Design of Concrete Structures PDF

DESIGN of CONCRETE STRUCTURES Fifteenth Edition David Darwin Ph.D., P.E., Distinguished Member of ASCE Fellow of ACI, Fellow of SEI Deane E. Ackers Distinguished Professor and Chair of Civil, Environmental & Architectural Engineering University of Ka...

DESIGN of CONCRETE STRUCTURES Fifteenth Edition David Darwin Ph.D., P.E., Distinguished Member of ASCE Fellow of ACI, Fellow of SEI Deane E. Ackers Distinguished Professor and Chair of Civil, Environmental & Architectural Engineering University of Kansas Charles W. Dolan Ph.D., P.E., Honorary Member of ACI Fellow of PCI H. T. Person Professor of Engineering, Emeritus University of Wyoming Arthur H. Nilson Ph.D., P.E., Honorary Member of ACI Fellow of ASCE Late Professor of Structural Engineering Cornell University DESIGN OF CONCRETE STRUCTURES, FIFTEENTH EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2016 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2010, 2004, and 1997. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 QVS/QVS 1 0 9 8 7 6 5 ISBN 978-0-07-339794-8 MHID 0-07-339794-6 Senior Vice President, Products & Markets: Kurt L. Strand Full-Service Manager: Faye Schilling Vice President, General Manager, Products & Markets: Michael Ryan Content Project Managers: Melissa M. Leick & Sandy Schnee Vice President, Content Design & Delivery: Kimberly Buyer: Laura M. Fuller Meriwether David Content Licensing Specialist: Deanna Dausener Managing Director: Thomas Timp Cover Image: Escala image: © Structural Engineer: Cary Kopczynski/ Director, Product Development: Rose Koos photographer: Michael Walmsley; Marriott image: © Magnusson Brand Manager: Thomas Scaife, Ph.D. Klemencic Associates Product Developer: Lorraine Buczek Compositor: Laserwords Private Limited Marketing Manager: Nick McFadden Typeface: 10.5/12 Times Lt std Director, Content Design & Delivery: Terri Schiesl Printer: Quad/Graphics All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Darwin, David. Design of concrete structures / David Darwin, Deane E. Ackers Distinguished Professor and chair of Civil, Environmental & Architectural Engineering, University of Kansas, Charles W. Dolan, H.T. Person Professor of Engineering, emeritus, University of Wyoming, Arthur H. Nilson, late professor of structural engineering, Cornell University.—Fifteenth edition. pages cm Primary author of previous edition: Arthur H. Nilson. Includes bibliographical references and index. ISBN 978-0-07-339794-8 (alk. paper) 1. Reinforced concrete construction. 2. Prestressed concrete construction. I. Dolan, Charles W. (Charles William), 1943- II. Nilson, Arthur H. III. Title. TA683.2.N55 2016 624.1'834—dc23 2014038966 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. www.mhhe.com David Darwin has been a member of the faculty at the University of Kansas since 1974, where he has served as director of the Structural Engineering and Materials Laboratory since 1982 and currently chairs the Department of Civil, Environmental, and Architectural Engineering. He was appointed the Deane E. Ackers Distin- guished Professor of Civil Engineering in 1990. Dr. Darwin served as President of the American Concrete Institute in 2007–2008 and is a member and past chair of ACI Committees 224 on Cracking and 408 on Bond and Development of Rein- forcement. He is also a member of ACI Building Code Subcommittee 318-B on Anchorage and Reinforcement and ACI-ASCE Committee 445 on Shear and Torsion. Dr. Darwin is an acknowledged expert on concrete crack control and bond between steel reinforcement and concrete. He received the ACI Arthur R. Anderson Award for his research efforts in plain and reinforced concrete, the ACI Structural Research Award, the ACI Joe W. Kelly Award for his contributions to teaching and design, and the ACI Foundation – Concrete Research Council Arthur J. Boase Award for his research on reinforcing steel and concrete cracking. He has also received a number of awards from the American Society of Civil Engineers, including the Walter L. Huber Civil Engineering Research Prize, the Moisseiff Award, and the State-of-the-Art of Civil Engineering Award twice, the Richard R. Torrens Award, and the Dennis L. Tewksbury Award, and has been honored for his teaching by both undergraduate and graduate students at the University of Kansas. He is past editor of the ASCE Journal of Structural Engineering. Professor Darwin is a Distinguished Member of ASCE and a Fellow of ACI and the Structural Engineering Institute of ASCE. He is a licensed professional engineer and serves as a consultant in the fields of concrete materials and structures. He has been honored with the Distinguished Alumnus Award from the University of Illinois Civil and Environmental Engineering Alumni Association. Between his M.S. and Ph.D. degrees, he served four years with the U.S. Army Corps of Engineers. He received the B.S. and M.S. degrees from Cornell University in 1967 and 1968 and the Ph.D. from the University of Illinois at Urbana-Champaign in 1974. Charles W. Dolan is a consulting engineer and emeritus faculty member of the University of Wyoming. At the University of Wyoming from 1991 to 2012, he served as Department Head from 1998 to 2001 and as the first H. T. Person Chair of Engi- neering from 2002 to 2012, for which he received the University of Wyoming’s John P. Ellbogen lifetime teaching award. A member of American Concrete Institute (ACI) Committee 318 Building Code for Concrete Structures for 12 years, he has chaired the Building Code Subcommittees on Prestressed Concrete and Code Reorganization. He has served as chair of the ACI Technical Activities Committee, ACI Committee iii iv About the Authors 358 on Transit Guideways, and ACI-ASCE Committee 423 on Prestressed Concrete. A practicing engineer for over 40 years, including 20 years at Berger/ABAM, he was the project engineer on the Walt Disney World Monorail, the Detroit Downtown Peoplemover guideway, and the original Dallas–Fort Worth Airport transit system guideway. He developed the conceptual design of the Vancouver BC SkyTrain struc- ture and the Dubai Palm Island monorail. He received the ASCE T. Y. Lin Award for outstanding contributions to the field of prestressed concrete, the ACI Arthur R. Anderson award for advancements in the design of reinforced and prestressed con- crete structures, and the Prestress/Precast Concrete Institute’s (PCI) Martin P. Korn award for advances in design and research in prestressed concrete. An Honorary Member of ACI and a Fellow of PCI, he is internationally recognized as a leader in the design of specialty transit structures and development of fiber-reinforced polymers for concrete reinforcement. Dr. Dolan is a registered professional engineer and lec- tures widely on the design and behavior of structural concrete. He received his B.S. from the University of Massachusetts in 1965 and his M.S. and Ph.D. from Cornell University in 1967 and 1989. The late Arthur H. Nilson was engaged in research, teaching, and consulting relat- ing to structural concrete for over 40 years. He was a member of the faculty of the College of Engineering at Cornell University from 1956 to 1991 when he retired and was appointed professor emeritus. At Cornell he was in charge of undergraduate and graduate courses in the design of reinforced concrete and prestressed concrete struc- tures. He served as Chairman of the Department of Structural Engineering from 1978 to 1985. Dr. Nilson served on many professional committees, including American Concrete Institute (ACI) Building Code Subcommittee 318-D. His pioneering work on high-strength concrete has been widely recognized. He was awarded the ACI Wason Medal for materials research in 1974, the ACI Wason Medal for best technical paper in 1986 and 1987, and the ACI Structural Research Award in 1993. Professor Nilson was an Honorary Member of ACI and a Fellow in the American Society of Civil Engineers (ASCE). He was honored by the civil engineering student body at Cornell for outstanding teaching. Professor Nilson was a registered professional engineer in several states and, prior to entering teaching, was engaged in full-time professional practice. He received the B.S. degree from Stanford University in 1948, the M.S. from Cornell in 1956, and the Ph.D. from the University of California at Berkeley in 1967. About the Authors iii Preface xii Chapter 1 Introduction 1 1.1 Concrete, Reinforced Concrete, and Prestressed Concrete 1 1.2 Structural Forms 2 1.3 Loads 8 1.4 Serviceability, Strength, and Structural Safety 12 1.5 Design Basis 15 1.6 Design Codes and Specifications 16 1.7 Safety Provisions of the ACI Code 17 1.8 Developing Factored Gravity Loads 18 References 22 Problems 22 Chapter 2 Materials 24 2.1 Introduction 24 2.2 Cement 24 2.3 Aggregates 25 2.4 Proportioning and Mixing Concrete 27 2.5 Conveying, Placing, Compacting, and Curing 29 2.6 Quality Control 30 2.7 Admixtures 34 2.8 Properties in Compression 36 2.9 Properties in Tension 42 2.10 Strength under Combined Stress 45 2.11 Shrinkage and Temperature Effects 46 2.12 High-Strength Concrete 49 2.13 Reinforcing Steels for Concrete 51 2.14 Reinforcing Bars 52 2.15 Welded Wire Reinforcement 57 2.16 Prestressing Steels 58 2.17 Fiber Reinforcement 60 References 62 Problems 63 v vi Contents Chapter 3 Design of Concrete Structures and Fundamental Assumptions 65 3.1 Introduction 65 3.2 Members and Sections 67 3.3 Theory, Codes, and Practice 67 3.4 Fundamental Assumptions for Reinforced Concrete Behavior 69 3.5 Behavior of Members Subject to Axial Loads 70 3.6 Bending of Homogeneous Beams 76 References 78 Problems 78 Chapter 4 Flexural Analysis and Design of Beams 80 4.1 Introduction 80 4.2 Reinforced Concrete Beam Behavior 80 4.3 Design of Tension-Reinforced Rectangular Beams 90 4.4 Design Aids 104 4.5 Practical Considerations in the Design of Beams 107 4.6 Rectangular Beams with Tension and Compression Reinforcement 109 4.7 T Beams 118 References 125 Problems 126 Chapter 5 Shear and Diagonal Tension in Beams 130 5.1 Introduction 130 5.2 Diagonal Tension in Homogeneous Elastic Beams 131 5.3 Reinforced Concrete Beams without Shear Reinforcement 134 5.4 Reinforced Concrete Beams with Web Reinforcement 141 5.5 ACI Code Provisions for Shear Design 146 5.6 Effect of Axial Forces 155 5.7 Beams with Varying Depth 160 5.8 Alternative Models for Shear Analysis and Design 161 5.9 Shear-Friction Design Method 170 References 174 Problems 176 Chapter 6 Bond, Anchorage, and Development Length 179 6.1 Fundamentals of Flexural Bond 179 6.2 Bond Strength and Development Length 183 6.3 ACI Code Provisions for Development of Tension Reinforcement 187 6.4 Anchorage of Tension Bars by Hooks 191 6.5 Anchorage in Tension Using Headed Bars 195 6.6 Anchorage Requirements for Web Reinforcement 199 6.7 Welded Wire Reinforcement 200 6.8 Development of Bars in Compression 201 6.9 Bundled Bars 202  Contents vii 6.10 Bar Cutoff and Bend Points in Beams 202 6.11 Structural Integrity Provisions 209 6.12 Integrated Beam Design Example 210 6.13 Bar Splices 215 References 218 Problems 219 Chapter 7 Serviceability 224 7.1 Introduction 224 7.2 Cracking in Flexural Members 224 7.3 ACI Code Provisions for Crack Control 227 7.4 Control of Deflections 230 7.5 Immediate Deflections 231 7.6 Deflections Due to Long-Term Loads 234 7.7 ACI Code Provisions for Control of Deflections 236 7.8 Deflections Due to Shrinkage and Temperature Changes 242 7.9 Moment vs. Curvature for Reinforced Concrete Sections 244 References 248 Problems 249 Chapter 8 Analysis and Design for Torsion 251 8.1 Introduction 251 8.2 Torsion in Plain Concrete Members 252 8.3 Torsion in Reinforced Concrete Members 255 8.4 Torsion Plus Shear 259 8.5 ACI Code Provisions for Torsion Design 260 References 270 Problems 270 Chapter 9 Short Columns 273 9.1 Introduction: Axial Compression 273 9.2 Transverse Ties and Spirals 276 9.3 Compression Plus Bending of Rectangular Columns 280 9.4 Strain Compatibility Analysis and Interaction Diagrams 281 9.5 Balanced Failure 284 9.6 Distributed Reinforcement 287 9.7 Unsymmetrical Reinforcement 289 9.8 Circular Columns 290 9.9 ACI Code Provisions for Column Design 292 9.10 Design Aids 293 9.11 Biaxial Bending 296 9.12 Load Contour Method 298 9.13 Reciprocal Load Method 299 9.14 Computer Analysis for Biaxial Bending of Columns 302 9.15 Bar Splicing in Columns and Ties Near Beam- Column Joints 303 9.16 Transmission of Column Loads through Floor Systems 305 References 305 Problems 306 viii Contents Chapter 10 Slender Columns 310 10.1 Introduction 310 10.2 Concentrically Loaded Columns 311 10.3 Compression Plus Bending 314 10.4 ACI Criteria for Slenderness Effects in Columns 319 10.5 ACI Criteria for Nonsway vs. Sway Structures 321 10.6 ACI Moment Magnifier Method for Nonsway Frames 322 10.7 ACI Moment Magnifier Method for Sway Frames 330 10.8 Second-Order Analysis for Slenderness Effects 336 References 338 Problems 339 Chapter 11 Analysis of Indeterminate Beams and Frames 343 11.1 Continuity 343 11.2 Loading 345 11.3 Simplifications in Frame Analysis 347 11.4 Methods for Elastic Analysis 349 11.5 Idealization of the Structure 350 11.6 Preliminary Design and Guidelines for Proportioning Members 355 11.7 Approximate Analysis 357 11.8 ACI Moment Coefficients 362 11.9 Limit Analysis 365 11.10 Conclusion 376 References 377 Problems 377 Chapter 12 Analysis and Design of One-Way Slabs 380 12.1 Types of Slabs 380 12.2 Design of One-Way Slabs 382 12.3 Temperature and Shrinkage Reinforcement 385 Reference 388 Problems 388 Chapter 13 Analysis and Design of Two-Way Slabs 390 13.1 Behavior of Two-Way Edge-Supported Slabs 390 13.2 Two-Way Column-Supported Slabs 393 13.3 Direct Design Method for Column-Supported Slabs 397 13.4 Flexural Reinforcement for Column-Supported Slabs 402 13.5 Depth Limitations of the ACI Code 405 13.6 Equivalent Frame Method 411 13.7 Shear Design in Flat Plates and Flat Slabs 419 13.8 Transfer of Moments at Columns 434 13.9 Openings in Slabs 437 13.10 Deflection Calculations 439 13.11 Analysis for Horizontal Loads 446 References 447 Problems 449  Contents ix Chapter 14 Walls 453 14.1 Introduction 453 14.2 General Design Considerations 454 14.3 Simplified Method for Axial Load and Out-of-Plane Moment 456 14.4 Alternative Method for Out-of-Plane Slender Wall Analysis 457 14.5 Shear Walls 458 References 462 Chapter 15 Footings and Foundations 463 15.1 Types and Functions 463 15.2 Spread Footings 463 15.3 Design Factors 464 15.4 Loads, Bearing Pressures, and Footing Size 465 15.5 Wall Footings 467 15.6 Column Footings 469 15.7 Combined Footings 477 15.8 Two-Column Footings 479 15.9 Strip, Grid, and Mat Foundations 486 15.10 Pile Caps 487 References 490 Problems 491 Chapter 16 Retaining Walls 492 16.1 Function and Types of Retaining Walls 492 16.2 Earth Pressure 492 16.3 Earth Pressure for Common Conditions of Loading 496 16.4 External Stability 497 16.5 Basis of Structural Design 500 16.6 Drainage and Other Details 501 16.7 Example: Design of a Gravity Retaining Wall 502 16.8 Example: Design of a Cantilever Retaining Wall 504 16.9 Counterfort Retaining Walls 511 16.10 Precast Retaining Walls 513 References 514 Problems 515 Chapter 17 Strut-and-Tie Models 516 17.1 Introduction 516 17.2 Development of Strut-and-Tie Models 516 17.3 Strut-and-Tie Design Methodology 520 17.4 ACI Provisions for Strut-and-Tie Models 526 17.5 Applications 531 References 540 Problems 541 x Contents Chapter 18 Design of Reinforcement at Joints 542 18.1 Introduction 542 18.2 Beam-Column Joints 543 18.3 Strut-and-Tie Model for Joint Behavior 555 18.4 Beam-to-Girder Joints 557 18.5 Ledge Girders 558 18.6 Corners and T Joints 561 18.7 Brackets and Corbels 564 References 568 Problems 569 Chapter 19 Concrete Building Systems 571 19.1 Introduction 571 19.2 Floor and Roof Systems 572 19.3 Precast Concrete for Buildings 584 19.4 Diaphragms 600 19.5 Engineering Drawings for Buildings 605 References 605 Chapter 20 Seismic Design 607 20.1 Introduction 607 20.2 Structural Response 609 20.3 Seismic Loading Criteria 614 20.4 ACI Provisions for Earthquake-Resistant Structures 619 20.5 ACI Provisions for Special Moment Frames 620 20.6 ACI Provisions for Special Structural Walls, Coupling Beams, Diaphragms, and Trusses 633 20.7 ACI Provisions for Shear Strength 638 20.8 ACI Provisions for Intermediate Moment Frames 642 References 644 Problems 644 Chapter 21 Anchoring to Concrete 646 21.1 Introduction 646 21.2 Behavior of Anchors 648 21.3 Concrete Breakout Capacity 649 21.4 Anchor Design 651 21.5 ACI Code Provisions for Concrete Breakout Capacity 651 21.6 Steel Strength 653 21.7 Concrete Breakout Capacity of Single Cast-In and Post-Installed Anchors 655 21.8 Pullout Strength of Anchors 662 21.9 Side-Face Blowout 663 21.10 Pryout of Anchors 664 21.11 Combined Shear and Normal Force 664 21.12 Anchor Reinforcement 667 21.13 Adhesive Anchors 667 21.14 Earthquake Design 671 References 672 Problems 673  Contents xi Chapter 22 Prestressed Concrete 677 22.1 Introduction 677 22.2 Effects of Prestressing 678 22.3 Sources of Prestress Force 682 22.4 Prestressing Steels 685 22.5 Concrete for Prestressed Construction 687 22.6 Elastic Flexural Analysis 688 22.7 Flexural Strength 694 22.8 Partial Prestressing 699 22.9 Flexural Design Based on Concrete Stress Limits 700 22.10 Shape Selection 711 22.11 Tendon Profiles 712 22.12 Flexural Design Based on Load Balancing 714 22.13 Loss of Prestress 719 22.14 Shear, Diagonal Tension, and Web Reinforcement 723 22.15 Bond Stress, Transfer Length, and Development Length 730 22.16 Anchorage Zone Design 731 22.17 Deflection 735 22.18 Crack Control for Class C Flexural Members 739 References 739 Problems 740 Chapter 23 Yield Line Analysis for Slabs (http://highered.mheducation.com:80/sites/0073397946) Chapter 24 Strip Method for Slabs (http://highered.mheducation.com:80/sites/0073397946) Appendix A Design Aids 743 Appendix B SI Conversion Factors: Inch-Pound Units to SI Units 776 Author Index 777 Subject Index 780  The fifteenth edition of Design of Concrete Structures continues the dual objectives of establishing a firm understanding of the behavior of structural concrete and of devel- oping proficiency in the methods of design practice. It is generally recognized that mere training in special design skills and codified procedures is inadequate for a suc- cessful career in professional practice. As new research becomes available and new design methods are introduced, these procedures are subject to frequent changes. To understand and keep abreast of these rapid developments and to engage safely in inno- vative design, the engineer needs a thorough grounding in the fundamental perfor- mance of concrete and steel as structural materials and in the behavior of reinforced concrete members and structures. At the same time, the main business of the structural engineer is to design structures safely, economically, and efficiently. Consequently, with this basic understanding as a firm foundation, familiarity with current design procedures is essential. This edition, like the preceding ones, addresses both needs. The text presents the basic mechanics of structural concrete and methods for the design of individual members subjected to bending, shear, torsion, and axial forces. It additionally addresses in detail applications of the various types of structural members and systems, including an extensive presentation of slabs, beams, columns, walls, footings, retaining walls, and the integration of building systems. The 2014 ACI Building Code, which governs design practice in most of the United States and serves as a model code in many other countries, is significantly reorganized from previous editions and now focuses on member design and ease of access to code provisions. Strut-and-tie methods for design and anchoring to concrete have been moved from the appendixes into the body of the Code. The Code emphasis on member design reinforces the importance of understanding basic behavior. To meet the challenges of a revised building Code and the objectives listed above, this edition is revised as follows: Every chapter is updated to account for the reorganization of the 2014 Ameri- can Concrete Institute Building Code. The opening chapters explore the roles of design theory, codes, and practice. The process of developing building design and the connection between the chapters in the text and the ACI Code is added. A new chapter on anchoring to concrete is included. A chapter on walls is added, doubling the coverage and adding design examples. Diaphragms are included for the first time. Coverage of seismic design is updated. xii  Preface xiii In addition to changes in the ACI Code, the text also includes the modified compression field theory method of shear design presented in the 2012 edition of the American Association of State Highway and Transportation Officials (AASHTO) LRFD Bridge Design Specifications. Chapters on yield line and strip methods for slabs are moved to the McGraw-Hill Education Website (www.mhhe.com/darwin15e). A strength of the text is the analysis chapter, which includes load combinations for use in design, a description of envelope curves for moment and shear, guidelines for proportioning members under both gravity and lateral loads, and procedures for developing preliminary designs of reinforced concrete structures. The chapter also includes the ACI moment and shear coefficients. Present-day design is performed using computer programs, either general-purpose commercially available software or individual programs written for special needs. Proce- dures given throughout the book guide the student and engineer through the increasingly complex methodology of design, with the emphasis on understanding the design process. Once mastered, these procedures are easily converted into flow charts to aid in preparing design aids or to validate commercial computer program output. The text is suitable for either a one or two-semester course in the design of con- crete structures. If the curriculum permits only a single course, probably taught in the fourth undergraduate year, the following will provide a good basis: the introduction and treatment of materials found in Chapters 1 through 3; the material on flexure, shear, and anchorage in Chapters 4, 5, and 6; Chapter 7 on serviceability; Chapter 9 on short columns; the introduction to one-way slabs found in Chapter 12; and footings in Chapter 15. Time may or may not permit classroom coverage of frame analysis or building systems, Chapters 11 and 19, but these could well be assigned as independent reading, concurrent with the earlier work of the course. In the authors’ experience, such complementary outside reading tends to enhance student motivation. The text is more than adequate for a second course, most likely taught in the senior year or the first year of graduate study. The authors have found that this is an excellent opportunity to provide students with a more general understanding of reinforced concrete structural design, often beginning with analysis and building systems, Chapters 11 and 19, followed by the increasingly important behavioral topics of torsion, Chapter 8; slender columns, Chapter 10; the strut-and-tie method of Chapter 17; and the design and detailing of joints, Chapter 18. It should also offer an opportunity for a much-expanded study of slabs, including Chapter 13, plus the methods for slab analysis and design based on plasticity theory found in Chapters 23 and 24 (available online), yield line analysis and the strip method of design. Other topics appropriate to a second course include retaining walls, Chapter 16, and the introduction to earthquake-resistant design in Chapter 20. Prestressed concrete in Chapter 22 is sufficiently important to justify a separate course in conjunction with anchoring to concrete, Chapter 21, and strut-and-tie methods, Chapter 17. If time constraints do not permit this, Chapter 22 provides an introduction and can be used as the text for a one-credit-hour course. At the end of each chapter, the user will find extensive reference lists, which pro- vide an entry into the literature for those wishing to increase their knowledge through individual study. For professors, the instructor’s solution manual is available online at the McGraw-Hill Education Website. A word must be said about units. In the United States customary inch-pound units remain prominent. Accordingly, inch-pound units are used throughout the text, although some graphs and basic data in Chapter 2 are given in dual units. Appendix B xiv Preface gives the SI equivalents of inch-pound units. An SI version of the ACI Building Code is available. A brief historical note may be of interest. This book is the fifteenth edition of a textbook originated in 1923 by Leonard C. Urquhart and Charles E. O’Rourke, both professors of structural engineering at Cornell University. Over its remarkable 92-year history, new editions have kept pace with research, improved materials, and new methods of analysis and design. The second, third, and fourth editions firmly estab- lished the work as a leading text for elementary courses in the subject area. Professor George Winter, also of Cornell, collaborated with Urquhart in preparing the fifth and sixth editions. Winter and Professor Arthur H. Nilson were responsible for the sev- enth, eighth, and ninth editions, which substantially expanded both the scope and the depth of the presentation. The tenth, eleventh, and twelfth editions were prepared by Professor Nilson subsequent to Professor Winter’s passing in 1982. Professor Nilson was joined by Professor David Darwin of the University of Kansas and by Professor Charles Dolan of the University of Wyoming beginning with the thirteenth edition. All three have been deeply involved in research and teaching in the fields of reinforced and prestressed concrete, as well as professional Code-writing committees, and have spent significant time in professional practice, invaluable in developing the perspective and structural judgment that sets this book apart. Special thanks are due to McGraw-Hill Education project team, notably, Lorraine Buczek, Developmental Editor, Melissa Leick, Project Manager, Thomas Scaife, Brand Manager, and Ramya Thirumavalavan, Full Service Project Manager. We gladly acknowledge our indebtedness to the original authors. Although it is safe to say that neither Urquhart or O’Rourke would recognize much of the detail and that Winter would be impressed by the many changes, the approach to the subject and the educational philosophy that did so much to account for the success of the early editions would be familiar. With the passing of Arthur Nilson in the spring of 2014, we have lost a long-standing mentor, colleague, and friend. David Darwin Charles W. Dolan  Introduction 1.1 CONCRETE, REINFORCED CONCRETE, AND PRESTRESSED CONCRETE Concrete is a stonelike material obtained by permitting a carefully proportioned mixture of cement, sand and gravel or other coarse aggregate, and water to harden in forms of the shape and dimensions of the desired structure. The bulk of the material consists of fine and coarse aggregate. Cement and water interact chemically to bind the aggregate particles into a solid mass. Additional water, over and above that needed for this chemical reaction, is necessary to give the mixture the workability that enables it to fill the forms and surround the embedded reinforcing steel prior to hardening. Concretes with a wide range of properties can be obtained by appropriate adjustment of the proportions of the constituent materials. Special cements (such as high early strength cements), special aggregates (such as various lightweight or heavyweight aggregates), admixtures (such as plasticizers, air-entraining agents, silica fume, and fly ash), and special curing methods (such as steam-curing) permit an even wider vari- ety of properties to be obtained. These properties depend to a very substantial degree on the proportions of the mixture, on the thoroughness with which the various constituents are intermixed, and on the conditions of humidity and temperature in which the mixture is maintained from the moment it is placed in the forms until it is fully hardened. The process of control- ling conditions after placement is known as curing. To protect against the unintentional production of substandard concrete, a high degree of skillful control and supervision is necessary throughout the process, from the proportioning by weight of the individual components, through mixing and placing, until the completion of curing. The factors that make concrete a universal building material are so pronounced that it has been used, in more primitive kinds and ways than at present, for thousands of years, starting with lime mortars from 12,000 to 6000 BCE in Crete, Cyprus, Greece, and the Middle East. The facility with which, while plastic, it can be deposited and made to fill forms or molds of almost any practical shape is one of these factors. Its high fire and weather resistance is an evident advantage. Most of the constituent materials, with the exception of cement and additives, are usually available at low cost locally or at small distances from the construction site. Its compressive strength, like that of natural stones, is high, which makes it suitable for members primarily subject to compression, such as columns and arches. On the other hand, again as in natural stones, it is a relatively brittle material whose tensile strength is low compared with its compressive strength. This prevents its economical use as the sole building material in structural members that are subject to tension either entirely (such as in tie-rods) or over part of their cross sections (such as in beams or other flexural members). 1 2 DESIGN OF CONCRETE STRUCTURES Chapter 1 To offset this limitation, it was found possible, in the second half of the nineteenth century, to use steel with its high tensile strength to reinforce concrete, chiefly in those places where its low tensile strength would limit the carrying capacity of the member. The reinforcement, usually round steel rods with appropriate surface deformations to provide interlocking, is placed in the forms in advance of the concrete. When completely surrounded by the hardened concrete mass, it forms an integral part of the member. The resulting combination of two materials, known as reinforced concrete, combines many of the advantages of each: the relatively low cost, good weather and fire resistance, good compressive strength, and excellent formability of concrete and the high tensile strength and much greater ductility and toughness of steel. It is this combination that allows the almost unlimited range of uses and possibilities of rein- forced concrete in the construction of buildings, bridges, dams, tanks, reservoirs, and a host of other structures. It is possible to produce steels, at relatively low cost, whose yield strength is 3 to 4 times and more that of ordinary reinforcing steels. Likewise, it is possible to produce concrete 4 to 5 times as strong in compression as the more ordinary concretes. These high-strength materials offer many advantages, including smaller member cross sec- tions, reduced dead load, and longer spans. However, there are limits to the strengths of the constituent materials beyond which certain problems arise. To be sure, the strength of such a member would increase roughly in proportion to those of the mate- rials. However, the high strains that result from the high stresses that would otherwise be permissible would lead to large deformations and consequently large deflections of such members under ordinary loading conditions. Equally important, the large strains in such high-strength reinforcing steel would induce large cracks in the surrounding low tensile strength concrete, cracks that not only would be unsightly but also could significantly reduce the durability of the structure. This limits the useful yield strength of high-strength reinforcing steel to 100 ksi† according to many codes and specifica- tions; 60 ksi steel is most commonly used. Construction known as prestressed concrete, however, does use steels and con- cretes of very high strength in combination. The steel, in the form of wires, strands, or bars, is embedded in the concrete under high tension that is held in equilibrium by compressive stresses in the concrete after hardening. Because of this precompression, the concrete in a flexural member will crack on the tension side at a much larger load than when not so precompressed. Prestressing greatly reduces both the deflections and the tensile cracks at ordinary loads in such structures and thereby enables these high- strength materials to be used effectively. Prestressed concrete has extended, to a very significant extent, the range of spans of structural concrete and the types of structures for which it is suited. 1.2 STRUCTURAL FORMS The figures that follow show some of the principal structural forms of reinforced con- crete. Pertinent design methods for many of them are discussed later in this volume. Floor support systems for buildings include the monolithic slab-and-beam floor shown in Fig. 1.1, the one-way joist system of Fig. 1.2, and the flat plate floor, without beams or girders, shown in Fig. 1.3. The flat slab floor of Fig. 1.4, frequently used for more heavily loaded buildings such as warehouses, is similar to the flat plate floor, but makes use of increased slab thickness in the vicinity of the columns, as well as †Abbreviation for kips per square inch, or thousands of pounds per square inch.  INTRODUCTION 3 FIGURE 1.1 One-way reinforced concrete floor slab with monolithic supporting beams. (Portland Cement Association.) FIGURE 1.2 One-way joist floor system, with closely spaced ribs supported by monolithic concrete beams; transverse ribs provide for lateral distribution of localized loads. (Portland Cement Association.) flared column tops, to reduce stresses and increase strength in the support region. The choice among these and other systems for floors and roofs depends upon functional requirements, loads, spans, and permissible member depths, as well as on cost and esthetic factors. Where long clear spans are required for roofs, concrete shells permit use of extremely thin surfaces, often thinner, relatively, than an eggshell. The folded plate roof of Fig. 1.5 is simple to form because it is composed of flat surfaces; such roofs have been employed for spans of 200 ft and more. The cylindrical shell of Fig. 1.6 is also relatively easy to form because it has only a single curvature; it is similar to the folded plate in its structural behavior and range of spans and loads. Shells of this type were once quite popular in the United States and remain popular in other parts of the world. Doubly curved shell surfaces may be generated by simple mathematical curves such as circular arcs, parabolas, and hyperbolas, or they may be composed of com- plex combinations of shapes. The hyperbolic paraboloid shape, defined by a concave downward parabola moving along a concave upward parabolic path, has been widely 4 DESIGN OF CONCRETE STRUCTURES Chapter 1 FIGURE 1.3 Flat plate floor slab, carried directly by columns without beams or girders. (Portland Cement Association.) FIGURE 1.4 Flat slab floor, without beams but with slab thickness increased at the columns and with flared column tops to provide for local concentration of forces. (Portland Cement Association.) used. It has the interesting property that the doubly curved surface contains two sys- tems of straight-line generators, permitting straight-form lumber to be used. The com- plex dome of Fig. 1.7, which provides shelter for performing arts events, consists essentially of a circular dome but includes monolithic, upwardly curved edge surfaces to provide stiffening and strengthening in that critical region.  INTRODUCTION 5 FIGURE 1.5 Folded plate roof of 125 ft span that, in addition to carrying ordinary roof loads, carries the second floor as well using a system of cable hangers; the ground floor is kept free of columns. (Photograph by Arthur H. Nilson.) FIGURE 1.6 Cylindrical shell roof providing column-free interior space. (Photograph by Arthur H. Nilson.) Bridge design has provided the opportunity for some of the most challenging and creative applications of structural engineering. The award-winning Napoleon Bonaparte Broward Bridge, shown in Fig. 1.8, is a six-lane, cable-stayed structure that spans St. John’s River at Dame Point, Jacksonville, Florida. It has a 1300 ft 6 DESIGN OF CONCRETE STRUCTURES Chapter 1 FIGURE 1.7 Spherical shell in Lausanne, Switzerland. Upwardly curved edges provide stiffening for the central dome. (Photograph by Arthur H. Nilson.) FIGURE 1.8 Napoleon Bonaparte Broward Bridge, with a 1300 ft center span at Dame Point, Jacksonville, Florida. (HNTB Corporation, Kansas City, Missouri.) center span. Figure 1.9 shows the Bennett Bay Centennial Bridge, a four-span con- tinuous, segmentally cast-in-place box girder structure. Special attention was given to esthetics in this award-winning design. The spectacular Natchez Trace Parkway Bridge in Fig. 1.10, a two-span arch structure using hollow precast concrete elements, carries a two-lane highway 155 ft above the valley floor. This structure has won many honors,  INTRODUCTION 7 FIGURE 1.9 Bennett Bay Centennial Bridge, Coeur d’Alene, Idaho, a four-span continuous concrete box girder structure of length 1730 ft. (HNTB Corporation, Kansas City, Missouri.) FIGURE 1.10 Natchez Trace Parkway Bridge near Franklin, Tennessee, an award-winning two-span concrete arch structure rising 155 ft above the valley floor. (Designed by Figg Engineering Group.) 8 DESIGN OF CONCRETE STRUCTURES Chapter 1 FIGURE 1.11 The Dakota Dome is a 10,000 seat multipurpose stadium with a concrete frame and ring girder to support the roof. (Photograph by Charles W. Dolan.) including awards from the American Society of Civil Engineers and the National Endowment for the Arts. The durability and sustainability of concrete structures is evident in the 10,000 seat multipurpose Dakota Dome shown in Fig. 1.11. Originally constructed in 1979 as an inflatable roof dome, the entire concrete structure was retained and a steel roof installed in 2001. Concrete structures may be designed to provide a wide array of surface textures, colors, and structural forms. Figure 1.12 shows a precast concrete building containing both color changes and architectural finishes. The forms shown in Figs. 1.1 to 1.12 hardly constitute a complete inventory but are illustrative of the shapes appropriate to the properties of reinforced or pre- stressed concrete. They illustrate the adaptability of the material to a great variety of one-dimensional (beams, girders, columns), two-dimensional (slabs, arches, rigid frames), and three-dimensional (shells, tanks) structures and structural components. This variability allows the shape of the structure to be adapted to its function in an eco- nomical manner, and furnishes the architect and design engineer with a wide variety of possibilities for esthetically satisfying structural solutions. 1.3 LOADS Loads that act on structures can be divided into three broad categories: dead loads, live loads, and environmental loads. Dead loads are those that are constant in magnitude and fixed in location throughout the lifetime of the structure. Usually the major part of the dead load is the weight of the structure itself. This can be calculated with good accuracy from the design configuration, dimensions of the structure, and density of the material. For buildings, floor fill, finish floors, and plastered ceilings are usually included as dead loads, and an allowance is made for suspended loads such as piping and lighting fixtures. For bridges, dead loads may include wearing surfaces, sidewalks, and curbs, and an allowance is made for piping and other suspended loads. Live loads consist chiefly of occupancy loads in buildings and traffic loads on bridges. They may be either fully or partially in place or not present at all, and may also change in location. Their magnitude and distribution at any given time are uncer- tain, and even their maximum intensities throughout the lifetime of the structure are not known with precision. The minimum live loads for which the floors and roof of  INTRODUCTION 9 FIGURE 1.12 Concrete structures can be produced in a wide range of colors, finishes, and architectural detailing. (Courtesy of Rocky Mountain Prestress, LLC.) a building should be designed are usually specified in the building code that governs at the site of construction. Representative values of minimum live loads to be used in a wide variety of buildings are found in Minimum Design Loads for Buildings and Other Structures (Ref. 1.1), a portion of which is reprinted in Table 1.1. The table gives uniformly distributed live loads for various types of occupancies; these include impact provisions where necessary. These loads are expected maxima and consider- ably exceed average values. In addition to these uniformly distributed loads, it is recommended that, as an alternative to the uniform load, floors be designed to support safely certain concen- trated loads if these produce a greater stress. For example, according to Ref. 1.1, office floors are to be designed to carry a load of 2000 lb distributed over an area 2.5 ft square (6.25 ft2), to allow for the weight of a safe or other heavy equipment, and stair treads must safely support a 300 lb load applied on the center of the tread. Certain reductions are often permitted in live loads for members supporting large areas with the understanding that it is unlikely that the entire area would be fully loaded at one time (Refs. 1.1 and 1.2). Tabulated live loads cannot always be used. The type of occupancy should be considered and the probable loads computed as accurately as possible. Warehouses for heavy storage may be designed for loads as high as 500 psf or more; unusually heavy operations in manufacturing buildings may require an increase in the 250 psf value specified in Table 1.1; special provisions must be made for all definitely located heavy concentrated loads. 10 DESIGN OF CONCRETE STRUCTURES Chapter 1 TABLE 1.1 Minimum uniformly distributed live loads in pounds per square foot (psf) Live Load, Live Load, Occupancy or Use psf Occupancy or Use psf Apartments (see Residential) Hospitals Access floor systems Operating rooms, laboratories 60 Office use 50 Patient rooms 40 Computer use 100 Corridors above first floor 80 Armories and drill roomsa 150 Hotels (see Residential) Assembly areas and theaters Libraries Fixed seats (fastened to floor)a 60 Reading rooms 60 Lobbiesa 100 Stack roomsa,e 150 Movable seatsa 100 Corridors above first floor 80 Platforms (assembly)a 100 Manufacturing Stage floorsa 150 Lighta 125 Balconies and decksb Heavya 250 Catwalks for maintenance access 40 Office buildings Corridors File and computer rooms shall be designed for First floor 100 heavier loads based on anticipated occupancy Other floors, same as occupancy served Lobbies and first floor corridors 100 except as indicated Offices 50 Dining rooms and restaurantsa 100 Corridors above first floor 80 Dwellings (see Residential) Penal institutions Fire escapes 100 Cell blocks 40 On single-family dwellings only 40 Corridors 100 Garages (passenger vehicles only)a,c,d 40 Recreational uses Trucks and busesc Bowling alleysa 75 (continued ) Live loads for highway bridges are specified by the American Association of State Highway and Transportation Officials (AASHTO) in its LRFD Bridge Design Specifications (Ref. 1.3). For railway bridges, the American Railway Engineering and Maintenance-of-Way Association (AREMA) has published the Manual of Railway Engineering (Ref. 1.4), which specifies traffic loads. Environmental loads consist mainly of snow loads, wind pressure and suction, earthquake load effects (that is, inertia forces caused by earthquake motions), soil pressures on subsurface portions of structures, loads from possible ponding of rain- water on flat surfaces, and forces caused by temperature differentials. Like live loads, environmental loads at any given time are uncertain in both magnitude and distribu- tion. Reference 1.1 contains much information on environmental loads, which is often modified locally depending, for instance, on local climatic or seismic conditions. Figure 1.13, from the 1972 edition of Ref. 1.1, gives snow loads for the con- tinental United States and is included here for illustration only. The 2010 edition of Ref. 1.1 gives much more detailed information. In either case, specified values represent not average values, but expected upper limits. A minimum roof load of 20 psf is often specified to provide for construction and repair loads and to ensure reasonable stiffness. Much progress has been made in developing rational methods for predicting horizontal forces on structures due to wind and seismic action. Reference 1.1  INTRODUCTION 11 TABLE 1.1 (Continued) Live Load, Live Load, Occupancy or Use psf Occupancy or Use psf Dances halls 100 All other construction 20 Gymnasiumsa 100 Schools Reviewing stands, grandstands, and bleachersa, f 100 Classrooms 40 Stadiums and arenas with fixed seats Corridors above first floor 80 (fastened to floor)a, f 60 First floor corridors 100 Residential One- and two-family dwellings Sidewalks, vehicular driveways, and yards, subject Uninhabitable attics without storageg 10 to truckinga,k 250 Uninhabitable attics with storageh 20 Stairs and exit-ways 100 Habitable attics and sleeping areas 30 One- and two-family residences only 40 All other areas except stairs and balconies 40 Storage areas above ceilings 20 All other residential occupancies Storage warehouses (shall be designed for Hotels and multifamily houses heavier loads if required for anticipated storage) Private rooms and corridors serving them 40 Lighta 125 Public rooms and corridors serving them 100 Heavya 250 Reviewing stands, grandstands, and bleachersc 100 Stores Roofs Retail Ordinary flat, pitched, and curved roofsi 20 First floor 100 Roofs used for roof gardens 100 Roofs used for assembly purposes Upper floors 73 Roofs used for other occupanciesj Wholesale, all floorsa 125 Awnings and canopies Walkways and elevated platforms (other than Fabric construction supported by a exit-ways) 60 lightweight rigid skeleton structure 5 Yards and terraces, pedestrians 100 a Live load reduction for this use is not permitted unless specific exceptions apply. b 1.5 times live load for area served. Not required to exceed 100 psf. c Floors of garages or portions of a building used for storage of motor vehicles shall be designed for the uniformly distributed live loads of this table or for concentrated loads specified in Ref. 1.1. d Design for trucks and buses shall be in accordance with Ref. 1.3; however, provisions for fatigue and dynamic load are not required. e The loading applies to stack room floors that support nonmobile, double-faced library book stacks subject to the following limitations: (1) The nominal book stack unit height shall not exceed 90 in.; (2) the nominal shelf depth shall not exceed 12 in. for each face; and (3) parallel rows of double-faced book stacks shall be separated by aisles not less than 36 in. wide. f In addition to the vertical live loads, the design shall include horizontal swaying forces applied to each row of seats as follows: 24 lb per linear seat applied in the direction parallel to each row of seats and 10 lb per linear ft of seat applied in the direction perpendicular to each row of seats. The parallel and perpendicular horizontal swaying forces need not be applied simultaneously. g See Ref. 1.1 for description of uninhabitable attic areas without storage. This live load need not be assumed to act concurrently with any other live load requirement. h See Ref. 1.1 for description of uninhabitable attic areas with storage and where this provision applies. i Where uniform roof live loads are reduced to less than 20 psf in accordance with Section 4.8.2 of Ref. 1.1 and are applied to the design of structural members arranged so as to create continuity, the reduced roof live load shall be applied to adjacent spans or to alternate spans, whichever produces the greatest unfavorable load effect. j Roofs used for other special purposes shall be designed for appropriate loads as approved by the authority having jurisdiction. k Other uniform loads in accordance with an approved method that contains provisions for truck loadings shall also be considered where appropriate. Data Source: From Ref. 1.1. summarizes current thinking regarding wind forces and has much information pertain- ing to earthquake loads as well. Reference 1.5 presents detailed recommendations for lateral forces from earthquakes. Reference 1.1 specifies design wind pressures per square foot of vertical wall sur- face. Depending upon locality, these equivalent static forces vary from about 10 to 50 psf. 12 DESIGN OF CONCRETE STRUCTURES Chapter 1 FIGURE 1.13 80 Snow load in pounds per 10 10 70 10 20 30 square foot (psf) on the 40 50 60 80 70 Sn alys ground, 50-year mean 50 60 an 50 ow is o recurrence interval. (From 50 20 40 40 40 loa f lo Minimum Design Loads for 30 30 30 d i ca 20 20 20 n t l cl Buildings and Other Structures, 10 20 his im ANSI A58.1–1972, American 10 are ate National Standards Institute, 5 10 a m and New York, NY, 1972.) us top tb 10 e b ogra 10 as phy ed 5 on 55 Factors include basic wind speed, exposure (urban vs. open terrain, for example), height of the structure, the importance of the structure (that is, consequences of failure), and gust effect factors to account for the fluctuating nature of the wind and its interaction with the structure. Seismic forces may be found for a particular structure by elastic or inelastic dynamic analysis, considering expected ground accelerations and the mass, stiffness, and damping characteristics of the construction. In less seismically active areas, the design is often based on equivalent static forces calculated from provisions such as those of Refs. 1.1 and 1.5. The base shear is found by considering such factors as location, type of structure and its occupancy, total dead load, and the particular soil condition. The total lateral force is distributed to floors over the entire height of the structure in such a way as to approximate the distribution of forces obtained from a dynamic analysis. 1.4 SERVICEABILITY, STRENGTH, AND STRUCTURAL SAFETY To serve its purpose, a structure must be safe against collapse and serviceable in use. Serviceability requires that deflections be adequately small; that cracks, if any, be kept to tolerable limits; that vibrations be minimized; etc. Safety requires that the strength of the structure be adequate for all loads that may foreseeably act on it. If the strength of a structure, built as designed, could be predicted accurately, and if the loads and their internal effects (moments, shears, axial forces) were known accurately, safety could be ensured by providing a carrying capacity just barely in excess of the known loads. However, there are a number of sources of uncertainty in the analysis, design, and construction of reinforced concrete structures. These sources of uncertainty, which require a definite margin of safety, may be listed as follows: 1. Actual loads may differ from those assumed. 2. Actual loads may be distributed in a manner different from that assumed. 3. The assumptions and simplifications inherent in any analysis may result in calcu- lated load effects—moments, shears, etc.—different from those that, in fact, act in the structure.  INTRODUCTION 13 4. The actual structural behavior may differ from that assumed, owing to imperfect knowledge. 5. Actual member dimensions may differ from those specified. 6. Reinforcement may not be in its proper position. 7. Actual material strength may be different from that specified. In addition, in the establishment of safety requirements, consideration must be given to the consequences of failure. In some cases, a failure would be merely an inconvenience. In other cases, loss of life and significant loss of property may be involved. A further consideration should be the nature of the failure, should it occur. A gradual failure with ample warning permitting remedial measures is preferable to a sudden, unexpected collapse. It is evident that the selection of an appropriate margin of safety is not a simple matter. However, progress has been made toward rational safety provisions in design codes (Refs. 1.6 to 1.11). a. Variability of Loads Since the maximum load that will occur during the life of a structure is uncertain, it can be considered a random variable. In spite of this uncertainty, the engineer must provide an adequate structure. A probability model for the maximum load can be devised by means of a probability density function for loads, as represented by the frequency curve of Fig. 1.14a. The exact form of this distribution curve, for a particu- lar type of loading such as office loads, can be determined only on the basis of statisti- cal data obtained from large-scale load surveys. A number of such surveys have been completed. For types of loads for which such data are scarce, fairly reliable informa- tion can be obtained from experience, observation, and judgment. For such a frequency curve (Fig. 1.14a), the area under the curve between two abscissas, such as loads Q1 and Q2, represents the probability of occurrence of loads Q of magnitude Q1 < Q < Q2. A specified service load Qd for design is selected conser- vatively in the upper region of Q in the distribution curve, as shown. The probability of occurrence of loads larger than Qd is then given by the shaded area to the right of Qd. It is seen that this specified service load is considerably larger than the mean load Q acting on the structure. This mean load is much more typical of average load condi- tions than the design load Qd. b. Strength The strength of a structure depends on the strength of the materials from which it is made. For this purpose, minimum material strengths are specified in standardized ways. Actual material strengths cannot be known precisely and therefore also consti- tute random variables (see Section 2.6). Structural strength depends, furthermore, on the care with which a structure is built, which in turn reflects the quality of supervision and inspection. Member sizes may differ from specified dimensions, reinforcement may be out of position, poorly placed concrete may show voids, etc. The strength of the entire structure or of a population of repetitive structures, such as highway overpasses, can also be considered a random variable with a probability density function of the type shown in Fig. 1.14b. As in the case of loads, the exact form of this function cannot be known but can be approximated from known data, such as statistics of actual, measured materials and member strengths and simi- lar information. Considerable information of this type has been, or is being, devel- oped and used. 14 DESIGN OF CONCRETE STRUCTURES Chapter 1 FIGURE 1.14 Frequency curves for (a) loads Q, (b) strengths S, f (Q ) and (c) safety margin M. Q Q1Q 2Qd f (S ) (a ) Load Q Sd Sn S (b ) Strength S f (M  S  Q) m 0 M (c) Safety margin M  S  Q c. Structural Safety A given structure has a safety margin M if M=S−Q>0 (1.1) that is, if the strength of the structure is larger than the load acting on it. Since S and Q are random variables, the safety margin M = S - Q is also a random variable. A plot of the probability function of M may appear as in Fig. 1.14c. Failure occurs when M is less than zero. Thus, the probability of failure is represented by the shaded area in the figure. Even though the precise form of the probability density functions for S and Q, and therefore for M, is not known, much can be achieved in the way of a rational approach to structural safety. One such approach is to require that the mean safety margin M be a specified number β of standard deviations σm above zero. It can be demonstrated that this results in the requirement that − −− ψsS ≥ ψLQ (1.2) where ψs is a partial safety coefficient smaller than 1.0 applied to the mean strength − −− S and ψL is a partial safety coefficient larger than 1.0 applied to the mean load Q. The magnitude of each partial safety coefficient depends on the variance of the quan- tity to which it applies, S or Q, and on the chosen value of β, the reliability index of the structure. As a general guide, a value of the safety index β between 3 and 4 cor- responds to a probability of failure of the order of 1:100,000 (Ref. 1.8). The value of β is often established by calibration against well-proved and established designs.  INTRODUCTION 15 In practice, it is more convenient to introduce partial safety coefficients with respect to code-specified loads which, as already noted, considerably exceed aver- age values, rather than with respect to mean loads as in Eq. (1.2); similarly, the par- tial safety coefficient for strength is applied to nominal strength† generally computed somewhat conservatively, rather than to mean strengths as in Eq. (1.2). A restatement of the safety requirement in these terms is ϕ Sn ≥ γ Qd (1.3a) in which ϕ is a strength reduction factor applied to nominal strength Sn and γ is a load factor applied to calculated or code-specified design loads Qd. Furthermore, recogniz- ing the differences in variability between, say, dead loads D and live loads L, it is both reasonable and easy to introduce different load factors for different types of loads. The preceding equation can thus be written ϕ Sn ≥ γd D + γl L (1.3b) in which γd is a load factor somewhat greater than 1.0 applied to the calculated dead load D and γl is a larger load factor applied to the code-specified live load L. When additional loads, such as the wind load W, are to be considered, the reduced probabil- ity that maximum dead, live, and wind or other loads will act simultaneously can be incorporated by using modified load factors such that ϕ Sn ≥ γdi D + γli L + γwiW +... (1.3c) Present U.S. design codes follow the format of Eqs. (1.3b) and (1.3c). 1.5 DESIGN BASIS The single most important characteristic of any structural member is its actual strength, which must be large enough to resist, with some margin to spare, all foreseeable loads that may act on it during the life of the structure, without failure or other distress. It is logical, therefore, to proportion members, that is, to select concrete dimensions and reinforcement, so that member strengths are adequate to resist forces resulting from certain hypothetical overload stages, significantly above loads expected actually to occur in service. This design concept is known as strength design. For reinforced concrete structures at loads close to and at failure, one or both of the materials, concrete and steel, are invariably in their nonlinear inelastic range. That is, concrete in a structural member reaches its maximum strength and subsequent fracture at stresses and strains far beyond the initial elastic range in which stresses and strains are fairly proportional. Similarly, steel close to and at failure of the member is usually stressed beyond its elastic domain into and even beyond the yield region. Consequently, the nominal strength of a member must be calculated on the basis of this inelastic behavior of the materials. A member designed by the strength method must also perform in a satisfactory way under normal service loading. For example, beam deflections must be limited to acceptable values, and the number and width of flexural cracks at service loads must be controlled. Serviceability limit conditions are an important part of the total design, although attention is focused initially on strength. † Throughout this book quantities that refer to the strength of members, calculated by accepted analysis methods, are furnished with the subscript n, which stands for “nominal.” This notation is in agreement with the ACI Code. It is intended to convey that the actual strength of any member is bound to deviate to some extent from its calculated, nominal value because of inevitable variations of dimensions, materials properties, and other parameters. Design in all cases is based on this nominal strength, which represents the best available estimate of the actual member strength. 16 DESIGN OF CONCRETE STRUCTURES Chapter 1 Historically, members were proportioned so that stresses in the steel and concrete resulting from normal service loads were within specified limits. These limits, known as allowable stresses, were only fractions of the failure stresses of the materials. For members proportioned on such a service load basis, the margin of safety was provided by stipulating allowable stresses under service loads that were appropriately small frac- tions of the compressive concrete strength and the steel yield stress. We now refer to this basis for design as service load design. Allowable stresses, in practice, were set at about one-half the concrete compressive strength and one-half the yield stress of the steel. Because of the difference in realism and reliability, the strength design method has displaced the older service load design method. However, the older method pro- vides the basis for some serviceability checks and is the design basis for many older structures. Throughout this text, strength design is presented almost exclusively. 1.6 DESIGN CODES AND SPECIFICATIONS The design of concrete structures such as those of Figs. 1.1 to 1.12 is generally done within the framework of codes giving specific requirements for materials, structural analysis, member proportioning, etc. The International Building Code (Ref. 1.2) is an example of a consensus code governing structural design and is often adopted by local municipalities. The responsibility of preparing material-specific portions of the codes rests with various professional groups, trade associations, and technical institutes. In contrast with many other industrialized nations, the United States does not have an official, government-sanctioned, national code. The Americ