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

276 Pages·2010·3.87 MB·English

by  Jeroen Hoefakker

Theory Review for Cylindrical Shells

and

Parametric Study of Chimneys and Tanks

Theory Review for Cylindrical Shells

and

Parametric Study of Chimneys and Tanks

P
ROEFSCHRIFT

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
[email protected] / 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.
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
v
vi
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
vii
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

viii
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
ix
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.

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