Pre Stressed Concrete Design and Practice_SA

PRESTRESSED CONCRETEDESIGN AND PRACTICE

VERNON MARSHALL

Concrete Society of Southern AfricaPrestressed Concrete Division

Midrand, South Africa

JOHN M. ROBBERTS

TABLE OF CONTENTS

PREFACE v

1 INTRODUCTION 1-1

1.1 THE BASIC IDEA OF PRESTRESSED CONCRETE . . . . . . . . . . . . . . . . . . . . 1-1

1.2 EFFECTS OF PRESTRESSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

1.3 GENERAL PRINCIPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5

1.4 BASIC DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10

1.5 PRESTRESSED VERSUS REINFORCED CONCRETE . . . . . . . . . . . . . . . . . . . 1-12

1.6 HISTORY OF PRESTRESSED CONCRETE . . . . . . . . . . . . . . . . . . . . . . . . . 1-13

1.7 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15

2 MATERIAL PROPERTIES 2-1

2.1 CONCRETE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.1.1 Compressive strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.1.2 Stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

2.1.3 Modulus of elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7

2.1.4 Tensile strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10

2.1.5 Time-dependent behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13

2.1.6 Thermal properties of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

2.1.7 Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

2.1.8 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

2.2 STEEL REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20

2.2.1 Non-prestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21

2.2.2 Prestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-25

2.2.3 Relaxation of prestressing steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-31

2.2.4 Fatigue characteristics of reinforcement . . . . . . . . . . . . . . . . . . . . . . . . 2-35

2.2.5 Thermal properties of reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37

2.3 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37

3 PRESTRESSING SYSTEMS AND PROCEDURES 3-1

3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.2 PRETENSIONING SYSTEMS AND PROCEDURES . . . . . . . . . . . . . . . . . . . . 3-1

3.2.1 Basic principle and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.2.2 Stressing beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

3.2.3 Structural frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

3.3 POST-TENSIONING SYSTEMS AND PROCEDURES . . . . . . . . . . . . . . . . . . . 3-10

3.3.1 Basic principle and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10

3.3.2 Post-tensioning systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12

3.3.3 Post-tensioning operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-20

3.3.4 Ducting for bonded construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23

3.3.5 Grouting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-25

3.4 PRETENSIONING VERSUS POST-TENSIONING . . . . . . . . . . . . . . . . . . . . . . 3-28

3.5 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-29

TABLE OF CONTENTS i

4 DESIGN FOR FLEXURE 4-1

4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.2 SIGN CONVENTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

4.3 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.3.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.3.2 Flexural response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

4.3.3 Analysis of the uncracked section . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

4.3.4 Cracking moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

4.3.5 Ultimate moment: Sections with bonded tendons . . . . . . . . . . . . . . . . . . . 4-12

4.3.6 Analysis of beams with unbonded tendons . . . . . . . . . . . . . . . . . . . . . . 4-31

4.3.7 Flexural analysis of composite sections . . . . . . . . . . . . . . . . . . . . . . . . 4-36

4.4 DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-51

4.4.1 Limit states design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-51

4.4.2 Design for the serviceability limit state . . . . . . . . . . . . . . . . . . . . . . . . 4-54

4.4.3 Design for the ultimate limit state . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-69

4.4.4 Limits on steel content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-74

4.4.5 Flexural design of composite sections . . . . . . . . . . . . . . . . . . . . . . . . . 4-75

4.4.6 Partial prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-82

4.5 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-87

5 PRESTRESS LOSSES 5-1

5.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.2 METHODS FOR CALCULATING PRESTRESS LOSSES . . . . . . . . . . . . . . . . . 5-1

5.2.1 Total loss in pretensioned members . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

5.2.2 Total loss in post-tensioned members . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

5.2.3 Methods for calculating prestress losses . . . . . . . . . . . . . . . . . . . . . . . . 5-3

5.3 ELASTIC SHORTENING OF THE CONCRETE . . . . . . . . . . . . . . . . . . . . . . . 5-5

5.3.1 Pretensioned concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5

5.3.2 Post-tensioned concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7

5.4 TIME-DEPENDENT LOSSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8

5.4.1 Loss due to relaxation of the steel . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8

5.4.2 Loss due to shrinkage of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . 5-9

5.4.3 Loss due to creep of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10

5.5 LOSSES DURING POST-TENSIONING . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28

5.5.1 Friction losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28

5.5.2 Anchorage seating losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-35

5.6 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-40

6 EFFECTS OF CONTINUITY 6-1

6.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.2 ELASTIC ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

6.2.1 Eccentricity of the prestressing force . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

6.2.2 Force (flexibility) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

6.2.3 Fixed-end moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

6.2.4 Displacement (stiffness) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

6.2.5 Concept of equivalent loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17

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6.2.6 Effects of losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-23

6.2.7 Concordancy and linear transformation . . . . . . . . . . . . . . . . . . . . . . . . 6-25

6.3 DESIGN AT SERVICEABILITY LIMIT STATE . . . . . . . . . . . . . . . . . . . . . . . 6-28

6.4 ANALYSIS AT ULTIMATE LIMIT STATE . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

6.4.1 Secondary moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

6.4.2 Moment redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-29

6.5 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-32

7 SHEAR 7-1

7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2 BEAMS WITHOUT WEB REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.1 Cracking behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.2 Shear capacity of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

7.3 BEAMS WITH WEB REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . 7-13

7.4 DESIGN PROCEDURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16

7.5 COMPOSITE BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25

7.6 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-30

8 DEFLECTIONS 8-1

8.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

8.2 UNCRACKED BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2

8.2.1 Instantaneous deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2

8.2.2 Long-term deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5

8.3 CRACKED BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

8.3.1 Instantaneous deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

8.3.2 Long-term deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-19

8.4 DEFLECTION LIMITATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31

8.5 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-32

9 ANCHORAGE ZONE DESIGN 9-1

9.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

9.2 TRANSFER LENGTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2

9.3 ANCHORAGE ZONE REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6

9.3.1 Spalling Stress Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8

9.3.2 Bursting Stress Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-15

9.4 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-21

10 PRESTRESSED CONCRETE SLABS 10-1

10.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

10.2 EFFECTS OF PRESTRESS ON STRUCTURAL BEHAVIOUR . . . . . . . . . . . . . . 10-3

10.2.1 Flexural behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4

10.2.2 Restraint to axial shortening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-5

10.3 STRUCTURAL ANALYSIS BY THE EQUIVALENT FRAME METHOD . . . . . . . . 10-8

10.4 DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11

10.4.1 Design codes of practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-12

10.4.2 Preliminary value for the slab thickness . . . . . . . . . . . . . . . . . . . . . . . . 10-13

TABLE OF CONTENTS iii

10.4.3 Prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14

10.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-19

10.4.5 Serviceability limit states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-21

10.4.6 Ultimate limit states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-26

10.5 DETAILING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-30

10.5.1 Prestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-30

10.5.2 Non-prestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-31

10.5.3 Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-33

10.6 DESIGN EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-33

10.6.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-34

10.6.2 Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-35

10.6.3 Balanced load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-35

10.6.4 Check the preliminary value for the slab thickness . . . . . . . . . . . . . . . . . . 10-35

10.6.5 Minimum cover to tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-35

10.6.6 Design: North-South direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-36

10.6.7 Design: East-West direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-59

10.6.8 Punching shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-60

10.6.9 Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-65

10.6.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-71

10.7 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-71

11 DETAILING 11-1

11.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

11.2 COVER TO TENDONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

11.2.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

11.2.2 Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6

11.3 LIMITATIONS ON PRESTRESSING STEEL CONTENT . . . . . . . . . . . . . . . . . . 11-6

11.3.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6

11.3.2 Minimum steel content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7

11.3.3 Maximum steel content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7

11.4 LIMITATIONS ON SPACING OF TENDONS . . . . . . . . . . . . . . . . . . . . . . . . 11-7

11.5 EFFECTS OF TENDON CURVATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8

11.5.1 In-plane normal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8

11.5.2 Out-of-plane multistrand effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-9

11.5.3 Minimum radius of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11

11.5.4 Minimum tangent length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12

11.5.5 Code requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13

11.6 LONGITUDINAL NON-PRESTRESSED REINFORCEMENT . . . . . . . . . . . . . . . 11-14

11.7 DRAWINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-18

11.8 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-19

APPENDIX A: LIST OF SYMBOLS

APPENDIX B: DRAWINGS

FLAT SLAB: REINFORCEMENT LAYOUTTENDON LAYOUT

BRIDGE DECK: PRESTRESSING DETAILS

iv TABLE OF CONTENTS

PREFACE

The content of this book was initially written and issued as a set of notes for the course PrestressedConcrete: Design and Practice, commissioned by the Prestressed Concrete Division of the ConcreteSociety of Southern Africa. The course was aimed at young engineers and technologists with littleor no experience in the design of prestressed concrete structures, and it was the intention that itshould serve as a vehicle for providing bridging education between tertiary training and designpractice. Although the objective and intended audience of the book is the same as that of the course,it can also serve as a useful reference text for undergraduate students, post-graduate students andpractising designers. It is important to note that a unique feature of the book is that current SouthAfrican practice is emphasised throughout the text.

Basic background information, essential for the design of prestressed concrete structures, is presentedin the first three chapters. These cover the material properties of concrete, prestressing steel andnon-prestressed reinforcement as well as the various prestressing systems and procedures generallyused in South Africa. These chapters also cover relevant specifications.

The basic concepts and procedures required for the analysis and design of a prestressed concreteflexural member are presented in Chapters 4 to 9 as follows:

• Chapter 4: Analysis and design of a section for flexure at the serviceability and ultimate limitstates. Composite sections, unbonded construction and partially prestressed sections are alsocovered.

• Chapter 5: Procedures for estimating the instantaneous and long-term loss of prestress inpretensioned and in post-tensioned construction.

• Chapter 6: The effects of continuity in prestressed concrete members.

• Chapter 7: Design for shear, including composite beams.

• Chapter 8: Procedures for calculating the instantaneous and long-term deflections of prestressedconcrete flexural members. Both uncracked and cracked beams are considered.

• Chapter 9: Design of the anchorage zone. The design considerations, applicable to bothpretensioned and post-tensioned construction are covered.

The analysis and design of post-tensioned flat plates and flat slabs are covered by Chapter 10. Thismaterial is limited to slabs using unbonded tendons and levels of prestress at which the slabs willbe cracked under the design service loads because most of the post-tensioned flat plates and flatslabs constructed in South Africa are of this type.

Various aspects, peculiar to prestressed concrete members, which affect detailing are presented inChapter 11. The details of both the prestressed and non-prestressed reinforcement are covered inthis chapter, and a number of local effects, induced by tendon curvature, are also discussed. Workingdrawings of the prestressing details of a flat slab and of a highway bridge are presented inAppendix B.

Generally, the procedures for simulating the various aspects of behaviour are developed from thebasic principles of structural mechanics. However, in a case where a semi-empirical approach isfollowed, the relevant experimental work on which such a procedure is based is presented anddiscussed. The manner in which these aspects of behaviour are reflected in the various design codesof practice, commonly used in South Africa, are also explained. Each chapter contains comprehensiveexamples that illustrate the analytical concepts and design procedures covered.

PREFACE v

1 INTRODUCTION

1.1 THE BASIC IDEA OF PRESTRESSED CONCRETE

In its general form, the term prestressing means the deliberate creation of permanent stresses in astructure before it is subjected to any imposed load. Because the object of prestressing a structureis to improve its performance, the stresses resulting from prestressing are designed to counteractthose induced by the acting loads.

As an example, consider the case of a simply supported beam made from an elastic material whichis equally strong in compression and in tension. The deflected shape of the beam and the stressdistribution over the depth of the midspan section, which result from the application of a uniformlydistributed load w, are shown in Fig. 1-1a. The principle of prestressing can subsequently be usedto counteract this response by applying an eccentric compression force P to each end of the beam.The prestressing forces are shown in Fig. 1-1b together with the resultant deflected shape of thebeam and the stress distribution over the midspan section. Figure 1-1c shows the response to thecombined application of the load w and the prestressing forces P, which is obtained by thesuperposition of the response to the load w (Fig. 1-1a) and the response to the prestressing forcesP (Fig. 1-1b).

A comparison of the deflected shapes and mid-span stresses shown in Figs. 1-1a and 1-1c illustratesthe effects of prestressing on the structural behaviour of the beam: Not only can both the compressiveand tensile stresses (and hence, the corresponding strains) in the top and bottom fibres of themid-span section be reduced, but the beam deflection can also be reduced. It should be noted thatalthough the stress in the bottom fibre (fwb – fpb) resulting from the combined action of the load wand the prestressing forces P is shown to be compressive in Fig. 1-1c, it could be tensile dependingon the relative magnitudes of fwb and fpb. Similarly, the resultant deflection (�w � �p) shown inFig. 1-1c to be upward, could be downward.

Given the fact that concrete is strong in compression and weak in tension, it seems natural that oneof the most successful applications of the principle of prestressing has been the development ofprestressed concrete. A simply supported plain, unreinforced concrete beam subjected to anincreasing load will fail immediately after the development of cracks when the induced flexuraltensile stress fwb (Fig. 1-1a) exceeds the tensile strength of the concrete. In the case of a reinforcedconcrete beam, suitable steel reinforcement is provided in the tension zone of the section to carrythe tensile forces required for equilibrium of the cracked section. For this reason, a reinforcedconcrete beam can carry loads which exceed the cracking load by a considerable margin.

As opposed to reinforced concrete, where the concrete is allowed to crack under service loads, theoriginal development of prestressed concrete was based on the prevention of flexural cracks formingunder service loads. This was achieved by applying the criterion of no tensile stress, because it isgenerally accepted that if there are no tensile stresses present in the concrete it will not crack.However, this criterion has been relaxed with the subsequent development of prestressed concreteand it is currently common practice to allow some tension to develop in the concrete. As shown inFig. 1-1, the tensile stresses induced by the load can be neutralised to any desired degree byproviding suitable prestressing.

With the subsequent development of the concept of partial prestressing significant tension andcontrolled cracking are allowed to develop at service load levels, in much the same way as inreinforced concrete. The latest schools of thought on prestressed concrete embodies the view thatpartially prestressed concrete occupies the range between reinforced concrete and fully prestressedconcrete (i.e. no tension is allowed to develop at service load levels). From this viewpoint reinforcedconcrete and fully prestressed concrete represent the two boundaries of the complete range of

THE BASIC IDEA OF PRESTRESSED CONCRETE 1-1

possibilities which exist for partially prestressed concrete and, as such, are two special cases ofpartially prestressed concrete.

In prestressed concrete, the most commonly used method of applying the prestressing force to theconcrete is by tensioning high-strength reinforcement, commonly referred to as tendons, against theconcrete prior to the application of imposed loads. Two different processes can be distinguished inthis regard:

• Pretensioning: In these prestressing methods, the prestressed reinforcement is tensioned beforethe concrete is placed.

w

w

�w

�p

� � �w p

fwt (compression)

fpt (tension)

f fwt pt–

f fwb pb–

fwb (tension)

fpb (compression)

Stresses at midspansection



(a) Response to uniformly distributed load

(b) Response to prestressing forces

(c) Response to uniformly distributed load and prestressing forces

e

e

e

e

P

P

P

P

Section

Section

Section

Figure 1-1: General effects of prestressing.

1-2 INTRODUCTION

• Post-tensioning: In these prestressing methods, the prestressed reinforcement is tensioned afterthe concrete has been placed and has developed sufficient strength to sustain the induced stresses.

The definition of prestressed concrete as given by the ACI Committee on Prestressed Concrete (takenfrom Ref. 1-1) is quoted here for completeness:

Prestressed concrete: Concrete in which there have been introduced internal stresses of suchmagnitude and distribution that the stresses resulting from given external loadings arecounteracted to a desired degree. In reinforced-concrete members the prestress is commonlyintroduced by tensioning the steel reinforcement.

It is apparent from Fig. 1-1 that the use of prestressing will enable a designer to provide a structureof which the deflections at service load levels can be made much less than those of its reinforcedconcrete counterpart. This benefit is obtained in addition to the bonus of being in a position toprovide a structure which is relatively crack-free at service load levels.

1.2 EFFECTS OF PRESTRESSING

The effects of prestressing are dictated by the fundamental reason for applying it in the first place:Prestressing is simply a means by which a controllable set of forces are applied to a structure tocounteract the stresses induced by loads (e.g. dead loads and live loads).

The effects of prestressing with regard to the development of stresses are illustrated by consideringthe rectangular beam section shown in Fig. 1-2a. If a moment M = 286 kN.m is applied to thesection, the resulting stresses at the top and bottom of the section can be calculated from f = My/I,where y is the distance from the top (or bottom) fibre to the centroidal axis and I is the secondmoment of area of the section about the centroidal axis. Taking tension positive and compressionnegative, this calculation yields a stress of –5.94 MPa at the top and a stress of +5.94 MPa at thebottom, as shown in Fig. 1-2b. The concrete can easily carry the compressive stress at the top ofthe section, but will most probably crack under the tensile stress at the bottom because it cracks ata much lower stress, which lies in the range of 50% to 75% of this value.

As a first attempt to neutralise the tensile stresses in the section, an axial compression forceP = 2258 kN is taken to act at the same time as the moment of 286 kN.m (see Fig. 1-2c). Thisaxial force induces an additional uniform compressive stress of –5.94 over the section, which iscalculated from f = –P/A, where A is the area of the section. The total stresses resulting from thesimultaneous application of M and P are obtained by adding the stresses which are separatelyproduced by each of these actions. As shown in Fig. 1-2c, a total stress of –11.88 MPa is obtainedat the top and a zero stress is obtained at the bottom. The concrete will be able to carry thesestresses for the strengths normally used in prestressed concrete structures.

The fairly large force of 2258 kN may be reduced by applying it eccentrically. Therefore, as a nextstep, a force P = 1127 kN is applied at an eccentricity of 127 mm, measured from the centroid ofthe section, as shown in Fig. 1-2d. The additional stress which arises from the eccentricity iscalculated from f = Pey/I, where e is the eccentricity as defined above. The stresses at the topand bottom of the section as produced by the various components of load are summarised inFig. 1-2d, from which it may be seen that P causes a zero stress at the top and a compression of–5.94 MPa at the bottom. The total stresses, which include those produced by M, are seen to be–5.94 MPa at the top and zero at the bottom. When these results are compared to those obtained inthe previous case, the beneficial effect of applying P eccentrically becomes clear: The tensile stressesin the section can still be completely neutralised even though the magnitude of P has been reducedby half, and in the process the total compressive stress in the top fibre has also been reduced by ahalf.

�

�

EFFECTS OF PRESTRESSING 1-3

M = 286 kNm–5.94 MPa –5.94 MPa

StressLoading Condition

(a) Section Properties

(b)

(c)

(d)

(e)

–5.94 MPa –11.88 MPa–5.94 MPa

–5.94 MPa –2.97 MPa +2.97 MPa –5.94 MPa

–5.94 MPa –1.98 MPa +3.96 MPa –3.96 MPa

+5.94 MPa +5.94 MPa

+5.94 MPa –5.94 MPa 0

+5.94 MPa –2.97 MPa –2.97 MPa 0

+5.94 MPa –1.98 MPa –3.96 MPa 0

M = 286 kNm

P = 2258 kN

P = 1127 kN

P = 751 kN

M = 286 kNm

e = 127

e = 254

M = 286 kNm

y = 380

500

760

y = 380

A = 500 760 = 380.0 10 mm3 2� �

= 380 mmy =760

2

= 182.9 10 mm8 4�I =500 7603�

12

P

A–

M y+ I– P e y

I

+–

M y P e yP

I IA+– +

––

Figure 1-2: Effects of prestressing on stresses.

1-4 INTRODUCTION

As a final example in this regard, consider the case where the compression force P is further reducedto 751 kN but its eccentricity is increased to 254 mm, as shown in Fig. 1-2e. Also shown are thestresses produced in the top and bottom fibres of the section by the various components of load.Once again, a total bottom fibre stress of zero is obtained while a total compression of –3.96 MPais obtained at the top, which is even smaller than before. This result is consistent with the previousfinding that an increased eccentricity has a beneficial effect as far as the total stresses are concerned.However, it may be seen that the eccentric force acting on its own causes a tension of (3.96 � 1.98)= 1.98 MPa at the top. Although this tension is probably not large enough to cause the concrete tocrack, it serves to illustrate that a larger eccentricity can be detrimental in the absence of externalload (represented here by M), even though it is beneficial when the external load is present. Thisfinding is important for design because it clearly shows that the critical stresses may arise either inthe loaded or in the unloaded structure.

These examples are intended to illustrate the effects of prestressing on the development of stress inthe section, and are not intended to show that limiting the total tensile stress in the section to zerois necessarily beneficial or not.

Another important effect of prestressing on structural behaviour is its impact on deflections at serviceload levels. This effect can be qualitatively investigated with reference to Fig. 1-1. In the case ofthe simply supported beam considered here, the externally applied load w will produce a downwarddeflection (see Fig. 1-1a) while the prestressing force P, which is applied at an eccentricity e, willcause an upward deflection (see Fig. 1-1b). The total deflection of the beam under the combinedactions of the external load and the prestressing force is obtained by adding the deflections yieldedby each load acting separately (see Fig. 1-1c). Because the deflections caused by the two componentsof load are opposite, it is clear that the downward deflection produced by the external load is alwaysreduced by the presence of prestressing and, depending on the relative magnitudes of the twocomponents of deflection, the resultant deflection can be upward. This observation, once again,points to the fact that the designer is working between various limits, and that he may find thatalthough the deflection of the loaded structure is small, the upward deflection of the unloadedstructure is unacceptably large. Such a situation can arise in cases where the live load to dead loadratio is large.

1.3 GENERAL PRINCIPLES

There are three different concepts which can be used to approach the simulation of the behaviourof a prestressed concrete member (Ref. 1-1). Each approach can be used for design provided thatit is properly understood by the designer, and provided that the limitations of each are realized. Inthe following, each approach is briefly described.

First approach: Prestressing transforms concrete into an elastic material.

The fundamental idea behind this approach is that the pre-compression applied during prestressingtransforms the concrete into an elastic material. The brittle behaviour of concrete arises from thefact that when its tensile strength, which is much less than its compressive strength, is reached itcracks and subsequently cannot carry any tensile stress. If it is accepted that concrete will not crackif there are no tensile stresses present, then it can be concluded that the removal of tensile stressesby prestressing will remove the source of its brittle behaviour and, in so doing, will transform itinto an elastic material.

Using this approach, it is convenient to view the concrete as being subjected to two sets of forces:

• The external load which induces tensile stresses.

• The internal prestress which sets up the compression required for neutralising any tension.

GENERAL PRINCIPLES 1-5

If the pre-compression induced by the prestress prevents the concrete from cracking, then thestresses, strains and deflections caused by each of the sets of forces can be considered separatelyand superimposed as required. The examples considered in Fig. 1-2 (see Section 1.2) serve as anillustration of how this approach can be used to calculate stresses in a beam section.

This approach is credited to Freyssinet and is the source of the zero tensile stress criterion whichhas been applied over many years.

Second approach: Prestressed concrete is a type of reinforced concrete.

Prestressed concrete can be viewed as a type of reinforced concrete in which high-strengthreinforcement has been tensioned against the concrete before any imposed load is applied. Followingthis approach, prestressed concrete is considered as a combination of concrete and steel, in whicha resisting internal couple must be developed to equilibrate an external moment. The internal couplearises from the compression supplied by the concrete and the tension supplied by the steel, as isthe case for reinforced concrete. From this point of view, the primary difference between thebehaviour of prestressed concrete and reinforced concrete lies in the increased cracking load andthe possibility of actively controlling the deformations of the structure. The fundamental principle,however, remains the same.

To illustrate the use of this approach to analysing a prestressed concrete beam section, consider thesection shown in Fig. 1-3 subjected to a moment M = 286 kN.m. The prestressed reinforcement isplaced at an eccentricity e = 254 mm and carries a tension T = 751 kN. This example is the sameas that shown in Fig. 1-2e.

380

Section Properties

�3.96 MPa

Stress distribution

0

T = 751 kN

C = 751 kN

e = 254

ec = 127

la = 381

M = 286 kNm

500

760

380254

A = 380.0 10 mm3 2�

I = 182.9 10 mm8 4�

Figure 1-3: Prestressed concrete considered as a type of reinforced concrete.

1-6 INTRODUCTION

Horizontal equilibrium yields T = C = 751 kN. The internal couple provided by the compression inthe concrete C and the tension in the steel T must be equal to the external moment M = 286 kN.mto satisfy moment equilibrium. The lever arm at which these forces are acting is given by

la=

286751

× 103 = 381 mm

Therefore C is acting at an eccentricity eC

= 381 � 254 = 127 mm. The stress distribution in theconcrete is obtained by considering the compression C = 751 kN acting on the concrete at aneccentricity of 127 mm. Using elastic theory

f = − CA

−+C e

Cy

I

= − 751× 103

380 × 103−+ 751× 103 × 127× 380

182.9 × 108

= − 1.98 −+ 1.98

So that ftop

= – 3.96 MPa (top fibre, compression)

fbot

= 0 (bottom fibre)

These results are shown in Fig. 1-3 and are the same as obtained before in Fig. 1-2e.

Third approach: Prestressing balances a part of the applied load.

In this approach the view is adopted that the forces exerted by the prestressed reinforcement(tendons) on the concrete balances the applied loads to some desired degree. Consider the simply

L / 2 L / 2

L

h

P

wb

P

(a) Parabolic tendon profile

(b) Tendon forces acting on the concrete

Figure 1-4: Simply supported beam with parabolic tendon.


supported beam shown in Fig. 1-4a which has a parabolically curved tendon. It can be seen fromFig. 1-4b that the tendon applies the following forces to the concrete:

• The prestressing force P at each end of the beam where the tendon is anchored.

• An upward uniformly distributed load wb acting over the span of the beam. This load arisesbecause the concrete prevents the tendon from straightening under the action of the prestressingforce. It can be shown that for the tendon profile considered here

wb= 8 P h

L2

where h = sag of the tendon

L = span of the beam

If the beam is subjected to a downward uniformly distributed load w, it is clear that the portion ofthe load which is not balanced by the action of the prestress is given by (w – w

b). Using this

approach, the beam is subsequently analysed by considering it as being subjected to the prestressingforce P applied at the anchor positions at the ends of the beam and the unbalanced load (w – wb)acting over its span.

As an example of how this approach can be used to analyse a prestressed concrete beam, considerthe simply supported beam shown in Fig. 1-5a, which is subjected to a uniformly distributed loadw = 42.9 kN/m. The prestressing force P = 751 kN and the tendon profile is parabolic, with aneccentricity e = 254 mm at midspan and zero eccentricity at the ends. Since the bending momentat midspan M = 42.9 � 7.32/8 = 286 kN.m, it is clear that this example is the same as that shownin Fig. 1-2e if the midspan section is considered.

The upward uniformly distributed load applied by the tendon is given by

wb= 8 P h

L2

= 8 × 751× 0.254

7.32

= 28.6 kN/m

The loads and forces acting on the concrete are shown in Fig. 1-5b, from which it is clear that theunbalanced load is (42.9 – 28.6) = 14.3 kN/m acting downward. The midspan bending momentinduced by this unbalanced load is

M =(w − w

b) L2

8

= 14.3 × 7.32

8

= 95.3 kN.m

The stress produced by this moment in the extreme fibres of the midspan section is given by

f = M yI

= 95.3 × 106 × 380

182.9 × 108

= 1.98 MPa

1-8 INTRODUCTION

3650 3650

7300

7300

h = 254 380 254760

760

500

500

–1.98 MPa –1.98 MPa –3.96 MPa

+1.98 MPa

Stress due tounbalanced load(= 14.3 kN/m)

Stress due toprestressing forceapplied at ends ofbeam

Total stress

–1.98 MPa 0

Section at midspan

380

P = 751 kN

wb = 28.6 kN/m

w = 42.9 kN/m

w = 42.9 kN/m

P = 751 kN

(a) Simply supported beam

(b) Loads and forces acting on the concrete

(c) Concrete stress in midspan section

A = 380.0 10 mm3 2�

I = 182.9 10 mm8 4�

Figure 1-5: Analysis using load balancing approach.


So that ftop

= – 1.98 MPa (top fibre, compression)

fbot

= + 1.98 MPa (bottom fibre, tension)

The stress induced by the prestressing force acting at the ends of the beam is, with negligible error,calculated from

f = − PA

= − 751× 103

380× 103

= – 1.98 MPa (compression)

Finally, the total stress in the top and bottom fibres of the midspan section are given by

ftop

= – 1.98 – 1.98 = – 3.96 MPa (top fibre, compression)

fbot

= – 1.98 + 1.98 = 0 (bottom fibre)

These results are shown in Fig. 1-5c and are the same as obtained before in Fig. 1-2e.

1.4 BASIC DEFINITIONS

Some of the most commonly encountered prestressing techniques and features of construction ofprestressed concrete structures are introduced in the following (Ref. 1-1). The descriptions are briefbecause the techniques and procedures covered here are more expansively dealt with in subsequentChapters.

The most commonly used prestressing method is to tension high-strength reinforcement against theconcrete. Hence the definition of tendon:

• Tendon: A tendon is the prestressed reinforcement used to apply the prestress to the concrete.This steel reinforcement may either be high-strength wires, bars or strand.

Prestressing methods can be classified either as being a pretensioning method or as being apost-tensioning method, depending on whether the concrete has not been placed or whether it hasbeen placed at the time of tensioning of the reinforcement. Although the terms pretensioning andpost-tensioning have been adequately defined in Section 1.1, their definitions are repeated here forconvenience:

• Pretensioning: In these prestressing methods, the prestressed reinforcement is tensioned beforethe concrete is placed.

• Post-tensioning: In these prestressing methods, the prestressed reinforcement is tensioned afterthe concrete has been placed and has developed sufficient strength to sustain the induced stresses.

The definitions given in the following are all concerned with special features or attributes relatedto the construction of prestressed concrete structures.

Internal and External Prestressing

Internal prestressing refers to prestressed concrete structures in which the tendons are containedwithin the concrete, while external prestressing implies that the prestressing force is appliedexternally. External prestressing can be achieved either by placing the tendons outside the memberor by applying external prestressing forces using jacks. Internal prestressing is by far the most

1-10 INTRODUCTION

commonly used method, although external prestressing by means of external tendons has recentlygained some popularity for use in bridge construction, particularly in Europe.

Jacks can be used to externally prestress a simply supported beam, as shown in Fig. 1-6. If thejacks are properly placed, the pre-compression which they produce can neutralise any tension causedby the applied load. However, this procedure is of little practical importance because thetime-dependent strains resulting from shrinkage and creep of the concrete soon reduce the strains.Hence, the stresses induced by the prestressing force are reduced to levels at which the prestressingbecomes ineffective, unless the jacks can be readjusted. Shrinkage can be viewed as thetime-dependent strain which develops in the absence of load, while creep may be seen as thetime-dependent strain which develops in the presence of load. These phenomena are moreexpansively dealt with in Section 2.1.5.

Linear and Circular Prestressing

Linear prestressing refers to elongated elements such as beams and slabs, even though the tendonsmay be curved and not straight. Circular prestressing, on the other hand, refers to circular structuressuch as silos, pressure vessels, tanks and pipes where the circular shape of the tendons is dictatedby the shape of the structural element.

Bonded and Unbonded Tendons

When tendons are bonded to the surrounding concrete, they are referred to as bonded tendons. Apretensioned tendon is bonded to the concrete by virtue of the construction method, although it canbe debonded over a portion of its length by taking appropriate steps to accomplish this.Post-tensioned tendons are encased in a duct so that they can be tensioned after the surroundingconcrete has hardened sufficiently. Bonding is subsequently accomplished by injecting grout intothe duct.

Tendons not bonded to the concrete over their entire length are referred to as unbonded tendons,and can only be accomplished with post-tensioning. Unbonded tendons require corrosion protection,which is commonly provided by placing them in grease filled plastic tubes.

Stage Stressing

It sometimes happens that, by the nature of the construction procedure, the dead load is applied instages. In such cases the prestressing may also be applied in appropriate stages to avoid overstressingthe concrete. This technique is referred to as stage stressing.

Jack Jack

Figure 1-6: External prestressing using jacks.

BASIC DEFINITIONS 1-11

Partial and Full Prestressing

When a prestressed concrete member is designed in compliance with the zero tensile stress criterion,i.e. not to develop any tensile stress under service loads, it is referred to as being fully prestressed.On the other hand, tension and cracking are allowed to develop in partially prestressed members atservice load levels. Additional ordinary non-prestressed reinforcement is usually provided in partiallyprestressed members to control the cracking and to ensure adequate ultimate strength.

1.5 PRESTRESSED VERSUS REINFORCED CONCRETE

One of the major differences between prestressed concrete and reinforced concrete, with regard totheir physical attributes, is that higher strength materials (for both concrete and steel) are used forprestressed concrete. In prestressed concrete the high-strength steel is tensioned and anchored againstthe concrete, which produces a number of desirable effects:

• The high strength of the steel can be properly used, even at service load levels.

• The prestressing tends to neutralise tensile stresses and strains induced by the load, so thatcracking of the section is eliminated and, as a result, the full concrete section becomes active inresisting the load. This mechanism is much more effective than is the case for reinforced concretewhere only the uncracked part of the section in the compression zone participates in resistingthe load.

• The deformations induced by the prestressing serve to offset those produced by the load, andcan be used by the designer to control deflections.

Higher strength concrete may be used to obtain more economic sections than with reinforcedconcrete.

The following advantages of prestressed concrete are often put forward when compared to reinforcedconcrete (Ref. 1-2):

• Prestressed concrete requires smaller quantities of material than reinforced concrete becausehigh-strength materials are efficiently and effectively used and because it uses the entire sectionto resist the load. This means that prestressed concrete members are lighter and more slenderthan their reinforced concrete counterparts.

• The fact that members are lighter and more slender if prestressed concrete rather than reinforcedconcrete is used, leads to other advantages:

- Savings can be realised in the reduced cost of lighter supporting structures and, in the caseof precast elements, in the reduced handling and transportation costs.

- Aesthetically pleasing structures are more readily provided.

- Longer spans are possible because of the reduced self weight.

- Innovative construction methods are facilitated.

- Thinner slabs result in reduced building heights and consequent savings in the cost of finishes.

These advantages are particularly evident in the case of long span bridges and multi-storeybuildings.

• Prestressed concrete generally provides better corrosion protection to the reinforcement than doesreinforced concrete. This advantage is significant for structures subjected to aggressive environ-ments and for fluid-retaining structures.

• Improved deflection control is possible with prestressed concrete.

• Prestressed concrete members will require less shear reinforcement than reinforced concretemembers. This follows from the fact that the shear capacity of a prestressed member is increased

1-12 INTRODUCTION

by curved tendons, which carry some of the shear, and by the pre-compression, which reducesthe principal tension.

• It often happens that the worst service load condition for a prestressed concrete structure occursduring the prestressing operation. In such a case, it can be claimed that the safety of the structurehas been partially tested: If the structure successfully withstands the effects of the prestressingoperation, chances are good that it will perform well during its service life.

A comparison of the economic advantages or disadvantages of prestressed concrete with those ofreinforced concrete is complicated by the fact that each has a range of applicability, depending onthe type of structure and the specific design requirements. However, if such a comparison is madewhere the ranges of applicability overlap, care must be taken to include not only the cost of thematerials but also to include the additional costs associated with prestressed concrete, such as theuse of specialised equipment and hardware, greater design effort, more supervision and the use ofspecialised personnel. Such a comparison should also reflect the relative performance and costadvantages inherent in each type of structure. For example, since the decking for post-tensionedslabs can be stripped after tensioning, shorter construction times are realized together with all therelated savings in construction and financing costs.

If the view is taken that prestressed concrete and reinforced concrete represent the two boundariesof the range of possibilities which exist for partially prestressed concrete, they form part of thesame system and cannot be considered as being in competition with each other. A comparison, asgiven above, can therefore be seen to be inappropriate because a specific prestressing level canalways be found within the spectrum of possibilities to yield the best solution to a given problem.From this viewpoint, it would seem much more appropriate to compare prestressed reinforcedconcrete to structural steel.

1.6 HISTORY OF PRESTRESSED CONCRETE

A brief overview of the history of the development of prestressed concrete, as taken from Refs. 1-1to 1-7, is presented in the following. It is interesting to note that the development of prestressedconcrete is characterised by its individualistic nature, even though it took place simultaneously inseveral countries. A possible reason for this is the lack of communication which existed betweenthe countries during World War II.

The first application of the principle of prestressing to concrete is credited to P. H Jackson, of SanFrancisco, who in 1886 applied for a patent for Constructions of Artificial Stone and ConcretePavements in which steel tie rods, passed through concrete blocks and concrete arches, weretightened by nuts. These structures served as slabs and roofs. An application for a patent, whichcan also be related to prestressing, was made in 1888 by the German C. E. W. Doehring. This patentcovered the manufacture of mortar slabs containing tensioned wires.

The purpose of the work done by the Austrian engineer J. Mandl was aimed at using the strengthof the concrete in a beam as effectively as possible. To achieve this he, in 1896, became the firstperson to clearly articulate the purpose of prestressing as the need to counteract the tension producedby the load with compression induced by an applied prestressing force. The German engineerM. Koenen developed this idea and in 1907 derived an expression from which the requiredprestressing force could be calculated. The loss of prestressing force resulting from elastic shorteningwas accounted for in these proposals.

In 1907 the Norwegian J. G. F. Lund suggested the construction of prestressed vaults usingprefabricated concrete blocks jointed in mortar. The prestressing was applied by tensioned tie rodswhich transmitted the compression to the blocks by bearing plates at the ends. Bond between thetie rods and the mortar was destroyed at stretching. A similar prestressing procedure was suggestedby the American engineer G. R. Steiner in the following year. This procedure consisted of initiallytightening the reinforcing rods against the green concrete to destroy bond and to subsequently

HISTORY OF PRESTRESSED CONCRETE 1-13

complete the tensioning operation once the concrete has hardened. These two procedures appear tobe the first applications of post-tensioning.

In the procedures outlined above mild steel was tensioned to the permissible stress prescribed at thetime (i.e. approximately 110 MPa), which corresponds to a strain of 0.00055 in the steel. Becausethis strain is comparable to the magnitude of the strain induced by shrinkage and creep of theconcrete, most of the prestressing would have been lost with time. Therefore, these early attemptswere bound to give unsatisfactory results because shrinkage and creep of the concrete were notaccounted for.

The American engineer R. H. Dill appears to have been the first, in 1923-25, to suggest that fullprestressing can be provided by post-tensioning high-strength steel, instead of mild steel. Dill coatedthe reinforcement with a plastic substance to prevent bond, and tensioned the reinforcement aftermost of the shrinkage in the concrete had taken place. The effects of creep were accounted for byoccasionally tightening the nuts used for stretching the reinforcement. However, it should be notedthat Dill did not actually say that high-strength steel was required for maintaining full prestressafter losses. In 1922, W. H. Hewett, also of America, successfully applied prestressing to circularconcrete tanks using an idea similar to that used by Dill.

E. Freyssinet of France was the first engineer to fully grasp the importance of the effects of shrinkageand creep of the concrete, and is credited with the development of prestressed concrete as we knowit today. In 1928, he introduced the use of high-strength steel bonded to the concrete, together withthe requirement that a high tensioning stress be applied to the steel. The significance of theseproposals is demonstrated by the fact that shrinkage and creep can together induce a strain ofapproximately 0.001 in the concrete, while a strain of approximately 0.007 can be induced inhigh-strength steel reinforcement during the prestressing operation. This means that, in this case,shrinkage and creep will reduce the prestressing force only by about 14%. Thus, by usinghigh-strength steel for prestressing, it is still possible to completely neutralise any tension inducedby the load in the concrete, even after losses. Freyssinet also demonstrated that a considerable savingin the required quantity of steel may be achieved by using high-strength reinforcement.

The large scale use of prestressed concrete only became possible after the development of reliableand economical methods of carrying out the tensioning operation. The first practical implementationof pretensioning was made by E. Hoyer of Germany who, in 1938, introduced a procedure wherebypiano wire was tensioned over a large distance, after which the concrete was cast. The prestresswas transferred to the concrete by cutting the wires after hardening of the concrete. Although Hoyerwas granted a patent for the long-line pretensioning method, it should be pointed out that the ideadid not originate with him, but rather with Freyssinet, whose proposal for the long-line process hecombined with Wettstein’s (1919) experience with the use of piano wire. The large scale use ofpost-tensioning started with the introduction, in 1939, of Freyssinet’s system whereby a double-act-ing jack was used to tension and to anchor 12 wire cables in conical wedges, which served asanchors.

Since this time prestressed concrete has been widely accepted and used, as revealed by the fact that:

• Many prestressing systems and techniques have been developed.

• A large number of books covering the design and construction of prestressed concrete structureshave been published.

• Numerous technical societies have been established who, through their activities and publications,have greatly contributed to the progress of prestressed concrete.

Some of the engineers and researchers who have made significant contributions to the subsequentdevelopment of prestressed concrete include: G. Magnel of Belgium (Ref. 1-8), Y. Guyon of France(Ref. 1-9), P. W. Abeles of England (Ref. 1-4 and 1-5), F. Leonhardt of Germany (Ref. 1-10),V. V. Mikhailov of Russia, and T. Y. Lin of America (Ref. 1-1 and 1-11).

1-14 INTRODUCTION

F. V. Emperger is credited with being the first to use the concept of partial prestressing when, in1939, he suggested that pretensioned wires be added to conventionally designed non-tensionedreinforcement to reduce the extent of cracking. This idea was further developed by Abeles who, in1940, suggested the use of non-tensioned high-strength steel together with pretensioned orpost-tensioned tendons. Apart from the recommendation that solely high-strength steel be used, thisproposal also differed from Emperger’s in that a prestressing force of a definite designed magnitudebe applied. The acceptance of partial prestressing was at first retarded, perhaps by the oppositionto this concept by Freyssinet (Ref. 1-12), who stated (Ref. 1-13) “... there is no half-way housebetween reinforced and prestressed concrete; any intermediate systems are equally bad as reinforcedor prestressed structures, and are of no interest.” However, partial prestressing has made enormousprogress through the efforts and contributions of many eminent engineers and researchers, and iscommonly used today.

1.7 REFERENCES

1-1 Lin, T. Y., and Burns, N. H., Design of Prestressed Concrete Structures, 3rd ed., John Wiley& Sons, New York, 1981.

1-2 Naaman, A. E., Prestressed Concrete Analysis and Design: Fundamentals, McGraw-Hill BookCompany, New York, 1982.

1-3 Abeles, P. W., The Principles and Practice of Prestressed Concrete, Crosby Lockwood &Son, London, 1949.

1-4 Abeles, P. W., An Introduction to Prestressed Concrete, Volume I, Concrete Publications Ltd.,London, 1964.

1-5 Abeles, P. W., An Introduction to Prestressed Concrete, Volume II, Concrete PublicationsLtd., London, 1966.

1-6 Collins, M. P., and Mitchell, D., Prestressed Concrete Structures, Prentice-Hall, EnglewoodCliffs, New Jersey, 1991.

1-7 Khachaturian, N., and Gurfinkel, G., Prestressed Concrete, McGraw-Hill Book Company, NewYork, 1969.

1-8 Magnel, G., Prestressed Concrete, Concrete Publications Ltd., London, 1948.

1-9 Guyon, Y., Prestressed Concrete, John Wiley & Sons, New York, Vol. 1, 1953, Vol. 2, 1960.

1-10 Leonhardt, F., Prestressed Concrete Design and Construction, English translation, WilhelmErnst und Sohn, Berlin 1964, (1st ed., 1955, 2nd ed., 1962 in German).

1-11 Lin, T. Y., Design of Prestressed Concrete Structures, John Wiley & Sons, New York, 1955.

1-12 Cohn, M. Z., “Some Problems of Partial Prestressing,” Partial Prestressing, from Theory toPractice. Volume I: Survey Reports, Edited by M. Z. Cohn, Chapter 2, NATO ASI Series,Series E, No. 113a, Martinus Nijhoff Publishers, Dordrecht, 1986, pp. 15-63.

1-13 Freyssinet, E., “Prestressed Concrete, Principles and Applications,” ICE Proceedings, Vol. 33,No. 4, February 1950, pp. 331-380.

REFERENCES 1-15

2 MATERIAL PROPERTIES

Prestressed concrete combines high quality concrete and prestressed steel, as well as non-prestressedordinary reinforcing steel. Before considering the behaviour of the materials in combination, it isessential that the designer is familiar with the relevant properties of each of these materials. Thistopic is extensively covered in the technical literature and this chapter summarizes the most importantproperties required for the design of prestressed concrete structures.

2.1 CONCRETE

The mechanical properties of concrete under uniaxial stress are considered in this Section. Althoughconcrete is usually subjected to a three dimensional state of stress in practical structures, theassumption of a uniaxial stress condition can very often be justified. Where the effects of amulti-axial state of stress are significant, these will be dealt with in the appropriate Chapters.

Concrete technology is not considered here. This topic is extensively covered in many textbooks,e.g. Ref. 2-1.

2.1.1 Compressive strength

The single most important mechanical property of concrete is its compressive strength because it isextensively used in quality control and because many other mechanical properties required for thedesign of prestressed concrete structures can be expressed in terms of this property. The compressivestrength can be obtained from standard tests using either cubes or cylinders loaded to failure(Refs. 2-2 to 2-4). The maximum load sustained during such a test, divided by the cross sectionalarea of the specimen yields the compressive strength.

It is extremely important to note that the compressive strength must be determined in strictcompliance with the requirements of a standard testing procedure because the measured resultsdepend on the test method and also because it is primarily used as an index of strength in itsapplication in structural design. The standard specification generally used in South Africa isSABS 863 (Ref. 2-2), according to which 150 mm cubes are loaded to failure in a calibrated testingmachine at a loading rate of approximately 15 MPa/min.

Apart from intrinsic factors, which cover the composition of the concrete, the following externalfactors influence the compressive strength:

• Age of concrete: The compressive strength increases with time, provided the concrete is properlycured. The development of strength with time is shown in Fig. 2-1 (Ref. 2-5) for a typical concreteusing ordinary Portland cement, where it may be seen that the rate at which the strength developsreduces with time. As a percentage of the value at 28 days, the strength will generally varybetween 33 and 50%, and between 60 and 75% after 3 and 7 days, respectively (Ref. 2-6). Thevalues listed in Table 2-1 for the characteristic strength at various other ages are suggested byTMH7 (Ref. 2-7) for use in structural design.

It is common practice to base the design of reinforced concrete structures on the 28-day strength,and to ignore any subsequent strength increase. However, in prestressed concrete, high stressesmay be induced prior to 28 days, e.g. high anchor zone stresses and high flexural stresses whichoccur at transfer. For such cases, the time dependence of strength must properly be accountedfor in the design.

• Shape and size of the specimens: Standard testing procedures which use 150 mm diameter and300 mm long cylinders are also used to determine the compressive strength (Ref. 2-4).Unfortunately, the magnitude of the compressive strength obtained from cylinders differs from

CONCRETE 2-1

values obtained from tests on cubes. This directly stems from the fact that the measuredcompressive strength is dependent on the shape of the specimen.

The cylinder strength is generally between 70 and 90% of the cube strength, and an averagevalue of 80% is widely accepted. Research has also shown that the ratio of cylinder strength tocube strength tends to increase as the strength of the concrete increases (Ref. 2-8). This trend isclearly demonstrated in Table 2-2 and Fig. 2-2, which show the relationship between cylinderstrength and cube strength. Note that the data shown in Fig. 2-2 applies to concretes with veryhigh strength, also referred to as high performance concrete.

The size of the specimen also has an influence on the magnitude of the measured compressivestrength as shown in Figs. 2-3 and 2-4 for cubes and cylinders, respectively. The general trendis that larger specimens yield lower compressive strengths. It should be noted that the datashown in Fig. 2-4 was obtained from cylinders with a height to diameter ratio of 2, which is thevalue normally used.

Among the various reasons put forward to explain the trend that the strength of a specimenincreases as it becomes smaller, the following seems reasonable for the size of specimensnormally tested. The testing machine provides some lateral restraint to the specimen because of

30

20

10

40

0

Com

pres

sive

stre

ngth

(MPa

)

Age of concrete (Log scale)

1 day 7 days 28 days 3 months 1 year 5 years

Figure 2-1: Increase of concrete strength with time. Typical curve for concrete made withordinary Portland cement (Ref. 2-5).

Grade Characteristicstrength fcu

(MPa)

Characteristic strength atother ages

(MPa)

28 days 7 days 2 months 3 months 6 months 1 year

20 20.0 13.5 22.0 23.0 24.0 25.0

25 25.0 16.5 27.5 29.0 30.0 31.0

30 30.0 19.0 32.0 34.0 35.0 36.0

40 40.0 27.0 42.5 44.0 46.0 48.0

50 50.0 35.0 52.5 54.0 56.0 58.0

Table 2-1: Characteristic strength of concrete (ordinary Portland cement) at other ages(Ref. 2-7).

2-2 MATERIAL PROPERTIES

friction which develops between the platen plates and the contact faces of the specimen. Forsmaller specimens, the restraint will be effective over a larger portion of its total height thanwill be the case for larger specimens, where its effect will be limited to the end regions. Sincethe stress field induced by the restraint tends to confine the concrete and hence, increase thestrength of the concrete, smaller specimens tend to be stronger than larger specimens.

An alternative explanation for this trend, which should be mentioned, is based on the assumptionthat failure is caused by the propagation of small cracks and that the largest crack is responsiblefor complete failure and fracture, similar to the weakest link in a chain. Since the probabilitythat such a flaw, which will induce failure at a given load, is contained in a specimen canreasonably be expected to increase with specimen size, the compressive strength of a largerspecimen tends to be smaller than that of a smaller specimen (see Ref. 2-1).

• Applied load rate: By increasing the loading rate beyond that prescribed by a standard test (15MPa/min, Ref. 2-2), the measured compressive strength can be increased by up to 20%. On theother hand, the compressive strength can be reduced by as much as 20% if the load is appliedover several months (Ref. 2-13).

The reduction of strength caused by the long-term loading is usually ignored in design becausethe unconservative consequence of this assumption is more than offset by the usual design practiceaccording to which a design is based on the 28 day strength, which ignores the significanttime-dependent strength increase.

Cylinder strength (MPa)150 × 300 mm

12 20 30 40 50 60 70 80

Cube strength (MPa)150 mm cubes

15 25 37 50 60 70 85 95

Table 2-2: Relationship between cylinder and cube strengths (Ref. 2-9).

30 50 70 90 110 130

Cube strength (MPa) 100 mm cubesfcu

Cyl

inde

rst

reng

th(M

Pa)

150

300

mm

cylin

ders

f c�

�

110

100

90

80

70

60

50

40

30

20

Held (Ref. 2-10)

f fc cu� = 0.8Smeplas (Ref. 2-11)

Figure 2-2: Influence of the specimen shape on the compressive strength (Ref. 2-12).

CONCRETE 2-3

Experience has shown that the measured compressive strengths obtained from specimens taken fromthe same mix can show a significant variation, even if the specimens are made under strict laboratorycontrol. It is therefore not a practical approach to specify a single precise value for compressivestrength. Instead, a statistical approach is followed by most of the modern design codes of practice(Refs. 2-7 and 2-14) whereby the strength is specified in terms of the characteristic strength fcu ,which is defined as the strength below which not more than 5% of the measured results may beexpected to fall. If it is assumed that the measured values of strength are normally distributed, thisdefinition can be expressed as follows:

(2-1)f fcu m� � 164. �

0 50 100 150 200 250 300

110

105

100

95

90

Nominal cube size (mm)

Rel

ativ

e st

reng

th(%

)

AkroydHarmanNeville

Figure 2-3: Influence of the cube size on the compressive strength (Ref. 2-15).

Nominal diameter of cylinder (in)

Nominal diameter of cylinder (mm)

Rel

ativ

est

reng

th(%

)

0 105

100 200 300 400 500 600 700 800 900 10000

2015 25 30 4035

110

100

105

90

95

80

85

Figure 2-4: Influence of the cylinder size on the compressive strength (Ref. 2-15).


where fcu

= characteristic compressive strength

fm= mean compressive strength

� = standard deviation

Note that fm

and � are obtained from test results.

High strength concrete is usually specified for prestressed concrete because of its improvedperformance, not only with regard to compressive strength but also with regard to increased tensilestrength, increased modulus of elasticity and reduced creep. Concrete strengths which range between30 to 60 MPa are usually specified for prestressed concrete in South Africa, while strengths of upto 70 MPa can be used in precast pretensioned applications. When specifying concrete it is importantto bear in mind the strength requirements at transfer, which can be the governing consideration.

The minimum characteristic strengths recommended by SABS 0100 (Ref. 2-14) for prestressedconcrete are shown in Table 2-3, while both SABS 0100 and TMH7 (Ref. 2-7) require that onlyconcrete with a characteristic strength of 30, 40, 50, or 60 MPa be used. When specifying theminimum compressive strength of the concrete at transfer in the case of post-tensioning, properinformation must be obtained from the supplier of the system because this value depends on theparticular system being used and can be higher than the values listed in Table 2-3.

2.1.2 Stress-strain relationship

The stress-strain behaviour of concrete loaded in uniaxial compression is shown in Fig. 2-5(Ref. 2-13) together with the stress-strain curves for the constituent materials of the concrete, namelyaggregate and the hardened cement paste. A comparison of these curves reveal the following:

• The stress-strain response of the aggregate and the paste are both more or less linear up to failure,whereas the concrete has a non-linear response over the entire load spectrum.

• The stress-strain response of the concrete falls between that of the aggregate and that of thecement paste.

The non-linear response of the concrete is caused by micro-cracking which occurs at theaggregate-paste interfaces (Ref. 2-16). These cracks are often only visible close to failure whenconsiderable lateral expansion occurs.

Figure 2-6 shows typical experimentally obtained stress-strain curves for normal weight concretehaving strengths which vary from 20 to approximately 85 MPa. Each curve is characterized by anascending portion followed by a descending portion.

The ascending portion is initially almost straight, becoming flatter, and hence progressively morenonlinear, with increasing load. The slope of this portion also tends to increase with an increase incompressive strength (this property is more expansively covered under Section 2.1.3).

At 28 days (fcu) At transfer

Pre-tensioned 40 MPa Bonded 25 MPa

Post-tensioned 30 MPa Unbonded 18 MPa

Table 2-3: Minimum recommended characteristic strength according to SABS 0100 (Ref. 2-14).

CONCRETE 2-5

The maximum stress, which separates the ascending and descending portions of the curve, is definedas the compressive strength. The strain corresponding to this stress is about 0.002 for normal strengthconcrete, while indications are that it increases with an increase in strength, particularly in the caseof high strength concrete. The transition from the ascending portion to the descending portionbecomes sharper as the compressive strength increases, thus indicating that a more brittle behaviouris associated with stronger concrete.

The slope of the descending portion of the stress-strain curve as well as the strain correspondingto failure of the specimen (often referred to as the ultimate strain) both change with a change incompressive strength, the slope becoming steeper and the ultimate strain becoming smaller with anincrease in strength. This reduction of the ultimate strain, once again, indicates that the uniaxialcompressive behaviour of concrete becomes more brittle with increasing strength. It should be noted

Coarseaggregate

rock Concrete

Hardenedcementpaste

Stre

ssf c

Strain �c

Cracks at interfaceof aggregate

Figure 2-5: Uniaxial stress-strain response of concrete and its constituent materials (Ref. 2-13).

00 0

30

20

102

40

4

50

6

608

70 10

80

90

12

0.001 0.002 0.003 0.004

Strain �c

Com

pres

sive

stre

ss(M

Pa)

f c

(ksi

)

Figure 2-6: Typical stress-strain curves for normal weight concrete in uniaxial compression (Ref.2-17).


that it is very difficult to experimentally determine the descending portion of the stress-strain curvebecause the failure mode of a compressive specimen is brittle. Consequently, special experimentaltechniques need to be resorted to for determining this portion of the curve.

The influence of load rate on the compressive stress-strain behaviour of concrete is shown inFig. 2-7 (Ref. 2-18), which reveals the following trends for a decrease in load rate:

• the strength decreases (see Section 2.1.1),

• the slope of the ascending portion is decreased, and

• the descending portion becomes flatter.

It is important to note that the stress-strain curve for concrete in uniaxial compression differs fromthat in flexure because the stress distribution in the specimens are different. This aspect is coveredin Chapter 4.

2.1.3 Modulus of elasticity

A material is defined as being elastic when the deformations induced by an applied load is completelyrecovered immediately after the load is removed. In the case of a linear elastic material, therelationship between the applied stress and the resulting strain can be expressed as follows:

f = E � (2-2)

where f = applied stress

� = resulting strain

E = modulus of elasticity (Young’s modulus) of the material

Inspection of Fig. 2-6 clearly shows that the stress-strain relationship for concrete is non-linear overthe complete load spectrum. However, the initial portion of the ascending branch may be seen tobe approximately linear. This feature makes it possible to approximate concrete as a linear elastic

0.00100

0.25

0.50

0.75

1.00

0.002 0.003 0.004 0.005 0.006 0.007Concrete strain �c

Rat

ioof

conc

rete

stre

ssto

cylin

der

stre

ngth

/f

fc

c�

Cylinder strength= 20.7 MPa (3000 psi)

at 56 daysfc�

Strain rate0.001 per 100 days

0.001 per day

0.001 per hr.

0.001 per min.

Figure 2-7: Influence of the loading rate on the stress-strain curve (Ref. 2-18).

CONCRETE 2-7

material at service load levels, because the magnitude of the induced stress generally falls withinthis quasi-linear range of the stress-strain curve.

The modulus of elasticity of concrete can be defined in various ways because linear elasticity is anapproximation of the actual non-linear behaviour in this range. Three possible definitions arepresented in Fig. 2-8:

• The initial tangent modulus Eci

is defined as the slope of the tangent to the stress-strain curveat its origin, and is often used as a parameter for the mathematical description of the stress-straincurve.

• The slope of a tangent at an arbitrary point P is defined as the tangent modulus Ect

at that point.

• The secant modulus Ec

is defined as the slope of a straight line drawn from the origin of thestress-strain curve to a specified point P on the curve.

The secant modulus is commonly used in prestressed concrete design, whereas the initial tangentmodulus and the tangent modulus are not commonly used in day-to-day design.

The modulus of elasticity of concrete must be determined in strict accordance with standard testingprocedures which have been developed for this purpose. These procedures include both staticmethods (Ref. 2-19) and dynamic methods (Ref. 2-20). The testing procedure prescribed by BS 1881:Part 121 (Ref. 2-19) requires that the static modulus be determined from tests on standard 150 mmdiameter by 300 mm high cylinders loaded at a rate of 0.6 ± 0.4 MPa/min. This test defines thestatic modulus as the secant modulus corresponding to a stress equal to a third of the strength.

Dynamic methods for determining the modulus of elasticity have been developed in recent years.In these methods the magnitude of the stresses induced by the dynamically applied loads are verysmall so that the dynamic modulus is often taken as an approximation of the initial tangent modulus.The effects of any creep are also negligible in these tests because loads are rapidly applied andreleased. SABS 0100 (Ref. 2-14) suggests that the following expression can be used to obtain anestimate of the static secant modulus E

cfrom the dynamic modulus E

cqto within 5 GPa:

Ec= 1.25E

cq− 19 GPa (2-3)

Strain εc

Stre

ssf c

P

1

1

1Ect

Ec

Eci

= Initial tangent modulus

= Secant modulus= Tangent modulus at point PEct

Ec

Eci

Figure 2-8: Definitions of the modulus of elasticity of concrete.


The type of aggregate used for the concrete appears to be the most important factor influencing themodulus of elasticity. Other factors which can also have an influence are mix proportions, shape ofthe aggregate, age of the concrete and moisture condition.

For a given aggregate type, the modulus of elasticity increases with an increase in compressivestrength. This finding, together with the fact that the compressive strength is often the only propertyof the concrete available at the design stage, has led to many attempts being made to correlate themodulus of elasticity with the compressive strength. This approach has often been questioned andrecent research (Refs. 2-21 and 2-22) has shown that no single expression can be used to relate themodulus of elasticity to the compressive strength only. The relationships between the modulus ofelasticity and the compressive strength for various South African aggregates shown in Fig. 2-9 (Ref.2-22) clearly illustrate this point, because they show that, for a given strength, the modulus ofelasticity varies widely depending on the aggregate type used.

The values for the static modulus of elasticity recommended by SABS 0100 (Ref. 2-14) for concreteusing normal-density aggregates are given in Table 2-4. SABS 0100 also suggests that the followingexpression may be used to estimate the magnitude of the static modulus from the 28-day cubestrength:

(2-4)

where Ec

= static secant modulus of elasticity

Ko = a constant closely related to the modulus of elasticity of the aggregate

fcu = characteristic cube strength at 28 days, in MPa

When the properties of the aggregate are unknown, SABS 0100 suggests that Ko

can be taken as20 GPa for normal weight concrete. In the case of low-density aggregate concrete, with a densityof between 1400 to 2300 kg/m3, the values in Table 2-4 should be multiplied by (�/2300)2, where� is the density of the concrete in kg/m3.

E K fc o cu� � 0 2. GPa

20

25

30

35

40

45

50

55

20 30 40 50 60 65

Dolomite(Olifantsfontein)

Dolerite(Ngagane) (Newcastle)

Andesite (Eikenhof) (Jhb)Greywacke(Malmesbury shale) (Peninsula)

Wits Quartzite (Vlakfontein)

Granite (Jukskei) (Midrand)

Siltstone(Leach & Brown)(Ladysmith)

Cube strength (MPa)

Stat

icel

astic

mod

ulus

(GPa

)

Figure 2-9: Relationship between static modulus of elasticity and compressive strength for agesfrom three days to 28 days (Ref. 2-22).

CONCRETE 2-9

The recommended values for the static modulus Ec

given in Table 2-4 are the same as those givenby TMH7 (Ref. 2-7). References 2-22 to 2-24 recommend that more accurate estimates of E

ccan

be obtained from Ec = Ko + fcu, where Ko

and are coefficients depending on the aggregate type.These references list values of K

oand for a fairly wide range of South African aggregates.

SABS 0100 (Ref. 2-14) recommends the following expression for estimating modulus of elasticityat any time t ≥ 3 days:

Ec, t

= Ec, 28

0.4 + 0.6

fcu, t

fcu, 28

(2-5)

where Ec, t

= modulus of elasticity at time t

Ec, 28

= modulus of elasticity at 28 days, obtained from Table 2-4

fcu, t

= characteristic strength at time t

fcu, 28

= characteristic strength at 28 days

t = time in days, ≥ 3 days

The ratio fcu, t

/ fcu, 28

can be estimated from Table 2-1.

It should be kept in mind that the expressions given above yield results which, at best, must beviewed as being approximate. It is therefore recommended that the modulus of elasticity should bedetermined from tests on concrete specimens made from the actual aggregates to be used in caseswhere structural deformations are important. Where such tests are not feasible, a reasonable approachto be followed in design would be to consider a range of values (as given in Table 2-4) whichwould bracket the expected deformations.

It is generally assumed that the modulus of elasticity in tension before cracking is the same as incompression. However, it should be pointed out that some researchers question the validity of thisassumption.

2.1.4 Tensile strength

The stress-strain diagram given in Figure 2-10 (Ref. 2-25) was obtained from a direct tensile test,and shows that the response is almost linear up to cracking. This diagram also shows that the tensilestrength of concrete is considerably smaller than its compressive strength.

Characteristiccube strength,

fcu(MPa)

Static modulus Ec(GPa)

Dynamic modulus Ecq(GPa)

Mean value Typical range Mean value Typical range

20 25 21 - 29 35 31 - 39

25 26 22 - 30 36 32 - 40

30 28 23 - 33 38 33 - 43

40 31 26 - 36 40 35 - 45

50 34 28 - 40 42 36 - 48

60 36 30 - 42 44 38 - 50

Table 2-4: Modulus of elasticity of concrete (Ref. 2-14).


Standard testing procedures from which the tensile strength may be indirectly measured have beendeveloped because of the practical difficulties associated with the direct tensile test. Two methodsare commonly used: The split cylinder test and the modulus of rupture test.

The split cylinder test, also known as the indirect tension test, is described in BS 1881: Part 117(Ref. 2-26). According to this test a 150 mm diameter cylinder, 300 mm long, is loaded across adiameter until failure occurs (see Fig. 2-11). If it is assumed that the cylinder behaves as an elasticbody, the resulting horizontal stress across the vertical diameter will be found to be uniformlydistributed over most of the depth of the cylinder, as shown in Fig. 2-11. The magnitude of thisstress at splitting is defined as the splitting tensile strength f

ct, and is given by:

(2-6)

where P = the measured compression force at splitting

L = length of cylinder (300 mm)

D = diameter of cylinder (150 mm)

The modulus of rupture test is described by SABS Method 864 (Ref. 2-27) (also BS 1881: Part 118(Ref. 2-28)) and consists of loading a simply supported beam of square cross section to failure. Thedimensions of the beam cross section are 100 × 100 mm (or 150 × 150 mm) while the span length

fP

L Dct �2

00 0.01 0.02

Con

cret

est

ress

(MPa

)f c

Elongation (mm)�

� �cr = 0.12 10-3

0.03 0.04 0.05 0.06 0.07

1

2

3

4

Specimen:76 19 305 mm

= 43.9 MPamax. aggregate size = 10 mmwater cured 28 days

� �

�fc

13 mmnotch

�83

mm

gaug

e

76 mm

Figure 2-10: Stress-strain response of concrete in uniaxial tension (Ref. 2-25).

P

P

LD

fct

fct

Figure 2-11: Split cylinder (indirect tension) test.

CONCRETE 2-11

is 300 mm (or 450 mm). The load is applied at the third-span points (see Fig. 2-12). The flexuraltensile stress in the bottom fibre of the section at failure, calculated on the basis of ordinary beamtheory, is defined as the modulus of rupture f

rand is given by:

fr= P L

b h2 (2-7)

where P = the measured load at failure

b = width of the section (100 or 150 mm)

h = height of the section (100 or 150 mm)

L = span (300 or 450 mm)

The modulus of rupture will overestimate the actual flexural tensile strength of concrete and shouldbe viewed as a hypothetical strength to be used as a comparative measure for practical purposesonly.

Figure 2-13 (Ref. 2-29) shows typical relationships between the various measures of tensile strengthand the compressive strength. From this figure it may be seen that the tensile strength can be relatedto the compressive strength and that the modulus of rupture is approximately 1.65 times the splittingtensile strength.

P / 2

L / 3 L / 3 L / 3

P / 2

b

h

L h= 3

fr

Figure 2-12: Modulus of rupture test.

2

4

6

8

00 10 20 30

Compressive strength (MPa)fcu

Tens

ilest

reng

th(M

Pa)

40 50 60 70

Modulusof rupture fr

Split cylinder fct

Direct tension

Figure 2-13: Relationships between tensile and compressive strengths of concrete (Ref. 2-29).


SABS 0100 (Ref. 2-14) and TMH7 (Ref. 2-7) do not give explicit characteristic values for thesplitting tensile strength nor the modulus of rupture. Where these properties are required for designthey are included as allowable stresses. The ACI code (Ref. 2-30) suggests the following relationshipbetween modulus of rupture f

rand cylinder strength:

fr= 0.63√f

c′ MPa (2-8)

where fc′ = Cylinder strength in MPa

Naaman (Ref. 2-8) gives the following expression for the splitting tensile strength fct

:

fct= 0.5√f

c′ MPa (2-9)

It is important to note that, because the splitting tensile strength and the modulus of rupture differnot only from each other but also from the direct tensile strength, these quantities represent differentmeasures of tensile strength. As such, care should be exercised to ensure that they are properly usedin design. Equations 2-8 and 2-9 should also be used with care because it appears that there is nosimple relationship between tensile and compressive strength, the reason being that factors such aswater-cement ratio, curing conditions, age, mix proportions and properties of the aggregate do notaffect these properties to the same degree (see Ref. 2-1).

2.1.5 Time-dependent behaviour

Definitions

When concrete is subjected to a sustained stress, the resulting strain can be divided into the followingthree components:

• Instantaneous elastic strain: When the stress is applied to the concrete it causes an instantaneouselastic strain, which can be expressed as follows (see Section 2.1.3):

εc=

fc

Ec

(2-10)

where fc= applied stress

εc= instantaneous elastic strain

Ec= Young’s modulus of the concrete

• Shrinkage strain: In the absence of temperature variations, shrinkage is defined as that part ofthe time-dependent strain which is independent of stress. Shrinkage therefore corresponds to thetime-dependent strain which occurs in the absence of stress.

• Creep strain: Creep is defined as the component of the time-dependent strain which is dependenton the applied stress. Although this definition has been used for many years, it is important topoint out that, strictly speaking, it is not correct because it implies that creep and shrinkage areindependent phenomena which are additive when they occur simultaneously (Ref. 2-31). It iswell known that creep and shrinkage are not independent, the effect of shrinkage on creep beingto increase its magnitude. In order to use the mass of experimental data obtained on the basisof the assumption that creep and shrinkage are independent, Neville (Ref. 2-31) suggests thatcreep should be defined as the time-dependent strain which takes place in excess of shrinkage.The consequence of this definition is that the total creep must be considered as consisting oftwo components:

- Basic creep, which is the component of creep which occurs under conditions where there isno moisture exchange with the ambient medium.

CONCRETE 2-13

Shrinkagefrom t0

Shrinkage of anunloaded specimen

Shrinkage

Nominal elasticstrain



True elastic strain

Creep on the basis ofadditive definition

Drying creep

Creep

Basic creep

t0

t0

t0

t0

1

1

1

2

3

3

Totalcreep

2

Age t

Time ( )t t� 0

Time ( )t t� 0

Time ( )t t� 0

Stra

in

Stra

inSt

rain

Stra

in

(a) Shrinkage of an unloaded companion specimen

(b) Change in strain of a loaded and drying specimen

(c) Creep of a loaded specimen in hygral equilibrium with the ambient medium

(d) Change in strain of a loaded and drying specimen

Figure 2-14: Definition of time-dependent deformations of concrete (Ref. 2-31).


- Drying creep, which is the component of creep influenced by the drying process.

These definitions are illustrated in Fig. 2-14, in which the various components of strain are shownfor a concrete specimen subjected to a sustained low-level compressive stress (i.e. less than 40% ofits short-term compressive strength). It should be noted that the elastic strain of the concrete reduceswith time because the elastic modulus increases with age. Strictly speaking, creep should bedetermined on the basis of the elastic strain at the time under consideration and not the time atwhich the load is applied. Although both methods can be used, the change in elastic strain is notaccounted for under normal circumstances because the difference is usually small and because thisapproach is more convenient for structural analysis.

Factors which Influence Creep and Shrinkage

Creep and shrinkage of concrete can be ascribed to the movement of water within the crystallinestructure of the cement paste and loss of water to the surrounding environment by evaporation. Thefactors which influence creep and shrinkage can be grouped into two broad categories: Intrinsicfactors, which deal with the actual composition of the concrete as well as the influence of stress,and extrinsic factors, which account for the state of the environment to which the concrete is exposed.A partial list of these factors includes (Ref. 2-32):

• Water-cement ratio: Both creep and shrinkage are increased by an increase in the water-cementratio, partially because the evaporable water is increased, and because of more and larger capillarypores.

• Aggregate: Since the seat of creep and shrinkage is to be found in the cement paste, the aggregatetends to restrain the deformation of the paste induced by creep and shrinkage. Hence, an increasein the aggregate-cement ratio will lead to lower values of creep and shrinkage. Aggregates whichhave higher values for the modulus of elasticity can offer greater restraint to potential creep andshrinkage of the paste and therefore tend to yield concrete which creeps and shrinks less. Theuse of more porous aggregates leads to increased creep and shrinkage, possibly because anincrease of porosity can facilitate moisture transfer within the concrete. However, it should benoted that aggregates with higher porosity tend to have a lower modulus of elasticity.

• Cement type: The influence of the type of cement on creep appears to be related in part to itseffect on the rate of strength development which, in turn, depends on the composition and finenessof grinding (Ref. 2-1). The magnitude of the creep of concrete made with the following cementsoccurs in an increasing order: high-aluminium, rapid-hardening, ordinary Portland, Portlandblast-furnace, low-heat and Portland-pozzolana.

Reference 2-1 suggests that the type of cement affects shrinkage mainly through variations inC3A content, and that fineness of grinding has a negligible effect on shrinkage, except when thecement is extremely fine or extremely coarse. It appears that concretes containing Portland blastfurnace cement (PBFC) and rapid-hardening Portland cement generally tend to shrink more thanconcrete containing ordinary Portland cement.

• Admixtures: The effect of admixtures on creep and shrinkage appears to be highly variable,depending on the specific admixture and cement used, as well as a number of other factors whichinclude exposure conditions, age at loading and time under load (Ref. 2-1). It is important tonote that the use of certain admixtures can significantly increase the creep and shrinkage ofconcrete.

• Member size and shape: The volume to exposed surface ratio of a member can be used as ageneral parameter for describing the influence of the size and shape of the member on creep andshrinkage. A larger value of this ratio represents a thicker (larger) member which has a longerdiffusion path for moisture loss. Consequently, creep and shrinkage reduce with an increase inthe volume to surface ratio, i.e. as the member becomes larger, with creep approaching the valueof basic creep for very large members. As far as creep is concerned, it is most probably onlydrying creep which is affected by a variation of the size and shape of the member because basiccreep remains unaffected by loss of moisture from the concrete and, as such, is independent of

CONCRETE 2-15

the size and shape of the member. Evidently shrinkage is affected to a greater extent than creepby the size and shape of the member.

• Magnitude of the applied stress: Creep strains are approximately proportional to the magnitudeof the applied sustained stress for values less than 50% of the cube strength. For most practicalstructures, creep may therefore be considered to be linearly related to stress within the serviceload range.

• Age of loading: The age of the concrete when it is loaded has an important influence on themagnitude of creep, the effect being to increase creep with earlier ages at loading. The mannerin which the age at loading influences creep seems to be related to the manner in which it affectsthe development of strength and the degree of hydration. For these reasons, creep has been foundto correlate well with maturity.

• Temperature: Creep is apparently not a monotonic function of temperature and passes a maximumin the vicinity of 50°C. Beyond this point creep reduces with temperature up to about 120°Cafter which it, once again, increases with temperature (Ref. 2-1). It also appears that the creepof specimens heated just prior to loading is more significantly influenced by temperature thanthat of specimens cured at the test temperature, because of improved hydration in the latter case.Tests by England and Ross (Ref. 2-33) indicated that the effect of temperature on creep is greaterin the range of 20-60°C than in the range 100-140°C. Shrinkage is also increased at highertemperatures during drying.

• Relative humidity: Both creep and shrinkage are increased with a decrease of the ambient relativehumidity. It appears that it is not the relative humidity which is the influencing factor with regardto creep, but rather the process of drying while under load. This is confirmed by the fact thatthe effect of relative humidity is much smaller if the concrete has already reached hygralequilibrium before loading and, furthermore, that creep is strongly dependent on relative humiditywhen the concrete is allowed to dry while under load. At 100% relative humidity the concreteabsorbs water and swells slightly (as opposed to shrinking).

Creep: behaviour and prediction

The development of creep with time is shown in Fig. 2-15, which shows that most of the creepdevelops within a fairly short time period after the application of the load. SABS 0100 (Ref. 2-14)suggests that, under conditions of constant relative humidity, 40, 60 and 80% of the final creepdevelops during the first month, the first 6 months and the first 30 months under load, respectively.It should be noted that the final creep is defined by SABS 0100 as the creep strain after 30 years.Evidently creep continues for a very long time, and even at ages of the order of 30 years small,but measurable, creep rates have been reported (Ref. 2-34).

Instantaneousrecovery

Creep recovery

Residualdeformation

Strain on applicationof load

Creep

Time since application of load (days)

Specimen under constant load Load removed

Stra

in(1

0)

�6

500

00 50 100 150 200

1000

1500

Figure 2-15: Creep and creep recovery of concrete (Ref. 2-31).


Removal of the sustained stress is accompanied by an instantaneous strain recovery in the concrete,which is normally smaller than the instantaneous elastic strain associated with the application of thestress. As shown in Fig. 2-15, the instantaneous recovery is followed by a time-dependent recoveryof strain, termed creep recovery, which tends to a finite value. The magnitude of the creep recoveryis usually smaller than that of the creep at the time of removal of the stress. An exception occursif the concrete is old when the stress is applied, in which case the creep recovery can have the samemagnitude as the creep.

Linear creep theory can be applied to most practical structures within the service load range. Thistheory leads to the conclusion that creep strain is linearly related to the instantaneous elastic strainunder constant sustained stress and under constant environmental conditions. Using this approach,the creep strain is given by

(2-11)

where �cr

(t) = creep strain, as a function of time t

�c = instantaneous elastic strain, given by Equation (2-10)

�(t) = creep coefficient, as a function of time t

t = time, measured from the time at which the sustained stress is applied t0

For most practical cases, the long-time value of �(t) can vary between 1.5 and 3.5. The 30 yearcreep coefficient �30 can be obtained from Fig. 2-16, which is taken from SABS 0100 (Ref. 2-14).This figure gives �30 as a function of the ambient relative humidity, the age at loading and theeffective thickness of the section which, for the purposes of Fig. 2-16, is defined as twice thecross-sectional area of the member divided by the exposed perimeter. More comprehensiveprocedures for determining �(t), which explicitly include a greater number of factors that influencecreep, are given in Ref. 2-7 and Refs. 2-35 through 2-37.

� � �cr ct t( ) ( )�

20 30

150 300 Air

cond

ition

edar

ea(o

ffic

es)

Coa

stal

area

Inla

nd

3.0

3.0

4.0

3.5

2.0

2.0

2.0

1.01.0

1.0

2.5

2.5

2.5

1.51.5

1.5

0.50.50.5

600

40 50 60

Ambient relative humidity (%)*

70 80 90 100

* Relevant values for outdoor exposure may be determined throughthe Weather Bureau, Department of Environmental Affairs

30 Year creepcoefficient for aneffective section

thickness (mm) of

Age of loading(days)

13728

90

365

Figure 2-16: Effects of relative humidity, age of concrete at loading and section thickness on thecreep coefficient (Ref. 2-14).

CONCRETE 2-17

Equation 2-11 expresses the creep strain as a linear function of the instantaneous elastic strain which,in turn, is dependent on the magnitude of the modulus of elasticity E

cof the concrete (see Eq. 2-10).

It is therefore clear that �(t) is implicitly defined in terms of Ec. Because some of the procedures

for estimating �(t) base the calculation of the instantaneous elastic strain on the magnitude of Ec

atthe time at which the concrete is loaded (Refs. 2-14 and 2-37) while others base it on the magnitudeat 28 days (Refs. 2-35 and 2-36), great care should be exercised to determine exactly which valueof E

cshould be used. This observation also emphasizes the fact that different procedures should

never be combined to estimate creep strains.

The creep strain is often expressed in terms of specific creep (defined as the creep strain per unitstress) as follows:

(2-12)

where C (t) = specific creep, as a function of time t

fc= sustained concrete stress

The specific creep can be expressed in terms of the creep coefficient by equating Equations (2-11)and (2-12), and using Eq. (2-10). Thus

so that

(2-13)

Shrinkage: behaviour and prediction

The development of shrinkage with time is shown in Fig. 2-17 where it may be seen that, as in thecase of creep, the rate of shrinkage reduces with time, and that a measurable rate can still be obtainedafter 20 years. The rate at which shrinkage develops depends on the conditions of drying: Most ofthe shrinkage can take place within a period of 3 months under adverse drying conditions, whilethe concrete may not shrink at all if it always remains wet. It is reported in Ref. 2-34 that forconcrete stored in air at 50% relative humidity and at 21°C (70°F) there are indications that creepand shrinkage develop at similar rates. For the purpose of estimating prestressing losses, SABS 0100(Ref. 2-14) suggests that 50% and 75% of the total shrinkage takes place within the first month andwithin the first six months after the transfer of prestress, respectively. Note that the total shrinkage,referred to by SABS 0100 above, excludes the shrinkage which takes place before transfer. Althoughthe time period associated with the total shrinkage is usually ill-defined, it appears reasonable totake it as the design life of the structure.

For the types of concrete generally used for prestressed concrete, the magnitude of the shrinkagestrain will normally vary between 0.0002 and 0.0006. Figure 2-18 gives the shrinkage strain after6 months and after 30 years as function of the ambient relative humidity and the effective sectionthickness (defined as for creep, see Fig. 2-16), as recommended by SABS 0100. These values applyto concrete with an original water content of 190 �/m3. If the concrete has a water content whichdiffers from this value, but which lies within the range 150 to 230 �/m3, then the shrinkage obtainedfrom Fig. 2-18 must be adjusted in proportion to the water content.

More comprehensive procedures for determining the shrinkage strain are presented in Ref. 2-7 andRefs. 2-35 through 2-37. These procedures explicitly include a greater number of factors whichinfluence shrinkage.

�cr ct C t f( ) ( )�

C t f tt f

Ec cc

c

( ) ( )( )

� ��

C tt

Ec

( )( )

��


1200

800

400

10 28 90

100%

70%

50%

1

Time (log scale)

Days Years

Shri

nkag

e10

�6

2 5 10 20 30- 400

0

Relative humidity:

Time reckoned since end of wet curingat the age of 28 days

Figure 2-17: Development of shrinkage with time for concretes stored at different relativehumidities (Ref. 2-1).

20

0 0

5

10

15

20

25

30

35

40

45

0

12.5

25.0

37.5

50.0

62.5

75.0

87.5

100200

175

150

125

100

75

50

25

300

600 600

30 Year shrinkage 10for an effective

section thickness(mm) of

� 6 6 Month shrinkage 10for an effective

section thickness(mm) of

� 6

300 300150 150

250

200

150

100

50

0

–200

350

300

250

200

150

100

50

0

–200

350

400

300

250

200

150

100

50

0

–200 –100–100 –100

30 40 50 60 70 80 90 100

Ambient relative humidity (%)

Inla

nd

Coa

stal

area

Air

cond

ition

edar

ea(o

ffic

es)

Shrinkage

Swelling

Figure 2-18: Drying shrinkage for normal density concrete (Ref. 2-14).

CONCRETE 2-19

2.1.6 Thermal properties of concrete

As is the case for most materials, concrete will expand when heated and shrink when cooled. Thestrain in unconfined concrete induced by a change in temperature is expressed as follows:

(2-14)

where �cth

= strain in concrete induced by a change in temperature

c = coefficient of thermal expansion

∆ T = change in temperature

The coefficient of thermal expansion for concrete c

is strongly dependent on the aggregate typeand can vary from 7.5 to 11.5 × 10− 6 / °C for South African aggregates (Ref. 2-22). SABS 0100(Ref. 2-14) recommends an average value of 10 × 10− 6/°C.

At temperatures in excess of 300°C, the strength of concrete can be significantly reduced, whileindications are that the stiffness can be reduced at temperatures as low as 100°C (Ref. 2-13). Thereis some evidence that E

cat 400°C can be as low as one-third the value at 20°C. However, it is

important to note that the effect of temperature on the mechanical properties of concrete is stronglydependent on the aggregate type.

2.1.7 Poisson’s ratio

When concrete is uniaxially loaded, strains develop both in the direction of the applied load and ina direction perpendicular to it. Poisson’s ratio is defined as the ratio of the perpendicular strain tothe strain in the direction of the load. For concrete in compression, Poisson’s ratio ranges between0.15 and 0.2 (Ref. 2-8). SABS 0100 (Ref. 2-14) and TMH7 (Ref. 2-7) recommend a value of 0.2for design.

2.1.8 Fatigue

In prestressed concrete members failure of the concrete in fatigue is not very common because thestress range and number of load cycles to which such members are subjected to in practice arenormally less than that which causes failure. It appears that, in direct compression, concrete cansustain about ten million cycles of load which fluctuate between 0 and 50% of its static compressivestrength (Ref. 2-8).

2.2 STEEL REINFORCEMENT

In most applications, prestressed concrete members will contain non-prestressed reinforcement inaddition to the prestressed reinforcement. The non-prestressed reinforcement is normally includedas shear reinforcement, as supplementary reinforcement for crack control and, particularly in thecase of partially prestressed concrete, to satisfy strength requirements. Hence, the following typesof reinforcement may be found in a prestressed concrete member:

• non-prestressed reinforcement, which consists of hot-rolled mild steel bars, hot-rolled high yieldstress bars, cold-worked high yield stress bars or welded steel fabric.

• prestressed reinforcement, which consists of high strength wires, strand or alloy bars.

The properties of the various types of reinforcement mentioned above are described in the followingSections. It should be noted that the material properties to be used for design can be determinedfrom tests on axially loaded specimens because the steel reinforcement in prestressed concretemembers is usually subjected to an almost uniaxial state of stress.

� cth c T� �


2.2.1 Non-prestressed reinforcement

Typical stress-strain curves for hot-rolled mild steel and hot-rolled high yield steel reinforcing barstested in tension are presented in Fig. 2-19a. For the sake of clarity, Fig. 2-19b shows the initialportion of these stress-strain curves with the strain axis enlarged.

The characteristics of the stress-strain behaviour are subsequently discussed with reference to thecurve for mild steel bars. Initially the response is linearly elastic up to point a, beyond which ayield plateau develops. There is little or no increase in stress for a corresponding increase in strainon the yield plateau, which is bounded by the onset of a region of strain hardening at point c. Thestrain hardening region is characterized by an increase in stress with an increase in strain until amaximum value of stress is reached at point d. Any subsequent increase in strain beyond this pointis accompanied by a stress reduction until fracture finally occurs at point f.

The slope of the initial linear elastic portion gives the modulus of elasticity, which generally variesbetween 200 and 210 GPa. The stress at which yielding occurs is referred to as the yield stress andis an important property of steel reinforcement. A sudden reduction of stress, from point a to pointb, often occurs immediately after first yielding. In such a case point a is referred to as the upperyield point and point b as the lower yield point. The upper yield point is strongly dependent on thespeed of testing, the section shape and form of the specimen and is usually of little interest. Hence,the lower yield point is taken as the yield strength of the material. The yield plateau for mild steelextends to a strain approximately equal to 10 times the strain at first yield. The maximum stresssustained by the specimen at point d is referred to as the ultimate stress.

A comparison of the stress-strain curve for the hot-rolled high yield bars with the curve for themild steel bars (see Fig. 2-19) reveals that, apart from the obvious difference of having higher yieldand ultimate strengths, the behaviour of the high yield bars is significantly less ductile than that ofthe mild steel bars. This feature is characterized by the smaller extent of the yield plateau as wellas the smaller elongation at fracture.

Typical stress-strain curves for cold-worked and hot-rolled high yield reinforcing bars are shown inFig. 2-20. It should be noted that these curves were taken from Ref. 2-39 and apply to Dutch steelwith a specified yield stress of 400 MPa. This figure clearly shows that the stress-strain behaviourof cold-worked reinforcement does not exhibit a definite yield point as in the case of hot-rolledbars, but rather shows a gradual transition from linear elastic to non-linear behaviour. Because of

0.10 0.0100 00.20 0.020

100 100

200 200

300 300

400 400

500 500

0 00.30 0.030

a

aa

a

b

b

bc

c

c

c

d

d

f

f

Strain �s

Stre

ss(M

Pa)

f s

Stre

ss(M

Pa)

f s

Strain �s

Mild steel barsMild steel bars

Hot-rolled high yield stress barsHot-rolled

high yield stress bars

(a) (b)

Figure 2-19: Stress-strain curve for normal reinforcing bars (Ref. 2-38).

STEEL REINFORCEMENT 2-21

this feature, the yield stress for cold-worked reinforcement must be defined. This is generally donein one of two ways, as shown in Fig. 2-21:

• The yield stress fy1

can be defined as the stress corresponding to a specified strain εy1

under load.

• The yield stress fy2

can also be defined as the stress corresponding to a specified plastic strainε

offset. This method is referred to as the offset strain method and the yield stress so determined

is known as the proof stress.

Another important feature of the stress-strain response of cold-worked reinforcement is that it issignificantly less ductile than that of hot-rolled reinforcement, as revealed by the reduced elongationat fracture (see Fig. 2-20).

In South Africa, the nominal sizes in which reinforcing bars can be supplied are listed in Table 2-5.Other geometric properties as well as the mass of the bars are also presented herein. Thereinforcement must also conform to the requirements of SABS 920 (Ref. 2-40), of which the requiredtensile properties are summarized in Table 2-6. The strength of the reinforcing bars is specified in

00

100

200

300

400

500

600

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

FeB 400HK(cold-worked)

FeB 400HWL (hot-rolled)

Strain �s

Stre

ss(M

Pa)

f s

Figure 2-20: Stress-strain curve for cold-worked and hot-rolled reinforcing bars (Ref. 2-39).

Stress fs

fy2

fy1

Strain �s

�y1�offset

Es Es

1 1

Figure 2-21: Definitions of yield-strength for gradual yielding steel.


terms of the characteristic strength, which is defined as the value of the yield stress or the proofstress, as appropriate, below which not more than 5% of the measured results may be expected tofall (Ref. 2-40). The reasons for following this approach corresponds to those given for concretestrength (see Section 2.1.1). It should be noted that SABS 920 (Ref. 2-40) does not explicitly specifythe offset strain which defines the proof stress. However, an offset strain of 0.2% seems appropriate.

The stress-strain curve recommended by SABS 0100 (Ref. 2-14) for use in design is presented inFig. 2-22 and the following aspects should be noted:

• The recommended design value for the modulus of elasticity is 200 GPa.

• The actual stress-strain behaviour is approximated by a bi-linear relationship which ignores strainhardening. In most cases this approximation will lead to a conservative result in reinforcedconcrete and partially prestressed concrete members. However, in cases where large strains canoccur in the steel, ignoring the effect of strain hardening may change the intended failure modeto one which is undesirable. Consider, for example, the case of a beam in which large strainsdevelop in the flexural reinforcement at failure: In such a case the effects of strain hardening

Diameter(mm)

Area(mm2)

Perimeter(mm)

Mass(kg/m)

6 28.27 18.85 0.222

8 50.27 25.13 0.395

10 78.54 31.42 0.617

12 113.1 37.70 0.888

16 201.1 50.27 1.578

20 314.2 62.83 2.466

25 490.9 78.53 3.853

32 804.2 100.5 6.313

40 1256.6 125.7 9.865

Table 2-5 : Nominal geometric properties of normal reinforcing bars (Refs. 2-40 and 2-41).

Type of steel Identifyingsymbol

(Refs. 2-42and 2-43)

Characteristicstrengthfy (MPa)

Minimumultimate

tensile strength(MPa)

Minimumelongation

at fracture*(%)

Min. Max.

Hot-rolled mildsteel

R 250 400 15% greaterthan the

measured yieldstress or 0.2%proof stress, as

appropriate

22

Hot-rolled high-yield steel andc o l d - w o r k e dhigh-yield steel

Y 450 – 14

* Measured on a gauge length of 5.65 √So, where S

ois the original equivalent cross-sectional area.

Table 2-6: Tensile properties of reinforcement to SABS 920 (Ref. 2-40).


can increase the flexural capacity to such an extent that the accompanying increase in shear forcemay lead to an undesirable brittle shear failure in the actual structure, rather than the intendedductile flexural failure.

• A maximum strain at fracture is not given.

• Although it is commonly accepted that the stress-strain response of steel in compression is similarto that in tension, the design curve indicates a smaller yield strength in compression than intension. Evidently the reason for this is that the restraint offered by the concrete to buckling ofthe reinforcing bar is significantly reduced under conditions of yielding in compression.

• The design curve includes a partial safety factor for material strength m. This aspect is discussedin Section 4.4.1.

Steel reinforcement is normally detailed in compliance with the recommendations of SABS 82(Ref. 2-42) and SABS 0144 (Ref. 2-43).

Welded steel fabric, which consists of a grid of cold-drawn steel wire placed at right angles andwelded at the intersections, can often be used advantageously because of improved crack controland a reduction in fixing time. Wires can either be smooth or deformed, and standard dimensionsare listed in Table 2-7 (Ref. 2-44). If required for a particular design, fabric can be supplied witha pitch and wire diameter different from the standard sizes.

Welded steel fabric used in South Africa must conform to the requirements of SABS 1024(Ref. 2-44) according to which the minimum required tensile properties are as follows:

• The yield stress measured at 0.43% total elongation under load should be at least 485 MPa.

• The tensile strength should be at least 510 MPa.

• In addition, the tensile strength must be at least 5 % greater than the yield stress, or the elongationat fracture must be at least 12 % measured on a gauge length of , where S

ois the initial

cross-sectional area.

It is interesting to note that the test specimen must contain at least one welded intersection withinits length. The reason for this most probably follows from the fact that the wire fabric is notstress-relieved, which can lead to the occurrence of a failure near welded intersections at relativelysmall strains (Refs. 2-13 and 2-39). This fact must be kept in mind if significant ductility is aprimary design requirement.

565. So

Es = 200 GPa

Es

Tension

Actual

Compression

1

1

fy m

εy =fy

m sEεyc =fyc

Es

fyc =fy

m y+ / 2000f

Stress fs

Strain �s

fy = Characteristic yield strength(in MPa)

Figure 2-22: Design short-term stress-strain relationship for non-prestressed reinforcement(Ref. 2-14).


Standard diameters and pitch of welded steel fabric commonly available in South Africa are givenin Table 2-7. Fabric with a pitch other than that specified in the Table is also available on request.

2.2.2 Prestressed reinforcement

The material almost universally used for prestressing is high tensile strength steel, and an obviousapproach to producing this material is by alloying. The tensile strength of the steel can be furtherincreased by cold-drawing, usually followed by a stress-relieving process.

Prestressing tendons take the form of either wires, strand or bars:

• Wires are typically manufactured by cold-drawing high-tensile steel bars through successive diesto obtain the required strength characteristics. Subsequent mechanical processes can be used toindent or crimp the wire. Finally, the wire is stress-relieved by a suitable heat treatment, carriedout either in the absence or presence of an applied tension.

Fabricreferencenumber

Nominal pitchof wires

(mm)

Nominal diameterof wires

(mm)

Nominal crosssectional area

of wires(mm2/m of width)

Nominalmass perunit area(kg/m2)*

Longitu-dinal

Cross Longitu-dinal

Cross Longitu-dinal

Cross

617 200 200 10.0 10.0 393 393 6.17

500 200 200 9.0 9.0 318 318 5.00

395 200 200 8.0 8.0 251 251 3.95

311 200 200 7.1 7.1 197 197 3.11

245 200 200 6.3 6.3 156 156 2.45

193 200 200 5.6 5.6 123 123 1.93

100 200 200 4.0 4.0 63 63 1.00

772 100 200 10.0 7.1 786 197 7.72

655 100 200 9.0 7.1 636 197 6.55

517 100 200 8.0 6.3 503 156 5.17

433 100 200 7.1 6.3 396 156 4.33

341 100 200 6.3 5.6 312 123 3.41

289 100 200 5.6 5.6 246 123 2.89

278 100 300 6.3 4.0 312 42 2.78

226 100 300 5.6 4.0 246 42 2.26

133 100 300 4.0 4.0 126 42 1.33

* For information only. These values are based on the wires having a mass of 0.00785 kg/mm2

per metre length.

Table 2-7: Standard dimensions of welded steel fabric (Refs. 2-43 and 2-44).


• In the manufacturing process of 7-wire strand, six similar peripheral wires are spun over a centralwire which has a slightly larger diameter. The complete process is summarized in Fig. 2-23 whereit may be seen that there are two possibilities with regard to stress-relieving, as in the case ofthe manufacture of wire. The impact of the stress-relieving process on the properties of the steelare discussed later in this Section. In South Africa three types of 7-wire strand are commonlyproduced: standard, super and drawn. The tensile strength of super strand is higher than that ofstandard strand, while the mechanical properties of drawn strand are enhanced by an additionaldrawing process.

• Prestressing bars are generally manufactured from alloy steel heat treated to obtain the requiredproperties, and can be supplied either as smooth bars or as bars with a ribbed surface whichserves as a continuous screw thread.

Typical stress-strain curves for prestressing wire, strand and bar are given in Fig. 2-24 together witha typical curve for a hot-rolled high yield reinforcing bar. The following observations can be madefrom this figure:

• Prestressing steel has a much higher tensile strength than the reinforcing bar. This is accompaniedby a significant reduction in the elongation at fracture.

• The stress-strain response of prestressing steel does not show a definite yield point, so that theyield strength must be defined in terms of either a proof stress or a stress corresponding to atotal strain under load, as for cold-worked reinforcing bars (see Section 2.2.1). However, it shouldbe noted that some high strength prestressing bars may have a short, but detectable, yield plateau.

• The stress-strain curve for prestressing steel can conveniently be divided into three portions: aninitial linear elastic portion, followed by a region containing a fairly sharp non-linear transitionto the final almost linear strain-hardening portion, which is bounded by fracture.

Significant residual stresses arise in wire and strand because of the various mechanical processesinvolved in their manufacture. The presence of these residual stresses leads to a very roundedstress-strain curve as shown in Fig. 2-25. Stress-relieving has the effect of removing the residualstresses and also of increasing the proportional limit of the steel. By carrying out stress-relievingunder tension (strain tempering), the proportional limit is increased even more. It is very importantto note that strain tempering has the additional benefit of substantially reducing the relaxation losseven more than with ordinary stress-relieving.

Although the modulus of elasticity of steel is independent of strength, the values for prestressingsteels can vary slightly depending on the form of the steel:

• Wires generally have the highest value.

• The modulus of elasticity for strand will be lower than for wire because they consist of spunwire.

• Prestressing bars usually have a lower modulus than wire because of alloying.

Although typical values for the modulus of elasticity to be used for design are presented later inthis Section, care should be taken because their magnitude can be influenced by the manufacturingprocess, and data supplied by the manufacturer should be used when available.

In South Africa, prestressing wire and strand must conform to the requirements of BS 5896 (Ref.2-45) while prestressing bars must conform to those of BS 4486 (Ref. 2-46). The dimensions andrequired minimum tensile properties of the standard prestressing wires, strand and bars, as given bythese specifications, are listed in Tables 2-8 to 2-11. It should be noted that the characteristic loadslisted in these tables are defined as the value of the appropriate load below which not more than5% of the measured results may be expected to fall.


Cold drawing:Pull throughsuccessively smallerdies to increasestrength

Stranding:Spin 6 helical wiresaround a straightcentral wire

Patenting:Heat to about800°C (1470 °F) thencool slowly to makehomogeneous

Base material:Round, plain,hot-rolled,non-alloyed,high carbon steel rod

Stress relieving: Heat to about350°C and cool slowlySTRESS-RELIEVED STRAND

Strain tempering: Heat to about350°C while strand is under tensionLOW RELAXATION STRAND

Figure 2-23: Production of seven-wire strand (Ref. 2-13).


2000

1500

1000

500

0.05 0.10 0.1500

Stress relieved wire (1620 MPa)

High strength prestressing bars (1103 MPa)

Hot-rolled high yield reinforcing bars (450 MPa)

Assuming same elastic modulus

Prestressing Strand (1860 MPa)

Stre

ss(M

Pa)

f p

Strain �p

Figure 2-24: Stress-strain curves for prestressed reinforcement (Ref. 2-8).

Strain �p

Stre

ssf p

Stress relieved

Untreated

Strain tempered(Low relaxation)

Figure 2-25: Stress-relieving and strain tempering of prestressing wire (Ref. 2-13).


Nominaldiameter

Nominaltensile

strength

Nominal0.1%proofstress

Nominalcross

section

Nominalmass

Specifiedcharac-teristic

breakingload

Specifiedcharac-teristic0.1%

proof load

Load at1%

elongation

(mm) (MPa) (MPa) (mm2) (g/m) (kN) (kN) (kN)

7 1570 1300 38.5 302 60.4 50.1 51.3

7 1670 1390 64.3 53.4 54.7

6 1670 1390 28.3 222 47.3 39.3 40.2

6 1770 1470 50.1 41.6 42.6

5 1670 1390 19.6 154 32.7 27.2 21.8

5 1770 1470 34.7 28.8 29.5

4.5 1620 1350 15.9 125 25.8 21.4 21.9

4 1670 1390 12.6 98.9 21.0 17.5 17.9

4 1770 1470 22.3 18.5 19.0

Note: Minimum elongation at maximum load must be 3.5% measured on a gauge length of 200 mm

Table 2-8: Dimensions and properties of cold-drawn wire to BS 5896 (Ref. 2-45).

Nominal di-ameter

Nominaltensile

strength

Nominalcross-section

Nominalmass

Specifiedcharacteristic

breakingload

Specifiedcharacteristic

load at 1%elongation

(mm) (MPa) (mm2) (g/m) (kN) (kN)

5 1570 19.6 154 30.8 24.6

5 1670 32.7 26.2

5 1770 34.7 27.8

4.5 1620 15.9 125 25.8 20.6

4 1670 12.6 98.9 21.0 16.8

4 1720 21.7 17.4

4 1770 22.3 17.8

3 1770 7.07 55.5 12.5 10.0

3 1860 13.1 10.5

Table 2-9: Dimensions and properties of cold-drawn wire in mill coil to BS 5896 (Ref. 2-45).


Type ofstrand

Nominaldiameter

Nominaltensile

strength

Nominalsteelarea

Nominalmass

Specifiedcharac-teristic

breakingload

Specifiedcharac-teristic0.1%

proof load

Load at1%

elongation

(mm) (MPa) (mm2) (g/m) (kN) (kN) (kN)

7-wireStandard

15.2 1670 139 1090 232 197 204

12.5 1770 93 730 164 139 144

11.0 1770 71 557 125 106 110

9.3 1770 52 408 92 78 81

7-wireSuper

15.7 1770 150 1180 265 225 233

12.9 1860 100 785 186 158 163

11.3 1860 75 590 139 118 122

9.6 1860 55 432 102 87 90

8.0 1860 38 298 70 59 61

7-wireDrawn

18.0 1700 223 1750 380 323 334

15.2 1820 165 1295 300 255 264

12.7 1860 112 890 209 178 184

Note: Minimum elongation at maximum load must be 3.5% measured on a gauge length ≥ 500 mm

Table 2-10: Dimensions and properties of seven-wire strand to BS 5896 (Ref. 2-45).

Type ofbar

Nominalsize

Nominaltensile

strength

Nominal0.1%proofstress

Nominalcross-

sectionalarea

Nominalmass

Specified properties

Charac-teristic

breakingload

Charac-teristic0.1%proofload

Min.elonga-tion at

fracture*

(mm) (MPa) (MPa) (mm2) (kg/m) (kN) (kN) (%)

Hotrolled or

hotrolled

and proc-essed

26.5 1030 835 522 4.33 568 460 6

32 804 6.31 830 670

36 1018 7.99 1048 850

40 1257 9.86 1300 1050

* Measured on a gauge length of 5.65 √So, where S

ois the original cross-sectional area.

Table 2-11: Dimensions and properties of hot-rolled and hot-rolled and processed high tensilealloy steel bars to BS 4486 (Ref. 2-46).


Wire and strand which satisfy the requirements of ASTM Specification A-421 (Ref. 2-47) and ASTMSpecification A-416 (Ref. 2-48), respectively, are also produced in South Africa.

The design stress-strain diagram for prestressing steel acting in tension, as recommended by SABS0100 (Ref. 2-14), approximates the actual behaviour by the tri-linear curve shown in Fig. 2-26. Itshould further be noted that a maximum strain at fracture is not given and that the design curveincludes a partial safety factor for material strength m. This partial safety factor is discussed inSection 4.4.1.

SABS 0100 suggests the following design values for the modulus of elasticity of prestressingtendons:

Ep= 205 GPa for high tensile steel wire (wire to Section 2 of BS 5896: 1980)

= 195 GPa for 7-wire strand (strand to Section 3 of BS 5896: 1980)

= 165 GPa for high tensile alloy bars.

It is important to note that the recommended value of 165 GPa for the modulus of elasticity of hightensile alloy bars most probably only applies to as-rolled and stretched bars conforming to BS 4486(Ref. 2-46). In the case of as-rolled and as-rolled stretched and tempered bars, the value of 205GPa recommended by BS 4486 seems more appropriate. However, it is strongly recommended that,whenever possible, values supplied by the manufacturer should be used because the magnitude ofthe modulus of elasticity can be significantly influenced by the manufacturing process.

2.2.3 Relaxation of prestressing steel

The time-dependent loss of tensioning force required for maintaining a constant strain in a highlystressed steel tendon is defined as relaxation. Creep, which is defined as the time-dependent changein strain under constant stress may be considered as another consequence, under different conditions,of the same phenomenon described by relaxation. Although the strain in a prestressing tendoncontinually changes with time because of shrinkage and creep of the concrete, it is generallyacknowledged that these conditions approach those to be found in a relaxation test rather than in acreep test.

As discussed in Section 2.2.2, the relaxation properties of prestressing steel is significantlyinfluenced by the particular stress-relieving process used. The ordinary stress-relieving process,

Ep Ep

1 1

0.8 /fpu m

fpu m/

Stress fp

Strain �p0.005

fpu = Characteristic strength

Figure 2-26: Design stress-strain relationship for prestressed reinforcement acting in tension(Ref. 2-14).


which involves a heat treatment only, yields normal relaxation steel while strain-tempering, whichinvolves heat treatment under tension, yields low relaxation steel.

Relaxation appears to be primarily influenced by the ratio of initial stress to yield stress, the typeof steel, temperature and time. Magura, Sozen and Siess (Ref. 2-49) proposed the followingexpression for predicting the stress in stress-relieved wire and strand at any time:

fs(t)fsi

= 1 − log t10

fsi

fy

− 0.55

(2-15)

where fs(t) = steel stress at time t

fsi= initial steel stress immediately after tensioning

fy= yield stress of the steel, measured at an offset strain of 0.001

log t = logarithm of time to the base 10

t = time after tensioning, in hours

The above equation is based on data obtained from 501 relaxation tests on stress-relieved wire. Ithas been suggested that this expression can also be applied to low-relaxation strand and prestressingbars if the denominator 10 under the log t term is replaced by 45 (see Refs. 2-8, 2-13 and 2-50).Typical relaxation curves are shown in Fig. 2-27 for normal relaxation and low relaxation SouthAfrican prestressing strand.

Both Fig. 2-27 and Eq. 2-15 clearly show that a large part of the relaxation loss occurs within arelatively short time period after application of the load and that relaxation proceeds with time, butat a decreasing rate. It is also evident that the relaxation loss of low relaxation steel is significantlysmaller than that of normal relaxation steel, it being generally accepted that the relaxation loss oflow relaxation steel is 20 to 25% that of normal relaxation steel.

The effect of initial stress on the relaxation loss after 1000 hours is shown in Fig. 2-28 for lowrelaxation strand tested at various temperatures. The figure clearly demonstrates that the relaxationloss is increased if the initial stress is increased. This trend is confirmed by Eq. 2-15, which predictszero relaxation loss for values of the initial stress smaller than or equal to 55 percent of the yield

1 10 100 1000 104 105 106 107

Hours (log scale)

1 year 10 years 50 years6 months

%St

ress

rela

xati

on(l

ogsc

ale)

0.1

1

10

100

Normal relaxation

Low relaxation

Test temperature: 20°CInitial load: 70% of nominal breaking load

Figure 2-27: Relaxation of prestressing strand (courtesy Haggie Rand Ltd.).


stress. It is generally accepted that relaxation losses are insignificant for initial stresses smaller than50 percent of the yield stress.

The magnitude of relaxation is strongly influenced by the temperature of the steel, the effect beingthat it is increased by an increase in temperature. This trend is demonstrated in Fig. 2-29 whichgives relaxation curves for normal relaxation and low relaxation strand tested at various temperatures.Care should therefore be taken to make proper allowance for the increased relaxation which willoccur in cases where the prestressing tendons are subjected to temperatures significantly higher than20°C for extended periods of time.

50 60

20°C

40°C

60°C80°C

70 80 90

Initial Load (as % of nominal breaking load)

%St

ress

rel

axat

ion

(log

scal

e)

0.1

1

10

100

1000 Hour testsLow relaxation strand

Test temperature:

Figure 2-28: Effect of initial stress on relaxation of low relaxation prestressing strand at varioustemperatures (courtesy Haggie Rand Ltd.).

1 10 100 1000 10 000

Hours (log scale)

%St

ress

rela

xatio

n(l

ogsc

ale)

0.1

1

10

1001000 Hour testsInitial load: 70% of nominal breaking load

20°C

20°C

40°C

40°C

60°C

60°C

80°C

80°C

100°C

100°C

Test temperature:Low relaxationNormal relaxation

Figure 2-29: Effect of temperature on the relaxation of strand (courtesy Haggie Rand Ltd.).


In practice, the relaxation loss is usually experimentally determined after 1000 hours at 20°C andmultipliers are subsequently used to estimate the long-term values required for design. The maximumrelaxation loss after 1000 hours for various prestressing tendons as specified by BS 5896 and BS4486 (Refs. 2-45 and 2-46), are listed in Table 2-12.

It is important to note that a prestressing tendon in a prestressed concrete member will not besubjected to a constant strain because the member, and hence the tendon, will shorten as a resultof the effects of creep and shrinkage of the concrete. This has the effect of reducing the initialstress level of the tendon, so that the relaxation loss in an actual member is less than the loss whichwould be obtained in a relaxation test where a constant strain is maintained in the steel for theduration of the test. This effect must be accounted for in design and must therefore be reflected inthe magnitude of the multipliers used for estimating long-term relaxation losses from experimentaldata obtained from relaxation tests at 1000 hours.

SABS 0100 (Ref. 2-14) recommends that the relaxation loss to be allowed for in design should betaken as twice the loss at 1000 hours for an initial force taken equal to the tendon force at transfer.In the absence of experimental data, SABS 0100 suggests that the relaxation loss for normalrelaxation strand or wire may be assumed to vary linearly from 10% for an initial stress of 80% ofthe characteristic strength of the tendon to 3% for an initial stress of 50% of the characteristicstrength. If the creep plus shrinkage strain of the concrete is greater than 500 × 10-6 then the lossfor an initial stress of 80% of the characteristic strength should be taken as 8.5%. The relaxationloss for low relaxation strand may be taken as half the above values. Although TMH7 (Ref. 2-7)also requires that the relaxation loss to be allowed for in design should be estimated from the lossat 1000 hours for an initial force taken equal to the tendon force at transfer, no guidance is givenon how this is to be done.

It is of some interest to note that the multiplying factors specified by BS 8110 (Ref. 2-51) forestimating the long-term relaxation loss from the 1000 hour test value account for the reinforcementtype (wire and strand, or bar), the relaxation properties of the steel (normal or low relaxation) andthe prestressing procedure (pre- or post-tensioning). This code explicitly states that the recommendedmultiplying factors account for the effects of creep and shrinkage of the concrete and, in the caseof pretensioning, the effects of elastic shortening of the concrete at transfer. It also carefully definesthe initial force for post-tensioning as the prestressing force immediately after transfer and forpretensioning as the force immediately after tensioning.

Initial loadas % ofbreaking

load

Maximum relaxation after 1000 hours

Cold-drawn wire 7-Wire strand Cold-drawnwire in

mill coil

High tensilealloy

steel barsNormalrelaxation

Lowrelaxation

Normalrelaxation

Lowrelaxation

(%) (%) (%) (%) (%) (%)

60 4.5 1.0 4.5 1.0 8 1.5

70 8.0 2.5 8.0 2.5 10 3.5

80 12 4.5 12 4.5 – 6.0

Table 2-12: Maximum specified relaxation at 1000 hours, BS 5896 (Ref. 2-45) and BS 4486 (Ref.2-46).


2.2.4 Fatigue characteristics of reinforcement

When steel is subjected to a fluctuating stress its mechanical properties will deteriorate, dependingon the minimum and maximum values of the fluctuating stress, by a process known as fatigue. Theresistance of reinforcement to fatigue is often defined in terms of a S-N curve, in which the stressrange S is plotted as a function of the corresponding number of load cycles N required to causefailure. Figure 2-30 shows several experimentally obtained S-N curves for deformed reinforcing bars(Ref. 2-13). Inspection of this figure will reveal that the numbers of cycles which the reinforcingbars can sustain without failure increases as the stress range is decreased until a limiting value ofthe stress range is reached, below which it can be assumed that the bars can sustain an indefinitenumber of cycles of load. This region is characterized by the almost horizontal portion of the S-Ncurve which, for the reinforcing bars considered here, commences after about one to two millioncycles. The stress range corresponding to the horizontal portion of the S-N curve is known as thefatigue limit or the endurance limit.

TMH7 (Ref. 2-7) recommends that the stress range should be limited to 250 MPa for mild steelreinforcing bars, and to 300 MPa for high-yield strength bars. These values apply to a maximum of2 � 105 cycles of load, while the stress range should be limited to 60% of these values for 2 � 106

cycles. The CEB-FIP code (Ref. 2-36) limits the characteristic fatigue strength to 250 MPa forsmooth bars, and to 150 MPa for deformed bars. In this code, the characteristic fatigue strength isdefined as the stress range which nine reinforcing bars out of ten can resist for 2 � 106 cycles ifthe maximum stress is 70% of the yield strength.

The various types of prestressing steel do not appear to have a fatigue limit (Ref. 2-8). A singleS-N curve cannot show the effect of the magnitude of the minimum stress on fatigue failure. Instead,a modified Goodman diagram, which represents the relationship between the maximum and minimumcyclic stress at a particular number of cycles corresponding to failure, can be used for this purpose.For design, it is common to consider a minimum of 2 � 106 cycles, while a maximum of 10 � 106

can be considered in exceptional cases. A typical modified Goodman diagram corresponding to2 � 106 cycles is given in Fig. 2-31 for prestressing wires and strand (Ref. 2-8). This figure showsthat for the minimum stress normally encountered in prestressed members (50% to 60% of theultimate tensile strength) a stress range of approximately 13% of the ultimate tensile strength canbe resisted for 2 � 106 cycles. This stress range is substantially larger than that encountered inuncracked fully prestressed members, with the result that fatigue is not normally critical for designin this case. It should, however, be noted that the stress range in cracked partially prestressed

500

400

300

200

100

0104 105 106 107 108

Stre

ssra

nge

(MP

a)S

Cycles to failure N

Figure 2-30: S-N curves for deformed reinforcing bars (Ref. 2-13).


members can be significantly larger than in fully prestressed members so that fatigue can becomean important design consideration in such cases.

TMH7 (Ref. 2-7) recommends the following maximum values for the stress range in prestressingtendons in partially prestressed members, provided that the minimum stress does not exceed 50%of the ultimate tensile strength:

• Tendons, not deformed 200 MPa

• Tendons, deformed 150 MPa

• Strand 200 MPa

• High-strength bars 200 MPa

These values apply to a maximum of 2 × 105 cycles of load, while 60% of these values should beused for 2 × 106 cycles.

The FIP Recommendations (Ref. 2-52) suggest that the characteristic fatigue strength of wires andstrand may be taken as 200 MPa, and that it may be taken as 80 MPa for high-strength bars. Thisdocument defines the characteristic fatigue strength of prestressing steel as the stress range whichcan be resisted for 2 × 106 cycles, with a probability of failure of 0.01, if the maximum stress is85% of the yield strength.

It is important to note that although there is ample experimental evidence of the fatigue life oftendons in beams being shorter than that of similar tendons tested in air, the recommendations relatedto fatigue strength made by the majority of the codes of practice are based on data obtained fromtendons tested in air. A designer is therefore well advised to exercise caution and to take aconservative approach when considering fatigue.

00 0.5 1.0

0.5

1.0

Minimum stress ( ) / Strengthf fps min pu

Max

imum

stre

ss(

)/S

tren

gth

f

f

psm

axpu

Stressrange

Maximumstress limit

Minimumstress limit

2 10 cycles� 6

Usual stressrange forprestressedconcrete

Figure 2-31: Typical Goodman diagram for prestressing wires and strand (Ref. 2-8).


2.2.5 Thermal properties of reinforcement

The strain induced by a change in temperature in unconfined steel reinforcement can be expressedas follows:

(2-16)

where �sth = strain in steel induced by a change in temperature

s

= coefficient of thermal expansion

�T = change in temperature

Evidently, the actual value of the coefficient of expansion for steel is about 11.5 � 10-6/°C(Ref. 2-13). However, a value of 10 � 10-6/°C is usually taken for design, which is equal to thevalue taken for concrete (see Section 2.1.6).

Although the mechanical properties of the reinforcement are not significantly affected by normalvariations of the ambient temperature, they can be significantly affected by extreme temperatureconditions. If the temperature is increased beyond a value of approximately 200°C both the stiffnessand strength will be substantially reduced. More specifically, the tensile strength of wire or strandat 400°C is about 50% of its value at room temperature (Ref. 2-13). Reducing the temperatureproduces opposite effects, with stiffness and strength being increased. However, these improvementsare accompanied by a reduction in ductility. If the temperature is reduced from 20°C to �200°Cthe yield and tensile strengths will be increased by about 20% (Ref. 2-8).

2.3 REFERENCES

2-1 Portland Cement Institute, Fulton’s Concrete Technology, 7th ed., Edited by B. J. Addis, PCI,Midrand, South Africa, 1994.

2-2 South African Bureau of Standards, “Compressive Strength of Concrete (Including Makingand Curing of the Test Cubes),” SABS Method 863, SABS, Pretoria, 1976.

2-3 British Standards Institution, “Method for Determination of Compressive Strength of ConcreteCubes,” BS 1881: Part 116: 1983, BSI, London, 1983.

2-4 American Society for Testing Materials, “Standard Test Method for Compressive Strength ofCylindrical Concrete Specimens,” ASTM C 39-86, ASTM, Philadelphia, 1986.

2-5 Mosley, W. H., and Bungey, J. H., Reinforced Concrete Design, 4th ed., MacMillan EducationLtd., London, 1990.

2-6 Portland Cement Institute, Cement and Concrete, 9th ed., PCI, Midrand, South Africa, 1992.

2-7 Committee of State Road Authorities, “Code of Practice for the Design of Highway Bridgesand Culverts in South Africa,” TMH7 Part 3, CSRA, Pretoria, 1989.


2-9 CEB-FIP, “Model Code for Concrete Structures,” First Draft, Bulletin d’Information No.195,Comité Euro-International du Béton - Fédération Internationale de la Précontrainte, Paris,March, 1990.

2-10 Held, M., “Research Results Concerning the Properties of High Strength Concrete,” DarmstadtConcrete, Vol. 5, Technishe Hochschule Darmstadt, Alexanderstrasse 5 Darmstadt, Germany,1990.

2-11 Smeplass, L., “High Strength Concrete,” SP4 - Material Design, Report 4.4, Mechanicalproperties - normal density concrete, STF65 F89020 - FCB-SINTEF 7034, Trondheim, Norway.

� sth s T� �

REFERENCES 2-37

2-12 FIP-CEB, High Strength Concrete - State of the Art Report, Fédération Internationale de laPrécontrainte - Comité Euro-International du Béton Bulletin d’Information No. 197, Chame-leon Press, London, 1990.


2-14 South African Bureau of Standards, “The Structural Use of Concrete,” SABS 0100: 1992,Parts 1 and 2, SABS, Pretoria, 1992.

2-15 Neville, A. M., “Some Aspects of the Strength of Concrete,” Civil Engineering and PublicWorks Review, Vol. 54, Part 2, No. 640, Nov. 1959, pp. 1308-1310.

2-16 Carrasquillo, R. L., Slate F. O., and Nilson A. H., “Micro Cracking and Behaviour of HighStrength Concrete Subject to Short-term Loads,” ACI Journal, Vol. 78, No. 3, May-June 1981,pp. 179-186.

2-17 Nilson, A. H., “High Strength Concrete - An Overview of Cornell Research,” Proceedings ofthe Symposium “Utilization of High Strength Concrete”, Stavanger, Norway, June 1987, Tapir,Trondheim, 1987, pp. 27-38.

2-18 Rüsch, H., “Researches Toward a General Flexural Theory for Structural Concrete,” ACIJournal, Vol. 57, No. 1, July 1960, pp. 1-28.

2-19 British Standards Institution, “Methods for Determination of Static Modulus of Elasticity inCompression,” BS 1881: Part 121: 1983, BSI, London, 1983.

2-20 British Standards Institution, “Recommendations for the Measurement of Dynamic Modulusof Elasticity,” BS 1881: Part 209: 1990, BSI, London, 1990.

2-21 Alexander, M. G., “Prediction of Elastic Modulus for Design of Concrete Structures,” TheCivil Engineer in South Africa, Vol. 27, No. 6, June 1985, pp. 313-324.

2-22 Alexander, M. G., and Davis, D. E., “The Influence of Aggregates on the Compressive Strengthand Elastic Modulus of Concrete,” The Civil Engineer in South Africa, Vol. 34, No. 5,May 1992, pp. 161-170.

2-23 Alexander, M. G., and Davis, D. E., Properties of Aggregates in Concrete, Part 1, HippoQuarries Technical Publication, 1989.

2-24 Alexander, M. G., and Davis, D. E., Properties of Aggregates in Concrete, Part 2, HippoQuarries Technical Publication, 1992.

2-25 Gopalaratnam, V. S., and Shah, S. P., “Softening Response of Plain Concrete in DirectTension,” ACI Journal, Vol. 82, No. 3, May-June, 1985, pp. 310-323.

2-26 British Standards Institution, “Method for Determination of Tensile Splitting Strength,”BS 1881: Part 117: 1983, BSI, London, 1983.

2-27 South African Bureau of Standards, “Flexural Strength of Concrete (Including Making andCuring of the Test Specimens),” SABS Method 864, SABS, Pretoria, 1980.

2-28 British Standards Institution, “Method for Determination of Flexural Strength,” BS 1881:Part 118: 1983, BSI, London, 1983.

2-29 Illston, J. M., Construction Materials: Their nature and behaviour, Second ed., Edited byJ. M. Illston, E & FN Spon, London, 1994.

2-30 ACI Committee 318, “Building Code Requirements for Reinforced Concrete (ACI 318-89) andCommentary - ACI 318 R-89, ” American Concrete Institute, Detroit, 1989.

2-31 Neville, A. M., Dilger, W. H., and Brooks, J. J., Creep of Plain and Structural Concrete,Construction Press, London, 1983.

2-32 Marshall, V., and Gamble, W. L., “Time-Dependent Deformations in Segmental PrestressedConcrete Bridges,” Structural Research Series No. 495, Civil Engineering Studies, Universityof Illinois, Urbana, October 1981.


2-33 England, G. L., and Ross, A. D., “Reinforced Concrete under Thermal Gradients,” Magazineof Concrete Research, Vol. 14, No. 40, March 1962, pp. 2-12.

2-34 Troxell G. D., Raphael J. M., and Davis R. E., “Long Time Creep and Shrinkage Tests ofPlain and Reinforced Concrete,” Proc. ASTM, Vol. 58, 1958, pp. 1101-1120.

2-35 CEB-FIP, “International Recommendations for the Design and Construction of ConcreteStructures - Principles and Recommendations,” Comité European du Béton - FédérationInternationale de la Précontrainte, FIP Sixth Congress, Prague, June 1970; published byCement and Concrete Association, London, 1970.

2-36 CEB-FIP, “Model Code for Concrete Structures,” Comité Euro-International du Béton -Fédération Internationale de la Précontrainte, Paris, 1978.

2-37 ACI Committee 209, “Prediction of Creep, Shrinkage, and Temperature Effects in ConcreteStructures,” ACI Report 209R-82 (Reapproved 1986), ACI, in ACI Manual of ConcretePractice, Part 1.

2-38 Case, J., and Chilver A. H., Strength of Materials and Structures, 2nd ed., Edward ArnoldPublishers Ltd., 1971.

2-39 Bruggeling, A. S. G., Structural Concrete: Theory and its Application, A. A. Balkema,Rotterdam, 1991.

2-40 South African Bureau of Standards, “Steel Bars for Concrete Reinforcement,” SABS 920:1985, SABS, Pretoria, 1985.

2-41 ISCOR Ltd., Data Sheet: Availability and Properties, Reinforcing Steel Bars. ISCOR Ltd.,Pretoria, 1993.

2-42 South African Bureau of Standards, “Bending Dimensions of Bars for Concrete Reinforce-ment,” SABS 82:1976, SABS, Pretoria, 1976.

2-43 South African Bureau of Standards, “Detailing of Steel Reinforcement for Concrete,” SABS0144: 1978, SABS, Pretoria, 1978.

2-44 South African Bureau of Standards, “Welded Steel Fabric for Reinforcement of Concrete,”SABS 1024: 1991, SABS, Pretoria, 1991.

2-45 British Standards Institution, “Specification for High Tensile Steel Wire and Strand for thePrestressing of Concrete,” BS 5896:1980, BSI, London, 1980.

2-46 British Standards Institution, “Specification for Hot Rolled and Hot Rolled and ProcessedHigh Tensile Alloy Steel Bars for the Prestressing of Concrete,” BS 4486:1980, BSI, London,1980.

2-47 American Society for Testing and Materials, “Standard Specification for Uncoated Stress-Re-lieved Steel Wire for Prestressed Concrete,” ASTM A421-80, ASTM, Philadelphia, 1980.

2-48 American Society for Testing and Materials, “Specification for Uncoated 7-wire Stress-Re-lieved Steel Strand for Prestressed Concrete,” ASTM A416-85, ASTM, Philadelphia, 1985.

2-49 Magura D. D., Sozen M. A., and Siess C. P., “A Study of Stress Relaxation in PrestressingReinforcement,” PCI Journal, Vol. 9, No. 2, April 1964, pp. 13-57.

2-50 OHBDC, “Ontario Highway Bridge Design Code,” 2nd ed., Ontario Ministry of Transportationand Communications, Toronto, 1983.

2-51 British Standards Institution, “Structural Use of Concrete, Part 1, Code of Practice for Designand Construction,” BS 8110: Part 1: 1985, BSI, London, 1985.

2-52 FIP Commission on Practical Design, FIP Recommendations-Practical Design of Reinforcedand Prestressed Concrete Structures, Thomas Telford Ltd., London, 1984.

REFERENCES 2-39

3 PRESTRESSING SYSTEMS AND PROCEDURES

3.1 INTRODUCTION

A designer must be familiar with the technology and techniques associated with prestressed concretenot only to ensure that the design requirements are properly satisfied, but also to ensure that membersare detailed to satisfy the practical requirements associated with the construction of such members.The importance of the latter consideration is emphasised by the fact that, in the interests ofcompetitive tendering, it is common practice in South Africa to dimension members in such a waythat several prestressing systems can be accommodated.

Almost all the commonly used prestressing systems involve the tensioning of high-strength steeltendons and can be classified either as being pretensioning or post-tensioning systems. Inpretensioning systems the tendons are tensioned before the concrete is placed, while in post-tension-ing systems the tendons are tensioned after the concrete has been placed and has developed sufficientstrength to sustain the induced loads. By the nature of the procedure, pretensioned elements arealways precast and the method usually requires a substantial capital investment in prestressingequipment and stressing beds. Post-tensioned elements can either be precast or cast in situ, and theprestressing operation requires much less equipment and facilities than is the case for pretensioning.Structures and structural elements which cannot feasibly be prefabricated in a precasting yard andtransported to site, such as shells, building slabs, large building frames, large bridge decks andcontinuous bridge decks, can only be prestressed by post-tensioning.

The purpose of this Chapter is to give a brief description of the types of prestressing systems mostcommonly used, including some detail regarding procedure. Because of the large number of systemsavailable, it is not feasible to present specific details of each system here. Complete details may beobtained from suppliers. The descriptions are limited to linear prestressing systems commonly usedbecause linear systems for special applications, circular prestressing systems, electrical prestressingand chemical prestressing fall beyond the scope of the work covered herein. References 3-1 to 3-3may be consulted for information on these specialized prestressing systems and procedures.

3.2 PRETENSIONING SYSTEMS AND PROCEDURES

3.2.1 Basic principle and procedure

The basic principle of pretensioning involves the tensioning of the tendons to a predetermined level,after which the concrete is placed (see Fig. 3-1a). The resulting elongation of the tendons ismaintained at a constant level while the concrete hardens. After the concrete has developed sufficientstrength the tendons are released and, because they are now bonded to the concrete, their shorteningis resisted by the concrete. In this way the concrete is prestressed by the action of bond when thetendons are released (see Fig. 3-1b).

It is important to ensure that the elongation of the tendons is maintained at a constant level whilethe concrete is allowed to harden, and this can be achieved by each of the following two methods(Refs. 3-1, 3-2 and 3-4):

• Pretensioning with individual moulds: According to this method the tendons are anchored directlyto the individual steel moulds in which the concrete is cast. In this case, the moulds must bedesigned and constructed to withstand the additional forces induced by the tendons.

• Pretensioning on stressing beds: When pretensioning on a stressing bed, the tendons are tensionedbetween and subsequently anchored to the rigid vertical steel anchor columns, called uprights,placed at each end of the bed (see Fig. 3-1a). In this manner the tension is maintained in the

INTRODUCTION 3-1

tendons while the concrete is placed and cured. The stressing bed also serves as a casting andcuring bed.

With the exception of railway sleeper production, pretensioning with individual moulds is notcommonly used (Refs. 3-1, 3-2 and 3-4). An apparent advantage of this method is that, in the caseof small products, the individual moulds can be can be moved through the plant on a mass productionline instead of having to move the materials and the process to the moulds, as is the case whenstressing beds are used (Refs. 3-1 and 3-2). Because of its limited use, this method is not discussedhere in any further detail.

Pretensioning on stressing beds is by far the most common method used today, and a typicalarrangement is shown in Fig. 3-1a. This method, often referred to as the long-line or Hoyer method,lends itself to efficient mass production because a number of similar elements can be manufacturedin a single tensioning operation if the bed is made long enough. The length of stressing beds variesbetween 25 m and 200 m, and long beds can be provided with removable intermediate uprights (seeFig. 3-1a) so that shorter tendons can also be tensioned (Refs. 3-2 and 3-5).

Tendons are tensioned by means of hydraulic jacks, and can either be stressed individually orsimultaneously from one end of the stressing bed. Special jacks with a ram stroke of at least 750to 1200 mm must be used if the strands are to be tensioned in a one-step operation (Ref. 3-1). Afterbeing tensioned, wires and strand are usually anchored by means of frictional split-cone wedges.Efficient quick-release grips are also available for this purpose (see Fig. 3-2).

Tendons can be released individually or simultaneously. Tendons are released individually either byflame cutting, sawing or by hydraulic cutters, and a strict cutting sequence which minimises eccentricloading on the concrete, must be adhered to when carrying out this operation. It is also importantto avoid the situation where too many tendons are cut at a single location because this can resultin the failure of the remaining tendons at that location. Tendons must be cut gradually and as closeto the ends of the members as possible to avoid large impact loads from being imparted to theconcrete. These precautions will prevent excessive damage to the concrete at the ends of themembers, and so will ensure that the bond between the concrete and the tendons in this zone is notimpaired.

Stressingjack Original length = L

L – ∆

Removableintermediate

upright Formwork

UprightUpright

(a) Tendons tensioned between uprights

(b) Tendons detensioned (elastic shortening = )�

Stressing bed

Figure 3-1: Pretensioning on a stressing bed (adapted from Ref. 3-5).

3-2 PRESTRESSING SYSTEMS AND PROCEDURES

Tendons are released simultaneously by making use of hydraulic rams. The principle advantages ofthis method are that the prestressing force is gradually transferred to the concrete so that impactloading is avoided, and that the tendons can subsequently be cut between the members withoutfollowing a strict cutting sequence. A disadvantage of this method is that the precast members closeto the releasing end will experience relatively large movements away from the releasing end becauseall the strain is released at that end.

In pretensioned construction, the prestress is transferred to the concrete by bond and, therefore,particular care must be taken to ensure that the bond strength of the concrete is not exceeded. Itcan be shown that the bond stress induced by a tendon will decrease as its diameter decreases, fora given stress in the tendon. For this reason small-diameter wires and strand are used inpretensioning. Wire is often indented or crimped to improve its bond properties while, in the caseof strand, 12.9 mm seven-wire strand is most commonly used.

It is often necessary to deflect some of the tendons to obtain the desired cable profile, particularlyin the case of long span members (see Fig. 3-3). These deflected tendons are often referred to asdraped or harped tendons, and are held in their deflected position by special hold-down devices atthe lower deflection points (also called hold-down or draping points) and by hold-up devices at thehigh positions. Depending on the design requirements, deflected tendons can be provided with eitherone or two hold-down points, as shown in Fig. 3-3. Tendons can only be deflected if the stressingbed has been properly reinforced to sustain the vertical forces imposed by the hold-down devices.

Chuck

Seven-wire strand

Retaining ring

Jaw assembly

Spring

Cap

Body

Figure 3-2: Typical quick release grip (adapted from Ref. 3-2).

Hold-updevice Double hold-down pointSingle hold-down point

Figure 3-3: Pretensioning with deflected tendons (Ref. 3-5).

PRETENSIONING SYSTEMS AND PROCEDURES 3-3

Deflected tendons are usually tensioned straight and then deflected by a hydraulic ram, after whichthe hold-down devices are installed to keep them in their deflected shape. Two methods of deflectingtendons in this way are shown in Figs. 3-4 and 3-5. In the method shown in Fig. 3-4, which iscommonly used for the manufacture of double-tee beams, the deflected tendons are pushed down totheir lower position by means of a temporarily installed hydraulic ram. These tendons aresubsequently held in position by the hold-down pins which bear against the hold-down reactionbeam. After the concrete has developed the required strength, the hold-down pins are removed andthe tendons are released. The method shown in Fig. 3-5 makes use of a centre hole jack to deflectthe tendons, while a strand chuck bearing against the hold-down anchors is used to anchor thetendons in their deflected positions. In this procedure, the strand chuck and the hold-down anchorscannot be recovered (Ref. 3-1).

The actual length of a deflected tendon, measured along its deflected path, can be significantlygreater than the horizontal distance between its ends, particularly in the case of deeper memberssuch as bridge girders. If such a tendon is initially tensioned straight, the subsequent deflectingoperation will increase the tension in the tendon and, hence, the prestressing force. It is importantto consider this effect when determining the initial tension to be applied to the tendons, particularlyif the increase in tension is significant.

Deflected tendons can also be tensioned in their deflected shape, in which case the hold-down devicesmust be capable of permitting the tendons to move longitudinally during the tensioning operationand provision must be made to reduce the friction between the tendons and the hold-up andhold-down devices. The various techniques which have been used to reduce the effects of frictioninclude tensioning the tendons from both ends, using rollers with needle bearings at the hold-upand hold-down points, and vibrating the tendons while they are being tensioned (Ref. 3-1).

As previously mentioned, the precast members will move longitudinally when the tendons arereleased so that it is essential to release the hold-down devices before releasing the tendons.However, when the hold-down devices are released before releasing the tendons, the undesirablesituation arises in which concentrated upward vertical forces are imposed on the beam at the positionsof the hold-down points before any prestress has been transferred to the beam. If these effects arenot properly accounted for in the design or in the releasing procedure (e.g. by partially releasingthe tendons to transfer some prestress to the beams before releasing the hold-down devices) crackscan develop in the top of the beam.

Tendons in beams are deflected to reduce the cable eccentricity in the support regions which, inturn, prevents flexural cracks from developing at the top of the beam in these regions. This objectivecan also be achieved by debonding some of the tendons over a distance at the ends of the beam.Such tendons are referred to as blanketed tendons (see Fig. 3-6).

Hydraulic rampushes pin down

Hold-down pin(removed fromhardened concrete)

Hold-downreaction beam

Double-tee formHold-downdevice

Ratchetadjustment

Figure 3-4: Deflecting tendons in a double-tee beam (Ref. 3-5).


12.5 mm diameter strand

Strand chuck

Strand chuck

Center holehydraulic jack

Hold-downanchors

Deflected strandgroup

Strand chuck

Figure 3-5: Tendon hold-down device for use with a centre hole jack (Ref. 3-1).

Plastic tubeover strandsin bottom

Blanketed strand length(debonded by plastic tubes)

Figure 3-6: Blanketed strands (Ref. 3-5).


Typically, a stressing bed must allow a daily production cycle so that members can be produced inlarge numbers. Under such conditions the use of steel forms or moulds is preferable for the followingreasons: Steel forms are durable and perform well under repeated use; they can be manufactured toa high degree of precision; they are easy to handle when being erected or stripped; they can bemade adjustable to easily accommodate variations in member shape; and they can easily be madestrong enough to allow form vibration (Refs. 3-1 and 3-4). Figure 3-7 shows a steel mould withremovable side forms and a form vibrator for a bridge girder.

The forms are removed after curing the concrete and before releasing the tendons. They should beloosened or stripped in such a way that they do not restrain any longitudinal movement or verticaldeflection of the member which may take place when the tendons are released.

To maintain a daily production cycle, the concrete must develop sufficient strength to allow thetendons to be released within about 16 hours after casting. This high early strength can be obtainedeither by using high early strength cement, by curing the concrete at an elevated temperature, or bycombining these two options (Refs. 3-1 and 3-5). Curing at an elevated temperature is often doneby steam curing, which involves the application of wet heat in the form of live steam under aconfining cover. Steam curing generally commences 2 to 3 hours after casting and continues for 12to 14 hours (Ref. 3-4). Other processes which can also be used to apply heat during curing includeelectrical-resistance heating and heating by circulating hot fluids through pipes contained in theforms or in the stressing bed (Ref. 3-5).

The long-line method is also used for the production of hollow core slab units. In this method, lowslump concrete is extruded around the pretensioned tendons by an extruding machine which travelsalong the stressing bed to form a long hollow core strip. The tendons are released once the concretehas developed sufficient strength, after which the long hollow core strip is sawed into the requiredlengths. References 3-2 and 3-5 can be consulted for further information on this procedure.

Transverse sleepers

Form underties

External vibratorsled and track

Figure 3-7: Steel mould for a bridge I-beam (Ref. 3-5).


3.2.2 Stressing beds

Several different types of stressing bed are available, each with its own range of applicationdepending on the product being produced, the site conditions and the requirements of the productionprocess. The following types are generally used (Ref. 3-1):

• Column type.

• Independent-abutment type.

• Abutment-and-strut type.

• Tendon-deflecting type.

• Portable benches.

It is possible to make a further distinction between stressing beds on the basis of whether they havebeen designed to produce one specific product or to produce many different types of product. Theformer are referred to as fixed beds while the latter are termed universal beds. The brief descriptionof each type of bed given below is a summary of the material presented in Ref. 3-1 on this topic.

Column beds

In a column bed the prestressing force is carried directly by the bed acting as a column. The principleis illustrated by Fig. 3-8, which shows an example of a column-type bed used for manufacturingdouble-tee beams. Clearly this type of bed can only be used as a fixed bed and, in the interests ofeconomy and efficiency, can only accommodate small eccentricities of the prestressing force withrespect to the bed. The primary considerations for the design of a column-type bed are crushing ofthe concrete and buckling of the bed.

Column-type stressing beds are primarily used for producing single-tee beams, double-tee beamsand piles.

Stressing mechanism

Section -A A

Elevation

Stessing end Releasing end

A

A

Metal form liners

Pipes for hot-water curing

Figure 3-8: Column-type stressing bed used for producing double-tee beams (Ref. 3-1).


Independent-abutment beds

The primary components of an independent-abutment bed are two large structurally independentabutments, together with the paved casting surface between the abutments. The abutments can eitherbe embedded in soil, supported on piles or founded on rock (see Fig. 3-9). The manner in whichstability against sliding and overturning of the abutments is provided depends on the foundingconditions as follows:

• Embedded in soil: In this case, stability is exclusively provided by the self weight of the abutmentand passive soil pressure (see Fig. 3-9a).

• Supported on piles: The structural action of the piles provides the required stability in this case(see Fig. 3-9b).

• Founded on sound rock: In this case, stability against sliding can be provided by keying into therock while the stability against overturning can be enhanced by anchoring the abutments to therock with ties or prestressed anchors, as shown in Fig. 3-9c.

Independent-abutment beds are commonly used and can provide a fairly cheap solution in the caseof long beds.

Abutment-and-strut beds

An abutment-and-strut stressing bed basically consists of an abutment at each end joined by aconcrete slab or strut, as shown in Fig. 3-10. It is clear from this figure that the overturning actionof the prestressing force on the abutments is counteracted by the self weight of the abutments, andthat its sliding action is resisted by the slab acting as a strut. It should also be noted that theconcrete hinges provided between the abutments and the slab ensure that no bending is induced inthe slab. In this way, the slab is subjected to an axial load only. Evidently, the abutment-and-strutbed is the type most commonly used for short beds.

Prestressingforce

Prestressingforce

W

Passive soilpressure

Tensionpiling

Compressionpiling

Prestressingforce

Soil overburden

RockConcrete

Steel dowels anchoredin drilled holes orprestressed anchors

(c) Founded on rock

(a) Embedded in soil

(b) Supported on piles

Figure 3-9: Independent abutments (Ref. 3-1).


Tendon-deflecting beds

After tensioning, deflected tendons are held in position by hold-down devices, anchored to the bedat the hold-down points, and by hold-up devices which bear on the bed at the high positions (seeSection 3.2.1). Consequently, the slab component of the bed is subjected to large vertical loads inaddition to the axial load associated with the structural action of an abutment-and-strut bed. Becauseof this loading condition, tendon-deflecting stressing beds, of which a typical layout is given inFig. 3-11, differ from abutment-and-strut beds as follows:

• The use of concrete hinges between the abutments and slab is no longer feasible.

• The slab must be reinforced or prestressed to sustain the combined flexure and thrust to whichit is subjected.

• Tendon-deflecting beds are much more expensive than abutment-and-strut beds of equal capacity.

Portable beds

Portable beds are usually made of structural steel, but are seldom used.

The efficiency of a universal stressing bed, with regard to the waste of pretensioning reinforcementor with regard to the production process, can be improved by providing some means by which itslength can be adjusted. This can be done either by providing the dead-end abutment with removableuprights which can be fitted in several positions (see Fig. 3-12a), or by providing an intermediateabutment with removable uprights (see Fig. 3-12b). When fitted with an intermediate abutment, thepotential efficiency of the stressing bed can be improved even more by designing the bed so thatthe prestressing force can be applied from either end. An alternative solution, which is often simplerand cheaper, is to splice the strand at the required length.

3.2.3 Structural frames

The hardware used for prestressing the products can conveniently be viewed as consisting of thestructural frame together with the hydraulic rams and pumping unit. Whereas rams are used to applythe prestressing force, structural frames are required to transfer this force to the abutments and to

StrutAbutment

PConcrete hinge

Recess to accomodate stressing mechanismPrestressingforce P

Figure 3-10: Abutment-and-strut stressing bed (Ref. 3-1).

P

This portion of bed is subjected tocombined axial load and bending

Vertical forces induced bydeflected tendonsPrestressing

force P

Figure 3-11: Tendon-deflecting stressing bed (Ref. 3-1).


maintain a constant strain in the tendons during the production cycle. They also provide the meansof tensioning and anchoring the tendons and, depending on the design of the system, of releasingthem.

Although the specific details of the structural configuration of frames will differ, the followingstructural steel components can usually be identified: Uprights, pull rods, cross beams and templates.The design of a structural frame for a particular situation must reflect the requirements of theinstallation and should, at least, cover the following considerations (Ref. 3-1):

• The capacity of the bed.

• The range and types of product to be manufactured on the bed. This consideration also coversthe aspect of whether the bed is fixed or universal.

• The method to be used for tensioning the tendons.

• The method to be used for releasing the prestressing force.

3.3 POST-TENSIONING SYSTEMS AND PROCEDURES

3.3.1 Basic principle and procedure

In post-tensioning systems the concrete is first cast and allowed to harden, after which the tendonsare tensioned and anchored. The prestressing force is transferred to the concrete by the anchorageassemblies which bear against the concrete. Obviously, the prestressing operations are carried outonly after the concrete has developed sufficient strength to sustain the induced loads.

The most common method of tensioning the tendons in post-tensioning systems is the mechanicalprestressing by means of hydraulic jacks which react against the concrete. The required capacity ofthe jack depends on the type, size and number of wires, strand or bars in each tendon andconsequently shows a large variation, from as low as 40 kN to as high as 10000 kN. It is essentialthat jacks can easily accommodate the specific technical requirements of the prestressing system,particularly with regard to the details of the anchorages and the tendons. Consequently, speciallydesigned jacks are usually supplied with each particular type of post-tensioning system.

After a tendon has been tensioned, it is anchored to the concrete by a purpose made anchoragesupplied with the system. Although the specific details of anchorages vary from system to system,essentially two types of anchorage are commonly used for anchoring the tendon at its stressing end:

Alternate upright positionsFixed upright

Fixed upright Fixed upright

Stressing end

Stressing end Alternate stressing end

Dead end

Removable upright

Removable uprights

(a) Stressing bed with removable uprights

Intermediate abutment

(b) Stressing bed with intermediate abutment

Figure 3-12: Stressing beds with adjustable length (Ref. 3-1).


• Those which produce a frictional grip on the individual wires, strand or bar by means of wedges.

• Those which anchor the tendon by direct bearing. In the case of wires, this is achieved by meansof cold formed rivet heads or so-called buttonheads while, in the case of a bar, a nut whichthreads onto the end of the bar is used.

The anchorages used in South Africa usually conform to the requirements of BS 4447 (Ref. 3-6).

Tendons can only be post-tensioned if they are not bonded to the concrete at the time of tensioning.This is usually accomplished by placing mortar-tight metal or plastic tubes (also referred to as ductsor sheaths) along the intended profiles of the tendons before the concrete is cast. Thus, ducts areformed in the hardened concrete through which the tendons can be passed and subsequentlytensioned. The tendons can either be pre-placed in the sheath prior to casting, or can be threadedthrough the ducts after the concrete has hardened, depending on the system being used.

After the tendons have been anchored, the completion of the post-tensioning operation depends onwhether the tendons are bonded or unbonded. In the case of bonded tendons, cement grout is injectedinto the duct to fill the void between each tendon and its duct (see Fig. 3-13a). Upon hardening,the grout effectively bonds the tendon to the surrounding concrete and also provides protectionagainst corrosion of the prestressing steel.

In the case of unbonded tendons, the anchoring of the tendons represents the final step in thepost-tensioning operation because the ducts are not subsequently filled with grout. A typical singlestrand unbonded tendon is shown in Fig. 3-13b and consists of a grease coated single seven-wirestrand encased in a plastic sheath. This type of tendon is prefabricated in the factory by extrudingthe plastic sheath over the strand after the strand has been coated with a layer of grease, whichprovides corrosion protection for the strand. Unbonded tendons remain unbonded over their entirelength for the service life of the structure, and it is important to note that they are attached to theconcrete only at their ends by the anchorages. These tendons are primarily used in the post-tensionedslab systems found in building construction because of the considerable economies offered by thistechnique under these circumstances.

StrandGreasePlastic tube

Filled with grout

Corrugated metal or plastic sheath

(a) Bonded multistrand tendon

(b) Unbonded monostrand tendon

Figure 3-13: Bonded and unbonded tendons (Ref. 3-5).

POST-TENSIONING SYSTEMS AND PROCEDURES 3-11

3.3.2 Post-tensioning systems

Several post-tensioning systems are available in South Africa. Although the basic principle used inthese systems are essentially similar, they differ in the type of tendon which is used, the details ofthe anchorages and the method used for tensioning the tendons.

Post-tensioning systems can conveniently be divided into four general categories (Refs. 3-1 and 3-5):Multi-strand systems, multi-wire systems, monostrand systems and high strength bar (includingmulti-bar) systems. The tendon forces that can be achieved with each of these types of system arecompared in Fig. 3-14.

It is usual for designers in South Africa to furnish designs which can reasonably accommodate anumber of the available post-tensioning systems, to encourage competitive tendering. For this reason,a designer must be familiar with the available post-tensioning systems to ensure that members aredetailed to satisfy the practical requirements of these systems in terms of housing the tendons andanchorages, and receiving the jacks used for tensioning the tendons. Since a detailed description ofall the available post-tensioning systems is beyond the scope of this book, a generalised descriptionof the typical features of post-tensioning systems which fall within each of the above-mentionedcategories is given in the following. It is strongly recommended that the details of a particularsystem should be obtained from the pamphlets issued by the company or to consult its repre-sentatives.

1 000

2 000

3 000

4 000

5 000

6 000

7 000

8 000

9 000

10 000

11 000

Tend

on f

orce

, 0.7

(kN

)A

fps

pu

Mon

ostr

and

12.9

mm

and

15.7

mm

Thr

eadb

ar 1

5 m

mto

36

mm

Mul

ti-w

ire

7 m

m

Mul

ti-st

rand

12.

9 m

m Mul

ti-st

rand

15.

7 m

m

1 Wire 1 Strand 1 Strand

19 Strands15 Strands

55 Strands

55 Strands

13 Wires

55 Wires

Aps≡ Area of prestressing steel

fpu ≡ Characteristic tensile strength

Figure 3-14: Ranges of tendon force for various tendon types.


Multi-strand systems

The details of a typical multi-strand system are shown in Fig. 3-15. In these systems each tendonis made up of a number of strands, either 12.9 mm or 15.7 mm seven-wire strand usually beingused. Some systems of this type offer tendons with up to 55 strands.

The most commonly used anchorages make use of the principle of wedge action, with the strandsusually being individually gripped by two- or three-piece conical wedge grips which seat in taperedholes contained in the anchorage block (see Fig. 3-15). When a tendon is tensioned the wedge gripsare inserted in the tapered holes around each strand and, upon release of the jack, the subsequentpull-in of the strand seats the grips which anchor the strand. The effects of the loss of tendonelongation resulting from seating of the anchorage must be accounted for in design. Some systemsinclude special devices for ramming the grips to reduce the anchorage seating loss as well as thescatter of the individual anchorage pull-in values for the strands contained in the tendon.

Tendons can be tensioned from one end only or, in the case of long or appreciably curved tendons,from both ends to reduce friction losses. When tensioned from one end only, a tendon is anchoredat its other end by a dead-end anchorage which can either be cast directly into the concrete or

Tendon

Sheath

Grout injection point

Grey cast ironor fabricated cone

Permanent anchorage block/head

Rubber springs

Pressure plate

Jack foot

Hydraulic ports

Jack piston

Temporary wedge grips

Wedge grips

Steel anchorage block

Figure 3-15: Typical multi-strand post-tensioning system.


Block Guide

DuctSwage holder plate

Swages

Wire ties

(a) Swaged anchorage

Spiral

(b) Splayed strand anchorage

Splayed end

Grout tube

Spiral

Reinforcement grid

Top view

Side view

(c) Looped anchorage

Grout tube

Spiral

Side view

Top view

U-plate

Duct

Tension ring

Figure 3-16: Typical dead-end anchorages for multi-strand post-tensioning systems.


mounted on the surface of the concrete. Typical examples of dead-end anchorages for multi-strandtendons are shown in Fig. 3-16.

The jacks used for the tensioning operations are supplied by the manufacturer and are designed tosuit the tendons and anchorages of the particular system. Most systems use hydraulic centre-holejacks capable of simultaneously tensioning all the strands in a particular tendon. Purpose mademulti-use jaws, which are self releasing after completion of the tensioning operation, are used toattach the strand to the jack (see Fig. 3-15).

Some types of construction procedure, such as the segmental construction of a box-girder bridge,require that tendons be joined to form a continuous tendon even though the member is constructedand post-tensioned in a number of phases or segments. This can be achieved by making use ofcouplers of which a typical example for a multi-strand system is shown in Fig. 3-17. In this figurephase 1 of the construction contains the tendons which have already been tensioned and anchored,while phase 2 contains the coupled tendons which are still to be tensioned.

In South Africa, multi-strand systems are by far the most commonly used systems for bondedconstruction because of their versatility and economy.

Multi-wire systems

The components and basic principle of a typical multi-wire system which anchors the wires by meansof buttonheads are illustrated in Fig. 3-18. These systems use tendons which each consist of a numberof smooth high-strength steel wires. The specific system shown in Fig. 3-18 uses 7 mm wire andcan be supplied with tendons containing up to 55 wires each.

The wires are anchored by buttonheads, formed at their ends, which bear directly onto the anchorhead(see Figs. 3-18a and b). The buttonheads are cold-formed with a special head-forming machine afterthe wires have been threaded through the anchorheads at each end. Tendons can either be completelyprefabricated in a factory or they can be made up on site with field buttonheading equipment.

All wires in a tendon are tensioned simultaneously using a hydraulic jack attached to the anchorhead.After the required elongation has been reached, the anchorhead is locked in the stressed positionwith packing pieces inserted between the anchorhead and the bearing plate (see Fig. 3-18d). It isessential that the length of the tendon as well as its elongation be estimated as accurately as possibleto ensure that the tendon elongation at anchoring corresponds to the desired prestressing force. Thinshims are usually available, in addition to the packing pieces, for making fine adjustments to theanchoring force to accommodate discrepancies between the estimated and measured elongations. Oneof the advantages of this type of anchorage is that the anchorage seating loss is negligible. Deadend anchorages and couplers are available for these systems.

Phase 1Phase 2

Figure 3-17: Typical tendon coupler for multi-strand post-tensioning systems.


The hydraulic jacks used for tensioning the tendons of these systems are supplied with the hardwarerequired for attaching them to the anchorhead. The particular system shown in Fig. 3-18 uses acentre-hole hydraulic jack to apply the tensioning force to the anchorhead by means of a pull rodwhich threads into a stressing sleeve which, in turn, threads onto the anchorhead. The jack bearson a stressing bridge which transfers the jack reaction to the concrete (see Figs. 3-18c and e).

It is important to note that multi-wire systems are usually not used in South Africa any more becauseof their higher cost. These systems are only used in situations where multi-strand systems or anyof the other available systems cannot offer a satisfactory solution.

Monostrand systems

The distinguishing feature of monostrand post-tensioning systems is the fact that each tendoncomprises a single seven-wire strand, with 12.9 mm and 15.7 mm being the most commonly usedsizes. These systems are usually unbonded and a typical tendon is shown in Fig. 3-13b.

The details of a typical monostrand system are shown in Fig. 3-19 together with the constructionsequence for a post-tensioned slab. As in the case of multi-strand systems, the anchorages used inmonostrand systems make use of two- or three-piece conical wedge grips which seat in tapered holes

Figure 3-18: Typical multi-wire post-tensioning system (Ref. 3-5).

7 mmdiameterwire

Cold-formed buttonhead

(a) Buttonhead anchor

(b) Multi-wire tendon

Buttonhead

Threadedanchorhead

Stressing bridgeStressing sleeve

Pull rod

Bearing plate

(c) Tendon tensioned by pull rod

(d) Locking the anchorhead into position

Packingpieces

Pull rod

Cylinder

Stressing bridge

SleeveBearing plate

(e) Jack details


in the anchorage body to anchor the strand (see Figs. 3-19a and c). However, the anchorages aredesigned to anchor only one strand and are therefore small. Plastic elements, referred to as grommets,are usually supplied with the anchorages for fastening them to the forms and for forming tensioningvoids, or pockets, in the concrete (see Fig. 3-19).

(1) Placement of monostrand andanchorage nailed to formwork

(2) Remove grommet

Tendon profile support

(3) Place wedge grips

(4) Tension and anchor strand

Hydraulic pump

(c) Construction sequence for post-tensioned slab

(5) Cut excess strand,cap end and fill inhole with weatherresistant grout

Figure 3-19: Typical monostrand post-tensioning system (Ref. 3-5).

Plastic former(grommet)

Wedge grips

Anchorage body

(a) Anchorage details

(b) Monostrand system for slab post-tensioning

Dead-end anchorage Intermediate anchorage Stressing anchorage

FormCorrosion protection cap

Corrosion protection sleeve

Corrosion protection sleeveGrommet

Monostrand


The various steps involved in installing and tensioning a monostrand tendon is summarised inFig. 3-19, which is self explanatory. It is to be noted that the hydraulic jacks used for tensioningthe tendons are small and light so that the tensioning equipment can usually be operated by oneperson.

Couplers are not required for monostrand systems because tendons can be supplied to almost anyrequired length, while successive sections of the same continuous tendon can be tensioned separatelyby using an open throat jack to stress at intermediate points between partial slabs. Dead-endanchorages are normally installed in the factory and, in the case of wedge anchorages, the wedgesare hydraulically seated in the factory.

Unbonded monostrand systems are particularly well suited for the post-tensioning of thin slabs andnarrow members because the small tendon diameter allows optimum eccentricities and because thecompact anchorages can be accommodated by such thin members. These factors, together with theelimination of the grouting operation as well as the simplicity and efficiency of the tensioningoperation all offer considerable economies. Hence, practically all cast-in-place prestressed slabs, flatplates and flat slabs, encountered in building structures in South Africa, are post-tensioned byunbonded monostrand systems.

Bar systems

Bar systems are characterized by the fact that high strength bar is used for the tendons. The barcan be supplied in most of the standard sizes (see Table 2-11) either as smooth bar or as threadbar,which has deformations rolled on over the entire length of the bar to form a continuous screw thread.Although single bar tendons are most commonly used, multiple bar tendons are possible. Bar systemsusually use bonded tendons, but unbonded tendons provided with a corrosion protection system areavailable.

A typical single bar post-tensioning system is shown in Fig. 3-20. The bar is anchored by a nutwhich threads onto the end of the bar and seats into either a bell-shaped or plate anchorage. The

Figure 3-20: Typical threadbar post-tensioning system.

(a) Bell and plate anchorages

Grout tube

Sheathing

Grout sleeve

Bell anchorage

(b) Tendon assembly at stressing end

End shutter Removable pocket former

Removable plastic nut

Ratchet

(c) Jack Details


seat is spherical to prevent the development of significant bending moments in the bar when theaxis of the bar is not perpendicular to the anchor plate (Ref. 3-1). The prime advantage of the bellanchorage is that the anchor ring contains the splitting forces induced by the anchorage after thetendon is tensioned. Both anchorage types shown in Fig. 3-20a exhibit negligible anchorage seatinglosses when properly installed.

When smooth bar is used, threads must be provided at the ends of the bar for the anchor nuts. Itis important to note, in this regard, that the elongation of the tendon must be estimated as accuratelyas possible to ensure that the threaded length is such that the nut is turned to its end at anchorage,after tensioning, so that the full strength of the bar can be developed (Ref. 3-2). This difficulty, ofcourse, does not exist for threadbar, in which case the thread runs over the complete length of thebar. It should be noted that wedge anchorages have been developed for use with smooth bar tendons.

A centre-hole hydraulic jack is commonly used for the tensioning operation. In some systems, thejack is provided with a ratchet which is used to tighten the anchor nut against the anchor platewhile the bar is being tensioned (see Fig. 3-20c). Any special hardware required for attaching thejack to the tendon, such as pulling bars and pulling nuts, is supplied with the jack.

The length in which the bar can be supplied is often limited by production practice as well astransportation and storage requirements. However, couplers can be used to provide tendons of almostany length. These couplers, of which an example for a threadbar system is shown in Fig. 3-21a, areusually of the sleeve type which simply screw onto each of the bars to be spliced. The couplers canalso be used to extend a previously tensioned bar, a situation which, for example, arises in segmentaland phase construction. When bars need to be coupled, the use of threadbar is particularly attractivebecause the continuous thread makes it possible to cut the bar to any required length, and alsobecause the coupling hardware and operation are simple and relatively low in cost. It is importantto note that when these sleeve couplers are used to splice bars, sufficient space must be providedin the concrete surrounding the couplers to allow the movement which takes place during tensioning.An example of a dead-end anchor for a threadbar system is shown in Fig. 3-21b.

Grout tube

Sheathing

Bell anchorage

(b) Bell anchorage

Anchor nut

Figure 3-21: Typical dead-end anchorage and coupler for a threadbar post-tensioning system.

(a) Coupler


Choice of system

Since the basic principle of the available post-tensioning systems is essentially the same, thefundamental difference between the various systems lies in the type of tendon which is used, thedetails of the anchorages and the method used for tensioning the tendons. Consequently the selectionof a specific system for a particular application will depend on how well these features of the systemwill meet the requirements of the job. It often happens that a number of systems will work equallywell for a particular application. Hence, the choice is often an economic one, that is, which systemwill be the cheapest in terms of the cost of the materials, equipment and the labour required toinstall, tension and grout the tendons.

3.3.3 Post-tensioning operations

After the concrete has reached the specified strength the tendons can be tensioned but, as a generalrule, the full prestressing force should be applied as late as is practically feasible to minimise theeffects of shrinkage and creep of the concrete (Ref. 3-3). However, certain conditions which inducesignificant tensile stresses in the concrete at an early age may prevail, particularly in larger members.Examples of such conditions are the development and subsequent decrease of the heat of hydration,temperature differentials induced by variations in the external temperature, and shrinkage of theconcrete. These tensile stresses can lead to the development of visible cracks. A possible remedyfor this problem is to apply a moderate prestress at a very early age and then to apply the fullprestress at a later age when the specified concrete strength has developed. It is important to ensurethat the compressive stress permitted in the concrete when it is initially prestressed, is based on itsstrength at the time of tensioning (Ref. 3-3).

It is essential that all side forms and other obstructions which may restrain the deflections andshortening of the member, induced by prestressing, be removed or loosened before tensioning thetendons. It is particularly important in this regard to ensure that, if present, sliding bearings arecleaned and that any devices used for temporarily fixing the bearings are released prior to tensioning.

When a member contains a number of tendons, the sequence in which they are tensioned must ensurethat severe eccentric loading is avoided at all stages of the tensioning operation. Sometimes it maybe necessary to tension some of the tendons in two steps to meet this requirement (Ref. 3-2). Thefollowing aspects regarding the sequence of tensioning should also be noted (Ref. 3-3):

• Tensioning should commence with tendons which are not located close to the edge of the section.

• When a member is to be prestressed transversely as well as longitudinally, the transverse tendonsshould be tensioned first.

• Before a tendon, which does not extend over the full length of a member is tensioned, the concretesurrounding its internal anchorage must first be subjected to compression. This is achieved byfirst tensioning a sufficient number of tendons which extend over the entire length of the member.

The tensioning force applied to a tendon is monitored in two ways: Firstly, by measuring thehydraulic pressure applied to the jack using a pressure gauge and, secondly, by measuring theelongation of the tendon. The measured elongations are used to check the pressure gauge readingsand to give an indication of the average force over the length of the tendon. The pressure gauge,on the other hand, provides the tendon force at the anchorage.

The theoretical hydraulic pressure required for a given applied force (as obtained by dividing theforce by the ram area) will always be less than the measured pressure because of the internal frictionof the jack. For this reason, and to ensure that the measurements of pressure taken from the gaugeare accurately translated into force, the stressing equipment must be calibrated. Jack calibration isusually accomplished by jacking against a laboratory calibrated load cell or proving ring, placed inthe load path of the jack, to produce a calibration curve of hydraulic pressure versus applied force.The stressing equipment should be calibrated to an accuracy of at least ± 2% (Ref. 3-7) before


tensioning any of the tendons. When a large number of tendons are to be tensioned, the calibrationof the stressing equipment should, at least, be repeated after completion of the tensioning operationswhile, on very large projects, the calibration should be repeated at regular intervals. It should benoted that some post-tensioning systems supply dynamometers which work in series with the jack,so that the tendon force can be directly monitored during the tensioning operation.

It is difficult to establish the zero point for the measurement of tendon elongation because of slack,which has to be taken up before a tension is induced in the tendon. However, this problem can besolved by making use of the fact that the material behaviour of both the concrete and steel remainslinear elastic at the stress levels induced by the prestress (Ref. 3-3). The normal procedure is tostress the tendon to between 5 and 10% of the full tensioning force and to use this position as thestarting point for measuring the elongation (Ref. 3-7). A load-elongation diagram can be constructedby plotting measurements of load against elongation, taken at regular load increments as tensioningproceeds. Because the material behaviour is essentially linear elastic, these points should plot as analmost straight line, so that the zero point can be obtained by extrapolating the load-elongationdiagram to the value of zero load (see Fig. 3-22). An alternative approach, which is often followedin practice, is simply to consider the calculated and measured increments of elongation beyond theinitially applied prestress, used as the starting point for measuring elongation.

Tendon elongations recorded during the tensioning operation provide a check on the appliedtensioning force by plotting them as a load-elongation diagram, which can be directly compared tothe calculated diagram. It is generally required that the measured tendon elongation should agreewith the calculated value to within ± 5% (Refs. 3-5 and 3-8). Specifically, Ref. 3-7 requires thatthe measured elongation of an individual tendon must agree with the calculated value to within± 6%, while the average difference between the measured and calculated elongations for all thetendons in a member must be less than ± 3%. Note that the calculated tendon elongation shouldinclude a correction which accounts for the elongation of the length of the tendon that extends fromthe anchor to the jack grip position.

Any of the following causes will individually, or in combination, result in the measured tendonelongation being larger than the calculated value (Ref. 3-3):

• The assessment of the effects of tendon friction is too conservative.

• The value assumed for the modulus of elasticity of the prestressing steel for the calculation ofelongation is larger than the actual value.

• The actual steel cross-sectional areas are smaller than assumed for the calculations.

P1

P2

∆1 ∆2 ∆3

P3

Tens

ioni

ngfo

rce

P

Measured points

Tendon elongation

Zero point formeasuring elongation

Initial starting point formeasuring elongation

Extrapolation of load-elongationdiagram to = 0P

Figure 3-22: Determining the zero point for measuring tendon elongation.


• An anchorage or the concrete surrounding the anchorage has failed. This event is usuallycharacterised by an increase in elongation without the associated increase in tensioning force.

• A wire or strand in the tendon has fractured. This event is identified by a loud cracking soundand a sudden drop in the applied tendon force.

The calculated tendon elongation may not be achieved during tensioning for the following reasons(Ref. 3-3):

• The actual tendon friction is higher than assumed in the calculations because of, for example,rust or the ingress of grout into the duct during casting of the concrete. In extreme cases of groutingress, the tendon can actually be jammed in the duct so that only the portion of the tendonwhich extends from the obstruction to the point of tensioning is stressed. In this event, thetensioning force should not simply be increased to obtain the required elongation because of thedanger of overstressing the tendon in the tensioned portion, while the level of the tensioningforce in the remainder of the tendon is essentially unknown. It is far better practice to overcomethe obstruction by repetitively releasing and re-applying the tensioning force. During such anoperation, care must be taken to ensure that the permissible tendon stress is not exceeded. Insome cases, excessive friction can be overcome by injecting water-soluble oil into the duct. Ifthis step is taken, the oil must be removed after tensioning by flushing the duct with water.

• The value assumed for the modulus of elasticity of the prestressing steel for the calculations issmaller than the actual value.

• The actual steel cross-sectional areas are larger than assumed for the calculations.

It is generally recommended that tendon elongation be measured to an accuracy of � 2%. Reference3-7 requires that the elongation be measured to an accuracy of � 2% or 2 mm, whichever is themost accurate.

After the prescribed tensioning force has been reached, the tendons are anchored when the hydraulicpressure on the jack is released. If wedge-type anchorages are being used, a loss of elongation takesplace because of the pull-in of the strand when the wedge grips are seated. It is important to notethat when the tensioning force is transferred from the jack to the anchorage, a further loss ofelongation takes place because of the resulting deformation of the anchorage components. Themagnitude of the anchorage deformation, which can be appreciable for some systems, appears to bedependent not only on the anchorage type, but also on the quality of workmanship (see Refs. 3-1and 3-3).

The total loss of elongation which takes place when a tendon is anchored is often referred to as theanchorage set (also pull-in or anchorage seating), and must be considered in design. For strandanchored by wedge grips, the anchorage set is of the order of 6 mm, while the anchorages commonlyused for threadbar systems do not show appreciable anchorage set if properly installed. Clearly,post-tensioning systems using anchorages which yield a significant anchorage set are not suitablefor use with short tendons. Note that anchorage set can be compensated for by installing shimsbehind the anchorhead.

Anchorage set must be recorded in the field to ensure that the values being obtained agree withthose assumed for design. Reference 3-7 specifies that the anchorage set should be measured to anaccuracy of � 2 mm and requires that the measured values must agree with the values assumed fordesign to within � 2 mm.

When a long or appreciably curved tendon is tensioned from one end only, the effects of frictionwill lead to a considerable loss of force along its length. This loss of force can be reduced bytensioning the tendon from both ends. An additional, or alternative, procedure which can be followedis to retension the tendon after it has been temporarily overstressed (see Refs. 3-2, 3-3 and 3-5).When overstressing a tendon, the temporary tension thus applied must not exceed 80% of its specified


characteristic tensile strength and, after anchoring, the tension in the tendon must not exceed 70%of its characteristic tensile strength (Refs. 3-9 and 3-10).

It is essential that the tensioning operation be supervised and carried out by experienced personnelwho are familiar with the post-tensioning system, equipment and the procedures to be used. It isalso strongly recommended that the tensioning equipment be power driven, and that provision bemade for an alternative power source which can be used in the case of a breakdown. The tensioningequipment must be capable of applying the load in a controlled manner without imposing significantsecondary stresses on the tendon, anchorage or the concrete. The schedule which gives the sequencein which the tendons are to be tensioned, as well as those which give the tensioning forces andcorresponding anticipated pressure gauge measurements, the calculated elongations, and the antici-pated anchorage set for each tendon must be available before commencing the tensioning operation(Refs. 3-7 and 3-8).

A considerable amount of strain energy is stored in the tendon during the tensioning operation andfailure of the tendon, jack or an anchorage can lead to a sudden uncontrolled release of this energy.Such an occurrence may cause serious injury to any person standing in line with the jack or theanchorage at the opposite end of the tendon. It is therefore important to exercise extreme cautionwhen tensioning the tendons by taking a number of safety precautions, such as:

• Making sure that personnel are kept away from the back of the tensioning equipment and theanchorage at the opposite end of the tendon.

• Ensuring that the tensioning equipment is properly maintained and assembled, and ensuring thatit is not misused.

• Immediately stopping the tensioning operation in the event of an unusual occurrence such as, forexample, a sharp noise being heard or a bearing plate receding into the concrete.

The list given above is by no means exhaustive, and further information regarding safety precautionsin post-tensioning can be obtained from Refs. 3-11 and 3-12.

3.3.4 Ducting for bonded construction

In post-tensioned construction, the tendons are tensioned after the concrete has been cast and afterit has developed the specified strength. This is accomplished by placing ducts along the specifiedtendon profiles to form conduits in the hardened concrete through which the tendons can be passedand subsequently tensioned. Therefore, ducts must satisfy the following requirements (Refs. 3-7, 3-8and 3-13):

• They must be of a type that does not permit the ingress of cement paste during casting.

• They must be flexible enough to be placed on the required profile without buckling.

• They must be rigid enough to maintain the profile on which they are placed during casting.

• They must be strong enough to resist damage during handling and casting, and to maintain theirshape under the weight of the fresh concrete.

In bonded construction, the shape of the ducts must be of a type which will enhance the transferof bond from the grout to the surrounding concrete, and the material used for the duct must nothave any adverse chemical reaction with the concrete, tendons or grout. In South Africa, the ductsmost commonly used are made of spirally wound steel strip which forms flexible corrugatedsheathing (see Fig. 3-13a). The corrugations are required for bond strength, while the thickness ofthe strip steel generally ranges between 0.2 mm, for small tendons, and 0.6 mm, for large tendons(Ref. 3-13). Although not commonly used in South Africa, polyethylene and polypropylene tubinghas successfully been used in Europe for many years. A primary advantage of the polyethylene andpolypropylene ducts is that they offer improved corrosion protection when compared to ducts made


of strip steel. The internal diameter of the duct is generally required to be large enough to yield aduct area of at least 2.5 times the area of the prestressing steel.

The ducts must be installed to the specified alignment by securely tying them to closely spacedsuppports. If the tendons are preplaced in the ducts prior to casting, the supports and ties must beable to support the tendon weight. When only the duct is placed before casting, with the tendonbeing installed prior to tensioning, the supports and ties must adequately resist the bouyancy forceswhich arise during casting.

It is important that ducts are installed on smooth curves without kinks to minimize the frictionlosses which arise during tensioning. This requirement can be satisfied by spacing the duct supportsclosely enough to prevent the ducts from sagging between them. A further consideration, whichrequires the use of closely spaced supports, is that the ducts must not be displaced during casting.It is difficult to give a general rule for the maximum spacing of supports because it depends onwhether or not the tendons are preplaced prior to casting as well as on the rigidity of the duct, thetype and size of the duct, and the tendon profile. Reference 3-7 recommends that a spacing ofbetween 1.0 m and 1.5 m should generally not be exceeded.

It is essential that each duct be fixed to its anchorage in such a way that the tendon axis isperpendicular to the bearing surface of the anchorage. Each anchorage must also be installed in sucha way that it will not be displaced during casting, and steps must be taken to ensure that the bearingplate is uniformly supported over its complete surface by the concrete onto which it bears.

The ducts must be carefully inspected after installation to ensure that they have been securely tiedinto position, that they have not been damaged during installation, and that grout cannot leak intothem during casting. Particular care should be taken to ensure that all joints in, and connections tothe ducts are completely grout-tight. The importance of this inspection is underscored by the factthat the cost associated with clearing the areas affected by the ingress of grout into the ducts oftenexceeds the cost of properly sealing the ducting before casting.

When casting the concrete, care must taken to ensure that the ducts are not damaged by, for example,internal vibrators. Such damage can lead to the ingress of grout into the duct or to a reduction ofthe duct diameter to such an extent that the tendons cannot be inserted. Areas congested byreinforcement and other embedded materials are particularly prone to this problem. Since voids inthe concrete behind the anchorage bearing plates, or insufficient concrete strength can lead to failureof the concrete in these regions during tensioning, the concrete at the anchorages must be properlyvibrated to ensure maximum density, free of voids.

When a tendon is grouted, air tends to be trapped at positions where there is a sudden change inthe cross-section of the duct and, if it is draped, at the high points of the duct (see Fig. 3-23).Water and bleed water which can accumulate in the resulting air pockets can lead to corrosion ofthe prestressing steel and, in so doing, seriously impair the durability of the structure. This situationcan be prevented by providing vents, through which trapped air and water can escape, at thefollowing positions:

• At the high points of the duct if the drape of the tendon, measured from the highest point to thelowest point exceeds 500 mm (see Fig. 3-24c) (Ref. 3-13). Reference 3-8 suggests that in caseswhere the tendon curvature is small, such as in continuous slabs, high point vents are not required.

• At significant changes in the duct cross-section, such as at anchorages and at couplers where theduct is enlarged in the region of the anchorage (see Figs. 3-24a and b).

The recommended minimum size of the inner diameter of the vent tubes ranges between 20 mm and25 mm (Refs. 3-7, 3-13 and 3-14). It is also recommended that the vent tubes should extend adistance of at least 500 mm above the surface of the concrete (Ref. 3-7).


Anchorages are provided with grout holes or grout tubes which serve as inlets or outlets for thegrout. In the case of long tendons, provision should be made for intermediate inlets which can beused if difficulties or blockages should develop at the main inlet (Ref. 3-13).

The ducts cannot be grouted if the temperature of the concrete falls below about 5ºC. If freezingtemperatures are likely to occur when the tendons are to be tensioned, the situation can arise whereducts, containing tensioned tendons, are left ungrouted for a considerable period of time. It isessential to ensure that water should not be allowed to collect in the ducts under such conditions,and drain tubes should be installed at all the low points of the ducts to ensure that they are properlydrained (Refs. 3-1 and 3-5).

3.3.5 Grouting

In bonded construction the ducts containing the tendons are filled with cement grout as soon aspossible after tensioning. The primary reasons for grouting the ducts are to provide corrosionprotection for the prestressing steel and to provide a means of bonding the prestressing steel to thesurrounding concrete. If these objectives are borne in mind, it should be clear that a suitable groutmust comply with the following requirements:

• Since the grout must flow over long distances in a fairly confined space, it must maintain itsfluidity during the grouting operation to ensure that all voids in the duct are completely filled.

1

1

Section 1-1Tendon profile near high point

Air pocket

Grout

Figure 3-23: Air pocket at tendon high point resulting from inadequate venting (Ref. 3-5).

(a) Ref. 3-13 (b) Ref. 3-13

(c) Ref. 3-5

Grout outlet Vent

Duct

Grout inlet

Figure 3-24: Placing of vents.


• The amount of sedimentation as well as the resulting contraction of the grout must be kept assmall as possible. It is generally recommended that the contraction be limited to 2% (Refs. 3-3and 3-13).

• The grout should exhibit minimum bleeding and, after setting, all bleed water must bere-absorbed. The amount of bleed water should not exceed 2% by volume 3 hours after the grouthas been mixed, and it should be completely re-absorbed after 24 hours (Refs. 3-7 and 3-13).

• The hardened grout must have adequate bond and shear strength. This requirement is deemed tobe satisfied if the compressive strength of 100 mm cubes, tested at 20ºC, exceeds 20 MPa after7 days and 30 MPa after 28 days (see Refs. 3-7 and 3-13).

• The grout should not contain excessive quantities of chlorides, nitrates, sulfides, or otheringredients harmful to the grout itself or to the prestressing steel (Refs. 3-5, 3-7 and 3-13).

• In freezing climates it is essential that steps be taken to ensure that the grout is resistant to frostso that it does not lose strength nor fracture as a result of freezing. Further information on thisaspect may be obtained from Refs. 3-3 and 3-13.

Cement and water are the primary constituents of grout, and admixtures are sometimes used toimprove its properties. Fine aggregate may be added under special circumstances such as, forexample, when grouting large diameter ducts.

Ordinary Portland cement is most commonly used for grout but, if conditions require their use, othertypes of cement can be used provided their suitability has been established by tests. It is importantthat the cement must be fresh and that it should not contain lumps or any other indications ofhydration (Ref. 3-8) and, for this reason, Ref. 3-7 recommends that the cement should not be olderthan one month. The water used for grout must not contain deleterious quantities of substancesharmful to the grout or to the prestressing steel and, therefore, should not contain more than 500mg of chloride ions per litre (Refs. 3-7 and 3-13). The minimum value of the water-cement ratio isusually controlled by the required fluidity of the grout while the maximum value is usually governedby the fact that the grout should not exhibit excessive bleeding. The water-cement ratio must bekept as low as possible within a range of 0.38 to 0.43, bearing in mind the above considerations.

Aggregates are usually not added to the mix and are only used under special circumstances, suchas when grouting ducts which contain large cavities. Fine aggregate can consist of siliceous granules,finely ground limestone, trass or very fine sand, all of which must be fine enough to pass througha 0.600 mm sieve (Ref. 3-7). It is recommended that the aggregate content should not exceed 30%of the weight of the cement (Ref. 3-7).

Admixtures are used only when the desired properties of the grout cannot be obtained by a suitablemixture of cement and water. Therefore, the objectives of adding admixtures are to improve theproperties of the grout such as improved fluidity, reduced bleeding and retarded setting time, andto impart other properties to the grout such as expansion, to compensate for contraction, and airentrainment, to improve freeze resistance. Only well-proven admixtures, which do not containchemical substances in quantities which are liable to damage the grout or the prestressing steel,should be used (Refs. 3-7 and 3-13). When using an expanding agent, the unrestrained expansionof the grout should be limited to 5% (Ref. 3-13).

The grouting equipment usually consists of a mixer, a holding reservoir and a pump together withall the connection hoses and valves. Mechanical mixers are used to consistently produce ahomogeneous and stable grout which is free of lumps. The two types of mixers used are: vanemixers, having a speed of approximately 1000 rev/min, and high speed compulsory mixers, with aspeed of about 1500 rev/min (Ref. 3-13). After mixing, the grout is usually passed through a screenwith openings not exceeding 5 mm into a holding reservoir equipped with an agitator, whichmaintains the colloidal condition of the grout during the grouting operation. The capacity of themixer and the reservoir must be sufficient to allow the duct to be filled without interruption at thespecified speed. The grout is delivered at the duct, from the reservoir, by a pump capable ofproviding a steady flow of grout. The pump must be able to maintain a pressure of at least 1.0 MPa


on a completely filled duct, and it should be equipped with a pressure gauge as well as a safetydevice which will prevent pressures exceeding 2.0 MPa from developing. The system should alsobe capable of recirculating the grout if pumping is interrupted. Grouting hoses and all hoseconnections must be airtight and must be of a size and type which will prevent the build-up ofpressure during grouting (Refs. 3-7 and 3-13). Further information on grouting equipment may beobtained from Refs. 3-3, 3-7, 3-13 and 3-14.

Equipment required for providing compressed air and for flushing grout out a duct, if the groutingoperation is interrupted for some reason (e.g. if a blockage is encountered or if a breakdown of thegrouting equipment occurs during grouting), must be at hand. It is extremely important that allgrouting equipment must be in good working order.

The ducts should be grouted within 7 days after tensioning, and should the grouting operation bedelayed for a longer period, specific steps must be taken to protect the prestressing steel fromcorrosion. Before grouting, the ducts should be checked for obstructions by water injection and allexcess water should afterwards be displaced by the grout.

The grouting operation begins by adding the constituent materials of the grout to the mixer. Thesequence in which this takes place depends on the type of mixer, and the following is recommendedby Ref. 3-13:

• For vane mixers: all the water, approximately two thirds of the cement, the admixture (if used),the remaining cement.

• For high speed compulsory mixers: Water, cement, admixture (if used).

The mixing time of the grout also depends on the type of mixer being used and should not exceed4 minutes for vane mixers or 2 minutes for high speed compulsory mixers (Ref. 3-13).

The grout is usually injected at the lowest inlet and, in the initial stages of the grouting operation,is wasted at the vents and at the outlet. Grouting should proceed continuously in one direction ata rate which is slow enough to prevent segregation of the grout. Reference 3-13 suggests that a rateof between 5 and 15 m per minute should be used. The first vent tube after the inlet is closed oncethe grout flowing out of it does not contain visible slugs of water or air, and is of the sameconsistency as at the inlet. The remaining vent tubes and the outlet are closed in sequence in thesame manner, while the duct progressively fills. After the outlet has been closed, the final groutingpressure or a pressure of at least 0.5 MPa, whichever is the greater, must be maintained on the groutfor at least 5 minutes before closing the inlet (Ref. 3-7). If an expanding agent is included in thegrout mix, the vent tubes must be re-opened shortly after grouting to allow bleed water to escape,after which the tubes must be closed. In all cases, the vent tubes should be opened after the grouthas hardened and inspected to establish the extent of the grout fill. If such an inspection revealsthe presence of voids, the problem can be remedied by topping up the vent tubes with grout or byundertaking a regrouting operation, depending on the specific circumstances (see Refs. 3-13 and3-14).

If a blockage is encountered in a duct, the grouting operation must immediately be stopped to preventa large pressure, which can damage the structure, from developing in the duct. The grout shouldalso immediately be flushed out of the duct by injecting water, against the direction of grouting,into the nearest vent tube. After removing the obstruction which caused the blockage, the groutingoperation can be restarted. It should be noted that excessive pressures which develop during groutingcan lead to water segregation and can also cause cracking or damage to the structural element, andshould be avoided.

The grout should be discarded after 30 minutes unless a retarder is used. Grouting should not beundertaken if the ambient temperature drops below 5ºC, and care should be taken to ensure that theducts are free from ice or frost before grouting commences in cold weather (Ref. 3-7).


The fluidity of the grout should be tested immediately after mixing and at regular intervals duringthe grouting operation. A flow cone test is normally used for testing fluidity on site (Ref. 3-7).Sufficient quantities of grout must be taken during grouting, usually at each vent, to enable testingof the other properties of the grout such as bleeding, strength and volume change. Appropriatetesting procedures for determining these properties are described in Refs. 3-3, 3-7, 3-8, 3-13 and3-14.

It is extremely important to ensure that the grouting operation is carried out properly because thedurability of a bonded post-tensioned structure depends on how sucessfully this operation has beencompleted. It is, therefore, strongly recommended that this highly specialised and critical operationbe carried out only by appropriately trained and experienced personnel.

3.4 PRETENSIONING VERSUS POST-TENSIONING

Pretensioning and post-tensioning systems each have specific theoretical and practical advantagesand disadvantages. These must be considered in conjunction with the particular technical require-ments and the prevailing economic considerations for a specific job, before a decision can be maderegarding which method of prestressing is to be used. It is useful to remember, in this regard, thatalthough pretensioning is generally perceived as being limited to permanent precasting factories, itcan be economically feasible for the contractor to set up a temporary prestensioning yard at, orclose to, the site on projects where a large number of pretensioned elements are to be used. On theother hand, post-tensioned members, which are usually constructed and tensioned in situ, can bemanufactured in a precasting plant and subsequently transported to site (e.g. precast segmentalpost-tensioned bridges).

Some of the differences between pretensioning and post-tensioning, which should also be consideredwhen comparing the two methods of prestressing a member, are listed in the following:

• The capital investment in the equipment and facilities required for post-tensioning is considerablyless than for the equipment and industrial layout needed for pretensioning.

• The efficiency of pretensioning, measured in terms of cost per unit of tensioning load, is greaterthan that of post-tensioning because of the additional material and labour costs associated withthe ducts, anchorages and grouting required for post-tensioning.

• Structural elements can be prestressed in situ only by post-tensioning.

• It is impractical to post-tension very short elements because any anchor set will lead to a largepercentage loss of tensioning force, and also because the small elongation of the short tendonrequires a high accuracy of measurement. Clearly these difficulties are non-existent if thelong-line method of pretensioning is used.

• Long and very large members may be more conveniently and economically cast in place andpost-tensioned, because the cost of transporting and handling large pretensioned members, whichare cast off site, can become excessive. When the precasting plant is situated too far away fromthe site, the transportation cost can also become prohibitive.

• The tendons in post-tensioned elements can easily be placed on smooth curves along the desiredprofile. Although pretensioned tendons can be deflected, the procedure remains costly and limited.In the case of continuous elements, such as continuous bridge beams, pretensioning becomesimpractical.

• The loss of prestressing force associated with tendon friction during tensioning is significant inpost-tensioned tendons and must be considered in design as well as during construction.


3.5 REFERENCES

3-1 Libby, J. R., Modern Prestressed Concrete: Design Principles and Construction Methods, 4thed., Van Nostrand Reinhold, New York, 1990.


3-3 Leonhardt, F., Prestressed Concrete Design and Construction, English translation, WilhelmErnst & Sohn, Berlin, 1964.

3-4 Khachaturian, N., Gurfinkel, G., Prestressed Concrete, McGraw-Hill Book Company, NewYork, 1969.


3-6 British Standards Institution, “British Standard Specification for The Performance of Prestress-ing Anchorages for Post-Tensioned Construction,” BS 4447:1973 (1990), BSI, London, 1973,reaffirmed 1990.

3-7 Committee of State Road Authorities, “Standard Specifications for Road and Bridge Works,”1st ed., The CSRA Secretariat: Division of Roads and Transport Technology, Council forScientific and Industrial Reseach, Pretoria, 1987.

3-8 Post-Tensioning Manual, Post-Tensioning Institute, Glenview, Illinois, 1976.

3-9 South African Bureau of Standards, “The Structural Use of Concrete,” SABS 0100: 1992,Part 1, SABS, Pretoria, 1992.


3-11 FIP Commission on Practical Construction, FIP Guide to good practice-Prestressed Concrete:Safety Precautions in Post-Tensioning, Thomas Telford Ltd, London, 1989.

3-12 Recommendations for Safety Precautions in Post-Tensioning Operations, Concrete Society ofSouthern Africa, Halfway House.

3-13 FIP Commission on Practical Construction, FIP Guide to good practice-Grouting of Tendonsin Prestressed Concrete, Thomas Telford Ltd, London, 1990.

3-14 "Grouting Specifications," CONCRETE, The Concrete Society Journal, Vol. 27, No. 4,July/August 1993.

REFERENCES 3-29

4 DESIGN FOR FLEXURE

4.1 INTRODUCTION

It is important that the difference between analysis and design for flexure be clearly understood.Analysis includes the processes required for assessing the response of the section to the appliedloadings and, therefore, implies that the configuration of the section and the properties of thematerials used are known. Design, on the other hand, involves the selection of a suitable sectionand suitable materials out of many possibilities. The process of design is more complex than thatof analysis because, on the one hand, it deals with unknowns while, on the other hand, a largenumber of combinations of possibilities exist. Analysis usually forms an integral part of the designprocess because once a section has been designed it must be analysed to check if it satisfies thespecified design criteria.

In this Chapter, the flexural behaviour of a prestressed concrete beam section over the completeloading spectrum, from zero load to failure, is first discussed. This is followed by a presentation ofmethods of analysis, after which various design procedures are dealt with. Sections subjected toflexure only are considered here, that is, only the effects of moment are considered. The materialpresented in this Chapter covers pretensioned and post-tensioned members, and includes bothcomposite and partially prestressed concrete sections. In the case of post-tensioned members, bothbonded and unbonded construction are considered.

4.2 SIGN CONVENTION

Any systematic structural analysis requires a consistent sign convention. The analysis of prestressedconcrete sections for flexure is no exception to this rule, even though the sense of some variables,such as stress and strain, can easily be determined by inspection. It is also important to realise thata computer implementation of any of the analytical procedures considered here should not becontemplated without the use of such a sign convention.

The sign convention followed in these notes conforms to that commonly used in structural mechanics,and is defined by the rules listed below. Any deviation from these rules is either self evident orclearly indicated in the text.

• Stress and force: Stress and force are both taken positive when tensile and negative whencompressive. It should be noted that many authors use the opposite convention, i.e. tensionnegative and compression positive, the reason being that since prestressed concrete beams arenormally under compression the sense of the stress which occurs most often is positive.

• Bending moment: Ordinary beam convention is applied to bending moment, according to whichpositive moment corresponds to a concave deflected shape of the beam while negative momentcorresponds to a convex deflected shape, as shown in Fig. 4-1.

Positive bending moment

+M +M

-M -M

Negative bending moment

Figure 4-1: Sign convention for bending moment.

INTRODUCTION 4-1

• Section properties: The dimensions, area A and the second moment of area about the centroidalaxis I of the section are always taken positive. The eccentricity e of the prestressing force isalways measured from the centroid of the section and is taken positive below the centroidal axis(see Fig. 4-2). The sign of the section modulus Z=I/y with respect to a particular fibre isdetermined by the distance y of the fibre measured from the centroidal axis. This distance istaken positive for fibres located below the centroidal axis (see Fig. 4-2).

4.3 ANALYSIS

4.3.1 Basic assumptions

The following basic assumptions are required for the analysis of a prestressed concrete beam section:

• Plane sections before bending remain plane after bending.

• The stress-strain relationships of the materials are known.

• The relationship between the strain in the steel and the strain in the surrounding concrete isknown.

Each of the three basic assumptions are discussed in the following with regard to their impact onthe analysis of prestressed concrete beam sections.

Plane sections before bending remain plane after bending.

The strain distribution over the depth of a beam in bending varies as a function of the distance fromthe neutral axis. The first assumption implies that a linear relationship exists between the strain ata fibre in the concrete and its distance from the neutral axis as shown in Figure 4-3. A large number

x

e, y

Centroidal axis

Figure 4-2: Axial system for section properties.

M M

�c

y�

Neutralaxis

(a) Beam subjected to flexure (b) Strain distribution with depth

Figure 4-3: Plane sections remain plane during bending.

4-2 DESIGN FOR FLEXURE

of tests on reinforced concrete members (Ref. 4-1) indicate that this assumption is very nearly correctat all stages of loading up to failure, provided that good bond exists between the concrete and thesteel.

The assumption proves to be accurate for the concrete in the compression zone even at high loadsclose to the ultimate load. After cracks have developed in the tension zone, the tensile strain in theuncracked concrete between cracks is known to vary from zero at the crack to some non-zero valueat positions located some distance away from the crack because of the action of bond between thesteel and the surrounding concrete. Consequently the assumption that plane sections remain planecannot be true in a cracked member when considering individual sections. However, if the gaugelength for measuring strain is large enough to include a number of cracks, this assumption will holdfor this “average” tensile strain (Ref. 4-1).

The first assumption does not hold for deep beams and regions of high shear. According toSABS 0100 (Ref. 4-2) a simply supported beam should be considered as being deep when the ratioof the height of the section to the effective span length exceeds ½.

Note that the validity of the first assumption has often been questioned for a number of reasons(Ref. 4-3):

• Most of the conclusions were derived from the results of tests on beams with rectangular crosssections and measurements were made in a region of constant moment.

• The strains are usually measured on the outside of the beam and it could be argued that thissituation is not representative of conditions inside the beam.

• For non-rectangular sections, disturbances occur at points where the width of the member abruptlychanges.

In spite of these objections, application of this assumption by many researchers has shown that agood correlation can be obtained between calculated and measured results so that it can be consideredto be accurate enough for design purposes. In lieu of an alternative, this assumption will be used.

The stress-strain relationships of the materials are known.

The stress-strain relationships of both the prestressed and non-prestressed steel, as presented anddiscussed in section 2.2, can be used. However, it is important to note that the concrete is actingin flexure and not in direct compression or tension, and the relationship used for the purposes offlexural analysis must therefore take this into account.

A great deal of research has been carried out to determine the stress-strain relationship of concreteflexural elements. The most notable research was carried out by Hognestad et al (Ref. 4-4) andRüsch (Ref. 4-5), and the following results were obtained:

• A similarity exists between the stress-strain relationships for concentrically loaded cylinders andthe stress-strain relationship for eccentrically loaded beam specimens.

• The maximum stresses reached in the beam specimens were lower than the cylinder strengths,with the difference increasing with an increase in cylinder strength.

• The stress-strain relationship for beam specimens could be determined for strains much largerthan the strain at which the maximum stress occurs. The determination of the stress-strainrelationship for concentrically loaded cylinders beyond the cylinder strength is complicated bythe fact that special testing equipment is required.

• The maximum strain �cu reached in the extreme compression fibre in bending is a function ofthe concrete strength, decreasing with an increase in cylinder strength. Rüsch (Ref. 4-5) hasshown that the strain at the extreme compression fibre at maximum moment is also a functionof the shape of the cross-section.

ANALYSIS 4-3

If the above points are kept in mind, the stress-strain relationships obtained for concrete in directcompression can be applied to beams in bending. The parabolic-rectangular stress-strain relationshiprecommended by the design codes of practice commonly used in South Africa (Refs. 4-2, 4-6, 4-7and 4-8) is shown in Figure 4-4. The purpose of the 0.67 factor is to take into account the differencesbetween the cube strength fcu and the experimentally obtained results for beams in bending. Aconstant value of 0.0035 is recommended for �cu and the partial factor of safety �m is discussedlater.

Because the stress-strain relationship is usually difficult to determine and to deal with computation-ally, much research has been carried out to represent the stress distribution in the compression zoneof a beam at ultimate as an equivalent rectangular stress-block. The recommendations given bySABS 0100 (Ref 4-2) and BS 8110 (Ref. 4-7) are summarized in Figure 4-5. It should be noted thatthe equivalent rectangular stress-block is only valid at ultimate, and not when considering flexuralresponse at other levels of loading.

When calculating the response of the section at ultimate, the tensile strength of the concrete isusually ignored because its influence on the moment of resistance is small. This follows becausethe concrete in the tension zone is usually cracked at ultimate, so that the remaining area in tensionis small with a correspondingly small lever-arm. The tensile strength becomes more important whencalculating deformations at loadings appropriate to the serviceability limit state, and its influenceon behaviour should be accounted for at these load levels.

ParaboliccurveStress

Strain

�c0

Eci

�m

0. 67 fcu

�cu = 0.0035

��c

cu

m

f0

42 4 10� ��.

Ef

cicu

m

� 55.�

GPa

fcu in MPa

Figure 4-4: Parabolic-rectangular stress-strain relationship for concrete in flexure (Refs. 4-2, 4-6,4-7 and 4-8).

�c0

�cu = 0.0035�m

0. 67 fcu�m

0. 67 fcu

s x= 0.9x

Parabolic-rectangularstress block

Equivalent rectangularstress block

Straindistribution

Neutral axis

Figure 4-5: Rectangular stress-strain relationship for concrete in flexure (Refs. 4-2 and 4-7).


The relationship between the strain in the steel and the strain in the surrounding concrete is known.

The strain in the concrete at the level of the steel is calculated by making use of the assumptionthat plane sections remain plane, and the change in strain in the steel is subsequently obtained byassuming that it is equal to the calculated strain in the concrete at the level of the steel. Thisapproach applies to all bonded steel.

The distribution of the strain in unbonded tendons is assumed to be uniform along the length of themember, even though it actually varies to some degree because of the effects of friction. Underthese conditions, the total change in length of the concrete at the level of the prestressing steel isassumed to be equal to the change in length of the prestressing steel. The implications of thisassumption are discussed in Section 4.3.6.

4.3.2 Flexural response

By making use of equilibrium and the basic assumptions (see Section 4.3.1), the moment-curvaturerelationship of a given beam section can be calculated over the full range of loading. This is aparticularly useful relationship because it only considers the response at a section and is independentof the response of the member as a whole.

Curvature � is defined as the angle between two faces of an element of unit length after deformation,and is determined by the following expression (see Fig. 4-3)

(4-1)

where �c is the strain in the concrete at a distance y from the neutral axis. The curvature is positivewhen the bottom of the section has an algebraically larger strain than the top of the section, and itis zero when y tends to infinity, i.e. the strains at the top and bottom of the section are equal.

Consider a typical beam section as shown in Figure 4-6a. The prestressing tendons are bonded tothe concrete and have material properties as shown in Figure 4-6b. A bi-linear stress-strainrelationship is assumed for the concrete as shown in Figure 4-6c. It is interesting to note that theexact shape of the concrete stress-strain relationship has little influence on the behaviour of aunderreinforced section (see Section 4.3.5 for definition), as is considered in this example.

The calculated moment-curvature diagram is shown in Figure 4-6e with the initial portion of thediagram enlarged in Figure 4-6d. As the externally applied moment increases from zero to failure,several important points can be identified and are denoted by capital letters A to H. At each of thesepoints the corresponding stress distribution is also shown. It should be noted that the shape of thisdiagram may differ according to the choice of material properties and level of prestressing.

Point A on the moment-curvature relationship indicates the point of zero moment where only theeffective prestressing force, including all losses, is acting on the section with no applied loading.This case corresponds to the fictitious case of a weightless beam. However, it is a convenient pointfrom which to start the calculations and the self weight will be taken into account as an externalmoment applied to the section.

As the externally applied moment increases, the strain at the top of the section will change fromtension to compression until point B is reached where the stress at the top of the section will beequal to the stress at the bottom of the section, i.e. a point of zero curvature. With a further increasein moment the stress decreases at the bottom until a point C is reached where it is zero, and thecorresponding moment is often refered to as the decompression moment.

When point D is reached the strain in the concrete at the level of the prestressing steel is zero.With a further increase in moment the tensile stress at the bottom of the section will increase untilpoint E is reached where the tensile strength of the concrete fr is exceeded and the concrete cracks.At this stage, a point of instability is reached where the curvature will increase with an accompanying

��

� c

y

ANALYSIS 4-5

Curvature ( )�� 10 3 1m

0 2-2 4 6 8 10

30

25

20

15

10

5

0

Mom

ent

(kN

.m)

A

B

C

D

E

(d) Initial moment curvature response

fr

Concretecracks

Decompression

Zerocurvature

Onlyprestressing

force

Zero concrete strainat level of steel

-10 100 3020 5040 7060 9080

40

35

30

25

20

15

10

5

0

Mom

ent

(kN

.m)

Curvature ( )�� 10 3 1m

A

B

C

D

E

FG

H

(e) Complete moment curvature response

Steel yields

Concrete becomesplastic ( = 0.0015)�c

Concrete fails

Enlarged abovein Figure (d)

ElasticCracked

ElasticUncracked

Range ofservice load

PlasticCracked

(c) Idealized concrete material properties

41.4 MPa

4.14 MPa(= )fr

0.0015 0.003

0.00015

Strain �c

Stre

ssf c

Ec = 27.6 GPa

155

307 230

Aps

fse = 813.6 MPaAps = 100.6 mm2

(a) Cross-section (b) Idealized prestressing steel material properties

� pu

=0.

06

Ep = 207 GPa

fpu = 1710 MPa

fpy = 1489 MPa

Stre

ssf p

Strain �p

2000

1500

1000

500

00 0.02 0.04 0.06

Figure 4-6 : Moment curvature response of an underreinforced beam with bonded tendons(Ref. 4-3).


reduction of moment. Stability is subsequently regained once an increase in curvature is accompaniedby an increase in moment, and beyond this point the section can sustain moments larger than thecracking moment.

At point F the stress in the prestressing steel is equal to the yield stress, and at point G the stressin the concrete will reach the end of its elastic range. The maximum moment that the section canresist corresponds to the moment at point H where the concrete fails in compression. Failure of theconcrete is defined as the point where the maximum concrete strain �cu is reached in the top fibre.

As indicated in Fig. 4-6e, the section behaves as an elastic uncracked section between points A andE, and as an elastic cracked section between points E and F. Beyond point F, up to failure at pointH, the response corresponds to that of a cracked plastic section.

The theoretical moment-deflection curve presented in Fig. 4-7 was calculated on the basis of themoment curvature relationship of Fig. 4-6 which, in turn, makes use of the basic assumptionsdiscussed in Section 4.3.1. The excellent agreement with the experimentally obtained curve clearlyshows that the basic assumptions can provide reasonable results.

4.3.3 Analysis of the uncracked section

A prestressed concrete beam section usually remains uncracked over a wide range of moment (seeFig. 4-6e). As discussed in Section 4.3.2, the material response remains essentially linear elasticwithin this range. If the Bernoulli/Navier hypothesis that plane sections remain plane is combinedwith the assumption that the material behaviour is linear elastic, stresses and strains in the uncrackedsection may be calculated on the basis of ordinary engineering beam theory.

Following this approach, the tensile force P in the tendon is taken to induce an equal and oppositecompressive force P in the concrete acting at the same position. The stresses induced in the concreteby the prestressing force alone are subsequently calculated by considering the section to be subjected

0 10 30 5020 40 60

40

35

30

25

20

15

10

5

0

Deflection calculated from moment curvature relationship

Deflection from experimental results (Ref. 4-9)

Deflection � (mm)

Mid

span

mom

ent

(kN

.m)

914 914 914

2 742

�

Figure 4-7: Moment-deflection response of an underreinforced beam with bonded tendons (Ref.4-3).

ANALYSIS 4-7

to an axial load P acting at its centroid together with a moment Pe acting about its centroidal axis,where e is the eccentricity of the tendon measured from the centroid of the section. Hence, the stressinduced by prestressing only on a fibre located a distance y from the section centroid is given by(see Fig. 4-8a)

(4-2)

where A = area of the section

I = second moment of area of the section about its centroidal axis

When applying the above equation, the sign convention must be carefully observed: The force Pwill carry a negative sign because it acts as a compressive force on the concrete. The moment Pewill also be negative (indicating a negative moment) if the cable is located below the centroidalaxis of the section because e carries a positive sign in this case. Also note that y is positive forfibres located below the centroidal axis while it is negative for fibres located above the centroidalaxis.

The stress induced by an externally applied moment M in the uncracked concrete section is alsocalculated by ordinary beam theory. Using this approach, the stress induced by M on a fibre locateda distance y from the section centroid is given by (see Fig. 4-8a)

(4-3)

fP

A

P e y

IP � �

fM y

IM �

P

M

e

P e M

MP e

Prestressing only

c.g.s.

Centroidalaxis

Beamsection

Prestressing forceand externally

applied moment

(a) At a fibre distance from the section centroidy

(b) At the outer fibres of the section

External loadings Total stresses

P

Ztop Ztop

ZbotZbot

A

P e+

P

ZbotA

P e+

P

ZtopA

P e M+ +

P

Ztop ZtopA

P e M+ +

P

Zbot ZbotA

+ += =

P

y

M

e

M yM yP e y

Prestressing only

Beamsection

c.g.s.

Centroidalaxis

Prestressing forceand externally

applied moment

External loadings Total stresses

P

IIIA+

P e yP

IA+

P e yP

IA+

+ += =

ytop

ybot

Figure 4-8: Calculation of stresses in the concrete due to prestressing and an externally appliedmoment.


It must be noted that Eq. 4-3 is applicable to any externally applied moment, irrespective of whetherit arises from the beam self weight or an externally applied load. As in the case of Eq. 4-2, thesign convention must be properly observed.

The total concrete stress resulting from both the prestressing force and the loads is subsequentlyobtained by superimposing the stresses induced by each of these effects acting on their own (seeFig. 4-8a). Thus, by combining Eqs. 4-2 and 4-3

(4-4)

Specifically, the extreme top and bottom fibre stresses are given by (see Fig. 4-8b)

(4-5)

(4-6)

where

ftop, fbot = stress in the extreme top and bottom fibres, respectively

Ztop = I/ytop = section modulus with respect to the extreme top fibre, located a distance ytop fromthe section centroid

Zbot = I/ybot = section modulus with respect to the extreme bottom fibre, located a distance ybotfrom the section centroid

Note that Ztop carries a negative sign because the extreme top fibre lies above the centroidal axisso that ytop is negative. On the other hand, Zbot can be shown to be positive because the extremebottom fibre lies below the centroidal axis, which means that ybot is positive.

EXAMPLE 4-1

The post-tensioned simply supported concrete beam shown in Fig. 4-9 is subjected to a uniformlydistributed load of 15 kN/m, including self weight. Calculate the extreme top and bottom fibrestresses at midspan if the tendon force is 1334 kN.

The beam section properties are calculated below. Please note the use of the sign convention.

f f fP

A

P e y

I

M y

IP M� � � � �

fP

A

P e

Z

M

Ztoptop top

� � �

fP

A

P e

Z

M

Zbotbot bot

� � �

A b h

I b h

yh

yh

top

bot

� � � � �

� � � � � �

� � � � � �

� � �

300 600 180 10

1

12

1

12300 600 5 4 10

2

600

2300

2

600

2300

3 2

3 3 9 4

mm

mm

mm

mm

.

L = 12000

300180

h = 600

b = 300

Section at midspan

Centroidal axis

c.g.s.

300

w = 15 kN/m

Figure 4-9: Example 4-1.

ANALYSIS 4-9

The bending moment at the midspan section is given by andthe prestressing force P = �1334 kN acts at an eccentricity e = 300 � 180 = 120 mm. Equations 4-5and 4-6 are subsequently used to calculate the stresses in the extreme top and bottom fibres of themidspan section, respectively:

The section properties used in the above example for calculating the stresses were based on thegross concrete section. Although this approach is convenient from a practical point of view, it isnot theoretically correct. Consider, for example, the case of a post-tensioned bonded prestressedconcrete beam: At transfer and up to the time at which the grout has hardened and become effectivethe tendons are not bonded to the concrete so that any loads applied to the beam at this stage, suchas the prestressing force and self weight, will act on the net concrete section. Hence, the stressesinduced by these loads in the concrete should be calculated on the basis of the properties of the netconcrete section which take the presence of the preformed ducts, within which the tendons arecontained, into account. After the grout has hardened, the tendons are effectively bonded to theconcrete so that the transformed section properties must be used for calculating the stresses inducedin the concrete by loads applied at this stage, such as the superimposed dead load and the live load.

By the nature of the procedure, the tendons in pretensioned beams will always be bonded to theconcrete. This means that the transformed section properties should be used for calculating thestresses in the concrete induced by all the loads, including the prestressing force. On the other hand,the properties of the net concrete section should be used for all stress calculations in the case ofunbonded construction because, in this case, the tendons are never bonded to the concrete.

The correct section to be used in the various situations described above are summarised in Table 4-1.It should also be noted that a distinction must be made between the transformed section propertiesused for calculating the stresses induced by short-term loadings and those used for calculating thestresses induced by the long-term loads. In the case of long-term loads, the transformed sectionproperties should, in some way, reflect the effects of creep of the concrete. This is commonly doneby making use of an effective modulus of elasticity for the concrete, which includes creep strain,for assessing the modular ratio used in the calculation of the transformed section properties.

Although it is theoretically more correct to base the calculation of stress on the section propertiesas outlined above, this is not frequently done in practice. Stress calculations are usually carried outusing the properties of the gross concrete section only. This approach greatly simplifies thecalculations and, under normal circumstances, provides a close approximation. However, incircumstances where the area of the ducts forms a significant part of the cross-section and/or if alarge quantity of steel is contained in the section, the section properties should be based on thesections as indicated in Table 4-1 to ensure that stresses are estimated with sufficient accuracy.

It is important to note that the magnitude of the prestressing force used for a stress calculation mustreflect the loss of prestress appropriate to the age of the beam at the time under consideration. Thetotal loss of prestress can conveniently be divided into instantaneous losses, which take place at

ZI

y

ZI

y

toptop

botbot

� ��

��

� ��

� �

54 10

30018 10

54 10

30018 10

96 3

96 3

.

.

mm

mm

M wL� � � �2 28 15 12 8 270/ / kN. m

fP

A

Pe

Z

M

Z

fP

A

Pe

Z

M

Z

toptop top

botbot bot

� � � ��

��

� � �

� ��

�

� ��

� � � ��

��

� � �

��

�

��

1334 10

180 10

1334 10 120

18 10

270 10

18 101352

1334 10

180 10

1334 10 120

18 10

270 10

18 10130

3

3

3

6

6

6

3

3

3

6

6

6

.

.

MPa

MPa


the time of transfer, and time-dependent losses, which gradually develop with time. The instantaneouslosses are attributed to the following sources:

• Elastic shortening of the concrete. When the prestressing force is transferred to the concrete, theconcrete shortens so that tendons already bonded or anchored to the concrete also shorten by thesame amount. This leads to a reduction of the stress in the tendons and, hence, a loss ofprestressing force. Although this loss occurs in both pretensioned and post-tensioned members,they are not affected to the same degree.

• Friction. When a tendon in a post-tensioned member is tensioned, friction is induced betweenthe sliding tendon and the surrounding duct material. This friction reduces the tensioning force,and the magnitude of the reduction, which increases for sections further away from the jackingend, represents the friction loss.

• Anchorage seating. When a post-tensioned tendon is anchored to the concrete after tensioning,the components of the anchorage will deform slightly and, if the anchorages make use of wedgegrips, a certain amount of slip must take place to seat the grips. The resulting loss of elongationof the tendon, termed anchorage seating, leads to a reduction of the tensioning force. This lossoccurs only in post-tensioning systems.

The time-dependent losses, which occur in both pretensioned and post-tensioned members, developwith time and are attributed to the time-dependent behaviour of the concrete and the steel as follows:

• Relaxation of the tendons. Because the tensioned tendons in a prestressed concrete member arecontinuously subjected to a large strain over the life of the member, a time-dependent loss oftensioning force, and hence prestress, takes place as a result of relaxation of the steel.

• Creep and shrinkage of the concrete. Creep and shrinkage of the concrete in a prestressed concretemember each induce a time-dependent shortening in the concrete which, in turn, leads to ashortening of the attached tendons. This action results in a time-dependent reduction of the stressin the tendons and, hence, a loss of prestressing force.

4.3.4 Cracking moment

The moment at which the section first cracks is referred to as the cracking moment. It is usuallytaken as the moment which, by elastic theory, induces a tensile stress in the extreme fibre equal tothe modulus of rupture fr. Although this approach has often been questioned, available experimentaldata indicate that it is sufficiently accurate (Ref. 4-10). The cracking moment Mcr with respect tothe bottom fibre is therefore determined by setting fbot = fr and M = Mcr in Eq. 4-6, and solvingthe resulting expression for Mcr. Hence,

so that

(4-7)

f fP

A

P e

Z

M

Zbot rbot

cr

bot

� � � �

M f Z PZ

Aecr r bot

bot� � �FHG

IKJ

Load Pretensionedbonded

Post-tensionedbonded

Post-tensionedunbonded

Prestress Transformed Net Net

Self weight Transformed Net Net

Superimposed dead load Transformed Transformed Net

Live load Transformed Transformed Net

Table 4-1: Correct sections for stress calculations.

ANALYSIS 4-11

The modulus of rupture, used in this way, merely serves as an index for measuring the load at whichhair cracks, often invisible to the naked eye, start to develop. A higher load is usually necessaryfor visible cracks to form. However, it is important to note that once the section has cracked it canno longer be analysed as an uncracked elastic section, but that the cracked section must be consideredinstead. The cracking moment is usually used to mark the end of uncracked section behaviour andthe onset of cracked section behaviour. Note that if the section has already been cracked in a previousloading, the cracking moment can no longer be used as the limit of uncracked section behaviour.In such a case, the decompression moment, defined as the moment which induces a zero stress inthe extreme fibre, must be used.

EXAMPLE 4-2

Consider the post-tensioned concrete beam of example 4-1. Determine the cracking moment of thesection at midspan if the cube strength of the concrete is fcu = 45 MPa.

According to the SABS 0100 (Ref. 4-2) the modulus of rupture is given by= 4.360 MPa. The cracking moment Mcr is calculated using Eq. 4-7. Note the proper use

of the sign convention and, in particular, that fr is assigned a positive value because it representsa tensile stress.

4.3.5 Ultimate moment: Sections with bonded tendons

The ultimate moment of a prestressed concrete beam section is, by definition, the maximum momentwhich it can resist. In the case of the section considered in Fig. 4-6 it is represented by point H onthe moment-curvature diagram. The mode in which a given prestressed concrete section with bondedtendons fails in flexure depends on the amount of steel provided, and one of the three typesillustrated in Fig. 4-10 is possible:

• Failure induced by fracture of the steel immediately after the concrete has cracked. This failuremode is brittle and occurs in very lightly reinforced sections in which insufficient steel isprovided to carry the additional tensile force which is transferred from the concrete to the steelupon cracking. This type of failure is highly undesirable and such sections are not commonlyencountered in practice.

• Failure induced by crushing of the concrete compression zone after the steel has yielded andundergone a large non-linear elongation. Sections which fail in this manner are referred to asunderreinforced sections. This failure mode is ductile because the section can sustain a momentclose to the ultimate moment over a wide range of deformations. Because of its ductility, thistype of failure is highly desirable and most sections encountered in practice are proportioned asunderreinforced sections.

• Failure induced by crushing of the concrete prior to yielding of the steel. Sections which fail inthis manner are heavily reinforced and are referred to as overreinforced sections. This failuremode is brittle and takes place suddenly because, once the ultimate moment has been reached,the section suddenly loses its ability to sustain moment with any further increase in deformation.Because of their brittle nature, overreinforced sections are undesirable and should be avoided.

Because the stress-strain response of prestressing steel does not show a definite yield point as inthe case of hot-rolled steel reinforcing bars (see Fig. 2-24), it is not possible to define a precise

f fr cu� 0 65.� 0 65 45.

M f Z PZ

Aecr r bot

bot� � �FHG

IKJ

� � � � � � ��

��

F

HGI

KJ

� � �

4 360 18 10 1334 1018 10

180 10120

372 0 10 372

6 36

3

6

. ( )

. N. mm .0 kN. m


limit to the percentage of reinforcement required for underreinforced failure as is possible forordinary reinforced concrete beam sections.

The most general approach to calculating the ultimate moment of a prestressed concrete beam sectionis to directly apply the basic assumptions listed in Section 4.3.1. In the following, this approach isdeveloped for the calculation of the ultimate moment of a rectangular bonded prestressed concretebeam section. Figure 4-11 shows the assumed strain distribution as well as the stress distributionin such a section when the ultimate moment has just been reached, and the following importantaspects must be noted:

• The ultimate condition is defined in terms of a limiting strain �cu being reached in the concreteat the extreme compression fibre.

• Although the compressive stress distribution in the concrete at ultimate is approximated by anequivalent rectangular stress block, the principle of the analytical procedure remains unaltered ifa more exact, but more complicated, approximation is used. The codes of practice commonlyused in South Africa (Refs. 4-2, 4-6, 4-7 and 4-8) set � = 0.45 or 0.4 and � = 0.9 or 1.0 (Notethat the values of � given here include a partial safety factor �m = 1.5).

• The tensile strength of the concrete is neglected. This means that any concrete which falls belowthe neutral axis is assumed to offer no resistance to bending.

Moment

Curvaturecr

Amount ofreinforcementincreases

Ultimate( )f fps py

Underreinforced( )f f fpy ps pu�

Ultimate momentless than cracking moment( = )f fps pu

Overreinforced

Cracking

Steel yields

f

f

f

pu

py

ps

= Characteristic strengthof the prestressing steel

= Defined yield stress of theprestressing steel

= Stress in the prestressing steelat ultimate

Figure 4-10: Moment-curvature behaviour for increasing reinforcement.

�cu

�s

� fcu

fc ( )�� x

Aps

x

b

dh

Assumed straindistribution

Assumed stressdistribution

Neutral axis �

C

T

z

Figure 4-11: Analysis of a rectangular bonded prestressed concrete beam section at ultimate.

ANALYSIS 4-13

• Because the tendons are bonded to the concrete, the changes in strain in the steel are taken tobe the same as in the adjacent concrete after bonding.

The total strain in the steel must include the strain induced by the effective prestress, including alllosses which have taken place at the time under consideration. Figure 4-12a shows that the initialstrain induced by the effective prestress alone (i.e. no moment from external loads acting on thesection) consists of two components:

• A tensile strain �se induced in the steel by the effective tensioning stress fse acting in the tendon,including all losses.

• A compressive strain �ce in the concrete at the level of the steel, induced by the effective prestressacting on its own.

Hence the total strain in the prestressing steel is given by

(4-8)

The various components of strain are calculated as follows:

• �se is simply taken as the elastic extension of the steel acting under the effective prestress fse.Thus

(4-9)

where Ep = Modulus of elasticity of the steel

• �ce is calculated by considering the prestress to be acting on the elastic uncracked section. Hence,at the level of the steel

(4-10)

where Ec = Modulus of elasticity of the concrete

Note that because �ce is a compressive strain, it will carry a negative sign if the sign conventionis properly applied to Eq. 4-10. This means that when �ce is substituted into Eq. 4-8 it will be

� � � �ps s ce se� � �

� sese

p

f

E�

�cec

P

A

P e

I E� �

L

NMM

O

QPP

2 1

�cu

�ce

�ce

�se �s �se

� � � � �ps s ce se= +

Aps

x

b

d

e

h

(a)Strain induced byeffective prestress

only (i.e. zero moment)

(b)Strain distribution

at ultimate conditions

Centroidalaxis

Figure 4-12: Strain in the steel.


added to the other strain components, as expected (see Fig. 4-12b). It is also important to notethat the value for P in Eq. 4-10 must include all prestress losses at the time under consideration.

• The change in strain �s induced by the ultimate moment is obtained by applying the compatibilityassumption that plane sections remain plane and that, since the steel is bonded to the concrete,the change in strain in the steel is the same as in the concrete at the level of the steel. Thus,considering similar triangles (see Figs. 4-11 and 4-12b),

(4-11)

where d = effective depth of the steel, always taken positive

x = depth to the neutral axis, always taken positive

�cu = limiting strain in the concrete at the extreme compression fibre, specified as0.0035 by the codes of practice commonly used in South Africa (Refs. 4-2, 4-6,4-7 and 4-8)

Once the total strain �ps in the prestressing steel has been determined by Eq. 4-8, used in conjunctionwith with Eqs. 4-9 through 4-11, the steel stress at ultimate fps is obtained from the stress-strainrelationship of the steel as the stress corresponding to the strain �ps (see Fig. 4-13). The designstress-strain curve recommended by SABS 0100 (Ref. 4-2) for prestressed reinforcement is shownin Fig. 2-26, and the other design codes of practice commonly used in South Africa also prescibesimilar design stress-strain curves. Although actual, experimentally determined stress-strain diagramsmay be used instead, care must be taken to properly include the partial safety factor �m, dependingon whether a nominal or design value of the ultimate moment is being calculated.

The total tensile force T acting in the steel at ultimate is subsequently calculated from (see Fig. 4-11)

(4-12)

where Aps = cross-sectional area of the prestressing steel

The total compressive force acting in the uncracked compression zone of the concrete at ultimateis calculated simply by evaluating the volume of the compressive stress prism (see Fig. 4-11). Thus,

(4-13)

where fc(�) = compressive stress in the fibre located a distance � from the neutral axis

b = width of the section

� �s cu

d x

x�

�FHG

IKJ

T A fps ps�

C f b dc

x� z ( )� �

0

Stress

fps

Strain�ps

Figure 4-13: Determining fps from the stress-strain curve for the steel with �ps known.

ANALYSIS 4-15

Note that fc(�) is negative because it represents a compressive stress and that, as for Eq. 4-11, x istaken positive. This means that C will carry a negative sign, which is consistent with the signconvention.

Equation 4-13 is general and can be applied to any assumed stress-strain relationship for the concrete.If an equivalent rectangular stress block is used, Eq. 4-13 can be simplified as follows (seeFig. 4-11):

(4-14)

where fcu = characteristic compressive strength of the concrete, taken negative because itrepresents a compressive stress

�, � = stress block parameters

The ultimate moment is calculated by considering moment equilibrium of the section. Takingmoments either about the line of action of C or T yields the following expressions for the ultimatemoment Mu (see Fig. 4-11):

(4-15)

where z = internal lever arm

The position of the line of action of C must first be determined before the internal lever arm z canbe calculated. Considering the compressive stress distribution (see Fig. 4-14), the followingexpression can be written:

Substituting for C from Eq. 4-13 and rearranging terms yields the following expression for thedistance from the neutral axis to the line of action of C:

From Figs. 4-11 and 4-14 it is clear that the internal lever arm z is given by

(4-16)

C f b xcu� � �

M T z C zu � � �

C f b dc

x� � � �� z ( )b g

0

��

� �

�zz

f bd

f bd

c

x

c

x

( )

( )

b g0

0

z d x d xf bd

f bd

c

x

c

x� � � � � �

zz

��

� �

( )

( )

b g0

0

fc ( )�x

C

Neutral axis

� �

Figure 4-14: Line of action of resultant compression force C.


Thus, to summarize:

These expressions are general and can be applied to any assumed stress-strain relationship for theconcrete. If an equivalent rectangular stress block is used they can be simplified by substitutingEq. 4-14 for C in Eq. 4-15 and by recognising that, in this case, z = d � � x / 2 (see Fig. 4-11).Hence,

(4-17)

An inspection of Eqs. 4-8 and 4-11 through 4-17 will reveal that all the quantities represented bythese equations, which includes the ultimate moment, can be directly calculated if the value of theneutral axis depth x at ultimate is known. Any solution technique should therefore initially be aimedat calculating x, after which T, C, z and Mu can be calculated. The iterative procedure presentedbelow follows this approach, and is recommended for use when more complicated approximationsof the stress-strain relationships for the concrete and steel are used.

(a) Assume a value for the depth to neutral axis x.

(b) Calculate the total strain in the prestressing steel �ps using Eqs. 4-8 to 4-11.

(c) Obtain the magnitude of the steel stress fps corresponding to the strain �ps using the stress-strainrelationship for the steel.

(d) Calculate the magnitude of T using Eq. 4-12.

(e) Determine the magnitude of C from Eqs. 4-13 or 4-14, as appropriate.

(f) The correct value of x will ensure that horizontal equilibrium is satisfied. Therefore, if therelationship T + C = 0 is satisfied, the value of x currently selected is correct and the ultimatemoment Mu can be calculated as indicated in the next step. However, if this expression is notsatisfied, a revised value must be selected for x and steps (b) through (f) repeated.

(g) Calculate the ultimate moment Mu using Eqs. 4-15 and 4-16 together with the current valuesof T, C and x, or by using Eq. 4-17 together with the current value of x, as appropriate.

If an equivalent rectangular stress block is used for the concrete together with a simple approximationof the stress-strain curve for the steel, then it is possible to find a closed form solution for x.However, it is important to note that the complexity of the resulting expression for x is dependenton the complexity of the stress-strain curve assumed for the steel. Consider, for example, the casewhere an equivalent rectangular stress block is used in conjunction with a tri-linear approximation

M T z C z

T A f

C f bd

z d x d xf bd

f bd

u

ps ps

c

x

c

x

c

x

� � �

�

�

� � � � � �

zzz

(

(

( ) (

( )

( )(

from Eq. 4-15)

with from Eq. 4-12)

from Eq. 4-13)

from Eq. 4-16)

� �

��

� �

0

0

0

b g

M A f dx

f b x dx

u ps ps cu� �FHG

IKJ � � �

FHG

IKJ� � � �

2 2

ANALYSIS 4-17

of the stress-strain curve for the steel (see Fig. 4-15). Horizontal equilibrium provides the followingexpression

Substituting Eqs. 4-12 and 4-14 into the above expression yields

(4-18)

Inspection of Eq. 4-18 will reveal that if fps is either known or expressed as a function of x, thena closed formed solution can be found for x. It is also clear from Fig. 4-15 that the particularexpression to be used for fps depends on whether �ps is smaller than �p1, whether it is larger than�py, or whether it lies between �p1 and �py, and that fps can be expressed in terms of �ps as follows:

(4-19a)

(4-19b)

(4-19c)

where Ep = Modulus of elasticity of the steel

Ep2 =

Equations 4-19b and 4-19c express fps in terms of �ps and must therefore be expanded to expressfps in terms of x. This is done by writing �ps as a function of x and substituting the result into eachof 4-19b and 4-19c. Combining Eqs. 4-8 and 4-11 yields

(4-20)

Substitution into 4-19b and 4-19c, and rearranging terms lead to

(4-21a)

(4-21b)

where

T C� � 0

A f f b xps ps cu� �� 0

f

f

f E

Eps

py ps py

p p ps p p ps py

p ps ps p

�

�

� � � �

R

S|

T|

for

for

for

� �

� � � � �

� � �1 2 1 1

1

d i

f fpy p

py p

�

�

1

1� �

� � � � � � � � � � �ps s ce se se ce cu se ce cu cu

d x

x

d

x� � � � � �

�FHG

IKJ � � � �c h

ff E

d

x

f Ed

x

ps

s p cu p ps py

s p cu ps p

�

�FHG

IKJ � �

�FHG

IKJ

R

S||

T||

1 2 1

1 1

� � � �

� � �

for

for

ff E

Es

p p se ce cu p p ps py

p se ce cu ps p1

1 2 1 1

1

��

� �

RS|

T|

� � � � � � �

� � � � �

c hb g

for

for

Stress, fps

fpy

Ep

fp1

Strain, �ps�py�p1

f = Eps p ps�

Ef f

ppy p

py p2

1

1

��

��

f f Eps p p ps p� � �1 2 1� �d i

Figure 4-15: Tri-linear approximation of the stress-strain curve for the steel.


Since the expression to be used for fps depends on the magnitude of �ps relative to �p1 and �py, thesolution for x will also depend on the magnitude of �ps. Thus

For �ps � �py

Substituting Eq. 4-19a into Eq. 4-18 and solving for x yields

(4-22a)

For �ps � �py

The following quadratic equation, which can be directly solved for x, is obtained if either ofEqs. 4-21a or 4-21b is substituted into Eq. 4-18:

(4-22b)

where E = Ep2 for �p1 < �ps < �py

= Ep for �ps � �p1

fs1 is defined in Eq. 4-21a for �p1 � �ps � �py and in Eq. 4-21b for �ps � �p1

Equations 4-22a and 4-22b are used as follows to calculate the correct value of x:

(a) Make an assumption about the range within which �ps lies and calculate x using either Eq. 4-22aor 4-22b, as appropriate.

(b) Using this value of x calculate �ps

(Eqs. 4-8 through 4-11).

(c) If �ps

calculated in step (b) falls within the range assumed in step (a), then the calculated valueof x is correct. Otherwise, the assumption made in step (a) is incorrect and the process mustbe repeated from step (a) with a revised assumption regarding the range within which �ps falls.

Once the correct value of x has been determined, as outlined above, the ultimate moment can bedirectly calculated from the second part of Eq. 4-17.

EXAMPLE 4-3

The rectangular prestressed concrete beam section shown in Fig. 4-16 contains six 12.9 mm 7-wiresuper grade strand, the centre of gravity of which is located 60 mm above the beam soffit. Thematerial properties are:

Concrete: fcu = 50 MPa Ec = 34 GPa

Steel: fpu = 1860 MPa Ep = 195 GPa

The properties of the uncracked beam section are listed in Fig. 4-16 and Aps = 6 � 100 = 600 mm2.At the time under consideration, fse = 1150 MPa. Make use of the equivalent rectangular stress blockas well as the design stress-strain curve for strand as prescribed by SABS 0100 (Ref. 4-2) to calculatethe design ultimate moment of the section using (a) the iterative procedure, and (b) the approachwhereby the depth to neutral axis x is directly calculated in closed form.

For the equivalent rectangular stress block prescribed by SABS 0100, � = 0.45 and � = 0.9, asshown in Fig. 4-16, while the design stress-strain curve recommended for the strand considered hereis shown in Fig. 4-17 for �m = 1.15. Referring to Figs. 2-26 and 4-15,

xA f

f bps py

cu

� �

� ��

f b

Ax f x E dcu

pss cu

F

HGI

KJ� � 2

1 0a f b g

ff

pypu

m

�

1860

1151617

.MPa

ANALYSIS 4-19

(a) Iterative approach.

Assume x = 300 mm

The magnitude of �ce is calculated on the basis of the effective prestress P = fse Aps = 1150 �600 � 103 = 690 kN acting on the elastic uncracked section (including all losses at the timeunder consideration). Therefore, from Eq. 4-10

The elastic extension of the steel acting under the effective prestress is given by

while the change in strain �s induced by the ultimate moment is obtained from Eq. 4-11

�

�

�

pypy

p

ppu

m

pp

p

f

E

ff

f

E

� �

�

�

�

0 0051617

195 100 005 0 01329

08 08 1860

1151294

1294

195 100 00664

3

1

11

3

. . .

. .

.

.

MPa

�cec

P

A

Pe

I E �

F

HGI

KJ

��

�

�

F

HGI

KJ

2

3

2

9

1 690

210 10

690 240

6 3 10

1

340 000282

..

� sese

p

f

E

�

1150

195 100 005897

3.

� �s cu

d x

x

FHG

IKJ

FHG

IKJ

540 300

3000 0035 0 0028. .

h = 600 d = 540

Aps = 600 mm2

0.45 fcu

0.9 xx

b = 350

Neutral axis

60

�cu = 0.0035

�s

C

T

e

A

I

�

�

240

210 10

6 3 10

3 2

9 4

mm

= mm

mm.


Stress, fps

fpy = 1617 MPa

fp1 = 1294 MPa

Strain, �ps�py =

0.01329�p1 =

0.00664

Figure 4-17: Stress-strain curve for the steel.


Therefore, the total strain in the steel at ultimate is given by Eq. 4-8 as

The steel stress at ultimate is subsequently obtained from Fig. 4-17 as the stress corresponding toa strain �ps = 0.00898. Hence, fps = 1408 MPa. Note that, in this particular case, fps could also havebeen directly calculated from Eq. 4-19b.

T can now be determined by Eq. 4-12

Because an equivalent rectangular stress block is being used for the concrete, C is calculated byEq. 4-14

Therefore, T + C = 1281 kN � 0 which means that horizontal equilibrium is not satisfied for theselected value of x (= 300 mm). Because the magnitude of C is larger than that of T, a smallervalue of x should be selected and the above computations repeated.

Assume x = 100 mm

If the above calculations are repeated for x = 100 mm the following results are obtained:

These results show that horizontal equilibrium is not satisfied by the selected value of x (= 100 mm),and that the magnitude of C is smaller than that of T. Consequently, a larger value must be selectedfor x. The results obtained for various selected values of x are presented in Table 4-2 while Fig. 4-18shows a plot of T, C and T + C versus x.

Inspection of Table 4-2 and Fig. 4-18 reveals that for x = 136.9 mm the condition T + C = 0 issatisfied for all practical purposes, which means that this value of x is correct because it satisfieshorizontal equilibrium. Substituting x = 136.9 mm and fps = 1617 MPa (see Table 4-2) into thefirst of Eq. 4-17 finally yields the magnitude of the design ultimate moment. The reader shouldverify that the second of Eq. 4-17 yields the same result.

(b) Using the closed form solution of x.

Assume �p1 � �ps � �py

For this case, x is calculated from Eq. 4-22b. Before this equation can be set up, Ep2 and fs1 haveto be calculated from Eqs. 4-19 and 4-21a, respectively. Note that �se = 0.005897 and �ce = 0.000282,as calculated in part (a) of this example. Thus

� � � �ps s ce se � � � � 0 0028 0 000282 0 005897 0 00898. . . .

T A fps ps � � 600 1408 10 844 83 . kN

C f b xcu � � � � � � � 0 45 50 10 350 0 9 300 21263. .d i kN

� ps

psf

T

C

T C

0 02158

1617

970 4

708 8

2617

.

.

.

MPa

kN

. kN

+ kN

M A f dx

u ps ps FHG

IKJ �

�FHG

IKJ � �

2600 1617 540

0 9 136 9

210 464 36. .

. kN. m

Ef f

ppy p

py p2

1

1

31617 1294 10

0 01329 0 0066448 58=

−

−=

− ×−

=−

ε εb g

. .. GPa

ANALYSIS 4-21

Substituting these results into Eq. 4-22b, the following expression is obtained:

f f Es p p se ce cu p1 1 2 1

31294 48 58 10 0 005897 0 00028 0 0035 0 00664

1102

�

� � � �

� � � �c h

b g. . . . .

MPa

0

0 45 50 350 0 9

6001102 4858 10 0 0035 540

1181 1102 91814

21 2

2 3

2

F

HGI

KJ� �

� � �F

HGIKJ � � � � �

� �

� ��

f b

Ax f x E d

x x

x x

cu

pss p cua f c h

c h. ( ) .

( ) . .

.

x(mm)

�ps fps(MPa)

T(kN)

C(kN)

T � C(kN)

100.0 0.02158 1617 970.4 708.8 261.7

150.0 0.01528 1617 970.4 1063 92.69

125.0 0.01780 1617 970.4 885.9 84.50

138.0 0.01638 1617 970.4 978.1 7.640

136.9 0.01649 1617 970.4 970.3 0.1560

Table 4-2: Calculation of x.

100 120 140 160 180 200 220 240 260 280 300-1500

-1000

-500

0

500

1000

1500

2000

2500

T

T C+

� �C

x (mm)

Inte

rnal

for

ces

(kN

)

x = 136.9 mm

Figure 4-18: Variation of the internal forces as a function of x.


Solving for x yields x = 146.4 mm. Check if �ps corresponding to this value of x falls within therange assumed. Using Eqs. 4-8 and 4-11

This value of �ps is larger than �py = 0.01329. The assumption that �ps is less than �py is thereforeincorrect,which means that the calculated value of x is also incorrect.

Assume �ps � �py

If �ps � �py then x can be directly calculated from Eq. 4-22a. Thus

As before, �ps corresponding to this value of x must be checked. Hence,

This value of �ps is larger than �py = 0.01329, as assumed, and therefore the calculated value of xis correct. The ultimate moment is subsequently calculated from the second of Eq. 4-17.

Note that these results are exactly the same as obtained in part (a) of this example. It should alsobe noted that a substantial amount of numerical work can be avoided if these calculations are startedby assuming �ps to be larger than �py, because this condition is often satisfied by the beam sectionsencountered in practice. The solution presented here initially considered the case where�p1 < �ps < �py simply to give a more complete illustration of the solution procedure.

Example 4-3 amply demonstrates that the computational effort required for the calculation of theultimate moment capacity Mu of a prestressed concrete beam section can be significantly reduced ifthe procedure for determining the steel stress at ultimate fps can be simplified because once fps isknown, the depth to neutral axis x and, hence, Mu can be directly calculated. Most design codes ofpractice provide a simplified approximate procedure for estimating fps, and the procedure recom-mended by SABS 0100 (Ref. 4-2), which is the same as that recommended by BS 8110 (Ref. 4-7),is presented in the following and illustrated by example 4-4. The method uses the equivalentrectangular stress block prescribed by SABS 0100 and assumes that the effective prestress fse doesnot exceed 0.6fpu. It is directly applicable to sections of which the compression zone, measured toa depth of 0.9x, is rectangular. The design ultimate moment is calculated by the following expression(given in the notation used in these notes):

(4-23)

where dn = 0.45x

x = depth to neutral axis, as obtained from Table 4-3

fps = design tensile stress in the tendons at failure, as obtained from Table 4-3

� � � �ps cu ce se

d x

x�

�FHG

IKJ

� �

��F

HGIKJ

� � � �

�

540 146 4

146 40 0035 0 00028 0 005897

0 01559

.

.. ( . ) .

.

xA f

f bps py

cu

� �F

HGI

KJ� �

�

� � � �

FHG

IKJ

��

600 1617

0 45 50 350 0 9136 9

. ( ) .. mm

� � � �ps cu ce se

d x

x�

�FHG

IKJ

� � ��F

HGIKJ

� � � �

�

540 136 9

136 90 0035 0 00028 0 005897

0 01648

.

.. ( . ) .

.

M f b x dx

u cu� � �FHG

IKJ

� � � � � � � ��F

HGIKJ

� ��

� ��

2

0 45 50 350 0 9 136 9 5400 9 136 9

210 464 26. ( ) . .

. .. kN. m

M A f d du ps ps n� �b g

ANALYSIS 4-23

EXAMPLE 4-4

Use the approximate method recommended by SABS 0100 to calculate the design ultimate momentof the prestressed concrete beam section of example 4-3.

For the beam of example 4-3 , which is slightly larger than 0.6.Therefore, when using Table 4-3 fse/fpu is set to 0.6, which is the maximum value provided for bythe method. Also,

Interpolating between the values given in Table 4-3 for fpu Aps / fcu b d equal to 0.1 and 0.15 atfse/fpu = 0.6, the following results are obtained:

Therefore, fps = 1.0 (0.87 � 1860) = 1618 MPa and x = 0.26 � 540 = 140.4 mm. The design ultimatemoment is calculated from Eq. 4-23

This result is very close to the ultimate moment Mu = 464.3 kNm obtained by the more elaborateprocedures employed in example 4-3.

So far, only rectangular sections have been considered. In the analysis of flanged sections (I- orT-sections) a distinction is made between the case where the compression zone falls entirely withinthe flange and the case where it extends down into the web (see Fig. 4-19). If the compression zone

f fse pu/ / .� �1150 1860 0 6183

f A

f bd

pu ps

cu

��

� ��

1860 600

50 350 54001181.

f

f

x

dps

pu0 8710 0 26

.. .� �and

M A f d xu ps ps� � � � � � � ��0 45 600 1618 540 0 45 140 4 10 463 06. . . .a f kN. m

Design stress in tendons as aproportion of the design strength,

Ratio of depth of neutral axis to thatof the centroid of the tendons in the

tension zone, x/d

0.6 0.5 0.4 0.6 0.5 0.4

0.05 1.00 1.00 1.00 0.11 0.11 0.11

0.10 1.00 1.00 1.00 0.22 0.22 0.22

0.15 0.99 0.97 0.95 0.32 0.32 0.31

0.20 0.92 0.90 0.88 0.40 0.39 0.38

0.25 0.88 0.86 0.84 0.48 0.47 0.46

0.30 0.85 0.83 0.80 0.55 0.54 0.52

0.35 0.83 0.80 0.76 0.63 0.60 0.58

0.40 0.81 0.77 0.72 0.70 0.67 0.62

0.45 0.79 0.74 0.68 0.77 0.72 0.66

0.50 0.77 0.71 0.64 0.83 0.77 0.69

f A

f bdpu ps

cuf fps pu0 87.

f fse pu = f fse pu =

Table 4-3: Conditions at the ultimate limit state for rectangular beams with pre-tensionedtendons or post-tensioned tendons having effective bond (Ref. 4-2).


is entirely contained within the flange (see Fig. 4-19a), then the analysis is exactly the same as fora rectangular section of the same width b and the equations developed above for a rectangular sectionapply without modification. This follows because the tensile strength of the concrete is neglected,which means that the concrete which falls below the neutral axis is ignored in the analysis and canbe of any shape.

However, if the compression zone extends into the web the analysis must account for the fact that,in this case, the compression zone is no longer rectangular. Since the principles on which the analysisis based remain unaltered, the equations developed above for a rectangular section, which do notinvolve the assumption that the compression zone is rectangular, remain valid. Specifically, Eqs. 4-8to 4-12, 4-15 and 4-19 to 4-21 remain valid because their derivation is not dependent on the shapeof the compression zone. The remaining expressions must be modified to account for thenon-rectangular compression zone. The solution procedure, which is illustrated by example 4-5, isexactly the same as for a rectangular section: The depth to neutral axis x is initially determined,after which the ultimate moment is calculated.

It should be noted that, when working with an equivalent rectangular stress block, the situation mayarise where the depth to neutral axis x is larger than the flange thickness hf, but that the depth ofthe stress block �x is less than hf. It is suggested that such cases be analyzed as rectangular sections,even though the magnitude of x indicates that the compression zone extends into the web. Thisapproach implies that the depth of the equivalent rectangular stress block �x is taken as the depthof the compression zone for the purpose of calculating the compressive force in the concrete atultimate.

EXAMPLE 4-5

Determine the design ultimate moment of the I-shaped prestressed concrete beam section shown inFig. 4-20. The section contains eight 12.9 mm 7-wire super grade strand, of which the centre ofgravity is located 60 mm above the beam soffit. The material properties are:

Concrete: fcu = 50 MPa Ec = 34 GPa

Steel: fpu = 1860 MPa Ep = 195 GPa

Use the equivalent rectangular stress block as well as the design stress-strain curve for strand asprescribed by SABS 0100 (Ref. 4-2), and assume that fse = 1060 MPa at the time under consideration.The properties of the uncracked beam section are listed in Fig. 4-20 and Aps = 8 � 100 = 800 mm².

For the equivalent rectangular stress block prescribed by SABS 0100, � = 0.45 and � = 0.9, asshown in Fig. 4-20. Since fpu and Ep of the strand considered here are the same as for that used inexample 4-3, the stress-strain curve for the steel is also the same, and is as shown in Fig. 4-17.

x

x

b b

hf

bw bw

�x�x

(a)Rectangular section

behaviour

(b)Flanged section

behaviour

Figure 4-19: Flanged section at ultimate.

ANALYSIS 4-25

Assume �ps > �py

If �ps > �py (= 0.01329) then, from Fig. 4-17, fps = fpy = 1617 MPa, and the total tensile force actingin the steel at ultimate T is given by Eq. 4-12:

In order to check whether or not the compression zone extends into the web, the maximumcompression force Cfmax which can be supplied by the flange only is calculated and compared to T.

Therefore, the magnitude of Cfmax is less than that of T, which means that the compression zonemust extend into the web to satisfy horizontal equilibrium. For convenience, the total compressiveforce acting in the concrete is divided into a part Cf which acts in the overhanging portion of theflange and a part Cw which acts in the web, as shown in Fig. 4-20. Thus,

(4-24)

The following condition must be satisfied to ensure horizontal equilibrium:

Solving for x yields x = 203.8 mm. Therefore s = � x = 0.9 � 203.8 = 183.4 mm is greater thanhf = 150 mm, as expected. Before the ultimate moment can be calculated, the validity of the initialassumption that �ps is greater than �py must be checked. �ps is calculated by combining Eqs. 4-8through 4-11, and by noting that the effective prestress acting on the section (including all lossesat the time under consideration) is given by P = � Aps fse = � 800 � 1060 � 10�3 = � 848 kN.Hence,

T A fps ps� � � � ��800 1617 10 12943 kN

C f bhfmax cu f= = × − × × × = −−α 0 45 50 10 350 150 11813. e j kN

C f b b hf cu w f= − = × − × × − × = −−α c h e j b g0 45 50 10 350 150 150 6753. kN

C f b x

x

x

w cu w�

� � � � � �

� �

�

� �

0 45 50 10 150 0 9

3 038

3. .

.

d i

T C C xf w+ + = − − =1294 675 3 038 0.

�

�

sese

p

cec

f

E

P

A

Pe

I E

� ��

�

� �FHG

IKJ

��

��

� �

�

FHG

IKJ

� �

1060

195 100 005436

1 848

165 10

848 290

8 938 10

1

340 000386

3

2

3

2

9

.

..

x

150

60

�cu = 0.0035

�s

Neutral axis

d = 640h =700

Aps = 800 mm2

hf =150

bw = 150

Cf

Cw

0.45 fcub = 350

350

s x= 0.9

T

e

A

I

=

= − =

= ×

= ×

eccentricity of the tendon

mm

mm

mm

700

260 290

165 10

8 938 10

3 2

9 4.



This value of �ps is larger than �py = 0.01329, as assumed, and therefore the calculated value of xis correct. The ultimate moment is subsequently calculated by considering moment equilibrium aboutthe line of action of T. Thus,

The magnitude of Cw to be used in the above expression is found by substituting x = 203.8 mminto Eq. 4-24, while the magnitude of Cf = � 675 kN remains unchanged. Therefore,

If the section contains non-prestressed reinforcement As (often referred to as slack reinforcement),the procedure for calculating the ultimate moment remains exactly the same as for the sectionsconsidered above, the only difference being that the two types of steel are considered separately asshown in Fig. 4-21. When calculating the tension in the prestressing steel Tps and in thenon-prestressed steel Ts, the difference in the strain histories of the two types of steel must beaccounted for in the analysis. Equation 4-12 can be used for calculating Tps, where fps correspondsto the total strain �ps = �s1 � �ce + �se (see Fig. 4-21 and Eq. 4-8). On the other hand, Ts is calculatedfrom the stress fs, corresponding to the strain �s2 (see Fig. 4-21), acting on the area As. Therefore,

(4-25)

It is important to note that fps and fs cannot be obtained from the same stress-strain relationship,but that they must be determined from the stress-strain curves which apply to the prestressing steeland to the non-prestressed steel, respectively. Figure 2-22 shows the design stress-strain relationshipprescribed by SABS 0100 (Ref. 4-2) for non-prestressed reinforcement. The procedure for calculatingthe ultimate moment of a prestressed concrete beam section containing non-prestressed reinforcementis illustrated by example 4-6.

� �

� � � �

s cu

ps s ce se

d x

x�

�FHG

IKJ �

�FHG

IKJ � �

� � � � � ��

640 2038

20380 0035 0 007493

0 007493 0 000386 0 005436 0 01332

.

.. .

. . . .

M C dh

C dx

u ff

w�� FHG

IKJ

� �FHG

IKJ2 2

�

Mu � � �� FHG

IKJ � � � ��

�FHG

IKJ �

�

� �675 640150

210 3038 2038 640

0 9 2038

210

720 7

3 3. .. .

. kN. m

T A fs s s=

x

b

d1d2

�x

Strain distribution Resultant forces

�cu

�s2

�s1

��se ce

Neutral axis

ApsAs

C

Tps

Ts

� fcu

Figure 4-21: Analysis at ultimate of a prestressed concrete beam section containing non-prestressed reinforcement.

ANALYSIS 4-27

EXAMPLE 4-6

Determine the design ultimate moment of the I-shaped prestressed concrete beam section shown inFig. 4-22. The dimensions of the section as well as the material properties of the concrete and theprestressing steel are exactly the same as for the section of example 4-5. However, this sectioncontains two Y20 non-prestressed reinforcing bars in addition to five 12.9 mm 7-wire super gradestrand, the position of which is shown in Fig. 4-22. Take fy = 450 MPa and Es = 200 GPa for thenon-prestressed reinforcement. Use the equivalent rectangular stress block as well as the designstress-strain curves for strand and for non-prestressed reinforcement as prescribed by SABS 0100(Ref. 4-2), and assume that fse = 1116 MPa at the time under consideration. Aps = 5 � 100 = 500 mm²and As = 628 mm².

As for example 4-5, � = 4.5 and � = 0.9 while the stress-strain curve for the prestressing steel isas shown in Fig. 4-17. The design stress-strain curve for the non-prestressed reinforcement is shownin Fig. 4-23, and is obtained by setting fy = 450 and �m = 1.15 MPa in Fig. 2-22. Therefore,

Assume �ps > �py and �s2 > �sy

If �ps > �py (= 0.01329) then fps = fpy = 1617 MPa (see Fig. 4-17), while fs = fsy = 391.3 MPa for�s2 > �sy (= 0.00196) (see Fig. 4-23). Tps and Ts are subsequently calculated from Eqs. 4-12 and4-25, respectively:

ff f

Esyy

msy

sy

s

� � � � ��

��

�450

1153913

3913

200 100 00196

3..

..MPa and

x

d 1=

640

d 2=

650

�s2

�s1

Neutral axis

Tps

Ts

150

5060

�cu = 0.0035

h=

700

Aps = 500 mm2As = 628 mm2

hf =150

bw = 150

b = 350

350

C

0.45 fcu

0.9 x

e

A

I

=

= − =

= ×

= ×

eccentricity of the tendon

mm

mm

mm

700

260 290

165 10

8 938 10

3 2

9 4.


Stress, fs

fsy = 391.3 MPa

Strain, �s�sy =

0.00196

Figure 4-23: Stress-strain curve for the non-prestressed reinforcement.


The magnitude of the maximum compression force which can be supplied by the flange

is larger than the total tensile force which can be provided by the prestressed and non-prestressedreinforcement . This means that the entire compression zoneis contained in the flange, and that C can be expressed as a function of x as follows (see Eq. 4-14):

As in the previous examples, x is calculated by considering horizontal equilibrium, according towhich the following condition must be satisfied:

Solving for x yields x = 148.8 mm. Therefore s = 0.9 x = 0.9 � 148.8 = 133.9 mm is less thanhf = 150 mm, as expected. Before the ultimate moment can be calculated, the validity of the initialassumption that �ps > �py and that �s2 > �sy must be checked. �ps is calculated by combining Eqs. 4-8through 4-11, and by noting that the effective prestress acting on the section (including all lossesat the time under consideration) is given by . Hence,

�s2 is calculated by considering the strain distribution (see Fig. 4-22). Thus, considering similartriangles:

From the above it is clear that �ps is larger than �py = 0.01329 and that �s2 is larger than �sy =0.00196, as assumed, and therefore the calculated value of x is correct. The ultimate moment isfinally calculated by considering moment equilibrium about the line of action of C. Thus,

The magnitudes of Tps and Ts to be used in the above expression are as calculated above. Uponsubstitution of these values:

As previously discussed in this Section, underreinforced sections are desirable because they exhibita gradual ductile failure with large accompanying deformations, as opposed to overreinforcedsections which fail suddenly in a brittle manner with small accompanying deformations. Ductile

T A f

T A f

ps ps ps

s s s

� � � � �

� � � � �

�

�

500 1617 10 808 7

628 3913 10 245 7

3

3

.

. .

kN

kN

C f bhfmax cu f= = × − × × × = −−α 0 45 50 10 350 150 11813. e j kN

T T Tps s= + = + =808 7 245 7 1054. . kN

C x f b x x xcu( ) . . .� � � � � � � � � �� 0 45 50 10 350 0 9 7 0883d i kN

T C x x� � � �( ) .1054 7 088 0

P f Ase ps� � � � � � � ��1116 10 500 5583 kN

�

�

� �

� � � �

sese

p

cec

s cu

ps s ce se

f

E

P

A

Pe

I E

d x

x

� ��

�

� �FHG

IKJ

��

��

� �

�

FHG

IKJ

� �

��F

HGIKJ �

�FHG

IKJ � �

� � � � � ��

1116

195 100 005723

1 558

165 10

558 290

8 938 10

1

340 000254

640 1488

14880 0035 0 01156

0 01156 0 000254 0 005723 0 01753

3

2

3

2

9

11

1

.

..

.

.. .

. . . .

� �s cu

d x

x22 650 148 8

148 80 0035 0 01179�

�FHG

IKJ �

�FHG

IKJ � �

.

.. .

M T dx

T dx

u ps s� �FHG

IKJ � �

FHG

IKJ1 22 2

� �

Mu � ��F

HGIKJ � � �

�FHG

IKJ �

�

� �808 7 6400 9 1488

210 2457 650

0 9 1488

210

606 7

3 3.. .

.. .

. kN. m

ANALYSIS 4-29

behaviour is generally characterised by a large curvature at failure �u relative to the curvature atfirst yielding of the steel �y (see Fig. 4-10). The conditions under which a prestressed concrete beamsection will exhibit a large value of �u and, hence, a more ductile behaviour, can be identified byrealising that the magnitude of �u is increased if the magnitude of x is reduced. This trend becomesapparent when considering the expression �u = �cu / x: If x is decreased, then �u must increaseprovided �cu is taken to remain constant. Figure 4-24 illustrates this effect and also demonstratesthat the change in strain �s induced by the ultimate moment and, therefore, that the total strain inthe steel at ultimate is increased when x is decreased. Note that although �cu generally does not varymuch for normal strength concrete, its magnitude can vary significantly in the case of very highstrength concrete.

The manner in which the steel content, the strength of the steel and the concrete strength influencethe magnitude of x can, in turn, be demonstrated by considering Eq. 4-22a (which was derived byconsidering horizontal equilibrium):

(4-22a)

Note that although this equation was derived on the basis of an equivalent rectangular stress blockand on the assumption that the stress-strain relationship of the steel shows a definite yield plateau,the trends identified below remain true in general. It should also be noted that the equation assumesthat stress in the steel at ultimate is equal to the yield stress and, hence, that the strain in the steelat ultimate exceeds the yield strain. Inspection of Eq. 4-22a reveals that:

• The magnitude of x is increased if the amount of steel Aps is increased.

• The magnitude of x is increased if the strength of the steel, as reflected by fpy, is increased.

• The magnitude of x is decreased if the concrete strength fcu is increased.

Bearing these trends in mind, together with the fact that the magnitude of �u is increased if themagnitude of x is reduced, it follows that the steel content, strength of the steel and the concretestrength influence the magnitude of �u as summarised below:

• If either the steel content or the strength of the steel is increased, x is increased and �u isdecreased. Therefore, the ductility of the section is reduced under these conditions.

• If the concrete strength is increased, x is decreased and �u is increased. Under these conditionsthe ductility of the section is, therefore, increased.

The importance of providing a section which is ductile cannot be over-emphasised and, for thisreason, ductility should always be a prime design consideration. The manner in which provision is

xA f

f bps py

cu

� ��

�sa

�ua

�ub

�cu

�sb

xa

xbx xa b

ua ub

sa sb

��

Figure 4-24: Influence of depth to neutral axis x on the curvature at ultimate �u.


made for sufficient ductility of a prestressed concrete beam section in design is covered inSection 4.4.4.

4.3.6 Analysis of beams with unbonded tendons

There is a significant difference between the behaviour of prestressed concrete beams with bondedtendons and that of beams with unbonded tendons. It is instructive to investigate some of thesedifferences before presenting the procedures for analysing beams with unbonded tendons.

Since the tendons in an unbonded beam are not bonded to the concrete, the compatibility assumptionthat the changes in strain in the steel are the same as in the adjacent concrete is no longer valid.Instead, any change in strain in an unbonded tendon will be distributed over its entire length. Thechange in steel strain resulting from an applied load can be calculated by making use of the factthat the total change in the elongation of the steel and of the concrete adjacent to the steel must beequal because the steel is anchored to the concrete at the ends of the beam (see Fig. 4-25). If thebeam remains uncracked, the change in strain in the concrete at the level of the steel at any sectionalong the span is given by (see Fig. 4-25)

where M(x) = moment at the section under consideration

e(x) = eccentricity of the tendon at the section under consideration

The total change in elongation of the concrete adjacent to the steel, which is equal to the totalelongation of the steel, is

If the effects of friction are ignored, then the steel strain must be uniformly distributed over thelength of the tendon. Therefore, the change in steel strain is given by

(4-26)

where L = original length of the tendon

The effect of bond, or the lack thereof, on the change in steel strain induced by external load in anuncracked beam is illustrated by example 4-7.

�cc

M x e x

E I�

� �

�LM x e x

E Idx

c

L�

� � z0

� sc

LL

L

M x e x

LE Idx� �

� � z�

0

x

x

e x( )

M x( )

L

e, y

Centroidal axis

Bending moment

Figure 4-25: Change in steel strain in an unbonded prestressed concrete beam.

ANALYSIS 4-31

EXAMPLE 4-7

The simply supported prestressed concrete beam shown in Fig. 4-26 carries a uniformly distributedload w over a span L. The cable is straight and is placed at an eccentricity e. Assume that the beamremains uncracked to compare the change in steel strain induced by the load in the case where thesteel is unbonded to the change in strain in the case where the steel is bonded.

If the cable is unbonded and free to slip, the change in steel strain is the same over the entire lengthof the cable, and is given by Eq. 4-26:

For the beam considered here:

Substitution into Eq. 4-26 yields

Therefore,

If the steel is bonded to the concrete, the maximum change in steel strain induced by the load willoccur at the midspan section, where the moment is given by M = wL²/8. Therefore, in this case, themaximum change in steel strain is calculated from

Thus,

Therefore, it can be concluded that the change in steel strain in the unbonded beam is 2/3 of thechange in steel strain at the midspan section of the bonded beam, in the case considered here.

� sc

LL

L

M x e x

LE Idx� �

� � z�

0

e x e

M xwx

L x

b g

b g b g

=

= −2

� s unbondedc

L

c

Le

LE I

wxL x dx

we

LE I

Lx x� ��

L

NMM

O

QPPz 2 2 2 30

2 3

0

ε s unbondedc

wL e

E I= 1

12

2

� s bondedc c

M e

E I

wL e

E I� �

1

8

2

ε εs unbonded s bonded= 2

3

x

w

e

L

e, y

Centroidal axis



Example 4-7 clearly demonstrates that if the beam remains uncracked the relative movement betweenthe unbonded steel and the concrete leads to a lower change in steel strain than is the case at thecritical section of a bonded beam. Inspection of Eq. 4-26 also reveals that the magnitude of thisdifference is dependent on the shape of the bending moment diagram and on the cable profile.

It should be noted that since the change in steel stress, which arises from the change in steel strain,is normally small and therefore ignored in stress calculations before cracking, the relative movementbetween the unbonded steel and the concrete is not of much practical significance at this stage.However, after the section cracks the magnitude of this relative movement appears to increasesignificantly with increasing load so that the steel strain and, hence, the steel stress increases muchmore gradually than would be the case at the critical section of a bonded beam. Because of thisbehaviour, the situation often arises in unbonded beams that the steel stress fps is much less thanits ultimate strength fpu when the limiting crushing strain �cu is reached in the concrete.

Bearing in mind that the ultimate moment is calculated from Mu = Aps fps(d � �x/2), it is clear that,all other things being equal, a reduction in fps will reduce the flexural capacity of the section. Sincethe stress in the steel at ultimate fps is significantly less in an unbonded beam than in a bondedbeam, it is reasonable to expect the ultimate moment of resistance Mu of an unbonded beam to beless than that of the corresponding bonded beam. This is indeed the case, and the difference appearsto range between 10 and 30% (Ref. 4-10).

Another major difference between the post-cracking behaviour of a beam containing no bonded steeland a bonded beam is illustrated in Fig. 4-27. If the beam contains only properly detailed bondedsteel, many evenly distributed cracks will develop in the region of maximum moment (seeFig. 4-27a). At flexural failure none of the cracks will be particularly wide and the concretecompression zone will tend to fail over a relatively large length of the member, often at least equalto the effective depth of the prestressing steel (Ref. 4-12). This type of failure is ductile with largeaccompanying curvatures, rotations and deflections.

If, on the other hand, the beam contains no bonded reinforcement then there is a tendency for it todevelop a single large crack, or only a few large cracks (see Fig. 4-27b). Major stress and strainconcentrations occur at the top of these large cracks so that flexural failure tends to be localized ata section. This behaviour reduces the ultimate moment capacity of an unbonded beam and leads tosmaller average concrete strains at failure than is the case in bonded beams.

The presence of non-prestressed bonded reinforcement tends to spread the flexural cracks and tolimit their size, and can therefore significantly improve this undesireable behaviour. Such non-prestressed reinforcement will increase the flexural capacity of an unbonded beam, not only becauseof its contribution to the tensile force in the ultimate resisting couple, but also because of theadvantages to be gained from the resulting improved crack control. Some design codes of practicespecify minimum amounts of non-prestressed bonded reinforcement to be included in unbondedprestressed concrete beams.

Many small, evenly distributed cracks Single large crack

Crushing Crushing

{(b) Unbonded(a) Bonded

{

Figure 4-27: Flexural failure of bonded and unbonded prestressed concrete beams.

ANALYSIS 4-33

Analysis of the uncracked section

Before cracking, the change in stress induced in the tendons of an unbonded beam by the externalload is usually small: In fact, this stress change is even smaller than in the case of a bonded beam,as illustrated by example 4-7. Since, in the case of uncracked bonded sections, the change in steelstress resulting from the application of external load is normally small enough to be safely ignoredin the calculation of concrete stresses, there is usually no reason why this change in steel stressshould be accounted for in the case of unbonded sections. Therefore, concrete stresses in uncrackedunbonded sections are calculated in exactly the same way as in uncracked bonded sections (seeSection 4.3.3). However, if the total area of the preformed ducts in which the unbonded tendonsare contained forms a significant part of the cross-section, the section properties to be used for thecalculation of concrete stress should be based on the net concrete section instead of the gross section,as discussed in Section 4.3.3.

Ultimate moment

A rational procedure for the flexural analysis of a cracked unbonded prestressed concrete beamsection is complicated by difficulties associated with quantifying the various factors which influenceflexural behaviour after cracking, and is generally much more complex than that of a cracked bondedsection. It appears that the flexural strength of an unbonded section depends on the following factors(Ref. 4-13):

• Magnitude of the effective stress in the tendons.

• Span-to-depth ratio of the beam.

• Properties of the materials used in the member.

• Shape of the bending moment diagram.

• Cable profile.

• Coefficient of friction between the tendon and the duct.

• Amount of non-prestressed bonded reinforcement.

Because of the difficulties encounted in analytically treating the influence of these factors on themagnitude of the stress fps in the steel at ultimate, the tendency has been to make use of empiricaland semi-empirical expressions for estimating fps. These expressions are usually of the followinggeneral form:

where fse = effective prestress in the steel, including all losses

�fs = additional stress induced in the steel by bending of the beam under the ultimateload

Examples of such expressions may be found in SABS 0100 (Ref. 4-2), BS 8110 (Ref. 4-7) andACI 318-89 (Ref. 4-11). The expression prescribed by SABS 0100, which is the same as thatrecommended by BS 8110, is an example of a semi-empirical equation, and is presented below inthe notation used here:

(4-27)

where � is normally taken as the length of the tendon between end anchorages. This equation appliesto rectangular sections and flanged sections in which the compression zone is entirely contained inthe flange, and was derived on the basis of an assumed length of the zone of inelasticity within theconcrete of 10x. Further guidance on the reduction of � in the case of continuous multi-span memberscan be found in SABS 0100. If the section contains non-prestressed bonded reinforcement As,

f f fps se s= + ∆

f fd

f A

f bdfps se

pu ps

cupu= + −

LNM

OQP ≤7000

1 17 0 7� /

. .MPa


SABS 0100 suggests that the effect of this reinforcement can be approximately accounted for byadding to Aps an equivalent area of prestressing steel equal to As fy / fpu.

Once fps is known, the depth to neutral axis x is calculated by considering horizontal equilibriumof the section, and the ultimate moment Mu is subsequently calculated by considering momentequilibrium. The procedure is illustrated by example 4-8. It is extremely important to realize thatwhen an expression for fps recommended by a particular design code of practice is used, thesubsequent calculations for estimating the ultimate moment must be based on the provisions of thatcode. Simply mixing the provisions of various codes can lead to totally misleading results.

EXAMPLE 4-8

The simply supported concrete beam shown in Fig. 4-28 is post-tensioned by unbonded tendons.The properties of the materials and the section at midspan are exactly the same as the sectionconsidered in example 4-6 (see Fig. 4-22), the only difference being that the tendons are unbondedin this case. Make use of the appropriate provisions of SABS 0100 to calculate the design ultimatemoment of the midspan section.

The stress-strain curve for the non-prestressed steel is as shown in Fig. 4-23 while � = 0.45 and� = 0.9 for the equivalent rectangular stress block. The effect of the non-prestressed reinforcementon the magnitude of fps is accounted for by converting As to an equivalent area of prestressing steel

Recognizing that the length of the tendon between the end anchorages � is virtually equal to thespan of the beam (= 12 m), fps can be directly calculated from Eq. 4-27:

which is greater than 0.7 fpu = 1302 MPa. Therefore fps = 1302 MPa.

Assume �s2 > �sy (= 0.00196) so that, from Fig. 4-23, fs = fsy = 391.3 MPa. With fps and fs known,the magnitudes of Tps and Ts can be calculated from Eqs. 4-12 and 4-25, respectively:

� � ��

�AA f

fpss y

pu

628 450

18601519 2. mm

f fd

f A A

f bdps se

pu ps ps

cu

� � �� L

NMM

O

QPP

� � ��

� � �

LNM

OQP

�

70001 17

11167000

12000 6401 17

1860 500 1519

50 350 640

1421

1 1� /.

/.

.

d iMPa

MPa

T A f

T A f

ps ps ps

s s s

� � � � �

� � � � �

�

�

500 1302 10 6510

628 3913 10 2457

3

3

.

. .

kN

kN

12 m

Centroidal axis

w

e = 290

6 m 6 m


ANALYSIS 4-35

The entire compression zone is contained in the flange because the magnitude of the maximumcompression force is larger than thetotal tensile force which can be provided by the prestressed and non-prestressed reinforcement

. Therefore C is given by Eq. 4-14 as

The depth to neutral axis is subsequently calculated by considering horizontal equilibrium, fromwhich

Solving for x yields x = 126.5 mm. Therefore s = 0.9 x = 0.9 � 126.5 = 113.9 mm is less thanhf = 150 mm, as expected. The validity of the assumption that �s2 > �sy must also be checked. Asfor example 4-6, �s2 is calculated by considering the strain distribution (see Fig. 4-22). Thus,

It is therefore clear that �s2 is larger than �sy = 0.00196, as assumed, so that the calculated valueof x is correct. The ultimate moment is finally calculated by considering moment equilibrium aboutthe line of action of C. Thus,

The magnitudes of Tps and Ts to be used in the above expression are as calculated above. Uponsubstitution of these values:

Note that the ultimate moment is less than Mu = 606.7 kN.m obtained for the bonded beam sectionof example 4-6, as expected.

4.3.7 Flexural analysis of composite sections

A composite structure is defined as a structure composed of structural elements using materials withdifferent material properties. Composite structures in prestressed concrete typically consist of precastconcrete beams with an in situ concrete slab. The beams would normally be prestressed and the slabwould be of reinforced concrete. Although both elements are made of concrete, their materialproperties are likely to differ: The precast beams can be manufactured in a casting yard where highcontrol of quality is possible and a strength of 50 - 60 MPa is feasible, while a strength higher than30 MPa is probably not economical for the slab.

A cross-section of the precast beam and the slab is called a composite section and several examplesare shown in Fig. 4-29. It is also possible to post-tension the composite element longitudinally asshown in Fig. 4-29f, or transversely to increase the flexural resistance in that direction.

Making use of composite construction can result in savings in both construction cost and time. Atypical application is that of a road over rail bridge where the interruption of the railway line mustbe limited. The precast beams are erected first and can be used to support the formwork for theslab. Once the beams are in place, permanent formwork can be placed and the slab can be cast withminimum interruption to the rail traffic since scaffolding is not required for these stages ofconstruction.

C f bhfmax cu f� � � � � � � � �� 0 45 50 10 350 150 11813. d i kN

T T Tps s� � � � �651 245 7 896 7. . kN

C x f b x x xcu� � � � � � � � � � �� 0 45 50 10 350 0 9 7 0883. . .d i

T C x x� � � � �896 7 7 088 0. .

� �s cu

d x

x22 650 1265

126 50 0035 0 01448�

�FHG

IKJ �

�FHG

IKJ � �

.

.. .

M T dx

T dx

u ps s� �FHG

IKJ � �

FHG

IKJ1 22 2

� �

Mu � ��F

HGIKJ � � �

�FHG

IKJ �

�

� �651 6400 9 126 5

210 245 7 650

0 9 126 5

210

525 3

3 3. ..

. .

. kN. m


The analysis of a composite section can be carried out as for non-composite sections if the followingfour main differences are taken into account (Ref. 4-14):

1. The loading stage under consideration will determine if it is the precast section only or thecomposite section resisting the loads.

2. A transformed effective flange width must be determined for the composite section to accountfor the difference in the stiffnesses of the materials used for the slab and for the beam.

3. The analysis of the composite section is based on the assumption that the horizontal shearresistance at the interface between the precast beam and the in situ slab is sufficient to ensurecomposite action.

4. Differential shrinkage takes place between the precast beam and the in situ slab, and the tensilestresses induced at the bottom of the precast member may need to be accounted for.

These differences will be addressed in each of the following sections as they arise.

Analysis of the uncracked section

Consider a composite section consisting of a precast beam section supporting an in situ slab asshown in Fig. 4-30. Assume that no temporary supports are used during construction so that thebeams support the formwork for the slab as well as the slab itself. The different loading stages andthe corresponding section resisting the loads can best be determined by considering the constructionprocedure (see Table 4-4):

(a) At transfer, when only the initial prestressing force Pt

and the moment induced by the selfweight of the beam Mb are acting on the precast beam section.

In situ concreteslab

Precast prestressedconcrete beam

(a)

(b)

(c)

(d)

(e)

(f)

Post-tensionedtendon

Figure 4-29: Typical cross sections of composite beams (Ref. 4-14).

ANALYSIS 4-37

(b) After the prestress losses have taken place, when the effective prestressing force Pe togetherwith the moment induced by the self weight of the beam M

bare acting on the precast beam

section.

(c) Directly after the slab has been cast, when the slab self weight moment Mf (including the weightof any formwork), the beam self weight moment Mb and the effective prestressing force Pe areacting on the precast beam section.

(d) After the concrete in the slab has hardened any additional loads, such as live loads, act on thecomposite section. The stresses caused by the additional loads must be added to the existingstresses in the precast beam.

The stress distributions at each of these load stages are shown in Fig. 4-30, from which it can beseen that the critical stages are (a) and (d). Loading stage (a) occurs at transfer when the maximumprestressing force acts together with minimum loading to induce maximum compressive and tensilestresses in the bottom and top fibres of the precast beam, respectively. This loading stage has beendiscussed in Section 4.3.3. Loading stage (d), where the minimum prestressing force is presenttogether with the maximum external loading, is the other critical stage. Here, the top and bottomfibres of the precast beam are subjected to maximum compressive and tensile stresses, respectively.At this loading stage the in situ slab will also be subjected to high compressive stress.

Although several different combinations of composite construction exist, as shown in Fig. 4-29, onlythe simple case shown in Fig. 4-29a using unpropped construction is considered here. In certaincases the precast beam and the in situ portions can overlap (see Fig. 4-29c), with the result thattwo different stresses can occur at the same level in the composite section, as shown in Fig. 4-31.However, the principles of analysis, as presented in this Section, can still be applied.

When a flanged beam is subjected to an applied loading, the compressive stresses in the flange willvary over the width of the flange, as illustrated in Fig. 4-32 for a simply supported beam. Thevariation is caused by shear lag effects and is dependant on several factors such as the type ofloading, dimensions of the cross-section and time dependant properties of the concrete (including

In situconcrete slab


P Mt b+ P Me b+

(a) (b) (c) (d)

P M Me b f+ +Mf

+ =

P M M Me b f L+ + +ML

+ =

Figure 4-30: Distribution of stress in a composite section during service (Ref. 4-10).

Loading stage Loads Resisting section

(a) Tensioning of precast beam Pt + Mb Precast beam

(b) Precast beam after losses Pe + Mb Precast beam

(c) Casting of slab Pe + Mb + Mf Precast beam

(d) Live and superimposed dead load Pe

+ Mb

+ Mf

+ ML

Composite section

Table 4-4: Loading stages on a composite section during service.


creep and shrinkage). Since an exact theoretical analysis is usually not justified, the followingsimplified approach is followed. The actual flange is replaced by a fictitious flange having aneffective flange width be so that it carries the same load as the actual flange. The following valuesfor be are recommend by local design codes (Ref. 4-2, 4-6, 4-7 and 4-8)

where Lz = distance between points of zero moment. For continuous beams Lz may be takenas 0.7 times the effective span.

S = the spacing of the webs, i.e. the actual flange width (See Fig. 4-33a)

It should be noted that TMH7 (Ref. 4-6) places the point of zero moment at a distance of 0.15 timesthe effective length of the span from the support. This implies that for the end span of a continuousbeam, as shown in Fig. 4-33b, the distance between points of zero moment Lz = 0.85 L.

For the analysis, the plane sections assumption is used to determine the distribution of strain in thesection. The difference between the modulus of elasticity of the concrete used for the precast beam

bb L S

b L Sew z

w z

��

� �

RS|

T|0 2

01

.

.

b gb g

for T - sections

for L - sections

(4-28a)

(4-28b)

In situconcrete slab


P Mt b+

(a) (b) (c)

P M Me b f+ + P M M Me b f L+ + +

Precastbeam

In situslab

Figure 4-31: Stress distribution for overlapping a composite section.

L

Bending moments

Figure 4-32: The effects of shear lag on the distribution of compressive stress in a flanged beam.

ANALYSIS 4-39

and that used for the in situ slab leads to different stresses being induced in the section by equalstrain. It is recommended that this difference be taken into account when the concrete strength differsby more than 10 MPa (Ref. 4-2 and 4-6). The calculations can be simplified by transforming onematerial to the other and it is generally more convenient to transform the slab material to the beammaterial. This is accomplished by replacing the effective width be of the slab by a transformed widthbft as follows:

(4-29)

where nc = modular ratio =

Ec,f = modulus of elasticity of the concrete in the in situ slab

Ec,b = modulus of elasticity of the concrete in the precast prestressed beam

After the section has been transformed to one material, a T-section is used for the analysis, whichproceeds as discussed in previous Sections. It is important to note that the stresses calculated in theslab on the basis of the transformed section must be transformed back to the slab material to obtainthe actual stresses in the slab. This is done by multiplying the stresses in the transformed slab bythe modular ratio nc.

At transfer, the stresses in the concrete can be calculated by Eqs. 4-5 and 4-6 using the beam sectionproperties because all the loads, including prestress, are resisted by the beam section only. However,at loading stage (d) (Fig. 4-30), superimposed loads applied after the slab concrete has hardenedare resisted by the composite section, and this fact must be accounted for in the calculation of stress.Hence, the concrete stresses at this stage can be calculated as follows (see Fig. 4-38):

(4-30a)

(4-30b)

where

Ab = Area of the precast beam

Ztop,b = Ib / ytop,b = Section modulus of the beam section with respect to the extreme top fibreof the precast beam, located a distance ytop,b from the centroid of the precastbeam

Zbot,b = Ib / ybot,b = Section modulus of the beam section with respect to the extreme bottomfibre of the precast beam, located a distance ybot,b from the centroid of theprecast beam

b n bft c e�E

Ec f

c b

,

,

fP

A

Pe

Z

M M

Z

M

Z

fP

A

Pe

Z

M M

Z

M

Z

top bb top b

b f

top b

L

top cb

bot bb bot b

b f

bot b

L

bot cb

,, , ,

,, , ,

� � ��

�

� � ��

�

be

bw

Actual stressdistribution

Equivalent stressdistribution

L

L = Lz 0.85

(a) (b)

S

S

Figure 4-33: Effective flange width be.


Ib = Second moment of area of the precast beam

Ztop,cb = Ic / ytop,cb = Section modulus of the composite section with respect to the extreme topfibre of the precast beam, located a distance ytop,cb from the centroid of thecomposite section

Zbot,cb = Ic / ybot,cb = Section modulus of the composite section with respect to the extreme bottomfibre of the precast beam, located a distance ybot,cb from the centroid of thecomposite section

Ic = Second moment of area of the composite section

EXAMPLE 4-9

A simply supported composite beam has a span of 15 m and the cross section shown in Fig. 4-34.The precast prestressed beams are spaced at a distance of 1200 mm. In addition to self weight, thebeam must support an additional uniformly distributed load of 16 kN/m. Assume unproppedconstruction and determine the elastic stresses at the following stages:

(a) At transfer of prestress. Take Pt

= 1500 kN.

(b) Just before the slab is cast, with P1 = 1350 kN (assuming some loss has taken place).

(c) Just after the slab has been cast, with P2 = 1350 kN.

(d) After a long time with no additional loads. Take P3 = 1200 kN (assuming all the losses havetaken place).

(e) After a long time with additional loads. Take P4 = 1200 kN.

The loading and corresponding maximum midspan moments are calculated in the following table:

Beam self-weight Slab self-weight Additional load

w (kN/m) 5.88 4.32 16.0

M (kN.m) 165.4 121.5 450.0

The effective flange width can be calculated from

This is greater than the actual flange so that bf = 1 200 mm is used.

b b Le w z� � � � � �0 2 350 0 2 15 000 3 350. . mm

hb = 700

hf = 150

d = 630

bf = 1200

bb = 35070

Concrete material properties:

fcu,b = 50 MPa Ec,b = 34 GPa

Ec,f = 28 GPafcu,f = 30 MPa

Precast beam

In situ slab

Unit weight �c = 24 kN/m3

Cross section at midspan

In situ slab

Precast beam

Figure 4-34: Composite cross section for Example 4-9.

ANALYSIS 4-41

To determine the section properties of the composite section the modular ratio is required

The transformed flange width is calculated from

The distances to the various fibres of importance, measured from the centroid of the transformedsection are given in Fig. 4-35.

The eccentricity of the prestressing force with regard to the precast beam e = 630 −700/2 = 280 mm,and the section properties are summarized as follows:

In situ slab(transformed)

Precast beam Compositesection

Area (� 103 mm2) 148.2 245.0 393.2

Second moment of area (� 109 mm4) � 10.00 26.96

(a) Stresses at transfer:

(b) Stresses in the beam just before casting the slab:

nE

Ecc f

c b

� � �,

,

.28

3408235

b n bft c e� � � �08235 1200 988 2. . mm

fP

A

P e M h

Itop btt

b

t b b

b,

/

.

. . .

� ��

��

��

�

� � � �

b gb g

d i

2

1500

245

1500 10 280 1654 10 350

10 106122 8 908 2 79

3 6

9

MPa

fP

A

P e M h

Ibot btt

b

t b b

b,

/. . .� �

��

b gb g26122 8 908 15 03 MPa

fP

A

P e M h

Itop bb

b b

b,

/

.

. . .

11 1

3 6

9

2

1350

245

1350 10 280 165 4 10 350

10 10551 7 439 193

� ��

��

��

�

� � � �

b gb g

d i

MPa

ytop,b = 350�

ytop,cf = 339.8�ytop,cb = 189.8�

ybot,cb = 510.2ybot,b = 350

Centroid ofcompositesection

Centroidof beamsection

Figure 4-35: Locations of section centroids for Example 4-9.


(c) Stresses in the precast beam when slab is cast:

(d) Stresses in the composite section after a long time without additional load:

(e) Stresses in the composite section after a long time with additional load:

In the slab

In the beam

fP

A

P e M h

Ibot bb

b b

b,

/. . .1

1 1 2551 7 439 12 95� �

��

b gb gMPa

fP

A

P e M M h

Itop bb

b f b

b,

/

. .

. . .

22 2

3 6

9

2

1350

245

1350 10 280 1654 1215 10 350

10 10551 3188 2 32

� ��

��

��

�

� � � � �

d ib g

d i

MPa

fP

A

P e M M h

Ibot bb

b f b

b,

/. . .2

2 2 2551 3188 8 70� �

� ��

d ib gMPa

fP

A

P e M M h

Itop bb

b f b

b,

/

. .

. . .

33 3

3 6

9

2

1200

245

1200 10 280 1654 1215 10 350

10 104 898 1719 318

� ��

��

��

�

� � � � �

d ib g

d i

MPa

fP

A

P e M M h

Ibot bb

b f b

b,

/. . .3

3 3 24 898 1719 6 62� �

� ��

d ib gMPa

fM y

Intop f

L top cf

cc,

, .

..

. . .

4

6

9

450 10 339 8

26 96 1008235

5 671 08235 4 67

�FHG

IKJ

��

�

F

HG

I

KJ �

� � � � � MPa

fM y

Inbot f

L top cb

cc,

, .

..

. . .

4

6

9

450 10 189 8

26 96 1008235

3167 08235 2 61

�FHG

IKJ

��

�

F

HG

I

KJ �

� � � � � MPa

fP

A

P e M M h

I

M y

Itop bb

b f b

b

L top cb

c,

,/

..

.. . .

44 4

6

9

2

3179450 10 189 8

26 96 103179 3167 6 35

� ��

�

� � ��

�

� � � � �

d ib g

MPa

ANALYSIS 4-43

The calculated stresses are summarized in Fig. 4-36.

Differential shrinkage

If the precast beam is relatively old when the in situ slab is cast, much of the creep and shrinkageof the precast beam has already taken place. Therefore, the shrinkage of the in situ slab is greaterthan the magnitude of the remaining creep and shrinkage of the precast beam. The resultingshortening of the slab relative to the precast beam is called differential shrinkage.

Figure 4-37 shows the strains and stresses caused by differential shrinkage in the composite section.These can be determined by making use of compatibility and equilibrium (Ref. 4-16). Assume thatthe beam and the slab act independently since casting of the in situ concrete, and that the followingstrains have occurred (see Fig. 4-37a):

�sf = unrestrained shrinkage of the in situ slab

�top,b �bot,b = unrestrained creep and shrinkage at the top and bottom of the precast beam

A position is required where compatibility can initially be established. Since any position may bechosen that will correspond to the deformed shape of the beam, a vertical position is selected, asshown in Fig. 4-37b, to simplify the problem. To achieve this position of compatibility, a momentMb must be applied to the beam, and a tensile force F must be applied to the in situ slab. Themoment Mb can be calculated as follows:

(4-31)

where �b = curvature of the beam section =

hb = height of the precast beam section

fP

A

P e M M h

I

M y

Ibot bb

b f b

b

L bot cb

c,

,/

..

.. . .

44 4

6

9

2

6 617450 10 510 2

26 96 106 617 8515 190

� ��

�

� � ��

��

d ib g c h

MPa

M E Ib b c b b= −� ,

1

hbbot b top b� �, ,�d i

ytop,c =339.8

ybot,c =510.2

bft = 988.2

bf = 1200

Centroid ofcompositesection

2.79

�15.03

1.93

�12.95

�2.32

�8.70

�3.18

�6.62

�4.67

�2.61�6.35

1.90

(a) (b) (c) (d) (e)

Figure 4-36: Stress distribution (in MPa) in the composite section of Example 4-9.


The force in the slab required to take up the differential strain can be determined from

(4-32)

where Af = cross sectional area of the in situ slab

�diff = differential shrinkage strain

= �sf � �avg,b

�avg,b = average strain in the beam

= for a symmetric beam

The beam and the slab can now be joined and the applied force F and moment Mb cancelled byapplying equal and opposite forces and moments to the composite section, as shown in Fig. 4-37c.It is customary to use a set of equivalent forces as shown in Fig. 4-37d, where

(4-33)

F E Adiff c f f� �� ,

� �bot b top b, , /�d i 2

M F yh

Mc top cf

b� �FHG

IKJ

�, 2

In situ slab

Precastprestressedbeam

�sf

�sf

�diff

�top,b

�bot,b �avg,b

Positionat casting

(b)Forces required for

compatibility

(a)Positions following unrestrained

shrinkage and creep

F

Mb

Final position

(c)Forces on composite

section

F

Mb

ytop,c

hf /2

Centroidal axisof the compositesection

hf

(d)Equivalent forces

on composite section

FMc

Figure 4-37: Differential shrinkage in a composite section.

ANALYSIS 4-45

The resulting stresses can now be determined as follows

(4-34a)

(4-34b)

(4-34c)

(4-34d)

where Aft = transformed cross sectional area of the in situ slab

Ac = cross sectional area of the composite section

The notation used for the distances to extreme fibres used in these equations are defined in Fig. 4- 38.It is important to note the sign convention assumed for determining these stresses. For a beamprestressed by a tendon located below the centroidal axis of the section, as shown in Fig. 4-37, thecreep and shrinkage strains will be negative with the corresponding curvature �b negative andmoment Mb (Eq. 4-31) positive. The differential shrinkage will usually be a negative value as thiswould indicate a shortening of the slab relative to the beam. The force F (Eq. 4-32) must reversethis shortening and is therefore a tensile (positive) force.

Creep itself tends to relieve the stresses caused by differential shrinkage, and the following reductionfactor can be applied to these stresses to account for this effect (Ref. 4-6 and 4-8):

(4-35)

where cc = ratio of creep strain to the elastic strain

The value of cc can range between 1.5 and 2.5 and an average value of 2, which results in areduction factor of 0.43, is often used for design. However, if high creep is expected because of,for example, a very dry environment, this value of cc must be increased to reflect the increasedcreep.

fF

A

F

A

M y

In

fF

A

F

A

M y

In

fM y

I

F

A

M y

I

fM y

I

F

A

M y

I

top fft c

c top cf

cc

bot fft c

c bot cf

cc

top bb top b

b c

c top cb

c

bot bb bot b

b c

c bot cb

c

,,

,,

,, ,

,, ,

� � �F

HG

I

KJ

� � �F

HG

I

KJ

� � �

� � �

��

�1 e cc

cc

ytop,c

ybot,c ybot,cbybot,b

ytop,b

ytop,cb ybot,cf

ytop,cf

Centroid of thecomposite section

Centroid ofthe precast beam

(b)Stresses caused by

differential shrinkage

(a)Definition of symbols

ftop,f

ftop,bfbot,f

fbot,b

Figure 4-38: Definition of symbols and stresses for differential shrinkage in a composite section.


EXAMPLE 4-10

Calculate the differential shrinkage stresses in the composite section of example 4-9. Because thein situ slab is cast 6 months after the beam is cast, assume that 60% of the creep and shrinkagehave already taken place in the beam at the time of casting of the slab. The following values forcreep and shrinkage apply:

�cr = �48 � 10�6 MPa�1 for creep of the precast beam

�sh = 310 � 10�6 for shrinkage of both the beam and slab

cc = 1.6

Assume that the remaining creep in the precast beam takes place under the stresses present at thetime the slab is cast. The stresses in the top and bottom of the beam are �2.322 MPa and�8.698 MPa, respectively. The creep strains in the top and bottom fibres of the beam will then be

The total strain in the precast beam including the remaining portion of the shrinkage is

The curvature of the beam caused by these strains is

while the moment required to rotate the beam through this curvature is

The average strain in the precast beam is given by

so that the differential shrinkage strain is

The tension force applied to the slab for compatibility is given by

The moment Mc is determined from

Finally, the reduction factor for creep is

� �

� �

cr top cr top b

cr bot cr bot b

f

f

, ,

, ,

. . . .

. . . .

� � � � ��

� � � � ��

� �

� �

0 4 0 4 48 10 2 322 44 58 10

0 4 0 4 48 10 8 698 167 0 10

26 6

26 6

� � �

� � �

top b cr top sh

bot b cr bot sh

, ,

, ,

. . . .

. . . .

� � � � � � � � � � � �

� � � � � � � � � � � �

� � �

� � �

0 4 44 58 10 0 4 310 10 168 6 10

0 4 167 0 10 0 4 310 10 2910 10

6 6 6

6 6 6

d i

d i

� � �bb

bot b top bh� � � � � �� 1 1

0 72910 168 6 10 174 9 106 6 1

, , .. . .c h b g m

M E Ib b c b b�� , ( . ) . .174 9 10 34 10 10 00 10 59 496 6 3 kN.m

� � �avg b bot b top b, , , . . .� � � � �� 1

2

1

22910 168 6 10 229 8 106 6d i

� � �diff sf avg b� � � � � � � � � � �� , . .310 10 229 8 10 80 21 106 6 6d i

F E Adiff c f f� � � � � � � � � �� , . .80 21 10 28 1200 150 404 26d i kN

M F yh

Mc top cf

b� �FHG

IKJ

� � � �FHG

IKJ � � ��

, . . . .2

404 2 339 8150

210 59 49 47 563 kN. m

��

��

�� 1 1

160 4988

1 6e ecc

cc

.

..

ANALYSIS 4-47

The stresses caused by differential shrinkage can now be calculated as follows

These stresses must be added to those caused by all the loadings after losses, as shown in Fig. 4-39.It can be seen from the final stresses that it is the tensile stress in the bottom of the precast beamthat is most significantly affected by differential shrinkage from the point of view of design.

fF

A

F

A

M y

Intop f

ft c

c top cf

cc,

,

.

.

.

.

. .

..

.

� � �F

HG

I

KJ

� � ��

�

FHG

IKJ

�

�

404 2

148 2

404 2

3933

47 56 339 8

26 96 10

28

340 4988

0 45

3

MPa

fF

A

F

A

M y

Inbot f

ft c

c bot cf

cc,

,

.

.

.

.

. .

..

.

� � �F

HG

I

KJ

� � ��

�

FHG

IKJ

�

�

404 2

148 2

404 2

3933

47 56 189 8

26 96 10

28

340 4988

056

3

MPa

fM y

I

F

A

M y

Itop bb top b

b c

c top cb

c,

, ,

. .

.

. .

..

.

� � �FHG

IKJ

��

��

� ��

�

FHG

IKJ

�

� �

59 49 350

10 10

404 24

3933

47 56 189 8

26 96 100 4988

172

3 3

MPa

fM y

I

F

A

M y

Ibot bb bot b

b c

c bot cb

c,

, ,

. .

.

. .

..

.

� � �FHG

IKJ

��

��

�

�

FHG

IKJ �

�

59 49 350

10 10

404 2

393 3

47 56 510 2

26 96 100 4988

0 97

3 3

MPa

�4.67

�2.61�6.35

1.90

(a)Stresses causedby all loadings

after losses

(b)Stresses causedby differential

shrinkage

(c)Total stress

0.45

0.56�1.72

0.97

�4.22

�2.05

��.07

2.87

+ =

Figure 4-39: Total stresses Example 4-10.


Ultimate moment

The flexural capacity of a composite section can be determined in the same way as for a flangedsection if provision is made for the difference in strength fcu of the concrete in the precast beamand in the in situ slab. This difference is generally only considered if it is more than 10 MPa.

The only way that the difference between fcu for the slab and for the beam impacts on the analysisof flexural strength is that it influences the calculation of the compressive force in the concrete.This can be accounted for either by transforming the slab concrete to the beam concrete on the basisof the strength ratio ncu (transformed section) or by making use of basic principles (untransformedsection) (see Fig. 4-40).

Consider the case where the compression zone extends into the precast beam, as shown in Fig. 4-40.The compression force in the slab Cf and in the beam Cw can be determined from the following:

(4-36a)

(4-36b)

where bft = transformed slab width = ncu be

ncu = strength ratio = fcu,f / fcu,b

fcu,f, fcu,b = characteristic concrete strength of the slab and beam, respectively

For the case where the compression zone is entirely contained in the slab (x < hf), Eq. 4-36 becomes

(4-37a)

(4-37b)

Horizontal shear

In both the preceding sections dealing with the analysis of the uncracked section and with theultimate strength of the composite section, the assumption was made the the section acts compositely.However, composite action is only possible if the induced horizontal shear can be transmitted acrossthe interface between the precast beam and the in situ slab.

Cf h b

f h b

C f x h b

fcu f f e

cu b f ft

w cu b f b

�RST

� �

�

�

�

,

,

,

for an untransformed slab width

for a transformed slab width

c h

Cf xb

f xb

C

fcu f e

cu b ft

w

�RST

�

�

�

,

,

for an untransformed slab width

for a transformed slab width

0

(a)Original slab width

(b)Transformed slab width

bb

bft

T

�fcu,b

Cf

Cw

hf

bb

be

x

T

x

�fcu,f

�fcu,b

Cf

Cw

Figure 4-40: Flexural capacity of a composite section.

ANALYSIS 4-49

There are two commonly used methods of calculating the horizontal shear stress at the precast-insitu concrete interface: The first method makes use of elastic theory while the second methodconsiders conditions at ultimate when plastic deformations have taken place in the section. It mustbe noted that the restrictions on shear stresses as recommended by design codes are dependent onthe method used for obtaining the interface shear stresses.

Elastic theory yields the following equation for determining the horizontal shear stress vhe at theinterface of an uncracked section

(4-38)

where V = shear force at the section where the shear stress is required

Q = first moment of area of the concrete on either side of the interface about theneutral axis of the transformed composite section

Ic = second moment of area of the transformed composite section

bv = width of the interface

The horizontal shear stress vhu at ultimate can be determined by dividing the horizontal force in theslab that has to be transmitted across the interface by the area of the interface. Figure 4-41a showsthe case where the compression zone is entirely contained in the slab, in which case the horizontalshear stress at the interface vhu is given by

(4-39)

where Lv = distance over which the force must be transmitted

For the case shown in Fig. 4-41b, the distance Lv extends from the section of maximum moment tothe point of zero moment, i.e. the support. In the case where the compression zone extends intothe precast beam, the compression force to be transmitted across the interface is taken as

.

It should be noted that the shear stress vhu is an average stress while the limiting values recommendedby design codes are given with regard to a maximum value. Recommendations exist (Refs. 4-7 and4-23) which suggest that the maximum value of the horizontal shear stress can be obtained bydistributing the average shear stress vhu along the length Lv in proportion to the vertical design shearforce diagram.

vV Q

I bhec v

�

vC

b L

f xb

b Lhuf

v v

cu f e

v v

� �� ,

� f h bcu f f e,

(a)Cross section at postionof maximum moment

(b)Beam elevation

hf

bv

Lv

be

x

T T

x

�fcu,f

Cf Cf0

Maximummomentsection

Minimummomentsection vhu

Figure 4-41: Horizontal shear in a composite beam.


4.4 DESIGN

4.4.1 Limit states design

All the design codes of practice normally used in South Africa for the design of prestressed concretestructures (Refs. 4-2, 4-6, 4-7 and 4-8) are based on the so called limit states design approach,which offers a rational and practical procedure for ensuring that there is an acceptable probabilitythat the structure will remain fit for its intended use during its design life. Any condition at whicha structure may become unfit for use constitutes a limit state, and the objective of the designprocedure is to ensure that such a limit state is not reached. Obviously, this approach requires thateach limit state must be examined separately to make sure that it has not been reached. These checkscan be made either on a deterministic or on a probabilistic basis, and the codes currently used inSouth Africa for the design of prestressed concrete structures all adopt a probabilistic basis. Thismeans that if the provisions of these codes are followed, each limit state is examined to establishif there is an acceptable probability of it not being reached (see Refs. 4-16, 4-17 and 4-24).

The various limit states can be placed in one of the two following categories:

• Ultimate limit states, which are concerned with the maximum load-carrying capacity of thestructure.

• Serviceability limit states, which are concerned with the normal use and durability of thestructure.

The limit states listed below are those applicable to SABS 0100 (Ref. 4-2), and are essentially thesame as those applicable to BS 8110 (Ref. 4-7).

Ultimate limit states

• Stability: The structure must remain stable under all the critical combinations of the designultimate loads. This requirement implies that no ultimate limit state is to be reached by ruptureof any section, by overturning or by buckling.

• Robustness: The design must be robust, in the sense that the failure of a single element or damageto a small area of the structure must not lead to the collapse of a major part of the structure.The structure should, therefore, not be unreasonably susceptible to the effects of accidents.

• Special hazards: If a potential hazard exists due to the nature of the occupancy, location or useof a structure (e.g. flour mill or chemical plant) the design must ensure that there is a reasonableprobability that the structure will survive an accident, even though it may be damaged.

Serviceability limit states

• Deflection: The deformation of the structure or any part thereof must be limited to ensure thatneither its appearance nor its performance is adversely affected. Possible damage to otherelements such as finishes, services, partitions, glazing and cladding, as well as to adjacentstructures is also a consideration in this regard.

• Cracking: The width of cracks must be controlled to ensure that the appearance, efficiency anddurability of the structure is not impaired.

• Vibration: Where there is a likelihood of the structure being subjected to excessive vibrations,appropriate measures must be taken to prevent discomfort or alarm to occupants, damage to thestructure, or interference with its proper function.

DESIGN 4-51

Other considerations

In addition to the limit states listed above, the following aspects must be considered in the design:

• Fatigue: The effects of fatigue must be considered if the nature of the imposed load on thestructure is predominantly cyclic.

• Durability: The durability of the structure must be considered in terms of its conditions ofexposure and its design life. Compliance with code recommendations regarding minimum concretecover to the reinforcement, minimum concrete strength and permissible crack width is intendedto ensure that the durability requirements of most structures are satisfied.

• Fire resistance: When a structural element may be exposed to fire its retention of structuralstrength, resistance to flame penetration and resistance to heat transmission must be considered.

• Lightning: Reinforcement may be used as part of a lightning protection system.

For a satisfactory design, the design resistance must exceed the design load effect at the ultimatelimit states and the design criteria must be satisfied at the serviceability limit states. In order tocarry out the necessary calculations to verify compliance with these requirements, the design materialstrengths and the design loads must be known. The approach followed by the limit states designmethod to determine these quantities, as implemented in the design codes of practice (Refs. 4-2,4-6, 4-7 and 4-8) and loading codes (Refs. 4-18, 4-19, 4-20 and 4-24) commonly used in SouthAfrica, is briefly described in the following.

Design material strengths

Material strengths are specified in terms of their characteristic values, which are defined as thestrength below which not more than 5% of the test results may be expected to fall (see Section 2.1.1for the specific case of the characteristic compressive strength of concrete). If it is assumed thatthe measured values of strength are normally distributed, this definition can be expressed as follows:

(4-40)

where fk = characteristic strength

fm = mean strength

� = standard deviation

The design material strength applicable to each limit state is derived from the characteristic strengthby dividing it by a partial safety factor for material strength �m. Thus,

design strength = (4-41)

The factor �m is intended to account for the following (Ref. 4-2):

• Possible reductions in the strength of the materials used in the actual structure as compared withthe characteristic values obtained from laboratory tested specimens.

• Local weaknesses.

• Inaccuracies in the assessment of the resistance of sections.

The value of �m depends on the material: By the nature of its manufacturing process, concrete is amore variable material than steel. It also depends on the importance of the limit state beingconsidered and, therefore, higher values are used for ultimate limit states than for serviceabilitylimit states. Values recommended by SABS 0100 (Ref. 4-2) for �m are listed in Table 4-5.

f fk m� � 164. �

f k

m�


Design loads

If sufficient statistical data were available, it would be possible to account for the variability of theloads acting on a structure by defining them in terms of characteristic values which have a 5%chance of being exceeded. Unfortunately, the data required to establish characteristic values for thevarious loads are not available at the present time, so that values are usually based on experienceand, possibly, on forecasts of the implications of future developments. Therefore, the values usuallygiven in the various loading codes are not characteristic values but are nominal values.

A design load, applicable to a particular limit state, is obtained by multiplying the correspondingnominal load by the appropriate partial safety factor �f. Thus,

design load = nominal load � �f (4-42)

The factor �f is intended to account for (Ref. 4-2):

• The possibility of unfavourable deviation of the loads from their nominal values.

• Inaccurate assesment of load effects.

• Unforeseen redistribution of stress within the structure.

• Variations in dimensional accuracy achieved during construction.

The value of �f depends on the following factors:

• Type of load: Higher values of �f are associated with loading types which, by their nature, aremore variable. Thus, for example, the value of �f for dead load will be less than the value forsuperimposed live load, because the variability of the dead load is less than that of the live load.

• Number of loads acting together: As the number of loads acting together increase, the value of�f for a particular load decreases because of the reduced probability that the various loads willall reach their nominal values simultaneously.

• Importance of the limit state: Higher values of �f are used for ultimate limit states than forserviceability limit states because of the requirement of having a smaller probability of the formerbeing reached.

Limit State Concrete Steel

Ultimate

Flexure or axial load 1.50 1.15

Shear 1.40 1.15

Bond 1.40

Others (eg. bearing stresses) � 1.50

Serviceability

Deflection 1.0 1.0

Cracking strength of prestressed concreteelements using tensile stress criteria

1.3 1.0

Table 4-5: Partial safety factors for material strength �m as recommended by SABS 0100(Ref. 4-2).

DESIGN 4-53

The combinations listed below for self-weight Dn, imposed loads Ln and wind loads Wn comply withthe recommendations of SABS 0160 (Ref. 4-18) at the ultimate limit state:

1.5Dn

1.2Dn + 1.6Ln

1.2Dn + 0.5Ln + 1.3Wn

0.9Dn + 1.3Wn

The following combinations comply with recommendations of SABS 0160 at the serviceability limitstate:

1.1Dn + 1.0Ln

1.1Dn + 0.3Ln + 0.6Wn

It should be noted that if a particular load in a load combination has a relieving effect on the loadeffect being considered, most loading codes will provide a reduced value of �f to be applied to thatparticular load.

The descriptions of the material and load factors given above are, strictly speaking, only applicableto the recommendations of SABS 0100 (Ref. 4-2) in this regard, which are, in principle, the sameas those of BS 8110 (Ref. 4-7). The codes for bridge loadings commonly used in South Africa(Refs. 4-20 and 4-24), however, differ slightly from SABS 0100 and BS 8110 in the manner inwhich the various factors which impact on the magnitude of the design material strengths and loadsare assigned to the material and load factors. The bridge loading codes also specify an additionalfactor �f3 with which the effects of the design loads must be multiplied to obtain the design loadeffects. However, the factors which all the different loading codes, referred to above, provide forin this regard are essentially the same. It is extremely important to note that the loads and loadfactors to be used must be obtained from the particular loading code specified by the design codeof practice being followed because the provisions of a code of practice is always dependent on thoseof a particular loading code.

The normal procedure followed in limit states design is to design on the basis of the expected criticallimit state and then to examine the remaining limit states to check that they are not reached. In theflexural design of prestressed concrete members, the critical limit state depends on the limitationsimposed by the limit state of cracking, which also provides the basis on which prestressed concreteelements are classified by the design codes of practice commonly used in South Africa. Thisclassification is essentially the same for all these codes of practice and is summarised below.

• Class 1 (full prestress): No tensile stress permitted.

• Class 2 (limited prestress): Tensile stresses permitted, but limited to the extent that no visiblecracks develop.

• Class 3 (partial prestress): Tensile stresses permitted, but with surface crack widths limited tovalues prescribed by the particular code being used.

Generally, the design of class 1 and class 2 members is governed by the serviceability limit stateof cracking, while the design of class 3 members tends to be controlled by the ultimate limit stateor the serviceability limit state of deflection.

4.4.2 Design for the serviceability limit state

Since the serviceability limit state of cracking governs the flexural design of class 1 and class 2members, the design procedure developed in this Section only covers class 1 and 2 pretensioned aswell as bonded and unbonded post-tensioned members. The flexural design of class 3 (partiallyprestressed) members is covered in Section 4.4.6.


The criteria which limit crack width are stated in terms of limiting concrete flexural tensile stressesby the design codes of practice used in South Africa (Refs. 4-2, 4-6, 4-7 and 4-8). In addition tothe limiting tensile stresses, these codes also specify maximum compressive stresses in the concretewhich may not be exceeded at the serviceability limit state. Although the purpose of the compressivestress limitations are not explicitly stated in the codes, it seems reasonable to assume that they areintended to prevent the development of excessive creep strains in the concrete under serviceabilityconditions, and to prevent micro-cracking and spalling of the concrete in the compression zone underserviceability conditions. The concrete stress limitations specified by SABS 0100 (Ref. 4-2) arelisted in Table 4-6. It should be noted that the concrete stress limitations are divided into two sets:One corresponding to conditions at transfer and another which applies to the serviceability limitstate. This follows because these conditions can normally be identified as being the most critical.

The design process for prestressed concrete members differs from that used for other constructionmaterials because a number of critical stages in the life of the structure, all related to the presenceof the prestressing force, can be identified. Of these, the stage corresponding to transfer of theprestressing force and the stage corresponding to maximum load at the serviceability limit state,after all the losses have occurred, generally appear to be the most important. These stages areillustrated in Fig. 4-42 for a simply supported beam subjected to a uniformly distributed load:

• At transfer of prestress (Fig. 4-42a), the prestress will be acting at its maximum value becausethe long-term losses have not yet taken place, while the applied external load will be acting atits minimum value because only the self weight of the beam will be present at this stage. Underthese conditions the beam will tend to deflect upwards and the stress distribution at the midspan

Class 1 members Class 2 members

Pretensioned Post-tensioned

At transfer

1. Compression

• Triangular or near triangu-lar distribution of prestress

0.45 fci

0.45 fci

0.45 fci

• Near uniform distributionof prestress

0.3 fci 0.3 fci 0.3 fci

2. Tension 1.0 MPa 0.45√fci

0.36√fci

Serviceability limit state

1. Compression

• Design load in bending 0.33 fcu * 0.33 fcu * 0.33 fcu *

• Design load in direct com-pression

0.25 fcu

0.25 fcu

0.25 fcu

2. Tension 0 0.45√fcu

** 0.36√fcu

**

fci = Concrete compressive strength at transfer

* Within range of support moments in continuous beams and other statically indeterminatestructures this limit may be increased to 0.4 fcu

** These stresses may be increased under certain conditions, as specified in SABS 0100

Table 4-6: Limiting concrete stresses in prestressed members, SABS 0100 (Ref 4-2).

DESIGN 4-55

section will show a small compression or even a tension in the top fibre and a large compressionin the bottom fibre.

• At the serviceability limit state the maximum service load will be acting together with theeffective prestressing force, which represents a minimum value because all the long-term losseswould have taken place at this stage. Under these conditions the beam will deflect downwardsand the stress distribution at the midspan section will show a large compression in the top fibreand a small compression or tension in the bottom fibre.

Since the stages described above are usually critical they will serve as the point of departure fordeveloping a design procedure. The magnitude of the stress limitations imposed by the design codesof practice also make it possible to assume a linear elastic uncracked section for purposes of analysis.Therefore, the criteria for design at the serviceability limit state can be stated as follows:

• At transfer of prestress: Ensure that the top and bottom fibre concrete stresses under maximumprestress and minimum applied moment Mmin do not exceed the allowable values for tension andcompression, respectively. The maximum prestressing force Pt corresponds to the value directlyafter transfer and includes all the instantaneous losses but excludes all time-dependent losses.The minimum moment is usually equal to the dead load moment.

• At the serviceability limit state: Ensure that the top and bottom fibre concrete stresses underminimum prestress and maximum applied moment Mmax do not exceed the allowable values forcompression and tension, respectively. The minimum prestressing force �Pt corresponds to thefinal value after the all the losses (instantaneous and time-dependent) have taken place. The factor� is the ratio of the final prestressing force to the initial value Pt which includes only theinstantaneous losses. It is also given by � = 1 � %long-term loss/100.

(a) Transfer of prestress

(b) At the serviceability limit state

Minimum loading

Maximum loading

Prestress at a maximum (before losses)

Prestress at a minimum (after losses)

�

�

Stress distributionat midspan section

Stress distributionat midspan section

Figure 4-42: Critical stages for a simply supported prestressed concrete beam.


Equations 4-5 and 4-6 can be used to express these criteria in terms of the following four stressinequality equations:

(4-43a)

(4-43b)

(4-43c)

(4-43d)

where

ftop,t, fbot,t = stress in the extreme top and bottom fibres, respectively, at transfer

ftop,s, fbot,s = stress in the extreme top and bottom fibres, respectively, at the serviceabilitylimit state

ftt, fct = allowable tensile and compressive stresses, respectively, at transfer

fts, fcs = allowable tensile and compressive stresses, respectively, at the serviceability limitstate

The design process basically involves a manipulation of these four stress inequality equations.However, before the design process can be developed expressions and procedures must be devisedto

• estimate the minimum required section properties in terms of Ztop and Zbot,

• establish the feasibility domain of Pt and e, and

• determine the so called permissible cable zone which delimits the zone along the span in whichthe cable may be placed.

Minimum required section properties

The purpose of the following derivation is to determine the minimum section properties which willsimultaneously satisfy the four stress inequality equations. This means that, from the design pointof view, the objective is to find the most efficient beam section. The derivation given here wasadapted from Ref. 4-14 and is credited to Guyon (Ref. 4-21).

Assume that the two allowable stresses ftt and fct are both attained at the critical beam section attransfer, as shown in Fig. 4-43a. If the effect of the prestress losses, which take place with thepassage of time, is superimposed on the stresses at transfer, then the stresses in the top and bottomfibres will be reduced to �1 and �2, respectively, as shown in Fig. 4-43b. It should, therefore, benoted that �1 and �2 correspond to the combined action of �Pt and Mmin. The stress changes ��topand ��bot induced in the top and bottom fibres by the application of an additional moment �M areshown in Fig. 4-43c, and the final stress condition at the serviceability limit state (see Fig. 4-43d)is reached when Mmin + �M = Mmax. If any of the allowable stresses at the serviceability limit statefcs or fts is exceeded, then the corresponding section modulus Ztop or Zbot is smaller than required(�� = �M / Z), and the opposite is true if an allowable stress is not attained. Therefore, themagnitudes of Ztop and Zbot for which the allowable stress requirements at the serviceability limitstate are exactly satisfied, as shown in Fig. 4-43d, represent minimum required values.

Referring to Fig. 4-43, the top fibre stress at transfer can be expressed as (Eq. 4-43a)

(4-44)

fP

A

P e

Z

M

Zf

fP

A

P e

Z

M

Zf

fP

A

P e

Z

M

Zf

fP

A

P e

Z

M

Zf

top tt t

top

min

toptt

bot tt t

bot

min

botct

top st t

top

max

topcs

bot st t

bot

max

bott s

,

,

,

,

�

� �

� �

�

� �

� �

fP

A

P e

Z

M

Zttt t

top

min

top

�

DESIGN 4-57

The top fibre stress under the action of Mmin, after all the prestressing losses have developed, isgiven by

which can be rewritten as

(4-45)

Substituting Eq. 4-44 into 4-45 yields

(4-46)

The stress increment induced by the additional moment �M in the top fibre is

(4-47)

The following condition must be satisfied to ensure that the allowable stress in the top fibre at theserviceability limit state is not exceeded:

(4-48)

Note that in Eq. 4-48 fcs will appear as a negative quantity because it represents a compressivestress. Substitution of Eqs. 4-46 and 4-47 into Eq. 4-48 yields

which, by noting that Mmax = Mmin + �M, can be solved to yield the following expression for Ztop:

(4-49)

A similar analysis of the state of stress in the bottom fibre will show that

(4-50)

Equations 4-49 and 4-50 can also be written in the following convenient form by using Mmax =Mmin + �M:

(4-51)

��

1 � P

A

P e

Z

M

Zt t

top

min

top

� ��

1 � F

HG

I

KJ �

P

A

P e

Z

M

Z

M

Z

M

Zt t

top

min

top

min

top

min

top

� � �1 1� �fM

Zt tmin

top

a f

��

� toptop

M

Z�

� �1 �� top csf

� �fM

Z

M

Zftt

min

top topcs � �1a f �

ZM M

f ftop

max min

cs tt

�

�

�

�

b g

d i

ZM M

f fbot

max min

t s ct

��

�

�

�

b g

d i

ZM M

f ftop

min

cs tt

�

�

1 �

�

a f

d i

�

(a) At transfer( + )P Mt min

(d) At the serviceabilitylimit state

( + )�P Mt max

(b)( + )�P Mt min

(c)( )�M

ftt

ft sfct

fcs

+ =Time

�1 ��top

��bot�2

+ =

Figure 4-43: Evolution of stress in a prestressed concrete beam.


(4-52)

Equations 4-49 and 4-50 or, alternatively, Eqs. 4-51 and 4-52 can be used to find a beam sectionwhich will satisfy the limiting stress criteria. Note, however, that these equations are functions ofMmin which, itself, is dependent on the cross-section so that the procedure by which a suitable sectionis found must involve some form of iteration. The process is usually started by assuming a valuefor Mmin based on experience, after which the minimum required values for Ztop and Zbot can becalculated and a suitable section subsequently selected. The validity of the assumed value of Mmincan then be checked and, if required, an improved value of Mmin can be used to calculate revisedminimum required values for Ztop and Zbot. This process normally converges rapidly, particularly asthe experience of the designer and, hence, the accuracy of the initial assumption for Mmin increases.

The following considerations are important when designing a beam section using the approachoutlined above:

• Although the section moduli may satisfy Eqs. 4-49 and 4-50, it is possible that the requiredeccentricity of the prestressing force may be larger than ybot, which means that the cable fallsoutside the section. In such a case the section must be revised to satisfy the practicalityconsideration that the cable must fall inside the section.

• Practical considerations, such as for example a required type of shape, usually lead to a sectionwhich can, at best, only satisfy one of Eqs. 4-49 and 4-50. Therefore, the normal situation isthat the section will have one section modulus which is approximately equal to the minimumrequired value while the other one is larger than required.

It is, once again, emphasised that the objective of the design approach followed here is to providethe most efficient cross-section and, hence, a least weight beam.

Magnel diagram

Once the section has been selected, the magnitude and corresponding eccentricity of the prestressingforce must be determined. Although numerous procedures for accomplishing this exist, the methodoriginally developed by Magnel (Ref. 4-22) still represents an extremely useful technique and ispresented here.

The procedure is basically a geometric interpretation of the four stress inequality equations,Eqs. 4-43, which are rewritten in the following form:

(4-53a)

(4-53b)

(4-53c)

ZM M

f fbot

min

t s ct

��

�

1 �

�

a f

d i

�

1

P

Z

Ae

f Z Mt

top

tt top min

L

NM

O

QP

�d i

1

P

Z

Ae

f Z Mt

bot

ct bot min

LNM

OQP

�d i

1

P

Z

Ae

f Z Mf Z M

Z

Ae

f Z Mf Z M

t

top

cs top max

cs top max

top

cs top max

cs top max

�

L

NM

O

QP

��

L

NM

O

QP

��

R

S

||||

T

||||

�

�

d i

d i

for

for

DESIGN 4-59

(4-53d)

At this stage of the design, the only two unknown quantities in the inequality Eqs. 4-53 are Pt ande if an a priori value has been taken for �. These equations can therefore be plotted at equality onthe e-1/Pt plane, in which case they will each plot as a straight line, as shown in Fig. 4-44. Eachline serves as a boundary which divides the plane into a part in which the inequality relationshiprepresented by the line is satisfied and another part in which it is not. Consider, for example, theline in Fig 4-44 which represents the inequality 4-53a at equality and, therefore, the stress inequality4-43a from which it was derived. All the points with coordinates e and 1/Pt which fall below theline will satisfy the stress inequality 4-43a while it is not satisfied by the points which lie abovethe line. If all the lines are examined in the same way, it can be concluded that the region boundedby the quadrilateral ABCD contains points with coordinates e and 1/Pt which satisfy all four thestress inequality equations and, therefore, represents a feasibility domain. It is important in thisregard to note that the so called Magnel diagram shown in Fig. 4-44 was constructed on thepresumption that fcsZtop < Mmax and ftsZbot < Mmax. The maximum practical eccentricity epl is alsoplotted on the Magnel diagram as a vertical line (Ref. 4-14), and is shown to intersect thequadrilateral ABCD. In this case the feasible domain is reduced to the region bounded by ABEFDwhich contains points which have e and associated Pt values satisfying not only the stress inequalitiesbut also the practicality requirements. If epl lies to the left of point A then no practical solutionexists and a revised section with larger section moduli must be selected. If, on the other hand, epllies to the right of point C then any point contained in ABDC will yield practically feasible valuesof e and Pt.

1

P

Z

Ae

f Z Mf Z M

Z

Ae

f Z Mf Z M

t

bot

t s bot max

t s bot max

bot

t s bot max

t s bot max

�

LNM

OQP

��

LNM

OQP

��

R

S

||||

T

||||

�

�

d i

d i

for

for

e

1

Pt

Domain offeasibility

A

O

B

D E

FC

epl

Eq. 4-53b

Eq. 4-53c Eq. 4-53dEq. 4-53a

�Z

Abot

�Z

Atop

Figure 4-44: The Magnel diagram.


Once the Magnel diagram has been constructed for a section, the required value of Pt andcorresponding value of e can be selected. Usually the value which leads to the smallest possiblevalue of Pt is selected. This point will correspond to the largest value of 1/Pt at the largest practicallyfeasible value of e (point F, Fig. 4-44).

The number of strands required for the selected value of Pt will depend on the maximum permissiblejacking force. The design codes of practice commonly used in South Africa (Refs. 4-2, 4-6, 4-7 and4-8) all recommend that the jacking force should normally not exceed 75% of the characteristicstrength of the tendon, but that it may be increased to 80% provided that special consideration isgiven to safety, to the stress-strain characteristics of the tendon, and to the assessment of the frictionlosses. BS 8110 (Ref. 4-7) also requires that the initial prestress at transfer should normally notexceed 70% of the characteristic strength of the tendon and that it must not exceed 75% under anycircumstances. The bridge codes (Refs. 4-6 and 4-8) specifically state that immediately aftertransferring the prestress, the prestressing force must not exceed 70% of the characteristic strengthof the tendons for post-tensioned tendons, or 75% for pretensioned tendons.

It should be noted that although the Magnel diagram is presented here as a design tool, it can alsoserve as a powerful analytical tool.

Permissible cable zone

The design of a beam section as well as the calculation of the prestressing force and its eccentricityis based on the conditions at the critical section, e.g. at the midspan section, in the case of asymmetrically loaded simply supported beam. Since the bending moment varies over the span, theeccentricity must normally be varied along the span to ensure that the stress limitations are notexceeded at other sections, if the same prestressing force is to be used over the entire span. Aprocedure is developed herein to determine the zone within which the cable can be placed so thatthe stress inequality equations are satisfied at each section over the span. This feasibility zone isoften referred to as the permissible cable zone.

The development presented here assumes that all the quantities contained in the four stress inequalityequations (Eqs. 4-43) are known at all sections of the beam, with the exception of the eccentricitye which is the variable to be determined. Equations 4-43 can be solved for e as follows:

(4-54a)

(4-54b)

(4-54c)

(4-54d)

These equations can be applied to any section of the beam to determine the permissible range of eand, hence, the region in which the cable may be placed so that all the stress inequalities for thatparticular section are satisfied. The maximum permissible value of e at a section, referred to as thebottom cable limit, is given by the smallest value yielded by Eqs 4-54a and 4-54b at equality.Similarly, the largest value of e obtained from Eqs. 4-54c and 4-54d at equality represents theminimum permissible value of e, referred to as the top cable limit. It is interesting to note that thebottom cable limit is governed by conditions at transfer while the top cable limit is determined bythe stress limitations imposed at the serviceability limit state.

The top and bottom cable limits can be determined at each section along the span and plotted onan elevation of the beam, as shown in Fig. 4-45. The region between the two cable limits clearly

ef

P AZ

M

P

ef

P AZ

M

P

ef

P AZ

M

P

ef

P AZ

M

P

tt

ttop

min

t

ct

tbot

min

t

cs

ttop

max

t

t s

tbot

max

t

�FHG

IKJ

�

�FHG

IKJ

�

� �FHG

IKJ

�

� �FHG

IKJ

�

1

1

1

1

� �

� �

DESIGN 4-61

represents a feasibility zone within which the cable may be placed so that the stress inequalityequations are satisfied at each section over the entire span.

Figure 4-46 presents three types of cable zone which may be obtained during the course of a design(Ref. 4-14). The cable zone shown in Fig. 4-46a is most commonly obtained and represents the casewhere the bottom cable limit falls outside the maximum available eccentricity epl, but the cable zoneis wide enough accommodate the cable. Figure 4-46b shows the cable zone which is obtained for aoptimum design where only one combination of Pt and e is possible at the critical section. The cablezone shown in Fig. 4-46c is characterised by the fact that a part of it lies outside the section, andis obtained when an insufficient concrete section is used. This problem, which will not arise if theMagnel diagram has been properly used in the design process, can only be overcome by using arevised section with larger section properties.

Cable zoneTop cable limit (largest of Eqs. 4-54c and 4-54d)

Bottom cable limit (smallest of Eqs. 4-54a and 4-54b)

Centroidal axis

Figure 4-45: The permissible cable zone.

Cable zone

Cable zone

Cable zone

Centroidal axis

Centroidal axis

Centroidal axis

(a) Common design

(b) Optimum design

(c) Insufficient concrete section

epl

CL

CL

CL

Figure 4-46: Typical types of cable zone (adapted from Ref. 4-14).


Design procedure

The objective of the flexural design of a prestressed concrete beam at the serviceability limit state,as presented herein, is to find a least weight concrete section, together with the magnitude andposition of the minimum required corresponding prestressing force, which will ensure that thespecified concrete stress limitations are satisfied at transfer and at the serviceability limit state. Thisaim can be achieved by the following design steps:

(a) Determine a satisfactory concrete section using Eqs. 4-49 and 4-50 or, alternatively, Eqs. 4-51and 4-52.

(b) Use the Magnel diagram (Eqs. 4-53) to determine the magnitude of the prestressing force andits eccentricity at the critical section.

(c) Calculate the permissible cable zone using Eqs. 4-54 and place the cable so that it falls withinthis zone.

(d) All the calculations required for steps (a) through (c) require a value for � which must initiallybe assumed because the prestress losses can only be evaluated after completing step (c). In thisstep, the prestress losses are calculated at each section of interest, as described in Chapter 5.

(e) The concrete stresses must always be checked at a representative number of sections along thespan to ensure that none of the specified permissible values are exceeded. The prestress lossescalculated in step (d) must be used in these calculations.

(f) If the stress check of the previous step reveals that the design is unsatisfactory, either becausesome of the permissible stress limitations are exceeded or because the design proves to beunacceptably conservative, the design must be suitably revised and the appropriate previoussteps repeated. Otherwise the design can be accepted.

Although the procedure outlined above applies to the design of the section as well as the prestress,it can easily be adapted to accommodate other circumstances. For example, if the section has alreadybeen selected on the basis of other specific requirements, the design procedure will then simplycommence at step (b).

The design process can be greatly facilitated by assuming reasonable values for the various initiallyunknown quantities, and many handbooks provide useful guidance in this regard. One of the mostimportant assumptions which must be made is that of the magnitude of �. Since the time-dependentlosses in prestensioned members tend to be larger than in post-tensioned members the magnitude of�, which can be taken between 0.75 and 0.85, is usually smaller for pretensioned than forpost-tensioned members. Another initially unknown quantity of which the magnitude must beassumed is the area of the concrete cross-section A, and Lin (Ref. 4-10) suggests the following inthis regard:

(4-55)

where

Obviously, the quality of the initial assumptions made by a designer will improve with experienceover time.

AA f

fps se

cs

05.

A

M

h fM M

M

h fM M

M

h

ps

max

semin max

L

semin max

L

�

�

R

S||

T||

�

�

0 650 2 0 3

050 2 0 3

.( . . )

.( . . )

for to

for to

superimposed dead and live load moment applied to the section

depth of the section

DESIGN 4-63

EXAMPLE 4-11

Make use of the provisions of SABS 0100 (Ref. 4-2) for flexural design at the serviceability limitstate to design a class 2 pretensioned concrete T-beam which is simply supported over a span of21 m. The beam must be designed to support a uniformly distributed live load of 5.8 kN/m and asuperimposed dead load of 0.6 kN/m. Assume fcu = 45 MPa and fci = 35 MPa.

The permissible stresses for fcu = 45 MPa and fci = 35 MPa, as obtained from SABS 0100, are asfollows:

At transfer:

At the serviceability limit state:

For the purposes of these calculations � is assumed to be 0.83.

Select a suitable section at midspan

The midspan moment due to the design superimposed load �w = 1.1 wsdl + wL = 1.1 � 0.6 + 5.8= 6.46 kN/m is given by . In order to obtain an initialestimate of Mmax, it is assumed that Mmin = �M, so that Mmin = 356.1 kN.m and Mmax = 2�M= 2 � 356.1 = 712.2 kN.m

Equations 4-49 and 4-50 can now be used to obtain an initial estimate of the minimum requiredvalues for the section moduli. Hence,

Figure 4-47 shows the selected T-section together with its section properties. If the self weight ofthe concrete is �c = 24 kN/m3, the self weight of the beam is given by wD = �c A = 24 � 345 � 10�3

= 8.28 kN/m. Therefore, the design loadings are obtained from SABS 0160 as follows:

For calculating Mmin:

For calculating Mmax:

Using these design loads, the minimum and maximum moments at the midspan section are calculatedas

f f

f f

tt ci

ct ci

� � � �

� � � ��

0 45 0 45 35 2 662

0 45 0 45 35 1575

. . .

. . .

MPa

MPa

f f

f f

ts cu

cs cu

� � � �

� � � ��

0 45 0 45 45 3019

0 33 0 33 45 14 85

. . .

. . .

MPa

MPa

� �M wL� � � �2 28 6 46 21 8 3561/ . / . kN. m

ZM M

f f

ZM M

f f

top minmax min

cs tt

bot minmax min

ts ct

,

,

. . .

. . ..

. . .

. . ..

��

��

� ��

� � ��

��

��

� ��

� � ��

�

�

�

�

712 2 083 3561 10

14 85 083 2 66224 42 10

712 2 083 3561 10

3019 083 15 752589 10

66 3

66 3

mm

mm

w wmin D� � � �10 10 8 28 8 28. . . . kN/m

w w w wmax D sdl L� � � � � �11 10 11 8 28 0 6 10 58 1557. . . . . . . .b g kN/m

Mw L

Mw L

minmin

maxmax

� ��

�

� ��

�

2 2

2 2

8

8 28 21

8456 4

8

1557 21

8858 2

..

..

kN. m

kN. m


Revised minimum required values can now be calculated for the section moduli, based on the abovemagnitudes of Mmin and Mmax.

Clearly, Ztop,min > Ztop = (�99.77 � 106 mm3) and Zbot,min < Zbot (= 47.34 � 106 mm3) so that thesection should be satisfactory. Note that since the magnitudes of Ztop,min and Zbot,min are almostequal, a symmetric section such as, for example, an I-section would be more efficient. However, aT-section is specified as a design requirement in this case. It should also be noted that although thesection can be optimised to a further degree, limitations imposed by deflection control and practicalconsiderations influenced this choice.

Determine the prestressing force at the midspan section

The next step is to determine the magnitude of the prestressing force and its eccentricity at themidspan section by making use of the Magnel diagram. This diagram is constructed by plottingEqs. 4-53 on the e-1/Pt plane. Since fcs Ztop (= 1481 kN.m) > Mmax (= 858.2 kN.m) andftsZbot (= 142.9 kN.m) < Mmax = 858.2 kN.m the second of Eqs. 4-53c and the first of Eqs. 4-53dare applicable. Hence,

(4-56a)

(4-56b)

ZM M

f f

ZM M

f f

top minmax min

cs tt

bot minmax min

ts ct

,

,

. . .

. . ..

. . .

. . ..

��

��

� ��

� � ��

��

��

� ��

� � ��

�

�

�

�

858 2 0 83 456 4 10

14 85 0 83 2 6622810 10

858 2 083 456 4 10

3019 083 15 7529 79 10

66 3

66 3

mm

mm

199 77 10

345 10

2 662 10 99 77 10 456 4 10

4005 10 1385 10

6

3

3 6 3

6 6

P

Z

Ae

f Z M

e

e

t

top

tt top min

��

� �

�

� � � � �

� � � �

�

� �

.

. . .

. .

d i

147 34 10

345 10

15 75 10 47 44 10 456 4 10

114 2 10 08319 10

6

3

3 6 3

6 6

P

Z

Ae

f Z M

e

e

t

bot

ct top min

��

�

�

� � � � � �

� � � � �

�

� �

.

. . .

. .

d i

1000

90

110

200

1200

A

I

y

y

Z

Z

top

bot

top

bot

� �

� �

� �

�

� � �

� �

345 10

3211 10

3218

678 2

99 77 10

47 34 10

3 2

9 4

6 3

6 3

mm

mm

mm

mm

mm

mm

.

.

.

.

.

Figure 4-47: Section for example 4-11.

DESIGN 4-65

(4-56c)

(4-56d)

where Pt is in kN and e is in mm.

If the cover is taken as 35 mm and it is assumed that the tendons are placed in three evenly spacedlayers at a vertical centre to centre spacing of 40 mm, then the maximum possible eccentricity eplis approximately 595 mm. The above four inequalities are plotted at equality in Fig. 4-48 togetherwith epl. Selecting e = 570 mm, it is clear that values of 1/Pt which range between 1/Pb =�0.5884 � 10�3 kN�1 and 1/Pd = �0.8206 �10�3 kN�1 all fall within the feasibility domain and,therefore, satisfy the four stress inequlity equations. These values correspond to permissible valuesof Pt which range between Pd = �1219 kN and Pb = �1700 kN, and a value of Pt = �1280 kN isselected.

Assume that 12.9 mm 7-wire super grade strand, jacked to 75% of its characteristic strength, isused. Since the characteristic strength per strand is 186 kN, the jacking force per strand is0.75 � 186 = 139.5 kN. If the loss of prestress due to elastic shortening is assumed to be 8.0%,

1083

99 77 10

345 10

14 85 10 99 77 10 858 2 10

385 0 10 1331 10

6

3

3 6 3

6 6

P

Z

Ae

f Z M

e

e

t

top

cs top max

FHG

IKJ

��

��

�

FHG

IKJ

� � � � � �

� � � �

�

� �

� ..

. . .

. .

d id i

1083

47 34 10

345 10

3 019 10 47 34 10 858 2 10

159 2 10 1160 10

6

3

3 6 3

6 6

P

Z

Ae

f Z M

e

e

t

bot

ts bot max

�

FHG

IKJ

��

��

�

FHG

IKJ

� � � � �

� � � � �

�

� �

� ..

. . .

. .

�1

0

1

0.8

0.6

0.4

0.2

�0.2

�0.4

�0.6

�0.5884

�0.8

��.2

��.4

0 1000200 400 600 800 1200�200

Eq. 4-56a

Eq. 4-56b

Eq. 4-56d

Eq. 4-56c

Feasibilitydomain

epl = 595 mme = 570 mm

e (mm)

Pt

(10

kN)

��

31

�1

�0.8206

Figure 4-48: Magnel diagram for the midspan section.


then the initial force per strand at transfer is (1 � 0.08) 139.5 = 128.3 kN. Therefore, �1280 /128.3= 9.974, say 10 strands are required. Under these conditions 10 strands will provide Pt = 1283 kN.

Determine the cable zone

Before the cable limits can be calculated, Mmin and Mmax must be expressed as functions of x. Thus,

Substitution of these expressions into Eqs. 4-54 give

(4-57a)

(4-57b)

(4-57c)

(4-57d)

Note that in Eqs. 4-57 e is in mm and x is in m. Comparing the right hand sides of Eqs. 4-57a and4-57b clearly shows that Eq. 4-57b governs the bottom cable limit. A similar examination ofEqs. 4-57c and 4-57d reveals that the top cable limit is controlled by Eq. 4-57d. The values of both

Mw x

L x x x

Mw x

L x x x

minmin

maxmax

� ��

� ��

24 140 21

27 784 21

.

.

kN. m

kN. m

ef

P AZ

M

P

x x

x x

tt

ttop

min

t

�FHG

IKJ

�

��

��

LNM

OQP

� � ��

� ��

� ��

1

2 662

1283 10

1

345 1099 77 10

4140 10

1283 1021

496 2 3 226 21

3 36

6

3

..

.

. .

d id i

ef

P AZ

M

P

x x

x x

ct

tbot

min

t

�FHG

IKJ

�

��

� ��

�

LNM

OQP� � �

�

� ��

� ��

1

1575

1283 10

1

345 1047 34 10

4140 10

1283 1021

443 7 3226 21

3 36

6

3

..

.

. .

d i

ef

P AZ

M

P

x x

x x

cs

ttop

max

t

� �FHG

IKJ

�

��

� � ��

�

L

N

MM

O

Q

PP� � � �

�

� � ��

� � ��

� �

1

14 85

0 83 1283 10

1

345 1099 77 10

7 784 10

0 83 1283 1021

1102 7 307 21

3 36

6

3

.

..

.

.

.

d id i

d i

ef

P AZ

M

P

x x

x x

ts

tbt

max

t

� �FHG

IKJ

�

��

��

L

N

MM

O

Q

PP� � �

�

� � ��

� � ��

� �

1

3 019

083 1283 10

1

345 1047 34 10

7 784 10

0 83 1283 1021

2714 7 307 21

3 36

6

3

.

..

.

.

. .

d i d i

To summarize: 10 @ 12.9 mm 7-wire super grade strand are required.

Pt = �1283 kNe = 570 mm at midspan

DESIGN 4-67

cable limits are listed in Table 4-7 at span/12 points along the span of the beam, and the resultingcable zone is drawn in Fig. 4-49a for the half span because of symmetry. Note that, since the centralregion of the bottom cable limit falls outside the beam, the cable zone is also limited by the maximumpractical eccentricity epl.

The cable profile is selected to lie within the cable zone and has draping points each located at adistance of span/ 3 = 7.0 m from a support, as shown in Fig. 4-49a. The eccentricity of the resultingcable profile can be expressed as follows for the half span:

where e(x) is in mm and x is in m. The magnitudes of the eccentricity at span/12 points are listedin Table 4-7, while Figs. 4-49b and 4-49c show possible tendon layouts at midspan and at thesupport, respectively.

The next step in the design process is the calculation of the prestress losses, after which a stresscheck can be made. These steps are illustrated in example 5-1, where this example is concluded. Itis also important to check the ultimate moment of resistance of the critical section, which in this

e xx x

x� � �

RST222 49 71 0 7 0

570 7 0 105

. .

. .

for m

for m

e pl=

595

e = 570

e = 222

e xb( )

e x( )

e xd( )

Centroidal axis

222

570Cable zone

7.0 m 3.5 m

L/2 = 10.5 m

(a) Cable zone

(b) Tendon layout at midspan (c) Tendon layout at support

108

456C.G.S

C.G.S

C.G.C C.G.C

6060

430

150

150

60 6060

x (m)

CL

Figure 4-49: Cable zone.


case is located at midspan. The reader should verify that the approximate procedure recommendedby SABS 0100 yields Mu = 1388 kN.m for the midspan section, and that a midspan moment of1099 kN.m is induced by the ultimate design loads prescribed by SABS 0160, i.e. 1.2Dn + 1.6Ln.Consequently, the design is also satisfactory with respect to flexural strength.

4.4.3 Design for the ultimate limit state

Flexural design for the ultimate limit state essentially provides a means of determining the concretearea under compression, the effective depth to the prestressing steel and the area of steel needed tomeet the requirements of flexural strength at the ultimate limit state. This follows because thecontribution of the tensile strength of the concrete is neglected, so that the tensile zone of the sectionis of no importance with regard to flexural strength and merely serves to contain the tendons.

The design procedure is based on Eqs. 4-17 and 4-18, which are expressions of moment andhorizontal equilibrium, respectively, and the required computations can be simplified by making useof a suitable approximate procedure for estimating the steel stress at ultimate fps. Equations 4-17and 4-18 can be restated in the following more general form, so that they apply to any cross-sectionalshape:

(4-58)

(4-59)

where z = internal lever arm

= concrete area under compression at ultimate

These equations can be rearranged to the following forms, which are more useful for design:

(4-60)

(4-61)

The design process is summarised in the following:

(a) Assume values for fps, z and for the overall section depth h.

• For bonded tendons it is suggested that fps

initially be taken equal to the design value offpu

because the large steel strains associated with the flexural failure of an underreinforced

M A f zu ps ps�

A f f Aps ps cu c � �� 0

�Ac

AM

f z

AA f

f

psu

ps

cps ps

cu

�

� � ��

x(m)

Bottom cable limiteb(x)(mm)

Top cable limited(x)(mm)

Selected eccentricitye(x)

(mm)

0 444 �271 2221.750 552 �25 3093.500 641 176 3965.250 710 333 4837.000 760 445 5708.750 789 512 570

10.500 799 534 570

Table 4-7: Cable limits and cable profile.

DESIGN 4-69

bonded section will result in these high steel stresses. However, the initial guess of fps forunbonded tendons must reflect the fact that it is normally significantly less than the designvalue of fpu, and a maximum value of 0.7fpu is suggested in this case.

• Since z ranges between 0.6h and 0.9h, depending on the section shape, it seems reasonableto accept a value of 0.8h as an initial guess (Ref. 4-10).

• The limitations imposed by deflection control at the serviceability limit state can be usedfor guidance when selecting an initial value for h.

(b) Obtain an initial estimate of the required area of prestressing steel Aps

from Eq. 4-60 using theassumed values of fps, z and h.

(c) The required concrete compression area is subsequently calculated from Eq. 4-61. At this

stage sufficient information is available for selecting a suitable section.

(d) Once a preliminary section has been selected, the actual values of fps and z corresponding tothis section can be determined either by the strain compatibility approach or by a suitableapproximate procedure, whichever is most convenient. These results can be used together withEq. 4-60 to calculate an improved value for Aps and, if required, Eq. 4-61 can be used to obtaina revised value for A’c. This process is continued until a satisfactory section has been obtained.Note that the calculation of f

psrequires a value for the effective prestress f

seincluding all

losses. This means that the magnitude of the prestress losses must be assumed.

(e) The final step in the design procedure is to verify that the ultimate moment of resistance ofthe section is larger than the moment produced by the design ultimate loads.

After completing the flexural design at the ultimate limit state, the section must be examined tocheck that the concrete flexural stress limitations are satisfied at transfer and at the serviceabilitylimit state.

When unbonded tendons are used, a minimum amount of bonded reinforcement should always beplaced to improve behaviour at ultimate (see Section 4.3.6). SABS 0100 (Ref. 4-2) and BS 8110(Ref. 4-7) do not specifically require such reinforcement, but ACI 318-89 (Ref. 4-11) requires thata minimum amount of bonded reinforcement equal to 0.004A be provided for this purpose, whereA denotes the area of that part of the section which lies between the tension face and the centroidof the gross concrete section.

EXAMPLE 4-12

Make use of the provisions of SABS 0100 (Ref. 4-2) for flexural design at the ultimate limit stateto design the midspan section of a 700 mm deep pretensioned concrete I-beam which is simplysupported over a span of 14 m. The beam is subjected to an imposed nominal live load of 9.0 kN/m.Assume fcu = 50 MPa and Ec = 34 GPa for the concrete. Use 12.9 mm 7-wire super grade strand,for which fpu = 1860 MPa and Ep = 195 GPa. For the equivalent rectangular stress block prescribedby SABS 0100, � = 0.45 and � = 0.9.

Assume fps = 0.87 fpu = 0.87 � 1860 = 1618 MPa and z = 0.8 h = 0.8 � 700 = 560 mm, so that aninitial estimate of Aps can be obtained from Eq. 4-60. However, before these calculations can bemade a value must be estimated for the ultimate design moment which, in turn, requires an assumedvalue for the self weight of the beam. If it is assumed that the nominal self weight of the beam iswD = 4.5 kN/m, the design ultimate load, using the load factors prescribed by SABS 0160, and thedesign ultimate moment at the midspan section are given by

�Ac

12 16 12 4 5 16 9 0 19 8. . . . . . .w wD L � � � � kN/m

MwL

� ��

�2 2

8

19 8 14

84851

.. kN. m


Using Eq. 4-60

The required concrete compression area is subsequently obtained from Eq. 4-61

Thus the preliminary section shown in Fig. 4-50 can be selected. It can be shown that this sectioncan accommodate the entire concrete compression zone within the top flange.The self weight of this section is given by wD = �c A = 24 � 165 � 10�3 = 3.96 kN/m, so that thedesign ultimate load is and the design ultimatemoment at the midspan section is .

The approximate procedure recommended by SABS 0100 for calculating the ultimate moment ofresistance of the section will be used to re-calculate fps and z for the preliminary section. In orderto do this it is assumed that fse = 1116 MPa.

For and

Table 4-3 gives and

Therefore, fps = 1.0 (0.87 fpu) = 1618 MPa and x = 0.20 d = 0.20 � 640 = 128 mm. Since thecompression zone falls entirely within the flange, the internal lever arm is given by z = d � 0.45x= 640 � 0.45 � 128 = 582.4 mm. Substituting the revised values for M, fps and z into Eqs. 4-60and 4-61 yields the following values for Aps and :

These values are fairly close to the values obtained in the initial trial, so that the section can beaccepted as it stands and the minimum required value of Aps can be taken as 497.9 mm2. Since thecross-sectional area provided by one 12.9 mm 7-wire super grade strand is 100 mm2, five strandswill provide an Aps = 500 mm2, which is sufficient.

As a final check, the ultimate moment of resistance of the section is calculated and compared withthe moment induced by the design ultimate loads.

AM

f zpsps

� ��

��

4851 10

1618 5605353

62.

. mm

�Ac

� � � � ��

� �� A

A f

fcps ps

cu�

535 3 1618

0 45 5038 5 103 2.

.. mm

� � �Ac 385 103 2. mm

12 16 12 3 96 16 9 0 1915. . . . . . .w wD L � � � � kN/mM wL� � � �

2 28 1915 14 8 469 2/ . / . kN. m

f

fse

pu

� �1116

18600 6.

f A

f b d

pu ps

cu

��

� � ��

1860 5353

50 350 6400 08890

..

f

fps

pu0 8710

..�

x

d� 0 20.

�Ac

AM

f zpsps

� ��

��

469 2 10

1618 582 4497 9

6.

.. mm2

� � � � ��

� �� A

A f

fcps ps

cu�

497 9 1618

0 45 503581 103 2.

.. mm

150

60

d = 640h =700

Aps

150

150

b = 350

350

e

A

I

�

� �

� �

290

165 10

8 938 10

3 2

9 4

mm

mm

mm.

Figure 4-50: Section for example 4-12.

DESIGN 4-71

For fse = 1116 MPa, and ,

so that and from Table 4-3.

Consequently, fps = 1.0 (0.87 fpu) = 1618 MPa and x = 0.18 d = 0.18 � 640 = 115.2 mm. Since thecompression zone falls entirely within the flange, the internal lever arm is given by z = d � 0.45x= 640 � 0.45 � 115.2 = 588.2 mm and the ultimate moment of resistance is given by Eq. 4-58 as

which is larger than the applied moment M = 469.2 kN.m.

When prestressed concrete members are designed for the serviceability limit state, the situation oftenarises where the ultimate moment of resistance of the section is less than the moment imposed bythe design ultimate loads. This problem can usually be rectified by providing a sufficient quantityof additional non-prestressed reinforcement, which is designed as follows:

(a) Assume a value for the stress in the prestressing steel at ultimate fps and set the sum of themoments of the internal forces, taken about the position of the non-prestressed steel, equal tothe moment induced by the design ultimate loads. The resulting expression can be directlysolved for the depth to neutral axis x, because it will be the only unknown variable containedin this expression.

(b) Using the value of x determined in step (a) together with an assumed value for the stress inthe non-prestressed steel at ultimate fs, the required area of non-prestressed steel As can becalculated directly by considering horizontal equilibrium of the section.

(c) At this stage the actual values of fps and fs, corresponding to As as determined in step (b), canbe calculated and compared to the assumed values. If the actual and assumed values differsignificantly, steps (a) and (b) must be repeated using improved assumptions for the magnitudesof fps and fs. This process is repeated until the assumed and calculated values of fps and fs agreeto within an acceptable tolerance, and the corresponding magnitude of A

sis then accepted.

This procedure is illustrated by example 4-13.

EXAMPLE 4-13

Provide suitable non-prestressed reinforcement for the section obtained in example 4-12 and shownin Fig. 4-50 so that it can sustain a total applied ultimate design moment of 600 kN.m. Make useof the provisions of SABS 0100 (Ref. 4-2) for flexural design, and take fy = 450 MPa andEs = 200 GPa for the non-prestressed reinforcement. Assume fse = 1116 MPa.

For the equivalent rectangular stress block prescribed by SABS 0100, � = 0.45 and = 0.9, whilethe design stress-strain curves for the prestressed and non-prestressed steel are as shown in Figs. 4- 17and 4-23, respectively. The effective depth to the prestressing steel d1 = 640 mm (see Fig. 4-50)and the effective depth to the non-prestressed steel is taken as d2 = 650 mm. The section nowcorresponds exactly to that shown in Fig. 4-22 example 4-6, except that As is unknown.

Assume fps = fpy and fs = fsy, so that the tensile force in the prestressing steel Tps and in thenon-prestressed steel Ts can be calculated from Eqs. 4-12 and 4-25, respectively:

If it is assumed that the entire compression zone is contained in the flange, the compression forcein the concrete can be expressed as a function of x as follows (see Eq. 4-14):

f

fse

pu

� 0 6.f A

f b d

pu ps

cu

��

� � ��

1860 500

50 350 6400 08304.

f

fps

pu0 8710

..�

x

d� 018.

M A f zu ps ps� � � � � ��500 1618 588 2 10 475 96. . kN. m

T A fps ps ps� � � � ��500 1617 10 808 73 . kN

T A f As s s s� � ��3913 10 3. kN


Moment equilibrium about the position of the non-prestressed steel provides the following expression(see Fig. 4-22):

Solving for x yields x = 146.9 mm. Therefore s = x = 0.9 � 146.9 = 132.2 mm is less thanhf = 150 mm, which means that the entire compression zone is contained in the flange, as assumed.Horizontal equilibrium requires that the following condition must be satisfied:

Solving this expression for As yields As = 594.9 mm2.

The validity of the assumption that fps = fpy and fs = fsy must be checked. This is done by calculating�ps and �s2 using the strain compatibility approach. Accordingly, �ps is calculated by combiningEqs. 4-8 through 4-11, and by noting that the effective prestress acting on the section is given by

. Hence,

The strain in the non-prestressed steel �s2 is calculated by considering the strain distribution (seeFig. 4-22). Thus, considering similar triangles:

Referring to Fig. 4-17, it is clear that since �ps is larger than �py (= 0.01329), fps = fpy (= 1617 MPa),as assumed. Figure 4-23 also confirms that the assumed value of fs = fsy (= 391.3 MPa) is correctbecause �s2 is larger than �sy (= 0.00196). Therefore, the calculated value of As = 594.9 mm2 iscorrect. This area of steel can be provided by 2 @ Y20 mm bars, for which As = 628 mm2, so thatthe section corresponds exactly to that shown in Fig. 4-22, as expected. It is of interest to note thatthe ultimate moment of resistance of this section is Mu = 606.7 kN.m (see example 4-6), which isslightly larger than the required value of M = 600 kN.m.

Note that although only flanged sections in which the compression zone at ultimate is entirelycontained in the top flange were considered in the examples presented in this Section, the designprocedures presented herein apply equally to flanged sections in which the compression zone extendsinto the web. The only difference lies in the formulation of the equations of equilibrium which must,in the latter case, account for the non-rectangular shape of the compression zone. However, notethat Eqs. 4-58 through 4-61 are general because they apply to any cross-sectional shape.

C f b x x xcu� � � ��

� 0 45 50 350 0 9 10 7 0883. . . kN

M C dx

T d d

xx

ps� � �FHG

IKJ

� �

� � � �� FHG

IKJ

� ��

2 2 1

3

2

600 10 7 088 6500 9

2808 7 650 640

b g

..

.

T T C

A x

A

ps s

s

s

�

� � �

� � � �

�

�

0

808 7 3931 10 7 088 0

808 7 3913 10 7 088 146 9 0

3

3

. . .

. . . .

P f Ase ps� � � � � � � ��1116 500 10 5583 kN

�

�

� �

� � � �

sese

p

cec

s cu

ps s ce se

f

E

P

A

Pe

I E

d x

x

� ��

�

� F

HG

I

KJ �

�

�

� �

�

F

HG

I

KJ � �

��F

HGIKJ

��F

HGIKJ

� �

� � � � ��

1116

195 100 005723

1 558

165 10

558 290

8 938 10

1

340 000254

640 146 9

146 90 0035 0 01174

0 01174 0 000254 0 005723 0 01772

3

2

3

2

9

11

1

.

..

.

.. .

. . . .

� �s cu

d x

x22 650 146 9

146 90 0035 0 01198�

�FHG

IKJ

��F

HGIKJ

� �.

.. .

DESIGN 4-73

4.4.4 Limits on steel content

As discussed in Section 4.3.5, a given section can fail in flexure in one of three modes, dependingon the amount of steel provided:

• In very lightly reinforced sections, an extremely brittle type of flexural failure can occur in whichthe steel fractures immediately after the concrete has cracked. This failure mode is highlyundesirable.

• In moderately reinforced (uderreinforced) sections, failure is induced by crushing of the concretecompression zone after the steel has yielded and undergone a large non-linear elongation. Becauseof its ductility, this type of failure is highly desirable.

• In heavily reinforced (overreinforced) sections, failure is induced by crushing of the concreteprior to yielding of the steel and takes place suddenly once the ultimate moment has been reached.Because of its brittle nature, this type of failure is undesirable.

The steel content of a beam section must therefore be controlled to avoid the undesirable failuremodes. For this reason, any design code of practice should provide limits on the maximum andminimum amounts of steel to be provided in a section to ensure ductile behaviour (see Section 4.3.5for a more expansive discussion on flexural ductility).

SABS 0100 (Ref. 4-2) and BS 8110 (Ref. 4-7) both limit the minimum amount of prestressing steelby requiring that the ultimate moment of resistance of a beam section must exceed its crackingmoment. According to SABS 0100 this requirement is deemed to be satisfied if the percentage ofreinforcement, calculated on an area equal to the width of the beam soffit multiplied by its overalldepth, is at least 0.15, while BS 8110 requires that the cracking moment be calculated on the basisof an assumed value of the modulus of rupture equal to 0.6√fcu. The intention of this provision isto ensure that cracking will precede flexural failure, thus avoiding the situation where the beamsuddenly fails when the concrete cracks. The bridge codes TMH7 (Ref. 4-6) and BS 5400 (Ref. 4-8)do not have any specific recommendations regarding the minimum amount of prestressing steel.

The maximum steel content of a prestressed concrete beam section is limited by TMH7 and BS 5400through the requirement that the strains in the outermost tendon must not be less than0.005 + fpu/(Ep�m). If the outermost tendon, or layer of tendons forms less than 25% of the totaltendon area, this requirement must also be satisfied within the outermost 25% of the tendon area.As an alternative, the strain at the centroid of the outermost 25% of the tendon area must be greaterthan the above value. This limitation is only applicable to cases where the ultimate moment ofresistance of the section is less than 1.15 times the required value. By limiting the steel strainsdeveloped in such sections to values greater than the value at “yielding” (as defined by the designstress-strain curve) it is obviously aimed at ensuring a ductile mode of flexural failure.

The maximum permissible values of the neutral axis depth x and steel ratio � = Aps / bd correspond-ing to the steel strain limit prescribed by TMH7 are listed in Table 4-8 for fpu = 1860 MPa. Notethat these values were derived for rectangular compression zones on the basis of the designstress-strain curves and material properties prescribed by TMH7, and on the assumption that all thesteel is concentrated at the centroid of the tendons.

Neither SABS 0100 nor BS 8110 contain any requirements which can obviously be related to limitingthe maximum steel content of a prestressed concrete beam. However, Ref. 4-25 infers that the codeprovision which requires the ultimate moment of resistance of a beam section to exceed its crackingmoment may serve this purpose because it is possible for a heavily overreinforced beam to fail inflexure before cracking. This inference is not really acceptable because the limitation on maximumsteel content is essentially related to the ductility of the section and must, therefore, be eitherexplicitly or implicitly expressed in terms of a limiting steel strain.


4.4.5 Flexural design of composite sections

The design of a composite section generally follows the same procedures used for non-compositesections. However, the design procedure must accomodate the following additional considerations,which arise from the characteristics of the construction procedure:

• The analysis must reflect the construction procedure and possible differences which may existbetween the properties of the materials used in the in situ and precast components.

• The effects of differential shrinkage must be accounted for under certain circumstances.

• Sufficient shear capacity must be provided at the interface between the preast and in situ concreteto ensure composite action.

In the following, the flexural design of composite prestressed beam sections at both the serviceabilityand ultimate limit states are discussed.

Serviceability limit states

The following stages of loading were identified as being critical with regard to stress in the concrete(see Section 4.3.7):

• At transfer of prestress when the minimum moment and the maximum prestressing force areacting on the beam section only.

• At the serviceability limit state when the minimum prestressing force, including all losses, andthe total self weight (beam plus slab) are acting on the beam section only, and the maximumsuperimposed loading is acting on the composite section.

A consideration of the stress conditions in the precast beam corresponding to these loading stages,in terms of permissible stresses, will provide the following four stress inequality equations (similarto Eqs. 4-43):

(4-62a)

(4-62b)

(4-62c)

(4-62d)

fP

A

P e

Z

M

Zf

fP

A

P e

Z

M

Zf

fP

A

P e

Z

M M

Z

M

Zf

fP

A

P e

Z

M M

Z

M

Zf

top bt

b

t

top b

b

top btt

bot bt

b

t

bot b

b

bot bct

top bt

b

t

top b

b f

top b

L

top cbcs

bot bt

b

t

bot b

b f

bot b

L

bot cbts

,, ,

,, ,

,, , ,

,, , ,

�

� �

�

�

�

� �

� �

fse / fpu (x / d)max Values of �max for fcu =

30 MPa 40 MPa 50 MPa 60 MPa

0.4 0.2720 0.002018 0.002691 0.003364 0.004036

0.5 0.2932 0.002175 0.002901 0.003626 0.004351

0.6 0.3180 0.002359 0.003146 0.003932 0.004718

Table 4-8: Limiting values of x / d and � = Aps / bd corresponding to steel strain limits of TMH7for fpu = 1860 MPa.

DESIGN 4-75

As in the case of non-composite sections, the design process basically involves the rationalmanipulation of these four inequality equations. Following the same procedures outlined inSection 4.4.2 the following expressions, which may be used for establishing the feasibility domainof Pt and e (Magnel diagram) as well as the permissible cable zone, can be derived.

Magnel diagram:

(4-63a)

(4-63b)

(4-63c)

(4-63d)

Cable zone:

(4-64a)

(4-64b)

(4-64c)

(4-64d)

where MD = Mb + Mf

1

P

Z

Ae

f Z Mt

top b

b

tt top b b

�

�LNM

OQP

�

,

,c h

1

P

Z

Ae

f Z Mt

bot b

b

ct bot b b

�

�LNM

OQP

�

,

,b g

1

P

Z

Ae

f Z M MZ

Z

f Z M MZ

Z

Z

Ae

f Z M MZ

Z

f Z M MZ

Z

t

top b

b

cs top b D Ltop b

top cb

cs top b D Ltop b

top cb

top b

b

cs top b D Ltop b

top cb

cs top b D Ltop b

top cb

�

�LNM

OQP

� �L

NMM

O

QPP

� �F

HGI

KJ

�

�LNM

OQP

� �L

NMM

O

QPP

� �F

HGI

KJ

R

S

|||||

T

|||||

�

�

,

,,

,

,,

,

,

,,

,

,,

,

for

for

1

P

Z

Ae

f Z M MZ

Z

f Z M MZ

Z

Z

Ae

f Z M MZ

Z

f Z M MZ

Z

t

bot b

b

ts bot b D Lbot b

bot cb

ts bot b D Lbot b

bot cb

bot b

b

ts bot b D Lbot b

bot cb

ts bot b D Lbot b

bot cb

�

�LNM

OQP

� �L

NM

O

QP

� �FHG

IKJ

�

�LNM

OQP

� �L

NM

O

QP

� �FHG

IKJ

R

S

|||||

T

|||||

�

�

,

,,

,

,,

,

,

,,

,

,,

,

for

for

ef

P AZ

M

P

ef

P AZ

M

P

ef

P AZ

PM M

Z

Z

ef

P AZ

PM M

Z

Z

tt

t btop b

b

t

ct

t bbot b

b

t

cs

t btop b

tD L

top b

top cb

ts

t bbot b

tD L

bot b

bot cb

� �FHG

IKJ

�

� �FHG

IKJ

�

� �FHG

IKJ

� �F

HGI

KJ

� �FHG

IKJ

� �FHG

IKJ

1

1

1 1

1 1

,

,

,,

,

,,

,

� �

� �


The discussion of the equivalent forms of these equations for non-composite sections given in Section4.4.2 are equally applicable here. Note that no equivalent forms of Eqs. 4-49 and 4-50 were derivedfor estimating the minimum required values of the section moduli, because the usefulness of suchequations are limited by the fact that the ratio of the section modulus of the beam to thecorresponding section modulus of the composite section must also be known. Moreover, a numberof useful design aids are available for obtaining an initial estimate of the section, when usingstandard precast sections.

After initially selecting a suitable section, the flexural design of a composite section at theserviceability limit state proceeds as described in Section 4.4.2 for non-composite sections, bearingin mind that Eqs. 4-62 through 4-64 must be appropriately substituted. Note that although the effectsof differential shrinkage were not included in the expressions above, these effects can easily beaccommodated simply by reducing the magnitudes of fcs and fts by the differential shrinkage stresses(either calculated or assumed, as appropriate) at the top and bottom of the beam section, respectively.

In general, the effects of differential shrinkage are not important, and may be neglected. TheHandbook to BS 8110 (Ref. 4-25) recommends that it is only necessary to include the effects ofdifferential shrinkage if all of the following conditions exist:

• If the stress in the top fibre of the precast beam under prestressing and permanent load is small.This condition leads to small creep in the precast beam and, hence, to large differential shrinkage.

• The difference between the strength of the concrete in the precast beam and in the in situ slabis more than 10 MPa.

• The time interval between casting of the precast beam and casting of the in situ slab is morethan 8 weeks.

The type of cross section also influences the importance of differential shrinkage: The effects ofdifferential shrinkage are more significant for sections of the type shown in Fig. 4-51a than forsections of the type shown in Fig. 4-51b.

Although the horizontal shear capacity is checked at the ultimate limit state, this check must alwaysbe carried out to ensure composite action. This is true even if the flexural design was carried outat the serviceablity limit state.

EXAMPLE 4-14

The composite section shown in Fig. 4-52 consists of precast pretensioned beams supporting an insitu slab. The beams are simply supported, having a span of 15 m, and spaced at a distance of1200 mm. Make use of the provisions of SABS 0100 for the flexural design at the serviceabilitylimit state to provide suitable class 2 prestressing for the beam. Assume unpropped construction anddesign the beam to support a uniformly distributed live load of 12 kN/m.

In situ concrete slab


(a) (b)

Figure 4-51: Influence of cross section on differential shrinkage.

DESIGN 4-77

The permissible stresses prescribed by SABS 0100 for the concrete in the beam are as follows:

ftt = 2.846 MPa fct = �18.00 MPa

fts = 3.182 MPa fcs = �16.50 MPa

The design self weight of the beam and slab at the serviceability limit state is

The design midspan moment induced by the beam self weight at transfer is

The design midspan moment caused by the self weight of the beam and the slab at the serviceabilitylimit state is

and the design midspan live load moment is

The modular ratio yields a transformed flange width of

where the effective flange width .

The section properties of the beam and the transformed composite sections are as follows

ytop,b = �350 mm ytop,cb = −164.9 mm ytop,c = �344.9 mm

ybot,b = 350 mm ybot,cb = 535.1 mm ybot,c = 535.1 mm

Ib = 10 � 109 mm4 Ztop,cb = �184.6 � 106 mm3 Ic = 30.44 � 109 mm4

Ztop,b = �28.58 � 106 mm3 Zbot,cb = 56.88 � 106 mm3

Zbot,b = 28.58 � 106 mm3

The Magnel diagram can now be constructed using Eqs. 4-63. To account for the loss of prestressat the serviceability limit state, � is assumed as 0.8.

w w wD b f � � � 11 11 5880 5184 1217. . . . .d i kN / m

M w Lb b � � 1

8

1

8588 15 16542 2. . kN. m

M w LD D � � 1

8

1

81217 15 342 32 2. . kN. m

M w LL L � � 1

8

1

812 15 337 52 2 . kN. m

n E Ec c f c b , ,/ / .28 34 08235

b n bft c e � 0 8235 1200 998 2. . mm

be 1200 mm

hb = 700

hf = 180

bf = 1200

bb = 350

In situ slab

Precast beam Concrete material properties:

fcu,b = 50 MPa

fci,b = 40 MPa

Ec,b = 34 GPa

Ec,f = 28 GPafcu,f = 30 MPa

Precast beam

In situ slab

Unit weight �c = 24 kN/m3

Figure 4-52: Composite cross section for Example 4-14.


An eccentricity of 260 mm is selected, which is less than the practical limit epl = 290 mm. Theprestressing force is taken as Pt = �1500 kN, which falls between the limits of �1397 kN and�1721 kN at e = 260 mm.

1472 9 10 4 053 106 6

P

Z

Ae

f Z Me

t

top b

b

tt top b b

�

�LNM

OQP

� � � ��

,

,

. .c h

11716 10 1471 106 6

P

Z

Ae

f Z Me

t

bot b

b

ct bot b b

�

�LNM

OQP

��

,

,

. .b g

11211 10 10 38 106 6

P

Z

Ae

f Z M MZ

Z

et

top b

b

cs top b D Ltop b

top cb

�

�LNM

OQP

� �L

NMM

O

QPP

� � � ��

�,

,,

,

.

12217 10 1901 106 6

P

Z

Ae

f Z M MZ

Z

et

bot b

b

ts bot b D Lbot b

bot cb

�

�LNM

OQP

� �L

NM

O

QP

� � � ��

�,

,,

,

. .

�0.5

0.1

0

�0.1

�0.2

�0.3

�0.4

�0.5809�0.6

�0.7�0.7159

�0.8

�0.90 25050 100 150 200 300� 0

Eq. 4-63a

Eq. 4-63b

Eq. 4-63d

Eq. 4-63c

Feasibilitydomain

epl = 290 mm

e = 260 mm

e (mm)

P t(

10kN

)�

�3

1�

1

Figure 4-53: Magnel diagram for Example 4-14.

DESIGN 4-79

Assuming the following conditions for the prestressing tendons

- 12.9 mm 7-wire strand (super grade)

- Tensioned to 75% of the characteristic strength (= 186 kN)

- Elastic losses of 7%

the force per strand at transfer is (1 � 0.07) � 0.75 � 186 = 129.7 kN. Therefore the number ofstrands required is 1500/129.7 = 11.56. Selecting 12 strands yields a prestressing force ofPt = �12 � 129.7 = �1557 kN, which still falls within the limits calculated above. At midspan,the strands can be placed as shown in Fig. 4-54.

The concrete stresses at the midspan section at the serviceability limit state are presented inFig. 4-55a, using the selected prestressing layout. The differential shrinkage stresses are also shownin this figure together with the resulting final stresses, which all comply with the specifiedpermissible values. Note that the following information was assumed for calculating the differentialshrinkage stresses by the method described in Section 4.3.7:

�cr = �48 � 10�6 MPa�1 for creep of the precast beam

�sh = 310 � 10�6 for shrinkage of both the beam and slab

�cc = 1.6

� = 10% at the time the slab is cast

60 % of the creep and shrinkage of the beam has taken place at the time of casting theslab

e = 260

ytop,b = 350�

ybot,b = 350

90C.G.S

C.G.B

6060

Figure 4-54: Cable layout at midspan, Example 4-14.

��.15

��.51��.56

1.50(a)

Stresses causedby external loading

plus prestressing

(b)Stresses caused by

differential shrinkage

(c)Total stress

0.31

0.27�1.22

0.71

��.84

��.24��.78

2.20

+ =

Figure 4-55: Final stresses, Example 4-14.


Ultimate limit states

Since the flexural analysis of composite sections at ultimate is exactly the same as of non-compositesections, the design procedure is also the same. Therefore, the procedures of Section 4.4.3 also applyto the flexural design of composite sections at the ultimate limit state. A slight difference ariseswhen the depth to neutral axis lies below the in situ slab, in which case the difference in strengthof the precast and in situ concrete must be accounted for as indicated in Section 4.3.7.

An additional consideration, perculiar to the design of a composite section, is the horizontal shearcapacity of the interface between the preast and in situ concrete. The general approach followed indesign is to calculate the magnitude of the horizontal shear stress at the interface as indicated inSection 4.3.7. This shear stress is subsequently compared to a permissible value. According to theprocedure recommended by BS 8110 (Ref. 4-7) and SABS 0100 (Ref. 4-2), the average horizontalshear stress at the interface is calculated by Eq. 4-39 and the maximum value is obtained bydistributing the average value in proportion to the vertical design shear force diagram. This maximumvalue is subsequently compared to the permissible values listed in Table 4-9. BS 5400 (Ref. 4-8)and TMH7 (Ref. 4-6) use Eq. 4-38 to calculate the horizontal shear stress at the interface. Obviously

Precast unit Surface type Design ultimate horizontal shearstresses at interface

Grade of in situ concrete (MPa)

25 30 � 40

Without links As-cast or as-extruded 0.4 0.55 0.65

Brushed, screeded or rough-tamped

0.6 0.65 0.75

Washed to remove laitance ortreated with retarder andcleaned

0.7 0.75 0.80

With nominal linksprojecting into in situconcrete

As-cast or as-extruded 1.2 1.8 2.0

Brushed, screeded or rough-tamped

1.8 2.0 2.2

Washed to remove laitance ortreated with retarder andcleaned

2.1 2.2 2.5

NOTES:

1. The description “as-cast” covers those cases where the concrete is placed and vibrated,leaving a rough finish. The surface is rougher than would be required for finishes to beapplied directly without a further finishing screed but not as rough as would be obtainedif tamping, brushing or other artificial roughening had taken place.

2. The description “as-extruded” covers those cases in which an open-textured surface isproduced direct from an extruding machine.

3. The description “brushed, screeded or rough-tamped” covers those cases where some formof deliberate surface roughening has taken place but not to the extent of exposing theaggregate.

4. For structural assessment purposes, it may be assumed that the appropriate value of �m(included in the table) is 1.5.

5. Where nominal links are provided, they should be of a cross section at least 0.15% of theinterface contact area.

Table 4-9: Design ultimate horizontal shear stresses at interface to BS 8110 (Ref. 4-7) andSABS 0100 (Ref. 4-2).

DESIGN 4-81

the permissible values prescribed by these codes also differ from those provided by BS 8110 andSABS 0100. If the permissible value is exceeded, steel crossing the interface must be provided tocarry the horizontal shear. The required amount of steel is calculated by following the provisionsof the particular code being used.

EXAMPLE 4-15

Determine the horizontal shear stress at the interface of the precast beam and the in situ slab of thecomposite section shown in Example 4-14.

The design load at the ultimate lmit state is given by

and the corresponding midspan moment is

It can be shown that the prestressing steel yields at ultimate, so that the stress in the steelfps = fpu/1.15 = 1617 MPa. The force in the prestressing steel will then be

The depth of the stress block s can be calculated from horizontal equilibrium

Since the stress block falls entirely in the slab, the horizontal force that has to be transmitted bythe interface C = Tps = 1941 kN. The average horizontal shear stress at the interface is therefore(see Eq. 4-39)

The vertical design shear force diagram varies linearly over the span of the beam in such a mannerthat it is zero at midspan and attains a maximum value at the support. Therefore, if the averagehorizontal shear stress is distributed in proportion to the vertical shear force diagram, it is obviousthat vhu,max = 2 � vhu = 2 � 0.7394 = 1.48 MPa.

If nominal links crossing the interface are provided, the permissible horizontal shear stress is2.0 MPa for a brushed, screeded or rough-tamped surface, according to SABS 0100 (see Table 4-9).Since this value is greater than vhu,max = 1.48 MPa, no additional links need to be provided underthese conditions.

4.4.6 Partial prestressing

Prestressed concrete members in which flexural tensile cracks are allowed to develop at service loadlevels are referred to as being partially prestressed. A partially prestressed member is reinforcedby a combination of prestressed and non-prestressed reinforcement, which both contribute to theultimate strength and serviceability behaviour of the member. Although the non-prestressed steelcan be either ordinary reinforcing bars or non-prestressed prestressing steel, the former is usuallyused for this purpose.

w w wu D L � � � � 12 16 12 11 06 16 12 32 48. . . . . . kN/m

M w Lu � � 1

8

1

832 48 15 91342 2. . kN. m

T f Aps ps ps � � �1617 1200 10 19413 kN

sT

f b

ps

cu f f

�

� � �

0 45

1941 10

0 45 30 1200119 8

3

. ..

,

mm

vC

Lbhub

�

� �

0 5

1941 10

0 5 15000 3500 7394

3

. .. MPa


When compared to ordinary reinforced concrete, partial prestress offers the advantage of improveddeflection control as well as the advantages to be gained from the fact that the member is usuallycrack free under long-term loads, depending on the degree of prestress. Partial prestressing alsooffers some advantages over full prestressing (Ref. 4-10):

• Improved control of camber.

• Savings in the cost of prestressing. Since a smaller prestressing force is required, the use ofpartial prestressing usually leads to savings in the amount of prestressing steel required, theanchorages required and the cost of the work associated with tensioning and grouting (in thecase of bonded post-tensioning) the tendons.

• Economical use of ordinary reinforcing steel.

• Possible improved ductility.

The most often quoted disadvantage of partial prestressing, when compared to full prestressing, isthat such members can be cracked at service load levels. However, ample evidence exists that ifappropriate steps are taken to control flexural cracks in terms of crack width and spacing, thepresence of these cracks will not adversely effect the durability of a partially prestressed concretemember (Ref. 4-26).

In addition to providing adequate flexural capacity, together with the prestressed steel, thenon-prestressed reinforcement performs the following functions (Ref. 4-10):

• Properly detailed non-prestressed steel can effectively control both crack width and spacing atservice load levels.

• In an unbonded member, some non-prestressed bonded reinforcement should be provided toprevent the development of a single large crack at ultimate and, thereby, to increase the flexuralcapacity of the member.

• If, at transfer, large tensile stresses are induced in the compression flange, non-prestressedreinforcement can be provided to prevent possible fracture, e.g. in the top flange over the midspanregion of a simply supported beam in which the live load is large in comparison to the selfweight of the beam.

• In the case of precast beams, properly placed non-prestressed reinforcement will ensure that thebeam is sufficiently robust with regard to unexpected stresses which may arise during handlingand erection.

Numerous procedures have been developed for the flexural design of partially prestressed concretebeams and a comprehensive discussion of a number of these can be found in Ref. 4-26. The variousmethods can usually be grouped into one of the following three categories, depending on the limitstate which the design procedure initially satisfies:

• Methods which initially satisfy the serviceability limit state. The British design codes BS 8110(Ref. 4-7) and BS 5400 (Ref. 4-8) as well as the South African code SABS 0100 (Ref. 4-2)recommend a method which is based on a limiting hypothetical tensile stress. The hypotheticaltensile stress in a cracked prestressed concrete beam section is defined as the flexural stresswhich would occur in the extreme tension fibre of the uncracked section. Leonhardt (Ref. 4-27)and Menn (Ref. 4-28) each outline procedures which use a crack width limitation as the pointof departure for design.

• Methods which initially satisfy the ultimate limit state. The method proposed by Naaman(Ref. 4-29) makes use of the partial prestressing ratio, while the procedure developed byBachmann (Refs. 4-30 and 4-31) employs the concept of the degree of prestress.

• Methods which simultaneously satisfy the ultimate and serviceability limit states. The procedureproposed by Huber (Ref. 4-32) is an example of such a method.

DESIGN 4-83

Over the years, a number of indices have been developed for quantifying the extent of prestressingin a partially prestressed concrete beam section (see Ref. 4-26). The partial prestressing ratio (PPR)and the degree of prestress � are two such indices: The partial prestressing ratio is defined as theratio of the ultimate moment of resistance provided by the prestressing steel only to the ultimatemoment of resistance provided by all the steel (i.e. the prestressed plus non-prestressed steel). Thedegree of prestress is defined as the ratio of the decompression moment (i.e. the moment whichinduces a zero stress in the extreme tension fibre of the section) to the total service load momentand therefore represents the fraction of the total service moment which is counteracted byprestressing effects. Consequently, a value of zero for the degree of prestress corresponds toreinforced concrete while a value of one applies to fully prestressed concrete.

A reasonable basis for any procedure for the flexural design of a partially prestressed concrete beamsection is to adopt a unified approach in the sense that the method must apply to the completespectrum of possible levels of prestress, from fully prestressed concrete through to reinforcedconcrete. The method should also provide a smooth transition from fully prestressed concrete toreinforced concrete. The design procedure proposed by Bachmann (Refs. 4-30 and 4-31) satisfiesthese requirements and is presented in the following. The method assumes that all the dimensionsof the concrete section are known, that all the material properties are known, and that the bendingmoments due to all dead and live loads can be determined. The section is initially designed toprovide the required ultimate strength and is subsequently checked for serviceability as follows:

(a) Select a suitable value for the degree of prestress � or, alternatively, the decompression moment.This choice is strongly dependent on engineering judgment to suit a given consideration suchas durability, deflection, fatigue, crack control and cost. The decompression moment iscommonly chosen at least equal to the dead load moment (Ref. 4-31) and it is suggested thatin the case of bridges a value larger than the dead load moment is appropriate (Refs. 4-31 and4-33). Wiessler (Ref. 4-33) recommends that the decompression moment should be taken equalto the dead load moment plus 33% of the live load moment for bridges in South Africa.

(b) Determine the required amount of prestressing steel. The prestressing force Pt

required fordeveloping the decompression moment selected in step (a) is determined by making use of theexpression for calculating the flexural stress in an uncracked section. The following expressionfor calculating P

tis derived by setting f

bot,sequal to zero and M

maxequal to the decompression

moment Mdec in Eq. 4-43d, and solving the resulting expression for Pt:

(4-65)

The required amount of prestressing steel can subsequently be determined from the calculatedvalue of Pt.

(c) Determine the required amount of non-prestressed steel. Non-prestressed reinforcement mustbe provided to ensure that the ultimate moment of resistance of the section exceeds the requiredvalue. The procedure outlined in Section 4.4.3 can be used to design this reinforcement.

(d) Detail the non-prestressed reinforcement carefully. In addition to its contribution to the ultimatestrength, soundly detailed non-prestressed reinforcement can effectively control both crackwidth and spacing at service load levels. This is often the last step in the design procedureunder normal circumstances.

(e) Check compliance with other limit states. Other limit states such as, for example, crack width,fatigue and deflection can be specifically examined, as required. These aspects are covered inlater Chapters.

It should be noted that since design is essentially an iterative procedure, the section is usuallydetermined on a trial and error basis, bearing in mind any specific design requirements. Generally,the number of iterations required for convergence to a solution rapidly reduces with experience.

PM

Z

Ae

tdec

bot

� �

�LNM

OQP

�


More expansive discussions on this design procedure can be found in Refs. 4-26 and 4-34. It shouldbe noted that some of the practical advantages of the method are that it is code independent, thatit treats uncertainties in a rational manner and that it does not preclude any serviceability check.The design procedure is illustrated by example 4-16.

EXAMPLE 4-16

The midspan section of a pretensioned partially prestressed concrete beam, which is simply supportedover a span of 14 m, is shown in Fig. 4-56. In addition to its self weight, the beam must supporta uniformly distributed superimposed dead load of 4.9 kN/m and a live load of 8.8 kN/m. Make useof the provisions of SABS 0100 (Ref. 4-2) to design suitable prestressed and non-prestressedreinforcement for the midspan section so that the decompression moment Mdec is equal to thepermanent load moment. Use 12.9 mm 7-wire super grade strand, for which fpu = 1860 MPa andEp = 195 GPa, for the prestressed reinforcement and take fy = 450 MPa and Es = 200 GPa for thenon-prestressed reinforcement. Take fcu = 50 MPa and Ec = 34 GPa for the concrete.

If the self weight of the concrete is taken as �c = 24 kN/m3, the self weight of the beam is givenby wD = �c A = 3.96 kN/m. Therefore the total permanent load is wPerm = wD + wsdl = 3.96 + 4.9= 8.86 kN/m. Using the load factors of SABS 0160, the design value of the permanent load momentappropriate to the serviceability limit state is given by

Since the decompression moment Mdec is taken to be equal to the permanent load moment, and thetotal design moment at the serviceability limit state is given by

the degree of prestress � corresponding to this choice of Mdec is

Assuming � = 0.84 (i.e. 16% losses), the prestressing force required for a decompression momentMdec = 238.8 kN.m is subsequently obtained from Eq. 4-65

If 12.9 mm 7-wire super grade strand, jacked to 75% of characteristic strength (= 186 kN per strand),is used the jacking force per strand is 0.75 � 186 = 139.5 kN. Assuming the loss of prestress due

Mw L

PermPerm

� ��

�11

8

11 886 14

8238 8

2 2. . ..

a fkN.m

Mw w L

maxPerm L

��

��

�11 10

8

11 8 86 10 8 8 14

8454 4

2 2. . . . . ..

a fkN.m

� � � �M

Mdec

max

238 8

454 40 5255

.

..

PM

Z

Ae

tdec

bot

� �

�L

NMO

QP

� ��

�

��

L

NM

O

QP

� �

�

238 8 10

0 842554 10

165 10290

639 13

6

3

.

..

. kN

d 1=

640

d 2=

650

150

5060

h=

700

Aps

As

hf =150

bw = 150

b = 350

350

e

A

I

Z

Z

top

bot

�

� �

� �

� � �

� �

290

165 10

8 938 10

2554 10

2554 10

3 2

9 4

6 3

6 3

mm

mm

mm

mm

mm

.

.

.

Figure 4-56: Midspan section for example 4-16.

DESIGN 4-85

to elastic shortening to be 5.0%, the initial force per strand at transfer is (1 � 0.05) � 139.5= 132.5 kN (= Pt,strand). Therefore, �639.1/132.5 = 4.823, say 5 strands are required, for whichAps = 500 mm2.

The non-prestressed reinforcement is designed by considering the required ultimate moment, inexactly the same manner as in example 4-13. Following the requirements of SABS 0160, the designload appropriate to the ult imate l imit state is given by wu = 1 . 2 wPerm + 1 . 6 wL =1.2 � 8.86 + 1.6 � 8.8 = 24.71 kN/m, so that the required ultimate moment of resistance of thesection is

For the equivalent rectangular stress block prescribed by SABS 0100, � = 0.45 and � = 0.9, whilethe design stress-strain curves for the prestressed and non-prestressed steel are as shown in Figs. 4- 17and 4-23, respectively. Assuming fps = fpy = 1617 MPa, fs = fsy = 391.3 MPa and that the entirecompression zone is contained in the flange, the depth to neutral axis is determined by takingmoments about the position of the non-prestressed steel and solving the resulting expression for x.Thus,

Solving for x yields x = 148.4 mm. Therefore s = �x = 0.9 � 148.4 = 133.6 mm is less thanhf = 150 mm, which means that the entire compression zone is contained in the flange, as assumed.Horizontal equilibrium requires that the following condition must be satisfied:

Solving this expression for As yields As = 621.8 mm2. This area of steel can be provided by2 @ Y20 mm bars, for which As = 628 mm2.

The validity of the assumption that fps = fpy and that fs = fsy must be checked. This is done bycalculating ps and s2 using the strain compatibility approach, as in example 4-13. Before this canbe done, fse must be estimated so that its value is consistent with the assumptions made with regardto the various losses. Since the cross sectional area per strand Aps,strand = 100 mm2

If the effective prestress acting on the section is taken as P = �fse Aps = �1113 � 500 � 10�3

= �556.6 kN, it can be shown that ps = 0.01752 and s2 = 0.01179. Since these values of ps and s2 are larger than py = 0.01329 and sy = 0.00196, respectively, fps = fpy and fs = fsy (see Figs 4-17and 4-23) as assumed.

Note that any other limit state can now be examined, as required. For example, consider the stressesin the concrete at transfer. For the losses assumed here, Pt = �5 Pt,strand = �5 � 132.5 = �662.6 kN,while so that, at transfer, the stresses in thetop and bottom fibres of the section are given by

Mw Lu� �

��

2 2

8

24 71 14

8605 4

.. kN. m

M f b x dx

A f d d

xx

cu ps ps� � �FHG

IKJ

� �

� � � � �� FHG

IKJ

� � � ��

� ��

b g b g

b g

2 2 1

6

2

605 4 10 0 45 50 350 0 9 6500 9

2500 1617 650 640. . .

.

A f A f f b x

A

ps ps s s cu

s

� � �

� � � � � ��

� � 0

500 1617 3913 0 45 50 350 0 9 148 4 0. . . .

fP

Aset strand

ps strand

� ��

�� ,

,

. .0 84 132 5 10

1001113

3

MPa

M w Lmin D� � � � �10 8 10 3 96 14 8 97 022 2. / . . / . kN. m

To summarize: 5 @ 12.9 mm 7-wire super grade strands tensioned to75% of their strength are required together with2 @ Y20 mm bars.


If the concrete strength at transfer is taken as fci = �40 MPa then the permissible tensile and

compressive stresses at transfer are and fct = 0.45 fci = 0.45

� (�40) = �18 MPa, respectively. A comparison of the calculated concrete stresses with thepermissible values clearly demonstrates that the stress limitations prescribed by SABS 0100 aresatisfied at transfer.

4.5 REFERENCES

4-1 Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley & Sons, 1975.

4-2 South African Bureau of Standards, “The Structural Use of Concrete,” SABS 0100: 1992,Parts 1 and 2, SABS, Pretoria, 1992.

4-3 Khachaturian, N., Gurfinkel, G., Prestressed Concrete, McGraw-Hill Book Company, NewYork, 1969.

4-4 Hognestad, E., Hanson N. W. and McHenry D., “Concrete Stress Distribution in UltimateStrength Design”. ACI Journal, Vol. 52, No. 6, December 1955.

4-5 Rüsch, H., “Researches Toward a General Flexural Theory for Structural Concrete”. ACIJournal, Vol. 57, No. 1, July 1960.

4-6 Committee of State Road Authorities, “Code of Practice for the Design of Highway Bridgesand Culverts in South Africa,” TMH7 Part 3, CSRA, Pretoria, 1989.


4-8 British Standards Institution, “Steel, Concrete and Composite Bridges. Part 4: Code of Practicefor Design of Concrete Bridges,” BS 5400: Part 4: 1984, BSI, London, 1984.

4-9 Warwaruk, J., Sozen, M. A., and Siess, C. P., “Strength Behaviour in Flexure of PrestressedConcrete Beams,” University of Illinois Engineering Experiment Station, Bulletin No. 464,1962.


4-11 ACI Committee 318,"Building Code Requirements for Reinforced Concrete (ACI 318-89) andCommentary - ACI 318 R-89," American Concrete Institute, Detroit, 1989.

4-12 Gamble, W. L., “Prestressed Concrete,” Lecture Notes for Prestressed Concrete: CE 368,Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, October1991.

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REFERENCES 4-87

4-13 Libby, J. R., Modern Prestressed Concrete: Design Principles and Construction Methods,4th ed., Van Nostrand Reinhold, New York, 1990.


4-15 Kajfasz, S., Somerville, G., and C., Rowe, R. E., “An investigation of the behaviour ofcomposite concrete beams,” Cement and Concrete Association, Research Report 15, November1963.

4-16 Clark, L. A., Concrete Bridge Design to BS 5400, Construction Press, London, 1983.

4-17 Kong, F. K., and Evans, R. H., Reinforced and Prestressed Concrete, 3rd ed., Van NostrandReinhold (UK), Workingham, 1987.

4-18 South African Bureau of Standards, “The General Procedures and Loadings to be Adopted inthe Design of Buildings,” SABS 0160: 1989, SABS, Pretoria, as amended 1990.

4-19 British Standards Institution, “Dead and Imposed Loads,” CP 3: Chapter V: 1967. Loading.Part I, BSI, London, 1967.

4-20 British Standards Institution, “Steel, Concrete and Composite Bridges. Part 2: Specificationfor Loads,” BS 5400: Part 2: 1978, BSI, London, 1978.

4-21 Guyon, Y., Prestressed Concrete, John Wiley & Sons, New York, Vol. 1, 1960.

4-22 Magnel, G., Prestressed Concrete, 3rd ed., revised and enlarged, Concrete Publications Ltd.,London, 1954.

4-23 FIP “Shear at the interface of precast and in situ concrete,” Technical Report FIP/9/4, August1978.

4-24 Committee of State Road Authorities, “Code of Practice for the Design of Highway Bridgesand Culverts in South Africa,” TMH7 Parts 1 and 2, CSRA, Pretoria, 1989.

4-25 Handbook to British Standard BS 8110: 1985: Structural Use of Concrete, PalladianPublications Ltd., London, 1987.

4-26 Olivier, J. J., The Use of Partial Prestressing for Road Bridges in South Africa, MEng Thesis,Department of Civil Engineering, University of Pretoria, Pretoria, May 1993.

4-27 Leonhardt, F., “To New Frontiers for Prestressed Concrete Design and Construction,” PCIJournal, Vol. 19, No. 5, September 1974, pp. 54-69.

4-28 Menn, C., “Partial Prestressing from the Designer’s Point of View,” Concrete International,Vol. 5, No. 3, March 1983, pp. 52-59.

4-29 Naaman, A. E., “Partially Prestressed Concrete: Review and Recommendations,” PCI Journal,Vol. 30, No. 6, November/December 1985, pp. 30-71.

4-30 Bachmann, H., “Partial Prestressing of Concrete Structures,” IABSE Surveys S-11/79,International Association for Bridge an Structural Engineering, Zürich, 1979.

4-31 Bachmann, H., “Design of Partially Prestressed Concrete Structures based on Swiss Experi-ences,” PCI Journal, Vol. 29, No. 4, July/August 1984, pp. 84-105.

4-32 Huber, A., “Practical Design of Partially Prestressed Concrete Beams,” Concrete International,Vol. 5, No. 4, April 1983, pp. 49-54.

4-33 Wiessler, H. H., “Partial Prestress for Bridges,” International Concrete Symposium, ConcreteSociety of Southern Africa, Portland Park, Halfway House, May 1984.

4-34 Marshall, V., Wium, D. J. W., and Olivier, J. J., “Use of Partial Prestressing for RoadBridges,” Annual Transportation Convention, Session 5D: Structures, Pretoria, August 1991.


The authors gratefully acknowledge:

• The support of the Concrete Society of Southern Africa for publishing this text as a book.

• The support and encouragement of the committee of the Prestressed Concrete Division of theSociety. The contributions made by various members of this committee in terms of planning thetext and in terms of their review comments were particularly useful.

The authors are particularly indebted to Michael A. Vasarhelyi of the Prestressed Concrete Divisionfor his careful review of the entire text. His comments and suggestions contributed significantly tothe value of the book.

Vernon MarshallJohn M. Robberts

vi PREFACE

Documents

Pre Stressed Concrete Design and Practice_SA