>·.­« rj^j'jXj­'­r^'^Tt^m

ίί^ί·ίί·4?»?Η Ti 'j ■

î'-i'i^r.fjr^K,^

■ί ¿Λ. ■

i?'i?'*."ït­ Η

&}£&&»«■

uC'r­B

. . . . . . . . H

¡J_£££££ol

Figure E6 ­ Finite Element mesh of a cylinder with an opening

(3500 elements for one half of the shell)

228

12,0.

Β

•i ιο,α Q.

ε S 8,0

6,0

4,0.

2,0-

0 O

First local buckling \\ Collapse

buckling

— r = l,47 — r = 2,49

r = 4,40 F = 6,61

^ r = 8,81

20 40 60 80 100 120 140 160 180 End shortening (μπι)

Figure E7 - Compression of load-deflection curves for cases 1 and 2 showing an effect of the hole size

ε «a 0)

Ό

ε <

360 Angle (Degree)

Initial 30% (L.B.) 50%(L.B.)

— - 80%(L.B.) L.B.

Figure E8 - Growth of circumferential profil accross the opening. Note large displacement around the rim of the hole

229

Figure £9 - Evolution of buckling forms (prebuckling, local buckling (LB), global buckling (CB). White represents peaks and blue represents valeys as viewed from inside

230

. /*

Cu 0.8

1 1 0.6 ..

IM

υ

1 0.4 .§

IM O

& 0.2

First local buckling

+ + ^ 8 9 10

Opening angle r

Figure E10 - Reduction of critical loads with the size of an opening (case 2)

% i *

Ό es

_o

"a o •■ss u u

a

0.9

0.8--

0 . 7 -

0.6

0.5

0.4-.

0.3 ■

0.2

0.1

0

0

Circumferential equi-distant

opening (2,3,4 openings)

1 opening

10 12 14 16 18 20

Added opening angle Σγ

Figure E l l - Reduction of the first local buckling load with the size and number

of openings (case 2 and 6)

231

"O a o

c ε

1 * ./ι

0.8 -

0.6 . .

•C 0.4 Ο.

0.2

First local buckling

+ + H

8 9 10 opening angle 7

Figure E12 - Reduction of critical loads with an opening angle showing weak effect of the height and position of a hole in the meridional direction

o Ρ*

•α ra o

u

υ

e ε 'E tu η . χ

0.8 1

0.6

0.4

0.2 --

Χ

Collapse buckling (square and circular

openings )

First local buckling (square openings)

8 9 10 opening angle Τ

Figure E13 - Dimensionnel critical load versus and opening angle for square and circular holes. Note that there is no local buckling for a circular opening and a generalised buckling load is the same for both shapes.

232

(a) prescribed parallel displacement of the cross-section A

(b) displacement and rotation allowed

GNA(a) GMNA(a)

L/A. (a)

0 1 2 3 4 5 6 7 8 9 1

Opening angle F

Figure E14 - Finite element results for two types of boundary and loading conditions (a) and (b) and linear (LA) and non-linear (GNA and GMNA) formulation

9.50

s o co Ol

β .

ε ο υ « 3

9 00

7.50

6.00

4.50

3.00

1.50

Collapse buckling

First local buckling

Experiment

Numeric Analysis GMNA(b

0 20 40 60 80 100 120 End-shortening (μιη)

Figure E15 - Axial compression versus end-shortening measured at three equi-distant points in the meridional direction. Comparison of experimental and numerical results. Note the effect of internal rotation.

233

GMNA (b) GMNA (b)

« Experiment (J)

L.A. * (J & K)

(a) prescribed parallel displacement

of the cross ­ section A

(b) displacement and rotation

allowed

­1800 ­1200 ­600 600 1200

radial displacements (μπι)

Figure E16 ­ Evolution of radial displacement at two points (J and K) with increasing

load. Comparison of experimental results and linear and non­linear analysis.

T3 a o

o

1 •β

CD

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1 ­

0

9 10

Figure El 7 ­ Effect of dimensionless length Üv on critical buckling load

(numerical study GMNA (b))

234

P<r_1

„° 0.9

CS _o

Έ u 'S c

0.8

0.7 -

0.6 -

0.5 -

0.4 * .

cu CU α °·3

0.2

0.1 -I

0

0

J. H. STARNES r/t = 400

INSA LYON r/t = 283

Φ First local buckling * A Collapse buckling a

2c = diameter or length

** ■ A *

I

■

1

♦ 5

1

4

1

è ■ ♦

' 1

9

1 1 1

8 9 1

opening angle r

Figure E18 ­ Comparison of INSA tests with earlier results by STARNES [2]

_ i

u 0.9 υ

PH

"­ 0.8 Ό ca å 0.7 ca

Έ 0.6 u

0.5

ΡοΌ.4

0.3

0.2

0.1

0

Per

\ O

\

Numerical analysis

- with geometrical imperfection GMNIA(b)

^ / - perfect shell GMNA(b)

coupling Κ > <-

(First local buckling)

no coupling (Θ L 450)

3 4 5 6 7 8 9 1 Opening angler

Figure E19 - Comparison between experimental results and numerical calculation for perfect (GMNA) and imperfect (GMNIA) shells. Note the existence of two ranges. For small r there is a coupling between geometrical imperfection and hole size. For medium opening, there is no coupling

235

Figure E20 - Conceptual sketch of the "opening reduction factor" function

236

Figure 21 - Global buckled forms of cylinders for different cases of opening .White represents peaks and blue represents valeys as viewed from inside

237

Figure E22 - Circumferential and meridional profil of initial geometrical imperfection and sizes of an opening superimposed on diamond buckle pattern

Pn— ' π 0

0.9 -

2; 0.8 -Ό

3 0.7 -73 ν ξ 0.6 -CI

3 05 -E "E 0.4 -α. χ w 0.3 -

0.2 -

0.1 -

0

*ν- s —. ^

Ν

1 —

Α,/Α„ 1 0

* ^ _ * V ^^"^*»^^s

1 opening

1 1—

" " - - - _ _ "- ·— — ___

~~ — ~-

^ ν ^ * ^ ^ ^ ^ <Γ^ ^ _ Ν ^ ^ * * « « ^ ^

^ ^ Ì 5 ­

s

/ " ­ s

Α, Ι, ν0/ΑοΙον, 1 1 υ υ u ι

1 1 1 1—

___ ' —.

Circumferential equi­distant opening

(2,3,4 openings)

1 1 1

10 12 14 16 18 20

Figure Ε23 ­ Measured (full line) and calculated (dotted line) slopes for shell

with and without « internal » rotation

238

Pc r ­ 1 *

ca o

0.8 ­­

ca u

2 0.6 u u

0.4

0.2 ?<* ­ ♦

Asymptotic limit (damaged shell)

First local buckling

1 1 1 1 1 1 1 1 1 1

0 1 2 3 4 5 6 7 8 9 1 opening angler

Figure E24 ­ Critical buckling load of predamaged shells with and without

opening compared with vergin shells. Experimental results

L.A. ι

GMNA (b)

Numerical

calculation

(Perfect shell)

Numerical

calculation GMNIA (b)

imperfect shell (w = 2t)

experiment (damaged shell)

150 200 250

End shortening (μπι)

Figure E25 ­ Correlation between calculation (medium imperfection) and

experimental (very large imperfection) load shortening curves

239

S 0.8 .2 e o

­4­»

O 3

•o 0.6

α

ir

0.4 ­­·ν·. . .

r/í = 283

0.2

10

Figure E26 ­ Comparison between critical load of shell (full line and ir )

and proposed design rule for classical construction (broken line) and for

quality construction (dotted line)

240

REFERENCES

[1] ­ TENNYSON R.C. ­ «The effect of unreinforced circular cutouts on the buckling of

circular cylindrical shells under axial compression» ­ Journal of Engineering for

Industry, November 1968, pp. 541­546

[2] ­ STARNES J.H. ­ «The effects of cutouts on the buckling of thin shells. Thin shell

structures», Ed. Y.C. Fung E.E., Sechler, Englewood Cliffs (New York) : Prentice Hall,

Inc., 1974, pp. 289­304

[3] ­ TODA S. ­ «Buckling of cylinders with cutouts under­axial compression» ­ Journal of

Experimental Mechanics, December 1983, Vol. 3, pp. 414­417

[4] ­ KNODEL P., SCHULZ U. ­ «Stabilité de cheminées d'acier à ouvertures dans les

tuyaux» ­ Stahlbau (Der), 1988, Vol. 57, n° 1, pp. 13­21

[5] ­ ALMROTH B.O., HOLMES A.M. ­ «Buckling of shells with cutouts. Experimental and

analysis» ­ Journal of Solids Structures, August 1972, Vol. 8, pp. 1057­566

[6] ­ SAMUELSON A.L., EGGWERTZ S. ­ «Shell stability handbook» ­ London : Elsevier

Applied Science, 1992, 278 pages

[7] ­ ECCS/TWG8.4 ­ «Buckling of Steel Shells. European Recommendations», fourth

Edition, n° 56, 1988

[8] ­ AL SARRAJ M. ­ «Effets des ouvertures sur la stabilité des coques cylindriques minces

soumises à compression axiale» ­ Thèse de Doctorat, INSA de LYON (France),

5 octobre 1995, 260 pages

[9] ­ AL SARRAJ M., LIMAM Α., JULLIEN J.F. ­ «A study of the effects of opening on the

stability of thin cylindrical shells under axial compression» ­ Nordic Steel Construction

Conference '95, MALMÖ (Suède), June 19­21, 1995

[10] ­ JULLIEN J.F., LIMAM Α., AL SARRAJ M. ­ «Effect of openings on the buckling of

cylindrical shell subjected to axial compression» ­ EUROMECH Colloquium 345

«Stability and Bifurcation in Solids Mechanics», PARIS, May 29­31, 1996

[11] ­ DUMAS­ROSSIGNOL Ch., JULLIEN J.F. ­ «Research on stability of shell structures

and design of tins cans» ­ International Conference on «Stability of structures», ICSS

95, PSG College of Technology, COIMBATORE (Inde), June 7­9, 1995, Invited Paper

[12] ­ Draft Eurocode 3 Part 3.2 «Chimney» ­ Annexe C «Stability of shells», August 1994

241

3. Overall concluding remarks The common overall objective of the project was to enhance the knowledge and

understanding concerning some practical shell buckling problems, where clear gaps in currently available design recommendations had been identified, and to use the results developed in the course of this research to develop appropriate guidance. A well defined strategy involving combinations of experimental and numerical/analytical studies was envisaged and this was largely followed as originally planned, see Parts A to E of this report. Most strands of structural analysis are represented in the range of studies undertaken across the different sub-projects, and this is, in itself, an important contribution which can be of use to shell buckling designers and/or specialists who need to appreciate the capabilities of various methods, as well as their limitations.

As can be seen from individual sub-project conclusions presented in Parts A to E, the overall objective has been largely met, since, in all cases, there is a variety of supporting studies leading to concise guidelines and/or proposed procedures for design against buckling, closely linked to the philosophy and format of the ECCS recommendations and the corresponding EC3 parts. The comments in this section only serve to highlight the design related conclusions.

Thus, the range of applicability of available rules on unstiffened cones in compression is now more clearly defined and the precise specification of the equivalent cylinder needed for imperfection sensitivity and slenderness calculations has been further validated. On the corresponding stringer-stiffened conical shell problem, a design procedure is in place which could be introduced into the recommendations.

On locally axial loaded cylinders, a design procedure for a range of support width parameters has been proposed which could supplement available guidance on uniformly compressed cylinders. In addition, the extent of end reinforcement to enable more efficient design and the important effect of axial load and internal pressure interaction were also addressed in this study from a practical viewpoint.

The problem of designing shell assemblies, in particular cone/cone or cone/cylinder combinations, had to be tackled in a different way to that employed for basic shell geometries. Design-orientated results include the specification of the type of numerical analysis required in order to estimate the buckling strength. As shown, the onerous (and open-ended) task of analysing imperfect geometries can be circumvented by the design engineer through less demanding numerical analyses coupled with reduction factors specified in current rules.

The conditions leading to favourable post-buckling behaviour in certain cylinder geometries under wind loading were studied and the potential for increased design efficiency was demonstrated through wind tunnel testing. The data collected are valuable for validation work in this challenging area, which is still in need of development.

The design of cylinders with openings up to l/8th of the circumference can now be based on rules which subdivide the problem into three ranges, i.e. opening dominated response, imperfection dominated response and intermediate response. Limits regarding opening sizes are given in order to distinguish between these three ranges and appropriate expressions for estimating the buckling strength have been proposed. The influence of other spatial characteristics of openings has also been quantified.

Finally, it should be stated that, although significant progress has been made in the areas prioritized within this project, shell buckling design remains comparatively under­developed. It is clear from the results obtained herein and other recent studies that outstanding work is primarily related to the relaxation of idealised or conservative assumptions which, in the end, penalise the actual design or created difficult construction conditions. Boundary effects, including the interaction with end-stiffeners, has been highlighted in some of the sub-projects and although the behaviour is now better understood, there is still some way before cases can be codified for design purposes. Over-conservatism in imperfection reduction factors can also be addressed in a more rationale way given the confidence gained with numerical tools.

243

European Commission

EUR 18460 — Properties and in-service performance Enhancement of ECCS design recommendations and development of Eurocode 3 parts related to shell buckling

R. Saikin

Luxembourg: Office for Official Publications of the European Communities

1998 — 243 pp. — 21 χ 29.7 cm

Technical steel research series

ISBN 92-828-4414-5

Price (excluding VAT) in Luxembourg: ECU 41.50

Buckling is an important limit state for the design of all thin-walled steel shell structures, e.g. tanks, silos, chimneys, tubular towers, pipelines, etc. Guidance for the buckling design check is given in the European recommendations on buckling of steel shells, edited by ECCS. These design rules are being worked into the relevant parts of Eurocode 3. However, the ECCS recommendations still have a number of gaps where for particular shell buckling problems no design recommendations are available. Of these gaps, the following five were identified as most essential: (a) stability and strength of stiffened conical shells; (b) local loads on cylindrical structures; (c) shells of revolution with arbitrary meridionial shapes — buckling design by use of computer analysis; (d) thin-walled shells under wind loading; (e) effects of cut-outs and openings in shells.

Research was focused on these topics within five subprojects handled by separate teams but coordinated by ECCS-TWG 8.4. The common objective was to gain deeper knowledge about the buckling characteristics of the particular case. Experimental and numerical (except (d)) investigations were carried out. From the results, guidelines and procedures have been developed and are proposed for design against shell buckling, closely linked to the philosophy and format of the ECCS recommendations and the corresponding EC 3 parts.

ELQIQUE/BELGIÉ NEDERLAND CYPRUS INDIA

MI I O· Lannoy

/enue du Rol 202/Koningslaan 202 1190 Bruxellos/Brussel

IL (32­2) 538 43 08 ix ¡32­2) 538 08 41 mail: lean.de.lannoyOlnfoboard.be RL: http://www.jean­de­lannoy.bo

ι librairie européenne/De Europee« »khan del

je de la Lol 244/Wetstraat 244 1040 Bruxollos/Brussol il. (32­2) 295 28 39 IX (32­2) 735 08 80 mail: mallOllbeurop.be RL· http://www.llbeurop.be

onlteur belge/Belglech Staatsblad

je de Louvain 40­42/Leuvenseweg 40­42 1000 Bruxolles/arussel il. (32­2)552 2211 ix (32­2) 511 01 84

WMARK

H. Schultz Information A/S irstedvang 10­12 (­2620 Albertslund .(45)43 63 23 00 «(45)43 6319 69 mall: schultzOschultz.dk IL· http://www.schultz.dk

•UTSCHLAND

Indesanzeiger Verlag GmbH

rtrtebsabtellung isterdamer Straße 192 50735 Köln 1.(49­221)97 66 80 χ (49­221 97 66 82 78 Mall: vertrlebObundesanzelger.de IL· http://www.bundesanzeiger.de

ΛΑΔΑ/GREECE

C. Eleftheroudakti SA

smattonai Bookstore nepistimiou 17 1­10564 Albina I. 30­1) 331 41 80/1/2/3/4/5 χ (30­1)323 98 21 nail: elebooksOnetor.gr

PAÑA

Μ η Oficial del Estado

ifalgar, 27 18071 Madrid .(34)915 38 21 11 (Ubros)/

913 84 17 15 (Suscripciones) < (34) 915 38 21 21 (Libros)/

913 84 1714 (Suscripciones) nail: clientesOcom.boe.es IL· http://www.boe.es

ndl Prensa Libro«, SA

Steno, 37 18001 Madrid .(34 914 36 37 00 c (34 915 75 39 98 nail: librerlaOmundiprensa.es L: http://www.nundiprensa.com

ANCE

jmal officiel

vice des publications des CE rue Desaix 5727 Paris Cedex 15 .(33)140 58 77 31 I 33 140 58 77 00

iLAND

vernment Supplies Agency )lications Section Harcourt Road )lin2 (353­1)66131 11

1(353­1)47527 60

LIA

osaSpA

Duca dl Calabria. 1/1 sella postale 552 1125 Firenze (39­55)64 54 15

: (39­55) 64 12 57 nail: licosaeitbcc.il L: http://www.ftbcc.lt/llcosa

KEMBOURG

•sagerles du livre SARL

uè Raiffeisen 411 Luxembourg (352)4010 20

: (352) 49 06 61 iaii:mdlOpLlu L: http://www.­ndl.lu

»moments:

•sagerles Paul Kraus

rue Christophe Plantin 339 Luxembourg . (352) 49 98 88­8 : (352) 49 98 88­444 Tall: mpkOptlu L: http://www.mpk.lu

SDU Servicecentrum Uitgevers Cbristoffel Plantijnstraal 2 Postbus 20014 2500 EA Den Haag Tel. (31­70) 378 98 80 Fax (31­70) 378 97 83 E­mail: sduOsdu.nl URL· http://www.sdu.nl

ÖSTERREICH

PORTUGAL

BÄLGARIJA

Europress Euromedia Ltd

59, blvd Vitosha BG­1000 Sofia Tel. (359­2) 980 37 68 Fax (359­2) 980 42 30 E­mail: Milena® mbox cit.bg

CESKAREPUBLIKA

USIS

NIS­prodejna Havelkova 22 CZ­130 00Praha3 Tel. (420­2)24 2314 86 Fax (420­2) 24 23 11 14 E­mail: nkpospOdec.nls.cz URL· http://www.nis.cz

Cyprus Chamber of Commerce and Industry

PO Box 1455 CY­1509 Nicosia Tel. (357­2) 66 95 00 Fax (357­2) 66 10 44 E­mail: info®ceci.org.cy

EESTI

Manz'sche Verlags­ und Universitätsbuchhandlung GmbH Kohlmarkt 16 A­1014Wien Tel. (43­1)5316 1100 Fax (43­1) 53 16 11 67 E­Mail: bestellen O manz.co.at URL· http://www.austria.EU.net:81/manz

Eesti Kaubandus­Tööstuskoda (Estonian Chamber of Commerce and Industry)

Toom­Kooll 17 EE­0001 Tallinn Tel. (372) 646 02 44 Fax (372) 646 02 45 E­mail: elnfoOkoda.ee URL· http://www.kodaee

MAGYARORSZAG

Distribuidora de Livros Bertrand Ld."

Grupo Bertrand, SA Rua das Terras dos Vales, 4­A Apartado 60037 P­2700 Amadora Tel. (351­2) 495 90 50 Fax (351­2) 496 02 55

Imprensa Nacional­Casa da Moeda, EP

Rua Marques Sá da Bandeira, 16­A Ρ­1050 Lisboa Codex Tel. (351­1) 353 03 99 Fax (351 ­1)353 02 94 E­mail: del.incm θ mail.telepac.pt URL: http://www.lncm.pt

SUOMI/FINLAND ~

Akateeminen Kirjakauppa/Akademlska Bokhandeln

Keskuskatu 1 /Centralgatan 1 PL/PB 128 FIN­00101 Helsinki/Helsingfors pytfn (358­9) 121 44 18 Fyiax (358­9) 121 44 35 Sähköposti: akatilausOstockmann.fi URL· http://www.akateeminen.com

SVERIGE

BTJAB

Traktorvägen 11 S­221 82 Lund Tfn. (46­46)18 00 00 Fax (46­46) 30 79 47 E­post btjeu­pubObtj.se URL: http://www.btj.se

UNITED KINGDOM ~

The Stationery Office Ltd

International Sales Agency 51 Nine Elms Lane London SWS 5DR Tel. (44­171)873 90 90 Fax (44­171) 873 84 63 E­mail: ipaenquiriesOtheso.co.uk URL: http://www.the­stationery­otfice.co.uk

ISLAND ~

Bokabud Larusar Blõndal Skólavördustig, 2 IS­101 Reykjavik Tel. (354) 551 56 50 Fax (354) 552 55 60

NORGE ~

Swets Norge AS 0stenjoveien18 Boks 6512 Etterstad N­0606 Oslo Tel. (47­22) 97 45 00 Fax (47­22) 97 45 45

SCHWEIZ/SUISSE/SVIZZERA ~

Euro Info Center Schwebt

c/oOSEC Stampfenbachstraße 85 PF 492 CH­8035 Zurich Tel. (41­1) 365 53 15 Fax (41­1) 365 54 11 E­mail: eicsOosec.ch URL: http://www.osec.ch/eics

Euro Info Service

Europa Haz vlaraitszigct PO Box 475 H­1396 Budapest 62 Tel. (36­1) 350 80 25 Fax (36­1) 350 90 32 E­mail: euroinfoOmall.matav.hu URL· http://www.euroinfo.hu/index.htrn

MALTA

Miller Distributors Ltd

Malta International Airport PO Box 25 Luqa LQA 05 Tel.(356)6644 88 Fax(356) 67 67 99 E­mail: gwirthOusa.net

POLSKA

Ars Polona

Krakowskie Przedmiescie 7 Skr. pocztowa 1001 PL­00­950 Warszawa Tel. (48­22) 826 12 01 Fax (48­22) 826 62 40 E­mail: ars_polObevy.nsn.com.pl

ROMANIA

Euromedia

Str. G­ral Berthelot Nr 41 RO­70749 Bucuresti Tel. (40­1)315 44 03 Fax (40­1) 315 44 03

SLOVAKIA

Centrum VU SR

Nam. Slobody, 19 SK­81223 Bratislava Tel. (421­7)53183 64 Fax (421­7) 531 83 64

■E­mail: europOtbbl.sltk.stuba.sk URL: http://www.sltk.stuba.sk

SLOVENIA

GospodarsM Vestnlk

Dunajska cesta 5 S LO­1000 Ljubljana Tel.(386)61133 03 54 Fax (386) 611 33 9128 E­mail: repansekjOgvestnik.si URL: http://www.gvestnik.si

TURKIYE

DQnya Infotel AS 100, Yil Mahallessi 34440 TR­80050 Bagcilar­lstanbul Tel. (90­212)629 46 89 Fax 90­212 629 4627

AUSTRALIA

Hunter Publications

PO Box 404 3067 Abbotsford, Victoria Tel. (61­3)94 17 53 61 Fax (61­3) 94 19 71 54 E­mail: jpdaviesOozemail.com.au

CANADA

Renouf Publishing Co. Ltd

5369 Chemin Canotek Road Unit 1 KU 9J3 Ottawa, Ontario Tel. (1­613)745 26 65 Fax (1­613) 745 76 60 E­mail: order.deptOrenoufbooks.com URL: http://www.renoufbooks.com

EGYPT

The Middle East Observer

41 Sherif Street Cairo Tel. (20­2) 393 97 32 Fax (20­2) 393 97 32

HRVATSKA

Mediatrade Ltd Pavia Hatza 1 HR­10000 Zagreb Tel. (385­1) 43 03 92 Fax 385­1 43 03 92

EBIC India

3rd Floor, Y. B. Chavan Centre Gen. J. Bhosale Marg. 400 021 Mumbai Tel. (91­22)282 60 64 Fax (91­22) 285 45 64 E­mail: ebicOgiasbmOl .vsnl.netin URL· http://www.obicindia.com

ISRAEL

ROY International

PO Box 13056 61130 Tel Aviv Tel. (972­3) 546 14 23 Fax (972­3) 546 14 42 E­mail: royil@netvision.net.il

Sub­agent for the Palestinian Authority:

Index Information Services

PO Box 19502 Jerusalem Tel. (972­2) 627 16 34 Fax (972­2 627 12 19

JAPAN

PSI­Japan

Asahi Sanbancho Plaza #206 7­1 Sanbancho, Chiyoda­ku Tokyo 102 Tel. (81­3)32 34 69 21 Fax ¡81­3) 32 34 69 15 E­mail: booksOpsi­japan.co.jp URL· http://www.psi­japan.com

MALAYSIA

EBIC Malaysia

Level 7, Wlsma Hong Leong 18 Jalan Perak 50450 Kuala Lumpur Tel. (60­3) 262 62 98 Fax (60­3) 262 61 98 E­mail: ebic­klOmol.netmy

PHILIPPINES

EBIC Philippines

19th Floor, PS Bank Tower Sen. Gil J. Puyat Ave. cor. Tindalo St. Makati City Metro Manilla Tel. (63­2) 759 66 80 Fax (63­2) 759 66 90 E­mail: eccpcom®globe.com.ph URL· http://virww.eccp.com

RUSSIA

CCEC

60­letiya Oktyabrya Av. 9 117312 Moscow Tel. (70­95)135 52 27 Fax (70­95) 135 52 27

SOUTH AFRICA

Sarto

Saffo House NO 5 Esterhyzen Street PO Box 782 706 2146 Sandten Tel. (27­11) 883 37 37 Fax (27­11)883 65 69 E­mail: emalstarOide.co.za URL· http:Zwww.safto.co.za

SOUTH KOREA

Information Centre for Europe (ICE)

204 Woo Sol Parkte! 395­185 Seogyo Dong, Mapo Ku 121­210 Seoul Tel. (82­2) 322 53 03 Fax (82­2) 322 53 14 E­mail: euroinloOshinbiro.com

THAILAND

EBIC Thailand

29 Vartissa Building, 8th Floor Soi Chidlom Ploenchit 10330 Bangkok Tel. (66­2)655 0627 Fax (66­2) 655 06 28 E­mail: ebicbkkOksc15.lh.com URL· http7www.eblcbkk.org

UNITED STATES OF AMERICA

Beman Associates

4611 ­F Assembly Drive Lanham MD20706 Tel. (1­800) 274 44 47 (toll free telephone) Fax (1­800) 865 34 50 (toll free fax) E­mail: quetyObeman.com URL· http://www.beman.com

ANDERE LANDER/OTHER COUNTRIES/ AUTRES PAYS

Bitte wenden Sie sich an ein Büro Direr Wahl / Please contact the sales office of

our choice / Veuillez vous adresser au ureau de vent» de votre choix

you bun

NOTICE TO THE READER

Information on European Commission publications in the areas of research and innovation can be obtained from:

♦ CORDIS, the Community R & D Information Service

For more information, contact: CORDIS Customer Service, BP 2373, L­1023 Luxembourg Tel. (352) 44 10 12­2240; fax (352) 44 10 12­2248; e­mail: helpdesk@cordis.lu or visit the website at http://www.cordis.lu/

♦ Euroabstracts

The European Commission's periodical on research publications, issued every two months. For more information, contact: RTD help desk, European Commission, DG XIII, L­2920 Luxembourg Fax (352) 43 01­32084; e­mail: rtd­helpdesk@lux.dg13.cec.be

Price (excluding VAT) in Luxembourg: ECU 41.50 I S B N T S ­ ß E f l ­ L m m ­ s

• * • OFFICE FOR OFFICIAL PUBLICATIONS

• ~&f> • O F T H E E U R O P E A N

COMMUNITIES

• * • L­2985 Luxembourg 9 7 8 9 2 8 2 " 8 4 4 U 4 >