Guidebook-1 Load Effects on Structures

FOREWORD The Leonardo da Vinci Project CZ/08/LLP-LdV/TOI/134020 “Transfer of Innovations Provided in Eurocodes” addresses the urgent need to implement the new system of European documents related to design of construction works and products. These documents, called Eurocodes, are systematically based on the recently developed Council Directive 89/106/EEC “The Construction Products Directive” and its Interpretative Documents ID1 and ID2. Implementation of Eurocodes in each Member State is a demanding task as each country has its own long-term tradition in design and construction. The project should enable an effective implementation and application of the new methods for designing and verification of buildings and civil engineering works in all the partner countries (CZ, DE, ES, IT, NL) and in other Member States. The need to explain and effectively use the latest principles specified in European standards is apparent from various enterprises, undertakings and public national authorities involved in construction industry and also from universities and colleges. Training materials, manuals and software programmes for education are urgently required. The submitted Guidebook 1 is one of 2 upcoming guidebooks intended to provide required manuals and software products for training, education and effective implementation of Eurocodes: Guidebook 1: Load Effects on Buildings Guidebook 2: Load Effects on Bridges It is expected that the Guidebooks will address the following intents in further harmonisation of European construction industry: - reliability improvement and unification of the process of design; - development of a single market for products and for construction services; - new opportunities for trained primary target groups in the labour market. The Guidebook 1 is focused on determining load effects on buildings and industrial structures. The following main topics are treated in particular: - basic requirements on structures, - basis of structural design, - actions on buildings including accidental actions, - combination rules for load effects, - examples and case studies. Annex A to the Guidebook 1 provides a review of the basic statistical concepts used in design assisted by testing, Annex B a short description of general procedures used for assessment of existing structures and Annex C provides latest information on further development of Eurocodes. The Guidebook 1 is written in a user-friendly way employing only basic mathematical tools. Attached software products supplemented by a number of examples enable direct applications of general rules in practice. A wide range of potential users of the Guidebooks and other training materials includes practising engineers, designers, technicians, experts of public authorities, young people - high school and university students. The target groups come from all territorial regions of the partner countries. However, the dissemination of the project results is foreseen to be spread into all Member States of CEN and other interested countries. Prague 2009

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Contents

GUIDEBOOK 1 – BASIS OF DESIGN AND ACTIONS ON BUILDINGS CONTENTS Page FOREWORD CONTENTS 1 BASIC REQUIREMENTS Summary 1 Introduction 2 Basic requirements 2.1. Principal requirements 2.2 Requirements related to the permanent design situations 2.2 Requirements related to the accidental design situations 3 Reliability management 4 Design working life 5 Durability 6 Quality management References Appendix A Reference documents A.1. Introduction A.2. Construction Product Directive A.3. Interpretative document No. 1 Mechanical resistance and stability A.4. Guidance Paper L

3 5 11 11 11 12 12 13 13 14 15 16 17 18 19 19 19 20 23

2 BASIS OF DESIGN – GENERAL PRINCIPLES Summary 1 Introduction 1.1. Background documents 1.2 General principles 2 Historical development 2.1 Uncertainties 2.2 Definition of reliability 2.3 Development of design methods 3 Basic concepts of EN 1990 3.1 Design working life and design situation 3.2 Limit states 3.3 Ultimate limit states 3.4 Serviceability limit states 4 Verification of limit states 4.1 Verification of static equilibrium and strength 4.2 Verification of the serviceability limit states 5 Concluding remarks Reference Appendix A A reinforced concrete slab – various design concepts A.1 Introduction A.2 A reinforced concrete slab A.3 Design and reliability consideration A.4 Concluding remarks

32 32 32 32 32 33 33 34 35 37 37 37 40 41 42 42 43 43 44 45 45 45 45 47

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Contents

3 RELIABILITY DIFFERENTIATION Summary 1 Introduction 1.1. Background documents 1.2 General principles 2 Basic reliability elements 3 Target reliability in the Eurocodes 3.1 General 3.2 Reliability classes 3.3 Variation with time – Discussion 3.4 Global failure – robustness 3.5 Existing structures 4 Partial safety factors 4.1 Derivation based on reliability methods 4.2 Simplified reliability differentiation (Annex B of EN 1990) 5 Examples 5.1 Residential steel building 5.2 Agricultural steel building 5.3 Agricultural concrete building 6 Concluding remarks References Appendix A: Risk Acceptance Approaches in Codes A.1 General A.2 Human safety A.3 Calibration A.4 Cost – benefit approach References

48 48 48 48 48 49 50 50 50 51 53 53 51 54 55 56 56 56 56 57 57 58 58 58 59 60 61

4 ACTIONS Summary 1 Introduction 1.1 Background documents 2 Actions and effects of actions 2.1 Definitions of actions 2.2 Effect of actions 3 Classification of actions 3.1 General 3.2 By their variation in time 3.3 By their origin 3.4 By their variation in space 3.5 By their nature or structural response 3.6 Bounded and unbounded actions 3.7 Environmental influences 4 Reference period and distribution of maxima 4.1 Climatic actions 4.2 Imposed actions 5 Characteristic values 5.1 General 5.2 Permanent actions

62 62 62 62 62 62 63 65 65 65 65 65 65 66 66 66 66 67 69 69 69

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7 8 9

5.3 Variable actions 5.4 Imposed loads 5.5 Snow loads 5.6 Wind loads 5.7 Thermal actions Representative values 6.1 General 6.2 The combination value of a variable action 6.3 The frequent value of a variable action 6.4 The quasi-permanent value of a variable action Representation of dynamic actions Representation of fatigue actions Representation of environmental influences References

70 71 73 76 81 83 83 83 83 83 84 85 85 85

5 ACCIDENTAL ACTIONS Summary 1 Introduction 1.1. History 1.2 Background documents 2 Eurocode approach 3 Design for impact and explosion loads 3.1 Impact from vehicles 3.2 Loads due to explosions 3.3 Design example of a column in a building for an explosion 4 Robustness of building 4.1 Background 4.2 Summary of design rules 4.2.1 Design Rules for Class 2, Lower Group, Framed structures 4.2.2 Rules for Class 2, Lower group, Load-bearing wall construction 4.2.3 Rules for Class 2, Upper Group, Framed structures 4.2.4 Rules for Class 2, Upper Group, Load-bearing wall construction 4.3 Example structures 4.3.1 Framed structure, Consequences class 2, Upper Group 4.3.2 Wall structure, Consequences class 2, Upper Group 5 Conclusions References Appendix A: Methodology related to robustness assessment A.1 Conditional probability of collapse A.2 Quantification of robustness A.3 Basic design philosophy References Appendix B: Impact Force Analysis

86 86 86 86 87 87 90 90 90 91 94 94 94 94 95 95 95 96 96 96 97 97 98 98 99 100 101 102

6 COMBINATION RULES IN EUROCODES Summary 1 Introduction 1.1. Background documents 1.2 General principles 2 Combination of actions

103 103 103 103 103 104

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Contents 2.1 General 2.2 Combinations of actions in persistent and transient design situations 2.3 Combination of actions for accidental and seismic design situations 2.4 Combination of actions for serviceability limit states 3 Examples 3.1 Cantilivered beam 3.2 Continuous beam of three spans 3.3 Cantilivered frame 3.4 Three bay two-dimensional frame 4 Concluding remarks References Appendix A: Alternative load combinations for the cantilevered beam

104 104 105 106 106 106 110 113 118 122 122 123

7 ACTIONS IN TRANSIENT DESIGN SITUATIONS Summary 1 Introduction 1.1. Background documents 1.2 General principles 2 Design situations during execution 2.1. Design situations 2.2 Nominal duration of design situations 3 Representative and design values of actions during execution 4 Combinations of actions 5 Actions during execution 6 Annex A for buildings and bridges 6.1. Annex A1 Supplementary rules for buildings 6.2 Annex A2 Supplementary rules for bridges 7 Annex B for Actions on structures during alteration, rehabilitation or demolition 8 Concluding remarks References

125 125 125 125 125 125 125 126 128 129 129 131 131 131

8 ACTIONS AND COMBINATION RULES FOR SILOS AND TANKS Summary 1 Introduction 1.1. Background documents 1.2 Basic principles 2 Design situations 3 Actions on silos and tanks 3.1 Types of actions 3.2 Actions specific on silos 4 Classification of silos 5 Combinations of actions for silos 5.1 Combinations of actions in persistent design situations 5.2 Combinations of actions in accidental design situations 5.3 Combinations of actions in seismic design situations 5.4 Combinations of actions in serviceability limit states 6 Combinations of actions for tanks 6.1 Actions 6.2 Combinations of actions

134 134 134 134 134 135 136 136 136 137 137 137 138 139 139 140 140 140

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Contents 7

An example of slender silo 7.1 Introduction 7.2 Symmetrical filling loads on vertical walls 7.3 Filling patch load 7.4 Symmetrical discharge load 7.5 Discharge patch load Concluding remarks References

140 140 141 143 144 145 145 145

9 LOAD EFFECTS IN STRUCTURAL MEMBERS Summary 1 Introduction 1.1 Background documents 1.2 General principles 2 Verification of static equilibrium and strength 3 Verification of serviceability limit states 4 Examples 4.1 Cantilevered beam 4.2 Continuous beam of three spans 4.3 Cantilevered frame 5 Concluding remarks References

147 147 147 147 147 148 149 149 149 153 156 161 161

10 DESIGN OF A REINFORCED CONCRETE BUILDING ACCORDING TO EUROCODES Summary 1 Introduction 2 The building 3 Actions, loadings and load combinations 3.1 Density and self-weight 3.2 Imposed loads 3.3 Snow load 3.4 Wind actions 3.5 Load combinations for ULS verifications 3.6 Load combinations and limitations for SLS verifications 4 Materials 4.1 Stress-strain diagrams 5 Results of the structural analysis 6 Static verification examples 6.1 Verification of the first floor beams 6.2 Verification of the corner column 7 Concluding remarks References Appendix to Chapter 10 A.1 Basic structural drawings of the building

162 162 162 162 164 164 164 164 165 167 168 169 170 170 172 172 174 174 175 176 176

ANNEX A: DESIGN ASSISTED BY TESTING Summary 1 Introduction 2 Statistical determination of a single property

185 185 185 185

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2.1 General principles 2.2 Assessment based on the characteristic value 2.3 Direct assessment of the design value Statistical determination of resistance models 3.1 General procedure 3.2 An example of a concrete slab References

185 186 187 188 188 191 192

ANNEX B: ASSESSMENT OF EXISTING STRUCTURES Summary 1 Introduction 2 Principles and general framework of assessment 3 Investigation 4 Basic variables 5 Evaluation of inspection results 6 Structural analysis 7 Verification 8 Assessment in the case of damage 9 Final report and decision 10 Numerical example 10.1 Updating of failure probability 10.2 Bayesian method for fractile estimation 11 Concluding remarks References

193 193 193 194 197 198 199 200 202 202 203 204 204 205 207 208

ANNEX C: FURTHER DEVELOPMENT OF EUROCODES Summary 1 Introduction 2 New CPR and sustainable constructions 3 Evolution of Eurocodes 3.1 Maintenance 3.2 Harmonization 3.3 Promotion 3.4 Further developments 4 Research for further development of Eurocodes References

209 209 209 210 211 211 211 212 213 214 214

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Chapter 1: Basic requirements

CHAPTER 1: BASIC REQUIREMENTS Angel Arteaga1, Ana de Diego1 and Albert Alzate1 1

E. Torroja Institute of Construction Sciences, CSIC. Madrid. Spain

Summary The Eurocodes system determines a set of basic requirements that all the structures have to fulfil in order to adequate the structure to its foreseen use and expected live. These requirements, based on European Commission Directives and other documents, are revised and explained in this chapter.

1.

INTRODUCTION

When a Member State joints the European Union (EU), it transfers some of its competencies to the European Commission. The European Commission publishes a lot of Directives that the Member States should adopt. The competencies referring to the level of safety at each country in all the fields, and in construction works in particular, are not transferred to EU; that means that each state is allowed to determine the level of safety applicable inside the country, and, therefore, the reliability level of its construction works. In the field of construction, the European Commission delivered the Construction Products Directive (CPD) [1], compulsory to the Member States, indicating the conditions needed to facilitate the free circulation of the construction products in the European market (not only for European products). Furthermore, there is a wish in harmonizing all the design procedures and values in a way that all the construction products and building companies can move easily all around the EU. The CPD states in its Annex I the six so-called essential requirements: 1. Mechanical resistance and stability 2. Safety in case of fire 3. Hygiene, health and the environment 4. Safety in use 5. Protection against noise 6. Energy economy and heat retention The Eurocode structural system only deals with the two firsts of these six requirements. In this Guidebook only the first, Mechanical resistance and stability, will be treated. These requirements are complemented by the other EC documents known as Interpretative Documents No. 1 (ID-1) [2] to Interpretative Documents No 6 (ID-6), each one dealing with and explaining in detail the corresponding requirement, and the Guidance Paper L [3], which defines the use of EN Eurocodes for structural design of works and in technical specifications for structural products, and also the future actions related to the Eurocode Programme.

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Chapter 1: Basic requirements

The main points of the CPD, ID-1, and the Guidance Paper L mentioned above are summarized in the Annex A of this Chapter.

2.

BASIC REQUIREMENTS

2.1.

Principal requirements

The essential Requirements 1 and 2, stated in the CPD and developed in the Interpretative Documents ID-1 and ID-2, indicated above, are translated into clauses in the Section 2 Requirements of the Eurocode EN 1990: Basis of structural design [9]. The first requirement: Mechanical resistance and stability is summarized in the first two clauses: (1)P A structure shall be designed and executed in such a way that it will, during its intended life, with appropriate degrees of reliability and in an economical way sustain all actions and influences likely to occur during execution and use, and remain fit for the use for which it is required. (2)P A structure shall be designed to have adequate: structural resistance, serviceability, and durability. The third clause corresponds to the second Essential Requirement: Safety in case of fire. (3)P In the case of fire, the load-bearing capacity of the structure shall be assured for the required period of time. In order to understand adequately this content, it should be distinguished, and so does the Eurocode, between what is referred to the transient, permanent and accidental design situations. Permanent design situations are those affecting the structure at most part of its working life, taking into account aspects related with the safety –structural resistance–, i.e.: Ultimate Limit State (ULS); and with serviceability; i.e.: Serviceability Limit State (SLS); and also with the durability; that is, the structural conditions that limit the deterioration of the structure not influencing its performance. Transient design situations are relevant during a period much shorter than the design working life of the structure. They refer to temporary conditions of the structure, of use, or exposure, e.g. during construction, repair or upgrading. Therefore, taking into account the permanent and transient design situations, a structure shall be designed, in a way that: – sustains all actions and influences likely to occur during execution and use, and – remains fit for the use for which it is required. Accidental situations are those situations which are foreseeable to occur, but with a small probability, during the working life of the structure. In case of accidental design situations the EN 1990 [9] states: A structure shall be designed and executed in such a way that it will not be damaged by events such as: – explosion, – impact, – and the consequences of human errors, to an extent disproportionate to the original cause.

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Chapter 1: Basic requirements

2.2.

Requirements related to the permanent design situations

First, it should be noted that the Eurocode system is mainly focused on the conditions that the designer of the structures must take into account. For this reason the Clause 2.1 reads that the structure shall be designed, and executed; nothing is indicated in this clause about the way the structure should be maintained. But the adequate use and maintenance of the structure is fundamental for fulfilling these requirements. Therefore, other documents related with the buildings or civil works must deal with these aspects. Other important point is what means the so-called intended life, that is: the period of time the structure is designed to fulfil its functions. This point is treated later in more detail. The terms with appropriate degrees of reliability and in an economical way will indicate the necessary trade-off between safety and economy. That is: the probability that the structure fails to fulfil the requirements and the cost of building and maintenance of the structure. As a first approach, a safer structure will require more detailed design, more material (bigger sections), and/or more quality control in all the stages, therefore an increment in the cost of execution of the work. The designer and owner have to balance this increment of cost with the increase of safety, taking into account, in any case, the minimum safety requirement determined in the country (not in the Eurocodes). The designer must take into account that the structure has to sustain all actions and influences likely to occur during execution and use; that is: the adequate behaviour taking into account the Ultimate limit states, and at the same time has to remain fit for the use for which it is required, the Serviceability Limit States. Each type of foreseeable failure of the structure and/or its members will correspond to different ULS: bending, shear, buckling, etc.; or SLS: cracks, deformations, sensibility to vibrations, etc.; and will correspond to a different Limit State Function; that is, the relationship (in general in mathematical form) between the actions or the effect of actions and the resistances of the structural material at the member considered that divides the safe and unsafe states of the structure or member. The Eurocodes give the way to determine the equations and the design values of the actions and resistances to be applied. These two requirements are interrelated: traditionally one structure safe enough – fulfilling the ULS requirements – was, normally, also stiff enough and it did not have serviceability problems. Nowadays, the situation is just the opposite: new materials with much higher resistance, and no stiffer in the same proportion and new design methods, yield to slender elements, therefore more deformable structures, what is the origin of frequent pathologies in the structures. It could be said now that one structure stiff enough is, probably, safe enough. 2.3. Requirements related to the accidental situations For the design of a structure, an accidental action is a type or value of one action that is foreseeable to occur in the lifetime of the structure, but not likely; let´s say, it has a probability of 10-4 of occurrence during its working life. Possible accidental actions are not only those considered in the EN 1990 [9]: gas explosion, impact, and consequences of human gross errors. The above-indicated ones are those for which the Eurocode 1 Part 1-7: General Actions - Accidental actions [10] gives further guidance. Other possible accidental actions are, for instance: winds or snows in a magnitude greater than foreseeable taking account the existing statistics, other type of explosions, plane crash, etc. In any case, voluntary actions as arson, terrorism, etc., are excluded from the Eurocodes. Nevertheless, the general guidance given in EN 1991-1-7 [10] can be useful even in these cases.

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Chapter 1: Basic requirements

Conceptually, seismic actions or fire actions are also accidental actions, but due to their particular importance and specific way of making calculations there are specific Eurocodes devoted to them. The philosophy of the Eurocodes to deal with the accidental actions is that it is not needed to design the structure in a way to sustain these extreme actions without damage. Some damage is acceptable, but not to an extent disproportionate to the original cause. That is: for the sake of example, an impact of a vehicle to a column could cause the failure of this column and the surrounding floors, but not of the whole structure. Different strategies can be used to face accidental situations. Recommendations are given in EN 1990 and also in EN 1991-1-7 [10] to avoid excessive damage. Potential alternative or concurrent strategies include: - avoiding, eliminating or reducing the hazards to which the structure can be subjected; - selecting a structural form which has low sensitivity to the hazards considered; - selecting a structural form and design that can survive adequately the accidental removal of an individual member or a limited part of the structure, or the occurrence of acceptable localised damage; - avoiding as far as possible structural systems that can collapse without warning; - tying the structural members together. Chapter 5 of this Guidebook deals with these situations in detail.

3. RELIABILITY MANAGEMENT Evidently neither all the structures, nor every part of a structure, have the same level of reliability, and even, for each member it also will depend on the type of the studied effect; i.e.: the different limit state. It is not the same to analyze the failure due to the buckling of columns (ULS) or the apparition of a crack of determined size (SLS). Indications of the adequate level of reliability for different circumstances are given in Section 2 of EN 1990 [9]. The concept of risk analysis is not highlighted in this section, but it is clear that it is behind all that was indicated here. The term risk is assumed to be the product of the consequences derived from an event and the probabilities that the event occurs; i.e. the probabilities of reaching that limit state or, in other words, the level of reliability. The important point is to keep the risk at an acceptable level. Unfortunately, it is easy to say ‘to keep the risk at acceptable level’, but not so easy to verify it in practical applications. Because both terms of the statement present important uncertainties. Firstly, as it is difficult to quantify the existing risk, a comprehensive set of scenarios encompassing all significant events has to be analysed. And, secondly, the exact knowledge of what is a quantified ‘acceptable’ level of risk is hardly to be achieved. In practical applications, the codes, and so do the Eurocodes, assume an implicit acceptable risk and, for each Limit State, assume, also, a level of average foreseeable consequences. With these assumptions in mind, the codes determine the ‘acceptable’ probabilities of failure for each Limit State (both ULS and SLS); that is: the level of reliability. For these considerations it is clear that the higher are the consequences of the failure the higher has to be the level of reliability. In all the calculations it is assumed that the design is developed following the Eurocodes 1990 to 1999 and taking care of appropriate execution and quality management measures.

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Chapter 1: Basic requirements

The cost and procedures necessary to reduce the risk of failure are other important factors to be taken into account for the choice of the adequate level of reliability. An example will clarify this concept: If we are evaluating an existing structure and we work out that a statically determined beam, according to the normal design procedures, would need to be reinforced with six steel bars diameter of 25 mm, but actually it has only five bars of that diameter; that is, it would need to be supplemented with one more bar. The evaluator, due to the fact that supplementing a beam with one additional bar is quite expensive - not for the cost of the steel, but for the cost of the implementation - could probably accept the small decrease of the reliability level. Exactly the same design problem, but in a case when dealing with a structure in the design phase, still not built, would probably lead to a modification of the drawing by adding a new bar, because the cost is much lower in this case. The adequate reliability is not only reached by means of designing structures with great resistance, but also by means of preventative and protective measures. Those are measures that decrease the probability of occurrence of the event taken in consideration or the consequences of the failure of the structure (e.g. implementation of safety barriers, active and passive protective measures against fire, protection against corrosion such as painting or cathodic protection, etc.) or an adequate quality management for reducing errors in design and execution of the structure, and gross human errors. In order to decrease the consequences of the failure other aspects can be taken into account: – the degree of robustness (structural integrity): interactions between the members of the structure in such a way that the failure of one member does not yield to the failure of the structure as a whole. See Chapter 5 of this Guidebook; – the choice of the design working life of the structure and its elements that could be different, with possible different plans for maintenance and replacement; – the extent and quality of preliminary investigations of soils and possible environmental influences; – the accuracy of the mechanical models used; – the detailing; – adequate inspection and maintenance according to procedures specified in the project documentation; – measures relating to design calculations: translating the adequate probability of failure into the choice of: – representative values of actions; – the values of the partial factors. This subject is more deeply treated and focused in practical applications in the Chapters 2 and 3 of this Guidebook: ‘Basis of design - General principles’ and ‘Reliability differentiation’.

4. DESIGN WORKING LIFE Design working life is the assumed period for which a structure or part of it is to be used for its intended purpose with anticipated maintenance but without major repair being necessary. All the structural calculations based on probabilities must be referred, implicitly or explicitly, to relevant design working life. The choice of the value of design working life is basic for the design process of the structure. It will affect not only aspects of the required durability of the structure, but also the design values of the actions to be considered. The longer is the time period the bigger probabilities of reaching higher values of the actions should be considered.

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Chapter 1: Basic requirements

The choice of design working life will depend on the economic or social importance of the structure. Indicative values are given for different types of civil and building structures in EN 1990 [9]. More detailed information may be given in the National Annex. For each particular case, the design working life can be established by an agreement between the owner, the National Authorities and the designer. Structures not included in the scope of the Eurocodes (e.g. dams, tunnels, nuclear power plants, etc.) could be designed for different design working lives, even longer than those indicative values given in Table 1. Table 1: Indicative design working life Design working life category 1 2

Indicative design working life (years) 10 10-25

Examples

Temporary structures (1) Replaceable structural parts, e.g. gantry girders, bearings 3 15-30 Agricultural and similar structures 4 50 Building structures and other common structures 5 100 Monumental building structures, bridges, and other civil engineering structures (1) Structures or parts of structures that can be dismantled with a view of being re-used should not be considered as temporary.

5. DURABILITY The ISO/DIS 13823 [8] defines the term durability as: the capability of a structure or any component to satisfy with planned maintenance the design performance requirements over a specified period of time under the influence of the environmental actions, or as a result of a self-ageing process. Structures, as everything, deteriorate with time adversely influencing their performance. There are multiple actions affecting the durability of the structure depending mainly on its materials. The most important of them refer to presence of water moisture with or without contaminants. The rate of deterioration depends on the environmental conditions, the chosen materials and the quality in the design, execution and maintenance. The requirement in this point is that the structure shall be designed so that deterioration over its design working life does not impair the performance of the structure below that intended, having due regard to its environment and the anticipated level of maintenance. In this point, maintenance has to be understood as the total set of activities (including inspection, cleaning and repair) performed during the design service life of a structure to preserve the appropriate structural performance [8]. In order to achieve an adequately durable structure, the following should be taken into account: – the intended or foreseeable use of the structure; – the required design criteria; – the expected environmental conditions; – the composition, properties and performance of the materials and products; – the properties of the soil;

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Chapter 1: Basic requirements

– – – – –

the choice of the structural system; the shape of members and the structural detailing; the quality of workmanship, and the level of control; the particular protective measures; the intended maintenance during the design working life. The Figure 1, adopted from [6] gives an indication on how a structure can perform in time. We can take the performance as a quantitative variable defining the behaviour of the structure. Once the structure is in use (even before), it starts to deteriorate, and, therefore, to decrease the value of the chosen variable ‘performance’. If we do not take any other measure that the normal maintenance, the performance after a time, longer or shorter, will reach the nominal Serviceability Limit State (SLS), defining what would be the actual working life of the structure. If, even in this point, no measures are taken, the structure will keep on deteriorating reaching the Ultimate Limit State (ULS) and, eventually, the actual failure of the structure. If repairs are undertaken in some points in time of the working life of the structure, before the SLS is reached, punctual increases of the performance value can be obtained, in general the performance will keep under the original value, allowing to have a longer working life for the structure.

Figure 1. Working life with and without repairs

6. QUALITY MANAGEMENT Quality management has three main components: quality control, quality assurance and quality improvement. Quality management is focused not only on product quality, but also the means to achieve it. The structure as built has to fulfil all the requirements and the assumptions adopted in the design phase. To assure this, appropriate quality management measures should be in place. These measures comprise:

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Chapter 1: Basic requirements

– definition of the reliability requirements, – organisational measures and – controls at the stages of design, execution, use and maintenance.

REFERENCES [1] Construction Products Directive (Council Directive 89/106/EEC) . European Commission, Enterprise Directorate-General, 2003 http://ec.europa.eu/enterprise/construction/internal/cpd/cpd.htm [2] Interpretative document No. 1: Mechanical resistance and stability. European Commission, Enterprise Directorate-General, 2004 http://ec.europa.eu/enterprise/construction/internal/intdoc/idoc1.htm [4] Guidance Paper L (concerning the Construction Products Directive - 89/106/EEC) Application and Use of Eurocodes: European Commission, Enterprise Directorate-General, 2004. http://ec.europa.eu/enterprise/construction/internal/guidpap/europart1.htm [5] Handbook 1. Basis of Structural Design. Leonardo da Vinci Pilot Project CZ/02/B/F/PP134007. Gaston, UK. 2004 [6] Gulvanessian, H., Calgaro, J.-A., Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, 192 pp. [7] ISO 834 Part 1 [8] ISO/DIS 13823, general Principles on the Design of Structures for Durability, Geneva 2006 [9] EN 1990 Eurocode: Basis of structural design. European Committee for Standardisation, 04/2002. [10] EN 1991-1-7 Eurocode 1: Actions on structures – Part 1-1: General actions – Accidental actions, European Committee for Standardisation, 2006.

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Chapter 1: Basic requirements

APPENDIX A: REFERENCE DOCUMENTS A.1 Introduction In order to better understand what the Essential Requirements express, the most important items of the documents and their backgrounds are described as follows. The full text of the documents can be easily obtained from the European Committee web page indicated in the references. A.2 Construction Products Directive The construction products directive (Council Directive 89/106/EEC) Council Directive 89/106/EEC of 21 December 1988 on the approximation of laws, regulations and administrative provisions of the Member States relating to construction products (89/106/EEC) (OJ L 40, 11.2.1989, p.12) amended by: Council Directive 93/68/EEC of 22 July 1993 (OJ L 220, 30.8.1993, p.1) and Regulation (EC) No 1882/2003 of the European Parliament and of the Council of 29 September 2003 (OJ L 284, 31.10.2003, p.1) A.2.1 Annex I: Essential requirements The products must be suitable for construction works, which (as a whole and in their separate parts) are fit for their intended use, account being taken of economy, and in this connection satisfying the following essential requirements where the works are subjected to regulations containing such requirements. Such requirements shall, under normal maintenance, be satisfied for an economically reasonable working life. The requirements generally concern actions which are foreseeable. Mechanical resistance and stability The construction works must be designed and built in such a way that the loadings that are liable to act on it during its constructions and use will not lead to any of the following: (a) collapse of the whole or part of the work; (b) major deformations to an inadmissible degree; (c) damage to other parts of the works, fittings or installed equipment due to deformation of the load-bearing structures; (d) damage by an event to an extent disproportionate to the original cause. Safety in case of fire The construction works must be designed and built in such a way that in the event of an outbreak of fire: - the load-bearing capacity of the construction can be assumed for a specific period of time, - the generation and spread of fire and smoke within the works are limited, - the spread of the fire to neighbouring construction works is limited, - occupants can leave the works or be rescued by other means. - the safety of rescue teams is taken into consideration. Hygiene, health and the environment The construction work must be designed and built in such a way that it will not be a threat to the hygiene or health of the occupants or neighbours, in particular as a result of any of the following: - the giving-off toxic gas, - the presence of dangerous particles or gases in the air. - the emission of dangerous radiation

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Chapter 1: Basic requirements

- pollution or poisoning of water or soil, - faulty elimination of waste water, smoke, solid or liquid wastes, - the presence of damp in parts of the works or on surfaces within the works. Safety in use The construction work must be designed and built in such a way that it does not present unacceptable risks of accidents in service or in operation such as slipping, falling, collision, burns, electrocution, injury from explosion. Protection against noise The construction works must be designed and built in such a way that noise perceived by the occupants or people nearby is kept down to a level that will not threaten their health and will allow them to sleep, rest and work in satisfactory conditions. Energy economy and heat retention The construction works and its heating, cooling and ventilation installations must be designed and built in such a way that the amount of energy required in use shall be low, having regard to the climatic conditions of the location and the occupants. A.3 Interpretative document No. 1 Mechanical resistance and stability A.3.1 Purpose and scope of Interpretative document No. 1 (1) This Interpretative Document relates to Council Directive 89/106/EEC of 21 December 1988 on the approximation of laws, regulations and administrative provisions of the Member States relating to construction products, hereinafter referred to as “the Directive”. (2) Article 3 of the Directive stipulates that the purpose of the Interpretative Documents is to give concrete form to the essential requirements for the creation of the necessary links between the essential requirements set out in Annex I to the Directive and the mandates for the preparation of harmonized standards and guidelines for European technical approvals or the recognition of other technical specifications within the meaning of Articles 4 and 5 of the Directive. Where considered necessary, the provisions of this Interpretative Document will be further specified in each particular mandate. In drafting the mandates, account will be taken, if necessary, of the other essential requirements of the Directive, as well as of other relevant Directives concerning construction products. (3) This Interpretative Document deals with the aspects of the works where “Mechanical resistance and stability” may be concerned. It identifies products or product families and characteristics relating to their satisfactory performance. “The construction works must be designed and built in such a way that the loadings that are liable to act on it during its construction and use will not lead to any of the following: a) collapse of the whole or part of the works; b) major deformations to an inadmissible degree; c) damage to other parts of the works or to fittings or installed equipment as a result of major deformation of the load-bearing construction; d) damage by an event to an extent disproportionate to the original cause.”

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Chapter 1: Basic requirements

4) In accordance with the Council Resolution of 7 May 1985 (New Approach) and the preamble of the Directive, this interpretation of the essential requirement is intended not to reduce the existing and justified levels of protection for works in the Member States. A.3.2 Meaning of the general terms used in the Interpretative documents. Construction works Construction works means everything that is constructed or results from construction operations and is fixed to the ground. This term covers both buildings and civil engineering works. In the Interpretative Documents "construction works" are also referred to as the "works". Construction works include for example: dwellings; industrial, commercial, office, health, educational, recreational and agricultural buildings; bridges; roads and highways; railways; pipe networks; stadiums; swimming pools; wharfs; platforms; docks; locks; channels; dams; towers; tanks; tunnels; etc. Construction products (1) This term refers to products which are produced for incorporation in a permanent manner in the works and placed as such on the market. The terms "construction products" or "products", where used in the Interpretative Documents, include materials, elements and components (single or in a kit) of prefabricated systems or installations which enable the works to meet the essential requirements. (2) Incorporation of a product in a permanent manner in the works means: - that its removal reduces the performance capabilities of the works; and - that the dismantling or the replacement of the product are operations which involve construction activities. Normal maintenance (1) Maintenance is a set of preventive and other measures which are applied to the works in order to enable the works to fulfil all its functions during its working life. These measures include cleaning, servicing, repainting, repairing, replacing parts of the works where needed, etc. (2) Normal maintenance generally includes inspections and occurs at a time when the costs of the intervention which has to be made are not disproportionate to the value of the part of the works concerned, consequential costs being taken into account. Intended use The intended use of a product refers to the role(s) that the product is intended to play in the fulfilment of the essential requirements. Economically reasonable working life (1) The working life is the period of time during which the performance of the works will be maintained at a level compatible with the fulfilment of the essential requirements. (2) An economically reasonable working life presumes that all relevant aspects are taken into account, such as: - costs of design, construction and use; - costs arising from hindrance of use; - risks and consequences of failure of the works during its working life and costs of insurance covering these risks; - planned partial renewal; - costs of inspections, maintenance, care and repair; - costs of operation and administration;

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Chapter 1: Basic requirements

- disposal; - environmental aspects. Actions Actions which may affect the compliance of the works with the essential requirements are brought about by agents acting on the works or parts of the works. Such agents include mechanical, chemical, biological, thermal and electro-magnetic agents. Performance Performance is a quantitative expression (value, grade, class or level) of the behaviour of a works, part of the works or product, for an action to which it is subject or which it generates under the intended service conditions (for the works or parts of works) or intended use conditions (for products). A.3.3. Basis for verification of the satisfaction of the essential requirement "Mechanical resistance and stability". General (1) This chapter identifies basic principles prevailing in Member States for the verification of the satisfaction of the essential requirement "Mechanical resistance and stability". These principles are currently complied with when and where the works are subject to regulations containing this essential requirement. (2) The essential requirement, as far as applicable, is satisfied with acceptable probability during an economically reasonable working life of the works. (3) The satisfaction of the essential requirement is assured by a number of interrelated measures concerned in particular with: - the planning and design of the works, the execution of the works and necessary maintenance; - the properties, performances and use of the construction products. (4) It is up to the Member States, when and where they feel it necessary, to take measures concerning the supervision of planning, design and execution of the works, and concerning the qualifications of parties and persons involved. Where this supervision and this control of qualifications are directly connected with the characteristics of products, the relevant provisions shall be laid down in the context of the mandate for the preparation of the standards and guidelines for European technical approval related to the products concerned. A.3.4. Working life and durability. 1 Treatment of working life of construction works in relation to the essential requirement (1) It is up to the Member States, when and where they feel it necessary, to take measures concerning the working life which can be considered reasonable for each type of works, or for some of them, or for parts of the works, in relation to the satisfaction of the essential requirements. (2) Where provisions concerning the durability of works in relation to the essential requirement are connected with the characteristics of products, the mandates for the preparation of the European standards and guidelines for European technical approvals, related to these products, will also cover durability aspects.

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Chapter 1: Basic requirements

A.4 GUIDANCE PAPER L A.4.1. Eurocodes Part 1: General 1.1 Aims and benefits of the Eurocode programme The Eurocodes provide common design methods, expressed in a set of European standards, which are intended to be used as reference documents for Member States to: – prove the compliance of building and civil engineering works or parts thereof with Essential Requirement n°1 Mechanical resistance and stability (including such aspects of Essential Requirement n°4 Safety in use, which relate to mechanical resistance and stability) and a part of Essential Requirement n°2 Safety in case of fire, including durability, as defined in Annex 1 of the CPD – express in technical terms these Essential Requirements applicable to the works and parts thereof; – determine the performance of structural components and kits with regard to mechanical resistance and stability and resistance to fire, insofar as it is part of the information accompanying CE marking (e.g. declared values). 1.1.2 EN Eurocodes are intended by the European Commission services, and the Member States, to become the European recommended means for the structural design of works and parts thereof, to facilitate the exchange of construction services (construction works and related engineering services) and to improve the functioning of the internal market. 1.1.3 The intended benefits and opportunities of Eurocodes are to: provide common design criteria and methods to fulfil the specified requirements for mechanical resistance, stability and resistance to fire, including aspects of durability and economy, provide a common understanding regarding the design of structures between owners, operators and users, designers, contractors and manufacturers of construction products facilitate the exchange of construction services between Members States, facilitate the marketing and use of structural components and kits in Members States, facilitate the marketing and use of materials and constituent products, the properties of which enter into design calculations, in Members States, be a common basis for research and development, in the construction sector, allow the preparation of common design aids and software, increase the competitiveness of the European civil engineering firms, contractors, designers and product manufacturers in their world-wide activities. 1.2 Background of the Eurocode programme 1.3 Objectives of the Guidance Paper 1.3.1 This Guidance Paper expresses, with the view of achieving the aims and benefits of the Eurocode programme mentioned in 1.1, the common understanding of the Commission and the Member States on: The application of EN Eurocodes in the structural design of works (chapter 2). The use of EN Eurocodes in harmonised standards and European technical approvals for structural construction products (chapter 3). A distinction is made between: 23

Chapter 1: Basic requirements

a) products with properties which enter into structural calculations of works, or otherwise relate to their mechanical resistance and stability, including aspects of durability and serviceability, and which for this reason should be consistent with the assumptions and provisions made in the EN Eurocodes ("structural materials" are the most concerned - see chapter 3.2) b) products with properties which can directly be determined by methods used for the structural design of works, and thus should be determined according to the EN Eurocode methods (prefabricated "structural components and kits" are the most concerned - see chapter 3.3). A.4.2. Eurocodes Part 2: Use of EN Eurocodes for structural design of works 2.1 National Provisions for the structural design of works 2.1.1 The determination of the levels of safety [The word safety is encompassed in the Eurocodes in the word reliability] of buildings and civil engineering works and parts thereof, including aspects of durability and economy [The introductory provisions of Annex I of the CPD lay down: "The products must be suitable for construction works which (as a whole and in their separate parts) are fit for their intended use, account being taken of economy, and in this connection satisfy the following essential requirements where the works are subject to regulations containing such requirements. Such requirements must, subject to normal maintenance, be satisfied for an economically reasonable working life. The requirements generally concern actions which are foreseeable." Economic aspects remain within the competence of the Member States. 2.1.2 Possible differences in geographical or climatic conditions (e.g. wind or snow), or in ways of life, as well as different levels of protection that may prevail at national, regional or local level in the sense of article 3.2 of the CPD. 2.1.3 When Member States lay down their Nationally Determined Parameters, they should: choose from the classes included in the EN Eurocodes, or use the recommended value, or choose a value within the recommended range of values, for a symbol where the EN Eurocodes make a recommendation, or when alternative methods are given, use the recommended method, where the EN Eurocodes make a recommendation, take into account the need for coherence of the Nationally Determined Parameters laid down for the different EN Eurocodes and the various Parts thereof. Member States are encouraged to co-operate in minimising the number of cases where recommendations for a value or method are not adopted for their nationally determined parameters. By choosing the same values and methods, the Member States will enhance the benefits listed in 1.1.3. 2.1.4 The Nationally Determined Parameters laid down in a Member State should be made clearly known to the users of the EN Eurocodes and other parties concerned, including manufacturers. 2.1.5 When the EN Eurocodes are used for the design of construction works, or parts thereof, the Nationally Determined Parameters of the Member State on whose territory the works are located shall be applied.

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Chapter 1: Basic requirements

Note: Any reference to an EN Eurocode design should include the information on which set of Nationally Determined Parameters was used, whether or not the Nationally Determined Parameters that were used correspond to the recommendations given in the EN Eurocodes (see 2.1.3). 2.1.6 National Provisions should avoid replacing any EN Eurocode provisions, e.g. Application Rules, by national rules (codes, standards, regulatory provisions, etc.). When, however, National Provisions do provide that the designer may – even after the end of the coexistence period - deviate from or not apply the EN Eurocodes or certain provisions thereof (e.g. Application Rules), then the design will not be called "a design according to EN Eurocodes". 2.1.7 When Eurocode Parts are published as European standards, they will become part of the application of the Public Procurement Directive. In all cases, technical specifications shall be formulated in public tender enquiries and public contracts by referring to EN Eurocodes, in combination with the Nationally Determined Parameters applicable to the works concerned, apart from the exceptions expressed in article 10.3 (Directive 93/37, article 10.2). However, in application of the PPD, and following the spirit of the New Approach, the reference to EN Eurocodes is not necessarily the only possible reference allowed in a Public contract. The PPD foresees the possibility for the procuring entity to accept other proposals, if their equivalence to the EN Eurocodes can be demonstrated by the contractor. Consequently, the design of works proposed in response to a Public tender can be prepared according to: EN Eurocodes (including NDPs), which give a presumption of conformity with all legal European requirements concerning mechanical resistance and stability, fire resistance and durability, in compliance with the technical specifications required in the contract for the works concerned; Other provisions expressing the required technical specification in terms of performance. In this case, the technical specification should be detailed enough to allow tenders to know the conditions on which the offer can be made and the owner to choose the preferred offer. This applies, in particular, to the use of national codes, as long as Member States maintain their use in parallel with EN Eurocodes (e.g. a Design Code provided by National Provisions), if also specified to be acceptable as an alternative to an EN Eurocode Part by the Public tender. 2.2 Indications to writers of EN Eurocodes 2.3 National Annexes of the EN Eurocode Parts 2.3.1 When a Eurocode Part is circulated by CEN for publication as an EN, the final text of the approved EN, according to CEN rules, is made available by CEN Management Centre to CEN members (the NSBs) in the 3 official languages (English, French and German).[This step corresponds to the DAV – Date of Availability] Each NSB shall implement this EN as a national standard by publication of an equivalent text (i.e. a version translated into another language) or by endorsement of one of the 3 language versions provided by CEN Management Centre (by attaching an "endorsement sheet"), within the timescale agreed for publication. The National standard transposing the EN Eurocode Part, when published by a National Standards Body (NSB), will be composed of the EN Eurocode text (which may be preceded by a National title page and by a National Foreword), generally followed by a National Annex. 25

Chapter 1: Basic requirements

2.3.2 The National Standards Bodies should normally publish a National Annex, on behalf of and with the agreement of the national competent authorities. A National Annex is not necessary if an EN Eurocode Part contains no choice open for Nationally Determined Parameters, or if an EN Eurocode Part is not relevant for the Member State (e.g. seismic design for some countries). Note: As stated by the CEN Rules, the National Annex is not a CEN requirement (a NSB can publish an EN Eurocode Part without one). However, in the context of this Guidance Paper, the National Annex serves for NSBs to publish the Nationally Determined Parameters, which will be essential for design. 2.3.3 The National Annex may contain [See EN 1990 and EN 1991 Part 1-1 – Foreword – National standards implementing Eurocodes], directly or by reference to specific provisions, information on those parameters which are left open in the Eurocodes for national choice, the Nationally Determined Parameters, to be used for the design of buildings and civil engineering works to be constructed in the country concerned, i.e: values and/or classes where alternatives are given in the EN Eurocode, values to be used where a symbol only is given in the EN Eurocode, country specific data (geographical, climatic, etc.), e.g. a snow map, the procedure to be used where alternative procedures are given in the EN Eurocode. It may also contain the following: decisions on the application of informative annexes, and, reference to non-contradictory complementary information to assist the user in applying the Eurocode. 2.3.4 A National Annex cannot change or modify the content of the EN Eurocode text in any way other than where it indicates that national choices may be made by means of Nationally Determined Parameters. 2.3.5 The National Annex of an EN Eurocode Part will normally be finalised when the safety and economy levels have been considered, i.e. at the end of the period allocated for the establishment of the Nationally Determined Parameters (see Annex A). 2.3.6 If a Member State does not choose any NDPs, the choice of the relevant values (e.g. the recommended value), classes or alternative method will be the responsibility of the designer, taking into account the conditions of the project and the National provisions. 2.3.7 The National Annex has an informative status. The content of a National Annex can be the basis for a national standard, via the NSB, and/or can be referred to in a National Regulation. 2.3.8 The National Annex can be amended, if necessary, according to CEN rules. 2.4 Packages of EN Eurocode Parts 2.4.1 The purpose of defining Packages, by grouping Parts of EN Eurocode, is to enable a common date of withdrawal (DoW) [At the date of withdrawal related to a new standard, all the specifications existing previously in the National collection of standards conflicting with the new standard have to be withdrawn and the national provisions have to be adapted to allow the legitimate use of EN Eurocodes] for all of the relevant parts that are needed for a particular design. Thus conflicting national standards shall have been withdrawn at the end of the coexistence period, after all of the EN Eurocodes of a Package are available, and National

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Chapter 1: Basic requirements

Provisions will have been adapted by the end of the National Calibration period, as described in Annex A. Publication of the individual Parts in a Package is likely to occur over a long period of time, so that, for many Parts, the coexistence period will be much longer than the minimum given in 2.5.5. When a National standard has a wider scope than the conflicting Eurocode Package, only that part of the National standard whose scope is covered by the Package has to be withdrawn. When more than one Package of EN Eurocodes is likely to be needed for the design of works, the dates of withdrawal of the related Packages can be synchronised. 2.4.2 No Parts from EN 1990 or the EN 1991, EN 1997 or EN 1998 series form a Package in themselves; those Parts are placed in each of the Packages, as they are material independent. 2.4.3 The list of the EN Eurocode Parts contained in the various Packages for each of the main materials, i.e. concrete, steel, composite concrete and steel, timber, masonry and aluminium, and their respective target dates, will be updated and made available through the CEN/MC web-site (see Annex C which presents the Packages as they are currently foreseen) 2.5 Arrangements for the implementation of EN Eurocodes and period of co-existence with national rules for the structural design of works 2.5.1 The arrangements for the implementation of an EN Eurocode Part include, from the time the final draft of the EN Eurocode is produced by the CEN/TC250, five periods: - Two periods before the date of availability (DAV): Examination period, CEN process period. - Three periods after the date of availability: Translation period, National calibration period, Coexistence period. The detailed content of each of the five periods is given in the table and chart in Annex A. The progress of each EN Eurocode (or Package), within these periods, will be provided by CEN/MC on their web-site. 2.5.2 The following basic requirements need to be fulfilled by the EN Eurocode Parts in order to be referred to in the national provisions: - Calculations executed on the basis of the Eurocode Part, in combination with the Nationally Determined Parameters, shall provide an acceptable level of safety. - The use of the EN Eurocode Part, in combination with the Nationally Determined Parameters, does not lead to structures that cost significantly more, over their working life [see Interpretative Document 1, clause 1.3.5], than those designed according to National standards or provisions, unless changes in safety have been made and agreed. 2.5.3 The European Commission encourages Member States to implement EN Eurocodes in the framework of their National Provisions. During the coexistence period, the construction regulation authorities should accept the use of EN Eurocodes, as an alternative to the previous rules (e.g. National codes, standards or other technical rules included, or referred to, in national provisions) for the design of construction works. Member States are also encouraged to adapt their national provisions to withdraw conflicting national rules before the end of the co-existence period. 2.5.4 When an EN Eurocode Part is made available, the Member States should: - set officially, before the end of the National calibration period (see Annex A), the Nationally Determined Parameters to be applied on their territory. In the event of any unexpected 27

Chapter 1: Basic requirements

obstacles to carrying out the calibration of an EN Eurocode Part, the Member State shall inform the Commission, when an extension of the period could be agreed by the SCC. - adapt, as far as necessary, their National Provisions so that the EN Eurocode Part can be used on their territory: - as a means to prove compliance of construction works with the national requirements for "mechanical resistance and stability" and "resistance to fire", in the sense of Annex I of the CPD, and - as a basis for specifying contracts for the execution of public construction works and related engineering services. If no NDPs are to be produced for an EN Eurocode Part, the coexistence period begins at DAV and ends at DoW. Thus the EN Eurocode is available and any existing national standard is still available, so that both can be used during this period. At the end of the "coexistence period" of the last EN Eurocode Part of a Package, the Member States should have adapted all their National Provisions which lay down (or refer to) design rules within the scope of the relevant Package. 2.5.5 Owing to the need for operational Packages (as defined in 2.4), the reference to the coexistence period of a Package is defined as the coexistence period of the last Eurocode Part of that Package. In Member States intending to implement EN Eurocodes, the coexistence period of this last part should be three years. After the three years coexistence period of the last EN Eurocode Part of a Package, the whole Package-related former conflicting national standards will be withdrawn, i.e. 5 years maximum after DAV [It is intended that the end of the coexistence period for each Package will be laid down by the Commission after consultation of Member States]. Conflicting National Provisions that would not allow the use of the first parts of a Package should be arranged, in order to allow the legitimate use of those Parts. 2.5.6 In order to increase the overall transparency of the implementation of the EN Eurocodes, the Commission wishes to be informed, by the Member States, of the main phases: translation, national calibration and coexistence Period, for each EN Eurocode Part, and the adaptation of National Provisions. Note: the Commission intends to prepare, for this purpose, a "test reporting form" on the basis of the items mentioned in the Annex B. A.4.3. Eurocodes Part 3: Use of EN Eurocodes in technical specifications for structural products This part of the Guidance Paper deals with structural products specified in the CPD as construction products: 3.1 Distinction is made between specifications for materials to be determined by test and specifications for components to be determined by calculation. 3.1.1 It follows from the CPD [Article 2.1 and 3.3] and the Interpretative Documents that there is a need for consistency between the technical specifications for construction products (hEN and ETA) and the technical rules for works. 3.1.2 For construction products, which contribute to the mechanical resistance and stability and/or fire resistance of works, two types of properties are distinguished, according to the validation method: Properties to be determined by testing (generally in the case of structural materials and constituent products, such as concrete, reinforcing steel for concrete, fire protection material, etc.), and Properties to be determined by calculation following methods, which are also used for the structural design of works (generally for prefabricated structural components and kits,

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Chapter 1: Basic requirements

consisting of structural components, such as prefabricated concrete components, prefabricated stairs, timber frame buildings kits, etc.). For both types of product properties the resulting values are to be "declared" in the information accompanying the CE marking [by application of CPD and in conformity with the mandate given by the Commission] of the product and used in the structural design of works or parts thereof. 3.1.3 For the reference to, or use of, EN Eurocodes in harmonised product specifications a distinction is made in this Part 3 between: - structural materials and constituent products with properties to be determined by testing, and - prefabricated structural components and kits consisting of structural components with properties to be calculated according to EN Eurocode methods. A.4.4. Eurocodes Part 4: Future actions related to the Eurocode Programme 4.1 Education 4.1.1 To build on the strong pedigree of the EN Eurocodes described above, the Commission recognises the importance of building on this with programmes of education to help the professions to implement the EN Eurocodes. 4.1.2 Aspects of education that need to be covered, include: informing and making the profession as a whole aware of the EN Eurocodes • providing continuing professional development and training to the profession • encouraging the production of handbooks, design aids, software etc. to facilitate the implementation of the EN Eurocodes • encouraging Universities and Technical Colleges to base their teaching of civil and structural engineering design on the EN Eurocodes. 4.1.3 The Commission, in liaison with industry and Member States, will encourage: • publication of easily understandable "jargon free" booklets covering the EN Eurocodes; • holding of European seminars aimed at the profession as a whole as key EN Eurocodes become available as ENs (e.g. EN 1990:Basis of Design); • publication of documents on the adoption of the EN Eurocodes through Government or on behalf of Government; • holding of meetings organised by professional and industry bodies to inform construction professionals and university teachers, to listen to and discuss their concerns, and to promote the opportunities offered by the EN Eurocodes; • the arrangement of continuing professional development and training courses; • the development of aids to implementation. 4.1.4 Central to any initiatives taken on education is the production of : • handbooks, worked examples and background documents; • software; • guides for everyday structures (e.g. normal buildings) based on the EN Eurocodes. Publishing companies, software houses and trade organisations will carry out these important activities, mainly as commercial ventures. Encouragement to these bodies can be given by a strong commitment to implementation of the EN Eurocodes both by the EC and the Member States. 4.1.5 Member States should encourage the use of the EN Eurocodes in private contracts, particularly through education and information campaigns, regardless of what may be requested by National provisions.

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Chapter 1: Basic requirements

4.2. Research with regard to EN Eurocodes 4.2.1 The Commission services recognises that, for the Construction sector to remain competitive in the world construction industry, it is essential that the EN Eurocodes, once published, should remain the most up to date, useable International Codes of Practice, meeting the requirements for a profession practising in a competitive environment. 4.2.2 The EN Eurocodes should be further developed taking into account the innovative pressures of the market and the progress of scientific knowledge. 4.2.3 The pressures from the market are generated by: • new material and new products; • new ways for procurement and execution of works; • needs for economy whilst maintaining acceptable levels of safety. The progress of the scientific knowledge and methods are generated by: • the need to avoid disasters in the area of safety (e.g. seismic, fire); • a knowledge of phenomena acquired in other domains (e.g. aeronautics for wind action); • the answer to new economic or social needs (e.g. High Speed Railways, nuclear plants); • the availability of powerful and widely-distributed tools for calculation (computers and software). 4.2.4 Initiatives for research arise from • the industry or the users concerned; • public authorities in charge of safety, economy, scientific development and education (for example, the development of NDPs) • universities and research organisations experienced from their involvement as third parties. 4.2.5 In many cases there will be a mutual interest for both industry and public authorities (including the European Commission) in research and this should be reflected by agreements on common funding according to the following criteria: • Industrial and user's sources - the main funding for research whose objectives are short-term benefits or particular advantages for special innovative companies and associated industries and users (e.g. unique verifications and ETA's). • EC or National public funding - the main funding for research whose objectives are medium to long term benefits for the European construction industry (e.g. for improving technical specifications and design codes, harmonising models for actions and resistances, improving safety aspects). 4.3. Maintenance of EN Eurocodes 4.3.1 The maintenance of the EN Eurocodes is essential; the need for updating, revision and completion is strongly recognised so that an improved second generation of EN Eurocodes can evolve. However, a period of stability should be observed before embarking on changes other than to correct errors. 4.3.2 Maintenance work will involve: • reducing open choices (NDPs); • urgent matters of health and safety; • correcting errors;

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Chapter 1: Basic requirements • • •

ensuring the most up to date information is in the EN Eurocodes, recognising recent proven innovations and improvements in construction technology; feedback from use of the EN Eurocodes in the various Member States through CEN; requests from industrial organisations or public authorities to CEN members for revision.

4.3.3 The organisation of maintenance should start after the receipt of a positive vote on a draft EN Eurocode, a Maintenance Group should be formed by the relevant CEN/TC250 SC to: • give further consideration of co-ordination items arising from the work of other Project Teams (this is necessary as the various parts of the EN Eurocodes are not being prepared simultaneously); • provide explanations to questions arising from the use of the EN Eurocode, e.g. on background and interpretation of rules; • collect comments and requests for amendment; • prepare action plans for urgent revision in the case of safety related matters, or future systematic revisions according to the CEN procedure and as decided by CEN/TC250. 4.3.4 The strategy to provide adequate resources to support the maintenance of the EN Eurocodes should be decided by the European Commission, Member States, Industry and CEN seeking to find a balance between: • the requirements for public safety; • the competitive demands of industry; • the availability of funds.

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Chapter 2: Basis of design – general principles

CHAPTER 2: BASIS OF DESIGN – GENERAL PRINCIPLES Milan Holický1, Jana Marková1 1

Klokner Institute, Czech Technical University in Prague, Czech Republic

Summary Construction works are complicated technical systems suffering from a number of significant uncertainties in all stages of execution and use. Reliability is therefore an important aspect of their design. The most important historical methods include method of permissible stresses and method of global and partial factors. Present European standard EN 1990 Eurocode - Basis of structural is based on the concept of limit states in conjunction with partial factor method. Probabilistic approach and methods of risk assessment are used as scientific bases of the partial factor method and as an alternative design method.

1

INTRODUCTION

1.1

Background documents

The standard EN 1990 Basis of structural design [1] and EN 1991-1-1 Actions on structures [2] are the fundamental documents for the whole system of Eurocodes. These documents are available since April 2002 and at present are implemented into the systems of national standards. An important background document is the International Standard ISO 2394 [3], Probabilistic Model Code [5], Designer's Guide to EN 1990 [5] and other literature (for example [6]). 1.2

General principles

Two basic sets of limit states should be considered in accordance to the design principles of EN 1990 [1]. Particularly it should be verified that the load effect E does not exceed the resistance of the structure in ultimate limit states and the relevant criteria for serviceability limit states. In common cases the general condition of structural performance with respect to ultimate or serviceability limit states may be expressed by the following inequality E < R,

(1)

Here R denotes the resistance. Note that in EN 1990 [1] the design value Ed and Rd are used in the condition (1) to represent the load effect E and resistance R. When probabilistic design is used, then the load effect E and resistance R are considered as random variables and probability that the condition (1) is valid or violated is analysed. For the selected design situations and identified limit states, the critical load cases should be determined. In accordance with EN 1990 [1], a load case is a compatible load arrangement, sets of deformations and imperfections considered simultaneously with fixed variable actions and permanent actions. Document [1] is primarily based on the partial factor method, called also semi-probabilistic method. However, as an alternative, a design directly based on probabilistic methods is also allowed.

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Chapter 2: Basis of design – general principles

Basic concepts of historical development and various design methods are shortly described in the following section.

2

HISTORICAL DEVELOPMENT

2.1

Uncertainties

It is well recognised that construction works are complicated technical systems suffering from a number of significant uncertainties in all stages of execution and use. Some uncertainties can never be eliminated absolutely and must be taken into account when designing or verifying construction works. Depending on the nature of a structure, environmental conditions and applied actions, some types of uncertainties may become critical. The following types of uncertainties can usually be identified: - natural randomness of actions, material properties and geometric data; - statistical uncertainties due to a limited size of available data; - uncertainties of theoretical models caused by a simplification of actual conditions; - vagueness due to inaccurate definitions of performance requirements; - gross errors in design, execution and operation of the structure; - lack of knowledge of the behaviour of new materials in real conditions. Note that the order of the listed uncertainties corresponds approximately to the decreasing amount of current knowledge and available theoretical tools to analyse them and to take them into account in design. The natural randomness and statistical uncertainties may be relatively well described by available methods of the theory of probability and mathematical statistics. In fact the Eurocode [1] and the International Standard [3] provide some guidance on how to proceed. However, lack of reliable experimental data, i.e. statistical uncertainty, particularly in case of new materials, some actions including environmental influences and also some geometrical data, cause significant problems. Moreover, the available data are often inhomogeneous and obtained under different conditions (for example for material properties, imposed loads, environmental influences, but also for internal dimensions of reinforced concrete crosssections). Then it may be difficult, if not impossible, to analyse such data and to use them in design. Uncertainties of theoretical models may be, to a certain extent, assessed on the basis of theoretical and experimental research. Again the Standards [1, 3] provide some guidance on how to proceed. The vagueness caused by inaccurate definitions (in particular of serviceability and other performance requirements) may be partially described by the theory of fuzzy sets. However, up to now these methods have a little practical significance, as convincing theoretical models are rarely available. The knowledge of the behaviour of new materials and structures may gradually increase due to newly developed theoretical tools and experimental research. The lack of available theoretical tools is obvious in the case of gross errors and lack of knowledge, which are nevertheless often the decisive causes of structural failures. To limit gross errors due to human activity a quality management system including the methods of statistical inspection and control may be effectively applied. Several design methods and operational techniques have been proposed and worldwide used to control the unfavourable effect of various uncertainties during a specified working life. Simultaneously the theory of structural reliability has been developed to describe and analyse the above-mentioned uncertainties in a rational way and to take them into account in design and verification of structural performance. In fact the development of the whole theory was initiated by observed insufficiencies and structural failures caused by 33

Chapter 2: Basis of design – general principles

various uncertainties. At present the theory of structural reliability is extensively used to calibrate reliability elements of newly proposed standards (partial and various reduction factors). The term "reliability" itself is, however, often used in a very broad sense and may need some clarification. 2.2

Definition of reliability

The term reliability is often simplified and used very vaguely and inaccurately. The concept of reliability is sometimes approached in an absolute (black and white) way – the structure either is or isn’t reliable. In accordance with this approach the positive statement is understood as “the failure of the structure will never occur“. This is of course an incorrect oversimplification, the failure may occur even though the structure is correctly declared as reliable. The interpretation of the complementary (negative) statement is usually understood more correctly: failures are admitted and the probability or frequency of their occurrence is then discussed. Thus according to this simplified approach there should be a certain set of structural conditions determining an area of “absolute reliability“, where any possibility of failure occurrence is excluded. Only when exceeding this limit the emergence of failure would be admitted. In general, such a simplified interpretation is incorrect. Although it may be unpleasant and for many perhaps unacceptable, the hypothetical area of “absolute reliability” for most structures (apart from exceptional cases) simply does not exist. On the contrary, in the design it is necessary to admit a certain small probability that the failure may occur within the intended life of the structure. Otherwise it would not be possible at all to design civil structures. What is then the correct interpretation of the keyword “reliability” and what sense has the generally used statement “the structure is safe”? In structural design a number of similar definitions of the term reliability or their interpretations are used in literature and in national and international documents. ISO 2394 [2] provides a definition of reliability, which is similar to the approach of other national and international standards: reliability is the ability of a structure to comply with given requirements under specified conditions during the intended life, for which it was designed. In Eurocode [1] no definition is offered and it is only noted that reliability covers the load-bearing capacity, serviceability as well as the durability of a structure. In the Fundamental requirements it is then stated that ”a structure shall be designed and executed in such a way that it will, during its intended life with appropriate degrees of reliability and in an economic way: - remain fit for the use for which it is required; and - sustain all actions and influences likely to occur during execution and use.” Generally a different level of reliability for load-bearing capacity and for serviceability may be accepted. In the document [1] the probability of failure pf and the reliability index β are related to failure consequences. Note that the above definition of reliability includes four important elements: - given (performance) requirements – the definition of the structural failure, - time period – the assessment of the required service-life T, - reliability level – the assessment of the probability of failure pf, - conditions of use – limiting input uncertainties. An accurate determination of performance requirements and thus an accurate specification of the term failure is of primary importance. In many cases, mainly when considering the requirements for the stability and collapse of a structure, the specification of this term is not very complicated. In many other cases, in particular when dealing with various requirements of occupants’ comfort, appearance and characteristics of the environment, the appropriate definitions of failure are dependent on vaguenesses and inaccuracies. The 34

Chapter 2: Basis of design – general principles

transformation of these occupants' requirements into appropriate technical quantities and precise criteria is very hard and often leads to considerably vague conditions. In the following the term failure is being used in a very general sense denoting simply any undesirable state of a structure (e.g. collapse or excessive deformation), which is unambiguously given by structural conditions. 2.3

Development of design methods

During their historical development the design methods have been closely linked to the available empirical, experimental as well as theoretical knowledge of mechanics and the theory of probability. The development of various empirical methods for structural design gradually stabilised in the twentieth century on three generally used methods, which have been, in various modifications, applied in standards for structural design till these days. In the context of efforts to simplify the computational procedures some of these methods are sometimes modified or rehabilitated. That is why it is useful to briefly mention these three basic design methods and to indicate those explicit measures, which may affect the probability of failure and structural reliability. The first worldwide design method for civil structures is the method of permissible stresses. It is based on the condition

σmax < σper, where σper = σcrit / k

(2)

where the coefficient k is assessed with regard to uncertainties in the determination of local load effect σmax and of resistance σcrit, and therefore may ensure with an appropriate level of security the reliability of the structure. The main insufficiency of this method is perhaps the local verification of reliability (in the elastic range) and the impossibility to consider separately the uncertainties of basic quantities and the uncertainties of computational models for the assessment of action effects and structural resistance. In this method, the probability of failure is controlled by one quantity only, the coefficient k. The second widespread method of structural design is the method of global safety factor. It is based on the condition s = Xcrit / Xmax > s0

(3)

according to which the calculated safety factor s must be greater than its specified value s0. It is a method which attempts mainly to give a truer picture of the behaviour of elements and their cross-sections, in particular through the aggregate quantities of structural resistance Xcrit and action effect Xmax. Like in the case of the permissible stresses method the main insufficiency of this method remains the impossibility to consider the uncertainties of particular basic quantities and theoretical models. The probability of failure can, again, be controlled by one quantity only, by the global safety factor s. At present, the most advanced operational method of structural design [1, 2] is the partial factor format (often inaccurately denoted as the limit states method). This method is based on the condition Ed (Fd, fd, ad, θd) < Rd (Fd, fd, ad, θd)

(4)

where the action effect Ed and the structural resistance Rd are assessed according to the design values of basic quantities describing the action Fd, material properties fd, dimensions ad and model uncertainties θd. The design values of these quantities are determined (taking into account their uncertainties as well as the uncertainties of computational models) using their characteristic values (Fk, fk, ak, θk), partial safety factors γ, combination factors ψ and other measures of reliability [1, 3, 5, 6]. Thus a whole system of various partial factors and other reliability elements may be used to control the probability of structural failure. 35

Chapter 2: Basis of design – general principles

Obviously the greatest possibility to harmonise structural reliability of different structures made of different materials is offered by the partial factor method. However, in any of the listed methods the probability of failure is not applied directly. Among standards for structural design the recent document ISO [3] was the first one to include probabilistic methods. Probabilistic design methods [4] are based on the condition that the probability of failure pf does not exceed during the service life of a structure T a specified target value pt pf ≤ pt

(5)

It is usually possible to assess the probability of failure pf using a computational structural model, defined through basic quantities X [X1, X2, ... , Xn] for actions, mechanical properties and geometrical data. The limit state of a structure is defined by the limit state function (the performance function) g(X), for which, according to the definition, in case of a favourable (safe) state of the structure the limit state function is positive; it holds that g(X) ≥ 0

(6)

and the unfavourable state (failure) of the structure occurs when the limit state function is negative, i.e. when g(X) < 0 (a more detailed explanation is given in Annex N). Basic quantities are generally time-dependent (stochastic) functions. Even so, in most cases it is sufficient to describe them by time-independent models having, however, their characteristics deduced for extreme (maximal or minimal) values of the appropriate quantity (action or resistance) during the specified design life T. For most ultimate limit states and serviceability limit states the probability of failure can be expressed by the equation pf = P{g(X) < 0}

(7)

More complicated procedures need to be used when some of the quantities are timedependent. Some details concerning theoretical models for time-dependent quantities (mainly actions) and their use for the structural reliability analysis are given in Annex B. However, in many cases the problem may be transformed to a time-independent one, for example by considering in equation (7) a minimum of the function g(X) over the time period T. The assessment of various reliability measures (characteristic values, partial and combination factors) in the new structural design standards [1, 2] is partially based on probabilistic considerations, but to a great extent they are based on historical and empirical experience. Moreover, the choice of these reliability measures is, in Eurocode 1 [1], affected by the intention to simplify the calculation in practical design. This intention, however, leads sometimes to oversimplification and, consequently, to the increase of material consumption. In this connection there is an obvious question how to harmonise the new design codes for various structures on the basis of general principles of the theory of reliability if the current intention to simplify computational procedures suppresses the advantages of the partial factors method. Material consumption is only one criterion of evaluation and need not be the most important one. A broad discussion of experts clearly shows that for a critical analysis of the new codes other criteria should be considered as well. Besides the material consumption they include laboriousness of design and construction, maintenance and repairs, service life, insurance, material recycling, possibility of changing the occupancy and other aspects. A further improvement of current methods will be based on calibration procedures, optimisation methods and risk assessment. These rational approaches use the methods of the theory of probability, mathematical statistics and the theory of reliability. The keyword of all these procedures is the probability of failure pf, and although it has a limited informative ability, it remains the most general, common measure of structural reliability. Methods of the 36

Chapter 2: Basis of design – general principles

theory of reliability then provide the most important tool of gradual improving and harmonisation of design for various structures from different materials. The theory of structural reliability also makes it possible to extend the general methodology for new structures and materials. Annex A to this chapter shows a simple example of a reinforced concrete slab designed in accordance with the above-mentioned techniques. This example will be also used in the following chapters to illustrate application of more advanced probabilistic approaches. Some parts of newly developed European standards for structural design often refer to methods of risk assessment, particularly in case of structures exposed to accidental actions when the consequences of adverse events are significant. Accidental design situations seem to occur more and more frequently and extents of unfavourable events are of increasing importance. Methods of risk assessment may be also used for calibration of the method of partial factor.

3

BASIC CONCEPTS OF EN 1990

3.1

Design working life and design situation

The design working life denotes the period for which a structure or part of it is to be used for its intended purpose with anticipated maintenance, but without a major repair being necessary. Table 1.1 taken from EN 1990 [1] gives categories together with the indicative design working life for a number of common types of construction works. It should be noted that some CEN Member States have adjusted some of the indicative values of design working life in their National Annexes (e.g. 80 years is presently considered in the National Annex of the Czech Republic). Table 1. Indicative design working life. Design Working Life Category

Notional Design Working Life (years)

Examples

1

10

Temporary structures (e.g. scaffolding)

2

10-25

Replaceable structural parts, e.g. gantry girders, bearings (see appropriate standards)

3

15-30

Agricultural and similar structures (e.g. buildings for animals like stables where people do not normally enter)

4

50

Building structures and other common structures (e.g. hospitals, schools etc)

5

100

Monumental building structures, bridges and other civil engineering structures (e.g. churches)

37

Chapter 2: Basis of design – general principles

The present state of knowledge is insufficient to enable precise prediction of the life of a structure. The behaviour of materials and structures over extended periods of time can only be estimated. The likely period of maintenance of the structure or time of replacements of the various components of a structure can, however, be determined. However, the notion of a design working life is useful for: • the selection of design actions (e.g. imposed load, wind, earthquake etc.) and the consideration of material property deterioration (e.g. fatigue, creep) in reliability verification. • comparison of different design solutions and choice of materials, each of which will give a different balance between the initial cost and cost over an agreed period - life cycle costing will need to be undertaken to evaluate the relative economics of the different solutions. • evolving management procedures and strategies for systematic maintenance and renovation of structures. In design the variation of actions, environmental influences and structural properties, which occur throughout the design working life of a structure, should be considered by selecting distinct situations representing a certain time interval with associated hazards. Four design situations are classified in EN 1990 [1] as follows: (a) persistent situations refer to conditions of normal use. These are generally related to the design working life of the structure. Normal use can also include possible extreme loading conditions from wind, snow, imposed loads, etc. (b) transient situations refer to temporary conditions of the structure, in terms of its use or its exposure, e.g. during construction or repair. This implies the use of a time period much shorter than the design working life; one year may be adopted in most cases. (c) accidental situations refer to exceptional conditions of the structure or of its exposure, e.g. due to fire, explosion, impact, local failure. This implies the use of a relatively short period, but not for those situations, where a local failure may remain undetected. (d) seismic situations refer to exceptional conditions applicable to the structure, when subjected to seismic events. These design situations should be selected so as to encompass all conditions which are reasonably foreseeable or occurring during the execution and use of the structure. For example a structure after an accidental design situation due to actions like fire or impact may need a repair (short time period of about one year), for which the transient design situation should be considered. In general, a lower reliability level and lower partial factors than those used for persistent design situations might be applicable for this period of time. However, it should be mentioned that the repair should be designed considering all the other foreseeable design situations. 3.2

Limit states

Traditionally, according to the fundamental concept of limit states it is considered that the states of any structure may be classified as either satisfactory (undamaged, serviceable) or unsatisfactory (failed, unserviceable). Distinct conditions separating satisfactory and unsatisfactory states of a structure are called limit states. Thus, the limit states are those beyond which the structure no longer satisfies the performance criteria. Each limit state is therefore associated with a certain performance requirement imposed on a structure. Often, however, these requirements are not formulated sufficiently clearly so as to allow for precise (sharp) definition of appropriate limit states. Generally, it may be difficult to express the performance criteria qualitatively and to define the limit states unambiguously (particularly the ultimate limit states of structures made of ductile materials, and also the serviceability limit states, typically those affecting user comfort). 38

Chapter 2: Basis of design – general principles

In these cases, only a suitable approximation is available (e.g. the conventional yield point of metals, or a limiting value for vertical deflection). The principles of this are indicated in Figure 1 and provided here as a background to the uncertainties of the limit state concept. According to the traditional (sharp) concept of limit states described above, a given structure is assumed to be fully satisfactory up to a certain value of the load effect E0 and beyond this value the structure is assumed to be fully unsatisfactory (see Figure 1 (a)). However, it may be very difficult to define precisely such a distinct value E0, separating the desired and undesired structural conditions, and the simplification in Figure 1 (a) may not be adequate. In these cases a transition region , in which a structure is gradually losing its ability to perform satisfactorily, provides a more realistic (vague) concept (see Figure 1 (b)). Uncertainties in the vague concept of limit states may be taken into account only in reliability analyses using special mathematical techniques, which are not covered in the present generation of Eurocodes. (a) Sharp Degree of Fully satisfactory

Fully unsatisfactory

Load effect E E0 (b) Vague Degree of Fully satisfactory

Transition region

Fully unsatisfactory

Load effect E E1

E2

Figure 1. Sharp and vague definition of a limit state. In order to simplify the design procedure two fundamentally different types of limit states are generally recognised: (a) ultimate limit states, (b) serviceability limit states. Ultimate limit states are associated with collapse or other similar forms of structural failure. Serviceability limit states correspond to conditions of normal use (deflections, vibration, cracks, etc.). In general, the design should include both safety and serviceability, including durability in both cases. The nature of ultimate limit states is essentially different from the nature of serviceability limit states and this should be taken into account in reliability verification. There are two main reasons of this distinction: (a) While the infringement of ultimate limit states leads almost always to the overall loss of structural integrity and to the removal or fundamental repair of the structure, the infringement 39

Chapter 2: Basis of design – general principles

of serviceability limit states does not usually lead to such fatal consequences for the structure, and the structure may normally be used after the removal of those actions, which caused the infringement. (b) While the criteria of ultimate limit states involve parameters of the structure and appropriate actions only, the criteria of serviceability limit states are also dependent on the requirements of the client and users (sometimes very subjective), and on the characteristics of the installed equipment or non-structural elements. The differences between the ultimate limit states and serviceability limit states result in a separate formulation of reliability conditions, and in dissimilar reliability levels assumed in the verification of both types of limit states. However, verification of one of the two limit states may be omitted if sufficient information is available to assure that the requirements of one limit state are met by the other limit state. For example, in case of reinforced concrete beams designed for ultimately limit state it is allowed to omit the verification of deflection provided that the span/effective depth ratio is less than 18 for highly stressed concrete or less than 25 in case of lightly stressed concrete. Variation of actions, environmental influences and structural properties, which occur throughout the life of the structure, should be considered in design by selecting distinct situations (persistent, transient, accidental and seismic) representing a certain time interval with associated hazards. The ultimate and serviceability limit states should be considered in all these design situations, which should be selected so as to encompass all conditions that are reasonably foreseeable or occurring during the execution and use of the structure. Within each load case, a number of realistic arrangements should be assumed to establish the envelope of action effects that should be considered in the design. If the limit states considered in design are dependent on time variant effects (described by action and/or resistance variables), the reliability verification of a structure should be related to the design working life. It should be mentioned that most time dependent effects (e.g. fatigue) have a cumulative character that should be taken into account. 3.3

Ultimate limit states

The ultimate limit states are associated with collapse and other similar forms of structural failure and directly concern the safety of the structure and the safety of people. However, in some cases the ultimate limit states may concern also the protection of the contents, for example of some chemicals or nuclear or other waste materials. In almost all cases which concern the ultimate limit states the first passage of the limit state is equivalent to failure. In some cases, e.g. when excessive deformations are decisive, states prior to structural collapse can, for simplicity, be considered in place of the collapse itself and treated as ultimate limit states. These important circumstances should be taken into account when specifying the reliability parameters of structural design and quality assurance. For example, in case of foundations of rotating machinery used in power plants excessive deformation is decisive and govern entirely the design. The following list provides the most typical ultimate limit states that may require consideration in the design: (a) loss of equilibrium of the structure or any part of it, considered as a rigid body; (b) failure of the structure or part of it due to rupture, fatigue or excessive deformation; (c) instability of the structure or one of its parts; (d) transformation of the structure or part of it into a mechanism; (e) sudden change of the structural system to a new system (e.g. snap through). Time dependent structural properties, such as fatigue and other time dependent deterioration mechanisms reduce the strength of a structure and can initiate one of the above mentioned ultimate limit states. In this respect it is useful to distinguish two types of structures: 40

Chapter 2: Basis of design – general principles

damage tolerant (i.e. robust) and damage intolerant (sensitive to minor disturbance or construction imperfections). Effects of various deteriorating mechanisms on the ultimate limit states should then be taken into account according to the type of the structure. Adequate reliability level of damage intolerant structures can also be assured by an appropriate quality control programme. In the case of damage tolerant structures, fatigue damage may be regarded as a serviceability limit state. Note that different sets of partial factors may be associated with various ultimate limit states. (a) Irreversible limit state Deformation

First passage Limiting value Failure domain

Time (b) Reversible limit state Deformation

First passage Limiting value

Failure domain Time

Figure 2. Irreversible and reversible limit states. 3.4

Serviceability limit states

The serviceability limit states are associated with conditions of normal use. In particular, they concern the functioning of the structure or structural members, comfort of people and appearance of the construction works. Taking into account the time dependency of load effects it is useful to distinguish two types of serviceability limit states, which are illustrated in Figure 2. (a) Irreversible serviceability limit states (see Figure 2(a)), which are those limit states that remain permanently exceeded even when the actions that caused the infringement are removed (e.g. a permanent local damage, permanent unacceptable deformations); the failure domain is the total time following the first passage of the limiting value. (b) Reversible serviceability limit states (see Figure 2(b)), which are those limit states that will not be exceeded when the actions which caused the infringement are removed (e.g. cracks of prestressed components, temporary deflections, excessive vibration); the failure domain consists of all parts where the response is above the limiting value. For irreversible limit states the design criteria are similar to those of ultimate limit states, but with reduced reliability. The first passage of the limit state is decisive (see Figure 2). This 41

Chapter 2: Basis of design – general principles

important aspect of irreversible limit states should be taken into account when determining the serviceability requirements in the contract or design documentation. For reversible limit states the first infringement (first passage) does not necessarily lead to the loss of serviceability. Various serviceability requirements can be formulated taking into account the acceptance of infringements, their frequency and their duration. Generally, three types of serviceability limit states are applicable as follows: (a) no infringement is accepted, (b) specified duration and frequency of infringements are accepted, (c) specified long term infringement is accepted. The correct serviceability criteria are then associated as appropriate with the characteristic, frequent and quasi-permanent values of variable actions. The following combinations of actions corresponding to the above three types of limit states are generally used in verification of serviceability limit states for different design situations: (a) the rare (characteristic) combination if no infringement is accepted, (b) the frequent combination if the specified time period and frequency of infringements are accepted, (c) the quasi permanent combination if the specified long term infringement is accepted. The list of serviceability limit states affecting the appearance or effective use of the structure, which may require consideration in the design, may be summarized as follows: (a)excessive deformation, displacement, sag and inclination, which can affect for example the appearance of the structure, comfort of users, functioning of the structure and can cause damages of finishes and non-structural members; (b) excessive vibration (acceleration, amplitude, frequency), which can for example cause discomfort to people and limit the functioning of the structure; (c) damage that is likely to adversely affect the appearance (local damage and cracking), durability or functioning of the structure. Depending on the type of structure additional requirements related to serviceability limit states may be found in material oriented codes. For example, in case of concrete structures the ultimate limit states may be induced by structural deformation.

4

VERIFICATION OF LIMIT STATES

4.1

Verification of static equilibrium and strength

Four types of ultimate limit states are distinguished in EN 1990 [1], denoted by symbols EQU, STR, GEO and FAT: - EQU covers loss of static equilibrium of a structure as a rigid body, in which: - minor variations in the value of the spatial distribution of action from a single source are significant, and - the strength of construction materials or ground are generally not governing; - STR covers internal failure or excessive deformation of the structure or structural members, in which the strength of construction materials of the structure governs; - GEO covers failure or excessive deformation of the ground, in which the strengths of soil and rock are significant in providing resistance; - FAT covers fatigue failure of the structure or structural elements. The limit states GEO and FAT are not considered in the three examples introduced in this Chapter. In the case of ultimate limit state of the type EQU (static equilibrium) the design values Ed and Rd may be symbolically written as

42

Chapter 2: Basis of design – general principles

Ed = Ed,dst, Rd = Ed,stb

(8)

where Ed,dst denotes the design value of the destabilising actions, Ed,stb denotes the design value of the stabilising actions. The verification of ultimate limit states of type EQU is possible to be expressed as (EN 1990, expression (6.7)) Ed,dst < Ed,stb

(9)

The partial factors for permanent and variable actions are given in EN 1990 [1], Annex A1, Table A1.2(A). This table indicates a recommended set of partial factors (for favourable and unfavourable permanent actions, leading and accompanying actions) considering the load combination rule given in EN 1990 [1] by expression (6.10) only (see the combination rule A in the following Section 3). It should be noted that the other combination rules given in EN 1990 [1] by expressions (6.10a) and (6.10b) are not allowed for the verification of static equilibrium. In addition to the γ values recommended in Note 1 of Table A1.2(A) in EN 1990 [1], an alternative set of γ values described in Note 2 may be allowed by the National Annex. The Note 2 refers to the case, when the verification of static equilibrium also involves the resistance of structural members. Application of both these sets of γ values is illustrated in the example 1 described in Chapter 6. In the case of the limit state of the type STR (internal failure) the design values Ed and Rd may be written as Ed = γEd E(Fd, Xd, ad), Rd = R(Fd, Xd, ad)/γRd,

(10)

where γEd denotes a partial factor accounting for uncertainties in the action effect model E, γRd denotes a partial factor accounting for uncertainties in the resistance model R, Fd denotes design values of actions F, Xd design values of material properties X, and ad design values of geometric data a (often equal to the nominal values). Note, that the load effect E generally depends also on material properties Xd including strength and stiffness (for example in the cases of indirect actions due to imposed deformations). 4.2

Verification of serviceability limit state

The verification of serviceability limit states, which presently becomes more and more important, is based in common cases (e.g. in evaluating of deflection or crack width) on inequality (EN 1990 [1], expression (6.13)) Cd ≥ Ed

(11)

where Cd is the serviceability constraint, for example admissible deflection, crack width, local stress or acceleration. The serviceability constraints Cd are often specified on the basis of empirical observations and past experience.

5

Concluding remarks

Partial factor method is the most effective operational technique accepted for the design of common construction works in a number of national codes and international standards. Probabilistic design and methods of risk assessment provide more sophisticated approaches, which are primarily applied to calibrate the partial factor method and, in some cases, to design specific structures as mass produced components and structures with particular reliability requirements.

43

Chapter 2: Basis of design – general principles

REFERENCES [1] EN 1990 Eurocode - Basis of structural design. European Committee for Standardisation, 04/2002. [2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. European Committee for Standardisation, 04/2002. [3] ISO 2394, General principles on reliability for structures. 1998. [4] Probabilistic Model Code, Parts 1 to 4, Basis of design, Load and resistance models, Examples, JCSS, 2001-2002. [5] Gulvanessian, H., Calgaro, J.-A., Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, 192 pp. [6] Holický M., Marková J: Reliability of Concrete Elements Designed for Alternative Load Combinations Provided in Eurocodes, Prague, Acta Polytechnica 2003/1.

44

Chapter 2: Basis of design – general principles

Appendix to Chapter 2: A reinforced concrete slab – various design concepts

A.1

Introduction

Various design methods used in history may be well illustrated considering a simple reinforced concrete slab in an office building. Without going into technical details the example also indicates the advantages of the reliability based partial factor method and the significance of the reliability theory in structural design compared to other design methods.

A.2

A reinforced concrete slab

A simply supported slab having the span of 6 m is exposed to a permanent load (the self-weight of the slab and other fixed parts of the building), which is estimated by the characteristic value (equal to the mean value) gk=7 kN/m2. In accordance with the EN 1990 [1] the imposed load in an office area may be considered by the characteristic value qk=3 kN/m2. It is, however, well known that the mean value of this imposed load is much lower, about 1,8 kN/m2. Further, the concrete C20/25 having the characteristic strength fck=20 MPa (the mean 30 MPa) and reinforcement bars having the characteristic strength fyk=500 MPa (the mean 560 MPa) are to be used. Using previous experience the height of the slab 0,2 m has been estimated in advance. Given the above data concerning the preliminary specifications an estimate of the necessary reinforcement of the slab should be done.

A.3

Design and reliability consideration

To simplify computational procedures it is assumed that the same assumption concerning stress distribution in the slab, shown in Figure A.1, can be accepted for all design methods. Note that the design methods based on permissible stresses assume the linear stress distribution in the compression zone of the slab, but the accepted assumption of the rectangular stress distribution (indicated in Figure A.1) is an acceptable approximation for illustration of the main features of the design methods. The basic variables are defined as follows: d denotes the effective depth, x the depth of the compression zone, b the width of the slab (considered as 1 m), As the area of the reinforcement, fc the concrete strength and fy the reinforcement strength (yield point). fc bxfc

0,8x x d

As

Asfy

Figure A.1. Slab cross-section.

45

Chapter 2: Basis of design – general principles

The following equilibrium conditions follow from Figure A.1 fc x b = As fy

(A.1)

As fy(d – x/2) = M

(A.2)

Here M = (g+q)L2/8 denotes the bending moment due to the permanent and imposed loads g and q. Using conditions (A.1) and (A.2) the following formula for the area As may be derived As =

f cbd ⎛⎜ 2M 2 1 1 f y ⎜⎝ f cbd 2

⎞ ⎟ ⎟ ⎠

(A.3)

Up to now all the variables have been considered as deterministic quantities without taking into account uncertainties that may potentially affect their actual values. Obviously, the basic variables entering equation (A.3) may have a considerable scatter that should be taken into account. A first estimate of the area As might be obtained considering the mean (average) values of all the basic variables involved in Figure A.1 and equation (A.3). This absurd attempt will provide a very small reinforcement area As ~ 0,000379 m2. The outcomes of various “code methods” (the permissible stresses, the global factor and the partial factor method) are summarised in Table A.1. Note that equation (A.3) may be used generally for any design method indicated in Table A.1.

Table A.1. A reinforced concrete slab specified by different design methods for the span L = 6 m, the height h = 0,2 m (effective depth d = 0,17 m) and the characteristic loads gk= 7 kN/m2, qk= 3 kN/m2, C25/20 (fck = 20MPa), fyk=500 MPa. Design method ME [kNm] The permissible stresses (CP114) 45,0 45,0 The global safety factor (γ=1,9) The partial factors method (CEN) 62,8

As [m2] μMR [kNm] 0,001161 103,6 0,001094 97,7 0,000933 84,1

β 6,2 5,9 4,9

Pf 2,8×10-10 1,8×10-9 4,8×10-7

Without going into technical details (that are offered later in the guidebook) Table A.2 indicates the mean values and the “design values” of loads and material strengths used according to the methods included in Table A.1 when determining the reinforcement area As. Table A.2. The mean values and the design values of the loads and material strengths. Basic variable

The mean values

Permissible stresses

Global Partial safety factor factors

Permanent load g [kN/m2] Imposed load q [kN/m2] Concrete strength fc [MPa] Reinforcement strength fy [MPa]

7 1,8 30 560

7 3 5,5 275

7 3 20 500

9,45 4,5 13,3 435

Table A.3 shows the theoretical models of basic variables used in probabilistic analysis. Note that the symbol N means a normal distribution, LN lognormal distribution, GUM Gumbel distribution and DET the deterministic quantity (having zero standard deviation).

46

Chapter 2: Basis of design – general principles

Table A.3. The theoretical models of basic variables. Basic variable

Characteristic values Permanent load g [kN/m2] 7 Imposed load q [kN/m2] 3 Concrete strength fc [MPa] 20 Reinforcement strength fy [MPa] 500 Effective depth of the slab d [m] 0,2 Width of the slab b [m] 1 Span of the beam L [m] 6 2 Reinforcement area As [m ] specified

Distributions N GUM LN LN N DET DET DET

The mean values 7 1,8 30 560 0,20 1,0 6 specified

Standard deviations 0,7 0,63 5 30 0,01 0 0 0

It follows from table A.1 that the “code methods” included in TableA.1 lead to different reinforcement areas As and, consequently to different failure probabilities Pf. The most conservative (and perhaps less economical) seems to be the permissible stresses method, less conservative (and perhaps more economical) is the global safety method and partial factor method.

A.4

Concluding remarks

It follows from Table A.1 that the permissible stresses method (e.g. CP 114) seems to lead to rather conservative (and perhaps uneconomic) results similarly as the method of the global safety factor. The partial safety factors method, accepted in the EN documents, appears to provide a less conservative and most likely more economic design format. The reliability index β = 4,9 obtained by this method is reasonably close to the target value β = 3,8. However, the most important advantage of the partial safety factors method is the possibility to take into account uncertainties of individual basic variables by adjusting (calibrating) relevant partial factors and other reliability elements. It is the aim of this guidebook to explain the fundamental principles of this method and to show how appropriate design values of basic variables could be specified using the theory of structural reliability in order to achieve an adequate reliability level.

47

Chapter 3: Reliability differentiation

CHAPTER 3: RELIABILITY DIFFERENTIATION Dimitris Diamantidis1 1

University of Applied Sciences, Regensburg, Germany

Summary Reliability is a major issue in structural engineering. Reliability is a property of the structure that can be achieved or assured and is quantified by the probability of failure or the associated reliability index as illustrated in Chapter 2. A more complete parameter to approach reliability is risk, which is a combination of the probability of failure and the consequences of failure. In general it is important to distinguish between various types of consequences, i.e. human losses, environmental damage and economic losses. Structural codes traditionally have been concerned foremost with public safety preventing loss of life or injury. Structural codes such as the Eurocodes aim at delivering structures with appropriate degree of reliability and in an economic way. Procedures and current codified criteria for reliability acceptance inherent in standards and especially in the Eurocodes are reviewed in this Chapter. The dependence between target reliability design lifetime and partial safety factors is illustrated in example cases. General approaches for selecting target and acceptable reliability levels, i.e. the human safety approach, the calibration and the optimization (costbenefit) are discussed also in the Appendix A to this Chapter.

1

INTRODUCTION

1.1

Background documents

European codes [1], [2], national [3, 4] and international documents [5, 6] provide general principles and guidance for the application of probabilistic methods to structural designs. The latest European document [2] and international standards [5] and [6] also indicate a theoretical basis of the so called “partial factor method” and procedures for determination of partial factors of material properties and actions using probabilistic principles. 1.2

General principles

Criteria for reliability acceptance of structures are based on present guidelines and modern methodologies. Therefore the acceptability limits (or targets) for probabilities of failure should take into account, implicitly or explicitly, for probabilities of failure, potential losses, amount of investments necessary to improve reliability, and possibly a combination of all these factors. Such targets have been developed for various industrial activities including new civil engineering structures. The present approaches to set acceptability limits for structural design include (see also Appendix A): • •

48

Derivation of acceptable failure probabilities from observed fatality rates and from reported economic consequences. Calibration to present practice assuming that the reliability implied by present practice is acceptable.

Chapter 3: Reliability differentiation

•

Optimization of generalized benefits and cost including expected failure cost (e.g. structural, loss of function, environmental damage, and human losses).

In the following the principles of reliability differentiation specified in current international documents [5,6,7,8] and related procedures for determining reliability measures to be applied in verification cases considering various design-working lives are discussed. Appropriate reliability elements (characteristic values and partial factors) are derived for design variables taking into account time dependence of failure probability and the related reliability index. 2

BASIC RELIABILITY ELEMENTS

The basic reliability measures include the probability of failure and reliability index as introduced in Chapter 2 and Annex B in this Handbook. The probability of structural failure Pf can be generally defined as pf = P{Z(X) < 0} (1) The limit state (performance) function Z(X) is formulated in such a way that the reliable (safe) domain of a vector of basic variables X = X1, X2, ... , Xn corresponds to the inequality Z(X) > 0, while the failure domain to the complementary inequality Z(X) < 0. A simple example of Z(X) describes the basic relationship between the resulting load effect E and resistance R Z(X) = Z = R − E (2) The random variable Z in equation (2) is often called the reliability margin; its mean

μZ, standard deviation σZ and skewness ωZ may be derived from corresponding characteristics of resulting variables R and E as indicated in Chapter II. Instead of the failure probability pf, the reliability index β is frequently used in reliability consideration as an equivalent quantity to pf. The reliability index β is related to the failure probability pf as already indicated in Chapter 2 pf = Φ(−β) ≈10−β

(3)

In this equation, Φ( ) denotes the distribution function of standardised normal distribution. Note that, if the safety margin Z has normal distribution, then the reliability index may be determined simply as the ratio of μZ and σZ, thus β = μZ /σZ (in this case β denotes the distance of the mean μZ from the origin taking the standard deviation σZ as a unit). Chapter 2 and Annex B show the numerical relationship of both quantities. It should be emphasized that the failure probability Pf and the reliability index β represent fully the equivalent reliability measures with one to one mutual correspondence given by equation (3). In the European document [2] a design working life for common structures is considered as Td = 50 years, the reliability index for ultimate limit states βd = 3,8 corresponds to the design failure probability Pd = 7,2 × 10-5, for serviceability limit states βd = 1,5 and pd = 6,7 × 10-2 (a more appropriate term is the “target probabilities” used in ISO documents [5] and [8]). These quantities are recommended as reasonable minimum requirements and it is emphasized that Pd and βd are formal conventional quantities only and may not correspond to actual frequency of failures. In design analysis of a structure it is generally required that pf ≤ pd

(4)

49

Chapter 3: Reliability differentiation

or equivalently in terms of reliability index

β ≥ βd

(5)

where pd denotes specified design (target) failure probability corresponding to the target reliability index βd. Conditions (4) or (5) have to be used by designers when probabilistic methods are applied for verification of structural reliability. Indicative target values pd and βd are declared in some national standards (e.g. [2] and [3]) and recently also specified in international documents (e.g. [4] to [7]) for various design conditions (limit states, failure consequences and economic aspects).

3

TARGET RELIABILITY IN THE EUROCODES

3.1

General

Risk acceptance criteria are introduced in the Eurocodes in terms of target and acceptable (i.e. design) failure probabilities and associated reliability indices. They are used in order to obtain safety factors for design purposes. The values have been derived through long studies by combining the various approaches reviewed in the previous paragraph. The values reflect the possible failure consequences, the reference time period (usually 1-year is used) and are valid for component failures. Special attention must be given to global failure conditions and to target reliability criteria for existing structures. 3.2

Reliability classes

Design failure probabilities pd are usually indicated in relation to the expected social and economical consequences in order to reflect the aforementioned risk acceptance criteria. Table 1 shows classification of target reliability levels provided in EN 1990 [1]. Reliability indexes β are given for two reference periods T (1 year and 50 years) only, without any explicit link to the design working life Td. The values are based on calibration and optimization and reflect results from several studies. The 1-year values are recommended for time independent (time invariant) limit state conditions, for example for component failures governed by dead load and permanent live load. It is noted that similar β-values as in Table 1 are given in other national and international guidelines (see for example [3], [4], [5]). It should be underlined that a couple of β values (βa and βd) specified in Table 1 for each reliability class (for 1 year and 50 years) correspond to the same reliability level. Practical application of these values depends on the time period Ta considered in the verification, which may be connected with available information concerning time variant vector of basic variables X = X1, X2, ..., Xn. For example, if the reliability class 2 and 50 years design working period is considered, then the reliability index βd = 3,8 should be used in the verification of structural reliability together with the statistical parameters for the time variant dominating action effects over the same period of 50 years. The same reliability level corresponding to class 2 is achieved when the time period Ta = 1 year and βa = 4,7 are used in the verification. It is recommended here to use the 50-years target values of Table 1 together with the associated statistical parameters of the variable action effects over the same time period. By applying such a verification procedure the compatibility in results can be reached.

50

Chapter 3: Reliability differentiation

Table 1. Reliability classification in accordance with EN [1] Reliability classes

RC3 – high RC2 – normal

Consequences for loss of human life, economical, social and environmental consequences High Medium

RC1 – low

Low

Reliability index β βa for Ta= βd for Td= 1 year 50 years 5,2 4,7

4,3 3,8

4,2

3,3

Examples of buildings and civil engineering works

Bridges, public buildings Residential and office buildings Agricultural buildings, greenhouses

Similar target βd values are provided in ISO 2394 [5] for the design working life Td (called in ISO “life time”) without specification of any particular value of Td. As indicated in Table 2, two factors are considered for reliability differentiation in [5]: relative costs of safety measures and consequences of failure. Table 2. Target reliability index βd for the design working life Td given in ISO 2394 [5] Relative costs of safety measures High Moderate Low

Consequences of failure some moderate 1,5 2,3 2,3 3,1 3,1 3,8

small 0 1,3 2,3

great 3,1 3,8 4,3

It appears that available documents do not provide an explicit guidance on how to take into account the design working life Td. Both international documents EN [1] and ISO [5] give the target value βd for specific reference periods T, however, no explicit rule is offered for adjustment of target value βd to different working design lives Td recommended for various types of construction works. Nevertheless, some indication is provided in another ISO document [8] for assessment of existing structures where it is recommended that reliability levels for any residual lifetime could be similar to those considered for the design working life Td in the case of a new structure. Consequently, similar reliability levels (expressed in terms of probability pd or reliability index βd) may be considered when designing structures for different design working lives Td, for example for Td = 50 and Td = 25 years. 3.3

Variation with time - Discussion

When the vector of basic variables X = X1, X2, ... , Xm is time variant for example in case of wind or earthquake dominated limit states, then the failure probability pf is also time variant and should always be related to a certain reference period T, which may be generally different from the design working life Td. Considering a structure of a given reliability level, the design failure probability pd = pn related to a reference period Tn = n T1 can be derived from the alternative probability pa = p1 corresponding to Ta = T1 (to simplify notation note that previously used subscript "d" corresponds now to "n" and subscript "a" to "1") using approximate relationship given in [5], [7] Pn = 1 − (1 − P1)n

(6)

51

Chapter 3: Reliability differentiation

For very small probabilities, this relationship could be further simplified as pn = p1 Tn / T1. Time periods T1 and Tn may have an arbitrary length and n = Tn / T1 may not be an integer; T1 is, however, often one year. Probability pn increases (almost linearly) with Tn. It follows from equation (6) that reliability indexes β1 = βa and βn = βd, given in accordance to equation (3) as p1 = Φ(−β1) and pn = Φ(−βn) are related as follows [5] Φ(βn) = [Φ(β1)]n

(7)

Here Φ(.) denotes the distribution function of standardised normal distribution. Figure 1 shows variation of βn with β1 for n = 5, 25, 50 and 100. Note that, if the reference period T1 is one year, then n indicates the number of years of the reference period Tn (n = Tn). Figure 1 confirms data indicated in Table 1. For example, if the target reliability level of a structure is specified by β50 = 3,8 for the design working life Td = Tn = 50 years, then it could be verified using reference period Ta = T1 =1 year and βa = β1 =4,7. When, however, the same reliability index 3,8 is specified for a structure having a design working life Tn = 25 years only, thus β25 = 3,8, then the reliability of this structure could be verified using an alternative reference period T1 = 1 year and reliability index β1 = 4,5, similarly when β5 = 3,8 then β1 = 4,2 (see Figure 1).

βn

5 4,5 3,8 n = 5 25 4• 50 3,5 100 3 2,5 2 1,5 1 0,5 0 2,5 3 3,5

•

4,7 4

4,5

5

5,5

β1 Figure 1: Variation of βn with β1 for n = 5, 25, 50 and 100 Note that, if 1-year period is used for specification of the target reliability level of a structure, then Figure 1 provides information on the resulting failure probability corresponding to a given working life Tn. For example, if the target reliability level is specified by the reliability index β1 = 4,7 (corresponding to the probability p1 = 1,3 × 10-6), then (as already mentioned) the reliability level of a structure having a working life, Tn = 50 years is characterised by β50 = 3,8. Similarly when Tn = 5 year period is used then β5 = 4,3 or when Tn = 100 years then β100 = 3,6. So, the reliability level of a structure can be specified using different time periods T, which may not necessarily coincide with the design working life Td. This is the case for time variant limit state conditions (dominated by wind or earthquake loads). As mentioned before, compatibility between: •

52

Selected time period for the target reliability level,

Chapter 3: Reliability differentiation

•

Selected time period for the derivation of the statistical parameters of the time varying action effects

is necessary. In general it is recommended to use the 50 year values representing the usual desired working lifetime of buildings. 3.4

Global failure – robustness

Structures are composed of various elements such as columns, beams, plates etc. The aforementioned target reliability values are valid for components, since structural design is based on design of components. The global reliability i.e. the reliability against collapse of the entire system or a major part of it is a function of the reliability of all the elements against local failure but also of the system response to local failure. The assumption that a consistent level of reliability of a structural system is reached by an adequate reliability of its members is not generally valid. Especially for structures subjected to extreme environmental loads such as earthquake or waves or accidental loads such as explosion or impact is not sufficient (s. [9]). Therefore structural codes for specific structures have additional requirements regarding global failure or progressive failure of the structure. The associated requirements in the Eurocodes are discussed in Chapter 5. 3.5

Existing Structures

Due to the social and economic need of utilizing existing structures, their reliability assessment is of major concern. In principle reliability acceptance of existing structures should be based on present guidelines, standards and methodologies. The mere fact that the structure fulfils the code of its time of construction cannot be decisive. Codes have changed over time due, for example, to technology development and experience gained with the performance of structures when struck by past events. This does not mean, however, that if a new code with more severe requirements than old ones comes into practice, old buildings should necessarily be deemed unsafe. In summary acceptable limits for existing structures are not necessarily the same as those for the new ones, because influencing factors such as • • • • • •

lifetime of the structure, degree of available technical knowledge (codes, methods, etc.), relative effort (costs) to control reliability, time constraints (for engineering and eventual repair), consequences of potential failures, socio-economical and political preference

at the time of the decision - and thus the degree of available information on the above items may have changed over time. Therefore, a “discount” in the reliability requirements for existing structures may be simply unavoidable due to economical and legal constraints. The discount of reliability acceptance criteria for existing structures has been discussed in [10] and [11] and associated target values lower than those for new structures have been proposed. In the Eurocodes no specific targets for existing structures are specified but based on the aforementioned ideas a discount of the target reliability appears reasonable due to the reduced failure consequences compared to a new structure.

53

Chapter 3: Reliability differentiation

4

PARTIAL SAFETY FACTORS

4.1

Derivation based on reliability methods

The influence of the target reliability index into practical design is highlighted next. It is reflected in the partial safety factors which are a function of the reliability level given by the reliability index. The one year reference period is usually used for the derivation of safety factors. Consider for example a resistance variable R (strength) having lognormal distribution. It is assumed that the characteristic value Rk of R is defined as its 5% fractile [5], [6] a [7]. Then of the resistance variable R, the characteristic value Rk and design value Rd are defined as [4], [5] Rk = μR × exp (− 1,645 × VR)

(8)

Rd = μR × exp (−αR × βd × VR)

(9)

Taking into account equations (8) and (9) it follows that the partial factor is given as

γR = Rk / Rd = exp (− 1,645 × VR) / exp (−αR × βd × VR)

(10)

2,5

γR 2

VR = 0,20 0,15 0,10 0,05

1,5

1

0,5 0

1

2 βd

3

4

5

Figure 2: Variation of γR with βd Considering selected values of the coefficient of variation VR, Figure 2 shows the partial factor γR for lognormal distribution of R (equation (10)) as a function of the reliability index βd. It follows from Figure 2 that when reliability of a structure is verified using βd = 4,7, the partial factor γR should generally be greater than in the case when the design reliability is lower for example βd = 3,8. Similar conclusions can be expected for partial factors of other basic variables, in particular for partial factors of permanent actions. Consider as another example a self-weight G having normal distribution. The characteristic value Gk of G is defined as the mean μG [5], [6] and [7]: Gk = μG 54

(11)

Chapter 3: Reliability differentiation

The design value Gd is given as [4], [5] Gd = μG − αG × β × σG = μG + 0,7× βd × σG = μG(1 + 0,7× βd× VG)

(12)

In equation (11) and (12) μG denotes the mean, σG the standard deviation, VG the coefficient of variation and αG = − 0,7 the sensitivity factor of G. The partial factor γG of G is defined as [5], [6] a [7]

γG = Gd / Gk

(13)

Taking into account equations (11) and (12) it follows from (13) that

γG = (1 + 0,7× βd × VG)

(14)

Figure 3 shows variation of the partial factor γG with the reliability index βd for selected values of the coefficient of variation VG = 0,05; 0,10; 0,15 and 0,20. Note that γG = 1,35 (recommended in EN 1990 [1]) corresponds approximately to the reliability index βd = 3,8 if the coefficient of variation is about 12% (the value in EN 1990 [1] was increased by 5% to take into account model uncertainty). 2,5

γG 2

VG = 0,20 0,15 0,10 0,05

1,5

1

0,5 0

1

2 βd

3

4

5

Figure 3: Variation of γG with βd and coefficient of variation VG (G normal). Assuming the coefficient of variation 0,1 for both the resistance R and the self weight G Figures 5 and 6 indicate that the partial factor of self-weight γG varies slightly more significantly with βd - values than with the partial factor γR of resistance variable R. This finding is, however, dependent on the distributions assumed for both variables. 4.2

Simplified reliability differentiation (Annex B of the Eurocodes)

Simplified procedure to account for the reliability classes is proposed in Annex B of the Eurocodes document [1]. The following possible approaches are proposed: a) One way of achieving reliability differentiation is by distinguishing classes of γF factors to be used in fundamental combinations for persistent design situations. For example,

55

Chapter 3: Reliability differentiation

for the same design supervision and execution inspection levels, a multiplication factor KFI, see Table 3, may be applied to the partial factors. Table 3 - KFI factor for actions according to [1] Annex B KFI factor for actions KFI

Reliability class RC1 RC2 RC3 0,9 1,0 1,1

Note: In particular, for class RC3, other measures as described in this Annex are normally preferred to using KFI factors. KFI should be applied only to unfavourable actions. b) Reliability differentiation may also be applied through the partial factors on resistance γM. However, this is not normally used. An exception is in relation to fatigue verification. Accompanying measures, for example the level of quality control for the design and execution of the structure, may be associated to the classes of γF. In the Annex B of [1], a three level system for control during design and execution has been adopted. Design supervision levels and inspection levels associated with the reliability classes are suggested.

5

EXAMPLES

5.1

Residential steel building

Consider a steel structure having the design working life Td = 50 years, for which the target failure probability is specified as pd = 7,2 × 10− 5 (βd = 3,8) associated to a reliability class RC2 in Table 1. Using equation (10) the partial safety factor γR assuming the coefficient of variation VR = 0,08 (corresponding to the common variability of the resistance of structural steel member) is given as (see also Figure 2)

γR = exp (− 1,645 × 0,08) / exp (− 0,8 × 3,8 × 0,08) = 1,17 The safety factor γR is about 1,15 and corresponds to the factor used in the Eurocode steel design. 6.2

Agricultural steel building

A different task is reliability verification of an agricultural structure, for which the target reliability index can be decreased to βd = 3,3 associated to a reliability class RC1 (see Table 1). It follows from equation (10) that the partial factor γR is

γR = exp (− 1,645 × 0,08) / exp (− 0,8 × 3.3 × 0,08) = 1,08 6.3

Agricultural concrete building

Consider now that the agricultural structure is a concrete structure. The variability of the reliability of the concrete member is significantly higher than the one for steel member and can be reflected through a coefficient of variation VR = 0,20. It follows from equation (10) that the partial factor γR is

γR = exp (− 1,645 × 0,20) / exp (− 0,8 × 3,3 × 0,20) = 1,22 56

Chapter 3: Reliability differentiation

The partial factor γR is increased from about 1,1 to about 1,4 due to the higher variability of concrete strength compared to the variability of the steel strength.

7

CONCLUDING REMARKS

Risk acceptance criteria inherent in the Eurocodes have been presented in this Chapter. The following conclusions can be drawn: (1) A reliability class differentiation scheme is proposed in the Eurocodes with basically three different reliability classes. A measure of reliability is the probability of failure or the associated reliability index, which depend on the reliability class (type of structure, consequences of failure) and on the lifetime of the structure. The reliability provisions are based on modern risk analysis and risk appraisal criteria. (2) Reliability in design (limit state design) is reflected through the design values of the design parameters and consequently through the partial safety factors and characteristic values for load and resistance parameters. (4) A simplified procedure for reliability differentiation through partial safety factors is provided (see Table 3) (5) The reliability classes are reflected according to Annex B of [1] also in the design supervision levels and the inspection levels during execution. (6) Limit state design is based on the consideration of local and not global failure, since design equations are usually defined and applied on a local level only. The global reliability i.e. the reliability against collapse of the entire system is discussed in Chapter 5.

REFERENCES [1] EN 1990 Eurocode - Basis of structural design. European Committee for Standardisation, 04/2002. [2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. European Committee for Standardisation, 04/2002. [3] NaBau, Grundlagen zur Festlegung von Sicherheitsanforderungen für bauliche Anlagen, DIN Deutsches Institut für Normung, Berlin, Köln, Beuth Verlag, 1981. [4] NKB The Nordic Committee on Building Regulations, Recommendations for Loading and Safety Regulations for Structural Design, NKB-Report No. 36, Copenhagen, Denmark, 1984. [5] ISO 2394, General principles on reliability for structures. 1998. [6] Probabilistic Model Code, Parts 1 to 4, Basis of design, Load and resistance models, Examples, JCSS, 2001-2002. [7] Gulvanessian, H., Calgaro, J.-A., Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, 192 pp. [8] ISO 13822, Basis for design of structures - Assessment of existing structures, ISO 2001. [9] Starossek, U. “Progressive Collapse of Structures: Nomenclature and Procedures”, Structural Engineering International Vol. 2, 2006. [10] Diamantidis, D., P. Bazzurro, Target Safety Criteria for Existing Structures, Workshop on Risk Acceptance and Risk Communication, Stanford University, CA, USA, March 2007. [11] Joint Committee on Structural Safety (JCSS), Assessment of Existing Structures, RILEM Publications S.A.R.L., 2000. 57

Chapter 3: Reliability differentiation

Appendix to Chapter 3: Risk Acceptance Approaches in Codes

A.1

General

The concept of risk acceptance criteria is well established in many industrial sectors. Comparative risk thresholds are implemented which allow a responsible organisation (or regulator) to identify activities which impose an unacceptable level of risk on the participating individuals or society as a whole. Risk acceptance can be defined by two different methods: implicitly or explicitly. Implicit criteria often involve reliability equivalence with other industrial sectors (e.g. stating that a certain activity must impose risk levels at most equivalent to those imposed by another similar activity). In the past, this approach was very common because some industrial sectors (for example nuclear and offshore) developed quantitative risk criteria well before others, and thus also constituted a basis for comparison. While this methodology has been surpassed by more refined techniques, it is still used occasionally today. Explicit criteria are now applied in many industrial sectors, as they tend to provide either a quantitative decision tool to the regulator or a comparable requirement for the industry when dealing with the certification / approval of a particular structure or system. In particular the human safety approach and the cost-benefit approach analytically described in the following paragraphs. A.2

Human Safety

Acceptable risk levels cannot be defined in an absolute and strict sense. As mentioned above, each individual has their own perception of acceptable risk which, when expressed in decision theory terms, represents their own “preferences”. In order to define what is meant by “acceptable risk levels”, a framework for risk acceptability is adopted as shown in Figure A1. It is well known that some risks are so high that they are unacceptable. Therefore risks should be reduced to a level that is “as low as reasonably practicable” (ALARP). In principle, there is also a level of risk that is negligible and needs no further risk reduction effort.

Figure A.1: Framework for Risk Acceptability

In order to define the criteria in Figure A.1 in more tangible terms many different aspects need to be taken into consideration and it is important to incorporate them into a consistent framework. In most practical studies the societal risk of an installation is given in 58

Chapter 3: Reliability differentiation

the form of a numerical F-N-curve. An F-N-curve (N represents the number of fatalities, F the frequency of accidents with more than N fatalities) shows the relationship between the annual frequency F of accidents with N or more fatalities. Upper and lower bound curves are recommended based on gained experience with similar projects/activities and the ALARP (As Low As Reasonably Practical) acceptability criterion is obtained as the domain between the aforementioned limits. The upper limit represents the risk that can be tolerated in any circumstances while below the lower limit the risk is of no practical interest. Such acceptability curves have been developed for various industrial fields including the chemical and the transportation industry. The ALARP method is illustrated here in Figure A.2 in an example.

Figure A.2. F-N curve and illustration of the ALARP range

A.3

Calibration

There is a large number of so-called calibration studies and a sufficiently long history of almost about 30 years of such studies. They aim, firstly, at re-computing designs according to present practice in terms of probabilistic safety measures such as the reliability index or the failure probability and, secondly, at determining probability-based safety elements in design codes such that the traditionally accepted reliability level is maintained. They all are based implicitly on the above-mentioned postulate formalized by Lind [A1], i.e. present design practice results in sufficiently safe structures and is economical. These studies vary in the selected case studies, in the sophistication in stochastic modelling, in the assumptions for the parameters, the structural codes used for calibration and in the reliability methods used. Most studies used modern First-Order-reliability-Methods (FORM) and, thus contain the possible bias implied by this method although the bias may rarely be so strong as to make results worthless. Almost all of them are related to specific failure modes of simple structural components. Non-structural failure modes and human error are not considered. The results related to yearly quantities show a considerable scatter ranging from β ≈ 3 (pf ≈ 10-3) to β > 7 (pf > 10-12). The smaller reliabilities are calculated for columns of medium slenderness, for some extraordinary design situations (accidental situations) and for fatigue or other deterioration causes. They depend to a certain degree on the set of probabilistic and physical models. Therefore, the great advantage of limits derived from code

59

Chapter 3: Reliability differentiation

calibration is that they are consistent in the sense that the models (physical and probabilistic) used in code calibration are likely to be used also in probabilistic design. It also follows that a realistic but also operational set of models needs to be agreed upon. Despite the large scatter in the results several observations appear important in the light of Lind’s postulate: •

Large reliability indices are observed whenever it is relatively inexpensive to control reliability.

•

Whenever brittle failure occurs, reliability requirements are strengthened.

•

Whenever deterioration or fatigue is present and thus failure, if it occurs, occurs at a later time, the present reliability requirements are weakened.

•

Reliability requirements frequently are higher for (important) details. This questions the call for uniform reliability seen in many modern codes. It appears as if present codes at least aimed at optimality with respect to benefits, measures to control reliability and expected cost even if the search for the optimum has been largely by trial and error and in a deterministic context. Code calibration can also be used to determine the failure cost including all “intangibles” implied by present designs. If according to Lind’s postulate present design rules are already optimal and the cost parameters controlling reliability can be determined the failure cost can be inferred from a cost benefit analysis. This is an especially attractive feature which can help in further cost benefit analysis. A.4

Cost - Benefit Approach

The decision to accept risk is not based on the absolute notion of one acceptable risk level but has some flexibility as the judgement depends on the cost/benefit ratio. It is always possible to reduce the risk of a hazardous facility. But the incremental costs needed for reducing risk by an additional unit increase as the risk becomes smaller. With resources always being limited, money spent at one place will be lacking at another. Hence, the limited funds for reliability measures must be used in such a way that a maximum level of reliability is achieved. The optimal allocation of funds is a classical optimisation problem. In order to deal with the optimization problem one has to define: B(p) is the benefit derived from the existence of the structure, C(p) are the construction cost and D(p) the expected cost of failure, p is a parameter vector with which cost and reliability can be controlled. Statistical decision theory dictates that the expected values for B(p), C(p) and D(p) have to be taken. B(p), in general, will be unaffected or slightly decrease with each component of p but this can be neglected without substantial error so that B = B(p). The optimization is illustrated in Figure A.3 and discussed in [A2]. Much debate has been going on whether to include human lives into cost benefit analyses and whether it is at all admissible to perform cost benefit analysis when human lives or injuries are involved in case of structural failures. This requires the implementation of a monetary equivalent to save human life and limb into the analysis.

60

Chapter 3: Reliability differentiation

Cost B C(p) D(p) Optimum

Design parameter p

reasonable domain

Figure A.3: Cost and benefit over design parameter p

More recent studies on behalf of the public use so-called compound social indicators [A3]. Social indicators are statistics that reflect some aspect of the quality of life in a society or group of individual. More specifically, they aim to reflect broadly accepted goals that may carry labels such as national development, high expectancy of quality-adjusted life, the common good or the public interest. Any undertaking (project, program or regulation, adoption of new therapy, etc.) that affects the public by changing health or risk and expenditure will have an expected impact on a compound social indicator. A positive net impact of an undertaking on the accepted social indicator will lend to support the undertaking. The optimisation problem can be consequently solved using the Life Quality Index (LQI) approach. The strategy is based on a social indicator that describes the quality of life as a function of the gross domestic product, life expectation, and the life working time. The LQI defined as L (see [A3]) is a compound societal indicator, which is defined as a monotonously increasing function of two societal indicators: the gross domestic product per person per year, g, and the life expectancy at birth, e. L=g w e1− w

(A1)

The exponent w is the proportion of life spent in economic activity. By applying the reliability vs. cost-benefit approach risk acceptability criteria are indirectly applied by evaluating each investment into reliability.

References [A1] Lind, N.C., Formulation of Probabilistic Design, Journ. of Eng. Mech. Div., ASCE, Vol. 103, EM2, 1977, pp. 273-284 [A2] Rosenblueth, E., Mendoza, E., Realibility Optimization in Isostatic Structures, Journ. of the Eng. Mech. Div., ASCE, 97, EM6, 1971, pp. 1625-1642 [A3] Rackwitz R., Optimization and Risk Acceptability based on the Life Quality Index, Journal of Structural Safety, Vol 24., 2002, pp. 297-331.

61

Chapter 4: Actions

CHAPTER 4: ACTIONS Angel Arteaga1 , Ana de Diego1 1

E. Torroja Institute of Construction Sciences -CSIC, Madrid, Spain

Summary Principles for specifications of different types of actions on structures are illustrated on practical examples. In general, the characteristic, design and representative values are defined as fractiles of appropriate theoretical models, taking into account their variation in time. The way to obtain the characteristic values of different permanent and variable actions including wind, snow and temperature is shown.

1

INTRODUCTION

1.1

Background materials

Basic principles and rules for specification of representative and design values of actions and their effect on structures are given in EN 1990 [1], in the International Standard ISO 2394 [2] and introduced in the Designer's Guide to EN 1990 [3]. Methods for obtaining representative values of various types of actions are given in different Parts of EN 1991 devoted to actions and effects of actions [4-7], in the CIB documents Actions on structures [8] and ISO Standards [9-10]. Additional information may be found in the material oriented Eurocodes EN 1992 to EN 1999. Probabilistic models of actions can be found in the Model Code of the JCSS [11]

2

ACTIONS AND EFFECT OF ACTIONS

2.1

Definition of actions EN 1990 [1] defines actions as:

a) Set of forces (loads) applied to the structure, as for example the self-weight of structure itself or the wind pressure on a structural surface (direct action). b) Set of imposed deformations or accelerations caused for example, by temperature changes, moisture variation, uneven settlement or earthquakes (indirect action). In general, an action is described by a theoretical model; in most cases a single scalar variable is sufficient to represent the action, which may have several representative values. For example the imposed load on a floor is described by a vertical uniform load expressed in kN/m2 or the wind actions are represented by forces applied on a vertical surface expressed in kN. For some actions, and for verifying some aspects, a more complex representation of actions may be necessary, for example an action evoking fatigue must be represented, at least, by the number of cycles, the mean value of action and its amplitude. The actions acting on a structure may have some mutual statistical correlation and also with variables of structural resistance. When they correspond to different sources, the error of considering statistical independence is not too significant and they may be taken into account 62

Chapter 4: Actions

as independent. However, in some cases the dependence of actions is significant and it should be considered. For instance, self-weight and resistances are always correlated through the dimensions, but commonly it is not important. Other example could be the case when the maximum wind actions in a place are always seasonal, occurring in summer. Thus, it could make no sense to combine the maximum wind actions, if they do not occur in winter, together with snow loads. The actions that may be assumed to be statistically independent in time and space of any other action acting on the structure are called single actions. 2.2

Effect of actions

The effects of actions (or action effects) are the internal forces, moments, stresses, strains, etc. of structural members, or their deflections, rotations, etc., caused by the actions on the structure. Each limit state needs to be described quantitatively by the formulation of the actions and resistances in comparable terms. It means that the actions and resistances are expressed both as the forces or moments applied to the structure and the forces or moments that the structure can sustain. The action effect is related to the action and the properties of the structure. In general the design action effect can be expressed as: (1) where

is the partial factor for the action Fi is the relevant representative value of the action Fi is the design value of the relevant geometric dimension

In the case of non-linear analysis (i.e. when the relationship between actions and their effects is not linear), two simplified rules may be considered: 1. If the action effect increases more than the action, the partial factor should be applied to action. It occurs in most structures. 2. If the action effect increases less than the action, the partial factor should be applied to the action effect of the representative value of the action. Example In order to help to understand the difference between action and action effects indicated above a simple example is presented here. Two identical structural members are loaded by the same action, however, different structural conditions are considered: in the first case, a simply supported beam, and in the second, a fully fixed beam. Note that in the ideal case when the beam is submitted to a uniform increase of the temperature, ΔT, in all its length and cross-sections, the beam will tend to elongate given by ΔL= α L ΔT, where α is the coefficient of thermal expansion, (steel is considered here). In the statically determinate case there is no constraint and therefore, the beam will expand without stresses. Where the double fixed beam is considered, its expansion is constrained in supports leading to uniform compression stresses. The stress in the cross-section is given by σ /E = ΔL/L.

63

Chapter 4: Actions

a) Simply supported beam: IPE 240 S235 g

b) Double fixed beam IPE 220 S235

q

L

g

q

L

Span L = 6,0 m Cross section area: A = 39,12 ·10-4 m2 Moment of inertia Iy = 3 892·10-8 m4 Yield stress fy = 235 MPa Elastic modulus E = 210 000 MPa Thermal expansion coef.: α = 12·10-6 / ºC

Span L = 6,0 m Cross section area: A = 33,37 ·10-4 m2 Moment of inertia Iy = 2 772·10-8 m4 Yield stress fy = 235 MPa Elastic modulus E = 210 000 MPa Thermal expansion coef.: α = 12·10-6 / ºC

Actions, characteristic value: Direct: Permanent load: gk = 7,0 kN/m Variable load: qk = 3,0 kN/m Indirect: Uniform temperature increase: ΔT = 20ºC Settlement at one support: δ = 12mm

Actions, characteristic value: Direct: Permanent load: gk = 7,0 kN/m Variable load: qk = 3,0 kN/m Indirect: Settlement at one support: δ = 12mm Uniform temperature increase: ΔT = 20ºC

Effects of actions, characteristic value: Permanent loads: Mid span moment 1/8 gk L 2 = 31,5 kNm

Effects of actions, characteristic value: Permanent loads: Mid span moment 1/24 gk L 2 = 10,5 kNm Moment at supports -1/12 gk L 2 = -91,0 kNm Variable loads Mid span moment 1/24 qk L 2 = 4,5 kNm Moment at supports -1/12 qk L 2 = -9,0 kNm Indirect: Settlement at one support: δ = 12 mm Mid span moment 0 kNm Moment at supports ±δ 6 EI / L 2 = ± 11,64 kNm

Variable loads Mid span moment 1/8 qk L 2 = 13,5 kNm Indirect: Settlement at one support: δ = 12 mm No effects

Uniform temperature increase: ΔT = 20ºC no effects*

64

Uniform temperature increase: ΔT Uniform compression stress* σ = α E ΔT = 50,4 MPa

Chapter 4: Actions

3

CLASSIFICATION OF THE ACTIONS

3.1

General

The actions on structures are classified according to different aspects related to the design situation considered in reliability verification. Actions are classified by their: – variation in time (section 3.2) – origin (section 3.3); – spatial variation (section 3.4); – nature and/or the structural response (section 3.5). – bounded and unbounded actions (section 3.6) In addition, environmental influences are indicated in section 3.7 3.2

By their variation in time

The most important classification of actions is referred to the time when the action is acting compared with the reference period or the design working life. The actions are classified as: • permanent action (G), those likely to act throughout a given reference period and for which the variation in time is negligible, or for which the variation is always in the same direction (monotonic) until the action attains a certain limit value; e.g. selfweight of structures, fixed equipment and road surfacing, and indirect actions caused by shrinkage and uneven settlements • variable action (Q), those likely to act throughout a given reference period for which the variation in magnitude with time is neither negligible nor monotonic, e.g. imposed loads on building floors, wind actions or snow loads • accidental action (A), usually of short duration, that is unlikely to occur with a significant magnitude on a given structure during the design working life, but its consequences might be catastrophic, e.g.: earthquakes, fires, explosions, or impact from vehicles. Chapter 4 of this Guidebook deals with these kinds of actions. The concept of reference period will be explained later. 3.3

By their origin

Two classes are distinguished: direct actions consisting of forces or moments applied to the structure and indirect actions consisting of imposed deformations or accelerations caused, for example, by temperature changes, moisture variation, uneven settlement or earthquakes. 3.4

By their variation in space

When an action has a fixed distribution and position over the structure or structural member so that the magnitude and direction of the action is determined unambiguously for the whole structure or structural member then it is considered as a fixed action. If the action may have various spatial distributions over the structure then it is considered as a free action. 3.5

By their nature or structural response

The static actions are those that do not cause significant acceleration of the structure or structural members. The dynamic actions cause significant accelerations of the structure or structural members. In most cases for dynamic actions it is enough to consider only the static component that may be multiplied by a coefficient to take account of the dynamic effects.

65

Chapter 4: Actions

3.6

Bounded and unbounded actions

In some cases, an upper (or lower) bound of the action can be found and then this bound can be established as representative value. For instance, the load due to water in a tank; the water weight is limited by the height of the tank, and, therefore also the maximum load will be bound, and this value should be taken as the representative value. In other cases a possible bound could be found, but it should be much higher than the load obtained by statistic assessment and therefore not suitable as representative value. For instance, in a warehouse the load given by the material of maximum density stocked up to the maximum possible height. 3.7

Environmental influences

The environmental influences may have a physical, chemical or biological character and may deteriorate the material of the structure affecting the safety and serviceability of the structure. For instance, the presence of chlorides or carbon dioxide and humidity, which will lead to the corrosion of the reinforcement; fire that deteriorates the resistance of the materials; etc. These influences can be classified depending on the time variability and also considered as permanent, variable or accidental ones. Actions and action effects may also be distinguished.

4

THE REFERENCE PERIOD AND DISTRIBUTION OF MAXIMA

The definitions given for permanent and variable actions include the “reference period” what is the time used as a basis for the statistical assessment of actions and the timevarying resistances. For each type of action, depending on its characteristics, the whole working life of the structure is split in one or various reference periods T, of the same or different (random) length. In each of these reference periods the action varies following a more or less similar pattern and, therefore the same independent, identical distributions could be adopted for the action in any such period. This means that the set of maxima coming from the maximum of one of each of such periods will form a sample of maxima, from which an extreme distribution function could be derived. The maximum in any period will correspond to an independent realization of such a distribution of maximum values. The adequate reference period for defining the characteristic value will depend on the type of variable action. 4.1

Climatic actions

For these actions – snow, wind, etc. - a period of a year is generally adequate; i.e.: it can be assumed that the statistics characterizing the climatic actions in one year do not vary and that the actions are independent every year. That means the annual maximum is independent of the maxima of previous and following years. If the distribution function of maxima of a climatic action on a reference period (one year) is known, the distribution function of maxima in the whole design working life, T, assuming the same distribution function for every reference periods is given by: (2)

66

Chapter 4: Actions

Where FQ,max(x) is the distribution function of annual maxima and [FQ(x)]T is the distribution of maximum values in the design working life. See Figure 4.1

Figure 4.1. Model and distribution for a variable climatic action, e.g. wind 4.2

Imposed loads

For these actions, a reference period corresponding to the change of owner or the change of use in the building or part of it is generally adopted. In [11] an average value of 510 years for reference periods of random length is indicated. This means from 5 to 10 changes of use in a working life of 50 years. See Figure 4.2

Figure 4.2. Model and distribution for an imposed load For imposed loads, the adopted characteristic value corresponds to a 95 % percentile of the distribution of maximum values during all the working life. Take note that in this case the characteristic value is not defined on the basis of the reference period, but referred to the whole design working life. If the distribution function of the imposed loads on one reference period in buildings with similar use is known (for instance, by a survey of imposed loads at a point in time), assuming that the distribution function does not change with time for the following reference periods, this distribution function can be adopted for all the different periods included in the working life. It is assumed in [11] that the time length of the reference periods is exponentially distributed, and that then the number of changes in the working life is Poisson’s distributed. With these assumptions, the distributions of the maxima and minima for the design working life are obtained as: (3a) 67

Chapter 4: Actions

(3b) where FQ (x) is the distribution function of maxima of the imposed load action in each reference period, FQ,min(x) and FQ,max(x) the distribution function of maxima and minima for the whole design working life, λ is the average rate of changes in use (changes per year) and T the design working life. Therefore λT represents the expected number of reference periods, (changes of use/owner) in the working life. From these expressions, the characteristic lower and upper values, corresponding to a 5% and 95% of not being reached or not being over passed, respectively, as function of the fractiles of the distribution of Q(x), are given in the Table 4.1, depending on the mean number of changes: Table 4.1 Fractiles corresponding to the characteristic values

λT

5 0.010 0.990

Qk,inf Qk,sup

7 0.007 0.993

10 0.005 0.995

The fractiles in Table 4.1 mean that the characteristic value to be adopted, for instance, Qk,sup, will correspond to the fractile 0.993 of the distribution in each reference period, assuming an average numbers of changes of 7. Assuming a Normal distribution for Q(x), the characteristic values can be obtained from the parameters of this distribution as: Qk,inf = μQ – k σ Q = μQ ( 1 – k VQ)

(4a)

Qk,sup = μQ + k σ Q = μQ ( 1 + k VQ)

(4b)

where μQ and σ Q are the mean value and coefficient of variation of Q and k is obtained from the standardized Normal distribution values (p-fractile). Examples of the values of k as function of the mean number of changes in the working life are given in the Table 4.2: Table 4.2 Coefficients for Normal distributions

λT k

5 2,32

7 2,44

10 2,57

Example: Consider a building where it is foreseen that changes of use or owner can modify the imposed load given by the weight of some non-structural elements (heavy partitions, p.e.) or equipments. It is assumed that this part of the imposed load has Normal distribution with a mean value of 1,0 kN/m2 and a coefficient of variation of 0,15. If a mean number of changes is 7, i.e.: changes with a rate of approximately 7 years in the 50 years working life of the structure, from the equation (4a) the following characteristic value is obtained: Qk,sup = μQ ( 1+ k VQ) = 1,0 ( 1 + 2,44 × 0,15) = 1,36 kN/m2 which is 36 % higher than the mean value μQ and 9 % higher than the 95 % percentile of the distribution function of the imposed load in one reference period.

68

Chapter 4: Actions

5

CHARACTERISTIC VALUES

5.1

General

The characteristic value Fk of an action F is its basic representative value. When there is enough data to assess its value on statistical bases, the characteristic value corresponds to a prescribed probability of not being exceeded on the unfavourable side during the “reference period” taking into account the design working life of the structure and the duration of the design situation. In case there is not available data enabling statistical evaluation, a nominal value or a value fixed in the project documentation is chosen, provided that consistency is achieved with methods given in EN 1991. 5.2

Permanent actions

The characteristic value of a permanent action shall be assessed as follows: - if the variability of G during the working life can be considered small, one single value Gk equal to the mean value may be used; - if the variability of G cannot be considered small, two values shall be used: an upper value Gk,sup and/or a lower value Gk,inf. The 5% fractile is adopted generally for Gk,inf and the 95% fractile for Gk,sup considering the statistical distribution for G, which may be assumed to be Gaussian. See Figure 4.3. With these assumptions Gk,inf and Gk,sup can be obtained from: Gk,inf = μG –– 1,64 σ G = μG ( 1 – 1,64 VG)

(5a)

Gk,sup = μG + 1,64 σ G = μG ( 1 + 1,64 VG)

(5b)

where μG is the mean value, σ G the standard deviation and VG the coefficient of variation of the distribution of G. The Eurocode EN 1991-1-1 [4] gives in its Annex A values of densities (actually unit weights) for the most common materials to be used to calculate the mean value of the permanent loads. In some cases, when the density is very dependent on the conditions of the material (e.g. the humidity or occluded air) a range is given instead of a single value. Values of the coefficient of variation are not given in this Eurocode; indicative values can be found in the Probabilistic Model Code [11]. The mean value of the self-weight of one element is calculated on the basis of the nominal dimensions of the element and its mean density.

Figure 4.3 Characteristic values of permanent actions.

69

Chapter 4: Actions

Example 1: Consider, for instance, a beam of normal weight concrete: the mean value of the density can be taken as γ = 24 kN/m3 as given in EN 1991-1-1. In normal situations the characteristic value of the permanent load due to the self-weight of the beam is obtained by multiplying this value by the nominal dimensions of the cross section. That is: gk = 24 a b kN/m, where a and b are the nominal dimensions of the cross section in metres. Consider now that, due to any circumstance, the structure is very sensitive to this selfweight and it is therefore necessary to take into account the lower and upper characteristic values. In [7] a coefficient of variation of 0.04 is given for the density of concrete. Introducing this value in the formula (5a) and (5b), with the nominal dimensions of the cross section gk,inf = μG ( 1- 1,64 VG) = 24 a b (1-1,64×0,04) = 22,4 a b kN/m gk,sup = μG ( 1+ 1,64 VG) = 24 a b (1+1,64×0,04) = 25,6 a b kN/m Example 2: Consider now that the material of the beam is glulam of type GL 36h. The mean density of this material given in EN 1991-1-1 is 4,4 kN/m3 and in [11] a coefficient of variation is 0.1. The coefficient of variation is just in the limit indicated in EN 1990 for considering a unique (mean) characteristic value or two values (upper and lower values) . The need to take the mean value or the upper and/or lower value for the characteristic value, in this case, will depend on the ratio between the self-weight of the element and other loads. Substituting the mean and coefficient of variation values in (5a) and (5b) we obtain: gk,inf = μG ( 1- 1,4 VG) =4,4 a b (1-1,64 × 0,1) = 3,7 a b kN/m gk,sup = μG ( 1+ 1,64 VG) = 4,4 a b (1+1.64 × 0,1) = 5,1 a b kN/m where a and b are the nominal dimensions of the cross section in metres. We can see that now the characteristic values are lower or upper by 16% to the mean value, while in the case of concrete the variation amounted only to 7%. 5.3

Variable actions -

For variable actions, the characteristic value (Qk) shall correspond to either: a higher value with an intended probability of not being exceeded or a lower value with an intended probability of being achieved, during some specific reference period;

or: a nominal value, which may be specified in cases where a statistical distribution is not known. For the characteristic value of climatic actions (wind, snow, etc.) a year reference period is generally chosen with an annual probability of exceedence taken as 0,02. This is equivalent to a mean return period of 50 years. It is worth noting that this chosen return period of 50 years is not related with the generally adopted 50 years as design working life of common structures. The return period just gives a probability of exceedence. That is, on the average, every 50 years there will be an action exceeding the value given statistically to this 50 years return period. -

70

Chapter 4: Actions

Assuming that the actions in all the years are independent and identically distributed, and considering an adopted return period R and design working life of T years, the probability r of exceedence during the design working life is given by: r = (1 – 1/R)T

(6)

In this formula, if the values of R = 50 years and T = 50 years are introduced, a value of r = 0,63 is obtained. That means that the probability that the characteristic value will be exceeded during the design working life is approximately two thirds. A Gumbel distribution is generally accepted as the theoretical model for distribution of maximum values [11]. In the following paragraphs the procedure for determination of characteristic values of different variable actions is given. 5.4

Imposed loads

In the Part 1.1 of Eurocode 1 [4] two different types of imposed loads are considered: for global effects, a uniformly distributed load is applied; for local effects, concentrated load acting alone may be used. The characteristic values of these distributed imposed loads, qk, and concentrated loads, Qk, are given in Tables 6.2 and 6.4 of EN1991-1-1 [4] for various categories of intended use of the building or sector. Category A for residential, B for offices, C for areas where people can concentrate, with some subdivisions, D for shopping uses, and so on. For the assessment of the imposed load affecting one horizontal structural member in a storey-distributed load, a free uniform distributed applied action is considered in the unfavourable part of the influence area of the action effect for this member. This can lead to the need of studying a large number of cases in complex structures. As a simplification for the assessment of the action effects for this member, the load coming from other stories may be treated as fixed uniform load. When the influence area for the studied element is big, it is not likely that all the area may have a high imposed load at the same time. In order to take account of this fact a reduction factor αA due to the size of the area and a factor αn due to the number of stories over the element can be used. The factor αA is given by:

αA = ψ0 x 5/7 + A0/A ≤ 1 where:

ψ0 A0 A

(7)

is the combination factor given in EN 1990 (= 0.7 for most common uses of buildings) is a reference area equal to 10 m2 the loaded area

For vertical members (columns, walls) the imposed load on the upper floors may be considered as uniform distributed load acting in the area affecting the member, reduced by the factor αn given by

αn = (2 + (n – 2) ψ0) / n

(8)

where n is the number of stories (> 2).

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Chapter 4: Actions

The simultaneous use of these two α factors is not allowed. Also, when the imposed load enters in a combination of actions as an accompanying action where multiplied by the corresponding ψ factor, then the simultaneous reduction by α factor is not allowed either. Example: Consider a building with six storeys used as offices, each an independent one, an inaccessible roof and a ground floor with retail shops. The building structure is formed by four vertical and plane rectangular frames, at 5 m each, connected by beams and floors. Each frame forms four 7 m long bays and six storeys (seven floors including roof). The building is shown in the following Figure 4.4.

21,5 m

28 m

15 m

5

C

B

5 B

5

C

7

7

7

7

Figure 4.4 Building layout and frames and floors distributions Beam B Roof: Category H, qk = 0.4 kN/m2 ; ψ0 = 0.0 (recommended values [1] ) Storeys: Category B, qk = 3.0 kN/m2; ψ0 = 0.7 (recommended values [1] ) Beams: Tributary area 7 × 5 = 35 m2 αA = ψ0 × 5/7 + A0/A ≤ 1 αA =0.7 × 5/7 + 10/35 = 0.79 qk = 0.79 × 3.0 = 2.37 kN/m2 Uniform load due to imposed loads beam B : 2.37 × 5 = 11,75 kN/m Column C Columns: Tributary area 7 × 5 = 35 m2 Load from roof: 35 × 0.4 = 14 kN Load from every office floor: 35 · 3.0 = 105,0 kN αn = (2 + (n – 2) ψ0) / n αn = (2 + (6– 2) × 0,7) / 6 = 0,80 (n=6, because the roof is different category and it is not included in the reduction) Load due to imposed loads in the column C : 14 + 6 × 105,0 × 0,82 = 530,6 kN The imposed load transmited from other floors is the total imposed load not reduced by the αA factor.

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Chapter 4: Actions

5.5

Snow loads

The characteristic load on a roof due to snow, s, is given in the Eurocode 1, part 1.3 [5] by the equation: s = ηi · Ce · Ct ·sk

(9)

where

ηi is the roof shape coefficient

Ce is the exposure coefficient Ct is the thermal coefficient sk is the characteristic value of the snow load on the ground for the site of the building The exposure coefficient takes into account the situation of the referred roof’s facility to accumulate snow or not. It is generally taken as unity, where Ce = 1,0. Only when the roof is on open terrain – windswept, resp. sheltered by higher buildings, it may be taken by smaller, Ce = 0,8, or bigger values, Ce = 1,2, respectively. The thermal coefficient expresses the possibility that the snow melt due to the heat transmission from the building in case of poor thermal isolation, for instance glass roofs. The usual value is Ct = 1,0. Roof shape coefficient The distribution of snow on a roof is generally not uniform. The wind, even of low velocity, causes the snow drift from one point of the roof to another, from more exposed to more sheltered parts of the roof due to the particular direction of the wind. The EN 1991-1-3 [5] gives values of the snow load shape coefficient ηi depending on the angle of the roof and the different shapes: mono-pitched or pitched, single-span or multispan, cylindrical or roofs abutting and close to taller structures. Consideration has to be given in the cases where there are systems to sweep the snow or where snow fences exist. Characteristic value of snow load on the ground This Eurocode part provides in the Annex C the European maps for various climatic regions: Alpine Region; Mediterranean Region; Norway, Sweden & Finland; UK & Republic of Ireland; Central East; Central West; Greece; and the Iberian Peninsula. Each Region map indicates the snow load at the sea level for each of the zones. Moreover, expressions given for these regions make it possible to determine the load at the altitude level of building site from the snow load at the sea level. These expressions are functions of region, the zone corresponding to the site in the map and the site level. Special cases Information is also given for special cases where the standard information is not sufficient, e.g. for the purpose of taking account of the local effects due to the snow overhanging at the edge of the roof. Indications are also given on how to deal with exceptional snows or exceptional accumulations of snow causing unfavourable local effects on the structure. Programs The spreadsheet snow.xls added to this Guidebook facilitates the obtaining of the snow loads on common buildings.

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Chapter 4: Actions

Example 1: Consider a building with a multi-span roof as given in the figure with the dimensions in the table (α1 = 45°, α2 = 26,5°).

h h a

b

a

a [m] 2

b

b[m] h1 [m] h2 [m] 4 2 3

The building is situated near Milan in Zone 3 (altitude 400 m). From [5] we obtain the map of Italy, the characteristic snow load at sea level and the formula how to obtain the characteristic value at the site elevation: Mediterranean Region

⎡ ⎛ A ⎞2 ⎤ sk =(0,498Z − 0,209) ⎢1+⎜ ⎟ ⎥ ⎣⎢ ⎝ 452 ⎠ ⎦⎥

Then, the characteristic snow load on the ground, sk at the site is: sk = (0,498 ×3 – 0,209) [1 + (400/452)2] = 1,121 kN/m2

The distribution of snow on the roof is defined by Figure 5.4 of [5].

Case (i)

µ1(α1)

µ1(α2)

µ1(α1) µ2(α)

Case (ii)

µ1(α2)

α = (α1+ α2)/2 µ1(α2)

µ1(α1)

α1

α2

α1

α2

The values of the roof shape coefficients are based on table 5.1 [5]:

74

Chapter 4: Actions

Angle of pitch α1= 45º α2 = 26º (α1 +α2)/2 =35,5 μ1 0,8 0,8(60 – α)/30 = 0,4 0,8(60 – α)/30 = 0,65 μ2 1,6 1,6 0,8 + 0,8 α/30 = 1,51 The following snow loads on the roof are obtained for the considered snow shape coefficients : a)

Undrifted case s1

b)

s2

s3

s4

s1 s2 s3 s4 0,448 kN/m2 0,897 kN/m2 0,448 kN/m2 0,8 kN/m2 s6 Drifted case s5

s7

s5 s6 s7 2 2 0,448 kN/m 1,794 kN/m 0,729 kN/m2 Example 2: Consider now the same building in a town with the same elevation (400 m) near the Madrid Zone 4 of the Snow map of Spain. The characteristic snow load on the ground, sk for this Zone and the altitude of 400 m is sk = 0,3 kN/m2. The distribution of the snow on the roof and the shape factors are the same; therefore, the loads on the roof will be obtained multiplying the same shape factors by the new characteristic snow load on the ground, resulting in: s1 s2 s3 s4 0,12 kN/m2 0,24 kN/m2 0,12 kN/m2 0,21 kN/m2 s5 s6 s7 0,12 kN/m2 0,48 kN/m2 0,20 kN/m2

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Chapter 4: Actions

5.6

Wind loads

The Part 1.4 of EN 1991 [9] deals with the effects of wind on structures. The scope of this part covers up to 200 m height buildings for the common effects on all parts of the building: components, claddings and fixings, etc. Other effects, as thermal effects on winds, vibrations where more than a relevant fundamental mode needs to be considered, the torsional vibrations due to transverse winds, etc. are not covered. Three models of response are given: the quasi-static response, the dynamic and the aeroelastic. The effect of the wind on the structure (i.e. the response of the structure) depends on the size, shape and dynamic properties of the structure. Wind fluctuates with time and this fluctuation can originate different effects depending on the building characteristics. For most buildings, only a quasi-static response structure needs to be considered. Dynamic structural responses are needed to be considered only in the cases with very low natural frequency (lower than 1 Hz) and low damping. Aeroelastic response should be considered for flexible structures such as cables, masts, chimneys and bridges. Therefore, the quasi-static response is treated here only. The wind acts directly as a pressure on the external surfaces of enclosed structures and, because of the porosity of the external surface it also acts indirectly on the internal surfaces. It may also act directly on the internal surface of open structures. Pressures act on areas of the surface resulting in forces normal to the surface of the structure or of individual cladding components. Additionally, when large areas of structures are swept by the wind, friction forces acting tangentially to the surface may be significant. The quasi-static action of the wind is represented by a simplified set of pressures or forces whose effects are equivalent to the extreme effects of the turbulent wind. The fundamental value of the basic wind velocity, vb,0, is the main variable used to define the wind in a site. It is defined as the characteristic 10 minutes mean wind velocity at 10 m height on a terrain category II. The terrain category II is defined as an area with low vegetation such as grass and isolated obstacles (trees, buildings) with separations of at least 20 obstacle heights. The Part [6] does not provide maps of fundamental wind velocity; they shall be given in the National Annexes of CEN Member States to be operative in design procedure. From the fundamental wind velocity the basic wind velocity is derived, vb as: vb = cdir·cseason· vb,0

(10

where cdir and cseason are the directional and seasonal factors, taking into account that the wind in some directions could be reduced and that temporary structures spanning a few months might have a lower probability of high winds. These two factors are usually taken as the unity. The basic wind pressure, qb is derived from the basic wind velocity as: qb = ρ/2 · vb2

(11)

where ρ is the density of the air (it can be set to 1,25 kg/m3). The basic wind pressure represents the mean value of the pressure on a building placed at a site in a terrain category II and with a reference height of 10 m. The transformation of this value to the pressure at a building at the actual terrain category in reference height is carried out by the mean wind velocity at the relevant height by: vm (z) = cr(z)· c0(z)· vb

76

(12)

Chapter 4: Actions

where: vm (z) is the mean wind velocity at z reference height cr(z) is the roughness factor, and c0(z) is the orography factor The orography factor takes into account the fact that for buildings placed on elevations, valleys, etc. the wind could be increased. Usually, it is considered as the unity. The roughness factor is derived as: cr(z) = kr ln (z/z0)

for z ≥ zmin ;

kr = 0,19 (z0/z0,II)0,07 ; where: kr z0 zmin

(13) (14)

the terrain factor, the roughness length given for every terrain category, the minimum height given for relevant terrain category.

There are five different terrain categories defined in Annex A of [6]: from category 0, corresponding to sea or coastal areas exposed to open sea to category IV, where areas with buildings of average height exceeding 15 m should be considered.

Figure 4.4 The peak and mean velocities The value of peak velocity represents the effect of the average velocity in 10 minutes plus the effect of the gust, see Figure 4.4, it is obtained from the mean wind velocity by multiplication by the gust factor G: vp (z) = G ·vm (z); (15) where: G =

for z ≥ zmin , and kI is the turbulence factor

The peak velocity pressure in the relevant height is finally: (16) In common cases it may be taken c0(z ) = kI = 1.0, then it can be simplified to: (17)

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Chapter 4: Actions

It can also be expressed in the form: (18) where

is the exposure factor (19)

This exposure factor then gives the relationship between the peak velocity pressure for the building reference height and the basic velocity pressure of the site, depending on the different terrain categories. See Figure 4.5.

Figure 4.5 Exposure factor for the reference height and terrain categories

Quasi-static wind response: pressure and wind forces The wind forces can be determined with the help of pressure or force coefficients: a) Pressure coefficients The force on the whole structure is determined by the vectorial summation of the external, internal and friction forces on all the surfaces of the building: (20)

(21)

(22) where: Fw,e, Fw,i and Afr are the external, internal and friction forces and

78

Chapter 4: Actions

where is the structural factor and are the pressure coefficients for external or internal pressure is the friction coefficient and are the external and internal pressure on the individual surface is the peak velocity pressure at the reference height is the reference area for an individual surface is the area of the external surface parallel to the wind

b) Force coefficients The force on the whole structure or on one member can be calculated by the expression: (23) where:

is the force coefficient for the member or the whole structure given for different shapes in section 7 of [8] Section 7 of [6] indicates the way to obtain all these coefficients for common types of buildings or members. Programs The supplementary spreadsheet wind.xls to this Guidebook facilitates to obtain the wind loads on common buildings. Example 1: Consider the case of the industrial hall shown in the figure to be built in Alicante (Spain) by the seashore.

Wind

30 m m

2m

8m

5m

15 m

From the wind map of Spain we obtain: The basic wind velocity, vb = cdir ·cseason· vb,0 = 1 × 1 × 27 = 27 m/s. Considering a terrain roughness of category 0 (faced to open sea) z0 = 0,003 m, zmin = 1 m

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Chapter 4: Actions

The basic wind pressure, qb qb = ρ/2 · vb2 = 1,25 kg/m3 × (27 m/s )2/2 = 0,455 kN/m2 z = 10 m z0 = 0,003 zmin = 1 m Roughness factor, cr(z) cr(z) = 0,19 (z0/z0,II)0,07 ln (z/z0)

= 1,26572

the mean wind velocity vm (z) = cr(z)· c0(z)· vb = 34.1744 m/s c0(z) =1. the peak velocity vp (z) = G ·vm (z) = where

G=

46,65 m/s

= 1,3649

The peak velocity pressure for c0(z ) = kI = 1.0; = 1,360 kN/m2

where

= 2,98453

Quasi-static wind response: pressure and wind forces The wind forces can be determined with the help of pressure or force coefficients. The pressure coefficients are used in this case. Pressure coefficients: The force on the whole structure is determined by the vectorial summation of the external, internal and friction forces on all the surfaces of the building:

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Chapter 4: Actions

The friction forces can be disregarded due to the small area on which they are applied. There is no dominant face in this building and, besides, no special characteristics are given to the calculation of the μ coefficient, so two hypotheses have to be analysed for the internal force coefficients: cpi = +0,2 and cpi = -0,3. Taking account of the dimensions of the building: e = min[ b, 2 h]= min[30, 2×10] = 20 m the pressures on the faces and roof can be taken from Figure 7.8 and Table 7.4a of [5].

F

G 2m

D

G

H

J

I

H 5.5 m m

J 2m

I 5.5 m m

E

D 5m

E

F

One central frame is considered here. The influence area for each frame is 5 m wide. Surface Cpe we wi cpi=0,2 wi cpi= -0,3 Aref [m2/m] Fwe Fwi (cpi= 0,2) Fwi (cpi=-0,3) Total force Fw (cpi=-0,3) [kN/m] Total force Fw (cpi= 0,2) [kN/m]

D +0,8 1,088 0,271 -0,408 10 10,88 -2,72 4,08 14,96

E -0,5 -0,680 0,271 -0,408 10 -6,80 -2,72 4,08 -2,72

F -0,9 -1,224 0,271 -0,408 1 -1,22 -0,27 0,41 -0,82

G -0,3 -0,408 0,271 -0,408 2 -4,90 -3,26 4,90 0

H -0,3 -0,408 0,271 -0,408 5,5 -2,24 -1,50 2,24 0

I -0,4 -0,544 0,271 -0,408 5,5 -2,24 -1,50 2,24 -0,75

J -1,0 -1,360 0,271 -0,408 20 -2,72 -0,54 0,815 -1,90

8,16

-9,52

-1,50

-8,16

-3,74

-4,49

-3,26

The signs of the internal forces are changed with respect to the internal coefficients to be added algebraically to the external forces; therefore, the sign of the total forces has to be considered from the point of view of the external surfaces. From the two hypotheses ((cpi= 0,2 or cpi= -0,3) the one most critical for the frame has to be chosen. 5.7

Thermal actions

The part 1.5 of EN 1991 [7] deals with thermal actions. Variable and indirect actions must be considered. The members of load-bearing structures shall be checked to ensure that

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thermal movement will not cause overstressing of the structure, either by the provision of movement joints or by including the effects in the design. The fundamental quantities for thermal actions are the extreme air temperatures, that is, the maximum and minimum, in the shade at the building site. The thermal actions on a structural member can be split in three basic quantities: 1. A uniform temperature component, given by the difference between the average temperature T, in summer or winter (or due to operational temperatures) of the member and its initial temperature T0; 2. A linearly varying temperature, given by the difference ΔΤΜ between the temperatures of the external and internal surfaces of a cross section or layers; 3. A temperature difference ΔΤp between different parts of the structure, given by the difference between the mean temperatures of the parts in question. The average temperature T should be determined using a temperature profile. Determination of the temperature profile The Annex D of [7] gives expressions to obtain the temperature profiles taking account the inner and outer environmental temperature. Table 4.3 Temperature in the inner environment, Tin Season Summer Winter

Temperature Tin T1 (recommended value 20 °C) T2 (recommended value 25 °C)

Table 4.4 Temperature in the outer environment, Tout Season

Summer

Winter

Significant factor Relative absorptivity depending on surface colour

0,5 bright light surface 0,7 light coloured surface 0,9 dark surface

Temperature Tout in 0C Tmax + T3 Tmax + T4 Tmax + T5 Tmin

Tmax, Tmin, and T3, T4, and T5 may be specified in the National Annex. Recommended values are; T3 = 0°C, T4 = 2°C, and T5 = 4°C , for North-East facing members; T3 = 18°C, T4 = 30°C, and T5 = 42°C for South-West or horizontal facing members. Once the temperature profiles are determined, the effect of the thermal actions can be specified taking into account the coefficients of thermal expansion of the materials involved.

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6

REPRESENTATIVE VALUES OF ACTIONS

6.1

General

A series of variable actions can generally act simultaneously on one structure; in order to obtain the maximum effect, the design combinations (see Chapter 6 of this Guidebook) include a main variable action accompanied with the other of variable actions. At the point in time of the working life of the structure when the main variable action could reach its maximum value (let us say, its design value), the accompanying actions will most probably have a lower value than its design value. The accompanying value of a variable action (ψ Qk) is defined as the value of a variable action that accompanies the leading action in a combination. This value may be the combination value, the frequent value or the quasi-permanent value, obtained from the characteristic value by multiplying by a factor ψ. 6.2

The combination value of a variable action (ψ0 Qk)

The combination value ψ0 Qk is presented as a product of the characteristic value multiplied by the coefficient ψ0 (ψ0 ≤ 1). It is used for the verification of ultimate limit states and irreversible serviceability limit states. The combination value is chosen – as far as it can be fixed on statistical bases – so that the probability of the combination value being exceeded is approximately the same as that one taken for the characteristic value of an individual action. 6.3

The frequent value of a variable action (ψ1 Qk )

The frequent value is represented as the product ψ1Qk, it is used for the verification of the ultimate limit states involving accidental actions and for verifications of reversible serviceability limit states. The frequent value is determined – also if it can be fixed on statistical bases – so that either the total time, within the reference period, during which it is exceeded, is only a small given part of the reference period, or that the frequency of it being exceeded is limited to a given value. For buildings, for example, the frequent value is chosen so that the time when it is exceeded is 0,01 of the reference period; for road traffic loads on bridges, the frequent value is assessed on the basis of a return period of one week. It may be expressed as a determined part of the characteristic value by using a factor ψ1 ≤ 1. 6.4

The quasi-permanent value of a variable action (ψ2Qk)

The quasi-permanent value is represented as a product ψ2Qk, it is used for the verification of ultimate limit states involving accidental actions and for the verification of reversible serviceability limit states. Quasi-permanent values are also used for the verification of long-term effects; their value is determined so that the total period of time in which it will be exceeded is a large fraction of the reference period. It may be expressed as a determined part of the characteristic value by using a factor ψ2 ≤ 1. For loads on building floors, the quasi-permanent value is usually chosen so that the proportion of the time it is exceeded is 0,50 of the reference period. The quasi-permanent value can alternately be determined as the value averaged on a chosen period of time. In the case of wind actions or road traffic loads, the quasi-permanent value is generally taken as zero. A schematic representation of the meaning of these accompanying values for a variable action along the working life of the structure is illustrated in Figure 4.4. 83

Chapter 4: Actions

Figure 4.6

7

Schematic representation of a variable action and its representative values

REPRESENTATION OF THE DYNAMIC ACTIONS

In common cases, the dynamic actions can be treated as static actions, i.e.: quasi-static actions, taking into account the equivalent static action obtained by multiplying the magnitude of the static part of the action by an adequate coefficient. In most cases this coefficient is higher than one, but if the time of application of the dynamic action is short, e.g. impacts from vehicles, this coefficient can be lower than one. The influence of the dynamic actions of fatigue of the structural material has to be considered. The dynamic effects of the action are generally taken into account by means of the characteristic values and fatigue load models given in EN 1991. These effects are considered well implicitly in the characteristic loads, or, well explicitly by applying dynamic enhancement factors to characteristic static loads. When dynamic actions cause significant acceleration of the structure, and the simplification of the quasi-static approach is no longer valid, dynamic analysis of the system should be used to assess the response of the structure. The model shall describe the time variation of the action in such a way so as to give results accurate enough. The description can be done in the time domain, which is the time history of the action, or in the frequency domain (the last one the Fourier's Transformed of the former one). It is necessary to take into account the mutual influence of loads and structures. For instance, in the case of lightweight structures loads may depend on the natural Eigen-frequency of the structure. The models of dynamic analysis include:

84

-

a stiffness model, similar to the static one

-

a damping model, due to different sources, and

-

an inertia model, taking account of the masses of the structural and non-structural elements.

Chapter 4: Actions

8

REPRESENTATION OF FATIGUE ACTIONS

When the actions may cause fatigue of the structural material, it shall be verified that the reliability with respect to fatigue is sufficient. The models for fatigue actions are strongly dependent on the type of structural material and should be those that have been established in the relevant parts of EN 1991 from evaluation of structural responses to fluctuations of loads performed for common structures (e.g. for simple span and multi-span bridges, tall slender structures for wind, etc.). In many cases, the models are based on empirically known relations between the stresses and the number of cycles to failure (S-N curves) or in considerations of the mechanics of the fracture.

9

REPRESENTATION OF THE ENVIRONMENTAL INFLUENCES

The effects of environmental influences should be taken into account, and where possible, be described quantitatively in the same way as for actions. The effects of the environmental influences are strongly dependent on the type of structural material. When a model of structural deterioration related to the in situ environmental conditions can be established it is possible to define a limit state in accordance with it. In this case the environmental influences are treated exactly in the same way as actions. This model could be deterministic with the uncertainties introduced via some selected random parameters or taking into account the model uncertainty. Unfortunately, up to now there are no accepted models for most of these environmental influences, due to the lack of data, which essentially depend very sharply on the location of the site. Up to now, most of these influences have to be considered using empirical formulae.

REFERENCES [1] EN 1990 Eurocode - Basis of structural design. European Committee for Standardisation, 04/2002. [2] ISO 2394, General principles on reliability for structures. 1998. [3] Gulvanessian, H. – Calgaro, J.-A. – Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, ISBN: 07277 3011 8, 192 pp. [6] [4] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. European Committee for Standardisation, 04/2002. [5] EN 1991-1-3 Eurocode 1: Actions on structures – Part 1-3: General actions – SnowLoads. European Committee for Standardisation, 07/2003. [6] EN 1991-1-4 Eurocode 1: Actions on structures – Part 1-4: General actions – Wind Loads. European Committee for Standardisation, 04/2005. [7] EN 1991-1-5 Eurocode 1: Actions on structures – Part 1-1: General actions – Thermal actions. European Committee for Standardisation, 11/2003. [8] CIB Actions on Structures. Reports of the CIB Working Commission W81. 1989-95 [9] ISO 4355, Basis for design of structures- Determination of snow loads on roofs. 1998. [10] ISO 4354, Wind actions on structures. 2009 [11] Probabilistic Model Code, Parts 1 to 4, Basis of design, Load and resistance models, Examples, JCSS, http://www.jcss.ethz.ch/ 2001-2002.

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Chapter 5: Accidental actions

CHAPTER 5: ACCIDENTAL ACTIONS Ton Vrouwenvelder1, Dimitris Diamantidis2 1

2

TNO, Delft, Netherlands University of Applied Sciences, Regensburg, Germany

Summary Structural engineers have become concerned with the phenomenon of accidental actions and progressive collapse since the collapse of the Ronan Point apartment building in the UK in 1968. In other industrial sectors such as offshore industry and nuclear industry the consideration of accidental actions and related structural resistance has been taken into account since the early seventies. Since the aforementioned accidents the issue of robustness as a property of structural systems has been recognised as an important topic. National and international codes have developed guidelines to consider accidental actions and progressive collapse. In the Eurocodes the accidental actions are covered by the Eurocode EN 1991-1-7. The document describes the principles and application rules for the assessment of accidental actions on buildings and civil engineering works. Such aspects are reflected in this Chapter 5. Parameters influencing the robustness are discussed as well as the probabilistic formulation of progressive collapse. Practical design examples are provided.

1

INTRODUCTION

1.1

History

Structural engineers have become concerned with the phenomenon of accidental actions and progressive collapse since the collapse of the Ronan Point apartment building in the UK in 1968. The trigger was a gas explosion in a corner flat that initiated a collapse of a major portion of the building. Several accidents followed since then, including: • • • • •

the collapse of the Beirut Embassy in 1973 caused by a bomb blast, the World Trade Center progressive damage due to a bomb in 1993, the bombing of the Murrah Federal Building in Oklahoma City in 1995 (Fig. 1), the bombing of the US Embassies in Nairobi and Dar Es Salaam in 1998, the collapse of the World Trade Center towers in New York in 2001 (Fig. 2).

In other industrial sectors such as offshore industry and nuclear industry the consideration of accidental actions and related structural resistance has been taken into account since the early seventies. Soon after the aforementioned accidents the issue of robustness as a property of structural systems has been recognised as an important topic. A variety of research efforts has been performed in the past and methods for assessing robustness have been proposed. National and international codes have developed guidelines to consider accidental loads and progressive collapse. However, further developments are necessary to deal with robustness and progressive collapse especially in view of contemporary

86

Chapter 5: Accidental actions

reliability acceptance criteria. The background of the Eurocodes with respect to accidental actions and robustness is reflected in this Chapter 5.

Figure 1: Collapse of the Murrah Federal Building in Oklahoma City (1995 in the USA)

Figure 2: Collapse of the World Trade Center towers (Sept. 11, 2001)

1.2 Background documents The topic of accidental actions is essentially covered by two Eurocodes, EN 1990: Basis of Structural Design [1], which provides the high level principles for achieving robustness and EN 1991-1-7 Eurocode 1: Part 1-7 Accidental Actions [2], which provides strategies, and methods to obtain robustness and the actions to consider. Most of the definitions given in the Eurocodes are based on ISO 2394 [3] and ISO 3898 [4].

2

EUROCODES APPROACH

The basic European document for structural design is the EN 1990. This Code indicates that sufficient structural reliability can be achieved by suitable measures including 87

Chapter 5: Accidental actions

ensuring an appropriate degree of structural integrity, i.e. structural robustness. In the EN 1991-1-7, 2006 robustness is defined as the ability of a structure to withstand events like fire, explosions, impact or the consequences of human error, without being damaged to an extent disproportionate to the original cause. In general, the Code following the aforementioned basic design philosophy states two strategies for the extraordinary design condition. The first strategy is based on identified extreme events (internal explosions, impact) and includes: a) design the structure to have sufficient robustness, b) prevent or reduce the action (protective measures), c) design the structure to sustain the action. The second strategy is based on the limiting extent of local failure, i.e: a) enhanced redundancy (alternative load paths), b) key element designed to sustain additional accidental load, c) prescriptive rules (integrity, ductility). For these strategies the Eurocode EN 1990 provides three failure effect categories for the design of structures under extraordinary events as illustrated in Table 1. They are compatible to the safety class differentiation presented in Chapters 2 and 3. Table 1: Definition of Consequence classes in EN 1990 Consequence Class CC1

CC2

CC3

Description

Examples of buildings and civil engineering works Low consequences for loss of Agricultural buildings, silos, human life, social and greenhouses environmental consequences small or negligible Medium consequences for Residential and office loss of human life, economic, building, public buildings social or environmental where consequences of consequences considerable failure are medium High Consequences for loss Grandstands, public of human life, economic, buildings, where social or environmental consequences of failure are consequences very great high

By consideration of these failure effect categories the design strategies lead to an adequate robustness of structures to minimize the extent of damage and failure without collapse. Thus the structure can withstand the effects of undefined extraordinary events. Thereby sets the code the minimum period of time that the structure must resist after such an event as the time necessary to safely evacuate persons from the damaged building and its surrounding area. For structures with dangerous goods, public affairs, or for reason of public security longer evacuation terms are required. Furthermore the Code [2] provides some constructional measures to obtain robustness in buildings in accordance to Section 3.1 of this Chapter. These measures are for example active vertical and horizontal traction anchors. For main structural members, that are capable of carrying an extraordinary action, the verification should be done under consideration of the effect for the main member and the adjacent components and their joints. For this case it is

88

Chapter 5: Accidental actions

thus necessary to consider the entire structure and not only the single member. The extraordinary design load according to EN 1990 should be applied as single load or uniformly distributed load. For structures in CC3 group a systematical risk assessment is required under consideration of predictable and unpredictable hazards. For this case an analytical model for damaged structures is recommended (see Fig.3).

Figure 3: Recommended limit of acceptable damage Legend (A) Local damage less than 15 % of floor area but not more than 100 m2 simultaneously in two adjacent floors (B) Column, deleted for analysis a) Floor plan b) Elevation with vertical section Also the Code provides rules for risk analysis, the qualitative and quantitative analysis. The three steps of the risk analysis are as based on the methodology of Section 2 as follows and consistent to the methodology reflected in Annex A of the Eurocodes and also in the Appendix A of this Chapter 5: (a) Assessment of the probability of occurrence of various hazards involving their intensity; (b) Assessment of the probability of various states of damage and of the associated consequences of failure for the given hazards; (c) Assessment of the probability for insufficient behavior of the damaged structure together with the associated consequences of failure. It is noted here that if the probability of occurrence of the accidental event is small (smaller than the target failure probability, i.e. for most cases smaller than 10-6 per year), steps (b) and (c) do not need to be performed. Methodological aspects regarding risk and robustness analysis can be found in Appendix A of this Chapter. Measures are also proposed to minimize the risk as shown below: a) Avoiding or decrease of the hazard; b) Avoidance of the hazard by changing the structural system or the use; c) Monitoring of the hazard; d) Overcoming of the hazard by enhanced strength and robustness, availability of alternate load paths by redundancies or resistance against abrasion, and so on;

89

Chapter 5: Accidental actions

e) Approval of controlled failure of the structure, if the hazards for human life are minimized, e.g. for impact on signs or pylons.

3

DESIGN FOR IMPACT AND EXPLOSION LOADS

The design philosophy necessitates that accidental actions are treated in a special manner with respect to load factors and load combinations. Partial load factors to be applied in analysis for identified accidental actions are defined in Eurocode, Basis of design, to be 1,0 for all loads (permanent, variable and accidental) with the following qualification in: "Combinations for accidental design situations either involve an explicit accidental action A (e.g. fire or impact) or refer to a situation after an accidental event (A = 0)". After an accidental event the structure will normally not have the required strength in persistent and transient design situations and will have to be strengthened for a possible continued application. In temporary phases there may be reasons for a relaxation of the requirements, e.g. by allowing wind or wave loads for shorter return periods to be applied in the analysis after an accidental event. 3.1

Impact from vehicles

In the case of hard impact, design values for the horizontal actions due to impact on vertical structural elements (e.g. columns, walls) in the vicinity of various types of internal or external roads may be obtained from Table 2. The forces Fdx and Fdy denote respectively the forces in the driving direction and perpendicular to it. There is no need to consider them simultaneously. The collision forces are supposed to act at height between 0,5 m to 1,5 m above the level of the carriegeway (0,5 m for cars). The force application area may be taken as 0,5 m (height) by 1,50 m (width) or the member width, whichever is the smallest. Table 2. Horizontal static equivalent design forces due to impact on supporting substructures of structures over roadways Type of road Motorway Country road Urban area Courtyards/garages Courtyards/garages

Type of vehicle Truck Truck Truck Passengers cars only Trucks

Force Fd,x [kN] 1000 750 500 50 150

Force Fd,y [kN] 500 375 250 25 75

In addition to the values in Table 2, the code specifies more advanced models for nonlinear and dynamic analysis in an informative annex. For impact force calculation the reader is referred to Annex B of this chapter. 3.2

Loads due to explosions

Explosions need to be taken into account in the design of all parts of the building and other civil engineering works where gas is applied. Key elements of a structure should be designed to withstand the effects of an internal natural gas explosion, using a nominal equivalent static pressure, given by: pd= 3 + pv

90

(1)

Chapter 5: Accidental actions

or

pd = 3 + 0,5 pv+0,04/(Av/V)2

(2)

whatever is the greater, where pv is the uniformly distributed static pressure in kN/m2 at which venting components will fail, Av is the area of venting components and V is the volume of the room. The venting components represents here the non-structural part of the enclosure (e.g. wall, floor, ceiling) with limited resistance that is intended to relieve the developing pressure from deflagration in order to reduce pressure on structural parts of the building. The explosive pressure acts effectively simultaneously on all of the bounding surfaces of the room. The expressions are valid for rooms up to a volume of 1000 m3 and venting areas over volume rations of 0,05 m-1 < Av / V < 0, 15 m-1. An important issue is further raised in Clause 5.2 of EN 1991-1-7 [2]. It states that the peak pressures in the main text may be considered as having a load duration of 0,2 s. The point is that in reality the peak will generally be larger, but the duration is shorter. So combining the loads from the above equations with a duration of 0,2 s seems to be a reasonable approximation. 3.3

Design example of a column in a building for an explosion

Consider a living compartment in a multi-storey flat building. Let the floor dimensions of the compartment be 8 x 14 m and let the height be 3 m. The two small walls (the facades) are made of glass and other light materials and can be considered as venting members. These walls have no load bearing function in the structure. The two long walls are concrete walls; these walls are responsible for carrying down the vertical loads as well as the lateral stability of the structure. This means that the volume V and the area of venting components Av for this case are given by: Av = 2 × 8 × 3 = 48 m2 V = 3 × 8 × 14 = 336 m3 So the parameter Av / V can be calculated as: Av / V = 48 / 336 = 0,144 m-1 As V is less then 1000 m3 and Av / V is well within the limits of 0,05 m-1 and 0,15 m-1 it is allowed to use the loads given in the code. The collapse pressure of the venting panels pv is estimated as 3 kN/m2. Note that these panels normally can resist the design wind load of 1,5 kN/m2. The equivalent static pressure for the internal natural gas explosion is given by: pEd = 3 +pv = 3 + 3 = 6 kN/m2 or

pEd = 3 + pv/2+0,04/(Av/V)2 =3 + 1,5 + 0,04 / 0,1442 = 3 + 1,5 + 2,0 = 6,5 kN/m2

This means that we have to deal with the latter. The load arrangement for the explosion pressures is presented in Figure 4. According to Eurocode EN 1990, Basis of design [1], these pressures have to be combined with the selfweight of the structure and the quasi-permanent values of the variables loads. Let us consider the design consequences for the various structural elements.

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Chapter 5: Accidental actions

H = 3m

pd

B=8m

Figure 4: Load arrangement for the explosion load

Bottom floor Let us start with the bottom floor of the compartment. Let the self-weight be 3 kN/m2 and the imposed load 2 kN/m2 (ψ1 = 0,5 for considered category A). This means that the design load for the explosion is given by: pda = pSW + pE + ψ1LL pLL = 3,00 + 6,50 + 0,5 ⋅ 2,00 = 10,50 kN/m² The design for normal conditions is given by: pd = γG ξ pSW + γQ pLL = 0,85 ⋅ 1,35 ⋅ 3,00 + 1,5 ⋅ 2,00 = 6,4 kN/m² We should keep in mind that for accidental actions there is no need to use a partial factor on the resistance side. So for comparison we could increase the design load for normal conditions by a factor of 1,2. The result could be conceived as the resistance of the structure against accidental actions, if it designed for normal loads only: pRd = 1,2 ⋅ 6,4 = 7,7 kN/m² So a floor designed for normal conditions only should be about 30 % too light. It is now time to remember the clause in Annex B of Eurocode EN 1991-1-7 [2]. If we take into account the increase in short duration of the load we may increase the load bearing capacity by a factor ϕd given by (see Annex 1):

ϕd = 1 +

pSW pRd

2umax g (Δt ) 2

where Δt = 0,2 s is the load duration, g = 10 m/s2 is the acceleration of the gravity field and umax is the design value for the midspan deflection at collapse. This value of course depends on the ductility properties of the floor slab and in particular of the connections with the rest of the structure. It is beyond the scope of this paper to discuss the details of that assessment, but assume that umax = 0,20 m is considered as being a defendable design value. In that case the resistance against explosion loading can be assessed as: pREd = ϕd pRd = [1 +

3 2 ⋅ 0,20 ] ⋅ 7,7 = 12,5 kN/m² 7,7 10 ⋅ (0,2) 2

So the bottom floor system is fulfilling the requirements. 92

Chapter 5: Accidental actions

Upper floor Let us next consider the upper floor. Note that the upper floor for one explosion could be the bottom floor for the next one. The design load for the explosion in that case is given by (upward value positive!): pda = pSW + pE + γQ ψ pLL = - 3,00 + 6,50 + 0 = 3,50 kN/m² So the load is only half the load on the bottom floor, but will give larger problems anyway. The point is that the load is in the opposite direction of the normal dead and live load. This means that the normal resistance may simply be close to zero. What we need is top reinforcement in the field and bottom reinforcement above the supports. The required resistance can be found by solving pRd from:

ϕd pRd = [1 +

pSW pRd

2umax g (Δt ) 2

] pRd = 3,50

Using again pSW = 3 kN/m2, Δt = 0,2 s, g = 10 m/s2 we arrive at pRd = 1,5 kN/m2. This would require about 25 % of the reinforcement for normal conditions on the opposite side. An important additional point to consider is the reaction force at the support. Note that the floor could be lifted from its supports, especially in the upper two stories of the building where the normal forces in the walls are small. In this respect edge walls are even more vulnerable. The uplifting may change the static system for one thing and lead to different load effects, but it may also lead to freestanding walls. We will come back to that in the next paragraph. If the floor to wall connection can resist the lift force, one should make sure that the also the wall itself is designed for it. Walls Finally we have to consider the walls. Assume the wall to be clamped in on both sides. The bending moment in the wall is then given by: m = 1/16 p H2 = 1/16 × 6,5 × 32 = 4 kNm/m If there is no normal force acting in the wall this would require a central reinforcement of about 0,1%. The corresponding bending capacity can be estimated as: mp = ω 0,4 d2 fy = 0,001 × 0,4 × 0,22 × 300000 = 5 kNm/m Normally, of course normal forces are present. Leaving detailed calculations as being out of the scope of this document, the following scheme looks realistic. If the explosion is on a top floor apartment and there is an adequate connection between roof slab and top wall, we will have a tensile force in the wall, requiring some additional reinforcement. In our example the tensile force would be (pE – 2 pSW) B/2 = (6,5 – 2 × 3) × 4 = 2 kN/m for a middle column and (pE – pSW) B/2 = (6,5 – 3) × 4 = 14 kN/m for an edge column. If the explosion is on the one but top story, we usually have no resulting axial force and the above mentioned reinforcement will do. Going further down, there will probably be a resulting axial compression force and the reinforcement could be diminished or even left out completely.

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Chapter 5: Accidental actions

4.

ROBUSTNESS OF BUILDINGS (Annex A of EN 1991-1-7)

4.1

Background

The rules drafted for Annex A of EN 1991-1-7 were developed from the UK Codes of Practice and regulatory requirements introduced in the early 70s following the partial collapse of a block of flats in east London caused by a gas explosion. The rules have changed little over the intervening years. They aim to provide a minimum level of building robustness as a means of safeguarding buildings against a disproportionate extent of collapse following local damage being sustained from an accidental event. The rules have proved to be satisfactory over the past 3 decades. Their efficacy was dramatically demonstrated during the IRA bomb attacks that occurred in the City of London in 1992 and 1993. Although the rules were not intended to safeguard buildings against terrorist attack, the damage sustained by those buildings close to the seat of the explosions that were designed to meet the regulatory requirement relating to disproportionate collapse was found to be far less compared with other buildings that were subjected to a similar level of abuse. The Annex A of the code also provides an example of how the consequences of building failure may be classified into “Consequences Classes” in accordance with Table 1 corresponding to reliability levels of robustness. Examples for the consequence classes are given below, by considering two subclasses for class 2: Consequences class class 1 class 2, lower group class 3, upper group class 4

Example structures low rise buildings where only few people are present most buildings up to 4 storeys most buildings up to 15 storeys high rise building, grandstands etc.

Proposals for quantification of robustness through an associated index are summarized in here in the Annex B of this Chapter 5. 4.2

Summary of design rules

In this section the rules will be summarised for every consequences class. Note that for consequence class 1 there are no special considerations except to ensure that the robustness and stability rules given in EN 1990 to EN 1999 are satisfied. For consequence class 2, depending upon the specific circumstances of the structure, a simplified analysis by static equivalent action models may be adopted or prescriptive design/detailing rules may be applied. For consequence class 3 a risk analysis is recommended and the use of refined methods such as dynamic analyses. The depth of the risk analysis is up to the local authorities. A distinction is made between framed structures and load-bearing wall construction. 4.2.1. Design Rules for Class 2, Lower Group, Framed structures: Horizontal ties should be provided around the perimeter of each floor (and roof) and internally in two right angle directions to tie the columns to the structure (Figure 5). Each tie, including its end connections, should be capable of sustaining the following force in [kN]: internal ties:

94

Ti = 0,8 (gk + ψ1 qk) s L (but > 75kN)

(3)

Chapter 5: Accidental actions

Tp = 0,4 (gk + ψ1qk) s L (but > 75kN).

perimeter ties:

(4)

In here gk and qk are the characteristic values in [kN/m2] of the self weight and imposed load respectively; Ψ is the combination factor, s [m] is the spacing of ties and L [m] is the span in the direction of the tie, both in m. Edge columns should be anchored with ties capable of sustaining a tensile load equal to 1 % of the vertical design load carried by the column at that level. 4.2.2

Rules for Class 2, Lower group, Load-bearing wall construction:

A cellular form of construction should be adopted to facilitate interaction of all components including an appropriate means of anchoring the floor to the walls.

internal ties

perimeter tie L

s Figure 5: Example of effective horizontal tying of a framed office building.

4.2.3

Rules for Class 2 - Upper Group, Framed structures: Horizontal ties as above; in addition one of the following measures:

–

–

–

Effective vertical ties: Columns and walls should be capable of resisting an accidental design tensile force equal to the largest design permanent and variable load reaction applied to the column from any storey. Ensuring that upon the notional removal of a supporting column, beam or any nominal section of load-bearing wall, the damage does not exceed 15% of the floor in each of 2 adjacent storeys. The nominal length of load-bearing wall construction referred to above should be taken as a length not exceeding 2,25H; for an external masonry, timber or steel stud wall, the length measured between vertical lateral supports. Key elements designed for an accidental design action Ad = 34 kN/m2.

4.2.4 Rules for Class 2 - Upper Group, Load-bearing wall construction. Rules for horizontal ties similar to those for framed buildings except that the design tensile load in the ties shall be as follows: For internal ties For perimeter ties

Ti =

Ft ( g k + ψ ⋅ qk ) z kN/m but > Ft 7,5 5 Tp = Ft

(5) (6)

95

Chapter 5: Accidental actions

Where Ft = (20 + 4 n) kNm with a maximum of 60 kNm, in which n represents the number of storeys; g, q and ψ1 have the same meaning as before, and z = 5 h or the length of the tie in [m], whichever is smallest. In vertical direction of the building the following expression is presented: 2

Tv =

For vertical tie

34 A ⎛ h ⎞ ⎜ ⎟ N 8000 ⎝ t ⎠

(7)

but at least 100 kN/m times the length of the wall. In this equation (7) A is the load bearing area of the wall, h is the storey height and t is the wall thickness. Load bearing wall construction may be considered to have effective vertical ties if (in the case of masonry) their thickness is at least 150 mm and the height of the wall h < 20 t, where t is wall thickness. 4.3

Example structures

4.3.1

Framed structure, Consequence class 2, Upper Group

Consider a 5 storey building with storey height h = 3,6 m. Let the span be L = 7,2 m and the span distance s = 6 m. The loads are qk = gk = 4 kN/m2 and ψ = 1,0. In that case the required internal tie force may be calculated as: Ti = 0,8 (4 + 4) (6 × 7,2) = 276 kN > 75 kN For steel quality FeB 500 this force corresponds to a steel area A = 550 mm2 or 2 ø18 mm. The perimeter tie is simply half the value. Note that in continuous beams this amount of reinforcement usually is already present as upper reinforcement anyway. For the vertical tying force we find: Tv = (4 + 4) (6 × 7,2) = 350 kN/column This corresponds to A = 700 mm2 or 3 ø18 mm. 4.3.2

Load bearing wall type of structure, Consequence class 2, Upper Group

For the same starting points we get Fb = min (60, 40) = 40 and z = L = 12 m, whichever is the smallest and from there for the internal and perimeter tie forces: Ti = 40

4 + 4 7.2 = 61 kN/m 7,5 5 Tp = 40 kN/m

The vertical tying force is given by: 2

34 ⋅ 0,2 ⎛ 3,6 ⎞ Tv = ⎜ ⎟ = 300 kN/m 8 ⎝ 0,2 ⎠ For many countries this may lead to more reinforcement then usual designed for this type of structural members.

96

Chapter 5: Accidental actions

5

CONCLUSIONS

Robustness and progressive collapse criteria have been reviewed in this contribution, especially those implemented in the Eurocodes. From the presented discussion and results the following conclusions can be drawn: 1. Robustness is a property of the structure and must be quantified in such a way, that it can be verified in practical design situations. 2. Performance objectives in damaged conditions must be specified in terms of a) degree of local damage and b) survival time and associated actions. 3. Rules and procedures in line of the aforementioned aspects are provided in the Eurocodes and illustrated by practical design examples.

REFERENCES [1] EN 1990 Eurocode - Basis of structural design. European Committee for Standardisation, 04/2002. [2] EN 1991-1-7 Eurocode1: Actions on structures – Part 1-1: General actions – Accidental actions, European Committee for Standardisation, 2006. [3] ISO 2394, General principles on reliability for structures. 1998. [4] ISO 3898, Bases for design of structures – Notations – General Symbols, 1997.

97

Chapter 5: Accidental actions

Appendix A: Methodology related to robustness assessment A.1: Conditional probability of collapse The methodology related to accidental actions and hazard scenarios is based on the assessment of the probability of progressive failure of a structure due to an accidental load. Progressive failure or here global failure follows a local damage or failure due to the accidental load. Consequently the problem of global failure due to an accidental load can be formulated in a probabilistic way by expressing the probability, P(C), of a progressive collapse, C, due to an event, E (see also for example Annex B in the Eurocodes EN 1991-1 [2] or in Ellingwood and Dusenberry, [A1]) as follows: P(C) = P(C|LE) P(L|E) P(E) Fy during a period of time Δt. The velocity of the mass achieved during this time interval is equal to: v = (F - Fy) Δt / m The corresponding kinetic energy of the mass is then equal to: E = 0,5 m v2 = 0,5 (F-Fy)2 Δt2 / m By equating this energy to the plastic energy dissipation, that is we put E = F Δu , we may find the increase in plastic deformation Δu. Δu = 0,5 (F - Fy)2 Δt2 / m Fy As mg = FSW we may also write: Δu = 0,5 (F-Fy)2 g Δt2 / FSW Fy Finally we may rewrite this formula in the following way: F = Fy (1 +

FSW FRd

2umax g (Δt )2

)

For the slab structure in section 5.1 we have replaced the forces F by the distributed loads p.

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Chapter 6: Combination rules in Eurocodes

CHAPTER 6: COMBINATION RULES IN EUROCODES Milan Holický1, Jana Marková1 1

Klokner Institute, Czech Technical University in Prague, Czech Republic

Summary Different combination rules for actions provided in EN 1990 are illustrated by examples of simple building structures. Partial factors and ψ factors for actions are used for the verification of the ultimate limit states including static equilibrium. For the verification of the serviceability limit states are considered constraints as recommended in material oriented Eurocodes. Resulting load effects based on alternative combination rules are presented as bending moment envelopes, in the case of static equilibrium also as shear forces. Comparison of obtained action effects indicates that alternative combination rules may lead to considerably diverse load effects. It appears that further investigation concerning alternative combination rules should take into account economic, commercial, societal and other aspects including laboriousness of design analysis.

1

INTRODUCTION

1.1

Background documents

EN 1990 [1] provides principles for design and verification of structures with regards to safety, serviceability and durability. The aim of this Chapter is to describe principles of structural design including load combination rules provided in EN 1990 [1]. Permanent loads, imposed loads and climatic actions due to wind and snow are considered in accordance with EN 1991-1-1 [2] and other parts of EN 1991. The alternative combination rules for ultimate limit states and serviceability limit states are compared using examples of simple structures. 1.2

General principles

For the selected design situations and identified limit states, critical load cases should be determined. In accordance with EN 1990 [1], a load case is a compatible load arrangement, sets of deformations and imperfections considered simultaneously with fixed variable actions and permanent actions. Document [1] is primarily based on the partial factor method, called also semi-probabilistic method. However, as an alternative, a design directly based on probabilistic methods is also allowed. Alternative combination rules for the ultimate limit states provided in EN 1990 [1] may lead to considerably diverse load effects. Decision concerning the choice of appropriate combination rules should take into account economic, commercial, societal and other aspects including laboriousness of design analysis.

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Chapter 6: Combination rules in Eurocodes

2

COMBINATION OF ACTIONS

2.1

General

To verify structural reliability, the design situations and relevant limit states shall be specified first. Then the load arrangements (the position, magnitude and direction) of free actions and the critical load cases (combination of compatible load arrangements) shall be determined. The critical load cases obviously depend on the type and location of structural shape (column, beam, slab) and on the overall configuration of structure. Assuming that a preliminary design of a structure is available (i.e. basic topology and structural materials are proposed) practical procedure to verify structural reliability (strength and serviceability) may follow four steps: 1) Selection of relevant design situations and limit states. 2) Determination of compatible load arrangements and critical load cases. 3) Calculation of the design values of action effects for relevant ultimate and serviceability limit states. 4) Verification of structural resistance (for specified reliability conditions). The detail procedure of the first three steps is illustrated by examples given in this contribution. The last fourth step (verification of structural resistance), which concerns material oriented Eurocodes EN 1992 to EN 1999, is not discussed here. 2.2

Combinations of actions in persistent and transient design situations Combinations of actions in persistent and transient design situations are based on : • •

design value of leading variable action, design value of accompanying variable action.

The fundamental combination of actions A for ultimate limit states (STR) is given in EN 1990 [1] by expression (6.10):

∑γ j ≥1

G, j

Gk , j " +" γ P P " +" γ Q ,1 Qk ,1" +"

∑γ i >1

ψ 0 ,i Qk ,i

Q ,i

(1)

Alternative combination B is composed of two expressions (6.10a), (6.10b) in EN 1990 [1]:

∑γ j ≥1

G,j

∑ξ γ j ≥1

Gk , j " +" γ P P " +" γ Q ,1 ψ 0 ,1 Qk ,1" +"

G,j

Gk , j " +" γ P P " +" γ Q ,1 Qk ,1" +"

∑γ i >1

∑γ j ≥1

ψ 0 ,i Qk ,i

(2)

ψ 0 ,i Qk ,i

(3)

Q ,i

Q ,i

Combination C is given in EN 1990 [1] by two expressions (6.10amod), (6.10b), thus, expression (3) and modified relationship (4), where only permanent loads are considered:

∑γ j ≥1

G, j

Gk , j

(4)

In expression (3) ξ denotes the reduction factor for adverse permanent actions G. Alternative combination of actions should be determined by CEN Member States in the National annex. Standard EN 1990 [1] gives no recommendations with respect to choice of one of these three

104

Chapter 6: Combination rules in Eurocodes alternatives. The following examples clearly show that resulting effects of actions, which were determined according to particular approaches, might considerably differ. Recommended values of partial factors for actions γ and reduction factors ψ are given in EN 1990 [1]. When the alternative combinations B or C are used, it may be useful to know which of the twin of expressions (6.10a), (6.10b) or (6.10amod), (6.10b) is decisive. For example when one permanent action G and two variable actions (e.g. imposed load Q and wind W) are considered only, the limit value of the load ratio χ = (Qk + Wk)/(Gk + Qk + Wk) may be determined [5,6]. Assuming that Gk, Qk and Wk denote the load effects of the characteristic actions (not the actions themselves) then the limit (boundary) value χlim for combinations B and C are given as

χ lim, B =

γ G (1 − ξ )(1 + k ) for combination B γ G (1 − ξ )(1 + k ) + γ Q (a − ψ Q ) + γ W k (b − ψ W )

(5)

γ G (1 − ξ )(1 + k ) for combination C γ G (1 − ξ )(1 + k ) + γ Q a + γ W kb

(6)

χ lim, C =

In equations (5) and (6) ξ denotes the reduction coefficient (usually ξ = 0,85) and k = Wk/Qk is the ratio between variable actions Wk and Qk. For the ratio k ≤ (1–ψQ)/(1–ψW) the auxiliary parameters a = 1 and b = ψW (action Q is leading) and for k > (1–ψQ)/(1–ψW) the parameters a = ψQ and b = 1 (wind W is leading). Relationships (5) or (6) may be used to determine, which of the twin of expressions (6.10a), (6.10b) or (6.10amod), (6.10b) is decisive: if χ < χlim then (6.10a) or (6.10amod) should be used, if χ > χlim then (6.10b) should be decisive. Application of equations (5) and (6) is shown in the example described in Section 3.2. 2.3

Combination of actions for accidental and seismic design situations

The load combination for verification of structure in accidental design situation may be symbolically written as (EN 1990, expression (6.12))

∑G j ≥1

k, j

"+" P "+" Ad "+" (ψ 1,1 or ψ 1, 2 ) ∑ψ 2,i Qk ,i

(7)

i >1

The choice between ψ1,1Qk,1 or ψ2,1Qk,1 depends on the type of accidental design situation (impact, fire or survival after an accidental event or situation). Further guidance may be found in the relevant Parts of EN 1991 to EN 1999. Combinations of actions for accidental design situations should either • •

involve an explicit accidental action A (fire or impact), or refer to a situation after an accidental event (A = 0).

For fire situations, apart from the temperature effect on the material properties, Ad should represent the design value of the indirect thermal action due to fire. The load combination for the verification of structure in seismic design situation may be symbolically expressed as

∑G j ≥1

k, j

"+" P "+" AEd "+" ∑ψ 2,i Qk ,i

(8)

i >1

where Aed is a seismic action arising due to earthquake ground motions.

105

Chapter 6: Combination rules in Eurocodes 2.4

Combination of actions for serviceability limit states

Combinations of actions that should be applied for verification of the serviceability limit states depend on a character of action effects. Three different types of load effects are recognised in EN 1990 [1]: irreversible, reversible and long-term effects. The corresponding load combinations are symbolically written as a) characteristic combination of actions (EN 1990, expression (6.14))

∑G

k, j

j ≥1

"+" Pk "+" Qk,1 "+"∑ψ 0,i Qk,i

(9)

i >1

normally used for verification of irreversible limit states; b) frequent combination (EN 1990 (expression (6.15))

∑G

k, j

j ≥1

"+" Pk "+" ψ 1,1Qk,1"+"∑ψ 2,i Qk,i

(10)

i >1

normally used for verification of reversible limit states; c) quasi-permanent combination (EN 1990 (expression (6.16))

∑G

k, j

j ≥1

"+" Pk "+" ∑ψ 2,i Qk,i

(11)

i ≥1

normally used for verification of long term effects, and appearance of the structure, e.g. when creep of concrete is considered. In accordance with Annex A1 of EN 1990 [1] all partial factors for serviceability limit states are equal to unity. The above mentioned load combinations differ by diverse use of ψ0, ψ1 and ψ2 factors. For example ψ0 is applied to reduce non-leading actions in the characteristic combinations, ψ1 and ψ2 are applied in the frequent combinations and ψ2 is used in the quasipermanent combinations. Note that depending on the verified structural property (deflection, crack width) and number of independent actions, each load combination may lead to several load cases. Following examples (analysed using the software Amses) show practical applications of the above described combination rules. 3

EXAMPLES

3.1

Cantilevered beam

Geometry and material. Cantilevered beam considered in the first example is indicated in Figure 1. The reinforced concrete beam having cross-section 0,30×0,40 m (width × depth) is made of concrete C20/25. q1

g1

(a)

g2

(b)

(c)

A

q2

B l1 = 4,5 m

l2 = 3,0 m

Figure 1. Cantilevered beam.

106

G

(d)

g1 = 15 kN/m g2 = 15 kN/m q1 = 9 kN/m q2 = 9 kN/m G = 6 kN

Chapter 6: Combination rules in Eurocodes

Uniform permanent load of the beam g1 and cantilever g2 (assumed to be from one source or independent), concentrated permanent load G, and imposed loads q1 and q2 (Category B - office areas) are considered only. The quantities indicated in Figure 1 denote the characteristic values (in order to simplify notation the subscript "k" is left out). Whether the permanent actions g1 and g2 are from one source or not (i.e. they are dependent or independent) should be verified considering particular conditions of the structure (weight of structural and nonstructural components acting on both parts of the beam). Nevertheless, it will be shown that mutual independence of g1 and g2 is a safe assumption that leads to a considerably greater bending moment in midspan point (c) than the assumption that g1 and g2 are from one source. Ultimate limit states (static equilibrium EQU and limit state of rupture STR), and serviceability limit states (characteristic, frequent and quasi-permanent combinations) are to be verified. Table 1 shows the critical load cases and appropriate factors (γG, γQ, γQ ×ψ or ξ × γG) assuming γG = 1,35, γQ =1,50, ψ = 0,70 (for considered office areas) and ξ = 0,85 relevant to verification of the equilibrium, bending resistance (shear is not considered) and deflection of the beam. If the permanent loads g1 and g2 could be considered as being from one source, then the factors of both actions would be the same as indicated in Table 1 by the values in brackets (when these are different from the case of independent permanent actions). Note that the assumption of g1 and g2 being from one source (and both q1 and q2 acting) would lead to the maximum shear forces at point (b) (not shown here). Table 1. Load cases and factors γG, γQ, γQ×ψ or ξ×γG corresponding to relevant expressions in EN 1990 [1] indicated in brackets, if g1 and g2 are actions from one source then factors in brackets should be applied. Load Bending Limit state FactorsγG, γQ, γQ ×ψ or ξ ×γG assuming γG = 1,35, case moment γQ =1,50, ψ = 0,70 and ξ = 0,85 for actions in *) g1 g2 q1 q2 G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

(c) (b) (c) (c) (b) (b) (c) (b) -

Equilibrium, exp. (6.7), (6.10) Equilibrium, exp. (6.7), (6.10) Equilibrium, exp. (6.7), (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10amod) Ultimate, exp. (6.10amod) Serviceability, exp. (6.14) Serviceability, exp. (6.14) Serviceability, exp. (6.15) Serviceability, exp. (6.15) Serviceability, exp. (6.16) Serviceability, exp. (6.16)

0,90 1,15 1,00 1,35 1,00 (1,35) 1,35 1,15 1,00 (1,35) 1,00 (1,15) 1,35 1,00 (1,35) 1,00 1,00 1,00 1,00 1,00 1,00

1,10 1,35 1,00 1,00 (1,35) 1,35 1,00 (1,35) 1,00 (1,15) 1,35 1,15 1,00 (1,35) 1,35 1,00 1,00 1,00 1,00 1,00 1,00

1,10 1,50 0 1,35 1,50 0 1,00 1,50 0 1,50 0 1,00 0 1,50 1,35 0 1,00 1,50×0,7 1,50 0 1,00 0 1,50×0,7 1,35 1,15 1,50 0 1,00 0 0 1,35 0 0 1,00 0 1,00 0 1,00 1,00 0 1,00 1,00×0,5 0 1,00×0,5 1,00 0 1,00 1,00×0,3 0 1,00×0,3 1,00

Note: *) Only the load cases 4 to 11 are directly related to a bending moment in a particular point (and its vicinity) of the beam.

107

Chapter 6: Combination rules in Eurocodes Load effects. If the support (a) of the beam shown in Figure 1 can not transmit tensile forces, static equilibrium EQU of the beam should be checked using equation (3) (expression (6.7) in EN 1990 [1]). In accordance with this equation the following condition should be satisfied

γg1 g1 l12/2 > γg2 g2 l22/2 +γq2 q2 l22/2 + γG G l2 As already indicated in Section 2, two alternative sets of partial factors are provided in EN 1990, Annex A1 [1], Table A1.2(A) as illustrated below. Both these sets are independent of the assumption concerning dependency of the permanent actions g1 and g2. Thus, the sets of partial factors provided in Table A1.2(A) do not distiquish between the case when g1 and g2 are from one source and the case when g1 and g2 should be considered as independent. In the load case 1 (see Table 1) factors 0,9 for favourable and 1,1 for unfavourable permanent actions are considered (as indicated in Note 1 in Table A1.2(A), Annex A1 of EN 1990 [1]). In the load case 2 (see Table 1) factors 1,15 and 1,35 are used (in accordance with Note 2 in Table A1.2(A), Annex A1 of EN 1990 [1]) provided that applying γG = 1 to both the favourable and the unfavourable parts of permanent actions do not give a more unfavourable load effect (verified in this example 1 by the load case 3, see Table 1). Figure 2 shows results obtained for the ultimate limit states EQU. It appears that the cantilevered beam should be provided by an anchor at the point (a). The load case 1 seems to be more severe (tensile force 4,03 kN) than load case 2 (tensile force 0,34 kN, not indicated in Figure 2). Note that the load case 3 leads to more favourable effect than the cases 1 and 2 (compressive force 1,25 kN, indicated in Figure 2). Thus, in the alternative approach indicated in the Note 2 in Table A1.2(A), Annex A1 of EN 1990 [1], the load case 2 is decisive. -77.97 -64.78

1

2

-4.03

1 1.25

Load case 1

3 2

6 8.1

Load case 3

91.5 109.4

Figure 2. Shear forces [kN] according to expression (6.7) in EN 1990 [1] for equilibrium verification (load cases 1and 3).

108

Chapter 6: Combination rules in Eurocodes From the comparison of bending moments for ultimate limit states STR (Figures 3 and 4) it follows that the assumption of independent g1 and g2 leads to a considerably greater positive moments (negative moments are not affected) than the assumption of g1 and g2 being from one source (for the combination A by more than 20 %, see Figures 3a and 3b). Assuming the independent g1 and g2, Figures 3b and 4b indicate that the positive moments for the combination A (Figure 3b) are about 18 % greater than those for the combination B or C (Figure 4b). The difference between the negative moments of the combinations A and B in point (b) is about 11 %. Load combinations B and C are in this example identical because expression (6.10b) in EN 1990 [1] is decisive in both cases while expressions (6.10a) and (6.10amod) are not effective.

1

-176.2 -176.2

-159.1 -159.1

-109.1 -109.1

-95.63 -95.62

2

1

3

1

2

3

1

2

2

36.69

39.02

Figure 3a. Bending moment envelopes [kNm] according to expression (6.10) in EN 1990 [1] assuming g1, g2 being from one source.

Figure 4a. Bending moment envelopes [kNm] according to exp. (6.10a), (6.10b) and (6.10amod), (6.10b) in EN 1990 [1] assuming g1, g2 being from one source.

-176.2

-159.1

-85.5

-85.5

1

2 1

3 2

47.81

Figure 3b. Bending moment envelopes [kNm] according to expression (6.10) in EN 1990 [1] assuming g1, g2 independent.

1

2 1

3 2

40.52

Figure 4b. Bending moment envelopes [kNm] according to exp. (6.10a), (6.10b) and (6.10amod), (6.10b) in EN 1990 [1] assuming g1, g2 independent. 109

Chapter 6: Combination rules in Eurocodes Deflections. Three combinations (called in EN 1990 [1] characteristic, frequent and quasi-permanent) of serviceability limit states are considered in Table 1. The characteristic load combination is described in EN 1990 [1] by expression (14) (load cases 12 and 13), the frequent combination is described in EN 1990 [1] by equation (15) (load cases 14 and 15), the quasi-permanent combination described in EN 1990 [1] by expression (16) (load cases 16 and 17). Deflection lines and the extreme deflections at a midspan point (c) and at the end point (d) due to characteristic and quasi-permanent load combinations are shown in Figure 5. Deflection lines were determined assuming the modulus of elasticity 29 GPa and creep coefficient 2,5 (in case of quasi-permanent load cases 16 and 17). 1 1

2

2

3

1

3

1

2

- 2,8 - 1,4

2

- 1,3 - 0,1

20,6 26,6

4,8 10,5

Figure 5. Deflection lines [mm] corresponding to the characteristic load cases 12 and 13 (left) and quasi-permanent cases 16 and 17 (right). Figure 5 indicates that the deflection at the cantilever end (d) may violate criteria for structural performance. If, for example, the cantilever supports a brittle cladding components, cracks and other performance deficiencies may occur. Note that slightly lower deflection, as that due to characteristic combination, were obtained for the frequent combination described in EN 1990 [1] by equation (15) and covered by load cases 14 and 15 (see Table 1). 3.2

Continuous beam of three spans

Geometry and material. A three span continuous beam of the cross-section 0,25×0,40 m made of concrete C 20/25 (modulus of elasticity 29 GPa) is loaded by permanent g (a single action of one origin) and imposed load q as indicated in Figure 6.

q1

q2

g

1

q3

2

3

1

(e)

(a) 5m

4

2

(b)

(f) 5m

3

(c)

(g)

(d)

g = 30 kN/m q1 = 18 kN/m q2 = 18 kN/m q3 = 18 kN/m

5m

Figure 6. Continuous beam. Load cases. The uniform permanent action g (considered as a single action from one source for the whole beam) and three independent imposed actions q1, q2 and q3 are considered. In the example the load effects needed for verification of ultimate and serviceability limit states (characteristic and quasi-permanent combinations) are analysed. The quantities indicated in Figure 6 denote the characteristic values (similarly as in example 1 the subscript "k" is omitted).

110

Chapter 6: Combination rules in Eurocodes Ultimate limit states (of the type STR) verified using general expression (6.8) and load combinations (6.10) given in EN 1990 [1] is checked using the total of seventeen load cases, for which appropriate factors γ are indicated in Table 2. Equations (5) and (6) may be used to identify the decisive expression in load combinations B and C introduced above. For the case of one variable action (imposed load Q only) the relationships (5) and (6) may be simplified as follows:

γ G (1 − ξ ) γ G (1 − ξ ) + γ Q (a − ψ Q ) γ G (1 − ξ ) χ lim, C = γ G (1 − ξ ) + γ Q a

χ lim, B =

Here a = 1 (the auxiliary quantity); γG = 1,35; γG = 1,5; ψQ = 0,7;ξ = 0,85. The load ratio

χ becomes

χ= Qk/(Gk+Qk)

where Qk and Gk denote action effects due to the characterstic values of permanent and variable actions g and q. The following criteria apply for application of twin expressions (6.10a) and (6.10b) in EN 1990 [1]: if χ < χlim,B or χ < χlim,C, then expression (6.10a) or (6.10amod) is to be used, if χ > χlim,B or χ > χlim,C, then expression (6.10b) is decisive.

• •

Evaluation of these criteria is shown in Table 3. Table 2. Load cases and factors γQ × ψ or ξ × γG for continuous beam of three spans, expressions given in EN 1990 are indicated in brackets. Load Bending Limit state Factors γQ × ψ or ξ × γG case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

moment in point *) (e) (f) (b) (b) (b) (e) (e) (f) (f) -

Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10amod) Serviceability (6.14) Serviceability (6.14) Serviceability (6.16) Serviceability (6.16)

g 1,35 1,35 1,35 1,35 1,35 0,85×1,35 1,35 0,85×1,35 1,35 0,85×1,35 1,35 0,85×1,35 1,35 1,00 1,00 1,00 1,00

for actions q2 q1 1,50 1,50 1,50 1,50 1,50 1,50 0,7×1,50 0,7×1,50 1,50 1,50 0,7×1,50 1,50 0,7×1,50 0,7×1,50 1,50 1,50 0,7×1,50 1,50 1,00 1,00 0,3×1,00 0,3×1,00

q3 1,50 1,50 0,7×1,50 1,50 0,7×1,50 1,50 1,00 0,3×1,00 -

Note: *) Only some load cases are directly related to a bending moment in a particular point (and its vicinity) of the beam.

111

Chapter 6: Combination rules in Eurocodes

Table 3. The limit values χlim,B and χlim,C for load combinations B and C. Decisive χlim,B χlim,C χ

M in point: Moment due to Moment due (see Fig. 6.6) Gk [kNm] to Qk [kNm] B 75 52,5 E 60 42 F 18,75 33,75

0,412 0,412 0,643

eq. (10) 0,31 0,31 0,31

eq. (11) 0,119 0,119 0,119

expression (6.10b) (6.10b) (6.10b)

Thus in the alternative combination rules B and C expression (6.10b) in EN 1990 [1] is decisive. Note that the ratio of the characteristic values of the original actions q/(g+q) = 0,375 is considerably different from the ratios of the load effects indicated in Table 3. Bending moments. The resulting bending moments of the beam are shown in Figures 7 and 8. -180 -135

-168.8 -123.8

-8.438

67.5

67.5

75.94 148.5

148.5

Figure 7. Bending moment envelope [kNm] according to combination A (expression (6.10) in EN 1990 [1]).

-164.9 -164.8 -119.8

-153.6 -108.6

-12.22

55.38

55.38 72.15 136.4

136.4

Figure 8. Bending moment envelope [kNm] according to combination B and C (expressions (6.10b) in EN 1990 [1]).

112

Chapter 6: Combination rules in Eurocodes The results of analysis considering ultimate limit states STR indicate that the internal moment according to the combination A (Figure 7) is in points (e) and (b) (see Figure 6) greater about 11 % than according to the combination B (Figure 8), resp. C (Figure 9). The numerical values in point (f) are for the combination A greater about 5 % than according to the combinations B and C. The combinations B and C are also equal in this case, the expressions (6.10a) and (6.10amod) are not expressed in envelope. The results of analysis taking into account serviceability limit states are shown in Figures 9 and 10. Considering characteristic load combination, the deflection lines corresponding to the load cases 14 and 15 (indicated in Table 2) are shown in Figure 9. Both deflection lines were determined for the modulus of elasticity 29 GPa. 1

.2,5

2 1

2,5

-1,6

3

2

4 3

2,2 6,2

6,2

Figure 9. Deflection lines [mm] corresponding to the load cases 14 and 15 specified in Table 2. Considering quasi-permanent load cases 16 and 17 (see Table 2) the extreme deflection lines are indicated in Figure 10. Both deflection lines were determined for the modulus of elasticity 29 GPa and creep coefficient 2,5. 1

-1,7

2 1

2

3

4 3

10,7

10,7 3,0 14,5

14,5

Figure 10. Quasi-permanent deflection lines [mm] corresponding to the load cases 16 and 17 specified in Table 2.

Note that the maximum deflection 14,5 mm is about L/340 (where L is the length of one span of the beam), which seems to be quite satisfactory (serviceability constraint L/250 is normally considered as sufficient). However, in some cases a more detailed analysis of the deflection may be required taking into account specific conditions (type of reinforcements, creep, performance requirements). 3.3

Cantilevered frame

Geometry and material properties. The cantilevered frame indicated in Figure 11 is exposed to five independent actions: permanent load g, imposed loads q1 and q2 (Figure 12) and climatic actions due to wind W and snow s (Figure 13). It is assumed that the identical frame is located every 6 m along the longitudinal direction of a building. The total height of the frame is 15 m, foundations are 3 m below the terrain, and the top of the frame is 12 m above the terrain. In a preliminary design of the frame two types of cross-sections are considered: • •

columns in the first storey, middle columns in the second to fourth storey, and all beams 0,30 × 0,60 m, edge columns of the second to fourth storey 0,30 × 0,30 m. 113

Chapter 6: Combination rules in Eurocodes

The frame is made of concrete C 20/25 (modulus of elasticity 29 GPa). A creep coefficient 2,5 is considered when determining long term deflection under quasi-permanent load combination. The ultimate limit state of structural resistance (STR) and serviceability limit states (characteristic and quasi permanent combination) shall be verified. Note that other actions (imposed load in cantilevered part of the frame only) may be needed when limit state of static equilibrium (EQU) of the frame should be verified (in the considered frame in Figure 11 the limit state of static equilibrium EQU is obviously satisfied). Load cases. The characteristic value of permanent load g imposed on beams is determined assuming equivalent thickness of the floor slab 0,20 m (representing the slab about 0,16 m, beams, floor and other permanent loads). Thus, for loading width of 6 m the characteristic value of the uniform load of the beam is gk = 0,20 × 25 × 6 = 30 kN/m Note that possible reduction factors αA and αn, which may be used when designing particular structural elements to reduce imposed load, are not considered here (their effect in this simple example is insignificant). The characteristic value qk of imposed load for office areas (3 kN/m2) and loading width of 6 m is qk = 3 × 6 =18 kN/m The characteristic value of wind load is derived assuming the wind speed v = 26 m/s, thus the reference pressure is qref = 1,25×v2/2 = 1,25×262/2 = 422,5 N/m2 In addition the following parameters are assumed: the exposure coefficient Ce = 2,5 (corresponding to the height of the structure 12 m above the terrain of category II), the external pressure coefficient cpe,10 = 0,8 on the pressure side and the factor cpe,10 = −0,3 on the suction side. Thus, for the loading width 6 m and height 3 m (one storey) we get the following pressure force Wkp and suction force Wks acting at the frame nodes as indicated in Figure 13: Wkp = 0,4225 × 2,5 × 0,8 × 6 × 3 ≅ 15,2 kN Load effects. In the following text load effects for the verification of ultimate limit states (STR) and serviceability limit states (characteristic and quasi-permanent combination of actions) are analysed. The total of 16 load cases, indicated in Table 4, are considered. It should be noted that additional load cases might be needed for the verification of the ultimate limit state of equilibrium (EQU), which are not considered here (the imposed load should be considered in the cantilevered part of the frame). Table 4. Load cases and appropriate factors γQ × ψ or ξ × γG , expressions given in EN 1990 are indicated in brackets. Load Limit state Factors γQ × ψ or ξ × γG for actions case g q1 q2 W s 1 2 3 4 5 6

114

Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b)

1,35 1,35 1,35 1,35 1,35 0,85×1,35

1,50 1,50 0,7×1,50 0,7×1,50 1,50

1,50 1,50 0,7×1,50 -

0,6×1,5 0,6×1,5 0,6×1,5 1,50 0,6×1,5 0,6×1,5

0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50

Chapter 6: Combination rules in Eurocodes Load case

Limit state

7 8 9 10 11 12 13 14 15 16

Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10b) Ultimate, exp. (6.10amod) Serviceability, exp. (6.14) Serviceability, exp. (6.14) Serviceability, exp. (6.16) Serviceability, exp. (6.16)

g 1,35 0,85×1,35 1,35 0,85×1,35 0,85×1,35 1,35 1,00 1,00 1,00 1,00

Factors γQ × ψ or ξ × γG for actions q1 q2 W 0,7×1,50 1,50 0,7×1,50 1,00 0,3×1,00 -

0,7×1,50 1,50 0,7×1,50 1,50 0,7×1,50 1,00 0,3×1,00

0,6×1,5 0,6×1,5 0,6×1,5 0,6×1,5 1,50 0,6×1,00 0,6×1,00 -

s 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,00 0,5×1,00 -

The resulting bending moments envelopes determined using load combination A, B and C are shown in Figures14 to 16. To achieve better legibility of these figures numerical values of bending moments are indicated for all horizontal beams, and for columns in the lowest floor only. It follows from Figures 14 to 16 that the bending moments obtained in some crosssections from combination A (Figure 14) are greater (up to 15 %) than those obtained from combination B (Figure 15) or combination C (Figure 16). It is interesting to note that in this example the bending moments corresponding to combinations B and C are almost identical. The only exception is the upper horizontal member where the extremes corresponding to combination B (Figure 15) are slightly greater than those corresponding to combination C (Figure 16). This difference is due to the following reason: in the case of combination B, expression (6.10a) in EN 1990 [1] is decisive (load case 9), while in the case of combination C, expression (6.10b) is decisive (load case 11). However, in the remaining members of the frame combinations B and C lead to the same bending moments given by load cases 6, 8, 10 and 11, all of them corresponding to expression (6.10b) in EN 1990 [1].

3m

3m

3m

6m

6m

3m

Figure 11. Cantilevered frame - permanent load g. 115

Chapter 6: Combination rules in Eurocodes

Imposed load q1

Imposed load q2

Figure 12. Cantilevered frame – imposed load q

0,3×0,3 m

0,3×0,6 m

Wind actions W

Snow loads s

Figure 13. Cantilevered frame – climatic actions 116

Chapter 6: Combination rules in Eurocodes

-305.2 15.49

-39

-399.3 96.11

35.26

63.05

-435.4 141.4

47.29

53.44

-504.2 147.6 137.5 28.5

157.2

156.9 196

-264.3

-274.4

Figure 14. Bending moment envelope [kNm] for combination A (expression (6.10) in EN 1990 [1]). -284.1

17.22

35.8

-367.1

96.01

37.17

58.06 -400.7 129.4 41.66

54

-468.5 135.5 136.9

24.73

156.7

154.1 185.2

-256.1

-265.2

Figure 15. Bending moment envelope [kNm] for combination B (expressions (6.10a),(6.10b) in EN 1990 [1]).

117

Chapter 6: Combination rules in Eurocodes

-274.6 17.22

-35.56

-367.1 85.12

37.17

56.89

-400.7 129.4 54

41.66

-468.5 135.5 136.9

156.7

154.1

24.73

185.2

-256.1

-265.2

Figure 16. Bending moment envelope [kNm] for combination C (exp. (6.10amod),(6.10b) in EN 1990 [1]).

3.4

Three bay two-dimensional frame

Geometry. A simple reinforced concrete frame analysed below is shown in Figure 17. It is a typical transverse frame located every 5 m in the longitudinal direction of a building. The total width and height of the frame is 15 m, foundations are 3 m below the terrain, and the top of the frame is 12 m above the terrain. In a preliminary design of the frame three types of cross-sections are considered: - external columns 0,50 × 0,25 m, - internal columns 0,25 × 0,25 m, - beams 0,40 × 0,25 m. Load cases. Similarly as in the case of cantilevered frame five independent actions are considered in this example: - permanent load g, - imposed load q1, - imposed load q2, - wind load W, - snow load s.

118

Chapter 6: Combination rules in Eurocodes

3m 0,40×0,25 m

3m

3m 3m

0,50× 0,25 m

0,50× 0,25 m

0,25× 0,25 m

3m

5m

5m

5m

Figure 17. Two dimensional frame. The characteristic values of actions shown in Figure 18 are determined similarly as in previous example of cantilevered frame assuming that identical transverse frames are located every 5 m along the structure. Permanent load g is considered as one action only. Note that the self-weight of beams is included in the uniform vertical load of beams. The self-weight of columns is introduced explicitly as uniform vertical load along the columns taking into account their dimensions. The total of 12 load cases are considered in the verification of the ultimate limit state of structural resistance (STR), and serviceability limit states (characteristic and quasipermanent combinations) of the frame. Load cases and appropriate partial factors are summarised in Table 5, which is similar to Table 4 in previous example of cantilevered frame. Table 5. Load cases and appropriate factors (γ × ψ). Load case 1 2 3 4 5 6 7 8 9 10 11 12

Limit state Ultimate, eq. (6.10) Ultimate, eq. (6.10) Ultimate, eq. (6.10) Ultimate, eq. (6.10) Ultimate, eq. (6.10) Ultimate, eq. (6.10) Ultimate, eq. (6.10) Ultimate, eq. (6.10) Serviceability, eq. (6.14) Serviceability, eq. (6.14) Serviceability, eq. (6.17) Serviceability, eq. (6.17)

g 1,35 1,35 1,35 1,35 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

Factors γQ × ψ or ξ × γG for actions q1 q2 W 1,50 1,5×0,6 1,50 1,5×0,6 1,50 1,50 1,5×0,6 1,5 1,5×0,7 1,5×0,7 1,50 1,5×0,6 1,50 1,5×0,6 1,50 1,50 1,5×0,6 1,5 1,5×0,7 1,5×0,7 1,00 1,0×0,6 1,00 1,0×0,6 0,30 0,30 -

s 1,5×0,6 1,5×0,6 1,5×0,6 1,5×0,6 1,5×0,6 1,5×0,6 1,5×0,6 1,5×0,6 1,0×0,5 1,0×0,5 -

119

Chapter 6: Combination rules in Eurocodes

17

19

25

27

20

1

1

2

3

3

15

5

23

4

-3.125 k N /m 8

6

3

3

4

5

7

(2) Imposed load q1

6

5

22 18 29 15 k N /m

2

1

(1) Permanent action g 2

27 21 28

1

16 14 21

25 20 26 15 k N /m

7

19 16 20 15 k N /m

19

9 10 14 15 k N /m

17 15 18

4

12 12 13

13

29 30 k N /m

8 7

10 11 11 15 k N /m

18

28 30 k N /m

-1.623 k N /m

26 30 k N /m

8

22

21

-1.623 k N /m

19

21 30 k N /m

6 6

9

14

-3.125 k N /m

23

20 30 k N /m

9

16

16

-1.623 k N /m

15 18 30 k N /m

14 30 k N /m

4 5

10

-1.623 k N /m

13 30 k N /m

2

-3.125 k N /m

12

-1.623 k N /m

-3.125 k N /m

15

13

-3.125 k N /m

12

11 11 30 k N /m

8 7 30 k N /m

-3.125 k N /m

10

9

-3.125 k N /m

-1.623 k N /m

8

6 6 30 k N /m

-1.623 k N /m

4 5 30 k N /m

-1.623 k N /m

-3.125 k N /m

2

2

7

4

6.3 kN

6

5

8

6

7 2.4 kN

8

10

9

12

11

9

12

11

8

10

13 15 k N /m

10

9

14

12

11 11

12.7 kN

9

12

10

13

14 4.8 kN

15

17

13

19

15

16

16

18 15 k N /m

15

14

20

17

13

21 15 k N /m

19

15 18

12.7 kN

16

16

14

20

21 4.8 kN

23

25

19

27

20

22

21

26

23

18

28 15 k N /m

25

19

29

27

20

19 kN

22

21

26

18

28

29 7.2 kN

1

1

2

3

3

4

5

1

7

1

(3) Imposed load q2

3

(4) Wind action W

2

4

6

5 6 k N /m

7 6 k N /m

10

12

11

14

17

19

15

14

20

23

21

25

27

20

18

28

29

2

3

22

21

26

1

16

16

18

19

10

13

15

9

12

11

13

8

6 6 k N /m

8 9

1

2

3

5

4

7

(5) Snow load s Figure 18. Actions considered for two-dimensional frame. 120

3

5

4

7

Chapter 6: Combination rules in Eurocodes

2

4

6

5

8

6

7

2

4

6

5 8

10

9

12

11

8

10

13

14

10

9

12

11

17

13

19

15

13

15

14

20

21

25

19

27

20

19

15

29

2

19

27

20

3

3

22

21

18

28

29

4 1

1

21

25

26

1

14

20

23

132,5 kNm

18

28

16

16

22

21

26

14

17

13 18

23

10

16

16

18

9

12

11 15

7

9

12

11

8

6

5

2

3

4

7 1

3

5

7

671 kN

(1) Bending moment envelope due to load cases 1, 2, 3 and 4. 2

4

6

5

8

15

23

14

21

18

28

1

2

3

5

4

7

(3) Deflection due to load cases 9 and 10.

14

1

14 21

25

27

20

18 29

2

3

22

21 28

1

16

16 20

23

9 10

19

15

26

3

12

17

19

29

10,8 mm 13

15

22

8 7

12

18

27

20 26

10 11

13

21

25

6 6

8 9

16

16 20

4

11

19

15 18

19

10 14

17

13

9

12 13

2 5

12

11 11

1

7

10

9

11,8 mm

8

6

(2) Axial forces corresponding to the bending moment envelope.

3

5

4

7

(4) Deflection due to load cases 11 and 12.

Figure 19. Load effects.

Load effects. Resulting load effects determined for the load cases indicated in Table 5 are indicated in Figure 19. Similarly as in previous example of cantilevered frame the load cases 5 to 8 differ from cases 1 to 4 only by the coefficient γg of permanent load g, which is considered as one independent action for the whole frame (an action from one source). The coefficient γg = 1,00 is applicable in those cases when effect of permanent load is favourable, for example when considering edge columns exposed to combination of bending moment and axial force (lower axial force may be unfavourable). It should be also mentioned that Table 5 includes the most typical actions and load cases only. Depending on actual structural geometry, nature of permanent and imposed load (for example of additional snow or imposed load on the roof), other independent actions and their combinations may be required in order to verify all possible limit states.

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Chapter 6: Combination rules in Eurocodes 4

CONCLUDING REMARKS

EN 1990 provides two sets of partial factors for the ultimate limit states of static equilibrium EQU regardless the origin of permament actions (whether they are from one source or not). The obtained results obtained indicate that the partial factors 0,9 for favourable and 1,1 for unfavourable permanent actions lead to more severe load effects than the alternative set of partial factors 1,15 and 1,35. The examples of selected structural members verified for the limit states of rupture STR indicate that the combination rule A (corresponding to expression (6.10) in EN 1990) is easier to apply than the combinations B and C (corresponding to twin expressions (6.10a), (6.10b) and (6.10amod), (6.10b) in EN 1990). However, selected examples show that the design procedure A leads to considerable greater load effects (up to 18 % greater) than the procedures B and C. Thus, the combination rule A will ensure greater reliability of structures than the combinations B and C. Nevertheless, the design procedure A would increase the material consumption compared with the procedures B and C and, therefore, would unfavourably affect the initial costs of structures and the overall economy of the building. On the other hand the application of the combination rules B and C might be more complicated than the use of the combination A. It appears that decisions concerning the recommendation on combination of actions, which have been prepared by national authorities (and should be provided in National annexes to EN 1990), represent demanding tasks. Obviously, in addition to structural reliability several other aspects should be taken into account as well. For example, due attention shoud be payed to economical, ecological, and societal consequences. In addition laboriousness, time consumption, and transparency of design analysis should be considered. It is expected that within next few years the selected National Determined Parameters (NDPs) including alternative combinations of actions and values of partial factors and other reliability elements will be analysed in co-operation of JRC, CEN/TC 250 and CEN Member States focused on the reduction of the parameters and on further harmonisation.

REFERENCES [1] EN 1990 Eurocode - Basis of structural design. European Comittee for Standardisation, 04/2002. [2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. European Comittee for Standardisation, 04/2002. [3] ISO 2394, General principles on reliability for structures. 1998. [4] Probabilistic Model Code, Parts 1 to 4, Basis of design, Load and resistance models, Examples, JCSS, 2001-2002. [5] Gulvanessian, H., Calgaro, J.-A., Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, 192 pp. [6] Holický M., Marková J: Reliability of Concrete Elements Designed for Alternative Load Combinations Provided in Eurocodes, Prague, Acta Polytechnica 2003/1.

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Chapter 6: Combination rules in Eurocodes Appendix A to Chapter 6: Alternative load combinations for the cantilevered beam.

A.1 Introduction As described in the main part of this chapter, various load combination rules are open for national choice. Two alternative combinations for static equilibrium (EQU) and three alternative combinations for ultimate limit states (STR/GEM) are provided in EN 1990 [1]. In case of the cantilevered beam described in section 3.1 all possible load cases generated by the alternative rules are included in Table 1. In a particular verification relevant load cases should be selected taking into account a particular national choice. Assuming the load case 1 for EQU, each of the following tables A.1 to A.3 includes only those load cases that are relevant to one of the three alternative combination rules provided in EN 1990 [1] for ultimate limit states (STR/GEO).

Table A.1. Load cases assuming alternative 1 for EQU and expression (6.10) for STR/GEO. Load Bending Limit state FactorsγG, γQ, γQ ×ψ assuming γG = 1,35, γQ =1,50 case moment In *) g1 g2 q1 q2 G 1 4 5 12 13 14 15 16 17

(c) (b) -

Equilibrium, exp. (6.7), (6.10) 0,90 Ultimate, exp. (6.10) 1,35 Ultimate, exp. (6.10) 1,00 (1,35) Serviceability, exp. (6.14) 1,00 Serviceability, exp. (6.14) 1,00 Serviceability, exp. (6.15) 1,00 Serviceability, exp. (6.15) 1,00 Serviceability, exp. (6.16) 1,00 Serviceability, exp. (6.16) 1,00

1,10 1,00 (1,35) 1,35 1,00 1,00 1,00 1,00 1,00 1,00

0 1,50 1,10 1,50 0 1,00 0 1,50 1,35 1,00 0 1,00 0 1,00 1,00 0 1,00 1,00×0,5 0 1,00×0,5 1,00 0 1,00 1,00×0,3 0 1,00×0,3 1,00

Table A.2. Load cases assuming alternative 1 for EQU and expressions (6.10a,b) for STR/GEO. Load Bending Limit state FactorsγG, γQ, γQ ×ψ or ξ ×γG assuming γG = 1,35, case moment γQ =1,50, ψ = 0,70 and ξ = 0,85 for actions In *) g1 g2 q1 q2 G 1 6 7 8 9 12 13 14 15 16 17

(c) (c) (b) (b) -

Equilibrium, exp. (6.7), (6.10) 0,90 Ultimate, exp. (6.10a) 1,35 Ultimate, exp. (6.10b) 1,15 Ultimate, exp. (6.10a) 1,00 (1,35) Ultimate, exp. (6.10b) 1,00 (1,15) Serviceability, exp. (6.14) 1,00 Serviceability, exp. (6.14) 1,00 Serviceability, exp. (6.15) 1,00 Serviceability, exp. (6.15) 1,00 Serviceability, exp. (6.16) 1,00 Serviceability, exp. (6.16) 1,00

1,10 0 1,50 1,00 (1,35) 1,50×0,7 0 1,00 (1,15) 1,50 0 1,35 0 1,50×0,7 1,15 0 1,50 1,00 1,00 0 1,00 0 1,00 1,00 0 1,00×0,5 1,00 0 1,00×0,5 0 1,00 1,00×0,3 1,00 0 1,00×0,3

1,10 1,00 1,00 1,35 1,15 1,00 1,00 1,00 1,00 1,00 1,00

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Chapter 6: Combination rules in Eurocodes

Table A.3. Load cases for alternative 1 for EQU and expressions (6.10amod,b) for STR/GEO. Load Bending Limit state FactorsγG, γQ, γQ ×ψ or ξ ×γG assuming γG = 1,35, case moment γQ =1,50, ψ = 0,70 and ξ = 0,85 for actions In *) g1 g2 q1 q2 G 1 7 9 10 11 12 13 14 15 16 17

(c) (b) (c) (b) -

Equilibrium, exp. (6.7), (6.10) 0,90 Ultimate, exp. (6.10b) 1,15 Ultimate, exp. (6.10b) 1,00 (1,15) Ultimate, exp. (6.10amod) 1,35 1,00 (1,35) Ultimate, exp. (6.10amod) Serviceability, exp. (6.14) 1,00 Serviceability, exp. (6.14) 1,00 Serviceability, exp. (6.15) 1,00 Serviceability, exp. (6.15) 1,00 Serviceability, exp. (6.16) 1,00 Serviceability, exp. (6.16) 1,00

1,10 1,00 (1,15) 1,15 1,00 (1,35) 1,35 1,00 1,00 1,00 1,00 1,00 1,00

0 1,50 1,10 1,50 0 1,00 1,15 1,50 0 1,00 0 0 1,35 0 0 1,00 0 1,00 0 1,00 1,00 0 1,00 1,00×0,5 0 1,00×0,5 1,00 1,00 0 1,00×0,3 0 1,00×0,3 1,00

Depending on the national choice one of the above tables A.1 to A.3 may be used for the verification of the cantilevered beam (assuming the load case 1 for the verification of static equilibrium EQU). Similar tables can be compiled when the verification of EQU is based on the load cases 2 and 3 given in Table 1 of section 3.1.

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Chapter 7: Actions in transient design situations

CHAPTER 7: ACTIONS IN TRANSIENT DESIGN SITUATIONS Milan Holický1, Jana Marková1 1

Klokner Institute, Czech Technical University in Prague, Czech Republic

Summary Eurocode EN 1991-1-6 [5] provides rules for the determination of actions to be used for the design of structures in transient design situations, such as executions, repairs, upgrading, partial or total demolition. The principles of transient design situations of structures are given in EN 1990 [1], supplementary rules for building and bridges in document [5]. Amongst all the many defined actions given in EN 1991-1-6 [5], particular attention is paid to the introduction of a number of different types of construction loads, which may typically be present during execution stages, but which are unlikely to be present after completion of the structure.

1

INTRODUCTION

1.1

Background documents

Eurocode EN 1991-1-6 [5] is based on the preliminary standard ENV 1991-2-6 [6], comments of CEN Member States and Background document [13]. It is the first code describing actions in transient design situations in Europe. 1.2

General principles

Principles and general rules on the choice of design situations, characteristic and design values of actions are given in EN 1991-1-6 [5] in accordance with EN 1990 [1]. In general, transient, accidental or seismic design situations should be considered for each part of the execution process. In general, the values of safety elements, such as partial factors γ and reduction factors ψ are based on recommended values given in EN 1990 [1] and EN 1991-1-6 [5]. The safety elements may be selected nationally through Nationally Determined Parameters (NDPs), or in some cases also determined for a specific individual project, taking into account the nominal values of construction loads and considering specific conditions, including changes during particular phases of execution as well as specific execution events that may occur. Additionally, planned duration of relevant phase of execution can be associated with a theoretical nominal duration, which may be used as a basis for the determination of characteristic values of climatic actions for return periods shorter than 50 years.

2

DESIGN SITUATIONS DURING EXECUTION

2.1

Design situations

The transient, accidental or seismic design situations are to be taken into account as appropriate. Selected design situations should be sufficiently severe and varied so as to encompass all conditions that can be reasonably foreseen to occur during the execution and 125

Chapter 7: Actions in transient design situations

use of a structure, including auxiliary structures. Where the conditions during execution stages may change, the critical load cases, which give the worst effects, should be verified. For the verification of execution stages, it is necessary to consider the conditions that can change during execution, including e.g. the shape of the structure, the structural system and especially the extent and degree of structural completeness. Usually, the servicebility criteria associated with the serviceability limit states during execution are the same ones as selected for the completed structure. The criteria may be modified nationally. 2.2

Nominal duration of design situations

The expected time period or duration of a particular stage of execution may be associated with a nominal duration of the selected design situation, thus enabling different return periods of climatic actions to be taken into account. The nominal duration is intended to be equal to or greater than the anticipated duration of the stage of execution under consideration. Four ranges of return periods are recommended in EN 1991-1-6 [5] as indicated in Table 1. Table 1 Recommended return periods for the assessment of the characteristic values of climatic actions Qk depending on nominal duration of execution phase Nominal duration of execution phase t

t ≤ 3 days 3 days < t ≤ 3 months 3 months < t ≤ 1 year t > 1 year

Return period R

2 years 5 years 10 years 50 years

p = 0,5 p = 0,2 p = 0,1 p = 0,02

The characteristic value of a climatic action Qk may be determined on the basis of assumed probability distribution and selected return period R related to the probability p of its possible exceeding. The probability distribution of the basic variable Q may be derived on the basis of known data from the locality of site. According to recommendations given in EN 1991-1-6 [5] and other Parts of Eurocode EN 1991, the characteristic value Qk,R of a variable action for the return period of R years may be determined on the basis of the characteristic value Qk,50 for a variable action for a 50 years return period. This may be determined from the general relationship given as: Qk,R = k Qk,50

(1)

where k is the reduction coefficient of a variable action based on the extreme-value distribution as shown below. The following relationships for thermal, snow and wind actions, respectively, are recommended in appropriate Parts of Eurocode EN 1991: (a) Thermal actions in EN 1991-1-5 [4] : Tmax,R = k Tmax,50 , for k = {k1 – k2 ln[–ln(1 – 1/R)]}

(2)

Tmin,R = k Tmin,50, for k = {k3 + k4 ln[–ln(1 – 1/R)]}

(3)

where Tmax,50/Tmin,50 is the maximum/minimum shade air temperature for 50 years return period and Tmax,R/ Tmin,R for n years return period, and coefficients k1 = 0,781, k2 = 0,056, k3 = 0,393, k4 = – 0,156 might be used (based on data of UK [13]),

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Chapter 7: Actions in transient design situations

(b)

Snow actions in EN 1991-1-3 [2] :

According to the Gumbel distribution [9] given as: ⎧ ⎫ 6 [ln(− ln(1 − p)) + 0,57722]⎪⎪ ⎪⎪1 − V π sk,R = k sk,50, for k = ⎨ ⎬ (1 + 2,5923V ) ⎪ ⎪ ⎪⎩ ⎪⎭

(4)

where sk,50 is the characteristic snow load on the ground for 50 years return period and sk,R for n years return period and V is the coefficient of variation of annual maximum snow load. (c) Wind actions in EN 1991-1-4 [3] : ⎡1 − K ln(− ln(1 − p )) ⎤ vb,R = k vb,50, for k = ⎢ ⎥ ⎣ 1 − K ln(− ln(0,98)) ⎦

R

(5)

where vb,R is the basic wind velocity for n years return period, vb,50 is for 50 years return period, K is the shape parameter depending on the coefficient of variation of the extremevalue distribution, and n is the exponent. The calculated reduction coefficients k indicating the amount of reduction of the characteristic values of climatic actions Qk,R for different return period R is shown in Table 2. Table 2 Reduction coefficient k of actions Qk,R for different return periods R Return period

p

2 years 5 years 10 years 50 years

0,5 0,2 0,1 0,02

for Tmax,R 0,8 0,86 0,91 1

Reduction coefficient k for Tmin,R for sn,R 0,45 0,64 0,63 0,75 0,74 0,83 1 1

for vb,R 0,77 0,85 0,90 1

An example of the determination of characteristic values of variable actions Qk,R for four different return periods (R) are shown in Table 3 based on the selected characteristic values of maximum shade air temperature Tmax,50 = 32 °C, minimum shade air temperature Tmin,50 = – 30 °C, snow load on the ground sn,50 = 1,5 kN/m2 and basic wind velocity vb = 26 m/s for 50 years return period. Table 3 Example of determination of climatic actions Qk,R for return periods R Return period R 2 years 5 years 10 years 50 years

p 0,5 0,2 0,1 0,02

Characteristic values of actions Qk,R for Tmax,R for Tmin,R for sn,R for vb,R o o 2 25,6 C -13,5 C 0,96 kN/m 20,2 m/s o o 2 27,7 C -18,8 C 1,13 kN/m 22,2 m/s o o 2 29 C -22,3 C 1,25 kN/m 23,5 m/s o o 2 32 C -30 C 1,50 kN/m 26 m/s

Tables 2 and 3 indicate that the reduction of the characteristic value of a climatic action Qk,R associated with short expected durations of the execution activities under

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Chapter 7: Actions in transient design situations

consideration, may not be appropriate. In such cases short term meteorological predictions may be more appropriate to serve as a basis for estimation of climatic actions. Recommended reductions of characteristic values of climatic actions may be specified in the National Annex. Examples are given in EN 1991-1-6 [5] (e.g. a minimum wind speed of 20 m/s). Characteristic values of some other variable actions associated with different duration periods during execution, (e.g. water actions), may be determined on the basis of Table 1 using appropriate probability distribution based on known statistical data.

3

REPRESENTATIVE AND DESIGN VALUES OF ACTIONS DURING EXECUTION

Characteristic and other representative values of actions are to be determined according to Eurocodes EN 1990 [1] for basis of design, EN 1991 for actions, EN 1997 for geotechnical design and EN 1998 for seismic actions. The characteristic value Fk in design situations during execution is a main representative value of a permanent or variable action. Following EN 1990 [1], the characteristic value of Fk shall be specified: – as a mean, an upper or lower value, or a nominal value, – in the project documentation provided that consistency is achieved with methods given in Eurocodes. For variable actions, two other representative values of actions Frep are, in common cases, distinguished during execution. These are: – the combination value, represented as a product ψ0Qk, used for the verification of the ultimate limit states and irreversible serviceability limit states, – the quasi-permanent value, represented as a product ψ2Qk, used for the verification of ultimate limit states, involving accidental actions and for the verification of reversible serviceability limit states. The frequent value of action is not needed for execution in common cases, but in specific cases it may be defined for the particular project. For buildings and bridges, respectively, the following ψ factors are relevant: – buildings, Annex A1 to EN 1991-1-6 [5] provides minimum recommended values of ψ factors of construction loads (ψ0 = 0,6 to 1, ψ2 = 0,2); – bridges, Annex A2 to EN 1990 [1] gives a unique value for both the combination and quasi-permanent values of ψ factors for construction loads (ψ0 = ψ2 = 1). As the determination of the representative values of construction loads often depends on the specific process of execution, it is better to specify the values of ψ factors, in some cases, for the individual project. The design value of action Fd can be determined from the following expression [1] Fd = γf Frep

(6)

where γf is the partial factor of action. In EN 1990 [1] three different sets of γ factors are recommended depending on the ULS and type of structure under consideration. For example, the following values of partial factors of construction loads Qc are recommended for the verification of static equilibrium (EQU): – γf = 1,5 for buildings, – γf = 1,35 for bridges.

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Chapter 7: Actions in transient design situations

As well as other partial factors given in EN 1990 [1], the values of partial factors γf for Qc may be altered nationally. The design value of construction loads should take into account uncertainties in the modelling of action or action effects. As characteristic values of construction loads are in many cases specified by the nominal values given in technical specifications, the uncertainties in the determination of the load effect may be considered as low, provided they are not contravened during the execution of the works on site. Thus, it might be reasonable to reduce the above mentioned γ factors for actions Qc from value 1,5 to 1,35 not only for bridges, but also for buildings. However, it is also necessary to take into account other reductions, e.g. magnitude of ψ factors influencing representative values of construction loads, and also reduction of characteristic values of other variable actions Q based on different return periods. The values of γ and ψ values may be altered in the National Annex depending on the allowed limits in the modelling of actions, on tolerances in the specific project and on the degree of approximation.

4

COMBINATIONS OF ACTIONS

General principles and rules for the combination of design values of actions during execution required for the verification of structures with respect to the ultimate and serviceability limit states are covered in EN 1990, Section 6 [1] and described in Chapter 6 of this Guidebook. Additional rules for the combination of actions during the execution stages, including rules for construction loads, are provided for all construction works in EN 1991-1-6 [6]. Rules for the combination of actions for bridges are provided in Annex A2 to EN 1990 [1], where additional combination of actions is recommended for execution phases in an accidental design situation, during which there is a risk of loss of static equilibrium. The rules introduced for construction processes may be nationally modified or defined for a specific project.

5

ACTIONS DURING EXECUTION

The standard EN 1991-1-6 [5] gives rules for the determination of common types of actions that should be taken into account during execution, including specific construction loads and methods for establishing their values. Representative values of actions during execution may be different from those used in the design of completed structure and they should be taken into account. Additional rules specific for actions during execution and some general rules are provided in EN 1991-1-6 [5], Section 4. Accidental actions Ad and seismic actions AEd should be identified for execution activities, and should be specified for individual projects. Indicative values of vehicle impacts are given in EN 1991-2 Traffic loads on bridges [8] and in EN 1991-1-7 [7] Accidental actions. Actions that can be present due to the execution activities, but are not present when the execution activities are completed, are called construction loads and are described in detail in EN 1991-1-6 [6]. Construction loads Qc, which may be represented in combinations of actions by a single variable action, as appropriate, can include the following different components: − working personnel, staff and visitors, possibly with hand tools or other small site equipment (Qca); 129

Chapter 7: Actions in transient design situations

−

storage of movable items (e.g. building and construction materials, precast elements and equipment) (Qcb); − non permanent equipment in position for use during execution, either static (e.g. formwork panels, scaffolding, falsework, machinery, containers) or during movement (e.g. travelling forms, launching girders and nose, counterweight) (Qcc); − moveable heavy equipment, usually wheeled or tracked (e.g. cranes, lifts, vehicles, lift truck, power installation, jacks, heavy lifting devices) (Qcd); − accumulation of waste materials (e.g. surplus construction materials, excavated soil, or demolition materials) (Qce), − parts of the structure in temporary states (e.g. under execution) before the final design actions take effect (e.g. additional loads due to concrete being fresh, loads and reverse load effects due to particular process of construction such as assemblage, loads from lifting operations etc.) (Qcf). The groups of loads to be taken into account depend on the specific project. The characteristic values of construction loads, including their vertical and horizontal components, where relevant, should be determined according to the technical requirements for the execution process and the requirements of EN 1990 [1]. Construction loads can be modelled by uniformly distributed loads or concentrated loads. The possible directions of loads, locations and contact areas should be taken into account. The limits of the area where actions may be moved (e.g. cranes, vehicles) or the tolerances for possible deviations from the theoretical and designed position should be defined. Effective quality control management on the site is necessary to ensure that the actual position of construction loads, the value of the loads and their movement comply with the design criteria. EN 1991-1-6 [5] gives in notes indicative values of actions for working personnel Qca, for storage of movable items Qcb and also recommends loads and loaded areas for construction activities due to the casting of concrete. Other information may be found for cranes in EN 1991-3 [10], for forklifts and some vehicles EN 1991-1-1 [2], and for other traffic load models in EN 1991-2 [9]. An example of the arrangement of construction loads due to working personnel (Qca) and storage of movable materials or equipments (Qcb), applied for the verification of the stability of bridge structure in the phase of execution, is illustrated in Fig. 1. The values of actions in Fig. 1 are based on EN 1991-1-6 [5]. Qcb qcb,k

qca,k

Fig. 1 Construction load Qcb,k = 100 kN, qca,k = 1 kN/m2 and qcb,k = 0,2 kN/m2 for the verification of bridge stability

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Chapter 7: Actions in transient design situations

6

ANNEX A FOR BUILDINGS AND BRIDGES

6.1

Annex A1 Supplementary rules for buildings

Annex A1 provides supplementary rules for application of EN 1991-1-6 [5] for the design of structures during execution. The same combination rules are applied for the verification of structures in transient design situations as for persistent design situations. The frequent combination of actions is not applied for the verification of structures in transient design situations. Therefore, the value of factor ψ1 for the frequent combination of construction loads during execution is not given in EN 1991-1-6 [5]. Recommended values of reduction factors ψ for construction loads Qc according to EN 1991-1-6 [5]: ψ0 = 0,6 to 1 and ψ2 = 0,2. It may be noted that for example in the National annex to [5] of the Czech Republic ψ0 = 0,6 is selected and the recommended value ψ2 is accepted. 6.2

Annex A2 Supplementary rules for bridges

Annex A2 provides a guidance for the application of EN 1991-1-6 [5] for the design of bridges. Supplementary rules for snow and construction loads are introduced. For the incremental launching of bridges the design values for vertical deflections (see Figure A2) may be defined in the National Annex or for the individual project. The recommended values in EN 1991-1-6 [5] are ± 10 mm longitudinally for one bearing, the other bearings being assumed to be at the theoretical level (Figure 2a), in transverse direction ± 2,5 mm, the other bearings being assumed to be at the theoretical level (Figure 2b). Recommendations for partial factors γ of actions and reduction factors ψ of bridges were transferred to EN 1990/A1 [1].

a) Longitudinal deflection

b) Differential deflection in the transverse direction

Fig. 2 Deflections of bearings during execution for bridges built by the incremental launching method

7

ANNEX B FOR ACTIONS ON STRUCTURES DURING ALTERATION, REHABILITATION OR DEMOLITION

General guidance for the determination of actions during structural repairs, partial or total demolitions is introduced in the informative Annex B. It is recommended here that for 131

Chapter 7: Actions in transient design situations

the verification of construction phases of reconstructions or demolitions, the actual behaviour of a structure should be taken into account including influences of degradation. All imposed loads, including traffic loads, should be considered if part of the structure remains in use during its reconstruction or partial demolition. These loads may vary at different transient stages. Traffic loads should include, for example, impact and horizontal forces from vehicles, wind actions on vehicles, and aerodynamic effects from passing vehicles and trains, where relevant. It is expected that Annex B will be in future extended to encompass guidance for repairs and strengthening of structures.

8

CONCLUDING REMARKS

The main objective of EN 1991-1-6 is to be helpful to – identify actions and critical design situations during execution; – classify actions; – define characteristic, other representative values and design values of actions during execution, giving recommended values of ψ and γ factors to be used in combination of actions for ULS and SLS verifications; – define supplementary combination rules for actions during execution. These objectives have been extended, as general rules, for the identification of actions to be taken into account for the design of auxiliary construction works and for design situations such as refurbishment, reconstruction, total or partial demolition, and particularly for where there may be structural alterations. EN 1991-1-6 has been drafted having in mind its use in close conjunction with EN 1990, with other parts of EN 1991 and with EN 1992 to EN 1999, inclusive. The main aim of this Part of Eurocode EN 1991 is thus achieved. The EN 1991-1-6 provides rules for the determination of actions for the design of structures in transient design situations, leaving the selection of numerical values of actions, partial factors and other safety elements to the decision of National Standards Institutions through National Determined Parameters. It is expected therefore that National Annexes to EN 1991-1-6 will be prepared, providing information needed for the design of construction works in transient design situations in each Member State of CEN. Moreover, it is necessary to have in mind that the design of a structure for transient design situation should take into account the specific construction loads given by producers (e.g. technical parameters of using machinery including tolerances), and including the methods of execution, the order of assemblage, and the detailed knowledge of all planned phases of execution. Therefore, it is expected that the user of EN 1991-1-6 will need additional information about the real site conditions and techniques to be used for execution of the individual building or civil engineering work. EN 1991-1-6 contributes to improving the appropriate application of actions in the design for execution including a quite wide set of recommendations helping the designer. Due to many different possible variations in execution processes and techniques, however, the application of competent engineering judgement remains the mainstay of good safe design. It is envisaged that there will be a future maintenance programme for the development and updating of the document, possibly taking into account scientific investigations focussed on the definition of actions such as construction loads, and giving more specific guidance on the identification of individual design situations of structural refurbishment.

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Chapter 7: Actions in transient design situations

REFERENCES [1]

EN 1990 Basis of structural design and Amendment 1. European Committee for Standardisation, CEN Brussels, 04/2002. [2] EN 1991-1-1 Eurocode 1 Actions on structures. Part 1-1: Densities, self-weight and imposed loads for buildings. CEN, Brussels, 06/2002. [3] EN 1991-1-3 Eurocode 1 Actions on structures. Part 1-3: Snow actions. CEN, Brussels, 06/2002. [4] EN 1991-1-4 Eurocode 1 Actions on structures. Part 1-4: Wind loads. CEN, Brussels, 06/2002. [5] EN 1991-1-5 Eurocode 1 Actions on structures. Part 1-5: Thermal actions. European Committee for Standardisation, CEN, Brussels, 06/2002. [6] EN 1991-1-6 Eurocode 1 – Actions on structures. Part 1.6: General actions – Actions during execution, Final PT Stage 34 Draft, CEN, Brussels, 08/2002. [7] ENV 1991-2-6 Eurocode 1 – Actions on structures. Part 2.6: General actions – Actions during execution, CEN, Brussels, 1999. [8] EN 1991-1-7 Eurocode 1 Actions on structures. Part 1-7: Accidental actions, working PT draft, CEN, Brussels, 11/2002 [9] EN 1991-2 Eurocode 1 - Actions on structures. Part 2: Traffic loads on bridges, CEN, Brussels, 01/2002. [10] EN 1991-3 Eurocode 3 - Actions on structures. Part 3: Cranes and machinery, CEN, Brussels, 01/2002. [11] G. König et al: New European Code for Thermal Actions, Background document, Report N.6, University of Pisa, 1999 [12] Haldar A., Mahadevan S., Probability, reliability, and statistical methods in engineering design, John Wiley & Sons, New York, 2000 [13] Background document to ENV 1991-2-6, 1996 [14] BS 6187:2000 Code of practice for demolition, British Standards Institution, London, 2000

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Chapter 8: Actions and combination rules for silos and tanks

CHAPTER 8: ACTIONS AND COMBINATION RULES FOR SILOS AND TANKS Milan Holický1, Jana Marková1 1

Klokner Institute, Czech Technical University in Prague, Czech Republic

Summary The loads due to stored solids or liquids are the specific actions that need to be considered in the design of silos or tanks. For the selected design situations and identified limit states, the critical load cases should be determined. The combination rules for actions and reliability elements provided in EN 1990 are supplemented for the purposes of silos and tanks in EN 1991-4, Annex A. For the verification of the limit states of type STR, the fundamental combination of actions given by two expressions (6.10a, 6.10b) in EN 1990 is recommended in EN 1991-4. Assessment of pressures due to stored particulate solids is illustrated on an example of a slender steel silo.

1

INTRODUCTION

1.1

Background documents

The aim of this Chapter is to introduce an overview of actions and supplementary rules for their combinations needed for the design of silos and tanks. The basic procedures for the determination of characteristic and design values of actions are given in EN 1990 [1] and in various Parts of EN 1991. EN 1991-4 [2] provides specific models for particulate solids stored in various types of silos and for liquids in tanks. Supplementary information concerning actions on silos and tanks is given e.g. in the Handbooks [6,7,8]. 1.2

Basic principles

EN 1990 [1] gives basic principles for the design of construction works. EN 1991-4 [2] provides supplementary rules for the design of silos and tanks, which are commonly subjected to full loads from particulate solids or liquids for most of their design working life. The limitations in geometry, in stored materials and types of filling or discharge arrangements are given within the scope of EN 1991-4 [2]. The dimensions of silos should be in a range hb/dc < 10, hb < 100, dc < 60, where hb is the total height of the silo and dc internal perimeter. Shapes of the silos with illustration of stresses are shown in Fig. 1. Three types of silos are distinguished: slender ((h/dc ≥ 2), intermediate (1,0 < hc/dc < 2,0), squat (0,4 < hc/dc ≤ 1,0). The maximum particle diameter of the stored solid should not be greater than 0,03dc. Outside the scope of the standard is the design of silo against shocks, quacking or silo music.

134

Chapter 8: Actions and combination rules for silos and tanks

straightened surface

internal diameter

hopper

Fig. 1 Shapes of silos with dimensions and pressures

2

DESIGN SITUATIONS

The general procedure for the determination of actions on silos and tanks for individual design situation is based on EN 1990 [1]. For selected design situations and identified limit states, the critical load cases need to be determined and verified. EN 1991-4 [2] gives basic rules for specification of loads due to particulate solids and arrangement of load patterns. The design for particulate solid fillings and discharge need to take into account the following basic load cases: – maximum normal pressure on the silo vertical wall, – maximum vertical frictional drag on the silo vertical wall, – maximum vertical pressure on a silo bottom, – maximum load on a silo hopper. For the verification of each load case, the single set of consistent values of solid properties, a wall friction coefficient μ, a lateral pressure ratio K and an angle of internal friction φi need to be used. Because each of these load cases attains the most critical values, when the stored solid properties μ, K and φi take characteristic values at different extremes of their statistical range, the relevant upper or lower characteristic values of the material properties should be considered to ensure that the design is appropriately safe for all the limit states.

135

Chapter 8: Actions and combination rules for silos and tanks

The upper characteristic value of the bulk unit weight γ is recommended to be used in EN 1991-4 [2] for all load calculations. It should be noted that EN 1991-1-1, Annex A [3], provides the mean values of the bulk unit weight only. Different silo slenderness, hopper geometries and discharge arrangements lead to the different design situations that have to be considered. For example, when the solids falling into a silo leads to an eccentric pile at some level, different packing densities can occur in different parts of the silo inducing unsymmetrical pressures. Thus, the design should consider the unsymmetrical pressures that may develop. EN 1991-4 [2] provides recommendations for design situations and specific construction forms of silos. For example, in concrete silos being designed for the serviceability limit states, the cracking needs to be limited to prevent water ingress. The crack control should comply with the crack width limitations of EN 1992-3 [4] appropriate for the environment, in which the silo is situated. EN 1991-4 [2] gives the basis for the design of silos and tanks against explosions. The potential damage should be limited or avoided by appropriate measures including – sufficient pressure relief area, – appropriate explosion suppression systems, – designing the structure to resist the explosion pressure. EN 1998-4 [5] gives provisions for seismic design situations.

3

ACTIONS ON SILOS AND TANKS

3.1

Types of actions

Silos and tanks are loaded by stored solids or liquids and also by other actions like climatic actions, geotechnical actions, accidental and seismic actions. The actions to be considered in the design of the silo include – filling and storage of particulate solids (filling loads), – discharge of particulate solids (discharge loads), – imposed loads, – snow loads, wind actions for full or empty silo, – climatic or technological thermal loads (temperature differences between the stored solid and the silo structure or between the external environment and the silo structure), – imposed deformations due to foundation settlement, – loads due to stored liquids, – accidental actions due to vehicle impact, dust explosion and fire, – seismic loads. The upper characteristic values of the bulk unit weight γ are recommended for the application in structural analyses. The assessment of each load case should be made using a single set of consistent values of the solid properties μ, K and φi. 3.2

Actions specific for silos

The silos are designated for the storage of particulate solids. The loads due to stored solids are classified as variable actions. Symmetrical loads on silos and eccentric loads due to eccentric filling or discharge processes are classified as variable fixed actions. Patch loads associated with filling and discharging processes in silos are classified as variable free actions.

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Chapter 8: Actions and combination rules for silos and tanks

The characteristic values of actions as given in EN 1991-4 [2] are intended to correspond to values having 2% probability to be exceeded within a reference period of one year. However, as it is noted in EN 1991-4 [2], the values of actions due to filling or discharge are commonly based on historical values only, which are given in earlier codes. The loads due to the storage of particulate solids are represented by a symetrical load and an assymetrical patch load. The symmetrical loads on silos are expressed in terms of a horizontal pressure ph on the inner surface of the vertical silo wall, a normal pressure pn on an inclined wall, tangential frictional tractions pw and pt on the walls, and a vertical pressure pv in the stored solid. Unsymmetrical loads on the vertical walls of silos with small eccentricities of filling and discharge are represented by patch loads causing a local horizontal pressure ph on the inner surface of the silo. Unsymmetrical loads on the vertical walls of silos with larger eccentricities of filling and discharge are represented by unsymmetrical distributions of horizontal pressure ph and wall frictional traction pw.

4

CLASSIFICATION OF SILOS

The reliability differentiation of silos is based on their capacities. Different levels of rigour need to be used in the design of silo structures, depending on the reliability of the structural arrangement and the susceptibility to different failure modes. Silos are categorized into three Action Assessment Classes 1 to 3 (AAC1 to AAC3) with respect to the capacity of the individual storage unit. Detailed description is given in EN 1991-4 [2], Table 2.1. The silos with the capacity exceeding 10000 tonnes are classified to the class AAC3, the silos with the capacity below 100 tonnes to class AAC1. For the design of silos in class AAC1, the simplified provisions may be applied including the load magnifiers C for the representation of unfavourable additional loads. The load magnifiers are intended to account for uncertainties concerning the flow patterns, the influence of the eccentricities of inlet and outlet on the filling and discharge processes, the influence of the silo shape on the type of flow pattern, and approximations used in transforming the time-dependent filling and discharge pressures into time-independent models. For the design of silos in classes AAC2 and AAC 3, the variability of the design parameters used to represent the stored solid is taken into account by means of characteristic values for the stored material properties χ, μ, K and φi.

5

COMBINATIONS OF ACTIONS FOR SILOS

5.1

Combinations of actions in persistent design situations

Combinations of actions in persistent design situations are based on the design value of a leading variable action and design values of accompanying variable actions. The fundamental combination of actions for ultimate limit states (STR) is based on two expressions (6.10a), (6.10b) given in EN 1990 [1] ∑ γ G , j Gk , j " +" γ Q ,1ψ 0 ,1 Qk ,1" +" ∑ γ Q ,iψ 0 ,i Qk ,i

(1)

∑ ξ j γ G , j Gk , j " +" γ Q ,1 Qk ,1" +" ∑ γ Q ,i ψ 0 ,i Qk ,i

(2)

j ≥1

j ≥1

i >1

j ≥1

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Chapter 8: Actions and combination rules for silos and tanks

In expression (2) ξ denotes the reduction factor (ξ = 0,9) for unfavourable permanent action G. The combinations of actions to be considered for persistent design situations are given in Table 1. The name of load combination indicates the leading variable action 1 jointly with accompanying action 2, which are combined with other accompanying variable actions. Thus, the leading variable action represented by solids filling, full silo or solids discharge (ψ0,1 = 1) is recommended to be combined with first accompanying variable action like climatic actions, technological temperatures, imposed deformations (ψ0,2), and other accompanying variable actions (ψ0,i). The values of partial factors given in EN 1990 [1], Table A.2 are recommended to be used. When the maximum depth of liquid and the unit weight of the heaviest stored liquid are defined, the value of the partial factor γF may be reduced from 1,50 to 1,35. The values of combination factors ψ are given in EN 1991-4 [2], Tables A.1 to A.5. It should be noted that the load combinations recommended in EN 1991-4 [2] seem to be inconsistent with the combination rules provided in EN 1990 [2] as only the loads due to storage are considered as leading variable actions. However, as it is explained in Chapter 3 that the critical load combinations on silos should be identified, it should also be checked in relevant cases whether some accompanying action like wind, temperature or unevent settlements may also need to be considered as the leading variable action. In such a case the actions due to stored solids should be represented by their accompanying value (apparently non-reduced, ψ0,2 = 1) as it is indicated in Table A.1 of EN 1991-4 [2]. It appears that Tables A.1 and A.2 of EN 1991-4 [2] are rather inconsistent and should be explained in more detail in the National annex. Table 1. Load combinations for ultimate limit states, persistent design situation Name of load combination

Permanent actions

ξ

Leading variable action 1

ψ0,1

Accompanying variable action 2

ψ0,2

Solids discharge

Solids discharge

1,0 Foundation settlement

Imposed deformation Self Snow weight Wind and full silo Wind and empty silo Temperature

Solids filling

1,0

5.2

0,9

Solids filling 1,0 Solids filling, full 1,0 silo Solids, empty silo 0,0 Solids filling

Accompanying variable action i

0,7 Snow, wind, temperature imposed loads, imposed deformation Imposed 0,7 Snow, wind, temperature deformation imposed loads Snow Wind 0,6 Imposed loads Wind

ψ0,i 0,6 0,7 0,7 0,6 0,7

0,7

1,0 Temperature

Combinations of actions in accidental design situations

The actions of Table 2 should be used with expression (6.11b) given in EN 1990 [1] for load combinations in the accidental design situations.

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Chapter 8: Actions and combination rules for silos and tanks

Table 2. Load combinations for accidental design situations Load combination

Explosion Vehicle impact

5.3

Accompanying variable action 1 (main) ψ1,1 or ψ2,1 0,9 Blast pressure Solid fillings or Self weight Vehicle 0,8 impact Permanent actions

Accompanying Accompanying variable variable action 2 action i

Leading accidental action

Imposed deformation

ψ2,2

ψ2,i

0,3 Imposed loads

0,3

Combinations of actions in seismic design situations

The actions in Table 3 should be used with expression (6.12b) given in EN 1990 [1] for the load combinations in the seismic design situations. Table 3 Load combinations for seismic design situations Load combination

Permanent actions

Leading Accompanying Accompanying accidental action variable variable action 2 action 1 (main)

ψ2,1

Self weight Seismic action and full silo Seismic Self weight action and empty silo

5.4

Seismic action (earthquake) Seismic action (earthquake)

Accompanying variable action i

ψ2,2

Imposed Imposed Solids filling, full 0,8 deformation 0,3 loads silo Solids, Imposed Imposed empty silo 0,8 deformation 0,3 loads

ψ2,i 0,3

0,3

Combinations of actions in serviceability limit states

For verification of silos or tanks in serviceability limit states, the actions of Table 4 should be used with expressions (6.14b) for characteristic combination of actions, (6.15b) for frequent combination of actions and (6.16b) for quasi-permanent combination of actions as given in EN 1990 [1].

139

Chapter 8: Actions and combination rules for silos and tanks

Table 4. Load combinations for serviceability limit states Load combination

Permanent actions

Accompanying variable action 1 ψ1,1 or ψ2,1

Accompanying variable action 2 ψ0,2 or ψ2,2

Accompanying variable action i

ψ0,i

Snow, wind, temperatures Solids discharge

Solids discharge

Imposed deformation

Foundation settlement

0,9 Imposed or deformation 0,8 Solids filling Snow Solids filling, full Wind silo Solids, empty Wind silo Temperatures Solids filling Solids filling

Self weight

Snow Wind and full silo Wind and empty silo Temperatures

0,7 or 0,3 0,7 or 0,3

0,6 or 0

Imposed loads, imposed deformation Snow, wind, temperatures

Imposed loads

0,6 or 0,0 0,7 or 0,3 0,6 or 0,0 0,7 or 0,3

Simplified design situations may be considered for silos in the class AAC 1: filling, discharge, wind when silo is empty, filling with wind, snow on roof. A simplified wind loading is permitted to be used according to EN 1991-4 [2].

6

COMBINATIONS OF ACTIONS FOR TANKS

6.1

Actions

For the design of tanks, the following actions need to be considered: self-weight, liquids, thermal actions, imposed load, snow, wind, suction due to inadequate venting, seismic and accidental actions, uneven settlements and loads resulting from pipes, valves and other items connected to the tank. The partial factors given in EN 1990 [1] are recommended to be applied. EN 1991-4 [2], Annex B recommends the partial factors for the liquid induced loads during operation γF = 1,20 and during tests γF = 1,00. 6.2

Combinations of actions

The combination of actions are based on EN 1990 [1], Section 6. The imposed load and snow actions need not to be considered simultaneously. The accidental and seismic actions need not to be applied during test conditions.

7

AN EXAMPLE OF SLENDER SILO

7.1

Introduction

The steel silo illustrated in Fig. 2 is of a circular cross-section with the internal diameter dc = 7,5 m and wall thickness t = 10 mm. The total height of silo is 37 m, the height

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Chapter 8: Actions and combination rules for silos and tanks

of a vertical-walled segment hc = 30 m and the height of a hooper hh = 4 m. The silo is used for the storage of wheat.

z

phf

phf

pwf

hc hb

dc pvf hh

pressure in vertical segment

Figure 2. Scheme of a silo The silo fulfils geometrical limitations (hb/dc < 10, hb < l00 m, dc < 60 m) given in the scope of EN 1991-4 [2]. Firstly the silo capacity needs to be determined for the specification of its class. The volume V of a silo is given as V=

π d c2 4

hc +

π d c2 3× 4

hh =

π 7 ,52 4

30 +

π 7 ,52 3× 4

4 = 1384 m3

and its capacity for the storage of wheat (using the upper characteristic value of the bulk unit weight γup = 9 kN/m3) is specified as Gk,up = V γup = 1384 × 9 = 12456,03 kN m = 1245,6 t ≥ 100 t The silo capacity exceeds 100 t, however fulfilling the requirements for eccentricities in EN 1991-4, Table 2.1 [2]. Therefore, the silo may be classified to the class AAC2. The loads on silo vertical walls need to be evaluated with respect to its slenderness. The silo belongs to the class of slender silos as the slenderness ratio fulfils the condition hc/ dc =30/7,5 = 4 ≥ 2 7.2

Symmetrical filling loads on vertical walls

Symmetrical filling loads lead to the horizontal pressure phf, the wall frictional traction pwf and vertical pressure pvf at the depth z below the solid surface given as phf ( z ) = phoYJ ( z )

(3)

pwf ( z ) = μ phoYJ ( z )

(4)

pvf ( z ) =

pho YJ ( z ) K

(5)

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Chapter 8: Actions and combination rules for silos and tanks

in which pho = γ K zo

(6)

1 A Kμ U

(7)

zo =

YJ ( z ) = 1 − e − z / zo

(8)

where γ is the characteristic value of bulk unit weight, μ is the characteristic value of the wall friction coefficient for solids sliding on the vertical wall, K is the characteristic value of the lateral pressure ratio, z0 is Janssen characteristic depth, pho is the asymptotic horizontal pressure at great depth due to stored particulate solids, A is the plan cross-sectional area and U is the internal perimeter of the plan cross-section of the silo. It is considered here that the properties of stored wheat are known and may be taken from Table E.1 of EN 1991-4 [2], where the mean lateral pressure ratio Km = 0,54 for wheat, the factor aK = 1,11 and wall friction coefficient μ = 0,24 for the wall type D1 are given. The upper and lower characteristic values of the lateral pressure ratio K are given as Kup = aK Km = 1,11×0,54 = 0,599 Klow = Km/aK = 0,54/1,11 = 0,486 and the Janssen characteristic depth z0 for the upper value of the lateral pressure ratio Kup z0 =

1 44 ,17 = 13,04 m 0,599 × 0,24 23,56

The Janssen pressure depth variation function Yj(z) in depth of 30 m is determined as YJ (30 ) = 1 − e − z / zo = 1 – e-30/13,04 = 0,9 and the cross-sectional area A of the silo and its internal perimeter U A = π d c2 / 4 = 3,141×7,52/4 = 44,17 m2 U = π dc = 3,141×7,5 = 23,56 m The horizontal pressure phf is determined for the height z = 30 m phf (30) = phoYJ (30) = γ K

1 44 ,17 1 A × × 0,9 = 63,27 kPa YJ(30) = 9 × Kμ U 0,24 23,56

and the wall frictional traction pwf pwf (30) = μ phoYJ (30) = μ γ K =9×

1 A A (1 − e − z / zo ) = γ (1 − e − z / zo ) = Kμ U U

44,17 × 0 ,9 = 15,19 kPa 23,56

and the vertical pressure pvf given as pvf (30) = = γ

142

1

A 1 44,17 (1 − e − z / zo ) = 9× × 0,85 = 122,3 kPa K low μ U 0,486 × 0 ,24 × 23,56

Chapter 8: Actions and combination rules for silos and tanks

7.3

Filling patch load

The filling patch load represents accidental asymmetries of loading due to eccentricities and imperfections in the filling process. The magnitude of the filling outward patch pressure ppf should be determined from the maximum eccentricity of the top pile throughout the filling process. Eccentric filling of a silo is illustrated in Fig. 3. The reference magnitude of the filling patch pressure ppf is given as ppf = Cpf phf

(9)

Cpf = 0,21 Cop (1+2E2) [1 – e −1,5 ( hc / d c −1) ]

(10)

E = 2 ef / dc

(11)

where

where ef

is the maximum eccentricity of the surface pile during filling

phf is the local filling pressure at the height at which the patch load is applied Cop is the patch load solid reference factor.

et

ef

Figure 3. Silo with eccentric filling The patch load should be applied in the zone of height s given as s = πdc/16 ≅ 0,2dc

(12)

The filling patch pressure ppf in selected height zp = 20 m for the assumed example (Cop = 0,5, see Table E.1 of EN 1991-4) is determined as ppf = 0,21 Cop [1 + 2(2ef/dc)2] [1 – e −1,5 ( hc / dc −1) ] phf = = 0,21×0,5 [1 + 2×(2×0,8/7,5)2] [1 – e–1,5(20/7,5–1)]×63,3 = 6,65 kPa The height of the zone, which the patch load should be applied on, is given as s = πdc/16 = 3,141×7,5/16 = 1,47 m For presented example of thin walled circular silo (dc/t = 7,5/0,01 > 200) in class AAC 2, the filling patch pressure should be taken to act over a height s from a maximum outward pressure on one side to an inward pressure on the opposite side (see Figure 4). 143

Chapter 8: Actions and combination rules for silos and tanks

The circumferential variation of pressure should be taken as ppfs = ppf cosθ

(13)

where ppf

is the outward patch pressure θ is the circumferential coordinate. The total horizontal force Fpf due to the filling patch load on a thin-walled circular silo should be determined as Fpf = Fpf =

π 2

π 2

s dc ppf

(14)

×1,47 ×7,5×6,65 = 78,3 kN

Figure 4. Side elevation and plan view of the filling patch load for a circular silo

7.4

Symmetrical discharge load

Symmetrical increases in the discharge load shall be used to represent the possible transitory increases in pressure that occur on silo walls during the discharge process. For silos in all classes AAC1 to 3, the symmetrical discharge pressures phe and pwe should be determined as phe = Ch phf

(15)

pwe = Cw pwf

(16)

where Ch Cw

is the discharge factor for horizontal pressure, is the discharge factor for wall frictional traction.

For fully unloaded silos, the values of discharge factors Ch and Cw may be taken as Ch = Cw = 1. For slender silos in classes AAC2 and 3, the discharge factors should be considered as Ch = C0 = 1,15 and Cw = 1,10.

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Chapter 8: Actions and combination rules for silos and tanks

The resulting characteristic value of the vertical compressive force in the wall nzSk per unit length of perimeter during discharge at any depth z is given as nzSk = Cw μ pho [z – zo YJ (z)]

(17)

For the considered example in depth z = 30 m nzSk = 1,1×0,24×70,3×(30 – 13,04×0,90) = 338,9 kN 7.5

Discharge patch load

The discharge patch load expresses accidental asymmetries of loading during discharge, as well as inlet and outlet eccentricities. The silos in Classes AAC2 and 3 should be verified for discharge patch loads. The circular silos should be verified for discharge patch loads, when – the eccentricity of the outlet e0 exceeds the critical value e0,cr = 0,25 dc, – the maximum filling eccentricity ef exceeds the critical value ef,cr = 0,25 dc and the slenderness of the silo is greater than the limiting value hc/dc = 4. The reference magnitude of the discharge outward patch pressure ppe is given as ppe = Cpe phe

(18)

where the shape of the formula for Cpe is based on the ratio hc/dc and provided in EN 1991-4 [2]. For the considered example, the maximum filling eccentricity ef and the silo slenderness fulfils the limiting conditions (ef = 0,8 m < ef,cr = 1,875 m) and, therefore, the discharge patch load is not necessary to be determined.

8

CONCLUDING REMARKS

The design of silos and tanks are based on the same principles as given in EN 1990. Supplementary rules for the specification of loads and load effects due to stored solids and liquids are provided in EN 1991-4. For the verification of the limit states of type STR, the fundamental combination of actions given by two expressions (6.10a, 6.10b) is recommended in EN 1991-4 only. However, a slightly increased value of the factor ξ (0,9) used for the reduction of unfavourable permanent actions in expression (6.10b) approximates the results obtained by the recommended combination of actions given by two expressions (6.10a, 6.10b) to expression (6.10) of EN 1990. It is considered in EN 1991-4 that the main variable actions arise due to the stored particulate solids. However, in some cases also other variable action may also need to be considered as a leading variable action. The critical load cases should be determined in those cases.

REFERENCES [1] EN 1990 Eurocode – Basis of design. European Committee for Standardisation, 2006. [2] EN 1991-4 Eurocode 1 – Actions on structures Part 4 : Silos and tanks. European Committee for Standardisation, 2006.

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Chapter 8: Actions and combination rules for silos and tanks

[3] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. European Committee for Standardisation, 04/2002. [4] EN 1992-3 Eurocode 2 - Design of concrete structures - Part 3: Liquid retaining and containment structures, European Committee for Standardisation, 11/2007. [5] EN 1998-4 Eurocode 8: Design of structures for earthquake resistance - Part 4: Silos, tanks and pipelines, European Committee for Standardisation, 03/2008. [6] Brown C.J., Nielsen J: Silos, Fundamentals of theory, behaviour and design; E&FN Spon, London, 1998, 836 pp. [7] Blight G.: Assessing loads on silos and other bulk storage structures, Taylor&Francis group, Balkema, Netherlands, 2006. [8] Poukhonto L.M.: Durability of concrete structures and constructions, Balkema, India, 2003.

146

Chapter 9: Load effects in structural members

CHAPTER 9: LOAD EFFECTS IN STRUCTURAL MEMBERS Milan Holický1, Miroslav Sýkora1 1

Klokner Institute, Czech Technical University in Prague, Czech Republic

Summary Newly available European standard EN 1990 Eurocode - Basis of structural design provides alternative combination rules for actions, illustrated by examples of analysis of simple building structures in this Chapter 9. Partial factors for actions and ψ factors used for verification of the ultimate limit states including static equilibrium and for verification of the serviceability limit states are considered in accordance with the recommendations provided in EN 1990. Resulting load effects due to alternative combination rules are presented as bending moment envelops, in the case of static equilibrium also as shear forces. Comparison of obtained action effects indicates that alternative combination rules may lead to considerably diverse load effects differing up to 18 %. It appears that further investigation concerning decision about the preferable alternative, to be made in the National Annexes to EN 1990, should take into account economic, commercial and other aspects including laboriousness of design analysis.

1

INTRODUCTION

The standard EN 1990 Basis of structural design [1] is the fundamental document for the whole system of Eurocodes. The document has been available since April 2002 and its implementation into the systems of national standards is expected within one or two years. EN 1990 [1] provides principles for design and verification of structures with regards to safety, serviceability and durability as described in Chapter 2. The aim of this Chapter is to illustrate combination rules provided in EN 1990 [1] for both the ultimate limit states and serviceability limit states considering permanent loads and imposed loads in accordance with EN 1991-1-1 [2] and climatic actions due to wind and snow. Alternative combination rules for ultimate limit states are compared using examples of simple structures.

2

EXAMPLES

2.1

Cantilevered beam Geometry and material. Cantilevered beam considered in the first example is indicated in Figure 1. The reinforced concrete beam having cross-section 0,30×0,40 m (width × depth) is made of concrete C20/25.

147

Chapter 9: Load effects in structural members

q1

g1

(a)

g2

(b)

(c)

A

q2

G

(d)

B l1 = 4,5 m

g1 = 15 kN/m g2 = 15 kN/m q1 = 9 kN/m q2 = 9 kN/m G = 6 kN/m

l2 = 3,0 m

Figure 1. Cantilevered beam Uniform permanent load of the beam g1 and cantilever g2 (alternatively assumed to be from one source or independent), concentrated permanent load G, and imposed loads q1 and q2 (Category B - office areas) are considered. Whether the permanent actions g1 and g2 are from one source or not (i.e. they are independent) should be verified considering particular conditions of the structure (weight of structural and nonstructural components acting on both parts of the beam). Nevertheless, it will be shown that mutual independence of g1 and g2 is a safe assumption that leads to a considerably greater bending moment in midspan point (c) than the assumption that g1 and g2 are from one source (fully dependent). Ultimate limit states (EQU and STR), and serviceability limit states (characteristic, frequent and quasi-permanent combinations) are to be verified. Table 1 shows the load cases and appropriate factors (γG, γQ, γQ ×ψ or ξ × γG) assuming γG = 1,35, γQ =1,50, ψ = 0,70 and ξ = 0,85 relevant to verification of the equilibrium, bending resistance (shear is not considered) and deflection of the beam. If the permanent loads g1 and g2 could be considered as being from one source, then the factors of both actions would be the same as indicated in Table 1 by the values in brackets (when these are different from the case of independent permanent actions). Note that the assumption of g1 and g2 being from one source (and both q1 and q2 acting) would lead to the maximum shear forces at point (b) (not shown here). Table 1. Load cases and factors γG, γQ, γQ×ψ or ξ×γG corresponding to relevant expressions in EN 1990 [1], when g1 and g2 are actions from one source then the factors in brackets should be applied

Load Bending case moment at *) 1 2 3 4 5 6 7 8 9 10 11

148

(c) (b) (c) (c) (b) (b) (c) (b)

Limit state

Equilibrium, exp. (6.7), (6.10) Equilibrium, exp. (6.7), (6.10) Equilibrium, exp. (6.7), (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10amod) Ultimate, exp. (6.10amod)

FactorsγG, γQ, γQ ×ψ or ξ ×γG assuming γG = 1,35, γQ =1,50, ψ = 0,70 and ξ = 0,85 for actions g2 q1 q2 G g1 0,90 1,15 1,00 1,35 1,00 (1,35) 1,35 1,15 1,00 (1,35) 1,00 (1,15) 1,35 1,00 (1,35)

1,10 1,35 1,00 1,00 (1,35) 1,35 1,00 (1,35) 1,00 (1,15) 1,35 1,15 1,00 (1,35) 1,35

1,10 1,50 0 1,35 1,50 0 1,00 1,50 0 1,50 0 1,00 0 1,50 1,35 0 1,00 1,50×0,7 1,50 0 1,00 0 1,50×0,7 1,35 1,15 1,50 0 1,00 0 0 1,35 0 0

Chapter 9: Load effects in structural members

12 13 14 15 16 17

-

Serviceability, exp. (6.14) Serviceability, exp. (6.14) Serviceability, exp. (6.15) Serviceability, exp. (6.15) Serviceability, exp. (6.16) Serviceability, exp. (6.16)

1,00 1,00 1,00 1,00 1,00 1,00

1,00 1,00 1,00 1,00 1,00 1,00

1,00 0 0 1,00 0 1,00×0,5 0 1,00×0,5 0 1,00×0,3 0 1,00×0,3

1,00 1,00 1,00 1,00 1,00 1,00

Note: *) Only the load cases 4 to 11 are directly related to a bending moment at a particular point of the beam. Load effects. If the support (a) of the beam shown in Figure 1 can not transmit tensile forces, static equilibrium EQU of the beam should be checked using equation (3) (expression (6.7) in EN 1990 [1]). In accordance with this equation the following condition should be satisfied

γg1g1l12 / 2 > γg2g2l22 / 2 + γq2q2l22 / 2 + γGG l2

(6)

Two alternative sets of partial factors are provided in EN 1990, Annex A1 [1], Table A1.2(A). Both these sets are independent of the assumption concerning dependency of the permanent actions g1 and g2. Thus, the verification of the static equilibrium does not distiquish between the case when g1 and g2 are from one source and the case when g1 and g2 should be considered as independent. In the load case 1 (Table 1) the factors 0,9 for favourable and 1,1 for unfavourable permanent actions are considered (as indicated in Note 1 in Table A1.2(A), Annex A1 of EN 1990 [1]). In the load case 2 (Table 1) the factors 1,15 and 1,35 are used (in accordance with Note 2 in Table A1.2(A), Annex A1 of EN 1990 [1]) provided that applying γG = 1 to both the favourable and the unfavourable parts of permanent actions does not give a more unfavourable load effect (verified in example 1 by the load case 3, see Table 1). Figure 2 shows results obtained for the ultimate limit states EQU. It appears that the cantilevered beam should be provided by an anchor at the point (a). The load case 1 seems to be more severe (tensile force 4,03 kN) than load case 2 (tensile force 0,34 kN, not indicated in Figure 2). Note that the load case 3 leads to more favourable effect than the cases 1 and 2 (compressive force 1,25 kN, indicated in Figure 2). Thus, in the alternative approch indicated in the Note 2 in Table A1.2(A), Annex A1 of EN 1990 [1], the load case 2 is decisive.

149

Chapter 9: Load effects in structural members

-77.97 -64.78

1

2

-4.03

1

3 2

1.25

6 8.1

91.5 109.4

Figure 2. Shear forces [kN] according to expression (6.7) for equilibrium verification

The comparison of bending moments for ultimate limit states STR (Figures 3 and 4) indicates that the assumption of independent g1 and g2 leads to considerably greater positive moments (negative moments are not affected) than the assumption of g1 and g2 being from one source (for the combination A by more than 20 %, see Figures 3a and 3b). Assuming the independent g1 and g2 Figures 3b and 4b indicate that the positive moments for the combination A (Figure 3b) are about 18 % greater than those for the load combination (6.10a) and (6.10b) (for convenience of notation hereafter denoted as “B”) and for the load combination (6.10amod) and (6.10b) (denoted as “C”) as shown in Figure 4b. The difference between the negative moments of the combinations A and B in point (b) is about 11 %. Load combinations B and C are in this example identical because expression (6.10b) is decisive in both cases while expressions (6.10a) and (6.10amod) are not effective.

150

Chapter 9: Load effects in structural members

1

-176.2 -176.2

-159.1 -159.1

-109.1 -109.1

-95.63 -95.62

2

1

3

1

2

3

1

2

2

36.69

39.02

Figure 3a. Bending moment envelopes [kNm] according to expression (6.10) assuming g1, g2 being from one source

Figure 4a. Bending moment envelopes [kNm] according to exp. (6.10a), (6.10b) and (6.10amod), (6.10b) assuming g1, g2 being from one source

-176.2

-159.1

-85.5

-85.5

1

2 1

3 2

47.81

Figure 3b. Bending moment envelopes [kNm] according to expression (6.10) assuming g1, g2 independent

1

2 1

3 2

40.52

Figure 4b. Bending moment envelopes [kNm] according to exp. (6.10a), (6.10b) and (6.10amod), (6.10b) assuming g1, g2 independent

Deflections. Three combinations of serviceability limit states (called in EN 1990 [1] characteristic, frequent and quasi-permanent) are considered in Table 1. The characteristic load combination is described in EN 1990 [1] by expression (6.14) (load cases 12 and 13), the frequent combination is described in EN 1990 [1] by equation (6.15) (load cases 14 and 15), the quasi-permanent combination described in EN 1990 [1] by expression (6.16) (load cases 16 and 17). Deflection lines and the extreme deflections at a midspan point (c) and at the end point (d) due to characteristic and quasi-permanent load combinations are shown in Figure 5. Deflection lines were determined assuming the modulus of elasticity 29 GPa and creep coefficient 2,5 (in case of quasi-permanent load cases 16 and 17). 151

Chapter 9: Load effects in structural members

1 1

2

2 1

3

1

3 2

- 2,8 - 1,4

2

- 1,3 - 0,1

20,6 26,6

4,8 10,5

Figure 5. Deflection lines [mm] corresponding to the characteristic load cases 12 and 13 (left) and quasi-permanent cases 16 and 17 (right)

Figure 5 indicates that the deflection at the cantilever end (d) may violate criteria for structural performance. When, for example, the cantilever supports a brittle cladding components, cracks and other performance deficiencies may occur. Note that slightly lower deflection, as that due to characteristic combination, were obtained for the frequent combination described in EN 1990 [1] by equation (6.15) covered by the load cases 14 and 15 (see Table 1). 2.2

Continuous beam of three spans Geometry and material. Three span continuous beam of the cross-section 0,25×0,40 m made of concrete C 20/25 (modulus of elasticity 29 GPa) is loaded by permanent g and imposed load q as indicated in Figure 6.

q1

q2

g

1

q3

2

3

1

(e)

(a) 5m

4

2

(b)

3

(f)

(c)

5m

(g)

(d)

g = 30 kN/m q1 = 18 kN/m q2 = 18 kN/m q3 = 18 kN/m

5m

Figure 6. Continuous beam Load cases. The uniform permanent action g (considered as a single action from one source for the whole beam) and three independent imposed actions q1, q2 and q3 are considered in the analysis of load effects needed for verification of ultimate and serviceability limit states (characteristic and quasi-permanent combinations). Ultimate limit states (of the type STR) verified using general expression (6.8) and load combinations (6.10) given in EN 1990 [1] is checked using the total of seventeen load cases, for which appropriate factors γ are indicated in Table 2. A decisive expression in the load combinations B and C may be identified as follows. In case of a single variable action (as an example imposed load Q is taken into account) the critical load ratio χ = Qk / (Gk + Qk) is:

χ lim, B =

γ G (1 − ξ ) γ G (1 − ξ ) + γ Q (a − ψ Q )

(7)

γ G (1 − ξ ) γ G (1 − ξ ) + γ Q a

(8)

χ lim, C =

152

Chapter 9: Load effects in structural members

where γG = 1,35; γQ = 1,5; ψQ = 0,7;ξ = 0,85. Qk and Gk denote action effects due to the variable and permanent loads. The following criteria are applied: - if χ > χlim,B or χ > χlim,C, then expression (6.10b) is decisive, - if χ < χlim,B or χ < χlim,C, then expression (6.10a) or (6.10amod) is to be used. Evaluation of these criteria is shown in Table 3. Table 2. Load cases and factors γQ × ψ or ξ × γG for continuous beam of three spans, expressions given in EN 1990 are indicated in brackets Load case

Bending moment in point (e) (f) (b) (b) (b) (e) (e) (f) (f) -

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Limit state

Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10amod) Serviceability (6.14) Serviceability (6.14) Serviceability (6.16) Serviceability (6.16)

g 1,35 1,35 1,35 1,35 1,35 0,85×1,35 1,35 0,85×1,35 1,35 0,85×1,35 1,35 0,85×1,35 1,35 1,00 1,00 1,00 1,00

Factors γQ × ψ or ξ × γG for actions q1 q2 1,50 1,50 1,50 1,50 1,50 1,50 0,7×1,50 0,7×1,50 1,50 1,50 0,7×1,50 1,50 0,7×1,50 0,7×1,50 1,50 1,50 0,7×1,50 1,50 1,00 1,00 0,3×1,00 0,3×1,00

q3 1,50 1,50 0,7×1,50 1,50 0,7×1,50 1,50 1,00 0,3×1,00 -

Table 3. The limit values χlim,B and χlim,C for load combinations B, C M in point: (see Fig. 6) b e f

Moment from Gk [kNm] 75 60 18,75

Moment from Qk [kNm] 52,5 42 33,75

χ

χlim,B

χlim,C

Decisive expression

0,412 0,412 0,643

0,31 0,31 0,31

0,119 0,119 0,119

(6.10b) (6.10b) (6.10b)

It follows from Table 3 that in the alternative combinations B and C expression (6.10b) is decisive. Bending moments. The resulting bending moments of the beam are shown in Figures 7 and 8.

153

Chapter 9: Load effects in structural members

-180 -135

-168.8 -123.8

-8.438

67.5

67.5

75.94 148.5

148.5

Figure 7. Bending moment envelope [kNm] according to combination A (expression (6.10))

-164.9 -164.8 -119.8

-153.6 -108.6

-12.22

55.38

55.38 72.15 136.4

136.4

Figure 8. Bending moment envelope [kNm] according to combination B and C (expressions (6.10b)) The results of analysis considering ultimate limit states STR indicate that the internal moment according to the combination A (Fig. 7) is at points (e) and (b) (see Fig. 6) greater about 11 % than according to the combination B (Fig. 8), resp. C. The numerical values in point (f) are for the combination A greater about 5 % than according to the combinations B and C. The combinations B and C are also equal in this case, the expressions (6.10a) and (6.10amod) are not expressed in envelope. The results of analysis taking into account serviceability limit states are shown in Figures 9 and 10. Considering characteristic load combination, the deflection lines corresponding to the load cases 14 and 15 (indicated in Table 2) are shown in Figure 9. Both deflection lines were determined for the modulus of elasticity 29 GPa (without time dependent creep effect).

154

Chapter 9: Load effects in structural members

1

.2,5

2 1

2,5

-1,6

3

2

4 3

2,2 6,2

6,2

Figure 9. Deflection lines [mm] corresponding to the load cases 14 and 15 specified in Table 2 Considering quasi-permanent load cases 16 and 17 (see Table 2) the extreme deflection lines are indicated in Figure 10. Both deflection lines were determined for the modulus of elasticity 29 GPa and creep coefficient 2,5. 1

-1,7

2 1

2

3

4 3

10,7

10,7 3,0 14,5

14,5

Figure 10. Quasi-permanent deflection lines [mm] corresponding to the load cases 16 and 17 specified in Table 2 Note that the maximum deflection 14,5 mm is about L/340 (where L is the length of one span of the beam), which seems to be quite satisfactory (serviceability constraint L/250 is normally considered as sufficient). However, in some cases more detail analysis of deflection may be required taking into account specific conditions (type of reinforcements, creep, performance requirements). 2.3

Cantilevered frame Geometry and material properties. The cantilevered frame indicated in Figure 11 is exposed to five independent actions: permanent load g, imposed load q1 and q2 (Figure 12) and climatic actions due to wind W and snow s (Figure 13). It is assumed that the identical frame is located every 6 m along the longitudinal direction of a building. The total height of the frame is 15 m, foundations are 3 m below the terrain, and the top of the frame is 12 m above the terrain. In a preliminary design of the frame two types of cross-sections are considered: - columns in the first storey, middle columns in the second to fourth storey, and all beams 0,30 × 0,60 m, - edge columns of the second to fourth storey 0,30 × 0,30 m. The frame is made of concrete C 20/25 (modulus of elasticity 29 GPa). A creep coefficient 2,5 is considered when determining long term deflection under quasi-permanent load combination. The ultimate limit state of structural resistance (STR) and serviceability limit states (characteristic and quasi permanent combination) shall be verified. Note that other actions (imposed load in cantilevered part of the frame only) may be needed when limit state of static equilibrium (EQU) of the frame should be verified (in the considered frame in Figure 11 the limit state of static equilibrium EQU is obviously satisfied). Load cases. The characteristic value of permanent load g imposed on beams is determined assuming equivalent thickness of the floor slab 0,20 m (representing the slab 155

Chapter 9: Load effects in structural members

about 0,16 m, beams, floor and other permanent loads). Thus, for loading width of 6 m the characteristic value of the uniform load of the beam is gk = 0,20 × 25 × 6 = 30 kN/m Note that possible reduction factors αA and αn, which may be used when designing particular structural elements to reduce imposed load, are not considered here (their effect in this simple example is insignificant). The characteristic value qk of imposed load for office areas (3 kN/m2) and loading width of 6 m is qk = 3 × 6 =18 kN/m The characteristic value of wind load is derived assuming the wind speed v = 26 m/s, thus the reference pressure is qref = 1,25×v2/2 = 1,25×262/2 = 422,5 N/m2 In addition the following parameters are assumed: the exposure coefficient Ce = 2,5 (corresponding to the height of the structure 12 m above the terrain of category II), the external pressure coefficient cpe,10 = 0,8 on the windward side and the factor cpe,10 = −0,3 on the leeward side. Thus, for the loading width 6 m and height 3 m (one storey) we get the following pressure force Wkp and suction force Wks acting at the frame nodes as indicated in Figure 13: Wkp = 0,4225 × 2,5 × 0,8 × 6 × 3 ≅ 15,2 kN Load effects. In the following load effects for verification of ultimate limit states (STR) and serviceability limit states (characteristic and quasi-permanent combination of actions) are analysed. The total of 16 load cases, indicated in Table 4, are considered. It should be noted that additional load cases might be needed for the verification of the ultimate limit state of equilibrium (EQU), which are not considered here (the imposed load should be considered in the cantilevered part of the frame). Table 4. Load cases and appropriate factors γQ × ψ or ξ × γG , expressions given in EN 1990 are indicated in brackets Load case

Limit state

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10a) Ultimate, exp. (6.10b) Ultimate, exp. (6.10b) Ultimate, exp. (6.10amod) Serviceability, exp. (6.14) Serviceability, exp. (6.14) Serviceability, exp. (6.16) Serviceability, exp. (6.16)

156

g 1,35 1,35 1,35 1,35 1,35 0,85×1,35 1,35 0,85×1,35 1,35 0,85×1,35 0,85×1,35 1,35 1,00 1,00 1,00 1,00

Factors γQ × ψ or ξ × γG for actions q1 q2 W 1,50 1,50 0,7×1,50 0,7×1,50 1,50 0,7×1,50 1,50 0,7×1,50 1,00 0,3×1,00 -

1,50 1,50 0,7×1,50 0,7×1,50 1,50 0,7×1,50 1,50 0,7×1,50 1,00 0,3×1,00

0,6×1,5 0,6×1,5 0,6×1,5 1,50 0,6×1,5 0,6×1,5 0,6×1,5 0,6×1,5 0,6×1,5 0,6×1,5 1,50 0,6×1,00 0,6×1,00 -

s 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,50 0,5×1,00 0,5×1,00 -

Chapter 9: Load effects in structural members

The resulting bending moments envelopes determined using load combination A, B and C are shown in Figures 14 to 16. To achieve better legibility of these figures numerical values of bending moments are indicated for all horizontal beams, and for columns in the lowest floor only. It follows from Figures 14 to 16 that the bending moments obtained in some crosssections from combination A (Figure 14) are greater (up to 15 %) than those obtained from combination B (Figure 15) or combination C (Figure 16). It is interesting to note that in this example the bending moments corresponding to combinations B and C are almost identical. The only exception is the upper horizontal member where the extremes corresponding to combination B (Figure 15) are slightly greater than those corresponding to combination C (Figure 16). This difference is due to the following reason: in the case of combination B, expression (6.10a) is decisive (load case 9), while in the case of combination C, expression (6.10b) is decisive (load case 11). However, in the remaining members of the frame combinations B and C lead to the same bending moments given by load cases 6, 8, 10 and 11, all of them corresponding to expression (6.10b).

3m

3m

3m

6m

6m

3m

Figure 11. Cantilevered frame - permanent load g

157

Chapter 9: Load effects in structural members

Imposed load q1

Imposed load q2

Figure 12. Cantilevered frame – imposed load q

0,3×0,3 m

0,3×0,6 m

Wind actions W

Snow loads s

Figure 13. Cantilevered frame – climatic actions

158

Chapter 9: Load effects in structural members

-305.2 15.49

-39

-399.3 96.11

35.26

63.05

-435.4 141.4

47.29

53.44

-504.2 147.6 137.5 28.5

157.2

156.9 196

-264.3

-274.4

Figure 14. Bending moment envelope [kNm] for combination A (expression (6.10))

-284.1

17.22

35.8

-367.1

96.01

37.17

58.06 -400.7 129.4 41.66

54

-468.5 135.5 136.9

24.73

156.7

154.1 185.2

-256.1

-265.2

Figure 15. Bending moment envelope [kNm] for combination B (expressions (6.10a),(6.10b)).

159

Chapter 9: Load effects in structural members

-274.6 17.22

-35.56

-367.1 85.12

37.17

56.89

-400.7 129.4 54

41.66

-468.5 135.5 136.9

156.7

154.1

24.73

185.2

-256.1

-265.2

Figure 16. Bending moment envelope [kNm] for combination C (exp. (6.10amod),(6.10b)) 3

CONCLUDING REMARKS

Results obtained for the ultimate limit states EQU indicate that the partial factors 0,9 for favourable and 1,1 for unfavourable permanent actions lead to more severe load effects than the alternative partial factors 1,15 and 1,35. The examples of selected structural members verified for the limit states of rupture STR indicate that combination rule A (corresponding to expression (6.10) in EN 1990) is easier to apply than combinations B and C (corresponding to twin expressions (6.10a), (6.10b) and (6.10amod), (6.10b) in EN 1990). However, selected examples show that the design procedure A leads to considerable greater load effects (up to 18 % greater) than procedures B and C. Thus, the combination rule A will ensure greater reliability of structures than combinations B and C. Nevertheless, the design procedure A would increase the material consumption compared with the procedures B and C and, therefore, would unfavourably affect the initial costs of structures. It appears, that further harmonisation of alternative combination of actions is needed. It is expected that CEN/TC 250 together with JRC will analyse database of NDPs and reach decision about the fundamental combination of actions within next few years.

160

Chapter 9: Load effects in structural members

REFERENCES [1] EN 1990 Eurocode - Basis of structural design. European Comittee for Standardisation, 04/2002. [2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self weight, imposed loads for buildings. European Comittee for Standardisation, 04/2002.

161

Chapter 10: Design of a reinforced concrete building according to Eurocodes

CHAPTER 10: DESIGN OF A REINFORCED CONCRETE BUILDING ACCORDING TO EUROCODES Pietro Croce1 1

Department of Civil Engineering – Structural Section, Pisa, Italy

Summary The design of a reinforced concrete building according Eurocodes is illustrated in detail. In the worked example, starting from the relevant actions and loading given in EN 1991, the most sever effects for ULS and SLS in each structural members are derived according to EN1990. Some relevant static verification, performed according EN1992-1-1, is finally illustrated.

1

INTRODUCTION

Aim of the paper is to illustrate the design of a reinforced concrete building according to Eurocodes, with reference to a relevant worked example concerning a real case. Starting from the relevant actions and loadings determined according to EN 1991 – Actions on structures and using the suitable combination formulae given in EN 1990 – Basis of design for SLS and ULS, the most severe effects in the various structural members are determined in order to perform the pertinent verifications. In the worked example some relevant static verification is finally illustrated, according to EN 1992-1-1 – Design of concrete structure – General rules and rules for building.

2

THE BUILDING

The reinforced concrete building is a four-storied residential building, 22.00x10.00 m in plan, with a total height of 16.25 m above the ground. The building is located in the Central part of Italy, in a town located at altitude of 624 m above the sea level. The structural scheme of the building is a 3-D r.c. frame. The columns are rectangular with a cross section of 300x300 mm, except the corner ones whose section is 300x400 mm; perimeter r.c beams are 300 mm wide and 450 mm height, while internal beams, 400 mm wide, have the same thickness of the floors. Resistance against horizontal forces is entrusted to r.c. shear walls. Foundations are made with a multi-cell r.c. stiffened plate, lightened with polystyrene blocks: the r.c. intrados slab is 250 mm thick, while the 200 mm thick r.c. extrados slab makes up also the basement floor. Floor slabs of elevation floors as well as pitches of the pavilion roof are composed of prefabricated lattice joists infilled with brick elements, 200 mm thick, topped and completed with a cast in-situ reinforced-concrete slab, 40 mm thick, so that their total thickness is 240 mm. Cantilevered balconies are realized with a r.c. slab., 150 mm thick. Steps of the r.c. stairs are conceived as cantilevers restrained on the adjacent r.c. shear walls.

162

Chapter 10: Design of a reinforced concrete building according to Eurocodes

A typical floor plan is represented in fig. 1, while more general information can be derived from the drawings reported in the annex to the present chapter. 0,35

2,61

0,62 1,08

9,84 0,3

1,03

2,8

0,4

2,51

2,52

2,57

8,69

3

0,4

3,49 0,3

1,38 0,95

2,64

2,64

3,04

2,64

21,84 1,18

5,65

0,62

4,78

3,37

0,4

0,3

4,06

2,64 21,84

0,3

0,4

2,8

3,63

3,38

1,76

0,3

9,1

2,3

2,2

0,4

4,93

4,11

2,3

0,3

1,08

2,87 0,3

0,89 0,3

4,01 0,3

0,4

2,44

3,49

8,69

3

2,2 0,4

0,3

2,3

9,1

40

0,3

1,76

3,38

3,58

2,8

0,62

0,4

0,3

1,53 1,42

0,4

4,78

0,3

2,51

4,06

4,01

0,3

1,02

2,8

1,08

0,4

0,35

2,61

0,3

0,3

0,3

2,87

2,52

0,3

0,91

0,4

1,08

0

0,62

Figure 1. Typical floor plan

163

Chapter 10: Design of a reinforced concrete building according to Eurocodes

3

ACTIONS, LOADINGS AND LOAD COMBINATIONS

The relevant elementary actions and loadings considered are: 1. Density and self weights (EN 199-1-1 – Eurocode 1: Actions on structures - Part 1-1: General actions - Densities, self-weight, imposed loads for buildings) 2. Imposed loads (EN 199-1-1 – Eurocode 1: Actions on structures - Part 1-1: General actions - Densities, self-weight, imposed loads for buildings) 3. Snow loads (EN 199-1-3 – Eurocode 1: Actions on structures - Part 1-3: General actions – Snow loads) 4. Wind actions (EN 199-1-4 – Eurocode 1: Actions on structures - Part 1-3: General actions – Wind actions) Due to the light color of the building and to its very efficient thermal insulation thermal actions result not significant and therefore they have been disregarded. 3.1

3.2

Density and self weight Steel Reinforced concrete Floor slabs (20+4 cm) Flooring and floor foundation Finishing of balconies Equivalent UDL for partition walls Curtain walls

78.5 25.0 3.2 1.0 2.4 1.4 2.8

kN/m3; kN/m3; kN/m2; kN/m2; kN/m2; kN/m2; kN/m2.

Uniformly distributed loads Rooms in residential building (Category A) Attic (not suitable for residential use) Maintenance load on roof (covered by snow load) Stairs in residential building (Category A) Balconies in residential building (Category A)

2.0 1.0 1.0 4.0 4.0

kN/m2; kN/m2; kN/m2; kN/m2; kN/m2.

Concentrated loads Rooms in residential building (Category A) Attic (not suitable for residential use) Maintenance load on roof (covered by snow load) Stairs in residential building (Category A) Balconies in residential building (Category A)

3.0 1.5 1.5 4.0 3.0

kN; kN; kN; kN; kN.

Imposed loads

The above mentioned loads values are the maxima foreseen in EN 1991-1-1 and, except the UDL for category A, are higher than the recommended ones, according to Italian national determination. 3.3

Snow load Being the altitude as=624 m above sea level, the snow load on the ground, according to the Italian snow map, it results: ⎡ ⎛ as ⎞ 2 ⎤ qsk = 1.35 ⎢1 + ⎜ 2.80 kN/m2; ⎥ ⎟ ⎣⎢ ⎝ 602 ⎠ ⎦⎥ shape coefficient μ=0.8

164

Chapter 10: Design of a reinforced concrete building according to Eurocodes

CE=1.0 Ct=1.0 qs = μi ⋅ qsk ⋅ CE ⋅ Ct = 2.24 kN/m2 .

exposure coefficient thermal coefficient snow load on the roof 3.4

Wind actions

The terrain category around the building, according to EN1991-1-4, is III (see table 1), while the reference wind velocity vref is 29 m/s, so that exposure coefficient ce(z) is equal to 1.703 for z=10 m and equal to 2.028 for z=16.25 m (fig. 2). Table 1. Terrain categories according EN1991-1-4 Terrain category

z0 [m]

zmin [m]

0

Sea or coastal area exposed to the open sea

0.003

1

I

Lakes or flat and horizontal area with negligible vegetation and without obstacles

0.01

1

II

Area with low vegetation such as grass and isolated obstacles (trees, buildings) with separations at least 20 obstacle heigths

0.05

2

Area with regular cover of vegetation or buildings or with isolated obstacles with separations of maximum 20 obstacle heights (such as villages, suburban terrain, permanent forest)

0.3

5

III

IV

Area in which at least 15% of the surface is covered with buildings and their average heights exceeds 15 m

1.0

10

Figure 2. Determination of exposure coefficient The peak pressure distribution qp(z) can be assumed uniformly distributed and equal to 1.07 kN/m2 for wind blowing in the transverse direction (h≤b) or equal to 0.9 kN/m2 below 10 m and equal to 1.07 kN/m2 between 10 m and 16.25 m, for wind blowing in the longitudinal direction (fig. 3).

165

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With reference to the relevant key zones of the building shown in figure 4, shape coefficients cp are summarized in table 2.

Figure 3. Peak wind pressure

Figure 4. Key zone for buildings Table 2. Shape coefficients for relevant key zones of the building Zone A B C D E

166

LT [m] 4.4 5.6

Transversal wind -1.2 -0.8 0.8 -0.531

LL [m] 2 8 12

Longitudinal wind -1.2 -0.8 -0.5 0.765 -0.365

Chapter 10: Design of a reinforced concrete building according to Eurocodes

3.5

Load combinations for ULS verifications The most severe effects are evaluated, alternatively, using formula (6.10) of EN 1990

∑γ j ≥1

G, j

Gk, j + γ P P +γ Q,1Qk,1 + ∑ γ Q,iψ 0,i Qk,i ,

(1)

i≥2

or the less favourable of the two expressions (6.10a) and (6.10b) ⎧∑ γ G, jGk, j + γ P P +γ Q,1ψ 0,1Qk,1 + ∑ γ Q,iψ 0,i Qk,i ⎪ j ≥1 i≥2 , ⎨ + + + ξ γ G γ P γ Q γ ψ Q ∑ P Q,1 k,1 Q,i 0,i k,i ⎪∑ j G, j k, j i≥2 ⎩ j ≥1

(2)

being the values of ψ0 factors those reported in table 3. For STR/GEO verifications in buildings, γG,j=1.35 when unfavourable or γG,j=1.00 when favourable and γQ,i=1.50 when unfavourable or γQ,i=0 when favourable, while ξj=0.85. Table 3. Values of ψ factors

Adoption of expressions (6.10a) and (6.10b) (2) instead of (6.10) (1) determines little difference in the action effects, as summarized in table 4, where the most adverse values of the bending moments in the second floor beams of the third transverse frame of the considered building, evaluated with the two alternative approaches are compared, considering imposed load as leading action. Imposed load is considered here as a free action applied only on the unfavourable part of the influence surface. Nevertheless, the action effects determined using equations (2) are lower than those obtained applying equation (1) and considering imposed load as a fixed action, reported in table 5. This implies that the literary interpretation of what stated in paragraph 6.2 of EN

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1991-1-1, reported in note 1, could determine a considerable increment of the number of load cases to consider in the analysis, which often is not justified. Clearly, it is designer responsibility to decide if this simplification is reasonable or not. In the present example no reduction has been considered for imposed loads, i.e. αA=αn=1. Table 4. Bending moments in 2nd floor beams of the 3rd frame

Beam 1 (L=4.40 m)

Beam 2 (L=5.05 m)

Section support A span support B support B span support C

Mmax [kNm] (6.10) -52.3 31.1 -59.7 -76.1 41.4 -68.6

Mmax [kNm] (6.10a and b) -47.9 28.3 -54.8 -69.8 37.8 -62.8

Table 5. Bending moments in 2nd floor beams of the 3rd frame considering imposed load as a fixed action and combining actions according to equation (6.10)

Beam 1 (L=4.40 m)

Beam 2 (L=5.05 m)

Section support A span support B support B span support C

M [kNm] (6.10) -51.4 29.9 -59.7 -76.1 40.6 -67.9

3.6

Load combinations and limitations for SLS verifications The SLS verifications for a reinforced concrete building are: a. limitation of stresses under serviceability conditions; b. crack control (LS for cracking); c. deflection control (LS of deformation). Assessments a. and b. depends on the exposure class of the environment, on the load combination to be considered, on the sensitivity of the reinforcement to the corrosion. For SLS three different load combinations are given in EN1990:

1

“6.2 Load arrangements (EN1991-1-1) 6.2.1 Floors, beams and roofs (1)P For the design of a floor structure within one storey or a roof, the imposed load shall be taken into account as a free action applied at the most unfavourable part of the influence area of the action effects considered. (2) Where the loads on other storyes are relevant, they may be assumed to be distributed uniformly (fixed actions). (3)P To ensure a minimum local resistance of the floor structure a separate verification shall be performed with a concentrated load that, unless stated otherwise, shall not be combined with the uniformly distributed loads or other variable actions. (4) Imposed loads from a single category may be reduced according to the areas supported by the appropriate member, by a reduction factor αA according to 6.3.1.2(10). 6.2.2 Columns and walls (1) For the design of columns or walls, loaded from several storeys, the total imposed loads on the floor of each storey should be assumed to be distributed uniformly. (2) Where imposed loads from several storeys act on columns and walls, the total imposed loads may be reduced by a factor αn according to 6.3.1.2(11) and 3.3.1(2)P.”

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(characteristic load combination)

∑G

+ P +Qk,1 + ∑ ψ 0,i Qk,i ,

(3)

∑G

k,j

+ P + ψ1,1Qk,1 + ∑ ψ 2,i Qk,i ,

(4)

∑G

k,j

+ P + ∑ ψ 2,i Qk,i ,

(5)

k,j

j ≥1

(frequent load combination)

j ≥1

(quasi-permanent load combination)

j ≥1

i≥2

i≥2

i ≥1

being the ψ values given in table 3. As the building is in the exposure class XC1 according to EN206-1, the environment can be classified as normally aggressive. a. limitation of stresses under serviceability conditions; According to EN 1992-1-1, stresses in concrete σc and in steel reinforcement σs should be limited as follows: in normal exposure classes: under the quasi-permanent load combination; σc≤0.45⋅fck σs≤0.8⋅fyk under the characteristic load combination; and in exposure classes XD (chlorides) XF (freeze) and XS (sea water): σc≤0.6⋅fck under the characteristic load combination; under the characteristic load combination. σs≤0.8⋅fyk b. crack control (LS for cracking); According to EN 1992-1-1, the maximum crack width should be limited as follows: in exposure classes XC0 and XC1 wmax≤0.4 mm under the quasi-permanent load combination; in exposure classes XC2, XC3, XC4, XD1, XD2, XS1, XS2 and XS3 wmax≤0.3 mm under the quasi-permanent load combination. c. deflection control (LS of deformation). According to EN 1992-1-1, the maximum deflection due to the quasi-permanent load combination should be limited to l/250, being l the span of the beam. This verification can be omitted when the span to effective depth ratio of the beam is below the limits given in table 7.4N of EN1992-1-1. 4

MATERIALS

Concrete:

C30/37 fck=30 MPa γc=1.5 fcd=20 MPa αcc=0.85 αcc⋅fcd =1.7

Steel reinforcement

B450C fyk=450 MPa γs=1.15 fyd=391.3 MPa

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Chapter 10: Design of a reinforced concrete building according to Eurocodes

4.1

Stress-strain diagrams SLS verifications For SLS verifications linear stress-strain diagrams are assumed for concrete in compression and for steel, setting Es/Ec=15. ULS verifications For ULS verifications non-linear stress-strain diagrams are assumed. An elastic-plastic diagram has been considered for reinforcing steel (figure 5), while the parabola-rectangle diagram shown in figure 6 has been assumed for concrete in compression.

Figure 5. Stress-strain diagram for reinforcing steel

Figure 6. Stress-strain diagram for concrete in compression 5

RESULTS OF THE STRUCTURAL ANALYSIS

The structural analysis has been performed via the finite element method using the SAP2000 v. 12.0 programme. Beams and columns have been modelled using 3D frame elements, while shear walls have been meshed with four nodes 3D shell elements. The multi cell box foundations have been also modelled using 3D shell elements for the slabs and the webs. The intrados slab, which transfers the load to the soil, has been considered supported by elastic springs, according to the Winkler model. Some example of the results for the ULS are summarized in figure 7, in terms of bending moments in the 2nd transversal frame, and in figure 8, in terms of normal force in the

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Chapter 10: Design of a reinforced concrete building according to Eurocodes

intermediate longitudinal frame, when imposed load is the leading action and transversal wind and snow are accompanying actions.

Figure 7. Bending moment diagram [kNm] (Leading action – imposed load)

Figure 8. Normal force diagram [kN] (Leading action – imposed load) 171

Chapter 10: Design of a reinforced concrete building according to Eurocodes

6

STATIC VERIFICATION EXAMPLES

6.1

Verification of the first floor beams The beams is rectangular, 400 m wide and 240 mm high. The 16 mm diameter reinforcing steel are 8 in total and they are symmetrically distributed ULS verification a. Bending moment The M-N interaction domain for the section is shown in figure 9. MSd=58.97 kNm – Af= A’f=4 φ16=804 mm2 – MSd66 kN, is attained for cot θ=2.5. Anyhow, shear assessment is satisfied if verifications are performed considering cot θ>1.3 (see figure 10). 400

300

VR[kN]

Vsd Stirrup failure

200

Concrete strut failure 100

0 1

1.5

cot θ

2

2.5

Figure 10.– VRd – cot θ curves for the beam SLS verifications a. limitation of stresses under serviceability conditions - SLS stress limitation in compressed concrete quasi-permanent load combination - M=35.01 kNm – Af= A’f=4 φ16=804 mm2 σc = 8.65 MPa 0) = P(g(X) < 0 ∩ H > 0) / P(H > 0) (4) As an example, consider a simply supported steel beam of the span L exposed to permanent uniform load g and variable load q. The beam has the plastic section modulus W and the steel strength fy. Using the partial factor method, the design condition Rd − Sd > 0 between the design value Rd of the resistance R and the design value Sd of the load effect S may be written as W fyk /γm - (γg gkL2/8 + γq qkL2/8) > 0 (5) where fyk denotes the characteristic strength, gk the characteristic (nominal) value of permanent load g, qk the characteristic (nominal) value of permanent load q, γm partial factor of the steel strength, γg the partial factor of permanent load and γq the partial factor of variable load. In analogy with (5) the limit state function g(X) follows as g(X) = R - S = W fy - ( gL2/8 + qL2/8) (6) where all the basic variables are generally considered as random variables described by appropriate probabilistic models. To verify its reliability the beam was investigated and a proof loading up to the level qtest was carried out. It is assumed that gact is the actual value of the permanent load g. If the beam resistance is sufficient, the information I obtained is described as I = {H > 0} = {W fy - (gact L2/8 + qtest L2/8) > 0} (7) where fy is the actual steel strength, gact the actual permanent load determined reasonably accurately by non-destructive methods). To determine the updated probability of failure P(F|I) using equation (4), it is necessary to assess the following two probabilities: P(g(X) < 0|H > 0)=P(W fy -(gL2/8 + qL2/8) < 0 ∩ W fy-(gact L2/8 + qtest L2/8) > 0) (8) 2 2 (9) P(H > 0) = P(W fy - (gact L /8 + qtest L /8) > 0) Additional assumptions concerning the basic variables are needed. Having the results of (8) and (9), the updated probability of failure P(g(X) < 0| H > 0) follows from (5). Alternatively, considering results of the proof test, the probability density function fR(r) of the beam resistance R = Wfy may be truncated below the proof load, as indicated in Figure 3.

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Annex B: Assessment of existing structures

fR(r)

updated resistance R

prior resistance R

r resistance adequate to proof load Figure 3. Effect of the proof loading on structural resistance Obviously, the truncation of structural resistance R decreases the updated probability of structural failure defined as pf = P(R − S < 0)

(10)

and increases, therefore, the updated value of structural reliability. 10.2

Bayesian method for fractile estimation

An important part of the deterministic assessment of existing structures is estimation of representative values of material properties defined as appropriate fractiles of an underlying theoretical distribution. Particularly in case of a limited number of test results, fractiles can be effectively estimated considering previous (prior) experience. Bayesian approach provides a consistent framework for updating the previous experience with test results. In the following the procedure described in ISO 12491 [5] is applied only. More general information can be found elsewhere [6, 11, 15]. The procedure accepted here is limited to a normal variable X, for which the prior distribution function Π’(μ,σ) of μ and σ is given as

[

]

⎧ 1 2 2 ⎫ Π′(μ , σ ) = Cσ −(1+ν ′+δ ( n′ ))exp⎨− ν ′(s′) + n′(μ − m′) ⎬ 2 2 σ ⎩ ⎭

(11)

where C is the normalising constant, δ (n') = 0 for n' = 0 and δ (n') = 1 otherwise. The prior parameters m', s', n', ν' are parameters asymptotically given as E(μ) = m', E(σ) = s', V(μ) =

s′ m′ n ′

, V (σ ) =

1 2ν ′

(12)

The parameters n' and ν' are independent and may be chosen arbitrarily (it does not hold that ν' = n' – 1). In equations (12) E(.) denotes the expectation and V(.) the coefficient of variation.

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Annex B: Assessment of existing structures

Equations (12) may be used to estimate unknown parameters n' and ν' provided the values V(μ) and V(σ) are estimated using experimental data or available experience. The posterior distribution function Π"(μ,σ) of μ and σ is of the same type as the prior distribution function, but with parameters m", s", n" and ν", given as n" = n' + n ν" = ν' +ν + δ (n’) m"n"= n'm' + nm 2 ν"(s") + n"(m")2 = ν'(s')2 + n'(m')2 + νs2 + nm2

(13)

where m and s are the sample mean and standard deviation, n is the size of the observed sample and ν = n − 1. The predictive value xp,pred of a fractile xp is then x p , Bayes = m′′ + t p s′′ 1 + 1 / n′′

(14)

where tp is the fractile of the t-distribution (see Table 2) with ν" degrees of freedom. If no prior information is available, then n'= ν'= 0 and the characteristics m", n", s", ν" equal the sample characteristics m, n, s, ν. Then equation (14) can formally be simplified to so called prediction estimates of the fractile given as x p ,pred = m + t p s 1 + 1 / n

(15)

where tp denotes again the fractile of the t-distribution (Table 2) with ν degrees of freedom. Furthermore, if the standard deviation σ is known (from the past experience), then ν = ∞ and s shall be replaced by σ. Table 2. Fractiles − tp of the t-distribution with ν degrees of freedom

ν 3 4 5 6 7 8 9 10

0.90 1.64 1.53 1.48 1.44 1.42 1.40 1.38 1.37

0.95 2.35 2.13 2.02 1.94 1.89 1.86 1.83 1.81

1−p 0.975 3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23

ν 0.99 4.54 3.75 3.37 3.14 3.00 2.90 2.82 2.76

0.995 5.84 4.60 4.03 3.71 3.50 3.36 3.25 3.17

12 14 16 18 20 25 30 ∞

0.90 1.36 1.35 1.34 1.33 1.32 1.32 1.31 1.28

0.95 1.78 1.76 1.75 1.73 1.72 1.71 1.70 1.64

1−p 0.975 2.18 2.14 2.12 2.10 2.09 2.06 2.04 1.96

0.99 2.68 2.62 2.58 2.55 2.53 2.49 2.46 2.33

0.995 3.06 2.98 2.92 2.88 2.85 2.79 2.75 2.58

In a numerical example a sample of n = 5 concrete strength measurements having the mean m = 29.2 MPa and standard deviation s = 4.6 MPa is used to assess the characteristic value of the concrete strength fck = xp, where p = 0.05. If no prior information is available, then n'= ν'= 0 and the characteristics m", n", s", ν" equal the sample characteristics m, n, s, ν. The predictive value of xp then follows from (15) as 1 + 1 × 4.6 = 18.5 MPa x p,pred = 29.2 - 2.13 × (16) 5 where the value tp = − 2.13 is taken from Table C.1 for 1 − p = 0.95 and ν = 5 − 1 = 4. When information from previous production is available, the Bayesian approach can be effectively used. Assume the following prior information:

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Annex B: Assessment of existing structures

m’ = 30.1 MPa, V(m’) = 0.50, s’ = 4.4 MPa, V(s’) = 0.28

(17)

It follows from equation (12) 2

1 1 ⎛ 4.6 1 ⎞ n′ = ⎜ ≈ 6 ⎟