Stress, Strain, and Structural Dynamics ppt

Stress, Strain, and Structural Dynamics An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes Bingen Yang University of Southern California Amsterdam Boston Heidelberg

Stress, Strain, and Structural Dynamics

The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs

to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs and/or the documentation

MATLAB is a registered trademark of The Mathworks, Inc

Stress, Strain, and

Structural Dynamics

An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes

Bingen Yang

University of Southern California

Amsterdam Boston Heidelberg London New York Oxford

Paris San Diego San Francisco Singapore Sydney Tokyo

Elsevier Academic Press

30 Corporate Drive, Suite 400, Burlington, MA 01803, USA

525 B Street, Suite 1900, San Diego, California 92101-4495, USA

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Library of Congress Cataloging-in-Publication Data

Yang, Bingen

Stress, strain, and structural dynamics

p cm

Includes bibliographical references

1 Strains and stresses 2 Structural dynamics I Title

TA648.3.Y36 2004

624.1'7-dc22

2004022861

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN: 0-12-787767-3

For all information on all Elsevier Academic Press publications

visit our Web site at www.books.elsevier.com

Printed in the United States of America

05 06 07 08 09 10 9 8 7 6 5 4 3 2 1

Working together to grow

libraries in developing countries

www.elsevier.com | www.bookaid.org | www.sabre.org

To my parents,

My wife Haiyan, and

My daughters Sonia and Tanya

This Page Intentionally Left Blank

Contents

Preface xv Acknowledgements xvii

Getting Started Beam Theory 2.2.1 Static Problem 2.2.2 Solution Methods 2.2.3 Fundamentals of Euler-Bernoulli Beams Static Analysis by the Toolbox

2.3.1 System Setup 2.3.2 Response to External Forces 2.3.3 Response to Boundary Disturbances 2.3.4 Total Response

2.3.5 Exact Analytical Solutions 2.3.6 Other Useful Functions Moments of Inertia of Beam Cross-Section Area Quick Solution Guide

References Static Analysis of Bars, Shafts, and Strings 3.1

3.2 3.3

Getting Started System Description Static Analysis by the Toolbox 3.3.1 System Setup

3.3.2 Response to External Forces 60 3.3.3 Response to Boundary Disturbances 63 3.3.4 Total Response 65 3.4 Stepped Bars and Shafts 67 3.4.1 System Description 67 3.4.2 Solution by the Toolbox 68 3.5 The Distributed Transfer Function Method 74

3.6 Quick Solution Guide 78 3.7 References 83

C h a p t e r 4 Buckling Analysis of Columns 85

4.1 Getting Started 85 4.2 Uniform Columns 87 4.2.1 Column Buckling Theory 87

4.2.2 Solution by the Toolbox 91 4.2.3 Eccentric Loading 96 4.2.4 Beam-Column Problem 100 4.2.5 Geometric Imperfection 104 4.3 Stepped and Nonuniform Columns 109 4.3.1 Constrained Stepped Columns 109 4.3.2 Nonuniform Columns 116 4.4 The Distributed Transfer Function Method 123

4.5 Quick Solution Guide 127 4.6 References 133

C h a p t e r 5 Stress Analysis in Two-Dimensional Problems 135

5.1 Getting Started 135 5.2 Plane Stress and Strain 136 5.2.1 In-Plane Stresses 136 5.2.2 In-Plane Strains 142 5.2.3 Strain Rosette 147 5.2.4 Hooke's Law 149 5.2.5 Criteria of Failure 151 5.3 Quick Solution Guide 155 5.4 References 156

Part II Structural Mechanics 157

Chapter 6 Static Analysis of Constrained Multispan Beams

6.1 Getting Started 6.2 System Description 6.3 Solution by the Toolbox 6.3.1 System Setup 6.3.2 Response to External Forces 6.3.3 Response to Boundary Disturbances 6.3.4 Response to Support Settlement 6.3.5 Total Response

6.3.6 Influence Lines 6.4 The Distributed Transfer Function Method

6.5 Quick Solution Guide

7.4 7.5 7.6 Stati 8.1 8.2 8.3

8.4 8.5 8.6

Getting Started System Description Solution by the Toolbox 7.3.1 System Setup 7.3.2 Response to External Forces 7.3.3 Support Settlement

7.3.4 Fabrication Errors and Thermal Effects 7.3.5 Total Response

The Stiffness Method Quick Solution Guide References

c Analysis of Plane Frames Getting Started

System Description Solution by the Toolbox 8.3.1 System Setup 8.3.2 Response to External Forces 8.3.3 Response to Settlement of Supports 8.3.4 Total Response

8.3.5 Response of Frame Members The Distributed Transfer Function Method Quick Solution Guide

Part III Dynamics and Vibrations 277

C h a p t e r 9 Dynamics of Particles and Rigid Bodies 279

9.1 Getting Started 279 9.2 Dynamics of Particles 280 9.2.1 Preliminaries 280 9.2.2 Kinematics 284 9.2.3 Kinetics of Single Particle 286

9.2.4 Particle Motion via Numerical Integration 289 9.2.5 Systems of Particles 297 9.2.6 Central Force Motion 303 9.3 Dynamics of Rigid Bodies in Plane Motion 307

9.3.1 Plane Motion 307 9.3.2 Mass Moments of Inertia 310

9.3.3 Kinetics 314 9.3.4 Simulation of Dynamic Response 317

9.4 Rigid Body Dynamics in Three Dimensions 323 9.4.1 Kinematics 323 9.4.2 Inertia Properties 327 9.4.3 Energy and Momentum 335

9.4.4 Equations of Motion 336 9.4.5 Rotation of Axisymmetric Bodies 340

9.5 Quick Solution Guide 349 9.6 References 350

C h a p t e r 1 0 Vibration Analysis of One-Degree-of-Freedom Systems 351

10.1 Getting Started 351 10.2 System Description 353 10.3 Time Response 357 10.3.1 Free Response 357 10.3.2 Forced Response 361 10.3.3 Specific Forcing Functions 365

10.3.4 Total Response 374 10.3.5 Delayed Forcing Functions 376

10.3.6 Solution by Superposition 377 10.3.7 Mechanical Energy 379 10.3.8 Plotting Computed Response 382

10.4 Analytical Vibration Solutions 384 10.4.1 Analytical Methods 384 10.4.2 Solution by the Toolbox 389 10.5 Frequency Response 394 10.5.1 Harmonic Excitation 394 10.5.2 Base Excitation 400 10.5.3 Response under Rotating Unbalance 403

10.5.4 Vibration Isolation 406 10.6 Response to Periodic Excitation 410 10.6.1 Steady-State Solution in Fourier Series 410

10.6.2 Solution by the Toolbox 412 10.6.3 Fourier Coefficients of Certain Loads 419

10.7 Nonlinear Vibration 421 10.8 Quick Solution Guide 431 10.9 References 435

C h a p t e r 1 1 Vibration and Control of Multiple-Degree-of-Freedom

Systems 437 11.1 Getting Started 437

11.2 System Description 439 11.2.1 Introduction 439 11.2.2 Modes of Vibration 442 11.2.3 Solution by the Toolbox 444 11.3 Dynamic Response 450 11.3.1 Modal Analysis 450 11.3.2 Laplace Transform Method 458

11.3.3 Plotting Analytical Solutions 462 11.3.4 Runge-Kutta Algorithm 465 11.3.5 Solution by Superposition 468 11.3.6 Harmonic Excitation 472 11.4 Dynamic Vibration Absorption 475 11.4.1 Undamped Vibration Absorbers 476

11.4.2 Damped Vibration Absorbers 481

x Contents

11.5 Transfer Function Formulation 11.5.1 Transfer Function and Green's Function 487 11.5.2 Open-Loop Transfer Function 491 11.6 Feedback Control 497 11.6.1 Vibration Control System 497

11.6.2 Position Control System 501 11.6.3 Solution by the Toolbox 506 11.7 Quick Solution Guide 515 11.8 References 518

Part IV Structural Dynamics 519

13.2.2 Eigenvalue Problem 622 13.2.3 System Setup by the Toolbox 624

13.2.4 Display of Modes of Vibration 627 13.3 Dynamic Response 630 13.3.1 Modal Expansion 631 13.3.2 Free Vibration 632

Chapter 12 Dynamics and Control of Euler-Bernoulli Beams

12.1 12.2

12.3

12.4

12.5

12.6 12.7

Getting Started System Description 12.2.1

12.2.2 12.2.3 12.2.4 12.2.5

Governing Equation Eigenvalue Problem System Setup by the Toolbox Animation of Modes of Vibration Distributed Transfer Function Method Dynamic Response

12.3.1 12.3.2 12.3.3 12.3.4 12.3.5 12.3.6 12.3.7

Modal Expansion Specification of Damping Free Vibration

Forced Vibration Total Response Animation of Time Response Frequency Response Feedback Control

12.4.1 12.4.2 Dynami 12.5.1 12.5.2 12.5.3 12.5.4 12.5.5

Control System Formulation Solution by the Toolbox

cs and Control of Nonuniform Beams Problem Statement

Rayleigh-Ritz Discretization Modes of Vibration

Time Response Control System Formulation Quick Solution Guide

References

13.3.3 Forced Vibration 635 13.3.4 Total Response 639 13.3.5 Animation of Time Response 642

13.3.6 Frequency Response 643 13.4 Free Vibration of Stepped Systems 647 13.4.1 System Description 647 13.4.2 Free Vibration Analysis 651 13.4.3 Solution by the Toolbox 655 13.5 Quick Solution Guide 661 13.6 References 665

C h a p t e r 1 4 Dynamic Analysis of Constrained, Combined, and Stepped

Beams 667 14.1 Getting Started 667

14.2 Constrained Beams 670 14.2.1 System Description 670

14.2.2 System Setup and Eigensolutions 675 14.2.3 Eigenvalue Locus 683 14.2.4 Frequency Response 687 14.2.5 Multispan Beam Structures 689 14.2.6 Transient Response 691 14.3 Combined Beams 691 14.3.1 System Description 692

14.3.2 System Setup and Eigensolutions 696 14.3.3 Transient Response 698 14.3.4 Frequency Response 707 14.3.5 Oscillators for Vibration Absorption 712

14.4 Stepped Beams 715 14.4.1 Free Vibration Analysis 715

14.4.2 Solution by the Toolbox 721 14.5 Quick Solution Guide 729 14.6 References 736

Part V Two-Dimensional Elastic Continua 737

;er 1 5 Static Analysis of Linearly Elastic Bodies

15.1 15.2

15.3

Getting Started Theory of Linear Elasticity 15.2.1

15.2.2 15.2.3 15.2.4 15.2.5

Stress and Strain Basic Equations of Elasticity Conversion of Elasticity Constants Strain Energy

Principal of Minimum Potential Energy Elasticity Problems in Two Dimensions

15.3.1 15.3.2 15.3.3 15.3.4 15.3.5 15.3.6

Plane Stress and Plane Strain Governing Equations Solution by Stress Function Thermal Stresses

Elasticity Problems in Polar Coordinates Stress Concentrations

15.4 Finite Element Method for 2-D Elasticity Problems 770 15.4.1 Finite Element Formulation 770 15.4.2 MATLAB Solutions in Rectangular Regions 777

15.4.3 MATLAB Solutions in Arbitrary-Shaped

Regions 792 15.5 Quick Solution Guide 802 15.6 References 805

C h a p t e r 1 6 Free Vibration of Membranes and Plates

16.1 Getting Started 16.2 Free Vibration of Membranes 16.2.1 Rectangular Membranes 16.2.2 Circular Membranes 16.3 Free Vibration of Rectangular Plates 16.3.1 Plate Theory

16.3.2 Eigenvalue Problem 16.3.3 Solution by the Toolbox 16.4 Free Vibration of Circular Plates 16.4.1 Equations in Polar Coordinates 16.4.2 Modes of Vibration

Appendix A

Appendix B

16.5 16.6

Quick Solution Guide References

Commonly Used Mathematical Formulas

A.l A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.ll A.12

Algebraic Formulas Areas and Volumes of Common Shapes Trigonometry

Hyperbolic Functions Derivatives and Integration Series Expansion

Analytical Geometry Vector Analysis Matrix Theory Complex Numbers and Complex Functions Laplace Transforms

Inverse Laplace Transform via Partial Fraction Expansion

MATLAB Basics

B.l B.2 B.3 B.4 B.5 B.6 B.7

Getting Started Matrix and Vector Manipulations Graphics

M-Files Control Flow Solution of Algebraic and Differential Equations Control System Toolbox

Appendix C The Distributed Transfer Function Method 913

C.l DTFM for One-Dimensional Continua 913 C.2 Transfer Function Synthesis of Multibody Structures 917

C.3 DTFM for Two- and Three-Dimensional Problems 919 C.4 References 920

Appendix D Conversion of Units 923

Appendix E Mechanical Properties of Engineering Materials 925

Index 929

xiv Contents

This book is intended to supply engineering professionals and students with a comprehensive and definitive reference to statics and dynamics of solids and structures The book is for use

as a resource and design tool in research and development, and for use as a study guide and learning aid in engineering education

The book is written to meet the needs for interactive computing in technical referencing and engineering education These needs result from the design requirements for high-accuracy and high-performance structures, machines and devices in a variety of engineering applications, and from the trend of engineering curriculum development in response to today's environment

of fast-changing technologies Chapter 1 further explains the purpose and philosophy of this writing

A unique feature of this book is the integration of the development of principles, formulas, and solutions with user-friendly interactive computer programs, written in the powerful and popular software MATLAB These programs produce instant engineering solutions, which cover pages of contents contained in a conventional handbook, and beyond Different from and complementary to general-purpose numerical codes, these programs deliver analytical results pertaining to many topics covered in the book Furthermore, with the rich resources

of MATLAB, these programs allow in-depth exploration of the physics of deformation, stress and motion by analysis, simulation, graphics, and animation

This book contains five main parts: strength of materials, structural mechanics, dynamics and vibrations, structural dynamics, and two-dimensional elastic continua Besides, the book presents feedback control of mechanical systems and flexible structures to a great extent These subjects are covered in 15 chapters (Chapters 2 to 16), each being a self-contained package

of subject review, fundamental theories, formulas, and a toolbox of MATLAB functions (computer programs) for numerical and analytical solutions The MATLAB functions are stored in a CD-ROM that is attached to the book In addition, Appendix A collects commonly used mathematical formulas for engineering analysis and Appendix B gives a tutorial on MATLAB-based computing and programming

There are two important points regarding this book First, the book keeps a good ance between fundamental theory and technical computation Such a balance, in the author's opinion, will best serve design engineers who often need a quick subject review, and instant engineering solutions at the same time Second, the organization of the text closely follows

bal-a curriculum on solid mechbal-anics bal-and structurbal-al dynbal-amics bal-adopted by bal-a typicbal-al engineering school This naturally renders the book a useful study guide and learning aid for many courses

All the numerical results in the book are obtained through use of the attached MATLAB functions All the artwork and tables in the book are created by the author; all the MATLAB functions of the book are generated by the author For theories and formulas that are generally available, a list of references is provided at the end of each chapter Specific formulas or results from certain resources have been duly acknowledged

Although much effort has been made to avoid errors, some of them are bound to escape detection The author welcomes and appreciates any comments and suggestions from readers for necessary corrections and for further improvement of the quality of this work

Bingen Yang Los Angeles, California December 2004

Acknowledgements

I would like to express my appreciation to the staff at Elsevier, especially Joel Stein, Shoshanna Grossman, and Carl Soares, for their assistance, guidance, and high professional competence Certain results and formulas presented in this book are the outcome of my previous research partially sponsored by the National Science Foundation, the US Army Research Office, and NASA's Jet Propulsion Laboratory

Finally, I wish to thank my wife Haiyan and my daughters Sonia and Tanya, for their love, dedication and encouragement Without their endless support, this work could never have been completed

This Page Intentionally Left Blank

1 Introduction

Inside

• The Making of This Book

• How to Use This Book

1.1 The Making of This Book

Computing is essentially important in both engineering education and engineering practice The use of mathematical software packages can greatly enhance the learning of various topics

in science and technology, and surely help increase the efficiency and accuracy of engineering designs The recent technological advancements in computers, software, and telecommuni-cations make it possible for individuals to enjoy an interactive and mobile computing environment in their study and work

Taking advantage of such a fast-changing environment, this book is intended to provide a new type of reference to statics and dynamics of solids and structures, with unique interactive computing capabilities The purpose, approach, scope, and features of this writing are described under the following headlines

Who Will Find the Book Useful

This book is ideal for both professionals and students dealing with aerospace, mechanical, and civil engineering, as well as naval architecture, biomechanics, robotics, and mechtronics For engineers and specialists, the book is a valuable resource and handy design tool in research and development For engineering students at both undergraduate and graduate levels, the book serves as a useful study guide and powerful learning aid in many courses And for instructors, the book offers an easy and efficient approach to curriculum development and teaching innovation

Uniqueness

This book differs from standard handbooks in that it integrates the development of las, fundamental theories, mathematical models, and solution methods with user-friendly interactive computer programs Unlike the commonly adopted approach of "finish-the-book-first-and-add-software-later," the text-software integration is harmonically fabricated in the

formu-book writing from scratch This unique merger of technical referencing and interactive computing allows instant solution of a variety of engineering problems and in-depth explo-ration of the physics of deformation, stress, and motion by analysis, simulation, graphics, and animation

Interactive Computing with MATLAB

The computer programs for the book are written in the powerful and popular MATLAB, which is a premier software package that provides an interactive environment for techni-

cal computation Different from many books that teach people how to use MATLAB in engineering analysis, this book shows how to obtain instant engineering solutions by hun-

dreds of preprogrammed MATLAB functions from a CD-ROM that is attached to the book These functions permit easy generation of data, figures, animation, and even analytical expres-sions, and produce results that are equivalent to the contents covered in pages of a conventional handbook and beyond

Motive for Writing

This writing is motivated by the following two needs in engineering education and technical referencing

Need for Interactive Computing Capabilities in Engineering Education The solution

of a problem in an undergraduate engineering course usually requires knowledge in the following four areas:

• The background material of the problem in consideration, such as strength of materials, vibrations, and structural dynamics;

• Mathematical physics, including differential equations, linear algebra, and matrix theory;

• Solution algorithms, which can be either analytical or numerical; and

• Computer coding in programming languages like C, Fortran, and MATLAB

While computer coding is normally introduced in the freshman year, adequate knowledge in mathematical physics and solution algorithms is not available until the senior year or later This lack of mathematical skills often limits undergraduate teaching to a few "classroom problems."

Of course, commercial computer programs may be used for solution of complicated problems The usage of those codes, however, still requires a background in mathematical physics and numerical algorithms

The current book fills this gap by offering adequate computing capabilities to many neering courses With this book, an undergraduate student in earlier years can solve various engineering problems without worrying about numerical algorithms This allows the student to focus on important aspects of fundamental principles in engineering science and to explore the physical insight of practical problems Moreover, with the interactive computing capabilities provided by this book, more advanced topics can be introduced to adapt an undergraduate or graduate curriculum to today's environment of emerging technologies

engi-Need for Interactive Computing Capabilities in Technical Referencing A standard

ref-erence collects formulas and tables that normally cover a number of simple cases Although general-purpose computer codes are available, they usually only deliver numerical results and are not integrated with many analytical formulas given in a standard reference Quite often,

an engineer or specialist would like to get a quick solution for verifying a design concept

or a research idea In this case, a reference with attached computer programs, which yield numerical or analytical solutions according to user-selected parameters, boundary conditions, and loads, would definitely be desirable

2 STRESS, STRAIN, AND STRUCTURAL DYNAMICS

This book provides such needed interactive computing capabilities to technical referencing With hundreds of preprogrammed MATLAB functions, numerical and analytical solutions of various engineering problems can be easily obtained Besides facilitating quick concept proof

in design and research, these solutions can serve as a benchmark for verification of numerical algorithms and computer codes developed by the user

Scope

This book covers basic topics regarding solids and structures, including strength of als, structural mechanics, elasticity, particle and rigid-body dynamics, vibrations, structural dynamics, and structural controls As indicated by the table of contents, these topics are pre-sented in five parts with a total of 15 chapters Each chapter deals with a type of problem or

materi-a clmateri-ass of systems encountered in engineering Emateri-ach chmateri-apter is materi-a self-contmateri-ained pmateri-ackmateri-age of subject review, fundamental theories, formulas, and a set (toolbox) of MATLAB functions for numerical and analytical solutions

For efficient utility of this book, no attempt has been made to include every topic in such a wide range of subjects Instead, the following three criteria have been applied in selecting the book materials:

(a) The problem in consideration is fundamentally important to engineering education and engineering practice Examples include static analysis of Euler-Bernoulli beams (Chapter 2), stress and deformation of elastic bodies (Chapters 5 and 15), and

rigid-body dynamics (Chapter 9)

(b) The problem in consideration is representative of a wide class of engineering

applications Examples include columns (Chapter 4), trusses (Chapter 7), frames (Chapter 8), and multispan beam structures (Chapters 6 and 14)

(c) The problem in consideration requires substantial analytical and numerical efforts for better understanding of its physics Examples include vibration of multiple-degree-of-freedom systems (Chapter 11), dynamics and control of Euler-Bernoulli beams

(Chapter 12), and vibration of plates (Chapter 16)

Special Features

Besides its unique interactive computing capabilities, this book has several special features, some of which are not available in the existing references

New Formulas and Solutions This book contains many new formulas and analytical

solutions Examples include

• Analytical expressions of static response of beams subject to general external loads and arbitrary boundary disturbances;

• Exact static deflections and stresses of flexible frames under arbitrary external loads and support settlement;

• Influence lines of statically indeterminate multispan beam structures;

• Exact vibration solutions of one-degree-of-freedom systems subject to general forcing functions;

• Exact expressions of normalized mode shapes (eigenfunctions) of beams, bars, shafts, and strings, under arbitrary boundary conditions;

• Eigenfrequency loci of constrained beam structures;

• Control system formulation and design for beams with feedback controllers; and

• Exact free vibration solutions of plates with various boundary conditions

What makes these new results more useful is that they can be obtained easily through use

of the attached MATLAB toolboxes

Exact Solution via the Distributed Transfer Function Method This book presents exact

static and dynamic responses of beams, bars, shafts, columns, and frames, which are mined by the Distributed Transfer Function Method (DTFM) The DTFM is a closed-form analytical solution technique for modeling, analysis, and control of flexible structures The DTFM is flexible in dealing with different geometric configurations and boundary conditions, and convenient in computer coding The DTFM is introduced in Appendix C and its application

deter-to specific problems is given in related chapters

Instant Animation of Motion and Vibration The MATLAB toolboxes of the book have

functions for animating the modes of vibration and transient response of beams, bars, shafts, constrained and combined beam structures, and plates, the motion of rigid bodies in two and three dimensions, and the response of lumped parameter systems and flexible beams under feedback control This animation functionality, which takes advantage of the rich resources of MATLAB, helps better understand the physics of motion and vibration and makes learning of difficult subjects a fun experience

Integrated System Modeling and Controller Design Feedback control has wide

applica-tions in machines and structures This important topic is addressed in the current book, for lumped dynamic systems (Chapter 11) and flexible beams (Chapter 12) For these systems, the book presents major steps in control system design, including system modeling, dynamic analysis, control system formulation, controller design and numerical simulation, and pro-vides MATLAB functions for each of the steps This integration of modeling, analysis, design, and simulation for feedback control of machines and structures is not available in any other reference on structural dynamics

1.2 How to Use This Book

Chapters 2 to 16 cover topics in strength of materials, structural mechanics, elasticity, particle and rigid-body dynamics, vibrations, structural dynamics, and structural controls Each of the chapters has the following parts:

Each chapter has a toolbox of MATLAB functions contained in the attached CD-ROM

In addition, Appendices A to E will be useful for engineering design and analyses

Some key points in using this book are given below

Getting Started

To start, find the right chapter from the Contents for the problem or system in consideration,

then go to the first section of the chapter, titled Getting Started This section tells what the

chapter is about, how to install the MATLAB Toolbox, how to use the Toolbox through a tutorial example (in most chapters), and what to do next

Fundamental Principles

The fundamental principles of each subject covered are briefly reviewed Some derivations

of theories and mathematical models are provided For detailed information on these basic issues, a list of references is given at the end of each chapter

4 STRESS, STRAIN, AND STRUCTURAL DYNAMICS

Formulas and Solutions

Formulas and solutions in a few special cases can be directly found from the text of a chapter Formulas and solutions in general cases of geometric configurations, boundary conditions, and loadings can be obtained through use of the MATLAB toolbox for the chapter This requires computing and programming with MATLAB

MATLAB

MATLAB is a software product of The MathWorks, Inc., headquartered in Natick, sachusetts MATLAB is a registered trademark of The MathWorks, Inc To obtain MATLAB, visit the company's Web site http://www.mathworks.com/ A quick tutorial and brief summary

Mas-of computing and programming with MATLAB is given in Appendix B Mas-of this book, which is useful for both beginners and advanced users

MATLAB Functions in CD-ROM

Attached to this book is a CD-ROM with hundreds of preprogrammed MATLAB functions These functions form 15 toolboxes, one for each of Chapters 2 to 16 The license agreement and limited warranty about the software package is given at the end of the book For possible updated versions of the MATLAB toolboxes contained in the CD-ROM, check the publisher's Web site

Windows

"Windows" are used to summarize the purpose and utility of the MATLAB functions from the attached CD-ROM The windows are distributed in the text flow so that they are naturally related to the formulas and solutions presented

Examples

The windows are normally followed by step-by-step examples demonstrating how the MATLAB functions can be used in analysis, simulation, graphics, and animation Further-more, each toolbox has a function Run Ex, which, when launched, displays all the numerical examples contained in the chapter

Quick Solution Guide

Each chapter has a section titled Quick Solution Guide This section briefly describes the

problem or system in consideration, lists the MATLAB functions from the toolbox with window or section numbers for easy reference, and outlines the solution procedure in the MATLAB-based computation This section is especially convenient to those who are familiar with the material covered in the chapter and would like to engage in technical computation directly

References

At the end of each chapter is a list of references for further reading These references are mostly textbooks and monographs

Unit Conversion

In all the examples, quantities are given in either the standard international system (SI) of units

or nondimensional units For conversion between SI system and the U.S customary system, refer to Appendix D

Mathematical Formulas

For convenience in engineering analyses, this book collects commonly used mathematical mulas in algebra, trigonometry, analytical geometry, calculus, vector analysis, matrix theory, complex analysis, differential equations, and Laplace transforms; see Appendix A

for-Mechanical Properties of Engineering Materials

For convenience in engineering designs, the mechanical properties of selected engineering materials are given in Appendix E

Comments and Technical Questions

For comments on this book and technical questions about the attached MATLAB toolboxes, please contact the author by the following mail and e-mail address:

Professor Bingen Yang

Department of Aerospace and Mechanical Engineering

University of Southern California

3650 McClintock Avenue, Room 430

Los Angeles, CA 90089-1453

E-mail: bingen@usc.edu

6 STRESS, STRAIN, AND STRUCTURAL DYNAMICS

PARTI

Strength of Materials

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2 Static Analysis of Euler-Bernoulli

Beams

Inside

• Getting Started

• Beam Theory

• Static Analysis by the Toolbox

• Moment of Inertia of Beam Cross-Section Area

• Quick Solution Guide

• References

2.1 Getting Started

What Is in This Chapter

This chapter is a package of subject review, fundamental theories, formulas, solution methods, and a set (toolbox) of MATLAB functions for static analysis of Euler-Bernoulli beams

System Requirements for the MATLAB Toolbox

• PC with Win 98SE/NT/2000 and XP or Mac with OS 9.x and up

• The software MATLAB (version 5.x and up) installed on computer

Software Installation and Test

(i) Drag the Toolbox folder from the CD onto a hard disk of your computer;

(ii) Launch MATLAB and set a path to the Toolbox folder on your hard disk;1 and

(iii) Test the toolbox by typing TBdemo in the MATLAB command window, which will

launch a demo program showing how the Toolbox works The demo ends with a message: "The Toolbox works properly."

At this stage, the Toolbox is properly installed, and it is ready for use

If the M-files of the toolbox are put in the MATLAB work folder, there is no need to set a path

Quick Tutorial

To show how to use the Toolbox, consider a simply-supported beam in Fig 2.1.1, which has

length L = 1 and bending stiffness EI = 25, and is subject to a pointwise forcefy = 1.2 at its

midpoint In the MATLAB command window, type:

where » i s the prompt in the MATLAB command window This yields the spatial distributions

of the displacement (transverse deflection), rotation, bending moment, and shear force of the beam as plotted in Fig 2.1.2, and the maximum response and the reactions as shown below

t /o

3& EI

F I G U R E 2.1.1 A simply-supported beam under a pointwise force

Maximum Beam Response in Absolute Value

Max displacement = 0.001, location x = 0.5

Max rotation (degrees) = 0.17189, location x = 0

Max bending moment = -0.3, location x = 0.5

Max shear force = -0.6, location x = 0

x i g ' 3

displacement, w(x)

0 0.5 1 bending moment, M(x)

-0.1 -0.2 -0.3 -0.4

x10" 3

slope, theta(x)

0 0.5 1 shear force, Q(x)

FIGURE 2.1.2 Beam response distribution

10 STRESS, STRAIN, AND STRUCTURAL DYNAMICS

Reactions at Two Ends of the Beam

Sign convention for support reactions:

* Positive moment Mc: counterclockwise

* Positive force Re: upward

Bending moment M(x) = 0.6*x + (-0.6)

Shear f o r c e Q(x) = 0.6

In the above static analysis, four functions from the Toolbox have been used:

(a) Function setbeam that assigns beam parameters and sets up beam boundary

conditions;

(b) Function beamf that computes beam response under an external force and puts the

computed results in a matrix y;

(c) Function pi otbeam that plots the computed beam response versus the spatial

coordinate x and shows the maximum beam response and reaction forces at beam

supports; and

(d) Function mathf that delivers the analytical expressions of beam response

These functions, along with others from the Toolbox, will be described in sequel

For a quick solution, the reader may refer directly to the Quick Solution Guide (Section 2.5),

or get the information on the Toolbox by typing TBinfo in the MATLAB command window

To run the examples contained in this chapter, type Run Ex in the MATLAB command window

To understand better how the toolbox works, the reader is encouraged to go through the entire chapter For further reading on the subject, refer to Section 2.6

2.2 Beam Theory

2.2.1 Static Problem

Governing Equation The transverse displacement (deflection) w(x) of a uniform

Euler-Bernoulli beam under static loads, as shown in Fig 2.2.1, is governed by the differential

where EI and L are the bending stiffness and length of the beam, respectively The forcing

function for the loads shown in Fig 2.2.1 has the form

d f{x) = q(x) +fo8(x - x f ) - T—8(X ~ *T)

where q(x) is a distributed external load, fo is a pointwise force applied at Xf, r is a torque applied at x T9 and <$(•) is the Dirac delta function Refer to Section 2.2.3 for the derivation of

Eq (2.1) The sign convention for beam displacement and forces is as follows The positive direction of beam displacement and transverse forces (either pointwise or distributed) is upward (f); the positive direction of a torque is counterclockwise (O)

Boundary Conditions The boundary conditions of the beam can be written as

At the left end #oi [w(x)] x=0 = a\, £02 [w(x)] x=0 = oti

At the right end B LX [w(x)] x=L = 0i, B L2 [w(x)] x=L = fa

(2.2)

where #oi, #02 > Bn, and BL2 are spatial differential operators, and a/ and fr represent boundary

disturbances Seven types of beam boundary conditions with associate boundary disturbances

are given in Table 2.2.1 In the table, wo and ui are transverse boundary displacements, #0 and

OL rotations, Mo and ML moments, and Qo and Qi transverse forces The positive direction of

boundary displacements and transverse forces is upward; the positive direction of boundary rotations and moments is counterclockwise

Static Response The static response of a beam includes:

Displacement (transverse deflection) w(x)

(2.3)

(2.4)

12 STRESS, STRAIN, AND STRUCTURAL DYNAMICS

TABLE 2.2.1 Boundary Conditions and Boundary Disturbances

Boundary Type Boundary Conditions

(Bl) Pinned or hinged end

Right end: w(L) = UL, = #L

Here M and Q are called the internal forces of the beam The positive direction of w(x) is

upward; the positive direction of rotation is counterclockwise; and the sign convention of bending moment and shear force is given in Fig 2.2.2

oi x = 0

M M

x=L D

R c K

FIGURE 2.2.3 Sign convention of reactions

Reactions at Supports All the boundaries listed in Table 2.2.1, except for B3 (free ends),

exert reaction moments and/or reaction forces to the beam when it is subject to external loads These reactions are to support the beam by balancing the external forces For this reason, these

boundaries are also called supports The reactions are represented by the following equations:

At the left end (x = 0)

At the right end (JC = L)

M c = -£/w"(0), R c = Elw'"(0)

M c = EIw"(L\ R c = -EIw"\L) where w' = dw/dx The sign convention for the reactions is the same as the boundary

disturbances (see Fig 2.2.3); that is,

• A positive reaction moment (M c ) is in the counterclockwise direction; and

• A positive reaction force (R c ) is in the upward direction

A fundamental problem in static analysis of Euler-Bernoulli beams is stated as follows:

Given external loads and boundary disturbances, determine the beam response (displacement, rotation, bending moment, and shear force) that is governed by Eq (2.1) and the boundary conditions (2.2)

Listed in Table 2.2.2 is the static beam deflection in some cases of boundary conditions and loads The bending moment and shear force in each case can be derived according to Eqs (2.4) and (2.5) The deflection solutions in the table can be obtained by the methods given in Section 2.2.2 Also, refer to Section 2.3.5 for analytical solutions with general beam configurations and arbitrary external loads

14 STRESS, STRAIN, AND STRUCTURAL DYNAMICS

TABLE 2.2.2 Static Beam Deflection

Beam Type and Load Transverse Deflection

W(JC) = ^2_a2 (3JC - a) for a<x<L

6EI

i

i t — r _

-W(JC) = - ^ - (x 4 - 4Lx 3 + 6L 2 x 2 ) for 0 < x < L 24EI V / - ~

W(JC) = - - ^ - (x4 - 2Lx 3 + L2x2) for 0 < x < L

24£/ V / ~ ~

(continued)

TABLE 2.2.2 Static Beam Deflection (Continued)

Beam Type and Load Transverse Deflection

^

< x < a w(x) = ^ Q * 0 *3 ~ \M 0 X 2 j for 0 :

1 /l , 1 9 r 9\ w(*) = ™ ( 5^°* ~ 2M0* "" 2 ^ ~ / f°r ° ~ * ~ L with J?0 = ^ ( V - M2) , M Q = r -{l- 3M2) , fi = l-a/L

In the next two subsections, three solution methods for beam static analysis are presented

and the fundamentals of the Euler-Bernoulli beam in bending are reviewed The reader who

is only interested in using the MATLAB toolbox can directly move to Section 2.3

2.2.2 Solution Methods

Many solution methods are available for static analysis of Euler-Bernoulli beams This

sec-tion introduces three analytical solusec-tion methods for the beam problem, namely the method

of singularity functions, the boundary value approach, and the distributed transfer function method

Method of Singularity Functions This method is often presented in textbooks on strength

of materials or solid mechanics; see References 3 and 4 for instance In the method, Eq (2.1)

is integrated four times, yielding

where the initial parameters

wo = w(0), 0 O = w'(0), Mo = - £ / M / ' ( 0 ) , R 0 = Elw"'(0) (2.7)

The sign convention of Ro and Mo follows that in Fig 2.2.3 Physically, Ro and Mo can be

known boundary disturbances, or unknown support reactions

The integral in Eq (2.6) can be expressed by the singularity function (JC — xo) n

Definition: The singularity function (x — xo) n is defined as follows:

From the above definition, {x — xo) l actually is the delta function 8(x — xo), and (x — xo)°

is the unite step function

With the singularity function, the integral / / / ffix) dx 4 for several types of external

o o o o forces is given in Table 2.2.3 The solution of a beam problem can be obtained according to

Eq (2.6) and Table 2.2.3, as shown in the following example

EXAMPLE 2.1

Consider a clamped-pinned beam subject to a torque at the midpoint; see Fig 2.2.4

The beam response, by Eq (2.6) and Table 2.2.3 is

EIw(x) = -R 0 x 3 - -M 0 x 2 + EI(Oox + w0) - \ {x - LIT) 2

TABLE 2.2.3 External Forces and Integrals

External Force Integral

and the rotation, bending moment, and shear force are

0(jt) = — ( x 2 x-{x- L/2)1 )

v }

EI \16L 8 X ' ) M(x) = x(^x- l {x-LI2f\

0.1 0.05

0 -0.05 -0.1

i

I — i

0.5

FIGURE 2.2.5

Boundary Value Approach

Response to boundary disturbances The response of a beam under boundary disturbances is

governed by

d 4 EI—jw(x) = 0, 0 < x < L

subject to boundary conditions as described in Table 2.2.1 The solution of the above equation

is of the form

where yo, yi, yi, and 5/3 are coefficients to be determined by the beam boundary conditions

Response to a pointwise force The response of a beam under a pointwise force q at x ~ Xf is

governed by

d 4 El-^wix) = q8(x - Xf\ 0<x<L (2.12)

The solution of Eq (2.12) is

The solution of Eq (2.15) is

w\(x) — ao~\- a\x + a 2 x 2 + fl3* 3 for 0 < x < xT

W2(x) = bo + £1* + &2*2 + b$x 3 for JCT < x < L (2.16) where the coefficients aj and ^ are determined by the beam boundary conditions and the

following compatibility conditions

20 STRESS, STRAIN, AND STRUCTURAL DYNAMICS

The solution of Eq (2.19) is of the form

w\(x) = ao 4- a\x 4- a 2 x 2 + a^x 3 for 0 < x < x\

W2(x) = do 4- di* 4- d^*2 + d^x 3 4- w>p(jc) for xi < x < X2

wi(x) = bo 4- ^ i * + b2X 2 4- ^3^3 for *2 < x < L

(2.20)

where the coefficients ay, &£, and d\ are determined by the boundary conditions of the beam

and the compatibility conditions

w\(x\) = w 2 (xi), w\(x\) = w f 2 (xi\ vS[(x\) = w^ixi), W[\x\) = wZ'ixi)

Distributed Transfer Function Method The MATLAB functions of this chapter are

devel-oped based on the Distributed Transfer Function Method (DTFM), which is an exact analytical modeling and solution method for many one-dimensional distributed systems including bars, shafts, strings, and beams Refer to Appendix C for more information on the DTFM

Spatial state formulation In the DTFM, Eq (2.1) is first cast into an equivalent spatial state

where [M b ] and [Nb] are four-by-four boundary matrices and {yb} is a boundary disturbance

vector For instance, for a clamped-free (cantilever) beam subject to an end moment Mi, the