Design of reinforced concrete

Design of Reinforced Concrete Design of Reinforced Concrete ACI 318-11 Code Edition Jack C McCormac Clemson University Russell H Brown Clemson University NINTH EDITION VP & EXECUTIVE PUBLISHER MARKETING MANAGER ACQUISITIONS EDITOR SENIOR PRODUCTION EDITOR CREATIVE DIRECTOR SENIOR DESIGNER PHOTO EDITOR COVER PHOTO Don Fowley Christopher Ruel Jennifer Welter Sujin Hong Harry Nolan Thomas Nery Sheena Goldstein Frank Leung/iStockphoto This book was set in 10/12 Times by Laserwords Private Limited and printed and bound by Courier The cover was printed by Courier This book is printed on acid free paper ∞ Founded in 1807, John Wiley & Sons, Inc has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work In 2008, we launched a Corporate Citizenship Initiative, a 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permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, website www.wiley.com/go/permissions Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year These copies are licensed and may not be sold or transferred to a third party Upon completion of the review period, please return the evaluation copy to Wiley Return instructions and a free of charge return mailing label are available at www.wiley.com/go/returnlabel If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy Outside of the United States, please contact your local sales representative ISBN: 978-1-118-12984-5 ISBN: 978-1-118-43081-1 (BRV) Printed in the United States of America 10 Brief Contents Preface Introduction Flexural Analysis of Beams Strength Analysis of Beams According to ACI Code Design of Rectangular Beams and One-Way Slabs Analysis and Design of T Beams and Doubly Reinforced Beams Serviceability Bond, Development Lengths, and Splices Shear and Diagonal Tension Introduction to Columns 10 Design of Short Columns Subject to Axial Load and Bending 11 Slender Columns 12 Footings 13 Retaining Walls 14 Continuous Reinforced Concrete Structures 15 Torsion 16 Two-Way Slabs, Direct Design Method 17 Two-Way Slabs, Equivalent Frame Method 18 Walls 19 Prestressed Concrete 20 Reinforced Concrete Masonry A Tables and Graphs: U.S Customary Units B Tables in SI Units C The Strut-and-Tie Method of Design D Seismic Design of Reinforced Concrete Structures Glossary Index xv 35 65 82 112 154 184 223 263 281 317 347 394 431 470 492 532 547 567 602 631 669 675 683 699 703 v Contents Preface Introduction xv 1.1 Concrete and Reinforced Concrete, 1.2 Advantages of Reinforced Concrete as a Structural Material, 1.3 Disadvantages of Reinforced Concrete as a Structural Material, 1.4 Historical Background, 1.5 Comparison of Reinforced Concrete and Structural Steel for Buildings and Bridges, 1.6 Compatibility of Concrete and Steel, 1.7 Design Codes, 1.8 SI Units and Shaded Areas, 1.9 Types of Portland Cement, 1.10 Admixtures, 1.11 Properties of Concrete, 10 1.12 Aggregates, 18 1.13 High-Strength Concretes, 19 1.14 Fiber-Reinforced Concretes, 20 1.15 Concrete Durability, 21 1.16 Reinforcing Steel, 22 1.17 Grades of Reinforcing Steel, 24 1.18 SI Bar Sizes and Material Strengths, 25 1.19 Corrosive Environments, 26 1.20 Identifying Marks on Reinforcing Bars, 26 1.21 Introduction to Loads, 28 1.22 Dead Loads, 28 1.23 Live Loads, 29 1.24 Environmental Loads, 30 1.25 Selection of Design Loads, 32 1.26 Calculation Accuracy, 33 1.27 Impact of Computers on Reinforced Concrete Design, 34 Problems, 34 Flexural Analysis of Beams 35 2.1 Introduction, 35 2.2 Cracking Moment, 38 2.3 Elastic Stresses—Concrete Cracked, 41 2.4 Ultimate or Nominal Flexural Moments, 48 2.5 SI Example, 51 2.6 Computer Examples, 52 Problems, 54 vii viii CONTENTS Strength Analysis of Beams According to ACI Code 65 3.1 Design Methods, 65 3.2 Advantages of Strength Design, 66 3.3 Structural Safety, 66 3.4 Derivation of Beam Expressions, 67 3.5 Strains in Flexural Members, 70 3.6 Balanced Sections, Tension-Controlled Sections, and Compression-Controlled or Brittle Sections, 71 3.7 Strength Reduction or φ Factors, 71 3.8 Minimum Percentage of Steel, 74 3.9 Balanced Steel Percentage, 75 3.10 Example Problems, 76 3.11 Computer Examples, 79 Problems, 80 Design of Rectangular Beams and One-Way Slabs 82 4.1 Load Factors, 82 4.2 Design of Rectangular Beams, 85 4.3 Beam Design Examples, 89 4.4 Miscellaneous Beam Considerations, 95 4.5 Determining Steel Area When Beam Dimensions Are Predetermined, 96 4.6 Bundled Bars, 98 4.7 One-Way Slabs, 99 4.8 Cantilever Beams and Continuous Beams, 102 4.9 SI Example, 103 4.10 Computer Example, 105 Problems, 106 Analysis and Design of T Beams and Doubly Reinforced Beams 112 5.1 T Beams, 112 5.2 Analysis of T Beams, 114 5.3 Another Method for Analyzing T Beams, 118 5.4 Design of T Beams, 120 5.5 Design of T Beams for Negative Moments, 125 5.6 L-Shaped Beams, 127 5.7 Compression Steel, 127 5.8 Design of Doubly Reinforced Beams, 132 5.9 SI Examples, 136 5.10 Computer Examples, 138 Problems, 143 Serviceability 6.1 6.2 6.3 Introduction, 154 Importance of Deflections, 154 Control of Deflections, 155 154 708 INDEX Lateral loads: (continued ) design of two-way slabs for, 496 in seismic design, 688–691 Lateral pressure, on retaining walls, 399–404, 408 Lateral support, for rectangular beams, 95 Laurson, P G., L beams, 127 Le Brun, F., Leet, K., 15 Length effect, 582 Leyh, George F., 212 Lightweight aggregate concrete, 224 Lightweight concrete, 18 Lightweight concrete modification factor, 192 Limit design: continuous structures, 434–444 under ACI Code, 442–444 collapse mechanism, 436–437 plastic analysis (equilibrium method), 438–441 plastic design vs., 435 Limit states, 154 Lintels, masonry, 611–616 cracking moment, 613, 616 deflections, 613–616 shear design of, 612–613 Live loads, 29–30 Loads, 28–36 axial: columns, 266, 274–277 footings subjected to, 380–382 short columns subject to bending and, 281–316 balanced, 288 dead, 28 design, 32–33 environmental, 30–32 factored, 65 on footings, 378–382 ice, 30–31 impact, 29 lateral: for continuous structures, 454–458 design of two-way slabs for, 496 in seismic design, 688–691 live, 29–30 longitudinal, 29 miscellaneous, 29 rain, 31 seismic (earthquake), 32, 84 in seismic design, 687–691 service, 36, 41, 65 snow, 30–31 traffic, 29 vertical: for continuous structures, 445–454 in seismic design, 687–688 wind, 31–32, 84 working, 36, 65 Load and resistance factor design (LRFD), 435 Load-bearing walls: empirical design method, 549–551 masonry walls with out-of-plane loads, 616–623 rational design method, 552–554 Load factors: and effective moment of inertia, 160 rectangular beams, 82–84 Long columns, 263 Longitudinal loads, 29 Long-term deflections, 160–162, 589 LRFD (load and resistance factor design), 435 M MacGregor, J G., 36, 127 Masonry, 602–630 concrete masonry units, 602–603 flexural tensile reinforcement of, 607 grout, 605 lintels, 611–616 mortar, 603–605 reinforcing bars in, 605–606 shear walls with in-plane loading, 623–628 specified compressive strength of, 606–607 walls with out-of-plane loads: load-bearing, 616–623 non–load-bearing, 607–611 Mass density, 13 Mat (raft, floating foundation) footings, 348, 349 Maximum considered earthquake, 684 Maximum steel percentage, 73 Mechanically anchored bars, 214–215 Middle strips (two-way slabs), 496–497 Minimum steel percentage, 74–75 Modular ratio, 41 Modulus of elasticity, 12, 13, 16, 25 apparent, 12 dynamic, 13 initial, 12 long term, 12 secant, 12 slender columns, 324 static, 12–13 tangent, 12 Modulus of rupture, 16, 35, 38–39 Moment frames, classes of, 691–698 Moment magnifier procedure (slender columns), 328 INDEX nonsway (braced) frames, 328–333 sway (unbraced) frames, 333–337 Moments of inertia: effective, 158–160 slender columns, 324 Moment strength, torsional, 477 Monier, J, Mortar, 603–605 M¨uller-Breslau, Heinrich, 432 N National Concrete Masonry Association (NCMA), 602 Nawy, E G., 13 Nilson, A H., 13 Nominal strength, 48–51 Nominal values (CMUs), 603 Nonlinear second-order analysis (slender columns), 328 Non–load-bearing walls, 547–548, 607–611 Nonsway (braced) frames (slender columns), 317–318, 320, 328–333 O One-way slabs, 99–102 defined, 492 simple-span, 99–102 Ordinary moment frames, 691–692 P Partially prestressed members, 596 Pedestals, 263 Pile caps, 348, 349 Plain concrete footings, 383–386 Plastic analysis (equilibrium method), 438–441 Plastic centroid, 282–284 Plastic design, limit design vs., 435 Plastic hinge, 435–440 Points of inflection, assumed, 454, 455 Poisson’s ratio, 13–14 Ponding, 31 Portal method, 454, 456–458 Portland cement, 7–9 Posttensioning, 570, 575 Pozzolana, Precast walls: non-prestressed, 548 retaining walls, 397 Prestressed concrete, 567–601 advantages of, 569 for composite construction, 595 continuous members, 596 deflections of, 586–590 disadvantages of, 569 709 elastic shortening in, 580–581 friction along ducts, 582 materials used for, 570–572 partial prestressing, 596 posttensioning, 570, 575 prestress losses, 579–582 pretensioning, 569–570 relaxation and creep in tendons, 581 shapes of prestressed sections, 576–579 shear in, 590–595 approximate method, 590 design of reinforcement, 591–595 shrinkage and creep in, 581 stress calculations, 572–576 stresses in end blocks, 595 ultimate strength of sections, 582–587 Pretensioning, 569–570 Primary moments (columns), 263 Principal stresses, 223–224 Proportions (rectangular beams), 85 Q Qualitative influence lines, 431–434 R Raft footings, 348, 349 Rain loads, 31 Ransome, E L., Rational design method (load-bearing walls), 552–554 Rectangular beams, 82–111 bundled bars, 98 cantilever, 102 continuous, 102 design of, 85–94 lateral support, 95 load factors, 82–84 one-way slabs as, 99–102 sizes of, 96 skin reinforcement, 95 steel area for predetermined dimensions, 96–98 Rectangular isolated footings, 369–372 Reinforced concrete: advantages, 1–2 defined, disadvantages, 2–3 history, 3–5 use of structural steel vs., 5–6 Reinforcement location factor, 191 Reinforcement size factor, 192 Reinforcing bars: bond stresses, 187–189 bundled, 98 710 INDEX Reinforcing bars: (continued ) development lengths for, 197–199 lap splices for, 212 cutting off or bending, 184–187, 208–211 headed, 214–215 in masonry, 605–606 mechanically anchored, 214–215 rectangular beams: minimum spacing of, 87–89 selection of, 86 splices: compression, 213–214 in flexural members, 211–212 tension, 213 Reinforcing steel, 22–26 axle, 24, 27 billet, 24, 27 coatings, 26 compatibility of concrete and, corrosion, 9, 25, 26 deformed, 4, 22–25 epoxy coated, 26 grades, 24–25 identifying marks, 26–27 maximum percentage of, 73 minimum percentage of, 74–75 plain, 22–23 rail, 26–27 SI sizes, 25–26 welded wire fabric, 22–25 Relaxation, in prestressed concrete tendons, 581 Retaining walls, 394–430 bridge abutments, 396, 397 buttress, 395, 396 cantilever, 394–396 design procedure for, 413–424 estimating sizes of, 409–413 without heel, 396 without toe, 396, 397 counterfort, 395, 396 cracks in, 424–425 drainage for, 397–398 failures of, 398 footing soil pressures for, 404–405 gravity, 394, 395 joints in, 424–426 lateral pressure on, 399–404 precast, 397 semigravity, 394, 395, 405–407 surcharge on, 408 types of, 394–397 Retrofitting, 32 Righting moment (retaining walls), 396 Roman cement, Rusch, H., 15 S Safety, 65–67 with cantilever retaining walls, 414–415 with columns, 271–272 Salmon, C G., Schlaich, J., 681 SDC (seismic design category), 683, 687 Secondary moments (columns), 263 Seismic design, 683–698 categories of, 683, 687 for classes of moment frames, 691–698 loads, 687–691 maximum considered earthquake, 684 risk and importance factors, 686–687 soil site class, 684–685 Seismic design category (SDC), 683, 687 Seismic (earthquake) loads, 32, 84 Self-consolidated concrete, 10 Semigravity retaining walls, 394, 395, 405–407 Service, 36 Serviceability, 154–183 cracks, 170–176 ACI Code provisions, 175–176 flexural, 170–175 types of, 170–171 deflections, 154–164 calculation of, 157–160 continuous-beam, 164–170 control of, 155–156 importance of, 154–155 long-term, 160–162 simple-beam, 162–164 effective moments of inertia, 158–160 Serviceability limit states, 154 Service loads, 36, 41, 65 Settlement (footings), 378–379 Shear: ACI Code requirements for, 232–237 column, 457 in column footings, 359–364 design for, 231–232 deep beams, 253–254 design problems, 237–247 stirrup spacing, 237–242, 247–249 and development length, 206–207 girder, 458 in prestressed concrete, 590–595 INDEX approximate method, 590 design of reinforcement, 591–595 in short columns subject to axial load and bending, 301–302 and tensile strength, 223 two-way slabs: shear resistance, 497–500 transfer between slabs and columns, 522–528 Shear cracking (reinforced concrete beams), 226–230 Shear friction, 249–251, 382–383 Shearheads, 492, 498, 499 Shear spans, 675–676 Shear strength: of concrete, 17–18, 225–226 and lightweight aggregate concrete, 224 of members subjected to axial forces, 251–253 Shear stresses: in concrete beams, 223–224 in two-way slabs, 497–500 Shear walls, 554–562 ACI provisions for, 558–559 arrangements of, 556–557 masonry, with in-plane loading, 623–628 Short columns: compression blocks and pedestals, 263 subject to axial load and bending, 281–316 biaxial bending, 302–309 capacity reduction factors, 309–311 interaction diagrams, 284–301 plastic centroid, 282–284 shear in, 301–302 Shotcreting, 8, 21 Shrinkage, 14–15 SI examples: axially loaded columns, 277–278 beam analysis, 51 cracking, 176 development length, 215–216 rectangular beam design, 103–104 SI units, 7, 25 stirrup spacing, 256–257 T beams and doubly reinforced beams, 136–138 torsion, 483–486 wall footings, 386–388 Silica fume, 19–20 Simple-beam deflections, 162–164 Single-column footings, see Isolated footings, Skin reinforcement (deep rectangular beams), 95 Slabs: continuous, 446–450 minimum thicknesses for, 155 711 one-way: defined, 492 simple-span, 99–102 two-way, 492–531 analysis of, 495, 517–522 with beams, 492, 494, 517–522 columns, 522–528 column strips, 496–497 defined, 492 depth limitations and stiffness, 500–505 design of, 495–496 direct design method, 495–531 distribution of moments in, 506–511 equivalent frame method, 495–496 factored moments in columns and walls, 528 flat plates, 492, 493, 511–514 flat slabs, 492, 493 for lateral loads, 496 live load placement, 514–517 middle strips, 496–497 openings in slab systems, 528 shear resistance, 497–500 transfer of moments and shear between slabs and columns, 522–528 waffle slabs, 492–493 Slate, F., 13 Sleeve splices, 212 Slender columns, 263, 317–346 ACI Code treatment of slenderness effects, 328 analyses of: first-order, 323–324 second-order, 328 avoiding, 325–326 effective length factors, 318–323 determined with alignment charts, 321–322 determined with equations, 322–323 in nonsway (braced) frames, 317–318, 320 magnification of column moments in, 328–333 sway (unbraced) frames vs., 324–327 in sway (unbraced) frames, 317–318, 320 analysis of, 336–341 magnification of column moments in, 333–337 nonsway (braced) frames vs., 324–327 unsupported lengths, 318 Slippage, in prestressed concrete, 582 Smith, Albert, 454 Snow loads, 30–31 Soil pressures: active, 401 actual, 350–351 allowable, 351–352 712 INDEX Soil pressures: (continued ) for footings: actual, 350–351 allowable, 351–352 of retaining walls, 404–405 on retaining walls, 399–404 Soil site class, 684–685 Special moment frames, 692–693 Spiral columns, 264 ACI Code requirements for, 270–271 failure of, 266–268 Splices: compression, 213–214 in flexural members, 211–212 tension, 213 Split-cylinder tests, 16 Spreadsheets: beam analysis, 52–54 columns, 278–279 deflection calculator, 177–178 development length, 216–217 doubly reinforced beams, 141–143 footings, 388–390 masonry shear walls, 628–629 prestressed concrete, 597 rectangular beams, 105–106 shear design, 257–258 short columns, 311–312 slender columns, 342–343 T beams, 138–141 torsion, 487 two-way slabs, 528–530 walls, 564–565 Square isolated footings, 357–364 Stability index, 317–318 Statically determinate torsion, 474 Statically indeterminate torsion, 474 Static moment, 506 Steel: prestressed, 571–572 reinforcement with, see Reinforcing steel, Steel area, for rectangular beams of predetermined dimensions, 96–98 Stems: cantilever retaining walls, 409–410, 413–414, 424 T-beams, 112 Stirrups, 201, 207 ACI Code requirements, 232–235 and design for shear, 231–235 in footings, 353 purpose of, 231 spacing of, 233–234, 237–246 ACI Code requirements, 232–235 economical, 247–249 torsional reinforcing, 471–474 for web reinforcement, 227–230 Straight-line design, 65 Strains in flexural members, 70–71 Straub, H., Strength analysis (beams), 65–81 balanced sections, 71 balanced steel percentage, 75–76 brittle sections, 71 compression-controlled sections, 71 derivation of beam expressions, 67–70 design methods, 65–66 minimum percentage of steel, 74–75 strains in flexural members, 70–71 strength design advantages, 66 strength reduction factors, 67, 71–74 structural safety, 66–67 tension-controlled sections, 71 Strength design: advantages, 66 defined, 65 Strength limit states, 154 Strength reduction factors, 67, 71–74 Stresses: bond, 187–189 in prestressed concrete: calculation of, 572–576 in end blocks, 595 principal, 223–224 shear, 223–224 torsional, 475–476 Stress-strain curves, 11–12 Strut and tie design, 675–682 Superplasticizers, 10, 20 Surcharge, on retaining walls, 408 Sway (unbraced) frames (slender columns), 317–318, 320 analysis of, 336–341 magnification of column moments in, 333–337 T Tables: balanced ratios of reinforcement: SI units, 672–674 U.S customary units, 636–645 circular column properties, 647 column properties, 655 live loads (typical), 29 minimum web widths for beams: SI units, 671 U.S customary units, 635 INDEX moduli of elasticity: SI units, 669 U.S customary units, 631 moment distribution constants for slabs, 648–654 reinforcing bar tables (areas, diameters, etc.): SI units, 669–671 U.S Customary units, 631, 634, 635 spirals for columns (size and pitch), 646 weights of common building materials, 28 welded wire reinforcement, 633 welded wire reinforcement sheets: U.S customary units, 632 T-beams, 112–127 analysis of: general method, 114–117 specific method for T beams, 118–119 deflections, 164–170 design of, 120–127 Tensile strength of concrete, 16–17, 223 modulus of rupture, 16, 18 split cylinder test, 16–17 Tension, diagonal, 224 Tension controlled section, 71 Tension reinforcing: development lengths for, 189–197, 203–204 hooks for, 199–203 Tension splices, 213 Ties, 201 circular, 271 spacing of, 270 Tied columns, 264 ACI Code requirements for, 270 as economical, 274 failure of, 266–268 Toe (retaining walls), 394, 414, 416–418 Torsion, 254–255, 470–491 ACI design requirements, 479–480 compatibility, 474–475 design of, 478–479 equilibrium, 474 moment strength, 477 reinforcing, 471–474 required by ACI, 476–480 using U.S customary units, 480–483 stresses, 475–476 and toughness of concrete, 21 using SI units, 483–486 Torsional moment strength, 477 Torsional reinforcing, 471–474 required by ACI, 476–480 using U.S customary units, 480–483 713 Torsional stresses, 475–476 Torsion cracks, 171 Traffic loads, 29 Transformed area, 36, 41 Transverse reinforcement index, 192 Trial-and-error (iterative method), 97–98 Truss analogy, 229, 677 Truss models, 679–681 Two-way slabs, 492–531 analysis of, 495, 517–522 with beams, 492, 494 direct design method, 517–522 equivalent frame method, 535–537 columns: equivalent frame method, 538–540 factored moments in, 528 transfer of shear and moments between slabs and, 522–528 column strips, 496–497 defined, 492 depth limitations and stiffness, 500–505 with interior beams, 503–505 without interior beams, 500–502 design of, 495–496 direct design method, 495–531 with beams, 517–522 depth limitations and stiffness with interior beams, 503–505 depth limitations and stiffness without interior beams, 500–502 factored moments in columns and walls, 528 interior flat plates, 511–514 live load placement, 514–517 transfer of shear and moments between slabs and columns, 522–528 equivalent frame method, 495–496, 532–546 properties of columns, 538–540 properties of slab beams, 535–537 flat plates, 492, 493, 511–514 flat slabs, 492, 493 for lateral loads, 496 live load placement, 514–517 middle strips, 496–497 moments in: distribution for nonprismatic members, 532–533 distribution of, 506–511 transfer between slabs and columns, 522–528 openings in slab systems, 528 shear: shear resistance, 497–500 transfer between slabs and columns, 522–528 714 INDEX Two-way slabs, (continued ) waffle slabs, 492–493 walls, factored moments in, 528 U Ultimate strength, of prestressed concrete sections, 582–587 Ultimate-strength design, 65, 66 See also Strength design Ultimate-strength stage (flexural analysis), 36–38, 48–51 Unbraced frames, see Sway (unbraced) frames, Uncracked concrete stage (flexural analysis), 35, 38–40 U.S customary units: tables and graphs, 631–668 torsional reinforcing using, 480–483 United States Geological Service (USGS), 684 V Van Ryzin, G., 31 Vertical loads: for continuous structures, 445–454 in seismic design, 687–688 Vibrations, 154 W Waffle slabs, 492–493 Walls, 547–566 economy in construction of, 563 load-bearing: empirical design method, 549–551 masonry walls with out-of-plane loads, 616–623 rational design method, 552–554 masonry: load-bearing walls with out-of-plane loads, 616–623 non–load-bearing walls with out-of-plane loads, 607–611 shear walls with in-plane loading, 623–628 non–load-bearing, 547–548, 607–611 shear, 554–562 Wall footings, 347, 348, 352–357 Wang, C K., Ward, W E., Wayss, G., Web reinforcement: ACI Code requirements for, 232–237 behavior of beams with, 229–230 defined, 224 for prestressed sections, 590 for shear cracking in beams, 227–229 T-beams, 112 Web–shear cracks, 171, 226, 227 Weep holes, 398 Weischede, D., 681 Welded wire fabric, 22–25 and shear cracks, 228–229 in tension, development lengths for, 203–204 Whitney, C S., 68, 289 Wilkinson, W B., Wind loads, 31–32, 84 Wire Reinforcement Institute, 23 Wobble effect, 582 Working loads, 36, 65 Working stress design (WSD), 65–67 McCormac bend01.tex V2 - January 10, 2013 6:43 P.M Frequently Used Notation α Ab Ac Ag Al As As Ask Ast At Av A1 A2 b b bo bw c cb = = = = = = = = = = = = = = = = = = = Cm d d db dc Dc Ec Es fc fc fm fct fr fs fs ft fy h hf Icr Ie Ig = = = = = = = = = = = = = = = = = = = = = = depth of equivalent compression rectangular stress block for flexural members cross-sectional area of a reinforcing bar area of core of a spirally reinforcing column measured out to out of spiral gross cross-sectional area of a concrete member total area of longitudinal reinforcing to resist torsion area of nonprestressed tensile reinforcing area of compression reinforcement area of skin reinforcement for a deep beam per unit of height on one side face of beam total area of nonprestressed longitudinal reinforcing (bars or steel shapes) area of one leg of a closed stirrup resisting torsion within a distance s cross-sectional area of shear reinforcing in a distance s in a flexural member loaded area maximum area of a supporting surface that is geometrically similar and concentric with the loaded area A1 width of the compression face of a flexural member effective width of the flange of a T or L beam perimeter of critical section for shear for slabs or footings web width or diameter of a circular section distance from extreme compression fiber to neutral axis smaller of: (a) the distance from center of a bar or wire to nearest concrete surface, and (b) one-half the center-to-center spacing of bars or wires being developed a factor relating an equivalent uniform moment diagram to the actual diagram effective depth of a section measured from extreme compression fiber to centroid of tensile reinforcement distance from extreme compression fiber to centroid of compression steel bar diameter concrete cover thickness measured from extreme tensile fiber to closed reinforcing bar or wire diameter of core of spiral column measured out to out of column modulus of elasticity of concrete modulus of elasticity of steel computed compression flexural fiber stress at service loads specified compression strength of concrete specified compressive strength of masonry average splitting tensile strength of lightweight aggregate concrete modulus of rupture of concrete computed flexural stress in tensile steel at service loads computed flexural stress in compression steel computed tensile flexural stress in concrete specified yield strength of nonprestressed reinforcing total thickness of member thickness of compression flange of a T, L, or I section transformed moment of inertia of cracked concrete section effective moment of inertia of a section used for deflection calculations gross moment of inertia of a section about its centroidal axis Page McCormac bend01.tex V2 - January 10, 2013 6:43 P.M k ld ldb ldh ldt lhb ln lu Mcr M1 M2 M1ns M2ns Ma Mo Mu n Pc Pno P0 qa qe Rn s Vnm Vns Vtu w wc yt z β = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = βdns = βds = β1 = δ δs λ = = = = = = λ μ ξ ρ ρ ρb φ = = = = = = = c s s effective length factor for a compression member development length of a straight bar embedded in confined concrete basic development length development length of a bar with a standard hook development length in tension of headed deformed bar basic development length of a standard hook in tension clear span measured face to face of supports unsupported length of a compression member cracking moment of concrete smaller end factored moment in a compression member, negative if double curvature larger end factored moment in a compression member smaller factored end moment in a compression member due to loads that result in no appreciable sidesway larger factored end moment in a compression member due to loads that result in no appreciable sidesway maximum moment in member due to service loads at stage deflection is computed total factored static moment factored moment at section modular ratio (ratio of modulus of elasticity of steel to that of concrete) Euler buckling load of column pure axial load capacity of column nominal axial load strength of a member with no eccentricity allowable soil pressure effective soil pressure a term used in required percentage of steel expression for flexural members (Mu /φbd 2) spacing of shear or torsional reinforcing parallel to longitudinal reinforcing nominal shear strength provided by masonry (see Chapter 20) nominal shear strength provided by shear reinforcement (see Chapter 20) torsional stress crack width unit weight of concrete distance from centroidal axis of gross section to extreme fiber in tension a term used to estimate crack sizes and specify distribution of reinforcing ratio of long to short dimensions: clear spans for two-way slabs; sides of column, concentrated load or reaction area; or sides of a footing ratio used to account for reduction of stiffness of columns due to sustained axial loads ratio used to account for reduction of stiffness of columns due to sustained lateral loads a factor to be multiplied by the depth d of a member to obtain the depth of the equivalent rectangular stress block moment magnification factor to reflect effects of member curvature between ends of compression member moment magnification factor for slender columns in frames not braced against sidesway strain in compression concrete strain in tension reinforcement strain in compression reinforcement modification factor reflecting the reduced mechanical properties of lightweight concrete; all relative to normal weight concrete of the same compressive strength a multiplier used in computing long-term deflections coefficient of friction a time-dependent factor for sustained loads used in computing long-term deflections ratio of nonprestressed reinforcement in a section ratio of compression reinforcing in a section ratio of tensile reinforcing producing balanced strain condition capacity reduction factor Page McCormac bend02.tex V2 - January 9, 2013 8:08 P.M Typical SI Quantities and Units Quantity Unit Length Area Quantity Symbol Unit (N/m2 ) Symbol meter m Stress pascal square meter m2 Moment newton meter N•m Pa Volume cubic meter m Work newton meter Nm Force newton N Density kilogram per cubic meter kg/m3 Weight newton per cubic meter N/m3 Mass kilogram kg SI Prefixes Prefix Symbol tera T Multiplication Factor 10 12 = 000 000 000 000 giga G 10 = 000 000 000 mega M 106 = 000 000 k 10 = 000 h 10 = 100 da 10 = 10 kilo hecto deca deci centi d c 10 10 −1 = 0.100 −2 = 0.010 −3 milli m 10 = 0.001 micro μ 10−6 = 0.000 001 nano n 10−9 = 0.000 000 001 pico p −12 10 = 0.000 000 000 001 Conversion of U.S Customary Units to SI Units U.S Customary Units in in.2 SI Units 25.400 mm = 0.025 400 m 645.16 mm2 = 6.451 600 m2 × 10−4 ft 304.800 mm = 0.304 800 m lb 4.448 222 N l kip 448 222 N = 4.448 222 kN psi 6.894 757 kN/m2 = 0.006 895 MN/m2 = 0.006 895 N/mm2 psf 47.880 N/m2 = 0.047 800 kN/m2 ksi 6.894 757 MN/m2 = 6.894 757 MPa in-lb 0.112 985 N • m ft-lb 1.355 818 N • m in-k 112.985 N • m ft-k 355.82 N • m = 1.355 82 kN • m Page McCormac bend02.tex V2 - January 9, 2013 8:08 P.M Page ... Prestressed Concrete 20 Reinforced Concrete Masonry A Tables and Graphs: U.S Customary Units B Tables in SI Units C The Strut-and-Tie Method of Design D Seismic Design of Reinforced Concrete Structures... Contents Preface Introduction xv 1.1 Concrete and Reinforced Concrete, 1.2 Advantages of Reinforced Concrete as a Structural Material, 1.3 Disadvantages of Reinforced Concrete as a Structural Material,... well-cured concretes very much 1.11 Properties of Concrete A thorough knowledge of the properties of concrete is necessary for the student before he or she begins to design reinforced concrete