Introduction

The dynamic plastic progressive buckling of thin-walled tubes subjected to axial impact loads was examined in Chapter 9. The impact loads were applied sufficiently slowly so that neither the axial nor the lateral inertia effects of the tubes played a significant role during the response. The duration of the impact loading was much longer than the transit time of an elastic stress wave which propagates along the length of a tube, as shown in § 9.8.3. A tube was unable, therefore, to support a mean dynamic axial load which was larger than the corresponding static value when disregarding the influence of material strain rate sensitivity examined in Chapter 8. Thus, the deformed profile of a tube is similar in this case for both static buckling and dynamic progressive buckling and a quasi-static theoretical analysis gave satisfactory agreement with the corresponding experimental results, as discussed in Chapter 9.

If a thin-walled tube, or other structural member, is subjected to a sufficiently severe dynamic axial load, then structural inertia effects produce the phenomenon of dynamic plastic buckling. In this circumstance, the deformed shape of the structure may be quite different from the corresponding progressive buckling profile, as illustrated in Figure 10.1 for an axially loaded circular tube. The shell is wrinkled over the entire length when buckled dynamically, unlike the dynamic progressive buckling case with wrinkling confined to one end. This situation should be contrasted with Figure 10.2, which shows the dynamic plastic buckling of a rod subjected to an axial impact load. The wrinkling is confined to the impacted end in this case, whereas a lateral deformation profile with a low mode number would be likely to develop over the entire length for static axial loads.

Introduction

An introduction to the static plastic behaviour of beams was presented in the preceding chapter, together with some theoretical solutions for several problems. It was observed that the idealisation of a perfectly plastic material, which is shown in Figures 1.3 and 1.4, is particularly attractive and simplifies considerably any theoretical calculations for the static plastic collapse load of a beam. Moreover, Figure 1.14 shows that the theoretical predictions for the static plastic collapse load give reasonable agreement with the corresponding experimental results on freely supported steel beams subjected to a central concentrated load. Good agreement between experimental results and the corresponding theoretical predictions for the static plastic collapse loads of many beams and frames may be found in the articles cited in the previous chapter.

The general concepts introduced in the previous chapter for beams are now used to study the static plastic collapse behaviour of plates and shells, which are important practical structures found throughout engineering. However, the theoretical analyses are more complex than for beams because the plastic flow in plates and shells is controlled by multi-dimensional yield criteria.

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.

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.

Introduction

The theoretical solutions for the static and dynamic plastic behaviour of beams, plates and shells were developed in the previous chapters for infinitesimal displacements. In other words, the equilibrium equations, which govern the response of these structures, were obtained using the original undeformed configuration. For example, the equilibrium equations (1.1) and (1.2) were derived for the beam element shown in Figure 1.1 by ignoring any deformations which would develop under the action of the external loads. It turns out that theoretical analyses, which incorporate this simplification, are often capable of predicting static plastic collapse loads which agree with the corresponding experimental results, as indicated in Figure 1.14 for the transverse loading of metal beams without any axial restraints at the supports.

It may be recalled from § 1.9 that the concentrated load which may be supported by an axially restrained rigid, perfectly plastic beam exceeds the associated static plastic collapse load for finite transverse displacements. In fact, it is evident from Figure 1.15 that the external load is about twice the associated static plastic collapse value when the maximum permanent transverse displacement equals the beam thickness. However, an element of the original beam cross-section is deformed severely and displaced considerably from its initial position, as sketched in Figure 7.1. This change of geometry is ignored in the theoretical methods which are presented in the previous chapters. It is evident that the equilibrium equations (1.1) and (1.2) no longer control the behaviour of a deformed beam element, and different equations are required. It transpires that the most important effect of this change is the development of the membrane, or in-plane, force, N, which is shown in Figure 7.1.

Introduction

Many ductile materials which are used in engineering practice have a considerable reserve capacity beyond the initial yield condition. The uniaxial yield strain of mild steel, for example, is 0.001 approximately, whereas this material ruptures, in a standard static uniaxial tensile test, at an engineering strain of 0.3, approximately. This reserve strength may be utilised in a structural design to provide a more realistic estimate of the safety factor against failure for various extreme loads. Thus, the static plastic behaviour of structures has been studied extensively and is introduced in many textbooks. An interested reader is referred to these textbooks for a deeper presentation of the subject than is possible in this book, which is concerned primarily with the influence of dynamic loadings. However, the methods of dynamic structural plasticity presented in this book owe a substantial debt to the theoretical foundation of static structural plasticity, which is, therefore, reviewed briefly in this chapter and the following one.

A considerable body of literature is available on the static behaviour of structures made from ductile materials which may be idealised as perfectly plastic. This simplification allows the principal characteristics and overall features of the structural response to be obtained fairly simply for many important practical cases. Moreover, the static collapse loads predicted by these simplified methods often provide good estimates of the corresponding experimental values. Indeed, the design codes in several industries now permit the use of plasticity theory for the design of various structures and components. The theoretical background of these methods, which were developed primarily to examine the static loading of structures made from perfectly plastic materials, are valuable for studies into the response of structures subjected to dynamic loads. Thus, this chapter and the next focus on the static behaviour of structures which are made from perfectly plastic materials.

Introduction

The two previous chapters have examined the response of beams and plates when made from rigid, perfectly plastic materials and subjected to large dynamic loads. This chapter employs similar methods of analysis to study the dynamic stable response of shells.

Shells are thin-walled structural members having either one non-zero curvature (e.g., cylindrical and conical shells) or two non-zero curvatures (e.g., spherical and toroidal shells). They are used throughout engineering for storage (e.g., gas storage tanks), transportation (e.g., pipelines and railway tank cars) and for protection purposes (e.g., crash helmets), and are vital components of submersibles, offshore platforms, chemical plant and many other applications.

Introduction

The testing of small-scale models is indispensable for complex structural systems which are difficult to analyse theoretically and numerically or to study experimentally. The dynamic response of underground structures, impact of nuclear fuel capsules, missile impact of nuclear power installations and collision protection of ships illustrate several areas which have been studied with the aid of small-scale models.