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Appendix 2 statistical assessment of test results
66
Appendix 2 Statistical Assessment of Test Results CONTENTS 1. Scope . . . . . . . . . . . . . . . . . . . . . . . . . 2. Object of Testing . . . . . . . . . . . . . . . . 3. Statistical Aspects within Assessment Method 1 . . . . . . . . . . . . . . . . . . . . . . 4. Statistical Aspects within Assessment Method 2 . . . . . . . . . . . . . . . . . . . . . . 5. Tests Accompanied by Calculation Models . . . . . . . . . . . . . . . . . . . . . . . . 6. References . . . . . . . . . . . . . . . . . . . . . 7. List of Symbols . . . . . . . . . . . . . . . . . Supplements: Probabilities Associated with the Test Quantity . . . . . . . . . . . .
2. OBJECT OF TESTING 66 66 69 70 71 72 72 72
1. SCOPE This appendix is concerned with particular statistical aspects of the experimentally determined fire resistance time. These aspects are of concern, - - i f within assessment m e t h o d 1, there is an option of either an experimental or analytical determination; both should be compatible by rendering the same degree of reliability. This is achieved by appropriate specification of safety factors for the analytical procedure. - - i f within assessment m e t h o d 2, safety elements are derived for experimentally evaluated components, which may or may not be classified. Statistical aspects are likewise of concern if different testing procedures are compared, with respect to different numbers of test results, different calculation rules for nominal values, etc. They are of some significance if experimental evidence is sought for calculation models, thus providing an estimate for correction factors or functions.
2.1. Definitions
Within test procedures, the fire resistance time of components is defined as the minim u m length of time during which components comply with given performance criteria under specified testing conditions. If, for example, these criteria refer to the load bearing capacity of a component, then this is the length of time during which components sustain a specified load s k. Since the resistance capacity r changes with increasing temperature and hence, with the proceeding duration t of the test, the c o m p o n e n t complies with the given criteria as long as r ( t ) > Sk and fails if r ( t ) - - Sk = 0
~ tf
(la)
presuming a temperature-invariant specification of loading (Sk(t) = Sk). The length of time until failure occurs (or more usually -- until infringement of any performance criteria), identifies the fire resistance time t~ of the c o m p o n e n t which obviously depends on the level sk of loading {Fig. la). The level of mechanical loading in testing may either - - b e prescribed and, thus, belongs to the package of testing conditions, or
kr.s r(o)
Fig. l a . D e f i n i t i o n o f fire resistance t i m e tf in testing for t e m p e r a t u r e - i n v a r i a n t loading.
67
--be a testing parameter, thus rendering the fire resistance time as a function of the load level: tf(s). It is recognized that the herewith defined fire resistance time is exclusively related to the mechanical and thermal conditions of testing. Certain conditions may be standardized, e.g. according to ISO 834 or according to other standards developed for particular applications. A thus established standard fire resistance time will nevertheless depend on residual conditions prevailing during the individual test and/or at the particular furnace.
Another source of variations of the experimental fire resistance time are unintentional variations of the testing conditions referring to a single furnace, and even more so, when referring to different furnaces (see section 2.4). It is understood that the quantities R(o), R(t) and T F a r e deterministic quantities for a particular c o m p o n e n t subject to particular testing conditions. The notion of randomness arises herein when predicting the experimental fire resistance time of a population of components from particular test results.
2.3. Evaluation of test results 2.2. Variability of test quantity The original resistance capacity r(o), i.e. the resistance prior to testing, of components of the same design is subject to unintentional -- random -- variations. Components manufactured in the workshop of the laboratory (for a hypothetical large test series) may display a smaller variation than components taken at random from the production process or at site. Eventually, their average resistance may differ as well. At any rate, R(o) and, consequently, the resistance R(t) after a certain duration of test, must be regarded as random quantities. Eventually, the alteration of the resistance may be influenced by structural or material properties, not (or only insignificantly) affecting R(o); hence, R(t) may display a greater variation than the original resistance capacity (see Fig. l b ) . From these considerations it follows that the length of time until failure occurs, i.e. until
R(t) --Sk = 0
From the test results or observations t = tl, t2, ... t,* of a sample of n components, a sample mean, variance and coefficient of variation (c.o.v.) V are c o m p u t e d according to ref. [1 ] 1
n
= -- ~, ti n
(2a)
1 tl
s 2 - n--ll ~ ( t i - - ~ ) 2 ;
V =s/t
(2b)
If the fire resistance time is assumed to be log-normally distributed, it is convenient to relate the sample mean and variance to the respective parameters of a normal distribution by appropriate transformation t'i = In ti, rendering: ~,
1~
,
= __ t i n 1
(lb)
>TF
is a random quantity as well, the variation of which may also depend on the level of loading. A temperature-dependent loading m a y additionally affect the variation.
s '2 - - n--ll
(t'i -- t ' ) and s '2 = ln(1 + V 2)
The following presentation is written in terms of normally distributed variables. By the aforementioned transformation all relations can be rewritten for log-normally distributed variables. Evaluation of test results generally involves calculation of a nominal value regarded as relevant to the fire resistance time of the components in the population in question.
tN = h(tl, t2 .... , t,)
fTF(t)
B
t
Fig. l b . R a n d o m realization o f fire resistance T F for t e m p e r a t u r e - i n v a r i a n t loading.
* I n d e x f in tf resp. T F o m i t t e d if self-evident; n o t e t h a t capital letters (R, S, T) r e p r e s e n t r a n d o m variables and small letters (r, s, t) r e p r e s e n t d e t e r m i n i s t i c quantities.
68 for instance tN = [
(3a)
tN = m i n ( t l , t2 . . . . . t , )
(3b)
t N = t -- At
(3c)
t s = [ - - ho
or
tN = t - - ks
(3d)
Obviously, a nominal value according to eqn. (3d) utilizes the available information on the mean and the variance as well. This t y p e of nominal value is recommended, provided the factor k is specified appropriately.
2.4. Uncertainties and compatibility of test results Statistical uncertainties: [ and s 2 are estimators for the actual distribution parameters (mean value 12 and variance 02) of the fire resistance time of a hypothetically infinite sample (n ~ oo) of equally designed components. If test series with only small sample sizes were repeated several times, then t as well as s 2 would vary between the different series. Hence, mean value and variance, estimated from small sample sizes, are accompanied by "statistical uncertainties" [2, 3 ].
Lack o f representativeness: If the components used in testing cannot be regarded as a representative sample with respect to the standard of quality to be expected at site, then consideration of additional uncertainties and possibly corrections would be necessary. However, assessment of these uncertainties is more or less confined to a subjective estimation -- at least as concerns a pure experimental evaluation of the fire resistance w i t h o u t support by accompanying calculation models. In the following it is, therefore, assumed t h a t the sample of components -regardless of sampling or manufacturing procedure -- can be regarded as representative of the population for which experimental evidence is sought. Deviations in testing conditions: Generally, deviations in testing conditions, especially when comparing test results from different furnaces, are regarded as a more severe impairment of experimental evidence than a lack of representativeness in the sample
quality. Concerning unavoidable variations of the conditions of testing at a particular furnace, it may be presumed t h a t these variations are inherent in the definition of the test quantity; consequently, t h e y are not to be extricated from the test results. As concerns variations of the conditions of testing at different furnaces, it may be argued that very large deviations suggest that actually different (populations of) test quantites are observed. Specification of a reference furnace and evaluation of correction factors by calibration seem to be the only means for providing compatibility of test results from different furnaces at present. Possibly, specification of thermal testing conditions by control of heat flux may highlight smaller residual variations [4 ].
Choice of experimental model: The experimental model comprising the thermal and mechanical testing conditions may be specified with the intention to either --grade components for certain reference conditions, or - - m o d e l natural conditions as closely as possible (either for grading or for using the test results directly). Often a mixed approach is employed: e.g. thermal conditions are specified by a reference or standard t e m p e r a t u r e - t i m e curve and mechanical conditions are chosen to comply with the actual conditions expected in the structure. Mixed approaches, representing transitional stages from a pure grading system to a rational fire design, require a deliberate interpretation of the requirements to be fulfilled, in order to avoid unintentional increases or decreases in levels of reliability and economy. Model uncertainties: Recognizing that the experimental fire resistance time is defined with respect to a package of specified testing conditions, i.e. with respect to a specified experimental model, uncertainties arising from deviating conditions in the actual structure (compared with those in testing) are actually of no concern when evaluating the experimental fire resistance time. This is fairly obvious for conditions which can be established irrespective of the particular type of c o m p o n e n t (e.g. thermal conditions by a standard t e m p e r a t u r e - t i m e curve, level of
69
loading -- if not a testing parameter). As for conditions which have to be fixed individually for the particular component, the basic consideration should be, to provide comparability with tests on other components (and not to comply with reality, although evidently comparability will often be by reference to likewise probable conditions in actual structures). From this reasoning, it follows that it would be appropriate to consider uncertainties arising from insufficiently ensured comparability.
3. S T A T I S T I C A L A S P E C T S WITHIN A S S E S S M E N T METHOD 1
3.1. Calibration Calibration involves an analysis of an existing design procedure with respect to the inherently provided degree of reliability. Calibration is of concern if modifications (or alternatives) to existing design procedures are considered which should render the same degree of reliability. Within the assessment m e t h o d 1, this may be the case - - i f analytical determination o f the fire resistance time is envisaged which should be compatible with the experimental determination; - - i f changes in the testing program are envisaged in terms of sample size, calculation rule for the nominal value, or if different testing programs (different with respect to these statistical aspects) are compared; and - - i f different classifications or procedures without classification are compared. As code requirements presumably are based on specific testing programs and testing conditions, statistical aspects may be of some significance also for comparing different national code requirements. More vital differences resulting from different testing conditions (including loading) are primarily a problem of physical modelling with minor statistical aspects. According to Fig. 1.1 in Section 1.1 of the general introduction, adequacy of the design - - within assessment m e t h o d 1 -- is presumed, if a structural member has a fire resistance t~ which meets the required time of sustained fire exposure tfd: tf/> tfd
(4a)
Referring to the traditional classification system, tf corresponds to the fire resistance of a graded structural member, and ted represents the lower limit tu of the required fire resistance class (see eqn. (5a)). If a c o m p o n e n t falls to meet the requirement, i.e. if the random fire resistance T F is shorter than the class limit tu = tfd TF - - tu ~< 0
(4b)
then this event does not necessarily cause any physical consequences. Strictly speaking, eqn. (4b) does n o t correspond to a limit state as defined in Section 6.1 of Part A. However, the equivalent formulation according to eqn. ( l b ) , R ( t u ) - Sk ~< 0
(4C)
suggests an experimental limit state (with a " b o u n d e d reference duration"), which may -by agreement -- be regarded as relevant. The degree of reliability associated with this experimental limit state corresponds to the probability (1 - - P u ) that the fire resistance time as a random quantity exceeds the class limit tu. Hence, for calibration purposes, we are concerned with the conditional probability Pu = P(TF • t~lgraded in class X(t~))
(4d)
requiring an evaluation of the distribution of the random quantity TF, i.e. the fire resistance of specifically graded members.
3.2. Distribution of the fire resistance time The distribution of T F , i.e. the probability to observe a fire resistance smaller than a certain value is governed by: - - t h e sample size (n) for determining the fire resistance of a certain population; -- the calculation rule for the nominal value tN decisive for classification tu ~< tN < tv
* class X(tu)
(5a)
-- any prior information available. Moreover, agreement is necessary on the population considered as representative for calibration. Reference can be made either --to the entity of populations produced, tested and classified as X(tu), or --to populations meeting minimum class requirements with t s = tu. In view of the increasing tendency to adapt production to minimum requirements, populations meeting minimum requirements could
70 be accepted as representative. In this case, the calibrated degree of reliability corresponds to the probability that the fire resistance time (under testing conditions) exceeds the nominal value tN = tu ; thus
As outlined in Parts A and B, practical verification in the mechanical strength domain is by comparing
Pu = PN = P(TF <, tN[graded in class X(t~))
where R k and Sk are appropriately defined characteristic values. The partial safety factors ~/m and 7f are derived from the probability
(55) As an example: if experimental evidence is only by a single test result with tN = tt, then P~ = PN = 0.50; if evidence is by the minimum of two test results with t N = m i n ( t l , t:), then P~ = PN = 0.33. If reference is made to the entity of populations classified as X(tu), then the frequency of nominal values tN (or mean values t) and the associated variances s 2 per class have to be considered in the evaluation of the distribution of T F. The resulting degree of reliability will consequently be higher. As an example: if evidence is by the minim u m of two test results with tN = min(t~, t2), then p~ ~ 0.10 for equally distributed t i within class and prior knowledge about the variance. (For the evaluation of probabilities see also the supplements to this appendix.) 3.3. Compatibility between analytical and experimental results Recognizing the character of a gradeverification within assessment m e t h o d 1, an analytical determination should, basically, be directed at reproducing test results by modelling the expected performance o f the c o m p o n e n t under testing conditions. In the analytical determination o f the fire resistance, it is assumed generally that all basic variables X in the calculation model R ( t ) = gR(t, X_, Zrd )
(6)
are known in terms of their characteristic values (and/or distribution functions). A model uncertainty may have to be considered which reflects the uncertainty in the calculation model as a correct representation of the expected performance of the c o m p o n e n t under testing conditions (at the time t = tu). This m a y be done by introducing an additional basic variable Xrd. The calculation model may also incorporate correction factors or functions. They are often combined with the model uncertainty (cf. Xc in Section 5).
Rk(tu)/Tm ~ Sk~/f
P(R(tu) < S) = Pu
(7a)
(75)
referring to the previous considerations on the appropriate population for calibration. If, according to the prescribed testing conditions, the level of loading is fixed at Sk (and is temperature-invariant), then analytical evidence requires verification for the same fixed loading and eqn. (7a) is reduced to Rk(tu)/'Ym <~ Sk
(7C)
In this case 3'm can be simply evaluated from (see also Appendix 1) : 1 --kV 7m-
1 --/JuV
where flu ....... CN I(Pu)
(7d)
For a characteristic value R k corresponding to, e.g. the 5% fractile (k = 1.64), a c.o.v, of V = 0.10 and for probabilities Pu = PN = 0.5 (one test) . . . . . 7m = 0.84 Pu = PN = 0.33 (rain. of two tests) ---~ "~m = 0.87 is obtained. Hence, in analytical analysis the mechanical strength should be multiplied by, for example, 1/0.84 = 1.2 in order to be consistent with experimental verification [ 5 ]. If the level of loading is accepted as a testing parameter, then consideration of the random load (-effect) acting in an accidental situation is appropriate. The possible decrease of ~/m in this case is generally negligible, and it may suffice to determine '~m likewise from eqn. (7d). The load factors are specified in accordance with Appendix 1
4. STATISTICAL ASPECTS WITHIN ASSESSMENT METHOD 2 The necessary fire resistance time derived from an equivalent fire exposure t e can be regarded as a quantitative requirement based
71 on an actual -- if only idealized - limit state
involving the corresponding random variables:
TF-TE-O
(Sa)
or R(te) -- s k
is available but not sufficiently accepted. The relevant basic variables _X are fairly well
established with respect to their characteristic values (and/0r distribution functions). There may be some uncertainties with respect to
= 0
(8b)
and equivalent formulations for a random and/or temperature
It is assumed that a calculation model R ( t ) = gft(t, X )
their temperature dependency, some thermal properties, etc. However, the t y p e of failure, i.e. the failure m o d e of the c o m p o n e n t subject to testing conditions, is well known. Account of this insufficiency in the calculation model may be taken by a correction factor Xc, which is assumed to be a random quantity [ 11 ]. (9)
R ( t ) = gR(t, X_, Xc)
(Ideally, X c and Xrd in eqn. (6) are identical.} With a calculation model available, tests may be basically directed at evaluating the variable X¢. If all variables _X (strength properties, geometrical data, ...) can be numerically determined for the particular c o m p o n e n t subject to testing, then gk(t, X)
- - Sk = 0
~ if
renders a particular fire resistance time t~l. The corresponding test result tf, will differ from the calculated value. Thus, a correction factor x¢1 can be determined. Repetition of the procedure will render a second factor Xc2 and so on. According to eqn. (2) a sample mean (Xc) and variance (S~c) are determined. Without a calculation model, the mean and variance of the fire resistance time are associated with statistical uncertainties (Section 2.4). Now these uncertainties only affect the correction factor Xc. The theoretical distribution function of Xc can be determined according to the supplements. Generally, only a nominal (or characteristic value or design value) is required for R ( t ) o r T F and, thus for X¢, which is associated with a certain probability. As the statistics of the basic variables _X are assumed to be known with reference to the population in question, the requirement on the representativeness of a sample may be less severe or even superfluous. Eventually, as insufficient knowledge, i.e. uncertainty, is confined to the correction factor, the effort employed in estimating this factor (sample size, sophistication in statistical modelling) may depend on the degree of confidence associated with the calculation model.
72
Standardized t
6. REFERENCES 1 A. Hald, Statistical Theory in Engineering Application, Wiley, New York, 1952. 2 G. Box and C. Tiao, Bayesian Inference in Statis. tical Analysis, Addison-Wesley, 1973. 3 J. R. Benjamin and C. A. Cornell, Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, New York, 1970. 4 J. Keough, CIB report. Van Keulen, CIB report. 5 O. Petterson and J. Witteveen, On the fire resistance of structural steel elements derived from standard fire tests or by calculation, Fire Safety J., 2 (1979/80) 73. 6 M. Kersken-Bradley, Einfluss von Annahmekennlinien auf die ZuverIdssigkeit yon Bauteilen beiBrandbeanspruchung, Research report, Institut fiir Bautechnik, 1979. 7 R. Rackwitz, Theoretische Grundlagen fiir die Bestimmung des Bemessungswertes yon Bauteilwidersth'nden aus Versuchen, Berichte zur Zuverl~ssigkeitstheorie der Bauwerke, Technische Universit/it, Miinchen, Heft 38, 1979. 8 JCSS, Basic Notes on Resistances, Draft R-01, 1981. 9 M. Zelen and N. C. Severo, in Abramowitz (ed), Handbook of Mathematical Functions, Dover, New York, 1970. 10 JCSS, General Principles on Reliability for Structures or Revision of IS 2394, draft, 1982. 11 W. Struck, Die traditionelle Praxis der experimentellen Bauteilpriifung aus der Sicht des probabilistischen Sicherheitskonzepts, Bautechnik, 2 (1981).
7. LIST OF SYMBOLS (INCLUDING THE SUPPLEMENTS TO THIS APPENDIX) r
R Sk
S tf TF
&(x) Fx(X) P O2 82
V n
ON(.)
Resistance c a p a c i t y Resistance c a p a c i t y as r a n d o m variable Level o f m e c h a n i c a l load in testing Mechanical load as r a n d o m variable Fire resistance t i m e Fire resistance time as r a n d o m variable Distribution density of a random variable X Distribution function of a random variable X Mean value Variance E s t i m a t e f o r m e a n value E s t i m a t e f o r variance C o e f f i c i e n t o f variation (c.o.v.) N u m b e r o f test results Standardized normal distribution f u n c t i o n (see Table 1 )
t u ~ tv tfd
k=rkJ
with v = n - - 1 degrees o f f r e e d o m (see Table 2) L o w e r a n d u p p e r class limits R e q u i r e d t i m e o f sustained fire exposure F a c t o r for calculating n o m i n a l or characteristic values (t N = [ - ks)
SUPPLEMENTS: PROBABILITIES ASSOCIATED WITH THE TEST QUANTITY
$1 D i s t r i b u t i o n o f the fire resistance t i m e Given a certain p o p u l a t i o n with a m e a n p and variance 02 o f the fire resistance t i m e T, t h e n t h e p r o b a b i l i t y t h a t a fire resistance t i m e can be observed w h i c h is smaller t h a n a certain value t ( P ( T <~ t)), is identified b y t h e d i s t r i b u t i o n f u n c t i o n Fw(t ) or m o r e precisely: Fw(t[p, 0). In o t h e r c o n t e x t s t h e c o n d i t i o n (.[#, o) is o f t e n o m i t t e d as self-evident. i
fT
tlIT~, .)FT~
P( T <:_
~"
Fig. 2a. Density function fT and distribution function F w for a given mean value g and variance o 2. F o r a n o r m a l d i s t r i b u t i o n f u n c t i o n , the respective probabilities are o b t a i n e d f r o m
F r ( t r p , o) = P ( T <~ t i p , o) = ON
(-
($1)
w h e r e CS d e n o t e s the s t a n d a r d i z e d (0, 1) n o r m a l d i s t r i b u t i o n f u n c t i o n with values listed in Table 1. If o n l y the variance o 2 within a certain p o p u l a t i o n can be assumed t o be k n o w n , whereas t h e m e a n p is e s t i m a t e d b y [, t h e n p r o b a b i l i t y s t a t e m e n t s are c o n d i t i o n e d b y t h e estimate [ and e v i d e n t l y b y o.
73 TABLE 1 Standard normal distribution function distribution function: ~(z)-- - .
~
(1)
exp --
u 2 du = p
X/27rl exp ( -12) z distribution density: ~(z) = ~
-z
0
~.
0
Z
Z
~
0
z
p = ¢(--z)
p = ¢(z)
~(z)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.5 4.0 4.5 5.0
0.500000 0.460173 0.420740 0.382089 0.344578 0.308535 0.274253 0.241964 0.211855 0.184060 0.158655 0.135666 0.115070 0.096805 0.080757 0.066807 0.054799 0.044565 0.035931 0.028716 0.022750 0.017864 0.013903 0.010724 0.008197 0.006209 0.004661 0.003467 0.002505 0.001866 0.001350 0.000232 0.000032 0.000004 0.0000003
0.500000 0.539827 0.579260 0.617911 0.655422 0.691465 0.725747 0.758036 0.788145 0.815940 0.841345 0.864334 0.884930 0.903195 0.919243 0.933193 0.945201 0.955435 0.964069 0.971284 0.977250 0.982136 0.986097 0.989276 0.991803 0.993791 0.995339 0.996533 0.997495 0.998134 0.998650 0.999768 0.999968 0.999996 0.9999997
0.398942 0.396952 0.391043 0.381388 0.368270 0.352065 0.333225 0.312254 0.289692 0.266085 0.241971 0.217852 0.194186 0.171369 0.149727 0.129518 0.110921 0.094049 0.078950 0.065616 0.053991 0.043984 0.035475 0.028327 0.022395 0.017528 0.013583 0.010421 0.007915 0.005952 0.004432 0.000873 0.000134 0.000016 0.0000015
The corresponding distribution function [ 3 , 7, 8 ] , c o n s i d e r i n g all p o s s i b l e e s t i m a t e s [ {and t h e i r a p p r o p r i a t e w e i g h t i n g ) , is F w ( t ] t , o) = P ( T <~ tit, o) = CN - - ~ (S2a)
Z
Z
and identifies an increase of the variance from 02 t o o2(n + 1)/n d u e t o t h e u n c e r t a i n t y i n the m e a n value, n d e n o t e s the n u m b e r of test r e s u l t s a v a i l a b l e f o r t h e e s t i m a t e a n d CN is d e f i n e d as i n e q n . ( S 1 ) . If both distribution parameters p and a 2 are e s t i m a t e d b y [ a n d s 2 o n t h e basis o f n t e s t
74 fT
Dt
ll-!
F-r
FT
] ~
i,ll,,
i t
t
Fig. 2b. D e n s i t y f u n c t i o n s fT a n d d i s t r i b u t i o n funct i o n s F T for d i f f e r e n t e s t i m a t e s t w i t h variance 02 ( t h i n lines) a n d for all possible e s t i m a t e s c o m b i n e d - w i t h variance a2(n + 1 )In ( t h i c k lines).
Fig. 2c. D e n s i t y f u n c t i o n fT a n d d i s t r i b u t i o n f u n c t i o n F w for d i f f e r e n t e s t i m a t e s t a n d s ( t h i n lines) a n d for all possible c o m b i n e d e s t i m a t e s ( t h i c k lines).
results, then probability statements are conditioned by the estimates t and s. The corresponding distribution function [ 3, 7, 8 ], considering all possible estimates and s (and their appropriate weighting), is
the uncertainty in the variance itself changes the t y p e of function from a normal distribution to a so-called "central t-distribution". qxv in eqn. (S2b) denotes the standardized t-distribution function with v = n - 1 degrees of freedom; values are listed in Table 2. With increasing sample size, i.e. as v -~ ~o and, thus decreasing statistical uncertainty, the central t-distribution converges towards the presumed normal distribution. Incorporation of appropriate prior information will
/
F w ( t l [ , s ) : P ( T < . t l t , s)
=
q%(ts/-
(S2b) and again identifies an increase of the (now estimated) variance b y sZ(n + 1)/n, whereas TABLE 2 Inverse c e n t r a l t - d i s t r i b u t i o n u = '#v~(p)
V p
0.25
0.10
0.05
0.025
0.01
0.005
0.0025
0.001
0.0005
1 2 3 4 5
1.000 0.816 0.765 0.741 0.727
3.078 1.886 1.638 1.533 1.476
6.314 2.920 2.353 2.132 2.015
12.706 4.303 3.182 2.776 2.571
31.821 6.965 4.541 3.747 3.365
63.657 9.925 5.841 4.604 4.032
127.321 14.089 7.453 5.598 4.773
318.309 22.327 10.214 7.173 5.893
636.619 31.598 12.924 8.610 6.869
6 7 8 9 10
0.718 0.711 0.706 0.703 0.700
1.440 1.415 1.397 1.383 1.372
1.943 1.895 1.860 1.833 1.812
2.447 2.365 2.306 2.262 2.228
3.143 2.998 2.896 2.821 2.764
3.707 3.499 3.355 3.250 3.169
4.317 4.029 3.833 3.690 3.581
5.208 4.785 4.501 4.297 4.144
5.959 5.408 5.041 4.781 4.587
11 12 13 14 15
0.697 0.695 0.694 0.692 0.691
1.363 1.358 1.350 1.345 1.341
1.796 1.782 1.771 1.761 1.753
2.201 2.179 2.160 2.145 2.131
2.718 2.681 2.650 2.624 2.602
3.106 3.055 3.012 2.977 2.947
3.497 3.428 3.372 3.326 3.286
4.025 3.930 3.852 3.787 3.733
4.437 4.318 4.221 4.140 4.073
75
reduce the statistical uncertainty associated with /~ and o. Prior information m a y be
expressed according to eqn. (S4b) with k = V ~ / 2 ~ 0.7*:
available concerning, e.g. the variance or the c.o.v., allowing assumptions on upper bounds or informative prior distributions. Prior information can also be modelled by considering additional fictitious samples [7] or by a fictitious increase of the sample size. However, rational criteria for the assessment of prior information are generally n o t available and, thus, assessment is basically by engineering judgement.
tN = min(t,, t2) = t - 0.7s
The probability associated with this nominal value excluding any prior information amounts to (see eqn. (S4b) and Table 2) PN
t N = t--
kO
or
tN =
~1(--0.7~)
t N = t --
[--
kS
(S3a, S3b)
is, thus, associated with the probability (see eqn. $2) PN
=
~ 0.33
It should be noted that, if it can be assumed that the variance is known, a nominal value of e.g.
$2 Nominal values The probability to observe a fire resistance time smaller than an arbitrary nominal value tN is obtained from the distribution functions as defined in the previous section. A nominal value, e.g. of the type
(S5b)
0.70
likewise based on two test results -- is associated with a smaller probability -
-
PN = CN(--0.7X/-~) ~ 0.29 Vice versa, if the nominal value of the fire resistance is required to correspond to a preselected probability (p-fractile), then the factor k in eqn. ($4) depends on the number of test results and should be introduced as
=
or
pN = ~,,(--k ~ n ~ l )
(S4b)
If only a single test result is available and is employed as nominal value, (S5a)
tN = tl = t
this is equivalent to operating with an estimate for the mean value. The probability that the fire resistance in subsequent tests (or ideally within the population) is smaller than this nominal value is PN = 0.5
(n = 1)
Since the t-distribution degenerates for v = n - 1 = 0, inference from a single test result for u n k n o w n variance is strictly speaking n o t possible. The probability PN can be derived only with reference to the identity of mean value and expected value for normal distributions. The probability for any other type of nominal value (e.g. tN = t l - At) remains indeterminable. In the case of two test results, the minimum of both is often specified as the nominal value which, however, can equivalently be
(S6a, S6b) thus, rendering nominal values which correspond to a constant p-fraetile for an arbitrary number of test results. The notion of equal probability PN (regardless of available information) presumes that costs are independent of the a m o u n t of inherent uncertainties. It may, however, prove to be economical to have reliability degrees depending on the available information.
$3 Classification The nominal value t N may be used for attributing components to fire resistance classes tu ~< tN < tv
" class
X(tu)
($7)
where t u and t v represent the lower and upper limits of the respective class (e.g. 30 and 60 min). The probability p~ of observing a fire resistance time smaller than the lower limit t~ within the population of components *t N = min(tl,
t2) = (t I + t2)/2 -- It 1 --
= [-
~ ( t l -- t2)2/4
t2[/2
76 in question is again obtained from the distribution functions defined in section $1 for t = t , ; e.g. for a nominal value according to eqn. (S3a): P , = CS
•
a
(S8a)
whereby Pu ~< PNOnly populations of components which just slipped into class X ( t , ) with t , = tN will fall short of the lower limit tu by the probability P , = PN, e.g. according to eqn. ($4). If we are concerned with the statistical properties of components of a certain fire resistance class X ( t , ) , all populations of components attributed to class X ( t , ) have to be considered, i.e. all populations, the nominal value of which lies between aforementioned limits. Hence, general probability statements on components of fire resistance class X ( t , ) require consideration of the distribution of the overall supply of components by design and production. This information can be derived by investigating the frequency for, e.g. nominal values t s ( o r mean values t) and variances s 2 (if available) from records of test results. If design and production are accomplished irrespective of any fire classification, the supply in terms of the nominal value t s ( o r [) may be uniformly distributed (see Fig. 3a). When design and production, however, are directed at a specific fire resistance class, the overall supply may then be distorted towards the lower limit t , (see Fig. 3b). It must be recognized that, generally, there is an in-
creasing tendency to adapt production to m i n i m u m requirements, i.e. to lower class limits. This t e n d e n c y is, however, inherently associated with an increasing knowledge on the attainable fire resistance of a particular product. Mathematically precise formulation of the distribution of classified components is rather cumbersome. Therefore, the following simplifications are suggested: (a) accounting for considerable adaption of production to minimum requirements. Since this adaption requires some prior information, the variance 0 2 can be assumed to be known. Since reference is made to all populations within a certain class, the variance is introduced as the average variance within a certain class. This simplification, which is not necessarily conservative, may be encountered by considering only populations with a nominal value of t N = t u + A t f with an appropriate choice for Atf (0 <~ Atf < (t, -- tu)/2). Hence, the distribution characterizing classified components in a simplified manner is given by (compare eqn. (S8a)): F T ( t ] X ( t u ) ) ~ C~N
(~
(SSb) (b) assuming no adaption of production to requirements. Consistently, the variance is only introduced by its estimate s 2, again referring to the average of all populations within the class. The nominal value is introduced as t N = tu + Atf, where Atf may now be closer to the half-class interval (Atf ~% (tv -- tu)/2). Hence, the corresponding distribution is given by Fw(tlX(t.))
~ q2v t .... t . - - k s .... Atf
(SSc)
[
$4 Correction factors
t
Fig. 3a. Uniform supply.
lftN
(l) u ~ t v
t t Filz. 3b. Sure, Iv directed at minimum reauirements.
If test results are.employed to establish a correction factor Xc in a calculation model, this factor can be interpreted as a random q u a n t i t y with a distribution function,according to eqn. (S2b). A nominal value for this factor may be determined from
XCN = Xc -- ]~Sxc with k according to eqn. (S6b), with a possibly increased probability p = ~bN(OtcONI(pN)) where a~ is an influence coefficient (compare o~. and (x¢ in Aooendix 1).
Appendix 2 Statistical Assessment of Test Results CONTENTS 1. Scope . . . . . . . . . . . . . . . . . . . . . . . . . 2. Object of Testing . . . . . . . . . . . . . . . . 3. Statistical Aspects within Assessment Method 1 . . . . . . . . . . . . . . . . . . . . . . 4. Statistical Aspects within Assessment Method 2 . . . . . . . . . . . . . . . . . . . . . . 5. Tests Accompanied by Calculation Models . . . . . . . . . . . . . . . . . . . . . . . . 6. References . . . . . . . . . . . . . . . . . . . . . 7. List of Symbols . . . . . . . . . . . . . . . . . Supplements: Probabilities Associated with the Test Quantity . . . . . . . . . . . .
2. OBJECT OF TESTING 66 66 69 70 71 72 72 72
1. SCOPE This appendix is concerned with particular statistical aspects of the experimentally determined fire resistance time. These aspects are of concern, - - i f within assessment m e t h o d 1, there is an option of either an experimental or analytical determination; both should be compatible by rendering the same degree of reliability. This is achieved by appropriate specification of safety factors for the analytical procedure. - - i f within assessment m e t h o d 2, safety elements are derived for experimentally evaluated components, which may or may not be classified. Statistical aspects are likewise of concern if different testing procedures are compared, with respect to different numbers of test results, different calculation rules for nominal values, etc. They are of some significance if experimental evidence is sought for calculation models, thus providing an estimate for correction factors or functions.
2.1. Definitions
Within test procedures, the fire resistance time of components is defined as the minim u m length of time during which components comply with given performance criteria under specified testing conditions. If, for example, these criteria refer to the load bearing capacity of a component, then this is the length of time during which components sustain a specified load s k. Since the resistance capacity r changes with increasing temperature and hence, with the proceeding duration t of the test, the c o m p o n e n t complies with the given criteria as long as r ( t ) > Sk and fails if r ( t ) - - Sk = 0
~ tf
(la)
presuming a temperature-invariant specification of loading (Sk(t) = Sk). The length of time until failure occurs (or more usually -- until infringement of any performance criteria), identifies the fire resistance time t~ of the c o m p o n e n t which obviously depends on the level sk of loading {Fig. la). The level of mechanical loading in testing may either - - b e prescribed and, thus, belongs to the package of testing conditions, or
kr.s r(o)
Fig. l a . D e f i n i t i o n o f fire resistance t i m e tf in testing for t e m p e r a t u r e - i n v a r i a n t loading.
67
--be a testing parameter, thus rendering the fire resistance time as a function of the load level: tf(s). It is recognized that the herewith defined fire resistance time is exclusively related to the mechanical and thermal conditions of testing. Certain conditions may be standardized, e.g. according to ISO 834 or according to other standards developed for particular applications. A thus established standard fire resistance time will nevertheless depend on residual conditions prevailing during the individual test and/or at the particular furnace.
Another source of variations of the experimental fire resistance time are unintentional variations of the testing conditions referring to a single furnace, and even more so, when referring to different furnaces (see section 2.4). It is understood that the quantities R(o), R(t) and T F a r e deterministic quantities for a particular c o m p o n e n t subject to particular testing conditions. The notion of randomness arises herein when predicting the experimental fire resistance time of a population of components from particular test results.
2.3. Evaluation of test results 2.2. Variability of test quantity The original resistance capacity r(o), i.e. the resistance prior to testing, of components of the same design is subject to unintentional -- random -- variations. Components manufactured in the workshop of the laboratory (for a hypothetical large test series) may display a smaller variation than components taken at random from the production process or at site. Eventually, their average resistance may differ as well. At any rate, R(o) and, consequently, the resistance R(t) after a certain duration of test, must be regarded as random quantities. Eventually, the alteration of the resistance may be influenced by structural or material properties, not (or only insignificantly) affecting R(o); hence, R(t) may display a greater variation than the original resistance capacity (see Fig. l b ) . From these considerations it follows that the length of time until failure occurs, i.e. until
R(t) --Sk = 0
From the test results or observations t = tl, t2, ... t,* of a sample of n components, a sample mean, variance and coefficient of variation (c.o.v.) V are c o m p u t e d according to ref. [1 ] 1
n
= -- ~, ti n
(2a)
1 tl
s 2 - n--ll ~ ( t i - - ~ ) 2 ;
V =s/t
(2b)
If the fire resistance time is assumed to be log-normally distributed, it is convenient to relate the sample mean and variance to the respective parameters of a normal distribution by appropriate transformation t'i = In ti, rendering: ~,
1~
,
= __ t i n 1
(lb)
>TF
is a random quantity as well, the variation of which may also depend on the level of loading. A temperature-dependent loading m a y additionally affect the variation.
s '2 - - n--ll
(t'i -- t ' ) and s '2 = ln(1 + V 2)
The following presentation is written in terms of normally distributed variables. By the aforementioned transformation all relations can be rewritten for log-normally distributed variables. Evaluation of test results generally involves calculation of a nominal value regarded as relevant to the fire resistance time of the components in the population in question.
tN = h(tl, t2 .... , t,)
fTF(t)
B
t
Fig. l b . R a n d o m realization o f fire resistance T F for t e m p e r a t u r e - i n v a r i a n t loading.
* I n d e x f in tf resp. T F o m i t t e d if self-evident; n o t e t h a t capital letters (R, S, T) r e p r e s e n t r a n d o m variables and small letters (r, s, t) r e p r e s e n t d e t e r m i n i s t i c quantities.
68 for instance tN = [
(3a)
tN = m i n ( t l , t2 . . . . . t , )
(3b)
t N = t -- At
(3c)
t s = [ - - ho
or
tN = t - - ks
(3d)
Obviously, a nominal value according to eqn. (3d) utilizes the available information on the mean and the variance as well. This t y p e of nominal value is recommended, provided the factor k is specified appropriately.
2.4. Uncertainties and compatibility of test results Statistical uncertainties: [ and s 2 are estimators for the actual distribution parameters (mean value 12 and variance 02) of the fire resistance time of a hypothetically infinite sample (n ~ oo) of equally designed components. If test series with only small sample sizes were repeated several times, then t as well as s 2 would vary between the different series. Hence, mean value and variance, estimated from small sample sizes, are accompanied by "statistical uncertainties" [2, 3 ].
Lack o f representativeness: If the components used in testing cannot be regarded as a representative sample with respect to the standard of quality to be expected at site, then consideration of additional uncertainties and possibly corrections would be necessary. However, assessment of these uncertainties is more or less confined to a subjective estimation -- at least as concerns a pure experimental evaluation of the fire resistance w i t h o u t support by accompanying calculation models. In the following it is, therefore, assumed t h a t the sample of components -regardless of sampling or manufacturing procedure -- can be regarded as representative of the population for which experimental evidence is sought. Deviations in testing conditions: Generally, deviations in testing conditions, especially when comparing test results from different furnaces, are regarded as a more severe impairment of experimental evidence than a lack of representativeness in the sample
quality. Concerning unavoidable variations of the conditions of testing at a particular furnace, it may be presumed t h a t these variations are inherent in the definition of the test quantity; consequently, t h e y are not to be extricated from the test results. As concerns variations of the conditions of testing at different furnaces, it may be argued that very large deviations suggest that actually different (populations of) test quantites are observed. Specification of a reference furnace and evaluation of correction factors by calibration seem to be the only means for providing compatibility of test results from different furnaces at present. Possibly, specification of thermal testing conditions by control of heat flux may highlight smaller residual variations [4 ].
Choice of experimental model: The experimental model comprising the thermal and mechanical testing conditions may be specified with the intention to either --grade components for certain reference conditions, or - - m o d e l natural conditions as closely as possible (either for grading or for using the test results directly). Often a mixed approach is employed: e.g. thermal conditions are specified by a reference or standard t e m p e r a t u r e - t i m e curve and mechanical conditions are chosen to comply with the actual conditions expected in the structure. Mixed approaches, representing transitional stages from a pure grading system to a rational fire design, require a deliberate interpretation of the requirements to be fulfilled, in order to avoid unintentional increases or decreases in levels of reliability and economy. Model uncertainties: Recognizing that the experimental fire resistance time is defined with respect to a package of specified testing conditions, i.e. with respect to a specified experimental model, uncertainties arising from deviating conditions in the actual structure (compared with those in testing) are actually of no concern when evaluating the experimental fire resistance time. This is fairly obvious for conditions which can be established irrespective of the particular type of c o m p o n e n t (e.g. thermal conditions by a standard t e m p e r a t u r e - t i m e curve, level of
69
loading -- if not a testing parameter). As for conditions which have to be fixed individually for the particular component, the basic consideration should be, to provide comparability with tests on other components (and not to comply with reality, although evidently comparability will often be by reference to likewise probable conditions in actual structures). From this reasoning, it follows that it would be appropriate to consider uncertainties arising from insufficiently ensured comparability.
3. S T A T I S T I C A L A S P E C T S WITHIN A S S E S S M E N T METHOD 1
3.1. Calibration Calibration involves an analysis of an existing design procedure with respect to the inherently provided degree of reliability. Calibration is of concern if modifications (or alternatives) to existing design procedures are considered which should render the same degree of reliability. Within the assessment m e t h o d 1, this may be the case - - i f analytical determination o f the fire resistance time is envisaged which should be compatible with the experimental determination; - - i f changes in the testing program are envisaged in terms of sample size, calculation rule for the nominal value, or if different testing programs (different with respect to these statistical aspects) are compared; and - - i f different classifications or procedures without classification are compared. As code requirements presumably are based on specific testing programs and testing conditions, statistical aspects may be of some significance also for comparing different national code requirements. More vital differences resulting from different testing conditions (including loading) are primarily a problem of physical modelling with minor statistical aspects. According to Fig. 1.1 in Section 1.1 of the general introduction, adequacy of the design - - within assessment m e t h o d 1 -- is presumed, if a structural member has a fire resistance t~ which meets the required time of sustained fire exposure tfd: tf/> tfd
(4a)
Referring to the traditional classification system, tf corresponds to the fire resistance of a graded structural member, and ted represents the lower limit tu of the required fire resistance class (see eqn. (5a)). If a c o m p o n e n t falls to meet the requirement, i.e. if the random fire resistance T F is shorter than the class limit tu = tfd TF - - tu ~< 0
(4b)
then this event does not necessarily cause any physical consequences. Strictly speaking, eqn. (4b) does n o t correspond to a limit state as defined in Section 6.1 of Part A. However, the equivalent formulation according to eqn. ( l b ) , R ( t u ) - Sk ~< 0
(4C)
suggests an experimental limit state (with a " b o u n d e d reference duration"), which may -by agreement -- be regarded as relevant. The degree of reliability associated with this experimental limit state corresponds to the probability (1 - - P u ) that the fire resistance time as a random quantity exceeds the class limit tu. Hence, for calibration purposes, we are concerned with the conditional probability Pu = P(TF • t~lgraded in class X(t~))
(4d)
requiring an evaluation of the distribution of the random quantity TF, i.e. the fire resistance of specifically graded members.
3.2. Distribution of the fire resistance time The distribution of T F , i.e. the probability to observe a fire resistance smaller than a certain value is governed by: - - t h e sample size (n) for determining the fire resistance of a certain population; -- the calculation rule for the nominal value tN decisive for classification tu ~< tN < tv
* class X(tu)
(5a)
-- any prior information available. Moreover, agreement is necessary on the population considered as representative for calibration. Reference can be made either --to the entity of populations produced, tested and classified as X(tu), or --to populations meeting minimum class requirements with t s = tu. In view of the increasing tendency to adapt production to minimum requirements, populations meeting minimum requirements could
70 be accepted as representative. In this case, the calibrated degree of reliability corresponds to the probability that the fire resistance time (under testing conditions) exceeds the nominal value tN = tu ; thus
As outlined in Parts A and B, practical verification in the mechanical strength domain is by comparing
Pu = PN = P(TF <, tN[graded in class X(t~))
where R k and Sk are appropriately defined characteristic values. The partial safety factors ~/m and 7f are derived from the probability
(55) As an example: if experimental evidence is only by a single test result with tN = tt, then P~ = PN = 0.50; if evidence is by the minimum of two test results with t N = m i n ( t l , t:), then P~ = PN = 0.33. If reference is made to the entity of populations classified as X(tu), then the frequency of nominal values tN (or mean values t) and the associated variances s 2 per class have to be considered in the evaluation of the distribution of T F. The resulting degree of reliability will consequently be higher. As an example: if evidence is by the minim u m of two test results with tN = min(t~, t2), then p~ ~ 0.10 for equally distributed t i within class and prior knowledge about the variance. (For the evaluation of probabilities see also the supplements to this appendix.) 3.3. Compatibility between analytical and experimental results Recognizing the character of a gradeverification within assessment m e t h o d 1, an analytical determination should, basically, be directed at reproducing test results by modelling the expected performance o f the c o m p o n e n t under testing conditions. In the analytical determination o f the fire resistance, it is assumed generally that all basic variables X in the calculation model R ( t ) = gR(t, X_, Zrd )
(6)
are known in terms of their characteristic values (and/or distribution functions). A model uncertainty may have to be considered which reflects the uncertainty in the calculation model as a correct representation of the expected performance of the c o m p o n e n t under testing conditions (at the time t = tu). This m a y be done by introducing an additional basic variable Xrd. The calculation model may also incorporate correction factors or functions. They are often combined with the model uncertainty (cf. Xc in Section 5).
Rk(tu)/Tm ~ Sk~/f
P(R(tu) < S) = Pu
(7a)
(75)
referring to the previous considerations on the appropriate population for calibration. If, according to the prescribed testing conditions, the level of loading is fixed at Sk (and is temperature-invariant), then analytical evidence requires verification for the same fixed loading and eqn. (7a) is reduced to Rk(tu)/'Ym <~ Sk
(7C)
In this case 3'm can be simply evaluated from (see also Appendix 1) : 1 --kV 7m-
1 --/JuV
where flu ....... CN I(Pu)
(7d)
For a characteristic value R k corresponding to, e.g. the 5% fractile (k = 1.64), a c.o.v, of V = 0.10 and for probabilities Pu = PN = 0.5 (one test) . . . . . 7m = 0.84 Pu = PN = 0.33 (rain. of two tests) ---~ "~m = 0.87 is obtained. Hence, in analytical analysis the mechanical strength should be multiplied by, for example, 1/0.84 = 1.2 in order to be consistent with experimental verification [ 5 ]. If the level of loading is accepted as a testing parameter, then consideration of the random load (-effect) acting in an accidental situation is appropriate. The possible decrease of ~/m in this case is generally negligible, and it may suffice to determine '~m likewise from eqn. (7d). The load factors are specified in accordance with Appendix 1
4. STATISTICAL ASPECTS WITHIN ASSESSMENT METHOD 2 The necessary fire resistance time derived from an equivalent fire exposure t e can be regarded as a quantitative requirement based
71 on an actual -- if only idealized - limit state
involving the corresponding random variables:
TF-TE-O
(Sa)
or R(te) -- s k
is available but not sufficiently accepted. The relevant basic variables _X are fairly well
established with respect to their characteristic values (and/0r distribution functions). There may be some uncertainties with respect to
= 0
(8b)
and equivalent formulations for a random and/or temperature
It is assumed that a calculation model R ( t ) = gft(t, X )
their temperature dependency, some thermal properties, etc. However, the t y p e of failure, i.e. the failure m o d e of the c o m p o n e n t subject to testing conditions, is well known. Account of this insufficiency in the calculation model may be taken by a correction factor Xc, which is assumed to be a random quantity [ 11 ]. (9)
R ( t ) = gR(t, X_, Xc)
(Ideally, X c and Xrd in eqn. (6) are identical.} With a calculation model available, tests may be basically directed at evaluating the variable X¢. If all variables _X (strength properties, geometrical data, ...) can be numerically determined for the particular c o m p o n e n t subject to testing, then gk(t, X)
- - Sk = 0
~ if
renders a particular fire resistance time t~l. The corresponding test result tf, will differ from the calculated value. Thus, a correction factor x¢1 can be determined. Repetition of the procedure will render a second factor Xc2 and so on. According to eqn. (2) a sample mean (Xc) and variance (S~c) are determined. Without a calculation model, the mean and variance of the fire resistance time are associated with statistical uncertainties (Section 2.4). Now these uncertainties only affect the correction factor Xc. The theoretical distribution function of Xc can be determined according to the supplements. Generally, only a nominal (or characteristic value or design value) is required for R ( t ) o r T F and, thus for X¢, which is associated with a certain probability. As the statistics of the basic variables _X are assumed to be known with reference to the population in question, the requirement on the representativeness of a sample may be less severe or even superfluous. Eventually, as insufficient knowledge, i.e. uncertainty, is confined to the correction factor, the effort employed in estimating this factor (sample size, sophistication in statistical modelling) may depend on the degree of confidence associated with the calculation model.
72
Standardized t
6. REFERENCES 1 A. Hald, Statistical Theory in Engineering Application, Wiley, New York, 1952. 2 G. Box and C. Tiao, Bayesian Inference in Statis. tical Analysis, Addison-Wesley, 1973. 3 J. R. Benjamin and C. A. Cornell, Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, New York, 1970. 4 J. Keough, CIB report. Van Keulen, CIB report. 5 O. Petterson and J. Witteveen, On the fire resistance of structural steel elements derived from standard fire tests or by calculation, Fire Safety J., 2 (1979/80) 73. 6 M. Kersken-Bradley, Einfluss von Annahmekennlinien auf die ZuverIdssigkeit yon Bauteilen beiBrandbeanspruchung, Research report, Institut fiir Bautechnik, 1979. 7 R. Rackwitz, Theoretische Grundlagen fiir die Bestimmung des Bemessungswertes yon Bauteilwidersth'nden aus Versuchen, Berichte zur Zuverl~ssigkeitstheorie der Bauwerke, Technische Universit/it, Miinchen, Heft 38, 1979. 8 JCSS, Basic Notes on Resistances, Draft R-01, 1981. 9 M. Zelen and N. C. Severo, in Abramowitz (ed), Handbook of Mathematical Functions, Dover, New York, 1970. 10 JCSS, General Principles on Reliability for Structures or Revision of IS 2394, draft, 1982. 11 W. Struck, Die traditionelle Praxis der experimentellen Bauteilpriifung aus der Sicht des probabilistischen Sicherheitskonzepts, Bautechnik, 2 (1981).
7. LIST OF SYMBOLS (INCLUDING THE SUPPLEMENTS TO THIS APPENDIX) r
R Sk
S tf TF
&(x) Fx(X) P O2 82
V n
ON(.)
Resistance c a p a c i t y Resistance c a p a c i t y as r a n d o m variable Level o f m e c h a n i c a l load in testing Mechanical load as r a n d o m variable Fire resistance t i m e Fire resistance time as r a n d o m variable Distribution density of a random variable X Distribution function of a random variable X Mean value Variance E s t i m a t e f o r m e a n value E s t i m a t e f o r variance C o e f f i c i e n t o f variation (c.o.v.) N u m b e r o f test results Standardized normal distribution f u n c t i o n (see Table 1 )
t u ~ tv tfd
k=rkJ
with v = n - - 1 degrees o f f r e e d o m (see Table 2) L o w e r a n d u p p e r class limits R e q u i r e d t i m e o f sustained fire exposure F a c t o r for calculating n o m i n a l or characteristic values (t N = [ - ks)
SUPPLEMENTS: PROBABILITIES ASSOCIATED WITH THE TEST QUANTITY
$1 D i s t r i b u t i o n o f the fire resistance t i m e Given a certain p o p u l a t i o n with a m e a n p and variance 02 o f the fire resistance t i m e T, t h e n t h e p r o b a b i l i t y t h a t a fire resistance t i m e can be observed w h i c h is smaller t h a n a certain value t ( P ( T <~ t)), is identified b y t h e d i s t r i b u t i o n f u n c t i o n Fw(t ) or m o r e precisely: Fw(t[p, 0). In o t h e r c o n t e x t s t h e c o n d i t i o n (.[#, o) is o f t e n o m i t t e d as self-evident. i
fT
tlIT~, .)FT~
P( T <:_
~"
Fig. 2a. Density function fT and distribution function F w for a given mean value g and variance o 2. F o r a n o r m a l d i s t r i b u t i o n f u n c t i o n , the respective probabilities are o b t a i n e d f r o m
F r ( t r p , o) = P ( T <~ t i p , o) = ON
(-
($1)
w h e r e CS d e n o t e s the s t a n d a r d i z e d (0, 1) n o r m a l d i s t r i b u t i o n f u n c t i o n with values listed in Table 1. If o n l y the variance o 2 within a certain p o p u l a t i o n can be assumed t o be k n o w n , whereas t h e m e a n p is e s t i m a t e d b y [, t h e n p r o b a b i l i t y s t a t e m e n t s are c o n d i t i o n e d b y t h e estimate [ and e v i d e n t l y b y o.
73 TABLE 1 Standard normal distribution function distribution function: ~(z)-- - .
~
(1)
exp --
u 2 du = p
X/27rl exp ( -12) z distribution density: ~(z) = ~
-z
0
~.
0
Z
Z
~
0
z
p = ¢(--z)
p = ¢(z)
~(z)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.5 4.0 4.5 5.0
0.500000 0.460173 0.420740 0.382089 0.344578 0.308535 0.274253 0.241964 0.211855 0.184060 0.158655 0.135666 0.115070 0.096805 0.080757 0.066807 0.054799 0.044565 0.035931 0.028716 0.022750 0.017864 0.013903 0.010724 0.008197 0.006209 0.004661 0.003467 0.002505 0.001866 0.001350 0.000232 0.000032 0.000004 0.0000003
0.500000 0.539827 0.579260 0.617911 0.655422 0.691465 0.725747 0.758036 0.788145 0.815940 0.841345 0.864334 0.884930 0.903195 0.919243 0.933193 0.945201 0.955435 0.964069 0.971284 0.977250 0.982136 0.986097 0.989276 0.991803 0.993791 0.995339 0.996533 0.997495 0.998134 0.998650 0.999768 0.999968 0.999996 0.9999997
0.398942 0.396952 0.391043 0.381388 0.368270 0.352065 0.333225 0.312254 0.289692 0.266085 0.241971 0.217852 0.194186 0.171369 0.149727 0.129518 0.110921 0.094049 0.078950 0.065616 0.053991 0.043984 0.035475 0.028327 0.022395 0.017528 0.013583 0.010421 0.007915 0.005952 0.004432 0.000873 0.000134 0.000016 0.0000015
The corresponding distribution function [ 3 , 7, 8 ] , c o n s i d e r i n g all p o s s i b l e e s t i m a t e s [ {and t h e i r a p p r o p r i a t e w e i g h t i n g ) , is F w ( t ] t , o) = P ( T <~ tit, o) = CN - - ~ (S2a)
Z
Z
and identifies an increase of the variance from 02 t o o2(n + 1)/n d u e t o t h e u n c e r t a i n t y i n the m e a n value, n d e n o t e s the n u m b e r of test r e s u l t s a v a i l a b l e f o r t h e e s t i m a t e a n d CN is d e f i n e d as i n e q n . ( S 1 ) . If both distribution parameters p and a 2 are e s t i m a t e d b y [ a n d s 2 o n t h e basis o f n t e s t
74 fT
Dt
ll-!
F-r
FT
] ~
i,ll,,
i t
t
Fig. 2b. D e n s i t y f u n c t i o n s fT a n d d i s t r i b u t i o n funct i o n s F T for d i f f e r e n t e s t i m a t e s t w i t h variance 02 ( t h i n lines) a n d for all possible e s t i m a t e s c o m b i n e d - w i t h variance a2(n + 1 )In ( t h i c k lines).
Fig. 2c. D e n s i t y f u n c t i o n fT a n d d i s t r i b u t i o n f u n c t i o n F w for d i f f e r e n t e s t i m a t e s t a n d s ( t h i n lines) a n d for all possible c o m b i n e d e s t i m a t e s ( t h i c k lines).
results, then probability statements are conditioned by the estimates t and s. The corresponding distribution function [ 3, 7, 8 ], considering all possible estimates and s (and their appropriate weighting), is
the uncertainty in the variance itself changes the t y p e of function from a normal distribution to a so-called "central t-distribution". qxv in eqn. (S2b) denotes the standardized t-distribution function with v = n - 1 degrees of freedom; values are listed in Table 2. With increasing sample size, i.e. as v -~ ~o and, thus decreasing statistical uncertainty, the central t-distribution converges towards the presumed normal distribution. Incorporation of appropriate prior information will
/
F w ( t l [ , s ) : P ( T < . t l t , s)
=
q%(ts/-
(S2b) and again identifies an increase of the (now estimated) variance b y sZ(n + 1)/n, whereas TABLE 2 Inverse c e n t r a l t - d i s t r i b u t i o n u = '#v~(p)
V p
0.25
0.10
0.05
0.025
0.01
0.005
0.0025
0.001
0.0005
1 2 3 4 5
1.000 0.816 0.765 0.741 0.727
3.078 1.886 1.638 1.533 1.476
6.314 2.920 2.353 2.132 2.015
12.706 4.303 3.182 2.776 2.571
31.821 6.965 4.541 3.747 3.365
63.657 9.925 5.841 4.604 4.032
127.321 14.089 7.453 5.598 4.773
318.309 22.327 10.214 7.173 5.893
636.619 31.598 12.924 8.610 6.869
6 7 8 9 10
0.718 0.711 0.706 0.703 0.700
1.440 1.415 1.397 1.383 1.372
1.943 1.895 1.860 1.833 1.812
2.447 2.365 2.306 2.262 2.228
3.143 2.998 2.896 2.821 2.764
3.707 3.499 3.355 3.250 3.169
4.317 4.029 3.833 3.690 3.581
5.208 4.785 4.501 4.297 4.144
5.959 5.408 5.041 4.781 4.587
11 12 13 14 15
0.697 0.695 0.694 0.692 0.691
1.363 1.358 1.350 1.345 1.341
1.796 1.782 1.771 1.761 1.753
2.201 2.179 2.160 2.145 2.131
2.718 2.681 2.650 2.624 2.602
3.106 3.055 3.012 2.977 2.947
3.497 3.428 3.372 3.326 3.286
4.025 3.930 3.852 3.787 3.733
4.437 4.318 4.221 4.140 4.073
75
reduce the statistical uncertainty associated with /~ and o. Prior information m a y be
expressed according to eqn. (S4b) with k = V ~ / 2 ~ 0.7*:
available concerning, e.g. the variance or the c.o.v., allowing assumptions on upper bounds or informative prior distributions. Prior information can also be modelled by considering additional fictitious samples [7] or by a fictitious increase of the sample size. However, rational criteria for the assessment of prior information are generally n o t available and, thus, assessment is basically by engineering judgement.
tN = min(t,, t2) = t - 0.7s
The probability associated with this nominal value excluding any prior information amounts to (see eqn. (S4b) and Table 2) PN
t N = t--
kO
or
tN =
~1(--0.7~)
t N = t --
[--
kS
(S3a, S3b)
is, thus, associated with the probability (see eqn. $2) PN
=
~ 0.33
It should be noted that, if it can be assumed that the variance is known, a nominal value of e.g.
$2 Nominal values The probability to observe a fire resistance time smaller than an arbitrary nominal value tN is obtained from the distribution functions as defined in the previous section. A nominal value, e.g. of the type
(S5b)
0.70
likewise based on two test results -- is associated with a smaller probability -
-
PN = CN(--0.7X/-~) ~ 0.29 Vice versa, if the nominal value of the fire resistance is required to correspond to a preselected probability (p-fractile), then the factor k in eqn. ($4) depends on the number of test results and should be introduced as
=
or
pN = ~,,(--k ~ n ~ l )
(S4b)
If only a single test result is available and is employed as nominal value, (S5a)
tN = tl = t
this is equivalent to operating with an estimate for the mean value. The probability that the fire resistance in subsequent tests (or ideally within the population) is smaller than this nominal value is PN = 0.5
(n = 1)
Since the t-distribution degenerates for v = n - 1 = 0, inference from a single test result for u n k n o w n variance is strictly speaking n o t possible. The probability PN can be derived only with reference to the identity of mean value and expected value for normal distributions. The probability for any other type of nominal value (e.g. tN = t l - At) remains indeterminable. In the case of two test results, the minimum of both is often specified as the nominal value which, however, can equivalently be
(S6a, S6b) thus, rendering nominal values which correspond to a constant p-fraetile for an arbitrary number of test results. The notion of equal probability PN (regardless of available information) presumes that costs are independent of the a m o u n t of inherent uncertainties. It may, however, prove to be economical to have reliability degrees depending on the available information.
$3 Classification The nominal value t N may be used for attributing components to fire resistance classes tu ~< tN < tv
" class
X(tu)
($7)
where t u and t v represent the lower and upper limits of the respective class (e.g. 30 and 60 min). The probability p~ of observing a fire resistance time smaller than the lower limit t~ within the population of components *t N = min(tl,
t2) = (t I + t2)/2 -- It 1 --
= [-
~ ( t l -- t2)2/4
t2[/2
76 in question is again obtained from the distribution functions defined in section $1 for t = t , ; e.g. for a nominal value according to eqn. (S3a): P , = CS
•
a
(S8a)
whereby Pu ~< PNOnly populations of components which just slipped into class X ( t , ) with t , = tN will fall short of the lower limit tu by the probability P , = PN, e.g. according to eqn. ($4). If we are concerned with the statistical properties of components of a certain fire resistance class X ( t , ) , all populations of components attributed to class X ( t , ) have to be considered, i.e. all populations, the nominal value of which lies between aforementioned limits. Hence, general probability statements on components of fire resistance class X ( t , ) require consideration of the distribution of the overall supply of components by design and production. This information can be derived by investigating the frequency for, e.g. nominal values t s ( o r mean values t) and variances s 2 (if available) from records of test results. If design and production are accomplished irrespective of any fire classification, the supply in terms of the nominal value t s ( o r [) may be uniformly distributed (see Fig. 3a). When design and production, however, are directed at a specific fire resistance class, the overall supply may then be distorted towards the lower limit t , (see Fig. 3b). It must be recognized that, generally, there is an in-
creasing tendency to adapt production to m i n i m u m requirements, i.e. to lower class limits. This t e n d e n c y is, however, inherently associated with an increasing knowledge on the attainable fire resistance of a particular product. Mathematically precise formulation of the distribution of classified components is rather cumbersome. Therefore, the following simplifications are suggested: (a) accounting for considerable adaption of production to minimum requirements. Since this adaption requires some prior information, the variance 0 2 can be assumed to be known. Since reference is made to all populations within a certain class, the variance is introduced as the average variance within a certain class. This simplification, which is not necessarily conservative, may be encountered by considering only populations with a nominal value of t N = t u + A t f with an appropriate choice for Atf (0 <~ Atf < (t, -- tu)/2). Hence, the distribution characterizing classified components in a simplified manner is given by (compare eqn. (S8a)): F T ( t ] X ( t u ) ) ~ C~N
(~
(SSb) (b) assuming no adaption of production to requirements. Consistently, the variance is only introduced by its estimate s 2, again referring to the average of all populations within the class. The nominal value is introduced as t N = tu + Atf, where Atf may now be closer to the half-class interval (Atf ~% (tv -- tu)/2). Hence, the corresponding distribution is given by Fw(tlX(t.))
~ q2v t .... t . - - k s .... Atf
(SSc)
[
$4 Correction factors
t
Fig. 3a. Uniform supply.
lftN
(l) u ~ t v
t t Filz. 3b. Sure, Iv directed at minimum reauirements.
If test results are.employed to establish a correction factor Xc in a calculation model, this factor can be interpreted as a random q u a n t i t y with a distribution function,according to eqn. (S2b). A nominal value for this factor may be determined from
XCN = Xc -- ]~Sxc with k according to eqn. (S6b), with a possibly increased probability p = ~bN(OtcONI(pN)) where a~ is an influence coefficient (compare o~. and (x¢ in Aooendix 1).