Structural Analysis, 6th Edition PDF

 BENDING MOMENTS, SLOPES, AND DEFLECTIONS OF PRISMATIC BEAMS UNDER VARIOUS LOADING CONDITIONS P a...

 BENDING MOMENTS, SLOPES, AND DEFLECTIONS OF PRISMATIC BEAMS UNDER VARIOUS LOADING CONDITIONS P a B A yB x L θB −Pa a M B A yB x L θB −M a w B A yB x L θB M = w2 (2ax − a2 − x 2) 2 − wa2 for 0 # x # a a w B A yB x θB L w M = − 6a (a − x) 3 2 − wa6 for 0 # x # a 30931_end_hr_02-04.indd 2 10/27/18 3:13 PM  P L L 2 2 A B θA θB ymax x PL 4 P a b A B θA θB x ymax L Pab L M A B θA ymax θB x L M 30931_end_hr_02-04.indd 3 10/27/18 3:13 PM  a M θA A B θB x L M (1 − a L ) − Ma L w A B θA θB ymax x L wL2 8 M = wx 2 (L − x) a w A B θA θB x L M = wax 2L (2L − a − Lx a ) 0#x#a M = wa2 (1 − Lx ) 2 a#x#L a w A B θA θB ymax x L wL2 M = wx 6L (L2 − x 2) 9 3 L 3 30931_end_hr_02-04.indd 4 10/27/18 3:13 PM  Structural Analysis Sixth Edition 30931_fm_hr_i-xviii.indd 1 10/26/18 10:18 AM 30931_fm_hr_i-xviii.indd 2 10/26/18 10:18 AM  Structural Analysis Sixth Edition Aslam Kassimali Southern Illinois University-Carbondale Australia Brazil Mexico Singapore United Kingdom United States 30931_fm_hr_i-xviii.indd 3 10/26/18 10:18 AM  Structural Analysis, Sixth Edition © 2020, 2015, 2011 Cengage Learning, Inc. Aslam Kassimali Unless otherwise noted, all content is © Cengage ALL RIGHTS RESERVED. No part of this work covered by the copyright herein Product Director, Global Engineering: may be reproduced or distributed in any form or by any means, except as Timothy L. Anderson permitted by U.S. copyright law, without the prior written permission of the copyright owner. Senior Product Assistant: Alexander Sham Associate Marketing Manager: Tori Sitcawich For product information and technology assistance, contact us at Content Manager: Samantha Gomez Cengage Customer & Sales Support, 1-800-354-9706 or support.cengage.com. IP Analyst: Nancy Dillon IP Project Manager: Jillian Shafer For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Production Service: RPK Editorial Services, Inc. Compositor: SPi Global Library of Congress Control Number: 2018958091 Senior Designer: Diana Graham Student Edition: Cover Image: Felix Lipov/ShutterStock.com ISBN: 978-1-337-63093-1 Manufacturing Planner: Doug Wilke Loose-leaf Edition: ISBN: 978-0-35703096-7 Cengage 20 Channel Center Street Boston, MA 02210 USA Cengage is a leading provider of customized learning solutions with employees residing in nearly 40 different countries and sales in more than 125 countries around the world. Find your local representative at www.cengage.com. Cengage products are represented in Canada by Nelson Education, Ltd. To learn more about Cengage platforms and services, register or access your online learning solution, or purchase materials for your course, visit www.cengage.com. Printed in the United States of America Print Number: 01 Print Year: 2018 30931_fm_hr_i-xviii.indd 4 10/26/18 10:18 AM  IN MEMORY OF AMI AND APAJAN 30931_fm_hr_i-xviii.indd 5 10/26/18 10:18 AM 30931_fm_hr_i-xviii.indd 6 10/26/18 10:18 AM  Contents Preface xiii About the Author xvii Part 1 Introduction to Structural Analysis and Loads  1 1 Introduction to Structural Analysis 3 1.1 Historical Background 3 1.2 Role of Structural Analysis in Structural Engineering Projects 5 1.3 Classification of Structures 7 1.4 Analytical Models 12 Summary 16 2 Loads on Structures 17 2.1 Structural Systems for Transmitting Loads 18 2.2 Dead Loads 29 2.3 Live Loads 31 2.4 Classification of Buildings for Environmental Loads 34 2.5 Wind Loads 35 2.6 Snow Loads 41 2.7 Earthquake Loads 44 2.8 Hydrostatic and Soil Pressures 45 2.9 Thermal and Other Effects 45 2.10 Load Combinations 46 Summary 46 Problems 48 vii 30931_fm_hr_i-xviii.indd 7 10/26/18 10:18 AM  viii   Contents Part 2 Analysis of Statically Determinate Structures  51 3 Equilibrium and Support Reactions 53 3.1 Equilibrium of Structures 53 3.2 External and Internal Forces 56 3.3 Types of Supports for Plane Structures 56 3.4 Static Determinacy, Indeterminacy, and Instability 58 3.5 Computation of Reactions 69 3.6 Principle of Superposition 85 3.7 Reactions of Simply Supported Structures Using Proportions 86 Summary 88 Problems 89 4 Plane and Space Trusses 97 4.1 Assumptions for Analysis of Trusses 99 4.2 Arrangement of Members of Plane Trusses—Internal Stability 103 4.3 Equations of Condition for Plane Trusses 107 4.4 Static Determinacy, Indeterminacy, and Instability of Plane Trusses 107 4.5 Analysis of Plane Trusses by the Method of Joints 113 4.6 Analysis of Plane Trusses by the Method of Sections 126 4.7 Analysis of Compound Trusses 132 4.8 Complex Trusses 137 4.9 Space Trusses 138 Summary 147 Problems 148 5 Beams and Frames: Shear and Bending Moment 161 5.1 Axial Force, Shear, and Bending Moment 161 5.2 Shear and Bending Moment Diagrams 168 5.3 Qualitative Deflected Shapes 172 5.4 Relationships between Loads, Shears, and Bending Moments 173 5.5 Static Determinacy, Indeterminacy, and Instability of Plane Frames 192 5.6 Analysis of Plane Frames 200 Summary 213 Problems 215 30931_fm_hr_i-xviii.indd 8 10/26/18 10:18 AM  Contents   ix 6 Deflections of Beams: Geometric Methods 224 6.1 Differential Equation for Beam Deflection 225 6.2 Direct Integration Method 227 6.3 Superposition Method 231 6.4 Moment-Area Method 231 6.5 Bending Moment Diagrams by Parts 243 6.6 Conjugate-Beam Method 247 Summary 262 Problems 262 7 Deflections of Trusses, Beams, and Frames: Work–Energy Methods 268 7.1 Work 268 7.2 Principle of Virtual Work 270 7.3 Deflections of Trusses by the Virtual Work Method 274 7.4 Deflections of Beams by the Virtual Work Method 283 7.5 Deflections of Frames by the Virtual Work Method 295 7.6 Conservation of Energy and Strain Energy 306 7.7 Castigliano’s Second Theorem 309 7.8 Betti’s Law and Maxwell’s Law of Reciprocal Deflections 317 Summary 319 Problems 320 8 Influence Lines 329 8.1 Influence Lines for Beams and Frames by Equilibrium Method 330 8.2 Müller-Breslau’s Principle and Qualitative Influence Lines 344 8.3 Influence Lines for Girders with Floor Systems 356 8.4 Influence Lines for Trusses 366 8.5 Influence Lines for Deflections 377 Summary 380 Problems 380 9 Application of Influence Lines 387 9.1 Response at a Particular Location Due to a Single Moving Concen- trated Load 387 9.2 Response at a Particular Location Due to a Uniformly Distributed Live Load 389 30931_fm_hr_i-xviii.indd 9 10/26/18 10:18 AM  x   Contents 9.3 Response at a Particular Location Due to a Series of Moving Concentrated Loads 393 9.4 Absolute Maximum Response 400 Summary 405 Problems 406 10 Analysis of Symmetric Structures 408 10.1 Symmetric Structures 408 10.2 Symmetric and Antisymmetric Components of Loadings 414 10.3 Behavior of Symmetric Structures under Symmetric and Antisymmetric Loadings 424 10.4 Procedure for Analysis of Symmetric Structures 428 Summary 435 Problems 436 Part 3 Analysis of Statically Indeterminate Structures  439 11 Introduction to Statically Indeterminate Structures 441 11.1 Advantages and Disadvantages of Indeterminate Structures 442 11.2 Analysis of Indeterminate Structures 445 Summary 449 12 Approximate Analysis of Rectangular Building Frames 450 12.1 Assumptions for Approximate Analysis 451 12.2 Analysis for Vertical Loads 454 12.3 Analysis for Lateral Loads—Portal Method 458 12.4 Analysis for Lateral Loads—Cantilever Method 473 Summary 480 Problems 480 13 Method of Consistent Deformations—Force Method 483 13.1 Structures with a Single Degree of Indeterminacy 484 13.2 Internal Forces and Moments as Redundants 504 13.3 Structures with Multiple Degrees of Indeterminacy 515 13.4 Support Settlements, Temperature Changes, and Fabrication Errors 537 13.5 Method of Least Work 545 Summary 551 Problems 552 30931_fm_hr_i-xviii.indd 10 10/26/18 10:18 AM  Contents   xi 14 Influence Lines for Statically Indeterminate Structures 559 14.1 Influence Lines for Beams and Trusses 560 14.2 Qualitative Influence Lines by Müller-Breslau’s Principle 575 Summary 579 Problems 580 15 Slope-Deflection Method 583 15.1 Slope-Deflection Equations 584 15.2 Basic Concept of the Slope-Deflection Method 591 15.3 Analysis of Continuous Beams 598 15.4 Analysis of Frames without Sidesway 617 15.5 Analysis of Frames with Sidesway 625 Summary 643 Problems 643 16 Moment-Distribution Method 648 16.1 Definitions and Terminology 649 16.2 Basic Concept of the Moment-Distribution Method 657 16.3 Analysis of Continuous Beams 665 16.4 Analysis of Frames without Sidesway 678 16.5 Analysis of Frames with Sidesway 681 Summary 696 Problems 697 17 Introduction to Matrix Structural Analysis 702 17.1 Analytical Model 703 17.2 Member Stiffness Relations in Local Coordinates 707 17.3 Coordinate Transformations 714 17.4 Member Stiffness Relations in Global Coordinates 719 17.5 Structure Stiffness Relations 721 17.6 Procedure for Analysis 728 Summary 745 Problems 745 APPENDIX A Areas and Centroids of Geometric Shapes 747 30931_fm_hr_i-xviii.indd 11 10/26/18 10:18 AM  xii   Contents APPENDIX B Review of Matrix Algebra 749 B.1 Definition of a Matrix 749 B.2 Types of Matrices 750 B.3 Matrix Operations 752 B.4 Solution of Simultaneous Equations by the Gauss-Jordan Method 758 Problems 762 APPENDIX C Computer Software 763 C.1 Starting the Computer Software 763 C.2 Inputting Data 763 C.3 Results of the Analysis 769 Problems 774 APPENDIX D Three-Moment Equation 775 D.1 Derivation of Three-Moment Equation 775 D.2 Application of Three-Moment Equation 780 Summary 786 Problems 787 Bibliography 789 Answers to Selected Problems 791 Index 799 30931_fm_hr_i-xviii.indd 12 10/26/18 10:18 AM  Preface The objective of this book is to develop an understanding of the basic principles of structural analysis. Emphasizing the intuitive classical approach, ­Structural Analysis covers the analysis of statically determinate and indeterminate beams, trusses, and rigid frames. It also presents an introduction to the matrix analysis of structures. The book is divided into three parts. Part One presents a general intro- duction to the subject of structural engineering. It includes a chapter devoted entirely to the topic of loads because attention to this important topic is gen- erally lacking in many civil engineering curricula. Part Two, consisting of Chapters 3 through 10, covers the analysis of statically determinate beams, trusses, and rigid frames. The chapters on deflections (Chapters 6 and 7) are placed before those on influence lines (Chapters 8 and 9) so that influence lines for deflections can be included in the latter chapters. This part also contains a chapter on the analysis of symmetric structures (Chapter 10). Part Three of the book, Chapters 11 through 17, covers the analysis of statically indeterminate structures. The format of the book is flexible to enable instructors to emphasize topics that are consistent with the goals of the course. Each chapter of the book begins with an introductory section defining its objective and ends with a summary section outlining its salient features. An important general feature of the book is the inclusion of step-by-step proce- dures for analysis to enable students to make an easier transition from theory to problem solving. Numerous solved examples are provided to illustrate the application of the fundamental concepts. A computer program for analyzing plane framed structures is available on the publisher’s website at https://login.cengage.com. It is also available at https://www.cengage.com/engineering/kassimali/software. This interactive software can be used to simulate a variety of structural and loading configu- rations and to determine cause versus effect relationships between loading and various structural parameters, thereby enhancing the students’ understanding of the behavior of structures. The software shows deflected shapes of struc- tures to enhance students’ understanding of structural response due to vari- ous types of loadings. It can also include the effects of support settlements, xiii 30931_fm_hr_i-xviii.indd 13 10/26/18 10:18 AM  xiv   Preface temperature changes, and fabrication errors in the analysis. A solutions man- ual, containing complete solutions to over 600 text exercises, is also available for the instructor. New to the Sixth Edition Building upon the original theme of this book, which is that detailed explanations of concepts provide the most effective means of teaching structural analysis, the following improvements and changes have been made in this sixth edition: Over 20 percent of the problems from the previous edition have been replaced with new ones. The chapter on loads has been revised to meet the provisions of the ASCE/SEI 7-16 Standard and the latest AASHTO-LRFD Specifications. The content of the chapter on the application of influence lines has been updated to incorporate the current HL-93 truck/tandem loadings as per AASHTO-LRFD Specifications. Throughout the book, there are numerous other revisions to enhance clarity and reinforce concepts. These include several new and upgraded examples in Chapters 3, 5, 10, 13, 15, 16 and in Appendix C, as well as an expanded discussion of the analysis of plane frames via classical versus matrix methods (Chapter 17). Some photographs have been replaced with new ones, and the page layout has been redesigned to enhance clarity. Finally, the computer software has been upgraded and recompiled to make it compatible with the latest versions of Microsoft Windows. Ancillaries for the Sixth Edition Worked-out solutions to all end-of-chapter problems are provided in the Instructors Solutions Manual and are available digitally to registered instruc- tors on the instructor resources web site. Image Banks containing every figure in the book are also available at https://login.cengage.com. The computer pro- gram for analyzing plane framed structures is available for both students and instructors through either https://login.cengage.com or https://www.cengage.com/engineering/kassimali/software. Acknowledgments I wish to express my thanks to Timothy Anderson, Mona Zeftel, and ­Alexander Sham of Cengage Learning for their constant support and encouragement throughout this project, and to Rose Kernan for all her help during the 30931_fm_hr_i-xviii.indd 14 10/26/18 10:18 AM  Preface   xv production phase. The comments and suggestions for improvement from colleagues and students who have used previous editions are gratefully acknowledged. All of their suggestions were carefully considered, and implemented whenever possible. Thanks are due to the following reviewers for their careful reviews of the manuscripts of the various editions, and for their constructive suggestions: Ayo Abatan George Kostyrko Virginia Polytechnic Institute and California State University State University E. W. Larson Riyad S. Aboutaha California State University/ Syracuse University Northridge Osama Abudayyeh Yue Li Western Michigan University Michigan Technological Thomas T. Baber University University of Virginia Roberto Lopez-Anido Gordon B. Batson University of Maine Clarkson University Eugene B. Loverich George E. Blandford Northern Arizona University University of Kentucky Lee L. Lowery, Jr. Ramon F. Borges Texas A&M University Penn State/Altoona College Eric M. Lui Kenneth E. Buttry Syracuse University University of Wisconsin L. D. Lutes Steve C. S. Cai Texas A&M University Louisiana State University David Mazurek William F. Carroll US Coast Guard Academy University of Central Florida Ghyslaine McClure Malcolm A. Cutchins McGill University Auburn University Ahmad Namini Jack H. Emanuel University of Miami University of Missouri—Rolla Farhad Reza Fouad Fanous Minnesota State University, Iowa State University Mankato Leon Feign Dominik Schillinger Fairfield University University of Minnesota Robert Fleischman Arturo E. Schultz University of Notre Dame North Carolina State University Ahmed Ibrahim Jason Stewart University of Idaho Arkansas State University Robert I. Johnson Elaina J. Sutley Colorado State University University of Kansas Changhong Ke Kassim Tarhini SUNY, Binghamton Valparaiso University 30931_fm_hr_i-xviii.indd 15 10/26/18 10:18 AM  xvi   Preface Robert Taylor Nicholas Willems Northeastern University University of Kansas Jale Tezcan John Zachar Southern Illinois University Milwaukee School of C. C. Tung Engineering North Carolina State Mannocherh Zoghi University University of Dayton Finally, I would like to express my loving appreciation to my wife, Maureen, for her constant encouragement and help in preparing this manu- script, and to my sons, Jamil and Nadim, for their love, understanding, and patience. Aslam Kassimali 30931_fm_hr_i-xviii.indd 16 10/26/18 10:18 AM  About the Author Aslam Kassimali was born in Karachi, Pakistan. He received his Bachelor of Engineering (B.E.) degree in civil engineering from the University of Karachi (N.E.D. College) in Pakistan in 1969. In 1971, he earned a Master of Engineering (M.E.) degree in civil engineering from Iowa State University in Ames, Iowa, USA. After completing further studies and research at the University of Missouri at Columbia in the USA, he received Master of Science (M.S.) and Ph.D. degrees in civil engineering in 1974 and 1976, respectively. His practical experience includes work as a Structural Design Engineer for Lutz, Daily and Brain, Consulting Engineers, Shawnee Mission, Kansas (USA), from January to July 1973, and as a Structural Engineering Specialist and Analyst for Sargent & Lundy Engineers in Chicago, Illinois (USA) from 1978 to 1980. He joined Southern Illinois University—Carbondale (USA) as an Assistant Professor in 1980, and was promoted to the rank of Professor in 1993. Consistently recognized for teaching excellence, Dr. Kassimali has received over 20 awards for outstanding teaching at Southern Illinois University— Carbondale, and was awarded the title of Distinguished Teacher in 2004. He is currently a Professor and Distinguished Teacher in the Department of Civil & Environmental Engineering at Southern Illinois University in Carbondale, Illinois (USA). He has authored and co-authored four textbooks on structural analysis and mechanics, and has published a number of papers in the area of nonlinear structural analysis. Dr. Kassimali is a life member of the American Society of Civil Engineers (ASCE) and has served on the ASCE Structural Division Committees on Shock and Vibratory Effects, Special Structures, and Methods of Analysis. xvii 30931_fm_hr_i-xviii.indd 17 10/26/18 10:18 AM  xviii   MindTap Reader Available via our digital subscription service, Cengage Unlimited, MindTap Reader is Cengage’s next-generation eBook for engineering students. The MindTap Reader provides more than just text learning for the student. It offers a variety of tools to help our future engineers learn chapter concepts in a way that resonates with their workflow and learning styles. Personalize their experience Within the MindTap Reader, students can high- light key concepts, add notes, and bookmark pages. These are collected in My Notes, ensur- ing they will have their own study guide when it comes time to study for exams. Flexibility at their fingertips With access to Merriam-Webster’s Dictionary and the book’s internal glossary, students can personalize their study experience by creating and collating their own custom flashcards. The Readspeaker feature reads text aloud to students, so they can learn on the go—wherever they are. 30931_fm_hr_i-xviii.indd 18 10/26/18 10:18 AM  Part One Introduction to Structural Analysis and Loads 30931_ch01_hr_001-016.indd 1 10/26/18 7:37 AM 30931_ch01_hr_001-016.indd 2 10/26/18 7:37 AM  1 Introduction to Structural Analysis 1.1 Historical Background 1.2 Role of Structural Analysis in Structural Engineering Projects 1.3 Classification of Structures 1.4 Analytical Models Summary Marina City District, Chicago Hisham Ibrahim/Photographer’s Choice RF/Getty Images Structural analysis is the prediction of the performance of a given structure under prescribed loads andyor other external effects, such as support move- ments and temperature changes. The performance characteristics commonly of interest in the design of structures are (1) stresses or stress resultants, such as axial forces, shear forces, and bending moments; (2) deflections; and (3) support reactions. Thus, the analysis of a structure usually involves deter- mination of these quantities as caused by a given loading condition. The objec- tive of this text is to present the methods for the analysis of structures in static equilibrium. This chapter provides a general introduction to the subject of structural analysis. We first give a brief historical background, including names of people whose work is important in the field. Then we discuss the role of structural analysis in structural engineering projects. We describe the five common types of structures: tension and compression structures, trusses, and shear and bend- ing structures. Finally, we consider the development of the simplified models of real structures for the purpose of analysis. 1.1 Historical Background Since the dawn of history, structural engineering has been an essential part of human endeavor. However, it was not until about the middle of the seven- teenth century that engineers began applying the knowledge of mechanics 3 30931_ch01_hr_001-016.indd 3 10/26/18 7:37 AM  4   CHAPTER 1 Introduction to Structural Analysis (mathematics and science) in designing structures. Earlier engineering struc- tures were designed by trial and error and by using rules of thumb based on past experience. The fact that some of the magnificent structures from earlier eras, such as Egyptian pyramids (about 3000 b.c.), Greek temples (500–200 b.c.), Roman coliseums and aqueducts (200 b.c.–a.d. 200), and Gothic cathedrals (a.d. 1000–1500), still stand today is a testimonial to the ingenuity of their builders (Fig. 1.1). Galileo Galilei (1564–1642) is generally considered to be the origi- nator of the theory of structures. In his book entitled Two New Sciences, which was published in 1638, Galileo analyzed the failure of some simple structures, including cantilever beams. Although Galileo’s predictions of strengths of beams were only approximate, his work laid the foundation for future developments in the theory of structures and ushered in a new era of structural engineering, in which the analytical principles of mechanics and strength of materials would have a major influence on the design of structures. Following Galileo’s pioneering work, the knowledge of structural mechanics advanced at a rapid pace in the second half of the seventeenth century and into the eighteenth century. Among the notable investigators of that period were Robert Hooke (1635–1703), who developed the law of linear relationships between the force and deformation of materials (Hooke’s law); Sir Isaac Newton (1642–1727), who formulated the laws of motion and devel- oped calculus; John Bernoulli (1667–1748), who formulated the principle of virtual work; Leonhard Euler (1707–1783), who developed the theory of FIG. 1.1 The Cathedral of Notre Dame in Paris Was Completed in the Thirteenth Century Ritu Manoj Jethani/Shutterstock.com 30931_ch01_hr_001-016.indd 4 10/26/18 7:37 AM  Section 1.2 Role of Structural Analysis in Structural Engineering Projects   5 buckling of columns; and C. A. de Coulomb (1736–1806), who presented the analysis of bending of elastic beams. In 1826 L. M. Navier (1785–1836) published a treatise on elastic behavior of structures, which is considered to be the first textbook on the modern theory of strength of materials. The development of structural mechanics continued at a tremendous pace throughout the rest of the nineteenth century and into the first half of the twentieth, when most of the classical methods for the analysis of structures described in this text were developed. The important contributors of this period included B. P. Clapeyron (1799–1864), who ­formulated the three-moment equation for the analysis of continuous beams; J. C. Maxwell (1831–1879), who presented the method of consistent deformations and the law of reciprocal deflections; Otto Mohr (1835–1918), who developed the conjugate-beam method for calculation of deflections and Mohr’s circles of stress and strain; Alberto Castigliano (1847–1884), who formulated the the- orem of least work; C. E. Greene (1842–1903), who developed the moment- area method; H. Müller-Breslau (1851–1925), who presented a principle for constructing influence lines; G. A. Maney (1888–1947), who developed the slope-deflection method, which is considered to be the precursor of the matrix stiffness method; and Hardy Cross (1885–1959), who developed the moment-distribution method in 1924. The moment-distribution method provided engineers with a simple iterative procedure for analyzing highly statically indeterminate structures. This method, which was the most widely used by structural engineers during the period from about 1930 to 1970, contributed significantly to their understanding of the behavior of statically indeterminate frames. Many structures designed during that period, such as high-rise buildings, would not have been possible without the availability of the moment-distribution method. The availability of computers in the 1950s revolutionized structural anal- ysis. Because the computer could solve large systems of simultaneous equa- tions, analyses that took days and sometimes weeks in the precomputer era could now be performed in seconds. The development of the current computer- oriented methods of structural analysis can be attributed to, among oth- ers, J. H. Argyris, R. W. Clough, S. Kelsey, R. K. Livesley, H. C. Martin, M. T. Turner, E. L. Wilson, and O. C. Zienkiewicz. 1.2 Role of Structural Analysis in Structural Engineering Projects Structural engineering is the science and art of planning, designing, and constructing safe and economical structures that will serve their intended purposes. Structural analysis is an integral part of any structural engineering project, its function being the prediction of the performance of the proposed structure. A flowchart showing the various phases of a typical structural engi- neering project is presented in Fig. 1.2. As this diagram indicates, the process is an iterative one, and it generally consists of the following steps: 1. Planning Phase The planning phase usually involves the establish- ment of the functional requirements of the proposed structure, the general layout and dimensions of the structure, and consideration of the possible types of structures (e.g., rigid frame or truss) that may be feasible and the types of materials to be used (e.g., structural steel 30931_ch01_hr_001-016.indd 5 10/26/18 7:37 AM  6   CHAPTER 1 Introduction to Structural Analysis FIG. 1.2 Phases of a Typical Structural Engineering Project or reinforced concrete). This phase may also involve consideration of nonstructural factors, such as aesthetics, environmental impact of the structure, and so on. The outcome of this phase is usually a struc- tural system that meets the functional requirements and is expected to be the most economical. This phase is perhaps the most crucial one of the entire project and requires experience and knowledge of construction practices in addition to a thorough understanding of the behavior of structures. 2. Preliminary Structural Design In the preliminary structural design phase, the sizes of the various members of the structural system selected in the planning phase are estimated based on approximate analysis, past experience, and code requirements. The member sizes thus selected are used in the next phase to estimate the weight of the structure. 3. Estimation of Loads Estimation of loads involves determination of all the loads that can be expected to act on the structure. 4. Structural Analysis In structural analysis, the values of the loads are used to carry out an analysis of the structure in order to determine the stresses or stress resultants in the members and the deflections at various points of the structure. 5. Safety and Serviceability Checks The results of the analysis are used to determine whether or not the structure satisfies the safety 30931_ch01_hr_001-016.indd 6 10/26/18 7:37 AM  Section 1.3 Classification of Structures   7 and serviceability requirements of the design codes. If these require- ments are satisfied, then the design drawings and the construction specifications are prepared, and the construction phase begins. 6. Revised Structural Design If the code requirements are not sat- isfied, then the member sizes are revised, and phases 3 through 5 are repeated until all the safety and serviceability requirements are satisfied. Except for a discussion of the types of loads that can be expected to act on structures (Chapter 2), our primary focus in this text will be on the analysis of structures. 1.3 Classification of Structures As discussed in the preceding section, perhaps the most important decision made by a structural engineer in implementing an engineering project is the selection of the type of structure to be used for supporting or transmitting loads. Commonly used structures can be classified into five basic categories, depending on the type of primary stresses that may develop in their members under major design loads. However, it should be realized that any two or more of the basic structural types described in the following may be combined in a single structure, such as a building or a bridge, to meet the structure’s func- tional requirements. Tension Structures The members of tension structures are subjected to pure tension under the action of external loads. Because the tensile stress is distributed uniformly over the cross-sectional areas of members, the material of such a structure is utilized in the most efficient manner. Tension structures composed of flexible steel cables are frequently employed to support bridges and long-span roofs. Because of their flexibility, cables have negligible bending stiffness and can develop only tension. Thus, under external loads, a cable adopts a shape that enables it to support the load by tensile forces alone. In other words, the shape of a cable changes as the loads acting on it change. As an example, the shapes that a single cable may assume under two different loading conditions are shown in Fig. 1.3. Figure 1.4 shows a familiar type of cable structure—the suspension bridge. In a suspension bridge, the roadway is suspended from two main cables by means of vertical hangers. The main cables pass over a pair of towers and are anchored into solid rock or a concrete foundation at their ends. Because suspension bridges and other cable structures lack stiffness in lateral directions, they are susceptible to wind-induced oscillations (see Fig. 1.5). Bracing or stiffening systems are therefore provided to reduce such oscillations. Besides cable structures, other examples of tension structures include vertical rods used as hangers (for example, to support balconies or tanks) and membrane structures such as tents and roofs of large-span domes (Fig. 1.6). 30931_ch01_hr_001-016.indd 7 10/26/18 7:37 AM  8   CHAPTER 1 Introduction to Structural Analysis FIG. 1.3 FIG. 1.4 Suspension Bridge FIG. 1.5 Tacoma Narrows Bridge Oscillating before Its Collapse in 1940 Smithsonian Institution Photo No. 72-787. Division of Work & Industry, National Museum of American History, Smithsonian Institution. 30931_ch01_hr_001-016.indd 8 10/26/18 7:37 AM  Section 1.3 Classification of Structures   9 FIG. 1.6 The Fabric (membrane) Roof of the Tokyo Dome Is Tensioned (inflated) by Air Pressure from Inside the Stadium Gavin Hellier/Alamy Stock Photo Compression Structures Compression structures develop mainly compressive stresses under the action of external loads. Two common examples of such structures are columns and arches (Fig. 1.7). Columns are straight members subjected to axially compressive loads, as shown in Fig. 1.8. When a straight member is sub- jected to lateral loads andyor moments in addition to axial loads, it is called a beam-column. An arch is a curved structure, with a shape similar to that of an inverted cable, as shown in Fig. 1.9. Such structures are frequently used to FIG. 1.7 Columns and Arches of the Segovia (Roman) Aqueduct Bridge in Spain (constructed in the first or the second centuries) Bluedog423 30931_ch01_hr_001-016.indd 9 10/26/18 7:37 AM  10   CHAPTER 1 Introduction to Structural Analysis support bridges and long-span roofs. Arches develop mainly compressive stresses when subjected to loads and are usually designed so that they will develop only compression under a major design loading. However, because arches are rigid and cannot change their shapes as can cables, other load- ing conditions usually produce secondary bending and shear stresses in these structures, which, if significant, should be considered in their designs. Because compression structures are susceptible to buckling or instability, the possibility of such a failure should be considered in their designs; if neces- sary, adequate bracing must be provided to avoid such failures. Trusses Trusses are composed of straight members connected at their ends by hinged connections to form a stable configuration (Fig. 1.10). When the loads are applied to a truss only at the joints, its members either elongate or shorten. Thus, the members of an ideal truss are always either in uniform FIG. 1.8 Column tension or in uniform compression. Real trusses are usually constructed by connecting members to gusset plates by bolted or welded connections. Although the rigid joints thus formed cause some bending in the mem- bers of a truss when it is loaded, in most cases such secondary bending stresses are small, and the assumption of hinged joints yields satisfactory designs. FIG. 1.9 Arch FIG. 1.10 Plane Truss Trusses, because of their light weight and high strength, are among the most commonly used types of structures. Such structures are used in a variety S of applications, ranging from supporting roofs of buildings to serving as sup- port structures in space stations and sports arenas. Shear Structures Shear structures, such as reinforced concrete shear walls (Fig. 1.11), are used in multistory buildings to reduce lateral movements due to wind loads and earthquake excitations (Fig. 1.12). Shear structures develop mainly in-plane S shear, with relatively small bending stresses under the action of external FIG. 1.11 Shear Wall loads. 30931_ch01_hr_001-016.indd 10 10/26/18 7:37 AM  Section 1.3 Classification of Structures   11 FIG. 1.12 The Shear Wall on the Side of This Building Is Designed to Resist Lateral Loads Due to Wind and Earthquakes NISEE, University of California, Berkeley Bending Structures Bending structures develop mainly bending stresses under the action of exter- nal loads. In some structures, the shear stresses associated with the changes in bending moments may also be significant and should be considered in their designs. Some of the most commonly used structures, such as beams, rigid frames, slabs, and plates, can be classified as bending structures. A beam is a straight member that is loaded perpendicular to its longitudinal axis (Fig. 1.13). Recall from previous courses on statics and mechanics of materials that the bending (normal) stress varies linearly over the depth of a beam from the maximum compressive stress at the fiber farthest from the neutral axis on the concave side of the bent beam to the maximum tensile stress at the outermost fiber on the convex side. For example, in the case of a horizontal beam subjected to a vertically downward load, as shown in Fig. 1.13, the bending stress varies from the maximum compressive stress at the top edge to the maximum tensile stress at the bottom edge of the beam. To utilize the material of a beam cross section most efficiently under this varying stress distribution, the cross sec- tions of beams are often I-shaped (see Fig. 1.13), with most of the material in the top and bottom flanges. The I-shaped cross sections are most effective in resisting bending moments. FIG. 1.13 Beam Rigid frames are composed of straight members connected together either by rigid (moment-resisting) connections or by hinged connections to form stable configurations. Unlike trusses, which are subjected only to joint loads, 30931_ch01_hr_001-016.indd 11 10/26/18 7:37 AM  12   CHAPTER 1 Introduction to Structural Analysis the external loads on frames may be applied on the members as well as on the joints (see Fig. 1.14). The members of a rigid frame are, in general, subjected to bending moment, shear, and axial compression or tension under the action of external loads. However, the design of horizontal members or beams of rectangular frames is often governed by bending and shear stresses only, since the axial forces in such members are usually small. Frames, like trusses, are among the most commonly used types of struc- tures. Structural steel and reinforced concrete frames are commonly used in multistory buildings (Fig. 1.15), bridges, and industrial plants. Frames are also FIG. 1.14 Rigid Frame used as supporting structures in airplanes, ships, aerospace vehicles, and other aerospace and mechanical applications. FIG. 1.15 Skeletons of Frame Buildings Racheal Grazias/Shutterstock.com It may be of interest to note that the generic term framed structure is fre- quently used to refer to any structure composed of straight members, including a truss. In that context, this textbook is devoted primarily to the analysis of plane framed structures. 1.4 Analytical Models An analytical model is a simplified representation, or an ideal, of a real struc- ture for the purpose of analysis. The objective of the model is to simplify the analysis of a complicated structure. The analytical model represents, as accurately as practically possible, the behavioral characteristics of the struc- ture of interest to the analyst, while discarding much of the detail about the members, connections, and so on that is expected to have little effect on the desired characteristics. Establishment of the analytical model is one of the most important steps of the analysis process; it requires experience and knowledge of design practices in addition to a thorough understanding of the behavior of structures. Remember that the structural response predicted from the analysis of the model is valid only to the extent that the model represents the actual structure. 30931_ch01_hr_001-016.indd 12 10/26/18 7:38 AM  Section 1.4 Analytical Models   13 Development of the analytical model generally involves consideration of the following factors. Plane Versus Space Structure If all the members of a structure as well as the applied loads lie in a sin- gle plane, the structure is called a plane structure. The analysis of plane, or two-dimensional, structures is considerably simpler than the analy- sis of space, or three-dimensional, structures. Fortunately, many actual three-dimensional structures can be subdivided into plane structures for analysis. As an example, consider the framing system of a bridge shown in Fig. 1.16(a). The main members of the system, designed to support verti- cal loads, are shown by solid lines, whereas the secondary bracing mem- bers, necessary to resist lateral wind loads and to provide stability, are represented by dashed lines. The deck of the bridge rests on beams called stringers; these beams are supported by floor beams, which, in turn, are con- nected at their ends to the joints on the bottom panels of the two longitudi- nal trusses. Thus, the weight of the traffic, deck, stringers, and floor beams is transmitted by the floor beams to the supporting trusses at their joints; the trusses, in turn, transmit the load to the foundation. Because this applied loading acts on each truss in its own plane, the trusses can be treated as plane structures. As another example, the framing system of a multistory building is shown in Fig. 1.17(a). At each story, the floor slab rests on floor beams, which transfer any load applied to the floor, the weight of the slab, and their own weight to the girders of the supporting rigid frames. This applied loading acts on each frame in its own plane, so each frame can be analyzed as a plane structure. The loads thus transferred to each frame are further transmitted from the ­girders to the columns and then finally to the foundation. Although a great majority of actual three-dimensional structural systems can be subdivided into plane structures for the purpose of analysis, some struc- tures, such as latticed domes, aerospace structures, and transmission towers, cannot, due to their shape, arrangement of members, or applied loading, be subdivided into planar components. Such structures, called space structures, are analyzed as three-dimensional bodies subjected to three-dimensional force systems. Line Diagram The analytical model of the two- or three-dimensional body selected for analysis is represented by a line diagram. On this diagram, each member of the structure is represented by a line coinciding with its centroidal axis. The dimensions of the members and the size of the connections are not shown on the diagram. The line diagrams of the bridge truss of Fig. 1.16(a) and the rigid frame of Fig. 1.17(a) are shown in Figs. 1.16(b) and Fig. 1.17(b), respectively. Note that two lines ( ) are sometimes used in this text to represent members on the line diagrams. This is done, when necessary, for clarity of presen- tation; in such cases, the distance between the lines does not represent the member depth. 30931_ch01_hr_001-016.indd 13 10/26/18 7:38 AM  14   CHAPTER 1 Introduction to Structural Analysis FIG. 1.16 Framing of a Bridge Connections Two types of connections are commonly used to join members of structures: (1) rigid connections and (2) flexible, or hinged, connections. (A third type of connection, termed a semirigid connection, although recognized by structural steel design codes, is not commonly used in practice and, therefore, is not considered in this text.) A rigid connection or joint prevents relative translations and rotations of the member ends connected to it; that is, all member ends connected to a rigid joint have the same translation and rotation. In other words, the origi- nal angles between the members intersecting at a rigid joint are maintained after the structure has deformed under the action of loads. Such joints are, therefore, capable of transmitting forces as well as moments between the 30931_ch01_hr_001-016.indd 14 10/26/18 7:38 AM  Section 1.4 Analytical Models   15 FIG. 1.17 Framing of a Multistory Building connected members. Rigid joints are usually represented by points at the intersections of members on the line diagram of the structure, as shown in Fig. 1.17(b). A hinged connection or joint prevents only relative translations of mem- ber ends connected to it; that is, all member ends connected to a hinged joint have the same translation but may have different rotations. Such joints are thus capable of transmitting forces but not moments between the con- nected members. Hinged joints are usually depicted by small circles at the intersections of members on the line diagram of the structure, as shown in Fig. 1.16(b). The perfectly rigid connections and the perfectly flexible frictionless hinges used in the analysis are merely idealizations of the actual connec- tions, which are seldom perfectly rigid or perfectly flexible (see Fig. 1.16(c)). However, actual bolted or welded connections are purposely designed to behave like the idealized cases. For example, the connections of trusses are designed with the centroidal axes of the members concurrent at a point, as shown in Fig. 1.16(c), to avoid eccentricities that may cause bending of mem- bers. For such cases, the analysis based on the idealized connections and supports (described in the following paragraph) generally yields satisfactory results. 30931_ch01_hr_001-016.indd 15 10/26/18 7:38 AM  16   CHAPTER 1 Introduction to Structural Analysis Supports Supports for plane structures are commonly idealized as either fixed ­supports, which do not allow any movement; hinged supports, which can prevent trans- lation but permit rotation; or roller, or link, supports, which can prevent translation in only one direction. A more detailed description of the charac- teristics of these supports is presented in Chapter 3. The symbols commonly used to represent roller and hinged supports on line diagrams are shown in Fig. 1.16(b), and the symbol for fixed supports is depicted in Fig. 1.17(b). Summary In this chapter, we learned about structural analysis and its role in structural engineering. Structural analysis is the prediction of the performance of a given structure under prescribed loads. Structural engineering has long been a part of human endeavor, but Galileo is considered to be the originator of the theory of structures. Following his pioneering work, many other people have made significant contributions. The availability of computers has revolutionized structural analysis. Structural engineering is the science of planning, designing, and con- structing safe, economical structures. Structural analysis is an integral part of this process. Structures can be classified into five basic categories, namely, tension structures (e.g., cables and hangers), compression structures (e.g., columns and arches), trusses, shear structures (e.g., shear walls), and bending structures (e.g., beams and rigid frames). An analytical model is a simplified representation of a real structure for the purpose of analysis. Development of the model generally involves (1) determination of whether or not the structure can be treated as a plane structure, (2) construction of the line diagram of the structure, and (3) ideal- ization of connections and supports. 30931_ch01_hr_001-016.indd 16 10/26/18 7:38 AM  2 Loads on Structures 2.1 Structural Systems for Transmitting Loads 2.2 Dead Loads 2.3 Live Loads 2.4 Classification of Buildings for Environmental Loads 2.5 Wind Loads 2.6 Snow Loads 2.7 Earthquake Loads 2.8 Hydrostatic and Soil Pressures 2.9 Thermal and Other Effects 2.10 Load Combinations Summary Problems Earthquake-Damaged Building Ints Vikmanis/Shutterstock.com The objective of a structural engineer is to design a structure that will be able to withstand all the loads to which it is subjected while serving its intended purpose throughout its intended life span. In designing a structure, an engi- neer must, therefore, consider all the loads that can realistically be expected to act on the structure during its planned life span. The loads that act on common civil engineering structures can be grouped according to their nature and source into three classes: (1) dead loads due to the weight of the struc- tural system itself and any other material permanently attached to it; (2) live loads, which are movable or moving loads due to the use of the structure; and