Analysis of Thin-Walled Beams 9789811977718, 9789811977725

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Solid Mechanics and Its Applications

Yoon Young Kim Gang-Won Jang Soomin Choi

Analysis of Thin-Walled Beams

Solid Mechanics and Its Applications Founding Editor G. M. L. Gladwell

Volume 257

Series Editors J. R. Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Anders Klarbring, Mechanical Engineering, Linköping University, Linköping, Sweden

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professors Barber and Klarbring welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Dr. Mayra Castro, Senior Editor, Springer Heidelberg, Germany, email: [email protected] Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink.

Yoon Young Kim · Gang-Won Jang · Soomin Choi

Analysis of Thin-Walled Beams

Yoon Young Kim Department of Mechanical Engineering Seoul National University Seoul, Korea (Republic of)

Gang-Won Jang Department of Mechanical Engineering Sejong University Seoul, Korea (Republic of)

Soomin Choi School of Mechanical Engineering Kyungpook National University Daegu, Korea (Republic of)

ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-981-19-7771-8 ISBN 978-981-19-7772-5 (eBook) https://doi.org/10.1007/978-981-19-7772-5 © Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book presents an extensive, thorough treatment for a structural analysis of thin-walled beam structures using an advanced beam theory. Beams, slender structural members having a high length-to-thickness ratio, can be efficiently analyzed even with one-dimensional beam theories such as the Euler and Timoshenko beam theories. However, these theories cannot describe sectional deformations that are non-negligible in thin-walled beams because they employ only three rigid-body displacements and three rigid-body rotations of the cross-section. Therefore, a higher-order beam theory, which includes additional degrees of freedom representing the deformation of the beam section in addition to the six rigid-body degrees of freedom, is needed to analyze thin-walled beams accurately. This book presents a newly developed higher-order beam theory. It allows the order of the beam theory to be hierarchically increased as desired. It can predict the structural behavior of thin-walled beams nearly as accurately as shell-based analysis. Currently, there are several higher-order beam theories available. Compared with existing theories, the theory and analysis presented in this book may be unique in some aspects. First, the advanced theory presented in this book is entirely consistent with the well-known Vlasov beam theory for beams with rectangular cross-sections. Accordingly, the Vlasov theory can be viewed as the first-order version of the presented higher-order theory. Second, the order of the beam theory presented in this book can be hierarchically and recursively increased, and all equations are derived in a closed form for beams with rectangular cross-sections. The same principle is carried over for beams with arbitrary cross-sections. For beam joint structures in which multiple beams meet at certain angles, the “exact” joint matching conditions, not explored elsewhere, are derived for beams with rectangular cross-sections. Also, what is termed a joint-section matching technique based on displacement approximation is developed for thin-walled beams with arbitrarily shaped cross-sections. We believe that this book will be helpful to graduate students and senior undergraduate students who are interested in learning advanced structural analysis. It will also be useful for field engineers/scientists who wish to use beam models for efficient analyses and designs. For instance, vehicle or aircraft body designers and civil engineers may greatly benefit from using the higher-order beam theory presented in v

vi

Preface

this book, as the time required to design new structures such as vehicle bodies can be substantially reduced. The minimum prerequisite before reading this book is a good understanding of undergraduate solid mechanics and basic finite element analysis. Although this book covers an advanced theory with many equations, we attempt to provide as much detail as possible to help readers follow the presented materials without explicitly resorting to other references. This book was an outcome of our 25 years of research on thin-walled structures. We became interested in this subject because we knew that fast yet accurate predictions of the structural behaviors of vehicle body structures with many thin-walled structures and their joints are critical when the goal is to reduce the time needed to develop new vehicles. Without an advanced beam theory like the one presented in this book, there is no reliable way to use beam modeling for structural analysis of thin-walled structures. Currently, the automobile industry seeks to incorporate advanced beam theories. From this perspective, we believe that publishing a book on a specific advanced beam theory with a joint matching method for thin-walled structures is very timely. To obtain the results in this book, many former students in our group at Seoul National University contributed; we want to thank Dr. J. H. Kim, Dr. Y. Kim, H. S. Kim, S. H. Song, Dr. M. J. Kim, Dr. D. M. Kim, and Dr. D. I. Shin. In particular, we appreciate the significant contribution of Dr. J. Y. Kim, who helped write Chaps. 9 and 11. We also thank the help of S. J. Choi of Sejong University. Finally, the first author expresses his gratitude to Prof. JM Lee at Seoul National University for providing an opportunity to study this subject and Prof. HJ Lim at Kookmin University for bringing technical issues in the classical-beam-based analysis of vehicle bodies to his attention. He also thanks the late Prof. CR Steele at Stanford University for cultivating my interest in thin-walled structures and Prof. S. Nishiwaki at Kyoto University for providing the encouragement to continue working on higher-order beam analysis. Seoul, Korea (Republic of) Seoul, Korea (Republic of) Daegu, Korea (Republic of)

Yoon Young Kim Gang-Won Jang Soomin Choi

Contents

1

2

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Beam Analysis for a Conceptual Design . . . . . . . . . . . . . . . . . . . . . 1.2 Vlasov Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Vlasov Beam Theory for Thin-Walled Open-Section Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Vlasov Beam Theory for Thin-Walled Closed-Section Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Higher-Order Beam Theory (HoBT) . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Other Higher-Order Beam Theories . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsional Warping and Torsional Distortion as Fundamental Deformable Section Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Vlasov Beam Theory for Thin-Walled Open Cross-Sectioned Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Kinematic Assumptions, Displacements, and Strains . . . 2.1.2 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Vlasov Beam Theory for Thin-Walled Closed Cross-Sectional Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Torsional Warping and Torsional Distortion of the HoBT . . . . . . . 2.3.1 Simple Torsion Problem of a Thin-Walled Rectangular Cross-Sectional Beam . . . . . . . . . . . . . . . . . . 2.3.2 Derivation of the Shape Functions of Torsional Warping and Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Governing Equations and Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes θ, W and χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Three-Dimensional Displacements, Strains, and Stresses at a General Point . . . . . . . . . . . . . . . . . . . . . .

1 2 7 7 9 10 14 16 21 23 23 25 29 33 33 36 44 45 46 46 vii

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3.1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Examples of Finite Element Solutions . . . . . . . . . . . . . . . 3.1.5 Analytic Solutions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Finite Element Implementation Using an Arbitrary Number of Section Mode Shapes for General Loading . . . . . . . . . 3.2.1 Three-Dimensional Displacements, Strains, and Stresses at a General Point for General Cases . . . . . 3.2.2 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Governing Equations* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General Field Relationship for the Higher-Order Deformable Section Modes of a Wall-Membrane Field . . . . . . . . 4.2 Generalized Force-Stress Relationship for Zeroth-Order Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Derivation of Higher-Order Deformable Section Modes by Means of a Recursive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . χ 4.3.1 Derivation of ψs 1 (Shape Function of the First-Order Distortion Mode) . . . . . . . . . . . . . . . . . 4.3.2 Derivation of ψzW1 (Shape Function of the First-Order Warping Mode) . . . . . . . . . . . . . . . . . . . χ 4.3.3 Derivation of ψs 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Derivation of ψzW2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sectional Shape Functions for a Box Beam Under Torsion: Wall-Bending Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General Field Relationship for Higher-Order Deformable Section Modes of a Wall-Bending Field . . . . . . . . . . . . . . . . . . . . . 5.2 Generalized Force-Stress Relationship . . . . . . . . . . . . . . . . . . . . . . . 5.3 Derivation of Sectional Shape Functions ψnxk of Unconstrained Distortion Mode χk . . . . . . . . . . . . . . . . . . . . . . . χ 5.3.1 Derivation of ψn k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Relationship Between the Generalized Force (Sk ) and Stress (σ zz ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4

Derivation of the Sectional Shape Functions

η {ψn k ,

48 53 58 62 64 64 71 77 80 83 88 92 94 96 101 108 110 113 115 117 121 122 124 128

ηˆ ψn k }

of Constrained Distortion Modes {ηk , ηˆ k } . . . . . . . . . . . . . . . . . . . . η 5.4.1 Derivation of ψn 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . η 5.4.2 Derivation of ψn 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . η 5.4.3 Derivation of ψn N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134 135 140 143

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5.4.4 Correction for Corner Conditions . . . . . . . . . . . . . . . . . . . 5.4.5 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Case Study 1: Static Wall-Membrane Response by Torsional Moment Mz . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Case Study 2: Coupled Response of a Wall Membrane and Wall Bending by Surface Traction {tzz , tzn , tzs } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Case Study 3: Free Vibration Response . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 151 151

Sectional Shape Functions for a Box Beam Under Extension . . . . . . 6.1 General Field Relationships for Higher-Order Deformable Section Modes of a Wall-Membrane Field . . . . . . . . . . . . . . . . . . . 6.1.1 Displacements, Stress, Strain Fields, and Generalized Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Generalized Force-Stress Relationship for the Zeroth-Order Mode . . . . . . . . . . . . . . . . . . . . . . . . . χ 6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 6.2.1 Derivation of ψs 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Derivation of ψzW1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 6.2.3 Derivation of ψs 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Derivation of ψzW2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 6.2.5 Derivation of ψs k and ψzWk for k ≥ 3 . . . . . . . . . . . . . . . . 6.3 n-Directional Displacements and Resulting Stress and Strain Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Generalized Force-Stress Relationships for Constrained Distortion Modes {ηk , ηˆ k } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 6.5 Derivation of ψn k of Distortion Mode χk and Related Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 6.5.1 Derivation of ψn k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Relationship Between the Generalized Force (Sk ) and Stress (σ zz ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ηˆ η 6.6 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained Distortion Modes {ηk , ηˆ k } . . . . . . . . . . . . . . . . . . . . 6.7 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Case Study 1: Static Wall-Membrane Response by Axial Force Fz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Case Study 2: Mixed Response of Wall-Membrane and Wall-Bending Deformations by Surface Traction {tzz , tzn } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

153

153 162 164

167 167 170 170 172 176 180 182 184 186 190 191 191 195 199 202 203 204

206

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6.8.3 Case Study 3: Free Vibration Response . . . . . . . . . . . . . . 211 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7

Sectional Shape Functions for a Box Beam Under Flexure . . . . . . . . . 7.1 General Field Relationships for Higher-Order Deformable Section Modes of a Wall-Membrane Field . . . . . . . . . . . . . . . . . . . 7.1.1 Displacements, Stress, Strain Fields, and Generalized Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Generalized Force-Stress Relationship of the Zeroth-Order Modes . . . . . . . . . . . . . . . . . . . . . . . . . χ 7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 7.2.1 Derivation of ψs 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Derivation of ψzW1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 7.2.3 Derivation of ψs 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Derivation of ψzW2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 n-Directional Displacements and Resulting Stress and Strain Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 7.4 Derivation of ψn k and the Generalized Force-Stress Relationship for Mode χk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . χ 7.4.1 Derivation of ψn k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Relationship Between the Generalized Force (Sk ) and Stress (σ zz ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

of Mode {ηk , ηˆ k } . . . . . . . . . . . . . . . . . . . Derivation of Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Case Study 1: Static Wall-Membrane Response by Vertical Force Fy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Case Study 2: Mixed Response of Wall-Membrane and Wall-Bending by Surface Tractions {tzz , tzn } . . . . . . 7.6.3 Case Study 3: Free Vibration Response . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

250 253

7.5 7.6

8

η {ψn k ,

ηˆ ψn k }

Bridging Between Rectangular Cross-Sections and Generally Shaped Cross-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Higher-Order Modes for Cross-Sections with General Thin-Walled Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Recursive Equations to Derive Sectional Shape Functions . . . . . . 8.2.1 Recursive Relationships to Derive Distortion Modes χk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Recursive Relationships to Derive Warping Modes Wk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Recursive Relationships to Derive Wall-Bending Modes ηk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

216 216 219 220 221 226 230 233 235 241 241 245

254 256 259 261 263 263 265 266 267 268

Contents

9

Sectional Shape Functions of Thin-Walled Beams with General Cross-Section Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Displacement, Strain, and Stress Fields at a Generic Point . . . . . . 9.2 Generalized Force-Stress Relationships . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Axial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Wall-Bending Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Non-Deformable Section Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Fundamental Deformable Section Modes: Linear Warping and Inextensional Distortion . .0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 χ 9.4.1 Deriving ψzW and ψs as Solutions

xi

271 272 277 278 279 280 281 284

to an Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . χ0 9.4.2 n-directional Shape Function ψn . . . . . . . . . . . . . . . . . . . 9.5 Higher-Order Deformable Section Modes . . . . . . . . . . . . . . . . . . . . 9.5.1 Higher-Order Unconstrained Distortion Modes (χ ) Involving Wall Extension . . . . . . . . . . . . . . . . 9.5.2 Non-Linear Higher-Order Warping Modes (W) . . . . . . . . 9.5.3 Higher-Order Constrained Distortion Modes (H) Not Involving Wall Extension . . . . . . . . . . . . . . . . . . . . . . 9.6 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Static Analysis: A Cantilever Beam with an Open Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Static Analysis: A Simply Supported Beam with an Open Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Static Analysis: A Cantilever Beam with a Closed Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.4 Modal Analysis: A Beam with a Flanged Cross-Section with a Free-free Support Condition . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285 293 296

10 Joint Structures of Box Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Higher-Order Beam Theory for Out-of-Plane Bending Problems of a Box Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Sectional Resultants and Edge Resultants . . . . . . . . . . . . . . . . . . . . 10.3 Generalized Force Equilibrium Conditions . . . . . . . . . . . . . . . . . . . 10.4 Field Variable Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Verification with Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Finite Element Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Case Study 1: Two-Box Beam Joint System . . . . . . . . . . 10.5.3 Case Study 2: T-Joint System . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Case Study 3: N Box Beam-Joint System . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319

296 299 302 306 306 309 313 315 317

322 325 326 334 338 338 341 341 344 346

xii

Contents

11 Joint Structures of Thin-Walled Beams with General Section Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Joint Section and Connection Points . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Kinematics of a Joint Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Rotations Calculated on a Joint Section . . . . . . . . . . . . . . 11.2.2 Displacements Calculated on a Joint Section . . . . . . . . . . 11.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Two-Beam-Joint Structure with a Uniform Rectangular Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Two-Beam-Joint Structure Having an I-shaped Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 T-joint Structure with Mixed Cross-Section Shapes . . . . 11.4.4 Simplified Vehicle Frame . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 348 351 351 354 356 358 358 361 363 364 366 370

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Chapter 1

Introduction

This book presents an advanced beam theory for accurate and efficient analyses of thin-walled beam structures, focusing primarily on thin-walled closed beams but including other types. Beam members exhibit non-negligible sectional deformations such as warping and distortion if they consist of thin-walled sections. Because classical beam theories, such as the Euler and Timoshenko beam theories [see, e.g., Gere and Timoshenko (1997)], use only six degrees of freedom (DOFs) representing three rigid-body translations and three rigid-body rotations of a beam cross-section, the aforementioned non-rigid sectional deformations cannot be depicted at all by them. Therefore, additional DOFs corresponding to non-rigid sectional deformations must be incorporated for an accurate analysis of a thin-walled beam, even when a beam theory is used. However, it is difficult to derive the sectional deformations systematically, and it is much more difficult to establish matching conditions among the corresponding degrees of freedom at a joint of multiply-connected thin-walled beams. It may be apparent that the standard field matching conditions established for classical beam theories are no longer useful if the field variables include DOFs representing non-rigid sectional deformations in addition to conventional six DOFs. In this case, therefore, an alternative field matching approach for these field variables should be established for an analysis of a thin-walled beam-joint structure. To address this issue, a beam theory which can include rigid-body section modes to as many deformable section modes as desired has been developed by the authors, with the theory simply being referred to here as the high-order beam theory (HoBT).1 The motivation to establish the HoBT is that if a one-dimensional (1D) structural analysis by the HoBT is sufficiently reliable, it can be indispensable to reduce the overall design time, especially during the conceptual design phase of a complicated structural system composed of thin-walled beams. Clearly, a shell-based analysis is 1

In the literature, there are several higher-order beam theories, but for the sake of notation, the HoBT in this book is used to denote our theory as presented here. (Sect. 1.3 briefly explains the essence of the HoBT and the works related to it.).

© Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_1

1

2

1 Introduction

more accurate than a beam-based analysis, but the use of a shell analysis may be overshadowed by its computational inefficiency. In this respect, the computationally efficient HoBT-based analysis can be greatly appreciated as long as its accuracy is guaranteed; the HoBT presented in this book can indeed yield results nearly as accurate as shell results in predicting not only displacement but also stress. The name HoBT is derived from the fact that it includes higher-order DOFs associated with non-rigid sectional deformations in addition to the six DOFs commonly used in the Euler and Timoshenko beam theories. Historically, Vlasov (1961) pioneered a higher-order beam theory. He proposed to employ a few additional non-rigid sectional DOFs to account for certain crosssectional deformations in addition to the six classical DOFs. Although his theory, known as the Vlasov theory, provides insight into the complicated structural behaviors of thin-walled box beams, an analysis based on the Vlasov theory alone cannot capture the structural responses of thin-walled beams accurately enough, especially when the thin-walled beams are connected at a joint to form a framed structure. The main reason is that there occur significant, complicated local sectional deformations near joints, which cannot be fully described by the Vlasov theory. In this book, we present a theory using more DOFs than those used in the Vlasov theory. The difficulty in establishing such a theory lies in how to derive the desired sectional deformation shapes systematically. Especially in the case of thin-walled beams of rectangular cross-sections, which will be simply referred to as box beams throughout this book, the HoBT gives “closed-form” cross-sectional shape functions that describe the sectional deformation patterns associated with higher-order DOFs.2 Because the developed approach is recursive and hierarchical, any order of a sectional deformation shape can be systematically derived. We also present the new concept of “edge forces,” which is very useful for analytical field matching at a joint of a thin-walled box beam structure. After presenting the motivation for the development of the HoBT with an industrial design example in Sect. 1.1, the key aspects of the Vlasov theory, which serves as a bridging theory between the classical beam theories and the present HoBT, will be briefly outlined in Sect. 1.2. Then, we will present an overview of the HoBT in Sect. 1.3. Other higher-order theories will be briefly reviewed in Sect. 1.4.

1.1 Beam Analysis for a Conceptual Design Computer-aided engineering (CAE) is actively used in industry to reduce the overall design time from a very early stage of the design. Figure 1.1 compares the time cost of the conventional computer-aided design (CAD)-led design process with that of the CAE-led design process (Lake et al. 2005). The CAD-led design starts with initial CAD drawings mostly obtained from design experience, which are revised using CAE to produce a better design. Every time the design is updated, its structural performance 2

It should be noted that no other available theory to date gives “closed-form” shape functions.

1.1 Beam Analysis for a Conceptual Design

3

Fig. 1.1 CAD-led design versus CAE-led design

is checked by CAE, with this process repeated until the final design is obtained. On the other hand, the CAE-led design can greatly shorten design time, especially if the design is based on a simplified CAE model employing low-dimensional finite elements such as beam finite elements. Thin-walled members are preferred loadcarrying members in most engineering structures because they have a high stiffnessto-mass ratio. However, the beam-element-based simplified CAE-led concept can be viable only if the beam-based one-dimensional analysis is sufficiently accurate. Figure 1.2 illustrates an example of a simplified body-in-white (BIW) of a passenger car used in the concept design stage. The main objective in the concept design is to determine “quickly” how much the structural performance indices of the BIW (such as the stiffness and eigenfrequency) change if the BIW’s layout and dimensions change. Therefore, the use of lowest-dimensional finite elements, i.e., one-dimensional beam finite elements, would clearly be the first choice as long as the predicted structural responses are reliable. If the members in the BIW in Fig. 1.2 are modeled by shell elements, the CAD model and the corresponding finite-element meshes should be repetitively regenerated during the design process for every design change, including changes in the dimensions of the beam sections. This process is inevitably very time-consuming, but this inefficiency can be avoided if a reliable and accurate beam-based analysis is employed. As classical beam theories are incapable of accurately predicting the structural responses of thin-walled beam members, a new higher-order beam theory that predicts accurate structural responses comparable to those by shell-based theories is crucial. Such a theory should include not only the six classical DOFs representing rigid sectional movements but also additional DOFs representing non-rigid (i.e., deformable) sectional deformations. For actual calculations, one-dimensional finite elements based on a higher-order beam theory should

4

1 Introduction

Cross section Thin-walled beams B-pillar

Joint

Cross section

Fig. 1.2 BIW (body-in-white) of a passenger car and a one-dimensional beam model

also be developed. The developed theory and resulting finite element should be useful for thin-walled beam members of arbitrarily shaped cross-sections. Despite the great advantages of beam models for rapid model changes and efficient analyses, the beam-based modeling of a thin-walled member generally predicts inaccurate structural responses, especially in stress prediction. The main reason for the inaccuracy is that conventional beam finite elements based on the Timoshenko beam theory cannot depict sectional deformations of thin-walled members. Because the Timoshenko beam theory does not have any DOF representing sectional deformations (such as warping and distortion), it models a thin-walled beam overly stiffly. We will refer to any beam model based on a classical beam theory (the Euler or Timoshenko beam theory) as the classical beam model in the subsequent discussion; this model should be distinguished from a higher-order beam model based on a higher-order beam theory that includes DOFs representing sectional deformations. It is also important to note that the geometric complexity of a thin-walled beam, such as rapid axial variations in the beam sectional shapes, makes it more difficult to obtain accurate results with a classical beam model. In fact, noticeable geometric variations can be found in a tapered curved center pillar of a passenger car (referred to as a B pillar in the automobile industry), as illustrated in Fig. 1.2. The classical beam model yields worse results near the joints of thin-walled beam structure. Because the classical beam theory does not take sectional deformations into account, it can by no means depict significant sectional deformations occurring near a joint. Therefore, the classical beam model predicts much stiffer structural behavior than the actual behavior because the flexibility resulting from sectional deformations dominant near a joint cannot be represented. For example, if a T-joint structure with a rectangular thin-walled cross-section of the type shown in Fig. 1.3a is analyzed using Timoshenko beam elements, its overall stiffness is evaluated as being too high compared to that by the shell model. As clearly shown in Fig. 1.3b, the vertical displacement by the classical beam model is only 33% of that by the shell model; this result suggests that the classical beam theory overestimates the stiffness near a joint. To illustrate further that the greatest source of inaccuracy in the classical beam theory is in its inability to represent cross-sectional deformations near a joint, we consider the fixed-free two-beam L-joint structure shown in Fig. 1.4. This structure

1.1 Beam Analysis for a Conceptual Design

5

Fig. 1.3 a Deformed shape of a T-joint thin-walled box beam structure subjected to shear force (calculated by the shell-based analysis) and b the deflection and rotation calculated along AOB marked in (a)

is analyzed by a shell model, and the predicted lowest torsional vibration mode of the structure is also sketched in Fig. 1.4. The Timoshenko beam finite element analysis predicts its lowest torsional eigenfrequency as 451.5 Hz, while the shell finite element analysis yields a result of 228.4 Hz. Because the classical beam model assumes that all cross-sections behave rigidly, the L-joint structure is evaluated to be overly stiff, yielding an eigenfrequency twice as high as the correct value predicted with the shell model. Although not shown here, some eigenmodes are often missing in the modal analysis of a thin-walled beam structure if the classical beam model is used. Therefore, it is crucial to include additional kinematic DOFs describing non-rigid cross-sectional deformations when developing a higher-order beam theory; otherwise, accurate predictions of the structural behaviors of thin-walled beam structures by one-dimensional beam theories would be impossible. Despite the aforementioned intrinsic difficulties associated with the classical beam model, there were a number of earlier efforts to improve the accuracy of results predicted by the classical beam model. A typical attempt was to introduce spring

6

1 Introduction

free end

fixed end

Lowest torsional eigenfrequency: 228.8 Hz by shell elements 451.2 Hz by classical beam elements (6DOF) Fig. 1.4 First torsional vibration mode shape of a two-beam-joint structure with a joint angle of 60° under a fixed-free boundary condition calculated using a shell model showing significant sectional deformation at the joint

elements to account for the flexibility effect occurring at a joint of a thin-walled structure. The spring stiffness is precalculated using a shell model (Chang 1974; El-Sayed 1989; Lee and Nikolaidis 1992, 1998; Moon et al. 1999; Nakagawa et al. 2004; Tsurumi et al. 2004; Zuo et al. 2012; Na et al. 2014; Kameyama et al. 2015), with the stiffness adjusted to match the stiffness of a beam-spring model and that of the shell model. Instead of using a shell model, other studies have utilized an artificial joint spring, the stiffness of which was estimated based on experimental studies (AI-Bermani et al. 1994; Zhu et al. 1995) or Eurocode 3 (Galvao et al. 2010; Sophianopoulos 2003). Although this modeling technique appears to be easy and intuitive, it is almost impossible to obtain a universal formula to predict the spring stiffness with moderate accuracy because the spring stiffness behaves in a highly nonlinear fashion with respect to every geometric parameter defining the joint. As an alternative, a shell model can be employed only in the vicinity of a joint, while other parts of the structure are modeled using classical beam elements (Donders et al. 2009; Gaetano 2014; Mundo et al. 2009; Mihaylova et al. 2010; Maressa et al. 2011; Mundo et al. 2011; Moroncini et al. 2012; Mihaylova et al. 2012) or advanced beam elements (Bianco et al. 2019; Manta et al. 2020, 2021a, 2021b; Nguyen et al. 2018). However, the results of this hybrid modeling approach do not show much of an improvement in the accuracy unless a large region covering the joint is modeled by shell elements. More importantly, this approach significantly sacrifices the benefits of the beam modeling method, i.e., modeling simplicity and good numerical efficiency.

1.2 Vlasov Beam Theory

7

y y s

un

us P

n

O z

h(s)

x

r(s) O

(a)

A

s

x

(b)

Fig. 1.5 Thin-walled beam with an open cross-section: a a three-dimensional view with the global coordinates (x, y, z) where the local coordinates n and s are the normal and tangential coordinates defined on the mid surface of the beam cross-section and b a two-dimensional view of the midline of the beam section with in-plane displacements u n and u s . Symbols O and A denote the centroid of the cross-section and the center of rotation (or torsional center) about the z axis, respectively

1.2 Vlasov Beam Theory Because the Vlasov beam theory serves as the basis of the higher-order beam theory presented in this book, we briefly present its core aspects. More details will be given in Chap. 2. Vlasov (1961) considered a few additional DOFs representing the sectional deformations of a thin-walled beam in addition to six rigid-body (non-deformable) section motions (three translations and three rotations).

1.2.1 Vlasov Beam Theory for Thin-Walled Open-Section Beams Let us consider the open-section thin-walled beam shown in Fig. 1.5. In an opensection beam, the midline of the sectional wall of the beam does not form any closed path (see Fig. 1.5 for example). To analyze this beam using a one-dimensional analysis, the Vlasov theory employs the following two kinematic assumptions: Assumption 1 A cross-section is rigid in its in-plane deformation.3

3

In-plane deformation is the deformation occurring on the plane of a cross-section, i.e., the x–y plane in Fig. 1.5. Vlasov assumed that the cross-section of a thin-walled open-section beam deforms rigidly on the plane of the cross-section while undergoing warping deformation (higher-order deformation) in the axial direction.

8

1 Introduction

Assumption 2 Shear strain γzs is zero on the midsurface of the wall (the surface where n = 0) forming the beam cross-section. Assumption 1 assumes that the beam cross-section undergoes x- and y-directional rigid-body translations and z-directional rigid-body rotation4 (see Fig. 1.5 for the coordinate system), and that higher-order in-plane cross-sectional deformation, hereafter referred to as distortion, does not occur. For out-of-plane deformation, however, higher-order deformation, hereafter referred to as warping, is considered. It should be noted that zero shear strain on the midsurface of the wall as stated in the second assumption does not imply zero shear stress; the shear stress is calculated by considering the force equilibrium, which will be discussed in Chap. 2. To explain how warping deformation can occur, we consider an I-section beam subjected to an eccentric axial force, as shown in Fig. 1.6a. The eccentric axial load in Fig. 1.6a can be decomposed into the four load sets in Fig. 1.6b–e. Consider the first three load sets in Fig. 1.6b–d, which denote the cases with non-zero resultants, i.e., the axial load, the x-directional bending moment, and the y-directional bending moment, respectively. On the other hand, the load set in Fig. 1.6e represents a state of self-equilibration with no net resultant. It is said that this state of loading produces a bimoment, with the deformation induced by it referred to as warping. In static problems, the resulting warping effect due to a bimoment should decay along the z direction from the loaded end because the bimoment is self-equilibrated. (The decaying phenomenon is due to the principle of Saint Venant). It is important to note that the warping deformation by the bimoment shown in Fig. 1.6e consists of bending deformations of the two flanges of the I-section beam in the opposite directions. It was shown in Vlasov (1961) that because warping can be coupled with torsion, the bimoment can twist a beam (We will discuss more general cases later in the main part of this book.). An important message to gain from the analysis in Fig. 1.6 is that the beam cross-sections, especially thin-walled cross-sections, can non-rigidly deform under a general load. This means that the classical beam theory using only rigidbody sectional translations and rotations cannot capture the non-rigid deformation of cross-sections. Furthermore, there can be different non-rigid deformation modes due to different types of applied loads. In theory, there will be an infinite number of deformation modes. In the Vlasov beam theory, as mentioned earlier, in-plane non-rigid deformations are not considered due to Assumption 1; only a single warping mode is employed for an open-section thin-walled beam. As we proceed through this book, we will study how to find all deformation modes including in-plane deformation modes, systematically for a given thin-walled cross-section using a higher-order beam theory.

4

The rotation implies rigid-body rotation about the z axis.

1.2 Vlasov Beam Theory

9

(a)

(d)

(b)

(e)

(c)

Fig. 1.6 Cantilevered I-section beam subjected to an eccentric concentrated axial force at its free end. The applied eccentric load in (a) can be decomposed into the four sets of loads sketched in (b–e), where the load set in (e) corresponds to a bimoment having no resultant force and moment over the beam cross-section

1.2.2 Vlasov Beam Theory for Thin-Walled Closed-Section Beams In the case of thin-walled closed cross-sectional beams, the two kinematic assumptions stated in Sect. 1.2.1 for open cross-sectional beams are no longer applicable. As a representative example of a closed cross-sectional beam, we consider the thinwalled rectangular cross-sectional beam (i.e., box beam) shown in Fig. 1.7. For an analysis of the box beam, the six classical DOFs of three rigid-body sectional translations and three rigid-body sectional rotations are used initially; they are shown in Fig. 1.7a–f. In addition to these six DOFs, Vlasov (1961) considered two additional DOFs representing sectional warping and distortion, as shown in Fig. 1.7g, h, respectively. While warping represents the out-of-plane deformation of a section, distortion describes its in-plane deformation. The deformations occurring on the x–y plane are in-plane deformations. As sketched in Fig. 1.7g, Vlasov assumed edgewiselinear warping by extending the linear behavior of warping in an open-section beam. In addition, he approximated in-plane non-rigid distortional deformation using a hinged-beam frame model; the members forming the beam frame model are regarded

10

1 Introduction

y

x

(a)

(b)

(d)

(c)

(e)

(f)

n2

s2 n2 s2

s1 n1

n3

y

s1

x

s3

n3

n1

s3 s4 n4

(g)

n4

s4

(h)

Fig. 1.7 Sectional shape functions for a thin-walled beam of a rectangular cross-section: a–f six non-deformable section modes (rigid-body degrees of freedom), g warping, and h distortion obtained using a hinged-beam frame model

in-extensional, meaning that their lengths do not change for any distortion of a crosssection. This intuitive approach to utilize a hinged-beam frame model has several limitations. This model can only represent edgewise-constant shear stress and does not allow extensional or bending behavior of the walls; therefore, it is difficult to extend this model to find higher-order sectional deformation modes.

1.3 Higher-Order Beam Theory (HoBT) The higher-order beam theory presented in this book is a generalization of the Vlasov theory in that it also considers sectional deformation modes as additional degrees of freedom. The main difference between our higher-order beam theory (HoBT) and the Vlasov theory is that the HoBT can include as many sectional deformation modes as desired in a systematic and hierarchical manner. It should be noted here that the extension of the Vlasov theory using only a limited number of DOFs5 to establish a higher-order beam theory employing as many DOFs as desired is not straightforward while retaining all key mechanics aspects of the Vlasov theory (such as orthogonality among sectional deformation modes and explicit stress-generalized force relations). A key advantage of the HoBT presented in this book is that as many higher-order modes as desired can be derived consistently and recursively for box beams. In fact, they are analytically derived in a closed form. Furthermore, exact field matching conditions consistent with the HoBT are established at a joint of thin-walled box 5

As will be examined in Sect. 2.2, the numbers of warping and distortion modes by the Vlasov beam theory for a closed thin-walled cross-section are determined if the number of nodes and the connectivity of the cross-section frame are given.

1.3 Higher-Order Beam Theory (HoBT)

11

beams for the first time. The HoBT presented in this book maintains the mechanics aspects of the Vlasov theory and is based on the studies by the authors’ group (Kim and Kim 1999a, b, 2000, 2002, 2003; Jang and Kim 2009a, b, 2010; Jang et al. 2008, 2012; Jung et al. 2018; Choi et al. 2012; Choi and Kim 2016a, b, 2019, 2020, 2021a, b; Kim and Jang 2017; Kim et al. 2021, 2022; Shin and Kim 2020). Table 1.1 shows a few non-rigid sectional deformations derived by the HoBT for a thin-walled rectangular cross-section. Sectional deformations belonging only to lower-order mode sets are listed in the table, although an infinite number of sectional deformation modes can be hierarchically derived in a recursive manner. The first warping mode in the first row denotes the lowest warping mode, which is equivalent to the warping mode shown in Fig. 1.6e for an I-section beam. This mode is usually referred to as a torsional warping mode, as it typically couples with torsional rotation. The second to third modes in the first row of the table denote the next higherorder warping modes. Note that they represent edgewise-nonlinear deformations. The second and third rows in Table 1.1 correspond to distortion modes, having nonlinear thickness-directional (n-directional) displacements which account for wall bending. At this point, we can consider non-uniform wall-bending stress (σss ) using the HoBT, which cannot be expressed by the Vlasov beam theory because the Vlasov theory employs a “hinged” beam frame model. In addition, wall extension by distortion is also considered in the HoBT. The HoBT considers that the bending moments at the corners of a cross-section are non-zero and should be continuous. The derivation of the cross-sectional shape functions of higher-order modes will be presented in detail for a rectangular cross-section in Chaps. 4–8. The shape functions for arbitrarily shaped cross-sections will be derived in Chap. 9. To demonstrate why a beam theory using higher-order cross-sectional modes, such as the HoBT, is so critical for an accurate structural analysis of a thin-walled beam structure, we consider the structural analysis of a T-joint structure with a rectangular thin-walled cross-section, as shown in Fig. 1.8a. Two load cases are considered: Load case 1 with shear force Fz (out-of-plane6 bending load) and Load case 2 with bending moment Mz (in-plane bending load). Figure 1.8b shows numerical results by the HoBT-based one-dimensional finite element employing two higher-order modes, i.e., the torsional distortion (χ0 ) and torsional warping (W0 ) modes. These modes correspond to the first mode in the second row and the first mode in the first row in Table 1.1 in addition to six non-deformable section modes. These results are compared with those by the shell finite element for the Load case 1. As shown in Fig. 1.8b, the results by the HoBT are in good agreement with the shell results, indicating that the inclusion of the torsional warping and torsional distortion modes in the section mode set is critical for an accurate prediction of the stiffness of a Tjoint structure under an out-of-plane load. On the other hand, the Timoshenko beam theory using six non-deformable section modes predicts much stiffer results.

6

We often denote the bending direction of a jointed beam structure (or a beam frame structure) as the in-plane direction or out-of-plane direction. Plane here refers to the plane on which the beam structure lies. For the T-joint in Fig. 1.14a, the bending direction is denoted with respect to the

12

1 Introduction

Table 1.1 A few sectional shape functions representing cross-sectional deformations derived by the HoBT. These shape functions are needed for an analysis of a thin-walled beam of a rectangular cross-section subjected to a torsional load Fundamental warping (typically First-order warping (W1 ) called torsional warping, W0 )

Second-order warping (W2 )

Fundamental distortion (typically called torsional distortion χ0 )

First-order unconstrained distortion (χ1 )

Second-order unconstrained distortion (χ2 )

First-order constrained distortion of type 1 η1

First-order constrained distortion of type 2 ηˆ 1

1.3 Higher-Order Beam Theory (HoBT)

C

Load case 1

13

Load case 2

z x

O

y

A

B

(a)

(b) Load case 2

Shell Timoshenko HoBT (2 Higher-order modes) HoBT (11 Higher-order modes) Shell Timoshenko HoBT (11 Higherorder modes)

Poisson distortion

(c) Fig. 1.8 a A T-joint structure consisting of rectangular thin-walled cross-sectional beams, b displacement along AOB calculated by different approaches for Load case 1, and c displacement along AOB calculated for Load case 2. In (c), the contribution of what is termed the Poisson’s mode is also examined along AOB

For the in-plane load denoted as the Load case 2, on the other hand, the finite element result based on the HoBT using the two higher-order modes mentioned above does not improve the solution accuracy much in comparison with that by the Timoshenko beam theory. This occurs because torsional warping and torsional distortion do not contribute much to the structural response of a T-joint under an inplane load. Obviously, the beam members in Fig. 1.8a are not in a state of torsion when bending moment Mz is applied at end B. This observation indicates that additional higher-order modes which can be excited by the load prescribed in the Load case 2 are needed. If higher-order modes for bending loads including those known as Poisson modes7 are employed as additional DOFs, the HoBT can yield displacement xy plane. Because the bending deflection of the beam structure is z-directional, it is out-of-plane bending and the shear force is an out-of-plane bending load. 7 A detailed description of the Poisson modes will be given later in this book. .

14

1 Introduction

comparable to that calculated by the shell theory; see Fig. 1.8c. To address the significance of the Poisson modes especially near the joint of the structure, the second figure in Fig. 1.8c plots the contribution of one of the Poisson modes along AOB; see the figure on the right in Fig. 1.8c for the shape of the Poisson mode. It should be noted that the Poisson mode involves a large amount of deformation related to the wall extension. Clearly, the results based on the classical beam theory using six non-deformable section modes significantly deviate from the shell results; the use of higher-order beam theories is thus justified. One final remark: In dealing with a jointed beam structure such as the T-joint shown in Fig. 1.8a, special attention must be paid to find relations among generalized 1D displacements or generalized 1D forces at a joint. The main difficulty arises because the resultant forces/moments by higher-order section modes calculated over a cross-section are identically zero. This means that the desired relations cannot be found by simply considering equilibrium conditions of the resultant forces/moments over the beam cross-sections at the joint because they are selfequilibrated. This problem has remained unsolved for decades, but recent studies by the authors have derived explicit relationships for the joints of thin-walled box beams and have proposed a reliable method for joints of thin-walled beams of arbitrarily shaped cross-sections. We will elaborate on this issue in Chaps. 10 and 11.

1.4 Other Higher-Order Beam Theories There are other higher-order beam theories considering higher-order sectional nonrigid deformation modes. A major difference among different theories may lie in how the higher-order sectional deformation shapes are constructed. To compare our HoBT with other existing higher-order beam theories, we will briefly review available theories, specifically the generalized beam theory (GBT), the variational asymptotic beam sectional analysis (VABS), the method of generalized eigenvectors (GE), and the Carrera unified formulation (CUF). The GBT originally proposed by Schardt (1989, 1994a, b) calculates sectional deformation shapes using a plane beam frame model that discretizes the walls of a beam section into one-dimensional beam segments (Camotim et al. 2010; Silvestre and Camotim 2002; Goncalves et al. 2010, 2014; Goncalves and Camotim 2015, 2016; Silvestre et al. 2008, 2011; Bebiano et al. 2015, 2018a, b; Martins et al. 2020; Vieira et al. 2021). The concept of the plane beam frame model has been extended to analyze beams of other cross-sections (Kim and Jang 2017; Vieira et al. 2013, 2014, 2015). Methods that use a two-dimensional cross-sectional analysis extracted from a three-dimensional (3D) continuum theory are also available. For instance, the variational asymptotic beam section analysis (VABS) (Cesnik and Hodges 1997; Yu et al. 2002a, b, 2012; Hodges 2006, 2015) was developed from the 3D energy functional by applying the variational asymptotic method (Berdichevskii 1979). On the other hand, the method of generalized eigenvectors (GE) (Genoese et al. 2013, 2014a, b; Garcea et al. 2016) uses 3D equilibrium equations based on an extension

1.4 Other Higher-Order Beam Theories

15

of the St. Venant rod theory. The Carrera unified formulation (CUF) (Carrera and Zappino 2016; Carrera and Pagani 2013, 2016; Carrera et al. 2011, 2012, 2015, 2017, 2019) represents the complex 3D beam behavior by generic expansion of 1D kinematic variables defined at specific section points. Similarly, a node-based degreeof-freedom approach (Goncalves and Camotin 2017) utilizing GBT elementary sectional deformation shapes was also proposed. In comparison with the higher-order beam theories reviewed above, the HoBT presented in this book derives higher-order cross-sectional shape functions hierarchically. Especially for thin-walled box beams (of a constant thickness), they are expressed in a “closed form,” which is not possible when using other theories. In the HoBT, a new set of higher-order warping and distortion modes is derived hierarchically from lower-order mode sets by considering the consistency between the strain field and the stress field generated by the modes in the lower-order sets. In establishing a hierarchical relationship, it is important to observe that the distortion mode in the current-order set is induced by the longitudinal stress of the out-ofplane deformations belonging to lower-order sets via Poisson’s effect and that the warping mode in the current-order set is influenced by the shear stress of the in-plane modes (The detailed physics will be discussed in the next few chapters.). Thereby, it will be shown that higher-order modes can be expressed as a linear combination of the integrated functions of lower-order modes. It will be also demonstrated that one can determine the combination coefficients using certain explicit conditions, including the orthogonality condition among the modes. Because this hierarchical derivation method does not require any approximation of the cross-sectional shape of a beam, no cross-sectional discretization, commonly used in existing studies, is needed. Interestingly, the deformable section modes can be grouped into families: warping modes involving out-of-plane displacements of the wall centerline of a crosssection, unconstrained distortion modes involving in-plane displacements of the wall centerline with non-zero wall corner displacement, and constrained distortion modes involving in-plane displacement of the wall centerline without wall corner displacement. The MATLAB codes for the higher-order sectional shape functions of the HoBT and beam sample problems can be downloaded from GitHub (https://github. com/SChoiKNU/Codes-for-HoBT-based-Finite-Element-Analysis). Another critical advantage of the HoBT in comparison with other theories is that the HoBT allows explicit relationships between 1D generalized forces and stresses and between 1D generalized forces and 1D generalized displacements. These relationships are fully consistent with similar relationships established by the Vlasov theory. For thin-walled box beams, explicit relationships are derived in Choi and Kim (2019, 2020, 2021a, b) and for arbitrarily-sectioned beams, they are given in Kim et al. (2021). It should also be noted that no such relationships fully consistent with those of Vlasov can be established in other higher-order beam theories for thin-walled beams under general types of loadings, including torsional loading. If these relationships can be explicitly described in a manner consistent with those of the Vlasov torsion theory, the physical significance of the generalized forces, especially self-equilibrated forces, can be directly understood (Choi and Kim 2016a, b). Moreover, these relationships can be crucial when deriving “explicit” equilibrium

16

1 Introduction

conditions among the generalized forces, including self-equilibrated forces at the joints of multiple thin-walled box beams (Choi and Kim 2016a). In the remaining parts of this book, a theory based on the HoBT will be presented in detail with numerical examples.

References AI-Bermani FGA, Li B, Zhu K, Kitipornchai S (1994) Cyclic and seismic response of flexibly jointed flames. Eng Struct 16:249–255 Bebiano R, Basaglia C, Camotim D, Gonçalves R (2018a) GBT buckling analysis of generally loaded thin-walled members with arbitrary flat-walled cross-sections. Thin-Walled Struct 123:11–24 Bebiano R, Camotim D, Gonçalves R (2018b) GBTul 2.0− A second-generation code for the GBTbased buckling and vibration analysis of thin-walled members. Thin-Walled Struct 124:235–257 Bebiano R, Goncalves R, Camotim D (2015) A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses. Thin-Walled Struct 92:29–47 Berdichevskii VL (1979) Variational-asymptotic method of constructing a theory of shells. J Appl Math Mech 43:664–687 Bianco MJ, Habtemariam AK, KönkeC ZV (2019) Analysis of warping and distortion transmission in mixed shell–GBT (generalized beam theory) models. Int J Adv Struct Eng 11:109–126 Carrera E, de Miguel AG, Pagani A (2017) Hierarchical theories of structures based on Legendre polynomial expansions with finite element applications. Int J Mech Sci 120:286–300 Carrera E, Fiordilino GA, Nagaraj M, Pagani A, Montemurro M (2019) A global/local approach based on CUF for the accurate and efficient analysis of metallic and composite structures. Eng Struct 188:188–201 Carrera E, Giunta G, Petrolo M (2011) Beam structures: classical and advanced theories. Wiley Carrera E, Zappino E (2016) Carrera unified formulation for free-vibration analysis of aircraft structures. AIAA J 54:280–292 Carrera E, Pagani A (2013) Analysis of reinforced and thin-walled structures by multi-line refined 1D/beam models. Int J Mech Sci 75:278–287 Carrera E, Pagani A (2016) Accurate response of wing structures to free-vibration, load factors, and nonstructural masses. AIAA J 54:227–241 Carrera E, Pagani A, Petrolo M, Zappino E (2012) A component-wise approach in structural analysis. Comput Methods Eng Sci 4:75–115 Carrera E, Pagani A, Petrolo M, Zappino E (2015) Recent developments on refined theories for beams with applications. Mech Eng Rev 14:00298 Camotim D, Basaglia C, Silvestre N (2010) GBT buckling analysis of thin-walled steel frames: a state-of-the-art report. Thin-Walled Struct 48:726–743 Cesnik CES, Hodges DH (1997) VABS: a new concept for composite rotor blade cross-section modeling. J Am Helicopter Soc 42:27–38 Chang DC (1974) Effects of flexible connections on body structural response. SAE Trans 83:233– 244 Choi S, Jang GW, Kim YY (2012) Exact matching condition at a joint of thin-walled box beams under out-of-plane bending and torsion. J Appl Mech 79:051018 Choi S, Kim YY (2016a) Exact matching at a joint of multiply-connected box beams under out-ofplane bending and torsion. Eng Struct 124:96–112 Choi S, Kim YY (2016b) Analysis of two box beams-joint systems under in-plane bending and axial loads by one-dimensional higher-order beam theory. Int J Solids Struct 90:69–94

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Choi S, Kim YY (2019) Consistent higher-order beam theory for thin-walled box beams using recursive analysis: membrane deformation under doubly symmetric loads. Eng Struct 197:109430 Choi S, Kim YY (2020) Consistent higher-order beam theory for thin-walled box beams using recursive analysis: edge-bending deformation under doubly symmetric loads. Eng Struct 206:110129 Choi S, Kim YY (2021a) Higher-order Vlasov torsion theory for thin-walled box beam. Int J Mech Sci 195:106231 Choi S, Kim YY (2021b) Higher-order beam bending theory for static, free vibration, and buckling analysis of thin-walled rectangular hollow section beams. Comput Struct 248:106494 Donders S, Takahashi Y, Hadjit R, Langenhove TV, Brughmans M (2009) A reduced beam and joint concept modeling approach to optimize global vehicle body dynamics. Finite Elem Anal Des 45:439–455 El-Sayed MEM (1989) Calculation of joint spring rates using finite element formulation. Comput Struct 33:977–981 Gaetano GD, Mundo D, Cosco FI, Maletta C, Donders S (2014) Concept modelling of vehicle joints and beam-like structures through dynamic FE-based methods. Shock Vib 2014:303567 Galvao AS, Silva ARD, Silveira RAM, Goncalves PB (2010) Nonlinear dynamic behavior and instability of slender frames with semi-rigid connections. Int J Mech Sci 52:1547–1562 Garcea G, Goncalves R, Bilotta A, Manta D, Bebiano R, Leonetti L, Magisano D, Camotim D (2016) Deformation modes of thin-walled members: a comparison between the method of generalized eigenvectors and generalized beam theory. Thin-Walled Struct 100:192–212 Gere JM, Timoshenko SP (1997) Mechanics of materials. PWS Pub Co, Boston Genoese A, Genoese A, Bilotta A, Garcea G (2013) A mixed beam model with non-uniform warpings derived from the Saint Venànt rod. Comput Struct 121:87–98 Genoese A, Genoese A, Bilotta A, Garcea G (2014a) A generalized model for heterogeneous and anisotropic beams including section distortions. Thin-Walled Struct 74:85–103 Genoese A, Genoese A, Bilotta A, Garcea G (2014b) A geometrically exact beam model with non-uniform warping coherently derived from the Saint Venant rod. Eng Struct 68:33–46 Goncalves R, Bebiano R, Camotim D (2014) On the shear deformation modes in the framework of generalized beam theory. Thin-Walled Struct 84:325–334 Goncalves R, Camotim D (2015) On distortion of symmetric and periodic open-section thin-walled members. Thin-Walled Struct 94:314–324 Goncalves R, Camotim D (2016) GBT deformation modes for curved thin-walled cross-sections based on a mid-line polygonal approximation. Thin-Walled Struct 103:231–243 Goncalves R, Camotim D (2017) Improving the efficiency of GBT displacement-based finite elements. Thin-Walled Struct 111:165–175 Goncalves R, Ritto-Corrêa M, Camotim D (2010) A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory. Comput Mech 46:759–781 Hodges DH (2006) Nonlinear composite beam theory. AIAA, Washington Hodges DH (2015) Unified approach for accurate and efficient modeling of composite rotor blade dynamics The Alexander A. Nikolsky honorary lecture. J Am Helicopter Soc 60:1–28 Jang GW, Kim KJ, Kim YY (2008) Higher-order beam analysis of box beams connected at angled joints subject to out-of-plane bending and torsion. Int J Numer Meth Eng 75:1361–1384 Jang GW, Kim MJ, Kim YY (2012) Analysis of thin-walled straight beams with generally shaped closed sections using numerically determined sectional deformatin functions. J Struct Eng 138:1427–1435 Jang GW, Kim YY (2009a) Higher-order in-plane bending analysis of box beams connected at an angled joint considering cross-sectional bending warping and distortion. Thin-Walled Struct 47:1478–1489 Jang GW, Kim YY (2009b) Vibration analysis of piecewise straight thin-walled box beams without using artificial joint springs. J Sound Vib 326:647–670

18

1 Introduction

Jang GW, Kim YY (2010) Fully coupled 10-degree-of-freedom beam theory for piecewise straight thin-walled beams with general quadrilateral cross sections. J Struct Eng 136:1596–1607 Jung JH, Jang GW, Shin D, Kim YY (2018) One-dimensional analysis of thin-walled beams with diaphragms and its application to optimization for stiffness reinforcement. Comput Mech 61:331–349 Kameyama S, Fujita S, Kaji S (2015) Elementary body structure analysis. SAE Int J Passenger Cars Mech Syst 8(2015-01-1321):73–81 Kim H, Jang GW (2017) Higher-order thin-walled beam analysis for axially varying generally shaped cross sections with straight cross-section edges. Comput Struct 189:83–100 Kim J, Choi S, Kim YY, Jang G-W (2021) Hierarchical derivation of orthogonal cross-section modes for thin-walled beams with arbitrary sections. Thin-Walled Struct 161:107491 Kim J, Jang GW, Kim YY (2022) Joint modeling method for higher-order beam-based models of thin-walled frame structures. Int J Mech Sci 220:107132 Kim JH, Kim YY (1999a) Analysis of thin-walled closed beams with general quadrilateral cross sections. J Appl Mech 66:904–912 Kim YY, Kim JH (1999b) Thin-walled closed box beam element for static and dynamic analysis. Int J Numer Meth Eng 45:473–490 Kim JH, Kim YY (2000) One-dimensional analysis of thin-walled closed beams having general cross-sections. Int J Numer Meth Eng 49:653–668 Kim Y, Kim YY (2003) Analysis of thin-walled curved box beam under in-plane flexure. Int J Solids Struct 40:6111–6123 Kim YY, Kim Y (2002) A one-dimensional theory of thin-walled curved rectangular box beams under torsion and out-of-plane bending. Int J Numer Meth Eng 53:1675–1693 Lake KL, Thomas R, Gambling M, Lawson T (2005) The application of FEA in the optimisation of die cast components & the consequent reduction in development costs & time. Swansea Metropolitan University, U.K Lee K, Nikolaidis E (1992) A two-dimensional model for joints in vehicle structures. Comput Struct 45:775–784 Lee K, Nikolaidis E (1998) Effect of member length on the parameter estimates of joints. Comput Struct 68:381–391 Manta D, Goncalves R, Camotim D (2020) Combining shell and GBT-based finite elements: Linear and bifurcation analysis. Thin-Walled Structures 152:106665 Manta D, Goncalves R, Camotim D (2021a) Combining shell and GBT-based finite elements: vibration and dynamic analysis. Thin-Walled Structures 167:108187 Manta D, Goncalves R, Camotim D (2021b) Combining shell and GBT-based finite elements: plastic analysis with adaptive mesh refinement. Thin-Walled Struct 158:107205 Maressa A, Mundo D, Donders S, Desmet W (2011) A wave-based substructuring approach for concept modeling of vehicle joints. Comput Struct 89:2369–2376 Martins A, Silvestre N, Bebiano R (2020) A new modal theory for wrinkling analysis of stretched membranes. Int J Mech Sci 175:105519 Mihaylova P, Baldanzini N, Pratellesi A, Pierini M (2010) On the improvement of concept modeling of joints within simplified finite element models with application to structural dynamics. In: Proceedings of the international conference on noise and vibration engineering—ISMA Mihaylova P, Baldanzini N, Pratellesi A, Pierini M (2012) Beam bounding box–a novel approach for beam concept modeling and optimization handling. Finite Elem Anal Des 60:13–24 Moon YM, Lee TH, Park YP (1999) Development of an automotive joint model using an analytically based formulation. J Sound Vib 220:625–640 Moroncini A, Cremers L, Baldanzini N (2012) Car body concept modeling for NVH optimization in the early design phase at BMW: A critical review and new advanced solutions. In: Proceedings of the international conference on noise and vibration engineering ISMA, pp 3809–3824 Mundo D, Donders S, Stigliano G, Auweraer HVD (2011) Concept design of vehicle bodies using reduced models of beams, joints and panels. Int J Veh Des 57:71–83

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Mundo D, Hadjit R, Donders S, Brughmans M, Mas P, Desmet W (2009) Simplified modelling of joints and beam-like structures for BIW optimization in a concept phase of the vehicle design process. Finite Elem Anal Des 45:456–462 Na W, Lee S, Park J (2014) Body optimization for front loading design process, SAE Technical Paper, 2014-01-0388 Nakagawa T, Nishigaki H, Tsurumi Y, Kikuchi N (2004) First order analysis for automotive body structure design-Part 4: Noise and vibration analysis applied to a subframe, SAE Technical Paper, 2004-01-1661 Nguyen NL, Jang GW, Choi S, Kim J, Kim YY (2018) Analysis of thin-walled beam-shell structures for concept modeling based on higher-order beam theory. Comput Struct 195:16–33 Schardt R (1989) Verallgemeinerte Technische Biegetheorie. Springer-Verlag, Berlin Schardt R (1994a) Generalised beam theory—an adequate method for coupled stability problems. Thin-Walled Struct 19(2–4):161–180 Schardt R (1994b) Lateral torsional and distortional buckling of channel and hat- sections. J Constr Steel Res 31(2–3):243–265 Shin D, Kim YY (2020) Data-driven approach for a one-dimensional thin-walled beam analysis. Comput Struct 231:106207 Silvestre N, Camotim D (2002) First-order generalised beam theory for arbitrary orthotropic materials. Thin-Walled Struct 40:755–789 Silvestre N, Camotim D, Silva NF (2011) Generalised Beam Theory Revisited: from the kinematical assumptions to the deformation mode determination. Int J Struct Stab Dyn 11:969–997 Silvestre N, Young B, Camotim D (2008) Non-linear behaviour and load-carrying capacity of CFRP-strengthened lipped channel steel columns. Eng Struct 30:2613–2630 Sophianopoulos DS (2003) The effect of joint flexibility on the free elastic vibration characteristics of steel plane frames. J Constr Steel Res 59:995–1008 Tsurumi Y, Nishigaki H, Nakagawa T, Amago T, Furusu K, Kikuchi N (2004) First order analysis for automotive body structure design—part 2: joint analysis considering nonlinear behavior, SAE technical paper series, 2004-01-1659 Vieira L, Goncalves R, Camotim D, Pedro JO (2021) Generalized beam theory deformation modes for steel–concrete composite bridge decks including shear connection flexibility. Thin-Walled Struct 169:108408 Vieira RF, Virtuoso FBE, Pereira EBR (2013) A higher order thin-walled beam model including warping and shear modes. Int J Mech Sci 66:67–82 Vieira RF, Virtuoso FBE, Pereira EBR (2014) A higher order model for thin-walled structures with deformable cross-sections. Int J Solids Struct 51:575–598 Vieira RF, Virtuoso FBE, Pereira EBR (2015) Definition of warping modes within the context of a higher order thin-walled beam model. Comput Struct 147:68–78 Vlasov VZ (1961) Thin-walled elastic beams. Israel Program for Scientific Translations Ltd. Yu W, Volovoi V, Hodges DH, Hong X (2002a) Validation of the variational asymptotic beam sectional analysis. AIAA J 40:2105–2112 Yu W, Hodges DH, Volovoi V, Cesnik CE (2002b) On Timoshenko-like modeling of initially curved and twisted composite beams. Int J Solids Struct 39:5101–5121 Yu W, Hodges DH, Ho JC (2012) Variational asymptotic beam sectional analysis—an updated version. Int J Eng Sci 59:40–64 Zhu K, AI-Bermani FGA, Kitipornchai S, Li B (1995) Dynamic response of flexibly jointed frames. Eng Struct 17:575–580 Zuo W, Li W, Xu T, Xuan S, Na J (2012) A complete development process of finite element software for body-in-white structure with semi-rigid beams in .NET framework. Adv Eng Softw 45:261–271

Chapter 2

Torsional Warping and Torsional Distortion as Fundamental Deformable Section Modes

As can be observed from the T-joint problem considered in Fig. 1.8, the effects of section deformations, i.e., torsional warping and torsional distortion, on the overall stiffness of a thin-walled beam are significant due to their coupling behavior with torsional rotation when the beam is subjected to a torsional load or a more general load. The torsional warping and torsional distortion modes involve deformation of the cross-section of a beam, but they will be treated as modes belonging to the fundamental mode set in the HoBT; these two modes as well as the six rigid-body section modes are treated as fundamental modes in this book. As shall be shown later, none of the higher-order section-deformable modes, such as torsional warping and torsional distortion modes (see Table 1.1), produce net non-zero resultant force or moment. Therefore, the stress and displacement of these modes should decay along the axis of a beam under a static load. However, their decay rates can be quite different depending on the mode type and order. For instance, the stress and displacement of torsional warping and torsional distortion modes can survive even several times longer than the cross-sectional width, while those of other higher-order modes decay rapidly. It should also be noted that due to geometric characteristics, the effects of the torsional warping mode for open-section beams generally survive longer than those for closed-sectioned beams. Because the warping effect in open-section beams is generally significant, a structural analysis of such beams using field variables having non-zero resultant forces and moments only produces unacceptably inaccurate results. For this reason, Vlasov (1961) mostly focused his analyses on the torsional warping mode in open-section beams in his beam theory. In the case of closed thinwalled cross-sections, the torsional distortion mode can be induced by torsional warping (Kim and Kim 1999a, b, 2000, 2003; Choi and Kim 2021; Camotim et al. 2010; Goncalves et al. 2010; Yu et al. 2012). Therefore, both torsional warping and torsional distortion modes should be used simultaneously to yield accurate results. (Note that torsion warping is generally coupled with torsion in thin-walled closedsection beams.) In this chapter, we present some details of the Vlasov beam theory for thin-walled open-section beams in Sect. 2.1 and for thin-walled closed-section beams in Sect. 2.2. © Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_2

21

22

2 Torsional Warping and Torsional Distortion as Fundamental …

y y s z

O

x

(a)

un

un P

n O

r(s) A

h(s) s

x

(b)

Fig. 2.1 Thin-walled beam with an open cross-section: a a three-dimensional view with the global coordinates (x, y, z), where n and s are correspondingly the normal and tangential coordinates defined on the middle surface of the beam cross-section, and b a two-dimensional view of the midline of the beam section with in-plane displacements un and us . Symbols O and A denote the centroid of the cross-section and the center of rotation (or torsional center) about the z axis, respectively

The materials covered in Sects. 2.1 and 2.2 are used as building blocks to establish the HoBT, a thin-walled beam theory developed mainly by the authors. Owing to the difference in the mechanics involved, the warping modes for thin-walled open- and closed-section beams are differently derived in Vlasov (1961) and thus two separate sections are utilized here. By adopting some important kinematics of the Vlasov beam theory, we will derive the sectional shape functions of the torsional warping and torsional distortion of a rectangular thin-walled cross-section. A sectional shape function refers to a function that describes the deformation of its corresponding section mode on a cross-section (See Table 1.1 or Fig. 1.7 for examples of sectional shape functions.). Based on the Vlasov beam theory, an edgewise-linear function is adopted to describe the sectional shape function of torsional warping having the z-directional displacement, and an edgewise-constant function is adopted for the sectional shape function of torsional distortion having the s-directional displacement. See Figs. 2.1 and 2.7 for the definition of the z and s coordinates. The rationale to determine the total number of sectional shape functions will be discussed in Sect. 2.3. As mentioned in Chap. 1, the distortion modes of the HoBT are chosen to have nonlinear n-directional (thickness-directional) displacements to account for wall bending. Cubic polynomials are employed to represent the sectional shape function of the torsional distortion for the n direction. Unknown constants which appear in these shape functions can be found using continuity conditions at the corners of a cross-section. The detailed process to derive the sectional shape functions of torsional warping and torsional distortion will be presented in Sect. 2.3.

2.1 Vlasov Beam Theory for Thin-Walled Open Cross-Sectioned Beams

23

2.1 Vlasov Beam Theory for Thin-Walled Open Cross-Sectioned Beams 2.1.1 Kinematic Assumptions, Displacements, and Strains Figure 2.1 shows a thin-walled beam with an open cross-section. In the figure, symbol z denotes the axial coordinate while x and y denote the sectional coordinates on the plane normal to the z axis. On the normal plane, it is often convenient to use s and n, the tangential and normal coordinates defined on the midline of the sectional wall. If the unit base vectors along coordinate p (p = z, n, s) are denoted by ep , they are related via ez = en × es where en and es are mutually orthogonal unit vectors. Because the sectional wall can be assumed to be very thin, it can be assumed that σn = τns = τzn = 0, where σ and τ denote the normal and shear stresses, respectively. Furthermore, σz and σs are assumed to be constant in the thickness direction (n-direction) while the shear stress, τzs , varies linearly (Timoshenko and Goodier 1970). The two kinematic assumptions for open cross-sectional beams discussed in Sect. 1.2 are repeated here to facilitate subsequent discussions: (1) A cross-section is rigid in terms of in-plane deformation. (2) Shear strain γzs is zero on the midsurface of the wall (the surface where n = 0). Under the assumption of small deformation, the s- (us ) and z-directional (uz ) displacements at point P on the midline of the cross-sectional wall in Fig. 2.1b can be written as us (z, s) = −Ux (z) sin α(s) + Uy (z) cos α(s) + θz (z)h(s),

(2.1)

un (z, s) = Ux (z) cos α(s) + Uy (z) sin α(s) − θz (z)r(s),

(2.2)

where Ux and Uy are the translational beam displacements of a thin-walled open beam due to rigid-body sectional motion in the x and y directions, respectively, and θz denotes the rotation of the cross-section with respect to the rotation center A. The distances from A to the tangent and normal lines for the generic point P are denoted by h(s) and r(s), respectively. As Ux , Uy , and θz are functions of only z, they will be referred to as generalized 1D displacements in this book. Using the second kinematic assumption that the shear strain γzs on the midsurface of the wall is zero, we have γzs =

∂us ∂uz + = 0. ∂s ∂z

(2.3)

One can express the axial displacement uz (x, y, z) by integrating Eq. (2.3) from point P1 corresponding to s = 0 in Fig. 2.1b to point P as

24

2 Torsional Warping and Torsional Distortion as Fundamental …

P uz = Uz1 − P1

∂us ds, ∂z

(2.4)

where Uz1 is the axial displacement at P1 (x1 , y1 , z). Substituting Eq. (2.1) into Eq. (2.4) yields the z-directional displacement uz at point P as uz (x, y, z) = Uz1 (x1 , y1 , z) − = =

P [

Ux, (z)dx + Uy, (z)dy + θz, (z)dw

]

P1 , Uz1 − Ux (x − x1 ) − Uy, (y − y1 ) − θz, w (Uz1 + Ux, x1 + Uy, y1 ) − Ux, x − Uy, y − θz, w

 Uz (z) − Ux, (z)x − Uy, (z)y − θz, (z)w

(2.5)

with Uz (z) ≡ Uz1 + Ux, (z)x1 + Uy, (z)y1 , where ds cos α = dy, ds sin α = −dx, hds = dw. In Eq. (2.5) and subsequent equations, a short-hand notation ( ), is used, which is defined as ( ), = ∂( )/∂z. In Eq. (2.5), x1 and y1 are measured from the centroid O of the cross-section and P w(x, y, z) = P1 hds is twice the sectorial area denoted in gray in Fig. 1.5b lying between AP1 and AP. To simplify the subsequent development, we will assume that the beam section does not vary along z. If the shape of the beam section does not vary along the z axis, w becomes a function of only (x, y), i.e., w = w(x, y). In Eq. (2.5), Uz (z) corresponds to the generalized z-directional displacement at the centroid. The last expression in Eq. (2.5) suggests that uz at point P consists of the axial extension/contraction field, Uz (z), the bending field due to rigid-body rotations, −Ux, (z)x − Uy, (z)y, and the higher-order displacement field due to warping, −θz, (z)w(x, y). The deformation pattern due to w(x, y) is usually called torsional warping because it is generally coupled with torsion. Using Eq. (2.5), the axial strain is calculated as εz (x, y, z) =

∂uz = Uz, (z) − Ux,, (z)x − Uy,, (z)y − θz,, (z)w(x, y). ∂z

(2.6)

2.1 Vlasov Beam Theory for Thin-Walled Open Cross-Sectioned Beams

25

2.1.2 Stresses Two important stress components considered in the Vlasov beam theory are the normal stress σz acting in the axial direction and the shear stress τzs acting on the wall of the cross-section on the sectional plane. Note that σz and τzs are assumed to be constant and linear, respectively, across the wall thickness; see Fig. 2.2. Moreover, the linear shear stress τzs can decomposed into constant and anti-symmetric stress fields. While the constant shear stress arises due to the longitudinal variation of the normal stress, the anti-symmetric shear stress appears even under pure torsion (corresponding to uniform torsional moment along the axial direction). Because stress components in the n direction (thickness direction) can be ignored, the plane stress condition on the z-s plane can be assumed. Accordingly, the following strain–stress relationships can be employed: εz =

1 1 τzs , (σz − νσs ), εs = (σs − νσz ), γzs = E E G

(2.7)

where E, G, and ν are Young’s modulus, the shear modulus, and Poisson’s ratio, respectively. From the first kinematic assumption, one can ignore the in-plane higherorder deformation (εs = 0). Accordingly, the relationship between the normal stress σx and normal strain εz is expressed as

τ zs = τ zsc + τ zsa s

y s O

z

n

n

x z

σ zz

t

n s

n

s

τ zsc

τ zsa

Fig. 2.2 Normal and shear stress profiles in the thickness direction of an open-section beam (t: wall thickness)

26

2 Torsional Warping and Torsional Distortion as Fundamental … σz

y s z

O

τ zs + dτ zs

n

σ z + dσ z

x

n

s

pz

τ zs

z ds

t

Fig. 2.3 Free-body diagram of an infinitesimal segment of a thin-walled beam used to derive the force equilibrium equation in the axial direction

σz =

E εz = E1 εz , 1 − ν2

(2.8)

where E1 is the reduced Young’s modulus1 defined as E1 =

E . 1 − ν2

(2.9)

Substituting Eq. (2.6) into Eq. (2.8) leads to σz = E1 (Uz, − Ux,, x − Uy,, y − θz,, w).

(2.10)

Because the shear strain is assumed to be zero according to the second kinematic assumption, the shear stress cannot be calculated directly from the shear strain through the constitutive relationship. Instead, it should be derived by considering the force equilibrium. To this end, we consider the axial force equilibrium for an infinitesimal segment of a thin-walled beam, as shown in Fig. 2.3. d(σz t)ds + d(τzs t)dz + pz dzds = 0,

(2.11)

where pz (z, s) is the external surface force per unit area. As was shown in Fig. 2.2, the shear stress τzs can be decomposed into the anti-symmetric part τzsa (resulting from uniform torsion) and the symmetric part τzsc 2 (resulting from warping). However, τzsa does not contribute to the axial force equilibrium (The calculation of τzsa will be given later.). Therefore, τzs in Eq. (2.11) is represented only by τzsc . Dividing Eq. (2.11) by zds leads to 1

Note that bending or axial stiffness of a beam can be overestimated if Eq. (2.8) is used. For example, in the case of steel, E1 is 10% larger than E. To avoid this, E rather than E1 is used in Eq. (2.8) in many higher-order beam theories assuming εs = 0. 2 The superscript c is used to denote a “constant” field.

2.1 Vlasov Beam Theory for Thin-Walled Open Cross-Sectioned Beams

27

∂(σz t) ∂(τzs t) + + pz = 0. ∂z ∂s

(2.12)

If there is no surface force and the edge at P1 (see Fig. 2.1b for the position of P1 ) is traction-free ( τzs |P1 = 0), Eq. (2.12) can be integrated as 1 τzs = − t

P P1

∂(σz t) ds. ∂z

(2.13)

The substitution of Eq. (2.10) into Eq. (2.13) yields τzs explicitly as: τzs = −

E t

P

[ ,, ] Uz − Ux,,, x − Ux,,, y − θz,,, w tds

P1

=−

EF(s) ,, ESx (s) ,,, ESy (s) ,,, ESw (s) ,,, Uz + Ux + Ux + θz . t t t t

(2.14)

Here, F(s), Sx (s), and the remaining variables are defined as P F(s) =

P tds, Sx (s) =

P1

P xtds, Sy (s) =

P1

P ytds, Sw (s) =

P1

wtds.

(2.15)

P1

Using the normal and shear stresses in Eqs. (2.10) and (2.14), the generalized 1D forces can be defined as the work conjugates of the generalized 1D displacements. The virtual work done by the axial stress at an arbitrary cross-section A is expressed as   [ ] σz δuz dA = σz δUz − δUx, x − δUy, y − δθz, w dA A

A

= Nz δUz + My δUx, − Mx δUy, + Bδθz, .

(2.16)

The quantities with the symbol δ denote virtual displacements. In Eq. (2.16), Nz , My , Mx , and B are defined as  Nz =

 σz dA, My = −

A

 σz xdA, Mx =

A

 σz ydA, B = −

A

σz wdA,

(2.17)

A

representing the axial force, y-directional bending moment, x-directional bending moment, and bimoment, respectively. They are collectively referred to as the generalized 1D forces. The bimoment B defined at the end of Eq. (2.17), which is the work

28

2 Torsional Warping and Torsional Distortion as Fundamental …

conjugate of the torsional warping, does not appear in the classical 6-DOF beam theory. Similarly, the virtual work by the shear stress can be expressed as 

 τzsc δus dA = A

] [ τzsc −δUx sin α + δUy cos α + δθz h dA

A

= Qx δUx + Qy δUy + H δθz ,

(2.18)

where the generalized forces are defined as  Qx = −

 τzsc

sin αdA, Qy =

A

 τzsc

cos αdA, H =

A

τzsc hdA.

(2.19)

A

In Eq. (2.19), Qx and Qy are the x- and y-directional shear forces, respectively, and H is the torsional moment. These generalized forces are due to τzsc , which is the shear stress caused by warping. Note that the torsional moment H should be distinguished from the torsional moment contributed by τzsa (illustrated in Fig. 2.2b), which is the shear stress induced by pure torsion. The moment due to τzsa is defined as Ha :  Ha =

τzsa ndA.

(2.20)

A

It is related to the twist rate θz, as Ha = GJ θz, ,

(2.21)

where GJ is called the torsional rigidity, which is given by GJ = Glt 3 /3 for a thin-walled open cross-section (l: length of the midline of the sectional wall; t: wall thickness). The one-dimensional governing equations can be derived by considering the force equilibrium conditions for an infinitesimal segment of a beam subjected to general loads (axial, bending, and torsional loads). The resulting equations will be four fourth-order ordinary differential equations for four field variables, Ux , Uy , Uz , and θz . Vlasov (1961) also derived analytic solutions to these equations for some boundary and load conditions.

2.2 Vlasov Beam Theory for Thin-Walled Closed Cross-Sectional Beams

29

2.2 Vlasov Beam Theory for Thin-Walled Closed Cross-Sectional Beams As was discussed in Sect. 1.2.2, the two kinematic assumptions for open-section beams are not valid for closed-section beams; in closed-section beams, the deformations of their cross-sections on the cross-sectional plate are not negligible, causing non-zero shear strain on the midsurface of the wall under torsion. In the Vlasov theory, torsional warping and torsional distortion are assumed to be edgewise-linear based on a hinged-beam frame model; the corresponding section deformations are sketched in Figs. 1.7g, h for a rectangular thin-walled cross-section. Because the members of the beam frame model are regarded as in-extensional, their lengths do not change for any cross-sectional distortion. For subsequent discussions, a generalized 1D displacement will be denoted as ξi , where the index i covers all generalized 1D displacements. Specifically, ξ1 = Ux , ξ2 = Uy , ξ3 = Uz , ξ4 = θx , ξ5 = θy , and ξ6 = θz and the corresponding motions of the section are shown in Fig. 2.4a–d. On the other hand, ξ7 and ξ8 are the generalized 1D displacements representing the torsional warping in Fig. 2.4g and torsional distortion in Fig. 2.4h, respectively. Because ξi represents the magnitude of the overall cross-sectional deformation3 as functions of ξ ξ only z, ξi = ξi (z), the sectional shape functions ψs i (s) and/or ψz i (s) are introduced to represent the deformation of the cross-section on the midline of the wall associated with ξi (z). The subscript α in ψαξi (s) (α = r, z) denotes the direction along which the displacement of the mode ξi occurs. In defining the cross-sectional shape functions representing torsional warping and torsional distortion, they are chosen to  ξ ξ  ξ ξ be orthogonal to each other: A ψz i ψz j dA = 0 and A ψs i ψs j dA = 0 for i /= j, where A denotes the cross-sectional area. Using the sectional shape functions and generalized 1D displacements, one may express the axial and transverse displacements uz (z, s) and us (z, s) on the midline of the sectional wall as uz (z, s) =

ND 

ψzξi (s)ξi (z),

(2.22a)

ψsξi (s)ξi (z).

(2.22b)

i=1

us (z, s) =

ND  i=1

In these equations, N D denotes the total number of section modes considered. As was discussed earlier, the first six modes in Eqs. (2.22a, b) (from i = 1 to 6) are selected to denote non-deformable section modes, ξ1 = Ux , ξ2 = Uy , ξ3 = Uz , ξ4 =

Note that the symbol ξi will represent the type of mode as well as the corresponding axial variation along z.

3

30

2 Torsional Warping and Torsional Distortion as Fundamental …

y

x

(a)

(b)

(d)

(c)

(e)

(f)

n2

s2 n2 s2

s1 n1

n3

y

s1

x

s3

n3

n1

s3 s4 n4

(g)

n4

s4

(h)

Fig. 2.4 Sectional shape functions of a rectangular thin-walled cross-section: a–f six nondeformable section modes, and g torsional warping and h torsional distortion modes obtained using a hinged-beam frame model (Vlasov 1961). The two modes in g and h are deformable section modes considered in the Vlasov theory

θx , ξ5 = θy , and ξ6 = θz . The warping and distortion modes4 are treated as higherorder modes denoted correspondingly by ξ7 and ξ8 . If mode i corresponds to a warping mode, its cross-sectional shape functions depict the out-of-plane deformation of ξ ξ the cross-section, i.e., ψz i (s) /= 0 and ψs i (s) = 0. If mode j corresponds to a distortion mode, its cross-sectional shape functions depict the in-plane deformation ξ ξ of the cross-section, i.e., ψz j (s) = 0 and ψs j (s) /= 0. Figure 2.5 illustrates how Eq. (2.22a) is calculated; the axial displacement on the midline of a sectional wall can be represented by summing up those caused by out-of-plane section modes (Uz , θx , θy , and W ). The transverse displacement on the midline in Eq. (2.22b) can be represented in the same way. To establish the HoBT, we also use this approach; we represent three-dimensional displacements on the midline of a sectional wall as the product of the sectional shape functions and generalized 1D displacements. To give an overview of the procedure used to find the sectional shape functions ξ ξ ψz i (s) and ψs i (s), let us start with the derivation of the shape function for a torsional warping mode. As mentioned earlier, Vlasov assumed an edgewise-linear function as the cross-sectional deformation of torsional warping. For a cross-section with m nodes (intersections and end points), we can find m edgewise-linear shape functions ξ ξ ξ ψz i (s) satisfying ψz i = 1 only at one node while ψz i = 0 at the other nodes. For ξi example, four edgewise-linear shape functions ψz (s) are illustrated for a rectangular thin-walled cross-section (m = 4) in Fig. 2.5a–d. Because the shape functions in Fig. 2.5a–d are linearly independent, their associated generalized 1D displacements 4

Note that these two modes are treated as fundamental modes when deriving all other higher-order modes in the HoBT.

2.2 Vlasov Beam Theory for Thin-Walled Closed Cross-Sectional Beams Generalized 1D displacement ( sectional mode)

Sectional shape function

ψ U (s)

X

U z ( z)

ψ θ (s)

X

θ x ( z)

z

x

31

3D displacement on the midline

uz ( z, s)

y

Σ

=

x z

θy

ψ ( s)

X

θ y ( z)

ψ W ( s)

X

W ( z)

Fig. 2.5 Illustrative explanation of Eq. (2.22a) for the calculation of the axial (z-directional) displacement on the midline of a sectional wall. (The three-dimensional displacements on the midline of a sectional wall are represented as the product of the sectional shape functions and the generalized 1D displacements in developing higher-order beam theories, including the HoBT)

can be employed as independent field variables for the beam analysis. However, it is not useful to treat the sectional shape functions in Fig. 2.5a–d as those of warping modes because the shape functions in Fig. 2.5a–d are coupled with non-deformable section modes. For example, the shape function for the z-directional translation can be obtained by adding the four shape functions in Fig. 2.5a–d. ξ Alternatively, one may find m edgewise-linear shape functions ψz i (s) (i = 1, 2, . . . , m) for out-of-plane deformation by imposing their mutual orthogonality: 

ξ

ψzξi ψz j dA = 0 for i /= j.

A

For a rectangular thin-walled cross-section, Fig. 2.5e–h show sectional shape functions that are orthogonal to each other. Note that the first three shape functions in Fig. 2.5e–g denote non-deformable section modes as used in the classical beam theories (i.e., the Euler and Timoshenko beam theories), while the shape function in Fig. 2.6h represents a torsional warping mode. Therefore, for a cross-section with m nodes, the number of Vlasov warping modes becomes m − 3. ξ Now, will explain how to find the shape functions of distortion modes ψs j (s) by considering the in-plane motion of the beam cross-section. To deal with a crosssection made of thin walls, Vlasov (1961) assumed that the cross-section itself can

32

2 Torsional Warping and Torsional Distortion as Fundamental …

Node 3

Node 3

Node 3

Node 2

Node 2

Node 2

Node 1

Node 1

Node 4

Node 4

Node 4

(a)

(b)

Node 1

(d) Node 2

Node 2

Node 2

Node 3

Node 3

Node 3

Node 1

Node 1

Node 1

Node 4

Node 4

Node 4

(e)

Node 4

(c)

Node 2

Node 4

Node 3

Node 1

Node 1

Node 3

Node 2

(f)

(g)

(h)

Fig. 2.6 Edgewise-linear shape functions for out-of-plane deformation: a–d non-orthogonal shape functions and e–h orthogonal shape functions

be viewed as a two-dimensional frame structure consisting of in-extensional bar members which can be regarded as hinge-connected. If the number of bar members is c, the number of sectional shape functions defining us (i.e., in-plane deformation) is n = 2m − c considering two in-plane degrees of freedom of each node and one inextension constraint of each bar. For the rectangular cross-section shown in Fig. 2.4, for example, the number of shape functions for in-plane deformation is n = 2m − c= 4 (m = 4 and c = 4). If the shape functions are orthogonalized using the relationship  ξi ξj A ψs ψs dA = 0 for i / = j, one can find the shape functions shown in Fig. 2.4b, d, f, h. Among them, the first three shape functions represent the rigid-body motions of a cross-section, while the last one in Fig. 2.4h represents torsional distortion representing an in-plane deformable motion. We can now consider a polygonal crosssection with m nodes, where all nodes are assumed to connect two adjacent bar members. In this case, the number of the Vlasov distortion modes is obtained via n = 2m−c −3. . If a cross-section has nodes connecting more than two bar members (a multicell cross-section) or only one bar member (a cross-section with flanges), the number c of constraints should change accordingly. Using the displacements in Eq. (2.22a, b), the axial normal (εz ) and wall shear (γzs ) strains on the midline of a cross-sectional wall are calculated as D  ∂uz ψzξi (s)ξi, (z), = ∂z i=1

N

εz (z, s) =

(2.23a)

2.3 Torsional Warping and Torsional Distortion of the HoBT D D   ∂uz ∂us γzs (z, s) = ψsξi (s)ξi, (z), + = ψ˙ zξi (s)ξi (z) + ∂s ∂z i=1 i=1

N

33

N

(2.23b)

where (· ) = ∂( )/∂s. Note in Eqs. (2.23a, b) that the normal and shear strains are assumed to be uniform in the wall thickness direction. Using the strains in Eqs. (2.23a, b) and the corresponding stresses, one can derive the governing one-dimensional differential equations from the principle of virtual work. Vlasov (1961) suggested that the strain energy by bending moments at the corners of the beam frame be additionally considered when the principle of virtual work is applied. Given that this book will cover a more comprehensive treatment of thin-walled beams in later chapters, further explanations of the Vlasov theory apart from the associated kinematic assumptions are omitted.

2.3 Torsional Warping and Torsional Distortion of the HoBT 2.3.1 Simple Torsion Problem of a Thin-Walled Rectangular Cross-Sectional Beam The HoBT is a generalization of the Vlasov theory in that it considers additional 1D degrees of freedom representing deformable section modes. A salient feature of the HoBT is that all deformable section modes are derived in a closed form for box beams using a systematic hierarchical and recursive method. Therefore, as many higher-order modes as desired can be included in the HoBT consistently. In this section, the torsional warping and torsional distortion shape functions of the rectangular cross-section shown in Fig. 2.7 will be derived. They will be used as the fundamental set (or lowest set5 ) of the higher-order modes of the HoBT. As in the Vlasov beam theory, the HoBT expresses the displacements up (z, s) (p = z, n, s) on the midline of a cross-sectional wall as the sum of the products of the generalized 1D displacements ξ(x) and corresponding sectional shape functions ψ ξ (s): uz (z, s) =

ND 

ψzξi (s)ξi (z),

(2.24a)

i=1

5

Due to the hierarchical nature of higher-order modes, a set number is assigned to modes according to the order in which they are derived. The torsional warping and torsional distortion that are discussed in this section are the fundamental modes; they are considered as the modes belonging to the lowest (or zeroth or fundamental) set in subsequent chapters.

34

2 Torsional Warping and Torsional Distortion as Fundamental …

y

Fig. 2.7 Edgewise local coordinates of a rectangular cross-section (b: width; h: height)

corner 2

n2

Edge 2 s2

corner 1 z

Edge 1 s1

Edge 3 h n3 s3

z

Edge 4 z

n4

un (z, s) =

ND 

x

z

z

corner 3

n1

s4

wall midline corner 4

b

ψnξi (s)ξi (z),

(2.24b)

ψsξi (s)ξi (z).

(2.24c)

i=1

us (z, s) =

ND  i=1

In Eqs. (2.24a, b, c), the subscript i (i = 1, 2, …, N D ) in ξi and ψ ξi denotes the ith generalized 1D displacement or the ith cross-sectional mode, while N D denotes the total number of section modes considered. For instance, ND = 6 for the classical Euler/Timoshenko beam theory, ND = 8 for a box beam according to the Vlasov theory (see Fig. 2.4), and ND ≥ 8 for the HoBT dealing with a thin-walled beam subject to a general load set (consisting of simultaneously applied axial, bending, and torsional forces/moments). Here, ξi can represent both out-of-plane modes6 and ξ in-plane modes. The subscript k in ψk i represents the component along coordinate k (k = z, n, s) of the section shape function ψ ξi . Comparing Eqs. (2.24a, b, c) for the HoBT and Eqs. (2.22a, b) for the Vlasov theory shows that unlike the Vlasov theory, the HoBT considers the n-directional (or thickness-directional) displacement un to account for wall bending. Accordingly, the HoBT, unlike the Vlasov theory, can consider non-uniform bending stress (σss ) varying along the n coordinate. For an analysis of the in-plane deformations of a beam section, the HoBT does not assume that a beam cross-section behaves as a hinged frame, as does the Vlasov beam 6

The plane in “out-of-plane modes” or “in-plane modes” denotes the plane when the cross-section is defined, i.e., the x–y plane in the case shown in Fig. 2.7.

2.3 Torsional Warping and Torsional Distortion of the HoBT

35

theory. If the HoBT is used, therefore, warping can be edgewise-nonlinear, walls can be extensional by distortion, and bending moments at the corners of the crosssection are non-zero and can be continuous. The cross-sectional shape functions by the HoBT will be derived in detail for a rectangular cross-section from Chaps. 4–7; those for a cross-section with a general shape will be derived in Chap. 9. To better address the difference between the HoBT and the Vlasov theory, let us consider a thin-walled rectangular cross-sectional beam (or a box beam) under torsion. Because we consider only the torsion load case here, ND = 1 for the classical beam theory, ND = 3 for the Vlasov theory, and ND ≥ 3 for the HoBT. Although the HoBT is valid for ND ≥ 3, we will consider the HoBT for the case of ND = 3 to compare the HoBT and the Vlasov theory. With ND = 3, the displacements on the midline of the wall can be expressed [using Eqs. (2.24a, b, c)] as uz (z, s) = ψzW (s)W (z),

(2.25a)

un (z, s) = ψnθ (s)θ (z) + ψnχ (s)χ (z),

(2.25b)

us (z, s) = ψsθ (s)θ (z) + ψsχ (s)χ (z).

(2.25c)

In Eqs. (2.25a, b, c), the ξi variables in Eqs. (2.24a, b, c) are replaced by θ , W and χ to clarify the meaning of these variables, where θ , W , and χ correspond to torsional rotation, torsional warping, and torsional distortion, respectively. Two nonzero cross-sectional shape functions for torsional rotation are denoted by ψsθ (s) and ψnθ (s) in Eqs. (2.25a, b, c). They can be obtained by simple rigid-body kinematics associated with θ , the rotation of the cross-section about the z axis. Thus, ψsθ denotes the distance from the shear center to the corresponding wall, and ψnθ is the distance from the center ( of the ) wall to the point of interest. Using the edgewise-defined local coordinates z, nj , sj (j: edge index) in Fig. 2.7, ψsθ and ψnθ can be found as ψsθ (s1 ) =

h b h b , ψsθ (s2 ) = , ψsθ (s3 ) = , ψsθ (s4 ) = 2 2 2 2 ψnθ (sj ) = −sj ,

(2.26) (2.27)

where b and h correspondingly denote the width and height of the cross-section (see Fig. 2.7).

36

2 Torsional Warping and Torsional Distortion as Fundamental …

2.3.2 Derivation of the Shape Functions of Torsional Warping and Distortion In this subsection, we will present the method to derive the sectional shape functions χ of torsional warping ψzW (s) and torsional distortion ψs (s). The derivation of these functions requires more considerations involving equilibrium equations, constitutive equations, and kinematic assumptions for thin plates. First, we consider the state of the edgewise-uniform shear stress of a box beam, which can be expressed as ∂τzs = 0, ∂s

(2.28)

τzs = fj (z) on edge j, (j = 1, 2, 3, 4)

(2.29)

or

where fj (z) denotes a function of z only. Equations (2.28) or (2.29) represents the state of uniform shear stress on the entire cross-section in the Saint Venant torsion problem, where torsional moments of the same magnitude are applied at both free ends of a beam, i.e., the beam is in a state of uniform torsion. Using Eqs. (2.7) and (2.25a, b, c), one can write τzs in Eq. (2.29) as ( τzs = G

∂uz ∂us + ∂z ∂s

)

= G(ψsθ θ , + ψsχ χ , + ψ˙ zW W ) = fj (z) onedgej. (j = 1, 2, 3, 4)

(2.30)

Derivation of ψzW (s) (shape function of torsional warping) In Sect. 2.1.1, torsional warping in an open-section beam arises when torsional rotation varies along the axial direction; see Eq. (2.5). This observation suggests that the state of uniform shear stress due to pure torsion in Eq. (2.28) cannot be obtained without an s-directional change of the z-directional deformation, i.e., the torsional warping deformation. As in an open-section beam should also be valid for a rectangular cross-section, torsional warping in a rectangular cross-section should develop by the axial variation of torsional rotation. In addition, one can assume that torsional warping can also be induced by the axial variation of torsional distortion. Accordingly, torsional warping for a rectangular cross-section can be related to the axial variations of torsional rotation and torsional distortion as W = cθ

dθ dχ + cχ , i.e.,W =cθ θ , + cχ χ , , dz dz

(2.31)

2.3 Torsional Warping and Torsional Distortion of the HoBT

37

where cθ and cχ are constants. The contributions of θ , and χ , to W can be treated independently because they are independent of each other. However, the simultaneous consideration of these contributions as in Eq. (2.31) can facilitate the subsequent analysis. In Eq. (2.31), one can simply choose cθ = cχ = 17 in Eq. (2.31) without a loss of generality. Thus, we write Eq. (2.31) as W =

dθ dχ + , i.e.,W =θ , + χ , dz dz

(2.32)

Substituting Eq. (2.32) into Eq. (2.30) yields [( ) ( ) ] G ψsθ (s) + ψ˙ zW (s) θ , (z) + ψsχ (s) + ψ˙ zW (s) χ , (z) = fj (z).

(2.33)

Equation (2.33) is valid if ψsθ (s) + ψ˙ zW (s) = cj , ψsχ (s) + ψ˙ zW (s) = dj , (j = 1, 2, 3, 4)

(2.34)

where cj and dj are edgewise-defined constants. Because ψsθ (s) is constant for each edge in a rectangular cross-section (see Eq. (2.26)), the first condition from Eq. (2.34) requires that ψzW (s) be edgewise-linear. For a rectangular cross-section, the warping shape function ψzW (s) can be expressed as ψzW (s1 ) = a10 + a11 s1 ,

(2.35a)

ψzW (s2 ) = a20 + a21 s2 ,

(2.35b)

ψzW (s3 ) = a30 + a31 s3 ,

(2.35c)

ψzW (s4 ) = a40 + a41 s4 ,

(2.35d)

where nj and sj denote the local sectional coordinates illustrated in Fig. 2.7. To determine the constants aj0 and aj1 (j = 1, 2, 3, 4) in Eqs. (2.35a, b, c, d), we use the following conditions: (1) The z-directional displacements should be continuous at cross-sectional corners. (2) The sectional shape function of a warping mode is set to be orthogonal to the sectional displacement field due to out-of-plane non-deformable section modes.

7

In Eq. (2.25a, b, c), a displacement component is expressed as the product of a 1D generalized displacement variable and its sectional shape function. Therefore, the coefficients of the section modes in Eq. (2.31) can be chosen arbitrarily depending on the scaling of their sectional shape functions.

38

2 Torsional Warping and Torsional Distortion as Fundamental …

From the first condition stated above, one can write the following four explicit conditions at the corners for ψzW (see Fig. 2.7 for the corner location indices): ( ( ) ) b h W s1 = = ψz s2 = − at corner 1, 2 2 ( ( ) ) b h = ψzW s3 = − at corner 2, ψzW s2 = 2 2 ( ( ) ) b h = ψzW s4 = − at corner 3, ψzW s3 = 2 2 ( ( ) ) b h W W = ψz s1 = − at corner 4. ψz s4 = 2 2 ψzW

(2.36a) (2.36b) (2.36c) (2.36d)

To determine the relative ratios among aj0 and aj1 (j = 1, 2, 3, 4), we employ the following three equations based on Condition (2): 

 ψzW ψzUz dA = 0, A



θ

ψzW ψz y dA = 0, A

ψzW ψzθx dA = 0,

(2.37a–c)

A

where the non-zero shape functions of the non-deformable section modes are simply given by θ

ψzUz = 1, ψzθx = y and ψz y = −x ( α ) ψn = ψsα = 0 for α = Uz , θx , and θy .

(2.38)

The substitution of Eq. (2.38) into Eq. (2.37a–c) yields  · (1)dA =

j=1 −s

A



ψzW (sj )tdsj = 0,

(2.39a)

ψzW (sj )xtdsj = 0,

(2.39b)

j

4  

sj

ψzW

· (x)dA =

j=1 −s

A

j

s¯j

 ψzW A

4  

sj

ψzW

· (y)dA =

4  

j=1 −¯s

ψzW (sj )ytdsj = 0, j

with − s¯ j ≤ sj ≤ s¯ j (j = 1, 2, 3, 4).

(2.39c)

2.3 Torsional Warping and Torsional Distortion of the HoBT

39

The eight constants, aj0 and aj1 (j = 1, 2, 3, 4) in Eqs. (2.35a, b, c, d), can be determined using the seven conditions given in Eqs. (2.36a, b, c, d) and (2.39a, b, c) and a scaling condition. If we scale the warping displacement at the section corner as ψzW (s1 = b/2) = bh/4, the warping shape function wzW is obtained as ψzW (s1 ) =

b · s1 , 2

h ψzW (s2 ) = − · s2 , 2

(2.40a) (2.40b)

b · s3 , 2

(2.40c)

h ψzW (s4 ) = − · s4 . 2

(2.40d)

ψzW (s3 ) =

The sectional shape function ψzW (s) given by Eqs. (2.40a, b, c, d) is plotted in Fig. 2.8a. It represents the lowest-order warping function. Higher-order warping functions are shown in the first row of Table 1.1 (W1 and W2 ). χ

Derivation of ψz (s) (shape function of torsional distortion) The torsional distortion denoted by the field variable χ (z) represents the deformation of a beam cross-section on its plane. Thus, the non-zero components of χ χ the corresponding shape functions are ψs (s) and ψn (s), which represent the s-

Fig. 2.8 Sectional shape functions of a torsional warping (W ) and b torsional distortion (χ ) of the HoBT for a rectangular thin-walled cross-section

40

2 Torsional Warping and Torsional Distortion as Fundamental …

and n-directional displacements, respectively (As shall be apparent, the displacement continuities of the torsional distortion shape function cannot be satisfied if χ χ either ψs or ψn is zero.). From the second relationship in Eq. (2.34), which is χ W ψs (s) = −ψ˙ z (s) + dj (j = 1, 2, 3, 4), and given that ψzW is edgewise-linear from χ χ Eqs. (2.40a, b, c, d), ψs should be edgewise-constant. Therefore, one can write ψs (s) 8 using four unknown constants cj (j = 1, 2, 3, 4) as ψsχ (s1 ) = c1 , ψsχ (s2 ) = c2 , ψsχ (s3 ) = c3 , ψsχ (s4 ) = c4 . χ

(2.41a–d) χ

On the other hand, there is no available condition for ψn (s). Here, we assume ψn as edgewise-cubic functions as (the rationale applied to choose cubic functions will be explained below): ψnχ (s1 ) = c10 + c11 s1 + c12 s12 + c13 s13 ,

(2.42a)

ψnχ (s2 ) = c20 + c21 s2 + c22 s22 + c23 s23 ,

(2.42b)

ψnχ (s3 ) = c30 + c31 s3 + c32 s32 + c33 s33 ,

(2.42c)

ψnχ (s4 ) = c40 + c41 s4 + c42 s42 + c43 s43 ,

(2.42d)

where cjk (j = 1, 2, 3, 4; k = 0, 1, 2, 3) are unknown constants. Before discussing a method to determine the 20 unknown constants appearing in Eqs. (2.41a–d) and χ (2.42a, b, c, d), we note that while ψn is assumed be a linear function in the Vlasov theory (shown in Fig. 1.7), we represent it as a cubic polynomial function. χ The motivation behind the use of cubic functions for ψn is that the linear approximation violates the continuities in rotation (about the z axis) at every corner of the cross-section. In other words, Vlasov assumed moment-free hinge conditions at every corner, but this assumption does not satisfy the condition that corner angles should remain unaltered after any deformation to ensure the continuity of rotation angles. To ensure continuity in the rotation angles at cross-sectional corners, therefore, non-zero moment should be applied to each of the edges meeting at a corner, causing in-plane edge bending. Because the general solution to the bending of a beam (representing each edge) is given by a cubic function,9 it is necessary to use a cubic polynomial to represent the edgewise n-directional displacement in Eqs. (2.42a, b, c, d). Now, we will explain the procedure to determine the 20 constants in Eqs. (2.41a–d) and (2.42a, b, c, d) using arguments similar to those used to determine ψzW (s): 8

This is different from cj in Eq. (2.34). To avoid the use of too many symbols in this book, we will not employ a new symbol for every new constant unless it is necessary for clarity. 9 The homogeneous governing equation for bending deflection ψ (s) of a beam in the Euler beam n theory is given by d 4 ψn (s)ds4 = 0 (Beer et al. 2020). The general solution has a form identical to those in Eqs. (2.42a, b, c, d).

2.3 Torsional Warping and Torsional Distortion of the HoBT

41

(1) The x- and y-directional displacements (i.e., in-plane displacements) and rotations about the z axis should be continuous at all corners of the beam cross-section. (2) The sectional shape function of a distortion mode is set to be orthogonal to the sectional displacement field due to in-plane non-deformable section modes. Condition (1) pertaining to the continuities of in-plane displacements at crosssectional corners can be written as ( ( ( ( ) ) ) ) h b h b = ψnχ s2 = − , ψnχ s1 = = −ψsχ s2 = − at corner 1, ψsχ s1 = 2 2 2 2 (2.43a) ) ) ) ) ( ( ( ( b h b h = ψnχ s3 = − , ψnχ s2 = = −ψsχ s3 = − at corner 2 ψsχ s2 = 2 2 2 2 (2.43b) ( ( ( ( ) ) ) ) h b h b = ψnχ s4 = − , ψnχ s3 = = −ψsχ s4 = − at corner 3 ψsχ s3 = 2 2 2 2 (2.43c) ( ( ( ( ) ) ) ) b h b h = ψnχ s1 = − , ψnχ s4 = = −ψsχ s1 = − at corner 4 ψsχ s4 = 2 2 2 2 (2.43d) Note that the continuities both in the n- and s-directions are considered at each corner in Eqs. (2.43a, b, c, d). In addition to the displacement continuity conditions given by Eqs. (2.43a, b, c, d), we consider the continuities in the rotation angles and bending moments (or curvatures) at the corners; see Fig. 2.9. Here, we assume that all edges have the same thickness (t) and are made of the same material (Young’s modulus E). Using the kinematic assumption of Kirchhoff’s thin plate theory, we assume zero shear strain (γns ) due to the thinness of the wall of every edge. In this case, the rotation angle and curvature at a point on the midline of the wall can be calculated as u˙ n and u¨ n , respectively where (·) = ∂( )/∂s. Therefore, the continuity conditions for the rotation angles and curvatures by torsional distortion at the corners are given as ( ( ) ) b h χ χ ˙ ˙ ψn s1 = = ψn s2 = − , 2 2 ( ( ) ) h b = ψ˙ nχ s3 = − , ψ˙ nχ s2 = 2 2 ) ) ( ( b h χ χ ˙ ˙ = ψn s4 = − , ψn s3 = 2 2

(2.44a) (2.44b) (2.44c)

42

2 Torsional Warping and Torsional Distortion as Fundamental …

Fig. 2.9 Consideration of continuities in the rotation angle and moment at a corner connecting edges 1 and 2 (t: edge thickness, E: Young’s modulus, r = b/2 for edges 1 and 3 and r = h/2 for edges 2 and 4)

( ( ) ) b h = ψ˙ nχ s1 = − , ψ˙ nχ s4 = 2 2

(2.44d)

and ( ( ) ) h b ψ¨ nχ s1 = = ψ¨ nχ s2 = − , 2 2 ( ( ) ) b h χ χ ¨ ¨ = ψn s3 = − , ψn s2 = 2 2 ( ( ) ) h b = ψ¨ nχ s4 = − , ψ¨ nχ s3 = 2 2 ( ( ) ) h b χ χ ¨ ¨ ψn s4 = = ψn s1 = − . 2 2

(2.45a) (2.45b) (2.45c) (2.45d)

Now we consider Condition (2) which can be explicitly written as 

ψsχ ψsθz dA = 0,

A

 A

ψsχ ψsUx dA = 0,



ψsχ ψs y dA = 0 U

(2.46)

A

U

where ψsθz , ψsUx , and ψs y are the s-directional shape functions of the torsional rotation θz , the x-directional translation Ux , and the y-directional translation Uy , respectively. U Referring to Fig. 2.9, one can write ψsθz , ψsUx , and ψs y explicitly as ψsθz = r with r =

h b for edges 1 and 3 and r = for edges 2 and 4 2 2

(2.47a)

2.3 Torsional Warping and Torsional Distortion of the HoBT

43

ψsUx = −1 for edge 2, ψsUx = 1 for edge 4, ψsUx = 0 for edges 1 and 3(∵ ψxUx = 1 &ψyUx = 0)

(2.47b)

ψsUx = 1 for edge 1, ψsUx = −1 for edge 3, U

ψsUx = 0 for edges 2 and 4 (∵ ψxUx = 0 & ψy y = 1)

(2.47c)

Substituting Eqs. (2.47a, b, c) into Eq. (2.46) yields  A

ψsχ rdA

h/ 2 =

b ψsχ |edge 1 tds1 2

−h/ 2

h/ 2 +

+ −b/ 2

b ψsχ |edge 3 tds3 2

−h/ 2



b/ 2

ψsχ ψsUx dA

h ψsχ |edge 2 tds2 2

b/ 2 + −b/ 2

b/ 2 =−

h ψsχ |edge 4 tds4 = 0, 2

(2.48a)

ψsχ |edge 2 tds2

−b/ 2

A

b/ 2 +

ψsχ |edge 4 tds4 = 0,

(2.48b)

−b/ 2



U ψsχ ψs y dA

h/ 2 =

ψsχ |edge 1 tds1

−h/ 2

A

h/ 2 −

ψsχ |edge 3 tds3 = 0.

(2.48c)

−h/ 2

Using the eight corner continuity conditions in Eqs. (2.43a, b, c, d) for in-plane displacements, the four continuity conditions in rotation angles in Eqs. (2.44a, b, c, d), the four continuity conditions of curvatures in Eqs. (2.45a, b, c, d), and the three orthogonality conditions in Eqs. (2.48a, b, c) with an arbitrarily selected scaling χ condition such that ψs (s1 ) = bh/(b + h), the distortion shape functions are obtained as ψsχ (sj ) =

bh for j = 1, 3, b+h

ψsχ (sj ) = −

bh for j = 2, 4, b+h

(2.49a) (2.49b)

44

2 Torsional Warping and Torsional Distortion as Fundamental …

and ψnχ (sj ) = − ψnχ (sj ) =

2b + h 4 sj3 + sj for j = 1, 3, h(b + h) b+h

4 b + 2h sj3 − sj for j = 2, 4. b(b + h) b+h

(2.50a) (2.50b)

The resulting shape function ψ χ of torsional distortion is sketched in Fig. 2.8b. Note that the torsional distortion shape function has an anti-symmetric shape, as does the torsional warping function.

References Beer FP, Johnston Jr. ER, DeWolf JT, Mazurek DR, Sanghi S (2020) Mechanics of materials, 8th edn. McGraw-Hill, New York Camotim D, Basaglia C, Silvestre N (2010) GBT buckling analysis of thin-walled steel frames: a state-of-the-art report. Thin-Walled Struct 48:726–743 Choi S, Kim YY (2021) Higher-order Vlasov torsion theory for thin-walled box beam. Int J Mech Sci 195:106231 Goncalves R, Ritto-Corrêa M, Camotim D (2010) A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory. Comput Mech 46:759–781 Kim JH, Kim YY (1999a) Analysis of thin-walled closed beams with general quadrilateral cross sections. J Appl Mech 66:904–912 Kim YY, Kim JH (1999b) Thin-walled closed box beam element for static and dynamic analysis. Int J Numer Meth Eng 45:473–490 Kim JH, Kim YY (2000) One-dimensional analysis of thin-walled closed beams having general cross-sections. Int J Numer Meth Eng 49:653–668 Kim Y, Kim YY (2003) Analysis of thin-walled curved box beam under in-plane flexure. Int J Solids Struct 40:6111–6123 Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York Vlasov VZ (1961) Thin-walled elastic beams. Israel Program for Scientific Translations Ltd. Yu W, Hodges DH, Ho JC (2012) Variational asymptotic beam sectional analysis–an updated version. Int J Eng Sci 59:40–64

Chapter 3

One-Dimensional Governing Equations and Finite Element Formulation

Chapter 1 suggests that the three-dimensional displacements on the midline of a crosssectional wall can be expressed as products of the generalized 1D displacements and sectional shape functions. The displacements at a general point on a cross-section are calculated using those on the midline based on the kinematic assumption of Kirchhoff’s thin plate theory, i.e., the assumption of no shear and normal strains in the thickness direction (εnn = γns = γzn = 0). The local coordinates (z, n, s) defined on the midline of a cross-sectional wall will be used to facilitate the expression of the strain field from the three-dimensional displacements. A two-dimensional stress field can be derived assuming a plane-stress state (i.e., σnn = τns = τzn = 0) for thin walls. Once the expressions for stress and strain everywhere in a beam are available, the one-dimensional governing equations of the HoBT can be derived using the principle of the minimum total potential energy and then integrating the results over a beam cross-section. The potential energy consists of the strain energy stored in a beam and the potential energy by external forces (The strain energy is expressed using the threedimensional stress and strain components.). By minimizing the total potential energy (or using the principle of virtual work) in which the generalized 1D displacements are the field variables, the one-dimensional governing equations can be derived for the HoBT. The explicit procedure to derive one-dimensional governing equations will be given below. To facilitate the explanation of the derivation procedure, we consider a simple case of beam torsion (Kim and Kim, 1999a, b, 2000), which was considered in Sect. 2.3. In this case, the derivation procedure can be explained using three generalized 1D displacements only: torsional rotation (θ (z)), torsional warping (W (z)), and torsional distortion (χ (z)). The symbol z denotes the beam axial direction, which is perpendicular to the plane of a beam cross-section. Note that W (z) and χ (z) represent 1D generalized displacements which involve the deformation of the cross-section, while θ (z) is the 1D generalized displacement representing the rigid-body rotation of the cross-section about z axis. In fact, Vlasov’s theory considered these three generalized displacements. Therefore, the resulting 1D Vlasov theory can be viewed as a © Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_3

45

46

3 One-Dimensional Governing Equations and Finite Element Formulation

special case of the HoBT because the HoBT allows as many higher-order warping and distortion displacements as desired beyond θ , W , and χ . Accordingly, the subsequent analysis can be directly extended to derive the necessary 1D field equations for the HoBT. To derive 1D equilibrium equations, three-dimensional displacements, strains and stresses at a general point (z, n, s) should be explicitly related to θ (z), W (z), and χ (z). The derived equations will show how W (z) and χ (z) are coupled with θ (z). . In the process of the derivation, the key rigidity measures relevant to rotation (θ ), warping (W ) and distortion (χ ) can be naturally identified. For example, the torsional rigidity relating the torsional rotation θ and the torsional moment can be identified. For a box beam subject to a simple boundary condition, an analytic solution will be presented. Otherwise, the finite element method will be used to solve general static load cases as well as to perform a modal analysis. After presenting the procedure to derive the 1D governing equations for the three generalized 1D displacements (θ (z), W (z) and χ (z)), we will demonstrate how to extend the procedure to deal with a case involving any number of generalized 1D displacements (i.e., cross-sectional modes).

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes θ, W and χ This section is concerned with an analysis of thin-walled box beams using the HoBT employing only three 1D generalized displacements, θ (z), W (z), and χ (z), which represent torsional rotation, torsional warping, and torsional distortion, respectively. The use of “torsional” in front of rotation, warping and distortion serves to emphasize that these variables are needed for the analysis of the torsion of a beam (This also implies that other field variables should be used for the HoBT analysis of beam structures under bending and extensional loads.).

3.1.1 Three-Dimensional Displacements, Strains, and Stresses at a General Point To derive the governing one-dimensional higher-order beam equations, the threedimensional displacements, strains, and stresses at a general point of a thin-walled box beam should be expressed in terms of 1D-variables, θ (z), W (z), and χ (z). Referring to Fig. 3.1, one can assume εnn = γns = γzn = 0 by following the kinematics of Kirchhoff’s thin plate theory because the walls forming the cross-section of a rectangular box beam are assumed to be thin. Under these assumptions, the threedimensional displacements u˜ z (z, n, s), u˜ n (z, n, s), and u˜ s (z, n, s) at a general point P (z, n, s), where n and s denote the normal and tangential coordinates, can be described in terms of the displacement of the midline given in Eq. (2.25) as

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

47

un′ corner 1 corner 2 segment A

−nun′

corner 4

−nun

un

corner 3 Fig. 3.1 Deformed shape of segment A by wall bending assuming εnn = γns = γzn = 0

∂un (z, s) , ∂z

(3.1a)

u˜ n (z, n, s) = ψnθ (s)θ (z) + ψnχ (s)χ (z),

(3.1b)

u˜ z (z, n, s) = ψzW (s)W (z) − n

u˜ s (z, n, s) = ψsθ (s)θ (z) + ψsχ (s)χ (z) − n

∂un (z, s) , ∂s

(3.1c)

where un (z, s) is the n-directional displacement at a point on the wall midline. The last terms on the right sides of Eqs. (3.1a, c) correspondingly represent the contributions1 of the bending rotations, ∂un /∂z and ∂un /∂s, of the midline; only the z- and sdirectional displacements are affected by the bending of the walls of a beam crosssection. To facilitate writing these expressions, we also use the following shorthanded notations: ( ') =

∂( ) ∂( ) , ( ·) = ∂z ∂s

(3.2a, b)

For a beam under torsion, the contribution of wall bending (−nun' ) to the u˜ z (z, n, s) field can be assumed to be negligible.2 Consequently, Eqs. (3.1a, b, c) can be expressed as

1

u˜ z (z, n, s) = ψzW (s)W (z),

(3.3a)

u˜ n (z, n, s) = ψnθ (s)θ (z) + ψnχ (s)χ (z),

(3.3b)

The z- and s-directional displacements due to the bending rotations of a cross-section wall will be denoted as uz and us , respectively, in later chapters, i.e., uz ≡ −n∂un /∂z and us ≡ −n∂un /∂s. 2 This is valid for a beam problem under a torsional load. The term (−nu' ) will be considered for n a beam under a general load in Sect. 3.2.

48

3 One-Dimensional Governing Equations and Finite Element Formulation

u˜ s (z, n, s) = ψsθ (s)θ (z) + ψsχ (s)χ (z) − nψ˙ nχ (s)χ (z).

(3.3c)

To obtain Eq. (3.3c), un (z, s) in Eq. (2.25b) is used. Also, ψ˙ sθ = 0 is utilized (see Eq. (2.26) for the explicit expression of ψsθ ). From the three-dimensional displacements in Eqs. (3.3a, b, c), the strain can be expressed as εzz (z, n, s) =

∂ u˜ z = ψzW (s)W ' (z), ∂z

(3.4a)

εss (z, n, s) =

∂ u˜ s = −nψ¨ nχ (s)χ (z), ∂s

(3.4b)

∂ u˜ s ∂ u˜ z + ∂s ∂z = ψ˙ zW (s)W (z) + ψsθ (s)θ ' (z) + ψsχ (s)χ ' (z) − nψ˙ nχ (s)χ ' (z),

γzs (z, n, s) =

εnn (z, n, s) = γns (z, n, s) = γzn (z, n, s) = 0.

(3.4c) (3.4d)

χ χ Note that ψ˙ s (s) = 0 was used to derive Eq. (3.4b) because ψs is edgewise constant, as given by Eq. (2.49). Equation (3.4d) represents the kinematic assumption used for thin plates forming the beam walls. Using the conditions of plane stress, non-zero stress components can be expressed in terms of the generalized displacements as

σzz (z, n, s) = E1 (εzz + νεss ) = E1 ψzW (s)W ' (z) − νE1 nψ¨ nχ (s)χ (z),

(3.5a)

σss (z, n, s) = E1 (εss + νεzz ) = −E1 nψ¨ nχ (s)χ (z) + νE1 ψzW (s)W ' (z),

(3.5b)

τzs (z, n, s) = Gγzs = G ψ˙ zW (s)W (z) + Gψsθ (s)θ ' (z) + Gψsχ (s)χ ' (z) − Gnψ˙ nχ (s)χ ' (z).

(3.5c)

/ In Eqs. (3.5a, b, c), G is the shear modulus of the material and E1 = E (1 − ν 2 ), where E is Young’s modulus and ν is Poisson’s ratio.

3.1.2 Governing Equations To derive the governing equilibrium equation, the variational principle is used. To this end, first we express the total potential energy, ∏, of a thin-walled beam:

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …



=U +Ω=

1 2

49

 (σss εss + σzz εzz + τzs γzs )dV V



(fs u˜ s + fn u˜ n + fz u˜ z )dV ,

−

(3.6)

V

where U is the internal strain energy and Ω is the external work done by body forces, fs , fn , and fz , acting in the s, n and z-directions, respectively. The volume of the beam is denoted by V. In Eq. (3.6), boundary forces applied to the end sections of a beam are omitted for simplicity. In the following discussions, we will assume fn = 0 as we are primarily concerned with the case of torsional loads. Before further manipulating Eq. (3.6), it should be noted that the volume integral over a beam in Eq. (3.6) can be expressed as the product of an area integral on the cross-section A and a line integral in the z-direction. Keeping this in mind, we substitute Eqs. (3.3a, b, c–3.5a, b, c) into Eq. (3.6) and integrate the resulting expression over the beam cross-section A to derive the following one-dimensional form of ∏: ∏(z; W , θ, χ ) =

1 2



[E1 aW '2 + E1 cχ 2 + G(b1 W 2 + b∗1 θ '2 + b5 χ '2 )

z

+ 2G(b2 W θ ' + b3 W χ ' + b4 θ ' χ ' )]dz  − (p1 W + q1 θ + q2 χ )dz,

(3.7)

z

where p1 , q1 , and q2 are one-dimensional body forces defined as  p1 (z) =

fz (z, s)ψzW (s)dA, A



fs (z, s)ψsθ (s)dA,

q1 (z) = A



fs (z, s)ψsχ (s)dA.

q2 (z) =

(3.8)

A

Other symbols appearing in Eq. (3.7), such as a and b1 , are sectional constants that depend on the beam cross-sectional shape. They are defined as  a=

(ψzW )2 dA,

(3.9a)

A

b∗1 =

 A

(ψsθ )2 dA, b1 =

 ( A

dψzW ds

)2 dA,

(3.9b)

50

3 One-Dimensional Governing Equations and Finite Element Formulation

 b2 = A

ψsθ

dψzW dA, b3 = ds



ψsχ

dψzW dA, ds

(3.9c)

A



ψsθ ψsχ dA, b5 =

b4 = A



(ψsχ )2 dA,

(3.9d)

A

(

 c=

n2

χ )2

d2 ψn ds2

dA.

(3.9e)

A

Not surprisingly, the sectional constants in Eqs. (3.9a, b, c, d, e) represent the stiffness of the cross-section for various modes; a, b∗1 , and b5 represent the torsional warping, torsional rotation, and torsional distortion stiffness, respectively. Because the constant c in Eq. (3.9e) is related to mode χ , it also represents the torsional distortion stiffness but is associated with the bending energy of the cross-sectional wall. The sectional constants b2 and b3 involving the product of two dissimilar crosssectional modes in Eqs. (3.9a, b, c, d, e) denote the coupling stiffness of the two modes involved. For a rectangular thin-walled cross-section with width b, height h, and thickness t, the sectional constants defined in Eqs. (3.9a, b, c, d, e) can be explicitly found via b2 h2 t(b + h) , 24

(3.10a)

bht(b + h) bht(b + h) , b1 = , 2 2

(3.10b)

2b2 h2 t bht(b − h) , b3 = , 2 b+h

(3.10c)

a= b∗1 =

b2 =

b4 = 0, b5 = c=

2b2 h2 t , b+h

8t 3 . b+h

(3.10.d)

(3.10.e)

To derive the equilibrium equations, we employ the variational principle [see, e.g., Cassel (2013)]. It states that if the first variation δ∏ of the functional ∏ taken over field variables θ , W , and χ is set to vanish, the equilibrium equations3 can be obtained (more generally, the Euler–Lagrange equations). Introducing the variable vector U = {θ, W , χ }T , δ∏ can be written as 3

A functional is a definite integral involving an unknown function and its derivatives. The total potential energy is the integral involving unknown field variables U = {θ, W , χ}T , which are also the functions of the z coordinate. Therefore, the total potential energy is a functional.

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

) ∂∏ T ' δ∏(θ, W , χ ) = δU + δU ∂U' ∂∏ ∂∏ ∂∏ ∂∏ ∂∏ ∂∏ ' δθ + δW + δχ + ' δθ ' + = δW ' + δχ = 0, ∂θ ∂W ∂χ ∂θ ∂W ' ∂χ ' (

∂∏ ∂U

)T

51

(

(3.11)

where δθ (or δW , δχ ), the variation of θ (or W , χ ), denotes a kinematically admissible displacement.4 Substituting Eq. (3.7) into Eq. (3.11) yields 

[E1 aW ' δW ' + E1 cχ δχ + G(b1 W δW + b∗1 θ ' δθ ' + b5 χ ' δχ ' )

δ∏ = z

+ G(b2 δW θ ' + b2 W δθ ' + b3 δW χ ' + b3 W δχ ' + b4 δθ ' χ ' + b4 θ ' δχ ' )]dz  − (p1 δW + q1 δθ + q2 δχ )dz = 0, (3.12) z

Integrating Eq. (3.12) by parts yields  δ∏ =

[(−Gb∗1 θ '' − Gb2 W ' − Gb4 χ '' − q1 )δθ

z

+ (−E1 aW '' + Gb1 W + Gb2 θ ' + Gb3 χ ' − p1 )δW + (−Gb3 W ' − Gb4 θ '' − Gb5 χ '' + E1 cχ − q2 )δχ ]dz ( ) |L + G b∗1 θ ' + b2 W + b4 χ ' δθ |z=0 ( ( ) |L ) |L + E1 aW ' δW |z=0 + G b5 χ ' + b3 W + b4 θ ' δχ |z=0 = 0.

(3.13)

where the one-dimensional domain of a beam is defined as 0 ≤ z ≤ L. The boundary terms appearing in the third line in Eq. (3.13) represent the virtual work done by external generalized forces at the boundaries of a beam. Because the virtual work by external forces should be the generalized boundary forces and the corresponding virtual displacements (δθ , δW , δχ ), one can naturally identify the following generalized forces: Mz = G(b∗1 θ ' + b2 W + b4 χ ' ) (work conjugate of θ ),

(3.14a)

B = E1 aW ' (work conjugate of W ),

(3.14b)

Q = G(b5 χ ' + b3 W + b4 θ ' )(work conjugate of χ ).

(3.14c)

The symbol δθ is often denoted as δθ = εη, where ε is a small number close to zero and η is an arbitrary function that vanishes at points where kinematic boundary conditions (such as a fixed condition) are prescribed.

4

52

3 One-Dimensional Governing Equations and Finite Element Formulation

The generalized forces Mz , B, and Q in Eqs. (3.14a, b, c) denote the torsional moment, bimoment, and transverse bimoment, respectively. Using Mz , B, and Q, the boundary terms in Eq. (3.13) can be rewritten as ) |L ( ) |L ( Boundary terms= G b∗1 θ ' + b2 W + b4 χ ' δθ |0 + E1 aW ' δW |0 ) |L ( + G b5 χ ' + b3 W + b4 θ ' δχ |0 = Mz δθ |L0 + BδW |L0 + Qδχ |L0

(3.15)

These boundary terms vanish either because δθ , δW , and δχ are zero at a fixed boundary or because Mz , B, and Q are zero at a traction-free boundary.5 Because the boundary terms are equal to zero, Eq. (3.13) yields −Gb∗1 θ '' − Gb2 W ' − Gb4 χ '' − q1 = 0,

(3.16a)

−E1 aW '' + Gb1 W + Gb2 θ ' + Gb3 χ ' − p1 = 0,

(3.16b)

−Gb3 W ' − Gb4 θ '' − Gb5 χ '' + E1 cχ − q2 = 0.

(3.16c)

Equations (3.16a–c) are the governing equations based on the HoBT for a thinwalled beam under torsion. It will be also instructive to derive the relationships (3.14a, b, c) starting from the three-dimensional stresses and the corresponding displacements. To this end, we consider the virtual work δWA done by the surface traction depicted in Fig. 3.2 through virtual displacements δ u˜ z (z, n, s) and δ u˜ s (z, n, s)6 :  δWA =

 σzz (z, n, s)δ u˜ z (z, n, s)dA +

A

τzs (z, n, s)δ u˜ s (z, n, s)dA.

(3.17)

A

Note that σss does not appear in Eq. (3.17) because it is not a component of the surface traction vector on cross-section A. Substituting Eqs. (3.3a, b, c) into Eq. (3.17) yields the following equation:  δWA =

 σzz ψzW dA · δW +

A

A

 BδW + Mz δθz + Qδχ .

5

τzs ψsθz dA · δθz +



τzs ψsχ dA · δχ

A

(3.18)

When non-zero generalized forces are imposed at a free end of a beam, the boundary terms in (3.15) do not vanish. In this case, they can be treated as prescribed generalized forces when the governing equations in Eqs. (3.16a, b, c)) are solved. 6 Because τ = 0, there is no need to consider δ u ˜ n. zn

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

53

τ zs σ zz

Cross section A Fig. 3.2 Surface tractions (σzz , τzs , τzn ) acting on cross-section A (τzn ≈ 0)

From the expression in Eq. (3.18), the generalized forces Mz , B, and Q can be identified as  Mz = τzs ψsθz dA, (3.19a) A

 B=

σzz ψzW dA,

(3.19b)

τzs ψsχ dA.

(3.19c)

A



Q= A

Substituting the stress expressions given in Eqs. (3.5a, b, c) into Eqs. (3.19a, b, c) yields the same expressions given in Eqs. (3.14a, b, c).

3.1.3 Finite Element Formulation In this subsection, we will present a finite element formulation based on the higherorder beam theory derived above for the torsion of a thin-walled beam. Specifically, the application of finite elements using one-dimensional two-node finite elements will be presented. An extension to finite element formulations with more nodes can easily be carried out once the two-node finite element implementation procedure is understood.

54

3 One-Dimensional Governing Equations and Finite Element Formulation

For the finite element operation, the field variables U(z) = {θz (z), W (z), χ (z)}T should be interpolated when the governing Eqs. (3.16a–c) are examined. Considering the eth thin-walled beam finite element defined over z1 ≤ z ≤ z2 (or −1 ≤ η ≤ 1 in terms of the natural coordinate η), we introduce the nodal variables die = {θi , Wi , χi }T (i = 1, 2) at node i. The relationship between z and η is expressed as z=

(z2 − z1 ) (z2 + z1 ) η+ . 2 2

(3.20)

If linear interpolation functions Ni (depicted in Fig. 3.3) are used, U can be approximated as ⎧ ⎫ ⎡ ⎤ N1 0 0 N2 0 0 ⎨ θz ⎬ U = W = ⎣ 0 N1 0 0 N2 0 ⎦de = Nde , ⎩ ⎭ χ 0 0 N1 0 0 N2

(3.21a)

with N1 =

1+η 1−η , N2 = , 2 2

(3.21b)

de = {θ1 , W1 , χ1 , θ2 , W2 , χ2 }T .

(3.21c)

In Eqs. (3.21a, b, c), N1 and N2 denote the linear shape functions and de is the nodal generalized displacement vector of element e. For the subsequent analysis, it will be convenient to write the interpolation of the field variables as ] [ θ = N1 0 0 N2 0 0 de ≡ Nθ de ,

(3.22a)

] [ W = 0 N1 0 0 N2 0 de ≡ NW de ,

(3.22b)

Fig. 3.3 Two-node thin-walled beam element

N1 =

1 −η 2

N2 =

node 2

node 1

η = −1 z = z1

d1e = {θ1 , W1 , χ1}T

1 +η 2

η =0

η =1

z = z2 d e2 = {θ 2 , W2 , χ 2 }T

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

] [ χ = 0 0 N1 0 0 N2 de ≡ Nχ de .

55

(3.22c)

By employing the same shape functions (Nθ , NW ,Nχ ) used to interpolate the true displacements, one can approximate the virtual displacements as δθ = Nθ δde = (δde )T NθT ,

(3.23a)

T δW = NW δde = (δde )T NW ,

(3.23b)

δχ = Nχ δde = (δde )T NχT .

(3.23c)

To find the field variables, U = {θz , W , χ }T , or their nodal values de , Eqs. (3.20), (3.22a, b, c), and (3.23a, b, c) are inserted into Eq. (3.12) to obtain the following result: ⎛  ( e )T T' ' δ∏ = δd ⎝ [E1 aNW NW + E1 cNχT Nχ z '

'

T + G(b1 NW NW + b∗1 NθT Nθ' + b5 NχT Nχ' ) '

T T Nθ' + b2 NθT NW + b3 NW Nχ' + G(b2 NW

) ' ' +b3 NχT ' NW + b4 NθT Nχ' + b4 NχT Nθ' )]dz · de  e T T − (δd ) (p1 NW + q1 NθT + q2 NχT )dz = 0.

(3.24)

z

Note that the work by internal forces acting at the nodes of the element is not considered in Eq. (3.24) because the internal forces should be cancelled out by the internal forces at the node of the corresponding adjacent elements when the entire beam consisting of multiple elements is considered, i.e., when the resulting finite element equation is assembled for the whole beam. In Eq. (3.24), Nθ' is calculated as Nθ' (z) =

2[ 1 dNθ (η) dη = −2 0 0 dη dz l

1 2

] 00 ,

(3.25)

' where z = N1 z1 + N2 z2 and l = z2 − z1 are used. The terms NW and Nχ' are similarly calculated. Because Eq. (3.24) should be valid for any virtual nodal displacement δde , the following equilibrium equation in the element level can be obtained:

k e de = f e , where the element stiffness matrix ke is defined as

(3.26)

56

3 One-Dimensional Governing Equations and Finite Element Formulation

ke =

 [ ) ( T ' T' ' T E1 aNW NW + E1 cNχT Nχ + G b1 NW NW + b∗1 NθT Nθ' + b5 NχT Nχ' z

] ( T T T +G b2 NW Nθ' + b2 NθT NW + b3 NW Nχ' )] T T +b3 NχT NW + b4 NθT Nχ' + b4 NχT Nθ' dz

(3.27)

and the element load vector7 f e is defined as  ( ) e T p1 NW f = + q1 NθT + q2 NχT dz.

(3.29)

z

For actual calculation, the integration in Eq. (3.27) should be rewritten in terms of the natural coordinate η: z2

1 ( )dz = −1

z1

dz ( ) dη = dη

1 −1

l ( ) dη. 2

If Eq. (3.27) is integrated over z (or η), ke is explicitly obtained as ⎡ ⎢ ⎢ ⎢ ⎢ e k =⎢ ⎢ ⎢ ⎣

G

b∗1 l

b

−G 22 b l E1 al + 31

sym

b

b∗

b

b

G l4 −G l1 −G 22 −G l4 b3 b2 b l b −G 2 G 2 −E1 al + G 61 G 23 b b G bl5 + E1 cl3 −G l4 −G 23 −G bl5 + E1 cl6 b∗1 b2 b Gl G2 G l4 b l b G 23 E1 al + G 31 b5 G l + E1 cl3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. (3.30) ⎥ ⎥ ⎦

To derive the finite element equation for a dynamic analysis, one may use D’Alembert’s principle; the inertial forces are additionally considered as the body forces by replacing fz , fn , and fs by fz − ρ∂ 2 u˜ z /∂t 2 ,fn − ρ∂ 2 u˜ n /∂t 2 , and fs − ρ∂ 2 u˜ s /∂t 2 (ρ: density of a material) in the expression of the total potential energy in Eq. (3.6): 

=

1 2



 (σss εss + σzz εzz + τzs γzs )dV − V

 +

(

)

ρ u¨˜ z u˜ z + u¨˜ n u˜ n + u¨˜ s u˜ s dV ,

(fs u˜ s + fz u˜ z )dV V

(3.31)

V

7

Generally, the contribution from the internal forces should also be included in f e , but this is not expressed explicitly in Eq. (3.29) because it will be cancelled out in the final assembled full finite element equation.

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

57

where u¨˜ p = ∂ 2 u˜ p /∂t 2 (p = z, n, s) and the work by the n-directional inertial force is also considered when the potential energy8 is calculated. Following a process similar to that used to derive the finite element equation for static problems, we set the first variation of to be equal to zero: δ∏ = (δde )T ke de − (δde )T f e  T + (δde )T ρ[aNW NW + b∗1 NθT Nθ + b4 NθT Nχ + +

z T b4 Nχ Nθ d3 NθT Nχ

+ b5 NχT Nχ + d1 NθT Nθ + d2 NχT Nχ , + d3 NχT Nθ ]dz · d¨ e = 0,

(3.32a)

with  d1 =

(ψnθ )2 dA, d2 =

A



(ψnχ )2 dA, d1 =

A



ψnθ ψnχ dA,

(3.32b)

A

where ke and f e are defined in Eq. (3.27). Note that terms not listed in the integrand in Eqs. (3.32a, b) are neglected. Because Eq. (3.32a) should be valid for any virtual nodal displacement δde , the following finite element equation of motion in the element level can be derived: me d¨ e + ke de = f e ,

(3.33)

where the element mass matrix me is given by  me =

T ρ[aNW NW + b∗1 NθT Nθ + b4 NθT Nχ + b4 NχT Nθ

z

+ b5 NχT Nχ + d1 NθT Nθ + d2 NχT Nχ + d3 NθT Nχ + d3 NχT Nθ ]dz.

(3.34)

One can express me explicitly as ⎡ (b∗1 +d1 )l ⎢ ⎢ ⎢ ⎢ e m = ρ⎢ ⎢ ⎢ ⎣

8

3

0 al 3

sym

(b4 +d3 )l (b∗1 +d1 )l 3 6

0

0

(b5 +d2 )l (b4 +d3 )l 3 6 (b∗1 +d1 )l 3

0 al 6

0 0 al 3

(b4 +d3 )l 6

0

(b5 +d2 )l 6 (b4 +d3 )l 3

0

(b5 +d2 )l 3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(3.35)

Hamilton’s principle should be used as a variational method to deal directly with elastodynamics. In a strict sense, ∏ in Eq. (3.31) does not represent potential energy because it includes the kinetic energy stemming from dynamic motion.

58

3 One-Dimensional Governing Equations and Finite Element Formulation

Assembling Eq. (3.33) for the whole beam, the dynamic equation in the finite element context is given by Md¨ + Kd = F,

(3.36)

with M=

∑

me , and K =

e

∑

ke .

(3.37)

e

In Eq. (3.36), M and K are the global mass and stiffness matrices, respectively, and ∑ is an operator to assemble the element matrices. Although damping is not considered in the present formulation, it may be also included in Eq. (3.36) using conventional damping models, such as the Rayleigh damping model (Bathe 1996). For a free vibration analysis, the force term is set to be zero and the nodal displacement is assumed to oscillate harmonically, d = ϕ cos ωt,

(3.38)

where ϕ is the amplitude of the nodal displacement vector d and ω is angular frequency. Because d¨ = −ω2 ϕ cos ωt = −ω2 d and F = 0 for the free vibration analysis, Eq. (3.36) is converted to an eigenvalue problem: Kϕ = ω2 Mϕ.

(3.39)

The eigenvalues and eigenvectors of Eq. (3.39) represent the natural frequencies and corresponding mode shapes, respectively.

3.1.4 Examples of Finite Element Solutions In this section, we will present some numerical examples calculated by the finite element equation derived in Sect. 3.1.3. We begin with the cantilevered thin-walled box beam shown in Fig. 3.4. Figure 3.4a shows a thin-walled box beam (L = 500 mm, b = 25 mm, h = 50 mm, t = 1 mm, E = 200 GPa, and ν = 0.3) with one end fixed, uz = 0,9 and the other end subjected to prescribed warping displacement uz = t at corners 2 and 4. Using the displacement approximation in Eq. (3.3a) and the warping shape function ψzW in Eq. (2.40), uz at corner 2 of the end cross-section of the beam is expressed as ) ) ( ( bh h h = ψzW s1 = W (z = L) = uz z = L, s1 = W (z = L) ≡ t. 2 2 4

(3.40)

Note that uα (z, s) = u˜ α (z, s, n = 0) (α = z, n, s; or x, y, z). This relationship implies that uα is the displacement calculated along the centerline of the walls forming the beam cross-section.

9

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

59

corner 3 n3

corner 2

s2

n2

s3 s1

corner 4

s4

corner 1

n1

n4

(a)

(b)

Fig. 3.4 Cantilevered thin-walled beam with a rectangular cross-section under a prescribed warping displacement and b a torsional couple

/ From Eq. (3.40), W (z = L) = 4t bh. Therefore, the boundary conditions of the beam in Fig. 3.4a can be written as θz = 0, W = 0, χ = 0 at z = 0, Mz = 0, W =

(3.41a)

4t , Q = 0 at z = L. bh

(3.41b)

Note that the prescribed torsional moment Mz and transverse bimoment Q are zero at z = L because only the displacement (uz |z=L ) representing warping is prescribed. Obviously, this problem cannot be solved with the Timoshenko or Euler beam elements because they cannot represent the prescribed warping displacement. Figure 3.5 plots the displacements uz and ux along the edge of the beam corresponding to corner 2. They represent the warping and distortional displacements, respectively. For the calculation, 15 HoBT elements were used. The converged results by the HoBT elements are in good agreement with those obtained using the shell elements of NASTRAN (QUAD4 elements). 1

5

HoBT

Present Beam Element

0.8

HoBT

4

Plate (ABAQUS)

Shell (ABAQUS)

Shell (ABAQUS)

3

0.6

2

0.4 1

0.2

0

0

-1 -2

-0.2 0

0.1

0.2

0.3

z coordinate (mm)

(a)

0.4

0.5

0

0.1

0.2

0.3

0.4

z coordinate (mm)

(b) /

/ Fig. 3.5 Displacements on corner 2 of the beam shown in Fig. 3.4a: a uz t, and b ux t

0.5

60

3 One-Dimensional Governing Equations and Finite Element Formulation 11

HoBTHoBT (linear) (linear

10

interpolation)

HoBT (Hermite)

HoBT (Hermite cubic interpolation)

9

Plate Analysis by ABAQUS

Shell elements (ABAQUS)

8

7

6

5

4 0

10

20

30

40

50

60

70

80

90

100

Number of nodes

/ Fig. 3.6 Convergence of ux t at corner 2 of the beam end (z = L) in Fig. 3.4a

To examine the convergence behavior when the analysis is based on the linearfunction-interpolated HoBT elements, the corner displacement ux at the beam end (z = L) is investigated with a varying degree of finite element refinement. The result in Fig. 3.6 shows that although the solution converges, the convergence rate is not rapid. The slow convergence appears to be due to the end effect causing the displacements to vary exponentially near the beam end (or near the joint in the case of a jointed beam). Recall that the wall-bending term (−nun' ) was ignored when expressing u˜ z , as in Eq. (3.3a). This simplification appears also to be responsible for the slow convergence. The use of Hermite cubic interpolations guaranteeing C 1 continuity10 is definitely preferred. We will discuss the convergence issue related to the finite element implementation of the HoBT for thin-walled beams under general loading in more detail in the next section. In Fig. 3.4b, a box beam is subjected to a torsional couple with F = 100 N at the free end (z = L). The same geometry and materials used in the previous example are used. Equation (3.19) is used to calculate the applied generalized forces at the free end where the torsional couple is prescribed. In doing this, first we express the couple in terms of the s-directional surface forces acting on edges 1 and 3 of the cross-section: ( ( ) ) h h , and fs (s3 ) = Fδ s3 − , (3.42) fs (s1 ) = Fδ s1 + 2 2 where δ is the Dirac delta function. The prescribed shear stress by the surface force in Eq. (3.42) can then be written as 10

If the displacement and its derivative are continuous between two adjacent elements, the corresponding displacement field is said to be C 1 continuous. If only the displacement is continuous, it is said to be C 0 continuous.

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

τzs (s1 ) =

( ) ( ) F h F h δ s1 + and τzs (s3 ) = δ s3 − . ht 2 ht 2

61

(3.43)

Substituting Eq. (3.43) into Eqs. (3.19a, b, c) allows us explicitly to write the applied generalized forces as: h/ 2 Mz =

θz τzs,1 ψs,1 tds1

h/ 2 +

−h/ 2

θz τzs,3 ψs,3 tds3

−h/ 2

( )] ( )] h/ 2 [ F F h h θz θz δ s1 − ψs,1 tds1 + δ s3 + ψs,3 = tds3 ht 2 ht 2 −h/ 2 −h/ 2 ( ( ) ) h h + Fψsθz s3 = − = bF, (3.44a) = Fψsθz s1 = 2 2 h/ 2

[

B = 0, h/ 2 Q=

χ τzs,1 ψs,1 tds1

−h/ 2

h/ 2 +

(3.44b)

χ

τzs,3 ψs,3 tds3

−h/ 2

] ] h/ 2 [ F F h h χ χ δ(s1 − ) ψs,1 tds1 + δ(s3 + ) ψs,3 tds3 = ht 2 ht 2 −h/ 2 −h/ 2 ( ( ) ) h h 2bhF = Fψsχ s1 = , (3.44c) + Fψsχ s3 = − = 2 2 b+h h/ 2

[

/ χ χ θ θ where / ψs (s1 ) = ψs (s1 ) = b 2 [see Eqs. (2.26a, c)] and ψs (s1 ) = ψs (s1 ) = bh (b + h) [see Eq. (2.49a)]. To solve the problem depicted in Fig. 3.4b using the HoBT employing three field variables θ , W , and χ , the following boundary conditions are used: θz = 0, W = 0, χ = 0 at z = 0, Mz = bF, B = 0, Q =

2bhF at z = L. b+h

(3.45a) (3.45b)

Figure 3.7 shows the displacements along corner 4 of the beam shown in Fig. 3.4b. Good agreement is observed between the results by the linear HoBT-based finite elements and those by the shell elements. Rapid variations in both the distortional (ux ) and warping (uz ) displacements are observed near the end of the beam, indicating significant end effects. Certainly, the use of more HoBT finite elements improves the

62

3 One-Dimensional Governing Equations and Finite Element Formulation 0.01

10

1

-3

0

0

-1

HoBT (Ne=5)

-0.01

-0.02

HoBT (Ne=10)

-2

HoBT (Ne=25)

-3

HoBT (Ne=5) HoBT (Ne=10) HoBT (Ne=25)

HoBT (Ne=50)

-0.03

HoBT (Ne=50) -4

Shell elements (ABAQUS)

Shell elements (ABAQUS)

-0.04

-5 0

0.1

0.2

0.3

0.4

0.5

0

0.1

z coordinate (mm)

0.2

0.3

0.4

0.5

z coordinate (mm)

(a)

(b)

/ / Fig. 3.7 Displacements on corner 4 of the beam under a torsional couple: a ux t, and b uz t

accuracy of the solution. As was discussed in the previous example, the convergence of the HoBT elements in Fig. 3.7 can be improved by employing higher-order shape functions.

3.1.5 Analytic Solutions* To find analytic solutions to Eqs. (3.16a, b, c), we introduce the function f (z) (Vlasov 1961), where W (z) =

) df (z) ( = f ' (z) . dz

(3.46)

Using Eqs. (3.16a, b, c), we can express χ and θ in terms of f (z) as G b∗1 (b1 b5 − b23 ) − b22 b5 (2) ab5 (4) f (z) − f (z), (3.47a) cb3 E1 cb3 b∗1 ] [ E1 a G b∗1 (b1 b5 − b23 ) − b22 b5 (2) b1 −ab5 (4) f (z) − f (z). f (z) + − θ (z) = ∗ cb2 G b2 E1 cb2 b1 b2 (3.47b) χ (z) =

In Eqs. (3.47a, b), f (n) (z) denotes the nth-order derivative of f (z) with respect to z. Substituting Eq. (3.46) into Eqs. (3.16b, c) yields the following sixth-order differential equation: f (6) (z) − 2r 2 f (4) (z) + s4 f (2) (z) = 0, where

(3.48)

3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes …

63

[ ] c E1 1 b∗1 (b1 b5 − b23 ) − b22 b5 G , + r = 2 ab5 b∗1 E1 b5 G 2

s4 =

(3.49b)

cb1 cb22 − . ab5 ab5 b∗1

(3.49c)

The general solution to Eq. (3.48) can be written as f (z) = A1 ϕ1 (z) + A2 ϕ2 (z) + A3 ϕ3 (z) + A4 ϕ4 (z) + A5 z + A6 ,

(3.50)

with ϕ1 (z) = cosh αz sin βz, ϕ2 (z) = cosh αz cos βz,

(3.51)

ϕ3 (z) = sinh αz sin βz, ϕ4 (z) = sinh αz cos βz, where α and β are defined as / α=

s2 + r 2 and β = 2

/

s2 − r 2 , 2

and Ai (i = 1, 2, ..., 6) are unknown coefficients to be determined based on the boundary conditions. In Fig. 3.8, the analytic solution using the HoBT is compared with the converged solution determined by means of shell finite elements for the problem of a beam under unit warping in Fig. 3.4. The good agreement found between the HoBT and shell results confirms the validity of the proposed HoBT. 1

HoBT analytic solution Shell elements (ABAQUS)

Analytic Solution

0.8

Plate (ABAQUS)

0.6 0.4 0.2 0 -0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

z coordinate (mm)

Fig. 3.8 z-directional displacement on corner 2 of the beam under unit warping (θ = 0, χ = 0, W = 1 at x = L) in Fig. 3.4 calculated by the analytic solution of the HoBT and shell elements

64

3 One-Dimensional Governing Equations and Finite Element Formulation

3.2 Finite Element Implementation Using an Arbitrary Number of Section Mode Shapes for General Loading The previous section demonstrated a simplest application of a higher-order beam theory; we presented a theory (governing equations and others) and its finite element implementation for a box beam under torsion using only three section modes. Specifically, we use one rigid-body torsional rotation mode θ , two higher-order sectiondeformable modes, and the warping mode W and distortion mode χ . In this section, we will present the finite implementation procedure for more general cases dealing with non-rectangular cross-sections as well as general loads requiring possibly as many deformable-section modes as desired. For the implementation, the sectional shape functions of desired higher-order modes should be available, but they will be derived in Chaps. 4–7 for rectangular cross-sections and in Chap. 9 for cross-sections with general shapes. Generally, it can be said that an analysis based on a higher-order beam theory depends on the derivation of higher-order sectional shape functions. While the sectional shape functions of higher-order modes will be presented in the next few chapters, it is assumed in the meantime that the desired numbers of higherorder sectional functions are available. Under this assumption, we will explain the finite element implementation procedure for general cases.

3.2.1 Three-Dimensional Displacements, Strains, and Stresses at a General Point for General Cases To consider the displacement, strain and stress fields in arbitrary-sectioned beams under general loading, we begin with a description of the three-dimensional displacement (uz (z, s), un (z, s), and us (z, s)) of a point on the centerline of the walls forming the cross-section of a thin-walled beam. If ND modes that include six non-deformable section modes and higher-order deformable section modes are used, the displacement components can take the following form: uz (z, s) =

ND ∑

ψzξi (s)ξi (z),

(3.52a)

ψnξi (s)ξi (z),

(3.52b)

ψsξi (s)ξi (z),

(3.52c)

i=1

un (z, s) =

ND ∑ i=1

us (z, s) =

ND ∑ i=1

where ξi denotes the ith mode or the corresponding field variable, which is a function ξ of z, and ψp i (s) is the sectional shape function representing the displacement shape

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

65

of up (p = z, n, s) associated with mode ξ i (i = 1, 2, ···, N D ). Among ND modes, six rigid-body (non-deformable sectional) modes are associated with indices i = 1 to 6 and higher-order deformable section modes are associated with i > 6. To distinguish the rigid-body modes from the higher-order modes, the following dedicated symbols are used: Ux = ξ1 , Uy = ξ2 , Uz = ξ3 , θx = ξ4 , θy = ξ5 , and θz = ξ6 ,

(3.53)

where Uq and θq (q = x, y, z) represent the q-directional translation and rotation, respectively. There are several different types of higher-order modes, such as warping modes denoted by W and distortional modes denoted by χ . If displacements up (z, s) (p = z, n, s) are written in terms of Ux , W , etc., instead of ξi , one can better understand how the displacements are related to specific modes. For example, the axial displacement in Eq. (3.52a) can be written as θ

uz (z, s) = ψzUz (s)Uz (z) + ψzθx (s)θx (z) + ψz y (s)θy (z) + ψzW0 (s)W0 (z) +

NS [ ∑ We Wv ψz k (s)Wke (z) + ψz k (s)Wkv (z) k=1

] Wh Wt +ψz k (s)Wkh (z) + ψz k (s)Wkt (z) ,

(3.54)

where W0 denotes the lowest torsional warping mode in Eq. (2.40), and Wk represents the kth higher-order torsional warping mode. In Eq. (3.54), NS is the number of higher-order mode sets employed for the analysis. Note that for each set k (k ≥ 1), there are four higher-order modes of warping, {Wke , Wkv , Wkh , Wkt }, which are warping modes mainly induced by extension (Uz ), x-directional (or vertical) bending rotation (θx ), y-directional (or horizontal) bending rotation (θy ), and torsion (θz ), respectively. The superscripts e, v, h, and t in {Wke , Wkv , Wkh , Wkt } stand for the deformation types, i.e., extension, vertical bending, horizontal bending, and torsion, respectively. The higher-order mode Wkα (α = e, v, h, t) can be recursively derived using lower-order modes of the same deformation type. The detailed derivations will be presented for each warping mode in Chaps. 4–7 for thin-walled beams of rectangular crosssections. It is emphasized that the two deformable section modes, torsional warping W0 and torsional distortion χ0 (as explained in Chap. 2) are classified as the modes of the lowest set (or zeroth set); they will be treated as fundamental modes along with the six rigid-body section modes. Figure 3.9 presents the axial displacement on the midline of a section using the modes belonging to the zeroth and first set for a beam subjected to a torsional load. Because the effects of the six non-deformable section modes on the structural response of a beam are generally more significant than those of the higher-order

66

3 One-Dimensional Governing Equations and Finite Element Formulation Sectional shape function

3D displacement on the midline

Generalized 1D displacement (sectional mode)

ψ U ( s)

X

U z ( z)

ψ θ ( s)

X

θ x ( z)

z

x

Σ

Zeroth set θy

ψ (s)

ψ W ( s) 0

X

θ y ( z)

X

W0 ( z )

uz ( z, s)

Σ e 1

ψ W ( s)

v 1

ψ W ( s)

X

W1e ( z )

X

W1v ( z )

=

y x z

Σ

First set h 1

ψ W (s)

t 1

ψ W ( s)

X

W1h ( z )

X

W1t ( z )

Fig. 3.9 Representation of the axial (z-directional) displacement uz (z, s) on the midline of the cross-sectional wall of a box beam subjected to a torsional load as the product sum of the sectional shape functions and generalized 1D displacements (Here, NS = 1 in Eq. (3.54).)

modes, only the six non-deformable section modes defined in Eq. (3.53) are considered in Euler–Bernoulli theory (Truesdell 1960) and Timoshenko beam theory (Timoshenko 1921, 1922). For the rectangular cross-section in Fig. 3.10, we explicitly give the shape functions ψpξi (s) (p = z, n, s) of the six non-deformable section modes as ψnUx (sj ) = sin(αj ), ψsUx (sj ) = cos(αj ), U

U

ψn y (sj ) = − cos(αj ), ψs y (sj ) = sin(αj ),

(3.55a) (3.55b)

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

67

Fig. 3.10 Edgewise local coordinates of a rectangular cross-section (b: width; h: height)

ψzUz (sj ) = 1, ψzθx (sj ) = θ

yj2 − yj1 sj + yj1 − yc , lj

ψz y (sj ) = −

xj2 − xj1 sj − xj1 + xc , lj

(3.55c) (3.55d) (3.55e)

ψnθz (sj ) = sj − lj , ψsθz (sj ) = rj ,

(3.55f)

all other ψp (sj ) = 0(p = z, n, s).

(3.55g)

In Eqs. (3.55a, b, c, d, e, f, g), subscript j denotes the edge (or wall) index and αj is the angle of edge j with respect to the x axis (see Fig. 3.10). The symbols (xj1 , yj1 ) and (xj2 , yj2 ) denote the coordinates of the two end points of edge j where the coordinate sj starts from (xj1 , yj1 ), one end of edge j. In addition, rj is the distance from the rotation center11 to edge j and lj is the distance from the point at (xj1 , yj1 ) (or sj = 0) to the intersection point Nj between edge j and the normal line from the rotation center to edge j (see Fig. 3.12). We represent the shape functions in Eqs. (3.55a, b,

11

In general, a rotation center is different from the shear center of a cross-section. The calculation for the rotation center will be discussed in Sect. 9.3. For a rectangular cross-section, a rotation center coincides with a geometric center.

68

3 One-Dimensional Governing Equations and Finite Element Formulation

c, d, e, f, g) in terms of general geometric parameters, in this case αj , (xj1 , yj1 ), rj , etc., so that we can use them for cross-sections with general shapes in Chap. 9.12 The shape functions expressed as ψ in Eqs. (3.55a, b, c, d, e, f, g) can be easily found using rigid-body kinematics. First, the shape functions ψpUx for x-directional U

translation mode Ux and ψp y (p = n, s) for y-directional translation mode Uy are obtained by calculating the s- and n-directional components of the rigid-body movements along the x- and y-directions, respectively; see Fig. 3.11 (To be able to deal with more general section, we considered a non-rectangular cross-section in Figs. 3.11 and 3.12.). To calculate the shape functions for torsional rotation θ , the displacement due to small rigid-body torsional rotation at a point on edge j is considered (see Fig. 3.12): us = un = −

cθz v = = rj θz , cos βj cos βj

cθz v =− = d θz = (sj − lj )θz . sin βj sin βj

(3.56a) (3.56b)

Noting that the s- and n-directional displacements by torsional rotation are us = ψsθz θz and un = ψnθz θz , respectively, the sectional shape functions in Eq. (3.55f) can θ be obtained. To find the shape functions ψzθx and ψz y in Eqs. (3.55d, e) defined for bending rotation modes θx and θy , we calculate the out-of-plane displacement (uz ) due to the rotation of the cross-section about the x and y axes, respectively. As a θ result, one can derive ψzθx and ψz y , which are linear functions of si . To facilitate the subsequent analysis, it is convenient to write Eqs. (3.52a, b, c) in a compact matrix form as up (z, s) =

ND ∑

ψpξi (s)ξi (z) = ψp (s)ξ(z), (p = z, n, s),

(3.57)

i=1

where row vector ψp and column vector ξ are defined as ξN

ψk (s) = [ψpξ1 (s), ψpξ2 (s), ..., ψp D (s)], (p = z, n, s)

(3.58)

ξ(z) = {ξ1 (z), ξ2 (z), ..., ξND (z)}T .

(3.59)

At this point, we express the three-dimensional displacements u˜ p (z, n, s) (p = z, n, s) at a general point (z, n, s) on a cross-section using the displacements up (z, s) (p = z, n, s) defined on the midline of the cross-sectional walls of a beam, which are given in Eq. (3.57). Assuming that the walls are sufficiently thin, one can use 12

In fact, the shape functions of non-deformable section modes for a generally shaped cross-section should be modified from Eqs. (3.55a, b, c, d, e, f, g) considering orthogonality conditions. See Eq. (9.37) for the corresponding closed forms.

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

69

Ux =1

ψU

x

ψ

Ux s

=ψ

Ux x

ψ nU = ψ xU sin α 2 x

cos α 2

α2

Edge 3

Edge 4

x

ψ Ux = 1 x

Edge 2 Edge 1

Edge 2

ψ

Uy =1

Uy

Uy

Uy

Uy

ψ s = ψ y sin α 2

Uy

ψ y =1

Edge 3

Uy

ψ n = −ψ y cos α 2

α2

Edge 4 Edge 2 Edge 1

Edge 2

Fig. 3.11 Calculation of the section shape functions for x- and y-directional translation modes on edge 2

Edge 3

Edge 4

ψθ

un = − z

v = cθ z

θz S Edge 2 Edge 1

us =

α2

c

θz S

β2

β2 r2 s2 = 0

Edge 2

N2

v sin β 2

d

s2

l2

Fig. 3.12 Calculation of the shape functions for torsional rotation mode θ on edge 2

v cos β 2

70

3 One-Dimensional Governing Equations and Finite Element Formulation

the Kirchhoff kinematics (or the Euler–Bernoulli kinematics) imposing εnn = γns = γzn = 0. With this assumption, one can write u˜ z (n, z, s) = uz (z, s) − nun' (z, s),

(3.60a)

u˜ n (n, z, s) = un (z, s),

(3.60b)

u˜ s (n, z, s) = us (z, s) − n˙un (z, s).

(3.60c)

Note that the term (−nun' ) in Eq. (3.60a), which represents the contribution of wall bending, was dropped in the case of the beam torsion problem considered earlier in Sect. 2.1 [see Eq. (3.3a)]. With the plane-stress assumption, the stress and strain are calculated as εss (z, n, s) = u˙˜ s (z, n, s) = u˙ s (z, s) − n¨un (z, s),

(3.61a)

εzz (z, n, s) = u˜ z' (z, n, s) = uz' (z, s) − nun'' (z, s),

(3.61b)

γzs (z, n, s) = u˜ s' (z, n, s) + u˙˜ z (z, n, s) = us' (z, s) + u˙ z (z, s) − 2n˙un' (z, s),

(3.61c)

and σss (z, n, s) = E1 (εss (z, n, s) + νεzz (z, n, s)),

(3.62a)

σzz (z, n, s) = E1 (νεss (z, n, s) + εzz (z, n, s)),

(3.62b)

τzs (z, n, s) = Gγzs (z, n, s).

(3.62c)

Substituting Eq. (3.57) into Eqs. (3.61a, b, c) and (3.62a, b, c) yields ˙ s − nψ ¨ n )ξ, εss = (ψ '

''

(3.63a)

εzz = ψz ξ − nψ n ξ ,

(3.63b)

˙ z ξ − 2nψ ˙ n ξ' , γzs = ψs ξ' + ψ

(3.63c)

˙ s − nψ ¨ n )ξ + νψz ξ' − nνψn ξ'' ], σss = E1 [(ψ

(3.64a)

˙ s − nψ ¨ n )ξ + ψz ξ' − nψn ξ'' ], σzz = E1 [ν(ψ

(3.64b)

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

˙ z ξ − 2nψ ˙ n ξ' ). τzs = G(ψs ξ' + ψ

71

(3.64c)

Because thin-walled beams of arbitrarily shaped cross-sections subjected to general loads will be solved numerically by a finite element method, the finite element formulation will be presented first. Then, the governing differential equations will be explicitly derived.

3.2.2 Finite Element Formulation For a thin-walled beam under a general loading condition, the contribution of the wall-bending terms (−nun' ) to the z-directional displacement [see Eq. (3.60a)] cannot be neglected near end-effect regions where stress and displacement vary rapidly. To be able to capture the rapid variation accurately by the finite element method, un' = ∂un /∂z as well as un must be continuous at the element borders, i.e., nodes. In the HoBT, un and us are mainly associated with in-plane cross-sectional modes, i.e., three rigid-body modes (Ux , Uy , θz ) and higher-order distortion modes. To ensure continuity in both un and un' between two adjacent finite elements, the variables ξi (z) belonging to ξI , the set of in-plane section modes, should have C 1 -continuity13 and thus are interpolated by Hermite cubic interpolation functions. On the other hand, no such step is necessary for the ξi (z) variables belonging to ξO , the set of out-of-plane section modes. Therefore, linear interpolation will be used for the variables related to three rigid-body modes (Uz , θx and θy ) and warping modes (W ). Based on the aforementioned interpolation argument, the following interpolations are employed: for ξi ∈ ξO (set of out-of-plane modes), ξi = N1 ξi,1 + N2 ξi,2 ,

(3.65a)

for ξi ∈ ξI (set of in-plane modes), ' ' ξi = H1 ξi,1 + M1 ξi,1 + H2 ξi,2 + M2 ξi,2 ,

(3.65b)

with 1+η 1−η , N2 = , 2 2

(3.65c)

1 1 (1 − η)2 (2 + η), H2 = (1 + η)2 (2 − η), 4 4

(3.65d)

N1 = H1 =

13

All nth-order differentiable functions belong to the function space C n .

72

3 One-Dimensional Governing Equations and Finite Element Formulation

M1 =

l l (1 − η)2 (1 + η), M2 = (1 + η)2 (η − 1), 8 8

(3.65e)

where the subscript i in ξi denotes the mode index [i = 1, 2, ..., ND ; ND is the number of section modes employed in the analysis in Eqs. (3.52a, b, c)]. In Eqs. (3.65a, b, c, d, e), N1 and N2 are linear shape functions of a two-node finite element (see Fig. 3.13a), and H1 , H2 , M1 , and M2 denote Hermite cubic interpolation functions (see Fig. 3.13b). The symbols ξi,1 and ξi,1 denote the values of ξi at nodes 1 and 2, ' ' and ξi,1 are correspondingly the values of ξi' at nodes 1 and respectively, while ξi,1 2. In Eq. (3.65e), l is the element length. The interpolation given by Eqs. (3.65a, b, c, d, e) can be put into a compact matrix form, ξ(z) = N(z)de ,

(3.66)

where N and de denote the matrix of the interpolating shape functions and the nodal vector for element e. For example, N and de can be explicitly written out in terms of their components for a special case considering only three rigid-body modes of an element for which ξ = {ξ2 , ξ3 , ξ4 }T = {Uy , Uz , θx }T : ⎡

⎤ H1 (z) M1 (z) 0 0 H2 (z) M2 (z) 0 0 N(z) = ⎣ 0 0 0 N2 (z) 0 ⎦, 0 N1 (z) 0 0 0 N2 (z) 0 0 0 N1 (z) 0

N1 =

1 −η 2

N2 =

1 +η 2

node 2

node 1

η = −1

(3.67)

η =0

η =1

z = z2

z = z1

(a) H1 =

1 H 2 = (1 + η ) 2 (2 − η ) 4

1 (1 − η ) 2 (2 + η ) 4

l M 1 = (1 − η ) 2 (1 + η ) 8

l M 2 = (1 + η ) 2 (η − 1) 8

node 2

node 1

η = −1

η =0

η =1

z = z2

z = z1

(b) Fig. 3.13 Two-node thin-walled beam element for a general loading case: a linear interpolation for out-of-plane displacements and b Hermite cubic interpolation for in-plane displacements

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

73

}T { ' ' de = Uy,1 , Uy,1 , Uz,1 , θx,1 , Uy,2 , Uy,2 , Uz,2 , θx,2 .

(3.68)

In Eqs. (3.67) and (3.68), only Uy is approximated with the Hermite cubic functions because it is the only in-plane section mode among the three field variables considered in this case (Uy , Uz , θx ). When the one-dimensional displacement variable ξ(z) is interpolated by Eq. (3.66), the displacements on the midline of the wall in Eq. (3.57) are given as ⎫ ⎡ ⎧ ⎤ ψz (s) ⎨ uz (z, s) ⎬ (3.69) u (z, s) = ⎣ ψn (s) ⎦ξ(z) = ψ(s)ξ(z) = ψ(s)N(z)de , ⎭ ⎩ n us (z, s) ψs (s) where ψ(s) is the matrix consisting of the three sectional shape functions ψ z (s), ψ s (s), and ψ n (s) defined in Eq. (3.58). Using Eq. (3.69), the three-dimensional displacements on a general point on the wall in Eqs. (3.60a, b, c) can then be expressed as ⎫ ⎧ ⎫ ⎡ ⎫ ⎧ ⎤⎧ 0 0 0 ⎨ u˙ z (z, s) ⎬ ⎨ u˜ z (z, n, s) ⎬ ⎨ uz (z, s) ⎬ u˜ (z, n, s) = un (z, s) + ⎣ 0 0 0 ⎦ u˙ n (z, s) ⎭ ⎩ ⎭ ⎭ ⎩ n ⎩ u˜ s (z, n, s) us (z, s) u˙ s (z, s) 0 −n 0 ⎫ ⎡ ⎤⎧ 0 −n 0 ⎨ uz' (z, s) ⎬ + ⎣ 0 0 0 ⎦ un' (z, s) , ⎭ ⎩ ' 0 0 0 us (z, s) e ˙ = ψ(s)N(z)de + A1 (n)ψ(s)N(z)d + A2 (n)ψ(s)N' (z)de (3.70) where A1 (n) and A2 (n) are 3 × 3 matrices having only one non-zero component (-n): ⎡

⎡ ⎤ ⎤ 0 0 0 0 −n 0 A1 (n) = ⎣ 0 0 0 ⎦ and A2 (n) = ⎣ 0 0 0 ⎦. 0 −n 0 0 0 0

(3.71)

The substitution of Eq. (3.70) into Eqs. (3.61a, b, c) yields the strain in matrix form, as ⎫ ⎫ ⎡ ⎧ ⎧ ∂ ⎤ 0 0 ∂s ⎨ u˜ z (z, n, s) ⎬ ⎨ εss (z, n, s) ⎬ ∂ 0 0 ⎦ u˜ n (z, n, s) ε (z, n, s) = ⎣ ∂z ⎭ ⎭ ⎩ ⎩ zz ∂ ∂ γzs (z, n, s) u˜ s (z, n, s) 0 ∂z ∂s

74

3 One-Dimensional Governing Equations and Finite Element Formulation

⎧ ⎫ ⎡ ⎫ ⎤⎧ ∂ ⎤ 0 0 ∂s 0 0 0 ⎨ u˜ z' (z, n, s) ⎬ ⎨ u˜ z (z, n, s) ⎬ = ⎣ 0 0 0 ⎦ u˜ n (z, n, s) + ⎣ 1 0 0 ⎦ u˜ n' (z, n, s) ⎩ ⎭ ⎭ ⎩ ' ∂ 0 0 u˜ s (z, n, s) u˜ s (z, n, s) 001 ∂s ⎧ ⎫ ⎧ ' ⎫ ⎨ u˜ z (z, n, s) ⎬ ⎨ u˜ z (z, n, s) ⎬ ≡ L u˜ n (z, n, s) + B u˜ n' (z, n, s) ⎩ ⎭ ⎩ ' ⎭ u˜ s (z, n, s) u˜ s (z, n, s) ˙ = L[ψ(s)N(z) + A1 (n)ψ(s)N(z) + A2 (n)ψ(s)N' (z)]de ⎡

' ˙ + B[ψ(s)N' (z) + A1 (n)ψ(s)N (z) + A2 (n)ψ(s)N'' (z)]de

= [S1 (n, s)N(z) + S2 (n, s)N' (z) + S3 (n, s)N'' (z)]de

(3.72)

with ⎡

⎡ ⎤ ∂ ⎤ 0 0 ∂s 000 L = ⎣ 0 0 0 ⎦, B = ⎣ 1 0 0 ⎦, ∂ 0 0 001 ∂s and S1 , S2 , and S3 are defined as ˙ S1 (n, s) = L[ψ(s) + A1 (n)ψ(s)],

(3.73a)

˙ S2 (n, s) = LA2 (n)ψ(s) + B[ψ(s) + A1 (n)ψ(s)],

(3.73b)

S3 (n, s) = BA2 (n)ψ(s).

(3.73c)

Finally, the stress can be expressed as ⎫ ⎡ ⎫ ⎧ ⎤⎧ E1 νE1 0 ⎨ εss (z, n, s) ⎬ ⎨ σss (z, n, s) ⎬ σ (z, n, s) = ⎣ νE1 E1 0 ⎦ εzz (z, n, s) ⎭ ⎭ ⎩ zz ⎩ τzs (z, n, s) 0 0 G γzs (z, n, s) ⎧ ⎫ ⎨ εss (z, n, s) ⎬ = C εzz (z, n, s) ⎩ ⎭ γzs (z, n, s) = C[S1 (n, s)N(z) + S2 (n, s)N' (z) + S3 (n, s)N'' (z)]de

(3.74)

The total potential energy in Eq. (3.6) can be written as 

=

1 2





u˜ T fdV ,

εT σdV − V

V

(3.75)

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

75

where ε = {εss , εss , εss }T , σ = {σss , σzz , τzs }T , and u˜ = {˜uz , u˜ n , u˜ s }T . In Eq. (3.75), f = {fz , fn , fs }T denotes the vector of the body forces. Using Eqs. (3.70–3.74), the total potential energy can be written as 

=



1 e T (d ) 2

(NT ST1 + N' ST2 + N'' ST3 )C(S1 N + S2 N' + S3 N'' )dV de T

T

V



˙ T AT1 + N' T ψ T AT2 )fdV , (NT ψ T + NT ψ

− (de )T

(3.76)

V

To find the equilibrium equation at the element level, the first variation of set to be equal to zero: δ∏ = (δde )T



is

∂∏ e ∂d 

(NT ST1 + N' ST2 + N'' ST3 )C(S1 N + S2 N' + S3 N'' )dV de T

= (δde )T V



− (δd )

e T

T

˙ T AT1 + N' T ψ T AT2 )fdV = 0. (NT ψ T + NT ψ

(3.77)

V

From Eq. (3.77), the following equilibrium equation can be derived at the element level: k e de = f e ,

(3.78)

where the element stiffness matrix ke and load vector fe are defined as  T T ke = (NT ST1 + N' ST2 + N'' ST3 )C(S1 N + S2 N' + S3 N'' )dV ,

(3.79)

V

and  fe =

˙ AT1 + N' ψ T AT2 )fdV . (NT ψ T + NT ψ T

T

(3.80)

V

To calculate ke in Eq. (3.79), it may be convenient to expand the integrand and treat the integral of each resulting term as a product of an integral over the beam cross-section A and a line integral along the z-direction. These results can be written as ke =

3 ∑ 3 ∑ i=1 j=1

kij ,

(3.81a)

76

3 One-Dimensional Governing Equations and Finite Element Formulation

with  k11 =

NT D11 Ndz,

(3.81b)

N' D12 Ndz,

(3.81c)

N'' D13 Ndz,

(3.81d)

N' D22 N' dz,

(3.81e)

N'' D23 N' dz,

(3.81f)

N'' D33 N'' dz,

(3.81g)

z



T

k12 = z



T

k13 = z



T

k22 = z



T

k23 = z



T

k33 = z

where Dij denotes the matrices of the sectional constants, which can be obtained from the integrals over the beam cross-section A:  Dij =

STj CSi dA, (i, j = 1, 2, 3).

(3.82)

A

Note that the components of the Dij matrices correspond to E1 a, E1 c, Gb1 , etc., appearing in Eq. (3.27), as obtained for beam torsional problems involving three section modes (θ , W , χ ). Thus, each component of Dij physically represents the stiffness of the section modes. A numerical integration method such as the Gaussian quadrature approach can be employed to perform the integrations in Eqs. (3.82) and (3.81b–g). The integral of the element force vector in Eq. (3.80) can also be separated as   T e T f = N f1 dz + N' f2 dz, (3.83) z

z

where f1 and f2 are generalized (or one-dimensional) body forces defined as  f1 = A

˙ T AT1 )fdA, (ψ T + ψ

(3.84a)

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

77

 f2 =

ψ T AT2 fdA.

(3.84b)

A

Following a procedure similar to that used in the torsion beam problem considered in Sect. 3.1.3, the element mass matrix me can be derived as  ˙ + A2 ψN' )T (ψN + A1 ψN ˙ + A2 ψN' )dV me = ρ(ψN + A1 ψN V

= m11 + m12 + m21 + m22 ,

(3.85)

where the submatrices mij (i, j = 1, 2) are given by  m11 =

 T NT E11 Ndz, m12 = m21 =

z

NT E12 N' dz, m22 =

z



N' E22 N' dz. T

z

(3.86) The matrices Eij (i, j = 1, 2) appearing in Eq. (3.86) represent sectional constants:  E11 =

˙ T AT1 )(ψ ˙ + A1 ψ)dA, ˙ ρ(ψ T + ψ

A

 E12 =

T ˙ AT1 )A2 ψdA, E21 = E12 ρ(ψ T + ψ , T

A

 E22 =

ρψ T AT2 A2 ψdA.

(3.87)

A

3.2.3 Governing Equations* The governing differential equations can be derived using a procedure identical to that used in the torsional beam problem. To facilitate the derivation, the three-dimensional strain and stress given in matrix form in Eqs. (3.63a, b, c, 3.64a, b, c) will be used. Accordingly, the resulting governing equations can also be obtained in matrix form. We consider the total potential energy, ∏, for a thin-walled beam: 

 1 =U +Ω= (σss εss + σzz εzz + τzs γzs )dV 2 V   − (fn u˜ n + fs u˜ s + fz u˜ z )dV − (tzs u˜ s + tzz u˜ z )dA, V

A

(3.88)

78

3 One-Dimensional Governing Equations and Finite Element Formulation

where U is the internal strain energy and Ω is the potential due to body forces (f z , f n , f s ) and the surface traction (t zz , t zs ). The volume of the beam and the area of the cross-section of the free end are denoted by V and A, respectively (dV = dAdz). If the strain energy U is expressed using the strain and stress in Eqs. (3.63a, b, c, 3.64a, b, c), one can write U as U = U1 + U2 + U3 + U4 + U5 ,

(3.89)

where  [ ( ) ] ˙ s + n2 ψ ¨ Ts ψ ¨ n + Gψ ˙ z dAξdz ¨ Ts ψ ˙ Tz ψ E1 ψ ξ



1 U1 = 2

T

z

A



U2 = U3 =

1 2

 (

ξT z

 ξ

'

) ˙ Ts ψz + G ψ ˙ Tz ψs dAξ' dz, νE ψ

z

A





U4 =

ξT z

U5 =

(3.90b)

A

 [ )] ( ˙ Tn ψ ˙ n dAξ ' dz, E1 ψTz ψz + G ψsT ψs + 4n2 ψ

T

(3.90a)

1 2

 z

A

ξ'' T

¨ s ψn dAξ'' dz, νE1 n2 ψ T



E1 n2 ψTn ψn dAξ'' dz,

(3.90c)

(3.90d)

(3.90e)

A

with the external work then expressed as   Ω=− z

( ) fn ψn + fs ψs + fz ψz dAξdz −

A



( ) tzs ψs + tzz ψz dSξ.

(3.91)

S

In Eq. (3.91), the body force and surface traction are assumed to be constant in the thickness direction. To find the governing equilibrium equation, the first variation of ∏ is set equal to zero: δ



= δU1 + δU2 + δU3 + δU4 + δU5 + δΩ = 0.

(3.92)

Performing integration by parts, each term appearing in Eq. (3.92) can be rewritten as  δU1 =

 [ ( ) ] ˙ s + n2 ψ ¨ Tn ψ ¨ n + Gψ ˙ z dAξdz, ˙ Ts ψ ˙ Tz ψ E1 ψ δξ T

z

A

(3.93a)

3.2 Finite Element Implementation Using an Arbitrary Number of Section …

 δU2 =

 (

δξT z

A



−

δξ z



T

) ˙ Ts ψz + G ψ ˙ Tz ψs dAξ' dz νE1 ψ ( ) ˙ s + GψTs ψ ˙ z dAξ' dz νE1 ψTz ψ

A



( ) ˙ s + GψTs ψ ˙ z dSξ, νE1 ψTz ψ

+ δξT

 [ )] ( ˙ Tn ψ ˙ n dAξ'' dz E1 ψTz ψz + G ψTs ψs + 4n2 ψ δξT

δU3 = − z

A

 [ )] ( T ˙ Tn ψ ˙ n dSξ' , E1 ψTz ψz + G ψTs ψs + 4n2 ψ + δξ 



A

¨ n ψn dAξ'' dz + νE1 n2 ψ T

δξT z

 δU5 =

 S

¨ n dSξ − δξT νE1 n2 ψTn ψ 

δξT z



− δξ

T



 δξT A

z

A

+ δξ' T

(3.93b)

S



δU4 =

79



(3.93c)

¨ n dAξ'' dz νE1 n2 ψTn ψ

¨ n dSξ' , νE1 n2 ψTn ψ

(3.93d)

S

E1 n2 ψTn ψn dAξ''' dz

A 2

E1 n S

ψTn ψn dSξ'''





δΩ = −

δξT z

− δξ



+ δξ

'T



E1 n2 ψTn ψn dSξ'' ,

( T ) fn ψn + fs ψTs + fz ψTz dAdz

A

(tzz ψTz + tzs ψTs )dSξ,

T

(3.93e)

S

(3.93f)

S

where δ(·) in the right-hand sides of Eqs. (3.93a–f) denotes the virtual field of (·). The virtual field can be arbitrary except at displacement-specified boundaries where δ(·) = 0. Specifically, δ u˜ p = 0 (p = z, n, s) are used at z = 0, where the displacement boundary conditions are assumed to be given. Because Eq. (3.92) should be valid for any δ(·), the following governing equations must hold: C1 ξ(z) + C2 ξ' (z) + C3 ξ'' (z) + C4 ξ''' (z) = F,

(3.94)

where the coefficient matrices Ci (i = 1, 2, 3, 4) and force vector F are given as

80

3 One-Dimensional Governing Equations and Finite Element Formulation

C1 =

 [ ( ) ] ˙ s + n2 ψ ¨ Tn ψ ¨ n + Gψ ˙ z dA ˙ Ts ψ ˙ Tz ψ E1 ψ

(3.95a)

A

 [ ( T ) ( T )] ˙s +G ψ ˙ z dA, ˙ s ψz − ψTz ψ ˙ z ψs − ψTs ψ νE1 ψ C2 = A

C3 = −

(3.95b)

 [ )] ( ˙ Tn ψ ˙ n dA E1 ψTz ψz + G ψTs ψs + 4n2 ψ A

 +

( T ) ¨ n dA, ¨ n ψn + ψTn ψ νE1 n2 ψ

A

(3.95c)

 C4 = 

F=

E1 n2 ψTn ψn dA,

(3.95d)

( T ) fn ψn + fs ψTs + fz ψTz dA.

(3.95e)

A

A

At a boundary where surface traction (tzs ,tss ) is prescribed, the following conditions should be satisfied:   ( T ) ( ) T ˙ s +GψTs ψ ˙ z dS ψz tzz + ψs tzs dS = ξ νE1 ψTz ψ S

S

+ξ

'

−ξ

'

 [

)] ( ˙ Tn ψ ˙ n dS E1 ψTz ψz + G ψTs ψs + 4n2 ψ

S

 ξ A

 νE1 n

2

ψTn

A

¨ n dA + ξ'' νE1 n2 ψTn ψ

¨ n dS − ξ''' ψ



 E1 n2 ψTn ψn dS,

(3.96a)

A

E1 n2 ψTn ψn dA = 0.

(3.96b)

A

References Bathe (1996) Finite element procedures. Prentice Hall Cassel K (2013) Variational methods with applications in science and engineering. Cambridge University Press Kim JH, Kim YY (1999a) Analysis of thin-walled closed beams with general quadrilateral cross sections. J Appl Mech 66:904–912 Kim JH, Kim YY (2000) One-dimensional analysis of thin-walled closed beams having general cross-sections. Int J Numer Meth Eng 49:653–668

References

81

Kim YY, Kim JH (1999b) Thin-walled closed box beam element for static and dynamic analysis. Int J Numer Meth Eng 45:473–490 Timoshenko SP (1921) On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section. Phil Mag 41:744–746 Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross-section. Phil Mag 43:125–131 Truesdell C (1960) The rational mechanics of flexible or elastic bodies, 1638–1788, Venditioni Exponunt Orell Fussli Turici Vlasov VZ (1961) Thin-walled elastic beams. Israel Program for Scientific Translations Ltd.

Chapter 4

Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

The higher-order sectional deformations of a box beam differ depending on the type of an applied load (see Vlasov (1961), Schardt (1994), Cesnik and Hodges (1997), Kim and Kim (1999), Carrera et al. (2011), Genoese et al. (2014), Bebiano et al. (2018)). Different section deformation modes, which may be classified as torsional, extensional, and bending modes, are needed to deal with a thin-walled beam subjected to different load types. This chapter and Chap. 5 are devoted to detailed derivations of higher-order sectional shape functions corresponding to torsional modes (Table 4.1). Before presenting the detailed procedure1 used to derive higher-order deformable section modes for torsion, it is important to clarify the types of section modes and the corresponding mode symbols. First, it should be noted that either the s-directional (u s (z, s)) or z-directional (u z (z, s)) displacement is non-vanishing on the wall centerline (for n = 0) for the warping and unconstrained distortion modes; this observation may be apparent from Eqs. (2.25a, c) in Sect. 2.3. Also note that non-vanishing ndirectional displacement (u n (z, s)) is accompanied by the unconstrained distortion mode, which can also be confirmed by Eq. (2.25b). Although Eq. (2.25) is written for the fundamental warping and distortion modes, these observations are valid for all higher-order modes. On the other hand, it will be argued that constrained distortion modes, both Type 1 and Type 2, have only non-zero n-directional displacement on the wall centerline while u s (z, s) = u z (z, s)= 0. If u n (z, s) /= 0 for modes χ , η, and η, ˆ walls forming the cross-section of a box beam can experience bending deformation in the cross-sectional plane (n − s plane), producing non-zero s- and z-directional displacements off the wall centerline2 (i.e., u˜ s (z, n /= 0, s) /= 0, u˜ z (z, n /= 0, s) /= 0). In this case, non-zero u˜ z (z, n, s) or u˜ s (z, n, s) will occur for

1

They are summarized in Table 4.1. In the table, the characteristics of the modes are also briefly described, but their origins will be clarified as the mode derivation proceeds in the coming sections. 2 More details will be given in Chap. 5. © Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_4

83

Fundamental warping mode W 0 (torsional warping mode)

• Deformable section modes A. Fundamental deformable section modes (by Vlasov) Out-of-plane field

Rigid-body torsional rotation θ Used for torsion problems

• Non-deformable (rigid-body) section modes

Table 4.1 Types of section modes and mode symbols

Fundamental unconstrained distortional mode χ0 (torsional distortion mode)

In-plane field

Not used for torsion problems

(continued)

84 4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

Higher-order constrained distortion modes of type 1 ηk (k = 1, 2, . . .)

Constrained distortiona Type 1b

Higher-order constrained distortion modes of type 2 ηˆ k (k = 1, 2, . . .)

Type 2b

a If s-directional displacement along the wall centerline is zero, the in-plane distortion is called “constrained distortion.” Otherwise, it is called “ unconstrained distortion.” or simply “distortion” b If rotation at every corner of the beam cross section identically vanishes, the corresponding mode is called Type 1

Higher-order warping modes W k (k = 1, 2, …)

Higher-order distortion modes χk (k = 1, 2, . . .)

B. Higher-order deformable section modes Out-of-plane field In-plane field Unconstrained distortiona

Table 4.1 (continued)

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field 85

86

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

Table 4.2 Non-zero wall centerline displacements and sectional shape functions depending on the mode types

Mode type

Wall centerline displacements

Non-zero sectional shape functions

Warping, W

u z (z, s) /= 0, u s (z, s) = u n (z, s) = 0

ψzW (s)

(Unconstrained) distortion, χ

u z (z, s) = 0, u s (z, s) /= 0, u n (z, s) /= 0

ψs (s), ψn (s)

Constrained distortion u z (z, s) = u s (z, s) = 0, η and ηˆ In short, denoted by u n (z, s) /= 0 C-distortion

ψn (s), ψn (s)

χ

χ

η

ηˆ

n /= 0 due to wall bending.3 Table 4.2 explains the non-vanishing field components depending on the mode types. Note that because wall bending will not occur due to ψzW (s) of warping mode W χ and ψs (s) of unconstrained distortion mode χ , one can focus only on the deformation of the wall centerline (n = 0) when deriving these shape functions. Therefore, ψzW (s) χ and ψs (s) can be derived independently from other n-directional sectional shape χ η ηˆ functions ψn (s), ψn (s), and ψn (s). . Accordingly, we will derive (for a box beam subjected to a torsional ⎧ Wload): ψz (s) of warping mode W in Chapter 4 : χ ψs (s) of distortion mode χ → Wall-membrane field (involving centerline stretch or shrinkage) ⎫ χ ψn (s) of distortion mode χ In Chapter 5 : η¯ ηˆ ¯ ηˆ ψn (s), ψn (s) of C-distortion mode η, → Wall-bending field Throughout this book, we may distinguish the different characteristics between the χ χ η shape functions in z and s (ψzW (s), ψs (s)) and the shape functions in n (ψn (s), ψn (s), ηˆ ψn (s)) by referring the former and the latter as the membrane and wall-bending fields, respectively. χ When deriving the sectional shape functions ψzW (s) and ψs (s), it is convenient to use both the local (n and s) and global (x and y) coordinate systems shown in

3

If a flexural displacement on the centerline (known more commonly as the neural axis) of a cantilevered beam (with a solid beam cross-section) occurs, non-vanishing longitudinal (or axial) displacements occur at points away from the centerline due to beam bending. In the present case, u n corresponds to flexural displacement and u˜ z (z, n, s) or u˜ s (z, n, s) corresponds to longitudinal displacement.

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field Fig. 4.1 Coordinate systems used to describe the geometry of a thin-walled box beam (wall thickness: t)

87

y n2

corner 2 Edge 2 s2

h

corner 1 z

s1

Edge 3 n3 s3

n1

x

z

z Edge 4 s z4

n4 corner 3

Edge 1

z wall midline corner 4

b

Fig. 4.1. To begin the derivation of higher-order deformable section modes Wk and χk (k: mode set number, k = 1, 2, . . .), we start with the following three modes: the rigid-body section rotation mode θz and two fundamental (lowest-order) deformable section modes W0 and χ0 , as shown in Fig. 4.2. For the subsequent derivation, we explicitly write the corresponding non-zero sectional shape functions ψzW0 (s) and χ ψs 0 (s) derived earlier in Sect. 2.3 (see Eqs. (2.26), (2.40), (2.49), and (2.50)) as follows: ⎫ θ ψs z (s j ) = r = b2 for edges 1 & 3, (4.1a) θ ψs z (s j ) = r = h2 for edges 2 & 4 ψnθ (s j ) = −s j , ⎧

⎫

ψzW0 (s j ) = (b/2)s j for j = 1 & 3, ψzW0 (s j ) = −(h/2)s j for j = 2 & 4, ⎫ χ bh for j = 1 & 3, ψs 0 (s j ) = b+h χ0 bh for j = 2 & 4 ψs (s j ) = − b+h

(4.1b)

(4.2)

(4.3a)

χ

4 s 3j + 2b+h s for j = 1 & 3, ψn 0 (s j ) = − h(b+h) b+h j χ0 4 b+2h 3 ψn (s j ) = b(b+h) s j − b+h s j for j = 2 & 4.

(4.3b)

88

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

y

y z

x

y x

z

z

x

Rotation mode (θ ) Fundamental warping mode ( W0 ) Fundamental distortion mode ( χ 0 ) Fig. 4.2 Three fundamental modes needed to derive the (next) higher-order deformable section modes

4.1 General Field Relationship for the Higher-Order Deformable Section Modes of a Wall-Membrane Field For a box beam subject to a torsional load, the following modes corresponding to the wall-membrane field will be considered in this chapter: {θz , χ0 , W0 }: zeroth-order (or fundamental) modes for torsion problems. {Wk , χk } (k ≥ 1): higher-order modes for torsional problems. While the symbols θz , χ0 , W0 , and {Wk , χk }k=1,2,... are used to denote the mode types, they will also be used to represent the one-dimensional field variables denoting their axial variations along the z axis. To facilitate the subsequent notation, we introduce the following one-dimensional (1D) displacement variable vector ξ, defined as ξ = {θz (z), χ0 (z), W0 (z), χ1 (z), W1 (z), χ2 (z), W2 (z), ...}T }T { = θz (z), χ0 (z), W0 (z), {χk (z), Wk (z)}k=1,2,...,Ns ,

(4.4)

where N S denotes the number of higher-order mode sets employed for the analysis. Note that constrained distortion modes {ηk , ηˆ k }k=1,2,... are not included in the expression (4.4) because they will be derived separately from the modes considered in this chapter. The next chapter will present the derivation of constrained distortional modes. To derive the sectional shape functions for modes {Wk , χk } (k ≥ 1), first we consider how these modes contribute to the displacements on the midline of a crosssectional wall. Using the modes in Eq. (4.4), one can write u z (z, s) = ψzW0 (s)W0 (z) +

NS  k=1

ψzWk (s)Wk (z),

(4.5a)

4.1 General Field Relationship for the Higher-Order Deformable Section …

u s (z, s) =

ψsθz (s)θz (z)

+

ψsχ0 (s)χ0 (z)

+

NS 

89

ψsχk (s)χk (z).

(4.5b)

k=1 χ

χ

Note that although mode χ has two non-zero components ψs and ψn (see Table χ 4.2), the n-directional component ψn generating wall-bending deformation is not considered in Eq. (4.5) because this chapter is concerned with only the membrane field. The same mode number Ns is used in both Eqs. (4.5a) and (4.5b) because we arrange one instance of torsional warping and one instance of torsional distortion to form one mode set {Wk , χk } for a consistent analysis of thin-walled box beams. For thin-walled beams consisting of cross-sections with general shapes, on the other hand, multiple warping and distortion modes can be adopted for each mode set; these are discussed in Chap. 9. To facilitate the determination of higher-order sectional shape functions, we impose the conditions that a higher-order mode should be orthogonal to the fundamental deformable modes and other higher-order modes. These conditions can be stated as  W (4.6) ψzWi (s)ψz j (s)dA = 0 if i /= j, and 

χ

ψsχi (s)ψs j (s)dA = 0 if i /= j.

(4.7)

χ

Because ψsWi = ψnWi = 0 and ψz i = 0, only ψzWi appears in Eq. (4.6) and only appears in Eq. (4.7).4 In addition, they are also set to be orthogonal to the shape functions of all non-deformable section modes (see Eqs. (2.37) and (2.46)):

χ ψs i



ψzWi (s)ψzα (s)dA = 0 (α = Uz , θx , and θ y ),

(4.8)

and 

ψsχi ψsβ dA = 0 (β = θz , Ux , and U y ),

(4.9)

A

where Uq and θq (q = x, y, z) denote the rigid-body modes of translation along the q axis and rotation about the q axis, respectively. The aforementioned orthogonality conditions significantly facilitate the derivation of higher-order modes. More importantly, they actually serve to decouple the 1D generalized forces (which are the work conjugates of each component of ξ) from each other in the generalized forcestress relationships, as shall be seen later. This decoupling is critical when analyzing 4

χ

χ

The orthogonality for ψn i will be considered when the ψn i (s) values are derived in Chap. 5.

90

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

the effect of each mode on a structural response and very useful in establishing the matching conditions among the 1D field variables at a joint of multiply connected thin-walled beams. The matching condition will be presented in Chap. 10. Let us consider strain due to displacement in Eq. (4.5). Using Eq. (4.5), one can calculate non-zero strains on the wall centerline (at n = 0) associated with the Wk and χk modes (k = 0, 1, 2, . . .) as S  ∂ uz = ψzW0 (s)W0, (z) + ψzWk (s)Wk, (z), ∂z k=1

N

εzz (z, s) =

S  ∂ us ψ˙ sχk (s)χk (z), = ∂s k=1

(4.10a)

N

εss (z, s) =

(4.10b)

∂ uz ∂ us + ∂z ∂s = ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψ˙ zW0 (s)W0 (z)

γzs (z, s) =

+

NS  (

) ψsχk (s)χk, (z) + ψ˙ zWk (s)Wk (z) .

(4.10c)

k=1

As introduced in the previous chapters, ( ), = d( )/dz and (˙) = d( )/ds. In θ θ χ χ Eq. (4.10), the terms ψ˙ s z θz and ψ˙ s 0 χ0 are omitted because ψ˙ s z = ψ˙ s 0 = 0 (see Eq. (2.26) and Eq. (2.49)). Employing the plane stress assumption, the stresses on the wall centerline can be calculated from the strains in Eq. (4.10) as σzz (z, s) = E 1 (εzz + νεss ) = E 1 [ ψzW0 (s)W0, (z) +

NS 

(ψzWk (s)Wk, (z) + ν ψ˙ sχk (s)χk (z))],

(4.11a)

k=1

σss (z, s) = E 1 (εss + νεzz ) = E 1 [νψzW0 (s)W0, (z) +

NS 

(ψ˙ sχk (s)χk (z) + νψzWk (s)Wk, (z))],

(4.11b)

k=1

τzs (z, s) = Gγzs = G[ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψ˙ zW0 (s)W0 (z) +

NS 

( ψsχk (s)χk, (z) + ψ˙ zWk (s)Wk (z))],

(4.11c)

k=1

where E 1  E/(1 − ν 2 ) (E: Young’s modulus; ν: Poisson’s ratio; G: shear modulus). The strains in Eqs. (4.10) and stresses in Eq. (4.11) can be viewed as wall-membrane components because they involve only the stretch of the wall centerline in the z and s directions.

4.1 General Field Relationship for the Higher-Order Deformable Section …

91

To derive explicit expressions for the 1D generalized forces that are the work conjugates of θz , Wk , and χk , one can consider the virtual work δW A done on the surface of the beam cross-section A by τzs and σzz , the components of the surface traction. Following the same procedure used in Sect. 3.1.2,5 we express the virtual work δW A as6   δW A (z) = τzs (z, s)δu s (z, s)dA + σzz (z, s)δu z (z, s)dA, (4.12) A

A

where δu z and δu s are the three-dimensional virtual displacements in the z and s directions, respectively, and dA = tds. In Eq. (4.12), cross-section A is assumed to be uniform throughout the beam. Therefore, the virtual work can be regarded as a function of z only. Substituting Eq. (4.11) into Eq. (4.12) yields 

 τzs (z, s)ψsθz (s)dA · δθz (z) + τzs (z, s)ψsχ0 (s)dA·δχ0 (z)  + σzz (z, s)ψzW0 (s)dA · δW0 (z)

δW A (z) =

+

N S ( 

τzs (z, s)ψsχk (s)dA

)

 · δχk (z) +

σzz (z, s)ψzWk (s)dA

· δWk (z)

k=1

 Mz (z)δθz (z) + Q 0 (z)δχ (z) + B0 (z)δW (z) +

NS 

(Q k (z)δχk (z) + Bk (z)δWk (z)),

(4.13)

k=1

where the quantity δ(·) denotes the virtual displacement corresponding to the 1D displacement variable (·). Examining Eq. (4.13), one can define the following form of the generalized forces: { }T F(z) = Mz (z), Q 0 (z), B0 (z), {Q k (z), Bk (z)}k=1,2,...,N S ,

(4.14)

where  Mz = B0 = Q0 = 5

 

τzs ψsθz dA, σzz ψzW0 dA, Bk = τzs ψsχ0 dA, Q k =

 

σzz ψzWk dA τzs ψsχk dA

(k = 1, 2, · · · , N S ), (k = 1, 2, · · · , N S ),

(4.15)

The generalized forces can also be derived from boundary terms by integrating by parts the virtual work of an entire beam domain. See Sect. 3.1.2 for details. 6 The argument n is not used to describe the traction or virtual displacements because only wallmembrane components are considered here.

92

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

where Mz (z) is the torsional moment, which is the work conjugate of the torsional rotation θz (z), and Bk (z) (k = 0, 1, . . . , N S ) is termed the bimoment, which is the work conjugate of Wk (z) representing the 1D displacement measure of the k th warping mode. Finally, Q k (k = 0, 1, . . . , N S ) is known as the transverse bimoment, which is the work conjugate of χk (z) representing the 1D displacement measure of the k th distortion mode.

4.2 Generalized Force-Stress Relationship for Zeroth-Order Modes In this subsection, explicit relationships between stresses and generalized forces will be derived. To this end, first we examine σzz generated by the zeroth-order modes in Eq. (4.11a): σzz (z, s) = E 1 ψzW0 (s)W0, (z).

(4.16)

Using σzz in Eq. (4.16), the generalized force B0 (z), which is the work conjugate of W0 (z), is calculated as   W0 B0 (z) = σzz (z, s)ψz (s)dA = E 1 (ψzW0 (s))2 dA · W0, (z) = E 1 JW0 W0, (z), (4.17) where JW0 is defined as  JW0 =

( ψzW0 )2 dA =

b2 h 2 t (b + h) . 24

(4.18)

The symbol JW0 denotes the second moment of inertia of mode W0 . Generally, the second moment of inertia of mode α will be defined as  Jα  ( ψ α (s))2 dA (α = θz , Wk, χk ; k = 0, 1, 2, . . .). (4.19) If W0, (z) in Eq. (4.17) is eliminated using Eq. (4.16), one can find the following explicit relationship between σzz and B0 7 : σzz (z, s) =

7

B0 (z) W0 ψ (s) ≡ σzzW0 (z, s), JW0 z

(4.20)

This equation provides the direct relationship between B0 (the generalized force of the zerothorder torsional warping mode) and the stress. Note that unless the orthogonality relationships, Eqs. (4.4) and (4.5), are used, no such direct relationship can be possible for higher-order modes.

4.2 Generalized Force-Stress Relationship for Zeroth-Order Modes

93

where the superscript W0 in σzzW0 is used to indicate that σzzW0 is induced by mode W0 . Generally, we will use the subscript or superscript α to denote the quantity of interest related to mode α. Using Eq. (4.11c), the shear stress generated by the zeroth-order modes can be expressed as [ ] τzs (z, s) = G ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψ˙ zW0 (s)W0 (z) .

(4.21)

To simplify the expression in Eq. (4.21), the following relationship, which can be derived from Eqs. (4.1a), (4.2), and (4.3a), will be used: ψ˙ zW0 (s) = c1 ψsθz (s) + c2 ψsχ0 (s),

(4.22)

where c1 =

b−h ; c2 = 1. b+h

(4.23)

Substituting Eq. (4.22) into Eq. (4.21) yields [ ] τzs (z, s) = G ψsθz (s){θz, (z) + c1 W0 (z)} + ψsχ0 (s){χ0, (z) + c2 W0 (z)} .

(4.24)

Using τzs in Eq. (4.24), the generalized force Mz defined in the first equation of Eq. (4.15) becomes 



G[ψsθz (θz, + c1 W0 ) + ψsχ0 (χ0, + c2 W0 )]ψsθz dA   = G(ψsθz )2 dA·(θz, + c1 W0 ) + Gψsχ0 ψsθz dA·(χ0, + c2 W0 )  = G(ψsθz )2 dA·(θz, + c1 W0 ) = G Jθz (θz, + c1 W0 ). (4.25)

Mz =

τzs ψsθz dA

=

the process of deriving Eq. (4.25), the orthogonality relationship,  During θ χ ψs 0 ψs z dA = 0, given in Eq. (4.9), is utilized. (Note that Mz in Eq. (4.25) is the work conjugate of the torsional rotation θz .) Using Eqs. (4.24) and (4.15), the generalized force Q 0 , which is the work conjugate of the zeroth-order torsional distortion χ0 , can be similarly found as:  Q0 = =



τzs ψsχ0 dA =



G[ψsθz (θz, + c1 W0 ) + ψsχ0 (χ0, + c2 W0 )]ψsχ0 dA

G(ψsχ0 )2 dA·(χ0, + c2 W0 ) = G Jχ0 (χ0, + c2 W0 ).

(4.26)

Using Eqs. (4.24), (4.25), and (4.26), one can decompose the shear stress τzs into two parts as

94

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

τzs (z, s) = τzsMz (z, s) + τzsQ 0 (z, s),

(4.27)

M

where τzs z and τzsQ 0 are defined as [ ] Mz (z) θz τzsMz (s, z) ≡ Gψsθz (s) θz, (z) + c1 W0 (z) = ψs (s), Jθz

(4.28)

[ ] Q 0 (z) χ0 τzsQ 0 ≡ Gψsχ0 (s) χ0, (z) + c2 W0 (z) = ψs (s). Jχ0

(4.29)

and

The significance of the results given by Eqs. (4.27–4.29) is that the shear stress can be completely decoupled into two contributions, one from the torsional moment Mz and the other from the transverse bimoment Q 0 . We will show later that this decoupling remains valid even when the stress is calculated using all higher-order modes as long as the sectional shape functions are defined as orthogonal to each other.

4.3 Derivation of Higher-Order Deformable Section Modes by Means of a Recursive Analysis χ

In this section, the non-zero sectional shape functions ψs k of higher-order distortion modes and ψzWk of higher-order warping modes (k ≥ 1) will be derived using a recursive analysis initially developed in Choi and Kim (2019, 2020, 2021). The mechanism of the recursive derivation of the higher-order modes will be outlined first using Fig. 4.3 before the detailed derivation procedure is presented. As the first step of the derivation procedure, we start with the known displacement and stress fields (Part A in Fig. 4.3) associated with the zeroth-order torsional warping W0 and its work conjugate B0 ; the sectional shape functions of the W0 mode are given in Eq. (4.2), and the corresponding non-zero stress σzzW0 (z, s) is given by Eq. (4.20). If σzzW0 is present, εss (z, s) can be induced due to Poisson’s effect (Part B in Fig. 4.3). Here, the induced strain εss (z, s) can be viewed as a type of secondary strain, and σzzW0 (z, s) can be viewed as the primary stress because εss (z, s) is caused by σzzW0 (z, s) due to Poisson’s ratio. However, the existence of a non-zero εss cannot be explained without an additional u s (z, s) displacement field; the corresponding deformation mode will be called χ1 , the first-order distortion mode. To derive in closed form the sectional shape function χ ψs 1 (s) that describes the displacement of the midline of the sectional wall associated with the χ1 mode (Part C in Fig. 4.3), some fundamental conditions are considered. Specifically, the geometric symmetry of a rectangular cross-section and the continuity of field quantities at every corner of the section are considered. If applicable, the

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

A

B

95

C

y x z

u z = ψ zW0W0

χ1 generated

ε ss by Poisson’s effect E

D

Next set

W1

γ zs by χ1′

generated

Fig. 4.3 Sequential generation of the first-order modes (χ1 and W1 ) by zeroth-order warping. After χ the sectional shape function ψs 1 of mode χ1 is determined using the W0 field, the sectional shape W1 function ψz of mode W1 is then determined. This process is recursively used to derive the sectional shape functions of all higher-order modes χ

orthogonality conditions are also considered. Once ψs 1 (s) representing the u s field in the cross-section is determined, one can calculate the shear strain and stress (Part D in Fig. 4.3), which in turn induces the next higher-order mode W1 , called the χ first-order warping mode. Following a procedure similar to how ψs 1 is derived, the W1 shape function ψz (s) of the first-order warping mode W1 can be derived (Part E χ in Fig. 4.3). Because the explicit expressions of ψs 1 and ψzW1 are available, the of χ1 (z) first-order generalized forces Q 1 (z) and B1 (z)that are the work conjugates  χ and W1 (z) can be explicitly found as Q 1 = τzs ψs 1 dA and B1 = σzz ψzW1 dA, respectively. A more detailed account of the recursive derivation process is outlined in Fig. 4.4. The recursive derivation procedure described above may be summarized into the following three steps: Step 1: Identification of the secondary strain field due to a known (primary) stress field. Step 2: Derivation of secondary displacement consistent with the secondary strain. Step 3: Stress field update due to secondary displacement. We will show below how these three steps are used to derive the sectional shape functions of the desired higher-order modes.

96

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

Fig. 4.4 Recursive procedure to derive the sectional shape functions ψzWk (u z component) and χ ψs k (u s component) of the distortion and warping modes

k =1 A B

C

D E

u z by warping of the (k-1)th order ε ss by Poisson’s effect Distortion of the k-th order generated

k = k +1

Shear strain γ zs generated by us Warping of the k-th order generated

k≤N End

χ

4.3.1 Derivation of ψs 1 (Shape Function of the First-Order Distortion Mode) χ

To derive the shape function ψs 1 of mode χ1 , we start with the stress induced by the zeroth-order warping mode, W 0 . The step-by-step procedure is discussed below in detail. Step 1. Identification of the secondary normal strain field due to the W 0 mode We begin with the stress field σzz induced by the zeroth-order torsional warping mode, which is explicitly given in Eq. (4.20). The non-zero σzz (z, s)8 is related to the normal stress εzz via εzz (z, s) =

σzz (z, s) . E

(4.30)

As indicated in Eq. (4.5a), ψzW0 (s)W0 (z) is the displacement inducing the nonzero εzz due to the zeroth-order warping mode W0 . The normal stress appearing in Eq. (4.30) also generates the non-zero normal strain εss in the s direction due to Poisson’s effect (see Part B in Fig. 4.3 and Part B in Fig. 4.4). This is expressed as 8

Note that other normal stress components, σss and σnn , are zero for the W 0 mode.

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

εss (z, s) = −ν

ν B0 (z) W0 σzz (z, s) =− ψ (s). E E JW0 z

97

(4.31)

As noted earlier, this type of strain (e.g., strain due to Poisson’s effect) will be called the “secondary,” strain while the strain field directly generated by the stress field in the same direction is referred to as the “primary” strain. The stress generating the primary and secondary strains is also referred to as the “primary stress.” The appearance of εss in Eq. (4.31) implies that the cross-sectional wall of a box beam can be stretched or shrunken on the midline of the section wall. Therefore, a non-zero displacement field (u s (z, s)) is needed in order to account for this wall stretching or shrinkage, i.e., membrane deformation. This displacement field constitutes a higher-order mode. The mode representing the u s displacement needed to account for εss in Eq. (4.31) will be called the first-order distortion mode, χ1 . χ If the s-directional section shape function of χ1 is defined as ψs 1 (s), which is yet undetermined, u s (z, s) due to mode χ1 can be expressed as u s (z, s) = ψsχ1 (s)χ1 (z).

(4.32)

In the next step, we will describe the procedure to derive the sectional shape χ function ψs 1 (s) for mode χ1 . Step 2. Derivation of the secondary displacement consistent with the secondary strain Considering the contribution of mode χ1 to u s (z, s), as expressed by Eq. (4.32), the total displacement u s (z, s) by the modes considered thus far can be written as u s (z, s) = ψsθz (s)θz (z) + ψsχ0 (s)χ0 (z) + ψsχ1 (s)χ1 (z).

(4.33)

Using Eq. (4.33), εss becomes εss (z, s) ≡ θ

∂u s (z, s) = ψ˙ sχ1 (s)χ1 (z), ∂s

(4.34)

χ

where ψ˙ s z (s) = 0 and ψ˙ s 0 (s) = 0 were used (see Eqs. (4.1a) and (4.3a)). The strain in Eq. (4.34) should be equal to εss in Eq. (4.31), which is induced by Poisson’s effect. Therefore, the following relationship must hold: χ1 (z) ψ˙ sχ1 (s) ≡ −

ν B0 (z)ψzW0 (s). E JW0

(4.35)

Because the left and right sides of Eq. (4.35) are written as products of a function of z and another function of s, Eq. (4.35) can be valid if ψ˙ sχ1 (s) = − p1∗ ψzW0 (s),

(4.36)

98

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

χ1 (z) =

ν/E JW0 B0 (z), p1∗

(4.37)

where p1∗ is an arbitrary scaling constant. If p1∗ = v/E JW0 were chosen, Eq. (4.37) would reduce simply to χ1 (z) = B0 (z). To facilitate the subsequent calculations, however, p1∗ is chosen as p1∗ = −

6 . 25bh 2

(4.38) χ

In actuality, the p1∗ value in (4.38) is selected to normalize ψs 1 as ( ) 1 h = . ψsχ1 s1 = 2 100

(4.39)

For the subsequent analysis, it is convenient to write Eq. (4.36) for every edge in the following form: ψ˙ sχ1 (s j ) = − p1∗ ψzW0 (s j ), ( j = 1, 2, 3, 4).

(4.40)

Integrating both sides of Eq. (4.40) with respect to s j gives [ ] χ ψsχ1 (s j ) = p1∗ −ψzW0 (s j ) + C j 1 ,

(4.41)

where ψzW0 (s j ) represents the indefinite integral of ψzW0 (s j )9 : ψzα (s j )

 =

[ ] −ψzα (s j ) ds j , (α = W0 , W1 , W2 , . . .),

(4.42)

χ

and C j 1 (j = 1, 2, 3, 4) is the integration constant. Using ψzW0 in Eq. (4.2), ψzW0 is found using the equations below: ⎫

ψzW0 (s j ) = b4 s 2j for j = 1, 3, ψzW0 (s j ) = − h4 s 2j for j = 2, 4.

(4.43)

χ

To determine four unknown coefficients C j 1 appearing in Eq. (4.41), we will consider the following: (1) The symmetry conditions of the displacement field. (2) The orthogonality conditions given by Eq. (4.7).

9

The symbols W 1 and W 2 are used similarly to how W 0 is used. They represent the first- and second-order warping modes, respectively, and the 1D generalized displacements that are functions of z.

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

99

To identify the symmetry conditions, we note that because σzzW0 given by Eq. (4.20) is anti-symmetrically distributed with respect to the x and y axes (see Fig. 4.3a), the secondary displacement field induced by the anti-symmetric stress σzzW0 should be χ also anti-symmetric. Accordingly, ψs 1 should satisfy ψsχ1 (s j ) = ψsχ1 (−s j ) ψsχ1 (s j ) = ψsχ1 (s j+2 )

( j = 1, 2, 3, 4),

(4.44a)

( j = 1, 2).

(4.44b)

Because the origin of s j is at the center of edge j, two points denoted by s j and −s j are located the same distance away from the origin of the s j coordinate. χ Equation (4.44a) implies that the u s displacement by ψs 1 on each edge is antiχ1 symmetric, i.e., that ψs (s j ) is a symmetric function of s j . Equation Eq. (4.44b) indicates that two walls facing each other deform anti-symmetrically. Using Eqs. χ (4.41) and (4.43), one can see that ψs 1 (s j ) is an even function of s j satisfying the condition stated by Eq. (4.44a). χ In determining the four constants C j 1 ( j = 1, 2, 3, 4), two additional conditions are needed in addition to the two conditions given in (4.44b). They are orthogonality conditions (see Eqs. (4.9) and (4.7)), which can be written as follows:  

ψsχ1 ψsθz ds = 0,

(4.45a)

ψsχ1 ψsχ0 ds = 0.

(4.45b)

Using the four equations in Eqs. (4.44) and (4.45) and p1∗ in Eq. (4.38), one can χ explicitly determine ψs 1 (s j ) as ( ) b 2 bh 2 6 − sj + ( j = 1, 3), =− 25bh 2 4 48 ( ) h 2 b2 h 6 − s ( j = 2, 4). + ψsχ1 (s j ) = 25bh 2 4 j 48

ψsχ1 (s j )

(4.46a)

(4.46b)

χ

Fig. 4.5(a) plots ψs 1 (s) derived for a box beam. Step 3. Stress field update due to the secondary displacement χ

As the secondary displacement ψs 1 (s)χ1 (z) in the direction of s due to χ1 (z) can now be known, the displacement, strain, and stress should be updated as u s (z, s) = ψsθz (s)θz (z) + ψsχ0 (s)χ0 (z) + ψsχ1 (s)χ1 (z), u z (z, s) = ψzW0 (s)W0 (z),

(4.47)

100

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

y z

y

y x

x

z

y x

x

z

(a)

(b)

z

(c)

(d) χ

Fig. 4.5 Sectional shape functions of torsion: a first-order distortion (ψs 1 ), b first-order warping χ (ψzW1 ), c second-order distortion (ψs 2 ), and d second-order warping (ψzW2 )

εzz (z, s) = ψzW0 (s)W0, (z), εss (z, s) = ψ˙ sχ1 (s)χ1 (z), γsz (z, s) = ψsθz (s)θz, + ψsχ0 (s)χ0, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zW0 (s)W0 (z),

(4.48)

] E [ W0 ψz (s)W0, (z) + ν ψ˙ sχ1 (s)χ1 (z) , 2 1−ν ] E [ χ1 σss (z, s) = ψ˙ s (s)χ1 (z) + νψzW0 (s)W0, (z) , 2 1−ν [ ] τzs (z, s) = G ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zW0 (s)W0 (z) . (4.49)

σzz (z, s) =

χ At this point, we use the relationship ψ˙ s 1 = − p1∗ ψzW0 given by Eq. (4.36) and the θ χ relationship ψ˙ zW0 = c1 ψs z + c2 ψs 0 in Eq. (4.22) to rewrite σzz and τzs as

σzz =

E ψ W0 (W0, − p1∗ νχ1 ), 1 − ν2 z

τzs = G[ψsθz (θz, + c1 W0 ) + ψsχ0 (χ0, + c2 W0 ) + ψsχ1 χ1, ].

(4.50a) (4.50b)

Compared to Eq. (4.49), Eq. (4.50) does not involve the derivatives of the shape functions with respect to s. It will be soon shown that the stress in the form of Eq. (4.50), not in the form of Eq. (4.49), is critical when deriving explicit generalized force-stress relationships. If the stresses given by Eq. (4.50) are substituted into Eqs. (4.15), the generalized forces can be written as  B0 = σzz ψzW0 dA = E 1 JW0 (W0, − p1∗ νχ1 ), (4.51a)

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

 Mz =  Q0 =

101

τzs ψsθz dA = G Jθz (θz, + c1 W0 ),

(4.51b)

τzs ψsχ0 dA = G Jχ0 (χ0, + c2 W0 ),

(4.51c)

 Q1 =

τzs ψsχ1 d A = G Jχ1 χ1, .

(4.51d)

Note that the bimoment B0 of the zeroth-order warping is updated from Eq. (4.20) due to the presence of the χ1 term. Using the generalized forces given in Eq. (4.51), σzz and τzs in Eq. (4.50) can be written as σzz (z, s) =

B0 (z) W0 ψ (s) ≡ σzzW0 (z, s), JW0 z

τzs (z, s) = τzsMz (z, s) + τzsQ 0 (z, s) + τzsQ 1 (z, s),

(4.52) (4.53)

M

where τzs z (z, s) and τzsQ 0 (z, s) were defined in Eqs. (4.28) and (4.29). The new term τzsQ 1 (z, s) is defined as τzsQ 1 =

Q 1 (z) χ1 ψs (s). Jχ1

(4.54)

Compared to Eq. (4.27), the stress in Eq. (4.53) has an additional term, τzsQ 1 , due to the inclusion of the χ1 field. On the other hand, the normal stress remains unaltered, as given by Eq. (4.20). An important observation to make from Eq. (4.53) (along with Eqs. (4.28), / (4.29), and (4.54)) is that the relationship σ α = F α ψ α Jα is valid for any α (α = Mz , Q 0 , Q 1 , . . .). These explicit relationships between generalized forces and stresses will turn out to be critically useful when matching field variables at a joint of multiply connected thin-walled beams, as shall be discussed in Chap. 10.

4.3.2 Derivation of ψ zW1 (Shape Function of the First-Order Warping Mode) The strain field γzs (z, s) corresponding to shear stress τzs in Eq. (4.53) can be written as ] 1 [ Mz τzs (z, s) + τzsQ 0 (z, s) + τzsQ 1 (z, s) G Q 0 (z) χ0 Q 1 (z) χ1 Mz (z) θz ψ (s) + ψ (s) + ψ (s). = G Jθz s G Jχ0 s G Jχ1 s

γzs (z, s) =

(4.55)

102

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

If only 1D generalized displacements up to the first-order distortion, ξ = {θz , χ0 , W0 , χ1 }T , are used, the shear strain γzs (z, s) can be expressed as ∂u z ∂u s + ∂z ∂s θz , = ψs (s)θz (z) + ψsχ0 (s)χ0, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zW0 (s)W0 (z).

γzs (z, s) =

(4.56)

Because the strain field in Eq. (4.55) calculated by the stress field should be identical to the strain field in Eq. (4.56) expressed in terms of the displacement, the following relationship should hold: ψsθz θz, + ψsχ0 χ0, + ψsχ1 χ1, + ψ˙ zW0 W0 =

M z θz Q 0 χ0 Q 1 χ1 ψ + ψ + ψ . (4.57) G Jθz s G Jχ0 s G Jχ1 s

θ χ If the relationship ψ˙ zW0 = c1 ψs z + c2 ψs 0 , which is given in Eq. (4.22), is used, Eq. (4.57) becomes

[ , ] [ ] θz (z) + c1 W0 (z) ψsθz (s) + χ0, (z) + c2 W0 (z) ψsχ0 (s) + χ1, (z)ψsχ1 (s) Mz (z) θz Q 0 (z) χ0 Q 1 (z) χ1 = ψs (s) + ψs (s) + ψ (s). G Jθz G Jχ0 G Jχ1 s (4.58) θ

χ

χ

Because ψs z , ψs 0 , and ψs 1 are independent, the following relationship should hold assuming that the relationship in Eq. (4.58) is valid: θz, + c1 W0 =

Mz , G Jθz

(4.59a)

χ0, + c2 W0 =

Q0 , G Jχ0

(4.59b)

χ1, =

Q1 . G Jχ1

(4.59c)

Among others, Eq. (4.59c) states that the first-order transverse bimoment Q 1 affects only its work conjugate χ1 . However, this statement cannot be valid because a warping mode is shown to be coupled with χ1 due to Poisson’s effect (see Eq. (4.35)). This contradiction indicates that the present 1D generalized displacement field consisting of ξ = {θz , χ0 , W0 , χ1 }T alone cannot generate the desired strain field; thus, the warping mode of the next order (denoted by W1 ) is required. Step 1. Identification of the secondary shear strain field due to the χ1 mode In order to generate γzs (z, s) from a displacement field, which is consistent with γzs (z, s) in Eq. (4.55), we need to consider a new warping mode, denoted here

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

103

by W1 . When the W1 mode is additionally considered, the displacements given in Eq. (4.47) are updated as u s (z, s) = ψsθz (s)θz (z) + ψsχ0 (s)χ0 (z) + ψsχ1 (s)χ1 (z),

(4.60)

u z (z, s) = ψzW0 (s)W0 (z) + ψzW1 (s)W1 (z),

(4.61)

and the corresponding shear strain γzs (z, s) becomes ∂u z ∂u s + ∂z ∂s = ψsθz θz, + ψsχ0 χ0, + ψsχ1 χ1, + ψ˙ zW0 W0 + ψ˙ zW1 W1 .

γzs =

(4.62)

Compared to γzs in Eq. (4.56), ψ˙ zW1 (s)W1 (z) is now newly added to express γzs (z, s) in terms of the displacement. Step 2. Derivation of secondary displacement consistent with the secondary strain To make the strain field in Eq. (4.62) equal to the strain field caused by stress in Eq. (4.55), the following relationship should hold: ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zW0 (s)W0 (z) + ψ˙ zW1 (s)W1 (z) Mz (z) θz Q 0 (z) χ0 Q 1 (z) χ1 = ψ (s) + ψ (s) + ψ (s). (4.63) G Jθz s G Jχ0 s G Jχ1 s θ χ Using ψ˙ zW0 = c1 ψs z + c2 ψs 0 in Eq. (4.22), Eq. (4.63) can be written as

(

) Mz (z) − θz, (z) − c1 W0 (z) ψsθz (s) G Jθ ) ( z Q 0 (z) − χ0, (z) − c2 W0 (z) ψsχ0 (s) + G Jχ0 ( ) Q 1 (z) , + − χ1 (z) ψsχ1 (s). G Jχ1

ψ˙ zW1 (s)W1 (z) =

(4.64)

Equation (4.64) implies that the derivative ψ˙ zW1 (s) = dψzW1 (s)/ds (not the funcθ χ χ tion ψzW1 (s) itself) is a linear combination of ψs z (s), ψs 0 (s), and ψs 1 (s). Because θz χ0 χ1 ψs (s), ψs (s), and ψs (s) are orthogonal to each other, ψ˙ zW1 (s) can be explicitly θ χ χ expressed as a linear combination of ψs z (s), ψs 0 (s), and ψs 1 (s). To derive the explicit θz χ χ relationship, we multiply both sides of Eq. (4.64) by ψs , ψs 0 , and ψs 1 and integrate the resulting expressions over the beam cross-section A: ∗ q11 W1 (z) =

(

) Mz (z) − θz, (z) − c1 W0 (z) , G Jθz

(4.65a)

104

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

(

) Q 0 (z) , = − χ0 (z) − c2 W0 (z) , G Jχ0 ( ) Q 1 (z) ∗ , q13 W1 (z) = − χ1 (z) , G Jχ1

∗ q12 W1 (z)

(4.65b) (4.65c)

where ∗ q11 =

1 Jθz

∗ q12 =

1 Jχ0

∗ q13 =

1 Jχ1

  

ψ˙ zW1 ψsθz dA,

(4.66a)

ψ˙ zW1 ψsχ0 dA,

(4.66b)

ψ˙ zW1 ψsχ1 dA.

(4.66c)

Substituting Eqs. (4.65) into Eq. (4.64) yields ∗ ∗ ∗ ψ˙ zW1 (s) = q11 ψsθz (s) + q12 ψsχ0 (s) + q13 ψsχ1 (s) [ ]  q1∗ ψsθz (s) + D1W1 ψsχ0 (s) + D2W1 ψsχ1 (s) ,

(4.67a)

or, in edgewise form, [ ] ψ˙ zW1 (s j ) = q1∗ ψsθz (s j ) + D1W1 ψsχ0 (s j ) + D2W1 ψsχ1 (s j ) ,

(4.67b)

where ∗ q1∗ = q11 , D1W1 =

∗ ∗ q13 q12 W1 , D = 2 ∗ ∗ . q11 q11

(4.68)

Note that q1∗ is treated only as a scaling parameter. The next step is to integrate Eq. (4.67b) with respect to s j :  [ ] θ χ χ ψs,z j (s j ) + D1W1 ψs,0j (s j ) + D2W1 ψs,1j (s j ) ds j [ ] θ χ χ 1 , = q1∗ ψs,z j (s j ) + D1W1 ψs,0j (s j ) + D2W1 ψs,1j (s j ) + C W j

ψz,Wj1 (s j ) = q1∗

(4.69)

α α 1 where C W j is an integration constant, and ψ p, j (s j ) and ψs, j (s j ) are defined as α ψ p, j (s j ) = ψ p (s j ) ( p = z, s),

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

ψs,α j (s j ) =



105

ψs,α j (s j )ds j (excluding the integration constant),

where α = θz , Wk , χk , and k = 0, 1. The distribution of the shear strain (or shear stress) by Q 1 is anti-symmetric χ because it is identical to the distribution of ψs 1 (s) (see Eq. (4.54)). Therefore, the W1 1 shape function ψz of mode W induced by the shear strain should be also antisymmetric. Based on this observation, the integration constants in Eq. (4.69) can be found using the following anti-symmetry conditions: ψzW1 (s j = s ∗ ) = −ψzW1 (s j = −s ∗ ) ( j = 1, 2, 3, 4),

(4.70a)

ψzW1 (s j ) = ψzW1 (s j+2 ) ( j = 1, 2).

(4.70b)

Note that the anti-symmetricity for the z-directional shape functions in Eq. (4.70a) is given differently from that for the s-directional shape function in Eq. (4.44a). Using θz χ0 1 Eq. (4.70a), one can conclude that C W j = 0 ( j = 1, 2, 3, 4). Because ψs, j , ψs, j , and χ ψs,1j in Eq. (4.69) have the same form for facing (or opposite) edges, Eq. (4.70b) is automatically satisfied. Therefore, we need additional conditions to determine D1W1 and D2W1 . The additional conditions come from the continuity of ψzW1 at the corners of a 1 beam cross-section. Due to the corner continuity, ψz,Wj1 and ψz,Wj+1 defined on adjacent edges should be continuous at their shared corner: ψzW1 (s j = +s j ) = ψzW1 (s j+1 = −s j+1 ) ( j = 1, 2, 3, 4),

(4.71)

( ) where s j denotes the s j coordinates of the ends of the j th edge −s j ≤ s j ≤ s j . The continuity conditions in Eq. (4.71) are defined at four corners of a cross-section, but if the anti-symmetry conditions in Eq. (4.70) are considered, only one independent equation survives from Eq. (4.71). Therefore, the continuity condition will be considered only at one corner of a cross-section; the corner can be selected arbitrarily. As the last condition needed to determine D1W1 and D2W1 , we impose the orthogonality of the shape function ψzW1 with respect to the shape functions of other out-ofplane modes in lower-order sets. Because the zeroth-order warping W0 is the only out-of-plane mode in the lower-order set in the present case, the following condition is used:  ψzW1 ψzW0 ds =

4   j=1

sj

−s j

ψz,Wj1 ψz,Wj0 ds j = 0.

(4.72)

Given that W1 is the out-of-plane deformation, the orthogonality with respect to instances of in-plane deformation such as χ1 is automatically satisfied.

106

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

Using six independent conditions in Eqs. (4.70–4.72), the unknowns in Eq. (4.69) are determined as

D1W1 =

C1W1 = C3W1 = 0,

(4.73a)

C2W1 = C4W1 = 0,

(4.73b)

b+h 250bh 2 (b + h) and D2W1 = . b−h (b − h)(b2 − bh + h 2 )

(4.73c)

Using the results in Eq. (4.73), one can find the sectional shape function of the first-order torsional warping mode W1 as ψz,Wj1 =

50h(b2

ψz,Wj1 =

[ 3 ] 1 20s j + (2b2 − 2bh − 3h 2 )s j ( j = 1, 3), (4.74a) 2 − bh + h )

[ ] 1 −20s 3j + (3b2 + 2bh − 2h 2 )s j ( j = 2, 4), 50h(b2 − bh + h 2 ) (4.74b)

where the scaling parameter q1∗ is set to q1∗ =

2(b − h) , 25bh(b + h)

which normalizes ψzW1 as ψzW1 (s1 = h/2) = 1/50. Figure 4.5b shows the shape of ψzW1 . Step 3. Stress field update due to secondary displacement Before proceeding to this step, we write the displacement, strain, and stress fields due to ξ = {θz , χ0 , χ1 , W1 }T : u s (z, s) = ψsθz (s)θz (z) + ψsχ0 (s)χ0 (z) + ψsχ1 (s)χ1 (z), u z (z, s) = ψzW0 (s)W0 (z) + ψzW1 (s)W1 (z),

(4.75a)

εzz (z, s) = ψzW0 (s)W0, (z) + ψzW1 (s)W1, (z), εss (z, s) = ψ˙ sχ1 (s)χ1 (z), γsz (z, s) = ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zW0 (s)W0 (z) + ψ˙ zW1 (s)W1 (z), σzz (z, s) =

E (ψ W0 (s)W0, (z) + ψzW1 (s)W1, (z) 1 − ν2 z

(4.75b)

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

107

+ ν ψ˙ sχ1 (s)χ1 (z)), E (ψ˙ χ1 (s)χ1 (z) + νψzW0 (s)W0, (z) σss (z, s) = 1 − ν2 s + νψzW1 (s)W1, (z)), τzs (z, s) = G(ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zW0 (s)W0 (z) + ψ˙ zW1 (s)W1 (z)).

(4.75c)

θ χ χ By substituting ψ˙ s 1 = − p1∗ ψzW0 in Eq. (4.36), ψ˙ zW0 = c1 ψs z +c2 ψs 0 in Eq. (4.22), θ χ χ and ψ˙ zW1 = q1∗ (ψs z + D1W1 ψs 0 + D2W1 ψs 1 ) in Eq. (4.67a) into Eq. (4.75c) and following a procedure similar to those used to obtain Eqs. (4.50, 4.51), the stresses can be represented in terms of the generalized forces as

σzz (z, s) ≡ σzzB0 (z, s) + σzzB1 (z, s) B0 (z) W0 B1 (z) W1 ≡ ψz (s) + ψ (s), JW0 JW1 z τzs (z, s) ≡ τzsMz (z, s) + τzsQ 0 (z, s) + τzsQ 1 (z, s) Mz (z) θz Q 0 (z) χ0 Q 1 (z) χ1 ≡ ψs (s) + ψs (s) + ψs (s). Jθz Jχ0 Jχ1

(4.76)

(4.77)

where the bimoment B1 (z) of the first-order torsional warping is defined as  B1 =

σzz ψzW1 dA = E 1 JW1 W1, .

(4.78)

As observed earlier in the previous subsections, the stresses σzz and τzs are expressed as the sum of the decoupled contributions from each of the generalized forces B0 , B1 , Mz , Q 0 , and Q 1 . Because σzzB1 has now appeared as a new component in the normal stress σzz in Eq. (4.76), one can consider the wall-extensional strain ε due to Poisson’s effect, which in turn requires the introduction of an additional distortion mode. The additional distortion mode is referred to as the second distortion mode, χ2 . If the shape function of χ2 is found, one can continue to derive an additional warping mode W2 following the procedure given in Sect. 4.3.2. When these two recursive procedures are completed, the shape functions for the second mode set {χ2 , W2 } can be deterχ mined in closed form. We will derive the shape functions ψs 2 (s) and ψzW2 (s) in the following subsections. Because the procedures used to derive the shape functions for the { χ2 , W2 } modes are identical to those used to derive the shape functions for the { χ1 , W1 } modes, we will not repeat all of the details of the derivations.

108

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

χ

4.3.3 Derivation of ψs 2 The strain εss induced by σzz in Eq. (4.76) due to Poisson’s effect becomes εss (z, s) = −ν

ν B1 (z) W1 σzz (z, s) ν B0 (z) W0 ψz (s) − ψ (s). =− E E JW0 E JW1 z

(4.79)

As indicated by Eq. (4.35), the first term on the right side of Eq. (4.79) is already associated with mode χ1 . Therefore, we need to introduce a new wall-extension mode χ2 only to explain the second term on the right side of Eq. (4.79). If the shape function χ ψs 2 associated with χ2 is considered, the u s displacement can be now written as (see Eq. (4.75)): u s (z, s) = ψsθz (s)θz (z) + ψsχ0 (s)χ0 (z) + ψsχ1 (s)χ1 (z) + ψsχ2 (s)χ2 (z).

(4.80)

Using Eq. (4.80), the wall-extending strain εss can be found via εss (z, s) =

∂u s (z, s) = ψ˙ sχ1 (s)χ1 (z) + ψ˙ sχ2 (s)χ2 (z). ∂s

(4.81)

Equating Eq. (4.79) and Eq. (4.81) yields ψ˙ sχ1 (s)χ1 (z) + ψ˙ sχ2 (s)χ2 (z) = −

ν B0 (z) W0 ν B1 (z) W1 ψz (s) − ψ (s). E JW0 E JW1 z

(4.82)

Substituting the relationship given in Eq. (4.36) into Eq. (4.82) yields ( ) ν B0 (z) ν B1 (z) W1 ∗ χ2 ˙ ψs (s)χ2 (z) = − − p1 χ1 (z) ψzW0 (s) − ψ (s), E JW0 E JW1 z

(4.83)

where p1∗ is defined by Eq. (4.38). After multiplying both sides of Eq. (4.83) by ψzW0 and ψzW1 and integrating them over A, we have ) ( ν B0 (z) ∗ − p1∗ χ1 (z) , (4.84a) p2,1 χ2 (z) = E JW0 ∗ p2,2 χ2 (z) =

ν B1 (z) , E JW1

(4.84b)

∗ ∗ where p2,1 and p2,2 are defined as ∗ p2,1

1 =− JW0



∗ ψ˙ sχ2 ψzW0 dA, p2,2 =−

1 JW1



ψ˙ sχ2 ψzW1 dA.

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

109

( ) In Eq. (4.84), the orthogonality of ψzW0 and ψzW1 A ψzW0 ψzW1 d A = 0 is used. The next step is to rewrite Eq. (4.83) using Eq. (4.84) as [ ] χ ∗ ∗ ψ˙ sχ2 (s) = − p2,1 ψzW0 (s) − p2,2 ψzW1 (s) ≡ p2∗ −ψzW0 (s) − D1 2 ψzW1 (s)

(4.85a)

or equivalently, [ ] χ ψ˙ sχ2 (s j ) = p2∗ −ψzW0 (s j ) − D1 2 ψzW1 (s j ) ( j = 1, 2, 3, 4),

(4.85b)

∗ where p2∗ = p2,1 is a scaling parameter which will be used to adjust the magnitude χ2 χ χ ∗ ∗ / p2,1 , which is a constant to be of ψs . The symbol D1 2 is defined as D1 2 = p2,2 determined. χ Edgewise integration of Eq. (4.85b) yields ψs 2 (s j ) as

[ ] χ χ ψsχ2 (s j ) = p2∗ −ψzW0 (s j ) − D1 2 ψzW1 (s j ) + C j 2 ,

(4.86)

where ψzW0 (s) (or ψzW0 (s j )) and ψzW1 (s) (or ψzW1 (s j )) are defined in Eq. (4.42). χ χ χ χ χ Because Eq. (4.86) involves five unknown constants (D 1 2 , C1 2 , C2 2 , C3 2 , C4 2 ), ∗ when excluding the scaling constant p2 , five conditions are needed to determine χ ψs 2 . From the anti-symmetry conditions of the distortion shape function given by Eq. (4.44), two independent conditions are available. This anti-symmetry condition can be justified because the primary stress σzz in Eq. (4.76) of the χ2 mode is antisymmetric due to the anti-symmetry of ψzW0 (s) and ψzW1 (s) (see Fig. 4.5b and d). χ Because ψz,Wj0 (s j ) and ψz,Wj1 (s j ) are even functions,10 ψs 2 in Eq. (4.86) automatically satisfies Eq. (4.44a). Therefore, only Eq. (4.44b) must be considered, resulting in two independent conditions. Three additional conditions are the following orthogonality conditions:    χ2 θ z χ2 χ0 (4.87) ψs ψs ds = 0, ψs ψs ds = 0, ψsχ2 ψsχ1 ds = 0. Using Eq. (4.44b) and Eq. (4.87), the five unknown constants are determined as follows: χ

bh(b2 − bh + h 2 )(175b2 − 175bh + 175h 2 ) , (6b4 + 8b3 h − 22b2 h 2 + 8bh 3 + 6h 4 ) bh 2 (−40b4 + 40b3 h + 9b2 h 2 − 9bh 3 + 9h 4 ) χ = C3 2 = , 192(3b4 + 4b3 h − 11b2 h 2 + 4bh 3 + 3h 4 ) b2 h(−40h 4 + 40bh 3 + 9b2 h 2 − 9b3 h + 9b4 ) χ . = C4 2 = − 192(3b4 + 4b3 h − 11b2 h 2 + 4bh 3 + 3h 4 )

D1 2 = χ

C1 2 χ

C2 2

(4.88)

ψzW0 (s) and ψzW1 (s) are odd functions because they are anti-symmetric. Therefore, their indefinite integrals, ψzW0 (s) and ψzW1 (s), are even functions.

10

110

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field χ

Figure 4.5c shows the derived ψs 2 in Eq. (4.86) with the constants given by Eq. (4.88). The scaling factor p2∗ is set to p2∗ = − χ

18b4 + 24b3 h − 66b2 h 2 + 24bh 3 + 18h 4 , 25bh 2 (10b4 − 10b3 h + 3b2 h 2 − 3bh 3 + 3h 4 ) χ

which normalizes ψs 2 as ψs 2 (s1 = h/2) = 1/100. Following Step 3 used in the previous derivations, the generalized forces are updated to include the contributions from mode χ2 . We will use the symbol Q 2 to denote the generalized force of mode χ2 . Now we can update the shear stress as τzs (z, s) = τzsMz (z, s) + τzsQ 0 (z, s) + τzsQ 1 (z, s) + τzsQ 2 (z, s),

(4.89)

where the newly introduced term τzsQ 2 (z, s) is related to the generalized force Q 2 (z) as τzsQ 2 (z, s) =

Q 2 (z) χ2 ψs (s). Jχ2

(4.90)

4.3.4 Derivation of ψ zW2 We use Eq. (4.89) to calculate the shear strain γzs : γzs =

M z θz Q 0 χ0 Q 1 χ1 Q 2 χ2 ψ + ψ + ψ + ψ . G Jθz s G Jχ0 s G Jχ1 s G Jχ2 s

(4.91)

Following the argument used to derive ψz,Wj1 , we examine the updated displacement field: u s (z, s) = ψsθz (s)θz (z) + ψsχ0 (s)χ0 (z) + ψsχ1 (s)χ1 (z) + ψsχ2 (s)χ2 (z), u z (z, s) = ψzW0 (s)W0 (z) + ψzW1 (s)W1 (z) + ψzW2 (s)W2 (z),

(4.92) (4.93)

where W2 denotes the new warping mode needed to make γzs as calculated from Eqs. (4.92) and (4.93) consistent with γzs as determined using Eq. (4.91). Using Eqs. (4.92) and (4.93), one can write γzs in terms of the 1D displacement measures {θz , χ0 , χ1 , χ2 , W0 , W1 , W2 } as γzs (z, s) = ψsθz (s)θz, (z) + ψsχ0 (s)χ0, (z) + ψsχ1 (s)χ1, (z)

4.3 Derivation of Higher-Order Deformable Section Modes by Means …

111

+ ψsχ2 (s)χ2, (z) + ψ˙ zW0 (s)W0 (z) + ψ˙ zW1 (s)W1 (z) + ψ˙ zW2 (s)W2 (z). (4.94) Equating Eq. (4.94) to Eq. (4.91) and using Eq. (4.22) and Eq. (4.67a) yield ) Mz − θz, − c1 W0 − q1∗ W1 ψsθz G Jθ ) ( z Q0 + − χ0, − c2 W0 − q1∗ D1W1 W1 ψsχ0 G Jχ0 ) ) ( ( Q2 Q1 , , ∗ W1 χ1 ψsχ2 . − χ1 − χ2 − q1 D2 W1 ψs + + G Jχ1 G Jχ2

ψ˙ zW2 W2 =

(

(4.95)

Employing the procedure used to derive ψzW1 , Eq. (4.95) can be organized as [ ] θ χ χ χ ψ˙ z,Wj2 (s j ) = q2∗ ψs,z j (s j ) + D1W2 ψs,0j (s j ) + D2W2 ψs,1j (s j ) + D3W2 ψs,2j (s j ) , ( j = 1, 2, 3, 4),

(4.96)

∗ where q2∗ = q2,1 is a scaling parameter, and the values of DkW2 (k = 1, 2, 3) are given by

DkW2 =

∗ q2,(k+1) ∗ q2,1

(k = 1, 2, 3),

with   1 1 ∗ ψ˙ zW2 ψsθz dA, q2,2 ψ˙ zW2 ψsχ0 dA, = Jθz Jχ0   1 1 ∗ ψ˙ zW2 ψsχ1 dA and q2,4 ψ˙ zW2 ψsχ2 dA. = = Jχ1 Jχ2

∗ q2,1 =

∗ q2,3

Integrating Eq. (4.96) with respect to s j yields [ θ χ χ ψz,Wj2 (s j ) = q2∗ ψs,z j (s j ) + D1W2 ψs,0j (s j ) + D2W2 ψs,1j (s j ) ] χ 2 +D3W2 ψs,2j (s j ) + C W (s ) . j j To determine the unknown constants

} { 2 CW j

{ j=1,2,3,4

and

DkW2

(4.97) } k=1,2,3

, four

anti-symmetry conditions given by Eq. (4.70), one continuity condition given by Eq. (4.71), and the following two orthogonality conditions are used:

112

4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field



 ψzW2 ψzW0 ds = 0, { } 2 Using these conditions, C W j

j=1,2,3,4

ψzW2 ψzW1 ds = 0. { } and DkW2

k=1,2,3

are found via

2 CW j = 0 ( j = 1, 2, 3, 4),

D1W2 =

(4.98)

D2W2 =

b+h , b−h 250(b2 h 2 +bh 3 ) , (b−h)(b2 −bh+h 2 )

D3W2 =

50bh 2 (10b4 −10b3 h+3b2 h 2 −3bh 3 +3h 4 )(3b5 +7b4 h−7b3 h 2 −7b2 h 3 +7bh 4 +3h 5 ) . (b−h)(b2 −bh+h 2 )(b8 −2b7 h+3b6 h 2 +17b5 h 3 −37b4 h 4 +17b3 h 5 +3b2 h 6 −2bh 7 +h 8 )

(4.99)

(4.100)

One can choose q2∗ appearing in Eq. (4.97) as q2∗ =

2 , 25bh(b + h)

which normalizes ψ zW2 as ( ) 1 h = . ψzW2 s1 = 2 50 The shape of ψzW2 is illustrated in Fig. 4.5d. The next step is to update the z-directional normal stress σzz : σzz (z, s) =

2  k=0

σzzBk (z, s)

2  Bk (z) Wk = ψ (s), JWk z k=0

(4.101)

where the contribution from the new warping function ψzW2 (s) is included. The symbol B2 denotes the bimoment of the second-order warping:  B2 =

σzz ψzW2 dA = E 1 JW2 W2,

It should be noted again that the contribution of each of the generalized forces to σzz is decoupled, as apparent from Eq. (4.101). As mentioned earlier, this decoupling is the consequence of imposing orthogonality among the sectional shape functions of all warping modes. The mode derivation procedure explained above can also be applied recursively to derive the shape functions of higher-order modes {χk , Wk } for 3 ≤ k ≤ N , as suggested in Fig. 4.4. The shape function of the Nth-order distortion mode χ N can be derived to ensure that the resulting strain field due to {χk }k=0,1,2,··,N matches the extensional strain generated by the warping mode of the lower-order modes,

References

113

{Wk }k=0,1,2,···,N −1 . By extending the result given by Eq. (4.82), this relationship can be written as N 

N −1  ν Bk (z)

ψ˙ sχk (s)χk (z) = −

k=0

k=0

E JWk

ψzWk (s).

(4.102) χ

Then, one can follow the procedure presented in Sect. 4.3.2 to derive ψs N using the equation [ χ ψs,Nj (s j )

=

p ∗N

−ψz,Wj0 (s j )

−

N −1 

] χ Dk N

ψz,Wjk (s j )

+

χ Cj N

(N ≥ 3),

(4.103)

k=1 χ

χ

where the unknown constants {Dk N }k=1,...,N −1 and {C j N } j=1,2,3,4 are determined using the anti-symmetry conditions in Eqs. (4.44b) and the following orthogonality conditions:   ψsχ N ψsχk ds = 0 (k = 0, 1, 2, . . . , N − 1). ψsχ N ψsθz ds = 0 and Finally, the shape function ψzW N for warping mode W N can be expressed by extending Eq. (4.97) as [ ψz,WjN (s j )

=

q N∗

θ ψs,z j (s j )

+

χ D1W N ψs,0j (s j )

+

N 

] χ WN Dk+1 ψs,kj (s j )

+

N CW j

(N ≥ 3).

k=1

(4.104) N To determine the unknown constants {DkW N }k=1,...,N +1 and {C W j } j=1,2,3,4 , four anti-symmetry conditions in Eq. (4.70), one continuity condition in Eq. (4.71), and the following orthogonality conditions can be used:

 ψzW N ψzWk ds = 0 (k = 0, 1, . . . , N − 1).

References Bebiano R, Camotim D, Gonçalves R (2018) GBTul 2.0—a second-generation code for the GBTbased buckling and vibration analysis of thin-walled members. Thin-Walled Struct 124:235–257 Carrera E, Giunta G, Petrolo M (2011) Beam structures: classical and advanced theories. Wiley Cesnik CES, Hodges DH (1997) VABS: a new concept for composite rotor blade cross-section modeling. J Am Helicopter Soc 42:27–38

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4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field

Choi S, Kim YY (2019) Consistent higher-order beam theory for thin-walled box beams using recursive analysis: Membrane deformation under doubly symmetric loads. Eng Struct 197:109430 Choi S, Kim YY (2020) Consistent higher-order beam theory for thin-walled box beams using recursive analysis: edge-bending deformation under doubly symmetric loads. Eng Struct 206:110129 Choi S, Kim YY (2021) Higher-order Vlasov torsion theory for thin-walled box beam. Int J Mech Sci 195:106231 Genoese A, Genoese A, Bilotta A, Garcea G (2014) A generalized model for heterogeneous and anisotropic beams including section distortions. Thin-Walled Struct 74:85–103 Kim JH, Kim YY (1999) Analysis of thin-walled closed beams with general quadrilateral cross sections. J Appl Mech 66:904–912 Schardt R (1994) Generalised beam theory—an adequate method for coupled stability problems. Thin-Walled Struct 19(2–4):161–180 Vlasov VZ (1961) Thin-walled elastic beams. Israel Program for Scientific Translations Ltd.

Chapter 5

Sectional Shape Functions for a Box Beam Under Torsion: Wall-Bending Field

χ

In Chap. 4, the sectional shape functions (ψzW , ψs ) for a box beam under torsion, which correspond to the wall-membrane field, were derived. This section is devoted η ηˆ χ to the derivation of the sectional shape functions (ψn , ψn , ψn ) corresponding to the wall-bending field. (See Ferradi and Cespedes (2014), Bebiano et al. (2015), and χ Choi et al. (2017) for earlier developments.) To argue for the co-existence of ψn with χ χ ψs , we observe that if the distortion mode χ has a non-zero ψs (the s-directional displacement component) only (see Fig. 5.1a) without its n-directional counterpart, χ ψn , two adjacent sectional edges cannot remain connected at the corners. Therefore, χ ψn cannot be zero. It was shown in Chap. 2 that the zeroth-order distortion mode χ χ χ0 has a non-zero ψn 0 , as given by Eq. (2.50). If ψn (z, s) does not vanish, it will induce a non-zero u˜ s (z, n, s) for n /= 0, as expressed by Eq. (3.3c) and thus causes χ the bending of cross-sectional walls. The sectional shape functions ψn k for k ≥ 1 χk χ will be derived in Sect. 5.3 identically to how ψs was derived in Chap. 4; ψn k can be obtained as the secondary deformation of the axial stress through Poisson’s effect. χ To argue for the presence of an additional wall-bending field besides ψn k , we note that wall bending can take place on the n−s plane (the plane of the beam cross-section) even when the s-directional displacement vanishes. The corresponding displacements are sketched in Fig. 5.1d and e. They are the shape functions of newly introduced ˆ Because no s-directional displacement is associated with modes denoted by η and η. these modes, the n-directional displacement at every corner vanishes. Accordingly, they will be referred to as constrained distortional modes.1 The difference between mode η and mode ηˆ is that none of the corners experience any rotation for mode η, whereas they do for mode η. ˆ For example, the mode in Fig. 5.1d corresponds to η because its corner rotations are zero, while the mode in Fig. 5.1e does so for ηˆ because its corner rotations are not zero.

1

The term “distortion” is used as long as a section deformation mode has only in-plane deformation on the x−y or n−s plane. The term “constrained” is used if a section deformation mode has no in-plane displacement at any corner of the beam cross-section of a box beam.

© Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_5

115

116

5 Sectional Shape Functions for a Box Beam Under Torsion: … y

u%s

n2

corner 2 Edge 2 s2

corner 1 z

s3

Edge 1 s1

Edge 3 n3

n1

z

Edge 4 n4

s z4

p x

z

z

u%n

wall midline corner 4

corner 3

wall midline

(a)

z

(b)

x

z

(c)

y

y

y

y

x

z

(d)

x

z

x

(e)

Fig. 5.1 a Coordinate systems used to describe the geometry of a thin-walled box beam (wall thickness: t), b the zeroth-order distortion χ0 mode, c the first-order distortion χ1 mode, d the firstorder constrained distortion mode η1 of type 1, and e the first-order constrained distortion mode ηˆ 1 of type 2

Based on the arguments above, the {χ0 , χk , ηk , ηˆ k }k=1, ..., N modes are found to generate non-zero wall-bending fields. It should be emphasized again that the η and ηˆ modes do not undergo wall-membrane (s-directional) deformation, unlike the ηˆ η distortion modes χk . The sectional mode shapes ψn of mode η and ψn of mode ηˆ will be derived in Sect. 5.4. For this derivation, we will follow the derivation procedure used in Chap. 4. In Sect. 5.5, we will consider some box beam examples in which the modes derived in Chaps. 4 and 5 are used.

5.1 General Field Relationship for Higher-Order Deformable Section …

117

5.1 General Field Relationship for Higher-Order Deformable Section Modes of a Wall-Bending Field The n-directional displacement2 at a generic point P on a sectional wall (see Fig. 5.1a) is expressed as u˜ n (z, n, s) = u˜ n (z, s) = ψnθz (s)θz (z) + ψnχ0 (s)χ0 (z) N ( ) Σ ψnχk (s)χk (z) + ψnηk (s)ηk (z) + ψnηˆ k (s)ηˆ k (z) +

(5.1)

k=1

where θz , χ0 , χk , and (ηk and ηˆ k ) denote the 1D generalized displacements of the torsional rotation, the zeroth-order distortion, the higher-order unconstrained distortions, and the constrained distortions, respectively. The sectional shape functions of the torsional rotation mode and the zeroth-order distortion are given in Eqs. (2.26), (2.27), (2.49), and (2.50), and those for higher-order unconstrained and constrained distortions will be derived in the following sections. Because u˜ n given in Eq. (5.1) is uniform in the thickness (n) direction, u˜ n is regarded as a function of only z and s. Employing the kinematic assumption of Kirchhoff’s thin plate theory3 (εnn = γns = γzn = 0), the wall-bending displacements {uz , us } can be expressed as (see Fig. 3.1) uz (z, n, s) = −n˜un, (z, s) [ = −n ψnθz (s)θz, (z) + ψnχ0 (s)χ0, (z) ] N ( ) Σ ηk , ηˆ k , χk , ψn (s)χk (z) + ψn (s)ηk (z) + ψn (s)ηˆ k (z) +

(5.2a)

k=1

us (z, n, s) = −nu˙˜ n (z, s) [ = −n ψ˙ nθz (s)θz (z) + ψ˙ nχ0 (s)χ0 (z)

] N ( ) Σ η χ η ˆ ψ˙ n k (s)χk (z) + ψ˙ n k (s)ηk (z) + ψ˙ n k (s)ηˆ k (z) +

(5.2b)

k=1

Here, up (p = z,(s))represents the thickness-directional variation part of displacement u˜ p . The symbol − is used to denote the quantities associated with the wall-bending field based on Kirchhoff’s thin plate theory. The wall-bending strains {εzz , εss , γ zs } calculated from {uz , us } in Eq. (5.2) can be written as

Three-dimensional displacements at a generic point are denoted as {˜uz , u˜ n , u˜ s }T , whereas those defined on the midline are denoted as {uz , un , us }T . 3 Based on this assumption, the 3D bending displacement is linear in n and proportional to the ( slopes u˜ n, (z, s) and u˙˜ n (z, s)) of the middle line. 2

118

5 Sectional Shape Functions for a Box Beam Under Torsion: …

εzz (z, n, s) = u,z (z, n, s) [

] N ( ) Σ ψnχk χk,, + ψnηk η,,k + ψnηˆ k ηˆ k,, , = −n ψnθz θz,, + ψnχ0 χ0,, + k=1

[ ε ss (z, n, s) = u˙ s (z, n, s) = −n

ψ¨ nχ0 χ0

+

N Σ

(5.3a)

] (ψ¨ nχk χk

+

ψ¨ nηk ηk

+

ψ¨ nηˆ k ηˆ k )

(5.3b)

k=1

γ zs (z, n, s) = u,s (z, n, s) + u˙ z (z, n, s) [ ] N ( ) Σ ˙ η θ , χ , , η ˆ , k z 0 k 2ψ˙ n ηk + 2ψ˙ n ηˆ k ≈ −n 2ψ˙ n θz + 2ψ˜ n · χ0 +

(5.3c)

k=1

In Eq. (5.3b), the term −n ψ¨ nθz θz is omitted because ψ¨ nθz = 0 (see Eq. (2.27)). In ) Σ ( χ Eq. (5.3c), Nk=1 −2nψ˙ n k χk, is omitted because it is very small compared to other χ terms.4 The function ψ˜ n 0 denotes the linearized part of the sectional shape function χ0 ψn , which is given as ψ˜ nχ0 (s1 ) = 2b/(b + h)s1 , ψ˜ nχ0 (s2 ) = −2h/(b + h)s2 , ψ˜ nχ0 (s3 ) = 2b/(b + h)s3 , ψ˜ nχ0 (s4 ) = −2h/(b + h)s4 ,

(5.4)

χ χ χ Note that ψn 0 as ψ˜ n 0 can be approximated because ψ˙˜ n 0 of the linearized distortion η ηˆ is always orthogonal to ψ˙ n k and ψ˙ n k (see Eq. (5.71b) for example). This approximation makes it simple to find orthogonal wall-bending shape functions. The linearized χ χ χ ψ˜ n 0 for ψn 0 used here can be justified because the shear strain due to ψ˜ n 0 is nearly χ0 identical to the shear strain due to ψn . Using the strains {εzz , εss , γ zs } in Eqs. (5.3) and the plane stress assumption, one can express {σ zz , σ ss , τ zs } as

[ σ zz (z, n, s) = E1 (εzz + νε ss ) = E1 (−n) ψnθz θz,, + ψnχ0 χ0,, + ν ψ¨ nχ0 χ0 ] N ( ) Σ ηk ,, η ηˆ k ,, χ η ˆ χk ,, k k ψn χk + ψn ηk + ψn ηˆ k + ν ψ¨ n χk + ν ψ¨ n k ηk + ν ψ¨ n ηˆ k + k=1

(5.5a)

Because {χk }k=1,...,N accompanies wall stretch or shrinkage, the corresponding deformations require higher strain energy levels than the deformations associated with other sectional modes. Therefore, the magnitudes of {χk }k=1,...,N are smaller than those of other modes. This can be confirmed in the results in Fig. 5.11e, where the difference from HoBT (N1 = 4, N2 = 0) and VTT ( ) Σ ˙ χk , (Vlasov’s thin-walled theory) is N k=1 −2nψn χk .

4

5.1 General Field Relationship for Higher-Order Deformable Section …

119

[ σ ss (z, n, s) = E1 (ε ss + νε zz ) = E1 (−n) ψ¨ nχ0 χ0 + νψnθz θz,, + νψnχ0 χ0,, ] N ( ) Σ χ η ˆ χ ,, ,, η ˆ ,, η η k k k k ψ¨ n χk + ψ¨ n k ηk + ψ¨ n ηˆ k + νψn χk + νψn k ηk + νψn ηˆ k + k=1

[

N ( Σ 2ψ˙ nηk η,k + 2ψ˙ nηˆ k ηˆ k, τ zs (z, n, s) = Gγ zs = G(−n) 2ψ˙ nθz θz, + 2ψ˙˜ nχ0 χ0, +

(5.5b) ] )

k=1

(5.5c) To define the generalized forces for a wall-bending field, the internal virtual work in a box beam (z1 ≤ z ≤ z2 ) associated with the aforementioned strain and stress fields is considered: ∫z2 {∫ δU =

} (σ zz δε zz + σ ss δε ss + τ zs δγ zs ) dA dz

z1

Substituting Eqs. (5.3) into Eq. (5.6) gives ∫z2 {∫

} (σ zz δε zz + σ ss δε ss + τ zs δγ zs ) dA dz

z1

[∫

∫ θz ˙ = τ zs (−2n ψn ) dA δθz + τ zs (−2n ψ˙˜ nχ0 ) dA δχ0 ]z2 ∫ ∫ θz , χ0 , + σ zz (−n ψn ) dA δθz + σ zz (−n ψn ) dA δχ0 +

N [∫ Σ

τ zs (−2n ψ˙ nηk ) dA δηk +

k=1

∫ + ∫ +

∫

σ zz (−n ψnχk ) dA δχk, σ zz (−n ψnηˆk ) dA δ ηˆ k,

∫z2 [∫ +

+ ]z2

∫

z1

τ zs (−2n ψ˙ nηˆk ) dA δ ηˆ k

σ zz (−n ψnηk ) dA δη,k

z1

− ∂τ∂zzs (−2n ψ˙ nθz ) dA δθz

z1

∫

+ +

σ ss (−n ψ¨ nχ0 ) −

N ∫ Σ k=1

∂τ zs (−2n ψ˙˜ nχ0 ) dA δχ0 ∂z

σ ss (−n ψ¨ nηk ) −

+ ···

∂τ zs (−2n ψ˙ nηk ) dA δηk ∂z

(5.6)

120

5 Sectional Shape Functions for a Box Beam Under Torsion: …

∫ + ··· +

] − ∂σ∂zzz (−n ψnηˆk ) dA δ ηˆ k, dz

(5.7)

where integration by parts is used. An examination of the second to fourth lines of Eq. (5.7) allows us to define the following 1D generalized displacement U and force F: }T { } { U = θz , χ0 , θz, , χ0, , ηk , ηˆ k , χk, , η,k , ηˆ k, k=1,...,N

(5.8a)

F = {M z , Q0 , Pz , S0 , {Rk , Rˆ k , Sk , T k , Tˆ k }k=1,...,N }T

(5.8b)

{ } In Eq. (5.8a), the derivatives θz, , χ0, , η,k , ηˆ k, in addition to {θz , χ0 , ηk , ηˆ k } are included in U because the wall-bending field is modeled using Kirchhoff’s assumption, as indicated in Eq. (5.2). Here, the relationship between mode α (α = θz , χ0 , χk , ηk , ηˆ k ) and its derivative α , is analogous to that between the vertical displacement and the bending rotation in the classical Euler beam theory. Thus, the derivatives in U represent the bending rotations of cross-sectional walls with respect to the contour coordinate (s), contributing to displacement uz , as expressed in Eq. (5.2a). Meanwhile, the 1D generalized forces F in Eq. (5.8b) are identified as the work conjugates of U. Each component of F is defined as shown below. ∫ ∫ M z = τ zs (−2n · ψ˙ nθz ) dA; Q0 = τ zs (−2n · ψ˙˜ nχ0 ) dA; ∫ Pz = σ zz (−n · ψnθz ) dA ∫ ∫ S0 = σ zz (−n · ψnχ0 ) dA; Rk = τ zs (−2n · ψ˙ nηk ) dA; ∫ ˆRk = τ zs (−2n · ψ˙ nηˆ k ) dA ∫ ∫ Sk = σ zz (−n · ψnχk ) dA; T k = σ zz (−n · ψnηk ) dA; ∫ ˆ Tk = σ zz (−n · ψnηˆ k ) dA (k = 1, 2, . . . , N )

(5.9)

In Eq. (5.9), M z represents the twisting moment induced by τ zs . Note that M z can be regarded as a higher-order version of Mz in Eq. (3.14) because it can be shown that Mz ≫ M z . Likewise, Q0 also represents a higher-order term of Q0 defined in Eq. (3.14). The symbols {Rk , Rˆ k } represent the work conjugates of the 1D ˆ generalized displacements {ηk , ηˆ k }, { while {Pz , S0 , S}k , T k , Tk } represent those of the 1D generalized displacements θz, , χ0, , χk, , η,k , ηˆ k, consisting of the derivatives of the displacements. In Eq. (5.9), all terms in F except M z are self-equilibrated forces.

5.2 Generalized Force-Stress Relationship

121

5.2 Generalized Force-Stress Relationship At this stage, we will derive generalized force-stress relationships for a wall-bending field. To this end, first we note that the stress-displacement relationship in Eq. (5.5) is described only in terms of ψn (n-directional displacement) because Kirchhoff’s assumption is employed in Eq. (5.2). On the other hand, the stress-displacement relationships in Eq. (4.11) for the wall-membrane field are expressed in terms of ψs and ψz (and their derivatives). Based on this observation, one may consider the following condition to derive ψnα (α = χ , etc.): ∫

ψnα1 ψnα2 ds = 0 (α1 /= α2 )

(5.10)

However, it can be shown that preferred explicit / forms of the generalized forcestress relationships, such as τ αzs = F α (−2nψ˙ α ) Jα 5 (F α : the generalized force of mode α), cannot be obtained if the condition given by (5.10) is used. If such explicit relationships are not available, it is difficult or impossible to match generalized force equilibriums at the joint of a multiply connected box beam in closed form. (This issue was addressed in Sect. 5 of Choi (2016) and will be discussed in Chap. 10 of this book). To establish explicit stress-generalized force relationships, we propose to η ηˆ impose the following forms of orthogonality conditions for ψn N and ψn N belonging to the N-th mode set: ∫ ∫ ∫ ∫ ψ˙ nθz ψ˙ nηN ds = ψ˙˜ nχ0 ψ˙ nηN ds = ψ˙ nηk ψ˙ nηN ds = ψ˙ nηˆk ψ˙ nηN ds = 0, (5.11a) ∫

ψ˙ nθz ψ˙ nηˆ N ds = =

∫ ∫

ψ˙˜ nχ0 ψ˙ nηˆ N ds =

∫

ψ˙ nηk ψ˙ nηˆ N ds =

∫

ψ˙ nηˆ k ψ˙ nηˆ N ds

ψ˙ nηN ψ˙ nηˆ N ds = 0, (1 ≤ k ≤ N − 1)

(5.11b)

η χ Equation (5.11a) states that ψ˙ n N is orthogonal to {ψ˙ nθz , ψ˙˜ n 0 } of the zeroth-order η η ˆ modes and also to {ψ˙ n k , ψ˙ n k }1≤k≤N −1 belonging to the lower-order sets of the wallηˆ bending modes. Similarly, Eq. (5.11b) represents the orthogonality of ψ˙ n N with η η χ η ˆ respect to {ψ˙ nθz , ψ˙˜ n 0 } and {ψ˙ n k , ψ˙ n k } of the lower-order sets as well as ψ˙ n N of the current set. Substituting the stress-displacement relationship in Eq. (5.5c) into {M z , Q0 , Rk , Rˆ k } in Eq. (5.9) and using the orthogonality conditions in Eqs. (5.11), one can find

M z = GIθz θz, , Q0 = GIχ0 χ0, , Rk = GIηk η,k , Rˆ k = GIηˆ k · ηˆ k, . 5

This relationship will be derived later; see Eqs. (5.14) and (5.15).

(5.12)

122

5 Sectional Shape Functions for a Box Beam Under Torsion: …

In these equations, ∫

∫ (−2n · ψ˙ nθz )2 dA , Iχ0 = (−2n · ψ˙˜ nχ0 )2 dA, ∫ ∫ = (−2n ψ˙ nηk )2 dA, Iηˆ k = (−2n ψ˙ nηˆk )2 dA.

I θz = Iη k

(5.13)

Physically, Iθz , Iχ0 , Iηk , and Iηˆ k denote the second ∫moments of inertia for modes θz , χ0 , ηk , and ηˆ k , respectively. Note that unlike Jα = ( ψ α )2 dA(α = θz , Wk, χk ; k = 0, 1, 2, . . .) defined in Chap. 4 for a wall-membrane field, Iα (α = θz , χk ; k = 0, 1, 2, . . .) appearing in Eq. (5.13) involve n terms in their definition because wall bending is considered. Finally, the generalized force-stress relationship between τ zs and {M z , Q0 , Rk , Rˆ k } can be obtained by substituting the generalized force– displacement relationship in Eq. (5.12) into the stress-displacement relationship in Eq. (5.5c): N [ ] ] Σ [ Rˆ k Q0 Rk z + τ zs = τ M τ τ τ + + zs , zs zs zs

(5.14)

k=1

where Q0 Mz 0 (−2n ψ˙˜ nχ0 ), (−2n ψ˙ nθz ), τ Q zs = Iχ0 I θz Rk Rˆ k ˆ = (−2n ψ˙ nηk ), τ Rzsk = (−2n ψ˙ nηˆk ). Iη k Iηˆ k

z τM zs =

τ Rzsk

(5.15)

5.3 Derivation of Sectional Shape Functions ψnxk of Unconstrained Distortion Mode χk The sectional shape functions of the unconstrained distortion mode χk and constrained distortion modes {ηk , ηˆ k } will be derived in this and the next section, respectively. For both cases, we will employ the three-step procedure used when deriving the sectional shape functions for the wall-membrane field above. This procedure is reiterated here: Step 1: Identification of the secondary strain field due to a known (primary) stress field. Step 2: Derivation of secondary displacement consistent with the secondary strain. Step 3: Stress field update due to secondary displacement (applies only to constrained distortion modes).

x

5.3 Derivation of Sectional Shape Functions ψn k of Unconstrained …

123

χ

We will now explain how the sectional shape function ψn k for distortion mode χk can be derived using the three steps mentioned above. First, we consider the normal stress σzzW0 of the zeroth-order warping mode W0 for a point on the wall midline (n = 0): σzzW0 (z, s) =

B0 (z) W0 ψ (s) JW0 z

(5.16)

which was given in Chap. 4 as Eq. (4.20). Substituting ψzW0 (s) = x(s)y(s) (see Eq. (2.40)) into Eq. (5.16) yields σzzW0 (z, s) =

B0 (z) x(s)y(s), JW0

(5.17)

where x and y represent the x and y coordinates of the wall midline. The symbol B0 (z) is the generalized force, which is the work conjugate of W0 (z). Extending the result in Eq. (5.17) for a generic point P that is away from the midline by distance n in the thickness direction (see Fig. 5.1a), the normal stress σzzW0 at P (z, n, s) can be written as: B0 (z) x(n, s)y(n, s) JW0 ] B0 (z) [ W0 ψz (s) + ψzW 0 (n, s)  JW0 B0 (z) W 0 B0 (z) W0 ψ (s) + ψ (n, s) = JW0 z JW0 z

σzzW0 (z, n, s) =

 σzzm (z, s) + σ zz (z, n, s)

(5.18)

where σzzm (z, s) and σ zz (z, n, s), which can be called the wall membrane and wallbending stress, respectively, are defined as σzzm (z, s) =

B0 (z) W0 B0 (z) W 0 ψz (s), σ zz (z, n, s) = ψ (n, s) JW0 JW0 z

(5.19a, b)

To obtain the second expression in Eq. (5.18), we used the fact that for any given edge, either x(n, s) or y(n, s) depends only on s, while the other depends only on n. For instance, x(n, s) = b/2 + n1 and y(n, s) = s1 for edge 1, where b is the width of the box beam in Fig. 5.1a. The symbol W 0 in Eq. (5.18) denotes the contribution from the wall-level (or edge-level) warping, i.e., the warping of the cross-sectional wall behaving as a thin plate. The explicit form of shape function ψzW 0 can be expressed as ψzW 0 (n, s) = nφ(s),

(5.20)

124

5 Sectional Shape Functions for a Box Beam Under Torsion: …

Fig. 5.2 Warping of ψzW 0 (n, s) on edge 1

t edge 2

edge 1 edge 3 edge 4

ψ Wz

ψ Wz

0

0

on edge 1

where the function φ(s) is defined as φ(s1 ) = s1 ; φ(s2 ) = −s2 ; φ(s3 ) = s3 ; φ(s4 ) = −s4

(5.21)

Figure 5.2 illustrates ψzW 0 (s) on edge 1 as an example.

χ

5.3.1 Derivation of ψn k χ

In this section, we will derive ψn k using the following steps: Step 1. Identification of the secondary normal strain field due to W 0 As was explained in Sect. 4.2.1, the first term in Eq. (5.18), σzzm = χ

B0 JW0

ψzW0 , induces

the wall-membrane deformation ψs 1 of χ1 . By the same reasoning, one can argue that the second term in Eq. (5.18), σ zz = JBW0 ψzW 0 = JBW0 nφ, induces the wall-bending 0

χ

0

χ

deformation ψn 1 of χ1 . To derive the explicit expression of ψn 1 , we examine ε ss that is induced as the secondary strain by σzz = σ zz due to Poisson’s effect: εss (z, n, s) = −ν

νB0 (z) σ zz (z, n, s) =− nφ(s) E EJW0

(5.22)

Equation (5.22) shows that the normal strain εss varies linearly in the thickness direction n, yielding zero strain on the wall midline (n = 0). Step 2. Derivation of secondary displacement consistent with the secondary strain The examination of the ε ss distribution given in Eq. (5.22) shows that extension occurs in the contour (s) direction over half of the thickness of the wall for 0 ≤ n ≤ t/2, while compression occurs for −t/2 ≤ n ≤ 0, or vice versa. This leads to in-plane

x

5.3 Derivation of Sectional Shape Functions ψn k of Unconstrained …

125

bending of the cross-sectional wall about the z axis. This secondary in-plane wallbending deformation can be depicted if the following s-directional displacement by the first-order distortion mode, χ1 , is introduced: us (z, n, s) = −n ψ˙ nχ1 (s)χ1 (z)

(5.23)

χ Here, u˜ n (z, s) = ψn 1 (s)χ1 (z) and us (z, n, s) = −nu˙˜ n (z, s) are used (see Eq. (5.1) and Eq. (5.2b)). One can now express ε ss in terms of the displacement given in Eq. (5.23) as

εss (z, n, s) =

∂us (z, n, s) = −n ψ¨ nχ1 (s)χ1 (z) ∂s

(5.24)

Equating Eq. (5.22) and Eq. (5.24) yields −nψ¨ nχ1 (s)χ1 (z) = −ν

B0 (z) nφ(s) EJW0

(5.25)

Equation (5.25) can be satisfied if the following relationship holds χ ψ¨ n 1 (s) νB0 (z) = ≡ p1∗ EJW0 χ1 (z) φ(s)

(5.26)

Note that the value of p1∗ appearing in Eq. (5.26) is identical to that appearing in Eq. (4.36) because νB0 /EJW0 χ1 appearing in Eq. (5.26) is identical to that appearing χ in Eq. (4.37). Therefore, the p1∗ value used to scale ψs 1 (see Eq. (4.38)) should also χ be used here. From Eq. (5.26), the relationship between ψn 1 and φ can be found: ψ¨ nχ1 (s) = p1∗ φ(s)

(5.27)

The next procedure is to integrate Eq. (5.27) twice to obtain the explicit formula χ for ψn 1 (s) and to determine the integration constants. Before the integration is performed, it will be shown that a relationship similar to Eq. (5.27) also holds for χ the sectional shape functions ψn k (s) belonging to higher-order distortional modes χk (k = 2, 3, . . . , N ). Because the process of integration is the same for the sectional χ shape functions ψn k (s) for the any-order distortional mode χk (k = 1, 2, . . . , N ), we will present a unified method applied to the any-order distortional mode. To show that a relationship similar to Eq. (5.27) holds for higher-order shape χ functions ψn k (s)(k = 2, 3, . . . , N ), we begin with the simplest case of N = 2. Extending Eq. (5.23) for the case of N = 2, the wall-bending deformation can be written as us (z, n, s) =

2 Σ k=1

[−n ψ˙ nχk (s)χk (z)]

(5.28)

126

5 Sectional Shape Functions for a Box Beam Under Torsion: …

Likewise, Eq. (5.25) can be extended as 2 Σ

[−n ψ¨ nχk (s)χk (z)] = −ν

k=1

B0 (z) nφ(s) EJW0

(5.29)

It appears that a term related to B1 (the first-order bimoment) should have appeared on the right-hand side of Eq. (5.29) for N = 2. However, there is no such term because B1 does not produce a non-zero σ zz (thickness-directional variation of σzz ). In fact, no higher-order bimoments Bk (k ≥ 1) produce a non-zero σ zz , and thus none of the Bk -related terms appears on the right-hand side of Eq. (5.29) for any instance for which k ≥ 1. χ Substituting ψn 1 in Eq. (5.27) into Eq. (5.29) yields φ(s)p1∗ χ1 (z) + ψ¨ nχ2 (s)χ2 (z) = ν

B0 (z) φ(s) EJW0

Equivalently, ( ) B0 (z) ∗ χ2 ν − p χ (z) 1 ¨ 1 EJW0 ψn (s) = φ(s) χ2 (z)

(5.30)

∗ If Eq. (4.84a) is substituted into Eq. (5.30) (with p2∗ = p2,1 ), it reduces to

ψ¨ nχ2 (s) = p2∗ φ(s)

(5.31)

Based on the earlier observation that none of the Bk -related terms (k ≥ 1) appears χ on the right side of Eq. (5.29),ψn k for any k can be related to φ(s) as ψ¨ nχk (s) = pk∗ φ(s) (k ≥ 1),

(5.32a)

ψ¨ nχk (sj ) = pk∗ φ(sj ) (j = 1, 2, 3, 4; k ≥ 1)

(5.32b)

or in an edgewise manner,

Note that the value of pk∗ appearing in Eq. (5.32) is identical to that used to scale (see Eq. (4.103)). χ If Eq. (5.32b) is integrated, ψn k can be expressed as

χ ψs k (s)

χ

χ

k k 0 ψnχk (sj ) = pk∗ [ΦW z (sj ) + Cj,1 sj + Cj,0 ] (j = 1, 2, 3, 4; k ≥ 1)

χ

χ

(5.33)

0 where {Cj,0k , Cj,1k } are the integration constants and ΦW z (sj ) is defined as shown below:

x

5.3 Derivation of Sectional Shape Functions ψn k of Unconstrained …

( ) 0 sj = ΦW z

¨

( ) φ sj dsj dsj (excluding integration constants).

127

(5.34)

If φ(sj ) in Eq. (5.21) is substituted into Eq. (5.34), one can find W0 W0 W0 1 3 1 3 1 3 1 3 0 ΦW z (s1 ) = 6 s1 ; Φz (s2 ) = − 6 s2 ; Φz (s3 ) = 6 s3 ; Φz (s4 ) = − 6 s4 (5.35) χ

χ

To determine the eight integration constants {Cj,0k , Cj,1k } (j = 1, 2, 3, 4) appearing in Eq. (5.33), we use the following conditions: ψnχk (sj = s∗ ) = −ψnχk (sj = −s∗ ) (j = 1, 2, 3, 4)

(5.36a)

ψnχk (sj = s∗ ) = −ψnχk (sj+2 = −s∗ ) (j = 1, 2)

(5.36b)

ψnχk (sj = +sj ) = − ψsχk (sj+1 = −sj+1 ), ψnχk (sj+1 = −sj+1 ) = ψsχk (sj = +sj ) (j = 1, 2, 3, 4)

(5.36c)

where sj denotes the bound values of sj such that −sj ≤ sj ≤ sj . Equations (5.36a, χ 5.36b) represent the anti-symmetric conditions of ψn k about the x and y axes, respec6 tively. Equation (5.36c) imposes the condition of the displacements of the crosssectional corners being continuous at a corner shared by two adjacent edges. Because χ χ mode χk has both s- and n-directional displacements (ψs k and ψn k , respectively), χk ψs must be considered in Eq. (5.36c) to satisfy the continuity rule. Note that only eight conditions are independent among all equations given by (5.36). This means that if conditions (5.36a, 5.36b) are used, Eq. (5.36c) can yield only two additional conditions and thus one can use Eq. (5.36c) only with j = 1. Using these equations, it becomes possible to find the following integration constants: χ

χ

2

h − C1,1k = C3,1k = − 24 χ

χ

C2,1k = C4,1k = χ

b2 24

−

χ

2 ψ χk pk∗ h s

(s2 = −b/2),

(5.37a)

2 ψ χk pk∗ b s

(s1 = h/2),

(5.37b)

χ

χ

C1,0k = C2,0k = C3,0k = C4,0k = 0

(5.37c) χ

Using Eqs. (5.33) and (5.37), one can explicitly obtain ψn k as χ

ψn,1k = pk∗

[

1 3 s 6 1

{ 2 h + − 24 −

χ

χ

2 pk∗ h

] ( )} · ψsχk s2 = − 2b · s1 ,

(5.38a)

In Chap. 4, the anti-symmetry of ψs 1 and ψs 2 is shown by using the anti-symmetry of their primary χ stress variables, σzB0 and σzB1 . The anti-symmetry of ψs k (k ≥ 3) can also be shown similarly using Bk−1 the anti-symmetry of the primary stress variable σz (e.g., see Eq. (4.101)).

6

128

5 Sectional Shape Functions for a Box Beam Under Torsion: …

[ { 2 χ b ψn,2k = pk∗ − 16 s23 + 24 − χ

ψn,3k = pk∗

[

1 3 s 6 3

{ 2 h + − 24 −

2 pk∗ h

[ { 2 χ b − ψn,4k = pk∗ − 16 s43 + 24

] ( )} · ψsχk s1 = 2h · s2 ,

(5.38b)

] ( )} · ψsχk s2 = − 2b · s3 ,

(5.38c)

] ( )} · ψsχk s1 = 2h · s4

(5.38d)

2 pk∗ b

2 pk∗ b

χ

χ

Using the values of ψs k (s1 = h/2) and ψs k (s2 = −b/2), which can be found χ χ from the function ψs k determined in Chap. 4, ψn,jk (j = 1, 2, 3, 4) can be explicitly χ1 obtained. For example, ψn is given as χ

ψn,11 = p1∗

[

1 3 s 6 1

−

(

b2 12

+

h2 24

[ ( 2 χ b ψn,21 = p1∗ − 16 s23 + 24 + χ

ψn,31 = p1∗

[

1 3 s 6 3

−

(

b2 12

+

)

h2 12

h2 24

[ ( 2 χ b + ψn,41 = p1∗ − 16 s43 + 24

] · s1 , )

)

h2 12

· s2

· s3

)

(5.39a)

] (5.39b)

]

· s4

(5.39c) ] (5.39d)

Note that the strain–stress relationship in Eq. (5.22) and the resulting relationship in Eq. (5.32) are not recursive; all expressions involve only W 0 -related terms. This χ means that ψn k for any k can be explicitly obtained without an updated stress field. χ Equation (5.38) also shows that ψn k always remains as a cubic function7 for any χ value of k. Therefore, the derivation of ψn k requires only Step 1 and Step 2, not Step 3.

5.3.2 Relationship Between the Generalized Force (Sk ) and Stress (σ zz ) In this section, the relationship8 between the generalized force Sk and the stress σ zz will be derived. To establish the Sk − σ zz relationship, we observe from Eq. (5.5a) that σ zz |χ1 ∼χN , the part of σ zz induced by χk (k = 1, 2, . . . , N ), can be identified as σ zz (z, n, s)|χ1 ∼χN = E1 (−n) ·

N Σ [ χk ] ψn (s)χk,, (z) + ν ψ¨ nχk (s)χk (z)

(5.40)

k=1

7

In general, the polynomial order increases as k increases in other sectional shape functions. η ηˆ In the next section, it will be shown that the derivation of the sectional shape functions {ψn k , ψn k } of modes ηk and ηˆ k also requires this relationship.

8

x

5.3 Derivation of Sectional Shape Functions ψn k of Unconstrained …

129

Using Eq. (5.32), Eq. (5.40) can be written as σ zz (z, n, s)|χ1 ∼χN = −E1 n

N Σ {

[ ]} ψnχk (s)χk,, (z) + φ(s) vpk∗ χk (z)

(5.41)

k=1

∫ χ Because Sk is defined as Sk = σ zz (−n · ψn k )dA(k = 1, 2, . . . , N ), Sk can be χ found by integrating Eq. (5.41) with a weight function (−n · ψn k (s)) over the beam χk cross-sectional area A. Given that ψn (s) for any k is expressed as a cubic function9 χ χ χ of s (for each edge), the {ψn 1 (s), ψn 2 (s), . . . , ψn N } terms are not independent of each other. χ Examining the functional form of ψn k (s)(see Footnote 8), one can express any of χk the ψn (s) values (k = 1, 2, . . . , N ) in terms of the following three basis functions: basis functions: φ(s), ψnχ (s), and ψnχˆ (s), χ

χˆ

where {ψn (s), ψn (s)} are the basis functions to be identified. To determine χ χˆ χ {ψn (s), ψn (s)}, first we observe that ψn k (s) for any k given by Eq. (5.38) (or Eq. (5.39)) can always be written as χ

χ

ψn,1k = ψn,3k = a1 (k)s3 + a2 (k)s (k = 1, 2, . . . , N ), χ

(5.42a)

χ

ψn,2k = ψn,4k = −a1 (k)s3 + a3 (k)s (k = 1, 2, . . . , N ).

(5.42b)

In Eq. (5.42), {a1 (k), a2 (k), a3 (k)} are known constants, but they can differ depending on the value of k. Note that φ(s) is given by ( ) φ(s) : φ1 = s, φ2 = −s, φ3 = s, and φ4 = −s, with φj  φ sj . χ

(5.43)

χˆ

If the basis functions {ψn (s), ψn (s)} are selected as, for instance, χ

χ

χ

χ

ψnχ : ψn,1 = ψn,3 = s3 and ψn,2 = ψn,4 = −s3 , χˆ

χˆ

χˆ

(5.44a)

χˆ

ψnχˆ : ψn,1 = ψn,2 = ψn,3 = ψn,4 = s χ

(5.44b) χ

χˆ

ψn k (s) for any k can be expressed by a linear combination of {φ, ψn , ψn } as: ψnχk (s) = a1 (k)ψnχ (s) + 21 [a2 (k) + a3 (k)]ψnχˆ + 21 [a2 (k) − a3 (k)]φ(s)

(5.45)

∫ χ However, the integration needed when calculating Sk = σ zz (−n · ψn k )dA χ χˆ becomes simpler if the selected {φ, ψn , ψn } values are mutually orthogonal to each 9

The cubic function considered here is composed of two terms, s3 and s, for each edge.

130

5 Sectional Shape Functions for a Box Beam Under Torsion: … χ

χˆ

other. Thus, we will modify {ψn , ψn } accordingly while keeping φ unchanged. Note for brevity that the resulting functions that are mutually orthogonal will be denoted χ χˆ χ χˆ by the same {ψn , ψn }. To this end, we add linear and cubic terms to {ψn , ψn } given in Eq. (5.44), respectively, to define the orthogonal basis functions: χ

χ

χ

χ

χˆ

χˆ

χˆ

χˆ

ψn,1 = ψn,3 = s3 + m1 s; ψn,2 = ψn,4 = −s3 + n1 s, ψn,1 = ψn,3 = s3 + m2 s; ψn,2 = ψn,4 = −s3 + n2 s

(5.46a) (5.46b)

where the m, s and n, s are constants. Note that s in Eq. (5.46) implies sj for edge j (j = 1, 2, 3, 4). The constants m, s and n, s should be so determined as to satisfy the following orthogonality relationships: ∫

φ · ψnχ ds = 0;

∫

φ · ψnχˆ ds = 0;

∫

ψnχ · ψnχˆ ds = 0

(5.47)

To determine the four constants (m1 , m2 , n1 , and n2 ), an additional condition, which may be quite arbitrary, is also needed. Here, we choose the following additional condition: m2 = 1

(5.48)

Substituting Eq. (5.46) into Eqs. (5.47) and (5.48) yields m1 = −

420h5 (b3 + h3 ) + 12b10 + 75b3 h7 + 63h10 , 140h3 (3h2 + 20)(b3 + h3 )

(5.49)

n1 =

h3 m b3 1

+

3(b5 +h5 ) 20b3

(5.50)

n2 =

h3 m b3 2

+

3(b5 +h5 ) 20b3

(5.51) χ

χˆ

χ

In terms of the mutually orthogonal basis functions {ψn , ψn , φ} of ψn k (k = χ 1, 2, . . . , N ) that are defined in Eqs. (5.46) and (5.43), ψn k can be now expressed as ψnχk (s) = Ak φ(s) + Bk ψnχ (s) + Ck ψnχˆ (s) (k = 1, 2, . . . , N ),

(5.52)

where {Ak , Bk , Ck } are found via ∫ Ak = ∫ Ck =

φψnχk

/∫

ψnχ ψnχk

/∫

ds φ ds, Bk = (ψnχ )2 ds, /∫ ds (ψnχˆ )2 ds, (k = 1, 2, . . . , N ).

ds

ψnχˆ ψnχk

∫

2

(5.53)

x

5.3 Derivation of Sectional Shape Functions ψn k of Unconstrained …

131

χ

If ψn k (s) is expressed as Eq. (5.52), σ zz (z, n, s)|χ1 ∼χN in Eq. (5.41) can be rewritten as shown below: N Σ

σ zz (z, n, s)|χ1 ∼χN = −E1 n +

{φ(s)[νpk∗ χk (z) + Ak χk,, (z)]

k=1 χ ψn (s)[Bk χk,, (z)]

{

= − E1 n φ(s)

N Σ

+ ψnχˆ (s)[Ck χk,, (z)]}

[νpk∗ χk (z) + Ak χk,, (z)]

k=1

+ψnχ (s)

N Σ

N Σ

Bk χk,, (z)+ψnχˆ (s)

k=1

} Ck χk,, (z)

.

(5.54)

k=1

Because σ zz (z, n, s)|χ1 ∼χN in Eq. (5.54) is expressed in terms of the orthogonal χ χˆ basis functions {ψn , ψn , φ}, the generalized force-stress relationships can be now written in a simpler, decoupled form. To this end, first we write S1 (z) defined in Eq. (5.9) as ∫ S1 (z) =

(

σ zz −n ·

ψnχ1

)

∫ dA =

( ) · −n · ψnχ1 (s) dA | ∫ | = A1 σ zz (z, n, s)||

χ1 ∼χN

| | σ zz (z, n, s)||

χ1 ∼χN

· (−n · φ)dA

| ∫ | ) ( + B1 σ zz (z, n, s)|| · −n · ψnχ dA χ ∼χ |1 N ( ∫ ) | + C1 σ zz (z, n, s)|| · −n · ψnxˆ dA χ1 ∼χN ] ] [ N [ N Σ( Σ( ) ) νpk∗ χk + Ak χk,, + E1 B1 Iχ Bk χk,, = E1 A1 IW 0 k=1

+ E1 C1 Iχˆ

[ N Σ(

Ck χk,,

)

k=1

]

(5.55)

k=1

where {IW 0 , Iχ , Iχˆ } are defined as ∫

∫ IW 0 =

(−n · φ)2 dA; Iχ =

(−n · ψnχ )2 dA; Iχˆ =

∫

(−n · ψnχˆ )2 dA (5.56)

Similarly, Sm of the m-th set can be expressed using the equations below (m = 1, 2, . . . , N ):

132

5 Sectional Shape Functions for a Box Beam Under Torsion: …

∫

( ) σ zz −n · ψnχm dA | ∫ | ( ) = −n · ψnχm dA σ zz (z, n, s)||

Sm (z) =

∫

= Am

χ1 ∼χN

| | σ zz (z, n, s)||

χ1 ∼χN

(−n · φ)dA

| | ( ) −n · ψnχ dA + Bm σ zz (z, n, s)|| χ ∼χ | 1 N( ∫ ) | + Cm σ zz (z, n, s)|| −n · ψnχˆ dA ∫

χ1 ∼χN

] [ N Σ( ) ∗ ,, = E1 Am IW 0 νpk χk (z) + Ak χk (z) k=1

] ] [ N [ N Σ( Σ( ) ) ,, ,, + E1 Bm Iχ Bk χk (z) + E1 Cm Iχˆ Ck χk (z) k=1

(5.57)

k=1

Using the results in Eqs. (5.55) and (5.57), one can now establish the following relationship: ⎤ ⎡ A1 IW 0 B1 Iχ S1 (z) ⎢ S2 (z) ⎥ ⎢ A2 IW B2 Iχ 0 ⎢ ⎥ ⎢ ⎢ S3 (z) ⎥ ⎢ A3 I W 0 B3 Iχ ⎢ ⎥=⎢ ⎢ . ⎥ ⎢ . .. ⎣ .. ⎦ ⎣ .. . SN (z) AN IW 0 BN Iχ ⎡

N ( ⎤⎡ Σ )⎤ E1 νpk∗ χk (z) + Ak χk,, (z) ⎥ ⎥⎢ k=1 ⎥ ⎥⎢ N ⎢ ⎥ ( ) Σ ⎥⎢ ,, ⎥ Bk χk (z) E1 ⎥⎢ ⎥ ⎥⎢ k=1 ⎥ ⎦⎣ ⎦ N ) Σ ( ,, C χ (z) E 1 k k CN Iχˆ

C1 Iχˆ C2 Iχˆ C3 Iχˆ .. .

(5.58a)

k=1

Equivalently in a symbolic form, S(z) = I · V(z)

(5.58b)

In Eq. (5.58), {S, V, I} are an N × 1 vector, a 3 × 1 vector, and an N × 3 matrix, respectively. Because matrix I is not a square matrix, V can be determined from Eq. (5.58) using the pseudo-inverse of I (i.e., using a least-square method) (Strang, 2016): V(z) = (IT I)−1 IT S(z)  I˜ S (z).

(5.59a)

Equation (5.59a) can be written out in terms of the corresponding components as

x

5.3 Derivation of Sectional Shape Functions ψn k of Unconstrained …

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

E1

N ( Σ k=1

E1 E1

νpk∗ χk

+

Ak χk,,

N ( Σ

)

k=1 N ( Σ

)

k=1

Bk χk,,

Ck χk,,

)⎤ ⎥ ⎡ ⎥ I˜11 ⎥ ⎥ = ⎣ I˜21 ⎥ ⎥ I˜31 ⎦

133

⎡

I˜12 I˜22 I˜32

S1 ⎤ S ˜I13 · · · I˜1N ⎢ ⎢ 2 ˜I23 · · · I˜2N ⎦⎢ ⎢ S3 ⎢ . I˜33 · · · I˜3N ⎣ ..

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(5.59b)

SN

By substituting Eq. (5.59) into Eq. (5.54), one can obtain the following result: ˆ

σ zz (z, n, s)|χ1 ∼χN = [σ Szz ] + [σ Szz ] + [σ Szz ],

(5.60)

where σ Szz =

S(z) [−n φ(s)], IW 0

S(z) [−n ψnχ (s)], Iχ ˆ S(z) = [−n ψnχˆ (s)], Iχˆ

σ Szz = ˆ

σ Szz

(5.61)

and S(z) = IW 0

N Σ

[I˜1k Sk (z)],

k=1

S(z) = Iχ

N Σ

[I˜2k Sk (z)],

k=1

ˆ S(z) = Iχˆ

N Σ

[I˜3k Sk (z)].

(5.62)

k=1

Note that Eq. (5.61) gives the explicit relationships between the generalized forces ˆ ˆ (S(z), S(z), and S(z)) and the stresses (σ Szz , σ Szz , σ Szz ). ˆ Figure 5.3 sketches the distributions of σ Szz , σ Szz , and σ Szz on edge 1. From Eq. (5.61), it is apparent that σ Szz varying as (−n φ(s)) equally behaves as σ zz in Eqs. (5.18) and (5.19), which originally developed due to the zeroth-order warping (W0 ). Apparently, the stress distribution of σ Szz induces the in-plane bending of cross-sectional walls ˆ due to the distortion modes χk (k = 1, 2, . . . , N ). Likewise, σ Szz and σ Szz behave χ χˆ as −n ψn (s) and −n ψn (s), respectively. They represent the wall-bending stresses, which can be accounted for by introducing new modes (ηk and ηˆ k ), correspondingly referred to as constrained distortion modes.

134

5 Sectional Shape Functions for a Box Beam Under Torsion: …

t

t

t

edge 2

edge 1 edge 3 edge 4

ψ Wz

σ zzS

0

σ zzS

ˆ

σ zzS

ˆ

ˆ

Fig. 5.3 Stress distributions of σ Szz , σ Szz , and σ Szz on edge 1. The stress components, σ Szz and σ Szz , η ηˆ induce wall bending, which can be accounted for by ψn k and ψn k , respectively

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained Distortion Modes {ηk , ηˆ k } As discussed at the end of Sect. 5.3, new modes ηk and ηˆ k are needed to account for ˆ wall-bending deformations due to σ Szz and σ Szz , respectively. Unlike the sectional mode shapes of χk , those of {ηk , ηˆ k } are represented only by the bending deformations η ηˆ of the walls of a box beam. Therefore, only their normal components (ψn k , ψn k ) ηk ηˆ k are non-vanishing while ψs = ψs = 0, as shown in Fig. 5.1. Accordingly, no displacement along the s-direction occurs at the sectional corners, but rotations can be non-zero. However, it may be quite cumbersome or difficult directly to find the shape functions that simultaneously satisfy all of the continuity conditions between two adjacent edges at every corner of the beam cross-section when the rotations η ηˆ are non-zero. To circumvent this difficulty, first we derive {ψn k , ψn k } assuming hinge conditions (possible discontinuities in moments and rotation angles between two edges meeting at a corner) at every corner. Then, we correct the resulting shape functions to satisfy the required continuities at the corner. Under the hinge conditions, ˆ η ηˆ the process of deriving ψn k is identical to that for ψn k except that σ Szz is used instead ηk S of σ zz . For brevity, only the derivation process for ψn with the hinge conditions will η ηˆ be presented, while corrections will be made for both ψn k and ψn k . ηk As illustrated in Fig. 5.4, ψn refers to the shape functions with zero rotation at ηˆ every corner and ψn k refers to the shape functions with non-zero rotation. In the subsequent discussions, modes ηk and ηˆ k are referred to as the type-1 and type-2 constrained distortion modes, respectively.

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

edge 2

135

Et 3 η ψ n ,2 12

ψ nη,2 = 0

Et 3 η ψ n ,1 12

η

edge 3

ψ n ,1 = 0

edge 1

zero angle edge 4

(a)

moment (curvature) continuity

edge 2 Et 3 ηˆ ψ n ,2 12

−ψ nηˆ,2 edge 3

Et 3 ηˆ ψ n ,1 12

edge 1

−ψ nηˆ,1

angle continuity edge 4

(b)

moment (curvature) continuity η

Fig. 5.4 Sketch of the sectional shape functions: a the section shape function ψn k of the typeηˆ 1 constrained distortion mode and b the sectional shape function ψn k of the type-2 constrained ηk distortion mode. While zero rotation is imposed at the corners for ψn , non-zero rotation is allowed ηˆ at the corners for ψn k . In either cases, continuities in the rotation angle and moment at the shared corner of two adjacent edges should be satisfied

η

5.4.1 Derivation of ψn 1 η

The sectional shape function ψn 1 of the lowest-order type-1 constrained distortion mode will be derived using the following three steps: Step 1. Identification of the secondary normal strain field due to χ The axial stress of σ Szz in Eq. (5.60) induces εss as a secondary strain due to Poisson’s effect. This is expressed as ε ss (z, n, s) = −ν

σ Szz (z, n, s) νS(z) n ψnχ (s) = E EIχ

(5.63)

136

5 Sectional Shape Functions for a Box Beam Under Torsion: …

Step 2. Derivation of secondary displacement consistent with the secondary strain To account for the secondary strain in Eq. (5.63) using the displacement field, the type-1 constrained distortion mode of the lowest order, η1 , is considered. The corresponding 1D generalized displacement is also denoted by η1 (z). Following the argument used to consider the us displacement field due to distortion χ1 as given in Eq. (5.23), the in-plane wall-bending displacement by η1 can be written as us (z, n, s) = −n ψ˙ nη1 (s)η1 (z)

(5.64)

/ Because the normal strain εss calculated from Eq. (5.64) (ε ss = ∂us ∂s = η −n ψ¨ n 1 η1 ) should be equal to the strain given in Eq. (5.63), the following relationship must hold: −nψ¨ nη1 (s)η1 (z) =

νS(z) n ψnχ (s) EIχ

(5.65)

Equation (5.65) can be valid if ψ¨ nη1 (s) = −r 1 ∗ ψnχ (s), η1 (z) =

(5.66)

ν/EIχ S(z) r1∗

(5.67)

where r ∗1 is an arbitrary scaling parameter. The double integration of Eq. (5.66) yields ) ( η η η χ ψn,j1 = r ∗1 −Φn,j + Cj,11 sj + Cj,01 , (j (edge number) = 1, 2, 3, 4) η

η

(5.68)

χ

where {Cj,01 , Cj,11 } (j = 1, 2, 3, 4) are integration constants, and Φn,j (sj ) is defined as χ

Φn,j =

¨

χ

ψn,j dsj dsj (excluding integration constants). χ

Using Eq. (5.46a), one can express Φn,j as χ

s5 m1 3 + s , 20 6 { 3 } s5 (b5 +h5 ) h s3 , = − + 6b 3 m1 + 3 40b 20 m1 3 s5 + s , = 20 6 { 3 } s5 (b5 +h5 ) h s3 . = − + 6b 3 m1 + 3 40b 20

Φn,1 = χ

Φn,2 χ

Φn,3 χ

Φn,4

(5.69)

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

137

Here, m1 is explicitly determined by Eq. (5.49). In Eq. (5.69), s stands for sj (j = 1, 2, 3, 4) in edge j. η η To determine the eight unknown coefficients {Cj,01 , Cj,11 } (j = 1, 2, 3, 4) appearing in Eq. (5.68), the following conditions are used: ψnη1 (sj = s∗ ) = − ψnη1 (sj = −s∗ ) (j = 1, 2, 3, 4),

(5.70a)

ψnη1 (sj = s∗ ) = − ψnη1 (sj+2 = −s∗ ) (j = 1, 2) ,

(5.70b)

ψnη1 (sj = +sj ) = 0 ψnη1 (sj+1 = −sj+1 ) = 0 (j = 1, 2, 3, 4), ∫

ψ˙ nθz · ψ˙ nη1 ds =

∫

ψ˙˜ nχ0 · ψ˙ nη1 ds = 0

(5.70c) (5.70d)

η

Equations (5.70a, 5.70b) denote the anti-symmetry conditions for ψn 1 , Eq. (5.70c) states the condition of zero translational displacement at the corners, and Eq. (5.70d) represents the orthogonality of mode η1 with respect to modes θz and χ0 . Note that χ ψ˜ n 0 defined in Eq. (5.4) is used in Eq. (5.70d), as discussed in relation to Eq. (5.4) and Eq. (5.11). A careful examination of Eq. (5.70) shows that there remain only two independent equations in Eq. (5.70c) (j = 1 for example) if all of the conditions stated in Eqs. (5.70a, 5.70b) are fulfilled. Furthermore, Eq. (5.70d) becomes automatically χ satisfied if Eq. (5.70c) is satisfied because ψ˙ nθz (s) and ψ˙˜ n 0 (s) are constant. (See χ0 θz ˜ Eq. (2.27) for ψn (s) and Eq. (5.4) for ψn (s).) This argument can be supported by ∫ ∫ η η the explicit calculations of ψ˙ nθz · ψ˙ n 1 ds and ψ˙ nθz · ψ˙ n 1 ds, as given below: ∫

⎡ ψ˙ nθz · ψ˙ nη1 (s) ds = ψ˙ nθz · ⎣ ⎡ = ψ˙ nθz · ⎣

4 ∫ Σ j=1 4 Σ {

⎤ ψ˙ nη1 (sj ) dsj ⎦

⎤ } ψnη1 (sj = +sj ) − ψnη1 (sj = −sj ) ⎦

j=1

⎫ ⎡⎧ 4 ⎬ ⎨Σ = ψ˙ nθz · ⎣ ψnη1 (sj = +sj ) ⎭ ⎩ j=1 ⎫⎤ ⎧ 3 ⎬ ⎨Σ ψnη1 (sj+1 = +sj+1 ) ⎦ − ⎭ ⎩ j=0 ⎫ ⎧ ⎫⎤ ⎡⎧ 4 3 ⎨Σ ⎬ ⎨Σ ⎬ = ψ˙ nθz · ⎣ 0 − 0 ⎦ = 0, ⎩ ⎭ ⎩ ⎭ j=1

j=0

(5.71a)

138

5 Sectional Shape Functions for a Box Beam Under Torsion: …

∫

ψ˙˜ nχ0 · ψ˙ nη1 (s) ds = ψ˙˜ nχ0

⎤ ⎡ 4 ∫ Σ ψ˙ nη1 (sj ) dsj ⎦ ·⎣ j=1

= ψ˙˜ nχ0

⎤ ⎡ 4 Σ { η } ψn 1 (sj = +sj ) − ψnη1 (sj = −sj ) ⎦ ·⎣ j=1

⎫ ⎡⎧ 4 ⎬ ⎨Σ = ψ˙˜ nχ0 · ⎣ ψnη1 (sj = +sj ) ⎭ ⎩ j=1 ⎧ ⎫⎤ 3 ⎨Σ ⎬ − ψnη1 (sj+1 = +sj+1 ) ⎦ ⎩ ⎭ j=0 ⎫ ⎧ ⎫⎤ ⎡⎧ 4 3 ⎨Σ ⎬ ⎨Σ ⎬ = ψ˙˜ nχ0 · ⎣ 0 − 0 ⎦ = 0. ⎩ ⎭ ⎩ ⎭ j=1

(5.71b)

j=0

η

As shall be seen in Sect. 5.4.4, the process used to determine the corrected ψn 1 (s) satisfying the rotation and moment continuities at the corners requires only the η functional form of ψn 1 (s) (e.g., a third-order polynomial having only odd orders). η Therefore, there is no need explicitly to determine the uncorrected ψn 1 (s) here. Step 3. Stress field update due to secondary displacement Once the constrained distortion mode η1 is included, the normal stress σ zz contributed from both the χ and the η1 modes can be written as (see Eq. (5.5a)): σ zz (z, n, s) = E1 (−n)[ψnχ (s)χ ,, (z) + ψnη1 (s)η,,1 (z) + ν ψ¨ nη1 (s)η1 (z)]

(5.72)

If Eq. (5.66) is used, Eq. (5.72) can be rewritten as σ zz (z, n, s) = E1 [(−n ψnχ (s))(χ ,, (z) − νr 1∗ η1 (z)) + (−n ψnη1 (s))η,,1 (z)]

(5.73)

Using σ zz in Eq. (5.73), one can express the generalized forces S and T 1 , the work conjugates of χ , and η,1 , as ∫ S(z) = ∫ T 1 (z) =

σ zz (−n · ψnχ ) dA = E1 Iχ [χ ,, (z) − νr ∗1 η1 (z)] + E1 Iχ η1 η,,1 (z)

(5.74a)

σ zz (−n ψnη1 ) dA = E1 Iχ η1 [χ ,, (z) − νr 1∗ η1 (z)] + E1 Iη η,,1 (z)

(5.74b)

where Iα and Iαβ denote the moments of inertia associated with the sectional shape functions of distortion modes α and β. They are defined as

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

∫ Iα =

(−n ψnα )2 dA and Iαβ =

∫

139

(−n ψnα )(−n ψnβ ) dA.

(α and β: specific modes) It may be more convenient to put Eqs. (5.74) into a matrix form as {

S(z) T 1 (z)

}

[ =

Iχ I χ η 1 Iχ η 1 Iη 1

]{

( )} E1 χ ,, (z) − νv ∗1 η,,1 (z) E1 η,,1 (z)

(5.75)

If Eq. (5.75) is solved for E1 (χ ,, − νr ∗1 η1 ) and E1 η,, in terms of S(z) and T 1 (z), one can write {

( )} [ ]−1 { } [ ∗ ∗ ]{ } Iχ Iχ η 1 L11 L12 S(z) S(z) E1 χ¯ ,, (z) − ν r¯ ∗1 η¯ ,,1 (z) =  T 1 (z) L∗21 L∗22 T 1 (z) Iχ η 1 Iη 1 E1 η¯ ,,1 (z) (5.76)

By substituting Eq. (5.76) into Eq. (5.73), one can write the axial stress due to the distortion modes as σ zz (z, n, s) = (L∗11 S(z) + L∗12 T 1 (z))(−n ψnχ (s)) + (L∗21 S(z) + L∗22 T 1 (z))(−n ψnη1 (s))

(5.77)

or ∗

T

∗

σ zz (z, n, s) = σ Szz (z, n, s) + σ zz1 (z, n, s)

(5.78)

where [ ∗

σ Szz

(z, n, s) =

∗

T σ zz1 (z, n, s)

] ∗ ] S (z) [ −n ψnχ (s) , Iχ ∗

] T (z) [ −n ψnη1 (s) , = 1 Iη 1

(5.79)

and ∗

S (z) = Iχ [L∗11 S(z) + L∗12 T 1 (z)], ∗

T 1 (z) = Iη1 [L∗21 S(z) + L∗22 T 1 (z)].

(5.80)

140

5 Sectional Shape Functions for a Box Beam Under Torsion: …

η

5.4.2 Derivation of ψn 2 η

To derive the sectional shape function ψn 2 of the second-order constrained distortion mode, we start with the initial strain field due to both χ and η1 . Step 1. Identification of the secondary strain field due to χ and η1 . The strain ε ss induced by the axial stress in Eq. (5.78) due to Poisson’s effect is σ zz (z, n, s) = ε ss (z, n, s) = −ν E

[

[ ∗ ] ] ∗ νS (z) νT 1 (z) χ n ψn (s) + n ψnη1 (z) EIχ EIη1

(5.81)

Step 2. Derivation of secondary displacement consistent with the secondary strain Extending the arguments used to write Eq. (5.64), the following in-plane wallbending displacement us (z, n, s) can be considered if the strain calculated from us (z, n, s) is made equal to the strain field due to σ zz (z, n, s) in Eq. (5.81): us (z, n, s) = [−n · ψ˙ nη1 (s)]η1 (z) + [−n · ψ˙ nη2 (s)]η2 (z)

(5.82)

/ Calculating the strain field by Eq. (5.82) using ε ss = ∂us ∂s and equating the resulting εss and the εss strain in Eq. (5.81), one can find [ −nψ¨ nη2 (s)η2 (z) =

] [ ∗ ] ∗ νS (z) νT 1 (z) − r 1∗ η1 (z) n ψnχ (s) + nψnη1 (s), EIη1 EIχ

(5.83)

η χ where the relationship ψ¨ n 1 = −r ∗1 ψn established in Eq. (5.66) is used. η χ Multiplying both sides of Eq. (5.83) by n ψn and nψn 1 and integrating the resulting equations over A yields

) ) ⎤ )( ⎡ (∫ ( η χ ][ ] [ ∗ ∗ nψn −nψ¨ n 2 dA η2 ⎣ (∫ ( )( ) ) ⎦ = Iχ Iχ η1 (νS /EJ∗ χ − r 1 η1 ) η η Iχ η 1 Iη 1 (νT 1 /EJη1 ) nψn 1 −nψ¨ n 2 dA η2

(5.84)

The relations given by Eq. (5.84) can be explicitly written as ∗

∗ r 2,1 η2 (z)

νS (z) = − r 1∗ η1 (z), EIχ

(5.85a)

∗

∗ r 2,2 η2 (z) =

where

νT 1 (z) EIη1

(5.85b)

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

141

) { (∫ (n ψnχ )(−n · ψ¨ nη2 ) dA r ∗2,1 = Iη1 )}/ { (∫ ( )2 } −Iχ η1 I χ I η 1 − Iχ η 1 , (n ψnη1 ) (−nψ¨ nη2 ) dA

(5.86a)

) { (∫ η1 η2 ¨ (n ψn ) (−n · ψn ) dA = Iχ )}/ { (∫ )2 } ( χ η2 ¨ (n ψn )(−nψn ) dA −Iχ η1 I χ I η 1 − Iχ η 1 .

(5.86b)

∗ r 2,2

∗ ∗ Note that r 2,1 and r 2,2 are yet unknown constants in Eqs. (5.86a, 5.86b). η If the relationships in Eq. (5.85) are substituted into Eq. (5.83), ψn 2 (s) is explicitly η χ related to ψn and ψn 1 as ∗ ∗ [− ψnχ (s)] + r 2,2 [− ψnη1 (s)] ψ¨ nη2 (s) = r 2,1 η

 r 2∗ [− ψnχ (s) − D1 2 ψnη1 (s)]

(5.87)

η

where the following new symbols r 2∗ and D1 2 are introduced: η

∗ r 2∗  r 2,1 , D1 2 

∗ r 2,2 ∗ r 2,1 η

Here, r 2∗ is an arbitrary scaling constant, and D1 2 is an unknown coefficient. An important observation which can be made from Eq. (5.87) is that it is a recursive η equation because a higher-order mode (ψn 2 (s)) can be expressed as the sectional η χ shape functions of lower-order modes, ψn (s) and ψn 1 (s). At this point, we perform the double integrations of Eq. (5.87) with respect to s to find ] [ η η ( ) η η η χ ( ) ψn,j2 (s) = r ∗2 −Φn,j sj − D1 2 Φn,j1 sj + Cj,12 sj + Cj,02 , (j = 1, 2, 3, 4),

(5.88)

where Φαn,j =

¨

( ) α ψn,j dsj dsj α = χ , η1 excluding the integration constants. η

η

(5.89)

η

To determine the nine unknown coefficients D1 2 and {Cj,02 , Cj,12 } (j = 1, 2, 3, 4) in Eq. (5.88), the following conditions are used: ψnη2 (sj = s∗ ) = − ψnη2 (sj = −s∗ ) (j = 1, 2, 3, 4),

(5.90a)

142

5 Sectional Shape Functions for a Box Beam Under Torsion: …

ψnη2 (sj = s∗ ) = − ψnη2 (sj+2 = −s∗ ) (j = 1, 2) ψnη2 (sj = +sj ) = 0, ψnη2 (sj+1 = −sj+1 ) = 0 (j = 1, 2, 3, 4) ∫

ψ˙ nθz · ψ˙ nη2 ds =

∫

ψ˙˜ nχ0 · ψ˙ nη2 ds =

∫

ψ˙ nη1 · ψ˙ nη2 ds = 0

(5.90b) (5.90c) (5.91)

Equations (5.90a, 5.90b) express the conditions of anti-symmetry, Eqs. (5.90c) denote the zero translational displacement at corners, and Eq. (5.91) represents the orthogonality of mode η2 with respect to mode {θz , χ0 , η1 }. As shall be seen in η Sect. 5.4.4, the process used to determine the corrected ψn 2 (s) satisfying the rotation η and moment continuities at corners requires only the functional form of ψn 2 (s) (e.g., a fifth-order polynomial having only odd orders). Therefore, there is no need explicitly η to determine the uncorrected ψn 2 (s) here. Step 3. Stress field update due to secondary displacement Using Eq. (5.5a), the wall-bending component of the axial stress due to modes χ, η1 , and η2 can now be expressed as ) ( η σ zz = E1 (−n) ψnχ χ ,, + ψnη1 η,,1 + ν ψ¨ nη1 η1 + ψnη2 η¨ 2,, + νψ n2 η2

(5.92)

If Eq. (5.66) and Eq. (5.87) are used, Eq. (5.92) can be rewritten as )( )( [( ) ( ) σ zz = E1 −nψnχ χ ,, − νr ∗1 η1 − νr ∗2,1 η2 + −nψnη1 η,,1 − νr ∗2,2 η2 ) ] ( (5.93) + −nψnη2 η,,2 . Using the stress field in Eq. (5.93), one can obtain the following generalized forces (S, T 1 , T 2 ): ∫ S = σ zz (−n · ψnχ ) dA = E1 Iχ (χ ,, − νr ∗1 η1 − νr ∗2,1 η2 ) + E1 Iχ η1 (η,,1 − νr ∗2,2 η2 ) + E1 Iχ η2 η,,2 ∫ T1 =

(5.94a)

σ zz (−n · ψnη1 ) dA

∗ η2 ) + E1 Iη1 (η,,1 − νr ∗2,2 η2 ) + E1 Iη1 η2 η,,2 (5.94b) = E1 Iχ η1 (χ ,, − νr 1∗ η1 − νr 2,1

∫ T2 =

σ zz (−n · ψnη2 ) dA

∗ ∗ = E1 Iχ η2 (χ ,, − νr ∗1 η1 − νr 2,1 η2 ) + E1 Iη1 η2 (η,,1 − νr 2,2 η2 ) + E1 Iη2 η,,2 (5.94c)

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

143

The relationships given in Eq. (5.94) can be represented in matrix form as ⎫ ⎧ ⎫ ⎡ ⎤⎧ Iχ Iχ η1 Iχ η2 ⎨ E1 (χ ,, − νr ∗1 η1 − νr ∗2,1 η2 ) ⎬ ⎨ S ⎬ = ⎣ Iχ η 1 Iη 1 Iη 1 η 2 ⎦ E1 (η,,1 − νr ∗2,2 η2 ) T ⎩ 1⎭ ⎭ ⎩ T2 Iχ η 2 Iη 1 η 2 Iη 2 E1 η,,2

(5.95)

Following the procedure used to obtain Eqs. (5.77–5.80), the wall-bending stress in Eq. (5.93) can be written in terms of the generalized forces as ∗

T

∗

T

∗

σ zz = σ Szz + σ zz1 + σ zz2

(5.96)

where ∗

∗

∗

∗

∗

∗

σ Szz = (S /Iχ )(−n ψnχ ), σ zz1 = (T 1 /Iη1 )(−n ψnη1 ), T

σ zz2 = (T 2 /Iη2 )(−n ψnη2 ) T

(5.97)

and ∗

S = Iχ (L∗11 S + L∗12 T 1 + L∗13 T 1 ), ∗

T 1 = Iη1 (L∗21 S + L∗22 T 1 + L∗23 T 2 ), ∗

T 2 = Iη2 (L∗31 S + L∗32 T 1 + L∗33 T 2 ).

(5.98)

The symbol L∗ij is related to Iij as L∗ij = [Iij ]−1 . (See Eq. (5.76) for more details.)

η

5.4.3 Derivation of ψn N η

η

In Sects. 5.4.1 and 5.4.2, we explained the procedures to derive ψn 1 and ψn 2 . As a η generalization of this procedure, we will present the procedure to derive ψn N while ηN −1 η1 assuming that all lower-order shape functions (from ψn to ψn ) are available. To this end, we extend the results given by Eqs. (5.96)–(5.98) and express the wallbending stress as S

∗

σ zz = σzz +

N −1 Σ k=1

T

∗

σzz k =

∗

N −1

∗

ΣT S k (−n ψnηk ) (−n ψnχ ) + Iη k Iχ

(5.99)

k=1

The secondary strain ε ss (z, n, s) caused by σ zz in Eq. (5.99) due to Poisson’s effect should be explained by the secondary in-plane wall-bending displacement

144

5 Sectional Shape Functions for a Box Beam Under Torsion: …

) Σ ( η us , which can be written as us = Nk=1 −nψ˙ n k ηk . Setting the secondary strain εss (z, n, s) according to σ zz in Eq. (5.99) (i.e., − νσ zz /E) equal to the strain by us Σ η (i.e., ∂us /∂s= Nk=1 (−nψ¨ n k )ηk ) yields N Σ

(−nψ¨ nηk )ηk =

k=1

∗

∗

N −1

Σ νT νS k (n ψnχ ) + (n ψnηk ) EIχ EIηk

(5.100)

k=1

Extending the procedure used to establish the relationship (5.87), one can derive η η χ the following general recursive relationship that relates ψn N to ψn and ψn k (k = 1, 2, . . . , N − 1): ψ¨ nηN = r N∗ ,1 (− ψnχ ) +

N −1 Σ

r N∗ ,k+1 (− ψnηk ),

(5.101)

k=1 ∗ ∗ where r N∗ ,k (k = 1, 2, · · · , N ) is defined similarly to how r 2,1 and r 2,2 were defined. The double integrations of Eq. (5.101) with respective to s yield

[ NN ψn,j

=

r ∗N

χ −Φn,j

+

N −1 Σ

η Dk N

( ) η η η −Φn,jk + Cj,1N sj + Cj,0N

] (j = 1, 2, 3, 4), (5.102)

k=1

where Φαn,j is defined in Eq. (5.89) and r N∗ is an arbitrary scaling constant.

η

Equation (5.102) involves a total of N + 7 unknowns (N − 1 unknowns from Dk N , η η four unknowns from Cj,1N , and four unknowns from Cj,0N ). They can be found by using six anti-symmetry conditions (Eq. (5.103a)), two corner continuity conditions η η (Eq. (5.103b)), and N −1 orthogonality conditions between ψn N and ψn k |k=1,2,...,N −1 . These conditions are given below. ψnηN (sj = s∗ ) = − ψnηN (sj = −s∗ ) (j = 1, 2, 3, 4), ψnηN (sj = s∗ ) = − ψnηN (sj+2 = −s∗ ) (j = 1, 2), ψnηN (sj = +sj ) = 0, ψnηN (sj+1 = −sj+1 ) = 0, (j = 1, 2, 3, 4) ∫

ψ˙ nθz · ψ˙ nηN ds = = η

∫ ∫

(5.103a) (5.103b)

ψ˙˜ nχ0 · ψ˙ nηN ds ψ˙ nηk · ψ˙ nηN ds = 0. (k = 1, 2, . . . , N − 1)

(5.103c)

Note that ψn N satisfying Eq. (5.103b) automatically satisfies the orthogonality ∫ ∫ χ η η conditions ψ˙ nθz · ψ˙ n N ds = ψ˙˜ n 0 · ψ˙ n N ds = 0 in Eq. (5.103c), as shown in

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

145

Eq. (5.71). If the conditions given by Eq. (5.103) are used, one can explicitly derive η η ψn k for any k ≥ 1. For example, ψn,j1 can be obtained as10 : η

ψn,j1 1 =

16sj1 (h2 −4sj2 ) 1

75h (32b −32b h+32b h −32b6 h3 +32b5 h4 −32b4 h5 +32b3 h6 +63b2 h7 +420b2 h5 −63bh8 −420bh6 +63h9 +420h7 ) 9 8 7 2 6 3 5 4 4 5 3

9

8

7 2

[(8b − 8b h + 8b h − 8b h + 8b h − 8b h + 8b3 h6 + 21b2 h7 + 140b2 h5 − 21bh8 − 140bh6 + 21h9 + 140h7 ) + (−84b2 h5 − 560b2 h3 + 84bh6 + 560bh4 − 84h7 − 560h5 )sj21 ] (j1 = 1, 3); η

ψn,j1 2 16sj (b2 −4s2 )

= − 75(32b9 −32b8 h+32b7 h2 −32b62h3 +32b5j2h4 −32b4 h5 +32b3 h6 +63b2 h7 +420b2 h5 −63bh8 −420bh6 +63h9 +420h7 ) 6 5 4 2 4

[(8b − 8b h − 13b h − 140b + 13b3 h3 + 140b3 h − 13b2 h4 + 84b2 h2 ss2 − 140b2 h2 − 8bh5 + 8h6 ) + (560b2 − 84bh3 − 560bh + 84h4 + 560h2 )sj22 ] (j2 = 2, 4).

(5.104)

η

The scaling parameter r 1∗ is chosen to make ψn 1 (s1 = h/4) = 1/100 without a ηˆ loss of generality. Similarly, one can explicitly derive ψn k for any k ≥ 1 using the same derivation procedure presented in Sects. 5.4.1–5.4.3. The only difference is ˆ ηˆ that σ Szz defined in Eq. (5.61) is used in Eq. (5.63) instead of σ Szz . For example, ψn,j1 is obtained as ηˆ

16sj1 (h2 −4sj2 ) 1 [(3b2 375h3 (3h2 +32)

ηˆ

16sj2 (b2 −4sj2 ) 2 [(3b5 375b3 h3 (3h2 +32)

ψn,j1 1 = ψn,j1 2 =

+ 40) + 12sj21 ] (j1 = 1, 3); + 6h5 + 40h3 ) − 12b3 sj22 ] (j2 = 2, 4).

(5.105)

5.4.4 Correction for Corner Conditions In this section, the sectional shape functions of the constrained distortion modes derived in Sect. 5.4.3 will be corrected to satisfy the field continuity conditions at 10

We calculate the unknowns of sectional shape functions including those in Eq. (5.102) using the Symbolic Math Toolbox of MATLAB so that we can present the closed forms of sectional shape functions in terms of the geometric parameters b and h.

146

5 Sectional Shape Functions for a Box Beam Under Torsion: …

the corners. Before proceeding with the correction, some important facts regarding η and η, ˆ as discussed in Sect. 5.4.3, are reviewed: ˆ (1) Constrained distortion modes are divided into two modes, η and η. η ηˆ (2) Neither mode carries wall-membrane deformations; i.e., ψs = ψs = 0. (3) Mode η always has zero rotation at every corner of a beam cross-section, while mode ηˆ does not. (4) Because hinge connections were assumed at the corners of a beam cross-section η when deriving ψn,jk in Eqs. (5.68), (5.88), and (5.102), they should be corrected η

to satisfy rotation and moment continuities at the corners. (Figure 5.4 shows ψn ηˆ and ψn that satisfy the corner continuity conditions.) 5.4.4.1

η

Correction of ψn N η

An examination of Eqs. (5.68) and (5.69) shows that ψn 1 is a fifth-order polynomial η having odd orders only. Likewise, an examination of Eq. (5.88) shows that ψn 2 is η a seventh-order polynomial having odd orders only. (The shape function ψn 2 is a η1 seventh-order polynomial because it involves the double integral of ψn , a fifthη order polynomial.) Similarly, it can be shown from Eq. (5.102) that ψn N is a (2N + ηN 3)-th-order polynomial having odd orders only. Thus, ψn can be expressed in an edgewise manner as η

N +2 Σ k=1

k=1

η

N +2 Σ

N +2 Σ

ψn,1N = ψn,3N =

η η C˜ k N · (s1 )2k−1 ; ψn,2N =

η η C˜ k N · (s3 )2k−1 ; ψn,4N =

k=1

N +2 Σ

˜ ηN · (s2 )2k−1 ; D k ˜ ηN · (s4 )2k−1 . D k

(5.106)

k=1 η

η

η

Owing to the anti-symmetry of ψn N , ψn,3N (or ψn,4N ) shares the same unknown η ηN η ˜ ηN ) (k = 1, . . . , N + 2) with ψn,1 constants C˜ k N (or D (or ψn,2N ). To facilitate the k η ˜ ηN }k=1,...,N +2 , we can set C˜ NηN+2 = r ∗N (r ∗N : scaling constant) determination of {C˜ k N , D k without a loss of generality. Accordingly, Eq. (5.106) can be rewritten as η ψn,1N

= r ∗N ·

[N +1 Σ

] η Ck N

k=1

η ψn,3N

= r ∗N ·

[N +1 Σ k=1

· (s1 )2k−1 + (s1 )2N +3 ,

η ψn,2N

= r ∗N ·

· (s3 )2k−1 + (s3 )2N +3 ,

] η Dk N

k=1

] η Ck N

[N +2 Σ

η ψn,4N

= r ∗N ·

[N +2 Σ

· (s2 )2k−1 , ]

η Dk N

· (s4 )2k−1 ,

k=1

(5.107) where

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

C ηN 

147

˜ ηN D C˜ ηN ηN and D  r ∗N r ∗N

Examining Eq. (5.107), it is clear that there are (2N + 3) unknowns η η η ({Ck N , Dk N }k=1,...,N +1 , DNN+2 ) to be determined. To determine these unknowns, the following conditions are considered:

∫

ψ˙ nηN (sj = +sj ) = 0, ψ˙ nηN (sj+1 = −sj+1 ) = 0 (j = 1, 2, 3, 4),

(5.108a)

ψ¨ nηN (sj = +sj ) = ψ¨ nηN (sj+1 = −sj+1 ) (j = 1, 2, 3, 4),

(5.108b)

ψnηN (sj = +sj ) = 0, ψnηN (sj+1 = −sj+1 ) = 0 (j = 1, 2, 3, 4),

(5.108c)

ψ˙ nθz · ψ˙ nηN ds = =

∫ ∫

ψ˙˜ nχ0 · ψ˙ nηN ds ψ˙ nηk · ψ˙ nηN ds =

∫

ψ˙ nηˆ k · ψ˙ nηN ds = 0 (1 ≤ k ≤ N − 1) (5.108d) η

Equation (5.108a) requires that the rotations of ψn N are zero at all corners, and Eq. (5.108b) represents the moment continuity conditions at the corners (see η Fig. 5.4). Equation (5.108c) states that the displacements by the shape function ψn N ηN at the corners are zero. Finally, Eq. (5.108d) denotes the orthogonality of ψn with η respect to all lower-order modes that are known before ψn N is determined. Note ηN that Eq. (5.108d) also includes the condition that ψn is orthogonal with respect to η ηˆ ηˆ ψn k |k=,2,...,N −1 . Because ψn k will be calculated directly after ψn k is determined using ηˆ the procedure described below, ψn k |k=,2,...,N −1 can be regarded as known at the time ηN η ηˆ when ψn is determined. Once all ψn k ’s (and ψn k ’s) (k = 2, . . . , N − 1) satisfying the corner continuities are determined, they should be used in all expressions given in the previous sections. Given the anti-symmetry, Eqs. (5.108a–5.108c) are not all independent if considered for all corners. In fact, one can find all independent conditions by considering Eqs. (5.108a–5.108c) only at one corner, as in Eqs. (5.108a–5.108c) only for j = 1, η χ yielding five independent conditions. Because ψ˙ nθz and ψ˙˜ n 0 are constant and ψn N is ∫ ∫ θ η η χ zero at the corners, the conditions ψ˙ n z · ψ˙ n N ds = ψ˙˜ n 0 · ψ˙ n N ds = 0 as stated in Eq. (5.108d) are automatically satisfied. Therefore, 2N − 2 independent conditions are obtained from the orthogonality conditions in Eq. (5.108d). By using 2N + 3 conditions (five from Eqs. (5.108a–5.108c) and 2N − 2 from Eq. (5.108d)), the unknowns in Eq. (5.107) can now be determined. η If we choose the scaling constant r N∗ to make ψn N (s1 = h/4) = 1/100, i.e.,

148

5 Sectional Shape Functions for a Box Beam Under Torsion: …

r ∗1 =

256 225h5

η

we obtain the following corrected ψn 1 : η

] 16 [ 5 2 3 4 16s (j1 = 1, 3); − (8h ) · s + (h ) · s j 1 j j 1 1 225h5 [ ] 16 5 2 3 4 16s (j2 = 2, 4). =− − (8b ) · s + (b ) · s j 2 j2 j2 225b3 h2

ψn,j1 1 = η

ψn,j1 2

η

(5.109)

η

Likewise, the corrected ψn 2 and ψn 3 are given by 64 η ψn,j2 1 = − 675h7 (11b2 − 11bh + 8h2 ) [ ] sj1 (h2 − 4sj21 )2 {(44b2 − 44bh − 352h2 )sj21 + h2 (−11b2 + 11bh + 16h2 )} (j1 = 1, 3); 64 η ψn,j2 2 = 675b7 (11b2 − 11bh + 8h2 ) [ ] sj2 (b2 − 4sj22 )2 {(44h2 − 44bh − 352b2 )sj22 + b2 (−11h2 + 11bh + 16b2 )} (j2 = 2, 4). η

(5.110)

256 675h9 (−65b2 + 65bh + 37h2 ) [sj1 (h2 − 4sj21 )2 {(−1872b2 + 1872bh + 9360h2 )sj41

ψn,j3 1 = −

+ h2 (520b2 − 520bh − 1352h2 )sj21 η

ψn,j3 2

+ h4 (−13b2 + 13bh + 41h2 )}] (j1 = 1, 3); 256 = 9 2 675b (−65b + 65bh + 37h2 ) [sj2 (b2 − 4sj22 )2 {(−1872h2 + 1872bh + 9360b2 )sj42 + b2 (520h2 − 520bh − 1352b2 )sj22 + b4 (−13h2 + 13bh + 41b2 )}] (j2 = 2, 4).

(5.111)

η

Similarly, one can explicitly derive ψn k for any k > 3.

5.4.4.2

ηˆ

Correction of ψn N η

ηˆ

As in the case of ψn N , we can approximate ψn N as a (2N + 3)-th-order polynomial having odd orders:

η

ηˆ

5.4 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

ψnηˆ N (s1 )

=

rˆN∗

·

[N +1 Σ

] ηˆ Ck N

· (s1 )

ηˆ Dk N

· (s2 )

ηˆ Ck N

· (s3 )

ηˆ Dk N

· (s4 )

2k−1

k=1

ψnηˆ N (s2 )

=

rˆN∗

·

[N +2 Σ

+ (s1 )

2N +3

=

rˆN∗

·

[N +1 Σ

, ]

2k−1

k=1

ψnηˆN (s4 )

=

rˆN∗

·

[N +2 Σ

,

] 2k−1

k=1

ψnηˆ N (s3 )

149

+ (s3 )

2N +3

,

] 2k−1

,

(5.112)

k=1 ηˆ

ηˆ

ηˆ

where rˆN∗ is a scaling constant. The unknowns {{Ck N , Dk N }k=1,...,N +1 , DNN+2 } appearing in Eq. (5.112) can be found using the following conditions:

∫

ψ˙ nηˆ N (sj = +sj ) = ψ˙ nηˆ N (sj+1 = −sj+1 ) (j = 1, 2, 3, 4),

(5.113a)

ψ¨ nηˆ N (sj = +sj ) = ψ¨ nηˆ N (sj+1 = −sj+1 ) (j = 1, 2, 3, 4),

(5.113b)

ψnηˆ N (sj = +sj ) = 0, ψnηˆ N (sj+1 = −sj+1 ) = 0 (j = 1, 2, 3, 4)

(5.113c)

ψ˙ nθz · ψ˙ nηˆ N ds = =

∫ ∫

ψ˙˜ nχ0 · ψ˙ nηˆ N ds = ψ˙ nηˆ k · ψ˙ nηˆ N ds =

∫ ∫

ψ˙ nηk · ψ˙ nηˆ N ds ψ˙ nηN · ψ˙ nηˆ N ds = 0 (1 ≤ k ≤ N − 1) (5.113d)

The rotation and moment continuities at the corners depicted in Fig. 5.4b are given by Eqs. (5.113a) and (5.113b), respectively. (Compare Eq. (5.108a) used for η ηˆ ψn N and Eq. (5.113a) used for ψn N .) Eq. (5.113c) states that the displacement of ηˆ N ηˆ ψn is zero at all corners. Due to the anti-symmetry of ψn N , Eqs. (5.113a–5.113c) establish non-trivial conditions only at one corner (e.g., corner 1), yielding only four ∫ χ ∫ ηˆ ηˆ independent conditions. The first two conditions ψ˙ nθz · ψ˙ n N ds = ψ˙˜ n 0 · ψ˙ n N ds = ηˆ N 0 in Eq. (5.113d) are automatically satisfied because ψn is set to zero at all corners. Therefore, there remain 2N − 1 non-trivial orthogonality conditions in Eq. (5.113d). By using 2N + 3 conditions in total, four conditions from Eqs. (5.113a–5.113c) and (2N − 1) conditions from Eq. (5.113d), the unknowns in Eq. (5.112) can be determined. ηˆ To express ψn N explicitly, we can arbitrarily set rˆN∗ to be equal to r N∗ , which is the η ηˆ scaling constant of ψn N . In this case, the corrected ψn 1 is obtained as ηˆ

ψn,j1 1 =

16 s (h2 675h5 (2b2 +3bh−3h2 ) j1

− 4sj21 )

150

5 Sectional Shape Functions for a Box Beam Under Torsion: …

[ ] (−24b2 − 36bh + 36h2 )sj21 + h2 (14b2 + bh − h2 ) (j1 = 1, 3); ηˆ

2 2 ψn,j1 2 = 675b4 h(2b16 2 +3bh−3h2 ) sj2 (b − 4sj2 ) [ ] (−24h2 − 36bh + 36b2 )sj22 + b2 (14h2 + bh − b2 ) (j2 = 2, 4). (5.114) ηˆ

ηˆ

Likewise, the corrected ψn 2 and ψn 3 are given by ηˆ

−45056b +45056bh+360448h ψn,j2 1 = − 1425600h7 (11b2 −11bh+8h 2 )(10b4 +51b3 h+43b2 h2 −33bh3 −11h4 ) 2

2

× [sj1 (h2 − 4sj21 ){h4 (176b4 + 563b3 h + 327b2 h2 − 151bh3 − 55h4 ) + h2 (−2376b4 − 10488b3 h − 5832b2 h2 + 3456bh3 + 1320h4 )sj21 + (5280b4 + 26928b3 h + 22704b2 h2 − 17424bh3 − 5808h4 )sj41 }] (j1 = 1, 3), ηˆ ψn,j2 2

−45056b +45056bh+360448h = − 1425600b6 h(11b2 −11bh+8h 2 )(10b4 +51b3 h+43b2 h2 −33bh3 −11h4 ) 2

2

[sj2 (b2 − 4sj22 ){b4 (176h4 + 563bh3 + 327b2 h2 − 151b3 h − 55b4 ) + b2 (−2376h4 − 10488bh3 − 5832b2 h2 + 3456b3 h + 1320b4 )sj22 + (5280h4 + 26928bh3 + 22704b2 h2 − 17424b3 h − 5808b4 )sj42 }] (j2 = 2, 4). ηˆ

(5.115)

−851968b +851968bh+4259840h ψn,j3 1 = − 1248000h9 (−65b2 +65bh+37h 2 )(154b4 +577b3 h+414b2 h2 −260bh3 −143h4 ) 2

2

[sj1 (h2 − 4sj21 ){h6 (−1066b4 − 3433b3 h − 1782b2 h2 + 716bh3 + 455h4 ) + h4 (39416b4 + 125708b3 h + 73512b2 h2 − 34096bh3 − 20020h4 )sj21 + h2 (−314080b4 − 1079728b3 h − 677664b2 h2 + 363584bh3 + 208208h4 )sj41 + (640640b4 + 2400320b3 h + 1722240b2 h2 − 1081600bh3 − 594880h4 )sj61 )}] (j1 = 1, 3), ηˆ ψn,j3 2

−851968b +851968bh+4259840h = − 1248000b8 h(−65b2 +65bh+37h 2 )(154b4 +577b3 h+414b2 h2 −260bh3 −143h4 ) 2

2

[sj2 (b2 − 4sj22 ){b6 (−1066h4 − 3433bh3 − 1782b2 h2 + 716b3 h + 455b4 ) + b4 (39416h4 + 125708bh3 + 73512b2 h2 − 34096b3 h − 20020b4 )sj22 + b2 (−314080h4 − 1079728bh3 − 677664b2 h2 + 363584b3 h + 208208b4 )sj42 + (640640h4 + 2400320bh3 + 1722240b2 h2 − 1081600b3 h − 594880b4 )sj62 )}] (j2 = 2, 4).

(5.116) ηˆ

Similarly, one can explicitly derive ψn k for any k > 3.

5.5 Case Studies

151

5.4.5 Finite Element Formulation Using the theory developed in Chaps. 4 and 5, torsional problems of a thin-walled box beam can be treated as one-dimensional problems using θ (z), W0 (z), χ0 (z), η(z), for instance, as 1D generalized displacements. To solve the resulting 1D problem numerically, we present a finite element formulation using a two-node finite element (Kim and Kim 1999). Depending on the load or deformation types, the generalized displacements for torsional problems can be divided into two groups: the out-of-plane field UO and the in-plane field UI 11 as UO (z) = {W0 (z), W1 (z), . . . , WN (z)} UI (z) = {θz (z), χ0 (z), {χk (z), ηk (z), ηˆ k (z)}k=1, ..., N }

(5.117a) (5.117b)

As mentioned in Sect. 3.2.2, we employ linear interpolation for the 1D out-ofplane field variables UO and Hermite cubic interpolation for the 1D in-plane field variables UI . The latter interpolation is needed to ensure that both the 1D variables and their z-directional derivatives are continuous at the element interfaces. Because un, (derivative with respect to z) induces wall-bending displacement uz (see Eq. (5.2a)), resulting in wall-bending stress σ zz and recursively generating wallbending modes, the continuity of the z-directional derivatives is needed to obtain accurate results. Once the 1D generalized displacements are interpolated in a finite element context using linear and Hermite interpolations, it becomes possible to derive the finite element equation in matrix form, as given Eq. (3.78). The stiffness and mass matrices needed to set up the equation can be calculated using Eq. (3.81) and Eqs. (3.85–3.87), respectively.

5.5 Case Studies Using the HoBT based on the modes presented in Chaps. 3 to 5, the structural responses of thin-walled box beams under torsion will be numerically investigated. The main objective of these case studies is to demonstrate the validity and accuracy of the HoBT. In the present case studies, we will investigate the wall-membrane responses for a box beam subjected to a torsional moment Mz at its end and the coupled responses of wall-membrane and wall-bending responses for a box beam subjected to certain forms of surface traction {tzz , tzn , tzs }. The free vibration response Here, we specifically consider the case of torsion. Therefore, other field variables, such as θx (z), needed to deal with bending and extensional problems, are not considered in Eq. (5.117).

11

152

5 Sectional Shape Functions for a Box Beam Under Torsion: …

of a box beam will also be considered. The material properties of the box beams considered in all examples are as follows: E (Young’s modulus) = 200 GPa, ν (Poisson’s ratio) = 0.3, ρ (density) = 7850 kg/m3 Unless otherwise specified, the dimensions of the considered box beams are L (length) = 1 m and t (thickness) = 0.002 m, with different sectional imensions (b, h) To check the validity and accuracy of the HoBT, the numerical results for the stress and/or displacement are compared with those obtained by other approaches, in this case an Abaqus shell (S4 element) analysis (Hibbett et al. 1998) and the Vlasov torsion theory,12 which are denoted here as Shell and VTT, respectively. Other higher-order beam-based results employing cross-sectional shape functions calculated by the GBTUL (Bebiano et al. 2018) are also provided. They are denoted as GBT. As reference results for all examples, we used fully converged results obtained from a shell finite element analysis using Abaqus S4 elements, where the square dimensions of each of the shell elements used are 6.25 mm × 6.25 mm. When this discretization is used, a box beam with dimensions L = 1 m, b = 0.05 m, and h = 0.1 m is discretized into 160 elements along the beam length and meshed by 7,680 shell elements. For the stress analysis, a more refined mesh is used; e.g., the aforementioned beam is discretized into 640 elements along the beam length and meshed by 122,880 elements. For all analyses based on 1D beam approaches (HoBT, VTT, and GBT), the same longitudinal discretization as used in the shell finite element discretization is employed for a fair comparison; 160 1D beam elements were used for the displacement and free vibration analysis; and 640 1D beam elements were used for the stress analysis. To catch rapidly varying local responses significant near the boundaries of a thin-walled box beam, i.e., the end effects, more efficiently, finer discretization near the boundaries may be preferred for practical applications. However, we used uniform meshing to ensure a fair comparison. To deal with box beams under general torsional loads, we used the HoBT employing three fundamental modes (θz , χ0 , W0 ), N1 sets of χ and W modes, i.e., {χk , Wk }k=1,...,N1 , and N2 sets of η and ηˆ k modes, i.e., {ηk , ηˆ k }k=1,...,N2 . As a result, 3 + 2N modes in total are used for the analysis, where N is calculated as N (total number of higher-order mode sets) = N1 + N2 .

This indicates an analysis employing zeroth-order modes only, {θz , W0 , χ0 }, as presented in Sect. 1.3.

12

5.5 Case Studies

153

5.5.1 Case Study 1: Static Wall-Membrane Response by Torsional Moment Mz The problem considered in Case Study 1 (Choi and Kim 2021) is shown in Fig. 5.5a, where one end A is fixed and the other end B is subjected to a twisting moment Mz = 100 Nm. The beam section is assumed to be rigid at end B with no sectional deformation (no warping or whatsoever) allowed. The {uz , ux } displacements and {σzz , σxx , τzx } stresses are calculated along a specific line with s2 = b/8 and are plotted in Fig. 5.5b–f. Note that the calculated values of {σzz , σxx , τzx } in this problem represent the membrane field. To solve this problem with the HoBT, N = N1 and N2 = 0 were used. Because no constrained distortion mode (representing pure wall bending) is needed, N2 = 0. Figure 5.5b, c show that the HoBT predicts both uz and ux accurately. Both the HoBT (even with N 1 = 1) and VTT approaches can capture the variations of {uz , ux } as accurately as Shell. However, Fig. 5.5d–f show that the rapid variations of the {σzz , σxx , τzx } stresses near both ends can be captured accurately by the HoBT using a large N1 value (e.g., N1 = 3). However, VTT is shown to be incapable of catching the end effect accurately. Next, a convergence study was conducted to determine how many 1D finite elements are needed to obtain fully converged HoBT results. To this end, we considered uz (z = 0.5 m, n2 = 0, s2 = b/8) and σzz (z = 1 m, n2 = 0, s2 = h/8) and varied the number of HoBT-based 1D finite elements. The corresponding results are plotted in Fig. 5.6, where the errors of the HoBT results are defined with respect to the converged shell results. The figure shows that fully converged HoBT results can be obtained within less than 1% error if more than 20 and 60 beam elements are used for the displacement and stress calculations, respectively.

5.5.2 Case Study 2: Coupled Response of a Wall Membrane and Wall Bending by Surface Traction {tzz , tzn , tzs } Figure 5.7 shows a cantilevered box beam subject to a set of complex traction values {tzz , tzn , tzs } at its free end (Choi and Kim 2021). The prescribed stress profiles are actually taken from stress distributions appearing at a joint section of the T-joint box beam structure shown in Fig. 1.8.13 By modeling the T-joint in Fig. 1.8 with Abaqus shell elements and solving for the stress distributions, we were able to extract the stress distributions sketched in Fig. 5.7 from the joint section. Then, the extracted stress distributions are approximated in polynomial functions of s and prescribed by Eq. (5.118) in terms of {tzz , tzn , tzs }: 13

Because multiply connected box beam systems, such as a T-joint box beam structure, are analyzed by the HoBT, this case study can demonstrate that the present HoBT is capable of dealing with multiply connected box beam systems.

154

5 Sectional Shape Functions for a Box Beam Under Torsion: …

Displacement uz (m)

measurement points (s2=b/8) A y

x z

Shell HoBT (N1 =1) VTT

B (a)

M z =100N m

Longitudinal coordinate (m)

(b)

Shell

σzz ( N/m2)

Displacement ux (m)

VTT

HoBT (N1 =1)

VTT

HoBT (N1=1) HoBT (N1=3)

Shell

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(c)

(d)

HoBT (N1=1)

Shell HoBT (N1=3)

τ zx ( N/m2)

σxx ( N/m2)

VTT

HoBT (N1=1)

HoBT (N1=3)

VTT

Shell

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(e)

(f)

Fig. 5.5 a Torsional problem considered here for Case Study 1. (HoBT, VTT, and Shell denote the results obtained by the present higher-order beam theory (HoBT), the Vlasov torsion theory, and the shell finite element analysis, respectively.) All HoBT results were obtained using N1 wall-membrane modes (N2 = 0) (Choi and Kim 2021)

155

Error (%)

5.5 Case Studies

Stress σzz Displacement uz

Number of the used beam elements Fig. 5.6 Convergence of displacement and stress for Case Study 1 (see Fig. 5.5a) by the HoBT with different numbers of 1D finite elements (Choi and Kim 2021)

measurement points (s2=b/4) A y

A y

A y

x

x

x

z

z

z B (a)

measurement points (s3=-2h/8)

measurement points (s3=-h/8, s3=-3h/8)

σ zz

B

σ zn

σ zs

(b)

B (c)

Fig. 5.7 Three torsional problems considered in Case Study 2, where thin-walled box beams are subjected to various forms of prescribed traction at its end (Choi and Kim 2021)

• tzz in Fig. 5.7a: tzz (z, sj1 ) = [(6530/47) · sj1 − {(588/47) × 104 } · sj31 ] MPa (j1 = 1, 3); tzz (z, sj2 ) = [(8990/47) · sj2 + {(1176/47) × 104 } · sj32 ] MPa (j2 = 2, 4) (5.118a) • tzs in Fig. 5.7b: tzs (z, sj1 ) = [(8/9) − (2000/3) · sj21 ] MPa (j1 = 1, 3); tzs (z, sj2 ) = [(116/90) + (4000/3) · sj22 ] MPa (j2 = 2, 4)

(5.118b)

156

5 Sectional Shape Functions for a Box Beam Under Torsion: …

• tzn in Fig. 5.7c: tzn (z, sj1 ) = [(88125/46000000) · sj1 − (1704/2300) · sj31 ] MPa (j1 = 1, 3); tzn (z, sj2 ) = [−(159375/184000000) · sj2 + (121875/23000) · sj32 ] MPa (j2 = 2, 4)

(5.118c)

Because the induced stresses along the axial z direction are expected to vary rapidly, this case study is useful to investigate the effects of the higher-order modes on the accuracy of the solution. The deformation shape and stress distributions in the beam subjected to the traction shown in Fig. 5.7a (i.e., the traction given by Eq. (5.118)) are presented in Fig. 5.8. Figure 5.8a and b show the deformed shapes of the box beam predicted by the HoBT and Shell results, respectively; the HoBT result is nearly identical to the Shell result. To obtain the displacement uz and {σzz , τzx } for s2 = b/4 in Fig. 5.8c–f using the HoBT, the η and ηˆ modes are not considered because they do not contribute to uz and {σzz , τzx } calculated at n = 0; N1 /= 0 and N2 = 0. Figure 5.8f examines how each of the generalized forces (B0 , {Bk |k=1,2,3 }) contributes to the stress σzz when s2 = b/4, where σzz = σzzB0 +

3 Σ

σzzBk ,

(5.119)

k=1

with σzzBk =

Bk Wk ψ (see Eq. (4.101)). JWk z

(5.120)

To calculate Bk appearing in Eq. (5.120), we note that Bk is the work conjugate of Wk . Therefore, it can easily be retrieved from the following element-level finite element equation because the component of the force vector f e corresponds to Wk in the displacement vector de : f e = k e de ,

(5.121)

where ke is the stiffness matrix for element e. From the results given in Fig. 5.8c–f, one can see that the load applied to the beam induces rapidly varying field distributions near end B. Clearly, the HoBT analysis (N 1 = 2, 3) can capture these end effects nearly as accurately as the Shell analysis. Also, the validity of the generalized force-stress relationships in Eq. (4.76) can be confirmed from the results in Fig. 5.8f. To observe a proper value of N 1 (the set number for the {χ , W } modes representing the wall-membrane field), the problem in Fig. 5.7a is solved using different values of N 1 (with N2 = 0). The errors by the HoBT results with respect to the converged shell results are plotted in Fig. 5.9. The uz and τzx responses are calculated at (z =

5.5 Case Studies

157

Deformed shape obtained by Shell (scale: x1e +03)

Deformed shape obtained by HoBT (scale: x1e+03)

y

y z

x

z

(a)

HoBT (N1=2)

Shell HoBT (N1=3) HoBT (N1=2)

Shell VTT

HoBT

σzz ( N/m2)

Displacement uz (m)

x

(b)

(N1=1)

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(c)

(d)

VTT

HoBT (N1=1)

HoBT (N1=3) Shell

σzz ( N/m2)

τ zx ( N/m2)

Shell

VTT HoBT (N1=2) Longitudinal coordinate (m)

Longitudinal coordinate (m)

(e)

(f)

Fig. 5.8 Results of Case Study 2 for the beam subjected to a load shown in Fig. 5.7a. a, b Deformation and c–f stress of a beam subjected to tzz . HoBT, VTT, and Shell correspondingly denote the results obtained by the present higher-order beam theory, the Vlasov torsion theory, and the shell finite element analysis. For the HoBT, N2 = 0 (Choi and Kim 2021)

1 m, s2 = b/4) and (z = 0.984 m, s2 = b/4), respectively, where uz and τzx reach their maximum values. The plot in Fig. 5.9 suggests that the use of N 1 ≥ 2 for the displacement analysis and N 1 ≥ 4 for the stress analysis provides converged and sufficiently accurate results. It is observed that the HoBT with N 1 ≥ 2 can predict uz

5 Sectional Shape Functions for a Box Beam Under Torsion: …

Fig. 5.9 Convergence of the displacement and stress for the beam subjected to a load shown in Fig. 5.7a. The number N1 (with N2 = 0) is varied for the HoBT-based finite element analysis. The error stands for the HoBT errors with respect to the Shell results (Choi and Kim 2021)

Error (%)

158

Stress τzx

Displacement uz

N1 (number of the used membrane mode sets)

with less than 3.6% error and that the HoBT with N 1 ≥ 4 can predict τzx with less than 4.8% error. Next, we will examine a box beam loaded with the shear traction described in Fig. 5.7b (Choi and Kim 2021). The deformed shapes of the box beam calculated by the Shell and HoBT approaches are shown in Fig. 5.10a, b, respectively. We calculated uz and {σzz , τzy } along lines (s3 = −3 h/8) and (s3 = −h/8), respectively, and plotted the results in Fig. 5.10c–f. In Fig. 5.10c–e, the result by the HoBT using N 1 sets of {χ , W } modes (N2 = 0) is denoted by HoBT. For comparison, the GBT results obtained by GBTUL (Bebiano et al. 2018) are also presented. Figure 5.10c–e show that complex responses including end effects can be accurately captured by the HoBT. In Fig. 5.10f, the shear stresses resulting from the generalized forces Mz and Qk (k = 0, 1, 2, 3) are separately plotted to show the effect of each of the generalized forces. The figure shows how each of the stress terms associated with the generalized forces contributes to the total stress. For this analysis, recall from Eq. (4.77) that the shear stress τzs can be decomposed as τzs (z, s) ≡ τzsMz (z, s) + τzsQ0 (z, s) + τzsQ1 (z, s).

(5.122)

with τzsMz (z, s) =

Q0 (z) χ0 Q1 (z) χ1 Mz (z) θz ψs (s), τzsQ0 (z, s) = ψ (s), τzsQ1 (z, s) = ψ (s) Jθz Jχ0 s Jχ1 s (5.123)

From the result in Fig. 5.10f, one can confirm the validity of the decoupled generalized force-stress relationship established in Chap. 4. The numerical results obtained for the beam subjected to traction illustrated in Fig. 5.7c are shown in Fig. 5.11. The deformed configuration of the beam shown in

5.5 Case Studies

159

Deformed shape obtained by Shell (scale: x5e +02)

Deformed shape obtained by HoBT (scale: x5e+02)

y

y z

x

z

(a)

x

(b)

Shell HoBT (N1=4)

σzz ( N/m2)

Displacement uz (m)

GBT (N1=4)

VTT

VTT

Shell GBT (N1=4)

HoBT (N1=2)

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(c)

(d)

τ zy ( N/m2)

Shell HoBT (N1=4)

τ zy ( N/m2)

GBT (N1=4)

Shell

VTT

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(e)

(f)

Fig. 5.10 Numerical results of Case Study 2 for the beam subjected to a load shown in Fig. 5.7b. a, b Deformed shapes by the Shell and HoBT analyses, c–e displacements and stresses, and f the decomposition of τzy into shear stresses induced by different generalized resultants by the HoBT approach. (For the HoBT results, N2 = 0.) (Choi and Kim 2021)

Fig. 5.11a, b reflects significant wall-bending deformation of the beam at its end. In Figs. 5.11c–f, ux and (σzz , τzy ) along the line of the inner wall14 (s3 = −2 h/8, n2 Because n /= 0 along this line, the effects of wall bending should appear in the calculated displacement and stress field.

14

160

5 Sectional Shape Functions for a Box Beam Under Torsion: …

Fig. 5.11 Numerical results of Case Study 2 for the beam subjected to a load shown in Fig. 5.7c. a, b Deformed shapes by the Shell and HoBT analyses, c–e displacements and stresses, and f the decomposition of τzy into shear stresses induced by different generalized resultants by the HoBT approach (Choi and Kim 2021)

5.5 Case Studies

161

= −t/2) are plotted. In Fig. 5.11c–e, the results by the HoBT-based finite element analysis employing N2 sets of the constrained distortion modes ({ηk , ηˆ k }k=1, ..., N2 ) are denoted by the HoBT. Note that the HoBT (N 2 = 0) denotes the results obtained with only {θz , χ0 , W0 } (i.e., without any of {ηk , ηˆ k } modes included). The number N1 representing the set number of χ and W modes is varied from N 1 = 2 to N 1 = 4. ˆ Figure 5.11f shows the distribution of the τzsRk and τzsRk stress components (k = 1, 2, 3), which can be calculated from Eq. (5.15) and Eq. (5.13), respectively. The generalized forces {Rk , Rˆ k } appearing in Eq. (5.15) are the work conjugates of {η k , ηˆ k }. Therefore, they can be calculated from the following element-level finite element equation because Rk and Rˆ k of the force vector f e correspond to {ηk , ηˆ k } in the displacement vector de : f e = k e de ,

(5.124)

Fig. 5.12 Convergence behavior of the displacement and stress for Case Study 2 for the beam shown in Fig. 5.7c for varying N2 values (N1 = 2 and 4 for the displacement and stress analysis, respectively). The error denotes the relative error by the HoBT result respect to the Shell result (Choi and Kim 2021)

Error (%)

where ke is the stiffness matrix for element e. The examination of Fig. 5.11(c–e) shows that the traction causing wall bending induces very localized effects near the end. Nonetheless, they are accurately estimated by the HoBT (N 2 = 1, 3). The results by the HoBT (N 2 = 1, 3) favorably compare with those by Shell, whereas the VTT and the HoBT with N 2 = 0 do not. To find the proper N 2 value, the problem described in Fig. 5.7c is solved using a different number (N 2 ) of constrained distortion mode sets {ηk , ηˆ k }k=1, ..., N2 (1 ≤ N 2 ≤ 10). The errors of the HoBT compared to the Shell results are plotted in Fig. 5.12. The displacement ux and stress τzy calculated at two points located at (z = 1 m, s3 = −2 h/8) and (z = 1 m, n3 = −t/2, s3 = −2 h/8), respectively, were used to obtain the results in Fig. 5.12. This figure shows that the use of N 2 ≥ 1 for the displacement analysis and N 2 ≥ 5 for the stress analysis provides sufficiently accurate results; the errors for the displacement and stress are within 1.4% and 1.6%, respectively.

Stress τzy Displacement ux

N2 (number of the used edge-bending mode sets)

162

5 Sectional Shape Functions for a Box Beam Under Torsion: …

5.5.3 Case Study 3: Free Vibration Response In the previous case studies, it was found that the static displacements of box beams can be accurately captured by the proposed HoBT when N 1 ≥ 2 and N 2 ≥ 1 are used. At this point, we also consider a free vibration analysis of box beams (Jang and Kim 2009; Choi and Kim 2021). With this example, the validity of the criteria using N 1 ≥ 2 and N 2 ≥ 1 may also be checked. The dimensions of the considered beam are as follows: b = 0.05 m and h = 0.125 m and L = 0.5 m and t = 0.002 m Both ends of the beam are considered to be free without any restraint on its sectional deformation. For the free vibration analysis, we used N 1 = 2 and N 2 = 1, implying that the {θz , χ0 , W0 }, {χk , Wk }k=1, 2 , and {ηk , ηˆ k }k=1 modes were considered for the analysis. The eigenfrequencies for the lowest five eigenmodes calculated by the HoBT (N 1 = 2, N 2 = 1) are given in Table 5.1. For comparison, those calculated by VTT, GBT, and Shell are also included in the table. The corresponding eigenmode shapes obtained by the HoBT (N 1 = 2, N 2 = 1) and Shell approaches are compared in Fig. 5.13. From the results in Table 5.1 and Fig. 5.13, it is apparent that the HoBT results (N 1 = 2, N 2 = 1) are nearly as accurate as those obtained by Shell. To investigate the contributions of the higher-order deformable section modes, {χk , Wk }k=1,...,N1 and {ηk , ηˆ k }k=1,...,N2 in the free vibration analysis of a box beam, we varied the (N 1 , N 2 ) values and calculated the eigenfrequencies. These results are shown in Fig. 5.14. Note that the result by the HoBT using only the modes {χk , Wk }k=1,...,N1 representing the membrane field is denoted by (N 1 = 2, N 2 = 0), while that by the Vlasov theory corresponds to the analysis with N 1 = 0 and N 2 = 0. Figure 5.14 shows that the use of N 1 ≥ 2 (N 2 = 0) yields a nearly converged result for the first eigenfrequency. However, it also shows that the constrained distortion mode {η, η} ˆ representing the pure wall-bending field plays a critical role in accurately Table. 5.1 Eigenfrequencies of the freely supported box beam (b = 0.05 m, h = 0.125 m, L = 0.5 m, and t = 0.002 m) considered in Case Study 3 (Choi and Kim 2021) Theory

Mode 1st

2nd

3rd

4th

5th

Shell (Abaqus)

343.3

435.6

1053.6

1646.8

1834.0

VTT

345.5 (0.64%)

437.1 (0.34%)

1193.0 (13.2%)

2840.6 (71.3%)

2893.9 (57.8%)

HoBT(N1 = 2, N2 = 1)

342.6 (0.20%)

434.3 (0.30%)

1050.5 (0.30%)

1640.3 (0.40%)

1827.9 (0.33%)

GBT

342.7 (0.17%)

434.4 (0.28%)

1052.5 (0.10%)

1633.8 (0.79%)

1818.3 (0.86%)

5.5 Case Studies

163

1st mode (Shell, 343.3 Hz)

1st mode (HoBT, 342.6 Hz)

3rd mode (Shell, 1053.6 Hz)

3rd mode (HoBT, 1050.5 Hz)

5th mode (Shell, 1834.0 Hz)

5th mode (HoBT, 1827.9 Hz)

Fig. 5.13 Three lowest modes of the box beam considered in Case Study 3 solved by the HoBT and Shell approaches (Choi and Kim 2021)

predicting the eigenfrequencies of the third and fifth modes because significant wallbending deformations arise in these modes (see Fig. 5.13).

Fig. 5.14 Convergence of eigenfrequencies with respect to the number of mode sets for Case Study 3. The error applies to the HoBT results errors with respect to the Shell results (Choi and Kim 2021)

5 Sectional Shape Functions for a Box Beam Under Torsion: …

Error (%)

164

5th mode

3rd mode

1st mode (N1 , N2 ) (number of the used higher-order mode sets)

References Bebiano R, Camotim D, Gonçalves R (2018) GBTul 2.0—a second-generation code for the GBTbased buckling and vibration analysis of thin-walled members. Thin-Walled Struct 124:235–257 Bebiano R, Goncalves R, Camotim D (2015) A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses. Thin-Walled Structures 92:29–47 Choi IS, Jang GW, Choi S, Shin D, Kim YY (2017) Higher order analysis of thin-walled beams with axially varying quadrilateral cross sections. Comput Struct 179:127–139 Choi S (2016) Unified higher-order beam analysis for multiply-connected thin-walled box beams. Ph.D. Thesis, Seoul National University Choi S, Kim YY (2021) Higher-order Vlasov torsion theory for thin-walled box beam. Int J Mech Sci 195:106231 Ferradi MK, Cespedes X (2014) A new beam element with transversal and warping eigenmodes. Comput Struct 131:12–33 Hibbett HD, Karlsson BI, Sorensen EP (1998) ABAQUS/standard: user’s manual. Hibbitt, Karlsson & Sorensen Jang GW, Kim YY (2009) Vibration analysis of piecewise straight thin-walled box beams without using artificial joint springs. J Sound Vib 326:647–670 Kim YY, Kim JH (1999) Thin-walled closed box beam element for static and dynamic analysis. Int J Numer MEth Eng 45:473–490 Strang G (2016) Introduction to linear algebra, 5th edn. Wellesley-Cambridge Press

Chapter 6

Sectional Shape Functions for a Box Beam Under Extension

In this chapter, we will derive the sectional shape functions for a box beam subjected to an extensional (or axial) load using a recursive and hierarchical method similar to that used to derive sectional shape functions for a box beam under torsion, as χ presented in Chaps. 4 and 5. In this chapter, the shape functions (ψzW , ψs ) correχ η ηˆ sponding to the wall-membrane field and those (ψn , ψn , ψn ) corresponding to the wall-bending field will be derived altogether. Although the applied load type considered in this chapter differs from the torsional loads considered in Chaps. 4 and 5, the characteristics of the sectional shape functions derived for extensional loads are identical to those derived for torsional loads. Therefore, the derivation procedures for both sets of shape functions are nearly identical. Accordingly, we will not present the details of the procedure used to derive the shape functions needed to deal with extensional loads. Guided by the procedure given in Chap. 4 for the torsional case, we begin with the axial stress field due to mode Uz representing rigid-body (non-deformable) sectional motion. Then, what is known as the secondary strain field εss induced by axial stress due to Poisson’s effect is calculated. To account for this strain from a displacement field, a new distortional mode χ1 having a non-zero s-directional displacement χ function ψs 1 (s) is introduced, after which we update the stress field considering the contribution of the χ1 mode and derive another secondary strain field, γzs . To account for this strain field, we introduce a new warping mode W1 having a non-zero sectional shape function ψzW1 (s) describing the axial deformation. The derived modes χ1 and W1 belong to the first-order set of high-order modes for a box beam subjected to extensional loads. These processes are applied recursively to find all higher-order wall-membrane distortion (χ2 , χ3 , . . .) and warping (W1 , W2 , . . .) modes. As all these modes involve the stretching of the walls of a beam cross-section, they correspond to the wall-membrane field. Figure 6.1 shows the two lowest-order sets (χ1 , χ2 , W1 , and W2 ) of the wall-membrane field for a box beam under extension. These modes will be derived in Sects. 6.1 and 6.2.

© Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_6

165

166

6 Sectional Shape Functions for a Box Beam Under Extension

y Cross section A

x z b

Fz h t

Fig. 6.1 Various section modes needed for the structural analysis of a box beam under an axial extensional load. Unlike the non-symmetric modes in Chap. 5, the illustrated distortion modes in this case exhibit symmetric deformations on the x−y plane. However, they are also referred to as distortional modes throughout this book

Figure 6.1 also shows the constrained distortion modes.1 Because the extensional load type considered here is symmetric with respect to the x and y axes, the constrained distortion modes should also be symmetric (Kim and Kim 2003; Jang and Kim 2009a). If the modes have zero corner rotation, they will be classified as type-1 constrained modes, denoted by η. If the modes have non-zero rotations, on the ˆ Like the other hand, they will be classified as type-2 constrained modes, denoted by η. constrained distortion modes derived for the torsion of a box beam, the constrained distortion modes for axial extension are not accompanied by s-directional deformation; see Fig. 6.1. The constrained distortion modes will be derived in Sects. 6.3–6.6. Some numerical analyses using higher-order deformable section modes derived in this chapter will be presented in Sect. 6.8 using the finite element formulation briefly explained in Sect. 6.7.

1

As explained in Chap. 5, the term “distortion” is used when a section deformation mode has only in-plane deformation on the x−y or n−s plane. The term “constrained” is used when a section deformation mode has no in-plane displacement at every corner of the beam cross-section of a box beam.

6.1 General Field Relationships for Higher-Order Deformable Section …

167

6.1 General Field Relationships for Higher-Order Deformable Section Modes of a Wall-Membrane Field χk

This section presents some of the preliminary analyses needed to derive ψs (s) and k ψzW (s), the sectional shape functions of the higher-order deformation modes corresponding to a wall-membrane field for a thin-walled beam under an axial extensional load.

6.1.1 Displacements, Stress, Strain Fields, and Generalized Forces For the analysis of a box beam subjected to an axial extensional load, one can use the following 1D generalized displacements ξ: ξ(z) = {Uz (z), χ1 (z), W1 (z), χ2 (z), W2 (z), . . .}T }T { = Uz (z), {χk (z), Wk (z)}k=1,2,... ,

(6.1)

where Uz (z) is the 1D field variable representing the rigid-body translation of the beam cross-section of a thin-walled box beam along the longitudinal or axial (z) direction. In Eq. (6.1), Wk and χk (k ≥ 1) are higher-order warping and distortion modes associated with extension, respectively.2 Here, we will follow the recursive derivation procedure established to derive higher-order deformable section modes {Wk , χk }k=1,2,..., for the case of torsion. To this end, one can start with the displacement and stress fields due to Uz . To find {Wk , χk }k=1,2,..., we also use the following orthogonality relationships: ∫

W ψzWi ψz j dA

∫ = 0,

χ

ψsχi ψs j dA = 0 if i /= j

(6.2a, b)

The imposition of the relationships (6.2) can greatly simplify the derivation procedure of the shape functions and can also contribute to the decoupling of the generalized forces appearing in the generalized force-stress relationships, as argued in the case of torsion. To derive the higher-order deformable section modes corresponding to the wallmembrane field, the displacements on the midline of a cross-section wall are expressed in terms of the 1D generalized displacement as Here, Wk and χk in Eq. (6.1) denote warping and distortion induced by an extensional load, different from those by a torsional load in Eq. (4.4), although the same notations are used here. To avoid confusion, we can introduce superscripts e and t to denote warping and distortion related to extension and torsion, respectively (see Eq. (3.54)); accordingly, Wke and χke represent the modes due to extension, and Wkt and χkt are those due to torsion. In this chapter, no superscript will be used because all instances of Wk and χk refer to warping and distortion related to extension.

2

168

6 Sectional Shape Functions for a Box Beam Under Extension

u z (z, s) = ψzUz (s)Uz (z) +

N Σ

ψzWk (s)Wk (z),

(6.3a)

k=1

u s (z, s) =

N Σ

ψsχk (s)χk (z),

(6.3b)

k=1 U

where ψz z (s j ) is simply given by (see the displacement field due to Uz in Fig. 6.1) ψzUz (s j ) = 1 ( j = 1, 2, 3, 4).

(6.3c)

The highest numbers of the shape functions needed to represent the u z and u s fields in Eq. (6.3) are set to be identical to N , implying that each mode set consists of one extensional warping mode and one extensional distortion mode. The strain field corresponding to the displacements in Eq. (6.3) can be found via Σ ∂ uz = ψzUz (s)Uz, (z) + ψzWk (s)Wk, (z), ∂z k=1

(6.4a)

Σ ∂ us ψ˙ sχk (s)χk (z) = ∂s k=1

(6.4b)

N

εzz (z, s) =

N

εss (z, s) =

γzs (z, s) =

N Σ ( χk ) ∂ uz ∂ us + = ψs (s)χk, (z) + ψ˙ zWk (s)Wk (z) , ∂z ∂s k=1

(6.4c)

U U where ψ˙ z z (s)Uz (z) is omitted because ψ˙ z z (s) = 0 (see Eq. (6.3c)). Using the strains in Eq. (6.4) with the plane-stress assumption, one can obtain the wall-membrane field stresses as

σzz (z, s) = E 1 (εzz + νεss ) = E 1 [ ψzUz (s)Uz, (z) +

N Σ

(ψzWk (s)Wk, (z) + ν ψ˙ sχk (s)χk (z))],

(6.5a)

k=1

σss (z, s) = E 1 (εss + νεzz ) = E 1 [νψzUz (s)Uz, (z) +

N Σ

(ψ˙ sχk (s)χk (z) + νψzWk (s)Wk, (z))],

(6.5b)

k=1

τzs (z, s) = Gγzs = G[

N Σ k=1

where E 1 = E/(1 − ν 2 ).

( ψsχk (s)χk, (z) + ψ˙ zWk (s)Wk (z))],

(6.5c)

6.1 General Field Relationships for Higher-Order Deformable Section …

169

To derive the generalized forces, we consider the virtual work done by the surface traction terms (σzz and τzs ) over the cross-section A, as was done in Chap. 4: ∫ δW A (z) =

∫ σzz (z, s)δu z (z, s)d A +

A

τzs (z, s)δu s (z, s)d A.

(6.6)

A

Equation (6.6) can be rewritten as, via Eqs. (6.3), ⎡ ∫ δW A (z) = ⎣ σzz (z, s) · ψzUz (s)dA · δUz (z) A

⎛ ∫ NS Σ ⎝ τzs (z, s)ψsχk (s)dA · δχk (z) + k=1

∫ +

A

⎞⎤

σzz (z, s)ψzWk (s)dA · δWk (z)⎠⎦

A

[

≡ Fz (z)δUz (z) +

N Σ

] (Q k (z)δχk (z) + Bk (z)δWk (z)) .

(6.7)

k=1

From Eq. (6.7), the generalized forces are derived as ∫ Fz = Bk =

∫

∫ σzz ψzUz dA; Q k =

τzs ψsχk d A;

σzz ψzWk dA (k = 1, 2, . . . , N ).

(6.8)

In Eq. (6.8), Fz is the axial force that is the work conjugate of the longitudinal displacement, Bk denotes the bimoment that is the work conjugate of the 1D warping displacement Wk , and Q k is the transverse bimoment that is the work conjugate of the 1D distortional displacement χk . For compact notation, the following force vector F is defined: { }T F(z) = Fz (z), {Q k (z), Bk (z)}k=1,2,...,N .

(6.9)

170

6 Sectional Shape Functions for a Box Beam Under Extension

6.1.2 Generalized Force-Stress Relationship for the Zeroth-Order Mode In this subsection, the relationship between the stress and the generalized force for the zeroth-order mode, i.e., the non-deformable section mode, will be established. U To this end, we use Eq. (6.5a) defining σzz z , the axial stress (σzz ) due to the Uz : σzzUz (z, s) = E 1 ψzUz (s)Uz, (z).

(6.10)

U

Then, the generalized force due to σzz z can be written as ∫ Fz (z) =

∫ σzzU (z, s)ψzUz (s)dA =

E 1 [ψzUz (s)]2 dA · Uz, (z) = E 1 AUz, (z), (6.11)

where A is the area of the rectangular cross-section of a box beam. To obtain the last expression in Eq. (6.11), the following result was used: ∫ ( ψzUz )2 dA = 2(b + h)t = A. Because stress fields due to other sectional modes (such as χ k and W k ) are orthogU U onal to ψz z (s), we may replace σzz z with σzz in Eq. (6.11) used to calculate Fz (z). U Using Eqs. (6.10) and (6.11), one can write σzz z as σzzUz (z, s) =

Fz (z) Uz ψz (s). A

(6.12) U

An important observation to make from Eq. (6.12) is that the stress σzz z by the axial displacement Uz is represented solely in terms of its corresponding generalized U force Fz . From Eq. (6.5c), it is clear that the shear stress τzs z generated by the non-deformable section mode (Uz mode) is zero: τzsUz = 0.

χ

Wk

6.2 Derivation of ψs k and ψ z Analysis

(6.13)

by Means of a Recursive

This section presents the recursive analysis for the derivation of the sectional shape χ functions, ψs k and ψzWk (k ≥ 1), of higher-order extensional distortion and extensional warping modes. These shape functions correspond to the wall-membrane field.

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

171

To this end, we start with displacement and stress by the non-deformable section mode Uz and the corresponding work conjugate Fz . The secondary strain field εss induced by Poisson’s effect is then considered. To account for the secondary strain εss using displacement, it is necessary to introduce the next higher-order mode having a nonzero u s field (on the wall midline), referred to here as the first-order distortion mode, χ χ1 . The corresponding sectional shape function will be denoted by ψs 1 . Using the symmetry conditions of a rectangular cross-section, corner continuity conditions, χ and orthogonality conditions if applicable, one can derive the closed form of ψs 1 for a box beam under an axial extensional load. The added u s field by χ1 then produces additional shear strain and stress, requiring the existence of a warping mode. This warping mode will be denoted by W1 . Figure 6.2 illustrates how new modes χ1 and W1 are sequentially introduced by Uz . The new generalized forces Q 1 and B1 corresponding to modes χ1 and W1 can be defined as ∫ Q 1 (z) =

τzs (z, s)ψsχ1 (s) dA and B1 (z) =

∫ σzz (z, s)ψzW1 (s) dA.

(6.14a, b)

Figure 6.3 shows the hierarchical and recursive procedure used to derive the distortion and warping modes corresponding to the wall-membrane field. Essentially, A

B

y

C

x z

u z = ψ Uz z U z

χ1 generated

ε ss by Poisson’s effect E

D

Next set

W1

generated

γ zs by χ1′

Fig. 6.2 Illustration of the sequential generation mechanism of the first-order unconstrained distortion mode χ1 and warping mode W1 from the Uz field

172

6 Sectional Shape Functions for a Box Beam Under Extension

Fig. 6.3 Recursive derivation of the shape χ functions (ψzWk and ψs k ) representing the wall-membrane field of higher-order unconstrained distortion and warping modes for a box beam subjected to an axial extensional load

k =1 A

B

u z by axial displacement U z ε ss by Poisson’s effect

C

D

Distortion of the k-th order generated

u z by warping of the (k-1)th order

Shear strain γ zs generated by us

E

k = k +1 Warping of the k-th order generated

k≤N End

the same three-step procedure established in Chap. 4 for the torsional loading case is also used here, as shown below. Step 1: Identification of a secondary strain field due to the known (primary) stress field Step 2: Derivation of secondary displacement consistent with the secondary strain Step 3: Stress field update due to secondary displacement. χ

6.2.1 Derivation of ψs 1 Step 1. Identification of a secondary normal strain field due to the Uz mode The stress σzz 3 in Eq. (6.12) associated with the axial displacement generates the εzz strain, but it also induces the εss strain due to Poisson’s effect (see figure B in Fig. 6.1 and block B in Fig. 6.2): εss (z, s) = −ν

3

ν Fz (z) Uz σzz (z, s) =− ψ (s). E EA z

Unless necessary, the superscript Uz in the stress field is dropped in this section.

(6.15)

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

173

The existence of εss implies that wall-extending or wall-shrinking deformation (i.e., s-directional wall-membrane deformation) occurs in the cross-section. The section mode of this in-plane wall-extending/shrinking deformation is denoted by χ1 . Step 2. Derivation of secondary displacement consistent with the secondary strain When χ1 is additionally considered, u s (the s-directional displacement) should be updated as (see the general form in Eq. (6.3b)) u s (z, s) = ψsχ1 (s)χ1 (z),

(6.16)

from which εss is calculated as εss (z, s) =

∂u s (z, s) = ψ˙ sχ1 (s)χ1 (z). ∂s

(6.17)

Because the strain induced by Poisson’s effect in Eq. (6.15) should be represented by the strain (Eq. (6.17)) due to the displacement field of χ1 , the following relationship must hold: ν Fz (z) Uz χ1 (z)ψ˙ sχ1 (s) = − ψ (s). EA z

(6.18)

U χ Note that ψ˙ s 1 and ψz z are functions of s while χ1 and ν Fz /E A are those of z. Because s and z are independent coordinates, the following relationships between U χ ψs 1 and ψz z must hold:

ψ˙ sχ1 (s) = − p1∗ ψzUz (s),

(6.19)

or ψ˙ sχ1 (s j ) = − p1∗ ψzUz (s j ),

( j = 1, 2, 3, 4),

(6.20)

where p1∗ is a scaling constant which can be arbitrarily selected. At this point, integrating both sides of Eq. (6.20) with respect to s j gives ψsχ1 (s j )

=

p1∗

∫

χ

− ψzUz (s j )ds j ≙ p1∗ [ −ΨzUz (s j ) + C j 1 ],

χ

U

where C j 1 (j = 1, 2, 3, 4) are integration constants and Ψz z is defined as ∫ ΨzUz (s j ) = U

ψzUz (s j ) ds j (no integration contant). U

Using ψz z in Eq. (6.3c), Ψz z is found to be

(6.21)

174

6 Sectional Shape Functions for a Box Beam Under Extension

∫ ΨzUz (s j ) =

ψzUz ds j = s j .

(6.22)

χ

To determine the four unknown coefficients C j 1 in Eq. (6.21), one can consider χ

U

symmetry conditions of ψs 1 . Because the distribution of σzz z in Eq. (6.12) is symmetric about the x and y axes (see figure A in Fig. 6.1), it is obvious that U χ χ1 induced by σzz z is also symmetric. Therefore, ψs 1 should satisfy the following symmetry conditions: ψsχ1 (s j = s ∗ ) = −ψsχ1 (s j = −s ∗ ) ( j = 1, 2, 3, 4),

(6.23a)

ψsχ1 (s j ) = − ψsχ1 (s j+2 ) ( j = 1, 2),

(6.23b)

where s ∗ represents the s coordinate at an arbitrary point on edge j. Because sj is measured from the center of edge j, s j = s ∗ and s j = −s ∗ represent two opposite χ points from the edge center. Equation (6.23a) describes the symmetry of ψs 1 on each χ1 edge while Eq. (6.23b) represents the symmetry of ψs between two facing edges of a rectangular cross-section. Note that because two of the six symmetry conditions in Eqs. (6.23) are found to be redundant, only four independent equations of Eqs. χ (6.23) are selected to determine C j 1 . For example, Eq. (6.23a) with j = 1, 2 and Eq. (6.23b) can be used. If we set the scaling parameter p1∗ such that p1∗ = − 1/10 χ χ χ (which normalizes ψs 1 (s1 ) as ψs 1 (s1 = h/2) = h/20), ψs 1 can be obtained as ψsχ1 (s j ) =

sj . 10

(6.24)

χ

The shape of ψs 1 is sketched in Fig. 6.4a. Note that the illustrated distortion mode, unlike those in Chap. 4, does not distort the cross-section of a beam, but expands it symmetrically.4 Step 3. Stress field update due to secondary displacement When the χ1 field is considered in addition to the Uz field, the displacements, stresses, and strains are updated as u s (z, s) = ψsχ1 (s)χ1 (z), u z (z, s) = ψzUz (s)Uz (z),

(6.25a)

εzz (z, s) = ψzUz (s)Uz, (z), εss (z, s) = ψ˙ sχ1 (s)χ1 (z), γsz (z, s) = ψsχ1 (s)χ1, (z), 4

(6.25b)

As mentioned earlier, this mode (as well as similar higher-order modes as shown in Fig. 6.3c) is called a “distortional” mode.

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

y z

y

y x

x

z

(a)

175

(b)

z

y x

x

z

(c)

(d) χ

Fig. 6.4 Sectional shape functions for extension: a first-order distortion (ψs 1 ), b first-order warping χ (ψzW1 ), c second-order distortion (ψs 2 ), and d second-order warping (ψzW2 )

[ ] σzz (z, s) = E 1 ψzUz (s)Uz, (z) + ν ψ˙ sχ1 (s)χ1 (z) , [ ] σss (z, s) = E 1 ψ˙ sχ1 (s)χ1 (z) + νψzUz (s)Uz, (z) , [ ] τzs (z, s) = G ψsχ1 (s)χ1, (z) .

(6.25c)

U χ If the relationship ψ˙ s 1 = − p1∗ ψz z in Eq. (6.19) is used, the stress σzz in Eq. (6.25c) can be rewritten as

[ ] σzz (z, s) = E 1 ψzUz (s) Uz, (z) − ν p1∗ χ1 (z) .

(6.26)

By using σzz in Eq. (6.26) and τzs in Eq. (6.25c), the generalized forces defined in Eq. (6.8) are calculated as ∫ Fz (z) =

[ ] σzz (z, s) · ψzUz (s)dA = E 1 A Uz, (z) − νp1∗ χ1 (z) , ∫ Q 1 (z) =

τzs (z, s)ψsχ1 (s)d A = G Jχ1 χ1, (z).

(6.27a) (6.27b)

Using Eq. (6.27), the shear stress in Eq. (6.25c) can be written as τzsQ 1 (z, s) =

Q 1 (z) χ1 ψs (s). Jχ1

(6.28)

The generalized force-stress relationship in Eq. (6.12) involving the Uz mode (with no sectional deformation) remains unchanged even if the χ1 field is newly considered. Equation (6.28) shows that there is no coupling between different modes force-stress relationship; generally, decoupled relationships, σ α = in the generalized / F α ψ α Jα (e.g., α = Uz , χ1 , W1 ), are universally valid in the HoBT for box beams.

176

6 Sectional Shape Functions for a Box Beam Under Extension

6.2.2 Derivation of ψ zW1 The first-order warping mode, W1 , can also be derived by following the three previously mentioned steps. This procedure is described in Fig. 6.3 (blocks C and D). Step 1. Identification of a secondary shear strain field due to the χ1 mode The shear stress τzs in Eq. (6.28) by generalized force Q 1 induces the following shear strain: γzs (z, s) =

1 Q1 Q 1 (z) χ1 τ (z, s) = ψ (s). G zs G Jχ1 s

(6.29)

In Chap. 4, a new deformable section mode describing the z-directional sectional deformation (i.e., warping mode W1 ) was required to represent the shear strain caused by Q 1 (see Sect. 4.3.2 for details). Likewise, the strain due to Q 1 in Eq. (6.29) cannot be represented only in terms of the existing field variables, ξ = {Uz , χ1 }T . Therefore, we need to introduce a new sectional mode, in this case, the first-order warping mode W1 . Step 2. Derivation of secondary displacement consistent with the secondary strain When an additional field by W1 is considered, the displacements and shear strain can be updated as follows (see Eqs. (6.3)): u s (z, s) = ψsχ1 (s)χ1 (z), u z (z, s) = ψzUz (s)Uz (z) + ψzW1 (s)W1 (z),

(6.30)

and γzs (z, s) =

∂u z ∂u s + = ψsχ1 (s)χ1, (z) + ψ˙ zUz (s)Uz (z) + ψ˙ zW1 (s)W1 (z). ∂z ∂s

(6.31)

To find ψzW1 (s), the strain in Eq. (6.29) is set equal to the strain in Eq. (6.31) obtained from the displacement field: ψsχ1 (s)χ1, (z) + ψ˙ zUz (s)Uz (z) + ψ˙ zW1 (s)W1 (z) =

Q 1 (z) χ1 ψ (s). G Jχ1 s

(6.32)

U Because ψ˙ z z = 0 from Eq. (6.3c), Eq. (6.32) can be simplified as

ψ˙ zW1 (s)W1 (z) =

(

) Q 1 (z) − χ1, (z) ψsχ1 (s). G Jχ1

(6.33)

By requiring that the functions of s appearing in Eq. (6.33) be equal to each other, χ the following relationship between ψ˙ zW1 and ψs 1 can be obtained:

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

177

ψ˙ zW1 (s) = q1∗ ψsχ1 (s),

(6.34a)

ψ˙ zW1 (s j ) = q1∗ ψsχ1 (s j ),

(6.34b)

or

where q1∗ is a scaling constant which can be arbitrarily selected. At this point, we integrate Eq. (6.34b) with respect to s j to obtain ψzW1 (s j ) = q1∗

∫

(

) 1 ψsχ1 (s j ) ds = q1∗ [Ψsχ1 (s j ) + C W j ],

(6.35)

1 W1 where C W j (j = 1, 2, 3, 4) are integration constants and Ψz is defined as

∫ ΨzW1 (s j )

=

ψzχ1 (s j ) ds j . (no integration constant).

χ

χ

Using ψs 1 (s j ) in Eq. (6.24), one can find Ψs 1 (s j ) as Ψsχ1 (s j ) =

s 2j 20

.

(6.36)

1 To find the integration constants C W j in Eq. (6.35), the symmetry of W1 is considered. Clearly, W1 induced by the shear strain should be symmetric because the distribution of the shear strain (or shear stress) by Q 1 in Eq. (6.29) is symmetric. Therefore, 1 the following conditions can be used to determine C W j :

ψzW1 (s j = s ∗ ) = ψzW1 (s j = −s ∗ ) ( j = 1, 2, 3, 4),

(6.37a)

ψzW1 (s j ) = ψzW1 (s j+2 ) ( j = 1, 2).

(6.37b)

Note that the symmetry condition for the z-directional deformation in Eq. (6.37a) is different from that for the s-directional deformation in Eq. (6.23a). Because ψzW1 (s j ) in Eq. (6.35) is an even function, the conditions for Eq. (6.37a) are automatically met. If Eq. (6.37b) is used, the following relationships can be found: C1W1 = C3W1 and C2W1 = C4W1 1 To find C W explicitly, two additional conditions must be provided. Because j ψzW1 should be continuous at cross-section corners, ψzW1 (s j ) and ψzW1 (s j+1 ) defined on adjacent edges should be identical at the shared corner of two adjacent edges. Therefore, the following conditions must hold:

178

6 Sectional Shape Functions for a Box Beam Under Extension

ψzW1 (s j = +s j ) = ψzW1 (s j+1 = −s j+1 ) ( j = 1, 2, 3, 4),

(6.38)

where s j represents the bound of s j (−s j ≤ s j ≤ s j ). Although the continuity conditions in Eq. (6.38) are defined at all four corners of the walls forming a cross-section, only one of them is independent due to the symmetry conditions in Eq. (6.37). Accordingly, the continuity condition in Eq. (6.38) at any one corner of a cross-section can be chosen. Because the continuity condition in Eq. (6.38) provides one independent condition, an additional condition must be considered. As the last∫ condition used to U 1 ψzW1 ψz z ds = 0. determine C W j , we use the orthogonality of the shape functions, Given that W1 represents the out-of-plane deformation, the W1 mode is automatically orthogonal to the in-plane deformation mode χ1 . The remaining orthogonality comes from the orthogonality between the z-displacement fields of the W1 and Uz modes: ∫ ψzW1 ψzUz ds =

4 ∫ Σ j=1

sj

−s j

ψzW1 (s j )ψzUz (s j ) ds j = 0.

(6.39)

1 Using the four independent conditions given by Eqs. (6.37–6.39), C W j (j = 1, 2, 3, 4) are obtained as

C 1W1 = C 3W1 =

2b2 − 2(b + h) − h 2 , 240

(6.40a)

C 2W1 = C 4W1 =

2h 2 − 2(b + h) − b2 . 240

(6.40b)

Using the results in Eq. (6.40), the shape function ψ zW1 of the first-order warping mode W1 is obtained as 1 ψ z,Wj1 (s j ) = {16(b + h)/ h 2 }[ 20 (s j )2

+ {2b2 − 2(b + h) − h 2 }/240 ] ( j = 1, 3)

(6.41a)

1 ψ z,Wj1 (s j ) = {16(b + h)/ h 2 }[ 20 (s j )2

+ {2h 2 − 2(b + h) − b2 }/240] ( j = 2, 4)

(6.41b)

where the following notation is used: ψ αi, j (s j ) ≡ ψ αi (s = s j ) (α: mode type, i = n, s, z).

(6.42)

To write ψ z,Wj1 (s j ) explicitly as in Eq. (6.41), the scaling factor q1∗ was set to W1 W1 q1∗ = 16(b + h)/ h 2 , making [ψ z,1 (s1 = h/2) − ψ z,1 (s1 = 0)] = (b + h)/5. W1 Figure 6.4b shows the shape of ψ z .

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

179

Step 3 Stress field update due to secondary displacement The displacement, strain, and stress fields approximated by ξ = {Uz , χ1 , W1 }T are u s (z, s) = ψsχ1 (s)χ1 (z), u z (z, s) = ψzUz (s)Uz (z) + ψzW1 (s)W1 (z),

(6.43a)

εzz (z, s) = ψzUz (s)Uz, (z) + ψzW1 (s)W1, (z), εss (z, s) = ψ˙ sχ1 (s)χ1 (z), γsz (z, s) = ψsχ1 (s)χ1, (z) + ψ˙ zW1 (s)W1 (z),

(6.43b)

[ ] σzz (z, s) = E 1 ψzUz (s)Uz, (z) + ψzW1 (s)W1, (z) + ν ψ˙ sχ1 (s)χ1 (z) , [ ] σss (z, s) = E 1 ψ˙ sχ1 (s)χ1 (z) + νψzUz (s)Uz, (z) + νψzW1 (s)W1, (z) , [ ] τzs (z, s) = G ψsχ1 (s)χ1, (z) + ψ˙ zW1 (s)W1 (z) . (6.43c) U χ χ By substituting ψ˙ s 1 = − p1∗ ψz z given in Eq. (6.19) and ψ˙ zW1 = q1∗ ψs 1 given in Eq. (6.34a) into Eq. (6.43c) and using a procedure similar to that used to derive Eqs. (6.26–6.27), one can express the stresses in terms of the generalized forces:

σzz (z, s) = σzzFz (z, s) + σzzB1 (z, s) =

B1 (z) W1 Fz (z) Uz ψ (s), ψz (s) + A JW1 z

τzs (z, s) = τzsQ 1 (z, s) =

Q 1 (z) χ1 ψs (s), Jχ1

(6.44a) (6.44b)

where the first-order warping bimoment B1 is defined as ∫ B1 (z) =

σzz (z, s)ψzW1 (s)dA = E 1 JW1 W1, (s).

(6.45)

As was noted with regard to the generalized forces of other modes, Eq. (6.44a) shows that the functional behavior of σzzB1 (the stress due to B1 ) is identical to that of its corresponding cross-section mode ψzW1 (s). One can expect that the second-order mode set {χ2 , W2 } can be derived from a recursive analysis by noting that σzzB1 generates its secondary strain due to Poisson’s effect. Because the derivation procedure for the second mode is generally equal to that used for the first mode set, it will be only briefly discussed.

180

6 Sectional Shape Functions for a Box Beam Under Extension

χ

6.2.3 Derivation of ψs 2 Due to Poisson’s effect, εss induced by σzz in Eq. (6.44a) becomes εss (z, s) = −ν

σzz (z, s) ν B1 (z) W1 ν Fz (z) Uz ψ (s). =− ψ (s) − E EA z E JW1 z

(6.46)

Using Eq. (6.18), the first term on the right-hand side of Eq. (6.46) can be idenχ tified as χ1 (z)ψ˙ s 1 (s). However, the second term cannot be explained without the introduction of a new wall-extension mode, χ2 . Thereby, u s is henceforth expressed as u s (z, s) = ψsχ1 (s)χ1 (z) + ψsχ2 (s)χ2 (z).

(6.47)

Using Eq. (6.47), the wall-extending strain is expressed as εss (z, s) =

∂u s (z, s) = ψ˙ sχ1 (s)χ1 (z) + ψ˙ sχ2 (s)χ2 (z). ∂s

(6.48)

Equating the strain in Eq. (6.46) and the strain in Eq. (6.48) yields ψ˙ sχ1 (s)χ1 (z) + ψ˙ sχ2 (s)χ2 (z) = −

ν Fz (z) Uz ν B1 (z) W1 ψz (s) − ψ (s). EA E JW1 z

(6.49)

If Eq. (6.19) is used, Eq. (6.49) is reduced to ( ) ν Fz (z) ν B1 (z) W1 ∗ χ2 ˙ ψs (s)χ2 (z) = − − p1 χ1 (z) ψzUz (s) − ψ (s). EA E JW1 z

(6.50)

U

Multiplying both sides of Eq. (6.50) by ψz z (s) and integrating the resulting equation over the beam cross-sectional area A yields ∗ p2,1 χ2 (z) =

(

) ν Fz (z) − p1∗ χ1 (z) , EA

(6.51)

where ∗ p2,1 =−

1 A

∫

ψ˙ sχ2 (s)ψzUz (s) dA

(6.52) χ

∗ Because p2,1 in Eq. (6.52) involves the unknown function ψs 2 (s) to be determined, is treated as a scaling factor and is be redefined as

∗ p2,1

∗ p2∗ ≡ p2,1 .

(6.53)

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

181

Multiplying both sides of Eq. (6.50) by ψzW1 (s) and integrating the resulting equation over A yields ∗ χ2 (z) = p2,2

ν B1 (z) , E JW1

(6.54)

where ∗ p2,2 =−

1 JW1

∫

χ ψ˙ sχ2 (s)ψzW1 (s)dA ≡ p2∗ D1 2 ,

(6.55)

with χ

(

D1 2 =

1 JW1

∫

ψ˙ sχ2 (s)ψzW1 (s)dA

)/ (

1 A

∫

χ

) ψ˙ sχ2 (s)ψzUz (s)dA . χ

Note that D1 2 is a constant that can be calculated once ψs 2 (s) is determined. Using Eq. (6.51), Eq. (6.50) can be rewritten as ] [ χ ∗ ψzW1 = − p2∗ ψzUz (s) − D1 2 ψzW1 (s) , ψ˙ sχ2 (s) = − p2∗ ψzUz − p2,2

(6.56a)

] [ U χ χ ψ˙ s,2j (s j ) = − p2∗ ψz, zj (s j ) + D1 2 ψz,Wj1 (s j ) ( j = 1, 2, 3, 4).

(6.56b)

or

χ

Equation (6.56) implies that the unknown function ψs 2 (s)(more specifically U χ2 ˙ ψs (s)) can be expressed as a linear combination of the known functions ψz z (s) χ2 W1 and ψz (s). By integrating Eq. (6.56) in an edgewise manner, ψs, j (s j ) is obtained as [ ] U χ χ χ ψs,2j (s j ) = p2∗ −Ψz, jz (s j ) − D 1 2 Ψz,Wj1 (s j ) + C j 2 . ( j = 1, 2, 3, 4) (6.57) χ

χ

To determine the five unknown coefficients D 1 2 and C j 2 ( j = 1, 2, 3, 4) appearing in Eq. (6.57) (except for the scaling factor p2∗ ), we use the symmetry χ condition for the deformation of ψs 2 (s) and the following orthogonality condition with respect to mode χ1 : ∫

χ

ψsχ1 ψsχ2 ds = 0.

(6.58)

The symmetry condition for ψs 2 (s) stems from that the symmetric stress field σzz in Eq. (6.44a), which always induces the symmetric deformation field of χ2 . Four independent conditions are given by Eqs. (6.23b), where χ1 should be replaced by χ2 in this case. Using these five conditions, the constants are obtained as

182

6 Sectional Shape Functions for a Box Beam Under Extension χ

2(b + h)(b4 + 4b3 h − 9b2 h 2 + 4bh 3 + h 4 ) , 75h 2 (b2 − bh + h 2 ) χ χ χ = C2 2 = C3 2 = C4 2 = 0.

D1 2 = χ

C1 2

(6.59)

χ

Figure 6.3c plots ψs 2 (s) defined in Eq. (6.57) with constants determined by Eq. (6.59). The scaling factor p2∗ is chosen as p2∗ = −15h 2 χ

b2 − bh + h 2 , 24b5 − 20b3 h 2 + 4h 5

(6.60)

χ

which normalizes ψs 2 via ψs 2 (s1 = h/2) = h/20. Because the sectional shape function of mode χ2 is determined, the shear stress τzs should be updated as τzs (z, s) = τzsQ 1 (z, s) + τzsQ 2 (z, s),

(6.61)

where the shear stress τzsQ 2 (z, s) due to the generalized force Q 2 can be expressed as τzsQ 2 (z, s) =

Q 2 (z) χ2 ψs (s). Jχ2

(6.62)

Equation (6.62) represents the generalized force-stress relationship for mode χ2 .

6.2.4 Derivation of ψ zW2 Employing Step 2 in Sect. 6.2.2 used to derive ψzW1 , one can describe the shear strain calculated from the updated shear stress in Eq. (6.61) by the displacement field of a newly introduced warping mode W2 . To this end, first we write the updated shear strain as γzs (z, s) =

Q 2 (z) χ2 Q 1 (z) χ1 ψ (s) + ψ (s). G Jχ1 s G Jχ2 s

(6.63)

At this stage, by updating the shear strain in Eq. (6.31) to include the secondorder warping mode W2 , the shear strain can be expressed in terms of the updated displacement field as γzs (z, s) = ψsχ1 (s)χ1, (z) + ψsχ2 (s)χ2, (z) + ψ˙ zW1 (s)W1 (z) + ψ˙ zW2 (s)W2 (z). Equating Eqs. (6.63) and (6.64) results in

(6.64)

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

183

) Q 1 (z) − χ1, (z) − q1∗ W1 (z) ψsχ1 (s) G Jχ ( 1 ) Q 2 (z) + − χ2, (z) ψsχ2 (s). G Jχ2 (

ψ˙ zW2 (s)W2 (z) =

(6.65)

Applying the procedure used to derive the first-order warping to the relationship in Eq. (6.65), one can find ψ˙ zW2 as ] [ χ χ ψ˙ z,Wj2 (s j ) = q2∗ ψs,1j (s j ) + D1W2 ψs,2j (s j ) , ( j = 1, 2, 3, 4)

(6.66)

where q2∗ is treated as a scaling parameter, which is defined as q2∗

1 = Jχ1

∫

ψ˙ zW2 ψsχ1 dA,

and D1W2 (treated as an unknown constant to be determined) is related to ψz,Wj2 (s j ) according to ( D1W2 =

1 Jχ2

∫

ψ˙ zW2 ψsχ2 dA

)/ (

1 Jχ1

∫

) ψ˙ zW2 ψsχ1 dA .

Integrating Eq. (6.66) with respect to s j yields χ

χ

2 ψz,Wj2 = q2∗ ( Ψs,1j + D1W2 Ψs,2j + C W j ), ( j = 1, 2, 3, 4).

(6.67)

2 To determine the five unknowns, D1W2 and C W j ( j = 1, 2, 3, 4), two symmetry conditions in Eq. (6.37), one continuity condition in Eq. (6.38), and the following two orthogonality conditions are used:

∫

∫ ψzW2 ψzUz

ds = 0 and

ψzW2 ψzW1 ds = 0.

(6.68)

Using these conditions, one can determine the unknowns and thus finally explicitly find ψzW2 as [ 2 −10bh+10h 2 ) 4 ψ z,Wj2 = q2∗ · 40(6b(10b 4 −6b3 h+b2 h 2 −bh 3 +h 4 ) s j { 4 −12b3 h−3b2 h 2 +3bh 3 −3h 4 ) + (12b 40(6b4 −6b3 h+b2 h 2 −bh 3 +h 4 )

}

+

3(b8 −2b7 h+3b6 h 2 +81b5 h 3 −165b4 h 4 +81b3 h 5 +3b2 h 6 −2bh 7 +h 8 ) 140(6b4 −6b3 h+b2 h 2 −bh 3 +h 4 )(b4 +4b3 h−9b2 h 2 +4bh 3 +h 4 )

+

(8b10 −24b9 h−132b8 h 2 +128b7 h 3 +211b6 h 4 −519b5 h 5 +456b4 h 6 −77b3 h 7 −72b2 h 8 +21bh 9 +3h 10 ) 2240(6b4 −6b3 h+b2 h 2 −bh 3 +h 4 )(b4 +4b3 h−9b2 h 2 +4bh 3 +h 4 )

( j = 1, 3)

s 2j

]

,

(6.69a)

184

6 Sectional Shape Functions for a Box Beam Under Extension

[ 2 −10bh+10h 2 ) 4 ψ z,Wj2 = q2∗ · 40(6b(10b 4 −6b3 h+b2 h 2 −bh 3 +h 4 ) s j { 4 +3b3 h−3b2 h 2 −12bh 3 +12h 4 ) + (−3b 40(6b4 −6b3 h+b2 h 2 −bh 3 +h 4 )

}

+

3(b8 −2b7 h+3b6 h 2 +81b5 h 3 −165b4 h 4 +81b3 h 5 +3b2 h 6 −2bh 7 +h 8 ) 140(6b4 −6b3 h+b2 h 2 −bh 3 +h 4 )(b4 +4b3 h−9b2 h 2 +4bh 3 +h 4 )

+

(3b10 +21b9 h−72b8 h 2 −77b7 h 3 +456b6 h 4 −519b5 h 5 +211b4 h 6 +128b3 h 7 −132b2 h 8 −24bh 9 +8h 10 ) 2240(6b4 −6b3 h+b2 h 2 −bh 3 +h 4 )(b4 +4b3 h−9b2 h 2 +4bh 3 +h 4 )

s 2j

( j = 2, 4)

] ,

(6.69b)

where q2∗ is determined as q2∗ =

2688b9 +10752b8 h−26432b7 h 2 +1792b6 h 3 +22848b5 h 4 −8512b4 h 5 −448b3 h 6 −4032b2 h 7 +1792bh 8 +448h 9 . 5h 2 (36b8 +96b7 h−431b6 h 2 +627b5 h 3 −480b4 h 4 +137b3 h 5 +24b2 h 6 −9bh 7 +h 8 )

(6.70) W2 W2 The value of q2∗ in Eq. (6.70) normalizes ψ zW2 (s) as [ψ z,1 (s1 = h/2) − ψ z,1 (s1 = 0)] = (b + h)/5. The function ψ zW2 is illustrated in Fig. 6.4d. We will now update the z-directional normal stress σzz by adding the new warping mode W2 as

σzz (z, s) =

σzzFz (z,

s) +

2 Σ

σzzBk (z, s)

k=1

Σ Bk (z) Fz (z) Uz ψz (s) + ψzWk (s), A J Wk k=1 2

=

(6.71)

where the bimoment B2 of the second-order warping is related to W2, as ∫ B2 (z) =

σzz (z, s) ψzW2 (s) d A = E 1 JW2 W2, (z).

(6.72)

Note again that the decoupled generalized force-stress relationship in Eq. (6.71) is possible because the sectional shape functions of the warping modes are set to be orthogonal to each other.

χ

Wk

6.2.5 Derivation of ψs k and ψ z

for k ≥ 3

The shape functions of higher-order modes with k ≥ 3 can be recursively derived similarly to those for k = 1 and k = 2 (see Fig. 6.2). The Nth-order distortion mode can be derived if the secondary wall-extending strain generated due to Poisson’s effect is set to be represented by the warping modes belonging to all of the lower-order sets.

χ

6.2 Derivation of ψs k and ψzWk by Means of a Recursive Analysis

185

χ

The relationship between the shape function ψs k of the Nth-order distortion mode and those of the warping modes in the lower-order sets can be obtained by extending Eq. (6.49) as N Σ

ψ˙ sχk (s)χk (z) = −

k=1

N −1

Σ ν Bk (z) ν Fz (z) Uz ψzWk (s). ψz (s) − EA E J W k k=1

(6.73)

χ

Following a procedure similar to that in Sect. 6.2.3, ψs N is derived as [ χ ψs,Nj (s j )

=

p ∗N

U −Ψz, jz (s j )

−

N −1 Σ

] χ Dk N

Ψz,Wjk (s j )

+

χ Cj N

k=1

(N ≥ 3),

(6.74) χ

where p ∗N is the scaling parameter. The unknown constants Dk N (k = 1, 2, . . . , N −1) χ and C j N ( j = 1, 2, 3, 4) can be determined by the four independent symmetry conditions given in Eqs. (6.23) and the following orthogonality conditions: ∫

ψsχ N (s) ψsχk (s) ds = 0 (k = 1, 2, . . . , N − 1).

The next step is to find the Nth-order warping mode W N . Following the procedure used to derive the shape function ψzW2 , one can find the following relationship: [ ψz,WjN (s j )

= q N∗

χ Ψs,1j (s j )

+

N −1 Σ

] χ DkW N Ψs,k+1 j (s j )

+

N CW j

(N ≥ 3).

(6.75)

k=1

where q N∗ is the scaling parameter. The unknown constants DkW N (k = 1, 2, . . . , N − N 1) and C W ( j = 1, 2, 3, 4) can be determined by two independent symmetry condij tions given in Eq. (6.37), a continuity condition given in Eq. (6.38), and the following orthogonality conditions: ∫

∫ ψzW N (s) ψzUz (s) ds = 0 and

(k = 1, 2, . . . , N − 1).

ψzW N (s) ψzWk (s) ds = 0

186

6 Sectional Shape Functions for a Box Beam Under Extension

y z

(a)

y

y x

z

(b)

x

z

y x

(c)

z

x

(d)

Fig. 6.5 Modes involving n-directional displacements needed for the structural analysis of a beam under an axial extensional load: a χ1 , b χ2 , c η1 (constrained distortion of type 1), and d ηˆ 1 (constrained distortion of type 2)

6.3 n-Directional Displacements and Resulting Stress and Strain Fields We derived the shape functions of the wall-membrane distortion modes in Sect. 6.2, which are primarily induced by Poisson’s effect from the warping modes belonging to the lower-order sets. If the distortion mode shape is represented by s-directional deforχ mation (ψs k ) alone, the displacements at the cross-section corners are not contin5 uous, as indicated by the deformation patterns in Fig. 6.5a, b. To ensure the displacement continuity, therefore, the s-directional displacement should be accompanied by χ n-directional displacement (ψn k ). It is apparent that the n-directional displacements of the unconstrained distortion modes are shown in Fig. 6.5a, b do not cause wall bending. In addition to this type of n-directional displacement, there can also be n-directional displacement which represents constrained distortion not involving any s-directional displacement. (Why constrained distortion modes are needed will be explained in Sect. 6.6.) The corresponding constrained distortion modes are sketched in Fig. 6.5c and d. As was done in relation to torsion problems in Chap. 5, the two types of constrained distorˆ Note that η(see Fig. 6.5c) represents a mode tion modes will be denoted by η or η. having zero corner rotation while η(see ˆ Fig. 6.5d) represents a mode having nonzero corner rotations. It is emphasized again that unlike unconstrained distortion modes χk , constrained distortion modes do not have any wall-membrane deformation (i.e., zero s- or z-directional displacement). This means that the s-directional displacements shown in Fig. 6.5a, b are accompanied only by unconstrained distortion modes {χk }k=1,··· ,N and not by constrained distortion modes {ηk , ηˆ k }k=1,··· ,N . The sectional shape functions of constrained distortion modes ({ηk , ηˆ k }k=1,··· , N ) will be derived in Sect. 6.6 by following the three-step derivation procedure discussed in Chap. 5. 5

The s-directional displacement of a vertical edge of a beam cross-section at its neighboring corners becomes the n-directional displacement of two horizontal edges. Therefore, no displacement continuity can arise if only the s-directional displacements are used.

6.3 n-Directional Displacements and Resulting Stress and Strain Fields

187

Based on the aforementioned arguments, the n-directional displacement due to modes {χk }k=1,...,N and {ηk , ηˆ k }k=1,..., N at an arbitrary point on a cross-section wall can be written as u˜ n (z, n, s) = u n (z, s) N [ ] Σ ψnχk (s)χk (z) + ψnηk (s)ηk (z) + ψnηˆ k (s) · ηˆ k (z) . =

(6.76)

k=1

In Eq. (6.76), χk represent the 1D generalized displacements associated with the unconstrained distortion modes while ηk and ηˆ k represent the 1D generalized displacements associated with the constrained distortion modes of types 1 and 2, respectively. The corresponding sectional shape functions will be derived in the following sections. Given that u˜ n (z, n, s) is uniform over the thickness direction, it is written as u n (z, s) without n-dependence in Eq. (6.76). Employing the kinematic assumption of the Kirchhoff thin plate theory (εnn = γns = γzn = 0), the three-dimensional wall-bending displacements {u z , u s } at a generic point (z, n, s) can be described as6 u z (z, n, s) = −nu ,n (z, s) N [ ] Σ = −n ψnχk (s)χk, (z) + ψnηk (s)η,k (z) + ψnηˆ k (s)ηˆ k, (z) ,

(6.77a)

k=1

u s (z, n, s) = −n u˙ n (z, s) = −n

N [ ] Σ ψ˙ nηk (s)ηk (z) + ψ˙ nηˆ k (s)ηˆ k (z) .

(6.77b)

k=1

ΣN χ χ In Eq. (6.77b), the terms −n [ k=1 (ψ˙ n k χk )] are omitted because ψ˙ n k = 0. (This will be shown in Sect. 6.5). Using u z and u s in Eq. (6.77), the wall-bending strains {ε zz , εss , γ zs } can be calculated as ε zz (z, n, s) = u ,z (z, n, s) [ N ] Σ ηk ,, ηˆ k ,, χk ,, = −n (ψn (s)χk (z) + ψn (s)ηk (z) + ψn (s)ηˆ k (z)) , (6.78a) k=1

εss (z, n, s) = u˙ s (z, n, s) [ N ] Σ ηˆ k ηk ¨ ¨ = −n (ψn (s)ηk (z) + ψn (s)ηˆ k (z)) ,

(6.78b)

k=1

The summation over k goes from 1 to N in Eq. (6.77), but the value of N for the χk modes and the value of N for (ηk , ηˆ k ) are not necessarily the same. However, we use the same N here to avoid lengthy expressions. In actual numerical implementations, different N values are typically used.

6

188

6 Sectional Shape Functions for a Box Beam Under Extension

γ zs (z, n, s) = u ,s (z, n, s) + u˙ z (z, n, s) [ N ] Σ = −n (2ψ˙ nηk (s)η,k (z) + 2ψ˙ nηˆ k (s)ηˆ k, (z)) .

(6.78c)

k=1

The wall-bending stresses {σ zz , σ ss , τ zs } at a generic point (z, n, s) can then be calculated from {ε zz , εss , γ zs } by employing the plane-stress assumption as σ zz (z, n, s) = E 1 (ε zz + νε ss ) [ N ] Σ ηˆ k ,, ηk ,, η χk ,, η ˆ = E 1 (−n) (ψn χk + ψn ηk + ψn ηˆ k + ν ψ¨ n k ηk + ν ψ¨ n k ηˆ k ) , k=1

(6.79a) σ ss (z, n, s) = E 1 (ε ss + νε zz ) [ N ] Σ η ˆ χ ,, ,, η ˆ ,, η η k k k k k = E 1 (−n) (ψ¨ n ηk + ψ¨ n ηˆ k + νψn χk + νψn ηk + νψn ηˆ k ) , k=1

(6.79b) [ τ zs (z, n, s) = Gγ zs = G(−n)

N Σ

] (2ψ˙ nηk η,k + 2ψ˙ nηˆ k ηˆ k, ) .

(6.79c)

k=1

To derive the generalized forces, the internal virtual work δU for a beam element (z1 ≤ z ≤ z2 ) is considered: ⎫ ⎧ ∫ z 2 ⎨∫ ⎬ (σ zz δε zz + σ ss δε ss + τ zs δγ zs )dA dz δU = (6.80) ⎭ z1 ⎩ A

Substituting Eq. (6.78) into Eq. (6.80) yields ∫ δU =

z2

{∫

} (σ zz δε zz + σ ss δεss + τ zs δγ zs ) dA } dz

z1

=

N [∫ Σ

τ zs (−2n ψ˙ nηk ) dA δ η˜ k +

∫

N [∫ Σ

σ zz (−n ψnχk ) dA δχk, +

k=1

∫

+

]z 2 z1

k=1

+

τ zs (−2n ψ˙ nηˆk ) dA δ ηˆ k

σ zz (−n

ψnηˆk ) dA δ ηˆ k,

]z 2 z1

∫

σ zz (−n ψnηk ) dA δ η˜ k,

6.3 n-Directional Displacements and Resulting Stress and Strain Fields

∫ +

z2

[ N ∫ Σ

z1

k=1

z1

k=1

σ ss (−n ψ¨ nηk ) −

∂τ zs (−2n ∂z

189

ψ˙ nηk ) dA δ η˜ k

] σ ss (−n ψ¨ nηˆk ) − ∂τ∂zzs (−2n ψ˙ nηˆk )d A δ ηˆ k dz ∫ z2 [Σ ∫ N ∫ + − ∂σ∂ zzz (−n ψnχk )dAδχk, + − ∂σ∂ zzz (−n ψnηk )dAδ η˜ k, ∫

+

∫ +

] − ∂σ∂ zzz (−n ψnηˆk )dAδ ηˆ k, dz

(6.81)

where integration by parts is conducted. To identify the generalized forces, we identify the 1D field variables that are components of the following vector U(z): U(z) = {{ηk (z), ηˆ k (z), χk, (z), η,k (z), ηˆ k, (z)}k=1,...,N }T .

(6.82)

In Eq. (6.82), the constrained distortion modes {ηk , ηˆ k } are included. Note that the derivatives of {χk , ηk , ηˆ k } are also included in U because Kirchhoff’s assumption is employed in Eq. (6.77). Here, the relationship between mode α (α = χk ,ηk , ηˆ k ) and its derivative α , is analogous to that between the vertical displacement and the bending rotation in the Euler beam theory. In fact, the derivatives in U represent the bending rotation of the section walls about the contour coordinate (s) and thus produce the three-dimensional displacement u z (z, n, s) at a generic point (z, n, s), as expressed in Eq. (6.77a). It now becomes possible to identify the 1D generalized forces considering the work conjugates of the generalized displacements defined in U(z): ∫ Rˆ k (z) = τ zs (−2n ψ˙ nηˆk ) dA ∫ ∫ χk Sk (z) = σ zz (−n ψn ) dA; T k (z) = σ zz (−n ψnηk ) dA; ∫ Tˆk (z) = σ zz (−n ψnηˆk ) dA ∫

R k (z) =

τ zs (−2n ψ˙ nηk ) dA;

(6.83)

In Eq. (6.83), the generalized forces {R k , Rˆ k }, associated with the shear stress τ zs , represent the work conjugates of the constrained distortion modes {ηk , ηˆ k } and the generalized forces {Sk , T k , Tˆk }, associated with the axial normal stress σ zz , represent the work conjugates of the derivatives {χk, , η,k , ηˆ k, }. It will be convenient to write the generalized forces in Eq. (6.83) as a vector F(z) F(z) = {{R k (z), Rˆ k (z), Sk (z), T k (z), Tˆk (z)}k=1,...,N }T

(6.84)

Note that all of the generalized forces in F are self-equilibrated, implying that none of them has a net resultant.

190

6 Sectional Shape Functions for a Box Beam Under Extension

6.4 Generalized Force-Stress Relationships for Constrained Distortion Modes {η k , ηˆ k } At this stage, we will derive generalized force-stress relationships for the constrained distortion modes {ηk , ηˆ k }. To this end, we consider the orthogonality conditions between modes α1 and α2 , where α1 , α2 ∈ {ηk , ηˆ k }k=1,2,...,N (α1 /= α2 ). Although these modes have non-zero displacement components, u s and u z (see Eq. (6.77)), we consider the orthogonality of the u s field only because it is the dominant displacement component7 producing in-plane wall-bending. The wall-bending deformations are illustrated in Figs. 6.5c, d. The imposition of orthogonality between u s for mode α1 and u s for mode α2 , where α1 , α2 ∈ {ηk , ηˆ k }k=1, 2,..., N (α1 /= α2 ), reduces to ∫

ψ˙ nα1 ψ˙ nα2 ds = 0 (α1 , α2 ∈ {ηk , ηˆ k }k=1, 2,..., N ; α1 /= α2 ).

(6.85)

If Eq. (6.85) is explicitly written out for modes η N and ηˆ N , it becomes ∫ ∫

ψ˙ nη N ψ˙ nηk ds = ψ˙ nηˆ N ψ˙ nηk ds = ∫

∫ ∫

ψ˙ nη N ψ˙ nηˆ k ds = 0 (1 ≤ k ≤ N − 1) for mode η N ,

(6.86a)

ψ˙ nηˆ N ψ˙ nηˆ k ds = 0 (1 ≤ k ≤ N − 1) for mode ηˆ N ,

(6.86b)

ψ˙ nηˆ N ψ˙ nη N ds = 0 between modes η N and ηˆ N .

(6.86c)

Substituting the stress-displacement relationship in Eq. (6.79c) into {R k , Rˆ k } in Eq. (6.83) and using the orthogonality conditions in Eqs. (6.86), one can find R k (z) = G Jηk η,k (z) and Rˆ k (z) = G Jηˆ k ηˆ k, (z),

(6.87)

where the second moments of inertia, Jηk for mode ηk and Jηˆ k for mode ηˆ k , are defined as ∫ ∫ ηk 2 ˙ (6.88) Jηk = (−2n ψn ) dA and Jηˆ k = (−2n ψ˙ nηˆk )2 dA. The generalized force-stress relationship between τ zs and {R k , Rˆ k } can be obtained by substituting the generalized force–displacement relationship in Eq. (6.87) into the stress-displacement relationship in Eq. (6.79c): τ zs (z, n, s) =

N [ Σ

ˆ

Rk Rk (z, n, s) + τ zs τ zs (z, n, s)

]

k=1

7

See the first paragraph of Sect. 6.6 for the justification of this argument.

χ

6.5 Derivation of ψn k of Distortion Mode χk and Related Analysis

=

N Σ

{

k=1

191

} ] ] Rˆ k (z) [ R k (z) [ η η ˆ −2n ψ˙ n k (s) . −2n ψ˙ n k (s) + Jηk Jηˆ k

(6.89)

The derivation of the sectional shape function of the constrained distortion modes (η N and ηˆ N ) will be presented in Sect. 6.6.

χ

6.5 Derivation of ψn k of Distortion Mode χk and Related Analysis This section is devoted to the derivation of the n-directional sectional shape function χ ψn k of mode χk and the related analysis, in this case the establishment of the generχ alized force-stress relationship. Recall that the s-directional shape function ψs k of mode χk was derived in Sects. 6.2.1, 6.2.3, and 6.2.5.

χ

6.5.1 Derivation of ψn k χ

We will use the following three-step approach to derive ψn k . Step 1. Identification of a secondary normal strain field due to Uz χ

Recall that in Sect. 5.3, we derived ψn 1 8 of the distortion modes for a box beam under a torsional load by considering the secondary strain induced from) the thickness( / W0 = B0 JW0 nψzW 0 (s) of the zerothwisely linearly varying axial stress field σ zz order torsional warping due to Poisson’s effect. For the current distortion mode for a box beam under an axial extensional load, we can follow the same approach used χ to derive ψn 1 of the χ1 mode for a box beam under a torsional load. To this end, we U U can extend the result in Eq. (6.12) (i.e., σzz z (z, s) = {Fz (z)/ A} ψz z (s)) so that it can be applied for a generic point P(z, n, s) with non-zero n as σ˜ zzUz (z, n, s) =

Fz (z) Uz ψ˜ z (n, s) ≙ σzzUz (z, s) + σ Uzzz (z, n, s), A

(6.90)

where ψ˜ z z (n, s) represents the shape function of Uz applicable to a point (n, s) of U U the cross-section. In Eq. (6.90), the stress σ˜ zz z defined at P(z, n, s) consists of σzz z and σ Uzzz , which represent the wall-membrane and wall-bending stress, respectively. U

χ

As mentioned earlier, we use the same notation ψn k to denote the n-directional shape functions of the distortion modes for a torsional and axial extensional load to avoid the use of an excess number of symbols. If necessary, we will refer to the unconstrained (or constrained) distortion mode needed to deal with a torsional load as the torsional unconstrained (constrained) distortion mode and will refer to that needed to deal with an axial extensional load as the extensional unconstrained (constrained) distortion mode.

8

192

6 Sectional Shape Functions for a Box Beam Under Extension U

Fig. 6.6 σ zzz on edges 1 and 2

t

σ zzU z = 0

on edge 2

edge 2

t

edge 1

edge 3 edge 4

σ% Uzzz

σ zzU z = 0

on edge 1

Because the rigid-body translation along the z-direction by Uz is uniform over the U U wall thickness, ψ˜ z z (n, s) in Eq. (6.90) should be identical to ψz z (s) in Eq. (6.12) Uz Uz ˜ (i.e., ψz (n, s) = ψz (s) = 1). From this observation, one can write σ˜ zzUz (z, n, s) = σzzUz (z, s) =

Fz (z) Uz ψz (s) , σ Uzzz (z, n, s) = 0. A

(6.91)

Equation (6.91) states that the linearly varying axial stress field σ Uzzz of the nominal Uz mode over the thickness direction is zero; see Fig. 6.6. Accordingly, the secondary strain ε ss stemming from Poisson’s effect is also zero: ε ss (z, n, s) = −ν

σ Uzzz (z, n, s) = 0. E

(6.92)

Despite the fact that the secondary strain εss is zero, this result will be utilized to χ derive ψn 1 , as presented in the next step. Step 2. Derivation of secondary displacement consistent with the secondary strain χ

χ

Even if ε ss (z, n, s) = 0, ψn 1 is not necessarily zero. To find ψn 1 , we follow the argument given in Step 2 in Sect. 5.3.1, in which the procedure used to derive the χ shape function ψn k 9 of the torsional distortion modes is presented. Following the procedure, the in-plane displacement u s (z, n, s) of the χ1 mode can be written as u s (z, n, s) = −n ψ˙ nχ1 (s)χ1 (z).

χ

(6.93)

Although the same symbol ψn for both the torsional and extensional distortion modes is used, its meaning should be interpreted correctly depending on the type of applied load, i.e., torsional or extensional.

9

χ

6.5 Derivation of ψn k of Distortion Mode χk and Related Analysis

193

/ Using the definitions of ε ss = ∂u s ∂s in Eq. (6.92) and u s (z, n, s) given by Eq. (6.93), one can write −n ψ¨ nχ1 (s)χ1 (z) = 0,

(6.94)

ψ¨ nχ1 (s) = 0.

(6.95)

i.e.,

The result in Eq. (6.95) is also valid for higher-order modes (χk , k = 2, . . . , N ). For example, for N = 2, u s can be written as u s (z, n, s) = −n

2 Σ

ψ˙ nχk (s)χk (z).

(6.96)

k=1

In this case, Eq. (6.94) should be replaced by 2 Σ

ψ˙ nχk (s)χk (z) = 0.

(6.97)

k=1

Note that in Eq. (6.97), the right-hand side is still zero because the generalized force Bk (k ≥ 1) does not produce a non-zero σ zz (thickness-directional variation of σzz ); i.e., σ zz = 0 due to Bk (k ≥ 1). χ

Because ψ¨ n 1 (s) = 0 according to Eq. (6.95), Eq. (6.97) yields ψ¨ nχ2 (s) = 0.

(6.98)

χ

Similarly, ψn k in any kth-order set can be obtained as χ ψ¨ nχk (s) = 0 or ψ¨ n,kj (s j ) = 0 ( j = 1, 2, 3, 4).

(6.99)

Integrating Eq. (6.99) twice yields χ

χ

χ

ψn,kj (s j ) = C j,1k s j + C j,0k ( j = 1, 2, 3, 4), χ

(6.100)

χ

where {C j,0k , C j,1k } are integration constants to be determined. χ χ To determine the eight unknown coefficients {C j,0k , C j,1k } (j = 1, 2, 3, 4) in Eq. (6.100), we use the following conditions: ψnχk (s j = s ∗ ) = ψnχk (s j = −s ∗ ) ( j = 1, 2, 3, 4),

(6.101a)

194

6 Sectional Shape Functions for a Box Beam Under Extension

ψnχk (s j ) = ψnχk (s j+2 ) ( j = 1, 2),

(6.101b)

ψnχk (s j = +s j ) = −ψsχk (s j+1 = −s j+1 ), ψnχk (s j+1 = −s j+1 ) = ψsχk (s j = +s j ) ( j = 1, 2, 3, 4),

(6.101c)

where s j is assumed to vary between −s j and s j ( −s j ≤ s j ≤ s j ). Equations (6.101a, χ 6.101b) represent the symmetric conditions of ψn k about the x and y axes, respectively. Equations (6.101c) require that the x- and y-directional displacements of χk χ be continuous at the cross-section corners. It should be noted that ψs k also appears χk χk in Eq. (6.101c) because the mode shape of χk consists of ψs and ψn . One can show that Eq. (6.101) has only eight independent conditions.10 Specifically, Eq. (6.101c) can yield only two additional conditions (e.g., Eq. (6.101c) with j = 1 can be used) if the conditions in Eqs. (6.101a, 6.101b) are used. Using the eight independent conditions from Eq. (6.101), therefore, one can find the following coefficients: χ

χ

χ

χ

C1,1k = C2,1k = C3,1k = C4,1k = 0, χ

(6.102a)

χ

C1,0k = C3,0k = −ψsχk (s2 = −b/2), χ

(6.102b)

χ

C2,0k = C4,0k = ψsχk (s1 = h/2).

(6.102c) χ

χ

Substituting (6.102) into Eq. (6.100) allows us to relate ψn k to ψs k as χ

ψn,1k = −ψsχk (s2 = −b/2), χ

ψn,2k = ψsχk (s1 = h/2), χ

ψn,3k = −ψsχk (s2 = −b/2), χ

ψn,4k = ψsχk (s1 = h/2). χ

(6.103a) (6.103b) (6.103c) (6.103d)

χ

If ψs 1 given in Eq. (6.24) is used, ψn 1 in Eq. (6.103) is explicitly written as χ

b , 20

(6.104a)

χ

h , 20

(6.104b)

χ

b , 20

(6.104c)

ψn,11 = ψn,21 = ψn,31 = 10

χ

Similar observations were made when deriving ψn k for the torsional distortion modes in Chap. 5.

χ

6.5 Derivation of ψn k of Distortion Mode χk and Related Analysis χ

ψn,41 =

195

h . 20

(6.104d)

χ

Because ψn k for any k (k = 1, . . . , N ) can be obtained as Eq. (6.103), Step 3 is no longer needed for the extensional unconstrained distortion χk (k = 1, . . . , N ).

6.5.2 Relationship Between the Generalized Force (Sk ) and Stress (σ zz ) χ

The ψn k field of the χk mode induces a non-zero σ zz , which represents the thicknessdirectional variation of σzz . This σ zz stress component causes the bending of the walls of a cross-section, deformation which cannot be explained without additional modes. The additional modes are represented by constrained distortion modes {ηk , ηˆ k }. To χ show why {ηk , ηˆ k } is needed, we will investigate how ψn k generates the non-zero σ zz stress field in detail. To this end, it is necessary to find the generalized force-stress relationships between Sk in Eq. (6.83), which is the work conjugate of χk, , and σ zz . This is done below. When a stress field only due to χk is considered, Eq. (6.99) becomes σ zz = −E 1 n

N Σ

(ψnχk χk,, ).

(6.105)

k=1 χ

Note that the shape {ψn k }k=1,2,...,N in Eq. (6.105) are not orthogonal ∫ functions χk χl to each other (i.e., A ψn ψn dA /= 0 for any k and l). Moreover, they are not χ independent of each other because the polynomial order of ψn k does not change with k. Therefore, the decoupled form of a generalized force-stress relationship cannot χ be directly obtained. To resolve this, we propose to split ψn k (for any k) into two χ χˆ independent mutually orthogonal functions {ψn , ψn }. This approach is possible χ χ χ because ψn k is piecewise constant for the s coordinate (e.g., ψn,11 = ψn,31 =b/20 and χ1 χ1 ψn,2 = ψn,4 = h/20, as given in Eq. (6.104)). Because b /= h in general, one can χ

χˆ

introduce the following two independent basis functions ψn and ψn : χ

χ

χ

χ

ψn,1 = ψn,3 = ψn,2 = ψn,4 = m 1 χˆ

χˆ

χˆ

χˆ

ψn,1 = ψn,3 = m 2 ; ψn,2 = ψn,4 = n 2 (n 2 /= m 2 )

(6.106) (6.107)

As one can choose m 1 and m 2 arbitrarily, we simply choose m 1 = m 2 = 1. To find n 2 , the following condition is used: ∫ A

ψnχ (s) · ψnχˆ (s)d A = 0,

(6.108)

196

6 Sectional Shape Functions for a Box Beam Under Extension

from which n 2 is determined as n2 = −

h hm 2 =− . b b

(6.109) χ

χˆ

Substituting Eq. (6.109) into Eqs. (6.106) and (6.107), one obtains ψn and ψn as χ

χ

χ

χ

ψn,1 = ψn,3 = ψn,2 = ψn,4 = 1,

(6.110)

h χˆ χˆ χˆ χˆ ψn,1 = ψn,3 = 1, ψn,2 = ψn,4 = − . b

(6.111)

χ

χˆ

χ

Using the two mutually orthogonal basis functions (ψn and ψn ) of {ψn k }k=1,···,N , is now rewritten as

χ ψn k

ψnχk = Ak ψnχ + Bk ψnχˆ ,

(6.112)

where the coefficients {Ak , Bk } can be explicitly determined as χ

χ

h · ψn k (s1 ) + b · ψn k (s2 ) , b+h χ χ b · ψn k (s1 ) − b · ψn k (s2 ) Bk = . (k = 1, . . . , N ) b+h

Ak =

(6.113)

χ

Using ψn k given in Eq. (6.112), σ zz in Eq. (6.105) can be written as σ zz = −E 1 n

N { } Σ ψnχ (Ak χk,, ) + ψnχˆ (Bk χk,, ) .

(6.114)

k=1

Because σ zz in Eq. (6.114) is expressed in terms of two orthogonal functions, the generalized force-stress relationships can now be obtained in a decoupled from. For example, S1 in Eq. (6.83) becomes ∫

σ zz (−n · ψnχ1 ) dA ∫ ∫ = A1 σ zz (−n · ψnχ ) dA + B1 σ zz (−n · ψnχˆ ) dA ] ] [ N [ N Σ Σ ,, ,, = E 1 A1 Jχ (Ak χk ) + E 1 B1 Jχˆ (Bk χk )

S1 =

k=1

where {Jχ , Jχˆ } are defined as

k=1

(6.115)

χ

6.5 Derivation of ψn k of Distortion Mode χk and Related Analysis

∫ Jχ =

(−n · ψnχ )2 dA,

∫ Jχˆ =

197

(−n · ψnχˆ )2 dA.

(6.116)

More generally, Sm can be calculated as ∫

σ zz (−n · ψnχm ) dA ∫ ∫ χ = Am σ zz (−n · ψn ) dA + Bm σ zz (−n · ψnχˆ ) dA ] ] [ N [ N Σ Σ = E 1 Am Jχ ( Ak χk,, ) + E 1 Bm Jχˆ (Bk χk,, )

Sm =

k=1

(m = 1, . . . , N ) (6.117)

k=1

which can be organized in matrix form as ⎡

⎤ A1 Jχ B1 Jχˆ ⎡ ⎤ N ⎢ ⎥ ⎢ A2 Jχ B2 Jχˆ ⎥ E Σ ,, (Ak χk ) ⎥ ⎥⎢ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ k=1 ⎥. S = J · V → ⎢ ⎥ = ⎢ A3 Jχ B3 Jχˆ ⎥⎢ N ⎦ Σ ⎥⎣ ⎢ ⎥ ⎢ . . ,, .. ⎦ E 1 (Bk χk ) ⎣ ⎦ ⎣ .. k=1 A N Jχ B N Jχˆ SN S1 S2 S3 .. .

⎤

⎡

(6.118)

In Eq. (6.118), {S, V, J} are an N × 1 vector, a 2 × 1 vector, and an N × 2 matrix, respectively. Because the matrix J is not square, V will be calculated using a least square method (or the pseudo inverse): ⎡

N Σ

⎤

( Ak χk,, ) ⎥

⎢ E 1 k=1 ⎥ V = (JT J)−1 JT S = J˜ S → ⎢ N ⎦ ⎣ Σ ,, E1 (Bk χk ) k=1

⎡ [ =

J˜11 J˜12 J˜13 · · · J˜1N J˜21 J˜22 J˜23 · · · J˜2N

]⎢ ⎢ ⎢ ⎢ ⎢ ⎣

S1 S2 S3 .. .

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

SN where J˜ =

[

J˜11 J˜21

J˜12 J˜22

J˜13 · · · J˜23 · · ·

] J˜1N =(JT J)−1 JT . J˜2N

(6.119)

198

6 Sectional Shape Functions for a Box Beam Under Extension

By substituting Eq. (6.119) into Eq. (6.114), the following generalized force-stress relationships can be obtained: ˆ

S S + σ zz σ zz ≡ σ zz

≡ −nψnχ E 1 ( = −n ψnχ

N Σ

( Ak χk,, ) − nψnχˆ E 1

k=1

S Jχ

)

( − n ψnχˆ

N Σ

(Bk χk,, )

k=1

)

Sˆ , Jχˆ

(6.120)

where S = Jχ

N Σ

( J˜1k Sk ), Sˆ = Jχˆ

k=1

N Σ

( J˜2k Sk ).

(6.121)

k=1 ˆ

χ

χˆ

S S Note in Eq. (6.120) that σ zz as (−n ψn (s)) and (−n ψn (s)) in the n-s and σ zz plane (or in the cross-section plane). The corresponding distributions are sketched in Fig. 6.7.

t

t

t σ

Fy zz

σ

S zz

ˆ

σ zzS

edge 2

t

t

t

edge 1

edge 3 edge 4

ψ Uz

z

Fy

σ zz ˆ

F

σ zzS

ˆ

σ zzS ˆ

S , and σ S on edges 1 and 2. Stresses σ S and σ S induce the pure Fig. 6.7 Distributions of σ zzz , σ zz zz zz zz

wall-bending modes of

η ψn k

and

ηˆ ψn k ,

respectively

η

ηˆ

6.6 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

η

199

ηˆ

6.6 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained Distortion Modes {η k , ηˆ k } To derive the sectional shape functions of the constrained distortion modes {ηk , ηˆ k }, Sˆ S it should be noted that the σ zz distributions given in Eq. (6.120) induce and σ zz secondary constrained distortion deformations, which require the introduction of new constrained distortion modes {ηk , ηˆ k } associated with an axial extensional load. To justify the necessity of introducing new modes, we examine the stress distribuSˆ S ) over the wall thickness in the n-direction in Fig. 6.7. (equivalently, σ zz tion of σ zz S S Considering edge 1, it is clear that σ zz < 0 for n > 0 and σ zz > 0 for n < 0. S Due to Poisson’s effect, the induced s-directional normal strain ε ss by σ zz becomes S εss = −νσ zz /E. Therefore, ε ss > 0 for n > 0 and εss < 0 for n < 0. The opposite sign of the ε ss field in the n-direction causes the bending of edge 1, which cannot be described without constrained distortion modes having a wall-bending u n field. To derive the sectional shape functions, a three-step approach is employed: Step 1: Identification of the secondary strain field due to the known (primary) stress field Step 2: Derivation of secondary displacement consistent with the secondary strain Step 3: Stress field update due to secondary displacement (only required for constrained distortion modes). η

ηˆ

Because the procedure used to derive {ψn k , ψn k } of the extensional constrained ηˆ η distortion modes is identical to that used to derive {ψn k , ψn k } of the torsional constrained distortion modes given in Chap. 5, we will not repeat the procedure ηˆ η here.11 Instead, we will directly start with the conditions that ψn k (s) and ψn k (s) ηk must meet. To determine ψn (s), the following conditions are used: ψ˙ nη N (s j = +s j ) = 0, ψ˙ nη N (s j+1 = −s j+1 ) = 0 ( j = 1, 2, 3, 4)

(6.122a)

ψ¨ nη N (s j = +s j ) = ψ¨ nη N (s j+1 = −s j+1 ) ( j = 1, 2, 3, 4)

(6.122b)

ψnη N (s j = +s j ) = 0, ψnη N (s j+1 = −s j+1 ) = 0 ( j = 1, 2, 3, 4)

(6.122c)

∫

ψ˙ nη N ψ˙ nηk ds =

∫

ψ˙ nη N ψ˙ nηˆ k ds = 0 (1 ≤ k ≤ N − 1),

(6.122d)

where Eqs. (6.122a) and (6.122b) denote the slope conditions and moment continuities at corners (see Fig. 6.7a). Because constrained distortion modes are considered,

11

If needed, more details are given in Choi and Kim (2019).

200

6 Sectional Shape Functions for a Box Beam Under Extension

edge 2

edge 3

Et 3 η ψ n ,2 12

ψ nη,2 = 0 ψ nη,1 = 0

edge 1

edge 4

Et 3 η ψ n ,1 12

moment (curvature) continuity

zero angle (a)

Et 3 ηˆ ψ n ,2 12

edge 2 −ψ nηˆ,2

edge 3

Et 3 ηˆ ψ n ,1 12

edge 1 −ψ nηˆ,1

edge 4

angle continuity

moment (curvature) continuity

(b) η

Fig. 6.8 Slope conditions and moment continuities for constrained distortion modes: a ψn k (type 1) and b

ηˆ ψn k

(type 2)

Eq. (6.122c) imposes zero translation at all corners. Equation (6.122c) represents the orthogonality conditions. η Because Eq. (6.122) states 2N + 3 conditions, one can assume ψn k (s) as η

N +1 Σ k=0

k=0

η

N +1 Σ

N +1 Σ

ψn,1N = ψn,3N =

ηN η · (s1 )2k ; ψn,2N = C˜ k+1

ηN η · (s3 )2k ; ψn,4N = C˜ k+1

k=0

N +1 Σ

ηN · (s2 )2k ; D˜ k+1

ηN · (s4 )2k D˜ k+1

(6.123)

k=0 η

where even functions of s j are used for ψn,k j (s j ) ( j = 1, 2, 3, 4) due to the symmetric nature of the deformation field observed in a box beam subjected to extensional η η η η loading. For the same reason, ψn,3k = ψn,1k and ψn,4k = ψn,2k . Note that the functions η ψn,k j (s j ) ( j = 1, 2, 3, 4) appearing in Eq. (6.123) are chosen to be polynomials of the same order, because the 2N + 4 coefficients associated with these polynomials can be uniquely determined using the conditions given in Eq. (6.122), except for an

η

ηˆ

6.6 Derivation of the Sectional Shape Functions {ψn k , ψn k } of Constrained …

201

η

arbitrarily selectable scaling coefficient. If C˜ NN+2 is chosen as a scaling constant and η η denoted by r ∗N (i.e., r ∗N ≙ C˜ NN+2 ), ψn N in Eq. (6.123) can be rewritten as [ η ψn,1N

=

r ∗N

·

N Σ

] ηN Ck+1

· (s1 ) + (s1 )

ηN Dk+1

· (s2 )

ηN Ck+1

· (s3 ) + (s3 )

ηN Dk+1

· (s4 )

2k

k=0

η ψn,2N

=

r ∗N

·

[ N +1 Σ [

η ψn,3N

=

r ∗N

·

=

r ∗N

·

;

k=0 N Σ

[ N +1 Σ

;

] 2k

] 2k

k=0

η ψn,4N

2N +2

2N +2

;

] 2k

(6.124)

k=0

where C η N ≙ (C˜ η N /r ∗N ) and D η N ≙ ( D˜ η N /r ∗N ). Thus, the 2N + 3 unknowns η η η {{CmN , DmN }m=1, 2, ..., N +1 , D NN+2 } can be determined using Eq. (6.122). For example, η ψn N for N = 1 can be explicitly written as η ψn,1j1 η

ψn,1j2

{ 1 4 16b s − = 2 · 5h h 2 j1 { 1 16b = 2 · 2 s j42 − 5h b

1 2 s + 2 j1 1 2 s + 2 j2

} h2 ( j1 = 1, 3); 16 } b2 ( j2 = 2, 4) 16 η

(6.125)

η

where the scaling constant r N∗ is chosen to normalize ψn N as ψn N (s1 = 0) = b/5. η Similarly, one can explicitly derive ψn k for any k ≥ 1.12 ηN ηˆ N Similarly as ψn , ψn can be expressed as even polynomial functions of order (2N + 2): [ ηˆ ψn,1N

=

rˆN∗

·

N Σ

] ηˆ N Ck+1

· (s1 ) + (s1 )

ηˆ N Dk+1

· (s2 )

2k

k=0

ηˆ ψn,2N

=

rˆN∗

·

[ N +1 Σ [

ηˆ ψn,3N

= rˆN∗ ·

,

]

k=0 N Σ

2N +2

2k

, ]

ηˆ N Ck+1

· (s3 )2k + (s3 )2N +2 ,

k=0 η

The computer code to calculate ψn k can be downloaded from https://github. com/SChoiKNU/Codes-for-HoBT-based-Finite-Element-Analysis; see the Matlab code, Sectional_Shape_Function_Derivation_Code.m, included in Codes_for_Box_Beam_Analysis \1_Mode _Shape_Derivation\2_Extensional_Modes\2nd_mode_set\ Type_1_Constrained_Distortion _Mode. 12

202

6 Sectional Shape Functions for a Box Beam Under Extension ηˆ ψn,4N

=

rˆN∗

·

[ N +1 Σ

] ηˆ N Dk+1

· (s4 )

.

2k

(6.126)

k=0

The (2N + 3) unknowns involved in Eq. (6.126) can be determined from the following conditions ψ˙ nηˆ N (s j = +s j ) = ψ˙ nηˆ N (s j+1 = −s j+1 ), ( j = 1, 2, 3, 4)

(6.127a)

ψ¨ nηˆ N (s j = +s j ) = ψ¨ nηˆ N (s j+1 = −s j+1 ), ( j = 1, 2, 3, 4)

(6.127b)

ψnηˆ N (s j = +s j ) = 0, ψnηˆ N (s j+1 = −s j+1 ) = 0, ( j = 1, 2, 3, 4)

(6.127c)

∫

ψ˙ nηˆ N ψ˙ nηk ds =

∫

ψ˙ nηˆ N ψ˙ nηˆ k ds =

∫

ψ˙ nηˆ N ψ˙ nη N ds = 0. (1 ≤ k ≤ N − 1). (6.127d)

Unlike Eq. (6.122a) imposing a zero slope at the wall corners, Eq. (6.127a) imposes slope continuities at the corners (see Fig. 6.7b). ηˆ If the scaling constant rˆN∗ is chosen to make ψn N (s1 = 0) = −b/5, one can find ηˆ 1 ψn as ηˆ

ψn,1j1 = − 5(10b2 h16bh 2 −bh 3 +h 4 ) [ 3 4 h · (2b2 +7bh−7h 2 ) s j1 − ηˆ

ψn,1j2 = ·

3(2b2 +bh−h 2 ) 2 s j1 2h

+

h(10b2 −bh+h 2 ) 16

] , ( j1 = 1, 3)

16bh 5(10b2 h 2 −bh 3 +h 4 )

[

(2h 2 +7bh−7b2 ) 4 s j2 b3

−

3(−b2 +bh+2h 2 ) 2 s j2 2b

+

b(b2 −bh+10h 2 ) 16

]

. ( j2 = 2, 4) (6.128)

ηˆ

The explicit expressions of ψn k for k ≥ 1 can be similarly found. For actual ηˆ numerical evaluations of ψn k at specific s values, the computer code mentioned in Footnote 12 can be used.

6.7 Finite Element Formulation Before presenting the finite element analysis using the higher-order section shapes derived above, it is convenient to group one-dimensional generalized displacements into out-of-plane (ξ O (z)) and in-plane (ξ I (z)) displacements as ξ O (z) = {Uz (z), W1 (z), . . . , W N (z)},

(6.129a)

6.8 Case Studies

203

ξ I (z) = {{χk (z), ηk (z), ηˆ k (z)}k=1,...,N },

(6.129b)

As mentioned in Sect. 3.2.2, linear interpolation for finite element implementation is employed to interpolate the out-of-plane displacement variables defined in ξ O (z). For the in-plane displacements, on the other hand, Hermite cubic functions are used to ensure that both the displacements and their z-directional derivatives can be continuous across element interfaces. Higher-order interpolation is needed because ∂u n /∂z induces wall-bending displacement u z (see Eq. (6.77a)), resulting in wall-bending stress σ zz which in turn is responsible for generating constrained distortional deformation. Therefore, accurate z-directional derivatives are important to obtain accurate HoBT-based results. Once the 1D generalized displacements are approximated using linear or Hermite interpolation, as expressed in matrix form by Eq. (3.66), the stiffness and mass matrices for the finite element analysis can be calculated using Eqs. (3.81) and (3.85), respectively (Bathe 1996; Kim and Kim 1999; Jang and Kim 2009b). Once the load vector in the form of Eq. (3.83) is determined, an element-level equilibrium equation of Eq. (3.78) can be obtained.

6.8 Case Studies The effectiveness of the HoBT for a box beam under an axial extensional load will be shown through numerical examples. The wall-membrane responses will be examined by solving a problem dealing with the applied axial force F z . The mixed responses of wall-membrane and wall-bending deformations will be also examined by solving a problem dealing with cases of applied surface traction {tzz , tzn }. In addition, a free vibration analysis based on the HoBT is performed. The following material properties are used for all examples considered here: Young’s modulus E = 200 GPa, Poisson’s ratio ν = 0.3, and density ρ = 7850 kg/m3 . Unless otherwise specified, the beam length and thickness are set to L = 1 m and t = 0.002 m, respectively, while different sectional dimensions (b, h) were used to examine the effects of section shapes on the accuracy of the solution. To check the validity and accuracy of the HoBT, the obtained results are compared to those by the Abaqus shell (S4 element) analysis (Hibbett et al. 1998) and the classical beam (or bar) theory,13 denoted here by Shell and CBT, respectively. Other higher-order beam-based results employing mode shapes calculated by the GBTUL (Bebiano et al. 2018) are also provided and are denoted by GBT. As reference results for all examples, we used fully converged shell analysis results obtained from square shell elements 6.25 mm × 6.25 mm in size. For example, a box beam with dimensions of L = 1 m, b = 0.05 m, and h = 0.1 m was discretized into 160 elements along the beam length for the HoBT-based displacement and stress 13

The classical theory considers only the non-deformable section mode (Uz ).

204

6 Sectional Shape Functions for a Box Beam Under Extension

analyses, respectively. For the shell-based finite element calculations, 7680 shell elements (160 elements along the beam axis) were for displacement analysis while 122,880 shell elements were used for stress analysis (640 elements along the beam axis). The numbers of the used elements for vibration analysis are the same as those for the stress analysis. Although rapidly varying local responses (or end effects), appearing near the beam boundaries, typically require finer discretization than other parts of a beam, a uniform mesh with sufficiently fine elements was adopted for convenience.

6.8.1 Case Study 1: Static Wall-Membrane Response by Axial Force Fz The problem considered in Case Study 1 is shown in Fig. 6.9a, where one end A is fixed while the other end B is subjected to Fz = 100 N. The beam section at end B is assumed to be rigid with no sectional deformation (or no higher-order modes) allowed. Wall-membrane displacements {u z (z, s), u x (z, s)} and stresses {σzz (z, s), σx x (z, s) and τzx (z, s)} are calculated along the measurement line of s2 = b/8 (sketched in Fig. 6.9a) and plotted in Fig. 6.9b–f. Because η and ηˆ are not needed in this calculation (N2 = 0, where N2 is the number of constrained distortion mode sets), the results obtained with m (m ≥ 1) sets of {χk , Wk }k=1,...,m corresponding to the wall-membrane field and Uz are denoted as HoBT (N1 = m), where N1 is the number of {χk , Wk } mode sets. Figure 6.9b, c show that u z as well as u x arise due to the effects of higher-order sectional deformations. These figures also show that HoBT (N 1 = 3) can capture the variations of {u z (z, s), u x (z, s)} nearly as accurately as Shell, while CBT cannot. Figures 6.9d–f demonstrate higher accuracy of HoBT compared to that of CBT; a rapid change of {σzz , σx x , τzx } near both ends is captured by HoBT (N 1 = 5) nearly as accurately as that by Shell. To examine the criteria to determine N 1 (the highest set number of the modes, {χk , Wk }k=1,...,N1 ), the problem in Fig. 6.9 is solved using different numbers of N 1 . As only wall-membrane responses are considered, the highest set number for constrained distortion modes, N 2 , is zero. The errors in the HoBT results with respect to the converged shell results are plotted in Fig. 6.10. The displacement u x and stress τzx are calculated at the point (z = 0.968 m, s2 = b/8) where the responses are maximal due to the end effect. The comparison suggests that the use of N 1 ≥ 3 for the displacement analysis and N 1 ≥ 5 for the stress analysis yield converged results with sufficient accuracy. Figure 6.10 shows that the HoBT with N 1 ≥ 3 can predict u x within error of 0.4% and that the HoBT with N 1 ≥ 5 can predict τzx within error of 1.6%.

6.8 Case Studies

205

measurement points (s2=b/8, n2=0)

y

HoBT (N1 =3) Shell

x z

A

HoBT (N1 =2)

HoBT (N1 =1) CBT

Fz=100 N

B

(b)

(a) Shell HoBT (N1 =3) HoBT (N1 =2) HoBT (N1 =1)

HoBT (N1 =3) Shell

σzz ( N/m2 )

HoBT (N1 =2) HoBT (N1 =5) Shell CBT HoBT (N1 =1)

HoBT (N1 =2)

HoBT (N1 =2) HoBT (N1 =5)

CBT

Shell

(d)

HoBT (N1 =2)

HoBT (N1 =1)

Shell HoBT (N1 =5)

HoBT (N1 =2) CBT

τ zx ( N/m2 )

σxx ( N/m2 )

(c)

HoBT (N1 =2) HoBT (N1 =1) CBT HoBT (N1 =5) HoBT (N1 =5) Shell Shell

HoBT (N1 =2)

HoBT (N1 =5) Shell

(e)

(f)

Fig. 6.9 Beam under axial force Fz at a cantilevered box beam (Only wall-membrane modes as for the higher-order modes are needed in this problem.) (Choi and Kim 2019)

6 Sectional Shape Functions for a Box Beam Under Extension

error (%)

206

Stress τzx Displacement ux

N1 (number of the used mode sets) Fig. 6.10 Convergence of displacement and stress in the beam problem shown in Fig. 6.8a (Choi and Kim 2019)

6.8.2 Case Study 2: Mixed Response of Wall-Membrane and Wall-Bending Deformations by Surface Traction {t zz , t zn } In Case Study 2, a cantilever box beam subject to complex forms of traction {tzz , tzn } at the free end is considered. The prescribed stress distributions are taken from the actual stress distribution occurring at the joint section of a T-joint structure (Figs. 6.11a, 6.12a, and 6.13a) or at a section of a beam with holes (Fig. 6.12b): • prescribed tzz in Fig. 6.11a: tzz (z, s j1 ) = [−{100/(b3 t + h 3 t)} · {s 2j1 − (h 2 /4)} + 100/(3bt + 3ht)] MPa ( j1 = 1, 3), tzz (z, s j2 ) = [−{100/(b3 t + h 3 t)} · {s 2j2 − (b2 /4)} + 100/(3bt + 3ht)] MPa ( j2 = 2, 4), • prescribed tzz in Fig. 6.12a: | | (tzz ) LC1 (z, s j ) = 0 MPa (0 ≤ |s j | < s j /4),

| | (tzz ) LC1 (z, s j ) = 100/(bt + ht) MPa (s j /4 ≤ |s j | < s j /2),

6.8 Case Studies

207

measurement points (s2=b/8, n2=0)

y

HoBT (N1 =1)

CBT Shell

x

HoBT (N1 =3)

z

A

HoBT (N1 =3) Shell HoBT (N1 =2) HoBT (N1 =1) CBT

tzz

B (a)

(b)

HoBT (N1 =2) HoBT (N1 =1) HoBT (N1 =2)

Shell HoBT (N1 =3)

CBT

σzz ( N/m2 )

Shell HoBT (N1 =3)

HoBT (N1 =2) HoBT (N1 =5) Shell CBT HoBT (N1 =1)

HoBT (N1 =2)

(d)

HoBT (N1 =2)

HoBT (N1 =5)

HoBT (N1 =2) CBT HoBT (N1 =5) Shell

(e)

Shell

τ zx ( N/m2 )

σxx ( N/m2 )

(c)

HoBT (N1 =1)

Shell HoBT (N1 =5)

HoBT (N1 =2)

Shell

HoBT (N1 =2)

HoBT (N1 =5)

HoBT (N1 =1) CBT HoBT (N1 =5) Shell

(f)

Fig. 6.11 a A beam subjected to applied traction tzz and b–f structural responses (displacement and stress) (Choi and Kim 2019)

208

6 Sectional Shape Functions for a Box Beam Under Extension y A

y

measurement points (s2=b/8, n2=0)

x z

17 mm

Location of the specific cross-section (z=983 mm)

A

measurement points (s2=b/8, n2=0)

x z

17 mm

( tzz ) LC2

Location of the specific cross-section (z=983 mm) B

B

( tzz ) LC1

(a)

(b)

Shell (LC2) HoBT CBT (LC1, 2) HoBT

(4sets, LC2)

(N =4, LC1)

Shell (LC2) HoBT (N =4, LC2)

Shell (LC1)

Shell (LC1, 2)

Shell (LC1)

HoBT (N =4, LC1, 2) CBT (LC1, 2)

(c)

(d)

HoBT (N =7, LC2) Shell (LC2)

CBT (LC1, 2) Shell (LC1)

τzx ( N/m2 )

Shell (LC2) HoBT (N =7, LC2) σzz ( N/m2 )

HoBT (N =4, LC1)

CBT (LC1, 2)

CBT (LC1, 2) HoBT (N =7, LC1) Shell (LC1)

HoBT (N =7, LC1)

(e)

(f)

Fig. 6.12 a, b Box beams subjected to applied tractions (tzz ) LC1 and (tzz ) LC2 , respectively. c, d Displacements and d, f stresses (Choi and Kim 2019)

6.8 Case Studies

209 measurement points (s2=b/8, n2=t/2)

y

scale: x5e−01

x A

z tzn tzz

B (a)

(b)

GBT (2 modes) Choi and Kim (1 mode)

CBT Shell

HoBT (2 modes)

HoBT (0 mode) Displacement uy (m)

Displacement uz (m)

HoBT (0 mode)

CBT Shell HoBT (2 modes) GBT (2 modes)

Choi and Kim (1 mode) Axial coordinate (m)

Axial coordinate (m)

(c)

(d)

GBT (6 modes) Shell HoBT (6 modes)

Choi and Kim (1 mode) Shell HoBT (6 modes) GBT (6 modes)

τzx ( N/m2)

σzz ( N/m2)

HoBT (0 mode) CBT

CBT Choi and Kim (1 mode)

HoBT (0 mode)

Axial coordinate (m)

Axial coordinate (m)

(e)

(f)

Fig. 6.13 a A box beam subjected to applied traction tzn , b the deformed shape, and c–f structural responses of the beam (Choi and Kim 2020)

210

6 Sectional Shape Functions for a Box Beam Under Extension

• prescribed tzz in Fig. 6.12b: (tzz ) LC2 (z, s j1 ) = 0 MPa ( j1 = 1, 3), (tzz ) LC2 (z, s j2 ) = 100/2bt MPa ( j2 = 2, 4), • prescribed tzz and tzn in Fig. 6.13a: tzz (z, s j1 ) = 0 MPa, tzn (z, s j1 ) = 0.5 MPa ( j1 = 1, 3), tzz (z, s j2 ) = 0.5/t MPa, tzn (z, s j2 ) = 0 MPa ( j2 = 2, 4). Because induced stresses vary rapidly along the axial direction (z-direction), these case studies are useful for investigating the effects of the number of mode sets employed on the accuracy of the solution. The structural responses of the beam for the applied traction case in Fig. 6.11a are given in Fig. 6.11b–f. Figures 6.11b, f show the wall-membrane displacements u z and {σzz , τzx } calculated along a line (s2 = b/8). These results show that the applied load induces rapidly varying fields near end B. More importantly, they show that HoBT (N 1 = 3, 5) can capture complex structural responses as accurately as Shell. Next, we examine the results for the cases of applied traction (tzz ) LC1 described in Fig. 6.12a and (tzz ) LC2 described in Fig. 6.12b. Although the load patterns of (tzz ) LC1 and (tzz ) LC2 are quite different, their axial resultant forces are identical. The {u z , u x } and {σzz , τzy } results calculated along the line of s2 = b/8 are given in Figs. 6.12c–f. More higher-order modes representing the wall-membrane field are clearly required to predict rapidly varying solution behaviors more accurately. The structural responses near end B are quite different depending on the load, being either (tzz ) LC1 or (tzz ) LC2 near end B. However, {u z , u x } and {σzz , τzy } including complicated end effects are accurately captured in the HoBT case (N 1 = 4, 7); the results by HoBT (N 1 = 7) are virtually identical to the Shell results. Figure. 6.13a depicts a cantilevered box beam subjected to the traction {tzz , tzn } at the free end. The deformed shape of the beam calculated by HoBT is shown in Fig. 6.13b. Figures 6.13c–f plot {u z , u y } and {σzz , τzx } along the line of the outer wall for s2 = b/8 and n 2 = t/2. In these figures, the results by HoBT employing m constrained distortion mode sets {ηk , ηˆ k }k=1,...,m are denoted as HoBT (2 m modes). Thus, the results by HoBT (0 mode) represent those obtained only with the Uz and {χk , Wk }1,2,...,N1 modes (N 1 = 3 or 5). For comparison, the results using the wallbending modes of GBT (Bebiano et al. 2018) and Choi and Kim (2016) are also provided. From Figs. 6.13c–f, it is readily apparent that the wall-bending traction induces a highly localized effect near the end, but it is accurately predicted by HoBT (2 or 6 modes). The results by HoBT (2 or 6 modes) are found to be comparable to those by the Shell and to the GBT results, whereas the results by CBT and by Choi and Kim (2016; 1 mode), and HoBT (0 mode) are not.

6.8 Case Studies

211

Table 6.1 Eigenfrequencies of a Box Beam (b = 0.125 m, h = 0.05 m, L = 0.5 m and t = 0.002 m)14 (Choi and Kim 2020) Theory

Mode 1st

2nd

3rd

4th

5th

Shell (Abaqus)

467.57

478.76

519.53

590.35

694.81

HoBT (N 1 = 3, N 2 = 1)

466.56 (0.22%)

477.96 (0.17%)

518.90 (0.12%)

589.99 (0.06%)

694.68 (0.02%)

GBT

470.8 8 (0.71%)

482.39 (0.76%)

523.64 (0.79%)

595.26 (0.83%)

700.50 (0.82%)

Choi and Kim

466.63 (0.20%)

478.08 (0.14%)

519.0 8(0.09%)

590.44 (0.02%)

695.65 (0.12%)

6.8.3 Case Study 3: Free Vibration Response In the previous case studies, it was shown that static displacements of box beams can be accurately captured by the proposed HoBT when N 1 ≥ 3 and N 2 ≥ 1 are used. This case study serves to confirm whether the criteria of N 1 ≥ 3 and N 2 ≥ 1 are also valid for a free vibration analysis of box beams. For this investigation, a beam with dimensions of b = 0.125 m and h = 0.05 m and L = 0.5 m and t = 2 mm is considered. Both ends of the beam are free from any restraints on sectional deformation. The free vibration responses of a box beam are analyzed first. In addition to the non-deformable section mode Uz , higher-order unconstrained distortion and warping modes {χk , Wk }k=1,2,3 and constrained distortion modes {ηk , ηˆ k }k=1 are employed for the analysis (i.e., N 1 = 3 and N 2 = 1). The eigenfrequencies for the lowest five modes calculated by HoBT (N 1 = 3, N 2 = 1) are given in Table 6.1. For comparison, those calculated by Choi and Kim (2016), the GBT, and by Shell are also included in the table. The corresponding mode shapes obtained by HoBT (N 1 = 3, N 2 = 1) and Shell are compared in Fig. 6.14. The results in Table 6.1 and Fig. 6.14 indicate that HoBT (N 1 = 3, N 2 = 1) can yield sufficiently accurate results for a vibration analysis comparable to those of Shell.

14

When the shell finite elements are used, not only the extension-related eigenfrequencies but also other eigenfrequencies representing torsion and bending are obtained. For the comparison here, only the extension-related eigenfrequencies are taken from the shell analysis.

212

6 Sectional Shape Functions for a Box Beam Under Extension

1st mode (Shell, 467.57 Hz)

1st mode (HoBT, 466.56 Hz)

3rd mode (Shell, 519.53 Hz)

3rd mode (HoBT, 518.90 Hz)

5th mode (Shell, 694.81 Hz)

5th mode (HoBT, 694.68 Hz)

Fig. 6.14. Three eigenmodes of the beam considered in Case Study 3. Among the five modes discussed in Table 6.1, the first, third, and fifth eigenmodes are plotted as the representative modes (Choi and Kim 2020)

References Bathe (1996) Finite element procedures. Prentice Hall Bebiano R, Camotim D, Gonçalves R (2018) GBTul 2.0—a second-generation code for the GBTbased buckling and vibration analysis of thin-walled members. Thin-Walled Struct 124:235–257 Choi S, Kim YY (2016) Analysis of two box beams-joint systems under in-plane bending and axial loads by one-dimensional higher-order beam theory. Int J Solids Struct 90:69–94

References

213

Choi S, Kim YY (2019) Consistent higher-order beam theory for thin-walled box beams using recursive analysis: membrane deformation under doubly symmetric loads. Eng Struct 197:109430 Choi S, Kim YY (2020) Consistent higher-order beam theory for thin-walled box beams using recursive analysis: edge-bending deformation under doubly symmetric loads. Eng Struct 206:110129 Hibbett HD, Karlsson BI, Sorensen EP (1998) ABAQUS/standard: user’s manual. Hibbitt, Karlsson & Sorensen Jang GW, Kim YY (2009a) Higher-order in-plane bending analysis of box beams connected at an angled joint considering cross-sectional bending warping and distortion. Thin-Walled Structures 47:1478–1489 Jang GW, Kim YY (2009b) Vibration analysis of piecewise straight thin-walled box beams without using artificial joint springs. J Sound Vib 326:647–670 Kim Y, Kim YY (2003) Analysis of thin-walled curved box beam under in-plane flexure. Int J Solids Struct 40:6111–6123 Kim YY, Kim JH (1999) Thin-walled closed box beam element for static and dynamic analysis. Int J Numer Meth Eng 45:473–490

Chapter 7

Sectional Shape Functions for a Box Beam Under Flexure

The sectional shape functions of a box beam subjected to a flexural load are derived in this chapter using a procedure similar to that presented in Chaps. 4–6. (Other approaches may be found in Ferradi and Cespedes (2014) and Bebiano et al. (2015)). As in the cases for torsional or extensional loads, three types of deformable section modes are considered in addition to rigid-body section modes: (1) warping modes {Wk }k=1,2,... , (2) unconstrained distortion modes {χk }k=1,2,... , and (3) constrained distortion modes {ηk , ηˆ k }k=1,2,... . The warping mode Wk has the z-directional shape function ψzWk (s) only, which depicts the wall-membrane deformations of a beam section. On the other hand, the unconstrained distortion mode χk has both the χ s-directional shape function ψs k (s) representing wall-membrane deformation and χ the n-directional shape function ψn k (s) representing wall-bending deformation. The constrained distortional modes ηk and ηˆ k have only the n-directional shape functions η ηˆ ψn k (s) and ψn k (s), respectively, representing wall-bending deformations. The shape χk functions ψs (s) and ψzWk (s) representing wall-membrane deformations are derived η ηˆ χ in Sect. 7.2 while ψn k (s), ψn k (s) and ψn k (s) representing wall-bending deformations are derived in Sects. 7.3–7.5. Section 7.6 presents numerical results using the derived modes. To give an idea of the shape functions to be derived, we begin with an illustration of some of the derived shape functions for a box beam subjected to a vertical flexural load in Fig. 7.1. Because the sectional shape functions of a box beam subjected to vertical and lateral flexural loads are derived in the same manner, we will mainly consider those for the vertical flexural loading case. Figure 7.1 shows these shape functions.

© Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_7

215

216

7 Sectional Shape Functions for a Box Beam Under Flexure

Uy

y z

θx

x

y

x z

y Rigid-body modes for vertical flexure

Cross section A

x z

Fy

χ1

b

y z

x

y

W1

x

y

χ2

z

z

W2 x

y x z

Unconstrained distortion and warping modes

η1

h t

ηˆ1

y z

x

y z

x

Constrained distortion modes

Fig. 7.1 Some of the sectional modes derived for the structural analysis of a box beam under a vertical bending load

7.1 General Field Relationships for Higher-Order Deformable Section Modes of a Wall-Membrane Field χ

This section presents the preliminary analysis needed to derive ψs k (s) and ψzWk (s), the section shape functions of the higher-order deformation modes corresponding to a wall-membrane field for a bar under a vertical flexural load.

7.1.1 Displacements, Stress, Strain Fields, and Generalized Forces For a thin-walled box beam under a vertical flexural load, there are two nondeformable section modes, U y and θx , representing the vertical displacement and rotation, respectively. If the wall-membrane displacements of higher-order modes {Wk , χk }k≥1 are also considered, we introduce the following 1D generalized displacement ξ(z) for the analysis of the box beam: { }T ξ(z) = U y (z), θx (z), χ1 (z), W1 (z), χ2 (z), W2 (z), . . . }T { = U y (z), θx (z), {χk (z), Wk (z)}k=1,2,... .

(7.1)

For the 1D generalized displacements considered in Eq. (7.1), one can express the corresponding wall-membrane displacement components describing the motion of the midline of the walls of the beam cross-section as

7.1 General Field Relationships for Higher-Order Deformable Section …

u z (z, s) = ψzθx (s)θx (z) +

N 

217

ψzWk (s)Wk (z),

(7.2a)

k=1 U

u s (z, s) = ψs y (s)U y (z) +

N 

ψsχk (s)χk (z).

(7.2b)

k=1

Note in Eq. (7.2) that ψzWk is the only non-zero shape function of mode Wk . χ χ Regarding mode χk , it has two non-zero components ψs k and ψn k , but the nχk directional component ψn is not considered here because it generates wall-bending deformation, which will be considered later. Because warping mode Wk and distortion mode χk are considered to be a pair,1 the highest number of these modes will be set to the same number N used in Eqs. (7.2a, 7.2b). The sectional shape functions of mode U y and mode θx are well known from the classical beam theory. They are expressed as U

U

U

U

ψs y (s1 ) = 1, ψs y (s2 ) = 0, ψs y (s3 ) = −1, ψs y (s4 ) = 0, ψzθx (s1 ) = s1 , ψzθx (s2 ) =

(7.3a)

h h , ψzθx (s3 ) = −s3 , ψzθx (s4 ) = − . 2 2

(7.3b)

Using Eq. (7.2) and Eq. (7.3), one can find the wall-membrane strains as  ∂ uz = ψzθx (s)θx, (z) + ψzWk (s)Wk, (z), ∂z k=1

(7.4a)

 ∂ us ψ˙ sχk (s)χk (z), = ∂s k=1

(7.4b)

N

εzz (z, s) =

N

εss (z, s) =

∂ uz ∂ us U + = ψs y (s)U y, (z) + ψ˙ zθx (s)θx (z) ∂z ∂s N  ( χk ) + ψs (s)χk, (z) + ψ˙ zWk (s)Wk (z) ,

γzs (z, s) =

(7.4c)

k=1 U U where ( ), = ∂()/∂z and (˙) = ∂()/∂s. Because ψ˙ s y = 0, the term ψ˙ s y U y is omitted in Eq. (7.4b). Using the strains in Eq. (7.4), one can express the membrane stresses as

σzz (z, s) = E 1 (εzz + νεss ) [ =

E 1 ψzθx (s)θx, (z)

+

N 

] (ψzWk (s)Wk, (z)

k=1

1

Refer to Sect. 7.2 to see why this can be justified.

+ ν ψ˙ sχk (s)χk (z)) ,

(7.5a)

218

7 Sectional Shape Functions for a Box Beam Under Flexure

σss (z, s) = E 1 (εss + νεzz ) [ =

E 1 νψzθx (s)θx, (z)

+

N 

] (ψ˙ sχk (s)χk (z)

+

νψzWk (s)Wk, (z))

,

(7.5b)

k=1

τzs (z, s) = Gγzs [ =G

U ψs y (s)U y, (z)

+ ψ˙ zθx (s)θx (z) +

N 

] (ψsχk (s)χk, (z) + ψ˙ zWk (s)Wk (z)) ,

k=1

(7.5c) where E 1 = E/(1 − ν 2 ). In Eq. (7.5), plane stress is assumed. To find the explicit expressions for generalized forces, the virtual work δW A done on the surface of a cross-section (A) by the surface traction types σzz and τzs is considered, δW A (z) = σzz (z, s)δu z (z, s)dA + τzs (z, s)δu s (z, s)dA. (7.6) A

A

Substituting Eq. (7.2) into u z and u s in Eq. (7.6) yields [ δW A (z) = +

U τzs ψs y d A

N ( 



σzz ψzθx d A · δθx

· δU y +

τzs ψsχk dA · δχk +

)]

σzz ψzWk dA · δWk

k=1

[

≡ Fy δU y + Mx δθx +

N 

] (Q k δχk + Bk δWk )

(7.7)

k=1

from which the generalized forces can be defined as }T { F(z) = Fy (z), Mx (z), {Q k (z), Bk (z)}k=1,2,...,N ,

(7.8)

where Fy (z) = Mx (z) = Q k (z) = Bk (z) =



U

τzs (z, s)ψs y (s)dA; σzz (z, s)ψzθx (s)dA; τzs (z, s)ψsχk (s)dA; σzz (z, s)ψzWk (s)dA (k = 1, 2, . . . , N ).

(7.9)

7.1 General Field Relationships for Higher-Order Deformable Section …

219

In Eq. (7.9), Fy and Mx are the vertical force and bending moment, which are the work conjugates of vertical displacement U y and bending rotation θx , respectively. The symbol Bk denotes the bimoment, which is the work conjugate of the warping Wk , and the symbol Q k represents the transverse bimoment, which is the work conjugate of the distortion χk .

7.1.2 Generalized Force-Stress Relationship of the Zeroth-Order Modes In this subsection, the relationship between the stress and the generalized force for the zeroth-order mode, i.e., the non-deformable section mode, will be established. To this end, we use Eq. (7.5a) to write σzzθx , which is the axial stress (σzz ) due to the θx field, as σzzθx (z, s) = E 1 ψzθx (s)θx, (z).

(7.10)

Using the second expression in Eq. (7.9), one can express the generalized force Mx , the work conjugate of θx , as Mx (z) =



=

σzzθx (z, s)ψzθx (s)dA E 1 [ψzθx (s)]2 dA · θx, (z) = E 1 Jθx θx, (z),

(7.11)

where the second moment of inertia Jθx is defined as Jθx =

(ψzθx )2 dA =

1 2 h t (3b + h). 6

Substituting Eq. (7.11) into Eq. (7.10), σzzθx can be rewritten as σzzθx (z, s) =

Mx (z) θx Mx (z) θx ψz (s) → σzzMx (z, s) = ψz (s) Jθx Jθx

(7.12)

where the symbol σzzMx is used instead of σzzθx because normal stress is generated by the applied bending moment Mx . Note that a direct relationship between σzzMx and Mx is possible because the sectional shape functions of all higher-order modes will be chosen to be orthogonal among themselves and to those of the U y and θx modes. From Eq. (7.5c), one can consider the following shear stress contribution only from the non-deformable section modes: τzs (z, s) = G(ψs y (s)U y, (z) + ψ˙ zθx (s)θx (z)). U

(7.13)

220

7 Sectional Shape Functions for a Box Beam Under Flexure

To simplify the right-hand side of Eq. (7.13), the following relationship, which can be found from the shape functions of the non-deformable section modes given in Eq. (7.3), can be used: U ψ˙ zθx (s) = ψs y (s).

(7.14)

Substituting Eq. (7.14) into Eq. (7.13) yields τzs = Gψs y (U y, + θx ). U

(7.15)

If one considers Fy as defined in Eq. (7.9) and utilizes the statement made pertaining to the orthogonality in the paragraph below about Eq. (7.12), one can write Fy as Fy (z) =



=

U

τzs (z, s)ψs y (s)dA G[ψs y (s)]2 dA·{U y, (z) + θx (z)} = G JU y {U y, (z) + θx (z)}. U

(7.16)

Substituting Eq. (7.16) into Eq. (7.15) makes it possible to express the shear stress associated with the non-deformable sectional modes (U y and θx ) as U &θx

τzs y

(z, s) =

Fy (z) U y Fy (z) U y F ψs (s) → τzsy (z, s) = ψs (s), JU y JU y

F

(7.17)

U &θ

where the symbol τzsy is used instead of τzs y x because shear stress is generated by the applied force Fy . The moment of inertia JU y is defined as JU y =

( ) U 2 ψs y dA = 2ht.

(7.18)

χ

7.2 Derivation of ψs k (s) and ψ zW k (s) by Means of a Recursive Analysis This section presents the results of a recursive analysis to derive the sectional shape χ functions of higher-order distortion and warping, ψs k and ψzWk (k ≥ 1). These functions represent a wall-membrane field for a box beam under a vertical flexural load. Starting with displacement and stress by the rigid-body bending rotation θx with its work conjugate Mx , one can find the secondary strain field εss using the stress σzzθx (z, s) = σzzMx (z, s) in Eq. (7.12) due to Poisson’s effect. This εss strain field can be explained in terms of displacement only when a new mode χ1 is introduced. The

χ

7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis

a

b

221

c

y x z

u z = ψ zθ x θ x

ε ss by Poisson’s effect e

χ1 generated d

Next set

W1 generated

γ zs by χ1′

Fig. 7.2 Illustration of the sequential generation mechanism of the first-order unconstrained distortion mode χ1 and warping mode W1 from the θx field χ

desired s-directional shape function ψs 1 representing wall-membrane deformation can be determined using symmetry, corner continuity, and orthogonality conditions. χ Then, shear strain and stress due to the added u s field of ψs 1 can be used to determine W1 the shape function ψz of a new warping mode, W1 , as was done to derive ψzW1 for torsional and extensional loads in Chaps. 4 and 6, respectively. Figure 7.2 illustrates the sequential generation of W1 and χ1 starting from θx , which is also explained in χ the flow chart shown in Fig. 7.3. Once ψs 1 and ψzW1 are found, Eq. (7.9) can be used to calculate the generalized forces Q 1 and B1 of the higher-order modes χ1 and W1 .

χ

7.2.1 Derivation of ψs 1 Step 1. Identification of a secondary normal strain field due to the θx mode The stress σzz in Eq. (7.12) stemming from the applied bending moment Mz generates secondary strain εss due to Poisson’s effect (see figure b in Fig. 7.2 and block B in Fig. 7.3):

222

7 Sectional Shape Functions for a Box Beam Under Flexure

k =1 A

B

C

D

E

u z by bending rotation θ x ε ss by Poisson’s effect Distortion of the k-th order generated

u z by warping

of the (k-1)th order

Shear strain γ zs generated by us

k = k +1 Warping of the k-th order generated

k≤N End χ

Fig. 7.3 Recursive derivation of the shape functions (ψzWk and ψs k ) representing the wallmembrane field of higher-order unconstrained distortion and warping modes for a box beam subjected to a vertical flexural load

εss (z, s) = −ν

σzz (z, s) ν Mx (z) θx =− ψz (s). E E Jθx

(7.19)

The existence of this strain implies that wall extension or wall shrinkage can occur in the cross-section. The section mode to be introduced to account for this in-plane wall-extension/shrinkage deformation is denoted by χ1 . Step 2. Derivation of secondary displacement consistent with the secondary strain If the χ1 mode is considered, the s-directional displacement can be written as (see Eq. (7.2b)): u s (z, s) = ψs y (s)U y (z) + ψsχ1 (s)χ1 (z). U

Using u s in Eq. (7.20), εss is updated as

(7.20)

χ

7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis

εss (z, s) =

223

∂u s (z, s) = ψ˙ sχ1 (s)χ1 (z), ∂s

(7.21)

U where ψ˙ s y = 0 is used. Matching the strains by Eqs. (7.19) and (7.21) yields

χ1 (z)ψ˙ sχ1 (s) = −

ν Mx (z) θx ψz (s). E Jθx

(7.22)

Equation (7.22) holds if the following relationships are met: χ ν Mx (z) ψ˙ s 1 (s) ≡ − p1∗ . = − θx E Jθx χ1 (z) ψz (s)

(7.23)

χ

where p1∗ is a scaling constant. From Eq. (7.23), ψs 1 and ψzθx are related as ψ˙ sχ1 (s) = − p1∗ ψzθx (s),

(7.24a)

which can also be written edgewise as ψ˙ sχ1 (s j ) = − p1∗ ψzθx (s j ) ( j = 1, 2, 3, 4),

(7.24b)

The integration of Eq. (7.24b) over s j yields χ

ψs,1j (s j ) = p1∗



( ) χ − ψz,θxj (s j ) ds j  p1∗ −ψz,θx j (s j ) + C j 1 .

(7.25)

χ

In Eq. (7.25), C j 1 (j = 1, 2, 3, 4) are integral constants and ψz,θx j is defined as. ψz,θx j (s j )

=

ψz,θxj (s j ) ds j (no integral constant).

If Eq. (7.3b) is employed, ψz,θx j can be found as ψz,θx 1 =

1 2 h 1 h s , ψz,θx 2 = s2 , ψz,θx 3 = − s32 , ψz,θx 4 = − s4 . 2 1 2 2 2

(7.26)

χ

Because σzzθx in Eq. (7.12) inducing ψs 1 is symmetric about the y axis and antiχ symmetric about the x axis (see figure A in Fig. 7.2), ψs 1 should have symmetric θx conditions identical to those of σzz : ψsχ1 (s j = s ∗ ) = ψsχ1 (s j = −s ∗ ) for j = 1, 3,

(7.27a)

ψsχ1 (s j = s ∗ ) = −ψsχ1 (s j = −s ∗ ) for j = 2, 4,

(7.27b)

224

7 Sectional Shape Functions for a Box Beam Under Flexure

ψsχ1 (s j ) = −ψsχ1 (s j+2 ) ( j = 1, 2),

(7.27c)

where s ∗ represents the s coordinate of an arbitrary point on edge j. Equation (7.27a) χ implies anti-symmetry of ψs 1 about the x axis on edges 1 and 3, while Eq. (7.27b) χ1 implies symmetry of ψs about the y axis on edges 2 and 4. Equation (7.27c) denotes χ the symmetric (j = 1) and anti-symmetric (j = 2) conditions of ψs 1 between two θx facing edges of a rectangular cross-section. Because ψz, j for j = 1 and j = 3 in χ Eq. (7.26) are even functions, ψs 1 in Eq. (7.25) automatically satisfies Eq. (7.27a). χ χ χ χ Using Eqs. (7.27b) and (7.27c), we can find C2 1 = C4 1 = 0 and C1 1 = −C3 1 . As χ1 χ an additional condition needed to determine C j , we use the orthogonality of ψs 1 U

with respect to ψs y :

ψsχ1 ψs y ds = 0. U

(7.28)

If p1∗ in Eq. (7.25) is set as p1∗ = − χ

1 , 25bh

χ

χ

(which normalizes ψs 1 as ψs 1 (s2 = −b/2) = −1/100), ψs 1 can be found via j−1

χ ψs,1j (s j )

(−1) 2 = 25bh

[

] 1 2 h2 ( j = 1, 3), s − 2 j 24

(7.29a)

j−2

χ ψs,1j (s j )

(−1) 2 h s j ( j = 2, 4). = 25bh 2

(7.29b)

χ

The shape function ψs 1 is plotted in Fig. 7.4a.

y z

(a)

y

y x

x

z

(b)

z

(c)

y x

x

z

(d)

Fig. 7.4 Sectional shape functions for a box beam under a vertical flexural load: a first-order χ unconstrained distortion (ψs 1 ), b first-order warping (ψzW1 ), c second-order unconstrained distortion χ2 (ψs ), and d second-order warping (ψzW2 )

χ

7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis

225

Step 3. Stress field update due to secondary displacement When the χ1 field is considered in addition to θx field, the displacements, stresses, and strains are updated as u s (z, s) = ψs y (s)U y (z) + ψsχ1 (s)χ1 (z), U

(7.30a)

u z (z, s) = ψzθx (s)θx (z), εzz (z, s) = ψzθx (s)θx, (z), εss (z, s) = ψ˙ sχ1 (s)χ1 (z), γsz (z, s) =

U ψs y (s)U y, (z)

(7.30b) + ψsχ1 (s)χ1, (z) + ψ˙ zθx (s)θx (z),

σzz (z, s) = E 1 (ψzθx (s)θx, (z) + ν ψ˙ sχ1 (s)χ1 (z)), σss (z, s) = E 1 ( ψ˙ sχ1 (s)χ1 (z) + νψzθx (s)θx, (z)), τzs (z, s) =

U G( ψs y (s)U y, (z)

+

ψsχ1 (s)χ1, (z)

+

(7.30c) ψ˙ zθx (s)θx (z)).

χ

If the relationships ψ˙ s 1 = − p1∗ ψzθx given in Eq. (7.24) and ψ˙ zθx = ψs y given in Eq. (7.14) are used, σzz and τzs can be simplified as U

σzz (z, s) = E 1 ψzθx (s)[ θx, (z) − p1∗ νχ1 (z)], [ ] U τzs (z, s) = G ψs y (s){U y, (z) + θx (z)} + ψsχ1 (s)χ1, (z) ,

(7.31a) (7.31b)

and the generalized forces defined in Eqs. (7.9) can be calculated as Mx (z) =

σzz (z, s)ψzθx (s)dA = E 1 Jθx [θx, (z) − p1∗ νχ1 (z)],

Fy (z) =

τzs (z, s)ψs y (s)dA = G JU y [U y, (z) + θx (z)], U

Q 1 (z) =

τzs (z, s)ψsχ1 (s)d A = G Jχ1 χ1, (z).

(7.32a) (7.32b) (7.32c)

Using Eq. (7.32), the shear stress in Eq. (7.31b) can be split into two components as F

τzs (z, s) = τzsy (z, s) + τzsQ 1 (z, s),

(7.33)

where τzsQ 1 is given by τzsQ 1 (z, s) =

Q 1 (z) χ1 ψs (s). Jχ1

(7.34)

226

7 Sectional Shape Functions for a Box Beam Under Flexure

Note that Eq. (7.34) represents the generalized force-stress relationship between Q 1 and the corresponding shear stress τzsQ 1 .

7.2.2 Derivation of ψ zW1 The first-order warping mode, W1 , can also be derived by following three steps previously mentioned, which is shown in Fig. 7.2 (blocks C and D) and Fig. 7.3 (blocks C and D). Step 1. Identification of a secondary shear strain field due to the χ1 mode }T { Shear stress τzs in Eq. (7.33) due to generalized forces F = Fy , Q 1 induces shear strain according to the equation below. γzs (z, s) =

] Fy (z) U y Q 1 (z) χ1 1 [ Fy τzs (z, s) + τzsQ 1 (z, s) = ψs (s) + ψ (s). G G JU y G Jχ1 s

(7.35)

Because the shear strain distribution in Eq. (7.35) cannot be represented only in terms of ξ = {U y , θx , χ1 }T , we need to introduce a new warping mode, in this case, the first-order warping mode, W1 . Step 2. Derivation of secondary displacement consistent with the secondary strain When an additional field by W1 is considered, the displacements and shear strain can be updated as u s (z, s) = ψs y (s)U y (z) + ψsχ1 (s)χ1 (z), U

u z (z, s) = ψzθx (s)θx (z) + ψzW1 (s)W1 (z),

(7.36)

and γzs (z, s) =

∂u ∂u s + z ∂z ∂s

= ψs y (s)U y, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zθx (s)θx (z) + ψ˙ zW1 (s)W1 (z). (7.37) U

Equating Eq. (7.35) and Eq. (7.37) yields U ψs y (s)U y, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zθx (s)θx (z) + ψ˙ zW1 (s)W1 (z) Fy (z) U y Q 1 (z) χ1 = ψs (s) + ψ (s). G JU y G Jχ1 s U If ψ˙ zθx = ψs y in Eq. (7.14) is utilized, Eq. (7.38a) reduces to

(7.38a)

χ

7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis

ψ˙ zW1 (s)W1 (z) =

(

227

) ( ) Fy (z) Q 1 (z) U − U y, (z) − θx (z) ψs y (s) + − χ1, (z) ψsχ1 (s). G JU y G Jχ1 (7.38b)

U χ To derive the relationship between ψ˙ zW1 (s) and {ψs y (s), ψs 1 (s)}, we multiply χ1 and ψs (s) by both sides of Eq. (7.38b) and integrate the resulting equations over the beam cross-sectional area A: ( ) Fy (z) , ∗ − U y (z) − θx (z) , (7.39a) q11 W1 (z) = G JU y ( ) Q 1 (z) ∗ , − χ1 (z) , (7.39b) q12 W1 (z) = G Jχ1 U ψs y (s)

where ∗ q11

1 = JU y



U ∗ = ψ˙ zW1 ψs y dA, q12

1 Jχ1



ψ˙ zW1 ψsχ1 dA

(7.40)

If the functions of z appearing on the right side of Eq. (7.38b) are eliminated using Eq. (7.39), Eq. (7.38b) reduces to: ] [ U ψ˙ zW1 (s) = q1∗ ψs y (s) + D1W1 ψsχ1 (s) ,

(7.41a)

or ] [ U ψ˙ zW1 (s j ) = q1∗ ψs y (s j ) + D1W1 ψsχ1 (s j ) ,

( j = 1, 2, 3, 4),

(7.41b)

∗ is a scaling constant that can be chosen arbitrarily and D1W1 is defined where q1∗ = q11 W1 ∗ ∗ /q11 . The coefficient D1W1 is an unknown constant to be determined. as D1 = q12 Integrating Eq. (7.41b) yields

[ ] U 1 ψzW1 (s j ) = q1∗ ψs y (s j ) + D1W1 ψsχ1 (s j ) + C W ( j = 1, 2, 3, 4), j

(7.42)

α 1 where C W j are integral constants ( j = 1, 2, 3, 4) and ψs, j is defined as

ψs,α j (s j ) =



ψs,α j (s j ) ds j (no integral constant) α = U y , χ1 .

W1 1 To determine C W j and D1 appearing in Eq. (7.42), we consider the symmetry W1 conditions of ψz first. To this end, it should be noted that the shear stress induced by Q 1 in Eq. (7.35) is symmetric about the y axis and anti-symmetric about the x

228

7 Sectional Shape Functions for a Box Beam Under Flexure

axis. Therefore, the shape function ψzW1 induced by shear stress should also have the same conditions. These can be stated as ψzW1 (s j = s ∗ ) = −ψzW1 (s j = −s ∗ ) for j = 1, 3,

(7.43a)

ψzW1 (s j = s ∗ ) = ψzW1 (s j = −s ∗ ) for j = 2, 4,

(7.43b)

ψzW1 (s j ) = −ψzW1 (s j+2 ) ( j = 1, 2).

(7.43c)

It should also be noted that the conditions in Eq. (7.43) for the z-direction displacement differ from those given in Eq. (7.27) for the s-directional displacement. An examination of Eq. (7.43) shows that Eq. (7.43b) is automatically satisfied because U χ ψs, yj and ψs,1j in Eq. (7.42) are even functions. Using Eqs. (7.43a) and (7.43c), one can find that C1W1 = C3W1 = 0 and C2W1 = −C4W1 . Still, two additional conditions are 1 needed to determine D1W1 and C W j (j = 2 or 4) in Eq. (7.42). Because ψzW1 should be continuous at every corner of the beam cross-section, W1 ψz (s j ) and ψzW1 (s j+1 ) defined on adjacent edges should have the same value at their shared corner: ψzW1 (s j = +s j ) = ψzW1 (s j+1 = −s j+1 ) ( j = 1, 2, 3, 4),

(7.44)

where s j denotes the bound of s j (−s j ≤ s j ≤ s j ). Because only one of the conditions in Eq. (7.44) is independent of Eq. (7.43), we use the following orthogonality as the last condition:

4 

sj

ψzW1 ψzθx ds =

j=1−s

ψz,Wj1 (s j )ψz,θx j (s j ) ds j = 0.

(7.45)

j

Equation (7.45) implies that the W1 mode is orthogonal to the θx mode. Using five independent conditions from Eqs. (7.43–7.45), the unknowns in Eq. (7.42) are found as C 1W1 = C 3W1 = 0, h(15b3 + 15b2 h − 2h 3 ) , 60b3 + 4h 3

(7.46b)

150h(3b + h) . 15b3 + h 3

(7.46c)

C 2W1 = −C 4W1 = − D 1W1 =

(7.46a)

Using Eq. (7.46), the non-zero shape function of the first-order warping mode W1 can be explicitly written as

χ

7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis

229

[ ] (−1) 2 (40h + 120b)s 3j + (60b3 − 30bh 2 − 6h 3 )s j 3 3 100h(15b + h ) ( j = 1, 3), (7.47a) j+1

ψz,Wj1 (s j ) =

[ ] (−1) 2 (60h 2 + 180bh)s 2j + (−15b3 h − 15b2 h 2 + 2h 4 ) 3 3 100h(15b + h ) ( j = 2, 4). (7.47b) j

ψz,Wj1 (s j ) =

Note that in writing ψ z,Wj1 (s j ) in Eq. (7.47), the following scaling constant q1∗ was used: q1∗ = −

1 , 25h

which normalizes ψ zW1 as ψzW1 (s1 = h/2) = −2/100. Figure 7.4b shows the shape of ψ zW1 . Step 3. Stress field update considering the secondary displacement If ξ includes four modes, i.e., ξ = { U y , θx , χ1 , W1 }T , the corresponding displacements, strains, and stresses become u s (z, s) = ψs y (s)U y (z) + ψsχ1 (s)χ1 (z), U

u z (z, s) = ψzθx (s)θx (z) + ψzW1 (s)W1 (z),

(7.48a)

εzz (z, s) = ψzθx (s)θx, (z) + ψzW1 (s)W1, (z), εss (z, s) = ψ˙ sχ1 (s)χ1 (z), γsz (z, s) = ψs y (s)U y, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zθx (s)θx (z) + ψ˙ zW1 (s)W1 (z), U

(7.48b)

σzz (z, s) = E 1 [ψzθx (s)θx, (z) + ψzW1 (s)W1, (z) + ν ψ˙ sχ1 (s)χ1 (z)], σss (z, s) = E 1 [ψ˙ sχ1 (s)χ1 (z) + νψzθx (s)θx, (z) + νψzW1 (s)W1, (z)], τzs (z, s) = G[ψs y (s)U y, (z) + ψsχ1 (s)χ1, (z) + ψ˙ zθx (s)θx (z) + ψ˙ zW1 (s)W1 (z)]. U

(7.48c)

U χ Substituting ψ˙ s 1 = − p1∗ ψzθx in Eq. (7.24), ψ˙ zθx = ψs y in Eq. (7.14), and ψ˙ zW1 = U χ q1∗ (ψs y + D1W1 ψs 1 ) in Eq. (7.41a) into Eq. (7.48c) and repeating a procedure similar to that used to derive Eqs. (7.31–7.33), one can obtain the following generalized force-stress relationships:

230

7 Sectional Shape Functions for a Box Beam Under Flexure

σzz (z, s) = σzzMx (z, s) + σzzB1 (z, s) ≡ F

τzs (z, s) = τzsy (z, s) + τzsQ 1 (z, s) ≡

Mx (z) θx B1 (z) W1 ψz (s) + ψ (s), Jθx JW1 z

(7.49a)

Fy (z) U y Q 1 (z) χ1 ψs (s) + ψs (s), JU y Jχ1

(7.49b)

where B1 , the bimoment of the first-order warping, is defined as B1 (z) =

σzz (z, s)ψzW 1 (s)dA = E 1 JW1 W1, (z).

(7.50)

As was expected, the s-directional behavior σzzB1 defined in Eq. (7.49a) is identical as that of its corresponding cross-section mode, ψzW1 .

χ

7.2.3 Derivation of ψs 2 The axial stress of σzzB1 causes wall extension (in the s direction) due to Poisson’s effect. According to the same argument used to derive the χ1 and W1 modes, one can derive the χ2 mode and then the W2 mode sequentially. Here, we examine the wall-extensional strain εss induced by σzz in Eq. (7.49a): εss (z, s) = −ν

σzz (z, s) ν Mx (z) θx ν B1 (z) W1 =− ψz (s) − ψ (s). E E Jθx E JW1 z

(7.51) χ

The first term on the right-hand side of Eq. (7.51) has been described by ψs 1 (see Eq. (7.22)), but the second term cannot be described without introducing a new χ mode, which will be defined as χ2 . If the s-directional displacement ψs 2 (s) of the shape function of the χ2 mode is used, u s can now be expressed as u s (z, s) = ψs y (s)U y (z) + ψsχ1 (s)χ1 (z) + ψsχ2 (s)χ2 (z), U

(7.52)

from which the wall-extending strain is calculated as εss (z, s) =

∂u s (z, s) = ψ˙ sχ1 (s)χ1 (z) + ψ˙ sχ2 (s)χ2 (z). ∂s

(7.53)

Equating Eq. (7.53) and Eq. (7.51) yields χ1 (z) ψ˙ sχ1 (s) + χ2 (z)ψ˙ sχ2 (s) = −

ν Mx (z) θx ν B1 (z) W1 ψz (s) − ψ (s), E Jθx E JW1 z

which can be rewritten, if Eq. (7.24) is used, as

(7.54)

χ

7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis

231

( ) ν Mx (z) ν B1 (z) W1 χ2 (z)ψ˙ sχ2 (s) = − − p1∗ χ1 (z) ψzθx (s) − ψ (s). E Jθx E JW1 z

(7.55)

Multiplying both sides Eq. (7.55) by ψzθx and ψzW1 and integrating the resulting equations will yield ∗ χ2 (z) = p2,1

(

) ν Mx (z) − p1∗ χ1 (z) , E Jθx

∗ p2,2 χ2 (z) =

ν B1 (z) , E JW1

(7.56a) (7.56b)

where ∗ p2,1

1 =− Jθx



∗ =− ψ˙ sχ2 ψzθx dA, p2,2

1 JW1



ψ˙ sχ2 ψzW1 dA.

(7.57)

If the functions of z appearing on the right side in Eq. (7.55) are eliminated using Eqs. (7.56a, 7.56b), Eq. (7.55) reduces to [ ] χ ψ˙ sχ2 (s) = p2∗ −ψzθx (s) − D1 2 ψzW1 (s) ,

(7.58a)

or ] [ χ χ ψ˙ s,2j (s j ) = p2∗ −ψz,θxj (s j ) − D1 2 ψz,Wj1 (s j ) , ( j = 1, 2, 3, 4)

(7.58b) χ

∗ where p2∗ = p2,1 is a scaling constant, which can be chosen arbitrarily, and D1 2 = ∗ ∗ p22 / p22 is an unknown constant to be determined. Integrating Eq. (7.58b) yields

[ ] χ χ χ ψs,2j (s j ) = p2∗ −ψz,θx j (s j ) − D 12 ψz,Wj1 (s j ) + C j 2 ( j = 1, 2, 3, 4),

(7.59)

χ

where C j 2 are integral constants ( j = 1, 2, 3, 4) and ψs,α j is defined as ψs,α j =



ψs,α j ds j (without an integral constant), α = θ x , W1 . χ

χ

To determine the unknown coefficients C j 2 ( j = 1, 2, 3, 4) and D1 2 , first χ we consider the symmetric characteristics of ψs 2 . Because σzz in Eq. (7.49a) is symmetric about the y axis and anti-symmetric about the x axis, χ2 introduced to explain the secondary strain field by σzz should also satisfy the same symmetry

232

7 Sectional Shape Functions for a Box Beam Under Flexure

and anti-symmetry conditions. Based on these observations, we can obtain three independent conditions as given by Eq. (7.27b) (for symmetry) and Eq. (7.27c)2 (for anti-symmetry). In addition, the following two orthogonality conditions are considered: χ2 U y (7.60) ψs ψs ds = 0, ψsχ2 ψsχ1 ds = 0. Using the aforementioned five conditions, one can obtain the following results: χ

D1 2 175b2 h(15b3 + h 3 )(3b + h) , 4(150b7 + 315b6 h + 210b5 h 2 − 70b4 h 3 − 70b3 h 4 + 10b2 h 5 + 11bh 6 + h 7 ) h 2 (−2520b6 − 840b5 h + 735b4 h 2 + 245b3 h 3 + 9bh 5 + 3h 6 ) χ , = −C3 2 = − 96(105b6 + 210b5 h − 70b3 h 3 + 10bh 5 + h 6 ) χ = C4 2 = 0. (7.61)

=− χ

C1 2 χ

C2 2

χ

Figure 7.4c illustrates the shape of the calculated ψs 2 . The scaling factor p2∗ used χ to write ψs 2 in Eq. (7.59) is set to p2∗ = −

315b6 + 630b5 h − 210b3 h 3 + 30bh 5 + 3h 6 , 125bh(63b6 + 21b5 h − 21b3 h 3 − 7b2 h 4 + 6bh 5 + 2h 6 ) χ

χ

which normalizes ψs 2 as ψs 2 (s2 = −b/2) = −1/100. At this point, we update the shear stress as F

τzs (z, s) = τzsy (z, s) + τzsQ 1 (z, s) + τzsQ 2 (z, s),

(7.62)

where the newly added term τzsQ 2 is given by τzsQ 2 (z, s) =

Q 2 (z) χ2 ψs (s). Jχ2

(7.63)

Equation (7.63) can be viewed as expressing the generalized force-stress relationship between Q 2 and τzsQ 2 .

χ

θx W1 2 Because ψz, j and ψz, j for j = 1, 3 are even functions, ψs in Eq. (7.59) automatically satisfies Eq. (7.27a).

2

χ

7.2 Derivation of ψs k (s) and ψzW k (s) by Means of a Recursive Analysis

233

7.2.4 Derivation of ψ zW2 To derive ψzW2 , we can use the same procedure that was used to derive ψzW1 . It is argued that the shear strain field directly calculated by the resultant forces should match that derived from the displacement field (which results the rigid-body mode and the higher-order modes including all modes up to the W2 mode): γzs (z, s) =

Fy (z) U y Q 1 (z) χ1 Q 2 (z) χ2 ψs (s) + ψs (s) + ψ (s) G JU y G Jχ1 G Jχ2 s

≡ ψs y (s)U y, (z) + ψsχ1 (s)χ1, (z) + ψsχ1 (s)χ2, (z) + ψ˙ zθx (s)θx (z) (7.64) + ψ˙ zW1 (s)W1 (z) + ψ˙ zW2 (s)W2 (z). U

We rewrite Eq. (7.64) using Eq. (7.14) and Eq. (7.41a): ) Fy (z) U , ∗ − U y (z) − θx (z) − q1 W1 (z) ψs y (s) G JU y ( ) Q 1 (z) , , ∗ W1 + − χ1 (z) − χ2 (z) − q1 D1 W1 (z) ψsχ1 (s) G Jχ1 ( ) Q 2 (z) + (7.65) ψsχ2 (s). G Jχ2

ψ˙ zW2 (s)W2 (z) =

(

Following the same argument used to write ψ˙ zW1 in Eq. (7.41), one can find the following relationship ] [ U χ χ ψ˙ z,Wj2 (s j ) = q2∗ ψs, yj (s j ) + D1W2 ψs,1j (s j ) + D2W2 ψs,2j (s j ) , ( j = 1, 2, 3, 4),

(7.66)

or its integration version [ ] U χ χ 2 . ψz,Wj2 (s j ) = q2∗ ψs, yj (s j ) + D1W2 ψs,1j (s j ) + D2W2 ψs,2j (s j ) + C W j

(7.67)

/ ∗ ∗ ∗ is a scaling constant and DkW2 = q2,(k+1) q2,1 In Eqs. (7.66) and (7.67), q2∗ = q2,1 ∗ ∗ ∗ (k = 1, 2) is treated as an unknown constant where q2,1 , q2,2 , and q2,3 are   U χ ∗ ∗ ∗ defined as q2,1 = (1/JU y ) ψ˙ zW2 ψs y dA, q2,2 = (1/Jχ1 ) ψ˙ zW2 ψs 1 dA, and q2,3 = { }  W χ2 W W 2 2 2 ( j = 1, 2, 3, 4; k = 1, 2), (1/Jχ2 ) ψ˙ z ψs dA. The six unknowns, C j , Dk can be determined using the three symmetry/anti-symmetry conditions in Eq. (7.43), the one continuity condition in Eq. (7.44), and the following two orthogonality conditions:

234

7 Sectional Shape Functions for a Box Beam Under Flexure



ψzW2 ψzθx ds = 0,

ψzW2 ψzW1 ds = 0.

(7.68)

Finally, one can obtain ψzW2 as (−1)( j+1)/2 + − 2520b5 h 6 + 315b3 h 8 + h 11 ) [(7560b10 − 56700b8 h 2 − 64260b7 h 3 + 15435b6 h 4 + 28350b5 h 5

ψ z,Wj2 =

200(945b10 h

5355b7 h 4

− 3570b3 h 7 + 210bh 9 + 15h 10 )s j + (226800b8 + 428400b7 h − 88200b6 h 2 − 246960b5 h 3 + 42000b3 h 5 − 3360bh 7 − 280h 8 )s 3j + (105840b6 + 211680b5 h − 70560b3 h 3 + 10080bh 5 + 1008h 6 )s 5j ] ( j = 1, 3)

(7.69a)

(−1)( j)/2 400(945b10 + 5355b7 h 3 − 2520b5 h 5 + 315b3 h 7 + h 10 ) [(2835b10 + 9450b9 h − 14490b7 h 3 + 8190b5 h 5 + 1575b4 h 6

ψ z,Wj2 =

− 1260b3 h 7 − 420b2 h 8 + 8h 10 ) + (−113400b8 − 302400b7 h + 317520b5 h 3 − 126000b3 h 5 − 7560b2 h 6 + 15120bh 7 + 1680h 8 )s 2j + (529200b6 + 1058400b5 h − 352800b3 h 3 + 50400bh 5 + 5040h 6 )s 4j ] ( j = 2, 4)

(7.69b)

where the following value of q2∗ is used, which normalizes ψzW2 as ψzW2 (s1 = h/2) = −2/100: q2∗ = −

1 . 25h

The shape of ψzW2 (s) is illustrated in Fig. 7.4d. Once the W2 mode is included, the stress should be updated as σzz (z, s) = σzzMx (z, s) +

2  k=1

 Bk (z) Mx (z) θx ψz (s) + ψzWk (s), Jθx J Wk k=1 2

σzzBk (z, s) ≡

(7.70) where the bimoment B2 of the second-order warping is given by (see Eq. (7.9)) B2 (z) =

σzz (z, s)ψzW2 (s)d A = E 1 JW2 W2, (z).

(7.71)

7.3 n-Directional Displacements and Resulting Stress and Strain Fields

235

The Nth-order modes χ N and W N can be similarly derived by following the procedure used to derive χ2 and W2 . To this end, we generalize Eq. (7.54) for the case where the highest order is N: N 

χk (z)ψ˙ sχk (s) = −

k=1

N −1

 ν B (z) ν Mx (z) θx k ψz (s) − ψzWk (s). E Jθx E J W k k=1

(7.72)

Similarly, one can generalize Eq. (7.59) as [ χ ψs,Nj (s j )

=

−ψz,θx j (s j )

p ∗N

−

N −1 

] χ Dk N ψz,Wjk (s j )

+

χ Cj N

k=1

j = 1, 2, 3, 4 (for N ≥ 3), χ

(7.73) χ

where p ∗N is a scaling constant and {Dk N }k=1,...,N −1 and {C j N } j=1,2,3,4 are unknowns χ χ to be determined. For consistency, p ∗N (N ≥ 3) that scales ψs N as ψs N (s2 = −b/2) = −1/100 is chosen. The unknowns can be determined using three independent symmetry/anti-symmetry conditions in Eqs. (7.27b) and (7.27c) and N orthogonality  χ U  χ χ conditions, ψs N ψs y ds = 0 and ψs N ψs k ds = 0 (k = 1, 2, . . . , N − 1). The shape function ψzW N (s) is similarly determined using the procedure used to derive ψzW2 . In this case, Eq. (7.67) is generalized to [ ψz,WjN (s j )

=

q N∗

U ψs, yj (s j )

+

N 

] χ DkW N ψs,kj (s j )

+

N CW j

k=1

( j = 1, 2, 3, 4) (N ≥ 3),

(7.74)

N where q N∗ is a scaling constant and {DkW N }k=1,...,N and {C W j } j=1,2,3,4 are unknowns to be determined. They can be explicitly determined using the three symmetry/antiin Eq. (7.44), and symmetry conditions in Eq. (7.43),  the one continuity condition  the N orthogonality conditions, ψzW N ψzθx ds = 0 and ψzW N ψzWk ds = 0 (k = 1, 2, . . . , N − 1).

7.3 n-Directional Displacements and Resulting Stress and Strain Fields By the same argument made in Sect. 6.3, it is apparent that in addition to the sχ directional shape function ψs k (s), the χk mode has the n-directional shape function χk ψn (s) representing wall-bending deformation. We should also consider non-zero ndirectional shape functions associated with constrained distortion modes, ηk and ηˆ k . These shape functions that are only non-zero shape functions of ηk and ηˆ k represent

236

7 Sectional Shape Functions for a Box Beam Under Flexure

y z

y x

y

z

(a)

x

z

(b)

y x

(c)

z

x

(d)

Fig. 7.5 Modes involving n-directional deformations needed to analyze a box beam under a vertical flexural load: a the χ1 mode, b the χ2 mode, c the η1 mode (constrained distortion of type 1), and d the ηˆ 1 mode (constrained distortion of type 2)

wall-bending deformations. Figure 7.5 Illustrates some modes having wall-bending deformations. Certain preliminary analyses needed when deriving the wall-bending parts of the shape functions of distortion, χk , and constrained distortion modes, ηk and ηˆ k , will be presented in this section. Based on the aforementioned arguments, the n-directional displacement due to modes U y , {χk }k=1,...,N , and {ηk , ηˆ k }k=1,...,N at an arbitrary point on a cross-section wall can be written as U

u˜ n (z, n, s) = u n (z, s) = ψn y (s)U y (z) N [ ]  ηˆ η ψnχk (s)χk (z) + ψn k (s)ηk (z) + ψn k (s)ηˆ k (z) . +

(7.75)

k=1

In Eq. (7.75), U y (z), χk (z), ηk (z), and ηˆ k (z) denote the 1D generalized displacements of the modes the symbols represent. Because u˜ n is uniform in the thickness (n) direction, it is simply written as u n (z, s) without n-dependence in Eq. (7.75). The U sectional shape function ψn y (s) of the U y mode, as easily predicted by the classical beam theory, is given by U

U

U

U

ψn,1y = 0, ψn,2y = 1, ψn,3y = 0, ψn,4y = −1.

(7.76)

The sectional shape functions of the unconstrained distortion (χ ) and constrained ˆ modes will be derived in the subsequent sections. distortion (η, η) When wall bending occurs by u n in Eq. (7.75), non-zero z- and s-directional displacements can occur at a generic point (z, n, s) of a wall that is apart by n from the wall midline. These displacements3 can be described based on the kinematic assumption of Kirchhoff’s thin plate theory (εnn = γns = γzn = 0). First, the z-directional displacement due to wall bending is denoted by

3

It should be noted that the displacements in considered here are only due to wall-bending deformations, not due to membrane deformations.

7.3 n-Directional Displacements and Resulting Stress and Strain Fields

237

u z (z, n, s) = −n u ,n (z, s) = −n ψn y (s)U y, (z) U

−n

N [ ]  ηˆ η ψnχk (s)χk, (z) + ψn k (s)η,k (z) + ψn k (s)ηˆ k, (z) .

(7.77)

k=1

Because −U y, (z) represents the rotation of the wall normal about the x axis according to Kirchhoff’s theory, it can be replaced by θx (z) in the present formulation, where the rotation is treated as an independent 1D variable. Thereby, u z in Eq. (7.77) is written as U

u z (z, n, s) = n ψn y (s)θx (z) N [ ]  ηˆ η −n ψnχk (s)χk, (z) + ψn k (s)η,k (z) + ψn k (s)ηˆ k, (z) .

(7.78)

k=1

The s-directional displacement u s caused by wall bending is given by u s (z, n, s) = −n u˙ n (z, s) N [ ]  ηˆ η ψ˙ nχk (s)χk (z) + ψ˙ n k (s)ηk (z) + ψ˙ n k (s)ηˆ k (z) , = −n

(7.79)

k=1

where the term ψ˙ n y U y is omitted in Eq. (7.79) because ψ˙ n y (s) = 0. From the displacements {u z , u s } in Eqs. (7.78) and (7.79), the strains {ε zz , εss , γ zs } representing wall-bending deformation can be found: U

U

ε zz (z, n, s) = u ,z (z, n, s) = n ψn y (s)θx, (z) N [ ]  ηˆ η −n ψnχk (s)χk,, (z) + ψn k (s)η,,k (z) + ψn k (s)ηˆ k,, (z) , U

(7.80a)

k=1

ε ss (z, n, s) = u˙ s (z, n, s) N [ ]  ηˆ η = −n ψ¨ nχk (s)χk (z) + ψ¨ n k (s)ηk (z) + ψ¨ n k (s)ηˆ k (z) ,

(7.80b)

k=1

γ zs (z, n, s) = u ,s (z, n, s) + u˙ z (z, n, s) N [ ]  ηˆ η ≈ −n 2ψ˙ n k (s)η,k (z) + 2ψ˙ n k (s)ηˆ k, (z) .

(7.80c)

k=1

N χ In Eq. (7.80c), k=1 (−2n ψ˙ n k χk, ) is omitted because it is very small compared to the other terms. To find the stresses {σ zz , σ ss , τ zs } from the strains {ε zz , ε ss , γ zs }

238

7 Sectional Shape Functions for a Box Beam Under Flexure

in Eqs. (7.80), the plane-stress assumption is used: [ U σ zz (z, n, s) = E 1 (ε zz + νε ss ) = E 1 (−n) −ψn y θx, +

N 

ηˆ

η

(ψnχk χk,, + ψn k η,,k + ψn k ηˆ k,, + ν ψ¨ nχk χk

k=1

] ηˆ η + ν ψ¨ n k ηk + ν ψ¨ n k ηˆ k ) ,

(7.81a)

[ U σ ss (z, n, s) = E 1 (εss + νε zz ) = E 1 (−n) −νψn y θx, +

N 

ηˆ η (ψ¨ nχk χk + ψ¨ n k ηk + ψ¨ n k ηˆ k + νψnχk χk,,

k=1

] ηˆ η +νψn k η,,k + νψn k ηˆ k,, ) , τ zs (z, n, s) = Gγ zs = G(−n)

(7.81b)

N ( )  ηˆ η 2ψ˙ n k η,k + 2ψ˙ n k ηˆ k, .

(7.81c)

k=1

Now, we consider the internal virtual work δU , from which the generalized forces can be identified: z2  δU =

 (σ zz δε zz + σ ss δε ss + τ zs δγ zs ) dA dz

z1

[

=

]z 2

U

σ zz (n ψn y ) dA δθx

+ z1

+

τ zs (−2n ψ˙ nηˆk ) dA δ ηˆ k σ zz (−n

z2 [ +

ψnηk ) dA δ η˜ k,

σ zz (−n

+ ]

ψnηˆk ) dA δ ηˆ k,

z2 [ N z1

σ ss (−n ψ¨ nηk ) −

σ zz (−n ψnχk ) dA δχk,

k=1



z1



N [ 

− ∂σ∂zzz (n ψn y ) dAδθx dz + U

τ zs (−2n ψ˙ nηk ) dA δ η˜ k

k=1

+ z1

+

]z 2

N [ 

k=1

ψ˙ nηk ) dA δ η˜ k ] ∂τ zs ηˆ k ηˆ k ¨ ˙ + σ ss (−n ψn ) − ∂ z (−2n ψn )dA δ ηˆ k dz +

∂τ zs (−2n ∂z

]z 2 z1

σ ss (−n ψ¨ nχk ) dAδχk

7.3 n-Directional Displacements and Resulting Stress and Strain Fields

239

z2  N ∂σ zz χk , + [ − ∂ z (−n ψn )dAδχk + − ∂σ∂ zzz (−n ψnηk )dAδ η˜ k, z1

+

k=1

] − ∂σ∂ zzz (−n ψnηˆk )dAδ ηˆ k, dz

(7.82)

The strains given in Eq. (7.80) were substituted into the virtual strains in Eq. (7.82), and the resulting equation was integrated by parts. If the following 1D field variable U is defined as U(z) = {θx (z), {ηk (z), ηˆ k (z), χk, (z), η,k (z), ηˆ k, (z)}k=1,...,N }T ,

(7.83)

the 1D generalized force variable F that is the work conjugate of U can be identified from Eq. (7.82) as F(z) = {M x (z), {R k (z), Rˆ k (z), Sk (z), T k (z), Tˆk (z)}k=1,...,N }T ,

(7.84)

where each component of F is identified as Mx = Rˆ k = Sk = Tˆk =



σ zz (n ·

U ψn y ) dA,

Rk =

ηˆ τ zs (−2n · ψ˙ n k ) dA,

σ zz (−n ·

ψnχk ) dA , ηˆ

σ zz (−n · ψn k ) dA.

η τ zs (−2n · ψ˙ n k ) dA,

Tk =

η

σ zz (−n · ψn k ) dA, (7.85)

A few remarks on U and F may be necessary. First, the derivative terms {χk, , η,k , ηˆ k, } are also treated as the components of the field variable U. As was discussed in Sect. 6.3, the derivatives represent the bending rotation of the walls of a beam cross-section with respect to the s coordinate, contributing to displacement u z in Eq. (7.79). The M x term represents the bending moment induced by σ zz . It can be regarded as a higher-order version of Mx in Eq. (7.9) because M x Mx . The R k and Rˆ k terms represent the work conjugates of the constrained distortion modes {ηk , ηˆ k }, and the terms {Sk , T k , Tˆk } represent the work conjugates of the derivative terms, {χk, , η,k , ηˆ k, }. Note that all terms in F except M x are self-equilibrated, producing no net resultant. Next, the generalized force-stress relationships will be derived for the {ηk , ηˆ k } modes. To establish explicit relationships, we impose the following orthogonality of the u s field4 : 4

Because u s is the dominant displacement component producing in-plane wall-bending as illustrated in Figs. 7.5c, d, only u s is considered.

240

7 Sectional Shape Functions for a Box Beam Under Flexure



) ( ψ˙ nα1 ψ˙ nα2 ds = 0 α1 , α2 ∈ {ηk , ηˆ k }k=1, 2, ..., N ; α1 /= α2 .

(7.86)

If Eq. (7.86) is explicitly written out for modes η N and ηˆ N , it becomes

η η ψ˙ n N ψ˙ n k ds =



ηˆ η ψ˙ n N ψ˙ n k ds

= 0 (1 ≤ k ≤ N − 1) for mode η N ,

ηˆ η ψ˙ n N ψ˙ n k ds =





ηˆ ηˆ ψ˙ n N ψ˙ n k ds = 0 (1 ≤ k ≤ N − 1) for mode ηˆ N ,

ηˆ η ψ˙ n N ψ˙ n N ds = 0 between modes η N and ηˆ N .

(7.87a) (7.87b) (7.87c)

Substituting the stress-displacement relationship in Eq. (7.81c) into {R k , Rˆ k } in Eq. (7.85) and using the orthogonality conditions in Eqs. (7.87), one can find R k (z) = G Iηk η,k (z), Rˆ k (z) = G Iηˆ k · ηˆ k, (z),

(7.88)

with Iη k =

η

(−2n ψ˙ n k )2 dA, Iηˆ k =



ηˆ

(−2n ψ˙ n k )2 dA

(7.89)

If η,k and ηˆ k, in Eq. (7.81c) are replaced with R k and Rˆ k using Eq. (7.88), the following form of the generalized force-stress relationship between τ zs and {R k , Rˆ k } can be established: τ zs (z, n, s) =

N  k=1

=

N  k=1

R

Rˆ

[τ zsk (z, n, s) + τ zsk (z, n, s)] ⎧

⎫ Rˆ k (z) R k (z) ηˆ k ηk [−2n ψ˙ n (s)] . [−2n ψ˙ n (s)] + Iη k Iηˆ k

(7.90)

χ

7.4 Derivation of ψn k and the Generalized Force-Stress Relationship …

241

χ

7.4 Derivation of ψn k and the Generalized Force-Stress Relationship for Mode χk χ

7.4.1 Derivation of ψn k χ

Recall that the wall-membrane component ψs 1 of mode χ1 was derived starting with the nominal axial stress σzzMx (z, s) = Mx (z)ψzθx (s)/Jθx , which is assumed to be χ uniform over the n-direction. Likewise, the wall-bending component ψn 1 of mode χ1 can be derived starting with the linearly varying axial stress over the n-direction. Therefore, it is important to find the exact distribution of σzz over the wall thickness. To this end, we observe that the axial wall-membrane component of the displacement u z (z, n, s) at a generic point on a box beam wall is due to the θx mode, which is given by ψzθx (s)θx (z) (see Eq. (7.2a)). On the other hand, its wall-bending component is U given by n ψn y (s)θx (z)(see Eq. (7.78)). Therefore, the total displacement u z (z, n, s) can be written as [ ] U u z (z, n, s)total field = ψzθx (s) + n ψn y (s) θx (z) = y(n, s)θx (z), (7.91) where y(n, s) denotes the vertical coordinate of a generic point. To obtain the last expression in Eq. (7.91), the following identity, which can be confirmed using Eq. (7.3b) and Eq. (7.76), is used: y(n, s) = ψzθx (s) + n ψn y (s). U

(7.92)

We now consider the axial stress σzzMx due to the applied moment Mx , which is given by σzzMx (z, n, s) =

Mx (z) y(n, s). Jθx

(7.93)

The expression in Eq. (7.93) is a well-known formula obtained from an elementary beam theory. If Eq. (7.92) is substituted into Eq. (7.93), σzzMx can be decoupled into two terms as σzzMx (z, n, s) =

Mx (z) U y Mx (z) θx ψz (s) + nψn (s) Jθx Jθx

≡ σzz (z, s) + σ zz (z, n, s).

(7.94) U

Figure 7.6 shows the σ zz distribution, equivalently, the n ψn y (s) distribution, on U edges 1 and 2 as an example (see Eq. (7.76) for the explicit expression of ψn y ). As the wall-bending stress σ zz is explicitly given by Eq. (7.94), we can now proceed to χ derive ψn k .

242

7 Sectional Shape Functions for a Box Beam Under Flexure

t U

nψ n y on edge 2 edge 2

t

edge 1

edge 3 edge 4

ψ θz

x

U nψ n y = 0 on edge 1 U

Fig. 7.6 n ψn y on edges 1 and 2

Step 1. Identification of a secondary normal strain field due to θx Because σ zz (the wall-bending stress) is given, one can calculate the secondary stress εss induced by Poisson’s effect: ε ss (z, n, s) = −ν

σ zz (z, n, s) ν Mx (z) [ U y ] =− nψn (s) . E E Jθx

(7.95)

As is evident from Eq. (7.95), ε ss is zero on the wall midline (n = 0) and varies linearly in the thickness direction. Step 2. Derivation of secondary displacement consistent with the secondary strain Owing to εss in Eq. (7.95), wall stretch and shrinkage can occur along the contour (s) direction on half of the wall (0 ≤ n ≤ t/2) and on the other half (−t/2 ≤ n ≤ 0), respectively (or vice versa). This strain distribution leads to in-plane wall-bending deformation of the beam wall. Considering the first term involving the distortion mode χ1 in Eq. (7.79), the in-plane wall-bending displacement can be expressed as5 u s (z, n, s) = −n ψ˙ nχ1 (s)χ1 (z)

(7.96)

Using Eq. (7.96), εss is expressed as 5

It is should be noted that the applied moment Mx causes both the membrane and wall-bending χ χ deformations, and ψn 1 (s) is related to the same χ1 used to define the membrane component ψs 1 (s).

χ

7.4 Derivation of ψn k and the Generalized Force-Stress Relationship …

ε ss (z, n, s) =

243

∂u s (z, n, s) = −n ψ¨ nχ1 (s)χ1 (z) ∂s

(7.97)

Equating Eqs. (7.95) and (7.97) yields −n ψ¨ nχ1 (s)χ1 (z) = −

ν Mx (z) U y nψn (s). E Jθx

(7.98)

Equation (7.98) can be satisfied if χ ν Mx (z) ψ¨ n 1 (s) ≡ p1∗ . = Uy E Jθx χ1 (z) ψn (s)

(7.99)

Note that p1∗ appearing in Eq. (7.99) has a value identical to that appearing in Eq. (7.23) because ν Mx /E Jθx χ1 appearing in Eq. (7.99) is identical to that in χ Eq. (7.23). Therefore, the p1∗ value used to normalize ψs 1 should also be used here. Uy χ1 From Eq. (7.99), the relationship between ψn and ψn can be found: U ψ¨ nχ1 (s) = p1∗ ψn y (s).

(7.100)

χ

If N higher-order modes of ψn k (k = 2, 3, . . . , N ) are used, the in-plane wallbending displacement should consider the contributions of all of these modes. When N = 2, for example, the wall-bending displacement can be written from Eq. (7.79): u s (z, n, s) =

2  [

] −n ψ˙ nχk (s)χk (z) .

(7.101)

k=1

Repeating the procedure used to derive Eq. (7.98), the following relationship can be found: 2  [ k=1

] ν Mx (z) U y −n ψ¨ nχk (s)χk (z) = − nψn (s). E Jθx

(7.102)

Note that the generalized force B1 that is the work conjugate of the higher-order warping W1 does not appear in Eq. (7.102) because B1 does not produce a non-zero σ zz (which varies along the thickness direction n). Likewise, σ zz due to Bk (k ≥ 2) also vanishes. χ Using Eq. (7.100) and Eq. (7.102), ψn 2 can be obtained as ψ¨ nχ2 (s)χ2 (z) =

(

) ν Mx (z) U ∗ − p1 χ1 (z) ψn y (s). E Jθx

(7.103)

∗ ), it reduces to If Eq. (7.56) is substituted into Eq. (7.103) (with p2∗ = p2,1

244

7 Sectional Shape Functions for a Box Beam Under Flexure U ψ¨ nχ2 (s) = p2∗ ψn y (s).

(7.104) χ

By repeating the procedure to derive Eq. (7.104), ψn k for the general case of k ≥ 2 can be found as U U χ ψ¨ nχk (s) = pk∗ ψn y (s) or ψ¨ n,kj (s j ) = pk∗ ψn,yj (s j )

( j = 1, 2, 3, 4).

(7.105)

Note that the value of pk∗ appearing in Eq. (7.105) is identical to that used to χ normalize ψs k (s) (see Eq. (7.73)). We will now integrate Eq. (7.105) to obtain [ ] U χ χ χ ψn,kj (s j ) = pk∗ Φn,yj (s j ) + C j,1k s j + C j,0k , χ

χ

(7.106) U

where {C j,0k , C j,1k } ( j = 1, 2, 3, 4) are the integration constants and Φn,yj is obtained using Eq. (7.76) as U

U

U

U

Φn,1y = 0; Φn,2y = 21 s22 ; Φn,3y = 0; Φn,4y = − 21 s42 . χ

(7.107)

χ

The eight unknowns {C j,0k , C j,1k } ( j = 1, 2, 3, 4) can be determined using the following conditions: ψnχk (s j = s ∗ ) = −ψnχk (s j = −s ∗ ) for j = 1, 3 and ψnχk (s j = s ∗ ) = ψnχk (s j = −s ∗ ) for j = 2, 4,

(7.108a)

ψnχk (s j ) = −ψnχk (s j+2 ) ( j = 1, 2),

(7.108b)

ψnχk (s j = +s j ) = −ψsχk (s j+1 = −s j+1 ), ψnχk (s j+1 = −s j+1 ) = ψsχk (s j = +s j ) ( j = 1, 2, 3, 4),

(7.108c)

where Eqs. (7.108a, 7.108b) represent the symmetric and anti-symmetric conditions χ of ψn k and Eq. (7.108c) states the continuities in the x- and y-directional displacements of mode χk at the corners of a beam cross-section. Note that only eight conditions are independent in Eq. (7.108); if all of the conditions stated in Eqs. (7.108a, 7.108b) are used, Eq. (7.108c) can yield only two additional independent conditions (for example, Eq. (7.108c) with j = 1 can be used). Using these conditions, one can determine the integration constants as follows: χ

χ

C1,1k = −C3,1k = − p2∗ h ψsχk (s2 = −b/2), k

χ

χ

2

C2,0k = −C4,0k = − b8 +

1 ψ χk (s1 pk∗ s

= h/2),

(7.109a) (7.109b)

χ

7.4 Derivation of ψn k and the Generalized Force-Stress Relationship … χ

χ

χ

χ

C2,1k = C4,1k = C1,0k = C3,0k = 0.

245

(7.109c)

χ

Using the values given in Eq. (7.109), ψn k is obtained as χ

ψn,1k (s1 ) = pk∗

} ] [{ − p2∗ h ψsχk (s2 = − b2 ) · s1 , k

{ 2 [ χ ψn,2k (s2 ) = pk∗ + 21 s22 + − b8 + χ

ψn,3k (s3 ) = pk∗

(7.110a)

}] ,

(7.110b)

} [{ ] + p2∗ h ψsχk (s2 = − b2 ) · s3

(7.110c)

1 ψ χk (s1 pk∗ s

= h2 )

k

{ 2 [ χ ψn,4k (s4 ) = pk∗ − 21 s42 − − b8 + χ

1 ψ χk (s1 pk∗ s

}] .

= h2 )

(7.110d)

χ

For example, ψn 1 can be explicitly written as (ψs 1 given in Eq. (7.29) is used): [ b ] χ 1 ψn,11 (s1 ) = − 25bh − 2 s1 ,

(7.111a)

( 2 2 )] [ χ +2h 1 ψn,21 (s2 ) = − 25bh + 21 s22 − 3b 24 ,

(7.111b)

[ b ] χ 1 ψn,31 (s3 ) = − 25bh + 2 s3 ,

(7.111c)

( 2 2 )] [ χ 1 +2h ψn,41 (s4 ) = − 25bh − 21 s42 + 3b 24 .

(7.111d)

Because the strain–stress relationship in Eq. (7.95) and the resulting recursive χ relationship in Eq. (7.105) are given only in terms of θx , ψn k can be calculated for all values of k without updating the stress field.

7.4.2 Relationship Between the Generalized Force (Sk ) and Stress (σ zz ) In this subsection, we will establish the explicit relationship between the σ zz field due to χk and Sk (see the definition of Sk in Eq. (7.85)). To this end, we examine σ zz stemming from the χk mode using Eq. (7.81a): σ zz = E 1 (−n) ·

N 

(ψnχk χk,, + ν ψ¨ nχk χk ).

k=1

Substituting Eq. (7.105) into Eq. (7.112) yields,

(7.112)

246

7 Sectional Shape Functions for a Box Beam Under Flexure

σ zz = −E 1 n

N 

{ψnχk χk,, + ψn y (νpk∗ χk )}. U

(7.113)

k=1 χ

Referring to Eq. (7.110), one can see that ψn k (for any k) can be written as χ

χ

ψn,1k = −ψn,3k = a1 (k)s, χ

χ

ψn,2k = −ψn,4k = a2 (k)s 2 + a3 (k),

(7.114a) (7.114b)

where a1 (k), a2 (k), and a3 (k) are constants. Equation (7.114) shows that the χ χ χ mode shapes {ψn k }k=1, 2, ..., N are neither orthogonal to each other ( A ψn k ψn l dA /= χ 0 for any k and l) nor independent. It also shows that the polynomial order of ψn k does not change for different k values. Because the polynomials have only constant, χ linear, and quadratic terms, one can express ψn k in terms of three independent Uy χ χˆ mutually orthogonal basis functions, {ψn , ψn , ψn }, as ψnχk = Ak ψn y + Bk ψnχ + Ck ψnχˆ , U

χ

U

(7.115)

χˆ

where ψn y is given in Eq. (7.76) and ψn and ψn are certain functions to be determined. The symbols Ak , Bk , and Ck are expansion coefficients. Although a function U U other than the piecewise constant function ψn y can be chosen, the choice of ψn y is important because the behavior of the resulting σ zz stress on the n−s plane can be U divided into two terms: (i) −n ψn y (s) previously considered in Eq. (7.94) and (ii) χ χˆ {−n ψn (s), −n ψn (s)} newly considered in this case (see Eq. (7.127)). χ Because the highest order in the polynomial ψn k is quadratic, one may assume χ χˆ ψn and ψn as6 χ

χ

χ

χ

ψn,1 = −ψn,3 = m 1 s; ψn,2 = −ψn,4 = 21 s 2 + n 1 χˆ

χˆ

χˆ

χˆ

ψn,1 = − ψn,3 = m 2 s; ψn,2 = − ψn,4 = 21 s 2 + n 2 .

(7.116a) (7.116b)

To determine the unknown coefficients (m 1 , n 1 , m 2 , n 2 ), the following orthogonality conditions are considered:

6

ψn y · ψnχ ds = 0

(7.117a)

ψn y · ψnχˆ ds = 0

(7.117b)

U

U

The coefficients of the quadratic terms can be arbitrarily selected, but ½ is used to facilitate the subsequent calculations.

χ

7.4 Derivation of ψn k and the Generalized Force-Stress Relationship …



ψnχ · ψnχˆ ds = 0

247

(7.117c)

Along with the conditions in Eq. (7.117), an additional condition, which may be quite arbitrary, is also needed. Here, we choose the following additional condition: m2 = 1

(7.118)

Substituting the m and n values determined using Eqs. (7.117) and (7.118) into Eq. (7.116) yields χ

χ

χ

5

χ

b 1 2 ψn,1 = − ψn,3 = − 60h 3 s; ψn,2 = − ψn,4 = 2 s − χˆ

χˆ

χˆ

χˆ

ψn,1 = − ψn,3 = s; ψn,2 = − ψn,4 = 21 s 2 − U

χ

b2 , 24

b2 . 24

(7.119a) (7.119b)

χˆ

Because the orthogonal functions, ψn y , ψn , and, ψn , are all determined, the expansion coefficients {Ak , Bk , Ck } in Eq. (7.115) can be found as / U U (ψn y )2 ds, ψn y ψnχk ds / (ψnχ )2 ds, Bk = ψnχ ψnχk ds / (ψnχˆ )2 ds. Ck = ψnχˆ ψnχk ds

Ak =

(7.120)

At this point, we substitute Eq. (7.115) into Eq. (7.113) to express σ zz as σ zz (z, n, s) = −E 1 n

N { 

[ ] U ψn y (s) νpk∗ χk (z) + Ak χk,, (z)

k=1

[

+ψnχ (s)

] [ ]} Bk χk,, (z) + ψnχˆ (s) Ck χk,, (z) .

Substituting σ zz in Eq. (7.121) into S1 in Eq. (7.85) yields

σ zz (−n · ψnχ1 ) dA Uy = A1 σ zz (−n · ψn ) dA + B1 σ zz (−n · ψnχ ) dA + C1 σ zz (−n · ψnχˆ ) d A

S1 =

(7.121)

248

7 Sectional Shape Functions for a Box Beam Under Flexure

[ = E 1 A1 IU y

N 

] (νpk∗ χk

k=1

[ + E 1 B1 Iχ

N 

+

Ak χk,, )

] (Bk χk,, )

+ E 1 C1 Iχˆ

[ N 

k=1

] (Ck χk,, )

,

(7.122)

k=1

where IU y , Iχ , and Iχˆ are defined as IU y = Iχ = Iχˆ =



U

(−n · ψz y )2 dA, (−n · ψnχ )2 dA, (−n · ψnχˆ )2 dA.

(7.123)

Likewise, Sm can be written similarly as

σ zz (−n · ψnχm ) dA Uy = Am σ zz (−n · ψn ) dA + Bm σ zz (−n · ψnχ ) dA + Cm σ zz (−n · ψnχˆ ) dA (m = 1, 2, . . . , N ) ] [ N  ∗ ,, = E 1 Am IU y (νpk χk + Ak χk )

Sm =

k=1

+ E 1 Bm Iχ

[ N 

] (Bk χk,, )

+ E 1 Cm Iχˆ

k=1

[ N 

] (Ck χk,, )

,

(7.124)

k=1

which can be organized in matrix form as ⎡ ⎢ ⎢ ⎢ S=I·V→⎢ ⎢ ⎣

S1 S2 S3 .. . SN

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣

A1 IU y A2 IU y A3 IU y .. . A N IU y

⎤ N ⎤⎡  ∗ ,, B1 Iχ C1 Iχˆ ⎢ E 1 k=1 (νpk χk + Ak χk ) ⎥ B2 Iχ C2 Iχˆ ⎥ ⎥ ⎥⎢ N ⎢ ⎥  ,, ⎢ ⎥. B3 Iχ C3 Iχˆ ⎥ (Bk χk ) E1 ⎥⎢ ⎥ .. .. ⎥ k=1 ⎢ ⎥ ⎦ N . . ⎦⎣  ,, (Ck χk ) E1 B N Iχ C N Iχˆ k=1

(7.125) Because matrix I in Eq. (7.125) is not square, V can be found using a least square method (or a pseudo inverse) as

χ

7.4 Derivation of ψn k and the Generalized Force-Stress Relationship …

⎡

N 

(νpk∗ χk

⎡

I˜11 = ⎣ I˜21 I˜31

I˜12 I˜22 I˜32

I˜13 · · · I˜23 · · · I˜33 · · ·

⎤ I˜1N ⎢ ⎢ ⎢ I˜2N ⎦⎢ ⎢ I˜3N ⎣

⎤

Ak χk,, ) ⎥

+ ⎢ E 1 k=1 ⎢ N ⎢  V = (IT I)−1 IT S = I˜ S → ⎢ E1 (Bk χk,, ) ⎢ k=1 ⎢ ⎣ N  E1 (Ck χk,, ) ⎡

249

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

k=1

S1 S2 S3 .. .

⎤

⎥ ⎥ ⎥ ⎥. ⎥ ⎦

(7.126)

SN By substituting Eq. (7.126) into Eq. (7.121), one can obtain the explicit relationship between σ zz and Sk (k = 1, 2, · · · , N ): ˆ

S S S σ zz (z, n, s) = σ zz (z, n, s) + σ zz (z, n, s) (z, n, s) + σ zz ] [ ] S(z) S(z) [ U ≡ −n ψnχ (s) −n ψn y (s) + IU y Iχ ] ˆS(z) [ −n ψnχˆ (s) , + Iχˆ

(7.127)

ˆ are related to Sk (z) (k = 1, 2, . . . , N ) as where S(z), S(z), and S(z) S(z) = IU y

N [ 

N [ ] ]  I˜2k Sk (z) ; I˜1k Sk (z) ; S(z) = Iχ

k=1

ˆ S(z) = Iχˆ

N [ 

k=1

] I˜3k Sk (z) .

(7.128)

k=1 U

S Equation (7.127) shows that σ zz (z, n, s) behaves as (−n ψn y (s)) on the n-s plane, which is identical to σ zz due to Mz , as expressed in Eq. (7.94). The remaining strain χ χˆ Sˆ S and σ zz , behave as (−nψn ) and (−nψn ), respectively. Figure 7.7 components, σ zz Sˆ Sˆ S S S , and σ zz and σ zz , σ zz , demonstrating that σ zz shows the stress distributions of σ zz η

ηˆ

cause pure bending of walls associated with ψn k of mode ηk and ψn k of mode ηˆ k , ηˆ η respectively. The shape functions ψn k and ψn k will be derived in the next section.

250

7 Sectional Shape Functions for a Box Beam Under Flexure

t

t

σ

t

σ

S zz

ˆ

S zz

σ zzS

edge 2

t

t

t

edge 1

edge 3 edge 4

ψ zθ

x

σ zzS

ˆ

σ zzS

σ zzS

ˆ

ˆ

S , σ S , and σ S on edges 1 and 2. Note that the σ S and σ S stresses Fig. 7.7 Distributions of σ zz zz zz zz zz

induce pure wall bending associated with

η

η ψn k

and

ηˆ ψn k

of mode ηk and ηˆ k , respectively

ηˆ

7.5 Derivation of {ψn k , ψn k } of Mode {η k , ηˆ k } As explained at the end of the previous section, wall-bending deformations due to S Sˆ σ zz can be described by new modes ηk and ηˆ k , respectively. Note that the and σ zz η

ηˆ

ηk and ηˆ k modes have only n-directional shape functions {ψn k , ψn k }. Moreover, the rotations at sectional corners are set to be zero for the ηk modes, while they are nonηˆ η zero for the ηˆ k modes. Because the procedure to derive {ψn k , ψn k } for a box beam subjected to a flexural load is identical to that given for a box beam subjected to a torsional load in Chap. 5, the detailed procedure will not be repeated here.7 Instead, ηˆ η we will directly start with the conditions that ψn k (s) and ψn k (s) must satisfy. ηk To determine ψn (s), the following conditions can be used: ψ˙ nη N (s j = +s j ) = 0, ψ˙ nη N (s j+1 = −s j+1 ) = 0 ( j = 1, 2, 3, 4),

(7.129a)

ψ¨ nη N (s j = +s j ) = ψ¨ nη N (s j+1 = −s j+1 ) ( j = 1, 2, 3, 4),

(7.129b)

ψnη N (s j = +s j ) = 0, ψnη N (s j+1 = −s j+1 ) = 0 ( j = 1, 2, 3, 4),

(7.129c)



7

ψ˙ nη N ψ˙ nηk ds =



ψ˙ nη N ψ˙ nηˆ k ds = 0 (1 ≤ k ≤ N − 1),

If necessary, one can find the details in Choi and Kim (2020) for this bending case.

(7.129d)

η

ηˆ

7.5 Derivation of {ψn k , ψn k } of Mode {ηk , ηˆ k }

251

edge 2

edge 3

edge 1

moment (curvature) continuity

zero angle edge 4

(a) edge 2

edge 3

edge 1

angle continuity edge 4

(b)

moment (curvature) continuity η

Fig. 7.8 Slope conditions and moment continuities for constrained distortion modes: a for ψn k (type 1) and b

ηˆ ψn k (type 2)

where Eq. (7.129a) denotes the zero-slope condition at the corners of the crosssectional walls of a box beam and Eq. (7.129b) defines the moment continuity at the corner; see Fig. 7.8a. Because pure wall bending is considered, the displacement of η ψn k should be zero at the wall corners, as given by Eq. (7.129c). Equation (7.129d) states the orthogonality conditions. Because there are 2N + 3 conditions available from Eq. (7.129), one can assume ηk ψn (s) as η

ψn,1N =

N +2 

η η C˜ j N · (s1 )2 j−1 , ψn,2N =

j=1 η

ψn,3N = −

N +2 

N +2 

η D˜ j N · (s2 )2 j−2 ,

j=1 η η C˜ j N · (s3 )2 j−1 , ψn,4N = −

j=1

N +2 

η D˜ j N · (s4 )2 j−2 .

(7.130)

j=1 η

η

where odd functions of si for ψn,ik (si ) (i = 1, 3) and even functions of si for ψn,ik (si ) (i = 2, 4) are used considering the symmetric nature of the deformation field of the cross-section of a box beam under vertical flexural loading. The polynomial orders in Eq. (7.130) are so determined that the involved 2N + 4 coefficients can be uniquely

252

7 Sectional Shape Functions for a Box Beam Under Flexure η

determined, except for an arbitrarily selectable scaling parameter. If C˜ NN+2 is selected as a scaling parameter, Eq. (7.130) can be put into the following form: η ψn,1N

= r ∗N ·

[ N +1 

] Cmη N · (s1 )2m−1 + (s1 )2N +3 ;

m=1

η ψn,2N

=

r ∗N

·

[ N +2 

] Dmη N

· (s2 )

2m−2

;

m=1 η ψn,3N

=

−r ∗N

·

[ N +1 

] Cmη N

· (s3 )

Dmη N

· (s4 )

+ (s3 )

2m−1

m=1

η ψn,4N

=

−r ∗N

·

[ N +2 

2N +3

;

] 2m−2

(7.131)

m=1 η η η η η η where r ∗N ≡ C˜ NN+2 and CmN and DmN are defined as CmN  (C˜ mN /r ∗N ) and DmN  η η η ( D˜ mN /r ∗N ). We can use Eq. (7.129) to determine CmN (m = 1, 2, . . . , N + 1) and DmN ηN (m = 1, 2, . . . , N + 2) and thus ψn . η As a specific case, ψn 1 is found using the equations η

ψn,1j1 = (−1)( j1 −1)/2 /(25b2 h 3 ) [ ] · 8s j51 − 4h 2 · s j31 + (h 4 /2) · s j1 ( j1 = 1, 3), η

ψn,1j2 = (−1)( j2 −2)/2 /(25b2 h 3 ) ] [ · (4h 3 /b2 ) · s j42 − 2h 3 · s j22 + (b2 h 3 /4) ( j2 = 2, 4), η

(7.132) η

in which r 1∗ is set to r 1∗ = 8/(25b2 h 3 ) to normalize ψn N (N ≥ 1) as ψn N (s2 = 0) = 1/100. ηˆ Likewise, ψn N is assumed as ψnηˆ N (s1 )

=

rˆN∗

·

[ N +1 

] Cmηˆ N

· (s1 )

Dmηˆ N

· (s2 )

2m−1

m=1

ψnηˆ N (s2 )

=

rˆN∗

·

[ N +2 

+ (s1 )

2N +3

;

] 2m−2

;

m=1

ψnηˆ N (s3 )

=

−ˆr N∗

·

[ N +1  m=1

ψnηˆ N (s4 ) = −ˆr N∗ ·

[ N +2  m=1

] Cmηˆ N

· (s3 )

2m−1

+ (s3 )

2N +3

;

] Dmηˆ N · (s4 )2m−2 .

(7.133)

7.6 Case Studies

253 η

If rˆN∗ in Eq. (7.133) is such that it is identical to r N∗ used to determine ψn 1 in Eq. (7.132), the remaining 2N + 3 coefficients can be determined using the following conditions: ψ˙ nη N (s j = +s j ) = ψ˙ nη N (s j+1 = −s j+1 ) ( j = 1, 2, 3, 4)

(7.134a)

ψ¨ nηˆ N (s j = +s j ) = ψ¨ nηˆ N (s j+1 = −s j+1 ) ( j = 1, 2, 3, 4)

(7.134b)

ψnηˆ N (s j = +s j ) = 0, ψnηˆ N (s j+1 = −s j+1 ) = 0 ( j = 1, 2, 3, 4)

(7.134c)



ψ˙ nηˆ N ψ˙ nηk ds =



ψ˙ nηˆ N ψ˙ nηˆ k ds =



ψ˙ nηˆ N ψ˙ nη N ds = 0 (1 ≤ k ≤ N − 1). (7.134d)

Compared to Eq. (7.129a) requiring a zero slope at the wall corners, Eq. (7.134a) simply stipulates slope continuities at the corners, as shown in Fig. 7.8b. Using the ηˆ ηˆ conditions given by Eq. (7.134), ψn k can be determined. For instance, ψn 1 (s)8 can be explicitly derived as ηˆ

ψn,1j1 =

72(b+h)2 · (−1)( j1 −1)/2 25(3b3 h 4 +15bh 6 +10h 7 )

[ · s j51 + · − 2b−h b

30b3 h 2 +27b2 h 3 −5h 5 18b(b+h)2

· s j31 −

42b3 h 4 +27b2 h 5 −h 7 144b(b+h)2

· s j1

]

( j1 = 1, 3); ηˆ ψn,1j2

= ·

72(b+h)2 · (−1)( j2 −2)/2 25(3b3 h 4 +15bh 6 +10h 7 )

[

−21b3 h 4 +15bh 6 +2h 7 18b3 (b3 +2b2 h+bh 2 )

· s j42 −

−3b2 h 4 +3bh 5 +2h 6 12b2 (b+h)

· s j22 +

(3b3 h 4 +15bh 6 +10h 7 ) 288(b+h)2

( j2 = 2, 4)

]

(7.135)

7.6 Case Studies We will consider three examples of a structural analysis of a box beam under vertical flexural loading using HoBT. All numerical analyses based on the HoBT were performed using the finite element method (Bathe 1996; Kim and Kim 1999; ηˆ

The computer code used to calculate ψn 1 can be downloaded from https://github. com/SChoiKNU/Codes-for-HoBT-based-Finite-Element-Analysis; see the Matlab code, Sectional_Shape_Function_Derivation_Code.m, included in Codes_for_Box_Beam_ Analysis\1_Mode_Shape_Derivation\3_Vertical_Flexural_Modes\1st_mode_set\Type_2_Constrained 8

η

ηˆ

_Distortion_Mode. Likewise, the computer code used to calculate ψn k and ψn k can also be found in Codes_for_Box_Beam_Analysis\1_Mode_Shape_Derivation\3_Vertical_ Flexural_Modes.

254

7 Sectional Shape Functions for a Box Beam Under Flexure

Jang and Kim 2009) in which the same interpolation scheme used in Chap. 6 was adopted; Hermite cubic interpolations were used for in-plane modes, allowing the zdirectional derivatives as well as instances of deformation to be continuous at element interfaces, while linear interpolation was employed for out-of-plane modes. Unless stated otherwise, the same material properties and geometric dimensions were used for all examples: Young’s modulus E = 200 GPa, Poisson’s ratio ν = 0.3, density ρ = 7850kg/m3 , beam length L = 1 m, width b = 50 mm, height h = 100 mm, and wall thickness t = 2 mm. The obtained results by HoBT will be compared with those by Abaqus shell elements (S4 elements; Hibbett et al. 1998) and the Timoshenko beam elements,9 denoted as “Shell” and “TBT,” respectively. The results by the generalized beam theory using GBTUL (Bebiano et al. 2015, 2018) are also compared. They are denoted as “GBT.” As the reference results, we used fully converged shell analysis results obtained using square elements of 6.25 mm × 6.25 mm. For all beam-based analyses, 160 beam elements for displacements and free vibration analyses and 640 beam elements for stress analyses were employed with a uniform mesh.

7.6.1 Case Study 1: Static Wall-Membrane Response by Vertical Force Fy Figure 7.9a shows a cantilever beam subjected to a tip shear force, Fy = 100 N. Because the beam is set to be rigid at its end, all higher-order modes are constrained at the end. Figures 7.9b–f show the displacements {u z , u x } along the line of s2 = b/4 and stresses {σzz , σ yy , τzy } along the line of s3 = −h/4. Because the displacements and ˆ modes are not stresses at the wall centerline (i.e., n = 0) are considered, the {η, η} involved in this problem. Therefore, we used N 2 = 0 (N 2 is the number of constrained distortion mode sets).10 The results obtained using m sets of {χk , Wk }k=1,...,m (m ≥ 1) and {U y , θx } are denoted as HoBT (N 1 = m). Figure 7.9c shows that HoBT (N 1 = 3) can capture the variations of u x as accurately as Shell, while the TBT approach cannot. In Figs. 7.9d–f, HoBT (N 1 = 5) can capture abrupt change in the stresses at the fixed end nearly as accurately as Shell. For convergence check, the problem depicted in Fig. 7.9a is solved using different values of N 1 . The errors by the HoBT results with respect to the converged shell results are plotted in Fig. 7.10, where u x and τzy are calculated at the points with (z = 3.75 mm, s2 = b/4) and (z = 1.959 mm, s3 = −h/4), respectively. The comparison suggests that the use of N 1 ≥ 3 for values displacement analysis and N 1 ≥ 5 for The analysis for TBT employs the non-deformable section modes (U x and θx ) only. Note that N2 should not be zero if the displacements or stresses at a generic point (z, n, s) of a wall, which is apart separate by n from the wall midline (i.e., n /= 0), are considered. 9

10

7.6 Case Studies

displacement measurement points (s2=b/4)

z

A

Displacement uz (m)

x

y

255

Fy=100 N

stress measurement points (s3=−h/4)

TBT HoBT (N1 =2) HoBT (N1 =1)

Shell

HoBT (N1 =3)

B

Longitudinal coordinate (m)

HoBT (N1 =1) HoBT (N1 =2)

(b) Shell HoBT (N1 =5) HoBT (N1 =2) HoBT (N1 =2)

TBT

HoBT (N1 =3) HoBT (N1 =2)

Shell

HoBT (N1 =1)

σzz ( N/m2)

Displacement ux (m)

(a)

HoBT (N1 =1)

HoBT

TBT

(N1 =5)

Shell HoBT (N1 =3)

HoBT (N1 =1)

TBT

Shell

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(c)

(d) HoBT (N1 =1)

( N/m2)

HoBT (N1 =1) HoBT (N1 =5) Shell

TBT

zy

σyy ( N/m2)

HoBT (N1 =2)

HoBT (N1 =1) HoBT (N1 =5) Shell HoBT (N1 =2)

TBT

Shell

HoBT (N1 =5) HoBT (N1 =2)

HoBT (N1 =5) Shell

HoBT (N1 =1)

HoBT (N1 =2)

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(e)

(f)

Fig. 7.9 a A cantilevered box beam under a vertical flexural load Fy at its end, b–c displacements along (s2 = b/4, n = 0), and d–f stresses along (s3 = −h/4, n = 0) (Choi and Kim 2021)

values stress analysis provides converged and sufficiently accurate results with less than 0.5% error.

256

7 Sectional Shape Functions for a Box Beam Under Flexure

Displacement ux

Error (%)

Fig. 7.10 Convergence of displacement and stress in the bending beam problem of Fig. 7.9a (No constrained distortion modes were used.) (Choi and Kim 2021)

Stress τzy

N1 (number of the used membrane mode sets)

7.6.2 Case Study 2: Mixed Response of Wall-Membrane and Wall-Bending by Surface Tractions {t zz , t zn } In Figs. 7.11a and 7.12a, a cantilever box beam is subject to somewhat arbitrarily distributed traction types {tzz , tzn } at its free end. Specifically, the following traction distributions are considered: • tzz in Fig. 7.11a: tzz (z, s1 ) = tzz (z, s2 ) = tzz (z, s3 ) = tzz (z, s4 ) =

1 t 1 t 1 t 1 t

) ( 6144 3 MPa; · − 24576 s 3 − 384 s 2 + 15680 s1 + 266 245 1 133 1 ) ( 100992 2 1578 · − 4655 s2 + 116375 MPa; ) ( 6144 3 MPa; · 24576 s 3 − 384 s 2 − 15680 s3 + 266 245 3 133 3 ) ( 74112 2 1158 · 4655 s4 − 116375 MPa,

(7.136)

• tzn in Fig. 7.12a: tzn (z, s1 ) = 0 MPa; s2 + tzn (z, s2 ) = − 32 45 2 tzn (z, s3 ) = 0 MPa; tzn (z, s4 ) =

32 2 s 45 4

−

1 1000

1 1000

MPa;

MPa.

(7.137)

7.6 Case Studies

A

displacement measurement points (s2=b/4)

x z

stress measurement points (s3=−h/8)

TBT Displacement uz (m)

y

257

Shell HoBT (12 modes) GBT (13 modes)

Shell HoBT (12 modes) GBT (13 modes)

tzz B

Longitudinal coordinate (m)

(b)

HoBT (12 modes) Shell GBT (13 modes) HoBT (12 modes)

(13 modes)

Shell HoBT (20 modes)

σzz ( N/m2)

Shell GBT

GBT (23 modes)

HoBT (20 modes) GBT (23 modes) Shell TBT

TBT

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(c)

(d)

GBT (23 modes) TBT

τ zy ( N/m2)

Displacement ux (m)

(a)

σyy ( N/m2)

TBT

TBT

Shell GBT (23 modes)

HoBT (20 modes)

Shell HoBT (20 modes) GBT (23 modes)

HoBT (20 modes)

Shell

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(e)

(f)

Fig. 7.11 a A cantilevered box beam under a distributed tzz (given in Eq. (7.136)) at its end, b, c displacements along (s2 = b/4, n 2 = 0), and d–f stresses along (s3 = −h/8, n 3 = 0). Here, the mode number in parentheses refers to 4N1 . (N2 = 0 because the displacements and stresses at the wall centerline (n = 0) are considered) (Choi and Kim 2021)

258

7 Sectional Shape Functions for a Box Beam Under Flexure

y

displacement measurement points (s2=0)

x z

A

scale: x1e+01

tzy

stress measurement points (s2=b/8, n2=t/2) B

(a)

(b)

Displacement uy (m)

Displacement uz (m)

Shell HoBT (0 mode) HoBT (2 modes)

TBT GBT (2 modes)

Shell

HoBT (0 mode) TBT GBT (2 modes) TBT

Shell

HoBT (2 modes) HoBT (0 mode) Longitudinal coordinate (m)

Longitudinal coordinate (m)

(c)

HoBT (2 modes) GBT (2 modes)

(d)

TBT

HoBT (0 mode)

τ zx ( N/m2)

σzz ( N/m2)

TBT Shell HoBT (6 modes) GBT (6 modes) GBT (6 modes) HoBT (6 modes)

HoBT (0 mode) HoBT (6 modes) GBT (6 modes) Shell HoBT (6 modes)

Shell

Shell GBT (6 modes)

Longitudinal coordinate (m)

Longitudinal coordinate (m)

(e)

(f)

Fig. 7.12 a A cantilevered box beam under a distributed tzy (given in Eq. (7.137)) at its end, b, c displacements along (s2 = 0, n 2 = 0), and d–f stresses along (s2 = b/8, n 2 = t/2). Here, the mode number in parentheses refers to 2N2 (N1 = 3 or 5) (Choi and Kim 2021)

7.6 Case Studies

259

In this example, we aim to examine the importance of employing the higher-order sectional modes in capturing rapidly varying stresses near the free end. The obtained results are illustrated in Figs. 7.11b–f. The figures plot the wall-membrane responses of {u z , u x } and {σzz , σ yy , τzy } calculated on the lines with s2 = b/4 and s3 = −h/8. In the figures, the result by HoBT using N 1 membrane mode sets only (i.e., N2 = 0) is denoted by HoBT (four N1 modes11 ). For comparison, the GBT results obtained by GBTUL (Bebiano et al. 2018) are also presented. Figures 7.11b–f demonstrate that HoBT (with 12 modes (N1 = 3) and 20 modes (N1 = 5)) can capture rapid field variations near the beam end nearly as accurately as the Shell and GBT approaches. Figure 7.12a describes a cantilevered box beam subjected to tzy .The displacements and stressed in the beam are plotted in Fig. 7.12b, c and Fig. 7.12e–f, respectively. The results by HoBT employing m constrained distortion mode sets {ηk , ηˆ k }k=1, ..., m are denoted by HoBT (2 × N 2 modes). Thus, HoBT (0 mode) denotes the results without using the constrained modes; {U y , θx } and wall-membrane mode sets (either with N 1 = 3 or 5) are used. Figures 7.12c–f show that the localized effect near the end can be well estimated in the HoBT case (with two modes (N2 = 1) and six modes (N2 = 3)). The results by HoBT (with two modes (N2 = 1) and six modes (N2 = 3)) are comparable to those by Shell and GBT, whereas the TBT and the HoBT (0 mode) outcomes are not. To examine the effect of the number of constrained distortion mode sets (N 2 ) on the solution accuracy, the problem shown in Fig. 7.12a is solved again with different N 2 values ranging from 1 to 8. The errors of the HoBT compared to the shell results are plotted in Fig. 7.13. The displacement u y and stress τzx calculated along the points at (z = 1 m, s2 = 0, n2 = 0) and (z = 1 m, s2 = b/8, n2 = t/2), respectively, were used for the results in Fig. 7.13. This figure shows that if N 2 ≥ 1 for the displacement analysis and N 2 ≥ 3 for the stress analysis, the obtained results by HoBT are accurate within error levels of 0.4% and 1.6%, respectively.

7.6.3 Case Study 3: Free Vibration Response The free vibration responses of a box beam were analyzed. For this analysis, nondeformable section modes {U y , θx },{χk , Wk }k=1,2,3 , and {ηk , ηˆ k }k=1 (thus, N1 = 3 and N2 = 1) were used. The eigenfrequencies for the lowest five modes by HoBT ,

11

If N 1 membrane mode sets are considered, the number of higher-order deformable sectional modes included in the HoBT is 4N 1 . Among 4N 1 modes, 2N 1 modes are unconstrained distortion and warping modes (i.e., N1 χ , s and N 1 W ’s.) derived for a beam under flexure. The other remaining 2N 1 modes are unconstrained distortion and warping modes (i.e., N1 χ , s and N 1 W ’s.) derived for a beam under extension, which were discussed in Chap. 6. Because the loading condition in Fig. 7.11a leads to the coupled behavior of extension and vertical bending, the extension-related higher-order deformation sectional modes are also considered in this case.

260

7 Sectional Shape Functions for a Box Beam Under Flexure

Error (%)

Stress τzx Displacement uy

N2 (number of the used edge-bending mode sets) Fig. 7.13 Convergence of displacement and stress for the beam problem depicted in Fig. 7.12a (Choi and Kim 2021)

Table 7.1 Eigenfrequencies of a Box Beam (b = 100 mm, h = 50 mm, L = 0.5 m, and t = 2 mm)12 (Choi and Kim 2021) Theory

Mode 1st

2nd

3rd

4th

5th

Shell (Abaqus)

815.4

949.5

1050.5

1179.7

1224.4

GBT

811.7 (0.5%) 944.3 (0.5%) 1045.5 (0.5%) 1174.8 (0.4%) 1213.9 (0.9%)

HoBT (N 1 = 3, 811.6 (0.5%) 944.2 (0.6%) 1045.1 (0.5%) 1173.9 (0.5%) 1222.4 (0.2%) N 2 = 1)

Shell, and GBT are listed in Table 7.1, and their mode shapes are shown in Fig. 7.14. The results in Table 7.1 and Fig. 7.14 show that the HoBT results with (N 1 = 3, N 2 = 1) are within error of 0.5% in comparison with those by the shell analysis.

12 If the shell finite elements are used, not only the bending-related eigenfrequencies but also other eigenfrequencies representing torsion and extension are obtained. Here, only the bending-related eigenfrequencies are taken from the shell analysis.

References

261

1st mode (Shell, 815.42 Hz)

1st mode (HoBT, 811.57 Hz)

3rd mode (Shell, 1050.5 Hz)

3rd mode (HoBT, 1045.1 Hz)

5th mode (Shell, 1224.4 Hz)

5th mode (HoBT, 1222.4 Hz)

Fig. 7.14 Mode shapes of the results in Table 7.1 Among the five modes discussed in Table 7.1, the first, third, and fifth eigenmodes are plotted as the representative modes (Choi and Kim 2021)

References Bathe (1996) Finite element procedures. Prentice Hall Bebiano R, Camotim D, Gonçalves R (2018) GBTul 2.0—a second-generation code for the GBTbased buckling and vibration analysis of thin-walled members. Thin-Walled Structures 124:235– 257 Bebiano R, Goncalves R, Camotim D (2015) A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses. Thin-Walled Structures 92:29–47 Choi S, Kim YY (2020) Consistent higher-order beam theory for thin-walled box beams using recursive analysis: edge-bending deformation under doubly symmetric loads. Eng Struct 206:110129 Choi S, Kim YY (2021) Higher-order beam bending theory for static, free vibration, and buckling analysis of thin-walled rectangular hollow section beams. Comput Struct 248:106494

262

7 Sectional Shape Functions for a Box Beam Under Flexure

Ferradi MK, Cespedes X (2014) A new beam element with transversal and warping eigenmodes. Comput Struct 131:12–33 Hibbett HD, Karlsson BI, Sorensen EP (1998) ABAQUS/standard: user’s manual. Hibbitt, Karlsson & Sorensen Jang GW, Kim YY (2009) Vibration analysis of piecewise straight thin-walled box beams without using artificial joint springs. J Sound Vib 326:647–670 Kim YY, Kim JH (1999) Thin-walled closed box beam element for static and dynamic analysis. Int J Numer Meth Eng 45:473–490

Chapter 8

Bridging Between Rectangular Cross-Sections and Generally Shaped Cross-Sections

Chapters 4, 5, 6, 7 presented procedures for deriving the shape functions of higherorder section-deformable modes of thin-walled rectangular cross-sections (or box beam cross-sections) recursively and hierarchically. To extend the procedure to thinwalled beams of generally shaped cross-sections, including open, closed, or open– closed sections, it is important to summarize the types of higher-order modes of a rectangular cross-section and the recursive relationships of their sectional shape functions. This summary will guide us to develop recursive relationships for the sectional shape functions of generally shaped cross-sections.

8.1 Higher-Order Modes for Cross-Sections with General Thin-Walled Shapes In Chaps. 4, 5, 6, 7, the following non-deformable (rigid-body) section modes and higher-order deformable section modes were identified for thin-walled box beams depending on the load type and/or deformation patterns: [Torsion] Nondeformable section mode: θz Fundamental deformable section modes: {χ0 , W0 } } ( ) { Higher - order deformable section modes: χkt , Wkt , ηtk , ηˆ kt k=1, 2, ··· , N t N t ≥ 1

(8.1a)

© Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_8

263

264

8 Bridging Between Rectangular Cross-Sections and Generally Shaped …

[Extension] Nondeformable section mode: Uz

{ ) } ( Higher - order deformable section modes: χke , Wke , ηek , ηˆ ke k=1, 2, ··· , N e N e ≥ 1

(8.1b) [Vertical Flexure]

{ } Nondeformable section modes: U y , θx

{ } ( ) Higher - order deformable section modes: χkv , Wkv , ηvk , ηˆ kv k=1, 2, ··· , N v N v ≥ 1

(8.1c) [Lateral Flexure]

{ } Nondeformable section modes: Ux , θ y

} { Higher - order deformable section modes: χkh , Wkh , ηkh , ηˆ kh

( k=1, 2, ··· , N h

) Nh ≥ 1 .

(8.1d) Although the modes for the case of lateral flexure have not been given explicitly, they can be readily found simply using the results given for the case of vertical flexure in Chap. 7. { } Recall that the Uq , θq (q = x, y, z) values represent the q-directional rigidbody sectional displacement and rotation, respectively. The zeroth-order distortion and warping modes {χ0 , W0 } involve sectional deformations just like other higherorder modes. Note that W0 describes a linear warping field. In Eq. (8.1a), the {χ0 , W0 } modes are treated separately from other higher-order modes {χk , Wk }k=1, 2,··· because they should be regarded as the fundamental modes needed to initiate a recursive analysis for the derivation of other higher-order deformable section modes. In Eq (8.1), the superscripts such as t, e, v, and h are used to denote different load types, in this case torsional, axial extension, vertical (or lateral) bending, ( bending, and horizontal ) respectively. Although each mode α α = χ , W, η, ηˆ is classified with different superscripts (t, e, v, h), modes α t , α e , α v , and α h having the same α represent the same deformation patterns. For instance, χ t , χ e , χ v , and χ h represent the in-plane distortion mode including s-directional wall extensions (i.e., extensional distortion), and W t , W e , W v , and W h denote the warping mode involving out-of-plane (zdirectional) non-linear wall deformations (i.e., non-linear warping). Likewise, ηβ and ηˆ β (β = t, e, v, h) represent in-plane constrained distortion modes with and without non-zero corner rotations, respectively. Based on the observations made for the rectangular cross-sections above, one can classify the section modes for thin-walled beams of generally shaped cross-sections as follows:

8.2 Recursive Equations to Derive Sectional Shape Functions

265

} { Nondeformable section motions: Ux , U y , Uz , θx , θ y , θz Fundamental deformable section modes { } : unconstrained distortion χk0 {

k=1, 2, ··· , Nχ 0

Wk0

}

k=1, 2, ··· , N W 0

: linear warping

(8.2)

Higher - order deformable section modes {χk }k=1, 2, ··· , Nχ : unconstrained distortion {Wk }k=1, 2, ··· , N W : nonlinear warping {ηk }k=1, 2, ··· , Nη : constrainted distortion

The deformable section modes in Eq. (8.2), in this case χ , W , and others, are not separately classified depending on the load type because thin-walled beams with general section shapes exhibit fully coupled behavior among the extension, bending, and torsion deformations. Non-deformable section modes Ux and U y representing rigid-body section translations are defined with respect to the x − y coordinate system, while the θx and θ y modes representing rigid-body section rotations are defined with respect to the x − y coordinate system. Different coordinate systems are useful because non-deformable section modes can be orthogonal to each other; the resultant forces and moments associated with Ux , U y , θx , and θ y become fully decoupled from each other. The origins and orientations of the (x, y) and (x, y) coordinate systems will be discussed in Sect. 9.3. It should be also noted that unlike a rectangular cross-section, an arbitrarily shaped cross-section can have multiple fundamental distortion and warping modes. For example, a polygonal thin-walled cross-section with 13 edges can have 20 fundamental deformable section modes—more details will be given in Chap. 9. Accord{ } ingly, new symbols χk0 , Wk0 are introduced in Eq. (8.2) to represent these fundamental modes. Lastly, constrained distortion modes are denoted solely as ηk , not classified into the type 1 and type 2 modes because they can be derived simultaneously from the same step in the recursive analysis procedure.

8.2 Recursive Equations to Derive Sectional Shape Functions In this section, recursive equations are established to derive the deformable section modes listed in Eq. (8.2) for thin-walled beams of generally shaped cross-sections. Here, the recursive equations are the equations that link the sectional shape functions of the current mode set to those of the lower-order mode sets. As shown in Chaps. 4, 5, 6, 7, recursive equations enable us to derive sectional shape functions through integration. In addition, explicit generalized force-stress relationships can also be derived using recursive equations. Below, the recursive equations used to derive the

266

8 Bridging Between Rectangular Cross-Sections and Generally Shaped …

sectional shape functions obtained in Chaps. 4, 5, 6, 7 are summarized and then extended for application to cross-sections with general shapes.

8.2.1 Recursive Relationships to Derive Distortion Modes χk { } The recursive relationships for distortion modes χkt , χke , χkv , χkh obtained in Chaps. 4, 5, 6, 7 for thin-walled beams of rectangular cross-sections were χ ψ˙ s N (s) = − p ∗N ,1 ψzW0 (s) − t

N −1

Wt

p ∗N ,k+1 ψz k (s),

(8.3a)

k χ ψ˙ s N (s) = − p ∗N ,1 ψzUz (s) − e

N −1

We

p ∗N ,k+1 ψz k (s),

(8.3b)

k χ ψ˙ s N (s) = − p ∗N ,1 ψzθx (s) − v

N −1

Wv

p ∗N ,k+1 ψz k (s),

(8.3c)

k=1 χ θ ψ˙ s N (s) = − p ∗N ,1 ψz y (s) − h

N −1

Wh

p ∗N ,k+1 ψz k (s),

(8.3d)

k=1 · / where ( ) = d ds. Among others, Eqs. (8.3) suggest that the sectional shape functions of the distortion modes χ in the current mode set (Nth-order set) are expressed in terms of the sectional shape functions of N − 1 lower-order mode sets (from order 1 to order N − 1). Note that the sectional shape functions of lower-order mode sets on the right sides of Eqs. (8.3) consist of z-directional shape functions only, i.e., the shape functions of the warping and z-directional non-deformable section modes; these modes represent out-of-plane modes. Based on this observation, the following χ relationship can be established for ψs k for arbitrarily shaped cross-sections,1

ψ˙ sχ N (s) = −

NO

pi ψzOi (s),

(8.4)

i=1

where Oi represents the ith element of the set of lower-order out-of-plane modes,

1

We can have multiple unconstrained and constrained distortion modes for each mode set for a crosssection of a general shape. Therefore, different notations will be used to denote the mode number and the mode set number, correspondingly indicated by subscript and superscript, respectively (details are given in Chap. 9.) If this rule applies, we should have denoted the mode set number N as the superscript of χ in the left side of Eq. (8.4). However, the rule was not applied here to avoid unnecessary complications.

8.2 Recursive Equations to Derive Sectional Shape Functions

267

}T { O = Uz , θx , θ y , W1 , · · · , W NW (N W : : total number of lower-order warping modes). { } Note that linear warping functions are included in W1 , · · · , W NW . In Eq. (8.4), N O is the total number of modes in set O. Therefore, N O = 3 + N W . The method to determine the coefficients pi in Eq. (8.4) and similar coefficients appearing below will be presented in Chap. 9.

8.2.2 Recursive Relationships to Derive Warping Modes Wk The warping mode Wk has a non-zero z-directional sectional shape function ψzWk . . The recursive equations used to derive the shape function of a rectangular crosssection were given as (see Chaps. 4, 5, 6, 7), t

W ψ˙ z N (s) = q N∗ ,N +1 ψsθz (s) + q N∗ ,N +2 ψsχ0 (s) +

N

χt

q N∗ ,k ψs k (s),

(8.5a)

k=1 e

W ψ˙ z N (s) =

N

χe

q N∗ ,k ψs k (s),

(8.5b)

k=1 v

W U ψ˙ z N = q N∗ ,N +1 ψs y +

N

χv

q N∗ ,k ψs k ,

(8.5c)

k=1 h

W ψ˙ z N = q N∗ ,N +1 ψsUx +

N

χh

q N∗ ,k ψs k .

(8.5d)

k=1

Note in Eqs. (8.5) that the shape functions on the right sides represent in-plane deformations only, i.e., in-plane non-deformable and deformable distortion modes where the fundamental distortion mode is also included. Extending these relationships for thin-walled beams of general cross-sections, the sectional shape function ψzW N of warping mode W N can be expressed as ψ˙ zW N (s) =

NI

qi ψsIi (s),

(8.6)

i=1

}T { where Ii denotes the ith element of set I = Ux , U y , θz , χ1 , · · · , χ Nχ of all inplane modes lower than the current mode set N (Nχ : : total number of lower-order distortion modes), and N I is the total number of modes in set {I. Therefore, N I = } 3+ Nχ . Note that inextentional distortion modes are included in χ1 , χ2 , · · · , χ Nχ .

268

8 Bridging Between Rectangular Cross-Sections and Generally Shaped …

8.2.3 Recursive Relationships to Derive Wall-Bending Modes ηk The constrained distortion mode n-directional sectional shape func{ t ηk hase a non-zero } η η ηh ηv η tion, ψn k . In Chaps. 5, 6, 7, ψn N , ψn N , ψn N , ψn N were derived as the constrained distortion modes of type 1 using the following recursive equations (similar recursive equations can also be found for constrained distortion modes of type 2 in Chaps. 5, 6, 7.): t η ψ¨ n N (s) = −r ∗N ,1 ψnχ (s) − t

N −1

ηt

r ∗N ,k+1 ψn k (s),

(8.7a)

k=1 e η ψ¨ n N (s) = −r ∗N ,1 ψnχ (s) − e

N −1

ηe

r ∗N ,k+1 ψn k (s),

(8.7b)

k=1 v η ψ¨ n N (s) = −r ∗N ,1 ψnχ (s) − v

N −1

ηv

r ∗N ,k+1 ψn k (s),

(8.7c)

k=1 h η ψ¨ n N (s) = −r ∗N ,1 ψnχ (s) − h

N −1

ηh

r ∗N ,k+1 ψn k (s).

(8.7d)

k=1

At this point, we generalize the recursive equations in Eq. (8.7) for generally shaped cross-sections as ψ¨ nη N (s) =

NI

I

r N , j ψn j (s),

(8.8)

j=1

where I j represents the jth element of set I, which is defined as I = I ∪ }T { η1 , η2 , ..., η Nη . Set I includes in-plane modes belonging to all mode sets of orders lower than N . Compared to set I, set I additionally includes constrained distortion modes that have only n-directional sectional shape function components. Note that unlike the case of a rectangular cross-section, index j in Eq. (8.8) does not indicate the mode set number because multiple unconstrained and constrained distortion modes can be obtained for each mode set for a cross-section of a general shape (details are given in Chap. 9). In addition, we assume that the recursive equation in Eq. (8.8) is also valid for fundamental distortion modes in the zeroth set2 :

2

For a box beam, the torsional distortion mode in Chap. 2 corresponds to the fundamental distortion mode.

8.2 Recursive Equations to Derive Sectional Shape Functions

ψ¨ nχ0 (s) =

NI

I

r0, j ψn j (s).

269

(8.9)

j=1

As will be explained in Chap. 9, the fundamental distortion mode χ0 in Eq. (8.9) is treated as the first mode among the derived deformable section modes. Therefore, I j }T { in Eq. (8.9) refers to non-deformable in-plane modes only; i.e., I = I 1 , I 2 , I 3 = }T { χ Ux , U y , θz . As will be shown in Chap. 9, ψn 0 for a cross-section with a general χ shape can be derived using continuity conditions at sectional corners once ψs 0 is χ0 determined. As a result, ψn is obtained as edgewise cubic polynomials, as in the case χ of a box beam (see Eq. (2.49)). However, this ψn 0 satisfying continuity conditions at sectional corners does not satisfy the recursive equation of Eq. (8.9) in general. Although both sides of Eq. (8.9) are edgewise linear, there is no unique r N , j that makes Eq. (8.9) hold for all edges. Nonetheless, the relationship in Eq. (8.9) is required to define general recursive relationships for n-directional sectional shape functions, expressed as follows: I ψ¨ n m (s) =

NI

I

rm, j ψn j (s).

(8.10)

j=1

The validity of using Eq. (8.10) will be confirmed through the various case studies χ considered in Chap. 9. Note that ψnI m on the left side of Eq. (8.10) can represent ψn 0 as η χ well as constrained distortions ψn . Moreover, ψn representing unconstrained distorχ tion modes of orders higher than mode ψn 0 of the lowest order can be represented χ by ψnI m in Eq. (8.10) because ψn will be also expressed as edgewise cubic polynomials (see details in Sect. 9.5.1). Therefore, the aforementioned recursive equation is assumed valid for all n-directional shape functions of in-plane sectional modes. According to the recursive equation in Eq. (8.10), the polynomial order of constrained distortion modes increases as the mode set order increases.

Chapter 9

Sectional Shape Functions of Thin-Walled Beams with General Cross-Section Shapes

In Chaps. 4, 5, 6, 7, the shape functions of the deformable section modes of a box beam were derived in an approach with three key steps. Recursive equations derived in a differential form were integrated edgewise to find sectional shape functions. To determine the unknown coefficients and integration constants of the sectional shape functions, the geometric symmetry of a rectangular cross-section, continuity of field quantities at every corner of the section, and orthogonality conditions were used. However, as the conditions of geometric symmetry are mostly not available for a cross-section with a general shape,1 we need to find new conditions to determine the unknowns. As in a box beam, new shape functions in a higher set for a beam with an arbitrarily shaped cross-section can be derived by integrating recursive equations. This means that they can be obtained by a linear combination of the integrated functions of the shape functions in lower-order sets. To determine the unknown coefficients and integration constants of new shape functions, continuity conditions at sectional corners as well as orthogonality conditions can be used as in the case of a box beam. To extend this idea to general cross-sections, we note that new shape functions in a higher set should be orthogonal to each other. Based on this observation, our proposition is to formulate an eigenvalue problem from which the coefficients and integration constants of new shape functions can be obtained as eigenvectors. Continuity conditions at sectional corners and additional orthogonality conditions with respect to shape functions in lower-order sets are imposed as constraints for the eigenvalue problem through Lagrange multipliers. In this approach, therefore, multiple shape functions can be simultaneously obtained in a higher set by finding multiple sets of the coefficient vector as the eigenvectors. If the eigenvalue problem is solved, one can find ψzW (s) (the z-directional shape functions of warping modes), χ ψs (s) (the s-directional shape functions of unconstrained distortion modes), and η ψn (s) (the n-directional shape functions of constrained distortion modes). On the 1

General shapes include cross-sections with various configurations, such as open, closed, singlecell, multicell, flanged, and non-flanged types.

© Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_9

271

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9 Sectional Shape Functions of Thin-Walled Beams with General … χ

other hand, ψn (s) (the n-directional shape functions of unconstrained distortion modes) can be calculated simply by considering continuity conditions at sectional χ χ corners once ψs (s) are derived. This step is needed because ψn (s) cannot appear explicitly in a recursive form.

9.1 Displacement, Strain, and Stress Fields at a Generic Point A thin-walled cross-section consisting of N E edges is illustrated in Fig. 9.1, where X, Y, and Z are the global coordinates and nj and sj , defined on the midline of edge j (j = 1, 2, ···, N E ), are local coordinates ( )representing the normal and tangential directions, respectively. The origin of nj , sj is located at one end of edge j (see Fig. 9.1), and the angle of edge j with respect to the X-axis is denoted as α j . As was examined in Sect. 3.2.1, the three-dimensional displacements on the midline of a cross-section are expressed as uz (z, s) =

ND Σ

ψzξi (s)ξi (z),

(9.1a)

i=1

(X4 , Y4) n4

Corner 2 4

s4 Midline Edge 4

Centroid & Principal axes (XC , YC)

y

Corner 3 Y

(X5 , Y5) n5

Z

Edge 2

s3

y

β

(X0 , Y0)

Corner 1

Edge 3

n3 3

x

β

Torsional center & Axes of deflections

x

s2

(X3 , Y3) s1

n2 2

(X2 , Y2)

Edge 1 1

s 5 5

Edge 5 5

X

(X1 , Y1) n1 Corner 4

5

Fig. 9.1 Geometry of a thin-walled (cross-section with various coordinate systems illustrated: ) (X , Y , Z) denote global coordinates, z, nj , sj represent edgewise coordinates, (x, y, z) are the principal axes, and (x, y, z) denote the axes of deflection

9.1 Displacement, Strain, and Stress Fields at a Generic Point

un (z, s) =

ND Σ

273

ψnξi (s)ξi (z),

(9.1b)

ψsξi (s)ξi (z),

(9.1c)

i=1

us (z, s) =

ND Σ i=1

where ξi (i = 1, 2. · · · , ND ) denotes the ith section mode (or field variable), which is a ξ function of z, and ψp i (s) is the sectional shape function representing the displacement shape of up (p = z, n, s) associated with mode ξ i . Equation (9.1) can be expressed more compactly in a vector form as up (z, s) =

ND Σ

ψpξi (s)ξi (z) = ψp ((s)ξ(z),

(p = z, n, s),

(9.2)

i=1

} { }T { ξN where ψp = ψpξ1 , · · · , ψp D and ξ = ξ1 , · · · , ξND . The section modes in Eq. (9.2) consist of both non-deformable (rigid-body) and deformable (higher-order) section modes: }T { ξ = {ξi }i=1,··· ,ND = Ux , Uy , Uz , θx¯ , θy¯ , θz , ξ7 , · · · , ξND }T { = Ux , Uy , Uz , θx¯ , θy¯ , θz , W1 , · · · , WNW , χ1 , · · · , χNχ , η1 , · · · , ηNη , (9.3) { } where the first six symbols Ux ,{Uy , Uz , θx , θy , θ}z denote the non-deformable section modes, and the symbols ξ7 , ξ8 , · · · , ξND represent deformable section modes can be further classified as warping }modes {modes. The deformable } { W1 , W2 , · · · , WNW , unconstrained χ distortion modes , 1 χ2 , · · · , χNχ , and { } constrained distortion modes η1 , η2 , · · · , ηNη . In Eq. (9.3), NW , Nχ , and Nη denote the numbers of warping, unconstrained distortion, and constrained distortion modes, respectively. Note that unlike the deformable section modes of a box beam as derived in Chaps. 4, 5, 6, 7, the subindices of W , χ , and η in Eq. (9.3) do not represent the corresponding mode set numbers. We use this rule here because multiple warping, unconstrained distortion, or constrained distortion modes ( can exist ) in the mode set and U U at the same level. Note also that the deflection modes x y and the bending ) ( rotation modes θx and θy are defined with respect to different coordinate systems, (x, y) and (x, y), respectively. Figure 9.1 shows how these two coordinate systems are defined for a given cross-section. We use two coordinate systems because doing so makes it more convenient to meet the orthogonality conditions among the nondeformable section modes. The use of two coordinate systems also decouples the resultant forces and moments from each other. The definitions for (x, y) and (x, y) will be explained in detail in Sect. 9.3.

274

9 Sectional Shape Functions of Thin-Walled Beams with General …

( ) The section modes in Eq. (9.3) can be classified into in-plane ξI and out-of-plane ( O) ξ deformation modes: ξ = ξI ∪ ξO

(9.4)

}T { ξI = Ux , Uy , θz , χ1 , · · · , χNχ , η1 , · · · , ηNη ,

(9.5)

}T { ξO = Uz , θx , θy , W1 , · · · , WNW ,

(9.6)

/{ }T ¯ η 1 , · · · , η Nη ξI = ξI }T { = Ux , Uy , θz , χ1 , · · · , χNχ ,

(9.7)

with

and

where ξI and ξO denote the sets of in-plane and out-of-plane sectional deformation modes, respectively. The symbol ξI in Eq. (9.7) represents the subset of ξI , consisting of modes with non-zero s-directional deformations. Using Eqs. (9.5), (9.6), (9.7), the three-dimensional displacements on the midline of a cross-section can be written as O uz (z, s) = ψO z (s)ξ (z),

(9.8a)

un (z, s) = ψIn (s)ξI (z),

(9.8b)

us (z, s) = ψIs (s)ξI (z),

(9.8c)

⎫ ⎧ } { IN IN I I1 ψsI1 , · · · , ψs I , and ψO = where = ψn , · · · , ψn , ψIs = z } { ON ψzO1 , · · · , ψz O are the shape function vectors of section modes ξI , ξI , and ξO , respectively. One can express the three-dimensional displacement u˜ p (p = z, n, s) at a generic point on the cross-section using the displacements on the midline in Eq. (9.8) as ψIn

u˜ z (z, n, s) = uz (z, s) − nun, (z, s) ¯

¯

= ψzO (s)ξ O (z) − nψnI (s)ξ I , (z) = uz (z, s) + u¯ z (z, s), u˜ n (z, s) = un (z, s)

(9.9a)

9.1 Displacement, Strain, and Stress Fields at a Generic Point ¯

275

¯

= ψnI (s)ξ I (z),

(9.9b)

u˜ s (z, n, s) =us (z, s) − n˙un (z, s) ¯

I ¯ =ψ Is (s)ξ I (z) − nψ˙ n (s)ξ I (z)

=us (z, s) + u¯ s (z, s),

(9.9c)

· / / where ( ) = ∂( ) ∂s and (), = ∂( ) ∂z. In Eqs. (9.9a, c), uz and us are displacements due to wall-bending deformation. Assuming a plane stress state, the strain (˜ε ) and stress (σ˜ ) are calculated as

ε˜ zz = u˜ z, ¯

¯

I O, I ,, = ψO z (s)ξ (z) − nψ n (s)ξ (z) = εzz + ε¯ zz ,

(9.10a)

ε˜ ss = u˙˜ s ¯

I I ¯ = ψ˙ s (s)ξ I (z) − nψ¨ n (s)ξ I (z)

= εss + ε¯ ss ,

(9.10b)

γ˜zs =˜us, + u˙˜ z O

I¯

¯

=ψ Is (s)ξ I , (z) + ψ˙ z (s)ξ O (z) − 2nψ˙ n (s)ξ I , (z) =γzs + γ¯ zs ,

(9.10c)

σ˜ zz = E1 (ν ε˜ ss + ε˜ zz ) ]⎫ ⎧ [ I I¯ I O O, I¯ I¯ I¯ ,, ˙ ¨ = E1 ν ψ s (s)ξ (z) + ψz (s)ξ (z) − n ν ψ n (s)ξ (z) + ψn (s)ξ (z) = σzz + σ¯ zz ,

(9.11a)

σ˜ ss = E1 (˜εss + ν ε˜ zz ) ]⎫ ⎧ [ I I¯ ¯ ¯ ¯ = E1 ψ˙ s (s)ξ I (z) + νψzO (s)ξ O, (z) − n ψ¨ n (s)ξ I (z) + νψnI (s)ξ I ,, (z) = σss + σ¯ ss ,

(9.11b)

τ˜zs = G γ˜zs ] [ O I¯ I I, O I¯ , ˙ ˙ = G ψ s (s)ξ (z) + ψ z (s)ξ (z) − 2nψ n (s)ξ (z) = τzs + τ¯ zs ,

(9.11c)

276

9 Sectional Shape Functions of Thin-Walled Beams with General …

where E 1 = E/(1 − ν 2 ), and E, ν, and G represent Young’s modulus, Poisson’s ratio, and the shear modulus, respectively. In Eqs. (9.10), (9.11), {εzz , εss , γzs } and {σzz ,{σss , τzs } correspondingly represent the strain and stress on the wall midline, } and εzz , ε ss , γ zs and {σ zz , σ ss , τ zs } representing the terms multiplied by n are the strain and stress generated by wall-bending deformation at a generic point of a cross-section. To derive the generalized forces, the virtual work δWA on the surface of crosssection A by the surface traction τ˜zs and σ˜ zz is considered:  δWA (z) =

 τ˜zs (z, s)δ u˜ s (z, s)dA +

A

σ˜ zz (z, s)δ u˜ z (z, s)dA A



[τzs (z, s) + τ¯ zs (z, s)]δ[us (z, s) + u¯ s (z, s)]dA

= A

 [σzz (z, s) + σ¯ zz (z, s)]δ[uz (z, s) + u¯ z (z, s)]dA

+ A

 =

 τzs (z, s)δus (z, s)dA +

A

 +

σzz (z, s)δuz (z, s)dA A



τ¯ zs (z, s)δ u¯ s (z, s)dA + A

σ¯ zz (z, s)δ u¯ z (z, s)dA,

(9.12)

A

where δ u˜ z and δ u˜ s are the three-dimensional virtual displacements in the z- and sdirections, respectively, and dA = dnds. In the last expression in Eq. (9.12), every product of a barred term and an unbarred term is dropped because the barred and unbarred terms are symmetric and anti-symmetric with respective to the wall midline, respectively. Substituting Eqs. (9.9) into Eq. (9.12) yields 



δWA (z) =

τzs (z, s)ψIs (s)δξI (z)dA + A



+

O σzz (z, s)ψO z (s)δξ (z)dA A

τ zs (z, s)(−n)ψ˙ nI (s)δξI (z)dA +



σ zz (z, s)(−n)ψIn (s)δξI , (z)dA,

A

A

(9.13) from which the following generalized forces can be defined: { } F(z) = FzO (z), FzI (z), FsI (z), FsI (z) ,

(9.14)

with  FzOk

= A

σzz ψzOk dA,

(9.15)

9.2 Generalized Force-Stress Relationships

 FzI m =

A

FsIl =  FsI m

= A

277

( ) σ zz −nψnI m dA,

(9.16)

 τzs ψsIl dA,

(9.17)

( ) τ zs −nψ˙ nI m dA,

(9.18)

A

where k, l, and m are indices associated with ξO , ξI , and ξI , respectively. For non-deformable section modes (k ≤ 3), FzOk in Eq. (9.15) can be the extensional force, the x-directional bending moment, and the y-directional bending moment, which are the work conjugates of Uz , θx , or θy , respectively. For higher-order section modes (k > 3), the FzOk symbols are bimoments, which are the work conjugates of the warping modes. In Eq. (9.17), the FsIl symbols are x- and y-directional shear forces and the torsional moment as the work conjugates of Ux , Uy , and θz , respectively, for non-deformable section modes (l ≤ 3). For higher-order modes (l > 3), on the other hand, the FsIl symbols denote transverse bimoments, the work conjugates of unconstrained distortion modes. The symbols FzI m and FsI m (m > 3) in Eqs. (9.16) and (9.18) represent generalized forces by wall-bending deformation which are the work conjugates of ξI m , and ξI m , respectively.

9.2 Generalized Force-Stress Relationships In Chaps, 4, 5, 6, 7, the recursive equations needed to derive sectional shape functions could be obtained using the generalized force-stress relationships, from which the sectional stresses could be expressed in term of the generalized forces and their shape functions. In this section, it will be shown that the relationships between the stresses and generalized forces for rectangular cross-sections can be seamlessly extended to the cross-sections of general shapes if the sectional shape functions are defined as orthogonal to each other, via  ψsIi (s)ψsIl (s)dA = 0 

O

ψz j (s)ψzOk (s)dA = 0

if i /= l,

(9.19a)

if j /= k,

(9.19b)

which can be rewritten as  ψsIi (s)ψsIl (s)dA = δil J Il ,

(9.20a)

278

9 Sectional Shape Functions of Thin-Walled Beams with General …



O

ψz j (s)ψzOk (s)dA = δjk J Ok .

(9.20b)

In Eqs. (9.20), δil and δjk are the Kronecker delta functions, and J Il and J Ok are the  ( I )2 second moments of inertia for modes ξ Il and ξ Ok , respectively: J Il = ψ l (s) dA  ( O )2 Ok k and J = ψ (s) dA.

9.2.1 Shear Stress In Eqs. (8.4) and (8.6), the derivatives of the z- and s-directional sectional shape functions were expressed as linear combinations of the s- and z-directional shape functions belonging to the lower-order sets. They are explicitly written as ψ˙ zOk =

NI Σ

ck,l ψsIl ,

(9.21)

dl,k ψzOk ,

(9.22)

l=1

ψ˙ sIl =

NO Σ k=1

where O and I are used as the superscripts instead of W and χ as used in Eqs. (8.4) and (8.6). The shear stress on the midline of a cross-section wall given in Eq. (9.11c) can now be expressed as ] [ O τzs =G ψ Is (s)ξ I , (z) + ψ˙ z (s)ξ O (z) ⎡ ⎤ NO NI Σ Σ =G ⎣ ψ˙ zOk (s)ξ Ok (z)⎦ ψsIl (s)ξ Il , (z) + l=1

k=1

l=1

k=1 l=1

⎡ ⎤ NO NI NI Σ Σ Σ =G ⎣ ψsIl (s)ξ Il , (z) + ck,l ψsIl (s)ξ Ok (z)⎦ Σ NI

=

ψsIl (s)α Il (z),

(9.23)

l=1

where α Il (z) represents the combination of ξ Il , (z) and ξ Ok (z). Substituting Eq. (9.23) into the generalized force in Eq. (9.17) yields  FsIl (z) =

A

τzs (z, s)ψsIl (s)dA

9.2 Generalized Force-Stress Relationships

=

279

NI  Σ



A p=1

= A

I

ψs p (s)α Ip (z)ψsIl (s)dA

( Il )2 ψs (s) dAα Il (z)

= J Il α Il (z),

(9.24)

where the orthogonality condition in Eq. (9.20a) is used. Using Eq. (9.24), the shear stress in Eq. (9.23) can be expressed as τzs =

NI Σ F Il (z) s

J Il

l=1

ψsIl (s).

(9.25)

9.2.2 Axial Stress The axial stress on the midline of a cross-section wall in Eq. (9.11a) can be rewritten using Eq. (9.22) as { I } O, σzz =E1 ν ψ˙ s (s)ξ I (z) + ψ O (s)ξ (z) z ⎡ ⎤ NO NI NO ΣΣ Σ dl,k ψzOk (s)ξ Il (z) + ψzOk (s)ξ Ok , (z)⎦ =E1 ⎣ν l=1 k=1

Σ

k=1

NO

=

ψzOk (s)β Ok (z),

(9.26)

k=1

where β Ok (z) represents the combination of ξ Il (z) and ξ Ok , (z). Substituting Eq. (9.26) into the generalized force in Eq. (9.15) and using the orthogonality condition in Eq. (9.20b) will give  FzOk (z) = =

A

σzz (z, s)ψzOk (s)dA

NO  Σ A q=1

 =

A

O

ψz q (s)β Oq (z)ψzOk (s)dA

( Ok )2 ψz (s) dAβ Ok (z)

= J Ok β Ok (z),

(9.27)

280

9 Sectional Shape Functions of Thin-Walled Beams with General …

from which the axial stress in Eq. (9.26) can be expressed as NO Σ F Ok (z)

σzz =

z

J Ok

k=1

ψzOk (s).

(9.28)

9.2.3 Wall-Bending Stress The axial stress by wall-bending deformation, σ zz in Eq. (9.11a), is ] [ I¯ ¯ ¯ ¯ σ¯ zz = −nE1 ν ψ¨ n (s)ξ I (z) + ψnI (s)ξ I ,, (z) .

(9.29)

To express σ zz in terms of the corresponding generalized forces, FzI m in Eq. (9.16), it is necessary to represent ψ¨ nI in terms of ψnI , which was presented in Eq. (8.10) as I ψ¨ n m =

NI Σ

I

rm,j ψn j ,

(9.30)

j=1 I

where ψn j on the right side denotes the n-directional shape functions of all lowerorder in-plane modes relative to the current mode set. The wall-bending stress in Eq. (9.29) can be rewritten if Eq. (9.30) is employed: ⎡ σ zz = − nE1 ⎣ν

NI NI Σ Σ

I

rj,m ψn m ξ I j (z) +

j=1 m=1

Σ

NI Σ

⎤ I ψn m (s)ξ I m ,, (z)⎦

m=1

NI

=−n

I

ψn m (s)μI m (z).

(9.31)

m=1

In Eq. (9.31), μI m (z) represents the combination of ξ I j (z) and ξ I m ,, (z). Substituting Eq. (9.31) into the generalized force in Eq. (9.16) yields I Fz m (z)

 = 

A

=

( ) I σ zz −nψn m dA 2

n A

NI Σ r=1

I

I

ψn r (s)μI r (z)ψn m (s)dA

9.3 Non-Deformable Section Modes

=

281

NI  Σ r=1

Σ

I

I

n2 ψn m (s)ψn r (s)dAμI r (z) A

NI

=

J I mr μI r (z),

(9.32)

r=1

where J I mr are the second moments of inertia:  ) ( I I J I mr = n2 ψn m (s)ψn r (s)dA, m, r = 1, 2, ..., NI .

(9.33)

A

Note that the J I mr values are non-zero for those involving the cross products of ) I ( shape functions because the orthogonality among ψn m m = 1, 2, . . . , NI ∗ does not hold. Therefore, it is shown in Eq. (9.32) that each generalized force is affected by all constrained distortion modes or that each constrained distortion mode can affect all generalized forces, which is not the case for the generalized forces of the shear stress in Eq. (9.25) and the axial stress in Eq. (9.28). Although we cannot express the normal stress in Eq. (9.31) in terms of decoupled generalized forces of constrained distortion modes as in Eqs. (9.25, 9.28), we can still derive the generalized force-stress relationships in a vector form, as follows: σ zz = −nψIn (s)μ(z),

(9.34)

Fz (z) = JI μ(z),

(9.35)

from which the wall-bending stress can be written as ( )−1 σ zz = −nψIn (s) JI Fz (z).

(9.36)

In Eqs. (9.34) and (9.35), ψIn (s), Fz (z), and μ(z) are corresponding vector repreI

sentations of ψn m (s), FzI m (z), and μI r (z), and JI is the matrix representation of ( ) J I mr m, r = 1, 2, ..., NI . The shear stress by wall-bending deformation, τ zs in Eq. (9.11c), is not considered because τ zs is not used for a recursive derivation of sectional modes.

9.3 Non-Deformable Section Modes The sectional shape functions for non-deformable section modes derived in Sect. 3.2.1 are generalized as

282

9 Sectional Shape Functions of Thin-Walled Beams with General …

( ) ( ) ψnUx sj = sin αj − β , ( ) ( ) ψsUx sj = cos αj − β ,

(9.37a)

( ) U ( ) ψn y sj = − cos αj − β , ) ( U ( ) ψs y sj = sin αj − β ,

(9.37b)

( ) ψzUz sj = 1,

(9.37c)

) ( ) ( ) ( ( ) ψzθx sj = − Xj − XC sin β + Yj − YC cos β + sj sin αj − β ,

(9.37d)

) ( ) ( ( ) θ ( ) ψz y sj = − Xj − XC cos β − Yj − YC sin β − sj cos αj − β ,

(9.37e)

( ) ( ) ) ( ψnθz sj = − Xj − X0 cos αj − Yj − Y0 sin αj − sj ,

(9.37f)

( ) ( ) ) ( ψsθz sj = Xj − X0 sin αj − Yj − Y0 cos αj ,

(9.37g)

( ) all other ψm sj = 0 (m = z, n, s),

(9.37h)

where j denotes the edge (or wall) index. In Eqs. (9.37), αj is the angle of edge j, and β and β are the orientations of the axes ( of )deflection and principal axes, respectively. The origin of sj is denoted by Xj , Yj , and the torsional center and centroid are denoted by (X ( ) ( 0 ,) Y0 ) and (XC , YC ), respectively (see Fig. 9.1). The functions ψnUx sj and ψsUx sj denote the n- and s-directional section deformations U ( ) U ( ) and ψn y sj and ψs y sj are those due to Uy . The bending due to Ux , respectively, ( ) rotation ψzθx sj at a point on a cross-section with respect to the x-axis is simply ( ) ( ) θ ( ) given by ψzθx sj = y sj . Similarly, ψz y sj at a point on a cross-section is given ( ) θ ( ) by ψz y sj = −x sj . If x and y are expressed in terms of Xj , Yj , s(j , β, ) etc., the θz s e) can be obtained. On the other hand, ψ expressions in Eqs. (9.37d, j = n ( ) ( sj − d)j and ψsθz sj = rj , where dj is the distance from the starting point of edge j sj = 0 to pj , pj is the closest point on edge j to (X0 , Y0 ), and rj is the distance from (X0 , Y0 ) to pj (see p5 and d5 in Fig. 9.1 for example.) If these functions are expressed in terms of Xj , Yj , sj , etc., Eqs. (9.37f) and (9.37g) can be obtained. The symbols β, β, (X0 , Y0 ), and (XC , YC ) appearing in Eq. (9.37) can be explicitly determined using the orthogonality conditions among the rigid-body modes. First,  U β can be calculated from the orthogonality relationships, A ψsUx ψs y dA = 0:

9.3 Non-Deformable Section Modes

283

⎛

⎞

NE 

⎜ j=1 lj sin 2αj ⎟ 1 ⎜ ⎟ tan−1 ⎜ N ⎟, ⎝ E ⎠ 2 lj cos 2αj

β=

(9.38)

e=1

where NE is the number of section edges and lj is the length of edge j. To determine   U (X0 , Y0 ), one can utilize A ψsUx ψsθz dA = 0 and A ψs y ψsθz dA = 0, from which the torsional center is obtained via ⎧ ⎫ X0 = B−1 (9.39a) 1 B2 , Y0 where ( ) ( )] [ NE Σ sin αj sin (αj − β ) − cos αj sin (αj − β ) , B1 = lj sin αj cos αj − β − cos αj cos αj − β

(9.39b)

( ) ( )⎫ ⎧ NE Σ Xj sin αj sin(αj − β ) − Yj cos αj sin (αj − β ) . B2 = lj Xj sin αj cos αj − β − Yj cos αj cos αj − β

(9.39c)

j=1

j=1

Note that β should be known to calculate B1 and B2 in Eq. (9.39). The centroid (XC , YC ) of a cross-section can be determined from A ψzUz ψzθx dA = 0  θ and A ψzUz ψz y dA = 0 as follows: NE ( 

XC =

j=1

Xj lj + 0.5lj2 cos αj NE 

) ,

(9.40a)

.

(9.40b)

lj

j=1

) NE (  Yj lj + 0.5lj2 sin αj

YC =

j=1

NE 

lj

j=1

Finally, one can consider



θ

A

ψzθx ψz y dA = 0 to determine β:

β= with

( ) 2A1 1 , tan−1 2 A2 − A3

(9.41a)

284

A1 =

9 Sectional Shape Functions of Thin-Walled Beams with General … NE ( Σ j=1

) 1 1 1 Xj∗ Yj∗ lj + Xj∗ lj2 sin αj + Yj∗ lj2 cos αj + lj3 cos αj sin αj , (9.41b) 2 2 3 ) NE ( ( )2 Σ 1 3 ∗ ∗ 2 2 Xj lj + Xj lj cos αj + lj cos αj , A2 = 3 j=1

(9.41c)

) NE ( ( )2 Σ 1 3 2 ∗ ∗ 2 Yj lj + Yj lj sin αj + lj sin αj , A3 = 3 j=1

(9.41d)

where Xj∗ = Xj − XC and Yj∗ = Yj − YC .

9.4 Fundamental Deformable Section Modes: Linear Warping and Inextensional Distortion In Sect. 2.3.2, we presented torsional warping and torsional distortion modes as the fundamental deformable section modes (or the zeroth-order modes). They express a general state of edgewise uniform shear stress. This shear stress distribution includes the state of uniform shear over an entire cross-section, such as the state observed in the Saint Venant uniform torsion problem. This edgewise uniform stress field can be obtained under edgewise linear z-directional sectional deformation by torsional warping and under edgewise-constant s-directional sectional deformation by torsional distortion. In this section, we extend the method utilized in Sect. 2.3.2 to define the fundamental deformable section modes for a cross-section with a general shape. Unlike a rectangular thin-walled cross-section, warping and distortion modes in the fundamental mode set are not necessarily associated with torsion in the case of a crosssection with a general shape. Therefore, the fundamental warping modes (denoted by W 0 ) and distortional modes (denoted by χ 0 ) will be referred to as linear warping modes and inextensional distortion modes, respectively.2 On the other hand, warping

2

In Eqs. (9.5) and (9.6), subscript characters for a higher-order mode denote the mode sequence. For example, χ2 is the second distortion mode. If superscript is used for a mode, it denotes the mode

9.4 Fundamental Deformable Section Modes: Linear Warping …

285

and distortion modes belonging to higher-order sets will be referred to as nonlinear warping and extensional distortion modes, respectively, because the former has non-linear edgewise deformations and the latter has extensional wall deformations.

χ0

0

9.4.1 Deriving ψzW and ψs as Solutions to an Eigenvalue Problem χ0

0

In this subsection, we will derive ψzW and ψs for a cross-section with a general shape. Noting that these fields correspond to the state of edgewise uniform shear stress on a beam cross-section, we begin with ∂τzs = 0, ∂s

(9.42)

or τzs = fj (z)

on edge j, (j = 1, 2, . . . , NE )

(9.43)

where fj (z) is a function of z only. If W 0 and χ 0 modes are considered in addition to the six non-deformable section modes for the shear stress calculation, τ can be split into two components as τzs = τzsND + τzsD ,

(9.44)

where τzsND is the shear stress by the non-deformable section modes and τzsD denotes the shear stress by deformable section modes (W 0 and χ 0 ). Let us consider the displacements on the wall midline by W 0 and χ 0 : usχ (z, s) = ψsχ (s)χ 0 (z), 0

0

0

0

uzW (z, s) = ψzW (s)W 0 (z).

(9.45) (9.46)

Using Eqs. (9.45) and (9.46), one can calculate τzsD as

set number. For example, χ20 is the second distortion mode belonging to the zeroth mode set. Like{ }T wise, superscript characters are used as in χ10 , χ20 , χ31 , χ42 , · · · , χNNχS (unconstrained distor{ }T { }T tion modes), W10 , W21 , W31 , W42 , · · · , WNNWS (warping modes), and η11 , η21 , η32 , · · · , ηNNηS (constrained distortion modes), where NS is the highest mode set number. In the modes listed above, there are two unconstrained distortion modes and one warping mode in the zeroth set. In many cases, we do not use superscript for brevity unless necessary.

286

9 Sectional Shape Functions of Thin-Walled Beams with General …

(

τzsD (z,

) χ0 ∂us (s, z) ∂uzW 0 (s, z) + s) = G ∂z ∂s ( χ0 ) 0, W = G ψs (s)χ (z) + ψ˙ z 0 (s)W 0 (z) .

(9.47)

Equation (9.25) is used to express the shear stress component τzsND explicitly: U

τzsND (z,

F Ux (z) Fs y (z) Uy Fsθz (z) θz ψ ψs (s). s) = s U ψsUx (s) + + (s) s J x J Uy J θz

(9.48)

Substituting Eqs. (9.47) and (9.48) into Eq. (9.44) and using Eq. (9.43) yield Uy Uy FsUx (z) Ux Fsθz (z) θz ψs ( (s) + FsJ U(z) ψs )(s) y ψs (s) + J Ux J θz +G ψsχ0 (s)χ 0, (z) + ψ˙ zW 0 (s)W 0 (z)

(9.49)

= fj (z) on edge j. (j = 1, 2, . . . , NE )

For Eq. (9.49) to be valid for any z, the following should hold: b1 ψsUx (s) + b2 ψs y (s) + b3 ψsθz (s) + b4 ψsχ (s) + b5 ψ˙ zW (s) = 1, U

0

0

(9.50)

where b1 , b2 , etc., represent certain constants. To facilitate the subsequent analysis, Eq. (9.50) is rewritten as 0 0 U ψ˙ zW (s) = h1 ψsUx (s) + h2 ψs y (s) + h3 ψsθz (s) + h4 ψsχ (s) + h5 ,

(9.51)

where h1 , h2 , etc., represent certain constants, as above. For the shear stress due to χ0 χ0 ψs (s) to be edgewise uniform (see Eq. (9.47)), ψs (s) should be the edgewiseχ0 χ0 constant. Because h4 ψs (s) + h5 is also edgewise-constant, we can set ψs (s) ≙ 0 χ h4 ψs (s) + h5 for simplification. Then, Eq. (9.51) reduces to 0 0 U ψ˙ zW (s) = h1 ψsUx (s) + h2 ψs y (s) + h3 ψsθz (s) + ψsχ (s).

(9.52)

χ0

The edgewise-constant function ψs (s) can be expressed as ⎧ ⎪ ⎨ c1 for edge 1 0 .. .. ψsχ (s) = , . . ⎪ ⎩ cNE for edge NE

(9.53)

or 0 ψsχ (s)

= c1 δ1 (s) + . . . + cNE δNE (s) =

NE Σ j=1

cj δj (s) = δ(s)c,

(9.54)

9.4 Fundamental Deformable Section Modes: Linear Warping …

287

where δj (s) is a unit edge function, defined as ⎧ δj (s) =

1 for edge j . 0 otherwise

(9.55)

Substituting Eq. (9.54) into Eq. (9.52) gives 0 U ψ˙ zW (s) = h1 ψsUx (s) + h2 ψs y (s) + h3 ψsθz (s) +

NE Σ

cj δj .

(9.56)

j=1

Integrating Eq. (9.56) with respect to s yields ψzW (s) = h1 ψsUx (s) + h2 ψs y (s) + h3 ψsθz (s) + U

0

NE Σ

cj δj s +

j=1 U

NE Σ

dj δj ,

(9.57)

j=1 U

where ψsUx , ψs y , and ψsθz are the integrals of ψsUx , ψs y , and ψsθz , respectively, with integration constants excluded. Note that δ j is used in the last term of Eq. (9.57) to express the edgewise integration constant d j . In vector form, Eq. (9.57) can be expressed as

ψzW

0

⎧ ⎫ ⎪ h1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪h ⎪ ⎪ ⎬ { }⎨ 2 ⎪ 0 Uy Ux θz = ψs , ψs , ψs , δs, δ h3 ≙ ψ W C, ⎪ ⎪ ⎪ ⎪ ⎪ c⎪ ⎪ ⎪ ⎪ ⎭ ⎩d⎪

(9.58)

where { } 0 U ψ W = ψsUx , ψs y , ψsθz , δs, δ ,

(9.59)

}T { C = h1 , h2 , h3 , cT , dT ,

(9.60)

with { { } { }T }T δ = δ1 , δ2 , · · · , δNE , c = c1 , c2 · · · , cNE , and d = d1 , d2 , · · · , dNE . 0

The symbol ψ W denotes a known basis function vector while the symbol C is an unknown coefficient vector which will be determined by considering the continuity conditions in the field quantities at cross-section corners and the orthogonality conditions.

288

9 Sectional Shape Functions of Thin-Walled Beams with General … W0

W0

Let ψz a and ψz b be the shape functions of linear warping modes 0 W0 W0 ψz a = ψ W Ca and ψz b = defined (by two different coefficient vectors, ) ◠

0

◠

ψ W Cb Ca , Cb ∈ C, C : the set of vectors . Because we are looking for shape functions orthogonal to each other, the coefficient vectors should be determined to meet the following orthogonality condition: 

W0

W0

ψz a ψz b dA = λδab

(9.61)

A

or 0

CTa PW Cb = λδab ,

(9.62)

0

where PW is defined as 0

PW =

 (

ψW

0

)T

( 0

ψ W dA

( 0 )T ) 0 PW = PW

(9.63)

A 0

In Eq. (9.61) or (9.62), λ represents the sectional moment of inertia, λ = J Wab , and δab is the Kronecker delta. A useful interpretation from Eq. (9.62) is that Ca 0 and Cb can be viewed as eigenvectors of the symmetric matrix PW (Golub and Loan (1996)). Therefore, Eq. (9.62) can be alternatively expressed as the following eigenvalue problem: 0

PW C = λC.

(9.64)

Once the eigenvectors C are obtained from Eq. (9.64), they become the coefficient vectors in Eq. (9.58). Therefore, one can determine multiple linear warping modes for the cross-section of a general shape simultaneously by solving the eigenvalue problem in Eq. (9.64). In solving the eigenvalue problem in Eq. (9.64), other conditions, such as orthogo0 χ0 nality and continuity conditions imposed on ψzW and ψs , should also be considered. (They are typically considered using the Lagrange multiplier; see Eq. (9.77).) 0 As the first condition, we consider that the linear warping modes ψzW should be orthogonal to non-deformable section modes, which can be written as ⎫ ⎧ Uz ⎪ ⎪ ψ ⎪ ⎪ z  {  ⎨ ⎬ 0 } 0 θ T θx ψz ψzW dA = ψzUz , ψzθx , ψz y ψzW dA = 0, ⎪ A A⎪ ⎪ ⎭ ⎩ ψ θy ⎪ z

Substituting Eq. (9.58) into (9.65) yields

(9.65)

9.4 Fundamental Deformable Section Modes: Linear Warping … 0

QW C = 0,

289

(9.66)

where W0

Q

 { } 0 θ T ψzUz , ψzθx , ψz y ψ W dA. =

(9.67)

A

Note in Eq. (9.65) or Eq. (9.66) ( ) that only non-deformable section modes belonging ( ) to the out-of-plane mode set ξO are considered because linear warping modes W 0 are out-of-plane modes. To find warping modes belonging to a higher-order set, the modes should be orthogonal to warping modes belonging to lower-order sets as well as the non-deformable section modes used in Eq. (9.65) 0 As the second condition, the displacement continuity of ψzW at cross-section corners is considered. At every corner (including intersection) of a cross-section, the 0 linear warping modes ψzW should be continuous. For the cross-section in Fig. 9.1, for example, the corner displacement continuity can be expressed as W0

W0

W0

W0

W0

W0

W0

W0

W0

W0

ψz (s1 = l1 ) = ψz (s2 = l2 ) ψz (s1 = l1 ) = ψz (s3 = 0) ψz (s3 = l3 ) = ψz (s4 = 0) ψz (s4 = l4 ) = ψz (s5 = 0) ψz (s5 = l5 ) = ψz (s1 = 0)

at Corner 1,

(9.68a)

at Corner 1,

(9.68b)

at Corner 2,

(9.68c)

at Corner 3,

(9.68d)

at Corner 4,

(9.68e)

which can be put into matrix–vector form using Eq. (9.58) as ⎤ 0 0 ψ W (s1 = l1 ) − ψ W (s2 = l2 ) ⎢ ψ W 0 (s = l ) − ψ W 0 (s = 0) ⎥ ⎥ ⎢ 1 1 3 0 0 ⎥ ⎢ W0 ⎢ ψ (s3 = l3 ) − ψ W (s4 = 0) ⎥C ≙ RW C = 0, ⎥ ⎢ W0 0 ⎣ ψ (s4 = l4 ) − ψ W (s5 = 0) ⎦ 0 0 ψ W (s5 = l5 ) − ψ W (s1 = 0) ⎡

(9.69)

where l j is the length of edge j (j = 1, ···, N E ). χ0 The third condition is the orthogonality of ψs (s), the shape function of an inextensional distortion to non-deformable section modes in the in( )mode, with respect χ0 plane mode set ξI . Using ψs (s) = δ(s)c in Eq. (9.54), this condition can be written as

290

9 Sectional Shape Functions of Thin-Walled Beams with General …

 {  { }T 0 }T U U ψsUx , ψs y , ψsθz ψsχ dA = ψsUx , ψs y , ψsθz δcdA A A { }T 0 U = ψsUx , ψs y , ψsθz δHdA C ≙ Qχ C = 0, A

(9.70) where H can be called the selection matrix, which extracts c from C: ψsχ (s) = δ(s)c = δ(s)HC. 0

The last condition is the compatibility conditions ensuring the uniqueness of at cross-section corners shared by more than two edges (or section walls), like corner 1 in Fig. 9.1. To explain why this condition is needed, we introduce a method for determining the displacement of a corner shared by two edges only, followed by a discussion of the compatibility issue at a corner shared by three edges. As a specific example, we consider the displacement of corner 1 in the crosssection shown in Fig. 9.1. In the first case, it is assumed that corner 1 is shared by two edges only, edges 1 and 2. In this case, the new location A of corner 1 is determined by the schematic shown in Fig. 9.2a, where the corner 1 point of( edge ) 1 moves by us,1 and the corner 1 point of edge 2 moves by us,2 , where us,j = us sj is us χ0 ψs (s)

χ0

on edge j. It should be noted that because ψs (s) has the s-directional displacement component only, no other displacement is considered here. The schematic procedure used to determine the location of corner 1 for the given values of us,1 and us,2 is valid with the assumption of minor displacement. Therefore, the in-plane X − and Y − directional displacements (uX and uY ) of corner 1 are calculated from us,1 and us,2 as ⎧

⎫| ]−1 ⎧ ⎫| [ uX || us,1 || cos α1 sin α1 = uY |corner 1 cos α2 sin α2 us,2 |corner 1

(9.71)

At this point, we consider the situation in which corner 1 is shared by three edges, as illustrated in Fig. 9.2b. Following the procedure used to determine the new corner point A from us,1 and us,2 , one can find the new corner point B due to us,2 and us,3 and the new corner point C due to us,1 and us,3 . As shown in Fig. 9.2b, new corner points B and C can merge to point A only if the displacements us,1 , us,2 , and us,3 are compatible. Otherwise, the situation sketched on the right side of Fig. 9.2b arises. Compatibility can be ensured if the following condition (which is graphically depicted on the left side of Fig. 9.2b) is met: ⎧ | } uX { | us,3 corner 1 − cos α3 sin α3 uY Substituting Eq. (9.71) into Eq. (9.72) gives

⎫| | | |

= 0. corner 1

(9.72)

9.4 Fundamental Deformable Section Modes: Linear Warping …

291

A

us,1 us,2

Edge 2

Edge 1 Corner 1

(a) B A

us,3

Edge 3 us,2

Compatible

C

us,1 Edge 2

Edge 1

A

Incompatible

us,3 Edge 3 us,2

us,1

Edge 1

Corner 1

Edge 2 Corner 1

(b) Fig. 9.2 Movement of corner 1 (of the cross-section shown in Fig. 9.1) due to the s− directional displacements of edges sharing corner 1: a the case when corner 1 is shared only by two edges, edges 1 and 2, and b the case when Corner 1 is shared by three edges, edges 1 to 3

(

] ⎧ ⎫)|| [ } cos α1 sin α1 −1 us,1 | us,3 − cos α3 sin α3 | | cos α2 sin α2 us,2 {

= 0.

(9.73)

corner 1

This relationship can be simplified to )| ( | sin(α1 − α3 ) sin(α3 − α2 ) us,3 − us,1 − us,2 || = 0. sin(α1 − α2 ) sin(α1 − α2 ) corner 1

(9.74)

If Eq. (9.54) is used, the compatibility condition given by Eq. (9.74) can be written as an equation involving C as ⎫ ⎧ sin(α3 − α2 ) sin(α1 − α3 ) 0 δ(s3 = 0) − δ(s1 = l1 ) − δ(s2 = l2 ) HC = Rχ C = 0. sin(α1 − α2 ) sin(α1 − α2 ) (9.75) χ0

where Rχ is the matrix of the compatibility condition imposed ( on )ψs at( corner) 1, and H is the selection matrix in Eq. (9.70). In Eq. (9.75), δ sj = 0 or δ sj = lj introduced in Eq. (9.59) takes a value of 1 for the component corresponding to edge 0

292

9 Sectional Shape Functions of Thin-Walled Beams with General …

j and 0 otherwise. If a beam cross-section has several corners shared by more than 0 two edges, Rχ should include all compatibility conditions of those corners. The four conditions discussed above can be put into matrix form as ⎡

0 ⎤ QW 0 ⎢ RW ⎥ 0 ⎢ ⎥ ⎣ Qχ 0 ⎦C = S C = 0. 0 Rχ

(9.76)

To solve Eq. (9.64) for C considering condition (76), the method of the Lagrange multiplier may be used: [

PW S0

[ ]⎧ ⎫ ( 0 )T ]⎧ ⎫ C I0 C S =λ , L 00 L 0

0

(9.77)

where λ is the eigenvalue in Eq. (9.64), L is the Lagrange multiplier vector, and I is the identity matrix. 0 One can find from Eqs. (9.57), (9.58), (9.63) that the size of PW is (3 + 2N E ) × (3 + 2N E ) (NE : the number of section edges). According to Eqs. (9.67), (9.70), 0 0 0 0 there are three rows in QW and Qχ . The sizes of RW and Rχ depend on the cross-section connectivity because the number of connected edges at a corner affects the continuity conditions (including the compatibility conditions.) To represent the number of continuity conditions at corner r, we define two functions as M1 (r) = NEr − 1, ⎧ M2 (r) =

( ) NEr − 2 (NEr > 2) , 0 NEr ≤ 2

(9.78)

(9.79)

where NEr is the number of connected edges at corner r. Using these functions, one  C  C 0 0 M1 (r) and Nr=1 M2 (r), can represent the number of rows of RW and Rχ as Nr=1 respectively, where NC is the number of cross-section corners. The number of total 0 0 0 0 0 constraints for PW is the sum of the numbers of rows of QW , RW , Qχ , and Rχ , as can be seen in Eq. (9.76). Noting that the number of eigenvalues in Eq. (9.77) is given 0 as the difference between the size of PW and the number of( the)total constraints, the number (NW 0 ) of linear warping modes, and the number Nχ 0 of inextensional distortion modes are given by NW 0 = 2NE − 3 −

NC Σ

(M1 (r) + M2 (r)),

(9.80)

r=1

N

χ0

= NE − 3 −

NC Σ r=1

M2 (r).

(9.81)

9.4 Fundamental Deformable Section Modes: Linear Warping …

293

χ0

9.4.2 n-directional Shape Function ψn

Once all C values are calculated from Eq. (9.77), the s-directional shape functions 0 χ0 ψs (s) and the z-directional shape functions ψzW are determined. The remaining 0 χ n-directional shape functions ψn can be found from the continuity conditions (to be explained below) at every section corner; because the movement of the corner has χ0 been described only by s− directional shape functions ψs explained in the previous 0 χ χ0 subsection, and ψn is needed to satisfy the continuity with ψs . χ0 To find ψn , it is approximated as edgewise polynomials: cubic functions for closed edges and quadratic functions for open edges. This approximation always allows the total number of unknown coefficients of the edgewise polynomials to be equal to the total number of available corner continuity conditions. This approach can be used for any arbitrarily shaped sections. Going back to the specific beam χ0 cross-section shown in Fig. 9.1, we approximate ψn in the following form: χ0

ψn,1 = c10 + c11 s1 + c12 s12 + c13 s13 ,

(9.82a)

χ0

ψn,2 = c20 + c21 s2 + c22 s22 ,

(9.82b)

ψn,3 = c30 + c31 s3 + c32 s32 + c33 s33 ,

(9.82c)

χ0

χ0

ψn,4 = c40 + c41 s4 + c42 s42 + c43 s43 ,

(9.82d)

χ0

ψn,5 = c50 + c51 s5 + c52 s52 + c53 s53 ,

(9.82e)

χ0 χ0 ( ) χ0 where ψn,j = ψn sj denotes ψn (s) on edge j, and cjk (j = 1, ···, N E ; k = 0, 1, 2, χ0

χ0

3) is an unknown coefficient of the polynomials defining ψn,j . Note that only ψn,2 is approximated as a quadratic function because edge 2 is an open-ended edge (see Fig. 9.1). Recall that for rectangular cross-sections (see Sect. 2.3.2, for instance), the continuity conditions at sectional corners were described in terms of the displacement and slope continuity conditions and the moment equilibrium conditions. These continuity conditions equally apply in the case of arbitrarily shaped cross-sections. We will begin with the continuity in displacement. Consider corner r. At this corner, the n-directional displacement un can be written using uX and uY : ⎧ | } uX { un,j |corner r = sin αj − cos αj uY

⎫| | | |

, corner r

(r = 1, · · · , Nr )

(9.83)

294

9 Sectional Shape Functions of Thin-Walled Beams with General …

where the subscript r refers to edges connected to corner r. Generalizing the relationship (71), one can relate uX and uY appearing in Eq. (9.83) to the s− directional displacements of edges j1 and j2 connected to corner r as ⎧

⎫| ]−1 ⎧ ⎫| [ uX || us,j1 || cos αj1 sin αj1 = , uY |corner r cos αj2 sin αj2 us,j2 |corner r

(r = 1, · · · , Nr ), (9.84)

where corner r is assumed to connect two edges, such as edge j1 and edge j2. Using | Eq. (9.83) and Eq. (9.84), un,j |corner r can be rewritten as ]−1 0 1| [ | } cos α sin α { | us,j1 | j1 j1 un,j |corner r = sin αj − cos αj | cos αj2 sin αj2 us,j2 |

,

(r = 1, · · · , Nr ).

corner r

(9.85)

| χ0 Because un,j |corner r is expressed in terms of us,j1 and us,j2 at corner r, ψn,j at sj = 0 χ0

and sj = lj can explicitly relate to ψs . For example, at corner 4 of the section shown χ0 χ0 in Fig. 9.1 (r = 4 and j1 = 1, j2 = 5), ψn,1 should relate to ψs as χ0 ψn,1 (s1

1 ] 0 χ0 [ } cos α1 sin α1 −1 ψs,1 (s1 = 0) = 0) = sin α1 − cos α1 χ0 cos α5 sin α5 ψs,5 (s5 = l5 ) {

cos(α5 − α1 ) χ 0 cos(α1 − α1 ) χ 0 ψs,1 (s1 = 0) + ψ (s5 = l5 ), sin(α1 − α5 ) sin(α1 − α5 ) s,5 (9.86a)

=−

χ0

χ0

and ψn,5 should relate to ψs as χ0

ψn,5 (s5 = l5 ) = −

cos(α1 − α5 ) χ 0 cos(α5 − α5 ) χ 0 ψs,1 (s1 = 0) + ψ (s5 = l5 ). sin(α1 − α5 ) sin(α1 − α5 ) s,5 (9.86b)

Similarly, the displacement continuity conditions at other corners of the crosssection in Fig. 9.1 are given as χ0

ψn,1 (s1 = l1 ) = −

χ0

ψn,2 (s2 = l2 ) = −

cos(α2 − α1 ) χ 0 cos(α1 − α1 ) χ 0 ψs,1 (s1 = l1 ) + ψ (s2 = l2 ), sin((α1 − α2 ) sin(α1 − α2 ) s,2 (9.86c) cos(α2 − α2 ) χ 0 cos(α1 − α2 ) χ 0 ψ (s1 = l1 ) + ψ (s2 = l2 ), sin(α1 − α2 ) s,1 sin(α1 − α2 ) s,2 (9.86d)

9.4 Fundamental Deformable Section Modes: Linear Warping … χ0

ψn,3 (s3 = 0) = −

cos(α2 − α3 ) χ 0 cos(α1 − α3 ) χ 0 ψs,1 (s1 = l1 ) + ψ (s2 = l2 ), sin(α1 − α2 ) sin(α1 − α2 ) s,2 (9.86e)

χ0

ψn,3 (s3 = l3 ) = −

χ0

ψn,4 (s4 = 0) = −

χ0

ψn,4 (s4 = l4 ) = −

χ0

ψn,5 (s5 = 0) = −

295

cos(α4 − α3 ) χ 0 cos(α3 − α3 ) χ 0 ψs,3 (s3 = l3 ) + ψ (s4 = 0), sin(α3 − α4 ) sin(α3 − α4 ) s,4 (9.86f) cos(α4 − α4 ) χ 0 cos(α3 − α4 ) χ 0 ψs,3 (s3 = l3 ) + ψ (s4 = 0), sin(α3 − α4 ) sin(α3 − α4 ) s,4 (9.86g) cos(α5 − α4 ) χ 0 cos(α4 − α4 ) χ 0 ψs,4 (s4 = l4 ) + ψ (s5 = 0), sin(α4 − α5 ) sin(α4 − α5 ) s,5 (9.86h) cos(α5 − α5 ) χ 0 cos(α4 − α5 ) χ 0 ψ (s4 = l4 ) + ψ (s5 = 0), sin(α4 − α5 ) s,4 sin(α4 − α5 ) s,5 (9.86i) χ0

Note that equations involving ψn,2 (s2 = 0) do not appear in the equations listed above because s2 = 0 corresponds to an open edge to which no other edge is connected. Next, the slope continuity and moment equilibrium are considered. For the crosssection in Fig. 9.1, the continuity conditions of rotation angles are given as χ χ ψ˙ n,1 (l1 ) − ψ˙ n,2 (l2 ) = 0

at corner 1

(9.87a)

χ χ ψ˙ n,1 (l1 ) − ψ˙ n,3 (0) = 0

at corner 1

(9.87b)

χ χ ψ˙ n,3 (l3 ) − ψ˙ n,4 (0) = 0

at corner 2

(9.87c)

χ χ ψ˙ n,4 (l4 ) − ψ˙ n,5 (0) = 0

at corner 3

(9.87d)

χ χ ψ˙ n,1 (0) − ψ˙ n,5 (l5 ) = 0

at corner 4

(9.87e)

0

0

0

0

0

0

0

0

0

0

Here and hereafter, sj = lj and sj = 0 are simply expressed as lj and 0. The continuity conditions of the moments are χ χ χ ψ¨ n,1 (l1 ) + ψ¨ n,2 (l2 ) − ψ¨ n,3 (0) = 0 0

0

0

χ χ ψ¨ n,3 (l3 ) − ψ¨ n,4 (0) = 0 0

0

at corner 1

at corner 2

(9.88a) (9.88b)

296

9 Sectional Shape Functions of Thin-Walled Beams with General … χ ψ¨ n,2 (0) = 0 0

at free end of edge 2

(9.88c)

χ χ ψ¨ n,4 (l4 ) − ψ¨ n,5 (0) = 0

at corner 3

(9.88d)

χ χ ψ¨ n,1 (0) − ψ¨ n,5 (l5 ) = 0

at corner 4

(9.88e)

0

0

0

0

Note that Eq. (9.88c) represents a moment-free condition at the end of an open edge (edge 2). Equation (9.88a) indicates that the sum of all bending moments calculated from three edges connected to corner 1 vanishes where the sign of each term is chosen considering the positive s-direction at each edge. Using the continuity conditions in χ0 Eqs. (9.86), (9.87), (9.88), ψn can now be uniquely defined. For the cross-section in Fig. 9.1, there are 19 continuity conditions in Eqs. (9.86), (9.87), (9.88) as well as 19 coefficients in Eq. (9.82).

9.5 Higher-Order Deformable Section Modes To derive the higher-order deformable section modes, we begin with the nondeformable modes, inextensional distortion modes, and linear warping modes of the fundamental mode set. In essence, this derivation process is recursive, as was the higher-order mode derivation for rectangular cross-sections. Three classes of higherorder modes, which are unconstrained distortion modes (involving wall extension), constrained distortion modes (not involving wall extension), and non-linear higherorder warping modes, will be derived using the following three-step procedure, which is the same procedure employed for rectangular cross-sections: Step 1 Identification of a secondary strain field due to a known primary stress field. Step 2 Derivation of secondary displacement consistent with the secondary strain. Step 3 Stress field update due to the secondary displacement.

9.5.1 Higher-Order Unconstrained Distortion Modes (χ ) Involving Wall Extension Step 1: Identification of a Secondary Normal Strain Field Due to Out-of-plane Modes χ The shape function ψs of the higher-order unconstrained distortion mode can be recursively derived using the axial stress (σzz ) induced by the out-of-plane modes of the lower-order sets, as expressed by Eq. (9.28). After identifying the secondary normal strain εss due to σzz in Eq. (9.28), we can find the displacement consistent χ with this εss , from which we can find the shape function ψs explicitly. Because the

9.5 Higher-Order Deformable Section Modes

297

secondary normal strain εss can be induced by the axial normal stress σzz in Eq. (9.28) due to Poisson’s effect, it can be written as εss (z, s) = −ν

NO FzOi (z) Oi σzz (z, s) ν Σ ψz (s). =− E E i=1 J Oi

(9.89)

Step 2: Derivation of Secondary Displacement Consistent With the Secondary Strain χ The next step is to find the s-directional displacement us of the higher-order χ unconstrained distortion mode consistent with εss in Eq. (9.89). If us is expressed as usχ (z, s) = ψsχ (s)χ (z),

(9.90)

χ

where ψs (s) is the shape function of the unconstrained distortion mode and χ (z) is the one-dimensional variable denoting the axial variation of the mode, the normal strain εss can be written as χ

εss (z, s) =

∂us = ψ˙ sχ (s)χ (z). ∂s

(9.91)

Equating Eq. (9.89) and (9.91) yields ψ˙ sχ (s) = −

O FzOi (z) Oi ν Σ ψz (s). Eχ (z) i=1 J Oi

N

(9.92)

χ Equation (9.92) indicates the functional behavior of ψ˙ s (s) can be expressed { that } χ as a linear combination of ψzOi (s) i=1,···,NO . Accordingly, one can write ψ˙ s (s) as

ψ˙ sχ (s) =

NO Σ

ci ψzOi (s),

(9.93)

i=1

Equation (9.93) can be written edgewise as NO ( ) Σ ( ) ci ψzOi sj ψ˙ sχ sj =

(9.94)

i=1

Integrating Eq. (9.94) yields ψsχ

NO ( ) Σ ( ) sj = ci ψzOi sj + cj,0 , i=1

(9.95)

298

9 Sectional Shape Functions of Thin-Walled Beams with General …

where ψzOi is the integral of ψzOi excluding the integration constant cj,0 (j = 1, ···, N E ). Equation (9.95) can be rewritten as ψsχ (s)

=

NO Σ

ci ψzOi (s)

+

i=1

NE Σ

cj,0 δj ,

(9.96)

j=1

where the symbol δ j was used in Eq. (9.55). For the subsequent analysis, it is convenient to write Eq. (9.96) in vector form as

{ ON ψsχ (s) = ψzO1 , · · · , ψz O , δ1 , · · · , δNE

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ }⎪

c1 .. .

cNO ⎪ c 1,0 ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩c

NE ,0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

≙ ϕ χ C,

(9.97)

where the row vector of known functions ϕ χ and the unknown coefficient column vector3 C are defined as } { { }T ON ϕ χ = ψzO1 , · · · , ψz O , δ1 , · · · , δNE ; C = c1 , · · · , cNO , c1,0 , · · · , cNE ,0 The first condition used to find C in Eq. (9.97) is the orthogonality condition:  A χ

ψsχa ψsχb dA = λδab ,

(9.98)

χ

where ψs a and ψs b are the shape functions associated with different coefficient vectors Ca and Cb , respectively. The symbol λ = J χab represents the sectional moment of inertia. Substituting Eq. (9.97) into Eq. (9.98) yields  CTa

A

(ϕ χ )T ϕ χ dA Cb ≙ CTa Pχ Cb = λδab .

(9.99)

where thesymmetric matrix Pχ is defined as Pχ = A (ϕ χ )T ϕ χ dA. Equation (9.99) can be formulated into a standard eigenvalue problem as Pχ C = λC,

(9.100)

where C is the coefficient vector defined in Eq. (9.97). 3

Although C in Eq. (9.97) is not the same coefficient vector as C used in Eq. (9.58), we use the same notation here and later to avoid introducing too many symbols.

9.5 Higher-Order Deformable Section Modes

299

The eigenvalue problem in Eq. (9.100) should be solved considering two additional constraints: orthogonality with respect to the in-plane modes in lower-order sets (similar to Eq. Equation (9.70)) and compatibility at corners shared by more than two edges (similar to Eq. (9.75)). The orthogonality condition can be written as  A



( I )T χ ψs ψs dA =

A

( I )T χ ψs ϕ dA C ≙ Qχ C = 0,

(9.101)

} { IN where ψIs = ψsI1 , · · · , ψs I is a vector containing the s-directional shape functions of the in-plane modes in the lower-order sets. The compatibility condition for the section shown in Fig. 9.1 having only one corner (corner 1) shared by more than two edges can be written as4 ⎧ ⎫ sin(α1 − α3 ) χ sin(α3 − α2 ) χ ϕ (s1 = l1 ) − ϕ (s2 = l2 ) C = Rχ C = 0. ϕ χ (s3 = 0) − sin(α1 − α2 ) sin(α1 − α2 )

(9.102)

One can put the two sets of conditions into matrix form as [

] Qχ C = Sχ C = 0, Rχ

(9.103)

with the coefficient vectors C discoverable using the following equation set up with the Lagrange multipliers L: [

Pχ (Sχ )T Sχ 0

]⎧

C L

⎫

[

I0 =λ 00

]⎧

⎫ C . L

(9.104) χ

Once the values of C are obtained by solving Eq. (9.104), ψs (s) can be explicitly found from Eq. (9.97).

9.5.2 Non-Linear Higher-Order Warping Modes (W) Step 1: Identification of the Secondary Shear Strain Field Due to the In-plane Modes As the unconstrained distortion modes (χ ) are found in Sect. 9.5.1, the shear stress should be updated to include the effects of these modes. The updated shear stress can be written as (see Eq. (9.25)), τzs =

NI Σ F Ii (z) s

i=1

J Ii

ψsIi (s).

(9.105)

If a beam cross-section has several corners shared by more than two edges, Rχ should include all compatibility conditions associated with these corners.

4

300

9 Sectional Shape Functions of Thin-Walled Beams with General …

The updated shear strain based on τzs in Eq. (9.105) becomes I Σ FsIi (z) Ii τzs = γzs = ψ (s). G GJ Ii s i=1

N

(9.106)

Step 2: Derivation of Secondary Displacement Consistent With the Secondary Strain To account for the updated shear strain in Eq. (9.106) using the displacement field, new warping modes should be considered. If the z-directional displacement of a new warping mode is written as uzW (z, s) = ψzW (s)W (z),

(9.107)

One can obtain the following expression for the shear strain based on Eq. (9.107): I Σ ∂u ∂uzW + s =ψ˙ zW (s)W (z) + ψsIi (s)ξ Ii , (z), ∂s ∂z i=1

N

γzs =

where us (z, s) =

NI  i=1

(9.108)

ψsIi (s)ξ Ii (z) (as given by Eq. (9.8c)) is used.

Equating Eq. (9.106) and Eq. (9.108) yields ψ˙ zW (s) =

) NI ( Ii Fs (z) 1 Σ Ii , ψsIi (s). − ξ (z) W (z) i=1 GJ Ii

(9.109)

Equation (9.109) indicates that the function behavior of the yet-unknown function { } ψ˙ zW (s) depends on the known functions ψsIi (s) i=1,··· ,NI . Therefore, one may expand { } ψ˙ zW (s) in terms of ψsIi (s) i=1,··· ,NI as ψ˙ zW (s) =

NI Σ

ci ψsIi (s),

(9.110)

i=1

Integrating Eq. (9.110) yields ψzW (s) =

NI Σ i=1

ci ψsIi (s) +

NE Σ

cj,0 δj ,

(9.111)

j=1

where ψsIi is the integral of ψsIi excluding the integration constant cj,0 that is constant on edge j. It is convenient to write Eq. (9.111) in vector form for the subsequent analysis:

9.5 Higher-Order Deformable Section Modes

301

{ IN ψzW (s) = ψsI1 , · · · , ψs I , δ1 , · · · , δNE

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ }⎪

c1 .. .

cNI ⎪ c 1,0 ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩c

NE ,0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

≙ ϕ W C,

(9.112)

where the row vector of known functions ϕ W and the unknown coefficient column vector C are defined as } { IN ϕ W = ψsI1 , · · · , ψs I , δ1 , · · · , δNE ; { }T C = c1 , · · ·, cNO , c1,0 , · · ·, cNE ,0 To determine the unknown coefficient vector C, we use the following orthogonality conditions:  A

 ψzWa ψzWb dA

=

CTa

A

( W )T W ϕ ϕ dA Cb ≙ CTa PW Cb = λδab ,

(9.113)

where the symmetric matrix PW is defined as 

( W )T W ϕ ϕ dA.

PW = A

In Eq. (9.113), ψzWa and ψzWb , respectively, denote the shape functions associated with different coefficient vectors Ca and Cb , and λ = J Wab is the sectional moment of inertia. Equation (9.113) can also be put into an equation for an eigenvalue problem as PW C = λC.

(9.114)

When solving Eq. (9.114), two additional conditions should be considered: (i) orthogonality between the current non-linear warping mode ψzW and the out-ofplane modes in lower-order sets, ψO z (introduced in Eq. (9.8a)), and (ii) the continuity (or compatibility) condition at all corners of a cross-section: (i) Orthogonality  A

( O )T W ψz ψz dA =

 A

( O )T W ψz ϕ dA C ≙ QW C = 0,

(9.115)

302

9 Sectional Shape Functions of Thin-Walled Beams with General …

(ii) Continuity (compatibility) (in the case of the cross-section in Fig. 9.1) ⎤ ϕ W (s1 = l1 ) − ϕ W (s2 = l2 ) ⎢ ϕ W (s = l ) − ϕ W (s = 0) ⎥ ⎥ ⎢ 1 1 3 ⎥ ⎢ W ⎢ ϕ (s3 = l3 ) − ϕ W (s4 = 0) ⎥C ≙ RW C = 0. ⎥ ⎢ W ⎣ ϕ (s4 = l4 ) − ϕ W (s5 = 0) ⎦ ϕ W (s5 = l5 ) − ϕ W (s1 = 0) ⎡

In Eq. (9.115), ψO z =

{

ONO

ψzO1 , · · · , ψz

(9.116)

} represents a vector consisting of the

shape functions of the out-of-plane modes in the lower-order sets, and QW is a matrix representing the orthogonality condition. In Eq. (9.116), RW is a constraint matrix representing the continuity (compatibility) condition. The combined form of QW and RW should be considered as the constraint condition when solving the eigenvalue problem in Eq. (9.114). As in the previous subsection, these two equations can be formulated into a single matrix equation, as [

] QW C = SW C = 0. RW

(9.117)

Finally, C can be determined by solving an equation identical to as Eq. (9.104) with Pχ and Sχ replaced by PW and SW , respectively.

9.5.3 Higher-Order Constrained Distortion Modes (H) Not Involving Wall Extension Step 1: Identification of Secondary Normal Strain Field Due to Wall-bending Modes As discussed in Chap. 5, constrained distortion modes are needed to account for the normal strain εss induced by the wall bending stress σ zz due to Poisson’s effect. Recall from Eq. (9.31) that } bending stress caused by wall-bending deformations { the I Ii is given by by in-plane modes ξ = ξ 1,··· ,NI

σ zz = −n

NI Σ

μI i (z)ψnI i (s),

(9.118)

i=1

where ψnI i is the n-directional shape function of the in-plane mode ξ I i . One can express the secondary strain caused by the axial stress σ zz in Eq. (9.118) due to Poisson’s ratio as

9.5 Higher-Order Deformable Section Modes

303

NI ν Σ σ zz =n μI i (z)ψnI i (s). ε ss = −ν E E i=1

(9.119)

Step 2: Derivation of Secondary Displacement Consistent With the Secondary Strain To account for the strain in Eq. (9.119) using displacement, the s-directional bending displacement us (z, s, n) is needed. It can be expressed by the constrained distortion mode η as η

us (z, s, n) = −n

∂un (z, s) = −nη(z)ψ˙ nη (s), ∂s

(9.120)

η

where un (z, s) is the n-directional displacement of the wall midline by the constrained distortion mode η. One can evaluate ε ss , the s-directional normal bending strain, using us in Eq. (9.120): ε ss =

∂us = −nη(z)ψ¨ nη (s). ∂s

(9.121)

Equating Eq. (9.119) and Eq. (9.121) yields I ν Σ μI i (z)ψnI i (s), Eη(z) i=1

N

ψ¨ nη (s) = −

(9.122)

η

As argued in the previous sections, ψ¨ n (s) can be expressed as a linear combination of {

} ψnI i (s)

i=1,··· ,NI

ψ¨ nη (s) =

NI Σ

ci ψnI i (s),

(9.123)

i=1

Integrating Eq. (9.123) twice over s yields ψnη (s)

=

NI Σ i=1

Ii

Ii ci ψ n (s)

NE Σ ( ) dj,1 s + dj,0 δj , +

(9.124)

j=1

where ψ n (s) is a twice-integrated function of ψnI i (s) excluding integration constants d j,0 and d j,1 (j = 1, ···, N E ) that are associated with edge j. As discussed in Sect. 4.2, the η n-directional shape functions ψn (s) should be determined considering the displacement continuity, slope continuity, and moment equilibrium at the corners of a beam

304

9 Sectional Shape Functions of Thin-Walled Beams with General …

cross-section. If one checks the actual continuity and equilibrium conditions given below, one may find that the number of unknown integration constants in Eq. (9.124) is smaller than the number of conditions. To resolve this, we use cubic polynomials η instead of linear functions in Eq. (9.124) and rewrite ψn (s) as ψnη (s)

=

NI Σ

Ii

ci ψ n (s) +

NE Σ ( ) cj,3 s3 + cj,2 s2 + cj,1 s + cj,0 δj ,

i=1

(9.125)

j=1

{ } where cj,m (m = 0, 1, 2, 3) are coefficients to be determined. To facilitate the subsequent analysis, Eq. (9.125) is put into vector form as ⎧ ⎫ ⎪ c⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c ⎨ ⎬ 3⎪ } { 3 2 η ψn (s) = ψ n , s δ, s δ, sδ, δ c2 ≙ ϕ η C, ⎪ ⎪ ⎪ ⎪ ⎪ c1 ⎪ ⎪ ⎪ ⎪ ⎩c ⎪ ⎭ 0

(9.126)

where the row vector of known functions ϕ η and the unknown coefficient column vector C are defined as { } { }T ϕ η = ψ n , s3 δ, s2 δ, sδ, δ ; C = cT , c3T , c2T , c1T , c0T and } { I { } IN 1 ψ n = ψ n , · · · , ψ n I , δ = δ1 , δ2 , · · · , δNE , { { }T }T c = c1 , c2 , · · · , cNI , cm = c1,m , c2,m , · · · , cNE ,m

(m = 0, 1, 2, 3).

To determine the unknown coefficient vector, one can set up an eigenvalue problem subjected to constraint equations to determine C in Eq. (9.126): Pη C = λC,

(9.127)

⎤ Qη ⎢ Rη ⎥ ⎢ 0η ⎥C = Sη C = 0. ⎣R ⎦ 1 η R2

(9.128)

subject to ⎡

This constrained eigenvalue problem can be solved using an equation identical to Eq. (9.104) with Pχ and Sχ replaced by Pη and Sη , respectively. The symmetric matrix η Pη is defined from the condition depicting orthogonality among the ψn functions:

9.5 Higher-Order Deformable Section Modes

 A η

ψ˙ nηa ψ˙ nηb dA = CTa

 A

(ϕ˙ η )T ϕ˙ η dA Cb ≙ CTa Pη Cb = λδab ,

305

(9.129)

η

where ψn a and ψn b are, respectively, the shape functions associated with the coefficient vectors Ca and Cb , and λ = J ηab is the sectional moment of inertia. The reason for using the derivatives of the shape functions in Eq. (9.129) instead of the functions themselves was explained in earlier chapters (see, e.g., Eq. (5.11)). In Eq. (9.128), Qη η is the constraint matrix for orthogonality conditions between ψ˙ n and the derivatives of constrained distortion modes in lower-order sets:   ( η )T η ( η )T η ψ˙ n ψ˙ n dA = ψ˙ n ϕ˙ dA C ≙ Qη C = 0, (9.130) A

A

} { ηN η where ψηn = ψn 1 , · · · , ψn η is the shape function vector of constrained distortion modes in lower-order sets. Other matrices appearing in the constraint Eq. (9.128) are used to represent the displacement and slope continuity and the moment equilibrium at the corners of the beam section. For the cross-section in Fig. 9.1, they are defined by the following equations: i. Displacement continuity ⎡

⎤ ϕ η (s1 = 0) ⎢ ϕ η (s1 = l1 ) ⎥ ⎢ ⎥ ⎢ ϕ η (s = l ) ⎥ 2 2 ⎥ ⎢ ⎢ ϕ η (s = 0) ⎥ ⎢ ⎥ 3 ⎢ η ⎥ η ⎢ ϕ (s3 = l3 ) ⎥C = R0 C = 0, ⎢ η ⎥ ⎢ ϕ (s4 = 0) ⎥ ⎢ η ⎥ ⎢ ϕ (s4 = l4 ) ⎥ ⎢ ⎥ ⎣ ϕ η (s5 = 0) ⎦

(9.131)

ϕ η (s6 = l5 )

ii. Slope continuity ⎡

⎤ ϕ˙ η (s1 = l1 ) − ϕ˙ η (s2 = l2 ) ⎢ ϕ˙ η (s = l ) − ϕ˙ η (s = 0) ⎥ ⎢ ⎥ 1 1 3 ⎢ η ⎥ η ⎢ ϕ˙ (s3 = l3 ) − ϕ˙ η (s4 = 0) ⎥C = R1 C = 0, ⎢ η ⎥ η ⎣ ϕ˙ (s4 = l4 ) − ϕ˙ (s5 = 0) ⎦ ϕ˙ η (s1 = 0) − ϕ˙ η (s5 = l5 )

(9.132)

306

9 Sectional Shape Functions of Thin-Walled Beams with General …

iii. Moment equilibrium ⎡

ϕ¨ η (s1 ⎢ ϕ¨ η (s ⎢ 3 ⎢ η ⎢ ϕ¨ (s2 ⎢ η ⎣ ϕ¨ (s4 ϕ¨ η (s1

⎤ = l1 ) + ϕ¨ η (s2 = l2 ) − ϕ¨ η (s3 = 0) ⎥ = l3 ) − ϕ¨ η (s4 = 0) ⎥ ⎥ η = 0) ⎥C = R2 C = 0. ⎥ η ⎦ = l4 ) − ϕ¨ (s5 = 0) η = 0) − ϕ¨ (s5 = l5 )

(9.133)

9.6 Case Studies Using the derivation procedure described in Sects. 9.3, 9.4, 9.5, the sectional shape functions for the thin-walled beams of open, closed, and flanged cross-sections shown in Fig. 9.3 are derived and shown in Figs. 9.4, 9.5, 9.6. The derived cross-section modes are employed in the HoBT to solve the static or modal analysis problems of thin-walled beams considered below. The numerical results predicted by the HoBT will be compared with those obtained by shell elements (ABAQUS S8R elements) and other beam-based approaches, in this case the Timoshenko beam theory (Timoshenko and Goodier (1970)), the generalized beam theory (GBT) (Schardt (1989, 1994a, 1994b); Goncalves et al. (2010); Silvestre et al. (2011)), and the method of generalized eigenvectors (GE) (Garcea et al. (2016); Genoese et al. (2014)). For numerical examples below, we use the following data: Young’s modulus E = 210 GPa (Sect. 9.6.2) or 200 GPa (Sects. 9.6.1, 9.6.3, and 9.6.4), Poisson’s ratio ν = 0.3, and density ρ = 7850 kg/m3 .

9.6.1 Static Analysis: A Cantilever Beam with an Open Cross-Section A static analysis is conducted for a thin-walled beam (length: 900 mm and thickness: 1 mm) with the open cross-section shown in Fig. 9.3a. One end of the beam (z = 0) is fixed, and the cross-section at the other end (z = 900 mm) is subjected to a set of distributed loads in the z- and s-directions, as described in Fig. 9.3a). For the analysis, 200 HoBT elements were used. To capture rapidly changing end effects, 100 elements were assigned near the loaded end between z = 800 and z = 900 mm. We tested the accuracy of the HoBT-based results using a different number of mode sets M = 2, 3, 4, and 9 (M: mode set number) corresponding to 26 modes, 44 modes, 65 modes, and 161 modes. The warping modes in the highest-order set were

9.6 Case Studies

307

(a)

Load: [N/m]

s5 (0, 100) (0, 90)

Thickness = 1

y x

(50, 50)

s1 (0, 10) P (12,0) s2 (0, 0)

Y

s4 (60, 100)

s3 (60, 0)

X

(b) (10, 40)

(c)

(40, 40) Thickness = 2

(0, 30)

(0, 30) (50, 30)

y

(-5, 20) (0, 20) (-10, 10)

172.1

Thickness = 1

(26.3, 20.1) x (10.6, 9.2)

(30, 10) (40, 10)

(30, 0) (40, 0)

(25, 10) (35, 10)

P (50, 6) (0, 0)

y

135.4

x

(50, 0) (0, 0)

(20, 0)

FY = -100N

Fig. 9.3 Various cross-section shapes of thin-walled beams: a an open section, b a closed section, and c a flanged section (unit: mm)

not employed because their effects on the solution accuracy were found to be less than those of the other modes. Figure 9.7a and b show the three-dimensional displacements and stresses, respectively, measured along the axial line corresponding to point P in Fig. 9.3a. In Figs. 9.7, the numbers in parentheses indicate the number of cross-section modes used for the analyzes. These figures show that excellent accuracy can be obtained for threedimensional displacements even with M = 2. The stresses, as shown in Fig. 9.7b, show that more mode sets are needed to capture rapidly changing stress variations accurately due to the end effect. The difference in the peak value of σ zz between the result by the HoBT and that of the shell elements is plotted in Fig. 9.8 with respect to the highest mode set number (and number of modes). In the figure, the use of M = 4 for the HoBT yields stress only within 4% error compared with the shell-elementbased calculation. These numerical tests suggest that a moderate level of accuracy can be expected when using M = 2 for the displacement calculations and M = 4 for

308

9 Sectional Shape Functions of Thin-Walled Beams with General …

Rigid-body mode 1st set

Linear warping Inextensional distortion Extensional distortion

2nd set

C-distortion

Nonlinear warping Extensional distortion 3rd set

θx

θy

θz

χ3

χ4

χ5

χ6

η2

η3

η4

η5

η6

η7

η8

η9

W1

W2

W3

W4

W5

W6

χ7

χ8

χ9

χ10

χ11

χ12

η10

η11

η12

η13

η14

η15

W7

W8

W9

W10

W11

W12

Ux

Uy

Uz

W10

W20

W30

χ10

χ 20

χ1

χ2

η1

C-distortion Nonlinear warping

Fig. 9.4 Sectional shape functions for the open section in Fig. 9.3a (Kim et al. (2021))

the stress calculations (within 4% error). If M ≥ 9 is used, the stress prediction by the HoBT is nearly as accurate as that by the shell analysis, within only 1% error. Figure 9.9 shows the contributions of the three dominant distortion modes of the HoBT to the shear stress (τZX ), as calculated using the generalized force-stress relationship in Eq. (9.25). Because the generalized forces are the work conjugates of onedimensional deformations, element force vectors associated with the points of interest are used to calculate the stress curves in Fig. 9.9. Note in the figure that the three most influential distortion modes, χ2 , χ3 , and χ10 , show wall-extending/shrinking deformations, especially on both horizontal cross-section edges. These modes can be restrained if the extension of the bottom edge is suppressed, for example, by connecting two corners of the bottom edge through a rigid bar. In this case, the peak stress is reduced from 112.3 Pa for the unconstrained state to 66.4 Pa (40.9% reduction). Figure 9.10 shows the stress results measured on the inner surface (n = −t/2), outer surface (n = t/2), and midline at point P along the axial direction. The HoBT predicts results nearly identical to those predicted by the shell elements.

9.6 Case Studies

Rigid-body mode

1st set

Linear warping

Inextensional distortion

Extensional distortion

309 Ux

Uy

Uz

θx

θy

θz

W10

W20

W30

W40

W50

W60

W70

W80

W90

W100

χ10

χ 20

χ 30

χ 40

χ 50

χ 60

χ 70

χ80

χ 90

χ100

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

χ10

χ11

χ12

η1

η2

η3

η4

η5

η6

η7

η8

η9

η10

η11

η12

η13

η14

η15

η16

η17

η18

η19

η20

η21

η22

η23

W1

W2

W3

W4

W5

W6

W7

W8

W9

W10

W11

W12

χ13

2nd set

C-distortion

Nonlinear warping

W13

Fig. 9.5 Sectional shape functions for the closed section in Fig. 9.3b (Kim et al. (2021))

9.6.2 Static Analysis: A Simply Supported Beam with an Open Cross-Section As the next example, we consider a simply supported beam of an open cross-section subjected to a half sinusoidal load, as illustrated in Fig. 9.11a. This problem was

310

9 Sectional Shape Functions of Thin-Walled Beams with General …

Rigid-body mode 1st set

Linear warping Inextensional distortion Extensional distortion

2nd set

C-distortion

Nonlinear warping Extensional distortion

3rd set

C-distortion

Nonlinear warping

Uy

Uz

χ1

χ2

χ3

χ4

η1

η2

η3

η4

η7

η8

η9

W1

W2

W3

W4

χ5

χ6

χ7

χ8

η10

η11

η12

η13

η16

η17

η18

W5

W6

W7

Ux

θx

θy

θz

η5

η6

η14

η15

W10

χ10

W8

Fig. 9.6 Sectional shape functions for the flanged section in Fig. 9.3c (Kim et al. (2021))

initially solved by Garcea et al. (2016), and we compare the numerical results by the HoBT, the GBT, and the GE methods. Figure 9.11b shows that the lateral displacements on the loaded line obtained by the three methods are nearly identical and are also in good agreement with the shell results. To obtain the results in Fig. 9.11b, the cross-section modes for M = 2 corresponding to the first 24 cross-section modes in Fig. 9.4 were used for the HoBTbased analysis. (Although there is a slight difference in the dimensions between the cross-sections in Figs. 9.3a and 9.11a, the section mode shapes for this case are virtually identical to those shown in Fig. 9.4. Therefore, they are not plotted here.) To obtain the GBT results, the modes were obtained using the GBTUL program with cross-section discretization of three intermediate nodes for the web and two intermediate nodes for each flange. Among the 39 modes obtained by GBTUL, the first 15 modes were employed in this analysis. Along the axial direction, 50 evenly distributed finite elements were used for the GBT and the HoBT methods. For the

9.6 Case Studies

311

(b)

(a)

Shell Timoshenko M = 2 (26) M = 4 (65) M = 9 (161) Shell Timoshenko M = 2 (26) M = 3 (44) M = 4 (65)

Shell Timoshenko M = 2 (26) M = 3 (44) M = 4 (65)

M = 4 (65) Shell & M = 9 (161)

Timoshenko M=2 (26)

Timoshenko Shell & M = 9 (161) M=4 (65)

τ

Shell & M = 4 (65)

M = 2 (26)

M = 3 (44)

M = 2 (26)

Timoshenko

Fig. 9.7 Static analysis results measured along point P in Fig. 9.3a: a displacement results and b stress results (M denotes the highest mode set number, and the numbers in parentheses indicate the numbers of employed cross-section modes for the HoBT-based analysis.) (Kim et al. (2021))

GE method, the results by Garcea et al. (2016), who used 19 in-plane modes and 19 out-of-plane modes, are presented for comparison. Figure 9.12 shows the contribution of each mode to the total strain energy and the displacement at the midpoint of the loading line. While the warping and in-plane modes are coupled in the GBT mode, as in modes 3 and 5, which are sketched in the inset of the illustration on the left in Fig. 9.12, they are separate in the HoBT. It is

9 Sectional Shape Functions of Thin-Walled Beams with General …

Fig. 9.8 Effect of the highest mode set on the relative error of the axial stress HoBT, ( HoBTby theshell )/ shell − σzz σzz σzz , at the peak stress point (z = 893.4 mm) (Kim et al. (2021))

Number of cross-section modes 70

26

44

65

2

3

4

83

103 122 140 161 179 198

60 50

Error (%)

312

40 30 20 10 5 0

5

6

7

8

9

10

11

M (highest mode set)

Fig. 9.9 Contribution of three dominant distortion modes to the shear stress (τZX ) in Fig. 9.7b (Kim et al. (2021))

shown in Fig. 9.12 that the modes with large contributions to the strain energy (gray bars) are obtained identically for the HoBT and GBT results. Due to the symmetry of the boundary and load conditions at the midpoint, it is obvious that only in-plane) ( deformation can arise at this point. Therefore, only in-plane modes Ux , χ10 , η1 make non-zero contributions to the displacement at the midpoint in the HoBT result.

9.6 Case Studies

313

Fig. 9.10 Stress results measured at different n coordinates (n = − t/2, 0, t/2) of point P for the thin-walled beam problem with the open cross-section in Fig. 9.3a (Kim et al. (2021))

9.6.3 Static Analysis: A Cantilever Beam with a Closed Cross-Section At this stage, we consider a clamped thin-walled beam with the cross-section in Fig. 9.3b, the free end of which is subject to vertical concentrated force at the lower right corner of the cross-section. The beam is 400 mm long, and its wall is 2 mm thick. The displacements and stresses at point P along the axial direction were calculated by the HoBT and are plotted in Fig. 9.13a and b, respectively. The HoBT results were

314

9 Sectional Shape Functions of Thin-Walled Beams with General …

(a)

(b)

Size: [mm]

0 Shell GBT GE (Garcea et al., 2016) Proposed

600

Load: [N/m]

uX (m)

60

2.5

FX = −1000 sin(

Z)

-0.05

-0.1

Y 10

120 Z

60

0

X

0.2

0.4

0.6

Axial coordinate (m)

Fig. 9.11 a Simply supported thin-walled beam with an open cross-section subjected to a half sinusoidal lateral load and b resulting lateral displacements on the loading line (Kim et al. (2021))

Displacement Energy 3

3

5

5 7

y

0 1

W20 Ux

1

Fig. 9.12 Contributions to the strain energy and displacement at the midpoint on the loading line of the GBT modes (left) and the HoBT modes (right) for the problem considered in Fig. 9.11a (Kim et al. (2021))

obtained using 200 finite elements, half of which were allocated near the loaded end from z = 350 mm to z = 400 mm. Figure 9.13a shows that for the HoBT, moderately accurate displacements can be obtained if M = 2 (62 modes), while accurate results even capturing the end effects of uz can be obtained if (M = 5 (213 modes). As observed in earlier problems, a large M value, such as M = 7 (311 modes), was needed to predict the stress sufficiently accurately. The contribution of the generalized forces to produce σ zz is shown in Fig. 9.14. This figure shows that W30 is the most dominant higher-order mode contributing to the axial stress. Figures 9.15a and b show the von-Mises stress for the cross-section midline calculated at z = 380 mm and the corresponding deformed shape. This location was selected because it is the location where the shear stress τZY reaches its maximum. These figures show that the HoBT results are in good agreement with the shell results, whereas those by the Timoshenko theory are not.

9.6 Case Studies

315

Fig. 9.13 Static analysis results measured along point P in Fig. 9.3b: a displacement results and b stress results (Kim et al. (2021))

9.6.4 Modal Analysis: A Beam with a Flanged Cross-Section with a Free-free Support Condition A modal analysis is conducted for a thin-walled beam with a flanged cross-section, as shown in Fig. 9.3c, with no end support. The beam is 500 mm long, and its wall is 1 mm thick. Figure 9.16 compares the free vibration mode shapes obtained using the shell elements and those obtained using the HoBT (50 finite elements along the

316

9 Sectional Shape Functions of Thin-Walled Beams with General …

Fig. 9.14 Contribution of three dominant warping modes to the axial stress (σZZ ) in Fig. 9.13b (Kim et al. (2021))

(a)

(b)

Shell Timoshenko M = 7 (311) Shell Timoshenko M = 2 (62)

Fig. 9.15 a Von Mises stress and b in-plane deformation calculated on the cross-section at z = 380 mm for the thin-walled beam problem with the cross-section in Fig. 9.3b (Kim et al. (2021))

beam length with M = 2 (21 modes)). The eigenfrequencies are compared in Table 9.1. Although relatively few cross-section modes were employed for the HoBT, the free vibration characteristics of the beam are accurately predicted by the higher-order beam theory.

References

317

(a)

1st

2nd

3rd

4th

5th

1st

2nd

3rd

4th

5th

(b)

Fig. 9.16 Free-vibration mode shapes for a thin-walled beam with the flanged cross-section in Fig. 9.3c: results by a the shell theory and b the HoBT (Kim et al. (2021))

Table 9.1 Eigenfrequencies (Hz) of a free-free beam with a flanged cross-section Mode

1

2

3

4

5

6

7

Shell

582.08

788.94

1482.1

1948.6

2098.0

2401.2

2850.9

Proposed

583.35 (0.22)

790.53 (0.20)

1492.5 (0.71)

1961.1 (0.64)

2100.8 (0.13)

2424.1 (0.95)

2873.2 (0.78)

Numbers in parentheses denote the percentage (%) error of the HoBT results in comparison with the shell results. (Kim et al. (2021))

References Garcea G, Goncalves R, Bilotta A, Manta D, Bebiano R, Leonetti L, Magisano D, Camotim D (2016) Deformation modes of thin-walled members: a comparison between the method of generalized eigenvectors and generalized beam theory. Thin-Walled Struct 100:192–212 Genoese A, Genoese A, Bilotta A, Garcea G (2014) A generalized model for heterogeneous and anisotropic beams including section distortions. Thin-Walled Struct 74:85–103 Golub GH, Loan CFV (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore Goncalves R, Ritto-Corrêa M, Camotim D (2010) A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory. Comput Mech 46:759–781 Kim J, Choi S, Kim YY, Jang G-W (2021) Hierarchical derivation of orthogonal cross-section modes for thin-walled beams with arbitrary sections. Thin-Walled Struct 161:107491 Schardt R (1989) Verallgemeinerte technische biegetheorie. Springer-Verlag, Berlin Schardt R (1994a) Generalised beam theory—an adequate method for coupled stability problems. Thin-Walled Struct 19(2–4):161–180 Schardt R (1994b) Lateral torsional and distortional buckling of channel and hat- sections. J Constr Steel Res 31(2–3):243–265 Silvestre N, Camotim D, Silva NF (2011) Generalised beam theory revisited: from the kinematical assumptions to the deformation mode determination. Int J Struct Stab Dyn 11(5):969–997 Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York

Chapter 10

Joint Structures of Box Beams

This chapter presents a higher-order beam analysis of a joint structure in which multiple straight box beam members are connected at a joint, as shown in Fig. 10.1a. Owing to extensive section deformation occurring near the joint, the overall structural behavior of the joint structure becomes considerably more flexible than predicted by the classical beam theory (Donders et al. (2009); Mundo et al. (2009)). One can certainly expect improved or more accurate predictions with a higher-order beam theory, but the field variables of multiple box beams at the joint are difficult to match (Basaglia et al. (2012); Basaglia et al. (2018); Choi et al. (2012); Choi and Kim (2016a, 2016b); Jang et al. (2008); Jang and Kim (2009); Jang et al. (2013)). Unless they are matched accurately, there is no way to make the use of the advantages of a higher-order beam theory which is shown to be accurate for straight box beams without any joints. The zoomed view in Fig. 10.1a shows how significant the local cross-sectional deformation is at a T-joint box beam structure where a force is applied to one end of the three box beams forming the structure. (This T-joint structure was also considered in Chap. 1). Figure 10.1b shows the y-directional bending deflection of the cross-section measured along AOB marked in Fig. 10.1a. In Fig. 10.1b, it is clearly shown that the flexibility of the joint is mainly caused by deformable section modes at the joint such that classical Euler/Timoshenko beam analyzes considering only non-deformable section modes cannot correctly capture the overall flexibility effect associated with deformation due to deformed section modes. If the higher-order beam analysis is properly used, the flexibility of the joint can be correctly represented by the combined effects of non-deformable and deformable sectional modes in the regions of the box beams near the joint. The coupling among these modes can be accounted for if the matching conditions among them at a joint are correctly established. These conditions will be referred to as the joint matching conditions hereafter. In classical beam theories, joint matching conditions are simply determined using a coordinate transformation matrix that can be obtained by matching the vector components of the generalized forces of the joining beams at the joint. For example, © Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_10

319

320

10 Joint Structures of Box Beams

(a)

(b) 10 -3

B

Beam 3

Y-directional displacement

Z X

Y

Beam 1

A O

Beam 2

3 2

Joint flexibility

1 0

A

C

Shell elements Timoshenko beam elements

Beam 1

O

Beam 3

B

Fig. 10.1 a Deformed shape of a T-joint thin-walled box beam structure subjected to shear force (as calculated by a shell analysis) and b the y-directional rigid-body displacement of a cross-section along AOB

y2

z2

Beam 2

x2

y1

φ Y

x1 z1

Beam 1

X Joint

Z Fig. 10.2 Two-beam joint structure

if the two-beam joint structure in Fig. 10.2 is analyzed using Euler/Timoshenko beam }T { elements with only non-deformable section modes of ξ = U y , θx , θz , the vectors of the generalized forces of the joining beams are matched in a component-wise manner at the joint as ⎫ ⎫ ⎧ ⎡ ⎤⎧ 1 ⎨ Fy ⎬ ⎨ Fy ⎬ = ⎣ cos φ − sin φ ⎦ Mx , M ⎭ ⎩ x⎭ ⎩ Mz Beam 2 Mz Beam 1 sin φ cos φ

(10.1)

from which the matching conditions among section modes can be obtained via

10 Joint Structures of Box Beams

321

Shared Top & Bottom Edge

Beam k+1

yi

Shared Side Edge k

b

Y

s2 h

X

Beam k-1

Shared Side Edge k-1

t

Z

n2 s1 n1

Edge 3

n3 s3

xi

Edge 1

n4

s4

zi

Edge 4

Beam k

Thin-Walled Box Beam

Coordinate systems

Fig. 10.3 Three or more thin-walled box beams connected at a joint. Only a portion of the structure consisting of Beam k – 1, Beam k, and Beam k + 1 (k ≥ 2) is depicted

⎧ ⎫ ⎡ ⎤⎧ ⎫ 1 ⎨ Uy ⎬ ⎨ Uy ⎬ = ⎣ cos φ sin φ ⎦ θx . θx ⎩ ⎭ ⎩ ⎭ θz Beam 2 θz Beam 1 − sin φ cos φ

(10.2)

The result in Eq. (10.2) can be obtained from the relationship given in Eq. (10.1) with the use of the principle of the virtual work (see detailed related analysis in Sect. 10.4). When both non-deformable and deformable section modes of a higher-order beam theory are considered, however, the procedure used to derive Eq. (10.2) cannot be applied, mainly because the generalized forces of deformable section modes (e.g., bimoments) do not produce any non-zero net resultant. In other words, the generalized forces of deformable section modes are self-equilibrated over a cross-section. To address this issue, a new concept called the “edge resultant” was introduced by Choi and Kim (2016a), representing an edgewisely defined resultant force or moment on a cross-section. Using this concept of an edge resultant, equilibrium conditions among generalized forces including those of deformable section modes can be found at a joint. In this chapter, the resultant forces and moments of non-deformable section modes will be referred to as “sectional resultants” to distinguish them from “edge resultants.” The joint structure consisting of multiple-connected box beams in Fig. 10.3 will be analyzed. Here, the beams are assumed to lie on the same plane (in this case the X − Z plane). Out-of-plane (Y-directional) bending deformation of the structure by an out-of-plane shear or torsion load is mainly examined in this chapter.1 For clarity, only five variables (or section modes) involved in out-of-plane bending { }T deformation will be used as ξ = U y , θx , θz , W0 , χ0 , where { the {W0}, χ0 } variables denote torsional warping and torsional distortion and the U y , θx , θz variables represent vertical displacement, bending rotation, and torsional rotation, respectively. Figure 10.4 explains the shapes of the sectional modes considered here. 1

One can refer to Choi and Kim (2016b) to investigate joint flexibility by in-plane bending or extension loads.

322

10 Joint Structures of Box Beams

Uy

θx

y

y

y x

x

z

z

z

x

θz χ0

W0

y

y x z

z

x

} { Fig. 10.4 Sectional deformations of ξ = U y , θx , θz , W0 , χ0

To present more clearly the matching approach based on the edge resultants, we construct this chapter as a self-contained chapter in which the higher-order beam { } theory for ξ = U y , θx , θz , W0 , χ0 is briefly reviewed and presented in Sect. 10.1. Edge resultants as well as sectional resultants will be derived in Sect. 10.2 using generalized force-stress relationships. Section 10.3 presents the equilibrium conditions among the generalized forces using the sectional and edge resultants, and Sect. 10.4 explains the joint matching conditions. To confirm the validity of the obtained joint matching conditions, numerical examples will be solved in Sect. 10.5.

10.1 Higher-Order Beam Theory for Out-of-Plane Bending Problems of a Box Beam As previously studied in Chaps. 4 and 7, the displacements {u s , u z } and stresses {σzz , τzs}} on the wall midline and generalized forces F = { Fy ,{Mx , Mz , B0 , Q 0 from } the higher-order beam theory using five field variables ξ = U y , θx , θz , W0 , χ0 as 1D variables can be written as u s (z, s) =ψs y (s)U y (z) + ψsθz (s)θz (z) + ψsχ0 (s)χ0 (z) U

u z (z, s) =ψzθx (s)θx (z) + ψzW0 (s)W0 (z) σzz (z, s) =Eψzθx (s)θx, (z) + E 1 ψzW0 (s)W0, (z)

(10.3a)

10.1 Higher-Order Beam Theory for Out-of-Plane Bending Problems …

⎡

ψs y (s)U y, (z) U

⎢ ⎢ +ψ θz (s)θ , (z) z s ⎢ ⎢ τzs (z, s) =G ⎢ +ψsχ0 (s)χ0, (z) ⎢ ⎢ +ψ˙ θx (s)θ (z) x ⎣ z

323

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

+ψ˙ zW0 (s)W0 (z) ∫ U Fy (z) = τzs ψs y d A; ∫ Mx (z) = σzz ψzθx d A; ∫ Mz (z) = τzs ψsθz d A; ∫ B0 (z) = σzz ψzW0 d A; ∫ Q 0 (z) = τzs ψsχ0 d A

(10.3b)

(10.3c)

where the sectional shape functions used in Eq. (10.3) are defined as {

{

{

U

U

ψs y = 1 for edge 1, ψs y = −1 for edge 3 U ψs y = 0 for edges 2 and 4

ψzθx = s1 for edge 1, ψzθx = h/2 for edge 2 ψzθx = −s3 for edge 3, ψzθx = −h/2 for edge 4 { θ ψs z = b/2 for edges 1 and 3 θ ψs z = h/2 for edges 2 and 4

ψzW0 = (b/2)s1 ψzW0 = (b/2)s3 { χ0 ψs χ ψs 0

for edge 1, ψzW0 = −(h/2) s2 for edge 2 for edge 3, ψzW0 = −(h/2) s4 for egde 4 / = bh/ (b + h) for edges 1 and 3 = −bh (b + h) for edges 2 and 4

(10.4a)

(10.4b)

(10.4c)

(10.4d)

(10.4e)

) / /( / where (), = d() dz, (·) = d() ds, and E 1  E 1 − ν 2 (E: Young’s modulus; ν: Poisson’s ratio) and G is the shear modulus. In Eq. (10.3b), the component of σzz by bending rotation θx, is expressed as Eψzθx θx, whereas the stress component is expressed as E 1 ψzθx θx, in Chap. 7 (see Eq. (7.5a)). These expressions are valid because no distortion mode χ for bending, which is required to consider Poisson’s effect, is included in the current mode set ξ. The generalized in Eq. (10.3c) are the work conjugates of ξ. { forces F defined } In this equation, Fy , Mx , Mz are the shear force, bending moment, and torsional moment, respectively, while {B0 , Q 0 } represent the axial bimoment and transverse

324

10 Joint Structures of Box Beams

bimoment, respectively. The generalized force-stress relationships can be written as (see Eqs. (4.20, 4.28–4.29, 7.12, and 7.17): ] [ ] [ x θx ] [ B0 W0 ] [ σzz (z, s) = σzzMx + σzzB0 = M ψz + JW ψz Jθ x

(10.5a)

0

[ ] [ ] [ ] [ F U ] [ z θ z ] [ Q 0 χ0 ] F (10.5b) τzs (z, s) = τzsy + τzsMz + τzsQ 0 = JUy ψs y + M ψs + Jχ ψs Jθ y

z

0

( ) where Jα α = U y , θx , θz , W0 , χ0 , the second moment of inertia of mode α is defined as ∫ ( ) U 2 JU y = ψs y d A = 2ht (10.6a) ∫ Jθx = ∫ Jθz = ∫

( θx )2 ψz d A =

h 2 t(3b+h) 6

(10.6b)

( θz )2 ψs d A =

bht(b+h) 2

(10.6c)

)2 ψzW0 d A =

b2 h 2 t(b+h) 24

(10.6d)

(

JW0 =

∫ Jχ0 =

( χ0 )2 ψs d A =

2b2 h 2 t b+h

(10.6e)

Although the procedure used to derive Eq. (10.5) is explained in detail in Chaps. 4 and 7, it may be helpful to briefly review it here. The stresses {σzz , τzs } in Eq. (10.3b) can be rewritten as σzz = Eψzθx θx, + E 1 ψzW0 W0, , [ ) ( U ( τzs = G ψs y U y, + θx + ψsθz θz, +

b−h W b+h 0

)

( )] + ψsχ0 χ0, + W0 ,

(10.7)

U θ χ where the relationships such that ψ˙ zθx = ψs y and ψ˙ zW0 = b−h ψ z + ψs 0 are used; b+h s see Eq. (10.4). The stresses {σzz , τzs } in Eq. (10.7) are then substituted into Eq. (10.3c) to find

∫ Fy =

( ) U τzs ψs y d A = G JU y U y, + θx ∫

Mx = ∫ Mz =

(10.8a)

σzz ψzθx d A = E Jθx θx,

( τzs ψsθz d A = G Jθz θz, +

b−h W b+h 0

(10.8b) )

(10.8c)

10.2 Sectional Resultants and Edge Resultants

∫

σzz ψzW0 d A = E 1 JW0 W0,

(10.8d)

( ) τzs ψsχ0 d A = G Jχ0 χ0, + W0

(10.8e)

B0 = ∫ Q0 =

325

∫ U θ where the orthogonality conditions among sectional shape functions ( ψs y ψs z d A = ∫ U y χ0 ∫ ∫ θ χ ψs ψs d A = ψs z ψs 0 d A = 0 and ψzθx ψzW0 d A = 0) are used. Lastly, substituting Eq. (10.8) into Eq. (10.7) yields the relationships given in Eq. (10.5).

10.2 Sectional Resultants and Edge Resultants The sectional resultants are calculated simply by performing surface integrals of the stresses over a beam cross-section as ∫ ∫ ∫ Fy = τzs d A, Mx = σzz yd A, Mz = τzs r d A. (10.9) These resultants correspond to generalized forces that are not in self-equilibrium the (see Fig. 10.5). The other generalized forces, {B0 , Q 0 }, appearing { when using } HoBT approach, are self-equilibrated. Therefore, the stresses σzzB0 , τzsQ 0 due to {B0 , Q 0 } contribute { nothing } to the sectional resultants. Although the σzzB0 , τzsQ 0 variables do not have any sectional resultant when they are evaluated over the entire cross-section, they can have non-zero forces or moments on each edge of the( cross-section, i.e., edge resultants. Using σ β in Eq. (10.5), edge ) resultants due to β β = Fy , Mx , Mz , B0 , Q 0 can be defined. We define the edge resultants on edge j (j = 1, 2, 3, 4) as β Fz ( j)

¨ =

σzzβ

dsdn,

β Fs ( j )

Edge j

¨ =

σzsβ

dsdn,

β Mn ( j)

Edge j

¨ =

s j · σzzβ dsdn,

Edge j

(10.10) Fy

y x

y

Mx

x

z

z

Fig. 10.5 Generalized forces with sectional resultants

y x z

Mz

326

10 Joint Structures of Box Beams β

β

β

where Fz ( j) is the axial force, Fs ( j ) is the tangential force, and Mn ( j) represents the } { β β β wall-normal moment on edge j. For example, Fz (1) , Fs (1) , Mn (1) calculated on edge 1 are F

F

y Fz(1) = 0,

Mx Fz(1) = 0,

y Fs(1) = 21 Fy ,

Mx Fs(1) = 0,

M

M

Fz(1)z = 0 , B0 Fz(1) = 0,

Fs(1)z =

Mx Mn(1) =

1 b+h

B0 Fs(1) = 0,

Q0 Fz(1) = 0,

Q0 Fs(1) =

F

y Mn(1) =0

Mz ,

h M 2 (h + 3b) x

1 2b

Q0 ,

(10.11b)

M

(10.11c)

h B b(b+h) 0

(10.11d)

z Mn(1) =0

B0 Mn(1) =

(10.11a)

Q0 Mn(1) =0

(10.11e)

where b and h denote the width and height of the{box beam cross-section, } respectively β β β (see Fig. 10.3). Using Eqs. (10.5 and 10.10), Fz ( j) , Fs ( j) , Mn ( j ) on edges 2, 3, and 4 can also be calculated. Figure 10.6 illustrates the calculated non-zero edge resultants, whose directions and magnitudes are denoted by the arrows and | · | symbols, respectively.

10.3 Generalized Force Equilibrium Conditions In the previous { section, we considered } sectional and edge resultants due to generalized forces F = Fy , Mx{, Mz , B0 , Q}0 . It is worth mentioning that {B0 , Q 0 } have nonQ0 B0 zero edge resultants Mn( j) , Fs( j)

j=1, 2, 3, 4

but zero sectional resultants. Using edge

resultants as well as sectional resultants, we will establish equilibrium conditions among the generalized forces in F at a joint. We begin with the equilibrium conditions for a two-beam joint structure (see Fig. 10.7) and then extend the obtained results for a joint structure consisting of three or more box beams, as shown in Fig. 10.3. We will use the beam model of a two-beam joint structure illustrated in Fig. 10.8 for the equilibrium analysis of the joint. We define the intersection of end sections as depicted in Fig. 10.8b as the joint location. The joint is assumed to pass through the centroids of the end sections. In Fig. 10.8, (X, Y, Z ) are the global coordinates, and (xk , yk , z k ) are the local coordinates of Beam k (k = 1, 2). For the joint structure, the generalized forces and the section modes for each beam are }T { F [k] = Fy[k] , Mx[k] , Mz[k] , B0[k] , Q [k] 0

(10.12a)

10.3 Generalized Force Equilibrium Conditions

327

FzM(2)x =

y

y x

x

F

Fs (1)y = 12 Fy

z F

FsM(2)z = b+1 h M z

y x Mz s (3)

F

Mz s (1)

= F

x

FsQ(3)0 = FsQ(1)0

B0 n (3)

=M

B0 n (1)

b h (b+ h )

B0

0 M nB(1) = b (bh+h ) B0

0 0 M nB(4) = M nB(2)

1 2 h Q0

FsQ(1)0 =

z

z

M

FsM(4)z = FsM(2)z

FsQ(2)0 =

y

x

h 2( h+3b ) M x

Fz M(4)x = Fz M(2)x 0 M nB(2) =

y

FsM(1)z = b+1 h M z

z

M nM(1)x =

z

M nM(3)x = M nM(1)x

F

Fs (3)y = Fs (1)y

3b h ( h+3b ) M x

1 2b Q0

FsQ(4)0 = FsQ(2)0

Fig. 10.6 Edge resultants by F = moment)

{

} Fy , Mx , Mz , B0 , Q 0 (Singe arrow: force, double arrow:

Beam 2

Shared top edge Shared side edge 2

Beam 1

Shared bottom edge

Shared side edge 1

Fig. 10.7 Two-beam joint structure (all beams are thin-walled rectangular cross-sectioned beams)

328

10 Joint Structures of Box Beams

(a)

y2

Y

Beam 2

z2

edge 2

edge 1

X

x2

edge 3 edge 4

Z

φ 2 − φ1

M1

y1 x1

z1

N2 M2 N 1 M 1′ N′2 M 2′ N′1

edge 3 edge 4

Beam 1

z2

(b) x2

X

Y

edge 1

φ2 Beam 2 y2

Z Joint

Shared side x 1 edge 1

M1

M2

N2

y1

φ1

z1

Beam 1 Shared side edge 2

N1

Fig. 10.8 Analysis model of a two-beam joint: a three-dimensional view and b top view

and }T { ξ[k] = U y[k] , θx[k] , θz[k] , W0[k] , χ0[k] ,

(10.12b)

respectively, where k is the beam index. First, we will consider equilibrium conditions among sectional resultants. Referring to the two-beam joint shown in Fig. } 10.8, we denote the sectional resultant force { and moment in Beam k as FP[k] , M P[k] (P = X, Y, Z; k = 1, 2), which act along the P-direction. Note that (X, Y, and Z) are the global coordinates. For a joint to be in equilibrium, it must satisfy the following conditions: FY[1] + FY[2] = 0

(10.13a)

M X[1] + M X[2] = 0

(10.13b)

10.3 Generalized Force Equilibrium Conditions

M Z[1] + M Z[2] = 0

329

(10.13c)

For the subsequent analysis, we use beam local coordinates (x, y, z) to express the sectional resultants as FY[k] = Fy[k]

(10.14a)

M X[k] = Mx[k] cos φk + Mz[k] sin φk

(10.14b)

M Z[k] = −Mx[k] sin φk + Mz[k] cos φk .

(10.14c)

In Eqs. (10.14b and c), φk represents the angle between the global coordinate Z and the axial coordinate z k of Beam k (see Fig. 10.8b for the positive direction). In addition to the three equilibrium equations in Eq. (10.13) for sectional resultants, we need two additional independent equations that include {B0 , Q 0 } to complete field matching at the joint because we consider five generalized forces. Because {B0 , Q 0 } are self-equilibrated, the force and moment equilibriums applied over the entire beam cross-section cannot be used to include them. Therefore, we need a new approach to accommodate these generalized forces and present a method using the edge resultants. Toward this direction, we consider equilibrium conditions along the shared edges of two beams in a two-beam joint. Referring to Fig. 10.8b, two beams meet at shared side edges 1 and 2. However, it is difficult to consider equilibrium conditions along these edges directly. Accordingly, we extend shared side edge 1 to edge M1 M1, for Beam 1 and to edge N2 N2, for Beam 2 (Fig. 10.8a explains the locations of M1, and N1, ). We then assume that edge M1 M1, of Beam 1 and edge N2 N2, of Beam 2 are connected to each other and consider the equilibrium conditions between the edge resultants. The same assumption is also applied to edge N1 N1, of Beam 1 and to edge M2 M2, of Beam 2. Based on this assumption, the following edgewise equilibrium equations can be established: [1] [2] Mn(2) + Mn(2) =0

( ) [1] [2] or Mn(4) + Mn(4) =0 ,

(10.15a)

[1] [2] Fs(1) − Fs(3) = 0,

(10.15b)

[2] [1] Fs(1) − Fs(3) = 0.

(10.15c)

In Eq. (10.15a), Mn( j ) (j = 2, 4) represents the wall-normal edge moment2 on edge j. Figure 10.6 shows that Mn( j) (j = 2, 4) is only induced by bimoment B0 as

2

Because the direction of this moment is normal to each edge, it is called the wall-normal edge moment.

330

10 Joint Structures of Box Beams B0 b Mn(2) = Mn(2) = − h (b+h) B0 ,

(10.16a)

B0 b Mn(4) = Mn(4) = − h(b+h) B0 .

(10.16b)

In Eq. (10.15b, c), Fs( j) (j = 1, 3) is the s-directional shear force on edge j, which can be written as (see Fig. 10.6) follows: F

M

Q0 y Fs(1) = Fs(1) + Fs(1)z + Fs(1) = 21 Fy + F

M

1 b+h

Q0 y Fs(3) = Fs(3) + Fs(3)z + Fs(3) = − 21 Fy +

Mz +

1 b+h

1 Q 2b 0

Mz +

1 Q 2b 0

(10.17a) (10.17b)

Equation (10.15a) represents the equilibrium condition between wall-normal moments of Beams 1 and 2 on edge 2 due to B0 . The same condition is obtained on edge 4 because Mn(2) = Mn(4) (see Eq. 10.16). Note in Fig. 10.6 that B0 also produces wall-normal moments on edges 1 and 3. However, the equilibrium conditions associated with the wall-normal moments on edges 1 {and 3 due to B0 cannot} be established because no generalized force among F = Fy , Mx , Mz , B0 , Q 0 induces z-directional wall moments on edges 1 and 3 (the counterpart moments for equilibrium), as shown in Fig. 10.6. Equation (10.15b) denotes the force equilibrium condition involving transverse bimoment Q 0 . In this equation, the Y-directional force on edge 1 of Beam 1 and that on edge 3 of Beam 2 are set to be equal, where care should be taken with regard to the sign because the positive s-direction on edge 3 is opposite to the Y-direction (see Fig. 10.3). This condition is reasonable because edge M1 M1, (i.e., edge 1 of Beam 1) and edge N2 N2, (i.e., edge 3 of Beam 2) in Fig. 10.8a are assumed to be connected to each other, as noted above. Likewise, Eq. (10.15c) expresses the equilibrium condition between the Y-directional force on edge 3 (edge N1 N1, ) of Beam 1 and that on edge 1 (edge M2 M2, ) of Beam 2. If Eq. (10.15b and c) are added, the following equation is obtained: {

} { } [1] [2] [2] [1] Fs(1) + Fs(1) − Fs(3) − Fs(3) ⎧( ⎫ ) 1 1 ⎪ ⎪ ⎨ 21 Fy[1] + b+h ⎬ Mz[1] + 2b Q [1] 0 ( ) = ⎪ ⎪ ⎩ − − 1 Fy[2] + 1 Mz[2] + 1 Q [2] ⎭ 2 b+h 2b 0 ⎧( ⎫ ) 1 1 ⎪ ⎪ ⎨ 21 Fy[2] + b+h ⎬ Mz[2] + 2b Q [2] 0 ( ) + ⎪ ⎪ ⎩ − − 1 Fy[1] + 1 Mz[1] + 1 Q [1] ⎭ 2 b+h 2b 0

(10.18)

= Fy[1] + Fy[2] = 0 This equation is identical to Eq. (10.13a), implying that one of the conditions in Eqs. (10.13a), (10.15b), and (10.15c) is redundant. From this observation, one can

10.3 Generalized Force Equilibrium Conditions

331

conclude that Eqs. (10.15b and c) provide only one independent condition for joint matching. Rather than Eqs. (10.15b, c), one may establish other equilibrium conditions involving Q 0 , as given below, because Q 0 also produces edge tangential forces on edges 2 and 4 (see Fig. 10.6): FX[1](2) + FX[2](2) = 0

( ) or FX[1](4) + FX[2](4) = 0

(10.19a)

FZ[1](2) + FZ[2](2) = 0

( ) or FZ[1](4) + FZ[2](4) = 0

(10.19b)

Here, FP( j) (P = X, Z; j = 2, 4) represents the sum of the P-directional edge forces on edge j and is written as (see Figs. 10.6 and 10.8) [k] [k] FX[k](2) = Fx(2) cos φk + Fz(2) sin φk ) ) ( ( [k] 1 3b 1 cos φ sin φk = − b+h Mz[k] + 2h Q [k] + M k x 0 h(h+3b)

(10.20a)

[k] [k] FZ[k](2) = −Fx(2) sin φk + Fz(2) cos φk ) ) ( ( 1 3b 1 sin φk + h(h+3b) = − − b+h Mz[k] + 2h Q [k] Mx[k] cos φk 0

(10.20b)

FX[k](4) = −FX[k](2)

(10.20c)

FZ[k](4) = −FZ[k](2)

(10.20d)

However, unlike Eqs. (10.15b, c), the conditions in Eqs. (10.19a, b) are incompatible with the sectional resultant equilibrium conditions given in Eq. (10.13) (especially with those in Eqs. (10.13b, c)). This can be demonstrated by considering an example case where φ1 = 60◦ and φ2 = 0. In such a case, the conditions in Eqs. (10.13b, c) and Eqs. (10.19a, b) are written as Mx[2] = − 21 Mx[1] − Mz[2] = 1 b+h

Mz[2] −

1 2h

3b M [2] h(h+3b) x

Q [2] 0 =

√ 3 Mx[1] 2

√ 3 3b M [1] 2h(h+3b) x

3b = − 2h(h+3b) Mx[1] −

−

√

3 Mz[1] 2

(10.21a)

− 21 Mz[1]

(10.21b)

1 M [1] 2(b+h) z

√ 3 M [1] 2(b+h) z

+

+

1 4h

Q [1] 0

√ 3 [1] Q0 4h

(10.21c) (10.21d)

where the conditions in Eqs. (10.21a, b) and Eqs. (10.21c, d) are obtained from Eqs. (10.13b, c) and Eqs. (10.19a, b), respectively. This example case shows that the condition in Eq. (10.21d) is generally incompatible with that in Eq. (10.21a); they are

332

10 Joint Structures of Box Beams √

3(h 2 −3b2 ) compatible with each other only when 6b(b+h) Mz[1] = (b+h)(h+3b) Q [1] 0 is satisfied, 2(h 2 −3b2 ) { } [1] [1] while Mz , Q 0 can have arbitrary values in general. Accordingly, Eqs. (10.19a, b) are not suitable to be used as the equilibrium conditions for the joint shown in Fig. 10.7. Based on the analysis above, the following equilibrium conditions can be stated for the joint in Fig. 10.7:

M X[1] + M X[2] = 0

(10.22a)

M Z[1] + M Z[2] = 0

(10.22b)

[1] [2] Mn(2) + Mn(2) =0

( ) [1] [2] or Mn(4) + Mn(4) =0

(10.22c)

[1] [2] Fs(1) − Fs(3) =0

(10.22d)

[2] [1] Fs(1) − Fs(3) =0

(10.22e)

Due to redundancy, the condition in Eq. (10.13a), FY[1] + FY[2] = 0, is omitted. The five conditions in Eq. (10.22) can be rewritten in terms of F [1] and F [2] if Eqs. (10.14), (10.16), and (10.17) are used: { {

} { } Mx[1] cos φ1 + Mz[1] sin φ1 + Mx[2] cos φ2 + Mz[2] sin φ2 = 0

(10.23a)

} { } −Mx[1] sin φ1 + Mz[1] cos φ1 + −Mx[2] sin φ2 + Mz[2] cos φ2 = 0

(10.23b)

{ } { } − h (bb+ h) B0[1] + − h (bb+ h) B0[2] = 0 { {

1 [1] F 2 y

+

1 b+h

Mz[1] +

1 Q [1] 2b 0

1 [2] F 2 y

+

1 b+h

Mz[2] +

1 Q [2] 2b 0

} }

{ − − 21 Fy[2] +

1 b+h

Mz[2] +

1 Q [2] 2b 0

{ − − 21 Fy[1] +

1 b+h

Mz[1] +

1 Q [1] 2b 0

(10.23c) } }

=0

(10.23d)

=0

(10.23e)

The equilibrium conditions in Eq. (10.22) pertain to a joint consisting of only twobox beams. They will be now extended to a joint connecting more than two-box beams in Fig. 10.9, where k is the beam index (k = 1, 2, ..., N ) numbered continuously in the Y-direction. To this end, the connectivity among the edges of N box beams at a joint should be clarified. In Fig. 10.9, edges j1 (j1 = 2, 4) of N box beams meet each } at the joint. This suggests that we should consider the equilibrium { other [k] of N box beams. In addition, we can consider the equilibrium among Mn( j) k=1, ··· ,N

[k] [k+1] between Fs(1) and Fs(3) because edge 1 of Beam k and edge 3 of Beam k + 1 are

10.3 Generalized Force Equilibrium Conditions

333

assumed{ to be } connected. Based on these arguments, the equilibrium conditions among F [k] k=1, ··· , N of N box beams (N ≥ 3) can be written by extending those in Eq. (10.22) as ΣN k=1

ΣN k=1

ΣN k=1

M X[k] = 0

(10.24a)

M Z[k] = 0

(10.24b)

( or

[k] Mn(2) =0

[k] [k+1] Fs(1) − Fs(3) =0

ΣN k=1

[k] Mn(4) =0

) (10.24c)

(k = 1, 2, · · · , N ).

(10.24d)

Equations (10.24a–d) consist of N + 3 equilibrium conditions for the case of N box beam joints. Note that Eq. (10.24d) reduces to Eqs. (10.22d, e) in the case of N = 2.

(a) xk +1

Y

yk +1 Beam k+1

X

yk −1

Nk

Z

N′k zk −1

M k −1

zk +1 edge 3

M k′ −1

yk

xk −1

edge 2 edge 1

xk

Beam k-1 edge 1

edge 3 edge 4

zk Beam k

(b) xk +1 X

Y

Joint

yk +1

Z

Nk φk −1

zk −1 Beam k-1

yk −1 xk −1

zk +1

Beam k+1

M k −1

Shared Side Edge k-1

φk +1

yk

φk

xk

zk Beam k

Fig. 10.9 a Beam modeling of a N box beam joint (N ≥ 3) and b corresponding top view

334

10 Joint Structures of Box Beams

10.4 Field Variable Matching Conditions Using the equilibrium conditions established in Sect. 10.3, we will derive the matching conditions among the field variables (ξ) of beams at a joint. To this end, the principle of virtual work will be used, noting that the ξ variables are the work conjugates of F. First, the matching conditions for the two-beam joint (Fig. 10.7) will be derived, and the results will then be extended to the case of an{ N box } beam joint (N ≥ 3) in Fig. 10.9; in theory, the matching conditions among ξ[k] k=1, ··· , N of N box beams may be derived directly from Eq. (10.24), but the derivation is too complex to employ. At the two-beam joint shown in Fig. 10.7, the principle of virtual work can be written as Σ2 k=1

)T )T ( ( δW , [k] = δ F [1] ξ[1] + δ F [2] ξ[2] = 0,

(10.25)

where δW , [k] represents the complementary virtual work for Beam k (k = 1, 2) while δ F [k] represents the admissible virtual force in Beam k. According to Eq. (10.23), δ F [1] and δ F [2] in Eq. (10.25) must satisfy the following relationship: [1] [2] M [1] + M [2] = 0, F δF F δF

(10.26a)

or ⎫ ⎤⎧ 0 cos φ1 sin φ1 0 0 ⎪ δ Fy[1] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ 0 − sin φ cos φ 0 0 ⎥ 1 1 ⎨ δ Mx[1] ⎪ ⎬ ⎥⎪ ⎢ ⎥ ⎢0 b [1] 0 0 − h(b+h) 0 ⎥ δ Mz ⎢ ⎢1 ⎪ 1 1 ⎥⎪ ⎪ ⎪ ⎦⎪ ⎣2 0 0 δ B0[1] ⎪ ⎪ ⎪ b+h 2b ⎪ ⎩ ⎭ [1] ⎪ 1 1 1 δ Q 0 − 0 − 0 2 b+h 2b ⎫ ⎧ ⎫ ⎡ ⎤⎧ 0 cos φ2 sin φ2 0 0 ⎪ ⎪0⎪ δ Fy[2] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0⎪ ⎪ ⎢ 0 − sin φ cos φ ⎪ ⎥⎪ [2] ⎪ ⎪ ⎪ 0 0 2 2 ⎨ ⎪ ⎬ ⎢ ⎥⎨ δ Mx ⎬ ⎪ ⎢0 ⎥ b [2] 0 0 − h(b+h) 0 ⎥ δ Mz + ⎢ = 0 , ⎢1 ⎪ ⎪ ⎪ 1 1 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣2 ⎦⎪ 0⎪ 0 − b+h 0 − 2b δ B0[2] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0⎪ ⎭ ⎩ [2] ⎭ 1 1 1 δ Q0 0 0 2 b+h 2b ⎡

(10.26b)

[2] where the matrices M [1] F and M F are invertible because Eq. (10.23) represents five [2] independent equilibrium conditions. Using the matrices M [1] F and M F in Eq. (10.26), Eq. (10.25) can be rewritten as

10.4 Field Variable Matching Conditions

335

Σ2

δW , [k] ( )T ( [1]T [1]−T ) [1] ( [2] )T ( [2]T [2]−T ) [2] MF MF ξ + δF MF MF ξ = δ F [1] )T ( ) ( )T ( ) ( [1]−T [1] [2] [2]−T [2] [1] [2] M + M M = 0. = M [1] δ F ξ δ F ξ F F F F k=1

(10.27)

[2] [1] Substituting the relationship M [2] = −M [1] given in Eq. (10.26a) into F δF F δF Eq. (10.27) yields

)T ( ) ( −T −T M [1] δW , [k] = M [1] δ F [1] ξ [1] − M [2] ξ [2] F F F k=1 )} ( )T { [1]T ( [1]−T [1] −T [2] = δ F [1] M F M F ξ − M [2] ξ F

Σ2

=0.

(10.28)

From Eq. (10.28), the following result can be obtained: −T

−T

M [1] ξ[1] = M [2] ξ[2] , F F

(10.29)

where we used the fact that Eq. (10.28) should be satisfied for an arbitrary δ F [1] and [1]T the fact that { [1]M F [2]is } invertible. Equation (10.29) represents the matching conditions when the equilibrium conditions in Eq. (10.22) are met at the between ξ , ξ two-beam joint shown in Fig. { 10.7. } [2] Using the definitions of M [1] given in Eq. (10.26), Eq. (10.29) can be F , MF explicitly written as: [1] [2] [2] ⊝[1] x cos φ1 + ⊝z sin φ1 = ⊝x cos φ2 + ⊝z sin φ2

(10.30a)

[1] [2] [2] −⊝[1] x sin φ1 + ⊝z cos φ1 = −⊝x sin φ2 + ⊝z cos φ2

(10.30b)

[2] ⊝[1] n( j1 ) = ⊝n( j1 )

( j1 = 2, 4)

(10.30c)

[1] [2] Us(1) = −Us(3)

(10.30d)

[1] [2] −Us(3) = Us(1)

(10.30e)

{ } where ⊝x , ⊝z , ⊝n(2) , ⊝n(4) , Us(1) , Us(3) are defined as ⊝x = θx ⊝z = θz −

2b χ b+h 0

(10.31a) (10.31b)

336

10 Joint Structures of Box Beams

⊝n(2) = ⊝n(4) = − h(b+h) W0 b

(10.31c)

Us(1) = U y + bχ0

(10.31d)

Us(3) = −U y + bχ0

(10.31e)

Equations (10.30a, b) represent the continuity conditions among the work conjugates of {Mx , Mz }. These equations can be found from the counterpart equilibrium conditions given in Eqs. (10.23a, b). Accordingly, ⊝[k] p (p = x, z; k = 1, 2) can be understood as the p-directional sectional effective rotation of Beam k at the joint, as illustrated in Fig. 10.10a, b. From this observation, Eqs. (10.30a, b) can be rewritten as [2] ⊝[1] X = ⊝X ,

(10.32a)

[2] ⊝[1] Z = ⊝Z ,

(10.32b)

where ⊝[k] P (P = X, Z; k = 1, 2) denoting the sectional effective rotation of Beam k in the P-direction is explicitly written as

(a)

(b)

yk

yk xk

] Θ[nk(2) = − h (bb+h ) W0[ k ]

(c)

xk

Θ[xk ] = θ x[ k ]

zk

Θ[zk ] = θ z[ k ] − b2+bh χ 0[ k ]

zk

(d) yk

yk xk

xk

zk [k ] n (4)

Θ

[k ] n (2)

=Θ

U s[ k(1)] zk

= U [yk ] + b χ 0[ k ]

U s[ k(3)] = −U [yk ] + b χ 0[ k ]

Fig. 10.10 Sectional/edge displacements of Beam k (k = 1, · · · , N ) associated with the field variable matching conditions: a sectional effective rotation ⊝[k] x in the x k direction, b sectional [k] effective rotation ⊝[k] x in the z k direction, c edge rotation ⊝n(2) of edge 2 in the yk direction and [k] edge rotation ⊝[k] n(4) of edge 4 in the −yk direction, d edge displacement Us(1) of edge 1 in the yk [k] direction and edge displacement Us(3) of edge 3 in the −yk direction

10.4 Field Variable Matching Conditions

337

[k] [k] ⊝[k] X = ⊝x cos φk + ⊝z sin φk ,

(10.33a)

[k] [k] ⊝[k] Z = −⊝x sin φk + ⊝z cos φk .

(10.33b)

For the same reason, Eq. (10.30c) represents{the continuity }condition between the [1] [2] work conjugates of the normal edge moments Mn( j1 ) , Mn( j1 ) in Eq. (10.22c). Thus, in Eq. (10.30c), ⊝[k] n( j1 ) (k = 1, 2; j 1 = 2, 4) represents the rotation at edge j 1 of Beam k in the normal direction (see Fig. 10.10c). Lastly, Eqs. (10.30d, e) represent the continuity conditions between the work conjugates of the tangential edge forces Fs([k]j2 ) (k = 1,2; j2 = 1, 3) in Eqs. (10.22d, e). Thus, Us([k]j2 ) used in Eqs. (10.30d, e) represents the s-directional displacement in edge j2 of Beam k, as depicted in Fig. 10.10d. With this interpretation, Eqs. (10.30d, e) are shown to represent the continuity conditions with respect to the Y-direction. Note that when interpreting Eqs. (10.30d, e), we should be careful about the signs because the positive s-directions are opposite to each other at edges 1 and 3 (Fig. 10.2). The aforementioned matching conditions can be extended to the N beam joint (N ≥ 3) shown in Fig. 10.9 as follows: [2] [N ] ⊝[1] X = ⊝X = · · · = ⊝X

(10.34a)

[2] [N ] ⊝[1] Z = ⊝Z = · · · = ⊝Z

(10.34b)

[2] [N ] ⊝[1] n( j1 ) = ⊝n( j1 ) = · · · = ⊝n( j1 ) [k] [k+1] Us(1) = −Us(3)

( j1 = 2, 4)

(1 ≤ k ≤ N )

(10.34c) (10.34d)

Equations (10.34a, b) represent the continuity conditions for the sectional effective rotations of N box beams in the X- and Z-directions, respectively. Equation (10.34c) represents the continuity condition for edge j1 ’s normal rotations of N box beams. Equation (10.34d) represents the continuity condition between edge 1’s Y-directional displacement of Beam k and edge 3’s Y-directional displacement of Beam k + 1 (1 ≤ k ≤ N ; Beam N + 1 refers to Beam 1.) Therefore, Eqs. (10.34a–d) express the 4N − 3 independent matching conditions; each equation from among Eqs. (10.34a– c) states N − 1 independent conditions, while Eq. (10.34d) expresses N independent conditions. For example, the joint matching conditions for a three-beam joint are obtained via [2] ⊝[1] X = ⊝X ;

[3] ⊝[2] X = ⊝X

(10.35a)

[2] ⊝[1] Z = ⊝Z ;

[3] ⊝[2] Z = ⊝Z

(10.35b)

338

10 Joint Structures of Box Beams [2] ⊝[1] n( j1 ) = ⊝n( j1 ) ; [1] [2] Us(1) = −Us(3) ;

[3] ⊝[2] n( j1 ) = ⊝n( j1 ) [2] [3] Us(1) = −Us(3) ;

( j1 = 2, 4)

(10.35c)

[3] [1] Us(1) = −Us(3)

(10.35d)

Note that when N = 2, Eq. (10.34) reduces to Eq. (10.30).

10.5 Verification with Numerical Examples Using the joint matching conditions in Eq. (10.34), this section will establish a higherorder beam analysis for N (N ≥ 2) box beam joint structures subjected to out-ofplane bending and torsion. To check the validity of the matching conditions presented in this chapter, the results obtained by the HoBT method with the matching conditions used will be compared with those from Abaqus shell analysis and Timoshenko beam analysis.

10.5.1 Finite Element Equations The one-dimensional for a box beam element (z 1 ≤ z ≤ z 2 ) } { finite element equation considering ξ = U y , θx , θz , W0 , χ0 can be written according to the procedure given in Chap. 3: K ·d = f

(10.36)

In Eq. (10.36), K, d, and f represent the 10 × 10 stiffness matrix, 10 × 1 nodal displacement vector, and 10 × 1 nodal force vector, respectively, which are explicitly written as { d = U y (z 1 ), θx (z 1 ), θz (z 1 ), W0 (z 1 ), χ0 (z 1 ), (10.37a) }T U y (z 2 ), θx (z 2 ), θz (z 2 ), W0 (z 2 ), χ0 (z 2 ) , { f = Fy (z 1 ), Mx (z 1 ), Mz (z 1 ), B0 (z 1 ), Q 0 (z 1 ), (10.37b) }T Fy (z 2 ), Mx (z 2 ), Mz (z 2 ), B0 (z 2 ), Q 0 (z 2 ) , ] [ K11 K12 , (10.37c) K= KT12 K22 where {K11 , K12 , K22 } are given by

10.5 Verification with Numerical Examples

⎡ G JU

K 11

339

GJ

y − Uy ⎢ LGe JU y G L e J2U y ⎢− 2 + 3 ⎢ =⎢ 0 ⎢0 ⎢ ⎣0 0 0 0

0 E Jθx Le

0 G Jθz Le (b−h)G J − 2(b+h)θz

0

0 0 (b−h)G J − 2(b+h)θz

(b−h)2 G L e Jθz 3(b+h)2 G Jχ0 − 2

⎡

K 12

GJ

− Uy ⎢ G JULy e ⎢ ⎢ 2 =⎢ ⎢0 ⎢ ⎣0 0

+

G L e Jχ0 3

G JU y 2 G L e JU y 6

E 1 JW0 Le

+

−

G Jχ0 2 E 1 L e C1 3

−

−

0 GJ − L eθz 0 0

0 0 (b−h)G J − 2(b+h)θz

G L e Jχ0 6

+

y G JU y Le 2 G JU y G L e JU y 2 3

K 22

−

E 1 JW0 G Jχ0 Le 2 E 1 L e C1 6

G J +C − ( χL0 e 2 )

+

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 G Jθz Le (b−h)G Jθz 2(b+h)

0

0 0

⎤

0 0 0 +

(10.38b)

0 E Jθx Le

0 0 0

(b−h)G Jθz 2(b+h) (b−h)2 G L e Jθz 6(b+h)2 G Jχ0 2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

0 0 0

⎡ G JU

⎢ ⎢ ⎢ =⎢ ⎢0 ⎢ ⎣0 0

G J +C + ( χL0 e 2 )

(10.38a)

0 E Jθx Le

0 0 0

(b−h)2 G L e Jθz 6(b+h)2 G Jχ0 − 2

⎤

0 0 0

G L e Jχ0 6

+

E 1 JW0 G Jχ0 Le 2 E 1 L e C1 3

G J +C + ( χL0 e 2 )

(10.38c)

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

In Eq. (10.38), L e (= z 2 − z 1 ) is the element length, and{ the values of {C1 , C2 } are } defined as (see Eq. (10.6) for the explicit expressions of JU y , Jθx , Jθz , JW0 , Jχ0 ): ∫ C1 =

(

n ψ¨ nχ0

)2

dA =

8t 3 b+h

(10.39a)

340

10 Joint Structures of Box Beams

∫ C2 =

( )2 2n ψ˙ nχ0 d A =

2t 3 (b2 +4bh+h 2 ) 15(b+h)

(10.39b)

Using the standard assembly procedure, the finite element equations for an N box beam joint structure (but not including the matching conditions of Eq. (10.34)) can be written as Ktotal · dtotal = ftotal ,

(10.40)

where Ktotal , dtotal , and ftotal denote a 5n × 5n stiffness matrix, a 5n × 1 nodal displacement vector, and a 5n × 1 nodal force vector, respectively (n: number of nodes). To impose the matching conditions in Eq. (10.34) when solving the system matrix Eq. (10.40), the method of Lagrange multipliers3 is used. First, we write Eq. (10.34) as an equality constraint equation: S · dtotal = 0,

(10.41)

where S is a (4N − 3) × (5n) matrix, as Eq. (10.34) represents 4N − 3 independent conditions. To this end, we note that Eq. (10.40) can be viewed as the stationarity condition of a functional4 ⊓(d total ) = 21 d Ttotal K total d total − d Ttotal f total : / ⊓ ∂ d total = 0 → K total d total − f total = 0 . Because S · dtotal should be always zero, ⊓(d total ) will not change even if S · d total is added to it. With this observation, one can replace ⊓ with a new functional called the Lagrangian ⊓ L , T T ⊓ L (dtotal , λ) = 21 dtotal Ktotal dtotal − dtotal ftotal + λT (S · dtotal ),

(10.42)

where λ is called the Lagrange multiplier (vector). For ⊓ L in Eq. (10.42), the stationarity conditions must be considered for both d total and λ. Imposing the stationarity of ⊓ L yields / ∂ ⊓ L ∂ dtotal = 0 → Ktotal dtotal − ftotal + λT S = 0, / ∂ ⊓ L ∂ λ = 0 → S · dtotal = 0.

(10.43a) (10.43b)

In matrix form, Eqs. (10.43a, b) can be compactly written as [

3

Ktotal ST S 0

]{

dtotal λ

}

{ =

} ftotal . 0

(10.44)

Lagrange multipliers are generally used to solve an optimization problem by identifying the maximum or minimum value of a functional subjected to equality constraints. 4 See the paragraph below Eq. (3.10) in Chap. 3.

10.5 Verification with Numerical Examples

341

One can determine dtotal (and λ) of Eq. (10.44) for the given boundary and loading conditions. Because the solving procedure is standard, no further discussion would be necessary.

10.5.2 Case Study 1: Two-Box Beam Joint System For all examples including this example, unless otherwise specified, the beam length and wall thickness are set to L = 1 m and t = 0.002 m, respectively, and the material properties are Young’s modulus E = 200 GPa and Poisson’s ratio ν = 0.3.. Because the joint flexibility is highly dependent upon the number of connected box beams (i.e., N), the joint angles (i.e., {φk }k=1,··· ,N ), and the aspect ratio of the beam cross-section (i.e., h/b; see Fig. 10.2), structural responses with different values of {N , φ, b, h} will be examined. For each beam member, 160 HoBT finite elements are employed. It was confirmed that converged results were obtained with this number of finite elements for the cross-sections considered in this section. For the Abaqus shell analysis, square shell elements of 6.25 × 6.25 mm are used. The two-beam joint structure in Fig. 10.11a is investigated. One end of the structure (denoted as A) is clamped, and the other end B is subjected to vertical force Fy = 100 N. The beam cross-section at end B is assumed to be rigid such that { W0 = 0 and χ0} = 0 at end B. The axial distributions of the field variables U y , θx , θz , W0 , χ0 due to the applied force are plotted in Fig. 10.11b–f, respectively; specifically, the distributions of Beam k’s field variables (k = 1, 2) along the z k direction are plotted from end A to end B. In Fig. 10.11, the results by the higher-order beam theory are denoted as HoBT, while those by the Abaqus shell analysis and Timoshenko beam theory are denoted as shell and TBT, respectively. For comparison, the results by Jang et al. (2008) are also plotted; Jang et al. (2008) employed the same higher-order beam theory but used different matching conditions obtained from a condition directly minimizing the mismatch in the joint displacements. Figure 10.11b–f show the effects of the joint matching condition presented in this chapter; the HoBT results based on the matching condition of this chapter are virtually identical to those by the shell analysis, while for the others, this is not the case.

10.5.3 Case Study 2: T-Joint System The T-joint structure in Fig. 10.12a is a special case of a three-beam joint problem. Beam 1 and Beam 3 are placed parallel to the Z-direction and φ2 = 90◦ (φ2 : joint angle of Beam 2). The ends of Beams 1 and 3 are fixed, and the end of Beam 2 (denoted as B) is subjected to vertical force Fy = 100 N. The beam section at end B is assumed to be rigid such that no sectional deformation occurs.

342

10 Joint Structures of Box Beams

(a)

(b)

X F

[2] y

Jang et al. (2008)

= 100N

Shell

z2 x2

HoBT

L2

φ2 z1

z2

Z

x1

TBT

L1

(c)

(d)

Jang et al. (2008) Shell HoBT

TBT

HoBT TBT Jang et al. (2008)

(e)

Shell

(f)

Jang et al. (2008) TBT

HoBT Shell

TBT

Shell Jang et al. (2008)

HoBT

Fig. 10.11 Numerical results for a two-box beam joint under vertical force Fy = 100 N: a problem description (L = 1000 mm, b = 50 mm, h = 100 mm, t = 2 mm, and φ2 = 150◦ ), b vertical bending deflection U y , c bending/shear rotation θx , d torsional rotation θz , e warping W0 , and f distortion χ0 (Choi et al. 2012)

Figure 10.12b–f show the variations of the vertical bending deflection U y , bending/shear rotation θx , torsional rotation θz , warping W0 , and distortion χ0 along the relevant lengthwise direction for every beam from k = 1 to k = 3. For instance, the horizontal axis from k − 1 and k in the figures denotes the axial range between 0 ≤ z k ≤ L k where z k is the axial coordinate of Beam k and L k denotes the length of Beam k(k = 1, 2, 3). In the figures, the results by a displacement matching method in Jang et al. (2013) are also plotted for comparison. Compared to the Abaqus shell

10.5 Verification with Numerical Examples

(a)

343

(b)

X F

[2] y

x2 z2

HoBT

= 100N

Shell

z2 L2

x1

φ2 z3

Jang et al. (2013)

z1

Z

x3

TBT

L1

L3

(c)

(d) HoBT Shell HoBT , Shell

Jang et al. (2013)

TBT

Jang et al. (2013) TBT

(e)

(f) HoBT, Shell TBT

Jang et al. (2013)

Jang et al. (2013)

TBT

HoBT, Shell

Fig. 10.12 Numerical results for a T-joint structure under vertical force Fy = 100 N: a problem description (L = 1000 mm, b = 100 mm, h = 50 mm, t = 2 mm, and φ2 = 90◦ ), b vertical bending deflection U y , c bending/shear rotation θx , d torsional rotation θz , e warping W0 , and f distortion χ0 (Choi and Kim (2016a))

results, the Timoshenko beam analysis overestimates the stiffness of the T-joint. In contrast, the HoBT-based analyzes (i.e., HoBT analyzes using the matching conditions in this chapter and in Jang et al. (2013)) provide more accurate results because the influence of the higher-order deformation on the joint flexibility is considered. Specifically, the HoBT approach, which employs the matching conditions in Eq.

344

10 Joint Structures of Box Beams

(a)

(b) HoBT

HoBT

Shell

Shell

TBT

TBT

Fig. 10.13 Numerical results for a T-joint with a different sectional height-to-width ratios (h/b) and b different value of φ2 (Choi and Kim (2016a))

(10.34), captures the responses of the T-joint nearly as accurately as does the Abaqus shell analysis. The structural responses of the T-joint in Fig. 10.13a are analyzed for different (b, h) and different φ2 values. Figure 10.13a shows the vertical tip displacement (or U y[2] ) at end B for various sectional height-to-width ratios, h/b (50 mm ≤ h, b ≤ 200 mm). In addition, the plots in Fig. 10.13b are the results for different joint angles, φ2 (b = 50 mm and h = 100 mm). The results in both Fig. 10.13a and b show that the HoBT method using the matching conditions given in this chapter provides accurate results comparable to those by shell elements for box beam joint structures with various sectional aspect ratios and joint angles.

10.5.4 Case Study 3: N Box Beam-Joint System Figure 10.14a illustrates a box beam joint structure consisting of eight ◦ beam ( (members / ) connected to each other ) at a uniform joint angle of 45 φk = 360 8 × (k − 1), k = 1, 2, ..., 8 . The end of Beam 1, denoted as A, is set to be rigid and is subjected to twisting moment Mz[1] = 100 N · m, while the other ends are clamped. Figure 10.14b–f show the variations of vertical bending deflection U y , bending/shear rotation θx , torsional rotation θz , warping W0 , and distortion χ0 along the lengthwise direction in this case for every beam from k = 1 to k = 8. The definition of the horizontal axis in the figure is identical to that in Fig. 10.12b–f. The numerical results in these figures show that the HoBT method using the matching conditions given in this chapter yields results nearly as accurate as those by the shell analysis, despite the fact that the number of joining beam members is significantly increased. Figure 10.15 shows the variation of the torsional rotation θz calculated at the end of Beam 1 when the number N of joining beams changes in a beam arrangement

10.5 Verification with Numerical Examples

345

X

(a)

(b) HoBT, Shell

x3

z3 x2

z4 x4

z5

z2

x1 z5

[1] z1 M z = 100N ⋅ m

Z

x5

z6

TBT

x8 z8

x6 z7

x7

(c)

(d) HoBT Shell

TBT

TBT Shell HoBT

(e)

(f) Shell HoBT HoBT TBT

Shell

TBT

Fig. 10.14 Numerical results for an eight-box beam joint structure under torsional moment Mz = 100 N(· m:/a )problem description (L = 1000 mm, b = 100 mm, h = 50 mm, t = 2 mm, and φk = 360 8 ×(k − 1) (k = 1, · · · , 8)), b vertical bending deflection U y , c bending/shear rotation θx , d torsional rotation θz , e warping W0 , and f distortion χ0 (Choi and Kim (2016a))

similar ( / to) that in Fig. 10.14a. In this case, the beam angle is expressed as φk = 360 N × (k − 1) (k = 1, 2, ..., N ). As expected, good agreement between the HoBT results and those from a shell analysis is shown in the plot.

346 Fig. 10.15 Torsional rotation θz calculated at the end of Beam 1 for a box beam joint structure similar to that shown in Fig. 10.14a with a different number N of joining beams (Choi and Kim (2016a))

10 Joint Structures of Box Beams

HoBT

Shell

TBT

References Basaglia C, Camotim D, Silvestre N (2012) Torsion warping transmission at thin-walled frame joints: kinematics, modelling and structural response. J Constr Steel Res 69(1):39–53 Basaglia C, Camotim D, Coda HB (2018) Generalised beam theory (GBT) formulation to analyse the vibration behaviour of thin-walled steel frames. Thin-Walled Struct 127:259–274 Choi S, Jang G-W, Kim YY (2012) Exact matching condition at a joint of thin-walled box beams under out-of-plane bending and torsion. J Appl Mech 79(5):051018 Choi S, Kim YY (2016a) Exact matching at a joint of multiply-connected box beams under out-ofplane bending and torsion. Eng Struct 124:96–112 Choi S, Kim YY (2016b) Analysis of two box beams-joint systems under in-plane bending and axial loads by one-dimensional higher-order beam theory. Int J Solids Struct 90:69–94 Donders S, Takahashi Y, Hadjit R, Langenhove TV, Brughmans M (2009) A reduced beam and joint concept modeling approach to optimize global vehicle body dynamics. Finite Elem Anal Des 45(6–7):439–455 Jang GW, Kim KJ, Kim YY (2008) Higher-order beam analysis of box beams connected at angled joints subject to out-of-plane bending and torsion. Int J Numer Meth Eng 75(11):1361–1384 Jang G-W, Kim YY (2009) Higher-order in-plane bending analysis of box beams connected at an angled joint considering cross-sectional bending warping and distortion. Thin-Walled Struct 47(12):1478–1489 Jang G-W, Choi S, Kim YY (2013) Analysis of three thin-walled box beams connected at a joint under out-of-plane bending loads. ASCE J Eng Mech 139(10):1350–1361 Mundo D, Hadjit R, Donders S, Brughmans M, Mas P, Desmet W (2009) Simplified modelling of joints and beam-like structures for BIW optimization in a concept phase of the vehicle design process. Finite Elem Anal Des 45(6–7):456–462

Chapter 11

Joint Structures of Thin-Walled Beams with General Section Shapes

In Chap. 10, we derived joint-matching conditions for a box beam by considering the equilibrium conditions of sectional and edge resultants, which are represented in terms of generalized forces. The principle of virtual work was used to derive the matching conditions in analytic form for the field variables. For a cross-section with a general shape, edge resultants may also be defined by following a procedure similar to that of a box beam. However, it is difficult to establish the matching conditions in analytic form because the relative positions of the matching edges of joining beams can be neither parallel nor intersecting. Furthermore, sectional edges are not always one-to-one matched when beams of different sectional shapes are connected at a joint. Therefore, an alternative method which does not explicitly deal with edge resultants should be developed. In this chapter, we directly impose the continuity of displacements at a joint of multiple thin-walled beams with different cross-sections to find the joint-matching conditions among the field variables of the beams involved. We will refer to this new approach as a displacement-based joint-matching method to distinguish it from the resultant-based method in Chap. 10. This displacement-based joint-matching method, although it involves some approximations, is much easier to implement than the resultant-based method, and it can be applied universally to any thin-walled crosssection types, i.e., open/closed, flanged/non-flanged, and single-cell/multicell types. Figure 11.1 illustrates the specific joint types that we consider in this chapter, i.e., L- and T-type joints. The L-type joint consists of two beams with the same sectional shape that are connected in series. For a T-type joint, one beam (a connected member) is connected to the side of another beam (a penetrating member). The cross-sectional shapes of the two beams of the T-type joint do not have to be identical. The joint-matching conditions in this chapter are defined using the continuities of displacements and rotations at a joint. However, it should be noted that the beam end sections of a one-dimensional beam model, denoted by S A and S B in Fig. 11.2a in the case of the L-type joint, are generally directly matched. Here, the beam end sections, S A and S B , are sections that are perpendicular to the longitudinal axes (or beam axes) of Beam A and Beam B, respectively. Because the beam axes are not parallel © Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5_11

347

348

11 Joint Structures of Thin-Walled Beams with General Section Shapes

(a)

L-type joint

One-dimensional model

Beam A Joint

Beam B

φ (b)

YA XA ZA

Joint

φ

YB ZB

T-type joint Beam A

One-dimensional model YA ZA

Joint

φ

XB

Beam B

Joint

XA

φ

XB

YB ZB

Fig. 11.1 Joint types and the corresponding one-dimensional models: a L-type and b T-type beamjoint structures

to each other, it is not possible directly to impose the continuities of displacements and rotations defined at the end sections (S A and S A ). To resolve this difficulty, we introduce a virtual intersection plane called a joint section, initially proposed by Jang et al. (2008); Jang and Kim (2009a, 2009b, 2010). The (blue) dotted regions denoted as S AB in Fig. 11.2a and b and b represent the joint sections of L-type and T-type joints, respectively. Because the joint section is defined as a virtual plane, the displacements and rotations on the joint section cannot be defined directly using the beam DOFs. To overcome this difficulty, Sect. 11.2 introduces certain kinematic assumptions that allow us to calculate the displacements and rotations on the joint section in terms of the field variables defined on the end sections of the beams forming the joint. The implementation details will be discussed in Sect. 11.3. The validity of the proposed joint connection conditions is presented in Sect. 11.4 by solving several L- and T-type joint structures, including a simplified vehicle frame. The results obtained using the proposed one-dimensional model are compared to those of shell elements as well as other one-dimensional models.

11.1 Joint Section and Connection Points The HoBT (higher-order beam theory) approximates the three-dimensional displacements of a thin-walled beam in terms of the DOFs of generalized 1D displacements. All field variables in the HoBT are defined on the cross-section of a beam that is

11.1 Joint Section and Connection Points

349

(a)

YA

ϕ

ϕ/2

Beam A

Beam B XA

Joint

ϕ

Beam B

ZB Beam B

SB : End section of beam B Joint axis

YA ZA

ϕ/2

YB

SAB : Joint section

ZA

Beam A

XB

SAB : Joint section

SA : End section of beam A

(b)

Joint axis

Joint axis

Beam A

XA

Joint axis

ϕ−90 Beam B

Joint

SB :

Beam A

SAB : Joint section

SAB : Joint section

End section of beam B XB

YB ZB

Fig. 11.2 Joint sections of a L-type and b T-type joints

perpendicular to its longitudinal axis of the beam. However, the DOFs of generalized 1D displacements cannot be directly defined on the intersecting cross-section of two beams at an angle joint. For instance, consider the L-type joint shown in Fig. 11.2a. In this case, the end section of Beam A and that of Beam B, denoted correspondingly as S A and S B , are the sections on which the field variables of each beam are defined. However, they are not coincident because two beams meet at a certain angle. Therefore, the relationship between the DOFs on S A and those on S B should be found to analyze a system of beams meeting at a joint of an arbitrary angle. The derived relationship should also be reflected in the finite element model in the assembly process of the system stiffness and mass matrices. (Refer to Chap. 10 to see why the joint-matching technique used for the Euler–Bernoulli and Timoshenko beam theories cannot be used when the HoBT is used.) Our approach to match the field variables at a joint considers the continuities of displacements and rotations on the joint section sketched in Fig. 11.2. Figures 11.2a and b illustrate how the joint sections are defined for L- and Ttype joints, respectively. For the L-type joint, the joint section, denoted as S AB in Fig. 11.2a, is defined as the natural intersection of two beams meeting at the joint. In Fig. 11.2a, the joining beams meeting at an L-type joint are assumed to have the same cross-section. In this case, the aspect ratio of the joint section is not identical to those of the end sections. Here, we introduce the joint axis which passes through the centroid of end sections S A and S B . Because the centroids of the end sections of the two joining beams are assumed to be coincident, the intersection points of the joining beams lie on the joint axis. It is also assumed that no stiffening member, such

350

11 Joint Structures of Thin-Walled Beams with General Section Shapes

as a diaphragm, is installed on the joint section of an L-type joint; i.e., the L-type joints here are assumed to be a hollow joint. For the T-type joint in Fig. 11.2b, the joint section S AB is defined as the intersection between the penetrating member (Beam A) and the connected member (Beam B). Unlike the L-type joint, the cross-sections of the joining beams of the T-type joint do not have to have the same geometry. It can be assumed that the end of Beam B is welded onto the sidewall of Beam A; because the sidewall area of Beam A holding the joint section acts as a diaphragm in this case, the joint has high rigidity against sectional distortion. To establish the joint conditions, the continuities of the displacements and rotations of the joining beams are considered at the “joint connection points, indicated by the red solid circles in Fig. 11.3, which lie on the joint section. Note that the displacements and rotations at a joint connection point cannot be directly calculated using Eq. (9.2) because the point is located on the virtual section of the joint. For example, Q ∗4 in Fig. 11.3a is a virtual point at which the displacement cannot be calculated by Eq. (9.2). Therefore, the displacements and rotations at a joint connection point should be approximated using those on the end sections. The corresponding points on the end sections, or the end connection points, are obtained by projecting the joint connection point on the end sections in the axial directions of the beams, as indicated by the squares or hollow circles in Fig. 11.3. For example, Q 4A and Q 4B in Fig. 11.3a are the end connection points, which are introduced to evaluate the displacements and rotations at joint connection point Q ∗4 . In Fig. 11.3a, the end connection points of the L-type joint are defined at the vertices of the end sections. The intersection points of the end sections on the joint axis are also selected as the end connection points. Because the intersection points lie on the joint section, they belong to the joint connection points as well. In addition to the end connection points shown in Fig. 11.3a, more points on the end sections can be selected as connection points, which may affect the joint stiffness of the one-dimensional model; the effect of the number of the end connection points on accuracy of the solution will be examined in Sect. 11.4.1 through numerical tests. In Fig. 11.3, the end connection points are grouped into “shared” and “unshared” points. Shared points, indicated by the squares in the figure, are shared by multiple sectional walls (or edges) while unshared points, denoted as circles in the figure, lie only on a single sectional wall (or edge). Because the rotations at a shared point can be calculated differently depending on the edges, some approximation or treatment would be needed. On the other hand, the displacements are uniquely calculated at a shared point because the sectional shape functions are defined as continuous on any cross-section. The rotations at an unshared point are calculated using the sectional shape functions of the corresponding edge. More detailed evaluations of the displacements and rotations at the joint connection points are provided in Sect. 11.2. For the T-type joint shown in Fig. 11.3b, the joint section lies on the sidewall of Beam A. In the figure, all of the joint connection points are on the sidewall of Beam A. In this case, the displacements and rotations of Beam A at the joint connection points can be calculated using Eq. (9.2). To facilitate the calculation, one can put the nodes for 1D higher-order finite element analysis at the sections indicated by

11.2 Kinematics of a Joint Section

(a)

351

(b)

Joint axis

Joint axis SA4

Joint connection point Unshared end connection point Shared end connection point

Centroid Y X

Intersection of Beams A and B

SA2

SA3

SA1

Beam B

Z Beam A SB Beam A

Joint axis Beam B

SAB

SAB SB

SA

Q4A

Q4∗

Q4B

Fig. 11.3 Joint connection points and end connection points for a an L-type joint and b a T-type joint

S A1 , S A2 , S A3 and S A4 because the nodal DOFs (ξ) of Beam A will be available at the nodes.

11.2 Kinematics of a Joint Section 11.2.1 Rotations Calculated on a Joint Section For the joining beams of an L-type joint and the connected member of a T-type joint (Beam B in Fig. 11.3b), displacements and rotations at the joint section cannot be directly obtained using the relationships expressed by Eq. (9.2). To evaluate them using 1D field variables of the HoBT, it is assumed that a joint connection point is connected to its corresponding point defined on the end section of a joining beam through a virtual bar, as illustrated in Fig. 11.4. ( ) Figure 11.4 shows that a joint connection point, Q ∗p p = 1, 2, . . . , N Q ; N Q : number of joint connection points), is assumed to be connected to the end connection point of a beam, Qp , through a virtual bar. With this virtual connection, we assume that the rotations at the end connection point of a joining beam are identical to those at the joint connection point via the concept of a virtual bar. Therefore, the following relationship is used: {⊝ X , ⊝Y , ⊝ Z } Q ∗p = {⊝ X , ⊝Y , ⊝ Z } Q p ,

(11.1)

where ⊝ denotes a rotation angle in the direction of a given subscript, and the symbols (X, Y, Z) denote the local beam coordinates. The rotation angles at a point on the end

352

11 Joint Structures of Thin-Walled Beams with General Section Shapes

(a)

(b)

Q1∗

Virtual bar

ΘY

Q1

ΘX

Q2 , Q2∗

Q3

Qp

∗ 3

Q

Y

uYadd = −rp Θ X

X Y

Virtual bar

Z

X

Q5 Q4

Virtual bar

Z

Q5∗ Q4∗

rp Virtual bar

u Xadd = rp ΘY Q

∗ p

( p=1, ···, NQ ; NQ : number of connection points)

Fig. 11.4 a Illustrations of the joint point on the joint section and end connection points on the joint section and b displacement mapping using the concept of a virtual bar

section of a joining beam can be calculated by differentiating the displacements: ⊝z = −

∂u n ; ∂s

∂u z ; ∂s ∂u n ⊝s = , ∂z

⊝n =

(11.2)

which can be rewritten in a matrix–vector form using Eq. (9.2), as ⎧ ⎫ ⎡ ⎡ ⎤⎧ ⎫ ⎤⎧ ⎫ 0 −1 0 ⎨ u˙ z ⎬ 0 0 0 ⎨ u ,z ⎬ ⎨ ⊝z ⎬ =⎣ 1 0 0 ⎦ u˙ n + ⎣ 0 0 0 ⎦ u ,n ⊝ ⎩ n⎭ ⎩ ⎭ ⎩ ,⎭ ⊝s Q p u˙ s Q p us Q p 0 0 0 010 ( ) ,( ) ( ) ( ) =Q1 ψ˙ s p ξ z p + Q2 ψ s p ξ z p ,

(11.3)

. ]T / / [ where ( ) = ∂( ) ∂s, ( ), = ∂( ) ∂z, and ψ = ψTz , ψTn , ψTs is the matrix of the sectional shape functions. In Eq. (11.3), Q1 and Q2 are the selection matrices, ξ is the DOF vector, and sp and zp are the s- and z-coordinates of Qp . The rotations in Eq. (11.3) can be expressed in terms of the local beam coordinates (X, Y, Z):

⎧ ⎫ ⎫ ⎡ ⎤⎧ 0 sin α j cos α j ⎨ ⊝z ⎬ ⎨ ⊝X ⎬ =⎣ 0 − cos α j sin α j ⎦ ⊝n ⊝ ⎩ ⎩ Y⎭ ⎭ ⊝Z Q 1 0 0 ⊝s Q p p ( ) ,( ) ( ) ( ) ˙ =T1 Q1 ψ s p ξ z p + T1 Q2 ψ s p ξ z p ,

(11.4)

11.2 Kinematics of a Joint Section

353

where T1 is a coordinate transformation matrix. In Eq. (11.4), α j is the angle of cross-section edge j on which Qp lies (see Fig. 9.1). Using Eq. (11.1), the rotations at the joint connection point are expressed as ⎧ ⎧ ⎫ ⎫ ⎨ ⊝X ⎬ ⎨ ⊝X ⎬ = ⊝Y ⊝ ⎩ ⎩ Y⎭ ⎭ ⊝ Z Q∗ ⊝Z Q p p

⎧ ( )⎫ [ ( )] ξ zp ( ) ( ) =T1 Q1 ψ˙ s p Q2 ψ s p ξ, zp ⎧ ( )⎫ ξ z =R p1 , ( p ) , ξ zp

(11.5)

where Rp1 is the joint rotation matrix. When calculating the displacements and rotation at a shared end connection point, it should be noted that the rotation angles are not uniquely defined because ⊝X and ⊝Y can be calculated differently on each connected edge of the point if they are calculated using Eq. (11.4). To resolve this difficulty, we propose to exclude ⊝s while retaining the contribution of ⊝n when we calculate ⊝X and ⊝Y at a shared end connection point. This approximation is reasonable because the effect of ⊝n on the structural stiffness is much greater than that of ⊝s . In Eq. (11.2), ⊝s is given as the z-directional derivative of un , obtained as the result of wall bending and thus represents the local wall-bending stiffness at a given point. On the other hand, ⊝n is obtained as the s-directional derivative of uz in Eq. (11.2), which generates the local membrane stiffness. Because the membrane stiffness of a thin plate is significantly greater than the bending stiffness, considering only ⊝n for the calculation of ⊝X and ⊝Y at a shared end connection point is a justifiable approximation for the joint analysis. For a connection point shared with more than two sectional edges, it can be shown that ⊝X and ⊝Y can be uniquely calculated using ⊝n of any two edges (see Appendix). Accordingly, ⊝X and ⊝Y at a shared end connection point Qp are calculated as [ ]⎧ ⎫ ⎧ ⎫ 1 cos α j2 − cos α j1 ⊝n( j1) ⊝X ) = ( ⊝n( j2) Q p ⊝Y Q p sin α j1 − α j2 sin α j2 − sin α j1 ⎧ ⎫ ⊝n( j1) =T2 ⊝n( j2) Q p )] [ ( ( ) ψ˙ s =T2 ˙ z ( j1, p ) ξ z p , (11.6) ψ z s j2, p where j1 and j2 are the edge indices for any two connected edges at Qp , α je is the angle of edge je (e = 1, 2), ⊝n(je) is ⊝n calculated on edge je , sje,p is the scoordinate of Qp on edge je , and T2 is the transformation matrix. Note in Eq. (11.6) | ( ) ( ) that ⊝n( je) | Q p = ψ˙ z s je, p ξ z p is obtained from Eq. (11.2).

354

11 Joint Structures of Thin-Walled Beams with General Section Shapes

Including the z-directional rotation, the rotations at the joint connection point linked to a shared end connection point can be now expressed as ⎧ ⎧ ⎫ ⎫ ⎨ ⊝X ⎬ ⎨ ⊝X ⎬ = ⊝Y ⊝ ⎩ ⎩ Y⎭ ⎭ ⊝ Z Q∗ ⊝Z Q p p )] ⎤ ⎡ [ ˙ ( ⎧ ( )⎫ ψ z s j1, p ( ) 0 T 2 ˙ ⎦ ξ (z p ) =⎣ ψ z s j2, p ( ) ξ, zp 0 −ψ˙ n s p ⎧ ( )⎫ ξ z =R p2 , ( p ) , ξ zp

(11.7)

where Rp2 is the joint rotation matrix for a shared connection point. Note that ( ) ( ) ⊝ Z | Q ∗p = −ψ˙ n s p ξ z p can be calculated on any connected edge of Qp because the z-directional rotation angle is continuous at a shared connection point according to Eqs. (9.87), which represent continuities in the rotation angles due to n-directional shape functions.

11.2.2 Displacements Calculated on a Joint Section Based on the kinematic assumption using a virtual rigid bar, the displacements at a joint connection point are approximated using the displacements at its corresponding end connection point, as ⎧ ⎫ ⎧ ⎫ ⎡ 0 ⎨ uX ⎬ ⎨ uX ⎬ = uY + r p ⎣ −1 uY ⎩ ⎭ ⎩ ⎭ u Z Q∗ uZ Qp 0

⎤ ⎫ 1 ⎧ ⊝X , 0⎦ ⊝Y Q p 0

(11.8)

p

where r p is the distance from Qp to Q ∗p (see Fig. 11.4b). In Eq. (11.8), the displacements at Qp are transferred to Q ∗p as a rigid-body. In the equation, the second term on the right side represents the displacements transferred by the rotations of the virtual bar, ⊝ X and ⊝Y . In Fig. 11.4b, the displacements by rigid rotations are denoted as u add X and u add Y . It is assumed that only the modes that have linear sectional deformations contribute to the rotations of the virtual bar: ⎧{ }T ⎨ Ux , U y , Uz , θx , θ y , θz , W10 , · · · , W N0 0 for open section , W (11.9) ξ= { ⎩ U , U , U , θ , θ , θ }T for closed section , x

y

z

x

y

z

11.2 Kinematics of a Joint Section

355

where ξ represents the sectional modes that contribute to the rotations of the virtual bar and Wi0 is the i-th linear warping mode (i = 1, ···, N W 0 ; N W 0 : : number of linear warping modes). Note in Eq. (11.9) that linear warping modes are considered for open sections but not for closed sections, as linear warping modes are mostly suppressed at the joints of closed-sectioned beams, as can be seen in Fig. 10.10e. (See static analysis results in Sect. 11.4.2 to check the validity of this statement.) Using Eqs. (11.4) and (11.6), ⊝ X and ⊝Y in Eq. (11.8) can be written as ⎧

¯X ⊝ ¯Y ⊝

⎫

] sin α j cos α j ( ¯ ˙¯ ( ) ( ) ¯ ¯ ( ) , ( )) = Q1 ψ s p ξ z p + Q2 ψ s p ξ z p − cos α j sin α j ⎧ ( )⎫ ( ) ξ zp ( ) , =H1 s p (11.10a) ξ, zp [

Qp

for unshared end connection points. They can be expressed as shown below for shared end connection points: ⎧

¯X ⊝ ¯Y ⊝

( )] ( ) ψ¯˙ z s j1, p =T2 ˙ ( ) ξ zp ¯ ψ z s j2, p ( ) ( ) =H2 s j1, p , s j2, p ξ z p . [

⎫ Qp

(11.10b)

In Eqs. (11.10), ψ¯ is the shape function matrix associated with ξ and ψ¯ z is the ¯ The symbols Q1 and Q2 correspond to the first two rows of Q1 and first row of ψ. Q2 in Eq. (11.3), and T2 is the transformation matrix in Eq. (11.6). Note that ψ¯ in Eqs. (11.10) is identical in size to ψ in Eq. (11.4) such that the components of ψ¯ corresponding to modes other than ξ are set to zero. Using Eqs. (11.10), the displacements at a joint connection point in Eq. (11.8) can be rewritten as ⎧ ⎫ ⎧ ( )⎫ ⎨ uX ⎬ ( ) ( ) ( ) ξ zp ( ) ξ z + r =T ψ s Q H s u 1 p p p 3 p ⎩ Y⎭ ξ, zp u Z Q∗ p ⎧ ( )⎫ ξ z (11.11) =S p , ( p ) , ξ zp ( ) ( ) where ψ s p ξ z p = {u z , u n , u s }TQ p , T1 is the transformation matrix in Eq. (11.4), Q3 is the selection matrix in Eq. (11.8), and Sp is the joint displacement matrix. Note in Eq. (11.11) that H is H1 for unshared connection points and [H2 , 0] for shared connection points.

356

11 Joint Structures of Thin-Walled Beams with General Section Shapes

11.3 Implementation As explained above, the joint connection conditions between two beams of arbitrarily shaped cross-sections are obtained by matching three-dimensional displacements and rotations at the joint connection points. For connection point Q ∗p on the joint section of Beam A and Beam B, the joint connection conditions can be written as u Q ∗p , A = u Q ∗p ,B ,

(11.12a)

 Q ∗p , A =  Q ∗p ,B .

(11.12b)

In Eqs. (11.12), u Q ∗p ,M and  Q ∗p ,M (M = A, B) are the displacements and rotations at joint connection point Q ∗p ; they are calculated using the displacements and rotations at the end connection point of Beam M. Using Eqs. (11.5, 11.7, 11.11) and expressing the displacements and rotations in the global coordinates, the joint-matching conditions expressed by Eqs. (11.12) can be rewritten as 

 B ξ p T A S Ap − T B S Bp = 0, ξ , Bp     A ξ pB A A ξp B B T Rp − T Rp = 0. ξ , Ap ξ , Bp ξ Ap ξ , Ap





(11.13a)

(11.13b)

The superscripts A and B in Eq. (11.13) represent the beam indices and the subscript p is the index of a connection point where the matrix or vector is calculated. In Eqs. (11.13), TM is the matrix used to transform displacements and rotations given in local coordinates of Beam M (M = A, B) to those in the global coordinates. The vectors of generalized 1D displacements at connection point Q M p and their derivaM ,M M tives are denoted by ξ p and ξ p , respectively. The symbol S p signifies the joint displacement matrix in Eq. (11.11) and the symbol R M p (M = A, B) denotes the joint rotation matrix such that  RM p

=

RM p1 for an unshared point , M R p2 for a shared point ,

M where R M p1 and R p2 are defined in Eqs. (11.5) and (11.7), respectively.

(11.14)

11.3 Implementation

357

If the high-order beam theory given in Chap. 9 is used for a finite element analysis, the resulting discrete system equation can be written as Kd + Md,tt = F,

(11.15)

where d is the one-dimensional nodal displacement vector and (·),tt denotes the second derivative of (·) with respect to time. The joint-matching conditions in Eqs. (11.13) can be imposed on the system equation in Eq. (11.15) in the form of a constraint matrix. For instance, for a connection point Q ∗p , one can define a matrix Lp that can be considered as constraint conditions for d as Lp d = 0, where the components of Lp are from Eqs. (11.13): [ ] ( A) ( B) T A S Ap L p :, c p = A A ; L p :, c p T Rp [ ] T B S Bp =− . T B R Bp

(11.16)

In Eq. (11.16), Lp (:, c M p ) (M = A, B) denotes the columns of Lp corresponding M ,M to c p , which is the DOF vector of the joining section of a beam, ξ M p and ξ p . Note M that Lp has six rows. For an L-type joint, c p denotes the DOFs at the end nodes of the joining beams because displacements and rotations on the joint section of an L-type joint are calculated from the field variables of the end sections of the joining beams. For a T-type joint, c M p can hold the DOFs of multiple nodes of a penetrating member (see Beam A in Fig. 11.3b), while it can hold the DOFs of the end node of a connected member (see Beam B in Fig. 11.3b). Therefore, the finite element model of the penetrating member of a T-type joint should be discretized so that all connection points on the joint section can be placed on the finite element cross-sections (or nodes) of the penetrating member. If a beam structure has N Q connection points, its total constraint matrix can ]T [ be written as L = L1T , L2T , · · · , LTN Q , which can be considered as a constraint condition Ld = 0. Obviously, the product of Ld and the Lagrange multiplier vector, dT LT λ, is always zero; therefore, dT LT λ can be directly added to the total potential energy equation. By taking the first variation of the constrained total potential energy equation, one can obtain a linearly constrained finite element equation, as follows: [

K LT L 0

⎫ ⎧ ⎫ ]⎧ ⎫ [ ]⎧ F d M0 d,tt = , + 0 λ 0 0 λ,tt

where λ is the Lagrange multiplier vector.

(11.17)

358

11 Joint Structures of Thin-Walled Beams with General Section Shapes

11.4 Numerical Examples 11.4.1 Two-Beam-Joint Structure with a Uniform Rectangular Cross-Section Figure 11.5a presents a cantilevered two-beam-joint structure with a rectangular cross-section (E = 200 GPa, and ν = 0.3). The structure is subjected to vertical force of 100 N at the center of the free end section of the joint structure, and the section is rigidly constrained against any sectional deformations. Although this problem dealing with rectangular cross-sections only can be solved by the exact matching method presented in Chap. 10, we solve this problem to check the validity of the method proposed in this chapter. Figure 11.5b shows the joint section and the connection points needed for joint matching when the HoBT is used for the analysis. The HoBT-based results when using the displacement-based joint-matching method of this chapter are compared with those by the shell theory (ABAQUS S8R elements), the Timoshenko beam theory, and the resultant-based method in Chap. 10. Figure 11.6 shows the vertical bending deflections and rotations calculated for different joint angles (φ = 30° , 60° , and 90° ). The results by the HoBT are classified into two categories in the figure, as shown below. HoBT (Exact): Analysis based on the HoBT using the exact matching condition in Chap. 10 HoBT (JSM): Analysis based on the HoBT using the displacement matching conditions on the joint section presented in this chapter

Figure 11.6 shows that the present results by the HoBT (JSM) are in good agreement with those by the shell theory and the HoBT using the exact joint-matching method in Chap. 10. To calculate the generalized 1D displacements used in the HoBT

(a)

(b)

SAB

SB 1m

100 N

Joint axis

Cross-section [mm] Joint axis

YA

1m

SA

XA ZA

2 Beam A

ϕ XB

Beam B

100 Centroid

YB ZB 50

Fig. 11.5 a L-type joint structure with a rectangular cross-section subjected to vertical force at the free end and b description of the joint section and the connection points

11.4 Numerical Examples

359

(a) Shell Timoshenko HoBT (Exact) HoBT (JSM)

Shell Timoshenko HoBT (Exact) HoBT (JSM)

Shell Timoshenko HoBT (Exact) HoBT (JSM)

Shell Timoshenko HoBT (Exact) HoBT (JSM)

Shell Timoshenko HoBT (Exact) HoBT (JSM)

Shell Timoshenko HoBT (Exact) HoBT (JSM)

(b)

(c)

Fig. 11.6 Analysis results for the problem depicted in Fig. 11.5 for a φ = 30° , b φ = 60° , and c φ = 90° . HoBT (Exact): HoBT using the exact joint-matching method, HoBT (JSM): HoBT using the displacement matching on the joint section, Shell: analysis by the shell theory, and Timoshenko: analysis by the Timoshenko theory. (Kim et al. (2022))

360

11 Joint Structures of Thin-Walled Beams with General Section Shapes

from the shell results, a least-square approximation method was used with the threedimensional displacements on the midlines of the walls of the cross-section, which could be obtained from the shell finite element analysis. Figure 11.7 shows the effect of the number of sectional modes employed ( )for the tip B analysis of the solution accuracy. The Y -directional displacement u HoBT at the loaded end calculated by the HoBT employing different numbers of sectional ( (JSM) ) tip modes is compared with that u shell by the shell analysis. The numerical test in ° Fig. 11.7 is conducted ( )/ for a joint angle of φ = 30 . The figure shows that the difference tip tip tip u shell − u HoBT u shell between the result by the HoBT (JSM) and that of the shell model substantially decreases as the number of sectional modes employed increases. The effects of the number of joint connection points used are also examined. As more connection points are used, the predicted results become stiffer and the rate of convergence slows. These observations indicate that the pointwise constraints in Eqs. (11.12) can overly constrain the local joint deformation if excessively many joint connection points are used. A series of numerical tests suggests that locating the joint connection points at the vertices of the joint section and the intersection points on the joint axis will yield satisfactory results, as suggested in Fig. 11.5b.

Case 2

Case 3

Case 4

Case 1

Fig. 11.7 The accuracy of the HoBT (JSM) in the prediction of the tip displacement for the ° cantilevered L-type ( )/ joint structure (φ = 30 ) in Fig. 11.5. “Difference” in the plot denotes tip tip tip tip tip u shell − u HoBT u shell × 100 [%], where u HoBT and u shell are the displacements at the free end of the structure as calculated by the HoBT (JSM) and shell methods, respectively. (Kim et al. (2022))

11.4 Numerical Examples

361

11.4.2 Two-Beam-Joint Structure Having an I-shaped Cross-Section A two-beam-joint structure having an I-shaped cross-section is shown in Fig. 11.8a. In this problem, we consider the two joint types sketched in Fig. 11.8b: one with a continuous flange and the other with a continuous web. This problem was initially solved by Basaglia et al. (2012, 2018) using the GBT. The exact joint condition derived in Chap. 10 cannot be directly used for this problem because the cross-sections are no longer rectangular. As indicated in Fig. 11.8a, both ends of the structure are fixed and the out-of-plane (Y A - or Y B -directional) displacement is constrained at the center of the joint section. The joint-matching models based on the HoBT (JSM) are illustrated in Fig. 11.8b. Figure 11.9 shows the analysis results for the two-beam-joint structure with a continuous flange-type joint shown in Fig. 11.8 (E = 205 GPa, ν = 0.3, L A = 4 m, and L B = 3 m). It is subjected to torsional moment of 1000 Nm in the middle of Beam A (or at Z A = 2 m). In the figures, the axial coordinate is measured from the fixed end of Beam A. The axial variations of the torsional rotation and linear warping degrees of freedom are accurately captured by the HoBT (JSM) as they match those predicted by the GBT (Basaglia et al. (2012)) and the shell-based analysis. Figure 11.9b shows that full transmission of the linear warping, which was discussed in Sharman (1985), is well captured by the proposed approach. Apparently, the prediction based on the Timoshenko beam theory neglecting the degrees of freedom involving sectional deformation appears nearly 100 times stiffer than those by the HoBT (JSM). In particular, it should also be noted that the axially decaying torsional rotation field from the joint section to the fixed end in Beam B is not captured by the Timoshenko beam

(a)

(b) Continuous flange type ZB

YB XB

Continuous web type

Cross-section [mm]

Beam B 5 LA 1000 Nm

LB

200

SA

Beam A ZA XA

YA

100

SA SAB

SB

SAB SB

Fig. 11.8 a Two-beam-joint structure with an I-shaped cross-section and b joint types considered here

362

11 Joint Structures of Thin-Walled Beams with General Section Shapes

theory. Likewise, the linear warping field cannot be represented by the Timoshenko theory as the theory does not include such sectional-deformation degrees of freedom. For the joint structure having a continuous web-type joint shown in Fig. 11.8b, a vibration analysis was conducted. The material and geometric data used for this problem are as follows: E = 210 GPa, ν = 0.3, ρ = 7850 kg/m3 , and L A = L B = 3 m. Table 11.1 lists the lowest 15 eigenfrequencies calculated by the HoBT (JSM), GBT (Basaglia et al. (2018)), and the shell analysis. The HoBT results are in good agreement with those by the GBT and the shell analysis.

(a)

(b) Shell Timoshenko ( 100) GBT (2012) HoBT (JSM)

ZA

Shell Timoshenko HoBT (JSM)

ZB

ZA

ZB

Fig. 11.9 Analysis results for a two-beam-joint structure with the continuous-flange-type joint shown in Fig. 11.8. The axial variations of a 1D torsional rotation and 1D linear warping variables are shown. (Kim et al. (2022))

Table 11.1 Eigenfrequencies (Hz) of a two-beam-joint structure having a continuous web-type joint in Fig. 11.8b. (Numbers in parentheses represent the differences (%) from the shell results.) (Kim et al. (2022))

GBT (2018)

HoBT (JSM)

1

27.92

28.21 (1.0)

27.43 (1.8)

2

28.90

29.23 (1.1)

28.40 (1.7)

3

38.24

39.02 (2.0)

37.83 (1.1)

4

40.63

42.23 (3.9)

41.06 (1.1)

5

77.67

79.45 (2.3)

78.34 (0.9)

6

90.98

91.87 (1.0)

90.06 (1.0)

7

95.93

99.49 (3.7)

95.62 (0.3)

8

108.06

112.42 (4.0)

108.14 (0.1)

9

118.68

122.43 (3.2)

118.69 (0.0)

10

144.94

149.09 (2.9)

142.59 (1.6)

11

147.80

153.34 (3.7)

148.06 (0.2)

12

160.07

164.61 (2.8)

166.33 (3.9)

13

179.67

185.93 (3.5)

176.31 (1.9)

14

223.83

231.94 (3.6)

216.99 (3.1)

15

234.54

241.16 (2.8)

232.44 (0.9)

Mode

Shell

11.4 Numerical Examples

363

(a) Joint axis

YA ZA

0.4 m

YB

Beam A

5 14.5

25

2

25

12

10

Beam B P3

(b)

2

ZB 0.2 m

ϕ

Cross-section of Beam B

Cross-section of Beam A [mm] 10

XB

P2

P1

XA

5

30

20

100 N

SA3 SA2

SA1

SAB

SB

Fig. 11.10 a T-type joint structure consisting of two beams, one beam (Beam A) made of a pentagonal cross-section and the other beam (Beam B) made of a rectangular cross-section. The structure is subjected to axial force at the end of Beam B. b Joint model by the HoBT (JSM)

11.4.3 T-joint Structure with Mixed Cross-Section Shapes In the T-joint structure shown in Fig. 11.10a (E = 200 GPa, and ν = 0.3), a box beam (Beam B) is connected to a pentagonal sectioned beam (Beam A) with joint angle φ. This structure may be practically useful, but it is chosen here to demonstrate the generality of the joint section matching method developed here. Both ends of Beam A are fixed while the free end of Beam B is constrained against higher-order modes and subjected to axial force. Figure 11.10b shows the joint model by the HoBT (JSM), where joint section S AB is included in the sidewall of Beam A. Note that the finite element model of Beam A is discretized such that its finite element nodes are placed on sections S A1 , S A2 , and S A3 . This is done to simplify the calculations of the displacements and rotations of Beam A at the joint connection points when constructing the joint condition matrix. For HoBT-based numerical analysis, 57 and 46 modes were used for Beams A and B, respectively. Here, the number of the modes used includes both rigid-body and deformable section modes. Figure 11.11 shows the deformed shapes of the joint structures for various joint angles (φ = 30° , 60° , 90° ), and Fig. 11.12 shows the

364

11 Joint Structures of Thin-Walled Beams with General Section Shapes

Fig. 11.11 Deformed shapes of the T-type joint structure in Fig. 11.10 for various joint angles (φ = 30° , φ = 60° , φ = 90° ): a comparison of the shell and Timoshenko beam results and b comparison of the shell models and HoBT (JSM) results (Kim et al. (2022)) 30 º

Shell Timoshenko HoBT (JSM)

60 º

Shell Timoshenko HoBT (JSM)

90 º

Shell Timoshenko HoBT (JSM)

Fig. 11.12 Vertical displacements along the line connecting P1 , P2 , and P3 marked in Fig. 11.10a (Kim et al. (2022))

vertical (Y A - or Y B -directional) displacements along the line connecting P1 , P2 , and P3 in Fig. 11.10a. These results show that the structural responses predicted by the HoBT (JSM) fairly accurately match those by the shell analysis, confirming that the joint connection conditions in Eqs. (11.12) can effectively handle joint structures consisting of thin-walled beams.

11.4.4 Simplified Vehicle Frame Figure 11.13a illustrates a beam model of a simplified vehicle frame (E = 200 GPa, ν = 0.3, and ρ = 7850 kg/m3 ). The structure is fixed at two rear points and subjected to a torsional couple at the front points. The beam sections under the

11.4 Numerical Examples

365

forces are set to be rigid. Figure 11.13b shows the cross-sections of the various beam members forming the vehicle frame. Although closed sections are mainly used in passenger cars, both closed and open sections are intentionally considered here to demonstrate the generality of the proposed displacement-based joint-matching method for general-sectioned beams when the HoBT is used for a structural analysis. The detailed modeling information of the beam members is given in Table 11.2. The frame consists of ten L-type joints and 34 T-type joints. Although some joints in Fig. 11.13a are not strictly classified as L- or T-type joints, joint modeling in these cases can be conducted by combining the techniques for the two joint types. For example, joint J indicated by the circle in Fig. 11.13a can be modeled using two T-type joints: one with Beam 10 and Beam 13 as joining beams and the other with Beam 10 and Beam 14 as joining beams. The deformed shape of the simplified vehicle frame analyzed by the HoBT (JSM) in (Fig. 11.14b) is shown to be in good agreement with that by the shell theory, while the Timoshenko beam model estimates the frame too stiffly, yielding a deformed shape with less deformation. Similar observations can be made from Fig. 11.15, which compares the vertical (or Y g -directional) displacement along the lower left edge of the frame structure from P1 to P2 marked in the figure. In Fig. 11.16, the vibration analysis of the same vehicle structure was conducted without any boundary conditions. Figure 11.16 shows that the HoBT (JSM) yields satisfactory results comparable to those by the shell theory. It is also apparent that the HoBT (JSM) outperforms the Timoshenko theory.

17 21

(a)

16 14 20

8

10

9

1.21 m

J

6 7

19

15 12

13 4

100 N

1.70 m 18

11 5

3 2 1

3.95 m Yg

100 N Zg

Xg

Global coordinates

Fig. 11.13 a Beam model of a simplified vehicle frame subjected to a torsional couple and b the cross-sections of the beam members (Kim et al. (2022))

366

11 Joint Structures of Thin-Walled Beams with General Section Shapes

(b) Cross-section A

[mm]

Cross-section B

Thickness: 2 140

Y

100

X

100

Cross-section C

100 Cross-section E

Cross-section D 45 50 45

120.7

50 50

50 42.7

70.7

100 39

14.6

50

50 Cross-section G

Cross-section F 20 100 20

20 80 20 140 73 40

70.7 40

37.8 30

30

Fig. 11.13 (continued)

Appendix Let us consider corner r of a cross-section connected with three edges: j1 , j2 , and j3 , as illustrated in Fig. 11.17. As shown in the figure, to obtain the uniquely defined ⊝X and ⊝Y as in Eq. (11.6) by using ⊝n of any two edges, the following continuity condition should hold ] ⎧ ⎫| [ | } sin α j1 − cos α j1 −1 ⊝n, j1 | { | = 0, (11.1A) | ⊝n, j3 r − sin α j3 − cos α j3 sin α j2 − cos α j2 ⊝n, j2 |r which can be rewritten as ( )| ) ) ( ( | sin α j1 − α j3 sin α j3 − α j2 | ( ( ) ⊝n, j1 − ) ⊝n, j2 | = 0. ⊝n, j3 − | sin α j1 − α j2 sin α j1 − α j2

(11.2A)

r

Note that Eqs. (11.1A) and (11.2A) represent compatibility condition for ndirectional rotation angles at a cross-section corner, which are derived in a proce| dure similar to that in Eqs. (9.72–9.75). In Eqs. (11.1A) and (11.2A), ⊝n, je |r are

Appendix

367

Table 11.2 Modeling Information for the Beam Members of the Vehicle Frame in Fig. 11.13 (Kim et al. (2022)) Section

Local X-direction

End coordinates 1

End coordinates 2

(DOFs)

Xg

Yg

Zg

Xg

Yg

Zg

Xg

Yg

Zg

1

A (53)

1

0

0

0.50

0

1.60

0.50

0

2.40

2

B (35)

0

0

−1

0.80

0.07

1.55

0.80

0.50

1.55

3

A (53)

0

0

−1

−0.85

0

1.55

0.85

0

1.55

4

E (28)

0

0

−1

−0.85

0.50

1.55

0.85

0.50

1.55

5

A (53)

1

0

0

0.80

0

−1.50

0.80

0

1.50

6

F (53)

1

0

0

0

0

0.05

0

0

1.50

7

C (53)

−0.94

−0.24

0.24

0.75

0.53

1.53

0.52

1

1.06

8

C (53)

−1

0

0

0.52

1

1.06

0.52

1.21

0.85

9

G (53)

0

0.71

−0.71

−0.47

1.07

0.93

0.47

1.07

0.93

10

C (53)

−1

0

0

0.52

1.21

0.85

0.52

1.21

−0.85

11

A (53)

0

0

−1

−0.75

0

0

0.75

0

0

12

D (53)

0

0

1

0.79

0.07

0

0.79

0.95

0

13

D (53)

0

0

1

0.79

0.95

0

0.54

1.19

0

14

G (53)

0

0

−1

−0.47

1.17

0

0.47

1.17

0

15

F (53)

1

0

0

0

−

−

0

0

−0.05

16

C (53)

−1

0

0

0.52

1.21

−0.85

0.52

1

−1.06

17

G (53)

0

−0.71

−0.71

−0.47

1.07

−0.93

0.47

1.07

−0.93

18

C (53)

−0.94

−0.24

−0.24

0.52

1

−1.06

0.75

0.53

−1.53

19

B (35)

0

0

−1

0.80

0.07

−1.55

0.80

0.50

−1.55

20

A (53)

0

0

−1

−0.85

0

−1.55

0.85

0

−1.55

21

E (28)

0

0

−1

−0.85

0.50

−1.55

0.85

0.50

−1.55

Beam

(a)

(b )

Meshed: Shell

Meshed: Shell

Filled: Timoshenko

Filled: HoBT (JSM)

Fig. 11.14 Deformed shapes of the vehicle frame in Fig. 11.13 compared with the shell results obtained by a the Timoshenko beam theory and b the higher-order beam theory (Kim et al. (2022))

368

11 Joint Structures of Thin-Walled Beams with General Section Shapes

P1

P2 Shell Timoshenko HoBT (JSM)

Fig. 11.15 Vertical displacement measured along the line connecting P1 and P2 (Kim et al. (2022)) 2nd mode

3rd mode

Shell

1st mode

38.97 Hz

51.77 Hz

52.44 Hz

67.39 Hz

69.92 Hz

32.18 Hz

39.88 Hz

52.45 Hz

HoBT (JSM)

Timoshenko

31.15 Hz

Fig. 11.16 Free vibration analysis results of the vehicle frame in Fig. 11.13 (Kim et al. (2022))

Appendix

369

Incompatible

Compatible Θn,j3 Θn,j2

Θn,j3 Θn,j2

A

Edge j3

Edge j3 Edge j2

Edge j1

Θn,j1

Edge j1

Corner r

A

Edge j2

B

C

Θn,j1 Corner r

Fig. 11.17 Compatible (left) and incompatible (right) cases for n-directional rotations of edges j1 , j2 , and j3 at corner r

n-directional rotation angles at corner r, as calculated on | edge je (e = 1, 2, 3). If | | one calculates the n-directional rotation ⊝n, je |r = ψ˙ z,Okje | Ok by differentiating the r z-directional displacement of the out-of-plane mode Ok , Eq. (11.2A) can be written in terms of the shape function ψ˙ z,Okje of the warping mode, as follows: ( ψ˙ z,Okj3

)| ) ) ( ( sin α j3 − α j2 Ok sin α j1 − α j3 Ok || ) ψ˙ ) ψ˙ ( ( − − | = 0. sin α j1 − α j2 z, j1 sin α j1 − α j2 z, j2 |

(11.3A)

r

According to Eq. (9.21), the derivatives of the z-directional shape function of an out-of-plane mode can be expressed as ψ˙ zOk =

NI %

ck,l ψsIl ,

(11.4A)

l=1

where ψsIl is the s-directional shape function of in-plane mode I l (l = 1, ···, N I ) and ck,l is a coefficient. According to Eq. (9.74), ψsIl satisfies the following corner displacement continuity (see also Kim et al. (2021)): ( ψs,Il j3

)| ) ) ( ( sin α j1 − α j3 Il || sin α j3 − α j2 Il )ψ )ψ ( ( − − | = 0. sin α j1 − α j2 s, j1 sin α j1 − α j2 s, j2 |

(11.5A)

r

Therefore, the following result is also valid: NI % l=1

( ck,l

ψs,Il j3

)| ) ) ( ( sin α j3 − α j2 Il sin α j1 − α j3 Il || )ψ )ψ ( ( − − | = 0. sin α j1 − α j2 s, j1 sin α j1 − α j2 s, j2 |

(11.6A)

r

Combining Eq. (11.6A) and Eq. (11.4A), we can derive Eq. (11.3A). Therefore, by | | Ok | | ˙ using ⊝n, je r = ψz, je | Ok , we can derive the compatibility condition in Eq. (11.2A), r proving that ⊝X and ⊝Y can be uniquely calculated using ⊝n of any two edges.

370

11 Joint Structures of Thin-Walled Beams with General Section Shapes

References Basaglia C, Camotim D, Silvestre N (2012) Torsion warping transmission at thin-walled frame joints: kinematics, modelling and structural response. J Constr Steel Res 69(1):39–53 Basaglia C, Camotim D, Coda HB (2018) Generalised beam theory (GBT) formulation to analyse the vibration behaviour of thin-walled steel frames. Thin-Walled Struct 127:259–274 Jang G-W, Kim KJ, Kim YY (2008) Higher-order beam analysis of box beams connected at angled joints subject to out-of-plane bending and torsion. Int J Numer Meth Eng 75(11):1361–1384 Jang G-W, Kim YY (2009a) Higher-order in-plane bending analysis of box beams connected at an angled joint considering cross-sectional bending warping and distortion. Thin-Walled Struct 47(12):1478–1489 Jang G-W, Kim YY (2009b) Vibration analysis of piecewise straight thin-walled box beams without using artificial joint springs. J Sound Vib 326(3–5):647–670 Jang G-W, Kim YY (2010) Fully coupled 10-degree-of-freedom beam theory for piecewise straight thin-walled beams with general quadrilateral cross sections. J Struct Eng 136(12):1596–1607 Kim J, Choi S, Kim YY, Jang G-W (2021) Hierarchical derivation of orthogonal cross-section modes for thin-walled beams with arbitrary sections. Thin-Walled Struct 161:107491 Kim J, Jang GW, Kim YY (2022) Joint modeling method for higher-order beam-based models of thin-walled frame structures. Int J Mech Sci 220:107132 Sharman PW (1985) Analysis of structures with thin-walled open sections. Int J Mech Sci 27(10):665–677

Index

A Axes of deflection, 282 Axial coordinate, 23

B Basis function vector, 287 Bimoment, 8, 92

C Carrera Unified Formulation (CUF), 15 Centroid, 282 Classical beam theory, 1 Coefficient vector, 271 Compatibility condition, 290, 299 C1 continuity, 60 Connected member, 350 Constrained distortion, 15, 84, 115 Constrained distortion modes, 273, 302 Constrained eigenvalue problem, 304 Constraint matrix, 302, 305 Continuities of displacements and rotations, 349 Continuity conditions at sectional corners, 293 Correction, 134 Cross-section shape function, see sectional shape function Curvature (or bending moment) continuity, 41

D D’Alembert’s principle, 56 Deformable section modes, 15, 273

Displacement-based joint-matching method, 347 Displacement continuity, 40, 41, 294 Displacements due to wall-bending deformation, 275 Distortion, 8

E Edge force, 331 Edge resultant, 321, 347 Eigenvalue problem, 288, 298, 301, 304 Element mass matrix, 57, 77 Element stiffness matrix, 55, 75 End connection point, 350, 351 End effects, 204, 306 End section, 347, 349 Equilibrium equations, 50, 55, 75, 78 Euler beam theory, 4 Euler-Lagrange equations, see equilibrium equations Extensional distortion modes, 284

F Finite element shape function, 53 Fundamental deformable section mode, 284 Fundamental mode set, 21

G GBTUL program, 310 Generalized 1D displacement, 23, 356 Generalized 1D force, 27 Generalized Beam Theory (GBT), 14, 152, 306 Generalized force, see generalized 1D force

© Springer Nature Singapore Pte Ltd. 2023 Y. Y. Kim et al., Analysis of Thin-Walled Beams, Solid Mechanics and Its Applications 257, https://doi.org/10.1007/978-981-19-7772-5

371

372 Generalized force-stress relationships, 100 Generally shaped cross-section, 263 Generic point, 123 Governing equations, see equilibrium equations

H Hermite cubic interpolation, 151 Hermite cubic interpolation functions, 71 Higher-Order Beam Theory (HoBT), 1, 10–13, 14 Higher-order modes, see deformable section modes, 117 Hinge conditions, 134 Hinged-beam frame model, 9

I Inextensional distortion, 284 In-plane modes, see in-plane sectional deformation, 280, 299 In-plane sectional deformation, 8

J Joint axis, 349 Joint condition, 350 Joint connection point, 350 Joint displacement matrix, 355, 356 Joint flexibility effect, 6 Joint matching conditions, 319 Joint rotation matrix, 353, 354, 356 Joint section, 348, 349 Joint spring element, 6 Joint structure, 319

K Kirchhoff’s kinematic assumption, 45 Kirchhoff’s thin plate theory, see Kirchhoff’s kinematic assumption

L Lagrange multiplier, 292 Lagrange multiplier vector, 357 Least-square approximation, 360 Linear warping, 284 L-joint, 5 Lowest mode set, see fundamental mode set L-type joint, 347, 349

Index M Method of generalized eigenvectors (GE), 306 Method of Lagrange multipliers, 340 Moment continuity condition, see Curvature continuity Moment equilibrium at the corners, see curvature (or bending moment) continuity

N Non-deformable section modes, 10, 273 Nonlinear higher-order warping modes, 299 Non-rigid degrees of freedom, see deformable section modes Non-rigid section modes deformable section modes, 1 Normal coordinate, 23

O One-dimensional body forces, 49 One-dimensional governing equations, see equilibrium equations One-dimensional nodal displacement vector, 357 1D displacement variable vector, 88 1D generalized forces, 89 Open section, 7 Orthogonality between in-plane sectional modes, 41 Orthogonality between out-of-plane sectional modes, 37 Orthogonality condition, 287 Out-of-plane modes, see out-of-plane sectional deformation, 301 Out-of-plane sectional deformation, 8

P Penetrating member, 350 Plane stress condition, 25 Plane stress state, 275 Poisson mode, 13 Primary strain, 97 Primary stress, 97 Primary stress field, 296 Principal axes, 282 Pure torsion, see Saint Venant torsion

R Rectangular cross-section, 263

Index Recursive analysis, see recursive derivation Recursive derivation, 94, 95 Recursive equation, 271, 277 Resultant-based method, 347 Rigid-body degrees of freedom, see non-deformable section modes Rigid-body modes, see non-deformable section modes, 89 Rigid-body section modes, see non-deformable, Rigid-body section modes Rotation angle continuity, 40 Rotation center, 67

S Saint Venant torsion, 36 Scaling parameter, 109 Secondary displacement, see secondary strain, 296 Secondary strain, 97 Secondary strain field, 296 Second moment of inertia, 92 Sectional constants, 49 Sectional coordinate, 23 Sectional effective rotation, 336 Sectional moment of inertia, 288 Sectional resultant, 321 Sectional shape function, 2, 22, 271, 273 Sectional wall, 117 Section deformation mode, see deformable section modes Selection matrix, 355 Self-equilibrated, 189 Shape function matrix, 355 Shared end connection point, 353 Shared point, 350 Shell, see Shell analysis Shell analysis, 156 Slope continuity, See rotation angle continuity

T Tangential coordinate, 23 TBT, 254 Timoshenko beam theory, 4

373 T-joint, 4 Torsional center, 282 Torsional distortion, 11, 39–44 Torsional rigidity, 28 Torsional warping, 11, 33–39 Transverse bimoment, 92 T-type joint, 347, 350 Two-beam joint structure, 319 Type-1 constrained distortion, 135, 136 Type-2 constrained distortion, 134, 135

U Unconstrained distortion, 15, 83, 84, 86 Unconstrained distortion modes, 271, 296 Unshared end connection point, 355 Unshared point, 350

V Variational Asymptotic Beam Section analysis (VABS), 14 Variational principle, 48 Virtual bar, 351 Virtual work, 91 Vlasov assumptions, 7 Vlasov beam theory, 2, 7, 10 Vlasov torsion theory, 152 VTT, see Vlasov torsion theory

W Wall bending, 34, 47 Wall-bending field, 86 Wall centerline, 83 Wall centerline displacements, 86 Wall-membrane field, 86 Wall midline, 123 Wall-normal edge moment, 329 Warping, 8 Work conjugate, 91

Z Zeroth mode set, see fundamental mode set Zeroth-order modes, see fundamental mode set