Structural steel designer’s handbook (third edition) part 1

STRUCTURAL STEEL DESIGNER’S HANDBOOK Roger L Brockenbrough Editor R L Brockenbrough & Associates, Inc Pittsburgh, Pennsylvania Frederick S Merritt Editor Late Consulting Engineer, West Palm Beach, Florida Third Edition McGRAW-HILL, INC New York San Francisco Washington, D.C Auckland Bogota´ Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto Library of Congress Cataloging-in-Publication Data Structural steel designer’s handbook / Roger L Brockenbrough, editor, Frederick S Merritt, editor.—3rd ed p cm Includes index ISBN 0-07-008782-2 Building, Iron and steel Steel, Structural I Brockenbrough, R L II Merritt, Frederick S TA684.S79 1994 624.1Ј821—dc20 93-38088 CIP Copyright ᭧ 1999, 1994, 1972 by McGraw-Hill, Inc All rights reserved Printed in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher DOC / DOC 9 ISBN 0-07-008782-2 The sponsoring editor for this book was Larry S Hager, the editing supervisor was Steven Melvin, and the production supervisor was Sherri Souffrance It was set in Times Roman by Pro-Image Corporation Printed and bound by R R Donnelley & Sons Company This book is printed on acid-free paper Information contained in this work has been obtained by McGraw-Hill, Inc from sources believed to be reliable However, neither McGraw-Hill nor its authors guarantees the accuracy or completeness of any information published herein and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services If such services are required, the assistance of an appropriate professional should be sought Other McGraw-Hill Book Edited by Roger L Brockenbrough Brockenbrough & Boedecker • HIGHWAY ENGINEERING HANDBOOK Other McGraw-Hill Books Edited by Frederick S Merritt Merritt • STANDARD HANDBOOK FOR CIVIL ENGINEERS Merritt & Ricketts • BUILDING DESIGN AND CONSTRUCTION HANDBOOK Other McGraw-Hill Books of Interest Beall • MASONRY DESIGN AND DETAILING Breyer • DESIGN OF WOOD STRUCTURES Brown • FOUNDATION BEHAVIOR AND REPAIR Faherty & Williamson • WOOD ENGINEERING AND CONSTRUCTION HANDBOOK Gaylord & Gaylord • STRUCTURAL ENGINEERING HANDBOOK Harris • NOISE CONTROL IN BUILDINGS Kubal • WATERPROOFING THE BUILDING ENVELOPE Newman • STANDARD HANDBOOK OF STRUCTURAL DETAILS FOR BUILDING CONSTRUCTION Sharp • BEHAVIOR AND DESIGN OF ALUMINUM STRUCTURES Waddell & Dobrowolski • CONCRETE CONSTRUCTION HANDBOOK CONTRIBUTORS Boring, Delbert F., P.E Senior Director, Construction Market, American Iron and Steel Institute, Washington, D.C (SECTION BUILDING DESIGN CRITERIA) Brockenbrough, Roger L., P.E R L Brockenbrough & Associates, Inc., Pittsburgh, Penn- sylvania (SECTION PROPERTIES OF STRUCTURAL STEELS AND EFFECTS OF STEELMAKING AND FABRICATION; SECTION 10 COLD-FORMED STEEL DESIGN) Cuoco, Daniel A., P.E Principal, LZA Technology/Thornton-Tomasetti Engineers, New York, New York (SECTION FLOOR AND ROOF SYSTEMS) Cundiff, Harry B., P.E HBC Consulting Service Corp., Atlanta, Georgia (SECTION 11 DESIGN CRITERIA FOR BRIDGES) Geschwindner, Louis F., P.E Professor of Architectural Engineering, Pennsylvania State University, University Park, Pennsylvania (SECTION ANALYSIS OF SPECIAL STRUCTURES) Haris, Ali A K., P.E President, Haris Enggineering, Inc., Overland Park, Kansas (SECTION DESIGN OF BUILDING MEMBERS) Hedgren, Arthur W Jr., P.E Senior Vice President, HDR Engineering, Inc., Pittsburgh, Pennsylvania (SECTION 14 ARCH BRIDGES) Hedefine, Alfred, P.E Former President, Parsons, Brinckerhoff, Quade & Douglas, Inc., New York, New York (SECTION 12 BEAM AND GIRDER BRIDGES) Kane, T., P.E Cives Steel Company, Roswell, Georgia (SECTION CONNECTIONS) Kulicki, John M., P.E President and Chief Engineer, Modjeski and Masters, Inc., Harris- burg, Pennsylvania (SECTION 13 TRUSS BRIDGES) LaBoube, R A., P.E Associate Professor of Civil Engineering, University of Missouri-Rolla, Rolla, Missouri (SECTION BUILDING DESIGN CRITERIA) LeRoy, David H., P.E Vice President, Modjeski and Masters, Inc., Harrisburg, Pennsylvania (SECTION 13 TRUSS BRIDGES) Mertz, Dennis, P.E Associate Professor of Civil Engineering, University of Delaware, New- ark, Delaware (SECTION 11 DESIGN CRITERIA FOR BRIDGES) Nickerson, Robert L., P.E Consultant-NBE, Ltd., Hempstead, Maryland (SECTION 11 DESIGN CRITERIA FOR BRIDGES) Podolny, Walter, Jr., P.E Senior Structural Engineer Bridge Division, Office of Bridge Technology, Federal Highway Administration, U.S Department of Transportation, Washington, D C (SECTION 15 CABLE-SUSPENDED BRIDGES) Prickett, Joseph E., P.E Senior Associate, Modjeski and Masters, Inc., Harrisburg, Penn- sylvania (SECTION 13 TRUSS BRIDGES) xv xvi CONTRIBUTORS Roeder, Charles W., P.E Professor of Civil Engineering, University of Washington, Seattle, Washington (SECTION LATERAL-FORCE DESIGN) Schflaly, Thomas, Director, Fabricating & Standards, American Institute of Steel Construc- tion, Inc., Chicago, Illinois (SECTION FABRICATION AND ERECTION) Sen, Mahir, P.E Professional Associate, Parsons Brinckerhoff, Inc., Princeton, New Jersey (SECTION 12 BEAM AND GIRDER BRIDGES) Swindlehurst, John, P.E Former Senior Professional Associate, Parsons Brinckerhoff, Inc., West Trenton, New Jersey (SECTION 12 BEAM AND GIRDER BRIDGES) Thornton, William A., P.E Chief Engineer, Cives Steel Company, Roswell, Georgia (SECTION CONNECTIONS) Ziemian, Ronald D., Associate Professor of Civil Engineering, Bucknell University, Lew- isburg, Pennsylvania (SECTION GENERAL STRUCTURAL THEORY) FACTORS FOR CONVERSION TO SI UNITS OF MEASUREMENT TO CONVERT FROM CUSTOMARY U.S UNIT TO METRIC UNIT MULTIPLY BY inch foot mm mm 25.4 304.8 Mass lb kg 0.45359 Mass/unit length plf kg/m 1.488 16 Mass/unit area psf kg/m2 4.882 43 Mass density pcf kg/m3 16.018 pound kip kip N N kN 4.448 22 4448.22 4.448 22 Force/unit length klf klf N/mm kN/m 14.593 14.593 Stress ksi psi MPa kPa 6.894 76 6.894 76 Bending Moment foot-kips foot-kips N-mm kN-m 355 817 1.355 817 Moment of inertia in4 mm4 416 231 Section modulus in3 mm3 16 387.064 QUANTITY Length Force xxi PREFACE TO THE THIRD EDITION This edition of the handbook has been updated throughout to reflect continuing changes in design trends and improvements in design specifications Criteria and examples are included for both allowable-stress design (ASD) and load-and-resistance-factor design (LRFD) methods, but an increased emphasis has been placed on LRFD to reflect its growing use in practice Numerous connection designs for building construction are presented in LRFD format in conformance with specifications of the American Institute of Steel Construction (AISC) A new article has been added on the design of hollow structural sections (HSS) by LRFD, based on a new separate HSS specification by AISC Also, because of their growing use in light commercial and residential applications, a new section has been added on the design of cold-formed steel structural members, based on the specification by the American Iron and Steel Institute (AISI) It is applicable to both ASD and LRFD Design criteria are now presented in separate parts for highway and railway bridges to better concentrate on those subjects Information on highway bridges is based on specifications of the American Association of State Highway and Transportation Officials (AASHTO) and information on railway bridges is based on specifications of the American Railway Engineering and Maintenance-of-Way Association (AREMA) A very detailed example of the LRFD design of a two-span composite I-girder highway bridge has been presented in Section 11 to illustrate AASHTO criteria, and also the LRFD design of a single-span composite bridge in Section 12 An example of the LRFD design of a truss member is presented in Section 13 This edition of the handbook regrettably marks the passing of Fred Merritt, who worked tirelessly on previous editions, and developed many other handbooks as well His many contributions to these works are gratefully acknowledged Finally, the reader is cautioned that independent professional judgment must be exercised when information set forth in this handbook is applied Anyone making use of this information assumes all liability arising from such use Users are encouraged to use the latest edition of the referenced specifications, because they provide more complete information and are subject to frequent change Roger L Brockenbrough xvii PREFACE TO THE SECOND EDITION This handbook has been developed to serve as a comprehensive reference source for designers of steel structures Included is information on materials, fabrication, erection, structural theory, and connections, as well as the many facets of designing structural-steel systems and members for buildings and bridges The information presented applies to a wide range of structures The handbook should be useful to consulting engineers; architects; construction contractors; fabricators and erectors; engineers employed by federal, state, and local governments; and educators It will also be a good reference for engineering technicians and detailers The material has been presented in easy-to-understand form to make it useful to professionals and those with more limited experience Numerous examples, worked out in detail, illustrate design procedures The thrust is to provide practical techniques for cost-effective design as well as explanations of underlying theory and criteria Design methods and equations from leading specifications are presented for ready reference This includes those of the American Institute of Steel Construction (AISC), the American Association of State Highway and Transportation Officials (AASHTO), and the American Railway Engineering Association (AREA) Both the traditional allowable-stress design (ASD) approach and the load-and-resistance-factor design (LRFD) approach are presented Nevertheless, users of this handbook would find it helpful to have the latest edition of these specifications on hand, because they are changed annually, as well as the AISC ‘‘Steel Construction Manual,’’ ASD and LRFD Contributors to this book are leading experts in design, construction, materials, and structural theory They offer know-how and techniques gleaned from vast experience They include well-known consulting engineers, university professors, and engineers with an extensive fabrication and erection background This blend of experiences contributes to a broad, well-rounded presentation The book begins with an informative section on the types of steel, their mechanical properties, and the basic behavior of steel under different conditions Topics such as coldwork, strain-rate effects, temperature effects, fracture, and fatigue provide in-depth information Aids are presented for estimating the relative weight and material cost of steels for various types of structural members to assist in selecting the most economical grade A review of fundamental steel-making practices, including the now widely used continuouscasting method, is presented to give designers better knowledge of structural steels and alloys and how they are produced Because of their impact on total cost, a knowledge of fabrication and erection methods is a fundamental requirement for designing economical structures Accordingly, the book presents description of various shop fabrication procedures, including cutting steel components to size, punching, drilling, and welding Available erection equipment is reviewed, as well as specific methods used to erect bridges and buildings A broad treatment of structural theory follows to aid engineers in determining the forces and moments that must be accounted for in design Basic mechanics, traditional tools for xix xx PREFACE analysis of determinate and indeterminate structures, matrix methods, and other topics are discussed Structural analysis tools are also presented for various special structures, such as arches, domes, cable systems, and orthotropic plates This information is particularly useful in making preliminary designs and verifying computer models Connections have received renewed attention in current structural steel design, and improvements have been made in understanding their behavior in service and in design techniques A comprehensive section on design of structural connections presents approved methods for all of the major types, bolted and welded Information on materials for bolting and welding is included Successive sections cover design of buildings, beginning with basic design criteria and other code requirements, including minimum design dead, live, wind, seismic, and other loads A state-of-the-art summary describes current fire-resistant construction, as well as available tools that allow engineers to design for fire protection and avoid costly tests In addition, the book discusses the resistance of various types of structural steel to corrosion and describes corrosion-prevention methods A large part of the book is devoted to presentation of practical approaches to design of tension, compression, and flexural members, composite and noncomposite One section is devoted to selection of floor and roof systems for buildings This involves decisions that have major impact on the economics of building construction Alternative support systems for floors are reviewed, such as the stub-girder and staggered-truss systems Also, framing systems for short and long-span roof systems are analyzed Another section is devoted to design of framing systems for lateral forces Both traditional and newer-type bracing systems, such as eccentric bracing, are analyzed Over one-third of the handbook is dedicated to design of bridges Discussions of design criteria cover loadings, fatigue, and the various facets of member design Information is presented on use of weathering steel Also, tips are offered on how to obtain economical designs for all types of bridges In addition, numerous detailed calculations are presented for design of rolled-beam and plate-girder bridges, straight and curved, composite and noncomposite, box girders, orthotropic plates, and continuous and simple-span systems Notable examples of truss and arch designs, taken from current practice, make these sections valuable references in selecting the appropriate spatial form for each site, as well as executing the design The concluding section describes the various types of cable-supported bridges and the cable systems and fittings available In addition, design of suspension bridges and cablestayed bridges is covered in detail The authors and editors are indebted to numerous sources for the information presented Space considerations preclude listing all, but credit is given wherever feasible, especially in bibliographies throughout the book The reader is cautioned that independent professional judgment must be exercised when information set forth in this handbook is applied Anyone making use of this information assumes all liability arising from such use Roger L Brockenbrough Frederick S Merritt CONTENTS Contributors xv Preface xvii Section Properties of Structural Steels and Effects of Steelmaking and Fabrication Roger L Brockenbrough, P.E 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 Structural Steel Shapes and Plates / 1.1 Steel-Quality Designations / 1.6 Relative Cost of Structural Steels / 1.8 Steel Sheet and Strip for Structural Applications / 1.10 Tubing for Structural Applications / 1.13 Steel Cable for Structural Applications / 1.13 Tensile Properties / 1.14 Properties in Shear / 1.16 Hardness Tests / 1.17 Effect of Cold Work on Tensile Properties / 1.18 Effect of Strain Rate on Tensile Properties / 1.19 Effect of Elevated Temperatures on Tensile Properties / 1.20 Fatigue / 1.22 Brittle Fracture / 1.23 Residual Stresses / 1.26 Lamellar Tearing / 1.28 Welded Splices in Heavy Sections / 1.28 k-Area Cracking / 1.29 Variations in Mechanical Properties / 1.29 Changes in Carbon Steels on Heating and Cooling / 1.30 Effects of Grain Size / 1.32 Annealing and Normalizing / 1.32 Effects of Chemistry on Steel Properties / 1.33 Steelmaking Methods / 1.35 Casting and Hot Rolling / 1.36 Effects of Punching Holes and Shearing / 1.39 Effects of Welding / 1.39 Effects of Thermal Cutting / 1.40 Section Fabrication and Erection Thomas Schflaly 2.1 2.2 2.3 2.4 1.1 2.1 Shop Detail Drawings / 2.1 Cutting, Shearing, and Sawing / 2.3 Punching and Drilling / 2.4 CNC Machines / 2.4 v 7.25 DESIGN OF BUILDING MEMBERS FIGURE 7.7 Transformed section of a composite beam Xϭ 21.52 ϫ 3.25 / ϩ 7.68(0.5 ϫ 15.69 ϩ 5.25) ϭ 4.64 in 21.52 ϩ 7.68 The elastic transformed moment of inertia for full composite action is Itr ϭ ͩ 90 ϫ 3.253 3.25 ϩ 21.52 4.64 Ϫ 13.6 ϫ 12 ͪ ϩ 7.68 ͩ 15.69 ϩ 5.25 Ϫ 4.64 ͪ ϩ 301 ϭ 1065 in4 Since partial composite construction is used, the effective moment of inertia is determined from Ieff ϭ Is ϩ (Itr Ϫ Is)͙͚Q n / Cƒ (7.32) where Cƒ ϭ concrete compression force based on full composite action Ieff ϭ 301 ϩ (1065 Ϫ 301) ͙104.5 / 276.5 ϭ 770.7 in4 Ieff is used to calculate the immediate deflection under service loads (without long-term effects) For long-term effect on deflections due to creep of the concrete, the moment of inertia is reduced to correspond to a 50% reduction in Ec Accordingly, the transformed moment of inertia with full composite action and 50% reduction in Ec is Itr ϭ 900.3 in4 and is based on a modular ratio 2n ϭ 27.2 The corresponding transformed concrete area is A1 ϭ 10.76 in2 The reduced effective moment of inertia for partial composite construction with longterm effect is determined from Eq (7.32): Ieff ϭ 301 ϩ (900.3 Ϫ 301) ͙104.5 / 276.5 ϭ 669.4 in4 Since unshored construction is specified, the deflection under the weight of concrete when placed and the steel weight is compensated for by the camber specified Long-term effect due to these weights need not be considered because the concrete is not stressed by them Deflection due to long-term superimposed dead loads is 7.26 SECTION SEVEN D1 ϭ ϫ 0.25 ϫ 304 ϫ 123 ϭ 0.235 in 384 ϫ 29,000 ϫ 669.4 Deflection due to short-term (reduced) live load is D2 ϭ ϫ 0.44 ϫ 304 ϫ 123 ϭ 0.358 in 384 ϫ 29,000 ϫ 770.7 Total deflection is D ϭ D1 ϩ D2 ϭ 0.235 ϩ 0.358 ϭ 0.593 in ϭ L / 607—OK Vibration Investigation The vibration study of composite beams is based on Wiss and Parmlee’s ‘‘drop-of-the-heel’’ method and Murray’s empirical equation (T M Murray, ‘‘Design to Prevent Floor Vibrations,’’ AISC Engineering Journal, third quarter, 1979, and ‘‘Acceptability Criterion for Occupant-Induced Floor Vibrations,’’ AISC Engineering Journal, second quarter, 1989 Also see T M Murray et al, ‘‘Floor Vibrations due to Human Activity,’’ AISC Steel Design Guide No 11, 1997.) The total dead load WD considered in the vibration equations consists of the weight of the concrete and steel beam plus a percentage of the superimposed dead load The percentage of superimposed dead load is 30% in this example: WD ϭ 0.50 ϫ 30 ϩ 0.026 ϫ 30 ϩ 0.30 ϫ 0.25 ϫ 30 ϭ 18.0 kips The frequency ƒ (Hz) of a composite simple-span beam is given by ƒ ϭ 1.57 ΊWgEIL t (7.33) D where g E It WD L ϭ ϭ ϭ ϭ ϭ gravitational acceleration ϭ 386.4 in / s2 steel modulus of elasticity, ksi transformed moment of inertia of the composite section, in4 total weight on the beam, kips span, in Substitution of previously determined values into Eq (7.33) yields ƒ ϭ 1.57 ϫ 29,000 ϫ 770.7 ϭ 5.03 Hz Ί386.418.0(30 ϫ 12) The amplitude Ao of a single beam is calculated by dividing the total floor amplitude Aot by the number Neff of effective beams: Ao ϭ Aot / Neff (7.34) For a constant to ϭ (1 / ␲ƒ ) tanϪ1 a Յ 0.05, Aot ϭ 0.246L3(0.10 Ϫ to) /EIt (7.35) For to Ͼ 0.05, Aot ϭ 0.246L3 ϫ ͙2(1 Ϫ a sin a Ϫ cos a) ϩ a2 EIt 2␲ƒ where a ϭ 0.1␲ƒ ϭ 0.1␲ ϫ 5.03 ϭ 1.58 radians (7.36) 7.27 DESIGN OF BUILDING MEMBERS The number of effective beams can be determined from Neff ϭ 2.967 Ϫ 0.05776(S / de) ϩ 2.556 ϫ 10Ϫ8L4 / It ϩ 0.0001(L / S)3 Ն 1.0 (7.37) where S ϭ spacing of beams in the floor, in dc ϭ effective depth of the slab, in ϭ average slab thickness when the metal deck ribs are perpendicular to the beam ϭ concrete thickness above the metal deck when the deck ribs are parallel to the beam With S ϭ 120 in and dc ϭ 4.25 in, the number of effective beams is Neff ϭ 2.967 Ϫ 0.05776 ϫ ͩ ͪ 120 3604 360 ϩ 2.556 ϫ 10Ϫ8 ϫ ϩ 0.0001 4.25 770.7 120 ϭ 1.90 For to ϭ (1 / 1.58␲) tanϪ1 1.58 Ͼ 0.05, the total floor amplitude is, from Eq (7.36), Aot ϭ 0.246 ϫ 3603 ϫ 29,000 ϫ 770.7 2␲ ϫ 5.04 ϫ ͙2(1 Ϫ 1.58 sin 1.58 Ϫ cos 1.58) ϩ 1.582 ϭ 0.188 in The amplitude of one beam then is, by Eq (7.34), Ao ϭ Aot / Neff ϭ 0.0188 / 1.9 ϭ 0.0099 in The mean response rating is given by R ϭ 5.08 ͩ ͪ 0.265 ƒAo D0.217 (7.38) where ƒ ϭ frequency of the composite beam, Hz Ao ϭ maximum amplitude of one beam, in D ϭ damping ratio For the following values of R, the rating denotes Imperceptible vibration Barely perceptible vibration Distinctly perceptible vibration Strongly perceptible vibration Severe vibration For the assumed 5% damping ratio, R ϭ 5.08 ͩ ͪ 5.03 ϫ 0.0099 0.050.217 0.265 ϭ 2.7 For 2.5 Ͻ 3.5, the vibration may be considered distinctly perceptible Murray’s equation gives the minimum acceptable damping (percent) as D ϭ 35Ao ƒ ϩ 2.5 ϭ 35 ϫ 0.0099 ϫ 5.03 ϩ 2.5 ϭ 4.2 Ͻ 5.0 (acceptable) 7.28 SECTION SEVEN FIGURE 7.8 Composite beam with overhang carries two concentrated loads and a uniformly decreasing load over part of the span Cantilever carries uniform loads 7.14 EXAMPLE—LRFD FOR COMPOSITE BEAM WITH CONCENTRATED LOADS AND END MOMENTS The general information for design of a floor system is the same as that given in Art 7.14 In this example, a girder of grade 50 steel is to support the floorbeams (Deck ribs are parallel to the girder.) The girder loads and span are shown in Fig 7.8 and Table 7.4 The spacing to the left adjacent girder is 30 ft and to the right girder 20 ft Dead-Load Moment for Unshored Beam The steel girder is to support construction dead loads, nonshored, with 30% additional dead load assumed applied during construction The girder is assumed to weigh 44 lb / ft The negative end moments are neglected for this phase of the design since the concrete may be placed over the entire span between the supports but not over the cantilever The factored dead loads are Pu ϭ 14.85 ϫ 1.30 ϫ 1.4 ϭ 27.03 kips WLu ϭ 0.5 ϫ 1.30 ϫ 1.4 ϭ 0.910 kips per ft WRu ϭ 0.2 ϫ 1.30 ϫ 1.4 ϭ 0.364 kips per ft WGu ϭ 0.044 ϫ 1.4 ϭ 0.062 kips per ft For the girder acting as a simple beam with a 30-ft span, the factored dead-load moment is TABLE 7.4 Concentrated and Partial Loads on Composite Beam Type of load Construction dead load Superimposed dead load Live load Concentrated load P, kips Negative moment ML, kip-ft Negative moment MR, kip-ft Partial-load start wL, kips per ft Partial-load end wR, kips per ft 14.85 22.5 7.5 0.50 0.20 7.5 7.5 2.5 0.75 0.30 15.0 20.0 7.0 0.50 0.20 DESIGN OF BUILDING MEMBERS 7.29 Mu ϭ 328.0 ft-kips, and the plastic modulus required is Z ϭ Mu / 0.9Fy ϭ 328 ϫ 12 / (0.9 ϫ 50) ϭ 87.5 in3 The least-weight section with larger modulus is a W21 ϫ 44, with Z ϭ 95.4 in3 Camber This is computed for maximum deflection attributable to full construction dead loads For this computation, the dead-load portion of the end moments is included The loads are listed under construction dead load in Table 7.4 The corresponding deflection is 1.09 in A camber of in may be specified Design for Maximum End Moment This takes into account the unbraced length of the girder For the maximum possible unbraced length of the bottom (compression) flange of the steel section, only the dead loads act between supports The factored dead loads are Pu ϭ 1.2 ϫ 14.85 ϭ 17.82 kips WLu ϭ 1.2 ϫ 0.5 ϭ 0.60 kips per ft WRu ϭ 1.2 ϫ 0.2 ϭ 0.24 kips per ft WGu ϭ 1.2 ϫ 0.044 ϭ 0.053 kips per ft MLu ϭ 1.2(22.5 ϩ 7.5) ϩ 1.6 ϫ 20 ϭ 68.0 kip-ft MRu ϭ 1.2(7.5 ϩ 2.50) ϩ 1.6 ϫ ϭ 23.2 kip-ft The unbraced length of the bottom flange is 2.9 ft The cantilever length is ft (governs) The design strength ␾Mn for a wide-flange section of grade 50 steel may be obtained from curves in the AISC ‘‘Steel Construction Manual—LRFD.’’ A curve indicates that the W21 ϫ 44 with an unbraced length of ft has a design strength ␾Mn ϭ 356 kip-ft Design for Positive Moment For this computation, the load factor used for the negative dead-load moments is 1.2, with only dead load on the cantilevers The load factor for live loads is 1.6 The factored loads, with live loads reduced 40% for the size of areas supported, are Pu ϭ 1.2(14.85 ϩ 7.5) ϩ 1.6 ϫ 9.0 ϭ 41.22 kips WLu ϭ 1.2(0.5 ϩ 0.75) ϩ 1.6 ϫ 0.30 ϭ 1.98 kips per ft WRu ϭ 1.2 (0.20 ϩ 0.30) ϩ 1.6 ϫ 0.12 ϭ 0.792 kips per ft WGu ϭ 1.2 ϫ 0.044 ϭ 0.053 kips per ft MLu ϭ 1.2 ϫ 22.5 ϭ 27.0 kip-ft MRu ϭ 1.2 ϫ 7.5 ϭ 9.0 kip-ft For these loads, the factored maximum positive moment is Mu ϭ 509.6 kip-ft For determination of the capacity of the composite beam, the effective concrete-flange width is the smaller of b ϭ 12(30 ϩ 20) / ϭ 300 in b ϭ 12 ϫ 30 / ϭ 90 in (governs) Design tables for composite beams in the AISC manual greatly simplify calculation of design strength For example, the table for the W21 ϫ 44 grade 50 beam gives ␾Mn for 7.30 SECTION SEVEN seven positions of the plastic neutral axis (PNA) and for several values of the distance Y2 from the top of the steel beam to the centroid of the effective concrete-flange force (͚Q n) (see Art 7.13) Try ͚Q n ϭ 260 kips The corresponding depth of the concrete compression block is aϭ 260 ϭ 1.133 in 0.85 ϫ 3.0 ϫ 90 From Eq (7.31), Y2 ϭ 5.25 Ϫ 1.133 / ϭ 4.68 in The manual table gives the corresponding design strength for case and Y2 ϭ 4.68 in, by interpolation, as ␾Mn ϭ 546 kip-ft Ͼ (Mu ϭ 509.6 kip-ft) The maximum positive moment Mu occurs 13.25 ft from the left support (Fig 7.8) The inflection points occur 0.49 and 0.19 ft from the left and right supports, respectively Shear Connectors Next, the studs required to develop the maximum positive moment and the moments at the concentrated loads are determined Welded studs 3⁄4 in in diameter are to be used As in Art 7.13, the nominal strength of a stud is Q n ϭ 17.7 kips For development of the maximum positive moment on both sides of the point of maximum moment, with ͚Q n ϭ 260 kips, at least 260 / 17.7 ϭ 14.69 studs are required Since the negative-moment region is small, it is not practical to limit the stud placement to the positivemoment region only Therefore, additional studs are required for placement of connectors over the entire 30-ft span Stud spacing on the left of the point of maximum moment should not exceed SL ϭ 12(13.25 Ϫ 0.49) / 14.69 ϭ 10.42 in Stud spacing on the right of the point of maximum moment should not exceed SR ϭ 12(30 Ϫ 13.25 Ϫ 0.19) / 14.69 ϭ 13.53 in For determination of the number of studs and spacing required between the concentrated load P 10 ft from the left support (Fig 7.8) and the left inflection point, the maximum load at that load is calculated to be MLu ϭ 502.1 kip-ft For the W21 ϫ 44 grade 50 beam, the manual table indicates that for ͚Q n ϭ 260 kips and Y2 ϭ 4.68 in, as calculated previously, the design strength is ␾Mn ϭ 546 kip-ft For 3⁄4-in studs and ͚Q n ϭ 260 kips, the required number of studs is 14.69 Spacing of these studs, which may not exceed 10.42 in, is also limited to SPL ϭ 12(10 Ϫ 0.49) / 14.69 ϭ 7.77 in Hence the number of studs to be placed in the 10 ft between P and the left support is 10 ϫ 12 / 7.77 ϭ 15.4 studs Use 16 studs For determination of the number of studs and spacing required between the concentrated load P 10 ft from the right support (Fig 7.8) and the right inflection point, the maximum moment at that load is calculated to be MRu ϭ 481.2 kip-ft For the W21 ϫ 44, the manual table indicates that, for case 7, ͚Q n ϭ 163 kips and ␾Mn ϭ 486 kip-ft The required number of studs for ͚Q n ϭ 163 kips is 163 / 17.7 ϭ 9.21 studs Spacing of these studs, which may not exceed 13.53 in, is also limited to SPR ϭ 12(10 Ϫ 0.19) / 9.21 ϭ 12.78 in The number of studs to be placed in the 10 ft between P and the right support is 10 ϫ 12 / 12.78 Use 10 studs The number of studs required between the two concentrated loads equals the sum of the number required between the point of maximum moment and P on the left and right On DESIGN OF BUILDING MEMBERS 7.31 the left, the required number of studs is 13.25 ϫ 12 / 10.42 Ϫ 16 ϭ Ϫ0.74 Since the result is negative, use on the left the maximum permissible stud spacing of 36 in On the right, the required number of studs is 16.75 ϫ 12 / 13.53 Ϫ 10 ϭ 4.85 Use studs The spacing should not exceed 12(16.75 Ϫ 10) / ϭ 16.2 in Specification of one spacing for the middle segment, however, is more practical Accordingly, the number of studs between the two concentrated loads would be based on the smallest spacing on either side of the point of maximum moment: 10 ϫ 12 / 16.2 ϭ 7.4 Use studs spaced 15 in center to center It may be preferable to specify the total number of studs placed on the beam based on one uniform spacing The spacing required to develop the maximum moment on either side of its location and between each concentrated load and a support is 7.77 in, as calculated previously For this spacing over the 30-ft span, the total number of studs required is 30 ϫ 12 / 7.77 ϭ 46.3 Use 48 studs (the next even number) Deflection Computations The elastic properties of the composite beam, which consists of a W21 ϫ 44 and a concrete slab 5.25 in deep (an average of 4.25 in deep) and 90 in wide, are as follows: Ec ϭ 1151.5͙3.0 ϭ 2136 ksi n ϭ Es / Ec ϭ 29,000 / 2136 ϭ 13.58 b / n ϭ 90 / 13.58 ϭ 6.63 in Itr ϭ 2496 in4 For determination of the effective moment of inertia Ieff at the location of the maximum moment, a reduced value of the transformed moment of inertia Itr is used based on the partial-composite construction assumed in the computation of shear-connector requirements For use in Eq (7.32), the moment of inertia of the W21 ϫ 44 is Is ϭ 843 in4, Q n ϭ 260 kips, and Cf is the smaller of Cƒ ϭ 0.85ƒ cЈ Ac ϭ 0.85 ϫ 3.0 ϫ 4.25 ϫ 90 ϭ 975.4 kips Cƒ ϭ AsFy ϭ 13.0 ϫ 50 ϭ 650 kips (governs) Ieƒƒ ϭ 843 ϩ (2496 Ϫ 843)͙260 / 650 ϭ 1888 in4 A reduced moment of inertia Ir due to long-time effect (creep of the concrete) is determined based on a modular ratio 2n ϭ ϫ 13.58 ϭ 27.16 and effective slab width b / n ϭ 90 / 27.16 ϭ 3.31 in The reduced transformed moment of inertia is 2088 in4 and the reduced effective moment of inertia is Ir ϭ 843 ϩ (2088 Ϫ 843)͙260 / 650 ϭ 1630 in4 The deflection computations for unshored construction exclude the weight of the concrete slab and steel beam Whether or not the steel beam is adequately cambered, the assumption is made that the concrete will be finished as a level surface Hence the concrete slab is likely to be thicker at midspan of the beams and deck For computation of the midspan deflections, the cantilevers are assumed to carry only dead load From Table 7.4, the superimposed dead loads are Ps ϭ 7.5 kips, wLs ϭ 0.75 kips per ft, and wRs ϭ 0.30 kips per ft The dead-load end moments are ML ϭ 22.5 kip-ft and MR ϭ 7.5 kip-ft For Ir ϭ 1630 in4, the maximum deflection due to these loads is Dϭ 15,865,000 ϭ 0.336 in 29,000 ϫ 1630 The deflection at the left concentrated load P is 0.296 in and at the second load, 0.288 in 7.32 SECTION SEVEN From Table 7.4, the live loads with a 40% reduction for size of area supported are PL ϭ 9.0 kips, wLL ϭ 0.30 kips per ft, and wRL ϭ 0.12 kips per ft The maximum deflection due to these loads and with an effective moment of inertia of 1888 in4 is 0.319 in The deflection at the left load is 0.282 in and at the second load, 0.275 in Total deflections due to superimposed dead loads and live loads are Maximum deflection ϭ 0.336 ϩ 0.319 ϭ 0.655 in Deflection at left load P ϭ 0.295 ϩ 0.282 ϭ 0.577 in Deflection at right load P ϭ 0.288 ϩ 0.275 ϭ 0.563 in 7.15 COMBINED AXIAL LOAD AND BIAXIAL BENDING Members subject to axial compression or tension and bending about one or two axes, such as columns that are part of rigid frames in two directions, are designed to satisfy the following interaction equations For symmetrical shapes when Pu / ␾Pn Ն 0.2, ͩ ͪ (7.39a) ͩ ͪ (7.39b) Pu Mux Muy ϩ ϩ Յ 1.0 ␾Pn ␾bMnx ␾bMny For Pu / ␾Pn Ͻ 0.2, Pu Mux Muy ϩ ϩ Յ 1.0 2␾Pn ␾bMnx ␾bMny where Pu Pn Mu Mn ␾t ␾b ϭ ϭ ϭ ϭ ϭ ϭ factored axial load, kips nominal compressive or tensile strength, kips factored bending moment, kip-in nominal flexural strength, kip-in resistance factor for tension ϭ 0.90 resistance factor for flexure ϭ 0.90 The factored moments Mux and Muy should include second-order effects, such as P Ϫ ⌬, for the factored loads If second-order analysis is not performed, the factored moments can be calculated with magnifiers as follows: Mu ϭ B1Mnt ϩ B2Mlt (7.40) where Mnt ϭ factored bending moment based on the assumption that there is no lateral translation of the frame, kip-in Mlt ϭ factored bending moment as a result only of lateral translation of the frame, kip-in Cm Ն1 B1 ϭ (1 Ϫ Pu / Pe) Pe ϭ AgFy / ␭c2 ␭c ϭ slenderness parameter (Art 7.4) with effective length factor K Յ 1.0 in the plane of bending Ag ϭ gross area of the member, in2 Fy ϭ specified minimum yield point of the member, ksi Cm ϭ coefficient defined for Eq (6.67) DESIGN OF BUILDING MEMBERS B2 ϭ ⌬ Ϫ ͚Pu oh ͚HL or 7.33 1Ϫ ͚Pu ͚Pe ͚Pu ϭ sum of factored axial loads of all columns in a story, kips ⌬oh ϭ translation deflection of the story under consideration, in ͚H ϭ sum of all horizontal forces in a story that produce ⌬oh, kips L ϭ story height, in Pe ϭ AgFy / ␭c2, where ␭c is the slenderness parameter with the effective length factor K Ն 1.0 in the plane of bending determined for the member when sway is permitted, kips Since several computer analysis programs are available with the P Ϫ ⌬ feature included, it is advisable to determine the P Ϫ ⌬ effects by second-order analysis of framing subject to lateral loads If the P Ϫ ⌬ effect is evaluated for frames subject to lateral as well as to vertical loads, the moment magnifier B2 can be considered to be unity 7.16 EXAMPLE—LRFD FOR WIDE-FLANGE COLUMN IN A MULTISTORY RIGID FRAME Columns at the ninth level of a multistory building are to be part of a rigid frame that resists wind loads Typical floor-to-floor height is 13 ft In the ninth story, a wide-flange column of grade 50 steel is to carry loads from a transfer girder, which supports an offset column carrying the upper levels Therefore, the lower column discontinues at the ninth level The loads on that column are as follows: dead load, 750 kips; superimposed dead load, 325 kips; and live load, 250 kips The moments due to gravity loads at the beam-column connection are Dead-load major-axis moment ϭ 180 kip-ft Live-load major-axis moment ϭ 75 kip-ft Dead-load minor-axis moment ϭ 75 kip-ft Live-load minor-axis moment ϭ 40 kip-ft The column axial loads and moments due to service lateral loads with P Ϫ ⌬ effect included are Axial load ϭ 600 kips Major-axis moment ϭ 1050 kip-ft Minor-axis moment ϭ 0.0 The beams attached to the flanges of the column with rigid welded connections are part of the rigid frame and have spans of 30 ft The following beam sizes and corresponding stiffnesses, at top and bottom ends of the column apply The beams at both sides of the column at the floor above and the floor below are W36 ϫ 300 The sum of the stiffnesses Ib / Lb of the beams is ͚(Ib / Lb) ϭ 20,300 ϫ / (30 ϫ 12) ϭ 112.8 in3 where Ib is the beam moment of inertia (in4) 7.34 SECTION SEVEN The effective length factor Kx corresponding to the case of frame with sidesway permitted is used in determining the axial-load capacity and the moment magnifier B1 The moment magnifier B2 is considered unity inasmuch as the P Ϫ ⌬ effect is included in the analysis Axial-Load Capacity Since the column is part of a wind-framing system, the K values should be computed based on column and beam stiffnesses To determine the major-axis Kx, assume that a W14 ϫ 426 with Icx ϭ 6600 in4 will be selected for the column At the top of the column, where there is no column above the floor, the relative column-beam stiffness is GA ϭ ͚(Ic / Lc) 6600 / 12(13 Ϫ 3) ϭ ϭ 0.49 ͚(Ib / Lb) 112.8 At the column bottom, with a W14 ϫ 426 column below, GB ϭ ͚(Ic / Lc ) ϫ 6600 / 12(13 Ϫ 3) ϭ ϭ 0.98 ͚(Ib / Lb ) 112.8 From a nomograph for the case when sidesway is permitted (Fig 7.9b), Kx ϭ 1.23 (at the intersection with the K axis of a straight line connecting 0.49 on the GA axis with 0.98 on the GB axis) Since the connection of beams to the column web is a simple connection with inhibited sidesway, Ky ϭ 1.0 The effective lengths to be used for determination of axial-load capacity are FIGURE 7.9 Nomographs for determination of the effective length factor for a column (a) For use when sidesway is prevented (b) For use when sidesway may occur DESIGN OF BUILDING MEMBERS 7.35 Kx Lx ϭ 1.23(13 Ϫ 3) ϭ 12.3 ft Ky Ly ϭ 1.0 ϫ 13 ϭ 13 ft The W14 ϫ 426 has radii of gyration rx ϭ 7.26 in and ry ϭ 4.34 in Therefore, the slenderness ratios for the column are KxLx / rx ϭ 12.3 ϫ 12 / 7.26 ϭ 20.3 KyLy / ry ϭ 13 ϫ 12 / 4.34 ϭ 35.9 (governs) Use of the AISC ‘‘Manual of Steel Construction—LRFD’’ tables for design axial strength of compression members simplifies evaluation of the trial column size For the W14 ϫ 426, grade 50 section, a table indicates that for Ky Ly ϭ 13 ft, ␾Pn ϭ 4830 kips Moment Capacity Next, the nominal bending-moment capacities are calculated For strong-axis bending moment, Ky Ly ϭ 13 ft is assumed for the flange lateral- buckling state The limiting lateral unbraced length Lp (in) for plastic behavior for the W14 ϫ 426 is Lp ϭ 300ry / ͙Fy ϭ 300 ϫ 4.34 / ͙50 ϭ 184 in 15.3 ft Ͼ 13 ft Since the unbraced length is less than Lp, ␾Mnx ϭ 0.9 ϫ 869 ϫ 50 / 12 ϭ 3259 kip-ft ␾Mny ϭ 0.9ZyFy ϭ 0.9 ϫ 434 ϫ 50 / 12 ϭ 1628 kip-ft Interaction Equation for Dead Load For use in the interaction equation for axial load and bending [Eq (7.39a) or (7.39b)], the factored dead load is Pu ϭ 1.4(750 ϩ 325 ϩ 0.426 ϫ 13) ϭ 1513 kips The factored moments applied to columns due to any general loading conditions should include the second-order magnification When the frame analysis does not include secondorder effects, the factored column moment can be determined from Eq (7.40) Computer analysis programs usually include the second-order analysis ( P Ϫ ⌬ effects) Therefore, the values of B2 for moments about both column axes can be assumed to be unity However, B1 should be determined for evaluation of the nonsway magnifications For a braced column (drift prevented), the slenderness coefficient Kx is determined from Fig 7.9a with GA ϭ 0.49 and GB ϭ 0.98, previously calculated The nomograph indicates that Ks ϭ 0.73 For determination of B1 in Eq (7.40), the column when loaded is assumed to have single curvature with end moments M1 ϭ M2 Hence Cm ϭ For determination of the elastic buckling load Pex, the slenderness parameter is ␭cx ϭ ϭ KLx rx␲ Ί E ϭ r Ί286,220 Fy KLx Fy x 0.73 ϫ 12(13 Ϫ 10) 7.26 Ί286,220 ϭ 0.159 50 and the elastic buckling load for the beam cross-sectional area Ag ϭ 125 in2 is Pex ϭ AgFy / ␭cx2 ϭ 125 ϫ 50 / 0.1592 ϭ 247,000 kips With these values, the magnification factor for Mux is 7.36 SECTION SEVEN B1x ϭ Cm 1.0 ϭ ϭ 1.006 Ϫ Pu / Pex Ϫ 1513 / 247,000 For determination of the elastic buckling load Pey , ␭cy ϭ ϫ 13 ϫ 12 4.34 Ί286,220 ϭ 0.475 50 The elastic buckling load with respect to the y axis is Pey ϭ AgFy / ␭cy2 ϭ 125 ϫ 50 / 0.4752 ϭ 27,700 kips With these values, the magnification factor for Muy is B1y ϭ Cm ϭ ϭ 1.058 Ϫ Pu / Pey Ϫ 1513 / 27,700 Application of the magnification factor to the dead-load moments due to gravity loads yields Mux ϭ 1.006 ϫ 1.4 ϫ 180 ϭ 253.5 kip-ft Muy ϭ 1.058 ϫ 1.4 ϫ 75 ϭ 111.1 kip-ft The interaction result, which may be considered a section efficiency ratio, is, from Eq (7.39a) for Pu / ␾Pn ϭ 1513 / 4830 ϭ 0.313 Ͼ 0.2, R ϭ 0.312 ϩ ϭ 0.313 ϩ ͩ ͪ 253.5 111.1 ϩ 3259 1628 (0.0778 ϩ 0.682) ϭ 0.443 Ͻ 1.0 Interaction Equation for Full Gravity Loading For use in the interaction equation based on factored loads and moments due to 1.2 times the dead load plus 1.6 times the live load, Pu ϭ 1.2(750 ϩ 325 ϩ 0.426 ϫ 13) ϩ 1.6 ϫ 250 ϭ 1697 kips Determined in the same way as for the dead load, the magnification factors are B1x ϭ 1.0 ϭ 1.007 Ϫ 1697 / 247,000 B1y ϭ 1.0 ϭ 1.065 Ϫ 1697 / 27,700 Application of the magnification factors to the factored moments yields Mux ϭ 1.007(1.2 ϫ 180 ϩ 1.6 ϫ 75) ϭ 338.4 kip-ft Muy ϭ 1.065(1.2 ϫ 75 ϩ 1.6 ϫ 40) ϭ 164.0 kip-ft With Pu / ␾Pn ϭ 1697 / 4830 ϭ 0.351 Ͼ 0.2, substitution of the preceding values in Eq (7.39a) yields DESIGN OF BUILDING MEMBERS R ϭ 0.351 ϩ ϭ 0.351 ϩ ͩ 338.4 164.0 ϩ 3259 1628 7.37 ͪ (0.1038 ϩ 0.1008) ϭ 0.533 Ͻ Interaction Equation with Wind Load For use in the interaction equation based on factored loads and moments due to 1.2 times the dead load plus 0.5 times the live load plus 1.3 times the wind load of 600 kips, including the P Ϫ ⌬ effect, P ϭ 1.2(750 ϩ 325 ϩ 0.426 ϫ 13) ϩ 0.5 ϫ 250 ϩ 1.3 ϫ 600 ϭ 2202 kips Under wind action, double curvature may occur for strong-axis bending For this condition, with M1 ϭ M2, Cmx ϭ 0.6 Ϫ 0.4 ϫ ϭ 0.2 In this case, the magnification factor for strong-axis bending is B1x ϭ 0.2 ϭ 0.202 Ͻ 1 Ϫ 2202 / 247,000 Use B1x ϭ 1.0 The magnification factor for minor axis bending is, with Cm ϭ for singlecurvature bending, B1y ϭ 1.0 ϭ 1.0864 Ϫ 2202 / 27,700 Application of the magnification factors to the factored moments yields Mux ϭ 1.0(1.2 ϫ 180 ϩ 0.5 ϫ 75 ϩ 1.3 ϫ 1050) ϭ 1618 kip-ft Muy ϭ 1.0864(1.2 ϫ 75 ϩ 0.5 ϫ 40) ϭ 119.5 kip-ft With Pu / ␾P*n ϭ 2202 / 4830 ϭ 0.456 Ͼ 0.2, substitution of the preceding values in Eq (7.39a) yields R ϭ 0.456 ϩ ϭ 0.456 ϩ ͩ 1618 119.5 ϩ 3259 1628 ͪ (0.496 ϩ 0.0734) ϭ 0.96 Ͻ This is the governing R value, and since it is less than unity, the column selected, W14 ϫ 426, is adequate 7.17 BASE PLATE DESIGN Base plates are usually used to distribute column loads over a large enough area of supporting concrete construction that the design bearing strength of the concrete will not be exceeded The factored load Pu is considered to be uniformly distributed under a base plate 7.38 SECTION SEVEN The nominal bearing strength ƒp (ksi) of the concrete is given by ƒp ϭ 0.85ƒ cЈ͙A1 / A1 and ͙A2 / A1 Յ (7.41) where ƒ Јc ϭ specified compressive strength of concrete, ksi A1 ϭ area of the base plate, in2 A2 ϭ area of the supporting concrete that is geometrically similar to and concentric with the loaded area, in2 In most cases, the bearing strength ƒp is 0.85ƒ Јc when the concrete support is slightly larger than the base plate or 1.7ƒ Јc when the support is a spread footing, pile cap, or mat foundation Therefore, the required area of a base plate for a factored load Pu is A1 ϭ Pu / ␾c 0.85ƒ cЈ (7.42) where ␾c is the strength reduction factor ϭ 0.6 For a wide-flange column, A1 should not be less than bƒ d, where bƒ is the flange width (in) and d is the depth of column (in) The length N (in) of a rectangular base plate for a wide-flange column may be taken in the direction of d as N ϭ ͙A1 ϩ ⌬ Ͼ d (7.43) ⌬ ϭ 0.5(0.95d Ϫ 0.80bƒ) (7.44) For use in Eq (7.43), The width B (in) parallel to the flanges, then, is B ϭ A1 / N (7.45) The thickness of the base plate (in) is the largest of the values given by Eqs (7.46) to (7.48): ϭ m Ί0.9F2P BN (7.46) Ί0.9F2P BN (7.47) u y ϭ n u y ϭ ␭nЈ Ί0.9F2P BN u (7.48) y where m ϭ projection of base plate beyond the flange and parallel to the web, in ϭ (N Ϫ 0.95d ) / n ϭ projection of base plate beyond the edges of the flange and perpendicular to the web, in ϭ (B Ϫ 0.80bƒ ) / nЈ ϭ ͙ (dbƒ) / ␭ ϭ (2͙X) / [1 ϩ ͙(1 Ϫ X)] Յ 1.0 X ϭ [(4 dbƒ ) / (d ϩ bƒ )2][Pu / (␾ ϫ 0.85ƒ Јc A1)] DESIGN OF BUILDING MEMBERS 7.18 7.39 EXAMPLE—LRFD DESIGN OF COLUMN BASE PLATE A base plate of A36 steel is to distribute the load from a W14 ϫ 233 column to a concrete pedestal whose size is slightly larger than that of the base plate The pedestal concrete strength ƒ Јc is 4.0 ksi The factored load on the column is 1731 kips From Eq (7.41), the nominal bearing strength of the concrete is ƒp ϭ 0.85 ϫ 4.0 ϭ 3.4 ksi The required area of the base plate is computed from Eq (7.42): A1 ϭ 1,731 / (0.6 ϫ 3.4) ϭ 848.5 in2 From Eq (7.44) for determination of the length N of the base plate, ⌬ ϭ 0.5(0.95 ϫ 16.04 Ϫ 0.80 ϫ 15.89) ϭ 1.26 in From Eq (7.43), N ϭ ͙848.5 ϩ 1.26 ϭ 30.4 in Use a 31-in length The width required then is B ϭ A1 / N ϭ 848.5 / 31 ϭ 27.4 in Use a 28-in-wide plate The area of the plate is A1 ϭ 868 in2 For determination of the base plate thickness, the projections beyond the column are m ϭ (31 Ϫ 0.95 ϫ 16.04) / ϭ 7.88 in (governs) n ϭ (28 Ϫ 0.8 ϫ 15.89) / ϭ 7.64 in Equation (7.46) yields a larger thickness than Eq (7.47) because m Ͼ n From Eq (7.46), ϭ 7.88 Ί0.9 ϫ 36 ϫ 28 ϫ 41 ϭ 2.76 in ϫ 1731 For use in Eq (7.48), nЈ ϭ ͙(16.04 ϫ 15.89) / ϭ 4.0 X ϭ [(4 ϫ 16.04 ϫ 15.89) / (16.04 ϩ 15.89)2][1731 / (0.6 ϫ 0.85 ϫ 4.0 ϫ 868)] ϭ 0.98 ␭ ϭ (2 ϫ ͙0.98) / [1 ϩ ͙(1 Ϫ 0.98)] ϭ 1.73 Ͼ 1.0 Then, the thickness given by Eq (7.48) is ϭ 4.0 ϫ 1.0 Ί0.9 ϫ 36 ϫ 868 ϭ 1.40 in Ͻ 2.76 in Use a base plate 28 ϫ 23⁄4 ϫ 31 in ϫ 1731 ... of Structural Steels and Effects of Steelmaking and Fabrication Roger L Brockenbrough, P.E 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 1. 10 1. 11 1 .12 1. 13 1. 14 1. 15 1. 16 1. 17 1. 18 1. 19 1. 20 1. 21 1.22 1. 23... Loads / 9.38 Section 10 Cold-Formed Steel Design R L Brockenbrough, P.E 10 .1 10.2 10 .3 10 .4 10 .5 10 .6 10 .7 10 .8 10 .9 10 .10 10 .11 10 .12 10 .13 10 .14 10 .15 10 .16 10 .17 10 .18 10 .1 Design Specifications... 11 .34 11 .35 11 .36 11 .37 11 .38 11 .39 11 .40 Standard Specifications / 11 .15 3 Design Method / 11 .15 3 Owner’s Concerns / 11 .15 3 Design Considerations / 11 .15 4 Design Loadings / 11 .15 5 Composite Steel