Crush MeChaniCs of Thin-Walled Tubes © 2016 by Taylor & Francis Group, LLC © 2016 by Taylor & Francis Group, LLC Crush MeChaniCs of Thin-Walled Tubes Dai-heng Chen Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business © 2016 by Taylor & Francis Group, LLC CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20151022 International Standard Book Number-13: 978-1-4987-5518-4 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2016 by Taylor & Francis Group, LLC Contents Preface vii Notation xi 1 Axial compression of circular tubes 1 1.1 Axisymmetric buckling of circular tubes . . . . . . . . . . . . 3 1.2 Axisymmetric crushing of circular tubes after buckling . . . 30 1.3 Non-axisymmetric crushing of circular tubes . . . . . . . . . 59 1.4 Deformation modes in the axial crushing of circular tubes . . 73 1.5 Effects of stiffeners . . . . . . . . . . . . . . . . . . . . . . . . 81 2 Axial compression of square tubes 91 2.1 Characteristics of axial compressive deformation of square tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.2 Buckling and collapse force of a plate . . . . . . . . . . . . . 97 2.3 Buckling and collapse force of square tubes . . . . . . . . . . 123 2.4 Average force in the axial crushing of square tubes . . . . . . 127 2.5 Influence of partition walls on the axial crushing of polygonal tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3 Axial compression of corrugated tubes 157 3.1 Axially groovedcircular tubes . . . . . . . . . . . . . . . . . 157 3.2 Axially corrugatedtubes . . . . . . . . . . . . . . . . . . . . 160 3.3 Circumferentially corrugated tubes . . . . . . . . . . . . . . . 190 4 Bending of tubes 201 4.1 Bending deformation of a cylinder . . . . . . . . . . . . . . . 201 4.2 Bending deformation of rectangular tubes . . . . . . . . . . . 230 5 Thin-walled structures with an open cross section 255 5.1 Basic mechanisms of plastic collapse of plates . . . . . . . . . 256 5.2 Plastic collapse during axial crushing . . . . . . . . . . . . . 261 5.3 Bending collapse due to buckling . . . . . . . . . . . . . . . . 264 5.4 Bending collapse due to cross-sectionalflattening . . . . . . . 274 v © 2016 by Taylor & Francis Group, LLC vi Contents 6 Torsion 283 6.1 Shear buckling of plates . . . . . . . . . . . . . . . . . . . . . 283 6.2 Torsion of cylinders . . . . . . . . . . . . . . . . . . . . . . . 284 6.3 Torsion of square tubes . . . . . . . . . . . . . . . . . . . . . 295 Bibliography 309 Index 327 © 2016 by Taylor & Francis Group, LLC Preface Many thin-walledstructures are used in vehicles as impact-energyabsorption components,anditis quiteimportanttocorrectlyevaluatecrushingphenom- ena exhibited by such structures during impact. In 2002, the author began a new research project aimed at development of novel collision devices for use in vehicles. These devices were developed within a consortium including a number of Japanese automotive parts suppliers. Since that time, there has beenextensiveinvestigationintheliteratureonthecrushingbehaviorsofthin- walledstructures,andmanyresearchresultshavebeenwrittenandpublished. These describe the current understanding of how such crushing deformations happen and which factors determine the crushing behaviors. Initially, these reports were used as seminar documents for graduate students and partners within the consortium.These reports provedpopular among the initial recip- ients,and itwas decidedthat commercialprinting wouldbe ofinterestto the broader scientific and engineering community. The purpose of this book is to provide a basic understanding of the fun- damental concepts and mechanisms of crushing deformations in thin-walled structures. The hope is that interested readers, particularly beginners, can easily and systematically grasp the knowledge required in the analysis and design of energy absorptioncomponents. Although there are many textbooks on buckling and post-buckling of thin-walled structures, there are few books of a similar nature aimedat those new to the field. In this book, the crushing deformations of circular and square tubes—the simplest hollow thin-walled structures—under axial compression, bending, and torsion are explained in detail. The book focuses on these simple and basic problems and attempts in this way to provide readers with a clear physical understanding and useful analytical techniques. When examining the deformation behaviors of thin-walled structures un- derdynamicloading,suchasinacollision,itisnecessarytoaccountforvarious effects, suchas inertialforce andthe strain-ratedependence ofthe dynamical characteristics of specific materials. However,in order to graspsuch dynamic behavior, it is illuminating to first investigate the deformation behaviors of thin-walled structures under a quasi-static load, for which the influences of inertial force, strain-rate dependence, and other such factors can be ignored. Moreover, the impact velocity during a vehicle crushing accident is about 40 m/satmost.Theinfluenceofthestress-wavespeedondeformationcharacter- isticsduringimpactweakensastheimpactvelocitydecreases.Thiswasshown in a study that the author conducted on the effect of impact velocity on the vii © 2016 by Taylor & Francis Group, LLC viii Preface peak load of circular tubes subjected to axial compression. Taking into con- sideration the aims mentioned above, this book discusses mainly quasi-static crushing behaviors. Thestructureofthisbookisasfollows.Chapter1addressesaxialcompres- sion of circular tubes, and Chapter 2 addresses axial compression of square tubes. Through these two chapters, the underlying concepts, such as features of buckling and post-buckling behavior of thin-walled hollow members, are shown in detail. In particular, the mechanism of collapse and associated cal- culationsforthe initialpeakforceandthe averagecompressiveforcearesum- marized. The geometry of a tube is an important factor in its energy absorption capacity. One well-known way, described by many researchers, to create a high-performanceenergyabsorptiondevice is to introduce corrugationsalong theaxisorcircumferenceofatube.Chapter3discusseshowintroducingaxial and circumferential corrugations affects the force-displacement and energy- absorption characteristics of tubes. In general, both an axial compressive load and a bending moment are applied to allmembers ofa vehicle when a collisionoccurs.It is therefore im- portant to understand the collapse behavior of members undergoing bending deformation when trying to evaluate strength and energy-absorption charac- teristics. Chapter 4 discusses the bending deformation of circular and square tubes. In particular, the characteristic flattening phenomenon, the maximum moment in bending deformation, and the moment-rotation relation during bending collapse are discussed in this chapter. Thin-walled members with an open cross section are commonly used as structural components. Chapter 5 discusses the collapse behavior of thin- walledstructureswithanopencrosssectionduringaxialcrushingandbending deformation. Pure torsional load is seldom applied to thin-walled members in practice, but both axial and bending deformations are frequently accompanied by tor- sional load. To understand deformation behavior under such combined loads, wemustfirstclarifythepropertiesofcollapsedeformationofthin-walledmem- bers under torsional loads. Chapter 6 discusses torsion of tubes. Data related to the topics of this book that have been made public are selected and summarized throughout the book. However, in order to obtain a more precise understanding, the author conducted a series of investigations about these issues. The book therefore contains many new results obtained by the author and members of the research consortium to which the author belongs. The investigations are carried out mainly by means of numerical simulation using the finite element method (so-called FEM analysis) because moderncomputershaveadvancedgreatly,andnowadays,computeranalysisis on equal footing with laboratory experiments. Therefore, to demonstrate the maximum load in collapse and the load-displacement relation after collapse for the circular and square tubes subjected to axial compression, bending, or torsion, the author also shows a comparison with numerical results at the © 2016 by Taylor & Francis Group, LLC Preface ix sametimethattheexistingtheoreticalapproachesareintroduced.Inaddition, some other originalresearchresults, beyond the comparisons,are included in the book. These results are summarized as follows. In Chapter 1, “Axial compression of circular tubes,” the effect of flanges in cylinder ends on the deformation mode of tubes, as well as conditions of changes in deformation modes, were investigated. Additionally, formulas are developed for evaluating the curvature radius ρ at the wrinkle tip, the f eccentricity ratio m and the maximum strain ε . θ max | InChapter2,“Axialcompressionofsquaretubes,”itisfoundthatthede- formationofsquaretubesischaracterizedbytwostages,whichisnotthecase with circular tubes. First, shallow buckles are created over the entire length of the square tube, after which they are crushedone by one.Additionally, for the collapse behavior of a plate subjected to axial compression, it is shown that the axial collapse of the plate is dominantly controlled by two factors: the stress limitations of the material and the limitations on in-plane lateral deformation at the side edges of the plate. Further, a simplified method is proposedfor predictingsuchcollapsestressesasa function ofthe two factors. In Chapter 3,“Crushing of corrugatedtubes,” the deformationcharacter- istics of tubes with a surface corrugatedin the axialor radialdirection under axial loading are investigated systematically on the basis of the author’s re- search results. In particular, it is shown that the axisymmetric deformations corresponding to the introduced axial corrugations can be classified into two modes: S-mode deformation and P-mode deformation. The crushing force in S-mode deformation continues increasing without fluctuation. Additionally, a simple method is proposed for predicting the average crushing force of a corrugated tube. In Chapter 4 “Bending collapseoftubes,” the flattening phenomenonand the maximum bending moment are investigated in detail. This includes the propositionofanewmethodforevaluatingthe maximumbending momentof square tubes with consideration of sidewall buckling. In Chapter 5, “Thin-walled structure with an open cross section,” the mechanism of collapse due to buckling is analyzed by using U-shaped and V-shaped beams (with compressive stress applied to the flange) as examples, andthe mechanismofcollapsedue tocross-sectionalflatteningis analyzedby usingV-shapedandW-shapedbeams(withtensilestressappliedtotheflange) asexamples.Forflattening-inducedcollapse,becauseitistypicallydifficultto evaluate the correspondence between cross-sectional flattening and rotation angle,theauthorproposesanewtechniquethatcanbeusedtodeterminethe relation between the bending moment M and the rotation angle θ. In Chapter 6, “Torsion,” it is found that there are two possible cases for torsion-inducedcollapse ofsquaretubes, as for bending collapse:collapse due tobucklingandcollapsedue toflattening,andanalysismethods areproposed for predicting the maximum torsion moment in each case. The author believes that this book will be helpful for deepening read- ers’ understanding of the basic concepts and analysis techniques of crushing © 2016 by Taylor & Francis Group, LLC x Preface deformation of thin-walled structures. The target audience for this book is engineering students, practicing mechanical and structural engineers, and re- searchers interested in analyzing energy absorption and designing structures that may undergo impacts. The book can be used for both teaching and self- study. TheauthorwouldliketothankDr.KuniharuUshijimaforhiscooperation in research,valuable advice and support in manuscript preparation. Jiangsu University, Professor Tokyo University of Science, Professor Emeritus Chen Daiheng © 2016 by Taylor & Francis Group, LLC