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  • Foreword

  • Preface

    • Information for Students and Instructors

  • Contents

  • About the Authors

  • Symbols, Acronyms and Abbreviations

    • Capital Roman Letters

    • Small Roman Letters

    • Small Greek letters

    • Subscripts

    • Frequently Used Symbols

    • Reinforced Concrete

    • Steel

    • Others

  • Short Notes on Limit State Method

    • Safety Verifications

    • Combination of Actions

  • 1 General Concepts on Reinforced Concrete

    • Abstract

    • 1.1 Mechanical Characteristics of Concrete

      • 1.1.1 Basic Properties of Concrete

      • 1.1.2 Strength Parameters and Their Correlation

      • 1.1.3 Failure Criteria of Concrete

    • 1.2 Creep

      • 1.2.1 Principles of Creep

      • 1.2.2 Creep with Variable Stresses

      • 1.2.3 Models of Linear Creep

    • 1.3 Structural Effects of Creep

      • 1.3.1 Resolution of the Integral Equation

      • 1.3.2 General Method

      • 1.3.3 Algebraic Methods

    • 1.4 Behaviour of Reinforced Concrete Sections

      • 1.4.1 Mechanical Characteristics of Reinforcement

      • 1.4.2 Basic Assumptions for Resistance Calculation

      • 1.4.3 Steel–Concrete Bond

    • Appendix: Characteristics of Materials

      • Table 1.1: Hardening Curves of Concrete

      • Table 1.2: Strength Classes of Concrete

      • Table 1.3: Deformation Parameters of Concretes

      • Table 1.4: Drying Shrinkage of Concrete

      • Table 1.5: Drying Shrinkage Curves of Concrete

      • Table 1.6: Autogenous Shrinkage of Concrete

      • Table 1.7: Autogenous Shrinkage Curves of Concrete

      • Chart 1.8: Concrete Shrinkage and Nominal Values

      • Table 1.9: Classes of Consistency of Fresh Concrete

      • Chart 1.10: Weight of Concrete Elements

      • Table 1.11: Concrete Production Control

      • Table 1.12: Creep: Class Coefficient

      • Table 1.13: Creep: Ambient Coefficient

      • Table 1.14: Creep: Reference Coefficient

      • Table 1.15: Creep: Effect of Temperature

      • Table 1.16: Creep: Nominal Coefficients

      • Table 1.17: Characteristics of Reinforcing Steel

      • Table 1.18: Bars and Wires: Commercial Diameters

      • Table 1.19: Bars for PC: Standard Diameters

      • Table 1.20: Cold-Drawn Wire: Standard Diameters

      • Table 1.21: Strands: Standard Diameters

      • Chart 1.22: Concrete σ–ε Models

  • 2 Centred Axial Force

    • Abstract

    • 2.1 Compression Elements

      • 2.1.1 Elastic and Resistance Design

      • 2.1.2 Effect of Confining Reinforcement

      • 2.1.3 Effects of Viscous Deformations

    • 2.2 Tension Elements

      • 2.2.1 Verifications of Sections

      • 2.2.2 Prestressed Tie Members

      • 2.2.3 Cracking in Reinforced Concrete Ties

    • 2.3 Cracking Calculations

      • 2.3.1 The Cracking Process

      • 2.3.2 Crack Width

      • 2.3.3 Verification Criteria

    • 2.4 Case A: RC Multi-storey Building

      • 2.4.1 Actions on Columns and Preliminary Verifications

      • 2.4.2 Notes on Reinforced Concrete Technology

      • 2.4.3 Durability Criteria of Reinforced Concrete Structures

    • Appendix: General Aspects and Axial Force

      • Table 2.1: Environmental Conditions: Exposure Classes

      • Chart 2.2: Concrete: Design Strength

      • Chart 2.3: Steel: Design Strength

      • Chart 2.4: Reinforcement: Shaping and Detailing

      • Chart 2.5: Reinforcement: Positioning Tolerances

      • Table 2.6: Bond: Design Strength

      • Chart 2.7: Reinforcement: Anchorages and Overlaps

      • Chart 2.8: Concrete Structures: Minimum Dimensions

      • Chart 2.9: Ordinary Columns: Formulas and Construction Rules

      • Chart 2.10: Confined Columns: Formulas and Construction Requirements

      • Chart 2.11: RC Walls: Construction Requirements

      • Table 2.12: Creep in Columns: Stress Redistribution

      • Table 2.13: Shrinkage in RC: Stress Effects

      • Chart 2.14: Ties in Reinforced and Prestressed Concrete

      • Chart 2.15: Cracking in RC and PC: Verification Scheme

      • Table 2.16: Cracking in RC and PC: Allowable Stresses

      • Table 2.17: Durability: Minimum Cover of Reinforcement

  • 3 Bending Moment

    • Abstract

    • 3.1 Analysis of Sections in Bending

      • 3.1.1 Elastic Design of Sections

      • 3.1.2 Resistance Design of Sections

      • 3.1.3 Prestressed Sections

    • 3.2 Flexural Cracking of Beams

      • 3.2.1 Crack Spacing

      • 3.2.2 Crack Width

      • 3.2.3 Verification Criteria

    • 3.3 Deformation of Sections in Bending

      • 3.3.1 Effects of Creep

      • 3.3.2 Moment-Curvature Diagrams

      • 3.3.3 Flexural Behaviour Parameters

    • 3.4 Case A: Design of Floors

      • 3.4.1 Analysis of Actions

      • 3.4.2 Service Verifications

      • 3.4.3 Resistance Verifications

    • Appendix: Actions and Bending Moment

      • Table 3.1: Partial Safety Factors For Actions

      • Chart 3.2: Formulas of Action Combination

      • Chart 3.3: Section in Bending: Elastic Design—Formulas

      • Chart 3.4: Section in Bending: Resistance Design—Formulas

      • Table 3.5: Section in Bending: Viscous Redistribution of Stresses

      • Chart 3.6: Section in Bending: Additional Formulas

  • 4 Shear

    • Abstract

    • 4.1 Behaviour of RC Beams in Shear

      • 4.1.1 Cracking of Beams

      • 4.1.2 Longitudinal Shear and Shear Reinforcement

      • 4.1.3 Mörsch Truss Model

    • 4.2 Beams Without Shear Reinforcement

      • 4.2.1 Analysis of Tooth Model

      • 4.2.2 Other Resistance Contributions

      • 4.2.3 Verification Calculations

    • 4.3 Beams with Shear Reinforcement

      • 4.3.1 The Modified Hyperstatic Truss Model

      • 4.3.2 The Variable Strut Inclination Truss Model

      • 4.3.3 Serviceability Verifications

    • 4.4 Case A: Beams Design

      • 4.4.1 Analysis of Actions

      • 4.4.2 Serviceability Verifications

      • 4.4.3 Resistance Verifications

    • Appendix: Shear

      • Chart 4.1: Beams Without Shear Reinforcement: Formulas

      • Chart 4.2: Resistance of Beams with Shear Reinforcement: Formulas

      • Chart 4.3: Beams with Shear Reinforcement: Service Conditions and Construction Rules

      • Table 4.4: Shear Cracking: Allowable Stresses in Stirrups

  • 5 Beams in Bending

    • Abstract

    • 5.1 Calculation Models of Beams in Bending

      • 5.1.1 Arch Behaviour

      • 5.1.2 Truss Model

      • 5.1.3 Standard Application Procedure

    • 5.2 Strut-and-Tie Balanced Schemes

      • 5.2.1 Support Details

      • 5.2.2 Corbels and Deep Beams

      • 5.2.3 Punching Shear in Slabs

    • 5.3 Flexural Deformations of Beams

      • 5.3.1 Curvature Integration

      • 5.3.2 Nonlinear Analysis of Hyperstatic Beams

      • 5.3.3 Collapse Behaviour of Hyperstatic Beams

    • 5.4 Case A: Shallow Rectangular Beam

      • 5.4.1 Serviceability Verifications

      • 5.4.2 Resistance Verifications

      • 5.4.3 Deflection Calculations

    • Appendix: Elements in Bending

      • Chart 5.1: Arch Behaviour: Formulas

      • Chart 5.2: Bearing Details: Formulas

      • Chart 5.3: Corbels: Formulas

      • Chart 5.4: Punching Shear in Slabs: Formulas

      • Chart 5.5: Coefficients for Moment Redistribution

      • Chart 5.6: Allowable Deformations of elements in Bending

  • 6 Eccentric Axial Force

    • Abstract

    • 6.1 Elastic Design of the Section

      • 6.1.1 Axial Compression Force with Small Eccentricity

      • 6.1.2 Compression and Tension with Uniaxial Bending

      • 6.1.3 Compression and Tension with Biaxial Bending

    • 6.2 Resistance Design of the Section

      • 6.2.1 Failure Mechanisms of the Section

      • 6.2.2 Resistance Verifications of the Section

      • 6.2.3 Design for Biaxial Bending

    • 6.3 Flexural Behaviour of Columns

      • 6.3.1 Design Models of Columns

      • 6.3.2 Moment-Curvature Diagrams

      • 6.3.3 Nonlinear Analysis of Frames

    • 6.4 Case A: Design of Columns

      • 6.4.1 Flexural Actions in Columns

      • 6.4.2 Serviceability Verifications

      • 6.4.3 Resistance Calculations

    • Appendix: Eccentric Axial Force

    • Chart 6.1: Eccentric Axial Force: Elastic Design––Formulas

    • Chart 6.2: Combined Axial Force and Uniaxial Bending: Resistance Design––Formulas

    • Chart 6.3: Columns Under Combined Compression and Bending: Shear Resistance

    • Chart 6.4: Combined Axial Force and Uniaxial Bending: Supplementary Formulas

    • Chart 6.5: Combined Compression and Biaxial Bending––Section with 4 Bars

    • Graph 6.6: Combined Biaxial Bending––Section with 8 Bars

    • Graph 6.7: Combined Biaxial Bending–Peripheral Reinforcement

    • Graph 6.8: Combined Biaxial Bending––Section with 6 Bars

    • Graph 6.9: Combined Biaxial Bending––2 Sides Reinforcement

    • Table 6.10 Combined Biaxial Bending––Analytical Verification

  • 7 Instability Problems

    • Abstract

    • 7.1 Instability of Reinforced Concrete Columns

      • 7.1.1 Analysis of Columns Under Eccentric Axial Force

      • 7.1.2 Methods of Concentration of Equilibrium

      • 7.1.3 Creep Effects

    • 7.2 Second-Order Analysis of Frames

      • 7.2.1 One-Storey Frames

      • 7.2.2 Multistorey Frames

      • 7.2.3 General Case of Frames

    • Appendix: Instability of Columns

      • Table 7.1: Instability Verification of Columns in Compression

      • Chart 7.2: Second Order Analysis of Frames

      • Chart 7.3: One-Storey Frames

  • 8 Torsion

    • Abstract

    • 8.1 Beams Subject to Torsion

      • 8.1.1 Peripheral Resisting Truss

      • 8.1.2 Improvement and Application of the Model

      • 8.1.3 Other Aspects of the Torsional Behaviour

    • 8.2 Case A: Stability Core

      • 8.2.1 Calculation of Internal Forces

      • 8.2.2 Verifications of the Current Section

      • 8.2.3 Verifications of Lintels and Stairs

    • Appendix: Torsion

      • Table 8.1: Torsion: Elastic Design—Formulas

      • Chart 8.2: Torsion: Resistance Design—Formulas

      • Chart 8.3: Torsion: Interaction Formulas

      • Chart 8.4: Torsion: Construction Requirements

  • 9 Structural Elements for Foundations

    • Abstract

    • 9.1 Isolated Foundations

      • 9.1.1 Massive Foundations

      • 9.1.2 Footing Foundations

      • 9.1.3 Pile Foundations

    • 9.2 Continuous Foundations

      • 9.2.1 Foundation Beams

      • 9.2.2 Structure–Foundation Interaction

      • 9.2.3 Foundation Grids and Rafts

    • 9.3 Retaining Walls

      • 9.3.1 Gravity Walls

      • 9.3.2 Foundation Retaining Walls

      • 9.3.3 Diaphragm Walls

    • 9.4 Case A: Foundation Design

      • 9.4.1 Verification of Footings

      • 9.4.2 Design of the Retaining Wall

      • 9.4.3 Design of the Corewall Foundation

    • Appendix: Data on Soils and Foundations

      • Table 9.1: Soil Parameters

      • Chart 9.2: Soil Resistance—Formulas

      • Table 9.3: Parameters of Resistance Formulas

      • Chart 9.4: Lateral Earth Pressure

      • Table 9.5: Active and Passive Pressure Coefficients

      • Table 9.6: Partial Safety Factors

      • Chart 9.7: Construction Requirements of Foundations

      • Chart 9.8: Verifications Against Overturning and Sliding

      • Chart 9.9: Reinforced Concrete Footings: Resistance Verifications

  • 10 Prestressed Beams

    • Abstract

    • 10.1 Prestressing: Technological Aspects

      • 10.1.1 Prestressing Systems

      • 10.1.2 Instantaneous Losses

      • 10.1.3 Long-Term Losses

    • 10.2 Tendons Profile

      • 10.2.1 Loads Equivalent to the Tendon

      • 10.2.2 Available Moment and Limit Points

      • 10.2.3 Hyperstatic Beams

    • 10.3 Resistance Calculations

      • 10.3.1 Verification of Prestressed Concrete Sections

      • 10.3.2 Resistance Models of Prestressed Beams

      • 10.3.3 Anchorage and Diffusion of Precompression

    • 10.4 Design Examples

      • 10.4.1 Pretensioned Concrete Element

      • 10.4.2 Post-tensioned Concrete Beam

      • 10.4.3 Prestressed Concrete Flanged Beam

    • Appendix: Data on Prestressing

      • Chart 10.1: Prestressing and Instantaneous Losses

      • Chart 10.2: Losses Due to Steel Relaxation

      • Chart 10.3: Experimental Evaluation of Relaxation

      • Chart 10.4: Total Prestressing Losses

      • Chart 10.5: Stages of Verification of Stresses

      • Chart 10.6: Verification of Resistance in Bending

      • Chart 10.7: Anchorage of Tendons

      • Chart 10.8: Additional Data

  • References

    • General Books

    • Specific Books

    • Handbooks

    • European CEN Standards

Nội dung

Springer Tracts in Civil Engineering Giandomenico Toniolo Marco di Prisco Reinforced Concrete Design to Eurocode English Edition by Michele Win Tai Mak Springer Tracts in Civil Engineering www.TechnicalBooksPDF.com Springer Tracts in Civil Engineering (STCE) publishes the latest developments in Civil Engineering—quickly, informally and in top quality The series scope includes monographs, professional books, graduate textbooks and edited volumes, as well as outstanding Ph.D theses Its goal is to cover all the main branches of civil engineering, both theoretical and applied, including: Construction and Structural Mechanics Building Materials Concrete, Steel and Timber Structures Geotechnical Engineering Earthquake Engineering Coastal Engineering Hydraulics, Hydrology and Water Resources Engineering Environmental Engineering and Sustainability Structural Health and Monitoring Surveying and Geographical Information Systems Heating, Ventilation and Air Conditioning (HVAC) Transportation and Traffic Risk Analysis Safety and Security To submit a proposal or request further information, please contact: Pierpaolo Riva at Pierpaolo.Riva@springer.com, or Li Shen at Li.Shen@springer.com More information about this series at http://www.springer.com/series/15088 www.TechnicalBooksPDF.com Giandomenico Toniolo Marco di Prisco • Reinforced Concrete Design to Eurocode English Edition by Michele Win Tai Mak 123 www.TechnicalBooksPDF.com Giandomenico Toniolo Department of Civil and Environmental Engineering Politecnico di Milano Milan Italy Marco di Prisco Department of Civil and Environmental Engineering Politecnico di Milano Milan Italy Publisher and Authors acknowledge the role and contribution of Michele Win Tai Mak, in translating into English the Italian language work, authoring the foreword and providing/ suggesting updates on the reference readings ISSN 2366-259X ISSN 2366-2603 (electronic) Springer Tracts in Civil Engineering ISBN 978-3-319-52032-2 ISBN 978-3-319-52033-9 (eBook) DOI 10.1007/978-3-319-52033-9 Library of Congress Control Number: 2017930409 © Springer International Publishing AG 2017 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland www.TechnicalBooksPDF.com Foreword This book on reinforced concrete design is unique for its comprehensive approach, as each topic is thoroughly analysed from more theoretical aspects, through the development of design formulas with their assumptions and justifications, and terminates with construction requirements and practical examples The textbook is primarily intended for undergraduate students and young practitioners However, the strong link between theory and practical applications makes it a valuable handbook that experienced engineers would also find useful As the complexity of projects increases, designers face progressively greater challenges, structural engineering deviates from standard solutions bringing the designers back to first principles; a thorough understanding of the theory and the structural fundamentals becomes extremely important to comprehend limits and worthiness of models The original book has been at the forefront of the development of the Limit State Design for the structural use of concrete in Italy and it has been a national reference for academics and practitioners for many years; since the first edition has been published, it has been continuously updated to incorporate the latest developments in reinforced concrete design Because of its validity, the preface to the original edition has been kept as a general introduction to the work, with few updates by the authors The terminology, definitions and explanations of the original text are remarkably rigorous, in line with a cultural tradition that values consistency and preciseness, and this aspect of the book has been retained as much as possible The need to make the English edition comply with a more practical nature of the industry made certain aspects of the translation particularly difficult, especially where theoretical rigour and preciseness had to be abandoned in favour of terms and expressions that are common in practice Conversely, when deemed important, consistency and accuracy have been retained at the cost of less immediate clarity I would like to apologize to the reader for any errors or mistakes in the text that may have inadvertently been made, despite the countless reviews of a perfectionist who probably will never learn that “Better is Enemy of Good” v www.TechnicalBooksPDF.com vi Foreword Finally, I wish to thank the authors, Proff Toniolo and di Prisco for giving me the opportunity to work on their book and bring it to a wider international audience, and for their continuous support and assistance Michele Win Tai Mak Michele Win Tai Mak is a Structural Engineer at Ove Arup & Partners His research and professional interests include the analysis and design of tall buildings, the assessment of existing reinforced concrete structures, seismic engineering, failure analysis and cementitious composites He also undertakes project consultations and tutorials with engineering and architecture students in several universities in the United Kingdom He holds a Master’s degree from Politecnico di Milano and a Diplôme d’Ingénieur from École Spéciale des Travaux Publics, du Bâtiment et de l’Industrie de Paris www.TechnicalBooksPDF.com Preface The present work derives from the university textbook originally drafted within the cultural tradition of the Structural Engineering School of the Politenico di Milano This English edition has been drafted following the publication of two fundamental documents: • Eurocode 2—Design of concrete structures; • fib Model Code, as better specified in References The first one represents the last amendment of the final version of the official EN design code collecting the consolidated principles and rules for concrete structures The second document represents the new edition of the design code issued by the International Association of Concrete Structures, collecting the latest innovative developments of the research proposed for possible future updating of the official regulations With respect to the original edition, the text has therefore been revised and extended, incorporating the most important technological-scientific innovations, which are the basis of the two aforementioned documents, to present a complete set of limit state design criteria of the modern theory of reinforced concrete, saving its educational purposes First of all, the completeness typical of a general treatise has been abandoned, incorporating the topics considered of fundamental educational value but leaving out many further developments and alternatives Specific references are reserved for those The intent has been to develop the textbook examining in depth methodological more than notional aspects of the presented topics, and focusing on the verification of assumptions, on the rigorousness of the analysis and on the consequent degree of reliability of results The textbook refers to part of the course of structural design and analysis for civil and building engineering students Form and extent of arguments are mainly driven by teaching needs, as developed throughout the weeks of the academic year vii www.TechnicalBooksPDF.com viii Preface About its field of competence, the course of structural design and analysis is placed as a logical development after the course of structural mechanics The fundamental models of structural behaviour are recalled from this discipline, fitting them out with the actual thicknesses due to the real construction materials The specific properties of these materials and their complex structural arrangement bring up the problem of the reliability of the model: not just one unique solution results, but a domain of possible solutions characterized by different degrees of refinement can be obtained and in any case influenced by the randomness of the input data Structural design and analysis is limited to problems of verifications related to simple structures for which the extraction of a model is simple The wider problem relative to the design choices and the analysis of real complex building arrangements is left to the subsequent specialized courses of the final academic year Information for Students and Instructors The organization of teaching activities has weekly cycles of exercise sessions devoted to numerical applications of the topics already discussed from the theoretical point of view during the lessons The structure of chapters in this text closely follows this organization Each chapter develops an organic topic, which is eventually illustrated by examples in each final paragraph containing the relative numerical applications The application paragraphs altogether follow an overall plan with the development of the design of principal structural elements in a typical construction ‘from roof, to foundations’ Other than being an opportunity for the application of single topics (e.g beam in bending, column in compression, foundation footing, etc.), the overall subject shows the first examples of extraction of calculation models from a real structural context and eventually gives the complete building arrangement on which the fundamental verifications of overall stability are to be carried Specific appendices are also reported at the end of each chapter, to be used for practical design applications, containing data about materials, formulas for verifications and auxiliary tables, in line with the latest European regulations Milan, Italy Giandomenico Toniolo Marco di Prisco www.TechnicalBooksPDF.com Contents General Concepts on Reinforced Concrete 1.1 Mechanical Characteristics of Concrete 1.1.1 Basic Properties of Concrete 1.1.2 Strength Parameters and Their Correlation 1.1.3 Failure Criteria of Concrete 1.2 Creep 1.2.1 Principles of Creep 1.2.2 Creep with Variable Stresses 1.2.3 Models of Linear Creep 1.3 Structural Effects of Creep 1.3.1 Resolution of the Integral Equation 1.3.2 General Method 1.3.3 Algebraic Methods 1.4 Behaviour of Reinforced Concrete Sections 1.4.1 Mechanical Characteristics of Reinforcement 1.4.2 Basic Assumptions for Resistance Calculation 1.4.3 Steel–Concrete Bond Appendix: Characteristics of Materials 1 10 18 22 23 26 28 33 35 37 38 40 41 46 52 57 Centred Axial Force 2.1 Compression Elements 2.1.1 Elastic and Resistance Design 2.1.2 Effect of Confining Reinforcement 2.1.3 Effects of Viscous Deformations 2.2 Tension Elements 2.2.1 Verifications of Sections 2.2.2 Prestressed Tie Members 2.2.3 Cracking in Reinforced Concrete Ties 83 83 87 91 96 101 102 104 108 ix www.TechnicalBooksPDF.com Design Examples Fig 10.54 Reinforcement details of the element 10.4 821 822 10 Prestressed Beams Appendix: Data on Prestressing Chart 10.1: Prestressing and Instantaneous Losses Beams in pressed reinforced concrete Symbols Po Npo = Po cos / Vpo = Po sin / P0o N 0po ¼ P0o cos / V 0po ¼ P0o sin / / e M 0g V 0g Ap Ep initial prestressing (at concrete decompression) axial component of Po transverse component of Po prestressing at tendons tensioning axial component of P0o transverse component of P0o angle of the tendon on the beam axis tendon eccentricity (positive towards the bottom) bending moment due to self-weight shear due to self-weigth area of prestressing reinforcement elastic modulus of prestressing reinforcement see also Charts 3.4 and 6.1 Initial Stresses “i” (post-tensioned tendons) N 0po M 0g À N 0poe yc rci ¼ À  À I i Ai N 0po V 0g À V 0poe rGi ¼ À  sG ¼ zbG Ai 0 N M À N e po g po e rci ẳ  ỵ I i Ai N 0po M 0g À N 0poe yc r0ci ¼ À  ỵ I i Ai P rpi ẳ o Ap upper edge centroid tendon level lower edge tendon with positive tensile stresses where  i ẳ Ac ỵ ae As A ae ¼ E s =E c and with I i ; yc ; y0c ; … calculated similarly to the homogenized concrete section with passive reinforcement The allowable stresses for the verification in service can be deduced from Charts 2.2 and 2.3 Appendix: Data on Prestressing 823 Losses Due to Friction (post-tensioned tendons) P ¼ P1 eÀlðw þ asÞ with P1 P s w l a tension applied at end (active side) force in the tendon at the abscissa s from end abscissa of the considered section (in m) progressive angle in rad between and s (absolute value) friction coefficient in the duct unintentional unit angular deviation Without more accurate data, for tendons against metallic ducts without rust one can assume: l ¼ 0:20 for wires or strands l ¼ 0:35 for smooth bars l ¼ 0:65 for ribbed bars a ¼ 0:01 rad/m Elastic Losses (pre-tensioned tendons) N po M g À N po e À yc Ii Ai V g À V po N po rGi ¼ À sG ¼ Ai zbG M À N e N po po g rci ẳ ỵ e Ii Ai N po M g À N po e r0ci ¼ À þ yc Ii Ai 0 Ã rpi ¼ rpo þ ae rci rci ¼ À upper edge centroid tendon level lower edge tendonrpo ẳ Po =Ap ị with Ai ẳ Ac ỵ ae As ỵ a0e Ap a0e ẳ E p =E c and with I i ; yc ; y0c ; … calculated similarly to the homogenized concrete section With passive and active reinforcement (with cos / ≅ 1, Npo ≅ Po) Chart 10.2: Losses Due to Steel Relaxation The tension loss due to steel relaxation, calculated after t hours from the load application in relation to the initial stress rpi, is given by 824 10 qs ¼ qsị ẳ Prestressed Beams Drps ẳ ql s0:751rị rpi with s = t/1000 and r = rpi/fptk The loss at 1000 h (s = 1) is given by 1 cðrÞ q1 ¼ q 1 is the loss measured at the time above for an initial stress correWhere q sponding to r = 0.7 and an average temperature of 20 °C For the following product classes Class 1—Ordinary wires and strands Class 2—Stabilized wires and strands Class 3—Prestressing bars In the absence of test results, one can assume and q1 ¼ 8:0% for class q1 ¼ 2:5% q1 ¼ 4:0% for class for class !4 r À 0:4 crị ẳ 0:3 !4 r 0:5 crị ¼ 0:2 for class for classes and The final value of the loss q1 ¼ Drp1 =rpi can be calculated with s = 500 (%57 years) The table gives the final loss for the different values of the initial stress rpi f ptk Class 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 rpi f ptk 100Drp1 =rpi 0.00 3.23 7.41 11.59 15.50 19.01 22.09 24.71 26.90 28.67 30.06 Class 0.00 1.09 2.50 3.90 5.22 6.40 Class 0.00 1.74 3.99 6.25 8.35 10.24 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 100Drp1 =rpi Class Class 32.25 32.36 32.43 32.43 32.39 32.29 32.15 31.98 31.76 31.51 9.06 9.37 9.65 9.90 10.12 10.31 10.47 10.60 10.71 10.79 Class 14.49 15.00 15.45 15.84 16.19 16.49 16.75 16.96 17.14 17.27 (continued) Appendix: Data on Prestressing 825 (continued) rpi 100Drp1 =rpi f ptk 0.61 0.62 0.63 0.64 0.65 Class Class Class 30.62 31.10 31.49 31.81 32.06 6.94 7.44 7.90 8.32 8.71 11.10 11.90 12.64 13.31 13.93 rpi f ptk 100Drp1 =rpi Class Class Class 0.76 0.77 0.78 0.79 0.80 31.22 30.91 30.57 30.21 29.82 10.86 10.90 10.92 10.92 10.90 17.37 17.43 17.47 17.47 17.45 Chart 10.3: Experimental Evaluation of Relaxation If the experimental value q1exp of the tension loss due to relaxation is available, measured at 1000 h from the load application for an initial stress Dri exp ¼ r exp f ptk , the final loss Drp1 ¼ q1 rpi can be evaluated with q1 q1 ẳ j where j ẳ crị5000:751rị 1 ẳ q1 exp =cðr exp Þ q (see Chart 10.2 for the expression c(r)) The following tables give the values of the factors j(r) and 1/c(rexp), respectively, for the different product classes as dened Chart 10.2 j ẳ jrị r ẳ rpi =f ptk Class Class and 0.60 2.962 2.561 1=cr exp ị r exp ẳ rpi exp =f ptk Class Class and 0.60 1.717 2.519 0.65 3.175 3.482 0.65 1.275 1.467 0.70 4.048 4.048 0.75 3.938 4.350 0.80 3.728 4.361 0.70 1.000 1.000 0.75 0.814 0.743 0.80 0.681 0.582 826 10 Prestressed Beams Chart 10.4: Total Prestressing Losses With respect to the initial values calculated with the formulas of Chart 10.1, the stresses in the materials are subject to the following long-term losses Symbols DPq DPs DPv DP DNp = DP cos / DVp = DP sin / loss due to steel relaxation loss due to concrete shrinkage loss due to concrete creep total prestressing loss axial component of DP transverse component of DP see also Chart 10.1 Long-Term Losses (with cos / ≅ 1, DNp ≅ DP) Without more precise data, the following final values can be assumed DPq ¼ Ap Drq1 DPs ¼ Ap Dr0p ¼ Ap E p ecs1 DPv ¼ Ap Dr00p ¼ Ap a0e /1 rÃci Total loss loss due toi relaxation (Chart 10.2) loss due to shrinkage (Chart 1.5) loss due to creep (Chart 1.14) DP ẳ DPq ỵ DPs ỵ DPv ẳ Ap Drp In order to take into account the interaction with the losses due to shrinkage and creep, the one due to relaxation can be reduced with the coefficient 0.8 Variations of stresses (positive in tension) DN p DN p e À y Ii c Ai DN p DV p DrG ẳ ỵ DsG ẳ ỵ Ai zbG DN p DN p e 0 Drc ẳ ỵ ỵ y Ii c Ai DN p DN p e2 Drc ẳ ỵ ỵ Ai Ii Drp ẳ Drp ỵ a0e Drc Drc ẳ ỵ upper edge centroid lower edge tendon level tendon Chart 10.5: Stages of Verification of Stresses The main stages of verification of stresses in the materials of a beam prestressed off-site and then transported to its final position are listed below The verification in particular concern: Appendix: Data on Prestressing 827 • the maximum compressions in concrete to avoid an excessive propagation of microcracking (see Chart 2.2); • the maximum tensions in steel, both passive and prestressed to avoid yielding (see Chart 2.3); • the maximum tensions in concrete for the verification of decompression and formation of cracks (see Chart 2.2); • the range of tensions in steel measured from decompression of concrete for the verifications of cracks opening (see Chart 2.16) The verifications related to cracking refer to the criteria of Chart 2.15 For the stresses due to shear see Charts 4.1, 4.5 and 4.6 The stresses in the section of the beam are assumed to be calculated with the load combinations related to the serviceability limit states (see Chart 3.2) The stages of verification of stresses can be articulated in various ways, depending of the type of structure, the construction methods and the refinement of the calculation itself The main ones, listed below, are illustrated with the schemes of the following figure and represent the incremental actions to be progressively cumulated to the previous ones A separate case is the transient condition (a′) of lifting that ends with no further developments Initial Stage (a): “Precompression” (with self-weight g1) The initial stresses rci, … are calculated with the formulas of Chart 10.1; the prestressing losses of the scheme (b) will then be evaluated based on these values 828 10 Prestressed Beams Transient Stage (a′): “Lifting” (change of supports and dynamic effects) Stresses are calculated with the same formulas of Chart 10.1 where the internal forces Mg0 and Vg0 due to the self-weight refer to the new configuration of supports with the action adg1 Without more accurate calculations, the dynamic effects can be calculated in the most unfavourable way with the coefficient ad ẳ ặ 0:15 The verications under minimum loads are usually carried on this stage Incremental stage (b): “Losses” The stress losses with respect to the initial values are calculated with the formulas of Chart 10.4 Incremental Stage (c): “Permanent” The effects of permanent loads g2 are calculated with Drc ¼ À M 00g Ii yc V 00g zbG M 00g Drp ¼ ae e Ii 00 Mg Dr0c ẳ ỵ y Ii c DsG ẳ upper edge centroid tendon lower edge The combination (a) + (b) + (c) for the verification under permanent loads is deduced at this point Incremental Stage (d): “Serviceability” The effects of variable loads wiq are calculated with Drc ¼ À Mq y Ii c Vq zbG Mq Drp ¼ a0e e Ii Mq Dr0c ẳ ỵ y Ii c DsG ẳ upper edge centroid tendon lower edge The combination (a) + (b) + (c) + (d) for the verifications under serviceability loads is deduced at this point, with the combination coefficient wi deduced from Chart 3.2 depending on the requirements (frequent or rare combination) Tensile stresses have been assumed positive in what mentioned above Appendix: Data on Prestressing Chart 10.6: Verification of Resistance in Bending Section in prestressed concrete subject to uniaxial bending Symbols MEd design value of applied bending moment MRd design value of the resisting bending moment As, A′s areas of passive reinforcement in tension and compression d, d′ distance of reinforcement from the edge in compression (see figure) Ap area of prestressing reinforcement dp distance of tendon from the edge in compression (see figura) b width of the edge in compression (see figure) bw web width (see figure) t flange thickness (see figure) epo initial strain of prestressing reinforcement see also Charts 2.2, 2.3 and 10.1 Verifications with Resisting Tendon (longitudinal tendon with initial strain epo) Rectangular section (or T-shaped with x t) À Á M Rd ẳ Ap f pd d p x=2 ỵ As f yd d x=2ị ỵ A0s f yd x=2 À d Þ ! M Ed 829 830 10 with x ẳ Ap f pd ỵ As f yd À A0s f yd =bf cd and with the limits dp x ecu ỵ epo x dx ecu es ¼ x xÀd ecu e0s ¼ x ep ¼  eÃyd À eyd À eyd ep es e0s  epd Á esd Á where x ffi x=0:8 ecu ẳ 0:35% up to class C50=60ị T-shaped (with x [ t) À Á À Á M Rd ¼ atf cd d p t=2 ỵ bwxf cd d p x=2 ỵ ỵ As f yd d d p ỵ A0s f yd d p À d ! M Ed with À Á x ẳ Ap f yd ỵ As f yd A0s f yd À atf cd =bw f cd a ¼ b À bw and with the same limits of the rectangular section Verifications with Prestressing Force (constant force Po = Ep epo in place of the tendon) Rectangular section (or T-shaped with x t) Prestressed Beams Appendix: Data on Prestressing 831 M Rd ẳ bxf cd yo x=2ị þ As f yd ðd À yo Þ þ A0s f yd ðyo À d Þ ! M ÃEd where M ÃEd ¼ M Ed À N pd e N pd ¼ Pd cos / Pd ¼ cP PO with x ẳ N pd ỵ As f yd À A0s f yd =bf cd and with the limits À Á dÀx ecu eyd es esd x À Á x À d0 ecu e0s ¼ eyd e0s x es ¼ where x ffi x=0:8 ecu ¼ 0:35% ðup to class C50=60Þ T-shaped section (with x [ t) M Rd ẳ atf cd yo t=2ị ỵ bwxf cd yo x=2ị ỵ ỵ As f yd d yo ị ỵ A0s f yd yo d Þ ! M ÃEd where M ÃEd ¼ M Ed À N pd e N pd ¼ Pd cos / Pd ¼ cP PO with À Á x ¼ N pd ỵ As f yd A0s f yd atf cd =bw f cd a ¼ b À bw and with the same limits of the rectangular section 832 10 Prestressed Beams Chart 10.7: Anchorage of Tendons End anchorages of unbonded post-tensioned or bonded pre-tensioned tendons Verification of Anchors of Post-Tensioned Tendons The verification of concrete bearing can be set with P d ¼ cP P PRd where the resisting value of prestressing force is given by PRd ¼ f Ãcd Ao with Ao ¼ a0 b0 a0 ; b0 gross area of loaded print sides of the anchorage plate and where f Ãcd ¼ f cdj pffiffiffiffiffiffiffiffiffiffiffiffiffi A1 =Ao with f cdj ¼ f ckj =cC A1 ¼ ab design resistance at tensioning involved loaded surface The sides a, b are calculated with a ẳ a0 ỵ 2d a 3a0 b ẳ b0 ỵ 2d b 3b0 based on the margins da, db of the loaded print with respect to the closest edges of the section In any case, the surfaces involved are to be cut halfway to the closest adjacent plates Anchorage Length of Bonded Tendons The end anchorage length of bonded pre-tensioned strand is calculated with lb ¼ / f pd f bd where / is the diameter of the bar, wire or strand, fpd = 0.9 fptk/cS is its design resistance, fbd = bbfctk/cC the design value of the bond resistance Without more accurate measurements, it can be assumed Appendix: Data on Prestressing bb ¼ 2:25 bb ¼ 3:00 833 for wires and ribbed bars for strands If the tendon is cut without a previous slow release, an uneffective end segment should be added, equal to lo ffi 7/ It can be assumed that the effectiveness of the strand vary linearly within the bonded segment x l b: f pd x=lb Chart 10.8: Additional Data Prestressed elements in bending with or without shear reinforcement Verifications in Shear with Resisting Tendons (bonded pre-tensioned strands along the edge in tension) What given in Charts 4.1, 4.2, 4.5 and 4.6, respectively, for elements without shear reinforcement and for resistance, serviceability and cracking of beams with stirrups is valid Verifications in Shear with Prestressing Force (Constant force in place of tendons) What given in Chart 6.21 is valid, provided No = Pd cos / is assumed and VEd is substituted with V ÃEd ¼ V Ed À Pd sin /: Construction Rules and Support Details What given in Chart 4.5 for shifting of moments, spacing and minimum shear reinforcement is valid, as well as what given in Chart 5.2 for support details, and what mentioned in Chart 6.22 for minimum longitudinal bonded reinforcement References General Books 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 T Y Lin, N H Burns: Design of Prestressed Concrete Structures, Wiley, 2004 P M Ferguson, J E Breen, J O Jirsa: Reinforced Concrete Fundamentals, Wiley, 1988 M K Hurst: Prestressed Concrete Design, Spon, 2007 J R Libby: Modern Prestressed Concrete, CBS Publishers, 2007 P Bhatt, T J MacGinley, B S Choo: Reinforced Concrete Design to Eurocodes: Design Theory and Examples, CRC Press, 2013 R Favre, J P Jaccoud, O Burdet, H Charif: Dimensionnement de structures en béton, Presses Plytech et Univ Romandes, 1990 R I Gilbert, N C Mickleborough, G Ranzi: Design of Prestressed Concrete to AS3600-2009, CRC Press, 2015 R Favre, J P Jaccoud, M Koprna, A Radojicic: Dimensionnement des structures en béton, Presses polytechniques universitaires et romandes, 1990 R Chaussin, A Fuentes, R Lacroix, J Perchat: La précontrainte, Ponts et chaussées, 1992 R Walther, M Miehlbradt: Dimensionnement des structures en béton: Bases et technologie, Presses Plytech et Univ Romandes, 1990 E O'Brien, A Dixon, E Sheils: Reinforced and Prestressed Concrete Design to EC2, CRC Press, 2012 M P Collins, D Mitchell: Prestressed Concrete Structures, Prentice-Hall, 1997 B G Gerwick: Construction of Prestressed Concrete Structures, Wiley, 1997 E G Nawy: Reinforced Concrete: a Fundamental Approach, Pearson, 2008 M G Richardson: Fundamentals of Durable Reinforced Concrete, Taylor and Francis, 2002 J McCormac, J H Brown: Design of Reinforced Concrete, Wiley, 2013 S Teng, F K Kong: Reinforced and Prestressed Concrete, Taylor & Francis 2007 R Park, T Paulay: Reinforced Concrete Structures, Wiley on line, 2009 fib: Structural concrete – Textbook on behaviour, design and performance, bull nn 51/52/53/54 & 62 (voll 1, 2, 3, & 5), 2009 to 2012 fib: Model Code 2010, bull nn 65/66 (voll & 2), 2012 W.H Mosley, R Hulse, J.H Bungey,: Reinforced Concrete Design to Eurocode 2, Palgrave, 2012 A.E Naaman: Prestressed Concrete Analysis and Design, Techno Press 3000, 2012 H Bachmann, A Steinle: Precast Concrete Structures, Ernst & Sohn, 2012 G F Limbrunner: Reinforced Concrete Design, Pearson, 2014 P Bhatt, T J MacGinley, B S Choo: Reinforced Concrete Design to Eurocodes: Design Theory and Examples, CRC Press, 2014 J K Wight: Reinforced Concrete: Mechanics and Design, Pearson, 2016 © Springer International Publishing AG 2017 G Toniolo and M di Prisco, Reinforced Concrete Design to Eurocode 2, Springer Tracts in Civil Engineering, DOI 10.1007/978-3-319-52033-9 835 836 References Specific Books 27 28 29 30 31 32 33 34 35 A C Liebenberg: Concrete Bridges: Design and Construction, Longman, 1992 J Moehle: Seismic Design of Reinforced Concrete Buildings, McGraw-Hill, 2014 T C Hsu, Y.-L Mo: Unified Theory of Concrete Structures, Wiley, 2010 J E Bowles: Foundation Analysis and Design, McGraw-Hill, 2001 R Park, W L Gamble: Reinforced Concrete Slabs, Wiley, 1999 E G Nawy: Fundamentals of High-Performance Concrete, Wiley 2001 P Mondorf: Concrete Bridges, Taylor & Francis, 2006 A Bentur, S Mindess, Routledge: Fibre Reinforced Cementitious Composites, CRC Press, 2006 A Peled, A Bentur, B Mobasher: Textile Reinforced Concrete, CRC Press, 2017 Handbooks 36 37 38 39 40 J Eibl (Ed.): Concrete Structures Euro-design Handbook, Ernst & Sohn 1995 C E Reynolds, J C Steedman: Reinforced Concrete Designer's Handbook, CRC Press, 2007 T Threlfall: Reinforced Concrete Designer Handbook, Taylor & Francis, 2007 C Sigmund: Cemento armato Manuale di calcolo agli stati limite – Handbook delle strutture, Flaccovio, 2008 SP-017(14): The Reinforced Concrete Design Handbook Volumes & Package European CEN Standards – EN 1992-1-1:2015 Eurocode – Design of concrete structures – Part 1-1: General rules and rules for buildings – EN 1992-1-2:2011 Eurocode – Design of concrete structures – Part 1-2: Structural fire design – EN 1992-2:2013 Eurocode – Design of concrete structures – Part 2: Concrete bridges – Design and detailing rules – EN 1992-3:2011 Eurocode – Design of concrete structures – part 3: Liquid retaining and containment structures – EN 206: 2014 Concrete – Specification, performance, production and conformity – EN 10080: 2006 Steel for the reinforcement of concrete – Weldable reinforcing steel – General – EN 10138-1: * Prestressing steel – Part 1: General requirements – EN 10138-2: * Prestressing steel – Part 2: Wire – EN 10138-3: * Prestressing steel – Part 3: Strand – EN 10138-4: * Prestressing steel – Part 4: Bars – EN 13670: 2013 Execution of concrete structures *to be published, existing draft is dated 2000

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