Structure Analysis by R.C. Hibbeler 8th Edition PDF

 L Table for Evaluating m mⴕ dx L0 L m m¿ dx L0 m1m¿1 + m¿22L...

 L Table for Evaluating m mⴕ dx L0 L m m¿ dx L0 m1m¿1 + m¿22L 1 1 2 mm¿L mm¿L mm¿L 2 2 3 m1m¿1 + 2m¿22L 1 1 1 5 mm¿L mm¿L mm¿L 2 3 6 12 m¿1m1 + m22L m¿1m1 + 2m22L 3m¿ 12m1 + m22 3m¿13m1 + 5m224L 1 1 1 1 2 6 6 1 12 + m¿21m1 + 2m224L a2 m3m¿11L + b2 + mm¿ a3 + - 2 bL 1 1 1 1 3a mm¿L mm¿1L + a2 m21L + a24 2 6 6 12 L L m12m¿1 + m¿22L 1 1 1 1 mm¿L mm¿L mm¿L 2 6 6 4 Beam Deflections and Slopes Loading v + c u + g Equation ⫹ c ⫹g PL3 PL2 1x3 - 3Lx22 P vmax = - umax = - v = 3EI 2EI 6EI at x = L at x = L MO L2 MO L MO vmax = umax = v = x2 2EI EI 2EI at x = L at x = L Beam Deflections and Slopes (continued) wL4 wL3 1x4 - 4Lx3 + 6L2x22 w vmax = - umax = - v = - 8EI 6EI 24EI at x = L at x = L PL3 PL2 14x3 - 3L2x2, P vmax = - umax = ⫾ v = 48EI 16EI 48EI at x = L>2 at x = 0 or x = L 0 … x … L>2 Pab1L + b2 1L2 - b2 - x22 Pbx uL = - v = - 6LEI 6LEI Pab1L + a2 u = 0…x…a 6LEI 5wL4 wL3 1x3 - 2Lx2 + L32 wx vmax = - umax = ⫾ v = - 384EI 24EI 24EI L at x = 2 116x3 - 24Lx2 + 9L32 wx v = - 384EI 3wL3 uL = - 0 … x … L>2 128EI 7wL3 18x3 - 24Lx2 + 17L2x - L32 wL uR = v = - 384EI 384EI L>2 … x … L MO L uL = - 6EI MO L2 1L2 - x22 MO x vmax = - v = - 923EI MO L 6EIL uR = 3EI STRUCTURAL ANALYSIS This page intentionally left blank STRUCTURAL ANALYSIS EIGHTH EDITION R. C. HIBBELER PRENTICE HALL Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Library of Congress Cataloging-in-Publication Data on File Vice President and Editorial Director, ECS: Marcia J. Horton Acquisitions Editor: Tacy Quinn/Norrin Dais Managing Editor: Scott Disanno Production Editor: Rose Kernan Art Director: Kenny Beck Managing Editor, AV Management and Production: Patricia Burns Art Editor: Gregory Dulles Media Editor: David Alick Director, Image Resource Center: Melinda Reo Manager, Rights and Permissions: Zina Arabia Manager, Visual Research: Beth Brenzel Manager, Cover Visual Research and Permissions: Karen Sanatar Manufacturing Manager: Alexis Heydt-Long Manufacturing Buyer: Lisa McDowell Senior Marketing Manager: Tim Galligan Cover Designer: Kenny Beck About the Cover: Background Image: Orange Steel girders/zimmytws/Shutterstock; Inset image: Building under construction/Vladitto/Shutterstock © 2012 by R. C. Hibbeler Published by Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice Hall™ is a trademark of Pearson Education, Inc. 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. Previous editions copyright © 2009, 2006, 2002, 1999, 1995, 1990, 1985 by R. C. Hibbeler. Pearson Education Ltd., London Pearson Education Australia Pty. Ltd., Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educación de Mexico, S.A. de C.V. Pearson Education—Japan, Tokyo Pearson Education Malaysia, Pte. Ltd. Pearson Education, Upper Saddle River, New Jersey Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 ISBN-10: 0-13-257053-X ISBN-13: 978-0-13-257053-4  To The Student With the hope that this work will stimulate an interest in Structural Analysis and provide an acceptable guide to its understanding. This page intentionally left blank  PREFACE This book is intended to provide the student with a clear and thorough presentation of the theory and application of structural analysis as it applies to trusses, beams, and frames. Emphasis is placed on developing the student’s ability to both model and analyze a structure and to provide realistic applications encountered in professional practice. For many years now, engineers have been using matrix methods to analyze structures. Although these methods are most efficient for a structural analysis, it is the author’s opinion that students taking a first course in this subject should also be well versed in some of the more important classicial methods. Practice in applying these methods will develop a deeper understanding of the basic engineering sciences of statics and mechanics of materials. Also, problem-solving skills are further developed when the various techniques are thought out and applied in a clear and orderly way. By solving problems in this way one can better grasp the way loads are transmitted through a structure and obtain a more complete understanding of the way the structure deforms under load. Finally, the classicial methods provide a means of checking computer results rather than simply relying on the generated output. New to This Edition Fundamental Problems. These problem sets are selectively located just after the example problems. They offer students simple applications of the concepts and, therefore, provide them with the chance to develop their problem-solving skills before attempting to solve any of the standard problems that follow. You may consider these problems as extended examples since they all have solutions and answers that are given in the back of the book. Additionally, the fundamental problems offer students an excellent means of studying for exams, and they can be used at a later time to prepare for the exam necessary to obtain a professional engineering license. Content Revision. Each section of the text was carefully reviewed to enhance clarity. This has included incorporating the new ASCE/ SEI 07-10 standards on loading in Chapter 1, an improved explanation of how to draw shear and moment diagrams and the deflection curve of a structure, consolidating the material on structures having a variable moment of inertia, providing further discussion for analyzing structures having internal hinges using matrix analysis, and adding a new Appendix B that discusses some of the common features used for running current structural analysis computer software. VIII P R E FA C E Example Changes. In order to further illustrate practical applications of the theory, throughout the text some of the examples have been changed, and with the aid of photos, feature modeling and analysis of loadings applied to actual structures. Additional Photos. The relevance of knowing the subject matter is reflected by the realistic applications depicted in many new and updated photos along with captions that are placed throughout the book. New Problems. There are approximately 70% new problems in this edition. They retain a balance between easy, medium, and difficult applications. In addition to the author, the problems have been reviewed and checked by four other parties: Scott Hendricks, Karim Nohra, Kurt Norlin, and Kai Beng Yap. Problem Arrangement. For convenience in assigning homework, the problems are now placed throughout the text. This way each chapter is organized into well-defined sections that contain an explanation of specific topics, illustrative example problems, and a set of homework problems that are arranged in approximate order of increasing difficulty. Organization and Approach The contents of each chapter are arranged into sections with specific topics categorized by title headings. Discussions relevant to a particular theory are succinct, yet thorough. In most cases, this is followed by a “procedure for analysis” guide, which provides the student with a summary of the important concepts and a systematic approach for applying the theory. The example problems are solved using this outlined method in order to clarify its numerical application. Problems are given at the end of each group of sections, and are arranged to cover the material in sequential order. Moreover, for any topic they are arranged in approximate order of increasing difficulty. Hallmark Elements Photographs. Many photographs are used throughout the book to explain how the principles of structural analysis apply to real-world situations. Problems. Most of the problems in the book depict realistic situations encountered in practice. It is hoped that this realism will both stimulate the student’s interest in structural analysis and develop the skill to reduce any such problem from its physical description to a model or symbolic representation to which the appropriate theory can be applied. Throughout the book there is an approximate balance of problems using either SI or FPS units. The intent has been to develop  P R E FA C E IX problems that test the student’s ability to apply the theory, keeping in mind that those problems requiring tedious calculations can be relegated to computer analysis. Answers to Selected Problems. The answers to selected problems are listed in the back of the book. Extra care has been taken in the presentation and solution of the problems, and all the problem sets have been reviewed and the solutions checked and rechecked to ensure both their clarity and numerical accuracy. Example Problems. All the example problems are presented in a concise manner and in a style that is easy to understand. Illustrations. Throughout the book, an increase in two-color art has been added, including many photorealistic illustrations that provide a strong connection to the 3-D nature of structural engineering. Triple Accuracy Checking. The edition has undergone rigorous accuracy checking and proofing of pages. Besides the author’s review of all art pieces and pages, Scott Hendricks of Virginia Polytechnic Institute, Karim Nohra of the University of South Florida, and Kurt Norlin of Laurel Technical Services rechecked the page proofs and together reviewed the entire Solutions Manual. Contents This book is divided into three parts. The first part consists of seven chapters that cover the classical methods of analysis for statically determinate structures. Chapter 1 provides a discussion of the various types of structural forms and loads. Chapter 2 discusses the determination of forces at the supports and connections of statically determinate beams and frames. The analysis of various types of statically determinate trusses is given in Chapter 3, and shear and bending-moment functions and diagrams for beams and frames are presented in Chapter 4. In Chapter 5, the analysis of simple cable and arch systems is presented, and in Chapter 6 influence lines for beams, girders, and trusses are discussed. Finally, in Chapter 7 several common techniques for the approximate analysis of statically indeterminate structures are considered. In the second part of the book, the analysis of statically indeterminate structures is covered in six chapters. Geometrical methods for calculating deflections are discussed in Chapter 8. Energy methods for finding deflections are covered in Chapter 9. Chapter 10 covers the analysis of statically indeterminate structures using the force method of analysis, in addition to a discussion of influence lines for beams. Then the displacement methods consisting of the slope-deflection method in Chapter 11 and moment distribution in Chapter 12 are discussed. Finally, beams and frames having nonprismatic members are considered in Chapter 13. X P R E FA C E The third part of the book treats the matrix analysis of structures using the stiffness method. Trusses are discussed in Chapter 14, beams in Chap- ter 15, and frames in Chapter 16. A review of matrix algebra is given in Appendix A, and Appendix B provides a general guide for using available software for solving problem in structural analysis. Resources for Instructors Instructor’s Solutions Manual. An instructor’s solutions manual was prepared by the author. The manual was also checked as part of the Triple Accuracy Checking program. 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 Prentice Hall representative. Video Solutions. Located on the Companion Website, Video Solutions offer step-by-step solution walkthroughs of representative homework problems from each chapter 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 Structural Analysis text. STRAN. Developed by the author and Barry Nolan, a practicing engineer, STRAN is a downloadable program for use with Structural Analysis problems. Access STRAN on the Companion Website, www. pearsonhighered.com/hibbeler and follow the links for the Structural Analysis text. Complete instructions for how to use the software are included on the Companion Website. Resources for Students Companion Website. The Companion Website provides practice and review materials including: ❍ Video Solutions—Complete, step-by-step solution walkthroughs of representative homework problems from each chapter. Videos offer: Fully worked Solutions—Showing every step of representative homework problems, to help students make vital connections between concepts.  P R E FA C E XI Self-paced Instruction—Students can navigate each problem and select, play, rewind, fast-forward, stop, and jump-to- sections within each problem’s solution. 24/7 Access—Help whenever students need it with over 20 hours of helpful review. ❍ STRAN—A program you can use to solve two and three dimensional trusses and beams, and two dimensional frames. Instructions for downloading and how to use the program are available on the Companion Website. An access code for the Structural Analysis, Eighth Edition Companion Website is included with this text. To redeem the code and gain access to the site, go to www.prenhall.com/hibbeler and follow the directions on the access code card. Access can also be purchased directly from the site. Acknowledgments Over one hundred of my colleagues in the teaching profession and many of my students have made valuable suggestions that have helped in the development of this book, and I would like to hereby acknowledge all of their comments. I personally would like to thank the reviewers contracted by my editor for this new edition, namely: Thomas H. Miller, Oregon State University Hayder A. Rasheed, Kansas State University Jeffrey A. Laman, Penn State University Jerry R. Bayless, University of Missouri—Rolla Paolo Gardoni, Texas A&M University Timothy Ross, University of New Mexico F. Wayne Klaiber, Iowa State University Husam S. Najm, Rutgers University Also, the constructive comments from Kai Beng Yap, and Barry Nolan, both practicing engineers are greatly appreciated. Finally, I would like to acknowledge the support I received from my wife Conny, who has always been very helpful in preparing the manuscript for publication. 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 [email protected] This page intentionally left blank  CREDITS Chapter 1 opener: © CJ Gunther/epa/Corbis Figure 1.6 (a), Page 7: Mark Harris/Photodisc/Getty Images Chapter 2 opener: Joe Gough/Shutterstock Chapter 3 opener: © Robert Shantz/Alamy Chapter 4 opener: Ralf Broskvar/123rf Chapter 5 opener: © Greg Balfour Evans/Alamy Chapter 6 opener: © Accent Alaska.com/Alamy Chapter 7 opener: © David R. Frazier Photolibrary, Inc./Alamy Chapter 8 opener: [Photographer]/Stone/Getty Images Chapter 9 opener: Alamy Images Chapter 10 opener: Shutterstock Chapter 11 opener: © 2011 Photos.com, a division of Getty Images. All rights reserved. Chapter 12 opener: Fotosearch/SuperStock Chapter 13 opener: iStockphoto.com Chapter 14 opener: © Corbis RF/Alamy Chapter 15 opener: © Paul A. Souders/CORBIS Chapter 16 opener: © Alan Schein/Corbis Cover 1: zimmytws\Shutterstock Cover 2: Vladitto\Shutterstock Other photos provided by the author, R. C. Hibbeler. This page intentionally left blank  CONTENTS 3.5 The Method of Sections 104 3.6 Compound Trusses 110 1 3.7 Complex Trusses 116 3.8 Space Trusses 120 Types of Structures and Problems 127 Loads 3 Chapter Review 130 1.1 Introduction 3 1.2 Classification of Structures 4 1.3 1.4 Loads 9 Structural Design 26 4 Problems 27 Internal Loadings Chapter Review 31 Developed in Structural Members 133 4.1 Internal Loadings at a Specified 2 Point 133 Analysis of Statically 4.2 Shear and Moment Functions 139 4.3 Shear and Moment Diagrams Determinate for a Beam 150 Structures 33 4.4 Shear and Moment Diagrams for a Frame 163 2.1 Idealized Structure 33 4.5 Moment Diagrams Constructed by 2.2 Principle of Superposition 46 the Method of Superposition 168 2.3 Equations of Equilibrium 47 Problems 173 2.4 Determinacy and Stability 48 2.5 Application of the Equations Chapter Review 178 of Equilibrium 59 Chapter Review 68 Fundamental Problems 70 Problems 72 5 Project Problem 77 Cables and Arches 181 3 5.1 Cables 181 Analysis of Statically 5.2 Cable Subjected to Concentrated Loads 182 Determinate 5.3 Cable Subjected to a Uniform Trusses 79 Distributed Load 184 5.4 Arches 194 3.1 Common Types of Trusses 79 5.5 Three-Hinged Arch 195 3.2 Classification of Coplanar Trusses 85 Problems 201 3.3 The Method of Joints 94 3.4 Zero-Force Members 98 Chapter Review 203 XVI CONTENTS 6 Influence Lines for Statically Determinate 8 Structures 205 Deflections 299 6.1 Influence Lines 205 8.1 Deflection Diagrams and the Elastic 6.2 Influence Lines for Beams 213 Curve 299 6.3 Qualitative Influence Lines 216 8.2 Elastic-Beam Theory 305 6.4 Influence Lines for Floor Girders 228 8.3 The Double Integration Method 307 6.5 Influence Lines for Trusses 232 8.4 Moment-Area Theorems 316 6.6 Maximum Influence at a Point due to a 8.5 Conjugate-Beam Method 326 Series of Concentrated Loads 240 Problems 335 6.7 Absolute Maximum Shear and Chapter Review 338 Moment 250 Problems 255 Chapter Review 260 9 7 Deflections Using Energy Approximate Analysis of Methods 341 Statically Indeterminate 9.1 External Work and Strain Energy 341 Structures 263 9.2 Principle of Work and Energy 345 9.3 Principle of Virtual Work 346 7.1 Use of Approximate Methods 263 9.4 Method of Virtual Work: Trusses 348 7.2 Trusses 264 9.5 Castigliano’s Theorem 355 7.3 Vertical Loads on Building 9.6 Castigliano’s Theorem for Trusses 356 Frames 270 9.7 Method of Virtual Work: Beams 7.4 Portal Frames and Trusses 273 and Frames 364 7.5 Lateral Loads on Building Frames: 9.8 Virtual Strain Energy Caused by Axial Load, Portal Method 282 Shear, Torsion, and Temperature 375 7.6 Lateral Loads on Building Frames: 9.9 Castigliano’s Theorem for Beams Cantilever Method 288 and Frames 381 Problems 294 Problems 388 Chapter Review 296 Chapter Review 392  CONTENTS XVII 10 12 Analysis of Statically Displacement Method Indeterminate Structures of Analysis: Moment by the Force Method 395 Distribution 487 10.1 Statically Indeterminate Structures 395 12.1 General Principles and 10.2 Force Method of Analysis: General Definitions 487 Procedure 398 12.2 Moment Distribution for 10.3 Maxwell’s Theorem of Reciprocal Beams 491 Displacements; Betti’s Law 402 12.3 Stiffness-Factor Modifications 500 10.4 Force Method of Analysis: Beams 403 12.4 Moment Distribution for Frames: 10.5 Force Method of Analysis: Frames 411 No Sidesway 508 10.6 Force Method of Analysis: Trusses 422 12.5 Moment Distribution for Frames: 10.7 Composite Structures 425 Sidesway 510 10.8 Additional Remarks on the Force Method Problems 518 of Analysis 428 Chapter Review 521 10.9 Symmetric Structures 429 10.10 Influence Lines for Statically Indeterminate Beams 435 10.11 Qualitative Influence Lines for Frames 439 13 Problems 446 Beams and Frames Chapter Review 448 Having Nonprismatic Members 523 11 13.1 Loading Properties of Nonprismatic Displacement Method of Members 523 13.2 Moment Distribution for Structures Having Analysis: Slope-Deflection Nonprismatic Members 528 Equations 451 13.3 Slope-Deflection Equations for Nonprismatic Members 534 11.1 Displacement Method of Analysis: Problems 536 General Procedures 451 Chapter Review 537 11.2 Slope-Deflection Equations 453 11.3 Analysis of Beams 459 11.4 Analysis of Frames: No Sidesway 469 11.5 Analysis of Frames: Sidesway 474 Problems 482 Chapter Review 485 XVIII CONTENTS 15.4 Application of the Stiffness Method for Beam Analysis 579 14 Problems 592 Truss Analysis Using the Stiffness Method 539 14.1 Fundamentals of the Stiffness 16 Method 539 14.2 Member Stiffness Matrix 542 Plane Frame Analysis 14.3 Displacement and Force Transformation Using the Stiffness Matrices 543 Method 595 14.4 Member Global Stiffness Matrix 546 14.5 Truss Stiffness Matrix 547 16.1 Frame-Member Stiffness Matrix 595 14.6 Application of the Stiffness Method 16.2 Displacement and Force Transformation for Truss Analysis 552 Matrices 597 14.7 Nodal Coordinates 560 16.3 Frame-Member Global Stiffness 14.8 Trusses Having Thermal Changes Matrix 599 and Fabrication Errors 564 16.4 Application of the Stiffness Method 14.9 Space-Truss Analysis 570 for Frame Analysis 600 Chapter Review 571 Problems 609 Problems 572 Appendices A. Matrix Algebra for Structural 15 Analysis 612 Beam Analysis Using the B. General Procedure for Using Structural Analysis Software 625 Stiffness Method 575 Fundamental Problems Partial Solutions 15.1 Preliminary Remarks 575 and Answers 628 15.2 Beam-Member Stiffness Matrix 577 Answers to Selected Problems 665 15.3 Beam-Structure Stiffness Matrix 579 Index 685 STRUCTURAL ANALYSIS The diamond pattern framework (cross bracing) of these high-rise buildings is used to resist loadings due to wind. Types of Structures and Loads 1 This chapter provides a discussion of some of the preliminary aspects of structural analysis. The phases of activity necessary to produce a structure are presented first, followed by an introduction to the basic types of structures, their components, and supports. Finally, a brief explanation is given of the various types of loads that must be considered for an appropriate analysis and design. 1.1 Introduction A structure refers to a system of connected parts used to support a load. Important examples related to civil engineering include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are important. When designing a structure to serve a specified function for public use, the engineer must account for its safety, esthetics, and serviceability, while taking into consideration economic and environmental constraints. Often this requires several independent studies of different solutions before final judgment can be made as to which structural form is most appropriate. This design process is both creative and technical and requires a fundamental knowledge of material properties and the laws of mechanics which govern material response. Once a preliminary design of a structure is proposed, the structure must then be analyzed to ensure that it has its required stiffness and strength. To analyze a structure properly, certain idealizations must be made as to how the members are supported and connected together. The loadings are determined from codes and local specifications, and the forces in the members and their displacements are found using the theory of structural analysis, which is the subject matter of this text. The results of this analysis then can be used to 3  4 CHAPTER 1 TYPES OF STRUCTURES AND LOADS redesign the structure, accounting for a more accurate determination of 1 the weight of the members and their size. Structural design, therefore, follows a series of successive approximations in which every cycle requires a structural analysis. In this book, the structural analysis is applied to civil engineering structures; however, the method of analysis described can also be used for structures related to other fields of engineering. 1.2 Classification of Structures It is important for a structural engineer to recognize the various types of elements composing a structure and to be able to classify structures as to their form and function. We will introduce some of these aspects now and expand on them at appropriate points throughout the text. rod bar Structural Elements. Some of the more common elements from which structures are composed are as follows. Tie Rods. Structural members subjected to a tensile force are often referred to as tie rods or bracing struts. Due to the nature of this load, these members are rather slender, and are often chosen from rods, bars, angle channel angles, or channels, Fig. 1–1. typical cross sections Beams. Beams are usually straight horizontal members used primarily to carry vertical loads. Quite often they are classified according to the way they are supported, as indicated in Fig. 1–2. In particular, tie rod when the cross section varies the beam is referred to as tapered or haunched. Beam cross sections may also be “built up” by adding plates to Fig. 1–1 their top and bottom. Beams are primarily designed to resist bending moment; however, if they are short and carry large loads, the internal shear force may become quite large and this force may govern their design. When the material simply supported beam used for a beam is a metal such as steel or aluminum, the cross section is most efficient when it is shaped as shown in Fig. 1–3. Here the forces developed in the top and bottom flanges of the beam form the necessary cantilevered beam couple used to resist the applied moment M, whereas the web is effective in resisting the applied shear V. This cross section is commonly referred to as a “wide flange,” and it is normally formed as a single unit in a rolling fixed–supported beam mill in lengths up to 75 ft (23 m). If shorter lengths are needed, a cross section having tapered flanges is sometimes selected. When the beam is required to have a very large span and the loads applied are rather large, the cross section may take the form of a plate girder. This member is fabricated by using a large plate for the web and welding or bolting continuous beam plates to its ends for flanges. The girder is often transported to the field in Fig. 1–2 segments, and the segments are designed to be spliced or joined together  1.2 CLASSIFICATION OF STRUCTURES 5 flange M V 1 web flange Fig. 1–3 The prestressed concrete girders are simply at points where the girder carries a small internal moment. (See the supported and are used for this highway photo below.) bridge. Concrete beams generally have rectangular cross sections, since it is easy to construct this form directly in the field. Because concrete is rather weak in resisting tension, steel “reinforcing rods” are cast into the beam within regions of the cross section subjected to tension. Precast concrete beams or girders are fabricated at a shop or yard in the same manner and then transported to the job site. Beams made from timber may be sawn from a solid piece of wood or laminated. Laminated beams are constructed from solid sections of wood, which are fastened together using high-strength glues. The steel reinforcement cage shown on the Shown are typical splice plate joints used right and left is used to resist any tension to connect the steel girders of a highway that may develop in the concrete beams bridge. which will be formed around it.  6 CHAPTER 1 TYPES OF STRUCTURES AND LOADS 1 Columns. Members that are generally vertical and resist axial compressive loads are referred to as columns, Fig. 1–4. Tubes and wide-flange cross sections are often used for metal columns, and circular and square cross sections with reinforcing rods are used for those made of concrete.  56 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S 2–11. Classify each of the structures as statically *2–12. Classify each of the frames as statically determinate determinate, statically indeterminate, or unstable. If or indeterminate. If indeterminate, specify the degree of indeterminate, specify the degree of indeterminacy. The indeterminacy. All internal joints are fixed connected. supports or connections are to be assumed as stated. 2 (a) (a) (b) (c) (b) (d) (c) (e) Prob. 2–11 (d) Prob. 2–12  2.4 DETERMINACY AND STABILITY 57 2–13. Classify each of the structures as statically determinate, statically indeterminate, stable, or unstable. fixed pin If indeterminate, specify the degree of indeterminacy. The supports or connections are to be assumed as stated. 2 pin roller fixed fixed fixed (a) (c) Prob. 2–14 fixed pin pin fixed 2–15. Classify each of the structures as statically (b) determinate, statically indeterminate, or unstable. If indeterminate, specify the degree of indeterminacy. pin pin (c) Prob. 2–13 2–14. Classify each of the structures as statically determinate, statically indeterminate, stable, or unstable. If indeterminate, specify the degree of indeterminacy. (a) The supports or connections are to be assumed as stated. pin fixed (b) rocker pin (a) roller pin roller pin fixed (c) Prob. 2–15 (b)  58 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S *2–16. Classify each of the structures as statically determi- 2–17. Classify each of the structures as statically nate, statically indeterminate, or unstable. If indeterminate, determinate, statically indeterminate, stable, or unstable. specify the degree of indeterminacy. If indeterminate, specify the degree of indeterminacy. 2 (a) (a) (b) (b) (c) (c) (d) (d) Prob. 2–16 Prob. 2–17  2.5 APPLICATION OF THE EQUATIONS OF EQUILIBRIUM 59 2.5 Application of the Equations of Equilibrium Occasionally, the members of a structure are connected together in such a way that the joints can be assumed as pins. Building frames and trusses 2 are typical examples that are often constructed in this manner. Provided a pin-connected coplanar structure is properly constrained and contains no more supports or members than are necessary to prevent collapse, the forces acting at the joints and supports can be determined by applying the three equations of equilibrium 1©Fx = 0, ©Fy = 0, ©MO = 02 to each member. Understandably, once the forces at the joints are obtained, the size of the members, connections, and supports can then be determined on the basis of design code specifications. To illustrate the method of force analysis, consider the three-member frame shown in Fig. 2–26a, which is subjected to loads P1 and P2. The free-body diagrams of each member are shown in Fig. 2–26b. In total there are nine unknowns; however, nine equations of equilibrium can be written, three for each member, so the problem is statically determinate. For the actual solution it is also possible, and sometimes convenient, to consider a portion of the frame or its entirety when applying some of these nine equations. For example, a free-body diagram of the entire frame is shown in Fig. 2–26c. One could determine the three reactions Ax, Ay, and Cx on this “rigid” pin-connected system, then analyze any two of its members, Fig. 2–26b, to obtain the other six unknowns. Furthermore, the answers can be checked in part by applying the three equations of equilibrium to the remaining “third” member.To summarize, this problem can be solved by writing at most nine equilibrium equations using free-body diagrams of any members and/or combinations of connected members. Any more than nine equations written would not be unique from the original nine and would only serve to check the results. By Dy Ay Bx Dx Ax Dy Dx By D A P1 Ay B Bx Ex Ax P1 Ey E Ex P1 P2 P2 Ey Cx P2 C Cx (a) (b) (c) Fig. 2–26  60 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S Consider now the two-member frame shown in Fig. 2–27a. Here the free-body diagrams of the members reveal six unknowns, Fig. 2–27b; however, six equilibrium equations, three for each member, can be written, so again the problem is statically determinate. As in the previous case, a free-body diagram of the entire frame can also be used for part 2 of the analysis, Fig. 2–27c. Although, as shown, the frame has a tendency to collapse without its supports, by rotating about the pin at B, this will not happen since the force system acting on it must still hold it in equilibrium. Hence, if so desired, all six unknowns can be determined by applying the three equilibrium equations to the entire frame, Fig. 2–27c, and also to either one of its members. The above two examples illustrate that if a structure is properly supported and contains no more supports or members than are necessary to prevent collapse, the frame becomes statically determinate, and so the unknown forces at the supports and connections can be determined from the equations of equilibrium applied to each member. Also, if the structure remains rigid (noncollapsible) when the supports are removed (Fig. 2–26c), all three support reactions can be determined by applying the three equilibrium equations to the entire structure. However, if the structure appears to be nonrigid (collapsible) after removing the supports (Fig. 2–27c), it must be dismembered and equilibrium of the individual members must be considered in order to obtain enough equations to determine all the support reactions. P1 D A B P1 P2 C Bx Ax (a) By Ay P1 Bx Ax B By Ay Cx P2 Cx P2 (b) Cy Cy (c) Fig. 2–27  2.5 APPLICATION OF THE EQUATIONS OF EQUILIBRIUM 61 Procedure for Analysis The following procedure provides a method for determining the joint reactions for structures composed of pin-connected members. 2 Free-Body Diagrams Disassemble the structure and draw a free-body diagram of each member. Also, it may be convenient to supplement a member free-body diagram with a free-body diagram of the entire structure. Some or all of the support reactions can then be determined using this diagram. Recall that reactive forces common to two members act with equal magnitudes but opposite directions on the respective free- body diagrams of the members. All two-force members should be identified. These members, regardless of their shape, have no external loads on them, and therefore their free-body diagrams are represented with equal but opposite collinear forces acting on their ends. In many cases it is possible to tell by inspection the proper arrowhead sense of direction of an unknown force or couple moment; however, if this seems difficult, the directional sense can be assumed. Equations of Equilibrium Count the total number of unknowns to make sure that an equivalent number of equilibrium equations can be written for solution. Except for two-force members, recall that in general three equilibrium equations can be written for each member. Many times, the solution for the unknowns will be straightforward if the moment equation ©MO = 0 is applied about a point (O) that lies at the intersection of the lines of action of as many unknown forces as possible. When applying the force equations ©Fx = 0 and ©Fy = 0, orient the x and y axes along lines that will provide the simplest reduction of the forces into their x and y components. If the solution of the equilibrium equations yields a negative magnitude for an unknown force or couple moment, it indicates that its arrowhead sense of direction is opposite to that which was assumed on the free-body diagram.  62 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S EXAMPLE 2.8 Determine the reactions on the beam shown in Fig. 2–28a. 60 k 60 sin 60 k 2 1 ft 60 50 kft 1 ft B 60 cos 60 k 50 k  ft Ax A A 10 ft 4 ft 10 ft 4 ft 7 ft Ay By (a) (b) Fig. 2–28 SOLUTION Free-Body Diagram. As shown in Fig. 2–28b, the 60-k force is resolved into x and y components. Furthermore, the 7-ft dimension line is not needed since a couple moment is a free vector and can therefore act anywhere on the beam for the purpose of computing the external reactions. Equations of Equilibrium. Applying Eqs. 2–2 in a sequence, using previously calculated results, we have + ©F = 0; : x Ax - 60 cos 60° = 0 Ax = 30.0 k Ans. d+ ©MA = 0; -60 sin 60°(10) + 60 cos 60°(1) + By(14) - 50 = 0 By = 38.5 k Ans. + c ©Fy = 0; -60 sin 60° + 38.5 + Ay = 0 Ay = 13.4 k Ans. EXAMPLE 2.9 15 kN/m Determine the reactions on the beam in Fig. 2–29a. 5 kN/m SOLUTION A 12 m Free-Body Diagram. As shown in Fig. 2–29b, the trapezoidal distributed loading is segmented into a triangular and a uniform load. (a) The areas under the triangle and rectangle represent the resultant 1 —(10 kN/m)(12 m)  60 kN forces. These forces act through the centroid of their corresponding 2 (5 kN/ m)(12 m)  areas. 60 kN 10 kN/m A Equations of Equilibrium Ax y 5 kN/m + ©F = 0; A = 0 MA A 4 m : x x Ans. 6m (b) + c ©Fy = 0; Ay - 60 - 60 = 0 Ay = 120 kN Ans. Fig. 2–29 d + ©MA = 0; -60142 - 60162 + MA = 0 MA = 600 kN # m Ans.  2.5 APPLICATION OF THE EQUATIONS OF EQUILIBRIUM 63 EXAMPLE 2.10 Determine the reactions on the beam in Fig. 2–30a. Assume A is a pin and the support at B is a roller (smooth surface). 2 B 500 lb/ ft 4 ft A 7 ft 3 ft (a) Fig. 2–30 SOLUTION Free-Body Diagram. As shown in Fig. 2–30b, the support (“roller”) at B exerts a normal force on the beam at its point of contact. The line of action of this force is defined by the 3–4–5 triangle. 5 3 3500 lb 4 ft 4 NB Ax A 3.5 ft 6.5 ft Ay (b) Equations of Equilibrium. Resolving NB into x and y components and summing moments about A yields a direct solution for NB. Why? Using this result, we can then obtain Ax and Ay. d+ ©MA = 0; -350013.52 + A 45 B NB142 + A 35 B NB1102 = 0 Ans. NB = 1331.5 lb = 1.33 k + ©F = 0; : x Ax - 4511331.52 = 0 Ax = 1.07 k Ans. + c ©Fy = 0; Ay - 3500 + 3511331.52 = 0 Ay = 2.70 k Ans.  64 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S EXAMPLE 2.11 The compound beam in Fig. 2–31a is fixed at A. Determine the reactions at A, B, and C. Assume that the connection at B is a pin and C is a roller. 2 400 lb/ft 6000 lbft A B C 20 ft 15 ft (a) Fig. 2–31 SOLUTION Free-Body Diagrams. The free-body diagram of each segment is shown in Fig. 2–31b. Why is this problem statically determinate? 8000 lb MA Bx Bx 6000 lb  ft Ax A B C Ay 10 ft 10 ft 15 ft By By Cy (b) Equations of Equilibrium. There are six unknowns.Applying the six equations of equilibrium, using previously calculated results, we have Segment BC: d+ ©MC = 0; -6000 + By1152 = 0 By = 400 lb Ans. + c ©Fy = 0; -400 + Cy = 0 Cy = 400 lb Ans. :+ ©F = 0; Bx = 0 Ans. x Segment AB: d+ ©MA = 0; MA - 80001102 + 4001202 = 0 MA = 72.0 k # ft Ans. + c ©Fy = 0; Ay - 8000 + 400 = 0 Ay = 7.60 k Ans. :+ ©F = 0; Ax - 0 = 0 Ax = 0 Ans. x  2.5 APPLICATION OF THE EQUATIONS OF EQUILIBRIUM 65 EXAMPLE 2.12 Determine the horizontal and vertical components of reaction at the pins A, B, and C of the two-member frame shown in Fig. 2–32a. 8 kN 3 kN/m 2 4 5 B C 3 2m 2m 1.5 m A 2m (a) By 6 kN 8 kN 4 5 C 3 Bx Bx Cx 1m 1m 2m By Cy 1.5 m A Ax (b) 2m Ay Fig. 2–32 SOLUTION Free-Body Diagrams. The free-body diagram of each member is shown in Fig. 2–32b. Equations of Equilibrium. Applying the six equations of equilibrium in the following sequence allows a direct solution for each of the six unknowns. Member BC: d+ ©MC = 0; -By122 + 6112 = 0 By = 3 kN Ans. Member AB: d+ ©MA = 0; -8122 - 3122 + Bx11.52 = 0 Bx = 14.7 kN Ans. + ©F = 0; A + 3182 - 14.7 = 0 : Ax = 9.87 kN Ans. x x 5 + c ©Fy = 0; Ay - 5 182 4 - 3 = 0 Ay = 9.40 kN Ans. Member BC: :+ ©F = 0; 14.7 - C = 0 Cx = 14.7 kN Ans. x x + c ©Fy = 0; 3 - 6 + Cy = 0 Cy = 3 kN Ans.  66 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S EXAMPLE 2.13 The side of the building in Fig. 2–33a is subjected to a wind loading that creates a uniform normal pressure of 15 kPa on the windward side and a suction pressure of 5 kPa on the leeward side. Determine the 2 horizontal and vertical components of reaction at the pin connections A, B, and C of the supporting gable arch. 2m 2m B 3m 3m C 4m A 3m 4m 3m wind (a) Fig. 2–33 SOLUTION Since the loading is evenly distributed, the central gable arch supports a loading acting on the walls and roof of the dark-shaded tributary area. This represents a uniform distributed load of 115 kN>m22 14 m2 = 60 kN>m on the windward side and 15 kN>m2214 m2 = 20 kN>m on the leeward side, Fig. 2–33b. B 60 kN/ m 20 kN/ m 3m 60 kN/ m 20 kN/ m 3 m A C 3m 3m (b)  2.5 APPLICATION OF THE EQUATIONS OF EQUILIBRIUM 67 Free-Body Diagrams. Simplifying the distributed loadings, the free- body diagrams of the entire frame and each of its parts are shown in Fig. 2–33c. B Bx Bx B 2 254.6 kN 84.9 kN 254.6 kN 2.12 m 84.9 kN 45 45 45 By By 45 4.5 m 3m 180 kN 60 kN 180 kN 60 kN 1.5 m 1.5 m Ax Cx Ax Cx A 3m C A C 1.5 m 1.5 m 1.5 m 1.5 m Ay Cy Cy Ay (c) Equations of Equilibrium. Simultaneous solution of equations is avoided by applying the equilibrium equations in the following sequence using previously computed results.* Entire Frame: d+ ©MA = 0; -1180 + 60211.52 - 1254.6 + 84.92 cos 45°14.52 - 1254.6 sin 45°211.52 + 184.9 sin 45°214.52 + Cy162 = 0 Cy = 240.0 kN Ans. + c ©Fy = 0; -Ay - 254.6 sin 45° + 84.9 sin 45° + 240.0 = 0 Ay = 120.0 kN Ans. Member AB: d+ ©MB = 0; -Ax162 + 120.0132 + 18014.52 + 254.612.122 = 0 Ax = 285.0 kN Ans. + ©F = 0; : -285.0 + 180 + 254.6 cos 45° - Bx = 0 x Bx = 75.0 kN Ans. + c ©Fy = 0; -120.0 - 254.6 sin 45° + By = 0 By = 300.0 kN Ans. Member CB: + ©F = 0; : -Cx + 60 + 84.9 cos 45° + 75.0 = 0 x Cx = 195.0 kN Ans. *The problem can also be solved by applying the six equations of equilibrium only to the two members. If this is done, it is best to first sum moments about point A on member AB, then point C on member CB. By doing this, one obtains two equations to be solved simultaneously for Bx and By.  68 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S CHAPTER REVIEW Supports—Structural members are often assumed to be pin connected if slight relative rotation can occur between them, and fixed connected if no rotation is possible. 2 weld stiffeners weld typical “pin-supported” connection (metal) typical “fixed-supported” connection (metal) ( ) Idealized Structures—By making assumptions about the supports and connections as being either roller supported, pinned, or fixed, the members can then be represented as lines, so that we can establish an idealized model that can be used for analysis. P P L –– L –– L L –– –– 2 2 2 2 actual beam idealized beam The tributary loadings on slabs can be determined by first classifying the slab as a one-way or two-way slab. As a general rule, if L2 is the largest dimension, and L2>L1 7 2, the slab will behave as a one-way slab. If L2>L1 … 2, the slab will behave as a two-way slab. L2 L2 L1 L1 ___ 2 L1 L1 ___ L1 2 one-way slab action two-way slab action requires L2 / L1  2 requires L2 / L1 2  CHAPTER REVIEW 69 Principle of Superposition—Either the loads or displacements can be added together provided the material is linear elastic and only small displacements of the structure occur. Equilibrium—Statically determinate structures can be analyzed by disassembling them and applying the equations of equilibrium to each member. The analysis of a statically determinate structure requires first drawing the free-body diagrams 2 of all the members, and then applying the equations of equilibrium to each member. g Fx = 0 g Fy = 0 g MO = 0 The number of equations of equilibrium for all n members of a structure is 3n. If the structure has r reactions, then the structure is statically determinate if r = 3n and statically indeterminate if r 7 3n The additional number of equations required for the solution refers to the degree of indeterminacy. Stability—If there are fewer reactions than equations of equilibrium, then the structure will be unstable because it is partially constrained. Instability due to improper constraints can also occur if the lines of action of the reactions are concurrent at a point or parallel to one another. P P P partial constraint concurrent reactions parallel reactions  70 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S FUNDAMENTAL PROBLEMS F2–1. Determine the horizontal and vertical components F2–4. Determine the horizontal and vertical components of reaction at the pins A, B, and C. of reaction at the roller support A, and fixed support B. 2 2m 2m 10 kN 8 kN/m 1m A B B 60 kNm A C 3m 4m 2m F2–4 C F2–5. Determine the horizontal and vertical components F2–1 of reaction at pins A, B, and C of the two-member frame. 300 lb B F2–2. Determine the horizontal and vertical components of reaction at the pins A, B, and C. 3 ft 10 kN/m C A B A 2m 2 ft 2 ft 4 ft 4m 45 C F2–5 F2–6. Determine the components of reaction at the roller support A and pin support C. Joint B is fixed connected. F2–2 6 kN F2–3. Determine the horizontal and vertical components 2m 2m of reaction at the pins A, B, and C. A B 2m C 2 kN 10 kN/m 2m 60 A C B 2m 2m F2–3 F2–6  FUNDAMENTAL PROBLEMS 71 F2–7. Determine the horizontal and vertical components F2–9. Determine the components of reaction at the fixed of reaction at the pins A, B, and D of the three-member support D and the pins A, B, and C of the three-member frame. The joint at C is fixed connected. frame. Neglect the thickness of the members. 8 kN 8 kN 2 2 k/ft 2m 2m 2m B C 3 kN/m B C 4 ft 0.5 k/ft 4m D D 2 ft A A 3m 8 ft F2–7 F2–9 F2–8. Determine the components of reaction at the fixed F2–10. Determine the components of reaction at the fixed support D and the pins A, B, and C of the three-member support D and the pins A, B, and C of the three-member frame. Neglect the thickness of the members. frame. Neglect the thickness of the members. 6 kN 6 kN 8 kN 8 kN 6 kN 6 kN 2m 2m 2m 2m 2m 2m B C B C 3m 4 kN 6m 1.5 kN/m 3m A D A D F2–8 F2–10  72 CHAPTER 2 A N A LY S I S OF S TAT I C A L LY D E T E R M I N AT E S T R U C T U R E S PROBLEMS 2–18. Determine the reactions on the beam. Neglect the 2–21. Determine the reactions at the supports A and B of thickness of the beam. the compound beam. Assume there is a pin at C. 2 18 kN 4 kN/m 20 kN 20 kN 26 kN 13 12 A C B 5 B A 6m 6m 3m 6m 2m 2m Prob. 2–18 Prob. 2–21 2–19. Determine the reactions on the beam. 2–22. Determine the reactions at the supports A, B, D, and F. 3 k/ft 2 k/ft 2 k/ft 8k 2 k/ ft B A C E 60 600 k · ft A F B D 8 ft 4 ft 4 ft 4 ft 4 ft 12 ft 12 ft 2 ft Prob. 2–19 Prob. 2–22 *2–20. Determine the reactions on the beam. 2–23. The compound beam is pin supported at C and supported by a roller at A and B. There is a hinge (pin) at D. Determine the reactions at the supports. Neglect the thickness of the beam. 2 k/ft B 8k 12 k 5 k/ft 10 ft 5 4 15 k · ft 3 A D B C A 30 6 ft 8 ft 8 ft 8 ft 24 ft 4k 4 ft 2 ft Prob. 2–20 Prob. 2–23  PROBLEMS 73 *2–24. Determine the reactions on the beam. The support 2–27. The compound beam is fixed at A and supported by at B can be assumed to be a roller. a rocker at B and C. There are hinges (pins) at D and E. Determine the reactions at the supports. 15 kN 2 k/ft 2 A D B E A C B 12 ft 12 ft 6m 6m 2m 2m 2m Prob. 2–24 Prob. 2–27 2–25. Determine the reactions at the smooth support C *2–28. Determine the reactions at the supports A and B. and pinned support A. Assume the connection at B is fixed The floor decks CD, DE, EF, and FG transmit their loads to connected. the girder on smooth supports. Assume A is a roller and B is a pin. C