Impact Tt

Contents Preface to the Second Edition page xi

Preface to the First Edition xiii

1 Static Plastic Behaviour of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction 1

1.2 Basic Equations for Beams 2

1.3 Plastic Collapse Theorems for Beams 5

1.4 Static Plastic Collapse of a Cantilever 8

1.5 Static Plastic Collapse of a Simply Supported Beam 10

1.6 Static Plastic Collapse of a Fully Clamped Beam 12

1.7 Static Plastic Collapse of a Beam Subjected

to a Concentrated Load 13

1.8 Static Plastic Collapse of a Partially Loaded Beam 14

1.9 Experiments on Beams 17

1.10 Final Remarks 19

problems 19 2 Static Plastic Behaviour of Plates and Shells . . . . . . . . . . . . . . . . . . . . 21 2.1 Introduction 21

2.2 Generalised Stresses and Strains 21

2.3 Basic Concepts 23

2.4 Plastic Collapse Theorems 26

2.5 Basic Equations for Circular Plates 27

2.6 Static Plastic Collapse Pressure of Circular Plates 30

2.7 Basic Equations for Rectangular Plates 33

2.8 Static Plastic Collapse Pressure of Rectangular Plates 34

2.9 Basic Equations for Cylindrical Shells 41

2.10 Static Collapse Pressure of a Long Reinforced Cylindrical Shell 45

2.11 Static Plastic Collapse of a Ring-Loaded Cylindrical Shell 48

2.12 Experiments on Plates and Shells 51


problems 57 vii in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01096-3 - Structural Impact: Second Edition Norman Jones Frontmatter More information viii Contents

3 Dynamic Plastic Behaviour of Beams . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Introduction 59

3.2 Governing Equations for Beams 61

3.3 Simply Supported Beam, pc p0 3pc 62

3.4 Simply Supported Beam, p0 > 3pc 66 3.5 Simply Supported Beam Loaded Impulsively 78

3.6 Fully Clamped Beam, pc p0 3 pc 83

3.7 Fully Clamped Beam, p0 > 3 pc 85 3.8 Impact of a Mass on a Fully Clamped Beam 86

3.9 Impact of a Cantilever Beam 94


problems 103 4 Dynamic Plastic Behaviour of Plates . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1 Introduction 105

4.2 Governing Equations for Circular Plates 107

4.3 Annular Plate Loaded Dynamically 107

4.4 Simply Supported Circular Plate Loaded Dynamically,

pc p0 2pc 115

4.5 Simply Supported Circular Plate Loaded Dynamically,

p0 > 2pc 120 4.6 Fully Clamped Circular Plate Loaded Impulsively 131

4.7 Governing Equations for Rectangular Plates 133

4.8 Simply Supported Square Plate Loaded Dynamically,

pc p0 2pc 135

4.9 Simply Supported Square Plate Loaded Dynamically, p0 > 2pc 140 4.10 Final Remarks 146

problems 149 5 Dynamic Plastic Behaviour of Shells . . . . . . . . . . . . . . . . . . . . . . . . 151 5.1 Introduction 151

5.2 Governing Equations for Cylindrical Shells 152

5.3 Long Cylindrical Shell 153

5.4 Long Reinforced Cylindrical Shell 158

5.5 Fully Clamped Short Cylindrical Shell 162

5.6 Elastic, Perfectly Plastic Spherical Shell Subjected to a

Spherically Symmetric Dynamic Pressure 170

5.7 Shallow Shells 190

5.8 Some Comments on Spherical Shells 196

5.9 Influence of Pressure Pulse Characteristics 197


problems 205 6 Influence of Transverse Shear and Rotatory Inertia . . . . . . . . . . . . . . 206 6.1 Introduction 206

6.2 Governing Equations for Beams 208 in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01096-3 - Structural Impact: Second Edition Norman Jones Frontmatter More information Contents ix

6.3 Transverse Shear Effects in a Simply Supported Beam

Loaded Impulsively 209

6.4 Impact of a Mass on a Long Beam 225

6.5 Transverse Shear Effects in a Simply Supported Circular Plate 231

6.6 Transverse Shear Effects in Cylindrical Shells 243


problems 265 7 Influence of Finite Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.1 Introduction 267

7.2 Static Plastic Behaviour of a Beam Subjected

to a Concentrated Load 269

7.3 Static Plastic Behaviour of Circular Plates 281

7.4 Static Plastic Behaviour of Rectangular Plates 283

7.5 Dynamic Plastic Behaviour of Rectangular Plates 291

7.6 Dynamic Plastic Behaviour of Beams 302

7.7 Dynamic Plastic Behaviour of Circular Plates 304

7.8 Dynamic Plastic Behaviour of a Circular Membrane 309

7.9 Mass Impact Loading of Plates 311


problems 326 8 Strain-Rate-Sensitive Behaviour of Materials . . . . . . . . . . . . . . . . . . 327 8.1 Introduction 327

8.2 Material Characteristics 329

8.3 Constitutive Equations 340

8.4 Theoretical Solutions of Idealised Models 347

8.5 Theoretical Behaviour of Strain-Rate-Sensitive

Structures 363


problems 375 9 Dynamic Progressive Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 9.1 Introduction 377

9.2 Static Axial Crushing of a Circular Tube 381

9.3 Dynamic Axial Crushing of a Circular Tube 385

9.4 Static Axial Crushing of a Square Tube 392

9.5 Dynamic Axial Crushing of a Square Tube 396

9.6 Comparison of the Axial Crushing Characteristics of Circular

and Square Tubes 397

9.7 Some Comments on Energy Absorption Systems 399

9.8 Structural Crashworthiness 406

9.9 Structural Protection 417


problems 423 in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01096-3 - Structural Impact: Second Edition Norman Jones Frontmatter More information x Contents

10 Dynamic Plastic Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 10.1 Introduction 425

10.2 Dynamic Elastic Buckling of a Bar 427

10.3 Dynamic Plastic Buckling of a Bar 437

10.4 Dynamic Plastic Buckling of a Circular Ring Subjected

to an External Impulse 442

10.5 Dynamic Axial Plastic Buckling of a Long Cylindrical Shell 457

10.6 Critical Impulsive Radial Velocity for Collapse of a Cylindrical

Shell without Buckling 472

10.7 Final Comments 473

problems 478 11 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 11.1 Introduction 479

11.2 Introduction to Geometrically Similar Scaling 479

11.3 Phenomena Which Do Not Scale Geometrically 484

11.4 Dimensional Analysis 486

11.5 Crack Propagation in Elastic Structures 492

11.6 DuctileBrittle Fracture Transitions 495 11.7 Experimental Results on the Scaling of Structures

Loaded Dynamically 499

11.8 Final Comments 508

problems 510 APPENDIX 1: Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

APPENDIX 2: Path-Dependence of an Inelastic Material . . . . . . . . . . . . . . . 514

APPENDIX 3: Principle of Virtual Velocities . . . . . . . . . . . . . . . . . . . . . . . . 516

APPENDIX 4: Consistent Sets of Equilibrium Equations

and Geometrical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 517

APPENDIX 5: Buckingham -Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 APPENDIX 6: Quasi-Static Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

APPENDIX 7: Martins Upper Bound Displacement Theorem . . . . . . . . . . . . 538 References 541

Answers to Selected Problems 571

Author Index 575

Subject Index 580 in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01096-3 - Structural Impact: Second Edition Norman Jones Frontmatter More information

Preface to the Second Edition The general field of structural impact has expanded significantly since the preparation

of the first edition of this book more than 20 years ago. This expansion is driven

partly by the quest for the design of efficient structures which require more accurate

safety factors against various types of dynamic loadings causing large plastic

strains. The enhancement of safety in many industries, including transportation,

has become more prominent in recent years, as well as the protection of structures

and systems against terrorist attacks. In tandem with these developments and

enhanced requirements, rapid advances have occurred in numerical analyses, which

have outpaced, in many ways, our understanding of structural impact. Nevertheless,

numerical schemes are used throughout design offices. This book emphasises the

basic mechanics of structural impact in order to gain some insight into its broad field.

It is important that an engineering designer has a good grasp of the mechanics which

underpin this highly nonlinear and complex engineering field.

The book attempts to achieve this aim through an analysis of simple models

which expose the basic aspects of the response, an understanding of which will pay

dividends when interpreting the results emanating from both experimental studies

and numerical calculations. For example, the issues raised in Chapters 8 and 11,

on material strain rate sensitivity and scaling, respectively, are certainly important

for both numerical calculations as well as experimental programmes. In some cases,

the equations presented in this book are suitable for preliminary design purposes,

particularly when bearing in mind frequent uncertainties in the input data. For

example, the values of coefficients and form of dynamic constitutive equations are

often approximate, and there are difficulties in specifying the correct details for the

boundary conditions at joints, etc., and in obtaining the characteristics of the external

dynamic loadings which arise from impact, explosive and large dynamic loadings.

The first five basic chapters of this book remain largely unchanged, except

for some slight improvements here and there to aid clarity and the addition of

two appendices, one on quasi-static behaviour and the other giving the proof of

a displacement bound theorem. Recent developments on these topics have been

confined largely to solutions for special cases and numerical studies. Considerable

research effort has been expended, over the last few decades, on the topics studied

in the last six chapters. Therefore, these chapters of the book have been updated xi in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01096-3 - Structural Impact: Second Edition Norman Jones Frontmatter More information xii Preface to the Second Edition

selectively. However, more research is required, particularly for Chapters 8 to 11,

before structural impact is a fully mature subject.

The author wishes to thank Mr Peter Gordon of Cambridge University Press

for his assistance and his invitation to prepare this second edition. I also wish to

thank the many people who have suggested improvements since the publication of

the paperback edition, particularly Professor M Alves and Dr Q. M. Li. I also wish

to thank Mrs I. Arnot for assistance with the new figures, and last but not least to

acknowledge the valuable support of my wife, Jenny.

Norman Jones

February 2011 in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01096-3 - Structural Impact: Second Edition Norman Jones Frontmatter More information

Preface to the First Edition Impact events occur in a wide variety of circumstances, from the everyday occurrence

of striking a nail with a hammer to the protection of spacecraft against meteoroid

impact. All too frequently, we see the results of impact on our roads. Newspapers and

television report spectacular accidents which often involve impact loadings, such as

the collisions of aircraft, buses, trains and ships, together with the results of impact

or blast loadings on pressure vessels and buildings due to accidental explosions

and other accidents. The general public is becoming increasingly concerned about

safety, including, for example, the integrity of nuclear transportation casks in various

accident scenarios involving impact loads.

Clearly, impact is a large field which embraces both simple structures (e.g., nails)

and complex systems, such as the protection of nuclear power plants. The materials

which are impacted include bricks, concrete, ductile and brittle metals, and polymer

composites. Moreover, on the one hand, the impact velocities may be low and give

rise to a quasi-static response, or, on the other hand, they may be sufficiently large

to cause the properties of the target material to change significantly.

In this book, I have concentrated on the impact behaviour of ductile structures

and, in particular, beams, plates and shells. Most complex engineering systems are

constructed largely of these simple structural members, so that an understanding of

their response is an essential prerequisite for revealing the dynamic behaviour of a

more complex system. The topic remains a large one, and so I have specialised it

further by focusing on large impact loads producing plastic strains which dominate

the elastic effects.

A dynamic load causes elastic and plastic stress waves to propagate through

the thickness of this class of structures, as well as produces an overall structural

response. The propagation of stress waves through the structural thickness can cause

failure by spalling when the shock or impact loads are sufficiently severe. This

phenomenon occurs in the same order of time that it takes a stress wave to propagate

through the thickness of the structure. Thus, this type of failure usually occurs

within microseconds of initial impact, and is sometimes referred to as the early time response to distinguish it from the gross structural behaviour which occurs at later times. The early time response of structural members is not considered further. xiii in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01096-3 - Structural Impact: Second Edition Norman Jones Frontmatter More information xiv Preface to the First Edition

This book focuses on the long-term behaviour of structures (typically of the order

of milliseconds for small structures), for which the external dynamic load is assumed

to impart momentum instantaneously to the middle surface of a structure (i.e.,

transverse wave propagation is disregarded). It is customary practice to uncouple

the early time wave propagation behaviour from the long-time or gross structural

response because the time durations of these two phenomena usually differ by a few

orders of magnitude. Obviously, a gross structural analysis cannot be used to predict

the detailed behaviour through the structural thickness. It is necessary, therefore, to

establish separately whether or not failure due to spalling can occur in a given case.

Although the static plastic behaviour of structures was first studied last century

(e.g., J. A. Ewing, The Strength of Materials, Cambridge University Press, 1899),

systematic investigations on the dynamic plastic behaviour are much more recent.

Serious studies appear to have commenced during the Second World War, when, for

example, J. F. Baker designed the Morrison air raid shelters to protect people from

falls of masonry in their own homes, G. I. Taylor studied the dynamic response of thin

plates, and Pippard and Chitty examined the dynamic behaviour of cylindrical shells

for research into submarine hulls. Considerable research activity and progress have

been reported over the past forty years, some of which is discussed in this book or

cited in the references. Generally speaking, this body of work seeks the response of a

structural member when subjected to a known impact load. However, the theoretical

solutions may also be used for diagnostic or forensic purposes. Lord Penney and his

colleagues, for example, estimated the nuclear explosive yields at Hiroshima and

Nagasaki by calculating the impact loads required to cause the permanent damage

which was observed for bent or snapped poles, squashed empty drums or cans, tops

of office cabinets pushed in by the blast, etc. Coupling between the external impact

loading and the structural response is a difficult topic, which is not well understood,

and is disregarded in this book.

Despite restricting our attention to the impact behaviour of beams, plates and

shells subjected to large impact loads, the field is still large, active and growing

rapidly. The results of studies in this area are being used to guide the development

of rational design procedures which avoid the destructive action of earthquakes on

buildings and to improve the collision protection of passengers in automobiles, trains,

buses and aircraft. The collision protection of automobiles and buses is achieved by

improving the interior and exterior energy-absorbing capabilities of the vehicles and

incorporating the principles of structural crashworthiness into the design of highway

safety systems and roadside furniture.

Theoretical methods have been used to design impact absorbers of various

types, as well as to assess the safety of reactor tubes subjected to violent transient

pressure pulses, which can arise in certain circumstances in sodium-cooled fastbreeder

reactors. The slamming damage which is sustained by the bottom plating

of ships and hydrovehicles has been estimated using these methods which have

also been employed to design buildings to withstand internal gaseous explosions.

The response of re-entry vehicles, structural crashworthiness of offshore platforms,

safety calculations for industrial plants, various military applications, interpretation

of constitutive equations from dynamic ring tests, and even the denting of aircraft

surfaces due to hail has been studied using the various methods discussed in this

book. Many other practical applications have been made, and, no doubt, more

Documents

Impact Tt