4 . b . 8

SOME PROBLEMS IN THE STATISTIC AL CALCULATION

OF SAFETY FACTORS

D , G, BEECH The British Ceramie Research AssoeiationJ Stoke-on~Trent J Great Britain

SOME PROBLEMS IN THE STATISTICAL CALCULATIOiV OF

SAFF:TY FAC'TORS

A simp Ze modeZ from whieh safety factors can be de­

duced is first deseribed and the eZaboration of this

modeZ to cory·espond mOl'e eZoseZy with pro.ctieaZ situ­

ations is diseussed. Difficulties aJ'e shown to arise

from the assumption that prope.rties are normaUy dis ­

tributed a~d the use of other distributions is demon­

st.rate1. Other probZems aY'ise from the faet that

functions of variabZes may have distributions that

cannot be expressed in simpZe terms . The ZoqnormaZ

distribution affords a pZausibZe method of m'oiding

this diffieulty. TabZes of ealeulated safety faetors

illustrate the methods used and the relation between

global and partiaZ safety factors is discussed .

QUELQUES PROBLEl1ES RELATIFS AU CALC'UL STATISTIQUE

DE FACTEURS DF: SECURITE

On déer,:t en prem":er lieu un modele simple à partir du ­

quel peuvent être déduits des faeteurs de séeurité

I' élaboy'ation de ee mode Ze est discutée afin qu' iZ eor­

responde le plus étroitement possible aux situations

pratiques. On montre les difficultés reneontrées pour

partir de l ' hypothese que les propriétés sont distri ­

buées normalement et on démontre l ' emploi d 'autres dis ­

tributions . D'autres problemes résultent du f ait que

les fonetions de variables peuvent avo ir des distribu­

tions qLo.i ne peuvent pas être exprimées en termes sim­

pl.es. La distribution logarithmique normaZe permet

d ' appz.iquer une métt-,ode pZausible pour éviter cette dif­

ficuUé. Des ta,bles de faeteurs de séeurité caleulés

illustrent les méthodes utilisées et on discute la re­

Zation entre faeteurs de sécurité globaux et partiels .

EINIGE PROBUNE BEI DER ST.4TISTISC'flEN

BERECHNUNG VON SIC'HEREEITSFAKTOREiV

Zunaehst wird eine einfaehe Methode beschrieben naeh

der Sic'zeY'heitsfa,ktoren abgeleitet werden kÓÍ1r~n , und

die Einze lheiten dieses Modd les , sowie seine Anwend­

barkeit unter PraxisbedingunJcn diskutiert . Es werden

die Schwierigkeite~ aufgezeigt die daraus entstehen ,

dass die Eigenschaften nonnalerweise stark streuen .

Einige di eser Streubere ie he werden darges te lU. Ande ­

r e Sch:..n:erigkeiten t r eten dadurch auf , &1.SS die Funk ­

tionen der verschiedenen Einflussgróssen zu Streube­

r eichen fú'h:ren , die nicht mit einfachqn Ferrr,eln aus­

ge:lrüé: k werden konnen . Die DarsteUung eines typischen

Stl'euber'eiches erforder t eine eingãngige ;\1gt'zode, die

dazu verhilft die oben angeführten SchwieriJkeiten zu

umge1ten . Tafeln mit auf diese Weise bereehneten

Sieherheitsfaktoren v erdeutlichen die dargestellte

Methode u'1.d die Verhiiltnisse, die sich a?!.s der Gegen­

ú'lxrstellung globaler und teillJeiser SwheY'heits­

fa,ktoren ergeben .

ENYELE PROBLEMEN IN VERliAND M5T DF:

STATISTISCHE BEREKENING VAN VEILIGHEIDS'FAKTOREN

Een eenvoudig model van waaruit veiligheidsfaktoren

kunnen afgeleidJ wordt beschr even, en de vraag of

di~ model nie~ nauwer ,in re lat'ie kan worden gebracht

met praktische situaties, wordt bediskussieerd. Er

wordt aangetoond wel ke moeilijkheden ontstaan uit de

veronderstelling dat eigenschappen een normale dis ­

tributie hebben . Het gebruik van andere distributies

wordt bediskussieerd. Bepaalde problemen ontstaan

daardoor, zodat sommige funkties van variabelen niet

in eenvoudige termen te vatten zijn . De llog- normale

distributie geeft een bruikbare methode om deze moei­

lijkheden te vermijden . Tabellen met berekende vei ­

ligheidsfaktoren illustreren de geb~~ikte methode en

de relatie tussen globale en partiine veiligheidsfak­

toren woy·dt bespreken .

1 . INTRODUCTION

The concept of factor of safety has a fairly long his ­tory and may be regarded as dating from the time when structure5 ceased to be designBd purely by trial and error methods and rules of thumb,. and began to be de­signed on tne basis of measuremen t s Df material strength made under controlled condi tions . rt ''Ias soon realised that in practice contingencies may arise which have eHects that may upset the simp licity of the design calculations . The procedure then vlas to carry outal l calculations , both Df load-bearing capacity and maxi.mulT~ l oad to bs carried , as precisely as practicable in the light of the knowl edge availab le artd then to ciloClse a design s uc n that the caL:ulated load-bsaring capacity was appreeiahly greater than the calculated maximum load . lhe ratio Clf load -bearing eapaeity to load ehosen '.vas termed t/le safety faetor . rts Inagni~ude is a meôsure Df trle likely effects Df thR cÜlltingencies th ôt may alise .

lhe fixing of safety factors ha s traditi onally relied on informed es timates of th e likely effects of th ese contingsGciss , in given circumstances and has there­fore bsen tG some 9xtent subjective . Over the years , however . the collectiva experience Df many eng in eers hôs led t~ :he adoption of stdnda r d values of safety fdetor for use il1 given cit"cumstances . This 1s de­sit'oble , ,JS it ena~ les desigl1ers to learn {rom the cxperience , both succeSS8S and failures . of others and avoids the necessicy to rp-tnink eech prc~lem completely afresh .

Th e growing interest in the us e of s tati stical melhod s i ll s::ientific research , especially since Lhe 1930 ' 5 . has led to attempts to calculate safety fa c tors from statistical princip I es . This idea seemed parlicularly apt , since safety faccors may be regarded as em~~dy­ing the key statistical concepts of variation and probability . In brief , given some kn ow l ed~e of the variotion around t~e estimate va lu es to which esti­matES of design paramsters are SUb j Bct . it ough t to bo possible to calculate a s~Fety factor which will ensure that the probability of failure does not exceed some ve ry low , and therefore acceptable . value . That 15 the ideal, if fully realiz ed it would cut out the subjective element from the ca lculation Df safety factors . Althou~h it cannot be c l aimed that this has been achieved , it is fair to claim that if the methods are pr operly used , he subjective element can be con­trolled .

In the development of these methods various problems and pitfa lls are encounte r ed , and it is the purpose of this paper to draw attention to some of them . This will be done by considering some approaches that have been made : the first essentially simple , others mo r e comclex , and by pointing out the consequences of possible assumptions and lhe problems that arise . The problems may have been recognised previously but in some cases they have not been adequa t e ly stressed . The paper also indicates the range of safety factor s that result from thS S8 simple mode l s and shows how the models may be extended to cover mo re complex assump­tions .

Elsewhere attenpts have been made to introduce the cost element into the situatio n . The object here is not to calculate a safety factor consistent with a given probability of failure uut to calculate one which 1s consistent with minimum expected cost . This entails some estimate of the cost Df failure , or par­tial failure , of a structure : such an estimate is not easy to derive . It is by no means generally accepted as a desirable app roach to the calc ul atio n of safety factors and will not be cünsidered further .

4 . b . 8-1

2 . SIMPLE TWO-PARAMETER MODEL

The two-parameter model analyses the design situation in terms of the resi stance Df the structure to stress , R, and the appl i ed stress , S . I n the real situation these are not known pr ec is ely , but are assumed to vary randomly arou nd mean val ues ,R and 5, which can be estimated .

It is clearly important that FI shall not be less than 5 , in statistical terms this is exp r essed by requiring that the probability that R-S ~O shall be very small . For brevity , here and hereafte r , the notation for this is

P [ p-S~O 1 p . .... . . ............. . . ... (1J

where P denotes the probability that the expression in t he brackets is true , the sign ~ denotes less than or eoual to , and p is a chosen small pr obabi lity , usually 1 in 100 . 000 or 1 in a mil lio n (10- 5 or 10 - 6 J .

It ',Iould noc do to choose a design such that R S , because in chat case FI wou ld be likely to be 185s than S roughly as often as not o The design equation is R/y , S . whe re y is th e safety factor . This ensures that R exceeds 5 ond decreBses the likelihood that R will be less than S . In orde r to find a suitable value for y , the equation invo lving y is cOlf,bined vl1th :nequality (1 J to give

(S -5J ~ O p ... •.... .. .. (ZJ

Th e solution of 12J is difficult in general , but if ie is assumeo that R and Sare distributed about thei r neans with known coefficients of variation and the forms of the dist ri butions are knO\~n , simple solutions are soretimes possible . ~Jot e thi3t no atto.mpt is made LO sU:J:Jiv ide the variability of R or 5 into components .

Among the dist ri but 10ns that have been assumed for R and/ o r Sare : laJ ~/orma l. (bJ Lognormal. (cJ No rmal truncated , and (dJ Extreme value distribution .

Assumption (a) for both R and S gives no t r ouble , as the distribu tion of t he exp r ession in brackets i n (ZJ is then also Normal . A direct cünsequence of the assurnptions is that its mean value is

Pro vided that R and Sa r e uncorrelated , as would usually be e xpected , a s tandard statistical theorem shows that the standard deviation of the expression in brackets is given by

v'( V(FlJ V(SJ J

where V(RJ and V[SJ are the vari a nces , or squared sta n­dard deviations Df li and S . 8ecause R/y 5, the e x-~ress ion for st an dard deviation

R ~/ [C 2 +

H

" 2 1 L- ~ o

y 2 J

may hR re -wri tten

whe r e C~ and Cs a r e the coefficients of va riation Df FI and S"e xpressed as fractions , that is standard deviation/mean .

In order to satisfy (2J it is merely necessary to write

o

where t is the value trom No r mal distribution tables

4 . b . B-2

corr espon ding t o probability p . After cancellation of R and re-a rrangeme nt this gives

This quadratic can be solved for y , the larg er r oot being taken . For probability 10 - 6 , t bec omes 4 . 75 .

Tab l e 1 gives so me values of y calc ul a t e d for co m­bination of C

R a nd Cs in the range O - 0 . 2 . I t may

be noted that an equation of type (3) was developed many years ago by Freudenthal a nd others l

, though th e t r eatment of the s ubject was at first geometric rather than algebraic in nat ure .

TABLE

Safety facto r y from Normal dist ribut i on hypothesis

Cs CR

O 0 . 05 0 . 1 0 . 15 0 . 2

O 1 . 00 1.24 1 . 47 1 . 71 1.95 0 . 05 1 . 31 1 . 41 1 . 61 1 . 64 2 . 07 0 . 1 1 . 90 1 . 96 2 . 11 2 . 31 2 . 53 0 . 15 3 . 4ll 3 . 52 3 . 63 3 . 60 4 . 01 0 . 2 20 . 00 20 . 03 20 . 12 20 . 26 20 . 46

It is clesr t ha t t he safety factdr is ve r y sensitive to the coe f f ic i e nt of va r iat ion of R wh e n i t e xc ee ds 0.1, but much less sensi tive to t he coeffic i e nt of variatjon of S . It can be shown th a t if the p rob ~b ­

illty i5 se t at 10 - 6 th e safety factor tends to in­finity when Cq e xceeds 0 . 21 . The latter va lu e is not unduly high , ~nd thi s tends to throw suspicion on the statistical approach .

Assumption (b) for both R and S a l so gives no trouble , as R - S ~ O i mplies log R - log S ~ O. If R and S are distributed l ogno r ma l ly , log R and log Sa r e dis­tributed No r mally a nd 50 is their diffe r ence . The sol ution Fo r y follow s simi l a r lines i nitial l y , but de v810ps di f fere nt l y .

P [lO g R - log S ~ O ] P

log R - l og Y l og 5

Therefo re P [llOg R - log R) + log Y - ( l og S -

log 5 ) ~ 01 p ... . ... . . . . ( 4)

It must be no t ed t hat if R varies lognormally rou nd mean R the mean val ue of log R is noL log R but log R - i V(log R) where log denotes logarithm to base e . V denotes var i an ce , as bofo r e .

Hence for t he e xpressi on in b rac ke t s i n (4),

Mean = log y + i V(log S ) - i V(log R) Standard devi at i on = I (V(log R) + V(log S])

By the same reasoning as before

log Y + i V(log S) - i V( l og R) -t/(V(log R ) +

V(log S)) O

Since it may be shown that V(log R ) and si milary fo r V(log S )

lo g y = tlI og [(1+ C/) (1 +C/ )1 + i

l og ( 1 + C/)

1 + C 2] lo g 1+C;2 . ( 5 )

It is clear that (5) i s quite different from (3) .

Table (2) gives values ca l culated f r om equation (5) ove r the same ra nge of values of C

R and Cs as for

Tab l e 1.

TABLE 2

Safe ty f ac t or y f rom Log norma l di st ri but io n hyp othesis

Cs CR

O 0 . 05 0 . 1 0 . 15 0 . 2

O 1 . 00 1 . 27 1 . 60 2 . 01 2 . 5 1 0 . 05 1 . 27 1 . 40 1 . 69 2 . 09 2 . 59 0 . 1 1 . 61 1 . 71 '1 . 95 2 . 33 2 . 83 0 . 15 2 . 05 2 . 13 2 . 36 2 . 72 3 . 22 0 . 2 2 . 61 2 . 69 2 . 91 3 . 27 3 . 76

It is clear t ha t a lth ough th e res ults i n Tab l e 2 a r e s imila r to those in Table 1 for low co effi cients of varia ti on , th ey are much less sensitive to changes in coe f fic i ent Df variation a t higher values . Calcul ation Df values at higher coefficients of variation , such as 0 . 3 , i s still feasible ; the value fo r C oC

S o 0 . 3 is

7 . 19 . The safety factor is about equalfy sensitive to changes in CR and C

S'

Assumpt ions (e) and (d) lead to diff icu lties , as do any assump t ion s involving diffe r ent dist r ibutions for R an d S . Eve n though the distribut i on functi on app li ­cab l e t o ( d) is kn own , the differe nce Df t wo s uch di s­t r i b ut io ns wi l l not have the same form o Solutions are possible i n sp ecific cases by graph i cal int eg ration or by numeri cal methods us ing a comp ute r, bu t these are troubl es ome . Truncated dist ri but i ons present the same problems . It is , howeve r , possible to obtain an estimat s Df the safety factor as fol l ows .

In p r act i ce the ~~ormal distribut i on i s rare ly ass umed to hold much beyond the 1 in 1000 pro babili ty leveI , tha t i s 3 . 09 standard deviationson sithe r side Df the mean . Truncat i on Df the dist r ibut ion at this point appea r s not unreasonable ; it ass umes that th e r e are no va lu es ou t side this r a nge . Fig . 1 i l l ustrat es the deriva tio n of a safety facto r fo r this si tu ation . The dist r i buti ons Df R and Sa r e ass umed to touch , but no t ova r lap ; in other words

nax (S) Min(R)

Hence S ( 1 + 3 . 09 CS

) R ( 1 - 3 . 09 CR

)

If R/y o 5 + 3 . 09 Cs

Y - 3 . 09 CR

If the dist r ib utions were tr uly truncat ed i n this way , t he probabili t y of failure wou ld be ze r o , so in one se nse t he me t hod i s not a p robabil i s ti c one . Howe ve r , knowledge of th e tails of dist ributi on s is usually imprec ise , so it is best to regard t he me th od a s giving a negligi b le , but not pr ecise l y de f ined , p r ob­abi lity of failure .

Val ues Df safety f acto r calcu l ated by th i s method are given in Table 3 . It i s clear that these results fo ll ow a pattern quite distinct from that Df Tables 1 and 2 , though on th e whole r esemb l ing the latter more closely .

TABLE 3

Sa f ety factor y from truncated Normal dist r i butio n

CR I O

Cs

I 0 . 05 I 0 . 1 O. '15 0 . 2

O 1 . 00 '1. 15 1 . 31 1 . ~6 1 . 62 0 . 05 1 . 18 1 . 37 1 . 55 1. 73 1 . 9'1 (J. 1 1. 45 1 . 67 1 . 89 2 . 12 2 . 34 0 . 15 ,'I . tJ6 : 2 . 15 2 . 44 2 . 73 3 . 02 0 . 2

12 . 62

I 3 . 02 3 . 43 3 . 83 4 . 24

The choies of distribution is an area in which tha r e is r oon for differ8nce of opinion . alld the foregoing results show that it can me rk e dly a f f~ ct ~hB rEs ultin g safsty facto r s . Assumpt i on Df thG No r ma l di stri butinrl wo~ld S38m to be ths Basiest c ourSB, as i t foll ow s well - trodden statisticBl paths . bu t un l es s cOB"fficisnts Df va!'iation 'lr'e lm./ , calcul a tion of sa f ety f ac t o r s bccomes i~practicable . That i5 a simp l e consequ e~ ce

Df t he fact that the No r mal dist ri buLicn is not a suitable mode l for a non~n8gative prope r ty unless the coefficient of variatton 1s low . The regular uS~ Df the Ncrrnal di3t ri bution i n s t atis ti ca l tB chniques based on means , where it i 5 adequ'lte , has lRd to its uncritical adcpticl1 il1 othe r cases .

Frsudenthdl i considared that for both st r eng th s and loads lognormal o r e xtreme vBl ue distributio ns might be suitabla choicss and stated that for most purposes the logn::lrmal distl'ibutiDn ',Ias preferable . This dis­tr i bution i3 pos it ively skew , that i s it has '1 longer taj] at higher values , and can accommodate a wjde r r Clng2 of coeffj.cients of variation t"'a r , the fJorrlal distrj but i on can o

3 . MACCHI ' S MOOE L

Tt18 next stage in complexity Df a na l ys8s i5 t o split t r, e pêrarrlsters into components , each h'ith i ts element Df va riab ility wh ich can be i dentified . In the ~ork so far 8xamined R has bEen spl it , t ho uEh S has been solit il1 other aporoaches to the de t a rminati on Df partial safety factors for l oad .

Ma cchi ' s mode1 2 i5 a good e xampl e Df thi s . He postulated

,~ /\0 -S ; O [6 )

whera ~ is a slenderness and e ccentrici t y reduction factor , A is tha area on which load S Bete , and a is the "unit st r ength of masonry tl . Thsse quantit i es a r e assumed to be random va r iab les wi th known mean values 4J J A , a and S . The v al ues Df a may presurnably b~ ta~en ~s calc~lable in s ome way ~r dsri ved fro m a t able in a Coda of P r act i ee .

I f it is a s sumed th a t ~ A o Iy S i t is po s s i ble t o calcul a te y to satis~ym[ ~) wi t hm known pr ob ab i l i ty by methods simi lar to those i n Sectio n 2 . Macchi assumed that alI tne distr i. butions VJer e ~ I ormal . It i 5 necessary in the derivation of t he 501 ut ion to use an approximete expression for the s t a ndard deviation Df 4'Ao - S , namely

The appro~imatio n is good provided t hat coe f f i cients of var1ation are less than 0 . 2 . Th1s leads to the quadratic equation

where C2 = C~2 + CA

2 + C0

2, and this is similar to [3) .

The diffi cu lty he r e is wi th the value of t . Macchi appears to assume t hat the di stributi on Df the e x­press i on in [6) wi l l be Normal if its component parts ar8 i~orma l , but this i s not so . The p roduct Df twCJ or more No rmal variabl es doas not have a No rmal dis­tribution . b ut a positively skewed one . Expressions can be de rived for t he sKewness and Kurtos i s Df this product ; they are eomp l e x , but for t wo No rmal dist r i ­b uti ons with eq ual coeffi cients Df variation C, an approximate e xorsssio n for the sKswness of the product 15

R

Althou g h f or coeffi c i e nts of variation of t he order of 0 . 2 thi s i s not a hi gh deg r ee Df skawness , the sffect on the l owa r t ail o f t hs distribut i on may be qui te Mark e d . Cal cul a t i ons we r e ca r ried out for ÜJO

1e53 seve re cases , whe r e t he coefficients we re 0 . 2 and 0 . 1 and 0 . 16 and O. oB , the 10- 6 point being estimated ~y graphical methods . I n both cases the t - value for thi5 ~rCJbabili ty was i n t he range 4 . 25 - 4 . 3 rather than the 4 . 75 expected from ~I orrnal theory . It i 5 e iair dedGctio n that s ubtrac ti on of another va riable , as in (6) , would t en d t Cl p u l l the value baC K towards 4 . 75 , but t~8 latisr figure would sti ll be an ove r­estimate .

Macchi appea r s t o have ass umed 4 . 75 for proba~ility 10- 6 , so nis safety faetors are li~ely to De on the hig~ side . Although the ef f Bct may not be large , it is ~ell to note this difficulty with th~ produets of f;orrnal dist!~ibutio n s . :lacchi assumed that C

A l'las

virtually zero and gave graphs of y fo r va rious ~aluas Df C" = and C ~ . SOMe of t nese hav8 beBn calculated af r~~h ~~d a re Bl ott ed in Fi gs . 2 and 3 for compa rison with ot~0 r r esu l ts . The di ff iculties that oceur when [~ and Co e xc0sd 0 . 15 ar e il lus trat2d .

l i alI rhe var'iab18s a r e assumed to be distr i buted log ­normally . th3 p r oble m ma y be solved oy transforming [6) to

\JJfVJ < - r'-

and taking l ogari thms . The method is me r e ly an exten­sion Df thet in Sac tion 2 and r esults in t lle equation

l og Y t vlog [ [ 1+[ 2 ) 11+C 2 ) (1+ C 2) 11 +C 2)] ~ A o 5

whe re logari thms a re to base e and t is legi t i mately taken as 4 . 75 fo r oro babil i t y- of failure 10 - 6

•

The results for t he ra nge Df va l ues covers d i n Figs . 2 an d 3 are g iven i n Figs . 4 a n d 5 t ogeths r with t hose calculat ed on t ha a s s umpt ion of truncated Normal distributions . The gra phs c on f irm the more g r adual rasponsa to c tl ôngi ng c o e ffici ents of variation shown by tha l o gnormal di s tr ib ut i on .

4 . ~ANY-PARAMETER MODELS

The object of thi s section is to extend t he model so as to make i t correspond to the situation where design 1s based on es timated values of t he mean unit st r ength rat her than on measu r ed values of test wa lls . In the

4 . b . E\-4

simo18st of t hese Mac chi ' s paramete r 0 i s r epl aced by uL . wh e r R u is th e compress ive s trengt h of Lhe units and L. 1= a r eduction facto r to give th e compressive st r ength of tlle ~ lason cy . lhe r ema j ning pa rame t ers are area A (ass umed o f negl igibl e variabilityl , a slendecness and ecce ntricity r eduction fact o r ~ , and the load S .

The approp ciate safety fac t or can be dete rmin ed by e xt ens i on of the previous methods by both t he log ­normal and the trullcated No r mal equations . The so lu tion based on the Normal distribution i s open t o th e ob jections noted e a r lier and will not be considered furt her .

One cr i ticism t hat may be made Df this model is that the assumption Df direct pr opo rtionBlity Df mason ry stnmgt!l élnd unit strenr;Ul is not 1J0 rne o ut by e x­pRriment 3 . \/ariOlls relaLionships have bee n proposed , i nc ludi ng linear r e lat i onships which when e xtrap ola t ed do not pass th r ough the o r i gin , but t he re l ationship vJ hich i 5 mos t arnenalJle to clfl a ly si s i~ Df th e f orm cr = unL . where n i s l es5 than unity . This f i t s e x­perimental results reasonably well and a l so fit s the ffiAsonry stre~gth - unit s tren gth tables given in t he British Cone of Practice 111 qu i te adequa tely : si nce this 1s an apprcvGd d8s ign method it is desira b l e t hat it s hou ld do so . \/a lu es of D. S a nd 0 . 6 hav e bBen take n as ty pica l f o r mortors norffia ll y used . TI12 value c,f y colcu13ted 15 not very sensi tive to cilélllges ill n . Th e r"orie l mdy iJe môc!t, mo r e genera l bj including n , since so lut i ons are sti ll feasible by tne lognormal ~n d t r'uncated Normal method,; . Oj.rect p r'oportioroal ity \Jould be given by puttill g n equal to 1 .

The design method i s to cho os e th e unit st r ength 50 that

whara cr is r e ad from taiJ l es for th e g iven va lue Df u and gi\llln rno rtar , or calr::u latecJ in all eq uiva l ent IPiJ"rle r·. For the detc rr rina tion of saf8ty factor y i t is assumed that t he equ ivd lent calculation i5

The eq uations f o r y a r e :

l og Y t i [n 2 10g (1+ C 21+1 0g (1+ CA 2)+10g(1 +C.2)+ LI tjJ

10 g( 1+ CS2 ~+ à [n l og (1+ Cu21+10g(1+ CL

21+

l o ~ ( 1 + C/ ) + lo g ( 1 + C'p 2 I - 1 DE ( 1 + Cs 2 ) J t = 4 . 75 for probabi lity of f ailu r e 10 - 6 .

Tr uncated No rmal mode l

for neg li gible prolJab ility Df failu r e .

It i5 pres umed that the est i mat es used i n design are t he best a va il abl e . For ~ t he value i 5 most like l y to be th 8 nominal s tre ng t h of the Ulli t , fo r exampl e the strength Df one c f the class es Df brick in r .. S . 39;!1 . It i 5 presurne rJ thiJt the man ufacturer adop t s rneiJsoJres for ens url.ng that b ri cks of no t l ess than tI> i s mca ll st rellgt ll are s upp lied .

This leads to t he p r oble m of dec iding what va lues of coeff i cie nt of varia t io n to take as r e asonable . It i s useless to argue ove r much abo ut t he subtleties of mathematical analysis i f t he coeff i ci en t s of variati on are unrealistic . I t has be en f oun d useful to cons i der t he e ffect i ve ran ge of a distribution as that of the Normal di s trib ution t runc ated at 3 . 09 t imes t he s tan­da r d deviati oll and to choose t he coefficient s o f variation t o give reason ab l e limitin g val ues . It sho ul d be noted that fo r uni t strength the variation i5 that of co nsi gnmen t means , not i ndivid ual bricks . \/al ues of 0 . 07 and 0 . 05 fo r normal and special contro l seem reasonab le f o r C . Variation of L i s due to variati on i n condjtioHs on sits ; C

L is take n as ze r o .

The s l enderness a nd ecce ntri city fa ct ors should be ~, nown fa i rly accurate l y , 50 C I =0 . 05 . The value Df Cs is take n as 0 . 2 to tie i n wi t~ p r a ct i cal values Df part i a l l o ad factor (se8 Section 51 . \/alues of n assumed were 1 , 0 . 8 and 0 . 6 .

In orde r t o placa the l ogn or mal and trun ca t e d N8rmal me t hods on a fair footi ng for compa r is on , coefficjent s 8f variation of the l og normal distribution were ad­justed t8 give roughly the same 1 in 1000 limiting values as th e limiting values of the truncated No rmal distribution . Fo r u , L and ~ lowe r li mits were con­s i dc r e d and for S , upper limits , si nce these are the op e r~ti v 8 on es in f a i l ur e . Th e act~al values use d f or C a nd Cr a r e 0 . 055 and 0 . 16 . The va lues f o r [ and ct a r e given in the correspon ding Tables . u

Table 4 gi ves the sa f e ty facto r s for various combin ­ations Df parameters calculat ed on a l ognormal basis , and Tab le 5 gives similar values cal cul ated on a trunca t e d Norma l bas is . It is clear t hat th e t run­cated No r ma l va lues a r e somewh a t higher . The e ffect of variat i on in n is of minor importance . The sirnp ler model , assuming that n is eq ual to un i ty , res ults i n a safety fact o r wh i ch e rrs on the conse r vative si de .

Additional ( 01' al t e rnative l p a rameters can be ac commo ­dat e d in th e equat jons f or y q uite essily . An a d~

ditional multipl ieron the st r e ngth side would appear as an additi ona l pos itive term in both e xpressions in the l ognu rmal equst i on and in the danomin atur of the truncated No rmal equa ti on . An addit i on a l mu l tipl ie r orl tc,e load side vJOu ld ap pear as an additiona l posi­ti va term i n th e first express io n and an additional negat i va t e r m i l1 the seco nd express i or l of t h!" log­norma l eqLlsti on : in the trun cated Normal eq uation it would éJppear in th e numerator .

TABLE 4

Gl oba l safety factor y f r om Lo gnorma l di s trib utio n hy ­po th es i s

Co nst r uction cont r o 1

Unit s t r e ngth C n Spec i a l Norma l va r ia tio n u

CL = 0 . 15 CL = 0 . 22

Lov/e r 0 . 055 1 3 . 01 3 . 84 0 . 8 2 . 97 3 . 80 0 . 6 2 . 95 3 . 78

No rmal 0 . 08 1 3 . 12 3 . 96 0 . 8 3 . 04 3 . 88 0 . 6 2 . 99 3 . 82

TA5LE 5

Global safety fa ctor Y from Truncated Normal distri­~ut ion hypo thesis

Con struction Contro 1

Unit st re ng t h C n Special No rmal variation u

I CL = 0 . 12 CL = 0 . 16

Lm·,er 0 . 05 1 3 . 60 4 . 48 0 . 6 3 . 48 4 . 33 0 . 6 J . JG 4 . 19

No r mal 0 . 07 1 3 . 88 4 . 83 0 . 8 3 . 70 4 . 60 0 . 6 3 . 52 4 . 38

5 . PARTIAL SAFE TY FACTORS

The previous discussion relates to t~e detarmination of a single safoty factor which allows for the vari­ation of alI the pa r ameters in the dosjgn R~ualion and which is therefore termed lhe glo~al safety facLor . There is , however , a tenuency nO~'i to USE par­tial factars , each céltering fOI' one aspect of the design o The number entel ing into a des i gn fo r mulation i5 quite large according to some proposals that have Geen made , out these factors appea r to have been de­ouced empiri cally rather than s atistically .

The statistical app r oach to partia l sôfety factors is best illustrated by t he simpIe case whe r e the re are two factors , one reloting to s t r ength and one to stress . For 8xamp l B, in th9 systt=.:m IJlith .r? end 5 desc r ibed in Section 2 , the dEs ig n eq uati on wou I d ne

5Y 5

where YR takes account Df var i ations in R onl y , S being presumeej to 02 kno\.rJn exaclly , \.r/r,ar8BS "'fI'" tak.es account of variations :n S only , R ~eing oresÔmed to De known exactly . Since in practice the design equation rearranges to give ~ _ 5, so thélt the

YpYs oroduct is actua l ly emoloyed , it is difficult at first sight to see what advélntage this has over the calculation of a g l ooal safety f acta r .

Partial safety factors may have come to be used be ­CAuse they were estimated smp iri cally ra ther than s t ôtistica ll y . vlllen statist i cal methads a r e use d Y

R and YS can be ca lculat ed from No rma l, l ogno rmal ,

extreme vél lue and some o t he r dist ri butions , and dis tri but ions ",h ich would ca LJse t r ouble ,·!hen combined in 3 global facta r can be handled quite successfully . Here , then , partial safe ty f3ctors ha ve an advantage .

Howeve r , the difficulty lies in the combination of the pr obabil ities Df f ailure associated with the partial s afety factors . It appears sometimes to be implied in the l ite r ature that if two safety fac to r s Y and Y

S are mulUplied t oge th e r , the prubability ofRfailure

associated with the product is the pr oduct Df the part ial pr obabilities . Th is argument invokes the product law fo r thE probab ility cf two eve nt s happen ­ing together . The fallacy i5 tha t partial safety factors a r e not events . A ~articular application Df t~is fa Ilacious argument might be to take YR and Y both with prebabi litv 10- 3 to derive a fa c tor Y Y S

vlith ;Jro~an:li,tv 10 - 6 Yet the sar:m fi nal orot'~b11-

4 . b . 8- 5

i ty would by this argument be achie ved by multiply i ng Y

R with orobability 10-' by Y

S wit h probability 10 - 2

•

The fal lacy is alI the more serious be cause it re­sults in a n unde r-estimate of t he global safety fa cto r , the d isc repancy depending on th e distri butions assumed and t he coeff icients Df variation .

For lognormal distributions , partial safety facto rs can be estimated from equation (5) . Yq by putting C = O an o Y

S by putting C,,= O. If t 1'5 ta ken as_3 . ~

in each case , to co rresp~nd with orobability 10 3

anc ~~,e product YqYS

is ca l culated , it ::1ay !:lo sub -s itut ed in t he comp lete equation (5) vli th appropriate CR ~nd Cc to givo a new va l ue Df t . Statistical tables enaole t~e true orobebility associated wit h Y YS to be found . .9.

In typical coses

0 . 2 P

Ci? O . 2 Cs = O. 1 :J = 2 x 10- 5

CR

= 0 . 1 Cs = 0 . 2 p = 2 x 10- 5

In the worst cases the ~ robabil i ty is 20 times that calculatdd fI'OM :he rallacious a r gument . \h th fJormal distributions the discrecancJes m3y be even graater .

On the nther hand , if t he probabilities associated with . Ya and Ys are ~oth made equal to the final prob­ablllty requl r ed to be associated with the com~ l ete

desi~n situatiofl it can De shm"n that Y Y may considerably overestimate the g l obal sa~e~y facto r re­quirpej .

r,i3 pro~le~ was conside r ed by A. L. L. Baker' , who pro­pose~ the f91lowin g approximate r ule :

Y = Y'r + Y S - 1

H~ stated tnat ~rovided that th e pro~abili ties in­v~lved arD no t muc h 18sB than 10 - 5 and the coefficients Df variation are not greater th a n 0 . 4 the ruI e works reasonably we ll , giv i ng r esults us ua lly on the con ­se~vat i ve slde . A test Df results at p ro~abi lity

10 6 assuming the l og norrna l d i st ri !:lucion an d C not ex­ceedinf 0 . 2 also Bdve co ns e rvat i ve estimates , ~he

rraxirnum o\lsr-p.stimate beirlg a~out 'i3'% .

If truncated distr i buti ons are used to ca l cu lat e safety facto rs , fo r e xa mpl e the truncated No rmal di s­tribut i on as described o revious ly , no speci fic prob ­ebilities are involved and it is clear from the method of calculatioll tllat the r elationshio Y = YRYs holds . To calculate global sa f ety factors for a varlety Df condi tions cf both !? and S oar'ameters it i s necessary only to w~rk out pa rti a l safety factors for ~ and S and com~ine them by mult i plica ti on , pai r bv pa ir o

The possibil i ty Df s imilar treatment of safety fac tors derived from t he logno ro.al dist ri bution has been i n ­v~~Li gd ted . For e xamole , Y, calculat ed by t he fu ll equat i on at probability of fôilu r e 10- 6

, may be par­titioned into Y" ano yc~ ea:::;h '.-I1th the same failure orobabiliUes . n ot - 10 3 bu t calculable if des ired . However , the partitioning is va lid only f o r the par ticular pai r of caefficients C and C assumed . The _ va lU:S Df Yi? and Y S ,:orked out fO? a se~~es whe re C

R - C" - 0 . 05 , 0 .1 , O . 1~ , 0 . 2 , etc . , cannoc be com­

bIned pair by pair to give global safety factors for othe r combinatlons of Ca and C

S.

Consequent ly this pa rtiti on in g of globa l safety factors in to partial safety factor8 appears ta have IjttlD ad­vantage . There would seem t o be little point ifl cal ­culating pa r tial sB f ety factors stôttstjçally if glO­bal Factors con be calculated as easily .

4 . b . 8-6

However . sometimes i nsufficient infofmat i o n i s available on one aspect Df the problem and cal­culation Df a partial safety factor may be the best solution , leaving the other factor to be esti­mated empirically. It i s advisable then to err on the safe side by calculati ng the pa~tial safety factor at the probability required for the comp lete system , that is 10- 6 in the present case . The following example deals with the calculation Df the factor which a l lows for variation in the estimate Df load-bear i ng capacity , customar ily termed Y . Equations for the calculation based on the log~ormal distribution a nd o n the truncated Normal di str ib ution are as follows .

Lognorma l

log Ym

= t ! [n 2 log(1 +C 2)+10g(1+C 2)+ 10g(1 +C 2 )+ U L A

10g( 1+CljJ2)J

r +1 Lnl og(1+C 2)+10g (1+C 2)+ 10g(1 +C 2 )+

U L A 10g( 1 +CljJ2~

t 4 . 75 for probabi l ity 10- 6 .

Truncated Normal

Ta bles 6 and 7 give t he safe t y factors ca l cula t ed on these assumptions for the values Df the parameters listad in Section 4. It may be notad t hat tha va l ues tally reasonably well with th058 customarily chosen . The effec t Df variat i o n in n , tha expone nt in t he power law relating masonry streng t h to unit s t rength , i5 5mal l. The va l ues gi ve n by the alternative hypo ­t hesis do not differ greatly . The construction con ­trol has a greater effe ct than unit strength var i ­ation for a compara ble cha nge in coefficients Df variation.

Such a facto r , Y , would be us e d in practice with a m " load safety f act o r Y

f WhlCh need not have been ca l -

cu l ated statistically .

TABLE 6

Partial safety fac t or Y from Lognormal dis t r i bu t ion h~po t hes i s

COlls t ruct i on cont r ol

Unit s t rength C n Specia l Normal variatio n

u C

L = 0. 15 r = 0 . 22 ~L

Lower 0 . 055 1 2 . 25 3 . 0B 0 . 8 2 . 22 3 . 04 0 . 6 2 . "19 3 . 01

Normal 0 . 08 1 2 . 37 3 . 19 0 . 8 2 . 29 3 . 11 0 . 6 2 . 23 3 . 05

TABLE 7

Partial safety fac t or y from Trunca t ed Normal hypo­mt hesis

Construction control

Unit streng t h C n Special Norma 1 vari ation u

CL = 0 . 12 C

L = 0 . 16

Lower 0 . 05 1 2 . 22 2 . 77 0 . 8 2 . 15 2 . 68 0 . 6 2 . 08 2 . 59

~ormal 0 . 07 1 2 . 40 2 . 98 0 . 8 2 . 28 2 . 84 0 . 6 2 . 18 2 . 71

6 . CHARACTER I STIC STRENGTH

In t he previous sections no mentio n has been made Df "cha r acteristic strength ", a term th at is frequently found in the litera t ure . This omiss i on was i n­tent i onal, as the object was t o de duce safety fact ors as far as poss i ble from conventional sta t istical para­meters such as mean and coefficien t Df variat i o n . However , it i 5 desira b le to consider how charac t er­istic strength fits into t he p i cture .

The de f inition Df charac t er i stic strength is somewhat loose , but the ge neral idea is that Df a lower li mi t for strength be l ow wh ich only a small proportion (usually 5%) Df values i s likely to fa l I . It is usually appl i ed to the st r e ngth Df br i ckwork (see , for example ISO 2394 ) 5 , but has also besn applied to the strength Df the units themselv8s .

When applied to the uni t strengt h, the inte nt i on is presumably to give some extra degree Df protection , but t he practice may be criticised oll at least t hree grounds . First , the degree of protection afforded by a 5% probabili t y is deriso r y in the contex t Df the f ailu r e Df a bu il ding or other large structure , und t he combi nation Df pro babilities in a safety factor is not easy , as noted ear li er . Secon dly , the met hod Df ca l culating character i stic strength usually adopted , involving subtraction of 1 . 64 times the standard devia t ion from th e sample mean , is strictly applicable only whe n the standa r d deviation is accurately known , and not when it is estimated from a small sample. (A pape r by de Gra ve and Motteu 6 recommends a more com­plicated method for dea l ing with small samples whi ch entai l s ca lcu l ation Df a lower confidence limit for t he mean (by a n i ncorrec t formula) and calculat i on from this limit of a characteristic st r ength by sub­tract i on Df 1 . 64 times the s t andard deviatioll . They admit t hat the statistical r easoning is no t rigorous . )

Thirdly , the important sou r ce Df variation that must be allowed for is no t the variation of individual uni t s from t he mea n, but variation be t ween consign­ments Df uni ts; lower strengths in individual units can be tolerated , whereas lower mean values may cause t rouble . I t i5 bett e r t o allow fo r this by using the guaranteed strength quoted by the manufacturer to­gethe r wi t h information f rom his q ual i ty control system .

Hence t he use Df a charact eri sti c strength at this stage does not fi t easily i nto th e pattern outlined in the previous sectio ns for t he calculation Df safety factors .

The cha r acterist i c sLrength of brickwo rk may , i n t heory , be derived in a similar way from means and stanuard deviat i ons of tests on walls , but in pr ac­tice this estimate is unlikely to be accu rate because of the smallness of tne test sample . Hence althoug h IS O 2394 definas cha racteris ti c st rength in this way , it indicates that the nominal values given in stan ­dard s , codes of practice or othe r r eg ulations may be t aken as the characteristic values provlded that they offer an equivalent gua r antee .

It may be doubted whether the des igne r is in a posi tion to know 'fhat guara ntee i s given by tabu l ar values in standards and codes , dS thei r origins are usually veiled in obscurity . Published work on the relationship between wall strength and bri ck strength , for example the work undartaken by B . Ceram . ~ . A . on wi r ecut and pressed bricks 3

, 7 , shows that the r esults obtained when wall strength is plotted against brick strength falI wit hi n a fairly linea r band rath er than on or very close to a st ra ight l ine or curva . The obvious coursa has been fo ll o'.ved '.v llen choosin g code values of taking a fairly linear low er bound t o the experimental values [all owing for any s lendernes s factors , etc . as necessa r yl . Superficially this con­flicts with the use Df mean values in the sect i ons concerned with :he dRrivation Df safoty facto r s . However , the situat:on here 1s that much of the va ri­ation within the band of wall stre ngth results for a given brick stre ngth is not random but systemat i c . In other words , bricks that give high wall strengths relative to others are likely to do 50 cO~5istent ly

and vice versa . Hence it is advisable to a ll ow the poorer performers to set the sta nda r d . The only ad­verse result is that buildings constructed Df b ricks t hat give relatively high wall st r engt hs may be made unnecessarily strong . This situatio n can be recti­fied by using wall test results as a basis for des i gn o

The va r iation in factor L, allowed for i n Section 4 , is that due to construction methods , not to variation in the test r esults which fo~m the basis of the code values .

Hence the use of cha rac te ri s t ic strengths for bric k­work , in this restricted sense , is not inc on sis t ent with the previous sections in which the calculation of safety {actors was considered . The pictu r e be ­comes somewhat blurre d as far as tha assessment Df probability of failu r e is concerned , but provided that experimental data are adequate and the co ­efficients of va riation a r e not greate r than those assumed , the probab ility of failure shou l d not ex­ceed that aimed at o

7. CONCLUSIONS

1 . Calculation of safety factors from stat i st i cal principIes using models of varyin g deg r ees of complexity is p05s1ble and is straightforwar o u~der a restricted r ange of condltions .

2 . Provided that the model i s pu r e l y addit i ve , Normel, lognormal, and truncated No rmal dis t ri ­butions gi ve little trouble .

3 . Othe r models , i nvolving products , give trouble with Normal distributio ns , but l ogno rmal and truncated No rma l distributions can still give st raightforwa rd answe rs .

4 . The No r mal distribution is not likely to give practically useful safet y fact ors unl ess very l ow coefficients of varia tion can be assumed .

5. The most generally useful distributions are the lognormal and truncated No rmal ; safety facto rs so calcu lated agree quit e we ll with values esti ­mated by other msans .

4 . b . 8- 7

6 . The explicit calculation of characteri st i c strength should not be necessa ry if statistically calculated safet y factors are emp l oyed .

ACKNOWLEOGEMENTS

The auth or wi shes to thank the other members of the Britis h Standards Institution Partial Safety Fact or Sub-Corrmittee for helpful discussions and his col l eague Mr G, N. Vaughan f or computational assistance .

Thi s paper is published by per mission of Mr A. Dinsdale , Oirect or of Research , British Ceramic Re­search Association .

REFERE"jCES

1 . FREUOEN THA L, A. M., Safety and the Probability Df St ructura l Fa il ure , Proc . Am .Soc . CE , BD [Separate No . 46 Bl , 1954 .

2 . MACCHI , G. , Safety Considerations fo r a Li mi t­State Oesi~n of Brick ~asonry . Proc . 2nd Inter­national Bri ck Masonry Conference , Ed . H. W.H. West and K.H. Speed , Stoke-on-Trent , S . Ceram . R. A. 1971 , 229 .

3 . WEST , H. W.H., HDDGKINSON , H. R. , SEECH , O. G., and DAVEr~PORT , S . T. E., The Performance of lolalls built Df Wirecut Bricks with and without Pe rforati ons . Trans . Brit . Ceram . Soc . 57 , 435 , 195B .

BAKER , A. L. L., quoted by ROSENBLEUTH , E., and ESTEVA , L. , Reliability Basis for some Mexican Codes . Ame rican Concrete I nstitute Publication SP-31 , 1972 .

5 . INTERNATIClr,,!\,L ORGANIZATI::Ji,j FOR STANDARDIZATIDN , Genera l Pr i ncip I es fo r the Verif i cat ion of Safety Df Structu r es , I SO 2394 : 1973 .

5 . DE GRAVE, A. , and MOTTEU , H. , Test ing and Ca l ­culation Df Masonry . Proc . 2nd International Brick Maso~ry Co nfe r e nc e , Stoke-on-Trent , S . Ceram . R. A. 19 71 , 266 .

7 . ' .. IEST , H. vI. H. , HOOGKINSOfJ, H. R., \'IEBB , \~ . F. and BEECH , D. G., The Comp r essive Strength of Walls Built of Frogged Bricks . B. Ceram . R. A. Tech . Note 194 , 1972 .

4 . b . 8 - 8

r i g .

"'I 4 -

MA CC HI

2 -

I 0.05--------~1------~-;-------0·1 0· 15

2

MACCH I

C A = o

Cs = 0·1 5

b~0~5-------~-------~~-------0·1 0· 15

c" Fig . 3

O·l

7

-- Logno r mal

--- T runcot qd

/

/ /

/ ..,.. ./ / 0·1

./ /" / ..,..

/' /' /' /'

/' /'

-------

Fig . 4

c..." 6, /0·15

,r CA ~ o -- Lo gnormal

/

Cs = 0·15

/

Truncated / / 0 ·1

/

/" /

/' /

/" /

/" .,-

1-./

/"

/'

./

/0·05

/' /

015

/' /0·0

....... ~O l ...... ..... ..... ./ 005

/" /" - 0·0

-- - -----

I 0{j0~5---------~-------~----0· 1 0·1 5

----'----0 2

C rI

Fig . 5