1 structural reliability theory and its applications

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, Pr of.:sso r. Ph, 0 ,
Institute ofSui!ding T-:chnology
and Struc'luml Engineering ..
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MICHAEL), BAKER. ii,s~, (Eng) "
Deparlment ol"Ci'ii Eng'in~erillg - •. ,"
'Imperial CoUege ofSdenciarid Te.:hno!ogy: ..
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ISBN 3-S-r.O-U73J-S Springe r~"er!ag Berlin Heidelberg New York
ISBN 0-387·1l731-8 SpringerNerlag New York Heidelberg Berlin
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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 ch·j! ~ 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 well·developea 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<.le without modelling them.:utd taking them into account. :Iany struClur.J.! ~n ·
Kineers are shielded irom ha'ing to think about such problems. ulleast when designing: ::.-nple'
s tructu~:>.lkoc:J.use ui lh~ prescriptive and essenti<lUr deterministic nature oi most .;odes of
pr:l!:tk~. This is an undesir.1ble situation. :lost loads and other structural design p:l.r:lm.!:~rs art
rardy known with c~rtaint>" 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 <lnulysis of load combimuions c:mnot even be fonnulated without re..:ourse to prc!)abili~tic
Tt':lsoning.
Th ... re is :J. Ilet'u for all stru.;tural en)l:ineers to ue'elop an underst3nding of ~tructurJl rel ~10il ity
th'ur'Y ami for this to he appli<.>d in desi~ and construction. "'hher indirectly through CO..les ;.J f.
by diflX"t :tpplil'ation in [he .;:lSe of sped.3i structures ha..-ing large failure consequences. :::e aim
in hoth c:tses l>t!ing to achie'e (>(-onomy together with :tIl appropri:ne d~ree of safety. T:;e sub·
jl..'ct is n~w ~ufficicntly well clevelopoo for it to be included a5 a formal part of the traim::'l oi
all ch·i[ anr.! structural engineers. both at !,mdergr.1duate :lnd pOH·gr:lduate It!'ek Cours!1-1 on
"tructural sai.,ty have lll!en ~h'en:1I some- uni'erst[ie~ [or a number uf Yl':l,rs.
In wruin~ thi~ book w.~ haw tri~d co brini! :1}1lC'ther u::.d~r ont' ~'o"t~r ti,le mqor cOr.:!;:lOn!':':ts
,If stnh:lUral wliahdity lh~ory with lh¥ :urn of makine Ie pU~iib!c (or;~ ne''omt!r til see ':"'1r.!

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 thO£e enginee!S in'otved in the deveiopment
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 ad1sed to tudy a more general"
text in addition. Topics such as the statistical theory of extremes. me~hqds_of parameter estimation
<lnd stochastic process theory ,ue inttoduced in later-chapters as and when they are required.
The mai~ core of the book is devoted to the so-called-leveI2 methods of analysis which
have provided the key to fast computational procedures ror -structUral reliability calculations.
Other chapters cover the reliability of structural systems. load combinations. iTOSS errors and
~ome major areas of application.
, :'.
ree work is set out in the form of a textbook with :t number of clCarnples and simple exercises.
The purpose of these is to illustrate the important principles and methods and to extend
the scope of the main" text with economy of space: The readeds ·.Yarned igainst n too literal in terpretation
of some of the simple examples as these were not inc!uded to provide insight into
particular pructical problems. In $ome examples. the parameters of t~e probability: distributions
used in the calculations h<lve been chosen quite arbitra.rily . ~~ in ~uch _il" way,as to demonstrate
the calculation procedure with maximum effect. This doe~ ~~t mea':!.that"th"e practical aspects
of nructural reliability theor~' have been overlooked - indeed. the theory ,":ould bEl of little
value if it could not be applied. Chapter 11, on the application_ oC reliabil,ity theory to "the de·
. velopment of level 1 codes. attemptS to address m;tny of the pr~~~I~:it "p~obI~~s faced by code
"!'iters in the selection of panial coefficienu (partial factors); ~d' in ~ha"pter 3. the modellinlJ
of load and resistance variables has been approachcd with' ap~iicati6ns strongly in mind. However.
3 complete book would be required to cover this subject-in ~'pth_ Chapte'r 12 on of{shore
structures should be of interest to those working in this field.
In compiting the bibliography OUt approach has been to list only a selection of the more im·
portant works 1n each subject area. 'along with other works to which specific reference is made.
Whilst many important contributions to the literature are tnus omi~ted_ it is considered that
this selective approach will ha of more help to the new reader:"
''''e should like to acknowfedge the major contributions to the field of structural reliability
theery that have been made by a relati'ely small number "ot peopi~.mainly during the last 10
to 15 years. and without which this book would not ha·e·"~~~ " p"asSi.b!~~ Th~ subject has benef:;::
ecl from a large degree of intermltional co·operation which lias p~en stimulated by various
~ociie~ - i;, particular. the Joint Committet' on Sttuctura(S3ie~·~:"~nder. the chairmanship of
J. :~:TY Borges. The respon~ibility lor thi3 hook must. nowo;!vet. rest vith the authors ,lnd we
ir.OU;U be plt'lSed to receh"e nOtiiicltion of l'orrections or omissions of any nature.

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 atemat{'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 Thoit·Christensen
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 l10:!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 RAl1)O:1 VARIABLES ..... _ .... . ... . .. ' • ••• • ..... 32
BIBLIOGRAPHy •...•...... .... " ...... , .•. , .....•..... , .... •• .. . . .. 35
3. PROBABILISTIC MODELS FOR LOADS AND RESISTANCE VARIABLES. . . • .. 3i
3.1 INTRODUCTION ............................... , ....•. ,......... 3i
3.2 STATISTICAL THEORY OF EXTREiIES ....... ,..................... 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.IE·YALUE 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 ~I od el 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 Monte·Carlo 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:<'IAL BASIC VARIABLES .........• ',' '.': :..'.! •••••• •• ••••• • • 108
BIBLIOGR.-PHY ...... .. ..... ....... , ......... _ ............• • ....... 110

7. RELLBILITY,OF STRUCTURAL SYSTE~(S ..•.•..•••...• , .•. .••• " ••...•• 113
7.1 L'''TRODUCTlON ......... ::' .......................... ; .. ~ ....... 113
7.2 PERFECTLY BRITTLE AND PERF:ECTJ,.Y Dl:CTILE·ELEMEN'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 SYSTEIS WITH UNEQUALLY CORRELATED ELEME:-..-rs . •••• • 136
BIBLIOGRAPHY ... .-:. ; .' ..... >.:; ; .:. : :,; .. ;'. ' .•. ': -~;. : .. -.... . : : .,;.: ,'. -;" ' . .. '~. '; 143
9. L~TROblicTION~TO STOCHASTIC PROCESS TH£bk'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 BORGES·CASTANHETA 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 F O R~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
',' . ,; ,J"
12.3.2 Morison':; coefficientS. 225
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~. 2·11

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 LOG·NOR!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<:ribed. procedures f~ computing
$tresses :!nd defle<:tions. This deterministic approach has almost certainly been reinforced by
the ';ery large extent to which structural enginccrin~ design is codified and t,M !ac~ of Ceedback
about the actuo.l performance of structures. For exo.mpl~ .. actual stresses are rarely known .
. deflections are rarely observed or monitored, and since most structures do not collapse the real
reserves of strengths arl! generally not known. In contrast, in the field-of hydr:lUlic systems,
much more is known about the actual performance of, say. pipe networks::wtin. spi1lways etc.,
3S their performance in service c:m be relatively easily observed Dr determined.'
The lack oC inCormation about the actual behaviour of structures combined ~h the use of
codes embodying rel:1tively high saCety factors can lead to the ;ew, 'still heid by some engi'
neers as well as by some members of the general public. that absolute safety C:an De achieved.
Absolute safety is of course unobtainable: ;lnd such a go:!1 is also ~ndesirnble, since absolute
S3J'ety could be achieved only by deplo~'ing infinile resources'. ..~', ~ "
It is now widely recognised. however. that some risk of un;:l.~ceptable st~ct~nal' performance
must be tolerated. The main object of structural desio;::n is therefore. to ensure;'at an acceptable
le'el of prohability. thnt each structure will not become unfit for its intended'purpose at any
cime during its sped£ied desi,~n lifE. )!ost structures. howc'er. hl1ve mUltiple ptrformancc ret;:
Jirements. commonly espressed in terms of a :set of serviceahility and ul.ornate limit states.
:nClst of which are not inuependent; and thus the problem is much-mort! cj:"tlpli:x. than the spe-
" . c:fication of just il sin!;Ic' probah~l.ity .

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 occurring
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 difference~'
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)' ,,10·100 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<rl ). In contrast. most pt'oplf' spend alleast 70% of their life indoon;
and are therefore e.'{posed to the possible errects of structural failure: but this leads to
an average annual risk ~ person of oLlly 10·~. Ne·ertheless. the only fair basis for comparing
this risk is comparison ith the inescapable minimum risk that has to be accepted by an indio
vidual member oC society as beyond his cOlltrol nnd for which 110 blame can be attributed to
other people. for example, the risk of d.cath by disease. Many people attept 'oluntary risks
., many orders of magnitude hi~ner. but these 3ho~ld not b~ take~ into account when consider-ing
structural safety.
!!10untaineeting (International I
.j Air, travel (Intemation31J
'Deep water leu'ling ,;
j Car travel ! Coal ~i.ni_n,g . ... : - . ! Cons~ru~.t1pn .. ~hes._
. : t.lanuf3!'!turing .
Accidents al.·~~~~ ,:alii.
Accidents at home (able.-b~died persons)
Fire at home
Naturnl causes 13'eraee. 31 ages)
Males aged 30.{alJ causes)
Females aged 30 (all'c:lllses) ,.! ~b.le!O aged 50 i,all causes I
! femaJes aged 50 Iail (';J.meH
; J:fumber oC deaths pl!r
hour pet 101> 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 masonry).
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 economic,
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<.:ERT,lSTJES 1:-: STRlil.Tt.:R.L ENGINEERING
Tb.e aims and benefits de~cribed above can only be achieved b}' a rutiollal .t5$essment ot the
various ~ncert.1irties :lsSocitlted with each type of structure tlnd :!.. study of their interactions.
This is the essence of structural reliability analysis - the CundamC!ntals of which will .be described
in the following ch~pte;';;: along with some recent applications.
1.2 UNCERTAINTY
1.2.1 General
Structural reliabili.ty an~lysls is concer'ned with the rational treatment pf uncertainties in struc·
tural engil?-eering desi;" and the associated problems'of rational decision making. Consider the
(ol/oIVing statement: " "
• All quantities (except physicaI and mathematical constants) that currently enter .. into en·
gineerinl1 cnlcul3tions are in reality associated with some uncertainty. This fnct hos been
implicitly recognised in .current and previous codes: If t.his· were ~ot 'the case.';' »S3iety
tactor"1t only, s~ightly in excess t;)f unity would suffice in 'all circumstahces. Thl! 'detf!rminatlon
of appropriate standards o"{ safety requires the quantification of these unc:eflainties
by some' appropriate means and a study of their interaction 'Cor the structur~ u.nder can-
. sideration. ,'.!.
Before continuing. it 15 worth noting that the argument is sometlmes ad~ance~ t~at the magni·
- tud'es of all variables are either bounded or can be restricted within specified ,1.imits-bY ~:<ercising
appropriate sundards of control. and that these bounding.v.alues shoul?, qe used as the
basis for design. In structural engineering. however, such allUments are inap'propriate .~or a
number of reasons:
• . upper limits to individual loads and lower limits to material strength are not cas!l}' identi·
fied in practice le.g. building occupancy}oads. wi~ !oads. t~e yield slress oC.steel. the
cube or cylinder strength of concrete): - .
. . , ; .r~ ;'.
• even if such natural limits exist. their.?irect use in design is likely. to be extremely un·
econom.ic;
• limits Imposed by quality control and testing can never be completely ef(~tive. particu.
larly in the case or properties whkh can be measured only by destructive tests or in circumstances
in which changes in the potential properties take place between sampling and
use oC the material (e.g. concretel:
• even if recognisable limits do exist. their use may not nlways be r~tional.
." : • > ' • ' .; : ~ • 1 . ,
Example 1.1. Consider:l column su~portinJfn' flc)ors of:l buildin~' on which the loads are
known to vary independently with time. Anumin" that the load on ea:h noor is physi·
cally restricted b~: .so me hypothetic:tl fai l-safe devJce so th:n unrier no circumst:lnces can
it exceed some speci(ied m~imum t':lIue, an? gi':en that each load stays (It this maximum
"alue for. say. 1::' of ~h~ eime. the "'tional desi~n load Cor the column can l1ener:ll1y he
shawl': to be lesi than the sum of the ma:<imum loads. This desu:!n ioad will. of course. depend
on t!le number of storeys supported and the design liCe DC the structure.

' :~, 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 independe nt ~, 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:<im~tely 1% of
the time). n~e figure shows that eyen ;th only !i:<: floors. :he probability that the maxi-
. l' ,. ', mum pos'sible' fcilumn Imid oceurs is"iifsn1all as :1b~ in: 50 ~·ears. 'Even 'if each floor is loaded
to its ma.,,<imum value for about' 10% ofthe time (p .. 0.1); the probability 'of the maxImum
possible column load occurring Is still very small if the number of floors supported is
10 or more.
,',I ' . ' :,' .. ' " ' , ' :~ l ,.,;:: ' ,';,.- , .. " "
In such cases, it would be irrational and uneconomic to design for this Vorst possible con-dition,
However, it should be noted th'at in practice the i:f~ee of conservatism depends on
whether the_ individual floor loads an In fact inde~ndent: .lnd the :l:ceeptable..risk level IS
" ~ '_ ' " - " r'" ,! " • " ', ", ,y, " " ", ' _ " "
govemea, bY,the cons~qiIen'ces oC faihire and the rapidit;· with which fallure OCCUrs. Some
". ' 1.."'nowledge"of th'EFp'rob'abilities 'C;! oc~urrence 'olto'ads Ie's:! than the inaXimum'l,.vould also be
. : ' required' (Or each noor for a rational uesign: ,'
;",;' " '::::-:.i":;: ,', ' '
,The precedini example demonstrates that although .. there is no (strictly, negligible) uncertainty
) n the magnitude o[ the,ma.,,<imum load intensit~' :on;each floor. ~ere may be appreciable un-
,- .certainty in the magnitude of the m:t.ximum combined load carried by the column.
'We retum:now to the 'quesiiori o( cl:lssif~;ins't1e 'aI;OU5 't)-p'es of uncertainty:that can arise in
structural reliobility analysis, Howe'er. beiore these.:ue discussed it is helpful to' introduce the
,yorc~p,t, ,()r~as!C ',;ar!ahles, Th,is..concept ii, tre~,ted in ::10~e: ~e,t~ in chapler, 5 .
,~ , " . " .1 > ,·,'
For ,the! purp9~es of quantifyin;r UI:cert~i~ties in the :Ield o[str..:ctural el')gineE¥'ing. ilnd for sub,~
eRu~wnt .re1~ll.iJ,i,~~; ~ry~,[y~is i~ is nen.'553ry, ;Cl define. a :e.t ,o~.ba$ic z,·ariabfe,$. The~ ;u:e defined

6 1. TH£TREAT:.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 use<! by designers 1ulc1 .an:i.l yslS in structural calculations.
Thus, the yield stress~o( steel can be considered as a ba5lc variable, althou!':h this property is
iUielf depende~i on chemical composition and·variOUs"rili,CrO-5tructuraJ p:ara~~ters. Mathematical
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~:-.ss ,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'/ u·ncertainty: 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. <.:ommon probability disuibutions h:l'e bet.ween one an(! four parameters which
immediately placts a lower bound on the sample size req~ired. but in practice very large sam·
o pies are required to establish reliable estimate!. of the :'J1:1fr!.erical ValU~5 of parameters. Fpr 11
given sCt of data. Iherefore. the distribution parameters may th~mStlves be considered (0 be
random variables. the uncenuint~· in 'which is dependent ~n ~he ~~ount of sample data· or.
in general, on the amount of data and any prior knowledge. This uncertainty is termed sta·
tisticoll.mcertainly and, unlike physical variability. arises solely as a result!lf I.ack of infor·
mation.
Model uncertainty: .Structural design and analysis make use of mathemati~al' 'm'odels relating
desired output quantities (e.g. the deflection at the centre o[ a re!nforc~ 'con~rele beam) to
~3 ': the values o[ a set of input quantities or basic variables (e.g. load intens!~ies; modulus of elastf~
' icity, duration of loading, etc.). These models are generally determitiistic in form (e.g. Iinear~'.~
·-;iasLiC structural analysis) although they lXIay be probabilistic.(e.g. calculation of the peak re-
" .sporu.e of an oUshore structure to' stochastic wave ioadina:). Furthermore. they may be based
on an intimate understanding of the mechanics of the problem (e.g. plastic collilpse analysis
oh steel portal irame) or they may be ~i8hly empiric~ (e.g. punching'shear at tubular joint
connections in offshore jacket stNctures). However. with very few exceptions, it is rarely pos·
sible to make hiehly accurate predictions about the magnitude of the respons~ of typical civil
engineer~ng structures to loading even when the governing input quantities are known exactly.
In other words; the response of typical structures and stntctural elements cont-ains a ~ampo·
nem. of uncertainty in addition to thai(! components arising {rom uncertaint!.es in t~e values
of the basic loadln~ anc. strength variables. This additional source o~ un.~ertaint~' is termed
model uncertaint)' aud occurs as a result of simplifyins assumptions, unknown boundary can·
ditions and as a result of the unknown effects of other variables and their interactions which
are not Included in the model. For example. the shear Strength of nominally similar reinforced
concrete beams exhibits considerable scatter even when due allowance has been made for the
various known dif!erences between iest specimens. ' r
The model uncertuinty associated with a particular mathematical model may be expressed in
terms of the probability distribution of a 'ariable Xm defined as
x . " actual strength (response)
m predicted strength (response) using model (Ll)
In many components and structures, mod!!l uncertainties have a large effect on structural reo
iiabiliL), anc! should not be neglected.
1:3 STRUCTURAL RELIABILITY _1!ALYSlS AND SAFETY CHECKING
.. ~i'ne F ",:·" linl!. rema.rks were concerned ..... ith th~ '~~riou5 types and sources of ~ncert.:.int)' thot
n('ed to be t:iken into acco .. nt in predicting the abilit.y' of D structure to withs~3.nd the actual
b~t u:lknown 10Jds th3.1·will h,: :).ppl.ied ·~"· it. . ..., tll"n no ..... comider the various WJ::s in which
lhe~ prt'dictions c:m bE' made and .the usc tnat can be- IIItlcle. 01 t j,.;.:I' . But first some definiliuns
und prelimir.:nies.

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 purpose
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 structures
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 ·taken·into 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;<a"i1!l1e ~vit~ care
exactly what-is meant by »the probability that a structure will not attai.t:i eac~ i;w=ified limit
state durin'g a specified reference periodll.
, f . • . ' ,.
Consider .CJr,st. t.~e need for defining a reference period. 8«nuse the.majority of structural loads
'nry with time in an uncertain manner." the probability that any selec~e? load btensitr will be
exceeded in a fL.. .. ed interval of time is II function of the length of that intetval,'.ll1d possibly the
time at 'which it begins). Hence. in general, struc~ural reliability is dep~ndent o,n time of expo-sure
to the loading environment. It is also affected it material properties chan3e with time.
Only (or the rare cues, when lauds and strength are constant. can the referent:'t period be ignored
, In .such cas:es the loads are applied 'once and the structure either does ·O! does not fail
(e.g, when the structure or component is loaded entirely by hs own self weightl.
The second que~tion is mo~ d~ding. 'What is meant by lithe probability tn3t .. . 11? This is
best explored by a simple example.
Example 1.2. Assume .that an offshore structure is idenlised as a unifon:l 'erticol cantilever
rigidly connected to the sea bed. The structure will fail when the ::1Clment 5 induced
at the root of the canlilever exceeds the flexural strength R, Assu:ne fUrther that
Rand S are random variables whose statistic.:LI distributions ure known ret)' precisely :lS
a result o[ a very long: series of measurements. R is a variable representin~ the varilltions
in strength between nominally identical structures. whereas S represent> the mu.imum
IOlld effects in 5UCcessn-e T year periods. The distributions of R .md S.:l..'"e both assumed
to be stationary with time. Under these assumptions •. the pr",balJiiity U::1t the StruCture
will collapse duri.n.II: an'l'1'f¥ronce }Jt:rtocJ of duouion T years will be siu:wn in chapter -I
~ c be given tiy
Pf - PDI.o;Ol - -· F'!'I.lxlfS,xklx t l.:!l
~-.-
where

.': 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 detCfmi'~
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 long·ntn 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, resistance
r ,,,ill be exceeded by the 3S yet' ~un~sampled .. reference period extreme 10!ld eH~ct S I note
that lover 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~bilty'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<,lreted as 'a conditional failure probability .wilh 3. relath'e fre·
. ql~ncy interpretation. i.e. :/,'
il1.6'
The symhol i may he read as .gl~:'1 thUll •.

l~
!
I. THE TRI::ATMENT OF L.'NCERTAI~TIES IN STRUCTURAL.£NGlNEERING
1,~,2 Methods of S3!ety Checking
tlany 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$ing. To help clarifying the
situation. the Joint Co~mi~tee an 'Structuia! Safety j'an inie~ational bOdy sponsored by such
international organis3tions as CEB, CIB, CECM, IABSE, lASS, FCP and RILEM) set up a sub·
commJttee in 1975.to prov!de a broad classification system for the diCierent method$ then being
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 pre·defined characteristic of'!
"(~ :norriinaJ values-of the major stnJ:Ctura! and loading 'ariabies, .j;
' I I •• , • • _ .,0< ~~
A level) struc!ural deSIgn, with the explicit ~o.n,s lderatlon of a number or'separate limit states, ~~ , - ' ,-" ~.
is what is now commonly ~alled limil·stat~ 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 methods
(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 fatton
· 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 Safety.
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" from·the 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 fi¥Ure 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
an·dE()E-0. ' . • . .. ,
Il is easy to show thai the 'i~ter$ectlon and union operations obey the following commutative,
associative. and distribuUye·hiws
.. " ~ .
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• Ej - 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;I£NTALS 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~ Il il> 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.dividual
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
accor-ding 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 '<; ' . ·i •. : .' _ , , ":. . -" ,:: _. :-. -; .. ... .
::'::
• PI {lliJure oi structure I '" 0.01 .;. 0.01 - ·1 . 0.01 .. 0.0100 . '. ':
:;." , "":.:)
"" Fin:lii;·. t~u/so"~:il1ed' &~'es' theorem ',vill be ceri"W. Let ttie sample space- n ':be divided into
:'l mutually cxclu$i'e e'ents E
1
, Ez, ... ·. E::: 'fS~ fi¥Ure·2.3. ~here' n • 4); Le't· .. Che on event
tn the same ~ample space. Then

18 2: : FUNi:iA}.i·E~TALS OPPROBABILin' THEOR'l' f
.. ;'
FiiUre 2.3
'" peA lEI )P(E1) + peA IE2)~(E2) + ... + P(A IEn)P(En )
.. IPIA!E1lP(Ej )
i"l
from the dt-finition (2.16) follow.1
so that
or by usinG' i2.22j
PI.'IEjIP(Ej ,
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. el~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) '" PlAIEPlE) .. 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 numerical
values artificiaUy.
In this wayan outcome or e'ent can be identified through the value of a function call£od 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 probabilities.
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 itl·values, namely X and x, .
respectively. It is a direct consequence.of the axioms (2.9) ~ i (~.: l1) that
.. ~tpX{x)-l
j-l
Pfa<;X<b}- ZPxlx,l- ZPxIX,)
lI"j"b 1[1""
~3:~1' ~.~:, ~:.~
,~ ;i.
":'i' .I(~ ,;
'iI: . 1:,.-
~~pr,o bability distribution function Px ; Rf""""'R is related to Px ~.i.
'. '. Px(x)::: PtX ~ xl '" .l'Px(xj )
'I1<1[
12.26)
12.27)
12.28)
12.29)
Sy the de(inition (2.29) the value Px(xI is the probability of the event that the random vari·
able X tak~s on values equal to or less than x. .

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) 1. 1.0 F'---~",
O. 0.5 ' ' ...->:
x ! ! I I , , I • I I X
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 <: xl xER (2.30)
It is often useful to use the derivative probability function. This function is called the pro·
bability density function fx :Rr"R aDd is defined by
(2.31)
assuming of course that th~ derivativeoists. ~ote that the symboJ Px (x) is used for .the. prob.
ability mass function and the symbol !xiX') for the probability density (unction.
Example 2.8. Figure 2.6 sho~s .. f:he probability density function fx and the probabUity
distribution function Fx for a conti.D.uous random variable X.
IF,,(x)
1.0+------------- ---
~~--------------------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-<iilCTete random uarilJbl~. i.e. " continuous
tandom variable admitting a countable number of discrete values with a non-uro probability
as shown in figure 2.7.
In this case the area under the curve in figure 2.7 15 equal to 1 - 0.2 - 0.1 • n.7.
oX
Figure 2 .•
~.5 MOMENTS
In this section a'number of important concepts will be introduced. Let. X be a continuous ran·
dam vari2ble. Then its probabilistic characteristic! are described by the distn'bution function
Fx' Ho~er. in manr applications the form of Fx is not known in all details. It 1& therefore
useful to have an approximate dc.!~rlp.tion of a random variable stressing its mast important
fearures. When F X (or!X) is completely known, however, it is also of interest to have lome
very simple way of der.cribing the probabillstic characteti;tie.~. For this puIpOlot the so-called
momtmt$ are introduced here.
When X is a random variable. Y '" Xk, where k is a positive integer, is also a random variable
because- PI {w : XII ...: y:·1 exists for every y. ln the following it is assumed that an random van·
2:bles are ;:ontinuous random variables. if not. otherwise stated. The ~%pcctf!!d volue of X is de·
~:ned as

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- 4·0.H 5·0.5 + 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.
: ".
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. FUNDAM£NTALS C?F PROBAIUUTY THEORY
41 .(x)· ' ' r1n : 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<--,,-.-) J (2.51)
where x;:' O.
: Exezcili<! 2.'7. Derive the lo&,-normal densit)' function'·{2.51),by the USE' of equatioD (2.38).
Let X ~ ]og·non:uii1ly distributee. with the parameten .u y' and ~y: Note that lAy and 0y are not
equal to.ux :md "x:ll can be shown that
(2.52)
EsetcisC! 2.8. Lei X be log·normally distributed with the parameters Ily and ay. Show
!~t .
. lnx-lJy
Fx (xl· P(X .;; x)· 4>(--,,-.- ) (2.54)
wh"ft' .]. is the 5c:mciard no~ttl cii3tribution function.

!! .6 UNIVARIATE DlSTR1BUTlONS
! fXj)n
0.0 1i
"0.41 T '~1 (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 Jog·normally 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. ')
I (k.~) · (1. 2) (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 real·valued function X :n ......... R mappin~ the
sample space n into the real line R. This definition can easily ~ extended to 3 vector·valued
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 '"
. (Xt , 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 sub,se t o( Etft.
____ C-___________ "
a'igure 2.11

:!.' RANDm,l VECTORS .-
c ,:.
Consider llgain two random "ariables X1 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::<l and x 2. To describe the joint behaviour of Xl and ,X'!. th~ft?i"t probabifi.
ty distribution. {unction FXI . x 2: Rlf'""' R is introduced and defined by .
(2.58)
It is often convenient to use the notation FX for FX1,xZ' where X .. (Xl' X2). The defini·
. tion 12.58) can be generalized to the n-dimensional case
,
FR(x)::: p( n (Xi C;; "i» (2.59)
ial
where X .. IX1,·, .• X"n)and X - (Xl"" .xn).
29
A random 'ector can be discrete or continuous. but the ~rp.sentation here will be confined to
continuous distributions. Only two-dimensional random 'ectors will be treated because genera-
. lization to n-dimensional rando~ vectors is straightforward.
The joint probability density {lInction (or the random vector X .. (Xl' X~) is dermed us
(2.60)
The inverse of {2.60) is
(2.61)
The distribution functions FXl and F:<2 for the single random variables Xt ~nd :<2 can be ob·
tained from (2.61) .
3nd similarly (or EX.,. By dirre~nti3tion of (2.62)
tx, (xt ) ,. )~~. !X:(x1• "2)dx'Z
~.nd correspondingly for (x
r 2
fX:!(:t:':!' "'" fX(X.l,x~)dxl
The density functions fXI and fX2 llre called marginal den.sity fUnctions.
(2.62)
(2.63)
.' .
2.64)

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 PX
1
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 probabill!)'
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<mdJlhmo' ·distribution function
rl .. (;~1':~/-"., 7·2)dx~ ..
FXIIX:(xllx2) - fXt(x2) (2.69)
"ext by integrating with respect to x2 the so-called IOhJl/"'lllUbillty theorem is shown
(2.70)
Example 2.14. ·Consider again the two jointly dislritJlJt.t:d discrete random variables Xl
and X2 from example 2.13. Note that
but for e);ample

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~ function! 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 (unctions
so that inverle 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 - IXI •. _ . . 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.
" " E(l";(Xill - IE!f;(X;lI (2.77)
i-I i-1
so that the'op'eration of expectations and summations can commut~'~~i£ompare with
exercise 2.5.
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[X1 ] ., Jlx
1
and E[X2, :a
IlX2
, The mixed central moment defined by
, (2.79)
-: '~ ,"!to ...... <':" .. - ••
"'-"" '.
is called the cOlJariancl! of Xl and X2. The ratio,
Cov(Xt , X.,]
PXl X~ " aX
t
aX
2
-
r, 'r;
(2.80)
: •. "
where aX t and aX
2 ~ the standard deviations'of the ran'dom variables Xi and X2'.' is called
the correlation coeffici.~nt. It can be used as a measure of mutual linear dependence between
a pair of random variables. It can be shown that - ~,;;; .oX1.X2·-';;; 1.
Two random variables Xl and X2 are said to be uncorrelated if .oX!::<2 .. O. It follows from
equation (2.76) that .
(2.81)
Therefore. for uncorrelatec random variables Xl and X2 wt' ha'e
E(X,·X,] = PX, JE(X,J (2.82)
!

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!X2 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 <lnd X2 , .. .... . .; . "1"," ' .:,
Exercise !U3, Show that the mo.r~inal density functio~li ~Xl (Xl) for (2,8?1 are
1 lx,-~~ f X (Xl) c -=- exp! - - (----.l) J (2.88)
• I , / 2:: a
l
2 0 1

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. H·S, & 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: Holden·Day. 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
:ime·varying. A5 .di.s.cl!s,s~ in !<~,,:p~rs 9 :md ~9 .. ti~e-v:a.I1:! .ng . loads ~ best m~71~ed as stochastic
processes, but,th}s i~: ~~t .3 c~~v,enient te~res,enta~ion COt use with the methods of reo
::ability analysis being presented here (chapters 5 and 6). L'utead. it is appropriate to usc the
i istribution of the'extremelvah.ie 'or tile !dad :n'the iefe~e:1ce" period :or which the reliability
37
!3 required; or, where there are two or more:!i.me-'arying .roads :lctin~.,n a structu~e together,
:he distribution of the extreme combined load or load effect. The particular problems associated
with the analys~ or combined loading are discussed in chapter 10.
The selection of probabilistic models (or basic random variables can he -:ti;ded into two parts •
the ehol~e'of 5uitabh~" prob3bility Clistnbutions:,vith which to cha.rac.t.:!rize ~~~~hysical uncertainty
in 'each c:ase and the C'hoice of-appropriate 'alues for the parameters of those distributions.
For most practical problems neither task is easy since there may be a number of distributions
which appear to fit the available data equally well. As mentioned above. loads and resistance
variables require different treatment and .will be discussed separately. Hovevet. it is first necessary
to introduce the i~po~ant subj~~:-~f ~h~ ~tatistic~i tbeo~y~ ~'{extTem~s '~hi'~ld~ of rele·
.... ance to both load and strength variables. This topic is disc':1s~~ ,in .tb~i next :~;'.~ ~.tio~.
3.2 STATISTICAL THEORY OF EXTRE~IES
In the modelling of loads and in the reliability analysis of SlrucrunJ systems it is necessar:y 1.0
deal with the theory of extreme values. For example. with tlme·v~ing loads. the analyst 1&
interested in the likely value of the greatest load during the life' oC the., stru:~iu're. To be more
,necise. he wishes to know the probability ci:5tribution oi th~ £-reate!t. 'road. This may be inter·
"reted physicaUy as the distrihut!on th~t. would-b.e obiain~ iJ. the ::o_a.xi~il!!l.Hre~ime I.oa~ were
:neasured in an 'iniinite set of nomin:llly icie:::ical structures.

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 information
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 distributed
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(x»n-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 (x»n - 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 vari£ble 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 non·occur·
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<-.,(--X) )dt
(
" 11 1 t - •• , '
• II • _ .. y'..!:J: Ox - Ox J (3.6)
It S~OLL;: i"~- ;;~l ~~<:l tOOlt Fx: is not normtJ~l)' distriO!.lICC.

3. PROBABILISTIC ;IODELS FOR LOADS A~D RESISTA:-:CE VARIABLES
1 1 .
, r)t"(S)
·l.5
Figura 3.2,
The probability density function fX." Z I ~ (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 independent
events (e.g. the annual maximum mean·hourly 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 iniormation
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 extreme·value 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 off·in 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:'.IE·VALUE 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'nj 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..'<.i:na genei~ted i'n n "i~d;epimdent
trials has it distribution function
FyA, ty)" ,exPI-=-: 1.1. expt.~, ~f.Y - .uHI
.. ' ,

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 texts·5ee (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 b.,e 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.xima·di·~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(y»n has the same coefficient of variation.
The type II_~~ d~tribu~ion is freqlJ.~ntly,used in modelling extreme.hY,drological and meterologica,
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:

1 structural reliability theory and its applications

1 structural reliability theory and its applications

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