Pressure Vessels: External Pressure Technology

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Pressure vessels

© Carl T. F. Ross, 2011

Related titles: Condition assessment of aged structures (ISBN 978-1-84569-334-3) In-service structural systems are subject to age-related deterioration, leading to concerns regarding maintenance, health and safety, environment and economics. This book provides an invaluable, single source of information on structural assessment techniques for marine and land-based structures and offers professionals from a broad range of disciplines a better understanding of current practice, advice on how to apply advanced design and development technologies in condition assessment, and describes likely future trends. Chapters are grouped in five broad themes: current practices and standards for structural condition assessment; fundamental mechanisms and advanced mathematical methods for predicting structural deterioration; residual strength assessment of deteriorated structures; reliability of aged structures; and survey, inspection and maintenance. Irradiation embrittlement of reactor pressure vessels in nuclear power plants (ISBN 978-1-84569-967-3) The reactor pressure vessel (RPV) is central to a nuclear reactor, and is therefore a life-determining structural component, i.e. RPV damage left unaddressed could jeopardise the continuing operation of the power plant. The main degradation mechanism affecting RPVs is irradiation embrittlement, where neutron bombardment from the nuclear reaction displaces atoms in the RPV steels. Inspection, testing, and appropriate maintenance are therefore required to ensure any damage is mitigated and thereby to allow the plant to continue generating power. This book discusses the irradiation embrittlement as it affects RPV steels, along with mitigation routes to manage the RPV lifetime. Finite element programs in structural engineering and continuum mechanics (ISBN 978-1-898563-28-0) Bridging the gap between theoretical texts with little software support and the giant finite element software packages, this modern-day handbook covers finite element programming in a comprehensive range of problems in civil, mechanical, structural, aeronautical and electrical engineering. Details of these and other Woodhead Publishing materials books can be obtained by: • •

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Pressure vessels External pressure technology Second edition

Carl T. F. Ross

Oxford

Cambridge

Philadelphia


New Delhi

Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 19102-3406, USA Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com First edition 2001, Horwood Publishing Limited Second edition 2011, Woodhead Publishing Limited © Carl T. F. Ross, 2011 The author has asserted his moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials. Neither the author nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 978-0-85709-248-9 (print) ISBN 978-0-85709-249-6 (online) The publisher’s policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publisher ensures that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Toppan Best-set Premedia Limited Printed by TJI Digital, Padstow, Cornwall, UK


Contents

Author contact details Preface Acknowledgements Notation 1

ix xiii xv xvii

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

An overview of pressure vessels under external pressure Pressure vessel types The spherical pressure vessel Cylinder/cone/dome pressure hulls Other vessels that withstand external pressure Weakening effect on ring-stiffeners owing to tilt Bulkheads Materials of construction Pressure, depth and compressibility

1 1 1 4 7 8 8 9 13

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Axisymmetric deformation of pressure vessels Axisymmetric yield failure Unstiffened circular cylinders and spheres Ring-stiffened circular cylinders Axisymmetric deformation of thin-walled cones and domes Thick-walled cones and domes Ring-stiffeners Plastic collapse Experimental procedure Theoretical plastic analysis Conclusions

15 15 15 16 30 52 77 83 85 95 96

3 3.1 3.2 3.3

Shell instability of pressure vessels Shell instability of thin-walled circular cylinders Instability of thin-walled conical shells Buckling of orthotropic cylinders and cones

100 100 111 117 v


vi

Contents

3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Buckling of thin-walled domes Boundary conditions The legs of off-shore drilling rigs Some buckling formulae for domes and cones Inelastic instability Higher order elements for conical shells Higher order elements for hemi-ellipsoidal domes Varying thickness cylinders

124 138 141 142 144 151 159 163

4 4.1 4.2

165 165

4.3

General instability of pressure vessels General instability of ring-stiffened circular cylinders Inelastic general instability of ring-stiffened circular cylinders General instability of ring-stiffened conical shells

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Vibration of pressure vessel shells Free vibration of unstiffened circular cylinders and cones Free vibration of ring-stiffened cylinders and cones Free vibrations of domes Higher order elements for thin-walled cones Higher order elements for thin-walled domes Effects of pressure on vibration Effects of added virtual mass Effects of damping

192 192 201 205 214 216 217 220 220

6 6.1 6.2 6.3 6.4

221 221 229 236

6.5

Vibration of pressure vessel shells in water Free vibration of ring-stiffened cones in water Free vibration of domes in water Vibration of domes under external water pressure Vibration of unstiffened and ring-stiffened circular cylinders and cones under external hydrostatic pressure Effect of tank size

7 7.1 7.2 7.3 7.4 7.5

Novel pressure hull designs Design of dome ends Design of cylindrical body Ring-stiffened or corrugated prolate domes A submarine for the oceans of Europa Conclusions

280 280 284 290 291 292

8 8.1

Vibration and collapse of novel pressure hulls Buckling of corrugated circular cylinders under external hydrostatic pressure

293


179 184

243 275

293

Contents 8.2 8.3 8.4 8.5 8.6 9 9.1 9.2 9.3 10 10.1 10.2 10.3 10.4 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 12 12.1

Buckling of a corrugated carbon-fibre-reinforced plastic (CFRP) cylinder Vibration of CFRP corrugated circular cylinder under external hydrostatic pressure Vibration and instability of tube-stiffened axisymmetric shells under external hydrostatic pressure Collapse of dome cup ends under external hydrostatic pressure A redesign of the corrugated food can

vii

303 316 324 334 346

Design of submarine pressure hulls to withstand buckling under external hydrostatic pressure Introduction The designs Conclusions

355 355 356 360

Nonlinear analyses of model submarine pressure hulls using ANSYS Introduction Experimental analysis Theoretical analysis Conclusions

361 361 364 368 372

Star wars underwater: deep-diving underwater pressure vessels for missile defence systems Introduction The design Manpower and living conditions Power requirements Environmental control and life support systems External requirements Size of elliptical structure Central spherical shell Connecting walkways Material property requirements Choice of material Pressure hull designs Required wall thickness Conclusions

375 375 377 379 380 381 384 385 385 385 386 386 390 390 391

Vibration of a thin-walled shell under external water pressure using ANSYS Introduction

393 393


viii

Contents

12.2 12.3 12.4 12.5 12.6

Experimental method Theoretical basis of the finite element method Vibration analysis of a prolate dome in air Vibration analysis of the prolate dome in water Vibration analysis of the prolate dome under external pressure Conclusions

12.7

394 395 399 406 415 418

References

419

Appendix I Computer program for axisymmetric stresses in circular cylinders stiffened by equal-strength ring frames

428

Appendix II Computer program for axisymmetric stresses in circular cylinders stiffened by unequal-strength ring frames

432

Appendix III Computer programs for shell instability

444

Appendix IV Computer programs for general instability

448

Appendix V Conversion tables of imperial units to SI

460

Index

463


Author contact details

Prof. Carl T. F. Ross 6 Hurstville Drive Waterlooville Hants PO7 7NB UK E-mail: [email protected]

ix © Carl T. F. Ross, 2011

Dedication

To my grandson Nathan

xi © Carl T. F. Ross, 2011

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Preface

This up-to-date second edition of the technology of external pressure vessels, covers problems that arise in submarine pressure hulls, torpedoes, aircraft fuselages, space shuttles, underwater storage vessels, intercontinental missiles, oil drilling rigs, grain storage silos, cooling towers, valves, tunnels, under-soil tubes, immersed tubes, medical equipment and tin can containers of food and everyday products. Stress analysts, designers, consultants and manufacturers need to take into account the effects of external pressures to prevent buckling of external walls of descending submarines, ascending or descending rockets, aircraft and space shuttles by devising countermeasures in their choice of structural design and materials. Computer programs (codes in QuickBASIC) are published in the Appendices. The present author, from long experience in engineering research and design, covers the problem, its remediation, and practical applications. He records experimental and theoretical work including plastic collapse, and provides many design charts. Chapter 1 introduces the subject of pressure vessels under external pressure and discusses the main modes of failure of these vessels. Chapter 2 gives theoretical and experimental solutions for the axisymmetric mode of failure, together with some of its history. Chapter 3 gives theoretical and experimental solutions for the shell instability mode of failure, also known as lobar buckling, together with some of its history. Chapter 4 provides a similar treatise on general instability, where the entire ring–shell combination buckles bodily in its flank. Chapter 5 is on the vibrations of shells, and Chapter 6 extends this work to vibrations in water; both experimental and theoretical analyses are considered. In Chapter 6, the theoretical and experimental ‘nonlinear’ effects of the vibration of shells under external water pressure are also considered, together with the possible effects of dynamic buckling. Chapter 7 introduces many novel pressure hull designs, some of which are structurally more efficient than conventional pressure hulls. Chapter 8 xiii © Carl T. F. Ross, 2011

xiv

Preface

gives some experimental and theoretical work on novel pressure ‘hull’ designs and food cans, including the use of composite materials. Chapter 9 is on the use of design charts to design pressure hulls, including the design charts of the present author and those of PD 5500 (previously BS 5500) and shows that the charts of the present author are more efficient and easier to use than PD 5500. Chapter 10 is on the detrimental effects of geometrical and material nonlinearity on these vessels; using the commercial computer package ANSYS. Chapter 11 is on a conceptual design of a ‘star wars underwater’ system and is of much strategic importance. Chapter 12 analyses the vibration of a dome shell in air and under external water pressure, using the commercial computer package ANSYS. There are also several Appendices, some of which provide ‘listings’ of computer programs, written in QuickBASIC. Carl T. F. Ross


Acknowledgements

The author would like to thank the following of his co-researchers for their valuable contributions: Drs Mike Mackney, Kevin Port, Andrew Little and Frank Abraham; Terry Johns, Grant Waterman, Bob Maguire, Emile, David Richards, Geoff Lafolley-Lane, David Sawkins, James Etheridge, Daniel Short, Astrit Spahiu, Terry Whittaker, Mohammad Al-Enezi, Chris Bull, Paul Smith, Angela Etheridge, Anouska Terry, and Philipp Köster. His thanks are extended to his sister, Helen Facey, for the considerable care and devotion she gave to the typing of this book. Without her help, the production of this book would not have been possible.

xv © Carl T. F. Ross, 2011

Notation

Unless otherwise stated the following notation is used: a Af bf [B] c [Cv] D [D] [DC] Df e E E1 E2 Ex Ey Ez f G [G] G12 h [H] I IZ K

radius of circular cylindrical shell cross-sectional area of a stiffener width of frame web in contact with shell plating a matrix relating strain to displacement, i.e. {ε} = [B]{Ui} speed of sound in water = % (K/ρF), or c = cos α a matrix of viscous damping terms Et 3/[12(1−ν 2)] a matrix of orthotropic material constants, which allow for the element’s material properties a matrix of directional cosines diameter of equivalent ring-stiffener combination distance of frame centroid from the mid-surface of the shell Young’s modulus of elasticity Young’s modulus in the direction of the fibres Young’s modulus perpendicular to the fibres Young’s modulus in the meridional direction Young’s modulus in the circumferential direction out-of-plane Young’s modulus frequency rigidity modulus Σ[g]e = a type of elemental ‘mass’ matrix for the fluid in-plane shear modulus thickness of shell plating Σ[h]e = a type of elemental ‘stiffness’ matrix for the fluid second moment of area of a ring’s cross-section about its centroid and an axis parallel to the axis of the vessel second moment of area of a ring’s cross-section about its centroid and an axis perpendicular to the axis of the vessel bulk modulus of water xvii © Carl T. F. Ross, 2011

xviii [K] [KG] [K*] L, l Lb [M] n N [N] [N ] P, p Pcr {Pi} {Po} R, r {R} Rf s se [S]

Notation

t t′ u ux {U 0} {Ui} {uo} v [ v] ve w wx,θ, etc. x z

stiffness matrix geometrical stiffness matrix dependent on external pressure overall stiffness matrix (= [K] + [KG]) length length between bulkheads mass matrix number of circumferential waves number of ring-stiffeners a matrix of shape functions normal component of the shape function (structural) pressure theoretical buckling pressure vector of nodal acoustic pressures vector of peak values of nodal acoustic pressures radius external vector of forcing functions radius of frame centroid length along a fluid–structure interface surface, or s = sin α elemental area of fluid/structure interface Σ[s]e = an elemental matrix for the fluid–structure interaction shell thickness or time shell thickness of equivalent ring–cylinder combination displacement in the x direction ∂u/∂x vector of nodal displacements (global) vector of nodal displacements (local) vector of peak values of nodal displacements displacement in the y direction v cos α volume of element displacement in the z direction ∂2w/∂x∂θ, etc. direction along the axis of the vessel radial direction of the vessel

α γ {ε} {εL} ζ θ λ

half cone angle (of conical element) shear strain vector of strains owing to small deflections vector of strains owing to large deflections x/P rotational displacement Windenburg’s thinness ratio


Notation λ′

xix

ν ν12 ρ ρF σ1 σ2 σx σϕ τ τϕx ϕ, θ χ ω

thinness ratio for general instability, corrugated and composite vessels Poisson’s ratio in-plane Poisson’s ratio material density fluid density principal longitudinal stresses principal transverse stresses in-plane meridional stress (of conical element) in-plane circumferential stress (of conical element) in-plane shear stress in-plane shear stress (of conical element) circumferential direction flexural or twisting (strain) radian frequency

CMC CON VMC

constant meridional curvature element conical element varying meridional curvature element

Superscripts e indicates an elemental property T indicates the transpose of the matrix


1 An overview of pressure vessels under external pressure

Abstract: This chapter gives an overview of pressure vessels under external pressure, highlighting their main modes of failure. These modes of failure, called shell instability (or lobar buckling), general instability and axisymmetric deformation, are described with the aid of photographs. Both circular cylinders and domes are considered, together with ring-stiffened circular cones. Material properties of suitable materials of construction, including a number of metals and composites, are presented. Key words: submarine pressure hulls, shell instability, deformation, construction materials.

1.1

Pressure vessel types

Structures designed to withstand external water pressure usually take the form of thin-walled curved shells constructed from metals because it is usually more efficient for such structures to withstand their pressure loading in a membrane manner rather than through bending. However, most underwater pressure vessels are not of this shape, because other shapes lend themselves more readily for other important purposes besides structural efficiency. For example, a submarine pressure hull of cylindrical shape would be more manoeuvrable at sea than one of spherical shape. Furthermore, the submarine pressure hull of cylindrical shape would present fewer difficulties during docking than a spherical one of the same volume and, in any case, a cylindrical pressure hull would be more efficient than a spherical one for housing large numbers of personnel, as the former can be made very long. Precise construction of a cylindrical vessel is usually more easily achieved than that of a similar spherical vessel. Nevertheless, the spherical pressure vessel is also considered in this chapter, as it is useful for the socalled miniature submarine, and also for deep-diving bathyscaphes.

1.2

The spherical pressure vessel

A spherical pressure vessel is usually constructed in the form of a thinwalled spherical shell with a pressure-tight hatch to allow access. In the case 1 © Carl T. F. Ross, 2011

2

Pressure vessels Hatch Pressure hull

Casing

Water

Water

Atmospheric pressure

1.1 The spherical pressure hull.

of mini-submarines, the pressure hull is usually covered with casing to improve hydrodynamic streamlining, as shown in Fig. 1.1. Whereas the pressure hull is under external pressure, the casing or hydrodynamic hull is only in a state of hydrostatic stress,1 so that the latter is unlikely to fail owing to static pressure. Thus, the casing can be constructed from plating that is relatively thin compared with the plating required for the pressure hull, although this casing must be strong enough to withstand hydrodynamic forces under motion and stresses owing to underwater currents and self-weight. For the pressure hull, however, the stresses caused by the external pressure can be very large. For example, a spherical pressure hull of diameter 5 m, diving to a depth of 1000 m, has to withstand a load of 786 MN (78 870 tonf). Perhaps the most famous spherically shaped pressure hulls are those developed by Auguste Picard, namely the Trieste and the FRNS-2, which operated at depths of 4.8 km (3 miles). On 22 January 1960, the Trieste dived to a depth of 11.52 km (7.16 miles), and stayed at that depth for the duration of about half an hour. Under uniform external pressure, a thin-walled spherical pressure hull can collapse either because of axisymmetric yield or by buckling in a lobar manner, as shown in Fig. 1.2. The mode of failure depends on a number of factors, including the thickness– radius ratio of the vessel and the mechanical properties of its material of construction. For example, for Picard’s Trieste, which was 15.24 cm (6 in) thick and 1.83 m (6 ft) diameter, the maximum membrane stresses at a depth of 11.52 km (7.16 miles) would have been about 347.5 MPa (50 400 lbf/in2), so that it is likely that its design was intended to eliminate lobar buckling. If Picard had used a safety factor of about 1.7, it would have been necessary to construct the vessel from a high-tensile steel with a 0.1% proof stress of about 590 MPa (38.26 tonf/in2). Another famous ‘spherical’ shell was that of Bushnell’s ‘Turtle’, Fig. 1.3. Bushnell produced this submarine in wood, in 1776 to fight the British in America’s War of Independence.


An overview of pressure vessels under external pressure

1.2 Lobar buckling of a thin-walled spherical shell.

1.3 Bushnell’s ‘Turtle’ (1776).


3

4

Pressure vessels

1.3

Cylinder/cone/dome pressure hulls

Most submarine pressure hulls take the form of a cylinder/cone/dome construction, surrounded by a casing, where the purpose of the latter is to improve the hydrodynamic streamlining, as shown in Fig. 1.4. As for the spherical pressure hull submarine, the casing is in a state of hydrostatic stress2 and it can therefore be constructed from relatively thin plating as the only loads likely to cause its failure are those caused by hydrodynamic forces, underwater currents and self-weight. However, for a pressure hull constructed from a combination of cylinders, cones and domes, these structures can fail either through axisymmetric yield or by buckling, as shown in Figs 1.5–1.8. Very often, the pressure required to cause shell instability of a thin-walled circular cylinder or cone is only a fraction of that necessary to cause axisymmetric yield resulting from a bulk stress. Thus, unstiffened thin-walled cirPressure hull Water Water Atmospheric pressure

Casing

1.4 Cylinder/cone/dome pressure hull.

1.5 Shell instability of thin-walled circular cylinders under external pressure.



5

cular cylinders and cones are structurally inefficient at withstanding external pressure, particularly if the vessels are long; one way of improving their structural efficiency is to stiffen them with suitably sized ring-stiffeners, spaced at suitable distances apart. The ring-stiffeners can be internal or external, or both, but theoretical studies3 have revealed that internal ringstiffeners are structurally more efficient than external ones, partly because

1.6 Lobar buckling of a hemi-ellipsoidal prolate or hemispherical dome under external pressure.

1.7 Axisymmetric buckling of an oblate dome under external pressure.


6

Pressure vessels

1.8 Axisymmetric buckling of a circular cylinder.

1.9 General instability of ring-stiffened circular cylinders.

of their increased curvature and partly because an internal ring-stiffener which otherwise had the same cross-sectional properties as an external ring-stiffener would weigh less. If, however, the ring-stiffeners were not strong enough to prevent structural instability, there is a possibility that the entire ring-shell combination could buckle bodily, as shown in Figs 1.9 and 1.10. This form of buckling is called general instability. Similarly, under external pressure, the buckling mode for thin-walled hemi-ellipsoidal prolate domes and thin-walled hemispherical domes tends to be of lobar form, and in the flanks of the vessels, as shown in Fig. 1.6, and the buckling mode for hemi-ellipsoidal oblate domes tends to be of axisymmetric form and in the noses of the vessels, as shown in Fig. 1.7. Thus, if the buckling resistance of hemi-ellipsoidal prolate domes and hemispherical domes is required to be improved, this can be achieved by introducing



7

1.10 General instability of ring-stiffened circular cones.

ring-stiffeners to the flanks of these vessels. Similarly, if the buckling resistance of hemi-ellipsoidal oblate domes is required to be improved, this can be achieved by introducing meridional stiffening, particularly to the noses of such vessels. It should be emphasised that owing to initial geometric imperfections the experimentally obtained buckling pressures can very often be considerably lower than predictions based on elastic theory, and therefore it is usually best to design the pressure hull so that buckling is eliminated and any likely failure will be caused by axisymmetric yield. Theoretical estimates of failure pressures based on axisymmetric yield are usually much better than those based on instability, providing the vessel is constructed properly. One question that readers who are not experts in the field may raise is why the ends of a submarine should be blocked off by doubly curved domes instead of, say, flat plates. The reason is that as flat plates have no meridional curvature, they will have to resist the effects of pressure in flexure and, because of this, in order for them to have equal strength to the circular cylindrical shell to which they are attached, their required thickness may be over ten times that of the circular cylindrical shell.

1.3.1 Thin-walled conical shells It should be pointed out that thin-walled conical shells of large apical angle can fail axisymmetrically, owing to either buckling or yield.

1.4

Other vessels that withstand external pressure

Other vessels that are required to withstand external pressure include the containment vessels of ships’ nuclear reactors, the legs of off-shore drilling


8

Pressure vessels

rigs, torpedoes, rockets, cooling towers, silos, aircraft fuselages, immersed tubes, tunnels, under-soil pipes, medical pressure vessels and food cans. If an accident takes place in a nuclear-powered ship and if the ship sinks, it is very important that the nuclear reactor is kept watertight, and this can best be achieved by surrounding the nuclear reactor with a containment vessel. Similarly, if a rocket or an aircraft ascends or descends in the atmosphere, the fuselage experiences external pressure. For off-shore drilling rigs, it is usually required that some sections of these structures withstand external water pressure, in addition to supporting the weight of the platform. The additional axial compression in the legs of the drilling rig can further weaken the legs, and much work has been done on this by Galletly et al.4

1.5

Weakening effect on ring-stiffeners owing to tilt

The weakening effect of tilt on ring-stiffeners results in localised failure if the stiffeners are not perpendicular to the axis of the cylinder. The strength of such stiffeners can be estimated by an extension of standard asymmetrical beam theory,1,2 or by Kendrick’s method.5

1.6

Bulkheads

As an additional safety precaution, it is usually desirable to introduce internal watertight bulkheads to submersibles, as shown in Fig. 1.11. These internal watertight bulkheads can be used to isolate a damaged compartment, in the event of an accident, so that personnel and perhaps even the vessel itself can be saved. Internal watertight bulkheads can appear in the form of cross-stiffened circular plates of variable thickness.

Pressure hull Bulkhead Bulkhead

Ring stiffeners

1.11 Pressure hull with internal watertight bulkheads.



9

External bulkhead

Ring stiffeners

Pressure hull

1.12 Pressure hull with an external bulkhead.

Sometimes, in addition to internal watertight bulkheads, it is desirable to introduce external bulkheads, as shown in Fig. 1.12. External bulkheads have two main purposes, namely to improve the general instability characteristics of the submersible and also as a means of attaching the casing to the pressure hull. The design of internal watertight bulkheads are not discussed here, but a computer program for designing a cross-stiffened plate is available.6

1.7

Materials of construction

The greatest depth of the ocean can be found in the Mariana’s Trench which is about 7.16 miles (11.52 km) deep. The average depth of the ocean is somewhere between 3 and 4 miles (4.83 to 6.44 km), but a large submarine can, at present, dive only to about 2000 ft (609.6 m). The reason for this is that as a submarine dives deeper into the ocean, the external pressure acting on its pressure hull increases. Thus, it is necessary to increase the wall thickness of the hull with an increase in its operating depth so that if the hull is made from a popular structural material, such as a high-strength steel, the vessel eventually has no reserve buoyancy and sinks like a stone. If the pressure vessel is not required to dive to great depths, a suitable material of construction could be mild steel, as it is relatively cheap and easy to weld. The main reason for its ease of welding is that, when it yields, it can strain to about 40 times the value of strain at first yield before it starts to strain-harden. This property obviates the necessity to carry out extensive stress-relieving of a welded pressure vessel. A more suitable alternative to mild steel is a high-tensile steel, but many high-tensile steels are difficult to weld, and all require extensive stress-relieving through heat treatment. Other materials with a better strength–weight ratio than mild


10

Pressure vessels

steel include aluminium alloy, titanium, glass-reinforced plastic (GRP), carbon fibre-reinforced plastic (CFRP), metal matrix composites (MMC), and ceramics. The materials for pressure vessels must not only be capable of withstanding the pressure at the required depth but must also have suitable characteristics to withstand the other factors that the operating environment imposes upon them. Such factors include: (a) (b)

(c) (d) (e) (f) (g)

Resistance to corrosion. High strength-to-density ratio; this is required for the structure to be able to obtain a positive buoyancy and therefore have a greater load carrying capacity. Cost of material. Fabrication properties: having chosen a specific material and a certain structural design, is it possible to manufacture it? Pressure hull design. Susceptibility to temperature and fire protection. Operating life span of material.

Unfortunately, as for most material requirements for specialised projects, there is not one material that satisfies all the particular requirements and therefore compromises have to be made in less critical areas of the materials properties, or its behaviour.

1.7.1 Possible materials and their problems Currently, the main materials used for specialised pressure hull designs are: (a) high-strength steels; (b) aluminium alloys; (c) titanium alloys; and (d) composites.

1.7.2 General corrosion In the marine environment, corrosion has been extensively studied and data generated regarding corrosion rates, making it relatively easy to predict and compensate for. The attack of submerged surfaces is governed principally by the rate of diffusion of oxygen through layers of rust and marine organisms. For steels the diffusion rate usually ranges from 3 to 6 mm per year and it is substantially independent of water temperature and tidal velocity, except where industrial pollution leads to higher rates. Certain marine organisms can also generate additional concentration cell and sulfur compound effects.6



11

1.7.3 Stress corrosion cracking Stress corrosion cracking is a form of localised failure which is more severe under the combined action of stress and corrosion than would be expected from the sum of the individual effects acting alone.7 There are many variables affecting the instigation of stress corrosion cracking and among these are alloy composition, tensile stress (internal or applied), corrosive environment, temperature and time. There are methods of relieving the internal stress and it is possible to solve the susceptibility of materials to stress corrosion cracking by using fracture mechanics. Therefore, although this is a problem, it is not one that cannot be predicted.

1.7.4 Other factors Other factors leading to failure are: (a) (b) (c)

brittle fracture; fatigue fracture; and fabrication-induced problems: i.e. stresses induced to welding and heat-affected zones around the weld area.

1.7.5 Material properties Tables 1.1 to 1.5 give material properties of various materials that are, or can be used for submersible pressure hulls. HY80 is the most commonly used steel for submarine hulls, its also commonly used for commercial applications including pressure vessels, storage tanks and merchant ships.8 Such alloys are attractive because of their availability, low cost, fabricability and high strength/density ratios. They have the disadvantage of being anodic to most other structural alloys and, therefore, vulnerable to corrosion when used in mixed structures, although these problems can be avoided by special design modifications.9 It is also difficult or impossible to obtain matching strength in weld metal and base metal and it is therefore necessary for the

Table 1.1 High tensile steels

Material

Specific density

Young’s modulus (GPa)

Compressive yield strength (MPa)

Heat treatment

HY80 HY100 HY130 HY180

7.8 7.8 7.8 7.8

207 207 207 207

550 690 890 1240

Q&T Q&T Q&T Q&T


12

Pressure vessels

Table 1.2 Aluminium alloys

Material

Specific density

Tensile strength (MPa)

Yield strength 0.2% (MPa)


5086-H116 6061-T6 7075-T6 7075-T73 L65

2.8 2.8 2.9 2.9 2.8

290 310 572 434 –

207 276 503 400 390

70 70 70 70 70

Table 1.3 Titanium alloys Material

UTS* (MPa)

Yield strength (MPa)

Density (g cm−3)

6-4 Alloy (annealed) 6-2-1-1 Alloy 6-4 STOA Alloy C.P. Grade 2

896 869 870 345

827 724 830 276

4.5 4.5 4.5 4.5

* Ultimate tensile strength. Table 1.4 Composites

Material GRP (epoxy/S-glass UD*) GRP (epoxy/S-glass filament wound) CFRP (epoxy/HS UD) CFRP (epoxy/HS filament wound) MMC (6061 Al/SiC fibre UD) MMC (6061 Al/ alumina fibre UD)

Density (g cm−3)

Fibre volume fraction

2.1

0.67

65

1200

1

2.1

0.67

50

1000

3.2

1.7 1.7

0.67 0.67

210 170

1200 1000

3.0 5.1

2.7

0.5

140

3000

11

3.1

0.5

190

3100

15


Compressive yield strength (MPa)

Relative cost

* Unidirectional. Table 1.5 Carbon nanotubes

Material

UTS* (GPa)


Density (kg m−3)

Cost

Carbon nanotubes

150

940

1350

?

* Ultimate tensile strength.



13

welds to be thicker than the surrounding base metal or for welds to be located in light stress areas.10 As shown in Table 1.3, titanium alloys have high strength/density ratios and are therefore ideal materials for pressure hull designs where payload is an important characteristic. Unfortunately, they are very high cost at about 5.5 times more expensive than aluminium alloys, and this constrains their use from a commercial point of view.10 The most commonly used composite for marine structures such as ships, is GRP-based. The main reason for this is not only its very high strength/ density ratio but also its low cost compared with other composites. Metal matrix composites have many advantages over both GRPs and FRPs but they are still in the development stages, their cost is high, and there are fabrication difficulties on large structures (presently limited to components up to about 500 mm in diameter).11,12 Wenk13 has designed a submarine to dive to a depth of 4.57 km (15 000 ft); to achieve this he had to specify aluminium alloy as the metal of construction. More research is required on the suitability of composites and ceramics for such vessels, together with the use of carbon nanotubes; details of these are given in Tables 1.4 and 1.5.

1.8

Pressure, depth and compressibility

The pressure P at a depth h of a liquid is given by: P = ρ gh

[1.1]

in SI units, or: P = ρw h

[1.2]

in Imperial gravitational units where ρ is density, 1020 kg m−3 (salt water) or 1000 kg m−3 (pure water) and ρw is specific weight, 64 lbf ft−3 (salt water) or 62.5 lbf ft−3 (pure water). Thus, at a depth of 500 m in salt water, the pressure is: P = 1020 × 9.81 × 500 = 5.0 MPa or P = 50 bar At the bottom of the Mariana’s Trench, where the depth of water is about 11.52 km, the pressure is 1152.7 bar or 16 718 psi or 7.46 tonf in−2. In salt water, a pressure of 1 bar is approximately equivalent to a 10 m depth of water.


14

Pressure vessels

1.8.1 Compressibility Although, for most purposes, water can be regarded as practically incompressible, it is in fact compressible, where:

εv =

δV P = V K

where εv is volumetric strain, δV is the change in volume over a volume (V), P is pressure and K is the bulk modulus (2.05 GPa). Thus, at a depth of 500 m, in salt water, the volumetric strain is about 0.24%, that is the density of water, because of compressibility, increases by 0.24%. Similarly, at the bottom of the Mariana’s Trench the density of water owing to compressibility increases by 5.6%. The density of the water also increases with increased salinity and with frigid temperatures which are just above the freezing point of water.


2 Axisymmetric deformation of pressure vessels

Abstract: Theoretical analyses are provided of the axisymmetric deformation of ring-stiffened circular cylinders, cones, and domes. Solutions for both thin-walled and thick-walled shells are presented. For a thin-walled ring-stiffened circular cylinder, the solution is based on beam-on-elastic-foundation theory, whereas for thin- and thick-walled cones and domes the solutions are based on the finite element method. Finite element solutions for thin-walled axisymmetric shells are extended to cater for orthotropic shells. Key words: axisymmetric deformation, cylinders, cones, domes, beam-onelastic-foundation theory, finite element analysis, orthotropic shells.

2.1

Axisymmetric yield failure

Failure owing to axisymmetric yield is perhaps the most important mode of failure of vessels under external pressure, as sensible design should prevent failure owing to instability. The difficulty of predicting the loss in buckling resistance of a vessel owing to the detrimental effects of its initial geometrical imperfections, which, in a vessel under external water pressure, can cause a catastrophic fall in its buckling resistance, leads to axisymmetric yield failure. In this chapter, theoretical analyses of the axisymmetric deformation of ring-stiffened circular cylinders, cones and domes are provided. Solutions for both thin-walled and thick-walled shells are presented. For a thin-walled ring-stiffened circular cylinder, the solution is based on beam-on-elasticfoundation theory, but for thin- and thick-walled cones and domes the solutions are based on the finite element method. Finite element solutions for thin-walled axisymmetric shells are extended to cater for orthotropic shells. Firstly, however, it is necessary to consider unstiffened circular cylinders and spheres.

2.2

Unstiffened circular cylinders and spheres

For an unstiffened thin-walled circular cylinder under uniform pressure:

σ H = hoop stress = pa / t σ L = longitudinal stress = pa /(2t ) 15 © Carl T. F. Ross, 2011

16

Pressure vessels

where p is pressure (positive if internal); a is internal radius if p is positive and external radius if p is negative (or external pressure); and t is wall thickness. For a thin-walled sphere:

σ = membrane stress = pa /(2t ) The above formula starts to breakdown when t/a > 1/30 For a thick-walled circular cylinder under internal pressure:

σ H (maximum ) =

p(R12 + R22 ) ( R22 − R12 )

For a thick-walled circular cylinder under external pressure:

σ H (maximum ) =

−2 pR22 ( R22 − R12 )

where R1 is the internal radius and R2 is the external radius. In both the above instances, the maximum value of hoop stress occurs on the internal surface of the thick-walled cylinder (r = R1). For a thick-walled sphere under uniform internal pressure and r = R1

σ H (maximum) =

p (R23 + 2 R13 ) 2 ( R23 − R13 )

and the maximum radial stress is:

σ R (maximum) = − p For a thick-walled sphere under uniform external pressure and r = R1

σ H (maximum ) =

−3 pR23 2 ( R23 − R13 )

σ R (maximum) = − p

2.3

Ring-stiffened circular cylinders

Thin-walled circular cylinders under uniform external pressure can collapse through nonsymmetric bifurcation buckling (lobar buckling) at a pressure that might only be a fraction of that needed to cause axisymmetric yield. To improve the structural efficiency of these vessels, ring-stiffeners are introduced. Ring-stiffeners, however, can cause large bending stresses near the ring-stiffener or bulkhead and, because of this, more elaborate solutions have been produced.


Axisymmetric deformation of pressure vessels

17

One of the earliest solutions presented for the axisymmetric deformation of ring-stiffened circular cylinders was that of von Sanden and Gunther in 1920,14 based on the differential equation [2.1]: d 4 w 12(1 − v 2 )w 12 (1 − v 2 ) p + = dx 4 t 2 a2 Et 3

[2.1]

where w = radial deflection (positive inwards) a = mean radius of cylinder t = wall thickness E = Young’s modulus ν = Poisson’s ratio p = pressure (positive external) Unfortunately, however, in presenting their solution, the authors accidentally interchanged two of the signs and consequently published an incorrect solution. Although the error was discovered in the following year by Hovgaard,15 a number of other authors have republished the original incorrect solution. The incorrect circumferential stress formula can be in error by as much as 20% for some cylindrical shells. Another, smaller, error in the von Sanden and Gunther solution is that their differential equation did not fully take into account the loading on the shell caused by the pressure normal to it, the so-called Viterbo effect.16 The Viterbo effect, however, is only about 1%, and its inclusion in the differential equation [2.1] is shown in equation [2.2]: d 4 w 12 (1 − v 2 ) w 12 (1 − v 2 ) p (1 − v / 2 ) = + dx 4 t 2 a2 Et 3

[2.2]

Faulkner17 publishes the corrected von Sanden and Gunther formula as: Py =

σ yp ( t / a) 1−γG

where Py = the pressure to cause yield at the mid-thickness of the shell at mid-bay σyp = yield stress or 0.2% proof stress A (1 − ν / 2 ) γ = A + bf t + 2 Nt / α A = Af(a/Rf)2 ν = Poisson’s ratio α L = 1.285L / at Af = cross-sectional area of frame bf = web thickness of the stiffener


18

Pressure vessels

L = unsupported length of shell = Lf − bf Lf = spacing of ring stiffeners 2(sinh (α L / 2 ) cos (α L / 2 ) + cosh (α L / 2 ) sin (α L / 2 )) G= sinh (α L) + sin (α L) Another deficiency with both equations [2.1] and [2.2] is that they do not include the beam-column effect, which causes the deformation to be nonlinear, and this can be quite large in certain instances. Salerno and Pulos18 introduced the beam-column effect in the original differential equation, which is shown in equation [2.3]. The beam-column effect can increase the maximum, longitudinal stress by about 10% for many vessels: d 4 w 6 (1 − v 2 ) pa d 2 w 12 (1 − v 2 ) w 12 (1 − v 2 ) p (1 − v / 2 ) + = + dx 4 dx 2 Et 3 t 2 a2 Et 3

[2.3]

A further improvement to the differential equation [2.3] was made by Wilson19 when he solved the differential equation [2.4] using a Fourier cosine transformation: d 4 w ⎡ v 6 (1 − v 2 ) pa ⎤ d 2 w 12 (1 − v 2 ) w 12 (1 − v 2 ) p (1 − v / 2 ) = + + ⎥ dx 2 + Et 3 dx 4 ⎢⎣ a 2 Et 3 t 2 a2 ⎦ [2.4] Similarly to Salerno and Pulos,18 Wilson19 solved this differential equation for a shell stiffened by equal-size ring-stiffeners, where the shell deformed symmetrically about mid-span, as shown by Fig. 2.1. The boundary conditions assumed by Wilson were:

Frame t

bf Shell Deformation line

Axis of cylinder

2.1 Ring-stiffened circular cylinder, stiffened by equal-size stiffening rings.


Axisymmetric deformation of pressure vessels (a) w is symmetrical about x = 0 (mid-span); dw = 0 at x = ± L / 2; (b) dx Et 3 d 3w (c ) = G1w − H1 at x = ± L / 2 6 (1 − v 2 ) dx 3

19

[2.5]

where G1 = E(Af/a2f + bft/a2) H1 = pbf(1 − ν/2) Af = cross-sectional area of a frame af = radius of centroid of frame bf = width of shell in direct contact with the frame’s web L = frame spacing Ross20 solved Wilson’s differential equation using the method of Salerno and Pulos as follows:

α 4 = 3 (1 − v 2 ) / ( a 2 t 2 )

β2 =

pa 3 t 2v + 2Et 12 (1 − v 2 )

C0 = pa 2 (1 − v / 2 ) / ( Et )

to give the differential equation [2.6]: 4 12 (1 − v 2 ) C0 d4w 4 2 d w 4 + + = 4 α β 4 α w t 2 a2 dx 4 dx 4

[2.6]

For F1 = α

(1 − α 2 β 2 )

and

F2 = α

(1 + α 2 β 2 )

the complete solution of equation [2.6] is w = A1 cosh F1 x cos F2 x + A2 sinh F1 x sin F2 x + A3 cosh F1 x sin F2 x + A4 sinh F1 x cos F2 x + C0

[2.7]

and some of its derivatives are dw = ( A1 F1 + A2 F2 )sinh F1 x cos F2 x dx + ( A2 F1 − A1 F2 )cosh F1 x sin F2 x + ( A3 F2 + A4 F1 )cosh F1 x cos F2 x + ( A3 F1 − A4 F2 )sinh F1 x sin F2 x


[2.8]

20

Pressure vessels d 2w = [ A1 (F12 − F22 ) + 2 A2 F1 F2 ]cosh F1 x cos F2 x dx 2 + [ A2 (F12 − F22 ) − 2 A1 F1 F2 ]sinh F1 x sin F2 x + [ A3 (F12 − F22 ) − 2 A4 F1 F2 ]cosh F1 x sin F2 x + [ A4 (F12 − F22 ) + 2 A3 F1 F2 ]sinh F1 x cos F2 x

[2.9]

d 3w = [ A1 F1 (F12 − 3F22 ) + A2 F2 (3F12 − F22 ]sinh F1 x cos F2 x dx 3 + [ A1 F2 (F22 − 3F12 ) + A2 F1 (F12 − 3F22 ]cosh F1 x sin F2 x + [ A3 F2 (3F12 − F22 ) + A4 F1 (F12 − 3F22 ]cosh F1 x cos F2 x + [ A3 F1 (F12 − 3F22 ) + A4 F2 (F22 − 3F12 ]sinh F1 x cos F2 x

[2.10]

The only unknown parts of the derivations shown above are the arbitrary constants, and these can be solved by assuming that certain conditions exist at the boundary, as described earlier in equation [2.5]. For the circular cylindrical shell element, stiffened by equal-strength frames, it can be seen from the boundary conditions (a) given in equation [2.5] that the animetric (asymmetric) terms must vanish because of the symmetry of w about midspan, i.e. A3 = A4 = 0 Thus, there are only two unknowns in equation [2.7], namely A1 and A2, and these are obtained from conditions (b) and (c) in equation [2.5] as follows: A1 = N1 / D

A2 = N 2 / D

where N1 = −(G1C0 − H1 ) × (F1 cosh 0.5F1L sin 0.5F2 L + F2 sinh 0.5F1L cos 0.5F2 L) N 2 = (G1C0 − H1 ) × (F1 sinh 0.5F1L cos 0.5F2 L − F2 cosh 0.5F1L sin 0.5F2 L) and D1 = D2 = {Et 3 / 12(1 − ν 2 )}{2 F1 F2 (F12 + F22 )(cosh F1L − cos F2 L)} + 0.5G1 (F1 sin F2 L + F2 sinh F1L) The stress distributions across the bay can be obtained by substituting A1 and A2 into equations [2.7] and [2.9], and then by substituting equations [2.7] and [2.9] into equations [2.11] and [2.12]:


Axisymmetric deformation of pressure vessels Hoop stress = −

21

Ew pav Et d2w ⎞ ⎛w − ± +v 2 ⎟ 2 ⎜ 2 a 2t 2 (1 − v ) ⎝ a dx ⎠

[2.11]

pa Et ⎛ vw d 2 w ⎞ ± + ⎜ ⎟ 2t 2 (1 − v 2 ) ⎝ a 2 dx 2 ⎠

[2.12]

Longitudinal stress = −

2.3.1 Circular cylinder stiffened by unequally sized rings Unfortunately, however, all the previous solutions are based on the assumption that the shell is supported by equal-strength ring-stiffeners and, as a result of this, the shell deforms symmetrically about mid-span. In practice, however, the shell may be stiffened by unequally sized stiffeners at unequal spacings and, because of the added effect of rigid bulkheads, the shell does not deform symmetrically between many pairs of adjacent stiffeners, as shown in Fig. 2.2. In 1970, Ross21 overcame this problem by publishing a solution for a multi-bay circular cylinder, stiffened by unequally sized stiffeners, which can be internal or external or any combination of the two, as shown in Fig. 2.3. The solution is based on the differential equation [2.13]:

Large stiffener

Shell plating

Deformation line Axis of cylinder

Small stiffener

w

N th ring

(i +1) ring

i th ring x

Bay 1

ti

Li a0

R1 a1

Bay i

Clamped end

Bay 0

Bay N

Clamped end

1st ring

2nd ring

2.2 Circular cylinder stiffened by unequal-size rings.

RN

Centre line x

2.3 Ring-stiffened cylinder.


22

Pressure vessels

d 4 w ⎡ v 6 (1 − v 2 ) ai Pi ⎤ d 2 w 12 (1 − v 2 ) w 12 (1 − v 2 ) + + = ( P − 0.5vP1 ) ⎥ dx 2 + Eti3 Eti3 ti2 ai2 dx 4 ⎢⎣ ai2 ⎦ [2.13] where P and P1 are the lateral and axial pressures, respectively, and the other terms refer to the ith bay. By substituting

α i4 = 3 (1 − v 2 ) / ( ai2 ti2 ) βi2 = {( P1ai3 / 2Eti ) + [vti2 / 12 (1 − v 2 )]} C0 i = ai2 ( P − 0.5vP1 ) / ( Eti ) then equation [2.13] takes the form of equation [2.14]: 2 d4w 4 2 d w + 4 α β + 4α i4 w = [12 (1 − v 2 ) C0 i / (ti2 ai2 )] i i dx 4 dx 2

[2.14]

and the complete solution of equation [2.14] is given by equation [2.15]: w = A1 cosh(F1i x)cos( F2 i x) + A2 sinh(F1i x)sin(F2 i x) + A3 cosh(F1i x)ssin( F2 i x) + A4 sinh(F1i x)cos(F2 i x) + C0 i

[2.15]

dw = ( A3 F2 i + A4 F1i )cosh(F1i x)cos(F2 i x) dx + ( A3 F1i − A4 F2 i )sinh( F1ii x)sin(F2 i x) + ( A2 F1i − A1 F2 i )cosh(F1i x)sin( F2 i x) + ( A1 F1i + A2 F2 i )sinh(F1i x)cos(F2 i x)

[2.16]

Now,

d 2w = [ A1 (F12i − F22i ) + 2 A2 F1i F2 i ]cosh(F1i x)cos(F2 i x) dx 2 + [ A2 (F12i − F22i ) − 2 A1 F1i F2 i ]sinh( F1i x)sin(F2 i x) + [ A2 (F12i − F22i ) − 2 A4 F1i F2 i ]cosh( F1i x)sin(F2 i x) + [ A1 (F12i − F22i ) + 2 A3 F1i F2 i ]sinh( F1i x)cos(F2 i x)

[2.17]

d 3w = [ A3 F2 i (3F12i − F22i ) + A4 F1i (F12i − 3F22i )]cosh(F1i x)cos(F F2 i x) dx 3 + [ A3 F1i (F12i − 3F22i ) + A4 F2 i (F22i − 3F12i )]sinh(F1i x)sin(F2 i x) + [ A1 F2 i (F22i − 3F12i ) + A2 F1i (F12i − 3F22i )]cosh(F1i x)sin(F2 i x) + [ A1 F1i (F12i − 3F22i ) + A2 F2 i (3F12i − F22i )]sinh(F1i x)cos(F2 i x) [2.18]



23

Now the ith bay is between the ith ring and the (i + 1)th ring and to solve the arbitrary constants for the ith bay the slope and deflection must be equated at the boundaries, i.e. w = wi ⎫ ⎬ at x = 0 dw / dx = θ i ⎭

and

w = w(i +1) ⎫ ⎬ at x = Li dw / dx = θ(i +1) ⎭

Substituting these boundary conditions into equations [2.15] and [2.16], the arbitrary constants can be obtained in terms of the slopes and deflections at the rings, as follows: – A1 = w1 − C0 i A2 = ψ 11wi + ψ 12θ i + ψ 13w(i +1) + ψ 14θ(i +1) + ψ 15 A3 = ( − F1i / F2 i ) [ψ 6 wi + (ψ 7 − 1 / F1i ) θ i + ψ 8(i +1) + ψ 9θ(i +1) + ψ 10 ] A4 = ψ 6 wi + ψ 7θ i + ψ 8 w(i +1) + ψ 9θ(i +1) + ψ 10 where F1i = αi(1 − α i2 − βi2)0.5 F2i = αi(1 + α i2βi2)0.5 C1i = cosh(F1iLi)cos(F2iLi) C2i = sinh(F1iLi)sin(F2iLi) C3i = cosh(F1iLi)sin(F2iLi) C4i = sinh(F1iLi)cos(F2iLi) Di = Eh3i /[12(1 − ν2)] ψ1 = −[F1iC4i − C1iF2i cot(F2iLi) − C1iF1i coth(F1iLi) − F2iC3i] ψ2 = −[−F1isin(F2iLi)]/[F2isinh(F1iLi)] ψ3 = −F2icot(F2iLi) − F1icoth(F1iLi) ψ4 = (F1iC3i/F2i − C4i) × [F2i cot(F2iLi) + F1i coth(F1iLi)] − (F 1i2 /F2i + F2i)C2i ψ5 = −C0i{−F1iC4i + (C1i − 1) × [F2i cot(F2iLi) + F1i coth(F1iLi)] + F2iC3i} ψ6 = ψ1/ψ4 ψ7 = ψ2/ψ4 ψ8 = ψ3/ψ4 ψ9 = 1/ψ4 ψ10 = ψ5/ ψ4 ψ11 = −C1i/C2i + ψ6 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ12 = −coth(F1iLi)/F2i + ψ7 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ13 = 1/C2i + ψ8 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ14 = ψ9 [F1i coth(F1iLi)/F2i − cot(F2iLi)] ψ15 = C0i (C1i − 1)/C2i + ψ10 [F1i coth(F1iLi)/F2i − cot(F2iLi)] The next step is to obtain the bending moments and shearing forces at the stations in terms of the slopes and deflections at these positions, and


24

Pressure vessels

this can be achieved by considering equilibrium at the ith stiffener, as shown in Fig. 2.4. In the ith bay the bending moment/unit length at any distance x from the ith ring is given by ⎛ d 2 w vw ⎞ M x = − Di ⎜ 2 + 2 ⎟ ⎝ dx ai ⎠ If x = 0: ⎛ d 2 w vw ⎞ Mi( 0 ) = − Di ⎜ 2 + 2 i ⎟ ⎝ dx ai ⎠ x = 0

= − Di [ A1 ( F12i − F22i ) + 2 A2 F1i F2 i + vwi / ai2 ]

= −γ 1wi − γ 2θ i − γ 3w(i +1) − γ 4θ(i +1) − γ 5

[2.20] In the ith bay, the shearing force/unit length at any distance x from the ith ring is given by Qx = − Di

d 3w ⎛ dw ⎞ − 0.5Pa i i⎜ ⎝ dx ⎟⎠ dx 3

Therefore Qi(0) = Qx at x = 0 ⎛ d 3w ⎞ ∴ Qx = − Di ⎜ 3 ⎟ − 0.5P1aiθ i ⎝ dx ⎠ = − Di [ A3 F2 i ( 3F12i − F22i ) + A4 F1i ( F12i − 3F22i )] − 0.5P1aiθ i = −γ 6 wi − γ 7θ i − γ 8 w(i +1) − γ 9θ(i +1) − γ 10

[2.21]

Now, ⎛ d 2 w vw(i +1) ⎞ Mi(Li ) = − Di ⎜ 2 + ⎟ ⎝ dx ai2 ⎠ x = Li

= − Di {[ A1 ( F12i − F22i ) + 2 A2 F1i F2 i ]C1i + [ A2 ( F12i − F22i ) − 2 A1 F1i F2 i ]C2 i + [ A3 ( F12i − F22i ) − 2 A4 F1i F2 i ]C3i

+ [ A4 ( F12i − F22i ) + 2 A3 F1i F2 i ]C4 i + vw(i +1) / ai2 } bi

Pi

Mi(0)

M i(Li–1)

[2.22] Q i(0)

Q i(Li–1)

i th bay (a)

i th bay (b)

2.4 Equilibrium at ith ring: (a) moments; and (b) shearing forces.



25

Mi ( Li ) = −φ1i wi − φ2 iθ i − φ3i w( i + 1) − φ4 iθ( i + 1) − φ5i and ⎛ d 3w ⎞ Qi (Li ) = − Di ⎜ 3 ⎟ − 0.5P1aiθ (i + 1) ⎝ dx ⎠ x = Li

= − Di {[ A1 F1i ( F12i − 3F22i ) + A2 F2 i ( 3F12i − F22i )]C4 i + [ A1 F2 i ( F22i − 3F12i ) + A2 F1i ( F12i − 3F22i )]C3i

+ [ A3 F2 i ( 3F12i − F22i ) + A4 F1i ( F12i − 3F22i )]C1i

+ [ A3 F1i ( F12i − 3F22i ) + A4 F2 i ( F22i − 3F12i )]C2 i } − 0.5P1aiθ (i + 1) Qi (Li ) = −φ6 i wi − φ7 iθ i − φ8 i w(i + 1) − φ9 iθ (i + 1) − φ10 i where γ1 = Di[(F 1i2 − F 2i2 ) + 2ψ11F1iF2i + ν/a2i ] γ2 = 2Diψ12F1iF2i γ3 = 2Diψ13F1iF2i γ4 = −2Diψ14F1iF2i γ5 = Di[2ψ15F1iF2i − C0i(F 1i2 − F 2i2 )] γ6 = −2DiF1i(F 1i2 + F 2i2 )ψ6 γ7 = Di[(3F 1i2 − F 2i2 ) − 2F1i(F 1i2 − F 2i2 )ψ7] + 0.5P1ai γ8 = −2DiF1i(F 1i2 + F 2i2 )ψ8 γ9 = −2DiF1i(F 1i2 + F 2i2 )ψ9 γ10 = −2DiF1i(F 1i2 + F 2i2 )ψ10 ⎧ φ1i = Di ⎨[ F12i − F22i + 2ψ 11 F1i F2 i ]C1i + [ψ 11 ( F12i − F22i ) − 2 F1i F2 i ]C2 i ⎩ ⎡ − F1i ( F12i − F22i ) ⎤ ⎫ + ψ6 ⎢ − 2 F1i F2 i ⎥ C3i − ψ 6 ( F12i + F22i ) C4 i ⎬ F 2i ⎣ ⎦ ⎭

φ2 i = Di {2 F1i F2 iψ 12C1i + ψ 12 ( F12i − F22i ) C2 i

⎛ − F1i ( F12i − F22i ) ⎞⎤ ⎡ ( F12i − F22i ) +⎢ +ψ7⎜ − 2 F1i F2 i ⎟ ⎥ C3i ⎝ ⎠⎦ F F 2i 2i ⎣ 2 2 + [ 2 F1i − ψ 7 ( F1i + F2 i )]C4 i }

φ3i = Di {2 F1i F2 iψ 13C1i + ψ 13 ( F12i − F22i ) C2 i

⎞⎤ ⎡ ⎛ − F1i ( F12i − F22i ) + ⎢ψ 8 ⎜ − 2 F1i F2 i ⎟ ⎥ C3i ⎝ ⎠⎦ F 2 i ⎣ 2 2 2 − ψ 8 ( F1i + F2 i ) C4 i + v / ai }


[2.23]

26

Pressure vessels

⎧ φ4 i = Di ⎨ 2 F1i F2 iψ 14C1i ( F12i − F22i ) ψ 14C2 i ⎩ ⎫ ⎡ − F1i ( F12i − F22i ) ⎤ + ψ9 ⎢ − 2 F1i F2 i ⎥ C3i − ψ 9 ( F12i + F22i ) C4 i ⎬ F 2 i ⎭ ⎣ ⎦ ⎧ φ5i = Di ⎨[ −C0 i ( F12i − F22i ) + 2 F1i F2 iψ 15 ]C1i ⎩ + [ψ 15 ( F12i − F22i ) + 2C0 i F1i F2 i ]C2 i

⎡ − F1i ( F12i − F22i ) ⎤ ⎫ + ψ 10 ⎢ − 2 F1i F2 i ⎥ C3i − ψ 10 ( F12i + F22i ) C4 i ⎬ F2 i ⎣ ⎦ ⎭ ⎧ φ6 i = Di ⎨[ Fi ( F12i − 3F22i ) + ψ 11 F2 i ( 3F12i − F22i )]C4 i ⎩ + [ F2 i ( F22i − 3F12i ) + ψ 11 F1i ( F12i − 3F22i )]C3i − 2ψ 6 F1i ( F12i + F22i ) C1i

F2 ⎡ ⎤ ⎫ + ψ 6 ⎢ F2 i ( F22i − 3F12i ) − 1i ( F12i − 3F22i )⎥ C2 i ⎬ F2 i ⎣ ⎦ ⎭ ⎧ φ7 i = Di ⎨ψ 12 F2 i ( 3F12i − F22i ) C4 i + ψ 12 F1i ( F12i − 3F22i ) C3i ⎩ + [( 3F12i − F22i ) − 2ψ 7 F1i ( F12i + F22i )]C1i

⎡ F1i ( F12i − 3F22i ) ⎫ F2 ⎛ ⎞⎤ + ψ 7 ⎜ F2 i ( F22i − 3F12i ) − 12i ( F12i − 3F22i )⎟ ⎥ C2 i ⎬ ⎢ ⎝ ⎠ F2 i F2 i ⎣ ⎦ ⎭

φ8 i = Di {ψ 13 F2 i ( 3F12i − F22i ) C4 i + ψ 13 F1i ( F12i − 3F22i ) C3i − 2ψ 8 F1i ( F12i + F22i ) C1i F2 ⎡ ⎤ ⎫ + ψ 8 ⎢ F2 i ( F22i − 3F12i ) − 1i ( F12i − 3F22i ) ⎥ C2 i ⎬ F2 i ⎣ ⎦ ⎭ ⎧ φ9 i = Di ⎨ψ 14 F2 i ( 3F12i − F22i ) C4 i + ψ 14 F1i ( F12i − 3F22i ) C3i ⎩ − 2ψ 9 F1i ( F12i + F22i ) C1i F2 ⎡ ⎤ ⎫ + ψ 9 ⎢ F2 i ( F22i − 3F12i ) − 1i ( F12i − 3F22i )⎥ C2 i ⎬ + 0.5P1ai F2 i ⎣ ⎦ ⎭ ⎧ φ10 i = Di ⎨[ −C0 i F1i ( F12i − 3F22i ) + ψ 15 F2 i ( 3F12i − F22i )]C4 i ⎩ + [ −C0 i F2 i ( F22i − 3F12i ) + ψ 15 F1i ( F12i − 3F22i )]C3i − 2ψ 10 F1i ( F12i + F22i ) C1i

F2 ⎡ ⎤ ⎫ + ψ 10 ⎢ F2 i ( F22i − 3F12i ) − 12i ( F12i − 3F22i )⎥ C2 i ⎬ F2 i ⎣ ⎦ ⎭



27

The equilibrium of the ith ring requires that clockwise moments − anticlockwise moments = zero; M(i − L )(Li ) a(i − L ) − Mi(0) ai + (EI i / Ri )θ i = 0

[2.24]

and outward forces − inward forces = zero; Q(i − L )(Li −1) a(i − L ) − Qi(0) ai + EAFi wi / Ri − Pbi ai + 0.5vP1bi ai = 0

[2.25]

Substitution of equations [2.20] to [2.23] into equations [2.24] and [2.25] leads to the following sets of simultaneous equations. −φ1(i −1) a(i −1)w(i −1) − φ2(i −1) a(i −1)θ(i −1) + ( −φ3(i −1) a(i −1) + γ 1ai ) wi

+ ( −φ4(i −1) a(i −1) + γ 2 ai − EI i / Ri ) θ i + γ 3 ai w(i +1) + γ 4 aiθ(i +1) = φ5(i −1) a(i −1) − γ 5 ai

[2.26]

−φ6(i − 1) a(i − 1)w(i − 1) − φ7(i − 1) a(i − 1)θ (i − 1) + ( −φ8(i − 1) a(i − 1) + γ 6 ai − EAFi / Ri ) wi

+ ( −φ9(i − 1) a(i − 1) + γ 7 ai ) θ i + γ 8 ai w(i + 1) + γ 9 aiθ (i + 1) = Pbi ai (1 − vPi / 2 P ) + φ10(i − 1) a(i − 1) − γ 10 ai

[2.27]

Solution of these simultaneous equations results in the slope and deflection of the ring-stiffeners and back-substitution of these into the appropriate equations results in the deflection and stress values. If the vessel is clamped at its end and there are N ring-stiffeners between the extremities, then, at i = 1:

φ1( 0 ) = φ2( 0 ) = φ6( 0 ) = φ7( 0 ) = 0 and at i = N:

γ3 =γ4 =γ8 =γ9 = 0 Equations for stress are given by: Hoop stress = − Longitudinal stress = −

Ew vP1a Ehi ⎛ w d2w ⎞ − ± +v 2 ⎟ 2 ⎜ 2 2 h 2 (1 − v ) ⎝ a a dx ⎠ P1a Ehi ⎛ vw d 2 w ⎞ ± + ⎜ ⎟ 2 h 2 (1 − v 2 ) ⎝ a 2 dx 2 ⎠

[2.28]

where hi = ti = shell thickness.

2.3.2 Comparisons between experiment and theory Comparisons between experiment and theory for model number 321 (Fig. 2.5 and Table 3.4) are shown in Figs 2.6–2.10, where they can be seen to be


Pressure vessels 0.75″

bF

L1

bf

L

0.7″

0.62″

t = 0.08″ 5.125″ Axis of cylinder

0.002

0.001

1st Bay

0

2nd Bay

3rd Bay

3rd Frame

0.002 2nd Frame

0.003

1st Frame

0.003

0.001 0

Closure plate

Inward radial deflection (in)

2.5 Model number 3.

2.6 Deflection of longitudinal generator at 100 lbf in−2 (Model number 3).

–6000

–6000

–4000

–4000

–2000

–2000

0

0

+6000 +8000

1st Bay

2nd Bay

3rd Bay

3rd Frame

+4000 2nd Frame

+2000

+4000 1st Frame

+2000 Closure plate

Stress (lbf in–2)

28

+6000 +8000

2.7 Longitudinal stress of the outermost fibre at 100 lbf in−2 (Model number 3).


–4000

–2000

–2000

+2000

1st Bay

2nd Bay

3rd Bay

0

3rd Frame

0

2nd Frame

–4000

1st Frame

–6000

Closure plate

Stress (lbf in–2)

–6000

+2000

2.8 Circumferential stress of the outermost fibre at 100 lbf in−2 (Model number 3). +2000

+2000

0

–6000

–8000

–8000

–10000

–10000

–12000

–12000

–14000

1st Bay

2nd Bay

3rd Bay

3rd Frame

–6000

2nd Frame

–4000

1st Frame

–4000

Closure plate

–2000

–14000

2.9 Longitudinal stress of the innermost fibre at 100 lbf in−2 (Model number 3).

–4000

–2000

–2000

0

1st Bay

2nd Bay

3rd Bay

3rd Frame

–4000

2nd Frame

–6000

1st Frame

–6000

Closure plate

Stress (lbf in–2)

Stress (lbf in–2)

0 –2000

0

2.10 Circumferential stress of the innermost fibre at 100 lbf in−2 (Model number 3).


30

Pressure vessels Table 2.1 Details of model No. 3: L1 = unsupported length of the end bays; L = unsupported length of a typical bay; bF = web width of the first and last stiffening ring; bf = web width of a typical ring-stiffener; E = Young’s modulus = 10.3 × 106 lbf in−2; ν = Poisson’s ratio = 0.32 (assumed); N = number of ring-stiffeners L1

L

bF

bf

N

2.5 in

2.25 in

0.4 in

0.325 in

5

in good agreement. Model number 3 was made from HE9/WP aluminium alloy, with the geometrical properties shown in Table 2.1 and Fig. 2.5.

2.3.3 Advantages of the beam-on-elastic-foundation theory Some submarine structural engineers might criticise the above solution, preferring one based on the finite element method. However, the author has found that for determining stresses in thin-walled circular cylinders, solutions based on the differential equation are much better than the finite element solutions, from the point of view of both precision and efficiency. For a finite element solution to be precise, it is necessary to take hundreds of elements for a typical vessel; as each elemental stiffness matrix is of order 6 × 6, computation is uneconomical when compared with the solution shown in this section, where only a few elements are required. It must be emphasised that the solution given in 2.3.1 is also a computational one, as it is based on determining slopes and deflections at the ringstiffeners (or nodes), and then through substitution obtaining the deflection and stress distributions across each bay. Computer programs for the stress analysis of thin-walled cylinders are given in Appendices I and II.

2.4

Axisymmetric deformation of thin-walled cones and domes

Solution of axisymmetric problems involving cones and domes by analytical methods is very difficult, and, in those instances, the finite element method provides a simpler and better solution. For thin-walled cones, the solution is based on the methods of Grafton and Strome22 and Zienkiewicz,23 except that the thin-walled conical element was allowed to have linear taper, as shown in Fig. 2.11. From Fig. 2.11, it can be seen that the element has two nodal circles at its ends, and that each nodal circle has three global degrees



31

w2 w 0,y 0

w

u2

w1 q1

u1

L

x=0 R1

q2

u,x x=1

x 0,u 0

t1 f

t2

R2

2.11 Thin-walled conical element.

of freedom, namely u0i , w0i and θi at node i. A brief derivation of the element is given below. In terms of the six degrees of freedom of the element, the u and the w displacements can be expressed as: 0 0 ξ ⎧ u ⎫ ⎡( 1 − ξ ) ⎨ ⎬=⎢ 2 3 2 3 (1 − 3ξ + 2ξ ) L (ξ − 2ξ + ξ ) 0 ⎩w ⎭ ⎣ 0 ⎧ui ⎫ ⎪w ⎪ ⎪ i⎪ ⎪θ i ⎪ × ⎨ ⎬ = [ N ] {ui } ⎪u j ⎪ ⎪w j ⎪ ⎪ ⎪ ⎩θ j ⎭

0 0 ⎤ 3 ( 3ξ − 2ξ ) L ( −ξ 2 + ξ 3 )⎥⎦ 2

[2.29]

where θ = dw/dx and ξ = x/L. According to Novozhilov,24 the four strain components are given by equation [2.30]. du / dx ⎧ ⎫ ⎧ε x ⎫ ⎪ ⎪ + w cos u sin / r φ φ ( ) ⎪ε ⎪ ⎪ ⎪⎪ ⎪ H⎪ ⎪ −d 2 w / dx 2 ⎨ ⎬=⎨ ⎬ ⎪χx ⎪ ⎪ ⎪ dw sin φ ⎪⎩ χ H ⎪⎭ ⎪ ⎪ − dx r ⎪⎩ ⎪⎭


[2.30]

32

Pressure vessels

so that the ‘strain’ matrix [B] becomes: 0 0 ⎡ −1 / L ⎢ sin φ cos φ cos φ ⎢ (1 − ξ ) L (ξ − 2ξ 2 + ξ 3 ) (1 − 3ξ 2 + 2ξ 3 ) r r r ⎢ [ B] = ⎢ 2 0 − ( −6 + 12ξ ) / L − ( −4 + 6ξ ) L ⎢ φ sin sin φ ⎢ 0 − ( −6ξ − 6ξ 2 ) − ( −1 + 4ξ − 3ξ 2 ) ⎢⎣ r (rL) 1/ L 0 0 ⎤ ⎥ sin φ cos cos φ φ ⎥ L ( −ξ 2 + ξ 3 ) ξ ( 3ξ 2 − 2ξ 3 ) r r r ⎥ [2.31] 0 − ( −6 − 12ξ ) / L2 − ( −2 + 6ξ ) L ⎥ ⎥ sin φ sin φ ⎥ − ( −6ξ + 6ξ 2 ) − ( −2 − 3ξ 2 ) 0 rL r ⎥⎦ ( ) The elemental stiffness matrix [k] is given by

[ k ] = ∫ [ B]T [ D][ B] d ( A)

[2.32]

where d(A) = 2πr dr; 0 0 ⎤ v 1 0 0 ⎥ ⎥ 0 t 2 / 12 vt 2 / 12 ⎥ ⎥ 0 vt 2 / 12 t 2 / 12 ⎦

⎡1 ⎢ Et ⎢v [ D] = 1 − v2 ⎢0 ⎢ ⎣0

[2.33]

for an isotropic material; and t = (1 − ξ ) t1 + ξt 2

[2.34]

The elemental stiffness matrix in global coordinates [k0] is given by

[k 0 ] = [Ξ]T [k ][Ξ]

[2.35]

where ⎡c ⎢− s ⎢ ⎢0 [Ξ ] = ⎢ ⎢0 ⎢0 ⎢ ⎣0

s c 0 0 0 0

0 0 0 0 1 0 0 c 0 −s 0 0

0 0 0 s c 0

0⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎦

and c = cos ϕ and s = sin ϕ.


[2.36]


33

Integration of equation [2.32] can be carried out using four Gauss points per element, although the author has found that by using three Gauss points no loss in precision was detected for a number of examples and, in certain instances, locking25 was avoided. The elemental load vector in global coordinates is given by: {qi } = [ Ξ]

T

∫ [N ]

T

⎣ 0 P ⎦ d ( A)

[2.37]

where P = pressure normal to surface. In the computer program, P is allowed to vary linearly between adjacent nodal circles, as follows: P = Pi (1 − ξ ) + Pj (ξ )

[2.38]

where Pi = lateral pressure at node i and Pj = lateral pressure at node j.

2.4.1 Comparisons with FEA and Woinowsky-Kreiger The comparison between the finite element solution of this section and the analytical solution of Timoshenko and Woinowsky-Kreiger26 for the thinwalled spherical shell cap of Fig. 2.12, is shown in Figs 2.13 and 2.14. To obtain the results of Figs 2.13 and 2.14, the shell was divided into 10 equal meridional length elements, where node 1 was at the nose of the shell and node 11 was at the encastré end (or wall). Figure 2.13 shows the distribution of hoop forces/unit length and Fig. 2.14 shows the distribution of meridional forces/unit length. A computer program in BASIC for analysing thin-walled axisymmetric shells, using the conical shell element, is published on the net, and other versions of this program are available directly from the author.

1 lbf in–2

Nose Wall

35° 35°

2.12 Spherical shell cap.


34

Pressure vessels 0 Internal Thin-walled External conical shell Timoshenko and Woinowsky-Kreiger (average)

lbf in–1

–10 –20 –30 –40 –50 11

10

Wall

9 Nodes

8

7

6

5

4

3

2

2.13 Hoop forces/unit length for cap.

40

lbf in–1

30 20 10 0

11

–10 Wall

10

9

8

7

6

5

4

3

2

Nodes

2.14 Meridional bending moment/unit length for cap.

2.4.2 Axisymmetric varying meridional curvature (AVMC) element The conical shell element described in Section 2.3 can also be used for domes, cylinders and cylinder/cone/dome combinations, but a more suitable element for an axisymmetric varying meridional curvature element (AVMC) was presented by Cook,27 as shown in Fig. 2.15. In this instance,

{ε }T = ⎣⎢ε s εφ χ s χφ ⎦⎥


[2.39]

Axisymmetric deformation of pressure vessels x=1

35

b2

s

w

bc b1

x = –1

R2

R1 Axis

2.15 Varying meridional curvature element.

where

ε s = ∂u / ∂s − w∂β / ∂s εφ = (1 / r ) (u sin β + w cos β )

[2.40]

χ s = −∂ 2 w / ∂s 2 − u∂ 2 β / ∂s 2 − ∂u / ∂s.∂β / ∂s

χφ = − (1 / r ) [1 / r∂ 2 w / ∂φ 2 + ( ∂w / ∂s + u∂β / ∂s ) sin β ]

Cook assumed that β was given by

β = α 0 + α 1 s + α 2 s2

[2.41]

and he determined these constants with the aid of the boundary conditions: at s = 0, β = β1, and at s = L, β = β2, where L is the arc length; and equation [2.42]:

∫

L

0

L

sin ( β − βc ) ds ≈ ∫ ( β − βc ) ds = 0

[2.42]

0

where it is assumed that the slope, which is small, is tan (β = βc) = dy/dx. Substituting the above boundary conditions into equation [2.41] results in the following values for the constants:

α 0 = β1 α 1 = (6βc − 4β1 + 2β2 ) α 2 =

( 3β1 + 3β2 − 6βc ) L2

[2.43]

If the arc length L is not known, Cook has shown that it can be approximated by equation [2.44]: L≈l+

l ⎡ 4 −1⎤ ⎧(β1 − βc ) ⎫ ⎢⎣(β1 − βc ) (β 2 − βc )⎥⎦ ⎢ ⎬ ⎥⎨ 60 ⎣ −1 4 ⎦ ⎩(β 2 − βc )⎭

[2.44]

where l = chord length. The radius r can be determined from equation [2.45]: s

r = R1 + ∫ sin βds

[2.45]

0


36

Pressure vessels

where

β ≈ β c + dy / dx

[2.46]

Equation [2.45] can be integrated explicitly for a hemi-ellipsoidal dome, and for an axisymmetric shell of arbitrary meridional shape it can be integrated numerically. In the meridional direction, it is convenient to assume that u = α3 + α4s. Cook27 shows that the rotational displacement θi is given by:

θ i = [ ∂w / ∂s + u.∂β / ∂s ]i

[2.47]

but if this is approximated by:

θ i = ∂wi / ∂s

[2.48]

then [k] can be calculated from equation [2.32] and [k0] from equation [2.35] where: ⎡ c1 s1 0 ⎤ ⎢− s c 0 ⎥ 03 ⎢ 1 1 ⎥ ⎢ 0 0 1 ⎥ [Ξ ] = ⎢ ⎥ c s 0 2 2 ⎢ ⎥ ⎢ 03 − s1 c2 0 ⎥ ⎢ ⎥ 0 0 1⎥⎦ ⎢⎣

[2.49]

and c1 = cos β1, s1 = sin β1; c2 = cos β2, s2 = sin β2 An alternative method of determining the arc length of the element, together with its associated parameters, is to introduce a mid-side node. This mid-side node need not be used for determining additional nodal displacements, but simply the shape of the element.

2.4.3 Axisymmetric constant meridional curvature (ACMC) element For the special case of the thin-walled axisymmetric shell of constant meridional curvature, ACMC28 (Figs 2.16 and 2.17), {ε} is given by ⎧ε s ⎫ ⎪ε ⎪ ⎪ φ⎪ {ε } = ⎨ ⎬ = [ B] {Ui } ⎪ χs ⎪ ⎪⎩ χφ ⎪⎭


[2.50]


Nodal circle 1

f

b1

r

s

w

v b

x

u

b2

Nodal circle 2

2.16 Axisymmetric shell element.

x=1 q2 2

w

b

s,x r

b2

u

x = –1 q1

b1

1 A

Ri

Rc1

r

Rc2

y

a a

x y

2.17 Longitudinal section through element.


37

38

Pressure vessels

where ⎧ ∂u + w ⎫ ⎪ ∂s R1 ⎪ ⎪ ⎪ ⎪ 1 [u sin β + w cos β ] ⎪ ⎪⎪ r ⎪⎪ {ε } = ⎨ 2 1 ∂w 1 ∂u ⎬ ⎪− ⎪ + ⎪ α 2 R12 ∂ξ 2 α R12 ∂ξ ⎪ ⎪ ⎪ ⎪ − 1 ⎛ ∂w − α u⎞ sin β ⎪ ⎜ ⎟ ⎠ ⎪⎩ α 1 Ri ⎝ ∂ξ ⎪⎭

[2.51]

The assumed displacement functions are:

(1 − ξ ) u1 + u2 2 2 (ξ 3 − 3ξ + 2 ) w + (1 + ξ ) (1 − ξ )2 α R θ w= 1 1 1 4 4 ( −ξ 3 + 3ξ + 2 ) w − (1 − ξ ) (1 + ξ )2 α R θ + 2 11 1 4 4 u=

(1 − ξ )

[2.52]

or ⎧u ⎫ ⎨ ⎬ = [ N ] {U1 } ⎩w ⎭ where [N] = a matrix of shape functions; and ⎧u1 ⎫ ⎪w ⎪ ⎪ 1⎪ ⎪θ1 ⎪ {Ui } = ⎨ ⎬ ⎪u2 ⎪ ⎪w2 ⎪ ⎪ ⎪ ⎩θ 2 ⎭ Hence, by substituting equations [2.52] into [2.51], [B] can be obtained, as shown in Table 2.2. Now,

[k ] = ∫ [ B]T [ D][ B] 2πr ds

and

[k 0 ] = [Ξ]T [k ][Ξ]

[2.53]

where [D] is obtained from equation [2.33] and [Ξ] is obtained from equation [2.36] and β=A−ψ r = Ri sin(90 − A + ψ) − Y



2rR1

2rR1 4r

− ( −1 − 2ξ + 3ξ 2 ) sin β

−3 (ξ 2 − 1) sin β

(1 − ξ ) sin β

4r − ( 3ξ − 1) 2αR1

4r

2r

(1 + ξ ) (1 − ξ )2αR1 cos β

−3ξ 2α 2R1

(ξ 3 − 3ξ + 2) cos β

(1 − ξ ) sin β

4

(1 + ξ ) (1 − ξ )2α

−1 2αR1

ξ 3 − 3ξ + 2 4R1

−1 2αR1

2rR1

(1 + ξ ) sin β

1 2αR12

2r

(1 + ξ ) sin β

1 2αR1

Table 2.2 [B] for axisymmetric constant meridional curvature (ACMC) element

4αrR1

−3 ( −ξ 2 + 1) sin β

3ξ 2α 2R12

4r

( −ξ 3 + 3ξ + 2) cos β

−ξ 3 + 3ξ + 2 4R1

4r

(1 − 2ξ − 3ξ2 ) sin β

− ( 3ξ + 1) 2αR1

2

− (1 − ξ ) (1 + ξ ) αR1 cos β 4r

2

− (1 − ξ ) (1 + ξ ) α 4

40

Pressure vessels

Y = Ri sin (90 − A + α) − Rc1 s = R1ψ ds = R1dψ but

ξ = ψ / α, therefore ds = R1α dξ. The assumed lateral pressure distribution is given by ps =

(1 − ξ ) 2

p1 +

(1 + ξ ) 2

p2

where, p1 = lateral pressure at ξ = −1 p2 = lateral pressure at ξ = +1 The vector of nodal forces is obtained from:

{q} = [Ξ]T ∫ [ N ]T ps 2πr ds

[2.54]

In a computer program for analysing axisymmetric elements of constant meridional curvature,28 four Gauss points were used in the meridian of each element. The use of three Gauss points is likely to give equally good results, with the added advantage of computational economy and the avoidance of locking25 for certain cases. Comparison of the computer solution with that of Timoshenko and Woinowsky-Kreiger26 for the spherical cap of Fig. 2.12 realised near-identical results with those of the conical element in Section 2.3 using a similar mathematical model.

2.4.4 Comparisons between the ACMC and AVMC elements To test the ACMC and the AVMC elements, experimental and theoretical investigations were carried out on two of the thin-walled hemi-ellipsoidal domes referred to in Section 3.4.2 and shown in Fig. 2.18. These domes were constructed in solid urethane plastic (SUP), in the manner described in Section 3.4.2, and their aspect ratios (AR) were 1.5 and 3, where: Aspect ratio =

Dome height Base radius



41

2.18 Hemi-ellipsoidal domes of aspect ratio: (left) 3; and (right) 1.5.

The domes were firmly clamped at their bases and they were subjected to external hydrostatic pressure in the tank shown in Fig. 2.19. Before carrying out the experimental investigation, the conforming properties of the two elements were examined, using these two domes as a standard. For both domes, the number of elements taken to model the vessel was varied from two to 16, in increments of two and, in Figs 2.20 and 2.21, plots are made of the variation of stress with mesh refinement. The stresses were calculated at the mid-meridian of each dome; from Figs 2.20 and 2.21, it can be seen that both elements appear to conform very rapidly. The figures also show better agreement for the hoop stresses than for the meridional stresses, but the latter were small in magnitude in any case. From these results, it can be seen that the stresses varied very little when 10 or more elements were adopted. Therefore 14 elements were used to model the domes for the comparisons between the theoretical and experimental results. For each dome, 10 strain gauges were placed at five positions on each dome, as shown in Figs 2.22 and 2.23. It was necessary to use one pair of strain gauges at each position, where the strain gauges were placed at 90° to each other and in the directions of the principal strains, because the problem was a two-dimensional one. Tables 2.3 and 2.4 show mean recorded strains at various pressures on the internal surfaces of these vessels. The slight nonlinearity that appeared to occur with these experimental results was attributed to the fact that the strain gauges used were intended for metals and not for plastics. Figures 2.24–2.27 show comparisons between the theoretical and the experimental stress distributions for these two domes when they were


Securing annulus

8 M12 securing bolts

Rubber annulus seal

Relief valve Dome aspect ratio 3.0

Pressure gauge Water

Pressure inlet

2.19 Pressure tank with test shell in position.

Internal stress (compressive) (MN m–2)

11

10

9

CMC hoop VMC hoop CMC meridional VMC meridional

8

7

6

0

2

4

8 10 12 6 Number of elements

14

16

2.20 Variation in stress with mesh refinement aspect ratio 1.5.



43

Internal stress (compressive) (MN m–2)

4

3

2

1 CMC hoop VMC hoop CMC meridional VMC meridional 0

2

4

10 12 6 8 Number of elements

14

16

150 111

69

35 11

2.21 Variation in stress with mesh refinement aspect ratio 3.0.

1 2 3

4 5

6A 7 8

9

B C

Circumferential gauges

D

10 E Meridional gauges

2.22 Strain gauge positions and directions on 1.5 aspect ratio dome (mm).

subjected to an external pressure of 45 lbf in−2 for the dome of 1.5 AR and to a pressure of 15 lbf in−2 for the dome of 3.0 AR. From these figures, it can be seen that both the ACMC and the AVMC elements tend to predict stresses of a higher magnitude than those obtained experimentally, although all the lines appear to follow similar patterns. The figures also show good


Pressure vessels 95 65 50 35

44

8 A 9 2 B 10 3 C 11 4 D

280

220

150

1

5

12 E

6

13 F

7

14

Circumferential gauges

G Meridional gauges

2.23 Strain gauge positions and directions on 3.0 aspect ratio dome.

Table 2.3 Experimental strains along dome meridian, aspect ratio 1.5 Mean recorded strain με at pressures of (lbf in−2) [MPa]: Gauge

10 [0.069]

20 [0.138]

30 [0.207]

40 [0.276]

45 [0.31]

1 2 3 4 5 6 7 8 9 10

−314 −534 −472 −416 −104 −96 −162 −270 −256 −140

−588 −1054 −914 −822 −214 −290 −296 −568 −548 −264

−850 −1600 −1360 −1249 −311 −603 −426 −878 −818 −368

−1105 −2199 −1838 −1677 −397 −950 −529 −1171 −1089 −457

−1198 −2465 −2067 −1910 −437 −1138 −570 −1315 −1215 −522

agreement between the predictions of the ACMC element and those of the AVMC element. It is believed that the main reason why the experimental stresses lay on lines ‘lower’ than the lines for the theoretical stresses was that the strain gauges used were meant for metals and not for plastics. The domes were constructed very precisely, as can be seen by the out-ofcircularity plots shown in Figs 2.28 and 2.29, where the meridians on which the gauges lay are also indicated.



45

Table 2.4 Experimental strains along dome meridian, aspect ratio 3.0 Mean recorded strain με at pressures of (lbf in−2) [MPa]: Gauge

7.5 [0.0517]

10 [0.069]

12.5 [0.0862]

15 [0.103]

1 2 3 4 5 6 7 8 9 10 11 12 13 14

−471 −489 −479 −492 −443 −345 −197 −12 −49 −77 −67 −56 −69 −71

−603 −633 −621 −653 −589 −457 −247 −10 −61 −91 −81 −65 −84 −91

−771 −803 −793 −831 −733 −573 −307 −6 −75 −107 −102 −97 −119 −129

−933 −978 −979 −1018 −876 −681 −352 −7 −96 −136 −123 −115 −140 −155

8

Stress (compressive) (MN m–2)

7

6

5

4

3

2

1

0

ACMC AVMC Experimental

100 50 Axial coordinate (mm)

150

2.24 Internal meridional stress distribution aspect ratio 1.5. Dashed curve shows another experimental possibility.


Pressure vessels


15

10

5

ACMC AVMC Experimental

0


150

2.25 Internal circumferential stress distribution aspect ratio 1.5.

3 Stress (compressive) (MN m–2)

46

2

1 ACMC AVMC Experimental 0


300

2.26 Internal meridional stress distribution aspect ratio 3.0.




47

5 4 3 2 ACMC AVMC Experimental

1

300


0

2.27 Internal circumferential stress distribution aspect ratio 3.0.

Aspect ratio 3.0

73 mm

Specimen 3.0 Mag Filter ×50 N Talyrond 10″Arm

Ra

nk

Tay lo

s D ice r H o bson Le

t er Location of meridional gauges

1 Division = 0.002″

2.28 Out-of-circularity plot for 3.0 aspect ratio dome.

2.4.5 Tapered cylindrical shell element Another useful element in this family of thin-walled axisymmetric elements is the tapered thin-walled circular cylindrical shell element of Fig. 2.30.29 Its usefulness is that, when used to model mathematically circular cylindrical shells, it is computationally more efficient than the elements described earlier. A brief description of the derivation of this element is given below. The cylinder is assumed to have a linear variation in thickness, as shown in Fig. 2.30, so that:


48

Pressure vessels

Aspect ratio 1.5

77 mm

Specimen 1.5 Mag Filter ×100 N Talyrond 5″Arm

Ra

nk

Tay lo

D st e ic e r H o bson Le

r

Location of meridional gauges

1 Division = 0.001″

2.29 Out-of-circularity plot for 1.5 aspect ratio dome.

z,w

w1

w2

q1

q2 a 2

1

x,u

l t1

t2

2.30 Tapered thin-walled circular cylinder.

t=

(t2 − t1 ) l

x + t1 = mx + c

[2.55]

Suitable displacement distributions for axial and radial displacements are: u = α1 + α 2 x

w = α 3 + α 4 x + α 5 x2 + α 6 x3

[2.56]

In terms of nodal displacements, these become: u = u1 + (u2 − u1 )( x / l )


[2.57]


49

−3w 2θ 3w θ 2w θ 2w θ w = w1 + θ1 x + ⎛⎜ 2 1 − 1 + 2 2 − 2 ⎞⎟ x 2 + ⎛⎜ 3 1 + 21 − 3 2 + 22 ⎞⎟ x 3 ⎝ l ⎝ l l ⎠ l l l ⎠ l l

[2.58]

and

For a circular cylinder of varying thickness, the bending strain energy is given by: Ub =

{

}

1 2 2 D ( χ x + χ y ) − 2 (1 − v) ( χ x χ y − χ xy ) dx dy 2 ∫∫

[2.59]

and the membrane strain energy by: Um =

1 (σ xε x + σ yε y + τ xyγ xy ) t dx dy 2 ∫∫

[2.60]

As σx and σy are principal stresses,

τ xy = γ xy = χ xy = 0 Furthermore, as deflections are small,

χy ≈ 0 so that 2

Ub =

1 ⎛ d 2w ⎞ D ⎜ 2 ⎟ dx dy ∫∫ ⎝ dx ⎠ 2

[2.61]

where, D = Et3/[12(1 − ν2)] σx = longitudinal stress εx = longitudinal strain σy = hoop stress εy = hoop strain τxy = shear stress in the plane of the cylinder’s shell γxy = shear strain in the plane of the cylinder’s shell χx = curvature in the longitudinal direction χy = curvature in the hoop direction χxy = twist in the plane of the cylinder’s shell For a two-dimensional system of stress,

σx =

Et

(1 − v 2 )

( ε x + vε y )

and

σy =


Et

(1 − v 2 )

( ε y + vε x )

50

Pressure vessels

but

εx =

du dx

and

εy =

w a

Therefore Um =

⎡⎛ du vw ⎞ 2 E ⎛ du w ⎞ ⎤ dx dy t + ⎟⎥ ⎟⎠ − 2 (1 − v ) ⎜⎝ 2 ∫∫ ⎢⎜ ⎝ 2 (1 − v ) dx a ⎠ ⎦ ⎣ dx a

[2.62]

Substituting the appropriate displacement configurations into the strain energy expression, the stiffness matrix of the element is given by

[ k ] = [ kA ] + [ k1 ] + [ k2 ] + + [ k8 ]

[2.63]

where [kA] = elemental stiffness matrix for the cylinder under axial load only. It can be seen that these matrices are of order 2 × 2, 4 × 4 and 6 × 6, and the displacement vectors corresponding to them are given by

{u1 u2 } {w1 θ1 w2 θ 2 } {u1 w1 θ1 u2 w2 θ 2 } Thus to obtain the stiffness matrix of the tapered cylinder, it will be necessary to construct a 6 × 6 matrix and add together the components of each of the following matrices in their appropriate positions:

[ kA ] =

2 πaE ⎛ ⎡ m / 2 − m / 2 ⎤ ⎡ c / l + (1 − v2 ) ⎜⎝ ⎢⎣− m / 2 m / 2 ⎥⎦ ⎢⎣−c / l

− c / l ⎤⎞ c / l ⎥⎦⎟⎠

[2.64]

sym. ⎤ ⎡ 42 / 5 2 ⎢ ⎥ πaEm ⎢ 12l / 5 4l / 5 ⎥ [ k1 ] = ⎥ 12 (1 − v 2 ) ⎢ −42 / 5 −12l / 5 42 / 5 ⎢ ⎥ 2 2 ⎣ 30l / 5 8l / 5 −30l / 5 22l / 5⎦

[2.65]

sym. ⎤ ⎡ 48 / 5l ⎢ ⎥ πaEm c ⎢ 14 / 5 16l / 15 ⎥ [ k2 ] = 2 ⎥ 4 (1 − v ) ⎢ −48 / 5l −14 / 5 48 / 5l ⎢ ⎥ ⎣ 34 / 5 26l / 15 −34 / 5 76l / 15⎦

[2.66]

3

2


Axisymmetric deformation of pressure vessels ⎡ 12 / l 2 ⎢ πaEmc 2 ⎢ 4 / l [ k3 ] = 4 (1 − v 2 ) ⎢ −12 / l 2 ⎢ ⎣ 8/l ⎡ 24 / l 3 ⎢ πaEc 3 ⎢ 12 / l 2 [ k4 ] = 12 (1 − v 2 ) ⎢ −24 / l 3 ⎢ 2 ⎣ 12 / l

2 −4 / l 12 / l 2 2 −8 / l

8l −12 / l 2 4/l

sym.⎤ ⎥ ⎥ ⎥ ⎥ 6 ⎦

24 / l 3 −12 / l 2

sym.⎤ ⎥ ⎥ ⎥ ⎥ 8/l ⎦

51

[2.67]

[2.68]

sym.⎤ ⎡ 0 ⎢ −3l 2 / 20 ⎥ 0 ⎢ ⎥ 3 0 0 ⎥ 2Emvπ ⎢ −l / 30 [ k5 ] = ⎢ ⎥ 2 3 2 3l / 20 l / 30 0 (1 − v ) l ⎢ 0 ⎥ ⎢ −7l 2 / 20 ⎥ 0 0 7l 2 / 20 0 ⎢ 3 ⎥ 3 0 0 −l / 20 0 0 ⎦ ⎣ l / 20 [2.69] sym.⎤ ⎡ 0 ⎢ −1 / 2 ⎥ 0 ⎢ ⎥ 0 ⎥ 2Ecvπ ⎢ −l / 12 0 [ k6 ] = ⎥ 2 ⎢ (1 − v ) ⎢ 0 1 / 2 l / 12 0 ⎥ ⎢ −1 / 2 ⎥ 0 0 1/ 2 0 ⎢ ⎥ 0 ⎦ −l / 12 0 0 0 ⎣ l / 12

[2.70]

sym. ⎤ ⎡ 6l 2 / 35 ⎢ l 3 / 30 ⎥ 4 l / 140 E πm ⎢ ⎥ [ k7 ] = ⎥ (1 − v2 ) a ⎢9l 2 / 70 l 3 / 30 4l 2 / 7 ⎢ 3 ⎥ 4 3 4 ⎣ −l / 35 −l / 140 −l / 14 l / 84 ⎦

[2.71]

sym. ⎤ ⎡ 26l / 35 2 3 ⎢ ⎥ 2l / 105 Ecπ ⎢ 22l / 210 ⎥ [ k8 ] = 2 2 ⎥ (1 − v ) a ⎢ 9l / 35 13l / 210 26l / 35 ⎢ ⎥ 2 3 2 3 ⎣ −13l / 210 −3l / 210 −22l / 210 2l / 105⎦ [2.72]


52

Pressure vessels

For a cylinder of uniform thickness, m = 0 and c = t = thickness. Hence, the elemental stiffness matrix is given by: ⎡ 2a / l ⎢ ⎢ −v ⎢ ⎢ −vl ⎢ πEt ⎢ 6 [k ] = (1 − v2 ) ⎢⎢ −2a / l ⎢ ⎢ −v ⎢ ⎢ vl ⎢ ⎣ 6

⎛ 24az + 26l ⎞ ⎜⎝ 3 ⎟ 35a ⎠ l ⎛ 12az 11l 2 ⎞ ⎜⎝ l 2 + 105a ⎟⎠

⎛ 8az 2l 3 ⎞ ⎜⎝ l + 105a ⎟⎠

v

vl / 6

⎛ −24az + 9l ⎞ ⎜⎝ 3 ⎟ l 35a ⎠

⎛ −12az 13l 2 ⎞ ⎜⎝ l 2 + 210a ⎟⎠

⎛ 12az 13l 2 ⎞ ⎜⎝ l 2 − 210a ⎟⎠

⎛ 4az 3l 3 ⎞ − ⎝⎜ l 210a ⎟⎠

2a / l ⎛ 24az + 26l ⎞ ⎜⎝ 3 ⎟ l 35a ⎠ ⎛ −12az 22l 2 ⎞ ⎜⎝ l 2 − 210a ⎟⎠

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎛ 8az 2l 3 ⎞ ⎥ ⎜⎝ l + 105a ⎟⎠ ⎥ ⎦ sym.

v − vl 6

[2.73]

where z = t2/12. Using a solution based on the above element, comparison is made in Fig. 2.31 with the semi-numerical solution30 for a vertical tank of varying wall thickness subjected to a linearly varying lateral pressure. The tank was assumed to be clamped at both ends. From Fig. 2.31 it can be seen that there is good agreement between the two sets of results.

2.5

Thick-walled cones and domes

The thin-walled solution described in the earlier sections in this chapter tends to break down if the wall thickness–radius ratio exceeds 1 : 30, and for these cases it is necessary to use a thick-shell theory. For both thick-walled cones and domes, the theory described here is that of Ahmad et al.31 For the thick-walled cone, the element is shown in Fig. 2.32, where it can be seen that it is described by two end nodal circles, and that its thickness is assumed to vary linearly with length. As there are two nodes, and the element is conical, the matrix shape functions can be assumed to be:

[ N ] = [ N1 N 2 ]

[2.74]



53

Radial deflection (in)

0.07 0.06 0.05 0.04 0.03

Numerical solution

0.02

Analytical solution Top

0.01 Bottom 20"

40"

0.8"

Shell

0.6" 120"

1"

6 p.s.i.

Lateral pressure

36 p.s.i.

0.4" 120"

2.31 Radial deflection of a longitudinal generator.

r ,w0 w0

Node 1 u10

t1 q1

Node 2

f t2

Axis

x

x0 , u0

2.32 Thick conical shell.

where, N1 =

1 (1 − ξ ) 2

N2 =

1 (1 + ξ ) 2

ξ, η = local coordinates (they can be curvilinear for some elements)


54

Pressure vessels

The radius r at ξ and η is given by r = N 1r1 + N 2 r2 + 0.5t1 N 1η cos φ + 0.5t2 N 2 η cos φ

[2.75]

Similarly, x 0 = N1 x10 + N 2 x20 + 0.5t1 N1η sin φ + 0.5t 2 N 2η sin φ

[2.76]

There are three nodal displacements at each node, making a total of six nodal displacements {U 0i } for an element, as follows: ⎧w10 ⎫ ⎪ 0⎪ ⎪u1 ⎪ ⎪θ ⎪ {Ui0 } = ⎪⎨ 10 ⎪⎬ ⎪w2 ⎪ ⎪u 0 ⎪ ⎪ 2⎪ ⎪⎩θ 2 ⎪⎭ The displacements w0 and u0 are given by: ⎧w 0 ⎫ ⎡wi0 ⎤ ti ⎧ − sin φi ⎫ = N ⎨ 0 ⎬ ∑ i ⎢ 0 ⎥ + ∑ Niηi ⎨ ⎬θi 2 ⎩ cos φi ⎭ ⎩u ⎭ ⎣ui ⎦

[2.77]

The derivatives w and u are given by: ⎡ ∂u 0 ⎢ ∂x 0 ⎢ 0 ⎢ ∂u ⎢⎣ ∂r

⎡ ∂u 0 ∂w 0 ⎤ ⎢ ∂ξ ∂x 0 ⎥ −1 ⎢ = J ⎥ [ ]⎢ 0 ∂w 0 ⎥ ∂u ⎢ ⎥ ∂r ⎦ ⎣ ∂η

∂w 0 ⎤ ∂ξ ⎥ ⎥ ∂w 0 ⎥ ⎥ ∂η ⎦

[2.78]

where the Jacobian [J] is defined as: ⎡ ∂x 0 ⎢ ∂ξ [J] = ⎢ 0 ⎢ ∂x ⎢ ⎣ ∂η

∂r ⎤ ⎡ ∂N 1 ∂N 2 ⎤ 0 ∂ξ ⎥ ⎢ ⎡ x1 r1 ⎤ ⎥ = ∂η ∂ξ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥ ∂r ⎣ x2 r2 ⎦ ⎢ ⎥ 0 0 ⎦ ⎥ ⎣ ∂η ⎦ ⎡0.5t η ∂N1 0.5t η ∂N 2 ⎤ 1 2 ⎡sin φ cos φ ⎤ ∂ξ ∂ξ ⎥ ⎢ +⎢ ⎢ ⎥ ⎣sin φ cos φ ⎥⎦ ⎢⎣ 0.5t1 N1 0.5t 2 N 2 ⎥⎦

or ⎡ x10 0 ⎣ x2

[ J ] = [ TT ] ⎢

r1 ⎤ ⎡sin φ cos φ ⎤ + [ TT 2 ] ⎢ ⎥ ⎥ r2 ⎦ ⎣sin φ cos φ ⎦


[2.79]


55

where, ⎡− 1 [ TT ] = ⎢ 2 ⎢ ⎣ 0

1⎤ 2⎥ ⎥ 0⎦

and ⎡ −t1η ⎢ [ TT2 ] = ⎢ 4 ⎢ t1 (1 − ξ ) ⎢⎣ 4

t 2η ⎤ 4 ⎥ ⎥ t 2 (1 + ξ ) ⎥ 4 ⎥⎦

and x01 r1 x02 r2 are nodal coordinates in global axes. Hence, ⎡ ∂u 0 ⎢ ∂ξ ⎢ ⎢ ∂u 0 ⎢ ⎣ ∂η

∂w 0 ⎤ ∂ξ ⎥ u0 ⎥ = [ TT ] ⎡⎢ 1 0 ∂w 0 ⎥ ⎣u2 ⎥ ∂η ⎦

w10 ⎤ ⎡ cos φθ1 + [ TT22 ] ⎢ 0⎥ w2 ⎦ ⎣cos φθ 2

− sin φθ1 ⎤ − sin φθ 2 ⎥⎦

The relationship between local and global derivatives is given by: ⎡ ∂u ⎢ ∂x ⎢ ⎢ ∂u ⎢⎣ ∂r 1

∂w ⎤ ⎡ ∂u 0 ⎢ 0 ∂x ⎥ T ∂x ⎥ = [ DC ] ⎢ 0 ∂w ⎥ ⎢ ∂u ⎢⎣ ∂r ∂r 1 ⎥⎦

∂w 0 ⎤ ∂x 0 ⎥ ⎥ [ DC ] ∂w 0 ⎥ ∂r ⎥⎦

[2.80]

−∂r ⎤ ∂ξ ⎥ ⎥ ∂x 0 ⎥ ⎥ ∂ξ ⎦

[2.81]

where,

[ DC ] =

⎡ ∂x 0 ⎢ ∂ξ 1 ⎢ 2 ∂x 0 ∂r 2 ⎢ ∂r ⎢ + ∂ξ ∂ξ ⎣ ∂ξ

= a matrix of directional cosines; u, w = local displacements ⎫ ⎬ see Fig. 2.33 x, r 1 = local axes ⎭


56

Pressure vessels w0, r

r1 , w0 x,u

Line η = constant

x0 , u0

2.33 Local and global axes.

⎡ ∂u ⎢ ∂x ⎢ ⎢ ∂u ⎢⎣ ∂r 1

∂w ⎤ ∂x ⎥ ⎥ ∂w ⎥ ∂r 1 ⎥⎦ A11 ( DC 12 u10 + DC 22 w10 ) ⎤ ⎡ A11 ( DC 11u10 + DC 21w10 ) ⎢ + A DC u0 + DC w0 + A DC u0 + DC w0 ⎥ 12 ( 11 2 21 2 ) 12 ( 12 2 22 2 ) ⎥ =⎢ 0 0 0 ⎢ A21 ( DC 11u1 + DC 21w1 ) A21 ( DC 12 u1 + DC 22 w10 ) ⎥ ⎢ 0 0 0 0 ⎥ ⎣ + A22 ( DC 11u2 + DC 21w2 ) + A22 ( DC 12 u2 + DC 22 w2 ) ⎦

C11θ 1 ( DC 12 cos φ − DC 22 sin φ ) ⎤ ⎡ C11θ 1 ( DC 11 cos φ − DC 21 sin φ ) ⎢ +C θ ( DC cos φ − DC sin φ ) +C θ ( DC cos φ − DC sin φ )⎥ 12 2 11 21 12 2 12 22 ⎥ +⎢ C21θ 1 ( DC 12 cos φ − DC 22 sin φ ) ⎥ ⎢ C21θ 1 ( DC 11 cos φ − DC 21 sin φ ) ⎢ ⎥ ⎣ +C22θ 2 ( DC 11 cos φ − DC 21 sin φ ) +C22θ 2 ( DC 12 cos φ − DC 22 sin φ )⎦

where,

[ A] = [ DC ]T [ J −1 ][ TT ] [C ] = [ DC ]T [ J −1 ][ TT 2 ] The matrix of strains {ε1} is given by: ⎧ ∂u ⎫ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ w0 0 1 {ε } = ⎨ ⎬ = [ B ′ ] {ui } r ⎪ ⎪ ⎪ ∂u ∂w ⎪ ⎪ ∂r 1 + ∂x ⎪ ⎩ ⎭ where [B′] is given in Table 2.5.


[2.82]


⎡ A11DC21 ⎢ ⎢ ⎢ N1 [B ′ ] = ⎢⎢ r ⎢ ⎢ A DC ⎢ 21 21 ⎢⎣ + A11DC22

w 10 A12DC21 N2 r

C11 (DC11 cos φ − DC21 sin φ )

−0.5t1ηN1 sin φ r

A11DC11

0

A21DC11 + A11DC12

C 21 (DC11 cos φ − DC21 sin φ ) A22DC21 +C11 (DC12 cos φ − DC22 sin φ ) + A12DC22

w 20

u10

θ1

Table 2.5 [B′] matrix for thick conical shell

A22DC11 + A12DC12

0

A12DC11

u20 C12 (DC11 cos φ − DC21 sin φ ) ⎤ ⎥ ⎥ ⎥ −0.5t 2ηN 2 sin φ ⎥ r ⎥ ⎥ C 22 (DC11 cos φ − DC21 sin φ ) ⎥⎥ +C12 (DC12 cos φ − DC22 sin φ )⎦⎥

θ2

58

Pressure vessels

The elemental stiffness matrix in global coordinates is obtained numerically from equation [2.83]:

[k 0 ] = ∫−1 ∫−1 [ B′ ]T [ D][ B′ ] 2πr (det dξ dη) 1

1

[2.83]

where, 0 ⎡1 v ⎤ E ⎢ ⎥ 0 v 1 [ D] = (1 − v2 ) ⎢⎢0 0 (1 − v) / 2k ⎥⎥ ⎣ ⎦ and k is a factor to account for the shear strain energy (taken as 1.2).31 The external load matrix for lateral pressure in global coordinates is obtained by equating work done, as follows:

{Ui0 }T { pi0 } = − ∫ {ux ,r }

T

0

= − {U

{ p( ) } d ( A) = ∫ {U } [ N ]{ p( ) } d ( A) 0 T i

x 0 ,r

x 0 ,r

} ∫ [ N ][ DC ]T { p1 } d ( A)

0 T i

or

{ p10 } = − ∫ [ N ]T [ DC ]T { p1 } d ( A)

[2.84]

where, ⎡ N1 ⎣0

[N ] = ⎢

0 N1

−0.5t1 N1η sin φ1 0.5t1 N1η cos φ1

N2 0

0 N2

−0.5t1 N1η sin φ1 ⎤ 0.5t 2 N 2η cos φ2 ⎥⎦

and p1 is the pressure perpendicular to either the outer surface or the inner surface.

2.5.1 Parabolic element A more suitable element for thick-walled cones and domes is that shown in Fig. 2.34. The element is of parabolic shape and of parabolic taper, and it is described by three nodal circles. As the element has three nodes and


Axisymmetric deformation of pressure vessels r ,w0

59

wi0

1

ui0 i=2

fi 3

x0 , u0

2.34 Three-node parabolic element.

is of parabolic shape, it is convenient to describe it with the following matrix of shape functions:

[ N ] = [ N1 N 2 N 3 ] where, 1 N 1 = − (1 − ξ ) ξ 2 N 2 = (1 − ξ 2 ) N3 =

1 (1 + ξ ) ξ 2

The radius r at ξ and η is given by: r = N1r1 + N 2 r2 + N 3r3 + 0.5t1 N1η cos φ1 + 0.5t 2 N 2η cos φ2 + 0.5t3 N 3η cos φ3 [2.85] and x 0 = N1 x10 + N 2 x20 + N 3 x30 + 0.5t1 N1η sin φ1 + 0.5t 2 N 2η sin φ2 + 0.5t3 N 3η sin φ3


[2.86]

60

Pressure vessels

There are a total of nine nodal displacements: ⎧w10 ⎫ ⎪ 0⎪ ⎪u1 ⎪ ⎪θ1 ⎪ ⎪ 0⎪ ⎪w2 ⎪ 0 {Ui } = ⎪⎨u20 ⎪⎬ ⎪θ ⎪ ⎪ 2⎪ ⎪w30 ⎪ ⎪ 0⎪ ⎪u3 ⎪ ⎪θ 3 ⎪ ⎩ ⎭ The displacements w0 and u0 are obtained from equation [2.77], and their derivatives from equation [2.78]. The Jacobian [J] is obtained from equation [2.79]: ⎡ x10 [ J ] = [ TT ] ⎢⎢ x20 ⎢⎣ x30

r1 ⎤ ⎡ sin φ1 r2 ⎥⎥ + [ TT 2 ] ⎢sin φ2 ⎢ ⎢⎣ sin φ3 r3 ⎥⎦

cos φ1 ⎤ cos φ2 ⎥ ⎥ cos φ3 ⎥⎦

[2.87]

where in this instance: ⎡ ∂N 1

[ TT ] = ⎢⎢ ∂ξ ⎢⎣ 0

∂N 2 ∂ξ 0

⎡0.5t η ∂N1 1 ∂ξ [ TT2 ] = ⎢⎢ ⎢⎣0.5t1 N1 ∂N 1 1 = − +ξ ∂ξ 2

∂N 3 ⎤ ∂ξ ⎥ ⎥ 0 ⎥⎦ ∂N 2 ∂ξ 0.5t 2 N 2 0.5t 2η

∂N 2 = −2ξ ∂ξ

∂N 3 ⎤ ∂ξ ⎥ ⎥ 0.5t3 N 3 ⎥⎦ 0.5t3η

∂N 3 1 = − +ξ ∂ξ 2

From equation [2.80], the derivatives with respect to local axes are given by:



61

⎡ ∂u ⎢ ∂x ⎢ ⎢ ⎢ ∂u ⎢ 1 ⎣ ∂r

∂w ⎤ ⎡u10 w10 ⎤ ⎡θ 1 cos φ1 −θ 1 sin φ1 ⎤ ∂x ⎥ ⎥ ⎢u0 w0 ⎥ DC + C ⎢θ cos φ −θ sin φ ⎥ DC = A ] [ ]⎢ 2 ] 2 [ 2 2 2 [ ⎥ [ ]⎢ 2 ⎥ ⎥ ⎢⎣u30 w30 ⎥⎦ ⎢⎣θ 3 cos φ3 −θ 3 sin φ3 ⎥⎦ ∂w ⎥ ⎥ ∂r 1 ⎦ ⎡⎛ A11 ( DC 11u10 + DC 21w10 ) ⎞ ⎛ A11 ( DC12 u10 + DC 22 w10 ) ⎞ ⎤ ⎢⎜ ⎜ ⎟⎥ 0 0 ⎟ C 12 u20 + DC 22 w20 )⎟ ⎥ ⎢⎜ + A12 ( DC 11u2 + DC 21w2 )⎟ ⎜ + A12 ( DC ⎢⎜⎝ + A DC u0 + DC w0 ⎟⎠ ⎜⎝ + A DC u0 + DC w0 ⎟⎠ ⎥ 13 ( 11 3 21 3 ) 13 ( 12 3 22 3 ) ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ 0 0 0 0 ⎢⎛ A21 ( DC11u1 + DC 21w1 ) ⎞ ⎛ A21 ( DC 12 u1 + DC 22 w1 ) ⎞ ⎥ ⎢⎜ ⎜ 0 0 ⎟ 0 0 ⎟⎥ ⎢⎜ + A22 ( DC 11u2 + DC 21w2 )⎟ ⎜ + A22 ( DC 12 u2 + DC 22 w2 )⎟ ⎥ ⎢⎜⎝ + A ( DC u0 + DC w0 )⎟⎠ ⎜⎝ + A ( DC u0 + DC w0 )⎟⎠ ⎥ 22 3 23 11 3 21 3 23 12 3 ⎣ ⎦ ⎡⎛ C11θ 1 ( DC 11 cos φ1 − DC 21 sin φ1 ) ⎞ ⎢⎜ +C θ ( DC cos φ − DC sin φ )⎟ 11 2 21 2 ⎢⎜ 12 2 ⎟ ⎝ ⎢ +C13θ 3 ( DC 11 cos φ3 − DC 21 sin φ3 ) ⎠ ⎢ +⎢ ⎢⎛ C21θ 1 ( DC 11 cos φ1 − DC 21 sin φ1 ) ⎞ ⎢⎜ ⎢⎜ +C22θ 2 ( DC 11 cos φ2 − DC 21 sin φ2 )⎟⎟ ⎢⎣⎝ +C23θ 3 ( DC 11 cos φ3 − DC 21 sin φ3 ) ⎠

⎛ C11θ 1 ( DC 12 cos φ1 − DC 22 sin φ1 ) ⎞ ⎤ ⎜ +C12θ 2 ( DC 12 cos φ2 − DC 22 sin φ2 )⎟ ⎥ ⎜ ⎟⎥ ⎝ +C13θ 3 ( DC 12 cos φ3 − DC 22 sin φ3 ) ⎠ ⎥ ⎥ ⎥ ⎛ C21θ 1 ( DC 12 cos φ1 − DC 22 sin φ1 ) ⎞ ⎥ ⎜ +C22θ 2 ( DC 12 cos φ2 − DC 22 sin φ2 )⎟ ⎥⎥ ⎜ ⎟ ⎝ +C23θ 3 ( DC12 cos φ3 − DC 22 sin φ3 ) ⎠ ⎥⎦

where [DC] is defined in equation [2.81]. The matrix of strains is defined in equation [2.82] and [B′] is given in Table 2.6. The matrix of nodal external forces owing to pressure is obtained from equation [2.84], where

(−t1 N1η sin φ1 ) / 2 N 2 0 (t1 N1η cos φ1 ) / 2 0 N 2 (−t2 N 2η sin φ2 ) / 2 N 3 0 (−t3 N 3η sin φ3 ) / 2 ⎤ (t2 N 2η cos φ2 ) / 2 0 N 3 (t3 N 3η cos φ3 ) / 2 ⎥⎦

⎡ N1 ⎣0

[N ] = ⎢

0 N1

For both instances of the thick-walled shell, numerical integration was carried out to determine [k0] and {Pi}. Four Gauss points were used in the ξ-direction and two Gauss points in the η-direction. The results obtained using computer programs for analysing thick-walled cones and domes28 are compared with the analytical solution of Timoshenko



A12DC21 N2 r

C11 (DC11 cos φ1 − DC21 sin φ1 )

−t1N1η sin φ1 2r

A11DC11

0

u30

A13DC11

0

A23DC11 + A13DC12

A13DC21

N3 r

A23DC21 + A13DC22

A21DC11 + A11DC12

C13 (DC11 cos φ3 − DC21 sin φ3 ) ⎤ ⎥ ⎥ ⎥ −t 3N 3η sin φ3 ⎥ 2r ⎥ ⎥ C 23 (DC11 cos φ3 − DC21 sin φ3 ) ⎥⎥ +C13 (DC12 cos φ3 − DC22 sin φ3 )⎦⎥

θ3

C 21 (DC11 cos φ1 − DC21 sin φ1 ) A22DC21 +C11 (DC12 cos φ1 − DC21 sin φ1 ) + A12DC22

w 20

θ1

u10

w 30

⎡ A11DC21 ⎢ ⎢ ⎢ N1 [B ′ ] = ⎢⎢ r ⎢ ⎢ A DC ⎢ 21 21 ⎢⎣ + A11DC22

w 10

−t 2N 2η sin φ2 2r


θ2

A22DC11 C 22 (DC11 cos φ2 − DC21 sin φ2 ) + A12DC12 +C12 (DC12 cos φ2 − DC22 sinφ2 )

0

A21DC11

u20

Table 2.6 [B′] matrix for the thick-walled three-node parabolic element


63

and Woinowsky-Kreiger26 for the shell cap of Fig. 2.12 (Fig. 2.35). The results from Fig. 2.35 show good agreement between the various solutions, and this was found to be particularly encouraging as the chosen shell was thin but the theories of the present section were based on thick-walled shells. Comparison of Ahmad and co-workers’ solution31 for the thick shell of Fig. 2.36 with the results obtained from the computer program for the three-

Angle f (degrees) 0

31.5

28.0 24.5 21.0 17.5 14.0 10.5

7.0

3.5

−1 −2 −3

Thick conical element Thick (three node) parabolic element

−4 −5

Timoshenko and Woinowsky-Kreiger26

−6

Mean hoop stress (Ibf in−2)

−7 −8 −9 −10 −11 −12 −13 −14 −15 −16 −17 −18

2.35 Mean hoop stress distribution for cap of 3 in. thickness.


0

64

Pressure vessels 100 lbf in−2

Wall Nose

35º

35º

2.36 Shell cap of 9 in. thickness.

node element is given in Fig. 2.37. These can be seen to be in good agreement.

2.5.2 Four-node element The results show that the three-node element gives better predictions than the two-node element, and to test whether four- or five-node elements would make even better predictions, the following cones were developed. For the four-node element of Fig. 2.38, the assumed displacement functions for u0 and w0 are: u 0 = a + bξ + cξ 2 + dξ 3

w0 = e + fξ + gξ 2 + hξ 3

[2.88]

which can be seen to be of cubic form. To determine the shape functions for u0 and w0, consider the boundary conditions for u0, which from Fig. 2.38 are: at ξ = −1, u 0 = u10 at ξ = − 13 , u 0 = u20 at ξ = 13 , u 0 = u30 at ξ = 1, u0 = u40

∴ u10 ∴ u20 ∴ u30 ∴ u40

= a−b+c−d = a − 13 b + 91 c − 271 d = a + 13 b + 91 c + 271 d = a+b+c+d

Solving the above simultaneous equations, the following expressions are obtained for the constants:


Axisymmetric deformation of pressure vessels Angle f (degrees)

Wall 100

31.5

28.0 24.5 21.0 17.5 14.0 10.5 7.0

65 Nose

3.5

0

External Internal

0

Ahmad et al.31 (average)

Hoop stress (Ibf in−2)

−100

−200

−300

−400

−500

−600

−700

−800

2.37 Hoop stresses for cap of 9 in. thickness.

a = − 161 (u40 + u10 ) + 169 (u30 + u20 ) b = − 161 ( u40 − u10 ) + 169 ( 3u30 − 3u20 )

c= d=

9 16 9 16

(u40 + u10 − u30 − u20 ) (u40 − u10 − 3u30 + 3u20 )

Similar expressions can be derived for the constants e to h, and substituting these constants back into the displacement functions, the shape functions for u0 and w0 are given by:


66

Pressure vessels r ,w0

1

wi0

•

2

•

3

ui0

•

4

•

fi

x0 , u0

2.38 Four-node cubic element.

N1 = N2 =

1 16 9 16

( −1 + ξ + 9ξ 2 − 9ξ 3 ) (1 − 3ξ − ξ 2 + 3ξ 3 )

N3 = N4 =

9 16 1 16

(1 + 3ξ − ξ 2 − 3ξ 3 ) ( −1 − ξ + 9ξ 2 + 9ξ 3 )

In a manner similar to that adopted for the two- and three-node elements, [B′] is obtained, as in Table 2.7, and [k0] can be calculated from equation [2.83].

2.5.3 Five-node element The five-node element is shown in Fig. 2.39, and the displacement function for u0, which is a quartic function, is given in equation [2.89]. It must be emphasised that as the shape function for w0 is the same as for u0, it will only be necessary to consider the latter: u 0 = a + bξ + cξ 2 + dξ 3 + eξ 4

[2.89]

From Fig. 2.39 it can be seen that the boundary conditions are as follows: at ξ = −1, u 0 = u10 at ξ = −0.5, u 0 = u20 at ξ = 0, u 0 = u30 at ξ = 0.5, u 0 = u40 at ξ = 1, u 0 = u50

∴ u10 = a − b + c − d + e b c d e ∴ u20 = a − + − + 2 4 8 16 ∴ u30 = a b c d e ∴ u40 = a + + + + 2 4 8 16 ∴ u50 = a + b + c + d + e



u30

A13DC11

0

A23DC11 + A13DC12

A13DC21

N3 r

A23DC21 + A13DC22

N4 r


A24DC11 C 23 (DC11 cos φ3 − DC21 sin φ3 ) +C13 (DC12 cos φ3 − DC22 sin φ3 ) + A14DC22

A14DC21

w 40


θ3

C 21 (DC11 cos φ1 − DC21 sin φ1 ) A22DC21 +C11 (DC12 cos φ1 − DC22 sin φ1 ) + A12DC22

N2 r


0

A21DC11 + A11DC12

A11DC21

w 20


θ1

A11DC11

u10

w 30

⎡ A11DC21 ⎢ ⎢ ⎢ N1 [B ′ ] = ⎢⎢ r ⎢ ⎢ A DC ⎢ 21 21 ⎢⎣ + A11DC22

w 10

Table 2.7 [B′] matrix for the four-node element

A24DC11 + A14DC12

0

A14DC11

u40

A22DC11 + A12DC12

0

A12DC11

u20

C14 (DC11 cos φ4 − DC21 sin φ4 ) ⎤ ⎥ ⎥ ⎥ −t 4N4η sin φ4 ⎥ 2r ⎥ ⎥ C 24 (DC11 cos φ4 − DC21 sin φ4 ) ⎥⎥ +C14 (DC12 cos φ4 − DC22 sin φ4 )⎦⎥

θ4

C 22 (DC11 cos φ2 − DC21 sin φ2 ) +C12 (DC12 cos φ2 − DC22 sinφ2 )



θ2

68

Pressure vessels r ,w0

1

wi0

•

2

•

3

•

4

ui0

•

5

•

fi

x0 , u0

2.39 Five-node quartic element.

Solving the above simultaneous equations, the following expressions are obtained for the constants: a = u30 b = − 61 (u50 − u10 ) + 43 (u40 − u20 ) c = − 61 (u50 + u10 ) − 5u30 + 83 (u40 + u20 ) d=

2 3

(u50 − u10 ) − 43 (u40 − u20 )

e=

2 3

(u50 + u10 ) + 4u30 − 83 (u40 + u20 )

Substituting these constants back into the displacement function for u0, the shape functions for both u0 and w0 are obtained as follows: N1 =

1 6

[ξ − ξ 2 − 4ξ 3 + 4ξ 4 ]

N2 =

1 3

[ −4ξ + 8ξ 2 + 4ξ 3 − 8ξ 4 ]

N 3 = [1 − 5ξ 2 + 4ξ 4 ] N4 =

1 3

[4ξ + 8ξ 2 − 4ξ 3 − 8ξ 4 ]

N5 =

1 6

[ −ξ − ξ 2 + 4ξ 3 + 4ξ 4 ]

In a manner similar to that adopted for the two- and three-node elements, [B′] is obtained, as is shown in Table 2.8, and [k0] can be obtained from



u40

A14DC11

0

A24DC11 + A14DC12

A14DC21

N4 r

A24DC21 +A A14DC22

A21DC11 + A11DC12

0

A11DC11

u10

w 40

⎡ A11DC21 ⎢ ⎢ ⎢ ⎢ ⎢ N1 ⎢ r [B ′ ] = ⎢ ⎢ ⎢ A21DC21 ⎢ ⎢ + A11DC22 ⎢ ⎢ ⎢⎣

w 10

N2 r

A12DC21

w 20

C 24 (DC11 cos φ4 −DC21 sin φ4 ) +C14 (DC12 cos φ4 −DC22 sin φ4 )

−t 4N4η sin φ4 2r

C14 (DC11 cos φ4 −DC21 sin φ4 )

θ4

A25DC21 + A15DC22

N5 r

A15DC21

w 50

C21 (DC11 cos φ1 A22DC21 −DC21 sin φ1 ) + A12DC22 +C11 (DC12 cos φ1 −DC22 sin φ1 )



θ1

Table 2.8 [B′] matrix for the five-node element

A25DC11 + A25DC12

0

A15DC11

u50

A22DC11 + A12DC12

0

A12DC11

u20

C15 (DC11 cos φ5 ⎤ −DC21 sin φ5 ) ⎥ ⎥ ⎥ ⎥ −t 5N5η sin φ5 ⎥ ⎥ 2r ⎥ ⎥ C 25 (DC11 cos φ5 ⎥ ⎥ −DC21 sin φ5 ) ⎥ +C15 (DC12 cos φ5 ⎥ ⎥ −DC22 sin φ5 ) ⎥⎦

θ5

C22 (DC11 cos φ2 −DC21 sin φ2 ) +C12 (DC12 cos φ2 −DC22 sin φ2 )



θ2

A23DC21 + A13DC22

N3 r

A13DC21

w 30

A23DC11 + A13DC12

0

A13DC11

u30

C 23 (DC11 cos φ3 −DC21 sin φ3 ) +C13 (DC12 cos φ3 −DC22 sin φ3 )



θ3

70

Pressure vessels 35.0 31.5 28.0 24.5 21.0 17.5 14.0 10.5 0

7.0

3.5

0

−1 −2 −3 −4 −5

Timoshenko and Woinowsky-Kreiger26 Five node quartic element Four node cubic element

−7 −8 3 in

−10 35°

−11

in

−9

90

Mean hoop stress (Ibf in−2)

Three node parabolic element −6

−12 −13 −14 −15 −16

2.40 Hoop stress distributions for 3-in. shell cap.

equation [2.83]. To compare the results predicted by the three-, four- and five-node elements, once again, the 3-in.-thick shell cap of Timoshenko and Woinowsky-Kreiger26 was used, and the results are shown in Fig. 2.40.

2.5.4 Comparisons between various elements It can be seen that the four- and five-node elements give slightly better results than the three-node element. One advantage of the four- and fivenode elements over the three-node element is that they require fewer elements to model the dome.



71

2.5.5 Comparison with a tapered dome A further test of the four-and five-node elements was carried out with the aid of the tapered Araldite models shown in Figs 2.41 and 2.42. The models were subjected to external hydrostatic pressure in the tank shown in Fig. 2.43, and 10 strain gauges were attached to the inner surface of each model in the positions shown in Figs 2.44 and 2.45. It was necessary to use two strain gauges at each position and to place the gauges perpendicular to each other, and in the directions of the principal strains, as the problem was of a plane stress type on the surfaces of the domes.

12.8

90.25

17.2

17.2 198.8

2.41 Experimental shell section: dome 1 (dimensions are in mm).

21.46

88.83

9.95

9.95 198.5

2.42 Experimental shell section: dome 2 (dimensions are in mm).


72

Pressure vessels Clamping flange Rubber seal Pressure tank

Araldite dome

Water Pressure gauge

Water pump connection

2.43 Method of pressurising dome.

Experimentally obtained strains are given in Tables 2.9 and 2.10. The observed strains show a little nonlinearity, which is attributed to the fact that the gauges were meant for metals and not for Araldite. The following material properties were assumed for Araldite: E = 2758 MPa and v = 0.345. Comparisons between experiment and the theoretical predictions from the five-node element are shown in Figs 2.46–2.49. Comparison is also made, in Figs 2.50–2.53, of the theoretical stresses in these domes as predicted by the three- and four-node elements; it can be seen that agreement between the two sets of results is very good.

2.5.6 Orthotropic element It is a relatively simple matter to extend the previous finite element theories for isotropic thin-walled axisymmetric shells to the orthotropic cases. For the finite element given in Section 2.4, the following matrix of material constants applies:



φ

1

6

7

2

8 3 9

10

4

f1−6

=

11°

f2−7

=

35°

f3−8

=

55°

f4−9

=

77°

f5−10

=

90°

5

2.44 Strain gauge positions: dome 1.

φ 1

7

2

3

4

5

6

8

9

10

f1−6

=

23°

f2−7

=

37°

f3−8

=

50°

f4−9

=

67°

f5−10

=

90°

2.45 Strain gauge positions: dome 2.


73

74

Pressure vessels

Table 2.9 Experimental strain results for dome 1

Gauge

345 kPa (50 lbf in−2)

517 kPa (75 lbf in−2)

690 kPa (100 lbf in−2)

862 kPa (125 lbf in−2)

Hoop strains, με

1 2 3 4 5

−234 −404 −347 −310.5 −305

−361 −582.5 −511.5 −470 −466

−477.3 −762 −670.7 −625.3 −618

−58 −950 −840 −778 −767

Meridional strains, με

6 7 8 9 10

448 −173.5 −357.5 −350.5 −344

483 −281.5 −529.5 −517 −502

602 −378 −687.3 −674 −647.3

688 −477 −851 −833 −790

D11 =

Ex t (1 − v x v y )

D33 =

Ex t 3 12 (1 − vx vy )

D22 =

Ey t (1 − v x v y )

D44 =

Ey t 3 12 (1 − vx vy )

D12 =

vy E x t (1 − v x v y )

D34

vy E x t 3 12 (1 − vx vy )

D21 =

vx E y t (1 − v x v y )

D43

vx E y t 3 12 (1 − vx vy )

All other Dij = 0 where, Ex = Young’s modulus in meridional direction Ey = Young’s modulus in hoop direction vx = Poisson’s ratio due to a direct stress in the x-direction vy = Poisson’s ratio due to a direct stress in the y-direction vxEy = vyEx t = shell thickness. Similarly, for the thick-walled axisymmetric shells in Section 2.5, D11 = Ex / (1 − vx vy )

D22 = Ey / (1 − vx vy )

D12 = D21 = vy Ex / (1 − vx vy ) = vx Ey / (1 − vx vy ) D33 = G / k All other Dij = 0



Meridional strains, με

Hoop strains, με

−239.5 −226.5 −215

−103 −93.5

10

−236

−99.5

7 −98.5

−178.5

−74.5

6

8

−232

−106.5

5

9

−218 −223.5

−99.5 −99.5

3

−189

−87

2

4

−190.5

345 kPa (50 lbf in−2)

−93.5

172.5 kPa (25 lbf in−2)

1

Gauge

Table 2.10 Experimental strain results for dome 2

−327

−341.5

−369

−364

−284

−352.5

−338

−332

−290.5

−284

517 kPa (75 lbf in−2)

−440

−461.5

−497

−485

−384

−472

−456.5

−441

−386.5

−348

690 kPa (100 lbf in−2)

−553

−581

−633

−633

−501

−595

−572

−553

−489

−390

862 kPa (125 lbf in−2)

Angle f (degrees)

85 75

0

65

55

45

35

25

15

5

0

Meridional stress (MN m−2)

−1

−2

−3

External stress

−4

Internal stress Five node element program Experimental results (internal)

−5

2.46 Meridional stress distribution for dome 1 at 690 kPa (100 lbf in−2). Angle f (degrees)

85 0

75

65

55

45

35

25

15

5 0

−0.5 External stress Hoop stress (MN m−2)

−1.0

Internal stress Five node element program Experimental results (internal)

−1.5

−2.0

−2.5

−3.0

2.47 Hoop stress distribution for dome 1 at 690 kPa (100 lbf in−2).


Axisymmetric deformation of pressure vessels Angle f (degrees)

80 0

77

70

60

50

40

30

20

10

0

−1


−2

−3

−4 External stress −5

Internal stress Five node element program

−6

Experimental results (internal)

−7

2.48 Meridional stress distribution in dome 2 at 690 kPa (100 lbf in−2).

2.6

Ring-stiffeners

The ring element consists of one nodal circle and shares a common node with the two adjacent axisymmetric shell elements on either side of it. The stiffness components of the ring element are partly flexural and partly inplane, and the flexural component of stiffness is now derived. Owing to the moment Mi on the ring-stiffener at node i, the stiffener rotates out of its plane (turned inside-out) by an angle θi, as shown in Fig. 2.54. According to Roark and Young,32

θi =

Mi Rf EI z

[2.90]


78

Pressure vessels Angle f (degrees) 80 0

70

60

50

40

30

20

10

0

−0.2 −0.4 −0.6

Hoop stress (MN m−2)

−0.8 −1.0

External stress Internal stress Five node element program Experimental results (internal)

−1.2 −1.4 −1.6 −1.8 −2.0 −2.2 −2.4 −2.6


where, Rf = radius of centroid of frame, together with the width of plate in direct contact with the frame’s web; Iz = second moment of area about the z–z plane of the frame, together with the width of plate in direct contact with the frame’s web; E = Young’s modulus of elasticity of the frame, or its equivalent if the shell has a different E. Therefore, Mi =

EI zθ i Rf


85 0

Angle f (degrees) 75

65

55

45

35

25

15

5 0


−1

−2

−3

−4

External stress Internal stress Three node element program Four node element program

−5

2.50 Meridional stress distribution for dome 1 at 690 kPa (100 lbf in−2). Angle f (degrees)

85 0

−0.5

75

65

55

45

35

25

15

5 0

External stress


Internal stress Three node element program −1.0

Four node element program

−1.5

−2.0

−2.5

−3.0



80

Pressure vessels Angle f (degrees) 80 0

70

60

50

40

30

20

10

0

−1


−2

−3

−4

−5 External stress Internal stress −6

Four node element program Three node element program

−7

2.52 Meridional stress distribution in dome 2 at 690 kPa (100 lbf in−2).

and the bending strain energy is: Ub =

πEI z2θ i2 Mi2 2πRf = 2E Rf

[2.91]

As previously reported26 U b = 12 kbθ i2 =

πEI zθ i2 Rf

[2.92]

Therefore, kb, the bending component of stiffness about the z–z plane is: kb =

2πEI z Rf

[2.93]



81

Angle f (degrees) 80 0

70

60

50

40

30

20

10

0

−0.2 −0.4 −0.6

External stress Internal stress

−1.0

Four node element program


−0.8

Three node element program

−1.2 −1.4 −1.6 −1.8 −2.0 −2.2 −2.4 −2.6

2.53 Hoop stress distribution in dome 2 at 690 kPA (100 lbf in−2).

The in-plane (circumferential) stiffness is derived as follows. The in-plane strain energy in the circumferential direction UH is given by: UH =

σ H2 × volume 2E

[2.94]

where σH, the hoop stress in the ring is EεH. However, the hoop strain in the ring is given by

εH =

wi Rf

σH =

Ewi Rf

[2.95]

so that


82

Pressure vessels z

Mi

qi Axis

Mi

z

2.54 Out-of-plane bending of a ring. Mi = couple at node i.

Therefore, UH =

E 2 wi2 1 πEAf wi2 1 × × R × A = = kH wi2 2 π f f Rf2 Rf 2E 2

[2.96]

where the in-plane (circumferential component of stiffness in the ring is given by: kH =

2πEAf Rf

[2.97]

i.e. the elemental stiffness matrix of the ring, in terms of wi and θi is: wi θ i 2 πE ⎡ Af 0 ⎤ wi [k ] = Rf ⎢⎣ 0 I z ⎥⎦ θ i

[2.98]

There is, however, another component of stiffness that is relatively small but which can be included for greater precision. This component results



83

from the change of curvature with circumferential direction owing to the deflection w, i.e. the curvature in the circumferential direction is given by:

χH =

wi Rf2

[2.99]

and the bending strain energy in the circumferential direction is given by: U bH = (EIχ H2 / 2)(2 πRf ) = 0.5kbH wi2

[2.100]

so that the component of flexural stiffness due to bending of the stiffener about its x–x axis is given by kbH =

2 πEI Rf3

[2.101]

where I is the second moment of area of stiffener about its x–x axis, which is an axis through the centroid of the ring-shell combination, parallel to the axis of the cylinder/cone/dome. The improved [k] for the ring is now given by: wi

[k ] = 2.7

2 πE ⎡( Af + I / R Rf ⎢⎣ 0

2 f

)

θi 0 ⎤ wi I z ⎥⎦ θ i

[2.102]

Plastic collapse

It is likely that an axisymmetric shell can withstand a pressure somewhat higher than that based on first yield.33 The theories provided in this chapter are based on failure at first yield, but the total plastic collapse load is likely to be higher than this, particularly for thick shells. For thin shells, which have negligible bending resistance, the plastic collapse pressure is likely to be only fractionally higher than the stress to cause yield in the circumferential direction. This is because, for thin shells, the maximum circumferential stress is a bulk stress, whereas the maximum meridional stress is only a local stress and will cease to resist bending once it becomes plastic (i.e. the shell will simply rotate at the points where it has become plastic owing to meridional bending). The designer must remember that, for thin shells, large bulk stresses, such as hoop stresses, are inherently more dangerous than large meridional bending stresses, which are local. In this chapter, a theoretical axisymmetric plastic buckling analysis is carried out based on an element similar to that of Grafton and Strome,22 and on the nonlinear finite element method of Turner et al.34 Also presented are experimental tests carried out on a number of thin-walled circular cylinders and cones,35 which were tested to failure under uniform external


84

Pressure vessels

pressure. Comparisons were made between experiment and theory, and good agreement was found.

2.7.1 Experimental apparatus This section describes tests carried out on two thin-walled circular conical shells, namely cone C and cone 9, and on three thin-walled circular cylindrical shells, namely cylinders 4, 5 and 6. All of the vessels were machined carefully from solid billets of EN1A steel, and the details of these vessels are now given.

2.7.2 Cone C The geometrical details of this vessel are given in Fig. 2.55 and Table 2.11, from which it can be seen that cone C was a thin-walled vessel that was stiffened by two quite heavy ring stiffeners. Cone C was from a series of three vessels shown in Fig. 2.56. The out-of-roundness of cone C was found to be 0.005 mm. This measurement was taken around the outer circumference of the vessel, in the region of the expected failure zone in the central bay. The out-of-roundness was defined as the difference between the maximum inward and outward radial deviations from the mean mid-length circumference, where the latter was obtained from a least-square fit.

2.7.3 Cone 9 The geometrical details of cone 9 are presented in Fig. 2.57. Cone 9 was one of a series of three thin-walled cones as shown in Fig. 2.58. The out-ofroundness for cone 9 was found to be 0.007 mm. This out-of-roundness was measured around the external circumference of the vessel at its mid-length in the central bay.

2.7.4 Cylinders 4, 5 and 6 The geometrical measurements for the circular cylinders 4, 5 and 6 are given in Fig. 2.59 and Table 2.12. A photograph illustrating the three circular cylinders is shown in Fig. 2.60. The out-of-roundness of these vessels was 0.0227, 0.0108 and 0.0205 mm for cylinders 4, 5 and 6, respectively. The out-of-roundness values were measured at the mid-lengths of the vessels on their external surfaces, in the central bays. The test tank used for all the models is shown in Fig. 2.61. The material properties of the vessels were found to be as follows: • Cones A, B and C: yield stress σyp is 288 MPa and Young’s modulus E is 190 GPa.



85

8 9

7 6

13 5 14

4 3 2

15 1 11 10 12

17

18

16

2.55 Geometrical measurements of cone C.

• Cone 9: σyp is 231 MPa and E is 193 GPa. • Cylinders 4, 5 and 6: σyp is 244 MPa and E is 200 GPa. • For all vessels: Poisson’s ratio v is 0.3 (assumed).

2.8

Experimental procedure

All five models were tested to destruction, under uniform external pressure, in the test tank shown in Fig. 2.61. Water was used as the pressure-raising liquid, and this was pumped into the test tank with the aid of a hand-


86

Pressure vessels Table 2.11 Geometrical details of cone C (mm) (average thickness = 1.082 mm; cone angle = 19.15°) Dimension

Measurement

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

151.867 19.057 7.631 31.755 7.590 25.427 15.874 76.230 38.112 139.693 101.650 40.358 49.343 74.423 97.320 103.733 75.949 103.283

2.56 Cones A, B and C (from right to left).



87

211.98

15.88 7.61

15.853

10 38.096

88.79 19.07

7.63

dia. 110.518

57.164 dia. 76.443

2.57 Geometrical details of cone 9 (mm).

2.58 Cones 7, 8 and 9 (from left to right).


dia. 76.2

dia. 101.66

dia. 139.67

0.79

88

Pressure vessels b Area Bay 0

Bay 1

Bay 2 d

t

r1

r2

r3

r4

Cylinder centre line

2.59 Geometrical details of cylinders 4, 5 and 6.

Table 2.12 Geometrical dimensions of cylinders 4, 5 and 6 (mm) Cylinder 4

Cylinder 5

Cylinder 6

Unsupported length Bay 0 Bay 1 Bay 2

9.254 25.416 9.154

9.296 18.966 9.389

9.621 12.661 9.348

Stiffener 1 Breadth Depth Area Radius

6.032 6.011 36.258 55.101

6.133 5.997 36.780 55.117

5.919 6.003 35.532 55.092

Stiffener 2 Breadth Depth Area Radius

5.956 6.012 35.807 55.101

6.083 5.991 36.443 55.114

6.024 6.000 36.144 55.090

Mean shell radius Shell thickness Internal radius External radius

51.472 1.245 50.850 52.095

51.484 1.267 50.851 52.118

51.467 1.246 50.844 52.090



89

2.60 Circular cylinders 4, 5 and 6 (from right to left).

Bolt

Hose from pump

Model

Closure plate

2.61 Test tank with model.

operated hydraulic pump. The hose connecting the hydraulic pump to the test tank was only about 2 m long; hence line losses were negligible.

2.8.1 Cone C All three cones in this series of vessels collapsed initially in their first bays through plastic axisymmetric buckling, as shown in Figs 2.62 and 2.63. The


90

Pressure vessels

vessels also collapsed through plastic lobar buckling in the second bay, but this was not of interest in this study. After cones A, B and C collapsed axisymmetrically in their first bays, the uniform pressure fell. On increasing this pressure, an experimentally obtained ring stiffener formed in the first bay of each vessel. This ring stiffener had the effect of increasing the strength of each vessel in its first bay,

2.62 The collapsed models: A, B and C (from left to right).

2.63 Bottom view of the collapsed cone C.



91

Table 2.13 Collapse pressures of cones A, B and C

Cone

Axisymmetric collapse pressure (MPa)

Lobar buckling pressure (MPa)

A B C

5.97 6.03 6.21

6.48 6.76 6.62

so that it was possible to increase the pressure to a value greater than that which caused plastic axisymmetric buckling. On achieving these higher pressures, each vessel failed by lobar buckling (or shell instability, in its central bay, as shown in Fig. 2.12. The two sets of collapse pressures are shown for each vessel in Table 2.13. From Table 2.13, it can be seen that the axisymmetric collapse pressure for cone B was slightly higher than for cone A, and that the axisymmetric collapse pressure for cone C was slightly higher than that for cone B. The reason for this was that the length of the mid-bay of cone C was shorter than that of cone B, and that the length of cone B was shorter than that of cone A. Thus, if the mid-bay is shorter, it gives more support to the first bay. The maximum values of measured hoop and meridional strains in the first bay were between −3000 and −4000 microstrain, thereby indicating that the vessels failed plastically.

2.8.2 Cone 9 Cone 9 collapsed through axisymmetric buckling at a uniform external pressure of 52.41 bar. This mode of collapse was completely unexpected, as the length of the mid-bay of cone 9 was about 4.66 times that of the first bay in which the first collapse took place. The reason the collapse took place in the first bay, rather than in the central bay, may have been because the mean diameter of the first bay was about 28.3% larger than the mean diameter of the central bay. After cone 9 had collapsed axisymmetrically in the first bay, the pressure fell to 31.03 bar. On increasing this pressure an experimentally obtained ring stiffener formed in the first bay, which strengthened the first bay, so that the pressure could be increased further. When the pressure reached 48.28 bar, the central bay collapsed through shell instability, as shown in Fig. 2.64. It was assumed, however, that the central bay collapsed at a pressure of 52.41 bar, because the vessel had previously withstood this pressure.


92

Pressure vessels

2.64 Collapsed modes for cones 7, 8 and 9 (from left to right).

Table 2.14 Experimental collapse pressures

Cylinder

Pressure (bar)

4 5 6

97.24 111.72 131.72

Figure 2.64 shows the collapsed modes of cones 7, 8 and 9, where it can be seen that cone 9 collapsed axisymmetrically in its first bay. Cone 9 also collapsed through lobar buckling in its second bay, as indeed did cones 7 and 8, but this mode of failure was not of interest in this study.

2.8.3 Circular cylinders 4, 5 and 6 All three cylinders collapsed in their central bays through plastic axisymmetric buckling. It was expected that these vessels would fail in their central bays and, for this reason, these bays were heavily strain gauged. The uniform external pressures that the vessels collapsed under are given in Table 2.14.



93

Ten linear strain gauges were attached to the inside surfaces of each of their central bays. Plots of the variation of hoop strain versus pressure are given for each of these vessels in Figs 2.65–2.67. From these figures it can be seen that the vessels collapsed through plastic axisymmetric deformation, the measured hoop strain reaching nearly 12 000 microstrain for cylinder 4 and 6000 microstrain for cylinders 5 and 6.

0

9.66

0

Pressure (MPa) 5.52

Hoop strain/microstrain

−2000 −4000 −6000

−8000

−10000 −12000

2.65 Pressure–hoop strain relationship for cylinder 4.

0

0

Pressure (MPa) 5.52


−1000 −2000 −3000 −4000 −5000 −6000



11.03

94

Pressure vessels

0

13.79

0

Pressure (MPa) 6.9 −1000


−2000 −3000 −4000 −5000 −6000 −7000 −8000 −9000 −10000


2.68 Collapsed modes of cylinders 4, 5 and 6 (from left to right).

The experimental observations for these vessels showed that they all suffered initial plastic axisymmetric deformation as a consequence of large hoop stresses but that, as the radial deflection increased, the axial loads became more and more significant until plastic axisymmetric buckling took place in the form of a three-hinge mechanism. The middle hinge was a sagging hinge and the two outer hinges were hogging hinges, as shown in Fig. 2.68.



2.9

95

Theoretical plastic analysis

The finite element adopted was based on the axisymmetric conical shell element of Grafton and Strome.22 It was of truncated form and had two end ring nodes, with six degrees of freedom per element. The computational analysis, which used the author’s computer program ‘PLASCONE’, allowed for both geometrical and material nonlinearity. It is based on the incremental method of analysis and is now described with the aid of Table 2.15. In the theoretical analysis, step 1 was to load the structure with a relatively small load that was well below the point where either geometrical or material nonlinearity would commence. After calculating the stresses and deflections at the end of step 1, the geometrical stiffness matrix was calculated as was the new geometry of the structure formed as a consequence of the resulting deflections. Similarly, in step 2, the structure was subjected to another small incremental load, and the stresses and deflections resulting from this load were calculated. They were added to the stresses and deflections at the end of step 1. If the resulting stress in any element exceeded its yield stress, based on the Hencky–von Mises theory of elastic failure,1,2 the Young’s modulus in that element was made equal to 1/50th of the elastic Young’s modulus. Furthermore, if the yield stress in any element exceeded its yield stress by a factor of 1.1, based on the Hencky–von Mises stress, the Young’s modulus in that element was made equal to 1/100th of the elastic Young’s modulus. The new stiffness and geometrical stiffness matrices were calculated, and the geometry of the structure updated. The process was repeated until the structure failed through plastic axisymmetric buckling, with the formation of three circumferential plastic hinges. Each vessel was assumed to be simply supported at one end, the larger end for the cones, and clamped at the other end. By the condition clamped, it was assumed that all the deflections, except for the axial displacement, were zero at that end. Plots of pressure versus axial displacement at the

Table 2.15 Incremental non-linear method ([K0] = stiffness matrix; [K G0 ] = geometrical stiffness matrix; {δq0} = a vector of nodal incremental loads; {δu0} = a vector of incremental nodal displacements) Step

{δq0} Stiffness matrix

{δu0} Displacements

1 2 3 ¯ n Σ

{δq 01}[K 00 (0)] + [K G0 (0)] {δq 02}[K 00 (u10)] + [K G0 (u01)] {δq03}[K 00 (u02)] + [K G0 (u02)] 0 0 {δq n0}[K 00 (un−1 )] + [K G0 (un−1 )] 0 {q n}

{δu 01}{u 10} = {δu 01} {δu 02}{u 02} = {u 01} + {δu 02} {δu 03}{u 03} = {u 02} + {δu 03} 0 {δu n0}{u n0} = {u n−1 } + {δu n0} 0 {un}


96

Pressure vessels

clamped end of the vessel are given for all the vessels in Figs 2.69–2.73. From these figures, it can be seen that there was good agreement between experiment and theory. The experimental tests showed that the collapse mechanism for the five vessels described in detail was through plastic axisymmetric buckling where, after initial yielding in the circumferential direction at mid-bay, the effects of axial load became more important with increasing radial deflection so that the axial load eventually caused a three-hinge plastic buckling mechanism to occur. Previous experimental work on the plastic axisymmetric collapse of thin-walled ring-stiffened cylinders was carried out by Lunchick,36 but it was not possible to analyse Lunchick’s models by the present analysis as that author omitted to give full details of his vessels.

2.10

Conclusions

The experimental tests showed that all five vessels collapsed through plastic axisymmetric buckling. The experimental observations also showed that these vessels initially suffered plastic axisymmetric deformation at their mid-bays, and that, as the inward radial deflections increased, the effect of

7

6

Pressure (MPa)

5

4

3

2

1

0

2

4

6 8 10 12 14 16 Axial deflection (mm x 10−3)

18

20

2.69 Theoretical plot of pressure versus axial deflection for cone C.



97

5

4

Pressure (MPa)

3

2

1

0

8

16

24

32

36

Axial deflection (mm x 10−3)

2.70 Theoretical plot of pressure versus axial deflection for cone 9.

11

Pressure (MPa)

9

7

5

3

1 0

1

2

3

4

5

6

7

8

9


2.71 Theoretical plot of pressure versus axial deflection for cylinder 4.


15

13

11

9

Pressure (MPa)

7

5

3

1 0

1

2

3

4

5

6

7

8

9



13

11

9

Pressure (MPa)

7

5

3

1 0

1

2

3 4 5 6 7 Axial deflection (mm x 10−2)

8

9




99

axial pressure became more and more significant, resulting in a three-hinge buckling mechanism for each vessel. Comparisons between experiment and the nonlinear axisymmetric finite element solution showed good agreement for plastic axisymmetric buckling. Snap-thru buckling of shallow domes is described in Section 3.8.1.


3 Shell instability of pressure vessels

Abstract: This chapter deals with the shell instability (or lobar buckling) of unstiffened circular cylinders and cones under external hydrostatic pressure, together with domes. Solutions for the instability of thin-walled circular cylinders, cones and domes under uniform external pressure are described. Some of the theories are based on an analytical approach and some on a numerical one, namely the finite element method. Experimental results are provided and compared with those obtained by theory. The effects of initial out-of-circularity and inelastic buckling are also considered and two design charts are provided, which are easier to use than PD 5500. Key words: shell instability, thin-walled cylinders, out-of-circularity, buckling, design charts.

3.1

Shell instability of thin-walled circular cylinders

Under uniform external pressure, a thin-walled circular cylinder may buckle in the manner shown in Fig. 1.5, usually at a fraction of that pressure required to cause axisymmetric yield. If the circular cylinder is very long, its buckling resistance is very small, the vessel suffering failure in a flattening mode (i.e. ovalling). According to Bryan,37 the elastic instability pressure for an infinitely long circular cylinder under uniform lateral pressure is given by equation [3.1]: Pcr =

E ⎛ t⎞ 4 (1 − v 2 ) ⎝ a ⎠

3

[3.1]

where Pcr = buckling pressure t = wall thickness of circular cylinder a = mean radius of circular cylindrical shell E = Young’s modulus ν = Poisson’s ratio In their famous paper, Windenburg and Trilling38 state that equation [3.1] applies to long, thin, accurately made tubes, under uniform lateral pressure, when l > 4.9a (a/t)0.5 and a/t > 10, where l = length of tube. 100 © Carl T. F. Ross, 2011

Shell instability of pressure vessels

101

To demonstrate the strength of a long thin-walled tube, under lateral pressure, consider a tube with the following properties: E = 2 × 1011 N m2; ν = 0.3; yield stress = σyp = 300 MPa; a = 2 m; t = 2 × 10−2 m; l = 500 m. Now, a 2 = 100 > 10 = t 2 × 10 −2 and 4.9a ( a / t )

1/ 2

= 4.9 × 2 (100 )

1/ 2

= 98

l = 500 > 98 i.e. the Bryan formula applies. From equation [3.1], the lateral pressure to cause buckling, Pcr is given by: 2 × 1011 ⎛ 1 ⎞ ⎜ ⎟ = 54 945 Pa = 0.0549 MPa 4 × 0.91 ⎝ 100 ⎠ 3

Pcr =

Based on axisymmetric yield, the resistance of the vessel to withstand uniform lateral pressure is given by the well-known expression:1 P=

σ yp × t 1 = 300 MPa × = 3 MPa 100 a

i.e. the elastic instability pressure is only about 1/55th of the pressure to cause axisymmetric yield.

3.1.1 Von Mises formula However, for shorter tubes firmly supported at their ends, the buckling resistance can be considerably increased. Von Mises39 gives equation [3.2], which is the elastic instability pressure for a thin-walled circular cylindrical shell, simply supported at its ends and subjected to the combined action of uniform lateral and axial pressure: Pcr =

E ( t / a) ⎡⎣ n2 − 1 + 0.5 ( πa / l )2 ⎤⎦ 2 2 ⎧⎪ ⎡ 2 t2 πa ⎞ ⎤ ⎫⎪ 1 ⎛ ×⎨ n − 1+ + 2 ⎢ 2 2 ⎝ l ⎠ ⎥⎦ ⎬ 12a2 (1 − v) ⎣ ⎪⎩ ⎡⎣ n2 ( l / πa) + 1⎤⎦ ⎪⎭

[3.2]

where l = length of shell between adjacent supports and n = number of circumferential waves or lobes into which the vessel buckles, as shown in Fig. 3.1. The normal method of using equation [3.2] is to calculate Pcr for various values of n, and to select the lowest or minimum value of Pcr. The


102

Pressure vessels

n=2

n=3

n=4

3.1 Circumferential wave patterns for buckling modes.

value of n corresponding to the minimum value of Pcr is said to be the buckling eigenmode. Although, with the aid of a computer, it is simple enough to use equation [3.2], Windenburg and Trilling38 have provided a simpler version that predicts similar buckling pressures to that of von Mises: Pcr =

(1 − v )

2 0.75

2.42 E ( t / 2a)

5/ 2

⎡⎣( l / 2a) − 0.447 ( t / 2a)1 / 2 ⎤⎦

[3.3]

If ν = 0.3, then 2.6 E ( t / d ) 1/ 2 l / d − 0.45 ( t / d ) 5/ 2

Pcr =

[3.4]

where d = 2a = mean shell diameter. Equation [3.3] is also known as the David Taylor Model Basin (DTMB) formula. If, for example, it is required to increase the buckling resistance of the vessel of Section 3.1, which is extremely low, it will be convenient to stiffen the tube with suitably sized rings at intervals of (say) 2 m. Hence, according to equation [3.4], the inter-bay elastic instability buckling pressure (shell instability) is given by: Pcr =

2.6 × 2 × 1011 × ( 2 × 10 −2 / 4)

(2 / 4 ) − 0.45 ( 2 × 10 / 4) −2

2.5

0.5

=

919 239 = 1.963 MPa 0.4682

From this calculation, it can be seen that the elastic instability resistance of the vessel has been increased by a factor of about 36, by introducing stiffening rings to the very long tube of Section 3.1. It must, however, be pointed out that if the ring-stiffeners are not strong enough, the entire ring-shell combination can buckle bodily, and this mode of failure is known as general instability (Fig. 1.9).

3.1.2 Sturm’s models Sturm40 produced a simply supported and a fixed-edges solution, based on elastic theory, together with some experimental results for eight carefully © Carl T. F. Ross, 2011


103

1'' 1/2'' Rubber tube seal

Model under external pressure

Aluminium alloy end disc cap

5/16'' 1/8''

3.2 End connection for Sturm’s models.

machined models which had initial out-of-roundness values less than 0.033t. The models were constructed in aluminium alloy and were sealed, as shown in Fig. 3.2. In order to determine the effects of end-conditions, Sturm,40 Nash41 and Ross42 produced fixed-edges solutions, and, in Table 3.1, comparison is made between these solutions and the simply supported solutions of Sturm, Reynolds, Ross and von Mises, together with the experimental results of Sturm. The solutions of Ross were based on the Kendrick strain energy43 expressions (see Section 4.1.1). In Table 3.1, comparison is also made with Windenburg’s thinness ratio λ, a parameter that can be used for detecting whether or not elastic instability solutions apply (see Section 3.1.3). From Table 3.1, it can be seen that the experimental results of Sturm agreed best with the simply supported solutions, but this was not altogether surprising, because the experimentally obtained boundary conditions were probably nearest the simply supported edges case. Table 3.1 also shows that the theoretical solutions, based on fixed ends, tended to agree with each other and predicted buckling pressures considerably higher than the simply supported edges solutions.

3.1.3 Windenburg’s models Experiments carried out earlier by Windenburg and Trilling38 on models soldered firmly at their ends (Table 3.2) do not, however, agree with these © Carl T. F. Ross, 2011


λ

4.897 0

3.720 5

3.064 3

2.129 1

1.827 3

1.457 7

1.457 7

3.671 7

No.

23

24

25

26

27

28

29

30

51.7 (4)

505 (6,7)

505 (6,7)

321 (5)

226 (4,5)

95 (3)

71.5 (3)

35.8 (2)

Sturm40

Fixed edges

50.4 (4)

463 (6)

463 (6)

294 (5)

230 (4)

106 (3)

74 (3)

38.5 (2)

Nash41

50 (4)

459 (6)

459 (6)

291 (5)

228 (4)

105 (3)

73.4 (3)

38 (2)

Ross42

34.2 (3)

358 (6)

358 (6)

216 (4)

152 (3)

62 (3)

45.2 (2)

26.3 (2)

Sturm40

36.4 (3)

346 (5)

346 (5)

217 (4)

159 (4)

76.2 (3)

51.6 (2)

29.7 (2)

Reynolds44

Simply supported edges

34.6 (3)

333 (5)

333 (5)

210 (4)

157 (4)

80 (3)

49.5 (2)

33.8 (2)

von Mises39

33.2 (3)

328 (5)

328 (5)

205 (4)

149 (4)

72.8 (3)

46.9 (2)

26.5 (2)

Ross42

34 (3)

339 (6)

330 (6)

240 (4,5)

155 (3,4)

81 (3)

53 (2,3)

29 (2)

Experimental

Table 3.1 Comparison of theoretical solutions with experimental observations from Sturm’s models40 (buckling pressures are in lbf in−2; the number of lobes is given in parentheses)


105

Table 3.2 Comparison of theoretical solutions with experimental observations from Windenburg and Trilling’s models38 (buckling pressures are in lbf in−2; the number of lobes is given in parentheses)

No.

Out-ofroundness

Thinness ratio λ

von Mises39

Nash41

Experimental

33 46 47 50 56 58 60 61 62 67 69 70 71

0.11t 0.13t 0.16t 0.16t 0.16t 0.16t 0.11t 0.14t 0.12t 0.15t 0.16t 0.15t 0.14t

1.229 1.334 2.427 1.080 2.368 1.632 1.123 3.205 2.219 1.646 1.569 2.147 1.182

193 (12) 209 (13) 56.6 (8) 301 (17) 28.2 (11) 165 (9) 351 (12) 42 (5) 84.6 (6) 205 (6) 313 (5) 153 (4) 579 (8)

266 (14) 288 (14) 80.6 (9) 463 (19) 40 (13) 229 (10) 489 (13) 60.8 (6) 122 (8) 290 (7) 440 (7) 224 (5) 795 (9)

139 (13,14) 163 (13) 65 (9) 195 (19) 31 (11) 159 (–) 199 (14) 48 (5, 6) 89 (6) 209 (6) 288 (5, 6) 149 (4) 327 (–)

3 9 2 4 3 9 9 8 4 3 8 4 9

theories. In general, these theories tend to overestimate the buckling pressures for many typical vessels. Windenburg and Trilling concluded that this was because many of the cylinders buckled inelastically owing to the initial out-of-circularity of the models, i.e. because of initial geometrical imperfections, various parts of the shell became plastic, triggering off inelastic instability at a pressure less than that required to cause elastic instability. For this reason, the models of Windenburg and Trilling are among the most important of their time, as fabricated pressure vessels, even those made today, may have appreciable out-of-circularity, and as a result are likely to suffer inelastic instability. Windenburg and Trilling38 introduced their thinness ratio λ, which can be used to determine experimentally the plastic reduction factor (PKD), where:

λ=

4

{(l / d )

2

(t / d )3 } × (σ yp / E )

[3.5]

Other experimental results on machined stiffened models were carried out by Ross3 and Reynolds44 (Table 3.3). Details of Ross’s models 1, 2 and 3, together with the models which failed by general instability (see 4.1.1), are shown on Table 3.4. These models were machined from a thick-walled tube, made from HE9-WP aluminium alloy, with the following properties: E = 10.3 × 106 lbf in−2; 0.1% proof stress = 21 700 lbf in−2;


106

Pressure vessels Table 3.3 Buckling pressures (lbf in−2) for models of Ross3 and Reynolds44 (the number of lobes is given in parentheses) No.

von Mises39

Nash41

Experimental

Thinness ratio, λ

1 2 3 U12 U22

363 (8) 443 (8) 709 (10) 1764 (13) 959 (14)

500 (9) 617 (9) 1041 (11) 3040 (16) 1569 (17)

335 350 402 975 735

1.184 1.081 0.882 0.853 1.034

(8) (9) (12) (–) (–)

2 0 6 7 9

Table 3.4 Dimensions of models 1 to 7 (inches) (internal diameter = 10.25 in; h = 0.08 in (models 1–6), 0.081 in (model 7); N = number of ring stiffeners) No.

L1

L

Lb

bF

bf

d

N

1 2 3 4 5 6 7

3.5 3.75 2.5 1.6 1.6 1.6 –

4.5 3.75 2.25 2.0 2.0 2.0 10.0

26.625 24.275 15.775 19.92 19.92 19.92 10.0

0.325 0.40 0.40 0.08 0.08 0.08 –

0.325 0.325 0.325 0.08 0.08 0.08 –

0.62 0.62 0.62 0.08 0.12 0.16 –

5 5 5 9 9 9 –

0.2% proof stress = 22 600 lbf in−2; nominal peak stress = 26 900 lbf in−2; ν = 0.32 (assumed). Table 3.4 also shows details of the machined model No. 7, which had no stiffeners and was made from mild steel with the following properties: E = 28 × 106 lbf in−2; σyp = 40 700 lbf in−2; UTS = 65 200 lbf in−2; ν = 0.32 (assumed). Measurements of the initial out-of-roundness of model Nos. 1–6 revealed a maximum value of 0.003 in, and similar measurements on model No. 7 revealed a maximum initial out-of-roundness of 0.004 in. The models were tested to destruction in the tank shown in Fig. 3.3.

3.1.4 Design charts To obtain the design charts for the inelastic instability of circular cylinders under uniform external pressure, Pcr was calculated by the von Mises


Shell instability of pressure vessels Scale (ins) 5

3

Outlet to pressure gauge

1 0

107

24 HT bolts 5/8" BSF

'O' Rings As below

Model No.1 Flexible hose from pump

1/4" BSF cap head screws every 20°

Centre spindle

Closure plate

'O' Rings

Pressure tight cable gland

3.3 Test tank with attachments.

formula.39 The formula assumed that the cylinder was supported at its ends. This assumption was probably quite reasonable for the practical case, as neither the web of a typical ring frame, nor the thin-walled shell could resist much bending; in practice there may have been some degree of rotational restraint at the ends of the shell, but this was not very much. A design chart for the shell instability of accurately machined circular cylinders under hydrostatic pressure is given in Fig. 3.4. This chart uses the experimental observations of Sturm,40 Reynolds,44 Seleim and Roorda45 and Ross,3 Ross and Kimber,46 and Ross et al.47


Pressure vessels 1\Lambda (thinness ratio)

108

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0

2.0

4.0

6.0

8.0

10.0

PKD (Pcr/Pexp)

3.4 Design chart for the shell instability of machined circular cylinders.

From Fig. 3.4, it can be seen that the vessels whose values of 1/λ were less than 0.9, probably failed by elastic instability as PKD was approximately one. The three vessels that had the largest values of 1/λ actually failed by axisymmetric deformation, but Ross and Johns35 have shown that there is a link between plastic axisymmetric deformation and inelastic shell instability. Recent tests carried out at the University of Portsmouth found that in some instances of inelastic shell instability, the cylinder initially suffers plastic axisymmetric deformation before failing by inelastic shell instability. The ability to withstand pressure is reduced if the cylinder is manufactured with an initial out-of-circularity. Additionally, by plotting 1/λ against (Pcr /Pexp), the possible asymptotic behaviour of the graph was avoided in the region of axisymmetric failure. Figure 3.5 provides a design chart for some soldered and welded vessels, which had moderate values of initial out-of-circularity and which failed through shell instability. The soldered models of Windenburg and Trilling38 had initial out-of-circularity values which did not exceed 0.16t, but the initial out-of-circularity values of the models of Reynolds44 were not given. Here again, it can be seen that vessels which had values of 1/λ < 0.6, had a PKD not very far from unity and failed elastically, where Pexp = experimentally obtained buckling pressure and PKD = Pcr /Pexp. The design process therefore calculates Pcr and either λ or λ′ for the appropriate vessel and obtains PKD from the design chart, so that the predicted buckling pressure Ppred is: Ppred =Pcr /PKD Previously, Ross48 and Ross et al.47 presented design charts that were obtained by plotting λ or λ′ against Pcr /Pexp. However, for shorter and thicker vessels, the developed design curves were practically asymptotic to the horizontal axis, thus making it very difficult to measure PKD.



109

1.6 1.4 1.2

1\Lambda

1.0 0.8 0.6

Safe side

0.4 0.2 0 0

0.5

1.0

1.5

2.0

2.5

3.0

PKD

3.5 Design chart for shell instability of soldered and welded circular cylinders.

By plotting 1/λ or 1/λ′ against Pcr /Pexp, the asymptotic behaviour of the design charts is avoided, so that more satisfactory design curves are obtained. Additionally, the design charts produced in the present report, use more experimental data than the design charts of Ross48 and Ross et al.47 and this makes them more reliable for design purposes. In order to demonstrate the use of Fig. 3.5, the vessel of Section 3.1 is assumed to be stiffened by suitably sized, equally spaced rings, 1 m apart. From equation [3.4]: Pcr = 4.213 MPa. From equation [3.5]:

λ=

4

{(1 / 4) / (2 × 10 2

−2

/ 4)

3

}

(300 × 106 / 2 × 1011 ) = 1.03

From the line of Fig. 3.5, the plastic reduction factor PKD is: PKD = Pcr /Pexp = 2.2 so that buckling pressure = 4.213/2.2 = 1.92 MPa For this vessel, therefore, the buckling pressure as calculated by equation [3.4] has to be reduced by the plastic reduction factor, which in this instance is 2.2. From Fig. 3.5, it can be seen that some models fail at an even lower pressure than that predicted by the von Mises formula and the plastic reduction


110

Pressure vessels

factor and, because of this, it is advisable to use a larger safety factor for guarding against instability than for guarding against axisymmetric yield. A computer program for calculating the shell instability buckling pressure of a thin-walled cylinder is provided in Appendix III. Failure against instability should be avoided at all costs, because of the difficulty of predicting buckling pressures. It is important to note that the plastic reduction factor must always be greater than one.

3.1.5 The ‘Holland’ The design chart of Fig. 3.5 can also be applied to the submarine ‘Holland’ (Fig. 3.6) and its 3/8th scale model; the latter was tested to destruction by the US Navy.49 The details are given in Table 3.5. The calculations for Ppred are given in Table 3.6, where the following assumptions are made: E = 30 × 106 lbf in−2; σyp = 34 160 lbf in−2; ν = 0.3.

3.6 The submarine ‘Holland’.

Table 3.5 Details of the submarine ‘Holland’ and its 3/8 scale model (inches)

t Size of frames Lf bf a Number of frames

Holland

Model

0.3 2 1/2 × 2 1/2 × 5/16 angles 18 2.5 48 –

0.109 1 × 1 × 1/8 angles 6.75 1.0 18 17



111

Table 3.6 Predicted collapse pressures (psi) for the submarine ‘Holland’ and its 3/8 scale model (psi) (figures in parentheses = n)

λ Pcr PKD Ppred Pexp

Holland

Model

1.026 311.6 (15) 2.22 140.4 –

1.045 290.7 (15) 2.12 137.1 150

Axis

3.7 Axisymmetric buckling of a cone of large apical angle.

From Table 3.6, it can be seen that Ppred was similar for the submarine ‘Holland’ and its model as required by the US Navy; this was a remarkable achievement for the year 1899. It can also be seen from Table 3.6, that the collapse depth for the ‘Holland’ was about 316 ft; thus, its maximum safety factor in its just-built condition, was about 6.32.

3.2

Instability of thin-walled conical shells

To determine by analytical methods the uniform external pressure required to cause the elastic instability of a thin-walled conical shell is very difficult, and for these cases it is more convenient to use the finite element method (FEM). If the cone is of small apical angle, then buckling will take place in a lobar manner, as shown in Fig. 1.5, but if the apical angle of the cone is large, the vessel can buckle axisymmetrically, as shown in Fig. 3.7. In this instance, both axisymmetric yield and axisymmetric buckling must be considered. The solution presented in this section is based on the elastic instability of thin-walled cones of small apical angle, and it must be emphasised that


112

Pressure vessels u,x

f a w

u0

0

Rj r

Ri

w

w

v

f

j

i

l

3.8 Truncated conical shell element.

as these vessels are sensitive to initial geometrical imperfections, they too can suffer inelastic instability at buckling pressures considerably less than that causing elastic instability. Thus, with the aid of a sufficient number of experimentally obtained results, the plastic reduction factor for a particular thin-walled cone, of a certain apical angle, can be determined in a manner similar to that adopted for circular cylinders. This plastic reduction factor must then be divided into the theoretical elastic buckling pressure for the conical shell in question, to obtain the (reduced) predicted inelastic buckling pressure. A brief description of the solution, based on small deflection elastic theory, is now presented. The element, which was first developed in 1974,50 is a truncated cone with two nodal circles at its ends, as shown by Fig. 3.8. Each node has four degrees of freedom (u0, v, w0 and θ), making a total of eight degrees of freedom per element. To obtain the stiffness matrix, a modified form of Novozhilov’s vector of strains24 was used, as follows: ⎧ux ⎫ ⎪1 ⎪ 1 ⎪ os α ) ⎧ε x ⎫ ⎪ r ν φ + r (u sin α − w co ⎪ ⎪ε ⎪ ⎪ ⎪ ⎪ φ ⎪ ⎪ 1 u + ν − 1 ν sin x ⎪ x ⎪⎪γ xφ ⎪⎪ ⎪⎪ r φ ⎪ r {ε } = ⎨ ⎬ = ⎨ ⎬ w χ ⎪ x ⎪ ⎪ xx ⎪ ⎪ χφ ⎪ ⎪ 1 ⎪ sin α 1 wx ⎪ ⎪ ⎪ 2 wφφ + 2 ν φ cos α + ⎪ r r r ⎪⎩ χ xφ ⎪⎭ ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎛1 ⎞ ⎪⎩2 ⎜⎝ r wxφ − r 2 wφ sin α + r ν x cos α − r 2 ν sin α cos α ⎟⎠ ⎪⎭


[3.6]


113

The assumed displacement functions were: u = [ui (1 − ξ ) + u j ξ ] cos nφ

ν = [ν i (1 − ξ ) + ν j ξ ] sin nφ

w = [(1 − 3ξ 2 + 2ξ 3 ) wi + l (ξ − 2ξ 2 + ξ 3 )θ i

[3.7]

+ ( 3ξ 2 − 2ξ 3 ) w j + l ( −ξ 2 + ξ 3 )θ j ] cos nφ

where

ξ = x/l or

{U } = [ N ]{U i }

[3.8]

and

[ N ] = a matrix of shape functions; ⎧ui0 ⎫ ⎪ν ⎪ ⎪ i ⎪ placements = ⎨ 0 ⎬ {Ui } = a vector of nodal disp ⎪wi ⎪ ⎪⎩θ i ⎪⎭

[3.9]

The assumed displacement functions for {U} included a sinusoidal distribution in the circumferential direction to cater for the lobes, and also to simplify computation. The stiffness matrix was given by T [k ] = lπ [ DC ]T ∫0 r [ B1 ] [ D][ B1 ] dξ [ DC ] 1

[3.10]

where r = Ri + (Rj − Ri)ξ; and [B] = [B1]*(either cos nϕ or sin nϕ) (Table 3.7). If c = cos α and s = sin α, then: ⎡c ⎢0 ⎢ ⎢s ⎢ 0 [ DC ] = ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

0 −s 1 0 0 c 0 0

04

0 0 0 1 c 0 s 0

0 1 0 0

⎤ ⎥ ⎥ 04 ⎥ ⎥ ⎥ −s 0⎥ ⎥ 0 0⎥ c 0⎥ ⎥ 0 1⎦⎥


[3.11]


0 0 −1 / l ⎧ 2 ⎪(1 − ξ ) sinα / r n (1 − ξ ) α / r − (1 − 3ξ + 2ξ 3 ) cos α / r ⎪ 0 ⎪ −n (1 − ξ ) / r [ −1 / l − (1 − ξ ) sinα / r ] ⎪ 0 0 6 12 ξ) / l 2 − + ( ⎪ 2 ⎪ 1 ⎡ n 2 3 ⎪ ⎢ − r 2 (1 − 3ξ + 2ξ ) + rl ⎪ ⎣ [B ′ ] = ⎨ 0 n (1 − ξ ) cos α / r 2 ⎤ ⎪ × ( −6ξ + 6ξ 2 ) sinα ⎥ ⎪ ⎦ ⎪ n (1 − ξ ) ⎡ n ⎪ ⎡ 2 2 ⎢ − cos α / rl − 2 ⎢ − ( −6ξ + 6ξ ) / rl + 2 ⎪ r2 r ⎣ r ⎣ ⎪ 0 ⎤ ⎤ ⎪ × (1 − 3ξ 2 + 2ξ 3 ) sinα ⎥ × cos α sin nα ⎥ ⎪ ⎦ ⎩ ⎦ 0 0 ⎫ − (3ξ 2 − 2ξ 3 ) cos α / r −l ( −ξ 2 + ξ 3 ) cos α / r ⎪ ⎪ 0 0 ⎪ ⎪ (6 − 12ξ ) / l 2 (−2 + 6ξ ) / l ⎪ 2 2 1 ⎪ 1 ⎡ n ⎡ nl 2 3 2 3 + − − − − + ξ + 3 ξ 2 ξ ξ ) ) ⎪ ⎢ r2 ( ⎢ r2 ( rl r ⎪ ⎣ ⎣ ⎬ ⎤ ⎤ × (6ξ − 6ξ 2 ) sinα ⎥ × ( −2ξ + 3ξ 2 ) sinα ⎥ ⎪⎪ ⎦ ⎦ ⎪ n nl ⎪ ⎡ n ⎡ n 2 2 2 ⎢ − (6ξ − 6ξ ) + 2 2 ⎢ − ( −2ξ + 3ξ ) + 2 ⎪ r r ⎣ rl ⎣ r ⎪ ⎤ ⎤ ⎪ 2 3 2 3 × (3ξ − 2ξ ) sinα ⎥ × ( −ξ + ξ ) sinα ⎥ ⎪ ⎦ ⎦ ⎭

Table 3.7 ‘Strain’ matrix for conical shell element

⎤ × (ξ − 2ξ 2 + ξ 3 ) sinα ⎥ ⎦

⎤ × (1 − 4ξ + 3ξ 2 ) sinα ⎥ ⎦ nl ⎡ n 2 2 ⎢ − (1 − 4ξ + 3ξ ) + 2 r ⎣ r

1 ⎡ n2 2 3 ⎢ − r 2 l (ξ − 2ξ + ξ ) + r ⎣

0 −l (ξ − 2ξ + ξ 3 ) cos α / r 0 (−4 + 6ξ ) / l 2

0

0

1/ l ξ sinα / r −nξ / r 0

2 (cos α / rl − ξ cos α sinα / r 2 )

nξ cos/ r

0 nξ / r (1 / l − ξ sinα / r ) 0


115

The relationship between local and global displacements was:

{U i } = [ DC ] {U 10 } = [u1v1w1θ 1 u2 v2w2θ 2 ] and ⎡1 ⎢v ⎢ Et ⎢0 [ D] = (1 − v2 ) ⎢⎢0 ⎢0 ⎢ ⎣0

0 0 0 0 ⎤ ⎥ 1 0 0 0 0 ⎥ 0 (1 − v) / 2 0 0 0 ⎥ ⎥ 2 2 t / 12 vt / 12 0 0 0 ⎥ ⎥ vt 2 / 12 t 2 / 12 0 0 0 ⎥ 2 0 0 0 0 (1 − v) t / 24 ⎦ v

[3.12]

where [DC] = a matrix of directional cosines; [D] = a matrix of material constants for the isotropic case; and α = cone angle. To obtain the geometrical stiffness matrix, it must be remembered that the prebuckling in-plane meridional stress is:

σx ≈

Pr 2t cos α

and the prebuckling in-plane hoop stress is:

σφ ≈

Pr cos α t

However, as σx and σϕ are principal stresses, then the shear stress in the x–ϕ plane

τ xφ = φ Thus, the additional strains owing to large displacements {εL} are a modified version of Stricklin et al.51 as follows: ⎡ wx 1⎢ {ε L } = 2⎢ 0 ⎣

⎤ ⎧ wx ⎫ ⎪ ⎪ (v + wφ ) ⎥⎥ ⎨ (v + wφ ) ⎬ ⎪ ⎪ R ⎦⎩ R ⎭ 0

[3.13]

where ¯v = v cos α. Using the same notation as Zienkiewicz:23 ⎧ wx ⎫ ⎪ ⎪ ⎨ v + wφ ⎬ = [G ] {U } ⎪⎩ R ⎪⎭

[3.14]


116

Pressure vessels

Therefore the geometrical stiffness matrix [k1] was obtained from:

[k1 ] = [ DC ]T ∫vol [G]T [σ ][G] d ( vol ) [ DC ] = [ DC ]

∫ [G] [σ ][G] Rdφdxdz [ DC ] = [ DC ] πRlt ∫ [G ] [σ ][G ] dξ [ DC ] T

T

vol

T

1

1 T

1

0

1 T ⎡σ x T = [ DC ] Rlt ∫ [G1 ] ⎢ 0 ⎣0

[3.15]

0⎤ 1 [G ] dξ [ DC ] σ φ ⎥⎦

where [G] = [G1] *(either cos nϕ or sin nϕ). 1 ⎡0 0 (−6ξ + 6ξ 2 ) (1 − 4ξ + 3ξ 2 ) 0 ⎢ l 1 [G ] = ⎢ (1 − ξ ) c n nl ⎢0 − (1 − 3ξ 2 + 2ξ 3 ) − (ξ − 2ξ 2 + ξ 3 ) 0 ⎢⎣ R R R 1 0 (6ξ − 6ξ 2 ) (−2ξ + 3ξ 2 ) ⎤⎥ l ⎥ ξc n nl − ( 3ξ 2 − 2ξ 3 ) − ( −ξ 2 + ξ 3 )⎥ ⎥⎦ R R R

[3.16]

where c = cos α. The elastic instability pressures for various values of n were obtained from the eigenvalue equation:

[ K ] − [ KG ] = 0

[3.17]

[ K ] = ∑ [k ]

[3.18]

[ KG ] = ∑ [k1 ]

[3.19]

where

A computer program for determining the buckling pressures of thinwalled cones under external pressure for various values of n is presented in reference 28. This program was used to analyse a thin-walled truncated conical shell with an apical angle of 10°. The FEM assumed that the cone was fixed at its left end and simply supported at its right end, and these assumptions yielded a buckling pressure of 340 lbf in−2, with four lobes. This result was similar to Volmir’s prediction52 of 350 lbf in−2, where both ends were assumed to be simply supported.

3.2.1 Niordson’s method Niordson53 provides a formula for the elastic instability of perfect conical shells of small taper ratio, where he applies the von Mises formula



117

y ,w y0 , w0 L/2

x ,u R

r j node

i node

x 0 , u0

R1 t

R2

3.9 Equivalent ‘cylinder’.

to an equivalent cylinder, as shown in Fig. 3.9, where a = R and l = L = generator length.

3.2.2 Truncated conical shells Short truncated conical shells can also fail by inelastic instability. To obtain the design chart for perfect vessels,50 Pcr was calculated using the FEM of Ross54 for the truncated circular conical shells. The left ends of the vessels were assumed to be fixed and their right ends were assumed to be clamped. In this instance, the calculation for λ assumed that the truncated conical shell was an equivalent circular cylindrical shell32 where the dimensions for R and L were as shown in Fig. 3.9. The design chart for this case is shown in Fig. 3.10, where the experimental observations are from the machined vessels of Ross48 and Ross and Kimber.46 There are few other published data on thin-walled truncated conical shells and most studies have vital information missing.

3.3

Buckling of orthotropic cylinders and cones

If a circular cylinder or cone is made from glass-reinforced plastic (GRP) or carbon fibre reinforced plastic (CFRP), it may be convenient for it to have orthotropic material properties, where the axes of orthotropy of the material lie in the axial and circumferential directions. This arrangement is particularly suitable for orthotropic circular cylinders and cones under uniform external pressure, as their buckling resistance depends to a large extent on the material properties of the vessel in its circumferential direction.


118

Pressure vessels 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1/l

1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Pcr/Pexp

3.10 Design chart for shell instability of machined truncated conical shells.

3.3.1 Orthotropic element The elemental stiffness and geometrical stiffness matrices55 can be obtained explicitly for an orthotropic circular cylinder under uniform external pressure, as follows.



119

The vector of strains in this instance was a modified form of Novozhilov,24 as shown by equation [3.20]:

ε x = ux εφ =

γ xφ =

1 w νφ − R R

1 uφ + ν x R

χ x = wxx wφφ ν φ + R2 R2 1 1 = 2 ⎛⎜ wxφ + ν x ⎞⎟ ⎝R R ⎠

χφ = χ xφ

[3.20]

where u, v, w = displacements in the x, ϕ and z directions, respectively (Fig. 3.11); R = mean shell radius; εx, εϕ = direct strains in the x and ϕ directions, respectively; γxϕ = shear strain in the x–ϕ plane; χx, χϕ = flexural strains in the x and ϕ directions, respectively; χxϕ = twist in the x–ϕ plane. The assumed displacement functions incorporated a sinusoidal variation in the circumferential direction to simplify computation, as follows: u = [ui (1 − ξ ) + u j ξ ] cos nφ

ν = [ν i (1 − ξ ) + ν j ξ ] sin nφ

w = [(1 − 3ξ 2 + 2ξ 3 ) wi + l (ξ 2 − 2ξ 2 + ξ 3 )θ i

+ ( 3ξ 2 − 2ξ 3 ) w j + l ( −ξ 2 + ξ 3 )θ j ] cos nφ

[3.21]

where ξ = x/l; θ = rotational displacement; and n = number of circumferential waves or lobes. The matrix of elastic constants allowed for orthotropic properties was as follows: ⎡E F 1 ⎢ F E1 ⎢ ⎢0 0 [ D] = ⎢ ⎢0 0 ⎢0 0 ⎢ ⎣0 0

0 0

0 0 0

A 0 B C1 0 C B1 0 0 0 f

2R

i

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ D⎦

0 0 0

u

v

w x

[3.22]

j

3.11 Circular cylindrical element.


f

w

120

Pressure vessels

where E = Ext/(1 − νxνϕ) F1 = νxE F = νϕE1 = F1 (assumed) A = Gt B = Ext3/[12(1 − νxνϕ)] C = νxB = C1 (assumed) B1 = Eϕt3/[12(1 − νxνϕ)] D = Gt3/12 G = rigidity modulus Ex = elastic modulus in x direction Eϕ = elastic modulus in ϕ direction νx = Poisson’s ratio owing to stress in x direction νϕ = Poisson’s ratio owing to stress in ϕ direction t = shell thickness Substituting equation [3.21] into equation [3.20] and combining with equation [3.22], the elemental stiffness matrix for an orthotropic cylinder is given by ⎡ K11 ⎢K 21 [K ] = ⎢ ⎢ ↓ ⎢ ⎣ K81

K12 K18 ⎤ K 22 K 28 ⎥ ⎥ ↓ ↓ ⎥ ⎥ K82 K88 ⎦

where K11 = E/l2 + An2/(3R2) K12 = −F1n/(2Rl) + An/(2Rl) K13 = F1/(2Rl) K14 = F1/(12R) K15 = −E/l2 + An2/(6R2) K16 = F1n/(2Rl) − An/(2Rl) K17 = F1/(2Rl) K18 = −F1/(12R) K22 = E1n2/(3R2) + A/l2 + 4D/(R2l2) K23 = −7E1n/(20R2) − 4Dn/(R2l2) K24 = −E1nl/(20R2) K25 = Fn/(2Rl) + An/(2Rl) K26 = E1n2/(6R2) − A/l2 − 4D/(R2l2) K27 = −3E1n/(20R2) + 4Dn/(R2l2) K28 = Enl/(30R2) K33 = 13E1/(35R2) + 12B/l4 − 6(C1+ C)ζ/(5R2l2) + 13B1ζ2/(35R4) + 24 Dn2/(5R2l2)


[3.23]


121

K34 = 11E1l/(210R2) + 6B/l3 − C1ζ/(10R2l) − 11Cζ/(10R2l) + 11B1ζ 2l/(210R4) + 2Dn2/(5R2l) K35 = −F/(2Rl) K36 = 3E1n/(20R2) + 4Dn/(R2l 2) K37 = 9E1/(70R2) − 12B/l4 + 6C1ζ/(5R2l 2) + 6Cζ/(5R2l 2) + 9B1ζ 2l/(70R4) − 24Dn2/(5R2l 2) K38 = −13E1/(420R2) + 6B/l 3 − C1ζ/(10R2l) − Cζ/(10R2l) − 13B1ζ 2l/(420R4) + 2Dn2/(5R2l) K44 = E1l 2/(105R2) + 4B/l 2 −2(C1+ C)ζ/(15R2) + B1l 2ζ 2/(105R4) + 8Dn2/(15R2) K45 = −F/(12R) K46 = −E1nl/(30R2) K47 = −13E1l/(420R2) − 6B/l 3 + C1ζ/(10R2l) + Cζ/(10R2l) + 13B1ζ 2l/(420R4) − 2Dn2/(5R2l) K48 = −E1l 2/(140R2) + 2B/l 2 + C1ζ/(30R2) + Cζ/(30R2) − B1l 2ζ 2l/(140R4) − 12Dn2/(15R2) K55 = E1/l 2 + An2/3R2 K56 = F1n/(2Rl) − An/(2Rl) K57 = −F1/(2Rl) K58 = F1/(12R) K66 = E1n2/(3R2) + 4D/(R2l 2) + A/l 2 K67 = −7E1n/(20R2) − 4Dn/(R2l 2) K68 = E1nl/(20R2) K77 = 13E1/(35R2) + 12B/l4 − 6(C1+ C)ζ/(5R2l 2) + 13B1ζ 2/(35R4) + 24Dn2/(5R2l 2) K78 = −11E1l/(210R2) − 6B/l3 + C1ζ/(10R2l) + 11Cζ/(10R2l) − 11B1ζ 2l/(210R4) − 4Dn2/(10R2l) K88 = E1l 2/(105R2) + 4B/l 2 − 2(C1 + C)ζ/(15R2) + B1ζ 2l 2/(105R4) + 8Dn2/(15R2) ζ = 1 − n2 To allow for geometrical nonlinearity, the additional strains resulting from large displacements were assumed to be as follows: in the x direction:

δε x =

wx2 2

in the circumferential direction:

δε φ =

1 ⎡ ν wφ ⎤ + 2 ⎢⎣ R R ⎥⎦

2

[3.24]

in the x–ϕ plane:

δγ xφ =

2wx wφ R


122

Pressure vessels

The matrix of stresses, just before buckling, was assumed to be: ⎡σ x ⎢0 ⎢ ⎣⎢ 0

0 σφ 0

0 ⎤ 0 ⎥ ⎥ τ xφ ⎥⎦

[3.25]

where σx = pR/(2t); σϕ = pR/t; τxϕ = 0; and p = uniform external pressure. From equations [3.21], [3.24] and [3.25], the geometrical stiffness matrix for the cylindrical shell element was found to be: ⎡ KG11 ⎢K G 21 [ KG ] = ⎢ ⎢ ↓ ⎢ ⎣ KG 81

KG12 KG18 ⎤ KG 22 KG 28 ⎥ ⎥ ↓ ↓ ⎥ ⎥ KG 82 KG 88 ⎦

[3.26]

where KG22 = σϕ / (3R2) KG23 = −7σϕ n / (20R2) KG24 = σϕ / (6R2) KG27 = σϕ nl / (30R2) KG28 = σϕ nl / (30R2) KG33 = 6σx / (5l 2) + 13 σϕ n2 / (35R2) KG34 = σx / (10l) + 11 σϕ n2l / (210R2) KG36 = −3σϕ n / (20R2) KG37 = −6σx / (5l 2) + 9σϕn2 / (70R2) KG38 = σx / (10l) – 13 σϕ n2l / (420R2) KG44 = 2σx / 15 + σϕ n2l 2 / (105 R2) KG46 = −σϕ nl / (30R2) KG47 = −σx / (10l) + 13 σϕ n2l / (420R2) KG48 = −σx / 30 – σϕ n2l 2 / (140R2) KG66 = σϕ / (3R2) KG67 = −7σϕ n / (20R2) KG68 = σϕ nl / (20R2) KG77 = 6σx / (5l 2) + 13 σϕ n2 / (35R2) KG78 = −σx / (10l) −11 σϕ n2l / (210R2) KG88 = 2σx / 15 + σϕ n2l 2 / (105R2) To test the theory used here, comparison is made in Table 3.8 of the experimental buckling pressure of the isotropic model No. 7 of Table 3.4 with the FEM solution and the David Taylor Model Basin (DTMB) formula of equation [3.4]. The experimental strain measurements of model No. 7 revealed that, at 95.1% of the collapse pressure, the maximum observed



123

Table 3.8 Buckling pressures (psi) and number of lobes (in parentheses) for model No. 7

λ

DTMB formula (simply supported)

FEM, top fixed and bottom clamped

FEM, top and bottom fixed

Experimental

1.424

427

533 (6)

596 (6)

568 (5)

Table 3.9 Buckling pressures (Pa) for orthotropic conical shells (Pcr / Ex) × 106 Model number

R2

l

Eϕ × 109

νϕx

Singer56

FEM

1 2 3 4 5

0.15 0.15 0.2 0.25 0.5

0.1 0.1 0.2 0.3 0.8

5 3.86 2.59 3.86 2.59

0.022 0.09 0.234 0.09 0.234

0.0409 0.2041 0.3048 0.0419 0.0307

0.0415 (16) 0.2426 (14) 0.466 (10) 0.0463 (12) 0.0467 (10)

circumferential stress was only about two-thirds of the yield stress, thereby indicating that the model failed elastically. Furthermore, as the model was firmly clamped around its edges, it is not surprising that the clamped- and fixed-edges solutions agreed best with the experimental buckling pressure.

3.3.2 Buckling of orthotropic conical shells For the buckling of thin-walled orthtropic conical shells under uniform external pressure, the element described in Section 3.2 can be adapted if the matrix of material constants given in equation [3.22] is used instead of the matrix of material constants for the isotropic case given in equation [3.12]. Comparison between the experiments of Singer56 and the FEM of this section is made in Table 3.9. Singer’s model No. 1 was constructed from plywood and his models Nos. 2–5 were made from fiberglass-reinforced epoxy. As Singer presented his results with the aid of a dimensionless parameter, it was necessary to make certain numerical assumptions for the FEM solution. In all instances the radius of the smaller end, R1 (Fig. 3.12), was assumed to be 0.1 m and Ex was assumed to be 1 × 1011 Pa. The wall thickness for


124

Pressure vessels

l x 30°

2R1

2R2

3.12 Truncated conical shell (Singer56).

model No. 5 was assumed to be 3.48 × 10−4 m and the wall thickness for all the other models was assumed to be 3.47 × 10−4 m. Where applicable, the units in Table 3.9 are in metres and Newtons, and the figures in parentheses represent the predicted number of circumferential waves or lobes (n) into which the vessels buckle. From Table 3.9, it can be seen that comparison of the theory of Singer and the FEM solution is reasonable for three of the five models. The slightly higher buckling pressures of the FEM solution, compared with those predicted by Singer, were probably because Singer assumed simply supported ends, but the FEM solution assumed that the smaller diameter was fixed and the larger diameter was simply supported.

3.4

Buckling of thin-walled domes

The dome ends of a submarine can take various forms, from oblate hemiellipsoids to prolate hemi-ellipsoids, and from hemispheres to torispherical caps. In this section, the elastic instability of hemi-ellipsoidal and hemispherical shells is considered.



125

3.4.1 Varying meridional curvature element If either a prolate hemi-ellipsoidal shell or a hemispherical shell is subjected to uniform external pressure, it can buckle in a lobar manner in its flank, as shown by Fig. 1.6. Similarly, if an oblate hemi-ellipsoidal shell is subjected to uniform external pressure, it can buckle axisymmetrically, in its nose, as shown by Fig. 1.7. The following theories,57 which are based on the FEM, are for a constant meridional curvature element (CMC) and also for a varying curvature elements (VMC). Novozhilov24 gives the vector of strains for a doubly curved axisymmetric element, as follows: ⎧ε s ⎫ ⎪ε ⎪ ⎪ φ ⎪ ⎪ ε sφ ⎪ {ε } = ⎨ ⎬ ⎪K s ⎪ ⎪Kφ ⎪ ⎪ ⎪ ⎩ K sφ ⎭

[3.27]

where 1 ∂u w + R1α ∂ξ R1

εs =

1 ⎛ ∂v ⎞ ε φ = ⎜ + u sin β + w cos β ⎟ ⎝ ⎠ r ∂φ 1 ⎛ r ∂v ∂u ⎞ − v sin β + ⎟ ε sφ = ⎜ r ⎝ R1α ∂ξ ∂φ ⎠ Ks =

1 ∂u −1 ∂ 2 w ∂2 β − u + 2 2 2 2 2 R1 α ∂ξ ∂s R1 α ∂ξ

[3.28]

The term u(∂2β/∂ξ 2) is zero for the constant curvature case: 1 ⎡ ∂ 2 w cos β ∂v 1 ⎛ 1 ∂w ⎤ ⎞ Kφ = ⎢ − + − u⎟ sin β ⎥ ⎠ r ⎣ r∂ φ 2 r ∂φ R1 ⎜⎝ α ∂ξ ⎦ 1 ∂u ⎞ 2 ⎛ −1 ∂ 2 w sin β ∂w cos β ∂v sin β cos β + + − v+ K sφ = ⎜ r ∂φ R1α ∂ξ r R1 ∂φ ⎟⎠ r ⎝ R1α ∂ξ ∂φ The assumed displacement functions were: u=

(1 − ξ ) 2

ui cos nφ +

(1 + ξ ) 2

u j cos nφ


126

Pressure vessels

(1 + ξ ) vi sin nφ + v j sin nφ 2 2 (ξ 3 − 3ξ + 2) w cos nφ + (1 + ξ ) (1 − ξ )2 R αθ cos nφ w= i i 1 4 4 ( −ξ 3 + 3ξ + 2) w cos nφ − (1 − ξ ) (1 + ξ )2 R αθ cos nφ + j 1 j 4 4 v=

(1 − ξ )

[3.29]

where

ξ=

s Rα

or, in matrix form, ⎧u ⎫ ⎪ ⎪ ⎨ v ⎬ = [ N ]{U i } ⎪w ⎪ ⎩ ⎭ where {Ui} = a matrix of nodal displacements [N] = a matrix of ‘shape functions’ = [N′] * (cos nϕ) or (sin nϕ) These displacement functions, which are with respect to the meridian of the shell, assume a linear variation for u and v, and a parabolic variation for w. The assumptions for the displacement functions in the circumferential direction allow for a variation in {U} to cater for the lobes, and also to simplify computation. Substitution of the displacement functions and their derivatives into the strain relationships yields the following expressions for strain: ⎡ −1 (ξ 3 − 3ξ + 2) (1 + ξ ) (1 − ξ )2 α 1 0 εs = ⎢ 4 R1 4 2 R1α ⎣ 2 R1α 2 3 ( −ξ + 3ξ + 2) − (1 − ξ ) (1 + ξ ) α ⎤ cos nφ U { i} ⎥ 4 R1 4 ⎦

0

(1 + ξ ) (1 − ξ 2) R1α 1 ⎡1− ξ n (1 − ξ ) (ξ 3 − 3ξ + 2) εφ = ⎢ cos β sin β cos β 2 2 r⎣ 2 4 n (1 + ξ ) ( −ξ 3 + 3ξ + 2) (1 + ξ ) sin β cos β 2 2 4 2 ⎤ − (1 − ξ ) (1 + ξ ) R1α cos β ⎥ cos nφ {U i } 4 ⎦


Shell instability of pressure vessels 1 ⎡ − n (1 − ξ ) ⎛ − (1 − ξ ) sin β r ⎞ ε sφ = ⎢ − ⎜⎝ ⎟ 0 0 r⎣ 2 2 2 R1α ⎠ r ⎞ ⎛ − (1 + ξ ) sin β ⎤ + ⎜⎝ ⎟⎠ 0 0 ⎥ sin nφ {U i } 2 2 R1α ⎦ ⎡ −1 Ks = ⎢ 2 ⎣ 2 R1 α

0

−3ξ 2 R12α 2

− ( −1 + 3ξ ) 0 2 R1α

1 2 R12α

127

− n (1 + ξ ) 2

3ξ 2 R12α 2

− (1 + 3ξ ) ⎤ cos nφ {U i } 2 R1α ⎥⎦

1 ⎡ − (1 − ξ ) sin β − n (1 − ξ ) cos β Kφ = − ⎢ r⎣ 2 R1 2r 2 3 2 ⎛ − n (ξ − 3ξ + 2) ( 3ξ − 3) sin β ⎞ + ⎜⎝ ⎟⎠ 4r 4R α 1

⎛ − n (1 + ξ ) (1 − ξ ) R1α (1 − 2ξ + 3ξ 2 ) sin β ⎞ + ⎜⎝ ⎟⎠ 4 4r 2

− (1 + ξ ) sin β 2 R1

2

n (1 + ξ ) cos β 2r

⎛ − n2 ( −ξ 3 + 3ξ + 2) 3 ( −ξ 2 + 1) sin β ⎞ + ⎜⎝ ⎟⎠ 4r 4 R1α ⎛ n 2 ( 1 − ξ ) (1 + ξ )2 (1 − 2ξ − 3ξ 2 ) sin β ⎞ ⎤ cos nφ U R − α { i} 1 ⎜⎝ ⎟⎠ ⎥ 4r 4 ⎦ [3.30] 1 ⎡ − n (1 − ξ ) ⎛ sin β cos β (1 − ξ ) cos β ⎞ K sφ = ⎢ − ⎜⎝ − ⎟ r ⎣ R1 r R1α ⎠ ⎛ − n sin β (ξ 3 − 3ξ + 2) n ( 3ξ 2 − 3) ⎞ + ⎜⎝ r 2 2 R α ⎟⎠ 1

⎛ − n sin β (1 + ξ ) (1 − ξ ) R1α n ( −1 − 2ξ + 3ξ 2 ) ⎞ + ⎜⎝ ⎟⎠ 2 2r 2

− n (1 + ξ ) ⎛ − sin β cos β (1 + ξ ) cos β ⎞ + ⎜⎝ ⎟ R1 r R1α ⎠

⎛ − n sin β ( −ξ 3 + 3ξ + 2) 3n ( −ξ 2 + 1) ⎞ + ⎜⎝ 2r 2 R1ξ ⎟⎠

⎛ n sin β (1 − ξ ) (1 + ξ )2 R1α n (1 − 2ξ − 3ξ 2 ) ⎞ ⎤ − ⎜⎝ ⎟⎠ ⎥ sin nφ {U i } 2r 2 ⎦ In matrix form, these appear as: T {ξ} = [ξs ξφ ξsφ Ks Kφ Ksφ ] = [ B] {U i }


128

Pressure vessels

The stiffness matrix is given by: 1

2n

[k ] = ∫∫ [ B]T [ D][ B] dx dy = ∫−1 ∫0 [ B]T [ D][ B] r dφ R1α dξ = πR1α ∫ r [ B1 ] [ D][ B1 ] dξ 1

T

−1

[3.31]

where [B1] = [B]/(either cos nϕ or sin nϕ); and [D] = matrix of elastic constants (equation [3.12]) In global coordinates,

[K 0 ] = [ DC ]T [k ][ DC ] where ⎡ζ 1 ⎣04

[ DC ] = ⎢

04 ⎤ ζ 2 ⎥⎦

⎡ c1 ⎢ 0 [ζ 1 ] = ⎢ ⎢ − s1 ⎢ ⎣ 0

0 s1 1 0 0 c1 0 0

0⎤ 0⎥ ⎥ 0⎥ ⎥ 1⎦

[3.33]

⎡ c2 ⎢ 0 [ζ 2 ] = ⎢ ⎢ − s2 ⎢ ⎣ 0

0 s2 1 0 0 c2 0 0

0⎤ 0⎥ ⎥ 0⎥ ⎥ 1⎦

[3.34]

[3.32]

and c1 = cos β1; c2 = cos β 2; s1 = sin β1; s2 = sin β 2. The wall thickness was assumed to vary linearly along the meridian of the shell, as follows: t=

(1 − ξ ) 2

t1 +

(1 + ξ ) 2

t2

where t1 = wall thickness at node 1; and t2 = wall thickness at node 2. To obtain the geometrical stiffness for this element, it was necessary to consider the additional strains resulting from large deflections,40 as follows:



129

⎧ 1 ⎛ ∂w u ⎞ 2 ⎫ − ⎟ ⎪ ⎜⎝ ⎪ 2 ∂s R1 ⎠ ⎪ ⎪ ⎧δε s ⎫ ⎪ 2 ⎪⎪ ⎞⎤ ⎪ ⎪ 1 ⎡ 1 ⎛ ∂w ⎪ {ε L } = ⎨δε φ ⎬ = ⎨ ⎢ ⎜ − ν cos β ⎟ ⎥ ⎬ ⎠⎦ ⎪δε ⎪ ⎪ 2 ⎣ r ⎝ ∂φ ⎪ ⎩ sφ ⎭ ⎪ ⎞ ⎤⎪ ⎛ ∂w u ⎞ ⎡ 1 ⎛ ∂w − ⎟⎢ ⎜ − ν cos β ⎟ ⎥ ⎪ ⎪⎜ ⎠ ⎦ ⎪⎭ ⎪⎩⎝ ∂s R1 ⎠ ⎣ r ⎝ ∂φ

[3.35]

Now the additional strain energy resulting from large deflections is given by: 1 2

{U i }T [G]T [G]{U i }

so that the geometrical stiffness matrix was given by 1

T

2π

[k1 ] = ∫ t [G] [σ ][G] R1α dξrdφ = R1α ∫−1 ∫0 t [G]T [σ ][G] rdξdφ

[3.36]

= πR1α ∫ t [G1 ] [σ ][G1 ] rdξ 1

T

−1

where [G ] = [G]/(either cos nϕ or sin nϕ as given in Table 3.10) 1

⎡σ s ⎣0

[σ ] = ⎢

0⎤ σ φ ⎥⎦

[3.37]

For hemi-ellipsoidal thin-walled domes, the prebuckling membrane stresses58 were approximated by:

σs = σφ =

{

− pa2

t 2 ( a2 cos2 β + b2 sin 2 β )

1/ 2

}

− pa2 {b2 − ( a2 − b2 ) cos2 β }

{

t 2b2 ( a2 cos2 β + b2 sin 2 β )

1/ 2

}

[3.38]

Table 3.10 [G1] for a doubly curved axisymmetric element

(3ξ2 − 3)

− (1 − ξ ) 2R1

0

0

− (1 − ξ ) cos β 2r

− (1 + ξ ) 2R1

0

0

− (1 + ξ ) cos β 2r

4R1α

−n (ξ − 3ξ + 2) 4r (−3ξ2 + 3) 4R1α 2

−n ( −ξ 3 + 3ξ + 2) 4r


(1 − ξ )2 − 2 (1 + ξ ) (1 − ξ ) 4 −n (1 + ξ ) (1 − ξ ) R1α 4r (1 + ξ )2 − 2 (1 − ξ ) (1 + ξ ) 4 2

−n (1 − ξ ) (1 + ξ ) R1α 4r 2

130

Pressure vessels

where p = uniform pressure (external positive); a = radius of hemiellipsoidal dome at base; and b = height of hemi-ellipsoidal dome. Integration in the circumferential direction was carried out explicitly, and in the meridional direction integration was carried out numerically, using four Gauss points per element. A computer program for determining the elastic instability pressures for thin-walled domes under uniform external pressure is available.9

3.4.2 Thin-walled domes To compare theory with experiment, 45 thin-walled domes were tested to destruction under uniform external pressure. The domes (Fig. 3.13) consisted of ten different profiles, varying from oblate hemi-ellipsoids of aspect ratio (AR) 0.25 to prolate hemi-ellipsoids of AR 4.0, where AR = dome height/base radius. The domes were constructed in solid urethane plastic (SUP), and their base diameters and wall thicknesses were 0.2 m and 2 mm, respectively. Each series of domes was constructed very precisely by thermosetting the liquid plastic between two carefully machined aluminium alloy male and female moulds, the out-of-roundness of some of the prolate domes being only about ±0.04 mm. By experiment, the material properties of SUP were found to be: E = 2.9 GPa; ρ = 1200 kg/m3; and ν = 0.3 (assumed). Table 3.9 shows a comparison between experiment and the FEM of this section, where the figures in parentheses represent the predicted number of circumferential waves or lobes into which the vessels buckle.

3.13 Ten hemi-ellipsoidal oblate/prolate domes.



131

Table 3.11 Buckling pressures for SUP domes Experimental buckling pressure Pcr (MPa) Aspect ratio

SUP 1

SUP 2

SUP 3

SUP 4

SUP 5

Mean

CMC

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.7 0.444 0.25

0.086 0.099 0.145 0.179 0.358 0.517 1.137 – – –

0.103 0.103 0.151 0.199 0.386 0.475 1.365 1.075 0.282 0.062

0.089 0.103 0.158 0.220 0.358 0.461 1.337 1.062 0.282 0.062

0.103 0.103 0.145 0.206 0.358 0.461 1.324 – 0.275 0.055

0.103 0.089 0.151 0.199 0.358 0.537 1.248 – 0.262 0.062

0.097 0.100 0.150 0.201 0.364 0.490 1.280 1.070 0.276 0.060

0.112 0.136 0.167 0.225 0.323 0.552 1.412 0.851 0.317 0.108

9 5 2 4 1 5

4 4 7 9 0 6

7 1

6 4 5 6 4 9

7 1

4 4 5 8 4 9

8 2

4 6 7 9 4 7

0 1

(5) (5) (6) (7) (8) (9) (10) (0) (0) (0)

CMC, constant meridional curvature.

From Table 3.11, it can be seen that comparison between experiment and theory is reasonable for these domes, and also that the hemispherical dome had the largest resistance to buckling. The very oblate and prolate domes had abysmal resistance to buckling, their buckling pressures being less than one-tenth of the buckling pressures of the hemispherical dome. The experimentally obtained buckling pressures were, in general, lower than their theoretical predictions, and this was attributed to geometrical imperfections in the vessels, the hemispherical and prolate hemi-ellipsoids being sensitive to out-of-circularity in their flanks, and the oblate hemiellipsoids being sensitive to the loss of meridional curvature in their noses. In addition to the tests carried out on the domes of Table 3.11, two more domes from this series were tested to destruction, in much more carefully observed experiments. These domes have already been described in Section 2.4 and their out-of-circularity plots are shown in Figs 2.28 and 2.29. The domes were expected to buckle in a lobar manner, and for this reason 20 strain gauges were attached to the inner surfaces of these vessels in the circumferential direction, in their flanks, at the position where it was expected that the maximum buckling deflection would take place. The strain gauge readings were taken at several pressures, and at small increments of pressure especially near the anticipated buckling pressures. Figures 3.14 and 3.15 show circumferential strain plots superimposed over the out-of-circularity plots, at pressures just below the experimentally obtained buckling pressures. These figures appear to show that the vessels buckled elastically, and the buckling pressures together with the number of


132

Pressure vessels 10 9

11 12

8

13

7

14

6 Specimen 1·5 Mag Filter x100 N Talyrond

15

5

5''arm 16

4 Ra D r nk ste Tayl or Hobson Leice 2000 3000

17

18

3

2 19 20

1

Gauge no. 1 Division=0.001''

3.14 Microstrain readings recorded at a pressure of 0.483 MPa for the 1.5 aspect ratio dome.

lobes (n) are shown in Table 3.12. For further details of the VMC element, see Section 3.4.3. Each model buckled with the appearance of a solitary lobe in its flank, and details of these lobes are shown in Fig. 3.16. On removal of the pressure, the models appeared to regain their original shapes.

3.4.3 VMC element For the VMC (Fig. 3.17), the angle β varies along the meridian, so that in equation [3.28]:



133

7 8

6

9

5

10

4

11

3

Specimen 3·0 Cone Filter Mag x50 N Talyrond

12

2

10''Arm 13

1 Ra

nk

Tay lor

r D ste H o b s o n L ei c e

14

20

1000 2000 15

3000 16

19 18

17 Gauge no. 1 Division=0·002''

3.15 Microstrain readings recorded at a pressure of 0.165 MPa for the 3.0 AR dome.

Table 3.12 Buckling pressures (MPa) and number of lobes n AR

Experimental

CMC

VMC

1.5 3.0

0.496 (10) 0.172 (6, 7)

0.552 (9) 0.167 (6)

0.548 (9) 0.167 (6)


134

Pressure vessels

(a)

(b)

3.16 Position and size of the buckled regions: (a) shell AR 1.5, meridional length 107 mm, circumferential length 83 mm, distance of top of lobe from base 27 mm; and (b) shell AR 3.0, meridional length 115 mm, circumferential length 75 mm, distance of top of lobe from base 50 mm.

A w

x=1

u

q

j

b

bj

v w

x = −1 i bi

r

Rs

Axis of symmetry

f

Rj

Part section on A–A

A

3.17 VMC element.

∂ 2 β / ∂s 2 ≠ 0 Thus, it is a simple matter to extend the solution of Section 3.4.1, by assuming that

β = a0 + a1 s + a2 s 2

[3.39]

Cook27 shows that the three constants in equation [3.39], namely a0, a1, a2, can be determined from the following three conditions:



135

β = β1 at s = 0 β = β2

at s = L

and

∫

L

0

L

sin ( β − βc ) ds ∫ ( β − βc ) ds = 0 0

so that a0 = β1

a1 = ( 6βc − 4β1 − 2β 2 ) a2 =

( 3β1 + 3β 2 − 6βc ) L2

[3.40]

where 1/R1 = −∂β/∂s; L = arc length of element; and βc = chord angle. Cook shows that: l ⎧(β1 − βc ) ⎫ ⎡ 4 −1⎤ ⎧(β1 − βc ) ⎫ ⎨ ⎨ ⎬ ⎬ 60 ⎩(β 2 − βc )⎭ ⎢⎣ −1 4 ⎥⎦ ⎩(β 2 − βc )⎭ T

Ll+

[3.41]

and s

r = R1 + ∫ sin β ds

[3.42]

β β c + dy / dx

[3.43]

⎧( β1 − βc ) ⎫ y = [ x (1 − x / l ) ( x 2 / l ) ( −1 + x / l )] ⎨ ⎬ ⎩( β 2 − βc )⎭

[3.44]

0

and

where

and l = chord length. For hemi-ellipsoids, it is a simple matter to carry out the necessary integrations, and for axisymmetric elements of more complex shape, integration can be carried out numerically with the aid of an additional mid-side node. The mid-side node need only be used to carry out the integrations and need not be used to determine the stiffness matrices. However, for a more comprehensive element, it may be advisable to include a mid-side node to determine the elemental stiffness matrix. Comparison is made between theory (VMC) and experiment in Table 3.13 for two SUP models, whose base diameters and wall thicknesses were 0.5 m and 4 mm, respectively. These models were constructed in a similar manner to those described in Section 3.4.2. The out-of-roundness of model L1 was 1.42 mm and that of model L2 was 1.51 mm. From Table 3.13, it can be seen that comparison between experiment and theory was reasonable, but that model L1 buckled at a pressure a little higher than model L2. This may be partly because of the larger


136

Pressure vessels Table 3.13 Buckling pressures and number of lobes n (in parentheses) of models L1 and L2 (MPa)

Model L1 Model L2

FEM

Experimental

0.202 (8) 0.202 (8)

0.214 (8) 0.203 (–)

3.18 Model L2, with rupture.

out-of-roundness of model L2 and partly because model L1 had 20 equally spaced resistance electrical strain gauges fitted in a circumferential direction around its flank to its inner surface, whereas model L2 had no strain gauges attached to it at all, i.e. the strain gauges on model L1 may have strengthened it a little in the circumferential direction. The important point about both models L1 and L2 is that they appeared to buckle elastically, with the appearance of a solitary lobe in their flanks, and that on removal of the pressure the models appeared to regain their original shapes. Further increases in pressure caused the vessels to buckle in the same positions, but at progressively lower pressures; the models eventually ruptured, as shown in Fig. 3.18.



137

3.4.4 Near hemispherical domes The investigation described in Section 3.4.2 shows that a hemispherical dome buckles in a lobar manner, in its flank, as shown in Fig. 3.19. However, Galletly et al.59 state that if there is flatness in the nose of a hemispherical dome, it can buckle axisymmetrically; in any case, most theories assume that a hemispherical dome buckles axisymmetrically. From Table 3.11, it can be seen that the hemispherical dome buckles asymmetrically (n > 1), and that the dome of AR = 0.7 buckles axisymmetrically (n = 0). To determine the transition point between axisymmetric buckling and lobar buckling, a theoretical investigation is now carried out on oblate hemi-ellipsoidal domes, whose AR varies between 0.7 and 1.0. These domes are assumed to be the same series as those referred to in Table 3.11 , and their theoretical buckling pressures Pcr, together with their associated values of the number of circumferential waves n, as predicted by the VMC element, are shown in Table 3.14. From Table 3.14, it can be seen that the transition point between lobar buckling (n > 1) and axisymmetric buckling (n = 0) is for a dome with an AR somewhere between 0.85 and 0.87. Table 3.14 also shows that when the AR is reduced from 1.0 to 0.95, the value of n decreases from 11 to 3.

3.19 Lobar buckling of a hemispherical dome.

Table 3.14 Buckling pressures (MPa) and number of lobes n (in parentheses) for oblate domes Aspect ratio

Pcr (n)

0.8

0.85

0.87

0.9

0.95

1.0

1.021 (0)

1.147 (0)

1.194 (2)

1.265 (2)

1.378 (3)

1.427 (11)


138

Pressure vessels

Inspection of the buckling eigenmodes for AR 0.87–0.95 showed that these were of the complex forms shown in Fig. 5.20. These findings appear to agree with the observations of Galletly et al.59 (see their Figure 12).

3.5

Boundary conditions

Boundary conditions play an important role in the experimentally obtained buckling pressures for many vessels, fixed boundaries usually resulting in a higher buckling pressure than simply supported edges. However, the effect of rotational restraint at a boundary for a thin shell is small, as thin shells have little resistance to bending. In this context, axial restraint at the boundary plays a much more significant role in increasing buckling pressures than does rotational restraint, i.e. a clamped boundary only yields a slightly higher buckling pressure than a simply supported one. Boundary conditions can be classified as follows: Simply supported: w = 0; u ≠ 0; v ≠ 0; θ ≠ 0 Clamped: u ≠ 0; v = w = θ = 0 Fixed: u = v = w = θ = 0 To determine the effects of stiffener size on shell instability, Ross and Johns60 tested to destruction three machine-stiffened circular cylinders under uniform external pressure. The three vessels had similar geometrical properties, the main difference being in the size of the ring-stiffeners, as shown in Table 3.15. The mechanical properties of TVR1 and TVR2 were found to be similar, but TVR3 proved to be slightly different. For TVR1 and TVR2: Young’s modulus = 29.3 × 106 lbf in−2; yield stress = 38 300 lbf in−2; and nominal peak stress = 79 300 lbf in−2. For TVR3: Young’s modulus = 29.3 × 106 lbf in−2; yield stress = 43 820 lbf in−2; and nominal peak stress = 84 690 lbf in−2. For all models, it is assumed v = 0.3. The models were very precisely made, as shown by the out-of-circularity plots for TVR2 in Fig. 3.20. They were tested in the tank shown in Fig. 3.21 and all three models buckled in a lobar manner, as shown in Fig. 1.5. Comparison is made in Table 3.16 of the experimentally obtained buckling pressures, the analytical solutions of von Mises and Ross, and the FEM solution described in Section 3.2. The von Mises solution assumes simply supported edges and the Ross solution,42 which was based on the elastic theory of Kendrick,43 assumed fixed edges. One FEM solution assumed that the left end was fixed and the right end was clamped, whereas the other one assumed that both edges of the shell were fixed at the ring-stiffeners. Sixteen equal length elements were used to model the shell for both FEM solutions. From Table 3.16 it can be seen that the buckling pressures of these vessels were larger than the simply supported case, and also that the experimental



139

Table 3.15 Dimensions of the models (inches)

4.000

0.75

0.675

L1

b1

d1

h1

b2

L2

L3 d2

h2

h3

Model Number

L1

L2

L3

TVR1

1.5

4.0

1.5

0.725 0.725

0.750

0.750 0.025 0.0251

0.025

TVR2

1.5

4.0

1.5

0.725 0.725

0.375

0.375 0.025 0.0250

0.025

TVR3

1.5

4.0

1.5

0.725 0.725 0.0625 0.0625 0.025 0.0250

0.025

d1

d2

b1

b2

h1

h2

h3

buckling pressures of the models increased in proportion to the sizes of the stiffening rings. It was also interesting to note from Table 3.16 that the relatively simple analytical fixed-edges solution compared favorably with the much more complex FEM solution.

3.5.1 End closures To test the effect of various types of end closures on thin-walled cylinders, Galletly and Aylward61 carried out a thorough experimental and theoretical investigation on several vessels. They sealed the ends of their machined circular cylinders with the following types of end closure: (a) (b) (c) (d) (e) (f)

conical (α = 75°); conical (α = 45°); torispherical; oblate ellipsoidal; toriconical (α = 45°), and hemispherical,

where α = cone angle. Galletly and Aylward found the experimental buckling pressures of these vessels to vary from 1.275 MPa for (a) to 3.86 MPa for (b). They also found that comparisons between the experimental buckling pressures and the theoretical predictions of BOSOR69 were good.


140

Pressure vessels

Specimen TVR 2 Filter Mag 4 Taylor-Hobson Talyrond

il Fa

Tra c

e e of inter nal surfac

1 Division = 0·0001''

Specimen TVR 2 Mag Filter 4 Taylor-Hobson Talyrond

il Fa

Tra c

e e of enter nal surfac

3.20 Out-of-circularity plots for TVR2 at mid-length.


Shell instability of pressure vessels 4 3 2 ins

141

16 HT bolts 5/8" BSF

1 0 Outlet to pressure gauge

'O' rings As below Model No. 3

Flexible hose from pump

1/4" BSF

'O' ring

bolts

Closure plate

3.21 Test tank with attachments.

Table 3.16 Comparison of theoretical and experimental buckling pressures (lbf in−2) (the number of lobes into which the vessels buckle are given in parentheses)

Model

λ

von Mises (simply Ross supported) (fixed)

TVR1 TVR2 TVR3

1.629 9 1.629 9 1.747 9

245 (6) 245 (6) 244 (6)

3.6

The legs of off-shore drilling rigs

345 (7) 345 (7) 343 (7)

FEM FEM (clamped/fixed) (fixed)

Experimental

305 (6) 305 (6) 303 (6)

320 (6) 316 (6) 304 (6)

343 (7) 343 (7) 340 (7)

It is possible for the legs of off-shore drilling rigs to be subjected to a uniform lateral pressure together with axial pressure of a different magnitude to the lateral pressure, because of the effects of the weight of the rig. Much work on this topic has been carried out by Galletly et al.4 and by Walker and co-workers63–65 and it is a simple matter to extend the theory given in Section 3.2 to cater for this effect. Care must, however, be taken


142

Pressure vessels

to ensure that the rig’s legs are not prone to suffer from other modes of failure, including buckling.

3.7

Some buckling formulae for domes and cones

3.7.1 Von Kármán and Tsien’s formula For the buckling of a perfect thin-walled sphere under uniform external pressure, von Kármán and Tsien66 give the following theoretical expression: Pcr =

2 Et 2 a2

[3.45]

[ 3 (1 − v 2 )]

However, experiments67 on the thin-walled spherical shells show that equation [3.45] overpredicts buckling pressures, and Roark and Young32 state that the ‘probable’ actual minimum buckling pressure is given by: Pcr = 0.365E ( t / a)

2

[3.46]

3.7.2 Spherical shell cap For a spherical shell cap (Fig. 3.22) with a half-centred angle between 20° and 60°, and where: 400 ≤ a / t ≤ 2000

t

R

q

q

0

3.22 Spherical shell cap.



143

Kloppel and Jungbluth67 give the following empirical formula: t (θ ° − 20°) ⎤ ⎡ (0.07a / t ) ⎤ ⎡ 1− Pcr = ⎢1 − 0.175 (0.3E ) ⎛⎜⎝ ⎞⎟⎠ 20° ⎥⎦ ⎢⎣ 400 ⎥⎦ a ⎣

2

[3.47]

3.7.3 Truncated conical shell For a perfect thin-walled truncated conical shell with closed ends and under uniform external pressure, Roark states that the von Mises formula for circular cylinders can be used, if the slant length of the cone is substituted for the length of the cylinder, and a is taken as: a = ( RA + RB ) / ( 2 cos a)

[3.48]

where the notation is shown in Fig. 3.23. Thus, if this approach is applied to the Windenburg and Trilling formula given in equation [3.4], the elastic buckling pressure for a perfect thinwalled conical shell is given by: 2.6 E ( t / dm ) ( cos α ) (l / dm ) − 0.45 (t / dm )1 / 2 (cos α )−0.5 5/ 2

Pcr =

3/ 2

where dm = RA + RB = mean diameter of the cone. The above formula can be used to analyse the thin-walled cone described in reference 31, whose details are: RA = 25 in, RB = 50 in, t = 1 in, l = 141.78 in, and α = 10°, so that dm = 75 in and: 2.6 × 10 × 106 (1 / 75) ( 0.9848 ) = 521.6 / 1.838 (141.78 / 75) − 0.45 (1 / 75)0.55 (0.9848 )−0.5 2.5

Pcr =

1.5

i.e. Pcr = 284 lbf in−2 which compares with Volmir’s prediction of 350 lbf in−2 and the FEM prediction of 340 lbf in−2.

u,x a RA

u0

θ w

0

w

RB

w φ

l

3.23 Truncated conical shell.


v

144

Pressure vessels

3.8

Inelastic instability

It must be emphasised that buckling formulae based on small-deflection elastic theory tend to predict higher buckling pressure than those obtained experimentally. The main reason for this is that, in practice, most pressure vessels have geometrical imperfections, and these geometrical imperfections can cause serious loss of buckling resistance, or ‘plastic knockdown’. This effect is worsened for thicker and shorter vessels and, conversely, is of less importance for long vessels of small thickness-to-radius ratios. However, sensible structural design should eliminate possible failure owing to instability because of the difficulty of predicting the effects of geometrical imperfections, if this is at all possible; but if it is not possible to eliminate failure caused by instability, the so-called safety factor against buckling should be made very large. One method of determining inelastic buckling pressures of imperfect vessels is to determine their plastic reduction factors from experimental results, in a manner similar to that described in Section 3.1.3. For the inelastic buckling of thin-walled domes under external pressure, such a method was produced by Galletly et al.,68 who used a factor ¯λ , where:

λ = 1.285

{(a / t ) (σ yp / E )}

[3.49]

Galletly et al., who compared experimental results with the famous shell buckling program BOSOR5,69 stated that if a hemispherical dome tended to be flat at its apex, then it was more likely to fail axisymmetrically, with its nose denting inwards, than through lobar buckling. Other studies on domes, including torispherical ones, were made by Galletly et al.59 and by Newland.70

3.8.1 Snap-thru buckling The investigation carried out in Section 3.4 on the collapse of hemiellipsoidal domes under uniform external pressure, considered nonsymmetric and symmetric bifurcation buckling. That investigation gave poor results for the oblate domes which may have failed through snap-thru buckling, where the curvature of the nose of the dome is reversed. In this instance, the compressive stresses increase nonlinearly with increase in external pressure, until the vessel fails in its nose through snap-thru buckling.54 For slightly less oblate domes, Galletly et al.68 and Blachut71 have shown that they can buckle in a lobar manner, where the lobe forms nonsymmetrically about the nose, but not necessarily in the flank. In this section, a theoretical and an experimental investigation is reported on seven hemi-ellipsoidal oblate domes, which were tested to destruction



145

under external hydrostatic pressure. Four of these domes were constructed in GRP and three in SUP. The domes were of AR 0.25, 0.444 and 0.7, where: AR =

dome height base radius

The nominal base radius of all the domes was 100 mm. The GRP domes were made from glass chopped strand mat, where the moulds used were some accurately made SUP domes. The SUP domes were made by injecting liquid SUP between carefully machined male and female aluminium alloy domes and then thermosetting the SUP domes. Thus the SUP domes were accurately made. The glass fibre domes were manufactured by wet laying chopped strand glass fibre mat around the external surfaces of the relevant SUP moulds. The base diameter of the SUP moulds was 0.2 m and the wall thicknesses of the SUP domes was 2 mm. The temperature of the workshop in which the GRP domes were manufactured, was kept between 15 and 20 °C. The materials were brought into the workshop at least one day before they were required, to allow the temperatures of the materials to stabilise. Two of the GRP domes, namely AR 0.25 and AR 0.444, are shown in Figs 3.24 and 3.25. The geometrical details of the GRP domes, including their values of initial out-of-roundness e were measured on a Mitutoyo co-ordinate measuring machine with a touch-trigger probe. The geometrical details of these

3.24 GRP dome (AR 0.25 and 0.444), top.

3.25 GRP dome (AR 0.25 and 0.444), bottom.


146

Pressure vessels

Table 3.17 Geometrical details of GRP domes

Model

Initial out-ofroundness e (mm)

Base diameter (mm)

Dome height (mm)

Wall thickness (mm)

AR AR AR AR

0.3144 0.2892 0.4805 0.2395

200.8 200.8 200.8 200.8

20.68 40.55 73.05 72.56

5.30 4.00 2.75 3.34

0.25 0.444 0.7A 0.7B

Securing bolts Securing annulus Rubber annulus seal Relief valve Dome Pressure inlet

3.26 Test tank with dome.

GRP domes are given in Table 3.17. Strain gauges were attached to the concave faces of the domes, as these were free of water pressure. The domes were tested under external water pressure in the test tank shown schematically in Fig. 3.26. From Fig. 3.26, it can be seen that the test rig allowed visual inspection of the concave surfaces of the domes and that as these surfaces were free of water, the strain gauges did not require water proofing. It must be emphasised that, initially, the GRP domes were porous to the effects of water, so that before testing them they needed water proofing. This was achieved by the application of several coats of marine varnish to the convex faces of the domes. Before applying any water pressure to the domes, a relief valve was left open to the atmosphere to allow any trapped air in the system to be pumped out. When the bulk of the trapped air was pumped out, the relief valve was made watertight and the experiment commenced. Each dome had six Cu/Ni linear strain gauges attached to its inner surface; three in the meridional direction and three in the circumferential direction. One pair of strain gauges was attached to the nose of each dome and one pair attached near the dome base in the flank of the vessel. The third pair of strain gauges was attached in between these two pairs. Each vessel in turn was subjected to a gradually increasing external water pressure. The pressure was applied by a hand-driven hydraulic pump. Strain gauge readings were taken at several values of external water pressure. Each



147

3.27 Snap-thru failure of AR 0.25.

vessel was tested to destruction. The vessels AR 0.25 and AR 0.444 clearly failed by axisymmetric snap-thru buckling, as shown in Figs 3.27 and 3.28. The vessel AR 0.7A, failed non-symmetrically in a lobar manner, as shown in Fig. 3.29, whereas the vessel AR 0.7B failed in the flange, as shown in Fig. 3.30. This may have been because the flange was machined flat to achieve water tightness and this may have weakened the flange in AR 0.7B. Flat tensile specimens were manufactured in a similar manner to the dome shells. These revealed the following material properties:GRP: Tensile modulus = E = 5.44 GPa; Tensile fracture stress = 141.9 MPa; and Poisson’s ratio = v = 0.3 (assumed). The volume fraction was found to be 6.37%. SUP: Tensile modulus = E = 2.9 GPa; Tensile fracture stress = 70 MPa; and Poisson’s ratio = v = 0.3 (assumed).


148

Pressure vessels

3.28 Snap-thru failure of AR 0.44.

3.29 Lobar failure of AR 0.7A.



149

3.30 Failure in flange of AR 0.7B.

Table 3.18 Buckling pressures of the hemi-ellipsoidal domes Dome

Material

E (GPa)

t (mm)

Pexp (bar)

AR AR AR AR AR AR AR

SUP SUP SUP GRP GRP GRP GRP

2.9 2.9 2.9 5.44 5.44 5.44 5.44

2.0 2.0 2.0 5.3 4.0 2.75 3.34

0.6 2.76 10.70 8.28 12.07 31.38 40.00

0.25 0.444 0.7 0.25 0.444 0.7A 0.7B

The experimentally obtained buckling pressures of the four GRP domes, together with the three SUP domes are shown in Table 3.18, where: t = wall thickness (mm) E = tensile modulus (GPa) Pexp = experimentally obtained buckling pressures (bar). The theoretical analysis was via FEM, which allowed for both material and geometrical nonlinearity, as described in Chapter 2. The finite element used for this analysis was the thin-walled truncated conical shell element described in Ross.50 Plots of pressure against the axial deflection at the nose of the vessel are given for all the vessels in Figs 3.31 to 3.37. From Figs 3.31 to 3.37, it can be seen there was some non-convergence, particularly for the AR 0.7 series. This may have been because this series of vessels did not fail axisymmetrically, but by a lobar mode, where the lobe was unsymmetrical about the nose of the vessel, as shown in Fig. 3.29. For


150

Pressure vessels 0.6

Pressure (bar)

0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

Deflection (mm)

3.31 Theoretical pressure–axial deflection plot for AR 0.25 (SUP).

4 3.5

Pressure (bar)

3 2.5 2 1.5 1 0.5 0 0

2

4

6

8

Deflection (mm)


AR 0.7B, it failed in the flange. Table 3.19 gives the theoretical buckling pressures (Pcr), as obtained from Figs 3.31 to 3.37. These buckling pressures are compared with the experimentally obtained buckling pressures (Pexp); convergence was not good for the AR 0.7 series. The flatter oblate hemiellipsoidal domes appeared to fail though a snap-thru axisymmetric buckling mode, in the noses of the vessels. The AR 0.7 series appeared to fail by a lobar mode, where the lobe was non-symmetrical about the nose of the



151

14

Pressure (bar)

12 10 8 6 4 2 0 0

2

4

6

8

10

Deflection (mm)


16 14 Pressure (bar)

12 10 8 6 4 2 0 0

2

4

6

8

10

Deflection (mm)

3.34 Theoretical pressure–axial deflection plot for AR 0.25 (GRP).

domes. The AR 0.7B model failed in the flange, near its flank. Comparisons between experiment and the nonlinear finite element theory were good.

3.9

Higher order elements for conical shells

A deficiency with the elements described in Sections 3.2 and 3.4 is that they assume a linear variation for the displacements in the meridional and


152

Pressure vessels 16 14

Pressure (bar)

12 10 8 6 4 2 0 0

2

4

6

8

10

Deflection (mm)

3.35 Theoretical pressure–axial deflection plot for AR 0.444 (GRP).

50 45

Pressure (bar)

40 35 30 25 20 15 10 5 0 0

1

2

3

4

5

Deflection (mm)

3.36 Theoretical pressure–axial deflection plot for AR 0.7A (GRP).

azimuthal directions. Rajagopalan72 suggested that the elements could be improved by using higher order polynomials in these two directions. Additionally, by using ‘mid-side’ nodes, together with Guyan reduction,73 he showed that the size of the system matrices need not be increased. In 1995, Ross74 presented two conical shell elements using Rajagopalan’s idea. One element used cubic relationships for all the u, v and w displacements and the other used quadratic relationships for u and v, and cubic relationship for w.



153

50 45

Pressure (bar)

40 35 30 25 20 15 10 5 0 0

1

3

2

4

5

Deflection (mm)

3.37 Theoretical pressure–axial deflection plot for AR 0.7B (GRP).

Table 3.19 Buckling pressures for the hemi-ellipsoidal domes (bar) Model

Material

Pexp

Pcr

AR AR AR AR AR AR AR

SUP SUP SUP GRP GRP GRP GRP

0.6 2.76 10.70 8.28 12.07 31.38 40.00

0.55 2.5 12.0 10.3 13.0 40.0 45.0

0.25 0.444 0.7 0.25 0.444 0.7A 0.7B

3.9.1 The element ALLCUBE For this element, the displacement functions for u, v and w were all assumed to be of cubic form. Now the displacement function for w is of the Hermitian form shown by equation [3.7], but the displacement functions for u and v are obtained as follows. If the nodal displacements for u are u1, u2, u3 and u4 and the nodal displacements for v are v1, v2, v3, and v4, whose positions are as shown in Fig. 3.38, then: u = α1 + α 2 x + α 3 x2 + α 4 x3 where αi = unknown constants, to be determined.


[3.50]

154

Pressure vessels l l/3

l/3 1

l/3 4

3

x, u 2

Axis

3.38 Nodal displacement positions for u and v of cubic form for conical shells.

The boundary conditions are: @ x = 0, @ x = l, @x = l/3 @ x = 2l / 3,

u = u1 u = u2 u = u3 u = u4

[3.51]

Substituting each of the above boundary conditions into equation [3.50], four simultaneous equations are obtained that, on solving, result in the following values for αi.

α 1 = u1 3 α 2 = ( −11u1 / 6 + u2 / 3 + 3u3 − 3u4 / 2 ) l 9 α 3 = 2 ( 2u1 − u2 − 5u3 + 4u4 ) 2l 9 α 4 = 3 ( −u1 + u2 + 3u3 − 3u4 ) 2l

[3.52]

Substituting equations [3.52] into equation [3.50] and rearranging, the following is obtained for u: u = u1 (1 − 11ξ / 2 + 9ξ 2 − 9ξ 3 / 2 ) + u2 (ξ − 9ξ 2 / 2 + 9ξ 3 / 2 )

+ u3 (9ξ − 45ξ 2 / 2 + 27ξ 3 / 2 )

+ u4 ( −9ξ / 2 + 18ξ 2 − 27ξ 3 / 2 ) u = u1 F1 + u2 F2 + u3 F3 + u4 F4 where

ξ=x l F1 = 1 − 11ξ 2 + 9ξ 2 − 9ξ 3 2


[3.53] [3.54]


155

F2 = ξ − 9ξ 2 2 + 9ξ 3 2 F3 = 9ξ − 45ξ 2 2 + 27ξ 3 2 F4 = −9ξ 2 + 18ξ 2 − 27ξ 3 2 Similarly,

ν = ν 1F1 + ν 2 F2 + ν 3 F3 + ν 4 F4

[3.55]

Hence, it is convenient to assume the following displacement functions for u, v and w: u = (u1F1 + u2 F2 + u3 F3 + u4 F4 ) cos nφ

ν = (ν 1F1 + ν 2 F2 + ν 3 F3 + ν 4 F4 ) sin nφ

w = [(1 − 3ξ 2 + 2ξ 3 ) w1 + l (ξ − 2ξ 2 + ξ 3 )θ 1

+ ( 3ξ 2 − 2ξ 2 ) w2 + l ( −ξ 2 + ξ 3 )θ 2 ] cos nφ

[3.56]

or

{U } = [ N ]{U i } The displacement functions for u and v are very different from those assumed by Rajagopalan72 in his Chapter 6. The stiffness matrix was obtained from the strain-displacement relationships of equation [3.10]. The mass matrix was obtained from equation [5.2]. It is evident that all these matrices are of order 12 × 12, but to simplify computations, Guyan reduction73 is used to eliminate the displacements u3, v3, u4, v4, in a manner similar to that of Rajagopalan.72 Thus, these matrices will now be of order 8 × 8, so that the original programs can be utilised with minimal changes.

3.9.2 The element QUQUCUBE For the QUQUCUBE element, the displacement functions for u and v were assumed to be of quadratic form in the meridional direction, whereas the displacement function for w was of the cubic form assumed in equation [3.7]. In this instance, the positions of the nodal displacements for u and v are shown in Fig. 3.39. Let: u = α1 + α 2 x + α 3 x2

[3.57]


156

Pressure vessels l/2

l/2

x, u 3

1

2

Axis

3.39 Nodal displacement positions for quadratic form.

having the boundary conditions: x = 0, u = u1 @ x = l, u = u2 @ x = l / 2 u = u3

[3.58]

By substituting equations [3.58] into equation [3.57], three simultaneous equations are obtained, the solution of which results in the following expressions for αi:

α 1 = u1 1 α 2 = ( −3u1 − u2 + 4u3 ) l 2 α 3 = 2 (u1 + u2 − 2u3 ) l

[3.59]

Substituting equations [3.59] into equation [3.57] and rearranging, the following is obtained: u = F1u1 + F2u2 + F3u3

[3.60]

where F1 = 1 − 3ξ + 2ξ 2 F2 = −ξ + 2ξ 2 F3 = 4ξ − 4ξ 2

[3.61]

Similarly,

ν = F1v1 + F2 v2 + F3v3

[3.62]

Hence, the displacement functions for u, v and w are: u = ( F1u1 + F2u2 + F3u3 ) cos nφ

ν = ( F1ν 1 + F2ν 2 + F3ν 3 ) sin nφ

w = [(1 − 3ξ 2 + 2ξ 3 ) w1 + l (ξ − 2ξ 2 + ξ 3 )θ 1

+ ( 3ξ 2 − 2ξ 3 ) w2 + l ( −ξ 2 + ξ 3 )θ 2 ] cos nφ


[3.63]


157

Substitution of the above displacement functions and their derivatives into equations [3.10] and [3.15] leads to the stiffness, geometrical and mass matrices for the QUQUCUBE element. Once again, Guyan reduction can be used to eliminate the u3 and v3 displacements, so that these matrices can be reduced from order 10 × 10 to order 8 × 8. This process enables the computer program for ALLCUBE to be modified quite easily to incorporate the QUQUCUBE element.

3.9.3 Computer analysis In this section, comparisons are made between various analyses for instability. The three elements of Ross, namely LILICUBE (linear–linear– cubic50), ALLCUBE and QUQUCUBE, are compared with Rajagopalan’s LILICUB (linear–linear–cubic) and REDCUBE (reduced-all-cubic), using Kendrick’s75 example in Table 3.20. Kendrick’s example is shown in Fig. 3.40, where four equal length elements were used to model this cylinder. From the results of Table 3.20, it can be seen that the simple LILICUBE element of Ross gives results very similar to the more sophisticated analyses, and better than the LILICUB element of Rajagopalan; this is despite the fact that only four equal length elements were used.

Table 3.20 Buckling pressures (lbf in−2) for Kendrick’s example

n

Kendrick75

Rajagopalan72 Ross von Mises39 LILICUB REDCUBE LILICUBE QUQUCUBE ALLCUBE

10 11 12 13

841 774 749 753

843 776 751 755

929 836 805 805

805 – 743 –

791 758 751 764

787.7 753.9 745.3 757.7

786.5 753.0 744.6 757.1

1''

106.5'' I

Axis 40''

5

3.40 Kendrick’s example: u1 = v1 = w1 = 0; E = 30 × 106 lbf in−2; and v = 0.3.


158

Pressure vessels

10°

Simply supported edge 50'' (u1 = v1 = w1 = 0)

Simply supported 100'' edge (w19 = 0)

141.78''

3.41 Simply supported cone.

Table 3.21 Buckling pressures (lbf in−2) for cone Ross n

LILICUBE

QUQUCUBE

ALLCUBE

3 4 5

425.0 339.3 403.5

421.8 337.2 400.9

422.7 337.4 401.0

Comparisons are also made between Ross’ solutions for the thin-walled circular cone shown in Fig. 3.41. For this cone, 18 elements were used with the mesh previously described.21 The assumed boundary conditions were simply supported at the ends and the results are shown in Table 3.21. The predicted buckling pressure by Volmir’s formula52 was 350 lbf in−2; this compares favourably with all three solutions and shows that the much simpler solution of Ross, namely LILICUBE is satisfactory. The buckling pressures predicted by QUQUCUBE are almost identical to those predicted by the more complicated ALLCUBE element.

3.9.4 Conclusions The results show that the simpler LILICUBE element of Ross is quite suitable for predicting the elastic instability buckling pressures of perfect thinwalled circular cylinders and cones of small apical angle. It must be emphasised that these theories do not take into account the effects of initial imperfections in the shells, which can cause a catastrophic decrease in the buckling pressures of these vessels. However, the theories can be used for such cases, providing a plastic reduction factor is divided into the theoretical elastic instability buckling pressures.



3.10

159

Higher order elements for hemi-ellipsoidal domes

Using the same method as for the cones, two higher order elements were developed by Ross76 for domes of varying individual curvature.

3.10.1 The all-cubic (CCC) element For this element, the displacement functions for u, v and w were all assumed to be of cubic form. Now the displacement function of w is of the Hermitian form shown by equation [3.42], but the displacement functions for u and v were obtained as follows. For the node element of Fig. 3.42, the assumed displacement functions for u and v are given by the following cubic forms: u = α + bζ + cζ 2 + dζ 3 v = e + fζ + gζ 2 + hζ 3 To determine the shape functions for u and v, consider the boundary conditions for u, as follows: at ζ = − 1, u = u1 at ζ = − 13 , u = u3 at ζ = 13 , u = u4 at ζ = 1, u = u2

∴u1 ∴u3 ∴u4 ∴u2

= = = =

− b + c − d; − b/3 + c/9 − d/27; + b/3 + c/9 + d/27; +b+c+d

a a a a

Solving these four simultaneous equations, the following expressions are obtained for the constants:

ς=1 ς = 1/3 2

ς = –1/3 4 3

ς = –1

1

3.42 Nodal displacement positions for u and v of cubic form for hemiellipsoidal domes.


160

Pressure vessels 1 9 (u2 + u1 ) + (u3 + u4 ) 16 16 1 9 b = − (u2 − u1 ) + ( 3u4 − 3u3 ) 16 16 9 c = (u1 + u2 − u3 − u4 ) 16 9 d = (u2 − u1 + 3u3 − 3u4 ) 16 a=−

[3.64]

Similar expressions can be obtained for the constants from e to h, and substituting these constants into the displacement functions, the shape functions for u and v are given by: 1 (−1 + ζ + 9ζ 2 − 9ζ 3 ) 16 1 N 2 = ( −1 − ζ + 9ζ 2 + 9ζ 3 ) 16 9 N 3 = (1 − 3ζ − ζ 2 + 3ζ 3 ) 16 9 N 4 = (1 + 3ζ − ζ 2 − 3ζ 3 ) 16 N1 =

[3.65]

i.e. u = ( N 1u1 + N 2u2 + N 3u3 + N 4u4 ) cos nφ

[3.66]

ν = ( N 1u1 + N 2u2 + N 3u3 + N 4u4 ) sin nφ

[3.67]

and These displacement functions are very different to those used by Rajagopalan72 in his chapter 6. The [B] matrix was obtained from ∂N 1 ∂ζ ∂N 2 DN 2 = ∂ζ ∂N 3 DN 3 = ∂ζ ∂N 4 DN 4 = ∂ζ DN 1 =

[3.68]

α = L / ( 2 R1 )

[3.69]

The geometrical stiffness matrix [k1] was obtained from equation [3.35], and the mass matrix [m] was obtained from equation [5.2].



161

It is evident that all these matrices are of order 12 × 12, but to simplify computation, Guyan reduction73 was used to eliminate the u3, v3, u4 and v4 displacements in a manner similar to that of Rajagopalan. Thus, these matrices were of order 8 × 8, so that the original programs for the VMC element could be used with minimal changes.

3.10.2 The quadratic-quadratic-cubic (QQC) element For this element, the displacement functions for u and v were assumed to be in quadratic form along the meridian of the element, as shown in Fig. 3.43. The positions of the nodal displacements for u and v are shown in Fig. 3.43. The assumed boundary conditions were: at ζ = −1, u = u1 and ν = ν 1 at ζ = 1, u = u2 and ν = ν 2 at ζ = 0, u = u3 and ν = ν 3

[3.70]

This resulted in the following displacement functions for u and v: u = N 1.u1 + N 2.u2 + N 3.u3

[3.71]

v = N 1.v1 + N 2.v2 + N 3.v3

[3.72]

where N 1 = − 21 (1 − ζ )ζ N 2 = 21 (1 + ζ )ζ N 3 = (1 − ζ 2 )

[3.73]

ς=1 ς=0

ς = –1

2

3

1

3.43 Three-node varying meridional curvature element.


162

Pressure vessels

and their derivatives with respect to ζ are ∂N 1 ∂ζ ∂N 2 DN 2 = ∂ζ ∂N 3 DN 3 = ∂ζ DN 1 =

[3.74]

Substitution of the above displacement functions, together with their derivatives into equations [3.10], [3.15] and [5.2], will lead to the stiffness, geometrical and mass matrices for the QQC element. Here, again, Guyan reduction was used to eliminate the u3 and v3 displacements, so that the 10 × 10 matrices became of order 8 × 8. Hence, it was a simple matter to modify the computer program for the CCC element to incorporate the QQC element.

3.10.3 Computational analysis In this section, comparison is made for buckling ten hemi-ellipsoidal domes (Fig. 3.13), with the theoretical predictions of three varying meridional curvature elements. The models were made in SUP, where the SUP liquid was poured between machined male and female aluminium alloy moulds and then thermoset. The models had an internal base diameter of 0.2 m and a wall thickness of 0.002 m. SUP was found to have the following properties: Young’s modulus = 2.89 × 109 N m−2; Poisson’s ratio = 0.3 assumed; and density = 1230 kg m−3. The three elements of Ross, the VMC element (linear-linear-cubic), the CCC element and the QQC element, are compared with the experimentally obtained buckling pressures for the ten hemi-ellipsoidal domes of Ross and Mackney57 in Table 3.22, where: AR =

dome height base radius

From Table 3.22, it can be seen that the CCC and the QQC elements give better results than the VMC element for all hemi-ellipsoidal domes, except for the oblate dome of AR 0.7. However, it must be emphasised that this dome is prone to even the slightest variations in meridional curvature. A slight increase in meridional curvature results in a much higher buckling pressure. The flatter oblate domes probably collapsed through snap-thru



163

Table 3.22 Buckling pressures (MPa) for hemi-ellipsoidal domes AR

Experiment

VMC

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.7 0.44 0.25

0.097 0.100 0.150 0.201 0.364 0.490 1.280 1.070 0.276 0.060

0.111 0.135 0.168 0.224 0.323 0.549 1.426 0.944 0.380 0.132

CCC (5) (5) (6) (7) (8) (9) (11) (0) (0) (0)

0.110 0.134 0.166 0.222 0.321 0.548 1.398 0.865 0.361 0.124

QQC (5) (5) (6) (7) (7) (9) (11) (0) (0) (0)

0.110 0.134 0.166 0.222 0.321 0.548 1.398 0.866 0.376 0.124

(5) (5) (6) (7) (7) (9) (11) (0) (0) (0)

buckling, and not through axisymmetric bifurcation buckling as assumed in the theory of this section.

3.11

Varying thickness cylinders

Thin-walled circular cylinders with a stepped variation in wall thickness, appear in various branches of engineering, including ocean engineering, agricultural engineering and aerospace engineering. In ocean engineering, such structures appear as the legs of off-shore structures and as underwater storage containers, whereas in aeronautical engineering, such structures appear as the fuselages of spacecraft. Under external hydrostatic pressure, such structures can fail through non-symmetric bifurcation buckling or shell instability at a pressure that might only be a small fraction of that needed to cause axisymmetric yield. If the unsupported meridional length of the vessel is small, then the cylinder can fail through axisymmetric collapse, as shown in Figs 1.7 and 1.8. For a varying thickness circular cylinder, the vessel can fail through either shell instability or axisymmetric collapse or, by a combination of these two modes of failure. Earlier work by Esslinger and Geier77 and also by Malik et al.78 were only concerned with the elastic instability of geometrically perfect vessels. From the theory of Esslinger and Geier and according to Rajagopalan,72 the former authors quoted the following semi-empirical formula to predict the elastic instability of varying thickness circular cylinders under hydrostatic pressure: L / Pcr = ∑ ( Li / Pi ) from i = 1 to N


164

Pressure vessels

where L = the entire length of the vessel between supports. Pcr = theoretical buckling pressure based on the von Mises theory. Pi = theoretical buckling pressure of the ith bay of the vessel, assuming that it is of length L and thickness ti. Li = length of the element i. The problem with this theory is that it does not take into account the distribution of wall thickness, nor does it allow for inelastic buckling. In 2000, Ross et al.79 presented a paper which successfully took these into account. They presented both theoretical and experimental results.


4 General instability of pressure vessels

Abstract: Mathematical theories for general instability, the process where the entire ring-shell combination of a circular cylinder or cone buckles bodily in its flank (between adjacent bulkheads) are described. Some theories are based on an analytical approach and some on a numerical one, namely the finite element method. Predictions of failure using theories are compared with experimental observations. The detrimental effects of initial out-of-circularity, together with inelastic buckling are also considered. Two design charts are provided; one for ring-stiffened circular cylinders and the other for ring-stiffened cones. Key words: general instability, buckling, initial out-of-circularity, circular cylinders, cones, design charts.

4.1

General instability of ring-stiffened circular cylinders

Tokugawa80 first identified the general instability mode of failure in 1929. His method of analysis was based on the elastic instability of an isolated ring, with no allowances for any support from bulkheads. He assumed that an effective width of shell plating buckled with the ring stiffener. One simple formula that can be used as a first stab on the design estimate for the ring-stiffener is that of Levy:81 p′ = 3EI / R3

[4.1]

where p′ = load per unit length; I = second moment of area of the ring’s cross-section about its centroidal axis, which is parallel to the axis of the cylinder; and R = radius of curvature of the ring centroid. This mode of failure, which is based on elastic instability, assumes that the ring buckles in a flattening mode (i.e. n = 2). It is a simple matter to extend Levy’s formula to the form shown by equation [4.2]: p′ = 3EI f / Rf3

[4.2]

where If = second moment of area of the ring-shell combination about its centroidal axis x–x, shown in Fig. 4.1; Rf = radius of curvature of the centroid of the ring-shell combination; Leff = effective width of the shell plating that 165 © Carl T. F. Ross, 2011

166

Pressure vessels Ring Shell x

x

Leff

Rf

Axis

4.1 Ring-shell combination.

buckles with the ring-stiffener. Several authors provide an estimate for the effective width of shell plating (Leff), including Sechler.82,83 The two main problems with equation [4.2] are that it allows neither for the increase in buckling resistance resulting from the support given by adjacent bulkheads nor for the decrease in buckling resistance caused by inelastic buckling and initial out-of-circularity. An important approximate formula for the general instability of ring-stiffened circular cylinders was presented by Bryant,84 giving a buckling pressure: Pcr = Pf + Ps where Pf = ( n2 − 1) EI c / ( R3 Lf ) ; ( Et / R ) λ 4 ; Ps = ( n 2 − 1 + λ 2 / 2 ) ( n 2 + λ 2 )2 n = number of circumferential waves or lobes; λ = πR/Lb; Ic = second moment of area of the cross-section of a ring-stiffener and the effective width of shell in contact with it; Lf = stiffener spacing; Lb = length between adjacent and bulkheads. It can be seen from the above formula for Pcr, that it has a frame component, namely Pf and a shell component, namely Ps; this is not unlike Tokugawa’s formula produced some 25 years earlier! Rajagopalan72 has shown that the Bryant formula can break down for certain geometries. In any case, since the advent of the digital computer, there is no need to use approximate formulae; even for design.


General instability of pressure vessels

167

4.1.1 Kendrick Part I A major breakthrough on the elastic instability of ring-stiffened circular cylinders was achieved by Kendrick.43 His solution, which assumed simply supported edges, was based on the Rayleigh–Ritz method. From first principles, Kendrick derived the following expressions for the bending strain energy of the shell: U b = {Eh3 / 24a (1 − v2 )} ×∫

2π

0

∫ {a w Lb

2

0

+ (wθθ + w ) / a2 + 2vwxx (wθθ + w ) 2

2 xx

}

+ 2 (1 − v) (wxθ + ν x / 2 − uθ / 2a) dθ dx 2

[4.3]

and for the extensional strain energy of the shell: U e = {Eah / 2 (1 − v2 )} ×∫

2π

×∫

2π

×∫

2π

0

∫ {u

+ (ν θ − w ) / a2 + 2vux (ν θ − w ) / a

∫ (ν

2 x

+ wx2 ) dθ dx + ( N oy / 2a)

2 θ

+ wθ2 − 2ν θw) dθ dx

Lb

2

2 x

0

+ (1 − ν ) (ν x + uθ / a2 ) 2} dθ dx + (aN ox / 2 ) 0

0

Lb

0

∫ (u Lb

0

[4.4]

where E = Young’s modulus; v = Poisson’s ratio; Nox = −0.5 pa; Noy = −pahLf/(Af + hLf); u, v, w are defined in Fig. 4.2; p = applied pressure; h = shell thickness, t; a = mean radius of shell; Af = cross-sectional area of a frame; Lf = frame spacing. The bending strain energy of the ring-stiffeners is: r=N

2π

Vb = ( EI / 2a3 ) ∑ ∫ (wθθ + w ) dθ 2

[4.5]

0

r =1

and the extensional strain energy of the ring-stiffeners is: r=N

Ve = ( EAf / 2a) ∑ r =1

+ ( N of / 2a) ∫

{∫

2π

0

2π

0

[(wθθ + w)(e / a) − (νθ − w)]2 dθ

(uθ2 + wθ2 − 2νθw) dθ

}


[4.6]

168

Pressure vessels U V

W φ

X

W

R R

L

4.2 Notation for the deflections of a circular cylinder.

Shell e Ring centroid

a Axis of cylinder

x

4.3 Ring-stiffener.

where N = number of ring stiffeners; Nof = −paLfAf/(Af + hLf); e = eccentricity of the ring centroid from the mid-surface of the shell (positive inwards) (Fig. 4.3); I = second moment of area of the ring’s cross-section about its centroid and parallel to the axis of the cylinder. The potential owing to radial pressure is: Wr = − ( pa / 2 ) ∫

2π

0

∫ ( 2wu Lb

x

0

+ 2wν θ / a − ν θ ux − w 2 / a) dθ dx

[4.7]

and the potential owing to axial pressure is: Wa = ( pa / 2 ) ∫

2π

0

{u( x =0) − u( x =L ) } dθ

[4.8]

b

Kendrick assumed the buckling configuration of equation [4.9], which corresponds to simply supported edges: w = A sin

πx cos nθ Lb

ν = B sin

πx sin nθ Lb

u = C cos


πx cos nθ Lb

[4.9]


169

where n = number of circumferential lobes or waves into which the vessels buckle; and A, B and C are constants. Substitution of equation [4.9] and its various derivatives into equations [4.3]–[4.8] results in the following expression for the total strain energy and potential, Ut: U t = U b + U e + Vb + Vc + Wr + Wa = {Eh3 / 24a (1 − ν 2 )}

× {π 5 a2 W14 A2 / L3b + π ( n2 − 1) LbW11 A2 / a2 + 2 π 3 n2 νW12 A2 / Lb 2

− 2 π 3νW12 A2 / Lb + 2 π 3 n2 (1 − ν ) W12 A2 / Lb + π 3 (1 − ν ) W12 B2 / 2 Lb + πn2 (1 − ν ) LbC 2 / 4a2 − 2 π 3 (1 − ν ) nW12 AB / Lb − 2 π 2 n2 (1 − ν ) W13 AC / a + π 2 n (1 − ν ) W13 BC / a} + {Eah / 2 (1 − ν 2 )}

× {0.5π 3 C 2 / Lb + πLbW11 ( nB − A) / a2 − 2 π 2 nνW10 BC / a 2

+ 2 π 2 νW10 AC / a + 0.5π 3 (1 − ν ) W12 B2 / Lb

+ 0.25πn2 Lb (1 − ν ) C 2 / a2 − π 2 n (1 − ν ) W13 BC / a} − 0.25π 3pa2 W12 B2 / Lb − 0.25π 3pa2 W12 A2 / Lb + 0.25πpn2Lb K0C 2 + 0.5πpn 2 Lb K0W11 A2 − πpnLb K0 W11 AB + ( 0.5EI / a3 ) π ( n2 − 1) W0 A2 + (0.5EAf / a) 2

× {( 2 πn3 e / a) AB − ( 2 πne / a) AB − ( 2 πn2 e / a) + ( 2 πe / a) A2

}

+ ⎡⎣ π ( n2 − 1) e 2 / a2 ⎤⎦ A2 + π ( nB − A) W0 2

2

+ 0.25πpn ( N − 1) K1C + 0.5πpnK1W0 ( nA2 − 2 AB) 2

2

+ π 2 paW10 AC − πpnLbW11 AB − 0.5π 2 panW10 BC + 0.5πpLbW11 A2

[4.10]

Now, according to the Rayleigh–Ritz theory, ∂U t ∂U t ∂U t = = =0 ∂A ∂B ∂C which, on application to equation [4.10], results in the following three simultaneous equations:

( a10 + pa11 ) A2 + ( a12 + pa13 ) B2 + ( a14 + pa15 ) C 2 = 0 ( a12 + pa13 ) A2 + ( a18 + pa19 ) B2 + ( a16 + pa17 ) C 2 = 0 ( a14 + pa15 ) A2 + ( a16 + pa17 ) B2 + ( a20 + pa21 ) C 2 = 0

[4.11]

As the solution A = B = C = zero is not of interest in this problem, the following determinant is obtained:


170

Pressure vessels ⎡ a10 + pa11 ⎢ a + pa 13 ⎢ 12 ⎢⎣ a14 + pa15

a12 + pa13 a18 + pa19 a16 + pa17

a14 + pa15 ⎤ a16 + pa17 ⎥ = 0 ⎥ a20 + pa21 ⎥⎦

[4.12]

The determinant expands to a32 p3 + a33 p2 + a34 p + a35 = 0

[4.13]

where the root of interest is the lowest positive one that can be calculated by the Newton–Raphson iterative process: W0 = W10 = W11 = W12 = W13 = W14 = 0.5 2 2 a10 = ( πEW0 / a ) ⎡⎣( n2 − 1) I / a 2 + Af ( n2 − 1) e 2 / a 2 + 1 + 2e / a − 2 n2 e / a ⎤⎦

{

{

}

+ {Eh3 / 12a (1 − ν 2 )} π 5 a 2W14 / L3b + ( 2 π 3W12 / Lb ) × [ν ( n − 1) + n (1 − ν )] 2

2

2 + πLbW11 ⎡⎣12 / h2 + ( n2 − 1) / a 2 ⎤⎦

}

a11 = πLbW11 (1 + n2 K0 ) − 0.5π 3 a 2W12 / Lb + πn2 K1W0 a12 = πnEAf W0 ( n2 e / a − e / a 2 − 1 / a )

− {πnEh / a (1 − ν 2 )} {π 2 h2 (1 − ν ) W12 / 12Lb + LbW11 }

a13 a14 a15 a16 a17 a18 a19 a20 a21 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33

= − πn {K1W0 + LbW11 (1 + K0 )} = {π2 Eh/(1 − ν 2)}{−h2n2(1 − ν)W13/12a2 + νW10} = π2 aW10 = {π2 Eh/(1 − ν 2)}{0.5n(1 − ν)W13(h2/12a2) − 1) − νnW10} = −0.5π2 anW10 = πn2 EAfW0/a + {Eh/(1 − ν 2)} × {π2a(1 − ν)W12(h2/24a2 + 0.5)/Lb + n2LbW11/a} = −0.5π3a2W12/Lb = {πEh/(1 − ν 2)}{0.25n2(1 − ν)Lb(h2/12a2 + 1)/a + 0.5π2 a/Lb} = 0.5πn2{K1(N − 1) + LbK0} = −a216 + a18a20 = −2a16a17 + a18a21 + a19a20 = −a217 + a19a21 = −a12a20 + a14a16 = −a12a21 + a13a20 + a14a17 + a15a16 = −a13a21 + a15a17 = a12a16 − a14a18 = a12a17 + a13a16 + a14a19 − a15a18 = a13a17 − a15a19 = a11a25 + a13a28 + a15a31 = a10a25 + a11a24 + a12a28 + a13a27 + a14a31 + a15a30



171

a34 = a10a24 + a11a23 + a12a27 + a13a26 + a14a30 + a15a29 a35 = a10a23 + a12a26 + a14a29 A computer program for determining the general instability pressure of a ring-stiffened cylinder is provided in Appendix IV.

4.1.2 Kendrick Part III In addition to this solution, Kendrick85,86 produced a number of other general instability solutions, some of which are believed to be more realistic than the solution presented in Section 4.1.1. For example, Kendrick assumed the buckling configuration given in equation [4.14], which also corresponds to simply supported edges: u = A1 cos nθ cos ( πx / Lb ) ν = B1 sin nθ sin ( πx / Lb ) + B2 sin nθ [1 − cos ( 2 πx / Lf )] w = C1 cos nθ sin ( πx / Lb ) + C2 cos nθ [1 − cos ( 2 πx / Lf )]

[4.14]

Equation [4.14] is shown diagrammatically in Fig. 4.4. This configuration assumes that the shell plating between adjacent ring-stiffeners suffers additional displacements, unlike his Part I buckling configuration, given in equation [4.9] and shown diagrammatically in Fig. 4.5. Neither Kaminsky87 nor Nash88,89 accepted Kendrick’s assumed buckling configuration, and produced clamped and fixed-edges solutions, where the former was based on Kendrick’s strain energy expressions.43

Axis

4.4 Kendrick Part III buckling configuration.

Axis

4.5 Kendrick Part I buckling configuration.


172

Pressure vessels

Kaminsky assumed the following buckling configuration: w = A [1 − cos ( 2 πx / Lb )] cos nθ

ν = B [1 − cos ( 2 πx / Lb )] sin nθ u = C cos ( πx / Lb ) cos nθ

[4.15]

and Nash,88,89 who used his own strain energy expressions, assumed the following buckling configuration: w = A [1 − cos ( 2 πx / Lb )] cos nθ

ν = B [1 − cos ( 2 πx / Lb )] sin nθ u = C sin ( 2 πx / Lb ) cos nθ

[4.16]

Nash’s energy expressions considered numerous extra terms, including twisting of the rings. The buckling configurations given by equations [4.9], [4.15] and [4.16] yield the following boundary conditions: for simply supported edges (equation [4.9]): w = 0 at x = 0 and x = Lb d 2 w / dx 2 = 0 at x = 0 and x = Lb dw / dx ≠ 0 at x = 0 and x = Lb u ≠ 0 at x = 0 and x = Lb for clamped edges (equation [4.15]):

ν = w = 0 at x = 0 and x = Lb d 2 w / dx 2 ≠ 0 at x = 0 and x = Lb dw / dx = 0 at x = 0 and x = Lb u ≠ 0 at x = 0 and x = Lb for fixed edges (equation [4.16]): u = ν = w = 0 at x = 0 and x = Lb d 2 w / dx 2 ≠ 0 at x = 0 and x = Lb dw / dx = 0 at x = 0 and x = Lb u = 0 at x = 0 and x = Lb

4.1.3 Galletly’s results To test the above solutions, Galletly et al.90 compared these solutions with their experimental results for six machine-stiffened models, as shown in Table 4.1, where the values given are in terms of the non-dimensional parameter Pcr/Pexp, where Pcr = theoretical buckling pressure and Pexp = experimental buckling pressure. From Table 4.1, it can be seen that comparison between experiment and Kendrick’s solutions is good, and that both Kaminsky87 and Nash88,89



173

Table 4.1 General instability results for the models of Galletly et al.90 Model no.a

Kendrick Part I43

Kendrick Part III85

Kaminsky87

Nash88,89

1 2 3 4 5 6

1.012 0.989 0.994 0.960 1.200 1.160

0.976 0.937 0.920 0.842 0.997 0.967

1.637 1.607 1.714 1.534 1.589 1.542

1.595 1.570 1.705 1.355 1.513 1.474

(4) (4) (3) (3) (3) (3)

a

The figures in parentheses represent the number of lobes (n) into which the vessels buckle.

overestimate the collapse pressures. It may be, however, that the main reason why the Kendrick solutions agreed best with experiment was that these models were not firmly clamped at their ends but simply secured by studs, where pressure tightness was achieved because the hydrostatic pressure caused the joints to be in compression. It appears that 12 studs were used on a diameter of about 8 in (i.e. the studs were spaced at over eight diameters apart). Kendrick argues that for ring-stiffened circular cylinders rotational restraint at the ends is small and localised because thin-walled shells have a relatively small resistance to bending, and thus solutions based on buckling configurations similar to those adopted by Kaminsky and Nash will overestimate buckling pressures. In this context, the present author agrees with Kendrick. However, later on in this chapter, it is shown that axial restraint at the edges increases the experimentally obtained buckling pressures because shells have a relatively high resistance to in-plane deformation. It must be emphasised that initial imperfections can cause experimental buckling pressures to be even lower than those predicted by Kendrick, particularly if the ring-stiffened cylinder is short and thick. Such vessels are said to suffer plastic knockdown and fail through inelastic instability, as discussed in chapter 3. The correlation between Kaminsky87 and Nash88,89 was poor, but this was probably caused by the different buckling configurations assumed for u, because when the present author applied Nash’s boundary conditions to the Kendrick strain energy expressions, resulting in solution (2b) of reference 42, he obtained good correlation between Nash and solution (2b). This finding appears to indicate that, whereas the strain energy expressions of Kendrick and Nash were different, they yielded similar results for the same boundary conditions.


174

Pressure vessels

4.1.4 Varying stiffener sizes One of the problems with Kendrick’s solutions is that they are for circular cylinders, stiffened by uniform-size stiffeners, at uniform spacing, and all the stiffeners are attached to either the internal surface or the external surface. In practice such symmetrical vessels are seldom encountered and, because of this, the present author91, 92 developed a finite element solution for the general instability of ring-reinforced circular cylinders under uniform external pressure. This solution is easily achieved by introducing a stiffness matrix for a ring-stiffener, which is used in conjunction with the stiffness matrix of the circular cylindrical shell element described in Section 3.3.1. The various components of the ring-stiffness matrix are simply added to the appropriate positions in the stiffness matrix of the circular cylindrical shell, in an eigenvalue problem similar to equation [3.17]. The ring-stiffener is shown in Fig. 4.6, and its matrix of shape functions [N] is given by:91 0 0 ⎤ ⎡cos nφ ⎢ sin nφ 0 ⎥ [N ] = ⎢ 0 ⎥ ⎢⎣ 0 0 cos nφ ⎥⎦

[4.17]

The strain matrix, which was based on the work of Kendrick43 is given by 2 ⎧ε φ ⎫ ⎧(νφ − w ) / R − (wφφ + w ) e / R ⎫ ⎪ ⎪ ⎪ ⎪ {ε } = ⎨ χφ ⎬ = ⎨ (wφφ + w ) / R2 ⎬ ⎪χ ⎪ ⎪ ⎪ θi / R ⎩ x⎭ ⎩ ⎭

[4.18]

x,u v φ

z

w

4.6 Buckling configuration for ring (n = 4).



175

or

{ε } = [ B] {u}. Therefore,

νi

wi

⎡ n cos φ / R 0 [ B] = ⎢⎢ ⎢⎣ 0

( −1 − e / R + n e / R ) cos nφ / R (1 − n2 ) cos nφ / R2 2

0

θi 0 ⎤ 0 ⎥⎥ 1 / R ⎥⎦

[4.19]

where θi = the rotation of the ith ring about the z axis; and [B] = a matrix relating strains and displacements. The relationship between the stress {σ} and strain {ε} matrices was taken as

{σ } = [ D] {ε } and a matrix of material constants: ⎡ Af ⎢0 D = E [ ] ⎢ ⎢⎣ 0

0 0⎤ I 0⎥ ⎥ 0 I z ⎥⎦

[4.20]

where Af = cross-sectional area of ring; I = second moment of area of ring cross-section about the x axis; and Iz = second moment of area about the z axis. The stiffness matrix for a ring of uniform section was determined from: 2π

[ k ] = ∫0 [ B]T [ D][ B] Rdφ

[4.21]

This was found to be: n2 Af / R 2 ⎡ ⎢ ⎡ nAf ( −1 − e / R + n2 e / R ) ⎤ [ k ] = πER ⎢⎢ ⎢ ⎥ R2 ⎣ ⎦ ⎢ 0 ⎢⎣

nAf ( −1 − e / R + n2 e / R ) / R 2

0 ⎤ ⎥ 1 A ⎡ f −1 − e / R + n 2 e / R 2 + 2 2⎤ ⎥ − 1 0 n ( ) ( ) ⎢⎣ R 2 ⎥⎦ R4 ⎥ 0 2 I z / R 2 ⎦⎥ The parameters used in the computation include the following: p = pressure (external positive); n = number of lobes in circumferential direction;


[4.22]

176

Pressure vessels

e x0

z0

4.7 Frame with shell.

t = shell thickness; R = radius of centroid of frame section; Af = cross-sectional area of frame; I = second moment of area of frame about the x0 axis (Fig. 4.7); Iz = second moment of area of frame about the z0 axis (Fig. 4.7); L = length between stiffeners; e = distance of centroid of frame section from mid-surface of shell, as shown in Fig. 4.7 (negative if frame is on outside surface of shell); u,v,w = displacement in x, ϕ and z directions; x,z = rectangular coordinates; ϕ = angular coordinate; E = elastic modulus; σ = stress. To obtain the geometrical stiffness matrix of the ring [kG], the additional strain owing to large displacements92 was considered: ⎧1 ν w 2 ⎫ {δε L } = ⎨ ⎛⎜ + φ ⎞⎟ ⎬ ⎩2 ⎝ R

R ⎠ ⎪⎭

[4.23]

Using the same notation as Zienkiewicz:23

{

}

ν wφ = [G ] {u} + R R

[4.24]

that is,

[G ] = (1/ R ) [sin nφ − n sin nφ ]

[4.25]

The geometrical stiffness matrix was obtained from:

[ kG ] = ∫vol [GT ][σ ][G ] d ( vol )

[4.26]

which for a ring of uniform cross-section was:

[ kG ] = Af ∫0 [GT ][σ ][G ] R dφ 2π


[4.27]


177

The following approximation was made for the hoop stress in a ring:

σ = pRL / ( Af + Lt )

[4.28]

that is,

[ kG ] =

pAf L 2π [sin nφ − n sin nφ ][sin nφ − n sin nφ ] dφ ( Af + Lt ) ∫0

⎡ ν i wi ⎤ πpAf L ⎢ 1 − n⎥ [ kG ] = ⎥ ( Af + Lt ) ⎢ ⎢⎣ − n n2 ⎥⎦

[4.29]

[4.30]

When no ring appeared at a particular nodal point, Af was set to zero. A computer program for the general instability of an orthotropic ringstiffened cylinder is given in Appendix IV.

4.1.5 End conditions To determine the effects of end conditions, comparison is made in Table 4.2 of the experimentally obtained collapse pressure of the ring-reinforced circular cylinder of reference 93 with various analytical and numerical solutions, where solution (2b) from reference 42 was a fixed-edges solution based on Kendrick’s method.43 The vessel was firmly clamped at its ends by 1 /4-inch bolts, spaced at about 0.547 in apart (i.e. 2.19 bolt diameters). From Table 4.2, it can be seen that the fixed-edges solutions agree best with experiment, but there is some doubt if these edges are realistic for the practical case. Comparison is also made in Table 4.3 of models 4, 5 and 6 from reference 3 with numerically and analytically determined buckling pressures. These models were secured at their ends by 1/4 inch bolts, spaced at about 1.93 in (i.e. a bolt spacing of 7.72 bolt diameters). Details of the buckling pressures of these models, which were machined from a thick-walled tube of HE9-WP aluminium alloy are given in Table 4.4, from which it can be seen that by having a slightly larger bolt spacing

Table 4.2 General instability collapse pressures of circular cylinder P3 (lbf in−2) Analytical

Finite element solution

Simply supported

Solution (2b) (fixed)

Simply supported

Clamped

Fixed

Experimental

218.1

338.4

229.6 (4)a

237.7 (4)

330.8 (4)

322

a

The figures in parentheses represent the number of lobes.


178

Pressure vessels

Table 4.3 General instability collapse pressures of circular cylinders (lbf in−2) Analytical

Finite element solutions

Kendrick Solution Model (simply Kaminsky (2b) Simply no. supported)a (clamped) (fixed) supported Clamped Fixed 4 5 6 a

94.6 (4) 126 (4) 170 (3)

156 (5) 189 (4) 241 (4)

147 (5) 182 (4) 233 (4)

95.6 (4) 121 (4) 160 (4)

99.7 (4) 125 (4) 164 (4)

Experimental

141 (5) 120 (4) 173 (4) 148 (4) 212 (4) 188 (4)

The number of nodes is indicated in parentheses. Table 4.4 General instability buckling pressures (lbf in−2) for DTMB machined models

Model no.

Kendrick (simply supported)a

Kaminsky (clamped)

Solution (2b) (fixed)

Solution (2a) (partially fixed)

Experimental

DD-6 DD-2A DD-3A DD-4A

558 331 232 197

910 576 536 485

969 524 488 362

636 467 322 260

700 409 378 289

(3) (2) (2) (2)

(3) (3) (3) (2)

(4) (3) (3) (2)

(3) (2) (2) (2)

(3) (3) (2) (2)

a

The figures in parentheses represent n, the number of waves or lobes into which the vessels buckle.

on the edge clamping rings than that for model P3,93 the fixed-edges solutions overestimated the experimental buckling pressures. It can also be seen that the finite element clamped solution predicted buckling pressures only fractionally higher than the simply-supported edges solutions, thereby indicating that axial restraint at the boundaries has a more significant effect on buckling resistance than does rotational restraint at these positions. Another conclusion that can be arrived from the observations of Table 4.3 is that the models must have been partially fixed at their boundaries, particularly in the axial direction. Similar conclusions are shown in Table 4.4, where comparison is made of the experimental results obtained from the DD8-2 series of machined models94 with various analytical solutions. Nevertheless, for vessels that buckle inelastically, the above arguments are purely academic, as such vessels often fail at pressures less than those predicted by even the simply supported edges solutions. Table 4.5 shows a comparison between theoretical and experimental buckling pressures for the machined models Pl, P2, P1/A and P2/A, all of



179

Table 4.5 Theoretical and experimental buckling pressures (lbf in−2) Model no.

Simply supported

Fixed

Experimental

P1 P2 P1/A P2/A

1401 1776 1488 1740

2284 2951 2421 2875

865 940 1005 1100

which failed inelastically. The models were firmly clamped at their edges by 1 /4-inch bolts, for P1 and P2 the spacing of the bolts was the same as that used for P3, but for P1/A and P2/A only four bolts were used at each end. From Table 4.5 it can be seen that the experimentally obtained buckling pressures are considerably lower than the theoretical ones, and also that edge conditions play a less significant role when vessels buckle plastically. Further geometrical details of models Pl, P2, P1/A, P2/A and P3 are given in Table 4.6. Careful measurements of the models found P1, P2 and P3 to have a maximum initial out-of-circularity of about 0.001 in, and P1/A and P2/A to have a maximum initial out-of-circularity of 0.002 in. The mechanical properties of these models were as follows: P1 and P2: E = 29 × 106 lbf in−2; σyp = 23 500 lbf in−2; and nominal peak stress = 48 700 lbf in−2. P1/A: E = 30 × 106 lbf in−2; σyp = 26 700 lbf in−2; and nominal peak stress = 51 000 lbf in−2. P2/A: E = 29 × 106 lbf in−2; σyp = 27 300 lbf in−2; and nominal peak stress = 51 000 lbf in−2. P3: E = 29.3 × 106 lbf in−2; σyp = 57 500 lbf in−2; and nominal peak stress = 99 200 lbf in−2. Despite the fact that the models were machined very precisely, as shown by a typical set of out-of circularity plots for model P1 in Fig. 4.8, the plastic knockdown was as high as 89% for one of these models when compared with a simply supported edges solution.

4.2

Inelastic general instability of ring-stiffened circular cylinders

In a manner similar to that adopted in Section 3.1.3, an attempt is now made to produce a chart which can be used to determine the plastic reduction factor for the general instability of a perfect ring-stiffened circular cylinder. In this instance, instead of using λ, a similar parameter, namely λ′, another thinness ratio, will be used:


180

Pressure vessels

Table 4.6 Dimensions of the P-series models (inches) xi

bi di h i=1

P1

i 1 2 3 4 5 6 7 8 9

i 1 2 3 4 5 6 7 8 9

i 1 2 3 4 5 6 7 8 9

i=2

P2

xi

di

bi

xi

di

bi

0.5010 1.2027 2.1451 3.0878 4.0296 4.9706 5.9117 6.8535 7.5555

0.0803 0.0603 0.0603 0.0604 0.0604 0.0604 0.0603 0.0603 0.0805

0.0620 0.0613 0.0635 0.0620 0.0615 0.0605 0.0617 0.0620 0.0620

0.5004 1.2008 2.1408 3.0813 4.0221 4.9628 5.9036 6.8447 7.5462

0.1203 0.1003 0.0801 0.0804 0.0804 0.0803 0.0800 0.1000 0.1200

0.0607 0.0601 0.0600 0.0609 0.0607 0.0608 0.0608 0.0613 0.0617

P1/A

P2/A

xi

di

bi

xi

di

bi

0.5017 1.2002 2.1416 3.0809 4.0209 4.9599 5.8994 6.8411 7.5384

0.0805 0.0606 0.0611 0.0611 0.0606 0.0604 0.0602 0.0601 0.0802

0.0622 0.0162 0.0609 0.0613 0.0609 0.0600 0.0615 0.0608 0.0605

0.5000 1.1996 2.1379 3.0780 4.0174 4.9573 5.8982 6.8381 7.5379

0.1201 0.1008 0.0803 0.0802 0.0802 0.0802 0.0800 0.0997 0.1194

0.0617 0.0609 0.0612 0.0604 0.0584 0.0610 0.0616 0.0611 0.0616

P3 xi

di

bi

0.6152 1.5253 2.4355 3.3428 4.2560 5.1645 6.0745 6.9850 7.8950

0.0304 0.0303 0.0304 0.0304 0.0301 0.0302 0.0301 0.0302 0.0302

0.0300 0.0325 0.0310 0.0320 0.0310 0.0310 0.0300 0.0310 0.0300

For P1 h = 0.0615 in, Lb = 8.0565 in internal diameter = 4.0003 in For P2 h = 0.0606 in, Lb = 8.0470 in internal diameter = 3.9990 in For P1/A h = 0.0608 in, Lb = 8.0390 in internal diameter = 3.9980 in For P2/A h = 0.05987 in, Lb = 8.0360 in internal diameter = 4.0006 in For P3 h = 0.0303 in, Lb = 8.5100 in internal diameter = 3.9997 in


and

and

and

and

and


SG no.7

l Fai

SG .6 no

Specimen P1 Bay no.7 Mag. Filter Taylor-Hobson Talyrond

Tra c

e of i n t e r n al s u rf a

ce

SG = strain gauge

SG no.7

l Fai

SG .6 no Specimen P1 Bay no.7 Filter Mag. Taylor-Hobson Talyrond

Tra

c e of e rf a c xtern al s u

e

4.8 Out-of-circularity plots for model P1 (one division = 0.0001 in).


181

182

Pressure vessels 0·06'' Ring

0·06''

0·06''

x

Shell x

0·94''

4.9 Ring-shell combination for P1 and P1/A.

λ′ =

4

{(L

b

/ Df ) / (t ′ / Df ) 2

3

}×

(σ yp / E )

[4.31]

where Df = diameter of the centroid of a typical ring-shell combination; and t′ = equivalent shell thickness. To demonstrate how λ′, Df and t′ can be determined, the models P1, P2 P1/A and P2/A, all of which failed inelastically, are considered.

4.2.1 P1 and P1/A A typical ring-shell combination for these vessels is shown in Fig. 4.9. From elementary theory: A = area of section = 0.06 in2; y¯ = 0.036 in; Ixx = 9.974 × 10−5 in4; I = Ixx − y¯2 × A = 3.02 × 10−5 in4; where: y¯ = distance of centroid of the ring-shell combination from x–x; Df = (4 + y¯) × 2 = 4.067 in; I = (0.94 × t′3)/12 = 3.02 × 10−5; ∴t′ = 0.073 in. Therefore, for P1, λ′ = 0.816 and Pcr / Pexp = 1.62; and for P1/A, λ′ = 0.856 and Pcr / Pexp = 1.48.

4.2.2 P2 and P2/A Similarly, for P2 and P2/A, A = 0.0612 in2; y¯ = 0.0355 in; Ixx = 1.182 × 10−4 in4 I = 4.11 × 10−5 in4 ∴t′ = 0.0807 in;



183

Df = 4.071 in; Lb = 8.04. Therefore, for P2, λ′ = 0.757 and Pcr / Pexp = 1.89; and for P2/A, λ′ = 0.816 and Pcr / Pexp = 1.58. A plot of 1/λ′ against Pcr /Pexp for these four models, together with the results of other machined models95–97 is shown in Fig. 4.10, which can be seen to be of similar form to Fig. 3.4. It should be noted that the plastic reduction factor Pcr /Pexp, must always be larger than unity, where: Pcr = theoretical general instability pressure, based on simply supported edges, calculated using ANSYS; and Pexp = experimentally obtained general instability pressure. It must be emphasised that for Fig. 4.10 only very precisely made machinestiffened models were used, with little out-of-circularity, and that, in practice, the out-of-circularity will be larger, resulting in a larger plastic reduction factor. That is, if λ′ were plotted against (Pcr/Pexp) for vessels with a larger out-of-circularity, the resulting graph would lie on a line somewhat higher

3.6 3.4 3.2 3.0 2.8 Unsafe side

2.6 2.4

1/λ'

2.2 2.0 1.8 Safe side

1.6 1.4 1.2 1.0

Ross cylinders P4, P5 and P6 Ross cylinders 4, 5 and 6 Ross cylinders P1, P2, P1/A and P2/A Ross cylinders PHR1, PHR2 and PHR3 Reynolds cylinders DD-6, DD-2A, DD-3A and DD-4A Seleim cylinders 1, 3, 5, 8 and 10

0.8 0.6 0.4 0.2 0.0 0

1

2

3

4

5

6

7

8

9

10

Pcr / Pexp

4.10 Design chart for the general instability of machined ring-stiffened circular cylinders using ANSYS.


184

Pressure vessels

3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 1/λ'

2.0 1.8 1.6 1.4 1.2 1.0


0.8 0.6 0.4 0.2 0.0 0

1

2

3

4

5 6 Pcr / Pexp

7

8

9

10

4.11 Design chart for the general instability of ring-stiffened cylinders using Kendrick Part I.

than that indicated in Fig. 4.10. It must also be emphasised that more experimental data are required to give Fig. 4.10 any validity, and also that the vessels used for such an investigation should include different geometries from those of this figure. In addition, two other charts are presented in Figs 4.11 and 4.12, where Pcr was calculated using Kendrick Part I and Kendrick Part III, respectively.

4.3

General instability of ring-stiffened conical shells

Theoretical analysis of these vessels by analytical methods is very difficult, and preferred solutions rely on the finite element method, which can be obtained by combining the stiffness matrices of the ring-stiffeners described in Section 4.1.3 with the stiffness matrices of the conical elements described in Section 3.2. Using such an approach, the present author analysed the ring-stiffened truncated conical shells of Singer,56 Nos. 6, 7 and 8 in Table 4.7. It should be brought to the attention of the reader that in Singer’s



185

3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 1/λ'

2.0 1.8 1.6 1.4 1.2 1.0


0.8 0.6 0.4 0.2 0.0 0

1

2

3

4

5

6

7

8

9

Pcr / Pexp

4.12 Design charts for the general instability of ring-stiffened cylinders, using Kendrick Part III.

Table 4.7 Buckling pressures (P/E × 106) for ring-stiffened cones Singer56 Model no. 6 7 8

Angle (deg) 30 30 20

Stiffened shell 1.263 0.0557 3.54

Finite element solution Equivalent cylinder 1.234 0.045 3.04

Simply supported a

1.104 (10) 0.0462 (9) 3.149 (5)

Clamped

Fixed

1.243 (11) 0.0484 (9) 3.415 (5)

1.423 (12) 0.0649 (11) 4.36 (6)

a The figures in parentheses are the numbers of circumferential lobes or waves, n. P = pressure; E = Young’s modulus.


186

Pressure vessels

original study,56 a small error appeared in the third column of Table 1b, where the dimensionless parameter for cone No. 8 should have been 155 and not 98.4. In addition, as Singer used a dimensionless parameter, and because the finite element solution was a numerical one, it was necessary to make some numerical assumptions. For the numerical solution, it was assumed that the slant length of the vessels was 100 in, and they were stiffened by ten equally spaced ring-stiffeners. The cone angle for model Nos. 6 and 7 was 30°, and for model No. 8 it was 20°. From Table 4.7 it can be seen that there is reasonable agreement between the simply-supported and clamped solutions, based on the finite element method, and Singer’s two solutions. The main reasons for discrepancy may have been partly that Singer presented his results with reference to a dimensionless parameter and partly because of Singer’s assumptions that his ring-stiffened cones were equivalent to simply supported orthotropic conical shells. It is interesting to note that Singer’s simpler equivalent cylindrical shell solution gave relatively good results. Nevertheless, the above arguments are academic for ring-stiffened cones that buckle inelastically, as shorter and thicker vessels are likely to suffer plastic knockdown. Details of experimental results for vessels that fall into this category are shown in Table 4.8, and details of the buckling pressures are shown in Table 4.9. As can be seen from the results of Table 4.9, despite the fact that the models were machined very precisely (Figs 4.13–4.15), the plastic knockdown was quite large for cone 3. It must be brought to the reader’s attention that the theoretical solution was based on the finite element method for Fig. 4.16 and this theory assumed that the larger ends of the cones were fixed whereas their smaller ends were clamped. For similar charts of the inelastic instability of ring-stiffened cones, Kendrick Parts I and III were used, respectively, for two more design charts, as shown in Figs 4.17 and 4.18, respectively. Before buckling the cones, vibration tests were carried out on these vessels and, for this reason, Figs 4.13–4.15 show out-of-circularity plots before and after vibration. Furthermore, according to the experimental observations of references 98 and 99, cone 1 appeared to buckle elastically, whereas cones 2 and 3 appeared to buckle inelastically. In order to determine the plastic reduction factor, it is suggested that a thinness ratio similar to λ′ as described in Section 4.2 is used in the same manner as that described in Section 3.1.3. In this instance, the finite element was used to calculate Pcr, where the circular truncated conical shell and the ring stiffener elements of Ross91 were used; the computer program was called RCONEBUR. These elements are described in much detail in this book. Once again, the boundary conditions for these vessels were assumed to be fixed at one end and clamped at the other end. The calculation for λ′



187

Table 4.8 Geometrical details of ring-stiffened cones Model no.

N

b (mm)

d (mm)

e (mm)

1 2 3

6 6 7

1.016 1.016 1.016

1.016 1.524 2.032

−0.826 −1.080 −1.333

101.6 mm

38.1 mm

211.0 mm

0.635 mm Cone no.

A

B

1

1.0

1.0

2

1.0

1.5

3

1.0

2.0

B A

d = depth of ring-stiffener or ‘rectangular’ cross-section. e = ring eccentricity (negative as the rings were external). R1 = radius of shell at small end = 1.905 cm. R2 = radius of shell at large end = 5.08 cm. h = shell thickness = 0.635 mm.

Table 4.9 Buckling pressures for the three cones Cone number

Experimental (MPa)

Elastic theory (MPa)

1 2 3

2.98 (4)a 3.93 (4) 4.10 (3, 4)

3.55 (4) 5.48 (4) 6.65 (3)

a

The number of lobes is given in parentheses.


188

Pressure vessels

Major buckle

0.0001''

0.0001''

(a)

(b)

4.13 Initial out-of-circularity plots for cone 1: (a) before vibration; and (b) after vibration.

Buckle

0.00005''

0.00005''

(a)

(b)




189

0.0001''

0.00005''

Buckle

(a)

(b)


2.0 1.8 1.6 1.4

1 / λ'

1.2 Safe side

1.0 0.8 0.6

Cones Cones Cones Cones

0.4 0.2

0.0

1.0

2.0

3.0

4.0

5.0

1, 2, 3 4, 5, 6 7, 8, 9 10, 11, 12

6.0

Pcr / Pexp

4.16 Design chart for the general instability of machined ring-stiffened circular conical shells using RCONEBUR.


190

Pressure vessels 2.0

1.8

1.6

1.4

1.2 1 / λ'

Safe side 1.0

0.8

0.6 Cones 1, 2, 3 Cones 4, 5, 6

0.4

Cones 7, 8, 9 Cones 10, 11, 12

0.2

0.0 0.0

1.0

2.0

3.0

4.0

5.0

PKD1 (Pcr1 / Pexp )

4.17 Design chart for the general instability of ring-stiffened cones using Kendrick Part I, buckling pressure, namely Pcr1.

was based on an equivalent ring-stiffened circular cylinder and it is described in detail by Ross et al.96,100–102 and Fig. 3.9. The design chart, shown in Fig. 4.16 is somewhat limited owing to the lack of available experimental data, but it appears that machined vessels which had a value of 1/λ′

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