The 2014 t6 aluminum rod ac is reinforced

Fundamental Equations of Mechanics of Materials Axial Load

Normal Stress

s = P A

Displacement

d = L L

0 P(x)dx A (x)E

d = ! PL AE

dT = a "TL

Torsion

Shear stress in circular shaft

t = Tr

J

where

J =

p

2 c4 solid cross section

J =

p

2 (co

4 - ci 4) tubular cross section

Power

P = Tv = 2pf T

Angle of twist

f = L L

0 T(x)dx J(x)G

f = !

TL JG

Average shear stress in a thin-walled tube

tavg =

T 2tA m

Shear Flow

q = tavg t =

T 2Am

Bending

Normal stress

s =

My

I Unsymmetric bending

s = -

Mz y

Iz +

Myz

Iy , tan a =

Iz Iy

tan u

Shear

Average direct shear stress

tavg = V A

Transverse shear stress

t =

VQ It

Shear flow

q = tt =

VQ I

Stress in Thin-Walled Pressure Vessel

Cylinder

s1 =

pr

t s2 =

pr

2t

Sphere

s1 = s2 =

pr

2t

Stress Transformation Equations

sx# = sx + sy

2 +

sx - sy 2

cos 2u + txy sin 2u

tx#y# = - sx - sy

2 sin 2u + txy cos 2u

Principal Stress

tan 2up = txy

(sx - sy)>2 s1,2 =

sx + sy 2

{ A asx - sy2 b2 + txy2 Maximum in-plane shear stress

tan 2us = - (sx - sy)>2

txy

tmax = A asx - sy2 b2 + t2xy savg =

sx + sy 2

Absolute maximum shear stress

tabs max

= smax

2 for smax, smin same sign

tabs max

= smax - smin

2 for smax, smin opposite signs

Geometric Properties of Area Elements Material Property Relations

Poisson’s ratio

n = - Plat

Plong

Generalized Hooke’s Law

ex =

1 E

3sx - n(sy + sz)4

ey = 1 E

3sy - n(sx + sz)4

ez = 1 E

3sz - n(sx + sy)4

gxy = 1 G

txy, gyz = 1 G

tyz, gzx = 1 G

tzx

where

G =

E 2(1 + n)

Relations Between w, V, M

dV dx

= w(x), dM dx

= V

Elastic Curve

1 r

= M EI

EI

d4v

dx4 = w(x)

EI

d3v

dx3 = V (x)

EI

d2v

dx2 = M(x)

Buckling Critical axial load

Pcr =

p2EI

(KL)2

Critical stress

scr =

p2E

(KL >r)2 , r = 2I>A Secant formula

smax =

P A

c 1 + ec r2

sec a L 2r

A PEA b d Energy Methods Conservation of energy

Ue = Ui

Strain energy Ui =

N2L 2AE

constant axial load

Ui = L

L

0

M2dx 2EI

bending moment

Ui = L

L

0

fsV 2dx

2GA transverse shear

Ui = L

L

0

T2dx 2GJ

torsional moment

xh

y A = bh

b

C

Rectangular area

Ix = bh 31

12

Iy = hb 31

12

Ix = bh 3x

h

A = bh

b

C

Triangular area

1 36h13

1 2

xh

A = h(a + b)

b

a

C

Trapezoidal area

h13 2a + b a + b

1 2

Ix = pr 4

x

y

C

Semicircular area

1 8

A = pr 2

2

4r 3p

Iy = pr 41

8

r

Ix = pr 4

x

y

C

Circular area

1 4

A = pr2

Iy = pr 41

4

r

A = ab

C

Semiparabolic area

2 3

a25

b38 a zero slope

b

Exparabolic area

a34 a

b310 zero slope

b C

A = ab3

Average Mechanical Properties of Typical Engineering Materialsa

(SI Units)

Materials Density R (Mg/m 3 )

Moduls of Elasticity E

(GPa)

Modulus of Rigidity G

(GPa)

Yield Strength (MPa) SY

Tens. Comp.b Shear

Ultimate Strength (MPa) Su

Tens. Comp. b Shear

%Elongation in 50 mm specimen

Poisson’s Ratio N

Coef. of Therm. Expansion A

(10–6)/°C

Metallic

Aluminum 2014-T6 Wrought Alloys 6061-T6

2.79 73.1 27 414 414 172 469 469 290 10 0.35 23

2.71 68.9 26 255 255 131 290 290 186 12 0.35 24

Cast Iron Gray ASTM 20 Alloys Malleable ASTM A-197

7.19 67.0 27 – – – 179 669 – 0.6 0.28 12

7.28 172 68 – – – 276 572 – 5 0.28 12

Copper Red Brass C83400 Alloys Bronze C86100

8.74 101 37 70.0 70.0 – 241 241 – 35 0.35 18

8.83 103 38 345 345 – 655 655 – 20 0.34 17

Magnesium [Am 1004-T61]

Alloy 1.83 44.7 18 152 152 – 276 276 152 1 0.30 26

Structural A-36

Steel Structural A992

Alloys Stainless 304

Tool L2

7.85 200 75 250 250 – 400 400 – 30 0.32 12

7.85 200 75 345 345 – 450 450 – 30 0.32 12

7.86 193 75 207 207 – 517 517 – 40 0.27 17

8.16 200 75 703 703 – 800 800 – 22 0.32 12

Titanium [Ti-6Al-4V]

Alloy 4.43 120 44 924 924 – 1,000 1,000 – 16 0.36 9.4

Nonmetallic

Concrete Low Strength

High Strength

2.38 22.1 – – – 12 – – – – 0.15 11

2.37 29.0 – – – 38 – – – – 0.15 11

Plastic Kevlar 49

Reinforced 30% Glass

1.45 131 – – – – 717 483 20.3 2.8 0.34 –

1.45 72.4 – – – – 90 131 – – 0.34 –

Wood Douglas Fir Select Structural White Spruce Grade

0.47 13.1 – – – – 2.1c 26d 6.2 d – 0.29e –

3.60 9.65 – – – – 2.5 c 36 d 6.7 d – 0.31 e –

a Specific values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment. For a more exact value reference books for the material should be consulted.

b The yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression. c Measured perpendicular to the grain. d Measured parallel to the grain. e Deformation measured perpendicular to the grain when the load is applied along the grain.

Average Mechanical Properties of Typical Engineering Materialsa

(U.S. Customary Units)

Materials Specific Weight (lb/in 3 )

Moduls of Elasticity E

(103) ksi

Modulus of Rigidity G (103) ksi

Yield Strength (ksi) SY

Tens. Comp.b Shear

Ultimate Strength (ksi) Su

Tens. Comp. b Shear

%Elongation in 2 in. specimen

Poisson’s Ratio N

Coef. of Therm. Expansion A

(10–6)/°F

Metallic

Aluminum 2014-T6 Wrought Alloys 6061-T6

0.101 10.6 3.9 60 60 25 68 68 42 10 0.35 12.8

0.098 10.0 3.7 37 37 19 42 42 27 12 0.35 13.1

Cast Iron Gray ASTM 20 Alloys Malleable ASTM A-197

0.260 10.0 3.9 – – – 26 96 – 0.6 0.28 6.70

0.263 25.0 9.8 – – – 40 83 – 5 0.28 6.60

Copper Red Brass C83400

Alloys Bronze C86100

0.316 14.6 5.4 11.4 11.4 – 35 35 – 35 0.35 9.80

0.319 15.0 5.6 50 50 – 35 35 – 20 0.34 9.60

Magnesium [Am 1004-T61]

Alloy 0.066 6.48 2.5 22 22 – 40 40 22 1 0.30 14.3

Structural A-36

Steel Structural A992

Alloys Stainless 304

Tool L2

0.284 29.0 11.0 36 36 – 58 58 – 30 0.32 6.60

0.284 29.0 11.0 50 50 – 65 65 – 30 0.32 6.60

0.284 28.0 11.0 30 30 – 75 75 – 40 0.27 9.60

0.295 29.0 11.0 102 102 – 116 116 – 22 0.32 6.50

Titanium [Ti-6Al-4V]

Alloy 0.160 17.4 6.4 134 134 – 145 145 – 16 0.36 5.20

Nonmetallic

Concrete Low Strength

High Strength

0.086 3.20 – – – 1.8 – – – – 0.15 6.0

0.086 4.20 – – – 5.5 – – – – 0.15 6.0

Plastic Kevlar 49

Reinforced 30% Glass

0.0524 19.0 – – – – 104 70 10.2 2.8 0.34 –

0.0524 10.5 – – – – 13 19 – – 0.34 –

Wood Douglas Fir Select Structural White Spruce Grade

0.017 1.90 – – – – 0.30c 3.78d 0.90 d – 0.29e –

0.130 1.40 – – – – 0.36 c 5.18 d 0.97 d – 0.31 e –

a Specific values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment. For a more exact value reference books for the material should be consulted.

b The yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression. c Measured perpendicular to the grain. d Measured parallel to the grain. e Deformation measured perpendicular to the grain when the load is applied along the grain.

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS NINTH EDITION

R. C. HIBBELER

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Copyright © 2014, 2011, 2008, 2005, 2003, 2001 by R. C. Hibbeler. Published by Pearson Prentice Hall. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1 Lake Street, Upper Saddle River, NJ 07458.

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10 9 8 7 6 5 4 3

Library of Congress Cataloging-in-Publication Data on File.

ISBN 10: 0-13-325442-9 ISBN 13: 978-0-13-325442-6

Prentice Hall is an imprint of

www.pearsonhighered.com

To the Student With the hope that this work will stimulate

an interest in Engineering Mechanics and provide an acceptable guide to its understanding.

PREFACE

It is intended that this book provide the student with a clear and thorough presentation of the theory and application of the principles of mechanics of materials. To achieve this objective, over the years this work has been shaped by the comments and suggestions of hundreds of reviewers in the teaching profession, as well as many of the author’s students. The eighth edition has been significantly enhanced from the previous edition, and it is hoped that both the instructor and student will benefit greatly from these improvements.

New to This Edition

• Preliminary Problems. This feature can be found throughout the text, and is given just before the Fundamental Problems. The intent here is to test the student’s conceptual understanding of the theory. Normally the solutions require little or no calculation, and as such, these problems provide a basic understanding of the concepts before they are applied numerically. All the solutions are given in the back of the text.

• Updated Examples. Some portions of the text have been rewritten in order to enhance clarity and be more succinct. In this regard, some new examples have been added and others have been modified to provide more emphasis on the application of important concepts. Included is application of the LRFD method of design, and use of A992 steel for structural applications. Also, the artwork has been improved throughout the book to support these changes.

• New Photos. The relevance of knowing the subject matter is reflected by the real-world applications depicted in over 30 new or updated photos placed throughout the book. These photos generally are used to explain how the relevant principles apply to real-world situations and how materials behave under load.

• Additional Fundamental Problems. These problem sets are located just after each group of example problems. In this edition they have been expanded. They offer students simple applications of the concepts covered in each section and, therefore, provide them with the chance to develop their problem-solving skills before attempting to solve any of the standard problems that follow. The fundamental problems may be considered as extended examples, since the key equations and answers are all listed in the back of the book. Additionally, when assigned, these problems offer students an excellent means of preparing for exams, and they can be used at a later time as a review when studying for the Fundamentals of Engineering Exam.

VI I I PREFACE

• Additional Conceptual Problems. Throughout the text, usually at the end of each chapter, there is a set of problems that involve conceptual situations related to the application of the principles contained in the chapter. These analysis and design problems are intended to engage the students in thinking through a real-life situation as depicted in a photo. They can be assigned after the students have developed some expertise in the subject matter and they work well either for individual or team projects.

• New Problems. There are approximately 31%, or about 460, new problems added to this edition, which involve applications to many different fields of engineering.

Contents The subject matter is organized into 14 chapters. Chapter 1 begins with a review of the important concepts of statics, followed by a formal definition of both normal and shear stress, and a discussion of normal stress in axially loaded members and average shear stress caused by direct shear.

In Chapter 2 normal and shear strain are defined, and in Chapter 3 a discussion of some of the important mechanical properties of materials is given. Separate treatments of axial load, torsion, and bending are presented in Chapters 4 , 5 , and 6 , respectively. In each of these chapters, both linear- elastic and plastic behavior of the material are considered. Also, topics related to stress concentrations and residual stress are included. Transverse shear is discussed in Chapter 7 , along with a discussion of thin-walled tubes, shear flow, and the shear center. Chapter 8 includes a discussion of thin-walled pressure vessels and provides a partial review of the material covered in the previous chapters, where the state of stress results from combined loadings. In Chapter 9 the concepts for transforming multiaxial states of stress are presented. In a similar manner, Chapter 10 discusses the methods for strain transformation, including the application of various theories of failure. Chapter 11 provides a means for a further summary and review of previous material by covering design applications of beams and shafts. In Chapter 12 various methods for computing deflections of beams and shafts are covered. Also included is a discussion for finding the reactions on these members if they are statically indeterminate. Chapter 13 provides a discussion of column buckling, and lastly, in Chapter 14 the problem of impact and the application of various energy methods for computing deflections are considered.

Sections of the book that contain more advanced material are indicated by a star (*). Time permitting, some of these topics may be included in the course. Furthermore, this material provides a suitable reference for basic principles when it is covered in other courses, and it can be used as a basis for assigning special projects.

PREFACE IX

Alternative Method of Coverage. Some instructors prefer to cover stress and strain transformations first, before discussing specific applications of axial load, torsion, bending, and shear. One possible method for doing this would be first to cover stress and its transformation, Chapter 1 and Chapter 9, followed by strain and its transformation, Chapter 2 and the first part of Chapter 10. The discussion and example problems in these later chapters have been styled so that this is possible. Also, the problem sets have been subdivided so that this material can be covered without prior knowledge of the intervening chapters. Chapters 3 through 8 can then be covered with no loss in continuity.

Hallmark Elements Organization and Approach. The contents of each chapter are organized into well-defined sections that contain an explanation of specific topics, illustrative example problems, and a set of homework problems. The topics within each section are placed into subgroups defined by titles. The purpose of this is to present a structured method for introducing each new definition or concept and to make the book convenient for later reference and review.

Chapter Contents. Each chapter begins with a full-page illustration that indicates a broad-range application of the material within the chapter. The “Chapter Objectives” are then provided to give a general overview of the material that will be covered.

Procedures for Analysis. Found after many of the sections of the book, this unique feature provides the student with a logical and orderly method to follow when applying the theory. The example problems are solved using this outlined method in order to clarify its numerical application. It is to be understood, however, that once the relevant principles have been mastered and enough confidence and judgment have been obtained, the student can then develop his or her own procedures for solving problems.

Photographs. Many photographs are used throughout the book to enhance conceptual understanding and explain how the principles of mechanics of materials apply to real-world situations.

Important Points. This feature provides a review or summary of the most important concepts in a section and highlights the most significant points that should be realized when applying the theory to solve problems.

Example Problems. All the example problems are presented in a concise manner and in a style that is easy to understand.

Homework Problems. Apart from the preliminary, fundamental, and conceptual problems, there are numerous standard problems in the

X PREFACE

book that depict realistic situations encountered in engineering practice. It is hoped that this realism will both stimulate the student’s interest in the subject and provide a means for developing the skill to reduce any such problem from its physical description to a model or a symbolic representation to which principles may be applied. Throughout the book there is an approximate balance of problems using either SI or FPS units. Furthermore, in any set, an attempt has been made to arrange the problems in order of increasing difficulty. The answers to all but every fourth problem are listed in the back of the book. To alert the user to a problem without a reported answer, an asterisk (*) is placed before the problem number. Answers are reported to three significant figures, even though the data for material properties may be known with less accuracy. Although this might appear to be a poor practice, it is done simply to be consistent, and to allow the student a better chance to validate his or her solution. A solid square (!) is used to identify problems that require a numerical analysis or a computer application.

Appendices. The appendices of the book provide a source for review and a listing of tabular data. Appendix A provides information on the centroid and the moment of inertia of an area. Appendices B and C list tabular data for structural shapes, and the deflection and slopes of various types of beams and shafts.

Accuracy Checking. The Ninth Edition has undergone a rigorous Triple Accuracy Checking review. In addition to the author’s review of all art pieces and pages, the text was checked by the following individuals:

• Scott Hendricks, Virginia Polytechnic University • Karim Nohra, University of South Florida • Kurt Norlin, LaurelTech Integrated Publishing Services • Kai Beng Yap, Engineering Consultant

Acknowledgments Over the years, this text has been shaped by the suggestions and comments of many of my colleagues in the teaching profession. Their encouragement and willingness to provide constructive criticism are very much appreciated and it is hoped that they will accept this anonymous recognition. A note of thanks is given to the reviewers.

S. Suliman, Penn State C. Valle, Georgia Institute of Tech C. Sulzbach, Colorado School of Mines K. Cook-Chennault, Rutgers University J. Ramirez, Purdue University J. Oyler, University of Pittsburg

PREFACE XI

P. Mokashi, Ohio State Y. Liao, Arizona State University P. Ziehl, University of South Carolina

There are a few people that I feel deserve particular recognition. A long- time friend and associate, Kai Beng Yap, was of great help to me in checking the entire manuscript and helping to prepare the problem solutions. A special note of thanks also goes to Kurt Norlin in this regard. During the production process I am thankful for the assistance of Rose Kernan, my production editor for many years, and to my wife, Conny, and daughter, Mary Ann, for their help in proofreading and typing, that was needed to prepare the manuscript for publication.

I would also like to thank all my students who have used the previous edition and have made comments to improve its contents; including those in the teaching profession who have taken the time to e-mail me their comments, notably S. Alghamdi, A. Atai, S. Larwood, D. Kuemmerle, and J. Love.

I would greatly appreciate hearing from you if at any time you have any comments or suggestions regarding the contents of this edition.

Russell Charles Hibbeler hibbeler@bellsouth.net Mastering Ad to come

your work...

your answer specific feedback

®

XIV PREFACE

Resources for Instructors • MasteringEngineering. This online Tutorial Homework program allows you to integrate dynamic homework with automatic grading and adaptive tutoring. MasteringEngineering allows you to easily track the performance of your entire class on an assignment-by- assignment basis, or the detailed work of an individual student. • Instructor’s Solutions Manual. An instructor’s solutions manual was prepared by the author. The manual includes homework assignment lists and was also checked as part of the accuracy checking program. The Instructor Solutions Manual is available at www.pearsonhighered.com . • Presentation Resources. All art from the text is available in PowerPoint slide and JPEG format. These files are available for download from the Instructor Resource Center at www.pearsonhighered.com . If you are in need of a login and password for this site, please contact your local Pearson representative. • Video Solutions. Developed by Professor Edward Berger, University of Virginia, video solutions located on the Companion Website offer step-by-step solution walkthroughs of representative homework problems from each section of the text. Make efficient use of class time and office hours by showing students the complete and concise problem solving approaches that they can access anytime and view at their own pace. The videos are designed to be a flexible resource to be used however each instructor and student prefers. A valuable tutorial resource, the videos are also helpful for student self-evaluation as students can pause the videos to check their understanding and work alongside the video. Access the videos at www.pearsonhighered.com/ hibbeler and follow the links for the Mechanics of Materials text.

Resources for Students • Mastering Engineering. Tutorial homework problems emulate the instrutor’s office-hour environment. • Companion Website —The Companion Website, located at www.pearsonhighered.com/hibbeler includes opportunities for practice and review including: • Video Solutions —Complete, step-by-step solution walkthroughs of representative homework problems from each section. Videos offer: students need it with over 20 hours helpful review.

An access code for the Mechanics of Materials, Ninth Edition companion website was included with this text. To redeem the code and gain access to the site, go to www.pearsonhighered.com/hibbeler and follow the directions on the access code card. Access can also be purchased directly from the site.

CONTENTS

1 Stress 3

Chapter Objectives 3 1.1 Introduction 3 1.2 Equilibrium of a Deformable Body 4 1.3 Stress 22 1.4 Average Normal Stress in an Axially

Loaded Bar 24 1.5 Average Shear Stress 32 1.6 Allowable Stress Design 46 1.7 Limit State Design 48

2 Strain 67

Chapter Objectives 67 2.1 Deformation 67 2.2 Strain 68

3 Mechanical Properties of Materials 83

Chapter Objectives 83 3.1 The Tension and Compression Test 83 3.2 The Stress–Strain Diagram 85 3.3 Stress–Strain Behavior of Ductile and

Brittle Materials 89 3.4 Hooke’s Law 92 3.5 Strain Energy 94 3.6 Poisson’s Ratio 104 3.7 The Shear Stress–Strain Diagram 106 *3.8 Failure of Materials Due to Creep

and Fatigue 109

4 Axial Load 121

Chapter Objectives 121 4.1 Saint-Venant’s Principle 121 4.2 Elastic Deformation of an Axially Loaded

Member 124 4.3 Principle of Superposition 138 4.4 Statically Indeterminate Axially Loaded

Member 139 4.5 The Force Method of Analysis for Axially

Loaded Members 145 4.6 Thermal Stress 153 4.7 Stress Concentrations 160 *4.8 Inelastic Axial Deformation 164 *4.9 Residual Stress 166

5 Torsion 181

Chapter Objectives 181 5.1 Torsional Deformation of a Circular

Shaft 181 5.2 The Torsion Formula 184 5.3 Power Transmission 192 5.4 Angle of Twist 204 5.5 Statically Indeterminate Torque-Loaded

Members 218 *5.6 Solid Noncircular Shafts 225 *5.7 Thin-Walled Tubes Having Closed Cross

Sections 228 5.8 Stress Concentration 238 *5.9 Inelastic Torsion 241 *5.10 Residual Stress 243

XVI CONTENTS

6 Bending 259

Chapter Objectives 259 6.1 Shear and Moment Diagrams 259 6.2 Graphical Method for Constructing Shear

and Moment Diagrams 266 6.3 Bending Deformation of a Straight

Member 285 6.4 The Flexure Formula 289 6.5 Unsymmetric Bending 306 *6.6 Composite Beams 316 *6.7 Reinforced Concrete Beams 319 *6.8 Curved Beams 323 6.9 Stress Concentrations 330 *6.10 Inelastic Bending 339

7 Transverse Shear 363

Chapter Objectives 363 7.1 Shear in Straight Members 363 7.2 The Shear Formula 365 7.3 Shear Flow in Built-Up Members 382 7.4 Shear Flow in Thin-Walled Members 391 *7.5 Shear Center for Open Thin-Walled

Members 396

8 Combined Loadings 409

Chapter Objectives 409 8.1 Thin-Walled Pressure Vessels 409 8.2 State of Stress Caused by Combined

Loadings 416

9 Stress Transformation 441

Chapter Objectives 441 9.1 Plane-Stress Transformation 441 9.2 General Equations of Plane-Stress

Transformation 446 9.3 Principal Stresses and Maximum In-Plane

Shear Stress 449 9.4 Mohr’s Circle—Plane Stress 465 9.5 Absolute Maximum Shear Stress 477

10 Strain Transformation 489

Chapter Objectives 489 10.1 Plane Strain 489 10.2 General Equations of Plane-Strain

Transformation 490 *10.3 Mohr’s Circle—Plane Strain 498 *10.4 Absolute Maximum Shear Strain 506 10.5 Strain Rosettes 508 10.6 Material-Property Relationships 512 *10.7 Theories of Failure 524

11 Design of Beams and Shafts 541

Chapter Objectives 541 11.1 Basis for Beam Design 541 11.2 Prismatic Beam Design 544 *11.3 Fully Stressed Beams 558 *11.4 Shaft Design 562

CONTENTS XVI I

12 Deflection of Beams and Shafts 573

Chapter Objectives 573 12.1 The Elastic Curve 573 12.2 Slope and Displacement by

Integration 577 *12.3 Discontinuity Functions 597 *12.4 Slope and Displacement by the Moment-

Area Method 608 12.5 Method of Superposition 623 12.6 Statically Indeterminate Beams

and Shafts 631 12.7 Statically Indeterminate Beams and

Shafts—Method of Integration 632 *12.8 Statically Indeterminate Beams and

Shafts—Moment-Area Method 637 12.9 Statically Indeterminate Beams and

Shafts—Method of Superposition 643

13 Buckling of Columns 661

Chapter Objectives 661 13.1 Critical Load 661 13.2 Ideal Column with Pin Supports 664 13.3 Columns Having Various Types

of Supports 670 *13.4 The Secant Formula 682 *13.5 Inelastic Buckling 688 *13.6 Design of Columns for Concentric

Loading 696 *13.7 Design of Columns for Eccentric

Loading 707

14 Energy Methods 719

Chapter Objectives 719 14.1 External Work and Strain Energy 719 14.2 Elastic Strain Energy for Various Types