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Lecture Notes in:
Mechanics and Design of
REINFORCED CONCRETE
CVEN4555
cVICTOR E. SAOUMA,
Fall 2001
Dept. of Civil Environmental and Architectural Engineering
University of Colorado, Boulder, CO 80309-0428
May 18, 2002
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Contents
1 INTRODUCTION 111.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.1.1 Mix Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1.1.1 Constituents . . . . . . . . . . . . . . . . . . . . . . . . 111.1.1.1.2 Preliminary Considerations . . . . . . . . . . . . . . . . 15
1.1.1.1.3 Mix procedure . . . . . . . . . . . . . . . . . . . . . . . 151.1.1.1.4 Mix Design Example . . . . . . . . . . . . . . . . . . . 18
1.1.1.2 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . 191.1.2 Reinforcing Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.2 Design Philosophy, USD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141.3 Analysis vs Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1151.4 Basic Relations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161.5 ACI Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2 FLEXURE 212.1 Uncracked Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
E 2-1 Uncracked Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Section Cracked, Stresses Elastic . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Basic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Working Stress Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24E 2-2 Cracked Elastic Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25E 2-3 Working Stress Design Method; Analysis . . . . . . . . . . . . . . . . . . . 26E 2-4 Working Stress Design Method; Design . . . . . . . . . . . . . . . . . . . 27
2.3 Cracked Section, Ultimate Strength Design Method . . . . . . . . . . . . . . . . . 282.3.1 Whitney Stress Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Balanced Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2102.3.3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.3.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
2.4 Practical Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2122.4.1 Minimum Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2122.4.2 Beam Sizes, Bar Spacing, Concrete Cover . . . . . . . . . . . . . . . . . . 2132.4.3 Design Aids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
2.5 USD Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215E 2-5 Ultimate Strength; Review . . . . . . . . . . . . . . . . . . . . . . . . . . 215E 2-6 Ultimate Strength; Design I . . . . . . . . . . . . . . . . . . . . . . . . . . 216E 2-7 Ultimate Strength; Design II . . . . . . . . . . . . . . . . . . . . . . . . . 217
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2.6 T Beams, (ACI 8.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2172.6.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2192.6.2 Design, (balanced) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219E 2-8 T Beam; Moment Capacity I . . . . . . . . . . . . . . . . . . . . . . . . . 220E 2-9 T Beam; Moment Capacity II . . . . . . . . . . . . . . . . . . . . . . . . . 221
E 2-10 T Beam; Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2222.7 Doubly Reinforced Rectangular Beams . . . . . . . . . . . . . . . . . . . . . . . . 223
2.7.1 Tests for fs and fs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
2.7.2 Moment Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226E 2-11 Doubly Reinforced Concrete beam; Review . . . . . . . . . . . . . . . . . 228E 2-12 Doubly Reinforced Concrete beam; Design . . . . . . . . . . . . . . . . . . 230
2.8 Bond & Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.8.1 Moment Capacity Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 235
3 SHEAR 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Shear Strength of Uncracked Section . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Shear Strength of Cracked Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 ACI Code Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
E 3-1 Shear Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Shear Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
E 3-2 Shear Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.7 Brackets and Corbels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.8 Deep Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
4 CONTINUOUS BEAMS 414.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.1 Detailed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 ACI Approximate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Effective Span Design Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Moment Redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.1 Elastic-Perfectly Plastic Section . . . . . . . . . . . . . . . . . . . . . . . . 444.4.2 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45E 4-1 Moment Redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 SERVICEABILITY 515.1 Control of Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
E 5-1 Crack Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.1 Short Term Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2.2 Long Term Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55E 5-2 Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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6 APPROXIMATE FRAME ANALYSIS 616.1 Vertical Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Horizontal Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2.1 Portal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64E 6-1 Approximate Analysis of a Frame subjected to Vertical and Horizontal
Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7 ONE WAY SLABS 717.1 Types of Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.2 One Way Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.3 Design of a One Way Continuous Slab . . . . . . . . . . . . . . . . . . . . . . . . 75
8 COLUMNS 81
9 COLUMNS 919.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.1.1 Types of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.1.2 Possible Arrangement of Bars . . . . . . . . . . . . . . . . . . . . . . . . . 919.2 Short Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.2.1 Concentric Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.2.2 Eccentric Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.2.2.1 Balanced Condition . . . . . . . . . . . . . . . . . . . . . . . . . 949.2.2.2 Tension Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959.2.2.3 Compression Failure . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.2.3 ACI Provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.2.4 Interaction Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.2.5 Design Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98E 9-1 R/C Column, c known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
E 9-2 R/C Column, e known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910E 9-3 R/C Column, Using Design Charts . . . . . . . . . . . . . . . . . . . . . . 9149.2.6 Biaxial Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915E 9-4 Biaxially Loaded Column . . . . . . . . . . . . . . . . . . . . . . . . . . . 918
9.3 Long Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9199.3.1 Euler Elastic Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9199.3.2 Effective Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9209.3.3 Moment Magnification Factor; ACI Provisions . . . . . . . . . . . . . . . 922E 9-5 Long R/C Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925E 9-6 Design of Slender Column . . . . . . . . . . . . . . . . . . . . . . . . . . . 926
10 PRESTRESSED CONCRETE 101
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.1.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.1.2 Prestressing Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.1.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.1.4 Tendon Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.1.5 Equivalent Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.1.6 Load Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10.2 Flexural Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
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E 10-1 Prestressed Concrete I Beam . . . . . . . . . . . . . . . . . . . . . . . . . 10810.3 Case Study: Walnut Lane Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010
10.3.1 Cross-Section Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101210.3.2 Prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101210.3.3 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013
10.3.4 Flexural Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013
11 FOOTINGS 111
12 DEEP BEAMS 121
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1.1 Schematic Representation of Aggregate Gradation . . . . . . . . . . . . . . . . . 121.2 MicroCracks in Concrete under Compression . . . . . . . . . . . . . . . . . . . . 1101.3 Concrete Stress Strain Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111.4 Modulus of Rupture Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111.5 Split Cylinder (Brazilian) Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111.6 Biaxial Strength of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121.7 Time Dependent Strains in Concrete . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.1 Strain Diagram Uncracked Section . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Transformed Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Stress Diagram Cracked Elastic Section . . . . . . . . . . . . . . . . . . . . . . . 232.4 Desired Stress Distribution; WSD Method . . . . . . . . . . . . . . . . . . . . . . 242.5 Cracked Section, Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Whitney Stress Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2102.7 Bar Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2152.8 T Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2182.9 T Beam as Rectangular Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2182.10 T Beam Strain and Stress Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 218
2.11 Decomposition of Steel Reinforcement for T Beams . . . . . . . . . . . . . . . . . 2192.12 Doubly Reinforced Beams; Strain and Stress Diagrams . . . . . . . . . . . . . . . 2242.13 Different Possibilities for Doubly Reinforced Concrete Beams . . . . . . . . . . . 2242.14 Strain Diagram, Doubly Reinforced Beam; is As Yielding? . . . . . . . . . . . . . 2252.15 Strain Diagram, Doubly Reinforced Beam; is A s Yielding? . . . . . . . . . . . . . 2262.16 Summary of Conditions for top and Bottom Steel Yielding . . . . . . . . . . . . . 2272.17 Bond and Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.18 Actual Bond Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2322.19 Splitting Along Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2322.20 Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2332.21 Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2332.22 Hooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2352.23 Bar cutoff requirements of the ACI code . . . . . . . . . . . . . . . . . . . . . . . 2362.24 Standard cutoff or bend points for bars in approximately equal spans with uni-
formly distributed load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2372.25 Moment Capacity Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
3.1 Principal Stresses in Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Types of Shear Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Shear Strength of Uncracked Section . . . . . . . . . . . . . . . . . . . . . . . . . 32
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3.4 Mohrs Circle for Shear Strength of Uncracked Section . . . . . . . . . . . . . . . 333.5 Shear Strength of Uncracked Section . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Free Body Diagram of a R/C Section with a Flexural Shear Crack . . . . . . . . 353.7 Equilibrium of Shear Forces in Cracked Section . . . . . . . . . . . . . . . . . . . 353.8 Summary of ACI Code Requirements for Shear . . . . . . . . . . . . . . . . . . . 37
3.9 Corbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.10 Shear Friction Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.11 Shear Friction Across Inclined Reinforcement . . . . . . . . . . . . . . . . . . . . 310
4.1 Continuous R/C Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Load Positioning on Continuous Beams . . . . . . . . . . . . . . . . . . . . . . . 414.3 ACI Approximate Moment Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 434.4 Design Negative Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Moment Diagram of a Rigidly Connected Uniformly Loaded Beam . . . . . . . . 444.6 Moment Curvature of an Elastic-Plastic Section . . . . . . . . . . . . . . . . . . . 454.7 Plastic Moments in Uniformly Loaded Rigidly Connected Beam . . . . . . . . . . 454.8 Plastic Redistribution in Concrete Sections . . . . . . . . . . . . . . . . . . . . . 46
4.9 Block Diagram for R/C Building Design . . . . . . . . . . . . . . . . . . . . . . . 48
5.1 Crack Width Equation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Uncracked Transformed and Cracked Transformed X Sections . . . . . . . . . . . 545.3 Time Dependent Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4 Time Dependent Strain Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 565.5 Short and long Term Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.1 Approximate Analysis of Frames Subjected to Vertical Loads; Girder Moments . 626.2 Approximate Analysis of Frames Subjected to Vertical Loads; Column Axial Forces636.3 Approximate Analysis of Frames Subjected to Vertical Loads; Column Moments 636.4 Approximate Analysis of Frames Subjected to Lateral Loads; Column Shear . . . 656.5 Approximate Analysis of Frames Subjected to Lateral Loads; Girder Moment . . 656.6 Approximate Analysis of Frames Subjected to Lateral Loads; Column Axial Force666.7 Example; Approximate Analysis of a Building . . . . . . . . . . . . . . . . . . . . 676.8 Approximate Analysis of a Building; Moments Due to Vertical Loads . . . . . . . 696.9 Approximate Analysis of a Building; Shears Due to Vertical Loads . . . . . . . . 6106.10 Approximate Analysis for Vertical Loads; Spread-Sheet Format . . . . . . . . . . 6126.11 Approximate Analysis for Vertical Loads; Equations in Spread-Sheet . . . . . . . 6136.12 Approximate Analysis of a Building; Moments Due to Lateral Loads . . . . . . . 6146.13 Portal Method; Spread-Sheet Format . . . . . . . . . . . . . . . . . . . . . . . . . 6166.14 Portal Method; Equations in Spread-Sheet . . . . . . . . . . . . . . . . . . . . . . 617
7.1 Types of Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.2 One vs Two way slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.3 Load Distribution in Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.4 Load Transfer in R/C Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.1 Types of columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919.2 Tied vs Spiral Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.3 Possible Bar arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.4 Sources of Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
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9.5 Load Moment Interaction Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 949.6 Strain and Stress Diagram of a R/C Column . . . . . . . . . . . . . . . . . . . . 959.7 Column Interaction Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.8 Failure Surface of a Biaxially Loaded Column . . . . . . . . . . . . . . . . . . . . 9159.9 Load Contour at Plane of ConstantPn, and Nondimensionalized Corresponding
plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9169.10 Biaxial Bending Interaction Relations in terms of . . . . . . . . . . . . . . . . . 9179.11 Bilinear Approximation for Load Contour Design of Biaxially Loaded Columns . 9179.12 Euler Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9199.13 Column Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9209.14 Critical lengths of columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.15 Effective length Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.16 Standard Alignment Chart (ACI) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9239.17 Minimum Column Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9239.18 P-M Magnification Interaction Diagram . . . . . . . . . . . . . . . . . . . . . . . 924
10.1 Pretensioned Prestressed Concrete Beam, (?) . . . . . . . . . . . . . . . . . . . . 102
10.2 Posttensioned Prestressed Concrete Beam, (?) . . . . . . . . . . . . . . . . . . . . 10210.3 7 Wire Prestressing Tendon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.4 Alternative Schemes for Prestressing a Rectangular Concrete Beam, (?) . . . . . 10510.5 Determination of Equivalent Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 10510.6 Load-Deflection Curve and Corresponding Internal Flexural Stresses for a Typi-
cal Prestressed Concrete Beam, (?) . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.7 Flexural Stress Distribution for a Beam with Variable Eccentricity; Maximum
Moment Section and Support Section, (?) . . . . . . . . . . . . . . . . . . . . . . 10710.8 Walnut Lane Bridge, Plan View . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101110.9 Walnut Lane Bridge, Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 1012
11.1 xxx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.2 xxx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.3 xxx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.4 xxx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11311.5 xxx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
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List of Tables
1.1 ASTM Sieve Designations Nominal Sizes Used for Concrete Aggregates . . . . . 131.2 ASTM C33 Grading Limits for Coarse Concrete Aggregates . . . . . . . . . . . . 131.3 ASTM C33 Grading Limits for Fine Concrete Aggregates . . . . . . . . . . . . . 131.4 Example of Fineness Modulus Determination for Fine Aggregate . . . . . . . . . 151.5 Recommended Slumps (inches) for Various Types of Construction . . . . . . . . 161.6 Recommended Average Total Air Content as % For Different Nominal Maximum
Sizes of Aggregates and Levels of Exposure . . . . . . . . . . . . . . . . . . . . . 161.7 Approximate Mixing Water Requirements, lb/yd3 of Concrete For Different
Slumps and Nominal Maximum Sizes of Aggregates . . . . . . . . . . . . . . . . . 171.8 Relationship Between Water/Cement Ratio and Compressive Strength . . . . . . 171.9 Volume of Dry-Rodded Coarse Aggregate per Unit Volume of Concrete for Dif-
ferent Fineness Moduli of Sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.10 Creep Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131.11 Properties of Reinforcing Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141.12 Strength Reduction Factors, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.1 Total areas for various numbers of reinforcing bars (inch2) . . . . . . . . . . . . . 2142.2 Minimum Width (inches) according to ACI Code . . . . . . . . . . . . . . . . . . 214
4.1 Building Structural Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.1 Columns Combined Approximate Vertical and Horizontal Loads . . . . . . . . . 6186.2 Girders Combined Approximate Vertical and Horizontal Loads . . . . . . . . . . 619
7.1 Recommended Minimum Slab and Beam Depths . . . . . . . . . . . . . . . . . . 74
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DraftLIST OF TABLES 03
Tentative ScheduleFall 1994
1 Aug. 28 Intro; MaterialAug. 30 Concrete mix design
2 Sep. 4 Elastic UncrackedSep. 6 WSD; USD singly reinforced
3 Sep. 11 USD singly, examplesSep. 13 T Beams
4 Sep. 18 T Beams, Doubly ReinfSep. 20 Doubly Reinf Development length
5 Sep. 25 ShearSep. 27 Shear
6 Oct. 2 TP LabOct. 4 Fall Break
7 Oct. 9 Crack width
Oct. 11 EXAM I8 Oct. 16 Deflection
Oct. 18 Crack Width-Defelction9 Oct. 23 Deflection, Continuous Systems
Oct. 25 Continuous Systems; One way slabs
10 Oct. 30 Columns; IntroNov. 1 Columns
11 Nov. 6 LAB
Nov. 8 Columns12 Nov. 13 Biaxial bending
Nov. 15 Long column
13 Nov. 20 LabNov. 22 Thanksgiving
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Chapter 1
INTRODUCTION
1.1 Material
1.1.1 Concrete
This section is adapted from Concrete by Mindess and Young, Prentice Hall, 1981
1.1.1.1 Mix Design
1.1.1.1.1 Constituents
1 Concrete is a mixture of Portland cement, water, and aggregates (usually sand and crushedstone).
2 Portland cement is a mixture of calcareous and argillaceous materials which are calcined ina kiln and then pulverized. When mixed with water, cement hardens through a process called
hydration.3 Ideal mixture is one in which:
1. A minimum amount of cement-water paste is used to fill the interstices between theparticles of aggregates.
2. A minimum amount of water is provided to complete the chemical reaction with cement.Strictly speaking, a water/cement ratio of about 0.25 is needed to complete this reaction,but then the concrete will have a very low workability.
In such a mixture, about 3/4 of the volume is constituted by the aggregates, and the remaining1/4 being the cement paste.
4 Smaller particles up to 1/4 in. in size are called fine aggregates, and the larger ones beingcoarse aggregates.
5 Portland Cement has the following ASTM designation
I Normal
II Moderate sulfate resistant, moderate heat of hydration
III High early strength (but releases too much heat)
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Draft1.1 Material 13
ASTM SizeDesign. mm in.
Coarse Aggregate
3 in. 75 3
21/2 in. 63 2.52 in. 50 2
11/2 in. 37.5 1.51 in. 25 1
3/4 in. 19 0.751/2 in. 12.5 0.503/8 in. 9.5 0.375
Fine Aggregate
No. 4 4.75 0.187No. 8 2.36 0.0937
No. 16 1.18 0.0469No. 30 0.60 (600m) 0.0234
No. 50 300 m 0.0124No. 100 150 m 0.0059
Table 1.1: ASTM Sieve Designations Nominal Sizes Used for Concrete Aggregates
Sieve Size % (Nominal Maximum Size)
11/2 in. 1 in. 3/4 in. 1/2 in.
11/2 in. 95-100 100 - -1 in. - 95-100 100 -3/4 in. 35-70 - 90-100 1001/2 in. - 25-60 - 90-1003/8 in. 10-30 - 20-55 40-70No. 4 0-5 0-10 0-10 0-15No. 8 - 0-5 0-5 0-5
Table 1.2: ASTM C33 Grading Limits for Coarse Concrete Aggregates
Sieve Size % Passing
3/4 in. 100
No. 4 95-100No. 8 80-100No. 16 50-85No. 30 25-60No. 50 10-30No. 100 2-10
Table 1.3: ASTM C33 Grading Limits for Fine Concrete Aggregates
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Draft14 INTRODUCTION
16 Moisture statesare defined as
Oven-dry (OD): all moisture is removed from the aggregate.
Air-dry (AD): all moisture is removed from the surface, but internal pores are partially full.
Saturated-surface-dry (SSD): All pores are filled with water, but no film of water on thesurface.
Wet: All pores are completely filled with a film of water on the surface.
17 Based on the above, we can determine
Absorption capacity (AC): is the maximum amount of water the aggregate can absorb
AC =WSS D WOD
WOD 100% (1.1)
most normal -weight aggregates (fine and coarse) have an absorption capacity in the rangeof 1% to 2%.
Surface Moisture (SM): is the water in excess of the SSD state
SM =WW et WSS D
WSS D 100% (1.2)
18 Thefineness modulusis a parameter which describe the grading curve and it can be usedto check the uniformity of the grading. It is usually computed for fine aggregates on the basisof
F.M. = cumulative percent retained on standard sieves
100
(1.3)
where the standard sieves used are No. 100, No. 50, No. 30, No. 16, No. 8, and No. 4, and3/8 in, 3/4 in, 11/2 in and larger.
19 The fineness modulus for fine aggregate should lie between 2.3 and 3.1 A small numberindicates a fine grading, whereas a large number indicates a coarse material.
20 Table 1.4 illustrates the determination of the fineness modulus.
21 Fineness modulus of fine aggregate is required for mix proportioning since sand gradationhas the largest effect on workability. A fine sand (low fineness modulus) has much higher pasterequirements for good workability.
22
The fineness modulus of coarse aggregate is not used for mix design purposes.23 no-fines concretehas little cohesiveness in the frsh state and can not be compacted to avoid-free condition. Hence, it will have a low strength, high permeability. Its only advantage islow density, and high thermal insulation which can be used if structural requirements are nothigh.
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Draft1.1 Material 15
Sieve Weight Amount Cumulative CumulativeSize Retained Retained Amount Amount
(g) (wt. %) Retained (%) Passing (%)
No. 4 9 2 2 98No. 8 46 9 11 89
No. 16 97 19 30 70No. 30 99 20 50 50No. 50 120 24 74 26No. 100 91 18 92 8
= 259
Fineness modulus=259/100=2.59
Table 1.4: Example of Fineness Modulus Determination for Fine Aggregate
1.1.1.1.2 Preliminary Considerations
24 There are two fundamental aspects to mix design to keep in mind:
1. Water/Cement ratio: where the strength in inversely proportional to the water to cementratio, approximately expressed as:
fc = A
B1.5w/c (1.4)
Forfc in psi,A is usually taken as 14,000 and B depends on the type of cement, but maybe taken to be about 4. It should be noted that w/c controls not only the strength, butalso the porosityand hence the durability.
2. Aggregate Grading: In order to minimize the amount of cement paste, we must maximizethe volume of aggregates. This can be achieved through proper packing of the granularmaterial. The ideal grading curve (with minimum voids) is closely approximated bythe Fuller curve
Pt=
d
D
q(1.5)
where Pt is the fraction of total solids finer than size d, and D is the maximum particlesize,qis generally taken as 1/2, hence the parabolicgrading.
1.1.1.1.3 Mix procedure
25 Before starting the mix design process, the followingmaterial propertiesshould be deter-mined:
1. Sieve analysis of both fine and coarse aggregates
2. Unit weight of the coarse aggregate
3. bulk specific gravities
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Draft16 INTRODUCTION
4. absorption capacities of the aggregates
1. Slump1 must be selected for the particular job to account for the anticipated methodof handling and placing concrete, Table 1.5 As a general rule, adopt the lowest possible
Type of Construction Max MinFoundation walls and footings 3 1Plain footings, caissons 3 1Beams and reinforced walls 4 1Building columns 4 1Pavement and slabs 3 1Mass concrete 3 1
Table 1.5: Recommended Slumps (inches) for Various Types of Construction
slump.
2. Maximum aggregate size: in general the largest possible size should be adopted.However, it should be noted that:
(a) For reinforced concrete, the maximum size may not exceed one-fifth of the mini-mum dimensions between the forms, or three-fourths of the minimum clear spacingbetween bars, or between steel and forms.
(b) For slabs on grade, the maximum size may not exceed one-third the slab depth.
In general maximum aggregate size is 3/4 in or 1 in.
3. Water and Air content Air content will affect workability (some time it is better to
increase air content rather than increasingw/cwhich will decrease strength). Air contentcan be increased through the addition of admixtures. Table 1.6 tabulates recommendedvalues of air content (obtained through such admixtures) for different conditions (forinstance under severe freezing/thawing air content should be high).
Recommended water requirements are given by Table 1.7.
Sizes of Aggregates
Exposure 3/8 in. 1/2 in. 3/4 in. 1 in. 11/2 in.
Mild 4.5 4.0 3.5 3.5 3.0Moderate 6.0 5.5 5.0 4.5 4.4Extreme 7.5 7.0 6.0 6.05 5.5
Table 1.6: Recommended Average Total Air Content as % For Different Nominal MaximumSizes of Aggregates and Levels of Exposure
1The slump test (ASTM C143) is a measure of the shear resistance of concrete to flowing under its own weight.It is a good indicator of the concrete workability. A hollow mold in the form of a frustum of a cone is filledwith concrete in three layers of equal volume. Each layer is rodded 25 times. The mold is then lifted vertically,and the slump is measured by determining the difference between the height of the mold and the height of theconcrete over the original center of the base of the specimen.
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Draft1.1 Material 17
Slump Sizes of Aggregates
in. 3/8 in. 1/2 in. 3/4 in. 1 in. 11/2 in.
Non-Air-Entrained Concrete
1-2 350 335 315 300 2753-4 385 365 340 325 300
6-7 410 385 360 340 315Air-Entrained Concrete
1-2 305 295 280 270 2503-4 340 325 305 295 2756-7 365 345 325 310 290
Table 1.7: Approximate Mixing Water Requirements, lb/yd3 of Concrete For Different Slumpsand Nominal Maximum Sizes of Aggregates
4. Water/cement ratio: this is governed by both strength and durability. Table 1.8
provides some guidance in terms of strength.
28 days w/c Ratio by Weightfc Non-air-entrained Air-entrained
6,000 0.41 -5,000 0.48 0.404,000 0.57 0.483,000 0.68 0.592,000 0.82 0.74
Table 1.8: Relationship Between Water/Cement Ratio and Compressive Strength
For durability, if there is a severe exposure (freeze/thaw, exposure to sea-water, sulfates),then there are severe restrictions on the W/C ratio (usually to be kept just under 0.5)
5. Cement Content: Once the water content and thew/cratio are determined, the amountof cement per unit volume of concrete is determined simply by dividing the estimatedwater requirement by the w/c ratio.
6. Coarse Aggregate Content: Volume of coarse aggregate required per cubic yard ofconcrete depends on its maximum size and the fineness modulus of the fine aggregate,Table 1.9. The oven dry (OD) volume of coarse aggregate in ft3 required per cubic yard
is simply equal to the value from Table 1.9 multiplied by 27. This volume can then beconverted to an OD weight by multiplying it by the dry-rodded2 weight per cubic foot ofcoarse aggregate.
7. The fine aggregate content can be estimated by subtracting the volume of cement,water, air and coarse aggregate from the total volume. The weight of the fine aggregatecan then be obtained by multiplying this volume by the density of the fine aggregate.
2Dry Rodded volume (DRV) is the normal volume of space a material occupies.
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Draft18 INTRODUCTION
Agg. Size Sand Fineness Moduliin 2.40 2.60 2.80 3.00
3/8 0.50 0.48 0.46 0.441/2 0.59 0.57 0.55 0.533/4 0.66 0.64 0.62 0.60
1 0.71 0.69 0.67 0.6511/2 0.76 0.74 0.72 0.70
Table 1.9: Volume of Dry-Rodded Coarse Aggregate per Unit Volume of Concrete for DifferentFineness Moduli of Sand
8. Adjustment for moisture in the aggregates: is necessary. If aggregates are airdry, they will absorb some water (thus effectively lowering the w/c), or if aggregates aretoo wet they will release water (increasing the w/c and the workability but reducing thestrength).
1.1.1.1.4 Mix Design Example
Concrete is required for an exterior column to be located above ground in an area wheresubstantial freezing and thawing may occur. The concrete is required to have an average 28-day compressive strength of 5,000 psi. For the conditions of placement, the slump should bebetween 1 and 2 in, the maximum aggregate size should not exceed 3/4 in. and the propertiesof the materials are as follows:
Cement: Type I specific gravity = 3.15
Coarse Aggregates: Bulk specific gravity (SSD) = 2.70; absorption capacity= 1.0%; Total
moisture content = 2.5%; Dry-rodded unit weight = 100 lb/ft3
Fine Aggregates: Bulk specific gravity (SSD) = 2.65; absorption capacity = 1.3 %; Totalmoisture content=5.5%; fineness modulus = 2.70
The sieve analyses of both the coarse and fine aggregates fall within the specified limits. Withthis information, the mix design can proceed:
1. Choice ofslump is consistent with Table 1.5.
2. Maximum aggregate size (3/4 in) is governed by reinforcing details.
3. Estimation of mixing water: Because water will be exposed to freeze and thaw, it must
be air-entrained. From Table 1.6 the air content recommended for extreme exposure is6.0%, and from Table 1.7 the water requirement is 280 lb/yd3
4. From Table 1.8, the water to cement ratio estimate is 0.4
5. Cement content, based on steps 4 and 5 is 280/0.4=700 lb/yd 3
6. Coarse aggregate content, interpolating from Table 1.9 for the fineness modulus ofthe fine aggregate of 2.70, the volume of dry-rodded coarse aggregate per unit volume ofconcrete is 0.63. Therefore, the coarse aggregate will occupy 0.63 27 = 17.01 ft3/yd3.
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Draft1.1 Material 19
The OD weight of the coarse aggregate is 17.01 100 = 1, 701 lb. The SSD weight is1, 701 1.01 = 1, 718 lb.
7. Fine aggregate content Knowing the weights and specific gravities of the water, cement,and coarse aggregate, and knowing the air volume, we can calculate the volume per yd 3
occupied by the different ingredients.
Water 280/62.4 = 4.49 ft3
Cement 700/(3.15)(62.4) = 3.56 ft3
Coarse Aggregate (SSD) 1,718/(2.70)(62.4) = 1.62 ft3
Air (0.06)(27) = 1.62 ft3
19.87 ft3
Hence, the fine aggregate must occupy a volume of 27.0 19.87 = 7.13 ft3. The requiredSSD weight of the fine aggregate is 7.13 2.65 62.4 = 1, 179 lb.
8. Adjustment for moisturein the aggregate. Since the aggregate will be neither SSD orOD in the field, it is necessary to adjust the aggregate weights for the amount of watercontained in the aggregate. Only surface water need be considered; absorbed water doesnot become part of the mix water. For the given moisture contents, the adjusted aggre-gate weights become:
Coarse aggregate (wet)=1,718(1.025-0.01) = 1,744 lb/yd3
Fine aggregate (wet)=1,179(1.055-0.013) = 1,229 lb/yd3
Surface moisture contributed by the coarse aggregate is 2.5-1.0 = 1.5%; by the fine ag-gregate: 5.5-1.3 = 4.2%; Hence the additional water required is then280-1,718(0.015)-1,179(0.042) = 205 lb/yd3.
Thus, the estimated batch weight per yd3 are
Water 205 lbCement 700 lbWet coarse aggregate 1,744 lbWet fine aggregate 1,229 lb
3,878 lb/yd33,878
27 143.6 lb/ft3
1.1.1.2 Mechanical Properties
26 Contrarily to steel to modulus of elasticity of concrete depends on the strength and is given
by
E= 57, 000
fc (1.6)
or
E= 331.5
fc (1.7)
where both fc and Eare in psi and is in lbs/ft3.
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Draft110 INTRODUCTION
27 Normal weight and lightweight concrete have equal to 150 and 90-120 lb/ft3 respectively.
28 Poissons ratio = 0.15.
29 Typical concrete (compressive) strengths range from 3,000 to 6,000 psi; However high strengthconcrete can go up to 14,000 psi.
30 Stress-strain curve depends on
1. Properties of aggregates
2. Properties of cement
3. Water/cement ratio
4. Strength
5. Age of concrete
6. Rate of loading, as rate
, strength
31 Non-linear part of stress-strain curve is caused by micro-cracking around the aggregates, Fig.1.2
~ 0.5 fc
u
fc
Linear
Non-Linear
Figure 1.2: MicroCracks in Concrete under Compression
32 Irrespective offc, maximum strain under compression is 0.003, Fig. 1.333 Full strength of concrete is achieved in about 28 days
fct= t4.0 + .85tfc,28
or
t (days) 1 2 4 7 10 15
%fc,28 20 35 54 70 80 90
34 Concrete always gain strength in time, but a decreasing rate
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Draft1.1 Material 111
u
=
fc
fc
0.003
/2
Figure 1.3: Concrete Stress Strain Curve
35 The tensile strength of concrete ft is very difficult to measure experimentally. Acceptedvalues
ft 0.07 0.11fc (1.8-a) 3 5
fc (1.8-b)
36 Rather than the tensile strength, it is common to measure the modulus of rupture fr, Fig.1.4
0 0
0 0
0 0
1 1
1 1
1 1
0 0
0 0
0 0
1 1
1 1
1 1
Figure 1.4: Modulus of Rupture Test
Figure 1.5: Split Cylinder (Brazilian) Test
fr 7.5
fc (1.9)
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Draft112 INTRODUCTION
fc
fc
~ 20% increase in strength
f
f
t
t
11
2
2
1
2
Figure 1.6: Biaxial Strength of Concrete
37 Using split cylinder (or brazilian test), Fig. 1.5ft 68
fc. For this test, a nearly uniformtensile stress
= 2P
dtwhere P is the applied compressive load at failure, d and t are diameter and thickness of thespecimen respectively.
38 In most cases, concrete is subjected to uniaxial stresses, but it is possible to have biaxial(shells, shear walls) or triaxial (beam/column connections) states of stress.
39 Biaxial strength curve is shown in Fig. 1.6
40 Concrete has also some time-dependent propertiesShrinkage: when exposed to air (dry), water tends to evaporate from the concrete surface,
shrinkage. It depends on thew/cand relative humidity. sh 0.00020.0007. Shrinkagecan cause cracking if the structure is restrained, and may cause large secondary stresses.
If a simply supported beam is fully restrained against longitudinal deformation, then
sh = Esh
= 57, 000
3, 000(0.0002) = 624 psi >3, 000
10 ft
if the concrete is restrained, then cracking will occur3.
Creep: can be viewed as the squeezing out of water due to long term stresses (analogous toconsolidation in clay), Fig. 1.7.
Creep coefficient, Table 1.10Cu =
ctci
2 3Ct =
t0.6
10+t0.6 Cu
3For this reason a minimum amount of reinforcement is always necessary in concrete, and a 2% reinforcement,can reduce the shrinkage by 75%.
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Draft1.1 Material 113
no load constant load
creepElastic recovery
Residual
Creep recovery
no load
Figure 1.7: Time Dependent Strains in Concrete
fc 3,000 4,000 6,000 8,000
Cu 3.1 2.9 2.4 2.0
Table 1.10: Creep Coefficients
41 Coefficient of thermal expansion is 0.65 105 /deg F for normal weight concrete.
1.1.2 Reinforcing Steel
42 Steel is used as reinforcing bars in concrete, Table 1.11.
43 Bars have a deformation on their surface to increase the bond with concrete, and usuallyhave a yield stress of 60 ksi.
44 Maximum allowable fy is 80 ksi.
45 Stirrups, used as vertical reinforcement to resist shear, usually have a yield stress of only 40ksi
46 Steel loses its strength rapidly above 700 deg. F (and thus must be properly protected fromfire), and becomes brittle at30 deg. F47 Prestressing Steel cables have an ultimate strength up to 270 ksi.
48 Welded wire fabric is often used to reinforce slabs and shells. It has both longitudinal and
transverse cold-drawn steel. They are designated by AAW BB, such as 66W1.41.4where spacing of the wire is 6 inch, and a cross section of 0.014 in2.
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Draft114 INTRODUCTION
Bar Designation Diameter Area Perimeter Weight(in.) ( in2) in lb/ft
No. 2 2/8=0.250 0.05 0.79 0.167No. 3 3/8=0.375 0.11 1.18 0.376No. 4 4/8=0.500 0.20 1.57 0.668
No. 5 5/8=0.625 0.31 1.96 1.043No. 6 6/8=0.750 0.44 2.36 1.5202No. 7 7/8=0.875 0.60 2.75 2.044No. 8 8/8=1.000 0.79 3.14 2.670No. 9 9/8=1.128 1.00 3.54 3.400
No. 10 10/8=1.270 1.27 3.99 4.303No. 11 11/8=1.410 1.56 4.43 5.313
No. 14 14/8=1.693 2.25 5.32 7.650
No. 18 18/8=2.257 4.00 7.09 13.60
Table 1.11: Properties of Reinforcing Bars
1.2 Design Philosophy, USD
49 ACI refers to this method as the Strength Design Method, (previously referred to as theUltimate Strength Method).
Rn iQi (1.10)
where
is a strength reduction factor, less than 1, and must account for the type of structuralelement, Table 1.12 (ACI 9.3.2)
Type of Member
Axial Tension 0.9Flexure 0.9Axial Compression, spiral reinforcement 0.75Axial Compression, other 0.70Shear and Torsion 0.85Bearing on concrete 0.70
Table 1.12: Strength Reduction Factors,
Rn is the nominal resistance (or strength).
Ru= Rd= Rn is the design strength.
i is the load factor corresponding to Qi and is greater than 1.
iQi is the required strengthbased on the factored load:
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Draft1.3 Analysis vs Design 115
i is the type of load
Mn MuVn VuPn
Pu
50 Note that the subscript d and u are equivalent.
51 The various factored load combinations which must be considered (ACI: 9.2) are
1. 1.4D+1.7L
2. 0.75(1.4D+1.7L+1.7W)
3. 0.9D+1.3W
4. 1.05D+1.275W
5. 0.9D+1.7H
6. 1.4D +1.7L+1.7H
7. 0.75(1.4D+1.4T+1.7L)
8. 1.4(D+T)
where D= dead; L= live; Lr= roof live; W= wind; E= earthquake; S= snow; T= temperature;H= soil. We must select the one with the largest limit state load.
52 Serviceability Limit States must be assessed under service loads (not factored). Themost important ones being
1. Deflections
2. Crack width (for R/C)
3. Stability
1.3 Analysis vs Design
53 In R/C we always consider one of the following problems:
Analysis: Given a certain design, determine what is the maximum moment which can be
applied.
Design: Given an external moment to be resisted, determine cross sectional dimensions (bandh) as well as reinforcement (As). Note that in many cases the external dimensions of thebeam (b and h) are fixed by the architect.
54 We often consider the maximum moment along a member, and design accordingly.
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Draft116 INTRODUCTION
1.4 Basic Relations and Assumptions
55 In developing a design/analysis method for reinforced concrete, the following basic relationswill be used:
1. Equilibrium: of forces and moment at the cross section. 1) Fx = 0 or Tension in thereinforcement = Compression in concrete; and 2) M= 0 or external moment (that is theone obtained from the moment envelope) equal and opposite to the internal one (tensionin steel and compression of the concrete).
2. Material Stress Strain: We recall that all normal strength concrete have a failure strainu= .003 in compression irrespective off
c.
56 Basic assumptionsused:
Compatibility of Displacements: Perfect bond between steel and concrete (no slip). Notethat those two materials do also have very close coefficients of thermal expansion under
normal temperature.Plane section remain planestrain is proportional to distance from neutral axis.Neglect tensile strength in all cases.
1.5 ACI Code
Attached is an unauthorizedcopy of some of the most relevant ACI-318-89 design code provi-sions.
8.1.1 - In design of reinforced concrete structures, members shall be proportioned for ad-equate strength in accordance with provisions of this code, using load factors and strength
reduction factors specified in Chapter 9.8.3.1- All members of frames or continuous construction shall be designed for the maximum
effects of factored loads as determined by the theory of elastic analysis, except as modifiedaccording to Section 8.4. Simplifying assumptions of Section 8.6 through 8.9 may be used.
8.5.1 - Modulus of elasticity Ec for concrete may be taken as W1.5c 33
fc ( psi) for values
of Wc between 90 and 155 lb per cu ft. For normal weight concrete, Ec may be taken as57, 000
fc.
8.5.2 - Modulus of elasticity Es for non-prestressed reinforcement may be taken as 29,000psi.
9.1.1- Structures and structural members shall be designed to have design strengths at allsections at least equal to the required strengths calculated for the factored loads and forces in
such combinations as are stipulated in this code.9.2- Required Strength9.2.1- Required strength U to resist dead load D and live load L shall be at least equal to
U= 1.4D+ 1.7L
9.2.2- If resistance to structural effects of a specified wind load W are included in design,the following combinations of D, L, and W shall be investigated to determine the greatestrequired strength U
U= 0.75(1.4D+ 1.7L + 1.7W)
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Draft1.5 ACI Code 117
where load combinations shall include both full value and zero value of L to determine the moresevere condition, and
U= 0.9D+ 1.3W
but for any combination of D, L, and W, required strength U shall not be less than Eq. (9-1).9.3.1 - Design strength provided by a member, its connections to other members, and its
cross sections, in terms of flexure, axial load, shear, and torsion, shall be taken as the nominalstrength calculated in accordance with requirements and assumptions of this code, multipliedby a strength reduction factor .
9.3.2- Strength reduction factor shall be as follows:9.3.2.1- Flexure, without axial load 0.909.4 - Design strength for reinforcement Designs shall not be based on a yield strength of
reinforcementfy in excess of 80,000 psi, except for prestressing tendons.10.2.2 - Strain in reinforcement and concrete shall be assumed directly proportional to
the distance from the neutral axis, except, for deep flexural members with overall depth toclear span ratios greater than 2/5 for continuous spans and 4/5 for simple spans, a non-lineardistribution of strain shall be considered. See Section 10.7.
10.2.3 - Maximum usable strain at extreme concrete compression fiber shall be assumedequal to 0.003.
10.2.4- Stress in reinforcement below specified yield strength fy for grade of reinforcementused shall be taken as Es times steel strain. For strains greater than that corresponding to fy,stress in reinforcement shall be considered independent of strain and equal to fy.
10.2.5 - Tensile strength of concrete shall be neglected in flexural calculations of reinforcedconcrete, except when meeting requirements of Section 18.4.
10.2.6 - Relationship between concrete compressive stress distribution and concrete strainmay be assumed to be rectangular, trapezoidal, parabolic, or any other shape that results inprediction of strength in substantial agreement with results of comprehensive tests.
10.2.7 - Requirements of Section 10.2.5 may be considered satisfied by an equivalent rect-
angular concrete stress distribution defined by the following:10.2.7.1- Concrete stress of 0.85fc shall be assumed uniformly distributed over an equiva-lent compression zone bounded by edges of the cross section and a straight line located parallelto the neutral axis at a distance (a= 1c) from the fiber of maximum compressive strain.
10.2.7.2 - Distance c from fiber of maximum strain to the neutral axis shall be measuredin a direction perpendicular to that axis.
10.2.7.3 - Factor 1 shall be taken as 0.85 for concrete strengths fc up to and including
4,000 psi. For strengths above 4,000 psi, 1 shall be reduced continuously at a rate of 0.05 foreach 1000 psi of strength in excess of 4,000 psi, but 1 shall not be taken less than 0.65.
10.3.2 - Balanced strain conditions exist at a cross section when tension reinforcementreaches the strain corresponding to its specified yield strength fy just as concrete in compressionreaches its assumed ultimate strain of 0.003.
10.3.3- For flexural members, and for members subject to combined flexure and compres-sive axial load when the design axial load strength (Pn) is less than the smaller of (0.10f
cAg)
or (Pb), the ratio of reinforcement p provided shall not exceed 0.75 of the ratiob that wouldproduce balanced strain conditions for the section under flexure without axial load. For mem-bers with compression reinforcement, the portion ofb equalized by compression reinforcementneed not be reduced by the 0.75 factor.
10.3.4 - Compression reinforcement in conjunction with additional tension reinforcementmay be used to increase the strength of flexural members.
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Draft118 INTRODUCTION
10.5.1 - At any section of a flexural member, except as provided in Sections 10.5.2 and10.5.3, where positive reinforcement is required by analysis, the ratio provided shall not beless than that given by
min =200
fy
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Draft
Chapter 2
FLEXURE
1 This is probably the longest chapter in the notes, we shall cover in great details flexuraldesign/analysis of R/C beams starting with uncracked section to failure conditions.
1. Uncracked elastic (uneconomical)
2. cracked elastic (service stage)
3. Ultimate (failure)
2.1 Uncracked Section
h d
b
A
c
s
s
Figure 2.1: Strain Diagram Uncracked Section
2 Assuming perfect bond between steel and concrete, we have s = c Fig. 2.1
s = c fsEs
= fcEc
fs = EsEc
fc fs= nfc (2.1)
wheren is the modular ration= EsEc
3 Tensile force in steelTs=Asfs= Asnfc
4 Replace steel by an equivalent area of concrete, Fig. 2.2.
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Draft22 FLEXURE
S(n-1)A
2
S(n-1)A
2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Figure 2.2: Transformed Section
5 Homogeneous section & under bending
fc =M c
I fs= nfc (2.2)
6 Make sure that +max < ft
Example 2-1: Uncracked Section
Givenfc = 4,000 psi; ft = 475 psi;fy = 60,000 psi; M= 45 ft-k = 540,000 in-lb; As = 2.35
in2
Determinef+max,fmax, and fs
s
2
t
bA = 2.35 in
25"23"
10"
y
y
Solution:
n = 29, 000
57
4, 000= 8 (n 1)As = (8 1)(2.35) = 16.45 in2 (2.3-a)
yb =
(10)(25)( 252) + (16.45)(2)
(25)(10) + 16.45 (2.3-b)
yb = 11.8 in (2.3-c)
yt = 25 11.8 = 13.2 in (2.3-d)I =
(10)(25)3
12 + (25)(10)(13.2 12.5)2 + (16.45)(23 13.2)2 (2.3-e)
= 14, 722 in2 (2.3-f)
fcc = M c
I =
(540, 000) lb.in(13.2)in
(14, 722) in4 = 484 psi (2.3-g)
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Draft2.2 Section Cracked, Stresses Elastic 23
fct = M c
I =
(540, 000) lb.in(25 13.2) in(14, 722) in4
= 433 psi fr, fcc
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Draft24 FLEXURE
2.2.2 Working Stress Method
12 Referred to as Alternate Design Method (ACI Code Appendix A); Based on WorkingStress Design method.
13 Places a limit on stresses and uses service loads (ACI A.3).
fcc .45fcfst 20 ksi for grade 40 or 50 steelfst 24 ksi for grade 60 steel
(2.5)
14 Location of neutral axis depends on whether we are analysing or designing a section.
Review: We seek to locate the N.A by taking the first moments:
= Asbd
b(kd)(kd)
2 = nAs(d kd)
k= 2n + (n)
2 n (2.6-a)
Design: Objective is to have fc & fs preset & determine As, Fig. 2.4, and we thus seek theoptimal value of k in such a way that concrete and steel reach their respective limitssimultaneously.
cf
fs
d
kd
kd/3C
T
s
c
(1-k/3)d=jd
Figure 2.4: Desired Stress Distribution; WSD Method
cs
= kddkdc =
fcEc
s = fs
Es
fcEc
Esfs
= k1kn = EsEc
r =
fs
fc
k= nn+r (2.7)
15 Balanced design in terms of: What is the value of such that steel and concrete will bothreach their maximum allowable stress values simultaneously
C = bkd2 fcT = AsfsC = T
= Asbd
fc2bkd = bfsbd
k = nn+r
b =
n2r(n+r) (2.8-a)
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Draft2.2 Section Cracked, Stresses Elastic 25
16 Governing equations
Review Start by determining,
If < b steel reaches max. allowable value before concrete, and
M=Asfsjd (2.9)
If > b concrete reaches max. allowable value before steel and
M=fcbkd
2 jd (2.10)
or
M=1
2
fcjkbd2 =Rbd2 (2.11)
where
k=
2n + (n)2 nDesign We define
R=1
2fckj (2.12)
solve for bd2 from
bd2 =M
R (2.13)
assumeb and solve for d. Finally we can determine As from
As = bbd (2.14)
17 Summary
Review Design
b,d,As
M
M? b ,d,As?
= Asbd k= nn+r
j = 1 k3k= 2n + (n)2 n r= fsfcr= fsfc R= 12 fckjb =
n2r(n+r) b =
n2r(n+r)
< b M=Asfsjd bd2 = MR
> b M= 12 fcbkd
2j As= bbd orAs = Mfsjd
Example 2-2: Cracked Elastic Section
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Draft26 FLEXURE
Same problem as example 2.1 fc = 4,000 psi; ft = 475 psi; fy = 60,000 psi; As = 2.35 in
2
however,M is doubled to M= 90 k.ft(instead of 45).Solution:
Based on previous example, fct would be 866 psi >> fr and the solution is thus no longer
valid.The neutral axis is obtained from
= As
bd =
2.35
(10)(23)= 0.0102 (2.15-a)
n = (0.010)(8) = 0.08174 (2.15-b)
k =
2n + (n)2 n (2.15-c)=
2(0.08174) + (0.08174)2 (0.08174) = 0.33 (2.15-d)
kd = (.33)(23) = 7.6 in (2.15-e)
jd =
1 0.33
3
(23) = 20.47 in (2.15-f)
fs = MAsjd
(2.15-g)
= (90)(1, 000)(12)
(2.35)(20.47) = 22, 400 psi (2.15-h)
fc = 2M
bjkd2 (2.15-i)
= (2)(90)(12, 000)
(10)(20.47) jd
(7.6)kd
= 1, 390 psi (2.15-j)
I = (10)(7.6)3
12
+ (10)(7.6)7.62 2
+ 8(2.35)(23
7.6)2 = 5, 922 in4 (2.15-k)
Uncracked Cracked Cracked/uncracked
M k.ft 45 90 2N.A in 13.2 7.6fcc psi 485 1,390 (< .5f
c) 2.9
I in4 14,710 5,910 .4 ( 1I)fs psi 2,880 22,400 ( 7 ) in 1 4 4
Example 2-3: Working Stress Design Method; Analysis
Same problem as example 2.1fc = 4,000 psi;ft = 475 psi; fy = 60,000 psi; As = 2.35 in
2.Determine Moment capacity.Solution:
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Draft2.2 Section Cracked, Stresses Elastic 27
= As
bd =
2.35
(10)(23)=.0102 (2.16-a)
fs = 24 ksi (2.16-b)
fc = (.45)(4, 000) = 1, 800 psi (2.16-c)k =
2n + (n)2 n=
2(.0102)8 + (.0102)2 (8)(.0102) =.331 (2.16-d)
j = 1 k3
=.889 (2.16-e)
N.A. @ (.331)(23) = 7.61 in (2.16-f)
b = n
2r(n + r)=
8
(2)(13.33)(8 + 13.33)=.014> Steel reaches elastic limit(2.16-g)
M = Asfsjd = (2.35)(24)(.889)(23) = 1, 154 k.in= 96 k.ft (2.16-h)
Note, had we used the alternate equation for moment (wrong) we would have overestimatedthe design moment:
M = =1
2fcbkd
2j (2.17-a)
= 1
2(1.8)(10)(0.33)(0.89)(23)2 = 1, 397 k.in> 1, 154 k.in (2.17-b)
If we define c = fc/1, 800 and s = fs/24, 000, then as the load increases both c and sincrease, but at different rates, one of them s reaches 1 before the other.
Load
1
s c
Example 2-4: Working Stress Design Method; Design
Design a beam to carry LL= 1.9 k/ft, DL = 1.0 k/ft with fc = 4, 000 psi, fy = 60, 000 psi,L= 32 ft.Solution:
fc = (.45)(4, 000) = 1, 800 psi (2.18-a)
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Draft28 FLEXURE
fs = 24, 000 psi (2.18-b)
n = Es
Ec=
29, 000
57
4, 000= 8 (2.18-c)
r = fs
fc=
24
1.8= 13.33 (2.18-d)
k = nn + r
= 88 + 13.33
=.375 (2.18-e)
j = 1 d3
= 1 .3753
=.875 (2.18-f)
R = 1
2fckj =
1
2(1, 800)(.375)(.875) = 295 psi (2.18-g)
b = n
2r(n + r)=
8
2(13.33)(8 + 13.33)=.01405 (2.18-h)
Estimate beam weight at .5 k/ft, thus
M = [(1.9) + (1.0 + .5)] (32)2
8 = 435k.ft (2.19-a)
bd2 = M
R =
435 k.ft in2(12, 000) lb.in
(295) lbs ft k= 17, 700 in3 (2.19-b)
Takeb= 18 in & d= 31.4 in h= 36 inCheck beam weight (18)(36)145 (.15)
in2 ft2
in2k
ft3 =.675 k/ft
As = (.01405)(18)(31.4) = 7.94 in2 use 8# 9 bars in 2 layersAs = 8.00 in2
2.3 Cracked Section, Ultimate Strength Design Method
2.3.1 Whitney Stress Block
a/2 = c
c
fs
fs
h d
b
As
c
c
Actual
c =
a
1
fc
c
a
C= fcb C= fab
Figure 2.5: Cracked Section, Limit State
Figure
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Draft2.3 Cracked Section, Ultimate Strength Design Method 29
18 At failure we have, linear cross strain distribution (ACI 10.2.2) (except for deep beams),non-linear stress strain curve for the concrete, thus a non-linear stress distribution.
19 Two options:
1. Analytical expression of
exact integration
2. Replace exact stress diagram with a simpler and equivalent one, (ACI 10.2.6)
Second approach adopted by most codes.
20 For the equivalent stress distribution, all we we need to know is C& its location, thus andWe adopt a rectangular stress, with depth a = 1c, and stress equal to f
c (ACI 10.2.7.1)
C = fcbc= fcab (2.20-a)
= fav
fc(2.20-b)
a = 1c (2.20-c)
Thus=
1(2.21)
But the location of the resultant forces must be the same, hence
1= 2 (2.22)
21 From Experiments
fc ( psi)
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Draft210 FLEXURE
C=0.85fabc
h d
b
A
s
s
0.85 fc
c
d
T
=0.003u
a= c1
Figure 2.6: Whitney Stress Block
2.3.2 Balanced Design
Tension Failure:
fs = fyA
sf
s = .85f
cab= .85f
cb
1c
= Asbd c= fy.85fc1 d (2.24-a)
Compression Failure:
c = .003 (2.25-a)
s = fs
Es(2.25-b)
c
d =
.003
.003 + s c= .003fs
Es+.003
d (2.25-c)
Balanced Design:
24 Balanced design occurs if we have simultaneous yielding of the steel and crushing of theconcrete. Hence, we simply equate the previous two equations
fy.85fc1
d = .003fsEs
+.003d
= b
bf2d
.85fc1= .003fs
Es+.003
d
Es = 29, 000 ksi
b = .851
fcfy
87,00087,000+fy
(ACI 8.4.3)(2.26-a)
25 To ensure failure by yielding,
< .75b (2.27)
26 ACI strength requirements
U = 1.4D+ 1.7L (ACI 9.2.1)U = 0.75(1.4D+ 1.7L + 1.7W) (ACI 9.2.2)
Md= Mu = Mn (ACI 9.1.1) = .90 (ACI 9.3.2.2)
(2.28)
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Draft2.3 Cracked Section, Ultimate Strength Design Method 211
27 Also we need to specify a minimum reinforcement ratio
min 200fy
(ACI 10.5.1) (2.29)
to account for temperature & shrinkage
28 Note, that need not be as high as 0.75b. If steel is relatively expensive, or deflection is ofconcern, can use lower .
29 As a rule of thumb, if b is not allowed by code, in this case we have an extra unknown fs.
31 We now have one more unknown fs, and we will need an additional equation (from strain
diagram).
c = Asfs.85fcb1Fx= 0
cd =
.003.003+s
From strain diagram
Md = Asfs(d 1c2 ) M= 0(2.32)
We can solve by iteration, or substitution and solution of a quadratic equation.
2.3.4 Design
32 We consider two cases:
I b d and As, unknown; Md known;
Fx= 0 a = Asfy
0.85fcb
= Asbd
a =
fy0.85fc
Md = Asfy
d a2
Md= fy
1 .59 fy
fc
R
bd2(2.33-a)
or
R= fy
1 .59 fy
fc
(2.34)
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Draft212 FLEXURE
which does not depend on unknown quantities. Then solve for bd2:
bd2 = MdR
(2.35)
Solve for b and d (this will require either an assumption on one of the two, or on theirratio).
As = bd
II b & d known & Md knownthere is no assurance that we can have a design with bIf the section is too small, then it will require too much steel resulting in an over-reinforcedsection.
Iterative approach
(a) Since we do not know if the steel will be yielding or not, use fs.
(b) Assume an initial value fora (a good start is a = d5 )
(c) Assume initially thatfs = fy(d) Check equilibrium of moments (M= 0)
As= Md
fs
d a2 (2.36)
(e) Check equilibrium of forces in the x direction (Fx= 0)
a= Asfs.85fcb
(2.37)
(f) Check assumption offs by either comparing with b, or from the strain diagram
sd c = .003c fs = Es d cc .003< fy (2.38)
wherec = a1 .
(g) Iterate until convergence is reached.
2.4 Practical Design Considerations
2.4.1 Minimum Depth
33 ACI 9.5.2.1 stipulates that the minimum thickness of beams should be
Simply One end Both ends Cantileversupported continuous continuous
Solid Oneway slab L/20 L/24 L/28 L/10
Beams orribbed One way slab L/16 L/18.5 L/21 L/8
where L is in inches, and members are not supporting partitions.
34 Smaller values can be taken if deflections are computed.
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Draft2.4 Practical Design Considerations 213
2.4.2 Beam Sizes, Bar Spacing, Concrete Cover
35 Beam sizes should be dimensioned as
1. Use whole inches for overall dimensions, except for slabs use 12 inch increment.
2. Ideally, the overall depth to width ratio should be between 1.5 to 2.0 (most economical).3. For T beams, flange thickness should be about 20% of overall depth.
36 Reinforcing bars
1. Minimum spacing between bars, and minimum covers are needed to
(a) Prevent Honeycombing of concrete (air pockets)
(b) Concrete (usually up to 3/4 in MSA) must pass through the reinforcement
(c) Protect reinforcement against corrosion and fire
2. Use at least 2 bars for flexural reinforcement
3. Use bars #11 or smaller for beams.
4. Use no more than two bar sizes and no more than 2 standard sizes apart (i.e #7 and #9acceptable; #7 and #8 or #7 and #10 not).
5. Use no more than 5 or 6 bars in one layer.
6. Place longest bars in the layer nearest to face of beam.
7. Clear distance between parallel bars not less that db (to avoid splitting cracks) nor 1 in.(to allow concrete to pass through).
8. Clear distance between longitudinal bars in columns not less that 1.5db or 1.5 in.
9. Minimum cover of 1.5 in.
10. Summaries in Fig. 2.7 and Table 2.1, 2.2.
2.4.3 Design Aids
37 Basic equations developed in this section can be easily graphed.
Review Givenb dand known steel ratio and material strength,Mncan be readily obtainedfromMn = Rbd
2
Design in this case
1. Set Md=Rbd22. From tabulated values, selectmax andmin often 0.5b is a good economical choice.
3. SelectR from tabulated values ofR in terms offy, fc and . Solve for bd
2.
4. Selectb and d to meet requirements. Usually depth is about 2 to 3 times the width.
5. Using tabulated values select the size and number of bars giving preference to largerbar sizes to reduce placement cost (careful about crack width!).
6. Check from tables that the selected beam width will provide room for the bars chosenwith adequate cover and spacing.
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Draft214 FLEXURE
Bar Nominal Number of Bars
Size Diam. 1 2 3 4 5 6 7 8 9 10#3 0.375 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.10#4 0.500 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00#5 0.625 0.31 0.62 0.93 1.24 1.55 1.86 2.17 2.48 2.79 3.10#6 0.750 0.44 0.88 1.32 1.76 2.20 2.64 3.08 3.52 3.96 4.40#7 0.875 0.60 1.20 1.80 2.40 3.00 3.60 4.20 4.80 5.40 6.00#8 1.000 0.79 1.58 2.37 3.16 3.95 4.74 5.53 6.32 7.11 7.90#9 1.128 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
#10 1.270 1.27 2.54 3.81 5.08 6.35 7.62 8.89 10.16 11.43 12.70#11 1.410 1.56 3.12 4.68 6.24 7.80 9.36 10.92 12.48 14.04 15.60#14 1.693 2.25 4.50 6.75 9.00 11.25 13.50 15.75 18.00 20.25 22.50
#18 2.257 4.00 8.00 12.00 16.00 20.00 24.00 28.00 32.00 36.00 40.00
Table 2.1: Total areas for various numbers of reinforcing bars (inch2)
Bar Number of bars in single layer of reinf.Size 2 3 4 5 6 7 8
#4 6.8 8.3 9.8 11.3 12.8 14.3 15.8#5 6.9 8.5 10.2 11.8 13.4 15.1 16.7#6 7.0 8.8 10.5 12.3 14.0 15.8 17.5#7 7.2 9.1 11.0 12.8 14.7 16.6 18.5#8 7.3 9.3 11.3 13.3 15.3 17.3 19.3#9 7.6 9.9 12.1 14.4 16.6 18.9 21.2
#10 7.8 10.3 12.9 15.4 18.0 20.5 23.0#11 8.1 10.9 13.7 16.6 19.4 22.2 25.0
#14 8.9 12.3 15.7 19.1 22.5 25.9 29.3#18 10.6 15.1 19.6 24.1 28.6 33.2 37.7
Table 2.2: Minimum Width (inches) according to ACI Code
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Draft2.5 USD Examples 215
Figure 2.7: Bar Spacing
2.5 USD Examples
Example 2-5: Ultimate Strength; Review
Determine the ultimate moment capacity of example 2.1 fc = 4,000 psi; ft = 475 psi; fy =
60,000 psi; As = 2.35 in2
s
2
t
bA = 2.35 in
25"23"
10"
y
y
Solution:
act = As
bd =
2.35
(10)(23)=.0102 (2.39-a)
b = .851fc
fy
87
87 + fy
= (.85)(.85)4
60
87
87 + 60
=.0285> act
(2.39-b)
a = Asfy
.85fcb=
(2.35)(60)
(.85)(4)(10)= 4.15 in (2.39-c)
Mn = Asfy
d a2
= (2.35)(60)
23 4.15
2
= 2, 950 k.in (2.39-d)
Md = Mn = 0.9(2, 950) = 2, 660 k.in (2.39-e)
Note:
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Draft216 FLEXURE
1. From equilibrium, Fx= 0 c= Asfy.851bfc = (2.35)(60)
(.85)(.85)(4)(10) = 4.87 in
2. Comparing with previous analysis
uncracked cracked ultimate
c 13.2 7.61 4.87M 45 90 2451.7 = 144
3. Alternative solution:
Mn = actfybd2(1 .59act fy
fc) (2.40-a)
= Asfyd(1 59act fyfc
) (2.40-b)
= (2.35)(60)(23)[1 (.59) 604
(.0102)] = 2, 950 k.in= 245 k.ft (2.40-c)
Md = Mn = (.9)(2, 950) = 2, 660 k.in (2.40-d)
Example 2-6: Ultimate Strength; Design I
Design a R/C beam with L= 15 ft; DL = 1.27 k/ft; LL = 2.44 k/ft; fc = 3,000 psi; fy =40 ksi; Neglect beam owns weight; Select = 0.75b
Solution:
wu = 1.4(1.27) + 1.7(2.44) = 5.92 k/ft Factored load (2.41-a)
Md = wuL
2
8 =
(5.92)(15)2
8 = 166.5 k.ft(12) in/ft= 1, 998 k.in (2.41-b)
= 0.75b = (0.75)(0.85)1fcfy
87
87 + fy(2.41-c)
= (0.75)(.85)23
40
87
87 + 40=.0278 (2.41-d)
R = fy1 .59 fy
fc (2.41-e)
= (.0278)(40)
1 (0.59)(.0278) 40
3
= 0.869 psi (2.41-f)
bd2 = Md
R =
1, 998
(0.9)(0.869)= 2, 555 in3 (2.41-g)
Takeb= 10 in,d= 16 in As = (.0278)(10)(16) = 4.45 in2 use 3 # 11
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Draft2.6 T Beams, (ACI 8.10) 217
Example 2-7: Ultimate Strength; Design II
Design a R/C beam for b= 11.5 in; d = 20 in; fc = 3 ksi; fy = 40 ksi;Md= 1, 600 k.inSolution:
Assumea= d5 = 20
5 = 4 in
As = Md
fy(d a2 )=
(1, 600)
(.9)(40)(20 42 )= 2.47 in2 (2.42)
check assumption,
a= Asfy(.85)fcb
= (2.47)(40)
(.85)(3)(11.5)= 3.38 in (2.43)
Thus take a= 3.3 in.
As = (1, 600)
(.9)(40)(20 3.32 )= 2.42 in2 (2.44-a)
a = (2.42)(40)(.85)(3)(11.5)
= 3.3 in
(2.44-b)
act = 2.42
(11.5)(20)=.011 (2.44-c)
b = (.85)(.85)3
40
87
87 + 40=.037 (2.44-d)
max = .75b = .0278> act
(2.44-e)
2.6 T Beams, (ACI 8.10)
38 Equivalent width for uniform stress, Fig. 2.8 must satisfy the following requirements (ACI8.10.2):
1. 12 (b bw) 8hf2. b bw
2
4. b < L4
39 Two possibilities:
1. Neutral axis within the flanges (c < hf)rectangular section of width b, Fig. 2.9.2. Neutral axis in the web (c > hf )T beam.
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Draft218 FLEXURE
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1
fh
b
wb
eb
Figure 2.8: T Beams
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
h d
As
b
fh
Figure 2.9: T Beam as Rectangular Section
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
As
s
0.85 fcu
1
s y
f
w
b
h
dh
b
=0.003
a= c
T=A f
d
cC=0.85fa
c
Figure 2.10: T Beam Strain and Stress Diagram
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Draft2.6 T Beams, (ACI 8.10) 219
40 For T beams, we have a large