Machanics and design of reinforced concrets

Draft DRAFT

Lecture Notes in:

Mechanics and Design of REINFORCED CONCRETE CVEN4555

c VICTOR


E. SAOUMA,

Fall 2002

Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-0428 December 26, 2002

Draft 0–2

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft Contents 1 INTRODUCTION 1.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Concrete . . . . . . . . . . . . . . . . . . . . 1.1.1.1 Mix Design . . . . . . . . . . . . . . 1.1.1.1.1 Constituents . . . . . . . . 1.1.1.1.2 Preliminary Considerations 1.1.1.1.3 Mix procedure . . . . . . . 1.1.1.1.4 Mix Design Example . . . 1.1.1.2 Mechanical Properties . . . . . . . . 1.1.2 Reinforcing Steel . . . . . . . . . . . . . . . . 1.2 Design Philosophy, USD . . . . . . . . . . . . . . . . 1.3 Analysis vs Design . . . . . . . . . . . . . . . . . . . 1.4 Basic Relations and Assumptions . . . . . . . . . . . 1.5 ACI Code . . . . . . . . . . . . . . . . . . . . . . . . 2 FLEXURE 2.1 Uncracked Section . . . . . . . . . . . . . . . . . . . E 2-1 Uncracked Section . . . . . . . . . . . . . . . 2.2 Section Cracked, Stresses Elastic . . . . . . . . . . . 2.2.1 Basic Relations . . . . . . . . . . . . . . . . . 2.2.2 Working Stress Method . . . . . . . . . . . . E 2-2 Cracked Elastic Section . . . . . . . . . . . . E 2-3 Working Stress Design Method; Analysis . . . E 2-4 Working Stress Design Method; Design . . . 2.3 Cracked Section, Ultimate Strength Design Method . 2.3.1 Whitney Stress Block . . . . . . . . . . . . . 2.3.2 Balanced Design . . . . . . . . . . . . . . . . 2.3.3 Review . . . . . . . . . . . . . . . . . . . . . 2.3.4 Design . . . . . . . . . . . . . . . . . . . . . . 2.4 Practical Design Considerations . . . . . . . . . . . . 2.4.1 Minimum Depth . . . . . . . . . . . . . . . . 2.4.2 Beam Sizes, Bar Spacing, Concrete Cover . . 2.4.3 Design Aids . . . . . . . . . . . . . . . . . . . 2.5 USD Examples . . . . . . . . . . . . . . . . . . . . . E 2-5 Ultimate Strength; Review . . . . . . . . . . E 2-6 Ultimate Strength; Design I . . . . . . . . . . E 2-7 Ultimate Strength; Design II . . . . . . . . .

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1–1 . 1–1 . 1–1 . 1–1 . 1–1 . 1–5 . 1–5 . 1–8 . 1–9 . 1–13 . 1–14 . 1–15 . 1–16 . 1–16

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2–1 . 2–1 . 2–2 . 2–3 . 2–3 . 2–4 . 2–5 . 2–6 . 2–7 . 2–8 . 2–8 . 2–10 . 2–11 . 2–11 . 2–12 . 2–12 . 2–13 . 2–13 . 2–15 . 2–15 . 2–16 . 2–17

Draft 0–2

2.6

2.7

2.8 2.9

CONTENTS

E 2-8 Exact Analysis . . . . . . . . . . . T Beams, (ACI 8.10) . . . . . . . . . . . 2.6.1 Review . . . . . . . . . . . . . . . 2.6.2 Design, (balanced) . . . . . . . . . E 2-9 T Beam; Moment Capacity I . . . E 2-10 T Beam; Moment Capacity II . . . E 2-11 T Beam; Design . . . . . . . . . . Doubly Reinforced Rectangular Beams . . 2.7.1 Tests for fs and fs
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3 SHEAR 3.1 Introduction . . . . . . . . . . . . . . 3.2 Shear Strength of Uncracked Section 3.3 Shear Strength of Cracked Sections . 3.4 ACI Code Requirements . . . . . . . 3.5 Examples . . . . . . . . . . . . . . . E 3-1 Shear Design . . . . . . . . . 3.6 Shear Friction . . . . . . . . . . . . . E 3-2 Shear Friction . . . . . . . . . 3.7 Brackets and Corbels . . . . . . . . . 3.8 Deep Beams . . . . . . . . . . . . . .

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. 2–17 . 2–20 . 2–21 . 2–22 . 2–22 . 2–23 . 2–24 . 2–26 . 2–27 . 2–29 . 2–30 . 2–32 . 2–33 . 2–35 . 2–39

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3–1 . 3–1 . 3–2 . 3–5 . 3–6 . 3–8 . 3–8 . 3–9 . 3–11 . 3–12 . 3–12

4 CONTINUOUS BEAMS 4.1 Continuity . . . . . . . . . . . . . . . . . 4.2 Methods of Analysis . . . . . . . . . . . 4.2.1 Detailed Analysis . . . . . . . . . 4.2.2 ACI Approximate Method . . . . 4.3 Effective Span Design Moment . . . . . 4.4 Moment Redistribution . . . . . . . . . 4.4.1 Elastic-Perfectly Plastic Section . 4.4.2 Concrete . . . . . . . . . . . . . E 4-1 Moment Redistribution . . . . . 4.5 Buildings . . . . . . . . . . . . . . . . .

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4–1 4–1 4–2 4–2 4–2 4–4 4–4 4–4 4–6 4–6 4–7

5 ONE WAY SLABS 5–1 5.1 Types of Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–1 5.2 One Way Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–4 5.3 Design of a One Way Continuous Slab . . . . . . . . . . . . . . . . . . . . . . . . 5–5

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft CONTENTS

6 SERVICEABILITY 6.1 Control of Cracking . . . . . E 6-1 Crack Width . . . . . 6.2 Deflections . . . . . . . . . . 6.2.1 Short Term Deflection 6.2.2 Long Term Deflection E 6-2 Deflections . . . . . .

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7 APPROXIMATE FRAME ANALYSIS 7.1 Vertical Loads . . . . . . . . . . . . . . . . . . . . . . . 7.2 Horizontal Loads . . . . . . . . . . . . . . . . . . . . . 7.2.1 Portal Method . . . . . . . . . . . . . . . . . . E 7-1 Approximate Analysis of a Frame subjected to Loads . . . . . . . . . . . . . . . . . . . . . . .

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6–1 6–1 6–3 6–3 6–4 6–5 6–7

7–1 . . . . . . . . . . 7–1 . . . . . . . . . . 7–4 . . . . . . . . . . 7–4 and Horizontal . . . . . . . . . . 7–6

8 COLUMNS

8–1

9 COLUMNS 9.1 Introduction . . . . . . . . . . . . . . . . . 9.1.1 Types of Columns . . . . . . . . . 9.1.2 Possible Arrangement of Bars . . . 9.2 Short Columns . . . . . . . . . . . . . . . 9.2.1 Concentric Loading . . . . . . . . . 9.2.2 Eccentric Columns . . . . . . . . . 9.2.2.1 Balanced Condition . . . 9.2.2.2 Tension Failure . . . . . . 9.2.2.3 Compression Failure . . . 9.2.3 ACI Provisions . . . . . . . . . . . 9.2.4 Interaction Diagrams . . . . . . . . 9.2.5 Design Charts . . . . . . . . . . . E 9-1 R/C Column, c known . . . . . . . E 9-2 R/C Column, e known . . . . . . . E 9-3 R/C Column, Using Design Charts 9.2.6 Biaxial Bending . . . . . . . . . . E 9-4 Biaxially Loaded Column . . . . . 9.3 Long Columns . . . . . . . . . . . . . . . 9.3.1 Euler Elastic Buckling . . . . . . . 9.3.2 Effective Length . . . . . . . . . . 9.3.3 Moment Magnification Factor; ACI E 9-5 Long R/C Column . . . . . . . . . E 9-6 Design of Slender Column . . . . .

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9–1 . 9–1 . 9–1 . 9–2 . 9–2 . 9–2 . 9–2 . 9–3 . 9–5 . 9–6 . 9–7 . 9–7 . 9–7 . 9–7 . 9–9 . 9–13 . 9–14 . 9–17 . 9–18 . 9–18 . 9–19 . 9–21 . 9–24 . 9–25

10 PRESTRESSED CONCRETE 10.1 Introduction . . . . . . . . . . 10.1.1 Materials . . . . . . . 10.1.2 Prestressing Forces . . 10.1.3 Assumptions . . . . . 10.1.4 Tendon Configuration

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10–1 . 10–1 . 10–1 . 10–4 . 10–4 . 10–4

Victor Saouma

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Mechanics and Design of Reinforced Concrete

Draft 0–4

CONTENTS

10.1.5 Equivalent Load . . . . . . . 10.1.6 Load Deformation . . . . . . 10.2 Flexural Stresses . . . . . . . . . . . E 10-1 Prestressed Concrete I Beam 10.3 Case Study: Walnut Lane Bridge . . 10.3.1 Cross-Section Properties . . . 10.3.2 Prestressing . . . . . . . . . . 10.3.3 Loads . . . . . . . . . . . . . 10.3.4 Flexural Stresses . . . . . . .

Victor Saouma

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. 10–4 . 10–4 . 10–6 . 10–8 . 10–10 . 10–12 . 10–12 . 10–13 . 10–13

Mechanics and Design of Reinforced Concrete

Draft List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7

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. 1–2 . 1–10 . 1–11 . 1–11 . 1–11 . 1–12 . 1–13

Strain Diagram Uncracked Section . . . . . . . . . . . . . . . . . . . . . . . . . Transformed Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Diagram Cracked Elastic Section . . . . . . . . . . . . . . . . . . . . . . Desired Stress Distribution; WSD Method . . . . . . . . . . . . . . . . . . . . . Cracked Section, Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . Whitney Stress Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bar Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T Beam as Rectangular Section . . . . . . . . . . . . . . . . . . . . . . . . . . . T Beam Strain and Stress Diagram . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition of Steel Reinforcement for T Beams . . . . . . . . . . . . . . . . Doubly Reinforced Beams; Strain and Stress Diagrams . . . . . . . . . . . . . . Different Possibilities for Doubly Reinforced Concrete Beams . . . . . . . . . . Strain Diagram, Doubly Reinforced Beam; is As Yielding? . . . . . . . . . . . . Strain Diagram, Doubly Reinforced Beam; is A
s Yielding? . . . . . . . . . . . . Summary of Conditions for top and Bottom Steel Yielding . . . . . . . . . . . . Bending of a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moment-Curvature Relation for a Beam . . . . . . . . . . . . . . . . . . . . . . Bond and Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . Actual Bond Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Splitting Along Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bar cutoff requirements of the ACI code . . . . . . . . . . . . . . . . . . . . . . Standard cutoff or bend points for bars in approximately equal spans with uniformly distributed load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.27 Moment Capacity Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 2–1 . 2–2 . 2–3 . 2–4 . 2–8 . 2–10 . 2–15 . 2–20 . 2–20 . 2–21 . 2–21 . 2–26 . 2–27 . 2–27 . 2–28 . 2–29 . 2–34 . 2–35 . 2–36 . 2–37 . 2–37 . 2–38 . 2–38 . 2–40 . 2–41

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26

3.1

Schematic Representation of Aggregate Gradation MicroCracks in Concrete under Compression . . . Concrete Stress Strain Curve . . . . . . . . . . . . Modulus of Rupture Test . . . . . . . . . . . . . . Split Cylinder (Brazilian) Test . . . . . . . . . . . Biaxial Strength of Concrete . . . . . . . . . . . . Time Dependent Strains in Concrete . . . . . . . .

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Principal Stresses in Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–1

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LIST OF FIGURES

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Types of Shear Cracks . . . . . . . . . . . . . . . . . . . . . . . . . Shear Strength of Uncracked Section . . . . . . . . . . . . . . . . . Mohr’s Circle for Shear Strength of Uncracked Section . . . . . . . Shear Strength of Uncracked Section . . . . . . . . . . . . . . . . . Free Body Diagram of a R/C Section with a Flexural Shear Crack Equilibrium of Shear Forces in Cracked Section . . . . . . . . . . . Summary of ACI Code Requirements for Shear . . . . . . . . . . . Corbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear Friction Mechanism . . . . . . . . . . . . . . . . . . . . . . . Shear Friction Across Inclined Reinforcement . . . . . . . . . . . .

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. 3–1 . 3–2 . 3–3 . 3–4 . 3–5 . 3–6 . 3–7 . 3–9 . 3–10 . 3–10

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Continuous R/C Structures . . . . . . . . . . . . . . . . . . . . . . Load Positioning on Continuous Beams . . . . . . . . . . . . . . . ACI Approximate Moment Coefficients . . . . . . . . . . . . . . . . Design Negative Moment . . . . . . . . . . . . . . . . . . . . . . . Moment Diagram of a Rigidly Connected Uniformly Loaded Beam Moment Curvature of an Elastic-Plastic Section . . . . . . . . . . . Plastic Moments in Uniformly Loaded Rigidly Connected Beam . . Plastic Redistribution in Concrete Sections . . . . . . . . . . . . . Block Diagram for R/C Building Design . . . . . . . . . . . . . . .

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4–1 4–1 4–3 4–4 4–5 4–5 4–5 4–6 4–8

5.1 5.2 5.3 5.4

Types of Slabs . . . . . . . . . One vs Two way slabs . . . . . Load Distribution in Slabs . . . Load Transfer in R/C Buildings

6.1 6.2 6.3 6.4 6.5

Crack Width Equation Parameters . Uncracked Transformed and Cracked Time Dependent Deflection . . . . . Time Dependent Strain Distribution Short and long Term Deflections . .

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14

Approximate Analysis of Frames Subjected to Vertical Loads; Girder Moments . 7–2 Approximate Analysis of Frames Subjected to Vertical Loads; Column Axial Forces7–3 Approximate Analysis of Frames Subjected to Vertical Loads; Column Moments 7–3 Approximate Analysis of Frames Subjected to Lateral Loads; Column Shear . . . 7–5 Approximate Analysis of Frames Subjected to Lateral Loads; Girder Moment . . 7–5 Approximate Analysis of Frames Subjected to Lateral Loads; Column Axial Force7–6 Example; Approximate Analysis of a Building . . . . . . . . . . . . . . . . . . . . 7–7 Approximate Analysis of a Building; Moments Due to Vertical Loads . . . . . . . 7–9 Approximate Analysis of a Building; Shears Due to Vertical Loads . . . . . . . . 7–10 Approximate Analysis for Vertical Loads; Spread-Sheet Format . . . . . . . . . . 7–12 Approximate Analysis for Vertical Loads; Equations in Spread-Sheet . . . . . . . 7–13 Approximate Analysis of a Building; Moments Due to Lateral Loads . . . . . . . 7–14 Portal Method; Spread-Sheet Format . . . . . . . . . . . . . . . . . . . . . . . . . 7–16 Portal Method; Equations in Spread-Sheet . . . . . . . . . . . . . . . . . . . . . . 7–17

9.1 9.2

Types of columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–1 Tied vs Spiral Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9–1

Victor Saouma

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5–1 5–2 5–2 5–3

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6–2 6–4 6–5 6–6 6–6

Mechanics and Design of Reinforced Concrete

Draft

LIST OF FIGURES 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18

0–3

Possible Bar arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources of Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Load Moment Interaction Diagram . . . . . . . . . . . . . . . . . . . . . . . . . Strain and Stress Diagram of a R/C Column . . . . . . . . . . . . . . . . . . . Column Interaction Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure Surface of a Biaxially Loaded Column . . . . . . . . . . . . . . . . . . . Load Contour at Plane of Constant Pn , and Nondimensionalized Corresponding plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biaxial Bending Interaction Relations in terms of β . . . . . . . . . . . . . . . . Bilinear Approximation for Load Contour Design of Biaxially Loaded Columns Euler Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Column Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical lengths of columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective length Factors Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Alignment Chart (ACI) . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Column Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . P-M Magnification Interaction Diagram . . . . . . . . . . . . . . . . . . . . . .

. 9–2 . 9–3 . 9–3 . 9–4 . 9–8 . 9–14 . 9–15 . 9–16 . 9–16 . 9–18 . 9–19 . 9–20 . 9–21 . 9–22 . 9–22 . 9–23

10.1 10.2 10.3 10.4 10.5 10.6

Pretensioned Prestressed Concrete Beam, (Nilson 1978) . . . . . . . . . . . . . . 10–2 Posttensioned Prestressed Concrete Beam, (Nilson 1978) . . . . . . . . . . . . . . 10–2 7 Wire Prestressing Tendon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–3 Alternative Schemes for Prestressing a Rectangular Concrete Beam, (Nilson 1978)10–5 Determination of Equivalent Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 10–5 Load-Deflection Curve and Corresponding Internal Flexural Stresses for a Typical Prestressed Concrete Beam, (Nilson 1978) . . . . . . . . . . . . . . . . . . . . 10–6 10.7 Flexural Stress Distribution for a Beam with Variable Eccentricity; Maximum Moment Section and Support Section, (Nilson 1978) . . . . . . . . . . . . . . . . 10–7 10.8 Walnut Lane Bridge, Plan View . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10–11 10.9 Walnut Lane Bridge, Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 10–12

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 0–4

Victor Saouma

LIST OF FIGURES

Mechanics and Design of Reinforced Concrete

Draft List of Tables 1.1 1.2 1.3 1.4 1.5 1.6

ASTM Sieve Designation’s Nominal Sizes Used for Concrete Aggregates . . . . ASTM C33 Grading Limits for Coarse Concrete Aggregates . . . . . . . . . . . ASTM C33 Grading Limits for Fine Concrete Aggregates . . . . . . . . . . . . Example of Fineness Modulus Determination for Fine Aggregate . . . . . . . . Recommended Slumps (inches) for Various Types of Construction . . . . . . . Recommended Average Total Air Content as % of Different Nominal Maximum Sizes of Aggregates and Levels of Exposure . . . . . . . . . . . . . . . . . . . . 1.7 Approximate Mixing Water Requirements, lb/yd3 of Concrete For Different Slumps and Nominal Maximum Sizes of Aggregates . . . . . . . . . . . . . . . . 1.8 Relationship Between Water/Cement Ratio and Compressive Strength . . . . . 1.9 Volume of Dry-Rodded Coarse Aggregate per Unit Volume of Concrete for Different Fineness Moduli of Sand . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Creep Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Properties of Reinforcing Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Strength Reduction Factors, Φ . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

1–3 1–3 1–3 1–5 1–6

. 1–6 . 1–7 . 1–7 . 1–8 . 1–13 . 1–14 . 1–14

2.1 2.2

Total areas for various numbers of reinforcing bars (inch2 ) . . . . . . . . . . . . . 2–14 Minimum Width (inches) according to ACI Code . . . . . . . . . . . . . . . . . . 2–14

4.1

Building Structural Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4–7

5.1

Recommended Minimum Slab and Beam Depths . . . . . . . . . . . . . . . . . . 5–4

7.1 7.2

Columns Combined Approximate Vertical and Horizontal Loads . . . . . . . . . 7–18 Girders Combined Approximate Vertical and Horizontal Loads . . . . . . . . . . 7–19

Draft 0–2

Victor Saouma

LIST OF TABLES

Mechanics and Design of Reinforced Concrete

Draft 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 1.1.1.1.1

Mix Design Constituents

Concrete is a mixture of Portland cement, water, and aggregates (usually sand and crushed stone).

1

Portland cement is a mixture of calcareous and argillaceous materials which are calcined in a kiln and then pulverized. When mixed with water, cement hardens through a process called hydration.

2

3

Ideal mixture is one in which: 1. A minimum amount of cement-water paste is used to fill the interstices between the particles 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 remaining 1/4 being the cement paste. Smaller particles up to 1/4 in. in size are called fine aggregates, and the larger ones being coarse aggregates.

4

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)

Draft 1–2

INTRODUCTION

0000 1111 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 111 1111 0000 1111 0000 1111 000 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 1111 0000111 000 0000 1111 000 111 0000 1111 0000 1111 1111 0000 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 0000 1111 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 000 111 0000 1111 000 111 0000 1111 0000 1111 0000 1111 000 111 000 111 0000 1111 0000 1111 0000 1111 000 111 000 111 0000 1111 0000 1111 000 111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 000 111 0000 111 1111 000 111 0000 1111 000 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 0001111 111 0000 1111 0000 111 1111 000 0000 0000 1111 000 0000 1111 0000 111 1111 000 111 0000 0000 1111 0001111 111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 1111 0000 0000 0000 1111 0000 1111 1111 0000 1111 0000 1111 1111 0000 0000 1111 0000 1111 0000 0000 1111 1111

11 00 00 11 000 111 000 111 000 111 00 11 00 000 11 111 00 11 000 000 111 000 111 00 11 00111 11 000 111 00 11 000 111 000 111 111 0001111 000 111 000 111 0000 000 111 00 11 0000 1111 000 111 0000 1111 0000 1111 00 11 00 11 00011 111 00 0000 1111 0000 11 1111 00 0000 1111 0000 1111 00 11 00 11 0000 00 11 0000 111 1111 000 000 00111 11 001111 11 00 11 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 111 000 000 111 000 111 00 11 000 111 000 111 000 111 000 111 000 111 00 11 000 000 111 111 0000 1111 000 111 00 11 0000111 1111 000 111 0000 1111 000 111 000 0000 1111 000 111 0000 1111 000 111 000 111 0000 1111 000 111 000 111 0000 1111 000 111 0000111 1111 000 111 0000 1111 000 000 111 0000 1111 000 111 000 111 111 000 0000 1111 0000 1111 000 111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 0000 1111 0000 111 1111 000 111 0000 1111 0000 1111 000 0000 1111 0000 1111 000 111 0000 1111 0000 1111 000 000 111 0000 111 1111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

111 000 00 11 000 111 000 111 00 11 000 111 000 111 00 000 11 111 00 11 00 11 00 11 00 11 00 11 00 11 000 111 00 11 000 111 00 11

111 000 000 111 0 1 0000 1111 000 111 00 11 0 1 0000 000 111 000 111 0 1111 1 000 111 00 11 00 11 0000 1111 0 1 000 111 0000 1111 0 1 000 111 000 111 0 1 000 111 00 11 0000 1111 0 1 0000 1111 000 111 00 11 0000 1111 0 1 0000 1111 0000 1111 00 11 0000 1111 00 11 0 1 00 11 0000 1111 0000 1111 00 11 0 1 0000 1111 0000 1111 000 111 00 11 0000 1111 00 11 0 1 00 11 0000 1111 0000 1111 00 11 0000 1111 0000 1111 0 1 000 111 00 11 0000 1111 00 11 0 1 0000 1111 0000 1111 0 1 0000 1111 0 1 000 111 00 11 00 11 0000 1111 0 1 00 11 0 1 0 1 0000 1111 0 1 0000 1111 0 1 00 1 11 0 0 1 00001 1111 0 0000 1111 000 111 0000 1111 0000 1111 000 111 0000 1111 0 1 0 1 000 111 11 00 00 000 111 0 11 1 0000 1111 0 1 0 1 11 00 000 111 0 1 0 111 1 000 111 0000 1111 00 11 000 0 1 0000 1111 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11 0000 1111 00 11 0000 1111 0 1 0000 1111 0000 1111 000 111 0000 1111 0000 1111 0 1 0000 1111 0000 1111 000 00 11 0000 1111 0000 1111 00111 11 00 11 0000 1111 000 111 0000 1111 01111 1 0000 00 11 00 11 000 111 0000 1111 0 1 0000 00111 11 11 00 000 0000 1111 0000 1111 0000 1111 01111 1 1 0 000 111 00 0000 1111 0000 1 11 1111 00 1 11 0 00 0 11 0000 1111 0000 1111 11 00 1 0 0000 1111 0000 1111 0000 1111 0000 1111 0 1 000 111 0 1 0000 1111 0000 1111 0 1 000 111 0 1 0 1 0 1 0 1111 1 000 0 1 0 1 000 1 0000 1111 0000 11111 000 111 0 1 0000 1111 0000 1111 11 00 000 111 0000 1111 0000 1111 000 111 0000 1111 0000 1111 00 11 0 1 0000 1111 00001111 1111 0000 00 11 11 00 11 00 0 1 0000 1111 00 11 00 1 11 0 0000 1111 00 11 0 1111 1 0000 0000 1 1111 0111 000 0 1 0000 1111 0000 1111 0 1 000 1111 111 0000 0000 1 1111 0111 000 0000 1111 0 1 0 1 000 111 0000 11 00 0 1 0 1 000 1111 111 11 00 0 1 0 1

11 00 00 11 000 111 00 11 00 11 000 111 00 11 00 00 11 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11

111 000 000 111 000 000111 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 111 000

11 00 00 11 00 11 000 111 000 111 000 111 00 11 000 111 00 111 11 000 000 111 11 00 00 11

00 11 0000 1111 000 111 0 1 00 11 0000 00 11 000 111 0 1111 1 0000 1111 00 11 111 000 0000 1111 11 00 00 11 000 111 0000 1111 00 11 0000 111 1111 0001111 0 1 0000 1111 00 11 0000 1111 0000 0 1 0000 1111 0 1 00 11 0000 1111 0000 1111 0 1 00 11 000 0000 1111 0000 111 1111 0 00 11 000 111 0000 1 1111 000 111 0000 1111 00 11 000 111 00 111 11 000 111 0000 1111 00 11 000 00 11 0000 1111 00 11 0 1 00 11 00 11 000 111 00 11 0000 1111 00 11 011 1 00 11 00 11 000 111 00 111 0000 1111 000 111 000 0000 1111 000 111 000 00 11 00 11 11 00 000 11 111 00111 000 111 00 11 00 11 11 00 000 111 00 11 00 111 11 000 111 000 00 11 000 111 0000 1111 0 1 000 111 0000 1111 000 111 0 1 000 111 00 11 0000 1111 0000 1111 0000 1111 000 111 000 00111 11 0000 1111 0000 1111 0000 000 111 0000 1111 0 1 00 1111 11 0000 1111 0000 1111 000 111 00 11 0000 1111 0 1 00 11 0000 1111 000 111 00 11 0000 0 1 00 11 0000 1111 000 111 0000 1111 000 001111 11 11 00 0000111 1111 0 1 000 111 0000 1111 11 1 1111 0 1 0 000 1 000 111 0000 1 0 1 0 000 111 0000 1111 000 111 0000 1111 00 11 0000 11 1111 00 11 000 111 0000 1111 00 0000 1111 00 11 00 11 00 11 00 11 0000 00 11 00 11 00 1111 11 000 111 000 111 1 0 0000 1111 00111 11 000 111 000 1 0 0000 1111 000 111 000 111 0000 1111 000 111 000 111 0 1 00 11 000 111 000 111 111 000 0 1 0 1 0 1 00 11 000 111 0 1 0 1 1 0 000 111 0 1 0 1 000 111 00011 111 00111 000 0 1 000 111 000 111 00 11 000 111 000 111 00011 111 00111 000 000 111 00 11 00 11 000 111 000 111 1 0 00 11 11 00 11 000 11 111 1 0 00 00

11 00 000 111 00 0011 11 000 00 111 11 00 11 000 00111 11 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111

Figure 1.1: Schematic Representation of Aggregate Gradation IV Low heat Portland cement, minimizes thermal cracking but must control initial temperature V Sulfate resistant (marine environment) Aggregate usually occupy 70% to 80% of the volume of concrete. They are granular material derived, for the most part, from natural rock, crushed stone, natural gravels and sands.

6

ASTM C33 (Standard Specifications for Concrete Aggregates) governs the types of rock which can produce aggregates.

7

8

The shape can be rounded, irregular, angular, flaky, or elongated.

9

The surface texture can be glassy, smooth, granular, rough, crystalline or honeycombed.

The particle size distribution or grading of aggregates is very important as it determines the amount of paste for a workable concrete, Fig. 1.1. Since cement is the most expensive component, proper gradation is of paramount importance. 10

The grading of an aggregate supply is determined by a sieve analysis. A representative sample of the aggregate is passed through a stack of sieves aranged in order of decreasing size opening of the sieve. 11

12

We divide aggregates in two categories

Coarse aggregate fraction is that retained on the No. 4 sieve, Table 1.1. Fine aggregate fraction is that passing the No. 4 sieve. 13

ASTM C33 sets grading limits for coarse and fine aggregates, Table 1.2 and 1.3 respectively.

If a concrete does not comply with these limits, than there will be a need for more paste, and there will be the possibility of aggregate segregation.

14

Since aggregates contain some porosity, water can be absorbed. Also water can be retained on the surface of the particle as a film or moisture. Hence, it is necessary to quantify the moisture content of the aggregates in order to make adjustments to the water. Because dry aggregates will remove water from the paste, then the w/c is effectively reduced. On the other hand moist aggregates may effectively increase the w/c ratio.

15

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1.1 Material

1–3

ASTM Design.

Size

mm Coarse Aggregate 3 in. 75 21/2 in. 63 2 in. 50 11/2 in. 37.5 1 in. 25 3/4 in. 19 1/2 in. 12.5 3/8 in. 9.5 Fine Aggregate No. 4 4.75 No. 8 2.36 No. 16 1.18 No. 30 0.60 (600 µm) No. 50 300 µm No. 100 150 µm

in. 3 2.5 2 1.5 1 0.75 0.50 0.375 0.187 0.0937 0.0469 0.0234 0.0124 0.0059

Table 1.1: ASTM Sieve Designation’s Nominal Sizes Used for Concrete Aggregates Sieve Size

11/2 in. 1 in. 3/4 in. 1/2 in. 3/8 in. No. 4 No. 8

% Passing Each Sieve (Nominal Maximum Size) 11/2 in. 1 in. 3/4 in. 1/2 in. 95-100 100 95-100 100 35-70 90-100 100 25-60 90-100 10-30 20-55 40-70 0-5 0-10 0-10 0-15 0-5 0-5 0-5

Table 1.2: ASTM C33 Grading Limits for Coarse Concrete Aggregates Sieve Size 3/4 in. No. 4 No. 8 No. 16 No. 30 No. 50 No. 100

% Passing 100 95-100 80-100 50-85 25-60 10-30 2-10

Table 1.3: ASTM C33 Grading Limits for Fine Concrete Aggregates Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1–4 16

INTRODUCTION

Moisture states are 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 the surface. 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 =

WSSD − WOD × 100% WOD

(1.1)

most normal -weight aggregates (fine and coarse) have an absorption capacity in the range of 1% to 2%. Surface Moisture (SM): is the water in excess of the SSD state SM =

WW et − WSSD × 100% WSSD

(1.2)

The fineness modulus is a parameter which describe the grading curve and it can be used to check the uniformity of the grading. It is usually computed for fine aggregates on the basis of
cumulative percent retained on standard sieves (1.3) F.M. = 100 where the standard sieves used are No. 100, No. 50, No. 30, No. 16, No. 8, and No. 4, and 3/8 in, 3/4 in, 11/2 in and larger. 18

The fineness modulus for fine aggregate should lie between 2.3 and 3.1 A small number indicates a fine grading, whereas a large number indicates a coarse material.

19

20

Table 1.4 illustrates the determination of the fineness modulus.

Fineness modulus of fine aggregate is required for mix proportioning since sand gradation has the largest effect on workability. A fine sand (low fineness modulus) has much higher paste requirements for good workability.

21

22

The fineness modulus of coarse aggregate is not used for mix design purposes.

no-fines concrete has little cohesiveness in the fresh state and can not be compacted to a void-free condition. Hence, it will have a low strength, high permeability. Its only advantage is low density, and high thermal insulation which can be used if structural requirements are not high. 23

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1.1 Material

1–5 Sieve Size No. No. No. No. No. No.

Weight Amount Cumulative Cumulative Retained Retained Amount Amount (g) (wt. %) Retained (%) Passing (%) 4 9 2 2 98 8 46 9 11 89 16 97 19 30 70 30 99 20 50 50 50 120 24 74 26 100 91 18 92 8
Sample Weight 500 g. = 259 Fineness modulus=259/100=2.59

Table 1.4: Example of Fineness Modulus Determination for Fine Aggregate 1.1.1.1.2 24

Preliminary Considerations

There are two fundamental aspects to mix design to keep in mind:

1. Water/Cement ratio: where the strength is inversely proportional to the water to cement ratio, approximately expressed as: fc
=

A B 1.5w/c

(1.4)

For fc
in psi, A is usually taken as 14,000 and B depends on the type of cement, but may be taken to be about 4. It should be noted that w/c controls not only the strength, but also the porosity and hence the durability. 2. Aggregate Grading: In order to minimize the amount of cement paste, we must maximize the volume of aggregates. This can be achieved through proper packing of the granular material. The “ideal” grading curve (with minimum voids) is closely approximated by the Fuller curve  q d Pt = (1.5) D where Pt is the fraction of total solids finer than size d, and D is the maximum particle size, q is generally taken as 1/2, hence the parabolic grading. 1.1.1.1.3

Mix procedure

Before starting the mix design process, the following material properties should be determined:

25

1. Sieve analysis of both fine and coarse aggregates 2. Unit weight of the coarse aggregate 3. Bulk specific gravities

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1–6

INTRODUCTION

4. absorption capacities of the aggregates 1. Slump1 must be selected for the particular job to account for the anticipated method of handling and placing concrete, Table 1.5 As a general rule, adopt the lowest possible Type of Construction Foundation walls and footings Plain footings, caissons Beams and reinforced walls Building columns Pavement and slabs Mass concrete

Max 3 3 4 4 3 3

Min 1 1 1 1 1 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 minimum dimensions between the forms, or three-fourths of the minimum clear spacing between 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 increasing w/c which will decrease strength). Air content can be increased through the addition of admixtures. Table 1.6 tabulates recommended values of air content (obtained through such admixtures) for different conditions (for instance under severe freezing/thawing air content should be high). Recommended water requirements are given by Table 1.7.

Exposure Mild Moderate Extreme

in. 4.5 6.0 7.5

3/8

Sizes of Aggregates 1/2 in. 3/4 in. 1 in. 4.0 3.5 3.5 5.5 5.0 4.5 7.0 6.0 6.05

11/2 in. 3.0 4.4 5.5

Table 1.6: Recommended Average Total Air Content as % of Different Nominal Maximum Sizes of Aggregates and Levels of Exposure 1 The 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 filled with 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 the concrete over the original center of the base of the specimen.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1.1 Material

1–7 Slump in. 1-2 3-4 6-7 1-2 3-4 6-7

Sizes of Aggregates 3/8 in. 1/2 in. 3/4 in. 1 in. Non-Air-Entrained Concrete 350 335 315 300 385 365 340 325 410 385 360 340 Air-Entrained Concrete 305 295 280 270 340 325 305 295 365 345 325 310

11/2 in. 275 300 315 250 275 290

Table 1.7: Approximate Mixing Water Requirements, lb/yd3 of Concrete For Different Slumps and 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 fc
6,000 5,000 4,000 3,000 2,000

w/c Ratio by Weight Non-air-entrained Air-entrained 0.41 0.48 0.40 0.57 0.48 0.68 0.59 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 the w/c ratio are determined, the amount of cement per unit volume of concrete is determined simply by dividing the estimated water requirement by the w/c ratio. 6. Coarse Aggregate Content: Volume of coarse aggregate required per cubic yard of concrete 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 be converted to an OD weight by multiplying it by the dry-rodded2 weight per cubic foot of coarse 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 aggregate can then be obtained by multiplying this volume by the density of the fine aggregate. 2

Dry Rodded Volume (DRV) is the normal volume of space a material occupies.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1–8

INTRODUCTION Agg. Size in 3/8 1/2 3/4

1 11/2

Sand 2.40 0.50 0.59 0.66 0.71 0.76

Fineness Moduli 2.60 2.80 3.00 0.48 0.46 0.44 0.57 0.55 0.53 0.64 0.62 0.60 0.69 0.67 0.65 0.74 0.72 0.70

Table 1.9: Volume of Dry-Rodded Coarse Aggregate per Unit Volume of Concrete for Different Fineness Moduli of Sand 8. Adjustment for moisture in the aggregates: is necessary. If aggregates are air dry, they will absorb some water (thus effectively lowering the w/c), or if aggregates are too wet they will release water (increasing the w/c and the workability but reducing the strength). 1.1.1.1.4

Mix Design Example

Concrete is required for an exterior column to be located above ground in an area where substantial freezing and thawing may occur. The concrete is required to have an average 28day compressive strength of 5,000 psi. For the conditions of placement, the slump should be between 1 and 2 in, the maximum aggregate size should not exceed 3/4 in. and the properties of 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 %; Total moisture content=5.5%; fineness modulus = 2.70 The sieve analyses of both the coarse and fine aggregates fall within the specified limits. With this information, the mix design can proceed: 1. Choice of slump 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 is 6.0%, 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/yd3 6. Coarse aggregate content, interpolating from Table 1.9 for the fineness modulus of the fine aggregate of 2.70, the volume of dry-rodded coarse aggregate per unit volume of concrete is 0.63. Therefore, the coarse aggregate will occupy 0.63 × 27 = 17.01 ft3 /yd3 . Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1.1 Material

1–9

The OD weight of the coarse aggregate is 17.01 ft3 /yd3 , × 100 lbs/ft3 =1,701 lb. The SSD weight is 1,701 × 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 yd3 occupied by the different ingredients. Water Cement Coarse Aggregate (SSD) Air

280/62.4 700/(3.15)(62.4) 1,718/(2.70)(62.4) (0.06)(27)

= = = =

4.49 3.56 1.62 1.62 19.87

ft3 ft3 ft3 ft3 ft3

Hence, the fine aggregate must occupy a volume of 27.0 − 19.87 = 7.13 ft3 . The required SSD weight of the fine aggregate is 7.13 ft3 (2.65)(62.4)lb/ft3 =1,179 lbs lb. 8. Adjustment for moisture in the aggregate. Since the aggregate will be neither SSD or OD in the field, it is necessary to adjust the aggregate weights for the amount of water contained in the aggregate. Only surface water need be considered; absorbed water does not become part of the mix water. For the given moisture contents, the adjusted aggregate weights become: Coarse aggregate (wet)=1,718(1.025-0.01) = 1,744 lb/yd3 of dry coarse Fine aggregate (wet)=1,179(1.055-0.013) = 1,229 lb/yd3 of dry fine Surface moisture contributed by the coarse aggregate is 2.5-1.0 = 1.5%; by the fine aggregate: 5.5-1.3 = 4.2%; Hence we need to decrease water to 280-1,718(0.015)-1,179(0.042) = 205 lb/yd3 . Thus, the estimated batch weight per yd3 are Water Cement Wet coarse aggregate Wet fine aggregate

205 lb 700 lb 1,744 lb 1,229 lb 3,878 lb/yd3 143.6 lb/ft3

3,878 27

1.1.1.2 26

Mechanical Properties

Contrarily to steel to modulus of elasticity of concrete depends on the strength and is given

by  E = 57, 000 fc


(1.6)

 fc


(1.7)

or E = 33γ 1.5

where both fc
and E are in psi and γ is in lbs/ft3 . Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1–10

INTRODUCTION

27

Normal weight and lightweight concrete have γ equal to 150 and 90-120 lb/ft3 respectively.

28

Poisson’s ratio ν = 0.15.

Typical concrete (compressive) strengths range from 3,000 to 6,000 psi; However high strength concrete can go up to 14,000 psi.

29

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 

Non-linear part of stress-strain curve is caused by micro-cracking around the aggregates, Fig. 1.2

31

f’

c

Non-Linear ~ 0.5 cf’ Linear

εu

Figure 1.2: MicroCracks in Concrete under Compression 32

Irrespective of fc
, maximum strain under compression is ≈ 0.003, Fig. 1.3

33

Full strength of concrete is achieved in about 28 days
fct =

t f
4.0 + .85t c,28

(1.8)

or t (days)
%fc,28 34

1 20

2 35

4 54

7 70

10 80

15 90

Concrete always gain strength in time, but a decreasing rate

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1.1 Material

1–11 σ f’ c

f’ / 2 c

εu =

ε

0.003

Figure 1.3: Concrete Stress Strain Curve The tensile strength of concrete ft
is very difficult to measure experimentally. Accepted values 35

ft
≈ 0.07 − 0.11fc
 ≈ 3 − 5 fc


(1.9-a) (1.9-b)

Rather than the tensile strength, it is common to measure the modulus of rupture fr
, Fig. 1.4 36

11 00 00 11 00 11

11 00 00 11 00 11

Figure 1.4: Modulus of Rupture Test

σ σ

Figure 1.5: Split Cylinder (Brazilian) Test

fr
≈ 7.5 Victor Saouma

 fc


(1.10)

Mechanics and Design of Reinforced Concrete

Draft 1–12

INTRODUCTION f’t 1

f’c f’t

σ2 σ1

σ1 σ2 f’c

~ 20% increase in strength 2

Figure 1.6: Biaxial Strength of Concrete  Using split cylinder (or brazilian test), Fig. 1.5 ft
≈ 6−8 fc
. For this test, a nearly uniform tensile stress 2P (1.11) σ= πdt where P is the applied compressive load at failure, d and t are diameter and thickness of the specimen respectively. 37

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.

38

39

Biaxial strength curve is shown in Fig. 1.6

40

Concrete has also some time-dependent properties

Shrinkage: when exposed to air (dry), water tends to evaporate from the concrete surface, ⇒ shrinkage. It depends on the w/c and relative humidity. εsh ≈ 0.0002 − 0.0007. Shrinkage can 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 σsh = Eεsh

 3, 000 = 57, 000 3, 000(0.0002) = 624 psi >  10  

(1.12-a) (1.12-b)

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 to consolidation in clay), Fig. 1.7. 3

For this reason a minimum amount of reinforcement is always necessary in concrete, and a 2% reinforcement, can reduce the shrinkage by 75%.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1.1 Material

1–13 ε

Elastic recovery

creep

Creep recovery

Residual

no load

constant load

no load

Figure 1.7: Time Dependent Strains in Concrete Creep coefficient, Table 1.10 Cu = Ct =

fc
Cu

3,000 3.1

4,000 2.9

εct ≈2−3 εci t0.6 Cu 10 + t0.6 6,000 2.4

(1.13-a) (1.13-b)

8,000 2.0

Table 1.10: Creep Coefficients

41

Coefficient of thermal expansion is 0.65 × 10−5 /deg F for normal weight concrete.

1.1.2 42

Reinforcing Steel

Steel is used as reinforcing bars in concrete, Table 1.11.

Bars have a deformation on their surface to increase the bond with concrete, and usually have a yield stress of 60 ksi.

43

44

Maximum allowable fy is 80 ksi.

Stirrups, used as vertical reinforcement to resist shear, usually have a yield stress of only 40 ksi

45

Steel loses its strength rapidly above 700 deg. F (and thus must be properly protected from fire), and becomes brittle at −30 deg. F

46

47

Prestressing Steel cables have an ultimate strength up to 270 ksi.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1–14

INTRODUCTION Bar Designation No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10 No. 11 No. 14 No. 18

Diameter (in.) 2/8=0.250 3/8=0.375 4/8=0.500 5/8=0.625 6/8=0.750 7/8=0.875 8/8=1.000 9/8=1.128 10/8=1.270 11/8=1.410 14/8 =1.693 18/8 =2.257

Area ( in2 ) 0.05 0.11 0.20 0.31 0.44 0.60 0.79 1.00 1.27 1.56 2.25 4.00

Perimeter in 0.79 1.18 1.57 1.96 2.36 2.75 3.14 3.54 3.99 4.43 5.32 7.09

Weight lb/ft 0.167 0.376 0.668 1.043 1.5202 2.044 2.670 3.400 4.303 5.313 7.650 13.60

Table 1.11: Properties of Reinforcing Bars Welded wire fabric is often used to reinforce slabs and shells. It has both longitudinal and transverse cold-drawn steel. They are designated by A×A−W B ×B, such as 6×6−W 1.4×1.4 where spacing of the wire is 6 inch, and a cross section of 0.014 in2 .

48

1.2

Design Philosophy, USD

ACI refers to this method as the Strength Design Method, (previously referred to as the Ultimate Strength Method).

49

ΦRn ≥ Σαi Qi

(1.14)

where Φ is a strength reduction factor, less than 1, and must account for the type of structural element, Table 1.12 (ACI 9.3.2) Type of Member Axial Tension Flexure Axial Compression, spiral reinforcement Axial Compression, other Shear and Torsion Bearing on concrete

Φ 0.9 0.9 0.75 0.70 0.85 0.70

Table 1.12: Strength Reduction Factors, Φ Rn is the nominal resistance (or strength).

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

1.3 Analysis vs Design

1–15

Ru = Rd = ΦRn is the design strength. αi is the load factor corresponding to Qi and is greater than 1. Σαi Qi is the required strength based on the factored load: i is the type of load ΦMn ≥ Mu

(1.15-a)

ΦVn ≥ Vu

(1.15-b)

ΦPn ≥ Pu

(1.15-c)

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). The most important ones being

1. Deflections 2. Crack width (for R/C) 3. Stability

1.3 53

Analysis vs Design

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 (b and h) as well as reinforcement (As ). Note that in many cases the external dimensions of the beam (b and h) are fixed by the architect. 54

We often consider the maximum moment along a member, and design accordingly.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1–16

1.4

INTRODUCTION

Basic Relations and Assumptions

In developing a design/analysis method for reinforced concrete, the following basic relations will be used:

55

1. Equilibrium: of forces and moment at the cross section. 1) ΣFx = 0 or Tension in the reinforcement = Compression in concrete; and 2) ΣM = 0 or external moment (that is the one obtained from the moment envelope) equal and opposite to the internal one (tension in steel and compression of the concrete). 2. Material Stress Strain: We recall that all normal strength concrete have a failure strain u = .003 in compression irrespective of fc
. 56

Basic assumptions used:

Compatibility of Displacements: Perfect bond between steel and concrete (no slip). Note that those two materials do also have very close coefficients of thermal expansion under normal temperature. Plane section remain plane ⇒ strain is proportional to distance from neutral axis. Neglect tensile strength in all cases.

1.5

ACI Code

Attached is an unauthorized copy of some of the most relevant ACI-318-89 design code provisions. 8.1.1 - In design of reinforced concrete structures, members shall be proportioned for adequate strength in accordance with provisions of this code, using load factors and strength reduction factors Φ 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 modified according 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 Wc1.5 33 fc
( psi) for values of Wc  between 90 and 155 lb per cu ft. For normal weight concrete, Ec may be taken as 57, 000 fc
. 8.5.2 - Modulus of elasticity Es for non-prestressed reinforcement may be taken as 29,000 psi. 9.1.1 - Structures and structural members shall be designed to have design strengths at all sections 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 Strength 9.2.1 - Required strength U to resist dead load D and live load L shall be at least equal to U = 1.4D + 1.7L

(1.16)

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 greatest required strength U U = 0.75(1.4D + 1.7L + 1.7W ) (1.17) Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

1.5 ACI Code

1–17

where load combinations shall include both full value and zero value of L to determine the more severe condition, and U = 0.9D + 1.3W (1.18) 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 nominal strength calculated in accordance with requirements and assumptions of this code, multiplied by a strength reduction factor Φ. 9.3.2 - Strength reduction factor Φ shall be as follows: 9.3.2.1 - Flexure, without axial load 0.90 9.4 - Design strength for reinforcement Designs shall not be based on a yield strength of reinforcement fy 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 to clear span ratios greater than 2/5 for continuous spans and 4/5 for simple spans, a non-linear distribution of strain shall be considered. See Section 10.7. 10.2.3 - Maximum usable strain at extreme concrete compression fiber shall be assumed equal to 0.003. 10.2.4 - Stress in reinforcement below specified yield strength fy for grade of reinforcement used 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 reinforced concrete, except when meeting requirements of Section 18.4. 10.2.6 - Relationship between concrete compressive stress distribution and concrete strain may be assumed to be rectangular, trapezoidal, parabolic, or any other shape that results in prediction 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 rectangular concrete stress distribution defined by the following: 10.2.7.1 - Concrete stress of 0.85fc
shall be assumed uniformly distributed over an equivalent compression zone bounded by edges of the cross section and a straight line located parallel to the neutral axis at a distance (a = β1 c) from the fiber of maximum compressive strain. 10.2.7.2 - Distance c from fiber of maximum strain to the neutral axis shall be measured in a direction perpendicular to that axis. 10.2.7.3 - Factor β1 shall be taken as 0.85 for concrete strengths fc
up to and including 4,000 psi. For strengths above 4,000 psi, β1 shall be reduced continuously at a rate of 0.05 for each 1000 psi of strength in excess of 4,000 psi, but β1 shall not be taken less than 0.65. 10.3.2 - Balanced strain conditions exist at a cross section when tension reinforcement reaches the strain corresponding to its specified yield strength fy just as concrete in compression reaches its assumed ultimate strain of 0.003. 10.3.3 - For flexural members, and for members subject to combined flexure and compressive axial load when the design axial load strength (ΦPn ) is less than the smaller of (0.10fc
Ag ) or (ΦPb ), the ratio of reinforcement p provided shall not exceed 0.75 of the ratio ρb that would produce balanced strain conditions for the section under flexure without axial load. For members with compression reinforcement, the portion of ρb equalized by compression reinforcement need not be reduced by the 0.75 factor. 10.3.4 - Compression reinforcement in conjunction with additional tension reinforcement may be used to increase the strength of flexural members. Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 1–18

INTRODUCTION

10.5.1 - At any section of a flexural member, except as provided in Sections 10.5.2 and 10.5.3, where positive reinforcement is required by analysis, the ratio ρ provided shall not be less than that given by 200 (1.19) ρmin = fy

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft Chapter 2

FLEXURE This is probably the longest chapter in the notes, we shall cover in great details flexural design/analysis of R/C beams starting with uncracked section to failure conditions.

1

1. Uncracked elastic (uneconomical) 2. cracked elastic (service stage) 3. Ultimate (failure)

2.1

Uncracked Section εc

h

d As

εs

b

Figure 2.1: Strain Diagram Uncracked Section 2

Assuming perfect bond between steel and concrete, we have εs = εc , Fig. 2.1 ε s = εc ⇒

where n is the modular ratio n =

fs fc Es = ⇒ fs = fc ⇒ fs = nfc Es Ec Ec

Es Ec

3

Tensile force in steel Ts = As fs = As nfc

4

Replace steel by an equivalent area of concrete, Fig. 2.2.

(2.1)

Draft 2–2

FLEXURE 11111111111111111 00000000000000000 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 (n-1)A S 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 2 00000000000000000 11111111111111111 00000000000000000 11111111111111111

(n-1)A S 2

Figure 2.2: Transformed Section 5

Homogeneous section & under bending fc =

6

Mc ⇒ fs = nfc I

(2.2)

+ Make sure that σmax < ft


Example 2-1: Uncracked Section in2

Given fc
= 4,000 psi; ft
= 475 psi; fy = 60,000 psi; M = 45 ft-k = 540,000 in-lb; As = 2.35 + , f − , and f Determine fmax s max

yt 25" 23" 2

As = 2.35 in

yb

10"

Solution:

29, 000 √ = 8 ⇒ (n − 1)As = (8 − 1)(2.35) = 16.45 in2 57 4, 000 (10)(25)( 25 2 ) + (16.45)(2) = (25)(10) + 16.45 = 11.8 in

n = yb yb

y t = 25 − 11.8 = 13.2 in (10)(25)3 I = + (25)(10)(13.2 − 12.5)2 + (16.45)(23 − 13.2)2 12 = 14, 722 in2 (540, 000) lb.in(13.2)in Mc = = 484 psi fcc = I (14, 722) in4 Victor Saouma

(2.3-a) (2.3-b) (2.3-c) (2.3-d) (2.3-e) (2.3-f) (2.3-g)

Mechanics and Design of Reinforced Concrete

Draft

2.2 Section Cracked, Stresses Elastic

2.2

2–3

√ Mc (540, 000) lb.in(25 − 13.2) in = 433 psi < 475 psi = 4 I (14, 722) in (540, 000)(23 − 13.2) in Mc = (8) = 2, 876 psi = n I (14, 722)

fct =

(2.3-h)

fs

(2.3-i)

Section Cracked, Stresses Elastic

This is important not only as an acceptable alternative ACI design method, but also for the later evaluation of crack width under service loads.

7

2.2.1

Basic Relations

If fct > fr , fcc <≈ .5fc
and fs < fy we will assume that the crack goes all the way to the N.A and we will use the transformed section, Fig. 2.3 8

11111111111111111 00000000000000000 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 (n-1)A S 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 2 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 (n-1)A S 2

fc C

kd/3

kd d

11111111111111111 00000000000000000 00000000000000000 11111111111111111

(1-k/3)d=jd T

b

Figure 2.3: Stress Diagram Cracked Elastic Section To locate N.A, tension force = compressive force (by def. NA) (Note, for linear stress distribution and with ΣFx = 0; σ = by ⇒ bydA = 0, thus b ydA = 0 and ydA = yA = 0, by definition, gives the location of the neutral axis)

 Es 10 Note, N.A. location depends only on geometry & n Ec 9

Tensile and compressive forces are equal to C = bkd 2 fc & T = As fs and neutral axis is determined by equating the moment of the tension area to the moment of the compression area   kd = nAs (d − kd) b(kd) 2nd degree equation (2.4-a) 2 M (2.4-b) M = T jd = As fs jd ⇒ fs = As jd bkd bd2 fc jd = kjfc ⇒ fc = 1 bdM2 kj M = Cjd = (2.4-c) 2 2 2 11

where j = (1 − k/3). Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–4

2.2.2

FLEXURE

Working Stress Method

Referred to as Alternate Design Method (ACI Code Appendix A); Based on Working Stress Design method.

12

13

Places a limit on stresses and uses service loads (ACI A.3). fcc ≤ .45fc
fst ≤ 20 ksi for grade 40 or 50 steel fst ≤ 24 ksi for grade 60 steel

14

(2.5)

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:  ρ = Abds   ⇒ k = 2ρn + (ρn)2 − ρn  b(kd) (kd) = nAs (d − kd) 2

(2.6)

Design: Objective is to have fc & fs preset & determine As , Fig. 2.4, and we thus seek the optimal value of k in such a way that concrete and steel reach their respective limits simultaneously. εc

fc C

kd/3

kd d (1-k/3)d=jd T

εs

fs

Figure 2.4: Desired Stress Distribution; WSD Method

εc εs

εc εs

= = =

kd d−kd fc Ec fs Es

    

f c Es Ec f s

n r

= = =

k 1−k Es Ec fs fc

        

k=

n n+r

(2.7)

Balanced design in terms of ρ: What is the value of ρ such that steel and concrete will both reach their maximum allowable stress values simultaneously  C = bkd 2 fc     fc T = As fs n 2 bkd = ρb fs bd ρb = 2r(n+r) (2.8) n C = T k =   n+r  ρ = Abds 15

16

Governing equations

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.2 Section Cracked, Stresses Elastic

2–5

Review Start by determining ρ, • If ρ < ρb steel reaches max. allowable value before concrete, and (2.9)

M = As fs jd • If ρ > ρb concrete reaches max. allowable value before steel and M = fc

bkd jd 2

(2.10)

or 1 M = fc jkbd2 = Rbd2 2

(2.11)

1 fc kj 2

(2.12)

where  k = 2ρn + (ρn)2 − ρn Design We define def

R = where k =

n n+r ,

solve for bd2 from

M R assume b and solve for d. Finally we can determine As from

17

bd2 =

(2.13)

As = ρb bd

(2.14)

Summary Review √ b, d, As M? ρ = Abds  k = 2ρn + (ρn)2 − ρn r = ffsc n ρb = 2r(n+r) ρ < ρb M = As fs jd ρ > ρb M = 12 fc bkd2 j

Design √ M b, d, As ? n k = n+r j = 1 − k3 r = ffsc R = 12 fc kj n ρb = 2r(n+r) bd2 = M R As = ρb bd or As =

M fs jd

Example 2-2: Cracked Elastic Section

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–6

FLEXURE

Same problem as example 2.1 fc
= 4,000 psi; ft
= 475 psi; fy = 60,000 psi; As = 2.35 in2 however, M is doubled to M = 90 k.ft (instead of 45). Determine concrete and steel stresses 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 2.35 = = 0.0102 bd (10)(23) ρn = (0.010)(8) = 0.08174  2ρn + (ρn)2 − ρn k =  = 2(0.08174) + (0.08174)2 − (0.08174) = 0.33 ρ =

kd = (.33)(23) = 7.6 in   0.33 (23) = 20.47 in jd = 1− 3 M fs = As jd (90)(1, 000)(12) = 22, 400 psi = (2.35)(20.47) 2M fc = bjkd2 (2)(90)(12, 000) = 1, 390 psi = (10) (20.47) (7.6)       jd

I =

kd

(10)(7.6)3 + (10)(7.6) 12

M N.A fcc I fs δ

k.ft in psi in4 psi in

Uncracked 45 13.2 485 14,710 2,880 1



7.6 2

(2.15-a) (2.15-b) (2.15-c) (2.15-d) (2.15-e) (2.15-f) (2.15-g) (2.15-h) (2.15-i) (2.15-j)

2 + 8(2.35)(23 − 7.6)2 = 5, 922 in4

Cracked 90 7.6 1,390 (< .5fc
) 5,910 22,400 ≈4

(2.15-k)

Cracked/uncracked 2 2.9 .4 (δα I1 ) (≈ 7 ) 4

Example 2-3: Working Stress Design Method; Analysis Same problem as example 2.1 fc
= 4,000 psi; ft
= 475 psi; fy = 60,000 psi; As = 2.35 in2 . Determine Moment capacity. Solution:

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.2 Section Cracked, Stresses Elastic

2–7

2.35 As = = .0102 bd (10)(23) = 24 ksi

(2.16-a)

ρ = fs

(2.16-b)

fc = (.45)(4, 000) = 1, 800 psi (2.16-c)   k = 2ρn + (ρn)2 − ρn = 2(.0102)8 + (.0102)2 − (8)(.0102) = .331 (2.16-d) k j = 1 − = .889 (2.16-e) 3 N.A. @ (.331)(23) = 7.61 in (2.16-f) 8 n = = .014 > ρ ⇒ Steel reaches elastic (2.16-g) limit ρb = 2r(n + r) (2)(13.33)(8 + 13.33) M

= As fs jd = (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 overestimated the design moment: M

1 = = fc bkd2 j 2 1 (1.8)(10)(0.33)(0.89)(23)2 = 1, 397 k.in > 1, 154 k.in = 2

(2.17-a) (2.17-b)

If we define αc = fc /1, 800 and αs = fs /24, 000, then as the load increases both αc and αs increase, but at different rates, one of them αs reaches 1 before the other.

1

αs

αc

Load

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 Victor Saouma

(2.18-a)

Mechanics and Design of Reinforced Concrete

Draft 2–8

FLEXURE fs = 24, 000 psi Es 29, 000 n = = √ =8 Ec 57 4, 000 24 fs = 13.33 = r = fc 1.8 8 n = = .375 k = n+r 8 + 13.33 d .375 j = 1− =1− = .875 3 3 n 8 ρb = = = .01405 2r(n + r) 2(13.33)(8 + 13.33) 1 1 fc kj = (1, 800)(.375)(.875) = 295 psi R = 2 2

(2.18-b) (2.18-c) (2.18-d) (2.18-e) (2.18-f) (2.18-g) (2.18-h)

Estimate beam weight at .5 k/ft, thus M

(32)2 = 435 k.ft 8 435 k.ft in2 (12, 000) lb.in M = = 17, 700 in3 R (295) lbs ft k

(2.19-a)

= [(1.9) + (1.0 + .5)]

bd2 =

(2.19-b)

in Take b = 18 in & d = 31.4 in ⇒ h = 36 2 2 (18)(36) in ft k = .675 k/ft√ Check beam weight 145 (.15) 2 3 in

As = (.01405)(18)(31.4) = 7.94

2.3

in2

ft

⇒ use 8# 9 bars in 2 layers ⇒ As = 8.00 in2

Cracked Section, Ultimate Strength Design Method

2.3.1

Whitney Stress Block σ

ε

γ f’c βc

c h

C=α f’cb c

c

a/2 = β c

a= β1c

C=γ f’ab c

d As

fs

ε b

fs

Actual

Figure 2.5: Cracked Section, Limit State Figure Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.3 Cracked Section, Ultimate Strength Design Method

2–9

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 σ ⇒ 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 need to know is C & its location, thus α and β. We adopt a rectangular stress, with depth a = β1 c, and stress equal to γfc
(ACI 10.2.7.1)

C = αfc
bc = γfc
ab fav α = fc
a = β1 c Thus γ=

(2.20-a) (2.20-b) (2.20-c)

α β1

(2.21)

But the location of the resultant forces must be the same, hence β1 = 2β 21

From Experiments fc
( psi) α β β1 = 2β γ = α/β1

22

(2.22)

<4,000 .72 .425 .85 0.85

5,000 .68 .400 .80 0.85

6,000 .64 .375 .75 0.85

8,000 .56 .325 .65 0.86

Thus we have, (ACI-318 10.2.7.3): β1 = .85 1 = .85 − (.05)(fc
− 4, 000) 1,000

23

7,000 .60 .350 .70 0.86

if fc
≤ 4, 000 if 4, 000 < fc
< 8, 000

(2.23)

Failure can occur by either

yielding of steel: εs = εy ; Progressive crushing of concrete: εc = .003; Sudden; (ACI 10.3.2).

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–10

FLEXURE ε u=0.003

0.85 f’c a= β1c

C=0.85f’c ab

c h

d d

As

εs

T

b

Figure 2.6: Whitney Stress Block

2.3.2

Balanced Design  fs = fy  As fs = .85fc
ab = .85fc
bβ1 c c=  As ρ = bd

Tension Failure:

ρfy .85fc
β1 d

(2.24)

Compression Failure:

εc = .003 fs εs = Es .003 c = ⇒ c= d .003 + εs

(2.25-a) (2.25-b) .003 fs +.003 Es

d

(2.25-c)

Balanced Design:

Balanced design occurs if we have simultaneous yielding of the steel and crushing of the concrete. Hence, we simply equate the previous two equations

24

25

ρfy .85fc
β1 d

=

ρ

= ρb

.003 fs +.003 Es



d

ρbf 2d .85fc
β1

=

.003 fs +.003 E−s

d



Es = 29, 000 ksi




87,000 ρb = .85β1 ffyc 87,000+f y

To ensure failure by yielding, (2.27)

ρ < .75ρb

26

(2.26)

(ACI 8.4.3)

ACI strength requirements U U Md = M u φ

Victor Saouma

= = = =

1.4D + 1.7L 0.75(1.4D + 1.7L + 1.7W ) φMn .90

(ACI (ACI (ACI (ACI

9.2.1) 9.2.2) 9.1.1) 9.3.2.2)

(2.28)

Mechanics and Design of Reinforced Concrete

Draft

2.3 Cracked Section, Ultimate Strength Design Method 27

2–11

Also we need to specify a minimum reinforcement ratio ρmin ≥

200 fy

(2.29)

(ACI 10.5.1)

to account for temperature & shrinkage Note, that ρ need not be as high as 0.75ρb . If steel is relatively expensive, or deflection is of concern, can use lower ρ. 28

29

As a rule of thumb, if ρ < 0.5ρb , there is no need to check for deflection.

2.3.3 30

Review

Given, b, d, As , fc
, fy , determine the moment capacity M . ρact = Abds
87 ρb = (.85)β1 ffyc 87+f y

(2.30)

• ρact < ρb : Failure by yielding and A f

s y a = .85f ΣFx = 0
cb a Md = φAs fy (d − 2 ) ΣM = 0

(2.31)

• ρact > ρb is not allowed by code, in this case we have an extra unknown fs . We now have one more unknown fs , and we will need an additional equation (from strain diagram). 31

As f s c = .85f
c bβ1 c .003 = d .003+εs Md = φAs fs (d −

ΣFx = 0 From strain diagram β1 c 2 ) ΣM = 0

(2.32)

We can solve by iteration, or substitution and solution of a quadratic equation.

2.3.4 32

Design

We consider two cases: I b d and As , unknown; Md known; Since design failure is triggered by fs = fy     As f y ρfy fy ΣFx = 0 a = 0.85f a = 0.85f
b
2 c c  Md = Φ ρfy 1 − .59ρ
bd (2.33-a) fc Md = As fy d − a2 ρ = Abds    R

where ρ is specified by the designer; or   fy R = ρfy 1 − .59ρ
fc Victor Saouma

(2.34)

Mechanics and Design of Reinforced Concrete

Draft 2–12

FLEXURE

which does not depend on unknown quantities. Then solve for bd2 : bd2 =

Md ΦR

(2.35)

Solve for b and d (this will require either an assumption on one of the two, or on their ratio). As = ρbd II b & d known & Md known ⇒ there is no assurance that we can have a design with ρb If the section is too small, then it will require too much steel resulting in an over-reinforced section. Iterative approach (a) Since we do not know if the steel will be yielding or not, use fs . (b) Assume an initial value for a (a good start is a = d5 ) (c) Assume initially that fs = fy (d) Check equilibrium of moments (ΣM = 0) As =

M  d  Φfs d − a2

(2.36)

(e) Check equilibrium of forces in the x direction (ΣFx = 0) a=

As fs .85fc
b

(2.37)

(f) Check assumption of fs by either comparing ρ with ρb , or from the strain diagram .003 d−c εs = ⇒ fs = Es .003 < fy d−c c c where c =

(2.38)

a β1 .

(g) Iterate until convergence is reached.

2.4 2.4.1 33

Practical Design Considerations Minimum Depth

ACI 9.5.2.1 stipulates that the minimum thickness of beams should be

Solid One way slab Beams or ribbed One way slab

Simply supported

One end continuous

Both ends continuous

Cantilever

L/20

L/24

L/28

L/10

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.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.4 Practical Design Considerations

2.4.2 35

2–13

Beam Sizes, Bar Spacing, Concrete Cover

Beam sizes should be dimensioned as 1. Use whole inches for overall dimensions, except for slabs use

1 2

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 #9 acceptable; #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 37

Design Aids

Basic equations developed in this section can be easily graphed.

Review Given b d and known steel ratio ρ and material strength, φMn can be readily obtained from φMn = φRbd2 Design in this case Set Md = φRbd2 From tabulated values, select ρmax and ρmin often 0.5ρb is a good economical choice. Select R from tabulated values of R in terms of fy , fc
and ρ. Solve for bd2 . Select b and d to meet requirements. Usually depth is about 2 to 3 times the width. Using tabulated values select the size and number of bars giving preference to larger bar sizes to reduce placement cost (careful about crack width!). 6. Check from tables that the selected beam width will provide room for the bars chosen with adequate cover and spacing. 1. 2. 3. 4. 5.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–14

Bar Size #3 #4 #5 #6 #7 #8 #9 #10 #11 #14 #18

FLEXURE

Nominal Diam. 0.375 0.500 0.625 0.750 0.875 1.000 1.128 1.270 1.410 1.693 2.257

1 0.11 0.20 0.31 0.44 0.60 0.79 1.00 1.27 1.56 2.25 4.00

2 0.22 0.40 0.62 0.88 1.20 1.58 2.00 2.54 3.12 4.50 8.00

3 0.33 0.60 0.93 1.32 1.80 2.37 3.00 3.81 4.68 6.75 12.00

Number of Bars 4 5 6 7 0.44 0.55 0.66 0.77 0.80 1.00 1.20 1.40 1.24 1.55 1.86 2.17 1.76 2.20 2.64 3.08 2.40 3.00 3.60 4.20 3.16 3.95 4.74 5.53 4.00 5.00 6.00 7.00 5.08 6.35 7.62 8.89 6.24 7.80 9.36 10.92 9.00 11.25 13.50 15.75 16.00 20.00 24.00 28.00

8 0.88 1.60 2.48 3.52 4.80 6.32 8.00 10.16 12.48 18.00 32.00

9 0.99 1.80 2.79 3.96 5.40 7.11 9.00 11.43 14.04 20.25 36.00

10 1.10 2.00 3.10 4.40 6.00 7.90 10.00 12.70 15.60 22.50 40.00

Table 2.1: Total areas for various numbers of reinforcing bars (inch2 )

Bar Size #4 #5 #6 #7 #8 #9 #10 #11 #14 #18

Number of bars 2 3 4 6.8 8.3 9.8 6.9 8.5 10.2 7.0 8.8 10.5 7.2 9.1 11.0 7.3 9.3 11.3 7.6 9.9 12.1 7.8 10.3 12.9 8.1 10.9 13.7 8.9 12.3 15.7 10.6 15.1 19.6

in single layer of reinf. 5 6 7 8 11.3 12.8 14.3 15.8 11.8 13.4 15.1 16.7 12.3 14.0 15.8 17.5 12.8 14.7 16.6 18.5 13.3 15.3 17.3 19.3 14.4 16.6 18.9 21.2 15.4 18.0 20.5 23.0 16.6 19.4 22.2 25.0 19.1 22.5 25.9 29.3 24.1 28.6 33.2 37.7

Table 2.2: Minimum Width (inches) according to ACI Code

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.5 USD Examples

2–15

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 yt 25" 23" 2

As = 2.35 in

yb

10"

Solution: 2.35 As = = .0102 bd (10)(23) √ 87 4 f
87 ρb = .85β1 c = .0285 > ρact = (.85)(.85) fy 87 + fy 60 87 + 60 As fy (2.35)(60) a = = = 4.15 in .85fc
b (.85)(4)(10)  

a 4.15 = (2.35)(60) 23 − = 2, 950 k.in Mn = As fy d − 2 2

ρact =

Md = φMn = 0.9(2, 950) = 2, 660 k.in

(2.39-a) (2.39-b) (2.39-c) (2.39-d) (2.39-e)

Note: Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–16

FLEXURE

1. From equilibrium, ΣFx = 0 ⇒ c =

As f y .85β1 bfc


=

(2.35)(60) (.85)(.85)(4)(10)

= 4.87 in

2. Comparing with previous analysis

c M

uncracked 13.2 45

cracked 7.61 90

ultimate 4.87 245 1.7 = 144

3. Alternative solution: Mn = ρact fy bd2 (1 − .59ρact = As fy d(1 − 59ρact

fy ) fc


fy ) fc


60 (.0102)] = 2, 950 k.in = 245 k.ft 4 = φMn = (.9)(2, 950) = 2, 660 k.in = (2.35)(60)(23)[1 − (.59)

Md

(2.40-a) (2.40-b) (2.40-c) (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 own’s weight; Select ρ = 0.75ρb Solution:

Factored load wu = 1.4(1.27) + 1.7(2.44) = 5.92 k/ft 2 2 wu L (5.92)(15) = = 166.5 k.ft(12) in/ft = 1, 998 k.in Md = 8 8 f
87 ρ = 0.75ρb = (0.75)(0.85)β1 c fy 87 + fy 87 3 = .0278 = (0.75)(.85)2 40 87 + 40   fy R = ρfy 1 − .59ρ
f   c 40 = 0.869 psi = (.0278)(40) 1 − (0.59)(.0278) 3 1, 998 Md = = 2, 555 in3 bd2 = φR (0.9)(0.869)

(2.41-a) (2.41-b) (2.41-c) (2.41-d) (2.41-e) (2.41-f) (2.41-g)

Take b = 10 in, d = 16 in ⇒ As = (.0278)(10)(16) = 4.45 in2 ⇒ use 3 # 11 Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.5 USD Examples

2–17

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.in Solution: Assume a =

d 5

=

20 5

= 4 in As =

(1, 600) Md = 2.47 in2 a = φfy (d − 2 ) (.9)(40)(20 − 42 )

(2.42)

As fy (2.47)(40) = = 3.38 in
(.85)fc b (.85)(3)(11.5)

(2.43)

check assumption, a= Thus take a = 3.3 in.

As = ⇒a = ρact = ρb = ρmax =

(1, 600) = 2.42 in2 (.9)(40)(20 − 3.3 ) 2 √ (2.42)(40) = 3.3 in (.85)(3)(11.5) 2.42 = .011 (11.5)(20) 87 3 = .037 (.85)(.85) 40 87 + 40 √ .75ρb = .0278 > ρact

(2.44-a) (2.44-b) (2.44-c) (2.44-d) (2.44-e)

Example 2-8: Exact Analysis As an Engineer questioning the validity of the ACI equation for the ultimate flexural capacity of R/C beams, you determined experimentally the following stress strain curve for concrete:


fc ε 2 εmax σ= 2

ε 1 + εmax

(2.45)

where fc
corresponds to εmax . 1. Determine the exact balanced steel ratio for a R/C beam with b = 10”, d = 23”, fc
= 4, 000 psi, fy = 60 ksi, εmax = 0.003. (a) Determine the equation for the exact stress distribution on the section. (b) Determine the total compressive force C, and its location, in terms of the location of the neutral axis c. Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–18

FLEXURE

(c) Apply equilibrium 2. Using the ACI equations, determine the: (a) Ultimate moment capacity. (b) Balanced steel ratio. 3. For the two approaches, compare: (a) Balanced steel area. (b) Location of the neutral axis. (c) Centroid of resultant compressive force. (d) Ultimate moment capacity. Solution:

1. Stress-Strain: σ=

2 4,000 2.667 × 106 ε .003 ε =  ε 2 1 + 1.11 × 105 ε2 1 + .003

(2.46)

2. Assume crushing at failure, hence strain distribution will be given by ε= 3. Combine those two equation: σ= 4. The total compressive force is given by  c   c dF = b σdy = b F = =

0.003 y c

(2.47)

8, 000 yc  2 1 + yc

(2.48)

 8, 000 yc 8, 000 c y dy = b  y 2  2 dy c 0 0 01 + c 01 + 1 c  c c  



2 2 y 2  y bc b 1  ln 1 + 8, 000  2 ln 1 +  = 8, 000   c2 1 c c 2 c 0 c

c

(2.49)

(2.50)

0

= 4, 000bc ln(2) = 2, 773bc

(2.51)

5. Equilibrium requires that T = C 2, 773bc = As fy

(2.52)

From the strain diagram: .003 c

=

c =

Victor Saouma

εy + .003 (.003)d ⇒c= d εy + .003 (.003)(23) = 13.6 in 60 29,000 + .003

(2.53) (2.54)

Mechanics and Design of Reinforced Concrete

Draft

2.6 T Beams, (ACI 8.10)

2–19

6. Combining Eq. 2.52 with Eq. 2.54 As =

(2, 773)(10)(13.6) = 6.28 in2 60, 000

(2.55)

7. To determine the moment, we must first determine the centroid of the compressive force measured from the neutral axis 

y = =

= =

 b yσdy



2

8, 000 yc  c b ydA 2 8, 000b y2 0 1 + (y/c) = = dy = dy (2.56)   A 2, 773bc 2, 773bc 2, 773bc2 0 1 + 1 2 y 2 c     c y 2.885 1 y2 dy 2.885 c (2.57)  2 dy =  2  1 2 −  1 2 c2 (13.61)2 01 + 1 01 + 1 y2 y2 c c c c    c 1  1 .01557 yc2 − c2   tan−1 y (2.58) c2  1 c2 0 ! 3 3 −1 .01557 c − c tan (1) = (.01557)(13.61)3 (1 − tan−1 (1)) = 8.43 in (2.59) c

8. Next we solve for the moment M = As fy (d − c + y) = (6.28)(60)(23 − 13.61 − 8.43) = 6, 713 k.in

(2.60)

9. Using the ACI Code ρb = .85β1

fc
87 4 87 = (.85)2 = .0285 fy 87 + 60 60 147

As = ρb bd = (.0285)(10)(23) = 6.55 in2 As fy (6.55)(60) = = 11.57 in a =
.85f cb (.85)(4)(10)  

a 11.57 = (6.55)(60) 23 − = 6, 765 k.in M = As fy d − 2 2 11.57 a = 13.61 = c = β1 .85

(2.61) (2.62) (2.63) (2.64) (2.65)

10. We summarize

As c y
M

Victor Saouma

Exact 6.28 13.6 5.18 6,713

ACI 6.55 13.6 5.78 6,765

Mechanics and Design of Reinforced Concrete

Draft 2–20

FLEXURE be b

11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111

hf

bw

Figure 2.8: T Beams

2.6

T Beams, (ACI 8.10)

Equivalent width for uniform stress, Fig. 2.8 must satisfy the following requirements (ACI 8.10.2): 38

1.

1 2 (b

− bw ) ≤ 8hf

2. b < 4bw for isolated T beams only 3. hf > 4. b < 39

bw 2

L 4

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. b h

f

00000000000000000 11111111111111111 11111111111111111 00000000000000000

h d As

Figure 2.9: T Beam as Rectangular Section For T beams, we have a large concrete area, start by assuming that failure will occur by steel yielding, Fig. 2.10.

40

41

The approach consists in decomposing As into 2 components Fig. 2.11.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.6 T Beams, (ACI 8.10)

2–21 b

hf hd

ε u =0.003 0.85 f’c

11111111111111111 00000000000000000 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111

a=β1 c C=0.85f’ a

c

c

d

As

εs

T=As yf

bw

Figure 2.10: T Beam Strain and Stress Diagram 1. Asf → resists compression force in (b − bw )hf 2. (As − Asf ) → resists compression force in bw c

2.6.1 42

Review

Given, b, d, hf , As , fc
, fy , determine the moment capacity M , Fig. 2.11. b

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111

hf

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111

+

=

A sf

bw

As

c

A s − A sf

(b−b w)h f

bwc

Figure 2.11: Decomposition of Steel Reinforcement for T Beams 43

The moment is obtained from

Flanges: Asf

=

Mn1 =

.85fc
(b−bw )hf fy h Asf fy (d − 2f )

ΣF = 0

(2.66)

ΣM = 0

Web: a =

(As − Asf )fy .85fc
bw

ΣF = 0

a Mn2 = (As − Asf )fy (d − ) 2

ΣM = 0

(2.67-a) (2.67-b)

Total moment: Mn = Mn1 + Mn2 Victor Saouma

(2.68)

Mechanics and Design of Reinforced Concrete

Draft 2–22

2.6.2 44

FLEXURE

Design, (balanced)

Let us derive an expression for ρb and use it for design c εu Strain Compatibility = d εu + εy As fy = .85fc
β1 cbw + .85fc
(b − bw )hf ΣF = 0   

(2.69-a) (2.69-b)

Asf fy

thus, As fy

= def

ρw

=

def

ρf

=

 .85fc
β1 cbw + Asf fy  

ρb

 εu fc
ρw = .85 β1 +ρf  fy εu + εy 

As bw d Asf bw d





(2.70)

Hence, ρwb = ρb + ρf

(2.71)

ρw,max = .75(ρb + ρf ) (2.72)

Example 2-9: T Beam; Moment Capacity I For the following beam: As = 8 # 11 ( 12.48 in2 ); fc
=3,000 psi; fy = 50,000 psi. Determine Mn 30" 7"

36"

11111111111111111 00000000000000000 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111

ε u=0.003

0.85 f’c a= β1c

c

C=0.85f’c ab

d

εs

T=Asfy

14"

Solution: 1. Check requirements for isolated T sections (a) bw = 30 in should not exceed 4bw = 4(14) = 56 in √ (b) hf ≥ b2u ⇒ 7 ≥ 14 2

√

2. Assume Rectangular section a=

Victor Saouma

As fy (12.48)(50) = = 8.16 in > hf
.85fc b (0.85)(3)(30)

(2.73)

Mechanics and Design of Reinforced Concrete

Draft

2.6 T Beams, (ACI 8.10)

2–23

3. For a T section Asf

= =

ρf

=

Asw = ρw = ρb = =

.85fc
hf (b − bw ) fy (.85)(3)(7)(30 − 14) = 5.71 in2 50 Asf 5.71 = .0113 = bwd (14)(36) As − Asf = 12.48 − 5.71 = 6.77 in2 12.48 Asw = = .025 bw d (14)(36) f
87 .85β1 c fy 87 + fy 87 3 = .0275 (.85)(.85) 50 87 + 50

(2.74-a) (2.74-b) (2.74-c) (2.74-d) (2.74-e) (2.74-f) (2.74-g)

4. Maximum permissible ratio ρmax = .75(ρb + ρf ) = .75(.0275 + .0113) = .029 > ρw

√

5. The design moment is then obtained from   7 = 9, 280 k.in Mn1 = (5.71)(50) 36 − 2 (As − Asf )fy a = .85fc
bw (6.77)(50) = 9.48 in = (.85)(3)(14) 9.48 ) = 10, 580 k.in Mn2 = (6.77)(50)(36 − 2 Md = (.9)(9, 280 + 10, 580) = 17, 890 k.in → 17, 900 k.in

(2.75-a) (2.75-b)

(2.76-a) (2.76-b) (2.76-c) (2.76-d) (2.76-e)

Example 2-10: T Beam; Moment Capacity II Determine the moment capacity of the following section, assume flange dimensions to satisfy ACI requirements; As = 6#10 = 7.59 in2 ; fc
= 3 ksi; fy =60 ksi.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–24

FLEXURE 28" 6"

26"

11111111111111111 00000000000000000 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111

ε u=0.003

0.85 f’c a= β1c

c

C=0.85f’c ab

d

εs

T=Asfy

10"

Solution: Assume rectangular beam ρ = ρb = a = = Asf

=

Asw

=

ρw = ρf

=

ρmax =

7.59 = .0104 (28)(26)    87 3 = .0214 > ρ ⇒ fs = fy (.85)(.85) 60 87 + 60 (As − Asf )fy .85fc
bw (7.59)(60) = 6.37 in > 6 in ⇒ T beam (.85)(3)(28) (.85)(3)(18)(6) = 4.59 in2 60 7.59 − 4.59 = 3.00 in2 7.59 = .0292 (26)(10) 4.59 = .0177 (26)(10) .75(.0214 + .0177) = .0294 > .0292 ⇒ Ductile failure

Mn1 = (4.59)(60)(26 − 3) = 6, 330 k.in As − Asf

= 7.59 − 4.59 = 3. in (3)(60) = 7.07 in a = (.85)(3)(10) 7.07 ) = 4, 050 k.in Mn2 = (3.00)(60)(26 − 2 Md = (.9)(6, 330 + 4, 050) = 9, 350 k.in 2

(2.77-a) (2.77-b) (2.77-c) (2.77-d) (2.77-e) (2.77-f) (2.77-g) (2.77-h) (2.77-i) (2.77-j) (2.77-k) (2.77-l) (2.77-m) (2.77-n)

Example 2-11: T Beam; Design given L = 24 ft; fy = 60 ksi; fc
= 3 ksi; Md = 6, 400 k.in; Design a R/C T beam.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.6 T Beams, (ACI 8.10)

2–25

3" 20"

11" 47" Solution: 1. Determine effective flange width: − bw ) ≤ 8hf 16hf + bw = (16)(3) + 11 = 59 in L 24 = 72 in 4 = 4 12 Center Line spacing = 47 in 1 2 (b

      

b = 47 in

(2.78-a)

2. Assume a = 3 in As = a =

6, 400 Md = 6.40 in2 a = φfy (d − 2 ) (0.9)(60)(20 − 32 ) As fy (6.4)(60) = = 3.20 in > hf
(.85)fc b (.85)(3)(47)

(2.79-a) (2.79-b)

3. Thus a T beam analysis is required. Asf Md1 Md2

.85fc
(b − bw )hf (.85)(3)(47 − 11)(3) = 4.58 in2 = fy 60 hf 3 ) = (.90)(4.58)(60)(20 − ) = 4, 570 k.in = φAsf fy (d − 2 2 = Md − Md1 = 6, 400 − 4, 570 = 1, 830 k.in =

(2.80-a) (2.80-b) (2.80-c) (2.80-d)

4. Now, this is similar to the design of a rectangular section. Assume a = As − Asf =

1, 830   = 1.88 in2 (.90)(60) 20 − 42

d 5

=

20 5

= 4. in (2.81)

5. check a =

(1.88)(60) = 4.02 in ≈ 4.00 (.85)(3)(11)

As = 4.58 + 1.88 = 6.46 in2 6.46 = .0294 ρw = (11)(20) Victor Saouma

(2.82-a) (2.82-b) (2.82-c)

Mechanics and Design of Reinforced Concrete

Draft 2–26

FLEXURE ρf ρb ρmax

4.58 = .0208 (11)(20)    87 3 = .0214 = (.85)(.85) 60 87 + 60 √ = (.0214 + .0208) = .042 > ρw

(2.82-d)

=

(2.82-e) (2.82-f)

6. Note that 6.46 in2 (T beam) is close to As = 6.40 in2 if rectangular section was assumed.

2.7 45

Doubly Reinforced Rectangular Beams

Negative steel reinforcement is needed to 1. Increase internal moment resistance capacity (not very efficient) 2. Support stirrups 3. Reverse moments (moving load) 4. Provide ductility (earthquake) 5. Reduce creep (long term deflections)

Approach will again be based on a strain compatibility analysis & equilibrium equation, Fig. 2.12.

46

0.85 f’c

ε u=0.003 A’s h

εs’

d’

A’s f’s

0.85 f’c

A’s f’s

a= β1c

a= β1c

c

d

d−d’

=

+

d As

εs

Asfs

Asfs

(As− A’s )f s

b

Figure 2.12: Doubly Reinforced Beams; Strain and Stress Diagrams 47

If ρ ≤ ρmax = .75ρb we can disregard compression steel

48

As for T beams, we decompose the tension steel into two components 1. A
s to resist the force in the top steel (assuming both yield) 2. As − A
s to resist compression in the concrete.

and we define ρ
=

Victor Saouma

A
s bd

(2.83)

Mechanics and Design of Reinforced Concrete

Draft

2.7 Doubly Reinforced Rectangular Beams

2.7.1 49

2–27

Tests for fs and fs


Different possibilities: Fig. 2.13 Yes

Yes

A’s yield?

No

As yield?

No

Not Allowed by ACI

Yes

A’s yield?

No

I

II

III

IV

f = fy

f = fy

f < fy

f < fy

f’ = f y

f’ < f

f’ = f

f’ < f y

s s

s

s

s

s

s

y

y

s

Figure 2.13: Different Possibilities for Doubly Reinforced Concrete Beams Test 1 fs = fy ? Assuming εs = εy , and fs
= fy , we have from the strain diagram, Fig. 2.14

d’ A’s h

ε u =0.003 εs’

d εs = εy

As

b Figure 2.14: Strain Diagram, Doubly Reinforced Beam; is As Yielding? εu d ε u + εy d
= εu − (εu + εy ) d = Es ε
s

c = ε
s fs
From equilibrium: Combining:

ρbdfy = ρ
bdfs
+ .85fc
β1 bc

(2.84-b) (2.84-c)

(2.85)

f
fs
εu + .85 c β1 (2.86) fy fy εu + εy    Mechanics andρbDesign of Reinforced Concrete

ρb = ρ1 = ρ
Victor Saouma

(2.84-a)

Draft 2–28

FLEXURE

thus fs
+ ρb fy f
= 0.75ρb + ρ
s fy

ρb = ρ1 = ρ
ρmax

(2.87) (2.88)

Note that 0.75 premultiplies only one term as in the other failure is ipso facto by yielding. We also note the similarity with ρmax of T Beams (where 0.75 premultiplied both terms). Test 2 fs = fy is fs
= fy ? We set ε
s = εy , and from the strain diagram

ε u =0.003 εs’ = εy

d’ A’s h

d εs > εy

As

b Figure 2.15: Strain Diagram, Doubly Reinforced Beam; is A
s Yielding? c= from equilibrium

εu d
εu − ε y

ρbdfy = ρ
bdfy + .85fc
β1 cb

(2.89)

(2.90)

combining ρmin ≡ ρ2 = ρ
+ .85β1

fc
d
87 fy d 87 − fy

(2.91)

which corresponds to the minimum amount of steel to ensure yielding of compression steel at failure. Thus, if ρ < ρmin then fs
< fy . Note that some times ρmin can be larger than ρ. Test 3 fs < fy , is fs
= fy (not allowed by ACI)? From strain diagram: c = εs =

Victor Saouma

εu d
εu − ε y d−c εy c − d


(2.92-a) (2.92-b)

Mechanics and Design of Reinforced Concrete

Draft

2.7 Doubly Reinforced Rectangular Beams

2–29

From equilibrium ρbdfs = ρ
bdfy + .85fc
β1 bc combining

(2.93-a)

  c − d

fc
c ρ = ρ3 = ρ + .85β1 d−c fy d

(2.94)

Summary of the tests are shown in Fig. 2.16

Test 1

Test 2

ρmin

II f’ < fy s

Test 3

ρ

ρ III

I f’ = f y

f=fy f
s

ρ IV

f’ < f y f’ = f y s

s

Figure 2.16: Summary of Conditions for top and Bottom Steel Yielding

2.7.2

Moment Equations

Case I fs = fy and fs
= fy , usually occur if we have small bottom and top reinforcement ratios.

As fy = A
s fy + .85fc
ab (As − A
s )fy a = .85fc
b

a + A
s fy (d − d
) MnI = .85fc
ab d − 2

(2.95-a) (2.95-b)

(2.96)

Case II We have fs = fy and fs
< fy (small bottom and large top reinforcement ratios, most common case) c − d
c
= E s εs

(2.97-b)

A
s fs


(2.97-c)

ε
s = εu fs


As fy =

+

.85fc
bβ1 c

(2.97-a)

solve for c and fs
by iteration. Alternatively, those equations can be combined yielding (0.85fc
b)a2 + (0.003Es A
s − As fy )a − (0.003Es A
s β1 d
) = 0 Victor Saouma

(2.98)

Mechanics and Design of Reinforced Concrete

Draft 2–30

Using a = β1 c

FLEXURE

a MnII = .85fc
ab d − + A
s fs
(d − d
) 2

(2.99)

Case III fs < fy and fs
= fy (large bottom and small top reinforcement ratios, rare) d−c c = E s εs

(2.100-a)

ε s = εu fs

As fs =

A
s fy

+

(2.100-b) .85fc
ab

a = β1 c

(2.100-c) (2.100-d)

solve for a

a + A
s fy (d − d
) MnIII = .85fc
ab d − 2

(2.101)

Case IV (not allowed by ACI) fs < fy and fs
< fy (large bottom and top reinforcement ratios, rare) c − d
c d−c = εu c = A
s fs
+ .85fc
ab

ε
s = εu

(2.102-a)

εs

(2.102-b)

As fs

a = β1 c solve for a

a MnIV = .85fc
ab d − + A
s fs (d − d
) 2

(2.102-c) (2.102-d)

(2.103)

Note that in most beams of normal size and proportions, it will be found that fs
< fy when fs = fy . We nevertheless use A
s in order to ensure ductility, stiffness and support for the stirrups. 50

Example 2-12: Doubly Reinforced Concrete beam; Review Given, fc
= 4, 000 psi, fy = 60,000 psi, A
s = 3 (1.56) = 4.68 in2 , As = 4 (1.56) = 6.24 in2 , determine the moment carrying capacity of the following beam.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.7 Doubly Reinforced Rectangular Beams 0.85 f’ c

ε u =0.003

A’s = 3 # 11

εs’

3"

2–31

A’s f’s

0.85 f’ c

A’s f’y

a=β1 c

a=β c

c

1

27.3"

=

d−d’

+

d As sf

εs

As yf

(As − A’ s s)f

16" As = 4 # 11

Solution:

1. Determine ρ: 4 fc
87 87 = (.85)(.85) = .0285 fy 87 + fy 60 87 + 60 6.24 = .0143 (16)(27.3) 4.68 = .0107 (16)(27.3)

ρb = (.85)β1 ρ = ρ
=

(2.104-a) (2.104-b) (2.104-c)

2. Check for ρmin fc
d
εu β1 fy d εu − εy 4 3 .003 = .0107 + (.85) (.85) = .0278 > ρ 60 60 27.3 .003 − 29,000

ρmin = ρ
+ .85

(2.105-a) (2.105-b)

Hence ρb ρ < ρmin < .0143 < .0278 < .0285

(2.106)

and thus fs = fy and fs
< fy and we have case II 3. We have two equations: strain compatibility (nonlinear equation) and summation of forces (linear equation), and two unknowns c and fs
c−3 c − d
= (29, 000)(.003) c c c−3 = 87 c

= As fs + .85fc
bβ1 c

fs
= Es εu

As fy

(6.24)(60) = 374.4 = fs


(4.68)fs
+ (.85)(4)(16)(.85)c 4.68fs
+ 46.24c

= −9.9c + 80.2

(2.107-a) (2.107-b) (2.107-c) (2.107-d) (2.107-e) (2.107-f)

Note that if we were to plott those two equations,

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–32

FLEXURE

50

25

2

3

4

5

6

-25

-50

-75

-100

We note that fs
increases with c from the strain diagram, but fs
decreases with c from equilibrium. If c increases, force in concrete increases too and force in steel decreases. Graphically the solution is around 4.9. 4. Combining those two equations1 c2 + .7085c − 26.42 = 0

(2.108)

we obtain c = 4.80 in a = 0.85(4.8) = 4.078 in, and fs
= (.003)(29, 000) 4.80−3 4.80 = 32.6 ksi 5. Substituting into the moment equation

a (2.109-a) Mn = .85fc
ab d − + A
s fs
(d − d
) 2   4.078 + (4.68)(32.62)(27.3 − 3) (2.109-b) = (.85)(4)(4.078)(16) 27.3 − 2 (2.109-c) = 9, 313 k.in Md = 0.9(9, 313) = 8, 382 k.in = 699 k.ft

(2.109-d)

6. Check ρmax = .75ρb +

fs

ρ fy

= (.75)(.0285) +

(2.110-a) √ 32.6 (.0107) =  .027 60

(2.110-b)

ρ

Example 2-13: Doubly Reinforced Concrete beam; Design Given Md = 505 k.ft, fc
= 4 ksi, fy = 60 ksi, b = 12 in, h = 24.5 in, d = 21 in, and = 2.5 in, determine the reinforcement As and possibly A
s . Solution: d


1

In this problem, unfortunately an iterative method diverges if we were to start with a = d5 .

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.8 Moment-Curvature Relations

2–33

1. Check if singly or doubly reinforced section: Md = (505)(12) = 6, 060 k.in 87 4 f
87 ρb = .85β1 c = .0285 = (.85)(.85) fy 87 + fy 60 87 + 60 ρmax = .75ρb = (.75)(.0285) = .0213

(2.111-b)

Amax s

(2.111-d)

2

= (.0213)(12)(21) = 5.37 in As fy (5.37)(60) = = 7.89 in a =
.85fc b (.85)(4)(12)  

a 7.89 = (.9)(5.37)(60) 21 − Mmax = (0.9)As fy d − 2 2 = 4, 943 k.in < 6, 060 k.in

(2.111-a)

(2.111-c)

(2.111-e) (2.111-f) (2.111-g)

Thus compression steel is required. 2. Assuming that fs
= fy Md2 = 6, 060 − 4, 943 = 1, 117 k.in 1, 117 Md2 = = 1.12 in2 A
s =
φfy (d − d ) (0.9)(60)(21 − 2.5) ⇒ A
s =

1.12 in2

As = 5.37 + 1.12 = 6.49 in2

(2.112-a) (2.112-b) (2.112-c) (2.112-d)

3. Check that fs
= fy 1.12 = .00444 (12)(21) 6.49 = .0257 ρ = (12)(21) f
d
εu ρmin = ρ
+ .85β1 c fy d εu − εy √ 87 4 2.5 = .0229 < ρ(.0257) = .00444 + (.85)(.85) 60 21.0 87 − 60 ρ
=

(2.113-a) (2.113-b) (2.113-c) (2.113-d)

Note that if it turned out that fs
< fy , then we will need to make an assumption on A
s (such as A
s = A2s , as we will have three equations (2 of equilibrium and one of strain compatibility) and four unknowns (As , A
s , fs
and c).

2.8

Moment-Curvature Relations

In ordinary reinforced concrete design, we need not be concerned by the moment curvature relation of a flexural member. Yet this relation is important to properly understand (in subsequent chapters)

51

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–34

FLEXURE

1. Redistribution of moments (reducing negative moments, increasing positive ones). 2. Short and long term deflections with the shifting of the neutral axis under service load. 3. Ductility in seismic design, i.e. the ability of a section to exhibit enough flexibility during seismic excitation, and thus absorbs enough energy. 52

Fig.2.17 shows portion of an originally straight beam which has been bent to the radius ρ ε ψ

O

fy y

ψ ρ

STEEL εy

ε

f’

c

C c d y b

A M

a B

M

CONCRETE

D

ε el

ε

Figure 2.17: Bending of a Beam by end couples M , thus the segment is subjected to pure bending. It is assumed that plane cross-sections normal to the length of the unbent beam remain plane after the beam is bent. Therefore, considering the cross-sections AB and CD a unit distance apart, the similar sectors Oab and bcd give y (2.114) ε= ρ where y is measured from the axis of rotation (neutral axis), ρ the radius of curvature. 53

Furthermore, we define the curvature Ψ as Ψ=

εc ε = y c

(2.115)

Next we seek to derive the moment curvature for a beam. This will clearly depend on the location of the neutral axis, and we identify the following key stages, Fig. ??:

54

Uncracked Elastic is the first stage Mcr = Ψcr =

fr Iut c2 εr fr = c2 E c c2

(2.116-a) (2.116-b)

Cracked Elastic: when the section cracks, the stiffness is immediately reduced and the curvature increases (the moment does not change). We will then have a linear response up

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.9 Bond & Development Length ε<ε

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 111111 000000 000000 111111 000000 111111 000000 111111 11111111 00000000

d

c

h

el

ψcr

1

c2 ε=ε

fr

cr

ε=ε c = kd

M el

d cS = d−kd

EI

ct

Failure

ut

M el

jd ε<ε

T=A

y

el

c

EI n

ψel

1

ε<ε <ε

111111 000000 111111 000000 111111 000000

2–35

f =f

u

1

S

εE S

M cr

S

Proportional Limit of Concrete Cracking

c

ψ inel

1

ψ

d Z ε=ε

cs

T=A

S

εE S

S

< A fy S

Figure 2.18: Moment-Curvature Relation for a Beam to the limit of elasticity of the concrete where, Eq. 2.4-c Mel = εel = Ψ1el = we should check that fs =

M As jd

bd2 kjfc 2 fc,el Ec εel c1

(2.117-a) (2.117-b) (2.117-c)

≤ fy .

Cracked Section, Inelastic Material: For this case 1. Select top face concrete strain εel ≤ ε ≤ εu 2. Assume the neutral axis depth to be at a distance c1 below top fibers. 3. From the strain diagram and similarity of triangles, determine εs = εcs . 4. Determine the tensile steel stress fs = Es εs ≤ fy , and T + As fs . 5. Check equilibrium of forces (C = T ), this requires computing C (area under the nonlinear stress curve). If equilibrium is not satified, adjust location of neutral axis upward or downward until equilibrium is reached. The internal lever arm z is then determined. 6. Solve for i = Ci zi Minel εi Ψiinel = i1 c1

(2.118-a) (2.118-b)

7. Plot the results.

2.9

Bond & Development Length

Considering the equilibrium of forces acting on an infinitesimal portion of a rebar, Fig. 2.19, and defining U as the force per unit length, we have

55

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–36

FLEXURE M

M +∆M

111 000 000 111 111 000

111 000

111 000

C

C + dC T

jd T

V+dV

V

T+dT

111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000

T+dT

dx dx

Figure 2.19: Bond and Development Length U dx = dT ⇒ U = 56

(2.119)

The tensile force is a function of the moment M

= T jd dM = jd

dT

57

dT dx

(2.120-a) (2.120-b)

But the shear is related to the moment V =

dM dx

(2.121)

V jd

(2.122)

Combing those equations together, we obtain U= 58

We define u as the bond stress, and is equal to u=

U Σ0

(2.123)

where Σ0 is the sum of all the bars perimeters. If plain bar → weak adhesion → slip → need end anchorage → no bond → u = 0 → dT = 0 → max steel stress is constant over entire length → T = Mjd → total steel elongation >than if bond present → large deflection and large crack width.

59

60 61 62

Actual stress distribution along steel bar is quite complex, Fig. 2.20. If bond stress is too large ⇒ splitting along reinforcement, Fig. 2.21.   Failure will initiate at points of high shear large dM dx .

It frequently starts at diagonal cracks ⇒ dowel action increases the tendancy of splitting ⇒ shear and bond failures are often interrelated. 63

Basedon tests with one single bar, ultimate average bond force/inch of length of bar is Un ≈ 35 fc
.

64

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.9 Bond & Development Length

M

2–37

u stresses on concrete

M

u stresses on rebar

1111111111111111111111111111111111 0000000000000000000000000000000000 0000000000000000000000000000000000 1111111111111111111111111111111111 1111111111111111111111111111111111 0000000000000000000000000000000000 Steel tension slope =dT dx

Bond stress u

Figure 2.20: Actual Bond Distribution

Figure 2.21: Splitting Along Reinforcement

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–38

FLEXURE

65 If we have several bars in one layer spaced 6 in or less, then the ultimate bond capacity is 80% of the single bar case. 66

In terms of bond stress, Fig. 2.22

 35 fc
un = Σ0

(2.124)

1111111111111111111111111111111111 0000000000000000000000000000000000 Ts = 0

Ld

Ts= A yf b

Figure 2.22: Development Length Putting it differently, the minimum length necessary to develop through bond a force As fy is, Fig. ??.  A f Ld = Ubny   0.028Ab fy Un = un√ Σ0  (2.125) ld =  fc
35 fc
 un = Σ0 67

U

A sf y

L

d

Figure 2.23: Development Length 68

For small bar spacing, we have to decrease the bond stress ld =

0.035Ab fy 1 0.028Ab fy   = 0.8 fc
fc


(2.126)

If actual development length l is smaller than ld , then we must provide anchorage in order to avoid a bond failure.

69

70

Note: 1. un is independent of diameter

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.9 Bond & Development Length 2. For a given fs T ld

2–39  = Ab fs   πd2 ld = = fs 4 b  Ab f s  = Un

fs πd2b 4Un

(2.127)

ld increases with the square of db ⇒ small bar diameters require shorter development length. Top bars, with more than 12 inch of concrete below them, will have a reduced bond stress (due to rise of water during vibration). This reduction in bond results in an increase of ld by 40% 71

72 ACI 12.2.2 may be obtained from above but rather than use φ we increase ld by 15% for safety.

ldb = .04 √b = =

A fy fc
fy .085 √
fc fy √ .125
fc

#11 or smaller; and deformed wire #14

(2.128)

#18

> 12 in. in all cases Consult ACI 12.5 code for hooks geometry, and corrections to this basic equation. Check ACI code for modifications related to top reinforcement, lightweight aggregate, high strength reinforcement, excess reinforcement, and spiral confinement.

73

(2.129)

ld = λd λdd ldb

74

If not enough development length can be provided ⇒ provide hooks, Fig. 2.24 at 1. 90 degrees: bar must extend by 12db 2. 180 degrees: see code.

where lhb = 1200 √db


fc

(2.130)

ldh = λd lhb and λd is given in the ACI code.

2.9.1

Moment Capacity Diagram

Ideally, the steel should be everywhere as nearly fully stressed as possible. Since the steel force is proportional to the moment, then the steel area is nearly proportional to the moment diagram.

75

76

Requirements include, Fig. 2.25:

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–40

FLEXURE db As in part (b)

Critical section

12db ldh (a)

db

Critical section

4db or 2 1/2 in. min.

ldh

4db

Nos. 3 through 8

5db

Nos. 9, 10, 11

6db

Nos. 14 and 18

(b)

Figure 2.24: Hooks 1. At least A3s in simple beams and in. into support.

As 4

for continuous beams should be extended at least 6

2. If negative bars are cut, they must extend at least ld beyond face of support. 3. Negative bars must extend d or 12db beyond theoretical cutoff point defined by moment diagram. 4. At least one third of top reinforcement at support must extend at least ld beyond theln oretical cutoff point of other bars, and d, 12db or 16 beyond the inflection point of the negative moment diagram. Determination of cutoff points can be rather tedious, for nearly equal spans uniformly loaded, in which no more than about one half the tensile steel is to be cut off or bent, locations shown in Fig. 2.26 are satisfactory (note that left support is assumed simply supported).

77

78

Fig. 2.27 is an illustration of the moment capacity diagram for a beam.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2–41

Theoretical positive moment Inflection point for (+As)

Theoretical negative moment

of span

Moment Capacity of bars O

Inflection point for (-As)

C L

Face of support

2.9 Bond & Development Length

Moment capacity of bars M

Greatest of d, 12 d , ln/16 b

d or 12 db

for at least 1/3 of (-AS)

ld Bars M

ld

Bars N

ld

Bars L

ld

Bars O

d or 12 db

6" for at least 1/4 of (+AS) (1/3 for simple spans)

Figure 2.25: Bar cutoff requirements of the ACI code

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–42

FLEXURE

L1 4

L1 3

0"

6"

L1

L1 8

L1 4

6"

L2 3

0"

6"

0"

L1

L1 4

L2 8

6"

L1 3

L1 7

L2 3

L2

L2 3

L2 3

0"

6"

6"

L2 4

L2 8

L2

L2 4

Figure 2.26: Standard cutoff or bend points for bars in approximately equal spans with uniformly distributed load

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

2.9 Bond & Development Length

A

2 bars

2–43

B

C

5 bars

4 bars

A

AA

B

C

BB Ld Ld

Mcap

of 5 bars

CC Mcap of 4 bars

Md=φMn Ld Mcap

of 2 bars

d or 12’’

Figure 2.27: Moment Capacity Diagram

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 2–44

Victor Saouma

FLEXURE

Mechanics and Design of Reinforced Concrete

Draft Chapter 3

SHEAR 3.1

Introduction

Beams are subjected to both flexural and shear stresses. Resulting principal stresses (or stress trajectory) are shown in Fig. 3.1.

1

45

α

90

τ

τ

τ τ

Tension trajectories Compression trajectories

τ

ο

45

τ

σ σ

τ

τ τ

σ1

σ2 α

τ σ

τ σ

τ

σ1

σ2

Figure 3.1: Principal Stresses in Beam 2

Due to flexure, vertical flexural cracks develop from the bottom fibers.

3

As a result of the tensile principal stresses, two types of shear cracks may develop, Fig. 3.2: Large V Small M

Web Shear Cracks

Small V Large M

Flexural Cracks

Large V Small M

Flexural Shear Cracks

Flexural Cracks

Figure 3.2: Types of Shear Cracks Web shear cracks: Large V, small M. They initiate in the web & spread up & down at ≈ 45o .

Draft 3–2

SHEAR

Flexural shear cracks: Large V, large M. They initiate as an extension of a pre-existing flexural crack, initially vertical, then curve. 4

Shear failure is sudden ⇒ φ = 0.85

5

Some of the important parameters controlling shear failure: 1. Shear span ratio

M Vd

2. Steel ratio ρ = Abds   3. ft
= 4 fc
note that fr
= 7.5 fc
We shall first examine the shear strength of uncracked sections, then the one of cracked sections (with shear reinforcement).

6

3.2

Shear Strength of Uncracked Section

Question: What is the maximum shear force which can be applied before a flexural crack develop into a flexural shear crack?

7

1. Apply M → flexural crack 2. Apply V → flexural shear crack Note that all shear resistance is provided by the concrete. As with flexural reinforcement, steel is ineffective as long as the section is uncracked.

8

vn

vc

C

fc

jd

+

Flexure

Shear

T τ

σ Shear

Figure 3.3: Shear Strength of Uncracked Section 9

Solution strategy: 1. Determine the flexural compressive stress fc in terms of M 2. Determine shear stress v in terms of V

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

3.2 Shear Strength of Uncracked Section

3–3

3. Compute the principal stresses 4. Equate principal tensile stress to the tensile strength 10

Using a semi-analytical approach 1. Assume that fc is directly proportional to steel stress  fc = α fns  f = α Mn  αMn Mn c nAs jd fc = = F1 A 2 s  ρ = bd nρjbd ρnbd2 n Mn = As fs jd ⇒ fs = AMs jd 2. Shear stress vn = F2

Vn bd

(3.1)

(3.2)

3. From Mohr’s circle, the tensile principal stress is

τ

vn f1 fc

σ

R

vn

Figure 3.4: Mohr’s Circle for Shear Strength of Uncracked Section

f1 =

fc + 2

"

fc 2

2 + vn2

(3.3)

4. Set f1 equal to the tensile strength f1 = ft
⇒ f1 Vn bd

= =

ft
Vn f1 bd ft
f1 bd Vn

Vn Vn = ft
bd bd

(3.4-a) (3.4-b) (3.4-c)

Combining Eq. 3.1, 3.2, and 3.3

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 3–4

SHEAR

Vn = bd



ft


2

1/2

(3.5)

 F E   F1 Ec Mn  1 c Mn  2  + + F    2   2 E ρVn d  2 Es ρVn d     s    C
C1


2

C1


 5. set ft
= 4 fc
Vn  = bd fc


6. Let the variables be

V √n bd fc


√ C1

fc


ρ

1 " √ Mn Vn d

+

C1

ρ

fc


(3.6)

2 Mn Vn d

+ C2

√

&

Mn fc
ρVn d

7. This is how far we can go analytically. To determine the exact factors associated with this equation, one has to undertake a series of tests. 8. From 440 tests, Fig. 3.5 it is found that

Vn bd f’c 3.5 2.0 1.9

Vn d M f’ n c Figure 3.5: Shear Strength of Uncracked Section

ρVn d Vn  ≤ 3.5  = 1.9 + 2, 500 bd fc
Mn fc
or if we set vc =

Vn bd ,

then

  ρVn d vc = 1.9 fc
+ 2, 500 ≤ 3.5 fc
Mn 9. Note that vc is in terms of Victor Saouma

(3.7)

Vn d M

(ACI 11.3.2.1)

(3.8)

or inverse of shear span ( M V ) Mechanics and Design of Reinforced Concrete

Draft

3.3 Shear Strength of Cracked Sections

3–5

10. This equation is usually found acceptable for predicting the flexure shear cracking load for shear span/depth ratio VMnnd of 2.5 to 6 & is found to be very conservative for lower values 11. Increasing ρ has a beneficial effort as a larger amount of steel results in narrower & smaller flexural tension cracks before formation of diagonal cracks ⇒ larger area of uncracked concrete can resist the shear. 12. Use of Vu & Mu instead of Vn =

3.3

Vu φ

& Mn =

Mu φ

Shear Strength of Cracked Sections

 If the shear stress exceeds 1.9 fc
+ 2, 500ρ VMndd , then the flexural crack will extend into a flexural shear crack, Fig. 3.6. and if 11

s

C Vc A v fv z Va

Vd

T=As fs p

Figure 3.6: Free Body Diagram of a R/C Section with a Flexural Shear Crack 1. No shear reinforcements ⇒ failure 2. Stirrups are present ⇒ stirrups will carry part of shear force 12

If the section is cracked, Fig. 3.7 Vext = Vc + Σn Av fv + Vd + Vay   

(3.9)

Vint

where Vc n Av fv Vd Va 13

Shear resisted by uncracked section # of stirrup traversing the crack n = Area of shear reinforcement Shear reinforcement stress Dowel force in steel Aggregate interlock

p s

We must determine the internal (resisting) shear forces at failure where fv = fy

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 3–6

SHEAR

V int

in t

V cz

ΣV

Vd V ay

Yield of stirups Failure

Inclined cracking

Flexural cracking

Vs

Vext

Figure 3.7: Equilibrium of Shear Forces in Cracked Section 1. Due to yielding → large separation between 2 sides of cracks ⇒ Va → 0 2. Neglect Vd 3. Vext = Vn =

Vc +nAv fy 

unknown

4. We will assume that at failure the shear force provided is equal to the one  by concrete Vn d
which caused the diagonal crack to form ⇒ va = 1.9 fc + 2, 500ρ Md . Thus, Vc = va bw d 5. Finally, if we assume p = d (implying a crack at 45◦ ) d Vn = Vc + Av fy   s

(ACI 11.1.1)

(3.10)

Vs

3.4 14

ACI Code Requirements

+ The ACI code requirements ( 11) are summarized by Fig. 3.8: 1. Design for Vu (factored shear) rather than Vn =

Vu φ

(ACI 11.1.1), ⇒ plot Vu diagram.

2. Determine φVc (nominal shear carried by the concrete) where  Vc = 2 fc
bw d (ACI 11.3.1.1) or   = [1.9 fc
+ 2, 500ρw VMuud ]bw d ≤ 3.5 fc
bw d (ACI 11.3.2.1) Vc where Vu d < 1 Mu

(3.11)

3. If Vu <0.5 φVc no shear reinforcement is needed (ACI 11.5.5.1) Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

3.4 ACI Code Requirements

3–7

4. If 0.5φVc < Vu ≤ φVc use minimum shear reinforcement; select Av (usually #3 bars) and determine Av f y s = 50b (ACI 11.5.5.3) w d (3.12) s < 2 (ACI 11.5.4.1) s < 24 in (ACI 11.5.4.1)

5. If Vu > φVc ⇒ provide stirrup such that Vu φ

= Vn = Vc + Vs = Vc + or

Av fy d s

(ACI 11.17)

φAv fy Av fy d = (v − Vc u − φvc )b

s =

(3.14)

Vu φ

 6. If Vu − Vc > 4 fc
bw d, then s < 7. Upper limit:

d 4

(3.13)

and s < 12 in, (ACI 11.5.4.3).

 Vu − Vc < 8 fc
bw d

(ACI 11.5.6.8)

(3.15)

8. fy ≤ 60 ksi, (ACI 11.5.2) 9. Critical section is at d from support (reduces design shear force), (ACI 11.1.3.1) d V b wd

Vu f ’c

6

f ’c

2

f ’c

Steel

f ’c 4

8

f ’c

φ 10

Concrete

s max=d/4 or 12"

s=

A vfy d Vu −Vc φ

s max=d/2 or 24"

=

s =

distance from support

φ Avfy (v u−φv c) b

no stirups

not allowable

A v fy min. stirups 50b w

f ’c

Figure 3.8: Summary of ACI Code Requirements for Shear

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 3–8

3.5

SHEAR

Examples

Example 3-1: Shear Design b = 12 in.; d=22 in.; wu = 8.8 k/ft; L= 20 ft.; As = 3# 11; fc
= 4 ksi; fy = 40 ksi; Design vertical stirrup Solution: 1. At support: Vu = 8.8 (20) 2 = 88 k and vu = 2. At d from support Vu = 88 −

22 12 (8.8)

88 (12)(22)

= .333 ksi

= 71.9 k and vu =

71.9 (12)(22)

= .272 ksi

 √ 3. vc = 2 fc
= 2 4, 000 = 126 psi; φvc = (0.85)(126) = 107.1 psi 4.

φvc 2

= 53.6 psi

 5. 4 fc
= 2(126) = 252 psi;

 √ 6. vu − vc = 272 − 126 = 146 psi < 4 fc
psi

d

333 Vu

272 107.1 φv c 53.6 φv c 2 x 19" min. reinforcement no reinforcement

38.6"

7. vu − φvc = 0 ⇒

333 (10)(12) x

= 107.1 ⇒ x = 38.6 in = 3.2 ft from mid-span

φvc 2

333 (10)(12) x

= 53.6 ⇒ x = 19.3 in = 1.6 ft

8. vu −

=0⇒

9. Selecting #3 bars, Av = 2(.11) = .22 in2 smax =

Victor Saouma

Av f y 50bw

=

(.22)(40,000) = 14.66 (50)(12) d 22 2 = 2 = 11

 in in

smax = 11 in

(3.16)

Mechanics and Design of Reinforced Concrete

Draft

3.6 Shear Friction

3–9

10. at support s =

Av fy d φAv fy = (v − Vc u − φvc )b

(3.17-a)

Vu φ

=

(.85)(.22)(40, 000) (272 − 107.1)(12)

(3.17-b)

=

3.78 in

(3.17-c) (3.17-d)

3.6

Shear Friction

Previous design procedure was applicable to diagonal tension cracks (where tension was induced by shear), for those cases where we do have large pure shear, Fig. 3.9, use shear friction concept.

15

An=

N uc φfy

#7

Vu weld Nuc

A n part of A v f

close sriru (usually #3)

Avf

assumed crack + shear plane remainder of A v f

assumed crack

Vu

Figure 3.9: Corbel The crack for which shear-friction reinforcement is required may not have been caused by shear. However once the crack has occurred a shear transfer mechanism must be provided for, Fig. 3.10. The shear friction theory is based on the assumption that a crack will occur and then reinforcement across it will resist relative displacement along the crack.

16

17

If we assume separation to be sufficient⇒ steel will yield Vn = µAvf fy

18

(3.18)

If the shear reinforcement is inclined with respect to the crack, Fig. 3.11

Component of tensile force in reinforcement gives rise to compression force at interface C ⇒ µc vertical force due to friction;   Vn = T (cos αf − µ sin αf ) Vn = T cos αf + µC Vn = Avf fy (cos αf + µ sin (3.19-a) αf C = T sin αf T = Avf fy

19

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 3–10

SHEAR

Vn

Vn

Vn

crack separation due to slip

crack

Vn

Vn

µ A vf f y

Shear−transfer reinforcement

A Avf f y 2

vf

fy

Avf f y 2

Figure 3.10: Shear Friction Mechanism

Tsin α f Tcos α f assumed crack

applied shear=Vn

A vf fy

µC

C=Tsin α f

αf T

Figure 3.11: Shear Friction Across Inclined Reinforcement

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

3.6 Shear Friction

3–11

20

Note: Vu = φVn and φ = 0.85

21

The preceding equation can be rewritten as

22

Avf

=

Vu φµfy

Avf

=

Vu ACI − 11.27 (3.21) φfy (cos αf + µ sin αf )

(3.20)

ACI-11.7.4.3 specifies µ as such that concrete cast monolithically µ = 1.4λ concrete against hardened concrete µ = 1.0λ concrete against steel µ = 0.7λ where λ = 1.0 for normal weight concrete and λ = .75 for lightweight concrete. and , 0.2fc
Ac (3.22) Vn < 800Ac

and Ac ( in2 ) is the area of concrete resisting shear. Example 3-2: Shear Friction Design reinforcement needed at the bearing region of a precast beam 14” wide & 28” deep supported on a 4” bearing pad. Vu = 105k, horizontal force due to restraint, shrinkage, creep is 0.3 Vu

possible crack

A

20

vf

3#6

15

N uc

2#6

15

N uc

Vu 4"

Vuc

24"

Solution: 1. Assume all the shear Vu will be acting parallel to crack (small angle 20◦ ) 2. Assume all Vu is parallel to crack ⇒ required Avf =

Vu φfy µ

=

105 (0.85)(60)(1.4)

= 1.47 in2

(0.3)(105) 2 ac 3. As = N φfy = (0.85)(60) = 0.62 in for horizontal force ⇒ As = Avf + An = 1.47 + 0.62 in2 = 2.09 in2 ⇒ use 5# 6 (As = 2.20 in2 )

4. Note: ACI 11-9-3-4 Nuc > 0.2 Vu for Corbels;

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 3–12

3.7

SHEAR

Brackets and Corbels

To be Edited 23

Nu might be due to shrinkage, prestressing · · ·

24

Design based on truss analogy

25

A.C.I. provisions (Chapter 11) 1. For

a d

< 12 , use shear friction theory

2. For

a d

> 1, use ordinary beam theory

3. For

1 2

≤

a d

≤1 

Vn = [6.5 − 5.1

   α Nu 3 Nuc fc
bw d ](1 − 0.5 ) 1 + [64 + 160 ( ) ]ρ Vu d Vu

(3.23)




u where ρ = A??s ; and ρ ≤ 0.13 ffyc ; N Vu not to be taken < 0.20 in calculating vu ; Nu = (+ve) compression, and (-ve) tension; Ah < As also Ah ≥ 0.50As distributed uniformly; thru fc
As 2 3 d adjacent to As ; ρ = bd ≥ .04 fy .

3.8

Deep Beams

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft Chapter 4

CONTINUOUS BEAMS 4.1

Continuity

R/C bldgs constructions commonly have floor slabs, beams, girders and columns continuously placed to form a monolithic system

1

Figure 4.1: Continuous R/C Structures +ve In a continuous system, load must be placed in such a way to maximize desired effect (Mmax −ve V Mmax max , Fig. 4.2 2

A

B

C

D

E

F

G

Max +ve M @ AB_CD_EF Max -ve M @ B

Min -ve @ B

Max -ve @ C

Min -ve @ C

Max -ve @ D

Min -ve @ D

Figure 4.2: Load Positioning on Continuous Beams Given the moment diagram for various load cases, a designer should draw the moment enveloppe and design for the maximum negative and positive moments (eventhough they may not be caused by the same load case).

3

Draft 4–2

4.2 4

CONTINUOUS BEAMS

Methods of Analysis

Two approaches: 1. Detailed analysis (a) Moment distribution (b) Computer analysis (such as RISA, SAP, etc...) 2. Approximate (but conservative) based on ACI 8.3.3 moment coefficients

4.2.1 5

Refer to CVEN3525/3535/4525

4.2.2 6

Detailed Analysis

ACI Approximate Method

This method, Fig. 4.3 can be used if: 1. 2 or more spans 2. Spans are approximately equals, and the larger of adjacent ones not greater than the shorter by more than 20% 3. Loads are uniformably distributed 4. LL < 3DL 5. Prismatic members Positive Moment End Spans Continuous end unrestrained Continuous end integral with support Interior spans Negative Moment Negative moment at exterior face of first of first interior support Two spans > Two spans Negative moment at other faces of interior support ...................................... Shear Shear in end member at face of first interior support Shear at face of all other supports

1 2 11 wu Ln 1 2 14 wu Ln 1 2 16 wu Ln

1 2 9 wu Ln 1 2 10 wu Ln 1 2 11 wu Ln

...... 1.15 wu2Ln wu Ln 2

where wu is the factored load, and Ln is the clear span. 7

These moment coefficients take into account some inelastic action (stress redistribution).

8

They are conservative compared to an exact analysis.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

4.2 Methods of Analysis

4–3

Figure 4.3: ACI Approximate Moment Coefficients

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 4–4

CONTINUOUS BEAMS

C L column

C L column

V

111 000 000 111 000 111 000 111 000 111 000 111

VaL 2

CL span

VaL 6

VaL 3

C L beam

aL 2 Column width aL

VaL 2

VaL 3

VaL 6

L 2 Moment curve based on prismatic member

aL 2

1111111111111111111111111 VaL 0000000000000000000000000 6 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 Adjusted Moment Curve 0000000000000000000000000 1111111111111111111111111 C beam L 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 C L beam 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111

Figure 4.4: Design Negative Moment

4.3 9 10

Effective Span Design Moment

Negative moments should be the one at the face of the columns, Fig. 4.4. We recall that the change in moment is equal to the area under shear diagram. M = Mcl −

V aL 2

(4.1)

but V and M vary in some unknown way between center line of column and edge, thus we can reduce M by V b/3 where b is the width of the column. Thus −ve Md−ve ≈ Mmax −

11

Vb 3

(4.2)

This can substantially reduce high M −ve .

4.4 4.4.1

Moment Redistribution Elastic-Perfectly Plastic Section

12

Let us consider a uniformly loaded rigidly connected beam, Fig. 4.5

13

The beam has an elastic plastic moment curvature relation, Fig. 4.6

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

4.4 Moment Redistribution

4–5 2

WL 24

W

1 0 0 1 0 1

11 00 00 11 00 11

+

2

WL 12

−

−

2

WL 12

Figure 4.5: Moment Diagram of a Rigidly Connected Uniformly Loaded Beam M Mp

X

Φy

Curvature

Φu

Figure 4.6: Moment Curvature of an Elastic-Plastic Section 14

|M −ve | > |M +ve | as w , M −ve → Mp first ⇒ 12Mp wL2 = Mp ⇒ w = 12 L2

15

Thus we will have a plastic hinge at the support however this is not synonymous with collapse.

Collapse or failure occurs when we have a mechanism or 3 adjacent hinges (plastic or otherwise). This can be easily determined from statics, Fig. 4.7

16

w 16 M

L M

p

12 M

L

M

p

M

p

2

p

2

∆ p

Figure 4.7: Plastic Moments in Uniformly Loaded Rigidly Connected Beam 2Mp = wu = 17

wu L2 8 16Mp L2

Thus capacity was increased 33% after first plastic hinge occurred.

This is accompanied by large rotation of the plastic hinges at the supports, and when compared with the linear elastic solution M −ve and M +ve 

18

19

The section must be designed to accomodate this rotation.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 4–6

4.4.2

CONTINUOUS BEAMS

Concrete

Concrete is brittle hence by itself no appreciable plastic deformation can occur, however in R/C, Fig. 4.8

20

ε ce

.003

fc kd

c

d−kd

d−c

θy

θu Asf y

ε s= ε y M

ε s> ε y

M

Asf y

.003 εc

u Steel yielding

M

fc

y First crack

cr

θcr θ y

θu

Strain caused by moment redistribution

θy

Unit rotation

εy

θu

Figure 4.8: Plastic Redistribution in Concrete Sections If certain rotation capacity exists (i.e., if ρ − ρ
is low) M is controlled by yielding of the steel while the concrete strain is still low compared to 0.003 ⇒ reserve rotation capacity θu − θy is then available for a redistribution of moment to occur before ε → 0.003 21

M −ve moment at support of continuous flexural members calculated by elastic theory can be decreased by no more than 22

∆M = 20(1 −

ρ − ρ
)% ACI 8.4.1 ρb

(4.3)




87 ) provided that where ρb = 0.85β1 ffyc ( 87+f y

1. Moments are exactly determined (i.e., not ACI coefficients) 2. ρ or ρ − ρ
< 0.5ρb 23

M +ve must be increased accordingly.

This capacity to redistribute moments (reduce M −ve and increase M +ve ) is a characterisitc of ductile members. 24

Earthquake resistant structures must have a certain ductility to absorb the lateral oscillating load ⇒ large amount of reinforcement at the joints.

25

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 4.5 Buildings

4–7

Example 4-1: Moment Redistribution Determine the moment redistribution for the following singly reinforced beam with ρ = 0.5ρb 2

WL 20

WL 24 +

2

WL 12

−

−

+

2

2

WL 12

0.9

WL 12

2

−

−

2

0.9 W L 12

Solution: From above, amout of redistribution ∆M M −ve M +ve

4.5 26

= 20(1 −

ρ−ρ
ρb )%

= 20(1 − 0.5) = 10% 2 = 0.9 wL 12 2 wL2 = 1.2 wL 24 = 20

Buildings

Building types, Table 4.1 Structural System Frame Shear Wall-Frame Single framed tube Tube in Tube

Number of Stories Up to 15 up to 40 up to 40 up to 80

Table 4.1: Building Structural Systems 27

We analyse separately for vertical and horizontal loads.

Vertical loads: DL and LL. This is typically done for a floor, through a grid analysis. No need to model the entire structure. We can use ACI Approximate equations Exact (Moment distribution, computer) Lateral laod: WL, EL. This requires the analysis of a 2D or 3D frame. Two approaches: Approximate method: Portal method, or cantilever method. Exact Moment distribution, computer. 28

Recommended analysis/design procedures

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 4–8

CONTINUOUS BEAMS

1. Use ACI approximate equations for the design of the slab. Then, there is no need to worry about optimal placement of load to maximize positive or negative moments, or moment redistribution. 2. Once the slab is designed, use exact method for beams, girders. Reduce negative moments. 3. Tabulate maximum +ve and -ve moments for each beam. 4. Determine the column loads, tabulate. 5. Can use approximate or “exact” method of analysis for frames. Tabulate results. 6. Add maximum positive and negative moments due to vertical and lateral loads. 7. Design accordingly. 29

A block diagram for the various steps is shown in Fig. 4.9

E-W SLAB

N-S BEAM

b

L

E-W GIRDER

b

L h

N-S GIRDER

L

L

h

h

b

hf DL

w0

w0

LL

wu

M L hf h M V R PW WL W0 Wa Col Fou

V Span Slab thickness Beam/girder depth Flexure Shear Reation Partition wall Wind load Self weight Total factored load Column Foundation

PW

PW

wu

R

M

WL

w0

PW

R

V

M

WL

wu

wu

V

w0

M

V

N

Col W

E

Fou S

R/C Bldg Design

Figure 4.9: Block Diagram for R/C Building Design

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft Chapter 5

ONE WAY SLABS 5.1 1

Types of Slabs

Types of slabs, Fig. 5.1

Beam

Beam

Beam

Beam

Beam

Beam

one−way slab

two−way slab

one−way slab

Flat plate slab

Grid slab

Flat slab

Figure 5.1: Types of Slabs 2

Two types of slabs, Fig. 5.2 1. One way slab: long span/short span > 2. Load is transmitted along the short span. 2. Two Way slab: Long span/short span <2. Load is transmitted along two orthogonal directions.

3

If

L s

> 2 than most of the load (≈ 95%) is carried in the short directions, Fig. 5.3

Load transfer in one way slabs is accomplished hierarchically through an interaction of slab, beam, girder, column and foundations, Fig. 5.4

4

Draft 5–2

ONE WAY SLABS

1111 0000 0000 1111 0000 1111 1111 0000

1111 0000 0000 1111 1111 0000 1111 0000

Strip B

S

S

L 1111 0000 0000 1111 0000 1111 0000 1111

0000 1111 0000 1111 0000 1111 0000 1111

11111 00000 00000 11111 00000 11111 00000 11111

11111 00000 00000 11111 11111 00000 11111 00000

Beam 2

B

1’−0"

Beam 2

Beam 1

Strip

Beam 1

B

Beam 1

B 1’−0"

11111 00000 00000 11111 00000 11111 00000 11111

Beam 1

11111 00000 00000 11111 11111 00000 11111 00000

Figure 5.2: One vs Two way slabs

ρB

B

111 000 000 111

P ρA A 101010

1 0 A 0 1 0 1

000 111

000 B111

Figure 5.3: Load Distribution in Slabs

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

5.1 Types of Slabs

5–3

Figure 5.4: Load Transfer in R/C Buildings

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 5–4

ONE WAY SLABS

Solid One way slab Beams or ribbed One way slab

Simply supported

One end continuous

Both ends continuous

Cantilever

L/20

L/24

L/28

L/10

L/16

L/18.5

L/21

L/8

Table 5.1: Recommended Minimum Slab and Beam Depths

5.2 5

One Way Slabs

Preliminary considerations for one way slabs: 1. Load on slabs ksf. 2. Design an imaginary 12 in strip. 3. The area of reinforcement is As /ft of width or   12 in As = Ab ft bar spacing in inches

(5.1)

where Ab is the area of one bar. or Bar spacing in inches =

12Ab As

(5.2)

4. Slab thickness t is usually assumed, and we design reinforcement. ACI 9.5.2.1 recommended minimum thickness of beams/slabs are given by Table 5.1. where L is in inches, and members are not supporting partitions. If a slab is so dimensioned (a) Deflection need not be checked (b) Usually, neither flexure, nor shear controls 5. In reinforcement design, a good initial guess for

a d

is 0.15.

6. Slab thickness are rounded to the neares 1/4 inch for slabs less than 6 inch, and 1/2 for thicker ones. 7. ACI Sect. 7.7.1 gives minimum cover for corrosion control (a) Concrete not exposed to weather or in contact with ground, No. 11 or smaller 3/4 inch. (b) Concrete exposed to weather or in contact with ground: i. No. 5 bars and smaller, 1.5 inch. ii. No. 6 and larger, 2. inch. 8. Transverse reinforcement (shrinkage, temperature) must be provided  0.002 Grade 40 and 50 bars As  ACI 7.12.2.1 =  bh 0.0018 Grade 60 and welded wire fabric Victor Saouma

(5.3)

Mechanics and Design of Reinforced Concrete

Draft

5.3 Design of a One Way Continuous Slab

5–5

 9. Shear does not usually control & no minimum reinforcement is needed (vc = 2 fc
)

10. Principal reinforcement shall not be spaced at more than 3 times the slab thickness nor 18 in (ACI 7.6.5). 11. Usually No. 4 and larger bars are used for flexural reinforcement, as No. 3 may be bent out of position by workers walking on it. This is more critical for top than bottom reinforcement. 12. Sometimes, No.3 is used for bottom, and No. 4 for top. 13. Shrinkage/temperature reinforcement shall not be spaced at more than 5 times the slab thickness nor 18 in (ACI 7.12.2.2).

5.3

Design of a One Way Continuous Slab

Design an 8 span floor slab. Each span is 15 ft long, fc
= 3, 750 psi, fy = 60 ksi, wl =100 psf, floor cover is 0.5 psf, mechanical equipment 4 psf, and ceiling 2 psf. Interior supporting beams have a width of 14 inch, and exterior ones 16 inches. First span is measured from exterior of exterior beam to center of first interior beam. Thickness: of the floor is based on ACI recommendation: 16 14 − = 165 in 2 2 14 = (15)(12) − 2 = 166 in 2 165 l = = 6.88 in = 24 24 166 l = = 5.93 in = 28 28

le = (15)(12) −

(5.4-a)

li

(5.4-b)

hemin himin

We round h up to h = 7.25 in. Assuming 3/4 in. cover and No. 4 bars   0.5 d = 7.25 − 0.75 + = 6.25 in 2 Factored Loads

(5.4-c) (5.4-d)

(5.5)

Slab wd =

7.25 (150) = 90.6 psf of floor surface 12

(5.6)

Total dead load 90.6 + 0.5 + 4 + 2 = 97.1 psf Factored load wu = 1.4(97.1) + 1.7(100) = 306 psf

(5.7)

The load per foot of strip is 306 lbs/ft Since wl < 3wd we can use the ACI 8.3.3 coefficients to compute the moments. Net spans 1. First interior span ln = (15)(12) − Victor Saouma

16 2

−

14 2

= 165 in = 13.75 ft

Mechanics and Design of Reinforced Concrete

Draft 5–6

ONE WAY SLABS

2. Second interior span ln = (15)(12) − 14 = 166 in = 13.83 ft 1 3. Average span ln = 12 (165 + 166) 12 = 13.79 ft

Flexural Design ai = 0.15d = 0.15(6.25) = 0.9375 in 12Mu 0.222 Mu As = = = Mu φfy (d − a2 ) 0.9(60)(6.25 − a2 ) 6.25 − a2 As fy 60 = As = 1.569As a =
0.85fc b (0.85)(3.75)(12)

(5.8-a) (5.8-b) (5.8-c)

= 0.0018bh = 0.0018(12)(7.25) = 0.157 in2 /f t Amin s

(5.8-d)

For maximum spacing, ACI specifies 3h = 3(7.25) = 21.75 in but no more than 18 in, ⇒ smax = 18 in.

ln , ft wu ln2 M Coeff. Mu ft-kip/ft a As a As a As Amin s Reinf. Aprov s

Support 13.75 57.85 1/24 2.41 0.937 0.092 0.145 0.087 0.136 0.087 √ 0.157 #4@15 0.16

Midspan 13.75 57.85 1/14 4.13 0.937 0.159 0.249 0.150 0.235 0.150 √ 0.157 #4 @15 0.16

Support 13.79 58.19 1/10 √ 5.82

5.82 0.937 0.223 0.351 0.213 0.334 √ 0.212 0.157 #4@12 0.20

1/11 5.29

Midspan 13.83 58.53 1/16 3.66 0.937 0.141 0.221 0.132 0.207 0.132 √ 0.157 #4@15 0.16

Support 13.83 58.53 1/11 5.32 0.937 0.204 0.320 0.194 0.304 √ 0.194 0.157 #4@12 0.20

Midspan 13.83 1/16 3.66

Shear Since we have unequal spans we must check at 1. Exterior face of the first interior support Vu = 1.15wu

(1.15)(306)(157) ln = = 2, 302 lb/ft of width 2 2

(5.9)

(1.0)(306)(166) ln = = 2, 117 lb/ft of width 2 2

(5.10)

2. Typical interior span Vu = 1.0wu

The shear resistance is   φVc = (0.85)2 f
cbw d = (0.85)(2) 3, 750(12)(6.25) = 7, 808 lb/ft

(5.11)

hence the slab is adequate for shear.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

5.3 Design of a One Way Continuous Slab

5–7

Shrinkage and Temperature Reinforcement must be provided perpendicular to the span of the slab (5.12) As = 0.0018bh = 0.0018(12)(7.25) = 0.157 in2 /f t and maximum spacing is 18 in. Therefore, we can provide # 4 bars at 15 in. as shrinkage and temperature reinforcement. They should be placed on top of the lower layer of steel. Note that in this problem a 6.5 in. thickness was acceptablee for the six interior spans, but a 7.25 in. thickness was required for the end spans. If the entire floor were made of 6. in. thick slab instead of 7.25 in. about 45 cubic yards of concrete could have been saved (for a total floor width of about 90 ft) per flor or 180 kips of dead load per floor. This would represent a considerable saving in say a 20 story building. In this case, it would be advisable to use 6., and check for delfections in the end spans.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 5–8

Victor Saouma

ONE WAY SLABS

Mechanics and Design of Reinforced Concrete

Draft Chapter 6

SERVICEABILITY So far we have focused on the ultimate structural behaviour (failure), Vu & Mu , i.e the strength of a member. 1

2 It is important to also control the behaviour of structural elements under service load (unfactored)

1. Cracking 2. Deflection

6.1

Control of Cracking

3 As σy , εy ⇒ larger crack width is associated with large fy . This is why the ACI code places a limitation on max fy = 80ksi. (ACI 9.4) 4

The concern is not the # of crack (we can not control it) but rather the crack width.

5

Crack width should be minimized because: 1. Appearance 2. Corrosion of steel 3. Redistributions of internal stresses 4. Effect on deflection

6

The controlling parameters are: 1. Surface of the reinforcing bar (a) Round & smooth ⇒ few wide cracks (bad) (b) Irregular & deformed ⇒ many small cracks (better) 2. Steel stress 3. Concrete cover

Draft 6–2

SERVICEABILITY

4. Distribution of steel over the tensile zone of concrete ←

Based on purely experimental research, the following emperical relation was determined, Fig. 6.1:  (6.1) w = .076βfs 3 dc A Gergely & Lutz Eq.

7

where w fs dc β A

width in 1/1,000 in Steel service stress ksi (if not computed can be assumed as 0.6 fy ) Thickness of concrete cover measured from tension face to center of bar closest to this face, in. h2 h1

Area of concrete surrounding one bar = Total effective tensile area in2 # of bars

Neutral Axis

2y y

111 000 000 111 000 111 000 111 000 111

h1

h

2

Steel Centroid

d

c

w Figure 6.1: Crack Width Equation Parameters 8

ACI 1. Expresses the crack width indirectly by z where z= and assumes β =

h2 h1

 w = fs 3 dc A .076β

(ACI 10.6.4)

(6.2)

= 1.2 ⇒ w = .091z

Interior beams z ≤ 175 (w = .016 in) Note that to reduce z (beneficial) we must Exterior beams z ≤ 145 (w = .013 in) reduce A or increase the number of bars. 2. Only deformed bars can be used 3. Bars should be well distributed in tension zone 4. fy < 80 ksi 5. In lieu of an accurate evaluation, fs = 0.6fy . 9

Maximum acceptable crack width (ACI Committee 224).

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

6.2 Deflections

6–3 Exposure dry air, or protective membrane humidity, moist air, soil deicing chemicals seawater, salt water retaining structures

wmax (in.) .016 .012 .007 .006 .004

Example 6-1: Crack Width

0000000000000000 1111111111111111 1111111111111111 0000000000000000

14.65"

00000000000 11111111111 11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111

12.15"

20" 2.5"

22.5"

7.85"

fc
= 3,000 ksi; fy = 40 ksi; As = 4 # 8; LL = 2.44 k/ft; DL = 1.27 k/ft; L = 15 ft.; Determine z and crack width

11.5"

Solution: √ 1. w = .076βfs 3 dc A √ 2. Ec = 57 3, 000 = 3, 120 ksi 3. n =

29×103 3,120

= 9.29

4. Taking first moment (Eq. 2.6) .869 ⇒ kd = 7.85 in

b(kd)2 2

− nAs (d − kd) = 0 ⇒ k = .393 ⇒ j = 1 −

k 3

=

2

2

(1.27+2.44)(15) (12) M 5. M = wL 8 and fs = As jd ⇒ fs = 8(3.14)(.869)(20) = 22.9 ksi Note that ACI allows 0.6fy = (0.6)(40) = 24 ksi conservative

6. β =

22.5−7.85 20−7.85

=

14.65 12.15

= 1.206 (note ACI stipulates 1.2)

(2.5)(2)(11.5) 4

= 14.38 in2  1 8. w = (.076)(1.206)(22.9) 3 (2.5)(14.38) 1,000 = .00696 in .

7. A =

 √ 9. or z = fs3 dc A = (22.9) 3 (2.5)(14.38) = 75.64

6.2

Deflections

10

ACI Code Sect. 9.5

11

Every structural design must satisfy requirements of strength, stiffness & stability

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 6–4

SERVICEABILITY

With the increased usuage of: a) high strength material (resulting in smaller cross section) & b) use of refined design methods, we can no longer rely on the factor of safety to “take care” of deflection, ⇒ we but must detemine it

12

13

Deflection should be controlled because of: 1. Visually unacceptable 2. Possible ponding of water 3. Cracking in partition walls 4. Functional difficulties (windows, doors, etc · · ·) 5. Machine misalignment 6. Vibration

14

Deflection are computed for service loads only

15

Both long term & short term deflection should be considered.

16

As a rule of thumb, deflections seldom control if ρ < 0.5ρb

6.2.1 17

Short Term Deflection

In general δ =

f (w,L) EI ,

i.e., uniform load over simply supported beam in

5wL4 384EI

f (w, l) and E are known, but how do we determine I? (uncracked transformed or cracked), Fig. 6.2

18

Ε cΙ ut

Ε cΙ e1 Ε cΙ e2

Ε cΙ cr

Μ2 Μ1 Μ cr Α

B Α B

Α B

Α

B

Α

B

∆

1

∆

2

Figure 6.2: Uncracked Transformed and Cracked Transformed X Sections 19 20

It would be too complicated to have I = I(M ) ACI recommends to use a weighted average expression for I → Ieff       Mcr 3 Mcr 3 Icr (ACI 9.5.2.3) Ieff = Ig + 1 − Ma Ma

Victor Saouma

(6.3)

Mechanics and Design of Reinforced Concrete

Draft

6.2 Deflections

6–5

where

Ieff ≤ Ig I Mcr = fr
ygb  fr
= 7.5 fc


and Ma is the maximum (service) moment at stage in which deflection is computed 21

For continuous beams average Ieff = 0.70Im + 0.15(Ie1 + Ie2 )

For beams with one end continuous Ieff = 0.85Im + 15(Icon ) where Im , Ie are the moment of inertia at the middle and the end respectively. 22

23

Note that Ig may be substituted for Iut

Deflection evaluation is a nonlinear problem, as w  M  ous beam , 5 wL4 w ∆ ∆= I ∆ 384 EI 24

6.2.2

Mcr Ma

Ieff and for a continu-

Long Term Deflection

δt δ inst. t Figure 6.3: Time Dependent Deflection 25

Creep coefficient: Cc = Ect =

εf εi σ σ ε = εi (1+Cc )

=

Ec 1+Cc

⇒ Creep tends to reduce the elastic modulus of concrete, Fig. 6.4 26

From Strain diagram: 1. Steel strain remains unchanged 2. As concrete undergoes creep, the N.A. moves down ⇒ larger area of concrete is under compression but since C = T ⇒ stress in concrete is slightly reduced

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 6–6

SERVICEABILITY

b Cracked elastic neutral axis

fci

εt

εi

fct

φi

kd

φt d

A s fs

As εs Figure 6.4: Time Dependent Strain Distribution

3. But since C is now lower and we still satisfy Mext = Mint both stresses in steel & concrete must increase with time 27

According to ACI section 9.5.2.5: 1. Additional long term deflection δ t δt = δi × λ where λ = ρ




=

(6.4)

ξ 1+50ρ

As bd

and 3 1.0

Time (months) ξ

6 1.2

12 1.4

≥ 60 2.0

Thus compressive reinforcement can substantially reduce long term deflections (6.5)

δtotal = δinitial (1 + λ)

A

B

C

LL short DL sustained

1111 0000 1111 0000

1111 0000 0000 1111

Figure 6.5: Short and long Term Deflections Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

6.2 Deflections 28

6–7

Short and long term deflections, Fig. 6.5 A → δi,sust B → δi,sust + δt,sust C → δsust + δi,short

Note that we are usually interested in the live load deflection (C-B), thus δi, short = δi, sust + short − δi, sust       Ieff (DL+LL)

29

(6.6)

Ieff (DL)

ACI max. deflections (ACI 9.5.2.6)

Flat roof not supporting nonstructural elements likely to be damaged Floors not supporting nonstructural elements likely to be damaged Roofs or floors supporting nonstructural elements likely to be damaged Floors not supporting nonstructural elements not likely to be damaged

δi,sh δi,sh δt,sust + δi,sh δt,sus + δi,sh

< < < <

Example 6-2: Deflections b = 11.5 in.; h = 22.5 in,; d = 20 in.; As = 4 # 8; fc
= 3,000 psi; fy = 40 ksi; DL = 1.27 k/ft; LL = 2.44 k/ft; L = 15 ft. 1. Determine the short term deflection 2. Find the creep portion of the sustained load deflection & immediate live load deflections Solution: 1. δi, short = δi,short + sust − δi, sust        2.44

1.27

1.27

2. Moment of inertias:

Ieff = Ig =

Mcr Ma

bh3 12

3

=

 Ig + 1 −

(11.5)(22.5)3 12



Mcr Ma

3  Ict

= 10, 916 in4

3. To find Ict , need to locate N.A @ service

Victor Saouma

Mechanics and Design of Reinforced Concrete

L 180 L 360 L 480 L 240

Draft 6–8

SERVICEABILITY

11.5"

7.85"

20" 12.15"

b(kd)2 − nAs (d − kd) = 0 ⇒ k = 2  2 (11.5)(7.85)3 + (11.5)(7.85) 7.85 12 2

Ict =

√ fr
= 7.5 3, 000 = 410.8 psi f
I Mcr = rybg (410.8)(10,916) 11.25

=

.393 ⇒ kd = 7.85 in + (9.29) (3.14)(12.152 ) = 6, 130 in4       n

As

= 33.2 k.ft = 399 k.in

4. Ma for sustained load (1.27)(15)2 (12) = 428.6 k.in = 35.72 k.ft 8

Masust =

5. Ma for sustained and short load Masust+short = 6. Moment of inertias Ieff, sust + short = Ieff, sust = 7. Deflections

(1.27 + 2.44)(15)2 (12) = 1, 252 k.in = 104 k.ft 8 2

1

  33.2 3 33.2 3 (6, 130) = 6, 209 in4 (10, 916) + 1 − 104.3 104.3 1  33.2 3  33.2 3 2 (6, 130) = 9, 993 in4 (10, 916) + 1 − 35.7 35.7 

√ E = 57 3, 000 = 3, 120 ksi 5 wL4 δ = 384 EI 5 (1.27+2.44)[(15)(12)]4 δi, short + sust = 384 = .218 in (3,120)(6,209) 4

5 (1.27)[(15)(12)] δi, sust = 384 (3,120)(9,993) = .046 in δi = .218 − .046 = .172 in

8. δcreep = λδi, sust λ=

Victor Saouma

2. = 2. ⇒ δcreep = (2)(.046) = .092 in 1+0

Mechanics and Design of Reinforced Concrete

Draft Chapter 7

APPROXIMATE FRAME ANALYSIS Despite the widespread availability of computers, approximate methods of analysis are justified by

1

1. Inherent assumption made regarding the validity of a linear elastic analysis vis a vis of an ultimate failure design. 2. Ability of structures to redistribute internal forces. 3. Uncertainties in load and material properties 2

Vertical loads are treated separately from the horizontal ones.

3

We use the design sign convention for moments (+ve tension below), and for shear (ccw +ve).

4

Assume girders to be numbered from left to right.

5

In all free body diagrams assume positivee forces/moments, and take algeebraic sums.

7.1

Vertical Loads

The girders at each floor are assumed to be continuous beams, and columns are assumed to resist the resulting unbalanced moments from the girders.

6

7

Basic assumptions 1. Girders at each floor act as continous beams supporting a uniform load. 2. Inflection points are assumed to be at (a) One tenth the span from both ends of each girder. (b) Mid-height of the columns 3. Axial forces and deformation in the girder are negligibly small. 4. Unbalanced end moments from the girders at each joint is distributed to the columns above and below the floor.

Draft 7–2

APPROXIMATE FRAME ANALYSIS

Based on the first assumption, all beams are statically determinate and have a span, Ls equal to 0.8 the original length of the girder, L. (Note that for a rigidly connected member, the inflection point is at 0.211 L, and at the support for a simply supported beam; hence, depending on the nature of the connection one could consider those values as upper and lower bounds for the approximate location of the hinge). 8

9

End forces are given by

Maximum positive moment at the center of each beam is, Fig. 7.1

w M

lft

M

V V

rgt

rgt

lft

0.1L

0.1L

0.8L L

0000 1111 0000 1111

111 000 000 111 1 0 0 1 0 1 0 1 0 1

11 00 00 11 00 11 00 11 00 11

Figure 7.1: Approximate Analysis of Frames Subjected to Vertical Loads; Girder Moments

1 1 M + = wL2s = w (0.8)2 L2 = 0.08wL2 8 8

(7.1)

Maximum negative moment at each end of the girder is given by, Fig. 7.1 w w M lef t = M rgt = − (0.1L)2 − (0.8L)(0.1L) = −0.045wL2 2 2

(7.2)

Girder Shear are obtained from the free body diagram, Fig. 7.2 V lf t =

wL 2

V rgt = −

wL 2

(7.3)

Column axial force is obtained by summing all the girder shears to the axial force transmitted by the column above it. Fig. 7.2 rgt P dwn = P up + Vi−1 − Vilf t

Victor Saouma

(7.4)

Mechanics and Design of Reinforced Concrete

Draft

7.1 Vertical Loads

7–3

P

above

rgt

lft

V i−1

Vi

below

P

Figure 7.2: Approximate Analysis of Frames Subjected to Vertical Loads; Column Axial Forces

h/2

h/2 above

M

M col

lft i−1

rgt

Mi−1

lft

rgt

Mi

M

rgt

Vlft

Vi−1

i−1

Li−1

V

Mbelow col

lft i

V

i

rgt i

Li h/2

h/2

Figure 7.3: Approximate Analysis of Frames Subjected to Vertical Loads; Column Moments

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 7–4

APPROXIMATE FRAME ANALYSIS

Column Moment are obtained by considering the free body diagram of columns Fig. 7.3 rgt bot M top = Mabove − Mi−1 + Milf t

M bot = −top

(7.5)

Column Shear Points of inflection are at mid-height, with possible exception when the columns on the first floor are hinged at the base, Fig. 7.3 V =

M top h 2

(7.6)

Girder axial forces are assumed to be negligible eventhough the unbalanced column shears above and below a floor will be resisted by girders at the floor.

7.2 10

Horizontal Loads

We must differentiate between low and high rise buildings.

Low rise buidlings, where the height is at least samller than the hrizontal dimension, the deflected shape is characterized by shear deformations. High rise buildings, where the height is several times greater than its least horizontal dimension, the deflected shape is dominated by overall flexural deformation.

7.2.1

Portal Method

Low rise buildings under lateral loads, have predominantly shear deformations. Thus, the approximate analysis of this type of structure is based on

11

1. Distribution of horizontal shear forces. 2. Location of inflection points. 12

The portal method is based on the following assumptions 1. Inflection points are located at (a) Mid-height of all columns above the second floor. (b) Mid-height of floor columns if rigid support, or at the base if hinged. (c) At the center of each girder. 2. Total horizontal shear at the mid-height of all columns at any floor level will be distributed among these columns so that each of the two exterior columns carry half as much horizontal shear as each interior columns of the frame.

13

Forces are obtained from

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

7.2 Horizontal Loads

7–5

H/2

H

H

H/2

Figure 7.4: Approximate Analysis of Frames Subjected to Lateral Loads; Column Shear Column Shear is obtained by passing a horizontal section through the mid-height of the columns at each floor and summing the lateral forces above it, then Fig. 7.4
V

ext

=

F lateral

V int = 2V ext

2No. of bays

(7.7)

Column Moments at the end of each column is equal to the shear at the column times half the height of the corresponding column, Fig. 7.4 M top = V

h 2

M bot = −M top

(7.8)

Girder Moments is obtained from the columns connected to the girder, Fig. 7.5 h/2

h/2 above

M col

lft

rgt

M i−1

M i−1

lft

rgt

rgt

Vlft

V

lft Vi−1

rgt

Mi

Mi

i−1

Vi

i

below

Li−1/2

Li−1/2

M col

Li /2

Li /2

h/2

h/2

Figure 7.5: Approximate Analysis of Frames Subjected to Lateral Loads; Girder Moment

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 7–6

APPROXIMATE FRAME ANALYSIS

rgt above below Milf t = Mcol − Mcol + Mi−1

Mirgt = −Milf t

(7.9)

Girder Shears Since there is an inflection point at the center of the girder, the girder shear is obtained by considering the sum of moments about that point, Fig. 7.5 V lf t = −

2M L

V rgt = V lf t

(7.10)

Column Axial Forces are obtained by summing girder shears and the axial force from the column above, Fig. ?? P

above

rgt

lft

V i−1

Vi

below

P

Figure 7.6: Approximate Analysis of Frames Subjected to Lateral Loads; Column Axial Force P = P above + P rgt + P lf t

(7.11)

Example 7-1: Approximate Analysis of a Frame subjected to Vertical and Horizontal Loads

Draw the shear, and moment diagram for the following frame. Solution: Vertical Loads

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

7.2 Horizontal Loads

7–7

0.25 k/ft 6 0.5 k/ft13 7

15 k 5 12 30 k 9 1

10

11 00

2

3

11 00

11 00

20’

30’

14 11

8 4

14’ 16’

11 00

24’

Figure 7.7: Example; Approximate Analysis of a Building 1. Top Girder Moments lf t M12 cnt M12 rgt M12 lf t M13 cnt M13 rgt M13 lf t M14 cnt M14 rgt M14

= = = = = = = = =

−0.045w12 L212 = −(0.045)(0.25)(20)2 0.08w12 L212 = (0.08)(0.25)(20)2 lf t M12 −0.045w13 L213 = −(0.045)(0.25)(30)2 0.08w13 L213 = (0.08)(0.25)(30)2 lf t M13 −0.045w14 L214 = −(0.045)(0.25)(24)2 0.08w14 L214 = (0.08)(0.25)(24)2 lf t M14

=− = =− =− = =− =− = =−

4.5 k.ft 8.0 k.ft 4.5 k.ft 10.1 k.ft 18.0 k.ft 10.1 k.ft 6.5 k.ft 11.5 k.ft 6.5 k.ft

=− = =− =− = =− =− = =−

9.0 k.ft 16.0 k.ft 9.0 k.ft 20.3 k.ft 36.0 k.ft 20.3 k.ft 13.0 k.ft 23.0 k.ft 13.0 k.ft

2. Bottom Girder Moments M9lf t M9cnt M9rgt lf t M10 cnt M10 rgt M10 lf t M11 cnt M11 rgt M11

= = = = = = = = =

−0.045w9 L29 = −(0.045)(0.5)(20)2 0.08w9 L29 = (0.08)(0.5)(20)2 M9lf t −0.045w10 L210 = −(0.045)(0.5)(30)2 0.08w10 L210 = (0.08)(0.5)(30)2 lf t M11 −0.045w12 L212 = −(0.045)(0.5)(24)2 0.08w12 L212 = (0.08)(0.5)(24)2 lf t M12

3. Top Column Moments M5top M5bot M6top M6bot M7top M7bot M8top M8bot Victor Saouma

= = = = = = = =

lf t +M12 −M5top rgt lf t −M12 + M13 = −(−4.5) + (−10.1) top −M6 rgt lf t −M13 + M14 = −(−10.1) + (−6.5) −M7top rgt −M14 = −(−6.5) −M8top

=− = =− = =− = = =−

4.5 4.5 5.6 5.6 3.6 3.6 6.5 6.5

k.ft k.ft k.ft k.ft k.ft k.ft k.ft k.ft

Mechanics and Design of Reinforced Concrete

Draft 7–8

APPROXIMATE FRAME ANALYSIS

4. Bottom Column Moments M1top M1bot M2top M2bot M3top M3bot M4top M4bot

= = = = = = = =

+M5bot + M9lf t = 4.5 − 9.0 −M1top lf t +M6bot − M9rgt + M10 = 5.6 − (−9.0) + (−20.3) top −M2 rgt lf t +M7bot − M10 + M11 = −3.6 − (−20.3) + (−13.0) top −M3 rgt +M8bot − M11 = −6.5 − (−13.0) −M4top

=− = =− = = =− = =−

4.5 4.5 5.6 5.6 3.6 3.6 6.5 6.5

k.ft k.ft k.ft k.ft k.ft k.ft k.ft k.ft

5. Top Girder Shear lf t V12 rgt V12 lf t V13 rgt V13 lf t V14 rgt V14

= w122L12 = lf t = −V12 = w132L13 = lf t = −V13 = w142L14 = lf t = −V14

(0.25)(20) 2 (0.25)(30) 2 (0.25)(24) 2

= =− = =− = =−

2.5 k 2.5 k 3.75 k 3.75 k 3.0 k 3.0 k

= =− = =− = =−

5.00 5.00 7.50 7.50 6.00 6.00

6. Bottom Girder Shear V9lf t V9rgt lf t V10 rgt V10 lf t V11 rgt V11

= w92L9 = (0.5)(20) 2 = −V9lf t = w102L10 = (0.5)(30) 2 lf t = −V10 = w112L11 = (0.5)(24) 2 lf t = −V11

7. Column Shears V5 = V6 = V7 = V8 = V1 = V2 = V3 = V4 =

M5top H5 2 M6top H6 2 M7top H7 2 M8top H8 2 M1top H1 2 M2top H2 2 M3top H3 2 M4top H4 2

=

−4.5

= − 0.64 k

=

−5.6

= − 0.80 k

=

3.6

=

0.52 k

=

6.5

=

0.93 k

=

−4.5

= − 0.56 k

=

−5.6

= − 0.70 k

=

3.6

=

0.46 k

=

6.5

=

0.81 k

14 2 14 2

14 2 14 2

16 2 16 2

16 2 16 2

k k k k k k

8. Top Column Axial Forces

Victor Saouma

P5 P6 P7 P8

= = = =

lf t V12 = 2.50 k rgt lf t −V12 + V13 = −(−2.50) + 3.75 = 6.25 k rgt lf t −V13 + V14 = −(−3.75) + 3.00 = 6.75 k rgt −V14 Mechanics and Design of=Reinforced 3.00 k Concrete

Draft

7.2 Horizontal Loads

7–9

0.25K/ft

12

5

6

13

9

10

1

20’

30’

+8.0’k

+18.0’k

-4.5’k +16.0’k

-9.0’k

+4.5’k

+5.6’k -4.5’k

+4.5’k

+5.6’k

k

14’ 16’

24’ +11.5’k -6.5’k

k

+23.0’k

+32.0’k

-20.2’

-4.5’k

4

-10.1’k -10.1’k -6.5’

-9.0’k

8

11 3

2

-4.5’k

14

7

0.50K/ft

k -13.0’ -20.2’ k

-13.0’k

-5.6’k

+3.6’k

-5.6’k

-3.6’k -6.5’k +3.6’k +6.5’k

+6.5’k

-3.6’k

-6.5’k

Figure 7.8: Approximate Analysis of a Building; Moments Due to Vertical Loads

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 7–10

APPROXIMATE FRAME ANALYSIS

+2.5K

+3.75K

+3.0K

-2.5K

-3.75K

K

+7.5

K

+5.0

K

+6.0

-5.0K

-0.64K

-0.56K

-3.0K

-6.0K

-7.5K

-0.80K

+0.51K

-0.70K

+0.45K

+0.93K

+0.81K

Figure 7.9: Approximate Analysis of a Building; Shears Due to Vertical Loads

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

7.2 Horizontal Loads

7–11

9. Bottom Column Axial Forces = = = =

P1 P2 P3 P4

P5 + V9lf t = 2.50 + 5.0 rgt P6 − V10 + V9lf t = 6.25 − (−5.00) + 7.50 rgt lf t P7 − V11 + V10 = 6.75 − (−7.50) + 6.0 rgt P8 − V11 = 3.00 − (−6.00)

= = = =

7.5 k 18.75 k 20.25 k 9.00 k

Horizontal Loads, Portal Method 1. Column Shears V5 V6 V7 V8 V1 V2 V3 V4

= = = = = = = =

15 (2)(3)

2(V5 ) = (2)(2.5) 2(V5 ) = (2)(2.5) V5 15+30 (2)(3)

2(V1 ) = (2)(7.5) 2(V1 ) = (2)(2.5) V1

= = = = = = = =

2.5 k 5k 5k 2.5 k 7.5 k 15 k 15 k 7.5 k

2. Top Column Moments M5top M5bot M6top M6bot

= V12H5 = = −M5top = V62H6 = = −M6top

(2.5)(14) 2 (5)(14) 2

= =− = =−

17.5 17.5 35.0 35.0

k.ft k.ft k.ft k.ft

M7top = 7 2 7 = M7bot = −M7top

V up H

(5)(14) 2

V up H

= 35.0 k.ft = − 35.0 k.ft

M8top = 8 2 8 = M8bot = −M8top

(2.5)(14) 2

= 17.5 k.ft = − 17.5 k.ft

3. Bottom Column Moments M1top = 1 2 1 = M1bot = −M1top

V dwn H

(7.5)(16) 2

= 60 k.ft = − 60 k.ft

M2top = 2 2 2 = M2bot = −M2top

V dwn H

(15)(16) 2

= 120 k.ft = − 120 k.ft

M3top = 3 2 3 = M3bot = −M3top

V dwn H

(15)(16) 2

= 120 k.ft = − 120 k.ft

M4top = 4 2 4 = M4bot = −M4top

V dwn H

(7.5)(16) 2

= 60 k.ft = − 60 k.ft

4. Top Girder Moments lf t M12 rgt M12 lf t M13 rgt M13 lf t M14 rgt M14

Victor Saouma

= = = = = =

M5top lf t −M12 rgt M12 + M6top = −17.5 + 35 lf t −M13 rgt M13 + M7top = −17.5 + 35 lf t −M14

= =− = =− = =−

17.5 17.5 17.5 17.5 17.5 17.5

k.ft k.ft k.ft k.ft k.ft k.ft

Mechanics and Design of Reinforced Concrete

Draft 7–12

APPROXIMATE FRAME ANALYSIS

Approximate Analysis Vertical Loads

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

A

B

Height 14 16

Span Load Load

C

D

APROXVER.XLS

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Victor E. Saouma

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L1 20 0.25 0.5

L2 L3 30 24 0.25 0.25 0.5 0.5 MOMENTS Bay 1 Bay 2 Bay 3 Col Beam Column Beam Column Beam Col Lft Cnt Rgt Lft Cnr Rgt Lft Cnt Rgt AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAA AAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA -10.1 18.0 -10.1AAAA 8.0 -4.5 AAAA 11.5 -6.5 AAAA A -4.5 AAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A -6.5 AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA -4.5 AAAA -5.6 3.6 6.5 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAA AA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA 4.5 AAAA 5.6 -3.6 -6.5 AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAAAAAAAAAAAAAA AAAA A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAA AAAAAAAA AAAAAAAA A -9.0 16.0 AAAAAAAAAAAA AAAAAAAAAAAA AAAA -20.3 36.0 -20.3AAAA AAAAAAAAAAAA AAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A -13.0 23.0 -13.0 AAAA -9.0 AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA -4.5 AAAA -5.6 3.6 6.5 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAAAAAA AA AAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAA 4.5 AAAA 5.6 -3.6 -6.5 AAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA SHEAR Bay 1 Bay 2 Bay 3 Col Beam Column Beam Column Beam Col Lft Rgt Lft Rgt Lft Rgt AAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAA AAAAA AAAA AAAAAAAAAAAA AAAAAAAAAAAA -2.50 AAAAAAAA AAAA 3.75 AAAA AAAAAAAAAAAA -3.00 AAAAAAAA AAAA AAAAAAAA AAAAA 2.50AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA -3.75AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A 3.00 AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAA A AAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA A AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA -0.64 AAAA -0.80 0.52 0.93 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAA AAAA AAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA A AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A AAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAA A 5.00AAAA AAAAAAAAAAAA AAAAAAAA AAAA -5.00 AAAAAAAA AAAA 7.50 AAAA AAAAAAAAAAAA -6.00 AAAAAAAA AAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA -7.50AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A 6.00 AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAAAAA AAAA AAAA A AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA A AAAAAAAAAAAA AAAAAAAA A AAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA -0.56 AAAA -0.70 0.46 0.81 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAA AXIAL FORCE Bay 1 Bay 2 Bay 3 Col Beam Column Beam Column Beam Col AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A AAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A 0.00 0.00 0.00 AAAA AAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA 2.50 6.25 6.75 3.00 AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA A AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAA A AAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA A 0.00 0.00 0.00 AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA 7.50 AAAA 18.75 20.25 9.00 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA

Figure 7.10: Approximate Analysis for Vertical Loads; Spread-Sheet Format

Victor Saouma

Mechanics and Design of Reinforced Concrete

Victor Saouma L1 20 0.25 0.5

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Victor E. Saouma

=-0.045*D5*D3^2 =0.08*D5*D3*D3 =+D13

=-F13+I13+G12 =-G14

=-0.045*I5*I3^2 =0.08*I5*I3*I3 =+I13

=-K13+N13+L12 =-L14

=-0.045*N5*N3^2 =0.08*N5*N3*N3 =+N13

=-P13+Q12 =-Q14

=+C28+D22

Bay 2 Beam 0

=+I3*I5/2

=-I22

Column

=2*L14/A5

Bay 3 Beam 0

=+N3*N5/2

=-N22

Col

=2*Q14/A5

0

=+G28-F22+I22

=-F20+I20

0

=+L28-K22+N22

=-K20+N20

0

=+Q28-P22

=-P20

AAAA AAA AAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAA AAAAAAAA AAAA AAAA AAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAA AAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAA AAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

=+D20

Column

AXIAL FORCE Bay 1 Col

Beam 0

=2*G14/A5

=-D22

=2*C14/A5

=+D3*D5/2

AAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA A AAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

Bay 2 Bay 3 Beam Column Beam Column Beam Col Lft Rgt Lft Rgt Lft Rgt AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA A A A AAAAAAAAAAAAAAAAAAAAAAA=-I20 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA =+N3*N4/2 AAAAAAAAAAAAAAAAAAAAAAAAA =+D3*D4/2 AAAAAAAAAAAAAAAAAAAAAAAAA=-D20 AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA=+I3*I4/2 AAAAAAAAAAAAAAAAAAAAAAAAA =-N20 AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA =2*C11/A4 AAAA AAAA =2*G11/A4 AAAA =2*L11/A4 AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA=2*Q11/A4 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AA

SHEAR Bay 1 Col

=+D13+C12 =-C14

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

Bay 2 Bay 3 Beam Column Beam Column Beam Col Lft Cnt Rgt AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAALft Cnr Rgt AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Lft Cnt Rgt AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA =-0.045*D4*D3^2 =0.08*D4*D3*D3 =+D10 =-0.045*I4*I3^2 =0.08*I4*I3*I3 =+I10 =-0.045*N4*N3^2 =0.08*N4*N3*N3 =N10 AAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A =+D10 =-F10+I10 =-K10+N10 =-P10 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A =-L11 =-C11 =-G11 =-Q11 AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA

MOMENTS Bay 1 Col

C

29 30

Span Load Load

B

24 25 26 27 28

22 23

16 17 18 19 20 21

13 14 15

A 1 2 3 Height 4 14 5 16 6 7 8 9 10 11 12

Approximate Analysis Vertical Loads

Draft 7.2 Horizontal Loads

7–13

Figure 7.11: Approximate Analysis for Vertical Loads; Equations in Spread-Sheet

Mechanics and Design of Reinforced Concrete

Draft 7–14

APPROXIMATE FRAME ANALYSIS

5. Bottom Girder Moments M9lf t M9rgt lf t M10 rgt M10 lf t M11 rgt M11

15K

= = = = = =

M1top − M5bot = 60 − (−17.5) −M9lf t M9rgt + M2top − M6bot = −77.5 + 120 − (−35) lf t −M10 rgt M10 + M3top − M7bot = −77.5 + 120 − (−35) lf t −M11

12

5

30K

13

6

9

10

1

20’

+17.5’K

+17.5’K

-35’K +60’K

-120’K +17.5’K

-120’K

+77.5’

+77.5’

-17.5’K

-60’K

+17.5’K

-17.5K K

16’

24’

-35’K +120’K

-60’K

k.ft k.ft k.ft k.ft k.ft k.ft

14’

4

+35’K

+35’K

77.5 77.5 77.5 77.5 77.5 77.5

11

30’

-17.5’K +120’K

+60’K

8

3

2

+17.5’K

14

7

= =− = =− = =−

K

+77.5’

-77.5’K

-17.5K

-17.5K

-77.5’K

-77.5’K

K

Figure 7.12: Approximate Analysis of a Building; Moments Due to Lateral Loads

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

7.2 Horizontal Loads

7–15

6. Top Girder Shear lf t 12 V12 = − L12 = − (2)(17.5) 20 rgt lf t V12 = +V12

2M lf t

= −1.75 k = −1.75 k

lf t 13 V13 = − L13 = − (2)(17.5) 30 rgt lf t V13 = +V13

2M lf t

= −1.17 k = −1.17 k

lf t 14 V14 = − L14 = − (2)(17.5) 24 rgt lf t V14 = +V14

2M lf t

= −1.46 k = −1.46 k

= − (2)(77.5) V9lf t = − L12 20 9 V9rgt = +V9lf t

2M lf t

= −7.75 k = −7.75 k

lf t 10 V10 = − L10 = − (2)(77.5) 30 rgt lf t V10 = +V10

2M lf t

= −5.17 k = −5.17 k

lf t 11 V11 = − L11 = − (2)(77.5) 24 rgt lf t V11 = +V11

2M lf t

= −6.46 k = −6.46 k

7. Bottom Girder Shear

8. Top Column Axial Forces (+ve tension, -ve compression) P5 P6 P7 P8

= = = =

lf t −V12 = −(−1.75) k rgt lf t +V12 − V13 = −1.75 − (−1.17) = −0.58 k rgt lf t +V13 − V14 = −1.17 − (−1.46) = 0.29 k rgt V14 = −1.46 k

9. Bottom Column Axial Forces (+ve tension, -ve compression) P1 P2 P3 P4

= = = =

P5 + V9lf t = 1.75 − (−7.75) rgt P6 + V10 + V9lf t = −0.58 − 7.75 − (−5.17) rgt lf t P7 + V11 + V10 = 0.29 − 5.17 − (−6.46) rgt P8 + V11 = −1.46 − 6.46

= = = =

9.5 k −3.16 k 1.58 k −7.66 k

Design Parameters On the basis of the two approximate analyses, vertical and lateral load, we now seek the design parameters for the frame, Table 7.2.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 7–16

APPROXIMATE FRAME ANALYSIS

Portal Method

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

PORTAL.XLS

B

C

D

E

F

G

H

I

J

K

Victor E. Saouma

L

M

N

O

P

Q

R

S

PORTAL METHOD # of Bays

3

# of Storeys 2 Force Shear H Lat. Tot Ext Int H1

14 15 15 2.5

5

H2

16 30 45 7.5 15

L1 20

L2 L3 30 24 MOMENTS Bay 1 Bay 2 Bay 3 Col Beam Column Beam Column Beam Col Lft Rgt AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Lft Rgt AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Lft Rgt AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAA AAAAAAAA AAAA AAAAAAAA A A AAAAAAAAAAAA AAAAAAAAAAAA AAA AAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAA 17.5 -17.5 17.5 -17.5AAAA A 17.5 -17.5 AAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAA AAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA 17.5 AAAA 35.0 35.0 17.5 AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA -17.5 AAAA -35.0 -35.0 -17.5 AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAA AAA 77.5 -77.5 AAAAAAAA AAAA 77.5 -77.5AAAAAAAA A 77.5 -77.5 AAAA A AAAAAAAAAAAA AAAAAAAA AAA AAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA 60.0 AAAA 120.0 120.0 60.0 AAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA -60.0 AAAA -120.0 -120.0 -60.0 AAAAAAAA AAAAAAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA SHEAR Bay 1 Bay 2 Bay 3 Col Beam Column Beam Column Beam Col Lft Rgt Lft Rgt Lft Rgt AAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAA AAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAA AAAAAAAA AAA AAAAAAAA AAAA AAAAAAAAAAAA A -1.75 -1.75 AAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAA -1.46 -1.46AAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA -1.17 -1.17 AAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA 2.50 AAAA 5.00 5.00 2.50 AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAA AAAA 2.50 AAAA 5.00 5.00 2.50 AAAAAAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAA AAAAAAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A -7.75 -7.75 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA -5.17 -5.17 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA -6.46 -6.46AAAA A AAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAA AAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA 7.50 AAAA 15.00 15.00 7.50 AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAA 7.50 AAAA 15.00 15.00 7.50 AAAA AAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AXIAL FORCE Bay 1 Bay 2 Bay 3 Col Beam Column Beam Column Beam Col AAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAA AAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAA AAAAAAAA AAAA AAAAAAAA A A AAAAAAAAAAAA AAAAAAAAAAAA AAA AAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA 0.00 0.00 0.00 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA 1.75 -0.58 0.29 -1.46 AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAA AAAAAAAA AAAA AAAAAAAA A A AAAAAAAAAAAA AAAAAAAAAAAA AAA AAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAAAAAA AAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAA 0.00 0.00 0.00 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA 9.50 AAAA -3.17 1.58 -7.92 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA

Figure 7.13: Portal Method; Spread-Sheet Format

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

7.2 Horizontal Loads Portal Method

A 1 PORTAL METHOD

A A A A A A

B

A A A A A A

7–17

PORTAL.XLS

C

A A A A A A

D

A A A A A A

E

A A A A A A

F

A A A A A

G

A A A A A

H

A A A A A A

I

A A A A A A

J

A A A A A A

K

A A A A A

L

AA AA AA AA AA AA

M

A A A A A A

N

A A A A A A

O

A A A A A

P

A A A A A A

Q

A A A A A

R

Victor E. Saouma

A A A A A A

S

AAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAA A3 AL1 A AAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAA AA 2 AAAA # ofAAAA Bays L2 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA L3 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAA AAAA AAAAAA AAAAAAAAA AAAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAA AAAAA A AAAA A AAAAAAAA A AAAAAAAAAAAAAAAAAAAAAAA AA AAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A A AA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA A A A A A A20 A A AA30 A A A 24 A 3 A A A A A A A A AA A A A A

A A A A A A A A A AA A A A A A A A A A A A A A A A AA A A A A A A A A A A A A A A A AA A A A A A A A A A A A MOMENTS A A A A AA A A A A A A AAAAAAAAAAA A A A A A A AA A A A A A A A A2 A A A AA A A A A Bay 1 Bay 2 AA Bay 3 AA A A A A A A A A A A A AAAAAAAAAAA A A AA A A A A A A A A A A A A A A A A A 6 Force AA Shear Col Beam Column Beam Column Beam Col A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA A A A A A AInt A Rgt A Rgt A Rgt 7 H Lat. AAA Tot Ext Lft Lft Lft A A A A A AAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA A A A AAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA A A AAAAAAAA AAAAAAAA AAAAAAAAAAAA=+H9 AAAAAA AAAA AAAA A A 8 =-I8 =+J8+K9 =-M8 =+N8+O9 =-Q8 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA A A AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA A A AAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAA AAAA A A AAAAAAAAAAAAAAAA=+F9*B9/2 AAAAAAAAAAAAAAAAAAAAAAA=+H9 AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAA =+K9 AAAAAAAA A=2*E9 9 AAAA H1 14 A15 AAAAAAAAAA =+C9 =+D9/(2*$F$2) AAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAA =+E9*B9/2 A AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA A AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAA AAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAA=-K9 AAAAAAAA AAA=+H10 AAAA AAAA AAAA AAAA AAAA AAAA A A AAAA AAAA AAAA AAAA AAAA AAAA AA 10 =-H9 =+K10 A A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA A A AAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAA AA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAA A A AAAAAAAAAAAAAAAAAAAA=+H12-H10 =-I11 AAAAAAAAAAAAAAAAAAAAAAAAAAA=+K12-K10+J11 =-M11 AAAAAAAAAAAAAAAAAAAAAAAAAAA =+O12-O10+N11 =-Q11 AAAAAAAAAAAAAA A A 11 AAAAAAAAAAAAAAAA AAAAAAAAAA AAAA AAAA AAAA A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAA A A AAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAA AAAA A A AAAAAAAAAAAAAAAA=+F12*B12/2 AAAAAAAAAAAAAAAAAAAAAAA=+H12 AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAA AA AAAAAAAA A A 12 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA H2 16 A30 =+E12*B12/2 AAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAA AAAA AAAAAAAAAAAAAAAA A =SUM($C$9:C12) A=+D12/(2*$F$2) A=2*E12 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA =+K12 AAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAA A A AAAA AAAAAAAAAAAAAAAAAAAA=-K12 AAAAAAAAAAAAAAAAAAAAAAAAAAA=+H13 A A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA =+K13 13 =-H12 AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA A A AAAA AAAA AAAA AAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA A A A A AA A AAAAAA A A A A AAAAAAA A A A A A A AAAAAAAAAAAA A A A A A AA A A A A A A A A A A A A A SHEAR A A A A AA A A A A A A A A A A A A 14 A A A A A AA A A A A A A A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A A A A A AA A A A A A A A A A A AA A A A A A A A 15 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Bay 1 Bay 2 AA Bay 3 A A A A A A A A A A A A A AA A A A A A A A A A A A A A A A A A A A 16 Col Beam Column Beam Column Beam Col A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A A A A A A A A A A A A A A A A 17 Lft Rgt Lft Rgt Lft Rgt A A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAA=-2*I8/I$3 =+I18 AAAA AAAAAAAAAA A A A A A A 18 =+M18AAAA =-2*Q8/Q$3 =+Q18AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA=-2*M8/M$3 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA A A A A A A AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAA A A A A A A AAAAAAAA AAAA AAAAAAAA AAA AAAA A A A A A A 19 =+E9 AAAAAAAA AAAAAAAA AAAAAAAA=+F9 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA =+E9 AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAA AA =+F9 AAAAAAAA A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA A A A A A A AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA A A A A A A 20 =+H19 =+K19 =+S19 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA =+O19 AAAAAAAA A A A A A A AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA A A A A A A AAAAAAAA AAAA=-2*I11/I$3 =+I21 AAAA AA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AAAA A A A A A A 21 =-2*M11/M$3 =+M21AAAA =-2*Q11/Q$3 =+Q21AAAAAAAA AAAAAAAA AAAA AAAA A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAA AAAA A A A A A A AAAAAAAAAAAAAAAAAAAA=+F12 AAAAAAAAAAAAAAAAAAAAAAAAAAA =+E12 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA =+F12 A A A A A A 22 =+E12 AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAA=+K22 AAAAAAAAAAAAAAAAAAAAAAAAAAA AAA=+S22 AAAA A A A A A A 23 =+H22 =+O22 AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AA AAAA A A A A A A AAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA A A A A AA A AAAAAA A A A A AAAAAAA A A A A A A A A A A A A AA A A A A A A A A A A A A A AXIAL FORCE A A A A AA A A A A A A A A A A A A 24 A A A A A AA A A A A A A A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A A A A A AA A A A A A A A A A A AA A A A A A A A A A A A A A A A A A 25 Bay 1 Bay 2 AA Bay 3 A A A AA A A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A A A A A A A 26 Col Beam Column Beam Column Beam Col A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA A A A A A A AAAA AAAA AAAAAAAA AAAA AAAA AAAA AAAAAA AAAA AAAA A A A A A A 27 0 0 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA0 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA A A A A A A AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA A A A AAAAAAAA AAAA=+J18-M18 AAAAAAAA AAA=+R18 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA =+N18-Q18 A AAAAAAAAAAAAAAAA AAAAAAAAA AAAAAAAAAAAA 28 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA =-I18 AAAA AAAAAAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAAAAAAAA AAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAA AAAA AAAA AAAA AAAA AAAA AAAA AAA A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAA AAAAAAAAA AAAAAAAAAAAAAAAAAAAAA A AAAAAAAAAAAAAAAA AAAAAAAAA AAAAAAAAAAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAAAA AAAA AAAA A A A A A 29 0 0 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA0 AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAA AAA AAAAAAAA A A A A A A AAAAAAAA AAAAAAAA AAAAAAAA AAAA AAAAAAAA AAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA AAAAAAAAAAAAAAAAAAAA AAAA AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA AAAA A A A A A A AAAA AAAA AAAA AAAA AAAA AAAA AA AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAA AAAAAAAA AAAAAAAAAAAA AAAAAAAAAAAAAAAAAAA A A A A A A AAAA 30 =+H28-I21 =+O28+N21-Q21 AAAAAAAAAAAAAAAA=+K28+J21-M21 AAAAAAAAAAAAAAAAAAAAAAA=+S28+R21 A A A A A A AAAA AAAA AAAA A A A A A A AAAAAAAAAAAAAAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAAAA AAAAAA AA AAAAAAAAAAAAAAAAAAAAAAAAAAA

4

5 # of Storeys

A A A A A A A A

A A A A A A A A

Figure 7.14: Portal Method; Equations in Spread-Sheet

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 7–18

APPROXIMATE FRAME ANALYSIS

Mem.

1

2

3

4

5

6

7

8

Moment Axial Shear Moment Axial Shear Moment Axial Shear Moment Axial Shear Moment Axial Shear Moment Axial Shear Moment Axial Shear Moment Axial Shear

Vert.

Hor.

4.50 7.50 0.56 5.60 18.75 0.70 3.60 20.25 0.45 6.50 9.00 0.81 4.50 2.50 0.64 5.60 6.25 0.80 3.60 6.75 0.51 6.50 3.00 0.93

60.00 9.50 7.50 120.00 15.83 15.00 120.00 14.25 15.00 60.00 7.92 7.50 17.50 1.75 2.50 35.00 2.92 5.00 35.00 2.63 5.00 17.50 1.46 2.50

Design Values 64.50 17.00 8.06 125.60 34.58 15.70 123.60 34.50 15.45 66.50 16.92 8.31 22.00 4.25 3.14 40.60 9.17 5.80 38.60 9.38 5.51 24.00 4.46 3.43

Table 7.1: Columns Combined Approximate Vertical and Horizontal Loads

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

7.2 Horizontal Loads

7–19

Mem.

9

10

11

12

13

14

-ve Moment +ve Moment Shear -ve Moment +ve Moment Shear -ve Moment +ve Moment Shear -ve Moment +ve Moment Shear -ve Moment +ve Moment Shear -ve Moment +ve Moment Shear

Vert.

Hor.

9.00 16.00 5.00 20.20 36.00 7.50 13.0 23.00 6.00 4.50 8.00 2.50 10.10 18.00 3.75 6.50 11.50 3.00

77.50 0.00 7.75 77.50 0.00 5.17 77.50 0.00 6.46 17.50 0.00 1.75 17.50 0.00 1.17 17.50 0.00 1.46

Design Values 86.50 16.00 12.75 97.70 36.00 12.67 90.50 23.00 12.46 22.00 8.00 4.25 27.60 18.00 4.92 24.00 11.50 4.46

Table 7.2: Girders Combined Approximate Vertical and Horizontal Loads

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 7–20

Victor Saouma

APPROXIMATE FRAME ANALYSIS

Mechanics and Design of Reinforced Concrete

Draft Chapter 8

COLUMNS

Draft Chapter 9

COLUMNS 9.1

Introduction

Columns resist a combination of axial P and flexural load M , (or M = P e for eccentrically applied load). 1

9.1.1

Types of Columns

Types of columns, Fig. 9.1 Composite colum

Tied column tie steel main longitudinal steel reinforcement

Pipe column Spiral column

Figure 9.1: Types of columns 2

Lateral reinforcement, Fig. 9.2 P Spiral X Tied

X

δ

Figure 9.2: Tied vs Spiral Reinforcement 1. Restrains longitudinal steel from outward buckling 2. Restrains Poisson’s expansion of concrete 3. Acts as shear reinforcement for horizontal (wind & earthquake) load 4. Provide ductility

Draft 9–2

COLUMNS

very important to resist earthquake load.

9.1.2 3

Possible Arrangement of Bars

Bar arrangements, Fig. 9.3

6 bars

4 bars

8 bars

Corner column

10 bars 12 bars

Wall column

16 bars 14 bars

Figure 9.3: Possible Bar arrangements

9.2

Short Columns

9.2.1 4

Concentric Loading

No moment applied,

Elastic Behaviour P

= fc Ac + fs As = fc (Ac + nAs )

Ultimate Strength Pd = φPn Pn = .85fc
Ac + fy As note: 1. 0.85 is obtained also from test data 2. Matches with beam theory using rect. stress block 3. Provides an adequate factor of safety

9.2.2 5

Eccentric Columns

Sources of flexure, Fig. 9.4 1. Unsymmetric moments M L = M R 2. Uncertainty of loads (must assume a minimum eccentricity)

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns

9–3 P

e

P M

ML

n

L

M

R

MR L

e= M − M P

R

n

Figure 9.4: Sources of Bending 3. Unsymmetrical reinforcement 6

Types of Failure, Fig. 9.5 1. Large eccentricity of load ⇒ failure by yielding of steel 2. Small eccentricity of load ⇒ failure by crushing of concrete 3. Balanced condition c

Pn ε cu P

ε cu

0

Compression failure range

e = 0; a = h; c =

8

Radial lines show constant e= e small

c

Mn

ε cu

Pn

ε

y

Balanced Failure Load path for givin e

c

Tension failure range

eb

e

8

c ~ h; e=

c

ε cu

e large M

Mn

ε

su

> εy

0

Figure 9.5: Load Moment Interaction Diagram 7

Assumptions A
s = As ; ρ =

9.2.2.1 8

As bd

=

A
s
bd ; fs

= fy

Balanced Condition

There is one specific eccentricity eb =

M P

such that failure will be triggered by simultaneous

1. Steel yielding 2. Concrete crushing Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–4

COLUMNS

From the strain diagram (and compatibility of concrete and steel strains), Fig. 9.6

9

d’ d h/2 A

A’

s

s

b

εcs

εs A sf y

ε’s Pn

A sf s

c

c

A sf s

0.85f’c A’sf s

A’sf y

a e e’

Figure 9.6: Strain and Stress Diagram of a R/C Column

εc = .003 fy εy = Es εu d= c = εu + εy

(9.1-a) (9.1-b) .003 fy Es

+ .003

d

(9.1-c)

Furthermore, ε
s c − d


=

⇒ ε
s =

εc c c − d
εc c

(9.2-a) (9.2-b)

thus the compression steel will be yielding (i.e. ε
s = εy ) for εc = .003 and d
= 2 in if c > 6 in 10

Equilibrium (neglecting ceter steel for now):  Pn Pn = .85fc
ab + A
s fy
− As fs  a = .85fc
b a = β1 cb fs = fy .003  c =
d b fy As = As +.003

    

Pn,b = .85β1 fc
bd fy Es

.003 + .003

(9.3)

Es

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns

9–5

or Pnb = .85β1 fc
bd

87, 000 fy + 87, 000

(9.4)

11 To obtain Mnb we take moment about centroid of tension steel As of internal forces, this must be equal and opposite to the externally applied moment, Fig. 9.6.

a Mnb = Pnb eb = .85fc
ab(d − ) + A
s fy (d − d
)    2    M ext

12

(9.5)

Mint

Note: Internal moments due to As fy and A
s fy cancel each other for symmetric columns.

9.2.2.2

Tension Failure

Case I, e is known and e > eb In this case a and Pn are unknowns, and for failure to be triggered by fy in As we must have e > eb . Can still assume As fy = A
s fy ΣFy = 0 ⇒ Pn = .85fc
ab ⇒ a =

Pn .85fc
b

a ΣM = 0 ⇒ Pn e
= Pn (d − ) + A
s fy
(d − d
) 2

(9.6-a) (9.6-b)

Two approaches 1. Solve iteratively for those two equations (a) (b) (c) (d) (e)

Assume a (a < h2 ) From strain compatibility solve for fsc , center steel stress if applicable. ΣFy = 0 ⇒ solve for Pn ΣM = 0 with respect to tensile reinforcement,⇒ solve for Pn If no convergence among the two Pn , iterate by solving for a from ΣFy = 0

2. Combine them into a quadratic equation in Pn   "   2   



e e d e 1− Pn = .85fc
bd −ρ − ( − 1) + + 2ρ µ 1 − +  d d d d  where ρ = µ =

Victor Saouma

(9.7)

A
s As bd = bd fy .85fc


Mechanics and Design of Reinforced Concrete

Draft 9–6

COLUMNS

Case II c is known and c < cb ; Pn is unknown In this case, we only have two unknown, Pn and fs
. a fs

= def

=

fs


=

C

=

Pn

=

Mn

=

e

=

β1 c

(9.8-a)

fy

(9.8-b)

c− c 0.85fc
ab

εu E s

d


≤ fy

(9.8-c) (9.8-d)

A
s fs


C+ − As fy     h h h−a


+ As fs − d + As fs d − C 2 2 2 Mn Pn

(9.8-e) (9.8-f) (9.8-g)

Note this approach is favoured when determining the interaction diagram. 9.2.2.3

Compression Failure

Case I e is known and e < eb ; Pn , a and fs are unknown Compression failure occurs if e
< e
b ⇒ εu = .003, assume fs
= fy , and fs < fy From geometry c = ⇒ fs

εu

d + εu d−c = E s εu c d − βa1 = E s εu a fs Es

(9.9-a)

β1

Pn = .85fc
ab + A
s fy − As fs a Pn e
= .85fc
ab(d − ) + A
s fy (d − d
) 2

(9.9-b) (9.9-c)

this would yield a cubic equation in Pn , which can be solved analytically or by iteration. 1. Assume a (a  h) 2. Solve for ΣM = 0 with respect to tensile reinforcement & solve for Pn 3. From strain compatibility solve for fs 4. Check that ΣFy = 0 & solve for a 5. If ai+1 = ai go to step 2 Case II: c is known and c > cb ; fs , fs
, and Pn are unknown

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns

9–7

In this case a = β1 c

(9.10-a)

d−c c c − d

fs = εc Es c
C = 0.85fc ab fs = εc Es

≤ fy

(9.10-b)

≤ fy

(9.10-c) (9.10-d)

A
s fs


Pn = C + As fs +     h h−a h + A
s fs
− d
+ As fs d − Mn = C 2 2 2

9.2.3

(9.10-e) (9.10-f)

ACI Provisions

1. Governing equations ρmin ρmax ρs φ φ

= = = = =

1% 8%
A 0.45( Agc − 1) ffyc 0.7 for tied columns 0.75 for spiral columns

ACI 10.9.1 ACI 10.5

(9.11)

where ρs minimum ratio of spiral reinforcement Ag gross area of section Ac area of core 2. A minimum of 4 bars for tied circular and rect 3. A minimum of 6 bars for spirals (ACI10.9.2) 4. φ increases linearly to 0.9 as φPn decreases from 0.10fc
Ag or φP0 , whichever is smaller, to zero (ACI 9.3.2). 5. Maximum strength is 0.8φP0 for tied columns (φ = 0.7) and 0.85φP0 for spirally reinforced columns (φ = 0.75).

9.2.4 13

Interaction Diagrams

Each column is characterized by its own interaction diagram, Fig. 9.7

9.2.5

Design Charts

To assist in the design of R.C. columns, design charts have been generated by ACI in term As +A
s χe Pn Mn of non dimensionalized parameters χ = bhf and
vs bh2 f
= h for various ρt where ρt = bh

14

µ=

fy .85fc


c

c

Example 9-1: R/C Column, c known Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–8

COLUMNS

Tied: Pn(max) = 0.80 P0 Spir. reinf: P n(max) = 0.85 P0

P

Compression control region

P0

A

P n(max)

1 e

P n−M n

(M

n’ Pn )

B

P d −M d (M

e

mi

n

e=0; a=h; c= infty

φ Pn(max)

eb

e

ilur

d fa

ce alan

Tension

nb’

Pnb ) control region

0.10f’ c A g

0

e~h; e = infty

φM n

Mn

M

Figure 9.7: Column Interaction Diagram A 12 by 20 in. column is reinforced with four No. 4 bars of area 1.0 in2 each, at each corner. fc
= 3.5 ksi, fy = 50 ksi, d
= 2.5 in. Determne: 1) Pb and Mb ; 2) The load and moment for c = 5 in; 3) load and moment for c = 18 in. Solution: Balanced Conditions is derived by revisiting the fundamental equations, rather than mere substitution into previously derived equation. d

=

cb

=

a

=

h − d
= 20 − 2.5 = 17.5 in .003 .003 17.5 = 11.1 in d = 50 fy + .003 + .003 29,000 E

(9.12-a) (9.12-b)

s

fs

def

=

fs


=

C

=

Pnb

=

Mnb

= =

eb

=

β1 cb = (0.85)(11.1) = 9.44 in

(9.12-c)

fy = 50 ksi (9.12-d)
c−d 11.1 − 2.5 εc = (29, 000)( (0.003) = 67.4 ksi > fy ⇒ fs
= 50(9.12-e) Es ksi c 11.1 0.85fc
ab = (0.85)(3.5)(9.44)(12) = 337 k (9.12-f) C + As fs − A
s fs = 337 + (2.0)(50) + (2.0)(−50) = 337 k (9.12-g) a Pnb e
= .85fc
ab(d − ) + A
s fy (d − d
) (9.12-h)  2  9.44 + (2.0)(50)(17.5 − 2.5) = 5, 807 k.in = 484 k.ft (9.12-i) 337 17.5 − 2 5, 807 = 17.23 in (9.12-j) 337

Tension failure, c = 5 in fs

def

=

Victor Saouma

fy = 50 ksi

(9.13-a) Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns fs


=

C Pn

Mn

c − d
c

≤ fy

(9.13-b)

=

(9.13-d)

=

0.85fc
ab

(9.13-e)

=

(0.85)(3.5)(4.25)(12) = 152 k

(9.13-f)

(0.003)(29, 000)

A
s fs


− As fy

(9.13-c)

=

C+

=

(9.13-h) 152 + (2.0)(43.5) − (2.0)(50) = 139 k     h h h−a + A
s fs
− d
+ As fs d − about section centroid (9.13-i) C 2 2 2       20 20 20 − 4.25 + (2.0)(43.5) − 2.5 + (2.0)(50) 17.5 − (9.13-j) (152) 2 2 2

= =

e

εc E s

5.0 − 2.5 = 43.5 ksi 5.0 β1 c = 0.85(5.0) = 4.25 in

= a

9–9

(9.13-g)

=

2, 598 k.in = 217 k.ft

(9.13-k)

=

2, 598 = 18.69 in 139

(9.13-l)

Compression failure, c = 18 in a = β1 c = 0.85(18) = 15.3 in d−c ≤ fy fs = εc Es c 17.5 − 18.0 = −2.42 ksi As is under compression = (0.003)(29, 000) 18.0 c − d
≤ fy fs
= εc Es c 18.0 − 2.5 = 75 ksi > fy ⇒ fs
= 50 ksi = (0.003)(29, 000) 18.0 C = 0.85fc
ab = (0.85)(3.5)(15.3)(12) = 546 k Pn = C +

Mn

A
s fs


− As fs

(9.14-a) (9.14-b) (9.14-c) (9.14-d) (9.14-e) (9.14-f) (9.14-g)

= 546 + (2.0)(50) − (−2.42)(2) = 650 k (9.14-h)     h h h−a + A
s fs
− d
+ As fs d − about section centroid (9.14-i) = C 2 2 2       20 − 15.3 20 20 = (546) + (2.0)(50) − 2.5 + (2.0)(−2.42) 17.5 − (9.14-j) 2 2 2 =

e =

2, 000 k.in = 167 k.ft

(9.14-k)

2, 000 = 3.07 in 650

(9.14-l)

Example 9-2: R/C Column, e known Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–10

COLUMNS

For the following column, determine eb , Pb , Mb ; Pn and Mn for e = 0.1h and e = h. fc
= 3, 000 psi and fy = 40, 000 psi. The area of each bar is 1.56 in2 . 12"

20"

3"

3"

24" c

εy

.003

Cc

Balanced Condition: fy 40 = .001379 = Es 29, 000 εu .003 .003 = 14.4 in d= cb = εu + εy .003 + .001379 a = β1 cb = (.85)(14.4) = 12.2 in

εy =

Cc = εsc fsc

(9.15-a) (9.15-b) (9.15-c)

.85fc
ab

= (.85)(3)(12.2)(20) = 624 k c − h/2 14.4 − 12 εu = .003 = .0005 = c 14.4 = (29, 000)(.0005) = 15 ksi center bars

(9.15-d) (9.15-e) (9.15-f)

Cs = (.0005)(29, 000)(2)(1.56) = 46.8 k

(9.15-g)

Pnb = 624 + 46.8 = 670.8 k

(9.15-h)

Note that the compression steel is yielding because d
> 2” and c > 6” (as previously proven) Taking moment about centroid of section Mnb = Pnb e



h a − 2 2









h h + As fy − d
+ A
s fy − d
2 2   12.2 + 4(1.56)(9)(40) = (.85)(3)(12.2)(20) 12 − 2 +4(1.56)(40 − .85 × 3)(12 − 3)

= .85fc
ab

(9.16-a) (9.16-b)

(9.16-c)

= 3, 671 + 2, 246 + 2, 246

(9.16-d)

=

8, 164 k.in; 680 k.ft

(9.16-e)

8, 164 = 12.2 in 670.8

(9.16-f)

eb = Victor Saouma



Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns

9–11

e= .1 h e = (.1)(24) = 2.4 in < eb ⇒ failure by compression. Pn , a and fs are unknown. Available equations: 1) ΣF = 0; 2) ΣM = 0; and 3) strain diagram; Solve by iterations.

12"

20"

3"

h−c−d’

3"

24" c=23.5"

εy

ε

s

9"

ε

sc

e=2.4"

ε

e’=11.4"

y

.003

Pn h/2=12" .85f’c

A sf

s

A scf sc

Cc

a=20" 1. Assume a = 20 in c=

A’ sf

y

a/2

a 20 = 23.5 in = β1 .85

(9.17)

2. For center steel (from geometry) εsc c − h2

=

.003 c

(9.18-a)

c − h2 .003 c = Es εsc

⇒ εsc = fsc

c− .003 c 23.5 − 12 .003 = 42.5 ksi > fy ⇒ fsc = fy = 29, 000 23.5 = Es

Victor Saouma

h 2

(9.18-b) (9.18-c) (9.18-d) (9.18-e)

Mechanics and Design of Reinforced Concrete

Draft 9–12

COLUMNS

3. Take moment about centroid of tensile steel bar a h (9.19-a) Pn e
= 0.85fc
ab(d − ) + A
s fy (h − 2d
) + Asc fy ( − d
) 2 2 20 (9.19-b) Pn (9 + 2.4) = (.85)(3)(20)(20)(21 − ) + 4(1.56)(40)(24 − 6) + 2(1.56)(40)(9) 2 = 11, 220 + 4, 493 + 259.7 (9.19-c) ⇒ Pn = 1, 476 k

(9.19-d)

4. Get εs in tension bar (from strain diagram) εs h − d
− 23.5

.003 c .003 (24 − 3 − 23.5) = 23.5 = −.000319 =

⇒ εs

fs = Eεs = (29, 000)(0.000319) = 9.25 ksi

(9.20-a) (9.20-b) (9.20-c) (9.20-d)

5. Take ΣF = 0 to check assumption of a Pn = 0.85fc
ab + A
s fy + Asc fsc + As fy

(9.21-a)

1, 476 k = (.85)(3)(a)(20) + (4)(1.56)(40) + (2)(1.56)(40) + (4)(1.56)(9.25) (9.21-b) 1, 476 = 51a + 432.1 √ ⇒ a = 20.4 in Pn =

(9.21-c) (9.21-d)

1, 476 k

(9.21-e)

Mn = (1, 476)(2.4) = 3, 542 k.in = 295 k.ft

e=h

(9.21-f)

1. In this case e = 24 in > eb ⇒ failure by tension. Pn and a are unknown. We have two equations: 1) ΣF = 0, and 2) ΣM = 0. 2. Assume a = 7.9 in ⇒ c =

a β1

=

7.9 .85

= 9.3 in

3. Steel stress at centroid c .003 ⇒ εsc ⇒ fsc

12 − c εsc 12 − 9.3 .003 = .00087 = 9.3 = (29, 000)(0.00087) = 25.3 ksi

=

(9.22-a) (9.22-b) (9.22-c)

4. Iterate ΣF = 0 ⇒ Pn = (.85)fc
ab + Asc fsc = (.85)(3)(7.9)(20) − 2(1.56)(25.3) = 403 − 79 = 324 k a ΣM = 0 ⇒ Pn (e + h/2 − d ) = .85fc
ab(d − ) + A
s fy (d − d
) 2 d − d
−Asc fsc ( ) 2

(9.23-a) (9.23-b) (9.23-c)




Victor Saouma

(9.23-d)

Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns

9–13 Pn (24 + 9) = (.85)(3)(7.9)(20)(21 − +2(1.56)(25.3)(9)

7.9 ) + 4(1.56)(40)(21 − 3) 2 (9.23-e)

Pn (33) = 6, 870 + 4, 493 − 710 = 10, 653 k.in √ ⇒ Pn = 323 k

(9.23-f) (9.23-g)

5. Determine Mn Mn = Pn e = (323)(24) = 7, 752 k.in = 646 k.ft

(9.24)

Example 9-3: R/C Column, Using Design Charts Design the reinforcement for a column with h = 20 in, b = 12 in, d
= 2.5 in, fc
= 4, 000 psi, fy = 60, 000 psi, to support PDL = 56 k, PLL = 72 k, MDL = 88 k.ft, MLL = 75 k.ft, Solution: 1. Ultimate loads 201 = 287 k 0.7 251 = 358 k.ft = (1.4)(88) + (1.7)(75) = 251 k.ft ⇒ Mn = 0.7

Pu = (1.4)(56) + (1.7)(72) = 201 k ⇒ Pn = Mu

(9.25-a) (9.25-b)

2. Chart parameters e h

=

γ = κ = κ

e h

(358)(12) = 0.75 (9.26-a) (287)(20) h − 2d
20 − (2)(2.5) = 0.75 ⇒ interpolate between A3 and A(9.26-b) 4 h 20 287 Pn = 0.3 (9.26-c) = bhfc
(12)(20)(4)

= (0.3)(0.75) = 0.225

(9.26-d)

3. Interpolating between A.3 and A.4 ⇒ ρt µ = 0.4 4. Reinforcement ρt = µ =

At bh fy .85fc


 At =

(0.4)(b)(h)(.85)(fc
) 1 2 = 5.45 in(9.27-a) = (0.4)(12)(20)(.85)(4) fy (60)

⇒ use 4 # 9 & 2 # 8, ⇒ At = 5.57 in2

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–14

9.2.6 15

COLUMNS

Biaxial Bending

Often columns are subjected to biaxial moments (such as corner columns)

An exact approach entails the trial and eror determination of an inclined neutral axis, this is an exact method but too cumbersome to use in practice.

16

17 Hence, we seek an approximate solution, the most widely used method is the load contour method or Bresler-Parme method. 18

The failure surface of a biaxialy loaded column is shown in Fig. 9.8, and the general nondi-

Pn

M0x

1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111

M0y

Mny

Mnx Figure 9.8: Failure Surface of a Biaxially Loaded Column mensional equation for the moment contour at a constant Pn may be expressed as     Mny α1 Mnx α1 + = 1.0 M0x M0y where Mnx Mny M0x M0y and α1

= = = = and

Pn ey Pn ex Mnx capacity at axial load Pn when Mny (or ex ) is zero Mny capacity at axial load Pn when Mnx (or ey ) is zero α2 are exponent which depend on geometry and strength.

19 Bresler suggested that we set α1 = α2 = α. For practical purposes, a value of α = 1.5 for rectangular columns, and between 1.5 and 2.0 for square sections has proven acceptable.

An improvement of Bresler equation was devised by Parme. The main assumption is that at any load Pn , Fig. 9.9 Mny M0y = Mnx M0x or Mnx = βM0x ; Mny = βM0y 20

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns

M 0y

9–15

M ny /M0y 1.0 C

M ny βM 0x C

β

βM 0y

B M 0y M0x

A M nx M0x

B β

45

o

A 1.0 Mnx /M0x

Figure 9.9: Load Contour at Plane of Constant Pn , and Nondimensionalized Corresponding plots Thus, β is the portion of the uniaxial moment strength permitted to act simultaneously on the column section. It depends on the cross section, strength, and layout.

21

22

The usual range is between 0.55 and 0.70, with a recommended value of 0.65 for design.

23

Hence, once β is selected, we can substitute in Bresler’s equation α



M0y α M0x + β = 1.0 βM M0y 0x β α = 12 α log β = log 0.5 0.5 α = log log β

thus,



Mnx M0x

log 0.5/logβ

 +

Mny M0y

log 0.5/logβ (9.28)

= 1.0

24

Effect of β is shown in Fig. 9.10.

25

Gouwens proposed to replace the above curves, by a bilinear model, Fig. 9.11

Review of a section   Mny Mny Mnx 1 − β Mnx = 1 If + ≥ M0y M0x β M0y M0x   Mny Mnx Mny 1 − β Mnx = 1 If + ≤ M0x M0y β M0y M0x Design of a column

(9.30)



  Mny M0y M0y 1−β = M0y If ≥ M0x β Mnx M0x    Mny M0y M0x 1−β = M0x If ≤ Mnx + Mny M0y β Mnx M0x Mny + Mnx

Victor Saouma

(9.29)

(9.31-a) (9.31-b)

Mechanics and Design of Reinforced Concrete

Draft 9–16

COLUMNS

Biaxial Bending Interaction Diagram 1.0

0.90

0. 0.8 85 0 0.7 5 0.7 0 0.6 5

0.8

0.6

be

ta

0.6

0

0.

55

=0

Nny/M0y

.5

0

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Mnx/M0x

Figure 9.10: Biaxial Bending Interaction Relations in terms of β

Mnx

β C

M ny /M 0y + M nx /M 0x (1− β/ β ) =1

1.0 α

α

(M nx /M 0x ) +(M ny /M 0y) =1

1−β β

M0y

B

0y

M0x

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111

M ny /M

Pn

M nx /M 0x + M ny /M 0y (1− β/ β) =1 β

Mny

β 1−β

45

o

A M nx /M 0x

1.0

Figure 9.11: Bilinear Approximation for Load Contour Design of Biaxially Loaded Columns

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

9.2 Short Columns

9–17

Note, circular or square columns with symmetric reinforcement should always be considered first for biaxially loaded columns.

26

Example 9-4: Biaxially Loaded Column Determine the adequacy of a 16 in. square tied column with 8 # 9 bars. d
= 2.5 in, and there are 3 bars on each side. The section is to carry factored loads of Pu = 144 k, Mux = 120 k.ft and Muy = 54 k.ft, fc
= 3 ksi and fy = 40 ksi. P0 = 952 k, M0x = M0y = 207 k.ft (we have a symmetrical reinforcement). Solution: ey = ex =

Mux Pu Muy Pu

(120)(12) = 10.0 in 144 (54)(12) = 4.5 in 144

= =

The interaction diagram for e = 10 in, e = 4.5 in and e = 0 will give Pn equal to 254, 486, and 952 kips respectively. The required load Pn = 144 0.7 = 205 k, the corresponding moments are M0x = M0y = 207 k.ft from the interaction diagram. Using β = 0.65 Required Mnx M0x

Required Mny M0y

= =

120 0.7

207

= 0.828

207

= 0.373

54 0.7

We shall use both solutions Bresler-Parme which is exact solution log(0.5)



Mnx M0x



log 0.5/logβ

Mny M0y 1.609 (0.828) + (0.373)1.609

+

Note that we could have first solved for This would have given is safe.

log β log 0.5/logβ

Mny M0y

Mnx M0x ,

=

log 0.5 log 0.65

= =

0.943

= 1.609 √

and then determined

Mny M0y

from Fig. 9.10.

≈ 0.45 which is greater than the actual value, hence the design

Gouwens which is an approximate solution

 Mny 1−β Mnx + ≤1 M0x M0y β  1−0.65  0.828 + 0.337 0.65 = 0.828 + 0.1815 = 1.0095 which indicates a slight overstress. We note that the approximate method is on the conservative side.

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–18

9.3 9.3.1

COLUMNS

Long Columns Euler Elastic Buckling

Column buckling theory originated with Leonhard Euler in 1744. An initially straight member is concentrically loaded, and all fibers remain elastic until buckling occur.

27

For buckling to occur, it must be assumed that the column is slightly bent as shown in Fig. 9.12. Note, in reality no column is either perfectly straight, and in all cases a minor imperfection

28

P

P

x

x and y are principal axes

x

Slightly bent position L

y

Figure 9.12: Euler Column is present. At any location x along the column, the imperfection in the column compounded by the concentric load P , gives rise to a moment

29

Mz = −P y

(9.32)

Note that the value of yis irrelevant. 30

Recalling that

d2 y Mz = 2 dx EI upon substitution, we obtain the following differential equation P d2 y − =0 2 dx EI 31

Letting k 2 =

P EI ,

(9.34)

the solution to this second-order linear differential equation is y = −A sin kx − B cos kx

32

(9.33)

(9.35)

The two constants are determined by applying the essential boundary conditions 1. y = 0 at x = 0, thus B = 0 2. y = 0 at x = L, thus A sin kL = 0

Victor Saouma

(9.36)

Mechanics and Design of Reinforced Concrete

Draft

9.3 Long Columns

9–19

This last equation can e satisfied if: 1) A = 0, that is there is no deflection; 2) kL = 0, that is no applied load; or 3) kL = nπ (9.37)   2 P = nπ or Thus buckling will occur if EI L P =

n2 π 2 EI L2

The fundamental buckling mode, i.e. a single curvature deflection, will occur for n = 1; Thus Euler critical load for a pinned column is

33

π 2 EI L2

(9.38)

π2E σcr =  2 L

(9.39)

Pcr = The corresponding critical stress is

r

where I = Ar. 34

Note that buckling will take place with respect to the weakest of the two axis.

9.3.2 35

Large

Effective Length kL r

kL r

column buckling, small

column crushing, Fig. 9.13. Pfail

f

Pn −1

tan E

t

fp

Pcr Crushing −1

tan

Buckling

E ε

(kl/r) lim

(kl/r)

Figure 9.13: Column Failures 36

Recall from strength of material slenderness ratio λ=

Le r



where Le is the effective length and is equal to Le = kL and r the radius of gyration (r = 37

I A ).

Le is the distance between two adjacent (fictitious or actual) inflection points, Fig. 9.13

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–20

COLUMNS

P cr

P cr

Pcr

i.p. i.p.

l/4

kl= l 2

l

l

l 2

kl=l


i.p.

i.p.

l/4

i.p.

i.p.

Pcr

P cr

k=1

P cr

k=1/2

P

1/2
P

Pcr

cr

cr

l

l kl=21

Pcr

l
l

i.p.

8

i.p.

kl=1

Pcr

Pcr

l
i.p.

k=2

8

i.p.

k=1 Figure 9.14: Critical lengths of columns

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

9.3 Long Columns

9–21

k is known for some simple highly idealized cases, but for most cases k depends on ΨA + ΨB (relative stiffnesses of columns to connected beams), Fig. 9.15 38

Ψ=

Σ( EI L )of columns EI Σ( L )of floor members

(9.40)

and k is then determined from the chart shown in Fig. 9.16. ( EI ln

1

P

2

∆

ψA

A

ψA ( EI ln

ψA

A

(

(

A

(

( EI ln

P

(

P ( EI ln

2

1

MA

MA

MA

B

MB

MB

MB

B

ψB

Single curvature

ψB

B

ψB

Double curvature Braced

Unbraced

Figure 9.15: Effective length Factors Ψ

9.3.3 39

Moment Magnification Factor; ACI Provisions

The critical stress in a column is given by   P π2E =  2 kL A cr r

Code recommends some minimum eccentricity to account for imperfectly placed load, Fig. 9.17

40

41

For an eccentrically placed load Mmax = M0

1 P 1 − 1−P cr    Moment magnification factor

(9.41)

42 The moment magnification factor reflects the amount by which the beam moment M0 is magnified by the presence of an axial load, Fig. 9.18

The previous equation assumes the presence of hinges at each end (Euler column). In the most general case we will have

43

Mmax = M0 Victor Saouma

Cm 1 − PPcr

(9.42-a)

Mechanics and Design of Reinforced Concrete

Draft 9–22

COLUMNS

Sidesway Inhibited ∞ 50. 10. 5. 3.

Ga

K

1.0

0.9

2.

Sidesway Uninhibited Gb ∞ 50. 10. 5. 3.

∞ 100. 50. 30. 20.

2. 0.8

1.

1.

0.8 0.7 0.6 0.5 0.4

0.8 0.7 0.6 0.5 0.4

0.7

0.3 0.2

Ga

0.3 0.6

K

∞ 20. 10.

5. 4.

10. 9. 8. 7. 6. 5.

3.

10. 9. 8. 7. 6. 5.

4.

2.

4.

3.

3.

2.

2. 1.5

0.2 1.

0.1

0.

Gb ∞ 100. 50. 30. 20.

1.

0.1

0.5

0.

0

1.

0

Figure 9.16: Standard Alignment Chart (ACI)

P

e

Figure 9.17: Minimum Column Eccentricity

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

9.3 Long Columns

9–23 P φ P

M

0

P 0(max)

P

C

P cr M

B

Pu in

Pn

∆

em

P

Pu ∆

Pu e

M 0

M

M c =δ M 2

0

M

kl/r

0

M

n

M

0

M

2

Mc

M

Figure 9.18: P-M Magnification Interaction Diagram Cm = .6 + .4 where M1 M1 M2 M1 M2

>0 <0 Cm < 1 Cm =1 44

45

M1 ≥ .4 M2

(9.42-b)

is numerically smaller than M2 (not algebracially) if single curvature if double curvature if members are braced against sidesway if members are not braced against sidesway

ACI Code Lu k ≤ 1.0 k ≥ 1.0 r = .3h r = .25d M1 kLu r < 34 − 12 M2 kLu r < 22

unsupported length ACI 10.11.1 braced columns ACI 10.11.2 unbraced columns ACI 10.11.2 rectangular x section ACI 10.11.3 circular cross section braced, neglect slenderness ACI 10.11.4 unbraced, neglect slenderness

From conventional elastic analysis get Pn &Mn Mc = δM2 Cm ≥ 1.0 δ = Pn 1 − φP cr π 2 EI 10.11.5 (kLu )2 M1 = .6 + .4 M2

(9.43) (9.44)

Pcr =

(9.45)

Cm

(9.46)

EI = or EI = βd =

Ec Ig 5

+ Es Is 1 + βd

Ec Ig 2.5

1 + βd MD 1.4PDL = MD + ML 1.4PDL + 1.7PLL

(9.47) (9.48) (9.49)

βd is the ratio of maximum design load moment to maximum design total load moment (always Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–24

COLUMNS

+ve) as β  EI ⇒ dead load has a detrimental effect (creep) Example 9-5: Long R/C Column

A 15 ft long, 14” circular column is connected to 40 ft long 14” by 22” beams. The column is on the last floor, below it the column is circular and has a 16” diameter. No sidesway. Given, Pn = 500 k, 14 × 22 has ρ = .015, fc
= 5, 000 psi, fy = 40, 000 psi Solution:

22 = 13.17 ft 12 r = .25d = (.25)(14 in) = 3.5 in   Ec = 57, 000 fc
= 57, 000 5, 000 = 4, 030 ksi π(14)4 πd4 = = 1, 886 in4 Ig = 64 64

Lu = 15 ft −

EIcol =



Ec Ig 2.5 1+βd

EIcol = (4, 030)(1, 886)

βd = 0 

EI L



1 = 3, 040, 000 k in2 2.5

3, 040, 000 = 16, 890 k.in (15)(12)

= c

Ig (14)(22)3 1 Ibeam = Icr  = = 6, 210 in4 2 12 2   (4, 030)(6, 210) EI = 52, 140 k.in = L beam (12)(40) Σ(EI/L)col (16, 890) = .162 = ΨA = Σ(EI/L)beam 2(52, 140) bottom column I =

π(16)4 64

(9.50-a) (9.50-b) (9.50-c) (9.50-d)

(9.51-a)

(9.52-a) (9.52-b) (9.52-c) (9.52-d)

= 3, 217 in4

EI =   EI = L col ΨB =

(4, 030)(3, 217) = 5, 186, 000 2.5 5, 186, 000 = 28, 800 k.in (15)(12) 16, 890 + 28, 800 = .438 2(52, 140)

(9.53-a) (9.53-b) (9.53-c)

From ACI commentary ΨA = .162, ΨB = .438, ⇒ k  .62 and kLu r M1 34 − 12 M2 kL r Victor Saouma

=

(.65)(13.16)(12) = 29.3 3.5

(9.54-a)

= 34 − 12 = 22 assuming M1 = M2

(9.54-b)

> 22 ⇒ consider column instability

(9.54-c)

Mechanics and Design of Reinforced Concrete

Draft

9.3 Long Columns

9–25 π 2 EI π 2 (3, 040, 000) = = 2, 848 k 2 (kl) [(.65)(13.16)(12)]2 M1 CM = .6 + .4 =1 M2 1 1 δ = = = 1.3 500 Pu 1 − (.75)(2,848) 1 − φPcr Pcr =

(9.54-d) (9.54-e) (9.54-f)

Example 9-6: Design of Slender Column Given: frame not braced, design AB as square column. PD = 46 k, MD = 92 k.ft, PL = 94 k, ML = 230 k.ft, fc
= 4 ksi, fy = 60 ksi

A

Lu =18’

L 3 l =43.3in

B

111 000 000 111 000 111

111 000 000 111 000 111

Solution:

Pu = 1.4 × 46 + 1.7 × 94 = 224 k Mu = 1.4 × 92 + 1.7 × 230 = 520 k.ft MDL (1.4)92 = .24 = βd = MDL + MLL 520

(9.55-a) (9.55-b) (9.55-c)

Assume a 22 × 22 inch column and ρt = .03 2.5 8.5 22"

22"

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 9–26

COLUMNS

If Is Ec

224 = 19, 500 in4 12 = (2)(.015)(22)2 (8.5)2 = 1, 050 in4  = 57, 000 4, 000 = 3.6 × 106 psi =

(9.56-a) (9.56-b) (9.56-c)

Es = 29 × 10 ksi 6

EI = = EIc L EIb L

kL r

+ Es Is 1 + βd

(3.6×106 )(19,500) 5

+ (29 × 106 )(1, 050) = 3.59 × 1010 1 + .24

(9.56-e)

3.59 × 1010 = 1.66 × 108 12 × 18

(9.56-f)

= (3.6 × 106 )(43.3) = 1.56 × 108

(9.56-g)

=

AtA&B Ψ = if

(9.56-d)

Ec Ig 5

2(1.66 × 108 ) = 2.13 from ACI commentary k = 1.65 1.56 × 108

(9.56-h)

= 22 neglect slenderness r = (.3)(22) = 6.6 in (1.65)(18)(12) kL = = 54 > 22 ⇒ r 6.6 π 2 EI π 2 (3.59 × 1010 ) Pcr = = = 279 × 106 lbs (kL)2 [(1.65)(18)(12)]2 Pu = 2.24 × 105 lb Cm = 1.0(unbraced) 1 = Moment Magnification δ = Pu 1 − φP 1− cr = 1.13

(9.57-a) (9.57-b) (9.57-c) (9.57-d) (9.57-e)

1 (2.24×105 ) (.7)(2.79×106 )

(9.57-f) (9.57-g)

⇒ Moment for which the column is to be designed (1.13) (520) = 587 k.ft and Pu = 224

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft Chapter 10

PRESTRESSED CONCRETE 10.1

Introduction

Beams with longer spans are architecturally more appealing than those with short ones. However, for a reinforced concrete beam to span long distances, it would have to have to be relatively deep (and at some point the self weight may become too large relative to the live load), or higher grade steel and concrete must be used.

1

2 However, if we were to use a steel with fy much higher than ≈ 60 ksi in reinforced concrete (R/C), then to take full advantage of this higher yield stress while maintaining full bond between concrete and steel, will result in unacceptably wide crack widths. Large crack widths will in turn result in corrosion of the rebars and poor protection against fire.

One way to control the concrete cracking and reduce the tensile stresses in a beam is to prestress the beam by applying an initial state of stress which is opposite to the one which will be induced by the load.

3

For a simply supported beam, we would then seek to apply an initial tensile stress at the top and compressive stress at the bottom. In prestressed concrete (P/C) this can be achieved through prestressing of a tendon placed below the elastic neutral axis.

4

Main advantages of P/C: Economy, deflection & crack control, durability, fatigue strength, longer spans.

5

6

There two type of Prestressed Concrete beams:

Pretensioning: Steel is first stressed, concrete is then poured around the stressed bars. When enough concrete strength has been reached the steel restraints are released, Fig. 10.1. Postensioning: Concrete is first poured, then when enough strength has been reached a steel cable is passed thru a hollow core inside and stressed, Fig. 10.2.

10.1.1

Materials

P/C beams usually have higher compressive strength than R/C. Prestressed beams can have fc
as high as 8,000 psi.

7

8

The importance of high yield stress for the steel is illustrated by the following simple example.

Draft 10–2

PRESTRESSED CONCRETE

Vertical bulkhead

Harping hold-up point

Harping hold-down point Jacks

Anchorage

Prestressing bed slab

Continuous tendon

Precast Concrete element Tendon anchorage

Jacks

Support force

Casting bed

Jacks

Casting bed

Hold-down force

Tendon

Figure 10.1: Pretensioned Prestressed Concrete Beam, (Nilson 1978)

Anchorage

Anchorage

Intermediate diaphragms

Jack

Beam

Jack

Tendon in conduct

Anchorage Jack

Slab

Wrapped tendon

Figure 10.2: Posttensioned Prestressed Concrete Beam, (Nilson 1978)

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

10.1 Introduction

10–3

If we consider the following: 1. An unstressed steel cable of length Ls 2. A concrete beam of length Lc 3. Prestress the beam with the cable, resulting in a stressed length of concrete and steel equal to L
s = L
c . 4. Due to shrinkage and creep, there will be a change in length ∆Lc = (εsh + εcr )Lc

(10.1)

we want to make sure that this amout of deformation is substantially smaller than the stretch of the steel (for prestressing to be effective). 5. Assuming ordinary steel: fs = 30 ksi, Es = 29, 000 ksi, εs =

30 29,000

= 1.03 × 10−3 in/ in

6. The total steel elongation is εs Ls = 1.03 × 10−3 Ls 7. The creep and shrinkage strains are about εcr + εsh  .9 × 10−3 8. The residual stress which is left in the steel after creep and shrinkage took place is thus (1.03 − .90) × 10−3 (29 × 103 ) = 4 ksi Thus the total loss is

30−4 30

(10.2)

= 87% which is unacceptably too high.

9. Alternatively if initial stress was 150 ksi after losses we would be left with 124 ksi or a 17% loss. 10. Note that the actual loss is (.90 × 10−3 )(29 × 103 ) = 26 ksi in each case 9

Having shown that losses would be too high for low strength steel, we will use

Strands usually composed of 7 wires. Grade 250 or 270 ksi, Fig. 10.3. 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 00000 11111 000000 000000 111111 00000111111 11111 000000 111111 000000 111111 00000 11111 000000 111111 000000 111111 00000 11111 000000 000000 111111 00000111111 11111 000000 111111 000000 111111 00000 11111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 00000 11111 000000 111111 000000 111111 000000 111111 00000 11111 00000 11111 000000 111111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 11111 00000 000000 111111

Figure 10.3: 7 Wire Prestressing Tendon Tendon have diameters ranging from 1/2 to 1 3/8 of an inch. Grade 145 or 160 ksi. Wires come in bundles of 8 to 52. Note that yield stress is not well defined for steel used in prestressed concrete, usually we take 1% strain as effective yield. Steel relaxation is the reduction in stress at constant strain (as opposed to creep which is reduction of strain at constant stress) occurs. Relaxation occurs indefinitely and produces significant prestress loss. If we denote by fp the final stress after t hours, fpi the initial stress, and fpy the yield stress, then   fp log t fpi =1− − .55 (10.3) fpi 10 fpy 10

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 10–4

10.1.2 11

PRESTRESSED CONCRETE

Prestressing Forces

Prestress force “varies” with time, so we must recognize 3 stages: 1. Pj Jacking force. But then due to (a) friction and anchorage slip in post-tension (b) elastic shortening in pretension is reduced to: 2. Pi Initial prestress force; But then due to time dependent losses caused by (a) relaxation of steel (b) shrinkage of concrete (c) creep of concrete is reduced to: 3. Pe Effective force

10.1.3 12

Assumptions

The following assumptions are made; 1. Materials are both in the elastic range 2. section is uncracked 3. sign convention: +ve tension, −ve compression 4. Subscript 1 refers to the top and 2 to the bottom 5. I, S1 =

I c1 ,

S2 =

I c2 ,

(section modulus)

6. e + ve if downward from concrete neutral axis

10.1.4

Tendon Configuration

Through proper arrangement of the tendon (eccentricity at both support and midspan) various internal flexural stress distribution can be obtained, Fig. 10.4.

13

10.1.5

Equivalent Load

An equivalent load for prestressing can be usually determined from the tendon configuration and the prestressing force, Fig. 10.5.

14

10.1.6 15

Load Deformation

The load-deformation curve for a prestressed concrete beam is illustrated in Fig. 10.6.

Victor Saouma

Mechanics and Design of Reinforced Concrete

P

000 111 000 111 000 111 000 111 111 000 f’y

fc =f t

=

2f c 000 111 000 111 000 111 000 111 000 111 111 000

fc 0000 1111 1111 0000

0000 1111 0000 + 1111 1111 0000 0000 1111

fc 000 111 000 111 000 111 000 111 000 111 000 111 fc 2f c 000 111 000 111 000 111 000 111 111 000 0

None P cos θ 2P sinθ

W

h

h/2

2f c 0 000 1111111 111 0000000 1111111 000 111 0000000 000 +1111111 111 0000000 = 000 111 0000000 1111111 111 000 0000000 1111111 2f c 2f =2f t c

P cos θ P cos θ

Q P

2h/3

2f c 000 111 000 111 000 111 000 111 000 111 000 111 0 fc 000 111 000 111 000 111 111 000 000 111 fc

fc

P

P

P h/2

f c 0000 1111 0000 1111 0000 1111 + 1111 0000 0000 1111 1111 0000 ft =f c

0

Midspan

fc 000 111 000 111 000 111 000 111 = 000 111 000 111

0 2f c 000 1111111 111 0000000 1111111 000 111 0000000 000 111 0000000 1111111 000 +1111111 111 0000000 = 1111111 000 111 0000000 000 111 0000000 1111111 2f c 2f =2f t c fc Midspan 00 11 00 11 = 0 00 + 11 00 11 00 11 Ends fc 0 000 111 000 111 000 111 000 111 000 111 111 000 2f c

+

fc 000 111 000 111 000 = 111 000 111 000 111 fc

(g) P

2Q

h/3

h/2

Ends

P

P

2Q

Q P

fc 000 111 000 111 000 111 111 000 000 111 fc

None

(f) P

P sinθ

P P sinθ P cosθ P (e) P

P

P h/3

P cosθ

M P sin θ M P sin θ Pθ (d) P e

P sin θ P sin θ (b)

P cos θ 2P sin θ P cos θ

P Pe Pe P P e (c) P

P cos θ P P θ

P sin θ P θ

Moment from prestressing Equivalent load on concrete from tendon

Member

P sinθ (a) P

Mechanics and Design of Reinforced Concrete

Victor Saouma

10–5

Draft

10.1 Introduction

Figure 10.4: Alternative Schemes for Prestressing a Rectangular Concrete Beam, (Nilson 1978)

Figure 10.5: Determination of Equivalent Loads

Draft 10–6

PRESTRESSED CONCRETE Load

Ru

ptu

Steel yielding Service load limit including tolerable overload

Overload

re

Tn

Service load range

First cracking load f cr

Decompression

or higher cgs (f=0)

Balanced Full dead load

∆

o

∆

∆ ∆

Deformation ∆ (deflection of camber)

∆ D

L

∆ pi = Initial prestress camber ∆ pe = Effective prestress camber ∆ O = Self−weight deflection ∆ D= Dead load deflection ∆ L= Live load deflection

pe

pi

Figure 10.6: Load-Deflection Curve and Corresponding Internal Flexural Stresses for a Typical Prestressed Concrete Beam, (Nilson 1978)

10.2 16

Flexural Stresses

We now identify the following 4 stages:

Initial Stage when the beam is being prestressed (recalling that r2 =

I Ac )

1. The prestressing force, Pi only Pi Pi Pi ec1 =− + Ac I Ac Pi Pi Pi ec2 =− = − − Ac I Ac

f1 = − f2



ec1  r2

ec2  1+ 2 r 1−

(10.4) (10.5)

2. Pi and the self weight of the beam M0 (which has to be acconted for the moment the beam cambers due to prestressing) Pi

ec1  M0 f1 = − (10.6) 1− 2 − Ac r S1 ec2  M0 Pi

1+ 2 + (10.7) f2 = − Ac r S2 Service Load when the prestressing force was reduced from Pi to Pe beacause of the losses, and the actual service (not factored) load is apllied 3. Pe and M0

Victor Saouma

Mechanics and Design of Reinforced Concrete

f1 = −

e

Stage 1

Stage 2

Stage 4

c1 c2

Pi Ac

00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 11 00

Pi Ac

e c1 ) r2

e c2 ) r2

e c1 Mo )− r2 S

Pi (1+ Ac

000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 111111111 000000000

Pi (1− Ac

Pe (1− Ac

Mo e c2 )+ r2 S2

1 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 111111 000000

Pe (1+ Ac

Pi e c 1 Ic

Pi e c 2 Ic

Mo S1

Pi (1− Ac

e c1 ) r2

000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 111111111 000000000

e c2 ) r2

Mo e c1 )− r2 S1

Pi (1+ Ac

Pi (1− Ac

000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 111111 000000

e c1 Mt )− r2 S1

e c2 Mo )+ r2 S2

000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 111 000

Pe (1+ Ac

e c2 Mt )+ r2 S2

00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 11111 00000

Pe (1− Ac

Pi (1+ Ac

Md + Ml S

Mo S2

−

0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 11111 00000

+

−

Md + Ml S2

1 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 1111111111 0000000000

+

Mechanics and Design of Reinforced Concrete

Victor Saouma

(10.10)

(10.11)

ec1  M0 + MDL + MLL − r2 S1

ec2  M0 + MDL + MLL 1+ 2 + r S2 1−

f1 = −

(10.8)

(10.9)

ec1  − r2

ec2  1+ 2 + r

M0 S1 M0 S2 1−

Pe Ac Pe = − Ac f2

Pe Ac Pe = − Ac f2

10–7 10.2 Flexural Stresses

Draft

4. Pe and M0 + MDL + MLL

The internal stress distribution at each one of those four stages is illustrated by Fig. 10.7.

Figure 10.7: Flexural Stress Distribution for a Beam with Variable Eccentricity; Maximum Moment Section and Support Section, (Nilson 1978)

Draft 10–8

PRESTRESSED CONCRETE

Those (service) flexural stresses must be below those specified by the ACI code (where the subscripts c, t, i and s refer to compression, tension, initial and service respectively):
fci permitted concrete compression stress at initial stage .60f ci fti permitted concrete tensile stress at initial stage < 3 fci

fcs permitted concrete compressive stress at service stage .45f c  fts permitted concrete tensile stress at initial stage 6 fc
or 12 fc
 Note that fts can reach 12 fc
only if appropriate deflection analysis is done, because section would be cracked. 17

18

Based on the above, we identify two types of prestressing:

Full prestressing (pioneered by Freysinet), no tensile stresses, no crack, but there are some problems with excessive camber when unloaded. Partial prestressing (pioneered by Leonhardt, Abeles, Thurliman), cracks are allowed to occur (just as in R/C), and they are easier to control in P/C than in R/C. 19 The ACI code imposes the following limits on the steel stresses in terms of fpu which is the ultimate strength of the cable: Pj < .80fpu As and Pi < .70fpu As . No limits are specified for Pe .

Example 10-1: Prestressed Concrete I Beam Adapted from (Nilson 1978) The following I Beam has fc
= 4, 000 psi, L = 40 ft, DL+LL =0.55 k/ft, concrete density γ = 150 lb/ft3 and multiple 7 wire strands with constant eccentricity e = 5.19 in. Pi = 169 k, and the total losses due to creep, shinkage, relaxation are 15%. 12" 4" 5"

2"

7" 4"

6" 24" 6"

7"

2" 5"

4"

The section properties for this beam are Ic = 12, 000 in4 , Ac = 176 in2 , S1 = S2 = 1, 000 in3 , = AI = 68.2 in2 . Determine flexural stresses at midspan and at support at initial and final conditions. Solution: r2

1. Prestressing force, Pi only f1 = − Victor Saouma

Pi

ec1  1− 2 Ac r

(10.12-a)

Mechanics and Design of Reinforced Concrete

Draft

10.2 Flexural Stresses

f2

10–9   (5.19)(12) 169, 000 1− = −83 psi = − 176 68.2 Pi

ec2  = − 1+ 2 Ac r   (5.19)(12) 169, 000 1+ = −1, 837 psi = − 176 68.21

(10.12-b) (10.12-c) (10.12-d)

2. Pi and the self weight of the beam M0 (which has to be acconted for the moment the beam cambers due to prestressing) w0 = M0 =

(176) in2 (.150) k/ ft3 = .183 k/ft (144) in2 / ft2 (.183)(40)2 = 36.6 k.ft 8

(10.13-a) (10.13-b)

The flexural stresses will thus be equal to: w0 f1,2 =∓

M0 (36.6)(12, 000) = ∓439 psi =∓ S1,2 1, 000 ec1  M0 Pi

1− 2 − Ac r S1 −83 − 439 = −522 psi  √ 3 fc
= +190 ec2  M0 Pi

1+ 2 + − Ac r S2 −1, 837 + 439 = −1, 398 psi √ .6fc
= −2, 400

(10.14)

f1 = −

(10.15-a)

=

(10.15-b)

fti = f2 = = fci =

(10.15-c) (10.15-d) (10.15-e) (10.15-f)

3. Pe and M0 . If we have 15% losses, then the effective force Pe is equal to (1 − 0.15)169 = 144 k ec1  M0 Pe

1− 2 − f1 = − (10.16-a) Ac r S1   (5.19)(12) 144, 000 1− − 439 (10.16-b) = − 176 68.2

f2

= −71 − 439 = −510 psi ec2  M0 Pe

1+ 2 + = − Ac r S2   144, 000 (5.19)(12) = − 1+ + 439 176 68.2

(10.16-d)

= −1, 561 + 439 = −1, 122 psi

(10.16-f)

(10.16-c)

(10.16-e)

note that −71 and −1, 561 are respectively equal to (0.85)(−83) and (0.85)(−1, 837) respectively. Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 10–10

PRESTRESSED CONCRETE

4. Pe and M0 + MDL + MLL (0.55)(40)2 = 110 k.ft 8

(10.17)

(110)(12, 000) = ∓1, 320 psi 1, 000

(10.18)

MDL + MLL = and corresponding stresses f1,2 = ∓ Thus,

ec1  M0 + MDL + MLL Pe

1− 2 − Ac r S1 −510 − 1, 320 = −1, 830 psi √ .45fc
= −2, 700 ec2  M0 + MDL + MLL Pe

1+ 2 + − Ac r S2 −1, 122 + 1, 320 = +198 psi  √ 6 fc
= +380

f1 = −

(10.19-a)

=

(10.19-b)

fcs = f2 = = fts =

(10.19-c) (10.19-d) (10.19-e) (10.19-f)

+198

10.3

2 -1122

3

1 -1398

4

-1837

-83 -510 -522

-1830

5. The stress distribution at each one of the four stages is shown below.

Case Study: Walnut Lane Bridge

Adapted from (Billington and Mark 1983) The historical Walnut Lane Bridge (first major prestressed concrete bridge in the USA) is made of three spans, two side ones with lengths of 74 ft and a middle one of length 160 feet. Thirteen prestressed cocnrete beams are placed side by side to make up a total width of 44 fet of roadway and two 9.25 feet of sidewalk. In between the beams, and cast with them, are transverse stiffeners which connect the beams laterally, Fig. 10.8

20

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft

10.3 Case Study: Walnut Lane Bridge

10–11

80 ft CENTER LINE

ELEVATION OF BEAM HALF

9.25’

44 ’

ROAD

9.25’

SIDEWALK

BEAM CROSS SECTIONS

TRANSVERSE DIAPHRAGMS

CROSS - SECTION OF BRIDGE

52" 10" 3" 7"

TRANSVERSE DIAPHRAGM 10"

7"

3’-3"

6’-7" SLOTS FOR CABLES

6 1/2" 3 1/2" 7" 30"

CROSS - SECTION OF BEAM

Figure 10.8: Walnut Lane Bridge, Plan View

Victor Saouma

Mechanics and Design of Reinforced Concrete

Draft 10–12

10.3.1 21

PRESTRESSED CONCRETE

Cross-Section Properties

The beam cross section is shown in Fig. 10.9 and is simplified 52"

8.9"

22.5"

7"

22.5" 6’-7" = 79"

61.2"

8.9"

SIMPLIFIED CROSS - SECTION OF BEAM

Figure 10.9: Walnut Lane Bridge, Cross Section Ac = 2(8.9)(52) + (7)(61.2) = 1, 354 in2     79 8.9 2 (7)(61.2)3 (52)(8.9)3 + (52)(8.9) − + I = 2 12 2 2 12

c 1 = c2 S1 = S2 r2

10.3.2

= = 1, 277 × 103 in4 79 h = = 39.5 in = 2 2 1, 277 × 103 I = = 32, 329 in3 = c 39.5 1, 277 × 103 I = = 943. in2 = A 1, 354

(10.20-a) (10.20-b) (10.20-c) (10.20-d) (10.20-e) (10.20-f)

Prestressing

Each beam is prestressed by two middle parabolic cables, and two outer horizontal ones along the flanges. All four have approximately the same eccentricity at midspan of 2.65 ft. or 31.8 inch.

22

Each prestressing cable is made up 64 wires each with a diameter of 0.27 inches. Thus the total area of prestressing steel is given by:

23

Awire = π(d/2)2 = 3.14( Victor Saouma

0.276 in 2 ) = 0.0598 in2 2

(10.21-a)

Mechanics and Design of Reinforced Concrete

Draft

10.3 Case Study: Walnut Lane Bridge

10–13

Acable = 64(0.0598) in2 = 3.83 in2 2

Atotal = 4(3.83) in = 15.32 in

2

(10.21-b) (10.21-c)

Whereas the ultimate tensile strength of the steel used is 247 ksi, the cables have been stressed only to 131 ksi, thus the initial prestressing force Pi is equal to

24

Pi = (131) ksi(15.32) in2 = 2, 000 k 25

The losses are reported ot be 13%, thus the effective force is Pe = (1 − 0.13)(2, 000) k = 1, 740 k

10.3.3 26

(10.22)

(10.23)

Loads

The self weight of the beam is q0 = 1.72 k/ft.

The concrete (density=.15 k/ ft3 ) road has a thickness of 0.45 feet. Thus for a 44 foot width, the total load over one single beam is 1 (10.24) qr,tot = (44) ft(0.45) ft(0.15) k/ ft3 = 0.23 k/ft 13 27

Similarly for the sidewalks which are 9.25 feet wide and 0.6 feet thick: 1 qs,tot = (2)(9.25) ft(0.60) ft(0.15) k/ ft3 = 0.13 k/ft (10.25) 13 We note that the weight can be evenly spread over the 13 beams beacause of the lateral diaphragms. 28

29

The total dead load is qDL = 0.23 + 0.13 = 0.36 k/ft

(10.26)

The live load is created by the traffic, and is estimated to be 94 psf, thus over a width of 62.5 feet this gives a uniform live load of 1 (10.27) wLL = (0.094) k/f t2 (62.5) ft = 0.45 k/ft 13

30

31

Finally, the combined dead and live load per beam is wDL+LL = 0.36 + 0.45 = 0.81 k/ft

10.3.4

(10.28)

Flexural Stresses

1. Prestressing force, Pi only

ec1  Pi

1− 2 Ac r   6 (31.8)(39.5) (2 × 10 ) 1− = 490. psi = − 1, 354 943. Pi

ec2  = − 1+ 2 Ac r   (31.8)(39.5) (2 × 106 ) 1+ = −3, 445. psi = − 1, 354 943.

f1 = −

f2

Victor Saouma

(10.29-a) (10.29-b) (10.29-c) (10.29-d)

Mechanics and Design of Reinforced Concrete

Draft 10–14

PRESTRESSED CONCRETE

2. Pi and the self weight of the beam M0 (which has to be acconted for the moment the beam cambers due to prestressing) (1.72)(160)2 = 5, 504 k.ft 8 The flexural stresses will thus be equal to: w0 =∓ f1,2

M0 =

(10.30)

M0 (5, 50.4)(12, 000) =∓ = ∓2, 043 psi S1,2 943.

(10.31)

ec1  M0 Pi

1− 2 − Ac r S1 490 − 2, 043 = −1, 553 psi  √ 3 fc
= +190 Pi

ec2  M0 1+ 2 + Ac r S2 −3, 445 + 2, 043 = −1, 402. psi √ .6fc
= −2, 400

f1 = −

(10.32-a)

=

(10.32-b)

fti = f2 = = fci =

(10.32-c) (10.32-d) (10.32-e) (10.32-f)

3. Pe and M0 . If we have 13% losses, then the effective force Pe is equal to (1−0.13)(2×106 ) = 1.74 × 106 lbs Pe

ec1  M0 f1 = − (10.33-a) 1− 2 − Ac r S1   (31.8)(39.5) 1.74 × 106 1− − 2, 043. = −1, 616 psi (10.33-b) = − 1, 354 943. ec2  M0 Pe

1+ 2 + (10.33-c) f2 = Ac r S2   (31.8)(39.5) 1.74 × 106 1+ + 2, 043. = −954. psi (10.33-d) = − 1, 354 943. 4. Pe and M0 + MDL + MLL MDL + MLL =

(0.81)(160)2 = 2, 592 k.ft 8

(10.34)

and corresponding stresses f1,2 = ∓ Thus,

ec1  M0 + MDL + MLL Pe

1− 2 − Ac r S1 −1, 616 − 962. = −2, 578. psi √ .45fc
= −2, 700 ec2  M0 + MDL + MLL Pe

1+ 2 + Ac r S2 −954 + 962. = +8. psi  √ 6 fc
= +380

(10.35)

f1 = −

(10.36-a)

=

(10.36-b)

fcs = f2 = = fts = Victor Saouma

(2, 592)(12, 000) = ∓962. psi 32, 329

(10.36-c) (10.36-d) (10.36-e) (10.36-f)

Mechanics and Design of Reinforced Concrete

Draft

10.3 Case Study: Walnut Lane Bridge

Victor Saouma

10–15

Mechanics and Design of Reinforced Concrete

Draft Bibliography Billington, D. and Mark, R.: 1983, Structural studies, Technical report, Department of Civil Engineering, Princeton University. Nilson, A.: 1978, Design of Prestressed Concrete, John Wiley and Sons.