Pre Stressed Concrete Design and Practice_SA
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PRESTRESSED CONCRETE DESIGN AND PRACTICE VERNON MARSHALL JOHN M. ROBBERTS
Concrete Society of Southern Africa Prestressed Concrete Division Midrand, South Africa
TABLE OF CONTENTS
i
TABLE OF CONTENTS
1
2
PREFACE
v
INTRODUCTION
11
1.1
THE BASIC IDEA OF PRESTRESSED CONCRETE . . . . . . . . . . . . . . . . . . . . 11
1.2
EFFECTS OF PRESTRESSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3
GENERAL PRINCIPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4
BASIC DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
1.5
PRESTRESSED VERSUS REINFORCED CONCRETE . . . . . . . . . . . . . . . . . . . 112
1.6
HISTORY OF PRESTRESSED CONCRETE . . . . . . . . . . . . . . . . . . . . . . . . . 113
1.7
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
MATERIAL PROPERTIES 2.1
2.2
2.3 3
21
CONCRETE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1
Compressive strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2
Stressstrain relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.3
Modulus of elasticity
2.1.4
Tensile strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
2.1.5
Timedependent behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
2.1.6
Thermal properties of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
2.1.7
Poisson’s ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
2.1.8
Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
STEEL REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 2.2.1
Nonprestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
2.2.2
Prestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
2.2.3
Relaxation of prestressing steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
2.2.4
Fatigue characteristics of reinforcement . . . . . . . . . . . . . . . . . . . . . . . . 235
2.2.5
Thermal properties of reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . 237
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
PRESTRESSING SYSTEMS AND PROCEDURES
31
3.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2
PRETENSIONING SYSTEMS AND PROCEDURES . . . . . . . . . . . . . . . . . . . . 31
3.3
3.2.1
Basic principle and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2
Stressing beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.3
Structural frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
POSTTENSIONING SYSTEMS AND PROCEDURES . . . . . . . . . . . . . . . . . . . 310 3.3.1
Basic principle and procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
3.3.2
Posttensioning systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
3.3.3
Posttensioning operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
3.3.4
Ducting for bonded construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
3.3.5
Grouting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
3.4
PRETENSIONING VERSUS POSTTENSIONING . . . . . . . . . . . . . . . . . . . . . . 328
3.5
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
ii
4
TABLE OF CONTENTS
DESIGN FOR FLEXURE 4.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2
SIGN CONVENTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3
ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4
4.3.1
Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.2
Flexural response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.3
Analysis of the uncracked section . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.4
Cracking moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
4.3.5
Ultimate moment: Sections with bonded tendons . . . . . . . . . . . . . . . . . . . 412
4.3.6
Analysis of beams with unbonded tendons . . . . . . . . . . . . . . . . . . . . . . 431
4.3.7
Flexural analysis of composite sections . . . . . . . . . . . . . . . . . . . . . . . . 436
DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 4.4.1
4.5 5
Limit states design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
4.4.2
Design for the serviceability limit state . . . . . . . . . . . . . . . . . . . . . . . . 454
4.4.3
Design for the ultimate limit state . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
4.4.4
Limits on steel content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
4.4.5
Flexural design of composite sections . . . . . . . . . . . . . . . . . . . . . . . . . 475
4.4.6
Partial prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
PRESTRESS LOSSES
51
5.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2
METHODS FOR CALCULATING PRESTRESS LOSSES . . . . . . . . . . . . . . . . . 51
5.3
5.4
5.5
5.6 6
41
5.2.1
Total loss in pretensioned members . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2.2
Total loss in posttensioned members . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2.3
Methods for calculating prestress losses . . . . . . . . . . . . . . . . . . . . . . . . 53
ELASTIC SHORTENING OF THE CONCRETE . . . . . . . . . . . . . . . . . . . . . . . 55 5.3.1
Pretensioned concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3.2
Posttensioned concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
TIMEDEPENDENT LOSSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.4.1
Loss due to relaxation of the steel . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4.2
Loss due to shrinkage of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4.3
Loss due to creep of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
LOSSES DURING POSTTENSIONING . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 5.5.1
Friction losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
5.5.2
Anchorage seating losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
EFFECTS OF CONTINUITY
61
6.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2
ELASTIC ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2.1
Eccentricity of the prestressing force . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.2
Force (flexibility) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2.3
Fixedend moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.4
Displacement (stiffness) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
6.2.5
Concept of equivalent loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
TABLE OF CONTENTS
8
6.2.7
Concordancy and linear transformation . . . . . . . . . . . . . . . . . . . . . . . . 625
DESIGN AT SERVICEABILITY LIMIT STATE . . . . . . . . . . . . . . . . . . . . . . . 628 ANALYSIS AT ULTIMATE LIMIT STATE . . . . . . . . . . . . . . . . . . . . . . . . . 629 6.4.1
Secondary moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
6.4.2
Moment redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
SHEAR
71
7.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.2
BEAMS WITHOUT WEB REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . 71 7.2.1
Cracking behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.2.2
Shear capacity of the concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.3
BEAMS WITH WEB REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . 713
7.4
DESIGN PROCEDURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716
7.5
COMPOSITE BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
7.6
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730
DEFLECTIONS
81
8.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.2
UNCRACKED BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.2.1
Instantaneous deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.2.2
Longterm deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
CRACKED BEAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 8.3.1
Instantaneous deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814
8.3.2
Longterm deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
8.4
DEFLECTION LIMITATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831
8.5
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832
ANCHORAGE ZONE DESIGN
91
9.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.2
TRANSFER LENGTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.3
ANCHORAGE ZONE REINFORCEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.4 10
Effects of losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
6.4
8.3
9
6.2.6 6.3
6.5 7
iii
9.3.1
Spalling Stress Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
9.3.2
Bursting Stress Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921
PRESTRESSED CONCRETE SLABS
101
10.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.2
EFFECTS OF PRESTRESS ON STRUCTURAL BEHAVIOUR . . . . . . . . . . . . . . 103 10.2.1 Flexural behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 10.2.2 Restraint to axial shortening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.3
STRUCTURAL ANALYSIS BY THE EQUIVALENT FRAME METHOD . . . . . . . . 108
10.4
DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 10.4.1 Design codes of practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 10.4.2 Preliminary value for the slab thickness . . . . . . . . . . . . . . . . . . . . . . . . 1013
iv
TABLE OF CONTENTS
10.4.3 Prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 10.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 10.4.5 Serviceability limit states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 10.4.6 Ultimate limit states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026 10.5
DETAILING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1030 10.5.1 Prestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1030 10.5.2 Nonprestressed reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 10.5.3 Openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033
10.6
DESIGN EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 10.6.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 10.6.2 Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 10.6.3 Balanced load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 10.6.4 Check the preliminary value for the slab thickness . . . . . . . . . . . . . . . . . . 1035 10.6.5 Minimum cover to tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 10.6.6 Design: NorthSouth direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036 10.6.7 Design: EastWest direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 10.6.8 Punching shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 10.6.9 Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 10.6.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
10.7 11
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
DETAILING
111
11.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
11.2
COVER TO TENDONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 11.2.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 11.2.2 Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
11.3
LIMITATIONS ON PRESTRESSING STEEL CONTENT . . . . . . . . . . . . . . . . . . 116 11.3.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 11.3.2 Minimum steel content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 11.3.3 Maximum steel content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.4
LIMITATIONS ON SPACING OF TENDONS . . . . . . . . . . . . . . . . . . . . . . . . 117
11.5
EFFECTS OF TENDON CURVATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 11.5.1 Inplane normal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 11.5.2 Outofplane multistrand effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 11.5.3 Minimum radius of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 11.5.4 Minimum tangent length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112 11.5.5 Code requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113
11.6
LONGITUDINAL NONPRESTRESSED REINFORCEMENT . . . . . . . . . . . . . . . 1114
11.7
DRAWINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118
11.8
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119
APPENDIX A: LIST OF SYMBOLS APPENDIX B: DRAWINGS FLAT SLAB: REINFORCEMENT LAYOUT TENDON LAYOUT BRIDGE DECK: PRESTRESSING DETAILS
PREFACE
v
PREFACE The content of this book was initially written and issued as a set of notes for the course Prestressed Concrete: Design and Practice, commissioned by the Prestressed Concrete Division of the Concrete Society of Southern Africa. The course was aimed at young engineers and technologists with little or no experience in the design of prestressed concrete structures, and it was the intention that it should serve as a vehicle for providing bridging education between tertiary training and design practice. 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, postgraduate students and practising designers. It is important to note that a unique feature of the book is that current South African practice is emphasised throughout the text. Basic background information, essential for the design of prestressed concrete structures, is presented in the first three chapters. These cover the material properties of concrete, prestressing steel and nonprestressed reinforcement as well as the various prestressing systems and procedures generally used in South Africa. These chapters also cover relevant specifications. The basic concepts and procedures required for the analysis and design of a prestressed concrete flexural 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 limit states. Composite sections, unbonded construction and partially prestressed sections are also covered.
• Chapter 5: Procedures for estimating the instantaneous and longterm loss of prestress in pretensioned and in posttensioned 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 longterm deflections of prestressed concrete flexural members. Both uncracked and cracked beams are considered.
• Chapter 9: Design of the anchorage zone. The design considerations, applicable to both pretensioned and posttensioned construction are covered. The analysis and design of posttensioned flat plates and flat slabs are covered by Chapter 10. This material is limited to slabs using unbonded tendons and levels of prestress at which the slabs will be cracked under the design service loads because most of the posttensioned flat plates and flat slabs constructed in South Africa are of this type. Various aspects, peculiar to prestressed concrete members, which affect detailing are presented in Chapter 11. The details of both the prestressed and nonprestressed reinforcement are covered in this chapter, and a number of local effects, induced by tendon curvature, are also discussed. Working drawings of the prestressing details of a flat slab and of a highway bridge are presented in Appendix B. Generally, the procedures for simulating the various aspects of behaviour are developed from the basic principles of structural mechanics. However, in a case where a semiempirical approach is followed, the relevant experimental work on which such a procedure is based is presented and discussed. The manner in which these aspects of behaviour are reflected in the various design codes of practice, commonly used in South Africa, are also explained. Each chapter contains comprehensive examples that illustrate the analytical concepts and design procedures covered.
THE BASIC IDEA OF PRESTRESSED CONCRETE
11
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 a structure before it is subjected to any imposed load. Because the object of prestressing a structure is to improve its performance, the stresses resulting from prestressing are designed to counteract those induced by the acting loads. As an example, consider the case of a simply supported beam made from an elastic material which is equally strong in compression and in tension. The deflected shape of the beam and the stress distribution over the depth of the midspan section, which result from the application of a uniformly distributed load w, are shown in Fig. 11a. The principle of prestressing can subsequently be used to counteract this response by applying an eccentric compression force P to each end of the beam. The prestressing forces are shown in Fig. 11b together with the resultant deflected shape of the beam and the stress distribution over the midspan section. Figure 11c shows the response to the combined application of the load w and the prestressing forces P, which is obtained by the superposition of the response to the load w (Fig. 11a) and the response to the prestressing forces P (Fig. 11b). A comparison of the deflected shapes and midspan stresses shown in Figs. 11a and 11c illustrates the effects of prestressing on the structural behaviour of the beam: Not only can both the compressive and tensile stresses (and hence, the corresponding strains) in the top and bottom fibres of the midspan section be reduced, but the beam deflection can also be reduced. It should be noted that although the stress in the bottom fibre (f wb – f pb ) resulting from the combined action of the load w and the prestressing forces P is shown to be compressive in Fig. 11c, it could be tensile depending on the relative magnitudes of f wb and f pb . Similarly, the resultant deflection (d w  d p ) shown in Fig. 11c to be upward, could be downward. Given the fact that concrete is strong in compression and weak in tension, it seems natural that one of the most successful applications of the principle of prestressing has been the development of prestressed concrete. A simply supported plain, unreinforced concrete beam subjected to an increasing load will fail immediately after the development of cracks when the induced flexural tensile stress f wb (Fig. 11a) exceeds the tensile strength of the concrete. In the case of a reinforced concrete beam, suitable steel reinforcement is provided in the tension zone of the section to carry the tensile forces required for equilibrium of the cracked section. For this reason, a reinforced concrete 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, the original development of prestressed concrete was based on the prevention of flexural cracks forming under service loads. This was achieved by applying the criterion of no tensile stress, because it is generally 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 concrete and it is currently common practice to allow some tension to develop in the concrete. As shown in Fig. 11, the tensile stresses induced by the load can be neutralised to any desired degree by providing suitable prestressing. With the subsequent development of the concept of partial prestressing significant tension and controlled cracking are allowed to develop at service load levels, in much the same way as in reinforced concrete. The latest schools of thought on prestressed concrete embodies the view that partially prestressed concrete occupies the range between reinforced concrete and fully prestressed concrete (i.e. no tension is allowed to develop at service load levels). From this viewpoint reinforced concrete and fully prestressed concrete represent the two boundaries of the complete range of
12
INTRODUCTION
w fwt (compression) dw
fwb (tension) Section
Stresses at midspan section
(a) Response to uniformly distributed load
fpt (tension) dp e
P
P
e fpb (compression) Section
Stresses at midspan section
(b) Response to prestressing forces
w fwt – fpt dw  dp e
P
P
e fwb – fpb Section
Stresses at midspan section
(c) Response to uniformly distributed load and prestressing forces
Figure 11:
General effects of prestressing.
possibilities which exist for partially prestressed concrete and, as such, are two special cases of partially prestressed concrete. In prestressed concrete, the most commonly used method of applying the prestressing force to the concrete is by tensioning highstrength reinforcement, commonly referred to as tendons, against the concrete prior to the application of imposed loads. Two different processes can be distinguished in this regard:
• Pretensioning: In these prestressing methods, the prestressed reinforcement is tensioned before the concrete is placed.
EFFECTS OF PRESTRESSING
13
• Posttensioning: In these prestressing methods, the prestressed reinforcement is tensioned after the 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 (taken from Ref. 11) is quoted here for completeness: Prestressed concrete: Concrete in which there have been introduced internal stresses of such magnitude and distribution that the stresses resulting from given external loadings are counteracted to a desired degree. In reinforcedconcrete members the prestress is commonly introduced by tensioning the steel reinforcement. It is apparent from Fig. 11 that the use of prestressing will enable a designer to provide a structure of which the deflections at service load levels can be made much less than those of its reinforced concrete counterpart. This benefit is obtained in addition to the bonus of being in a position to provide a structure which is relatively crackfree 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 to counteract 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 considering the rectangular beam section shown in Fig. 12a. If a moment M = 286 kN.m is applied to the section, the resulting stresses at the top and bottom of the section can be calculated from f = m My/I, where y is the distance from the top (or bottom) fibre to the centroidal axis and I is the second moment of area of the section about the centroidal axis. Taking tension positive and compression negative, this calculation yields a stress of –5.94 MPa at the top and a stress of +5.94 MPa at the bottom, as shown in Fig. 12b. The concrete can easily carry the compressive stress at the top of the section, but will most probably crack under the tensile stress at the bottom because it cracks at a 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 force P = 2258 kN is taken to act at the same time as the moment of 286 kN.m (see Fig. 12c). This axial force induces an additional uniform compressive stress of –5.94 over the section, which is calculated from f = –P/A, where A is the area of the section. The total stresses resulting from the simultaneous application of M and P are obtained by adding the stresses which are separately produced by each of these actions. As shown in Fig. 12c, a total stress of –11.88 MPa is obtained at the top and a zero stress is obtained at the bottom. The concrete will be able to carry these stresses 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 next step, a force P = 1127 kN is applied at an eccentricity of 127 mm, measured from the centroid of the section, as shown in Fig. 12d. The additional stress which arises from the eccentricity is calculated from f = ± Pey/I, where e is the eccentricity as defined above. The stresses at the top and bottom of the section as produced by the various components of load are summarised in Fig. 12d, 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 in the previous case, the beneficial effect of applying P eccentrically becomes clear: The tensile stresses in the section can still be completely neutralised even though the magnitude of P has been reduced by half, and in the process the total compressive stress in the top fibre has also been reduced by a half.
14
INTRODUCTION
500 y=
760 2
= 380 mm
y = 380 A = 500 ´ 760 = 380.0 ´ 103 mm2
760 y = 380
I=
500 ´ 7603 12
= 182.9 ´ 108 mm4
(a) Section Properties
Stress
(b)
–
P A
– +
– My + I
Pey I
– My – P + I A
– +
Loading Condition
–5.94 MPa
–5.94 MPa
+5.94 MPa
+5.94 MPa
Pey
M = 286 kNm
(c)
–5.94 MPa
–5.94 MPa
–11.88 MPa
+5.94 MPa
–5.94 MPa
–5.94 MPa
–2.97 MPa
+2.97 MPa
+5.94 MPa
–2.97 MPa
–2.97 MPa
–5.94 MPa
–1.98 MPa
+3.96 MPa
+5.94 MPa
–1.98 MPa
–3.96 MPa
M = 286 kNm P = 2258 kN
(d)
0
–5.94 MPa
M = 286 kNm e = 127 P = 1127 kN
(e)
0
–3.96 MPa
M = 286 kNm e = 254 P = 751 kN
Figure 12:
Effects of prestressing on stresses.
0
I
GENERAL PRINCIPLES
15
As a final example in this regard, consider the case where the compression force P is further reduced to 751 kN but its eccentricity is increased to 254 mm, as shown in Fig. 12e. Also shown are the stresses 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 MPa is obtained at the top, which is even smaller than before. This result is consistent with the previous finding 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 to crack, it serves to illustrate that a larger eccentricity can be detrimental in the absence of external load (represented here by M), even though it is beneficial when the external load is present. This finding is important for design because it clearly shows that the critical stresses may arise either in the loaded or in the unloaded structure. These examples are intended to illustrate the effects of prestressing on the development of stress in the section, and are not intended to show that limiting the total tensile stress in the section to zero is necessarily beneficial or not. Another important effect of prestressing on structural behaviour is its impact on deflections at service load levels. This effect can be qualitatively investigated with reference to Fig. 11. In the case of the simply supported beam considered here, the externally applied load w will produce a downward deflection (see Fig. 11a) while the prestressing force P, which is applied at an eccentricity e, will cause an upward deflection (see Fig. 11b). The total deflection of the beam under the combined actions of the external load and the prestressing force is obtained by adding the deflections yielded by each load acting separately (see Fig. 11c). Because the deflections caused by the two components of load are opposite, it is clear that the downward deflection produced by the external load is always reduced by the presence of prestressing and, depending on the relative magnitudes of the two components 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 that although the deflection of the loaded structure is small, the upward deflection of the unloaded structure is unacceptably large. Such a situation can arise in cases where the live load to dead load ratio is large.
1.3
GENERAL PRINCIPLES
There are three different concepts which can be used to approach the simulation of the behaviour of a prestressed concrete member (Ref. 11). Each approach can be used for design provided that it is properly understood by the designer, and provided that the limitations of each are realized. In the following, each approach is briefly described. First approach: Prestressing transforms concrete into an elastic material. The fundamental idea behind this approach is that the precompression applied during prestressing transforms the concrete into an elastic material. The brittle behaviour of concrete arises from the fact that when its tensile strength, which is much less than its compressive strength, is reached it cracks and subsequently cannot carry any tensile stress. If it is accepted that concrete will not crack if there are no tensile stresses present, then it can be concluded that the removal of tensile stresses by prestressing will remove the source of its brittle behaviour and, in so doing, will transform it into 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.
16
INTRODUCTION
If the precompression induced by the prestress prevents the concrete from cracking, then the stresses, strains and deflections caused by each of the sets of forces can be considered separately and superimposed as required. The examples considered in Fig. 12 (see Section 1.2) serve as an illustration 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 which has 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 highstrength reinforcement has been tensioned against the concrete before any imposed load is applied. Following this approach, prestressed concrete is considered as a combination of concrete and steel, in which a resisting internal couple must be developed to equilibrate an external moment. The internal couple arises from the compression supplied by the concrete and the tension supplied by the steel, as is the case for reinforced concrete. From this point of view, the primary difference between the behaviour of prestressed concrete and reinforced concrete lies in the increased cracking load and the 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 the section shown in Fig. 13 subjected to a moment M = 286 kN.m. The prestressed reinforcement is placed at an eccentricity e = 254 mm and carries a tension T = 751 kN. This example is the same as that shown in Fig. 12e. 500
380
A = 380.0 ´ 103 mm2
380
I = 182.9 ´ 108 mm4
760 254
Section Properties
3.96 MPa M = 286 kNm
C = 751 kN ec = 127 la = 381
e = 254 T = 751 kN
0 Stress distribution
Figure 13:
Prestressed concrete considered as a type of reinforced concrete.
GENERAL PRINCIPLES
17
Horizontal equilibrium yields T = C = 751 kN. The internal couple provided by the compression in the concrete C and the tension in the steel T must be equal to the external moment M = 286 kN.m to satisfy moment equilibrium. The lever arm at which these forces are acting is given by 286 3 la = × 10 = 381 mm 751 Therefore C is acting at an eccentricity e C = 381  254 = 127 mm. The stress distribution in the concrete is obtained by considering the compression C = 751 kN acting on the concrete at an eccentricity of 127 mm. Using elastic theory f = − = −
C − C eC y + I A 751 × 103 − 751 × 103 × 127 × 380 + 182.9 × 108 380 × 103
= − 1.98 +− 1.98 So that
ftop = – 3.96 MPa (top fibre, compression) fbot = 0 (bottom fibre)
These results are shown in Fig. 13 and are the same as obtained before in Fig. 12e. 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
h
L/2
L/2 L
(a) Parabolic tendon profile
P
P
wb (b) Tendon forces acting on the concrete Figure 14:
Simply supported beam with parabolic tendon.
18
INTRODUCTION
supported beam shown in Fig. 14a which has a parabolically curved tendon. It can be seen from Fig. 14b 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 w b acting over the span of the beam. This load arises because the concrete prevents the tendon from straightening under the action of the prestressing force. It can be shown that for the tendon profile considered here wb = where
8Ph L2
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 of the 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 prestressing force P applied at the anchor positions at the ends of the beam and the unbalanced load (w – w b ) acting over its span. As an example of how this approach can be used to analyse a prestressed concrete beam, consider the simply supported beam shown in Fig. 15a, which is subjected to a uniformly distributed load w = 42.9 kN/m. The prestressing force P = 751 kN and the tendon profile is parabolic, with an eccentricity e = 254 mm at midspan and zero eccentricity at the ends. Since the bending moment at midspan M = 42.9 ´ 7.3 2 /8 = 286 kN.m, it is clear that this example is the same as that shown in Fig. 12e if the midspan section is considered. The upward uniformly distributed load applied by the tendon is given by wb = =
8Ph L2 8 × 751 × 0.254 7.32
= 28.6 kN/m The loads and forces acting on the concrete are shown in Fig. 15b, from which it is clear that the unbalanced load is (42.9 – 28.6) = 14.3 kN/m acting downward. The midspan bending moment induced by this unbalanced load is M = =
(w − wb) 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 = =
My I 95.3 × 106 × 380 182.9 × 108
= 1.98 MPa
GENERAL PRINCIPLES
19
w = 42.9 kN/m
500
380 760 h = 254
3650
380
254
Section at midspan
3650
A = 380.0 ´ 103 mm2
7300
I = 182.9 ´ 108 mm4 (a) Simply supported beam
w = 42.9 kN/m
P = 751 kN
P = 751 kN
wb = 28.6 kN/m 7300
(b) Loads and forces acting on the concrete
500 –1.98 MPa
–1.98 MPa
+1.98 MPa
–1.98 MPa
–3.96 MPa
760
Stress due to unbalanced load (= 14.3 kN/m)
Stress due to prestressing force applied at ends of beam
(c) Concrete stress in midspan section Figure 15:
Analysis using load balancing approach.
0 Total stress
110
So that
INTRODUCTION
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 = − = −
P A 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. 15c and are the same as obtained before in Fig. 12e.
1.4
BASIC DEFINITIONS
Some of the most commonly encountered prestressing techniques and features of construction of prestressed concrete structures are introduced in the following (Ref. 11). The descriptions are brief because the techniques and procedures covered here are more expansively dealt with in subsequent Chapters. The most commonly used prestressing method is to tension highstrength reinforcement against the concrete. 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 highstrength wires, bars or strand. Prestressing methods can be classified either as being a pretensioning method or as being a posttensioning method, depending on whether the concrete has not been placed or whether it has been placed at the time of tensioning of the reinforcement. Although the terms pretensioning and posttensioning have been adequately defined in Section 1.1, their definitions are repeated here for convenience:
• Pretensioning: In these prestressing methods, the prestressed reinforcement is tensioned before the concrete is placed.
• Posttensioning: In these prestressing methods, the prestressed reinforcement is tensioned after the 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 related to the construction of prestressed concrete structures. Internal and External Prestressing Internal prestressing refers to prestressed concrete structures in which the tendons are contained within the concrete, while external prestressing implies that the prestressing force is applied externally. External prestressing can be achieved either by placing the tendons outside the member or by applying external prestressing forces using jacks. Internal prestressing is by far the most
BASIC DEFINITIONS
111
commonly used method, although external prestressing by means of external tendons has recently gained 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. 16. If the jacks are properly placed, the precompression which they produce can neutralise any tension caused by the applied load. However, this procedure is of little practical importance because the timedependent 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 prestressing becomes ineffective, unless the jacks can be readjusted. Shrinkage can be viewed as the timedependent strain which develops in the absence of load, while creep may be seen as the timedependent strain which develops in the presence of load. These phenomena are more expansively dealt with in Section 2.1.5. Jack
Figure 16:
Jack
External prestressing using jacks.
Linear and Circular Prestressing Linear prestressing refers to elongated elements such as beams and slabs, even though the tendons may be curved and not straight. Circular prestressing, on the other hand, refers to circular structures such as silos, pressure vessels, tanks and pipes where the circular shape of the tendons is dictated by 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. A pretensioned tendon is bonded to the concrete by virtue of the construction method, although it can be debonded over a portion of its length by taking appropriate steps to accomplish this. Posttensioned tendons are encased in a duct so that they can be tensioned after the surrounding concrete has hardened sufficiently. Bonding is subsequently accomplished by injecting grout into the duct. Tendons not bonded to the concrete over their entire length are referred to as unbonded tendons, and can only be accomplished with posttensioning. 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 in stages. In such cases the prestressing may also be applied in appropriate stages to avoid overstressing the concrete. This technique is referred to as stage stressing.
112
INTRODUCTION
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 at service load levels. Additional ordinary nonprestressed reinforcement is usually provided in partially prestressed 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 to their physical attributes, is that higher strength materials (for both concrete and steel) are used for prestressed concrete. In prestressed concrete the highstrength steel is tensioned and anchored against the 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 that cracking of the section is eliminated and, as a result, the full concrete section becomes active in resisting the load. This mechanism is much more effective than is the case for reinforced concrete where only the uncracked part of the section in the compression zone participates in resisting the load.
• The deformations induced by the prestressing serve to offset those produced by the load, and can be used by the designer to control deflections. Higher strength concrete may be used to obtain more economic sections than with reinforced concrete. The following advantages of prestressed concrete are often put forward when compared to reinforced concrete (Ref. 12):
• Prestressed concrete requires smaller quantities of material than reinforced concrete because highstrength materials are efficiently and effectively used and because it uses the entire section to resist the load. This means that prestressed concrete members are lighter and more slender than their reinforced concrete counterparts.
• The fact that members are lighter and more slender if prestressed concrete rather than reinforced concrete is used, leads to other advantages:
 Savings can be realised in the reduced cost of lighter supporting structures and, in the case of 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 multistorey buildings.
• Prestressed concrete generally provides better corrosion protection to the reinforcement than does reinforced concrete. This advantage is significant for structures subjected to aggressive environments and for fluidretaining structures.
• Improved deflection control is possible with prestressed concrete. • Prestressed concrete members will require less shear reinforcement than reinforced concrete members. This follows from the fact that the shear capacity of a prestressed member is increased
HISTORY OF PRESTRESSED CONCRETE
113
by curved tendons, which carry some of the shear, and by the precompression, which reduces the principal tension.
• It often happens that the worst service load condition for a prestressed concrete structure occurs during the prestressing operation. In such a case, it can be claimed that the safety of the structure has been partially tested: If the structure successfully withstands the effects of the prestressing operation, 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 of reinforced concrete is complicated by the fact that each has a range of applicability, depending on the type of structure and the specific design requirements. However, if such a comparison is made where the ranges of applicability overlap, care must be taken to include not only the cost of the materials but also to include the additional costs associated with prestressed concrete, such as the use of specialised equipment and hardware, greater design effort, more supervision and the use of specialised personnel. Such a comparison should also reflect the relative performance and cost advantages inherent in each type of structure. For example, since the decking for posttensioned slabs can be stripped after tensioning, shorter construction times are realized together with all the related savings in construction and financing costs. If the view is taken that prestressed concrete and reinforced concrete represent the two boundaries of the range of possibilities which exist for partially prestressed concrete, they form part of the same system and cannot be considered as being in competition with each other. A comparison, as given above, can therefore be seen to be inappropriate because a specific prestressing level can always 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 reinforced concrete 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. 11 to 17, is presented in the following. It is interesting to note that the development of prestressed concrete is characterised by its individualistic nature, even though it took place simultaneously in several countries. A possible reason for this is the lack of communication which existed between the countries during World War II. The first application of the principle of prestressing to concrete is credited to P. H Jackson, of San Francisco, who in 1886 applied for a patent for Constructions of Artificial Stone and Concrete Pavements in which steel tie rods, passed through concrete blocks and concrete arches, were tightened by nuts. These structures served as slabs and roofs. An application for a patent, which can also be related to prestressing, was made in 1888 by the German C. E. W. Doehring. This patent covered 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 strength of the concrete in a beam as effectively as possible. To achieve this he, in 1896, became the first person to clearly articulate the purpose of prestressing as the need to counteract the tension produced by the load with compression induced by an applied prestressing force. The German engineer M. Koenen developed this idea and in 1907 derived an expression from which the required prestressing force could be calculated. The loss of prestressing force resulting from elastic shortening was accounted for in these proposals. In 1907 the Norwegian J. G. F. Lund suggested the construction of prestressed vaults using prefabricated concrete blocks jointed in mortar. The prestressing was applied by tensioned tie rods which transmitted the compression to the blocks by bearing plates at the ends. Bond between the tie rods and the mortar was destroyed at stretching. A similar prestressing procedure was suggested by the American engineer G. R. Steiner in the following year. This procedure consisted of initially tightening the reinforcing rods against the green concrete to destroy bond and to subsequently
114
INTRODUCTION
complete the tensioning operation once the concrete has hardened. These two procedures appear to be the first applications of posttensioning. In the procedures outlined above mild steel was tensioned to the permissible stress prescribed at the time (i.e. approximately 110 MPa), which corresponds to a strain of 0.00055 in the steel. Because this strain is comparable to the magnitude of the strain induced by shrinkage and creep of the concrete, most of the prestressing would have been lost with time. Therefore, these early attempts were bound to give unsatisfactory results because shrinkage and creep of the concrete were not accounted for. The American engineer R. H. Dill appears to have been the first, in 192325, to suggest that full prestressing can be provided by posttensioning highstrength steel, instead of mild steel. Dill coated the reinforcement with a plastic substance to prevent bond, and tensioned the reinforcement after most of the shrinkage in the concrete had taken place. The effects of creep were accounted for by occasionally tightening the nuts used for stretching the reinforcement. However, it should be noted that Dill did not actually say that highstrength steel was required for maintaining full prestress after losses. In 1922, W. H. Hewett, also of America, successfully applied prestressing to circular concrete 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 shrinkage and creep of the concrete, and is credited with the development of prestressed concrete as we know it today. In 1928, he introduced the use of highstrength steel bonded to the concrete, together with the requirement that a high tensioning stress be applied to the steel. The significance of these proposals is demonstrated by the fact that shrinkage and creep can together induce a strain of approximately 0.001 in the concrete, while a strain of approximately 0.007 can be induced in highstrength 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 using highstrength steel for prestressing, it is still possible to completely neutralise any tension induced by the load in the concrete, even after losses. Freyssinet also demonstrated that a considerable saving in the required quantity of steel may be achieved by using highstrength reinforcement. The large scale use of prestressed concrete only became possible after the development of reliable and economical methods of carrying out the tensioning operation. The first practical implementation of pretensioning was made by E. Hoyer of Germany who, in 1938, introduced a procedure whereby piano wire was tensioned over a large distance, after which the concrete was cast. The prestress was transferred to the concrete by cutting the wires after hardening of the concrete. Although Hoyer was granted a patent for the longline pretensioning method, it should be pointed out that the idea did not originate with him, but rather with Freyssinet, whose proposal for the longline process he combined with Wettstein’s (1919) experience with the use of piano wire. The large scale use of posttensioning started with the introduction, in 1939, of Freyssinet’s system whereby a doubleacting jack was used to tension and to anchor 12 wire cables in conical wedges, which served as anchors. 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 structures have 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 subsequent development of prestressed concrete include: G. Magnel of Belgium (Ref. 18), Y. Guyon of France (Ref. 19), P. W. Abeles of England (Ref. 14 and 15), F. Leonhardt of Germany (Ref. 110), V. V. Mikhailov of Russia, and T. Y. Lin of America (Ref. 11 and 111).
REFERENCES
115
F. V. Emperger is credited with being the first to use the concept of partial prestressing when, in 1939, he suggested that pretensioned wires be added to conventionally designed nontensioned reinforcement to reduce the extent of cracking. This idea was further developed by Abeles who, in 1940, suggested the use of nontensioned highstrength steel together with pretensioned or posttensioned tendons. Apart from the recommendation that solely highstrength steel be used, this proposal also differed from Emperger’s in that a prestressing force of a definite designed magnitude be applied. The acceptance of partial prestressing was at first retarded, perhaps by the opposition to this concept by Freyssinet (Ref. 112), who stated (Ref. 113) “... there is no halfway house between reinforced and prestressed concrete; any intermediate systems are equally bad as reinforced or prestressed structures, and are of no interest.” However, partial prestressing has made enormous progress through the efforts and contributions of many eminent engineers and researchers, and is commonly used today.
1.7
REFERENCES
11
Lin, T. Y., and Burns, N. H., Design of Prestressed Concrete Structures, 3rd ed., John Wiley & Sons, New York, 1981.
12
Naaman, A. E., Prestressed Concrete Analysis and Design: Fundamentals, McGrawHill Book Company, New York, 1982.
13
Abeles, P. W., The Principles and Practice of Prestressed Concrete, Crosby Lockwood & Son, London, 1949.
14
Abeles, P. W., An Introduction to Prestressed Concrete, Volume I, Concrete Publications Ltd., London, 1964.
15
Abeles, P. W., An Introduction to Prestressed Concrete, Volume II, Concrete Publications Ltd., London, 1966.
16
Collins, M. P., and Mitchell, D., Prestressed Concrete Structures, PrenticeHall, Englewood Cliffs, New Jersey, 1991.
17
Khachaturian, N., and Gurfinkel, G., Prestressed Concrete, McGrawHill Book Company, New York, 1969.
18
Magnel, G., Prestressed Concrete, Concrete Publications Ltd., London, 1948.
19
Guyon, Y., Prestressed Concrete, John Wiley & Sons, New York, Vol. 1, 1953, Vol. 2, 1960.
110 Leonhardt, F., Prestressed Concrete Design and Construction, English translation, Wilhelm Ernst und Sohn, Berlin 1964, (1st ed., 1955, 2nd ed., 1962 in German). 111 Lin, T. Y., Design of Prestressed Concrete Structures, John Wiley & Sons, New York, 1955. 112 Cohn, M. Z., “Some Problems of Partial Prestressing,” Partial Prestressing, from Theory to Practice. Volume I: Survey Reports, Edited by M. Z. Cohn, Chapter 2, NATO ASI Series, Series E, No. 113a, Martinus Nijhoff Publishers, Dordrecht, 1986, pp. 1563. 113 Freyssinet, E., “Prestressed Concrete, Principles and Applications,” ICE Proceedings, Vol. 33, No. 4, February 1950, pp. 331380.
CONCRETE
21
2 MATERIAL PROPERTIES Prestressed concrete combines high quality concrete and prestressed steel, as well as nonprestressed ordinary reinforcing steel. Before considering the behaviour of the materials in combination, it is essential that the designer is familiar with the relevant properties of each of these materials. This topic is extensively covered in the technical literature and this chapter summarizes the most important properties required for the design of prestressed concrete structures.
2.1
CONCRETE
The mechanical properties of concrete under uniaxial stress are considered in this Section. Although concrete is usually subjected to a three dimensional state of stress in practical structures, the assumption of a uniaxial stress condition can very often be justified. Where the effects of a multiaxial 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. 21.
2.1.1 Compressive strength The single most important mechanical property of concrete is its compressive strength because it is extensively used in quality control and because many other mechanical properties required for the design of prestressed concrete structures can be expressed in terms of this property. The compressive strength can be obtained from standard tests using either cubes or cylinders loaded to failure (Refs. 22 to 24). The maximum load sustained during such a test, divided by the cross sectional area of the specimen yields the compressive strength. It is extremely important to note that the compressive strength must be determined in strict compliance with the requirements of a standard testing procedure because the measured results depend on the test method and also because it is primarily used as an index of strength in its application in structural design. The standard specification generally used in South Africa is SABS 863 (Ref. 22), according to which 150 mm cubes are loaded to failure in a calibrated testing machine at a loading rate of approximately 15 MPa/min. Apart from intrinsic factors, which cover the composition of the concrete, the following external factors influence the compressive strength:
• Age of concrete: The compressive strength increases with time, provided the concrete is properly cured. The development of strength with time is shown in Fig. 21 (Ref. 25) for a typical concrete using ordinary Portland cement, where it may be seen that the rate at which the strength develops reduces with time. As a percentage of the value at 28 days, the strength will generally vary between 33 and 50%, and between 60 and 75% after 3 and 7 days, respectively (Ref. 26). The values listed in Table 21 for the characteristic strength at various other ages are suggested by TMH7 (Ref. 27) for use in structural design. It is common practice to base the design of reinforced concrete structures on the 28day strength, and to ignore any subsequent strength increase. However, in prestressed concrete, high stresses may be induced prior to 28 days, e.g. high anchor zone stresses and high flexural stresses which occur at transfer. For such cases, the time dependence of strength must properly be accounted for in the design.
• Shape and size of the specimens: Standard testing procedures which use 150 mm diameter and 300 mm long cylinders are also used to determine the compressive strength (Ref. 24). Unfortunately, the magnitude of the compressive strength obtained from cylinders differs from
22
MATERIAL PROPERTIES
Compressive strength (MPa)
40
30
20
10
0 1 day
7 days
28 days 3 months
1 year
5 years
Age of concrete (Log scale) Figure 21:
Increase of concrete strength with time. Typical curve for concrete made with ordinary Portland cement (Ref. 25).
Table 21:
Characteristic strength of concrete (ordinary Portland cement) at other ages (Ref. 27).
Grade
Characteristic strength f cu (MPa)
Characteristic strength at other 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
values obtained from tests on cubes. This directly stems from the fact that the measured compressive strength is dependent on the shape of the specimen. The cylinder strength is generally between 70 and 90% of the cube strength, and an average value of 80% is widely accepted. Research has also shown that the ratio of cylinder strength to cube strength tends to increase as the strength of the concrete increases (Ref. 28). This trend is clearly demonstrated in Table 22 and Fig. 22, which show the relationship between cylinder strength and cube strength. Note that the data shown in Fig. 22 applies to concretes with very high strength, also referred to as high performance concrete. The size of the specimen also has an influence on the magnitude of the measured compressive strength as shown in Figs. 23 and 24 for cubes and cylinders, respectively. The general trend is that larger specimens yield lower compressive strengths. It should be noted that the data shown in Fig. 24 was obtained from cylinders with a height to diameter ratio of 2, which is the value normally used. Among the various reasons put forward to explain the trend that the strength of a specimen increases as it becomes smaller, the following seems reasonable for the size of specimens normally tested. The testing machine provides some lateral restraint to the specimen because of
CONCRETE
Table 22:
23
Relationship between cylinder and cube strengths (Ref. 29). 12
20
30
40
50
60
70
80
Cube strength (MPa) 150 mm cubes
15
25
37
50
60
70
85
95
Cylinder strength fc¢ (MPa) 150 ´ 300 mm cylinders
Cylinder strength (MPa) 150 × 300 mm
Figure 22:
110 100 90 80 70 60 50 40
Held (Ref. 210)
30
Smeplas (Ref. 211) fc¢ = 0.8 fcu
20 30
50
70 90 110 Cube strength fcu (MPa) 100 mm cubes
130
Influence of the specimen shape on the compressive strength (Ref. 212).
friction which develops between the platen plates and the contact faces of the specimen. For smaller specimens, the restraint will be effective over a larger portion of its total height than will be the case for larger specimens, where its effect will be limited to the end regions. Since the stress field induced by the restraint tends to confine the concrete and hence, increase the strength 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 assumption that failure is caused by the propagation of small cracks and that the largest crack is responsible for complete failure and fracture, similar to the weakest link in a chain. Since the probability that such a flaw, which will induce failure at a given load, is contained in a specimen can reasonably be expected to increase with specimen size, the compressive strength of a larger specimen tends to be smaller than that of a smaller specimen (see Ref. 21).
• Applied load rate: By increasing the loading rate beyond that prescribed by a standard test (15 MPa/min, Ref. 22), the measured compressive strength can be increased by up to 20%. On the other hand, the compressive strength can be reduced by as much as 20% if the load is applied over several months (Ref. 213). The reduction of strength caused by the longterm loading is usually ignored in design because the unconservative consequence of this assumption is more than offset by the usual design practice according to which a design is based on the 28 day strength, which ignores the significant timedependent strength increase.
24
MATERIAL PROPERTIES
Relative strength (%)
110 Akroyd Harman Neville
105
100
95
90 0
50
100
150
200
250
300
Nominal cube size (mm) Figure 23:
Influence of the cube size on the compressive strength (Ref. 215).
Nominal diameter of cylinder (mm) 0
100
0
5
200
300
400
500
600
700
800
900 1000
110
Relative strength (%)
105 100 95 90 85 80
Figure 24:
10 15 20 25 30 Nominal diameter of cylinder (in)
35
40
Influence of the cylinder size on the compressive strength (Ref. 215).
Experience has shown that the measured compressive strengths obtained from specimens taken from the same mix can show a significant variation, even if the specimens are made under strict laboratory control. It is therefore not a practical approach to specify a single precise value for compressive strength. Instead, a statistical approach is followed by most of the modern design codes of practice (Refs. 27 and 214) whereby the strength is specified in terms of the characteristic strength f cu , which is defined as the strength below which not more than 5% of the measured results may be expected to fall. If it is assumed that the measured values of strength are normally distributed, this definition can be expressed as follows:
f cu = f m  164 . s
(21)
CONCRETE
where
25
fcu = characteristic compressive strength fm = mean compressive strength s = standard deviation
Note that fm and s are obtained from test results. High strength concrete is usually specified for prestressed concrete because of its improved performance, not only with regard to compressive strength but also with regard to increased tensile strength, increased modulus of elasticity and reduced creep. Concrete strengths which range between 30 to 60 MPa are usually specified for prestressed concrete in South Africa, while strengths of up to 70 MPa can be used in precast pretensioned applications. When specifying concrete it is important to bear in mind the strength requirements at transfer, which can be the governing consideration. The minimum characteristic strengths recommended by SABS 0100 (Ref. 214) for prestressed concrete are shown in Table 23, while both SABS 0100 and TMH7 (Ref. 27) require that only concrete with a characteristic strength of 30, 40, 50, or 60 MPa be used. When specifying the minimum compressive strength of the concrete at transfer in the case of posttensioning, proper information must be obtained from the supplier of the system because this value depends on the particular system being used and can be higher than the values listed in Table 23. Table 23:
Minimum recommended characteristic strength according to SABS 0100 (Ref. 214).
At 28 days (f cu )
At transfer
Pretensioned
40 MPa
Bonded
25 MPa
Posttensioned
30 MPa
Unbonded
18 MPa
2.1.2 Stressstrain relationship The stressstrain behaviour of concrete loaded in uniaxial compression is shown in Fig. 25 (Ref. 213) together with the stressstrain curves for the constituent materials of the concrete, namely aggregate and the hardened cement paste. A comparison of these curves reveal the following:
• The stressstrain response of the aggregate and the paste are both more or less linear up to failure, whereas the concrete has a nonlinear response over the entire load spectrum.
• The stressstrain response of the concrete falls between that of the aggregate and that of the cement paste. The nonlinear response of the concrete is caused by microcracking which occurs at the aggregatepaste interfaces (Ref. 216). These cracks are often only visible close to failure when considerable lateral expansion occurs. Figure 26 shows typical experimentally obtained stressstrain curves for normal weight concrete having strengths which vary from 20 to approximately 85 MPa. Each curve is characterized by an ascending portion followed by a descending portion. The ascending portion is initially almost straight, becoming flatter, and hence progressively more nonlinear, with increasing load. The slope of this portion also tends to increase with an increase in compressive strength (this property is more expansively covered under Section 2.1.3).
26
MATERIAL PROPERTIES
Coarse aggregate rock
Stress fc
Concrete
Cracks at interface of aggregate
Hardened cement paste Strain ec Figure 25:
Uniaxial stressstrain response of concrete and its constituent materials (Ref. 213).
90 12
70
10
60 8 50 6
40 30
4
20 2
10 0 0
Figure 26:
(ksi)
Compressive stress fc (MPa)
80
0.001
0.002 Strain ec
0.003
0 0.004
Typical stressstrain curves for normal weight concrete in uniaxial compression (Ref. 217).
The maximum stress, which separates the ascending and descending portions of the curve, is defined as the compressive strength. The strain corresponding to this stress is about 0.002 for normal strength concrete, while indications are that it increases with an increase in strength, particularly in the case of high strength concrete. The transition from the ascending portion to the descending portion becomes sharper as the compressive strength increases, thus indicating that a more brittle behaviour is associated with stronger concrete. The slope of the descending portion of the stressstrain curve as well as the strain corresponding to failure of the specimen (often referred to as the ultimate strain) both change with a change in compressive strength, the slope becoming steeper and the ultimate strain becoming smaller with an increase in strength. This reduction of the ultimate strain, once again, indicates that the uniaxial compressive behaviour of concrete becomes more brittle with increasing strength. It should be noted
CONCRETE
27
that it is very difficult to experimentally determine the descending portion of the stressstrain curve because the failure mode of a compressive specimen is brittle. Consequently, special experimental techniques need to be resorted to for determining this portion of the curve. The influence of load rate on the compressive stressstrain behaviour of concrete is shown in Fig. 27 (Ref. 218), 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
Ratio of concrete stress to cylinder strength fc / fc¢
• the descending portion becomes flatter. 1.00 Strain rate 0.001 per 100 days 0.75
0.50 0.001 per day Cylinder strength fc¢ = 20.7 MPa (3000 psi) at 56 days
0.25
0.001 per min. 0
Figure 27:
0.001 per hr.
0
0.001
0.002
0.003 0.004 Concrete strain ec
0.005
0.006
0.007
Influence of the loading rate on the stressstrain curve (Ref. 218).
It is important to note that the stressstrain curve for concrete in uniaxial compression differs from that in flexure because the stress distribution in the specimens are different. This aspect is covered in Chapter 4.
2.1.3 Modulus of elasticity A material is defined as being elastic when the deformations induced by an applied load is completely recovered immediately after the load is removed. In the case of a linear elastic material, the relationship between the applied stress and the resulting strain can be expressed as follows: f = Ee where
(22)
f = applied stress e = resulting strain E = modulus of elasticity (Young’s modulus) of the material
Inspection of Fig. 26 clearly shows that the stressstrain relationship for concrete is nonlinear over the complete load spectrum. However, the initial portion of the ascending branch may be seen to be approximately linear. This feature makes it possible to approximate concrete as a linear elastic
28
MATERIAL PROPERTIES
material at service load levels, because the magnitude of the induced stress generally falls within this quasilinear range of the stressstrain curve. The modulus of elasticity of concrete can be defined in various ways because linear elasticity is an approximation of the actual nonlinear behaviour in this range. Three possible definitions are presented in Fig. 28:
• The initial tangent modulus Eci is defined as the slope of the tangent to the stressstrain curve at its origin, and is often used as a parameter for the mathematical description of the stressstrain curve.
• 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 the stressstrain curve to a specified point P on the curve.
1
Eci P
1 Stress fc
Ect
Ec 1 Eci = Initial tangent modulus Ect = Tangent modulus at point P Ec = Secant modulus Strain εc
Figure 28:
Definitions of the modulus of elasticity of concrete.
The secant modulus is commonly used in prestressed concrete design, whereas the initial tangent modulus and the tangent modulus are not commonly used in daytoday design. The modulus of elasticity of concrete must be determined in strict accordance with standard testing procedures which have been developed for this purpose. These procedures include both static methods (Ref. 219) and dynamic methods (Ref. 220). The testing procedure prescribed by BS 1881: Part 121 (Ref. 219) requires that the static modulus be determined from tests on standard 150 mm diameter by 300 mm high cylinders loaded at a rate of 0.6 ± 0.4 MPa/min. This test defines the static 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 very small 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 and released. SABS 0100 (Ref. 214) suggests that the following expression can be used to obtain an estimate of the static secant modulus Ec from the dynamic modulus Ecq to within 5 GPa: Ec = 1.25 Ecq − 19 GPa
(23)
CONCRETE
29
55
Dolomite (Olifantsfontein)
Static elastic modulus (GPa)
50 Dolerite (Ngagane) (Newcastle)
45
Andesite (Eikenhof) (Jhb) Greywacke (Malmesbury shale) (Peninsula) Wits Quartzite (Vlakfontein)
40 35
Granite (Jukskei) (Midrand) 30
Siltstone (Leach & Brown) (Ladysmith)
25 20 20
30
40
50
60
65
Cube strength (MPa) Figure 29:
Relationship between static modulus of elasticity and compressive strength for ages from three days to 28 days (Ref. 222).
The type of aggregate used for the concrete appears to be the most important factor influencing the modulus of elasticity. Other factors which can also have an influence are mix proportions, shape of the aggregate, age of the concrete and moisture condition. For a given aggregate type, the modulus of elasticity increases with an increase in compressive strength. This finding, together with the fact that the compressive strength is often the only property of the concrete available at the design stage, has led to many attempts being made to correlate the modulus of elasticity with the compressive strength. This approach has often been questioned and recent research (Refs. 221 and 222) has shown that no single expression can be used to relate the modulus of elasticity to the compressive strength only. The relationships between the modulus of elasticity and the compressive strength for various South African aggregates shown in Fig. 29 (Ref. 222) clearly illustrate this point, because they show that, for a given strength, the modulus of elasticity varies widely depending on the aggregate type used. The values for the static modulus of elasticity recommended by SABS 0100 (Ref. 214) for concrete using normaldensity aggregates are given in Table 24. SABS 0100 also suggests that the following expression may be used to estimate the magnitude of the static modulus from the 28day cube strength:
E c = Ko + 0.2 f cu GPa where
(24)
E c = static secant modulus of elasticity K o = a constant closely related to the modulus of elasticity of the aggregate f cu = characteristic cube strength at 28 days, in MPa
When the properties of the aggregate are unknown, SABS 0100 suggests that Ko can be taken as 20 GPa for normal weight concrete. In the case of lowdensity aggregate concrete, with a density of between 1400 to 2300 kg/m 3 , the values in Table 24 should be multiplied by (r/2300) 2 , where r is the density of the concrete in kg/m 3 .
210
MATERIAL PROPERTIES
Table 24:
Modulus of elasticity of concrete (Ref. 214). Static modulus E c (GPa)
Characteristic cube strength, f cu (MPa)
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
Dynamic modulus E cq (GPa)
The recommended values for the static modulus Ec given in Table 24 are the same as those given by TMH7 (Ref. 27). References 222 to 224 recommend that more accurate estimates of Ec can be obtained from E c = K o + a f cu , where Ko and a are coefficients depending on the aggregate type. These references list values of Ko and a for a fairly wide range of South African aggregates. SABS 0100 (Ref. 214) recommends the following expression for estimating modulus of elasticity at any time t ≥ 3 days: fcu, t Ec, t = Ec, 28 0.4 + 0.6 fcu, 28 where
(25)
Ec, t = modulus of elasticity at time t Ec, 28 = modulus of elasticity at 28 days, obtained from Table 24 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 21. It should be kept in mind that the expressions given above yield results which, at best, must be viewed as being approximate. It is therefore recommended that the modulus of elasticity should be determined from tests on concrete specimens made from the actual aggregates to be used in cases where structural deformations are important. Where such tests are not feasible, a reasonable approach to be followed in design would be to consider a range of values (as given in Table 24) which would bracket the expected deformations. It is generally assumed that the modulus of elasticity in tension before cracking is the same as in compression. However, it should be pointed out that some researchers question the validity of this assumption.
2.1.4 Tensile strength The stressstrain diagram given in Figure 210 (Ref. 225) was obtained from a direct tensile test, and shows that the response is almost linear up to cracking. This diagram also shows that the tensile strength of concrete is considerably smaller than its compressive strength.
211
ecr = 0.12 ´ 103
13 mm notch
3 D
Concrete stress fc (MPa)
4
83 mm gauge
CONCRETE
76 mm
Specimen: 76 ´ 19 ´ 305 mm fc¢ = 43.9 MPa max. aggregate size = 10 mm water cured 28 days
2
1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Elongation D (mm) Figure 210:
Stressstrain response of concrete in uniaxial tension (Ref. 225).
Standard testing procedures from which the tensile strength may be indirectly measured have been developed because of the practical difficulties associated with the direct tensile test. Two methods are 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. 226). According to this test a 150 mm diameter cylinder, 300 mm long, is loaded across a diameter until failure occurs (see Fig. 211). If it is assumed that the cylinder behaves as an elastic body, the resulting horizontal stress across the vertical diameter will be found to be uniformly distributed over most of the depth of the cylinder, as shown in Fig. 211. The magnitude of this stress at splitting is defined as the splitting tensile strength f ct , and is given by:
f ct = where
2P p LD
(26)
P = the measured compression force at splitting L = length of cylinder (300 mm) D = diameter of cylinder (150 mm) P D
L
fct fct
Figure 211:
P
Split cylinder (indirect tension) test.
The modulus of rupture test is described by SABS Method 864 (Ref. 227) (also BS 1881: Part 118 (Ref. 228)) and consists of loading a simply supported beam of square cross section to failure. The dimensions of the beam cross section are 100 × 100 mm (or 150 × 150 mm) while the span length
212
MATERIAL PROPERTIES
is 300 mm (or 450 mm). The load is applied at the thirdspan points (see Fig. 212). The flexural tensile stress in the bottom fibre of the section at failure, calculated on the basis of ordinary beam theory, is defined as the modulus of rupture fr and is given by: fr = where
PL b h2
(27)
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)
P/2
P/2
h
b
fr L/3
L/3
L/3
L=3h Figure 212:
Modulus of rupture test.
The modulus of rupture will overestimate the actual flexural tensile strength of concrete and should be viewed as a hypothetical strength to be used as a comparative measure for practical purposes only.
Tensile strength (MPa)
Figure 213 (Ref. 229) shows typical relationships between the various measures of tensile strength and the compressive strength. From this figure it may be seen that the tensile strength can be related to the compressive strength and that the modulus of rupture is approximately 1.65 times the splitting tensile strength. 8 Modulus of rupture fr
6 Split cylinder fct 4 Direct tension
2
0 0
10
20
30
40
50
60
70
Compressive strength fcu (MPa) Figure 213:
Relationships between tensile and compressive strengths of concrete (Ref. 229).
CONCRETE
213
SABS 0100 (Ref. 214) and TMH7 (Ref. 27) do not give explicit characteristic values for the splitting tensile strength nor the modulus of rupture. Where these properties are required for design they are included as allowable stresses. The ACI code (Ref. 230) suggests the following relationship between modulus of rupture fr and cylinder strength: fc′ MPa fr = 0.63 √
(28)
fc′ = Cylinder strength in MPa
where
Naaman (Ref. 28) gives the following expression for the splitting tensile strength fct : fc′ MPa fct = 0.5 √
(29)
It is important to note that, because the splitting tensile strength and the modulus of rupture differ not only from each other but also from the direct tensile strength, these quantities represent different measures of tensile strength. As such, care should be exercised to ensure that they are properly used in design. Equations 28 and 29 should also be used with care because it appears that there is no simple relationship between tensile and compressive strength, the reason being that factors such as watercement ratio, curing conditions, age, mix proportions and properties of the aggregate do not affect these properties to the same degree (see Ref. 21).
2.1.5 Timedependent behaviour Definitions When concrete is subjected to a sustained stress, the resulting strain can be divided into the following three components:
• Instantaneous elastic strain: When the stress is applied to the concrete it causes an instantaneous elastic strain, which can be expressed as follows (see Section 2.1.3): εc = where
fc Ec
(210)
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 of the timedependent strain which is independent of stress. Shrinkage therefore corresponds to the timedependent strain which occurs in the absence of stress.
• Creep strain: Creep is defined as the component of the timedependent strain which is dependent on the applied stress. Although this definition has been used for many years, it is important to point out that, strictly speaking, it is not correct because it implies that creep and shrinkage are independent phenomena which are additive when they occur simultaneously (Ref. 231). It is well known that creep and shrinkage are not independent, the effect of shrinkage on creep being to increase its magnitude. In order to use the mass of experimental data obtained on the basis of the assumption that creep and shrinkage are independent, Neville (Ref. 231) suggests that creep should be defined as the timedependent strain which takes place in excess of shrinkage. The consequence of this definition is that the total creep must be considered as consisting of two components:
 Basic creep, which is the component of creep which occurs under conditions where there is no moisture exchange with the ambient medium.
MATERIAL PROPERTIES
Strain
214
1
t0
Shrinkage from t0
Age t
Strain
(a) Shrinkage of an unloaded companion specimen
2
Creep on the basis of additive definition
1
Shrinkage of an unloaded specimen Nominal elastic strain
True elastic strain t0
Time (t  t0)
Strain
(b) Change in strain of a loaded and drying specimen
3
Creep Nominal elastic strain
t0
Time (t  t0)
Strain
(c) Creep of a loaded specimen in hygral equilibrium with the ambient medium
3
Drying creep Total creep Basic creep 2
1
Shrinkage Nominal elastic strain
t0
Time (t  t0) (d) Change in strain of a loaded and drying specimen
Figure 214:
Definition of timedependent deformations of concrete (Ref. 231).
CONCRETE
215
 Drying creep, which is the component of creep influenced by the drying process. These definitions are illustrated in Fig. 214, in which the various components of strain are shown for a concrete specimen subjected to a sustained lowlevel compressive stress (i.e. less than 40% of its shortterm compressive strength). It should be noted that the elastic strain of the concrete reduces with time because the elastic modulus increases with age. Strictly speaking, creep should be determined on the basis of the elastic strain at the time under consideration and not the time at which the load is applied. Although both methods can be used, the change in elastic strain is not accounted for under normal circumstances because the difference is usually small and because this approach 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 crystalline structure of the cement paste and loss of water to the surrounding environment by evaporation. The factors which influence creep and shrinkage can be grouped into two broad categories: Intrinsic factors, 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. 232):
• Watercement ratio: Both creep and shrinkage are increased by an increase in the watercement ratio, partially because the evaporable water is increased, and because of more and larger capillary pores.
• Aggregate: Since the seat of creep and shrinkage is to be found in the cement paste, the aggregate tends to restrain the deformation of the paste induced by creep and shrinkage. Hence, an increase in the aggregatecement ratio will lead to lower values of creep and shrinkage. Aggregates which have higher values for the modulus of elasticity can offer greater restraint to potential creep and shrinkage of the paste and therefore tend to yield concrete which creeps and shrinks less. The use of more porous aggregates leads to increased creep and shrinkage, possibly because an increase of porosity can facilitate moisture transfer within the concrete. However, it should be noted 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 its effect on the rate of strength development which, in turn, depends on the composition and fineness of grinding (Ref. 21). The magnitude of the creep of concrete made with the following cements occurs in an increasing order: highaluminium, rapidhardening, ordinary Portland, Portland blastfurnace, lowheat and Portlandpozzolana. Reference 21 suggests that the type of cement affects shrinkage mainly through variations in C 3 A content, and that fineness of grinding has a negligible effect on shrinkage, except when the cement is extremely fine or extremely coarse. It appears that concretes containing Portland blast furnace cement (PBFC) and rapidhardening Portland cement generally tend to shrink more than concrete 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 which include exposure conditions, age at loading and time under load (Ref. 21). It is important to note that the use of certain admixtures can significantly increase the creep and shrinkage of concrete.
• Member size and shape: The volume to exposed surface ratio of a member can be used as a general parameter for describing the influence of the size and shape of the member on creep and shrinkage. A larger value of this ratio represents a thicker (larger) member which has a longer diffusion path for moisture loss. Consequently, creep and shrinkage reduce with an increase in the volume to surface ratio, i.e. as the member becomes larger, with creep approaching the value of basic creep for very large members. As far as creep is concerned, it is most probably only drying creep which is affected by a variation of the size and shape of the member because basic creep remains unaffected by loss of moisture from the concrete and, as such, is independent of
216
MATERIAL PROPERTIES
the size and shape of the member. Evidently shrinkage is affected to a greater extent than creep by the size and shape of the member.
• Magnitude of the applied stress: Creep strains are approximately proportional to the magnitude of the applied sustained stress for values less than 50% of the cube strength. For most practical structures, creep may therefore be considered to be linearly related to stress within the service load range.
• Age of loading: The age of the concrete when it is loaded has an important influence on the magnitude of creep, the effect being to increase creep with earlier ages at loading. The manner in which the age at loading influences creep seems to be related to the manner in which it affects the development of strength and the degree of hydration. For these reasons, creep has been found to correlate well with maturity.
• Temperature: Creep is apparently not a monotonic function of temperature and passes a maximum in the vicinity of 50°C. Beyond this point creep reduces with temperature up to about 120°C after which it, once again, increases with temperature (Ref. 21). It also appears that the creep of specimens heated just prior to loading is more significantly influenced by temperature than that of specimens cured at the test temperature, because of improved hydration in the latter case. Tests by England and Ross (Ref. 233) indicated that the effect of temperature on creep is greater in the range of 2060°C than in the range 100140°C. Shrinkage is also increased at higher temperatures during drying.
• Relative humidity: Both creep and shrinkage are increased with a decrease of the ambient relative humidity. It appears that it is not the relative humidity which is the influencing factor with regard to creep, but rather the process of drying while under load. This is confirmed by the fact that the effect of relative humidity is much smaller if the concrete has already reached hygral equilibrium before loading and, furthermore, that creep is strongly dependent on relative humidity when the concrete is allowed to dry while under load. At 100% relative humidity the concrete absorbs water and swells slightly (as opposed to shrinking). Creep: behaviour and prediction The development of creep with time is shown in Fig. 215, which shows that most of the creep develops within a fairly short time period after the application of the load. SABS 0100 (Ref. 214) suggests that, under conditions of constant relative humidity, 40, 60 and 80% of the final creep develops 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. 234).
1500
Strain (106)
Specimen under constant load
Load removed
1000 Creep
Instantaneous recovery Creep recovery
Strain on application of load
Residual deformation
500
0 0
Figure 215:
50
100 150 Time since application of load (days)
Creep and creep recovery of concrete (Ref. 231).
200
CONCRETE
217
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 the stress. As shown in Fig. 215, the instantaneous recovery is followed by a timedependent recovery of strain, termed creep recovery, which tends to a finite value. The magnitude of the creep recovery is usually smaller than that of the creep at the time of removal of the stress. An exception occurs if the concrete is old when the stress is applied, in which case the creep recovery can have the same magnitude as the creep. Linear creep theory can be applied to most practical structures within the service load range. This theory leads to the conclusion that creep strain is linearly related to the instantaneous elastic strain under constant sustained stress and under constant environmental conditions. Using this approach, the creep strain is given by
e cr (t ) = f (t ) e c where
(211)
e cr (t) = creep strain, as a function of time t e c = instantaneous elastic strain, given by Equation (210) f(t) = creep coefficient, as a function of time t t = time, measured from the time at which the sustained stress is applied t 0
4.0
300
2.5 2.0
2.0
1.5
1.0 0.5
1 3 7 28
2.5
2.5 1.5
Age of loading (days)
3.0
3.5 3.0
600
Coastal area
150
Inland
30 Year creep coefficient for an effective section thickness (mm) of
Airconditioned area (offices)
For most practical cases, the longtime value of f(t) can vary between 1.5 and 3.5. The 30 year creep coefficient f 30 can be obtained from Fig. 216, which is taken from SABS 0100 (Ref. 214). This figure gives f 30 as a function of the ambient relative humidity, the age at loading and the effective thickness of the section which, for the purposes of Fig. 216, is defined as twice the crosssectional area of the member divided by the exposed perimeter. More comprehensive procedures for determining f(t), which explicitly include a greater number of factors that influence creep, are given in Ref. 27 and Refs. 235 through 237.
2.0 1.5
1.0
1.0
0.5
0.5
90 365
20
30
40
50
60
70
80
90
100
Ambient relative humidity (%)* * Relevant values for outdoor exposure may be determined through the Weather Bureau, Department of Environmental Affairs
Figure 216:
Effects of relative humidity, age of concrete at loading and section thickness on the creep coefficient (Ref. 214).
218
MATERIAL PROPERTIES
Equation 211 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 Ec of the concrete (see Eq. 210). It is therefore clear that f(t) is implicitly defined in terms of Ec. Because some of the procedures for estimating f(t) base the calculation of the instantaneous elastic strain on the magnitude of Ec at the time at which the concrete is loaded (Refs. 214 and 237) while others base it on the magnitude at 28 days (Refs. 235 and 236), great care should be exercised to determine exactly which value of Ec should 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 unit stress) as follows:
e cr (t ) = C (t ) f c
(212)
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 (211) and (212), and using Eq. (210). Thus
C (t ) f c = f (t ) e c =
f (t ) f c Ec
so that
C (t ) =
f (t ) Ec
(213)
Shrinkage: behaviour and prediction The development of shrinkage with time is shown in Fig. 217 where it may be seen that, as in the case of creep, the rate of shrinkage reduces with time, and that a measurable rate can still be obtained after 20 years. The rate at which shrinkage develops depends on the conditions of drying: Most of the shrinkage can take place within a period of 3 months under adverse drying conditions, while the concrete may not shrink at all if it always remains wet. It is reported in Ref. 234 that for concrete stored in air at 50% relative humidity and at 21°C (70°F) there are indications that creep and shrinkage develop at similar rates. For the purpose of estimating prestressing losses, SABS 0100 (Ref. 214) suggests that 50% and 75% of the total shrinkage takes place within the first month and within 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. Although the time period associated with the total shrinkage is usually illdefined, it appears reasonable to take it as the design life of the structure. For the types of concrete generally used for prestressed concrete, the magnitude of the shrinkage strain will normally vary between 0.0002 and 0.0006. Figure 218 gives the shrinkage strain after 6 months and after 30 years as function of the ambient relative humidity and the effective section thickness (defined as for creep, see Fig. 216), as recommended by SABS 0100. These values apply to concrete with an original water content of 190 l/m 3 . If the concrete has a water content which differs from this value, but which lies within the range 150 to 230 l/m 3 , then the shrinkage obtained from Fig. 218 must be adjusted in proportion to the water content. More comprehensive procedures for determining the shrinkage strain are presented in Ref. 27 and Refs. 235 through 237. These procedures explicitly include a greater number of factors which influence shrinkage.
CONCRETE
219
1200
Relative humidity: 50% 800
Shrinkage ´ 106
70%
400 Time reckoned since end of wet curing at the age of 28 days 0
100%
 400 10
28 Days
90
1
2
5
10
20 30
Years Time (log scale)
150
300
400
350
350 300
300
600
6 Month shrinkage ´ 106 for an effective section thickness (mm) of
Coastal area
30 Year shrinkage ´ 106 for an effective section thickness (mm) of
Airconditioned area (offices)
Development of shrinkage with time for concretes stored at different relative humidities (Ref. 21).
Inland
Figure 217:
150 200
300
100
45
87.5
250
150
40 35
75.0 250
200
125
200
100
150
200 150
75 100
100 50
30 62.5 25 50.0 20
150
50
600
175
250
100
300
37.5
15
50
25.0
10
25
12.5
5
0
0
0
–100
–100
–100
50
Shrinkage 0
0
0
–200
–200
–200
Swelling
20
30
40
50
60
70
80
90
100
Ambient relative humidity (%) Figure 218:
Drying shrinkage for normal density concrete (Ref. 214).
220
MATERIAL PROPERTIES
2.1.6 Thermal properties of concrete As is the case for most materials, concrete will expand when heated and shrink when cooled. The strain in unconfined concrete induced by a change in temperature is expressed as follows:
e cth = a c DT where
(214)
e cth = strain in concrete induced by a change in temperature a c = coefficient of thermal expansion ∆ T = change in temperature
The coefficient of thermal expansion for concrete a c is strongly dependent on the aggregate type and can vary from 7.5 to 11.5 × 10− 6 / °C for South African aggregates (Ref. 222). SABS 0100 (Ref. 214) recommends an average value of 10 × 10− 6/°C. At temperatures in excess of 300°C, the strength of concrete can be significantly reduced, while indications are that the stiffness can be reduced at temperatures as low as 100°C (Ref. 213). There is some evidence that Ec at 400°C can be as low as onethird the value at 20°C. However, it is important to note that the effect of temperature on the mechanical properties of concrete is strongly dependent 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 in a direction perpendicular to it. Poisson’s ratio is defined as the ratio of the perpendicular strain to the strain in the direction of the load. For concrete in compression, Poisson’s ratio ranges between 0.15 and 0.2 (Ref. 28). SABS 0100 (Ref. 214) and TMH7 (Ref. 27) recommend a value of 0.2 for design.
2.1.8 Fatigue In prestressed concrete members failure of the concrete in fatigue is not very common because the stress range and number of load cycles to which such members are subjected to in practice are normally less than that which causes failure. It appears that, in direct compression, concrete can sustain about ten million cycles of load which fluctuate between 0 and 50% of its static compressive strength (Ref. 28).
2.2
STEEL REINFORCEMENT
In most applications, prestressed concrete members will contain nonprestressed reinforcement in addition to the prestressed reinforcement. The nonprestressed reinforcement is normally included as shear reinforcement, as supplementary reinforcement for crack control and, particularly in the case of partially prestressed concrete, to satisfy strength requirements. Hence, the following types of reinforcement may be found in a prestressed concrete member:
• nonprestressed reinforcement, which consists of hotrolled mild steel bars, hotrolled high yield stress bars, coldworked 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 following Sections. It should be noted that the material properties to be used for design can be determined from tests on axially loaded specimens because the steel reinforcement in prestressed concrete members is usually subjected to an almost uniaxial state of stress.
STEEL REINFORCEMENT
221
2.2.1 Nonprestressed reinforcement Typical stressstrain curves for hotrolled mild steel and hotrolled high yield steel reinforcing bars tested in tension are presented in Fig. 219a. For the sake of clarity, Fig. 219b shows the initial portion of these stressstrain curves with the strain axis enlarged.
d c
Stress fs (MPa)
400
Mild steel bars
Hotrolled high yield stress bars
b
f
100 0
c
400
d
300 a bc 200
a
500
Hotrolled f high yield stress bars
Stress fs (MPa)
500
a
a
300
Mild steel bars
c
b 200 100
0
0.10
0.20 Strain es
0.30
0
0
0.010
(a) Figure 219:
0.020 Strain es
0.030
(b)
Stressstrain curve for normal reinforcing bars (Ref. 238).
The characteristics of the stressstrain behaviour are subsequently discussed with reference to the curve for mild steel bars. Initially the response is linearly elastic up to point a, beyond which a yield plateau develops. There is little or no increase in stress for a corresponding increase in strain on the yield plateau, which is bounded by the onset of a region of strain hardening at point c. The strain hardening region is characterized by an increase in stress with an increase in strain until a maximum value of stress is reached at point d. Any subsequent increase in strain beyond this point is 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 varies between 200 and 210 GPa. The stress at which yielding occurs is referred to as the yield stress and is an important property of steel reinforcement. A sudden reduction of stress, from point a to point b, often occurs immediately after first yielding. In such a case point a is referred to as the upper yield point and point b as the lower yield point. The upper yield point is strongly dependent on the speed 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 steel extends to a strain approximately equal to 10 times the strain at first yield. The maximum stress sustained by the specimen at point d is referred to as the ultimate stress. A comparison of the stressstrain curve for the hotrolled high yield bars with the curve for the mild steel bars (see Fig. 219) reveals that, apart from the obvious difference of having higher yield and ultimate strengths, the behaviour of the high yield bars is significantly less ductile than that of the mild steel bars. This feature is characterized by the smaller extent of the yield plateau as well as the smaller elongation at fracture. Typical stressstrain curves for coldworked and hotrolled high yield reinforcing bars are shown in Fig. 220. It should be noted that these curves were taken from Ref. 239 and apply to Dutch steel with a specified yield stress of 400 MPa. This figure clearly shows that the stressstrain behaviour of coldworked reinforcement does not exhibit a definite yield point as in the case of hotrolled bars, but rather shows a gradual transition from linear elastic to nonlinear behaviour. Because of
222
MATERIAL PROPERTIES
600
FeB 400HWL (hotrolled)
FeB 400HK (coldworked)
Stress fs (MPa)
500 400 300 200 100 0
0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Strain es Figure 220:
Stressstrain curve for coldworked and hotrolled reinforcing bars (Ref. 239).
this feature, the yield stress for coldworked reinforcement must be defined. This is generally done in one of two ways, as shown in Fig. 221:
• 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 strain εoffset . This method is referred to as the offset strain method and the yield stress so determined is known as the proof stress.
Stress fs fy1 fy2
Es Es 1
1
eoffset
Figure 221:
ey1 Strain es
Definitions of yieldstrength for gradual yielding steel.
Another important feature of the stressstrain response of coldworked reinforcement is that it is significantly less ductile than that of hotrolled reinforcement, as revealed by the reduced elongation at fracture (see Fig. 220). In South Africa, the nominal sizes in which reinforcing bars can be supplied are listed in Table 25. Other geometric properties as well as the mass of the bars are also presented herein. The reinforcement must also conform to the requirements of SABS 920 (Ref. 240), of which the required tensile properties are summarized in Table 26. The strength of the reinforcing bars is specified in
STEEL REINFORCEMENT
Table 25 :
223
Nominal geometric properties of normal reinforcing bars (Refs. 240 and 241).
Diameter (mm)
Area (mm 2 )
Perimeter (mm)
6
28.27
18.85
0.222
8
50.27
25.13
0.395
10
78.54
31.42
0.617
Mass (kg/m)
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
terms of the characteristic strength, which is defined as the value of the yield stress or the proof stress, as appropriate, below which not more than 5% of the measured results may be expected to fall (Ref. 240). The reasons for following this approach corresponds to those given for concrete strength (see Section 2.1.1). It should be noted that SABS 920 (Ref. 240) does not explicitly specify the offset strain which defines the proof stress. However, an offset strain of 0.2% seems appropriate.
Table 26:
Tensile properties of reinforcement to SABS 920 (Ref. 240).
Type of steel
Identifying symbol (Refs. 242 and 243)
Characteristic strength f y (MPa) Min.
Max.
Hotrolled mild steel
R
250
400
Hotrolled highyield steel and coldworked highyield steel
Y
450
–
Minimum ultimate tensile strength (MPa)
Minimum elongation at fracture* (%)
15% greater than the measured yield stress or 0.2% proof stress, as appropriate
22 14
* Measured on a gauge length of 5.65 √ So , where So is the original equivalent crosssectional area.
The stressstrain curve recommended by SABS 0100 (Ref. 214) for use in design is presented in Fig. 222 and the following aspects should be noted:
• The recommended design value for the modulus of elasticity is 200 GPa. • The actual stressstrain behaviour is approximated by a bilinear relationship which ignores strain hardening. In most cases this approximation will lead to a conservative result in reinforced concrete and partially prestressed concrete members. However, in cases where large strains can occur in the steel, ignoring the effect of strain hardening may change the intended failure mode to one which is undesirable. Consider, for example, the case of a beam in which large strains develop in the flexural reinforcement at failure: In such a case the effects of strain hardening
224
MATERIAL PROPERTIES
Actual
Stress fs fy gm
Tension Es = 200 GPa 1
εyc =
fyc Es
Compression
Figure 222:
1
εy =
Es fyc =
fy gm + fy / 2000
fy gm Es
Strain es
fy = Characteristic yield strength (in MPa)
Design shortterm stressstrain relationship for nonprestressed reinforcement (Ref. 214).
can increase the flexural capacity to such an extent that the accompanying increase in shear force may lead to an undesirable brittle shear failure in the actual structure, rather than the intended ductile flexural failure.
• A maximum strain at fracture is not given. • Although it is commonly accepted that the stressstrain response of steel in compression is similar to that in tension, the design curve indicates a smaller yield strength in compression than in tension. Evidently the reason for this is that the restraint offered by the concrete to buckling of the reinforcing bar is significantly reduced under conditions of yielding in compression.
• The design curve includes a partial safety factor for material strength g m . This aspect is discussed in Section 4.4.1.
Steel reinforcement is normally detailed in compliance with the recommendations of SABS 82 (Ref. 242) and SABS 0144 (Ref. 243). Welded steel fabric, which consists of a grid of colddrawn steel wire placed at right angles and welded at the intersections, can often be used advantageously because of improved crack control and a reduction in fixing time. Wires can either be smooth or deformed, and standard dimensions are listed in Table 27 (Ref. 244). If required for a particular design, fabric can be supplied with a 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. 244) 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 elongation at fracture must be at least 12 % measured on a gauge length of 5.65 So , where S o is the initial crosssectional area. It is interesting to note that the test specimen must contain at least one welded intersection within its length. The reason for this most probably follows from the fact that the wire fabric is not stressrelieved, which can lead to the occurrence of a failure near welded intersections at relatively small strains (Refs. 213 and 239). This fact must be kept in mind if significant ductility is a primary design requirement.
STEEL REINFORCEMENT
225
Standard diameters and pitch of welded steel fabric commonly available in South Africa are given in Table 27. Fabric with a pitch other than that specified in the Table is also available on request.
Table 27: Fabric reference number
Standard dimensions of welded steel fabric (Refs. 243 and 244). Nominal pitch of wires (mm)
Nominal diameter of wires (mm)
Nominal cross sectional area of wires (mm 2 /m of width)
Nominal mass per unit area (kg/m 2 )*
Longitudinal
Cross
Longitudinal
Cross
Longitudinal
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/mm 2 per metre length.
2.2.2 Prestressed reinforcement The material almost universally used for prestressing is high tensile strength steel, and an obvious approach to producing this material is by alloying. The tensile strength of the steel can be further increased by colddrawing, usually followed by a stressrelieving process. Prestressing tendons take the form of either wires, strand or bars:
• Wires are typically manufactured by colddrawing hightensile steel bars through successive dies to obtain the required strength characteristics. Subsequent mechanical processes can be used to indent or crimp the wire. Finally, the wire is stressrelieved by a suitable heat treatment, carried out either in the absence or presence of an applied tension.
226
MATERIAL PROPERTIES
• In the manufacturing process of 7wire strand, six similar peripheral wires are spun over a central wire which has a slightly larger diameter. The complete process is summarized in Fig. 223 where it may be seen that there are two possibilities with regard to stressrelieving, as in the case of the manufacture of wire. The impact of the stressrelieving process on the properties of the steel are discussed later in this Section. In South Africa three types of 7wire strand are commonly produced: standard, super and drawn. The tensile strength of super strand is higher than that of standard strand, while the mechanical properties of drawn strand are enhanced by an additional drawing process.
• Prestressing bars are generally manufactured from alloy steel heat treated to obtain the required properties, and can be supplied either as smooth bars or as bars with a ribbed surface which serves as a continuous screw thread. Typical stressstrain curves for prestressing wire, strand and bar are given in Fig. 224 together with a typical curve for a hotrolled high yield reinforcing bar. The following observations can be made from this figure:
• Prestressing steel has a much higher tensile strength than the reinforcing bar. This is accompanied by a significant reduction in the elongation at fracture.
• The stressstrain response of prestressing steel does not show a definite yield point, so that the yield strength must be defined in terms of either a proof stress or a stress corresponding to a total strain under load, as for coldworked reinforcing bars (see Section 2.2.1). However, it should be noted that some high strength prestressing bars may have a short, but detectable, yield plateau.
• The stressstrain curve for prestressing steel can conveniently be divided into three portions: an initial linear elastic portion, followed by a region containing a fairly sharp nonlinear transition to the final almost linear strainhardening portion, which is bounded by fracture. Significant residual stresses arise in wire and strand because of the various mechanical processes involved in their manufacture. The presence of these residual stresses leads to a very rounded stressstrain curve as shown in Fig. 225. Stressrelieving has the effect of removing the residual stresses and also of increasing the proportional limit of the steel. By carrying out stressrelieving under tension (strain tempering), the proportional limit is increased even more. It is very important to note that strain tempering has the additional benefit of substantially reducing the relaxation loss even more than with ordinary stressrelieving. Although the modulus of elasticity of steel is independent of strength, the values for prestressing steels 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 spun wire.
• 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 in this Section, care should be taken because their magnitude can be influenced by the manufacturing process, 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. 245) while prestressing bars must conform to those of BS 4486 (Ref. 246). The dimensions and required minimum tensile properties of the standard prestressing wires, strand and bars, as given by these specifications, are listed in Tables 28 to 211. It should be noted that the characteristic loads listed in these tables are defined as the value of the appropriate load below which not more than 5% of the measured results may be expected to fall.
STEEL REINFORCEMENT
227
Base material: Round, plain, hotrolled, nonalloyed, high carbon steel rod
Patenting: Heat to about 800°C (1470 °F) then cool slowly to make homogeneous
Cold drawing: Pull through successively smaller dies to increase strength
Stranding: Spin 6 helical wires around a straight central wire
Stress relieving: Heat to about 350°C and cool slowly STRESSRELIEVED STRAND Figure 223:
Strain tempering: Heat to about 350°C while strand is under tension LOW RELAXATION STRAND
Production of sevenwire strand (Ref. 213).
228
MATERIAL PROPERTIES
2000 Prestressing Strand (1860 MPa) Stress relieved wire (1620 MPa)
Stress fp (MPa)
1500
High strength prestressing bars (1103 MPa) 1000 Hotrolled high yield reinforcing bars (450 MPa)
500 Assuming same elastic modulus 0 0
Figure 224:
0.05
0.10 Strain ep
0.15
Stressstrain curves for prestressed reinforcement (Ref. 28).
Stress fp
Strain tempered (Low relaxation)
Stress relieved Untreated
Strain ep
Figure 225:
Stressrelieving and strain tempering of prestressing wire (Ref. 213).
STEEL REINFORCEMENT
Table 28:
229
Dimensions and properties of colddrawn wire to BS 5896 (Ref. 245).
Nominal diameter
Nominal tensile strength
Nominal 0.1% proof stress
Nominal cross section
Nominal mass
Specified characteristic breaking load
(mm)
(MPa)
(MPa)
(mm 2 )
(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
47.3
39.3
40.2
6
1770
1470
50.1
41.6
42.6
5
1670
1390
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
28.3
19.6
222
154
Specified Load at charac1% teristic elongation 0.1% proof load
Note: Minimum elongation at maximum load must be 3.5% measured on a gauge length of 200 mm
Table 29:
Dimensions and properties of colddrawn wire in mill coil to BS 5896 (Ref. 245).
Nominal diameter
Nominal tensile strength
Nominal crosssection
Nominal mass
Specified characteristic breaking load
Specified characteristic load at 1% elongation
(mm)
(MPa)
(mm 2 )
(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
12.5
10.0
3
1860
13.1
10.5
7.07
55.5
230
Table 210: Type of strand
7wire Standard
7wire Super
7wire Drawn
MATERIAL PROPERTIES
Dimensions and properties of sevenwire strand to BS 5896 (Ref. 245). Nominal diameter
Nominal tensile strength
Nominal steel area
Nominal mass
Specified characteristic breaking load
Specified Load at charac1% teristic elongation 0.1% proof load
(mm)
(MPa)
(mm 2 )
(g/m)
(kN)
(kN)
(kN)
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
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
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 211:
Type of bar
Hot rolled or hot rolled and processed
Dimensions and properties of hotrolled and hotrolled and processed high tensile alloy steel bars to BS 4486 (Ref. 246).
Nominal tensile strength
Nominal 0.1% proof stress
Nominal crosssectional area
Nominal mass
(mm)
(MPa)
(MPa)
(mm 2 )
(kg/m)
(kN)
(kN)
(%)
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
Nominal size
Specified properties Characteristic breaking load
Min. Characelongateristic tion at 0.1% fracture* proof load
* Measured on a gauge length of 5.65 √ So , where So is the original crosssectional area.
STEEL REINFORCEMENT
231
Wire and strand which satisfy the requirements of ASTM Specification A421 (Ref. 247) and ASTM Specification A416 (Ref. 248), respectively, are also produced in South Africa. The design stressstrain diagram for prestressing steel acting in tension, as recommended by SABS 0100 (Ref. 214), approximates the actual behaviour by the trilinear curve shown in Fig. 226. It should further be noted that a maximum strain at fracture is not given and that the design curve includes a partial safety factor for material strength g m . This partial safety factor is discussed in Section 4.4.1.
Stress fp fpu / gm 0.8 fpu / gm fpu = Characteristic strength Ep 1
Ep 1
0.005 Figure 226:
Strain ep
Design stressstrain relationship for prestressed reinforcement acting in tension (Ref. 214).
SABS 0100 suggests the following design values for the modulus of elasticity of prestressing tendons: Ep =
205 GPa for high tensile steel wire (wire to Section 2 of BS 5896: 1980)
=
195 GPa for 7wire 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 high tensile alloy bars most probably only applies to asrolled and stretched bars conforming to BS 4486 (Ref. 246). In the case of asrolled and asrolled stretched and tempered bars, the value of 205 GPa 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 of the modulus of elasticity can be significantly influenced by the manufacturing process.
2.2.3 Relaxation of prestressing steel The timedependent loss of tensioning force required for maintaining a constant strain in a highly stressed steel tendon is defined as relaxation. Creep, which is defined as the timedependent change in 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 tendon continually changes with time because of shrinkage and creep of the concrete, it is generally acknowledged that these conditions approach those to be found in a relaxation test rather than in a creep test. As discussed in Section 2.2.2, the relaxation properties of prestressing steel is significantly influenced by the particular stressrelieving process used. The ordinary stressrelieving process,
232
MATERIAL PROPERTIES
which involves a heat treatment only, yields normal relaxation steel while straintempering, which involves heat treatment under tension, yields low relaxation steel. Relaxation appears to be primarily influenced by the ratio of initial stress to yield stress, the type of steel, temperature and time. Magura, Sozen and Siess (Ref. 249) proposed the following expression for predicting the stress in stressrelieved wire and strand at any time: fs(t) fsi
= 1 −
log t 10
fsi f − 0.55 y
(215)
fs(t) = steel stress at time t
where
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 stressrelieved wire. It has been suggested that this expression can also be applied to lowrelaxation strand and prestressing bars if the denominator 10 under the log t term is replaced by 45 (see Refs. 28, 213 and 250). Typical relaxation curves are shown in Fig. 227 for normal relaxation and low relaxation South African prestressing strand.
% Stress relaxation (log scale)
100 Test temperature: 20°C Initial load: 70% of nominal breaking load 10 Normal relaxation
Low relaxation
1
6 months
1 year
10 years
50 years
0.1 1
10
100
1000
104
105
106
107
Hours (log scale)
Figure 227:
Relaxation of prestressing strand (courtesy Haggie Rand Ltd.).
Both Fig. 227 and Eq. 215 clearly show that a large part of the relaxation loss occurs within a relatively short time period after application of the load and that relaxation proceeds with time, but at a decreasing rate. It is also evident that the relaxation loss of low relaxation steel is significantly smaller than that of normal relaxation steel, it being generally accepted that the relaxation loss of low 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. 228 for low relaxation strand tested at various temperatures. The figure clearly demonstrates that the relaxation loss is increased if the initial stress is increased. This trend is confirmed by Eq. 215, which predicts zero relaxation loss for values of the initial stress smaller than or equal to 55 percent of the yield
STEEL REINFORCEMENT
233
stress. It is generally accepted that relaxation losses are insignificant for initial stresses smaller than 50 percent of the yield stress.
% Stress relaxation (log scale)
100 1000 Hour tests Low relaxation strand 10
Test temperature: 80°C 60°C 40°C 20°C
1
0.1 50
60
70
80
90
Initial Load (as % of nominal breaking load) Figure 228:
Effect of initial stress on relaxation of low relaxation prestressing strand at various temperatures (courtesy Haggie Rand Ltd.).
The magnitude of relaxation is strongly influenced by the temperature of the steel, the effect being that it is increased by an increase in temperature. This trend is demonstrated in Fig. 229 which gives 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 will occur in cases where the prestressing tendons are subjected to temperatures significantly higher than 20°C for extended periods of time. 100
% Stress relaxation (log scale)
1000 Hour tests Initial load: 70% of nominal breaking load Low relaxation Normal relaxation
Test temperature: 100°C 80°C
10
60°C 40°C 20°C
100°C 80°C 60°C 40°C 20°C
1
0.1 1
10
100
1000
10 000
Hours (log scale)
Figure 229:
Effect of temperature on the relaxation of strand (courtesy Haggie Rand Ltd.).
234
MATERIAL PROPERTIES
In practice, the relaxation loss is usually experimentally determined after 1000 hours at 20°C and multipliers are subsequently used to estimate the longterm values required for design. The maximum relaxation loss after 1000 hours for various prestressing tendons as specified by BS 5896 and BS 4486 (Refs. 245 and 246), are listed in Table 212.
Table 212:
Initial load as % of breaking load
Maximum specified relaxation at 1000 hours, BS 5896 (Ref. 245) and BS 4486 (Ref. 246). Maximum relaxation after 1000 hours Colddrawn wire
7Wire strand
Colddrawn High tensile wire in alloy mill coil steel bars
Normal relaxation
Low relaxation
Normal relaxation
Low relaxation
(%)
(%)
(%)
(%)
(%)
(%)
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
It is important to note that a prestressing tendon in a prestressed concrete member will not be subjected to a constant strain because the member, and hence the tendon, will shorten as a result of the effects of creep and shrinkage of the concrete. This has the effect of reducing the initial stress level of the tendon, so that the relaxation loss in an actual member is less than the loss which would be obtained in a relaxation test where a constant strain is maintained in the steel for the duration of the test. This effect must be accounted for in design and must therefore be reflected in the magnitude of the multipliers used for estimating longterm relaxation losses from experimental data obtained from relaxation tests at 1000 hours. SABS 0100 (Ref. 214) recommends that the relaxation loss to be allowed for in design should be taken 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 normal relaxation strand or wire may be assumed to vary linearly from 10% for an initial stress of 80% of the characteristic strength of the tendon to 3% for an initial stress of 50% of the characteristic strength. If the creep plus shrinkage strain of the concrete is greater than 500 × 10 6 then the loss for an initial stress of 80% of the characteristic strength should be taken as 8.5%. The relaxation loss for low relaxation strand may be taken as half the above values. Although TMH7 (Ref. 27) also requires that the relaxation loss to be allowed for in design should be estimated from the loss at 1000 hours for an initial force taken equal to the tendon force at transfer, no guidance is given on how this is to be done. It is of some interest to note that the multiplying factors specified by BS 8110 (Ref. 251) for estimating the longterm relaxation loss from the 1000 hour test value account for the reinforcement type (wire and strand, or bar), the relaxation properties of the steel (normal or low relaxation) and the prestressing procedure (pre or posttensioning). This code explicitly states that the recommended multiplying factors account for the effects of creep and shrinkage of the concrete and, in the case of pretensioning, the effects of elastic shortening of the concrete at transfer. It also carefully defines the initial force for posttensioning as the prestressing force immediately after transfer and for pretensioning as the force immediately after tensioning.
STEEL REINFORCEMENT
235
2.2.4 Fatigue characteristics of reinforcement When steel is subjected to a fluctuating stress its mechanical properties will deteriorate, depending on the minimum and maximum values of the fluctuating stress, by a process known as fatigue. The resistance of reinforcement to fatigue is often defined in terms of a SN curve, in which the stress range S is plotted as a function of the corresponding number of load cycles N required to cause failure. Figure 230 shows several experimentally obtained SN curves for deformed reinforcing bars (Ref. 213). Inspection of this figure will reveal that the numbers of cycles which the reinforcing bars can sustain without failure increases as the stress range is decreased until a limiting value of the stress range is reached, below which it can be assumed that the bars can sustain an indefinite number of cycles of load. This region is characterized by the almost horizontal portion of the SN curve which, for the reinforcing bars considered here, commences after about one to two million cycles. The stress range corresponding to the horizontal portion of the SN curve is known as the fatigue limit or the endurance limit.
Stress range S (MPa)
500 400 300 200 100 0 104
Figure 230:
105
106 Cycles to failure N
107
108
SN curves for deformed reinforcing bars (Ref. 213).
TMH7 (Ref. 27) recommends that the stress range should be limited to 250 MPa for mild steel reinforcing bars, and to 300 MPa for highyield strength bars. These values apply to a maximum of 2 ´ 10 5 cycles of load, while the stress range should be limited to 60% of these values for 2 ´ 10 6 cycles. The CEBFIP code (Ref. 236) limits the characteristic fatigue strength to 250 MPa for smooth bars, and to 150 MPa for deformed bars. In this code, the characteristic fatigue strength is defined as the stress range which nine reinforcing bars out of ten can resist for 2 ´ 10 6 cycles if the maximum stress is 70% of the yield strength. The various types of prestressing steel do not appear to have a fatigue limit (Ref. 28). A single SN 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 minimum cyclic 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 ´ 10 6 cycles, while a maximum of 10 ´ 10 6 can be considered in exceptional cases. A typical modified Goodman diagram corresponding to 2 ´ 10 6 cycles is given in Fig. 231 for prestressing wires and strand (Ref. 28). This figure shows that for the minimum stress normally encountered in prestressed members (50% to 60% of the ultimate tensile strength) a stress range of approximately 13% of the ultimate tensile strength can be resisted for 2 ´ 10 6 cycles. This stress range is substantially larger than that encountered in uncracked fully prestressed members, with the result that fatigue is not normally critical for design in this case. It should, however, be noted that the stress range in cracked partially prestressed
236
MATERIAL PROPERTIES
2 ´ 106 cycles
Maximum stress (fps)max /Strength fpu
1.0
Maximum stress limit 0.5 Stress range
Usual stress range for prestressed concrete
Minimum stress limit 0
Figure 231:
0
0.5 Minimum stress (fps)min / Strength fpu
1.0
Typical Goodman diagram for prestressing wires and strand (Ref. 28).
members can be significantly larger than in fully prestressed members so that fatigue can become an important design consideration in such cases. TMH7 (Ref. 27) recommends the following maximum values for the stress range in prestressing tendons 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
• Highstrength bars
200 MPa
These values apply to a maximum of 2 × 10 5 cycles of load, while 60% of these values should be used for 2 × 10 6 cycles. The FIP Recommendations (Ref. 252) suggest that the characteristic fatigue strength of wires and strand may be taken as 200 MPa, and that it may be taken as 80 MPa for highstrength bars. This document defines the characteristic fatigue strength of prestressing steel as the stress range which can be resisted for 2 × 10 6 cycles, with a probability of failure of 0.01, if the maximum stress is 85% of the yield strength. It is important to note that although there is ample experimental evidence of the fatigue life of tendons in beams being shorter than that of similar tendons tested in air, the recommendations related to fatigue strength made by the majority of the codes of practice are based on data obtained from tendons tested in air. A designer is therefore well advised to exercise caution and to take a conservative approach when considering fatigue.
REFERENCES
237
2.2.5 Thermal properties of reinforcement The strain induced by a change in temperature in unconfined steel reinforcement can be expressed as follows:
e sth = a s DT where
(216)
e sth = strain in steel induced by a change in temperature a s = coefficient of thermal expansion DT = change in temperature
Evidently, the actual value of the coefficient of expansion for steel is about 11.5 ´ 10 6 /°C (Ref. 213). However, a value of 10 ´ 10 6 /°C is usually taken for design, which is equal to the value taken for concrete (see Section 2.1.6). Although the mechanical properties of the reinforcement are not significantly affected by normal variations of the ambient temperature, they can be significantly affected by extreme temperature conditions. If the temperature is increased beyond a value of approximately 200°C both the stiffness and strength will be substantially reduced. More specifically, the tensile strength of wire or strand at 400°C is about 50% of its value at room temperature (Ref. 213). Reducing the temperature produces opposite effects, with stiffness and strength being increased. However, these improvements are accompanied by a reduction in ductility. If the temperature is reduced from 20°C to 200°C the yield and tensile strengths will be increased by about 20% (Ref. 28).
2.3
REFERENCES
21
Portland Cement Institute, Fulton’s Concrete Technology, 7th ed., Edited by B. J. Addis, PCI, Midrand, South Africa, 1994.
22
South African Bureau of Standards, “Compressive Strength of Concrete (Including Making and Curing of the Test Cubes),” SABS Method 863, SABS, Pretoria, 1976.
23
British Standards Institution, “Method for Determination of Compressive Strength of Concrete Cubes,” BS 1881: Part 116: 1983, BSI, London, 1983.
24
American Society for Testing Materials, “Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens,” ASTM C 3986, ASTM, Philadelphia, 1986.
25
Mosley, W. H., and Bungey, J. H., Reinforced Concrete Design, 4th ed., MacMillan Education Ltd., London, 1990.
26
Portland Cement Institute, Cement and Concrete, 9th ed., PCI, Midrand, South Africa, 1992.
27
Committee of State Road Authorities, “Code of Practice for the Design of Highway Bridges and Culverts in South Africa,” TMH7 Part 3, CSRA, Pretoria, 1989.
28
Naaman, A. E., Prestressed Concrete Analysis and Design: Fundamentals, McGrawHill Book Company, New York, 1982.
29
CEBFIP, “Model Code for Concrete Structures,” First Draft, Bulletin d’Information No.195, Comité EuroInternational du Béton  Fédération Internationale de la Précontrainte, Paris, March, 1990.
210 Held, M., “Research Results Concerning the Properties of High Strength Concrete,” Darmstadt Concrete, Vol. 5, Technishe Hochschule Darmstadt, Alexanderstrasse 5 Darmstadt, Germany, 1990. 211 Smeplass, L., “High Strength Concrete,” SP4  Material Design, Report 4.4, Mechanical properties  normal density concrete, STF65 F89020  FCBSINTEF 7034, Trondheim, Norway.
238
MATERIAL PROPERTIES
212 FIPCEB, High Strength Concrete  State of the Art Report, Fédération Internationale de la Précontrainte  Comité EuroInternational du Béton Bulletin d’Information No. 197, Chameleon Press, London, 1990. 213 Collins, M. P., and Mitchell, D., Prestressed Concrete Structures, PrenticeHall, Englewood Cliffs, New Jersey, 1991. 214 South African Bureau of Standards, “The Structural Use of Concrete,” SABS 0100: 1992, Parts 1 and 2, SABS, Pretoria, 1992. 215 Neville, A. M., “Some Aspects of the Strength of Concrete,” Civil Engineering and Public Works Review, Vol. 54, Part 2, No. 640, Nov. 1959, pp. 13081310. 216 Carrasquillo, R. L., Slate F. O., and Nilson A. H., “Micro Cracking and Behaviour of High Strength Concrete Subject to Shortterm Loads,” ACI Journal, Vol. 78, No. 3, MayJune 1981, pp. 179186. 217 Nilson, A. H., “High Strength Concrete  An Overview of Cornell Research,” Proceedings of the Symposium “Utilization of High Strength Concrete”, Stavanger, Norway, June 1987, Tapir, Trondheim, 1987, pp. 2738. 218 Rüsch, H., “Researches Toward a General Flexural Theory for Structural Concrete,” ACI Journal, Vol. 57, No. 1, July 1960, pp. 128. 219 British Standards Institution, “Methods for Determination of Static Modulus of Elasticity in Compression,” BS 1881: Part 121: 1983, BSI, London, 1983. 220 British Standards Institution, “Recommendations for the Measurement of Dynamic Modulus of Elasticity,” BS 1881: Part 209: 1990, BSI, London, 1990. 221 Alexander, M. G., “Prediction of Elastic Modulus for Design of Concrete Structures,” The Civil Engineer in South Africa, Vol. 27, No. 6, June 1985, pp. 313324. 222 Alexander, M. G., and Davis, D. E., “The Influence of Aggregates on the Compressive Strength and Elastic Modulus of Concrete,” The Civil Engineer in South Africa, Vol. 34, No. 5, May 1992, pp. 161170. 223 Alexander, M. G., and Davis, D. E., Properties of Aggregates in Concrete, Part 1, Hippo Quarries Technical Publication, 1989. 224 Alexander, M. G., and Davis, D. E., Properties of Aggregates in Concrete, Part 2, Hippo Quarries Technical Publication, 1992. 225 Gopalaratnam, V. S., and Shah, S. P., “Softening Response of Plain Concrete in Direct Tension,” ACI Journal, Vol. 82, No. 3, MayJune, 1985, pp. 310323. 226 British Standards Institution, “Method for Determination of Tensile Splitting Strength,” BS 1881: Part 117: 1983, BSI, London, 1983. 227 South African Bureau of Standards, “Flexural Strength of Concrete (Including Making and Curing of the Test Specimens),” SABS Method 864, SABS, Pretoria, 1980. 228 British Standards Institution, “Method for Determination of Flexural Strength,” BS 1881: Part 118: 1983, BSI, London, 1983. 229 Illston, J. M., Construction Materials: Their nature and behaviour, Second ed., Edited by J. M. Illston, E & FN Spon, London, 1994. 230 ACI Committee 318, “Building Code Requirements for Reinforced Concrete (ACI 31889) and Commentary  ACI 318 R89, ” American Concrete Institute, Detroit, 1989. 231 Neville, A. M., Dilger, W. H., and Brooks, J. J., Creep of Plain and Structural Concrete, Construction Press, London, 1983. 232 Marshall, V., and Gamble, W. L., “TimeDependent Deformations in Segmental Prestressed Concrete Bridges,” Structural Research Series No. 495, Civil Engineering Studies, University of Illinois, Urbana, October 1981.
REFERENCES
239
233 England, G. L., and Ross, A. D., “Reinforced Concrete under Thermal Gradients,” Magazine of Concrete Research, Vol. 14, No. 40, March 1962, pp. 212. 234 Troxell G. D., Raphael J. M., and Davis R. E., “Long Time Creep and Shrinkage Tests of Plain and Reinforced Concrete,” Proc. ASTM, Vol. 58, 1958, pp. 11011120. 235 CEBFIP, “International Recommendations for the Design and Construction of Concrete Structures  Principles and Recommendations,” Comité European du Béton  Fédération Internationale de la Précontrainte, FIP Sixth Congress, Prague, June 1970; published by Cement and Concrete Association, London, 1970. 236 CEBFIP, “Model Code for Concrete Structures,” Comité EuroInternational du Béton Fédération Internationale de la Précontrainte, Paris, 1978. 237 ACI Committee 209, “Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures,” ACI Report 209R82 (Reapproved 1986), ACI, in ACI Manual of Concrete Practice, Part 1. 238 Case, J., and Chilver A. H., Strength of Materials and Structures, 2nd ed., Edward Arnold Publishers Ltd., 1971. 239 Bruggeling, A. S. G., Structural Concrete: Theory and its Application, A. A. Balkema, Rotterdam, 1991. 240 South African Bureau of Standards, “Steel Bars for Concrete Reinforcement,” SABS 920: 1985, SABS, Pretoria, 1985. 241 ISCOR Ltd., Data Sheet: Availability and Properties, Reinforcing Steel Bars. ISCOR Ltd., Pretoria, 1993. 242 South African Bureau of Standards, “Bending Dimensions of Bars for Concrete Reinforcement,” SABS 82:1976, SABS, Pretoria, 1976. 243 South African Bureau of Standards, “Detailing of Steel Reinforcement for Concrete,” SABS 0144: 1978, SABS, Pretoria, 1978. 244 South African Bureau of Standards, “Welded Steel Fabric for Reinforcement of Concrete,” SABS 1024: 1991, SABS, Pretoria, 1991. 245 British Standards Institution, “Specification for High Tensile Steel Wire and Strand for the Prestressing of Concrete,” BS 5896:1980, BSI, London, 1980. 246 British Standards Institution, “Specification for Hot Rolled and Hot Rolled and Processed High Tensile Alloy Steel Bars for the Prestressing of Concrete,” BS 4486:1980, BSI, London, 1980. 247 American Society for Testing and Materials, “Standard Specification for Uncoated StressRelieved Steel Wire for Prestressed Concrete,” ASTM A42180, ASTM, Philadelphia, 1980. 248 American Society for Testing and Materials, “Specification for Uncoated 7wire StressRelieved Steel Strand for Prestressed Concrete,” ASTM A41685, ASTM, Philadelphia, 1985. 249 Magura D. D., Sozen M. A., and Siess C. P., “A Study of Stress Relaxation in Prestressing Reinforcement,” PCI Journal, Vol. 9, No. 2, April 1964, pp. 1357. 250 OHBDC, “Ontario Highway Bridge Design Code,” 2nd ed., Ontario Ministry of Transportation and Communications, Toronto, 1983. 251 British Standards Institution, “Structural Use of Concrete, Part 1, Code of Practice for Design and Construction,” BS 8110: Part 1: 1985, BSI, London, 1985. 252 FIP Commission on Practical Design, FIP RecommendationsPractical Design of Reinforced and Prestressed Concrete Structures, Thomas Telford Ltd., London, 1984.
INTRODUCTION
31
3 PRESTRESSING SYSTEMS AND PROCEDURES 3.1
INTRODUCTION
A designer must be familiar with the technology and techniques associated with prestressed concrete not only to ensure that the design requirements are properly satisfied, but also to ensure that members are 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 of competitive tendering, it is common practice in South Africa to dimension members in such a way that several prestressing systems can be accommodated. Almost all the commonly used prestressing systems involve the tensioning of highstrength steel tendons and can be classified either as being pretensioning or posttensioning systems. In pretensioning systems the tendons are tensioned before the concrete is placed, while in posttensioning systems the tendons are tensioned after the concrete has been placed and has developed sufficient strength to sustain the induced loads. By the nature of the procedure, pretensioned elements are always precast and the method usually requires a substantial capital investment in prestressing equipment and stressing beds. Posttensioned elements can either be precast or cast in situ, and the prestressing 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 and transported to site, such as shells, building slabs, large building frames, large bridge decks and continuous bridge decks, can only be prestressed by posttensioning. The purpose of this Chapter is to give a brief description of the types of prestressing systems most commonly used, including some detail regarding procedure. Because of the large number of systems available, it is not feasible to present specific details of each system here. Complete details may be obtained from suppliers. The descriptions are limited to linear prestressing systems commonly used because linear systems for special applications, circular prestressing systems, electrical prestressing and chemical prestressing fall beyond the scope of the work covered herein. References 31 to 33 may 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. 31a). The resulting elongation of the tendons is maintained at a constant level while the concrete hardens. After the concrete has developed sufficient strength the tendons are released and, because they are now bonded to the concrete, their shortening is resisted by the concrete. In this way the concrete is prestressed by the action of bond when the tendons are released (see Fig. 31b). It is important to ensure that the elongation of the tendons is maintained at a constant level while the concrete is allowed to harden, and this can be achieved by each of the following two methods (Refs. 31, 32 and 34):
• Pretensioning with individual moulds: According to this method the tendons are anchored directly to the individual steel moulds in which the concrete is cast. In this case, the moulds must be designed and constructed to withstand the additional forces induced by the tendons.
• Pretensioning on stressing beds: When pretensioning on a stressing bed, the tendons are tensioned between and subsequently anchored to the rigid vertical steel anchor columns, called uprights, placed at each end of the bed (see Fig. 31a). In this manner the tension is maintained in the
32
PRESTRESSING SYSTEMS AND PROCEDURES
tendons while the concrete is placed and cured. The stressing bed also serves as a casting and curing bed. With the exception of railway sleeper production, pretensioning with individual moulds is not commonly used (Refs. 31, 32 and 34). An apparent advantage of this method is that, in the case of small products, the individual moulds can be can be moved through the plant on a mass production line instead of having to move the materials and the process to the moulds, as is the case when stressing beds are used (Refs. 31 and 32). Because of its limited use, this method is not discussed here in any further detail. Pretensioning on stressing beds is by far the most common method used today, and a typical arrangement is shown in Fig. 31a. This method, often referred to as the longline or Hoyer method, lends itself to efficient mass production because a number of similar elements can be manufactured in a single tensioning operation if the bed is made long enough. The length of stressing beds varies between 25 m and 200 m, and long beds can be provided with removable intermediate uprights (see Fig. 31a) so that shorter tendons can also be tensioned (Refs. 32 and 35).
Stressing jack
Upright Original length = L
Removable intermediate upright
Upright Formwork
(a) Tendons tensioned between uprights
L–∆
Stressing bed
(b) Tendons detensioned (elastic shortening = D) Figure 31:
Pretensioning on a stressing bed (adapted from Ref. 35).
Tendons are tensioned by means of hydraulic jacks, and can either be stressed individually or simultaneously from one end of the stressing bed. Special jacks with a ram stroke of at least 750 to 1200 mm must be used if the strands are to be tensioned in a onestep operation (Ref. 31). After being tensioned, wires and strand are usually anchored by means of frictional splitcone wedges. Efficient quickrelease grips are also available for this purpose (see Fig. 32). Tendons can be released individually or simultaneously. Tendons are released individually either by flame cutting, sawing or by hydraulic cutters, and a strict cutting sequence which minimises eccentric loading on the concrete, must be adhered to when carrying out this operation. It is also important to avoid the situation where too many tendons are cut at a single location because this can result in the failure of the remaining tendons at that location. Tendons must be cut gradually and as close to the ends of the members as possible to avoid large impact loads from being imparted to the concrete. These precautions will prevent excessive damage to the concrete at the ends of the members, and so will ensure that the bond between the concrete and the tendons in this zone is not impaired.
PRETENSIONING SYSTEMS AND PROCEDURES
33
Sevenwire strand Chuck
Body
Jaw assembly Retaining ring Spring Cap Figure 32:
Typical quick release grip (adapted from Ref. 32).
Tendons are released simultaneously by making use of hydraulic rams. The principle advantages of this method are that the prestressing force is gradually transferred to the concrete so that impact loading is avoided, and that the tendons can subsequently be cut between the members without following a strict cutting sequence. A disadvantage of this method is that the precast members close to the releasing end will experience relatively large movements away from the releasing end because all 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. It can be shown that the bond stress induced by a tendon will decrease as its diameter decreases, for a given stress in the tendon. For this reason smalldiameter wires and strand are used in pretensioning. Wire is often indented or crimped to improve its bond properties while, in the case of strand, 12.9 mm sevenwire strand is most commonly used. It is often necessary to deflect some of the tendons to obtain the desired cable profile, particularly in the case of long span members (see Fig. 33). These deflected tendons are often referred to as draped or harped tendons, and are held in their deflected position by special holddown devices at the lower deflection points (also called holddown or draping points) and by holdup devices at the high positions. Depending on the design requirements, deflected tendons can be provided with either one or two holddown points, as shown in Fig. 33. Tendons can only be deflected if the stressing bed has been properly reinforced to sustain the vertical forces imposed by the holddown devices.
Single holddown point
Figure 33:
Holdup device
Double holddown point
Pretensioning with deflected tendons (Ref. 35).
34
PRESTRESSING SYSTEMS AND PROCEDURES
Deflected tendons are usually tensioned straight and then deflected by a hydraulic ram, after which the holddown devices are installed to keep them in their deflected shape. Two methods of deflecting tendons in this way are shown in Figs. 34 and 35. In the method shown in Fig. 34, which is commonly used for the manufacture of doubletee beams, the deflected tendons are pushed down to their lower position by means of a temporarily installed hydraulic ram. These tendons are subsequently held in position by the holddown pins which bear against the holddown reaction beam. After the concrete has developed the required strength, the holddown pins are removed and the tendons are released. The method shown in Fig. 35 makes use of a centre hole jack to deflect the tendons, while a strand chuck bearing against the holddown anchors is used to anchor the tendons in their deflected positions. In this procedure, the strand chuck and the holddown anchors cannot be recovered (Ref. 31).
Hydraulic ram pushes pin down Holddown pin (removed from hardened concrete) Holddown reaction beam
Ratchet adjustment Holddown device
Figure 34:
Doubletee form
Deflecting tendons in a doubletee beam (Ref. 35).
The actual length of a deflected tendon, measured along its deflected path, can be significantly greater than the horizontal distance between its ends, particularly in the case of deeper members such as bridge girders. If such a tendon is initially tensioned straight, the subsequent deflecting operation will increase the tension in the tendon and, hence, the prestressing force. It is important to consider this effect when determining the initial tension to be applied to the tendons, particularly if the increase in tension is significant. Deflected tendons can also be tensioned in their deflected shape, in which case the holddown devices must be capable of permitting the tendons to move longitudinally during the tensioning operation and provision must be made to reduce the friction between the tendons and the holdup and holddown devices. The various techniques which have been used to reduce the effects of friction include tensioning the tendons from both ends, using rollers with needle bearings at the holdup and holddown points, and vibrating the tendons while they are being tensioned (Ref. 31). As previously mentioned, the precast members will move longitudinally when the tendons are released so that it is essential to release the holddown devices before releasing the tendons. However, when the holddown devices are released before releasing the tendons, the undesirable situation arises in which concentrated upward vertical forces are imposed on the beam at the positions of the holddown points before any prestress has been transferred to the beam. If these effects are not properly accounted for in the design or in the releasing procedure (e.g. by partially releasing the tendons to transfer some prestress to the beams before releasing the holddown devices) cracks can develop in the top of the beam. Tendons in beams are deflected to reduce the cable eccentricity in the support regions which, in turn, prevents flexural cracks from developing at the top of the beam in these regions. This objective can 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. 36).
PRETENSIONING SYSTEMS AND PROCEDURES
35
12.5 mm diameter strand Strand chuck
Center hole hydraulic jack
Strand chuck Holddown anchors
Deflected strand group
Strand chuck
Figure 35:
Tendon holddown device for use with a centre hole jack (Ref. 31).
Plastic tube over strands in bottom Figure 36:
Blanketed strands (Ref. 35).
Blanketed strand length (debonded by plastic tubes)
36
PRESTRESSING SYSTEMS AND PROCEDURES
Typically, a stressing bed must allow a daily production cycle so that members can be produced in large numbers. Under such conditions the use of steel forms or moulds is preferable for the following reasons: Steel forms are durable and perform well under repeated use; they can be manufactured to a high degree of precision; they are easy to handle when being erected or stripped; they can be made adjustable to easily accommodate variations in member shape; and they can easily be made strong enough to allow form vibration (Refs. 31 and 34). Figure 37 shows a steel mould with removable side forms and a form vibrator for a bridge girder.
Form underties External vibrator sled and track
Transverse sleepers
Figure 37:
Steel mould for a bridge Ibeam (Ref. 35).
The forms are removed after curing the concrete and before releasing the tendons. They should be loosened or stripped in such a way that they do not restrain any longitudinal movement or vertical deflection 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 the tendons to be released within about 16 hours after casting. This high early strength can be obtained either by using high early strength cement, by curing the concrete at an elevated temperature, or by combining these two options (Refs. 31 and 35). Curing at an elevated temperature is often done by steam curing, which involves the application of wet heat in the form of live steam under a confining cover. Steam curing generally commences 2 to 3 hours after casting and continues for 12 to 14 hours (Ref. 34). Other processes which can also be used to apply heat during curing include electricalresistance heating and heating by circulating hot fluids through pipes contained in the forms or in the stressing bed (Ref. 35). The longline method is also used for the production of hollow core slab units. In this method, low slump concrete is extruded around the pretensioned tendons by an extruding machine which travels along the stressing bed to form a long hollow core strip. The tendons are released once the concrete has developed sufficient strength, after which the long hollow core strip is sawed into the required lengths. References 32 and 35 can be consulted for further information on this procedure.
PRETENSIONING SYSTEMS AND PROCEDURES
37
3.2.2 Stressing beds Several different types of stressing bed are available, each with its own range of application depending on the product being produced, the site conditions and the requirements of the production process. The following types are generally used (Ref. 31):
• Column type. • Independentabutment type. • Abutmentandstrut type. • Tendondeflecting type. • Portable benches. It is possible to make a further distinction between stressing beds on the basis of whether they have been designed to produce one specific product or to produce many different types of product. The former are referred to as fixed beds while the latter are termed universal beds. The brief description of each type of bed given below is a summary of the material presented in Ref. 31 on this topic. Column beds In a column bed the prestressing force is carried directly by the bed acting as a column. The principle is illustrated by Fig. 38, which shows an example of a columntype bed used for manufacturing doubletee beams. Clearly this type of bed can only be used as a fixed bed and, in the interests of economy and efficiency, can only accommodate small eccentricities of the prestressing force with respect to the bed. The primary considerations for the design of a columntype bed are crushing of the concrete and buckling of the bed. Columntype stressing beds are primarily used for producing singletee beams, doubletee beams and piles.
Stressing mechanism
A
Stessing end
Releasing end
A Elevation Metal form liners
Pipes for hotwater curing Section AA
Figure 38:
Columntype stressing bed used for producing doubletee beams (Ref. 31).
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PRESTRESSING SYSTEMS AND PROCEDURES
Independentabutment beds The primary components of an independentabutment bed are two large structurally independent abutments, together with the paved casting surface between the abutments. The abutments can either be embedded in soil, supported on piles or founded on rock (see Fig. 39). The manner in which stability against sliding and overturning of the abutments is provided depends on the founding conditions as follows:
• Embedded in soil: In this case, stability is exclusively provided by the self weight of the abutment and passive soil pressure (see Fig. 39a).
• Supported on piles: The structural action of the piles provides the required stability in this case (see Fig. 39b).
• Founded on sound rock: In this case, stability against sliding can be provided by keying into the rock while the stability against overturning can be enhanced by anchoring the abutments to the rock with ties or prestressed anchors, as shown in Fig. 39c. Independentabutment beds are commonly used and can provide a fairly cheap solution in the case of long beds.
Prestressing force
Passive soil pressure
Prestressing force
W (a) Embedded in soil Tension piling
Prestressing force Soil overburden Concrete
Compression piling
(b) Supported on piles
Rock
Steel dowels anchored in drilled holes or prestressed anchors (c) Founded on rock Figure 39:
Independent abutments (Ref. 31).
Abutmentandstrut beds An abutmentandstrut stressing bed basically consists of an abutment at each end joined by a concrete slab or strut, as shown in Fig. 310. It is clear from this figure that the overturning action of the prestressing force on the abutments is counteracted by the self weight of the abutments, and that its sliding action is resisted by the slab acting as a strut. It should also be noted that the concrete hinges provided between the abutments and the slab ensure that no bending is induced in the slab. In this way, the slab is subjected to an axial load only. Evidently, the abutmentandstrut bed is the type most commonly used for short beds.
PRETENSIONING SYSTEMS AND PROCEDURES
39
Prestressing force P
P
Recess to accomodate stressing mechanism Concrete hinge
Abutment Figure 310:
Strut
Abutmentandstrut stressing bed (Ref. 31).
Tendondeflecting beds After tensioning, deflected tendons are held in position by holddown devices, anchored to the bed at the holddown points, and by holdup devices which bear on the bed at the high positions (see Section 3.2.1). Consequently, the slab component of the bed is subjected to large vertical loads in addition to the axial load associated with the structural action of an abutmentandstrut bed. Because of this loading condition, tendondeflecting stressing beds, of which a typical layout is given in Fig. 311, differ from abutmentandstrut 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 which it is subjected.
• Tendondeflecting beds are much more expensive than abutmentandstrut beds of equal capacity. Vertical forces induced by deflected tendons
Prestressing force P
P
This portion of bed is subjected to combined axial load and bending Figure 311:
Tendondeflecting stressing bed (Ref. 31).
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 reinforcement or with regard to the production process, can be improved by providing some means by which its length can be adjusted. This can be done either by providing the deadend abutment with removable uprights which can be fitted in several positions (see Fig. 312a), or by providing an intermediate abutment with removable uprights (see Fig. 312b). When fitted with an intermediate abutment, the potential efficiency of the stressing bed can be improved even more by designing the bed so that the prestressing force can be applied from either end. An alternative solution, which is often simpler and 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 the structural frame together with the hydraulic rams and pumping unit. Whereas rams are used to apply the prestressing force, structural frames are required to transfer this force to the abutments and to
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PRESTRESSING SYSTEMS AND PROCEDURES
Fixed upright
Removable upright Alternate upright positions
Stressing end
Dead end (a) Stressing bed with removable uprights
Fixed upright
Removable uprights
Intermediate abutment
Stressing end
Fixed upright
Alternate stressing end
(b) Stressing bed with intermediate abutment Figure 312:
Stressing beds with adjustable length (Ref. 31).
maintain a constant strain in the tendons during the production cycle. They also provide the means of tensioning and anchoring the tendons and, depending on the design of the system, of releasing them. Although the specific details of the structural configuration of frames will differ, the following structural 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 the installation and should, at least, cover the following considerations (Ref. 31):
• The capacity of the bed. • The range and types of product to be manufactured on the bed. This consideration also covers the 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
POSTTENSIONING SYSTEMS AND PROCEDURES
3.3.1 Basic principle and procedure In posttensioning systems the concrete is first cast and allowed to harden, after which the tendons are tensioned and anchored. The prestressing force is transferred to the concrete by the anchorage assemblies which bear against the concrete. Obviously, the prestressing operations are carried out only after the concrete has developed sufficient strength to sustain the induced loads. The most common method of tensioning the tendons in posttensioning systems is the mechanical prestressing by means of hydraulic jacks which react against the concrete. The required capacity of the jack depends on the type, size and number of wires, strand or bars in each tendon and consequently shows a large variation, from as low as 40 kN to as high as 10000 kN. It is essential that 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, specially designed jacks are usually supplied with each particular type of posttensioning system. After a tendon has been tensioned, it is anchored to the concrete by a purpose made anchorage supplied 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:
POSTTENSIONING SYSTEMS AND PROCEDURES
311
• 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 means of cold formed rivet heads or socalled buttonheads while, in the case of a bar, a nut which threads onto the end of the bar is used. The anchorages used in South Africa usually conform to the requirements of BS 4447 (Ref. 36). Tendons can only be posttensioned if they are not bonded to the concrete at the time of tensioning. This is usually accomplished by placing mortartight metal or plastic tubes (also referred to as ducts or sheaths) along the intended profiles of the tendons before the concrete is cast. Thus, ducts are formed in the hardened concrete through which the tendons can be passed and subsequently tensioned. The tendons can either be preplaced in the sheath prior to casting, or can be threaded through the ducts after the concrete has hardened, depending on the system being used. After the tendons have been anchored, the completion of the posttensioning operation depends on whether the tendons are bonded or unbonded. In the case of bonded tendons, cement grout is injected into the duct to fill the void between each tendon and its duct (see Fig. 313a). Upon hardening, the grout effectively bonds the tendon to the surrounding concrete and also provides protection against corrosion of the prestressing steel.
Corrugated metal or plastic sheath
Filled with grout
(a) Bonded multistrand tendon
Plastic tube
Grease
Strand
(b) Unbonded monostrand tendon Figure 313:
Bonded and unbonded tendons (Ref. 35).
In the case of unbonded tendons, the anchoring of the tendons represents the final step in the posttensioning operation because the ducts are not subsequently filled with grout. A typical single strand unbonded tendon is shown in Fig. 313b and consists of a grease coated single sevenwire strand encased in a plastic sheath. This type of tendon is prefabricated in the factory by extruding the plastic sheath over the strand after the strand has been coated with a layer of grease, which provides corrosion protection for the strand. Unbonded tendons remain unbonded over their entire length for the service life of the structure, and it is important to note that they are attached to the concrete only at their ends by the anchorages. These tendons are primarily used in the posttensioned slab systems found in building construction because of the considerable economies offered by this technique under these circumstances.
312
PRESTRESSING SYSTEMS AND PROCEDURES
3.3.2 Posttensioning systems Several posttensioning systems are available in South Africa. Although the basic principle used in these systems are essentially similar, they differ in the type of tendon which is used, the details of the anchorages and the method used for tensioning the tendons.
Multistrand 15.7 mm
Posttensioning systems can conveniently be divided into four general categories (Refs. 31 and 35): Multistrand systems, multiwire systems, monostrand systems and high strength bar (including multibar) systems. The tendon forces that can be achieved with each of these types of system are compared in Fig. 314.
Aps ≡ fpu
Area of prestressing steel ≡ Characteristic tensile strength
Multistrand 12.9 mm
11 000
9 000 8 000
55 Strands
55 Strands
7 000
4 000 3 000 2 000 1 000
Figure 314:
Multiwire 7 mm
5 000
Threadbar 15 mm to 36 mm
6 000
Monostrand 12.9 mm and 15.7 mm
Tendon force, 0.7 Aps fpu (kN)
10 000
55 Wires
19 Strands
13 Wires 1 Wire
1 Strand
15 Strands
1 Strand
Ranges of tendon force for various tendon types.
It is usual for designers in South Africa to furnish designs which can reasonably accommodate a number of the available posttensioning systems, to encourage competitive tendering. For this reason, a designer must be familiar with the available posttensioning systems to ensure that members are detailed to satisfy the practical requirements of these systems in terms of housing the tendons and anchorages, and receiving the jacks used for tensioning the tendons. Since a detailed description of all the available posttensioning systems is beyond the scope of this book, a generalised description of the typical features of posttensioning systems which fall within each of the abovementioned categories is given in the following. It is strongly recommended that the details of a particular system should be obtained from the pamphlets issued by the company or to consult its representatives.
POSTTENSIONING SYSTEMS AND PROCEDURES
313
Multistrand systems The details of a typical multistrand system are shown in Fig. 315. In these systems each tendon is made up of a number of strands, either 12.9 mm or 15.7 mm sevenwire strand usually being used. Some systems of this type offer tendons with up to 55 strands.
Tendon Sheath Grey cast iron or fabricated cone
Wedge grips Pressure plate
Grout injection point Hydraulic ports Permanent anchorage block/head Rubber springs Jack foot
Jack piston Steel anchorage block Temporary wedge grips
Figure 315:
Typical multistrand posttensioning system.
The most commonly used anchorages make use of the principle of wedge action, with the strands usually being individually gripped by two or threepiece conical wedge grips which seat in tapered holes contained in the anchorage block (see Fig. 315). When a tendon is tensioned the wedge grips are inserted in the tapered holes around each strand and, upon release of the jack, the subsequent pullin of the strand seats the grips which anchor the strand. The effects of the loss of tendon elongation resulting from seating of the anchorage must be accounted for in design. Some systems include special devices for ramming the grips to reduce the anchorage seating loss as well as the scatter of the individual anchorage pullin 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 anchored at its other end by a deadend anchorage which can either be cast directly into the concrete or
314
PRESTRESSING SYSTEMS AND PROCEDURES
Spiral
Swages Swage holder plate
Wire ties
Block
Duct
Guide (a) Swaged anchorage Grout tube Spiral
Splayed end Reinforcement grid
Side view
Top view (b) Splayed strand anchorage Grout tube Tension ring
Spiral Uplate
Side view
Duct
Top view (c) Looped anchorage Figure 316:
Typical deadend anchorages for multistrand posttensioning systems.
POSTTENSIONING SYSTEMS AND PROCEDURES
315
mounted on the surface of the concrete. Typical examples of deadend anchorages for multistrand tendons are shown in Fig. 316. The jacks used for the tensioning operations are supplied by the manufacturer and are designed to suit the tendons and anchorages of the particular system. Most systems use hydraulic centrehole jacks capable of simultaneously tensioning all the strands in a particular tendon. Purpose made multiuse jaws, which are self releasing after completion of the tensioning operation, are used to attach the strand to the jack (see Fig. 315). Some types of construction procedure, such as the segmental construction of a boxgirder bridge, require that tendons be joined to form a continuous tendon even though the member is constructed and posttensioned in a number of phases or segments. This can be achieved by making use of couplers of which a typical example for a multistrand system is shown in Fig. 317. In this figure phase 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.
Phase 2 Phase 1
Figure 317:
Typical tendon coupler for multistrand posttensioning systems.
In South Africa, multistrand systems are by far the most commonly used systems for bonded construction because of their versatility and economy. Multiwire systems The components and basic principle of a typical multiwire system which anchors the wires by means of buttonheads are illustrated in Fig. 318. These systems use tendons which each consist of a number of smooth highstrength steel wires. The specific system shown in Fig. 318 uses 7 mm wire and can 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. 318a and b). The buttonheads are coldformed with a special headforming machine after the wires have been threaded through the anchorheads at each end. Tendons can either be completely prefabricated 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 position with packing pieces inserted between the anchorhead and the bearing plate (see Fig. 318d). It is essential that the length of the tendon as well as its elongation be estimated as accurately as possible to ensure that the tendon elongation at anchoring corresponds to the desired prestressing force. Thin shims are usually available, in addition to the packing pieces, for making fine adjustments to the anchoring force to accommodate discrepancies between the estimated and measured elongations. One of the advantages of this type of anchorage is that the anchorage seating loss is negligible. Dead end anchorages and couplers are available for these systems.
316
PRESTRESSING SYSTEMS AND PROCEDURES
Coldformed buttonhead
Buttonhead 7 mm diameter wire
(a) Buttonhead anchor
Threaded anchorhead
Stressing sleeve Stressing bridge
(b) Multiwire tendon
Packing pieces
Pull rod Bearing plate (c) Tendon tensioned by pull rod Stressing bridge
Pull rod
Figure 318:
Sleeve Bearing plate Cylinder (e) Jack details
(d) Locking the anchorhead into position
Typical multiwire posttensioning system (Ref. 35).
The hydraulic jacks used for tensioning the tendons of these systems are supplied with the hardware required for attaching them to the anchorhead. The particular system shown in Fig. 318 uses a centrehole hydraulic jack to apply the tensioning force to the anchorhead by means of a pull rod which threads into a stressing sleeve which, in turn, threads onto the anchorhead. The jack bears on a stressing bridge which transfers the jack reaction to the concrete (see Figs. 318c and e). It is important to note that multiwire systems are usually not used in South Africa any more because of their higher cost. These systems are only used in situations where multistrand systems or any of the other available systems cannot offer a satisfactory solution. Monostrand systems The distinguishing feature of monostrand posttensioning systems is the fact that each tendon comprises a single sevenwire strand, with 12.9 mm and 15.7 mm being the most commonly used sizes. These systems are usually unbonded and a typical tendon is shown in Fig. 313b. The details of a typical monostrand system are shown in Fig. 319 together with the construction sequence for a posttensioned slab. As in the case of multistrand systems, the anchorages used in monostrand systems make use of two or threepiece conical wedge grips which seat in tapered holes
POSTTENSIONING SYSTEMS AND PROCEDURES
317
in the anchorage body to anchor the strand (see Figs. 319a and c). However, the anchorages are designed 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 tensioning voids, or pockets, in the concrete (see Fig. 319).
Anchorage body
Plastic former (grommet)
Wedge grips
(a) Anchorage details Corrosion protection cap Form Corrosion protection sleeve Corrosion protection sleeve
Grommet
Monostrand Deadend anchorage
Intermediate anchorage
Stressing anchorage
(b) Monostrand system for slab posttensioning Hydraulic pump
(5) Cut excess strand, cap end and fill in hole with weather resistant grout Tendon profile support
(3) Place wedge grips (2) Remove grommet
(4) Tension and anchor strand
(1) Placement of monostrand and anchorage nailed to formwork (c) Construction sequence for posttensioned slab Figure 319:
Typical monostrand posttensioning system (Ref. 35).
318
PRESTRESSING SYSTEMS AND PROCEDURES
The various steps involved in installing and tensioning a monostrand tendon is summarised in Fig. 319, which is self explanatory. It is to be noted that the hydraulic jacks used for tensioning the tendons are small and light so that the tensioning equipment can usually be operated by one person. Couplers are not required for monostrand systems because tendons can be supplied to almost any required length, while successive sections of the same continuous tendon can be tensioned separately by using an open throat jack to stress at intermediate points between partial slabs. Deadend anchorages are normally installed in the factory and, in the case of wedge anchorages, the wedges are hydraulically seated in the factory. Unbonded monostrand systems are particularly well suited for the posttensioning of thin slabs and narrow members because the small tendon diameter allows optimum eccentricities and because the compact anchorages can be accommodated by such thin members. These factors, together with the elimination of the grouting operation as well as the simplicity and efficiency of the tensioning operation all offer considerable economies. Hence, practically all castinplace prestressed slabs, flat plates and flat slabs, encountered in building structures in South Africa, are posttensioned by unbonded monostrand systems. Bar systems Bar systems are characterized by the fact that high strength bar is used for the tendons. The bar can be supplied in most of the standard sizes (see Table 211) 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 systems usually use bonded tendons, but unbonded tendons provided with a corrosion protection system are available. A typical single bar posttensioning system is shown in Fig. 320. The bar is anchored by a nut which threads onto the end of the bar and seats into either a bellshaped or plate anchorage. The
(a) Bell and plate anchorages End shutter
Removable pocket former Grout tube Sheathing Ratchet
Grout sleeve Bell anchorage Removable plastic nut (b) Tendon assembly at stressing end
Figure 320:
Typical threadbar posttensioning system.
(c) Jack Details
POSTTENSIONING SYSTEMS AND PROCEDURES
319
seat is spherical to prevent the development of significant bending moments in the bar when the axis of the bar is not perpendicular to the anchor plate (Ref. 31). The prime advantage of the bell anchorage is that the anchor ring contains the splitting forces induced by the anchorage after the tendon is tensioned. Both anchorage types shown in Fig. 320a exhibit negligible anchorage seating losses when properly installed. When smooth bar is used, threads must be provided at the ends of the bar for the anchor nuts. It is important to note, in this regard, that the elongation of the tendon must be estimated as accurately as 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. 32). This difficulty, of course, does not exist for threadbar, in which case the thread runs over the complete length of the bar. It should be noted that wedge anchorages have been developed for use with smooth bar tendons. A centrehole hydraulic jack is commonly used for the tensioning operation. In some systems, the jack is provided with a ratchet which is used to tighten the anchor nut against the anchor plate while the bar is being tensioned (see Fig. 320c). Any special hardware required for attaching the jack 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 as transportation and storage requirements. However, couplers can be used to provide tendons of almost any length. These couplers, of which an example for a threadbar system is shown in Fig. 321a, are usually of the sleeve type which simply screw onto each of the bars to be spliced. The couplers can also be used to extend a previously tensioned bar, a situation which, for example, arises in segmental and phase construction. When bars need to be coupled, the use of threadbar is particularly attractive because the continuous thread makes it possible to cut the bar to any required length, and also because the coupling hardware and operation are simple and relatively low in cost. It is important to note that when these sleeve couplers are used to splice bars, sufficient space must be provided in the concrete surrounding the couplers to allow the movement which takes place during tensioning. An example of a deadend anchor for a threadbar system is shown in Fig. 321b.
(a) Coupler
Bell anchorage
Grout tube Sheathing
Anchor nut (b) Bell anchorage Figure 321:
Typical deadend anchorage and coupler for a threadbar posttensioning system.
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Choice of system Since the basic principle of the available posttensioning systems is essentially the same, the fundamental difference between the various systems lies in the type of tendon which is used, the details of the anchorages and the method used for tensioning the tendons. Consequently the selection of a specific system for a particular application will depend on how well these features of the system will meet the requirements of the job. It often happens that a number of systems will work equally well for a particular application. Hence, the choice is often an economic one, that is, which system will be the cheapest in terms of the cost of the materials, equipment and the labour required to install, tension and grout the tendons.
3.3.3 Posttensioning operations After the concrete has reached the specified strength the tendons can be tensioned but, as a general rule, the full prestressing force should be applied as late as is practically feasible to minimise the effects of shrinkage and creep of the concrete (Ref. 33). However, certain conditions which induce significant 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 the concrete. These tensile stresses can lead to the development of visible cracks. A possible remedy for this problem is to apply a moderate prestress at a very early age and then to apply the full prestress at a later age when the specified concrete strength has developed. It is important to ensure that the compressive stress permitted in the concrete when it is initially prestressed, is based on its strength at the time of tensioning (Ref. 33). It is essential that all side forms and other obstructions which may restrain the deflections and shortening of the member, induced by prestressing, be removed or loosened before tensioning the tendons. It is particularly important in this regard to ensure that, if present, sliding bearings are cleaned 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 ensure that severe eccentric loading is avoided at all stages of the tensioning operation. Sometimes it may be necessary to tension some of the tendons in two steps to meet this requirement (Ref. 32). The following aspects regarding the sequence of tensioning should also be noted (Ref. 33):
• 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 tendons should be tensioned first.
• Before a tendon, which does not extend over the full length of a member is tensioned, the concrete surrounding its internal anchorage must first be subjected to compression. This is achieved by first 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 the hydraulic pressure applied to the jack using a pressure gauge and, secondly, by measuring the elongation of the tendon. The measured elongations are used to check the pressure gauge readings and 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 the force by the ram area) will always be less than the measured pressure because of the internal friction of the jack. For this reason, and to ensure that the measurements of pressure taken from the gauge are accurately translated into force, the stressing equipment must be calibrated. Jack calibration is usually accomplished by jacking against a laboratory calibrated load cell or proving ring, placed in the 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. 37) before
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321
tensioning any of the tendons. When a large number of tendons are to be tensioned, the calibration of the stressing equipment should, at least, be repeated after completion of the tensioning operations while, on very large projects, the calibration should be repeated at regular intervals. It should be noted that some posttensioning 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 be solved by making use of the fact that the material behaviour of both the concrete and steel remains linear elastic at the stress levels induced by the prestress (Ref. 33). The normal procedure is to stress the tendon to between 5 and 10% of the full tensioning force and to use this position as the starting point for measuring the elongation (Ref. 37). A loadelongation diagram can be constructed by plotting measurements of load against elongation, taken at regular load increments as tensioning proceeds. Because the material behaviour is essentially linear elastic, these points should plot as an almost straight line, so that the zero point can be obtained by extrapolating the loadelongation diagram to the value of zero load (see Fig. 322). An alternative approach, which is often followed in practice, is simply to consider the calculated and measured increments of elongation beyond the initially applied prestress, used as the starting point for measuring elongation.
Tensioning force P
P3
P2 Measured points P1 Extrapolation of loadelongation diagram to P = 0 ∆1
∆2
∆3
Tendon elongation
Initial starting point for measuring elongation Zero point for measuring elongation Figure 322:
Determining the zero point for measuring tendon elongation.
Tendon elongations recorded during the tensioning operation provide a check on the applied tensioning force by plotting them as a loadelongation diagram, which can be directly compared to the calculated diagram. It is generally required that the measured tendon elongation should agree with the calculated value to within ± 5% (Refs. 35 and 38). Specifically, Ref. 37 requires that the 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 the tendons in a member must be less than ± 3%. Note that the calculated tendon elongation should include a correction which accounts for the elongation of the length of the tendon that extends from the anchor to the jack grip position. Any of the following causes will individually, or in combination, result in the measured tendon elongation being larger than the calculated value (Ref. 33):
• 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 of elongation is larger than the actual value.
• The actual steel crosssectional areas are smaller than assumed for the calculations.
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• An anchorage or the concrete surrounding the anchorage has failed. This event is usually characterised 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 sound and a sudden drop in the applied tendon force. The calculated tendon elongation may not be achieved during tensioning for the following reasons (Ref. 33):
• 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 grout ingress, the tendon can actually be jammed in the duct so that only the portion of the tendon which extends from the obstruction to the point of tensioning is stressed. In this event, the tensioning force should not simply be increased to obtain the required elongation because of the danger of overstressing the tendon in the tensioned portion, while the level of the tensioning force in the remainder of the tendon is essentially unknown. It is far better practice to overcome the obstruction by repetitively releasing and reapplying the tensioning force. During such an operation, care must be taken to ensure that the permissible tendon stress is not exceeded. In some cases, excessive friction can be overcome by injecting watersoluble oil into the duct. If this 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 is smaller than the actual value.
• The actual steel crosssectional areas are larger than assumed for the calculations. It is generally recommended that tendon elongation be measured to an accuracy of ± 2%. Reference 37 requires that the elongation be measured to an accuracy of ± 2% or 2 mm, whichever is the most accurate. After the prescribed tensioning force has been reached, the tendons are anchored when the hydraulic pressure on the jack is released. If wedgetype anchorages are being used, a loss of elongation takes place because of the pullin of the strand when the wedge grips are seated. It is important to note that when the tensioning force is transferred from the jack to the anchorage, a further loss of elongation takes place because of the resulting deformation of the anchorage components. The magnitude of the anchorage deformation, which can be appreciable for some systems, appears to be dependent not only on the anchorage type, but also on the quality of workmanship (see Refs. 31 and 33). The total loss of elongation which takes place when a tendon is anchored is often referred to as the anchorage set (also pullin or anchorage seating), and must be considered in design. For strand anchored by wedge grips, the anchorage set is of the order of 6 mm, while the anchorages commonly used for threadbar systems do not show appreciable anchorage set if properly installed. Clearly, posttensioning systems using anchorages which yield a significant anchorage set are not suitable for use with short tendons. Note that anchorage set can be compensated for by installing shims behind the anchorhead. Anchorage set must be recorded in the field to ensure that the values being obtained agree with those assumed for design. Reference 37 specifies that the anchorage set should be measured to an accuracy of ± 2 mm and requires that the measured values must agree with the values assumed for design to within ± 2 mm. When a long or appreciably curved tendon is tensioned from one end only, the effects of friction will lead to a considerable loss of force along its length. This loss of force can be reduced by tensioning the tendon from both ends. An additional, or alternative, procedure which can be followed is to retension the tendon after it has been temporarily overstressed (see Refs. 32, 33 and 35). When overstressing a tendon, the temporary tension thus applied must not exceed 80% of its specified
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323
characteristic tensile strength and, after anchoring, the tension in the tendon must not exceed 70% of its characteristic tensile strength (Refs. 39 and 310). It is essential that the tensioning operation be supervised and carried out by experienced personnel who are familiar with the posttensioning system, equipment and the procedures to be used. It is also strongly recommended that the tensioning equipment be power driven, and that provision be made for an alternative power source which can be used in the case of a breakdown. The tensioning equipment must be capable of applying the load in a controlled manner without imposing significant secondary stresses on the tendon, anchorage or the concrete. The schedule which gives the sequence in which the tendons are to be tensioned, as well as those which give the tensioning forces and corresponding anticipated pressure gauge measurements, the calculated elongations, and the anticipated anchorage set for each tendon must be available before commencing the tensioning operation (Refs. 37 and 38). A considerable amount of strain energy is stored in the tendon during the tensioning operation and failure 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 the anchorage at the opposite end of the tendon. It is therefore important to exercise extreme caution when 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 the anchorage at the opposite end of the tendon.
• Ensuring that the tensioning equipment is properly maintained and assembled, and ensuring that it is not misused.
• Immediately stopping the tensioning operation in the event of an unusual occurrence such as, for example, 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 precautions in posttensioning can be obtained from Refs. 311 and 312.
3.3.4 Ducting for bonded construction In posttensioned construction, the tendons are tensioned after the concrete has been cast and after it has developed the specified strength. This is accomplished by placing ducts along the specified tendon profiles to form conduits in the hardened concrete through which the tendons can be passed and subsequently tensioned. Therefore, ducts must satisfy the following requirements (Refs. 37, 38 and 313):
• 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 their shape under the weight of the fresh concrete. In bonded construction, the shape of the ducts must be of a type which will enhance the transfer of bond from the grout to the surrounding concrete, and the material used for the duct must not have any adverse chemical reaction with the concrete, tendons or grout. In South Africa, the ducts most commonly used are made of spirally wound steel strip which forms flexible corrugated sheathing (see Fig. 313a). The corrugations are required for bond strength, while the thickness of the strip steel generally ranges between 0.2 mm, for small tendons, and 0.6 mm, for large tendons (Ref. 313). Although not commonly used in South Africa, polyethylene and polypropylene tubing has successfully been used in Europe for many years. A primary advantage of the polyethylene and polypropylene ducts is that they offer improved corrosion protection when compared to ducts made
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of strip steel. The internal diameter of the duct is generally required to be large enough to yield a duct 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 spaced suppports. If the tendons are preplaced in the ducts prior to casting, the supports and ties must be able to support the tendon weight. When only the duct is placed before casting, with the tendon being installed prior to tensioning, the supports and ties must adequately resist the bouyancy forces which arise during casting. It is important that ducts are installed on smooth curves without kinks to minimize the friction losses which arise during tensioning. This requirement can be satisfied by spacing the duct supports closely enough to prevent the ducts from sagging between them. A further consideration, which requires 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 on whether or not the tendons are preplaced prior to casting as well as on the rigidity of the duct, the type and size of the duct, and the tendon profile. Reference 37 recommends that a spacing of between 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 is perpendicular to the bearing surface of the anchorage. Each anchorage must also be installed in such a way that it will not be displaced during casting, and steps must be taken to ensure that the bearing plate 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 tied into position, that they have not been damaged during installation, and that grout cannot leak into them during casting. Particular care should be taken to ensure that all joints in, and connections to the ducts are completely grouttight. The importance of this inspection is underscored by the fact that the cost associated with clearing the areas affected by the ingress of grout into the ducts often exceeds 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 of the duct diameter to such an extent that the tendons cannot be inserted. Areas congested by reinforcement and other embedded materials are particularly prone to this problem. Since voids in the concrete behind the anchorage bearing plates, or insufficient concrete strength can lead to failure of the concrete in these regions during tensioning, the concrete at the anchorages must be properly vibrated 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 in the crosssection of the duct and, if it is draped, at the high points of the duct (see Fig. 323). Water and bleed water which can accumulate in the resulting air pockets can lead to corrosion of the prestressing steel and, in so doing, seriously impair the durability of the structure. This situation can be prevented by providing vents, through which trapped air and water can escape, at the following positions:
• At the high points of the duct if the drape of the tendon, measured from the highest point to the lowest point exceeds 500 mm (see Fig. 324c) (Ref. 313). Reference 38 suggests that in cases where the tendon curvature is small, such as in continuous slabs, high point vents are not required.
• At significant changes in the duct crosssection, such as at anchorages and at couplers where the duct is enlarged in the region of the anchorage (see Figs. 324a and b). The recommended minimum size of the inner diameter of the vent tubes ranges between 20 mm and 25 mm (Refs. 37, 313 and 314). It is also recommended that the vent tubes should extend a distance of at least 500 mm above the surface of the concrete (Ref. 37).
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Air pocket 1
Grout
1 Tendon profile near high point Figure 323:
Section 11
Air pocket at tendon high point resulting from inadequate venting (Ref. 35).
(a) Ref. 313 Grout outlet
(b) Ref. 313 Vent
Grout inlet
Duct
(c) Ref. 35
Figure 324:
Placing of vents.
Anchorages are provided with grout holes or grout tubes which serve as inlets or outlets for the grout. In the case of long tendons, provision should be made for intermediate inlets which can be used if difficulties or blockages should develop at the main inlet (Ref. 313). The ducts cannot be grouted if the temperature of the concrete falls below about 5ºC. If freezing temperatures are likely to occur when the tendons are to be tensioned, the situation can arise where ducts, containing tensioned tendons, are left ungrouted for a considerable period of time. It is essential 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 properly drained (Refs. 31 and 35).
3.3.5 Grouting In bonded construction the ducts containing the tendons are filled with cement grout as soon as possible after tensioning. The primary reasons for grouting the ducts are to provide corrosion protection for the prestressing steel and to provide a means of bonding the prestressing steel to the surrounding concrete. If these objectives are borne in mind, it should be clear that a suitable grout must comply with the following requirements:
• Since the grout must flow over long distances in a fairly confined space, it must maintain its fluidity during the grouting operation to ensure that all voids in the duct are completely filled.
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• The amount of sedimentation as well as the resulting contraction of the grout must be kept as small as possible. It is generally recommended that the contraction be limited to 2% (Refs. 33 and 313).
• The grout should exhibit minimum bleeding and, after setting, all bleed water must be reabsorbed. The amount of bleed water should not exceed 2% by volume 3 hours after the grout has been mixed, and it should be completely reabsorbed after 24 hours (Refs. 37 and 313).
• The hardened grout must have adequate bond and shear strength. This requirement is deemed to be satisfied if the compressive strength of 100 mm cubes, tested at 20ºC, exceeds 20 MPa after 7 days and 30 MPa after 28 days (see Refs. 37 and 313).
• The grout should not contain excessive quantities of chlorides, nitrates, sulfides, or other ingredients harmful to the grout itself or to the prestressing steel (Refs. 35, 37 and 313).
• In freezing climates it is essential that steps be taken to ensure that the grout is resistant to frost so that it does not lose strength nor fracture as a result of freezing. Further information on this aspect may be obtained from Refs. 33 and 313. Cement and water are the primary constituents of grout, and admixtures are sometimes used to improve its properties. Fine aggregate may be added under special circumstances such as, for example, when grouting large diameter ducts. Ordinary Portland cement is most commonly used for grout but, if conditions require their use, other types of cement can be used provided their suitability has been established by tests. It is important that the cement must be fresh and that it should not contain lumps or any other indications of hydration (Ref. 38) and, for this reason, Ref. 37 recommends that the cement should not be older than one month. The water used for grout must not contain deleterious quantities of substances harmful to the grout or to the prestressing steel and, therefore, should not contain more than 500 mg of chloride ions per litre (Refs. 37 and 313). The minimum value of the watercement ratio is usually controlled by the required fluidity of the grout while the maximum value is usually governed by the fact that the grout should not exhibit excessive bleeding. The watercement ratio must be kept 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, such as 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 through a 0.600 mm sieve (Ref. 37). It is recommended that the aggregate content should not exceed 30% of the weight of the cement (Ref. 37). Admixtures are used only when the desired properties of the grout cannot be obtained by a suitable mixture of cement and water. Therefore, the objectives of adding admixtures are to improve the properties of the grout such as improved fluidity, reduced bleeding and retarded setting time, and to impart other properties to the grout such as expansion, to compensate for contraction, and air entrainment, to improve freeze resistance. Only wellproven admixtures, which do not contain chemical substances in quantities which are liable to damage the grout or the prestressing steel, should be used (Refs. 37 and 313). When using an expanding agent, the unrestrained expansion of the grout should be limited to 5% (Ref. 313). The grouting equipment usually consists of a mixer, a holding reservoir and a pump together with all the connection hoses and valves. Mechanical mixers are used to consistently produce a homogeneous and stable grout which is free of lumps. The two types of mixers used are: vane mixers, having a speed of approximately 1000 rev/min, and high speed compulsory mixers, with a speed of about 1500 rev/min (Ref. 313). After mixing, the grout is usually passed through a screen with openings not exceeding 5 mm into a holding reservoir equipped with an agitator, which maintains the colloidal condition of the grout during the grouting operation. The capacity of the mixer and the reservoir must be sufficient to allow the duct to be filled without interruption at the specified speed. The grout is delivered at the duct, from the reservoir, by a pump capable of providing a steady flow of grout. The pump must be able to maintain a pressure of at least 1.0 MPa
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on a completely filled duct, and it should be equipped with a pressure gauge as well as a safety device which will prevent pressures exceeding 2.0 MPa from developing. The system should also be capable of recirculating the grout if pumping is interrupted. Grouting hoses and all hose connections must be airtight and must be of a size and type which will prevent the buildup of pressure during grouting (Refs. 37 and 313). Further information on grouting equipment may be obtained from Refs. 33, 37, 313 and 314. Equipment required for providing compressed air and for flushing grout out a duct, if the grouting operation is interrupted for some reason (e.g. if a blockage is encountered or if a breakdown of the grouting equipment occurs during grouting), must be at hand. It is extremely important that all grouting equipment must be in good working order. The ducts should be grouted within 7 days after tensioning, and should the grouting operation be delayed for a longer period, specific steps must be taken to protect the prestressing steel from corrosion. Before grouting, the ducts should be checked for obstructions by water injection and all excess water should afterwards be displaced by the grout. The grouting operation begins by adding the constituent materials of the grout to the mixer. The sequence in which this takes place depends on the type of mixer, and the following is recommended by Ref. 313:
• 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 exceed 4 minutes for vane mixers or 2 minutes for high speed compulsory mixers (Ref. 313). 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 at a rate which is slow enough to prevent segregation of the grout. Reference 313 suggests that a rate of between 5 and 15 m per minute should be used. The first vent tube after the inlet is closed once the grout flowing out of it does not contain visible slugs of water or air, and is of the same consistency as at the inlet. The remaining vent tubes and the outlet are closed in sequence in the same manner, while the duct progressively fills. After the outlet has been closed, the final grouting pressure or a pressure of at least 0.5 MPa, whichever is the greater, must be maintained on the grout for at least 5 minutes before closing the inlet (Ref. 37). If an expanding agent is included in the grout mix, the vent tubes must be reopened 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 grout has hardened and inspected to establish the extent of the grout fill. If such an inspection reveals the presence of voids, the problem can be remedied by topping up the vent tubes with grout or by undertaking a regrouting operation, depending on the specific circumstances (see Refs. 313 and 314). If a blockage is encountered in a duct, the grouting operation must immediately be stopped to prevent a large pressure, which can damage the structure, from developing in the duct. The grout should also 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 grouting operation can be restarted. It should be noted that excessive pressures which develop during grouting can lead to water segregation and can also cause cracking or damage to the structural element, and should be avoided. The grout should be discarded after 30 minutes unless a retarder is used. Grouting should not be undertaken if the ambient temperature drops below 5ºC, and care should be taken to ensure that the ducts are free from ice or frost before grouting commences in cold weather (Ref. 37).
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The fluidity of the grout should be tested immediately after mixing and at regular intervals during the grouting operation. A flow cone test is normally used for testing fluidity on site (Ref. 37). Sufficient quantities of grout must be taken during grouting, usually at each vent, to enable testing of the other properties of the grout such as bleeding, strength and volume change. Appropriate testing procedures for determining these properties are described in Refs. 33, 37, 38, 313 and 314. It is extremely important to ensure that the grouting operation is carried out properly because the durability of a bonded posttensioned structure depends on how sucessfully this operation has been completed. It is, therefore, strongly recommended that this highly specialised and critical operation be carried out only by appropriately trained and experienced personnel.
3.4
PRETENSIONING VERSUS POSTTENSIONING
Pretensioning and posttensioning systems each have specific theoretical and practical advantages and disadvantages. These must be considered in conjunction with the particular technical requirements and the prevailing economic considerations for a specific job, before a decision can be made regarding which method of prestressing is to be used. It is useful to remember, in this regard, that although pretensioning is generally perceived as being limited to permanent precasting factories, it can be economically feasible for the contractor to set up a temporary prestensioning yard at, or close to, the site on projects where a large number of pretensioned elements are to be used. On the other hand, posttensioned members, which are usually constructed and tensioned in situ, can be manufactured in a precasting plant and subsequently transported to site (e.g. precast segmental posttensioned bridges). Some of the differences between pretensioning and posttensioning, which should also be considered when comparing the two methods of prestressing a member, are listed in the following:
• The capital investment in the equipment and facilities required for posttensioning is considerably less 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 greater than that of posttensioning because of the additional material and labour costs associated with the ducts, anchorages and grouting required for posttensioning.
• Structural elements can be prestressed in situ only by posttensioning. • It is impractical to posttension very short elements because any anchor set will lead to a large percentage loss of tensioning force, and also because the small elongation of the short tendon requires a high accuracy of measurement. Clearly these difficulties are nonexistent if the longline method of pretensioning is used.
• Long and very large members may be more conveniently and economically cast in place and posttensioned, because the cost of transporting and handling large pretensioned members, which are cast off site, can become excessive. When the precasting plant is situated too far away from the site, the transportation cost can also become prohibitive.
• The tendons in posttensioned elements can easily be placed on smooth curves along the desired profile. Although pretensioned tendons can be deflected, the procedure remains costly and limited. In the case of continuous elements, such as continuous bridge beams, pretensioning becomes impractical.
• The loss of prestressing force associated with tendon friction during tensioning is significant in posttensioned tendons and must be considered in design as well as during construction.
REFERENCES
3.5
329
REFERENCES
31
Libby, J. R., Modern Prestressed Concrete: Design Principles and Construction Methods, 4th ed., Van Nostrand Reinhold, New York, 1990.
32
Lin, T. Y., and Burns, N. H., Design of Prestressed Concrete Structures, 3rd ed., John Wiley & Sons, New York, 1981.
33
Leonhardt, F., Prestressed Concrete Design and Construction, English translation, Wilhelm Ernst & Sohn, Berlin, 1964.
34
Khachaturian, N., Gurfinkel, G., Prestressed Concrete, McGrawHill Book Company, New York, 1969.
35
Collins, M. P., and Mitchell, D., Prestressed Concrete Structures, PrenticeHall, Englewood Cliffs, New Jersey, 1991.
36
British Standards Institution, “British Standard Specification for The Performance of Prestressing Anchorages for PostTensioned Construction,” BS 4447:1973 (1990), BSI, London, 1973, reaffirmed 1990.
37
Committee of State Road Authorities, “Standard Specifications for Road and Bridge Works,” 1st ed., The CSRA Secretariat: Division of Roads and Transport Technology, Council for Scientific and Industrial Reseach, Pretoria, 1987.
38
PostTensioning Manual, PostTensioning Institute, Glenview, Illinois, 1976.
39
South African Bureau of Standards, “The Structural Use of Concrete,” SABS 0100: 1992, Part 1, SABS, Pretoria, 1992.
310 British Standards Institution, “Structural Use of Concrete, Part 1, Code of Practice for Design and Construction,” BS 8110: Part 1: 1985, BSI, London, 1985. 311 FIP Commission on Practical Construction, FIP Guide to good practicePrestressed Concrete: Safety Precautions in PostTensioning, Thomas Telford Ltd, London, 1989. 312 Recommendations for Safety Precautions in PostTensioning Operations, Concrete Society of Southern Africa, Halfway House. 313 FIP Commission on Practical Construction, FIP Guide to good practiceGrouting of Tendons in Prestressed Concrete, Thomas Telford Ltd, London, 1990. 314 "Grouting Specifications," CONCRETE, The Concrete Society Journal, Vol. 27, No. 4, July/August 1993.
INTRODUCTION
41
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 applied loadings and, therefore, implies that the configuration of the section and the properties of the materials used are known. Design, on the other hand, involves the selection of a suitable section and suitable materials out of many possibilities. The process of design is more complex than that of analysis because, on the one hand, it deals with unknowns while, on the other hand, a large number of combinations of possibilities exist. Analysis usually forms an integral part of the design process because once a section has been designed it must be analysed to check if it satisfies the specified design criteria. In this Chapter, the flexural behaviour of a prestressed concrete beam section over the complete loading spectrum, from zero load to failure, is first discussed. This is followed by a presentation of methods of analysis, after which various design procedures are dealt with. Sections subjected to flexure only are considered here, that is, only the effects of moment are considered. The material presented in this Chapter covers pretensioned and posttensioned members, and includes both composite and partially prestressed concrete sections. In the case of posttensioned members, both bonded and unbonded construction are considered.
4.2
SIGN CONVENTION
Any systematic structural analysis requires a consistent sign convention. The analysis of prestressed concrete 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 that a computer implementation of any of the analytical procedures considered here should not be contemplated 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 or clearly indicated in the text.
• Stress and force: Stress and force are both taken positive when tensile and negative when compressive. It should be noted that many authors use the opposite convention, i.e. tension negative and compression positive, the reason being that since prestressed concrete beams are normally 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 which positive moment corresponds to a concave deflected shape of the beam while negative moment corresponds to a convex deflected shape, as shown in Fig. 41.
+M
+M
M Positive bending moment Figure 41:
Sign convention for bending moment.
M
Negative bending moment
42
DESIGN FOR FLEXURE
• Section properties: The dimensions, area A and the second moment of area about the centroidal axis I of the section are always taken positive. The eccentricity e of the prestressing force is always measured from the centroid of the section and is taken positive below the centroidal axis (see Fig. 42). The sign of the section modulus Z=I/y with respect to a particular fibre is determined by the distance y of the fibre measured from the centroidal axis. This distance is taken positive for fibres located below the centroidal axis (see Fig. 42).
Centroidal axis
x
e, y Figure 42:
4.3
Axial system for section properties.
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 stressstrain relationships of the materials are known. • The relationship between the strain in the steel and the strain in the surrounding concrete is known. Each of the three basic assumptions are discussed in the following with regard to their impact on the 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 from the neutral axis. The first assumption implies that a linear relationship exists between the strain at a fibre in the concrete and its distance from the neutral axis as shown in Figure 43. A large number
ec M
k
M
y
Neutral axis
(a) Beam subjected to flexure Figure 43:
(b) Strain distribution with depth
Plane sections remain plane during bending.
ANALYSIS
43
of tests on reinforced concrete members (Ref. 41) indicate that this assumption is very nearly correct at all stages of loading up to failure, provided that good bond exists between the concrete and the steel. The assumption proves to be accurate for the concrete in the compression zone even at high loads close to the ultimate load. After cracks have developed in the tension zone, the tensile strain in the uncracked concrete between cracks is known to vary from zero at the crack to some nonzero value at positions located some distance away from the crack because of the action of bond between the steel and the surrounding concrete. Consequently the assumption that plane sections remain plane cannot be true in a cracked member when considering individual sections. However, if the gauge length for measuring strain is large enough to include a number of cracks, this assumption will hold for this “average” tensile strain (Ref. 41). The first assumption does not hold for deep beams and regions of high shear. According to SABS 0100 (Ref. 42) a simply supported beam should be considered as being deep when the ratio of 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. 43):
• Most of the conclusions were derived from the results of tests on beams with rectangular cross sections 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 this situation is not representative of conditions inside the beam.
• For nonrectangular sections, disturbances occur at points where the width of the member abruptly changes. In spite of these objections, application of this assumption by many researchers has shown that a good correlation can be obtained between calculated and measured results so that it can be considered to be accurate enough for design purposes. In lieu of an alternative, this assumption will be used. The stressstrain relationships of the materials are known. The stressstrain relationships of both the prestressed and nonprestressed steel, as presented and discussed in section 2.2, can be used. However, it is important to note that the concrete is acting in flexure and not in direct compression or tension, and the relationship used for the purposes of flexural analysis must therefore take this into account. A great deal of research has been carried out to determine the stressstrain relationship of concrete flexural elements. The most notable research was carried out by Hognestad et al (Ref. 44) and Rüsch (Ref. 45), and the following results were obtained:
• A similarity exists between the stressstrain relationships for concentrically loaded cylinders and the stressstrain 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 stressstrain relationship for beam specimens could be determined for strains much larger than the strain at which the maximum stress occurs. The determination of the stressstrain relationship for concentrically loaded cylinders beyond the cylinder strength is complicated by the fact that special testing equipment is required.
• The maximum strain e cu reached in the extreme compression fibre in bending is a function of the concrete strength, decreasing with an increase in cylinder strength. Rüsch (Ref. 45) has shown that the strain at the extreme compression fibre at maximum moment is also a function of the shape of the crosssection.
44
DESIGN FOR FLEXURE
If the above points are kept in mind, the stressstrain relationships obtained for concrete in direct compression can be applied to beams in bending. The parabolicrectangular stressstrain relationship recommended by the design codes of practice commonly used in South Africa (Refs. 42, 46, 47 and 48) is shown in Figure 44. The purpose of the 0.67 factor is to take into account the differences between the cube strength f cu and the experimentally obtained results for beams in bending. A constant value of 0.0035 is recommended for e cu and the partial factor of safety g m is discussed later. 0. 67 fcu gm
Eci Stress
E ci = 55 .
Parabolic curve
f cu GPa gm 4
e c 0 = 2.4 ´ 10
f cu gm
fcu in MPa ec0 Strain Figure 44:
ecu = 0.0035
Parabolicrectangular stressstrain relationship for concrete in flexure (Refs. 42, 46, 47 and 48).
Because the stressstrain relationship is usually difficult to determine and to deal with computationally, much research has been carried out to represent the stress distribution in the compression zone of a beam at ultimate as an equivalent rectangular stressblock. The recommendations given by SABS 0100 (Ref 42) and BS 8110 (Ref. 47) are summarized in Figure 45. It should be noted that the equivalent rectangular stressblock is only valid at ultimate, and not when considering flexural response at other levels of loading. 0. 67 fcu gm
ecu = 0.0035
ec0
x
0. 67 fcu gm
s = 0.9 x
Neutral axis
Strain distribution Figure 45:
Parabolicrectangular stress block
Equivalent rectangular stress block
Rectangular stressstrain relationship for concrete in flexure (Refs. 42 and 47).
When calculating the response of the section at ultimate, the tensile strength of the concrete is usually ignored because its influence on the moment of resistance is small. This follows because the concrete in the tension zone is usually cracked at ultimate, so that the remaining area in tension is small with a correspondingly small leverarm. The tensile strength becomes more important when calculating deformations at loadings appropriate to the serviceability limit state, and its influence on behaviour should be accounted for at these load levels.
ANALYSIS
45
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 assumption that plane sections remain plane, and the change in strain in the steel is subsequently obtained by assuming that it is equal to the calculated strain in the concrete at the level of the steel. This approach applies to all bonded steel. The distribution of the strain in unbonded tendons is assumed to be uniform along the length of the member, even though it actually varies to some degree because of the effects of friction. Under these conditions, the total change in length of the concrete at the level of the prestressing steel is assumed to be equal to the change in length of the prestressing steel. The implications of this assumption 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 momentcurvature relationship of a given beam section can be calculated over the full range of loading. This is a particularly useful relationship because it only considers the response at a section and is independent of the response of the member as a whole. Curvature k 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. 43)
k=
ec y
(41)
where e c is the strain in the concrete at a distance y from the neutral axis. The curvature is positive when the bottom of the section has an algebraically larger strain than the top of the section, and it is 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 46a. The prestressing tendons are bonded to the concrete and have material properties as shown in Figure 46b. A bilinear stressstrain relationship is assumed for the concrete as shown in Figure 46c. It is interesting to note that the exact shape of the concrete stressstrain relationship has little influence on the behaviour of a underreinforced section (see Section 4.3.5 for definition), as is considered in this example. The calculated momentcurvature diagram is shown in Figure 46e with the initial portion of the diagram enlarged in Figure 46d. 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 these points the corresponding stress distribution is also shown. It should be noted that the shape of this diagram may differ according to the choice of material properties and level of prestressing. Point A on the momentcurvature relationship indicates the point of zero moment where only the effective 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 point from which to start the calculations and the self weight will be taken into account as an external moment applied to the section. As the externally applied moment increases, the strain at the top of the section will change from tension to compression until point B is reached where the stress at the top of the section will be equal to the stress at the bottom of the section, i.e. a point of zero curvature. With a further increase in moment the stress decreases at the bottom until a point C is reached where it is zero, and the corresponding 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 until point E is reached where the tensile strength of the concrete f r is exceeded and the concrete cracks. At this stage, a point of instability is reached where the curvature will increase with an accompanying
46
DESIGN FOR FLEXURE
2000 fpu = 1710 MPa
230
307 Aps
fpy = 1489 MPa
epu = 0.06
Stress fp
1500 1000
Ep = 207 GPa 500
Aps = 100.6 mm2 fse = 813.6 MPa
0
155 (a) Crosssection
0
0.02
0.04 Strain ep
0.06
(b) Idealized prestressing steel material properties
Stress fc
30 41.4 MPa Ec = 27.6 GPa
Moment (kN.m)
20
0.00015 4.14 MPa (= fr)
Concrete cracks E fr
25
0.0015 Strain ec
0.003
Zero concrete strain D at level of steel
15
C Decompression
10
(c) Idealized concrete material properties B
5 Only prestressing force
0
Zero curvature
A0
2
2
4
6 3
Curvature (´10
8
10
1
m )
(d) Initial moment curvature response 40 35
Concrete becomes plastic (ec = 0.0015) G F
30
Steel yields
H
Plastic Cracked Elastic Cracked
25
Moment (kN.m)
Concrete fails
E 20 15
D
10
C
Elastic Uncracked
Range of service load
B
5 0 10
Enlarged above in Figure (d)
A0
10
20
30
40
50 3
Curvature (´10
60
70
80
90
1
m )
(e) Complete moment curvature response
Figure 46 :
Moment curvature response of an underreinforced beam with bonded tendons (Ref. 43).
ANALYSIS
47
40
Midspan moment (kN.m)
35 30 914
914
914
25 20
D
15
2 742
10 Deflection calculated from moment curvature relationship
5
Deflection from experimental results (Ref. 49)
0 0
10
20
30
40
50
60
Deflection D (mm) Figure 47:
Momentdeflection response of an underreinforced beam with bonded tendons (Ref. 43).
reduction of moment. Stability is subsequently regained once an increase in curvature is accompanied by an increase in moment, and beyond this point the section can sustain moments larger than the cracking moment. At point F the stress in the prestressing steel is equal to the yield stress, and at point G the stress in the concrete will reach the end of its elastic range. The maximum moment that the section can resist corresponds to the moment at point H where the concrete fails in compression. Failure of the concrete is defined as the point where the maximum concrete strain e cu is reached in the top fibre. As indicated in Fig. 46e, the section behaves as an elastic uncracked section between points A and E, and as an elastic cracked section between points E and F. Beyond point F, up to failure at point H, the response corresponds to that of a cracked plastic section. The theoretical momentdeflection curve presented in Fig. 47 was calculated on the basis of the moment curvature relationship of Fig. 46 which, in turn, makes use of the basic assumptions discussed in Section 4.3.1. The excellent agreement with the experimentally obtained curve clearly shows 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 (see Fig. 46e). As discussed in Section 4.3.2, the material response remains essentially linear elastic within this range. If the Bernoulli/Navier hypothesis that plane sections remain plane is combined with the assumption that the material behaviour is linear elastic, stresses and strains in the uncracked section 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 opposite compressive force P in the concrete acting at the same position. The stresses induced in the concrete by the prestressing force alone are subsequently calculated by considering the section to be subjected
48
DESIGN FOR FLEXURE
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 stress induced by prestressing only on a fibre located a distance y from the section centroid is given by (see Fig. 48a)
fP = where
A = area of the section
P Pe y + A I
(42)
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 P will carry a negative sign because it acts as a compressive force on the concrete. The moment Pe will also be negative (indicating a negative moment) if the cable is located below the centroidal axis of the section because e carries a positive sign in this case. Also note that y is positive for fibres located below the centroidal axis while it is negative for fibres located above the centroidal axis. The stress induced by an externally applied moment M in the uncracked concrete section is also calculated by ordinary beam theory. Using this approach, the stress induced by M on a fibre located a distance y from the section centroid is given by (see Fig. 48a)
My I
fM =
(43)
M Centroidal axis
e
c.g.s.
P Beam section
+
y
P A
Prestressing force and externally applied moment
=
+
=
P Pey + A I
Pey I
Prestressing only
My I
P Pey My + + I A I
External loadings
Total stresses
(a) At a fibre distance y from the section centroid
M
Pe M P + + Ztop Ztop A
M Ztop
ytop
Centroidal axis
e
c.g.s.
P Beam section
Pe P + A Ztop
Pe Ztop
P A
+
Prestressing force and externally applied moment
=
+
=
ybot Pe Zbot
Pe P + A Zbot
Prestressing only
M Zbot External loadings
Pe M P + + A Zbot Zbot Total stresses
(b) At the outer fibres of the section
Figure 48:
Calculation of stresses in the concrete due to prestressing and an externally applied moment.
ANALYSIS
49
It must be noted that Eq. 43 is applicable to any externally applied moment, irrespective of whether it arises from the beam self weight or an externally applied load. As in the case of Eq. 42, the sign convention must be properly observed. The total concrete stress resulting from both the prestressing force and the loads is subsequently obtained by superimposing the stresses induced by each of these effects acting on their own (see Fig. 48a). Thus, by combining Eqs. 42 and 43
f = fP + fM =
M y P Pe y + + A I I
(44)
Specifically, the extreme top and bottom fibre stresses are given by (see Fig. 48b)
where
f top =
Pe P M + + A Z top Z top
(45)
f bot =
Pe P M + + A Zbot Zbot
(46)
f top , f bot = stress in the extreme top and bottom fibres, respectively Z top = I/y top = section modulus with respect to the extreme top fibre, located a distance y top from the section centroid Z bot = I/y bot = section modulus with respect to the extreme bottom fibre, located a distance y bot from the section centroid Note that Z top carries a negative sign because the extreme top fibre lies above the centroidal axis so that y top is negative. On the other hand, Z bot can be shown to be positive because the extreme bottom fibre lies below the centroidal axis, which means that y bot is positive. EXAMPLE 41 The posttensioned simply supported concrete beam shown in Fig. 49 is subjected to a uniformly distributed load of 15 kN/m, including self weight. Calculate the extreme top and bottom fibre stresses at midspan if the tendon force is 1334 kN.
w = 15 kN/m
b = 300
300
Centroidal axis 300 c.g.s. 180 L = 12000 Figure 49:
h = 600
Section at midspan
Example 41.
The beam section properties are calculated below. Please note the use of the sign convention. 3
A = b h = 300 ´ 600 = 180 ´ 10 mm
1 1 3 3 9 4 bh = ´ 300 ´ 600 = 5.4 ´ 10 mm 12 12 h 600 =  == 300 mm 2 2 h 600 = = = 300 mm 2 2
I= ytop ybot
2
410
DESIGN FOR FLEXURE
Z top = Zbot =
I ytop I ybot
9
=
5.4 ´ 10 6 3 = 18 ´ 10 mm 300
=
5.4 ´ 109 = 18 ´ 10 6 mm3 300 2
2
The bending moment at the midspan section is given by M = w L / 8 = 15 ´ 12 / 8 = 270 kN. m and the prestressing force P = 1334 kN acts at an eccentricity e = 300  180 = 120 mm. Equations 45 and 46 are subsequently used to calculate the stresses in the extreme top and bottom fibres of the midspan section, respectively: 3
3
6
3
3
6
f top =
P Pe M 1334 ´ 10 1334 ´ 10 ´ 120 270 ´ 10 + + = . MPa + + = 1352 3 6 6 A Z top Z top 18 ´ 10 18 ´ 10 180 ´ 10
f bot =
P Pe M 1334 ´ 10 1334 ´ 10 ´ 120 270 ´ 10 . MPa + + = + + = 130 3 6 6 A Zbot Zbot 180 ´ 10 18 ´ 10 18 ´ 10
The section properties used in the above example for calculating the stresses were based on the gross concrete section. Although this approach is convenient from a practical point of view, it is not theoretically correct. Consider, for example, the case of a posttensioned bonded prestressed concrete beam: At transfer and up to the time at which the grout has hardened and become effective the tendons are not bonded to the concrete so that any loads applied to the beam at this stage, such as the prestressing force and self weight, will act on the net concrete section. Hence, the stresses induced by these loads in the concrete should be calculated on the basis of the properties of the net concrete section which take the presence of the preformed ducts, within which the tendons are contained, into account. After the grout has hardened, the tendons are effectively bonded to the concrete so that the transformed section properties must be used for calculating the stresses induced in 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 the concrete. This means that the transformed section properties should be used for calculating the stresses 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 of unbonded 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 41. It should also be noted that a distinction must be made between the transformed section properties used for calculating the stresses induced by shortterm loadings and those used for calculating the stresses induced by the longterm loads. In the case of longterm loads, the transformed section properties should, in some way, reflect the effects of creep of the concrete. This is commonly done by 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 properties as outlined above, this is not frequently done in practice. Stress calculations are usually carried out using the properties of the gross concrete section only. This approach greatly simplifies the calculations and, under normal circumstances, provides a close approximation. However, in circumstances where the area of the ducts forms a significant part of the crosssection and/or if a large quantity of steel is contained in the section, the section properties should be based on the sections as indicated in Table 41 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 must reflect the loss of prestress appropriate to the age of the beam at the time under consideration. The total loss of prestress can conveniently be divided into instantaneous losses, which take place at
ANALYSIS
Table 41:
411
Correct sections for stress calculations. Load
Pretensioned bonded
Posttensioned bonded
Posttensioned unbonded
Prestress
Transformed
Net
Net
Self weight
Transformed
Net
Net
Superimposed dead load
Transformed
Transformed
Net
Live load
Transformed
Transformed
Net
the time of transfer, and timedependent losses, which gradually develop with time. The instantaneous losses are attributed to the following sources:
• Elastic shortening of the concrete. When the prestressing force is transferred to the concrete, the concrete shortens so that tendons already bonded or anchored to the concrete also shorten by the same amount. This leads to a reduction of the stress in the tendons and, hence, a loss of prestressing force. Although this loss occurs in both pretensioned and posttensioned members, they are not affected to the same degree.
• Friction. When a tendon in a posttensioned member is tensioned, friction is induced between the 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 jacking end, represents the friction loss.
• Anchorage seating. When a posttensioned tendon is anchored to the concrete after tensioning, the components of the anchorage will deform slightly and, if the anchorages make use of wedge grips, a certain amount of slip must take place to seat the grips. The resulting loss of elongation of the tendon, termed anchorage seating, leads to a reduction of the tensioning force. This loss occurs only in posttensioning systems. The timedependent losses, which occur in both pretensioned and posttensioned members, develop with time and are attributed to the timedependent behaviour of the concrete and the steel as follows:
• Relaxation of the tendons. Because the tensioned tendons in a prestressed concrete member are continuously subjected to a large strain over the life of the member, a timedependent loss of tensioning 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 concrete member each induce a timedependent shortening in the concrete which, in turn, leads to a shortening of the attached tendons. This action results in a timedependent reduction of the stress in 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 usually taken as the moment which, by elastic theory, induces a tensile stress in the extreme fibre equal to the modulus of rupture f r . Although this approach has often been questioned, available experimental data indicate that it is sufficiently accurate (Ref. 410). The cracking moment M cr with respect to the bottom fibre is therefore determined by setting f bot = f r and M = M cr in Eq. 46, and solving the resulting expression for M cr . Hence,
f bot = f r =
M Pe P + + cr A Zbot Zbot
so that
M cr = f r Zbot  P
FG Z H A
bot
+e
IJ K
(47)
412
DESIGN FOR FLEXURE
The modulus of rupture, used in this way, merely serves as an index for measuring the load at which hair cracks, often invisible to the naked eye, start to develop. A higher load is usually necessary for visible cracks to form. However, it is important to note that once the section has cracked it can no longer be analysed as an uncracked elastic section, but that the cracked section must be considered instead. The cracking moment is usually used to mark the end of uncracked section behaviour and the onset of cracked section behaviour. Note that if the section has already been cracked in a previous loading, 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 in the extreme fibre, must be used. EXAMPLE 42 Consider the posttensioned concrete beam of example 41. Determine the cracking moment of the section at midspan if the cube strength of the concrete is f cu = 45 MPa. According to the SABS 0100 (Ref. 42) the modulus of rupture is given by f r = 0.65 f cu = 0.65 45 = 4.360 MPa. The cracking moment M cr is calculated using Eq. 47. Note the proper use of the sign convention and, in particular, that f r is assigned a positive value because it represents a tensile stress. Z M cr = f r Zbot  P bot + e A
FG H
IJ K
F 18 ´ 10 GH 180 ´ 10
6
= 4.360 ´ 18 ´ 106  (1334 ´ 103 ) ´
3
I JK
+ 120
= 372.0 ´ 106 N. mm = 372.0 kN. m
4.3.5 Ultimate moment: Sections with bonded tendons The ultimate moment of a prestressed concrete beam section is, by definition, the maximum moment which it can resist. In the case of the section considered in Fig. 46 it is represented by point H on the momentcurvature diagram. The mode in which a given prestressed concrete section with bonded tendons fails in flexure depends on the amount of steel provided, and one of the three types illustrated in Fig. 410 is possible:
• Failure induced by fracture of the steel immediately after the concrete has cracked. This failure mode is brittle and occurs in very lightly reinforced sections in which insufficient steel is provided to carry the additional tensile force which is transferred from the concrete to the steel upon cracking. This type of failure is highly undesirable and such sections are not commonly encountered in practice.
• Failure induced by crushing of the concrete compression zone after the steel has yielded and undergone a large nonlinear elongation. Sections which fail in this manner are referred to as underreinforced sections. This failure mode is ductile because the section can sustain a moment close to the ultimate moment over a wide range of deformations. Because of its ductility, this type of failure is highly desirable and most sections encountered in practice are proportioned as underreinforced sections.
• Failure induced by crushing of the concrete prior to yielding of the steel. Sections which fail in this manner are heavily reinforced and are referred to as overreinforced sections. This failure mode 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 stressstrain response of prestressing steel does not show a definite yield point as in the case of hotrolled steel reinforcing bars (see Fig. 224), it is not possible to define a precise
ANALYSIS
413
Moment
Ultimate ( fps £ fpy )
Overreinforced Underreinforced ( fpy < fps £ fpu )
Cracking Steel yields Amount of reinforcement increases
Ultimate moment less than cracking moment ( fps = fpu )
kcr
Figure 410:
fpu = Characteristic strength of the prestressing steel fpy = Defined yield stress of the prestressing steel fps = Stress in the prestressing steel at ultimate Curvature
Momentcurvature behaviour for increasing reinforcement.
limit to the percentage of reinforcement required for underreinforced failure as is possible for ordinary reinforced concrete beam sections. The most general approach to calculating the ultimate moment of a prestressed concrete beam section is to directly apply the basic assumptions listed in Section 4.3.1. In the following, this approach is developed for the calculation of the ultimate moment of a rectangular bonded prestressed concrete beam section. Figure 411 shows the assumed strain distribution as well as the stress distribution in such a section when the ultimate moment has just been reached, and the following important aspects must be noted:
• The ultimate condition is defined in terms of a limiting strain e cu being reached in the concrete at the extreme compression fibre.
• Although the compressive stress distribution in the concrete at ultimate is approximated by an equivalent rectangular stress block, the principle of the analytical procedure remains unaltered if a more exact, but more complicated, approximation is used. The codes of practice commonly used in South Africa (Refs. 42, 46, 47 and 48) set a = 0.45 or 0.4 and b = 0.9 or 1.0 (Note that the values of a given here include a partial safety factor g m = 1.5).
• The tensile strength of the concrete is neglected. This means that any concrete which falls below the neutral axis is assumed to offer no resistance to bending. a fcu
b
ecu x
h
C
x
Neutral axis
d
z Aps es Assumed strain distribution
Figure 411:
fc (x)
bx
T
Assumed stress distribution
Analysis of a rectangular bonded prestressed concrete beam section at ultimate.
414
DESIGN FOR FLEXURE
• Because the tendons are bonded to the concrete, the changes in strain in the steel are taken to be 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 all losses which have taken place at the time under consideration. Figure 412a shows that the initial strain induced by the effective prestress alone (i.e. no moment from external loads acting on the section) consists of two components:
• A tensile strain e se induced in the steel by the effective tensioning stress f se acting in the tendon, including all losses.
• A compressive strain e ce in the concrete at the level of the steel, induced by the effective prestress acting on its own. Hence the total strain in the prestressing steel is given by
e ps = e s  e ce + e se
(48)
The various components of strain are calculated as follows:
• e se is simply taken as the elastic extension of the steel acting under the effective prestress f se . Thus
e se = where
f se Ep
(49)
E p = Modulus of elasticity of the steel
• e ce is calculated by considering the prestress to be acting on the elastic uncracked section. Hence, at the level of the steel
LM P + P e OP 1 MN A I PQ E 2
e ce =
(410)
c
where
E c = Modulus of elasticity of the concrete
Note that because e ce is a compressive strain, it will carry a negative sign if the sign convention is properly applied to Eq. 410. This means that when e ce is substituted into Eq. 48 it will be
ecu
b x
h
d
Centroidal axis e
Aps ece
ese
es
ese
ece eps = es  ece + ese (a) Strain induced by effective prestress only (i.e. zero moment) Figure 412:
Strain in the steel.
(b) Strain distribution at ultimate conditions
ANALYSIS
415
added to the other strain components, as expected (see Fig. 412b). It is also important to note that the value for P in Eq. 410 must include all prestress losses at the time under consideration.
• The change in strain e s induced by the ultimate moment is obtained by applying the compatibility assumption 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. 411 and 412b),
es = where
FG d  x IJ e H x K
(411)
cu
d = effective depth of the steel, always taken positive x = depth to the neutral axis, always taken positive e cu = limiting strain in the concrete at the extreme compression fibre, specified as 0.0035 by the codes of practice commonly used in South Africa (Refs. 42, 46, 47 and 48)
Once the total strain e ps in the prestressing steel has been determined by Eq. 48, used in conjunction with with Eqs. 49 through 411, the steel stress at ultimate f ps is obtained from the stressstrain relationship of the steel as the stress corresponding to the strain e ps (see Fig. 413). The design stressstrain curve recommended by SABS 0100 (Ref. 42) for prestressed reinforcement is shown in Fig. 226, and the other design codes of practice commonly used in South Africa also prescibe similar design stressstrain curves. Although actual, experimentally determined stressstrain diagrams may be used instead, care must be taken to properly include the partial safety factor g m , depending on whether a nominal or design value of the ultimate moment is being calculated.
Stress fps
eps Figure 413:
Strain
Determining f ps from the stressstrain curve for the steel with e ps known.
The total tensile force T acting in the steel at ultimate is subsequently calculated from (see Fig. 411)
T = A ps f ps where
(412)
A ps = crosssectional area of the prestressing steel
The total compressive force acting in the uncracked compression zone of the concrete at ultimate is calculated simply by evaluating the volume of the compressive stress prism (see Fig. 411). Thus,
C= where
z
x
0
f c (x ) b dx
f c (x) = compressive stress in the fibre located a distance x from the neutral axis b = width of the section
(413)
416
DESIGN FOR FLEXURE
Note that f c (x) is negative because it represents a compressive stress and that, as for Eq. 411, x is taken positive. This means that C will carry a negative sign, which is consistent with the sign convention. Equation 413 is general and can be applied to any assumed stressstrain relationship for the concrete. If an equivalent rectangular stress block is used, Eq. 413 can be simplified as follows (see Fig. 411):
C = a f cub b x where
(414)
f cu = characteristic compressive strength of the concrete, taken negative because it represents a compressive stress a, b = stress block parameters
The ultimate moment is calculated by considering moment equilibrium of the section. Taking moments either about the line of action of C or T yields the following expressions for the ultimate moment M u (see Fig. 411):
M u = T z = C z where
(415)
z = internal lever arm
The position of the line of action of C must first be determined before the internal lever arm z can be calculated. Considering the compressive stress distribution (see Fig. 414), the following expression can be written:
zb x
Cx =
0
g
f c ( x ) b dx x
Substituting for C from Eq. 413 and rearranging terms yields the following expression for the distance from the neutral axis to the line of action of C:
zb z x
x =
g
f c (x )bdx x
0
x
0
f c (x )bdx
From Figs. 411 and 414 it is clear that the internal lever arm z is given by
zb z x
z = d  x+x = d  x+
g
f c (x )bdx x
0
x
0
(416)
f c (x )bdx
C
x
fc (x)
x
x
Neutral axis Figure 414:
Line of action of resultant compression force C.
ANALYSIS
417
Thus, to summarize:
M u = T z =  C z ( from Eq. 415) with
T = A ps f ps ( from Eq. 412) C=
z
x
0
f c (x )bdx ( from Eq. 413)
zb z x
z = d  x+x = d  x+
g
f c (x )bdx x
0
x
0
( from Eq. 416) f c (x )bdx
These expressions are general and can be applied to any assumed stressstrain relationship for the concrete. If an equivalent rectangular stress block is used they can be simplified by substituting Eq. 414 for C in Eq. 415 and by recognising that, in this case, z = d  b x / 2 (see Fig. 411). Hence,
FG H
M u = A ps f ps d  b
IJ K
FG H
x x = a f cub b x d  b 2 2
IJ K
(417)
An inspection of Eqs. 48 and 411 through 417 will reveal that all the quantities represented by these equations, which includes the ultimate moment, can be directly calculated if the value of the neutral axis depth x at ultimate is known. Any solution technique should therefore initially be aimed at calculating x, after which T, C, z and M u can be calculated. The iterative procedure presented below follows this approach, and is recommended for use when more complicated approximations of the stressstrain 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 e ps using Eqs. 48 to 411.
(c)
Obtain the magnitude of the steel stress f ps corresponding to the strain e ps using the stressstrain relationship for the steel.
(d)
Calculate the magnitude of T using Eq. 412.
(e)
Determine the magnitude of C from Eqs. 413 or 414, as appropriate.
(f)
The correct value of x will ensure that horizontal equilibrium is satisfied. Therefore, if the relationship T + C = 0 is satisfied, the value of x currently selected is correct and the ultimate moment M u can be calculated as indicated in the next step. However, if this expression is not satisfied, a revised value must be selected for x and steps (b) through (f) repeated.
(g)
Calculate the ultimate moment M u using Eqs. 415 and 416 together with the current values of T, C and x, or by using Eq. 417 together with the current value of x, as appropriate.
If an equivalent rectangular stress block is used for the concrete together with a simple approximation of the stressstrain 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 dependent on the complexity of the stressstrain curve assumed for the steel. Consider, for example, the case where an equivalent rectangular stress block is used in conjunction with a trilinear approximation
418
DESIGN FOR FLEXURE
of the stressstrain curve for the steel (see Fig. 415). Horizontal equilibrium provides the following expression
T +C = 0 Substituting Eqs. 412 and 414 into the above expression yields
A ps f ps + a f cu b b x = 0
(418)
Stress, fps fpy fp1
E p2 =
f py  f p1 e py  e p1
d
f ps = f p1 + E p 2 e ps  e p1
i
fps = Ep eps Ep ep1 Figure 415:
epy
Strain, eps
Trilinear approximation of the stressstrain curve for the steel.
Inspection of Eq. 418 will reveal that if f ps is either known or expressed as a function of x, then a closed formed solution can be found for x. It is also clear from Fig. 415 that the particular expression to be used for f ps depends on whether e ps is smaller than e p1 , whether it is larger than e py , or whether it lies between e p1 and e py , and that f ps can be expressed in terms of e ps as follows:
R f = Sf TE
py
f ps
where
d
+ E p 2 e ps  e p1 p e ps
p1
i
for e ps ³ e py
(419a)
for e p1 < e ps < e py for e ps £ e p1
(419b) (419c)
E p = Modulus of elasticity of the steel E p2 =
f py  f p1 e py  e p1
Equations 419b and 419c express f ps in terms of e ps and must therefore be expanded to express f ps in terms of x. This is done by writing e ps as a function of x and substituting the result into each of 419b and 419c. Combining Eqs. 48 and 411 yields
e ps = e s  e ce + e se = e se  e ce +
FG d  x IJ e H x K
cu
c
h
= e se  e ce  e cu +
d e x cu
(420)
Substitution into 419b and 419c, and rearranging terms lead to
R f =S  f T
f ps
where
f s1 =
s1 + E p 2 s1 + E p
R f + E ce  e  e SE be  e  e g T p1 p
p2
se
se
ce
ce
cu
e cu
cu
FG d IJ H xK FG d IJ H xK
e cu
 e p1
h
for e p1 < e ps < e py
(421a)
for e ps £ e p1
(421b)
for e p1 < e ps < e py for e ps £ e p1
ANALYSIS
419
Since the expression to be used for f ps depends on the magnitude of e ps relative to e p1 and e py , the solution for x will also depend on the magnitude of e ps . Thus For e ps ³ e py Substituting Eq. 419a into Eq. 418 and solving for x yields A ps f py x =a f cub b
(422a)
For e ps < e py The following quadratic equation, which can be directly solved for x, is obtained if either of Eqs. 421a or 421b is substituted into Eq. 418:
F a f b b I x +a f f x +b E e d g = 0 GH A JK cu
2
s1
cu
(422b)
ps
where
E = E p2 for e p1 < e ps < e py = E p for e ps £ e p1 f s1 is defined in Eq. 421a for e p1 < e ps < e py and in Eq. 421b for e ps £ e p1
Equations 422a and 422b are used as follows to calculate the correct value of x: (a)
Make an assumption about the range within which e ps lies and calculate x using either Eq. 422a or 422b, as appropriate.
(b)
Using this value of x calculate e ps (Eqs. 48 through 411).
(c)
If e ps calculated in step (b) falls within the range assumed in step (a), then the calculated value of x is correct. Otherwise, the assumption made in step (a) is incorrect and the process must be repeated from step (a) with a revised assumption regarding the range within which e ps falls.
Once the correct value of x has been determined, as outlined above, the ultimate moment can be directly calculated from the second part of Eq. 417. EXAMPLE 43 The rectangular prestressed concrete beam section shown in Fig. 416 contains six 12.9 mm 7wire super grade strand, the centre of gravity of which is located 60 mm above the beam soffit. The material properties are: Concrete:
f cu = 50 MPa
E c = 34 GPa
Steel:
f pu = 1860 MPa
E p = 195 GPa 2
The properties of the uncracked beam section are listed in Fig. 416 and A ps = 6 ´ 100 = 600 mm . At the time under consideration, f se = 1150 MPa. Make use of the equivalent rectangular stress block as well as the design stressstrain curve for strand as prescribed by SABS 0100 (Ref. 42) to calculate the design ultimate moment of the section using (a) the iterative procedure, and (b) the approach whereby the depth to neutral axis x is directly calculated in closed form. For the equivalent rectangular stress block prescribed by SABS 0100, a = 0.45 and b = 0.9, as shown in Fig. 416, while the design stressstrain curve recommended for the strand considered here is shown in Fig. 417 for g m = 1.15. Referring to Figs. 226 and 415, f pu 1860 f py = = = 1617 MPa gm 115 .
420
DESIGN FOR FLEXURE
ecu = 0.0035
b = 350
Neutral axis
0.45 fcu C
0.9 x
x
e = 240 mm 3
d = 540
h = 600
A = 210 ´ 10 mm
T
2
I = 6.3 ´ 109 mm 4
es
60 Aps = 600 mm2 Figure 416:
Example 43.
Stress, fps fpy = 1617 MPa fp1 = 1294 MPa
ep1 = 0.00664 Figure 417:
epy = 0.01329
Strain, eps
Stressstrain curve for the steel.
e py = f p1 = e p1 =
f py Ep
+ 0.005 =
0.8 f pu gm f p1 Ep
=
1617 195 ´ 10 3
+ 0.005 = 0.01329
0.8 ´ 1860 = 1294 MPa . 115
=
1294 195 ´ 10
3
= 0.00664
(a) Iterative approach. Assume x = 300 mm The magnitude of e ce is calculated on the basis of the effective prestress P =  f se A ps = 1150 ´ 3 = 690 kN acting on the elastic uncracked section (including all losses at the time 600 ´ 10 under consideration). Therefore, from Eq. 410
F P + Pe I 1 = F 690 GH A I JK E GH 210 ´ 10 2
e ce =
3
+
690 ´ 240 9
2
I 1 = 0.000282 JK 34
6.3 ´ 10 c The elastic extension of the steel acting under the effective prestress is given by
f se 1150 = = 0.005897 3 Ep 195 ´ 10 while the change in strain e s induced by the ultimate moment is obtained from Eq. 411 e se =
es =
FG d  x IJ e H x K
cu
=
FG 540  300 IJ 0.0035 = 0.0028 H 300 K
ANALYSIS
421
Therefore, the total strain in the steel at ultimate is given by Eq. 48 as
e ps = e s  e ce + e se = 0.0028  (0.000282) + 0.005897 = 0.00898 The steel stress at ultimate is subsequently obtained from Fig. 417 as the stress corresponding to a strain e ps = 0.00898. Hence, f ps = 1408 MPa. Note that, in this particular case, f ps could also have been directly calculated from Eq. 419b. T can now be determined by Eq. 412 3
T = A ps f ps = 600 ´ 1408 ´ 10
= 844.8 kN
Because an equivalent rectangular stress block is being used for the concrete, C is calculated by Eq. 414 C = a f cub b x = 0.45 ´ 50 ´ 103 ´ 350 ´ 0.9 ´ 300 = 2126 kN
d
i
Therefore, T + C = 1281 kN ¹ 0 which means that horizontal equilibrium is not satisfied for the selected value of x (= 300 mm). Because the magnitude of C is larger than that of T, a smaller value 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:
e ps = 0.02158 f ps = 1617 MPa T = 970.4 kN C = 708.8 kN . kN T + C = 2617 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 selected for x. The results obtained for various selected values of x are presented in Table 42 while Fig. 418 shows a plot of T, çCç and T + C versus x. Inspection of Table 42 and Fig. 418 reveals that for x = 136.9 mm the condition T + C = 0 is satisfied for all practical purposes, which means that this value of x is correct because it satisfies horizontal equilibrium. Substituting x = 136.9 mm and f ps = 1617 MPa (see Table 42) into the first of Eq. 417 finally yields the magnitude of the design ultimate moment. The reader should verify that the second of Eq. 417 yields the same result.
FG H
M u = A ps f ps d 
IJ K
FG H
IJ K
bx 0.9 ´ 136.9 6 = 600 ´ 1617 540 ´ 10 = 464.3 kN. m 2 2
(b) Using the closed form solution of x. Assume e p1 < e ps < e py For this case, x is calculated from Eq. 422b. Before this equation can be set up, E p2 and f s1 have to be calculated from Eqs. 419 and 421a, respectively. Note that e se = 0.005897 and e ce = 0.000282, as calculated in part (a) of this example. Thus
E p2 =
f py − f p1 ε py − ε p1
=
b1617 − 1294g × 10
−3
0.01329 − 0.00664
= 48.58 GPa
422
DESIGN FOR FLEXURE
Table 42:
Calculation of x.
x (mm)
e ps
f ps (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
2500 2000
 C
Internal forces (kN)
1500 T
1000 500 0 500
T+C
1000 x = 136.9 mm 1500 100
120
140
160
180
200
220
240
260
280
300
x (mm) Figure 418:
Variation of the internal forces as a function of x.
c
f s1 = f p1 + E p 2 e se  e ce  e cu  e p1
h
b
= 1294 + 48.58´103 0.005897  (0.00028) 0.0035  0.00664
g
= 1102 MPa Substituting these results into Eq. 422b, the following expression is obtained:
F a f b b I x +a f f x +c E GH A JK F 0.45´ (50)´ 350´ 0.9 IJ x =G H K 600
0=
cu
2
s1
p2
e cu d
h
ps
= 1181 . x 2 +1102 x + 91814
2
c
+ (1102 ) x + 48.58´103 ´ 0.0035 ´ 540
h
ANALYSIS
423
Solving for x yields x = 146.4 mm. Check if e ps corresponding to this value of x falls within the range assumed. Using Eqs. 48 and 411
FG d  x IJ e  e + e H x K F 540  146.4 IJ 0.0035  (0.00028) + 0.005897 =G H 146.4 K
e ps =
cu
ce
se
= 0.01559 This value of e ps is larger than e py = 0.01329. The assumption that e ps is less than e py is therefore incorrect,which means that the calculated value of x is also incorrect. Assume e ps > e py If e ps > e py then x can be directly calculated from Eq. 422a. Thus
FA GH a f
ps
x =
I = FG 600 ´ 1617 IJ = 136.9 mm J bb K H 0.45 ´ (50) ´ 350 ´ 0.9 K
f py
cu
As before, e ps corresponding to this value of x must be checked. Hence,
e ps =
FG d  x IJ e H x K
cu
 e ce + e se =
FG 540  136.9 IJ 0.0035  (0.00028) + 0.005897 H 136.9 K
= 0.01648 This value of e ps is larger than e py = 0.01329, as assumed, and therefore the calculated value of x is correct. The ultimate moment is subsequently calculated from the second of Eq. 417.
FG H
M u = a f cub b x d 
bx 2
IJ K
FG H
= 0.45 ´ (50) ´ 350 ´ 0.9 ´ 136.9 540 
IJ K
0.9 ´ 136.9 6 ´ 10 = 464.2 kN. m 2
Note that these results are exactly the same as obtained in part (a) of this example. It should also be noted that a substantial amount of numerical work can be avoided if these calculations are started by assuming e ps to be larger than e py , because this condition is often satisfied by the beam sections encountered in practice. The solution presented here initially considered the case where e p1 < e ps < e py simply to give a more complete illustration of the solution procedure.
Example 43 amply demonstrates that the computational effort required for the calculation of the ultimate moment capacity M u of a prestressed concrete beam section can be significantly reduced if the procedure for determining the steel stress at ultimate f ps can be simplified because once f ps is known, the depth to neutral axis x and, hence, M u can be directly calculated. Most design codes of practice provide a simplified approximate procedure for estimating f ps , and the procedure recommended by SABS 0100 (Ref. 42), which is the same as that recommended by BS 8110 (Ref. 47), is presented in the following and illustrated by example 44. The method uses the equivalent rectangular stress block prescribed by SABS 0100 and assumes that the effective prestress f se does not exceed 0.6f pu . It is directly applicable to sections of which the compression zone, measured to a depth of 0.9x, is rectangular. The design ultimate moment is calculated by the following expression (given in the notation used in these notes):
b
M u = A ps f ps d  d n
where
g
d n = 0.45x x = depth to neutral axis, as obtained from Table 43 f ps = design tensile stress in the tendons at failure, as obtained from Table 43
(423)
424
DESIGN FOR FLEXURE
Table 43:
Conditions at the ultimate limit state for rectangular beams with pretensioned tendons or posttensioned tendons having effective bond (Ref. 42).
f pu Aps f cu bd
Design stress in tendons as a proportion of the design strength, f ps 0.87 f pu
Ratio of depth of neutral axis to that of the centroid of the tendons in the tension zone, x/d
f se f pu =
f se f pu =
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
EXAMPLE 44 Use the approximate method recommended by SABS 0100 to calculate the design ultimate moment of the prestressed concrete beam section of example 43. For the beam of example 43 f se / f pu = 1150 / 1860 = 0.6183 , which is slightly larger than 0.6. Therefore, when using Table 43 f se /f pu is set to 0.6, which is the maximum value provided for by the method. Also,
f pu A ps f cu bd
=
1860 ´ 600 = 01181 . 50 350 ´ 540
Interpolating between the values given in Table 43 for f pu A ps / f cu b d equal to 0.1 and 0.15 at f se /f pu = 0.6, the following results are obtained:
f ps 0.87 f pu
= 10 .
x = 0.26 d
and
Therefore, f ps = 1.0 (0.87 ´ 1860) = 1618 MPa and x = 0.26 ´ 540 = 140.4 mm. The design ultimate moment is calculated from Eq. 423
a
f
M u = A ps f ps d  0.45 x = 600 ´ 1618(540  0.45 ´ 140.4) ´ 106 = 463.0 kN. m This result is very close to the ultimate moment M u = 464.3 kNm obtained by the more elaborate procedures employed in example 43.
So far, only rectangular sections have been considered. In the analysis of flanged sections (I or Tsections) a distinction is made between the case where the compression zone falls entirely within the flange and the case where it extends down into the web (see Fig. 419). If the compression zone
ANALYSIS
425
is entirely contained within the flange (see Fig. 419a), then the analysis is exactly the same as for a rectangular section of the same width b and the equations developed above for a rectangular section apply 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 can be of any shape.
b
b bx
x
hf
bw
(a) Rectangular section behaviour Figure 419:
bx x
bw
(b) Flanged section behaviour
Flanged section at ultimate.
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 analysis is based remain unaltered, the equations developed above for a rectangular section, which do not involve the assumption that the compression zone is rectangular, remain valid. Specifically, Eqs. 48 to 412, 415 and 419 to 421 remain valid because their derivation is not dependent on the shape of the compression zone. The remaining expressions must be modified to account for the nonrectangular compression zone. The solution procedure, which is illustrated by example 45, is exactly 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 may arise where the depth to neutral axis x is larger than the flange thickness h f , but that the depth of the stress block bx is less than h f . 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. This approach implies that the depth of the equivalent rectangular stress block bx is taken as the depth of the compression zone for the purpose of calculating the compressive force in the concrete at ultimate. EXAMPLE 45 Determine the design ultimate moment of the Ishaped prestressed concrete beam section shown in Fig. 420. The section contains eight 12.9 mm 7wire super grade strand, of which the centre of gravity is located 60 mm above the beam soffit. The material properties are: Concrete:
f cu = 50 MPa
E c = 34 GPa
Steel:
f pu = 1860 MPa
E p = 195 GPa
Use the equivalent rectangular stress block as well as the design stressstrain curve for strand as prescribed by SABS 0100 (Ref. 42), and assume that f se = 1060 MPa at the time under consideration. The properties of the uncracked beam section are listed in Fig. 420 and A ps = 8 ´ 100 = 800 mm². For the equivalent rectangular stress block prescribed by SABS 0100, a = 0.45 and b = 0.9, as shown in Fig. 420. Since f pu and E p of the strand considered here are the same as for that used in example 43, the stressstrain curve for the steel is also the same, and is as shown in Fig. 417.
426
DESIGN FOR FLEXURE
ecu = 0.0035
b = 350
Cf
hf = 150 h= 700
0.45 fcu
x
s = 0.9 x
Neutral axis
Cw
d = 640
bw = 150
e = eccentricity of the tendon 700 = − 60 = 290 mm 2
Aps = 800 mm2 T
150
es
350
Figure 420:
A = 165 × 10 mm 3
60
2
I = 8.938 × 10 mm 9
4
Example 45.
Assume e ps > e py If e ps > e py (= 0.01329) then, from Fig. 417, f ps = f py = 1617 MPa, and the total tensile force acting in the steel at ultimate T is given by Eq. 412:
T = A ps f ps = 800 ´ 1617 ´ 103 = 1294 kN In order to check whether or not the compression zone extends into the web, the maximum compression force C fmax which can be supplied by the flange only is calculated and compared to T.
C fmax = α f cubh f = 0.45 × −50 × 10 −3 × 350 × 150 = −1181 kN
e
j
Therefore, the magnitude of C fmax is less than that of T, which means that the compression zone must extend into the web to satisfy horizontal equilibrium. For convenience, the total compressive force acting in the concrete is divided into a part C f which acts in the overhanging portion of the flange and a part C w which acts in the web, as shown in Fig. 420. Thus,
C f = α f cu b − bw h f = 0.45 × −50 × 10 −3 × 350 − 150 × 150 = −675 kN
c
h
j b
e
g
Cw = a f cubw b x
(424) 3
d
= 0.45 ´ 50 ´ 10
i
´ 150 ´ 0.9 x
= 3.038 x The following condition must be satisfied to ensure horizontal equilibrium:
T + C f + Cw = 1294 − 675 − 3.038 x = 0 Solving for x yields x = 203.8 mm. Therefore s = b x = 0.9 ´ 203.8 = 183.4 mm is greater than h f = 150 mm, as expected. Before the ultimate moment can be calculated, the validity of the initial assumption that e ps is greater than e py must be checked. e ps is calculated by combining Eqs. 48 through 411, and by noting that the effective prestress acting on the section (including all losses 3 =  848 kN. at the time under consideration) is given by P =  A ps f se =  800 ´ 1060 ´ 10 Hence, f 1060 = 0.005436 e se = se = 3 Ep 195 ´ 10
F P + Pe I 1 = F 848 GH A I JK E GH 165 ´ 10 2
e ce =
c
3
+
848 ´ 2902 8.938 ´ 10
9
I 1 = 0.000386 JK 34
ANALYSIS
427
es =
FG d  x IJ e H x K
cu
=
FG 640  2038. IJ 0.0035 = 0.007493 H 2038. K
e ps = e s  e ce + e se = 0.007493  (0.000386) + 0.005436 = 0.01332 This value of e ps is larger than e py = 0.01329, as assumed, and therefore the calculated value of x is correct. The ultimate moment is subsequently calculated by considering moment equilibrium about the line of action of T. Thus,
FG H
M u =C f d 
IJ  C FG d  b x IJ H 2K 2 K
hf
w
The magnitude of C w to be used in the above expression is found by substituting x = 203.8 mm into Eq. 424, while the magnitude of C f =  675 kN remains unchanged. Therefore,
FG H
M u = (675) 640 
IJ K
FG H
IJ K
150 0.9 ´ 2038 . ´ 103  (3.038 ´ 2038 . ) 640 ´ 103 2 2
= 720.7 kN. m If the section contains nonprestressed reinforcement A s (often referred to as slack reinforcement), the procedure for calculating the ultimate moment remains exactly the same as for the sections considered above, the only difference being that the two types of steel are considered separately as shown in Fig. 421. When calculating the tension in the prestressing steel T ps and in the nonprestressed steel T s , the difference in the strain histories of the two types of steel must be accounted for in the analysis. Equation 412 can be used for calculating T ps , where f ps corresponds to the total strain e ps = e s1  e ce + e se (see Fig. 421 and Eq. 48). On the other hand, T s is calculated from the stress f s , corresponding to the strain e s2 (see Fig. 421), acting on the area A s . Therefore,
Ts = As f s
(425)
ecu
b
a fcu C
bx
x Neutral axis d2 d1 ese  ece Aps As
es1 es2 Strain distribution
Figure 421:
Tps Ts Resultant forces
Analysis at ultimate of a prestressed concrete beam section containing nonprestressed reinforcement.
It is important to note that f ps and f s cannot be obtained from the same stressstrain relationship, but that they must be determined from the stressstrain curves which apply to the prestressing steel and to the nonprestressed steel, respectively. Figure 222 shows the design stressstrain relationship prescribed by SABS 0100 (Ref. 42) for nonprestressed reinforcement. The procedure for calculating the ultimate moment of a prestressed concrete beam section containing nonprestressed reinforcement is illustrated by example 46.
428
DESIGN FOR FLEXURE
EXAMPLE 46 Determine the design ultimate moment of the Ishaped prestressed concrete beam section shown in Fig. 422. The dimensions of the section as well as the material properties of the concrete and the prestressing steel are exactly the same as for the section of example 45. However, this section contains two Y20 nonprestressed reinforcing bars in addition to five 12.9 mm 7wire super grade strand, the position of which is shown in Fig. 422. Take f y = 450 MPa and E s = 200 GPa for the nonprestressed reinforcement. Use the equivalent rectangular stress block as well as the design stressstrain curves for strand and for nonprestressed reinforcement as prescribed by SABS 0100 (Ref. 42), and assume that f se = 1116 MPa at the time under consideration. A ps = 5 ´ 100 = 500 mm² and A s = 628 mm². ecu = 0.0035
b = 350 Neutral axis
bw = 150
x
e = eccentricity of the tendon 700 = − 60 = 290 mm 2
150 60 50
A = 165 × 10 mm 3
es1 350
C
0.9 x
d2 = 650
d1 = 640
h = 700
hf = 150
0.45 fcu
Tps Ts
es2
2
I = 8.938 × 10 mm 9
4
mm2
As = 628 Aps = 500 mm2
Figure 422:
Example 46.
As for example 45, a = 4.5 and b = 0.9 while the stressstrain curve for the prestressing steel is as shown in Fig. 417. The design stressstrain curve for the nonprestressed reinforcement is shown in Fig. 423, and is obtained by setting f y = 450 and g m = 1.15 MPa in Fig. 222. Therefore, fy f sy 450 3913 . f sy = = = 3913 = = 0.00196 . MPa and e sy = gm 115 Es . 200 ´ 10 3
Stress, fs fsy = 391.3 MPa
esy = 0.00196 Figure 423:
Strain, es
Stressstrain curve for the nonprestressed reinforcement.
Assume e ps > e py and e s2 > e sy If e ps > e py (= 0.01329) then f ps = f py = 1617 MPa (see Fig. 417), while f s = f sy = 391.3 MPa for e s2 > e sy (= 0.00196) (see Fig. 423). T ps and T s are subsequently calculated from Eqs. 412 and 425, respectively:
ANALYSIS
429
Tps = A ps f ps = 500´1617 ´103 = 808.7 kN Ts = As f s = 628´ 3913 . ´103 = 245.7 kN The magnitude of the maximum compression force which can be supplied by the flange
C fmax = α f cubh f = 0.45 × −50 × 10
e
−3
j × 350 × 150 = −1181 kN
is larger than the total tensile force which can be provided by the prestressed and nonprestressed reinforcement T = Tps + Ts = 808.7 + 245.7 = 1054 kN . This means that the entire compression zone is contained in the flange, and that C can be expressed as a function of x as follows (see Eq. 414): C ( x ) = a f cubb x = 0.45 ´ 50 ´ 103 ´ 350 ´ 0.9 ´ x = 7.088 x kN
d
i
As in the previous examples, x is calculated by considering horizontal equilibrium, according to which the following condition must be satisfied:
T + C ( x ) = 1054  7.088 x = 0 Solving for x yields x = 148.8 mm. Therefore s = 0.9 x = 0.9 ´ 148.8 = 133.9 mm is less than h f = 150 mm, as expected. Before the ultimate moment can be calculated, the validity of the initial assumption that e ps > e py and that e s2 > e sy must be checked. e ps is calculated by combining Eqs. 48 through 411, and by noting that the effective prestress acting on the section (including all losses 3 at the time under consideration) is given by P =  f se A ps = 1116 ´ 10 ´ 500 = 558 kN . Hence,
e se =
f se 1116 = = 0.005723 3 Ep 195 ´ 10
F P + Pe I 1 = F 558 + 558 ´ 290 I 1 = 0.000254 GH A I JK E GH 165 ´ 10 8.938 ´ 10 JK 34 F d  x IJ e = FG 640  148.8 IJ 0.0035 = 0.01156 =G H 148.8 K H x K 2
2
e ce =
3
9
c
e s1
1
cu
e ps = e s1  e ce + e se = 0.01156  (0.000254) + 0.005723 = 0.01753 e s2 is calculated by considering the strain distribution (see Fig. 422). Thus, considering similar triangles: e s2 =
FG d H
2
IJ K
FG H
IJ K
x 650  148.8 e cu = 0.0035 = 0.01179 148.8 x
From the above it is clear that e ps is larger than e py = 0.01329 and that e s2 is larger than e sy = 0.00196, as assumed, and therefore the calculated value of x is correct. The ultimate moment is finally calculated by considering moment equilibrium about the line of action of C. Thus,
FG H
M u = Tps d1 
IJ K
FG H
bx bx + Ts d 2 2 2
IJ K
The magnitudes of T ps and T s to be used in the above expression are as calculated above. Upon substitution of these values:
FG H
M u = 808.7 640 
IJ K
FG H
IJ K
0.9 ´ 148.8 0.9 ´ 148.8 ´ 103 + 245.7 650 ´ 103 2 2
= 606.7 kN. m As previously discussed in this Section, underreinforced sections are desirable because they exhibit a gradual ductile failure with large accompanying deformations, as opposed to overreinforced sections which fail suddenly in a brittle manner with small accompanying deformations. Ductile
430
DESIGN FOR FLEXURE
behaviour is generally characterised by a large curvature at failure k u relative to the curvature at first yielding of the steel k y (see Fig. 410). The conditions under which a prestressed concrete beam section will exhibit a large value of k u and, hence, a more ductile behaviour, can be identified by realising that the magnitude of k u is increased if the magnitude of x is reduced. This trend becomes apparent when considering the expression k u = çe cu ç / x: If x is decreased, then k u must increase provided e cu is taken to remain constant. Figure 424 illustrates this effect and also demonstrates that the change in strain e s induced by the ultimate moment and, therefore, that the total strain in the steel at ultimate is increased when x is decreased. Note that although e cu generally does not vary much for normal strength concrete, its magnitude can vary significantly in the case of very high strength concrete. ecu xb
kub kua
xa
xa > xb kua < kub esa < esb
esa esb
Figure 424:
Influence of depth to neutral axis x on the curvature at ultimate k u .
The manner in which the steel content, the strength of the steel and the concrete strength influence the magnitude of x can, in turn, be demonstrated by considering Eq. 422a (which was derived by considering horizontal equilibrium): A ps f py x =(422a) a f cubb Note that although this equation was derived on the basis of an equivalent rectangular stress block and on the assumption that the stressstrain 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 assumes that stress in the steel at ultimate is equal to the yield stress and, hence, that the strain in the steel at ultimate exceeds the yield strain. Inspection of Eq. 422a reveals that:
• The magnitude of x is increased if the amount of steel A ps is increased. • The magnitude of x is increased if the strength of the steel, as reflected by f py , is increased. • The magnitude of x is decreased if the concrete strength f cu is increased. Bearing these trends in mind, together with the fact that the magnitude of k u is increased if the magnitude of x is reduced, it follows that the steel content, strength of the steel and the concrete strength influence the magnitude of k u as summarised below:
• If either the steel content or the strength of the steel is increased, x is increased and k u is decreased. Therefore, the ductility of the section is reduced under these conditions.
• If the concrete strength is increased, x is decreased and k u is increased. Under these conditions the ductility of the section is, therefore, increased. The importance of providing a section which is ductile cannot be overemphasised and, for this reason, ductility should always be a prime design consideration. The manner in which provision is
ANALYSIS
431
made for sufficient ductility of a prestressed concrete beam section in design is covered in Section 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 bonded tendons and that of beams with unbonded tendons. It is instructive to investigate some of these differences 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 assumption that 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. The change in steel strain resulting from an applied load can be calculated by making use of the fact that the total change in the elongation of the steel and of the concrete adjacent to the steel must be equal because the steel is anchored to the concrete at the ends of the beam (see Fig. 425). If the beam remains uncracked, the change in strain in the concrete at the level of the steel at any section along the span is given by (see Fig. 425)
ec = where
M ( x )e( x ) Ec I
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 total elongation of the steel, is
DL =
z
L
0
M ( x )e( x ) dx Ec I
If the effects of friction are ignored, then the steel strain must be uniformly distributed over the length of the tendon. Therefore, the change in steel strain is given by
es = where
DL = L
z
L
0
M ( x )e( x ) dx L Ec I
(426)
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 an uncracked beam is illustrated by example 47.
Centroidal axis e(x)
x
x L e, y
M(x) Bending moment Figure 425:
Change in steel strain in an unbonded prestressed concrete beam.
432
DESIGN FOR FLEXURE
EXAMPLE 47 The simply supported prestressed concrete beam shown in Fig. 426 carries a uniformly distributed load w over a span L. The cable is straight and is placed at an eccentricity e. Assume that the beam remains uncracked to compare the change in steel strain induced by the load in the case where the steel is unbonded to the change in strain in the case where the steel is bonded.
w
x
Centroidal axis e L e, y Figure 426:
Example 47.
If the cable is unbonded and free to slip, the change in steel strain is the same over the entire length of the cable, and is given by Eq. 426:
es =
DL = L
z
L
0
M ( x )e( x ) dx L Ec I
For the beam considered here:
bg wx M b xg = b L − xg 2 e x = e
Substitution into Eq. 426 yields
e s unbonded
e = L Ec I
z
L
0
wx we ( L  x )dx = 2 2 L Ec I
LM L x MN 2
2
3
x 3
OP PQ
L
0
Therefore,
ε s unbonded =
2 1 wL e 12 E c I
If the steel is bonded to the concrete, the maximum change in steel strain induced by the load will occur at the midspan section, where the moment is given by M = wL²/8. Therefore, in this case, the maximum change in steel strain is calculated from
e s bonded =
2 Me 1 wL e = Ec I 8 Ec I
Thus,
ε s unbonded =
2 ε 3 s bonded
Therefore, it can be concluded that the change in steel strain in the unbonded beam is 2/3 of the change in steel strain at the midspan section of the bonded beam, in the case considered here.
ANALYSIS
433
Example 47 clearly demonstrates that if the beam remains uncracked the relative movement between the unbonded steel and the concrete leads to a lower change in steel strain than is the case at the critical section of a bonded beam. Inspection of Eq. 426 also reveals that the magnitude of this difference 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 movement between 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 increase significantly with increasing load so that the steel strain and, hence, the steel stress increases much more gradually than would be the case at the critical section of a bonded beam. Because of this behaviour, the situation often arises in unbonded beams that the steel stress f ps is much less than its ultimate strength f pu when the limiting crushing strain e cu is reached in the concrete. Bearing in mind that the ultimate moment is calculated from M u = A ps f ps (d  bx/2), it is clear that, all other things being equal, a reduction in f ps will reduce the flexural capacity of the section. Since the stress in the steel at ultimate f ps is significantly less in an unbonded beam than in a bonded beam, it is reasonable to expect the ultimate moment of resistance M u of an unbonded beam to be less than that of the corresponding bonded beam. This is indeed the case, and the difference appears to range between 10 and 30% (Ref. 410). Another major difference between the postcracking behaviour of a beam containing no bonded steel and a bonded beam is illustrated in Fig. 427. If the beam contains only properly detailed bonded steel, many evenly distributed cracks will develop in the region of maximum moment (see Fig. 427a). At flexural failure none of the cracks will be particularly wide and the concrete compression zone will tend to fail over a relatively large length of the member, often at least equal to the effective depth of the prestressing steel (Ref. 412). This type of failure is ductile with large accompanying curvatures, rotations and deflections.
Crushing
Crushing
{
{ Many small, evenly distributed cracks (a) Bonded Figure 427:
Single large crack (b) Unbonded
Flexural failure of bonded and unbonded prestressed concrete beams.
If, on the other hand, the beam contains no bonded reinforcement then there is a tendency for it to develop a single large crack, or only a few large cracks (see Fig. 427b). Major stress and strain concentrations occur at the top of these large cracks so that flexural failure tends to be localized at a section. This behaviour reduces the ultimate moment capacity of an unbonded beam and leads to smaller average concrete strains at failure than is the case in bonded beams. The presence of nonprestressed bonded reinforcement tends to spread the flexural cracks and to limit their size, and can therefore significantly improve this undesireable behaviour. Such nonprestressed reinforcement will increase the flexural capacity of an unbonded beam, not only because of its contribution to the tensile force in the ultimate resisting couple, but also because of the advantages to be gained from the resulting improved crack control. Some design codes of practice specify minimum amounts of nonprestressed bonded reinforcement to be included in unbonded prestressed concrete beams.
434
DESIGN FOR FLEXURE
Analysis of the uncracked section Before cracking, the change in stress induced in the tendons of an unbonded beam by the external load is usually small: In fact, this stress change is even smaller than in the case of a bonded beam, as illustrated by example 47. Since, in the case of uncracked bonded sections, the change in steel stress resulting from the application of external load is normally small enough to be safely ignored in the calculation of concrete stresses, there is usually no reason why this change in steel stress should be accounted for in the case of unbonded sections. Therefore, concrete stresses in uncracked unbonded sections are calculated in exactly the same way as in uncracked bonded sections (see Section 4.3.3). However, if the total area of the preformed ducts in which the unbonded tendons are contained forms a significant part of the crosssection, the section properties to be used for the calculation 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 beam section is complicated by difficulties associated with quantifying the various factors which influence flexural behaviour after cracking, and is generally much more complex than that of a cracked bonded section. It appears that the flexural strength of an unbonded section depends on the following factors (Ref. 413):
• Magnitude of the effective stress in the tendons. • Spantodepth 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 nonprestressed bonded reinforcement. Because of the difficulties encounted in analytically treating the influence of these factors on the magnitude of the stress f ps in the steel at ultimate, the tendency has been to make use of empirical and semiempirical expressions for estimating f ps . These expressions are usually of the following general form:
where
f ps = f se + ∆f s f se = effective prestress in the steel, including all losses Df s = additional stress induced in the steel by bending of the beam under the ultimate load
Examples of such expressions may be found in SABS 0100 (Ref. 42), BS 8110 (Ref. 47) and ACI 31889 (Ref. 411). The expression prescribed by SABS 0100, which is the same as that recommended by BS 8110, is an example of a semiempirical equation, and is presented below in the notation used here:
f ps = f se +
LM N
f pu A ps 7000 . 1 − 17 l/d f cubd
OP MPa ≤ 0.7 f Q
pu
(427)
where l is normally taken as the length of the tendon between end anchorages. This equation applies to rectangular sections and flanged sections in which the compression zone is entirely contained in the flange, and was derived on the basis of an assumed length of the zone of inelasticity within the concrete of 10x. Further guidance on the reduction of l in the case of continuous multispan members can be found in SABS 0100. If the section contains nonprestressed bonded reinforcement A s ,
ANALYSIS
435
SABS 0100 suggests that the effect of this reinforcement can be approximately accounted for by adding to A ps an equivalent area of prestressing steel equal to A s f y / f pu . Once f ps is known, the depth to neutral axis x is calculated by considering horizontal equilibrium of the section, and the ultimate moment M u is subsequently calculated by considering moment equilibrium. The procedure is illustrated by example 48. It is extremely important to realize that when an expression for f ps recommended by a particular design code of practice is used, the subsequent calculations for estimating the ultimate moment must be based on the provisions of that code. Simply mixing the provisions of various codes can lead to totally misleading results. EXAMPLE 48 The simply supported concrete beam shown in Fig. 428 is posttensioned by unbonded tendons. The properties of the materials and the section at midspan are exactly the same as the section considered in example 46 (see Fig. 422), the only difference being that the tendons are unbonded in this case. Make use of the appropriate provisions of SABS 0100 to calculate the design ultimate moment of the midspan section.
w
Centroidal axis e = 290
6m
6m 12 m
Figure 428:
Example 48.
The stressstrain curve for the nonprestressed steel is as shown in Fig. 423 while a = 0.45 and b = 0.9 for the equivalent rectangular stress block. The effect of the nonprestressed reinforcement on the magnitude of f ps is accounted for by converting A s to an equivalent area of prestressing steel
A¢ps =
As f y f pu
=
628 ´ 450 . mm 2 = 1519 1860
Recognizing that the length of the tendon between the end anchorages l is virtually equal to the span of the beam (= 12 m), f ps can be directly calculated from Eq. 427:
f ps = f se +
LM MN
d
f pu A ps + A¢ps 7000 1  17 . f cubd1 l / d1
= 1116 +
i OP MPa PQ
LM N
1860 ´ (500 + 1519 . ) 7000 1  17 . 50 ´ 350 ´ 640 12000 / 640
OP Q
= 1421 MPa which is greater than 0.7 f pu = 1302 MPa. Therefore f ps = 1302 MPa. Assume e s2 > e sy (= 0.00196) so that, from Fig. 423, f s = f sy = 391.3 MPa. With f ps and f s known, the magnitudes of T ps and T s can be calculated from Eqs. 412 and 425, respectively:
Tps = A ps f ps = 500 ´ 1302 ´ 103 = 6510 . kN 3
Ts = As f s = 628 ´ 3913 . ´ 10
= 245.7 kN
436
DESIGN FOR FLEXURE
The entire compression zone is contained in the flange because the magnitude of the maximum compression force C fmax = a f cubh f = 0.45 ´ 50 ´ 103 ´ 350 ´ 150 = 1181 kN is larger than the total tensile force which can be provided by the prestressed and nonprestressed reinforcement T = Tps + Ts = 651 + 245.7 = 896.7 kN . Therefore C is given by Eq. 414 as
d
i
C( x ) = a f cub b x = 0.45 ´ 50 ´ 103 ´ 350 ´ 0.9 ´ x = 7.088 x
d
i
The depth to neutral axis is subsequently calculated by considering horizontal equilibrium, from which T + C( x ) = 896.7  7.088 x = 0
Solving for x yields x = 126.5 mm. Therefore s = 0.9 x = 0.9 ´ 126.5 = 113.9 mm is less than h f = 150 mm, as expected. The validity of the assumption that e s2 > e sy must also be checked. As for example 46, e s2 is calculated by considering the strain distribution (see Fig. 422). Thus,
e s2 =
FG d H
2
IJ K
FG H
IJ K
x 650  126.5 e cu = 0.0035 = 0.01448 x 126.5
It is therefore clear that e s2 is larger than e sy = 0.00196, as assumed, so that the calculated value of x is correct. The ultimate moment is finally calculated by considering moment equilibrium about the line of action of C. Thus,
FG H
M u = Tps d1 
IJ K
FG H
bx bx + Ts d 2 2 2
IJ K
The magnitudes of T ps and T s to be used in the above expression are as calculated above. Upon substitution of these values:
FG H
IJ K
FG H
IJ K
0.9 ´ 126.5 0.9 ´ 126.5 ´ 103 + 245.7 650 ´ 103 2 2 = 525.3 kN. m
M u = 651 640 
Note that the ultimate moment is less than M u = 606.7 kN.m obtained for the bonded beam section of example 46, as expected.
4.3.7 Flexural analysis of composite sections A composite structure is defined as a structure composed of structural elements using materials with different material properties. Composite structures in prestressed concrete typically consist of precast concrete beams with an in situ concrete slab. The beams would normally be prestressed and the slab would be of reinforced concrete. Although both elements are made of concrete, their material properties are likely to differ: The precast beams can be manufactured in a casting yard where high control of quality is possible and a strength of 50  60 MPa is feasible, while a strength higher than 30 MPa is probably not economical for the slab. A crosssection of the precast beam and the slab is called a composite section and several examples are shown in Fig. 429. It is also possible to posttension the composite element longitudinally as shown in Fig. 429f, 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. A typical application is that of a road over rail bridge where the interruption of the railway line must be limited. The precast beams are erected first and can be used to support the formwork for the slab. Once the beams are in place, permanent formwork can be placed and the slab can be cast with minimum interruption to the rail traffic since scaffolding is not required for these stages of construction.
ANALYSIS
437
In situ concrete slab Precast prestressed concrete beam (d) (a)
(e) (b)
Posttensioned tendon (c)
Figure 429:
(f)
Typical cross sections of composite beams (Ref. 414).
The analysis of a composite section can be carried out as for noncomposite sections if the following four main differences are taken into account (Ref. 414): 1.
The loading stage under consideration will determine if it is the precast section only or the composite section resisting the loads.
2.
A transformed effective flange width must be determined for the composite section to account for 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 shear resistance at the interface between the precast beam and the in situ slab is sufficient to ensure composite action.
4.
Differential shrinkage takes place between the precast beam and the in situ slab, and the tensile stresses 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 as shown in Fig. 430. Assume that no temporary supports are used during construction so that the beams support the formwork for the slab as well as the slab itself. The different loading stages and the corresponding section resisting the loads can best be determined by considering the construction procedure (see Table 44): (a)
At transfer, when only the initial prestressing force P t and the moment induced by the self weight of the beam M b are acting on the precast beam section.
438
DESIGN FOR FLEXURE
(b)
After the prestress losses have taken place, when the effective prestressing force P e together with the moment induced by the self weight of the beam M b are acting on the precast beam section.
(c)
Directly after the slab has been cast, when the slab self weight moment M f (including the weight of any formwork), the beam self weight moment M b and the effective prestressing force P e are acting on the precast beam section.
(d)
After the concrete in the slab has hardened any additional loads, such as live loads, act on the composite section. The stresses caused by the additional loads must be added to the existing stresses in the precast beam.
In situ concrete slab
=
+ Precast prestressed concrete beam
Pt + M b
Pe + M b
(a)
(b)
Mf
Pe + M b + M f (c)
=
+ ML
Pe + M b + M f + M L (d)
Figure 430:
Distribution of stress in a composite section during service (Ref. 410).
Table 44:
Loading stages on a composite section during service. Loading stage
Loads
Resisting section
(a) Tensioning of precast beam
Pt + Mb
Precast beam
(b) Precast beam after losses
Pe + Mb
Precast beam
Pe + Mb + Mf
Precast beam
Pe + Mb + Mf + ML
Composite section
(c) Casting of slab (d) Live and superimposed dead load
The stress distributions at each of these load stages are shown in Fig. 430, from which it can be seen that the critical stages are (a) and (d). Loading stage (a) occurs at transfer when the maximum prestressing force acts together with minimum loading to induce maximum compressive and tensile stresses in the bottom and top fibres of the precast beam, respectively. This loading stage has been discussed in Section 4.3.3. Loading stage (d), where the minimum prestressing force is present together with the maximum external loading, is the other critical stage. Here, the top and bottom fibres 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. the simple case shown in Fig. 429a using unpropped construction is considered here. cases the precast beam and the in situ portions can overlap (see Fig. 429c), with the two different stresses can occur at the same level in the composite section, as shown in However, the principles of analysis, as presented in this Section, can still be applied.
429, only In certain result that Fig. 431.
When a flanged beam is subjected to an applied loading, the compressive stresses in the flange will vary over the width of the flange, as illustrated in Fig. 432 for a simply supported beam. The variation is caused by shear lag effects and is dependant on several factors such as the type of loading, dimensions of the crosssection and time dependant properties of the concrete (including
ANALYSIS
439
In situ concrete slab
Precast beam
Precast prestressed concrete beam Figure 431:
In situ slab
Pt + M b
Pe + M b + M f
Pe + M b + M f + M L
(a)
(b)
(c)
Stress distribution for overlapping a composite section.
creep and shrinkage). Since an exact theoretical analysis is usually not justified, the following simplified approach is followed. The actual flange is replaced by a fictitious flange having an effective flange width b e so that it carries the same load as the actual flange. The following values for b e are recommend by local design codes (Ref. 42, 46, 47 and 48)
be = where
Rbb S bb T
g + 01 . L g£S
w
+ 0.2 Lz £ S
for T  sections
(428a)
w
z
for L  sections
(428b)
L z = distance between points of zero moment. For continuous beams L z may be taken as 0.7 times the effective span. S = the spacing of the webs, i.e. the actual flange width (See Fig. 433a)
L
Bending moments
Figure 432:
The effects of shear lag on the distribution of compressive stress in a flanged beam.
It should be noted that TMH7 (Ref. 46) places the point of zero moment at a distance of 0.15 times the effective length of the span from the support. This implies that for the end span of a continuous beam, as shown in Fig. 433b, the distance between points of zero moment L z = 0.85 L. For the analysis, the plane sections assumption is used to determine the distribution of strain in the section. The difference between the modulus of elasticity of the concrete used for the precast beam
440
DESIGN FOR FLEXURE
S be
Equivalent stress distribution Actual stress distribution
L
bw Lz = 0.85 L
S (a)
Figure 433:
(b)
Effective flange width b e .
and that used for the in situ slab leads to different stresses being induced in the section by equal strain. It is recommended that this difference be taken into account when the concrete strength differs by more than 10 MPa (Ref. 42 and 46). The calculations can be simplified by transforming one material to the other and it is generally more convenient to transform the slab material to the beam material. This is accomplished by replacing the effective width b e of the slab by a transformed width b ft as follows:
where
n c = modular ratio =
Ec, f
b ft = nc be
(429)
Ec,b
E c,f = modulus of elasticity of the concrete in the in situ slab E c,b = modulus of elasticity of the concrete in the precast prestressed beam After the section has been transformed to one material, a Tsection is used for the analysis, which proceeds as discussed in previous Sections. It is important to note that the stresses calculated in the slab on the basis of the transformed section must be transformed back to the slab material to obtain the actual stresses in the slab. This is done by multiplying the stresses in the transformed slab by the modular ratio n c . At transfer, the stresses in the concrete can be calculated by Eqs. 45 and 46 using the beam section properties because all the loads, including prestress, are resisted by the beam section only. However, at loading stage (d) (Fig. 430), superimposed loads applied after the slab concrete has hardened are 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. 438):
where
f top ,b =
Mb + M f ML Pe P + + + Ab Z top ,b Z top ,b Z top , cb
(430a)
f bot ,b =
Mb + M f ML Pe P + + + Ab Zbot ,b Zbot ,b Zbot , cb
(430b)
A b = Area of the precast beam Z top,b = I b / y top,b = Section modulus of the beam section with respect to the extreme top fibre of the precast beam, located a distance y top,b from the centroid of the precast beam Z bot,b = I b / y bot,b = Section modulus of the beam section with respect to the extreme bottom fibre of the precast beam, located a distance y bot,b from the centroid of the precast beam
ANALYSIS
441
I b = Second moment of area of the precast beam Z top,cb = I c / y top,cb = Section modulus of the composite section with respect to the extreme top fibre of the precast beam, located a distance y top,cb from the centroid of the composite section Z bot,cb = I c / y bot,cb = Section modulus of the composite section with respect to the extreme bottom fibre of the precast beam, located a distance y bot,cb from the centroid of the composite section I c = Second moment of area of the composite section EXAMPLE 49 A simply supported composite beam has a span of 15 m and the cross section shown in Fig. 434. The precast prestressed beams are spaced at a distance of 1200 mm. In addition to self weight, the beam must support an additional uniformly distributed load of 16 kN/m. Assume unpropped construction and determine the elastic stresses at the following stages: (a)
At transfer of prestress. Take P t = 1500 kN.
(b)
Just before the slab is cast, with P 1 = 1350 kN (assuming some loss has taken place).
(c)
Just after the slab has been cast, with P 2 = 1350 kN.
(d)
After a long time with no additional loads. Take P 3 = 1200 kN (assuming all the losses have taken place).
(e)
After a long time with additional loads. Take P 4 = 1200 kN. bf = 1200 Concrete material properties: Precast beam fcu,b = 50 MPa Ec,b = 34 GPa fcu,f = 30 MPa Ec,f = 28 GPa In situ slab Unit weight gc = 24 kN/m3
hf = 150 In situ slab Precast beam
d = 630
hb = 700
70 bb = 350 Cross section at midspan
Figure 434:
Composite cross section for Example 49.
The loading and corresponding maximum midspan moments are calculated in the following table: Beam selfweight Slab selfweight
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
be = bw + 0.2 Lz = 350 + 0.2 ´ 15 000 = 3 350 mm This is greater than the actual flange so that b f = 1 200 mm is used.
442
DESIGN FOR FLEXURE
To determine the section properties of the composite section the modular ratio is required nc =
Ec, f E c ,b
=
28 = 0.8235 34
The transformed flange width is calculated from
b ft = ncbe = 0.8235 ´ 1200 = 988.2 mm The distances to the various fibres of importance, measured from the centroid of the transformed section are given in Fig. 435.
ytop,b = 350
Centroid of composite section Centroid of beam section
ybot,b = 350
Figure 435:
ytop,cb = 189.8
ytop,cf = 339.8
ybot,cb = 510.2
Locations of section centroids for Example 49.
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:
3
In situ slab (transformed)
Precast beam
Composite section
148.2
245.0
393.2

10.00
26.96
2
Area (´ 10 mm ) 9
4
Second moment of area (´ 10 mm ) (a) Stresses at transfer:
f top ,bt =
b
gb
Pt e + M b hb / 2 Pt + Ab Ib
g
3 6 1500 ´ 10 ´ 280 + 165.4 ´ 10 (350) 1500 + 9 245 10 ´ 10 = 6122 + 8.908 = 2.79 MPa .
d
=
f bot ,bt =
b
i
gb
g
Pt e + M b hb / 2 Pt + = 6122 .  8.908 = 15.03 MPa Ab Ib
(b) Stresses in the beam just before casting the slab:
f top ,b1 =
b
gb
P1e + M b hb / 2 P1 + Ab Ib
g
3 6 1350 ´ 10 ´ 280 + 165.4 ´ 10 (350) 1350 + 9 245 10 ´ 10 = 5.51 + 7.439 = 193 . MPa
=
d
i
ANALYSIS
443
b
gb
g
P1e + M b hb / 2 P1 + = 551 .  7.439 = 12.95 MPa Ab Ib
f bot ,b1 =
(c) Stresses in the precast beam when slab is cast: f top ,b 2 =
ib
d
P2 e + M b + M f hb / 2 P2 + Ab Ib
g
1350 ´ 10 3 ´ 280 + (165.4 + 1215 . ) ´ 106 (350) 1350 = + 245 10 ´ 10 9 = 5.51 + 3188 = 2.32 MPa .
d
f bot ,b 2 =
i
ibh / 2g = 551 .  3188 . = 8.70 MPa
d
P2 e + M b + M f P2 + Ab Ib
b
(d) Stresses in the composite section after a long time without additional load: f top ,b 3 =
ib
d
P3e + M b + M f hb / 2 P3 + Ab Ib
g
1200 ´ 103 ´ 280 + (165.4 + 1215 . ) ´ 10 6 (350) 1200 = + 9 245 10 ´ 10 = 4.898 + 1719 . = 318 . MPa
d
f bot ,b 3 =
i
d
P3e + M b + M f P3 + Ab Ib
ibh / 2g = 4.898  1719 . = 6.62 MPa b
(e) Stresses in the composite section after a long time with additional load: In the slab
f top , f 4 =
FM GH
L
ytop , cf
I n = F 450 ´ 10 (339.8) I ´ 0.8235 JK GH 26.96 ´ 10 JK 6
c
Ic
9
= 5.671 ´ 0.8235 = 4.67 MPa f bot , f 4 =
FM GH
L
ytop , cb Ic
I n = F 450 ´ 10 (189.8) I ´ 0.8235 JK GH 26.96 ´ 10 JK 6
c
9
. = 3167 ´ 0.8235 = 2.61 MPa In the beam f top ,b4 =
d
ib
g
P4 e + M b + M f hb / 2 M L ytop , cb P4 + + Ab Ib Ic
= 3179 . +
6 450 ´ 10 (189.8) 9
26.96 ´ 10 = 3179 .  3167 . = 6.35 MPa
444
DESIGN FOR FLEXURE
f bot ,b 4 =
d
P4 e + M b + M f P4 + Ab Ib
ibh / 2g + M c y b
L
bot , cb
h
Ic
450 ´ 10 (510.2) 6
= 6.617 +
9
26.96 ´ 10 = 6.617 + 8.515 = 190 . MPa
The calculated stresses are summarized in Fig. 436.
bf = 1200 bft = 988.2 4.67 2.79
ytop,c = 339.8
2.32
1.93
3.18
2.61 6.35
Centroid of ybot,c = composite 510.2 section
12.95
15.03 (a)
Figure 436:
8.70
(b)
(c)
6.62 (d)
1.90 (e)
Stress distribution (in MPa) in the composite section of Example 49.
Differential shrinkage If the precast beam is relatively old when the in situ slab is cast, much of the creep and shrinkage of the precast beam has already taken place. Therefore, the shrinkage of the in situ slab is greater than the magnitude of the remaining creep and shrinkage of the precast beam. The resulting shortening of the slab relative to the precast beam is called differential shrinkage. Figure 437 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. 416). Assume that the beam and the slab act independently since casting of the in situ concrete, and that the following strains have occurred (see Fig. 437a): e sf = unrestrained shrinkage of the in situ slab e top,b e 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 be chosen that will correspond to the deformed shape of the beam, a vertical position is selected, as shown in Fig. 437b, to simplify the problem. To achieve this position of compatibility, a moment M b must be applied to the beam, and a tensile force F must be applied to the in situ slab. The moment M b can be calculated as follows: M b = −k b E c ,b I b
where
k b = curvature of the beam section = h b = height of the precast beam section
1 e  e top,b hb bot ,b
d
(431)
i
ANALYSIS
445
ediff
esf
esf
In situ slab
F etop,b
Position at casting Mb
Precast prestressed beam ebot,b
eavg,b
(a) Positions following unrestrained shrinkage and creep
ytop,c
hf
F
(b) Forces required for compatibility
hf /2 F
Mc
Mb
Centroidal axis of the composite section Final position (c) Forces on composite section
Figure 437:
(d) Equivalent forces on composite section
Differential shrinkage in a composite section.
The force in the slab required to take up the differential strain can be determined from
F = e diff E c, f A f where
(432)
A f = cross sectional area of the in situ slab e diff = differential shrinkage strain = e sf  e avg,b e avg,b = average strain in the beam =
de
bot , b
i
+ e top ,b / 2 for a symmetric beam
The beam and the slab can now be joined and the applied force F and moment M b cancelled by applying equal and opposite forces and moments to the composite section, as shown in Fig. 437c. It is customary to use a set of equivalent forces as shown in Fig. 437d, where
F GH
M c = F ytop , c 
I M 2 JK
hf
b
(433)
446
DESIGN FOR FLEXURE
The resulting stresses can now be determined as follows
FF GH A FF =G HA
f top , f =
M c ytop, cf F + nc Ac Ic

M c ybot , cf F + Ac Ic
ft
f bot , f
ft
f top ,b = f bot ,b =
where
I JK In JK

M b ytop ,b Ib M b ybot ,b Ib
(434a) (434b)
c

M c ytop , cb F + Ac Ic
(434c)

M c ybot , cb F + Ac Ic
(434d)
A ft = transformed cross sectional area of the in situ slab A c = cross sectional area of the composite section
ftop,f ytop,c
ybot,c
ytop,cb ybot,cf Centroid of the composite section Centroid of the precast beam
ytop,cf
fbot,f
ftop,b
ytop,b ybot,cb ybot,b fbot,b
(a) Definition of symbols
Figure 438:
(b) Stresses caused by differential shrinkage
Definition of symbols and stresses for differential shrinkage in a 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 beam prestressed by a tendon located below the centroidal axis of the section, as shown in Fig. 437, the creep and shrinkage strains will be negative with the corresponding curvature k b negative and moment M b (Eq. 431) positive. The differential shrinkage will usually be a negative value as this would indicate a shortening of the slab relative to the beam. The force F (Eq. 432) must reverse this shortening and is therefore a tensile (positive) force. Creep itself tends to relieve the stresses caused by differential shrinkage, and the following reduction factor y can be applied to these stresses to account for this effect (Ref. 46 and 48):
y= where
 b cc
1 e b cc
(435)
b cc = ratio of creep strain to the elastic strain
The value of b cc can range between 1.5 and 2.5 and an average value of 2, which results in a reduction 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 b cc must be increased to reflect the increased creep.
ANALYSIS
447
EXAMPLE 410 Calculate the differential shrinkage stresses in the composite section of example 49. Because the in situ slab is cast 6 months after the beam is cast, assume that 60% of the creep and shrinkage have already taken place in the beam at the time of casting of the slab. The following values for creep and shrinkage apply: e cr = 48 ´ 10 e sh = 310 ´ 10
6
6
MPa
1
for creep of the precast beam
for shrinkage of both the beam and slab
b cc = 1.6 Assume that the remaining creep in the precast beam takes place under the stresses present at the time 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 6 6 e cr , top = 0.4e cr f top ,b 2 = 0.4´ 48 ´ 10 (2.322) = 44.58 ´ 10 6 6 e cr ,bot = 0.4e cr f bot ,b 2 = 0.4´ 48 ´ 10 (8.698) = 167.0 ´ 10
The total strain in the precast beam including the remaining portion of the shrinkage is 6
e top ,b = e cr , top + 0.4e sh = 44.58 ´ 10
e bot ,b = e cr ,bot + 0.4e sh = 167.0 ´ 106
6
6
6
6
d i = 168.6 ´ 10 + 0.4´ d310 ´ 10 i = 2910 . ´ 10
+ 0.4´ 310 ´ 10
The curvature of the beam caused by these strains is
kb =
1 1 2910 .  (168.6) ´106 =174.9 ´106 m1 e bot ,b  e top ,b = 0.7 hb
c
b
h
g
while the moment required to rotate the beam through this curvature is
M b =k b E c,b I b =(174.9 ´106 ) ´ 34 ´10 6 ´10.00´103 = 59.49 kN.m The average strain in the precast beam is given by
e avg ,b =
1 1 6 6 e bot ,b + e top ,b = (2910 .  168.6) ´ 10 = 229.8 ´ 10 2 2
d
i
so that the differential shrinkage strain is e diff = e sf  e avg ,b = 310 ´ 106  229.8 ´ 106 = 80.21 ´ 106
d
i
The tension force applied to the slab for compatibility is given by F = e diff E c, f A f =  80.21 ´ 106 ´ 28 ´ 1200 ´ 150 = 404.2 kN
d
i
The moment M c is determined from
F GH
M c = F ytop , c 
hf 2
I M JK
b
Finally, the reduction factor for creep is
y=
 b cc
1 e b cc
=
1.6
1 e 16 .
FG H
= 404.2 339.8 
= 0.4988
IJ K
150 ´ 103  59.49 = 47.56 kN. m 2
448
DESIGN FOR FLEXURE
The stresses caused by differential shrinkage can now be calculated as follows
F F  F + M y In y GH A A JK I F 404.2  404.2 + 47.56 ´ (339.8) IJ 28 ´ 0.4988 =G H 148.2 393.3 26.96 ´ 10 K 34 c top , cf
f top , f =
c
ft
c
c
3
= 0.45 MPa
F F  F + M y In y GH A A JK I F 404.2  404.2 + 47.56 ´ (189.8) IJ 28 ´ 0.4988 =G H 148.2 393.3 26.96 ´ 10 K 34 c bot , cf
f bot , f =
c
ft
c
c
3
= 0.56 MPa
F M y  F + M y Iy GH I JK A I F 59.49 ´ (350)  404.24 + 47.56 ´ (189.8) IJ ´ 0.4988 =G 393.3 H 10 ´ 10 K 26.96 ´ 10
f top ,b =
b top , b
c top , cb
b
c
c
3
3
= 172 . MPa
F M y  F + M y Iy GH I JK A I F 59.49 ´ 350  404.2 + 47.56 ´ 510.2 IJ ´ 0.4988 =G H 10 ´ 10 393.3 26.96 ´ 10 K
f bot ,b =
b bot , b
c bot , cb
b
c
c
3
3
= 0.97 MPa These stresses must be added to those caused by all the loadings after losses, as shown in Fig. 439. It can be seen from the final stresses that it is the tensile stress in the bottom of the precast beam that is most significantly affected by differential shrinkage from the point of view of design. 4.67 2.61
0.56 1.72
6.35
=
+
1.90 (a) Stresses caused by all loadings after losses
Figure 439:
Total stresses Example 410.
4.22
0.45
0.97 (b) Stresses caused by differential shrinkage
2.87 (c) Total stress
2.05 8.07
ANALYSIS
449
Ultimate moment The flexural capacity of a composite section can be determined in the same way as for a flanged section if provision is made for the difference in strength f cu of the concrete in the precast beam and 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 f cu for the slab and for the beam impacts on the analysis of 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 basis of the strength ratio n cu (transformed section) or by making use of basic principles (untransformed section) (see Fig. 440). Consider the case where the compression zone extends into the precast beam, as shown in Fig. 440. The compression force in the slab C f and in the beam C w can be determined from the following:
Cf =
RSa f Ta f
h f be h cu ,b f b ft cu , f
for an untransformed slab width for a transformed slab width
(436a)
h
(436b)
c
Cw = a f cu,b b x  h f bb where
b ft = transformed slab width = n cu b e n cu = strength ratio = f cu,f / f cu,b f cu,f , f cu,b = characteristic concrete strength of the slab and beam, respectively
For the case where the compression zone is entirely contained in the slab (bx < h f ), Eq. 436 becomes
Cf =
RSa f Ta f
b xbe cu ,b b xb ft cu , f
for an untransformed slab width for a transformed slab width
(437a)
Cw = 0
(437b)
afcu,b afcu,f
be hf
bx
x
bft Cf
Cf
Cw
Cw
T bb (a) Original slab width Figure 440:
afcu,b
T bb (b) Transformed slab width
Flexural capacity of a composite section.
Horizontal shear In both the preceding sections dealing with the analysis of the uncracked section and with the ultimate 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 across the interface between the precast beam and the in situ slab.
450
DESIGN FOR FLEXURE
There are two commonly used methods of calculating the horizontal shear stress at the precastin situ concrete interface: The first method makes use of elastic theory while the second method considers conditions at ultimate when plastic deformations have taken place in the section. It must be noted that the restrictions on shear stresses as recommended by design codes are dependent on the method used for obtaining the interface shear stresses. Elastic theory yields the following equation for determining the horizontal shear stress v he at the interface of an uncracked section
vhe = where
VQ I c bv
(438)
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 the neutral axis of the transformed composite section I c = second moment of area of the transformed composite section b v = width of the interface
The horizontal shear stress v hu at ultimate can be determined by dividing the horizontal force in the slab that has to be transmitted across the interface by the area of the interface. Figure 441a shows the case where the compression zone is entirely contained in the slab, in which case the horizontal shear stress at the interface v hu is given by Cf a f cu , f b xbe vhu = = (439) bv Lv bv Lv where
L v = distance over which the force must be transmitted
afcu,f
be
Cf
x bx
hf
Minimum moment section 0
Maximum moment section Cf
vhu
bv T
T Lv
(a) Cross section at postion of maximum moment Figure 441:
(b) Beam elevation
Horizontal shear in a composite beam.
For the case shown in Fig. 441b, the distance L v extends from the section of maximum moment to the point of zero moment, i.e. the support. In the case where the compression zone extends into the precast beam, the compression force to be transmitted across the interface is taken as a f cu, f h f be . It should be noted that the shear stress v hu is an average stress while the limiting values recommended by design codes are given with regard to a maximum value. Recommendations exist (Refs. 47 and 423) which suggest that the maximum value of the horizontal shear stress can be obtained by distributing the average shear stress v hu along the length L v in proportion to the vertical design shear force diagram.
DESIGN
4.4
451
DESIGN
4.4.1 Limit states design All the design codes of practice normally used in South Africa for the design of prestressed concrete structures (Refs. 42, 46, 47 and 48) are based on the so called limit states design approach, which offers a rational and practical procedure for ensuring that there is an acceptable probability that the structure will remain fit for its intended use during its design life. Any condition at which a structure may become unfit for use constitutes a limit state, and the objective of the design procedure is to ensure that such a limit state is not reached. Obviously, this approach requires that each limit state must be examined separately to make sure that it has not been reached. These checks can be made either on a deterministic or on a probabilistic basis, and the codes currently used in South Africa for the design of prestressed concrete structures all adopt a probabilistic basis. This means that if the provisions of these codes are followed, each limit state is examined to establish if there is an acceptable probability of it not being reached (see Refs. 416, 417 and 424). The various limit states can be placed in one of the two following categories:
• Ultimate limit states, which are concerned with the maximum loadcarrying capacity of the structure.
• Serviceability limit states, which are concerned with the normal use and durability of the structure. The limit states listed below are those applicable to SABS 0100 (Ref. 42), and are essentially the same as those applicable to BS 8110 (Ref. 47). Ultimate limit states
• Stability: The structure must remain stable under all the critical combinations of the design ultimate loads. This requirement implies that no ultimate limit state is to be reached by rupture of any section, by overturning or by buckling.
• Robustness: The design must be robust, in the sense that the failure of a single element or damage to 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 use of a structure (e.g. flour mill or chemical plant) the design must ensure that there is a reasonable probability 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 that neither its appearance nor its performance is adversely affected. Possible damage to other elements such as finishes, services, partitions, glazing and cladding, as well as to adjacent structures is also a consideration in this regard.
• Cracking: The width of cracks must be controlled to ensure that the appearance, efficiency and durability 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 the structure, or interference with its proper function.
452
DESIGN FOR FLEXURE
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 the structure is predominantly cyclic.
• Durability: The durability of the structure must be considered in terms of its conditions of exposure and its design life. Compliance with code recommendations regarding minimum concrete cover to the reinforcement, minimum concrete strength and permissible crack width is intended to ensure that the durability requirements of most structures are satisfied.
• Fire resistance: When a structural element may be exposed to fire its retention of structural strength, 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 ultimate limit states and the design criteria must be satisfied at the serviceability limit states. In order to carry out the necessary calculations to verify compliance with these requirements, the design material strengths and the design loads must be known. The approach followed by the limit states design method to determine these quantities, as implemented in the design codes of practice (Refs. 42, 46, 47 and 48) and loading codes (Refs. 418, 419, 420 and 424) commonly used in South Africa, is briefly described in the following. Design material strengths Material strengths are specified in terms of their characteristic values, which are defined as the strength below which not more than 5% of the test results may be expected to fall (see Section 2.1.1 for the specific case of the characteristic compressive strength of concrete). If it is assumed that the measured values of strength are normally distributed, this definition can be expressed as follows:
f k = f m  164 . s where
(440)
f k = characteristic strength f m = mean strength s = standard deviation
The design material strength applicable to each limit state is derived from the characteristic strength by dividing it by a partial safety factor for material strength g m . Thus, design strength =
fk gm
(441)
The factor g m is intended to account for the following (Ref. 42):
• Possible reductions in the strength of the materials used in the actual structure as compared with the characteristic values obtained from laboratory tested specimens.
• Local weaknesses. • Inaccuracies in the assessment of the resistance of sections. The value of g m depends on the material: By the nature of its manufacturing process, concrete is a more variable material than steel. It also depends on the importance of the limit state being considered and, therefore, higher values are used for ultimate limit states than for serviceability limit states. Values recommended by SABS 0100 (Ref. 42) for g m are listed in Table 45.
DESIGN
453
Table 45:
Partial safety factors for material strength g m as recommended by SABS 0100 (Ref. 42). Limit State
Concrete
Steel
Flexure or axial load
1.50
1.15
Shear
1.40
1.15
Bond
1.40
Ultimate
Others (eg. bearing stresses)
³ 1.50
Serviceability Deflection
1.0
1.0
Cracking strength of prestressed concrete elements using tensile stress criteria
1.3
1.0
Design loads If sufficient statistical data were available, it would be possible to account for the variability of the loads 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 the various loads are not available at the present time, so that values are usually based on experience and, possibly, on forecasts of the implications of future developments. Therefore, the values usually given 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 corresponding nominal load by the appropriate partial safety factor g f . Thus, design load = nominal load ´ g f
(442)
The factor g f is intended to account for (Ref. 42):
• 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 g f depends on the following factors:
• Type of load: Higher values of g f are associated with loading types which, by their nature, are
more variable. Thus, for example, the value of g f for dead load will be less than the value for superimposed 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 g f for a particular load decreases because of the reduced probability that the various loads will all reach their nominal values simultaneously.
• Importance of the limit state: Higher values of g f are used for ultimate limit states than for serviceability limit states because of the requirement of having a smaller probability of the former being reached.
454
DESIGN FOR FLEXURE
The combinations listed below for selfweight D n , imposed loads L n and wind loads W n comply with the recommendations of SABS 0160 (Ref. 418) at the ultimate limit state: 1.5D n 1.2D n + 1.6L n 1.2D n + 0.5L n + 1.3W n 0.9D n + 1.3W n The following combinations comply with recommendations of SABS 0160 at the serviceability limit state: 1.1D n + 1.0L n 1.1D n + 0.3L n + 0.6W n It should be noted that if a particular load in a load combination has a relieving effect on the load effect being considered, most loading codes will provide a reduced value of g f to be applied to that particular load. The descriptions of the material and load factors given above are, strictly speaking, only applicable to the recommendations of SABS 0100 (Ref. 42) in this regard, which are, in principle, the same as those of BS 8110 (Ref. 47). The codes for bridge loadings commonly used in South Africa (Refs. 420 and 424), however, differ slightly from SABS 0100 and BS 8110 in the manner in which the various factors which impact on the magnitude of the design material strengths and loads are assigned to the material and load factors. The bridge loading codes also specify an additional factor g f3 with which the effects of the design loads must be multiplied to obtain the design load effects. However, the factors which all the different loading codes, referred to above, provide for in this regard are essentially the same. It is extremely important to note that the loads and load factors to be used must be obtained from the particular loading code specified by the design code of practice being followed because the provisions of a code of practice is always dependent on those of a particular loading code. The normal procedure followed in limit states design is to design on the basis of the expected critical limit state and then to examine the remaining limit states to check that they are not reached. In the flexural design of prestressed concrete members, the critical limit state depends on the limitations imposed by the limit state of cracking, which also provides the basis on which prestressed concrete elements are classified by the design codes of practice commonly used in South Africa. This classification 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 visible cracks develop.
• Class 3 (partial prestress): Tensile stresses permitted, but with surface crack widths limited to values prescribed by the particular code being used. Generally, the design of class 1 and class 2 members is governed by the serviceability limit state of cracking, while the design of class 3 members tends to be controlled by the ultimate limit state or 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 2 members, the design procedure developed in this Section only covers class 1 and 2 pretensioned as well as bonded and unbonded posttensioned members. The flexural design of class 3 (partially prestressed) members is covered in Section 4.4.6.
DESIGN
455
The criteria which limit crack width are stated in terms of limiting concrete flexural tensile stresses by the design codes of practice used in South Africa (Refs. 42, 46, 47 and 48). In addition to the limiting tensile stresses, these codes also specify maximum compressive stresses in the concrete which may not be exceeded at the serviceability limit state. Although the purpose of the compressive stress limitations are not explicitly stated in the codes, it seems reasonable to assume that they are intended to prevent the development of excessive creep strains in the concrete under serviceability conditions, and to prevent microcracking and spalling of the concrete in the compression zone under serviceability conditions. The concrete stress limitations specified by SABS 0100 (Ref. 42) are listed in Table 46. 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 limit state. This follows because these conditions can normally be identified as being the most critical.
Table 46:
Limiting concrete stresses in prestressed members, SABS 0100 (Ref 42).
Class 1 members
Class 2 members Pretensioned
Posttensioned
At transfer 1. Compression •
Triangular or near triangular distribution of prestress
0.45 f ci
0.45 f ci
0.45 f ci
•
Near uniform distribution of prestress
0.3 f ci
0.3 f ci
0.3 f ci
1.0 MPa
0.45 √ fci
0.36 √ fci
0.33 f cu *
0.33 f cu *
0.33 f cu *
0.25 f cu
0.25 f cu
0.25 f cu
0
0.45 √ fcu **
0.36 √ fcu **
2. Tension Serviceability limit state 1. Compression •
Design load in bending
•
Design load in direct compression
2. Tension
f ci = Concrete compressive strength at transfer *
Within range of support moments in continuous beams and other statically indeterminate structures this limit may be increased to 0.4 f cu
** These stresses may be increased under certain conditions, as specified in SABS 0100
The design process for prestressed concrete members differs from that used for other construction materials because a number of critical stages in the life of the structure, all related to the presence of the prestressing force, can be identified. Of these, the stage corresponding to transfer of the prestressing 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 are illustrated in Fig. 442 for a simply supported beam subjected to a uniformly distributed load:
• At transfer of prestress (Fig. 442a), the prestress will be acting at its maximum value because the longterm losses have not yet taken place, while the applied external load will be acting at its minimum value because only the self weight of the beam will be present at this stage. Under these conditions the beam will tend to deflect upwards and the stress distribution at the midspan
456
DESIGN FOR FLEXURE
section will show a small compression or even a tension in the top fibre and a large compression in the bottom fibre.
• At the serviceability limit state the maximum service load will be acting together with the effective prestressing force, which represents a minimum value because all the longterm losses would have taken place at this stage. Under these conditions the beam will deflect downwards and the stress distribution at the midspan section will show a large compression in the top fibre and a small compression or tension in the bottom fibre.
Minimum loading
Prestress at a maximum (before losses)
Stress distribution at midspan section
(a) Transfer of prestress Maximum loading 
Prestress at a minimum (after losses)
Stress distribution at midspan section
(b) At the serviceability limit state Figure 442:
Critical stages for a simply supported prestressed concrete beam.
Since the stages described above are usually critical they will serve as the point of departure for developing a design procedure. The magnitude of the stress limitations imposed by the design codes of 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 maximum prestress and minimum applied moment M min do not exceed the allowable values for tension and compression, respectively. The maximum prestressing force P t corresponds to the value directly after transfer and includes all the instantaneous losses but excludes all timedependent 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 under minimum prestress and maximum applied moment M max do not exceed the allowable values for compression and tension, respectively. The minimum prestressing force hP t corresponds to the final value after the all the losses (instantaneous and timedependent) have taken place. The factor h is the ratio of the final prestressing force to the initial value P t which includes only the instantaneous losses. It is also given by h = 1  %longterm loss/100.
DESIGN
457
Equations 45 and 46 can be used to express these criteria in terms of the following four stress inequality equations: P Pe M f top , t = t + t + min £ f tt (443a) A Z top Ztop f bot , t =
Pt Pe M + t + min ³ f ct A Zbot Zbot
(443b)
f top , s =
h Pt h Pt e M max + + ³ fcs A Z top Z top
(443c)
f bot , s =
h Pt h Pt e M max + + £ ft s A Zbot Zbot
(443d)
where
f top,t , f bot,t = stress in the extreme top and bottom fibres, respectively, at transfer f top,s , f bot,s = stress in the extreme top and bottom fibres, respectively, at the serviceability limit state f tt , f ct = allowable tensile and compressive stresses, respectively, at transfer f ts , f cs = allowable tensile and compressive stresses, respectively, at the serviceability limit state 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 devised to
• estimate the minimum required section properties in terms of Z top and Z bot , • establish the feasibility domain of P t and e, and • determine the so called permissible cable zone which delimits the zone along the span in which the cable may be placed. Minimum required section properties The purpose of the following derivation is to determine the minimum section properties which will simultaneously satisfy the four stress inequality equations. This means that, from the design point of view, the objective is to find the most efficient beam section. The derivation given here was adapted from Ref. 414 and is credited to Guyon (Ref. 421). Assume that the two allowable stresses f tt and f ct are both attained at the critical beam section at transfer, as shown in Fig. 443a. If the effect of the prestress losses, which take place with the passage of time, is superimposed on the stresses at transfer, then the stresses in the top and bottom fibres will be reduced to s 1 and s 2 , respectively, as shown in Fig. 443b. It should, therefore, be noted that s 1 and s 2 correspond to the combined action of hP t and M min . The stress changes Ds top and Ds bot induced in the top and bottom fibres by the application of an additional moment DM are shown in Fig. 443c, and the final stress condition at the serviceability limit state (see Fig. 443d) is reached when M min + DM = M max . If any of the allowable stresses at the serviceability limit state f cs or f ts is exceeded, then the corresponding section modulus Z top or Z bot is smaller than required (Ds = DM / Z), and the opposite is true if an allowable stress is not attained. Therefore, the magnitudes of Z top and Z bot for which the allowable stress requirements at the serviceability limit state are exactly satisfied, as shown in Fig. 443d, represent minimum required values. Referring to Fig. 443, the top fibre stress at transfer can be expressed as (Eq. 443a)
f tt =
Pt Pe M + t + min A Z top Z top
(444)
458
DESIGN FOR FLEXURE
s1
ftt
+
fct (a) At transfer (Pt + Mmin)
Figure 443:
Time
Dstop
=
fcs
=
+
s2 (b) (hPt + Mmin)
Dsbot
ft s (d) At the serviceability limit state (hPt + Mmax)
(c) (DM)
Evolution of stress in a prestressed concrete beam.
The top fibre stress under the action of M min , after all the prestressing losses have developed, is given by
s1 =
h Pt h Pt e M min + + A Z top Z top
which can be rewritten as
s1 = h
FP + Pe + M I hM GH A Z Z JK Z t
t
min
top
top
min
+
top
M min Z top
(445)
Substituting Eq. 444 into 445 yields
s 1 = h ftt +
M min 1 h Z top
a
f
(446)
The stress increment induced by the additional moment DM in the top fibre is
DM Z top
D s top =
(447)
The following condition must be satisfied to ensure that the allowable stress in the top fibre at the serviceability limit state is not exceeded:
s 1 + Ds top ³ f c s
(448)
Note that in Eq. 448 f cs will appear as a negative quantity because it represents a compressive stress. Substitution of Eqs. 446 and 447 into Eq. 448 yields
h f tt +
M min DM 1 h + ³ fcs Z top Z top
a
f
which, by noting that M max = M min + DM, can be solved to yield the following expression for Z top :
bM  h M g df h f i A similar analysis of the state of stress in the bottom fibre will show that bM  h M g Z ³ df h f i Z top £
max
cs
min
(449)
min
(450)
tt
max
bot
ts
ct
Equations 449 and 450 can also be written in the following convenient form by using M max = M min + DM:
Z top £
a1  h f M + D M df h f i min
cs
tt
(451)
DESIGN
459
a1  h f M + D M df h f i min
Zbot ³
ts
(452)
ct
Equations 449 and 450 or, alternatively, Eqs. 451 and 452 can be used to find a beam section which will satisfy the limiting stress criteria. Note, however, that these equations are functions of M min which, itself, is dependent on the crosssection so that the procedure by which a suitable section is found must involve some form of iteration. The process is usually started by assuming a value for M min based on experience, after which the minimum required values for Z top and Z bot can be calculated and a suitable section subsequently selected. The validity of the assumed value of M min can then be checked and, if required, an improved value of M min can be used to calculate revised minimum required values for Z top and Z bot . This process normally converges rapidly, particularly as the experience of the designer and, hence, the accuracy of the initial assumption for M min increases. The following considerations are important when designing a beam section using the approach outlined above:
• Although the section moduli may satisfy Eqs. 449 and 450, it is possible that the required eccentricity of the prestressing force may be larger than y bot , which means that the cable falls outside the section. In such a case the section must be revised to satisfy the practicality consideration that the cable must fall inside the section.
• Practical considerations, such as for example a required type of shape, usually lead to a section which can, at best, only satisfy one of Eqs. 449 and 450. Therefore, the normal situation is that the section will have one section modulus which is approximately equal to the minimum required 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 provide the most efficient crosssection and, hence, a least weight beam. Magnel diagram Once the section has been selected, the magnitude and corresponding eccentricity of the prestressing force must be determined. Although numerous procedures for accomplishing this exist, the method originally developed by Magnel (Ref. 422) still represents an extremely useful technique and is presented here. The procedure is basically a geometric interpretation of the four stress inequality equations, Eqs. 443, which are rewritten in the following form: Z top +e A 1 £ (453a) Pt f Z M
LM N
d
tt
OP Q
top
LM Z NA
bot
min
+e
i
OP Q
1 £ Pt f ct Zbot  M min
d
R h L Z  ³ MN A S d f Z  M i  h LM Z + eOP  £ N A Q T df Z  M i top
1 Pt
i O + eP Q
c s top
(453b)
for f cs Z top < M max
max
(453c)
top
c s top
max
for f cs Z top > M max
460
DESIGN FOR FLEXURE
R h L Z + eO  ³ MN A PQ S d f Z  M i  h LM ZA + eOP  £ d f ZN  M Q i T bot
1 Pt
t s bot
for f t s Zbot < M max
max
(453d)
bot
t s bot
for f t s Zbot > M max
max
At this stage of the design, the only two unknown quantities in the inequality Eqs. 453 are P t and e if an a priori value has been taken for h. These equations can therefore be plotted at equality on the e1/P t plane, in which case they will each plot as a straight line, as shown in Fig. 444. Each line serves as a boundary which divides the plane into a part in which the inequality relationship represented by the line is satisfied and another part in which it is not. Consider, for example, the line in Fig 444 which represents the inequality 453a at equality and, therefore, the stress inequality 443a from which it was derived. All the points with coordinates e and 1/P t which fall below the line will satisfy the stress inequality 443a while it is not satisfied by the points which lie above the line. If all the lines are examined in the same way, it can be concluded that the region bounded by the quadrilateral ABCD contains points with coordinates e and 1/P t which satisfy all four the stress inequality equations and, therefore, represents a feasibility domain. It is important in this regard to note that the so called Magnel diagram shown in Fig. 444 was constructed on the presumption that f cs Z top < M max and f ts Z bot < M max . The maximum practical eccentricity e pl is also plotted on the Magnel diagram as a vertical line (Ref. 414), and is shown to intersect the quadrilateral ABCD. In this case the feasible domain is reduced to the region bounded by ABEFD which contains points which have e and associated P t values satisfying not only the stress inequalities but also the practicality requirements. If e pl lies to the left of point A then no practical solution exists and a revised section with larger section moduli must be selected. If, on the other hand, e pl lies to the right of point C then any point contained in ABDC will yield practically feasible values of e and P t .
1 Pt

Z  bot A
Z top A
epl
O
e
Domain of feasibility
A
B
Eq
53 c
The Magnel diagram.
53 b
F
. 4
Figure 444:
Eq. 4
E
D
C Eq .4 Eq 53 d .4 5 3a
DESIGN
461
Once the Magnel diagram has been constructed for a section, the required value of P t and corresponding value of e can be selected. Usually the value which leads to the smallest possible value of P t is selected. This point will correspond to the largest value of 1/P t at the largest practically feasible value of e (point F, Fig. 444). The number of strands required for the selected value of P t will depend on the maximum permissible jacking force. The design codes of practice commonly used in South Africa (Refs. 42, 46, 47 and 48) all recommend that the jacking force should normally not exceed 75% of the characteristic strength of the tendon, but that it may be increased to 80% provided that special consideration is given to safety, to the stressstrain characteristics of the tendon, and to the assessment of the friction losses. BS 8110 (Ref. 47) also requires that the initial prestress at transfer should normally not exceed 70% of the characteristic strength of the tendon and that it must not exceed 75% under any circumstances. The bridge codes (Refs. 46 and 48) specifically state that immediately after transferring the prestress, the prestressing force must not exceed 70% of the characteristic strength of the tendons for posttensioned tendons, or 75% for pretensioned tendons. It should be noted that although the Magnel diagram is presented here as a design tool, it can also serve 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 eccentricity is based on the conditions at the critical section, e.g. at the midspan section, in the case of a symmetrically loaded simply supported beam. Since the bending moment varies over the span, the eccentricity must normally be varied along the span to ensure that the stress limitations are not exceeded at other sections, if the same prestressing force is to be used over the entire span. A procedure is developed herein to determine the zone within which the cable can be placed so that the stress inequality equations are satisfied at each section over the span. This feasibility zone is often referred to as the permissible cable zone. The development presented here assumes that all the quantities contained in the four stress inequality equations (Eqs. 443) are known at all sections of the beam, with the exception of the eccentricity e which is the variable to be determined. Equations 443 can be solved for e as follows:
F f  1 IZ GH P A JK F f  1 IZ e£G H P A JK F f  1 IZ e³G H h P A JK F f  1 IZ e³G H h P A JK
e£
top

M min Pt
bot

M min Pt
tt t
ct t
top

M max h Pt
bot

M max h Pt
cs
t
ts
t
(454a) (454b) (454c) (454d)
These equations can be applied to any section of the beam to determine the permissible range of e and, hence, the region in which the cable may be placed so that all the stress inequalities for that particular section are satisfied. The maximum permissible value of e at a section, referred to as the bottom cable limit, is given by the smallest value yielded by Eqs 454a and 454b at equality. Similarly, the largest value of e obtained from Eqs. 454c and 454d at equality represents the minimum permissible value of e, referred to as the top cable limit. It is interesting to note that the bottom cable limit is governed by conditions at transfer while the top cable limit is determined by the 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 on an elevation of the beam, as shown in Fig. 445. The region between the two cable limits clearly
462
DESIGN FOR FLEXURE
represents a feasibility zone within which the cable may be placed so that the stress inequality equations are satisfied at each section over the entire span.
Cable zone
Top cable limit (largest of Eqs. 454c and 454d) Centroidal axis
Bottom cable limit (smallest of Eqs. 454a and 454b) Figure 445:
The permissible cable zone.
Figure 446 presents three types of cable zone which may be obtained during the course of a design (Ref. 414). The cable zone shown in Fig. 446a is most commonly obtained and represents the case where the bottom cable limit falls outside the maximum available eccentricity e pl , but the cable zone is wide enough accommodate the cable. Figure 446b shows the cable zone which is obtained for a optimum design where only one combination of P t and e is possible at the critical section. The cable zone shown in Fig. 446c is characterised by the fact that a part of it lies outside the section, and is obtained when an insufficient concrete section is used. This problem, which will not arise if the Magnel diagram has been properly used in the design process, can only be overcome by using a revised section with larger section properties.
Cable zone
CL
Centroidal axis epl
(a) Common design Cable zone
CL
Centroidal axis
(b) Optimum design Cable zone
CL
Centroidal axis
(c) Insufficient concrete section Figure 446:
Typical types of cable zone (adapted from Ref. 414).
DESIGN
463
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 and position of the minimum required corresponding prestressing force, which will ensure that the specified concrete stress limitations are satisfied at transfer and at the serviceability limit state. This aim can be achieved by the following design steps: (a)
Determine a satisfactory concrete section using Eqs. 449 and 450 or, alternatively, Eqs. 451 and 452.
(b)
Use the Magnel diagram (Eqs. 453) to determine the magnitude of the prestressing force and its eccentricity at the critical section.
(c)
Calculate the permissible cable zone using Eqs. 454 and place the cable so that it falls within this zone.
(d)
All the calculations required for steps (a) through (c) require a value for h which must initially be assumed because the prestress losses can only be evaluated after completing step (c). In this step, 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 the span to ensure that none of the specified permissible values are exceeded. The prestress losses calculated 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 because some of the permissible stress limitations are exceeded or because the design proves to be unacceptably conservative, the design must be suitably revised and the appropriate previous steps 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 already been selected on the basis of other specific requirements, the design procedure will then simply commence at step (b). The design process can be greatly facilitated by assuming reasonable values for the various initially unknown quantities, and many handbooks provide useful guidance in this regard. One of the most important assumptions which must be made is that of the magnitude of h. Since the timedependent losses in prestensioned members tend to be larger than in posttensioned members the magnitude of h, which can be taken between 0.75 and 0.85, is usually smaller for pretensioned than for posttensioned members. Another initially unknown quantity of which the magnitude must be assumed is the area of the concrete crosssection A, and Lin (Ref. 410) suggests the following in this regard: A ps f se A» (455) 0.5 f cs where
R M 0.65h f »S  0.5Mh f T
max
A ps
for M min > (0.2 to 0.3) M max
se
L
for M min < (0.2 to 0.3) M max
se
M L = superimposed dead and live load moment applied to the section h = depth of the section Obviously, the quality of the initial assumptions made by a designer will improve with experience over time.
464
DESIGN FOR FLEXURE
EXAMPLE 411 Make use of the provisions of SABS 0100 (Ref. 42) for flexural design at the serviceability limit state to design a class 2 pretensioned concrete Tbeam which is simply supported over a span of 21 m. The beam must be designed to support a uniformly distributed live load of 5.8 kN/m and a superimposed dead load of 0.6 kN/m. Assume f cu = 45 MPa and f ci = 35 MPa. The permissible stresses for f cu = 45 MPa and f ci = 35 MPa, as obtained from SABS 0100, are as follows: At transfer:
f tt = 0.45 f ci = 0.45 35 = 2.662 MPa f ct = 0.45 f ci = 0.45 ´ (35) = 15.75 MPa At the serviceability limit state:
f ts = 0.45 f cu = 0.45 45 = 3.019 MPa f cs = 0.33 f cu = 0.33 ´ (45) = 14.85 MPa For the purposes of these calculations h is assumed to be 0.83. Select a suitable section at midspan The midspan moment due to the design superimposed load Dw = 1.1 w sdl + w L = 1.1 ´ 0.6 + 5.8 2 2 = 6.46 kN/m is given by DM = DwL / 8 = 6.46 ´ 21 / 8 = 3561 . kN. m . In order to obtain an initial estimate of M max , it is assumed that M min = DM, so that M min = 356.1 kN.m and M max = 2DM = 2 ´ 356.1 = 712.2 kN.m Equations 449 and 450 can now be used to obtain an initial estimate of the minimum required values for the section moduli. Hence, (712.2  0.83 ´ 3561 M  h M min . ) ´ 106 6 3 Z top , min = max = = 24.42 ´ 10 mm f cs  h f tt 14.85  0.83 ´ 2.662
Zbot , min =
(712.2  0.83 ´ 3561 M max  h M min . ) ´ 106 6 3 . ´ 10 mm = 2589 = ( ) f ts  h f ct 3.019  0.83 ´ 15.75
Figure 447 shows the selected Tsection together with its section properties. If the self weight of 3 3 the concrete is g c = 24 kN/m , the self weight of the beam is given by w D = g c A = 24 ´ 345 ´ 10 = 8.28 kN/m. Therefore, the design loadings are obtained from SABS 0160 as follows: For calculating M min :
wmin = 10 . w D = 10 . ´ 8.28 = 8.28 kN/ m For calculating M max :
b
g
wmax = 11 . wD + wsdl + 10 . w L = 11 . (8.28 + 0.6) + 10 . ´ 58 . = 1557 . kN/ m Using these design loads, the minimum and maximum moments at the midspan section are calculated 2 as wmin L 8.28 ´ 212 M min = = = 456.4 kN. m 8 8
M max =
wmax L2 1557 . ´ 212 = = 858.2 kN. m 8 8
DESIGN
465
1200 90 110
A = 345 ´ 10 3 mm 2 9
I = 32.11 ´ 10 mm . mm y top = 3218
4
y bot = 678.2 mm
1000
Z top = 99.77 ´ 10 6 mm 3 6
Z bot = 47.34 ´ 10 mm
3
200 Figure 447:
Section for example 411.
Revised minimum required values can now be calculated for the section moduli, based on the above magnitudes of M min and M max .
Z top , min =
(858.2  0.83 ´ 456.4) ´ 106 M max  h M min = = 2810 . ´ 106 mm 3 f cs  h f tt 14.85  0.83 ´ 2.662
Zbot , min =
(858.2  0.83 ´ 456.4) ´ 106 M max  h M min = = 29.79 ´ 106 mm 3 f ts  h f ct 3.019  0.83 ´ (15.75) 6
3
6
3
Clearly, Z top,min > Z top = (99.77 ´ 10 mm ) and Z bot,min < Z bot (= 47.34 ´ 10 mm ) so that the section should be satisfactory. Note that since the magnitudes of Z top,min and Z bot,min are almost equal, a symmetric section such as, for example, an Isection would be more efficient. However, a Tsection is specified as a design requirement in this case. It should also be noted that although the section can be optimised to a further degree, limitations imposed by deflection control and practical considerations 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 the midspan section by making use of the Magnel diagram. This diagram is constructed by plotting Eqs. 453 on the e1/P t plane. Since f cs Z top (= 1481 kN.m) > M max (= 858.2 kN.m) and f ts Z bot (= 142.9 kN.m) < M max = 858.2 kN.m the second of Eqs. 453c and the first of Eqs. 453d are applicable. Hence,
Z top
99.77 ´ 10
6
+e 3 1 345 ´ 10 A £ = 6 3 3 Pt f tt Z top  M min 2.662 ´ 10 99.77 ´ 10  456.4 ´ 10 +e
d
6
= 400.5 ´ 10
6
.  1385 ´ 10
i
(456a)
e 6
47.34 ´ 10 Zbot +e +e 3 1 A 345 ´ 10 £ = 6 3 3 f ct Z top  M min Pt 15.75 ´ 10 ´ 47.44 ´ 10  456.4 ´ 10
d
6
= 114.2 ´ 10
i
6
 0.8319 ´ 10
e
(456b)
466
DESIGN FOR FLEXURE
1 0.8 0.6 Eq. 456c 0.4
1 (´ 103 kN1) Pt
0.2 0 Feasibility domain
0.2 0.4
0.5884
Eq. 456a
0.6 0.8 1
Eq. 456b 0.8206
1.2
Eq. 456d e = 570 mm
1.4 200
200
0
epl = 595 mm
400
600
800
1000
1200
e (mm)
Figure 448:
Magnel diagram for the midspan section.
h
FZ GH A
top
+e
I JK
F 99.77 ´ 10 + eI GH 345 ´ 10 JK id99.77 ´ 10 i  858.2 ´ 10 6
0.83 ´
1 £ = f cs Z top  M max Pt 14.85 ´ 103
d
3
6
3
(456c)
. = 385.0 ´ 106 + 1331 ´ 106 e
FZ hG HA
bot
I + eJ K
0.83 ´
F 47.34 ´ 10 GH 345 ´ 10
3
6
+e
I JK
1 ³ = 3 6 3 f ts Zbot  M max Pt 3.019 ´ 10 ´ 47.34 ´ 10  858.2 ´ 10
(456d)
= 159.2 ´ 106  1160 ´ 106 e . where P t 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 spaced layers at a vertical centre to centre spacing of 40 mm, then the maximum possible eccentricity e pl is approximately 595 mm. The above four inequalities are plotted at equality in Fig. 448 together with e pl . Selecting e = 570 mm, it is clear that values of 1/P t which range between 1/P b = 3 1 3 1 kN and 1/P d = 0.8206 ´10 kN all fall within the feasibility domain and, 0.5884 ´ 10 therefore, satisfy the four stress inequlity equations. These values correspond to permissible values of P t which range between P d = 1219 kN and P b = 1700 kN, and a value of P t = 1280 kN is selected. Assume that 12.9 mm 7wire super grade strand, jacked to 75% of its characteristic strength, is used. Since the characteristic strength per strand is 186 kN, the jacking force per strand is 0.75 ´ 186 = 139.5 kN. If the loss of prestress due to elastic shortening is assumed to be 8.0%,
DESIGN
467
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 P t = 1283 kN. To summarize: 10 @ 12.9 mm 7wire super grade strand are required. P t = 1283 kN e = 570 mm at midspan Determine the cable zone Before the cable limits can be calculated, M min and M max must be expressed as functions of x. Thus,
wmin x ( L  x ) = 4.140 x (21  x ) kN. m 2 w x = max ( L  x ) = 7.784 x (21  x ) kN. m 2
M min = M max
Substitution of these expressions into Eqs. 454 give
F f  1IZ GH P A JK L 2.662 =M N1283 ´ 10
e£

M min Pt

1
tt
top
t
3
OPd99.77 ´ 10 i  4.140 ´ 10 x (21  x) Q d1283 ´ 10 i
(457a)
OP ´ 47.34 ´ 10  4.140 ´ 10 x (21  x) 345 ´ 10 Q d1283 ´ 10 i
(457b)
6
6
345 ´ 103
3
= 496.2 + 3.226 x (21  x )
F f  1IZ GH P A JK L (15.75) =M N1283 ´ 10
e£
ct
bot

t
3

M min Pt 6
1
6
3
3
= 443.7 + 3.226 x (21  x )
F f  1IZ  M GH hP A JK hP L (14.85) 1 =M MN 0.83 ´ d1283 ´ 10 i  345 ´ 10
e³
max
cs
top
t
t
3
OP 7.784 ´ 10 PQ ´ d99.77 ´ 10 i  0.83 ´ d1283 ´ 10 i x (21  x)
(457c)
OP PQ ´ 47.34 ´ 10
(457d)
6
6
3
3
= 1102 + 7.307 x (21  x )
F f  1IZ  M GH hP A JK hP L 3.019 1 =M MN 0.83 ´ d1283 ´ 10 i  345 ´ 10
e³
ts
max
bt
t
t
3
3
6

7.784 ´ 10
d
6
0.83 ´ 1283 ´ 10
3
i
x (21  x )
. + 7.307 x (21  x ) = 2714
Note that in Eqs. 457 e is in mm and x is in m. Comparing the right hand sides of Eqs. 457a and 457b clearly shows that Eq. 457b governs the bottom cable limit. A similar examination of Eqs. 457c and 457d reveals that the top cable limit is controlled by Eq. 457d. The values of both
468
DESIGN FOR FLEXURE
CL
x (m)
222
Centroidal axis
Cable zone
e(x)
570
epl = 595
ed(x)
eb(x)
7.0 m
3.5 m L/2 = 10.5 m (a) Cable zone
150
C.G.C
C.G.C
150 e = 222 C.G.S e = 570
430 456
60 60 60
C.G.S 108
(b) Tendon layout at midspan
Figure 449:
60 60 (c) Tendon layout at support
Cable zone.
cable limits are listed in Table 47 at span/12 points along the span of the beam, and the resulting cable zone is drawn in Fig. 449a for the half span because of symmetry. Note that, since the central region of the bottom cable limit falls outside the beam, the cable zone is also limited by the maximum practical eccentricity e pl . The cable profile is selected to lie within the cable zone and has draping points each located at a distance of span/ 3 = 7.0 m from a support, as shown in Fig. 449a. The eccentricity of the resulting cable profile can be expressed as follows for the half span:
e( x ) =
RS222 + 49.71x T570
for 0 £ x £ 7.0 m for 7.0 £ x £ 10.5 m
where e(x) is in mm and x is in m. The magnitudes of the eccentricity at span/12 points are listed in Table 47, while Figs. 449b and 449c show possible tendon layouts at midspan and at the support, respectively. The next step in the design process is the calculation of the prestress losses, after which a stress check can be made. These steps are illustrated in example 51, where this example is concluded. It is also important to check the ultimate moment of resistance of the critical section, which in this
DESIGN
469
Table 47:
Cable limits and cable profile.
x (m)
Bottom cable limit e b (x) (mm)
Top cable limit e d (x) (mm)
Selected eccentricity e(x) (mm)
0 1.750 3.500 5.250 7.000 8.750 10.500
444 552 641 710 760 789 799
271 25 176 333 445 512 534
222 309 396 483 570 570 570
case is located at midspan. The reader should verify that the approximate procedure recommended by SABS 0100 yields M u = 1388 kN.m for the midspan section, and that a midspan moment of 1099 kN.m is induced by the ultimate design loads prescribed by SABS 0160, i.e. 1.2D n + 1.6L n . 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 concrete area under compression, the effective depth to the prestressing steel and the area of steel needed to meet the requirements of flexural strength at the ultimate limit state. This follows because the contribution of the tensile strength of the concrete is neglected, so that the tensile zone of the section is of no importance with regard to flexural strength and merely serves to contain the tendons. The design procedure is based on Eqs. 417 and 418, which are expressions of moment and horizontal equilibrium, respectively, and the required computations can be simplified by making use of a suitable approximate procedure for estimating the steel stress at ultimate f ps . Equations 417 and 418 can be restated in the following more general form, so that they apply to any crosssectional shape:
where
M u = A ps f ps z
(458)
A ps f ps + a f cu Ac¢ = 0
(459)
z = internal lever arm Ac¢ = concrete area under compression at ultimate
These equations can be rearranged to the following forms, which are more useful for design: Mu A ps = (460) f ps z
Ac¢ = 
A ps f ps a f cu
(461)
The design process is summarised in the following: (a)
Assume values for f ps , z and for the overall section depth h.
• For bonded tendons it is suggested that f ps initially be taken equal to the design value of
f pu because the large steel strains associated with the flexural failure of an underreinforced
470
DESIGN FOR FLEXURE
bonded section will result in these high steel stresses. However, the initial guess of f ps for unbonded tendons must reflect the fact that it is normally significantly less than the design value of f pu , and a maximum value of 0.7f pu is suggested in this case.
• Since z ranges between 0.6h and 0.9h, depending on the section shape, it seems reasonable to accept a value of 0.8h as an initial guess (Ref. 410).
• The limitations imposed by deflection control at the serviceability limit state can be used for guidance when selecting an initial value for h. (b)
Obtain an initial estimate of the required area of prestressing steel A ps from Eq. 460 using the assumed values of f ps , z and h.
(c)
The required concrete compression area Ac¢ is subsequently calculated from Eq. 461. At this stage sufficient information is available for selecting a suitable section.
(d)
Once a preliminary section has been selected, the actual values of f ps and z corresponding to this section can be determined either by the strain compatibility approach or by a suitable approximate procedure, whichever is most convenient. These results can be used together with Eq. 460 to calculate an improved value for A ps and, if required, Eq. 461 can be used to obtain a revised value for A’ c . This process is continued until a satisfactory section has been obtained. Note that the calculation of f ps requires a value for the effective prestress f se including 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 of the 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 to check that the concrete flexural stress limitations are satisfied at transfer and at the serviceability limit state. When unbonded tendons are used, a minimum amount of bonded reinforcement should always be placed to improve behaviour at ultimate (see Section 4.3.6). SABS 0100 (Ref. 42) and BS 8110 (Ref. 47) do not specifically require such reinforcement, but ACI 31889 (Ref. 411) requires that a minimum amount of bonded reinforcement equal to 0.004A be provided for this purpose, where A denotes the area of that part of the section which lies between the tension face and the centroid of the gross concrete section. EXAMPLE 412 Make use of the provisions of SABS 0100 (Ref. 42) for flexural design at the ultimate limit state to design the midspan section of a 700 mm deep pretensioned concrete Ibeam which is simply supported over a span of 14 m. The beam is subjected to an imposed nominal live load of 9.0 kN/m. Assume f cu = 50 MPa and E c = 34 GPa for the concrete. Use 12.9 mm 7wire super grade strand, for which f pu = 1860 MPa and E p = 195 GPa. For the equivalent rectangular stress block prescribed by SABS 0100, a = 0.45 and b = 0.9. Assume f ps = 0.87 f pu = 0.87 ´ 1860 = 1618 MPa and z = 0.8 h = 0.8 ´ 700 = 560 mm, so that an initial estimate of A ps can be obtained from Eq. 460. However, before these calculations can be made a value must be estimated for the ultimate design moment which, in turn, requires an assumed value for the self weight of the beam. If it is assumed that the nominal self weight of the beam is w D = 4.5 kN/m, the design ultimate load, using the load factors prescribed by SABS 0160, and the design ultimate moment at the midspan section are given by
12 . w D + 16 . w L = 12 . ´ 4.5 + 16 . ´ 9.0 = 19.8 kN/ m
M=
19.8 ´ 14 2 wL2 = = 4851 . kN. m 8 8
DESIGN
471
Using Eq. 460
A ps =
M f ps z
=
. ´ 10 6 4851 = 535.3 mm 2 1618 ´ 560
The required concrete compression area Ac¢ is subsequently obtained from Eq. 461
Ac¢ = 
A ps f ps a f cu
=
535.3 ´ 1618 3 2 = 38.5 ´ 10 mm 0.45 ´ (50)
Thus the preliminary section shown in Fig. 450 can be selected. It can be shown that this section 3 2 can accommodate the entire concrete compression zone Ac¢ = 38.5 ´ 10 mm within the top flange. 3 The self weight of this section is given by w D = g c A = 24 ´ 165 ´ 10 = 3.96 kN/m, so that the design ultimate load is 12 . w D + 16 . w L = 12 . ´ 3.96 + 16 . ´ 9.0 = 19.15 kN /m and the design ultimate 2 2 moment at the midspan section is M = wL / 8 = 19.15 ´ 14 / 8 = 469.2 kN. m .
b = 350 e = 290 mm
150
3
A = 165 ´ 10 mm
h= 700
d = 640
150
150
I = 8.938 ´ 10 9 mm4
Aps 350
Figure 450:
2
60
Section for example 412.
The approximate procedure recommended by SABS 0100 for calculating the ultimate moment of resistance of the section will be used to recalculate f ps and z for the preliminary section. In order to do this it is assumed that f se = 1116 MPa.
f pu A ps f se 1116 1860 ´ 535.3 = = 0.6 and = = 0.08890 1860 f pu 50 ´ 350 ´ 640 f cu b d f ps x = 0.20 Table 43 gives = 10 . and 0.87 f pu d For
Therefore, f ps = 1.0 (0.87 f pu ) = 1618 MPa and x = 0.20 d = 0.20 ´ 640 = 128 mm. Since the compression 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, f ps and z into Eqs. 460 and 461 yields the following values for A ps and Ac¢ :
A ps =
M
=
469.2 ´ 10 6 = 497.9 mm 2 1618 ´ 582.4
f ps z A ps f ps 497.9 ´ 1618 3 2 Ac¢ = == 3581 . ´ 10 mm a f cu 0.45 ´ (50)
These values are fairly close to the values obtained in the initial trial, so that the section can be 2 accepted as it stands and the minimum required value of A ps can be taken as 497.9 mm . Since the 2 crosssectional area provided by one 12.9 mm 7wire super grade strand is 100 mm , five strands 2 will provide an A ps = 500 mm , which is sufficient. As a final check, the ultimate moment of resistance of the section is calculated and compared with the moment induced by the design ultimate loads.
472
DESIGN FOR FLEXURE
For f se = 1116 MPa, so that
f ps 0.87 f pu
f se = 0.6 and f pu
= 10 . and
f pu A ps f cu b d
=
1860 ´ 500 = 0.08304 , 50 ´ 350 ´ 640
x from Table 43. = 018 . d
Consequently, f ps = 1.0 (0.87 f pu ) = 1618 MPa and x = 0.18 d = 0.18 ´ 640 = 115.2 mm. Since the compression 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. 458 as
M u = A ps f ps z = 500 ´ 1618 ´ 588.2 ´ 106 = 475.9 kN. m 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 often arises where the ultimate moment of resistance of the section is less than the moment imposed by the design ultimate loads. This problem can usually be rectified by providing a sufficient quantity of additional nonprestressed reinforcement, which is designed as follows: (a)
Assume a value for the stress in the prestressing steel at ultimate f ps and set the sum of the moments of the internal forces, taken about the position of the nonprestressed steel, equal to the moment induced by the design ultimate loads. The resulting expression can be directly solved for the depth to neutral axis x, because it will be the only unknown variable contained in this expression.
(b)
Using the value of x determined in step (a) together with an assumed value for the stress in the nonprestressed steel at ultimate f s , the required area of nonprestressed steel A s can be calculated directly by considering horizontal equilibrium of the section.
(c)
At this stage the actual values of f ps and f s , corresponding to A s as determined in step (b), can be calculated and compared to the assumed values. If the actual and assumed values differ significantly, steps (a) and (b) must be repeated using improved assumptions for the magnitudes of f ps and f s . This process is repeated until the assumed and calculated values of f ps and f s agree to within an acceptable tolerance, and the corresponding magnitude of A s is then accepted.
This procedure is illustrated by example 413. EXAMPLE 413 Provide suitable nonprestressed reinforcement for the section obtained in example 412 and shown in Fig. 450 so that it can sustain a total applied ultimate design moment of 600 kN.m. Make use of the provisions of SABS 0100 (Ref. 42) for flexural design, and take f y = 450 MPa and E s = 200 GPa for the nonprestressed reinforcement. Assume f se = 1116 MPa. For the equivalent rectangular stress block prescribed by SABS 0100, a = 0.45 and b = 0.9, while the design stressstrain curves for the prestressed and nonprestressed steel are as shown in Figs. 4 17 and 423, respectively. The effective depth to the prestressing steel d 1 = 640 mm (see Fig. 450) and the effective depth to the nonprestressed steel is taken as d 2 = 650 mm. The section now corresponds exactly to that shown in Fig. 422 example 46, except that A s is unknown. Assume f ps = f py and f s = f sy , so that the tensile force in the prestressing steel T ps and in the nonprestressed steel T s can be calculated from Eqs. 412 and 425, respectively:
Tps = A ps f ps = 500 ´ 1617 ´ 103 = 808.7 kN 3 Ts = As f s = 3913 . ´ 10 As kN If it is assumed that the entire compression zone is contained in the flange, the compression force in the concrete can be expressed as a function of x as follows (see Eq. 414):
DESIGN
473
3
C = a f cubb x = 0.45 ´ (50) ´ 350 ´ 0.9 ´ 10
x = 7.088 x kN
Moment equilibrium about the position of the nonprestressed steel provides the following expression (see Fig. 422):
FG H
M = C d 2 
IJ K
b
bx  Tps d 2  d1 2
FG H
600 ´ 10 = (7.088 x ) 650 3
g
IJ K
0.9 x  808.7(650  640) 2
Solving for x yields x = 146.9 mm. Therefore s = b x = 0.9 ´ 146.9 = 132.2 mm is less than h f = 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:
Tps + Ts + C = 0 3
808.7 + 3931 . ´ 10 3
. ´ 10 808.7 + 3913
As  7.088 x = 0
As  7.088 ´ 146.9 = 0 2
Solving this expression for A s yields A s = 594.9 mm . The validity of the assumption that f ps = f py and f s = f sy must be checked. This is done by calculating e ps and e s2 using the strain compatibility approach. Accordingly, e ps is calculated by combining Eqs. 48 through 411, and by noting that the effective prestress acting on the section is given by P =  f se A ps = 1116 ´ 500 ´ 103 = 558 kN . Hence,
e se =
f se Ep
=
1116 195 ´ 10 3
= 0.005723
F P + Pe I 1 = F 558 + 558 ´ 290 I 1 = 0.000254 GH A I JK E GH 165 ´ 10 8.938 ´ 10 JK 34 F d  x IJ e = FG 640  146.9 IJ 0.0035 = 0.01174 =G H 146.9 K H x K 2
2
e ce =
c
e s1
3
9
1
cu
e ps = e s1  e ce + e se = 0.01174  (0.000254) + 0.005723 = 0.01772 The strain in the nonprestressed steel e s2 is calculated by considering the strain distribution (see Fig. 422). Thus, considering similar triangles:
e s2 =
FG d H
2
x x
IJ e K
cu
=
FG 650  146.9 IJ 0.0035 = 0.01198 H 146.9 K
Referring to Fig. 417, it is clear that since e ps is larger than e py (= 0.01329), f ps = f py (= 1617 MPa), as assumed. Figure 423 also confirms that the assumed value of f s = f sy (= 391.3 MPa) is correct 2 because e s2 is larger than e sy (= 0.00196). Therefore, the calculated value of A s = 594.9 mm is 2 correct. This area of steel can be provided by 2 @ Y20 mm bars, for which A s = 628 mm , so that the section corresponds exactly to that shown in Fig. 422, as expected. It is of interest to note that the ultimate moment of resistance of this section is M u = 606.7 kN.m (see example 46), which is slightly larger than the required value of M = 600 kN.m.
Note that although only flanged sections in which the compression zone at ultimate is entirely contained in the top flange were considered in the examples presented in this Section, the design procedures presented herein apply equally to flanged sections in which the compression zone extends into the web. The only difference lies in the formulation of the equations of equilibrium which must, in the latter case, account for the nonrectangular shape of the compression zone. However, note that Eqs. 458 through 461 are general because they apply to any crosssectional shape.
474
DESIGN FOR FLEXURE
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, depending on the amount of steel provided:
• In very lightly reinforced sections, an extremely brittle type of flexural failure can occur in which the steel fractures immediately after the concrete has cracked. This failure mode is highly undesirable.
• In moderately reinforced (uderreinforced) sections, failure is induced by crushing of the concrete compression zone after the steel has yielded and undergone a large nonlinear elongation. Because of its ductility, this type of failure is highly desirable.
• In heavily reinforced (overreinforced) sections, failure is induced by crushing of the concrete prior 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 failure modes. For this reason, any design code of practice should provide limits on the maximum and minimum amounts of steel to be provided in a section to ensure ductile behaviour (see Section 4.3.5 for a more expansive discussion on flexural ductility). SABS 0100 (Ref. 42) and BS 8110 (Ref. 47) both limit the minimum amount of prestressing steel by requiring that the ultimate moment of resistance of a beam section must exceed its cracking moment. According to SABS 0100 this requirement is deemed to be satisfied if the percentage of reinforcement, calculated on an area equal to the width of the beam soffit multiplied by its overall depth, is at least 0.15, while BS 8110 requires that the cracking moment be calculated on the basis of an assumed value of the modulus of rupture equal to 0.6 √ fcu . The intention of this provision is to ensure that cracking will precede flexural failure, thus avoiding the situation where the beam suddenly fails when the concrete cracks. The bridge codes TMH7 (Ref. 46) and BS 5400 (Ref. 48) 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 5400 through the requirement that the strains in the outermost tendon must not be less than 0.005 + f pu /(E p g m ). If the outermost tendon, or layer of tendons forms less than 25% of the total tendon 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 greater than the above value. This limitation is only applicable to cases where the ultimate moment of resistance of the section is less than 1.15 times the required value. By limiting the steel strains developed in such sections to values greater than the value at “yielding” (as defined by the design stressstrain 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 r = A ps / bd corresponding to the steel strain limit prescribed by TMH7 are listed in Table 48 for f pu = 1860 MPa. Note that these values were derived for rectangular compression zones on the basis of the design stressstrain curves and material properties prescribed by TMH7, and on the assumption that all the steel is concentrated at the centroid of the tendons. Neither SABS 0100 nor BS 8110 contain any requirements which can obviously be related to limiting the maximum steel content of a prestressed concrete beam. However, Ref. 425 infers that the code provision which requires the ultimate moment of resistance of a beam section to exceed its cracking moment may serve this purpose because it is possible for a heavily overreinforced beam to fail in flexure before cracking. This inference is not really acceptable because the limitation on maximum steel content is essentially related to the ductility of the section and must, therefore, be either explicitly or implicitly expressed in terms of a limiting steel strain.
DESIGN
Table 48:
f se / f pu
475
Limiting values of x / d and r = A ps / bd corresponding to steel strain limits of TMH7 for f pu = 1860 MPa.
Values of r max for f cu =
(x / d) max 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
4.4.5 Flexural design of composite sections The design of a composite section generally follows the same procedures used for noncomposite sections. 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 exist between 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 concrete to ensure composite action. In the following, the flexural design of composite prestressed beam sections at both the serviceability and 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 are acting on the beam section only.
• At the serviceability limit state when the minimum prestressing force, including all losses, and the total self weight (beam plus slab) are acting on the beam section only, and the maximum superimposed 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 (similar to Eqs. 443):
f top ,b =
Pt Pe Mb + t + £ f tt Ab Ztop ,b Z top ,b
f bot ,b =
Pt Pe Mb + t + ³ f ct Ab Zbot ,b Zbot ,b
(462b)
f top ,b =
ML h Pt h Pt e M b + M f + + + ³ f cs Ab Z top ,b Z top,b Z top ,cb
(462c)
f bot ,b =
ML h Pt h Pt e M b + M f + + + £ f ts Ab Zbot ,b Zbot ,b Zbot ,cb
(462d)
(462a)
476
DESIGN FOR FLEXURE
As in the case of noncomposite sections, the design process basically involves the rational manipulation of these four inequality equations. Following the same procedures outlined in Section 4.4.2 the following expressions, which may be used for establishing the feasibility domain of P t and e (Magnel diagram) as well as the permissible cable zone, can be derived. Magnel diagram:
LM Z NA
top ,b
+e
OP Q
1 b £ Pt f tt Z top ,b  M b
c
LM Z NA
bot ,b
+e
OP Q
1 b £ Pt f ct Zbot ,b  M b
b
R ³  LM f 1 MN S P  £  LM f T MN
h
top ,b
+e
b
OP Q
cs Z top ,b  M D  M L
L hM N
t
R ³ LM f  1  N S P  £  LM f T N
LM Z NA
Z top ,b Ab
LM Z NA
bot ,b
+e
b
bot ,b
t
b
Z top ,cb
Zbot ,b Zbot ,cb
O + eP Q
ts Zbot ,b  M D  M L
Cable zone:
Z top ,b
OP Q
ts Zbot ,b  M D  M L
LZ hM NA
Z top ,cb
O + eP Q
cs Z top ,b  M D  M L
h
Z top ,b
Zbot ,b Zbot ,cb
FG f  1 IJ Z HP A K F f  1 IJ Z e£G HP AK F f  1 IJ Z e³G HhP A K F f  1 IJ Z e³G HhP A K e£
tt t
t
t
where
MD = Mb + Mf
Z top ,cb
I JK (463c)
F GH
for f cs Z top ,b > M D + M L
OP PQ
F GH
for f ts Zbot ,b < M D + M L
OP Q
Z top ,b Z top ,cb
Zbot ,b Zbot ,cb
I JK
I JK (463d)
F GH
for f ts Zbot ,b > M D + M L
OP Q
Zbot ,b Zbot ,cb
I JK
bot ,b 
Mb Pt
(464b)
FM GH 1 F M h P GH
top ,b 
b
b
OP PQ
Z top ,b
(464a)
ts
t
F GH
for f cs Z top ,b < M D + M L
Mb Pt
b
cs
(463b)
g
top ,b 
b
ct
(463a)
h
bot ,b
1 h Pt
t
D +ML
D +ML
Z top ,b Z top ,cb Zbot ,b Zbot ,cb
I JK I JK
(464c) (464d)
DESIGN
477
The discussion of the equivalent forms of these equations for noncomposite sections given in Section 4.4.2 are equally applicable here. Note that no equivalent forms of Eqs. 449 and 450 were derived for estimating the minimum required values of the section moduli, because the usefulness of such equations are limited by the fact that the ratio of the section modulus of the beam to the corresponding section modulus of the composite section must also be known. Moreover, a number of useful design aids are available for obtaining an initial estimate of the section, when using standard precast sections. After initially selecting a suitable section, the flexural design of a composite section at the serviceability limit state proceeds as described in Section 4.4.2 for noncomposite sections, bearing in mind that Eqs. 462 through 464 must be appropriately substituted. Note that although the effects of differential shrinkage were not included in the expressions above, these effects can easily be accommodated simply by reducing the magnitudes of f cs and f ts 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. The Handbook to BS 8110 (Ref. 425) recommends that it is only necessary to include the effects of differential 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 slab is more than 10 MPa.
• The time interval between casting of the precast beam and casting of the in situ slab is more than 8 weeks. The type of cross section also influences the importance of differential shrinkage: The effects of differential shrinkage are more significant for sections of the type shown in Fig. 451a than for sections of the type shown in Fig. 451b.
In situ concrete slab
(a) Figure 451:
Precast prestressed concrete beam
(b)
Influence of cross section on differential shrinkage.
Although the horizontal shear capacity is checked at the ultimate limit state, this check must always be carried out to ensure composite action. This is true even if the flexural design was carried out at the serviceablity limit state. EXAMPLE 414 The composite section shown in Fig. 452 consists of precast pretensioned beams supporting an in situ slab. The beams are simply supported, having a span of 15 m, and spaced at a distance of 1200 mm. Make use of the provisions of SABS 0100 for the flexural design at the serviceability limit state to provide suitable class 2 prestressing for the beam. Assume unpropped construction and design the beam to support a uniformly distributed live load of 12 kN/m.
478
DESIGN FOR FLEXURE
bf = 1200 hf = 180 In situ slab Concrete material properties: Precast beam fcu,b = 50 MPa Ec,b = 34 GPa fci,b = 40 MPa fcu,f = 30 MPa Ec,f = 28 GPa In situ slab Unit weight gc = 24 kN/m3
Precast beam hb = 700
bb = 350
Figure 452:
Composite cross section for Example 414.
The permissible stresses prescribed by SABS 0100 for the concrete in the beam are as follows: f tt = 2.846 MPa
f ct = 18.00 MPa
f ts = 3.182 MPa
f cs = 16.50 MPa
The design self weight of the beam and slab at the serviceability limit state is w D = 11 . wb + w f = 11 . (5880 . + 5184 . ) = 12.17 kN / m
d
i
The design midspan moment induced by the beam self weight at transfer is
Mb =
1 1 2 2 wb L = ´ 588 . ´ 15 = 165.4 kN. m 8 8
The design midspan moment caused by the self weight of the beam and the slab at the serviceability limit state is
MD =
1 1 w L2 = ´ 12.17 ´ 152 = 342.3 kN. m 8 D 8
and the design midspan live load moment is
ML =
1 1 2 2 w L = ´ 12 ´ 15 = 337.5 kN. m 8 L 8
The modular ratio nc = E c, f / E c,b = 28 / 34 = 0.8235 yields a transformed flange width of
b ft = ncbe = 0.8235 ´ 1200 = 998.2 mm where the effective flange width be = 1200 mm . The section properties of the beam and the transformed composite sections are as follows y top,cb = −164.9 mm
y top,b = 350 mm y bot,b = 350 mm
y top,c = 344.9 mm
y bot,cb = 535.1 mm 9
I b = 10 ´ 10 mm
4
Z top,cb = 184.6 ´ 10 mm
6
Z top,b = 28.58 ´ 10 mm 6
Z bot,b = 28.58 ´ 10 mm
y bot,c = 535.1 mm 6
3
6
Z bot,cb = 56.88 ´ 10 mm
3
9
I c = 30.44 ´ 10 mm
4
3
3
The Magnel diagram can now be constructed using Eqs. 463. To account for the loss of prestress at the serviceability limit state, h is assumed as 0.8.
DESIGN
479
LM Z NA
top ,b
+e
OP Q
1 b £ = 472.9 ´106  4.053´106 e Pt f tt Z top ,b  M b
c
h
LM Z NA
bot ,b
+e
OP Q
1 b =1716 £ . ´106 1471 . ´106 e Pt f ct Zbot ,b  M b
b
1 £ Pt
1 ³ Pt
g
h
LM Z NA
top ,b
+e
b
LM f MN
cs Z top ,b  M D  M L
h
LM Z NA
bot ,b
+e
b
LM f N
OP Q
6
Z top ,b Z top ,cb
OP Q
ts Zbot ,b  M D  M L
Zbot ,b Zbot ,cb
OP =1211´10 PQ
OP Q
+10.38´106 e
=2217 . ´106 1901 . ´106 e
An eccentricity of 260 mm is selected, which is less than the practical limit e pl = 290 mm. The prestressing force is taken as P t = 1500 kN, which falls between the limits of 1397 kN and 1721 kN at e = 260 mm.
0.1 0 Eq. 463c
0.1
Eq. 463a
1 (´ 103 kN1) Pt
0.2 0.3
Eq. 463b
0.4
Feasibility domain
0.5 0.6 0.7
0.5809 Eq. 463d 0.7159
0.8 e = 260 mm 0.9 50
0
50
100
150 e (mm)
Figure 453:
Magnel diagram for Example 414.
200
250 epl = 290 mm
300
480
DESIGN FOR FLEXURE
Assuming the following conditions for the prestressing tendons  12.9 mm 7wire 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 of strands required is 1500/129.7 = 11.56. Selecting 12 strands yields a prestressing force of P t = 12 ´ 129.7 = 1557 kN, which still falls within the limits calculated above. At midspan, the strands can be placed as shown in Fig. 454.
ytop,b = 350 C.G.B e = 260
ybot,b = 350
C.G.S
60 60 Figure 454:
90
Cable layout at midspan, Example 414.
The concrete stresses at the midspan section at the serviceability limit state are presented in Fig. 455a, using the selected prestressing layout. The differential shrinkage stresses are also shown in this figure together with the resulting final stresses, which all comply with the specified permissible values. Note that the following information was assumed for calculating the differential shrinkage stresses by the method described in Section 4.3.7: e cr = 48 ´ 10 e sh = 310 ´ 10
6
6
MPa
1
for creep of the precast beam
for shrinkage of both the beam and slab
b cc = 1.6 h = 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 the slab 3.15 1.51
7.56
+
1.50 (a) Stresses caused by external loading plus prestressing
Figure 455:
Final stresses, Example 414.
2.84
0.31
1.24
0.27 1.22
=
0.71 2.20 (b) (c) Stresses caused by Total stress differential shrinkage
8.78
DESIGN
481
Ultimate limit states Since the flexural analysis of composite sections at ultimate is exactly the same as of noncomposite sections, the design procedure is also the same. Therefore, the procedures of Section 4.4.3 also apply to the flexural design of composite sections at the ultimate limit state. A slight difference arises when the depth to neutral axis lies below the in situ slab, in which case the difference in strength of 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 shear capacity of the interface between the preast and in situ concrete. The general approach followed in design is to calculate the magnitude of the horizontal shear stress at the interface as indicated in Section 4.3.7. This shear stress is subsequently compared to a permissible value. According to the procedure recommended by BS 8110 (Ref. 47) and SABS 0100 (Ref. 42), the average horizontal shear stress at the interface is calculated by Eq. 439 and the maximum value is obtained by distributing the average value in proportion to the vertical design shear force diagram. This maximum value is subsequently compared to the permissible values listed in Table 49. BS 5400 (Ref. 48) and TMH7 (Ref. 46) use Eq. 438 to calculate the horizontal shear stress at the interface. Obviously
Table 49:
Design ultimate horizontal shear stresses at interface to BS 8110 (Ref. 47) and SABS 0100 (Ref. 42).
Precast unit
Surface type
Design ultimate horizontal shear stresses at interface Grade of in situ concrete (MPa)
Without links
With nominal links projecting into in situ concrete
25
30
³ 40
Ascast or asextruded
0.4
0.55
0.65
Brushed, screeded or roughtamped
0.6
0.65
0.75
Washed to remove laitance or treated with retarder and cleaned
0.7
0.75
0.80
Ascast or asextruded
1.2
1.8
2.0
Brushed, screeded or roughtamped
1.8
2.0
2.2
Washed to remove laitance or treated with retarder and cleaned
2.1
2.2
2.5
NOTES: 1.
The description “ascast” 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 be applied directly without a further finishing screed but not as rough as would be obtained if tamping, brushing or other artificial roughening had taken place.
2.
The description “asextruded” covers those cases in which an opentextured surface is produced direct from an extruding machine.
3.
The description “brushed, screeded or roughtamped” covers those cases where some form of deliberate surface roughening has taken place but not to the extent of exposing the aggregate.
4.
For structural assessment purposes, it may be assumed that the appropriate value of g 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 the interface contact area.
482
DESIGN FOR FLEXURE
the permissible values prescribed by these codes also differ from those provided by BS 8110 and SABS 0100. If the permissible value is exceeded, steel crossing the interface must be provided to carry the horizontal shear. The required amount of steel is calculated by following the provisions of the particular code being used. EXAMPLE 415 Determine the horizontal shear stress at the interface of the precast beam and the in situ slab of the composite section shown in Example 414. The design load at the ultimate lmit state is given by
wu = 12 . w D + 16 . w L = 12 . ´ 11.06 + 16 . ´ 12 = 32.48 kN/ m and the corresponding midspan moment is
M=
1 1 2 2 w L = ´ 32.48 ´ 15 = 913.4 kN. m 8 u 8
It can be shown that the prestressing steel yields at ultimate, so that the stress in the steel f ps = f pu /1.15 = 1617 MPa. The force in the prestressing steel will then be Tps = f ps A ps = 1617 ´1200´103 = 1941 kN
The depth of the stress block s can be calculated from horizontal equilibrium
s=
Tps 0.45 f cu, f b f
=
1941´103 = 119.8 mm 0.45´ 30 ´1200
Since the stress block falls entirely in the slab, the horizontal force that has to be transmitted by the interface C = T ps = 1941 kN. The average horizontal shear stress at the interface is therefore (see Eq. 439) 3
vhu =
C 1941 ´ 10 = = 0.7394 MPa 0.5 L bb 0.5 ´ 15000 ´ 350
The vertical design shear force diagram varies linearly over the span of the beam in such a manner that it is zero at midspan and attains a maximum value at the support. Therefore, if the average horizontal shear stress is distributed in proportion to the vertical shear force diagram, it is obvious that v hu,max = 2 ´ v hu = 2 ´ 0.7394 = 1.48 MPa. If nominal links crossing the interface are provided, the permissible horizontal shear stress is 2.0 MPa for a brushed, screeded or roughtamped surface, according to SABS 0100 (see Table 49). Since this value is greater than v hu,max = 1.48 MPa, no additional links need to be provided under these conditions.
4.4.6 Partial prestressing Prestressed concrete members in which flexural tensile cracks are allowed to develop at service load levels are referred to as being partially prestressed. A partially prestressed member is reinforced by a combination of prestressed and nonprestressed reinforcement, which both contribute to the ultimate strength and serviceability behaviour of the member. Although the nonprestressed steel can be either ordinary reinforcing bars or nonprestressed prestressing steel, the former is usually used for this purpose.
DESIGN
483
When compared to ordinary reinforced concrete, partial prestress offers the advantage of improved deflection control as well as the advantages to be gained from the fact that the member is usually crack free under longterm loads, depending on the degree of prestress. Partial prestressing also offers some advantages over full prestressing (Ref. 410):
• Improved control of camber. • Savings in the cost of prestressing. Since a smaller prestressing force is required, the use of partial prestressing usually leads to savings in the amount of prestressing steel required, the anchorages required and the cost of the work associated with tensioning and grouting (in the case of bonded posttensioning) the tendons.
• Economical use of ordinary reinforcing steel. • Possible improved ductility. The most often quoted disadvantage of partial prestressing, when compared to full prestressing, is that such members can be cracked at service load levels. However, ample evidence exists that if appropriate steps are taken to control flexural cracks in terms of crack width and spacing, the presence of these cracks will not adversely effect the durability of a partially prestressed concrete member (Ref. 426). In addition to providing adequate flexural capacity, together with the prestressed steel, the nonprestressed reinforcement performs the following functions (Ref. 410):
• Properly detailed nonprestressed steel can effectively control both crack width and spacing at service load levels.
• In an unbonded member, some nonprestressed bonded reinforcement should be provided to prevent the development of a single large crack at ultimate and, thereby, to increase the flexural capacity of the member.
• If, at transfer, large tensile stresses are induced in the compression flange, nonprestressed reinforcement can be provided to prevent possible fracture, e.g. in the top flange over the midspan region of a simply supported beam in which the live load is large in comparison to the self weight of the beam.
• In the case of precast beams, properly placed nonprestressed reinforcement will ensure that the beam is sufficiently robust with regard to unexpected stresses which may arise during handling and erection. Numerous procedures have been developed for the flexural design of partially prestressed concrete beams and a comprehensive discussion of a number of these can be found in Ref. 426. The various methods can usually be grouped into one of the following three categories, depending on the limit state which the design procedure initially satisfies:
• Methods which initially satisfy the serviceability limit state. The British design codes BS 8110 (Ref. 47) and BS 5400 (Ref. 48) as well as the South African code SABS 0100 (Ref. 42) recommend a method which is based on a limiting hypothetical tensile stress. The hypothetical tensile stress in a cracked prestressed concrete beam section is defined as the flexural stress which would occur in the extreme tension fibre of the uncracked section. Leonhardt (Ref. 427) and Menn (Ref. 428) each outline procedures which use a crack width limitation as the point of departure for design.
• Methods which initially satisfy the ultimate limit state. The method proposed by Naaman (Ref. 429) makes use of the partial prestressing ratio, while the procedure developed by Bachmann (Refs. 430 and 431) employs the concept of the degree of prestress.
• Methods which simultaneously satisfy the ultimate and serviceability limit states. The procedure proposed by Huber (Ref. 432) is an example of such a method.
484
DESIGN FOR FLEXURE
Over the years, a number of indices have been developed for quantifying the extent of prestressing in a partially prestressed concrete beam section (see Ref. 426). The partial prestressing ratio (PPR) and the degree of prestress k are two such indices: The partial prestressing ratio is defined as the ratio of the ultimate moment of resistance provided by the prestressing steel only to the ultimate moment of resistance provided by all the steel (i.e. the prestressed plus nonprestressed steel). The degree of prestress is defined as the ratio of the decompression moment (i.e. the moment which induces a zero stress in the extreme tension fibre of the section) to the total service load moment and therefore represents the fraction of the total service moment which is counteracted by prestressing effects. Consequently, a value of zero for the degree of prestress corresponds to reinforced 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 beam section is to adopt a unified approach in the sense that the method must apply to the complete spectrum of possible levels of prestress, from fully prestressed concrete through to reinforced concrete. The method should also provide a smooth transition from fully prestressed concrete to reinforced concrete. The design procedure proposed by Bachmann (Refs. 430 and 431) satisfies these requirements and is presented in the following. The method assumes that all the dimensions of the concrete section are known, that all the material properties are known, and that the bending moments due to all dead and live loads can be determined. The section is initially designed to provide the required ultimate strength and is subsequently checked for serviceability as follows: (a)
Select a suitable value for the degree of prestress k or, alternatively, the decompression moment. This choice is strongly dependent on engineering judgment to suit a given consideration such as durability, deflection, fatigue, crack control and cost. The decompression moment is commonly chosen at least equal to the dead load moment (Ref. 431) and it is suggested that in the case of bridges a value larger than the dead load moment is appropriate (Refs. 431 and 433). Wiessler (Ref. 433) recommends that the decompression moment should be taken equal to 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 P t required for developing the decompression moment selected in step (a) is determined by making use of the expression for calculating the flexural stress in an uncracked section. The following expression for calculating P t is derived by setting f bot,s equal to zero and M max equal to the decompression moment M dec in Eq. 443d, and solving the resulting expression for P t :
Pt = 
M dec Z h bot + e A
LM N
OP Q
(465)
The required amount of prestressing steel can subsequently be determined from the calculated value of P t . (c)
Determine the required amount of nonprestressed steel. Nonprestressed reinforcement must be provided to ensure that the ultimate moment of resistance of the section exceeds the required value. The procedure outlined in Section 4.4.3 can be used to design this reinforcement.
(d)
Detail the nonprestressed reinforcement carefully. In addition to its contribution to the ultimate strength, soundly detailed nonprestressed reinforcement can effectively control both crack width and spacing at service load levels. This is often the last step in the design procedure under 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 in later Chapters.
It should be noted that since design is essentially an iterative procedure, the section is usually determined 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.
DESIGN
485
More expansive discussions on this design procedure can be found in Refs. 426 and 434. It should be noted that some of the practical advantages of the method are that it is code independent, that it treats uncertainties in a rational manner and that it does not preclude any serviceability check. The design procedure is illustrated by example 416. EXAMPLE 416 The midspan section of a pretensioned partially prestressed concrete beam, which is simply supported over a span of 14 m, is shown in Fig. 456. In addition to its self weight, the beam must support a uniformly distributed superimposed dead load of 4.9 kN/m and a live load of 8.8 kN/m. Make use of the provisions of SABS 0100 (Ref. 42) to design suitable prestressed and nonprestressed reinforcement for the midspan section so that the decompression moment M dec is equal to the permanent load moment. Use 12.9 mm 7wire super grade strand, for which f pu = 1860 MPa and E p = 195 GPa, for the prestressed reinforcement and take f y = 450 MPa and E s = 200 GPa for the nonprestressed reinforcement. Take f cu = 50 MPa and E c = 34 GPa for the concrete.
b = 350 e = 290 mm
bw = 150 150
A = 165 ´ 10 3 mm2 9
I = 8.938 ´ 10 mm
4
Z top = 2554 . ´ 106 mm3 Zbot = 2554 . ´ 106 mm 3
Aps As 350
Figure 456:
d2 = 650
d1 = 640
h = 700
hf = 150
60 50
Midspan section for example 416. 3
If the self weight of the concrete is taken as g c = 24 kN/m , the self weight of the beam is given by w D = g c A = 3.96 kN/m. Therefore the total permanent load is w Perm = w D + w sdl = 3.96 + 4.9 = 8.86 kN/m. Using the load factors of SABS 0160, the design value of the permanent load moment appropriate to the serviceability limit state is given by
a11. w f L =
2
M Perm
Perm
8
=
(11 . ´ 8.86)´14 2 = 238.8 kN.m 8
Since the decompression moment M dec is taken to be equal to the permanent load moment, and the total design moment at the serviceability limit state is given by
M max
a11. w =
. wL Perm +10
fL
2
8
(11 . ´ 8.86 +10 . ´ 8.8)´14 2 = = 454.4 kN.m 8
the degree of prestress k corresponding to this choice of M dec is
k =
M dec 238.8 = = 0.5255 M max 454.4
Assuming h = 0.84 (i.e. 16% losses), the prestressing force required for a decompression moment M dec = 238.8 kN.m is subsequently obtained from Eq. 465
Pt = 
3 M dec 238.8 ´ 10 == 639.1 kN 6 Z 2554 . ´ 10 h bot + e 0.84 + 290 3 A 165 ´ 10
LM N
OP Q
LM N
OP Q
If 12.9 mm 7wire 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
486
DESIGN FOR FLEXURE
to elastic shortening to be 5.0%, the initial force per strand at transfer is (1  0.05) ´ 139.5 = 132.5 kN (= P t,strand ). Therefore, 639.1/132.5 = 4.823, say 5 strands are required, for which 2 A ps = 500 mm . The nonprestressed reinforcement is designed by considering the required ultimate moment, in exactly the same manner as in example 413. Following the requirements of SABS 0160, the design lo ad ap pro priate to the u ltimate limit state is given by w u = 1 . 2 w Perm + 1 . 6 w L = 1.2 ´ 8.86 + 1.6 ´ 8.8 = 24.71 kN/m, so that the required ultimate moment of resistance of the section is 2
M=
wu L 24.71 ´ 14 2 = = 605.4 kN. m 8 8
For the equivalent rectangular stress block prescribed by SABS 0100, a = 0.45 and b = 0.9, while the design stressstrain curves for the prestressed and nonprestressed steel are as shown in Figs. 4 17 and 423, respectively. Assuming f ps = f py = 1617 MPa, f s = f sy = 391.3 MPa and that the entire compression zone is contained in the flange, the depth to neutral axis is determined by taking moments about the position of the nonprestressed steel and solving the resulting expression for x. Thus,
b
gFGH
M =  a f cu bb x d 2 
IJ K
b
bx  A ps f ps d 2  d 1 2
b
gFGH
605.4 ´ 10 =  0.45 ´ (50) ´ 350 ´ 0.9 x 650 6
g
IJ K
0.9 x  500 ´ 1617 ´ (650  640) 2
Solving for x yields x = 148.4 mm. Therefore s = bx = 0.9 ´ 148.4 = 133.6 mm is less than h f = 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: A ps f ps + As f s + a f cubb x = 0
500 ´ 1617 + As ´ 3913 . + 0.45 ´ (50) ´ 350 ´ 0.9 ´ 148.4 = 0 2
Solving this expression for A s yields A s = 621.8 mm . This area of steel can be provided by 2 2 @ Y20 mm bars, for which A s = 628 mm . The validity of the assumption that f ps = f py and that f s = f sy must be checked. This is done by calculating e ps and e s2 using the strain compatibility approach, as in example 413. Before this can be done, f se must be estimated so that its value is consistent with the assumptions made with regard 2 to the various losses. Since the cross sectional area per strand A ps,strand = 100 mm f se =
h Pt , strand A ps, strand
3
=
0.84 ´ 132.5 ´ 10 = 1113 MPa 100 3
If the effective prestress acting on the section is taken as P = f se A ps = 1113 ´ 500 ´ 10 = 556.6 kN, it can be shown that e ps = 0.01752 and e s2 = 0.01179. Since these values of e ps and e s2 are larger than e py = 0.01329 and e sy = 0.00196, respectively, f ps = f py and f s = f sy (see Figs 417 and 423) as assumed. To summarize: 5 @ 12.9 mm 7wire super grade strands tensioned to 75% of their strength are required together with 2 @ Y20 mm bars.
Note that any other limit state can now be examined, as required. For example, consider the stresses in the concrete at transfer. For the losses assumed here, P t = 5 P t,strand = 5 ´ 132.5 = 662.6 kN, 2 2 . w D L / 8 = 10 . ´ 3.96 ´ 14 / 8 = 97.02 kN. m so that, at transfer, the stresses in the while M min = 10 top and bottom fibres of the section are given by
REFERENCES
487
f top ,t = =
Pt Pe M + t + min A Z top Z top 662.6´103
165´10 = 0.2901 MPa f bot , t = =
3
+
662.6´103 ´ 290 2554 . ´10
6
+
97.02 ´10 6 2554 . ´106
Pt Pe M + t + min A Zbot Zbot 662.6 ´ 10
3
165 ´ 103 = 7.742 MPa
3
+
662.6 ´ 10 ´ 290 2554 . ´ 10 6
+
97.02 ´ 10
6
2554 . ´ 106
If the concrete strength at transfer is taken as f ci = 40 MPa then the permissible tensile and compressive stresses at transfer are f tt = 0.45 f ci = 0.45 40 = 2.846 MPa and f ct = 0.45 f ci = 0.45 ´ (40) = 18 MPa, respectively. A comparison of the calculated concrete stresses with the permissible values clearly demonstrates that the stress limitations prescribed by SABS 0100 are satisfied at transfer.
4.5
REFERENCES
41
Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley & Sons, 1975.
42
South African Bureau of Standards, “The Structural Use of Concrete,” SABS 0100: 1992, Parts 1 and 2, SABS, Pretoria, 1992.
43
Khachaturian, N., Gurfinkel, G., Prestressed Concrete, McGrawHill Book Company, New York, 1969.
44
Hognestad, E., Hanson N. W. and McHenry D., “Concrete Stress Distribution in Ultimate Strength Design”. ACI Journal, Vol. 52, No. 6, December 1955.
45
Rüsch, H., “Researches Toward a General Flexural Theory for Structural Concrete”. ACI Journal, Vol. 57, No. 1, July 1960.
46
Committee of State Road Authorities, “Code of Practice for the Design of Highway Bridges and Culverts in South Africa,” TMH7 Part 3, CSRA, Pretoria, 1989.
47
British Standards Institution, “Structural Use of Concrete, Part 1, Code of Practice for Design and Construction,” BS 8110: Part 1: 1985, BSI, London, 1985.
48
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PREFACE
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 the Society. The contributions made by various members of this committee in terms of planning the text and in terms of their review comments were particularly useful. The authors are particularly indebted to Michael A. Vasarhelyi of the Prestressed Concrete Division for his careful review of the entire text. His comments and suggestions contributed significantly to the value of the book. Vernon Marshall John M. Robberts
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