Theory Review for Cylindrical Shells

and

Parametric Study of Chimneys and Tanks

Theory Review for Cylindrical Shells

and

Parametric Study of Chimneys and Tanks

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben,

voorzitter van het College voor Promoties

in het openbaar te verdedigen

op maandag 22 maart 2010 om 15.00 uur

door

Jeroen Hendrik HOEFAKKER

civiel ingenieur

geboren te Amersfoort

Dit proefschrift is goedgekeurd door de promotor:

Prof. dr. ir. J. Blaauwendraad

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof.dr.ir. J. Blaauwendraad, Technische Universiteit Delft, promotor

Prof.dr.ir. L.J. Ernst Technische Universiteit Delft

Prof.dr. A. Metrikine Technische Universiteit Delft

Prof.dr.ir. L.J. Sluys Technische Universiteit Delft

Dr.ir. W. van Horssen Technische Universiteit Delft

Dr.ir. P. Liu INTECSEA

Ing. H. van Koten Gepensioneerd, eerder TNO Bouw

ISBN 978-90-5972-363-4

Eburon Academic Publishers

P.O. Box 2867

2601 CW Delft

The Netherlands

tel.: +31 (0) 15 - 2131484 / fax: +31 (0) 15 - 2146888

info@eburon.nl / www.eburon.nl

Cover design: J.H. Hoefakker

2010 J.H. HOEFAKKER. All rights reserved. No part of this publication may be reproduced,

stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical,

photocopying, recording, or otherwise, without the prior permission in writing from the

proprietor.

v

Acknowledgement

The majority of the research reported in this thesis was performed at Delft University

of Technology, Faculty of Civil Engineering and Geosciences under the supervision of

my promotor Prof. Johan Blaauwendraad in the Section of Structural Mechanics.

I am deeply indebted to Prof. Blaauwendraad for the journey we have travelled so far

together. I am really proud that I have been able to work with such an excellent mentor,

who in turn has been a challenging sparring partner and the source of much valuable

inspiration over these last few years. I am especially thankful for the chance to teach

students together with him on the application of shell theory, which has been of crucial

importance in my understanding of shell behaviour and in the focus of my research.

I am very grateful to Carine van Bentum for her valuable contribution to the

development of the computer program as part of her graduation project.

I would also like to thank my family, friends and colleagues at INTECSEA and the

Delft University of Technology for their interest, encouragement and support. Special

thanks go out to my colleague Pedro Ramos for the numerical simulations to validate

the computer program and to Frank van Kuijk for his help during the creation of the

cover design.

I am sincerely grateful for the sacrifices my parents have made and the possibilities

they have offered me. Dear Mother, I am sure that Dad would be as proud of this result

as you are!

Mirjam, my gratitude to you is beyond words. Your continual sacrifice, endurance and

cardinal support throughout these years have been truly admirable. At last I hope to

devote more time to you and our wonderful daughters, whom I daily thank for

enriching my world.

Utrecht, February 2010

vi

vii

Table of Contents

Acknowledgement v Summary ix Samenvatting xiii List of symbols xix 1 Introduction 1 1.1 Motive and scope of the research 1 1.2 Research objective and strategy 2 1.3 Outline of the thesis 3 1.4 Short review of the existing work within the scope 4

2 General part on shell theory 7 2.1 Introduction to the structural analysis of a solid shell 7 2.2 Fundamental theory of thin elastic shells 10 2.3 Principle of virtual work 21 2.4 Boundary conditions 26 2.5 Synthesis 28 2.6 Analysis by former authors 32 2.7 Proposed theory 42

3 Computational method and analysis method 51 3.1 Introduction to the numerical techniques for a solid shell 51 3.2 The super element approach 53 3.3 Calculation scheme 60 3.4 Introduction to the program CShell 60 3.5 Overview of the analysed structures 64

4 Circular cylindrical shells 65 4.1 Introduction 65 4.2 Sets of equations 66 4.3 The resulting differential equations 68 4.4 Full circular cylindrical shell with curved boundaries 71 4.5 Approximation of the homogeneous solution 84 4.6 Characteristic and influence length 89 4.7 Concluding remarks 92

5 Chimney Numerical results and parametric study 93 5.1 Wind load 93 5.2 Behaviour for a fixed base and free end 94 5.3 Influence of stiffening rings 112 5.4 Influence of elastic supports 137

6 Tank Numerical study 149 6.1 Introduction 149 6.2 General description of large liquid storage tanks 150 6.3 Load-deformation conditions and analysed cases 151 6.4 Content load cases 155 6.5 Wind load cases 159 6.6 Settlement induced load and/or deformation cases 166

viii

7 Conclusions 169 Appendices 175 Literature 245 Curriculum Vitae 250

List of Appendices

Appendix A Results from differential geometry of a surface 177 Appendix B Kinematical relation in orthogonal curvilinear coordinates 183 Appendix C Equilibrium equations in curvilinear coordinates 185 Appendix D Strain energy and Laplace-Beltrami operator 187 Appendix E Expressions and derivation of the stiffness matrix for the elastostatic behaviour of a circular ring 191 Appendix F Ring equations comparison 199 Appendix G Semi-membrane concept 203 Appendix H Solution to MK and SMC equations 215 Appendix I Back substitution for MK and SMC solutions 223 Appendix J Program solution for influence of stiffening rings 233

ix

Summary

Since the considerable effort in the development of rigorous shell theories dating

back to the early twentieth century many approximate shell theories have been

developed, mainly on the assumption that the shell is thin. With the development of the

numerical formulations and the continuously increasing computing power, a gradual

cessation of attempts to find closed-form solutions to rigorous formulations has taken

place. This has led to an increasing lack of understanding of the basic and generic

knowledge of the shell behaviour, the prevailing parameters and the underlying

theories, which is obviously required for the use of numerical programs and to

understand and validate the results.

Objective and scope of the research

This research project intended to combine the classic shell theories with the

contemporary numerical approach. The goal was to derive and employ a consistent and

reliable theory of shells of revolution and to present that theory in the context of

modern computational mechanics. The aim of the project was to derive an expeditious

PC-oriented computer program for that by reshaping the closed-form solutions to the

rigorous shell formulations into the well-known direct stiffness approach of the

displacement method. The objective was to conduct a generic study of the physically

and geometrically linear behaviour of the typical thin shells of revolution, i.e. circular

cylindrical, conical and spherical shells, under static loading by evaluating both the

closed-form solution to the thin shell equations and the output of the computer

program.

This research concentrated on the behaviour of circular cylindrical shells under

static loading while accounting for the axisymmetric, beam-type and non-axisymmetric

load-deformation conditions. Due to required effort identified during the development

of such a program for circular cylinders and upon inspection of the sets of equations for

conical and spherical shells, it has been decided to fully focus on circular cylindrical

shells as a first, but complete and successful step towards more applications.

Review of the first-order approximation theory for thin shells

Based on previous work, it was envisaged to employ the so-called Morley-Koiter

equation for thin circular cylindrical shells. The Morley-Koiter equation fits in the

category of the first-order approximation theory, viz. only first-order terms with respect

to the thinness of the shell are retained, resulting in an eighth order partial differential

equation. To understand the assumptions and simplifications, which are introduced to

obtain such a thin shell equation, the set of equations resulting from a fundamental

derivation for thin elastic shells is reproduced. The formulations for thin, shallow, non-

linear and cylindrical shells by some former authors are discussed and, as a result of

the comparison, a set of equations for thin elastic shells within the first-order

approximation theory is proposed. This set comprises kinematical and constitutive

relations that are complemented by the equilibrium relation and boundary conditions,

which are derived by making use of the principle of virtual work. To arrive at a

consistent and reliable theory of shells of revolution, the expansion of the strain

x

description that adopts the changes of curvature has been considered and, while

simultaneously approximating the constitutive relation, the combined internal stress

resultants of the boundary conditions are congruently approximated.

Computational method and expeditious PC-oriented computer program

The concept of generating the stiffness matrix of shell elements on basis of closed-form

solutions was already proposed as early as 1964 by Loof. Since then little effort with a

similar approach has been reported and to date the method has been employed only to

study axisymmetric structures subject to loads that are also axisymmetric with respect

to the axis of symmetry of the structure.

For shells of revolution with circular boundaries under general loading, the

numerical procedure to be performed by a digital computer is described. This approach

avoids the shortcomings of most existing element stiffness matrices and attempts to

minimise the number of elements needed to model a given problem domain. Similar to

the conventional method, the first and crucial step is to compute the element stiffness

matrix but for the super element, this is synthesized on the basis of an analytical

solution to the governing equation. The precise formulation of the classic approach is

reshaped into the well-known direct stiffness approach of the displacement method

enabling the calculation of combinations of elements and type of elements while the

valuable knowledge of the classic approach is preserved. In addition to the

conventional transition and end conditions, the method enables implementation of

stiffening rings, elastic support, prescribed displacement and various load types. Based

on the proposed solution procedure and with the mentioned functionalities, an

expeditious PC-oriented computer program has been developed using the Fortran-

package in combination with graphical software. The formulations that are

implemented in this program are based on the approximated solution to the Morley-

Koiter equation for circular cylindrical shells.

General solutions to the circular cylindrical shell equations

The proposed set of equations is formulated for circular cylindrical shells with circular

boundaries and the resulting single differential equation has been derived. An

approximation of this exact equation is introduced to arrive at mathematically the most

suitable equation for substitution with the same accuracy, i.e. the Morley-Koiter

equation.

The exact roots to the Morley-Koiter equation have been obtained and, albeit being

surplus to requirements, the presented solution is a unification of former results by

other authors. To progress towards generic knowledge of the shell behaviour based on

closed-form solutions, approximate roots have been derived for the axisymmetric,

beam-type, and non-axisymmetric load-deformation conditions. The associated

characteristic and influence lengths have been derived and discussed to facilitate

insight in the prevailing parameters of the shell response to the respective load-

deformation conditions.

xi

Parametric study of long circular cylindrical shells (chimneys)

Design formulas, based on closed-form solutions to the Morley-Koiter equation and an

equation derived by the semi-membrane concept, and numerical solutions obtained by

the developed program are provided for long circular cylindrical shell structures, i.e.

long in comparison with their radius (for example industrial, steel chimneys).

The design formula that describes the stress distribution at the fixed base of long

circular cylindrical shells without stiffening rings subject to wind load has been

derived, which is a marked improvement of the existing formula that is based on the

Donnell equation. This formula relates total membrane stress ,xx total to the beam

stress ,xx beam .

For the specified wind pressure distribution around the cylinder, this formula reads 2

2

, ,1 6.39 1

xx total xx beam

a a

l t

= +

in which the radius, length and thickness of the shell are represented by a , l and t ,

respectively, and denotes Poissons ratio of the shell material. Alternatively, this equation can be written as

4

2

, ,

2

1 6.39 1xx total xx beama

l

= +

in which the characteristic length 2l is defined by 4 2

2l atl= .

New design formulas, which describe the effect of (centric and eccentric)

stiffening rings and elastic supports (in the axial and planar directions), are presented

such that the respective influence is represented by inclusion of an additional factor in

the formula for the fixed base case without stiffening rings.

The formula for the case with stiffening rings reads 4

2

, ,

2

1 6.39 1xx total xx beam ra

l

= +

in which the stiffness ratio r represents the ratio of the bending stiffness of the

circular cylindrical shell only to the modified bending stiffness of the shell (with the

contribution of the ring stiffness per spacing).

It has been concluded that, in case of an elastic support to a long circular cylinder,

only the axial spring stiffness has to be taken into account. The formula for the case

with axial elastic supports reads 4

2

, ,

2

1 6.39 1xx total xx beam xna

l

= +

in which the normalised stress ratio xn is introduced, which depends on the respective

factors and mode numbers of the load and the parameter x , which in turn is mainly

described by the geometrical properties of the cylinder and the ratio of the axial elastic

support to the modulus of elasticity.

xii

From the comparison with the numerical results, the range of application of the

improved and new design formulas has been obtained within which a close agreement

is observed. These formulas have been shown to be applicable to cylinders for which

the characteristic length 2l is larger or equal to its radius. For ring-stiffened cylinders,

the formula has further been shown to be applicable to cylinders with ring spacing

shorter than half of the influence length of the long-wave solution for circumferential

mode number 2n = .

Numerical study of short circular cylindrical shells (tanks)

For short circular cylindrical shells (lengths in the range of 0.5 to 3 times the radius),

numerical solutions have been presented with the intention to demonstrate the

capability of the developed program to model the shell of large vertical liquid storage

tanks. Additionally, tentative insight into the response of such tank shells to the

relevant load and/or deformation conditions is provided, which is obtained by several

calculations (for the response to content or wind load or due to full circumferential

settlement) and by comparison with the insight as obtained for the behaviour of the

long cylinder.

Conclusions

This study has focused on a thorough analysis of the behaviour of circular cylindrical

shells with the following main results:

o The first-order approximation theory for thin shells and the various approaches

discussed in the literature have been reviewed and a consistent set of thin shell

equations has been proposed. On basis of the proposed set, the Morley-Koiter

equation has been identified as being the most suitable single differential

equation for deriving closed-form solutions.

o On basis of these closed-form solutions, an expeditious PC-oriented computer

program has been developed for first-estimate design of long and short circular

cylindrical shells, e.g. chimneys and tanks.

o In the literature, a design formula for the stress at the base of a chimney

subject to wind load has been developed by combining a solution obtained on

basis of the Donnell equation with finite element analysis. On basis of the

closed-form solutions to the Morley-Koiter equation, this formula has been

confirmed. As an advantage of the new solution, the design formula is

generalized with respect to the wind pressure distribution around the chimney.

o The above mentioned design formula has been extended for the influence of

elastic supports at the base of the chimney.

o The above mentioned design formula has been extended for the influence of

stiffening ring properties and spacing along the chimney.

o The range of application of these formulas has been conclusively and

conveniently obtained by comparison with results obtained with the developed

computer program.

xiii

Samenvatting

Sinds de aanzienlijke inspanningen in de ontwikkeling van strenge schaaltheorien

die teruggaan tot het begin van de twintigste eeuw zijn er veel benaderende

schaaltheorien ontwikkeld, voornamelijk op basis van de veronderstelling dat de

schaal dun is. Door de ontwikkeling van de numerieke formuleringen en de continu

toenemende rekenkracht is er geleidelijk mee gestopt om voor strenge formuleringen

oplossingen in gesloten vorm te vinden. Dit heeft geleid tot een toenemend gebrek aan

begrip van de fundamentele en algemene kennis van het schaalgedrag, de dominante

parameters en de onderliggende theorien. Dat is een spijtige ontwikkeling omdat juist

dat inzicht vereist is voor het gebruik van numerieke programmas en om de resultaten

te begrijpen en te valideren.

Doel en reikwijdte van het onderzoek

Dit onderzoeksproject beoogde om de klassieke schaaltheorien te combineren met de

hedendaagse numerieke benadering. Het aanvankelijke doel was het afleiden van een

consistente en betrouwbare theorie van omwentelingsschalen en deze theorie te

presenteren in de context van de moderne numerieke mechanica. Het project beoogde

de ontwikkeling van een snel PC-georinteerd computerprogramma door de

oplossingen in gesloten vorm voor de strenge schaalformuleringen te herstructureren

en onder te brengen in de bekende aanpak van de verplaatsingsmethode. De

doelstelling was de uitvoering van een generieke studie van het fysisch en geometrisch

lineaire gedrag van de meest voorkomende dunne omwentelingsschalen dat wil

zeggen de cirkelcilindrische, conische en bolvormige schalen onder statische

belasting door de beoordeling van zowel de oplossing in gesloten vorm van de dunne

schaalvergelijkingen en de uitvoer van het computerprogramma.

Het hier gerapporteerde onderzoek is afgebakend tot het gedrag van

cirkelcilindrische schalen onder statische belasting, waarbij drie specifieke

belastingstoestanden zijn betrokken: axiaalsymmetrie, liggerwerking en asymmetrie.

Gezien de inspanning die tijdens de ontwikkeling van een dergelijk

computerprogramma voor cirkelcilinders vereist bleek te zijn, en na beoordeling van de

sets van vergelijkingen voor de conische en bolvormige schalen is het besluit genomen

het onderzoek volledig te richten op cirkelcilindrische schalen als een eerste, maar

volledige en succesvolle stap naar andere toepassingen in de toekomst.

Terugblik op de eerste-orde benaderingstheorie voor dunne schalen

Op basis van eerder werk was de aanwending van de zogenaamde Morley-Koiter

vergelijking voor dunne cirkelcilindrische schalen beoogd. De Morley-Koiter

vergelijking past in de categorie van de eerste-orde benaderingstheorie waarin alleen

eerste-orde termen met betrekking tot de dunheid van de schaal worden meegenomen,

hetgeen resulteert in een achtste-orde partile differentiaalvergelijking. Om de

aannames en vereenvoudigingen, die tijdens de afleiding van een dergelijke dunne

schaalvergelijking ingevoerd zijn, te kunnen begrijpen is de set van vergelijkingen

gereproduceerd die uit een fundamentele afleiding voor dunne elastische schalen volgt.

De formuleringen van enkele eerdere auteurs voor dunne, licht gekromde, niet-lineaire

xiv

en cilindrische schalen worden besproken en, als gevolg van de vergelijking, een set

van vergelijkingen binnen de eerste-orde benaderingstheorie voor dunne elastische

schalen is voorgesteld. Deze set bestaat uit kinematische en constitutieve betrekkingen

die gecomplementeerd worden door de evenwichtsrelatie en randvoorwaarden, welke

door gebruik te maken van het principe van virtuele arbeid zijn afgeleid. Om tot een

consistente en betrouwbare theorie van omwentelingsschalen te komen is de

reeksontwikkeling van de rekbeschrijving op basis van de krommingveranderingen

beschouwd en, onder gelijktijdige benadering van de constitutieve relatie, zijn de

gecombineerde interne spanningsresultanten van de randvoorwaarden overeenkomstig

benaderd.

Numerieke methode en snel PC-georinteerd computerprogramma

Het genereren van de stijfheidsmatrix van schaalelementen op basis van oplossingen in

gesloten vorm werd in 1964 reeds voorgesteld door Loof. Sindsdien is er weinig

inspanning met betrekking tot een soortgelijke aanpak gemeld en tot op heden is de

methode slechts toegepast om axiaalsymmetrische structuren te bestuderen onder

belastingen die ook axiaalsymmetrisch zijn met betrekking tot de symmetrieas van de

structuur.

Voor omwentelingsschalen met cirkelvormige randen onder algemene belasting is

de numerieke procedure beschreven die door een digitale computer uitgevoerd moet

worden. Deze aanpak vermijdt de tekortkomingen van de meeste stijfheidmatrices van

bestaande elementen en beoogt om het aantal elementen dat nodig is om een bepaald

probleemdomein te modelleren tot het minimum te beperken. We noemen zulke

elementen super elementen. Net als in de standaard eindige-elementenmethode (EEM)

is de eerste en cruciale stap het berekenen van de stijfheidsmatrix per element, maar

voor het super element is deze synthese uitgevoerd op basis van een analytische

oplossing van de heersende differentiaalvergelijking. De precieze formulering van de

klassieke theorie is omgevormd tot de bekende aanpak van de verplaatsingsmethode

hetgeen het mogelijk maakt om combinaties van elementen en type elementen te

berekenen, terwijl de waardevolle kennis van de klassieke theorie bewaard is gebleven.

In aanvulling op de conventionele overgangsvoorwaarden en eindvoorwaarden maakt

de methode de implementatie van verstijvingsringen, elastische ondersteuningen,

voorgeschreven verplaatsingen en verschillende soorten belasting mogelijk. Op basis

van de voorgestelde oplossingsprocedure en met de genoemde functionaliteiten is, met

behulp van Fortran in combinatie met grafische software, een snel PC-georinteerd

computerprogramma ontwikkeld. De formuleringen in dit programma zijn gebaseerd

op de benaderde oplossing van de Morley-Koiter vergelijking.

Algemene oplossingen voor de cirkelcilindrische schaalvergelijkingen

De voorgestelde set van vergelijkingen is voor cirkelcilindrische schalen met

cirkelvormige randen geformuleerd en de daaruit voortvloeiende enkele

differentiaalvergelijking is afgeleid. Een benadering van deze exacte vergelijking is

ingevoerd om te komen tot de mathematisch meest geschikte vergelijking voor

terugsubstitutie met dezelfde nauwkeurigheid, dwz de Morley-Koiter vergelijking.

xv

De exacte wortels van de Morley-Koiter vergelijking zijn verkregen en hoewel

deze expressies de vereisten overtreffen is de gepresenteerde oplossing een unificatie

van eerdere resultaten van andere auteurs. Om te komen tot generieke kennis van het

schaalgedrag op basis van oplossingen in gesloten vorm zijn de benaderde wortels

afgeleid voor de drie eerder genoemde specifieke belastingstoestanden

(axiaalsymmetrie, liggerwerking, asymmetrie). Bijbehorende karakteristieke lengtes en

invloedslengtes vergemakkelijken het inzicht in de parameters die het schaalgedrag in

de betreffende belastingstoestanden bepalen.

Parametrische studie van lange cirkelcilindrische schalen (schoorstenen)

Ontwerpformules zijn verstrekt voor cilinders die lang zijn in vergelijking met hun

straal (bijvoorbeeld industrile, stalen schoorstenen). De formules zijn gebaseerd op de

oplossingen in gesloten vorm van de Morley-Koiter vergelijking en van een

vergelijking die is afgeleid met behulp van het semi-membraan concept. Ook

numerieke oplossingen hebben een bijdrage geleverd.

De ontwerpformule voor de spanningsverdeling aan de onderkant van lange

cirkelcilindrische schalen zonder verstijvingsringen onder windbelasting is afgeleid.

Deze is een duidelijke verbetering van de bestaande formule die op de Donnell

vergelijking gebaseerd is. De formule relateert de totale membraanspanning ,xx total aan

de spanning ,xx beam volgens de liggertheorie. Bij de gebruikte winddrukverdeling rond

de cilinder luidt de formule 2

2

, ,1 6.39 1

xx total xx beam

a a

l t

= +

waarbij de straal, lengte en dikte van de schaal door respectievelijk a , l en t worden

vertegenwoordigd en de dwarscontractiecofficint van het materiaal weergeeft (Poisson verhouding). Deze vergelijking kan tevens geschreven worden als

4

2

, ,

2

1 6.39 1xx total xx beama

l

= +

waarin de karakteristieke lengte 2l is gedefinieerd door 4 2

2l atl= .

Nieuwe ontwerpformules worden gegeven voor het effect van (centrische en

excentrische) verstijvingsringen en elastisch ondersteuningen (in axiale en

omtreksrichting). Het effect is beschreven met een extra factor in de formule voor de

spanning onderin de schaal bij afwezigheid van verstijvingsringen. De aangepaste

formule luidt: 4

2

, ,

2

1 6.39 1xx total xx beam ra

l

= +

waarin r de verhouding is tussen de buigstijfheid van alleen de cirkelcilindrische

schaal en de gewijzigde buigstijfheid van de schaal (met de bijdrage van de

ringstijfheid per afstand).

xvi

Voor het geval van een elastische ondersteuning van een lange cirkelvormige

cilinder hoeft alleen de axiale veerstijfheid in rekening gebracht te worden. De formule

luidt 4

2

, ,

2

1 6.39 1xx total xx beam xna

l

= +

waarin de genormaliseerde spanningsverhouding xn is ingevoerd, welke afhangt van

de respectieve factoren, het aantal golven (in omtreksrichting) van de belasting en de

parameter x ; deze is op zijn beurt vooral beschreven door de geometrische

eigenschappen van de cilinder en de verhouding tussen de axiale elastische

ondersteuning en de elasticiteitsmodulus.

Uit een vergelijking met numerieke resultaten is het toepassingsgebied van de

verbeterde en nieuwe ontwerpformules verkregen. De formules zijn van toepassing op

cilinders waarvoor de karakteristieke lengte 2l groter dan of gelijk aan de straal is. De

formule voor ring-verstijfde cilinders is van toepassing voor cilinders met een

ringafstand korter dan de helft van de invloedslengte van de lange golf in de oplossing;

bedoeld is de invloedslengte voor de belastingscomponent met twee golven in

omtreksrichting ( 2n = ).

Numerieke studie van korte cirkelcilindrische schalen (tanks)

Voor korte cirkelcilindrische schalen (lengtes 0,5 tot 3 maal de straal) zijn numerieke

oplossingen gepresenteerd om de geschiktheid van het programma te demonstreren

voor het modelleren van de schaalwand van grote opslagtanks. Daarnaast is inzicht

verkregen in de reactie van dergelijke tankwanden onder de beschouwde drie

specifieke belastingstoestanden. Dit is bereikt op basis van verscheidene berekeningen

en door vergelijking met het inzicht dat verkregen is voor het gedrag van de lange

cilinder. Voor de berekeningen is gewerkt met de belasting ten gevolge van de

tankinhoud of winddruk; ook is de response op een varirende zakking langs de

volledige omtrek onderzocht.

xvii

Conclusies

Dit onderzoek heeft zich gericht op een grondige analyse van het gedrag van

cirkelcilindrische schalen met de volgende resultaten:

o De eerste-orde benaderingstheorie voor dunne schalen en de verscheidenheid

in aanpak in de literatuur zijn in een terugblik gevalueerd, en een consistente

set van dunne schaalvergelijkingen is voorgesteld. Op basis van deze set is de

Morley-Koiter vergelijking gedentificeerd als de meest geschikte

differentiaalvergelijking voor het afleiden van oplossingen in gesloten vorm.

o Op basis van deze oplossingen is een snel PC-georinteerd

computerprogramma ontwikkeld voor een eerste ontwerp van lange en korte

cirkelcilindrische schalen zoals bijvoorbeeld schoorstenen en tanks.

o In de literatuur bestaat een ontwerpformule voor de spanningsverhoging aan

de voet van een schoorsteen. Deze is tot stand gekomen door het combineren

van een oplossing op basis van de (niet nauwkeurige) Donnell theorie en

EEM-berekeningen. De formule is bevestigd met de Morley-Koiter theorie.

Het voordeel van de nieuwe oplossing is dat de ontwerpformule

veralgemeniseerd is met betrekking tot de winddrukverdeling rond de

schoorsteen. Hij geldt ook voor andere verdelingen dan gebruikt in deze

studie.

o Bovengenoemde ontwerpformule is uitgebreid voor de invloed van een

elastische ondersteuning aan de voet van de schoorsteen.

o De ontwerpformule is ook uitgebreid voor de invloed van verstijvingsringen

langs de schoorsteen (ringeigenschappen en onderlinge afstand).

o Het toepassingsgebied van de formules is overtuigend en doelmatig verkregen

door vergelijking met resultaten van het computerprogramma.

xviii

xix

List of symbols

indices written as subscript with a specific range

( ) ( ), , 1,2,3i j k = in case of a single quantity with two indices, the following applies: - first index denotes fibre orientation or surface,

- second index denotes direction of subject quantity

( ) ( ), 1,2 = in case of a single quantity with two indices, the following applies: - first index denotes fibre orientation or surface,

- second index denotes direction of subject quantity

generic notation

A quantity within the shell space

a quantity on the reference surface or boundary line 1

,

A A matrix or vector with components A and its inverse, respectively

a vector with components a

da differential increment of quantity a

a virtual variation of quantity a a quantity a at an edge

a adjoint of quantity a ha homogeneous solution for quantity a ia inhomogeneous solution for quantity a

a amplitude of quantity a ca continuous expression for quantity a within the element ea expression for quantity a at the edges of the element na expression for quantity a at the nodes connecting the elements

a quantity a in the deformed state

...0a quantity a for mode number 0n =

...1a quantity a for mode number 1n =

...na quantity a for mode numbers 1n >

specific notation (in order of introduction)

Chapter 2

S (arbitrary or total boundary) surface

,i i

x rectangular and curvilinear coordinate system, respectively

1 2, orthogonal curvilinear coordinates of the reference surface

coordinate in the thickness direction, viz. normal to the reference surface

R position vector within the shell space

r position vector on the reference surface

n unit normal vector of the reference surface

xx

( )2ds line element on the reference surface ,

oP P point within the shell space and infinitesimal close point, respectively

iig metric coefficients along the orthogonal parametric lines

iA scale factors

1 2, Lam parameters of the reference surface

1 2,R R principal radii of curvature at the point on the reference surface

V volume

1 2,ds ds differential lengths of arc of the edge of an infinitesimal element

1 2,dS dS differential areas of a strip on the edge of an infinitesimal element

dV differential volume of a layer within an infinitesimal element

f scalar field

Laplace operator

iU displacements in the direction normal to the coordinate surfaces

i

,ii ije e extension and shear components of the strain tensor, respectively

U displacement in the thickness direction, viz. normal to the reference

surface

1 2, rotation in the

2 -direction of a fibre along the

1 -direction and

rotation in the 1 -direction of a fibre along the

2 -direction,

respectively

n rigid body rotation about the normal to the reference surface

iu displacement components at the reference surface

1 2, rotation of a normal to the reference surface in the direction of the

parametric lines 1 and

2 , respectively

u displacement components of the reference surface in the thickness

direction

11 22, normal strains of the reference surface

12 21, longitudinal shearing strains of the reference surface

1 2, transverse shearing strains

11 22, changes of rotation of the normal to the reference surface

12 21, torsion of the normal to the reference surface

,ii ij normal stress and shearing stress components, respectively

E modulus of elasticity, Youngs modulus

Poissons ratio G shear modulus

3 3 3, ,E G elastic constants specifically in the direction normal to the reference

surface

11 22, normal stresses

12 21, longitudinal shearing stresses

xxi

1 2, transverse shearing stresses

11 22,n n normal stress resultants

12 21,n n longitudinal shearing stress resultants

1 2,v v transverse shearing stress resultants

11 22,m m bending stress couples

12 21,m m torsional stress couples

t finite thickness of the thin shell

p surface force vector per unit area of the reference surface

1 2, ,p p p resultant components of the surface force vector

1 2,m m couple components of the surface force vector

f edge force vector per unit length of the boundary lines

,f uS S part of the boundary surface where the edge forces and edge

displacements are known, respectively

u displacement vector

pE potential energy

sE strain energy

PW work done by the surface force vector

FW work done by the edge force vector

sE strain energy density function

iP components per unit volume of the external force vector

iF components per unit area of the boundary surface of the external force

vector ( ) ( )1 21 1, pair of edges of constant

1

( ) ( )1 22 2, pair of edges of constant

2

1 2, ,f f f resultant components of the edge force vector

1 2,t t couple components of the edge force vector

nR point load at the corner of and in the direction normal to the reference

surface

e strain vector

s stress vector

p load vector, viz. equal to the external surface force vector

B differential operator matrix B transpose of the matrix B where the components are the adjoint

operators

,ij ijB B components of the differential operator matrix and its adjoint,

respectively

D rigidity matrix

12 12, alternative shearing strain angle quantities; shear strain and torsion of

the reference surface, respectively

xxii

12 12,n m alternative longitudinal shearing stress quantities; longitudinal

shearing stress resultant and torsional stress couple, respectively

11 22, changes of curvature, alternative deformation quantities for

11 22,

11 11,n m alternative stress quantities for

11 11,n m

22 22,n m alternative stress quantities for

22 22,n m

12 21, alternative deformation quantities for

12 21,

,m b

D D extensional (membrane) rigidity and flexural (bending) rigidity,

respectively

1 2,v v alternative transverse shearing stress resultants for

1 2,v v

12 1,n v combined internal stress resultants; the latter is similar to Kirchhoffs

effective shearing stress resultant

Chapter 3

, , orthogonal coordinate system for a shell of revolution, viz.

meridional, circumferential and normal to the reference surface,

respectively

also used as index for load, stress and strain quantities and rotations

of a shell of revolution

n mode number equal to the number of whole waves of a trigonometric

quantity in circumferential direction

also used as (additional) index to denote parameters typically

depending on the mode number

,h i

u u homogeneous and inhomogeneous displacement solutions,

respectively

hC arbitrary constant of the homogeneous solution, ( )1,2,3,...,8h = c vector containing the constants of the homogeneous solution

( ) cu continuous displacement vector ( ) ci u inhomogeneous part of the continuous displacement vector ( )cA continuous displacement matrix

( ) cn continuous stress quantity vector

( ) ci n inhomogeneous part of the continuous stress quantity vector

( )cB continuous stress quantity matrix ;

, e i e

u u element displacement vector and its inhomogeneous part, respectively eA element displacement matrix

; ,e i ef f element force vector and its inhomogeneous part, respectively

eB element stress quantity matrix ; prim ef element primary load vector

; tot ef total element load vector eK element stiffness matrix ;ext nf external nodal load vector

xxiii

; ; prim e nf nodal primary load vector ; tot nf total nodal load vector

;e nf nodal force vector

K global stiffness matrix totf global load vector

Chapter 4

a radius of a circular cylindrical reference surface

,x orthogonal coordinates of a circular cylindrical reference surface

z coordinate in the thickness direction of a circular cylindrical shell

, ,x z

u u u displacements at the reference surface of a circular cylindrical shell

,xx normal strains of a circular cylindrical shell

,xx changes of curvature of a circular cylindrical shell

,x x shear strain and torsion of a circular cylindrical shell, respectively

,xx

n n normal stress resultants of a circular cylindrical shell

,xx

m m bending stress couples of a circular cylindrical shell

,x x

n m longitudinal shearing stress resultant and torsional stress couple of a

circular cylindrical shell, respectively

,x

v v transverse shearing stress resultants of a circular cylindrical shell

, ,x z

p p p surface forces at the reference surface of a circular cylindrical shell

,xx normal stress of a circular cylindrical shell; axial and circumferential,

respectively

x longitudinal shearing stress of a circular cylindrical shell

, ,x z

f f f resultants of the edge forces at a circular cylindrical reference surface

xt couple of the edge forces at the circular edge of a circular cylindrical

shell

xv combined internal stress resultants of a circular cylindrical shell;

similar to Kirchhoffs effective shearing stress resultant

,x rotation of a normal to the circular cylindrical reference surface in the

x -direction and -direction, respectively

ijL components of a differential operator matrix

k dimensionless parameter used to describe the components ijL

dimensionless parameter used to describe differential equations of a circular cylindrical shell

0 0,a b dimensionless parameters of the homogeneous solution for 0n =

0 dimensionless parameter used to describe

0 0,a b

1 1,a b dimensionless parameters of the homogeneous solution for 1n =

1 dimensionless parameter used to describe

1 1,a b

1 1,n n

a b dimensionless parameters of the homogeneous solution for 1n >

describing the short edge disturbance

xxiv

2 2,n n

a b dimensionless parameters of the homogeneous solution for 1n >

describing the long edge disturbance

,n n dimensionless parameters used to describe 1

na , 1

nb , 2

na and 2

nb

cl characteristic length of an edge disturbance

il influence length of an edge disturbance

l length of a circular cylindrical shell

,1 ,2,c cl l characteristic length ( )1n > of the short edge disturbance and the long edge disturbance, respectively

,1 ,2,i il l influence length ( )1n > of the short edge disturbance and the long edge disturbance, respectively

Chapter 5

wp wind stagnation pressure

0 5,.., factors per mode number of the wind load distribution

2 5n

xx

axial stress at the base of a circular cylindrical shell due to the mode

numbers 2 5n , i.e. the self-balancing terms of the specified wind load

0 5n

xx

axial stress at the base of a circular cylindrical shell due to the mode

numbers 0 5n , i.e. all terms of the specified wind load 0 5

,

n

xx t

tensile axial stress at the base of a circular cylindrical shell 0 5

,

n

xx c

compressive axial stress at the base of a circular cylindrical shell 1n

xx

= axial stress at the base of a circular cylindrical shell due to the mode

number 1n = , i.e. the beam term of the specified wind load

1 2,l l characteristic lengths of a circular cylindrical shell introduced to

describe the axial stress ratio of the self-balancing terms to the beam

term 2

,2

n

il= influence length of the long edge disturbance specifically for 2n =

, ,r r r

A S I ring cross-sectional properties

ring ring parameter of the long edge disturbance

,SMC SMCn n

a b dimensionless parameters of the homogeneous solution for 1n >

describing the long edge disturbance within the SMC approach SMC

n dimensionless parameter used to describe ,SMC SMC

n na b

,modbD modified bending stiffness, viz. the bending stiffness of the stiffening

rings is smeared out along the bending stiffness of the circular

cylinder

rl ring spacing, along which the ring bending stiffness is smeared out

mod mod,k modified dimensionless parameters

r stiffness ratio of bending stiffness of the circular cylindrical shell only

to the modified bending stiffness of the shell

xxv

,b h width and height of a stiffening ring

fb width of the flange of a beam cross section

effb effective width of the flange of a curved beam

effl effective length of the circular cylindrical shell in axial direction

re eccentricity of the ring centre of gravity to the middle plane of the

cylinder

, ,x z

k k k axial, circumferential and radial spring stiffness, respectively

k rotational spring stiffness

x axial elastic support parameter

rotational elastic support parameter

z combined circumferential and radial elastic support parameter

,modx modified axial elastic support parameter

xn normalised stress ratio to account for influence of axial elastic support

,modz modified combined circumferential and radial elastic support

parameter

zn normalised stress ratio to account for influence of planar elastic

support

Chapter 6

w density of water

h height of shell course in a tank wall

g ratio of bending rigidity of the wind girder itself to the tank wall

,g gh t wind girder dimensions; plate width and thickness, respectively

gI wind girder circumferential bending rigidity

,maxsu maximum circumferential settlement of a tank shell

auxiliaries

,L L operator and its adjoint, respectively

,u v vectors

1 2, factors that account for the curvature of the parametric lines

, ii factors in differential operator matrices and their adjoint factors,

respectively

i order of differential equation, identification of a cylindrical

subdomain

,a b identifier for opposite circular edges

scalar function on the reference surface of a circular cylindrical shell

0 1, ,

n scalar functions for 0n = , 1n = and 1n > , respectively

( )q x alternative surface load on a circular cylindrical shell

1,

n Laplace operator for 1n = and 1n > , respectively

xxvi

1 2, parameters used to describe 1

na , 1

nb , 2

na and 2

nb

1 2, parameters used to describe

1 and

2

, , parameters used to describe 1 and

2

r root in trial solution to characteristic equation

0 1,r r expansions of the large roots in case of parameter perturbation

small parameter in case of parameter perturbation

0 1,s s expansions of the small roots in case of parameter perturbation

hS arbitrary constants in case of a rewritten homogeneous solution,

( )1,2,3,...,8h =

h phase angle, arbitrary constants in case of a rewritten homogeneous

solution, ( )1,2,3,4h =

cd distance to the centre across the profile of a circular

density

g gravitational acceleration

1 Introduction

1

1 Introduction

1.1 Motive and scope of the research In the field of structural mechanics the word shell refers to a spatial, curved structural

member. The enormous structural and architectural potential of shell structures is used

in various fields of civil, architectural, mechanical, aeronautical and marine

engineering. The strength of the (doubly) curved structure is efficiently and

economically used, for example to cover large areas without supporting columns. In

addition to the mechanical advantages, the use of shell structures leads to aesthetic

architectural appearance.

Examples of shells used in civil and architectural engineering are: shell roofs,

liquid storage tanks, silos, cooling towers, containment shells of nuclear power plants,

arch dams, et cetera. Piping systems, curved panels, pressure vessels, bottles, buckets,

parts of cars, et cetera are examples of shells used in mechanical engineering. In

aeronautical and marine engineering, shells are used in aircrafts, spacecrafts, missiles,

ships, submarines, et cetera.

Because of the spatial shape of the structure the behaviour of shell structures is

different from the behaviour of beam and plate structures. The external loads are

carried by both membrane and bending responses. As a result, the mathematical

description of the properties of the shell is much more elaborate than for beam and

plate structures. Therefore, many engineers and architects are unacquainted with the

aspects of shell behaviour and design.

In practice, many shell structures are single or combined shells of revolution (also

referred to as axisymmetric shells) and often they are stiffened by rings. The research

in this thesis focuses on the analyses of these shell structures, which find their

application in industries involved with structures like, for example, pipelines, liquid

storage tanks, chimneys and cooling towers.

The considerable effort in the development of rigorous shell theories dates back to the

early twentieth century. These shell theories reduce a basically three-dimensional

problem to a two-dimensional one. Nevertheless, the analysis of shells with the aid of

such theories involves complicated differential equations, which either cannot be

solved at all, or whose solution requires the use of high-level mathematics unfamiliar

to structural engineers. Therefore many approximate shell theories have been

developed, mainly on the assumption that the shell is thin, and to obtain generic

analysis tools obviously some accuracy had to be traded for convenience and

simplicity.

Hence, it is not surprising that the development of the numerical formulations

since the 1950s has led to a gradual cessation of attempts to find closed-form solutions

to rigorous formulations. But, with todays availability of greatly increased computing

power (also since the mid twentieth century), completeness rather than simplicity is

given more emphasis.

Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks

2

The drawback of the numerical methods is that they do not provide generic

knowledge of the shell behaviour and the prevailing parameters. Also the foundations

of the formulations that are used and thus their justification and validity are often not

completely understood, which has resulted in numerous finite element formulations

that work quite well for certain problems but do not work well in other problems. This

results from the sensitivity of the problem to the geometry and support conditions,

which characterises the complicated behaviour of shell structures under various loading

types. For the use of numerical programs and to understand and validate the results,

some basic knowledge of the underlying theories and the mechanical behaviour of the

structure is obviously essential.

These observations give rise to a need for a study that is not based on blunt

computer power but on the rigorous shell formulations obtained by the classic

approach. But, due to its highly mathematical character, this reappraisal is only useful

if this approach is combined with modern methods for handling complicated boundary

and transition equations in a stiffness method approach. Hereby a generic study of the

shell behaviour can be conducted by evaluating the solution to the general equations as

well as the output of the computer program.

1.2 Research objective and strategy This research project intends to combine the classic shell theories with the

contemporary numerical approach. The goal is to derive and employ a consistent and

reliable theory of shells of revolution and to present that theory in the context of

modern computational mechanics.

The contemplated set of equations concentrates on physically as well as

geometrically linear behaviour under static loading. A lot of basic and necessary

knowledge of this static and linear behaviour is lacking or not well understood and it is

this incomprehension that obstructs the shell analyst of gaining valuable insight into

the general shell behaviour.

This research not only focuses on the axisymmetric loading, but also on non-

axisymmetric loading, which means that for example a quasi-static wind load or non-

uniform settlements can be studied. The results from the studies of both bending and

membrane dominated responses will enable a better evaluation and interpretation of the

results from finite element studies regarding the same and the more complete

behaviour.

With the proper set of equations as a starting point, the following successive steps are

performed. For cylindrical shells with circular boundaries, which are the most

frequently used in structural application, it is possible to obtain a closed-form solution

or at least an approximate solution (within the assumptions of the theory) to the

rigorous shell formulations. Already from these solutions, valuable insight is gained

into the type of response to each type of load and the prevailing parameters describing

this response. By reshaping the precise formulation of the classic approach into the

well-known direct stiffness approach of the displacement method, the valuable

knowledge of the classic approach is preserved. The aim of the project is to derive a

fast PC-oriented computer program for that. This is done using the Fortran-package in

1 Introduction

3

combination with graphical software and has resulted in a stable and well-working tool

that can be used by structural analysts for rational first-estimate design of shells of

revolution.

The approach of the displacement method enables the calculation of combinations

of elements and type of elements, which makes the use of an electronic calculation

device more sensible in view of the increasing number of equations. Next to that, it is

fairly simple to implement stiffening rings in the formulation and hereby the influence

of the number and size of these members on the shell behaviour can be studied.

Similarly, the elastic supports and prescribed displacements can be easily implemented

and various load types can be described. Combined with the generic knowledge from

the closed-form solutions, appropriate design tables, graphs and formulas are properly

presented using the suitable parameters.

1.3 Outline of the thesis Chapter 2 deals with the fundamentals of the theory, the results by former authors and

the proposed set of equations. In chapter 3, the numerical solution procedure for this

set is introduced and this not often applied procedure is clarified. The formulations for

circular cylindrical shells that are implemented in this computational method are

derived in chapter 4. The combination of the generic knowledge from these two

chapters with the numerical results from the computer program enables a parametric

study of the geometrical properties of the shell types. These numerical results and

parametric study for long circular cylindrical shells (such as industrial chimneys) are

presented in chapter 5, while chapter 6 presents the numerical study for short circular

cylindrical shells (such as storage tanks). The conclusions from this study and

recommendation for further application of the proposed method are discussed in

chapter 7.

Introduction

CH1 Conclusions

CH7

Circular cylindrical

shells CH4

General part on shell theory

CH2

Computational

method CH3

Chimney

CH5

Tank CH6

Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks

4

1.4 Short review of the existing work within the scope In 2001, Van Bentum started a graduation project, which embodied a part of the tasks

of the present research. The main goal of that project was to show that, on basis of the

closed-form solutions to the Donnell equation for circular cylindrical shells, an exact

(within the theory) stiffness matrix could be synthesized. The resulting report was

published 2002 [1]. As Donnells solution is only applicable to the load-deformation

behaviour for circumferential modes with at least two whole waves in circumferential

direction, the solution for the axisymmetric and beam mode were implemented using

an alternative solution. For the response to axisymmetric loads, a simplified Donnell

solution was adopted using the displacement normal to the middle surface as the only

degree of freedom. For the response to beam loads, the membrane solution was

employed. In a successive step, the possible incompatibility between this membrane

solution and the requirements at the edges was compensated by an edge disturbance

congruent with the solution for the axisymmetric mode.

Although these solutions were successfully implemented and the result for the

study of rather long cylinders subject to a wind load were very satisfactory, the

following drawbacks can be noticed. Firstly, the axisymmetric mode can be better

described by using two independent degrees of freedom by taking into account the

longitudinal displacement in axial direction. Secondly, the approach for the beam mode

is only valid for a cylinder with rather large length-to-radius-ratio. For shorter

cylinders, the membrane behaviour and the edge disturbance resulting from the

complete differential equation should be described simultaneously. Thirdly, as it is well

known that Donnells solution does not describe the ring-bending behaviour, a better

description in circumferential direction should be adopted for the lower mode numbers

of the self-balancing modes (the modes with at least two whole waves in

circumferential direction).

The present study is restricted to closed circular cylindrical shells like long

industrial chimneys and storage tanks. The differential equations also facilitate

calculating cylindrical roof shells, but this study refrains from this type of structure.

Substantial research in this domain was performed by A.L. Bouma, H.W. Loof and H.

van Koten in The Netherlands, which was reported in [2]. This research was based on

the Donnell equation that sufficiently accurately describes the behaviour of this

structural type.

The concept of generating the stiffness matrix on basis of the closed-form solution

was already proposed as early as 1964 by Loof [3]. A number of systematically and

efficiently structured calculation schemes were developed, be it restricted to certain

load-deformation cases per shell structure due to the state of the programmable

electronic machines and available programming procedures of that period.

A literature study showed that Bhatia and Sekhon [4] recently applied the method

to axisymmetric structures. In their first paper of a series, the method is introduced and

applied to an annular plate element. Three follow-up papers [5-7] focus on the

generation of exact stiffness matrixes for a cylindrical, a conical and a spherical shell

element, respectively. However, Bhatia and Sekhon did only employ the method to

axisymmetric structures subject to loads that are also axisymmetric with respect to the

axis of symmetry of the structure. Hereby, the problem is reduced considerably, but the

application is rather limited and important engineering problems cannot be modelled.

1 Introduction

5

To study the influence of, e.g., elastic supports, stiffening rings and various load

types on the behaviour of circular cylindrical shells, these can be implemented into a

computer program as described above. With the same objective, Melerski [8] derived

solutions for beams, circular plates and cylindrical tanks, especially on elastic

foundations, and included a diskette with the resulting software. However, for circular

plates and cylindrical tanks the application of the in other aspects general approach is

limited to axisymmetric load cases.

Another interesting approach, which has the objective to obtain insight into the

load carrying behaviour of cylindrical shell structures, is the semi-membrane concept,

which is able to deal with non-axisymmetric load cases. The semi-membrane concept

assumes that, to simplify the initial equilibrium equations, the circumferential strain as

well as both the axial and torsional bending stiffness may be equated to zero. The

resulting equation exactly describes the ring-bending behaviour, but it can only be

applied to self-balancing modes. As shown by Pircher, Guggenberger and Greiner [9],

this concept can be applied to, e.g., a radial wind load, an axial elastic support and an

axial support displacement. However, not all load cases or support conditions can be

described. Moreover, the semi-membrane concept is only applicable to certain load-

deformation behaviours of cylindrical shell structures. Closely related to the

simplifications, it should be allowed to neglect the influence of the part of the solution

described by the short influence length in comparison to the part described by the long

influence length. In other words, the cylinder should be sufficiently long in comparison

to its radius and the boundary effects should mainly influence the more distant

material.

The present research overcomes the above-mentioned drawbacks of the solutions

used by Van Bentum and extends the results of that and the other mentioned research,

which is limited to either axisymmetric or non-axisymmetric load-deformation

behaviour. Instead of the Donnell equation, the Morley-Koiter equation is employed in

the present research. This equation is probably the best alternative, as it overcomes the

inaccuracy of Donnells simplifications in its inability to describe rigid-body modes but

preserves its elegance and simplicity.

The Morley-Koiter equation can be derived by using a so-called first-order

approximation theory. To understand the assumptions and simplifications, which are

introduced to obtain such an equation for a thin elastic shell, the set of equations

resulting from a fundamental derivation for thin elastic shells are reproduced. Since

these are well established, similar derivations can be found in many textbooks, which

are referenced in the text. However, the derivation in this research is set up as a more

integrated treatment of concepts by various authors. The objective of this treatment is

to correctly and consistently introduce the assumptions and simplifications throughout

the derivation of (i) the differential equations and boundary conditions, (ii) the single

differential equation and its solution and (iii) the expressions for all quantities obtained

by back substitution of this solution.

Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks

6

2 General part on shell theory

7

2 General part on shell theory This chapter deals with the fundamentals of the theory. The geometry of a thin elastic

shell is treated briefly and the equations that describe the shell behaviour are derived.

The formulations for thin, shallow, non-linear and cylindrical shells by some former

authors are discussed and as a result of the comparison a set of equations is proposed.

This set comprises kinematical and constitutive relations that are complemented by the

equilibrium relation and boundary conditions, which are derived by making use of the

principle of virtual work.

2.1 Introduction to the structural analysis of a solid shell

2.1.1 Geometrical interpretation

The primary purpose of a structure is to carry the applied external loading. Every

particle of this structure is a three-dimensional object on its own. In spite of this,

structural engineers (almost) never use the three-dimensional theory of elasticity, but

they model the structural elements as lines with a finite cross-sectional area, which has

become customary in the theory of structures.

The structural purpose of shell elements is to span a finite space. As a result of this,

a description of the structural element by one line is not possible and the stress analysis

has to be established with the concept of a physical surface. An important difference

has to be made to this: plates refer to flat surfaces and shells refer to curved surfaces.

Describing it, the shell element is interpreted as a materialisation of a curved

surface. This definition implies that the shell problem is reduced to the study of the

displacements of the reference (or middle) surface and that the thickness of the shell is

small in comparison to its other dimensions. The geometry of the shell is thus

completely described by the curved shape of the middle surface and the thickness of

the shell. In structural mechanics this geometrical description corresponds to the one of

the beam with a rectangular cross-section; the course of the middle axis in combination

with the accompanying cross-section. The shell thickness is henceforth kept constant

for convenience, but the analysis method and considerations are also applicable to

shells with a varying thickness.

The above-mentioned schematisation does not require that the shell be made of an

elastic material. Since most shells are made of a solid material, it will further be

assumed that the material behaves linear elastic conform Hookes law.

2.1.2 Generalised Hookes law

The first rough law of proportionality between the forces and displacements was

published by Hooke. The generalisation of Hookes law assumes that at each point of

the medium the strain components are linear functions of the stress components and

that it is possible to invoke the principle of superposition of effects. For many

engineering materials, the relation between stress and strain is indeed linear and the

Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks

8

deformation disappears during unloading. Obviously any material has its elastic limit,

viz. the greatest stress that can be applied and removed without permanent

deformation. Beyond this limit, which is nearly equal to the proportional limit, the

material behaves both elastic and plastic.

In this thesis, it will further be assumed that the material behaves conform a

generalised Hookes law, because we are interested in the general behaviour of shell

structures, especially since for rational first-estimate design it is naturally not advised

to rely on the plastic range of the structural capacity.

The assumption of homogeneity and isotropy of the material seems plausible for

most structural materials since we are interested in the global behaviour of an entire

body. It is not our objective to study the very small portions of material, which must be

regarded as orthotropic, but the chaotically distribution of the orthotropy over the entire

body allows the natural interpretation of a homogeneous and isotropic medium.

2.1.3 Mechanical behaviour of elastic shells

This research focuses on thin elastic shells. A thin shell has a very small thickness-to-

minimal-radius ratio, often smaller than 1 50 . Due to its initial curvature, a shell is able

to carry an applied load by in-plane as well as out-of-plane actions. Similar to the

behaviour of plates and beams, the resistance of a shell structure is optimally used if

bending actions are minimised as much as possible. A thin shell therefore mainly

produces in-plane actions, which are called membrane forces. These membrane forces

are actually resultants of the normal stresses and the in-plane shear stresses that are

uniformly distributed across the thickness. The corresponding theory of this membrane

behaviour is called the membrane theory.

However, the membrane theory does not satisfy all equilibrium and/or

displacement requirements in case of:

Boundary conditions and deformation constraints that are incompatible with the requirements of a pure membrane field, (b) and (c);

Concentrated loads (d); and Change in the shell geometry (e).

(a) (b) (c)

Membrane compatible Membrane incompatible Deformation constraint

2 General part on shell theory

9

(d) (e) Concentrated load Change in the geometry

In the regions where the membrane theory will not hold, some (or all) of the

bending field components are produced to compensate the shortcomings of the

membrane field in the disturbed zone. These disturbances have to be described by a

more complete analysis, which will lead to a bending theory of thin elastic shells.

If the bending field components are developed, it often has a local range of

influence. Theoretical calculations and experiments show that the required bending

field components attenuate and mostly this effect is confined to the vicinity of the

origin of the membrane nonconformity. In many cases, the bending behaviour is

restricted to an edge disturbance. Therefore, the undisturbed and major part of the shell

behaves like a true membrane. This unique property of shells is a result of the curvature

of the spatial structure. The efficient structural performance is responsible for the

widespread appearance of shells in nature.

2.1.4 Static-linear analysis of shells of revolution

Many shell theories have been developed to analyse the mechanical behaviour of shell

structures. To overcome the complexity of an exact theory assumptions are made

wherein the membrane theory is the most appealing. Because of its simplicity, the

membrane theory gives a direct insight into the structural behaviour and the order of

magnitude of the expected response without elaborate computations. But in the cases

where the membrane behaviour is not the dominant type of response, use is often made

of finite element packages.

The usefulness of the finite element approach for the initial design and analysis is

however doubtful and an intermediate approach between the contemporary and the

classic approach is recommendable. This intermediate approach is thus the main focus

of this study.

For shells of revolution with circular boundaries, which are the most frequently

used in structural application, the rigorous shell formulations have been well

established. Keeping in mind the objective of employing closed-form solutions,

attempting to investigate the linear models first seems to be the natural strategy. Hence,

the starting point is the analysis of the small deformation behaviour of shells of

revolution under static loading.

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2.2 Fundamental theory of thin elastic shells The set of equations resulting from a fundamental derivation for thin elastic shells are

well established. Consequently, the expressions derived in this section are probably

well known but they are stated without accurate reference here for further use. Similar

derivations can be found, e.g., in the books by Kraus [10] and Leissa [11] and the

report by Hildebrand, Reissner and Thomas [12]. However, the following derivation is

set up as an integrated treatment and complement of concepts by various authors.

2.2.1 Kirchhoff-Love assumptions

On the basis of the assumptions Kirchhoff introduced with the purpose of deriving a

theory of a thin plate, Love [13] was the first to derive a set of basic equations which

describe the behaviour of a thin elastic shell. Generally referred to as Loves first

approximation this classic small deformation theory of a thin shell is based on the

following postulates, which are also known as the Kirchhoff-Love assumptions:

1. The shell is thin.

2. Strains and displacements are sufficiently small so that the quantities of

second- and higher-order magnitude in the strain-displacements relations may

be neglected in comparison to the first-order terms.

3. The transverse normal stress is small in comparison to the other normal stress

components and may be neglected.

4. A normal to the reference surface before deformation remains straight and

normal to the deformed reference surface and suffers no extension.

Before utilising these assumptions, it is useful to discuss their implications

individually.

The assumption that the shell is thin is inevitable for the other assumptions as these

are only appropriate if the thickness of the shell is small in comparison to the other

dimensions. The thinness of a shell is often characterised by the ratio of the thickness

to the radius of curvature, but no precise definition is available and suggestions differ

largely. For the present discussion, the thinness will be such that the ratio mentioned is

negligible in comparison to unity.

The second assumption is necessary to keep the equations linear and to be allowed

to describe all resulting equations in the initial configuration. This assumption also

implies that the first derivatives of all displacements are negligible in comparison to

unity.

The assumption that the transverse normal stress is negligible seems plausible for a

thin shell except in the vicinity of highly localised loading.

The last assumption is a continuation of the well-known Bernoulli-Euler

hypothesis and implies that not only the transverse shear deformation but also the strain

components in the direction of the normal to the reference surface can be neglected.

Flgge [14] states that conclusions drawn from the last two assumptions can only

be exact if the shell be made of a non-existent anisotropic material for which the

modulus of elasticity in the direction normal to the reference surface and the shear

2 General part on shell theory

11

modulus for the transverse shearing strains are infinite, whereas two of the Poissons

ratios (that take into account the lateral contraction of a material) are equal to zero.

However, it is obvious that for a thin shell the assumptions are acceptable so that

whatever happens in the direction normal to the reference surface of the shell, stress or

strain, is of no significance to the solution.

2.2.2 Mathematical description of a shell surface

To describe the curved reference surface of a shell it is natural to use a curvilinear

coordinate system that coincides with the lines of principal curvature, which can be

shown to be orthogonal. The derivation and proof of this feature and all the other

expressions in this subsection are exemplified in Appendix A, which contains parts of

the well-documented study of the differential geometry of surfaces especially when

applied to the mathematical description of a shell surface.

A surface S in the rectangular coordinate system 1 2 3, ,x x x can be written as a function

of two parameters; viz. 1 2, , which are the curvilinear coordinates of the reference

surface. To describe the location of an arbitrary point within the two outer surfaces of

the shell a third coordinate is introduced in the thickness direction. The position vector R to this arbitrary point is described by

( ) ( ) ( )1 2 1 2 1 2, , , , = + R r n where r is the position vector of the corresponding point on the reference surface and

n is the unit normal vector.

The line element ( )2ds is calculated by taking the dot product of the differential change dR in the position vector from a point

oP to an infinitesimal close point P

within the shell space and hence is expressed by

( ) ( ) ( ) ( )2 2 2 211 1 22 2 33ds d d g d g d g d= = + + R R (2.1) where

iig ( )1,2,3i = are the metric coefficients along the orthogonal parametric lines.

These coefficients are defined by

1 11 1 2 22 2 3 33

1 2

1 , 1 , 1A g A g A gR R

= = + = = + = =

(2.2)

where i

A are the scale factors, 1 and

2 are the so-called Lam parameters of the

reference surface and 1

R and 2

R are the principal radii of curvature at the point on the

reference surface corresponding to point o

P . The Lam parameters and the principal

radii are related to the position vector and the unit normal vector by

2

1 2

1 1 1 1 1 1

2

2 2

2 2 2 2 2 2

1 1

1 1

R

R

= =

= =

r r r n

r r r n

Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks

12

There are three differential equations relating the parameters of the reference surface.

The two equations, which are known as the Codazzi conditions, are

1 1 2 2

2 2 2 1 1 1 1 2

1 1,

R R R R

= =

(2.3)

and the third is known as the Gauss condition, which is given by:

2 1 1 2

1 1 1 2 2 2 1 2

1 1

R R

+ =

An infinitesimal element within the volume V of the thin shell is obtained by making

four cuts perpendicular to the reference surface, which coincide with a pair of

differentially spaced parametric lines of the reference surface, and the space that is then

limited by two surfaces that are d apart (at distance from the reference surface) is the infinitesimal element. By evaluating the expressions for a line element (2.1), it is

obvious that the differential lengths of arc of the edges of the element are

( ) ( )1 1 2 1 1 2 1 2 2 21 2

, , 1 , , , 1ds d ds dR R

= + = +

(2.4)

and that the differential areas of a strip on the faces of the element are

( ) ( )1 1 2 1 1 2 1 2 2 21 2

, , 1 , , , 1dS d d dS d dR R

= + = +

(2.5)

Hence, the differential volume of a layer of the element bounded by these strips is

( )1 2 1 2 1 21 2

, , 1 1dV d d dR R

= + +

(2.6)

Finally, the Laplace-Beltrami operator of a scalar field f described in an orthogonal

curvilinear coordinate system is a scalar differential operator defined by

2 3 1 3 1 2

1 2 3 1 1 1 2 2 2 3 3 3

1 f f ff

= + +

as derived, for example, by Borisenko and Tarapov [15]. For the scalar field f that

acts on the reference surface within a shell space described by (2.2), the Laplace-

Beltrami operator, which is further referred to as the Laplace operator, is given by

2 1

1 2 1 1 1 2 2 2

1 f ff

= +

(2.7)

2.2.3 Kinematical relation

For a curvilinear coordinate system determined by the coordinate lines i , which are

assumed to be orthogonal, the metric coefficients along these parametric lines are

denoted by ii

g as shown in Appendix A. The displacements in the direction normal to

the coordinate surfaces 1 2 3, , are represented by

1 2 3, ,U U U respectively. By

applying the assumptions of infinitesimal deformations in this curvilinear coordinate

2 General part on shell theory

13

system as shown in Appendix B, the extension and shear components of the strain

tensor, iie and ije respectively, are obtained in the form

( ) ( )

3

1

1, 1,2,3

2

1, , 1,2,3 , if

2

i ii kii

ki ii kii kk

jiij ii jj

j iii jj ii jj

U g Ue i

gg g

UUe g g i j i j

g g g g

=

= + =

= + =

Hereby the extension iie is defined as the relative elongation in the i -direction of a

fibre in the i -direction and the shear component ije is defined as half of the angle with

which the originally perpendicular i - and j -directions decreases.

By substituting ( )2ii ig A= from (2.2), we get:

1 2 1 3 1 1 1 2 211 12

1 1 2 2 3 3 2 2 1 1 1 2

1 2 2 3 2 1 1 3 322 13

2 1 1 2 3 3 3 3 1 1 1 3

1 3 2 3 333

3 1 1 2 2

1, 2

1, 2

1

U U A U A A U A Ue e

A A A A A A A

U A U U A A U A Ue e

A A A A A A A

U A U A Ue

A A A

= + + = +

= + + = +

= + +

2 2 3 3

23

3 3 3 2 2 2 3

, 2A U A U

eA A A A

= +

(2.8)

In the case of the adopted coordinate system, the substitutions 3 = for the coordinate

and 3U U= for the displacement in the direction of the normal to the reference surface

are made.

By definition the in-plane shear angle 12

2e is defined by:

12 12 21 1 22

n ne e e= + = + +

Hereby the angle 1

is the rotation in the 2 -direction of a fibre along the

1 -direction

and the angle 2

is defined correspondingly. The angle n

is the rigid body rotation

about the normal to the reference surface, which is taken positive according to the

right-hand rule.

The introduction of the rotation n

is similar to the procedure that is well known

for a plate element. For that geometry, the shear strain is found by describing two

changes of the straight angle in the respective directions. These changes are then split

in a symmetric part (the shear strain) and a skew-symmetric part (the rigid body

rotation). This is exactly the procedure that is applied above.

Therefore, it is remarkable that this procedure is not widely applied in describing

the deformation of a shell element. Sanders [16] does introduce the rigid body rotation

n , but on a reverse consideration, which is discussed in subsection 2.6.3.

Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks

14

Following the above-mentioned procedure the shear angle 12

2e expressed by (2.8)

is also described by

1 1 2 212

2 2 1 1 1 2

1 1 2 1 2 2

1 2 2 1 1 2 2 1 2 1

2

1 1n n

A U A Ue

A A A A

U A U U U A

A A A A A A

= +

= + + +

Hence, it follows that

1 1 212 1

1 2 2 1 1

1 2 221 2

2 2 1 2 1

1

1

n n

n n

U A Ue

A A A

U U Ae

A A A

= = +

= + = +

From the definition 12 21

e e= it is obtained that the rigid body rotation n

is equal to

1 1 2 2

1 2 2 1

1

2n

AU AU

A A

= +

which is also shown in Appendix B.

To relate all components of strain to quantities of the reference surface the fourth

assumption of Loves postulates has to be employed.

The first part of that assumption which requires that a normal remains straight is

satisfied when the displacements are linearly distributed through the thickness of the

shell. Hence, the displacement components are represented by

( ) ( ) ( )1 2 1 2 1 2, , , , ,0ii iU

U u

= +

where i

u is the respective displacement component at the reference surface and iU

is

the change of the displacement component in the normal direction.

The second part of the fourth assumption requires inextensibility of a normal to the

reference surface, which implies that normal strain vanishes. By substituting 3

1A =

from (2.2) into (2.8) for the normal strain, we get

333

3

UUe

= =

and hence ( )1 2, , 0U =

to disregard the normal strain.

Since a normal to the reference surface remains straight, the derivatives 1U

and

2U

are equal to the respective rotations of the normal from its initial position to its

position after deformation. So, the rotations 1 and

2 are introduced, which denote

the rotations of a normal to the reference surface in the direction of the parametric lines

1 and

2 , respectively.

2 General part on shell theory

15

As a consequence of the above, the displacement components are represented by

1 1 1

2 2 2

U u

U u

U u

= +

= +

=

(2.9)

To relate the strain components to the displacements of the reference surface, the scale

factors (2.2) and the representation of the displacement components (2.9) are

substituted into (2.8). Making use of the Codazzi conditions (2.3) we arrive at the

following six expressions of the strain components related to ten deformation

quantities.

( ) ( )

( ) ( )

( ) ( )

11 11 11 22 22 22

1 2

12 12 12 21 21 21

1 2

1 1 2 2

1 2

1 1

1 1

1 1

1 1

1 12 2 2 2

1 1

e e

R R

e e

R R

e e

R R

= + = +

+ +

= + = +

+ +

= =

+ +

(2.10)

The ten deformation quantities are separated in four strains of the reference surface

denoted by 11 , 22 , 12 and 21 , in four changes of rotation of the normal to the

reference surface denoted by 11 , 22 , 12 and 21 , and in two transverse shearing

strains denoted by 1 and 2 . The ten deformation quantities of the kinematical

relation are related to the reference surface