Crush mechanics of thin-walled tubes

342 Pages·2016·8.327 MB·English

by  Chen, Dai-heng

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. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-
ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.
com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
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