Shell Stability Handbook

SHELL STABILITY HANDBOOK

SHELL STABILITY HANDBOOK Edited by

LARS Å.SAMUELSON The Swedish Plant Directorate, Stockholm, Sweden and

SIGGE EGGWERTZ Bloms Ingenjörsbyrå, Sundbyborg, Sweden

ELSEVIER APPLIED SCIENCE LONDON and NEW YORK

ELSEVIER SCIENCE PUBLISHERS LTD Crown House, Linton Road, Barking, Essex IG11 8JU, England This edition published in the Taylor & Francis e-Library, 2005. “ To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to http://www.ebookstore.tandf.co.uk/.” WITH 9 TABLES AND 192 ILLUSTRATIONS ENGLISH EDITION © 1992 ELSEVIER SCIENCE PUBLISHERS LTD British Library Cataloguing in Publication Data Shell Stability Handbook I. Samuelson, Lars A. II. Eggwertz, Sigge 624.1 ISBN 0-203-21367-X Master e-book ISBN

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PREFACE The Shell Stability Handbook is a handbook for calculation of the carrying capacity of shell structures with respect to buckling. The aim has been to develop a branch independent handbook which gives conservative estimates of the carrying capacity. The safety factors which should be used in a certain case must be taken from the applicable code for the considered structure, e.g. pressure vessel codes, steel structure codes etc. Initiator behind the handbook is Professor Lars Å.Samuelson who already in the seventies during his time at the Aeronautical Research Institute of Sweden felt a strong need for such a handbook. The development of the handbook, which also included substantial research in the field of shell stability, started in 1982. The first Swedish edition was published in 1990. The handbook has been developed by a working group including members from the pressure vessel, the building and the aeronautical branches: Professor Torsten Höglund, Chairman The Royal Institute of Technology, Stockholm

(2, 3.6, 3.7, 6.3.2)

Dr Lars Dahlberg The Swedish Plant Inspectorate, Stockholm

(5.1)

Professor Sigge Eggwertz Bloms Ingenjörsbyrå AB

(3.4, 8)

Mr Gert Hessling The Swedish Plant Inspectorate, Stockholm.

(3.5)

Professor Bernt Johansson Luleå Institute of Technology, Luleå

(3.1, 3.2.1, 3.2.2, 3.3)

Mr Kurt Lundi The Swedish Institute of Steel Construction, Stockholm

Notations

Professor Lars Å.Samuelson The Swedish Plant Inspectorate, Stockholm

(1, 3.2.2, 3.2.3, 3.7, 4, 5, 6, 7, 8)

The work in the group was carried out in close cooperation, but the responsibility for the different chapters was divided as indicated above. A reference group consisting of members from industries, authorities and universities gave technical support during the work: Mr Gunnar Emling, Chairman

The National Swedish Board of Occupational Safety and Health

Mr Kjell Andersson

Hedtank AB

Mr Bengt Andreason

Kockums Marine

Professor Bo Edlund

Chalmers Institute of Technology

Mr Lars Hammarström

Volvo AB

Mr Bengt Hellman

The National Swedish Board of Physical Planning and Building

Mr Gunnar Holmberg

Saab Scania AB

Mr Rolf Jarlås

The Aeronautical Research Institute of Sweden

Mr Karl-Arne Johansson

Amfibieteknik AB

Mr Åke Ringqvist

Flakt Industries AB

Professor Alf Samuels son

Chalmers Institute of Technology

Mr Lars Sjöström

Saab Scania AB

Mr Bengt Skanselid

ÅF Energikonsult Syd AB

Dr Per-Olof Thomasson

The Swedish Institute of Steel Construction

Mr Adam Zdunek

The Aeronautical Research Institute of Sweden

Professor Bengt Åkesson

Chalmers Institute of Technology

The reference group gave valuable support and many contributions during the work including examples from their own practice which increased the usefulness of the handbook. The work was financially supported by: The Swedish Work Environment Fund Bloms Ingenjörsbyrå AB The Swedish Council for Building Research Chalmers Institute of Technology Götaverken Arendal AB Götaverken Mekan AB The Swedish Institute of Plastics end Rubber The National Swedish Board of Physical Planning and Building The Swedish Institute of Steel Construction The Swedish Plant Inspectorate SSAB Association of Swedish Engineering Industries Tyréns Företagsgrupp AB 3K Akustikbyrån

The main financier was The Swedish Work Environment Fund, but all contributions were important for the first publication of the handbook. The figures were, with excellent result, drawn by Mr Anders Samuelson. Lars Dahlberg

CONTENTS PREFACE NOTATIONS

Chapter 1: INTRODUCTION

v ix 1

Chapter 2: DESIGN PHILOSOPHY

33

Chapter 3: ELEMENTARY CASES

55

Chapter 4: EFFECT OF LOCAL LOADS AND DISTURBANCES

163

Chapter 5: INFLUENCE OF TEMPERATURE

193

Chapter 6: STABILITY ANALYSIS BY USE OF NUMERICAL METHODS Chapter 7: SHELLS OF COMPOSITE MATERIALS

212

Chapter 8: EXAMPLES

237

REFERENCES

231

258

APPENDIX Brief review of the theory of stability for shell structures A

266

APPENDIX Elementary cases for stress and deformation analysis of rings B subjected to forces in their own plane

276

APPENDIX Diagrams for analysis of stringer stiffened cylindrical shells C under axial compression

283

F

Force, load.

Fk

Characteristic value of force.

G

Elastic shear modulus.

I

Moment of inertia of cross section.

M

Bending moment.

N

Normal force.

P

Force.

Q

Load.

T

Torsional moment.

V

Shear force, lateral force.

W

Section modulus.

b

Width.

d

Diameter.

e

Eccentricity.

f

Strength (in this handbook the symbol σ has been used synonymously with f).

fk

Characteristic value of strength.

h

Height.

l

Length.

m

Number of halfwaves of the buckling mode in the axial direction.

n

Number of complete waves of the buckling mode in the circumferential direction.

p

Pressure.

r

Radius.

s

Coordinate in the meridional direction.

t

Thickness.

u

Displacement in the x- or s-direction.

v

Displacement in the φ-direction.

w

Displacement in the z-direction (perpendicular to the middle surface of the shell).

x

Coordinate in the axial direction of a global system, Fig. 1.

y

Coordinate in a global system, Fig.1.

z

Coordinate perpendicular to the middle surface of the shell in a local system, Fig. 1.

α

Angle, reduction factor with reference to tolerance class.

β

Rotation of shell element, Fig. 1.23, spherical shell parameter, eqn (3.110).

γ

Load or strength factor (partial coefficient), shear deformation, gravity per unit volume.

γf

Load factor.

γm

Resistance factor related to the material properties.

γn

Resistance factor related to safety class.

γm n

Abbreviation for the product γmγn.

ε

Strain.

η

Reduction factor with reference to initial deflections etc.

λ

Slenderness ratio.

v

Poisson’s ratio.

φ

Angle, co-ordinate in the circumferential direction.

ψ

Angle, coordinate in the meridional direction of a spherical shell.

θ

Meridional rotation, Fig.3.37.

ω

Reduction factor for strength in the elasto-plastic region.

Subscripts R

Resistance, synonomous with carrying capacity.

S

Load effect (sollicitation).

r

Ring (example: Ar), circumferential direction.

x

Coordinate direction.

φ

Coordinate direction.

u

Ultimate strength, at collapse.

el

Buckling quantity according to linear (classical) theory.

elr

Reduced buckling quantity.

s

Stringer, coordinate direction.

b

Bending.

z

Coordinate direction.

y

Coordinate direction.

cr

Critical.

ef

Effective.

eq

Equivalent.

Differentials

NOTATIONS The notations of the Swedish Regulations for Steel Structures are used throughout this handbook. These are very similar to those internationally accepted and used for instance in the ECCS design rules and in the CEN Eurocodes 1 and 3. The symbol σ has been chosen, however, to denote stress whether it refers to the strength of the material or the load effect. The meaning should be evident from the context in all cases.

Coordinate system The most common shell category is, beyond comparison, the shell of revolution which may be composed of cylinders, spherical caps, cones etc. For such shells the symbols shown in Fig. 1 are generally used. In cases where other symbols are introduced, they will be defined in the section where they appear.

Fig. 1. Coordinate systems for cylindrical and axisymmetrical shells. Symbols A

Area.

C

Extensional stiffness, eqn (1.15).

D

Bending stiffness, eqn (1.15).

E

Young’s modulus of elasticity.

Units All formulas in this handbook are independent of the unit system used. SI units are used in some special applications and in the examples . Abbreviations ASME

American Society of Mechanical Engineers.

ASTM

American Standard for Testing Material.

BS

British Standard.

BSK

Swedish Regulations for Steel Structures.

DASt

Deutscher Ausschuss für Stahlbau (German Task Group for Steel Design).

DIN

Deutsche Industrienormen (German industrial code).

ECCS

European Convention for Constructional Steelwork.

EC 1&3

CEN Eurocode 1 for actions on structures and Eurocode 3 for steel structures.

NR

Swedish Building Code (new).

SBN

Swedish Building Code (old).

TKN

Swedish Pressure Vessel Code.

Definitions and concepts Bifurcation, branching into two or more states of equilibrium. The bifurcation load of a perfect structure is commonly called the classical buckling load. Branching point, see Bifurcation. Buckling is a phenomenon giving large deflections perpendicular to the shell surface. It is caused by forces acting in the shell middle surface where they are giving rise to compressive and/or shear stresses. The buckling behavior may be stable or unstable. Characteristic value is a value of the quantity considered which corresponds to a given fractile of its statistical distribution. For the resistance the 5 per cent fractile is normally chosen with the exception for steel materials where a value is chosen approximately equal to the 1 per cent fractile. For loads, the 98 per cent fractile of the distribution of the yearly maximum is usually chosen as representative for an extreme value of a variable load (see 2.4.2). Circumferential direction, see Fig. 1.

Classical buckling theory is a linear theory for analysis of structural stability under the action of a load. It is often based on a number of simplifications which have been introduced in order to obtain solutions in a closed form. The theory assumes that the material is linearly elastic, that the structure has an ideal geometrical shape (straight bar, plane plate) and further that the displacements are small. Collapse means such a decrease of the carrying capacity of a structure that it becomes unsuitable for its purpose. Collapse is characterized by the maximum of the loaddeflection diagram. Ultimate failure is used synonymously with collapse. Conical shell, see Shell. Crown ring, see Fig. 3.55. Cylindrical shell, see Shell. Effective area is a fictitious cross-section area which is used in design calculations instead of the actual area. The effective area can be used in compressed members with cross sections built up from slender elements, where it forms a means of considering the influence of local buckling of the elements on the carrying capacity of the member. The effective area of a slender, rectangular cross-section element can be evaluated either as an effective width multiplied by the actual thickness (see Dubas and Gehri, 1986) or as an effective thickness multiplied by the actual width. Formulas for calculating the effective width, or the effective thickness respectively, have been deduced on the basis of the behavior of slender plates in compression. Equilibrium can be stable, unstable or indifferent. Equilibrium branching (bifurcation) implies that two or more conditions of equilibrium exist at a given load (branching point). Equivalent wall thickness is a concept introduced to describe a shell with varying thickness, or provided with stiffeners, as a homogeneous shell with constant thickness. Failure. In this handbook the word is used as a definition of the state where the maximum load carrying capacity has just been passed under increasing deformation. In this context the word is synonymous with collapse. First-order theory implies that a model is used where the influence of the deflections on the internal forces and moments is neglected. The first-order theory yields first-order forces and moments etc. Generator, see Shell. Imperfection is a term for a deviation from the assumptions on which the classical theory is based. Examples of imperfections in shells are initial buckles, residual stresses and unintended load eccentricity. Initial buckles (often local) are deviations from the ideal geometrical shape in an unloaded shell or an unloaded plate. Interaction formulas are design criteria for such load cases which can easily be divided into simpler cases. Interaction formulas assume that for each one of the simpler load cases a measure of the load effect can be given as well as the corresponding carrying capacity. The terms of the interaction formula contain the ratio between the load effect and the capacity for each single load case. Examples of interaction formulas are found in 3.2.1.8. Limit state, see Serviceability limit state and Ultimate limit state respectively as well as subsection 2.2.4.

Limit state design (method of partial coefficients) is a method for structural design where the safety requirements are considered by introducing statistically determined quantities (characteristic values) and load and resistance factors (partial coefficients). Linear buckling theory (=classical buckling theory) is a theory for the stability of a structure subjected to compressive loads. It assumes that the deformations are small, that the material is a linearly elastic and that internal forces and moments are determined according to first-order theory. Load factor is a factor which takes into account the variability of the load, in addition to the variability considered by the characteristic value. Meridian, see Fig.3.37. Synonym for Generator. Membrane state is a state of loading where only in-plane internal forces in the middle surface of the shell exist. Method of allowable stresses is a method for structural design where the safety requirements are considered by applying a maximum allowable stress for the structure. The stresses in the structure are calculated for given values of loads and load combinations. The allowable stress is determined as a given fraction of the stress which leads to fracture or instability. Mode, shape of buckling deflection. Node line is a line, or curve, formed by the points on the shell which do not move during buckling. Partial coefficient, see Limit state design. Perfect shell is a shell which is assumed to have a perfect geometrical shape, e.g. without initial buckles. It is furthermore assumed to have such material properties that it is in complete accord with the linear theory of elasticity and that it is relieved from residual stresses. The boundary conditions are assumed to agree with a chosen idealized model, such as a hinge without friction, complete clamping, entirely fixed boundary etc. Plastic buckling implies that buckling occurs at a load level where at least part of the structure before or during buckling is subjected to plastic deformation, i.e. strain which does not return to zero on unloading. Polygonal ring, see Fig. 3.55. Postbuckling state. A structural member can, under certain conditions, carry a load which exceeds the buckling load according to the classical theory. This exceedance is a result of redistribution of stresses. When a structural member is subjected to a load which is higher than the buckling load the structure is said to be in a postbuckling state, characterized by loss of stiffness and large deformations. Prebuckling state is the state of the structural member when the load is lower than the buckling load. Residual stresses are stresses present in a structural member which is neither subject to external loads nor to other actions (not even gravity loads). Residual stresses arise from manufacturing procedures such as rolling, flame-cutting, welding, cold forming and straightening. Resistance, (used as a synonym for load carrying capacity) is a measure of the capacity of the structure to carry load. The resistance can be expressed as a load, as an internal cross-sectional quantity (normal force, bending moment, shear force), or as a stress. If not otherwise stated, the maximum resistance is referred to as the load which causes failure of the structure (collapse).

Resistance factor is a factor which takes into account the variability of the resistance in addition to the variability considered by the characteristic value. Safety class refers to a classification of building structures and structural members with regard to the consequences of a failure, considering mainly the risk of human injury. Safety factor is the ratio between the ultimate failure stress and the allowable stress. Second-order theory implies a computer model in which the effect of the deflection on the internal forces and moments is taken into account. The second-order theory yields second-order forces and moments etc. Serviceability limit state is a state where the structure is on the limit of not satisfying one of the requirements prescribed for its function under normal conditions. Shell, a thin-walled body shaped as a curved surface with the thickness measured along the normal to the surface being small compared with the dimensions in the other directions. The shell carries its load mainly by membrane forces.. The middle surface has a finite radius of curvature at each point. The radius of curvature can, however, be infinite in one direction (shell with single curvature, e.g. cylinder). Axisymmetric shells form a particularly important class of shells. They can, depending on the shape of the meridian, be divided into: – cylindrical shells – conical shells – spherical shells (domes, caps) – toroidal shells – torispherical shells. The middle surface of such shells is formed by the points on a so called generator (meridian) when the generator makes one revolution around the axis. The generator may be a straight line or a plane curve defined in the same plane as the axis of rotation, see e.g. Fig. 1. Single-curvature shell, see Shell. Spherical shell, see Shell. Torsional buckling is a stability phenomenon where a bar, loaded in compression, and possibly also in bending, is subject to deflection and simultaneous twisting. Torispherical shell,see Shell. Ultimate limit state is a state where the carrying structural member approaches the limit of some kind of failure.

1 INTRODUCTION 1.1 Scope Stability of structural members of various kinds in buildings, vehicles, tanks, process vessels, silos etc. constitutes one of the factors which must be considered in the design analysis. For straight compressed bars the Euler theory, well known to most engineers, may be applicable under certain conditions. A similar type of theory can be applied for plane plates and shells with results which are more or less accurate. Especially for some types of shells a design based on the classical theory, corresponding to the Euler theory, may imply a significant risk, since the actual carrying capacity may be only a fraction of the value obtained from this theory. The fact that this behavior of shell structures is not widely known and, furthermore, that simple procedures for analysis of the buckling of shells are available only to a very limited extent in design codes, has lead to the development work presented in this Shell Stability Handbook. The objectives of the work can be summarized as follows: – provide practical guidance for determination of the actual resistance of various types of shell elements subject to the most commonly occurring external loads which can cause buckling. – give advise and direction for the interpretation of buckling loads obtained by use of general computer programs based on FEM and similar methods. – discuss possible ways of considering the effect of disturbances and imperfections in geometry and loading etc. on the resistance of a shell loaded in compression. – constitute a supplement to existing design codes and standards. In most countries, shell buckling is treated to a very limited extent. In many cases problems occur which are entirely neglected in these documents. The assessment of structural safety varies today between different codes, which may result in misinterpretations. The Shell Stability Handbook provides routines for determining a safe, lower limit of the resistance of the shell (characteristic value). This resistance can be used as a basis for design. The design value is obtained by dividing the value by a safety factor or a partial coefficient (resistance factor). – form a textbook on shell stability, where different aspects of the basic principles of shell theory and analysis are described. The presentation focuses on the importance of the different parameters which affect the carrying capacity of the shell. Rules for the design of bars, beams and thin plates subjected to buckling are given in most national codes. A comprehensive treatment of plate stability problems was published by the ECCS, see Dubas and Gehri, 1986. Recommendations for design of shell structures were published by ECCS in 1988. The present handbook may be considered as a supplement to the kind of documents mentioned. While the ECCS

Shell stability handbook

2

recommendations are adapted for use in the design of steel structures, the Shell Stability Handbook is directed towards a more general description of the buckling behavior. Applications for different materials are described separately, in particular when the elasto-plastic properties differ. This edition of the handbook is obviously not complete. The intention is to continue the work by developing design recommendations for additional types of shells and loadings. These will be introduced into later editions. The stability problems addressed in this handbook mainly concern shells of metallic materials. Fiber reinforced plastics, reinforced concrete and sandwich shells are used to an increasing extent. Problems occurring with these materials are not treated in detail but reference to relevant literature is made, if available. It should be pointed out that the handbook is mainly concerned with factors which may influence the buckling behavior and, consequently, the buckling resistance. Local effects, such as the design of edge stiffeners around a hole, are considered in the traditional manner according to applicable standards, using the factors prescribed for the critical load case, which may be ultimate failure under internal pressure. Where coupling is possible between e.g. local stresses and the buckling behavior, an assessment of the influence is given. 1.2 Brief historical notes with references to the literature The theory of stability of shells is relatively young but, due to the extensive use of shells as structural elements in modern technical applications—aerospace vehicles, containment vessels, offshore and building structures etc.—very intensive research has been carried out since the thirties. Some of the most important milestones in the development of the shell buckling theory are presented in the following review. 1.2.1 Development of the shell theory Euler’s formulas for determining the critical load of a compressed straight bar were published in the middle of the 18th century. The theory was further developed during the latter half of the 19th century, when the stability of thin plates was also analyzed in an analogous way. A theory of shell buckling was first proposed in the beginning of the 20th century, by Lorenz, 1908, and Timoshenko, 1910, who presented solutions for axially compressed circular cylindrical shells. External lateral pressure on cylindrical shells was treated by Lorenz, 1911, Southwell, 1913, and by von Mises, 1914. More profound studies of the theory of stability for shell structures were, however, not initiated until the thirties. To a large extent, the theoretical and experimental data produced during that time were of a basic nature. The results are, however, still used in the design of shells, although the capabilities for performing accurate numerical analyses have improved drastically since then. Flügge, 1932, 1934, presented an extensive theory for cylindrical shells where he treated various load cases and load combinations. During the thirties the first edition of Timoshenko’s monograph “Theory of elastic stability” was also published, see Timoshenko, 1936, or the revised edition, Timoshenko-Gere, 1961. A linear buckling

Introduction

3

theory for spherical shells was first established by Zoelly, 1915. The stability of conical shells does not seem to have attracted attention during the first decades of the 20th century. The first paper on cones was published by Pflüger, 1937. The first attempts to develop a theory of stability all had in common that a linear theory was used which will be referred to as the classical theory in the text to follow. Donnell, 1934, investigated the influence of second order quantities and found that they could be of great importance in the estimate of the resistance. As a result he developed a nonlinear theory by which he could take second order effects into account, see Donnell and Wan, 1950. Donnell’s equations have, subsequently, been used by a large number of investigators, see e.g. Brush and Almroth, 1975. The nonlinear shell theory leads to very complicated systems of equations. General, closed form solutions of these equations have been achieved only for a few simple cases. In the design of shells against buckling, the methods developed during the thirties are, consequently, still in use. During the course of the years a large number of test series have been carried out in order to verify the theories. A classical work is the NACA Handbook of Structural Stability, Part I– VI, Gerard-Becker, 1957–58, in which all test results known up to that time were collected and compared to current theories. The NACA Handbook has been used as a basis for the design of several generations of aircraft and space vehicles. During the sixties, the awareness of the extreme sensitivity, in some cases, of shell structures to initial imperfections, residual stresses and other disturbances gradually increased, and extensive research work was performed in an effort to find a solution to the buckling problem which would give more realistic results. Many suggestions were made, but few of these lead to any significant improvement. The shape and size of initial deformations vary randomly, and it is now commonly agreed that the design procedure must consider the statistical nature of the disturbances. A comprehensive study of the buckling behavior of shells including their sensitivity to initial deformations was published in a thesis by Koiter, 1945. The research work, which was carried out during World War II, was written in the Dutch language. For that reason it did not become known internationally until the end of the sixties when translations into English were published, Koiter, 1967, 1970. Today it is considered to be a pioneering work. Koiter’s theory for evaluation of the sensitivity of a shell to initial deformations has been of great use in the further development of theories and calculation procedures during the last decade. Since the theory treats the condition at incipient buckling only, it does not enable a quantification of the reduction of the buckling resistance of the shell. It only provides an indication whether the shell is imperfection sensitive or not. Statistical studies of the influence of initial imperfections have been performed by several scientists, see e.g. Hansen, 1977, and Arbocz, 1987, 1991. In these papers, probabilistic methods are introduced in the evaluation of the risk that failure due to buckling will occur at a given load level. During the last years of the sixties, the electronic computer became more commonly available for research and development and several computer programs for buckling analysis were produced during this period. Examples of such programs can be found in : Bushnell, 1972, 1974, Cohen, 1968, Ball, 1968, Zienkiewics, 1971, Almroth et al, 1980, Samuelson, 1975. One of these programs, BOSOR, was developed by Bushnell as a special purpose program for the analysis of buckling and vibration of axisymmetric shells

Shell stability handbook

4

under otherwise rather general conditions. BOSOR determines the buckling load by use of eigenvalue theory and cannot be used to study the effect of nonsymmetric initial imperfections. Other similar programs have been developed, see Ball, 1968, Cohen, 1968, Samuelson, 1975, where nonsymmetric collapse can be evaluated. Use of such programs has made it possible to demonstrate a good agreement between theory and experiments for shells with known initial deformations. Another specialized program is STAGS, see Almroth et al., 1980, which was developed especially for solving general, nonlinear shell problems. It was, therefore, more efficient than many of the Finite Element (FEM) programs which were originally written for solution of linear problems of elasticity and later on were provided with routines for handling the nonlinear behavior. With the fast development of computer systems it has become feasible to solve more and more complicated problems, and today several efficient FEM-systems for nonlinear buckling analysis are available. In spite of this the computer capacity still poses practical limits to the size of the problems which can be analyzed. During the last few years, the development has thus been directed towards establishing efficient routines for numerical analysis of large systems of nonlinear equations with special reference to the behavior of shell structures in the postbuckling state and at failure. An efficient method for integration which is being introduced in many systems for structural analysis, was developed by Riks, 1979. 1.2.2 Current state of the art Shell stability research and development activities are probably even more intense today than ever before. The attention has changed to new fields of application. Optimization with respect to weight and manufacturing cost has become a major design factor. A brief summary of the present situation is given below. 1.2.2.1. Development of the shell theory The fundamental shell theory, the history of which has been briefly reviewed, is no longer subject to further development, since the parameters which govern the buckling process are known and sufficiently well described by existing theories. The emphasis is, today, rather on the development of numerical methods for solving the general shell equations and establishing solution routines for large highly nonlinear problems. The development of FEM-programs for different types of structures, e.g. shells, also proceeded in parallel to the evolution of the shell theory. In many cases it has been possible to solve extremely difficult shell problems by use of FEM-systems based on nonlinear theory. Today, a relatively large number of such systems are available for analysis of stability problems of various kinds. Examples of such systems are given in a collocation published by Fredriksson-Mackerle, 1980. Many shell structures belong to special classes, such as shells of revolution subjected to axisymmetric or nonsymmetric loads. In such cases it may be advantageous to develop special purpose programs, which would be faster and less expensive to use. Examples of such programs have been mentioned above.

Introduction

5

1.2.2.2. Methods of design Despite the fact that powerful computer programs are available for the analysis of buckling of shell structures it is seldom advisable to use these in the design. The reason is that the buckling resistance of the shell is determined to a high degree by other parameters than the nominal geometry and the material properties. Initial imperfections and external disturbances in the loading of a magnitude and form which cannot normally be predicted at the design stage, play a very important role. The design calculations will, thus, under the present circumstances have to be based on methods where the influence of disturbances etc. has been determined by comparison with test results. On the other hand, a subsequent check of the resistance might be performed for the completed structure where the actual geometry can be measured and be properly considered in the analysis. Examples of comparisons between theoretical buckling loads according to the classical theory and experimentally determined values are shown in Fig.1.12. Furthermore, test results are presented in most sections of Chapter 3. The scatter of the test results is often rather large and for that reason it does not appear to be sensible to provide strict recommendations for the buckling analysis. The simple classical theories are, consequently, still used for design calculations together with reduction factors obtained from comparison with experimental results. The classical buckling theory has resulted in solutions for a number of elementary cases, compare Chapter 3, but in practice many shells are built for which analytical solutions do not exist. In such cases the critical load may be calculated by use of a computer program. Such calculations are becoming more frequent as the access to programs and computers is growing steadily. If the buckling analysis is based on linear theory, the computer calculations give results which are equivalent to the classical theory, and it is necessary to interpret them correctly in order to attain an adequate safety with regard to buckling. The progress in this field has not reached very far, however, and general design rules do not yet exist. It does not seem to be feasible to determine theoretically an adequate reduction factor. Guidance may, however, in some cases be obtained by calculating Koiter’s buckling parameter b according to 1.3.3.3. Design with regard to buckling is mostly performed according to the following scheme: – Compute the theoretical buckling stress σel according to the classical theory, using handbook formulas or a computer program. – Define an empirical reduction factor η which represents the reduction of the resistance considering small initial deformations, disturbances in the loading etc. – Define a reduction factor α which reflects the manufacturing process, tolerance level etc. – Calculate the buckling resistance of the elastic shell

Under certain conditions an additional safety factor 0.75 is recommended, e.g. when the test results reveal a large scatter or are too few to form a satisfactory basis for the design. In that case σu is given by

Shell stability handbook

6

If buckling takes place in the elasto-plastic region, i.e, σelr>σy/3, a variable factor ωs, which takes the degree of plasticity into account, is introduced σu=ωsσy where ωs is a function of the slenderness ratio λs according to the expression:

This method of representing the buckling resistance is often used for comparison with experiments. It can be utilized within the plastic region as well as in the elastic region. The function ωs takes on different values depending on whether the extra factor 0.75 is introduced or not. See Chapter 3, eqns (3.10) and (3.37). This scheme of analysis is applied today in many design codes and recommendations and it will also form the basis for design as presented in this handbook. 1.2.2.3. Research activities Primarily, the main contributions to shell stability were made in the aerospace technology, but new applications in other fields have also required efforts within branches such as building construction and offshore. These branches are using other design materials e.g. steel, concrete, glass fiber reinforced plastics (GlRP) the properties of which differ in several respects from those of materials normally used in aircraft and space vehicles. The yield limit of the material, for instance, has a much stronger impact on the resistance of these shells, which, due to the new applications, has lead to an increase in the amount of investigations on plastic buckling. Experimental research is carried out by a number of companies and scientific institutes. Typical applications being investigated are the following (note that the list is only meant to supply a number of examples and does not claim to be complete). Circular cylindrical shells—with or without stiffeners—are studied by several institutions: Plastic buckling of cylinders under axial load was studied at the University of Stuttgart, see Häfner, 1982, Bornscheuer, 1984. The influence on the buckling resistance of a longitudinal weld under lateral external pressure may be considerable as shown in a test series performed at the University of Essen, Stracke-Schmidt, 1984. Plastic buckling of cylindrical shells subjected to edge shear loads has been studied at the University of Liverpool, Galletly-Blachut, 1985a. Buckling of stiffened cylindrical shells (ring and/or stringer stiffened) has attracted an increased interest lately in connection with applications in the offshore industry. Extensive testing has been performed at e.g.: Imperial College, London, Dowling-Harding 1982, and Ronalds-Dowling, 1986, University College, London, Walker et al., 1982, Det Norske Veritas, Valsgård-Steen, 1980,

Introduction

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Chicago Bridge & Iron Co., Chicago, Miller, 1982a, Technion, Israel, Singer, 1983, Los Alamos Nuclear Research Lab, Baker et al., 1983, University of Stuttgart, Diack, 1983. The influence of disturbing factors on the carrying capacity of shell structures has attracted increased attention during the last few years, compare Knoedel and Schulz, 1985, Teng-Rotter, 1990, Rotter et al, 1991 and Eggwertz-Samuelson, 1991. Finally, the problem of designing by use of computer analysis is being discussed. Suggestions on analysis procedures have been presented by Samuelson, 1987 and Schmidt, 1991. At an international conference, “Buckling of shells in the air, on land and in the sea”, Lyon, France, Sept 1991, Ed. J.F.Jullien, 1991, a large number of different aspects of shell stability were discussed. The proceedings form a valuable state of the art review of the current research activities. 1.2.2.4. Development of codes and recommendations The growing use of shell structures within industrial areas such as steel building, ship building, offshore and nuclear power production has resulted in a need for special codes and recommendations for shell design. At the present time, several working groups are active in the development of design recommendations. The following efforts can be mentioned: – Det Norske Veritas, 1984, has, for a long time, published rules for the design of ships with detailed recommendations for the structural analysis. At the start of the offshore operations rules were also developed for offshore structures. The DNV code includes rules for the design of different types of shell elements. – DASt, 1980, (Deutscher Ausschuss für Stahlbau, Richtlinie 013) treats a large number of shell elements and load cases common in building and construction applications. Rather comprehensive rules are given e.g. for cylindrical containment vessels with varying sheet thickness and subjected to wind loading. The rules can be applied to tanks and silos. Parts of these recommendations have been utilized in this handbook. – DIN 18800, Teil 4, Stahlbauten, Stabilitätsfälle, Beulen von Schalen (in preparation, available in a preliminary edition 1990). – API RP2A , 1982, American Petroleum Institute, Washington D.C. – API-SPEC, 1977, Specification for Fabricated Structural Steel Pipe. – BS 5500, Specification for Unfired Fusion Welded pressure Vessels. – ECCS, 1988, (European Convention for Constructional Steelwork) develops recommendations for design of shells with respect to buckling. The Technical Working Group TWG 8.4, is working continuously to update the contents and introduce new shell elements and load cases. The cooperation with the group and exchange of experience and information etc. has been of direct use in the work on the present handbook. – ASME Code Case N-284, 1980, has approximately the same range of contents as DASt Ri 013. Contacts with the authors of the ASME Code have given valuable input for the handbook. The ASME code is the only code, so far, which gives indications on how the results of computer programs for buckling analysis should be interpreted.

Shell stability handbook

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Many nations have developed their own rules and work is continuing in several organizations. The publications listed above have been studied closely during the preparation of this handbook since they are frequently referred to internationally in the design of shell structures. Although great efforts are being made to improve and extend the present design codes, solutions are still lacking for important cases that occur frequently in practical applications, such as the effect of local or concentrated loads on the shell, influence of cutouts and stiffeners, temperature gradients and variations, deep buckles (dents), or how to perform an analysis of the buckling resistance using a general FEM-program. The discussion of these questions will, hopefully, be of some value in similar cases for which no complete and final answer can be given. 1.3 Theory of stability In the science of applied mechanics three different forms of equilibrium are considered, stable, indifferent and unstable. These conditions can be illustrated simply by a ball on different beddings, see Fig.1.1, a)–c). The stable equilibrium, on a concave spherical surface, implies that a small disturbance (displacement) will bring the ball to return to its original position of equilibrium. In the unstable case, on a convex surface, a small disturbance leads to further increase of the distance from the position of equilibrium. Alternatively it can be stated that in the stable case the potential energy has a minimum in the position of equlibrium. A disturbance gives an increase of the potential energy. In the case of indifferent equilibrium, on a plane surface, the change in potential energy is zero, while it is negative for unstable equilibrium. Variations of the equilibrium conditions may be conceived according to Fig. 1.1, d)–f). The ball may have a stable position d) as long as the disturbance is very small. If the disturbance exceeds a certain limit the equilibrium will become unstable. Case g) illustrates a stable equilibrium state where the stability is extremely sensitive to small external disturbances. In many cases of shell buckling the state of equilibrium will adopt this character where the disturbances are produced by initial deformations, load eccentricities etc.

Fig.1.1. Illustration of the definiton of equilibrium conditions in mechanics. The conditions for stability are of great importance for loaded members and an analysis of the limit of stability can in principle be performed with the same methods as used in

Introduction

9

mechanics. A structure loaded by a system of external forces is remaining in a stable equilibrium if it tries to return to its original state after the introduction of a small disturbance. In the treatment of structural stability a number of conditions or limitations are usually introduced: – the material is linearly elastic, – the structural element has an ideal geometrical shape (straight bar, plane plate, shell with given radii of curvature), – constitutive relationships are based on the assumption of small displacements. A theory based on these assumptions will, subsequently, be referred to as the classical buckling theory. Structural components are designed with regard to several different failure mechanisms. For tension loading the yield, fatigue or ultimate limit of the material is normally critical. Compression may cause buckling necessitating introduction of special methods of analysis. The buckling behavior varies between different types of structural elements, and in many cases the theoretical buckling load turns out to be a very poor estimate of the actual resistance. Plane plates may have a resistance which is considerably higher than the value given by the classical theory—the postbuckling state is stable. For some types of shells the resistance may be much lower than the theoretical buckling load and in most cases, the postbuckling behavior is highly unstable. 1.3.1 Classical buckling theory Brief presentation for bars, plates, shells. A straight bar (column) which is loaded in compression in its longitudinal direction, Fig.1.2, may be subjected to buckling in the lateral direction. The load at which buckling occurs can be estimated for a slender bar by the following expression, where n=1 or, alternatively, lf=l, corresponds to Euler’s buckling case No 2. (1.1) When loaded by an axial force P