Palle Thaft-Christensen Michael 1. Baker

Structural Reliability Theory, and Its Applications With 107 Figures

.: ~.

Springer-Verlag Berlin Heidelberg New York 1982

- /~. "'- .. P.-\LLE THOIT-CHRJSTENSEN, Prof.:ssor. Ph, 0 , Institute ofSui!ding T-:chnology and Struc'luml Engineering ..

~~:~~~.~~::~t C~nlr~ .~t;; !;r1~X $I.S. ~ ~:~> r J .>~",;," :- .:' ~ i:' .' "f; "' .,,::, .::;'

MICHAEL), BAKER. ii,s~, (Eng) " Deparlment ol"Ci\'ii Eng'in~erillg - . ,"

' Imperial CoUege ofSdenciarid Te.:hno!ogy: .. 'L?ndon'~ ~-~gi~~.;,Lt:. ~;. ;.: H ~~ .~~ ~. ~~ .>~ }

,~ / !l:IlIuru. bro.W ....... inlf., r~I"\l..:n,~ ot"" 'I'~,;,li,; ;UI~"l~n" :n.OI.~.n n.",co.l: C'.:n'rl ll

pREFACE

Structural [\.'liability theory is con~meci with the r:ltjonalll1!3tment of uncert:lintie~ in Hrue (urOlI engin~ring and with the methods :'or a;sessing the :>nitty and sel":iceability of chj! ~n ' '.lin~erin~ and other structures. It i.3 :J. subject which has grown r3pidly during" the l:m de-::lde and has t!\'oh'ed from bt.'ing :J. topic {or :lC'3.riemic research to:1 set of welldevelopea or ::t\'elop ing nl'"'thuuologies with a wide r3.llge of ?tactical applications.

L'm:ertainties exist in most :ueas ot d;"jj :md structural engineeri!'l~ nnd rational desi'l c~isicns cannot LJe ma.lkoc:J.use ui lh~ prescriptive and essenti" and should be regarded as r.mdoro v:1riables 0\ stochastic prxC!5slIs. I:wn if in u('si!J:" calculations tll~y are c\'entually treated as deterministic. Some prohl~1lU such

,

~IS the

V/

stt:dy the suhject as a whole. The hook should be of value to those with no prior knowledge oi reliability theory. but it shoult1:also be of interest to thOe enginee!S in\'otved in the de-veiopment of structur31 and loading codes and to those concerned with the safety assessment oi compl~x s~ruc:ures. The .b~k does not try to caver all aspects of structural safety and no

at~mpt is made. for example. to discuss structural (:tilures except in generaistatisticai terms. It ',"'as the intention to make this book moderately self.(;ontained and ro~ .thi5 reason chapter 2 is de\'oted to the essential fundamentals of probability theory. H~we,:ei. readers.who have had no training in this branch of mathematics would be well ad\1sed to \tudy a more general" text in addition. Topics such as the statistical theory of extremes. me~hqds_of parameter esti-mation

VII

Thanks are due to our respective culleagues in Aalborg :md Lonuon for tnt-Ir helpful comments and contributions and in pnrticulu to MI'$. Kirsten Aakjrer :md :'-Irs. Norm3 Hornung who h,we undertaken the type.setting and drawing of fifPlres. respectively, with such skilJ and efficiency.

We conclude with some words of caution. Structural reliability theory should nol be thought of as the solution to all safety problems or as a procedure w~ich t."1n be applied in :l. mecha.ni~l fashion. In the right bnnds it is a.powerful tool to aid decision m3king ill matters of structural safety. but like other tools it c!l.n be misused. It should not be thought of !is an a\temat{\'c to more tr:aditional methods of safety analysis. because all the information that is currently used in other approaches can and should be incorporated within :d reliability annlysis. On OC(;nsions the theory may gi\'e results which seem to contradict '~xperience". In lhi~ case. either IICxperi. ence)t will be found to have been incorrectly interpreted or SOITll! part of the rnliahility analysis will be at fault. generally the modelling. The resolulion of these real or app:nent contr.lI..lictions will often pro\;de considerable insight. into the nature of the prohlem being examined. which can only be of benefit.

~1;uch. 1982

P:lUe ThoitChristensen Institute of Building Technology and Structural Engineering

.~alborg Unive~ity Centre AaJborg, Denmark

:'o.lichuel J. Baker Department. of Civil Enl;;ineering imperial Colle~e of Science and T8(:hnoto~y London. England

CONTENTS

1 . THE TREAiMENTOF UNCERTAINT1ES IN STRUCTu'RAL ENGINEERIN'G . ',.

1.1 INTRODUCTION .......................................... 1.1.1 Current risk levels, 2 1.1.2 Struciural codes, 3

IX

1

1.2 UNCERTAINTY ,'....................................... ........ 4 1.2.1 General. 4 1.2.2' Basic variables, 5 1.2.3 Types of un~ty. 6

1.3 STRUCTURAL RELIABILITY AN.-\LYSISAND SAFETY CHECKING .... ' 7 1 .3.1 Structural reliability I 8 1.3.2 Methods of safety checking, 10

BIBLIOGRAPHY .... , .. .... .............. .'. . . . . . . . . . .. . . .. .. 11

2. FUNDAMENTALS 9F PROBABILITY THEORy...... . .................... 13 2.1 INTRODUCTION...... ............... ................ . . . ... . . ... 13 2.2 SAMPLE SPACE .... . ..... , ................ ............ .. 13 2.3 AXIO~IS AND THEOREMS OF PROBABILITY THEORy ... ,.......... .. 15 2,4 RANDOM VARIABLES." .. ,-" ...... ,.,., .. ,., ... ... , .... , .... ,. 19

2.5 l\10:!l.1E:\'TS .............. , . .... , .. "., . .. ,., ... . , .. . .. . 22 2.6 UNIVARIATE DfSTRmUTIONS ......................... . . .... .. ... ' 25 2.7 RANDOl\! VEC~ORS ....... , .. .. : . . . . . . . . . . . . . . . . . . . . . .. 25 2.8 CONDITIONAL DlSTRIBUTlO:-.lS ............. .. .. . ........ . . . _. . . .. 31 2.9 FUNCTIONS OF RAl\1)O:\1 VARIABLES ..... _ .... . ... . .. ' ..... 32 BIBLIOGRAPHy ......... .... " ...... , .. , .......... , .... .. . . .. 35

3. PROBABILISTIC MODELS FOR LOADS AND RESISTANCE VARIABLES. . . .. 3i

3.1 INTRODUCTION ............................... , ..... ,......... 3i 3.2 STATISTICAL THEORY OF EXTREi\IES ....... ,..................... 37

3.2.1 Derivation of the eumulaLive distribution of the jth smaJlest value of n identically distributed independent randont variables Xi' 38

3.2.2 Normal exttemes. 39 3.3 ASYMPTOTIC EXTRE.\IEYALUE DISTRIBUTIONS ... , ... , . . . . 40

3.3.1 Type I extT'eme~alue distributions (Cumbel dinributions), 40 3.3.2 Type II exttem-value distributions. 42 3.3.3 Type 1II exueme-value dif.!!ibution . ;-1.2

x

3.4 ~IODELLING OF RESISTANCE VARIABLES ~IODELSELECTION ...... 44 3.4.1 General remarks. H 3.4.2 Choice of d~tributioru 'for resisUince variables. 52

3.5 ~[ODELLING OF LOAD VARIABLES MODEL SELECTlQN; . :.... ... .. 54 3,5.1 General rem:1rks. 54 . 3.5.2 Choice or distributions of loads and other actions, 58

3.6 ESTIMATION OF DISTRIBUTION PARAMETERS ...... .. ... . .... .. ... 59 3.6.1 Techniques (or parameter estimation, 59 3.6.2 ~Iodel verification, 63

3.7 INCLUSION OF STATISTICAL UNCERTAINTy............ . . . .. .. .... 63 BIBLlOGRAPHY ............... : ..... '::;:: ........ , .. ~ . ;' : .... ...... 64

4, FUNDAMENTALS OF STRUCTURAL R;ELIABI~lTY TJ,-E;ORY . , '... . . ........ 6i

4.1 INTRODUCTION ... : .......... >:: :,,;;r .. ~ : . . .. . ;;: ..... :: . ......... Gi 4.2 ELEMENTS OF CLASSICAL RELIABILITY THEORY ~ ; .. ~ '.: . , .. , ,. 67 4..3 STRUCTURAL R~LIABILITY .-\NALYSIS ..... .. _ ~: ... " . ,...... 70

.t.3.1 General. 70 4.3.2 The fundamental case.;1 4.3.3 Problems reducing to the (un'damental case. i5 ' 4,3.4 Treatment of a single time,varying load. 77 4.3.5 The ~eneral case, 71 4.3.6 MonteCarlo methods. 79

BIBLIOGRAPHy . .. , . .. , ....... ,........... . . . ...... .. ...... . . . . .... 80

a. LEVEL 2 METHODS ..... , , ....... '.' : - . , : . 81

5.1 INTRODUCTION ................. ............. ........... . ...... 81 5.2 BASIC VARIABLES AND FAILURE SURF ACES .. : . . .. . . . . . . . . . . . . . . .. 81 5.3 RELIABILITY INDEX FOR LINEAR FAILURE Fl.i'NCTlONS AND NOR ; , ~ ._ .: .

~1AL BASIC VARIABLES ............................. : .. 83 5.4 HASOFER AND LL.'lO'S RELL-\BILITY lNDEX ..... ......... , _ . .. 88 BIBLIOGRAPHy ............. , ... ,. ..... . .. ................. . 93

6. EXTENDED LEVEL 2 METHODS ..... . , . 95

6.1 INTR.ODUCTION................................... . ............ 95 6.2 CONCEPT OF CORRELATlO:s,.................................... 96 6.3 CORREL.-\TEO BASIC VA RIABLES ...................... , .......... 101 6A XON;-';OR:

7. RELL\BILITY,OF STRUCTURAL SYSTE~(S ........ , .. . " ... 113

7.1 L'''TRODUCTlON ......... ::' .......................... ; .. ~ ....... 113 7.2 PERFECTLY BRITTLE AND PERF:ECTJ,.Y Dl:CTILEELEMEN'TS .... 114 7.3 FUNDAMENTAL SYSTEMS :.:: . : ....... ... . : ....... ~ . : ............... 115 7.4 SYSTEMS WITH EQUALLY CORRELATED ELEMENTS .: : ... : .... 122 BIBUOCRAPHY .. ..... . .. : ... '; ... . : .. : ....... : ..... : .. :. ;.::-.' l ~ .~I.; .. ... 127

-. .

" ~

8. REL~~~.~"BOUNDS ~(}R STRUcrURAL SYSTEMS' . '. ~~: ~ . 129 8.11):TROO~PTIO~ .. ~ . ,,',, ................. : .. .. ~ .. ; ., ... .... : . : ... 129 8.2 $[\lPLE BOUNDS ...... .. .... . ............ ; .. , ' ... : . ' .. '. : .. ~ .. 130 8.3 OITLEVSEN BOUNDS ....... ' ......... ;': '. ' ..... :.::.:: ............ 133 8.4 PARALLEL SYSTEMS WITH UNEQUALLY CORRELATED EiE~IE~"S .. 134 8.5 SERlES SYSTE\IS WITH UNEQUALLY CORRELATED ELEME:-..-rs . 136 BIBLIOGRAPHY ... .-:. ; .' ..... >.:; ; .:. : :,; .. ;'. ' .. ': -~;. : .. -.... . : : .,;.: ,' . -;" ' . .. '~. '; 143

9. L~TROblicTION~TO STOCHASTIC PROCESS THbk'i" :~~~ Irs' 4SES ........ 145 9.11~TROOUCTION .. . ..................... ';- ...... . ;' ......... : .. .... 145 9.2 STOCHASTIC PROCESSES ......... , . '.' ...... . :: .... ',' ...... 145 9.3 GAUSSIAN PROCE,SSES ...... ...... .. . .... .. . ... :. ~ . ' ... .. ~ ; ....... 148 9.4 BARRIER CROSSING PROBLE~I. ... .. . . .. .......... ... .... . ........ 150 9.5 PEAK DiSTRIBUTION ....... ....... . . .... .. . .... ......... :: ....... 156 BIBLIOGRAPHY ......... , ..... , .... . ... : . :' .. -' .. .' ..... 159

'.,'

10. LOAD COMBINATIONS ....... 0 .............. : ... :' .. :~~':.; .. . " ... 0 .... 161 10.1 INTRODUC!ION .......................... '' , o' . , ;.~., ; . ;.: .... ' ... 161 10.2 THE LOAD COMBINATION PROBLEM .............. ~' ;':; . ' . :., . : ..... 162 10.3 THE FERRY BORGESCASTANHETA LOAD ~IODEL ~.: . ' . 0 ; 166 10.4 CO~;8INATION RULES ....... " ......... ::-: ...... ..... -:.: . : .. : : ...... 168 BIBLIOGR .. \PHY ............ .... ..... . . .............. ... . ::f. '~'. '~ ~ . .' ..... 175

11. APPLICATIONS TO STRUCTURAL CODES ............ ,." ............... 177

1l.II!'lTRODUCTION ..... ....... ..... : .............. . ... .;: .... . ; .. ...... 177 11.2 STRUCTURAL SAFETY AND LEVEL 1 CODES .. . ...... -.. . : ... , ...... 178

"

, XII

11.3 mjcO;\I:-.tENDED SAFETY FOR~t:.l,.TS FOR LEVEL 1 COnES, .. . . . .... 180 lli3.1 Limit ;;uu' (ullction; :md checkin,? equ:nions. 180 11.32 . Characteristic \':Uu(',;. of basi" \'3.r1ables. 1 S2 11.3.3 T~atment of geomeuical \'ariables~ IS3 11.3.4 Treatment of material propenies. 185 11.3.5 Trea.tment of loads and ,other'actions. 185

11,-\ :-.IETHODS FOR THE EVAL-VATIO!\" OF PARTIAL COEFFICiENTS.; ... 188 11A..1 Relationship of parcbJ coefficients to level 2 design point. ISS 11.L2 Approximate direct method for the e\'aluotion of panial coem

d ents.190 .. 1';" :~. 11.-1.3 General metho"d for ihe evaluation of parti:ll coefficients. 194

11.5 A." EXA:.iPLE OF PROB.-\BILlSTIC CODE CALIBRATION" .. .. ,.< ....... 196 11.5.1 Aims of calibration.l96

'-. 11.5.2 Results of calibration. 198 ~IQLlO.GR.~P~Y .. ...... . .. . .. . ...... : . ... :.: ' _" ':' :;. :: . . .. : . d 201

12. APPLICATIONS TO FIXED OFFSHORE STRUCTURES ... .- ... . ~ ... 203

12.1 INTRODUCTION ........ . ...................................... 203 12.2 M.OI?~LLIfJ.9. THE .~E~.P9~~E ~F J~-\'CKET ~-Z:~UCTURES. FOR RELIA-

BlUTY AN.-\!. YSIS ..... ...... . . ... .. .. .. , . .. . .. . . . . . ... ...... 203 1~2.l $ea-5late model. ~O; 1~.2.2 Wa\'e model, 215 12.2.3 Lo.3.di.~g model. 217 12.2A Natural frequency model. 219 12.2.5 E\'aluationof structural r~sponse. 219 12.:!.a E\'aluation ,f pe3k response. 220 12.2.7 Oti1er models, 2~2

12.3 PROBABILITY DlSTRI3lJTIO~S FOR L:\IPORTANT LOADING \' ARI _-\BLES . . .......... . .. ... ......... . .. . . .. ....... .. . . . . ...... 223 12.3.1 Wind speed, 223 12.3.2 Morison':; coefficientS. 225

',' . ,; ,J"

12.4 ;\IETIIODS OF RELIABILITY A:-.'ALYSI5 .. ' . . . ... :-: :.":; ;.: :.- . ".' : . ~ . ... 226 12.4.1 Geneal.226

. 12.-1.2 Lelle12 method. 227 .~ . 12.550)[ RESL'L TS FRO)} THE STUDY or .-\ JACKET SrRUcrURE . . . .... 232 BIBLIOGR.-\PHY ............ ... ............. . ......... '" . . ' ... ' .... 234

13. RELIABIL!TY THEORY AND QUALlTY ASSUR.-\NCE ......... 239 lS.1 )~"RODUCTION ...... ..... ... ~ . . .. ... ..... . . : .. . . . . . ... ~:. :.' .!: .... !239 lS.2GROSSERRORS .... ..... ........... . .. . . .. : .. -~ ~ .. : . :; ,': ....... 239

13..2.1 General. 239 13.2.2 Classification of gro$3 error~. 211

XIll

13.3 I:\TERACTlOX OF RELIABILITY A~D QL'ALlTY ASSUR.-\XCE ... ' " .. 2-13 13.3.1 General. 243 1Z.3.2 The effect of gross errors on the choice of p:ll'tiaJ coefficients. 244 ...

13..1 Ql}.-\LITY .-\SStJR.-\~CE ....................................... ' 247 BIBLIOGRAPHy ................................................. 247

APPENDIX A. RANDOM NID1BER GENERATORS ................. 249

1. GENERAL .......................................... 249 2. UNIFORM RANDO~l NU?l.1BER GENERATORS: .................. 249 3. MULTIPLICATIVE CONGRUENCE METHOD .................... 250 4. GENERATION OF RA..~DOM DEVIATES WITH A SPECIFIED PROB

ABILITY O'ISTRIBUTION FUNCTION Fx . . . . . . . . . . . . . . . . . . . . . . . .. 251 5. SPECIAL CASES: GENERATION OF RANDO:"! DEVIATES HAVING

NORMAL AND LOGNOR!o.lAL DISTRIBUTIONS ..................... 252 BIBLIOGRAPHY ......................................... 253

APPENDIX B. SPECTRAL ANALYSIS O~ WAVE FORCES ....... 255

1. INTRODUCTION ............................................... 255 2. GENERAL EQUATIONS OF MOTION .............................. 255 3. MODAL ANALySIS ............................................. 257 4. SOLUTION STRATEGy .......................................... 258 5. MULTIPLE PILES ........................ . ..................... 261 6. COMPUT.-\TIONALPROCEDURE .................................. 261 BIBLIOGRAPHY ........................................... 261

, INDEX . ~ ' .................... 263

Chapter 1

THE TREATMENT OF UNCERTAINTIES IN ST~WCTURAL ENGINEERING

....

' ;. "

1.1 INTRODUCTION:

Cntil fairly rec'entlY there has- been 3 tendency Cor Structural engineering to be dominated by ~eterministic thinking, characterised in design calculations by the use of s~ified minimum ml1terial properties. specified load intensities and by pres

I nn: TREA nl~;":T OF L1:-':CcnT,\!:-;TI~:$ 1:-: STRUCTURAL ENGI!'IF.t~RI~G 1".1.1 Current Risk Levels . .l,.s 01 me::uis of as':('ning Ini' rel:llh'f' imponanC'(' of structural f:lilun'~ a5 a caU~l' of Ol'alh. !;Olnt'

comparati\'~ SJ,atistics for the U.K. nre ~;\'en in tanl(> 1.1 for a numher of caus~~. 'ilws(' iiJrurf'f show that, at least for a typical Western Euwp,~al1 cou:1try. lh(' ri,;.k lO Hrl' from ,;.trUl:tur ... l failures is nCllligihle. For the 3 yr3l' period repont-d. Ihe :11'cr.Jj!l' numh!! or cle;lth~ per annum directly auributablt> 10 structural 3;lul(, I\'.u l~. divided almost L'qually between failures oc-curring during conruuction and the failures of completed StruClures. Other structur.ll failures occur in which there are no deaths or personal injuries: but data on such railures arc more difficult to assemble because in many counuie3 they do not have io be reported.

In comparine: th~ reilth'e risks given in tabie 1.1, account should be taken of differ-ence~' in ~~"po$Ure time i)-pical Cor the \'~ious activities. For example. although air tra ... el \S as.sd~iat.ed with 3 high ruk per hour, a typical passenger rna)' be exposed fur betwee~ only, 53)' ,,10100 hours per year,leading to a risk of death of between 10" and 10'" per year. ii.e. between 1 in 10J and 1 in 1 persons

2iOO 120

59 56 21

7:1 2.0 2.1 0.7

,I 0.1 I 0.002 , 129 15 13 8' 51

Tachr 1.1 Compan~~ Q~.th rilk. !Ayer~Gr li1:{hlS;3 ill t:.K. boiSed on CC'HflI SI"listic~ Off~t. Abnr:t.c; 19:..11.

I I ,

I.;.:? STRCCTl:RAL CODES 3

In assessing the imponance of structural failurei. account should also he- taken a! lbe economh: consequences of collapse and unserviC't:ability. In fact the- economic aspects m:JY be' rel!arded as dominanlsinct' the marrinal returns in terms oC Ih'es 5a\'ed for each additional 1 r.1i11ion in vested in impro\'ln~ the sarety of structures may be small in comparisor. willI the benefits of investint: the same sum in. say, road safet)' or health care. However. structures should. where possib!e. be designed in such a way thllt there is ample warning of impending failure and with brittle failure modes having l~er safety maTiins than ductile modes (Le. higher reliabilities).

1.1.2 Structural Codes Most structural design is undenaken in accordance with codes of practice. which In many coun tries have legal status: although in the U.K., structural codes for buildings are simply Ildeemed to satisfYI) the building regulations. meaning that compliance with the code automatically en sures compliance with the relevant clauses of the building laws. Structural codes typically and properly have a deterministic fonnat and describe what are considered 10 be the minimum standards for design, construction. and workmanship for each type of structure. Most codes can be seen to be evolutionary in nature, with changes being introduced or major re\'isions made at intervals of 3 10 y~s to allow for: new types of structural form (e.g. reinforced ma-sonry). the effects of improved understanding of structural behaviour le.g. of nif[ened plated structures), the effects of changes in manufacturing tolerances or quality control procedures. 3 better knowledge of loads, etc.

Until recently. structural codes could be considered to be documents in which current good practice was codified; and these documents could be relied upon to produce sate, if not econ-omic, structures. These high standards of safety were achieved for the majority of structures. not from an understanding of all the uncertainties that afff!('l the loading and response, but by codifying practice that was known by experience to be satisfactory. The recent generation of structuraJ codes, including the Euro-codes and the associated model cooes for steel and can crete, are however more scientific in nature. They typically cover a wider ra:lf!' of structural elements and incorporate the results of much experimental and theoretical research. They are also more complex documents to assimilate and to use and the associated desip1 costs are can s('quenl-ly highet, as are the risks of errors in interpretation. The benefits of these new codes mU51lherefore lie in the possibility of:

increased O\'er~1I safety for the same construction COStS;

the same or more consistent levels of safety 'with reduced construction COStsj

or, a combination of these two.

A rurther aim should be the trend. where appropriate. towards design procedures which can be applied with confidence to completely new forms of construction without t~e prior need fOI prototypt> ,esting.

1. THE TREATIIEXT OF L:N

' :~, 1.2:2' BASIC VARIABLES ,

':, : 8"' '1 I " -I " ,

,."',, , ,-\'

,

I, ' .

.;;, 1-: \0 . " .. 15 "''';1 ,'' , .. ;, ,, ' , SO. ,Qr .t~~~S ,~u"p~rt~,..;,

',,',

Fi~re 1.1 shows Ihe probabilitieS' p'that the ma.'I>lmum cOiu~~l~,~d:l.i. ground' t~\'el will reach the sum of the ma....:fina of the individual floor'io:1ds ' i:e. the ma....:imum''pOssible

' column 'loadra:t so'ine' time 'duiing a 50 year perioa. o'n'the MSumpticri tharthe ,(ioor loads are mutu:llly independent ~, that.,they remain con.Slant_for an hour and then change to some new random value. and that each has a 1% chance fp =- 0.01) of being at its mu:imum \'alue 'after each renewal (Le. each Ooor is loaded tJ its maxi!Dl.\~ '~~!IU~ (ot:, !lppro:

6 1. THTREAT:.IE. ... .,.Of UNCERTAINTIES IN STRUCTURAL NGlSEERING

as lhi set of basic quantities go\'erning the s:.atic or dynamic response of t;he structure. Basic \'aria~les are quantities such as mechanical propenie! of materials. dimensions, unit. weights, e:lvironment~H.OOIds-;.etc. They are basic in the .sen~e that they are the most fundamental quan ;:~ tit.ies normallY:recognised and useThus, the yield stress~o( steel can be considered as a ba5lc variable, althou!':h this property is iUielf depende~i on chemical composition andvariOUs"rili,CrO-5tructuraJ p:ara~~ters. Mathema-tical models involvina these latter parameters are often use~ by steel producers (or predicting the mechanical p~opertie.s of structural steei.:1i and for .the purposes of quali~y control. HoweVer It is generall)' seDsible to treat the mechanical propenies as basic variables for the purPoses of r

'; Structural reliability.analysis. One justific3tion is that more st:ltisticai data are available for the l-

mechanicaiproperi'r~s of, say, steels than for the mor~" basic metallurgical properties. .~ -... " " . .... " :.: It should also be ,mentioned that it is genernlly impractic3bl"irlo try to obtain sufficient statisti- L;

v cal 'data to model the variations in the saength of complete structural components directly. Re- ;~ li.:lnce must be plact!d on the abilit)~ of the analyst to synthesise this higher level information ~. 'when required.

Ideally, basic variables should be coasen so that they are statistically independent quantities, 'Ho;,vever, this ni'a):\iot ahy"1iys be possible if the' s'trerigth '~f a structure is k'no'wn to be depend- " em "an, fo~ example, any t~~~ ~echanical pro~erties 'that are known to be co~elated, e.g. the .\

tensile strength and the compressive suength of a batch of concrete . .~:

"1.2.3 Typei of Un'cer1:aiht), ;. For the purposes 'of 'si~ctural re~~il!~~' : ~.n~:-.s\s ,it is' necessB.l); .~o d.istinguish between at least It three types of uncenainty - physical uncena.! . .''lry. statistical uncertainty and model uncertain-ty. These are now described.

It should be nOled that in the foUowing. ramjom \'ariables will be den~ted by ~pper case letters~

' Physida'/ uncertainty: Whe'tllet';or"riot 'a suuet:o.lre 'or ~irUcturi.'l element: fails ;hen loaded depen~'~ , ~ p~ 'on the actua~ \:~l~~; ?~~the, ,~ete\'.~t m:neri'al prop~.rii~.s.that gO\'~Fn . i~~ ~t.rength. The reo : :

liability analyst must therefore be concemec:. with the nature of the actual variability of phys: } ical quantities. such as loads. material propen:ies and dimensions. This \'ariabilicy can be de

scribed in terms or probability distributions or stochastic proCesses and some typical examples '~ are discussed in detail in chapter 3. However. physical variabili.ty can be qUlntified only by .~ examining sample data; but. since sample sites are limited by practical and economic consider ~.~ ations. some uncertainty. must remain. This .p~cticallimit gives rise to so~alled statistical un . certain.ty . . :.::,

Statistical uncerla/my: As-,yill be discussed ~ iSler chapters, statisi.ics. ss'bpposea to proba~i: . . ~ .;.;;

ity, is concerned with inference. and in particular with tile inferences that can be drawn from ) s:!mpie observations. Data may be collected (or the purposes of buildi:-ig' -~ probabilistic model :J of the physical variahiliiy of s'ome quantify ,,'hieh will' irl'o'ol\'e , firstly the 'seiection of an ap H propri3te p~obatiil:ty distribution t)'pe. and !!len dete::ninarion 'of numeiYc'ai'\'alues for its pa \

"3

:~~~i.2:3 TYPES OF UNCERTA1NTY 7

r...r.lc~h.

s 1. THE TRE.\nJ~:~T OF UNCER:rAINTIES I~ STRCCTCR.\L E:-:GTNEERING

1.3.1 Structural Reliability The term :JCnlcturai reii(Jbility should !Je clJnsiclered as having two meaninll5 - 3 gl?nerai one and :1 mathematical one.

tn the most general sense. the reliD.bility of a structure is its ability to fulfil ;[5 design pur-pose ror some specified time.

In 3 narrow sense it is the probDbility that a structure will not attain crlch specified limit state (ultimate or serviceabUity)'during a specified ~e{erence period. .

- '-"

'-Cn this book we shall be con~e.med_~.th st~.ci~~r re~~~bil.ity in "the narrow sense and sho.ll gen-erally be treating each limit state or [:lilure mode separately o.nd explicitly. HOWever. most struc-tures and structural elements have a number of possible failure modes. and in determining the overall reliability at a stru~tural system this must t>e takeninto 3Ccount. makinf due allowance for th~ .correl:ltions ads'lng Crom'common sourees of loading and common m:lterial J:lroperties. These aspects of the problem are covered in chapters 7 and 8. However, Glthou~h" the ~defi~ition' above may seem clear. it is necessary to e;

.': 9

)'1 c R-S

. :md where FR is the ,t:'~o.b.abiiity distFibution ('t~c:i,~n' ~:C Ii m~~ .Cs i~e'p:o~~bility density , . . (unction.of .S, These term&:u? defined more Cu.Uy iri chapter 2. " .. . . .

" .' .. , ~ ,.,1.1

" .- ' .... . ... ,.

Because. pt thedeCinition of 'R d~ci S i~', ~e~~ ,of Crequentisl ~ro"babiliti~s. tne p,robability de-tCfmi'~ed i~o~' ~q~atio~ I 1.2) m3~' be i~te~r~~ed as 3 10n~~~.~~~aiiure fre.iuenCY:'Simil.arly the reliability ~\, defined as

", .

.Il = I- Pr ", ..

" \" -. '

m~y be interpreted 'as along-run-survival frequency or longntn reliabili.ty Jlnd is the percent-. age of a' ~~tionalJy Infinite set of :tominany~ldentical 5tiuCtures which survive Cor tlte durn

tion of the reference period T.'~ :::.ay therefore be c.alled a frequentisl reliability. If. however. 'we are Carced to focus 'our 3Uention 'on'one'particular structure (and this is genernlly the C3St! Cor Ii~.~~.offll ~ivij'~n~nee~ing,' structures)\:~ ina~' 0.150 be interpreted is a me;l5l1re of the relia bility' of t.hat p~rticular struct~r~ :': : . : p This interpretation of reliability ii (undnment3.lly different from that given above. becnuse. a1. though ttlc ~'tructure may 'be' s':lf~'p:ed at random Crom th'e' th~oreticnlly infinitc population de-~cribed b~Hhe random'varillb'le R, ',mce'ltle p:irticula'r structu'te bas been selected (and . in proc tice, constructed) the reliability b,;,:omes the probability that the fixed. but unknown, resist-ance r ,,,ill be exceeded by the 3S yet'~un~sampled .. reference period extreme 10!ld eH~ct S I note that lo\ver case r is used here to denote,the outcome of R J. ~The numerical value ,pf U~~ . failure probability.remains the same but ~ now dependent upon two rndically different types of un certainty . ii~s~iy. the physicaJ vat~bil\ty'of the' e~tre~e'!o~d ~r"f~t,:-3nd.se~ondIY;lack of knowledge about the true value oi,the f~~" b~~'~ '~.tnkJ!.~~n . (~~~t.a~ce. Tbi~- type of probability does not h.we a rellldve frequency interpretation and is commonly.c:dled ,a.subjecliV1! plI!babil. ity. The associated reliability C:1n be called a subjectl,-c or Bayesian reUability. ror a particular stru~t~;~:':the i1Ume~jc:iI';'alu~ c;r~his 're'iiabii'ityttiimies 3S the state of krlowledge 3bOut'~he mUfture' changes - for ex~~\ple'-if :iori~~srruai\'e tests were to be'c3i'ried'out on the structure to estimate the magnitude or r.rn:he,iriiit when i:b'ecomes known'e;,(3ctly, the probability or r .. ilure given hy equation i 1.2 ) cha.'\ges to .r - '.

. (1.5)

This special case may also he inte

l~ !

I . THE TRI::ATMENT OF L.'NCERTAI~TIES IN STRUCTURAL.NGlNEERING

1,~,2 Methods of S3!ety Checking t\lany de\'elopments ha\'e Laken place in tne field of stNctur:d reliabi.Jir.)." an;uysis during Ihe last 10 years ~rid to the n'e\~comer the Jiter",ture may seem confu$ing. To help clarifying the situation. the Joint Co~mi~tee an 'Structuia! Safety j'an in ie~ational bOdy sponsored by such international organis3tions as CEB, CIB, CECM, IABSE, lASS, FCP and RILEM) set up a sub commJttee in 1975.to prov!de a broad classification system for the diCierent method$ then be-ing p~oposed fO,r -cheCking Ui'e safety of structures and 'to establish tne main differences be-tween them~ This cl~ifica'~ion' is ~till usef;tl. ,- , Methods of structural reliability analysis can be dh'iried into two broad classes. These are:

Leuel3: Methods in which calculations aP. made to determine the JreX!lcu. probability of faillolre [or a structure or Structur.ll component, making use of a full probabilistic description of the joint' occurrence of the \'arious,quantities,which affect the reo sponse of the st~cture and takiag into accou,nt t~e true nature of ~~e failure do-,main.

Level 2: . Methods involving certain appronmate iterative calculation procedures to obtain . an,npproximatic!'.l to ,the failure pIObability of a structure or structuRt sy~tem.

generally requiring an ide31isation of faUW'e domain -and often'associated with a simplified representation of the joint probatiility distribution of the variables.

In theory, both level 3 and Jevel 2 methods can' be used for checking the safety of a design or direcdy in ,the design p;~cess. provided ,a ~get reliabi,I,I~y '~r re:lia.bii.r~~:inde~ 'has betn spe- f

,clfied ,

For the sake 'of completeness, some mention should also be made of level 1 methods at this stag;:: The~'are not 'm.!thocls of reliability anaJ}'sii, but are methods of design or safety checking.

,-

,Level,1: Desi~n "~e~h'~s in which appr~Priate depees of litrUClllraJ reliii6ilit}; 'areprovided on ~:. , 1\' stnid'uial elemern basis (occasionally on a structuri.! bash) tiy the use of a number ~

, :' "';o! partial 'o.fety'factotS ~' or pania! coefficients, related to predefined characteristic of'! "(~ :norriinaJ values-of the major stnJ:Ctura! and loading \'ariabies, .j;

' I I , _ .,0< ~~ A level) struc!ural deSIgn, with the explicit ~onslderatlon of a number or'separate limit states, ~~

., , - ' ,-" ~. is what is now commonly ~alled limilstat~ de~ign. It is pre!t!rablc. however, that this should be,::'': called le!:,cl1 design, . an~ that tbe ,term limit.st:ate should be usee solely ,to desc~ibe the separate", limiting performance requirements.

The terms, levell, 2 and 3 will be discussed in detail in chapten 5, 6, 7,8 and 11. The fundame~~aJ distinctions between levels 2 and 3 cannot ealiilr be understood at,this stage umil the neces- ' sery background in probability theory has been co\'ered (Chllptt!: 2). However, the three levels , ,~; of safety checking should be seen as a hierachy of methods in whi~hJev~1 2 methods are all "'lJ''f!: proximation to level 3 methods and in wi'Jch level 1 methods are a discretisation of level 2 meth-ods (i,e. ~\'in2 identical desiiTls 10 le\'e! 2 method~ ior only a iew discrett! sets of values of the) structural design parametersl. t

".1 for pra::tic:i.l purposes - for ex:amph:, Ic~ Citect use in d(>si~n or for el'3Juating level 1 partial fat-ton it is necessary to haVE: :. method of rtliabilily analysis which is computationally {~t alld'~~

;~,\ ,~.

BIBLIOGRAPHY 11

efiicient ano which produces results with tht- desired degree of acC'.lr.lcy. The only methods which currently sathi), these requirements are the level 2 method;;. although analysis by Montf.- Cilflo simulation is sometimes feasible. In this book emphasis is pJ.acec. on tn.? theory and appli cation of le\'el 2 methods and their use in the design of le\'ell structurAl codes.

BIBLIOGRAPHY

(1.11 CIRIA: Rationalisation of Safety and Seruiceability Factors.in Structural Codes. Can struction Industry Research and Information Association, Report. No. 63, 1977.

(1.2J Cornell, C. A.: Bayesian Statistical Decision Theory and RelUlbility.Based Desig/!. Inter national Conference on Struct~iaI Safety and Reliability, Wa;hington, 1969 .. (pub. Pergamon 1972):_.

11.3] Ditlevsen, 0.: Uncertaint:r /I~od,!lillg. Mcqraw Hill,19B1. (1.4J - Freudenthal. A. M . .-Garretts, J. M. and Shinoz.uka, M.: The Analysis of Structural Safe-

ty. Jownal of the Structural Division, ASCE, Vol. 9~, ~o. STi, Feb. 1966. 11.5)- Joint Committee on Structural Safety, CEB.- CECM CIB FIP . L0U3SE lASS RILEI"II:

First Order Reliability Concepts for Design Codes. CEB Bulletin No. 112, July 1976. 11.6J Joint Committee on Structural Safety, CEB CECM CIB FIP lA-BSE lASS . RILEM:

Interlltltiolltll System of Unified Standard Codes for Structures. Volume I: Common Uni fied Rules for Different Types of Construction and Material. ~EB/~IP 1978.

11.7] Joint Committee on Structural Safety, CEB CECt-,j CIB FIP - L-\.BSE . lASS RILEM: Genera! Principles on Reliabilit)' for Structural Design. Intemational.Association for Bridge .and Structural Engjneering,1981.

[1.8J Leporati, L: The Assessment of Structural Safety. Researeh StUdies Press, 1979. 11.9J Nordic Committee on Building Regulations: Recommendation for Loading and Safety

Regulatiolls for StructurtJI Desigll ... NKB.Report No. 36, No .... 1S7S.

,

Chapter 2

FUNDAMENTALS OF PROBABILITY THEOR:>:

: ~ . ","."

2.1 INTRODUCTION .~ ., ' / ... , ' " ,'X$ e~pbsiied in"~'liiip~er J."mOdtrn structural reEability anaiysis is ~sed on a pro'babilistic: point ~f v'jew', It is il;ereC~re imp6rt~nt to get a profound knowledge obt least some part of probll~ility theory. It is beyond the scope oC thii book to give a thorough presentation oC pr9b~bili~y ~e~~y o~ 'a "r.iOfOUs' axiomatic: bash;" \\'bat is needed to understand .lhe fol -lowing Chllpters is so~e knowied'ge 0'( the (undame:1ta1 assumptions of modem probability theorr. c~~bined with a n~~be'r O:(derfnitions:me theorems'; It will be assumed that !-he reader is f3.miliar with the terminology :md the al~eora of.simple set theory.

The purpose oi this ~haPter is th~r~(O~~ to gh:~c a ~lf-contain~d presenta.~ion cif p"rob'lbility thcor)-' with emphasis on concepts of importanCe :or structur::a\ reliabilit!-' analysis.

13

A SUlndard \\:,ay of delermin;n~ the yield stress of 3 material such as steel is to-perform a nu'~be~ O(~iT:ple. U!~~i~~. tes~~ with specimens mace" fromthe material in question. By each test a v~lu~ r~)I: t~e .y!el.d str~iS is d~termined but tbis value will probably be different from test to test. Therefore. tn this connection, the yie'le stress must be taken as an uncertain quan tity and it is in accordance with this point oi view said to be a random quantity. The set of 311 possible outcomes of such tests is called the sam'pI~.5pact! and eReh indi~.~ual outcome is a sample point. The sample space for the' yield stress. is the .open intel'!'al J 0 ;",,[ tha~ is the sct ..... oi all positive real numbers. :SOte th!lt this sample space has an infinite number of sample

points. [t is an example of a cOlltlmlOu.! sample !pace. A sample space can also be discr~te . . namely when the sample points are discret; and count~ble ~~tities .

Example 2.1~ Consider a simply .supp/?ned beam . -\B with t;.vo co.ncen~rated forces PI and P2 liS shown-on fiUre 2.1. funher.let the possible vlllues of Pl and P:! be 4. 5. 6. and 3. -t. respectively. In this ex::.mple all values are in kN. The sample space ior the

lOi1din~ will then be tl":e set .

(2.11

2. f'UNDA.\IE?o.'TALS or PROBABILITY THEORY

'-" - " ' -

r ! P2 B , , , ~ ,.

' :3" . '7777. , :>I' ,

' . ,

A

Figure 2.1

This sample space is discrete. Further, it ha,s a finite number oC sample points. There fore, it is caUed a finite sample space. A s:unpl~ space with the countable infinite num ber of sample poinu is called .an infinite 5:Imple spacp. ?\ot.e that the sample spaces for the loads P1 and P2 are 0 1 - {4,.S,6} ,"and:n 2 ~ . ~ '{3, 4}, respedivel)'. Also note that n ~.nl X: fi2,' ~~o~ as.ll~ e~~.rc,isf. ~!'Iat.~,~~mple space for the reacuon RA in point A.is ~A .. P]/~~ 121.3. 13/3, 1~/3, ' 1513; 16la}.

> . , .

A subset of a sample sp3ce is called an ~~en/. ~~yent is therefore a se~ or sample 'points. If It contains no sample paints. it is called an l~po~ble e~~nt, A cert(Ji~' e'uenf co'nt:ains ali the sample points in the sampJe ~pace that is, a c~~ eye~t is equal t'o' the sample space itself.

~ . . - ' Example 2.2. Consider again the be:lm in ligure 2.1. The sample space fo ... the reaction RA is Sl A - {lIla, 12/3. -13/3. 14/3, 1513.: 16/3). The. subset {IS/3, ,16/3) is the eventlh'at R . .a.' is equal to 1513 or 1613.

Let 1 and .2 be tWO events. The Imion of El and E2 ls an event d,e~,o~ ~~'~1? 2 and it Is the_subset. of sample points,~a~ ~Iong to E1 ,and/or E2, Tne interseclion of El and E:? Is an event denoted by l ' f'I 2 and ,is ~he sl;lbse1..oi ~ple"po~U::~I~nglng to b'ot.1l E} 'arid E2. The tWO events ~1 and 2 are said t~ be mUluall." ex'?-lusilJe if the)~ ace disjoint. ihlit'is if they ha"e no sample poinu in common. In this c~ 1 f". E2 '" C!'-,: where 0 'fi titi Impossible event (an-emplY set},' . -- ..... r .. ';. , .. , - ~. .. .. ..

. Let P. -bf-nample-'.ipace and E an event, The event com.aining all the sample'poini,s in n that . "an- o;'t in E is called the camp/eruen tary ellent and is denoled by E. Obviously. E u E n

andE()E-0. ' . . .. ,

Il is easy to show thai the 'i~ter$ectlon and union operations obey the following commutative, associative. and distribuUyehiws

.. " ~ .

El n (E2 n E31 '" (E1 n E2) n E3 } }

(2.2)

12,3)

(2.-11

i ~ "

2.3 AXIOMS ASD THEOREMS 0: PROBABrL1n' TkEORY 15

Due to these Jaws it mtlkes sensE' to t'onsicier the intersection or the union of tOI!' events E1, E2 ... En' These new l'\'ents are denoted

and

n E] - El '" E::!. n ... . "1 En i-I

U E j - El U E2 U ... 'J En i-I

Exercise 2.1. Prove thl: so-called De Morgan's laws

ElnE2c~

2.3 A."{IOMS AND THEOREMS OF PROBABILITY THEORY

(2.5/

12.6)

(2."} )

(2.8)

In this section is shown how a probability measure can be assigned to any e\ent . Such a proba . billty measure is a set function because an event is a subset of the sample space. Further. the prob:lbility of the certain e\'eot (the sample space itself) is unity. finally, it is reasonable to as sump that the prohability o~ the union of mutually exclusive events is equal 10 the sum of the probability of the individual events. These assumptions .re given a mathematical, precise formu lation by , he following funciament31 axioms of probability theory.

Axiom 1 For any event E

0< P(E) < 1 (2.9)

where lhe IUnction P is the probabilit), measure. peE) is the probabillty of the event E.

Axiom 2 Let the sample space be Po. Then

P(O) - ' 12 .10)

Axiom 3 U 1' E2~ ... En are mutually exclusive events then

P( U Eil"l~p(Ei)

i-I j_ ) 12.11 )

16 _. ~'l!:-;OA;\INTALS Of PROBABILITY THEORY

Exercise 2.2. Prove the following theorem~

PIE)' 1-PIE) (2.12)

'2.131

CU-II

Example 2.3. Consider the statically determinate structural system with 1 elements shown in figure 2.2. Lel the event that elemen~ Ilil> fnils be denoted by Fi and tet the probability oi failure or element ~hl he P(F;). Further aui.l,me that failuf(!S oi the ir.di-vidual members are ~tatistically independent. thnt is P(Flrl F\) PIF1}' PfFj ) for an:: pair of (i. j). The failure of any member will result in system lailure (or this natic3lly determinate stmcture', Thus.' .. ,~':

.. '

P(failure of structure) - PI Fl U ... U F-; I - P( U Fj ) ' ,- -. ,, ' ., . I-I.

-l-PI U F;l-i-PI ii F,) (:!.15J i-I i-I

according to De ~Iorgan's law (2.81. Becaus~ of statistical {ndependen'ce.12,15) C31l be - . " }yriuen

P/f.:ll)ure of stnlct.ure) .. 1'- P(F1) , P(F~) , , . P(~)

Let PIFl ) " PIF3J.P(Fs'" P{F;}" 0.02, P(F2)" P{Fs). "".(),~l.:.3:"d ?W,) .'" 0.03. Then P(Cailure o( structure) " 1 - 0.98~ 0.991 0.97" 1 - 0',876'9 ';:0,1231

; ' , I ....

. . " .

. .f;

: FiJ;Ure 2.2

In many prnctk:1.! applications the probahility of occutrence of 1!\'ent.E1 conditionai upon the tCcurrf>nce of en'nl E:!-i,; (.)( ~rt'al inll~r~t. This prohahili.ty called the cOllditioIlDI:~N?babmty is d.molt'd PI E: .. E:!., :llld is defineu by

ir p! F.:!, > O. The conditional probnbility ;s not defined (or PfE2) ~ O. !::\'ent E\ is said to be statistically independent of event E2 ir

that is. H the occurrence of E2 d.oes not affect the p~~b~bilitY of El . from eqv:ltion 12.16) the prob..,bility of the e\'ent E1 n"Ez-is-givim' by

If El nnd E2 are statisticnlly independent 12.181 hecomes

17

(2.1;1

~2.1S i

12.191

TIll' rule 12.19) is calloo the multiplicacior. rule lind has alrelld~' heen used in example 2.3.

Exercise 2.3. Show 1hat El and 2 are ,;tatistic3l1y independent, when ~l and I:::! are mltisticaJly independent.

Exercise 2..1. Show that

(2.20, . . , .

Eumple 2.4. Consider ngain the structure in figure 2.2. It is now assum.ed (or the sake of simplicity. however. that only element 2 and 6 can fa.il. Therefore.

P{Cailure or structure) P(F:! U Fs) "" P(F:1:) + P(Fs ' - Plf:!':' Fs)

(2.211

Ie f~ and FIS :lre statistically independent as in example 2.3 and if P(F:) ,. P(F'l):> 0.01 then

P(failure of muctureJ Q 0.01 ... 0.01 - 0.01 o.oi ,;, O.Ot9~ .~ ,

But if F:! and Ffi ine not independent then knowledge of P(FzIFIj) is required. If the two elements are fabricated ffom the same steel bar it is reasonable to expect them to h.l~e the same strength. Funher. they have th'e same loading. :ind'then!{oTe' iii this ip!Ci.31 casc. one can expect P(F2 ! F6 ) to be close to 1. With PIP::!! Fa) t one gets

,from 12,211 '

18 2: : FUNi:iA}.iE~TALS OPPROBABILin' THEOR'l' f

.. ;'

FiiUre 2.3

'" peA lEI )P(E1) + peA IE2)~(E2) + ... + P(A IEn)P(En )

.. IPIA!E1lP(E j ) i"l

from the dt-finition (2.16) follow.1

so that

or by usinG' i2.22j

PI.'\IEjIP(E j , P(E11..\t.. n

~P(AtEi)P(Ej) . j-I

Thls . i~ t~'e important Bay.;5 ~ , t.~~o~em .. ,',

(2.22)

, '; ' -

(2.23)

12.24)

~ . f .

c,

;,::.

' 0'

Example 2.5. Assume that a steel girder has to pass a given test before application. Fur- '~J ther, assump from elt.perience that 955'( of all girders are found to pass the test. b!Jt the :'':' test is assumed only 90% reliable. Therefore, z. eonclwion based on such ~ lest has a proba-:.t bility of 0.1 of being erroneous. The problem is now the following: What is the peoba. ."~ ;.~ bility ~hat a perieC't girder will pass th~ lest? Let E be the . e\l~n.t th.at the girder is perfect \ ..

: . a~d I~t A .bl? lhe> event In:n .il pas5~~ th~ lest. . . : . . &? G PIEiAj- 0.90 and ~~:,:

2..& RANDOM VARIABLES 19

so that

P(EIA):Ii: 1- 0.90 = 0.10

Flam experience P(M" 0.95. The problem is lo find PtA IE). The events A and A are mutually exclusive, so that. according to (2.22)

P(E) " P(EIA)P(A) + P(EIA)P(A) .. 0.90 0.95+ 0.10' 0.05 '" 0.860

Finally,

P(AIE) _ PiElA)' PlA) .. ~ c a 99~ ' PIE} 0.86

Example 2.6. ConSlder a number of tensile specimens ~esj~ed to su'pport a load of 2 kN. The problem is now to estimate the probability that a specimen can suppon a load of 2.5 kN. Based on previous experiments It Is estimated that thore is a proba. bility of 0.80 that a specimen can carry 2.5 kN. Further, it is known that 50% of those not able to support 2.5 kN fail at loads less than 2.3 kN. The probability of 0.80 mentioned above can now be' updated if the following test is successful. A single specimen is loaded to 2.3 kN. Let E be the event that the specimen can support 2.5 kN and A the event that the- test is successful (the specimens can support 2.3 kN). Then P(AIE) = 0.5, and P{E) " 0.80. Further P(AIE) " 1.0 so that Bayes' theorem gives

PIElA) '" PlAIE\PlE) .. 1.0 0.80 .. 0 89 . P(AJE)P{E) + P(AIE)P{E) 1.0'0.80+ 0.5'0.20 .

The previous value of 0.80 lor the probability that 8 specimen can carry 2.5 k!' is in this way updated to 0.89.

2.4 RANDOM VARIABLES

,

The outcome of experiments will in most cases be numerical values. But this will not always be true. If, lot example, one wants to check whether a given structure_can carry a given load the outcome may be yes or no. However. in such a case it-is possible,to'8ssign a numerical value to the outcome, ror example the number 1 to the event that the st;uc:ture can CaIY)' the !oad, and the number 0 to the event that the structure.cannot c~ ~,~.1oad. Note that the numbf!ll> 0 and 1 are artificially u signed numerical values and therefore. other \'alues could have been associated with the events in qUt:iti~n. !:. ~h;c W;':l.' H is possi_ble ~o identify p-o:sslble outcomes or a random phenomenon by numerical values. In most cases .thes,e ':'altJ..,:. .... m ~imo!v be the outcomes of the phenomenon but as mentioned it may be necessary to assign the numeri-cal values artificiaUy.

In this wayan outcome or e\'ent can be identified through the value of a function callod a ron dom LlQriable. A random variable is a function which maps. events in the sample spaCe!! into the real line R. Usually a random variable is denoted hy a capital letter such as X. To empha-&ize the domain of X the random variable is often writlen X: n~R_ The concept of a ~onti:l'.!o:J$ random vmable is illustrated in figure 2.4. The event E1 C n. where n Ii a continuous :;ampi~

20 '. FUNDAMENTALS OF PROBAB!UTY THEORY

sample space n __ ---_~"'.n.dom varillble X

---~~~~L-______ _ , R b

Figure 2.4

space, is mapped by the [unction X on to the interval (a ; bJ c R. If the. sample space i. discrete, the random variable is ca11ed a discrete random variable.

In section 2.3 the probability of an event E is introduced by the probability measure P. In this section. it is shown how a numerical value is associated with any event by the random variable. This permits a convenient analytical nnd graphical description of events and associated probabili-ties. Usually the argument to! in X(w) is omitted. Similarly, the abbreviation P(X C;; x) is used Cor P({w :X(w) < x}). First consider a diM!rete random uariahfe X. This is a function that takes on ont'l ::J. f~te or countably infinite number of discrete values. For such a ~ndom variable the probabilicy mass function Px is defmed by

px(x)" P(X = x) 12.25).

where X is the nndom variable, and x ... Xl' X2 ' Xli.' and where n can be finite or infmite. Note that difCermt symbols are used for the random variable and itlvalues, namely X and x, . respectively. It is a direct consequence.of the axioms (2.9) ~ i (~.:l1) that

.. ~tpX{x)-l j-l

Pfa

2.4 RANDOM VARIABLES

Example 2.7. Consider again example 2.1 and let P(PI .. 4) - 0.3, P(PI .5)" 0.5 and P(PI .. 6) .. 0.2. The probability mass function PPl and the p.robability distribution .. Cunction PP

1 Cor the random variable P 1 are shown on figure 2.5. Note that the circled

points are not included in PPI (s).

fpp, (x) F'---~", 1. 1.0 O. 0.5 ' '

...-\>: x

! ! I I , , X I I I 0 5 0 5

Figure 2.5

21

Next consider a continuous random mritlbte X. Thill is a Cunction which can take on any value within one or several intervals. For such a random variable the probability for it to assume a specific value is zero. Therefore. the ~babllitymass function deCined in (2.25) is of no in terest. However, the probability distribution /Unction Fx : R~R can still be defined by

FX (x) ., P(X

"2. FUNDAMENTALS OF PROBAmLlTY THEORY ";"

I~ follows directly from the axioms (2.9) (2.11) that for any probability distribution function

(2) FX is non-decreasing

Inversion of the equation (2.31J gives ..

FX(x) e \ f,,(tidt (2.32) '"--

for II continuous random variable. From (?,.32), it follows that

r fX (tjdt - Fx '-)-l (2.33) ."--

It is sometimes useCul to use a mixed continuoUl-

i .. 2.5 ~ MOMENTs .

(2.34)

The expected value is also called the ensemble average. mean or the first moment of X and the symbollJx is often used for it. By analogy with this the n'th moment of X is called Elxn 1 and is defined as

E(Xn ) - C xn fx(x)dx (2.35) '--

For discrete random variables the integrals in (2.34) and (2.35) must be replaced by summa tions.

Note that the flISt moment of X defmed by equation (2.34) Is analogous to the location of the centroid of a unit mass. Likewise, the second moment ean be compared with the mas~ moment of inertia.

: Example 2.9. Consi"-er the discrete random variable X defmed in example 2.7. The : discrete venion of (2.34) rives

E[XI- 40.H 50.5 + 6,0.2,4.9

The most probable value is called the mode and is in this ease equal to 5,0 (see figure 2.5). Further

E(Xl J ,. 16 0.3 + 25 0.5 + 36 0.2 '" 24.5

Above. a new random vuiable Y X wu considered. Tnis is a spe.cial case oC a random vari able which is a function of another random variable whose distribution function is known, Let Y l(X), where f is a function with at most a finite number of discontinuities. Then it is possible to Ihow tbat Y .is a ~dom variable according to the definition of a random variable. If the {unction f is monotonic the distribution function Fy II given by

Fy(Y) -P(Y" y)-P(X" f-' (YII-FXW' (y)) (2.36)

and the density function fy by

(2.37)

or simply

!y(Y)- !x(X)I~i (2.38)

!!.. FUNDA.\lENTALS OF PROBABILITY THSORY

Example 2.10. Let Y II aX + b. Then X - tV - b)/a:md

I IY).r IY -b). Illl Y X a a I

IL is important to note that the expected value oC Y .. [(X) can be computed in the following way without determining { '( ' .-

12.39)

Exercise 2.5. Show that . " "

E{ I fj{X)J .. I Elfj(X)J 12.40) i-I ;-1

SO that the operations of expectation and summation can commutate.

Returning to the momenu of a random variable X. the nthcentroi moment oCX is defined by EI (X - JI. X)n I. where JI. X E{ XI: ~ote that the first central moment of X is always equal to zero. The second central moment of X is c3.lled the t'ar;ance of X and Is denoted,by a~ or

~X) .

The positive square root aC t~e v~iance. aX' i~ ' caJled the ~t~ndard. ~~.U~tiO~ oC X.

Exercise 2.6. Show that : ".

12.U)

12.42)

The standard deviation aX is a measure of how closely the values of the random variable X are . con~ntrated around the expected. value EIXI. It is difficult only by knowledge?C aX to decide

whether the dispersion should be considered small or large heause this "ill depend on the ex . pecLed value. However; the coe{ficient of variationNx ' defined by

.--.- --vx .. .. ;; 12.43) ,

~ives better information rc"ardim; the dispersion.

r ," ;1' -:. J. ' . 2.6 UNIV.\RI.-\TE DISTRIBUTIONS

Example 2.11. Consider the same discrete random v3'rintlle X,as in example 2.9. where E(XI '" 4.9 and E(X1l = 24.5, The variance. there(~re. is''-' -, - '. I.

VariXI - 24.5 - 4.9' 0.49

and the standard deviation is

ax' -yQ.'49 .. 0.7 , ",;

Thus the coerficient of variation is

Vx . ::: ~:~::: 0.14

The third central moment is a measure of the asymmetry or s/~ewness of the distribution of a random variabie.:-F,~r a continuous random vOlriOlble it. is defined b~

(2.44) ...

2.6 UNIVARIATE DISTRIBUTIONS

In ,~h.is ,~tio~ ~!lle of most .. widely used probOlbilit~ distributions are introduced. Perhaps t~e most importa~t distr'ibution is the ~o~m~1 diitribuliori aJs'o called the Ga'ussian, distriqution. It . . '" -r ' , ' , f ' !"~' " . "'. . is a two-parameter distribution defined' by' the densft-y func'fion ,",i. ;,; I, :, . . ,: /..

'-";"!.' .,,1: '1',,"

(2.45)

where II. and a are par.ameters equal to.ux and ax - This normal distribution will be denoted N(Il,a).

',:\ 'The distribution functic:m c::on:espondin,~ ,t.o (2,t5),.:is ~.~en. by

(2.46)

This integral cannot be evaluated o~ a closed (o'nn, 'By the'substitution "'- ,.,':

s.t:;p ,dt - O'ds (2.47)

the equation 1_2,46) becomes ,_ "

(2.48)

. where '1>:< is the standard normal distr(/mtiol1 (ullction defined by :"

25

26 2. FUNDAMNTALS C?F PROBAIUUTY THEORY

\' 1 " 41 .(x) ' rn: eXPl- ,?ldt

X v~:: -'-00 . ' ..

(2.49)

The corresponding $tand"rd normal density functioll ill

(2.50)

Due to the important relation (2.4S) only a standud normal table is neces~. The functions ';x and 4lx are ",own in figure 2.8.

f.;x tx,

A -3-2-1 123

, 1.0 ---. .:::-:;-~-

x

Figure 2.8. .: , ; , ' r .

. Let the random variable Y .. tnX De normally distributed N{~y. 11~)' Then the"l'lndom variable X isS3.id to follow a Joprjthmic normal distribution with th~ paiimeters ~rE R and Oy > O. The " IOjl:.normal density function is . . . ; -. .,.

1 1 1 inx-,uy 2 'x(X.)~ ay$ xexP[-I

!!.6 UNIVARIATE DlSTR1BUTlONS

! fXj)n 001 . i

"'~ 0.41 1 T (2".1) 0.2 '--

0.0 I 1.0 2.0 3.0

Figure 2.9

x

~7 i

The log.normal density {"'nctions with the parameters (J.'y, (ly)- (0, 1) and (1/2, 1) are illu strated in figure 2.9. ,

Example 2.12. Let the compressive strength X lorconcret.e be Jognormally distributed with the parameters (j.lX' "y) " (3 MPa, 0.2 MPa). Then

-"x - exp(3 + t . 0.04} - 20.49 MPa .-- ok .. 20.49'(1.0408-1) - 17.14 (MPa)~

Ox z 4.14 MPa and

P(X," 10 MPa) at 0II'lnl0 - 3)(0.2) '" 4>(- 3.467) - 204 10~

An important distribution Is the so-called Weibull distribution with 3 parameters tI. (and k. The density function ex is defined by

(2.55)

where).:;;' r andtI > 1. k > E.

If r" 0 equation (2.55) is

~~(.X) : .. t (~)JI:.~ exp{- (I)JI) , x;> 0 (2.56) The density function (2.56) is called a two-parameter Weibull density function and is shown iii i1i:; ... ~ : t1 Tf F '" 0 and ~ - 2 in (2.S5)lhe density function is identical with the so-called Rayleigh density (unction

12.5'j)

28

Figure 2.10

2. FU!'lDA.'tENT.~LS OF PROBABILITY THEORY

\----- (k.~) (I. ') (k.~) (1. 2) I (k.~)=(2.2)

-""-:.-

2.7 RANDOM VECTORS

Until now, the concept of a mndom variable has been u~ed only in a one.(;jimensional sense. In section 2.4 a random variable is detined as a realvalued function X :n ......... R mappin~ the sample space n into the real line R. This definition can easily ~ extended to 3 vectorvalued random variable X :nARn called a mndom vector (random n.t"ple), where Rn '" R X R X ." X R. An n-dimensional random \'ector X:n'""'R" can be considered an ordered set X '"

. (X t , X2 ... Xfl) of one-dimensional random v31iables XI ;n,.-.,R. i = 1 .. 1 n. Note that Xl' X:!: .. . X" oue defined on the same sample space n.

Let Xl an? Xz be .. two random variables. The range of the random vector X'" (Xl' Xzi is then a subset of R~ as shown in figure 2.11. Likewise. the range of an n-dimen~ional random vector 15 a subset o( Etft. ,

____ C-___________ "

a'igure 2.11

:!.' RANDm,l VECTORS .-

c ,:.

Consider llgain two random "ariables X 1 and X~ and the corre~ponding distribution rune tions Fx\ and Fx

2 It Is clear that the latter give no information regarding the)oint

beha"jour or::

30 o F".OAMeNTALS OF fROBAlL/TY THEORY

E.umplc 2.13. Consider again e;!l:3mple 2.1 and let a 2-dimensional discr~\e random vec-tor X" (X~. :;\.21 be defined on r.! by .. . ,;"

P'(j, 3) ';' O~l ~ P( 4'~ ~i) - 0.1' P(5, 3}- 0.3 P(5, 4) ""0..2 P(6, 3}' 0.2 pte. 4) 0:0.1 .

.; '

". f

.....

'The mass [unaJon Pi is illuStrated in fil\.Il'e 2.12, and the mar&inal mass functions PX1

and PX:l in figure 2.13, .... Note that PR(x1 , "2}" PX1 (Xl) . px,(x2)

Figure 2.12

r>x,lSl)

0.5 j I ( . ~x 4 5 6 1

Figure 2.13

~ . 6 CONDITIONAL DISTRIBl.'TIONS

2,8 CONDITIONAL DlSTRIBUTION~ In equation (2,16) the probability of occurrence of foVt'ul 1.: 1 ,,lllditional upon the occurrenct' of event Ez was d~Iin~d by

(2.16)

In accord:tnce with this derinition the conditional prCllNlbi'.,y mass (unction for LWO jointly dis. tribui.ed discrete random variables Xl and X2 is ddinl!tJ U~

, . Px x (x"x2) . I' - '-'!L! .""''-;,:'-;...:-PX1IX:(xl x 2) - PX~(X2) (2.65)

A natural extension to the continuous case is the followinv. lh'firiition of the conditional proba-bill!)' density function

(2.66)

where fx:(x2) > 0 and where fX:l is defined by (2.64). N"",. that PX 1 IX: is a mass function in (2.65) and fXll X2 a density function in (2.66). The two random variables Xl ~d X2 are said to be im.lt:p.'"fJl'~t if .

~2.6i)

v .. hieh im~lies

(2.68)

By integrating (2.66) with respe.ct to xl one gets the c

32 _. FT.:NOAMENTALS OF PROBABILITY THEORY

Therefore. Xl and X2 are not independent.

Exerci~e 2.9. Consider two jointly distributed discrete random variables Xl and X2 with the probability mass functions PX

t and PX2 given in figure 2.13 and assume that Xl

and X2 are i~dependent. Determine the joint probability mass function Px for the ran dom vector X (Xl' X2)

2.9 FUNCfIONS OF RANDOM VARIABLES

In chapter 2.5 a random variable Y, . which is a function ((X) of anothe~ random variable X. was ' treated and it was shown how the density function fy could be deter:mi~ed on the basis of the

densit~ functi on! x' namely by equation (2.38)

(2.38)

where X" C-l (y) . This will now be generalized to random vectors, where the random vector y .. (VI- y2. . Yn) isa function1 - (f1, ., (n)or the rando,1Tt ve~to~ X " (X1.X2' ~ ,Xn ), ~at~ .. . "

(2 .71 )

where i .. 1, 2, ' . . , n. It is assumed that the functions fj,l .. 1, 2 .. , n are one-to-one (unc-tions so that inverl\e relations exist

(2.72)

It can then be shown that

(2.73)

(2.74)

'x_ 13V - I

is the Jacobian determinant.

Let the random variable Y be a function (of the random vector X - IX I . _ . . Xn). that is

f2.75)

:\. -2.9 FUNCTIONS OF RAl'Dm.t VARiABLES 33

It can be shown that

" (2.16)

where i ., (xl' ... xn) and f xCi) is the probability density function for the random vector x.

Exercise 2.10. Show that

" " E(l";(Xill - IE!f;(X;lI (2.77) i-I i-1

so that the'op'eration of expectations and summations can commut~'~~iompare with exercise 2.5.

Exercise 2.11. Show that

" " EliZ ';(X;J1~ II E!f;(X;)] (2.78) i-I i-I

when Xl' , , , X:1 are independent random variables,

_ ~et Xl and X2 be two random \-ariables with the expected values E[X 1 ] ., Jlx 1 and E[X 2, :a IlX2 , The mixed central moment defined by , (2.79)

-: '~ ,"!to ......

I 2, FUNDAMENTACs OF' PROBABILITY THEORY

I !

,

! 1: is imponant to note th:n indcpentlem random variables 3n~ uncorrelat!d. but uncorrelated \'anables art' not ir. i!eneml independem .

. :Note that

(2.83)

Therefore. the mutual correlation between random variables Xl' X! ... Xn can be expressed by the so-called COt'Griancc mcurix C defined by

.;', .

.. .. . . . . . . , c. ..',Co,jX,.X,:.) 1

.. " ,.: .: ... ~ CovIX2 Xnl

. .

I, . , Var!X21

.. ....... ':. V..{X,I J .... Exercise 2.1.2. Let the random \'ariable Y be defined by

where Xl ' X:: are random variables and ai' 32 constants. Show that

Varl \'1 .. ai VarlXl J + 3i Var!X 2 J' + 2a132 C~\'[Xl' x:!]

2.10 MULTIVARl.-\TE DISTRIBUTIONS

(2.84)

(2.85)

(2,86)

The most imponan~ joint density function of two continuous random variables Xl and X2 is the biLoariate norln~! d.~ns~t~.fu'}.rr.tion , j!:i\'en ,br

(2.87)

, .. here - ... " ~1 " ... - ... ..:;; x2 .:;: -. and,.: ';:2 are {h.e-means, 0l:~ 02 ~he standard deviations and p the correlation coefficient of Xl

BIBLIOCRAPHY

.. ' . ,; .

The multiloariale normal ciensilY {uncllon i;; defined aJ;

," .. '

" ',." ;~

BIBLIOGRAPHY

12.11

12.21

12.31 " ..

12.41 ' (2.5]

12.61

12.7J

An~, A. HS, & W. H. Tan(/:: Probabili~y Concepts in Enginacring Planning and Design. Vol.l, Wiley, N. Y.,197S.

Benja~in. {R: & C. A. Corn'ell: Probability, Statistics and Decision for Civil Engi. neers. Mcci~a\~.Hii1: N.Y' .. 1970. Bolotin. V. V.: Statistical Methods in Structural Alecharlics: HoldenDay. San Fran cisco,1969.

Ditlevsen, 0.: Uncertainty Modeling. McGra ..... HiII. N.V .. 1981.

Lin, Y. K.:'Probabilistic Tneory of Structural Dynamics. McGraw-HilL N.Y., 1967. Feller, W.: A~ Jntrod~ction to Probabilit) Theory and ils Applications. Wiley, N.Y., Vol. I, 1950, \'01. 1l,1966,

Larson, H. J. & 6. O. Shubert: Probabilistic Models in Engjllacring Sciencc . T. Wiley & Sam. N.)' .. Vol. I 6: fl. 1979.

35

.,: n

;s -,:-1::1':: .. .. . . ""! -

I.;' .. , :i. .;.,, : ~ ' -f " , ' t :.

. ... 'I'

:," ' :.'

Chapter 3 I::' :: . t-.

PRoilAiifLIsTiCMODELs FOR LO,\IiS AND RESISTANCE VARIABLES .' :.:I.Ji : .'~" ,;,;:. '::}:1' ',.::, ", : t - .TJ." , ~. :.' ',;, :' -,m:" .. l ... 1; ..... 1

~.o..: . " : ..

.. ... , !, .... : .

. :. !

.. , . ~r

,r. In .thls c;hapt.e~,.tpe a.i!1l)~;,~9. .~~.~.~jn.e ~he, w~y}n I\~h~ch .~.~~bl~. p:ro~~\l~~}i,c;: ~,~~~s:.,~,'r. ~e . developed t.o. represeq.t ~h,e .. ~l!~~r.tai"t!es. that.~xist,in typ.i.~, ~~je ~b~~5: ,}~~~.?aIlJi,9~ ; consider the problem of modelling physical \':uiability ,an.c;t IH~~n,turp t.~.t~~: ~~~s.~~C?~ .~~ I~~ corporating statistical uncertainty,

Load and ~islance parameters clearly require different treatment. ii."lCe loads are generally :imevarying. A5 .di.s.cl!s,s~ in !

38 3. PROB.ABILlSTICMqp~!-S FOR~OADS AND RESISTANCE VARIABLES ,

, In an analogous way, if the strength of a structure depenqs on the strength of the weakest 'Jf a number of elements ioc example, a statically dett:rmin~te truss one is concerned with the probability distribution of the minimum strength.

In g~J.l~ral. one car estimate, fr(lm, test r~sults, ~~ refo,~s ~he,~~meter5 of ~?~ ~t~i~l~~ion of the instantaneous \'aIues of load or of the strength of individual components, and from this in-formation the aim is to derive the distribution for the smallest or largest values.

3.2.1 Derivation of the cumulative distribution of the ith smallest value of n identically dis-tributed independent random variables Xi .wume the existence oi a random variable X (e.g. the maximum mean.hourly wind speed in consecutive yearly periods) having a cumulative distribution function .'~x ~~ a ~orresponding probability density function fx ' This is often referred'to as the paren-t ~istribution: Taking a Sample'size of '(((e.g. h'years'records and n values oftheiniiXimum niean-hoilrlywind speed) lE!t t~e c'liinUiatlve districlIiti'on 'function of the ith sm::iUest'i.lalue X!l in the sample be F X" and

_ ',' " I , I its correspoiuiing density function be fx~' - .. , ", " ",' .: ,J

Then

f~~ (x)dX '" co'nstant X probability that (i -1) values of X fall below :It ~- I ',1: ,", ,;- : _00',' _ -, ",:, ' , "-, "

~,probabi1ity that. (n -'i) values of X fall above i x. pro,bability that 1 valU!? of _X }ie~ in the range_l.: to, (x T d,~) ,

0; cFr1 cx)(l-:- Fx(xn-i,fx

(x)dx ,lr>! _ (3.1)

where

the numb~r of ways otch,?osing,,{i -:-:-J),val~j~ l~~ :~han x, together with (n - i) values greater than x, (3.2)

, , , -. 1 ", - -',

ThUs

--FX~(Y) '" r:f~n(xjdii,,; \Y ~Ftl (x){1 '- 'F ~(x))n"";i f>i(x)dx 1 I "0 - L ',. 0 . . ','

(33)

ThiS can be-sho~n to be equal t~

c

Figure 3.1

l .!! STATISTICAL THEORY OF XTRE~IES

[(F".(Y)Ji In-i\ 1Fxty))i+l I'n-i) -;-- ~ 1 ; (i'" 1) + , 2 x

(F ( }}i1'2 n "\ X Y _ (n_il(n-i)(fx ()')) ! (i+2) .. +( 1) n-i n J (3.4)

Exercise 3.1, Show that equation (3.4) can be derived from equat.ion (3.3) by expanding (1 - FX (xn - i an.d integrating by parts.

Equation (3.4) gives the probability distribution function for the jth smallest value of n values sampled at random from a varible X with a probability distribution F x.

Two special cases will now be considered in the following examples.

Example 3.1. For i = n equation (3.4) simplifies to:

FXn (x) .. (Fx(x))n

(3.5)

This is the distribution function for the muimum value in a sample size n.

Example 3.2. For i" 1 equation (3.4) simplifies to:

(3.6)

This is the distribution function for the minimum "alue in a sample size n.

It. should be noted that F X",(x) may also be interpreted as tht!. probahilit)' of the nonoccur renee co! the event (X > x) in any ofn independent triah.$O that equation (3.5) follows imme diatel)' from the multiplication rule for probabilities. Equation (3.6) mty be interpreted in an analogous manner. See also chapter 7 . .

3.2.2 Normal extremes . If a random variable is nonnally distributed with mean IlX with standasd deviation Ox the vari

able has a distribution function Fx (see (2.46

F (x.) - -- -exp(--(:.....!:.X. dt \

x 1 1 1 t-II- 2 x _oo..;z; Ox 2 Ox (3.7)

If we are interested in the distribution of the maximum \'alue of n identically distributed normal random variables with paramete:-s Px and Ox this has a distribution function

Fx"'(x) '" \ ~ - el:p

3. PROBABILISTIC ;\IODELS FOR LOADS A~D RESISTA:-:CE VARIABLES

, r)t"(S)

1 1 . l.5

Figura 3.2,

The probability density function fX " ZI ~ (Fx ") is shown in figure 3.2 (or various \4Iu~ of n . . and with X distributed N(O. 1).

3.3 ASY~fPTOTIC EXTRE~IE-VALl"E DISTRIBUTIONS

It is fortunate that for.:l very wide class of parent distributions. the distribution functions of the maximum or minimum values of large random samples taken from the parent distribution tend

tO~~lIds certain limitinl;l: distributions as tbe sample becomes large. These are called rJsYI'!!.ototic extreme-I:a{ue discrfbutions and are of three main types. 1. II and lIt. For eXa!),ple. if the particular .variable of interest is the mLximum of many similar but inde-pendent events (e.g. the annual maximum meanhourly wind speed at 3 particular site) there are generally good theoretical grounds for expeding the variable to have a distribution function which is very close to one of the asymptotic extreme value distributions. For detailed iniorma-tion on this subject the reader should refer to a specialist text. e.g. Gumbel [3.8J or Mann. Schafer and Slngpurwalla [3.111. Only the most frequently used extremevalue distributions will be referred to here.

3.3.1 Type ( extreme~value distributions (Gumbel di5tribitt~ons) Type {asymptotic distribution of the largest extreme: If the upper tail of the parent distribution falls offin an exponential manner. i.e.

(3.91

where g Is an incre3sing {unc~ian of x. then the distribution function F~ of the'la~est \'a!ue Y. from a large sample selected at random from the parent population. will be of the for~

Fy(Yi - expl-expt-o:(y -ullJ -"'''y''.

formally. F y will asymptojic311r 2pproach the dist:-ibution given by the right' hand side of ~qu.:ltion !3.10J as n - "".

3.3 ASYMPTOTIC EXTRE:'.IEVALUE DISTRIBUTIONS

fl' .

Figur.3.3

The parameters u and Q: are respectively ::1easures of location and dispersion. u is the mode of the asymptotic extrem'e.valuedistribU'tic:l (see' figure '3.3). The me~n and standard deviation of the :ype I ma:dma distribution (3.10) are related to the parameters u and 0 as {oUows

(3.11)

'nd

a .-'-Y .J6 fl (3.12)

_ or' where "1 is Euler's constant. This distribution is positively skew as shown in flgUfe 3.3. .

: . . . .. -- ~" . . . . . ' .: :: .. , ... ,:, .. .~ ,. . . A useful property oC the type I maxima distribution is that the distribution Cunction Fyn for the largest extreme in any s3mple of size n is also type I maxima distributed. Furthermo~e, the standard devi~tton '~emai'ns constant (is c:dependent'of n), i.e. .. . ,; .;..~ , " . . ~ :.

. ~ . ' ... ' . -. '. ' : '':'! : ~ ~"

(3.13)

This property is 'Of help in the anal~-sis :-o: load combinations when diCferent-num6e"rs of repe titions of loads'n j need to-be considered ' see 'chapter 10). In this connee'tion. t(is uSeCul"lo be

. 'able to calculate the parameters oTthe -eitreme vari.i!.ble y~ from a kri"owledge 'of the para~ . meters of Y.

IC Y is type I mtlxima distributed with u:s;:ribution (unction Fy given by equation f 3.10) and with p~rameters Q: and u. then the e~me;::~ distribution 01 ma..'

42 3. PRODAB.ILlSTIC MODELS FOIl LOADS AND RESISTANCE VARIABLES 1

with mean given by

(3.15)

Type I asymptotic distribution of the smallest extreme: This is of rather similar form to the, Type I maxima distribution. but will not be discussed here. The reader should refer to one of the standard texts5ee (3.81.13.111 or 13.51. '

3.3.2 Type U extRme-value distributions As with the type I e:.;:trem~va1ue distribullons, the type II distributions 'are of two types. Oldy tbe type II distribution of the largest extreme will be conside~ed here. Its di~tributionfu~ction

.,

Fy is given by

Fy(Y) s::: &p(- (u/y)") y;o. O. u > 0, k > 0 (3.16)

where the ~eterS u and k are related tO,the mean and ~~~~dde~iat~on by .. #J.~ =ur(i~l/k)

"',1. (3.17.) .

,

0y - u{f{l- 2/k) - r 1 (1 - l/k)]2 with k> 2 (3.18) .

where r is the gamma function defined by

,- -11 11:-1 r(k):\ e Il. du . 0

(3.19)

It should be noted that for k '" 2, the standard deviation Oy is not defined. It is also of i~terest that if Y is" type II maximi'distributed, then Z;;. .l!ny is type I ma.ximadi~tributed~ .

Elo:etcise 3.2. Let Y be type II maxima d.istribuied with ~istribu~~.o~,~uncti~~l'Y and '. coefficient o!nriation ay/Jl.y' Show that the variable representing the largest extreme with

distribution function (Fy(yn has the same coefficient of variation.

The type II_~~ d~tribu~ion is freqlJ.~ntly,used in modelling extreme.hY,drological and me-terologica,l, events. ~F ~~.as the limiting distri,bution of the largest valLIe .of manY.independent ident.ically distribute~ .~.9P~_var~ables,_ whe~. the parent distribution_is limited to ,values greater than zero and bas an infmite tail to the right of the form

3.3.3 Type III exbeme--\'alue distl'ibutions In this case only the t)'pe 1IJ asymptotic distribution of the smallest extreme will be considered. It arises when the parent distribution 15 of the form:

3.3 ASYMPTOTIC EXTREMEVALUE DISTRlBUTIONS 43

with x;' (3.21 )

i.e. the parent distribution is limited to the left at a value x .. '" E. In many practical cases f may be zero (i.e. representing a physical limitation on, say, slrengthj. The distribution of the minimum Y of n independent and identically distributed variables Xi asymptotically approaches the form

","'llh y;;;' E, P > 0, k > E;;;' (3.22)

as n .... ""'.

The mean and standard deviation of Y are:

(3.23)

and

(3.24)

The type III minima distribution (3.22) is often known as the 3parameter Vi'ejbufl distribu-tion and has fr.equently been used for the treatment of fatigue and fr~cture problems: For the special case f C 0, the distribution simplifies to the socalled 2pa-rameter Weibull distribution

(3.25)

-10-_

10-t. .

_10...,0; . : I -10~ I . -- -- I

_10-.11 rormal .. k1Y' ~_-'.:- 'i . .-..p. type II maxima i I , -10~ I i: 'k.: -- I '-+ type 1 maxim .. i I ........ ;

1 I )'1 U"olnonn~ 1 1 -10-1

II I i I ! I d I I ! I I I I I I 10-' ,., I , I 1

1O-1 j lAY I ! I I ! I iii/! I I i I 10-.1 I

. I

10""" 1 / .,' , I I , ,

10'" i , I : 10

3. PROBABILISTIC ~IODELS FOR LOADS A~D RESIST.-\:\CE VARIABLES

with

13.26)

'nd

(3.27)

Comparisons of the type I maxima and type II maxima distributions with the normal and log-normal distributions are shown in figure 3.4. The random variables in each case have the same mean and standard de\;ation. namely 1.0 and 0.2.

3.4 ~IODELLING OF RESIST_-\"'~CE VARIABLES - MODEL SELECTION

3.4.1 General remarks In this section some general guidelines are given for the selection of probability distributions to represent the physical ,uncertainty in variables which affect,the.strength of structural compo-nents and complete structures - for example, dimensions, geometrical imperfections and ma-terial properties. Since each material and mechanical property is different. each requires indivi-dual attention. Nevertheless. a number of general rules apply. Attention will be restricted here to the modelling of continuously distributed as opposed to discrete quantities.

The easiest starting point is to consider the probability density function fX of a random variable X as the limiting case of a histogram of sample observations as the number of sample elements is increased and the class interval reduced. However. for small sample sizes, the shape of the histo-gram varies somewhat from 'sample to sample. as a result of the random nature of the variable. Figure 3.5 shows two sets of 100 observations of the thickness T of reinforced concrete slabs having a nominal thickness of 150 m!", which illustrates this point. These data wer.e not. in fact, obtained by measurements in rea] structures but were randomly sampled from a logarithmic normal distriJution with a mean JlT"=' 150 mm and a coefficient of variation VT ". 0.15 (see ap-pendix A). The corresponding density function fT is also shown.in figure 3.S. For comparison. figure 3.6 shows data obtained from'a real construction sit~.

A clear disti!lction mus~ be made. however~ betwe-en a histo~am or a relative freguency dia-gram on the one hand and a probability density function on the other. Whereas the former is Simply a record of obsen-ations. the latter is intended for predicting the occurrence of future events e.g. a thickness less than 100 mm.

If the probability den,;ity function fx of a random variable X is interp_ret~d as the limiting case of a histogram or re!ati\'e frequency dia~m as the sample. size i~[Ids tojnii~ity. the probaoiiity P given by

,x:! P=>P(:

Ui 0,03'

10 0.02

r I ,

, 1

l--+---,.

160 L _ - .-

Fi~ur~ 3.6~ Ilistu~~:m~.or.5Iab ~hic~n~5.'i measurements.. " t:' .. ~,

. ;c

.. ,.'

.".'

.- .-. - ; .. :;.

45

" 'f'! .,.;

- , --. ~

.t(mml

46 : / . ~, 3. PROlU.BrLISTIC MOD~ts;FOR LOAi$s~ND R~s'i~A~~~ V~~IABLES

I

clearly has a relative'frequency interpreUltion: ~ :;. if ~ very large sample of varilible X is obtained at random, the proportion of \'alues within the s!.:npie fo.JIin{!. in the range1 < X " x~ is likely te. to be ver), close to P. Ho~ever. thi>; interpretatic:: may not in practice be too helpfuL All that ;;' can be said is that jf a variable X does in fact b~xe a known probability density' (unction lx-and if it is sampled at random an infinite numbe: of tim~~. the proportion in the range 1 );:1_ x!! [ wlll be P. .

The problem of modelling is completely di!ferer.:. In gene!.afthe engineer is likely to have only a relatively small sample of actual observations of X. along with some prior information' obtained from a different source. The problem then is ho~ best"io use aU'.this Information, Before this question can be answered it is necessary to deCin~ ex.actly what the variable X represents. This Is best explained by means of an elWDple.

( . . ( ..... . ". . .

Example 3.S. Consider the mE:Chanical pro~rtie5 of a single nominal size of continuously

~, .

cast hotrolled reinforc~ sWeI. Let. us rest.&.!t our attention to a single property, the dy ~ namic yield stress, 0yd,delennmed at a controlled strain rate oC 300 microstrain per minute ~ and deCined as the aVer.!ge height oC the stre.s-strain curve between strains of 0.003 and 0 .005. : i.e.

(3.29) .

where 0y(e) is the d}'namic yield stress at s~-in e . Let us assume that tM property can be me3:-.Jred with negligible ex.perimental error and tha: all the reinforcine: hars from a single cast of ;;eel are cut into test specimens 0.6 m long and then tested. If 0vd is ploned against Z. the pjsition in the bar, the outcome will be of the form r' shown in figure '2.7. This is an example of a i:ep..wise continuousstate/continuous-time Sto r chastic process X(tJ in which the parameter: alii:!! be interpreted as the distance Z'along the -reinforcing bar. (See chapter 9 Cor further dE.:ails of stochastic processes). ,:'. The process is interrupted approximately e\'r:.,-y 600 m because the continuously cast steel is cut into ingots and these are fe-heated and tc.Ued separately. The fluctuations in yield stress within each 600 m Jengtb are typically very S!D.all, i.e. in the order of 1 - 2 N/mm'. For each 600 m length f, the spatial average yield stresi a d is defined as

1 ~ ::: :~ . y 0yd ~2L o),ddE "_. :- (3.30)

The variations in ~ from one roUe?: le~gth. ~o'another: ~~ tYpi.cally I~~r, tp.an th~ within le~gth . variations and are ciused mainly by d:.:ierences iil ' the terop.erature of the ingot at the start of-rolling and by a number of other factors.. Some typical data giving values of Oyd for consecutive lengths of 20 mm diameter hotroU~ high-yield bins froin the same cast of steel are shown in figure 3.8 (along ,..;th values for the sl-Atic yield stress). These can be can sidered as a continuoll5state/discrete-time stochastic process. It can be seen that there is a fairly strong posith-e corrclatioo between aye for adjacent lengths, as might be expected. If c is the totalle11Jlh of reinforcement proe'.1eed Irom a single cast of steel then the average yield stress lor the cast can be d~Cimc as

(3.31)

; . .. ;

) .. ,. ..

J 'I :1

3A ~10DELLING or RESISTANCE VARIABLES MODEL SEl.ECTIO!'

t yd (NJmm 1) soo I :::Lj ~~~~ ~ .40

Zlm) '+I----r---~--~----+_--_+----~I--~~---0. 500 1000 1500. 2000 2500 3000 3GOO .

Fi,un 3.7 VUbtioJU in dynamic yiel4 stress alone a 20 rum diameter botrolled reinCorc:ine bar.

soo

.eo

...

10 "

20

, ,.

" 30 3S 40 .. 50 bar number

Fi,ure 3.b. Withil\~as\ var;llialU in the yield streu of. 20 .mm aliLI"~ _ ""Irolled reinfarcine bu.

47

-48

Provided that the \'3riattons in yield sttesS alohg ~a~h 600 m cl~~n~~h oC co~tin~~u~I}: rolled b:tr car'! be assumed to be small in comparison with \'ariations in Oyd. the average yield stress for the cast may DC expressed ~

~ -! . ~ . . uyd - n .::... q~'d(l) (3.32)

j"l . :"'. " ), .;:; where ayd(i) Is the yield stress oC the jth bar and n is the number eCbars [olled"from' the cast. ," .,.

It we are interested in the statistical dfs,tribution of the yield stress of reinforcing bars sup-plied to a construction site. accourt mu.s~ also !>e taken 9,' ~h~. Y~riati~n.s in ~d that occur from cast to cast. If the steel is to be supplied by a single manufacturer and very cloSe can trol is exercised over the chemical compoi!;'ition of each cast, variaUonS ln Uyd will lM!' very. ~mnll; but if the chemistry is not well controiled significant difference's bet\veencw can 'Jccur. If bars are supplied by a number of ,diCterent manufacturers. systematic-differences hetween manufacturers will be evident even for nominally identical products (e.g. 20.mm diameter bars) because of differences in rolling procedures. r\ final effect which must tle t:1ken into account is Uiesystematic 'change in mean yield :it,ress with bar diameter as illustrated in figure 3.9. This phenomenon is quite inarkeci"iind is rarely taken Inw acc,?unt in structl,lr.al design~ " ,- "" !~ !I :

Yield mc" (X!mm:)

'00

,,0 ~ ,00JI ______ -; ______ -; ______ -;cOcc,Cdc;cmc'+',c,_icmC'n~'

-. I I 1'1 10 20 30 '10 :..: Bako!r t.nd \\1ckh2m (1979) o nak~fj. 19.0 ) ~anniuer (l!!GS)

','

3.01 ~IODELLI~G OF RESISTA:-."CE VARIABLES - "IQOEL SELECTION

T . . . !, . ,t;

From the preceding example it is clear that there are mony sources of physical variability which contribute to the ovenil uncertainty in the yield $treS$ oC '" grade oC rei.nf~m:;ing steel. Le~ us now define the quantity X os the random variable representing the yield str_~ssp{a particular grade of-reinforcing steel irrespective oC source and where ~yie!d ~tressll is defill~.n a precise way. We now wish to es:uiblish a suitable probability density .function.Cor X to use' in further

calciila~~o!ls .. ~i"iS.clear ;:hat 'the mathe~'atical Corm oC fll. will depend on the p~i~C~,lar subset of X,e.g.:

Let ~I be t~~ event [b .. ars are suppti~a by manufacturer iJ 81 !1e th~ even.t (bars .~re o'i~~~t~r jJ C be the event' (t~ars aie (r.~~ a sin ale cast of steel)

The.ll"i.p ge~eral.the.de1"'~i~~' i~jl~tiOri's .~X' IXIA!' fxrS;. fxtAln 81.'. fxrA~ "'.Bj''"'I.C etc. will all be different: not onlythee parameter.s. but'also then- shapes .. It is also clear that the probability density functioitti rep:esenti~g" all b~, iire;~tive of size or manufacturer. will not be oC a simple or standard fom je~,. normal."i6gJ1o,,;a1. etc.J.lt will take the (orm

. . -. --. -- ~ . . . . .. " . - - -- . __ .

(3.33)

(3.34)

qj being the probability ,hat the bar is of diameter j. Equation 13.33) represents what is known as a mixed distribution model, ;..,,; H snould be-noted thatbeca~se .oc th; systematic decreaJIe in reinforcing bar yield ' ~t~ess with

incr:asi~g dia~eter, equation-(3.34) gives rise to a density runcd.on fXI AI- which ~.:n.~tter and

has less pr~mounced tails I platykurtic) than any of the compon~,:,:t distributions fXIAI n BI' Furthermore~it ,i~_~enerally Cound that .the dens:ity""functl?_n~~"iB~ representing bars of a par-ticular .si~e considered.Q\er:;l.1I manufac.turels is highly positi;,eIY's~ew; The 're

50

,.

,

,-., , i. 3. PROBABILISTIC MOPELS FOP.. LOADS AND RESISTANCE VARIABLES

; ., <

0 . 0.7 0.' 0.6 0.' 0.' .. 0.2

O.l

0 .05

0.02 0.01 0.005

- . . . - .. _j -. 'I ' I .

I I E I Y' i i

I, . ; . .. I.

I . . 1 i. 1 i ' I"~ "J. I I i 1

I' . I

1/1/ .. d i A A" i/VI

""

Mill i'i. , ;6, mm pi!lt,,:,: . . . '

fo' illure 3.10. Q.ainulativt! fr,quenc}' di~';;; (or yieid nrea of mild .t .... 1 plates.

0 .998 . . 0.995

0.99 0.98

0.95

0.9

0 . 0.7 0.' 0.'

00.4 , 0.3 .

0.2 ..

O.l

0,05

0.02

. 1

. I I ..

' ! " . . 1 '. . I

I 1/ I 1

' 1 0- I I I ,. I

i . "" . 1 i j.,," .. h ... ',/. i 1 1 .', ! I . " fl. I 1 I: I i I . 1

',t .1 , I I ! , 1 ' i I I I I 1 ,

i :i I !t'lmml

, 0.01 r::=+:=+t::i:===:+==t======+==::;:=t:~ 0.005 Lf_-'-_--'-_--'-___ -'--_-'---'_--'-_'---'-__ ..L---'_-'-_

220 24" 260 2.0 '00 320 , .. 'SO

Filure 3.11. Combined cumulative frequency dia\:f.m for 12 mm mUd steel platu (rom three mills.

3.'; !l.IODELLl:-lC OF RESISTANCE VARIABLES MODEL SELECTION 51 ,~ Ir"

. ,'.

We now retu:n to the queslior. oi selectin!!: a suitable probability distribution to model the un ce~~.i!lty in :r.e strength variable X. It should be clear from the preceding arg"Umt'nts tbat:l pfoc'edure 0: random sampling and testing of, say, reinforcing bars at a constructio~ site and attempts to fit a standard probability diitribution to th~ data will not lead to a sensible ou; come. In partIcular, such a distribution will behave poorly as a predictor of the occurrenc(> of

.'. values.of X.outside the range of the sample obtained. The only sensible approach is to synthp-sise the probability distribution~'of X from a kno~le'dge of the component sources o(un;:::er mintY. (as in:e'quatiori (,3.33. Admittedly thi!; ~pproach can be adopted only when such in

,' .~,) .:. I, . ' , .. . formati?n is available. Expressing this problem in another way. it is important thal [he sta tistical analysis of data should be restricted to samples which are homogeneous (O! more pre eisely:. for which there is no e\'idence of .~,on~hom~~e.~eitY). '!' ~urthe.r ~pect. o~: ~~em,,!g_ ?lust now be imrodue~. Models do not represent reality. they

" ~oq~f .. approDmate it. As .is ~ell1:mown in other branches of engineering, anyone of a number of different empirical models may often be equally satisfactory for'some particular purpose., e.g. finiteelement versus finitedlffer1!nce approaches. The same is true of prob.3bilistic models. The question that must be asked is whether the model is suitable for the particular application where it is to be used.

For most structural reliability calculations, the analyst is concerned with obtaining a good Cit in the lower tails of the strength distributions, but this may not always be important. for example, when the strength of a structural member is governed by the sum of the strengths of its components. This Is illustrated by the followin,e example . .

Example 3.3. Consider an axiallyloaded reinforced concrete column, a crosssection of which ii shown in figure 3.12, If, for the sake of simplicity. the Joadcarryin~ capacity of the column is assumed to be given exactly by:

12 R re + ~' Rj (3.35)

j"l where r~ is the loadcarrying capacity of the concrete (assumed known) and Rj is the ran dom loadcru:rying eapacil~' of the ith reinforcing bar at yie!d. Then, if the \'arious R j are statistically independent, .

12 12 [IRI - EIre + ~., Rjl- re + ~ ElRjJ (3.361

j"l j"l and

l!! 12 VartRJ::. Vartrc + ~ Rjl = :s VarlRiJ (3.37)

il i.l i.e .

~'''':'''.~r-

Ficut. J.i~. Cr~a'WClion of felnfornd Ci;ltIe,ete column.

52 J. PROBABILISTIC ).IODELS FOR LOADS A:-lD RESISTANCE VARIABLES

(3.38)

and 12

(1:.' )' R j .. 1 R.

. (3.39)

Assuming Curther that the various Rj are also identically distributed normal variables, N(100.20) with unitsofkN,and thatrc""500kN.then . "

~R ::: 500 + 12 X 100 ::: 1700 kN ' and (JR " 6'9.28 kN Since R is also normally distributed in this case, the value oC R which has. a 99.99% chance o( being exceeded is thus . , . ,

PR + 't>'l(O,QOOl)(JR -1700-3.719X 69.28 "'1442 k1'f This totalloadcarrying capacity corresponds to an Dtieraglnoad..canying capicity oC ' (1-142 -:- 500)/12 78.~ kN Cor the individual reinCorcing bars, i.e. only 1.07 standard de viations below the. mean .. For this type oC structural configuration (in fact. a parallel ductile structural"system in the reliabiiity sense see chapter 7) in which the structural stren~h is governed ~y the.average strength oC the components, it can be anticipated from the above ~ ,8.tthough it ,~lii not be Connally provoo ;,ere that the reliability of the structure is not sensitive to the extreme lower tails of the strength distributions of the components. Hence the lac:::,k . ~f,.,!~.iJabmt.y oC statistical data on ex~remely, low strengths is not too import.~t, Cor such .c~es. .,

'. Finally, jt,should be emphasised that th