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Nonlinear models of elastic and visco-elastic onedimensional continuous structures (beams and cables) are formulated by the authors of this title. Several models of increasing complexity are presented: straight/curved, planar/non-planar, extensible/inextensible, shearable/unshearable, warpingunsensitive/ sensitive, prestressed/unprestressed beams, both in statics and dynamics. Typical engineering problems are solved via perturbation and/or numerical approaches, such as bifurcation and stability under potential and/or tangential loads, parametric excitation, nonlinear dynamics and aeroelasticity.
Contents
1. A One-Dimensional Beam Metamodel. 2. Straight Beams. 3. Curved Beams. 4. Internally Constrained Beams. 5. Flexible Cables. 6. Stiff Cables. 7. Locally-Deformable Thin-Walled Beams. 8. Distortion-Constrained Thin-Walled Beams.
E-Book Content
Mathematical Models of Beams and Cables
To Fiorella (A.L.) To my parents Elio & Marisa (D.Z.)
Mathematical Models of Beams and Cables Angelo Luongo Daniele Zulli Series Editor Noël Challamel
First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK
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© ISTE Ltd 2013 The rights of Angelo Luongo and Daniele Zulli to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2013948060 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-421-7
Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY
Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction
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xi xiii
List of Main Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Chapter 1. A One-Dimensional Beam Metamodel . . . . . . . . . . . . . . .
1
1.1. Models and metamodel . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Internally unconstrained beams . . . . . . . . . . . . . . . . . . . . . 1.2.1. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3. The hyperelastic law . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4. The Fundamental Problem . . . . . . . . . . . . . . . . . . . . 1.3. Internally constrained beams . . . . . . . . . . . . . . . . . . . . . . 1.3.1. The mixed formulation for the internally constrained beam . . kinematics and constraints . . . . . . . . . . . . . . . . . . . . . 1.3.2. The displacement method for the internally constrained beam 1.4. Internally unconstrained prestressed beams . . . . . . . . . . . . . . 1.4.1. The nonlinear theory . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2. The linearized theory . . . . . . . . . . . . . . . . . . . . . . . 1.5. Internally constrained prestressed beams . . . . . . . . . . . . . . . . 1.5.1. The nonlinear mixed for