Design of Steel Structures for Buildings in Seismic Areas E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 653

Veröffentlichungsjahr: 2018

Beschreibung

This volume elucidates the design criteria and principles for steel structures under seismic loads according to Eurocode 8-1. Worked Examples illustrate the application of the design rules. Two case studies serve as best-practice samples.

Leseprobe

Table of Contents

Cover

Foreword

Preface

Chapter 1: Seismic Design Principles in Structural Codes

1.1 INTRODUCTION

1.2 FUNDAMENTALS OF SEISMIC DESIGN

1.3 CODIFICATION OF SEISMIC DESIGN

Chapter 2: EN 1998-1: General and Material Independent Parts

2.1 INTRODUCTION

2.2 PERFORMANCE REQUIREMENTS AND COMPLIANCE CRITERIA

2.3 SEISMIC ACTION

2.4 CHARACTERISTICS OF EARTHQUAKE RESISTANT BUILDINGS

2.5 METHODS OF STRUCTURAL SEISMIC ANALYSIS

2.6 STRUCTURAL MODELLING

2.7 ACCIDENTAL TORSIONAL EFFECTS

2.8 COMBINATION OF EFFECTS INDUCED BY DIFFERENT COMPONENTS OF THE SEISMIC ACTION

2.9 CALCULATION OF STRUCTURAL DISPLACEMENTS

2.10 SECOND ORDER EFFECTS IN SEISMIC LINEAR ELASTIC ANALYSIS

2.11 DESIGN VERIFICATIONS

Chapter 3: EN 1998-1: Design Provisions for Steel Structures

3.1 DESIGN CONCEPTS FOR STEEL BUILDINGS

3.2 REQUIREMENTS FOR STEEL MECHANICAL PROPERTIES

3.3 STRUCTURAL TYPOLOGIES AND BEHAVIOUR FACTORS

3.4 DESIGN CRITERIA AND DETAILING RULES FOR DISSIPATIVE STRUCTURAL BEHAVIOUR COMMON TO ALL STRUCTURAL TYPES

3.5 DESIGN CRITERIA AND DETAILING RULES FOR MOMENT RESISTING FRAMES

3.6 DESIGN CRITERIA AND DETAILING RULES FOR CONCENTRICALLY BRACED FRAMES

3.7 DESIGN CRITERIA AND DETAILING RULES FOR ECCENTRICALLY BRACED FRAMES

Chapter 4: Design Recommendations for Ductile Details

4.1 INTRODUCTION

4.2 SEISMIC DESIGN AND DETAILING OF COMPOSITE STEEL-CONCRETE SLABS

4.3 DUCTILE DETAILS FOR MOMENT RESISTING FRAMES

4.4 DUCTILE DETAILS FOR CONCENTRICALLY BRACED FRAMES

4.5 DUCTILE DETAILS FOR ECCENTRICALLY BRACED FRAMES

Chapter 5: Design Assisted by Testing

5.1 INTRODUCTION

5.2 DESIGN ASSISTED BY TESTING ACCORDING TO EN 1990

5.3 TESTING OF SEISMIC COMPONENTS AND DEVICES

5.4 APPLICATION: EXPERIMENTAL QUALIFICATION OF BUCKLING RESTRAINED BRACES

Chapter 6: Multi-storey Building with Moment Resisting Frames

6.1 BUILDING DESCRIPTION AND DESIGN ASSUMPTIONS

6.2 STRUCTURAL ANALYSIS AND CALCULATION MODELS

6.3 DESIGN AND VERIFICATION OF STRUCTURAL MEMBERS

6.4 DAMAGE LIMITATION

6.5 PUSHOVER ANALYSIS AND ASSESSMENT OF SEISMIC PERFORMANCE

Chapter 7: Multi-storey Building with Concentrically Braced Frames

7.1 BUILDING DESCRIPTION AND DESIGN ASSUMPTIONS

7.2 STRUCTURAL ANALYSIS AND CALCULATION MODELS

7.3 DESIGN AND VERIFICATION OF STRUCTURAL MEMBERS

7.4 DAMAGE LIMITATION

Chapter 8: Multi-storey Building with Eccentrically Braced Frames

8.1 BUILDING DESCRIPTION AND DESIGN ASSUMPTIONS

8.2 STRUCTURAL ANALYSIS AND CALCULATION MODELS

8.3 DESIGN AND VERIFICATION OF STRUCTURAL MEMBERS

8.4 DAMAGE LIMITATION

Chapter 9: Case Studies

9.1 INTRODUCTION

9.2 THE BUCHAREST TOWER CENTRE INTERNATIONAL

9.3 SINGLE STOREY INDUSTRIAL WAREHOUSE IN BUCHAREST

9.4 THE FIRE STATION OF NAPLES

References

End User License Agreement

List of Tables

Chapter 2: EN 1998-1: General and Material Independent Parts

Table 2.1 – Importance factors for each building category

Table 2.2 – EC8 recommended values of the parameters describing both Type 1 and Type 2 elastic response spectra

Table 2.3 – EC8 recommended values of parameters describing the vertical elastic response spectra

Table 2.4 – Recommended values for

ψ

2,

i

. coefficient used to determine the quasi-permanent fraction of the variable action according to EN 1990 (CEN, 2002a)

Table 2.5 – EC8 Recommended

φ

coefficient values used to determine

ψ

E,i

Table 2.6 – Consequences of structural regularity on seismic analysis

Chapter 3: EN 1998-1: Design Provisions for Steel Structures

Table 3.1 – Acceptance criteria for interstorey drift ratios according to FEMA 356

Table 3.2 – Upper limits of

q

factors for systems regular in elevation

Table 3.3 – Upper limits of

q

factors for dual systems regular in elevation

Table 3.4 – Requirements on cross-sectional class of dissipative elements depending on Ductility Class and reference behaviour factor

Chapter 4: Design Recommendations for Ductile Details

Table 4.1 – Resistance of components

Table 4.2 – Effective Length Factors

K

of continuous X-bracings

Chapter 5: Design Assisted by Testing

Table 5.1 – Annex D procedure

Table 5.2 – Tests for Displacement Dependant Devices (selected from EN 15129, Table 17)

Chapter 6: Multi-storey Building with Moment Resisting Frames

Table 6.1 – Material properties and partial factors

Table 6.2 – Characteristic values of vertical permanent and live loads

Table 6.3 – Combination coefficients for both loads and masses in seismic design condition

Table 6.4 – Seismic weights and masses

Table 6.5 – Coordinates of the centre of stiffness (CS), centre of mass (CM), structural eccentricity

e

o

, torsional radius (r) and verifications for each floor

Table 6.6 – Periods and participating mass per mode of vibration

Table 6.7 – Accidental torsional moments per floor

Table 6.8 – Global sway imperfections

Table 6.9 – Stability coefficients calculated in X direction

Table 6.10 – Stability coefficients calculated in Y direction

Table 6.11 – Classification of beam cross sections

Table 6.12 – Stable length of beam between the section at a plastic hinge location and the adjacent lateral restraint

Table 6.13 – Flexural checks for beams belonging to MRF in X direction

Table 6.14 – Flexural checks for beams belonging to MRF in Y direction

Table 6.15 – shear checks for beams belonging to MRF in X direction

Table 6.16 – shear checks for beams belonging to MRF in Y direction

Table 6.17 – Flexural checks for columns “A” in X direction

Table 6.18 – Flexural checks for columns “G” in Y direction

Table 6.19 – Shear checks for columms belonging to vertical “A” in X direction

Table 6.20 – Shear checks for columms belonging to vertical “G” in Y direction

Table 6.21 – Local hierarchy criterion for external and inner columns in X direction

Table 6.22 – Verification of shear panel zone and design of additional web plates for external joints of column A in X direction

Table 6.23 – Verification of shear panel zone and design of additional web plates for internal joints of column B in X direction

Table 6.24 – Damage limitation check for MRFs in X direction

Table 6.25 – Damage limitation check for MRFs in Y direction

Table 6.26 – Response parameters and acceptance criteria for plastic hinges

Table 6.27 – Yield rotation for plastic hinges of beams (MRF in X direction)

Table 6.28 – Yield rotation for plastic hinges of columns (MRF in X direction)

Table 6.29 – Lateral force distributions (pushover in X direction)

Table 6.30 – Overstrength ratios

a

u

/a

l

Table 6.31 – Yield rotation for plastic hinges of columns (MRF in X direction)

Chapter 7: Multi-storey Building with Concentrically Braced Frames

Table 7.1 – Material properties and partial factors

Table 7.2 – Characteristic values of vertical permanent and live loads

Table 7.3 – Seismic weights and masses

Table 7.4 – Global sway imperfections in the X and Y directions

Table 7.5 – Stability coefficients for X-CBFs (i.e. frame in X direction)

Table 7.6 – Stability coefficients for inverted V-CBFs (i.e. frame in Y direction)

Table 7.7 – Cross section properties of X-braces

Table 7.8 – Design checks of X-braces

Table 7.9 – Axial strength checks of beams in X-braced bays

Table 7.10 – Combined bending-axial strength checks of beams in X-braced bays

Table 7.11 – Shear strength checks of beams in X-braced bays

Table 7.12 – Axial strength checks for columns in + X direction

Table 7.13 – Axial strength checks for columns in – X direction

Table 7.14 – Inverted V-brace cross section properties

Table 7.15 – Design checks of Inverted V-braces (D1) in tension

Table 7.16 – Design checks of Inverted V-braces (D1) in compression

Table 7.17 – Design checks of Inverted V-braces (D2) in tension

Table 7.18 – Design checks of Inverted V-braces (D2) in compression

Table 7.19 – Seismic induced axial forces into brace-intercepted beams (see Figure 7.18)

Table 7.20 – Axial strength checks of brace-intercepted beams

Table 7.21 – Combined bending-axial strength checks of brace-intercepted beams

Table 7.22 – Shear checks of brace-intercepted beams

Table 7.23 – Axial strength checks for the external columns of the braced cantilever

Table 7.24 – Axial strength checks for the central column of the braced cantilever

Table 7.25 – Damage limitation check for X-CBFs

Table 7.26 – Damage limitation check for inverted V-CBFs

Chapter 8: Multi-storey Building with Eccentrically Braced Frames

Table 8.1 – Material properties and partial safety factors

Table 8.2 – Characteristic values of vertical permanent and live loads

Table 8.3 – Seismic weights and masses

Table 8.4 – Global sway imperfections in both X and Y direction

Table 8.5 – Stability coefficients for EBF in

X

direction

Table 8.6 – Stability coefficients for EBF in Y direction

Table 8.7 – Link sections, upper bound shear length and selected length per storey (X and Y direction)

Table 8.8 – Strength verifications of links of frames in X direction

Table 8.9 – Strength verifications of links of frames in Y direction

Table 8.10 – Verification of braces in compression (X direction)

Table 8.11 – Verification of braces in compression (Y direction)

Table 8.12 – Axial strength checks for columns of EBF in X direction

Table 8.13 – Axial strength checks for columns of EBF in Y direction

Table 8.14 – Damage limitation check for EBF in X direction

Table 8.15 – Damage limitation check for EBF in Y direction

Chapter 9: Case Studies

Table 9.1 – Comparison between the configurations in the preliminary analysis

Table 9.2 – Wind direction on the scaled model

Table 9.3 – Periods of vibration and modal participating mass ratio

Table 9.4 – Acceptance criteria

Table 9.5 – Plastic rotation in beams and columns (in rad) and plastic deformation in braces (in %) at SLS, ULS and CPLS, average values

Table 9.6 – Choice of quality class according to EN 10164

Table 9.7 – Deviations from the theoretical position

Table 9.8 – Main dimensions of the buildings

Table 9.9 – Recommended values of external pressure coefficients for vertical walls of rectangular plan buildings

Table 9.10 – External pressure coefficients for flat roofs and sharp eaves

Table 9.11 – Values of the parameters describing the Type 1 elastic response spectrum

Table 9.12 – Modal participating mass ratio, structure A1

Guide

Cover

Table of Contents

Begin Reading

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e1

ECCS Eurocode Design Manuals

ECCS Editorial Board

Luís Simões da Silva (ECCS

)

António Lamas (Portugal

)

Jean-Pierre Jaspart (Belgium

)

Reidar Bjorhovde (USA

)

Ulrike Kuhlmann (Germany

)

Design of Steel Structures – 2nd Edition

Luís Simões da Silva, Rui Simões and Helena Gervásio

Fire Design of Steel Strcutures – 2nd Edition

Jean-Marc Franssen and Paulo Vila Real

Design of Plated Structures

Darko Beg, Ulrike Kuhlmann, Laurence Davaine and Benjamin Braun

Fatigue Design of Steel and Composite Structures

Alain Nussbaumer, Luís Borges and Laurence Davaine

Design of Cold-formed Steel Structures

Dan Dubina, Viorel Ungureanu and Raffaele Landolfo

Design of Joints in Steel and Composite Structures

Jean-Pierre Jaspart and Klaus Weynand

Design of Steel Structures for Buildings in Seismic Areas

Raffaele Landolfo, Federico Mazzolani, Dan Dubina, Luís Simões da Silva and Mario D’Aniello

ECCS – SCI EUROCODE DESIGN MANUALS

Design of Steel Structures, UK Edition

Luís Simões da Silva, Rui Simões, Helena Gervásio

Adapted to UK by Graham Couchman

Design of Joints in Steel Structures, UK Edition

Jean-Pierre Jaspart and Klaus Weynand

Adapted to UK by Graham Couchman and Ana M. Girão Coelho

ECCS EUROCODE DESIGN MANUALS – BRAZILIAN EDITIONS

Dimensionamento de Estruturas de Aço

Luís Simões da Silva, Rui Simões, Helena Gervásio, Pedro Vellasco, Luciano Lima

Information and ordering details

For price, availability, and ordering visit our website www.steelconstruct.com.

For more information about books and journals visit www.ernst-und-sohn.de.

Design of Steel Structures for Buildings in Seismic Areas

Eurocode 8: Design of structures for earthquake resistancePart 1-1 – General rules, seismic actions and rules for buildings

Raffaele Landolfo

Federico Mazzolani

Dan Dubina

Luís Simões da Silva

Mario D’Aniello

Design of Steel Structures for Buildings in Seismic Areas

1st Edition, 2017

Published by:

ECCS – European Convention for Constructional Steelwork

[email protected]

www.steelconstruct.com

Sales:

Wilhelm Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin

All rights reserved. No parts of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,

recording or otherwise, without the prior permission of the copyright owner.

ECCS assumes no liability with respect to the use for any application of the material and information contained in this publication.

Copyright © 2017 ECCS – European Convention for Constructional Steelwork

ISBN (ECCS): 978-92-9147-138-6

ISBN (Ernst & Sohn): 978-3-433-03010-3

Legal dep.: 432199/17 Printed in Sersilito, Empresa Gráfica Lda, Maia, Portugal

Photo cover credits: Dan Dubina

Foreword

There are many seismic areas in Europe. As times goes by, regional seismicity is better known and the number of places where earthquake is an action to consider in design increases. Of course, there are substantial differences in earthquake intensity between regions and the concern is much greater in many areas of Italy, for instance, than in most places in Northern Europe. However, even in Northern Europe, for structures for which a greater level of safety is required, like Seveso industrial plants, hospitals and public safety facilities, seismic design can be the most requiring design condition.

Designing for earthquake has original features in comparison with design for classical loading like gravity, wind or snow. The reference event for Ultimate Limit State seismic design is rare enough for an allowance to permanent deformations and structural damages, as long as people’s life is not endangered. This means that plastic deformations are allowed at ULS, so that the design target becomes a global plastic mechanism. To be safe, the latter requires many precautions, on global proportions of structures and on local detailing. The seismic design concepts are completely original in comparison to static design. Of course, designing for a totally elastic behaviour even under the strongest earthquake remains possible but, outside of low seismicity areas, this option is generally left aside because of its cost.

This book is developed with a constant reference to Eurocode 8 or EN 1998-1:2004; it follows the organization of that code and provides detailed explanations in support of its rather dry expression. Of course, there are many other seismic design codes, but it must be stressed that there is nowadays a strong common thinking on the principles and the application rules in seismic design so that this book is also a support for the understanding of other continents codes.

Chapter 1 explains the principles of seismic design and their evolution throughout time, in particular the meaning, goals and conditions set forward by capacity design of structures and their components, a fundamental aspect of seismic design.

Chapter 2 explains the general aspects of seismic design: seismic actions, design parameters related to the shape of buildings, models for the analysis, safety verifications. Methods of analysis are explained in an exhaustive way: theoretical background, justifications of limits and factors introduced by the code, interest and drawbacks of each method, together with occasionally some tricks to facilitate model making and combination of load cases.

Chapter 3 focuses on design provisions specific to steel structures: ductility classes, requirements on steel material, structural typologies and design conditions related to each of them; an original insight on design for reparability is also included.

Chapter 4 provides an overview about the best practice to implement the requirements and design rules for ductile details, particularly for connections in moment resisting frames (MRF), concentrically braced frames (CBF) and eccentrically braced frames (EBF), and for other structural components like diaphragms.

Chapter 5 describes the guidance provided for design assisted by testing by EN 1990 and the specific rules for tests, a necessary tool for evaluating the performance characteristics of structural typologies and components in the plastic field and in cyclic/dynamic conditions.

Chapter 6 illustrates and discusses the design steps and verifications required by EN 1998-1 for a multi-storey Moment Resisting Frame.

Chapter 7 and 8 do the same respectively for buildings with CBF’s and EBF’s.

Chapter 9 presents three very different examples of real buildings erected in high seismicity regions: one tall building, one industrial hall and one design using base isolation. These examples are complete in the sense that they show the total design, where seismic aspects are only one part of the problem. These examples are concrete, because they illustrate practical difficulties of the real world with materials, execution, positioning…

The concepts, design procedures and detailing in seismic design may seem complex. This publication explains the background behind the rules, which clarify their objectives. Details on the design of the different building typologies are given, with reference to international practice and to recent research results. Finally, design examples and real case studies set out the design process in a logical manner, giving practical and helpful advice.

This book will serve the structural engineering community in expanding the understanding and application of seismic design rules, and, in that way, constitute a precious tool for our societies safety.

André Plumier

Honorary Professor, University of Liege

Preface

This manual aims to provide its readers with the background and the explanation of the main aspects dealing with the seismic design of steel structures in Europe. Therefore, the book focuses on EN 1998-1 (usually named part 1 of Eurocode 8 or EC8-1) that is the Eurocode providing design rules and requirements for seismic design of building structures. After 10 years from its final issue, both the recent scientific findings and the design experience carried out in Europe highlight some criticisms. In the light of such considerations, this book complements the explanation of the EC8-1 provisions with the recent research findings, the requirements of renowned and updated international seismic codes (e.g. North American codes and design guidelines) as well as the design experience of the Authors. Although the manual is oriented to EC8-1, the book aims to clarify the scientific outcomes, the engineering and technological aspects rather than sticking to an aseptic explanation of each clause of the EC8-1. Indeed, as shown in Chapter 4, the proper detailing of steel structures is crucial to guarantee adequate ductility of seismic resistant structures and the current codes does not give exhaustive guidelines to design ductile details since it only provides the fundamental principles. In addition, the practice of earthquake engineering significantly varies between European regions, reflecting the different layouts of each national seismic code as well as the level of knowledge and confidence with steel structures of each country. With this regard, a large number of European engineers believe that steel structures can withstand severe earthquakes without requiring special details and specifications as conversely compulsory for other structural materials like reinforced concrete and masonry. This belief direct results from the mechanical features of the structural steel, which is a high performance material, being stronger than concrete but lighter (if comparing the weight of structural members) and also very ductile and capable of dissipating large amounts of energy through yielding when subjected to cyclic inelastic deformations. However, although the material behaviour is important, the ductility of steel alone is not enough to guarantee ductile structural response. Indeed, as demonstrated by severe past earthquakes (e.g. Northridge 1994, Kobe 1995 and Christchurch 2011) there are several aspects ensuring good seismic behaviour of steel structures, which are related to (i) the conceptual design of the structure, (ii) the overall sizing of the member, (iii) the local detailing and (iv) proper technological requirements as well as ensuring that the structures are actually constructed as designed.

Therefore, this book primarily attempts to clarify all these issues (from Chapter 1 to 4) for European practising engineers, working in consultancy firms and construction companies. In addition, the examples of real buildings (see Chapter 9) are an added value, highlighting practical and real difficulties related to both design and execution.

This design manual is also meant as a recommended textbook for several existing courses given by the Structural Sections of Civil Engineering and Architectural Engineering Departments. In particular, this manual is oriented to advanced students (i.e. those attending MSc programmes) thanks to the presence of various calculation examples (see Chapter 6, 7 and 8) that simplify and speed up the understanding and the learning of seismic design of EC8 compliant steel structures. In addition, research students (i.e. those attending PhD programmes) can find useful insights for their experimental research activities by reading Chapter 5, which provides some guidance and discussion on how performing experimental tests of structural typologies and components in cyclic pseudo-static and dynamic conditions.

The Authors

Raffaele Landolfo

Federico Mazzolani

Dan Dubina

Luís Simões da Silva

Mario D’Aniello

Chapter 1Seismic Design Principles in Structural Codes

1.1 INTRODUCTION

Earthquake Engineering is the branch of engineering aiming at mitigating risks induced by earthquakes with two objectives: i) to predict the consequences of strong earthquakes on urban areas and civil infrastructures; ii) to design, build and maintain structures that are able to withstand earthquakes in compliance with building codes.

Researchers and experts working within emergency management organizations (e.g. the civil protection) actively work on the first issue. On the contrary, structural designers focus their attention and efforts on the second objective. With this regard, it should be noted that the seismic design philosophy substantially differs from the design approaches conventionally adopted for other types of actions, raising difficulties to structural engineers less confident with seismic engineering. Indeed, broadly speaking, for quasi-static loads (e.g. dead and live loads, wind, snow, etc.) the structure should behave mostly elastically without any damage until the maximum loads are reached, while in case of seismic design it is generally accepted that structures can experience damage because they should perform in the plastic range for seismic events. The philosophy of structural seismic design establishes the performance levels that properly engineered structures should satisfy for different seismic intensities, which can be summarized as follows:

– prevent near collapse or serious damage in rare major ground shaking events, which are called in the following Ultimate Limit State seismic action or ULS seismic action;

– prevent structural damage and minimize non-structural damage in occasional moderate ground shaking events;

– prevent damage of non-structural components (such as building partitions, envelopes, facilities) in frequent minor ground shaking events.

Hence, the most meaningful performance indexes for seismic resistant structures are the amount of acceptable damage and the repair costs. Owing to the unforeseeable nature of seismic actions, it is clear that damage control is very difficult to be quantified by code provisions, especially because it is related to acceptable levels of risk. The challenge for efficient design of seismic resistant structures is to achieve a good balance between the seismic demand (namely the effect that earthquakes impose on structures) and the structural capacity (namely the ability to resist seismic induced effects without failure). However, the quantification of different types of damage (structural and non-structural) associated to the reference earthquake intensity (e.g. frequent/minor, occasional/moderate, and rare/major) and the definition of relevant operational design criteria are still open issues that need clarification and further studies.

This chapter describes and discusses the concept of capacity design in the light of existing seismic codes, illustrating the evolution of seismic design principles throughout time, and explains the criteria that form the basis of EN 1998-1:2004 (CEN, 2004a), henceforth denoted as EC8-1.

1.2 FUNDAMENTALS OF SEISMIC DESIGN

1.2.1 Capacity design

It is generally acknowledged that structural safety depends on the ductility that the structural system can provide against the design loads. Indeed, ductility represents the capacity of a mechanical system (e.g. a beam, a structure, etc.) to deform in the plastic domain without substantially reducing its bearing capacity.

In seismic design of structures it is generally not economical or possible to ensure that all the elements of the structure behave in a ductile manner. Inevitably, a dissipative (ductile) structure comprises both dissipative (ductile) elements and non-dissipative (brittle) ones. In order to achieve a dissipative (ductile) design for the whole structure, the failure of the brittle elements must be prevented. This may be done by prioritizing structural elements strength, which will lead to the prior yielding of ductile structural elements, preventing the failure of brittle structural elements. This principle is known as ”capacity design”. Capacity design may be explained by considering the chain model, introduced by Paulay and Priestley (1992) and depicted in Figure 1.1a, whereby the chain represents a structural system made of both ductile elements (e.g. the ring “1”) and brittle zones (e.g. the ring “i”).

According to non-seismic design procedures for quasi-static loads (hereinafter referred to as “direct design”), the design force is the same for all elements belonging to the chain, because the applied force is equal for all rings, being a system in series. Hence, the design resistance Fy,i is the same for all elements. Under this assumption, the yield resistance of the ductile chain Fy,1 is equal or even slightly larger than Fy,i.

Figure 1.1 – Ductility of a chain with brittle and ductile rings

As shown in Figure 1.1b, with the direct design approach the system cannot develop strength larger than Fy and the ultimate elongation of the chain is given as

(1.1)

According to capacity design principles, in order to improve the ductility of the chain, some rings should be designed with ductile behaviour and lower strength, as is the case of ring “1” in Figure 1.1c. The remaining rings “i” that are brittle should be designed to provide a resistance Fy,i larger than the maximum resistance Fu,1 exhibited by the ring “1” beyond yielding. The ductile ring “1” behaves as a sacrificial element, i.e. a ductile fuse, which filters the external actions and limits the transfer of forces into the brittle elements. Hence, the maximum force that the chain can sustain is equal to the maximum resistance Fu,1 of the ductile ring “1”. It is interesting to observe that the beneficial improvement of the capacity design methodology is the increase of displacement capacity, given as follows:

(1.2)

Comparing equations (1.1) and (1.2), it can be easily recognized that the collapse displacement of the chain is significantly larger than that obtained by adopting the direct design approach.

This trivial example allows to understand that the brittle elements represent protected zones that must be designed to resist larger forces than those supported by the ductile elements. Those larger forces do not directly depend on the external applied loads but they are obtained from the maximum capacity of the connected ductile elements. However, it should be emphasized that the external forces are used to design the dissipative elements, which establish the threshold of structural strength.

Concerning the practical application to building structures, this methodology leads the structural designers to work on two different schemes for the same structure, as follows:

1) elastic behaviour with the calculation of the relevant internal forces

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to design the dissipative elements. Hence, following an elastic analysis, the ductile structural elements should satisfy the following check:

(1.3)