Nonlinear Behaviour and Stability of Thin-Walled Shells

Natalia I. ObodanOlexandr G. LebedeyevVasilii A. Gromov

Solid Mechanics and Its ApplicationsSeries Editor: G.M.L. Gladwell

Solid Mechanics and Its Applications

Volume 199

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Natalia I. Obodan • Olexandr G. LebedeyevVasilii A. Gromov

Nonlinear Behaviourand Stability ofThin-Walled Shells

123

Natalia I. ObodanVasilii A. GromovOles Honchar Dnepropetrovsk National

UniversityDnepropetrovskUkraine

Olexandr G. LebedeyevAtlantis Industrial SystemsDnepropetrovskUkraine

ISSN 0925-0042ISBN 978-94-007-6364-7 ISBN 978-94-007-6365-4 (eBook)DOI 10.1007/978-94-007-6365-4Springer Dordrecht Heidelberg New York London

Library of Congress Control Number: 2013933110

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Contents

1 In Lieu of Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Boundary Problem of Thin Shells Theory . . . . . . . . . . . . . . . . . . . 112.1 General Concepts and Hypotheses . . . . . . . . . . . . . . . . . . . . . . 112.2 Geometrical and Physical Relations . . . . . . . . . . . . . . . . . . . . . 142.3 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Shell Support Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Generalized Solution in Displacements. . . . . . . . . . . . . . . . . . . 22Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Branching of Nonlinear Boundary Problem Solutions . . . . . . . . . . 293.1 Branching Patterns and Types of Singular Points. . . . . . . . . . . . 293.2 Branching Points and Structural Behaviour . . . . . . . . . . . . . . . . 313.3 Bifurcation Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Linearized Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 General Criteria of Shell Stability . . . . . . . . . . . . . . . . . . . . . . 37

3.5.1 Static Stability Criterion . . . . . . . . . . . . . . . . . . . . . . . 373.5.2 Energy Stability Criterion . . . . . . . . . . . . . . . . . . . . . . 37

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Equivalent Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Determination of Cauchy Problem Initial Vector . . . . . . . . . . . . 454.4 Solution Parametric Continuation Algorithm. . . . . . . . . . . . . . . 464.5 Singular Points of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.6 Branching Pattern and Postcritical Branches Investigation . . . . . 48Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

v

5 Non-Axisymmetrically Loaded Cylindrical Shell . . . . . . . . . . . . . . 535.1 General Considerations. Linear Problem. . . . . . . . . . . . . . . . . . 535.2 Nonlinear Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2.1 ‘‘Wind’’-Type Pressure . . . . . . . . . . . . . . . . . . . . . . . . 555.2.2 Cyclic Pattern of External Pressure . . . . . . . . . . . . . . . . 615.2.3 Applicability of Simplified Models . . . . . . . . . . . . . . . . 655.2.4 Local External Pressure . . . . . . . . . . . . . . . . . . . . . . . . 675.2.5 Radial Concentrated Loads. . . . . . . . . . . . . . . . . . . . . . 74

5.3 Influence of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 765.4 Nonuniform Axial Compression . . . . . . . . . . . . . . . . . . . . . . . 78

5.4.1 Shell Subcritical State . . . . . . . . . . . . . . . . . . . . . . . . . 785.4.2 Buckling Modes and Critical Stresses . . . . . . . . . . . . . . 82

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Structurally Non-Axisymmetric Shell Subjectedto Uniform Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.1 Open Circular Cylindrical Shell Subjected

to Uniform Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.1.1 Uniform External Pressure . . . . . . . . . . . . . . . . . . . . . . 876.1.2 Uniform Axial Compression. . . . . . . . . . . . . . . . . . . . . 95

6.2 Closed Circular Cylindrical Shells with Big Cutouts . . . . . . . . . 1016.2.1 Shells Subjected to External Pressure . . . . . . . . . . . . . . 1016.2.2 Influence of Structural Parameters on Critical Loads . . . . 1036.2.3 Applicability of Simplified Models . . . . . . . . . . . . . . . . 1056.2.4 Influence of External Pressure Nonuniformity . . . . . . . . 106

6.3 Elliptic Shell Subjected to External Pressure. . . . . . . . . . . . . . . 1076.3.1 Uniform External Pressure . . . . . . . . . . . . . . . . . . . . . . 1076.3.2 Combined Loading of Elliptical Shell . . . . . . . . . . . . . . 111

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7 Postcritical Branching Patterns for Cylindrical Shell Subjectedto Uniform External Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.1 Postcritical Shell Behaviour for Arbitrary Pattern

of External Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.2 Closed Cylindrical Shell Subjected to Uniform External

Pressure: Primary and Secondary Bifurcation Paths . . . . . . . . . . 1187.3 Cylindrical Panel Subjected to Uniform External Pressure . . . . . 1257.4 Closed Cylindrical Shell Subjected to Uniform Axial

Compression: Primary, Secondary, and TertiaryBifurcation Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.4.1 Primary Bifurcation Paths: Regular Deformed Shapes . . . 1277.4.2 Secondary Bifurcation Paths: Longitudinally

Local Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.4.3 Tertiary Bifurcation Paths: Local Deformed Shapes . . . . 129

vi Contents

7.4.4 Tertiary Bifurcation Paths: Groups of Local Dents . . . . . 1347.5 Cylindrical Panel Subjected to Uniform Axial Compression . . . . 140Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8 Postbuckling Behaviour and Stability of Anisotropic Shells . . . . . . 1438.1 General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.2 Anisotropic Circular Cylindrical Shell: Critical Loads . . . . . . . . 145

8.2.1 Cyclic External Pressure . . . . . . . . . . . . . . . . . . . . . . . 1458.2.2 ‘‘Wind’’-Type Pressure . . . . . . . . . . . . . . . . . . . . . . . . 1478.2.3 Local External Pressure . . . . . . . . . . . . . . . . . . . . . . . . 148

8.3 Delaminated Thin-Wall Structures: General Featuresof Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.3.1 Delaminated Area Geometry . . . . . . . . . . . . . . . . . . . . 1508.3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8.4 Delaminated Spherical Segment Under External Pressure:Joint Deformation and Snapped-Through Deformed Shapes . . . . 1538.4.1 Branching Pattern: Primary and Secondary

Bifurcation Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.4.2 Critical Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.4.3 Influence of Initial Imperfections . . . . . . . . . . . . . . . . . 1628.4.4 Comparison of Buckling Analysis Models . . . . . . . . . . . 164

8.5 Delaminated Closed Cylindrical Shell Under External Pressure:Joint Deformation and Snapped-Through Deformed Shapes . . . . 1678.5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 1678.5.2 Branching Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Contents vii

Chapter 1In Lieu of Introduction

Abstract Thin shells are often the best and only choice for designers of flyingvehicles, naval vehicles and civil engineering structures. Shell behaviour under agrowing load demonstrates its essential nonlinearity, manifesting itself in buck-ling, in a variety of postbuckling shapes, and in rapid transitions from one shape toanother. Nonuniformity of shell structure and loading appeared to be the keyfactors influencing shell instability (that is the possibility for rapid change ofdeformed shape, for development of large deflections). The classical Eulerapproach to stability analysis presumes an ideal undeformed initial state andconsiders possibility of the solution non-uniqueness in its small vicinity, narrowingthe scope of analysis and often delivering improper critical loads. Full nonlinearanalysis and its efficient numerical implementation are needed for an investigationof shell behaviour. Typical shell behaviour patterns are studied and the compli-cated branching of the respective nonlinear boundary problems (including primary,secondary, and tertiary bifurcation paths) are revealed and analyzed. Suchimportant factors as nonuniformity of load and structure (non-symmetric loadpattern, structural defects and imperfections, anisotropy, etc.) are to be studied asthe causes of initially nonlinear behaviour, transformations of stress-strain stateduring shell uploading, and a variety of postbuckling forms. Such analysis isperformed on the basis of wide-scale numerical analysis. The technical progress ofrecent decades has placed before designers the paramount task of perceivingcomplicated loads for structures with minimal structural weight. For aerospace andnaval vehicles, as well as civil engineering structures, thin shells, mostly cylin-drical and spherical, have been accepted as the best solutions.

The first applications of shell structure brought up the problem of investigatingshell behaviour under compressive loading. For example, the cylindrical body of amissile, designed to withstand the compressive thrust of a rocket engine, suc-cessfully accelerated the useful load to the required velocity and travelled to atarget according to ballistic calculations. Nevertheless, the scattering around thetarget appeared to be unexpectedly high. It was found that the cylindrical shell,subjected to the pressure of high-speed airflow while descending through dense

N. I. Obodan et al., Nonlinear Behaviour and Stability of Thin-Walled Shells,Solid Mechanics and Its Applications 199, DOI: 10.1007/978-94-007-6365-4_1,� Springer Science+Business Media Dordrecht 2013

1

atmospheric layers, collapsed, drastically changing the aerodynamic properties ofthe originally symmetrical body. Unforeseen lateral aerodynamic forces draggedthe missile off the predesigned trajectory.

In that case it seemed reasonable to use the classical Euler formula of criticaluniform external pressure for an ideal shell. Unfortunately, most real-worldstructures and loads do not correspond to the ideal models. Say the rocket engine isattached to the hull of a space shuttle at several points, creating initially nonuni-form hull compression. Local compression can cause local concave dents dam-aging the fragile ceramic thermal coating and putting the ship at grave danger.

For an earthbound example, we consider the thin cylindrical shell of an oil tank(say, 100 feet in height and diameter). In certain typical situations—nearly emptytank, strong wind—the dents have appeared along the tank base. Detailed analysisshowed the appearance of critically compressed areas. Another common struc-ture—the roof in the form of a cylindrical panel—showed high flexibility and evensnapped into the new equilibrium form under snowfall or icing.

Thus, the maximal (critical) load concept was replaced by the residual opera-bility concept with predefined admissible ranges for geometrical and physicalsystem characteristics. This concept of load-carrying capability estimation doesnot allow for application of the classical Euler approach, as it substantially narrowsthe scope of investigation.

Consequently, investigations within the frameworks of the residual operabilityconcept are usually based on a parameter continuation approach for the nonlinearmodel that makes it possible to ‘‘observe’’ the structural model under operationalconditions.

The lessons of design practice encouraged the engineers to turn to unusual andunfamiliar problems:

• to consider initial bending and nonuniformity of the shell stress-strain state inorder to estimate shell stability;

• to investigate buckling as the process of transition from initial to new (oftenunexpectedly bent) forms of equilibrium;

• to investigate a plurality of possible postcritical forms and of conditions of itsmutual transition;

• to determine specific factors of shell structure and loading able to lower thecritical loads and to induce the rapid and undesirable shell shape change.

Further complications of modern shell structure design arose due to the wide useof composite and multilayered materials. The new factors—delaminations,incisions, and other damages caused by possible collisions at sea, in the air and inspace—attracted more attention to shell design, to the prediction of shell behaviour,and to its load-carrying capability.

Experimental investigations became the first matter-of-course instrument ofmechanical engineers. Load nonuniformity, structural imperfections and features(edge support, cut-outs, etc.) were modelled, and the strains, stresses and deflectionsrecorded. The experimental study of shell stability under nonuniform pressurebecame the bright pioneer example of ingenuity.

2 1 In Lieu of Introduction

Hutchinson and Koiter (1970) immersed the hermetically sealed shell into theliquid and placed it (off-centre) into a centrifuge. Centrifugal forces induced thepressure inside the liquid, and the pressure profile along the shell’s circumferentialcross-section was similar to the one for the flying vehicle inside high-speedairflow.

Chu and Turula (1970) simulated a complicated pressure profile through use ofa set of separately inflated elastic balloons. A cylindrical panel under gravity loadwas studied by Yang and Guralnick (1975) with the employment of stringsattached to the patches on the shell surface and driven hydraulically to create thenecessary tension.

Circumferentially nonuniform axial compression was studied experimentally byAndreev et al. (1988), Krasovsky (1990) using specially tooled flange edge plates.Experiments revealed strongly non-monotonous dependence of critical loads uponthe load nonuniformity. Similarly, non-monotonous dependence of a critical loadupon a cut-out aperture angle was found in the experiment.

These experimental data directly inspired analytical investigations. Experi-ments revealed (and required the investigation and theoretical explanations of) keyfeatures of the behaviour of compressed shells:

• non-homogeneity (non-axisymmetry) of a structure and its loading leads to theessential initial nonuniformity of stress-strain state and—of fundamentalimportance—to its transformation under a growing load;

• prominent load localization and local nonuniformity of a shell structure (say,edges, cut-outs, stiffeners, ledges, etc.) coinciding with asymptotic (for cylin-drical shells—with meridional) lines causes substantial (equal to shell thicknessor more) deflection and its nonlinear dependence upon the load level;

• nonlinear subcritical bending leads to strong nonlinear dependence of criticalloads upon the structural geometry and load profile, to a variety of bucklingmodes, and often to a significant drop of the critical load level;

• simplified stability analysis models based on the membrane model of a sub-critical state showed its fundamental inapplicability.

To summarize, the problems in estimating critical loads and analyzing thepostcritical behaviour of thin-walled shells under non-axisymmetric deformationare of great importance (Hunt 2006).

A diversity of recent researches concerning nonlinear shell analysis may besorted as follows:

1. Researches devoted to the influence of structural irregularities [shell thicknessand curvature variations of load (Babich et al. 2011; Ishinabea and Hayashib2012; Jamal et al. 2000; Jasion 2009; Goldfeld et al. 2005; Huang et al. 2011,2000; Khosravi et al. 2008; Lindgaard and Lund 2011a, b; Luongo 2010;Semenyuk and Zhukova 2011; Shariati and Rokhi 2008; Shkutin 2004; Singhet al. 2009; Wang and Koizumi 2010), cut-outs (Silvestre and Gardner 2011),cracks (Vaziri and Estekanchi 2006), incisions, delamination of layered shells(Biagi and del Medico 2008; Blachut 2009; Jabareen 2009; Houliara and

1 In Lieu of Introduction 3

Karamanos 2010; Huhnea et al. 2008; Li and Lin 2010; Liew et al. 2012;Lindgaard and Lund 2011a, b; Overgaard et al. 2010; Perret et al. 2011, 2012;Shen 2010a, b), presence of weld beads (Wang and Koizumi 2010), shellmaterial anisotropy (Shen 2010a, b; Zhang and Gu 2012; Semenyuk and Trach2007; Semenyuk et al. 2008; Li 2007; Li et al. 2011; Lindgaard and Lund2011b)];

2. Researches dedicated to the influence of loading non-axisymmetry [wind-typepressure, aerodynamic pressure at a nonzero angle of attack (Chen and Rotter2012; Huang and Han 2010; Li and Lin 2010; Ohga et al. 2005; Rodriguez andMerodio 2011; Schneider et al. 2005; Sosa and Godo 2009), combined axialand lateral loading (Batikha et al. 2009; Ohga et al. 2006; Vaziri and Estekanchi2006; Mathon and Limam 2006), nonuniform axial compression (Biagi and delMedico 2008; Blachut 2010; Kumarpanda and Ramachandra 2010; Liew et al.2012; Rodriguez and Merodio 2011), shear, bending, and torsion (Alinia et al.2009; Liew et al. 2012; Pirrera et al. 2012; Zhang and Han 2007)];

3. Studies devoted to the influence of initial structural imperfections (Barlag andRothert 2002; Bielewicz and Górski 2002; Cao and Zhao 2010; Campen et al.2002; Cederbaum and Touati 2002; Degenhardt et al. 2010; Ewert et al. 2006;Junior et al. 2006; Gavrilenko 2003; Grigorenko and Kas’yan 2001; Guggen-berger 2006; Hong and Teng 2008; Hunt and Lucena Neto 1993; Krasovskyand Varyanychko 2004; Kristanic and Korelc 2008; Lee et al. 2010; Legay andCombescure 2002; Mang et al. 2006; Nemeth et al. 2002; Obrecht et al. 2006;Papadopoulos et al. 2009; Schenk and Schuëller 2003; Schneider 2006;Schneider and Brede 2005; Schneider and Gettel 2006; Schneider et al. 2005;Vaziri and Estekanchi 2006; Waszczyszyn and Bartczak 2002; Wullschlegerand Meyer-Piening 2002; Wunderlich and Albertin 2002; Zhang and Han 2007;Zhang et al. 2000; Zhu et al. 2002);

4. Studies of subcritical geometrical nonlinearity and of its influence on criticalloads (Junior et al. 2006; Fujii et al. 2000; Gavrylenko 2007; Goncalves et al.2011; Grigoluk and Lopanicyn 2002, 2003; Guarracino and Walker 2008; Kimand Yang 1998; Lord et al. 1999a, b, 2000; Mang et al. 2011; Obodan andGromov 2006, 2013; Polat and Calayir 2010; Prabu et al. 2010; Silvestre 2007;Zhang and Gu 2012).

Modern practical stability analysis concepts result in some calculations for acertain structure and load in order to obtain an admissible load level—say, ifexceeding it can be dangerous due to the rapid change of the shell-deformed shape.But the real shell and load parameters, boundary conditions, and shell materialproperties under operational conditions may differ from the ones assumed in theanalysis, changing the expected structural behaviour. It is thus vitally important toinvestigate a complete picture of the possible versions of behaviour, and to esti-mate the bounds of existence for various equilibrium forms and their correlationwith the integral properties of load and structure (regardless of given load per-turbations and geometrical and physical shell characteristics).

4 1 In Lieu of Introduction

It was ascertained that such an integral structural parameter represents thesolution variability—the number of the Fourier expansion principal harmonic for aperturbed system model solution. It helps to investigate the influence of variousstructural properties—imperfections, load deviations, and unexpected impacts—which may substantially affect load-carrying capability (Andreev et al. 1988).

Thus, available investigations do not make it possible to estimate residualoperability if the system characteristic changes itself as the system is deformed.The investigation of a complete set of equilibrium paths makes it possible todiscover the load bounds for various deformed shapes, including the ‘‘energeticallyhigh’’ ones that are nonetheless realizable under certain conditions (perturbations)and dangerous for the low level of correspondent critical loads. Such a problem ofthe sophisticated analysis of possible deformation patterns is important for thedesign of aerospace vehicles and similar structures which demand weight opti-mization under the condition of securing the required reliability level.

Naturally, an infinity of possible perturbations exists and it is impossible toinvestigate all of them, but a finite number of postcritical deformed shapes doesexist and the number is equal to the number of respective boundary problemsolutions.

If one considers the initial imperfection influence, one may distinguish thefollowing models for stability problems (Mang et al. 2011):

1. nonlinear subcritical state and nonlinear stability problem (nonlinear equationsin variations);

2. linear subcritical state and nonlinear stability problem;3. nonlinear subcritical state and linear stability problem;4. linear subcritical state and linear stability problem.

Each model considered possesses a finite adequate application domain and thedomain boundaries depend on the external influence variability, geometrical andphysical properties as well.

One should emphasize again that it is an untrue assumption that the monoto-nous change of the parameters mentioned inevitably leads to a monotonous criticalload and postcritical behaviour change. Unfortunately, it is impossible to ascertainthat fact in the frameworks of a second and fourth model. Furthermore, it makes itnecessary to create a nonlinear deformation model and algorithm to solve it undernon-axisymmetric deformation.

The plurality of shell-buckling practical analyses, the contradictory dataobtained by the use of simplified models, the absence of a unified scientificallygrounded approach for such analyses, wide usage of applied computer packagesunable to reveal complete branching patterns—all that has inspired us

• to formulate the initial geometrically nonlinear problem using inevitablesimplifications;

• to draw up a numerical algorithm versatile enough to build and to investigate themost complete picture of postbuckling shapes, and to perform a wide-rangingsimulation in order to reveal the features of nonlinear solutions;

1 In Lieu of Introduction 5

• to classify the typical equilibrium forms, its interconnection, and mutualtransition;

• to study the dependence of critical loads upon the parameters of load andstructure, and to reveal the key integral parameters which describe the structuralbehaviour.

We very much hope that this book may be useful for theoretical investigationsand the practice of shell structure design, as well as for the training of mechanicalengineers.

The authors are sincerely indebted to Dr. Victor Ya. Adlutsky, Dr. Alexandr D.Fridman, Prof. Natalia A. Guk, and Dr. Igor P. Zhelezko for their helpful dis-cussions. The authors are deeply indebted to Mr. Marc Beschler and Mr. EdwinBeschler for the manuscript proof-reading and language editing. We are alsothankful to Galina M. Gavelya, Irina P. Shapoval, and Irina A. Shevchenko fortheir aid in the preparation of the manuscript.

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8 1 In Lieu of Introduction

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10 1 In Lieu of Introduction

Chapter 2Boundary Problem of Thin Shells Theory

Abstract A boundary problem for thin elastic shells is formulated. The generallyacceptable geometrical (straight normal) and physical (linear elasticity) hypotheseswhich underlie the relations are considered. Geometrical nonlinearity of shelldeformation is taken into account. Equilibrium equations expressed via the dis-placements of shell middle surface are presented as governing relations. Tangentand bending boundary conditions along the arbitrary shell contour (free andclamped edge, free-hinge and fixed-hinge, elastic support) are formulated. A set ofpreviously satisfied conditions (regularity of shell surface and material, piecewisecontinuity of shell contour and of boundary support parameters) is implementedand the concept of a generalized solution is introduced. The small perturbation of avector-function of a generalized solution is considered, becoming the basis ofinvestigation of non-uniqueness of the generalised solution, and of branching ofthe solutions.

2.1 General Concepts and Hypotheses

Thin shells are the bodies bounded by two curvilinear surfaces placed in such amanner that the distance h between such surfaces is sufficiently less than any otherspecific overall dimensions Li; i ¼ 1; 2. It is convenient to assign the spatialposition of a shell point to its �x ¼ �x1;�x2;�x3ð Þ coordinates referred to shell middlesurface X (see Fig. 2.1). Here coordinate lines �x1;�x2 coincide with the middlesurface curvature lines, and �x3 is normal to these lines.

A11; A22; A12 are the surface’s second fundamental form coefficients.D2 ¼ A11A22 � A2

12.Contravariant components of the respective tensor are given by A11 ¼ A22D�2;

A12 ¼ A21 ¼ �A12D�2; A22 ¼ A11D�2;

N. I. Obodan et al., Nonlinear Behaviour and Stability of Thin-Walled Shells,Solid Mechanics and Its Applications 199, DOI: 10.1007/978-94-007-6365-4_2,� Springer Science+Business Media Dordrecht 2013

11

�G111 ¼

A11; x1 A22 þ A11; x2 A12 � 2A12A12; x1

2D2;

�G122 ¼

A22 2A12; x2 � A22; x1

� �� A12A22; x2

2D2;

�G112 ¼ �G1

21 ¼A22A11; x2 � A12A22; x1

2D2; 1$ 2:

Contour C bounds shell middle surface X. Its segments are denoted as Ci. Thesegments Ci are considered to be finite and possibly disconnected sets. Ci [ 0means that contour segment Ci contains the connected piece of positive length.

mk and mk are covariant and contravariant components of unit normal vector mto C (belonging to X).

sk and sk are covariant and contravariant components of unit tangent vector to C.�Ri denote middle surface curvature radii in �xi directions i ¼ 1; 2.�Bij are principal surface curvatures, i; j ¼ 1; 2.Shell material is characterized by Young’s modulus E �xð Þ and Poisson’s ratio m.

The existence of shell thickness parameter h=Ri� 1

� �makes possible the

transition from a three-dimensional model for the shell body to a two-dimensionalmodel, presuming predefined strain-stress distribution pattern across the shell wall.

It is usually presumed that the rectilinear segments normal to the shell middlesurface remain normal to it and of the same length after deformation (Kirchhoffkinematic hypothesis), and stresses at area elements with �x3 normals are essentiallyless than stresses at area elements with �x1 and �x2 normals, so the influence offormer ones upon the tangential deformations can be neglected (Kirchhoff statichypothesis).

The induced error d depends on shell geometry parameters and of the stress-strain state developed. The error order can be estimated as

d�max h=Ri; h

2�l2i

� �; ð2:1Þ

where li––solution variability parameters in �xi directions, i.e. distances at whichthe stress-strain state changes slowly

x1

x2

x3u1

u2

L i

m

w

h

Ri

Fig. 2.1 Shell element

12 2 Boundary Problem of Thin Shells Theory

ouoxi

����

�����max uj j

li; i ¼ 1; 2;

(u — any component of solution)

Therefore, the error level for states of slow variability depends mainly upon therelative shell thickness, and, for rapidly varying states, upon its variability.Hereinafter the stress-strain states for d� 1 are considered and Kirchhoffhypotheses are valid.

In order to switch to a two-dimensional model, instead of stresses and dis-placements of shell points, the following parameters are introduced:

– middle surface displacement components �u1; �u2; �w in �x1; �x2; �x3 coordinatedirections (see Fig. 2.1);

– middle surface normal unit vector rotation angles �h1; �h2 in �x1; �x2 coordinatedirections;

– middle surface elongation �e11; �e22 and shear �e12; �e21 in �x1; �x2 coordinatedirections;

– middle surface curvature variation �v11; �v22 in �x1; �x2 coordinate directions andtwist �v12;

– projections of force factors which are statically equivalent to acting stresses in�x1 ¼ const;�x2 ¼ const cross-sections (see Fig. 2.2).

In the �x1 ¼ const cross-section, the full vector and full moment of internalforces, being the resultants of internal stresses, are decomposed to:

• tangent tension �T11 and shearing �T12 forces;• transverse force �Q1;• bending M11 and torsional �M12 moments.

In x2 ¼ const––to:

– tangent tension �T22 and shearing �T21 forces;– transverse force �Q2;– bending �M22 and torsional �M21 moments.

x1

x2

x3q

1

q2

q3

T11 T12

M11M12

Q1

T21T22

M21

M22Q2

Fig. 2.2 Force factors inshell cross-sections

2.1 General Concepts and Hypotheses 13

Vectors �x and �U are defined as �u1; �u2ð Þ and �x �u1; �u2ð Þ; �wð Þ, respectively.�am ¼ �akmk, �as ¼ �aksk where a is any variable considered.In particular, �hm ¼ o�w

om denotes an inclination of normal m to the deformed shellmiddle surface.rjðÞ � ðÞ; �xj

denotes oo�xj

.

rijðÞ � ðÞ; �xi �xjdenotes o2

o�xi�xj.

L2X, WijX are standard notations for the space of square-integrable functions and

Sobolev’s functional spaces, respectively.External loading in �xi coordinate direction is denoted as �kqi, i ¼ 1; 2; 3, where

qi ¼ qi �x1;�x2ð Þ are load functions qi �x1;�x2ð Þk k ¼ 1ð Þ, �k is a load parameter. q � q3.Subscript ‘‘cr’’ denotes critical load.

One should emphasize that all loads and stresses throughout the presentmonograph are rationed to respective estimates obtained within the frameworks ofthe linearized boundary problem (see Sect. 3.5 for the formulae).

The summation agreement over repeating indexes is applied throughout thebook unless otherwise explicitly specified.

The principal harmonic number of external load function Fourier expansion isdenoted as mxi (for �xi coordinate direction).

nxi stands for the solution variability (principal harmonic of solution Fourierexpansion) for �xi coordinate direction.

The above-mentioned characteristics for cylindrical and spherical shells con-sidered in the monograph are:

R is a shell radius; L is a shell length (for cylindrical shell).For a circular cylindrical shell �Gs

ij ¼ 0.

For a spherical shell �Gsij ¼ 0 but �G1

12 ¼ � 1R tg �x2

R ;�G2

11 ¼ � 12R sin 2�x2

R :

Principal curvatures:For a circular cylindrical shell: �B11 ¼ 0; �B22 ¼ 1=R; �B12 ¼ 0:For a spherical shell: �B11 ¼ 1=R; �B22 ¼ 1=R; �B12 ¼ 0:

The notations of non-dimensional functions are assigned similar to dimensionalones, omitting overline markings.

An asterisk denotes prescribed values on the boundary contour for respectivefunctions. A tilde denotes equilibrium functions variations.

2.2 Geometrical and Physical Relations

Nonlinear relations between middle surface deformations and displacements,based on the assumption of smallness of deformations and of rotation anglessquares in comparison with unity, are accepted as geometrical relations. So thecomponents of tangent �eij and bending �vij deformation tensor are defined nonlin-early as

14 2 Boundary Problem of Thin Shells Theory

�eij ¼12

�ui; �xj þ �uj; �xi

� � �Bij �w� �Gk

ij�uk þ12

�w; �xi �w; �xj ; ð2:2Þ

�vij ¼ �hi; �xjþ�Gkij �w; �xk ;

�hi ¼ ��w; �xi ; i; j ¼ 1; 2: ð2:3Þ

Transverse shears �ei3; i ¼ 1; 2 and transverse deformation are much smallerthan the other components of the deformation tensor and are accepted to be equalto zero. Let us assume that the kinematic hypothesis of rectilinear normals andlinear distribution of displacements and deformations through the shell wall arevalid for thin-wall structures.

According to the Kirchhoff–Love hypothesis, displacements and deformations�eij �x1;�x2;�x3ð Þ of the shell layer, distanced at �x3 from the middle surface, can beapproximated by linear functions:

�ui �x1;�x2;�x3ð Þ ¼ �ui x1; x2ð Þ þ �x3 �w; �xi �x1;�x2ð Þ; i; j ¼ 1; 2;

�w �x1;�x2;�x3ð Þ ¼ �w �x1;�x2ð Þ;�eij �x1;�x2;�x3ð Þ ¼ �eij �x1;�x2ð Þ þ �x3�vij �x1;�x2ð Þ; i; j ¼ 1; 2:

Taking into account possible inhomogeneity of the shell material, let uspresume the dependence of elastic constants of material upon �x1;�x2;�x3. Let uspresume as well that E �x1;�x2;�x3ð Þ can be described by a piecewise-smooth func-tion, and the shell material structure appears to be regular.

Taking into account the structural symmetry of the shell wall, let us assume theelastic constants E �x1;�x2;�x3ð Þ to be even functions of �x3. Hence the shell tangenttensile and bending stiffnesses can be presented by

�C 1ð Þ1111 �x1;�x2ð Þ ¼

Zh=2

�h=2

�D1111 �x1;�x2;�x3ð Þd�x3;

�C 1ð Þ1212 �x1;�x2ð Þ ¼

Zh=2

�h=2

�D1212 �x1;�x2;�x3ð Þd�x3;

�C 1ð Þ1122 �x1;�x2ð Þ ¼

Zh=2

�h=2

�D1122 �x1;�x2;�x3ð Þd�x3;

�C 2ð Þ1111 �x1;�x2ð Þ ¼

Zh=2

�h=2

�D1111 �x1;�x2;�x3ð Þ�x23 d�x3;

�C 2ð Þ1212 �x1;�x2ð Þ ¼

Zh=2

�h=2

�D1212 �x1;�x2;�x3ð Þ�x23 d�x3;

C 2ð Þ1122 �x1;�x2ð Þ ¼

Zh=2

�h=2

�D1122 �x1;�x2;�x3ð Þ�x23 d�x3;

ð2:4Þ

2.2 Geometrical and Physical Relations 15

where

�D1111 �x1;�x2;�x3ð Þ ¼ E �x1;�x2;�x3ð Þ1� m2

;

�D1122 �x1;�x2;�x3ð Þ ¼ mE �x1;�x2;�x3ð Þ1� m2

;

�D1212 �x1;�x2;�x3ð Þ ¼ E �x1;�x2;�x3ð Þ2 1þ mð Þ :

For instance, for a homogeneous circular cylindrical shell the rigidity coeffi-cients are

�C11111ð Þ ¼ Eh

1�m2; �C11221ð Þ ¼ Ehm

1�m2; �C12121ð Þ ¼ Eh

2 1þmð Þ;

�C11112ð Þ ¼ Eh3

12 1�m2ð Þ;�C1122

2ð Þ ¼ Eh3m12 1�m2ð Þ;

�C12122ð Þ ¼ Eh3

24 1þmð Þ:

Expressions for �Cijklsð Þ and similar expressions for �C sð Þ

ijkl for an arbitrary shell can

be found in the monograph (Vorovich 1999).Functions of forces �Tij and moments �Mij are introduced using integral charac-

teristics, so the relations between the forces and tangent deformation components,between the moments and bending deformation components, subordinate to Hooklaw, are expressed as

�Tij ¼ �Cijkl1ð Þ�ekl; ð2:5Þ

�Mij ¼ �Cijkl2ð Þ�vkl: ð2:6Þ

Here i; j; k; l ¼ 1; 2, if i 6¼ j, then k 6¼ l, and in turns k ¼ 1; l ¼ 2; k ¼ 2; l ¼ 1:Respectively,

�eij ¼ �C 1ð Þijkl

�Tkl; ð2:7Þ

�vij ¼ �C 2ð Þijkl

�Mkl: ð2:8Þ

In order to formulate the governing equations, let us introduce

– non-dimensional variables:

xi ¼ �xi=R; i ¼ 1; 2;

– non-dimensional functions:

Tij ¼ �Tij=Eh; Mij ¼ �Mij=EhR; k qi; qð Þ ¼ �k qi; qð ÞR=Eh;

u1; u2;wð Þ ¼ R�1 �u1; �u2; �wð Þ; vij ¼ R�vij;

16 2 Boundary Problem of Thin Shells Theory

– non-dimensional coefficients:

Bij ¼ R�Bij; Gsij ¼ R�Gs

ij; E x1; x2ð Þ ¼ �E �x1;�x2ð Þ=E;

Cijkl1ð Þ ¼ C

ijkl1ð Þ=Eh; Cijkl

2ð Þ ¼ Cijkl2ð Þ=Eh3;

C 1ð Þijkl ¼ C

1ð ÞijklEh; C 2ð Þ

ijkl ¼ C2ð Þ

ijklEh3;

– parameter e ¼ h2

R2.

2.3 Equilibrium Equations

The equilibrium of the thin-wall shell element is described by the system ofequations (Mushtari and Galimov 1957):

~rj DTij� �

þ kDqi ¼ 0; i ¼ 1; 2: ð2:9Þ

Hereinafter ~rj DTijð Þ � DTijð Þ;xj þ DTstGist.

Projection of forces Tij on the normal to the shell middle surface, taking intoaccount the moments Eq. (2.8) in respect to x1; x2 axes, yields

~rij DMij� �

þ Tijw; xi

� �; xjþDTijBij þ kDq ¼ 0: ð2:10Þ

Hereinafter ~rij DMijð Þ � DMijð Þ;xi xjþ DMijGs

ij

� �

; xs

.

2.4 Governing Equations

Substitution of (2.2), (2.3) into (2.5) and (2.6), respectively, gives us the expres-sions of forces Tij and moments Mij as functions of middle surface displacements

Tij ¼ Cijkl1ð Þ

12 uk;xl þ ul;xk

� �þ 1

2 w2;xkþ w2

;xl

� �� Bklw� Gs

klus

h i;

Mij ¼ e Cijkl2ð Þvkl ¼ eCijkl

2ð Þ �w;xkxl þ Gsklw;xs

� :

Substitution of these expressions into equilibrium Eqs. (2.9), (2.10) yields thesystem of three governing equations with three unknown functions (Vorovich1999).

2.2 Geometrical and Physical Relations 17

rj DCijkl1ð Þrkul

� �þ DGi

stCstkl1ð Þ rkul � DGi

stCstkl1ð Þ Bklw�

12

w; xk w; xl

��

DCijkl1ð Þ Bklw�

12

w; xk w; xl

��

; xj

þkDqi ¼ 0 i ¼ 1; 2;

ð2:11Þ

erij DCijkl2ð Þrklw

� �� DCijkl

1ð Þekl Bij þrijw� �

� kDq ¼ 0: ð2:12Þ

2.5 Shell Support Conditions

Governing Eqs. (2.11), (2.12) are to be completed with the conditions of boundarycontour support. Boundary contour C is considered to be partitioned in two dif-ferent manners: for bending boundary conditions (Vorovich 1999)

C1 ¼ C1 þ C2 þ C3 þ C4;

and for tangent ones,

C2 ¼ C5 þ C6 þ C7 þ C8:

In order to describe the shell boundary support, the elasticity support coeffi-

cients �kijl x1; x2ð Þ; i; j ¼ 1; 2; �kpn

2 x1; x2ð Þ; p; n ¼ 3; 4; are introduced. All functions�kab

l sð Þ; l ¼ 1; 2 are considered to be piecewise-continuous functions of therespective edge segment. In order to introduce dimensionless factors, the functions�kab

l sð Þ were divided into their average values.Let us consider four types of bending support conditions:

1. At C1––clamped edge:

wjC1¼ w�; ð2:13Þ

hmjC1¼ h�m: ð2:14Þ

Condition mostly used in practice––clamping of initially undeformed structures(w� ¼ 0; h�m ¼ 0).

2. At C2––elastic support in respect to rotation,

MmjC2¼ k44

2 hm;

deflection prescribed

18 2 Boundary Problem of Thin Shells Theory

wjC2¼ w�; ð2:15Þ

and energy accumulation along elastic edge support

PjC2¼ 1

2

Z

C2

k442 h2

mds: ð2:16Þ

3. At C3––elastic support in respect to deflection

QjC3¼ k33

2 w;

clamping in respect to edge rotation

hm C3j ¼ h�m; ð2:17Þ

and edge energy accumulation

PjC3¼ 1

2

Z

C3

k332 w2ds: ð2:18Þ

4. At C4 general elastic support takes place and edge energy accumulation ispresented by

PjC4¼ 1

2

Z

C4

kpn2 wpwnjp; n¼3;4 ds; ð2:19Þ

where w3 � w; w4 � hm; matrix kpn2 sð Þ is positively definite, i.e.

8 wp; wn kpn2 sð Þwpwnjp; n ¼ 3;4 � 0:

Tangent conditions of the shell edge support belong to a second type of contourpartitioning:

1. At C5 the shell is clamped tangentially:

usjC5¼ u�s ; umjC5

¼ u�m: ð2:20Þ

Clamping of initially undeformed structure implies u�s ¼ 0; u�m ¼ 0.

2. At C6 the mixed boundary conditions are assumed: deflection is prescribed, andtangentially the edge is supported elastically as

umjC6¼ u�; ð2:21Þ

PjC6¼ 1

2

Z

C6

kss1 u2

s ds: ð2:22Þ

2.5 Shell Support Conditions 19

3. At C7 the mixed boundary conditions are assumed as well: tangent displacementis prescribed, and normal direction displacement is supported elastically as

usjC7¼ u�s ; ð2:23Þ

PjC7¼ 1

2

Z

C7

kmm1 u2

m ds; ð2:24Þ

no summation over m.

4. At C8 Tangential elastic support for both displacements is assumed and edgeenergy accumulation can be calculated as

PjC8¼ 1

2

Z

C8

kij1 uiujji;j¼1;2ds; ð2:25Þ

where matrix kij1 sð Þ is positive definite, i.e., 8 ui; uj kij

1 ðsÞuiujji;j¼1;2� 0.The system of Eqs. (2.11), (2.12) in conjunction with geometrical boundary

conditions (2.13)–(2.25) describes the broad class of nonlinear problems of thin-wall structures.

The system of formulated Eqs. (2.11), (2.12) is essentially nonlinear in spite ofEq. (2.11) being linear in respect to ul; l ¼ 1; 2:

Boundary problem formulation for the system of Eqs. (2.11), (2.12) allowscombining various types of boundary conditions, including mandatory wjC0

¼ 0 ata certain boundary contour segment C0 [ 0. Thus, determination of the stress-strain state of thin-wall systems is reduced to generation and investigation of thenonlinear differential equations system (2.11), (2.12) with boundary conditionsproperly chosen from (2.13)–(2.25) according to the problem’s formulation.

2.6 Variational Principles

Variational methods are widely used in shell theory to build the governingequations and to establish numerical procedures.

Variational methods based on energetic principles make it possible to buildenergetically optimal solution approximations without resort to additionalhypotheses, frequently contradictory ones. The variational approach delivers auniform and consistent method for simplifying governing relations in order tobuild a soundly simplified model of analysis.

Solution functions appear in the variational method functionals with their lowerorder derivatives in comparison with their appearance in governing differentialequations, allowing wider approximative functions classes.

The variational approach does not make it necessary to consider naturalboundary conditions or the formulation of boundary conditions resulting from therespective functionals’ stationarity conditions. It is a matter of importance for

20 2 Boundary Problem of Thin Shells Theory

simplified models with a lower order of correspondent governing equations. As forboundary conditions for the simplified models, their simplification is connectedwith the difficult problem of distinguishing ‘‘principal’’ and ‘‘secondary’’ restraints,a problem that can not, actually, be resolved univocally. Implementation of themethods mentioned is usually based on some variational principle (Washizu 1982).

Total shell energy E consists of deformation potential energy P and of surfaceand boundary loads work A:

I�EhR2 ¼ Pþ A ¼ 1

2

ZZ

X

Cijkl1ð Þeijekl þ eCijkl

2ð Þvijvkl

h iDdx1 dx2

þ 12

Z

C2

k442 h2

m dsþ

8<

:

Z

C3

k332 w2 dsþ

Z

C4

kpn2 wpwnjp;n¼3; 4 dsþ

Z

C6

kss1 u2

s ds

þZ

C7

kmm1 u2

m dsþZ

C8

kij1 uiujji;j¼1; 2 ds

9=

;�

Z

C2

Mm�hm

2

64 dsþZ

C3

Q�w ds

þZ

C4

Mm�hm þ Q�w½ dsþZ

C6

Ts�ws dsþ�Z

C7

Tm�wm ds

þZ

C8

Tm�wm þ Ts�ws½ ds�ZZ

X

k qwþ qiui

� dx1 dx2: ð2:26Þ

Here w3 � w; w4 � hm. Hooke’s law relations are presumed to be satisfiedpreviously.

In order to express the governing relations in physical variables (vector-func-tion U x1; x2ð Þ ¼ u1;f u2; w; h1; h2; T11; T12; T22; M11; M22; Q11; Q22g) let ustransform the functional (2.26) (by joining the relations (2.2), (2.3) using Lagrangemultipliers):

I�EhR2¼PþA¼

Z Z

X

12

Cijkl1ð Þ

�eij Tij� �

ekl Tkl� �

�

þ eCijkl2ð Þvij Mij

� �vkl Mkl� ��

�Tijuij�Mij/ij�Qimfi

iDdx1dx2g

þ12

Z

C2

k442 h2

mdsþ

8<

:

Z

C3

k332 w2dsþ

Z

C4

kpn2 wpwnjp;n¼3;4ds

þZ

C6

kss1 u2

sdsþZ

C7

kmm1 u2

mdsþZ

C8

kij1 uiujji;j¼1;2ds

9=

;�

Z

C2

Mm�hm

2

64 dsþZ

C3

Q�wds

2.6 Variational Principles 21

þZ

C4

Mm�hmþQ�w½ dsþZ

C6

Ts�wsds

þZ

C7

Tm�wmdsþZ

C8

Tm�wmþTs�ws½ dsþZ

C1þC2

w�w�ð ÞQds

þZ

C5þC6

wm�w�m� �

TmdsþZ

C5þC7

ws�w�s� �

Tsds

þZ

C1

hm�h�m� �

Mmds

3

75�Z Z

X

k qwþqiui

� dx1dx2 ;

ð2:27Þ

where

uij � eij Tij� �

� 12

ui;xj þ uj;xi

� þ Bijwþ Gk

ijuk �12

hihj;

fi � hi þ w;xi ;

uij � vij Mij� �

� hi;xj�Gkijhk;

Qim � Qi þ Tijhj – generalized Kirchhoff transverse force;

eij Tijð Þ; vij Mijð Þ are given by dimensionless (2.7), (2.8), respectively.

2.7 Generalized Solution in Displacements

In order to apply a method for solving nonlinear boundary problem (2.11), (2.12),(2.13)–(2.25), one should prove the existence of a solution to the problem andjustify the method’s ability to converge to the solution. Standard variationalmethods are not always justifiable as they require the functional to meet specificdemands, particularly, it should be coercitive to ensure method convergence.Functional (2.27) does not meet this demand tacitly as it is not quadratic.

Meanwhile, the theorem (Vorovich 1999) states that the critical points set offunctional (2.27) (defined in special space Wtj) coincides with the generalizedsolution set of the problem in question (2.11), (2.12), (2.13)–(2.25).

Let us formulate the generalized nonlinear boundary problem of shell theory indisplacements presuming the following conditions to be satisfied previously:

(1) shell middle surface S 2 C2 is regular;(2) C is piecewise-smooth contour of C1

C; C ¼ C1 [ C2 class;(3) shell material is regular;

22 2 Boundary Problem of Thin Shells Theory