Handbook-1_heritage Indistrial Structures

Miroslav Sýkora et al.

>>> Handbook on STRUCTURAL ASSESSMENT OF INDUSTRIAL HERITAGE BUILDINGS

A/CZ0046/2/0013 A s s e s s m e n t o f h i s t o r i c a l i m m o v a b l e s w w w . h er er i t a g e. e. c v u t . c z

HANDBOOK

STRUCTURAL ASSESSMENT OF INDUSTRIAL HERITAGE BUILDINGS

PARTNERSHIP:

Czech Technical University in Prague, Klokner Institute: Assoc. Prof. Jana Marková Prof. Milan Holický Dr. Karel Jung Dr. Miroslav Sýkora Norwegian University of Live Sciences, Sciences, Institute for Mathematics Mathematics and Technology: Prof. Thomas Thiis Ing. Andreas Flo Assoc. Prof. Knut Kvaal

Prague/Aas, August 2010

The project is supported by a grant from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism, Mechanism, the the Norwegian Norwegian Financial Financial Mechanism Mechanism and the Czech Czech state budget. budget.

© Czech Technical University in Prague, Klokner Institute Šolínova 7, 16608 Prague 6, Czech Republic ISBN 978-80-01-04609-8 Pages: 155

Foreword FOREWORD The project Assessment of historical immovables

Protection, conservation or renewal of historical immovables is becoming an important task for art historians, architects and civil engineers in many European countries. Inevitable part of a preservation of many historical immovables, including heritage structures such as industrial buildings, bridges and folk architecture, is assessment of their reliability and design of adequate repairs taking into account the actual structural conditions and expected performance. The research project A/CZ0046/2/0013 Assessment of historical immovables is aimed at developing the general methodology for the complex assessment of heritage structures with a particular focus on industrial buildings and bridges. The main goal of the project is to provide operational tools and background information for decision making concerning the protection, conservation, renewal and extended use of historical immovables. The primary target group includes researchers, designers, practicing engineers, cultural heritage management, local authorities and other specialists interested in preservation of industrial heritage. Outcomes of the project include: - papers in prestigious journals - active participation in international conferences, - theoretical background documents for standardisation, - handbook, seminar and lectures for life-long education - software products and - project web sites . In the period 2009-2010 the project is partly supported by the Research Support Fund (EEA Grants / Norway Grants), Czech state budget and by the partners. Project partners

The project is based on the partnership of the Czech Technical University in Prague Klokner Institute and the Norwegian University of Live Sciences - Institute for Mathematics and Technology. Klokner Institute of the Czech Technical University in Prague, leader of the project, is a research institution with an outstanding position in the following fields: - Theory of structural safety and risk assessment, - Structural diagnostics based on experimental mechanics, numerical analysis and verification of numerical models, - Material research of concrete, steel, masonry and composite materials, optimisation of material properties and determination of their functional characteristics, - Experimental analysis of actual properties of existing construction materials, - Durability of structures, assessment of environmental degradation processes and optimisation of interventions. So far the researchers of the Klokner Institute have participated (as leaders or co-leaders) in several international projects in the framework of the Copernicus, Leonardo da Vinci, Jean Monnet and Growth programs, in more than 30 projects supported by the Czech Science Foundation, in 4 research plans of the Ministry of Education, Youth and Sports of the Czech Republic and in several research projects of the Ministry of Transport and Ministry of Industry and Trade of the Czech Republic. More than 800 scientific publications have been elaborated and several patents have been registered in the framework of these projects during the last 5 years. Most of the projects and publications have been evaluated by reviewers as outstanding. 3

Foreword

The Norwegian University of Live Sciences (UMB) comprises 8 departments. The University is recognised as a leading international centre of knowledge, focused on higher education and research within environmental- and biosciences. Together with other research institutes established at Aas, UMB provides state-of-the-art knowledge based on a broad range of disciplines. These include Applied Mathematics and Statistics, Physics, Spatial Planning, Environment and Natural Resources, Landscape Architecture, Civil Engineering and Building Science. In total, UMB has about 2 600 students of which about 300 are PhD students. Annually, the University confers about 40 PhD degrees upon successful candidates. Of the 870 University staff, more than half hold scientific positions. Recent research projects include: - Rural building heritage – transformation of old rural buildings, - Building modelling and climatic adaptation of buildings, - Farm buildings in the arctic – climatic adaptation of farm buildings. Handbook

The Handbook is focused on the complex methodology for structural assessment of industrial heritage buildings. The following main topics are treated in particular: - Basis of assessment - Actions - Material and geometric properties - Deterioration models - Reliability analysis - Decision on construction interventions. In addition theoretical procedures are supplemented by several case studies provided in Annex A. Annex B describes basic statistical concepts and techniques. The Handbook is written in a user-friendly way employing only basic mathematical tools. A wide range of potential users of the Handbook includes practising engineers, designers, technicians, experts of public authorities and students. Prague, 2010

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Contents

HANDBOOK STRUCTURAL ASSESSMENT OF INDUSTRIAL HERITAGE BUILDINGS

Page FOREWORD............................................................................................................................ 3 CONTENTS.............................................................................................................................. 5 I INTRODUCTION .............................................................................................................. 9

1. 2. 3.

Industrial heritage............................................................................................. 10 Criteria for listing industrial heritage............................................................... 10 Importance of protection.................................................................................. 12

3.1 Industrial heritage buildings 3.2 Industrial heritage bridges

4.

12 12

Initiatives concerning the industrial heritage ................................................... 13 4.1 Initiatives on an international level 4.2 Initiatives in the Czech Republic

13 14

References .................................................................................................................... 16 II

BASIS OF ASSESSMENT ............................................................................................. 17

1. Introduction...................................................................................................... 18 2. Principles.......................................................................................................... 18 3. Investigation..................................................................................................... 21 4. Basic variables ................................................................................................. 22 5. Structural analysis ............................................................................................ 23 6. Verification ...................................................................................................... 23 7. Assessment in case of damage ......................................................................... 25 8. Report............................................................................................................... 25 References .................................................................................................................... 27 III

ACTIONS....................................................................................................................... 29

1. 2.

Introduction...................................................................................................... 30 Actions and effect of actions............................................................................ 30

2.1 Definition of actions 2.2 Effect of actions

3.

Classification of the actions ............................................................................. 31 3.1 3.2 3.3 3.4 3.5 3.6

4.

30 30

General Variation in time Origin Variation in space Nature or structural response Bounded and unbounded actions

Reference period and distribution of extremes ................................................ 32 4.1 Climatic actions 4.2 Imposed loads

5.

31 31 31 32 32 32 32 33

Characteristic values ........................................................................................ 35 5.1 General 5.2 Permanent actions 5.3 Variable actions 5.4 Imposed loads 5.5 Snow loads

35 35 37 37 38

5

Contents 5.6 Wind actions 5.7 Thermal actions

6.

38 41

Representative values of actions ...................................................................... 41 6.1 General 6.2 Combination value of a variable action 6.3 Frequent value of a variable action 6.4 Quasi-permanent value of a variable action

7. 8. 9.

41 42 42 42

Representation of the dynamic actions ............................................................ 42 Representation of fatigue actions ..................................................................... 44 Probabilistic models of actions ........................................................................ 44 9.1 9.2 9.3 9.4 9.5 9.6

Permanent actions Imposed loads Snow loads Wind actions Time-dependency of the climatic actions Model uncertainties of load effects

44 45 47 47 48 48

References .................................................................................................................... 50 IV

MATERIALS AND GEOMETRY............................................................................... 53

1.

Introduction...................................................................................................... 54

1.1 Background materials 1.2 General principles

2.

54 54

Characteristic values of material properties..................................................... 54 2.1 General 2.2 Determination of the characteristic values

3.

Estimation of the characteristic and design material properties....................... 56 3.1 Estimation from a theoretical model 3.2 Estimation from limited experimental data

4.

56 59

Probabilistic models of material properties...................................................... 63 4.1 Strengths 4.2 Model uncertainties of resistance

5.

54 55

63 64

Geometrical data .............................................................................................. 65 5.1 General 5.2 Probabilistic models of geometrical data

65 66

References .................................................................................................................... 68 V

DETERIORATION ........................................................................................................ 71

1. 2.

Introduction...................................................................................................... 72 Overview of weathering effects on building materials .................................... 73 2.1 Environmental effects 2.2 Pollution effects

3.

73 76

Overview of damage functions ........................................................................ 78 3.1 3.2 3.3

Stone recession Concrete Steel

79 80 82

References .................................................................................................................... 84 VI

RELIABILITY ANALYSIS ......................................................................................... 89

1. 2.

General principles ............................................................................................ 90 Uncertainties .................................................................................................... 90 6

Contents

3.

Reliability......................................................................................................... 91 3.1 General 3.2 Definition of reliability 3.3 Probability of failure 3.4 Reliability index

4.

91 91 92 93

Reliability verification ..................................................................................... 93 4.1 Deterministic methods 4.2 Probabilistic methods 4.3 Probabilistic updating

93 94 101

References .................................................................................................................. 103 VII

DECISION ON CONSTRUCTION INTERVENTIONS....................................... 105

VIII

CONCLUDING REMARKS ................................................................................... 113

1. Design of construction interventions ............................................................. 106 2. Target reliability levels .................................................................................. 106 3. Principles of the total cost minimisation........................................................ 107 4. Simplified estimation of failure cost .............................................................. 108 References .................................................................................................................. 110

ANNEX A - CASE STUDIES.............................................................................................. 117

1.

Textile mill..................................................................................................... 118

1.1 Probabilistic reliability analysis 1.2 Cost optimisation

2.

Steel roof........................................................................................................ 120 2.1 Deterministic verification 2.2 Probabilistic reliability analysis 2.3 Parametric study

3. 4. 5.

118 119 120 121 122

Steel beam...................................................................................................... 123 Assessment of concrete strength .................................................................... 124 Masonry strength............................................................................................ 125 5.1 5.2 5.3 5.4 5.5

Motivation Evaluation of tests Masonry strength in accordance with present standards Probabilistic analysis Concluding remarks

125 125 129 130 132

References .................................................................................................................. 133 ANNEX B - BASIC STATISTICAL CONCEPTS AND TECHNIQUES ...................... 135

1.

Introduction.................................................................................................... 136 1.1 Background materials 1.2 General principles

2.

136 136

Population and samples.................................................................................. 136 2.1 General 2.2 Sample characteristics 2.3 Distribution function 2.4 Population parameters

3.

136 136 137 138

Selected models of random variables............................................................. 140 3.1 Normal distribution 3.2 Lognormal distribution

140 141

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Contents 3.3 3.4 3.5 3.6

Gamma distribution Beta distribution Gumbel and other distributions of extreme values Function of random variables

144 146 148 152

References .................................................................................................................. 153 Appendix 1 - Probabilistic models of basic variables Appendix 2 - Statistical parameters of functions of random variables

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I INTRODUCTION

Chapter I - Introduction According to the Convention concerning the protection of the world cultural and natural heritage, adopted in 1972 by UNESCO, the following shall be considered as cultural heritage: – Monuments: architectural works, works of monumental sculpture and painting, elements or structures of an archaeological nature, inscriptions, cave dwellings and combinations of features, which are of outstanding universal value from the point of view of history, art or science, – Groups of buildings: groups of separate or connected buildings which, because of their architecture, their homogeneity or their place in the landscape, are of outstanding universal value from the point of view of history, art or science, – Sites: works of man or the combined works of nature and man, and areas including archaeological sites which are of outstanding universal value from the historical, aesthetic, ethnological or anthropological point of view. 1.

INDUSTRIAL HERITAGE

A number of factories, warehouses, power plants and other industrial buildings, built since the beginning of the Industrial Revolution in the second half of the eighteenth century, has been registered as industrial cultural heritage in the Czech Republic and abroad. Such structures are mostly of significant architectural, historic, technological or social value [1]. They often form part of the urban landscape and provide the cityscape with visual historical landmarks. However, insufficient attention seems to be paid to systematic recognizing, declaring and protecting the industrial heritage in many countries including the Czech Republic. This is an alarming situation as the lack of attention and awareness of the industrial structures may gradually lead to their extinction [2]. When out of use, the industrial heritage buildings are degrading and often turning into ruins. Re-use and adaptation of such buildings allow for integration of the industrial heritage buildings into a modern urban lifestyle and help protect cities’ cultural heritage [2]. These buildings are often adapted to become hotels, museums, residential parks, commercial centres etc. Several examples of successful reconversions are shown in Figs. I-1 to I-3. The handbook attempts to provide the framework for complex reliability assessment of such structures. In addition the handbook is aimed to increase awareness of the high architectural and cultural significance of the industrial buildings and bridges, indicating positive influence of their preservation on the sustainable development and promoting discussion among experts on sustainable use of the industrial heritage structures. 2.

CRITERIA FOR LISTING INDUSTRIAL HERITAGE [6]

According to the Industrial Buildings Selection Guide [6] key issues to address when considering industrial structures for designation include: – The Wider Industrial Context . Industrial structures should be considered in their wider setting as a part of the system where each element (building) plays its role. – Regional Factors. Regional perspective in the selection of buildings and sites should be considered to achieve a representative sample for each sector of an industry. It also requires the identification of regional specialism, and the study of survivals related to these industries.

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Chapter I - Introduction

Fig. I-1. School building [3].

Fig. I-2. Residential houses in Manchester [4].

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Fig. I-3. Residential houses in Wien [5]. – Integrated Sites. If the process to which a building is related involved numerous components, then the issue of completeness becomes overriding. For instance an exceptionally complete site (perhaps with water systems and field monuments as well as buildings) may provide such an exceptional context that it raises the importance of buildings that might otherwise not be listable. – Architecture and Process. An industrial building should normally reflect in its design (plan form and appearance) the specific function it was intended to fulfil. – Machinery. The special interest of some sites lies in the machinery. Where the machinery makes a building special, the loss of this will reduce its eligibility for listing. – Technological Innovation. Some buildings may have been the site of the early use of important processes, techniques or factory systems (e.g. coke-based iron production, mechanised cotton spinning, steam power applied to pumping etc.). Technological significance may also reside in the building itself rather than the industrial process it housed, e.g., early fire-proofing or metal framing, virtuoso use of materials etc. The works of noteworthy wheelwrights or engineers will be of equal importance to major architects. – Historic Interest . Where physical evidence of important elements of industrial history survives well, a high grade may be justified. In some cases historical association with notable achievements may be sufficient to list: much will depend on the force of the historical claims, and the significance of the persons or products involved at the site in question. – Rebuilding and Repair . In assessments for listing, a high level of reconstruction is sometimes the basis for a decision not to list. With industrial buildings, partial rebuilding and repair is often related to the industrial process and provides evidence for technological change that may in itself be significant enough to warrant protection and alteration can thus have a positive value. Note that specific industries such as engineering works, factories, mines, mills, warehouses may be valuable for different features that have to be taken into account. 3.

IMPORTANCE OF PROTECTION

3.1.

Industrial heritage buildings

Protection (including adaptations and re-use) of the industrial heritage structures is an important issue of the sustainable development. More specifically, it has been recognised in [1,7-9] that the protection and re-use may positively contribute to the sustainable development by: – Preservation of the cultural values and identity of locations, 12


– Recycling of all potential resources and avoiding wasting energy, making use of existing infrastructures, – Facilitating the economic regeneration of regions in decline that may provide psychological stability for communities facing a sudden high rate of unemployment. It follows that the protection has considerable ecological and social contexts that are becoming even more important due to the global shortage of energy, economic crisis and environmental protection. The re-use of the industrial heritage is thus no longer an isolated issue, but is of general significance and may directly represent policy of the sustainable development. The protection of historical sites and the full use of limited resources deserve considerable public attention and participation. The following critical factors to assure the sustainable use of the industrial heritage buildings include [10]: – Multidisciplinary approach to analysis and decision-making, – Iterative, incremental process, – Public-private partnerships, – “Begin with the end” (end-user goals and end goals for environment, economy and society), – Integration of the natural, remediated and built environments, – Life-cycle economics (assessing long-term costs and benefits and the value of noneconomic benefits). 3.2.

Industrial heritage bridges

About 350 bridges are included in the Czech Register of the Industrial Heritage. Road bridges have become an essential part of cities’ transport infrastructure. However, unfavourable environmental effects and ever-increasing traffic loads may yield severe deterioration of existing road bridges. Therefore, rehabilitation of bridges is presently an urgent issue of bridge engineers and responsible authorities in many European countries. Present expenditures on maintenance and rehabilitation are limited and seem to be inadequate [11]. 4.

INITIATIVES CONCERNING THE INDUSTRIAL HERITAGE

4.1.

Initiatives on an international level

The protection of the industrial heritage is a multidisciplinary topic including historical, architectonic, civil engineering and ecological aspects. First initiatives aimed at research and conservation of industrial buildings started in 1950s on an amateur basis [7]. In 1973 the Society for Industrial Archeology was established and the First International Congress on the Conservation of Industrial Monuments took place. When the third congress was held in 1978, the International Committee on the Conservation of the Industrial Heritage (TICCIH) was founded to promote preservation, conservation, investigation, documentation, research and interpretation of industrial heritage. This wide field includes the material remains of industry - industrial sites, buildings and architecture, plant, machinery and equipment - as well as housing, industrial settlements, industrial landscapes, products and processes, and documentation of the industrial society. Members of TICCIH come from all over the world and include historians, conservators, museum curators, researchers, students, teachers, heritage professionals and anyone with an interest in the development of industry and industrial society.

13


The recent cooperation of TICCIH and the International Council on Monuments and Sites (ICOMOS) has resulted in registration of more than 40 industrial sites in the World Heritage List, such as Völklingen Ironworks in Germany shown in Fig. I-4. 4.2.

Initiatives in the Czech Republic

In the Czech Republic numerous industrial heritage structures including structures of railways infrastructures, breweries, sugar factories, and other industrial structures were built in the period from 1870 to 1930. Due to historical reasons, views of Czech architects and civil engineers on protection of the industrial heritage are often considerably different [4] and an important issue may then be to achieve consensus on significance of the heritage value to be preserved. To provide a desired coordinating platform for experts in the fields of historicalstructural research, monument conservation, and the reconstruction of the industrial heritage buildings and sites, the Research Centre for Industrial Heritage has been established as an independent research institution of the Czech Technical University in Prague. The Centre the Czech local representative in TICCIH - compiles findings from various research fields, maintains a database of the Czech industrial monuments (containing more then 10 000 monuments) and seeks for new uses of the industrial heritage buildings.

Fig. I-4. Völklingen Ironworks in Germany. To promote discussions among experts from various fields, the international conference Vestiges of Industry (www.industrialnistopy.cz) is organised every two years. On 14


the occasion of the 3rd biennial conference held in 2005, the international cooperation relating to the conservation, documentation, promotion and interpretation of a common European industrial heritage was declared. The activities which deserve special attention include: – The promotion of education, knowledge and a deeper understanding of the industrial heritage by conferences, seminars and educational programmes, – The evaluation and conservation of industrial heritage, – The conversion of industrial heritage to new uses as a positive form of cultural potential with the objective of revitalising industrial regions, towns and brownfields in decline. The declaration is available on the conference web sites. In 2009 the Czech Technical University in Prague and the University of Applied Sciences in Ås (Norway) launched the research project Assessment of historical immovables described in the Foreword.

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REFERENCES

[1] TICCIH: The Nizhny Tagil Charter for the Industrial Heritage, Nizhny Tagil: The International Committee for the Conservation of the Industrial Heritage, 2003. . [2] Läuferts, M. & Mavunganidze, J.: Ruins of the past: Industrial heritage in Johannesburg. In Proc. STREMAH XI, ed. C.A. Brebbia, Ashurst Lodge: WIT Press, 2009, pp. 533-542. [3] de Bouw, M., Wouters, I. & Lauriks, L.: Structural analysis of two metal de Dion roof trusses in Brussels model schools. In Proc. STREMAH XI, ed. C.A. Brebbia, Ashurst Lodge: WIT Press, 2009, pp. 121-130. [4] Fragner, B.: Př ístupy k záchran ě pr ůmyslového dědictví v České republice (Approaches to protection of the industrial heritage in the Czech Republic - in Czech), Stavebnictví Vol. IV, No. 01/2010 (2010), pp. 16-18. [5] Fragner, B.: „Brownfield“ v souvislostech pr ůmyslového dědictví (Brownfield in relation to industrial heritage - in Czech), Vesmír Vol. 84, No. leden 2005 (2005), pp. 58-60. [6] English Heritage, Heritage Protection Department: Industrial Buildings Selection Guide. March 2007, English Heritage, 2007, p. 16. [7] Zhang, S.: Conservation and adaptive reuse of industrial heritage in Shanghai, Frontiers of Architecture and Civil Engineering in China Vol. 1, No. 4 (2007), pp. 481-490. [8] Schneider, B. & Osika, K.: Inspired by the past, built for the future conversion of a former substation, a monumental listed building. In Proc. CESB10, eds. P. Hájek, J. Tywoniak, A. Lupíšek, J. R ůžička and K. Sojková, Prague: Grada Publishing, 2010, pp. 5. [9] Sýkora, M., Holický, M. & Marková, J.: Advanced assessment of industrial heritage buildings for sustainable cities’ development. In Proc. CESB10, eds. P. Hájek, J. Tywoniak, A. Lupíšek, J. R ůžička and K. Sojková, Prague: Grada Publishing, 2010, pp. 9. [10] Brebbia, C. A. & Beriatos, E. (eds.): Brownfields IV, Ashurst Lodge: WIT Press, 2008. [11] Marková, J., Holický, M., Sýkora, M. et al.: Assessment of Bridges Registered as Industrial Heritage. In Proc. 5th Int. ASRANet Conf. Anonymous ASRANet Ltd., 2010, pp. 10.

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II BASIS OF ASSESSMENT

Chapter II - Basis of assessment

1.

INTRODUCTION

Assessment of industrial heritage buildings and bridges preceding reconversions and re-use of these structures is becoming a more and more important and frequent engineering task. General principles of sustainable development regularly lead to the need for extension of the life of a structure, in majority of practical cases in conjunction with severe economic constraints. That is why the assessment of industrial heritage structures often requires application of sophisticated methods, as a rule beyond the scope of traditional design codes. The new European standards EN Eurocodes are intended for the design of new structures. Supplementary rules for the verification of existing structures including those registered as the industrial heritage are still missing. A new Part of the EN Eurocodes on the assessment and retrofitting of existing structures is to be prepared as indicated within a medium-term strategy of the development of the EN Eurocodes [1]. This document is intended to cover the following topics: – Methodology of collecting, evaluating and updating data, – Recommendations for the verification applying the partial factor method and/or using directly the probabilistic methods consistent with EN 1990 [2], – Target reliability level of existing structures taking into account the residual life time, consequences and costs of safety measures, – Assessment based on satisfactory past performance, – Recommendations concerning intervention and report. General requirements and procedures for the assessment of existing structures based on the theory of structural reliability are provided in the International Standard ISO 13822 [3]. For practical applications, information on properties of various construction materials may be provided in National Annexes. Supplementary guidance can also be found in ISO 2394 [4], ISO 12491 [5], and in numerous publications [6-9]. Specific issues related to heritage structures are detailed in [10]. 2.

PRINCIPLES AND GENERAL FRAMEWORK OF THE ASSESSMENT

In general industrial heritage structures are subjected to the reliability assessment before reconversions due to: – Rehabilitation of an existing structure during which new structural members are added to the existing load-carrying system, – Adequacy checking in order to establish whether the existing structure can resist loads associated with the anticipated change in use of the facility, operational changes or extension of its design working life, – Repair of an existing structure, which has deteriorated due to time-dependent environmental effects or which has suffered damage from accidental actions, – Doubts concerning actual reliability of the structure - it has been recognised that many heritage structures do not fulfil requirements of present codes of practice. Under some circumstances, assessments may also be required by authorities, insurance companies or owners, or may be demanded by a maintenance plan. However, it appears that insufficient attention has been paid by experts to specific issues of the reliability assessment of industrial heritage structures so far. Differences between the assessment of industrial heritage structures and design of new structures that should be considered in the assessment include: – Social and cultural aspects - loss of cultural and heritage values, 18


– Economic aspects - additional costs of measures to increase reliability of a heritage structure in comparison with a new structure (at a design stage cost of such measures is normally minor while cost of strengthening is much higher), – Principles of the sustainable development - waste reduction and recycling of materials (these aspects may be more significant in case of the assessment), – Lack of information for the assessment - commonly, testing of the properties of materials is difficult, expensive, but also very important due to variability of properties and changes that may have occurred during the working life of a structure (influence of deterioration and damage). As a consequence, it is often required to minimise construction interventions and use original materials in rehabilitation and upgrades of industrial heritage structures. Effects of the construction process and subsequent life of the structure, during which it may have undergone alteration, deterioration, misuse, and other changes to its as-built (as-designed) state, must be taken into account. However, even though an existing structure may be investigated several times, some uncertainty in behaviour of the basic variables shall always remain. Therefore, similarly as in design of new structures, actual variation of the basic variables describing actions, material properties, geometric data and model uncertainties are taken into account by partial factors or other code provisions. Two main principles are usually accepted when assessing industrial heritage structures: 1. Currently valid codes for verification of structural reliability should be applied, original codes valid in the period when the structure was designed should be used only as guidance documents. 2. Actual characteristics of construction materials, actions, geometric data and structural behaviour should be considered, the original design documentation including drawings should be used as guidance documents only. The first principle should be applied in order to achieve similar reliability level as in case of newly designed structures. The second principle should avoid negligence of any structural condition that may affect actual reliability (in favourable or unfavourable way) of a given structure. It follows from the second principle that a visual inspection of the assessed structure should be made whenever possible. Practical experience shows that inspection of the site is also useful to obtain a good feel for actual situation and state of the structure. As a rule, the assessment need not to be performed for those parts of the existing structure that will not be affected by structural changes, rehabilitation, repair, change in use or which are not obviously damaged or are not suspected of having insufficient reliability. In general, the assessment procedure consists of the following steps (see Fig. II-1): – Specification of the assessment objectives required by the client or authority, – Scenarios related to structural conditions and actions, – Preliminary assessment, – Study of available documentation (design documentation, construction methods and events that might have altered structural behaviour), – Preliminary inspection, – Preliminary checks, – Decision on immediate actions, – Recommendation for detailed assessment, – Detailed documentary search, – Detailed inspection, – Material testing, connections, effects of deterioration, determination of actions, irreversible deflections, – Determination of structural properties, 19


Requests/Needs

Specification of the assessment objectives and plan Scenarios Preliminary assessment Study of documents and other evidence Preliminary inspection Preliminary checks Decisions on immediate actions Recommendation for detailed assessment No Detailed assessment? Yes

-

Detailed documentary search and review Detailed inspection Material testing and determination of actions Determination of properties of the structure Structural analysis Verification of structural reliability Yes Further inspection? No -

Reporting results of assessment

-

Judgement and decision

Periodical inspection Maintenance

Yes Sufficient reliability? No Intervention Construction - Repair - Upgrading - Demolition

Operation Monitoring - Change in use

Fig. II-1. General flow of the assessment of industrial heritage structures.

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– Structural analysis, – Verification of structural reliability, – Report including proposal for construction intervention. The sequence is repeated if necessary. When the preliminary assessment indicates that the structure is reliable for its intended use over the remaining life, a detailed assessment may not be required. Conversely, if the structure seems to be in dangerous or uncertain condition, immediate interventions and detailed assessment may be necessary. 3.

INVESTIGATION

Investigation of an industrial heritage structure is intended to verify and update the knowledge of the present condition (state) of a structure with respect to a number of aspects. Often, the first impression of the structural condition will be based on visual qualitative investigation. The description of possible damage of the structure may be presented in verbal terms like unknown, none, minor, moderate, severe, or destructive. Very often the decision based on such observation will be made by experts in a purely intuitive way. A better judgement of the structural condition can be made on the basis of (subsequent) quantitative inspections. Typically, the assessment of existing structures is a cyclic process, when the preliminary inspection is supplemented by subsequent detailed investigations. The purpose of the subsequent investigations is to obtain better knowledge about the actual structural condition (particularly in the case of damage) and to verify information required for determination of the characteristic and design values of important basic variables. For all inspection techniques, information on the probability of detecting damage, if present, and the accuracy of the results should be given. The statement from the investigation contains, as a rule, the following data describing – Actual state of the structure, – Types of construction materials and soils, – Observed damage, – Actions including environmental effects, – Available design documentation. A proof loading is a special type of investigation. Based on such tests one may draw conclusions with respect to: – The bearing capacity of the tested member under the test load condition, – Other members, – Other load conditions, – The behaviour of the system. The inference in the first case is relatively easy; the probability density function of the load bearing capacity is simply cut off at the value of the proof load. The inference from the other conclusions is more complex. Note that the number of proof load tests need not to be restricted to one. Proof testing may concern one member under various loading conditions and/or a sample of structural members. In order to avoid an unnecessary damage to the structure due to the proof load, it is recommended to increase the load gradually and to measure the deformations. Measurements may also give a better insight into the behaviour of the system. In general, proof loads can address long-term or time-dependent effects. These effects should be compensated by calculation. A diagnostic test may be used to verify or refine analytical or predictive structural models. Diagnostic testing attempts to explain why the structure is performing differently than assumed. The disadvantage of this method, as compared with the proof loading, is that the results are determined for service loads and need to be extrapolated to ultimate load levels.

21


When the structural damage is small or it is in the interior of the system, its detection should not be carried out visually. A useful tool is then dynamic testing (e.g. horizontal or vertical vibration testing of structural members or a whole structure) that is based on the fact that the damage or loss of integrity in a structural system leads to changes in the dynamic properties of the structure such as natural frequencies, mode shapes and damping. Dynamic measurements can give information on the position and severity of the damage that occurred. Generally, the eigenfrequencies decrease while the damping increases. 4.

BASIC VARIABLES

In accordance with the above-mentioned general principles, characteristic and representative values of all basic variables shall be determined taking into account the actual situation and state of the structure. Available design documentation is used as a guidance material only. Actual state of the structure should be verified by inspection to an adequate extent. If appropriate, destructive or non-destructive inspections should be performed and evaluated using statistical methods. For verification of the structural reliability using partial factor method, the characteristic and representative values of basic variables shall be considered as follows: 1. Dimensions of the structural elements shall be determined on the basis of adequate measurements. However, when the original design documentation is available and no changes in dimensions exist, the nominal dimensions given in the documentation may be used in the analysis. 2. Load characteristics shall be introduced considering the values corresponding to the actual situation verified by destructive or non-destructive inspections. When some loads were reduced, or removed completely, the representative values can be reduced or appropriate partial factors can be adjusted. When overloading has been observed in the past and no damage has occurred, it may be appropriate to increase adequately the representative values. This may be important in particular for industrial buildings. For industrial heritage structures with significant permanent actions, the actual geometry should be verified by measurements and the characteristic values of weight densities of materials should be obtained from statistical evaluation of test results. 3. Material properties shall be considered according to the actual state of the structure verified by destructive or non-destructive testing. In many cases it may be appropriate to combine limited new information from inspection and tests with prior information available from previous experience (material properties known from long-term production, performance of similar structures with similar exposure levels). Bayesian techniques [5], [6] and [11] provide a consistent basis for this updating. Analytical or predictive approaches used to determine structural resistance may be overly conservative due to neglected system effects, load redistribution etc. In these cases, proof, diagnostic or dynamic load tests may help to update information on structural properties [8]. When the original design documentation is available and no serious deterioration, design errors or construction errors are suspected, the characteristic values given in original design may be used. 4. Model uncertainties shall be considered in the same way as in design stage unless previous structural behaviour (especially damage) indicates otherwise. In some cases model factors, coefficients and other design assumptions may be established from measurements on the existing structure (e.g. structural stiffness). Thus, the reliability verification of an existing structure should be backed up by inspection of the structure including collection of appropriate data. Evaluation of prior

22


information and its updating using newly obtained measurements is one of the most important steps of the assessment. 5.

STRUCTURAL ANALYSIS

Structural behaviour should be analysed using models that describe actual situation and state of an existing structure. Generally the structure should be analysed for ultimate limit states and serviceability limit states using basic variables and taking into account relevant deterioration processes. All basic variables describing actions, material properties, load and model uncertainties should be considered as mentioned above. The uncertainty associated with the validity and accuracy of the models should be considered during assessment, either by adopting appropriate factors in deterministic verifications or by introducing probabilistic model factors in reliability analysis. When an industrial heritage structure is analysed, conversion factors reflecting the influence of shape and size effect of specimens, temperature, moisture, duration-of-load effect, etc., should be taken into account. The level of knowledge of the condition of structural members and their connections should be also considered. This can be achieved by adjusting the assumed variability in either the load carrying capacity of the members or the dimensions of their cross-sections, depending on the type of structure. When deterioration of a structure is observed, the deterioration mechanisms shall be identified and a deterioration model should predict changes in structural parameters due to foreseen structural loading, environmental conditions, maintenance practices and past exposures, based on theoretical or experimental investigation, inspection, and experience. Examples of unfavourable environmental effects and defects of structures due to degradation, accepted with modifications from [12], are listed in Tab. II-1. 6.

VERIFICATION

Reliability verification of an industrial heritage structure shall be made using valid codes of practice, as a rule based on the limit state concept. Attention should be paid to both the ultimate and serviceability limit states. Verification may be carried out using partial safet y factor or structural reliability methods with consideration of structural system and ductility of components. The reliability assessment shall be made taking into account the remaining working life of the structure, the reference period, and changes in the environment in the vicinity of the structure associated with an anticipated change in use. The conclusion from the assessment shall withstand a plausibility check. In particular, discrepancies between the results of structural analysis (e.g. insufficient safety) and the real structural condition (e.g. no sign of distress or failure, satisfactory structural performance) must be explained. It should be kept in mind that many engineering models are conservative and cannot be always used directly to explain an actual situation. The target reliability level used for the verification can be taken as the level of reliability implied by acceptance criteria defined in design codes. The target reliability level shall be stated together with clearly defined limit state functions and specific models of the basic variables.

23


Tab. II-1. Examples of unfavourable environmental effects and defects of structures due to degradation. Defects of structures due to degradation Unfavourable environmental effects

Concrete

Structural steel, aluminium, iron

Erosion

Cracking

Fatigue cracking

Abrasion

Reinforcement corrosion

Fracture cracking

Scour

Honeycombing

Corrosion

Weathering

Scaling

Wetting

Spalling

Leaks

Delamination

Efflorescence

Disintegration

Vegetation

Alkali-silica reaction

Freeze-thaw

Breaking-away Deterioration of protective coatings Damage to mortar coatings Stratification Deformation Deflections

Masonry Scaling, spalling and delamination Falling-out of units

Timber Splitting Decay

Deterioration of impregnants Elongated Friability bolt holes Corrosion of Disintegration metallic of mortar connectors Detachment Corrosion of metallic connectors Peeling of mortar coating Deformation Cracking

Deflections

The target reliability level can also be established taking into account the required performance level for the structure, the reference period and possible failure consequences. In accordance with ISO 2394 [4] the performance requirements for assessment of existing structures are the same as for design of a new structure. Lower reliability targets for existing structures may be used if they can be justified on the basis of economical, social and sustainable consideration (see Annex F to ISO 13822 [3]). An adequate value of the target reliability index β should be, in general, determined considering appropriate reference period [3]. For serviceability and fatigue the reference period equals the remaining working life, while for the ultimate limit states the reference period is in principle the same as the design working life specified for new structures (50 years for buildings). This general approach should be, in specific cases, supplemented by detailed consideration of the character of serviceability limit states (reversible, irreversible), fatigue (inspectable, not inspectable) and consequences of ultimate limit states (economic consequences, number of endangered people). If the structure does not satisfy the reliability requirements, the construction interventions may become necessary. Decision-making concerning construction interventions may be based on a cost-benefit analysis. Note that particularly in case of heritage structures, the use of original materials is the preferable in design of rehabilitation or repairs.

24


A common approach is a step-by-step assessment from a crude visible inspection to a detailed structural reliability analysis. Several levels of numerical assessment can be distinguished: – Linear analysis and verification, – Non-linear analysis, – System reliability analysis. It is recommended to start from a very simple level with a little effort and crude assumptions, and systematically increase the level of detail. If the structure did not pass the requirements of the codes, a more advanced numerical assessment method has to be applied. 7.

ASSESSMENT OF A DAMAGED STRUCTURE

For the assessment of a damaged structure, the following stepwise procedure is recommended: 1. Visual inspection. It is always useful to make an initial visual inspection of the structure to get a feel for its condition. Major defects should be reasonably evident to an experienced eye. In the case of very severe damage, immediate measures (like abandonment of the structure, immediate structural interventions) may be taken. 2. Explanation of observed phenomena. In order to be able to understand the present condition of the structure, one should simulate the damage or the observed behaviour, using a model of the structure and the estimated intensity of various loads or physical/chemical agencies. It is important to have available documentation with respect to design, analysis and construction. If there is a discrepancy between calculations and observations, it might be worthwhile to look for design errors, errors in construction, etc. 3. Reliability assessment. Given the structure in its present state and given the present information, the reliability of the structure is estimated, either by means of a failure probability or by means of partial factors. Note that the model of the present structure may be different from the original model. If the reliability is sufficient (i.e. better than commonly accepted in design), one might be satisfied and no further action is required. 4. Additional information. If the reliability according to step 3 is insufficient, one may look for additional information from more advanced structural models, additional inspections and measurements or actual load assessment. 5. Final decision. If the degree of reliability is still too low, one might decide to: – accept the present situation for economical reasons; – reduce the load on the structure; – repair the building; – demolish the structure. The first decision may be motivated by the fact that the cost for additional reliability is much higher for an industrial heritage structure than for a new structure. This argument is sometimes used by those who claim that a higher reliability should be generally required for a new structure than for an existing one. However, if human safety is involved, economical optimisation has a limited significance. 8.

FINAL REPORT AND DECISION

The final report on structural assessment and possible interim reports (if required) should include clear conclusions with regard to the objective of the assessment based on 25


careful reliability assessment and cost of repair or upgrading. The report shall be concise and clear. A recommended report format is indicated in Annex G to ISO 13822 [3]. If the reliability of an existing structure is sufficient, no action is required. Otherwise appropriate interventions should be proposed. Temporary intervention may be recommended and proposed by an engineer if required immediately. The engineer should indicate a preferred solution as a logical follow-up to the whole assessment in every case. It should be noted that a client in collaboration with a relevant authority should make the final decision on possible interventions, based on engineering assessment and recommendations. The engineer performing the assessment might have, however, the legal duty to inform a relevant authority if a client does not respond in a reasonable time.

26


REFERENCES

[1] CEN/TC 250 Structural Eurocodes & JRC (Joint Research Centre): The Eurocodes and the construction industry (medium-term strategy 2008 – 2013). January 2009. [2] EN 1990: Eurocode - Basis of structural design, Brussels: CEN, 2002. [3] ISO 13822: Bases for design of structures - Assessment of existing structures, Geneve, Switzerland: ISO TC98/SC2, 2003. [4] ISO 2394: General principles on reliability for structures, 2nd edition, Geneve, Switzerland: ISO, 1998. [5] ISO 12491: Statistical methods for quality control of building materials and components, 1st edition, Geneve, Switzerland: ISO, 1997. [6] Diamantidis, D.: Probabilistic Assessment of Existing Structures, Joint Committee on Structural Safety, RILEM Publications S.A.R.L., 2001. [7] Institution of Structural Engineers: Appraisal of existing structures, 2nd edition, Institution of Structural Engineers, 1996. [8] Bucher, C., Brehm, M. & Sørensen, J. D.: Assessment of Existing Structures and Life Extension. Working documents of SAFERELNET, 2005, p. 25. [9] Finnish Ministry of the Environment, Housing and Building Department: Probabilistic Calibration of Partial Safety Factors (Eurocode and Finish proposal). 2000. [10] ISO 13822: Bases for design of structures – Assessment of existing structures. Annex I Heritage structures, Draft compiled on 17 October 2008, Geneve, Switzerland: TC98/SC2/WG6, 2008. [11] JCSS: JCSS Probabilistic Model Code, Zurich: Joint Committee on Structural Safety, 2006. . [12] COST 345: Procedures Required for the Assessment of Highway Structures. Final Report, Reports of Working Groups 1-6, COST 345, 2004,.

27

III ACTIONS

Chapter III - Actions

1.

INTRODUCTION

This chapter provides principles for specifications of different types of actions applied commonly in assessment of industrial heritage structures. In general, the characteristic, design and representative values are defined as fractiles of appropriate theoretical models, taking into account their variation in time. The procedure to obtain the characteristic values of permanent and variable actions including wind, snow and temperature is described in detail. Basic principles and rules for specification of representative and design values of actions and their effect on existing structures including those listed as the industrial heritage are given in ISO 13822 [1]. Supplementary guidance can also be found in EN 1990 [2], ISO 2394 [3] and in the Designer's Guide to EN 1990 [4]. 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 [5-8]. Additional information can also be obtained from the CIB documents on actions on structures and from the material oriented Eurocodes EN 1992 to EN 1999. Probabilistic models of actions can be found in the JCSS Probabilistic Model Code [9]. For guidance on load combinations reference is made to EN 1990 [2] (deterministic combinations) and to the document [9] (probabilistic combinations). 2.

ACTIONS AND EFFECT OF ACTIONS

2.1.

Definition of actions

EN 1990 [2] defines actions as: 1. Set of forces (loads) applied to the structure, as for example the self-weight of structure itself or the wind pressure on a surface (direct action). 2. 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 a more complex representation of actions may be necessary, for example an action with fatigue effects must be represented by the number of cycles, the mean value of action and its amplitude. The actions on a structure may be mutually correlated and may also be correlated with resistance variables. When actions are originated from different sources, they often can be taken into account 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 dimensions, but commonly this is less important. Other example could be the case when the maximum wind actions are seasonal, occurring for instance in summer. Thus, it could make no sense to combine the maximum wind actions with snow loads. The actions that can 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 30


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. The action effect is related to the action and the properties of the structure. In general the design action effect can be expressed as: E d = E(γ F , F i rep,i, ad)

(III-1)

where γ F ,i = partial factor for the action F i; F rep,i = relevant representative value of the action F i; and ad = design value of relevant geometric dimension(s). 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 is increasing more progressively than the action, the partial factor should be applied to the action. This occurs in most cases. 2. If the action effect is increasing less than the action, the partial factor should be applied to the action effect that corresponds to the representative value of the action. 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: – Variation in time (section 3.2), – Origin (3.3), – Spatial variation (3.4), – Nature and/or the structural response (3.5), – Bounds (bounded or unbounded actions - section 3.6). 3.2.

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 an anticipated working life. The actions are classified as: – Permanent 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. self-weight of structures, fixed equipment and road surfacing, and indirect actions caused by shrinkage and uneven settlements, – Variable 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 floors, wind actions or snow loads, – Accidental A, usually of short duration, that is unlikely to occur with a significant magnitude on a given structure during the working life, but its consequences might be severe, e.g. earthquakes, fires, explosions, or impacts from vehicles. The concept of reference period will be explained later. 3.3.

Origin

As already mentioned in Section 3.1, 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.

31

Chapter III - Actions 3.4.

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 a free action. 3.5.

Nature or structural response

The static actions are those that do not cause significant acceleration of the structure or structural members. The d ynamic 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 an appropriate coefficient to take account of the dynamic effects. 3.6.

Bounded and unbounded actions

In some cases, an upper (or lower) bound of the action can be found and then one of these can be established as a representative value. For instance for 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 bounded. 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 the load given by the material of maximum density stocked up to the maximum possible height. 4.

REFERENCE PERIOD AND DISTRIBUTION OF EXTREMES

The definitions given for permanent and variable actions include the term “reference period” that is the time used as a basis for the statistical assessment of actions and timevarying resistances. For each type of action, depending on its characteristics, the whole working life of the structure may be split in several 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 can be accepted for the action in such a period. This means that the set of extremes coming from the extreme of one of each of such periods will form a sample of extremes from which an extreme distribution function can be derived. The extreme in any period will correspond to an independent realization of such a distribution of extreme 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, thermal actions etc. - a period of a year is generally adequate; i.e. it can be assumed that the annual extremes are mutually independent. If the distribution function of extremes to related one year is known, the distribution function of maxima in the whole working life, T , assuming the same distribution function for each reference period is given by: FQ,max( x) = [FQ( x)]T or FQ,min( x) = 1 - [1 - FQ( x)]T

32

(III-2)

Chapter III - Actions x) = distribution function of working life maxima; F Q,min( x x) = distribution where FQ,max( x x) = distribution of annual extremes. Simplification function of working life minima; and FQ( x is illustrated in Fig. III-1. 4.2.

Imposed loads

For these actions, a reference period corresponding to the change of owner or the change of use of a structure or part of it is generally accepted. In [9] an average value of 5-10 years for reference periods is indicated. This means that about 5 to 10 changes of use may be commonly expected expected during a 50-year 50- year working life as shown in Fig. III-2. 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 accepted for all the different periods included in the working life. It is assumed in [9] that the duration is exponentially distributed, and that then the number of changes in the working life has a Poisson distribution. With these assumptions, the distributions of the maxima and minima related to the working life are obtained as: x) = exp{-λ T T [1 - FQ( x x)]} or FQ,min( x x) = 1 – exp[- λ T FQ( x x)] FQ,max( x

(III-3)

T thus where λ = = average rate of changes in use per year; and T = working life. The product λ T thus represents the expected number of changes of use during 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 exceeded, respectively, as function of the fractiles of the x), are given in Tab. III-1 as a function of the mean number of changes. distribution of Q( x

Tab. III-1. Probabilities of the fractiles corresponding to the characteristic values. 5 7 10 λ T Qk,inf 0.010 0.007 0.005 Qk,sup 0.990 0.993 0.995 The fractiles in Tab. III-1 mean that the characteristic value to be accepted, for instance, Qk,sup, will correspond to the fractile 0.993 of the distribution in each reference period, assuming assuming an average numbers numbers of changes changes of 7. x), the characteristic values can be obtained from Assuming a normal distribution of Q( x the mean and standard deviation (or coefficient of variation) as follows: Qk,max = μ Q + k σ k V Q) σQ = μ Q(1 + k V

k V Q) σQ = μ Q(1 - k V or Qk,min = μ Q - k σ

(III-4)

where μ Q = mean value of Q; σ Q = standard deviation of Q; V Q = coefficient of variation of Q; and k = coefficient obtained from the fractiles of the standardized normal distribution. distri bution. Examples of the values of k as as function of the mean number of changes in the t he working life are given in Tab. III-2.

33


Fig. III-1. Model and distribution for a variable climatic cli matic action.

Fig. III-2. Model and distribution for an imposed load.

T . Tab. III-2. Variation of the coefficient k with with the mean number of changes λ T 5 7 10 λ T T k 2,32 2,44 2,57

As an example it is considered a building for which it is foreseen that changes of use will likely modify the imposed load given by the weight of non-structural members (heavy partitions) and equipment. This part of the imposed load is assumed to have the normal distribution with the mean value of 1,0 kN/m 2 and coefficient of variation of 0,15. For the mean number of changes during a working life equal to 7, the characteristic value follows from (III-4): Qk,max = 1,0(1 + 2,44 × 0,15) = 1,36 kN/m 2 which is 36 % higher than the mean value μQ and 9 % higher than the 95% fractile of the imposed load in one reference period.

34


f G

1,64σ G

σ G

1,64σ G

σ G

5%

5%

G Gk,inf

Gm

Gk,sup

Fig. III-3. Characteristic values of permanent actions.

5.

CHARACTERISTIC VALUES

5.1.

General

Basic representative value of an action F is the characteristic value F k k . When sufficient data to assess the characteristic value on the statistical basis are available, the characteristic value corresponds to a prescribed probability of not being exceeded (for unfavourable effects) during the reference period. Otherwise a nominal value or a value given in the project documentation may be considered provided that it is consistent with methods given in EN 1991. 5.2.

Permanent actions

The characteristic value of a permanent action G shall be assessed as follows: – If the variability of G during the working life can be considered small, one single value Gk equal equal to the mean value may be used; – If the variability of G is not small, two values shall be used: an upper value Gk,sup and/or a lower value Gk,inf . Generally the 5% fractile is accepted for Gk,inf and and the 95% fractile for Gk,sup as indicated in G. With these assumptions Gk,inf and Fig. III-3. Commonly normal distribution is assumed for G Gk,sup can be obtained from: Gk,sup = μG + 1,64σ G = μG(1 + 1,64V G) or Gk,inf = = μG – 1,64σ G = μG(1 – 1,64V G)

(III-5)

where μG = mean value; σ G = standard deviation; and V G = coefficient of variation of the distribution of G. According to ISO 13822 [1] permanent actions on existing structures should be determined considering actual structural dimensions and material properties. In addition foreseen modifications should be taken into account. When the original documentation is unavailable or does not provide sufficient information for load specification, permanent actions should be verified on the basis of site surveys and measurements (densities, dimensions). Characteristic values should be then determined using statistical methods.

35


For n samples g 1, g 2,…, g n, sample mean mG and sample standard deviation sG, the characteristic value Gk of the permanent action is assessed using the following relationships: mG

g =∑ ; i

n

sG

=

∑ ( g − m i

G

)2

n −1

; Gk = mG ± k n sG

(III-6)

Variation of the coefficient k n on the sample size is indicated in Tab. III-3 given in the Czech National Annex to ISO 13822 [1]. For determination of unfavourable effect of Gk the positive sign is applied, the negative sign is used otherwise. Tab. III-3. Variation of the coefficient k n with the sample size. Sample size n Coefficient k n Sample size n Coefficient k n 5 0,69 15 0,35 6 0,60 20 0,30 7 0,54 25 0,26 8 0,50 30 0,24 9 0,47 40 0,21 12 0,39 >50 0,18 For intermediate values of n the coefficient k n can be interpolated. The coefficient k n is given assuming a normal distribution of the permanent action. At least 5 measurements are needed according to ISO 13822 [1]. In cases of a lower sample size, it is recommended to critically compare an estimated sample standard deviation sG with previous results. In these cases statistical assessment cannot be made directly and an estimated characteristic value is bounded by the minimum value taken as the maximum test value for an unfavourable permanent action. As an indication for the assessment of industrial heritage structures EN 1991-1-1 [5] 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 considerably dependent on the conditions of the material (e.g. effects of humidity), 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 [9]. The mean value of the self-weight of one member is calculated on the basis of the nominal dimensions of the member and its mean density. As an example it is considered a beam of normal weight of concrete: the mean density can be taken as γ = 24 kN/m3 as given in EN 1991-1-1 [5]. In common 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: g k = 24ab 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 its selfweight and it is therefore necessary to take into account the lower and upper characteristic values. In [9] a coefficient of variation of 0.04 is recommended for the density of concrete. It then follows from (III-5): g k,sup = 24ab (1 + 1,64 × 0,04) = 25,6 ab kN/m or g k,inf = 24ab (1 - 1,64 × 0,04) = 22,4ab kN/m

36


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 a 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 the mean return period of 50 years. It is worth noting that this return period is not related with a generally accepted 50 years as 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 characteristic value. Assuming that the actions in all the years are independent and identically distributed, and considering a return period R and working life T (in years), the probability r of exceedence during the working life is given by: r = (1 – 1/ R)T (III-7) For R = 50 years and T = 50 years the probability of exceedence is r = 0,63. 5.4.

Imposed loads

In EN 1991-1-1 [5] two different types of imposed loads are considered: – Uniformly distributed load for global effects, – Concentrated load acting alone for local effects. The characteristic values of the uniformly distributed imposed loads qk and concentrated loads Qk are given in Tables 6.2 and 6.4 of EN 1991-1-1 [5] for various categories of intended use (Category A - residential, B - offices, C - areas where people can concentrate, D – shops etc.). 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 load 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 member is large, 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, a reduction factor α A due to the size of the area and a factor α n with respect to a number of stories can be used. The factor α A is given by: α A = ψ 0 × 5/7 + A0 / A ≤ 1

(III-8)

where ψ 0 = combination factor given in EN 1990 [2] (commonly 0.7); A0 = reference area equal to 10 m 2; and A = 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 α n given by: α n = [2 + ( n – 2) ψ 0] / n

where n = number of stories (> 2).

37

(III-9)


The simultaneous use of the α -factors is not allowed. Also, when the imposed load enters in a combination of actions as an accompanying action being multiplied by the corresponding ψ -factor, then the simultaneous reduction by an α -factor is not allowed. 5.5.

Snow loads

The characteristic load on a roof due to snow s is given in EN 1991-1-3 [6] by the relationship: s = μ i C e C t sk

(III-10)

where μ i = roof shape coefficient; C e = exposure coefficient; C t = thermal coefficient; and sk = characteristic value of the snow load on the ground. The exposure coefficient takes into account the situation of the referred roof to accumulate snow or not. It is generally taken as C e = 1,0. Only when the roof is on an open terrain – windswept, it is considered as C e = 0,8. When the roof is sheltered by higher buildings, the coefficient increases to C e = 1,2. The thermal coefficient describes the possibility snow melting due to the heat transmission from the building. The usual value is C t = 1,0. In case of poor thermal isolation such as for glass roofs, the lower value may be reduced according to the guidance provided in ISO 4355 [10]. The distribution of snow load on the roof is generally not uniform. The wind, even of low velocity, causes the snow drift from more exposed to more sheltered parts of the roof. EN 1991-1-3 [6] gives values of the snow load shape coefficient μ i depending on the angle of the roof and the different shapes: mono-pitched or pitched, single-span or multi-span, 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. In Annex C of EN 1991-1-3 [6] the European snow maps for various climatic regions are provided including 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. It should be noted that, based on fifty year return period, new CEN Member States have developed their snow maps which are included in their National Annexes to EN 1991-1-3 [6]. 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. 5.6.

Wind actions

EN 1991-1-4 [7] deals with the effects of wind on structures. The scope of this standard covers buildings of height up to 200 m 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

38


fluctuation can originate different effects depending on the building characteristics. For most buildings, only a quasi-static response of 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 some bridges. 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 v b,0 is the main variable used to determine the wind in a site. It is defined as the characteristic 10-minute 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. EN 1991-1-4 [7] does not provide maps of fundamental wind velocity; they are 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 v b as: v b = cdir cseason v b,0

(III-11) where cdir = directional factor; and cseason = seasonal factor, taking into account that wind in some directions may 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 q b is derived from the basic wind velocity as: q b = ρv b2 / 2

(III-12)

where ρ = density of the air taken as 1,25 kg/m 3. 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 ) co( z ) v b (III-13) where vm ( z ) = mean wind velocity at z reference height; cr ( z ) = roughness factor; and co( z ) = 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 ) = k r ln ( z/z 0) for z ≥ z min (III-14) where k r = 0,19( z 0 /z 0,II)0,07 is the terrain factor; z 0 = roughness length given for a terrain category; and z min = minimum height given for a terrain category. Five terrain categories are defined in Annex A of EN 1991-1-4 [7]. The value of peak velocity represents the effect of the average velocity in 10 minutes and the effect of the gust as shown in Fig. III-4. It is obtained from the mean wind velocity by multiplication by the gust factor G:

39

Chapter III - Actions v p( z ) = G vm ( z )

(III-15)

Fig. III-4. The peak and mean velocities.

where: G = 1 +

7k I

for z ≥ z min; and k I is the turbulence factor. ⎛ z ⎞ co ( z ) log⎜⎜ ⎟⎟ ⎝ z 0 ⎠ The peak velocity pressure in the relevant height becomes: q p( z ) = cr 2( z ) co2( z ) G2 q b (III-16) In common cases c0( z ) = k I = 1 may be taken. The wind forces can be determined on the basis of pressure or force coefficients. In the first approach 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:

∑w

⋅ Aref

(III-17)

(III-18)

F fr = cfr ⋅ q p( z e) ⋅ Afr

(III-19)

F w,e

= cs cd ⋅

e

surfaces

F w,i

=

∑ w ⋅ A i

ref

surfaces

where F w,e = external force; F w,i = internal force; F fr = friction force; cscd = structural factor; we = external pressure on a surface; wi = internal pressure on a surface; Aref = reference area for a surface; Afr = area of the external surface parallel to the wind; q p( z e) = peak velocity pressure at the reference height z e. In the second approach based on the force coefficients, the force on the whole structure or on one member can be calculated from the relationship: F w,e

= cs cd ⋅

∑c

f

⋅ Aref

(III-20)

elements

Guidance on specification of the coefficients for common types of buildings or members is provided in EN 1991-1-4 [7].

40


Thermal actions

EN 1991-1-5 [8] deals with thermal actions which are classified as variable and indirect actions. The load-bearing structural members shall be checked to ensure that thermal movement will not cause overstressing of the structure, either by the provision of movement joints or by including the effects of thermal actions in the assessment. The fundamental quantities for thermal actions are extreme (maximum and minimum) air temperatures in the shade at a 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 T 0, 2. A linearly varying temperature, given by the difference ΔΤ Μ between the temperatures of the external and internal surfaces of a cross section or layers, 3. A temperature difference ΔΤ p 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. Annex D of EN 1991-1-5 [8] gives expressions for specifications of the temperature profiles taking account the inner and outer environmental temperature as indicated in Tab. III4 and III-5. 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.

Season Summer Winter

Tab. III-4. Temperature in the inner environment T in Temperature T in T 1 (recommended value 20 °C) T 2 (recommended value 25 °C)

Tab. III-5. Temperature in the outer environment T out Temperature T out in Season Significant factor °C 0,5 T + T bright light surface max 3 0,7 Relative absorptivity depending on Summer T max + T 4 light coloured surface colour surface 0,9 T max + T 5 dark surface T min Winter T max, T min, T 3, T 4 and T 5 may be specified in the National Annex. 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 load combinations include a main variable action accompanied with other variable actions. At the point in time of the working life of the 41


structure when the main variable action could reach its maximum value (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.

Combination value of a variable action (

0Q k)

The combination value ψ 0Qk is presented as a product of the characteristic value multiplied by the coefficient ψ 0 ≤ 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.

Frequent value of a variable action (

1Q k)

The frequent value is represented as the product ψ 1Qk , 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 ψ 1 ≤ 1. 6.4.

Quasi-permanent value of a variable action (

2Q k)

The quasi-permanent value is represented as a product ψ 2Qk , 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 ψ 2 ≤ 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 Fig. III-5. 7.

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.

42


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.

Q

Characteristic value Qk Δt 1

Δt 2

Combination value Frequent value

Quasi-permanent value

0

1

2

Δt 3

Qk

Qk

Qk

Time

Fig. III-5. Schematic representation of a variable action and its representative values. 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. 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 Eigenfrequency of the structure. The models of dynamic analysis include: – A stiffness model, similar to the static one, – A damping model due to different sources, – An inertia model, taking into account the masses of structural and non-structural members.

43

Chapter III - 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.

PROBABILISTIC MODELS OF ACTIONS

This section aims to summarize conventional probabilistic models of actions based on excessive review of available literature. Proposed models represent average values of action and common structural conditions. Recent scientific publications are taken into account [9,1121]. The following conventional models are intended to be used as prior theoretical models that could be accepted for general reliability studies and calibration procedures in connection with foreseen revisions of codes [2,3,5-7]. The theoretical models may be used in timeinvariant reliability analyses of simple structural members. In addition, parameters describing time-variant properties of selected actions are provided for time-variant reliability analyses. 9.1.

Permanent actions

The characteristic value of self-weight is in general determined from nominal dimensions (which are normally equal to the mean dimensions) and from the mean densities. Therefore, in common cases the mean of self-weight is approximately equal to its nominal (characteristic) value. The actual dimensions of structural members may, however, differ (by several percents) from their nominal values depending on construction material and production as indicated in the Probabilistic Model Code [9]. The coefficient of variation of weight density (and self-weight) varies in common cases from 0.03 up to 0.10 [9,12]. Selfweight G of structural members may usually be determined as the product of a volume Ω and density γ : G = Ω γ

(III-21)

Both the volume Ω and density γ are random variables that may be described by a normal distribution [9]. The mean of the volume Ω is approximately equal to the nominal value (as a rule slightly greater), the mean of the density γ is usually well defined by a producer. Informative coefficients of variation are indicated in Tab. III-6; more extensive data are available in the JCSS Probabilistic Model Code [9]. The coefficient of variation wG of the resulting self-weight may be estimated as: wG2 = wΩ 2 + wγ 2 + wΩ 2 wγ 2

(III-22) The last term in equation (III-22) may be usually neglected. The data in Tab. III-6 should be considered as informative only. The coefficients of variation of Ω for concrete and timber depend strongly on a size (increasing with decreasing thickness of members) and type of material. Note also that variability of non-structural members may be considerably greater than self-weight of structural members. 44


Tab. III-6. Examples of the coefficients of variation (indicative values only). G Material Ω γ Steel (rolled) 0.03 0.01 0.032 Concrete (plate 300 mm thick, ordinary) 0.02 0.04 0.045 Masonry unplastered 0.04 0.05 0.064 Timber (sawn beam 200 mm thick, dry) 0.01 0.10 0.101 9.2.

Imposed loads

Characteristic value of the imposed load models in office areas recommended in [5] is within the range from 2 to 3 kN/m 2. The experimentally determined mean of sustained imposed load is 0.5 kN/m 2 which is approximately 0.2 X k . The standard deviation may vary in a broad range depending on the loaded area, influence coefficients and other factors [9,13]. The coefficient of variation 1.1 corresponds approximately to the loaded area of 50 m 2 and influence coefficient 1.4. Note that the coefficient of variation decreases with an increasing area. These parameters are derived considering typical office areas. They may, however, be used as a first approximation (prior information) for other types of imposed loads. In the time-variant analysis behaviour of most actions including imposed loads can be described by jump process with intermittencies [9,22,23]. Fig. III-6 shows a possible approximation of the time-dependency of an action. The imposed load is usually split into a sustained (long-term) p and intermittent (short-term) q action. Parameters of both the components (including a jump rate λ - the expected number of load renewals within a time unit - and 1/ μ - mean duration of “on”-state) should be taken from available documents [9,13]. Imposed load Q is usually described by a Gumbel distribution. Gamma and exponential distributions are also used for the sustained and intermittent loads, respectively [9]. The sustained load q is always present while the intermittent component p may be absent and in fact may be active rarely (for example few days per year only). The parameters of both the components including the jump rate λ of the sustained load, jump rate v of the intermittent load and d duration of “on”-state of the intermittent load are indicated in Tab. III-7 [9]. In accordance with [9] the standard deviation of the sustained load q may be determined as: σ q2 = σ V 2 + σ U2 A0 κ / A

E (t )

(III-23)

Action approximation

1/λ

t

1/ μ

Fig. III-6. Jump process with intermittencies.

45

Chapter III - Actions κ = 1

i

κ = 1,4

i

ξ

κ=2 κ =2

ξ κ = 2,4

η

η

ξ 0

ξ 0

1

1

Fig. III-7. Typical influence lines and corresponding factors κ . where σ V = standard deviation of the overall load intensity; σ U = standard deviation of a random field describing space variation of the load; A0 = reference area (20 or 100 m 2); A = loaded area; and κ = influence factor depending on structural arrangements including boundary conditions. A relationship similar to equation (III-23) may be used to determine the standard deviation σ p of the intermittent load p. In common cases the factor κ is within the interval from 1 to 2.4 [9]. Fig. III-7 shows typical influence lines and corresponding factors κ (κ = 2 is considered in the following example as a representative value). Tab. III-7. Parameters of the imposed load for various loading areas. Category Sustained load q Intermittent load p A0 μ q 1/ λ μ p 1/v σ V σ U σ U 2 2 2 2 2 2 m kN/m kN/m kN/m year kN/m kN/m year Office 20 0.5 0.3 0.6 5 0.2 0.4 0.3 Lobby 20 0.2 0.15 0.3 10 0.4 0.6 1 Residence 20 0.3 0.15 0.3 7 0.3 0.4 1 Hotel rooms 20 0.3 0.05 0.1 10 0.2 0.4 0.1 Patient room 20 0.4 0.3 0.6 5-10 0.2 0.4 1 Laboratory 20 0.7 0.4 0.8 5-10 Libraries 20 1.7 0.5 1 10 Classroom 100 0.6 0.15 0.4 10 0.5 1.4 0.3 Stores - first floor 100 0.9 0.6 1.6 1-5 0.4 1.1 1.0 - upper floors 100 0.9 0.6 1.6 1-5 0.4 1.1 1.0 Storage 100 3.5 2.5 6.9 0.1-1 Industrial - light 100 1 1 2.8 5-10 - heavy 100 3 1.5 4.1 5-10 Concentration of people 20 1.25 2.5 0.02

46

d

days 1-3 1-3 1-3 1-3 1-3 1-5 1-14 1-14

0.5


Symb. X

μ 1C e C t sg,1 sg,50 s1 s50 *

Tab. III-8. Snow load models. s = μ 1 C e C t sg Name of basic variable

Dist. Dim.

Shape and exposure coefficients Thermal coefficient

– –

N D

Annual extremes of snow load on ground 50-year extremes of snow load on ground Annual extremes of snow load on roof 50-year extremes of snow load on roof

X k

0.8 1

μ X

μ X / X k σ X

0.8 1

w X

1 1

0.12 0.15 – –

G

kN/m2 1.33 0.47*

0.35

0.33 0.70

G

kN/m2

1.48

1.11

0.33 0.22

G

kN/m2 1.06 0.38

0.36

0.27 0.72

G

kN/m2

1.11

0.32 0.27

–

–

1.19

Determined assuming the probability 0.02 of the characteristic value being exceeded by annual extremes.

9.3.

Snow loads

The statistical parameters of a snow load given in Tab. III-8 are based on the model in [6]: s = μ 1 C e C t sg

(III-24)

where μ 1 = load shape coefficient for the uniform snow load covering a whole roof area and for roof slope about 15°; C e = exposure coefficient; and C t = thermal coefficient. The characteristic snow load on the ground at a weather station sg is specified in maps. Tab. III-8 shows the statistical parameters of these coefficients and resulting snow actions assuming the middle value 0.7 of the coefficient of variation of annual extremes of snow load on ground sg,1. It should be noted that the coefficient of variation of annual extremes of snow on the ground may vary in the broad range from 0.30 up to 1.15 [9] depending on local conditions (coastal, inland and mountain regions). 9.4.

Wind actions

The statistical parameters of wind pressure w( z ) indicated in Tab. III-9 are derived considering the model in [7] and recommendations in [9]. Assuming a unit orography factor, wind pressure w( z ) can be written as: w( z ) = c pcg( z )cr ( z )2mq q b, cg( z ) = 1 + 7 I v( z ), q b = 0.5 ρ v b2

(III-25) where c p = pressure coefficient, which depends on geometry of a structure and the loaded area; cg( z ) = gust factor, which depends on the turbulence intensity I v( z ) defined in [7] - for terrain category II cg( z ≅ 7.5 m) = 2.4; cr ( z ) = roughness factor defined in [7] - for terrain category II cr ( z ≅ 7.5 m) = 0.95; mq = model coefficient introduced in [9], which describes the ratio between expected and computed values of the basic wind pressure q b; ρ = 1,25 kg/m2 is the air density; and v b = reference wind speed specified in wind maps (assumed as v b = 26 m/s). Tab. III-9 shows probabilistic models of all the coefficients and resulting wind actions assuming the reference wind speed v b = 26 m/s and the coefficient of variation of annual maxima of the wind speed 0.2, which is a middle value - it may vary from 0.10 up to 0.35 [9].

47


Tab. III-9. Wind load parameters assuming the terrain category II and z = 7.5 m. w = c pcgcr 2mq q b Symb. X c p cg 2 cr mq q b,1 q b,50 w1 w50 *

Name of basic variable

Dist. Dim.

X k

μ X

μ X / X k σ X

w X

Pressure coefficient Gust factor Roughness factor Model coefficient Annual extremes of basic wind pressure 50-year extremes of basic wind pressure Annual extremes of wind pressure (c p = 1) 50-year extremes of wind press. (c p = 1)

N N N N

– – –

nom 2.4 0.91 1

nom 2.4 0.73 0.8

1 1 0.8 0.8

0.1nom 0.24 0.073 0.16

0.1 0.1 0.1 0.2

G

kN/m2 0.42 0.20* 0.47

0.085

0.43

G

kN/m2 –

0.46

1.09

0.085

0.19

G

kN/m2 0.92 0.28

0.30

0.15

0.50

G

kN/m2 –

0.70

0.211

0.33

0.64

determined assuming the probability 0.02 of the characteristic value being exceeded by annual extremes.

9.5.

Time-dependency of the climatic actions

Similarly as for the imposed loads, time-variant behaviour of the wind and snow loads may be described by jump processes with intermittencies shown in Fig. III-6. The jump rate λ and mean duration 1/ μ should be taken from available documents [9,14,15], considering available local information. Indicative values based on the data provided in [9] and judgement are given in Tab. III-10. However, these values should be considered as informative only. In the case of absence of representative information, the lower and upper bounds should be used to assess effect of this uncertainty on the resulting reliability level. Numerical experience reveals that changes in the mean 1/ μ have commonly a minor effect on the resulting reliability level. Tab. III-10. Indicative values of the jump rate λ and the mean duration 1/ μ . Wind Snow Type of climate λ [1/year] 1/ μ [hours] λ [1/year] 1/ μ [hours] Continental 5 to 15 4 to 24 2 to 5 24 to 144 Maritime 10 to 20 8 to 48 1 to 3 12 to 72 Mountainous 12 to 24 12 to 72 12 to 24 24 to 144 9.6.

Model uncertainties of load effects

Model uncertainties in action effects are conventionally expressed through the coefficients defined as the ratio of observed and model values [9]. In some cases little data are available and often professional judgments and experience are taken into consideration. The information given in Tab. III-11 is adopted from the latest documents of JCSS. In all the cases log-normal distribution having the origin at zero is recommended [9]. The data should be, however, considered as indicative only and ought to be verified taking into account structural conditions and available experimental data. It is expected that the coefficients of model uncertainties will be further developed by JCSS considering new experimental data.

48


Tab. III-11. Model uncertainties for load effects. Model type Mean CoV Axial force in frames 1.00 0.05 Moment and shear in frames, forces in 1.00 0.10 plates Moments in plates 1.00 0.20 Stresses in 2D and 3D solids 1.00 0.05 Deflection of steel structures 1.00 0.07 Deflection of concrete structures 1.00 0.10 Crack width in concrete 1.00 0.30

49


REFERENCES

[1] ISO 13822: Bases for design of structures - Assessment of existing structures, Geneve, Switzerland: ISO TC98/SC2, 2003. [2] EN 1990: Eurocode - Basis of structural design, Brussels: CEN, 2002. [3] ISO 2394: General principles on reliability for structures, 2nd edition, Geneve, Switzerland: ISO, 1998. [4] Gulvanessian, H., Calgaro, J. A.- Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design, London: Thomas Telford, 2002. [5] EN 1991-1-1: Eurocode 1: Actions on structures - Part 1-1: General actions; Densities, self-weight, imposed loads for buildings, Brussels: CEN, 2002. [6] EN 1991-1-3: Eurocode 1: Actions on structures - Part 1-3: General actions; Snow loads, Brussels: CEN, 2003. [7] EN 1991-1-4: Eurocode 1: Actions on structures - Part 1-4: General actions - Wind Actions, Brussels: CEN, 2005. [8] EN 1991-1-5: Eurocode 1: Actions on structures - Part 1-5: General actions; Thermal Actions, Brussels: CEN, 2003. [9] JCSS: JCSS Probabilistic Model Code, Zurich: Joint Committee on Structural Safety, 2006. . [10] ISO 4355: Basis for design of structures - Determination of snow loads on roofs, Geneve: ISO, 1994. [11] Vrouwenvelder, A. C. W. M.: JCSS Probabilistic Model Code. In Proc. of Safety, Risk and Reliability - Trends in Engineering, Rotterdam: Balkema, 2001, pp. 65-70. [12] CIB: Actions on Structures, Self-Weight Loads. Publication 115, CIB, 1989. [13] CIB: Actions on Structures, Live Loads in Buildings. Publication 116, CIB, 1989. [14] CIB: Actions on Structures, Snow Load. Publication 141, CIB, 1995. [15] CIB: Actions on Structures, Wind Load. Draft of Publication W81, CIB, 1995. [16] SAKO: Basis of Design of Structures. Proposal for Modification of Partial Safety Factors in Eurocodes. Joint Committee of NKB and INSTA-B, 1999. [17] Sørensen, J. D., Hansen, S.O. & Nielsen, T.A.: Partial Safety Factors and Target Reliability Level in Danish Codes. In Proc. of Safety, Risk and Reliability - Trends in Engineering, Rotterdam: Balkema, 2001, pp. 179-184. [18] Holický, M. & Marková, J.: Verification of load factors for concrete components by reliability and optimization analysis: Background documents for implementing Eurocodes, Progress in Structural Engineering and Materials Vol. 2, No. 4 (2000), pp. 502-507.

50


[19] Caramelli, S., Cecconi, A., Croce, P., Salvatore, W. & Sanpaolesi, L.: Partial safety factors for resistance of steel elements to EC3 and EC4. Calibration for various steel products and failure criteria. Commissione Europea, 1997.

[20] Fajkus, M., Holický, M., Rozlívka, L. et al.: Random Properties of Steel Elements Produced in Czech Republic. In Eurosteel 1999, 1999, pp. 657-660. [21] Finnish Ministry of the Environment, Housing and Building Department: Probabilistic Calibration of Partial Safety Factors (Eurocode and Finish proposal). 2000. [22] Wen, Y. K.: Structural load modeling and combination for performance and safety evaluation, 1st edition, Amsterdam: Elsevier, 1990. [23] Rackwitz, R.: Computational techniques in stationary and non-stationary load combination - A re-view and some extensions, J Structural Engr Vol. 25, No. 1 (1998), pp. 120.

51

IV MATERIALS AND GEOMETRY

Chapter IV - Materials and geometry

1.

INTRODUCTION

1.1.

Background materials

Basic principles and rules concerning structural resistance and geometric data are given in ISO 13822 [1], EN 1990 [2] and ISO 2394 [3]. Additional information may be found in the material oriented Eurocodes EN 1992 to EN 1999 and in working materials of JCSS [4] focused on material properties and geometric data. 1.2.

General principles

The reliability principles of present procedures of the assessment are based on a general principle that all the basic variables are considered as random quantities having an appropriate type of probability distribution. Characteristics of material properties and geometric data are defined as fractiles of appropriate distributions. Guidance concerning models for material properties and geometric data is provided in the JCSS Probabilistic model code [4]. Useful information on material properties of existing structures may be found in National Annexes of ISO 13822 [1]. For instance in the Czech Annexes the characteristic values of various grades of structural, reinforcement and prestressing steel as well as design values of iron etc. are described in detail. MATHCAD sheets that supplement described computational procedures can be effectively used in practical applications of computational procedures for determination determination of characteristic and design values. 2.

CHARACTERISTIC CHARACTERISTIC VALUES OF MATERIAL PROPERTIES

2.1.

General

Properties of materials and soils constitute an important group of basic variables that may significantly affect the structural reliability. In the assessment the properties of materials (including soil and rock) or products are represented by characteristic values, which correspond to the prescribed probability of not being infringed (exceeded in an unfavourable sense). When a material property X is is an extremely significant variable, both the lower and upper characteristic values X k,inf and X k,sup k,inf and k,sup should be taken into account (see Fig. IV-1). In most cases the lower value X k,inf of material or product property is unfavourable. k,inf of Then the 5% (lower) fractile is usually considered as the characteristic value. There are, however, cases when an upper estimate of strength is required (e.g. for the tensile strength of concrete when the effect of indirect actions is analysed). In these cases the use of the upper characteristic value of the strength X k,sup k,sup is needed. When the upper value is unfavourable, then the 95% (upper) fractile f ractile is usually considered as the characteristic value.

54


1,64σ X

1.64

σ X

X

σ X

5%

5%

X k,inf k,inf

X

X k,sup k,sup

Fig. IV-1. Lower X k,inf and upper X k,sup k,inf and k,sup characteristic values of a material property X .

2.2.

Determination Determination of the characteristic values

A material property shall normally be determined from standardised tests performed under specified conditions. It is sometimes necessary to apply a conversion factor to convert the test results into values which can be assumed to represent the behaviour of the structure or the ground. These factors and other details of standardised tests are given in EN 1992 to 1999. For traditional materials, e.g. steel and concrete, previous experience and extensive tests are available and appropriate conversion factors are well established and presented in EN 1992 and 1993. Properties of materials or the soil parameters should be verified by tests. Assuming that the theoretical model for random behaviour of a material property is known or sufficient data are available to determine such a model, basic operational rules to determine specified fractiles are described below in this chapter. If only limited test data are available, then statistical uncertainty due to limited number of data should be taken into account and the above mentioned operational rules should be substituted by more complicated statistical techniques (see for example Annex D of EN 1990 [2]). According to EN 1990 [2] whenever there is a lack of information about the statistical distribution of the property a nominal value may be used in the assessment. In the case of insignificant sensitivity to the variability of the property a mean value may be considered as the characteristic value. Relevant values of material properties and their definitions are available in ENs 1992 to 1999. Note that stiffness parameters are normally defined as the mean value. This can be explained in the following way. Very often, stiffness parameters are used in interaction models (e.g. ground-structure interaction) or in finite element models associating several materials. The stiffness properties of materials cannot be altered by b y partial factors because the results of the calculation would be distorted. This is the reason for which it is recommended not to alter stiffness parameters. Nevertheless, in some cases (e.g. for the calculation of piles subject to horizontal forces at the top), it may be necessary to take into account a lower and an upper value of these parameters, generally assessed from an engineering judgment.

55


lower grade upper grade

Fig. IV-2. Distortion of probability density function due to combination of two material grades. In general, when the lower or upper characteristic value is derived from tests, the available data should be carefully examined for cases where the material (e.g. timber and steel components) is classified using a grading system with a number of classes in order to avoid the possibility that the manufacturer may have included the specimens failed for the upper grade into the lower grade, thus distorting the statistical characteristics (including the mean, standard deviation and fractiles) of the lower grade, see Fig. IV-2. Obviously such a "mixture" of two grades may significantly affect both the lower and upper characteristic value. Specific values of material and product properties are given in material oriented EN 1992 to 1999 where also appropriate partial factors are specified. Unless suitable statistical information is available a conservative value of partial factors shall be normally used. 3.

ESTIMATION OF THE CHARACTERISTIC AND DESIGN MATERIAL PROPERTIES

3.1.

Estimation from a theoretical model

As most material properties are random variables of considerable scatter, applied characteristic values should always be based on appropriate statistical parameters or fractiles. Commonly, for a given material property X the following statistical parameters are considered: the mean μ X , standard deviation σ X , coefficient of skewness (asymmetry) α x. In some cases also other statistical parameters, e.g. the lower or upper distribution limit are taken into account. In case of a symmetrical distribution (e.g. the normal distribution) the coefficient α X = 0 and the normal distribution characterised by the mean μ X and standard deviation σ X is usually considered. This type of distribution is indicated in Fig. IV-3 by a solid line. The characteristic and design values of material properties are defined as specified fractiles of the appropriate distribution. Usually the lower 5% fractile is assumed for the characteristic strength X k and a smaller fractile probability (around 0,1 %) is considered for the design value X d. If the normal distribution is assumed, the characteristic value X k , defined as the 5% lower fractile, is derived from the statistical parameters μ X and σ X as: X k = μ X –

1,64 σ X

56

(IV-1)


X =

+1

1,64σ X

σ X

σ X

5%

α X = 0

X k X k’

μ X

Fig. IV-3. Normal and lognormal distribution. where the coefficient -1,64 corresponds to the fractile probability 5 %. The statistical parameters μ X , σ X and the characteristic value X k are shown in Fig. IV-3 together with the normal probability density function of the variable X (solid line). The coefficient -3,09 should be used when the 0,1% lower fractile (design value) is considered. Generally, however, the probability distribution of the material property X may have asymmetrical distribution, usually with positive or negative skewness α X . The dashed line in Fig. IV-3 shows the general three parameter (one-sided) lognormal distribution having a positive coefficient of skewness α X = 1. The lower limit of the definition domain is then x0 = μ x − 1,34 σ x. In case of asymmetric distribution a fractile X p corresponding to the probability P may be then calculated from the general relationship: X p= μ X + k P ,α σ X (IV-2) where the coefficient k P,α depends on the probability P and on the coefficient of skewness α X . Assuming the three parameter lognormal distribution, selected values of the coefficient k P,α for determination of the lower 5% and 0,1% fractiles are indicated in Tab. IV-1. It follows from Tab. IV-1 and equation (IV-2) that the lower 5% and 0,1% fractiles for the normal distribution (when α X = 0) may be considerably different from those corresponding to an asymmetrical lognormal distribution. When the coefficient of skewness is negative, α X < 0, the predicted lower fractiles for lognormal distribution are less (unfavourable) than those obtained from the normal distribution with the same mean and standard deviation. When the coefficient of skewness is positive, α X > 0 (see Fig. IV-3), the predicted lower fractiles for lognormal distribution are greater (favourable) than those obtained from the normal distribution. A popular lognormal distribution with the lower bound at zero, which is used frequently for various material properties, has always a positive skewness α X > 0 given as: α X = 3V X + V X 3 (IV-3)

57


Tab. IV-1. The coefficient k P,α for determination of the lower 5% and 0,1% fractile assuming three parameter lognormal distribution. Coefficient of skewness α X -2,0 -1,0 -0,5 0,0 0,5 1,0 2,0 Coefficient k P,α for P =5 % -1,89 -1,85 -1,77 -1,64 -1,49 -1,34 -1,10 Coefficient k P,α for P =0,1 % -6,24 -4,70 -3,86 -3,09 -2,46 -1,99 -1,42

1 X p / X

X 0,05 / X 0.8 X 0,001 / X 0.6 0.4

α X

0.2 V X 0 0

0.05

0.10

0.15

0.20

Fig. IV-4. The skewness α x and fractiles X 0,05 and X 0,001 (the characteristic and design values) as fractions of the mean μ X for lognormal distribution with lower bound at zero versus coefficient of variation V X . where V X denotes the coefficient of variation of X . When, for example, V X = 0,15 (a typical value for in situ cast concrete) then α X ≅ 0,45. For this special type of distribution the coefficients k P,α can be estimated from data indicated in Tab. IV-1 taking into account actual skewness α x given by equation (IV-3). However, in this case of the lognormal distribution with lower bound at zero the fractile can be determined from the following equation: exp(k P ,0 √ln(1+V X 2))/√(1+V X 2) which is often approximated (for V X < 0,2) by a simple formula: X P = μ X

X P = μ X exp(k P ,0 V X )

(IV-4) (IV-5)

Note that k P ,0 is the coefficient taken from Tab. IV-1 for the skewness α x = 0 (as for the normal distribution). As mentioned above usually the probability P = 0,05 is assumed for the characteristic value, thus X 0,05 = X k , and the probability P = 0,001 is approximately considered for the design value, thus X 0,001 ≅ X d. Relative values of these important fractiles (related to the mean μ X ) determined using equation (IV-5) are shown in Fig. IV-4, where the ratios X 0,05/ μ X and X 0,001/ μ X are plotted as functions of the coefficient of variation V X . Fig. IV4 also shows corresponding skewness α x given by equation (IV-3). The skewness α x shown in Fig. IV-4 should be used as a sensitive indicator for verification of suitability of the lognormal distribution with lower bound at zero. If the actual skewness determined from available data is considerably different from that indicated in Fig. IV-4 (which is given by equation (IV-3) for a given V X ), then a more general three parameter lognormal or other types of distribution (for example the distribution of minimum values, type III, called also Weibull distribution) should be used. Nevertheless simple expression (IV-2) with the coefficients k P,α taken from Tab. IV-1 may provide a good approximation or a control 58


check. If the actual skewness is small, say | α X | < 0,1, then the normal distribution may be used as an approximation (expression (IV-2) with the coefficients k P,0). However, when the normal distribution is used and the actual distribution has a negative coefficient of skewness, α X < 0, the predicted lower fractiles will then have an unfavourable error (i.e. will be greater than the correct values). For the case when the correct distribution has a positive coefficient of skewness, α X > 0, the lower fractiles, estimated using the normal distribution, will have a favourable error (i.e. will be less than the correct values). However, in the case of the 5% lower fractile value (commonly accepted for the characteristic value) with the coefficient of skewness within the interval <-1, 1> the error is relatively small (up to 6 % for a coefficient of variation less than 0,2). Considerably greater differences may occur for the 0,1% fractile value (which is approximately considered for design values) when the effect of asymmetry is more significant than in case of 5% fractile. For example, in the case of a negative asymmetry with α X = -0,5 (extreme case but still indicated by statistical data for strength of some grades of steel and concrete), and a coefficient of variation of 0,15 (adequate to concrete), the correct value of the 0,1% fractile value corresponds to 78 % of the value predicted assuming the normal distribution. When the coefficient of variation is 0,2, then the correct value decreases to almost 50 % of the value determined assuming the normal distribution. However, when the material property has a distribution with a positive skewness, then the estimated lower fractile values obtained from the normal distribution may be considerably lower (and therefore conservative and uneconomical) than the theoretically correct value corresponding to appropriate asymmetrical distribution. Generally, the consideration of asymmetry to determine properties is recommended whenever the coefficient of variation is greater than 0,1 or the coefficient of skewness is outside the interval <-0,5, 0,5>. This is one of the reasons why the design value of a material property should be preferably determined on the basis of the characteristic value which is not significantly sensitive to the distribution asymmetry. When the upper fractiles representing upper characteristic values are needed, equation (IV-2) may be used provided that all numerical values for the coefficient of skewness α X and k p,α given in Tab. IV-1 are taken with the opposite sign. However, in this case the experimental data should be carefully checked to avoid the possible effect of material not passing the quality test for the higher grade and affecting the lower grade when included in the experimental data (see Fig. IV-2). 3.2.

Estimation from limited experimental data

The above operational rules are applicable when the theoretical model for the probability distribution is known (for example based on extensive experimental data and previous experience). If, however, only limited experimental data are available (common situation for the assessment of industrial heritage structures), then a more complicated statistical technique should be used to take account of statistical uncertainty due to limited information. In general the statistical uncertainty leads to more conservative estimates. General pri nci ples of statisti cal evaluati ons

According to EN 1990 [2] the behaviour of test specimens and failure modes should be compared with theoretical predictions when evaluating test results. When significant deviations from a prediction occur, an explanation should be sought by additional testing (perhaps under different conditions), and/or modification of the theoretical model. The evaluation of test results should be based on statistical methods, with the use of available (statistical) information about the type of distribution to be used and its associated parameters.

59


The methods described in the following text may be used only when the following conditions are satisfied: – The statistical data (including prior information) are taken from identified populations which are sufficiently homogeneous, – A sufficient number of observations is available. At the level of interpretation of tests results, three main categories can be distinguished: – Where one test only (or very few tests) is (are) performed, no classical statistical interpretation is possible. Only the use of extensive prior information associated with hypotheses about the relative degrees of importance of this information and of the test results, make it possible to present an interpretation as statistical (Bayesian procedures, see ISO 12491 [5] or the JCSS publication [6]), – If a larger series of tests is performed to evaluate a parameter, a classical statistical interpretation might be possible. This interpretation will still need to use some prior information about the parameter; however, this will normally be less than above. – When a series of tests is carried out in order to calibrate a model (as a function) and one or more associated parameters, a classical statistical interpretation is possible. The result of a test evaluation should be considered valid only for the specifications and load characteristics considered in the tests. If the results are to be extrapolated to cover other design parameters and loading, additional information from previous tests or from theoretical bases should be used. It is hereafter assumed that larger series of tests are performed to assess a single material property. For other cases reference is made to Annex D of EN 1990 [2], ISO 13822 [1], ISO 12491 [5] and Designer’s Guide [7]. Deri vation of desi gn valu es

According to EN 1990 [2] the derivation from tests of the design values for a material property, a model parameter or a resistance should be carried out in one of the following ways: a) By assessing a characteristic value, which is then divided by a partial factor and possibly multiplied if necessary by an explicit conversion factor, b) By direct determination of the design value, implicitly or explicitly accounting for the conversion of results and the total reliability required. In general method a) is to be preferred provided the value of the partial factor is determined from the normal design procedure. The derivation of a characteristic value from tests (Method (a)) should take into account the scatter of test data, statistical uncertainty associated with the number of tests and prior statistical knowledge. The partial factor to be applied to a characteristic value should be taken from the appropriate Eurocode provided there is sufficient similarity between the tests and the usual field of application of the partial factor as used in numerical verifications. The single material property X may represent a resistance of a structural member or a property contributing to the resistance. It is further assumed that: – All variables follow either a normal or a lognormal distribution with the origin at zero, – There is no prior knowledge about the value of the mean, – For the case " V X unknown", there is no prior knowledge about the coefficient of variation, – For the case " V X known", there is full knowledge of the coefficient of variation. It is noted that adopting a lognormal distribution has the advantage that no negative values can occur. In practice, it is often preferable to use the case " V X known" together with a

60


conservative upper estimate of V X , rather than to apply the rules given for the case " V X unknown". Moreover V X , when unknown, should be assumed to be not smaller than 0,10. Assessment vi a the char acter isti c valu e

In accordance with the Annex D of EN 1990 [2] (see also ISO 13822 [1], ISO 2394 [3], ISO 12491 [5] and Designer’s Guide [7]), the design value of a property X assessed via the characteristic value should be found by using: for normal distribution: X d = η d

X k(n)

=

η d m X (1 - k n V X ) γ m

γ m η for lognormal distribution: X d = d exp(m y − k n s y ) γ m

(IV-6)

where η d = design value of the conversion factor; k n = coefficient given in Tab. IV-2; m X = Σ xi / n is the sample mean; and m y = Σln( xi) / n. When using Tab. IV-2, one of two cases should be considered as follows: – The row "V X known" should be used if the coefficient of variation V X , or a realistic upper bound of it, is known from prior knowledge that might come from the evaluation of previous tests in comparable situations, assessment of similar structures, long-term material production etc. – The row "V X unknown" should be used if the coefficient of variation V X is not known from prior knowledge and needs to be estimated from the sample as: 1 ( xi − m X )2 ∑ n −1 i 1 for lognormal distribution: s y = (ln xi − m y ) 2 ∑ n −1 for normal distribution: s X 2 =

(IV-7)

V X = s X / m X

(IV-8) If V X is known from prior knowledge, the standard deviation s y for lognormal distribution is estimated as: s y

= ln(V X 2 + 1) ≈ V X

(IV-9)

Note that the term “prediction method” is used in ISO 12491 [5] while “Bayesian procedures with vague prior distributions” are referred to as in EN 1990 [2]. Di r ect assessment of the desi gn valu e for UL S ver if icati ons

In Method b) it should be considered that the design value of the resistance also includes: – The effects of other properties, – The model uncertainty, – Other effects (scaling, volume, etc.). The design value X d should be found by using: for normal distribution: X d = η d m X (1 − k d,nV X ) for lognormal distribution: X d =

η d exp(m y − k d ,n s y ) γ m

(IV-10)

where k d,n = coefficient obtained from Tab. IV-3. It is noted that Tab. IV-3 is based on the assumption that the design value corresponds to a product α R β = 0,8 × 3,8 = 3,04 (see 61


EN 1990 [2]). This gives a probability of observing a lower value of about 0,1 %. In this case, η d should cover all uncertainties not covered by the tests. Tab. IV-2. Values of k n for the 5% characteristic value. n 1 2 3 4 5 6 8 10 20 30 ∞ V X known 2,31 2,01 1,89 1,83 1,80 1,77 1,74 1,72 1,68 1,67 1,64 V X unknown 3,37 2,63 2,33 2,18 2,00 1,92 1,76 1,73 1,64 Tab. IV-3. Values of k d,n for the ULS design value. n 1 2 3 4 5 6 8 10 20 30 ∞ V X known 4,36 3,77 3,56 3,44 3,37 3,33 3,27 3,23 3,16 3,13 3,04 V X unknown 11,40 7,85 6,36 5,07 4,51 3,64 3,44 3,04 Bayesi an method for fr actil e estim ation

Particularly in the 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 [4,6,8]. 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 Π′( μ , σ ) = C σ −(1+ν ′+δ (n′ ))exp⎧⎨− 2 [ν ′( s′)2 + n′( μ − m′)2 ]⎫⎬ ⎩ 2σ ⎭

(IV-11)

where C = normalising constant; and 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 ′

(IV-12)

The parameters n' and ν ' are independent and may be chosen arbitrarily (it does not hold that ν ' = n' – 1). In equations (IV-12) E(.) denotes the expectation and V (.) the coefficient of variation. Equations (IV-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 s2 + nm2 ν " (s" ) + n" (m" )2 = ν ' (s' )2 + n' (m' )2 + ν

(IV-13)

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

= m′′ + t p s′′ 1 + 1 / n′′

62

(IV-14)


where t p is the fractile of the t -distribution (see Tab. IV-4) with ν " degrees of freedom. If no prior information is available, then n'= ν '= 0 and the characteristics m", n", s", ν " are equal to the sample characteristics m, n, s, ν . Then equation (IV-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

(IV-15)

where t p denotes again the fractile of the t -distribution (Tab. IV-4) 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 and t p by the fractile of the standardised normal distribution u p.

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

Tab. IV-4. Fractiles −t p of the t -distribution with ν degrees of freedom. 1 − p 1 − p ν 0.95 0.975 0.99 0.995 0.90 0.95 0.975 0.99 0.995 2.35 3.18 4.54 5.84 12 1.36 1.78 2.18 2.68 3.06 2.13 2.78 3.75 4.60 14 1.35 1.76 2.14 2.62 2.98 2.02 2.57 3.37 4.03 16 1.34 1.75 2.12 2.58 2.92 1.94 2.45 3.14 3.71 18 1.33 1.73 2.10 2.55 2.88 1.89 2.36 3.00 3.50 20 1.32 1.72 2.09 2.53 2.85 1.86 2.31 2.90 3.36 25 1.32 1.71 2.06 2.49 2.79 1.83 2.26 2.82 3.25 30 1.31 1.70 2.04 2.46 2.75 1.81 2.23 2.76 3.17 ∞ 1.28 1.64 1.96 2.33 2.58

4.

PROBABILISTIC MODELS OF MATERIAL PROPERTIES

4.1.

Strengths

This section aims to summarize conventional probabilistic models in order to enable reliability studies of various structural members made of different materials (steel, concrete, composite). Proposed models represent average values of material properties, common structural conditions and normal quality control. Recent scientific publications are taken into account [4,9-19]. The following conventional models of basic variables are intended to be used as prior theoretical models that should be updated whenever information on actual structural conditions is available. The data indicated in Tab. IV-5 represent conventional models only, which may not be adequate in specific cases. The mean value of a material property (e.g. yield strength of common structural steel of the grades S235 and S355, compressive concrete strength, reinforcing bars) may be estimated using a simple expression: μ X = X k + k σ X = X k / (1− k V X )

(IV-16) where k (usually equal to 2) = coefficient taking into account a quality control procedure. An alternative expression for the mean of yield strength of structural steel (rolled sections) may be found in [4] and references indicated there: μ X = X k α exp (k V X ) − C

63

(IV-17)


where α = spatial position factor ( α = 1.05 for webs of hot rolled section, α = 1 otherwise); and C = constant reducing the yield strength obtained from usual mill tests to the static yield strength (the value 20 MPa is recommended, but a rate of loading should be considered). Tab. IV-5. Conventional models of resistance variables. Mean St.dev. Probab. Symb. Material Name Dimen. Distribut. References X Φ X ( X k ) μ X σ X 0.07Yield f y Steel MPa LN 0.02 f yk + 2σ strength 0.10 μ X Ultimate f u MPa LN κ μ f y* 0.05 μ X – strength [4,15-18] 0.10Compressive f c Concrete MPa LN 0.02 f ck + 2σ strength 0.18 μ X Reinforce30 f y MPa LN 0.02 f yk + 2σ ment MPa *

The coefficient κ can be considered as follows - κ = 1.5 for structural carbon steel; κ = 1.4 for low alloy steel; and κ = 1.1 for quenched and tempered steel [4].

Comparison of expressions (IV-16) and (IV-17) indicates that expression (IV-17) is more conservative (primarily due to the reduction constant C ). For steel S355 the differences may be about 5 %. Recent experience with a great number of experimental data clearly indicates that the second expression (IV-17) may be rather unrealistic for small coefficients of variation while the first expression (IV-16) appears to be more suitable and is thus recommended here. However, even expression (IV-16) may be conservative in some cases. Such an expression is always an approximation providing informative values (prior information) only, which should be updated for particular conditions by tests. The mean of ultimate strength is usually determined from the mean of yield strength using the factor κ as indicated in Tab. IV-5. The coefficient of variation V X for structural steel is within the range from 0.07 to 0.10, for the ultimate strength is around 0.05 [4]. For concrete strength and yield strength of reinforcing bars usually the standard deviation σ X (commonly 5 and 30 kN/m2 respectively) is considered instead of the coefficient of variation V X . However, the coefficient of variation of concrete strength may vary in the broad range 0.05 to 0.18 depending on a production procedure. 4.2.

Model uncertainties of resistance

Model uncertainties in resistances are conventionally expressed through the coefficients defined as the ratio of observed and model values [4]. In some cases little data are available and often professional judgments and experience are taken into consideration. The information given in Tab. IV-6 is adopted from the latest documents of JCSS. In all the cases log-normal distribution having the origin at zero is recommended [4]. The data indicated in Tab. IV-6 should be, however, considered as indicative only and ought to be verified taking into account structural conditions and available experimental data. It is expected that the coefficients of model uncertainties will be further developed by JCSS considering new experimental data.

64


Tab. IV-6. Model uncertainties for resistance of steel and concrete members. Material Model type Mean CoV Steel members Beam bending moment capacity 1.15 0.05 Column resistance 1.20 0.10 Welded connection capacity 1.15 0.20 Bolted connection capacity 1.25 0.08 Concrete members Bending moment capacity Shear capacity of beams Column resistance Capacity of connections, punching shear

5.

GEOMETRICAL DATA

5.1.

General

1.20 1.40 1.20 1.00

0.15 0.25 0.15 0.15

Geometrical data are generally random variables. In comparison with actions and material properties their variability can be considered small or negligible in most cases. Such quantities can be assumed to be non-random and as specified on the design drawings (e.g. effective span, effective flange widths). However, when the deviations of certain dimensions can have a significant effect on actions, action effects and resistance of a structure, the geometrical quantities should be considered either explicitly as random variables, or implicitly in the models for actions or structural properties (e.g. unintentional eccentricities, inclinations, and curvatures affecting columns and walls). Relevant values of some geometric quantities and their deviations are usually provided in Eurocodes EN 1992 to 1999. The manufacturing and the execution process (e.g. setting out and erection) together with physical and chemical causes will generally result in deviations in the geometry of a completed structure, compared to the design. Generally two types of deviations may occur: a) Initial (time independent) deviations due to loading, production, setting out and erection, b) Time dependent deviations due to loading and various physical, chemical causes. The deviations due to manufacturing, setting out and erection are also called induced deviations; the time dependent deviations due to loading and various physical and chemical causes (creep, effect of temperature and shrinkage) are called inherent deviations (or deviations due to the inherent properties of structural materials). For some building structures (particularly when large-span precast components are used) the induced and inherent deviations may be cumulative for particular components of the structure (e.g. joints and supporting lengths). In the assessment, the effects of cumulative deviations with regard to the reliability of the structure including aesthetic and other functional requirements should be taken into account. The initial deviations of a dimension may be described by a suitable random variable and the time dependent deviations may be described by the time dependent systematic deviations of the dimension. To clarify these fundamental terms Fig. IV-5 shows a probability distribution function of a structural dimension a, its nominal (reference) size anom, systematic deviation δasys (t ), limit deviation Δa and the tolerance width 2 Δa.

65

Chapter IV - Materials and geometry Actual size a

Nominal size anom

anom

δ asys(t ) Tolerance width 2Δa

Δa=1,64σa

Δa=1,64σa

5%

-3σ a

5%

-2σ a

-1σ a

a

+1σ a

+2σ a

+3σ a

Fig. IV-5. Characteristics of a dimension a. The nominal (reference) size anom is the basic size which is used in design drawings and documentation, and to which all deviations are related. The systematic deviation δasys(t ) is a time dependent quantity representing the time dependent dimensional deviations. In Fig. IV5 the limit deviation Δa is associated with the probability 0.05, which is the probability commonly used to specify the characteristic strength. In this case the limit deviation is given as Δa = 1,64 σa. In special cases, however, other probabilities may be applied and instead of the coefficient 1,64 other values should be used. Generally a fractile a p of a dimension a corresponding to the probability p may be expressed as: a p = anom+ δasys(t ) + k p σ a

(IV-18) where k p = coefficient dependent on the probability p and assumed type of distribution. 5.2.

Probabilistic models of geometrical data

Statistical characteristics of geometrical data indicated in Tab. IV-7 are taken from available data [4,18] and other measurements. The mean of steel sections in case of H-profiles is 0.99 X nom, in case of L-profiles and rods it is slightly greater (1.02 X nom); coefficient of variation is about 3 %. In general variation of dimensions of reinforced concrete sections (standard deviation from 5 to 10 mm) is more significant than variation of steel sections. In particular variation of the concrete cover of reinforcing bars may be important, depending on a type and size of the cross section. A bounded Beta distribution (or Gamma distribution) seems to be the most

66


suitable theoretical model in this case. The data indicated in Tab. IV-7 correspond to middle values of expected ranges [4]. A lower bound of the Beta distribution can be zero, a = 0, the mean equal to the nominal (design) value, μ x = xnom, and the upper bound equal to three times the mean, b = 3 μ x. This corresponds to a skewness α x = 2V x characteristic for a Gamma distribution.

Category Name Steel sections

Tab. IV-7. Conventional models of geometrical data. Mean St.dev. Probab. Symb. Dimen. Distribut. References X Φ X ( X k) μ X σ X

IPE profiles L-section, rods

m2,m3, A,W , I N m4

0.99 X nom

≅ 0.73

[4,18]

≅ 0.16

[4,18]

μ X

m2,m3, A,W , I N m4

Concrete Cross-section b, h sections Cover of a reinforcement Additional e eccentricity

0.010.04

1.02 X nom

0.010.02

μ X

m

N

bk , hk

m

BET

ak

m

N

0

67

0.0050.5 0.01 0.0050.5 0.015 0.003 – 0.01

[4] [4] [4]


REFERENCES

[1] ISO 13822: Bases for design of structures - Assessment of existing structures, Geneve, Switzerland: ISO TC98/SC2, 2003. [2] EN 1990: Eurocode - Basis of structural design, Brussels: CEN, 2002. [3] ISO 2394: General principles on reliability for structures, 2nd edition, Geneve, Switzerland: ISO, 1998. [4] JCSS: JCSS Probabilistic Model Code, Zurich: Joint Committee on Structural Safety, 2006. . >. [5] ISO 12491: Statistical methods for quality control of building materials and components, 1st edition, Geneve, Switzerland: ISO, 1997. [6] Diamantidis, D.: Probabilistic Assessment of Existing Structures, Joint Committee on Structural Safety, RILEM Publications S.A.R.L., 2001. [7] Gulvanessian, H., Calgaro, J. A., Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design, London: Thomas Telford, 2002. [8] Ang, A. H. S. & Tang, W. H.: Probabilistic Concepts in Engineering Emphasis on Applications to Civil and Environmental Engineering, 2nd edition, USA: John Wiley & Sons, 2007. [9] Vrouwenvelder, A. C. W. M.: JCSS Probabilistic Model Code. In Proc. of Safety, Risk and Reliability - Trends in Engineering, Rotterdam: Balkema, 2001, pp. 65-70. [10] CIB: Actions on Structures, Self-Weight Loads. Publication 115, CIB, 1989. [11] CIB: Actions on Structures, Live Loads in Buildings. Publication 116, CIB, 1989. Load. Publication 141, CIB, 1995. [12] CIB: Actions on Structures, Snow Load.

[13] CIB: Actions on Structures, Wind Load. Draft of Publication W81, CIB, 1995. [14] SAKO: Basis of Design of Structures. Proposal for Modification of Partial Safety Factors in Eurocodes. Joint Committee of NKB and INSTA-B, 1999. [15] Sørensen, J. D., Hansen, S.O. & Nielsen, T.A.: Partial Safety Factors and Target Reliability Level in Danish Codes. In Proc. of Safety, Risk and Reliability - Trends in Engineering, Rotterdam: Balkema, 2001, pp. 179-184. [16] Holický, M. & Marková, J.: Verification of load factors for concrete components by reliability and optimization analysis: Background documents for implementing Eurocodes, Progress in Structural Engineering and Materials Vol. 2, No. 4 (2000), pp. 502-507. [17] Caramelli, S., Cecconi, A., Croce, P., Salvatore, W. & Sanpaolesi, L.: Partial safety factors for resistance of steel elements to EC3 and EC4. Calibration for various steel products and failure criteria. Commissione Europea, 1997.

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[18] Fajkus, M., Holický, M., Rozlívka, L. et al.: Random Properties of Steel Elements Produced in Czech Republic. In Eurosteel 1999, 1999, pp. 657-660. [19] Finnish Ministry of the Environment, Housing and Building Department: Probabilistic Calibration of Partial Safety Factors (Eurocode and Finish proposal). 2000.

69

V DETERIORATION

Chapter V - Deterioration

1.

INTRODUCTION

Industrial heritage structures are subject to a wide array of climatic, physical, organic and pollution effects, causing costly and often irreparable damage. This Chapter focuses on the common degradation processes caused by environmental and pollution effects from the atmosphere, thus excluding biological damage from fungi, algae, moss, insects and animals, direct physical damage from abrasion and vandalism, and damage from natural disasters. The term for this kind of slow continuous damage is weathering, a process that affects all substances exposed to the atmosphere [1]. In addition to the limitation of damaging effects, only the most commonly used building materials are considered. Protecting the industrial heritage has substantial economic and ecological value, as well as cultural and historical importance. For political handling of this problem in regards to funding and preservation measures, methods for damage analysis and damage control need to be utilized. Mapping the impact of weathering effects, and predicting further development is the first step [1]. To predict the rate of future recession of building material due to the atmospheric effects, damage functions are employed. These are mathematical equations developed to predict damage of materials due to environmental influences. Damage in this context is usually defined as a loss of material, either in depth or in volume, but there are also damage effects due to soiling and blackening. The processes that deteriorate buildings are highly reliant on the weather and the compounds in the atmosphere. Weathering is therefore constantly fluctuating depending on the changes in climate and pollution. During the last century the recession of building materials has increased rapidly, mainly because of increasing emissions of sulfur dioxide and nitrogen oxide from industry. In recent years however, cleaner industry and pollution regulations have resulted in a decrease in sulfur emissions. The emissions of nitrogen oxides still increase, along with a massive increase in carbon dioxide emissions, mainly from combustion processes in cars. In addition to these changes in pollution, global warming may result in major climate changes in the near future. These changes will have consequences for the processes that affect weathering of building materials, both positive and negative. In general, the causes of deterioration of heritage structures include [2]: – Lack of appropriate consideration and errors in the original design, – Lack of scientific knowledge, – Use of structures beyond their working life, – Introduction of new conditions (change of use, environmental changes). Environmental parameters that affect all (or most) of the deterioration processes cover [3]: – Temperature, – Moisture/humidity, – Wind, solar radiation. Other environmental parameters influencing specific deterioration processes are [3]: – Chloride content in air or sea water, de-icing salts (affect corrosion of reinforcing steel in concrete and atmospheric corrosion of steel), – Concentration of carbon dioxide CO 2 (affects carbonation and consequently corrosion of reinforcing steel in concrete), – Concentration of sulfur dioxide SO 2 (affects atmospheric corrosion of steel), etc. The effects of environmental influences should be taken into account, and where possible, be described quantitatively in the same way as for actions. When a model of structural deterioration related to the in situ environmental conditions can be established it is possible to define a relevant limit state. In this case the environmental influences are treated exactly in the same way as actions, classified as permanent, variable or accidental actions or action 72


effects. This model could be deterministic with the uncertainties introduced via appropriate random parameters or coefficients of model uncertainty. Up to now, most of these influences are considered using empirical relationships. 2.

OVERVIEW OF WEATHERING EFFECTS ON BUILDING MATERIALS

2.1

Environmental effects

Salt deter ior ation

Salt deterioration is a result of crystallization and dissolution of salts on building materials, and often repeated cycles of this. It is one of the more dominant contributors to weathering of historical buildings and artefacts [4], since almost all common building materials are somewhat affected by damage from salt in one way or another [5]. Salt deterioration is either in the form of efflorescence or subflorescence. Efflorescence is the formation of crystals on a material surface. This leads to unattractive, usually white discoloration, but it is normally harmless to the construction. Efflorescence may occur in almost every part of a building depending on the microclimate inside and around the structure, but cellars and areas near the roof are usually more vulnerable due to higher humidity. The process of efflorescence is when water in a salt solution evaporates due to decreasing relative humidity, leaving the salt to crystallize. The less soluble salts are most likely to produce efflorescence [6]. Subflorescence is crystallization below the surface, causing different types of damage depending on the material. The exact mechanics on how damage is produced is not yet fully understood [7]. The most popular theoretical explanation is that salt crystallizing expands in volume, often several times, and that this growth often produces damage caused by physical stress. In mineral materials, this can lead to granular disintegration, also known as sugaring [8]. Salts can damage building materials through other mechanisms as well, such as differential thermal expansion, osmotic swelling of clays or hydration pressure [4]. For metals, contact with salt usually leads to corrosion and subsequent volume growth. This often causes a major problem in reinforced concrete with the steel expanding and dissolving the concrete surrounding it. Timber reacting with salt can cause delignification causing mainly visual nuisance, but may be critical for the structure if exposed for prolonged periods. Delignification means the breakdown of the timber cells. Chlorides, nitrates, sulfates and to some extent carbonates, are the most common salts found in building materials [6]. Chlorides come from deicing salts in northern regions or marine aerosol in coastal areas, and it is sometimes a compound of brick masonry. Chloride salts can absorb moisture from humidity and is therefore liable to cause damage by subflorescence. Nitrates origin mainly from organic sources like soils, soil fertilizers or decomposing organic materials. Crystallization of nitrates produce needle-like crystals, but these require constant water supply and is normally negligible when it comes to damage. Compounds of sulfate are the most common salt found in efflorescence, and one of the most damaging. This is due to the large and aggressive expansion sulfates experience during crystallization. These salts generally come from the building materials themselves, like portland cement, limestone, marble and concrete [8]. Carbonates have low water solubility, but are still sometimes included as soluble salts since they are dissolved in high carbon dioxide concentrations. Salt crystallization cycles are mainly governed by the relative humidity and the environment temperature. Environmental control is therefore proposed to prevent salt deterioration damage [9]. This is a challenging task since the correct application of this kind 73


of control is not straightforward. Setting the relative humidity too high can cause growth of bioorganisms, while setting it too low may lead to other salts crystallizing. The best method would most likely be to keep it in a constant environment to prevent cycles of crystallization which normally causes the most damage [5]. Building materials already infused by salt can often be cleaned by poulticing. This has proven to be effective for chloride and nitrate, but sulfate is more resistant to this treatment. When damage is excessive, replacement of building material or injection of waterproofing agents like siloxane, might be necessary [4]. F reezin g-thawing disin tegration

Frost damage on buildings is a result of moisture trapped inside the materials at the time of freezing temperatures. Water expands by roughly 9 % when transforming to a solid, and this can lead to rupturing of porous building materials when the tensile forces in the pore system exceeds the tensile strength in the material [10]. This is dependent on the degree of saturation and the properties of the pore system. The types of damage caused by freezing can be classified as either surface scaling or internal cracking [11]. Surface scaling is characterized as gradually degradation or disintegration of the material surface. Internal cracking is characterized as permanent volume expansion of a material due to cracks forming inside, often without visible surface damage [12]. Most critical damage is a result of repeated freeze-thaw cycles. Frost disintegration starts on the surface, but can progress through the material due to alternating freezing and thawing periods. This is due to formation of new cracks during freezing into which water in turn can migrate during the following thawing period [13]. Water driving mechanisms include diffusion and convection for vapour, and gravitation, wind pressure and capillary suction for liquid. The most common way for a material to absorb water is by capillarity which is the process of water progressing vertically through a porous media due to surface tension. Smaller pores have larger absorptive forces, as well as a lesser rate of ventilation to disperse water. For these reasons, materials with small pores are more likely to contain water and therefore suffer moist damage. The amount of water absorbed by a material relative to its porosity is known as the saturation coefficient [10]. This factor is 1 if the volume of water normally absorbed equals the volume of the pores. Low values of this factor therefore indicate good frost damage resistance. To prevent freezing disintegration from occurring it is vital that the pore system contains enough volume for expelled water close to capillary cavities, hence the air entrainment added in concrete. The best way to prevent freezing disintegration is to reduce the volume of capillary pores, increasing the materials tensile stress in the process. Frost damage is usually limited to alpine or northern areas, due to the need for both moist and frequent alternations above and below freezing temperature. Areas which experience frequent variations in temperature above and below freezing point combined with rainfall are more likely to promote damage from freeze-thaw cycles [13]. The materials affected the most are mineral building materials like concrete, masonry or stone. This is due to their both porous and brittle textures. Several models describing damage due to freezing and thawing have been derived, but most of them are theoretical. Practical use in calculating building deterioration due to freezethaw cycles is therefore limited. One of the more recent models describing rock decay has been postulated in [14]. This function also describes the impact of heating-cooling cycles: -λ N

I N = I 0e

(V-1)

where I N = rock integrity after N cycles; I 0 = initial integrity; and λ = decay constant. Integrity in this context is the hardness and the structural wholeness of the rock [14]. The decay

74


constant indicates the mean relative integrity loss by the action of any single cycle [15]. Its values vary depending on stone types and are calculated experimentally. Lifetime of a material can be estimated from a constant flaking rate, predicted number of cycles and allowed flaking depth [12]: lifetime = acceptable flaking depth / (rate o flaking × N a) (V-2) where N a = predicted number of cycles for a year based on climatic data. Lifetime is measured in years as well, acceptable flaking depth is in mm and rate of flaking is in mm/cycle. A connection between the decay constant derived in [14] and the rate of flaking in this equation is missing however, and experimental data is required for the flaking rate value. Determining deterioration due to cracking is more difficult due to the progressive behaviour of this process [12]. Since freezing-thawing cycles revolve around a fixed temperature it is the process most affected by climate variations. A few degrees change in temperature might seem insignificant, but it can be essential for rates of frost deterioration. The climate change experienced now suggests that the damage due to frost will decline in much of Europe in the coming century. However, some areas to the far north and in high altitudes will be likely to experience an increase in the number of cycles [16]. Kar st eff ect

The karst effect is defined as the dissolution of stones, usually carbonate rocks like limestone, due to clean rain. This is a result of erosion and reactions involving the chemicals naturally present in rainwater [17]. Dissolving due to this effect is naturally most present in areas with heavy rainfall, but not limited to these. Limestone is to a small degree soluble in rainwater due to the calcite that is slightly soluble even in pure water, and the natural amount of CO 2 found in rainwater accelerates this process. Even with acid rain present, this process of recession should not be ignored. Calcium carbonate reacts with the rain and form calcium bicarbonate, which in turn is about a hundred times more soluble then calcium carbonate [17]. An increase in concentrations of carbon dioxide in the atmosphere in the next century will expectedly lead to an increase in the karst recession of carbonate stone [18]. Karst also concerns the process of cavity gouging of bedrock which in rare cases can affect constructions [19]. Th er mal eff ects

Temperature is usually an indirect source of damage on buildings as a prerequisite of frost damage or salt crystallization, but it may also cause direct damage to certain materials. All building and construction materials are sensitive to changes in temperature because of the volume change it involves; hence temperature is one of the main contributors to material disintegration [20]. Historic environments are subjected to cyclic thermal changes depending on geographic situation and time of year. This may lead to cracking and changes in dimension, but often it helps stabilizing the building components rather than damaging them. This is due to the release of constraint stress in the material during the expansion process. However, cracking may still be a major threat. Temperature changes can also lead to cycles of wetting and drying in porous materials. The inhomogeneity created by this will often lead to cracking. The effects of thermal change are usually underestimated in practice, even if they are well know and published. Numerous studies on how materials relate to temperature change have been performed, but few damage models have been proposed [18]. The expression postulated in [14] which was described in the earlier segment about frost damage is one of the few.

75

Chapter V - Deterioration Wind-driven r ain

Wind-driven rain (WDR) is rain with a horizontal velocity component given by wind. It is the main source of moisture on building facades and therefore of great importance to the performance of most building materials [21]. The damage caused by this moisture can take many forms; including moisture induced salt migration, frost damage and thermal cracking. WDR is also responsible for the appearance of surface soiling patterns on many building facades. Experiments indicate that high amounts of driving rain can cause erosion and clean, light coloured surfaces, while low amounts can cause dirty, dark coloured surfaces [22]. A lot of research during the past 70 years has shed light on the effects of WDR impingement on building façades and the response of the buildings to the impinging water, but little is known about the exact mechanics of a raindrops impact on a surface [23]. The amount of WDR accumulating on a building façade is determined by precipitation intensity, wind velocity and direction, local topography, rain-drop size distribution and the building geometry [24]. All these factors combined with the effects of raindrop impacts like splashing, bouncing, evaporation, adhesion, runoff, impact-angle and absorption, makes quantification of WDR highly complex [21]. A result of this is that while experimental and numerical evaluations of WDR problems might be more accurate, they will also be too time-consuming in everyday use compared to semi-empirical evaluations. In standards, driving rain is calculated from hourly wind speed, wind direction and precipitation data, combined with terrain roughness, topography, obstructions and wall positions. The result from these equations is measured in l m -2 per year or per spell. A spell is defined as a period, or a sequence of periods of driving rain. 2.2

Pollution effects

Wet deposi ti on

Wet deposition of pollutants, commonly called acid rain, has historically been a major cause of recession on stone surfaces and to some degree corrosion on metals and deterioration of wood. Acid rain accelerates the rate of damage already being caused by natural weather effects like wind, rain, sun and frost, through a process of first cleaning the material surfaces from previously deposited substances, and afterwards transporting acidic chemicals directly to the surface. The most principal sources of acidic precipitation are sulfur dioxide and nitrogen oxides. The impact of nitrogen oxides on building materials is not yet fully understood, but experiments indicate damage to be much less harmful then from sulfur dioxide. Sulfur dioxide on the other hand is a major contributor to material degradation. Rainfall is naturally acidic due to normal levels of carbon dioxide in the atmosphere. Burning of fossil fuel like coal and oil in industry has added to this acidity due to impurities producing sulfur dioxide, while emissions of nitrogen oxides largely origin from automobiles. This pollution started with the industrial revolution and increased through the World War II. In recent decades however, the emissions of sulfur dioxide have decreased rapidly due to more effective combustion engines, cleaner industry and better environmental regulations. This in turn leads to the conclusion that the centuries where the effect of pollution damage to buildings seem to be over, and that weathering may have the greatest impact in the future [25]. Most materials are somewhat affected by acidic influence, but some of the more vulnerable are limestone, marble, carbon-steel, zinc, nickel, paint and some plastics. Limestone and marble primarily consist of calcite which dissolves easily in acids. Other commonly used stones like granite and sandstone are for the most part composed of silicate minerals which is more resistant to acidic damage. The damage on calcite stones occurs when the calcium compounds chemically react with the sulfuric acids and create gypsum, which in 76


turn dissolute and falls off. Gypsum is visible as a dark porous crust on the stone surface. The reaction between calcium carbonate and nitric acid is also shown [26]. Sulfuric acid also reacts with concrete in much the same way, which is especially relevant in regards to sewer systems and concrete structures in industrious areas. Sulfuric acid combines with the lime compounds in the concrete and form a skin on the concrete surface. The calcium sulfate then crystallizes under this skin and eventually leads to disintegration due to expansion and deterioration [27]. The effects of acid rain on metal works like any acid reacting with metal yield corrosion. This process is electrochemical degradation of the material, meaning a breakdown of the metal. This kind of reaction needs metal, oxygen and H + to work. The H + normally comes from acids forming naturally when CO 2 dissolves in water, but with an acid present the amount of H+ is multiplied meaning the reaction will run easier. Corrosion of metals due to contact with sulfur dioxide or nitrogen oxides from wet deposition is mainly a problem because of the water. Metals corroding produce layers of oxide which normally acts as barriers for further corrosion, but exposure to water may break down this protective shield, and accelerate the deterioration process. Little is known about nitrogen oxides effect on corrosion, but it seems likely that it may stimulate the corrosive effect of sulfur dioxide [28]. Wood exposed to severe acidic conditions can degrade twice as fast as wood exposed to normal distilled water and even brief periods of exposure will show signs of deterioration. Painted wood will be more protected when the timber is painted soon after being installed outdoors. Paint with calcium carbonate filler will deteriorate faster than paint with other fillers [29]. Dr y deposit ion

Dry deposition is the deposition of polluting substances in dry conditions, by absorption on the surfaces of building materials. It is separated from wet deposition by the absence of precipitation, meaning dry deposition in most cases takes place in a limited radius from the pollution source. Acid rain on the other side usually occurs further from the source. This leads to conclusions that dry deposition often is the predominant source of pollution deposition in urban areas and areas close to industry, as well as in dry areas [28]. This leads to dry deposition being the main agent for material recession regarding pollution. Another reason why dry deposition is more damaging is because it affects all surfaces of a construction, whereas acid rain only affects the exposed surfaces [30]. The process of dry deposition is the transfer of pollutant particles to wet or damp surfaces, causing the same chemical reactions as described for acid rain when the sulfur oxide meets a film of water [8]. The rate of absorption of the receptor surface depends highly on the relative humidity, surface moist, surface roughness and area, and the acid buffering capacities of the surface [31]. High relative humidity leads to high rates of dry deposition while lower humidity gives less. Blackening

Blackening is the visual effect of accumulation of dark particles on a lighter surface. Usually this is carbonaceous fine particles containing dark elemental carbon. Historically, blackening is a known problem with complaints about the soiling of buildings recorded from as far back as Roman times. It was with the industrial revolution and the widespread use of coal in 17th century London that it became a common problem however. Since burning of coal has been reduced largely since that period, the main source of blackening has shifted from coal smoke to mainly originate from diesel soot and nitrogen deposition. In addition to this very fine kind of soot, emissions of nitrogen oxides and volatile organic compounds have been dominating for much of the 20th century. This has resulted in production of

77


photochemically polluted urban air, where secondary pollutants like ozone are created. More familiarly known as smog, this kind of pollution is now a problem around the world [32]. The reduction of sulfuric pollution reduces the damage on building materials, making aesthetic considerations even more important. This makes blackening of surfaces one of the biggest issues regarding conservations of historical immovables, especially in urban areas. The rate in which a surface suffers from blackening is usually measured as reflectance change, but may also be measured with regards to colour change [33]. Determining aesthetic thresholds for blackening has proven to be a problem seeing as soil on a building façade produces different impressions on different buildings. A new building covered with dark soot will naturally be perceived as dirty, while older buildings might get more character and look more antique. There is however, a certain point where the degree of blackening will reach a point where it is unacceptable and cleaning will be necessary. Research done on public perception of blackening is scarce, and producing air pollution standards for this phenomenon will be a challenge. 3.

OVERVIEW OF DAMAGE FUNCTIONS

Damage functions in this context are mathematical expressions that determine material degradation as a function of time, based on various input parameters of pollution and weather effects. Damage functions express the relationship between pollution concentration, meteorological variables and material change [34]. There are functions for soiling of surfaces due to pollution as well. The damage functions used to calculate recession of building materials is individual depending on the material and to some degree individual for the different damaging effects. There exists a large variety of damage functions in literature today, most of which are regression functions. Material recession due to pollution and other weathering effects is a serious and costly threat to the rich European cultural heritage. For an effective policy to reduce pollution and indirectly reduce material deterioration, scientific approaches to the problem are needed [35]. Environmental impact assessments, cost benefits analysis and risk management are some of the aspects needed, and this is the primary use of damage functions. Producing damage functions have proven to be a challenge for most materials. The reason for this is the complexity of material degradation combined with the need for longterm accumulated data. Another problem is that extrapolation of short-term exposure trials is deemed inaccurate due to the ever-changing processes of decay [36]. This makes it hard to review prior recession and to predict future recession based on experimental data. Damage functions are normally in the form of dose-response functions, meaning that they describe the cause of corrosion of a material compared to the rate of corrosion. The former is often termed the “dose” and the latter the “response” [37]. The contents of damage functions vary heavily, depending on the available data and the conjecture of the author as to what might be important. A parameter related to water like relative humidity, time of wetness, rainfall, is almost universally chosen as a factor. The average concentration of sulfur dioxide is usually present as a factor as well. Regression equations have been favoured in much of the European work compared to theoretical approaches. The problem with regression analysis in the context of material recession is the difficulty of extrapolation, making them troublesome to use in long time-scale applications. Several projects have invested considerable efforts to produce these kinds of functions on a large scale. The most notable are the ICP materials and MULTI-ASSESS projects. ICP materials is one of several effect oriented International Co-operative Programmes (ICPs) within the Convention on Long-Range Transboundary Air Pollution (CLRTAP). The primary objective of ICP materials is to collect information on corrosion, 78


soiling and environmental data, in order to evaluate dose-response functions and trend effects [38]. The MULTI-ASSESS project is an extension of the ICP program, but with more focus on a multipollutant environment rather than heavy focus on the damage of sulfur dioxide. Notation and units for the following sections: – ML Mass loss (stone) [g/m2] – R Surface recession (metals) [µm] – LL Depth of leached layer (glass) [nm] – PM 10 Concentration of particles less than 10 µm [µg/m 3] – RH Relative humidity [%] – RH 60 Relative humidity subtracted by 60 [%] – Rn Rainfall per year [mm] – T Temperature [°C] – t Time [years] – [] Concentration of SO2, NO2 or O3 [µg/m3] – [H+] Acidity of precipitation [mg/l] 3.1

Stone recession

Stone is one of the most widely used materials in historical immovables, and there is a wide array of types with different geological properties like chemical composition and porosity. When studying damage it is appropriate to relate stone types to deterioration mechanisms rather than geological criteria. Therefore it is reasonable to think of building stones in categories like (i) carbonate stones like marble and limestone, (ii) high porosity stones like sandstone, and (iii) granite [39]. These categories require different damage functions. However, no dose-response function or damage function exist for granite [43]. Car bonate stones

Limestone and marble are historically the most widely used stones in buildings and statuary and is consequently the category where the most research concerning damage functions has been done. The most recognized functions for carbonate stones are the theoretical Lipfert function which has been proven accurate for limestone [40], in addition to the more recent MULTI-ASSESS (MA) and ICP functions. The difference is that the Lipfert function focuses a lot more on the damage caused by karst than the other two. The MULTI-ASSESS and ICP functions for carbonate stones are based on portland limestone which is a common building stone in a lot of historical buildings on the British isles. Although it is not entirely accurate these functions are considered to be representative for other carbonate stones: R = 3,1 + (0,85 +

0,0059[SO2] RH 60 + 0,054 Rn[H+] + 0,078[HNO2] RH 60 + 0,0258 PM 10)t (V-3)

or R = 2,7[SO2]0,48ef(T )t 0,96 + 0,019 Rn[H+]t 0,96

(V-4)

where f(T ) = -0,018T . The Lipfert function from 1989 takes into account three mechanics of material loss. These are karst effect, acid rain effect and dry deposition [41]: -d x/dt = Lv Rn + 0,016[H+] Rn + 0,18(V ds[SO2] + V dn[HNO3])

(V-5)

Lipfert uses a different form of notation and units. Here is –d x/dt the recession rate in µm a , Rn is rainfall in m a -1, and V dS and V dN are deposition velocities in cm s -1. Deposition velocities are dependent on the properties of the surface and type of exposure. It is defined as the number of particles depositing per unit time and area divided by the number of particles per unit volume in airborne state [42]. Lv is the Lipfert value as recession in µm per year. This -1

79


factor represents the solubility of CaCo 3 in relation to the amount of CO 2 in the atmosphere. Typical value for this parameter is 18.8. Some examples to give knowledge of the magnitude of limestone recession include the measurements of the St. Paul’s Cathedral of Portland limestone, which around 20 years ago reportedly experienced an average recession of 0.06 mm/year. Measurements of marble tombstones in USA varied from 3.6 mm/100year to 2.8 mm/100year in urban areas, with recession rates of 1.7 mm/100y in suburban areas [39]. The measurements vary depending on the method however, and much higher recession rates have been measured on the very same objects. H igh porosit y stones

Fewer damage functions for the more porous stones like sandstones have been made. There is ,however, an ICP function describing weathering for white Mansfield sandstone. 0,52 f( ) 0,91 + 0,91 R = 2,0[SO2] e T t + 0,028 Rn[H ]t (V-6) where f(T ) = 0 for T < 10 °C and f( T ) = -0,013(T - 10) otherwise. 3.2

Concrete

General

According to [3] corrosion of reinforcing steel is the main cause of deterioration of reinforced concrete structures. Under normal conditions concrete protects embedded reinforcing steel against corrosion. These protective properties of concrete are attributed to a passive oxide film which forms on the surface of steel in highly alkaline environment provided by the concrete pore solution. However, carbonation or penetration of chloride ions negates the protective properties of concrete and may lead, over time, to corrosion of reinforcing steel. After corrosion starts, its effects on reinforced concrete structures include cracking of the concrete cover, reduction and eventually loss of bond between concrete and corroding reinforcement, and reduction of cross-sectional area of reinforcing steel (i.e., corrosion affects both strength and serviceability of reinforced concrete structures). Thus, the deterioration of reinforced concrete structures due to corrosion can be divided into two stages – initiation and propagation, which is shown schematically in Fig. V-1 [44]. The time to corrosion initiation (or the initiation period) is denoted as t i; at this stage CO2 and/or chloride ions are penetrating into the concrete and there is no corrosion-induced damage. The time of the appearance of first corrosion-induced cracking (i.e., hairline cracks of width less than 0.05 mm) on the concrete surface is denoted as t cr1. The time of excessive cracking (i.e., cracking that is defined as serviceability failure) is denoted as t cr . As has been noted previously corrosion affects not only serviceability but also the strength of reinforced concrete structures. The time when the strength reduction is such that the reinforced concrete structure does not satisfy anymore an ultimate limit state is denoted as t u. Usually, as shown in Fig. V-1, t u is greater than t cr . However, in some cases, e.g., strongly localized corrosion or delamination occurring without surface cracking, t u may be smaller than t cr , or even than t cr1. It should be also noted that Fig. V-1 provides just a schematic description of the deterioration process, and the rates may be more complex than those presented in the figure: the rate within each stage is not necessarily constant, and they may increase when going from one stage to another to a greater extent than shown in Fig. V-1.

80


Fig. V-1. Schematic representation of the development of corrosion in reinforced concrete structures [44].

Carbonation

Modelling of carbonation induced corrosion of uncracked concrete usually starts with the limit state based on comparison of the carbonation depth xc(t ) with the concrete cover a of the reinforcement [45]: g[X(t)] = a - xc(t ) (V-7) Both carbonation depth and concrete cover are random variables of considerable scatter and, therefore, probabilistic methods should be used to verify reliability of reinforcement protection. Several empirical models based on experimental data have been proposed [46]. More complex model for the carbonation depth (in mm) is provided in [45]: xc(t ) = √[2k ek c(k tRACC,0-1 + ε t)C S]√t × W(t )

(V-8) where t = time in years; k e = environmental function; k c = execution transfer parameter; k t = regression parameter; RACC,0-1 = inverse effective carbonation resistance of concrete in (mm2/year)/(kg/m3); and ε t = error term. Indicative probabilistic models for basic variables included in equation (V-8) are provided in [3,45]. Chlor ide penetration

It has been recognised that deterioration of reinforced concrete structures occurs mainly due to chloride contamination. Chlorides may diffuse through a protective concrete cover. In this section, simplified deterioration model proposed in [47] is briefly described. In this model, penetration of chlorides is described by Fick’s second law of diffusion. The chloride content C ( x,t ) at a distance x from the concrete surface at a point in time t is: C ( x,t ) = C 0{1 – erf[ x/(2√tD)]} (V-9) where C 0 = surface chloride content; D = diffusion coefficient; and erf = error function. The model for the diffusion coefficient developed in [48] is applied, taking into account variations of the aggregate-to-cement and water-to-cement ratios and mass densities of cement and aggregates. Model uncertainties are described by the coefficient θ D. The chloride concentration must reach a critical threshold chloride concentration C r at the depth a (concrete cover), to initiate corrosion of reinforcement. Time to initiation of corrosion due to chlorides is then determined from equation (V-9). Indicative probabilistic models for basic variables are indicated in Tab. V-1. It is, however, emphasised that the models may be dependent on site conditions and may need to be updated considering results of surveys and tests. More complex models for chloride penetration are provided elsewhere [3,45]. 81

Chapter V - Deterioration Corr osion propagation

At time τ since the corrosion initiation (in years), the corrosion rate may be modelled by simplified relationship [47]: icorr (τ ) = θ icorr 0.85icorr0τ -0.29

(V-10)

where θ icorr = model uncertainties of the corrosion rate. The initial corrosion rate icorr0 (in μA/cm2) is given by: icorr0 = [37.8(1- w/c)-1.64] / a (V-11) where a = concrete cover (in cm); and w/c = water-to-cement ratio obtained as 27 / [ f c + 13.5] with f c being the concrete compressive strength in MPa. Due to corrosion, the diameter of reinforcement bars d (t ) is reduced at an arbitrary point in time t as follows: … t ≤ t i = max[d 0 – 2 × 0.0116 ×1.2 icorr0(t – t i)0.71; 0] … t > t i (V-12) where d 0 = initial diameter of bars. Indicative probabilistic models for basic variables are indicated in Tab. V-1. It is again emphasised that the models may need to be updated considering actual site conditions and results of surveys and tests. d (t ) = d 0

Tab. V-1. Indicative probabilistic models for basic variables of the deterioration models. Symbol Variable Unit Distr. Mean CoV C 0

Surface chloride content

kg/m3 LN0 3.0

D

Diffusion coefficient

cm2/s N

2e-8 0.45

θ D

Model unc. diffusion coeff.

-

1.0

0.2

C r

Critical threshold chloride conc. kg/m3 U

0.6*

1.2*

θ icorr

Model unc. of corrosion rate

1.0

0.2

-

N N

0.5

N – normal distribution, LN0 – lognormal distribution with the lower bound at the origin, U – uniform distribution; * lower/upper bound.

It is important to note that the mechanical properties of steel and concrete are assumed to be unaffected by the corrosion in the deterioration models. It has been recognised that significant uncertainties are related to models for reinforced concrete deterioration. Studies available in literature provide scattered data describing chloride penetration and reinforcement corrosion. Apparently, experimental data are needed for development of a realistic deterioration model to be applied in a specific case. 3.3

Steel

Structural steel is the most widely used metal in construction through history. Corrosion rates of structural steel and other selected metals may be estimated following ISO 9223 [49], see Tab. V-2.

82


Tab. V-2. Corrosion rates in μm/year for carbon steel, zinc and copper for different categories of corrosivity [49]. Corrosivity Steel Zinc Copper ≤1.3 ≤0.1 ≤0.1 Very low Low 1.3–25 0.1–0.7 0.1–0.6 Medium 25–50 0.7–2.1 0.6–1.3 High 50–80 2.1–4.2 1.3–2.8 Very high 80–200 4.2–8.4 2.8–5.6 As an example the dose-response functions based on T and Rh for the estimation of the annual corrosion rate of carbon steel in μm/year provided in [50] is shown: for T ≤ 10 °C … C = 1.77[SO2]0.52exp(0.020 Rh)exp{0.150(T – 10)} + 0.102[Cl–] 0.62exp(0.033 Rh + 0.040T ) for T > 10 °C … (V-13) 0.52 0.62 C = 1.77[SO2] exp(0.020 Rh)exp{–0.054(T – 10)} + 0.102[Cl–] exp(0.033 Rh + 0.040 T ) Damage functions exists for a lot of other metals such as copper and bronze, glass corrosion and also for surface blackening [28,35].

83


REFERENCES

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Doehne E. Salt weathering: a selective review. Natural Stone, Weathering Phenomena, Conservation Strategies and Case Studies , Geological Society, Special Publication No. 205 (2002), p. 51-64. [5] Bionda D. Salt deterioration and microclimate in historical buildings . Intermediary report of the project Sustained Care of Sensitive Historical Monuments (2004). [6]

Harris SY. Building pathology: deterioration, diagnostics, and intervention (2001).

[7] Bonn NS., Bertrand F. & Bonn D. Damage due to salt crystallization in porous media (2009). [8] Wheeler GS., Gale F., Kelly SJ. Stone Masonry. Historic Building Facades , The Manual for Maintenance and Rehabilitation (1997). [9] Arnold A. & Zehnder K. Monitoring wall paintings affected by soluble salts. Proceedings of a symposium organized by the Courtauld Institute of Art and the Getty Conservation Institute (1991). [10] Richardson BA. Defects and Deterioration in Buildings, 2nd edition (2001), p. 42-44. [11] Cho T. Prediction of cyclic freeze-thaw damage in concrete structures based on response surface method. Construction and Building Materials 21 (2007), p. 2031-2040. [12] Jacobsen S. Frostnedbrytning av betong og andre porøse byggematerialer. Byggforsk kunnskapssystemer , blad 520.067 (1999). [13] Lisø KR., Kvande T., Hygen HO., Thue JV. & Harstveit K. A frost decay exposure index for porous mineral building materials. Building and Environment 42 (2007), p. 35473555. [14] Mutlutürk M., Altindag R. & Türk G. A decay function for the integrity loss of rock when subjected to recurrent cycles of freezing-thawing and heating-cooling. International Journal of Rock Mechanics & Mining Sciences 41 (2004), p. 237-244.

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[15] Altindag R., Alyildiz IS. & Onargan T. Mechanical property degradation of ignimbrite subjected to recurrent freeze-thaw cycles. International Journal of Rock Mechanics & Mining Sciences 41 (2004), p. 1023-1028. [16] Grossi CM., Brimblecombe P. & Harris I. Predicting long term freeze-thaw risks on Europe built heritage and archaeological sites in a changing climate. Science of the Total Environment 377 (2007), p. 273-281. [17] Cardell-Fernández C., Vleugels G., Torfs K. & Van Grieken R. The processes dominating Ca dissolution of limestone when exposed to ambient atmospheric conditions as determined by comparing dissolution models. Environmental Geology (2002) 43, p. 160-171. [18] Bonazza A., Messina P., Sabbioni C., Grossi CM. & Brimblecombe P. Mapping the impact of climate change on surface recession of carbonate buildings in Europe. Science of the Total Environment 407 (2009), p. 2039-2050. [19] Tolmachev VV. & Neshchetkin OB. Evaluation of Karst Hazards for Civil and Industrial Buildings. Acta Geologica Sinica (2001). [20] Grøntoft T & Drdácký M. Effekter av klima og klimaendringer på den bygde kulturarven; Nedbrytningsmekanismer og sårbarhet . Norges forskningsråd – Strategisk instituttprogram, Adapting to extreme weather in municipalities ”Klima SIP”(2008). [21] Blocken B. & Carmeliet J. A review of wind-driven rain research in building science. Journal of Wind Engineering and Industrial Aerodynamics , 92 (2004), p. 1079-1130. [22] Tang W., Davidson CI., Finger S. & Vance K. Erosion of limestone building surfaces caused by wind-driven rain: 1. Field measurements. Atmospheric Environment 38 (2004), p. 5589-5599. [23] Blocken B, Abuku M, Roels S & Carmeliet J. Wind-driven rain on building facades: some perspectives. Proceedings of the 5th European & African Conference on Wind Engineering (2009), p. 249-252. [24] Jelle BP. & Lisø KR. Slagregn – klimadata og grunnlag for beregninger . Delrapport fra prosjekt 11 i FoU-programmet ”Klima 2000” (2003). [25] Brimblecombe P. & Grossi CM. Millennium-long recession of limestone facades in London. Environmental Geology (2008) 56, p. 463-471. [26] Bravo H., Soto R., Sosa R., Sánchez P., Alarcón AL., Kahl J. & Ruíz J. Effect of acid rain on buildingmaterial of the El Tajín archaeological zone in Veracruz, Mexico. Environmental Pollution 144 (2006), p. 655-660. [27] Dermirbaş A., Öztürk T., Karataş FÖ. Long-term wear on outside walls of buildings by sulfur dioxide corrosion. Cement and Concrete Research 31 (2001), p. 3-6. [28] Manning M. Corrosion of building materials due to atmospheric pollution in the United Kingdom. Air pollution, acid rain and the environment (2nd ed.), Watt Committee Report No. 18, London, U.K. (1988). [29] Williams RS. Effects of acid deposition on wood . Forest Products Laboratory (2002). 85


[30] Charola AE. & Ware R. Acid deposition and the deterioration of stone: a brief review of a broad topic. Natural Stone, Weathering Phenomena, Conservation Strategies and Case Studies, Geological Society Special Publication No. 205 (2002), p. 393-407. [31] Spiker EC., Hosker RP., Weintraub VC. & Sherwood SI. Laboratory study of SO 2 dry deposition on limestone and marble: Effects of humidity and surface variables. Water, air and soil pollution, Vol. 85 No. 4 (1995), p. 2679-2685. [32] Grossi CM. & Brimblecombe P. Effect of long-term changes in air pollution and climate on the decay and blackening of European stone buildings. Building Stone Decay, from Diagnosis to Conservation, Geological Society Special Publication 271 (2007). [33] Grossi CM., Esbert RM., Díaz-Pache F., Alonso FJ. Soiling of building stones in urban environment. Building and environment 38 (2003), p. 147-159. [34] Grossi CM., Bonazza A., Brimblecombe P., Harris I. & Sabbioni C. Predicting twentyfirst century recession of architectural limestone in European cities. Environmental Geology (2008) 56, p. 455-461. [35] Kucera V. MULTI-ASSESS modell . Deliverable 0.2 , Publishable Final report (2005). [36] Turkington AV., Martin E., Viles HA. & Smith BJ. Surface change and decay of sandstone samples exposed to a polluted urban atmosphere over a six-year period: Belfast, Northern Ireland. Building and environment 38 (2003), p. 1205-1216. [37] Graedel TE. & Leygraf C. Scenarios for Atmospheric Corrosion in the 21 st Century. Abstracted from Atmospheric Corrosion (2000). [38] Swerea Kimab homepage. www.corr-institute.se/ICP-materials (2009). [39] Sabbioni C. Mechanisms of air pollution damage to stone. The effects on air pollution in the built environment (2003), p. 63-106. [40] Delalieux F., Cardell-Fernandez C., Torfs G., Vleugels G. & Van Grieken R. Damage functions and mechanism equations derived from limestone weathering in field exposure

(2001). [41] Brimblecombe P, Grossi CM. Millennium-long damage to building materials in London. Science of the Total Environment 407 (2009), p. 1354-1361. [42] Horvath H., Pesava P., Toprak S. & Aksu R. Technique for measuring the deposition velocity of particulate matter to building surfaces. The science of the Total Environment 189/190 (1996), p. 255-258. [43] Knotkova D. & Kreislova K. Life time cycle estimates for materials and elements of exposed cultural heritage objects for later application in cost estimates and management strategies, deliverable 7. Assessment of Air Pollution Effects on Cultural Heritage – Management Strategies (2005). [44] Tuutti, K. Corrosion of steel in concrete, Report CBI fo 4.82, Swedish Cement and Concrete Research Institute, Stockholm (1982).

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[45] fib. Model Code for Service Life Design, bulletin 34, fib (2006). [46] Holický M. & Mihashi H. Stochastic Optimisation of Concrete Cover Exposed to Carbonation. In: Application of Statistics and Probability . A. A. Balkema Rotterdam (2000), pp. 279-284. [47] Vu KAT. & Stewart MG. Structural reliability of concrete bridges including improved chloride-induced corrosion models, Structural Safety Vol. 22, No. 4 (2000), pp. 313-333. [48] Papadakis VG., Roumeliotis, AP., Fardis, MN. et al. Mathematical modelling of chloride effect on concrete durability and protection measures. In Concrete repair, rehabilitation and protection, eds. R.K. Dhir and M.R. Jones, London: E&FN Spon (1996), pp. 165-174. [49] ISO 9223. Corrosion of metals and alloys - Corrosivity of atmospheres - Classification . ISO (1992). [50] Mikhailov AA., Tidblad J. & Kucera V. The Classification System of ISO 9223 Standard and the Dose–Response Functions Assessing the Corrosivity of Outdoor Atmospheres. Protection of Metals, Vol. 40, No. 6 (2004), pp. 541–550. [51] Brimblecombe P & Grossi CM. The rate of darkening of material surfaces. Air pollution and cultural heritage (2004), p. 193-198.

87

VI RELIABILITY ANALYSIS

Chapter VI - Reliability analysis

1.

GENERAL PRINCIPLES

General principles of structural reliability are described in the international documents ISO 13822 [1], EN 1990 [2] and ISO 2394 [3]. Basic requirements on structures are specified in Section 2 of EN 1990 [2]: a structure shall, during its intended life, with appropriate degrees of reliability and in an economic way: – Sustain all actions and influences likely to occur during possible repair and use, – Remain fit for the use for which it is required. It should be noted that two aspects are explicitly mentioned: reliability and economy. In the following reliability of structures is focused, including aspects of: – Structural resistance, – Serviceability; – Durability. Additional requirements may concern fire safety of structures or other accidental design situations. In particular it is required by EN 1990 [2] that in the case of fire, the structural resistance shall be adequate for the required period of time. To verify all the aspects of structural reliability implied by the above-mentioned basic requirements, an appropriate working life, design situations and limit states should be considered. Note that the basic working life for a common building is 50 years and that, in general, four design situations are identified: permanent, transient, accidental and seismic. Two types of limit states are normally verified: ultimate limit states and serviceability limit states. 2.

UNCERTAINTIES

It is well recognised that construction works are complicated technical systems suffering from a number of significant uncertainties in all stages of execution, use and repairs. Depending on the nature of a structure, environmental conditions and applied actions, various types of uncertainties become more significant than the others. For the industrial heritage structures, the following types of uncertainties can be identified: – Natural randomness of actions, material properties and geometric data, – Statistical uncertainties due to a limited size of available data, – Uncertainties of the resistance and load effect models due to simplifications of actual conditions, – Vagueness due to inaccurate definitions of performance requirements, – Gross errors in the assessment, during repair and use, – Lack of knowledge concerning behaviour of new materials and actions in actual conditions. The order of the listed uncertainties corresponds approximately to the decreasing level of current knowledge and available theoretical tools for their description and consideration in the assessment. It should be emphasized that most of the above listed uncertainties (randomness, statistical and model uncertainties) can never be eliminated absolutely and must be taken into account in the assessment. Natural randomness and statistical uncertainties may be relatively well described by available methods provided by the theory of probability and mathematical statistics. In fact EN 1990 [2] gives some guidance on available techniques. However, lack of credible experimental data (e.g. for new materials, some actions including environmental influences and also for some geometrical properties) causes significant problems. In some cases the 90


available data are inhomogeneous, obtained under different conditions (e.g. for imposed loads and environmental influences). Then it may be difficult, if not impossible, to analyse and use them in the assessment. The uncertainties of computational models may be to a certain extent assessed on the basis of theoretical and experimental research. EN 1990 [2] and materials of JCSS [4] provide some guidance. The vaguenesses caused by inaccurate definitions (in particular of serviceability and other performance requirements) may be partially described by means of the theory of fuzzy sets. However, these methods have a little practical significance, as suitable experimental data are rarely available. The knowledge of the behaviour of new materials and structures may be gradually increased through theoretical analyses verified by 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. Various methods of reliability verification and operational techniques, which take these uncertainties into account, have been developed and worldwide used. The theory of structural reliability provides background concept techniques and theoretical bases for description and analysis of the above-mentioned uncertainties concerning structural reliability. 3.

RELIABILITY

3.1.

General

The term "reliability" is often used very vaguely and deserves some clarification. Often the concept of reliability is conceived 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 in the sense that “a failure of the structure will never occur“. This interpretation is unfortunately an oversimplification. Although it may be unpleasant and for many people perhaps unacceptable, the hypothetical area of “absolute reliability” for most structures (apart from exceptional cases) simply does not exist. Generally speaking, any structure may fail (although with a small or negligible probability) even when it is declared as reliable. The interpretation of the complementary (negative) statement is usually understood more correctly: failures are accepted as a part of the real world and the probability or frequency of their occurrence is then discussed. In fact in the assessment it is necessary to admit a certain small probability that a failure may occur within the intended life of the structure. Otherwise designing of civil structures would not be possible at all. What is then the correct interpretation of the keyword “reliability” and what sense does the generally used statement “the structure is reliable or safe” have? 3.2.

Definition of reliability

A number of definitions of the term “reliability” are used in literature and in national and international documents. ISO 2394 [3] provides a definition of reliability, which is similar to the approach of national standards used in some European countries: 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 a quantitative sense reliability may be defined as the complement of the probability of failure. Note that the above definition of reliability includes four important elements: – Given (performance) requirements – definition of the structural failure, 91


– Time period – assessment of the required service-life T , – Reliability level – 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 are of uttermost importance. In many cases, when considering the requirements for stability and collapse of a structure, the specification of the failure 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 several vaguenesses and inaccuracies. The transformation of these occupants' requirements into appropriate technical quantities and precise criteria is very hard and often leads to considerably different 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. The same definition as in ISO 2394 [3] is provided in EN 1990 [2] including note that the reliability covers the load-bearing capacity, serviceability as well as the durability of a structure. Fundamental requirements include the statement (as already mentioned) 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 sustain all actions and influences likely to occur during execution, use and repairs, and remain fit for the use for which it is required”. Generally a different level of reliability for load-bearing capacity and for serviceability may be accepted for a structure or its parts. In the documents [1-3] the probability of failure pf (and reliability index β ) are indicated with regard to failure consequences. 3.3.

Probability of failure

The most important term used above (and in the theory of structural reliability) is evidently the probability of failure pf . In order to define pf properly it is assumed that structural behaviour may be described by a set of basic variables X = [ X 1, X 2, ... , X n] characterizing actions, mechanical properties, geometrical data and model uncertainties. Furthermore it is assumed that the limit state (ultimate, serviceability, durability or fatigue) of a structure is defined by the limit state function (or the performance function), usually written in an implicit form as: Z(X) = 0 (VI-1) The limit state function Z( X) should be defined in such a way that for a favourable (safe) state of a structure the function is positive, Z( X) ≥ 0, and for a unfavourable state (failure) of the structure the limit state function is negative, Z (X) < 0. For most limit states (ultimate, serviceability, durability and fatigue) the probability of failure can be expressed as: pf = P{Z( X) < 0}

(VI-2)

The failure probability pf can be assessed if basic variables X = [ X 1, X 2, ... , X n] are described by appropriate probabilistic (numerical or analytical) models. Assuming that the basic variables X = [ X 1, X 2, ... , X n] are described by time independent joint probability density function f X(x) then the probability pf can be determined from the integral: p f

=

∫ f (x)dx X

Z ( X)<0

92

(VI-3)

Chapter VI - Reliability analysis 3.4.

Reliability index

An equivalent term to the failure probability is the reliability index β , formally defined as a negative value of a standardized normal variable corresponding to the probability of failure pf . Thus, the following relationship may be considered as a definition: β = -Φ-1( pf )

(VI-4)

where Φ-1 = inverse standardised normal distribution function. At present the reliability index β is a commonly used measure of structural reliability in several international documents [1-4]. 4.

RELIABILITY VERIFICATION

4.1.

Deterministic methods

During their historical development the methods of reliability verification have been closely linked to the available empirical, experimental as well as theoretical knowledge of structural mechanics and the theory of probability. The development of various empirical methods for structural verification gradually crystallized in the twentieth century in three generally used methods, which are, in various modifications, still applied in standards for structural design until today: the permissible stresses method, the global factor and partial factor methods. All these methods are often discussed and sometimes reviewed or updated. Permissi ble str esses

The first of the worldwide-accepted methods of reliability verification is the method of permissible stresses that is based on linear elasticity theory. The basic design condition of this method can be written in the form: σ max < σ per , where σ per = σ crit / k

(VI-5) The coefficient k (greater than 1) is the only explicit measure supposed to take into account all types of uncertainties (some implicit measures may be hidden). Moreover, only a local effect (a stress) σ max is compared with the permissible stress σ per and, therefore, a local (elastic) behaviour of a structure is used to guarantee the reliability. No proper way is provided for treating geometric non-linearity, stress distribution and ductility of structural materials and members. For these reasons the permissible stress method leads usually to conservative and uneconomical verifications. However, the main insufficiency of the permissible stress method is lack of possibility to consider uncertainties of individual basic variables and computational models used to assess load effects and structural resistances. Consequently, reliability level of structures exposed to different actions and made of different material may be not only conservative (uneconomical), but also considerably different. Global safety factor

The second widespread method is the method of global safety factor. Essentially it is based on a condition relating the standard or nominal values of the structural resistance R and load effect E. It may be written as: s = R / E > s0 (VI-6) Thus the calculated safety factor s must be greater than its specified value s0 (for example s0 = 1,9 is commonly required for bending resistance of reinforced concrete members). The global safety factor method attempts to take into account realistic assumptions concerning structural

93


behaviour of members and their cross-sections, geometric non-linearity, stress distribution and ductility; in particular through the resulting quantities of structural resistance R and action effect E . However, as in the case of the permissible stresses method the main insufficiency of this method remains a lack of possibility to consider the uncertainties of particular basic quantities and theoretical models. The probability of failure can, again, be controlled by one explicit quantity only, by the global safety factor s. Obviously, harmonisation of reliability degree of different structural members made of different materials is limited. Partial factor method

At present, the most advanced operational method of reliability verification [1-3] accepts the partial factor format (sometimes incorrectly called the limit states method) usually applied in conjunction with the concept of limit states (ultimate, serviceability or fatigue). This method can be generally characterised by the inequality: E d( F d, f d, ad, θ d) < Rd( F d, f d, ad, θ d)

(VI-7) where the design values of action effect E d and structural resistance Rd are assessed considering the design values of basic variables describing the actions F d = ψ γ F F k , material properties f d = f k / γ m, dimensions ad + Δa and model uncertainties θ d. The design values of these quantities are determined (taking into account various uncertainties) using their characteristic values ( F k , f k , ak , θ k ), partial factors γ , reduction factors ψ and other measures of reliability [1-3,5]. Thus the whole system of partial factors and other reliability elements may be used to control the level of structural reliability. Compared with the previous methods the partial factor format obviously offers the greatest possibility to harmonise reliability of various types of structures made of different materials. Note, however, that in any of the above listed methods the failure probability is not applied directly. Consequently, the failure probability of different structures made of different materials may still considerably vary even though sophisticated calibration procedures were applied. Further desired calibrations of reliability elements on probabilistic bases are needed; it can be done using the guidance provided in ISO 2394 [3] and EN 1990 [2]. 4.2.

Probabilistic methods

Probabil istic methods in the assessment of i ndu str ial heri tage bui ldi ngs

Decisions about adequate construction interventions should be based on the complex assessment of an industrial heritage structure. It has been recognised that many heritage structures do not fulfil requirements of present codes of practice. Minimisation of construction interventions is required in rehabilitation and upgrades, but sufficient reliability should also be guaranteed. Application of simplified procedures used for design of new structures may lead to expensive repairs and losses of the cultural and heritage value. A general probabilistic procedure is thus proposed to be applied in the assessment of industrial heritage structures in order to: – Improve the reliability assessment of industrial heritage structures, – Describe better uncertainties related to the assessment and – Allow for inclusion of results of inspections and tests and the satisfactory past performance of a structure. It is recognised [6] that probabilistic methods may be useful for the assessment of existing structures where appropriate data can be obtained. According to [7] uncertainties that can be greater than in structural design (such as uncertainties related to inaccessible members and connections where construction details cannot be inspected and verified) may be adequately described by such methods. On the contrary, some of the uncertainties reflected 94


(often implicitly) in the load and resistance factors (modelling approximations, deviations from specified dimensions and strengths) may be less than in new construction, particularly when in-situ measurements are taken. The probabilistic methods introduced in ISO 2394 [3] are based on a requirement that during the service life of a structure T the probability of failure pf does not exceed the target value pt or the reliability index β is greater than its target value β t: pf ≤ pt or β > β t

(VI-8)

Probabil istic analysis f or a mul ti vari ate case

As a rule a number of basic variables X 1 , X 2 ,… X n have to be considered in the assessment. Variables X 1 , X 2 ,… X n are denoted as the vector X [ X 1 , X 2 ,… X n] and their realisations x1, x2, …, xn as the vector x [ x1, x2, …, xn]. In the multivariate case the reliability margin can be generalised as: G( X 1 , X 2 ,… X n) = G(X) (VI-9) The safe domain of the basic variables is described by the inequality: G( X 1 , X 2 ,… X n) = G(X) > 0

(VI-10)

The unsafe domain of the basic variables is described by the inequality: G( X 1 , X 2 ,… X n) = G(X) < 0

(VI-11)

The limit state function is thus given as: G( X 1 , X 2 ,… X n) = G(X) = 0 (VI-12) When a non-linear performance function G( X) and more basic variables X are considered, failure probability pf can be generally expressed using the limit state function G( X) as: pf = P[G(X) ≤ 0] =

∫ f (x)dx

(VI-13)

G ( X) ≤ 0

The integral in equation (VI-13) can also be written as a multiple integral: pf = P[G(X) ≤ 0] =

∫ f

X 1

( x1 ) f X 2 ( x 2 )...f Xn ( x n ) d x1 d x 2 ....d xn

(VI-14)

G ( X) ≤ 0

The integral in equation (VI-13) or (VI-14) indicates how the probability pf can be determined provided that the joint probability density function f X(X) and densities f Xi( xi) are known. In some special cases the integration indicated in equations (VI-13) and (VI-14) can be done analytically, in some other cases, when the number of basic variables is small (up to 5), various types of numerical integration may be effectively applied. In general the failure probability pf may be computed using [3]: – Exact analytical integration; – Numerical integration methods; – Approximate analytical methods (FORM, SORM, methods of moments); – Simulation methods; or – By a combination of these methods. Exact analytical methods can be applied only in exceptional academic cases. Numerical integration can be applied much more frequently. The most popular computational procedures to determine the failure probability constitute approximate analytical methods. In complicated cases simulation methods or their combination with approximate analytical methods are commonly applied. Most of the commercially available software products include approximate analytical methods and various types of simulation methods. 95

Chapter VI - Reliability analysis F ORM and SORM

The FORM (First Order Reliability Method) is one of the basic and very efficient reliability methods. The FORM method is used as a fundamental procedure by a number of software products for the reliability analysis of structures and systems. It is also mentioned in EN 1990 [2] that the design values are based on the FORM reliability method. Considering the multivariate case, the main steps of the FORM method can be summarised as follows: – The basic variables X are transformed into a space of standardised normal variables U , and the performance function G( X) = 0 is transformed into G’( U) = 0 (Fig. VI-1), – The failure surface G’( U) = 0 is approximated at a chosen given point by a tangent hyperplane (using Taylor expansion), – The design point, i.e. the point on the surface G’( U) = 0 closest to the origin, is found by iteration (see Fig. VI-1), – The reliability index β is determined as the distance of the design point from the origin (see Fig. VI-1) and then the failure probability pf is given as pf = Φ(– β ). The FORM method can be refined by approximating the failure surface G’( U) = 0 by a quadratic surface. Such a method is called The Second Order Reliability Method (SORM). The first step, transformation of the original variable X into a space of standardised normal variables U, is illustrated in Fig. VI-1 (a) showing the original basic variables R and E and Fig. VI-1 (b) showing the transformed variables U R and U E . The transformation into the equivalent normal variables at a given point x* is based on two conditions: – Equal distribution functions:

⎛ x * − μ X e ⎞ ⎟⎟ F X ( x*) = Φ⎜⎜ e σ ⎝ X ⎠

(VI-15)

– Equal probability increments:

⎛ x * − μ X e ⎞ ⎟ f X ( x*) = e ϕ ⎜⎜ σ X ⎝ σ X e ⎠⎟ 1

U E =

E

(VI-16)

E – μ R

σ E

Limit state function Limit state function

Design point (u E d, u Rd ) β

α E β

U R =

μ E α R β

R

R- μ R

σ R

R

(a) Original basic variables R and E. (b) Transformed variables U R and U E. Fig. VI-1. First Order Reliability Method.

96


The mean and the standard deviation of the equivalent normal distribution follow from equations (VI-15) and (VI-16): μ X e = x * −σ X e Φ −1 (F X ( x*)

(VI-17)

⎡ x * − μ X e ) ⎤ 1 1 = ) σ = ϕ ⎢ ϕ [Φ −1 (F X ( x*))] ⎥ e f X ( x*) ⎣ σ X ⎦ f X ( x*)

(VI-18)

e X

The whole computation iteration procedure of the FORM method can be summarised in the following ten steps. 1. The limit state function G( X) = 0 is formulated and theoretical models of basic variables X = { X 1, X 2, ... X n} are specified. 2. The initial assessment of the design point x* = { x1*, x2*, ..., xn*} is made; for example by the mean values of n – 1 basic variables and the last one is determined from the limit state function G(x*) = 0. 3. At the point x* = { x1*, x2*, ... xn*} equivalent normal distributions are found for all the basic variables using equations (VI-13) and (VI-14). 4. The transformed design point u* = {u1*, u2*, ...un*} of the standardised random variables U = {U 1, U 2, ...U n} corresponding to the design point x* = { x1*, x2*, ... xn*} is determined using equation: ui * =

xi

* − μ X e

i

e X i

σ

(VI-19)

5. Partial derivatives denoted as a vector D of the limit state function in respect of the standardised variables U = {U 1, U 2, ...U n} are evaluated at the design point: ⎡ D1 ⎤ ⎢ D ⎥ ⎢ 2⎥ ⎥ where Di = ∂G = ∂G ∂ X i = ∂G σ e D = ⎢ ∂U i ∂ X i U i ∂ X i X i ⎢ ⎥ ⎢ ⎥ ⎢ Dn ⎥ ⎣ ⎦

(VI-20)

For a linear limit state function a0 + ∑ a X i i = 0 the derivatives are Di= ai. 6. The reliability index β is estimated as: ⎧u1 * ⎫ ⎪u *⎪ T ⎪⎪ 2 ⎪⎪ { D} {u *} where {u *} = ⎨ ⎬ β = − { D}T { D} ⎪ ⎪ ⎪ ⎪ ⎪⎩un *⎪⎭

(VI-21)

For a linear limit state function a0 + ∑ a X i i = 0 the reliability index is given as e + ∑ a μ i Xi β = e 2 ∑ (aiσ Xi )

a0

(VI-22)

7. Sensitivity factors are determined as:

{α } =

{ D} T { D} { D}

97

(VI-23)


8. A new design point is determined for n – 1 standardised and original basic variables from: (VI-24)

ui* = α β i i e xi* = μ X i

e − ui*σ X i

(VI-25)

9. The design value of the remaining basic variable is determined from the limit state function G(x*) = 0. 10. The steps 3 to 9 are repeated until the reliability index β and the design point { x*} have the required accuracy. Note that different sign conventions are used in literature and software products concerning the FORM method. In particular, the sensitivity factors and the derivatives sometimes have opposite signs to those indicated in the above-mentioned equations. The signs of the sensitivity factors and the derivatives of the limit state function used here are consistent with those provided in EN 1990 [2]. Sim ul ation methods

Various simulation methods (direct, adaptive and allocated) are very popular and attractive for their simplicity and transparency. All the simulation methods are based on the generation of random variables of given distribution. Available software products (EXCEL, MATHCAD, MATLAB) include special subroutines for the generation of commonly used types of distributions (uniform, normal, lognormal, Gumbel). Simulation methods have a number of modifications that can be divided into two basic groups: – Zero-one indicator based methods, which operate in the original space of variables X, – Conditional expectation methods, which can be called semi-analytical methods. The first group of the zero-one indicators includes the direct Monte Carlo simulation (when the original probability density is applied), the method of importance sampling (when the original probability density close to the design point is applied) and the adaptive sampling (updated importance sampling). The second group of the conditional expectation consists of directional simulation (suitable in the case of a union of events) and axis orthogonal simulation (suitable in the case of an intersection of events). In the following the direct Monte Carlo method is described briefly. Information concerning more sophisticated simulation methods is available in a number of specialised references [8,9]. Simulation of a random variable X having an arbitrary distribution F X ( x) may be in general carried out provided that a generator of random numbers having the uniform distribution in the interval <0, 1> is available. If z j denotes realisation of a random Z having the uniform distribution in the interval <0, 1>, then the corresponding realisation x j of the variable X can be obtained using the inverse of the distribution function F X -1( z ) which has the definition domain interval <0,1>. Realisations xi of the random variable X can be therefore obtained from the relationship: x j = F X -1( z j) (VI-26) Using equation (VI-26), realisations xij of all basic variables X i can be generated and then it is verified whether a combination of obtained realisations leads to a failure or not. A failure occurs if: G( x1i ,x2i ,x3i...) < 0

98

(VI-27)


If the number of all realisations is n and the number of realisations which comply with inequality (VI-27) is nf , the failure probability pf may be assessed using the classical definition of probability based on the ratio pf ≈

nf n

(VI-28)

Obviously, the assessment of the probability pf is more accurate when the number of realisations n is sufficiently large. A general rule for the specification of the number n is relatively simple. If the expected failure probability is about 10 –5, i.e. from the number of realisations 105 on average just one should lead to a failure. Then n should be about two orders greater, thus n > 107. Note that the coefficient of variation V pf of the failure probability can be estimated using formula V pf = (1− pf )0,5(n pf ) –0,5 (VI-29) If pf = 10 –5 and n = 107, then it follows from (VI-29) that the coefficient of variation is V pf = 0,10, which is considered a reasonable accuracy [10]. Clearly, to realise n = 107 generations of all the basic variables is a time-consuming, cumbersome procedure. That is why a number of modifications of the direct Monte Carlo have been developed (zero-one indicator-based methods or conditional expectation methods, methods of Latin Hypercube Sampling, or their combination with FORM). These modifications significantly improve the assessment and decrease the number of required realisations. A detailed description of these methods is available in specialised literature and in manuals to the software products such as COMREL [11]. Ti me-var iant r eli abili ty under stationary conditions

The industrial heritage structures are as a rule exposed to actions that may be of a significant variability in time. It is assumed that: – Resistance, permanent actions and model uncertainties can be described by timeinvariant random variables (no degradation is considered), – Load effects of time-variant actions such as wind and snow loads are considered to be stationary and ergodic processes [4], – All the processes are mutually independent. The maximum effect E max(t D) of simultaneously acting time-variant loads over a working life t D is a time-invariant variable that can be estimated using the load combination rule proposed in [12] - the maximum effect occurs when the leading action takes its maximum max t D[S 1(t )] while other (accompanying) actions are at “arbitrary point-in-time values” S j(t ): E max (t D ) ≈ max t D [S 1 (t )] +

∑ S (t ) j

(VI-30)

j

More details on Turkstra’s rule are provided elsewhere [4,13]. The failure probability related to a working life t D can be obtained as follows: pf (t D) = P{Z[X(t D)] < 0} ≈ C SORM(t D)Φ[- β (t D)]

(VI-31)

where Z(·) = limit state function; X(t D) = vector of basic variables including model uncertainties, resistance, permanent actions and the maximum effect of the load combination; C SORM(t D) = curvature correction factor; Φ(·) = cumulative distribution function of the standardised normal variable; and β (t D) = FORM reliability index determined considering the maximum load effect E max(t D).

99


Resistance, traffic load

R[ R0, g(t )]

f R0(r 0)

Q(t ) f Q(q)

~ 1/λ

working life

Time

Fig. VI-2. Decreasing resistance and traffic load within a working life.

Ti me-var iant r eli abili ty under non-stationary conditi ons

For analysis of non-stationary cases, the following simplifying assumptions are made: – Resistance can be described by a monotonically decreasing function R[ R0, g(t )] where R0 denotes the random initial resistance and g( t ) is the degradation function, – Occurrence of a single time-variant load Q(t ) can be approximated by a stationary and ergodic sequence of identically distributed independent load intensities Qi, with the mean renewal rate λ . The simplified models for the time-variant resistance and traffic load are indicated in Fig. VI2. Note that extremes related to the reciprocal of the mean renewal rate are conservatively assumed to be present until a subsequent load renewal. Hence no intermittencies are taken into account. Other variable actions are not considered here. The instantaneous failure probability is defined as follows: pf (t ) = P{Z[X(t )] < 0} ≈ Φ[- β (t )]

(VI-32)

where t = point in time. An upper bound on the failure probability, related a remaining working life (0, t d), can be obtained as follows [14]: pf (0,t D) = P{min Z[ X(τ )] < 0 for 0 < τ < t D}

⎧⎪1 − exp[f β ′ (t d )t d ]⎫⎪ ≤ λ ⎨ ⎬Φ[− β (t d )] ′ ( ) t f ⎪⎩ ⎪⎭ β d

(VI-33)

where f β ’(t d) = time derivative of the function f β (t d) = ln{Φ[- β (t d)]}. Guidance on analysis including combinations of intermittent processes is provided in [14].

100


f X ( x), f X ( x |I )

updated distribution f X ( x |I )

prior distribution f X ( x)

prior xd

updated xd Fig. VI-3. Updating of probability density function.

4.3.

Probabilistic updating

Using additional information on structural conditions, the properties and reliability estimates of the structure may be updated. Two different procedures can be distinguished: 1. Updating of the failure probability, 2. Updating of the probability distributions of basic variables. Di rect updatin g

Direct updating of the structural reliability (procedure 1) can formally be carried out using the following basic formula of probability theory: P( F |I ) = P( F ∩ I ) / P( I ) (VI-34) where P = probability; F = local or global failure; I = inspection information; and ∩ is the intersection of two events. The information should be selected to maximise correlation between the events { F } and { I }. Strong correlation improves the posterior estimate of failure probability while weak correlation yields nearly the same estimates as based on equation (VI13) [7]. The new information may be based for instance on consideration of the satisfactory past performance that may be very important for the industrial heritage structures. For instance a structure, originally used as a factory, might have likely survived loads much greater than those expected for future use as e.g. a museum or gallery. The satisfactory past performance of a structure during a period t A till the time of assessment may be included in the reliability analysis considering the conditional failure probability pf ”(t D|t A) that a structure will fail during a working life t D given that it has survived the period t A. This probability may be estimated in several ways. When the load to which the structure has been exposed during the period t A is known with negligible uncertainties, the resistance or a joint distribution of time-invariant variables may be truncated (a lower bound is set to the value of load). Using the bounded distribution, the conditional (updated) probability pf ”(t D|t A) can be estimated. This approach, similar to the updating for proof load

101


testing described in [15] is exemplified elsewhere [16]. More generally, the updated failure probability may be determined using the following relationship: p f " (t D t A ) =

P{ F (t D ) ∩ F (t A )} P{ F (t D )} - P{ F (t D ) ∩ F (t A )} = P{ F (t A )} 1 - P{ F (t A )}

(VI-35)

where F = complementary event to the failure. The updated probability can be determined by standard techniques for reliability analysis such as the FORM/SORM methods or importance sampling. Updati ng of di str ibu ti ons of basic vari ables

The updating procedure of a univariate or multivariate probability distribution (procedure 2) is given formally as: f X ( x |I ) = C P( I |x) f X ( x)

(VI-36)

where f X ( x |I ) = updated probability density function of a basic variable or statistical parameter X ; f X ( x) = probability density function of X before updating (prior); C = normalising constant; and P( I |x) = likelihood function. New information can be obtained for instance from inspections providing data for updating of a deterioration model, material tests and in-situ measurements that may be taken to improve models of concrete compressive strength, steel yield strength, geometry etc. In these cases the new information is usually applied in the direct updating of (prior) distributions of relevant basic variables that are commonly based on experience from assessments of similar structures, long-term material production, findings reported in literature or engineering judgement. An illustration of use of relationship (VI-36) is presented in Fig. VI-3. In this example updating leads to a more favourable distribution with a greater design value than the prior design value. In general, however, the updated distribution might also be less favourable than the prior distribution. The updating procedure can be used to derive updated (posterior) characteristic and representative values (fractiles of appropriate distributions) of basic variables to be used in the partial factor method or to compare directly action effects with limit values (cracks, displacements). More information on the updating may be found in ISO 12491 [17]. Once the updated distributions for the basic variables f X ( x |I ) have been found, the updated failure probability P( F |I ) may be determined by performing a probabilistic analysis using a common method of structural reliability for new structures. Symbolically this can be written: P( F I ) = ∫ f X ( x I )d x (VI-37) G ( x )<0

where f X ( x |I ) denotes the updated probability density function and g( x) < 0 denotes the failure domain (g( x) being the limit state function). It should be verified that the probability P( F |I ), given the design values for its basic variables, does not exceed a specified target value.

102


REFERENCES

[1] ISO 13822: Bases for design of structures - Assessment of existing structures, Geneve, Switzerland: ISO TC98/SC2, 2003. [2] EN 1990: Eurocode - Basis of structural design, Brussels: CEN, 2002. [3] ISO 2394: General principles on reliability for structures, Geneve, Switzerland: ISO, 1998. [4] JCSS: JCSS Probabilistic Model Code, Zurich: Joint Committee on Structural Safety, 2006. . [5] Gulvanessian, H., Calgaro, J. A.- Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design, London: Thomas Telford, 2002. [6] Matthews, S. L.: Delivering long service life concrete structures by learning from throughlife performance, utilising service life design methods and adopting effective measures to control execution processes. In Proc. fib Symposium "Concrete is this century's superhero", The Concrete Society, 2009, pp. 18. [7] Ellingwood, B. R.: Reliability-based condition assessment and LRFD for existing structures, Struct.Saf. Vol. 18, No. 2-3 (1996), pp. 67-80. [8] Melchers, R. E.: Structural Reliability Analysis and Prediction, 2nd edition, Chichester, England: John Wiley & Sons Ltd., 2001. [9] Ditlevsen, O. & Madsen, H. O.: Structural Reliability Methods, Chichester (England): John Wiley & Sons, 1996. [10] Wen, Y. K.: Structural load modeling and combination for performance and safety evaluation, 1st edition, Amsterdam: Elsevier, 1990. [11] RCP (2003) Strurel (2003): A Structural Reliability Analysis Program System, Comrel & Sysrel User’s Manual 2003. Version 8.00, Munich. [12] Turkstra, C. J.: Theory of Structural Design Decisions. SM Studies Series No. 2, Ontario, Canada: Solid Mechanics Division, University of Waterloo, 1970. [13] Sýkora, M.: Accuracy of selected approaches to time-variant reliability analysis of serviceability limit states. In Proc. 6th Int. Probab. Workshop, eds. C.A. Graubner, H. Schmidt and D. Proske, Darmstadt: TU Darmstadt, 2008, pp. 483-496. [14] Rackwitz, R.: A Concept for Deriving Partial Safety Factors for Time-variant Reliability. In Proc. ESREL’ 97, Advances in Safety and Reliability, ed. C. Guedes Soares, Pergamon, 1997, pp. 1295-1305. [15] Diamantidis, D.: Probabilistic Assessment of Existing Structures, Joint Committee on Structural Safety, RILEM Publications S.A.R.L., 2001.

103


[16] Sýkora, M., Holický, M., Jung, K. et al.: Reliability Assessment of Industrial Heritage Structures and Application to a Light-Weight Steel Roof. In Proc. ICSA2010, ed. P.J.S. Cruz, Leiden: CRC Press/Balkema, 2010, pp. 605-612. [17] ISO 12491: Statistical methods for quality control of building materials and components, 1st edition, Geneve, Switzerland: ISO, 1997.

104

VII DECISION MAKING

Chapter VII - Decision making

1.

DESIGN OF CONSTRUCTION INTERVENTIONS

As a rule re-use and adaptation of the industrial buildings require assessment of structural reliability. It has been recognised that many heritage structures do not fulfil requirements of present codes of practice. Minimisation of construction interventions is required in rehabilitation and upgrades, but sufficient reliability should also be guaranteed. When dealing with the preservation of heritage buildings, it may be difficult to propose construction interventions that respect all requirements for preservation of the heritage value. According to [1,2] modern principles of interventions seem to include the following aspects: – Unobtrusiveness and respect of the original conception, – Safety of the construction, – Durability of materials, – Balance between costs and available financial resources and in some cases also: – Removability, – Compatibility of materials, – Indoor environment quality including aspects of comfort, security and accessibility. 2.

TARGET RELIABILITY LEVELS

Since construction interventions may be excessively expensive and may lead to losses of the cultural and heritage value, a framework based on the probabilistic optimisation is proposed to indicate optimum target reliability levels for the industrial heritage structures. Obtained results are compared with the target reliability levels indicated in [3] and those based on the empirical relationship proposed in [4]. Reliability verification may be based on the (equivalent) relationships (VI-8). The target reliability level can be taken as the level of reliability implied by acceptance criteria defined in proved and accepted design codes. The target level should be stated together with clearly defined limit state functions and specific models of basic variables. For the industrial heritage buildings, moderate consequences of failure and moderate costs of safety measures can often be assumed. In these cases ISO 2394 [3] indicates β t = 3.1. The target reliability level can also be established taking into account the required performance level of the structure, reference period, cost of upgrades (including potential losses of the cultural and heritage value) and possible consequences of failure or malfunction. Lower target levels can be used if they are justified on the basis of social, cultural, economical, and sustainable considerations as indicated in ISO 13822 [5]. A simple model for estimation of the target reliability level has been proposed in [4]: pt = S c t D Ac C f / (n p W ) × 10-4

(VII-1) where S c = social criterion factor (recommended value for listed historical buildings 0.05); t D = remaining working life (considered as 50 years); Ac = activity factor (recommended value for buildings 3); C f = economical factor (5 for a moderate consequences, recommended values: 10 - not serious, 1 - serious consequences of failure); n p = number of endangered persons (in accordance with [6] the most favourable and unfavourable estimates n p,min = 1 and n p,max = 10, respectively, are considered for significant risk of injury or fatalities - a middle class of consequences); and W = warning factor (1 - sudden failure without previous warning). Considering these indicative data, lower and upper estimates of the target reliability level are obtained from equation (VII-1) as follows: 106

Chapter VII - Decision making pt,max = 0.05×50×3×5/(1×0.3) × 10-4 ≈ 3.8 × 10-3; β t,min = 2.7 pt,min = 0.05×50×3×5/(10×0.3) × 10-4 ≈ 3.8 × 10-4; β t,max = 3.4

(VII-2) It appears that the target reliability is within the broad range from 2.7 to 3.4. The value recommended in ISO 2394 [3] is approximately in the middle of this range. 3.

PRINCIPLES OF THE TOTAL COST MINIMISATION

According to [7] the underlying economics is of concern and importance in the upgrading of existing structures. ISO 2394 [3] indicates that the target level of reliability should depend on a balance between the consequences of failure and the costs of safety measures. From an economic point of view, the objective may be to minimize the total working-life cost. Based on studies concerning existing structures [7,8], the objective function for the total cost C tot of the industrial heritage structures is proposed as follows: minimisep E[C tot(t D;p |I )] = E[C IM(t D;p |I )] + E[C R (t D;p |I )] + ΣiE[C f,i(t D;p |I )] (VII-3) where C tot = total cost over the working life; p = decision parameters specified in the assessment that may influence resistance, durability, maintenance, inspection, repair strategies etc.; C IM = preventative inspection and maintenance cost over t D; C R = repair cost over t D; and C f,i = failure cost over t D, dependent on the failure probability for a failure mode i. The summation is made over all (independent) failure modes and load combinations. Principles of cost optimization techniques are described in more details e.g. in [7,9,10] and [11]. The repair cost may include the cost of repair immediately taken after the assessment as well as costs of future repairs. These costs may cover: – Direct costs related to surveys, clean-up of the site, design and construction, and loss of the cultural heritage value, – Indirect costs associated with economic losses due to business interruption or replacement of users if relevant. The failure costs are related to consequences of structural failure (malfunction), including: – Direct costs related to structural damage (cost of repair or replacement) and loss of the cultural heritage value, – Indirect costs associated with economic losses, societal consequences (cost of injuries and fatalities), unfavourable environmental and psychological effects (release of dangerous substances, loss of reputation). The costs to be considered are of course case-specific and need to be assessed taking into account actual structural conditions and foreseen use. The first insight may be obtained from several studies published in [12]. For instance it follows that the ratio of clean-up costs over the repair costs is mostly up to 5 %, but sometimes may reach up to 20 % and in exceptional cases even 50 %. Detailed cost analysis revealed that the repair may include land purchase (10 %), assessment (5 %), clean-up costs (5 %), construction costs (55 %), financial costs (20 %), reserves and other costs (5 %) [12]. Decision in the assessment can result in the complete repair of a structure (to achieve a target reliability), minor repair to postpone the complete repair, or in acceptance of an actual state and postponement of the decision about repair. The target reliability is the reliability level corresponding to the optimum decision (optimum structural parameters popt): pt = pf ”(t D,popt |I ),

β t = β ”(t D,popt |I )

(VII-4) It is hereafter assumed that the decision concerns the immediate repair while inspection, maintenance and future repair strategies are influenced marginally. This may be a 107


reasonable assumption in many practical cases. The optimum decision can then be found by minimisation of the modified total cost C tot’(t D;p |I ): min.p E[C tot’(t D;p |I )] = E[C tot(t D;p |I ) - C IM(t D |I )] = C R (p |I ) + ΣiE[C f,i(t D;p |I )] (VII-5) For industrial heritage structures that are not in use, the immediate repair cost consists of the direct cost only. It is further assumed that the cost corresponding to an immediate repair strategy (decision on parameters p) can be reasonably well estimated using previous experience with repairs of similar structures. 4.

SIMPLIFIED ESTIMATION OF FAILURE COST

Estimation of the failure cost is a very important, but likely the most difficult step in the cost optimisation. For consistency, the repair and failure costs need to be expressed on a common basis. The repair cost is normally specified in a present value. All the expected failure costs that may occur within a working life should thus be likewise estimated in the present worth [7]. This leads to the expected failure cost as follows:

∑ E[C f ,i (t D , p I )] = ∑ ∫ i

i

t D

C f ,i (t )

r (t , p I )dt ≈ t i

(1 + q )

r i (t , p I )

∑ ∫ (1 + q ) C f ,i

i

t

dt

(VII-6)

t D

where C f,i = failure cost that can often be considered as time-independent; q = annual discount rate; and r i(·) = conditional failure rate given by the relationship: ”

”

r i(t ,p) = P{Fi(t ,t +Δt ,p |I )| F i (0,t ,p |I )} / Δt = [ pf,i (t ,p |I )]’ / [1 - pf,i (t ,p |I )]

(VII-7)

where (·)’ = time derivative. Estimation of the failure cost requires analysis of cultural, economic, societal and environmental consequences. It is further assumed that the environmental consequences can be neglected. All the other components of the failure cost should be preferably assessed in monetary terms, which may, however, be difficult. To facilitate this task, classification of the failure consequences is proposed in [13]. For three classes, the rate ρ between: – Societal and economic consequences plus construction cost over – Construction cost is indicated. The rate ρ is primarily dependent on the purpose of a structure. For many industrial heritage structures typically adapted to serve as office, residential buildings, or museums, Class 2 may be considered (moderate consequences - risk to life given a failure moderate and economic consequences considerable; ρ between 2 and 5; examples: office, industrial, residential buildings). The JCSS recommendations seem to be proposed primarily for new structures where construction cost may be assessed from previous experience. This technique may be adjusted for the industrial heritage structures as follows: – Consider a new structure and adapted industrial heritage structure of similar configuration, intended for the same purpose, – From the definition of the consequence classes, it follows that the rate ρ would be similar, – The societal and economic consequences would also be similar since they logically depend primarily on purpose of a structure, – This implies the societal and economic consequences of failure of the heritage structure be estimated using a relevant rate ρ and ‘equivalent construction cost’ C 0’

108


that approximately equals to the construction cost of the new structure C 0, ( ρ - 1) × C 0’ ≈ ( ρ - 1) × C 0. In addition the loss of a cultural heritage value needs to be quantified. In accordance with Annex I of ISO 13822 [14], the cultural heritage value includes authenticity and integrity of a historic structure and its character-defining elements (historic materials, forms, locations, spatial configurations, morphology, concept and details, and structural design). It is indicated that judgments about the cultural heritage value may differ from culture to culture and it is thus difficult to establish any fixed criteria. Several methods have been proposed for the assessment of an environmental value of assets, which may be a similar issue to the estimation of the cultural heritage value as indicated in [15,16]. However, in most applications the loss of a cultural heritage value of a structure and possibly its content C c is estimated by a qualitative expert judgement. In absence of any quantitative assessment of the cultural value, it is proposed to appropriately increase the rate ρ by Δ ρ c = C c / C 0’. For instance, assuming the adaptation of a heritage structure to an office building, the middle rate ρ = 3.5 might be considered. Depending on an estimated cultural value, the rate may be increased by say Δ ρ c ≈ 1.5 to cover the loss of the cultural value in case of failure. Equation (VII-6) can thus be rewritten as follows:

∑ E[C (t , p I )] ≈ C ∑ ( ρ + Δ ρ f ,i

i

D

' 0

i

i

c ,i

− 1)∫ t D

109

r i (t , p I )

(1 + q )

t

dt

(VII-8)


REFERENCES

[1] Lourenco, P. B.: Computations on historic masonry structures, Prog. Struct. Engng Mater. Vol. 4, No. 3 (2002), pp. 301-319. [2] Olivero, S., Huovila, P., Porkka, J. et al.: Managing the indoor security and safety in historical buildings. In Proc. CESB10, eds. P. Hájek, J. Tywoniak, A. Lupíšek, J. R ůžička and K. Sojková, Prague: Grada Publishing, 2010, pp. 11. [3] ISO 2394: General principles on reliability for structures, Geneve, Switzerland: ISO, 1998. [4] Schueremans, L. & Van Gemert, D.: Assessing the safety of existing structures: reliability based assessment framework, examples and application, Journal of Civil Engineering and Management Vol. X, No. 2 (2004), pp. 131-141. [5] ISO 13822: Bases for design of structures - Assessment of existing structures, Geneve, Switzerland: ISO TC98/SC2, 2003. [6] Trbojevic, V. M.: Another look at risk and structural reliability criteria, Struct.Saf. Vol. 31, No. 3 (2009), pp. 245-250. [7] Ang, A. H. S. & De Leon, D.: Determination of optimal target reliabilities for design and upgrading of structures, Struct.Saf. Vol. 19, No. 1 (1997), pp. 91-103. [8] Onoufriou, T. & Frangopol, D. M.: Reliability-based inspection optimization of complex structures: a brief retrospective, Comput.Struct. Vol. 80, No. 12 (2002), pp. 1133-1144. [9] Rackwitz, R.: Optimization and risk acceptability based on the Life Quality Index, Struct.Saf. Vol. 24, No. 2-4 (2002), pp. 297-331. [10] Rackwitz, R., Lentz, A. & Faber, M.: Socio-economically sustainable civil engineering infrastructures by optimization, Struct.Saf. Vol. 27, No. 3 (2005), pp. 187-229. [11] Holický, M.: Probabilistic risk optimization of road tunnels, Struct.Saf. Vol. 31, No. 3 (2009), pp. 260-266. [12] Brebbia, C. A. & Beriatos, E. (eds.): Brownfields IV, Ashurst Lodge: WIT Press, 2008. [13] JCSS Probabilistic Model Code, Zurich: Joint Committee on Structural Safety, 2006. . [14] ISO 13822: Bases for design of structures – Assessment of existing structures. Annex I Heritage structures, Draft compiled on 17 October 2008 edition, Geneve, Switzerland: TC98/SC2/WG6, 2008. [15] Sanz, J. Á., Herrero, L. C. & Bedate, A.: Contingent Valuation and Semiparametric Methods: A Case Study of the National Museum of Sculpture in Valladolid, Spain, Journal of Cultural Economics Vol. 27, No. 3-4 (2003), pp. 241-257.

110


[16] Bedate, A., Herrero, L. C. & Sanz, J. Á.: Economic valuation of the cultural heritage: application to four case studies in Spain, Journal of Cultural Heritage Vol. 5, No. 1 (2004), pp. 101-111. [17] de Bouw, M., Wouters, I. & Lauriks, L.: Structural analysis of two metal de Dion roof trusses in Brussels model schools. In Proc. STREMAH XI, ed. C.A. Brebbia, Ashurst Lodge: WIT Press, 2009, pp. 121-130.

111

VIII CONCLUSIONS

Chapter VIII - Conclusions

I mportance of the in dustri al heritage stru ctures

A number of factories, warehouses, power-plants and other industrial construction works, built since the beginning of the Industrial Revolution, has been registered as industrial cultural heritage worldwide. Such structures are mostly of significant architectural, historic, technological or social value. The industrial heritage structures often form part of the urban landscape and provide the cityscape with visual historical landmarks. Protection of the industrial heritage structures helps preserve cultural values, avoids wasting energy and facilitates economic regeneration of regions in decline. Present insufficient attention to systematic recognizing, declaring and protecting the industrial heritage may, however, lead to their extinction. Desired protection of the industrial heritage structures requires a public recognition of the industrial heritage to be equally important as other cultural heritage. Introduction of educational programs and relevant legislation is needed. General pri nci ples f or the assessment of the in dustr ial heri tage str uctur es

The main principles of the assessment of industrial heritage structures are: – Currently valid codes for verification of structural reliability should be applied, historic codes valid in the period when the structure was designed should be used as guidance documents only, – Actual characteristics of construction material, actions, geometric data and structural behaviour should be considered; the original design documentation including drawing should be used as guidance material only. The most important step of the whole assessment procedure is evaluation of inspection data and updating of prior information concerning strength and structural reliability for which Bayesian approach provides an effective tool. Typically, the assessment of the industrial heritage structures is a cyclic process in which the first preliminary assessment is often supplemented by subsequent detailed investigations and assessment. A report on the structural assessment prepared by an engineer should include a recommendation on possible interventions. However, a client in collaboration with a relevant authority should make the final decision concerning suitable interventions. Probabil istic assessment of the in dustr i al h er itage str uctur es

In many cases reliability assessments of the industrial heritage structures have to account for significant uncertainties related to actual structural conditions that can hardly be realistically described by simplified procedures used for design of new structures. The assessments may then lead to expensive repairs and losses of the cultural and heritage value. In the Handbook the use of probabilistic methods in the assessment is thus systematically recommended, facilitating: – Better description of uncertainties, – Inclusion of results of inspections and tests and the satisfactory past performance in the assessment. The following remarks should be considered in the probabilistic assessment of the industrial heritage structures: – Models for basic variables should be determined taking into account the actual situation and state of the structure. – Target reliability levels are primarily dependent on costs of safety measures and consequences of failure including loss of the cultural heritage value, and may be specified on the basis of the total working-life cost optimisation. Applications of the 114

Chapter VIII - Conclusions

cost optimisation in practice should, however, be based on carefully formulated objective functions, well assessed costs, specified reference period and discount rate. – Judgment about the cultural heritage value is one of the most difficult issues within the cost optimisation. – The target reliability might be selected lower than 3.8 recommended in EN 1990 for basis of structural design. It appears that the reliability index may vary in the broad range from 2.7 to 3.4 for moderate consequences of failure. The probabilistic updating, accounting for the satisfactory past performance, may substantially improve the estimates of the reliability level for structures exposed to dominant permanent actions.

115

ANNEX A CASE STUDIES

Annex A - Case studies

1.

TEXTILE MILL

Application of the proposed probabilistic procedure is illustrated by the reliability assessment of a steel member of a 100-year old building registered as the industrial heritage. The building, originally built as a part of a textile mill, is to be used as an office building. The selected structural member is exposed to bending moment due to permanent and imposed loads. An anticipated working life is 50 years. The reliability assessment is considerably simplified to illustrate general steps of the probabilistic verification and cost optimisation rather than to describe case-specific details. Initially, reliability of the member is verified by the partial factor method. Characteristic values of the resistance and permanent action, given in Tab. A-1, are specified considering results of on-site surveys and original design documentation. No significant degradation is observed. Characteristic value of the imposed load is determined in accordance with EN 1991-1-1 [1]. The deterministic verification reveals that reliability of the member is insufficient as the actual resistance is approximately by 40 % lower than required by Eurocodes. 1.1.

Probabilistic reliability analysis

The limit state function for the member exposed to bending can be written as follows: Z(X,t ) = K R R – K E [G + Q(t )] (A-1) where K R = model uncertainty of resistance; R = flexural resistance; K E = model uncertainty of load effects; G = permanent action; and Q = maxima of the imposed load related to a reference period t . The considered characteristic values and probabilistic models of the basic variables, based on recommendations of JCSS [2] and findings published e.g. in [3], are given in Tab. A-1. Note that for reference periods different from 50 years, the mean of the imposed load is modified as follows: μ Q,t = μ Q,50 + 0.78σ Q ln(t / 50)

(A-2)

where t = time in years. The standard deviation σ Q is constant for any reference period and the coefficient of variation V Q is adjusted accordingly. For convenience all the basic variables in Tab. A-1 are normalised by L2 / 8 ( L is a span of the member). The reliability verification is firstly based on equation (VI-13) (no new information). Using the FORM method, the reliability index is rather low, β ≈ 2.0. Considering the target reliability levels indicated in Chapter VII, the reliability of the member seems to be insufficient. Secondly, the reliability is updated considering the satisfactory past performance to improve this estimate. It is known from previous performance of the structure that the member has survived the load S equal to 1.2-times the characteristic value of the imposed load. Uncertainties in the survived load effect are described by the normal distribution with the mean equal to the observed value and coefficient of variation 0.05. Given the survival of the load S , the updated reliability index β ”(t D|S ) ≈ 2.6 follows from the conditional failure probability based on equation (VI-35): pf ”(t D|S ) = (A-3) 〈P{ K R R - K E (G + Q50) < 0} - P{ K R R - K E (G + min(Q50,S ) < 0}〉 / 〈1 - P{ K R R - K E (G +S ) < 0}〉 It appears that the predicted reliability is still rather low.

118


Variable Bending resistance Permanent load Imposed load (50 years) Resistance uncertainties Load effect uncertainties

Tab. A-1. Models for basic variables. Sym. Unit Dist. R kN/m LN G kN/m N Q50 kN/m GU K R LN K E LN

xk

5.21 3.06 3 1 1

μ X / xk

1.19 1 1.11 1.15 1

V X

0.08 0.05 0.27 0.05 0.1

xk =

characteristic value; μ X = mean; V X = coefficient of variation; LN = lognormal distribution; N = normal distribution; and GU = Gumbel distribution of maximum values. 5

0.025 C tot‘‘

Δ ρ c =

1.5

β (κ )

0.019 Δ ρ c =

0

4 β opt = 3.3

0.013

3 κ opt = 1

0.007 1

1.1

κ opt 1.34 ≈

1.2

1.3

1.4

1.5

2 1.6

1.7

Fig. A-1. Variation of the standardised cost and reliability index with the decision parameter.

1.2.

Cost optimisation

The total cost is further minimised to find the optimum decision on an immediate repair. The optimisation is based on the following assumptions: – The decision does not concern inspection, maintenance and future repair strategies and related costs are not included in the optimisation, – The immediate repair does not lead to the loss of cultural heritage value, – The repair cost can be approximated by C R (κ ) ≈ [0.01(κ - 1) + 0.0075]C 0’ where 1 < κ ≤ 1.5 is the ratio of the resistance after the repair over the actual resistance; if the actual state is accepted, the repair cost is C R (κ = 1) = 0, – The discount rate is q = 3 %, – The moderate societal and economic consequences are considered (the rate ρ = 3.5). Based on equations (VII-5) and (VII-8), the objective function for κ > 1 reads: min.κ E[C tot’’(κ |S )] = E[C tot’(κ |S ) / C 0’]

′

= 0.01(κ -1) + 0.0075 + (2.5 + Δ ρ c) ∫

p f " (κ , t S )

1.03t [1 − p f " (κ , t S )] t D

dt

(A-4)

where C tot’’(·) = standardised cost. The updated failure probability, dependent on the decision parameter κ , and its time derivative are obtained from equation (A-3). Fig. A-1 shows variation of the standardised cost (left-hand vertical axis) with the decision parameter for two alternatives: 119


Fig. A-2. An illustrative photo of the roof. 1. The loss of the cultural heritage value is taken into account ( Δ ρ c = 1.5), 2. The loss of the cultural heritage value is neglected ( Δ ρ c = 0). In addition the reliability index as a function of the ratio κ is plotted in Fig. A-1 (righthand vertical axis). It follows that the decision would be to accept the actual state when the loss of cultural heritage value is neglected. However, when considering the heritage value, the optimum decision is to repair the structure in order to achieve the optimum ratio κ opt ≈ 1.34. Note that the corresponding reliability index is about 3.3. It appears that the target reliability level depends on the cost of repair and consequences of failure including loss of the cultural heritage value. The target reliability index and the optimum ratio increase with the failure consequences. Complementary studies also indicate that the optimum reliability may also be dependent on a reference period and the discount rate. 2.

STEEL ROOF

In this section reliability assessment of a generic member of steel roof of a building registered as the industrial heritage is described. An illustrative photo of the roof is given in Fig. A-2. A roof of the building is exposed to permanent, snow and wind loads. An anticipated remaining working life is 50 years. The reliability assessment is based on the bending moment criterion. 2.1.

Deterministic verification

Initially, reliability of a generic member is verified by the partial factor method. Using the load combination (6.10) provided in EN 1990 [4], the deterministic reliability condition is: r k / γ M0 = γ G g k + γ Q sk + γ Qψ 0,wwk

(A-5) where r k = characteristic resistance obtained as the product of plastic section modulus and characteristic value of yield strength of steel; γ M0 = partial factor for resistance of a crosssection; γ G = partial factor for the permanent load; g k = characteristic value of the permanent action; γ Q = partial factor for variable actions; sk = characteristic snow load on roof; ψ 0,w = factor for combination value of a wind action; and wk = characteristic wind action.

120


Variable Resistance Permanent load Snow load (50 years) Wind action (1 year) Resistance uncertainties Load effect uncertainties *

Tab. A-2. Models for basic variables. Sym. Unit Dist. Partial factor R kN/m LN* 1.0 G kN/m N† 1.35 ‡ S 50 kN/m GU 1.5 W APT kN/m GU 1.5 K R LN K E LN -

xk

2.25 0.8 0.8 0.4 1 1

Lognormal distribution; † Normal distribution; ‡Gumbel distribution of maximum values.

μ X / xk

1.19 1 1.11 0.3 1.15 1

V X

0.08 0.05 0.27 0.5 0.05 0.1

For steel members not susceptible to stability phenomena, the partial factor for resistance is considered by the value 1.0 [5]. Assuming unfavourable effects of the actions, the partial factor for the permanent load is 1.35 and the partial factor for the variable actions 1.5; the combination factor of wind action is 0.6 [4]. Characteristic values of the resistance and permanent action are specified considering results of on-site surveys and original design documentation. No significant degradation is observed. Characteristic values of the climatic actions are determined considering models in appropriate Eurocodes [6,7]. For convenience all the basic variables in equation (A-5) are normalised by L2 / 8 ( L is a span of the member). Characteristic values of the basic variables are included in Tab. A-2. It follows from equation (A-5) that the minimum design resistance (according to the Eurocodes) is 2.64 kN/m. The deterministic verification thus reveals that reliability of the member is insufficient as the actual resistance is approximately by 15 % lower than required. 2.2.

Probabilistic reliability analysis

The limit state function for the member exposed to bending can be written as follows: Z(X) = K R R – K E (G + S 50 + W APT)

(A-6)

where K R = model uncertainty of resistance; K E = model uncertainty of load effects; S 50 = 50year maxima of the snow load; and W APT = arbitrary point-in -time value of the wind action. The combination of the time-variant loads is based on Turkstra’s rule - the snow load, considered here as the leading action, is described by maxima corresponding to a working life while the accompanying wind action is approximated by an arbitrary point-in-time value W APT. Duration of a 50-year maximum of the snow load may be assumed to be about one week [3] and weekly wind maxima thus should be taken into account. As these are often unavailable, annual maxima may be conservatively used. The considered probabilistic models of the basic variables, based on recommendations of JCSS [2] and findings published e.g. in [3], are given in Tab. A-2. Models for the climatic actions are based on available measurements (lowlands and continental climate) [3]. The reliability verification is initially based on equation (VI-13) (no account for the satisfactory past performance). The resulting reliability index is β = 2.68. Considering the target reliability levels indicated in Chapter 7, it seems that the reliability of the member is rather insufficient. To improve this estimate, the reliability is updated considering the satisfactory past performance. The available meteorological records (covering a period of 30 years only, however) indicate that the observed maximum of the snow load is equal to the characteristic value. It is conservatively assumed that no wind was present during this snow event. Neglecting uncertainties in the snow load effect, the updated failure probability may be obtained as:

121


5

ξ = 1.2 including updating

4

ξ = 1

ξ = 0.85 3.4 3.1 3 2.7

without updating 2 0

0.2

0.4

0.6

0.8

1

Fig. A-3. Variation of the reliability index with the load ratio for ξ = 0.85, 1 and 1.2. p fupd (t D t A ) =

∫ f (x)dx X

(A-7)

Z ( X )< 0 K R R − K E G ≥ sk

where the boundary condition for the variables K R, R, K E and G follows from the limit state function (A-6). Using equation (A-7), the reliability index slightly increases to β upd = 2.76. The predicted reliability level exceeds the minimum target reliability index 2.7, but it is still lower than the reliability levels 3.1 and 3.4. 2.3.

Parametric study

To generalise findings of the probabilistic analysis, parametric study is conducted for the load ratio given as the fraction of the characteristic variable actions over the total characteristic load: χ = ( sk + wk ) / ( g k + sk + wk ) = sk (1 + k ) / [ g k + sk (1 + k )]

(A-8)

where k = wk / sk = variable load ratio, considered here by the fixed value 0.5. The load ratio χ may vary within the interval from nearly 0 (underground structures, foundations) up to nearly 1 (local effects on bridges, crane girders). For steel roofs, the load ratio may be expected within the range from 0.4 up to 0.8, depending on a type of roofing [8]. For a given load ratio and characteristic snow load, the characteristic permanent load is obtained from equation (A-8) and characteristic resistance according to the Eurocodes from equation (A-5). The ratio between the actual characteristic resistance and characteristic resistance according to the Eurocodes is denoted ξ . Results of the reliability analysis are indicated in Fig. A-3 that shows variation of the reliability index with the load ratio for ξ = 0.85, 1 and 1.2 when the satisfactory past performance (updating) is taken into account. In addition, for ξ = 0.85 reliability indices determined without the updating are indicated. It follows from Fig. A-3 that the probabilistic updating improves the estimates of the reliability level particularly for structures exposed to dominant permanent actions (for low load ratios χ < 0.4). For light-weight roofs, the influence of updating is insignificant. It is indicated that a sufficient reliability level for any load ratio is obtained only when the actual characteristic resistance exceeds the resistance required by the Eurocodes by 20 % ( ξ = 1.2). In particular for ξ < 1 and χ > 0.6 the reliability level seems to be insufficient.

122

Annex A - Case studies 3.

STEEL BEAM

This example is adopted from [9] and [10]. Consider the limit state function G( X), where X is a vector of basic variables, and the failure F is described by the inequality G(X) < 0. The result of an inspection of the structure I is an event described by the inequality, H > 0. Using equation (VI-34), the updated probability of failure P( F |I ) may be written as: P( F |I ) = P[G(X) < 0 | H > 0] = P[G( X) < 0 ∩ H > 0] / P( H > 0) (A-9) 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 yield strength f y. Using the partial factor method, the design condition Rd − S d > 0 between the design value Rd of the resistance R and the design value S d of the load effect S may be written as: 2 W f yk /γ m - (γ G g k L L2 / 8) > 0 / 8 + γ Q qk

(A-10) where f yk = characteristic strength; g k = characteristic (nominal) value of permanent load g ; qk = characteristic (nominal) value of imposed load q; γ m = partial factor of steel, γ G = partial factor of permanent load; and γ Q = the partial factor of variable load. In analogy with (A-10) the limit state function G( X) follows as: G(X) = R - S = W f y - ( gL2 / 8 + qL2 / 8) (A-11) where all the basic variables are generally considered as random variables described by appropriate probabilistic models. To verify its reliability the beam has been investigated and a proof loading up to the level qtest has been applied. It is assumed that g act 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 f y - ( g act L2/8 + qtest L2/8) > 0}

(A-12) where f y = actual yield strength; and g act = actual permanent load determined reasonably accurately by non-destructive methods. To determine the updated probability of failure P( F |I ) using equation (A-9), it is necessary to assess the following two probabilities: P[G(X) < 0| H > 0] = P[ W f y - ( gL2/8 + qL2/8) < 0 ∩ W f y - ( g act L2/8 + qtest L2/8) > 0]

(A-13)

P( H > 0) = P[ W f y - ( g act L2/8 + qtest L2/8) > 0] (A-14) Additional assumptions concerning the basic variables are needed. Having the results of (A13) and (A-14), the updated probability of failure P(G( X) < 0| H > 0) follows from (A-10). Alternatively, considering results of the proof test, the probability density function f R(r ) of the beam resistance R = Wf y may be truncated below the proof load, as indicated in Fig. A-4. Obviously, the truncation of structural resistance R decreases the updated probability of structural failure defined as pf = P( R − S < 0)

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

123

(A-15)


f R(r )

updated resistance R

prior resistance R r

resistance adequate to proof load Fig. A-4. Effect of the proof loading on structural resistance.

4.

ASSESSMENT OF CONCRETE STRENGTH

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 f ck = x p 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 x p then follows from (IV-15) as: x p, pred = 29.2 - 2.13 ×

1 + 1 × 4.6 = 18.5 MPa 5

(A-16)

where the value t p = − 2.13 is taken from Tab. IV-4 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: m’ = 30.1 MPa, V (m’ ) = 0.50, s’ = 4.4 MPa, V ( s’ ) = 0.28

(A-17)

It follows from equation (IV-12): 2

⎛ 4.6 1 ⎞ < 1 , ν ′ = 1 1 ≈ 6 n′ = ⎜ ⎟ 2 0.28 2 ⎝ 30.1 0.50 ⎠

(A-18)

The following characteristics are therefore considered: n' = 0 and ν ' = 6. Taking into account that ν = n – 1 = 4, equations (IV-13) yield: n′′ = 5, ν ′′ = 10, m′′ = 29.2 MPa, s ′′ = 4.5 MPa

and finally it follows from equation (IV-14): 124

(A-19)


x p,Bayes = 29.2 - 1,81×

1 + 1 × 4.5 = 20.3 MPa 5

(A-20)

where the value t p = − 1.81 is taken from Tab. IV-4 for 1 − p = 0.95 and ν = 10. In this example the resulting characteristic strength is greater (by about 10 %) than the value obtained by prediction method without using prior information. Thus, when previous information is available the Bayesian approach may improve (not always) the fractile estimate, particularly in the case of a great variance of the variable. In any case, however, a due attention should be paid to the origin of the prior information with regard to the nature of considered variable. 5.

MASONRY STRENGTH

5.1.

Motivation

In the Czech Republic numerous industrial heritage structures are made of different types of masonry. Due to inherent variability of masonry, information on its actual mechanical properties has to be obtained from tests. Estimation of masonry strength from measurements may then be one of key issues in the assessment of an existing structure. Probabilistic framework for design and assessment of masonry structures has been suggested by Mojsilovic & Faber [11] to allow more consistent representation of the material characteristics, description of uncertainties and more economical designs or decisions about repairs. In this example the standard technique provided in Eurocode EN 1996-1-1 [12] is used to develop the probabilistic model of masonry compressive strength in the direction perpendicular to the bed joints (the key characteristic of masonry). An example of the assessment of a masonry building built in the 19 th century is used to clarify general concepts. Masonry strength is estimated from a limited number of destructive tests and series of nondestructive tests of its constituents. Probabilistic model for the model variable is based on experimental results reported in the literature. The characteristic and design values of masonry strength derived using principles of Eurocodes are compared with appropriate fractiles of a proposed probabilistic model. 5.2.

Evaluation of tests

The investigated structure, located in the downtown of Prague, was built in about 1890. Analysis of the six-storey masonry building is based on models for several parts of the structure. The example is focused on estimation of compressive strength of unreinforced masonry - the key issue of the assessment. Mechanical properties of masonry are strongly dependent on properties of its constituents. Commonly, there is a large variability of mechanical properties within a structure due to workmanship and inherent variability of materials as indicated by Lourenco [13] and Stewart and Lawrence [14]. In the present case information about material properties needs to be obtained from tests. Series of non-destructive tests was supplemented by few destructive tests. In addition previous experience on accuracy of applied testing procedures is taken into account in evaluation of test results.

125


Relative frequency

0.5 0.4 0.3 0.2 outlier

0.1 0

30

35

40

45

50

55

60

Non-destructive tests of masonry units in MPa

Fig. A-5. Histogram of the masonry unit strength obtained by non-destructive tests.

Tab. A-3. Statistical characteristics of variables influencing the masonry strength. Coefficient of Variable Symbol Mean Skewness variation Strength of masonry units (non43.1 f b’ 0.08 0.15 destructive tests) MPa Conversion factor - masonry units 0.45 0.2 unknown η b Strength of mortar (non-destructive 1.26 ’ f m 0.41 -0.06 tests) MPa Conversion factor - mortar 1 0.2 unknown η m Model variable 0.68 0.26 unknown Κ Str ength of masonr y uni ts

Non-destructive tests of strength of masonry units by Schmidt hammer were made in 33 selected locations all over the structure. Histogram of the obtained measurements is indicated in Fig. A-5. It appears that the sample includes an extreme measurement (maximum) that may result from an error within the measurement procedure. Therefore, the test proposed by Grubbs [15] is used to indicate whether the hypothesis that there is no outlier in the sample can be rejected. At the significance level 0.05 the test indicates that the hypothesis can be rejected and the measurement is deleted from the sample. Point estimates of the sample characteristics – mean, coefficient of variation and skewness – are then estimated by the classical method of moments described by Ang and Tang [16] for which prior information on the type of an underlying distribution is not needed. The sample characteristics are indicated in Tab. A-3. It appears that the sample coefficient of variation and skewness of the masonry unit strength estimated by the non-destructive tests are low. These characteristics may provide valuable information for the choice of an appropriate statistical distribution to fit the sample data. However, it is emphasized that the sample size may be too small to estimate convincingly the sample skewness.

126

Annex A - Case studies Relative frequency, probability density function 0.3

LN0 LN

0.2

N 0.1

0

30

35

40

45

50

55

60

Strength of masonry units in MPa (non-destructive tests)

Fig. A-6. Histogram of the masonry unit strength obtained by the non-destructive tests without the outlier and the considered theoretical models. The sample characteristics in Tab. A-3 indicate that the strength of masonry units estimated by the non-destructive tests might be described by a two-parameter lognormal distribution having the lower bound at the origin (LN0) or by a more general three-parameter shifted lognormal distribution having the lower bound different from zero (LN). Another possible theoretical model is the popular normal distribution. Probability density functions of these three theoretical models (considering sample characteristics) and a sample histogram without the outlier are shown in Fig. A-6. It follows that, due to the low sample coefficient of variation and skewness, all the considered models describe the sample data similarly. To compare goodness of fit of the considered distributions, Kolmogorov-Smirnov and chi-square tests described by Ang and Tang [16] are further applied. It appears that no distribution should be rejected at the 5% significance level; however, the lognormal distribution LN0 seems to be the most suitable model. Therefore, this distribution is considered hereafter. The conversion factor η b is further taken into account to determine normalised compressive strength of masonry units f b: η b = f b / f b’

(A-21)

where f b’ = strength of masonry units estimated from the non-destructive tests. Previous experience indicates that the coefficient of variation of the conversion factor may be assessed by the value 0.2. Using a limited number of measurements, the mean value of the conversion factor was estimated by the value 0.45. M ortar strength

Estimation of mortar strength may be a complicated issue since sufficiently large specimens for destructive tests can hardly be taken. Therefore, a non-destructive testing method based on a relationship between hardness and strength of mortar was developed in the Klokner Institute of the Czech Technical University in Prague.

127


0.5 Relative frequency, probability density 0.4 function LN0 LN

0.3

N 0.2 0.1 0

0

0.5

1

1.5 2 2.5 3 Mortar strength in MPa (non-destructive tests)

Fig. A-7. Histogram of the mortar strength obtained by the non-destructive tests and the considered theoretical models. This method is used in the assessment. Histogram of 29 measurements is indicated in Fig. A-7. Point estimates of the sample characteristics given in Tab. A-3 are estimated using the method of moments. The sample coefficient of variation of mortar strength is considerably greater than that of the strength of masonry units. The sample distribution seems to be nearly symmetric as the skewness is about zero. This indicates that a normal distribution might be a suitable model. However, normal distribution is not recommended for description of the variables with the coefficient of variation exceeding, say, 0.20 as negative values can be predicted. Due to the zero skewness, a three-parameter lognormal distribution yields the similar model as the normal distribution. Therefore, the lognormal distribution LN0 is assumed hereafter for the mortar strength estimated by the non-destructive tests. Probability density functions of the theoretical models are shown in Fig. A-7. The conversion factor η m is applied to derive compressive strength of masonry mortar f m from results of the non-destructive tests: η m = f m / f m’

(A-22) where f m’ = mortar strength estimated from the non-destructive tests. Previous experience indicates that the conversion factor has the unit mean and coefficient of variation 0.2 as indicated in Tab. A-3. M odel vari able

The EN 1996-1-1 model for the characteristic compressive strength of unreinforced masonry introduces also the model variable K (see equation (A-24) bellow). In the present study, the Group 1 of masonry units is assumed and the model variable is 0.55. The probabilistic model of K is assumed to include model uncertainties including lack of experimental evidence, simplifications related to the EN 1996-1-1 model and the probabilistic modelling, and unknown quality of the execution. Contrary to the models of the strengths of the constituents, it is hardly possible to obtain experimental data on the model variable in the assessment of a specific existing structure. Therefore, available previous experience and reported experimental data need to be used in the development of a probabilistic model. 128


Evaluation of 20 experimental results [17] reveals that the mean of the model variable is about 1.2-times the characteristic value given in EN 1996-1-1 [12] and the coefficient of variation is 0.2. Considering information provided in the JCSS background material [18], it is estimated that the mean of the model variable is about 1.3-times the characteristic value and the coefficient of variation is 0.34. The sample size is, however, unknown. Using an engineering experience, it is assumed that this information is relatively weak compared to the previous one [17] and the sample size is assumed to be 10. Combining these two samples yields: m K =

n = n1 + n2 = 20 + 10 = 30 n1 m1 + n2 m2 20 × 1.2 × 0.55 + 10 ×1.3 × 0.55

= 0.68 (A-23) 30 n1 s12 + n2 s 22 n1n2 20 × 0.132 + 10 × 0.24 2 20 ×10 2 (0.66 − 0.72)2 = 0.18 + 2 (m1 − m2 ) = + s K = 2 n n 30 30 5.3.

n

=

Masonry strength in accordance with present standards

Char acter i sti c value

According to EN 1996-1-1 [12] the characteristic compressive strength of unreinforced masonry made with general purpose mortar can be estimated as (see Table A-3): 0.7

0.3

f k = K f b f m

= K ( μ η b μ f b’)0.7 ( μ ηm μ f m’)0.3 = 0.55 × (0.45 × 43.1) 0.7 × (1 × 1.26) 0.3 = 4.7 MPa

(A-24)

where μ = mean value. Note that estimates of the mean values of f b’ and f m’, based on the coverage method and related to an appropriate confidence level, should rather be used in equation (A-24) than the point estimates determined by the method of moments. The difference may become significant particularly for small sample sizes. In the considered case, however, this influence is negligible and the point estimates of the mean values are applied. It is emphasized that a rather simplified empirical model for the masonry strength considered in EN 1996-1-1 [12] may not fit available experimental data properly. Other theoretical models may then be used to describe the compressive strength of a particular type of masonry. For instance, application of an exponential function similar to that in equation (A-24), but with general exponents, may improve estimation of the resulting strength [18]. More advanced models can be found in [19]. Desi gn value

Design value of the masonry strength is derived from the characteristic value using the partial factor γ M: f d = f k / γ M = 4.7 / 2.5 = 1.9 MPa

(A-25)

The partial factor is dependent on a category of masonry units and class that may be related to execution control. However, EN 1996-1-1 [12] provides insufficient guidance on classification of masonry into the proposed categories of a quality level. Following recommendations of the Czech National Annex to EN 1996-1-1 [12], the partial factor 2.5 seems to be appropriate in this case. Note that dependence of partial factors for masonry and execution control is thoroughly analysed in the previous study [20].

129

Annex A - Case studies Tar get r eli abili ty for exi stin g stru ctures

In the design of new structures, the design value of the masonry strength f d is the fractile corresponding to the probability, EN 1990 [4]: pd = Φ(-α R × β ) = Φ(-0.8 × 3.8) = 0.0012

(A-26)

where Φ(·) = cumulative distribution function of the standardised normal distribution; α R = FORM sensitivity factor approximated by the value -0.8 recommended for the leading resistance variable; and β = target reliability index 3.8 for a fifty-year reference period. The considered values of the sensitivity factor and reliability index are assumed to be implicitly represented by the partial factor 2.5. It is shown in Chapter VII that the target reliability index β = 3.1 can be considered for moderate consequences of failure and moderate costs of safety measures. The design value of the masonry strength is then the fractile corresponding to the probability: pd = Φ(-0.8 × 3.1) = 0.0066 5.4.

(A-27)

Probabilistic analysis

Probabil istic model

It has been recognised that present standards and professional codes of practice adopt a conservative approach including the partial factor method to take into account various uncertainties. This may be appropriate for new structures where safety can often be easily increased. However, such an approach may fail for existing structures where requirements to improve the strength may lead to demanding repairs. In case of historical structures repairs may additionally yield loss of a cultural and heritage value, ICOMOS [26]. Therefore, probabilistic model for the masonry strength is proposed to estimate the characteristic and design values from the statistical data obtained by the tests and from previous experience and reduce the uncertainties implicitly covered by the model in EN 19961-1 [12]. Considering equation (A-24), the compressive strength of masonry - random variable f , is given by: f = K (η b f b’)0.7 (η m f m’)0.3

(A-28)

All the variables in equation (A-28) are considered as random variables. Statistical characteristics are provided in Tab. A-3. In the previous section the lognormal distribution LN0 is proposed to describe variability of the strength of the constituents estimated by the non-destructive tests. In the absence of statistical data and considering general experience, the lognormal distribution LN0 is adopted also for the other variables influencing the strength of masonry. However, it is emphasized that if there is any evidence to support another distribution, then such a distribution should be preferably applied. When all the basic variables included in equation (A-28) are described by the lognormal distribution LN0, it can be easily shown that the strength of masonry has also the lognormal distribution LN0, which is in agreement with assumptions of previous studies [14,27]. The natural logarithm of the masonry strength is normally distributed with the mean and standard deviation: μ ln( f ) = μ ln( K ) + 0.7[ μ ln(η b) + μ ln( f b’)] + 0.3[ μ ln(η m) + μ ln( f m’)] σ ln( f ) = √{σ ln( K )2 + 0.72[σ ln(η b)2 + σ ln( f b’)2] + 0.32[σ ln(η m)2 + σ ln( f m’)2]}

130

(A-29)


0.3 Probability density function

0.2

1.2‰ fractile

6.6‰ fractile

f d

5% fractile

0.1 f k

0

0

3

6 9 12 15 Masonry strength in MPa

Fig. A-8. Probability density function of the masonry strength and the characteristic and design values. where μ ln( X ) = mean of ln( X ); and σ ln( X ) = standard deviation of ln( X ): μ ln( X ) = μ X – 0.5ln[1 + V X 2];

σ ln( X ) = √{ln[1 + V X 2]}

(A-30)

where μ X = mean X ; and V X = σ X / μ X coefficient variation of a variable X , given in Tab. A-3. Resul ts of the probabil istic anal ysi s

From equations (A-29) and (A-30), the mean 5.7 MPa and coefficient of variation 0.33 of the masonry strength are derived. Probability density function of the masonry strength and the characteristic and design values are indicated in Fig. A-8. In accordance with EN 1996-11 [12], the characteristic strength of masonry corresponds to the 5% fractile of the assumed statistical distribution. In the present case the fractile of the lognormal distribution 3.2 MPa is more than 30 % lower than the characteristic value estimated by equation (A-24) that seems to be considerably unconservative. Similar findings have been achieved earlier by Holicky et al. [17]. Considering the target reliability index 3.8, the 1.2‰ fractile of the probability distribution is 2.0 MPa and partial factor 1.6. For the lower target reliability index 3.1, the design value (6.6‰ fractile) increases to 2.4 MPa and the partial factor reduces to 1.3. Remarkably, the theoretical design value is by about 25 % greater than the design value estimated by equation (A-25). Thus significant economic effects may be achieved when the probabilistic model is used. Characteristic and design values and partial factors for the masonry strength are summarised in Tab. A-4. Sensitivity analysis is further conducted to investigate the importance of basic variables on the resulting probabilistic model. FORM sensitivity factors given in Tab. A-5 are evaluated by the software package Comrel. It follows that the model variable K is the most influencing variable and the proposed model may be improved particularly by reducing variability of this variable.

131


Tab. A-4. Characteristic and design values and partial factors for the masonry strength. Characteristic value or 5% Design value (1.2‰ or Partial Model fractile in MPa 6.6‰ fractile) in MPa factor Deterministic 4.7 1.9 2.5 Probabilistic (target 3.2 2.0 1.6 β = 3.8) Probabilistic (target 3.2 2.4 1.3 β = 3.1) Tab. A-5. FORM sensitivity factors of the variables influencing the masonry strength. Variable Symbol FORM sensitivity factor ’ f b Strength of masonry units (non-destructive tests) 0.17 Conversion factor - masonry units 0.42 η b ’ f m Strength of mortar (non-destructive tests) 0.36 Conversion factor - mortar 0.18 η m Model variable 0.79 Κ 5.5.

Concluding remarks

The following conclusions are drawn from the presented assessment of masonry strength of the industrial heritage structure: − Due to inherent variability of masonry, information on its actual mechanical properties has to be obtained from tests and estimation of masonry strength from measurements may be one of key issues in assessment of the industrial heritage structures. − Available samples should be verified by an appropriate test of outliers as extreme measurements, possibly due to an error, may significantly affect sample characteristics. − Appropriate models for basic variables influencing masonry strength should be selected on the basis of the statistical tests, taking into account general experience with distribution of masonry unit strength. − Lognormal distribution having the lower bound at the origin may be a suitable model for masonry strength. − 5% fractile of a proposed probabilistic model for masonry strength is more than 30 % lower than the characteristic value according to EN 1996-1-1. − The theoretical design value (6.6‰ fractile corresponding to the reliability index 3.1) is greater by about 25 % than the design value estimated in accordance with EN 1996-1-1. − Significant economic effects may be achieved when the probabilistic model of masonry strength is used. − The model for masonry strength may be improved particularly by reducing the variability of the model variable.

132


REFERENCES

[1] EN 1991-1-1: Eurocode 1: Actions on structures - Part 1-1: General actions; Densities, self-weight, imposed loads for buildings, Brussels: CEN, 2002. [2] JCSS: JCSS Probabilistic Model Code, Zurich: Joint Committee on Structural Safety, 2006. . [3] Holický, M. & Sýkora, M.: Stochastic Models in Analysis of Structural Reliability. In Proc. SMRLO’10, 2010. [4] EN 1990: Eurocode - Basis of structural design, Brussels: CEN, 2002. [5] EN 1993-1-1: EN 1993-1-1:2005 Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings, Brussels: CEN, 2005. [6] EN 1991-1-3: Eurocode 1: Actions on structures - Part 1-3: General actions; Snow loads, Brussels: CEN, 2003. [7] EN 1991-1-4: Eurocode 1: Actions on structures - Part 1-4: General actions - Wind Actions, Brussels: CEN, 2005. [8] Holický, M. & Sýkora, M.: Partial Factors for Light-Weight Roofs Exposed to Snow Load. In Supplement to Proc. ESREL 2009, eds. R. Bris, C. Guedes Soares and S. Martorell, Ostrava: VŠB Technical University of Ostrava, 2009, pp. 23-30. [9] Melchers, R. E.: Structural Reliability Analysis and Prediction, 2nd edition, Chichester, England: John Wiley & Sons Ltd., 2001. [10] Ellingwood, B. R.: Reliability-based condition assessment and LRFD for existing structures, Struct.Saf. Vol. 18, No. 2-3 (1996), pp. 67-80. [11] Mojsilovic, N. & Faber, M.H.: Probabilistic assessment of masonry compressive strength. In Proc. ICOSSAR 2009, eds. H. Furuta, D.M. Frangopol and M. Shinozuka, Leiden: CRC Press/Balkema, 2009, pp. 5. [12] EN 1996-1-1: Eurocode 6 - Design of masonry structures - Part 1-1: General rules for reinforced and unreinforced masonry structures, Brussels: CEN, 2005. [13] Lourenco, P. B.: Computations on historic masonry structures, Prog. Struct. Engng Mater. Vol. 4, No. 3 (2002), pp. 301-319. [14] Stewart, M. G. & Lawrence, S. J.: Model Error, Structural Reliability and Partial Safety Factors for Structural Masonry in Compression, Masonry International Vol. 20, No. 3 (2007), pp. 107-116. [15] Grubbs, F.: Procedures for Detecting Outlying Observations in Samples, Technometrics Vol. 11, No. 1 (1969), pp. 1-21. [16] Ang, A. H. S. & Tang, W. H.: Probabilistic Concepts in Engineering Emphasis on Applications to Civil and Environmental Engineering, 2nd edition, USA: John Wiley & Sons, 2007.

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[17] Holický, M., Pume, D. & Vorlí ček, M.: Masonry Strength Determination from Tests. In Computer Methods in Structural Masonry 3 - Proc. of the Third Int. Symp. on Computer Methods in Structural Masonry, eds. G.N. Pande and J. Middleton, Swansea: Books &

Journals International, 1997, pp. 107-116. [18] JCSS: JCSS Probabilistic Model Code (first draft of 3.2 masonry properties), Zurich: Joint Committee on Structural Safety, 1 April 2009. . [19] Stewart, M. G. & Lawrence, S.: Structural Reliability of Masonry Walls in Flexure, Masonry International Vol. 15, No. 2 (2002), pp. 48-52. [20] Holický, M., Middleton, J. & Vorlí ček, M.: Statistical Analysis of Partial Safety Factors for Structural Masonry. In Computer Methods in Structural Masonry 4: Proc. of the Fourth Int. Symp. on Computer Methods in Structural Masonry, eds. G.N. Pande, J. Middleton and B. Kralj, London: Taylor & Francis, 1998, pp. 325-338. [21] ISO 2394: General principles on reliability for structures, 2nd edition, Geneve, Switzerland: ISO, 1998. [22] ISO 13822: Bases for design of structures - Assessment of existing structures, Geneve, Switzerland: ISO TC98/SC2, 2003. [23] Allen, D. E.: Safety Criteria for the Evaluation of Existing Structures. In Proceedings IABSE Colloquium on Remaining Structural Capacity, 1993. [24] Schueremans, L. & Van Gemert, D.: Assessing the safety of existing structures: reliability based assessment framework, examples and application, Journal of Civil Engineering and Management Vol. X, No. 2 (2004), pp. 131-141. [25] Sýkora, M., Holický, M., Jung, K. et al.: Reliability Assessment of Industrial Heritage Structures and Application to a Light-Weight Steel Roof. In Proc. ICSA2010, ed. P.J.S. Cruz, Leiden: CRC Press/Balkema, 2010, pp. 605-612. [26] ICOMOS: Recommendations for the analysis, conservation and structural restoration of architectural heritage, Paris: International council on monuments and sites, 2003. . [27] Ellingwood, B. R. & Tallin, A.: Limit States Criteria for Masonry Construction, ASCE Journal of Structural Engineering Vol. 111, No. 1 (1985), pp. 108-122.

134

ANNEX B BASIC STATISTICAL CONCEPTS AND TECHNIQUES

Annex B - Basic statistical concepts and techniques

1.

INTRODUCTION

1.1.

Background materials

Elementary concepts and techniques of the theory of probability and mathematical statistics applicable to civil engineering are available in a number of standards [1-5], background materials [6,7], software products such as [8] and books [9-12]. Additional information may be found in the extensive literature listed in the books, see e.g. [9]. In particular, documents developed by JCSS [6] and handbook [7] are closely related to the statistical techniques described in this text. 1.2.

General principles

The theory of structural reliability is based on a general principle that all the basic variables are considered as random variables having appropriate type of probability distribution. Different types of distributions should be used for description of actions, material properties and geometric data. Prior theoretical models of basic variables and procedures for probabilistic analysis are indicated in JCSS documents. Sample characteristics are used as estimates of population parameters. In addition the population fractiles must be often assessed using small samples. 2.

POPULATION AND SAMPLES

2.1.

General

Actions, mechanical properties and geometric data are generally described by random variables (mainly by continuous variables). A random variable X , (e.g. concrete strength), is such a variable, which may take each of the values of a specified set of values (e.g. any value from a given interval), with a known or estimated probability. As a rule, only a limited number of observations, constituting a random sample x1, x2, x3,..., xn of size n taken from a population, is available for a variable X . Population is a general statistical term used for the totality of units under consideration, e.g. for all concrete produced under specified conditions within a certain period of time. The aim of statistical methods is to make decisions concerning the properties of the population using the information derived from one or more random samples. 2.2.

Sample characteristics

A sample characteristic is a quantity used to describe the basic properties of a sample. The three basic sample characteristics, which are most commonly used in practical applications, are: – The mean m representing the basic measure of central tendency; – The variance s2 describing the basic measure of dispersion; and – The coefficient of skewness ω giving the basic measure of asymmetry. The sample mean m (an estimate of the population mean) is defined as the sum m = (Σ xi) / n

(B-1)

with the summation over all the n values of xi. The sample variance s2 (an estimate of the population variance), is defined as: s2 = [Σ ( xi - m)2] / (n - 1)

136

(B-2)


The summation is again over all values xi. Sample standard deviation s is the positive square root of the variance s2. The sample coefficient of skewness ω (an estimate of the population skewness) characterising asymmetry of the distribution is defined as ω = [n (Σ ( xi - m)3) / (n-1) / (n-2)] / s3

(B-3)

Thus, the coefficient of skewness is derived from the central moment of order 3 divided by s3. If the sample has more distant values to the right from the mean than to the left, the distribution is said to be skewed to the right or to have a positive skewness. Otherwise it is said to be skewed to the left or to have a negative skewness. In some cases two different samples may be taken from one population and their combination is needed. If the original data are not available, then the characteristics of combined sample may be determined using the characteristics of both samples. If the sample sizes are n1, n2, the means m1, m2, standard deviations s1, s2 and skewnesses ω 1, ω 2, then the combined sample of the size n = n1 + n2 has the characteristics m= s 2

=

n1 s12

+ n

n1m1 + n2 m2 n

n2 s22

+

n1n2 n

2

(m1 − m2 )2

(B-4)

1 ⎡ n1 s13ω 1 + n2 s23ω 2 3n1n2 (m1 − m2 )( s13 − s23 ) n1n2 (n1 − n2 )(m1 − m2 )3 ⎤ + − ω = 3 ⎢ ⎥ s ⎣ n n2 n3 ⎦ Another important characteristic describing the relative dispersion of a sample is the coefficient of variation v, defined as the ratio of standard deviation s to the mean m v = s / m (B-5) The coefficient of variation v can be effectively used only if the mean m differs from zero. When the mean is much less than the standard deviation, then the standard deviation rather then the coefficient of variation should be considered as a measure of the dispersion. The coefficient of variation v is often used as a measure of production quality; for concrete strength may be expected within a broad range from 0,05 up to 0,20, for structural steel from 0,07 to 0,10. 2.3.

Distribution function

Probability distribution is a term generally used for any function giving the probability that a variable X belongs to a given set of values. The basic theoretical models used to describe the probability distribution of a random variable may be obtained from a random sample by increasing the sample size or by smoothing either the frequency distribution or the cumulative frequency polygon. An idealisation of a cumulative frequency polygon is the distribution function F( x) giving, for each value x, the probability that the variable X is less than or equal to x: F( x) = P( X ≤ x)

(B-6)

A probability density function f( x) is an idealisation of a relative frequency distribution. It is formally defined as the derivative (when it exists) of the distribution function: f( x) = dF( x) / d x

137

(B-7)


1

Φ( x) = ( x-a)/(b-a) Distribution function

a

b

ϕ( x) = 1/(b-a)

a

Probability density function b

Fig. B-1. Uniform distribution. Note that Appendix 1 to this Chapter provides review of selected theoretical models of continuous random variables that are most frequently used in reliability analysis of civil structures. Example

A continuous random variable, which may attain equally likely any point x within a two-sided interval < a, b> (each point x has the same probability density f( x)) is described by a so-called uniform distribution shown in Fig. B-1. The uniform distribution is a basic type of distribution used not only in simulation procedures but also in theoretical modelling of some actions and geometric data. Shapes of the distribution function F( x) and probability density function f( x) for the uniform distribution are shown in Fig. B-1. We can easily observe that it is a general property of the probability density function that the probability of a set of all values of any random variable is equal to 1: ∞

b

∫ f( x)d x = ∫ f( x)d x = 1

−∞

(B-8)

a

Thus, the surface bounded by the horizontal axis x and the curve of the density function f( x) has the area equal to unity. 2.4.

Population parameters

The population parameters are quantities used in describing the distribution of a random variable, as estimated from one or more samples. As in the case of random samples, three basic population parameters are commonly used in practical applications: – The mean μ representing the basic measure of central tendency, – The variance σ 2 as the basic measure of dispersion, and – The coefficient of skewness ω .giving the degree of asymmetry. The population mean μ , for a continuous variable X having the probability density f( x), is defined as:

138


∫

μ = x f( x)d x

(B-9)

the integral being extended over the interval of variation of the variable X . The population variance σ 2, for a continuous variable X having the probability density function f( x), is the mean of the squared deviation of the variable from its mean:

∫

σ 2 = ( x - μ )2 f( x)d x

(B-10)

The population standard deviation σ is the positive square root of the population variance σ 2. The population coefficient of skewness, characterising asymmetry of the distribution, is defined as: ω =

3 3 μ x x x σ ( ) f( )d / ∫

(B-11)

Another population parameter based on the fourth order moment is called kurtosis ε : ε =

∫ ( x - μ )4 f( x)d x / σ 4 - 3

(B-12)

Note that for normal distribution the kurtosis ε defined by equation (B-12) is zero. However, this parameter is used mainly in theoretical considerations. Another important parameter of the population is the coefficient of variation V defined similarly as the sample coefficient of variation: V = σ / μ (B-13) The same restriction on the practical use of V applies as in the case of samples. Geometrically μ is actually the x coordinate of the centre of gravity of the area bounded by the horizontal axis x and the curve of density function f( x). Fig. B-2 shows an example of probability density function of lognormal distribution illustrating the geometric interpretation of the mean μ and standard deviation σ . The measure of dispersion of a random variable X relative to the mean μ is given by the central moment of the second order (moment of inertia) of the area, and standard deviation σ is therefore the centroidal radius of gyration around the mean μ of the area bounded by the horizontal axis x and the curve of probability density function f( x). A very important population characteristic is the fractile x p. If X is a continuous variable and p is a probability (a real number between 0 and 1), the p-fractile x p is the value of the variable X for which the probability that the variable X is less than or equal to x p is p, and hence, for which the distribution function F( x p) is equal to p. Thus: P ( X ≤ x p) = F( x p) = p

(B-14)

In civil engineering the probabilities p = 0,001; 0,01; 0,05 and 0,10 are used most frequently. The probability p is often written as a percentage (e.g. p = 0,1 %; 1 %; 5 %; 10 %). If this is done, then x p is called a percentile, for example the 5th percentile is used when p = 5 %. If p=50 %, then x p is called the median. More details about the fractiles of continuous variables are given in the following sections.

139


Probability density ϕ( x) 2.5

2.0

1.5

ω =1.0 1.0

• 0.5

0.0 0.4

• σ

0.6

0.8

σ

1.0

1.2

1.4

1.6

1.8

Fig. B-2. Geometric illustration of the mean μ and standard deviation σ .

Example

Parameters of the uniform distribution from the previous example may be derived using equations (B-9) to (B-13) as: μ = (a+b) / 2, σ = (b-a) / 12 , ω = 0, ε = - 2,96, V = (b-a) / ((a+b) 3 )

The skewness of a uniform distribution is zero, kurtosis is negative (independent of the bounds a and b). Obviously the distribution is symmetric as the values of the random variable are distributed uniformly. If the lower bound of the distribution is zero, a = 0 (which is sometimes assumed in practical applications), then: μ = 0,5b, σ = 0,289b, ω = 0, ε = - 2,96, V = 0,577

Let us note that the coefficient of variation V in this case (when a = 0) is independent of b and its value is relatively high ( V = 0,577). 3.

SELECTED MODELS OF RANDOM VARIABLES

3.1.

Normal distribution

Most frequently used models of continuous random variables that are applied in reliability analysis of civil structures are reviewed in Appendix 1 of this Chapter. From a practical and theoretical point of view the most important type of distribution of a continuous random variable is the normal (Laplace-Gauss) distribution. Symmetric normal distribution of a variable X is defined on an unlimited interval - ∞ < x < ∞ (which may be undesirable in some practical applications) and depends on two parameters only – on the mean μ and on the standard deviation σ . Symbolically it is often denoted as N( μ ,σ ). The normal distribution is frequently used as a theoretical model of various types of random variables describing some loads (self-weight), mechanical properties (strengths) and

140


geometrical properties (outer dimensions). It is convenient for symmetric random variable with a relatively low variance (coefficient of variation V < 0,3). It fails when used for asymmetric variables with great variance and skewness ω > 0,5. The probability density function of a normal random variable X with a mean μ and standard deviation σ is given by the exponential expression:

⎡ 1 ⎛ x − μ ⎞ 2 ⎤ 1 f( x) = exp ⎢− ⎜ ⎟ ⎥ σ 2π ⎢⎣ 2 ⎝ σ ⎠ ⎥⎦

(B-15)

Skewness ω and kurtosis ε are zero for a normal distribution. Tables for normal distributions are commonly available for probability density function f(u) and distribution function F(u) of a standardized variable U [9], which is defined by a general transformation relation (used for any type of distribution): U =

X −

σ

(B-16)

The standardized random variable U has a zero mean and variance (standard deviation) equal to one; symbolically it is often denoted as N(0, 1). The probability density function of the standardized random variable U is then given as a function of u:

⎛ u 2 ⎞ 1 f(u) = exp⎜ − ⎟ 2π ⎝ 2 ⎠

(B-17)

The probability density function of a normal and lognormal distribution with a coefficient of skewness ω = 1,0 (described in the next section) of the standardized random variable u is shown in Fig. B-3. Note that the probability density function of the standardized normal distribution is plotted in Fig. B-3 for u in the interval < –3,+3>, which covers the standardised variable U with a high probability of 0,9973 (in engineering practice this interval is often called interval ±3σ ). 3.2.

Lognormal distribution

Generally one-sided limited asymmetric lognormal distribution is defined on a limited interval x0 < x < ∞ or -∞ < x < x0. Therefore it eliminates one of the undesirable properties of the normal distribution. A lognormal distribution is generally dependent on three parameters. Commonly the moment parameters are used: mean μ X , standard deviation σ X and skewness ω x. If the skewness ω x is unknown or uncertain, the lower or upper bound x0 is used. Random variable X has a lognormal (general three-parametric) distribution if the transformed random variable: Y = ln |X – x0|

(B-18) has a normal distribution. In this relation x0 denotes the lower or upper limit of distribution of a variable X , which depends on skewness ω x. If the variable has a mean μ x and standard deviation σ x, then the lower or upper limit can be expressed as: x0 = μ X - σ X /c

141

(B-19)


Probability density ϕ( x) 0.5

Lognormal distribution, ω = 1,0

0.4

Normal distribution N(0,1)

0.3

0.2 0.1 0.0 -3

-2

-1

0

1

2

3

4

5

Standardized random variable u Fig. B-3. Normal and lognormal distribution (skewness ω = 1,0). where the coefficient c is given by the value of skewness ω X according to the relation: ω X = c3 + 3c3

(B-20)

from which follows an explicit relation for c:

( ⎣

13 ) + 4 (B-21) − ω X ⎤⎥ 2 −1 3 X ⎦ The dependence of the limit x0 on the coefficient c is obvious from Tab. B-1 in which the lower bound u0= −1/c of the standardised random variable U =( X- μ X )/σ X are given for selected values of the coefficient of skewness ω X ≥ 0. For ω X ≤ 0 values of u0 with an inverse sign (i.e. positive) are considered. A lognormal distribution with the skewness ω x = 0 becomes a normal distribution (u0= −1/c → ± ∞). 2 c = ⎡ ω X + 4 + ω x ⎢

) − ( ω 13

2

Tab. B-1. The lower limit u0= −1/c for selected values of coefficient of skewness ω X ≥ 0. ω X

0

0,5

1,0

1,5

2,0

u0= −1/c

-∞

-6,05

-3,10

-2,14

-1,68

When creating a theoretical model it is therefore possible to consider, besides the mean μ X and standard deviation σ X , the coefficient of skewness ω X or alternatively the lower or upper bound of distribution x0. Generally the former possibility is preferred because more credible information is available about the coefficient of skewness, which better characterises the overall distribution of the population (particularly of large populations) compared to the lower or upper bounds. The probability density function and distribution function of the general three parameter lognormal distribution may be obtained from well known normal distribution using 142


modified (transformed) standardised variable u’ obtained from the original standardised random variable u = ( x- μ X )/σ X as:

⎛ 1 ⎞ ln⎜ u + ⎟ + ln(c 1 + c 2 ) c ⎠ u′ = ⎝ 2 ln(1 + c )

(B-22)

where (as above) u = ( x- μ X )/σ X denotes the original standardised variable. The probability density function f LN,U (u’ ) and the distribution function F LN,U (u’ ) = FLN, X ( x) of the lognormal distribution are then given as: f LN,U(u’) =

ϕ (u )

⎛ 1 ⎞ ⎜⎜ u + ⎟⎟ ln(1 + c 2 ) ⎝ c ⎠

FLN, X ( x) = FLN,U(u’) = Φ(u)

(B-23)

(B-24)

where φ(u) and Φ(u) denote the probability density and distribution function of the standardised normal variable. A special case is the popular lognormal distribution with a lower bound at zero ( x0 = 0), which depends on two parameters only – the mean μ X and the standard deviation σ X (symbolically it is denoted LN( μ , σ )). In such a case it follows from equations (B-19) that the coefficient c is equal to the coefficient of variation V X . It further follows from equation (B-20) that the skewness ω X of the lognormal distribution with a lower bound at zero is given by the value of the coefficient of variation V X as: ω X = 3V X + V X 3

(B-25)

Thus the lognormal distribution with the lower bound at zero ( x0 = 0) always has a positive skewness, which may have relatively high value (greater than 0,5); e.g. for the coefficient of variation equal to 0,30 a coefficient of skewness V x = 0,927 obtained from relation (B-25). Applications of the lognormal distribution with the lower limit at zero ( xo = 0) can thus lead to unrealistic theoretical models (usually underestimating the occurrence of negative and overestimating the occurrence of positive deviations from the mean), particularly for higher values of coefficient of variation V X . Although the occurrence of negative values can also be undesirable (unrealistic for most mechanical quantities), it is usually negligible from a practical point of view. Example

Reinforcement cover layer of a reinforced concrete cross-section X has the mean μ = 25 mm and standard deviation σ = 10 mm. The probability density function f( x) for a normal distribution and for a lognormal distribution with a lower limit at zero is shown in Fig. B-4.

143


Probability density ϕ ( x ) 0.05

Log-normal distribution LN(25,10) 0.04

Normal distribution N (25,10)

0.03

0.02

0.01 0.00 -1 0

0

10

20

30

40

50

60

70

80

90

Concrete cover X [mm]

Fig. B-4. Probability density function for the concrete cover. It follows from Fig. B-4 that the normal distribution leads to occurrence of negative values of the reinforcement cover layer, which obviously does not correspond to reality. On the other hand, the lognormal distribution with lower limit at zero overestimates the occurrence of positive deviations of the cover layer, which may not be realistic either and can further lead to unfavourable influences on the strength of the cross-section. The overestimation of occurrence of extreme positive deviations corresponds to a high skewness ω = 1,36 of the lognormal distribution, which follows from equation (B-25). The available experimental data on the concrete cover indicate that the skewness of the distribution is around ω ≈ 0,5, in most cases ω < 1,0. The lognormal distribution is widely applied in the theory of reliability. It is used as a model for various types of random variables describing some loads (self-weight of some materials), mechanical properties (strengths) as well as geometrical data (inner and outer dimensions of cross-sections). It can be used for general asymmetric random variables with both positive and negative skewness. The lognormal distribution with lower limit at zero ( x0 = 0) is very often used for description of mechanical properties (strengths) of various materials (concrete, steel, masonry). 3.3.

Gamma distribution

Another popular type of one-side limited distribution is the type III Pearson distribution. Its detailed description is e.g. in the book [9]. A special case of the type III Pearson distribution with lower limit at zero is the gamma distribution. The probability density function of this important distribution is dependent on two parameters only: on the mean μ and standard deviation σ . To simplify the notation two auxiliary parameters λ and k are often used: k

ϕ( x) = λ x

2

k −1

exp(−λ x) μ ⎛ μ ⎞ , λ = 2 , k = ⎜ ⎟ Γ(k ) ⎝ σ ⎠ σ

(B-26)

Γ(k ) is the gamma function of parameter k . For the moment parameters of the gamma distribution it holds that: 144

Annex B - Basic statistical concepts and techniques Probability density ϕ( x ) 0.05

Log-normal distribution

n = 157 m = 26,8 s = 11,1 v = 0,42 a = 0,40

0.04

Gamma distribution Normal distribution Beta distribution

0.03

0.02

0.01

0.00

0

10

20 30 40 Concrete cover [mm]

50

60

70

Fig. B-5. Histogram and theoretical models for concrete cover of reinforcement.

μ =

k

,σ =

λ

k

λ

,ω =

2 k

=

2σ μ

= 2V

(B-27)

The curve is bell shaped for k > 1, i.e. for skewness ω < 2 (in the inverse case it is a decreasing function of x). For k → ∞, the gamma distribution approaches the normal distribution with parameters μ and σ . The gamma distribution is applied similarly as the lognormal distribution with lower bound at zero. However, it varies from the lognormal distribution by its skewness, which is equal to twice the coefficient of variation ( ω = 2V ) and is thus lower than the skewness of lognormal distribution, which is more than 50% higher (according to equation (B-25) it is ω X = 3V X + V X 3 ). That is the reason why the gamma distribution is more convenient for describing some geometrical quantities and variable action that do not have a great skewness. Example

A sample of the size n = 157 of the experimental results of concrete cover of reinforcement measurements has these characteristics: m = 26,8 mm, s = 11,1 mm and v = 0,42. It is a relatively large sample, which can be used for the assessed skewness (furthermore a long-term experience is available). A histogram of the obtained values and theoretical models of normal distribution, lognormal distribution with origin at zero, gamma distribution and beta distribution are shown in Fig. B-5, with help of which the appropriateness of the individual models can be considered. According to Fig. B-5 it seems that the gamma distribution describes the histogram of obtained results better than the normal and lognormal distribution. But also the both-side limited beta distribution (described in the following section) seems to be an appropriate model. However, to choose an appropriate theoretical model for describing variables of interest is a complicated task, which can be treated in a theoretical way. Information about

145


some methods of mathematical statistics (about the so-called goodness of fit tests) can be found in specialised literature such as [9]. 3.4.

Beta distribution

An interesting type of distribution is the so-called beta distribution (also called Pearson‘s type I curve), which is defined on a both-side limited interval (but this interval can be arbitrarily extended and the distribution then approaches the normal distribution). Generally it is dependent on four parameters and it is used mainly in those cases when it is evident that the domain of the random variable is limited on both sides (some actions and geometrical data, e.g. weight of a subway car, fire load intensity, concrete cover of reinforcement in a reinforced concrete cross-section). The principal difficulty in practical application is the need to estimate all the four parameters for which credible data may not be available. The beta distribution is usually written in the form: ( x − a )c −1 ( x − b)d −1 ϕ( x) = B (c, d )(b − a)c + d −1

(B-28)

For the lower and upper limit of distribution it holds: a = μ - c g σ , b = μ + d g σ , g =

c + d + 1 cd

(B-29)

where g is an auxiliary parameter. From equations (B-29), relations for parameters c and d can be derived: c=

μ − a ⎛ ( μ − a )(b − μ )

⎜ b − a ⎝

2

σ

b − μ ⎛ ( μ − a )(b − μ ) ⎞ ⎞ − 1⎟ , d = − 1⎟ ⎜ 2 b − a ⎝ σ ⎠ ⎠

(B-30)

For the moment parameters of the beta distribution it holds that: μ =

a + (b − a)c

(c + d )

, σ =

(b − a) (cg + dg )

2 g (d − c) 3 g 2 (2(c + d ) 2 + cd (c + d − 6)) − 3 ω = , ε = (c + d + 2) (c + d + 2)(c + d + 3)

(B-31)

(B-32)

Note that skewness ω and kurtosis ε are dependent only on the parameters c and d (they are independent of the limits a and b). That is why the parameters c and d are called shape parameters. In practical applications the distribution is used for c > 1 and d > 1 (otherwise the curve is J or U shaped), for c = d = 1 it becomes a uniform distribution, for c = d = 2 it is the so-called parabolic distribution on the interval . When c = d , the curve is symmetric around the mean. When d → ∞, the curve becomes the type III Pearson distribution. If c = d → ∞, it approaches the normal distribution. Depending on the shape parameters c and d the beta distribution thus covers various special types of distributions. The location of the distribution is given by parameters a and b. The beta distribution can be defined in various ways. If the parameters a, b, c and d are given, then the moment parameters μ , σ , ω and ε can be assess using equations (B-31) to (B-32). In practical applications however, two other combinations of input parameters are often applied:

146


1. The input parameters are μ , σ , a and b. The remaining parameters c and d can be assessed from equations (B-30), the parameters ω and ε from equations (B-32). 2. The input parameters are μ , σ , ω and one of the limits a (for ω > 0) or b (for ω < 0); the parameters b (or a), c and d can be assessed using equations (B-30) to (B-31). In practical applications the distribution with lower limit a = 0 is often used. It can be shown that in such a case the beta distribution is defined if: ω ≤ 2V

(B-33)

where the coefficient of variation V = σ / μ . For ω = 2V the curve becomes the type III Pearson distribution. Therefore if the input parameters are the mean μ , standard deviation σ and skewness ω ≤ 2V , the beta distribution with the lower limit at zero ( a = 0) is fully described. The upper limit b of the beta distribution with the lower limit at zero follows from the relation (B-29): b=

μ (c + d ) c

=

(1 + V (2 + ω V )) (2V − ω )

(B-34)

In equation (B-34) the parameters c and d are substituted by the following expressions: ω (2V − ω )2 − (4 + ω 2 ) c=− 2V (V ω + 2) 2 − (4 + ω 2 )

(B-35)

ω (2V − ω ) 2 − (4 + ω 2 ) 2 + ω V d = 2 (V ω + 2)2 − (4 + ω 2 ) ω − 2V

(B-36)

which follow from general equations (B-30) to (B-32) for a = 0. Example

Given the mean μ = 25 mm, standard deviation 10 mm ( V = 0,40) and skewness ω = 0,5, assess the parameters of a beta distribution with the lower bound at zero ( a = 0) for a reinforcement cover layer. Equation (B-33) is satisfied (0,5 < 2 × 0,4). From equations (B-35) and (B-36) it follows that: 0,5 (2 × 0,4 − 0,5) 2 − (4 + 0,52 ) c=− = 4,407 2 × 0,4 (0,4 × 0,5 + 2) 2 − (4 + 0,52 ) 0,5 (2 × 0,4 − 0,5) 2 − (4 + 0,52 ) 2 + 0,5 × 0,4 = 12,927 d = 2 (0,4 × 0,5 + 2) 2 − (4 + 0,52 ) 0,5 − 2 × 0,4 The upper bound of the distribution b follows from equation (B-34) that: b=

25 × (4,407 + 12,927) = 98,325 4,407

147


Probability density ϕ( x) 0,05

Log-normal distribution LN(25;10) a = 0, ω = 1,264;

0,04

Normal distribution N(25;10), ω = 0 Gamma distribution Gamma(25;10) a = 0, ω = 0,8;

0,03

Beta distribution Beta(25;10) a = 0, b = 98,3, ω = 0,5;

0,02

0,01

0,00 -10

0

10

20

30

40

50

60

70

80

Concrete cover x [mm] Fig. B-6. Normal, lognormal, gamma and beta distributions for the concrete cover layer of reinforcement in a reinforced concrete member. The beta distribution having the assessed parameters is shown in Fig. B-6 together with a corresponding normal, lognormal and gamma distribution with the lower bound at zero and the same mean μ and standard deviation σ . Fig. B-6 further shows that the normal distribution (skewness ω = 0) leads to the occurrence of negative values, which may not correspond to the real conditions for the reinforcement cover layer. According to equation (B-25) the lognormal distribution with lower limit at zero has skewness ω = 1,264, which does not correspond to experimental results and leads to an overestimation of the occurrence of positive deviations (which may further lead to unfavourable consequences in the reliability analysis of the reinforced concrete member). The gamma distribution has, according to equation (B-27), a skewness ω = 2 V = 0,8 that is closer to the experimental value 0,5. The most convenient seems to be the beta distribution having the skewness ω = 0,5 corresponding exactly to the experimental results. It should be mentioned that mathematical statistics offers a number of “goodness of fit tests“ for evaluation of fitness of a distribution as a theoretical model for obtained experimental results (see for example [9] and a number recently developed ISO standards). The above discussion can therefore be supplemented by statistical tests. On the other hand it is essential to remark that goodness of fit tests very often fail and do not lead to an unambiguous result. In such a case the selection of a convenient model depends on the character of the basic variable, on available experience and on common experience. 3.5.

Gumbel and other distributions of extreme values

The extreme values (maximal or minimal) in a population of a certain size are random variables and their distribution is very important in the theory of structural reliability. Three types of extreme values distribution denoted as types I, II and III are usually covered in the specialised literature. Each of the types has two versions – one for the distribution of minimal 148


values, the second for maximal values distribution. All these types of distribution have a simple exponential shape and are convenient to work with. We will describe in detail the type I extreme value distribution, which is commonly called the Gumbel distribution. Description of other types of distribution can be found elsewhere [9-11]. The distribution function for the type I maximal values distribution (Gumbel distribution of maximum values) has the form: F( x) = exp(-exp(-c( x - xmod))) (B-37) It is a distribution defined on an infinite interval, which depends on two parameters: on mode xmod and parameter c > 0. By differentiating the distribution function the probability density function is obtained in the form: f( x) = c exp(-c ( x - xmod) - exp(-c( x - xmod))) (B-38) Both the parameters xmod, c of the Gumbel distribution can be assessed from the mean μ and standard deviation σ : xmod = μ − 0,577

c=

6σ π

π 6σ

(B-39) (B-40)

Skewness and kurtosis of the distribution are constant: ω = 1,14, ε = 2,4. An important characteristic of the Gumbel distribution is the simple transformation of the original distribution function F( x) to the distribution function F N ( x) describing the maxima of populations that are N times greater than the original population with mean μ and standard deviation σ . If the individual multiples of the original population are mutually independent, then it holds for the distribution function F N ( x): F N ( x) = (F( x)) N (B-41) By substitution of equation (B-37) into equation (B-41) we obtain the distribution function F N ( x) as: F N ( x) = exp(-exp(-c( x - xmod – ln N/c))) (B-42) so the mean μ N and standard deviation σ N of maxima of populations that are N times greater than the original population are: μ N = μ + ln N/c = μ + 0,78 ln N σ , σ N = σ

(B-43)

Thus the standard deviation σ N of the greater population is equal to the standard deviation of the original population, σ N = σ , but the mean μ N is greater than the original value μ by 0.78 ln N /c. Example

One-year maxima of wind pressure are described by Gumbel distribution with a mean μ 1 = 0,35 kN/m2, σ 1 = 0,06 kN/m 2. The corresponding parameters of 50 years maximum value distribution, i.e. parameters μ 50 and σ 50, follow from equation (B-43): μ 50 = 0,35 + 0,78 × ln (50 × 0,06) = 0,53 kN/m 2, σ 50 = 0,06 kN/m 2

149


ϕN ( x x )

Probability density 8

x) ϕ1( x)

7

x) ϕ 50( x)

6

5

4

3

2

1

0

0.2

0.3

0.4

0.5

0 .6

0.7

0.8

Wind pressure x pressure x

Fig. B-7. Distribution of maximum wind pressure over the periods of 1 year and 50 years. Fig. B-7 shows both distributions of one-year and fifty-year maxima of wind pressure described by the Gumbel distribution. The distribution function of type I minimal values distribution (Gumbel distribution distri bution of minimum values) has the form: x) = 1 - exp(-exp(-c( x xmod - x))) F( x (B-44) This distribution is symmetrical to the distribution of maximal values given by equation (B-37). It is therefore also defined on an open interval and depends on two parameters: on mode xmod and parameter c > 0. By differentiating the distribution function we obtain the probability density function in the form: x) = c exp(-c ( x xmod - x) - exp(-c ( x xmod - x))) f( x

(B-45)

Both these parameters can be assessed from the mean μ and and standard deviation σ xmod

= μ + 0,577 c=

π 6σ

6σ π

(B-46) (B-47)

The probability density function of the minimum values is symmetrical to the shape of maximal values relative to mode xmod, as it is apparent from Fig. B-8.

150


Probability density ϕ( x x) 0,4

0,3

Distribution of the minimum values

0,2

Distribution of the maximum values

0,1

0,0

4

6

7

9

10

12

13

15

16

Variable x

Fig. B-8. The Gumbel distribution di stribution of the minimum and maximum values.

Distribution of the maximum values type III

type I

0 1,14 Distribution of the minimum values type II type I type III

-1,14

0

type II

ω

ω

Fig. B-9. Types of distribution distri bution of extreme values versus the skewness ω . In a similar way the type II distribution, the so-called Fréchet distribution, and type III distribution, the so-called Weibull distribution, are defined. All the three types of distribution complement each other with respect to the possible values of skewness ω . Each type covers a certain area of skewnesses, as indicated in Fig. B-9. B- 9. The extreme values distributions of the type I and II I I are often used to describe random variables depending on the maximal values of populations (for example climatic actions). The > 1,14 (for example for type II is particularly convenient for variables with high skewness ω > flood discharge that have ω ~ 2). The extreme values distribution of the type III is usually applied for random variables depending on the minimal values of populations (e.g. strength >− 1,14. and other material properties) assuming that ω >

151

Annex B - Basic statistical concepts and techniques 3.6.

Function of random variables

In general many variables entering reliability analysis of structures may be considered as a function of basic variables X = [ X 1, X 2, ... , X n]. For example resistance R or load effect E may be given as a function: Z = = F(X)

(B-48)

where X = [ X 1, X 2, ... , X n] denotes a vector of basic variables. Then the resulting variable Z is is a random variable and its characteristics may be derived deri ved from relevant characteristics of basic X 1, X 2, ... , X n]. Usually three moment parameters, the mean μ , standard variables X = [ X deviation σ and skewness ω , are used for a first assessment of the resulting variable Z . Experience shows that using derived moment parameters ( μ , σ and ω ) three parameter lognormal distribution provides satisfactory approximation of Z . Appendix 2 of this Chapter provides approximate expressions for fundamental functions of two basic variables that can be used in assessment of failure probability in case of small number of basic variables.

152


REFERENCES

[1] EN 1990: Eurocode - Basis of structural design, Brussels: CEN, 2002. [2] EN 1991-1-1: Eurocode 1: Actions on structures - Part 1-1: General actions; Densities, self-weight, imposed loads for buildings, Brussels: CEN, 2002. [3] ISO 2394: General principles on reliability for structures, Geneve, Switzerland: ISO, 1998. [4] ISO 12491: Statistical methods for quality control of building materials and components, Geneve, Switzerland: ISO, 1997. [5] ISO 13822: Bases for design of structures - Assessment of existing structures, Geneve, Switzerland: ISO TC98/SC2, 2003. [6] JCSS Probabilistic Model Code, Zurich: Joint Committee on Structural Safety, 2006. . [7] Gulvanessian, H., Calgaro, J. A., Holický, M.: Designer's Guide to EN 1990, Eurocode: Basis of Structural Design, London: Thomas Telford, 2002. [8] RCP. Strurel (2003): A Structural Reliability Analysis Program System, Comrel & Sysrel User’s Manual 2003. Version 8.00, Munich. [9] Ang, A. H. S. & Tang, W. H.: Probabilistic Concepts in Engineering Emphasis on Applications to Civil and Environmental Engineering, USA: John Wiley & Sons, 2007. [10] Ditlevsen, O. & Madsen, H. O.: Structural Reliability Methods, Chichester (England): John Wiley & Sons, 1996. [11] Melchers, R. E.: Structural Reliability Analysis and Prediction, Chichester, England: John Wiley & Sons Ltd., 2001. [12] Schneider, J.: Introduction to Safety and Reliability of Structures, revised edition, IABSE, 2006.

153

ω

. w e 0 0 k S

3

3

c + c

k

V + V

3

√ /

2

3

σ 2 .

1 v e √ / d ) . a σ d − n b a ( t S μ 2 / n ) b a e + μ a M ( s r e t a e a μ σ > m a b r a P n b ≤ i a ≤ x m x ≤ ∞ o ≤ D a ∞ −

, ) 2 1 + c d

, ) 2 1 + c d

2 c

2 c

d − + + c d d ( + c g =

g

1

, λ / k

μ V

σ

√

+

g d d ) d c a + − + c

b ( g c

=

g

d − + + c d d ( + c g =

g

1

,

+

λ / k

μ

+ 0

x

) d a

+ + − c b

a (

g

c d + b c

+

σ c

μ /

−

μ σ c

= 0

x

μ σ = V

, 0 0 ∞ 0 x ∞ < < < > ≤ x x ω < ω x ≤ o o ≤ 0 r ∞ r x p − p 0 ⎞ ⎟ ⎟ ⎟ ⎠ ) )

0 1 1 > ≥ ≥ b c d

∞

b

b

x

≤

x

≤

≤

0

x

≤ ∞ ∞ −

≤

0

a

d 7 π o 7 m 5 =

0 c <

x

≤

− / σ 6 σ √ μ 6 / = √ (

x ,

k

<

c + / 7 d o 7 m 5 , x 0 π )

2 2 )

σ / σ / a 1 1 μ μ a > ≥ ≥ b c d = ( λ =

)

c g d d 6 d + c √ ( b + c / g π c =

c

σ c

4 1 , 1

2

c

+

1 ( n l 2 (

2 ⎞ ⎟ ⎟ ⎠ 2

n o i t c n u f y t i s n e d y t i l i b a b o r P

c

⎤ ⎥ ⎥⎦

2

⎞ ⎟ ⎠

)

a −

μ − σ

x

⎛ ⎜ ⎝

1 2

+

1 | c | | σ 0

x

−

) )

⎞ ⎟ ⎟ ⎟ ⎠

2

V +

1 ( n l 2 (

2 ⎞ ⎟ ⎟ ⎠ 2

V

+ 1 μ

x

s x e − n b l l | ⎡ ⎢ ⎢⎣ ( / b ⎛ ⎜ ⎜ ⎝ p n 1 a l x − i ⎛ ⎜ ⎜ ⎝ r e ⎛ ⎜ ⎜ ⎜ ⎝ a − π p v ⎛ ⎜ ⎜ ⎜ ⎝ x 1 2 p c e i x s σ e a π b π 2 2 f o ) ) s 2 l 2 e c 1 V d + + o 1 1 1 ( m ( n n l c l i t | s x 0 i l x i b − a x | b o r P , , n r - n ) ) l a l 0 l o i ) a a l l ω 1 i ) , x g σ t u ) , a a i m m , σ b r σ x u g , m , r μ σ r r i , , e b n o o o a r μ d i μ μ n ( a ( o ( n n r t ( ( n o e N g r R c e t N o g N N g s e L o p i e N L L L L z p D R A

) ) ) d

o m

x

) k ( Γ / ) x λ ( p x e

1 k

x λ

k

−

1

−

1

−

) d c x + ) −

d

a b − ( 1 b − (

) ) a d , c − ( x B ( c

1

−

1

−

) d +

d

x

c

− b

) b ( d , − c c ( ) x B (

1

x ( c

− ( p x e −

) d

o m

x − x ( c − ( p x e

c

) ) b , b ) , l a σ a ω , , , a , m ( a r μ t σ σ , e , m m e n μ μ a a B e ( ( a a t G G g t e e B B

) n ) b ω i , , g , σ i σ a r , , t μ μ ( e o ( a B o a t r e e t e z B B

) l e σ , b ( μ m m u u G G

Handbook-1_heritage Indistrial Structures

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