Handbook Reliability Backgrounds

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PARTNERSHIP:

The Klokner Institute of the Czech Technical University in Prague (KI CTU), convener, Prof. Milan Holicky, Czech Republic The Czech Chamber of Certified Engineers and Technicians Engaged in Construction

(CKAIT), Prof. Alois Materna, Czech Republic, The Institute of Steel Constructions of the University of Technology Aachen (RWTH),

Prof. Gerhard Sedlacek, Germany The Spanish Organisation for Scientific Research (IET), Spain, Dr. Angel Arteaga The University of Pisa (UOP), Prof. Luca Sanpaolesi, Italy

The Netherlands Organisation for Applied Scientific Research (TNO), Prof. Ton

Vrouwenvelder, The Netherlands The Institute for Metal Constructions (IMK), Dr. Igor Kovse, Slovenia The Building Research Establishment (BRE), Prof. Haig Gulvanessian, The United Kingdom

Prague 10.2005 Leonardo da Vinci Pilot Project CZ/02/B/F/PP-134007

DEVELOPMENT OF SKILLS FACILITATING IMPLEMENTATION OF

EUROCODES

HANDBOOK 2

BASIS OF STRUCTURAL RELIABILITY AND RISK ENGINEERING I BASIC CONCEPTS OF STRUCTURAL RELIABILITY (Pages I-1 to I-10)

Summary I-1

1 INTRODUCTION I-1

1.1 B ackground materi al s I-1

1.2 General principles I-1

2 UNCERTAINTIES I-2

2.1 Classification of uncertainties I-2

2.2 Available tools to describe uncertainties I-2

3 RELIABILITY I-3 3. 1 General I-3

3.2 Definition of reliability I-3

3.3 Probability of failure I-4

3.4 Reliability index I-5

3.5 Time variance of failure probability I-5

4 DESIGN TARGETS I-5

4.1 Indicative values of design working life I-5

4.2 Target reliability level I-6

5 DESIGN METHODS IN PRACTICE I-6

5.1 General I-6

5.2 Permissible stresses I-7

5.3 Global safety factor I-7

5.4 Partial factor method I-7

5.5 Probabilistic methods I-8

6 DESIGN ASSISTED BY TESTING I-8

7 CONCLUDING REMARKS I-9 REFERENCES I-10 ATTACHMENTS I-10

1. MATHCAD sheet "Beta-Time" I-11

II ELEMENTARY METHODS OF STRUCTURAL RELIABILITY I (Pages II-1 to II-17)

Summary II-1

1 INTRODUCTION II-1

1.1 B ackground materi al s II-1

1.2 General principles II-1

2 FUNDAMENTAL CASES OF STRUCTURAL RELIBILITY II-1

2.1 General II-1

2.2 Fundamental cases of one random variable II-3

2.3 Fundamental case of two random variables II-6

3 EXACT SOLUTION FOR TWO RANDOM VARIABLES II-8

4 CONCLUDING REMARKS II-10 REFERENCES II-11 ATTACHMENTS II-11

1. MATHCAD sheet "SteelRod.mcd" II-12

2. MATHCAD sheet "DesVRod.mcd" II-14

3. MATHCAD sheet "PrLnLn.mcd" II-16

III RELIABILITY DIFFERENTIATION (Pages III-1 to III-12)

Summary III-1

1 INTRODUCTION III-1 1. 1 Background documents III-1 1.2 General principles III-1

2 BASIC RELIABILITY ELEMENTS III-2

3 DESIGN WORKING LIFE AND RELIABILITY III-3

4 VARIATION OF FAILURE PROBABILITY WITH TIME III-4

5 PARTIAL FACTOR OF A MATERIAL PROPERTY III-5

6 PARTIAL FACTORS OF SELF-WEIGHT III-6

7 CLIMATIC ACTIONS AND IMPOSED LOADS III-7

8 EXAMPLES III-9

9 CONCLUDING REMARKS III-10 REFERENCES III-10 ATTACHMENTS III-11

1. MATHCAD sheet "GammaRG.mcd" III-12

2. MATHCAD sheet "PSIO.mcd" III-13

3. MATHCAD sheet "PSI12.mcd" III-15

IV DESIGN ASSISTED BY TESTING (PAGES IV-1 TO IV-24)

Summary IV-1

1 INTRODUCTION IV-1

1.1 Background documents IV-1

1.2 General principles IV-1

1.3 Preliminary statistical concepts IV-2

2 STATISTICAL DETERMINATION OF A SINGLE PROPERTY IV-4

2.1 General principles IV-4

2.2 Assessment via the characteristic value IV-6

2.3 Direct assessment of the design value IV-8

2.4 Approximation of the factors kn and kd,n IV-9

3 STATISTICAL DETERMINATION OF RESISTANCE MODELS IV-9 REFERENCES IV-14 APPENDIX A - THE DERIVATION OF THE EQUATION (50) IV-15

ATTACHMENTS IV-16

1. EXCEL worksheet from the workbook "dast.xls" IV-17

2. Source file "dast.c" to the program "dast.exe" IV-18

3. Input file "dast.i1" for the program "dast.exe" IV-21

4. Output file "dast.o1" produced by "dast.exe" from "dast.o1" IV-22

5. Input file "dast.i2" for the program "dast.exe" IV-23

6. Output file "dast.o2" produced by "dast.exe" from "dast.o2" IV-24

7. EXCEL chart showing the data from dast.o2 IV-24

V ASSESSMENT OF EXISTING STRUCTURES (PAGES V-1 TO V-15)

Summary V-1

1 INTRODUCTION V-1

1.1 Background documents V-1

1.2 General principles V-1

2 GENERAL FRAMEWORK OF ASSESSMENT V-2

2.1 Reasons for assessment V-2

2.2 Common rules V-2

2.3 General procedure V-2

3 INVESTIGATION V-3 3. 1 Purpose V-3 3.2 Statement V-3

4 BASIC VARIABLES V-4

4.1 General V-4

4.2 Characteristic values V-4

5 EVALUATION OF INSPECTION RESULTS V-5

5.1 Updating in general V-5

5.2 Updating of probability distribution V-5

5.3 Updating of failure probability V-6

5.4 Updating of characteristic and design values V-6

6 STRUCTURAL ANALYSIS V-7

7 VERIFICATION V-7

8 ASSESSMENT IN THE CASE OF DAMAGE V-8

9 FINAL REPORT AND DECISION V-9

10 CONCLUDING REMARKS V-9 REFERENCES V-10 APPENDIX A - GENERAL FLOW OF ASSESSMENT OF EXISTING

STRUCTURES V-11

APPENDIX B - PROBABILITY UPDATING V-12 APPENDIX C - BAYESIAN METHOD FOR FRACTILE ESTIMATION V-14

ATTACHMENTS V-16

1. MATHCAD sheet "Update.mcd" V-17

2. MATHCAD sheet "BayesFract.mcd" V-18

VI PRINCIPLES OF RISK ASSESSMENT (PAGES VI-1 TO VI-12)

Summary VI-1

1 INTRODUCTION VI-1

1.1 Background documents VI-1

1.2 General principles VI-1

2 HAZARD IDENTIFICATION VI-2

3 DEFINITION AND MODELLING OF RELEVANT SCENARIOS VI-3

4 ESTIMATION OF PROBABILITIES VI-4

5 ESTIMATION OF CONSEQUENCES VI-4

6 ESTIMATION OF RISK VI-5

7 LOGIC TREES VI-5

8 BAYESIAN NETWORK VI-8

9 DECISION-MAKING VI-10

10 THE IMPLIED COST OF AVERTING A FATALITY VI-10

11 CONCLUDING REMARKS VI-11 REFERENCES VI-11 NOTATION VI-12

ANNEX A - BASIC STATISTICAL CONCEPTS AND TECHNIQUES (PAGES A-1 TO A-42)

Summary A-1

1 INTRODUCTION A-1

1.1 B ackground materi al s A-1

1.2 General principles A-1

2 POPULATION AND SAMPLES A-1

2.1 General A-1

2.2 Sample characteristics A-2

2.3 Distribution function A-3

2.4 Population parameters A-4

3 SELECTED MODELS OF RANDOM VARIABLES A-6

3.1 Normal distribution A-6

3.2 Lognormal distribution A-6

3.3 Gamma distribution A-9

3.4 B eta di stributi on A-11

3.5 Gumbel and other distributions of extreme values A-13

3.6 Function of random variables A-17

4 ESTIMATION OF FRACTILES A-17

4.1 Fractile of a theoretical model A-17

4.2 Coverage method of fractile estimation A-21

4.3 Prediction method of fractile estimation A-22

4.4 Coefficients of the coverage and prediction methods A-23

4.5 Bayes' method of fractile estimation A-26

4.6 Estimation of fractiles according to Eurocodes A-28 REFERENCES A-30 APPENDIX 1 - PROBABILISTIC MODELS OF BASIC VARIABLES A-31 APPENDIX 2 - STATISTICAL PARAMETERS OF FUNCTIONS OF

RANDOM VARIABLES A-32

APPENDIX 3 - FRACTILE OF A RANDOM VARIABLE A-33

ATTACHMENTS A-34

1. MATHCAD sheet "DistFract.mcd" A-35

2. MATHCAD sheet "SampFract.mcd" A-37

3. MATHCAD sheet "Mod_est.mcd" A-3 9

ANNEX B - ELEMENTARY METHODS OF STRUCTURAL RELIABILITY II (PAGES B-1 TO B-42)

Summary B-1

1 INTRODUCTION B-1

1.1 Background materials B-1

1.2 General principles B-1

2 DESIGN POINT B-1

3 PARTIAL FACTORS B-4

3.1 Material properties B-4

3.2 Permanent load B-6

3.3 Variable load B-7

4 THE GENERAL CASE OF RELIABILITY ANALYSIS B-9 4.1 General B-9

4.2 Basic variables B-10

4.3 Tail sensitivity problem B-11

5 AN EXAMPLE OF REINFORCED CONCRETE SLAB B-11

5.1 General B-11

5.2 A reinforced concrete slab B-12

5.3 Design of a slab B-12

5.4 Reliability consideration B-14

6 ASSESMENT OF THE FAILURE PROBABILITY IN

GENERAL CASE B-15

6.1 General B-15

6.2 First and Second Order Reliability Methods (FORM and SORM) B-15

7 SYSTEM RELIABILITY B-19

7.1 General B-19

7.2 Parallel systems B-20

7.3 Series systems B-21

8 CONCLUDING REMARKS B-21 REFERENCES B-21

ATTACHMENTS B-22

1. MATHCAD sheet "GammaRGQ.mcd" B-24

2. MATHCAD sheet "Prindex.mcd" B-26

3. MATHCAD sheet "RCBeam.mcd" B-27

4. MATHCAD sheet "R elRCB m cd" B-29

5. EXCEL sheet "RCBeam.xls" B-32

6. MATHEMATICA notebook "Fit_distribution.nb " B -34

7. MATHEMATICA notebook "FORM.nb " B -40

8. MATHEMATICA package "Level2.m" B-41

9. MATHCAD sheet "FORM2.mcd" B-44

10. MATHCAD sheet "FORM7.mcd" B-45

11. EXCEL sheet "FORM7.xls" B-47

12. MATLAB package "FORM7.m" B-48

13. MATLAB function "Lndens (x,mu,sigma,sk)" B-50

14. MATLAB function "Lndist (x,mu,sigma,sk)" B-52

15. MATLAB function "Ndens (x, mu,sigma)" B-53

16. MATLAB function "Ndinv (p)" B-54

17. MATLAB function "Ndist (x, mu,sigma)" B-55

ANNEX C - CALIBRATION PROCEDURE (PAGES C-1 TO C-42)

Summary C-1

1 INTRODUCTION C-1

1.1 Background materials C-1

1.2 General principles C-1

2 FUNDAMENTAL LOAD COMBINATIONS C-2

3 GENERIC STRUCTURAL MEMBER C-4

4 PRINCIPLES OF RELIABILITY ANALYSIS C-5

4.1 Limit state function C-5

4.2 Probabilistic models of basic variables C-6

4.3 Reliability measures C-7

4.4 Sensitivity factors C-8

5 RESULTS FOR THE GENERIC CROSS-SECTION C-8

5.1 One variable action C-8

5.2 Two variable actions C-10

6 CONCLUDING REMARKS C-13 REFERENCES C-13 APPENDIX A - DIRECT COMPARISON OF LOAD EFFECTS C-15 APPENDIX B - EFFECT OF THE RESISTANCE VARIABILITY C-19 APPENDIX C - NOTATION C-20 ATTACHMENTS C-21

1. MATLAB function "generic(gR,wr,k) " C-23

2. MATLAB function "Action3(mr,sr,skr,Rd,k) " C-24

3. MATLAB function "Lnpf(mr,sr,skr,me,se,ske) " C-29

4. MATLAB function "Lndens(x,mu,sigma,sk)" C-31

5. MATLAB function "Lndist(x,mu,sigma,sk)" C-32

6. MATLAB function "Ndinv(p)" C-33

7. MATHCAD sheet "Generic.mcd" C-34

8. MATHCAD sheet "Load effect.mcd" C-39

FOREWORD

The Leonardo da Vinci Pilot Project CZ/02/B/F/PP-134007, "Development of Skills Facilitating Implementation of Structural Eurocodes" addresses the urgent need to implement the new system of European documents related to design of construction works and products. These documents, called Eurocodes, are systematically based on recently developed Council Directive 89/106/EEC "The Construction Products Directive" and its Interpretative Documents ID1 and ID2. Implementation of Eurocodes in each Member State is a demanding task as each country has its own long-term tradition in design and construction.

The project should enable an effective implementation and application of the new methods for designing and verification of buildings and civil engineering works in all the partner countries (CZ, DE, ES, IT, NL, SI, UK) and in other Member States. The need to explain and effectively use the latest principles specified in European standards is apparent from various enterprises, undertakings and public national authorities involved in construction industry and also from universities and colleges. Training materials, manuals and software programmes for education are urgently required.

The submitted Handbook 2 is one of 5 upcoming handbooks intended to provide required manuals and software products for training, education and effective implementation of Eurocodes:

Handbook 1: Basis of Structural Design

Handbook 2: Reliability Backgrounds

Handbook 3: Load Effect for Buildings

Handbook 4: Load Effect for Bridges

Handbook 5: Design of Buildings for Fire Situation

It is expected that the Handbooks will address the following intents in further harmonisation of European construction industry:

- reliability improvement and unification of the process of design;

- development of the single market for products and for construction services;

- new opportunities for the trained primary target groups in the labour market.

The Handbook 2 is focused on the basis of structural reliability and risk engineering related to Eurocodes. The following topics are treated in particular:

- basic concepts of structural reliability;

- elementary methods of the reliability theory;

- reliability differentiation and design working life;

- design assisted by testing;

- assessment of existing structures;

- basis of risk assessment.

Annex A to the Handbook 2 provides a review of "Basic Statistical Concepts and Techniques" frequently used in the text. Annex B provides an extension of elementary methods of structural reliability and annex C describes calibration procedures that may be used for specification of reliability elements. The Handbook 2 is written in a user-friendly way employing only basic mathematical tools. Attached software products accompanying a number of examples enable applications of general rules in practice.

A wide range of potential users of the Handbooks and other training materials includes practising engineers, designers, technicians, experts of public authorities, young people - high school and university students. The target groups come from all territorial regions of the partner countries. However, the dissemination of the project results is foreseen to be spread into all Member States of CEN and other interested countries.

Prague 05/2005

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